Commun. Math. Phys. 280, 1–25 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0464-y
Communications in
Mathematical Physics
Logarithmic Deformations of the Rational Superpotential/Landau-Ginzburg Construction of Solutions of the WDVV Equations James T. Ferguson, Ian A. B. Strachan Department of Mathematics, University of Glasgow, Glasgow G12 8QQ, UK. E-mail:
[email protected];
[email protected] Received: 23 June 2006 / Accepted: 5 December 2007 Published online: 15 March 2008 – © Springer-Verlag 2008
Abstract: The superpotential in the Landau-Ginzburg construction of solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations is modified to include logarithmic terms. This results in deformations - quadratic in the deformation parametersof the normal prepotential solutions of the WDVV equations. Such solutions satisfy various pseudo-quasi-homogeneity conditions, on assigning a notional weight to the deformation parameters. These solutions originate in the so-called ‘water-bag’ reductions of the dispersionless KP hierarchy. This construction includes, as a special case, deformations which are polynomial in the flat coordinates, resulting in a new class of polynomial solutions of the WDVV equations.
1. Introduction One of the most basic classes of Frobenius manifolds is comprised of those which are defined on orbit spaces Cn /W, W being a finite Coxeter group [9]. Following from the observation of Arnold that the three polynomial solutions in 3-dimensions were related to the Coxeter numbers of the Platonic solids it was realized that the earlier Saito construction [22] provided a construction of Frobenius manifolds and that the prepotentials (solutions to the WDVV-equations - see below) were automatically polynomial with respect to a distinguished coordinate system, the so-called flat coordinates {t i }. Such prepotentials are quasihomogeneous, a property that may be expressed in terms of an Euler vector field 1 i i ∂ E= d t i h ∂t i
as L E F = (2 +
2 )F, h
2
J. T. Ferguson, I. A. B. Strachan
where the d i are the degrees of the basic W -invariant polynomials and h is the Coxeter number of W. Such solutions are semi-simple and it was conjectured by Dubrovin that all semi-simple polynomial solutions arise from this construction for some Coxeter group. This was later proved by Hertling [15]. In this paper we construct a new class of semi-simple polynomial solutions to the WDVV equations. This does not contradict the result of Hertling as the solution does not satisfy the full set of axioms of a Frobenius manifold, in particular the solutions are not quasi-homogeneous. These solutions may be regarded as a deformation of the A N -polynomial solutions, in the sense that the prepotential takes the form F(t 1 , . . . , t N , b) = F (0) (t 1 , . . . , t N ) + k F (1) (t 1 , . . . , t N , b), where F (0) is the polynomial solution defining the Frobenius manifold structure on the space C N /A N and k is some deformation parameter. Such solutions satisfy a pseudoquasi-homogeneity condition. With the Euler vector field E=
N (N + 2 − i) i ∂ b ∂ t i + N +1 ∂t N + 1 ∂b
(1)
i=1
we define the degree of an invariant function f , denoted deg( f ), by (N + 1)L E f = deg( f ) f. With this, each part of F is separately quasi-homogeneous: deg F (0) = (2N + 4)F (0) , deg F (1) = (N + 3)F (1) . By assigning a fictitious scaling degree of (N +1) to the deformation parameter k the full solution may be thought of as pseudo-quasi-homogeneous. These solutions will appear as a special case of a more general construction. The Frobenius manifold structure on the orbit space C N /A N may also be derived [9,16,17] via a Landau-Ginzburg formalism as the structure on the parameter space of polynomials of the form λ( p) = p N +1 + s1 p N −1 + . . . + s N . More explicitly, the metric η(∂si , ∂s j ) = −
res
dλ=0
∂si λ( p) ∂s j λ( p) λ ( p)
(2) dp
is flat (though, in these variables, it does not have constant entries) and the tensor ∂si λ( p) ∂s j λ( p) ∂sk λ( p) c (∂si , ∂s j , ∂sk ) = − dp res dλ=0 λ ( p)
(3)
(4)
defines a totally symmetric (3, 0)-tensor which further satisfies various potentiality conditions from which one may construct a so-called prepotential F which satisfies the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations of associativity ∂3 F ∂3 F ∂3 F ∂3 F λµ λµ η − η = 0, α, β, γ , δ = 1 . . . , N , ∂t α ∂t β ∂t λ ∂t µ ∂t γ ∂t δ ∂t δ ∂t β ∂t λ ∂t µ ∂t γ ∂t α
Logarithmic Deformations of WDVV Solutions
3
where the coordinates {t i } are a set of flat coordinates for the metric η defined by (3). Geometrically, a solution defines a multiplication ◦ : T M × T M → T M of vector fields on the parameter space M, i.e.
∂3 F ησ γ α ∂t ∂t β ∂t σ γ := cαβ (t) ∂t γ ,
∂t α ◦ ∂t β =
∂t γ ,
the metric η being used to raise and lower indices. Example 1. With λ( p) = p 4 + s1 p 2 + s2 p + s3 the formula (3) gives the metric1 η=
1 1 s1 ds1 ds3 + ds22 − ds12 . 2 4 8
While this metric is flat, the s i are not flat coordinates. With 1 s3 = t1 + t32 , 8 s2 = t2 , s 1 = t3 , one obtains a metric with constant coefficients. The tensor given by the formula (4) may then be used to construct the prepotential F=
1 2 1 1 1 5 t1 t3 + t1 t22 − t22 t32 + t . 8 8 64 3840 3
Such polynomial solutions may be seen from a variety of different points of view (and part of the rich mathematical structure of Frobenius manifold arises as from the fact that it lies at the intersection of seemingly disconnected areas of mathematics): (i) (ii) (iii)
as a basic example of an orbit space construction (here the manifold is Cn /A N , where A N is a Coxeter group); as a topological Landau-Ginzburg field theory; as a reduction of the dispersionless KP hierarchy.
The point of view that will be taken in this paper is the last, i.e. that a solution to the WDVV equations may be obtained from a specific reduction of the dispersionless KP hierarchy [16,17]. In particular it will be shown that the so-called water-bag reduction of the KP hierarchy [14] (see also [3]) also results in solutions of the WDVV equations, though not, as in earlier examples, a full Frobenius manifold because of the non-existence of an Euler vector field. This builds on a recent preprint [5] where a 2-component system was studied. 1 In all examples indices are lowered for notational convenience.
4
J. T. Ferguson, I. A. B. Strachan
2. The Dispersionless KP Hierarchy The dispersionless KP (or dKP) hierarchy is defined in terms of a Lax function λ( p) = p +
∞
u n (x, t) p −n
n=1
by the Lax equation
∂Tn λ( p) = λ( p), λn ( p)+ , where { f, g} = f x g p − f p gx is the ordinary Poisson bracket and [ ]+ denotes the projection onto non-negative powers of p. Various reductions of this infinite component hierarchy have been studied, the most fundamental being the A N -reduction
1 N +1 λ( p) = p N +1 + s1 p N −1 + . . . + s N , and this leads to a Frobenius manifold structure, defined above, on the space of parameters {si }. More recently a so-called ‘water-bag’ reduction has been studied, where one takes N p − pi . λ( p) = p + ki log p − p˜ i i=1
In a recent preprint Chang [5] showed that in the N = 1 case one may construct a solution of the WDVV equation by analysing the recursion relations satisfied by the conservation laws of the associated 2-component dispersionless hierarchy. Here we generalise this setting and consider functions of the form λ( p) = (rational function) ( p) +
M
ki log( p − bi ).
i=1
Formally one may expand this function for large p as a series, but this will have terms of the form M ki log p
i=1
and the constraint ki = 0 is often imposed. Here we show that one still gets a solution without such a constraint. The logarithmic terms mean that λ( p) is multi-valued. However, since the constructions in Sect. 3 involve only the derivatives of λ, all physical quantities are well-defined. For simplicity we present proofs in the polynomial case, with λ( p) = p N +1 + s1 p N −1 + . . . + s N +
M
ki log( p − bi )
(5)
i=1
and state the result for the rational case - no essential new features will be present in the rational case that are not already present in the polynomial case. Note that without this
Logarithmic Deformations of WDVV Solutions
5
constraint the function is not technically a reduction of the dKP hierarchy, but one may associate a ‘regularised’ function M ki log p λreg ( p) = λ( p) −
(6)
i=1
which is [20]. For this reason we call the form (5) a generalised water-bag reduction. We denote the space of such superpotentials M(M,N ) or just M. 3. Solutions of the WDVV Equations from the Generalised Water-Bag Reduction of the Dispersionless KP Hierarchy We begin by proving that the formulae (3,4) with the function (5) define a commutative, associative, semi-simple multiplication on the tangent space to the manifold of parameters. This will be done using canonical coordinates - the critial values of λ (i.e. λ evaluated at its critical points). Since λ( p) only involves logarithms its derivative is a rational function which may be written in the form M+N (N + 1) i=1 ( p − ξi ) M j=1 ( p − b j )
λ ( p) =
(we assume that we are considering the generic case, where the poles and zeros are all distinct). The canonical coordinates are then u i = λ(ξi ),
i = 1 ..., N + M
(for such a formula to be single-valued, various cuts have to be made in the complex plane). The proof follows [9], Lemma 4.5. From the formulae ∂ λ( p) = δi j , i = 1 ..., N + M ∂u i p=ξ j and M −1 ∂ λ( p) = ( p − br ) Bi ( p) ∂u i r =1
(where Bi is a polynomial of degree N + M − 1) one obtains 0, i = j Bi (ξ j ) = M r =1 (ξi − br ), i = j. The Lagrange interpolation formula then gives Bi ( p) =
− ξ j ) rM=1 (ξi − br ) , j=i (ξi − ξ j )
j=i ( p
6
and hence
J. T. Ferguson, I. A. B. Strachan
M ∂λ( p) j=i ( p − ξ j ) r =1 (ξi − br ) , = M ∂u i j=i (ξi − ξ j ) r =1 ( p − br ) M 1 (ξi − br ) 1 r =1 = λ ( p) , (N + 1) ( p − ξi ) j=i (ξi − ξ j ) =
λ ( p) 1 . ( p − ξi ) λ (ξi )
(7)
Note that this is the same functional form as in the polynomial case. With this 1 λ ( p) η(∂u i , ∂u j ) = − dp , res dλ=0 ( p − ξi )( p − ξ j ) λ (ξi )λ (ξ j ) 1 δi j . = − λ (ξi ) Note that while log-terms appear in λ, the metric formula involves derivatives of λ and hence involves rational functions only. Similarly ⎧ ⎨− 1 , i = j = k, c(∂u i , ∂u j , ∂u k ) = λ (ξi ) ⎩ 0, otherwise. Collecting these results one arrives at the following: Lemma 2. The formulae (3) and (4) with λ given by (5) define, at a generic point, a semi-simple, commutative, associative multiplication ∂ ∂ ∂ ◦ = δi j , i j ∂u ∂u ∂u i
(8)
compatible with the metric η=−
M+N r =1
du i2 . λ (ξi )
(9)
This multiplication has an identity. Since e(λ) = 1, where the vector field e is defined to be ∂ e= N, ∂s it is immediate from Eqs. (3) and (4) that c(∂, ∂ , e) = η(∂, ∂ ). From this it follows that e is the identity for the multiplication. In semi-simple coordinates it follows from the multiplication (8) that e=
M+N r =1
∂ . ∂u i
Logarithmic Deformations of WDVV Solutions
7
We prove next that the metric is flat and Egoroff. In the pure-polynomial case (or A N -case) the flat coordinates are defined by an inverse series, using the so-called thermodynamic identity. The presence of the logarithms makes such an inversion problematical. However, it turns out that part of the flat-coordinates of the metric are exactly the same as in the polynomial case. Lemma 3. The formula (3) with λ given by (5) gives the following: ∂si λ+ ( p)∂s j λ+ ( p) η(∂si , ∂s j ) = − res dp , i, j = 1, . . . , N , dλ+ =0 λ+ ( p) where λ+ ( p) = p N +1 + s1 p N −1 + . . . + s N is a truncation of λ, and η(∂br , ∂s j ) = 0, η(∂bi , ∂b j ) = ki δi j ,
r = 1, . . . , M, j = 1, . . . , N , i, j = 1, . . . , M.
It follows from these formulae that the metric is flat. Proof. These formulae just involve the use of basic ideas from complex variable theory:
p 2N −i− j dp , res η(∂si , ∂s j ) = − dλ=0 λ ( p) 2N −i− j p = res dp . p=∞ λ ( p)
Now λ ( p) = λ+ ( p) + =
λ+ ( p)
M r =1
ki , ( p − bi )
M ki 1 1+ . λ+ ( p) ( p − bi ) r =1
Hence ⎧ −1 ⎫ M ⎬ ⎨ p 2N −i− j ki 1 dp , η(∂si , ∂s j ) = res 1+ p=∞ ⎩ λ+ ( p) ⎭ λ+ ( p) ( p − bi ) r =1
⎧ −1 ⎫ M ⎬ ⎨ p˜ i+ j−N −2 N +1 ki p˜ = − res d p˜ , 1+ ⎭ ⎩ µ( p) ˜ µ( p) ˜ 1 − pb ˜ i p=0 ˜ = − res
p=0 ˜
r =1
p˜ i+ j−N −2 d p˜ , µ( p) ˜
where p˜ = p −1 and λ+ ( p) = p˜ −N µ( p). ˜ Reversing the argument yields the result. Similarly,
8
J. T. Ferguson, I. A. B. Strachan
p N −i −kr dp , η(∂si , ∂br ) = − res dλ=0 λ ( p) ( p − br ) kr p N −i r =i ( p − br ) 1 =− res dp , M+N N + 1 p=∞ j=1 ( p − ξ j ) ˜ 1 r =i (1 − br p) i−1 d p˜ , = res kr p˜ M+N N + 1 p=0 ˜ ˜ j=1 (1 − ξ j p)
= 0. Finally, M kj 1 ki r =1 ( p − br ) η(∂bi , ∂b j ) = − res dp . M+N dλ=0 ( p − bi ) ( p − b j ) N +1 k=1 ( p − ξk ) For i = j this, on deforming the contour around the Riemann sphere, gives zero: there is no pole at infinity, and the simple poles cancel. For i = j, 1 1 η(∂bi , ∂bi ) = −ki2 res dp , dλ=0 ( p − bi )2 λ ( p) 1 k=i (bi − bk ) = ki2 . N +1 k (bi − ξk ) On evaluating the residue at the poles using the two different formulae for λ ( p), (N + 1) p + (N − 1)s1 p N
N −2
M
+ . . . s1 +
r =1
one obtains
M+N ( p − ξi ) kr = (N + 1) i=1 , M ( p − br ) j=1 ( p − b j )
(bi − ξk ) ki = (N + 1) k k=i (bi − bk )
from which the final formula follows.
Alternative proof via ‘thermodynamical identity’. Following the polynomial case in [9], invert λ+ ( p) as N t t N −1 1 1 t1 + 2 + ... + N + O , p+ (k) = k + N +1 k k k k N +1 where λ+ = k N +1 . Then λ( p+ (k, t), t, b) = λ+ ( p+ (k, t), t) +
M
ki log( p+ − bi ) ,
i=1
= k N +1 +
M i=1
ki log( p+ − bi ) .
Logarithmic Deformations of WDVV Solutions
9
Differentiating with respect to t α gives M ∂ p+ ∂λ ki ∂ p+ dλ + = . dp p= p+ (k) ∂t α ∂t α p+ − bi ∂t α i=1 1 . =O k N +2−α So we have as our thermodynamical identity in this case ∂ ∂ 1 dk . (λdp) + ( p dλ) = O + ∂t α ∂t α k N +1−α Although the right hand side is not zero as it is for polynomial λ, this identity is sufficient to give ∂ 1 α−1 dk (λdp) = −k dk + O ∂t α k (Eq. (4.68) in [9]), from which it follows, using 1 dλ = dλ+ + O dk, k that η(∂t α , ∂t β ) = −
δα+β,N +1 . N +1
The flat coordinates are therefore {t i , i = 1, . . . , N ; b j , j = 1, . . . , M}, where the t i are defined by the inverse series for the truncated function λ+ = λ+ ( p), expanded as a Puiseaux series as λ → ∞, N 1 t t N −1 t1 1 , (10) + 2 + ... + N + O p(k) = k + N +1 k k k k N +1 1
where k = (λ+ ) N +1 , in the standard way [9]. Note that each t i is a polynomial in the si and vice versa. Consider the diagonal metric (9). Its rotation coefficients βi j are defined by the formula βi j =
∂u i H j , Hi
Hi2 =
1 . λ (ξi )
Such a metric is said to be Egoroff if the rotation coefficients are symmetric. This then implies that the metric may be written in terms of a single potential function V (u), η=
M+N i=1
∂ V i 2 du . ∂u i
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J. T. Ferguson, I. A. B. Strachan
Lemma 4. The metric (9) is Egoroff. Proof. In canonical coordinates η is diagonal with i th entry −
1 . λ (ξi )
From (7) ∂λ 1 λ ( p) , = i ∂u p − ξi λ (ξi ) N + 1 r =i ( p − ξr ) = , M λ (ξi ) s=1 ( p − bs ) so we have m ∂λ N +1 ( p − bs ) = ( p − ξi ), i ∂u λ (ξi ) r =i
s=1
where each side is a polynomial of degree N + M − 1. Also ∂s1 ∂s2 ∂s N kr ∂br ∂λ = i p N −1 + i p N −2 + · · · + − , i ∂u ∂u ∂u ∂u i p − br ∂u i M
r =1
so M m M ∂λ ∂br ∂s1 N −1 ∂s N ( p − b ) = p + · · · + ( p − b ) − k ( p − bs ) . s s r ∂u i ∂u i ∂u i ∂u i s=1
r =1
s=1
Comparing coefficients of p N +M−1 in gives
∂λ ∂u i
M
s=1 ( p
s=r
− bs ) in these two expressions
∂s1 N +1 = i . λ (ξi ) ∂u Hence ηii = η
∂ ∂ , i i ∂u ∂u
=−
∂ 1 = i λ (ξi ) ∂u
−
1 s1 . N +1
(11)
This Egoroff property is equivalent to a potentiality condition on the (3, 0)-tensor c, namely that the tensor ∇c is totally symmetric. Since the metric is flat one may, in flat-coordinates, integrate by Poincaré’s lemma and express everything in terms of a prepotential F which satisfies the WDVV equations. Collecting these results together one obtains:
Logarithmic Deformations of WDVV Solutions
11
Proposition 5. The flat metric (3) and totally symmetric (3,0) tensor (4), with λ given by λ = p N +1 + s1 p N −1 + . . . + s N +
M
ki log( p − bi ), ki constant
i=1
define, on the space M(M,N ) a solution to the WDVV equations. Geometrically they define a semi-simple, associative, commutative algebra with unity on the tangent space T M compatible with the flat metric. Before giving some examples, it must be remarked that we do not have a Frobenius manifold, just a solution to the WDVV equations. As was remarked in one of the earliest papers on water-bag reductions, such reductions do not have a scaling symmetry and this fact manifests itself in the non-existence of an Euler vector field, the existence of which is part of the definition of a Frobenius manifold (though it should be remarked that some authors do not require such a field in their definition, denoting manifolds with such a field as a conformal Frobenius manifold). Example 6. N = 0, M = 2. In the above proofs it has been assumed that N = 0. However one may adapt these proofs to deal with this case. In particular, the identity field, normally associated to the variable s N , has to be carefully defined. With λ( p) = p + k1 log [ p − (t1 + t2 )] + k2 log [ p − (t1 − t2 )] one obtains the prepotential 1 F= k1 (t1 + t2 )3 + k2 (t1 − t2 )3 + 2k1 k2 t22 log t2 . 6 Note that if the condition k1 + k2 = 0 is imposed, one obtains, after some rescalings, the solution obtained by Chang [5]. This example was the original motivation of this work. Lemma 7. ∂ ∂ ∂ = 0, α, β, γ distinct, c , , ∂bα ∂bβ ∂bγ kα kβ ∂ ∂ ∂ c , , , α = β, = ∂bα ∂bα ∂bβ bβ − bα k α kr ∂ ∂ ∂ = kα λ+ (bα ) + , , , c ∂bα ∂bα ∂bα b − br r =α α ∂ ∂ ∂ = 0, α = β, c , , ∂bα ∂bβ ∂sγ ∂ ∂ ∂ c = kα (bα ) N −γ , , , ∂bα ∂bα ∂sγ ∂ ∂ ∂ = kα Sβ+γ (s1 , . . . , s N , bα ), c , , ∂bα ∂sβ ∂sγ M ∂ ∂ ∂ (0) (1) = Rα+β+γ c , , (s1 , . . . , s N ) + k j Rα+β+γ (s1 , . . . , s N , b j ), ∂sα ∂sβ ∂sγ j=1
12
J. T. Ferguson, I. A. B. Strachan (0)
(1)
where Sσ , Rσ and Rσ are polynomial functions of their respective variables, and independent of all ki ’s. (0) In particular, the term independent of k j , Rα+β+γ (s1 , . . . , s N ), is precisely the value of c(∂sα , ∂sβ , ∂sγ ) found from (4) using the polynomial λ+ ( p) as the Landau-Ginzburg potential (2). Proof. Here we write λ ( p) = M
ν( p)
j=1 ( p
− bj)
,
where ν( p) = λ+ ( p)
M
(p − bj) +
j=1
= (N + 1)
M+N
M j=1
kj
( p − bk ) ,
k= j
(p − ξj) .
j=1
After the substitution p → 1/ p˜ we will have cause to refer to the polynomial 1 N µ( p) ˜ = p˜ λ+ = (N + 1) + (N − 1)s1 p˜ 2 + (N − 2)s2 p˜ 3 + · · · + s N −1 p˜ N . p˜ (bbb) From the definition (4), M kα kβ kγ ∂ ∂ ∂ j=1 ( p − b j ) = res dp . , , c ν=0 ( p − bα )( p − bβ )( p − bγ ) ∂bα ∂bβ ∂bγ ν( p) This is evaluated by deforming the contour to encompass the poles at p = ∞ and possibly at p = bα if there is repetition in the b’s. The residue at infinity is zero, and so in particular c(∂bα , ∂bβ , ∂bγ ) = 0 for α, β, γ distinct. For the case (α, α, β), the pole at p = bα is simple, and the result follows immediately, noting that ν(bα ) = kα k=α (bα − bk ). For the case α = β = γ , the pole is second order, and is evaluated directly as ∂ kα3 ∂ ∂ k=α ( p − bk ) = − res dp , c , , 2 p=bα ( p − bα ) ∂bα ∂bα ∂bα ν( p) d k=α ( p − bk ) = −kα3 . dp p=bα ν( p) (bbs) p N −γ M kα kβ ∂ ∂ ∂ j=1 ( p − b j ) c =− res dp, , , ν=0 ( p − bα )( p − bβ ) ∂bα ∂bβ ∂sγ ν( p) kα kβ = res + res + res p=∞ p=bα p=bβ ( p − bα )( p − bβ ) p N −γ M j=1 ( p − b j ) × dp. ν( p)
Logarithmic Deformations of WDVV Solutions
13
Once again there is no pole at infinity, and there exists a (simple) pole at p = bα only if α = β. The result again follows from ν(bα ) = kα j=α (bα − b j ). (sss)
∂ ∂ ∂ c , , ∂sα ∂sβ ∂sγ
= res
p 3N −α−β−γ
p=∞ λ ( p) +
M
j=1 ( p
M
j=1 ( p
M
− bj) +
− bj)
j=1 k j
k= j ( p − bk ) ⎤−1
⎡ M kj p 3N −α−β−γ ⎣ ⎦ = res 1 + p=∞ λ+ ( p) λ+ ( p)( p − b j )
dp,
dp.
j=1
This is expanded as a Taylor series in x =
∂ ∂ ∂ , , c ∂sα ∂sβ ∂sγ
k j /λ+ ( p)( p − b j ) to give a series of terms
=
∞
(i) R˜ α+β+γ ,
i=0
where R˜ σ(i)
⎤i ⎡ M 3N −σ k 1 p j ⎦ dp. ⎣ = (−1)i res p=∞ λ+ ( p) λ+ ( p) p − bj j=1
(0) (0) So, in particular, Rα+β+γ := R˜ α+β+γ = res
∂sα λ+ ∂sβ λ+ ∂sγ λ+
p=∞
λ+
dp is cαβγ from the A N
orbit space corresponding to λ+ . M ki Rσ(1) (s1 , . . . , s N , bi ), R˜ σ(1) (s1 , . . . , s N , b1 , . . . , b M ) can be decomposed as i=1 where p 3N −σ dp, p=∞ ( p − b)(λ+ ( p))2 1 = res p˜ σ −N −1 d p. ˜ (1 − b p)(µ( ˜ p)) ˜ 2 p=0 ˜
Rσ(1) (s1 , . . . , s N , b) = − res
This is seen to be zero for σ ≥ N + 1, and 1/(N + 1)2 for σ = N . For σ < N it is a pole of order N + 1 − σ and can be evaluated as N −σ 1 d 1 . (12) (N − σ )! d p˜ (1 − b p)(µ( ˜ p)) ˜ 2 p=0 ˜
Clearly this evaluates to a polynomial in {s1 , . . . , s N , b}. Finally, by making the substi(i) tution p = 1/ p˜ it can be seen that R˜ σ = 0 for i ≥ 2. (bss) Proceeding as in the (sss) case, we are led to c
∂ ∂ ∂ , , ∂bα ∂sβ ∂sγ
= kα
∞ i=0
(i)
Sβ+γ ,
14
J. T. Ferguson, I. A. B. Strachan
where Sσ(i) = (−1)i+1
⎡ ⎤i M kj p 2N −σ 1 ⎣ ⎦ dp, res p=∞ p − bα (λ+ ( p))i+1 p − bj j=1
= (−1)i+1
⎡ ⎤i M kj p˜ σ −2N 1 d p˜ ⎣ ⎦ res −1 − 2 , p˜ − bα (λ+ ( p˜ −1 ))i+1 p˜ −1 − b j p˜ p=0 ˜
= (−1) res
p˜ σ −N −1+i(N +1)
i
p=0 ˜
(1 − bα p)(µ( ˜ p)) ˜ i+1
$
j=1
kj 1 − b j p˜
%i d p. ˜
(i)
From this we can see that Sσ = 0 for i ≥ 1. This leaves only p˜ σ −N −1 d p, ˜ (1 − bα p)µ( ˜ p) ˜ p=0 ˜
Sσ := Sσ(0) = res
which is zero for σ ≥ N + 1, and 1/(N + 1) for σ = N , whilst for σ ≤ N − 1 it may be evaluated as N −σ d 1 1 Sσ = . (13) (N − σ )! d p˜ (1 − bα p)µ( ˜ p) ˜ p=0 ˜
For the Frobenius structure on the space of polynomials λ( p) = p N +1 + s1 p N −1 + . . . + s N , the variables si inherit a scaling symmetry from the scaling of the polynomial. Namely if p → p and we ask λ → N +1 λ, then we require si → i+1 si . Thus we conclude si has degree i + 1. For the water-bag reduction λ( p) = p
N +1
+ s1 p
N −1
+ · · · + sN +
M
ki log( p − bi ),
i=1
the same degrees may be attached to the coefficients {si }, and to preserve homogeneity of the arguments of the logarithms, each bi is assigned degree 1. If, in addition, a non-geometrically justified degree of N + 1 is assigned to each ki , then the regularised function λreg = λ( p) − ki log p, introduced in (6), is homogeneous of degree N + 1. Lemma 8. Under the rescalings si → i+1 si i = 1 . . . N , bi → bi i = 1 . . . M , ki → N +1 ki i = 1 . . . M, the prepotential F associated to the water-bag reduction (5) is homogeneous of degree 2N + 4.
Logarithmic Deformations of WDVV Solutions
15
Proof. This may be verified from the explicit expressions for the components of the tensor c(∂, ∂ , ∂ ) obtained in Lemma 7, remembering to add the degrees lost from differentiating along ∂, ∂ , ∂ . kr In particular, for c(∂bα , ∂bα , ∂bα ) = kα λ+ (bα ) + r =α bkαα−b , we note that λ+ (bα ) = r (N + 1)(bα ) N + (N − 1)s1 (bα ) N −1 + · · · + s N −1 has degree N . (0) (1) The degrees of the polynomials Rσ , Rσ and Sσ , when they are not zero or constant, can de determined from the differential expressions (12), (13) and the corresponding expression for Rσ(0) , which is
Rσ(0)
=
⎧ ⎪ ⎪ ⎨ 1 ⎪ ⎪ ⎩ (2N +1−σ )!
0 −1/(N + 1) 2N +1−σ d d p˜
σ ≥ 2N + 2 σ = 2N + 1 p=0 ˜
1 µ( p) ˜
σ ≤ 2N
.
In this the degree of zero is undetermined, whilst for the middle case, the degree of a constant is 0. Integrating with respect to sα ,sβ and sγ adds to this degree (α + 1) + (β + 1) + (γ + 1) = σ + 3 = 2N + 4. In the final case, if p˜ = 1/ p is considered to have degree −1, then µ( p) ˜ has degree zero. Thus on differentiation we obtain the quotient of two homogeneous polynomials with relative degrees 2N +1−σ . Evaluation at p˜ = 0 merely makes this the ratio of constant terms, so that Rσ(0) has degree 2N + 1 − σ . Integrating (1) will add σ + 3 to this, making 2N + 4 as required. Sσ and Rσ proceed similarly. The degrees of the flat coordinates {t i , i = 1 . . . N } are inherited from the polynomial transformations rules relating them to the si . They can also be deduced from the Puiseaux series (10), in which we require both p and k to scale with degree 1, so that the degree of t i is N + 2 − i. We now draw together some simple observations, which follow immediately from Lemmas 3, 7 and 8. Proposition 9. The prepotential is at most quadratic in the parameters ki , that is, up to quadratic terms in the flat coordinates: F(t 1 , . . . , t N , b1 , . . . , b M ) = F (0) (t 1 , . . . , t N ) ki F (1) (t 1 , . . . , t N , bi ) + i
+
ki k j F (2) (bi , b j ),
i= j
where F (0) , F (1) , F (2) are independent of the parameters ki . F (0) is the prepotential for the C N /A N orbit space with λ+ as the Landau-Ginzburg potential, and as such is a polynomial in the flat coordinates {t 1 , . . . , t N }. F (1) is also a polynomial, and F (2) (bi , b j ) =
1 i (b − b j )2 log(bi − b j )2 . 8
16
J. T. Ferguson, I. A. B. Strachan
In place of quasi-homogeneity we have deg F (0) = 2N + 4, deg F (1) = N + 3, deg F (2) = 2, (modulo quadratic ter ms). The structure functions for the Frobenius algebra are always at most linear in the parameters ki , that is: (0) γ (i) γ cαβ γ = cαβ + ki cαβ , i
where the
(0) γ cαβ
and
(i) γ cαβ
are independent of the parameters.
An important class of solutions are polynomial in the flat coordinates. Corollary 10. For M = 1, the prepotential on the space of functions λ( p) = p N +1 + s1 p N −1 + · · · + s N + k log( p − b) is polynomial in the flat coordinates {t i , b} Conversely, if the prepotential is polynomial in the flat coordinates then M = 1 (or M = 0). Proof. This is an immediate consequence of the decomposition of F given in Proposition 9: the component F (2) contains all non-polynomial terms appearing in F, and is present if and only if M ≥ 2. We finish this main section with two simple examples. Example 11. • N = 2, M = 1. With λ( p) = p 3 + t2 p + t1 + k log( p − t3 ) one obtains the prepotential 1 1 1 4 1 1 t2 − k (t22 t3 + t2 t33 ) − k t35 . F = t12 t2 − k t1 t32 − 6 2 216 6 20 • N = 1, M arbitrary. In this case one has λ( p) = p 2 + t1 +
M
ki log( p − bi ).
i=1
With this, Lemmas 3 and 7 give, on integrating, the following prepotential: M t1 bi2 bi4 1 3 1 + ki ki k j (bi − b j )2 log(bi − b j )2 . F = − t1 + + 12 2 12 8 i=1
i= j
We note that the z 2 log z-type terms have appeared in the WDVV-literature before (see, for example, [13,18]) but one normally considers these as being derived as examples of dual Frobenius manifolds [10]. Their functional form suggests the type of term that may be present in a construction of deformed solutions for other Coxeter group orbit spaces.
Logarithmic Deformations of WDVV Solutions
17
4. Geometric and Algebraic Properties In this section we study certain geometric and algebraic properties of the manifold. 4.1. Geometric properties. An important additional structure on a Frobenius manifold is an additional flat metric known as the intersection form. It plays a vital role in the understanding of various properties of the manifold, such as the monodromy properties of the Gauss-Manin connection and associated bi-Hamiltonian structures. Following this, we define a second metric on manifold; while this is not flat, it shares many properties with the intersection form of a genuine Frobenius manifold. Definition 12. We define the metric g on M as g −1 (ω1 , ω2 ) = i E (ω1 , ω2 ), where E is the Euler vector field (1). It follows immediately from this that g(E ◦ u, v) = η(u, v) and, in components, ij
g i j = ck E k . To understand the scaling properties of this metric we introduce an extended Lie derivative Lext X , Lext X = LX +
M r =1
kr
∂ , ∂k r
i... j
so, for an arbitrary tensor ωa...b , M ' ext (i... j ∂ i... j i... j k r r ωa...b . L X ω a...b = (L X ω)a...b + ∂k r =1
This may be used to clarify the pseudo-quasi-homogeneity properties of the various structures, for example Lext E F = (3 − d)F,
d=
N −1 . N +1
Similarly the metrics g and η have various pseudo-quasi-homogeneity properties: Lemma 13. The following equations hold: [e, E] = e, −1 = (d − 1)g −1 , Lext η−1 = (d − 2)η−1 , Lext E g E −1 = η−1 , Lext e g
−1 = 0. Lext e η
18
J. T. Ferguson, I. A. B. Strachan
However, the metric g is not flat, and moreover, despite being linear in t 1 the pencil = g −1 + η−1 does not define an almost compatible pencil (the tensor E◦ : T M → T M fails to satisfy the Nijenhuis condition [8]), let alone a compatible pencil. The role of this second metric is therefore unclear. Given the origin of these structures in reductions of the dKP hierarchy one would expect bi-Hamiltonian structures of differential-geometric type. Non-local bi-Hamiltonian structures have been found recently [6]. However, these do not become the standard structures in the limit as ki → 0. Another possibility for the metric g is −1 g
g=
ηii ui
du i2 ,
where ηii are the components of the metric η in canonical coordinates, given in (11). This choice for g coincides with the previous case when one is dealing with a true Frobenius manifold. It does define a non-local bi-Hamiltonian structure [21] but finding its form in the flat-coordinate system for the metric η is problematic. A related problem is to relate the Euler vector field (1) with the vector field E =
M+N i=1
ui
∂ , ∂u i
the two being equal in the undeformed case. The various structures on the manifold may be encoded in the deformed (or Dubrovin) connection D
∇ X Y = ∇ X Y + z X ◦ Y,
z ∈ P1 .
For this connection to be torsion free and flat one requires commutativity and associativity of the multiplication, flatness of the Levi-Civita connection ∇ and potentiality, and visa-versa; and these properties are satisfied for water-bag models. This therefore establishes the integrability of the system D ∇α ζβ = 0. Solutions ζα are automatically gradients, ζα = ∂α t˜. Expanding t˜ = n ψ (n) z n yields the recursion relation (n−1) ∂ 2 ψ (n) k ∂ψ = c . ij ∂t i ∂t j ∂t k (0)
Starting with the seed solutions ψi = t i , i = 1, . . . dim M one may construct a fun(n) damental system of solutions. The resulting structure functions ψi are in involution with respect to the Dubrovin-Novikov type Poisson bracket associated with the metric η, and consequently form a set of conserved densities for an integrable system of hydrodynamic type. It was by reconstructing these structure functions that Chang [5] was led to the solution of the WDVV equations related to the two-dimensional reduction λ( p) = p − c
log( p − p1 ) log( p − p˜ 1 )
of the dispersionless KP hierarchy, which was the original motivation for the work in this paper.
Logarithmic Deformations of WDVV Solutions
19
4.2. Algebraic deformation theory. In this section we examine the linearity of the structure functions of the Frobenius algebra with respect to the parameters k i from the point of view of deformation theory (we follow the notation of [4]). Let . . × V, → V |m linear in each arguement}. M k (V ) = {m : )V × .*+ k
Recall that a bilinear map c ∈ M 2 (V ) defines an associative structure if and only if [c, c]G = 0, where [·, ·]G is the Gerstenhaber bracket. Owing to the super-Jacobi identity one has δc2 = 0, where δc = [c, ·]G
: M • (V ) → M •+1 (V ),
and this gives rise to the Hochschild complex of (V, c). From Proposition 9 we have the following structure: c(k) = c(0) + ki c(i) , i
that is, linearity of the structure functions of the associative algebra. Decomposing the condition [c(k), c(k)]G = 0 for all k one obtains the following conditions: [c(0) , c(0) ]G = 0,
[c(0) , c(i) ]G = 0, i = 1, . . . , M,
[c(i) , c( j) ]G = 0, i, j = 1, . . . , M. Thus each c(i) , i = 0, 1, . . . , M separately defines an associative structure on T M. Each of these define a map δc(i) and each c(i) is a cocycle with respect to each cohomology map δc( j) , that is: [c(i) , c(i) ]G = 0, i = 0, 1, . . . , M,
δc(i) c( j) = 0, i, j = 0, 1, . . . , M.
It is also interesting to note that the pair (◦, E) satisfies the conditions L X ◦Y (◦) = X ◦ LY (◦) + Y ◦ L X (◦) and Lext E (◦) = d ◦, the former following from the semi-simplicity of the multiplication, and the latter from the pseudo-scaling property of the free energy. If one had L E (◦) = d ◦, then one would have a F-manifold [15]. Here one has a modified version, where the scaling condition is replaced by the pseudo-scaling condition. One could also regard the multiplication as defining a deformation of the F-manifold based on the orbit space C N /A N .
20
J. T. Ferguson, I. A. B. Strachan
5. Further Results An immediate question these results raise is whether or not the ideas may be applied to other classes of Frobenius manifolds, the obvious potential generalization being to other Coxeter orbit spaces Cn /W , for an arbitrary Coxeter group W. By this we mean is there a prepotential schematically of the form F(t, b) = FW (t) + k F (1) (t, b) + k 2 F (2) (t, b) based on the Cn /W prepotential FW which is pseudo-quasi-homogeneous with respect to some suitable Euler field? For the group W = Bn this is immediate, using the idea originally due to Zuber [23], of embedding the group Bn as a subgroup of A2n+1 , or geometrically, of regarding the Bn Frobenius manifold as the induced manifold on certain hyperplanes submanifolds in the A2n+1 Frobenius manifold. This idea generalizes to water-bag type reductions and this will be presented in Sect. 5.1. Another possible generalization, already alluded to above, is to replace the polynomial part of λ by an arbitrary rational function, generalizing the construction of [1,2]. The Frobenius manifold structure on the space of such rational functions has been much studied and these results can be generalized to include logarithmic terms. These results are presented in Sect. 5.3
5.1. B N -type reductions. The Bn Frobenius manifold may be regarded as a submanifold in the A2n+1 Frobenius manifold [23]. This idea generalizes to water-bag type potentials. Proposition 14. On the space of functions λ( p) = p 2N +2 + s1 p 2N + s3 p 2N −2 + · · · + s2N +1 +
M
ki log( p 2 − bi2 ),
i=1
formulas (3) and (4) define a pseudo-quasi-homogeneous solution of the WDVV equations. Proof. The function λ above is obtained from the following water-bag deformation of the A2N +1 superpotential: λ A N ( p) = p 2N +2 + s1 p 2N + s2 p 2N −1 + s3 p 2N −2 + · · · + s2N +1 +
M
ki log( p − bi ) +
i=1
M
ki log( p − bi+M ).
i=1
We restrict this to the submanifold sr = 0 for r even, bi+M = −bi for 1 ≤ i ≤ M. The restriction of the sr may be achieved in flat coordinates by setting all t i of odd degree (i.e. even i) to zero. We introduce new flat coordinates b˜i = bi and d˜i = bi + bi+M
Logarithmic Deformations of WDVV Solutions
21
(i = 1, . . . , M), and restrict to d˜i = 0. We check the following components of the multiplication tensor restricted to zero on this hyperplane: ˜
r
cd˜ k ˜ ,
ct˜
bi b˜ j
bi b j
˜
cd˜ k r bi t
˜
ctdrkt s
s
for r odd, ct˜
bi t r
u
for r, s odd, ctt r t s
for r even,
for r odd, s even, for r, s odd, u even.
Polynomial terms arising in these components can be seen to vanish from consideration of their degrees; all polynomials in {t 1 , . . . , t 2N +1 } of odd degree must vanish when all t i of odd degree vanish, whereas polynomials in {t i } of even degree are always multiplied by (at least) a factor of bi +bi+M for some i, and hence vanish on di = 0. Non-polynomial terms are given explicitly in Lemma 7.
5.2. D N -Waterbag models. The D N orbit space can be formulated in terms of the Landau-Ginzburg formalism, as described in [12]. Here the superpotential is λ( p) = p 2(N −1) +
N −2
ar p 2r −
r =0
2 1 a−1 . 2 p2
(14)
As in the B N -Waterbag models, we aim to preserve the symmetry p → − p in the water-bag model by considering the superpotential (the resulting solutions have also been obtained independently in [24]): λ( p) = p 2(N −1) +
N −2
2 1 a−1 + cs log( p 2 − bs2 ). 2 2 p m
ar p 2r −
r =0
(15)
s=1
Proposition 15. On the space of functions (15) formulas (3) and (4) define a pseudoquasi-homogeneous solution of the WDVV equations. The canonical coordinates are u i = λ(±qi ), where λ (±qi ) = 0. The flat coordinates follow the pattern of the A N models: the poles bα of the logarithms are flat coordinates, and then we take the flat coordinates of the D N models: a−1 and {t0 , . . . , t N −2 } defined by the inverse series ∞
p=k−
tr 1 , 2(n − 1) k 2r +1 r =0
−2 where λ+ ( p) = p 2(N −1) + rN=0 ar p 2r = k 2(N −1) for p ∼ ∞. Thus the water-bag deformation can be extended to all classical Coxeter group orbit spaces. It would be of interest to see if these ideas can be applied to an arbitrary Coxeter group orbit space.
22
J. T. Ferguson, I. A. B. Strachan
5.3. Rational water-bag potentials. Proposition 16. On the space of functions λ( p) = p N +1 + s1 p N −1 + · · · + s N % K $ vi,1 vi,L i + ··· + + ( p − vi,0 ) ( p − vi,0 ) L i i=1
+
M
ki log( p − bi ),
i=1
the formulas (3) and (4) define a solution of the WDVV equations. Proof. Canonical coordinates are found as in Lemma 2. The flat coordinates are {b1 , . . . , b M } together with those obtained for the purely rational case [1]. Namely invert λ+ ( p) = p N +1 + s1 p N −1 + · · · + s N about p = ∞ using the Puiseaux series (10), and invert vLi vi,1 λ−i ( p) = + ··· + ( p − vi,0 ) ( p − vi,0 ) L i for p ∼ vi,0 as p=
xi,1 xi,L i 1 + ··· + L , xi,L i +1 + Li w w i
where λ−i = w L i , and xi,L i +1 = L i vi,0 . The flat coordinates are then {t α , xβ,γ , bδ }. In these coordinates the metric has only the following non-zero components: ∂ 1 ∂ η =− , δα+β,N +1 , α β ∂t ∂t N +1 1 ∂ ∂ = − δ j+k,L i +2 , , η ∂ xi, j ∂ xi,k Li ∂ ∂ = kα δαβ . η , ∂bα ∂bβ Note one may combine the results from Sects. 5.1 and 5.3 and consider Bn -type reductions of the rational case, where the superpotential, including logarithmic terms, is an even function. The Dn water-bag models are a special case of such superpotentials. In the above proposition the location of the poles {si } and the logarithmic poles {bi } were taken to be distinct. However, a modified version of the above proposition may be formulated which takes into account possible coincidences in these sets. Rather than state this we give an example. Example 17. The superpotential λ( p) = p 2 + t1 +
t2 + k log( p − t3 ) ( p − t3 )
leads to the following solution of the WDVV equation: F=
1 3 1 1 1 1 3 t + t1 t2 t3 − k t1 t32 − t22 + t22 log t2 + t2 t33 − k t34 . 12 1 2 4 2 3 12
(16)
Logarithmic Deformations of WDVV Solutions
23
This produces an interesting class of solutions, as no extra variables have had to be introduced, so in a sense they are true deformations of the original solution. The single pole case - generalizations of the above example - are isomorphic to deformations of the extended-affine-Weyl orbit space [11], since - (L) (A N +L ). H0,N +L+1 (N + 1, L) ∼ = C N +L+1 /W Explicitly this is given by a Legendre transformation (which acts on solutions of the WDVV equations, not just to those solutions which define Frobenius manifolds). Example 18. Applying the Legendre transformation S2 (using the notation of [9]) to the solution (16) yields the solution 1 1 1 1 1 1 2 1 Fˆ = tˆ1 + tˆ22 tˆ3 − k tˆ2 tˆ32 − tˆ14 + tˆ1 etˆ3 − k tˆ1 tˆ3 + tˆ2 tˆ32 + k 2 tˆ33 . 4 2 2 96 4 2 6 - 1 (A2 ). This defines a deformation of the extended-affine-Weyl space C3 /W One would expect that the associated dispersionless integrable systems would be related to water-bag type-reductions of the dispersionless Toda equations and their generalizations [3,7].
6. Open Problems Some open problems have already been outlined above; here we draw them together and raise some other open problems, potential generalizations and applications. •
•
• •
Can the construction be applied, independent of the Landau/Ginzburg construction, directly to an arbitrary Coxeter group orbit space, or more generally, to other orbit spaces? By this we mean, is there a Saito-type construction of these solutions? The absence of a flat ‘intersection form’ would seem problematic. A related question is whether one can formulate axiomatically a theory of pseudo-quasi-homogeneous solutions of the WDVV equations. The Frobenius manifold structure on the space of rational functions may be generalized to the space of branched coverings of an arbitrary Riemann surface (i.e. a Hurwitz space). All that is required for the direct calculation of the residues (3) and (4) is the meromorphicity of the derivatives of λ rather than the meromorphicity of λ itself. This suggests that one should look at generalizations where λ lies in some extension of the field of meromorphic functions. In a semi-simple Frobenius manifold there exist interesting submanifolds: discriminants and caustics [21]. What are the properties of such structures in the present case? What are the properties of the dispersionless integrable systems associated to such solutions of the WDVV equations, i.e. the water-bag reductions of the dKP hierarchy itself, and how are they encoded in the geometry of these pseudo-quasi-homogeneous manifolds? In particular, the (non-local) bi-Hamiltonian structure [7], especially in the flat coordinates system for the metric η is unknown in general. Can these dispersionless systems be deformed, and how does the form of such deformations follow from the geometry of the undeformed systems [5]?
24
•
J. T. Ferguson, I. A. B. Strachan
Finally, is there an algebraic description, say of the An -deformations, in terms of a deformed Milnor ring? Is there a field theoretic interpretation of these results in terms of a topological quantum field theory [16,17]. There also appears to be close connections, or at the very least, similarities with, the integrable hierarchies found by Milanov and Tseng in their study of the orbifold cohomology of the projective line [19].
We hope to address some of these problems in the future. Acknowledgements. James Ferguson would like to thank the Carnegie Trust for the Universities of Scotland for a research studentship.
References 1. Aoyama, S., Kodama, Y.: Topological conformal field theory with a rational W potential and the dispersionless KP hierarchy. Mod. Phys. Lett. A 9, 2481–2492 (1994) 2. Aoyama, S., Kodama, Y.: Topological Landau-Ginzburg with a rational potential, and the dispersionless KP hierarchy. Commun. Math. Phys. 182, 185–219 (1996) 3. Bogdanov, L.V., Konopelchenko, B.G.: Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type. Phys. Lett. A 330, 448–459 (2004) 4. Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. Berkeley Mathematics Lecture Notes, Vol. 10 Providence, RI: Amer. Math. Soc., 1999 5. Chang, J-H.: On the water-bag model of dispersionless KP hierarchy. J. Phys. A 39, 11217–11230 (2006) 6. Chang, J-H.: On the water-bag model of dispersionless KP hierarchy (II). J. Phys. A 40, 12973– 12985 (2007) 7. Chang, J-H.: Remarks on the waterbag model of dispersionless Toda hierarchy. http://arxiv.org/list/ 0709.3859v1, 2007 8. David, L., Strachan, I.A.B.: Compatible metrics on manifolds and non-local bi-hamiltonian structures. Int. Math. Res. Notices, 66, 3533–3557 (2004) 9. Dubrovin, B.: Geometry of 2D topological field theories in Integrable Systems and Quantum Groups. ed. Francaviglia, M., Greco, S., Springer Lecture Notes in Mathematics, 1620, Berlin-Heidelberg-New York: Springer,1996, pp. 120–348 10. Dubrovin, B.: On almost duality for Frobenius manifolds. In: Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 212, Providence, RI: Amer. Math. Soc., 2004, pp. 75–132 11. Dubrovin, B., Zhang, Y.: Extended affine Weyl groups and Frobenius manifolds. Comp. Math. 111, 167–219 (1998) 12. Eguchi, T., Yamada, Y., Yang, S.-K.: Topological field theories and the period integrals. Modern Phys. Lett. A 8, 1627–1637 (1993) 13. Feigin, M.V., Veselov, A.P.: Logarithmic Frobenius structures and Coxeter discriminants. Adv. in Math. 212(1), 143–162 (2007) 14. Gibbons, J., Tsarev, S.P.: Reductions of the Benney Equations. Phys. Lett. A 211, 19–24 (1996) 15. Hertling, C.: Frobenius manifolds and moduli spaces for singularities. Cambridge Tracts in Mathematics, 151, Cambridge: Cambridge University Press, 2002 16. Krichever, I.M.: The dispersionless equations and topological minimal models. Commun. Math. Phys. 143(2), 415–429 (1992) 17. Krichever, I.M.: The τ -function of the universal Whitham hierarchy, matrix models, and topological field theories. Comm. Pure Appl. Math 47, 437–475 (1994) 18. Martini, R., Hoevenaars, L.K.: Trigonometric Solutions of the WDVV Equations from Root Systems. Lett. Math. Phys. 65, 15–18 (2003) 19. Milanov, T.E., Tseng, H-H.: Equivariant orbifold structures on the projective line and integrable hierarchies. http://arxiv.org/list/math/0707.3172, 2007 20. Pavlov, M.V.: The Hamiltonian approach in classification and integrability of hydrodynamic chains. http://arxiv.org/list/nlin/0603057, 2006 21. Strachan, I.A.B.: Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures. Differential Geom. Appl. 20, 67–99 (2004) 22. Saito, K.: On a linear structure of a quotient variety by a finite reflection group. Preprint RIMS-288 (1979), Publ. RIMS Kyoto Univ. 29, 535–579 (1993) 23. Zuber, J.-B.: On Dubrovin Topological Field Theories. Mod. Phys. Lett. A 9, 749–760 (1994)
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24. Zuo, D.: Frobenius manifolds associated to Bl and Dl , revisited. International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm020, 24 pages, doi:10.1093/imrn/rnm020, published on May 24, 2007 Communicated by G.W. Gibbons
Commun. Math. Phys. 280, 27–76 (2008) Digital Object Identifier (DOI) 10.1007/s00220-007-0407-z
Communications in
Mathematical Physics
Counting BPS States on the Enriques Calabi-Yau Albrecht Klemm1 , Marcos Mariño2,3 1 UW-Madison Physics Department, 1150 University Avenue, Madison, WI 53706-1390, USA.
E-mail:
[email protected]
2 Department of Physics, Theory Division, CERN, CH-1211 Geneva, Switzerland.
E-mail:
[email protected]
3 Departamento de Matemática, IST, Lisboa, Portugal
Received: 25 August 2006 / Accepted: 7 August 2007 Published online: 7 March 2008 – © Springer-Verlag 2008
Abstract: We study topological string amplitudes for the FHSV model using various techniques. This model has a type II realization involving a Calabi-Yau threefold with Enriques fibres, which we call the Enriques Calabi-Yau. By applying heterotic/type IIA duality, we compute the topological amplitudes in the fibre to all genera. It turns out that there are two different ways to do the computation that lead to topological couplings with different BPS content. One of them gives the standard D0-D2 counting amplitudes, and from the other one we obtain information about bound states of D0-D4-D2 branes on the Enriques fibre. We also study the model using mirror symmetry and the holomorphic anomaly equations. We verify in this way the heterotic results for the D0-D2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms, and up to genus three. This model turns out to be much simpler than the generic B-model and might be exactly solvable. Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Heterotic/Type II Duality and Fg Couplings . . . . . . . . . . 2.1 The Fg couplings in heterotic string theory . . . . . . . . . 2.2 BPS content of the Fg couplings . . . . . . . . . . . . . . The FHSV Model . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The heterotic side of the FHSV model . . . . . . . . . . . 3.2 The type II side of the FHSV model . . . . . . . . . . . . Heterotic Computation of the Fg Couplings . . . . . . . . . . . 4.1 The geometric reduction . . . . . . . . . . . . . . . . . . 4.2 The BHM reduction . . . . . . . . . . . . . . . . . . . . . Geometric Computation of the BPS Invariants in the Fibre . . . The B-Model for an Algebraic Realization of the FHSV Model 6.1 The geometric description of the Enriques Calabi-Yau . . .
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6.1.1 Periods and prepotential . . . . . . . . . . . . . . . 6.1.2 Mirror symmetry on an algebraic realization . . . . . 6.1.3 Arithmetic expressions for the B-model quantities . . 6.2 Topological string amplitudes from the reduced B-model 6.2.1 Genus one amplitude . . . . . . . . . . . . . . . . . 6.2.2 Propagators and higher genus amplitudes . . . . . . 6.2.3 Comparison with the heterotic results. . . . . . . . . 6.2.4 Extending the results to the CY threefold. . . . . . . A. Theta Functions and Modular Forms . . . . . . . . . . . . . B. Lattice Reduction . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The solution of topological string theory on Calabi-Yau (CY) manifolds is an important problem with applications both in string theory and in enumerative geometry. Impressive tools have been developed in an extensive effort over the last twelve years, most notably mirror symmetry, localization, deformation, large N -dualities, cohomological calculations in D2-D0 brane moduli spaces, and heterotic/type II duality. Nevertheless, a complete solution for the topological string amplitudes on compact CY manifolds is presently out of reach.1 Applicable to the compact case are B-model calculations based on mirror symmetry and the holomorphic anomaly equation of [8,31], and A-model calculations based on deformation arguments and relative Gromov-Witten invariants, which are calculated by localization [20,44]. Both methods calculate the amplitudes genus by genus. The former provides contributions in all degrees at once, but only up to a holomorphic function (the so-called holomorphic ambiguity), whose determination requires a further finite amount of data, which in practice are provided in a rather unsystematic case by case analysis. The A-model calculation proceeds degree by degree and the combinatorial complexity is in general prohibitive. Many (potentially all) compact CY manifolds are connected by transitions through complex degenerations. The behaviour of the topological string amplitudes at these transitions is relatively well understood. In view of this situation it is important to identify the compact CY manifold where the topological string is most tractable. There is a compact example where topological string theory is exactly solvable, namely K3×T2 . The topological string amplitudes are all zero for genus g ≥ 2, at g = 0 one has just the classical piece of the prepotential, and for g = 1 one just has the elliptic η function typical of the two-torus [7]. Hence this example is too simple, and this is due to the extended N = 4 supersymmetry of the corresponding type II theory, related in turn to the SU (2) holonomy. N = 2 supersymmetry and the generic SU (3) holonomy can be obtained by fibering K3 over P1 . In these examples one can use heterotic/type II duality or special properties of the Hilbert scheme of complex surfaces to write down explicitly all genus topological amplitudes for the classes in the K3 fiber [43,33]. However the decisive step in going from the surface to the threefold, i.e. the inclusion of the base and mixed classes, is hard. Results up to g = 2 have been obtained in [33]. This motivates one to consider the problem on a special CY with intermediate holonomy SU (2) × Z2 constructed in [10,53,18], as an orbifold w.r.t. a free Z2 involution 1 In contrast, on non-compact toric Calabi-Yau manifolds the problem is completely solved by localization, and much more efficiently by using the topological vertex [1], which relates the genus expansion to a 1/N expansion of Chern-Simons theory.
Counting BPS States on the Enriques Calabi-Yau
29
of K3×T2 . The resulting space exhibits a K3 fibration with four fibres of multiplicity two over the four fixed points of the involution in the base, which are Enriques surfaces. A good deal of the nontrivial geometry of this CY comes from the geometry of the Enriques fibers, and we will call it the Enriques CY manifold. The string vacuum obtained by compactifying type II theory on the Enriques CY has N = 2 supersymmetry and is known as the FHSV model. The Z2 lifts instanton zero modes related to the T2 so that simple instanton effects which cancel in the N = 4 theory contribute to the N = 2 effective action. The Enriques CY seems to be the simplest CY compactification with nontrivial topological string amplitudes. Moreover it has a dual description as an asymmetric orbifold of the heterotic string [18]. Various aspects of the FHSV model have been studied in the past, see for example [4,13]. In particular, the genus one topological string amplitude of the FHSV model was determined by Harvey and Moore in [26]. This paper makes a first step to determine the topological string amplitudes of the FHSV model using heterotic/type II duality, B-model techniques and the cohomology of D2-D0 brane moduli spaces. Although we haven’t been able to solve the model in full, we will present various results which show that indeed the model has some simplifying features that might lead to a complete solution. The simplicity of the model is also apparent from a mathematical point of view, and it turns out that the techniques developed in [20,44] lead to simple recursive formulae for the Gromov-Witten invariants of the Enriques CY at low genera [45]. A heterotic one-loop calculation is used to determine the Fg couplings in the fibre direction, using the techniques developed in [24,3,43]. It turns out that this calculation can be made in two different ways, which we call the geometric reduction and the Borcherds-Harvey-Moore (BHM) reduction. The resulting expressions are appropriate for different regions in moduli space, and they turn out to have a different enumerative meaning. The result obtained in the geometric reduction corresponds to the large radius limit, reproduces the geometric expectations one has for a generating functional of Gromov-Witten invariants, and as explained in [21] counts D0-D2 bound states. We suggest that the result obtained in the BHM reduction is related to a counting of D0-D2-D4 bound states in a different region of moduli space. The result of [26] for the genus one amplitude was in fact implicitly obtained in the BHM reduction. We then study the model by using mirror symmetry and the holomorphic anomaly equations. To do that, we first find an algebraic realization of the CY manifold involved in the FHSV model, and we find its mirror by using standard techniques. This leads to a model which is still very difficult to solve due to the presence of ten deformation parameters. To avoid this problem, we find a reduced model with only two fibre parameters which is obtained by blowing down the E 8 part of the homology of the original type A model. This model turns out to be very tractable, and all relevant quantities can be expressed in closed form in terms of modular forms. We present explicit formulae for the topological string amplitudes in the fiber up to genus 3 which agree with the predictions of heterotic/type II duality. The holomorphic anomaly equations turn out to be extremely simple due to various exceptional properties of the model (like the absence of worldsheet instanton corrections for the prepotential, already pointed out in [18]). Although we do not have enough information to fix the holomorphic ambiguity in the base (except at genus 2, where explicit results have been obtained in [45]), we make a natural conjecture that leads to consistent results in genus three and four and might hold in general. The organization of this paper is as follows. In Sect. 2 we review the heterotic computation of topological string amplitudes in K3×T2 compactifications. In Sect. 3 we
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present various results on the FHSV model, both in its type IIA and its heterotic incarnations. In Sect. 4 we compute the topological string amplitudes in the heterotic theory, in both the geometric and the BHM reductions. In Sect. 5 we present an interpretation of the results in the geometric reduction in terms of BPS invariants associated to D2-D0 bound states, following [21,31]. In Sect. 6 we study the mirror B-model for a “reduced” version of the theory, and we compare the results with those obtained in the heterotic computation. Appendix A collects some useful formulae for modular forms. Appendix B summarizes some results about the lattice reduction technique which is used in the heterotic computation. 2. Heterotic/Type II Duality and Fg Couplings 2.1. The Fg couplings in heterotic string theory. The duality between heterotic compactifications on K3×T2 and type II theory on Calabi-Yau’s which are K3 fibrations [30] has been a source of very rich information in string theory (see for example [41] for a review of results). One of the most interesting applications of this duality is the computation of the topological string amplitudes Fg . These Fg couplings are F-terms for compactifications of type II theory on Calabi-Yau manifolds [8,2], and they give terms in the four-dimensional effective action of the form (2.1) Fg (t, t¯)T 2g−2 R 2 + · · · , where T is the graviphoton field strength and R is the Riemann curvature. It turns out that, on the heterotic side, all these couplings appear at one-loop [3] and can be computed in closed form [43]. In this section we will briefly review the computation of the Fg amplitudes by using heterotic/type II duality. One drawback of this duality is that it only enables us to compute this amplitude in the limit of infinite volume for the basis of the K3 fibration. This is due to the fact that, under heterotic/type IIA duality, the heterotic dilaton S is identified with the complexified area of the base P1 of the fibration, volC (P1 ) = 4π S =
4π . 2 ghet
(2.2)
Therefore the perturbative regime of the heterotic string corresponds to the limit volC (P1 ) → ∞. On the other hand, the duality gives closed, elegant expressions for all the Fg amplitudes restricted to fiber classes in terms of modular forms. These classes correspond to the Picard lattice of the K3 fibre, which will be denoted by Pic(K3) (more precisely, one has to consider the monodromy-invariant part of the Picard lattice). Before stating the main results, we introduce some notation on Narain lattices and Siegel-Narain theta functions. Given a lattice of signature (b+ , b− ), a projection P is an orthogonal decomposition of ⊗ R into subspaces of definite signature: P : ⊗ R Rb+ ⊥ Rb− . We will denote by p± = P± ( p) the projections onto the two factors. The Siegel-Narain theta function is defined as exp πiτ ( p + β/2)2+ + πiτ ( p + β/2)2− + πi( p + β/2, α) . (2.3) (τ, α, β) = p∈
Counting BPS States on the Enriques Calabi-Yau
31
When α = β = 0, we will simply write (τ ). As usual, we write q = exp(2πiτ ), and τ2 = Im τ . We will consider compactifications of the heterotic string on K3×T2 and orbifolds thereof. These compactifications lead to effective theories with N = 2 supersymmetry in four dimensions, and they involve Narain lattices with b+ = 2, which can be identified with the two right-moving directions along T2 . Therefore, we can identify Rb+ C, and we will represent p+ ∈ R2 by p R ∈ C, so that p+2 is given by | p R |2 . The general expression for the Fg couplings in these compactifications is given by the one-loop integral [3] (see [32] for an excellent introduction to one-loop corrections in string theory) ϑ ab (τ ) d 2τ 1 i a+b+ab Fg = (−1) (2.4) ∂τ Z gint ab . 4 η(τ ) F τ2 |η| even π In this equation, the integration is over the fundamental domain of the torus, Z gint ab = : (∂ X )2g :
(2.5)
is a correlation function evaluated in the internal conformal field theory, and X is the complex boson corresponding to the right-moving modes on the T2 . The evaluation of the correlation function reduces to zero modes [3], and the final result involves insertions of the right-moving momentum p R . For this reason, it is convenient to introduce the Narain theta function with an insertion, 2g−2 g (τ, α, β) = pR exp πiτ ( p + β/2)2+ + πiτ ( p + β/2)2− + πi( p + β/2, α) . p∈
(2.6) In general the internal CFT will be an orbifold theory and we will have to consider different orbifold blocks, which will be labelled by J . For each of these blocks there is a different Narain lattice J with different α J , β J , and we will denote g
g
J = J (τ, α J , β J ). The integral (2.4) can now be written as [3,48] g 2g−3 Fg = d 2 τ τ2 IJ , F
(2.7)
(2.8)
J
where g
IJ =
g (q) g P (τ ) f J (q). Y g−1 J
(2.9)
g (q) is a one-loop correlation function of the bosonic fields and is In this equation, P given by [37,3] e
−π λ2 τ2
2π η3 λ ϑ1 (λ|τ )
2 =
∞ g=0
g (q). (2π λ)2g P
(2.10)
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A. Klemm, M. Mariño
f J (q) is a modular form which depends on the details of the internal CFT. Finally, the quantity Y in (2.9) is a moduli-dependent function related to the Kähler potential as g (q) by K = − log Y . We will also define the holomorphic counterpart Pg (q) of P
2π η3 λ ϑ1 (λ|τ )
2 =
∞
(2π λ)2g Pg (q).
(2.11)
g=0
The quantities Pg (q) can be explicitly written in terms of generalized Eisenstein series. To do this, one uses the expansion ∞
ζ (2k) 2π η3 z 2k = − exp E 2k (τ )z . (2.12) ϑ1 (z|τ ) k k=1
If we now introduce the polynomials Sk through: ∞
∞ n exp xn z = Sn (x1 , . . . , xn )z n , n=1
(2.13)
n=0
we can easily check that Pg (q) is a quasimodular form of weight (2g, 0) given by
|B2k | E 2k (q) , Pg (q) = Sg xk = (2.14) k(2k)! where B2k are Bernoulli numbers, and E 2k (q) is the Eisenstein series introduced in (A.9). We have, for instance, P1 (q) =
1 1 E 2 (q), P2 (q) = (5E 22 + E 4 ). 12 1440
(2.15)
g are obtained by an equation identical to (2.14) The non-holomorphic modular forms P after changing E 2 by 2 (τ ) = E 2 (τ ) − 3 . E π τ2
(2.16)
The computation of the Fg amplitudes involves, then, the determination of the modular forms f J (q), and the evaluation of the integral over the fundamental domain. The first step is easy when there is an orbifold realization. The second step is more involved and requires the method first introduced in [17] in the context of string threshold corrections. This method was further refined and developed in [24,12,47]. We will refer to the approach presented in [24,12,47] to calculate these integrals as the lattice reduction technique. This technique, which was used to compute the Fg couplings in [43], computes the integral (2.8) iteratively by “integrating out” a sublattice of the Narain lattice of signature (1, 1), therefore reducing its rank at every step. In the cases considered in this paper, where one starts with lattices of signature 2,2+s , it is sufficient to perform the lattice reduction once by “integrating out” a sublattice 1,1 = z, z .
(2.17)
The generating vector z is called the reduction vector. The details of the lattice reduction procedure are rather intricate, and we present some of them in Appendix B. There are
Counting BPS States on the Enriques Calabi-Yau
33
two general important properties of the result for Fg which are worth pointing out. The first one is that different choices for the sublattices 1,1 to be integrated out in the process of lattice reduction lead in general to different results for the integral, and each of these expressions is valid in a different region of moduli space. The second property is that, although the result for the integral (2.8) is rather complicated, the holomorphic limit t¯ → ∞,
t fixed,
(2.18)
leads to a rather simple expression for Fg . This holomorphic limit is the one needed to extract the topological information of Fg [8]. We now present some general results on the holomorphic limit of Fg , obtained from a lattice reduction computation of the heterotic integral (2.8). For simplicity, we will restrict ourselves to the case in which one has a single lattice involved in the integrand (2.8), and α = β = 0. In general, the integrand will be a sum over different orbifold blocks and different lattices J with nonvanishing α, β. The final answer for Fg in these cases will be given by a sum over the different blocks. The presence of α, β leads however to nontrivial modifications of the result, as it has been already noticed in various papers [48,27,47,42,40]. We will consider these modifications when we analyze the FHSV model. Most of the results we will present have been obtained in [43], although we will consider a slightly more general situation which will be needed for the FHSV model. We first introduce some necessary ingredients to write down the answer. First of all, the norm |z + |2 of the projected reduction vector depends on the Narain moduli of the compactification as |z + |2 =
ν , Y
(2.19)
where Y = e−K is the moduli-dependent quantity introduced in (2.9), and ν is a real number related to the norm of z. In the STU model considered in [43], ν = 1, but as we will see in general it can take other values. We will label an element p K of the reduced lattice by a vector r of integer coordinates. The resulting expression for the holomorphic limit of Fg depends on the moduli through the combination [43]
+ ( p K ) 2π P . (2.20) exp 2π i( p K , µ/N ) + |z + | The different ingredients in this expression are explained in detail in Appendix B. The first term of (2.20) comes from the exponent in the second line of (B.13), and the second term comes from the argument of the Bessel function in (B.13). One can easily see, by using the explicit expressions for the different quantities involved, that the exponent in (2.20) can be written in the form 1
ν − 2 2π ir · y,
(2.21)
where y is a vector of holomorphic coordinates for the heterotic moduli space (which is related to t, the flat coordinates in the positive Kähler cone in the type II realization, in a simple way). We will see concrete examples of this in the calculation for the FHSV model. Finally, we define the coefficients cg (n) through cg (n)q n . (2.22) Pg (q) f (q) = n
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A. Klemm, M. Mariño
The final expression for Fg is: Fg (t) = ν 1−g
∞
cg (r 2 /2)
r >0
2g−3 e ν
− 21
2π ir ·y
.
(2.23)
=1
In this equation, r 2 is computed with the norm of the reduced lattice K , and the restriction r > 0 means that we consider vectors such that Im(r · y) > 0, as well as a finite number of boundary cases [24,43]. The sum over in (2.23) can be written as Li3−2g (eν
− 21
2π ir ·y
),
(2.24)
where Lin is the polylogarithm of index n defined as Lin (x) =
∞ xk . kn
(2.25)
k=1
In the above expression for Fg we are not taking into account constant terms as well as polynomial terms in y and Im y which also appear in the heterotic computation [24,43].
2.2. BPS content of the Fg couplings. As shown in [24,25], the couplings Fg are BPS-saturated amplitudes and they can be regarded as generating functions that count in an appropriate way the BPS states of the N = 2 compactification. The underlying structure of the couplings was further clarified in the work of Gopakumar and Vafa [21], who gave a precise formula for the BPS content of the Fg in terms of bound states of D0-D2 branes in a type IIA compactification on a CY threefold X . These bound states lead to BPS particles in four dimensions labelled by three quantum numbers. The first quantum number is the homology class r ∈ H2 (X, Z) of the Riemann surface wrapped by the D2. The other two quantum numbers are given by the off-shell spin content jL , j R with respect to the algebra su(2) L × su(2) R of the rotation group S O(4). Let us denote by N j 3 , j 3 (r ) the number of BPS states with these quantum numbers. This number is not L R invariant under deformations, therefore [21] considered the index n g (r ) defined by
3
(−1)2 j R (2 j R3 + 1)N j 3 ,J 3 (r )[jL ] = R
j L3 , j R3
∞
L
n g (r )Ig ,
(2.26)
g=0
⊗g 1 where Ig = + 2(0) . The integer numbers n g (r ), which characterize the L 2 L spectrum of D2-D0 bound states in CY compactifications of type IIA, are called Gopakumar-Vafa (GV) invariants. Let us now consider the generating functional of topological string amplitudes F(λ) =
∞ g=0
Fg (t)λ2g−2 .
(2.27)
Counting BPS States on the Enriques Calabi-Yau
35
According to [21], the worldsheet instanton corrections to F(λ) can be obtained by a Schwinger one-loop computation involving only the D2-D0 bound states: F(λ) =
∞
∞
n g (r )
∞
0
g=0 r ∈H2 (M,Z) m=−∞
s s 2g−2 ds 2 sin exp − (r · t + 2π im) . s 2 λ (2.28)
In this formula, the sum over m is over the number of D0 states bound to the D2s, and we have taken into account that the index n g (r ) is independent of m. After a Poisson resummation over m one finds [21]: F(λ) =
∞
∞
g=0 r ∈H2 (M,Z) d=1
dλ 2g−2 −dr ·t 1 2 sin n g (r ) e . d 2
(2.29)
Notice that the sum over d in (2.29) plays the same role as the sum over in the heterotic computation (2.23). This expression, which takes into account the spectrum of “electric” states associated to D2-D0 branes, is valid in the large radius limit of the CY compactification, since in this region the lightest states are indeed the D2 and D0 branes and their bound states, while the D4 and D6 “magnetic” states are heavy. Equation (2.29) leads to strong structural predictions for the topological string amplitudes Fg when written in terms of GV invariants. Up to genus 4, one finds (for the instanton part) n 0 (r )Li3 (e−r ·t ), F0 = r ∈H2 (M,Z)
n 0 (r ) + n 1 (r ) Li1 (e−r ·t ), 12 r ∈H2 (M,Z)
n 0 (r ) + n 2 (r ) Li−1 (e−r ·t ), F2 = 240 r ∈H2 (M,Z)
n 0 (r ) n 2 (r ) − + n 3 (r ) Li−3 (e−r ·t ), F3 = 6048 12 r ∈H2 (M,Z) n 0 (r ) n 2 (r ) n 3 (r ) + − + n 4 (r ) Li−5 (e−r ·t ). F4 = 172800 360 6 F1 =
(2.30)
r ∈H2 (M,Z)
In the simple case where ν = 1, the heterotic result (2.23) leads to a simple generating function for the GV invariants. To see this, notice that, if we write F(λ) =
∞
g (r )Li3−2g (e−r ·t )λ2g−2 , N
(2.31)
g=0 r ∈H2 (M,Z)
then from (2.29) one has the following relation for fixed r : ∞ g=0
∞ λ 2g−2 n g (r ) 2 sin = N g (r )λ2g−2 . 2 g=0
(2.32)
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A. Klemm, M. Mariño
Under heterotic/type II duality, and with an appropriate choice of lattice reduction, the reduced lattice K that appears in the heterotic computation becomes the Picard lattice of the K3 fiber Pic(K3), and the vectors r label homology classes in this lattice. According g (r ) = cg (r 2 /2), where the cg (n) are defined in (2.22). If we to (2.23), one has that N now use (2.11) and the product representation of ϑ1 (ν|τ ) given in (A.4), we find
∞
n g (r )z g q r
2 /2
= f (q)ξ 2 (z),
(2.33)
r ∈Pic(K3) g=0
where z = 4 sin2 (λ/2), and ξ(z) is the function that appears in helicity supertraces (see for example [32,15]) ξ(z) =
∞ n=1
∞ (1 − q n )2 (1 − q n )2 = . 1 − 2q n cos λ + q 2n (1 − q n )2 + zq n
(2.34)
n=1
A similar expression was written down in [29] in a particular example. Equation (2.33) applies to many different heterotic duals, like the ones studied in [43,33]. Notice that, if the modular form f (q) has an expansion in q with integer coefficients, then integrality of n g (r ) is manifest. The final result for the generating functional of the n g (r ) involves a model-dependent quantity (the modular form f (q)) as well as the universal factor ξ 2 (z). Therefore, in these heterotic models, the enumerative information of the Fg s is encoded in a single modular form f (q), and this leads to a powerful principle which can be used to determine these couplings in a variety of models [33]. 3. The FHSV Model In this section we will introduce and study the FHSV model of [18]. We will first discuss the heterotic side and give some details about the one-loop partition function which will be needed in the computation of the Fg couplings. Then we discuss the type IIA side and the geometry of the Calabi-Yau, which will be important to give an interpretation of the couplings and in the B-model analysis of Sect. 6. 3.1. The heterotic side of the FHSV model. The FHSV model is defined, on the heterotic side, by an asymmetric orbifold [18]. One first considers the splitting of the compactification lattice 6,22 as u = 11,9 ⊕ 21,9 ⊕ s1,1 ⊕ 2,2 ⊕ g1,1 ,
(3.1)
where each of the 1,9 can be further decomposed as 1,9 = d1,1 ⊕ E 8 (−1).
(3.2)
We now act with a Z2 symmetry as follows: | p1 , p2 , p3 , p4 , p5 → eπiδ· p3 | p2 , p1 , p3 , − p4 , − p5 ,
(3.3)
where δ = (1, −1) ∈ s1,1 , and δ 2 = −2. Therefore Z2 acts as an exchange symmetry in the direct sum 11,9 ⊕ 21,9 , as a shift in s1,1 , and as −1 in 2,2 ⊕ g1,1 . It is easy to
Counting BPS States on the Enriques Calabi-Yau
37
see [18] that this asymmetric orbifold leads to an heterotic string compactification with N = 2 supersymmetry in four dimensions. The massless spectrum consists of 11 vector multiplets, 11 hypermultiplets, and the supergravity multiplet. The vector multiplet moduli space for this compactification is given by SL(2, Z)\SL(2, R)/S O(2) × M,
(3.4)
M = O(1 )\O(2, 10)/[O(2) × O(10)],
(3.5)
where
and O(1 ) is the group of automorphisms of the lattice 1 = s1,1 ⊕ d1,1 (2) ⊕ E 8 (−2).
(3.6)
This is in fact the lattice associated to the untwisted, projected sector of the orbifold. As a warm up exercise, we will now compute the one-loop partition function of the FHSV orbifold, since the results will be useful for the computation of the Fg amplitudes (the helicity supertrace generating function of this model has been independently computed in the recent paper [15]). We will denote by Z [ hg ] the partition functions on the sector twisted by h and with the g element inserted. Here, g, h = 0, 1 in the usual way. Let us first consider the bosonic sector. In the untwisted, unprojected sector we simply have 1 Z b 00 = 24 u (τ ). (3.7) 2η¯ (τ )η8 (τ ) In order to consider the other sectors, we introduce the lattices J with J = 1, 2, 3: J = s1,1 ⊕ d1,1 (ζ J ) ⊕ E 8 (−ζ J ).
(3.8)
The values of ζ J , α J , β J are given in Table 1. The three different cases J = 1, 2, 3 correspond respectively to the orbifold blocks 01, 10 and 11. In the untwisted, projected sector we identify the two sets of bosonic excitations associated to the two 1,9 lattices. This amounts to a doubling of the τ parameter in the nonzero ‘modes [36]. We then find, η(τ ) 3 4 (τ, δ, 0). (3.9) Z b [01 ] = 9 1 η¯ (2τ )η(2τ )η¯ 3 (τ )η3 (τ ) ϑ 10 (τ ) For the 10 and 11 orbifold blocks we find
η(τ ) 3 4 (τ, 0, δ), = 9 2 η¯ (τ/2)η(τ/2)η¯ 3 (τ )η3 (τ ) ϑ 01 (τ ) η(τ ) 3
4 b 1 Z [1 ] = ϑ 0 (τ ) 3 (τ, δ, δ). τ +1 3 3 η¯ 9 ( τ +1 0 2 )η( 2 )η¯ (τ )η (τ )
Z b [10 ]
(3.10)
In the 11 block, the in the theta function indicates that the sum over lattice vectors includes an insertion of (−1)v , where v is the projection of p onto 1,1 21 ⊕ E 8 (− 21 ). 2
(3.11)
38
A. Klemm, M. Mariño Table 1. ζ J , α J and β J for the different blocks
J ζJ αJ βJ
1 2 δ 0
2 1/2 0 δ
3 1/2 δ δ
Let us now consider the fermionic sector in detail. The fermions in the s1,1 lattice do not change under the Z2 symmetry, so together with the fermions in the uncompactified directions we have a 3/2 a ϑ b (τ ) f Z 1,1 b = . (3.12) s η(τ ) The orbifold blocks for two complex fermions with symmetry ψ → −ψ are given by (see for example [32], Eq. (12.4.15)): a−h (τ )ϑ ϑ a+h b−g (τ ) b+g . (3.13) 2 η Therefore, for the fermions in 2,2 ⊕ g1,1 one finds
Z
h a
f 2,2 ⊕g1,1 g
b
⎛ ⎞3/4 a−h ϑ a+h b+g (τ )ϑ b−g (τ ) ⎠ . =⎝ η2
(3.14)
The treatment of the two fermions coming from 1,9 ⊕ 1,9 is slightly more delicate. The 00 block in the a, b sector is simply ϑ a (τ ) f b . (3.15) Z 1,9 ⊕ 1,9 00 ab = η(τ ) Let us now analyze the invariant states in the NS sector. A convenient basis for the (1) (2) Hilbert space HNS ⊗ HNS is given by (1) (1) (2) (2) ψ−n 1 · · · ψ−n 2k ψ−m 1 · · · ψ−m l ± (1 ↔ 2) |0 , (1) (1) (2) (2) ψ−n 1 · · · ψ−n 2k+1 ψ−m 1 · · · ψ−m 2l+1 ∓ (1 ↔ 2) |0 ,
(3.16)
where n i , m i > 0 are half-integers. The above states have the sign ±1, respectively, under the Z2 symmetry generator g which exchanges the two lattices. It is easy to see that in computing the trace over the Hilbert space with an insertion of g, the above states cancel except when the (1) and the (2) content is the same. Therefore, only the states (1) (1) (2) ψ−n · · · ψ−n ψ (2) · · · ψ−n |0 , 1 2k+1 −n 1 2k+1 (1)
(1)
(2)
(2)
ψ−n 1 · · · ψ−n 2k ψ−n 1 · · · ψ−n 2k |0
(3.17)
Counting BPS States on the Enriques Calabi-Yau
39
contribute to the trace, with signs −1 and +1 under g, respectively. An odd number of fermion oscillators leads to a −1 sign, but this is like having an insertion of (−1) F . We then find 1 ϑ 01 (2τ ) 2 L 0 −c/24 F 2L 0 −c/12 Tr H(1) ⊗H(2) g q = Tr H N S (−1) q = , (3.18) NS NS η(2τ ) where the doubling in τ is due to the doubling in the oscillator content. Notice that the insertion of (−1) F in the above trace does not change anything, since (−1) F1 and (−1) F2 cancel each other, therefore a
f Z 1,9 ⊕ 1,9 01
b
1 ϑ a1 (2τ ) 2 = , η(2τ )
(3.19)
and the expressions for the other blocks can be obtained by modular transformations. Putting all these results together, we can write up the one-loop partition functions for the different blocks. One finds, for example: 4 a 1 0 a+b+ab ϑ b (τ ) Z [0 ] = 24 (τ ) (−1) (3.20) 2η¯ (τ )η8 (τ ) u η(τ ) a,b
for the 00 block. For the 01 block, one finds Z [01 ] =
1 4 3 1 (τ, δ, 0) η¯ 9 (2τ )η(2τ )|η(τ )|3 1 ϑ 0 (τ ) 3/2 3/2 1/2 1/2 0 ϑ 00 (τ ) ϑ 01 (τ ) − ϑ 1 (2τ ) ϑ 00 (2τ ) × . (3.21) 1 η3 (τ ) (η(2τ )) 2
3.2. The type II side of the FHSV model. The dual type II realization of the FSHV model is a compactification on the Enriques CY M with holonomy SU (2) × Z2 . The two covariant constant spinors of opposite chirality on M lead to N = 2 supersymmetry in four dimensions, but many features of the model are between the SU (2) holonomy case with N = 4 supersymmetry and the generic situation with SU (3) holonomy and N = 2 supersymmetry. The compactification manifold M of the type II string is constructed as a free quotient of the manifold Y = K3 × T2 . The Z2 acts as the free Enriques involution [5] on the K3 and as inversion Z2 : z → −z on the coordinate z of the T2 . If T2 = C/Z2 is defined by z ∼ z + 1 ∼ z + τ , we have four Z2 fixed points at { p1 , p2 , p3 , p4 } = {0, 21 , τ2 , 21 + τ2 }. The geometry of the T2 /Z2 orbifold is that of a P1 with area Im(τ )/2 and four conical curvature singularities at the pi each of which has deficit angle π . The total space M is a K3 fibration over the P1 , and by construction it has Enriques fibres E of multiplicity two, over the four pi .2 2 It also exhibits an elliptic fibration over the Enriques E surface with four sections. An Enriques surface itself has two elliptic fibres with multiplicity two and can be obtained from d P9 –a P2 blown up in nine points– by a logarithmic transform on two fibres.
40
A. Klemm, M. Mariño
Every Enriques surface E = K3/Z2 is a free quotient of a K3 by the Enriques involution ρ : K3 → K3. In order to construct a type II realization of the FHSV model, one first notices that the two-cohomology lattice H 2 (K3, Z), K3 = 3,19 = 11,9 ⊕ 21,9 ⊕ g1,1
(3.22)
can be identified with the same blocks that appear in (3.1). The Enriques involution ρ ∗ ( p1 ⊕ p2 ⊕ p5 ) = p2 ⊕ p1 ⊕ (− p5 ) on K3 acts as in (3.3). The s1,1 lattice is spanned by H 0 (K3, Z) and H 4 (K3, Z) in the type II realization, and after quotienting by the involution ρ it can be identified as s1,1 = H 0 (E, Z) ⊕ H 4 (E, Z).
(3.23)
The shift on this lattice in the orbifold (3.3) corresponds to turning on a Wilson line expectation value for the RR U (1) fields [4]. Some properties of the model are most clearly seen in an algebraic realization. We realize the two-torus as a hyperelliptic branched twofold covering of P1 , with homogeneous coordinates denoted by w : x, and described by the equation y 2 = f 4 (w : x).
(3.24)
The Z2 acts as κ : y → −y, and the fixed points are the four branch points pi of the degree four polynomial f 4 (w : x) = 0. The holomorphic (1, 0) form ω1,0 =
dx y
(3.25)
is anti-invariant. A similar realization of a K3 admitting the Enriques involution ρ is as a double covering of P1 × P1 branched at the vanishing locus of a bidegree (4, 4) hypersurface in P1 × P1 [5,28]. The total space is a eighteen-parameter family of K3 surfaces Y 2 = f 4,4 (s : t, u : v) .
(3.26)
The Enriques involution acts freely as ρ : (Y, s : t, u : v) → (−Y, s : (−t), u : (−v))
(3.27)
on a symmetric but otherwise generic slice of the family. The holomorphic (2, 0) form is given by ω2,0 =
sudt ∧ dv . Y
(3.28)
Since ρ acts freely, the fundamental group of the Enriques surface E is Z2 and the Euler number is χ (E) = χ (K3)/2 = 12. As ω2,0 (and ω¯ 2,0 = ω0,2 ) is anti-invariant, the cohomology groups have dimensions h 00 = h 22 = 1, h 10 = h 01 = h 20 = h 02 = 0 and h 11 = χ (E) − 2 = 10. The canonical bundle is a two torsion class, i.e. K E⊗2 = O E , hence non-trivial: K E = O E . On the blow up of the special configuration with f 4,4 = (u − v)(u + v)(as 4 (u 2 − v 2 ) + bs 2 t 2 (u 2 − v 2 ) + t 4 (cu 2 + dv 2 )) (3.29)
Counting BPS States on the Enriques Calabi-Yau
41
+ and and with Picard number 18, one can explicitly check [5] that the invariant part K3 − 3,19 ∗ anti-invariant part K3 of under ρ are + = 1,1 (2) ⊕ E 8 (−2), K3
− K3 = [ 1,1 (2) ⊕ E 8 (−2)] ⊕ g1,1 .
(3.30)
The middle cohomology H 2 (E, Z) is isometric to the lattice E =
1 + = 1,1 (1) ⊕ E 8 (−1). 2 K3
(3.31)
The Calabi-Yau manifold M is constructed as M = (K3 × T2 )/Z2 , where the Z2 acts as (ρ, κ) : (Y, s : t, u : v, y, w : x) → (−Y, s : −t, u : −v, −y : w : x).
(3.32)
On the generic K3 fiber the Z2 acts as the monodromy ρ, when the corresponding base point is transported in a loop around the special points pi . The Z2 part of the holonomy is generated as follows. A tangent vector v ∈ TM transported over a nontrivial loop in the base (3.24) around two points of total deficit angle of π is inverted: v → −v. In the base direction the inversion occurs because of the deficit angles, and in the fiber direction due to the monodromy ρ. The cohomology of M is easy to find. = ω2,0 ∧ ω1,0 is invariant and becomes (i) the unique, nowhere vanishing (3, 0)-form on M. The 10 invariant (1, 1) forms ω1,1 , + , together with the volume form on P1 , ω , give 11 harmonic forms i = 1, . . . , 10 in K3 1,1 1,1 in H (M, Z). We will adopt the type IIA interpretation in which the vector multiplets are mapped to the complexified Kähler moduli. Notice that the heterotic moduli of the Narain compactification are mapped to the Kähler moduli of the fiber (as we will make explicit in the next section), while the heterotic dilaton S is mapped to the complexified Kähler modulus of the P1 base. Since χ (M) = 0, one has h 2,1 = 11. Ten of these forms − can be explicitly constructed by taking the ten forms in K3 of type (1, 1) and forming their wedge product with ω1,0 . The remaining (2, 1) form is ω2,0 ∧ ω0,1 . The moduli space of M has two different types of singular loci [18,4] which lead to conformal field theories in four dimensions. The first degeneration comes from the shrinking of a smooth rational curve e ∈ E with e2 = −2. Since P1 has no unramified cover, the preimage of e in K3 must be the sum e1 + e2 of two spheres e1 , e2 in K3 with ei ∈ i1,9 and ρ ∗ (e1 ) = e2 . If e goes to zero size so do e1 and e2 in K3 . The shrinking P1 leads to an SU (2) gauge symmetry enhancement: in type IIA theory, a D2-brane wrapping the P1 with two possible orientations leads to massless W ± bosons, which complete the corresponding U (1) vector multiplet to a vector multiplet in the adjoint representation. This is plainly visible in the spectrum of the perturbative heterotic string, where the gauge group is realized by a level 2 WZW current. For each vanish+ there is a vanishing e −e in the first summand of − which leads to a ing e1 +e2 in K3 1 2 K3 hypermultiplet, also in the adjoint representation of the gauge group. We then obtain for this point the massless spectrum of N = 4 supersymmetric gauge theory, which has a vanishing beta function and no Higgs branch. The second degeneration is again plainly visible in the perturbative heterotic string and arises if one goes to the selfdual point in the lattice s1,1 factor in (3.1). As usual one gets a SU (2) gauge symmetry enhancement at level 1. In addition one gets four hypermultiplets in the fundamental representation of SU (2), one from each fixed point of the T2 . The resulting gauge theory is N = 2, SU (2) Yang-Mills theory with four
42
A. Klemm, M. Mariño
massless hypermultiplets. This theory has a vanishing beta function and it is believed to be conformal [51]. It also has a Higgs branch which leads to a transition to a generic simply connected CY with SU (3) holonomy and Hodge numbers h 21 = 10 and h 11 = 16 [18,4]. An interesting difference between the two degenerations is that the first one occurs when a two-cycle of the covering K3 becomes small, while in the second one the full K3 surface has a volume of order the Planck scale [4]. The fact that these degenerations are associated to conformal theories indicates that there are no genus zero contributions to the Gromow-Witten invariants in type IIA theory on M [18]. For these degenerations, the Kähler class of the base is identified with the scale of the gauge coupling constant. An eventual scale dependence in N = 2 supersymmetric theories comes from a one-loop correction to the beta function, which corresponds to worldsheet instantons with degree zero in the base, and from space time instantons, which are in turn related to the growth of worldsheet instantons with non-vanishing degree in the base. Both contributions are expected to vanish for the conformal theories. Later we will check with explicit computations that indeed there are no worldsheet instanton corrections to the type IIA prepotential3 . 4. Heterotic Computation of the Fg Couplings In this section we compute the couplings Fg in the heterotic side. It turns out that there are two natural lattice reductions to perform the computation: the geometric reduction, and the Borcherds-Harvey-Moore (BHM) reduction. We will present the results for the couplings in both reductions and we will also propose a type IIA interpretation of these results. Before doing the lattice reduction, we have to evaluate the integrand (2.8) for the heterotic FHSV model. This is rather straightforward by using the results of the previous section. We have four orbifold blocks, but the first block (corresponding to h = g = 0) vanishes. The blocks (h, g) = (0, 1), (1, 0), (1, 1) will be labelled by J = 1, 2, 3, and an easy computation shows that the modular forms f J (q) in (2.9) are given by f 1 (q) = − f 2 (q) =
∞ 128 2 (1 − q 2n )−12 , = − q η6 (τ )ϑ26 (τ ) n=1
∞ 4 − 14 (1 − q n )−12 (1 − q n−1/2 )−12 , = 4q η6 (τ )ϑ46 (τ ) n=1
(4.1)
∞ 4 − 14 f 3 (q) = 6 (1 − q n )−12 (1 + q n−1/2 )−12 . = 4q η (τ )ϑ36 (τ ) n=1
The Narain lattices for J = 1, 2, 3 are given in (3.8), and the corresponding theta functions in (2.9) are the same ones that appear in the computation of the one-loop partition function in the previous section. The modular forms in (4.1) have the right modular weight: the conjugate Narain-Siegel theta function for a lattice of signature (2, 10) with g has modular 2g − 2 insertions has modular weight (5, 2g − 1), the modular form P 2g−1 weight (2g, 0), and the insertion τ2 has modular weights (−2g + 1, −2g + 1). Taking into account that the weight of the forms f J (q) is (−6, 0), we see that the integrand in 3 A direct A-model argument can be given using relative Gromov-Witten invariants [45].
Counting BPS States on the Enriques Calabi-Yau
43
(2.8) has zero modular weight, as it should. Notice that the Narain-Siegel theta functions involve nonzero α, β and lattices which are not self-dual. This means that we will have to modify in an appropriate way the computation in [43]. We will now present the computation of the couplings in both reductions. 4.1. The geometric reduction. In order to apply the reduction technique we need explicit formulae for the projections of the lattice, that in turn depend on the moduli. Let us write a vector p ∈ J as p = (n, m, n 2 , m 2 , q),
(4.2)
where (n, m), (n 2 , m 2 ) are integer coordinates on s1,1 and d1,1 (ζ J ), and q is a vector of integer coordinates in the E 8 lattice. The norm of p is given by p 2 = p 2R − p 2L = 2nm + ζ J (2n 2 m 2 − q 2 ).
(4.3)
Since the lattices J have two 1,1 factors, there are two natural reductions that one can use. The first one will be referred to as the geometric reduction. The reason for this name is that, as we will see, this reduction leads to an expression for Fg which is valid in the large volume limit of the Kähler moduli space and gives a generating functional of Gromov-Witten invariants, or equivalently, of BPS invariants that count D2-D0 bound states. In the geometric reduction, one chooses the reduction vector z = (1, 0) ∈ s1,1 .
(4.4)
We then have z = (0, 1) ∈ s1,1 . The reduced lattice is K J = E 8 (−ζ J ) ⊕ d1,1 (ζ J ),
J = 1, 2, 3.
(4.5)
Different choices of reduction vectors correspond to different choices of cusps in the moduli space, and in particular lead to different parameterizations of this space. To make this explicit, we recall that the exact moduli space of vector multiplets for the Kähler parameters of the fiber is the coset O(2, 10)/[O(2) × O(10)]. This coset is given by the following algebraic equations satisfied by the complex variables (w1 , · · · , w12 ) [19,39]: 10
|wi |2 − |w11 |2 − |w12 |2 = −2Y,
i=1 10
2 2 wi2 − w11 − w12 = 0.
(4.6)
i=1
The quantity Y above is the same one that appears in (2.9) and the mass formula gives | p R |2 =
|v · w|2 , Y
where the vector v is defined by
1 1 1 1 1 1 1 v = ζ J2 q, m − n, ζ J2 (m 2 − n 2 ), m + n, ζ J2 (m 2 + n 2 ) . 2 2 2 2
(4.7)
(4.8)
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A. Klemm, M. Mariño
For the reduction vector (4.4) it is convenient to parameterize the coset by ten independent complex coordinates y = (y + , y − , y),
(4.9)
which are defined as follows: wj = yj,
j = 1, · · · , 8, w9 = 1 +
1 2 y , 4
1 + 1 (y − 2y − ), w11 = −1 + y 2 , 2 4 1 + − = (y + 2y ), 2
w10 = w12
(4.10)
where the (complex) norm of the vector (4.9), y 2 , is defined by y 2 = 2y + y − − y 2 .
(4.11)
We will denote by y2± , y2 the imaginary parts of these moduli. From the above parameterization one finds, Y = (Im y)2 = 2y2+ y2− − y22 ,
(4.12)
1
and p R = v · w/Y 2 ∈ C is given by
1 1 1 1 2 1 + − 2 2 2 p R = 1/2 −n + my + ζ J m 2 y + ζ J n 2 y + ζ J q · y . Y 2
(4.13)
With this parameterization, the resulting topological couplings will have good convergence properties in the region Im y → ∞. Notice that |z + | =
1 1
Y2
,
(4.14)
therefore ν = 1. It is easy to evaluate the exponent of (2.20), which in this case it is equal to 1 2π ir · y = 2π iζ J2 m 2 y + + n 2 y − + q · y ,
(4.15)
We can now proceed to evaluate the integral (2.4). The first thing to observe is that, with the choice of reduction vector (4.4), only the untwisted sector J = 1 contributes. The reason for that is that in the twisted sectors J = 2, 3, the lattice s1,1 where the reduction is performed has the shift β J = δ = z − z . As shown in Sect. 5 of [47], in those cases the integral over the fundamental domain is zero. This is easy to understand by looking at the expression (B.10) in Appendix B. The effect of this nonzero β is to shift c → c − 1/2. As this integral is effectively evaluated when |z + | → 0, the integrand vanishes. On the other hand, the theta function associated to 1 in the untwisted, projected sector J = 1 includes a phase eπiδ· p . It was shown in [40] that the effect of this phase is to shift the integer in (B.10) as → − 21 . This means that the polylogarithm in (2.23) becomes Lim (x) =
1 ∞ ∞ 1 x x k+ 2 → = 2m Lim (x 2 ) − Lim (x), 1 m m
(k + 2 )
=1
k=0
(4.16)
Counting BPS States on the Enriques Calabi-Yau
45
with m = 3 − 2g. In order to write the argument of the polylogarithm, we will relabel m 2 , n 2 → m, n, and introduce the moduli parameters t = (t + , t − , t) as √ (4.17) − 2π iy = 2t. The integer vector r in (4.15) reads r = (n, m, q) ∈ E = 1,1 ⊕ E 8 (−1),
(4.18)
and it follows from (3.31) that it labels two-cohomology classes in the Enriques fiber. It has norm r 2 = 2mn − q 2 . The argument of the polylogarithm is 1
x 2 = exp(−r · t),
r · t = mt + + nt − + q · t.
The final formula for the Fg is then Fg (t) = cg (r 2 ) 23−2g Li3−2g (e−r ·t ) − Li3−2g (e−2r ·t ) ,
(4.19)
r >0
where
cg (n)q n = f 1 (q)Pg (q),
(4.20)
n
f 1 (q) is given in (4.1). The restriction r > 0 means that [24] n > 0, or n = 0, m > 0, or n = m = 0, q > 0. The reason that the above formula involves cg (r 2 ) instead of cg (r 2 /2) as in (2.23) is simply that the norm of the reduced lattice is twice the norm of the lattice in (4.18). Notice that due to the shift in there is no contribution from the “zero orbit” (B.12). The above expression is only valid in principle for g > 0, and the computation of the prepotential involves a somewhat different procedure explained in [24]. It is easy to check however that the worldsheet instanton corrections to the prepotential are given by (4.19) specialized to g = 0 (the same thing happens in the STU model analyzed in [43]). Since r 2 is always even and f 1 (q) has no even powers of q in its expansion, we conclude that the instanton corrections to F0 vanish along the fiber directions. This is in agreement with the analysis of [18]. The genus one amplitude can be written as follows: 2 1 − e−r ·t 2c1 (r ) 1 F1 (t) = − log . 2 1 + e−r ·t
(4.21)
r >0
The infinite product appearing in this equation was previously found by Borcherds in a related context [12]. In Example 13.7 of that paper, Borcherds considers two different expressions for the same automorphic form, obtained by expanding it around different cusps. Both expressions are denominator formulae for two different superalgebras. The second denominator formula in this Example is precisely the infinite product of (4.21) (to see this one notices that the part of 2 f 1 (q)P1 (q) involving even powers of q equals the modular form f 00 (2τ ) introduced by Borcherds). We now propose the following type IIA interpretation of this computation. As shown in (3.23), the reduction lattice s1,1 we are choosing here corresponds to the H 0 (E, Z) ⊕ H 4 (E, Z) cohomology of the Enriques surface, and z and z are integer generators of H 0 (E, Z) and H 4 (E, Z), respectively. The remaining lattice can be identified with
46
A. Klemm, M. Mariño
H 2 (E, Z), and indeed r in (4.18) is a set of integer coordinates for two-homology classes on the Enriques fiber. For this reduction, the region of moduli space where Im y → ∞ is the region where the D2s and the D0 are light (as shown in (4.13), their mass goes like 1 1 1 1/Y 2 and y/Y 2 , respectively) while the D4s are heavy (their masses go like y 2 /Y 2 ). Therefore, the region Im y where this reduction is appropriate is the region of moduli space where the D2 and D0 are the lighter states, and the D4s wrapping the Enriques fiber are heavy. This is the large volume limit, and we expect the answer for Fg to encode information about Gromov-Witten invariants of the Enriques fiber. The sum over in (2.23) can be interpreted as the Poisson resummation of a sum over D0 brane charges (notice that appears after Poisson resummation of the integer n in (4.13)), and the shift in (4.16) corresponds to the RR Wilson line background along the H 0 (E, Z) direction [18,4]. To substantiate this interpretation, we will see in the next section that (4.19) matches with the geometric computation of BPS invariants proposed in [31]. Moreover, in Sect. 6 we will find perfect agreement of the heterotic predictions with a B-model computation of Fg for g ≤ 4. Although (4.19) is similar to other results obtained for heterotic models, it has some additional properties that make it particularly simple. For example, one can show that Ci j
∂ 2 F1 = −16F2 , ∂ti ∂t j
(4.22)
where Ci j is the intersection matrix of 1,1 ⊕ E 8 (−1). This is a consequence of the following identity among the coefficients of the modular forms (4.20): n c1 (n) = −4 c2 (n), n even,
(4.23)
which can be proved by comparing the even part of the τ derivative of f 1 (q)P1 (q) with the even part of f 1 (q)P2 (q). We now discuss the other possible lattice reduction available on the heterotic side. 4.2. The BHM reduction. Since the lattices (3.8) include the sublattice d1,1 (ζ J ), it is natural to compute the topological string amplitudes by choosing the reduction vector z = (1, 0) ∈ d1,1 (ζ J ).
(4.24)
We call this the Borcherds-Harvey-Moore (BHM) reduction, since as we will see it is the choice of reduction made by Harvey and Moore in [26], and leads to the infinite product introduced by Borcherds in [11]. The reduced lattice is then K J = s1,1 ⊕ E 8 (−ζ J ). Since we have chosen a different reduction vector, the associated parameterization of the moduli space will be different from the one made in (4.10). We introduce, as before, ten independent complex coordinates y = (y + , y − , y) defined through: w j = y j , i = 1, · · · , 8, w9 =
1 2 1 y , w11 = (y + + 2y − ), 4 2 1 = −1 + y 2 . 4
w10 = 1 + w12
1 + (y − 2y − ), 2 (4.25)
Counting BPS States on the Enriques Calabi-Yau
47
Although we have used the same notation for the y coordinates, they are related to the w coordinates in a different way than in the geometric reduction. With this parameterization, we find
1 1 1 1 1 p R = 1/2 −ζ J2 n 2 + ζ J2 m 2 y 2 + ny − + my + + ζ J2 q · y . (4.26) Y 2 Notice that the reduction vector has the norm
1 ζJ 2 |z + | = , Y
(4.27)
hence the quantity ν introduced in (2.19) has the value ν = ζ J for each block. The exponent of (2.20) is now
1 − 21 − 12 − + 2 ny + my + ζ J q · y . ν 2π ir · y = 2π iζ J (4.28) The computation of the integral (2.8) is very similar to the one performed in [43], which we summarized in Sect. 2 and Appendix B. The answer in this case is a sum over the three orbifold blocks J = 1, 2, 3, involving different lattices. One has now to be careful with the effects of the shifts α, β. Since with this choice of reduction vector the shifts are orthogonal to z, z , they only lead to insertions of phases in the sum over the reduced lattice, as well as to shifts in their vectors, and their effect is easy to track. To write down the final answer, we first define the coefficients of modular forms: Pg (q) f J (q) = cgJ (n)q n . (4.29) n
Then, the couplings Fg are given by Fg = 21−g (−1)m+n cg1 (mn − q 2 )Li3−2g (e−r ·t ) r >0
1 − q 2 m+n 2 cg + mn + +2 4 2 r >0
2 m+n q 2 /2+m+n 3 1 − q + mn + −(−1) cg 4 2 ·Li3−2g (e−r t ). g−1
(4.30)
In this equation, r = (m, n, q), the coordinate t = (t ± , t) is defined in terms of y by √ (4.31) − 2π iy = ( 2t ± , t), and the inner products in (4.30) are given by r · t = mt + + nt − + q · t, r t = (2m + 1)t + + (2n + 1)t − + q · t.
(4.32)
The insertions (−1)m+n in the first and last block are due to the nonzero α = δ, while the shift in the second inner product in (4.32) is due to the shift by β = δ, and we have
48
A. Klemm, M. Mariño
relabelled n → n + 1. Finally, the insertion of (−1)q /2 in the J = 3 orbifold block comes from the insertion (3.11). The expression (4.30) can be simplified as follows. First, one notices that cg2 (1/4 + p/2) equals (−1) p+1 cg3 (1/4 + p/2). This is easy to see by noting that they are the coef2
1
1
ficients of modular forms related by q 2 → −q 2 . Therefore, the J = 2 and J = 3 contributions are equal and add up. We will call their contribution the contribution of the twisted sector, while the contribution from J = 1 will be called the contribution of the untwisted sector. It is easy to see that the polylogarithms whose argument involves a Kähler class of the form mt + + nt − + q · t with n and m both odd receive contributions from both the untwisted and twisted sector, while if m or n is even only the untwisted sector contributes. In the first case, the contributions come from the coefficients of odd powers in the modular form 21−g Pg (q) f 1 (q) + 2g Pg (q 4 ) f 2 (q 4 ),
(4.33)
while in the second case they come from the contributions of even powers in the first term in (4.33). However, since the second term in (4.33) has only odd powers of q, we can use the modular form (4.33) for both cases. As a last step, one notices by using doubling formulae (see Appendix A) that 64 1 = 6 , η6 (τ )ϑ26 (τ ) η (4τ )ϑ46 (4τ )
(4.34)
therefore f 2 (q 4 ) = −2 f 1 (q). We can finally write down a compact expression for Fg as follows: Fg (t) = cg (r 2 /2)(−1)n+m Li3−2g (e−r ·t ), (4.35) r >0
where the coefficients cg (n) are defined by cg (n)q n = f 1 (q) 21−g Pg (q) − 21+g Pg (q 4 ) ,
(4.36)
n
and in (4.35) we regard r as a vector in 1,1 ⊕ E 8 (−2), i.e. r 2 = 2nm − 2 q 2. We now consider some particular cases of (4.35) in more detail. Although the above expression is in principle valid for g ≥ 1, one can see again that the instanton corrections to the prepotential are given by its specialization to g = 0, and one finds F0 = 0 due to a cancellation between the untwisted and twisted sectors. Let us now look at g = 1. This involves the modular form P1 (q) given in (2.15). The doubling formulae (A.12) gives E 2 (τ ) − 4E 2 (4τ ) = −3ϑ34 (2τ ),
(4.37)
(−1)n+m cB (r 2 /2) 1 F1 = − log 1 − e−r ·t , 2
(4.38)
and one finds
r >0
where n
cB (n)q n =
η(2τ )8 . η(τ )8 η(4τ )8
(4.39)
Counting BPS States on the Enriques Calabi-Yau
49
This is the modular form introduced by Borcherds in [11], and the above expression for F1 agrees with that found by Harvey and Moore in [26] (up to a factor of 1/2 due to different choice of normalizations). The infinite product appearing in (4.38) is the denominator formula of a superalgebra, and it was pointed out in [12] that it is actually identical to the infinite product in (4.21), but expanded around a different cusp. This is of course expected, since in both cases we are evaluating the same integral, but with different choices of reduction vector. What is the interpretation of the Fg amplitudes in the BHM reduction? In the remainder of this subsection, we will make a proposal for what is the enumerative content of the topological string amplitudes in this reduction. The first thing to notice is that in the BHM reduction the reduced lattice is H 0 (E, Z) ⊕ H 4 (E, Z) together with the E 8 (−2) sublattice of H 2 (E, Z), and the integers n and m in (4.35) label zero and fourcohomology classes. The computation in this reduction is appropriate for the region Im y → ∞ in moduli space. However, one can see from (4.26) that this is the region where the light states are the D0, the D4 wrapping the Enriques surface, the D2s in E 8 (−2), and one of the D2s in d1,1 , while the other D2 in this sublattice (labelled by m 2 in (4.26)) is heavy. Therefore, we are not in the large radius regime, and the Fg amplitudes computed with this reduction do not have an enumerative interpretation in terms of Gromov-Witten theory. It is easy to see that they do not lead to integer GV invariants, and indeed we will see in the next sections that the usual Gromov-Witten/D2-D0 counting interpretation has to be reserved for the geometric reduction considered in the previous subsection. The Fg couplings in the BHM reduction must be counting bound states of the light states associated to this cusp. One hint about their BPS content comes from writing the generating functional as 2 ∞ ξ (λ+ , q) ξ 2 (λ− , q 4 ) 2 r 2 /2 2g−2 , (4.40) cg (r /2)q λ = f 1 (q) −4 4 sin2 (λ+ /2) 4 sin2 (λ− /2) g=0 r >0 where ξ(λ, q) is given in (2.34), and λ± =
λ
. (4.41) 2 The derivation of (4.40) is very similar to the derivation of (2.33). This expression suggests that there should be a formula similar to (2.28), in which one does a Schwinger computation including the light states appropriate for this region of moduli space. The role of the D0s in the computation of [21] is now played by the light D2s in the reduc1 tion lattice. Since the BPS masses of these states have an extra factor ν 2 , the Schwinger integral in (2.28) becomes ∞ ∞ s 1 s 2g−2 ds 2 sin exp − (r · y + 2πiν 2 n 2 ) = s 2 λ 0 n 2 =−∞
∞ 1
λ 2g−2
2 sin 1 exp − 1 r · y , (4.42)
2ν 2 ν2
=1 ± 21
where n 2 is the number of light D2 states. This fits with the heterotic expression given in (2.23), and has the effect of rescaling the string coupling constant by λ→
λ 1
ν2
,
(4.43)
50
A. Klemm, M. Mariño
which explains the overall factor ν 1−g in (2.23) as an effect of this rescaling, as well as the appearance of (4.41) in (4.40). In order for the logic of [21] to apply, however, it was important that the number of D2 branes bound to D0 branes is independent of the number of D0s. In our case, we would have a bound state problem involving D4, D2 and D0, and it is not clear that the number of bound states is independent of the number of D2s along the direction of the reduction vector z. On the other hand, we expect the number of BPS states to depend only on the norm of the vector of charges in the cohomology lattice of the fiber (4.3), as in the counting on K3 surfaces considered in [52,25]. Since the D2 charge n 2 we are summing over in (4.42) is part of the d1,1 sublattice, and we are setting m 2 = 0 in the complementary direction, the norm of the charge vector does not depend on n 2 , therefore the degeneracies are independent of n 2 as in the situation considered in [21]. We then propose that (4.40) is a generating functional for an index that counts BPS particles keeping track of their helicities, as in [21]. These BPS particles are obtained from bound states of D4s wrapping the Enriques fiber, D2s wrapped around the curves in the E 8 sublattice of the Enriques cohomology, and D0s. As we argued above, a formula similar to (2.33) should hold after taking into account the rescaling of the string coupling constant (4.43) and the fact that we have two different sectors (untwisted and twisted). The natural labels for D-brane charges in the untwisted and twisted sector are, respectively, ru = (n, m, q),
rt = (2m + 1, 2n + 1, q).
(4.44)
The difference between the D-brane charges in the two sectors is due to the fact that the D-branes in the untwisted sector are double covers of the D-branes in the twisted sector. For example [13], the twisted sector contains 4 D4 branes which have half the charge of a D4 brane in the untwisted sector. These “fractional” branes differ in their torsion charge. Bound states in the twisted sector are made out of one of these “fractional” D4 branes together with an arbitrary number of D4 branes with integer charge. This is why the D4-brane charge in the twisted sector is of the form 2m + 1. In order to count the bound states, we introduce two sets of BPS invariants for the twisted and untwisted sectors, n g (ru ) and n g (rt ). Our proposal for their generating functionals is the natural one from the above results: 2 (−1)n+m n g (ru )z g q ru /2 = f 1 (q)ξ 2 (z, q), ru
2 (−1)n+m n g (rt )z g q rt /2 = −4 f 1 (q)ξ 2 (z, q 4 ),
(4.45)
rt
where the norm of the vectors are computed as before in the lattice 1,1 ⊕ E 8 (−2). Notice that the extra factor of 4 in the twisted sector corresponds to the D-brane states that differ in torsion classes. This factor is in turn related to the four hypermultiplets in the fundamental representation of SU (2) that appear in the N f = 4 degeneration of the type IIA theory. We show some values of these BPS invariants in Tables 2 and 3. 5. Geometric Computation of the BPS Invariants in the Fibre In this section we will analyze the heterotic predictions for the Fg amplitudes in the geometric reduction. We will extract the GV invariants and show that they fit with the
Counting BPS States on the Enriques Calabi-Yau
51
Table 2. BPS invariants (−1)n+m n g (ru ) for the untwisted sector, counting bound states of D0-D2-D4 branes on the Enriques fibre g 0 1 2 3
ru2 /2 = −1 −2 0 0 0
0 0 4 0 0
1 −24 12 −6 0
2 0 64 −32 8
3 −180 172 −162 60
4 0 576 −576 336
5 −1040 1464 −1980 1420
6 0 3840 −5760 5280
7 −5070 9396 −16470 17340
8 0 21056 −42112 52640
Table 3. BPS invariants (−1)n+m n g (rt ) for the twisted sector. Notice that rt2 /2 is always odd, as follows from (4.44) g 0 1 2 3
rt2 /2 = −1 8 0 0 0
1 96 0 0 0
3 720 −16 0 0
5 4160 −192 0 0
7 20280 −1488 24 0
9 87264 −8896 288 0
11 340912 −44944 2288 −32
geometrical approach developed in [31]. This will support our interpretation of the geometric reduction as the one corresponding to the counting of D2/D0 bound states. As we already mentioned in Sect. 2, the free energy of the perturbative topological string can be written in terms of BPS or GV invariants n g (r ). The expansion (2.29) implies that the BPS invariants of non-constant maps r = 0 contribute to Fg as
Fg =
r
Li3−2g (e−r ·t )
g
ah,g n h (r ) ,
(5.1)
h=0
with ag,g = 1, ag−1,g = −(g − 2)/12, . . . , a2,g = 2(−1)g /(2g − 2)!, a1,g = 0, a0,g = |B2g |/(2g(2g − 2)!). With (5.1) we rewrite now (4.19) in terms of the BPS invariants. In the type II picture they are expected to correspond to an integer index in the cohomology of the moduli space of D2 branes wrapping curves in the fiber direction of M. Let us now extract these invariants from (4.19). If at least one entry in r ∈ E = 1,1 ⊕ E 8 (−1) is odd, the second term in (4.19) 2 does not contribute and we get the invariants n odd g (r ) listed in Table 4. Note that r ∈ 2Z odd because E is even. In particular the prediction that n g=0 (r ) = 0 follows from the fact that the modular form f 1 (q) in (4.1) has no even powers. The fact that all n odd g (r ) are integers is an important check on the consistency of the calculation. If all entries in r are even then r 2 ∈ 8Z and we call the class r even. In (4.19) the second term gives a subleading correction to n even g (r ), and we again find that all of them are integer. The first few are listed in Table 5. In [21] the numbers n g (r ) were given a geometrical interpretation. In simple cases they can be computed as an index on the cohomology H ∗ (M) of the moduli space M of D2 branes [31]. By “simple” we mean that the D2 wraps an irreducible and possibly mildly nodal curve C ∈ M in the class r . Infinitesimally, the moduli space is parameterized by the zero modes on the D2 brane. These form a supersymmetric spectrum with 2g zero modes of the flat U (1) connection on C parameterizing the Jacobian Jac(C) ∼ T2g . Furthermore, there are h 0 (O(C)) zero modes corresponding to the deformations MC of the curve C. For this reason the total moduli space M is expected to have a fibration
52
A. Klemm, M. Mariño Table 4. BPS invariants n odd g (r ) ∈ Z for the odd classes r in the fiber direction
g 0 1 2 3 4 5 6 7 8 9
r2 = 0 0 8 0 0 0 0 0 0 0 0
2
4
6
8
10
12
14
16
0 128 −16 0 0 0 0 0 0 0
0 1152 −288 24 0 0 0 0 0 0
0 7680 −2880 480 −32 0 0 0 0 0
0 42112 −21056 5264 −704 40 0 0 0 0
0 200448 −125280 41760 −8400 960 −48 0 0 0
0 855552 −641664 267360 −71872 12384 −1248 56 0 0
0 3345408 −2927232 1463616 −492800 113728 −17312 1568 −64 0
0 12166272 −12166272 7096992 −2872512 831960 −169920 23280 −1920 72
Table 5. BPS invariants n even g (r ) ∈ Z for the even classes r in the fiber direction g 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
r2 = 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0
8
16
24
32
0 42048 −21024 5256 −704 40 0 0 0 0 0 0 0 0 0
0 12165696 −12165696 7096656 −2872416 831948 −169920 23280 −1920 72 0 0 0 0 0
0 1242726144 −1864089216 1708748448 −1158884992 611668944 −254819136 83673040 −21406464 4174920 −598848 59472 −3648 104 0
0 69636018752 −139272037504 174090046880 −165915421248 127601309256 −80867605120 42545564896 −18592299200 6721882484 −1994908928 480175264 −92117568 13732280 −1531072
structure Jac(C) −→ M ↓ MC .
(5.2)
The cohomology H ∗ (M) has a natural su(2) L × su(2) R Lefshetz action which corresponds to the spacetime helicities of BPS bound states. The su(2) R is essentially generated by the Kähler form of the base and the su(2) L by the one of the fibre. The dimension of the cohomology group with eigenvalues jL3 , j R3 , N j 3 , j 3 (r ), is not invariant L R under complex structure deformations. However, the index n g (r ) defined in (2.26) is an invariant. In general, it is not clear how to define the Lefshetz actions on M. However, in [31] the problem was bypassed by using the Abel-Jacobi map, and the following formula for the n g (r ) was derived: n g−δ (r ) = (−1)(dim(MC )+δ)
δ
bg− p,δ− p χ (C ( p) ),
p=0
bg,k :=
2 k!
k−1 i=0
(2g − (k + 1) + i), bg,0 := 1 .
(5.3)
Counting BPS States on the Enriques Calabi-Yau
53
Here, C ( p) is the moduli space of the curve C in the class r together with a choice of p points, which correspond to nodes of C. In particular C (0) = MC . In (5.3) δ is the number of nodes and the formula is applied as follows. In the simplest situation C is a smooth curve of genus g in the class r , then δ = 0 and n g (r ) = (−1)dim(MC ) χ (MC ).
(5.4)
This can be understood directly as follows. If C is smooth the Jac(C) is non-degenerate and carries Ig as su(2) L Lefshetz representation of the fibre. The sum over j R3 in (2.26) gives –up to sign— the Euler number of the base. If the contribution to n g−δ (r ) comes only from an irreducible curve with δ nodes we can calculate in certain situations χ (C ( p) ) to obtain the BPS number. We now apply these ideas to D2 branes wrapping curves C in the fibre of the Calabi-Yau manifold M of the FHSV model. The moduli space MC factorizes for these curves into MC (F), parameterizing movements of C in the fibre, and P1 , parameterizing movements of C over the base of M. Along this P1 direction and outside the pi , the Jac(C) is constant. The P1 is therefore a component, whose contribution factors in (2.26). Moreover on this component the su(2) R Lefshetz action in (2.26) reduces its contribution to an integral over the Euler class P1 e. This integral localizes to the pi . The relevant part of the D2 brane moduli space to curves C in the fiber hence localizes to curves which sit in the Enriques fibre. It is therefore sufficient to consider curves in the four special Enriques fibres to explain the BPS numbers in the tables in Sect. 5. Let us first recall an important fact about curves in an Enriques surfaces. According to Proposition 16.1 in [5], for every such C in the class r in the Kähler cone there is a second curve C + K E in the class r up to torsion with |C + K E | = ∅ and r 2 = [C]2 = [C + K E ]2 . So each curve in the Enriques fibre is effectively doubled. Since we have four fibers we expect that the numbers in Tables 4 and 5 are divisible by eight, which is indeed the case. Let us now compute the moduli space of deformations MC for smooth curves of genus g. By the adjunction formula, for a curve C in the class r = [C] we have 2g − 2 = [C]2 + [C][K E ] ,
(5.5)
where the second term [C][K E ] = 0 on an Enriques surface. The moduli space of the curve C in a surface S is given by the projectivization of MC = PH 0 (O(C)), and the dimension h 0 (O(C)) can be calculated using the Riemann-Roch theorem χ (O(C)) =
[C]2 + [C][K S ] + χ (O(S)), 2
(5.6)
n where χ (O(C)) = i=1 (−1)i h i (O(C)). For smooth curves in an Enriques surface, 1 2 H (O(C)) = H (O(C)) = 0 and χ (O(C)) = h 0 (O(C)). Moreover from Sect. 3.2 we 2 h i,0 (E) = 1, and combining that with (5.5,5.6) yields know that χ (O(E)) = i=1
MC = Pg−1 .
(5.7)
We apply now (5.3) and get, for smooth curves in the class r of genus g = r2
r2
r2
n g (r ) = 8 · (−1) 2 χ (P 2 ) = 8 · (−1) 2
r2 +1 2
r2 2
+ 1, (5.8)
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A. Klemm, M. Mariño
Table 6. Differences between the heterotic BPS prediction in Table 2 and the geometric BPS calculation using (5.3,5.10) r2 = 0 0 0 0 0
g 0 1 2 3
2 0 0 0 0
4 0 0 0 0
6 0 0 0 0
8 0 24 0 0
10 0 288 0 0
12
14
16
0 2160 0 0
0 12544 −32 0
0 61608 −384 0
in agreement with Table 4. Let us now give a more detailed calculation involving the nodal curves. The task is to calculate the Euler numbers χ (C (δ) ). If we force the smooth curve C to pass through δ given points in E, corresponding to the locations of the nodes, we impose δ linear constraints on its moduli space M = Pg−1 . The moduli space of deformations is therefore reduced to Pg−δ−1 . On the other hand we are free to choose the position of the points, which are therefore part of the moduli space of the nodal curves. The freedom of choosing n-points on E is naively E n . Since the points are undistinguishable one considers the orbifold Symn (E) = E n /Sn by the symmetric group Sn . The relevant model for the moduli space of n points is the “free field” resolution [55] Mδ = Hilbδ (E) of this orbifold. The name comes from the fact that the Euler numbers of the resolved spaces are generated by a free field representation ∞ n=0
χ (Mn )q = n
∞
n=1
1 1 − qn
χ (E)
= 1 + 12q + 90q 2 + 520q 3 + 2535q 4 + . . . (5.9)
This is a special bosonic case of a formula [22], which gives the Poincaré polynomial of Mn in terms of bosonic and fermionic free fields. The reason that one needs only 12 bosons here is that E has only even cohomology. Since Hilbδ (E) fibers trivially over Pg−δ−1 we obtain χ (C (δ) ) = (g − δ)χ (Mδ ).
(5.10)
If we insert this result into (5.3) we reproduce immediately, and to a large extent, the heterotic predictions in Table 4. The deviations between the two calculations are given in Table 6. As we will see later, the heterotic predictions are in full agreement with the computation of the topological string amplitudes by using the B model, and the deviations recorded in Table 6 are due to the fact that for reducible curves with many nodes one has to refine the computation of BPS invariants in (5.3) as explained in [31]. It is instructive to compare this situation with the K3 curve counting of [55]. Because of χ (O(K 3)) = 2 we get in that case MC = Pg , instead of (5.7). On K3 we can force up to g nodes to get rational curves. For the Enriques surface, the smooth moduli of a genus g curve is too small to allow generically for g nodes. This also explains the absence of genus zero invariants. We close this section by noticing that the genus one BPS invariants for odd classes listed in Table 4 have an interesting algebraic interpretation. As we pointed out after (4.21), F1 (t) can be interpreted as the logarithm of a denominator formula of a superalgebra. The genus one BPS invariants c1 (r 2 ) are then multiplicities of the (super)root spaces of this superalgebra.
Counting BPS States on the Enriques Calabi-Yau
55
6. The B-Model for an Algebraic Realization of the FHSV Model In this section we find an algebraic realization for the double cover of the Enriques CY, and we study it and its Z2 quotient using mirror symmetry. To simplify the analysis we define a “reduced” FHSV model by blowing down an E 8 part of the Picard lattice. The reduced model has only three Kähler moduli, and its mirror can be analyzed in detail in the context of the B-model. We show that the topological string of the reduced model can be solved in terms of modular forms through the arithmetic properties of the mirror map. Using the holomorphic anomaly equations of [8] we find explicit, closed expressions for the topological string amplitudes up to genus 4. In this section we will denote genus g amplitudes by F (g) .
6.1. The geometric description of the Enriques Calabi-Yau. Here we describe the periods of the Enriques Calabi-Yau and we find an algebraic description of the double cover and its mirror. 6.1.1. Periods and prepotential If M is a d =dimC dimensional CY manifold and ωd,0 is its unique holomorphic (d, 0)-form, we may consider the quantities (k) W = ωd,0 ∂ k ωd,0 , (6.1) M k
ρ−2 where ∂ k = ∂zk11 . . . ∂z ρ−2 denotes derivatives w.r.t. to the complex moduli. If the order of the derivative operator is ki = |k| then Griffiths transversality [14] implies
W (k) = 0 , if |k| < d .
(6.2)
In particular, for d ∈ 2Z we get from W (0) = 0 an algebraic relation between the periods, while for d = 3 Eqs. (6.2) lead to N = 2 special geometry. Let us describe first properties of the periods of the manifold M = (K3 × T2 )/Z2 , which follow from the double cover. On the K3 covering of the Enriques surface we can choose a twelve-dimensional basis of two-forms, αi , i = 0, . . . , ρ − 1, in the anti− invariant lattice K3 and satisfying αi ∧ α j = ηi j . (6.3) K3
Here ηi j is the symmetric, even intersection form on − K 3 . We consider families of K3 surfaces covering the Enriques surface where ρ = 12 is the number of anti-invariant transcendental cycles, i.e. we choose a polarization so that the dual cycles, with basis j j such that j αi = δi , are transcendental. We have i ∩ j = ηi j with ηi j η jk = δik . The discussion below holds for general algebraic K3 surfaces with ρ transcendental cycles, and in particular for the reduced model for which ρ = 4. The holomorphic (2, 0) form can be expanded as ω2,0 =
ρ−1 i=0
Xˆ i αi
with Xˆ i =
i
ω2,0 .
(6.4)
56
A. Klemm, M. Mariño
The period integrals Xˆ i (z) depend on the complex structure deformation parameters z a , a = 1, . . . , ρ − 2, that appear in the algebraic definition of the model. Griffiths transversality (6.2) implies for |k| = 0, ρ−1
Xˆ i Xˆ j ηi j = 0,
(6.5)
i, j=0
and for |k| = 1 3 1 ∂1 W (2,0,0) = W (3,0,0) , ∂2 W (2,0,0) + ∂1 W (1,1,0) = W (2,1,0) , 2 2 1 (∂1 W (0,1,1) + ∂2 W (1,0,1) + ∂3 W (1,1,0) ) = W (1,1,1) . 2
(6.6)
These can be integrated using the Picard-Fuchs (PF) equations and yield rational expressions for the B-model two-point function W (k) , |k| = 2 in terms of the complex structure deformation parameters z k . We will denote them as C zi ,z j := W (k) , |k| = 2, where k has an entry 2 at the i th position if i = j and entries 1 at the i th and j th position otherwise. There is a maximal unipotent point in the moduli space of complex structures of the K3 at which one has one unique holomorphic solution Xˆ 0 , ρ − 2 single logarithmic solutions Xˆ a , a = 1, . . . , ρ − 2 and one double logarithmic solution Fˆ := Xˆ ρ−1 . We define the mirror map for the K3 as tˆa (z) =
Xˆ a (z), Xˆ 0
a = 1, . . . , ρ − 2.
(6.7)
The couplings C za ,z b transform like sections of the bundle Sym2 T ∗ M ⊗ L−2 , where M is the moduli space of complex structures on the K3 and L is the Kähler line bundle. The mirror map and the special gauge of the B-model w.r.t. to the Kähler line bundle relates then the B-model couplings to their A-model counterparts in the following way: Ct a t b =
1 ( Xˆ 0 (z(t)))2
C z c z d (z(t))
∂z c ∂z d (z(t)) b (z(t)) = ηˆ ab , a ∂t ∂t
(6.8)
where ηˆ ab are the classical intersection numbers of the generators of the Picard lattice of the mirror K3, related to the ηi j above by
ηi j = h 0,ρ−1 ⊕ ηˆ ab ,
h 0,ρ−1 =
01 . 10
(6.9)
Equation (6.8) is a consequence of (6.2) and reflects the absence of instanton corrections to the classical intersection ring of K3. It yields a simple relation between the quantity Xˆ 0 (z(t)), which corresponds to a gauge choice of ω2,0 in L−1 , and the derivatives ∂z i ∂ta (z(t)) of the mirror map. The latter is a total invariant under the subgroup X of the discrete automorphism group Aut( − K 3 ), that is realized as monodromy on the periods ( Xˆ 0 , . . . , Xˆ ρ−1 ) on the algebraic family X double covering the Enriques surface E. The ability to express Xˆ 0 (z(t)) using (6.8) as a modular form of X will become important
Counting BPS States on the Enriques Calabi-Yau
57
to solve for the F (g) in Sect. 6.2. We can write the periods as a vector of inhomogeneous coordinates ⎛ ˆ0 ⎞ ⎛ ⎞ X 1 ⎜ Xˆ 1 ⎟ ⎜ ⎟ t1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ . . 0 ˆ ˆ =⎜ . ⎟=X ⎜ . (6.10) ⎟ , . ⎜ . ⎟ ⎜ ⎟ ρ−2 ⎝ Xˆ ρ−2 ⎠ ⎝ ⎠ t − 21 ηˆ a,b t a t b − 1 Xˆ ρ−1 where we used the mirror map (6.7) as well as (6.5) and the explicit form of ηi j . Note X0 that the −1 in the last period is a specialization to d = 2 of the (2πi) d ζ (d)χ (M) term, which appears in d = 3, 4 CY manifolds. Similarly we define for the T2 a basis of anti-invariant one forms α, β with T2 α∧β = 1 and all other integrals are zero. We expand ω1,0 = x 0 α − x 1 β,
(6.11)
where the coefficients x 0 , x 1 are the periods 0 1 x = ω1,0 , x = ω1,0 a
(6.12)
b
and a α = − b β = 1. At the point of maximal unipotent monodromy for the T2 we have a regular solution for x 0 and a logarithmic solution for x 1 , and we define the mirror map as τ (z) =
x1 (z). x0
(6.13)
With similar definitions as above we get a one-point function from integrating ∂1 W (1) = W (2) . The analogue of (6.8) yields well known relations between the j-function and the Schwarz triangle functions for subgroups T2 of S L(2, Z) (see, for example, [35]). We combine information about T2 and K3 to write periods of M. The 3-cycles of M are A0 = a × 0 , B0 = b × ρ−1 , ρ−2 Ai = a × i , Bi = b × j=1 ηi j j , Aτ = b × 0 , Bτ = a × ρ−1 .
i = 1, . . . ρ − 2,
(6.14)
This basis is symplectic with Ai ∩ B j = δi j . The invariant holomorphic (3, 0)-form of M is given by = ω2,0 ω1,0 which in the algebraic model is realized as =
dx dt ∧ dv ∧ su . y Y
If we integrate this e.g. over A0 we get
dx dt ∧ dv 0 X = su = x 0 Xˆ 0 . = y Y A0 a 0
(6.15)
(6.16)
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We can write the period vector of the threefold in a symplectic basis ⎛ ⎞ ⎛ ⎞ −x 1 Xˆ 11 2F − t i ∂i F − τ ∂τ F ⎛ ⎞ ⎜ 1 a⎟ ˆ −x η X 1a ⎜ ⎟ ⎜ ⎟ ∂1 F B0 ⎟ ⎜ ⎟ .. ⎜ .. ⎟ ⎜ . ⎜ ⎟ ⎜ ⎟ . ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 11 ⎜ B ⎟ ⎜ x Xˆ F ∂ ⎟ ⎜ ⎟ τ 0 τ ⎟=⎜ = X =⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ ⎜ x 0 Xˆ 0 ⎟ 1 ⎜ ⎟ ⎜ A0 ⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜ . ⎟ ⎜ x 0 Xˆ 1 ⎟ t ⎜ ⎟ ⎝ .. ⎠ ⎜ ⎟ ⎜ ⎟ .. .. ⎜ ⎟ ⎝ ⎠ . ⎝ ⎠ . Aτ τ x 1 Xˆ 0
(6.17)
From the above we read off the prepotential 1 F0 = − τ ηˆ a,b t a t b − τ 2
(6.18)
and conclude that there are no instanton corrections at genus 0 in base, fibre and mixed directions. T2 acts on the periods (x 0 , x 1 ) a subgroup of S L(2, Z). Similarly, X acts on 0 ˆ ( X , . . . , Xˆ ρ−1 ). The action of these two groups on does not commute and generates a bigger discrete group M . From the point of view of the heterotic string dual, X generates T -dualities, T2 generates S-dualities, and M is called the U-duality group. 6.1.2. Mirror symmetry on an algebraic realization In order to calculate the periods discussed in the last section we need an algebraic realization and understand mirror symmetry on it. Let us first explain some features of mirror symmetry for K3 surfaces. Mirror pairs of K3 can be given by three dimensional reflexive polyhedra following Batyrev’s mirror symmetry construction [6]. Fortunately the double covering of Enriques that [28] uses is of this type. The small polyhedron is given by the convex linear hull of the corners {ν1 , . . . , ν5 } = {[1, 0, 1], [0, 1, 1], [−1, 0, 1], [0, −1, 1], [0, 0, −1]} .
(6.19)
Its points are on a lattice ∼ Z3 , which makes it an integral polyhedron. is obviously embedded like ∈ R = N ∼ R3 . The polyhedron is reflexive, which means that the dual ∗ := {x ∈ M ∗ |x, y ≥ −1, ∀ y ∈ } is an integral polyhedron in the dual lattice ∗ ∈ N ∗ . The ν1 , . . . , ν5 above are corners of the polyhedron, and the corners pi∗ of the large polyhedron ∗ are found by solving the equations νi∗ , ν jk = −1, k = 1, . . . , 3: {ν1∗ , . . . , ν5∗ } = {[2, 2, 1], [2, −2, 1], [−2, −2, 1], [−2, 2, 1], [0, 0, −1]} . (6.20) To the polyhedra Batyrev associate hypersurfaces p=
#ν i=1
∗
ai
#ν j=1
νi ,ν ∗ +1 xj j
∗
= 0,
∗
p =
#ν i=1
bi
#ν
νi ,ν ∗j +1
yj
= 0,
(6.21)
j=1
which describe mirror manifolds M and W . For example we get a special double cover of P1 × P1 , p = a1 s 4 u 4 + a2 s 4 v 4 + a3 t 4 v 4 + a4 t 4 u 4 + a5 y 2 + a0 ystuv = 0 .
(6.22)
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Note that we take the product only over the corners of ∗ , to which we associate the variables {x j } = {s, u, t, v, y}, while we include all points that are not inside codimension one faces in . Four of the ai are redundant, i.e. they can be set to say one by the automorphism of the ambient space P . Calabi-Yau threefolds correspond to 4d polyhedra and the cohomology groups of M and W are given by the formulas l ∗ (θ ) + l ∗ (θ ∗ )l ∗ (θ ), h 21 (M) = l() − 5 − codim(θ ∗ )=2
codim(θ)=1
h 11 (M) = l(∗ ) − 5 −
l ∗ (θ ∗ ) +
codim(θ ∗ )=1
l ∗ (θ ∗ )l ∗ (θ ) ,
(6.23)
codim(θ ∗ )=2
where θ is a face ( is the top dim face) and l(θ ) means all points in that face, while l ∗ (θ ) means interior points in that face. For K3 these formulas become l ∗ (θ ) + l ∗ (θ )l ∗ (θ ∗ ), ρ(M) − 2 = l() − 4 − codim(θ)=1
h(M) = l(∗ ) − 4 −
codim(θ)=2
l(θ ∗ ) +
codim(θ ∗ )=1
l ∗ (θ ∗ )l ∗ (θ ),
(6.24)
codim(θ)=2
and the interpretation is as follows. h(M) is the rank of the Picard group of the K3 M and ρ(M) is the number of transcendental cycles of M. For all 3d reflexive polyhedra one has h(M) + ρ(M) = h 11 (K 3) + h 20 (K 3 ) + h 02 = 22, and the different choices are merely different choices of the polarization. In our case we have ρ(M) = 4 and h(M) = 18. Here as above for the mirror W we just exchange with ∗ . For the polyhedra , i.e. the manifold M, we define the variables z1 =
a1 a3 a52 a04
, z2 =
a2 a4 a52 a04
.
(6.25)
In terms of these variables we have the following Picard-Fuchs equations: L1 = θ12 − 4(4θ1 + 4θ2 − 3)(4θ1 + 4θ2 − 1)z 1 , L2 = θ22 − 4(4θ1 + 4θ2 − 3)(4θ1 + 4θ2 − 1)z 2 ,
(6.26)
where we defined the logarithmic derivative θi = z i dzd i . The periods are linear solutions and in particular z i = 0 is a point of maximal unipotency. Around this point we have a pure power series and two single logarithmic solutions, which correspond to geometric periods. In particular the mirror map is given by z 1 = q1 − 40q12 + 1324q13 − 64q1 q2 + 2560q12 q2 + O(q 4 ), z 2 = q2 − 64q1 q2 + 2560q12 q2 − 40q22 + 2560q1 q22 + O(q 4 ),
(6.27)
with qi = e−ti , where ti , i = 1, 2 are complexified Kähler parameters of the mirror. Indeed, calculation of the intersection numbers in the polyhedron reveals that the corresponding Picard lattice is 1,1 . Using Griffiths transversality (6.2) and (6.26) we can evaluate the two-point functions C x1 x1 =
2 , x1
C x1 x2 =
1 − x1 − x2 , x1 x2
C x2 x2 =
2 . x2
(6.28)
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Here = 1 − 2(x1 + x2 + x1 x2 ) + x12 + x22
(6.29)
is the principal discriminant, and we rescaled the variables suitably z i = xi /64. Due to the above mentioned special properties of the mirror map of K3, we find that Ct1 t1 = Ct2 t2 = 0, Ct1 t2 = 1 in agreement with the identification of the Picard lattice 1,1 . So far we have calculated on the mirror of the double covering of the Enriques surface and we need a justification that the two parameter family treated above does appear as a subfamily of the family of K3 admitting the Enriques involution. We consider now this symmetric splice of the mirror polynomial in (6.21). Let us label the coordinates {yi } again by (s : t, u : v, y). The Z2 involution that we want to mod out (s : t, u : v, y) → (−s : t, −u : v, −y) will break some of the automorphisms of the ambient space. Therefore we can not follow Batyrev’s methods that restrict to the true deformation parameters, which amounts to dropping the codimension one points, as used above. The geometry of the mirror is again a double covering of P1 × P1 branched at the generic degree (4, 4)-hypersurface. The following expression keeps only monomials that are invariant under the Z2 : Y 2 = b1 t 4 v 4 + b2 t 4 u 2 v 2 + b3 t 4 u 4 + b4 u 4 t 2 s 2 + b5 s 4 u 4 + b6 s 4 u 2 v 2 +b7 v 4 s 4 + b8 s 4 t 2 s 2 + b9 t 3 u 3 sv + b10 s 3 u 3 tv +b11 v 3 s 3 tu + b12 t 3 v 3 us + b13 u 2 v 2 t 2 s 2 .
(6.30)
Normally the S L(2, Z) transformations of the two P1 eliminate each three complex parameters. However to be compatible with the Z2 we can only make a rescaling (s, t) → (µs, µ−1 t) and similarly for (u, v). The weighted transformation of y 2 → Y 2 + Y f 2,2 (s, t, u, v) has been used to eliminate linear terms in y, and an overall rescaling will eliminate a third bi . In total we have hence ten invariant bi , which are precisely the deformation parameters of the Enriques surface. It is still very tedious to derive a ten parameter PF system. Let us therefore look at a further symmetric restriction forced by the symmetry u → iu and s → is so that the Z2 × Z4 invariant subslice has the form b0 Y 2 = b1 t 4 v 4 + b3 t 4 u 4 + b5 s 4 u 4 + b7 v 4 s 4 + b13 u 2 v 2 t 2 s 2 .
(6.31)
We will argue now that this family is, up to a Z2 symmetry acting on their moduli space, identical to the family (6.22). At first glance this seems strange, because every monomial in the family (6.31) is invariant under the Enriques involution (s : t, u : v, y) → (−s : t, −u : v, −y), while the monomial ystuv is projected out from (6.22). We can keep this monomial by considering an induced action on the moduli space a0 → −a0 . Now recall, e.g. from the Landau-Ginzburg description of the CY manifold, that terms ∂∂xpi are trivial in the sense that they do not change the residua or periods in the compact case. a0 stuv in the last We can hence use ∂∂ py = 2a5 y − a0 stuv ∼ 0 and substitute y = 2a
5 2 a term of (6.22), which leads to the parameter identification b13 = 2a05 , establishing the equivalence of the two families.
Counting BPS States on the Enriques Calabi-Yau
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6.1.3. Arithmetic expressions for the B-model quantities As discussed in Sect. 6.1.1 we expect to be able to give arithmetic expressions for the mirror map and the fundamental period, similar to what is obtained in [34,38]. If we restrict the PF system (6.26) to one variable by setting z 2 = 0 or z 1 = 0 we find the PF operator L = θ 2 − 4(4θ − 3)(4θ − 1)z,
(6.32)
which corresponds to the (2) elliptic curve 1
x12 = x24 + x34 + z − 4 x1 x2 x3 .
(6.33)
After transforming this curve to the Weierstrass form we calculate its j-function as j (q) = 1728J (q) =
(1 + 192z)3 . z(1 − 64z)2
(6.34)
Let us now define K 2 = ϑ34 + ϑ44 , K 4 = ϑ28 .
(6.35)
In terms of these modular forms, the relation (6.34) can be inverted to obtain the Hauptmodul of (2) as z(q) =
K 4 (q) , 64K 22 (q)
(6.36)
which is the arithmetic expression of the mirror map for the curve (6.33). The triviality of the one-point coupling for the elliptic curve 1=
1 dz 1 , ω02 dt z(1 − 64z)
(6.37)
which is similar to the triviality of the two-couplings in the K3 case (6.8), leads to the following equation for the fundamental period ω0 of the PF equation (6.32) ω02 (q) =
1 K 2 (q) , 2
(6.38)
where we have used (6.36). The full PF system (6.26) can also be solved arithmetically in terms of (6.36) by using the techniques of [38]. It can be easily shown that the simple ansatz K4 K˜ 4 z 1 (q1 , q2 ) = z(q1 )(1 − 64 z(q2 )) = 1− 2 , 64K 22 K˜ 2 K˜ 4 K4 z 2 (q1 , q2 ) = z(q2 )(1 − 64 z(q1 )) = 1− 2 , (6.39) K2 64 K˜ 22 where we defined K 2 = K 2 (q1 ), K 4 = K 4 (q1 ) and K˜ 2 = K 2 (q2 ), K˜ 4 = K 4 (q2 ), provides an analytic expression for the mirror map (6.27). One can also find an analytic expression for the fundamental period of the system (6.26) using (6.38) ( Xˆ 0 )2 (q1 , q2 ) = ω02 (q1 )ω02 (q2 ) =
1 K 2 K˜ 2 . 4
(6.40)
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It is now easy to show that the discriminant (6.29) can be written as = (1 − 64 z(q1 ) − 64 z(q2 ))2 ,
(6.41)
where z(qi ) is the mirror map (6.36). In order to analyze the dependence of the model on the T2 in the base, it is convenient to realize it also algebraically, e.g. as the degree 6 curve x16 + x23 + x32 + z −1/6 x1 x2 x3 = 0
(6.42)
in P(1, 2, 3), which also can be solved arithmetically. The mirror map is determined by the equation J (q3 ) =
1 , z(q3 )(1 − 432z(q3 ))
(6.43)
which can be explicitly inverted to [35,34] z 3 (q3 ) =
J (q3 ) +
√
2 . J (q3 )(J (q3 ) − 1728)
(6.44)
Finally, the fundamental period is 1/4
x 0 (q3 ) = E 4 (q3 ).
(6.45)
Therefore, the fundamental periods x 0 , Xˆ 0 as well as the mirror map for the reduced model can be expressed as modular forms in the parameters t1 , t2 , t3 or functions of subgroups of S L(2, Z)3 , and this fact will become very useful in solving the B-model. It is instructive to compare the reduced model with the Enriques Calabi-Yau constructed as an orbifold of (T2 )3 by Z2K × Z2E , which acts of the coordinates of the three tori (z 1 , z 2 , z 3 ) as a Kummer involution and a free Enriques involution K : E:
− , − , + + 21 , − 21 , − (6.46) 1 1 KE : − 2 , + 2 , −. Here − indicates a sign change and 21 a shift of the coordinate z i . This model has three moduli from the invariant sector of the orbifold, which also have a natural S L(2, Z)3 acting on them. However the geometry is very different. The rank 18 Picard lattice of the mirror of the K 3 family X has intersection E 8 (−1) ⊕ E 8 (−1) ⊕ H (1). As we will verify in detail in Sect. 6.2, the reduced model is obtained by contracting the curves in the E 8 (−1) part of the Picard lattice, after the Enriques identification. If we denote their complex volumes by ti , i = 1, · · · , 8, this is achieved by setting ti = 0, i.e. qi = e−ti = 1. On the other hand the rank 18 Picard lattice of the Kummer K3 K that emerges after the Z2K orbifold in the first two coordinates is generated over Q by sixteen P1 ’s that resolve the A1 singularities at Pi j , i = 1, . . . , 4, j = 1, . . . , 4, and two invariant classes from the T4 . If we call the resolution map σ : K → T4 /Z2K and the exceptional divisors E i j := σ −1 (Pi j ) we get E i j ∩ E kl = −2δik δ jl . Obviously the lattices that get contracted to reach the orbifold point and the reduced model are quite different. Moreover due to the non-trivial B-field one expects that the complex volumes of the P1 ’s t˜i approach |t˜i | → π i at the orbifold limit so that q˜i = −1. This might explain why all information about the full lattice disappears in a type II one-loop computation of F1 in the invariant sector of the orbifold [7].
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6.2. Topological string amplitudes from the reduced B-model. In this subsection we use the holomorphic anomaly equations of [8] to compute the topological string amplitudes for the reduced model in the fiber directions, up to (and including) g = 4. 6.2.1. Genus one amplitude As explained in [8] the topological or holomorphic limit of the genus one free energy F (1) is related to the holomorphic Ray-Singer torsion [50]. The latter describes aspects of the spectrum of the Laplacians of V,q = ∂¯ V ∂¯ V† + ∂¯ V† ∂¯ V of a del-bar operator ∂¯ V : ∧q T¯ ∗ ⊗ V → ∧q+1 T¯ ∗ ⊗ V coupled to a holomorphic vector bundle V over M. More precisely with a regularized determinant over the non-zero mode spectrum of V,q one defines4 [50] I R S (V ) =
n
q (−1)q+1 det V,q 2 .
(6.47)
q=0
One case of interest, V = ∧ p T ∗ with p,q := ∧ p T ∗ ,q , leads to the definition of a family index F (1) =
n n
(−1) p+q pq 1 log det pq , 2
(6.48)
p=0 q=0
depending only on the complex structure of M. As was shown in [8] the holomorphic and antiholomorphic dependence of this object on the complex structure [9] can be integrated using special geometry to yield ∂z f (z) det 1 1 ∂t F (1) = log . (6.49) 2 (X 0 )κ In this expression, X 0 is the fundamental period of the PF system, ∂z/∂t is the Jacobian χ of the mirror map, κ = 3 + h 11 − 12 depends on global topological data, and f 1 (z) is the holomorphic ambiguity at genus one. Up to the normalization factor 21 this is the same expression that was derived in [7] using world-sheet arguments. The large volume behavior F
(1)
h 11 → ti c2 (T ) ∧ Ji , t → ∞, i=1
(6.50)
M
as well as local topological data of other singular limits in the complex structure moduli space, determine the leading behavior of F (1) and fix the holomorphic ambiguity f (z). We can now use our solution to the variations of Hodge structures for the family (6.22) to compute F (1) for this model. Indeed our variable choice at the point of maximal unipotent monodromy given in (6.25) is invariant under a0 → −a0 and the result above can be used verbatim as describing a subfamily of the Enriques surface. Let us first calculate in the Enriques fibre direction from (6.49), which we parameterize as
∂z i + r2 log( Xˆ 0 ) + r3 log(z 1 z 2 ) + r4 log . (6.51) F (1),E (q1 , q2 ) = r1 log det ∂t j 4 [49] reviews these facts and relates the Ray-Singer torsion to Hitchin’s generalized 3-form action at one loop.
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Notice that this can be also interpreted as a calculation for K3×T2 in which one would expect that F (1) vanishes. This is the case if we set r1 = −r2 , r2 = r3 = r4 . As we will explain in Subsect. 6.3.3, the heterotic prediction (4.19) is reproduced for the choice r1 = − 21 (2 + r2 ), r3 = 21 (2 + r2 ) and r4 = 41 (1 + r2 ). Comparing with (6.49) we set χ r2 = −3 to get r1 = 21 . This yields also the expected result κ = 3 + h 11 − 12 = 6 for the three parameter model. We can now use the arithmetic expressions for the mirror map (6.39) and fundamental period (6.40) to derive an exact analytic B-model expression for the Ray-Singer torsion F (1)E (q1 , q2 ) = F (1) (q1 , q2 , q3 = 0) in the Enriques direction (6.51), 1 F (1) E = − log(δ/16), 2
(6.52)
δ = K 22 K˜ 22 − K 4 K˜ 22 − K 22 K˜ 4 ,
(6.53)
where
and we used the fact that the discriminant (6.29) can be written as 2 δ . = K 2 K˜ 2 2
(6.54)
2
Indeed, we can write as well 1 F (1) E = − log() − 2 log( Xˆ 0 ). 4
(6.55)
On the B-model side it is easy to argue that, in genus one, there are no contributions from curves in classes with mixed degree in base and fiber. According to (6.7,6.13) the term det (∂t/∂z) will factorize. The same is true for the X 0 contribution due to (6.16). Finally the f 1 (z) is a product of discriminant factors. There will be two for the fibre and one for the base with no mixing between the base and fibre complex structure coordinates. Because of the logarithm we get a sum of two terms. The first one is the F (1)E in the fiber, and depends only on qi , i = 1, 2 while the second one depends only on the τ parameter of the base, q3 = exp(2πiτ ). We conclude that the total Ray-Singer torsion for the FHSV model is 1 F (1) (q1 , q2 , q3 ) = − log(δ/16) − 12 log(η(q3 )), 2
(6.56)
where the contribution of the base is the same one as for K3×T2 . This has also been argued in [26], however the expansion in [26] is not related to the instanton expansion of the type IIA string as explained in Sects. 4.1 and 4.2. 6.2.2. Propagators and higher genus amplitudes In order to compute the F (g) amplitudes for g > 1 we use the holomorphic anomaly equations of [8]. These equations lead to an expression for F (g) of the form (g) + (X 0 )2g−2 f g , F (g) = Frec (g)
(6.57)
where Frec is completely determined in a recursive way from the topological string
amplitudes at lower genera F (g ) , g < g, their derivatives w.r.t. the flat coordinates, and
Counting BPS States on the Enriques Calabi-Yau
65
the propagators of the Kodaira-Spencer theory introduced in [8]. f g is the holomorphic ambiguity at genus g. It is a rational function on the moduli space of complex structures and to determine it we need some extra data, like for example explicit values of Gromov-Witten invariants at low degree. If we apply this procedure to the reduced, three–parameter model, we find important (g) simplifications in the computation of Frec . This is due to the fact that, in flat coordinates, there is only one nonzero Yukawa coupling C123 which moreover does not receive any worldsheet instanton corrections and it is simply given by C123 = 1. Further derivatives (g) of the Yukawa coupling vanish, and this sets to zero many contributions to Frec . One of the fundamental ingredients of the holomorphic anomaly equations of [8] are the propagators S i j , S i and S of Kodaira-Spencer theory, where i, j are indices for the complex moduli. The procedure to find these propagators in the multiparameter case has been explained in [33]. It turns out that, in the case of the reduced model that we are studying, one can make a choice of gauge in which they have a particularly simple form. One first finds that it is possible to set S ii = 0, i = 1, 2, 3. To solve for the remaining propagators with two indices S i j , i = j, one can use the equation χ 1 ij S Ci jk = ∂k F (1) + − 1 ∂k (log X 0 + log f ), (6.58) 2 24 where f is a holomorphic function of the complex moduli which arises as an integration constant of the holomorphic anomaly equations. The equation (6.58) determines uniquely the propagators S 12 , S 13 , S 23 up to an integration constant. Using now (6.55), we see that the only piece of S 13 that cannot be absorbed into f is −∂ log Xˆ 0 /∂t2 . The same thing happens to S 23 , after exchanging t1 ↔ t2 . We then make the following choice of propagators: 1 d 1 log K˜ 2 = ∂t2 F (1) + A2 , 2 dt2 2 1 d 1 =− log K 2 = ∂t1 F (1) + A1 , 2 dt1 2
S 13 = − S 23
(6.59)
where Ai =
1 ∂t log , 8 i
i = 1, 2.
(6.60)
Finally, one can choose the remaining propagator S 12 to be −E 2 (q3 )/12. The final result for the propagators S i j of the reduced model is then 1 1 K˜ 4 1 ˜ E 2 (q2 ) + − K2, 12 8 K˜ 2 24 = S 13 (q1 ↔ q2 ), 1 = − E 2 (q3 ), 12 = S 33 = 0.
S 13 = S 31 = S 23 = S 32 S 12 = S 21 S 11 = S 22
(6.61)
Notice that, with this choice, S i j only depends on qk , where (i jk) is a permutation of 123. By using now the explicit expressions in [8], it is easy to see that the propagators S i , S can be chosen to be S i = S i j S ik , i = 1, 2, 3,
S = S 12 S 13 S 23 ,
(6.62)
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A. Klemm, M. Mariño
where (i jk) is a permutation of 123. Notice that the structure of the propagators of the reduced model is very similar to the case of toroidal orbifolds studied in [8]. Using the one-loop result (6.56), the recursive formula of [8] for F (2) leads to the expression (1) F (2) = S 12 (F12 + F1(1) F2(1) ) − 2S 12 (S 13 F1(1) + S 23 F2(1) ) + 4S 12 S 13 S 23 1 − E 2 (q3 )(S 13 F1(1) + S 23 F2(1) − 2S 13 S 23 ) + (X 0 )2 f 2 , (6.63) 2 (1) where Fi denote derivatives of F (1) w.r.t. ti , X 0 = Xˆ 0 (q1 , q2 )x 0 (q3 ), and f 2 is the holomorphic ambiguity. In the fibre limit
S 12 → −
1 , 12
X 0 → Xˆ 0 ,
f 2 → f 2E ,
(6.64)
(6.63) should become F (2)E , the genus two amplitude on the fiber. Here f 2E is simply the fibre limit of the holomorphic ambiguity. The heterotic prediction can be recovered by simply setting f 2E = 0.
(6.65)
After using the explicit expressions for the propagators (6.61) and taking into account that (1)
F12 = 8A1 A2 ,
(6.66)
where the Ai are defined in (6.60), we find that the genus two free energy for the Enriques fibre is simply given by 1 (1) (1) F (2) E = − F1 F2 , 4 (1)
where Fi
(6.67)
can be written in terms of K i , K˜ i , E 2 = E 2 (q1 ) and E˜ 2 = E 2 (q2 ), as (1) Fi
1 = 6
κ K 2 (qi ) E 2 (qi ) − , i = 1, 2, 2δ
(6.68)
and κ = δ + 3K 4 K˜ 4 . (g) In the computation of F (2) we have used the formulae for Frec obtained in [8], and we have applied them directly to the reduced model. However, for higher genus amplitudes this method is problematic. The reason for this is that, in order to obtain the correct result, we have to implement the reduction consistently, i.e. we have to first consider the for(g) mulae for Frec in the original model with 11 Kähler parameters and then set the E 8 (−1) parameters to zero. In general, the result of this will be different from the result obtained (g) by computing Frec directly in the reduced, three–parameter model. The two procedures lead to the same tensorial structures, but with different numerical coefficients. It can be seen that at genus 2 this is not a problem, but for higher genus we cannot use the reduced model to obtain the answer for F (g) .
Counting BPS States on the Enriques Calabi-Yau
67
Table 7. Genus one BPS invariants n 1 (m, n) for the two parameter subfamily. By construction there is a symmetry under exchange of (m, n) m 0 1 2 3 4 5 6
n=0 – 8 4 8 4 8 4
1
2
3
4
8 2048 49152 614400 5373952 37122048 216072192
4 49152 5372928 216072192 5061451776 83300614144 1063005671424
8 614400 216072192 21301241856 1063005978624 34065932304384 794110053826560
4 5373952 5061451776 1063005978624 100372720320512 5641848336678912 218578429867425792
One can still use the recursive formulae in the reduced model in order to obtain general properties of the amplitudes, as well as expressions for F (g)E in terms of modular forms. For g = 3 one finds, for example,
˜ 2 K˜ 2 ρ1 ρ 1 κ E2 K 2 κ E 2 (3) E 2 2 E2 − + + F = − 10 4 E˜ 2 − 2 3 δ (2δ)2 δ (2δ)2
1 3 +9µ 2 (E 2 K˜ 2 − E˜ 2 K 2 )2 − 4 ρ3 , (6.69) δ 4δ where µ = (K 22 − K 4 )( K˜ 22 − K˜ 4 )K 4 K˜ 4 and ρ1 = κ 2 K 22 − 9(K 22 − K 4 )K 4 (( K˜ 22 + 7 K˜ 4 )δ + 9 K˜ 22 K 4 K˜ 4 ), ρ2 = ρ1 (q1 ↔ q2 ), (6.70) 3 2 2 2 2 2 2 ρ3 = δ + (7δ + 72δ K 4 K˜ 4 )(K 2 K˜ 4 + K 4 K˜ 2 ) + 12K 4 K˜ 4 (5δ + 6K 2 K˜ 2 K 4 K˜ 4 ). We have also found explicit results for the genus four topological string amplitude, which are too long to be reported here but are available on request. Note that F (g)E exhibits a leading order pole at the discriminant of the form F (g)E ∼
bg b˜ g = 2g−2 , g−1 δ
(6.71)
which restricts the ansatz for the holomorphic ambiguity along the fiber f gE . 6.2.3. Comparison with the heterotic results. Let us now do a more detailed comparison of the B-model results with the heterotic predictions in the geometric reduction. Table 7 contains the B-model prediction for genus one BPS numbers in the classes (m, n) of 1,1 , the cohomology lattice of the reduced model. This is identified with the 1,1 sublattice inside the Picard lattice of the Enriques surface E = 1,1 ⊕ E 8 (−1). The total class in 1,1 ⊕ E 8 (−1) is labelled by a vector r = (n, m, v) with norm r 2 = 2mn − v 2 . The B-model for the subfamily calculates BPS invariants in which, for given (m, n), one sums over all possible vectors v in the E 8 (−1) lattice. Recall that the coefficients of q in the E 8 -theta function 2 2 E 8 (q) = q v /2 = m( v 2 /2)q v /2 v∈E 8 (1)
v2
= E 4 (q) = 1 + 240q + 2160q 2 + 6270q 3 + . . .
(6.72)
68
A. Klemm, M. Mariño
yield the total number m( v 2 /2) of vectors v with a given norm. This means that the results obtained from the reduced B model should correspond to a “reduced” heterotic theory in which we freeze the moduli of the E 8 (−1) lattice. Therefore, the topological string amplitudes of this “reduced” heterotic model will be given by Fgred (t) = cgred (2nm) 23−2g Li3−2g (q1n q2m ) − Li3−2g (q12n q22m ) , (6.73) n,m
where
cgred (n)q n = f 1 (q)E 4 (2τ )Pg (q).
(6.74)
n
The first thing we notice when we compare the heterotic theory and the B-model is that, in the results from the B model, the BPS invariants depend only on the product nm but also on whether the class (n, m) is even or odd. For (m, n) in 1,1 to be in a even class, m and n have to be even. In odd classes either m or n or both are odd. This is needed in order to match (6.73). One can easily see that indeed there is a precise agreement between (6.73) and the B-model results presented above. For example, the invariant 2048 at genus one in the B-model is given by 2048 = 128 + 240 · 8,
(6.75)
v2
where the number 240 counts the vectors of norm = 2. Notice that, as a by-product of this comparison at g = 1, we obtain the following Borcherds-type identity: 1 − q n q m c(2nm) 2 2 2 2 1 2 , K 2 (q1 )K 2 (q2 ) − K 4 (q1 )K 2 (q2 ) − K 2 (q1 )K 4 (q2 ) = 16 nqm 1 + q 1 2 n,m (6.76) where n
c(n)q n = −
64 3η6 (τ )ϑ26 (τ )
E 2 (τ )E 4 (2τ ).
(6.77)
One can also check that the above B-model expressions on the fiber, for 2 ≤ g ≤ 4, agree with the heterotic prediction for the reduced model (6.73). 6.2.4. Extending the results to the CY threefold. Let us now turn from the fibre limit to the full Enriques Calabi-Yau by including the base classes. We first notice one important general property of F (g) : it will be a modular form with respect to a modular subgroup5 in SL(2, Z) × SL(2, Z) × SL(2, Z), acting on the parameters (q1 , q2 , q3 ). The modular weight is given by (2g − 2, 2g − 2, 2g − 2).
(6.78)
This can be seen most easily by looking at the last term in (6.57). The holomorphic ambiguity has zero modular weight, since it is given by a rational function of the coordinates z 1 , z 2 , z, which are all modular forms of zero weight. The fundamental period X 0 has however modular weight (1, 1, 1), which leads to (6.78). 5 It was noticed in [54] that the F (g) calculated for the quintic in P4 in [31] can be written as polynomials of five generators. This is presumably a manifestation of modular properties of the correponding F (g) w.r.t. the modular group of the quintic in S P(4, Z).
Counting BPS States on the Enriques Calabi-Yau
69
In order to study the full dependence on the T2 on the base, we first notice that the F (g) amplitudes restricted to the base vanish, since they should be equal to the topological string amplitudes of K3×T2 on the base. If we denote F (g)B (q3 ) = F (g) (q1 = 0, q2 = 0, q3 ), we have, F (g)B = 0, g > 1.
(6.79)
F (g) (q
We now use this information to find expressions for 1 , q2 , q3 ). We first consider g = 2. Since S 13 = O(q2 ), S 23 = O(q1 ), the expression (6.63) reproduces the result (6.79) if we assume that the holomorphic ambiguity vanishes as well when restricted to the base. Given these facts, it is natural to assume that the total ambiguity f 2 vanishes for the choice of propagators (6.61). In this case, the only dependence of (6.63) on q3 is an overall factor E 2 (q3 ), and we finally obtain the remarkably simple expression for g = 2, F (2) = E 2 (q3 )F (2)E .
(6.80)
This simple form for the reduced model agrees with the results of Maulik and Pandharipande on the Gromov-Witten invariants at genus two for mixed classes of the full Enriques CY [45] (in fact, their results suggested to us the existence of a simple gauge for the propagators). Notice that (6.80) can be also written, after setting q3 = e−t3 , as 1 (1) (1) (1) F (2) = − F1 F2 F3 , 2
(6.81)
again with the same structure as the genus 2 amplitude for the toroidal orbifold considered in [8]. As is clear from (6.68), the expression (6.81) exhibits a pole at 1 ∼ δ12 . A remarkable fact is that it does not have such a pole at the discriminant b = 1 − 432z 3 of the base curve (6.42). It seems reasonable to assume that such poles do not occur at any genus and to refine (6.78) in that F (g) is a holomorphic, quasimodular function of q3 of weight 2g − 2, i.e. in what concerns the dependence on q3 it is generated by E 2 , E 4 , E 6 . In this respect, the modular properties of F (g) with respect to the modular parameter of the torus would be similar to those found in the case of the elliptic curve [16]. On the other hand, F (g) is a weakly holomorphic function of q1 , q2 with weights (2g − 2, 2g − 2), which in particular contains δ −1 . This assumption restricts the ambiguity considerably and leads uniquely to the following expression for F (3) : F (3) = E 22 (q3 )F (3)E + (E 22 (q3 ) − E 4 (q3 ))H (3) (q1 , q2 ),
(6.82)
1 1 H (3) (q1 , q2 ) = − F (3)E − (F1(1)E F2(2)E + F2(1)E F1(2)E ). 2 24
(6.83)
where
In the case g = 3 we don’t have results on the mixed classes to compare with and check in detail the conjectures (6.82) and (6.83). However, we have verified that they lead to results which are consistent with (6.79) and with integrality of the BPS numbers n g (r ) ∈ Z in the expansions (2.30). As expected from the discussion above (5.5), all of these numbers are divisible by eight. In the tables above we list some BPS invariants n g (r ) for base degree d equal to one. In [23], we will extend these ideas and present an alternative and more powerful method to derive (6.82) and (6.83) which makes it also possible to obtain results for F (g) to high genus.
70
A. Klemm, M. Mariño Table 8. Genus two BPS invariants n 2 (m, n, 1) for branes wrapping the base torus once
m 0 1 2 3 4
n=0 0 0 0 0 0
1
2
3
4
0 384 99072 2557440 34604544
0 99072 34604544 2425752576 82015423488
0 2557440 2425752576 399200753664 28156719273984
0 34604544 82015423488 28156719273984 3717898174470144
Table 9. Genus three BPS invariants n 3 (m, n, 1) for branes wrapping the base torus once m 0 1 2 3 4
n=0 0 0 0 0 0
1
2
3
4
0 128 33792 1052160 17047552
0 33792 25704448 2596196352 113305067520
0 10521600 2596196352 635491780608 58963231506432
0 17047552 113305067520 58963231506432 10321183934611456
Acknowledgements. We would like to thank Ignatios Antoniadis, Jim Bryan, Igor Dolgachev, Rajesh Gopakumar, Sheldon Katz, Wolfgang Lerche, Margarida Mendes Lopes, Boris Pioline, Emanuel Scheidegger, and Cumrun Vafa for useful conversations, and Ciprian Borcea and Elias Kiritsis for useful correspondence. We are specially grateful to Greg Moore for extensive discussions on various issues addressed in this paper, and for a detailed critical reading of the manuscript. Finally, our thanks to Davesh Maulik and Rahul Pandharipande for extensive discussions on the Gromov–Witten theory of the Enriques CY, and for sharing their results with us.
A. Theta Functions and Modular Forms Our conventions for the Jacobi theta functions are: 1 2 ϑ1 (ν|τ ) = ϑ[11 ](ν|τ ) = i (−1)n q 2 (n+1/2) eiπ(2n+1)ν , n∈Z
ϑ2 (ν|τ ) =
ϑ[10 ](ν|τ )
ϑ3 (ν|τ ) =
ϑ[00 ](ν|τ )
=
1
q 2 (n+1/2) eiπ(2n+1)ν , 2
n∈Z
=
1 2
q 2 n eiπ 2nν ,
(A.1)
n∈Z
ϑ4 (ν|τ ) = ϑ[01 ](ν|τ ) =
1 2 (−1)n q 2 n eiπ 2nν , n∈Z
where q = e2πiτ . When ν = 0 we will simply denote ϑ2 (τ ) = ϑ2 (0|τ ) (notice that ϑ1 (0|τ ) = 0). The theta functions ϑ2 (τ ), ϑ3 (τ ) and ϑ4 (τ ) have the following product representation: ϑ2 (τ ) = 2q 1/8
∞
(1 − q n )(1 + q n )2 ,
n=1
ϑ3 (τ ) = ϑ4 (τ ) =
∞ n=1 ∞ n=1
1
(1 − q n )(1 + q n− 2 )2 , 1
(1 − q n )(1 − q n− 2 )2 ,
(A.2)
Counting BPS States on the Enriques Calabi-Yau
71
and under modular transformations they behave as: # ϑ2 (−1/τ ) = τi ϑ4 (τ ), ϑ2 (τ + 1) = eiπ/4 ϑ2 (τ ), # ϑ3 (τ + 1) = ϑ4 (τ ), ϑ3 (−1/τ ) = τi ϑ3 (τ ), # τ ϑ4 (τ + 1) = ϑ3 (τ ). ϑ4 (−1/τ ) = i ϑ2 (τ ),
(A.3)
The theta function ϑ1 (ν|τ ) has the product representation 1 8
ϑ1 (ν|τ ) = −2q sin(π ν)
∞
(1 − q n )(1 − 2 cos(2π ν)q n + q 2n ).
(A.4)
n=1
We also have the following useful identities: ϑ34 (τ ) = ϑ24 (τ ) + ϑ44 (τ ),
(A.5)
ϑ2 (τ )ϑ3 (τ )ϑ4 (τ ) = 2 η3 (τ ),
(A.6)
and
where η(τ ) = q 1/24
∞
(1 − q n )
(A.7)
n=1
is the Dedekind eta function. One has the following doubling formulae: % $ ϑ32 (τ ) − ϑ42 (τ ) η(τ )ϑ2 (τ ) , ϑ2 (2τ ) = , η(2τ ) = 2 2 % & ϑ32 (τ ) + ϑ42 (τ ) , ϑ4 (2τ ) = ϑ3 (τ )ϑ4 (τ ), ϑ3 (2τ ) = 2 & η(τ/2) = η(τ )ϑ4 (τ ).
(A.8)
The Eisenstein series are defined by E 2n (q) = 1 −
∞ 4n k 2n−1 q k , B2n 1 − qk
(A.9)
k=1
where Bm are the Bernoulli numbers. The formulae for the derivatives of the theta functions are also useful: d 1 q log ϑ4 = E 2 − ϑ24 − ϑ34 , dq 24 d 1 q log ϑ3 = E 2 + ϑ24 − ϑ34 , (A.10) dq 24 d 1 q E 2 + ϑ34 + ϑ44 , log ϑ2 = dq 24
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A. Klemm, M. Mariño
and from these one finds, q
1 d log η = E 2 (τ ), dq 24
q
d 1 2 E2 = E2 − E4 . dq 12
(A.11)
The doubling formulae for E 2 (τ ), E 4 (τ ) are 1 1 E 2 (τ ) + (ϑ34 (τ ) + ϑ44 (τ )), 2 4 1 15 E 4 (2τ ) = E 4 (τ ) + ϑ34 (τ )ϑ44 (τ ). 16 16
E 2 (2τ ) =
(A.12)
B. Lattice Reduction In this Appendix we briefly review the computation of integrals of the form (2.8) with the technique of lattice reduction. This technique applies to integrals of the form
d 2τ f J (τ, τ¯ ) J (τ, α J , β J , P, φ). 2 F τ2 J
(B.1)
These integrals are sometimes called theta transforms of the (quasi)modular forms f J (τ, τ¯ ). The generalized Siegel-Narain theta function which appears in this integral is defined as (τ, α, β, P, φ) = exp − (φ(P(λ))) 8π τ2 p∈ × exp πiτ ( p + β/2)2+ + πiτ ( p + β/2)2− + πi( p + β/2, α) , (B.2) −
where is a lattice of signature (b+ , b− ), P is the projection, φ is a polynomial on Rb ,b of degree m + in the first b+ variables and of degree m − in the second b− variables, and + − is the (Euclidean) Laplacian in Rb +b . The rest of the notations were introduced in Sect. 2. The lattices involved in (B.1) all have the same signature, and they only differ in overall factors for their norms as well as in the shifts α J , β J . In the computation of these integrals by lattice reduction, one proceeds iteratively and in each step the rank of the lattice is reduced by two. Proceeding in this way, one can reduce the computation of (B.1) to evaluation of quantities associated to the reduced lattices. Let us consider the simple case in which there is only one term in the sum (B.1), with α = β = 0, and the lattice is even and self-dual. In this case we will denote (B.2) by (τ, P, φ). The theta transform is then given by +
(P, φ, F ) =
d 2τ (τ, P, φ)F (τ, τ¯ ), 2 F τ2
(B.3)
where b+ /2+m +
F (τ, τ¯ ) = τ2
F(τ )
(B.4)
Counting BPS States on the Enriques Calabi-Yau
73
is a (quasi)modular form with weight (−b− /2−m − , −b+ /2−m + ), constructed from the (quasi)modular form F(τ ) with weights (b+ /2 + m + − b− /2 − m − , 0). We will assume that F(τ ) is an almost holomorphic form, i.e. it has the expansion F(τ ) =
c(m, t)q m τ2−t ,
(B.5)
m∈Q t≥0
where c(m, t) are complex numbers which are zero for all but a finite number of values of t and for sufficiently small values of m. Lattice reduction is then implemented as follows. Let z be a primitive vector of of zero norm, and let K = ( ∩ z ⊥ )/Zz. This lattice, which has signature (b+ − 1, b− − 1), is called the reduced lattice. A typical situation when choosing a reduction vector occurs when the lattice has 1,1 as a sublattice, where 1,1 is the lattice of rank two and intersection form
01 . (B.6) 10 In this case, one can take z to be one of the vectors that generate 1,1 . In the reduced as follows: consider z ± ≡ P± (z), and lattice one can construct “reduced” projections P ± b ⊥ decompose R z ± ⊕ z ± . The projection on the orthogonal complement z ± ⊥ ± . It can be explicitly written in terms of P± as is the reduced projection P ± ( p) = P± ( p) − (P± ( p), z ± ) z ± . P 2 z±
(B.7)
Once this reduced projection has been constructed, we have to decompose the polynomial involved in (B.2) with respect to this projection, according to the expansion φ(P( p)) =
+ − p)), ( p, z + )h ( p, z − )h φh + ,h − ( P(
(B.8)
h + ,h −
where ph + ,h − are homogeneous polynomials of degrees (m + −h + , m − −h − ) on P(⊗R). We now write the vectors of the lattice as p = cz + mz + p K ,
(B.9)
where p K is a vector in the reduced lattice K and (z , z) = N . When the reduction vector belongs to a sublattice 1,1 , the vector z is the other generator of the sublattice and N = 1. One can now rewrite the Siegel-Narain theta function in terms of the reduced lattice, after a Poisson resummation of the integer m, as in [12]
h + − τ2 z +2 h h (τ, P, φ) = & − 2 π h h 2τ2 z + h≥0 h + ,h −
π |N cτ + |2 + − (N cτ¯ + )h −h (N cτ + )h −h exp − × 2τ2 z +2 c, 1
h! + − (−2iτ2 )h +h
φh + ,h − ). × K (τ, µ/N , −cµ, P,
(B.10)
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A. Klemm, M. Mariño
In writing this formula we have assumed that ( p K , z ) = (z , z ) = 0, and we have introduced the vector z z + − + 2 ∈ K ⊗ R. µ = −z + N (B.11) 2z +2 2z − Once this expression is inserted into the integral (B.1), one can apply the “unfolding” procedure, in which the integral over the fundamental domain F of SL(2, Z) becomes an integral over the domain [−1/2, 1/2]×(0, ∞). At the same time, one can set c = 0 in (B.10) by modular invariance. There are two types of contributions in the end. The first one comes from = 0 and it is sometimes called the contribution of the “zero orbit.” It is given by
h 1 z +2 φh,h , F K ). & K ( P, 2z +2 h≥0 4π
(B.12)
Notice that this is another theta transform, but for the reduced lattice, which is smaller. The contribution from the nonzero orbits comes from > 0, and it involves a sum over + (λ K ) = 0, it is given by the reduced lattice K . When P %
2 h + −
z+ h h j 2 h! 1 p K )) − p h + ,h − ( P( + − − π h h j! 8π (2i)h +h z +2 + − K h≥0 h ,h
∞
j
p ∈K
+ ( p K )| 2|z + || P t
=1
+ ( p K )| 2π | P + − + + . ·K h−h −h − j−t+b /2+m −3/2 |z + | ·
e2πi ( p
K ,µ)/N
2c( p 2 /2, t)
h−h + −h − − j−t+b+ /2+m + −3/2 (B.13)
Here, K ν (z) is the modified Bessel function, which comes from an integral over the strip + (λ K ) = 0, the integral has to be regularized and this leads to a slightly τ2 > 0. When P different expression which can be found in [12]. An important remark is that the above expressions are only valid if |z + |2 1.
(B.14)
For a fixed primitive vector z, the value of z +2
depends on the projection we choose for the lattice, which is in turn determined by the Narain moduli of the string compactification. Different choices of regions in moduli space will require different choices of vectors for the lattice reduction, and therefore to different expressions for the resulting integral. We have considered here the simplest case in which the theta transform (B.1) involves a single summand with α = β = 0. More general cases can also be analyzed with the technique of lattice reduction, see [42,40] for examples. References 1. Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005) 2. Antoniadis, I., Gava, E., Narain, K.S., Taylor, T.R.: Topological amplitudes in string theory. Nucl. Phys. B 413, 162 (1994)
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37. Lerche, W., Nilsson, B.E.W., Schellekens, A.N., Warner, N.P.: Anomaly Cancelling Terms From The Elliptic Genus. Nucl. Phys. B 299, 91 (1988) 38. Lian, B.H., Yau, S.-T.: Arithmetic properties of mirror map and quantum coupling. Commun. Math. Phys. 176, 163 (1996); Mirror maps, modular relations and hypergeometric series 1, & 2. http://arxiv.org/list/ hep-th/9507151, 1995; Nucl. Phys. Proc. Suppl. 46, 248 (1996) [arXiv:hep-th/9507153] 39. Lopes Cardoso, G., Lust, D., Mohaupt, T.: Threshold corrections and symmetry enhancement in string compactifications. Nucl. Phys. B 450, 115 (1995) 40. Lozano, C., Mariño, M.: Donaldson invariants of product ruled surfaces and two-dimensional gauge theories. Commun. Math. Phys. 220, 231 (2001) 41. Lüst, D.: String vacua with N = 2 supersymmetry in four dimensions. http://arxiv.org/list/hep-th/ 9803072, 1998 42. Mariño, M., Moore, G.W.: Donaldson invariants for non-simply connected manifolds. Commun. Math. Phys. 203, 249 (1999) 43. Mariño, M., Moore, G.W.: Counting higher genus curves in a Calabi-Yau manifold. Nucl. Phys. B 543, 592 (1999) 44. Maulik, D., Pandharipande, R.: A topological view of Gromov-Witten theory. http://arxiv.org/list/math. AG/0412503, 2004 45. Maulik, D., Pandharipande, R.: New calculations in Gromov-Witten theory. http://arxiv.org/list/math. AG/0601395 46. Mayr, P., Stieberger, S.: Threshold corrections to gauge couplings in orbifold compactifications. Nucl. Phys. B 407, 725 (1993) 47. Moore, G.W., Witten, E.: Integration over the u-plane in Donaldson theory. Adv. Theor. Math. Phys. 1, 298 (1998) 48. Morales, J.F., Serone, M.: Higher derivative F-terms in N = 2 strings. Nucl. Phys. B 481, 389 (1996) 49. Pestun, V., Witten, E.: The Hitchin functionals and the topological B-model at one loop. Lett. Math. Phys. 74, 21–51 (2005) 50. Ray, D.B., Singer, I.M.: Analytic Torsion for complex manifolds. Ann. of. Math. 98, 154 (1973) 51. Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nucl. Phys. B 431, 484 (1994) 52. Vafa, C.: Instantons on D-branes. Nucl. Phys. B 463, 435 (1996) 53. Voisin, C.: Miroirs et involutions sur les surfaces K3. Astérisque 218, 273 (1993) 54. Yamaguchi, S., Yau, S.-T.: Topological string partition functions as polynomials. JHEP 0407, 047 (2004) 55. Yau, S.-T., Zaslow, E.: BPS States, String Duality, and Nodal Curves on K3. Nucl. Phys. B 471, 503 (1996) Communicated by N.A. Nekrasov
Commun. Math. Phys. 280, 77–121 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0413-9
Communications in
Mathematical Physics
Critical Elastic Coefficient of Liquid Crystals and Hysteresis Xing-Bin Pan Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China. E-mail:
[email protected] Received: 7 December 2006 / Accepted: 16 July 2007 Published online: 7 March 2008 – © Springer-Verlag 2008
Abstract: P. G. de Gennes predicted the analogies between the effect of the elastic coefficients to liquid crystals and the effect of applied magnetic fields to superconductors, and predicted that all elastic coefficients diverge to infinity at smectic-C to nematic transition. One would expect quantitative comparison in the analogies. In the case of equal elastic coefficients (K 1 = K 2 = K 3 = K ), we define the critical value K c of the elastic coefficients and make comparison of it with the upper critical magnetic field HC3 for type II superconductors. We classify the smectic liquid crystals into subcritical, critical and supercritical cases according to the Ginzburg-Landau parameter κ, the wave number q and the boundary value of the director at the surface. We show that in the subcritical case the liquid crystal does not undergo phase transition; and in the supercritical case both phase transition and hysteresis occur. The prediction of de Gennes is true in the critical case where µπ (u0 , q) = κ 2 and K c = +∞.
Contents 1.
2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The variational problem, and the critical elastic coefficient K c 1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Physical explanation . . . . . . . . . . . . . . . . . . . . . . 1.5 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Properties of the Critical Elastic Coefficient . . . . . . . . Subcritical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . Supercritical Case . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Degenerate Minimizing Harmonic Maps . . . . . . . . . . . . Critical Case: Estimates of K c . . . . . . . . . . . . . . . . . . . . Critical Case: Behavior of Minimizers at K c . . . . . . . . . . . .
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78 78 79 81 86 86 87 91 92 97 100 105
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7.1 Three subcases . . . . . . . . . . . . . . . . . . . . . . 7.2 Sufficient conditions for hysteresis . . . . . . . . . . . . 8. Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Eigenvalues Continuously Depend on Magnetic Potential . B. Minimality of Trivial Critical Points . . . . . . . . . . . . C. Two Estimates of K c . . . . . . . . . . . . . . . . . . . . D. Proof of (1.11) . . . . . . . . . . . . . . . . . . . . . . .
X.-B. Pan
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1. Introduction 1.1. Motivations. The analogies between liquid crystals and superconductors were discovered by P. G. de Gennes in 1972 [dG1], where he compared the effects of magnetic fields to superconductivity with the effects of twist and bend to smectics. See also [dGP].1 In 1973 de Gennes further gave the following predictions [dG2, p.49]: (I) At smectic-C to nematic transition, all elastic coefficients diverge. (II) At smectic-A to nematic transition, K 2 and K 3 diverge, while K 1 remains bounded. In this paper we try to analyze prediction (I) by using of the Landau-de Gennes model, and leave the analysis of prediction (II) in future work [P4]. Let us begin with checking the analogies by comparing the Landau-de Gennes functional for liquid crystals ([dGP])2 κ2 |∇qn ψ|2 − κ 2 |ψ|2 + |ψ|4 + K 1 |div n|2 + K 2 |n · curl n|2 E[ψ, n] = 2 (1.1) 2 2 2 + K 3 |n × curl n| + (K 2 + K 4 )[tr(∇n) − (div n) ] d x with the Ginzburg-Landau functional for superconductivity ([GL]) κ2 |curl A − H|2 d x. G L[ψ, A] = {|∇κA ψ|2 − κ 2 |ψ|2 + |ψ|4 }d x + κ 2 3 2 R
(1.2)
In these formulas we use the notation ∇qn ψ = ∇ψ − iqnψ. In (1.2), A is the magnetic potential, and ψ is the order parameter of the superconductor; ψ = 0 if the superconductor is in the normal state, and ψ ≡ 0 if it is in a superconducting state. H is the applied magnetic field. κ > 0 is the Ginzburg-Landau parameter, √ which indicates the 2, and it is type II type of the superconductor: The superconductor is type I if κ < 1/ √ if κ > 1/ 2. In (1.1), n is a unit length vector field called director, and ψ is a complex-valued function called order parameter of the liquid crystal; ψ = 0 if the liquid crystal is in a nematic state, and ψ ≡ 0 if it is in a smectic state. K j ’s are called elastic coefficients, among them K 1 (splay constant), K 2 (twist constant) and K 3 (bend constant) are positive. q is a real parameter called wave number. Motivated by the analogies between (1.1) and (1.2), we shall also call the parameter κ in (1.1) the Ginzburg-Landau parameter of the 1 The impact of this theory to physics of liquid crystals has been well recognized. Here we would like to mention the discovery of the twist grain boundary, which was first predicted in 1988 by S. Renn and T. Lubensky [RL] using de Gennes theory, and observed in 1989 by J. Goodby et at. [GW1,GW2]. 2 Here we take the form of the Landau-de Gennes functional as given in [C]. See also [P1].
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liquid crystal. Our recent results suggest that liquid crystals with large κ exhibit type II behaviors (see [P1]), and those with small κ exhibit type I behaviors (see [P2]).3 It is interesting to us to find more quantitative comparisons between the effect of the elastic coefficients to liquid crystals and the magnetic effect to superconductivity. Let us recall that, for type II superconductors subjected to an applied magnetic field, there exists a critical number HC3 , called the upper critical field, such that as the applied field increases and reaches HC3 , the superconductor changes from the superconducting state to the normal state (see [SdG, LP1, LP2]).4 We wish to know if there also exists a critical value for the elastic coefficients, which is an analogue of HC3 , such that the liquid crystal changes from a smectic state to a nematic state as the elastic coefficients increase and reach the critical value. These considerations motivated us to introduce the critical elastic coefficient K c and to examine the phase transitions of liquid crystals as the elastic coefficients increase and pass by K c . We were not able to find any literature which discussed (finite) critical values for the elastic coefficients. 1.2. The variational problem, and the critical elastic coefficient K c . To make our question clear, let us consider the variational problem for the Landau-de Gennes functional where the director field n is subjected to the Dirichlet boundary condition: n = u0 on ∂, where u0 : ∂ → S2 is a given smooth vector field. Then the last term in (1.1) depends only on u0 and can be dropped ([HKL1]). Our results can be most simply represented in the case of equal elastic coefficients, namely when K 1 = K 2 = K 3 = K > 0. In this case the energy functional (1.1) has the form5 E K [ψ, n] =
{|∇qn ψ|2 − κ 2 |ψ|2 +
κ2 |ψ|4 + K |∇n|2 }d x. 2
(1.3)
Throughout this paper we assume that is a bounded domain in R3 with smooth surface, and u0 ∈ C 2+α (∂, S2 ). (1.4) We look for minimizers in H 1 (, C) × H 1 (, S2 , u0 ), where H 1 (, S2 , u0 ) = {n ∈ H 1 (, S2 ) : n = u0 on ∂}. The existence of the minimizers was proved in [P1]. In this paper the word “minimizer” always means a global minimizer. (ψ0 , n0 ) is called a local minimizer of E K if there exists a neighborhood U of (ψ0 , n0 ) in H 1 (, C) × H 1 (, S2 , u0 ) such that E K [ψ0 , n0 ] ≤ E K [ψ, n] for all (ψ, n) ∈ U. 3 See [P3] for more of our study on the analogies between liquid crystals and superconductors. 4 More precisely, there exist two critical values for the upper critical field, namely, H C3 and H C3 , with
H C3 ≤ H C3 , such that the superconductor is in the superconducting state if the applied magnetic field H is below H C3 , and it is in the normal state if H is above H C3 . It has been proved recently by Fournais and Helffer [FH] that under some additional condition on the geometry of the superconductor it holds that H C3 = H C3 if the Ginzburg-Landau parameter κ is large. 5 This can be verified directly using the fact |n(x)| = 1.
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The Euler-Lagrange equations for a critical point (ψ, n) of E K are6 ⎧ 2 −∇qn ψ = κ 2 (1 − |ψ|2 )ψ, ⎪ ⎪ ⎨ q ¯ ¯ − (n · ψ∇ψ)n} in , −n = |∇n|2 n + {ψ∇ψ ⎪ K ⎪ ⎩ ν · ∇qn ψ = 0, n = u0 on ∂,
(1.5)
where ν is the unit outer normal vector to ∂, and we use the notation ∇A2 ψ = ψ − i[2A · ∇ψ + ψdiv A] − |A|2 ψ. ¯ qn ψ)n}. ¯ ¯ ¯ qn ψ − (n · ψ∇ Note that {ψ∇ψ − (n · ψ∇ψ)n} = {ψ∇ E K has trivial critical points (0, n), where n are the minimizers of the Dirichlet energy F[n] = |∇n|2 d x,
namely they are minimizing weak harmonic maps into the sphere S2 , and hence satisfy the equation − n = |∇n|2 n in ,
n = u0 on ∂.
(1.6)
A trivial critical point (0, n) represents a nematic state. A minimizer (ψ, n) of E K with ψ ≡ 0 is called a nontrivial minimizer, which represents a smectic state. Notations 1.1. C(u0 , K ) ≡ C(u0 , K , κ, q) =
inf
(ψ,n)∈H 1 (,C)×H 1 (,S2 ,u0 )
E K [ψ, n],
M(E K ) = {all nontrivial minimizers of E K }. Notations 1.2. Cπ (u0 ) =
|∇n|2 d x, |∇n|2 d x = Cπ (u0 ) . (u0 ) = n ∈ H 1 (, S2 , u0 ) : inf
n∈H 1 (,S2 ,u0 )
Definition 1.3. We define the critical value of the elastic coefficients by K c ≡ K c (u0 , κ, q) = inf {K > 0 : E K has only trivial minimizers}. The (global) minimizers of E K are nontrivial (which describe the smectic states) when 0 < K < K c (if K c > 0), and they are trivial (which describe the nematic states) when K > K c (if K c < +∞), see Lemma 2.1. So K c has a clear physical meaning.7 Hence it is interesting to know whether K c is finite and how it depends on the parameters κ and q. Let us mention that the critical wave number Q c3 introduced in [P1] is the critical value of the wave number q, while the critical elastic coefficient K c introduced here is the critical value of the elastic coefficients. Based on de Gennes’ prediction (I), we may ask the following question: 6 Equation (1.5) can be derived by the standard method. See [P1, (3.1)] for a more general case. 7 Under some condition on the geometry of the superconductor, the upper critical field H C3 has a clear
physical meaning for large κ, see Footnote 4.
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Question 1. Given boundary data u0 , find the regime of parameters q and κ such that the following hold: (i) K c = +∞; (ii) as K → +∞, the nontrivial minimizers have a subsequence that converges to a trivial critical point. If (i) and (ii) in Question 1 happen, we may say that the nontrivial minimizers bifurcate from a trivial critical point at K = +∞; which means that the phase transition from the smectic state to the nematic state occurs at K = +∞. A necessary condition for (i) and (ii) to hold is that µπ (u0 , q) = κ 2 ,
(1.7)
see Theorems 2, 3 and 3.1, where the number µπ (u0 , q) will be defined in Notation 1.4 below. 1.3. Main results. Investigations on K c naturally lead to study of the following eigenvalue problem 2 φ = µφ in , − ∇qn
ν · ∇qn φ = 0
on ∂.
(1.8)
2 . We shall simply call an eigenfunction of (1.8) an eigenfunction of −∇qn
Notations 1.4. µ(qn) denotes the lowest eigenvalue of (1.8). 2 associated with µ(qn)}, S p(qn) = {all eigenfunctions of −∇qn ¯ S p + (qn) = {φ ∈ S p(qn) : (φ∇φ) × n ≡ 0 in },
µh (u0 , q) = µπ (u0 , q) =
inf
n∈H 1 (,S2 ,u0 )
inf
n∈ (u0 )
µ(qn),
µ(qn),
∗ (u0 , q) = {n ∈ (u0 ) : µ(qn) = µπ (u0 , q)}. Remark 1.5. (i) µπ (u0 , q) is achieved, and hence ∗ (u0 , q) = ∅. (ii) For generic domains , boundary data u0 and wave number q > 0 we have8 µh (u0 , q) < µπ (u0 , q).
(1.9)
Hence in the following we shall always assume that (1.9) holds. It implies in particular that u0 is not a constant vector. Hence Cπ (u0 ) > 0. We shall see that K c > 0 if and only if µh (u0 , q) < κ 2 (Lemma 2.2). For our convenience we give the following classification according to the value µπ (u0 , q) − κ 2 : (i) subcritical case: µπ (u0 , q) < κ 2 ; (ii) critical case: µπ (u0 , q) = κ 2 ; (iii) supercritical case: µπ (u0 , q) > κ 2 > µh (u0 , q). 8 However it will be interesting to find a criterion on , u and q such that µ (u , q) = µ (u , q), see π 0 0 h 0 Question 8.7.
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We shall see that, in the subcritical case K c = +∞ and the minimizers remain in a smectic state for all K (Theorem 3.1). In the supercritical case K c < +∞, and phase transitions and hysteresis occur (Theorem 2). The critical case is particularly interesting, where the finiteness of K c and the nature of phase transitions of the liquid crystals depend heavily on the non-degeneracy of the minimizing harmonic maps (the nematic states) n ∈ ∗ (u0 , q) and on the eigenfunctions associated with µ(qn) (Theorem 3). Definition 1.6. Let n ∈ (u0 ) and set V(n) = {v ∈ H01 (, R3 ) : v ≡ 0, n · v = 0 a.e. in , |∇n||v| ∈ L 2 ()}. (i) n ∈ (u0 ) is said to be non-degenerate if for any v ∈ V(n) it holds that (|∇v|2 − |∇n|2 |v|2 )d x > 0.
(ii) n ∈ (u0 ) is said to satisfy condition (A) if n ∈ W 1,3 (, S2 ) and it is non-degenerate. (iii) n ∈ (u0 ) is said to satisfy condition (B) if there exists c(n) > 0 such that (|∇v|2 − |∇n|2 |v|2 )d x ≥ c(n) v 2L 3 () , for all v ∈ H01 (, R3 ). (1.10)
Conditions (A) and (B) are not comparable, however we can show that, if n ∈ (u0 ) satisfies (A), then (1.10) holds for all v ∈ V(), see Lemma 5.1. To see the relation between non-degeneracy and minimality, we first note that if n ∈ (u0 ), then (|∇v|2 − |∇n|2 |v|2 )d x ≥ 0, for all v ∈ V(n0 ) ∩ L ∞ (, R3 ). (1.11)
The proof of (1.11) is given in Appendix D for the reader’s convenience. Also note that if n ∈ (u0 ) is degenerate, then V0 (n) = ∅, where V0 (n) = v ∈ V(n) : (|∇v|2 − |∇n|2 |v|2 )d x = 0 . (1.12)
Definition 1.7. Let n ∈ (u0 ). We define ¯ v · (φ∇φ)d x = 0 VS(n, q) = (v, φ) ∈ V(n) × S p(qn) : either or (|∇v|2 − |∇n|2 |v|2 )d x = 0 ,
2 ¯ qn φ)d x v · (φ∇ m[n, v, φ] = . 4
φ L 4 () (|∇v|2 − |∇n|2 |v|2 )d x
Let m(n, q) =
sup
(v,φ)∈V S(n,q)
m(n, q) = 0
m[n, v, φ] if V S(n, q) = ∅, if V S(n, q) = ∅.
(1.13)
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Finally we define m v (u0 , q) =
sup
n∈ ∗ (u0 ,q)
m(n, q).
(1.14)
Note that if there exists a degenerate map n ∈ ∗ (u0 , q) (hence V0 (n) = ∅), and if there exist v ∈ V0 (n) and φ ∈ S p(qn) such that ¯ v · (φ∇φ)d x = 0, (1.15)
then m v (u0 , q) = +∞. It is also easy to see that, if every n ∈ ∗ (u0 , q) is nondegenerate, then m v (u0 , q) =
sup
sup
n∈ ∗ (u0 ,q) v∈V (n), φ∈S p + (qn)
m[n, v, φ].
(1.16)
¯ ¯ qn φ). In (1.13), since n · v = 0, we have v · (φ∇φ) = v · (φ∇ Now we can state our main results. We begin with the finiteness of K c . Theorem 1. We have ⎧ =0 ⎪ ⎪ ⎨ ∈ (0, +∞) Kc ∈ [2q 2 κ −2 m v (u0 , q), +∞] ⎪ ⎪ ⎩ = +∞
if if if if
µh (u0 , q) ≥ κ 2 , µh (u0 , q) < κ 2 < µπ (u0 , q), µπ (u0 , q) = κ 2 , µπ (u0 , q) < κ 2 .
Next we examine the behavior of the minimizers of E K for K near K c . We need the following definition. Definition 1.8. We say that the nontrivial global minimizers of E K bifurcate from the trivial critical points at K c if the following is true: For any sequence {K j } increasing to K c , there exist a subsequence {K jl } and a nontrivial global minimizer (ψ K jl , nk jl ) of E K jl such that (ψ K jl , n K jl ) converges strongly in H 1 (, C) × H 1 (, R3 ) to a trivial critical point of E K c . Assume that K c < +∞ and that there exists K˜ c > K c such that E K has a nontrivial local minimizer for every K c < K < K˜ c . We say that the nontrivial local minimizers of E K bifurcate from the nontrivial global minimizers at K c if the following is true: For any sequence {K j } decreases to K c , there exist a subsequence {K jl } and a nontrivial local minimizer (ψ K jl , n K jl ) of E K jl such that (ψ K jl , n K jl ) converges strongly in H 1 (, C) × H 1 (, R3 ) to a nontrivial global minimizer of E K c . The subcritical case is simple, see Theorem 3.1. For the supercritical case we have: Theorem 2. In the case µh (u0 , q) < κ 2 < µπ (u0 , q), we have the following conclusions. (i) E K c has nontrivial minimizers. (ii) There exists K˜ c > K c such that, for every K ∈ (K c , K˜ c ), E K has nontrivial local minimizers (which are not global minimizers). The nontrivial local minimizers bifurcate from the nontrivial global minimizers at K c . (iii) The nontrivial global minimizers do not bifurcate from the trivial critical points at K c .
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Remark 1.9. Note that in the supercritical case a trivial critical point (0, n) is a global minimizer if K ≥ K c , and it is a local minimizer if 0 < K < K c (Lemma B.1). Let us define another critical value of the elastic coefficients by K˜ c = inf{K > 0 : E K does not have nontrivial local minimizers}.
(1.17)
Obviously K c ≤ K˜ c ≤ +∞.9 From Theorem 2 we get the following corollaries: (i) In the supercritical case we have K c < K˜ c . (ii) Let K increase from below K c . When K < K c , E K has nontrivial global minimizers, and hence the liquid crystal is in a smectic state. As K increases and passes by K c , the only global minimizers are the trivial critical points (0, n), n ∈ (u0 ), however the liquid crystal does not jump down to a nematic state; instead, it remains in the smectic state, which is a local minimizer, until K reaches some critical value, say, K 0 . If K 0 < +∞, then the liquid crystal stays in a nematic state for all K > K 0 . (iii) If a liquid crystal is in a nematic state for large K , then as K decreases it remains in the nematic state. Now we consider the critical case. From Theorem 1 there are three possibilities: (1) Subcase 1. K c = +∞. (2) Subcase 2. K c < +∞ and E K c has only trivial minimizers. (3) Subcase 3. K c < +∞ and E K c has nontrivial minimizers. In Subcase 1 the minimizers bifurcate from a trivial critical point at K = +∞ (Lemma 7.1). In Subcase 2 we have K c = 2q 2 κ −2 m v (u0 , q), and the minimizers bifurcate from a trivial critical point (0, n0 ) with n0 ∈ ∗ (u0 , q) as K decreases from K c (Lemma 7.2). In Subcase 3 there occurs hysteresis at K c (Lemma 7.3). Obviously we have Subcase 1 if m v (u0 , q) = +∞. Theorem 3. Consider the case µπ (u0 , q) = κ 2 . Assume that 0 < m v (u0 , q) < +∞, and every member in ∗ (u0 , q) satisfies condition (A). Then we have: (i) K c < +∞. (ii) If E K c has only trivial minimizers, then K c = 2q 2 κ −2 m v (u0 , q) < +∞,
(1.18)
and at K c the nontrivial global minimizers bifurcate from the trivial critical points. (iii) If m v (u0 , q) is achieved by (n0 , v, ξ ), where n0 ∈ ∗ (u0 , q) and (v, ξ ) ∈ V(n0 )× ∈ S p(qn0 ), and if v · (ξ¯ ∇ξ )d x ∇v · ∇(|v|2 n0 )d x
>
|v| n0 · (ξ¯ ∇ξ )d x
(1.19)
2
(|∇v| − |∇n0 | |v| )d x, 2
2
2
9 It is interesting to know when we will have K˜ c = +∞, and to investigate the behavior of the local minimizers of E K as K → K˜ c . We would like to mention that K˜ c is an analogy of the superheating field Hsh
for superconductors. For the recent research on mathematical theory of the superheating field see for instance [BH,BP,Ch,DP,PK].
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then 2q 2 κ −2 m v (u0 , q) < K c < +∞,
(1.20)
and there exists K˜ c > K c such that for every K ∈ (K c , K˜ c ), E K has nontrivial local minimizers (which are not global minimizers), and they bifurcate from the nontrivial global minimizers at K c . Remark 1.10. If instead of condition (A) we assume that every member in ∗ (u0 , q) satisfies condition (B), then we also have K c < +∞, see Lemma 6.2. 2 Remark 1.11. In condition (A) we require n ∈ W 1,3 (,
S ). This assumption implies the smallness of the normalized Dirichlet energy r −1 Br (x0 )∩ |∇n|2 d x, and then the Hölder continuity of n follows from the small energy regularity, see [SU1,SU2] for weak harmonic maps, and [HKL1] for the minimizers of Oseen-Frank energy. In the case of equal elastic coefficients, the continuity of W 1,3 -minimizers follows also from the classification of singularities of weak harmonic maps. In fact, every minimizing weak harmonic map from a bounded domain in R3 into S2 has only a finite number of singular (x−x0 ) points (see [SU1,SU2]), and it behaves near its singular point x0 as ± R|x−x , where 0| R is a rotation (see [BCL, Corollaries 7.2, 7.12]). Hence a weak minimizing harmonic map belonging to W 1,3 (, S2 ) is in fact continuous in .10 We use the condition n ∈ W 1,3 (, S2 ) instead of the continuity condition in order to generalize the conclusions of Theorem 3 to cases without small energy regularity.
Example 1.12. If = B1 (0), the unit ball in R3 , and if u0 = Rx, where R is a rotation x in S O(3), then n = R |x| is the unique minimizing weak harmonic map, which has a x singularity at x = 0. R |x| satisfies condition (B) (see Example 8.1). On B1 (0) we have x x ) = 0 and µ(±q R µ(±q |x| |x| ) > 0 if R = I , where I is the identity matrix. Thus ⎧ if u0 = ±x, ⎪ ⎨ = +∞ x 2 c = +∞ if u0 = ±Rx, R = I, µ(q R K (1.21) |x| ) < κ , ⎪ x 2. ⎩ < +∞ if u0 = ±Rx, R = I, µ(q R ) ≥ κ |x|
Remark 1.13. Bauman, Calderer, Liu and Phillips [BCLP] investigated the phase transition between a chiral nematic phase and a smectic A* phase using the functional (1.1) with the term K 2 |n · curl n| replaced by K 2 |n · curl n + τ |, where τ > 0 is the chirality parameter, and the boundary data of the director fields are not prescribed. In the case where K 2 = K 3 , it was proved in [BCLP] that there exist positive constants β1 , β2 , K ∗ , 2 K ∗ and λ such that, if min{qτ, (qτ )2 } ≤ κβ1 , q ≥ τ and K 2 ≥ K ∗ , then the minimizers
are non-trivial; and if min{qτ, (qτ )2 } ≥ κβ2 , q > λτ and K 2 > K ∗ , then the minimizers are trivial. These results were improved in [P1, Theorem 3.6], where the critical wave number Q c3 was introduced to identify the value of the wave number q at which the liquid crystal changes its phase from a chiral nematic to a smectic A* phase. In [P1, Sect. 5] we also discussed the asymptotic behavior of the minimizers as the elastic coefficients tend to infinity, where the director fields are subjected to the Dirichlet boundary condition, and the results in [P1] motivated us to introduce the critical elastic coefficients 2
10 However when K ’s are not all equal, singular points of a minimizer of the Oseen-Frank energy may not j be isolated ([HKL1,HKL2,LinL]), and there is no classification theory of singularities available yet.
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in the present paper. We would like to mention that in a very recent work [JP], Joo and Phillips investigated the Chen-Rubensky energy which describes the phase transitions from a chiral nematic phase to a smectic C* or a smectic A* phase. 1.4. Physical explanation. Theorems 1, 2 and 3 indicate that the diagram of phase transitions of liquid crystals depends on the material and temperature (through the parameters κ, q, and K j ’s), on the geometry and topology of the container (the domain ), and on its interaction with the surface of the container (the boundary condition and boundary data). (i) If µh (u0 , q) ≥ κ 2 then K c = 0. The liquid crystal remains in a nematic state for all values of K , and no phase transition to a smectic state will happen as K varies. (ii) If µh (u0 , q) < κ 2 < µπ (u0 , q), then 0 < K c < K˜ c ≤ +∞. Hysteresis occurs as K varies between K c and K˜ c . (iii) If µπ (u0 , q) = κ 2 , there are three possibilities depending on the physical conditions. (iii-1) K c = +∞, and the phase transition from a smectic state to a nematic state happens as K increases to +∞. This is exactly the phenomenon predicted by de Gennes. (iii-2) 2q 2 κ −2 m v (u0 , q) ≤ K c < K˜ c ≤ +∞, and both hysteresis and phase transitions happen. The global minimizers do not bifurcate from a trivial critical point at K c . (iii-3) K c = 2q 2 κ −2 m v (u0 , q) < +∞, and the phase transition happens at K c where the minimizers bifurcate from the trivial critical points. (iv) If µπ (u0 , q) < κ 2 , then K c = +∞. The liquid crystal remains in a smectic state for all K , and it approaches a smectic state as K → +∞. There does not exist the phase transition to a nematic state. 1.5. Notations. c p (n): C(K , R): C(u0 , K ): Cπ (u0 ): E[ψ, n]: E K [ψ, n]: F[n]: gπ (q): G[ψ, qn]: H 1 (, C): H 1 (, R3 ): H 1 (, S2 , u0 ): K c: K˜ c : m[n, v, φ]: m(n, q): m v (u0 , q): M(E K ): M(n):
(5.2) (4.15) minimum value of E K , Notations 1.1 minimum value of Dirichlet energy, Notations 1.2 Landau-de Gennes functional, (1.1) Landau-de Gennes functional of equal elastic coefficients, (1.3) Dirichlet energy, § 1.2 (2.7) reduced Ginzburg-Landau energy, proof of Lemma 2.1 Sobolev space of complex-valued functions Sobolev space of vector fields §1.2 critical elastic coefficient, Definition 1.3 Remark 1.9 Definition 1.7 Definition 1.7 Definition 1.7 set of minimizers of E K , Notations 1.1 Definition 7.6
Liquid Crystals
M(n): M+ (n): M[n, w, ξ ]: M(u, q): N (K ): S p(qn): S p + (qn): T (n): u⊥ : Ua (u0 ): U R (u0 ): V(n): V0 (n): V S(n, q): µ(qn): µh (u0 , q): µπ (u0 , q): (u0 ): ∗ (u0 , q):
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Definition 7.4 Definition 7.6 Definition 7.6 Definition 7.6 (4.13) set of eigenfunctions associated with µ(qn), Notations 1.4 Notations 1.4 Definition 7.4 (7.15) (4.11) (4.14) Definition 1.6 (1.12) Definition 1.7 2 , (1.8) lowest eigenvalue of −∇qn Notations 1.4 Notations 1.4 set of minimizing harmonic maps, Notations 1.2 Notations 1.4
2. General Properties of the Critical Elastic Coefficient Lemma 2.1. (i) If the strict inequality holds C(u0 , K ) < K Cπ (u0 ),
(2.1)
then E K has a nontrivial minimizer. (ii) If K 0 > 0 and E K 0 has a nontrivial minimizer, then for all K ∈ (0, K 0 ), E K has nontrivial minimizers and (2.1) holds. (iii) We have the following characterization of K c : K c = inf{K > 0 : C(u0 , K ) ≥ K Cπ (u0 )}.
(2.2)
If there exists K > 0 be such that C(u0 , K ) = K Cπ (u0 ), then K c = min{K > 0 : C(u0 , K ) = K Cπ (u0 )}.
(2.3)
(iv) The minimizers of E K are nontrivial when 0 < K < K c (if K c > 0), and they are trivial when K > K c (if K c < +∞). Proof. (i) is obvious because a trivial minimizer has energy K Cπ (u0 ). To prove (ii), let (ψ, n) be a nontrivial minimizer of E K 0 . Then ψ ≡ 0 and E K 0 [ψ, n] ≤ K 0 Cπ (u0 ).
(2.4)
First assume that the strict inequality in (2.4) holds. Then for any K ∈ (0, K 0 ), since
2 d x ≥ C (u ), we have |∇n| π 0 E K [ψ, n] =E K 0 [ψ, n] − (K 0 − K ) |∇n|2 d x
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Hence from (i), E K has a nontrivial minimizer. Next assume that the equality in (2.4) holds. Since
ψ ≡ 0, using the first equation in (1.5) and the boundary condition of ψ, we have {|∇qn ψ|2 − κ 2 |ψ|2 + k 2 |ψ|4 }d x = 0. Hence k2 κ2 G[ψ, qn] ≡ {|∇qn ψ|2 − κ 2 |ψ|2 + |ψ|4 }d x = − |ψ|4 d x < 0, 2 2 1 1 |∇n|2 d x = {E K 0 [ψ, n] − G[ψ, qn]} > E K 0 [ψ, n] = Cπ (u0 ). K K 0 0
For K ∈ (0, K 0 ),
E K [ψ, n] =K 0 Cπ (u0 ) − (K 0 − K )
|∇n|2 d x
K 0 , E K does not have nontrivial minimizers. In fact, if E K has a nontrivial minimizer, then C(u0 , K ) = K Cπ (u0 ). From (ii), for any K ∈ (0, K ) we have inequality (2.1), and hence E K has nontrivial minimizers. In particular C(u0 , K 0 ) < K 0 Cπ (u0 ), which contradicts the definition of K 0 . So K c ≤ K 0 . Hence (iii) is true. (iv) is a direct consequence of (iii). Lemma 2.2. K c > 0 if and only if µh (u0 , q) < κ 2 .
(2.5)
Proof. Assume (2.5) holds. Then there exists n ∈ H 1 (, S2 , u0 ) such that µ(qn) < κ 2 .
(2.6)
2 associated with µ(qn), and let ψ = tφ. Let φ ∈ H 1 (, C) be an eigenfunction of −∇qn t Using (2.6), we can choose t > 0 small such that G[ψt , qn] < 0. If n ∈ (u0 ), for any K > 0 we have E K [ψt , n] = G[ψt , qn] + K Cπ (u0 ) < K Cπ (u0 ), and hence (2.1) holds. If n ∈ (u0 ), then ∇n 2L 2 () > Cπ (u0 ), and for any K satisfying
0
−G[ψt , qn] ,
∇n 2L 2 () − Cπ (u0 )
we have E K [ψt , n] < K Cπ (u0 ), and again (2.1) holds. So in any case we have K c > 0. c Next assume K c > 0. Then for K ∈ (0, K has a nontrivial minimizer (ψ, n),
K ), E and from Lemma 2.1 (i), G[ψ, qn] + K |∇n|2 d x = E K [ψ, n] < K Cπ (u0 ). Thus
G[ψ, qn] < −K |∇n|2 d x − Cπ (u0 ) ≤ 0. Since ψ ≡ 0 we have
κ2 (|∇qn ψ| − κ |ψ| )d x = G[ψ, qn] − 2 2
2
2
So (2.6) holds and hence (2.5) is true.
|ψ|4 d x < 0.
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Let us define gπ (q) =
inf
(ψ,n)∈H 1 (,C)× (u0 )
G[ψ, qn].
(2.7)
It is easy to show that gπ (q) is achieved. Lemma 2.3. (i) C(u0 , K ) ≤ gπ (q) + K Cπ (u0 ). (ii) If gπ (q) < 0, then E K has nontrivial minimizers for any K > 0. (iii) gπ (q) < 0 if and only if µπ (u0 , q) < κ 2 . Proof. Let (ψ 0 , n0 ) ∈ H 1 (, C) × (u0 ) be a minimizer of gπ (q). Then C(u0 , K ) ≤ E K [ψ 0 , n0 ] = gπ (q) + K Cπ (u0 ). Hence (i) is true. If gπ (q) < 0, then (2.1) holds, hence from Lemma 2.1, E K has nontrivial minimizers for any K > 0. So (ii) is true. To prove (iii), assume first gπ (q) < 0. Then there exists (ψ 0 , n0 ) ∈ H 1 (, C) × (u0 ) such that G[ψ 0 , qn0 ] = gπ (q) < 0. So ψ 0 ≡ 0 and κ2 (|∇qn0 ψ 0 |2 − κ 2 |ψ 0 |)d x < − |ψ 0 |4 d x < 0, 2 which implies µ(qn0 ) < κ 2 , so µπ (u0 , q) < κ 2 . On the other hand, if µπ (u0 , q) < κ 2 , let n0 ∈ (u0 ) be such that µ(qn0 ) = µπ (u0 , q) < κ 2 , and let ψ 0 be an eigenfunction 2 associated with the lowest eigenvalue µ(qn0 ). As in the proof of Lemma 2.2 of −∇qn 0 we can show that G[tψ 0 , qn0 ] < 0 for small t > 0. So gπ (q) < 0. (iii) is proved.
Lemma 2.4. (i) For any 0 < K < +∞ and for any (ψ, n) ∈ M(E K ),
∇n 2L 2 () + ∇ψ 2L 2 () < Cπ (u0 ) +
κ2 1 + 2K gπ (q) + + κ 2 + 2q 2 ||. (2.8) K 2K
(ii) C(u0 , K ) is continuous in K ∈ (0, +∞).
2 Proof. (i) Note that G[ψ, qn] ≥ |∇qn ψ|2 d x − κ2 || for all ψ ∈ H 1 (, C) and n ∈ H 1 (, R3 ). Let (ψ K , n K ) be a minimizer of E K . From Lemma 2.3 (i), G[ψ K , qn K ] + K |∇n K |2 d x = E K [ψ K , n K ] = C(u0 , K ) ≤ gπ (q) + K Cπ (u0 ),
(2.9)
|∇n K |2 d x ≤ Cπ (u0 ) +
Now we write (2.9) as follows: G[ψ K , qn K ] ≤ gπ (q) − K So
1 κ2 gπ (q) + || . K 2
|∇n K |2 d x − Cπ (u0 ) ≤ gπ (q).
|∇qn K ψ K |2 d x ≤ gπ (q) +
κ2 ||. 2
(2.10)
(2.11)
(2.12)
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Applying the maximum principle to the first equation in (1.5) we have |ψ K (x)| ≤ 1. So 2
∇ψ K 2L 2 () ≤ ∇qn K ψ K L 2 () + q n K ψ K L 2 () ≤ 2 ∇qn K ψ K 2L 2 () + 2q 2 ψ K 2L 2 () ≤ 2gπ (q) + κ 2 || + 2q 2 ψ K 2L 2 () ≤ 2gπ (q) + (κ 2 + 2q 2 )||.
From this and (2.10) we get (2.8). To prove (ii), assume that 0 < K < +∞ and K j → K . Let (ψ K , n K ) be a minimizer of E K . Then C(u0 , K j ) ≤ E K j [ψ K , n K ] → E K [ψ K , n K ] = C(u0 , K ) and hence lim sup C(u0 , K j ) ≤ C(u0 , K ). j→+∞
On the other hand, let (ψ j , n j ) be a minimizer of E K j . From (2.8), {(ψ j , n j )} is bounded in H 1 (, C)×H 1 (, S2 , u0 ). Passing to a subsequence we may assume that (ψ j , n j ) → (ψ0 , n0 ) weakly in H 1 (, C) × H 1 (, R3 ), strongly in L 4 (, C) × L 4 (, R3 ), and a.e. in . Hence n0 ∈ H 1 (, S2 , u0 ). Moreover, C(u0 , K ) ≤ E K [ψ0 , n0 ] ≤ lim inf E K j [ψ j , n j ] = lim inf C(u0 , K j ). j→+∞
So lim j→+∞ C(u0 , K j ) = C(u0 , K ).
j→+∞
The following lemma is used frequently in this paper. Lemma 2.5. Let K j → +∞ and let (ψ j , n j ) be a minimizer of E K j . Then there exists a subsequence jl → +∞ such that (ψ jl , n jl ) → (ψ0 , n0 ) strongly in H 1 (, C) × H 1 (, S2 , u0 ),
(2.13)
where n0 ∈ (u0 ) and G[ψ0 , qn0 ] = gπ (q).
(2.14)
Proof. The major part of Lemma 2.5 follows from Lemma 5.1 in [P1]. Since Lemma 5.1 in [P1] was given without proof, for the reader’s convenience we include a brief proof here. From (2.10) 1 κ2 (2.15) gπ (q) + || . |∇n j |2 d x ≤ Cπ (u0 ) + Kj 2 So {n j } is bounded in H 1 (, S2 , u0 ), and we can find a subsequence, still denoted by {n j }, and n0 ∈ H 1 (, R3 ), such that n j → n0 weakly in H 1 (, R3 ) and pointwise a.e. in . Hence |n0 (x)| = 1 a.e., so n0 ∈ H 1 (, S2 , u0 ). Moreover, 2 Cπ (u0 ) ≤ |∇n0 | d x ≤ lim inf |∇n j |2 d x ≤ Cπ (u0 ).
j→+∞
Hence n0 ∈ (u0 ), and |∇n j |2 d x → |∇n0 |2 d x, which in turn implies that
∇n j − ∇n0 L 2 () → 0. Hence n j → n0 strongly in H 1 (, S2 , u0 ).
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Using (2.8) and the fact |ψ j (x)| ≤ 1, {ψ j } is bounded in H 1 (, C). So we can pass to a subsequence again and assume that ψ j → ψ0 weakly in H 1 (, C) and strongly in L 4 (, C). From this, the fact n0 ∈ (u0 ), and (2.11), we have gπ (q) ≤ G[ψ0 , qn0 ] ≤ lim G[ψ j , qn j ] ≤ gπ (q). j→+∞
So G[ψ0 , qn0 ] = lim G[ψ j , qn j ] = gπ (q), j→+∞
(2.16)
which proves (2.14). Since ψ j → ψ0 strongly in L 4 (, C), and using (2.16), we see that
|∇qn0 ψ0 | d x = lim 2
j→+∞
|∇qn j ψ j |2 d x.
(2.17)
Since ∇qn j ψ j → ∇qn0 ψ0 weakly in L 2 (, C3 ), from (2.17) we find ∇qn j ψ j → ∇qn0 ψ0 strongly in L 2 (, C3 ). Since n j ψ j → n0 ψ0 strongly in L 2 (, C3 ), we have ∇ψ j → ∇ψ0 strongly in L 2 (, C3 ). So (2.13) is true.
3. Subcritical Case In this section we examine the subcritical case, namely we assume µπ (u0 , q) < κ 2 .
(3.1)
Theorem 3.1. Assume (3.1) holds. We have: (i) K c = +∞. (ii) Let K j → +∞ and let (ψ j , n j ) be a minimizer of E K j . Then there exists a subsequence jl → +∞ such that (2.13) holds with n0 ∈ (u0 ), ψ0 = 0 and G[ψ0 , qn0 ] = gπ (q) < 0.
(3.2)
Proof. We first prove (i). Under condition (3.1), we can choose n ∈ ∗ (u0 , q) such 2 associated with µ(qn). Using (2.6) that (2.6) holds. Let φ be an eigenfunction of −∇qn we find that for small t > 0 we have G[tφ, qn] < 0. So for all K > 0, C(u0 , K ) ≤ E K [tφ, n] < K Cπ (u0 ). Using Lemma 2.1 (i) we have K c > K . This is true for any K > 0, so K c = +∞. To prove (ii), let K j → +∞ and let (ψ j , n j ) be a minimizer of E K j . From Lemma 2.5, there exists a subsequence {(ψ jl , n jl )} and (ψ0 , n0 ) with n0 ∈ (u0 ) such that (2.13) and (2.14) hold. From (3.1) and Lemma 2.3 (iii), gπ (q) < 0. Since G[ψ0 , qn0 ] = gπ (q) < 0 and G[0, qn0 ] = 0, we must have ψ0 = 0. So (ii) is true.
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4. Supercritical Case In this section we consider the supercritical case µh (u0 , q) < κ 2 < µπ (u0 , q).
(4.1)
Theorem 4.1. Under condition (4.1) we have the following conclusions. (i) 0 < K c < +∞. (ii) E K c has a nontrivial minimizer. (iii) There exists K˜ c > K c such that for every K ∈ (K c , K˜ c ) the functional E K has a nontrivial local minimizer (ψ K , n K ) which is not a global minimizer. (iv) The nontrivial local minimizers bifurcate from the nontrivial (global) minimizers at K c . We prove Theorem 4.1 in several lemmas. Lemma 4.2. Under condition (4.1) we have 0 < K c < +∞. Proof. From Lemma 2.2, if (4.1) holds then 0 < K c ≤ +∞. Now we show K c < +∞. Suppose K c = +∞. Then there exists a sequence K j → +∞ such that E K j has a nontrivial minimizer (ψ j , n j ). So E K j [ψ j , n j ] = G[ψ j , qn j ] + K j |∇n j |2 d x < K j Cπ (u0 ). (4.2)
Since |∇n j |2 d x ≥ Cπ (u0 ), we have G[ψ j , qn j ] < 0, so µ(qn j ) < κ 2 . From Lemma 2.5, there exists a subsequence, still denoted by {(ψ j , n j )}, such that n j → n0 strongly in H 1 (, R3 ), and n0 ∈ (u0 ). Then we use Lemma A.1 to get µ(qn j ) → µ(qn0 ), so µ(qn0 ) ≤ κ 2 , and hence µπ (u0 , q) ≤ κ 2 , which contradicts the assumption. Hence we must have K c < +∞. Lemma 4.3. Assume (4.1) holds. Then E K c has nontrivial minimizers. Proof. Suppose the conclusion were false. We shall find a contradiction. Let K j increase to K c , and assume E K j has a nontrivial minimizer (ψ j , n j ). Then (4.2) is still true. Since C(u0 , K j ) ≤ K j Cπ (u0 ) ≤ K c Cπ (u0 ) < +∞, we can show that {(ψ j , n j )} is bounded in H 1 (, C) × H 1 (, S2 , u0 ). After passing to a subsequence we may assume that (ψ j , n j ) → (ψ0 , n0 ) weakly in H 1 (, C) × H 1 (, R3 ) and strongly in L 4 (, C)×L p (, R3 ) for all 1 < p < +∞, and (ψ0 , n0 ) ∈ H 1 (, C)× H 1 (, S2 , u0 ) is a minimizer of E K c . By the assumption E K c has only trivial minimizers, ψ0 = 0. In the following we shall derive a contradiction. Step 1. We first show that (ψ j , n j ) → (0, n0 ) strongly in H 1 (, C) × H 1 (, S2 , u0 ), where n0 ∈ (u0 ). Note that 0 = G[0, qn0 ] ≤ lim inf G[ψ j , qn j ], j→+∞
|∇n0 |2 d x ≤ lim inf j→+∞
|∇n j |2 d x.
(4.3)
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Using the continuity of C(u0 , K ) (see Lemma 2.4 (ii)) we have lim K →K c C(u0 , K ) = C(u0 , K c ) = K c Cπ (u0 ). So
lim sup K j |∇n j |2 d x = lim sup C(u0 , K j ) − G[ψ j , qn j ]
j→+∞ c ≤K Cπ (u0 ) − G[0, qn0 ]
=
j→+∞ c K Cπ (u0 ).
So |∇n0 |2 d x = lim j→+∞ |∇n j |2 d x = Cπ (u0 ) and n0 ∈ (u0 ). Then it follows that ∇n j − ∇n0 L 2 () → 0, and hence n j → n0 strongly in H 1 (, R3 ). Moreover, G[ψ j , qn j ] = E K j [ψ j , n j ] − K j |∇n j |2 d x → C(u0 , K c ) − K c Cπ (u0 ) = 0.
From this and since ψ j → 0 strongly in L 4 (, C), we have |∇qn j ψ j |2 d x → 0. Thus ∇ψ j L 2 () ≤ ∇qn j ψ j L 2 () + q ψ j L 2 () → 0. Hence (4.3) is true. Step 2. Write ψj = sjφj,
n j = n0 + t j v j ,
(4.4)
where s j > 0, t j > 0, and φ j ∈ H 1 (, C), φ j H 1 () = 1, v j ∈ H01 (, R3 ), v j L 4 () = 1.
(4.5)
To verify that in (4.4) the number t j can be chosen to be positive, we need to show that n j = n0 . Since (ψ j , n j ) is a nontrivial minimizer, so ψ j ≡ 0, and hence µ(qn j ) < κ 2 < µπ (u0 , q), which implies n j ∈ (u0 ), and hence n j = n0 . From (4.3) we have s j → 0 and t j → 0. Since {φ j } is bounded in H 1 (, C), we can pass to a subsequence again and assume that φ j → φ0 weakly in H 1 (, C) and strongly in L 4 (, C). By computations |∇n j |2 = |∇n0 |2 + 2t j ∇n0 · ∇v j + t 2j |∇v j |2 , |∇qn j ψ j |2 = s 2j {|∇qn0 φ j |2 − 2qt j v j · (φ¯ j ∇qn0 φ j ) + q 2 t 2j |v j φ j |2 }. t
The unit length condition |n j |2 = 1 implies n0 · v j = − 2j |v j |2 . Since n0 is a weak solution of (1.6) and v j = 0 on ∂, we have tj 2 ∇n0 · ∇v j d x = |∇n0 | n0 · v j d x = − |∇n0 |2 |v j |2 d x. 2 Since n0 ∈ (u0 ) and n j = n0 = u0 on ∂, we have 2 2 0≤ |∇n j | d x − |∇n0 | d x = {2t j ∇n0 · ∇v j + t 2j |∇v j |2 }d x 2 2 2 2 = tj {|∇v j | − |∇n0 | |v j | }d x.
So
{|∇v j |2 − |∇n0 |2 |v j |2 }d x ≥ 0.
(4.6)
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Now we have the expansion of the energy E K j [ψ j , n j ] = K j Cπ (u0 ) + K j t 2j {|∇v j |2 − |∇n0 |2 |v j |2 }d x
κ 2 s 2j 2 + sj |∇qn0 φ j |2 − κ 2 |φ j |2 − 2qt j v j · (φ¯ j ∇qn0 φ j ) + q 2 t 2j |v j φ j |2 + |φ j |4 . 2 Recall that E K j [ψ j , n j ] < K j Cπ (u0 ). From the energy expansion we have (|∇v j |2 − |∇n0 |2 |v j |2 )d x K j t 2j
κ 2 s 2j
|φ j |4 }
(4.7)
From this and (4.6) we get (|∇qn0 φ j |2 − κ 2 |φ j |2 )d x < 2qt j v j · (φ¯ j ∇qn0 φ j )d x.
(4.8)
{|∇qn0 φ j | − κ |φ j | ≤ 2qt j s 2j v j · (φ¯ j ∇qn0 φ j )d x.
+ s 2j
2
2
2
+ q 2 t 2j |v j φ j |2
+
2
Since φ j H 1 () = 1, we have ∇qn0 φ j L 2 () ≤ 1 + q. Recall that t j v j = n j − n0 → 0 strongly in H 1 (, R3 ). So t j ≤ t j v j φ j L 2 () ∇qn φ j L 2 () ¯ v · ( φ ∇ φ )d x j j qn0 j 0
≤ C t j v j H 1 () φ j H 1 () ∇qn0 φ j L 2 () ≤ C(1 + q) t j v j H 1 () → 0. From this and (4.8) we have
{|∇qn0 φ j |2 − κ 2 |φ j |2 }d x → 0.
(4.9)
We claim that φ0 = 0. Otherwise we have φ j → 0 strongly in L 2 (, C). From (4.9) we find ∇qn0 φ j L 2 () → 0, then ∇φ j L 2 () ≤ ∇qn0 φ j L 2 () + qn j φ j L 2 () → 0. So φ j H 1 () → 0, which contradicts (4.5). Since φ j L 2 () → φ0 L 2 () = 0, from (4.9) we get |∇qn0 φ0 |2 d x ≤ κ 2 |φ0 |2 d x. (4.10)
Hence µ(qn0 ) ≤ κ 2 . This contradicts the assumption µπ (u0 , q) > κ 2 . Now the conclusion of Lemma 4.3 is proved. Lemma 4.4. Under condition (4.1), there exists a constant K˜ c > K c such that for every K ∈ (K c , K˜ c ) the functional E K has a nontrivial local minimizer (ψ K , n K ) which is not a global minimizer. Proof. In the proof of Lemma 4.3 we have actually showed that, as K increases and approaches K c , the nontrivial minimizers stay away from the set {0} × (u0 ). Let Ua (u0 ) = {(ψ, u) ∈ H 1 (, C) × H 1 (, S2 , u0 ) : ψ L 2 () ≤ a}.
(4.11)
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Claim 1. There exists a > 0 such that M(E K c ) ∩ Ua (u0 ) = ∅. Proof. We can show that there exist ε0 > 0 and a > 0 such that M(E K ) ∩ Ua (u0 ) = ∅ for all K ∈ [K c − ε0 , K c ]. Suppose it were false. We have a sequence K j → K c and, for each integer j, a nontrivial minimizer (ψ j , n j ) of E K j such that
ψ j L 2 () → 0, E K j [ψ j , n j ] = C(u0 , K j ) → C(u0 , K c ) = K c Cπ (u0 ). (4.12) From Lemma 2.4, {(ψ j , n j )} is bounded in H 1 (, C) × H 1 (, R3 ). After passing to a subsequence we may assume that (ψ j , n j ) → (ψ0 , n0 ) weakly in H 1 (, C) × H 1 (, S2 , u0 ) and strongly in L 4 (, C) × L 2 (, R3 ). From this and (4.12) we see that ψ0 = 0. Moreover,
K c Cπ (u0 ) ≤
hence
{|∇qn j ψ j |2 + K c |∇n j |2 }d x + o(1) = E K j [ψ j , n j ] + o(1) → K c Cπ (u0 ),
|∇qn j ψ j |
→ 0 and 2 Cπ (u0 ) ≤ |∇n0 | d x ≤ lim sup |∇n j |2 d x = Cπ (u0 ). 2d x
j→+∞
Thus n0 ∈ (u0 ), n j → n0 strongly in H 1 (, R3 ), and ψ j → 0 strongly in H 1 (, C). Now we repeat Step 2 of the proof of Lemma 4.3 to show that µ(qn0 ) ≤ κ 2 , which contradicts the assumption µπ (u0 , q) > κ 2 . Claim 2. Let a be the positive number given in Claim 1. For any R ∈ (0, a) and d > 0, there exists b(R, d) > 0 such that for any K ∈ [K c , K c + d], inf E K [ψ, n] : (ψ, n) ∈ H 1 (, C) × H 1 (, S2 , u0 ), ψ L 2 () = R ≥ C(u0 , K ) + b(R, d). Proof. Suppose the claim were not true. Then there exist R ∈ (0, a), d > 0, K j ∈ [K c , K c + d], and (ψ j , n j ) ∈ H 1 (, C) × H 1 (, S2 , u0 ), such that ψ j L 2 () = R and E K j [ψ j , n j ] − C(u0 , K j ) → 0. We may pass to a subsequence and assume that K j → K 0 , (ψ j , n j ) → (ψ0 , n0 ) weakly in H 1 (, C) × H 1 (, R3 ) and strongly in L 4 (, C) × L 2 (, R3 ), and n j (x) → n0 (x) a.e. in . Hence n0 ∈ H 1 (, S2 , u0 ) and we have E K 0 [ψ0 , n0 ] ≤ lim inf j→+∞ E K j [ψ j , n j ] = C(u0 , K 0 ). Thus (ψ0 , n0 ) is a minimizer of E K 0 . Moreover ψ0 L 2 () = R, hence (ψ0 , n0 ) is a nontrivial minimizer of E K 0 . So K 0 = K c , because E K has no nontrivial minimizers for K > K c , see Lemma 2.1 (iv). However from the choice of a in Claim 1, (ψ0 , n0 ) ∈ M(E K c ), which is a contradiction. When M(E K ) = ∅, let N (K ) ≡
sup
(ψ,n)∈M(E K )
|∇n|2 d x.
(4.13)
From (1.9) and Lemma 2.4 we have 0 < N (K ) < +∞. Let a be the positive number given in Claim 1. For R ∈ (0, a], set U R (u0 ) = {(ψ, u) ∈ H 1 (, C) × H 1 (, S2 , u0 ) : ψ L 2 () ≥ R}.
(4.14)
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It is a closed subset of H 1 (, C) × H 1 (, S2 , u0 ). From Claim 1, U R (u0 ) contains all nontrivial minimizers of E K c . For K > K c , let C(K , R) =
inf
(ψ,n)∈U R (u0 )
E K [ψ, n].
(4.15)
c c Claim 3. Let 0 < R < a and δ = min{ b(R,1) N (K c ) , 1}. If K < K ≤ K + δ, then C(K , R) is achieved on U R (u0 ) and the minimizers satisfy ψ L 2 () > R.
Proof. Let {(ψ j , n j )} ⊂ U R (u0 ) be a minimizing sequence of C(K , R). From the proof of Lemma 2.4 we see that {(ψ j , n j )} is bounded in H 1 (, C) × H 1 (, S2 , u0 ). After passing to a subsequence we may assume that (ψ j , n j ) → (ψ0 , n0 ) weakly in H 1 (, C) × H 1 (, R3 ) and strongly in L 4 (, C) × L 2 (, R3 ), n0 ∈ H 1 (, S2 , u0 ), and E K [ψ0 , n0 ] ≤ lim inf j E K [ψ j , n j ] = C(K , R). In particular ψ0 L 2 () ≥ R. Hence (ψ0 , n0 ) ∈ U R (u0 ). Thus (ψ0 , n0 ) achieves C(K , R). Now we show ψ0 L 2 () > R. Let (ψ K c , n K c ) be a nontrivial (global) minimizer of E K c . Then E K c [ψ K c , n K c ] = C(u0 , K c ) = K c Cπ (u0 ). Note that (ψ K c , n K c ) ∈ U R (u0 ). From Claim 2, for K ∈ (K c , K c + δ] we have C(K , R) ≤E K [ψ K c , n K c ] = E K c [ψ K c , n K c ] + (K − K c ) |∇n K c |2 d x
≤K c Cπ (u0 ) + (K − K c )N (K c ) ≤ K c Cπ (u0 ) + δ N (K c )
(4.16)
(ψ,n)∈∂ U R (u0 )
Here we have used Claim 2. Hence (ψ0 , n0 ), the minimizer of C(K , R) obtained above, does not lie on ∂U R (u0 ), and thus it is a local minimizer of E K . Lemma 4.4 follows from Claim 3.
Lemma 4.5. Assume condition (4.1). Let (ψ K , n K ) be the nontrivial local minimizer of E K given in Lemma 4.4. Then for any sequence {K j } decreasing to K c , there exists a subsequence {K jl } such that (ψ K jl , n K jl ) converges strongly in H 1 (, C) × H 1 (, S2 , u0 ) to a nontrivial global minimizer of E K c . Proof. From Claim 3 in the proof of Lemma 4.4, for K ∈ (K c , K c + δ] we have
ψ K L 2 () > R. As before we can show that {(ψ K , n K )} is bounded in H 1 (, C) × H 1 (, S2 , u0 ). If K j decreases to K c , we can pass to a subsequence and have (ψ K j , n K j ) → (ψ0 , n0 ) weakly in H 1 (, C) × H 1 (, R3 ) and strongly in L 4 (, C) × L 4 (, R3 ). From this, the fact that M(E K c ) ⊂ U R (u0 ), and Lemma 2.4, we have C(u0 , K c ) ≤ E K c [ψ0 , n0 ] ≤ lim inf E K j [ψ K j , n K j ] j→+∞
= lim inf
inf
j→+∞ (ψ,n)∈U R (u0 )
=
inf
(ψ,n)∈M(K c )
E K j [ψ, n] ≤ lim sup
inf
c j→+∞ (ψ,n)∈M(K ) c
E K j [ψ, n]
(4.17)
E K c [ψ, n] = C(u0 , K ).
Thus (ψ0 , n0 ) is a global minimizer of E K c . Since ψ0 L 2 () ≥ R, it is a nontrivial global minimizer of E K c .
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From (4.17) we also have lim j→+∞ E K j [ψ j , n j ] = E K c [ψ0 , n0 ]. Using this and the fact that (ψ j , n j ) → (ψ0 , n0 ) strongly in L 4 (, C) × L 4 (, S2 ), we have
lim
j→+∞
{|∇ψ j |2 + K j |∇n j |2 }d x =
{|∇ψ0 |2 + K c |∇n0 |2 }d x.
(4.18)
Since ∇ψ j → ∇ψ0 weakly in L 2 (, C3 ) and ∇n j → ∇n0 weakly in L 2 (, R9 ), we have 2 2 2 |∇ψ0 | d x ≤ lim inf |∇ψ j | d x, |∇n0 | d x ≤ lim inf |∇n j |2 d x. (4.19) j→+∞
j→+∞
Combining (4.18) and (4.19) we see that the equalities in (4.19) hold. Hence ∇ψ j → ∇ψ0 strongly in L 2 (, C3 ) and ∇n j → ∇n0 strongly in L 2 (, R9 ). Thus (ψ j , n j ) → (ψ0 , n0 ) strongly in H 1 (, C) × H 1 (, S2 , u0 ). Proof of Theorem 4.1. Theorem 4.1 follows from Lemmas 4.2-4.5.
5. Non-Degenerate Minimizing Harmonic Maps In this section we present several preliminary results related to non-degenerate minimizing harmonic maps. These results will be needed in Sects. 6 and 7. Lemma 5.1. Assume that n ∈ (u0 ) is non-degenerate and n ∈ W 1,3 (, S2 ). Then for any 1 < p < 6, there exists a constant c p (n) > 0 depending only on , u0 , n and p such that (|∇v|2 − |∇n|2 |v|2 )d x ≥ c p (n) v 2L p () , for all v ∈ V(n). (5.1)
¯ except at a finite number of Proof. Let n ∈ (u0 ). From [SU1,SU2], n is C 2+α on m points in , say, x1 , . . . , xm . Let U (r ) = ∪ j=1 Br (x j ) ∩ . For any ε > 0 we can find
r > 0 such that U (r ) |∇n|3 d x < ε. We prove (5.1) for 2 ≤ p < 6. Define
c p (n) = inf
(|∇v|
v∈V (n)
2
− |∇n|2 |v|2 )d x
v 2L p ()
.
(5.2)
We shall show that c p (n) is achieved. Let {v j } ⊂ V(n) be a minimizing sequence for c p (n) and assume v j L p () = 1. Then 2/3
1/3
|∇n| |v j | d x ≤ |∇n| d x |v j | d x U (r ) U (r ) 2/3 ≤ C() |∇n|3 d x |∇v j |2 d x ≤ C()ε2/3 |∇v j |2 d x. 2
2
3
6
U (r )
U (r )
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Choose ε > 0 small such that C()ε2/3 < 21 . Then 2 (c p (n) + o(1)) v j L p () = (|∇v j |2 − |∇n|2 |v j |2 )d x 1 2 ≥ |∇v j | d x − |∇n|2 |v j |2 d x 2 \U (r ) 2/ p 1 2 ≥ |∇v j | d x − sup |∇n|2 |v j | p d x ||( p−2)/ p . 2 ¯ x∈\U (r ) Hence ∇v j L 2 () is bounded. Passing to a subsequence we may assume that v j → v0 weakly in H01 (, R3 ) and strongly in L p (, R3 ). Then 2 |∇v0 | d x ≤ lim |∇v j |2 d x,
j→+∞
v0 L p () = lim v j L p () = 1, j→+∞ |∇n|2 |v0 |2 d x = lim |∇n|2 |v j |2 d x, j→+∞ \U (r ) \U (r ) 2/3 |∇n|2 |v0 |2 d x ≤ C() |∇n|3 d x lim inf |∇v j |2 d x ≤ Cε2/3 . U (r )
U (r )
j→+∞
Hence
c p (n) = c p (n) v0 2L p () ≤ (|∇v0 |2 − |∇n|2 |v0 |2 )d x 2 ≤ lim |∇v j | d x − |∇n|2 |v j |2 d x + O(ε2/3 ) j→+∞ \U (r ) = lim (|∇v j |2 d x − |∇n|2 |v j |2 )d x + O(ε2/3 ) = c p (n) + O(ε2/3 ). j→+∞
These inequalities are valid for all small ε. Hence v0 achieves c p (n). It follows that c p (n) > 0 because n is non-degenerate. Definition 5.2. Let and u0 satisfy condition (1.4). We say that (u0 , q) satisfies condition (C) if there exists n ∈ ∗ (u0 , q) such that S p + (qn) = ∅. Lemma 5.3. Assume that every n ∈ ∗ (u0 , q) is non-degenerate. (i) If (u0 , q) does not satisfy condition (C), then m v (u0 , q) = 0. (ii) If (u0 , q) satisfies condition (C), and if there exists c4 > 0 such that for p = 4 and for every n ∈ ∗ (u0 , q), (5.1) holds for some c4 (n) ≥ c4 , then 0 < m v (u0 , q) < +∞. (iii) If (u0 , q) satisfies condition (C), and if {|∇n|3 : n ∈ ∗ (u0 , q)} is uniformly integrable,11 then 0 < m v (u0 , q) < +∞ and it is achieved. Proof. (i) If condition (C) does not hold, then for every n ∈ ∗ (u0 , q) and every ¯ ¯ φ ∈ S p(qn), (φ∇φ) is parallel to n. Hence for any v ∈ V(n), v · (φ∇φ)d x = 0. So m v (u0 , q) = 0. 11 A set of functions { f } is called uniformly integrable on if for any ε > 0, there exists δ > 0 such that λ for any measurable set E ⊂ with |E| < δ, it holds that f λ L 1 (E) < ε for all λ.
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(ii) Assume that there exists n ∈ ∗ (u0 , q) such that S p + (qn) = ∅. Let φ ∈ ¯ ¯ S p + (qn) and set w = (φ∇φ)−(n· ( φ∇φ))n. Then w ≡ 0, and n·w = 0. Recall that n 2+α is a C map away from a finite number of points x1 , · · · , xm in . Applying the elliptic 3+α ( \ {x , · · · , x }, C). regularity theory ([GT, Chapter 6]) to (1.8) we see that φ ∈ Cloc 1 m 2+α Thus w is a C map away from x1 , · · · , xm . Therefore we can find a non-empty open set D \ {x1 , · · · , xm } such that w = 0 in D. Choose a smooth function η supported in D, η ≥ 0 and η ≡ 0, and let v = ηw. Then v ∈ V(n) and 2 2 ¯ ¯ ¯ v · (φ∇φ)d x = η{[ (φ∇φ)] − [n · (φ∇φ)] }d x > 0.
From this and (5.1) we see that m v (u0 , q) ≥ m(n, q) ≥ m[n, v, φ] > 0. On the other hand, for any n ∈ ∗ (u0 , q), v ∈ V(n), φ ∈ S p(qn),
∇qn φ 2L 2 () = µ(qn) φ 2L 2 () ≤ µπ (u0 , q)||1/2 φ 2L 4 () , 2 ¯ qn φ)d x ≤ v 2 4 φ 2 4 ∇qn φ 2 2 v · (φ∇ L () L () L ()
≤ µπ (u0 , q)||1/2 v 2L 4 () φ 4L 4 () . From this and the assumption that (5.1) holds for c4 (n) ≥ c4 > 0, we have µπ (u0 , q)||1/2 v 2L 4 () µπ (u0 , q)||1/2 ≤ . m[n, v, φ] ≤
2 2 2 c4 (|∇v| − |∇n| |v| )d x Hence 0 < m v (u0 , q) ≤ µπ (u0 , q)||1/2 /c4 < +∞. (iii) Assume that {|∇n| : n ∈ ∗ (u0 , q)} is uniformly integrable, and that there exists n ∈ ∗ (u0 , q) such that S p + (qn) = ∅. Let {(n j , v j , φ j )} be a maximizing sequence for m v (u0 , q), where n j ∈ ∗ (u0 , q) and (v j , φ j ) ∈ V S(n j , q). Without loss of generality we may assume that
φ j L 4 () = 1, (|∇v j |2 − |∇n j |2 |v j |2 )d x = 1. (5.3)
It is easy to see that {n j } is a pre-compact sequence in H 1 (, S2 , u0 ). After passing to a subsequence we may assume that n j → n0 strongly in H 1 (, S2 , u0 ), ∇n j (x) → ∇n0 (x) a.e., and n0 ∈ ∗ (u0 , q). By the assumption {|∇n j |3 } is uniformly integrable, and from the Convergence Theorem of Vitali (see [Ru, p.133]), we have n0 ∈ W 1,3 (, S2 ), and n j → n0 strongly in W 1,3 (, S3 ). Using again the uniform integrability assumption on {|∇n j |3 }, we can modify the proof of Lemma 5.1 to show that there exists c4 > 0 independent of j such that (|∇v|2 − |∇n j |2 |v|2 )d x ≥ c4 v 2L 4 () for all v ∈ V(n j ) and j ≥ 1. (5.4)
So v j 2L 4 () ≤ 1/c4 . As in the last part of the proof of (ii), we find m v (u0 , q) ≤ lim inf
j→+∞
µπ (u0 , q)||1/2 v j 2L 4 ()
(|∇v j |
2
− |∇n j |2 |v j |2 )d x
≤
µπ (u0 , q)||1/2 < +∞. c4
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Since {φ j } is bounded in H 1 (, C), after passing to a subsequence we may assume that φ j → φ0 weakly in H 1 (, C) and strongly in L 4 (, C), φ0 L 4 () = 1, and φ0 ∈ S p(qn0 ). Since ∇qn j φ j → ∇qn0 φ0 weakly in L 2 (, C3 ), and since
∇qn j φ j 2L 2 () = µπ (u0 , q) φ j 2L 2 () → µπ (u0 , q) φ0 2L 2 () = ∇qn0 φ0 2L 2 () , we
find that ∇qn j φ j → ∇qn0 φ0 strongly in L 2 (, C3 ), and hence φ j → φ0 strongly in H 1 (, C). Using the second equality in (5.3) and the uniform integrability assumption on {|∇n j |3 } we can show that {v j } is bounded in H01 (, R3 ). After passing to a subsequence again we may assume that v j → v0 weakly in H 1 (, R3 ), strongly in L 4 (, R3 ), and pointwise a.e. in , and v0 L 4 () = 1. Hence lim inf |∇v j |2 d x ≥ |∇v0 |2 d x, j→+∞ (5.5) lim v j · (φ¯ j ∇φ j )d x = v0 · (φ¯ 0 ∇φ0 )d x. j→+∞
Since {|∇n j |3 } is uniformly integrable and { v j L 6 () } is bounded, we see that {|∇n j |2 |v j |2 } is uniformly integrable. Applying the Convergence Theorem of Vitali again we have lim |∇n j |2 |v j |2 d x = |∇n0 |2 |v0 |2 d x. (5.6) j→+∞
Since n0 is non-degenerate and v0 ∈ V(n0 ), we have (|∇v0 |2 − |∇n0 |2 |v0 |2 )d x ≥ c4 (n0 ) v0 2L 4 () = c4 (n0 ) > 0.
From (5.5) and (5.6) we see that (n0 , v0 , φ0 ) achieves m v (u0 , q).
Using the variational formula for m v (u0 , q) we find the following identity. Lemma 5.4. Assume 0 < m v (u0 , q) < +∞ and is achieved by (n0 , v, φ), where n0 ∈ ∗ (u0 , q) and (v, φ) ∈ V S(n0 , q). Then for any z ∈ V(n0 ) it holds that ¯ v · (φ∇φ)d x (∇v · ∇z − |∇n0 |2 v · z)d x (5.7) 2 2 2 ¯ = z · (φ∇φ)d x (|∇v| − |∇n0 | |v| )d x.
6. Critical Case: Estimates of K c In this section we consider the case where (1.7) holds. Lemma 6.1. Under condition (1.7) we have 2q 2 κ −2 m v (u0 , q) ≤ K c ≤ +∞. In particular if m v (u0 , q) = +∞, then K c = +∞.
(6.1)
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Proof. If m v (u0 , q) = 0 then (6.1) is obviously true. In the following we assume 0 < m v (u0 , q) ≤ +∞. Hence there exists n0 ∈ ∗ (u0 , q) such that V S(n0 , q) = ∅. Let n0 ∈ ∗ (u0 , q) and (v, φ) ∈ V S(n0 , q) be such that ¯ qn0 φ)d x = 0. v · (φ∇ (6.2)
Replacing v by −v if necessary we may assume the integral in (6.2) is positive. We take a test function in the form n0 + tv ψ = sφ, nt = = n0 + tv + t 2 wt , (6.3) |n0 + tv| where wt = w + tzt , s=
√ at,
1 w = − |v|2 n0 , zt = O(1), 2
¯ qn0 φ)d x v · (φ∇ > 0. a = 2qκ −2
4 |φ| d x
(6.4)
We have |∇nt |2 =|∇n0 |2 + 2t∇n0 · ∇v + t 2 (|∇v|2 + 2∇n0 · ∇wt ) + 2t 3 ∇v · ∇wt + O(t 4 ),
¯ qn0 φ) + q 2 t 2 |v + twt |2 φ|2 |∇qnt ψ|2 =s 2 |∇qn0 φ|2 − 2qt (v + twt ) · (φ∇ ¯ qn0 φ) =at|∇qn0 φ|2 − 2qat 2 v · (φ∇ ¯ qn0 φ)} + O(t 4 ). + t 3 {q 2 a|vφ|2 − 2qawt · (φ∇ Since v = 0 on ∂, so wt = 0 on ∂. Since n0 is a weak solution of (1.6) and n0 · v = 0, we have ∇n0 · ∇vd x = |∇n0 |2 n0 · vd x = 0, 1 2 2 2 2 − |∇n0 | |v| + t|∇0 | n0 · zt d x. ∇n0 · ∇wt d x = |∇n0 | n0 · wt d x = 2 Using the fact φ ∈ S p(qn0 ), we have E K [ψ, nt ] = K Cπ (u0 ) − t 2 I2 + t 3 I3 + O(t 4 ), where
(6.5)
κ 2a2 4 ¯ qn0 φ)d x, I2 = −K (|∇v| − |∇n0 | |v| )d x − |φ| d x + 2qa v · (φ∇ 2 ¯ qn0 φ)}d x I3 = {2K [∇v · ∇w + |∇n0 |2 n0 · zt ] + q 2 a|vφ|2 − 2qaw · (φ∇ ¯ qn0 φ) d x. = K [−∇v · ∇(|v|2 n0 ) + |∇n0 |2 n0 · zt ]+q 2 a|vφ|2 + qa|v|2 n0 · (φ∇ 2
2
2
Let
c2 (n0 , v) =
(|∇v|2 − |∇n0 |2 |v|2 )d x.
(6.6)
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Case 1. For all such n0 and v we have c2 (n0 , v) > 0. In this case, from the choice of a (see (6.4)) we have
¯ v · (φ∇qn0 φ)d x
4 |φ| d x
2
(|∇v|2 − |∇n0 |2 |v|2 )d x + 2q 2 κ −2 2 −2 (|∇v|2 − |∇n0 |2 |v|2 )d x = 2q κ m[n0 , v, φ] − K 2 −2 = c2 (n0 , v) 2q κ m[n0 , v, φ] − K .
I2 = −K
If K < 2q 2 κ −2 m[n0 , v, φ], then I2 > 0, and E K [ψ, nt ] < K Cπ (u0 ) if t is small. So K c ≥ 2q 2 κ −2 m[n0 , v, φ]. Since this is true for all such n0 , v and φ, we find K c ≥ 2q 2 κ −2
sup
sup
n∈ ∗ (u0 ,q) (v,φ)∈V S(n,q)
m[n, v, φ] = 2q 2 κ −2 m v (u0 , q).
Case 2. For some n0 ∈ ∗ (u0 , q) there exist v ∈ V0 (n0 ) (hence c2 (n0 , v) = 0) and ¯ φ ∈ S p(qn0 ) such that v · (φ∇φ)d x = 0. Then m v (u0 , q) = +∞. We shall show that K c = +∞. For the above v and φ we take test functions ψ and nt as in (6.3), (6.4). The same computations yield E K [ψ, nt ] = K Cπ (u0 ) − I2 t 2 + O(t 3 ), where I2 = −
κ 2a2 2
= 2q 2 κ −2
|φ|4 d x + 2qa
¯ qn0 φ)d x v · (φ∇
¯ v · (φ∇qn0 φ)d x
4 |φ| d x
2 > 0.
Hence for any K > 0, we can choose t > 0 small so that E K [ψ, nt ] < K Cπ (u0 ). So K c > K . Thus K c = +∞. (6.1) is also true. Lemma 6.2. Under condition (1.7) we assume 0 ≤ m v (u0 , q) < +∞. If every n ∈ ∗ (u0 , q) satisfies either condition (A) or condition (B), then 2q 2 κ −2 m v (u0 , q) ≤ K c < +∞.
(6.7)
Proof. We prove K c < +∞. Then (6.7) follows from Lemma 6.1. Step 1. Suppose K c = +∞. Then there exists a sequence K j → +∞ such that E K j has a nontrivial minimizer (ψ j , n j ), and (4.2) holds. Using (1.7), we argue as in the proofs of Lemmas 4.2 and 4.3 to show that, after passing to a subsequence, (ψ j , n j ) → (0, n0 ) strongly in H 1 (, C) × H 1 (, S2 , u0 ), where n0 ∈ (u0 ).
(6.8)
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Step 2. Similar to (4.4), (4.5), we write ψj = sjφj,
n j = n0 + t j v j ,
(6.9)
where s j > 0, t j > 0, and φ j ∈ H 1 (, C), φ j H 1 () = 1, v j ∈ H01 (, R3 ), v j L 2 () = 1.
(6.10)
Note that in (4.5) we required v j L 4 () = 1 but now we require v j L 2 () = 1. We can require in (6.9) t j > 0 because n j = n0 . From (6.8), t j → 0 and s j → 0. Passing to a subsequence if necessary we assume φ j → φ0 weakly in H 1 (, C) and strongly in L 4 (, C). Now we repeat the argument in Step 2 of the proof of Lemma 4.3 to show that (4.7), (4.8), (4.9) and (4.10) hold, and φ0 = 0. From (4.10) we have µ(qn0 ) ≤ κ 2 . Since µ(qn0 ) ≥ µπ (u0 , q) = κ 2 , we must have µ(qn0 ) = κ 2 , and hence the equality in 2 associated (4.10) holds. Therefore n0 ∈ ∗ (u0 , q) and φ0 is an eigenfunction of −∇qn 0 with the eigenvalue µ(qn0 ). Moreover, using (4.9) we have 2 2 2 |φ0 | d x = |∇qn0 φ0 | d x ≤ lim inf |∇qn0 φ j |2 d x κ j→+∞ 2 2 ≤ lim sup |∇qn0 φ j | d x = lim κ |φ j |2 d x = κ 2 |φ0 |2 d x. j→+∞
j→+∞
So ∇qn0 φ j L 2 () → ∇qn0 φ0 L 2 () . Hence ∇qn0 φ j → ∇qn0 φ0 strongly in L 2 (, C3 ), and thus φ j → φ0 strongly in H 1 (, C). Step 3. We look for the limit of v j . Write φ j = φ0 + s j ξ j . Since φ0 is an eigenfunction, we have (|∇qn0 φ j |2 − κ 2 |φ j |2 )d x = |∇qn0 φ0 |2 − κ 2 |φ0 |2 + 2s j [∇qn0 φ0 ∇qn0 ξ − κ 2 φ0 ξ¯ j ] + s 2j (|∇qn0 ξ j |2 − κ 2 |ξ j |2 ) d x 2 = s j (|∇qn0 ξ j |2 − κ 2 |ξ j |2 )d x.
(6.11)
(6.12)
Plugging (6.12) into (4.7) we have, for any δ > 0, K j t 2j (|∇v j |2 − |∇n0 |2 |v j |2 )d x + s 4j (|∇qn0 ξ j |2 − κ 2 |ξ j |2 )d x
κ 2 s 4j
|φ j |4 d x + q 2 s 2j t 2j |v j φ j |2 d x 2 ≤ 2qt j s j v j · (φ¯ j ∇qn0 φ j )d x
+
≤
2
t 2j δ
|v j φ j |2 d x + δq 2 s 4j
|∇qn0 φ j |2 d x.
(6.13)
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Since |φ j |4 d x → |φ0 |4 d x > 0 and |∇qn0 φ j |2 d x → |∇qn0 φ0 |2 d x, we can choose δ > 0 small such that κ2 δq 2 |∇qn0 φ j |2 d x ≤ |φ j |4 d x. 2 Since µ(qn0 ) = κ 2 , the second integral in the left of (6.13) is non-negative. From (6.13) we get 1 K j (|∇v j |2 − |∇n0 |2 |v j |2 )d x ≤ |v j φ j |2 d x. (6.14) δ From Sobolev imbedding theorem, (6.10) and (6.14), C() (|∇v j |2 − |∇n0 |2 |v j |2 )d x ≤
∇v j 2L 2 () . δK j
(6.15)
Step 4. Let us assume n0 satisfies condition (A). Let {x1 , · · · , xm } be the set of singular points of n, and let U (r ) = ∪mj=1 B(x j , r ) ∩ . As in the proof of Lemma 5.1 we can choose r small such that
(|∇v j |2 − |∇n0 |2 |v j |2 )d x ≥
1 |∇v j |2 d x − sup |∇n0 |2 |v j |2 d x. 2 \U (r ) x∈\U (r ) (6.16)
From (6.10), (6.15), (6.16), and the assumption K j → +∞, we find that {v j } is bounded in H01 (, R3 ). Passing to a subsequence we may assume v j → v0 weakly in H01 (, R3 ) and strongly in L 4 (, R3 ), and v0 ∈ V(n), v0 L 2 () = 1. Since n0 is non-degenerate and n0 ∈ W 1,3 (, S2 ), we argue as in the proof of Lemma 5.1 to get |∇v0 |2 d x ≤ lim inf |∇v j |2 d x, |∇n0 |2 |v0 |2 d x = lim |∇n0 |2 |v j |2 d x.
j→+∞
Then we use (6.15) to find
j→+∞
0 < c2 (n0 ) = ≤ (|∇v0 |2 − |∇n0 |2 |v0 |2 )d x C() 2 2 ≤ lim inf (|∇v j | − |∇n0 | |v j |2 )d x ≤ lim inf
∇v j 2L 2 () = 0. j→+∞ j→+∞ δ K j c2 (n0 ) v0 2L 2 ()
This contradiction shows that the nontrivial minimizers can not exist for large K . This proves K c < +∞. Step 5. Now we assume that n0 satisfies condition (B). From (6.14) and condition (B) we have, for a constant c3 (n0 ) > 0, 1 c3 (n0 ) v j 2L 3 () ≤ (|∇v j |2 − |∇n0 |2 |v j |2 )d x ≤ |v j φ j |2 d x δ K j C() C() 2 2 2
v j L 3 () φ j H 1 () =
v j L 3 () . ≤ δK j δK j So c3 (n0 ) ≤ C()/(δ K j ) → 0, a contradiction. Again we get K c < +∞.
Proof of Theorem 1. Theorem 1 follows from Lemma 2.2, Theorems 3.1, 4.1 and Lemma 6.1.
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7. Critical Case: Behavior of Minimizers at K c 7.1. Three subcases. As mentioned in Introduction, the critical case can be further classified into three subcases. Lemma 7.1. (Subcase 1) Under condition (1.7), we assume that K c = +∞. Let K j → +∞ and let (ψ j , n j ) be a minimizer of E K j . Then there exists a subsequence jl → +∞ such that (ψ jl , n jl ) → (0, n0 ) strongly in H 1 (, C) × H 1 (, S2 , u0 ), where n0 ∈ ∗ (u0 , q). Proof. Let K j → +∞ and let (ψ j , n j ) be a minimizer of E K j . From Lemma 2.5, we can find a subsequence which converges to (ψ0 , n0 ) strongly in H 1 (, C) × H 1 (, S2 , u0 ), n0 ∈ (u0 ), and (2.14) holds. As in the proof of Lemma 4.2, we have µ(qn0 ) ≤ κ 2 . From this and (1.7) we have µ(qn0 ) = κ 2 , and hence n0 ∈ ∗ (u0 , q). It also implies that the only minimizer of G[·, qn0 ] is ψ = 0. Hence ψ0 = 0. Lemma 7.2. (Subcase 2) Under condition (1.7), we assume that (a) 0 < m v (u0 , q) < +∞; (b) every n ∈ ∗ (u0 , q) satisfies condition (A); (c) E K c has only trivial minimizers. Then equality (1.18) holds, namely K c = 2q 2 κ −2 m v (u0 , q) < +∞; and at K c the global minimizers bifurcate from the trivial critical points. More precisely, let K increase to 2q 2 κ −2 m v (u0 , q) from below, and let (ψ K , n K ) be a nontrivial minimizer of E K . Then there exists a subsequence K j such that the following expansions hold in the topology of H 1 (, C) × H 1 (, S2 , u0 ): ψ K j = a j t j φ0 + o(t j ), n K j = n0 + t j v0 + o(t j ), (7.1) where t j > 0, t j → 0, a j → a, n0 ∈ ∗ (u0 , q), φ0 ∈ S p(qn0 ), φ0 H 1 () = 1, v0 ∈ V(n0 ), v0 L 4 () = 1,
¯ −2 v0 · (φ0 ∇qn0 φ0 )d x ¯ , v0 · (φ0 ∇qn0 φ0 )d x > 0, a = 2qκ 4 |φ0 | d x
2 ¯ 0 ∇qn0 φ0 )d x v · ( φ 0
= m v (u0 , q).
φ0 4L 4 () (|∇v0 |2 − |∇n0 |2 |v0 |2 )d x
(7.2)
Proof. From conditions (a) and (b) and using Lemma 6.2 we have K c < +∞. Let {K j } be an increasing sequence converging to K c . E K j has a nontrivial minimizer (ψ j , n j ). In particular µ(qn j ) < κ 2 . As in the proofs of Lemmas 4.2, 4.3 and 6.2 we can pass to a subsequence and assume that (ψ j , n j ) → (ψ0 , n0 ) weakly in H 1 (, C) × H 1 (, R3 ) and strongly in L 4 (, C) × L p (, R3 ) for all 1 < p < 6, and (ψ0 , n0 ) ∈ H 1 (, C) × H 1 (, S2 , u0 ) is a minimizer of E K c . From condition (c), ψ0 = 0. Note that (4.2) is still valid, from which we have K c Cπ (u0 ) ≤K c
|∇n0 |2 d x ≤ lim inf E K j [ψ j , n j ] ≤ lim inf K j Cπ (u0 ) = K c Cπ (u0 ), j→+∞
j→+∞
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so n0 ∈ (u0 ). We can further show that, after passing to another subsequence, (6.8) holds. As in (4.4) we write ψ j = s j φ j , n j = n0 + t j v j , where φ j and v j satisfy (4.5). Again we have s j → 0 and t j → 0. The computations in Step 2 of the proof of Lemma 6.2 remain valid. In particular, we can pass to a subsequence to have φ j → φ0 2 associated with the lowest strongly in H 1 (, C), where φ0 is an eigenfunction of −∇qn 0 2 eigenvalue µ(qn0 ). Since κ = µπ (u0 , q) ≤ µ(qn0 ) = lim j→+∞ µ(qn j ) ≤ κ 2 , we have n0 ∈ ∗ (u0 , q). As in (6.11) we write φ j = φ0 + s j ξ j . Similar to (6.13) we now have 2 2 2 2 4 (|∇v j | − |∇n0 | |v j | )d x + s j (|∇qn0 ξ j |2 − κ 2 |ξ j |2 )d x K jtj
+
κ 2 s 4j 2
≤ 2qt j s 2j
|φ j |4 d x + q 2 s 2j t 2j
|v j φ j |2 d x
(7.3)
v j · (φ¯ j ∇qn0 φ j )d x
≤ 2qt j s 2j v j L 4 () φ j L 4 () ∇qn0 φ j L 2 () ≤ Ct j s 2j . Since µ(qn0 ) = κ 2 , each integral in the left side of (7.3) is non-negative, and we have κ 2 s 4j 2
|φ j |4 d x ≤ Ct j s 2j .
(7.4)
Since φ j L 4 () → φ0 L 4 () > 0, we see from (7.4) that s 2j = O(t j ). Set s 2j = a j t j ,
Aj =
κ2 2
|φ j |4 d x,
Bj = q
v j · (φ¯ j ∇qn0 φ j )d x,
where a j > 0 and is bounded. After passing to a subsequence we assume a j → a. From (7.3), B j > 0 and K j (|∇v j |2 − |∇n0 |2 |v j |2 )d x + a 2j (|∇qn0 ξ j |2 − κ 2 |ξ j |2 )d x B j 2 + q 2a j t j |v j φ j |2 d x + A j a j − Aj (7.5)
2 ¯ v j · (φ j ∇qn0 φ j )d x ≤ 2q 2 κ −2 ≤ C.
φ j 4L 4 () In particular Kj
(|∇v j |2 − |∇n0 |2 |v j |2 )d x ≤ C.
(7.6)
Since n0 ∈ W 1,3 (, S2 ) and {v j } is bounded in L 4 (, R3 ), we can use (7.6) and apply the argument used in the proof of Lemma 5.1 to show that {v j } is bounded in H01 (, R3 ).
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Moreover, after passing to a subsequence again, we have v j → v0 weakly in H01 (, R3 ) and strongly in L 4 (, R3 ), n0 (x) · v0 (x) = 0 a.e. in , and
|∇v0 |2 d x ≤ lim inf
j→+∞
|∇v j |2 d x,
|∇n0 |2 |v0 |2 d x = lim
j→+∞
|∇n0 |2 |v j |2 d x. (7.7)
In particular, v0 L 4 () = 1, so v0 ∈ V(n0 ). Since n0 is non-degenerate, we have 2 2 2 lim inf (|∇v j | − |∇n0 | |v j | )d x ≥ (|∇v0 |2 − |∇n0 |2 |v0 |2 )d x j→+∞
≥ c4 (n0 ) v0 2L 4 () = c4 (n0 ) > 0. Now from (7.5) we have
K j ≤ 2q 2 κ −2
vj
φ j 4L 4 ()
· (φ¯ j ∇qn0 φ j )d x
(|∇v j |
2
− |∇n0 |2 |v j |2 )d x 2 ¯ 0 ∇qn0 φ0 )d x v · ( φ 0
+ o(1) ≤ 2q 2 κ −2 4
φ0 L 4 () (|∇v0 |2 − |∇n0 |2 |v0 |2 )d x 2
≤ 2q 2 κ −2 m v (u0 , q) + o(1). Therefore K c ≤ 2q 2 κ −2 m v (u0 , q). Combining this with (6.1) we find that (1.18) holds. Moreover,
2 ¯ j ∇qn0 φ j )d x v · ( φ j
lim j→+∞ φ j 4 4 (|∇v j |2 − |∇n0 |2 |v j |2 )d x L () (7.8)
2 ¯ v · ( φ ∇ φ )d x 0 qn0 0 0
= = m v (u0 , q). 4
φ0 L 4 () (|∇v0 |2 − |∇n0 |2 |v0 |2 )d x Hence φ0 ∈ S p + (qn0 ) and m[n0 , v0 , φ0 ] = m v (u0 , q). From this and (7.8) we find that v0 · (φ¯ 0 ∇qn0 φ0 )d x > 0.
Moreover, from (7.8) and (7.7) we have lim j→+∞ |∇v j |2 d x = |∇v0 |2 d x, so
∇v j − ∇v0 L 2 () → 0, hence v j → v0 in H01 (, R3 ). Now κ2 A j → A0 = |φ0 |4 d x, B j → B0 = q v0 · (φ¯ 0 ∇qn0 φ0 )d x > 0. 2 Sending j → +∞ in (7.5) we find B0 2 a 2 lim sup (|∇qn0 ξ j |2 − κ 2 |ξ j |2 )d x + A0 (a − ) ≤ 0. A 0 j→+∞ Hence a = B0 /A0 . Thus (7.1) and (7.2) hold.
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Lemma 7.3. (Subcase 3) Under condition (1.7), we assume that (1.20) holds and that every n ∈ ∗ (u0 , q) satisfies condition (A). Then there exists K˜ c > K c such that E K has nontrivial local minimizers when K c < K < K˜ c (which are not global minimizers), and they bifurcate from the nontrivial global minimizers at K c . Proof. We first show that Claim 1 in the proof of Lemma 4.4 holds. Otherwise, there exists a sequence (ψ j , n j ) such that ψ j L 2 () → 0 and E K c [ψ j , n j ] = C(u0 , K c ) = K c Cπ (u0 ). Since every n ∈ ∗ (u0 , q) satisfies condition (A), we can repeat the proof of Lemma 7.2 with K j replaced by K c and find that K c ≤ 2q 2 κ −2 m v (u0 , q). But it contradicts (1.20). Next we see that Claims 2 and 3 in the proof of Lemma 4.4 remain true. Hence K˜ c exists. Then we repeat the proof of Lemma 4.5 and show that the local minimizers bifurcate from the global minimizers at K c . 7.2. Sufficient conditions for hysteresis. In order to find a sufficient condition for hysteresis, in Lemma 7.7 below we give another formula for K c . Definition 7.4. Given n0 ∈ (u0 ), let T (n0 ) = {w ∈ H01 (, R3 ) : w = 0, |n0 (x) + w(x)| = 1 a.e. on }, 1 2 2 2 (|∇q(n0 +w) ξ | − κ |ξ | )d x < 0 . M(n0 ) = (w, ξ ) ∈ T (n0 ) × H (, C) :
Lemma 7.5. If µh (u0 , q) <
κ2
then M(n0 ) = ∅.
Proof. Let n ∈ H 1 (, S2 , u0 ) be such that µ(qn) < κ 2 . Let w = n − n0 and ξ ∈ S p(qn). Then (w, ξ ) ∈ M(n0 ). Note that for any w ∈ T (n0 ) we have (|∇w|2 − |∇n0 |2 |w|2 )d x ≥ 0.
(7.9)
Moreover if (u0 ) consists of a unique minimizing weak harmonic map then the strict inequality in (7.9) holds for any w ∈ T (n0 ), w = 0 (Lemma 8.2). Definition 7.6. Let n0 ∈ (u0 ). If M(n0 ) = ∅, we set M+ (n0 ) = (w, ξ ) ∈ M(n0 ) : (|∇w|2 − |∇n0 |2 |w|2 )d x > 0 ,
2 2 − κ 2 |ξ |2 d x |∇ ξ | q(n +w) 0
, M[n0 , w, ξ ] =
ξ 4L 4 () |∇w|2 − |∇n0 |2 |w|2 d x if M(n0 ) = M+ (n0 ), sup M[n0 , w, ξ ] : (w, ξ ) ∈ M(n0 ) M(n0 ) = +∞ if M(n0 ) = M+ (n0 ). If M(n0 ) = ∅, we define M(n0 ) = 0. Finally we define M(u0 , q) =
sup
n0 ∈ ∗ (u0 ,q)
M(n0 ).
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Lemma 7.7. We have Kc =
M(u0 , q) . 2κ 2
(7.10)
0 ,q) Proof. Step 1. We first show K c ≥ M(u . If M(n) = ∅ for any n ∈ ∗ (u0 , q), then 2κ 2 M(u0 , q) = 0, and the inequality is obviously true. Now we assume that there exists n0 ∈ (u0 ) such that M(n0 ) = ∅. For any (ψ, n) ∈ H 1 (, C) × H 1 (, S2 , u0 ), write n = n0 + w and ψ = sξ , where s > 0. We have E K [ψ, n] − K Cπ (u0 ) = K (|∇w|2 − |∇n0 |2 |w|2 )d x + A[ξ ]s 4 − B[w, ξ ]s 2
B[w, ξ ] 2 = A[ξ ] s 2 − + K (|∇w|2 − |∇n0 |2 |w|2 )d x 2 A[ξ ] 2 1 2 2 2 − 2 (|∇q(n0 +w) ξ | − κ |ξ | )d x , 2κ ξ 4L 4 ()
(7.11)
where κ2 |ξ |4 d x, 2 B[w, ξ ] = − (|∇q(n0 +w) ξ |2 − κ 2 |ξ |2 )d x = − {|∇qn0 ξ |2 − κ 2 |ξ |2 + q 2 |wξ |2 − 2qw · (ξ¯ ∇qn0 ξ )}d x. A[ξ ] =
Note that when M(n0 ) = ∅, B[w, ξ ] > 0 for any (w, ξ ) ∈ M(n0 ). First assume M(n0 ) = M+ (n0 ). Let M[n0 , w j , ξ j ] → M(n0 ) and set ψ j = s j ξ j , n j = n0 + w j , s j =
B[w j , ξ j ] 2 A[ξ j ]
1/2 .
Plugging (ψ j , n j ) into (7.11) we have E K [ψ j , n j ] − K Cπ (u0 ) = K (|∇w j |2 − |∇n0 |2 |w j |2 )d x
2 1 2 2 2 (|∇q(n0 +w j ) ξ j | − κ |ξ j | )d x − 2 2κ ξ j 4L 4 () M(n0 ) = K− + o(1) (|∇w j |2 − |∇n0 |2 |w j |2 )d x. 2κ 2 0) 0) and if j is large. Thus K c ≥ M(n . The right side is negative if K < M(n 2κ 2 2κ 2 Next assume M(n0 ) = M+ (n0 ). Then M(n0 ) = +∞. Let K be any positive number. Take (w0 , ξ0 ) ∈ M(n0 ) \ M+ (n0 ) and let B[w0 , ξ0 ] 1/2 . ψ = sξ0 , n = n0 + w0 , s = 2 A[ξ0 ]
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Plugging (ψ, n) into (7.11) we have 1 E K [ψ, n] − K Cπ (u0 ) = − 2 2κ ξ0 4L 4 ()
2 (|∇q(n0 +w0 ) ξ0 | − κ |ξ0 | )d x 2
2
2
< 0.
So E K has a nontrivial minimizer. Hence K c = +∞ and the claimed inequality is still valid. 0 ,q) Step 2. Now we show K c ≤ M(u . If M(u0 , q) = +∞, then the inequality is obviously 2κ 2 true. Now assume M(u0 , q) < +∞. Let K < K c and let (ψ K , n K ) be a nontrivial minimizer of E K . Then µ(qn K ) < κ 2 . For any n0 ∈ (u0 ) and for any s > 0, let (w, ξ ) = (n K − n0 , 1s ψ). Then (w, ξ ) ∈ M(n0 ) and w = 0. Hence M(n0 ) = ∅. Moreover, since M(u0 , q) < +∞, we have M(n0 ) = M+ (n0 ). So w ∈ M+ (n0 ). From (7.11), 0 >E K [ψ K , n K ] − K Cπ (u0 ) ≥ K (|∇w|2 − |∇n0 |2 |w|2 )d x
1 − 2 2κ ξ 4L 4 () K < So K c ≤
(|∇q(n0 +w) ξ |2 − κ 2 |ξ |2 )d x
2 ,
1 1 1 M[n0 , w, ξ ] ≤ 2 M(n0 ) ≤ 2 M(u0 , q). 2 2κ 2κ 2κ
M(u0 ,q) . 2κ 2
Lemma 7.8. Assume that (1.7) holds. (i) If every n ∈ ∗ (u0 , q) satisfies either condition (A) or condition (B), then 4q 2 m v (u0 , q) ≤ M(u0 , q) < +∞.
(7.12)
(ii) If every n ∈ ∗ (u0 , q) satisfies condition (A), and if the following strict inequality holds 4q 2 m v (u0 , q) < M(u0 , q),
(7.13)
then K c < +∞, and E K c has nontrivial minimizers. The global minimizers of E K do not bifurcate from the trivial minimizers, and there exists K˜ c > K c such that E K has nontrivial local minimizers for all K ∈ [K c , K˜ c ). Proof. Under the conditions in (i), (7.12) follows from (6.7) and (7.10). Assume the conditions in (ii) hold. From (i) and (7.10) we see that (1.20) holds. Hence E K c has nontrivial minimizers, because otherwise we can use Lemma 7.2 to derive (1.18), which is contradictory to (1.20). Then we use Lemma 7.3 to conclude the existence of K˜ c . Lemma 7.9. Under condition (1.7), assume that 0 < m v (u0 , q) < +∞ and it is achieved by (n0 , v, ξ ), where n0 ∈ ∗ (u0 , q) and (v, ξ ) ∈ V S(n0 , q). Assume furthermore that (n0 , v, ξ ) satisfies (1.19). Then (7.13) is valid.
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Proof. Without loss of generality we may assume that m v (u0 , q) > 0. Let (n0 , v, ξ ) achieve m v (u0 , q), where n0 ∈ ∗ (u0 , q) and (v, ξ ) ∈ V S(n0 , q). We may replace v by −v if necessary and hence assume v · (ξ¯ ∇ξ )d x > 0. (7.14)
For any u ∈
H01 (, R3 ),
set nt =
n0 + tv + t 2 u . |n0 + tv + t 2 u|
We have nt = n0 + tv + t 2 n2 − t 3 n3 + O(t 4 ), |v|2 n0 , n2 = u⊥ − u⊥ = u − (u · n0 )n0 , 2 2 |v| + u · n0 v + (u · v)n0 . n3 = 2
(7.15)
Let wt = nt − n0 , and let B[w, ξ ] be the functional defined in the proof of Lemma 7.7. Since φ ∈ S p(qn0 ), we have B[wt , ξ ] = q [2wt · (ξ¯ ∇qn0 ξ ) − q|wt ξ |2 ]d x = qt[2α − β1 t − β2 t 2 + O(t 3 )], (|∇wt |2 − |∇n0 |2 |wt |2 )d x = t 2 [a − b1 t + b2 t 2 + O(t 3 )],
where
α=
a b1 b2
v · (ξ¯ ∇ξ )d x > 0,
[q|vξ |2 − 2n2 · (ξ¯ ∇qn0 ξ )]d x, = 2 [qv · n2 |ξ |2 + n3 · (ξ¯ ∇qn0 ξ )]d x, = (|∇v|2 − |∇n0 |2 |v|2 )d x, = 2 (|∇n0 |2 v · n2 − ∇v · ∇n2 )d x, = [|∇n2 |2 − 2∇v · ∇n3 − |∇n0 |2 (|n2 |2 − 2v · n3 )]d x.
β1 = β2
v · (ξ¯ ∇qn0 ξ )d x =
(7.16)
In the following computations let us write m = m v (u0 , q) and ξ = ξ L 4 () .
ξ 4 [2α − β1 t − β2 t 2 + O(t 3 )]2 M[n , w , ξ ] = 0 t q2 [a − b1 t + b2 t 2 + O(t 3 )] 4t 2 β12 b1 α 2 b12 4α 2 4t α 2 + b1 − αβ1 + − αβ2 − αβ1 + ( − b2 ) + O(t 3 ) = a a a a 4 a a a = 4m ξ 4 +
4t 2 4t I+ II + O(t 3 ), a a
(7.17)
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where α2 I = b1 − αβ1 , a
β2 b1 α 2 II = 1 − αβ2 − αβ1 + 4 a a
b12 − b2 . a
Using (7.15) and (7.16) we have
|∇n0 |2 v · n2 − ∇v · ∇n2 d x I = 2m ξ −α q|vξ |2 − 2n2 · (ξ¯ ∇qn0 ξ ) d x =2 αn2 · (ξ¯ ∇ξ ) − m ξ 4 ∇v · ∇n2 − |∇n0 |2 v · n2 d x. 4
Since (n0 , v, φ) achieves m v (u0 , q), we take z = u⊥ in (5.7) and find that αu⊥ · (ξ¯ ∇ξ ) − m ξ 4 ∇v · ∇u⊥ − |∇n0 |2 v · u⊥ d x = 0.
Hence I =−
α|v|2 n0 · (ξ¯ ∇ξ ) − m ξ 4 ∇v · ∇(|v|2 n0 ) d x.
(7.18)
Since α2 , 2 2 2 (|∇v| − |∇n0 | |v| )d x
m ξ 4 =
we have I 2 2 2 ¯ (|∇v| − |∇n0 | |v| )d x = v · (ξ ∇ξ )d x ∇v · ∇(|v|2 n0 )d x α (7.19) − |v|2 n0 · (ξ¯ ∇ξ )d x (|∇v|2 −|∇n0 |2 |v|2 )d x.
From (1.19) and (7.19) we have I > 0, and B[wt , ξ ] = 2qαt + O(t 2 ) > 0 if t > 0 is small. So (wt , ξ ) ∈ M(n0 ), and from (7.17), M[n0 , wt , ξ ] = 4mq 2 + Hence (7.13) is true.
4q 2 t I + O(t 2 ) > 4mq 2 . a ξ 4
Proof of Theorem 3. Conclusion (i) has been proved in Lemma 6.2. Conclusion (ii) has been proved in Lemma 7.2. Conclusion (iii) follows from Lemmas 7.8 and 7.9.
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8. Further Remarks Let us begin with an example in which condition (B) is satisfied. Example 8.1. Let = B1 = B1 (0) be the unit ball in R3 . Consider boundary data u0 = ±Rx, where R is a rotation in S O(3). Then n0 = ±Rx/|x| is the unique minimizing harmonic map with boundary data ±Rx ([BCL]). Hence (u0 ) = {n0 }. Now we show that n0 satisfies condition (B). Let r = |x|. We compute |∇n0 |2 = 2r −2 . Recall the Hardy inequality for u ∈ ([HLP], also see [CKN]) 1 |x|−2 |u|2 d x ≤ |∇u|2 d x. (8.1) 4 R3 R3
C01 (R3 , R3 )
For v ∈ H01 (B1 , R3 ), using (8.1) we have 1 2 2 2 (|∇v| − |∇n0 | |v| )d x ≥ |∇v|2 d x. 2 R3 B1
(8.2)
Hence n0 satisfies (5.1) for any 1 < p < 6. So n0 satisfies condition (B). In particular n0 is non-degenerate. Our next lemma shows that condition (B) implies the uniqueness of the minimizing weak harmonic maps. Lemma 8.2. Let n0 ∈ (u0 ). Then n0 is a unique minimizing weak harmonic map if and only if for any w ∈ T (n0 ) it holds that (|∇w|2 − |∇n0 |2 |w|2 )d x > 0, (8.3)
where T (n0 ) is the set defined in Definition 7.4. Proof. For any n ∈ H 1 (, S2 , u0 ), n − n0 ∈ H01 (, R3 ) and {|∇n − ∇n0 |2 − |∇n0 |2 |n − n0 |2 }d x = {|∇n|2 − |∇n0 |2 − 2∇n · ∇n0 + 2|∇n0 |2 n0 · n}d x.
Plugging n0 in (1.6) and multiplying it by n we have 2 |∇n0 | n0 · nd x = − n · n0 d x = ∇n · ∇n0 d x,
∂n0 ∂ν
0 = n0 · ∂n because n = n0 on ∂ and |n0 | = 1 implies n · ∂ν = 0. Hence 2 2 2 {|∇n − ∇n0 | − |∇n0 | |n − n0 | }d x = (|∇n|2 − |∇n0 |2 )d x.
Therefore n0 is the only minimizer if and only if {|∇n − ∇n0 |2 − |∇n0 |2 |n − n0 |2 }d x > 0 for all n ∈ H 1 (, S2 , u0 ), n = n0 ,
namely, if and only if (8.3) holds for all w ∈ T (n0 ).
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Next we examine condition (C). If n0 = x/|x|, then µ(q x/|x|) = 0, and it has only ¯ one linearly independent eigenfunction φ = ceiq|x| , and (φ∇φ) = q x/|x| = qn. ∇r So S p + (q x/|x|) = ∅. Also note that in this case n0 = r for x ∈ B1 \ {0} = {x : ¯ (φ∇φ) = 0}. This example carries a general feature. More precisely we have the following conclusion. Lemma 8.3. Let n ∈ H 1 (, S2 ) be a weak harmonic map and S p(qn) = S p + (qn). Let φ ∈ S p(qn)\S p + (qn) and set ¯ 0 = \{singular points of n}, U = {x ∈ 0 : (φ(x)∇φ(x)) = 0}.
(8.4)
Then for any simply-connected subdomain D of U , there exists a function u such that for x ∈ D, n=
∇u . |∇u|
(8.5)
Proof. From the regularity theory of harmonic maps (see [SU1], [SU2]) n has at most a finite number of singular points. So 0 is an open set and n is smooth in 0 . We 2 ( , C). In fact, for any x ∈ , take B (x ) . Then claim that φ ∈ Cloc 0 0 0 2R 0 0 3 n ∈ C (B2R (x0 ), R3 ). Applying the elliptic regularity theory to (1.8) (see [GT, Chapter 6]), we find φ ∈ C 2 (B R (x0 ), C). ¯ Hence U = {x ∈ 0 : (φ∇φ) = 0} is an open set. Let D be a simply-connected subdomain of U . We can write φ = f eiqu in D. Since φ ∈ S p + (qn), there exists a ¯ function g(x) such that q f 2 ∇u = (φ∇φ) = gn in D. Note that g(x) = 0 in D because the left side does not vanish in D. So n = h(x)∇u in D, where h(x) = q f 2 /g. Finally the equality |h(x)∇u| = 1 implies that (8.5) holds in D. Lemma 8.4. If u0 satisfies the condition ν · curl u0 = 0 on ⊂ ∂,
(8.6)
where is a non-empty open subset of ∂, then for any n ∈ (u0 ), {φ ∈ S p(qn) : φ ≡ 0 on } ⊂ S p + (qn). Proof. If φ ∈ S p(qn) and φ ≡ 0 on , there exists x0 ∈ such that φ(x0 ) = 0. We can find a small neighborhood U of x0 so that we can write φ = f eiqχ on U ∩ , where ¯ f and χ are real-valued functions. Then (φ∇φ) = q f 2 ∇χ . If φ ∈ S p + (qn), then ¯ ¯ (φ∇φ) is parallel to n, and we may write (φ∇φ) = q f 2 gn. So gn = ∇χ . From the boundary condition in (1.8) we have ¯ · (∇qn φ)] = ν · (φ∇φ) ¯ 0 = [φν − q|φ|2 ν · n = q f 2 (g − 1)ν · n. So g = 1 on U ∩ ∂. Thus ∇χ = n = u0 on U ∩ ∂. Since ν · curl u0 is completely determined by the tangential component of u0 (see for instance [BP, Lemma 2.4]), it holds that ν · curl u0 = 0 on U ∩ ∂. This violates condition (8.6). From Lemma 8.3, if n ∈ (n0 ) and S p + (qn) = ∅, then in an open set n has a local representation by (8.5), where the scalar function may not be globally defined on .
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Question 8.5. Let n ∈ ∗ (u0 , q) be such that S p + (qn) = ∅. Is it true that there exists a single-valued function u defined in except a set of zero Lebesgue measure, such that (8.5) holds a.e. in ? A related problem is: Given a function φ defined in a neighborhood of ∂ such that |∇φ| = 1 on ∂, does there exist a function that is a minimizer of the following problem ∇u 2 inf D |∇u| d x? u∈H 2 (), u=φ on ∂
(8.7)
For our convenience we introduce the following definition. Definition 8.6. Let and u0 satisfy condition (1.4). We say that (u0 , q) satisfies condition (D) if every member in ∗ (u0 , q) is non-degenerate. Note that in Example 8.1 condition (D) is satisfied. One may show using the idea of Uhlenbeck [U1,U2] that for generic boundary data u0 the corresponding minimizing weak harmonic maps are non-degenerate. Thus for generic u0 condition (D) is satisfied. Regarding the values of µπ (u0 , q) and µh (u0 , q), we have the following observations: (i) If there exists a function χ ∈ H 2 () such that |∇χ | = 1 a.e. in , ∇χ = u0 on ∂,
|D 2 χ |2 d x = Cπ (u0 ),
(8.8)
then µπ (u0 , q) = 0. In fact, ∇χ ∈ (u0 ) and µ(q∇χ ) = 0 with associated eigenfunction eiqχ . On the other hand, if µπ (u0 , q) = 0 and if n ∈ ∗ (u0 , q), then curl n = 0 except at finitely many points. In fact, let φ be an associated eigenfunction, then ∇qn φ = 0. As in the proof of Lemma 8.3, n and φ are smooth away from a finite number of points x1 , . . . , xm . By Kato’s inequality ∇|φ| L 2 () ≤ ∇qn φ L 2 () = 0, so |φ| = c, a constant. Near any point x ∈ {x1 , . . . , xm }, we can locally write φ = ceiqu . The equality ∇qn φ = 0 implies ∇u = n, hence curl n = 0 for x ∈ {x1 , . . . , xm }. If is simply-connected, then µπ (u0 , q) = 0 if and only if every n ∈ ∗ (u0 , q) is curl -free; and if and only if there exists χ ∈ H 2 () satisfying (8.8). However for a multi-connected domain these conditions are not equivalent (see [H]). (ii) If u0 allows a unit length gradient field extension, namely if there exists a function χ ∈ H 2 () such that |∇χ | = 1 a.e. in , ∇χ = u0 on ∂, then µh (u0 , q) = 0. Our final question is the following: Question 8.7. Find a general condition on and u0 under which the equality µh (u0 , q) = µπ (u0 , q) holds.
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Appendix A. Eigenvalues Continuously Depend on Magnetic Potential The following lemma concerns the continuous dependence on the vector field A of the lowest eigenvalue µ(A) of the magnetic Schrödinger operator −∇A2 in : − ∇A2 φ = µ(A)φ in ,
∇A φ · ν = 0 on ∂.
(A.1)
Lemma A.1. Let be a bounded and smooth domain in R N , N = 2, 3. Assume A j → A in L p (, R N ), where p = 3 if N = 3 and p > 2 if N = 2. Then µ(A j ) → µ(A). Proof. We prove the lemma in the case N = 3. Recall that µ(A) =
inf
φ∈H 1 (,C)
∇A φ 2L 2 ()
φ 2L 2 ()
.
Since A j and A are in L 3 (, R3 ), µ(A j ) and µ(A) are achieved. Note that {µ(A j )} is bounded. In fact, taking φ = 1 as a test function we see that µ(A j ) ≤ A j 2L 2 () /||. Let φ j ∈ H 1 (, C) be an eigenfunction associated with µ(A j ) and φ j L 2 () = 1. By Kato’s inequality and the Sobolev inequality,
∇|φ j | 2L 2 () ≤ ∇A j φ j 2L 2 () = µ(A j ),
φ j 2L 6 () ≤ C() ∇|φ j | 2L 2 () + φ j 2L 2 () ≤ C()[1 + µ(A j )],
(A j − A)φ j L 2 () ≤ A j − A L 3 () φ j L 6 () → 0. So µ(A) ≤ lim inf ∇A φ j 2L 2 () ≤ lim inf { ∇A j φ j L 2 () + (A j − A)φ j L 2 () }2 j→+∞
=
lim inf ∇A j φ j 2L 2 () j→+∞
j→+∞
= lim inf µ(A j ). j→+∞
On the other hand, let φ0 ∈ H 1 (, C) be an eigenfunction associated with µ(A) and
φ0 L 2 () = 1. Then
∇A j φ0 L 2 () ≤ ∇A φ0 L 2 () + (A − A j )φ0 L 2 () ≤ µ(A) + A − A j L 3 () φ0 L 6 () = µ(A) + o(1). So µ(A j ) ≤ ∇A j φ0 2L 2 () = µ(A) + o(1), and hence lim sup j→+∞ µ(A j ) ≤ µ(A).
Appendix B. Minimality of Trivial Critical Points In this section we examine the minimality of a trivial critical point (0, n0 ), where n0 ∈ (u0 ). Lemma B.1. Let n0 ∈ (u0 ) and µ(qn0 ) = κ 2 . (i) If µ(qn0 ) < κ 2 , then for any K > 0, (0, n0 ) is not a local minimizer of E K . (ii) If µ(qn0 ) > κ 2 , then (0, n0 ) is a local minimizer of E K when 0 < K < K c (if K c > 0) and it is a global minimizer when K ≥ K c .
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2 , we choose Proof. If µ(qn0 ) < κ 2 , and if φ0 is an associated eigenfunction of −∇qn 0 a test function (ψs , n) = (sφ0 , n0 ) and find that (0, n0 ) is not a local minimizer. If µ(qn0 ) > κ 2 , for any given (φ, n) ∈ H 1 (, C) × H 1 (, S2 , u0 ), write w = n − n0 . We have E K [φ, n] = K Cπ (u0 ) + K {|∇w|2 − |∇n0 |2 |w|2 }d x (B.1) κ2 2 2 2 ¯ qn0 φ) + q 2 |wφ|2 + |φ|4 }d x. + {|∇qn0 φ| − κ |φ| − 2qw · (φ∇ 2
Using the Sobolev inequality and Kato’s inequality we have
φ L 4 () ≤ C() φ L 2 () + ∇|φ| L 2 () ≤ C() φ L 2 () + ∇qn0 φ L 2 () . Hence
w · (φ∇ ¯ qn0 φ)d x ≤ φ L 4 () ∇qn0 φ L 2 () w L 4 () ≤ C() φ L 2 () + ∇qn0 φ L 2 () ∇qn0 φ L 2 () w L 4 () ≤ 2C() φ 2L 2 () + ∇qn0 φ 2L 2 () w L 4 () .
Let 0 < ε <
µ(qn0 )−κ 2 1+µ(qn0 ) .
If w L 4 () ≤
ε 4qC() ,
we have
¯ qn0 φ)}d x E K [ψ, n] − K Cπ (u0 ) ≥ {|∇qn0 φ|2 − κ 2 |φ|2 − 2qw · (φ∇ ≥ {(1 − ε)|∇qn0 φ|2 − (κ 2 + ε)|φ|2 }d x ≥ [(1 − ε)µ(qn0 ) − κ 2 − ε] |φ|2 d x > 0.
Hence (0, n0 ) is a local minimizer for all K > 0. By the definition of K c the trivial critical point is a global minimizer if K ≥ K c . Lemma B.2. Let n0 ∈ (u0 ) and µ(qn0 ) = κ 2 . Assume 0 < m(n0 , q) < +∞. (i) If 0 < K < 2q 2 κ −2 m(n0 , q), then (0, n0 ) is not a local minimizer of E K . (ii) If 2q 2 κ −2 m(n0 , q) < K < +∞ and if in addition n0 satisfies condition (A), then (0, n0 ) is a local minimizer of E K . Proof. Conclusion (i) follows from the proof of Lemma 6.1, Case 1. In fact if 0 < K < 2q 2 κ −2 m(n0 , q), we can choose (v, φ) such that 2q 2 κ −2 m[n0 , v, φ] > K . Then in any small neighborhood of (0, n0 ) we can take (ψ, nt ) such that I2 [ψ, nt ] > 0 and E K [ψ, nt ] < K Cπ (u0 ), see (6.5). So (0, n0 ) is not a local minimizer. Conclusion (ii) follows from the proof of Lemma 7.2. In fact, if (0, n0 ) is not a local minimizer of E K , then there exists a sequence (ψ j , n j ) → (0, n0 ) in H 1 (, C) × H 1 (, S2 , u0 ) such that E K [ψ j , n j ] < E K [0, n0 ]. Write ψ j = s j φ j and n j = n0 +t j v j . We argue as in the proof of Lemma 7.2 and find that
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2 ¯ j ∇qn0 φ j )d x v · ( φ j
K ≤ 2q 2 κ −2 ≤ 2q 2 κ −2 m(n0 , q) + o(1). 4
φ j L 4 () (|∇v j |2 − |∇n0 |2 |v j |2 )d x Therefore K ≤ 2q 2 κ −2 m(n0 , q).
Appendix C. Two Estimates of K c In this section we give estimates of K c when there exists n0 ∈ (u0 ) satisfying for some 1 ≤ p < +∞ the following (|∇v|2 − |∇n0 |2 |v|2 )d x ≥ c p (n0 ) v 2L p () for all v ∈ T (n0 ), (C.1)
where c p (n0 ) > 0 and T (n0 ) was defined in Definition 7.4. Lemma C.1. Assume that there exists n0 ∈ (u0 ) such that µ(qn0 ) > κ 2 and n0 satisfies (C.1) for p = 2 and c2 (n0 ) > 0. Then Kc ≤
q 2κ 2 . c2 (n0 )[µ(qn0 ) − κ 2 ]
(C.2)
Proof. Let (ψ, n) be a critical point of E K . Write n = n0 + v. Then v ∈ T (n0 ). We use the condition (C.1) and the assumption µ(qn0 ) > κ 2 to find 2 2 2 E K [ψ, n] − K Cπ (u0 ) = K {|∇v| − |∇n0 | |v| }d x + {|∇qn0 ψ|2 − κ 2 |ψ|2 }d x
κ2 ¯ qn0 ψ) d x q 2 |vψ|2 + |ψ|4 − 2qv · (ψ∇ + 2 κ2 2 ) ≥ K c2 (n0 ) v L 2 () + (1 − |∇qn0 ψ|2 d x µ(qn0 ) κ2 q2 q 2 |vψ|2 + |ψ|4 − δ|∇qn0 ψ|2 − |vψ|2 d x, + 2 δ where δ > 0 is an arbitrary number. Here we have used the Holder inequality. From the first equation in (1.5) and the boundary condition of ψ we see that |ψ(x)| ≤ 1 on .
Hence |vψ|2 d x ≤ |v|2 d x. Choosing δ ≤ 1, we have (1 − δ)q 2
v 2L 2 () E K [ψ, n] − K Cπ (u0 ) ≥ K c2 (n0 ) − δ κ2 κ2 + 1− |∇qn0 ψ|2 d x + |ψ|4 d x. −δ µ(qn0 ) 2 κ Let δ = 1 − µ(qn . If K ≥ 0) 2
q 2 (1−δ) c2 (n0 )δ
and if (ψ, n) is a nontrivial critical point of E K , then κ2 |ψ|4 d x > 0, E K [ψ, n] − K Cπ (u0 ) ≥ 2
and hence (ψ, n) is not a minimizer. (C.2) is not useful if µ(qn0 ) =
κ 2.
In this case we have the following conclusion.
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Lemma C.2. Assume that there exists n0 ∈ (u0 ) such that µ(qn0 ) ≥ κ 2 and n0 satisfies (C.1) for p = 4 and c4 (n0 ) > 0. Then 3q 2 ||1/2 . (C.3) c4 (n0 ) Proof. Let (ψ, n) be a critical point of E K . From the first equation of (1.5) we have |∇qn ψ|2 d x = κ 2 (1 − |ψ|2 )|ψ|2 d x ≤ κ 2 |ψ|2 d x ≤ κ 2 ||1/2 ψ 2L 4 () . Kc ≤
Using the maximum principle for elliptic equations we have |ψ(x)| ≤ 1 on . Write n = n0 + v. Then v ∈ T (n0 ). Note that ¯ qn0 ψ)d x = 2q 2 |vψ|2 + 2qv · (ψ∇ ¯ qn ψ). 2qv · (ψ∇ We carry out the computations as in the proof of Lemma C.1. Using the condition µ(qn0 ) = κ 2 and using (C.1) with p = 4, we have E K [ψ, n] − K Cπ (u0 ) = K {|∇v|2 − |∇n0 |2 |v|2 }d x + {|∇qn0 ψ|2 − κ 2 |ψ|2 }d x
2 κ ¯ qn ψ) − q 2 |vψ|2 d x + |ψ|4 − 2qv · (ψ∇ 2 κ2 ≥ K c4 (n0 ) v 2L 4 () + ψ 4L 4 () − 2q v L 4 () ψ L 4 () ∇qn ψ L 2 () 2 −q 2 v 2L 4 () ψ 2L 4 () ≥ K c4 (n0 ) v 2L 4 () +
κ2
ψ 4L 4 () − 2qκ||1/4 v L 4 () ψ 2L 4 () 2
−q 2 v 2L 4 () ψ 2L 4 () ≥ K c4 (n0 ) v 2L 4 () − 2q 2 ||1/2 v 2L 4 () − q 2 v 2L 4 () ||1/2 = (K c4 (n0 ) − 3q 2 ||1/2 ) v 2L 4 () . Hence the minimizers are trivial when K >
3q 2 ||1/2 c4 (n0 ) .
Appendix D. Proof of (1.11) Let n ∈ (u0 ) and v ∈ V(n) ∩ L ∞ (, R3 ). The proof of (1.11) is similar to the proof of (4.6) with a slight modification, because |n + tv| ≡ 1. Define nt as in (6.3) and (6.4) with n0 replaced by n. Then for all small t, F[nt ] = |∇n + t∇v + t 2 ∇wt |2 d x = {|∇n|2 + 2t∇n · ∇v + 2t 2 ∇n · ∇wt + t 2 |∇v|2 }d x + O(t 3 ) (D.1) = {|∇n|2 + 2t|∇n|2 n · v + 2t 2 |∇n|2 n · wt + t 2 |∇v|2 }d x + O(t 3 ) 2 =F[n] + t (|∇v|2 − |∇n|2 |v|2 )d x + O(t 3 ).
Since n is a minimizer of F, (1.11) follows from (D.1) immediately.
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Acknowledgements. We would like to thank the referees for carefully reading the manuscript, and for giving many valuable comments and suggestions. The research work was carried out with partial support from the National Natural Science Foundation of China grant no. 10471125, the Science Foundation of the Ministry of Education of China grant no. 20060269012, the National Basic Research Program of China grant no. 2006CB805902, and Shanghai Pujiang Program grant no. 05PJ14039.
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Communicated by H. Spohn
Commun. Math. Phys. 280, 123–144 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0421-9
Communications in
Mathematical Physics
Asymptotics of Spectral Clusters for a Perturbation of the Hydrogen Atom Alejandro Uribe1, , Carlos Villegas-Blas2, 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.
E-mail: [email protected]
2 Universidad Nacional Autónoma de México, Instituto de Matemáticas,
Unidad Cuernavaca, Mexico. E-mail: [email protected] Received: 18 January 2007 / Accepted: 28 June 2007 Published online: 6 March 2008 – © Springer-Verlag 2008
Abstract: We consider perturbations of the Schrödinger operator of the hydrogen atom, of size O(1+δ ) with δ > 0. We show that, if is restricted to take values along a certain sequence converging to zero, the multiple eigenvalue −1/2 of the unperturbed Hamiltonian breaks into a cluster of eigenvalues disjoint from the rest of the spectrum. Then we obtain a Szegö limit theorem for the spectral shifts in this cluster, as → 0. Specifically, we prove: The weak limit of the normalized spectral measure of the spectral shifts is the push-forward of Liouville measure of the unperturbed energy surface H = −1/2 by the averaged symbol of the perturbation.
Contents 1. 2.
3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . Eigenvalue Clusters . . . . . . . . . . . . . . . . . . . 2.1 Existence of clusters . . . . . . . . . . . . . . . . 2.2 Replacing W by a commuting perturbation of A . 2.3 Passing to the momentum representation . . . . . . Passage to S n . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 3.2 The Fock transformation . . . . . . . . . . . . . . 3.3 Reduction to band asymptotics on S n . . . . . . . . The Problem on S n . . . . . . . . . . . . . . . . . . . . 4.1 -admissible operators on S n . . . . . . . . . . . . 4.2 The singular case . . . . . . . . . . . . . . . . . . 4.3 Moser’s regularization and the symbolic calculation References . . . . . . . . . . . . . . . . . . . . . . . .
A. Uribe supported in part by NSF grant DMS-0401064. C. Villegas-Blas supported in part by PAPIIT-UNAM IN106106-2.
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124 126 126 127 131 132 132 135 136 136 136 140 141 144
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1. Introduction In this article we consider semi-classical Schrödinger operators of the form H = −
1 2 − + Q 2 |x|
(1)
on L 2 (Rn ), n ≥ 2 , where Q h is a selfadjoint -admissible pseudodifferential operator of order zero (for example, multiplication by a real valued bounded smooth function). Moreover, let us assume that the operator norm of Q h is uniformly bounded in . We will work in the regime = O(1+δ )
(2)
for some δ > 0. Recall that the point spectrum of the unperturbed operator (the quantum Hamiltonian for the hydrogen atom, corresponding to = 0) consists of the numbers 2 −1 n − 1 k = − 22 +k , k = 0 , 1, 2, . . . (3) 2 with multiplicity (n − 2 + k)!(n − 1 + 2k) . (n − 1)!k!
dk =
(4)
We will restrict our attention to values of of the form: =
1 n−1 2
+N
,
N = 0, 1, 2, , . . . .
(5)
This relationship between the variables and N will be assumed throughout the article. Notice that, with this choice of , −1/2 = N is always an eigenvalue of the unperturbed operator. As we will show below, the spectrum of H near −1/2 is of the form 1 λ N ,i = − + µ N ,i , 2
i = 1, . . . , d N ,
(6)
where |µ N ,i | = O() due to the assumption that Q is uniformly bounded in . Since the distance between −1/2 and N ±1 is O(), we have that, for small enough, the spectral shifts µ N ,i are well defined. Our main result is a formula for the weak limit of the scaled spectral cluster measures, lim
→0
dN 1 µ N ,i . δ µ− dN
(7)
i=1
The limit is given in terms of the average of the symbol of Q h along the (periodic) trajectories of the Kepler problem on the surface of energy −1/2. This is similar to results for eigenvalue clusters for perturbations of operators with periodic bicharacteristic flow, [3,10]. The present case is more subtle because of the existence of collision orbits. We will show, however, that a quantum analogue of Moser’s regularization of the Kepler
Spectral Clusters for a Perturbation of the Hydrogen Atom
125
problem (that goes back to V. Fock) reduces the current case to the “band asymptotics” case on S n . To state our result, let 1 2 1 ∗ n = −1/2 (8) = (x, p) ∈ T R ; | p| − 2 |x| be the (−1/2) energy surface of the Kepler problem. Note that there are two types of orbits in : (a) orbits with non-zero angular momentum and (b) orbits with zero angular momentum. Orbits of type (a) are periodic orbits with period t = 2π . Orbits of type (b) correspond to one dimensional collision trajectories having the collision with the center of attraction in a finite time and with the magnitude of the momentum vector going to infinity. J. Moser [6] has shown one way to regularize the Kepler problem in order that we can regard all of the orbits of both types (a) and (b) as closed t = 2π periodic orbits. Namely, by considering an extension of the stereographic projection from the momentum space Rn onto the n-sphere S n with the north pole removed (the Moser map) and a time re-parametrization t → s, we can map the orbits of type either (a) or (b) to geodesics on S n either not passing or passing through the north pole respectively (see Sect. 4.3 below). The new time parameter s corresponds to arc-length on S n . The energy surface corresponds to the unit cotangent bundle of the punctured n-sphere. By adding the north pole into the picture, we can make the convention that, in a collision orbit, the particle is reflected by the center of attraction, and retraces its incoming path backwards. We can therefore regard the collision orbits as 2π periodic orbits as well with respect to time t. Moreover, the unit cotangent bundle T1∗ S n of the n-sphere S n gives the regularized flow on through the inverse of the Moser map mentioned above. Let A be the set of points in α ∈ Cn+1 such that the real and imaginary parts of α have unit length and are orthogonal (the set A is actually a subset of the complex null quadric in Cn+1 , see Sect. 3 below). The set A can be identified with T1∗ S n through a map σ which assigns to each α ∈ A the point (α, −α) in T1∗ S n . Thus, a point α ∈ A determines a geodesic on S n and therefore an orbit (x(t, α), p(t, α)) on . We will state our main result in two different ways: one involving an integral over the set A with respect to the probability rotation invariant measure dµ on A and the other one involving an integral on the energy surface with respect to the normalized Liouville measure dλ on . Theorem 1. Let a denote the principal symbol of a selfadjoint pseudodifferential operator Q h and assume that a is a symbol in the class S2n (1) so that the operator norm of Q h is bounded uniformly with respect to (see e. g. [5]). Assume that 0 < = O(1+δ ), δ > 0. Then for every continuous function on the real line, f , as → 0 along the values (5),
2π dN µ
1 1 N ,i = f f a (x(t, α), p(t, α)) dt dµ(α) N →∞ d N 2π 0 A i=1
2π 1 ˜ = f a(φt (x, p)) dt dλ(x, p), (9) 2π 0 lim
where φ˜ t denotes the Hamiltonian flow of the Kepler problem on the surface with the convention mentioned above for the collision orbits.
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We can interpret the right-hand side of (9) as follows. Let X denote the space of orbits of the Kepler problem with energy −1/2. More precisely, let X be the quotient of A by the flow; notice that X is compact. This space inherits a measure, dλ X , which is the push-forward of Liouville measure. The function aˆ : X → R,
2π 1 a(γ ˆ α) = a (x(t, α), p(t, α)) dt 2π 0 (where γα denotes the trajectory corresponding to α ∈ A) can be thought of as a kind of Radon transform of the function a. On the other hand, by the Riesz representation theorem, there exists a measure dm a on the real line such that the right-hand side of (9) is equal to f dm a . In fact dm a is the push-forward by aˆ of the measure dλ X on X . Note that the support of dm a is compact since X is compact. Corollary 2. Suppose that for any set J ⊂ R of measure zero the inverse image aˆ 1 (J ) has dλ X measure zero. Then, for any interval I ⊂ R,
µN , j 1 ∈ I = dλ X (aˆ −1 (I )),
j; lim N →∞ d N that is, (9) holds with f a characteristic function of an interval. We give the simple proof of this corollary at the end of the paper. The proof of Theorem 1 is organized as follows. First we show (in §2) that the spectral clusters indeed exist, if is constrained to take the values above. In the same section we show that the spectral measures (7) can be approximated by the spectral measures of the operator N D2 Q h D−2 N , where N is the projection onto the N th eigenspace µ of − 21 − | x1 | and Dr are dilation operators. (Note that, since N ,i = O(1) and a is bounded, we only have to consider the case when f is a monomial.) We then show that N D2 Q h D−2 N is isospectral to an operator on the sphere S n , which roughly is the compression of an -admissible D O on S n to a space of spherical harmonics, of degree correlated with −1 . This is done by passing to the momentum representation and applying a transformation defined by Fock (see Sect. 3). The proof of isospectrality involves passing through a Berezin-type operator on the quadric A. Finally, in §4 we solve the problem on S n , and show that the answer is equal to the right-hand side of (9) because, at the classical level, the passage from the original configuration space, Rn \ {0}, to S n , corresponds to Moser’s regularization of the Kepler problem. We remark that in the case when n = 3 and the operator Q being multiplication by a bounded function of the position variable x, Theorem 1 is a consequence of Theorem 6 in this paper and a Szegö-type theorem proved by L. Thomas and C. Villegas-Blas (see Theorem 4.2 in [8]). Finally, we note that recently Laurent Charles and V˜u Ngo.c San have also reduced certain spectral perturbation problems to spectral problems of Toeplitz operators, [2], although their setting is different from ours. 2. Eigenvalue Clusters 2.1. Existence of clusters. We begin by showing that, under the previous assumptions, the spectrum of H near −1/2 indeed has the form (6). For r > 0, let us define the following dilation operator Dr : L 2 (Rn ) → L 2 (Rn ) by Dr (x) = r n/2 (r x).
(10)
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127
It can be easily shown that Dr is unitary and that D1/r is the inverse of Dr . It can also be checked that −1 1 r 1 1 D1/r = r −1 − . (11) Dr − − − 2 |x| 2 |x| Taking r = −2 , we obtain the following equality: 1 1 1 2 1 = 2 D−2 − − − − D2 . 2 |x| 2 |x|
(12)
Thus we obtain −
1 1 1 2 1 − + Q h = 2 D−2 − − + 2 D2 Q h D−2 D2 . 2 |x| 2 |x|
(13)
From this equation we conclude that the spectrum of H is equal to the spectrum of the operator 1 1 + 2 D2 Q h D−2 , S := − − 2 |x|
(14)
scaled by a factor of 12 . We think of this operator as having the form S = A + W, where 1 1 and W := 2 D2 Q h D−2 , A := − − 2 |x| that is, we think of S as a perturbation of the non-semiclassical hydrogen atom Hamiltonian. Recall that the discrete spectrum of the operator A is the union of the negative eigen−2 with k = 0, 1, 2, . . .. Moreover, each eigenvalue has values E k := − 21 ( n−1 2 + k) (n−2+k)!(n−1+2k) . Let us study the spectrum of S around the unpermultiplicity dk = (n−1)!k! −2 , which corresponds to the quantum number turbed eigenvalue E N = − 21 n−1 2 +N k = N . Notice that the distance between E N and its closest neighbors is O( N13 ). Moreover, if = O(1+δ ) = O( N 11+δ ), the L 2 norm of the perturbation is O( N 13+δ ). Thus we obtain from perturbation theory that the operator S has a well defined cluster of d N eigenvalues around E N (counting multiplicity). We conclude that the spectrum of the operator H around −1/2 is a cluster of d N eigenvalues.
2.2. Replacing W by a commuting perturbation of A. Let PN be the orthogonal projector associated to the d N -dimensional subspace of L 2 (Rn ) spanned by the eigenfunctions of S = A + W with corresponding eigenvalues in the cluster around E N . Let us denote by N the orthogonal projector of A associated to its eigenvalue E N . Let us denote by γ N ,i , 1 ≤ i ≤ d N , the eigenvalues of S in the cluster around E N . We will write γ N ,i = E N + ν N ,i .
(15)
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Theorem 3. Let {τ N ,i } be the spectrum of the operator N D2 Q h D−2 N . Let m be any positive integer. Then the following equation holds: dN dN 1 ν N ,i m 1 1 = (τ N ,i )m + O( δ ). dN 2 dN N i=1
(16)
i=1
Notice that, by their definitions, indeed |ν N ,i | = O(2 ) and |τ N ,i | = O(1). In order to prove the theorem, we first establish two lemmas (the proof follows [9]). Lemma 4. For some positive constant C, N ⊥ PN ≤ C N 3 W op ≤ C2 N 3 Q h op = O(2 N 3 ), ⊥
PN N ≤ C N W op ≤ C N Q h op = O( N ), 3
2
3
2
3
(17) (18)
where PN⊥ = 1 − PN and ⊥ N = 1 − N . Proof. Let f be an element of the domain of A and {d Fλ } be a resolution of the identity of A. From the spectral theorem we have
|λ − E N |2 d Fλ ( f ) 2 ≥ (A − E N ) f 2 = |λ − E N |2 d Fλ ( f ) 2 , (19) R J where J ⊂ R is the complement of the open interval (E N − C N −3 , E N + C N −3 ) and C is any positive constant such that the cluster of eigenvalues of S around E N is contained in that interval and |E N − E N ±1 | ≥ C N −3 . Thus we obtain (A − E N ) f 2 ≥ (C N −3 )2
J
2 d Fλ ( f ) 2 = (C N −3 )2 ⊥ N f
(20)
(21)
which in turn implies ⊥ N f ≤
1 (A − E N ) f . C N −3
(22)
Taking f ∈ PN L 2 (Rn ) we get the estimate ⊥ N PN op ≤
1 (A − E N )PN op . C N −3
(23)
In order to get an estimate of the right-hand side of the previous inequality, let us notice that (A − E N )PN op ≤ (S − E N )PN op + W PN op .
(24)
Now we claim that (S − E N )PN op ≤ W op .
(25)
Spectral Clusters for a Perturbation of the Hydrogen Atom
129
To show (25), we can take an orthonormal basis {φ1 , . . . , φd N } of the vector space d N αi φi be any element of PN L 2 (Rn ) with norm one. Since PN L 2 (Rn ). Let f = i=1 dN 2 2 (S − E N ) f = i=1 αi νi2 with |νi | ≤ W op , we obtain (25). From (24) and (25) we conclude the first inequality (17) of Lemma (4) by using (23). Here and in the sequel the letter C will denote different constant values. The proof of the second inequality (17) of Lemma (4) is similar. In this case we just need to consider a resolution of the identity of the operator S and integrate over the complement of the open interval J as above in order to show that for f in the domain of S we have (S − E N ) f ≥
C P ⊥ f op N3 N
(26)
which in turns implies (taking f ∈ N L 2 (Rn )) PN⊥ N op ≤ C N 3 (S − E N ) N op .
(27)
Since (A − E N ) N = 0, then we obtain PN⊥ N op ≤ C N 3 W N op . ≤ C N 3 W op .
(28)
Lemma 5. For any positive integer m, the following equation holds: m S − E N m N PN = N Dlog(2 ) Q h D− log(2 ) N PN + N Rm,N , (29) 2 where Rm,N = O(2 N 3 ). Proof. Let us notice that N (S − E N )PN = N (S − E N ) N PN + N (S − E N )⊥ N PN .
(30)
N (S − E N ) N = N (A − E N ) N + N W N = N W N , ⊥ ⊥ ⊥ ⊥ N (S − E N )⊥ N = N A N − N E N N + N W N = N W N ,
(31) (32)
Since
we have N (S − E N )PN = N W N PN + N W⊥ N PN .
(33)
⊥ 2 2 3 From Lemma (4), we obtain W⊥ N PN op ≤ W op N PN op = O( N ), which implies that (29) is certainly true for m = 1. From (33) we find
( N (S − E N )PN )m = ( N W N PN )m + N R,
(34)
where the operator R is a linear combination of terms which are products of operators involving both Wm and ⊥ N PN . Thus the norm of the operator R in the last equation
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⊥ can be estimated from above by K W m op N PN op , where K depends on m. Hence, from Lemma (4), we have for fixed m:
( N (S − E N )PN )m = ( N W N PN )m + (2 )m N O(2 N 3 ).
(35)
Lemma (5) is a consequence of the last equation and the following claims: ( N (S − E N )PN )m = N (S − E N )m PN + N (2 )m O(2 N 3 ), ( N W N PN )m = ( N W N )m PN + N (2 )m O(2 N 3 ).
(36) (37)
We will do the proof by induction. Assume that (36) and (37) hold for m − 1 (the case m = 1 is obvious). Let us prove (37) first. We begin by writing ( N W N PN )m − ( N W N )m PN = N W N PN ( N W N PN )m−1 − ( N W N )m−1 PN .
(38)
By the inductive assumption the operator in brackets is equal to PN⊥ ( N W N )m−1 PN + PN N (2 )m−1 O(2 N 3 ), and therefore (38) is equal to − N W N PN⊥ ( N W N )m−1 PN + N W N PN N (2 )m−1 O(2 N 3 ), which is of the form N T , where T op = (2 )m O(2 N 3 ) (because W op = O(2 ) and because of Lemma (4)). This proves (37). To prove (36), we use the fact (S − E N )m−1 PN = PN (S − E N )m−1 PN , and so ( N (S − E N )PN )m − N (S − E N )m PN = N (S − E N ) PN ( N (S − E N )PN )m−1 − (S − E N )m−1 PN . (39) By the inductive assumption, the operator in brackets is equal to PN N (S − E N )m−1 PN + N (2 )m−1 O(2 N 3 ) − (S − E N )m−1 PN , and therefore (39) is equal to N Z , where m−1 2 m−1 2 3 (S − E ) P + P ( ) O( N ) . (40) Z = (S − E N ) −PN ⊥ N N N N N m−1 , we Since (S − E N )PN op ≤ W op and (S − E N )m−1 PN op ≤ W op obtain from Lemma (4) the following estimate: 2 3 2 m−1 Z op ≤ W m O(2 N 3 ) op O( N ) + W op ( )
= (2 )m O(2 N 3 ), and the proof of the lemma is complete.
Spectral Clusters for a Perturbation of the Hydrogen Atom
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Let us go back to the proof of Theorem 3. Notice that m (S − E N )m PN = N (S − E N )m PN + ⊥ N (S − E N ) PN ,
( N W N )m = ( N W N )m PN + ( N W N )m PN⊥ .
(41)
Therefore, using Lemma (5) we obtain (S − E N )m PN − ( N W N )m m m ⊥ = (2 )m N Rm,N + ⊥ N (S − E N ) PN − ( N W N ) PN .
(42)
Now we claim that each term in the right-hand side of (42) has trace norm (denoted by 1 ) equal to d N (2 )m O(2 N 3 ). We estimate one term at a time: a) From Lemma (5) we have: (2 )m N Rm,N 1 ≤ (2 )m N 1 Rm,N op ≤ d N (2 )m O(2 N 3 ).
(43)
b) Using Lemma (5) again: ( N W N )m PN⊥ 1 ≤ ( N W N )m 1 N PN⊥ op ≤ N 1 W( N W N )m−1 op N PN⊥ op 3 2 m 2 3 ≤ d N W m op O( N ) = d N ( ) O( N ). ⊥ m m c) Writing ⊥ N (S − E N ) PN = ( N PN )((S − E N ) PN )PN , m m ⊥ ⊥ N (S − E N ) PN 1 ≤ N PN op PN 1 (S − E N ) PN op 2 m 2 3 ≤ d N O( N 3 ) W m op = d N ( ) O( N ).
Since the operator (S − E N )m PN − ( N W N )m is selfadjoint, then the absolute value of its trace is bounded by its trace norm which in turn is d N (2 )m O(2 N 3 ). This concludes the proof of Theorem 3. Since the spectrum of H is the spectrum of S scaled by a factor 12 , we can state Theorem (3) in the following way: Theorem 6. Let m be any positive integer. Then the following estimate holds: dN dN µ N ,i m 1 1 1 m . = (τ N ,i ) + O dN dN Nδ i=1
(44)
i=1
2.3. Passing to the momentum representation. By Theorem 6, our problem is equivalent to finding the weak asymptotics of the spectral measure of the operator R N = N D2 Q h D−2 N
(45)
restricted to the image of N . We will transform this problem into a problem on the n-sphere, using the stereographic projection, but first one has to pass to the momentum representation using the Fourier transform. Let
1 F( f )( p) = e−i p·x f (x) d x (2π )n/2
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A. Uribe, C. Villegas-Blas
denote the ordinary Fourier transform extended to be a unitary operator on L 2 (Rn ). Let us denote by ˆ N := F ◦ N ◦ F −1
(46)
the projection onto the E N eigenspace of the unperturbed hydrogen atom Hamiltonian in the momentum representation (and with = 1). We also let Rˆ N := F ◦ R N ◦ F −1 ,
(47)
ˆ N. ˆ N F D2 Q h D−2 F −1 Rˆ N =
(48)
which we write as
The following is an easy exercise: Lemma 7. (1) F := F ◦ Dh is the -Fourier transform:
1 −1 Fh ( f )( p) = e−i p·x f (x) d x. n/2 (2π ) (2) For any r > 0, F ◦ Dr = D1/r ◦ F. Using this lemma we can write Rˆ N in the form: ˆ N, ˆ N D−1 Qˆ h D Rˆ N =
(49)
Qˆ h = F Q h F−1 .
(50)
where
Since F is a Fourier integral operator (in the semiclassical sense), Qˆ is again an -admissible pseudodifferential operator. To proceed further with the analysis of the spectrum of Rˆ N , we need to recall how ˆ N can be described in terms of coherent states on S n . the projector 3. Passage to Sn 3.1. Preliminaries. Consider the n-dimensional sphere, S n = {(ω1 , . . . , ωn+1 ) ∈ Rn+1 ;
n+1
ω2j = 1}.
j=1
Let us denote by S n the Laplace-Beltrami operator on S n , and by Lk its k th eigenspace (restrictions to the sphere of the harmonic polynomials of degree k in (n + 1) variables). The cotangent bundle of S n with the zero section removed can be identified with the complex quadric Q n \ {0} with the origin removed: 2 = 0}. Q n := {α ∈ Cn+1 |α12 + · · · + αn+1
Spectral Clusters for a Perturbation of the Hydrogen Atom
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The identification occurs through the map ρ : Q n \ {0} → T ∗ S n − {0}, α ρ(α) = , −α . |α|
(51)
Under this identification, the unit cotangent bundle of S n corresponds to the set A := {α ∈ Cn+1 | |α| = |α| = 1 and α · α = 0}.
(52)
In this model of the unit cotangent bundle, geodesic flow is given by complex multiplication by eit . We will come back to the geodesic flow and the Kepler flow in the last section. For each α ∈ A and k a non-negative integer number, let us define the function α ,k (ω) = a(k)(α • ω)k ∈ Lk ,
(53)
with a(k) a normalization constant chosen such that α ,k (ω) L 2 (S n ) = 1. The constants a(k) are given by (see Eq. 2.11 in [8]) (n−1)/2 k 1 2 −1/2 + O(k a (k) = ) . (54) π 2π The functions α ,k are the “coherent states” on the sphere, see op. cit. and [9]. For k large, α ,k concentrates along the geodesic through α. (Notice that for α’s on a geodesic, the α ,k are equal up to a phase factor.) But the properties that we will use are intimately related to the fact that these functions provide a resolution of the identity for Lk . (These properties, in a general setting, were used by F. A. Berezin in the 1970’s in a general theory of quantization.) Specifically and as a consequence of Schur’s lemma we have: Proposition 8. The Schwartz kernel of the orthogonal projector PkS : L 2 (S n ) → Lk is the following function on S n × S n :
α ,k (ω) α ,k (ω ) dµ(α), PkS (ω, ω ) = dk α ∈A
(55)
is the dimension of Lk and dµ(α) is the S O(n +1) invariant where dk = (2k+n−1)(k+n−2)! k!(n−1)! probability measure on the space A. Another way of stating this proposition (using quantum mechanical language) is:
S Pk = dk |α ,k α ,k | dµ(α). (56) α ∈A As a corollary, for any bounded operator F : L 2 (S n ) → L 2 (S n ),
Tr PkS F PkS = dk F(α ,k ) , α ,k dµ(α). α ∈A
(57)
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For each integer k > 0, we introduce the following transform: Uk : L 2 (S n ) −→ L 2 (A), f → α → f , α ,k .
(58)
By Proposition 8, for every f ∈ Lk ,
f = dk
α ∈A
f , α ,k α ,k dµ(α)
(59)
(this is the so-called reproducing property of the coherent states), and therefore Uk is zero on L⊥ k and injective on Lk . Let us denote by Wk := Image (Uk ) ⊂ C ∞ (A) the image of Uk . Proposition 9. Let F : L 2 (S n ) → L 2 (S n ) be a bounded operator, and define the function on A × A, K F (α, β) := F(β ,k ) , α ,k .
(60)
Let F be the operator on L 2 (A, dµ) with Schwartz kernel K F . Then the following diagram commutes: Uk
L 2 (S n ) → L 2 (A) ↓ F
PkS F PkS ↓ Uk L 2 (S n ) →
(61)
L 2 (A)
k That is, F maps Wk into itself, and the restriction F |W Wk is similar to the operator PkS F PkS via the transform (58).
S S Proof. If f ∈ L⊥ k , Uk ( f ) = 0 = (Pk F Pk )( f ), assume that f ∈ Lk . Then
F Uk ( f )(α) = =
A
F(β ,k ) , α ,k f , β ,k dµ(β) f , β ,k F(β ,k ) dµ(β) , α ,k .
A
Applying F to both sides of (59), we see that the first entry in the last inner product is precisely F( f ), and therefore F Uk ( f )(α) = F( f ) , α ,k = PkS F( f ) , α ,k = Uk PkS F( f )(α).
Spectral Clusters for a Perturbation of the Hydrogen Atom
135
3.2. The Fock transformation. The Fock transformation maps the eigenspaces of A = − 21 Rn − | x1 | unitarily to the spaces Lk of spherical harmonics. We now recall its definition. Consider the inverse of the stereographic projection S : Rn → Son , from the momentum space Rn onto the punctured sphere Son (the n-sphere with the north pole removed). In coordinates, S is given by the equations: 2 pi , i = 1, . . . , n, | p|2 + 1 | p|2 − 1 = . | p|2 + 1
ωi = ωn+1
(62)
The physical motivation for considering the stereographic projection onto the momentum space is that the non-zero angular momentum Kepler orbits in momentum space are circles and those orbits in turn are mapped to great circles on S n through the stereographic projection S. Let K : L 2 (S n ) → L 2 (Rn ) be the operator given by n/2 2 2 n ∀ f ∈ L (S ) K ( f )( p) = f (S( p)). (63) | p|2 + 1 n/2 is equal to the square root of the Jacobian of the stereoThe factor 2/(| p|2 + 1) graphic projection, and therefore K is unitary. We will also need the multiplication operator, J , acting in momentum space, J : L 2 (Rn ) → L 2 (Rn ), ˆ p) = J ()(
2 ˆ p). ( | p|2 + 1
(64)
What we call the Fock transformation is described by the following theorem: Theorem 10. [1] Given a non-negative integer k, denote by Ek the eigenspace of 1 A = − 21 Rn − | x1 | with eigenvalue E k = − , k = 0, 1, 2, . . .. Then the 2 2(k+ n−1 2 ) operator F(Ek ) → L 2 (S n ) ˆ → K −1 J −1/2 D −1 (), ˆ r
(65)
k
where rk = k + (n − 1)/2, is a unitary isomorphism onto Lk . A puzzling aspect of this transformation is that it is defined “piece-wise”, that is, the k dependence in the definition (65) is non-trivial. No global definition of the Fock transformation is known to us. However, for our purposes we will only need the following corollary: ˆ α ,k ∈ L 2 (Rn ) denote Corollary 11 [8]. Given a positive integer k, for each α ∈ A let the function corresponding to the coherent state α ,k under the Fock transformation: ˆ α ,k = Drk J 1/2 K (α ,k ). ˆ α ,k }α ∈A is a resolution of the identity for F(Ek ) . In particular, the orthogonal Then { projector onto F(Ek ) is given by
ˆ k = dk ˆ α ,k ˆ α ,k | dµ(α). | (66) α ∈A
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ˆ α ,k α ∈ A play exactly the same role for the spaces F(Ek ) Therefore, the vectors as the α ,k play for the Lk . In particular, a proposition completely analogous to Proposition 9 holds for these vectors. 3.3. Reduction to band asymptotics on S n . Recall from the end of §2 that we are interested in the asymptotics of the spectral measure of the operator Rˆ N given by Eq. (49). Proposition 12. Let F := K −1 J 1/2 Qˆ J 1/2 K ,
(67)
considered as an operator on L 2 (S n ). Then, for each N , the operator Rˆ N restricted to ˆ N is isospectral to the operator the image of PNS F : L N → L N
(68)
n−1 . 2
(69)
with −1 = N +
Proof. Let us compute the kernel of Rˆ N , in the sense of Proposition 9: ˆ α ,N , ˆ K N (α, β) = Rˆ N β ,N = D−1 Qˆ h D Dr N J 1/2 K (α ,N ) , Dr N J 1/2 K (β ,N ),
(70)
ˆ α ,N . Substituting by the values (69), the dilawhere we have used the definition of the tion operators crucially and mysteriously cancel each other out. Using next the unitarity of K and the self-adjointness of J 1/2 , one finds: K N (α, β) = K −1 J 1/2 Qˆ J 1/2 K (α ,N ) , β ,N = F(α ,N ) , β ,N . Therefore Rˆ N and PNS F have the same kernel function K N (α, β) and therefore, by Proposition 9 are conjugate to the same operator on A, and therefore are conjugate to each other. The point of this proposition is that F is an -admissible operator on S0n . Considering this operator as an operator on L 2 (S n ), we are seeking the spectral asymptotics of its compression to the spherical harmonics of degree N . Were PNS to be an admissible operator on S n , the conclusion would be almost immediate. However, we have to deal with the incompleteness of the stereographic projection at the north pole which corresponds to the behavior of the perturbation Wh at infinity in momentum space. 4. The Problem on Sn 4.1. -admissible operators on S n . Throughout this section we consider a self-adjoint -admissible operator on S n of order zero, V, with principal symbol σ ∈ C ∞ (T ∗ S n ). Recall that A is identified with the unit cotangent bundle of S n .
Spectral Clusters for a Perturbation of the Hydrogen Atom
Proposition 13. Let c be a constant and fix α ∈ A. Then, as N → ∞,
1 1 V 1 (α ,N ) , α ,N = , σ ds + O N +c 2π γ (α ) N
137
(71)
where γ (α) ⊂ A is the geodesic through α and the integration is with respect to arc length s. Proof. This is a direct application of the method of stationary phase. We sketch the calculation. Using the natural √ action of SO(n + 1) on A, we can assume without loss of generality that α = e1 + −1e2 , where e1 , . . . , en+1 is the standard basis of Rn+1 . We then introduce the coordinate system (whose domain is open and dense), which is the inverse of the parametrization (θ, z 1 , . . . , z n−1 ) → (r cos(θ ), r sin(θ ), z 1 , . . . , z n−1 ), where r=
1 − |z|2 .
The Schwartz kernel of V in these coordinates is of the form
1 −1 V(θ, z, θ , z ) = ei [ pθ (θ−θ )+ pz (z−z )] a(θ, z; pθ , pz ; ) dp. n (2π ) In these coordinates, α ,N (θ, z) = a(N ) r N ei N θ . Moreover, in these coordinates the volume form on S n is simply dθ dz 1 · · · dz n−1 (exercise left to the willing reader). Thus the matrix coefficient V 1 (α ,N ) , α ,N is equal to N +c
2 a(N ) ei N f (θ,θ ,z,z , p) eic[ pθ (θ−θ )+ pz (z−z )] a(θ, z; pθ , pz ; ) dp dθ dθ dz dz , (2π )n (72) where the phase is f (θ, θ , z, z , p) = (−1 + pθ )(θ − θ ) + pz (z − z ) − i log(r (z)r (z )).
(73)
It is easy to check that the critical points of f are precisely the points of the form (θ, θ , z, z , pθ , pz ) = (θ, θ, 0, 0, 1, 0), and that the circle they form is a non-degenerate critical manifold (the Hessian of f in the normal directions is non–degenerate). Therefore, by writing the integral in (72) as an integral with respect to dθ and then a 3n − 1 dimensional integral with respect to dp dθ dz dz and applying the stationary phase method to the last integral we obtain that the matrix coefficient we are interested in admits an asymptotic expansion as N → ∞, in decreasing powers of N , and with leading term
2π 1 a(θ, 0, 1, 0) dθ, 2π 0 where a is the leading term in the semiclassical expansion of a(θ, z; pθ , pz ; ), we have used (54) and the fact that the absolute value of the determinant of the Hessian at the critical point is 2n−1 . Notice that the constant c plays no role in the leading term.
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In view of the previous result, it is natural to introduce the manifold O = A/S 1 of oriented closed geodesics. This manifold has a natural measure, dm, which is the pushforward of dµ under the natural projection P : A → O. Given a function σ : A → R, we define σ : O → R, (74) γ → γ σ ds. Note that for a given smooth function g on the real line,
g ◦ σ dm = g σ ds dµ, O A γ (α )
(75)
where γ (α) = P(α). We will denote the integral appearing in the right-hand side of (75) by A g ◦ σ dµ. Corollary 14. Let c be a constant. Then, as N → ∞,
1 1 S S Tr PN ◦ V 1 ◦ PN = σ dm + O . N +c dN N O
(76)
We now state and prove the main result of this section. The ideas in the proof are a combination of those used by Dozias in [4] and the averaging method for pseudodifferential operators, [10,3]. Theorem 15. The weak limit of the spectral measures of the operators PNS ◦ V
1 N +c
◦ PNS
as N → ∞ is the push-forward σ ∗ (dm). In fact, for any smooth function g on the real line,
1 Tr g PNS ◦ V 1 ◦ PNS = g ◦ σ dm + O(1/N ). (77) N +c dN O Proof. Fix a number 0 < δ << 1, and pick ϕ a smooth function on the line such that 0 if λ ≤ δ/2 ϕ(λ) = √ λ if λ ≥ δ. Using this function, define the operator n−1 . (78) B := ϕ 2 S n + 2 (n − 1)2 /4 − 2 Claim 1. B is an -admissible pseudodifferential operator on S n of order one, whose spectrum consists of the sequence n−1 , k = 0, 1, . . . , λk := ϕ (k + (n − 1)/2)2 − 2 and the eigenspace corresponding to the k th eigenvalue is precisely the k th order spherical harmonics, Lk . In particular, by the way we chose ϕ, √ δ λk = k if k > . (79)
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139
The first part of the claim follows from the functional calculus [7], which in fact shows that the principal symbol of B is the function b(ω, ξ ) = ϕ(|ξ |2 ). In particular, √ b(ω, ξ ) = |ξ | if |ξ | ≥ δ. (80) The eigenvalues of B are easily computed if we recall that the eigenvalue of S n associated with the k th order spherical harmonics is k(k + n − 1). Let us now define
2π 1 −1 −1 V˜ := χ (2 S n ) e−i t B V ei t B dt, (81) 2π 0 where χ (λ) is a compactly supported smooth function identically equal to one in a small neighborhood of λ = 1. By Egorov’s theorem, V is again an -admissible operator whose symbol we will partially analyze in a moment. Claim 2. For all integers N 1, [PNS , V˜
1 N +c
] = 0.
(82)
Here is a detailed proof of this claim. Let us introduce the notation √ k N := min{k ; k > δ(N + c)}. This number is relevant because if k ≥ k N the eigenvalue of (N + c)B 1 associated to N +c Lk is precisely k. (The ϕ term in λk for such k is just the square root of its argument.) Note that, since we took δ < 1, N ≥ k N if N is large enough. Now take φ ∈ L N and decompose V
1 N +c
(φ) =
∞
φk , where φk ∈ Lk .
(83)
k=0
Then a calculation shows that V˜
1 1 (φ) = φ N + N +c 2π
0≤k
2π
e
−it[(N +c)λk −N ]
dt χ ((N + c)−2 S n )(φk ).
0
However, by the way we picked the cut-off χ , it can be checked that k < k N ⇒ χ ((N + c)−2 S n )(φk ) = 0, and therefore V˜
1 N +c
(φ) = φ N = PNS (V˜
This proves the second claim. Actually, Eq. (84) also implies:
1 N +c
(φ)).
(84)
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Claim 3. For all integers N 1, PNS V˜
1 N +c
PNS = PNS V
PNS .
1 N +c
(85)
We can now finish the proof of Theorem 15. By Claims 2 and 3, for any smooth function g on the real line and N 1, PNS . g PNS V 1 PNS = PNS g V˜ 1 N +c
N +c
We now apply Corollary 14 to the operator on the right-hand side, which is possible since g(V˜) is an -admissible pseudodifferential operator whose symbol is g composed with the symbol of V˜. We claim that the symbol of V˜, restricted to the unit tangent bundle of S n , is precisely the average of the symbol of V. This follows from: (a) Egorov’s theorem, Eq. (79), and the fact that the Hamilton flow of the Hamiltonian |ξ | is geodesic flow parametrized by arc length, and (b) that the symbol of the cutoff χ (2 S n ) is identically equal to one in a neighborhood of the unit circle bundle. This finishes the proof.
4.2. The singular case. Theorem 15 does not directly apply to the operator F, because said operator might have a singularity along the cotangent space to S n at the north pole. We will now show how to circumvent this problem. Let a = a(x, p, ) be the symbol of the pseudodifferential operator Q , i.e. Q = O p(a). Moreover, let us assume that a is a classical symbol: a∼
∞
j a j (x, p).
(86)
j=0
Thus we have F = K −1 J 1/2 F O p (a − a0 )F −1 J 1/2 K + K −1 J 1/2 F O p (a0 )F −1 J 1/2 K (87) with O p (a − a0 ) = O() = O(1/N ). By following the transformations involved in the operator K −1 J 1/2 F O p (a0 )F −1 J 1/2 K at the symbolic level (see the next section and Lemma 17) we obtain: PNS F PNS = PNS O p (σ0 )PNS + O(1/N )
(88)
with σ0 (ω , ξ ) = (1 − ωn+1 ) a(− y(ω, ξ ) , p(ω)) and the operator O p (σ0 ) denoting the -admissible pseudodifferential operator of order zero with symbol the continuous extension of σ0 (also denoted by σ0 ) to all of T ∗ S n . Here (ω , ξ ) → ( p(ω) , y(ω, ξ )) is the canonical transformation T ∗ S0n → T ∗ Rn induced by the stereographic projection (see Eq. 62), and ω = (ω1 , . . . , ωn+1 ) ∈ S n . (Note a is bounded at infinity, so the limit of σ0 at any covector at the north pole is zero.) Let δ > 0, then there exist a smooth function f δ defined on T ∗ S n that approximates σ0 uniformly in a neighborhood of the unit cotangent bundle of the sphere: | f δ − σ0 | < δ on A. It is easy to see (for example using anti-Wick symbols) that then PNS O p ( f δ )PNS − PNS O p (σ0 )PNS < δ + O(1/N ).
Spectral Clusters for a Perturbation of the Hydrogen Atom
Thus we have: m m 1 1 Tr PNS F PNS = Tr PNS O p ( f δ )PNS + O(1/N ) + O(δ). dN dN Applying Theorem 15 we obtain: m
1 m S S Tr PN F PN = f δ dµ + O(δ). lim N →∞ d N A
141
(89)
(90)
By letting δ → 0 we conclude the following: Corollary 16. For any integer m ≥ 0 and with −1 = N + n−1 2 ,
m 1 Tr PNS Fh PNS = σm lim 0 dµ. N →∞ d N A 4.3. Moser’s regularization and the symbolic calculation. All that remains to finish the proof of the main theorem is to identify σ0 , to show that it can be extended continuously to T ∗ S n , and to express the integrals A σ 0 dµ, in terms of the symbol of the original perturbation, Q , of Eq. (1). Recall that F was constructed in three steps: (1) The perturbation, Q was conjugated by the -Fourier transform. The resulting operator was called Qˆ h , see Eq. (50). (2) The operator Qˆ h was multiplied on both sides by the operator, J 1/2 , of multiplication by 2(| p|2 + 1)−1 . (3) Finally, the resulting operator was pulled-back to the sphere using the stereographic projection. It is a simple matter to follow each of these steps symbolically, given that the F is an -Fourier integral operator associated to the canonical transformation (x , p) → (− p , x) : Lemma 17. Let a ∈ C ∞ (T ∗ R ) denote the principal symbol of Q . Then n
σ0 (ω , ξ ) = (1 − ωn+1 ) a(− y(ω, ξ ) , p(ω)),
(91)
where (ω , ξ ) → ( p(ω) , y(ω, ξ )) is the canonical transformation T ∗ S0n → T ∗ Rn induced by the stereographic projection, and ω = (ω1 , . . . , ωn+1 ). The factor (1−ωn+1 ) is exactly the pull-back of 2(| p|2 +1)−1 under the stereographic projection. This lemma implies in particular that σ can be extended continuously to T ∗ S n by defining it as zero at the cotangent space of the north pole (note that a is bounded by assumption and (1 − ωn+1 ) is zero at the north pole). To express the integrals A (σ )m dµ in terms of the original data, we need to recall Moser’s regularization of the Kepler problem. Let us consider the n-dimensional Kepler problem in T ∗ (Rn \{0}) = (Rn \ {0})×Rn given by the Hamiltonian H=
1 | p|2 − 2 |x|
(92)
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with x ∈ (Rn − {0}) the position of the particle and p ∈ Rn its momentum. Let us consider the stereographic projection S : Rn → Son from the momentum space n R onto the punctured sphere Son (the n-sphere with the north pole removed), given by Eqs. (62) where the vector ω = (ω1 , . . . , ωn+1 ) is in S n . Let us denote by ξ a vector in T ∗ S n at the point ω. The Moser map M : T ∗ (Rn ) → T ∗ (Son ) is obtained by requiring that S ∗ (ξ • dω) = y • d p, with y ≡ −x. Namely, the Moser map sends the point ( p, y) to the point (ω, ξ ) under (62), and the following equations hold: | p|2 + 1 yi − ( y • p) pi , i = 1, . . . , n, 2 = y • p.
ξi = ξn+1
(93)
The inverse map M−1 is determined by the equations ωi , i = 1, . . . , n, 1 − ωn+1 yi = (1 − ωn+1 )ξi + ξn+1 ωi . i = 1, . . . , n.
pi =
(94)
The Moser map is a canonical transformation (i.e. M is a diffeomorphism with the property M∗ (dξ ∧ dω) = d y ∧ d p). The physical relevance of the Moser map appears when we restrict ourselves to the surface of constant energy E = −1/2, or, equivalently, to the unit cotangent bundle of S n where the geodesic flow takes place. In this situation, the Moser map transforms the geodesic flow on the punctured sphere Son onto the Hamiltonian flow of the Kepler problem with a time re-parametrization s → t given by the equation d d = |x| . ds dt
(95)
More specifically, let us consider α ∈ A and regard it as an element of the unit cotangent bundle of S n under the identification given by the map ρ defined by (51). Let us consider in turn the geodesic spanned by α with parameter the arc length s and given by ω(s, α) = cos(s)α + sin(s)α, ξ (s, α) = − sin(s)α + cos(s)α.
(96)
Then, from the expression for the inverse Moser map (94) we obtain: α + (− cos(s) + αn+1 ) α, x(s, α) = (sin(s) − αn+1 ) cos(s) α + sin(s) α , p(s, α) = 1 − cos(s)αn+1 − sin(s)αn+1
(97)
where α = (α1 , . . . , αn ). From (97) we have |x| = |1 − cos(s)αn+1 − sin(s)αn+1 | which implies |x| = 0 iff | p| = ∞ and in such a case ddsx = 0. Additionally, one can check that the curves described by (97) have energy E = | p|2 /2 − 1/|x| = −1/2. Note that, from (97), the curves (x(s), d xds(s) ) are periodic functions with respect to the parameter s, which in particular implies that they are well-defined after the physical collision with the central mass. Thus the regularization extends the description of the motion after a collision: The regularized motion has the particle reflected back from the center of attraction after a collision.
Spectral Clusters for a Perturbation of the Hydrogen Atom
143
One can check by a straight computation that the orbits (x(s(t),α), p(s(t),α)) obtained from (97) and (95) are actually the Kepler orbits, i.e. they satisfy Hamilton’s equations dx ∂H = (x(t), p(t)), dt ∂p ∂H dp =− (x(t), p(t)). dt ∂x
(98)
Therefore, Lemma 17 holds under the identification (ω , ξ ) = ρ(α). Since the push-forward of the measure dµ on the space A is the Liouville measure, dλ, on , under the composition of Moser’s map and ρ, we obtain (9) for a monomial. Finally, because of the factor (1 − ωn+1 ) = 2(| p|2 + 1)−1 = |x| appearing in (91), the average of the symbol, a, appearing in (9) is the average with respect to the physical time t. This ends the proof of Theorem 1. Proof of Corollary 2. Let χ be the characteristic function of the interval I , and let χ±δ , δ > 0, be two families of continuous functions such that χ−δ ≤ χ ≤ χ+δ pointwise, and
lim
δ→0
χ±δ dm a =
χ dm a .
(99)
It is not hard to construct these families since the hypothesis of the corollary amounts to saying that dm a is absolutely continuous with respect to Lebesgue measure. Then, for each δ > 0, dN dN
µ µ µN , j 1 1 1 N, j N, j ≤ ∈I ≤ . χ−δ
j; χ+δ dN dN dN j=1
j=1
Therefore, by Theorem 1
µN , j 1 χ−δ dm a ≤ lim inf ∈ I ≤ χ+δ dm a ,
j; N →∞ d N
and similarly for lim sup. Letting δ → 0 and using (99) yields the result.
Acknowledgements. We wish to thank Lawrence E. Thomas for suggesting to us the investigation of the existence and asymptotics of eigenvalue clusters for the hydrogen atom, and for useful discussions. We are very grateful to Thierry Paul for observing that Theorem 1 holds with = O(1+δ ) for any δ > 0, as stated. (In the first version of this paper we had taken = O(2 ).) In fact he has pointed out that the result should hold with = O(), provided the implied constant is small enough so that the eigenvalue clusters are welldefined. A proof of this statement however will require replacing our methods in §2.2 by more sophisticated techniques.
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References 1. Bander, M., Itzykson, C.: Group Theory and the Hydrogen Atom. Rev. Mod. Phy. 38(3), 330–345 (1966) 2. Charles, L., San, V.N.: Spectral asymptotics via the semiclassical Birkhoff normal form. Preprint, May 2006, math.SP/0605096 3. Colin de Verdière, M.: Sur le spectre des oprateurs elliptiques bicaractristiques toutes priodiques. Comment. Math. Helv. 54(3), 508–522 (1979) 4. Dozias, S.: Clustering for the Spectrum of h-pseudodifferential Operators with Periodic Flow on an Energy Surface. J. Funct. Anal. 145, 296–311 (1997) 5. Martinez, A.: An introduction to semiclassical and microlocal analysis. Berlin-Heidelberg-New York: Springer, 2002 6. Moser, J.: Regularization of Kepler’s problem and the averaging method on a manifold. Comm. Pure Appl. Math. 23, 609–636 (1970) 7. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. I. New York: Academic Press, 1975 8. Thomas, L., Villegas-Blas, C.: Asymptotics of Rydberg States for the Hydrogen Atom. Commun. Math. Phys. 187, 623–645 (1997) 9. Uribe, A.: Band invariants with non-smooth potentials. J. Funct. Anal. 74(1), 1–9 (1987) 10. Weinstein, A.: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44(4), 883–892 (1977) Communicated by B. Simon
Commun. Math. Phys. 280, 145–205 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0427-3
Communications in
Mathematical Physics
A Centre-Stable Manifold for the Focussing Cubic NLS in R1+3 Marius Beceanu Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA. E-mail: [email protected] Received: 26 January 2007 / Accepted: 16 July 2007 Published online: 15 March 2008 – © Springer-Verlag 2008
Abstract: Consider the focussing cubic nonlinear Schrödinger equation in R3 : iψt + ψ = −|ψ|2 ψ.
(0.1)
It admits special solutions of the form eitα φ, where φ ∈ S(R3 ) is a positive (φ > 0) solution of − φ + αφ = φ 3 .
(0.2)
The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the 8-dimensional manifold that consists of functions of the form ei(v·+) φ(· − y, α). We prove that any solution starting sufficiently close to a standing wave in the = W 1,2 (R3 ) ∩ |x|−1 L 2 (R3 ) norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that N is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones [BatJon]. The proof is based on the modulation method introduced by Soffer and Weinstein for the L 2 -subcritical case and adapted by Schlag to the L 2 -supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in R3 for the nonselfadjoint Schrödinger operator obtained by linearizing (0.1) around a standing wave solution. All results in this paper depend on the standard spectral assumption that the Hamiltonian φ(·, α)2 + 2φ(·, α)2 − α (0.3) H= −φ(·, α)2 − − 2φ(·, α)2 + α has no embedded eigenvalues in the interior of its essential spectrum (−∞, −α) ∪ (α, ∞). This work is part of the author’s Ph. D. thesis at the University of Chicago.
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1. Introduction 1.1. Main result. For a parameter path π = (vk , Dk , α, ) such that π˙ ∞ +tπ˙ (t)1 < ∞, define the nonuniformly moving soliton W (π(t)) by W (π(t))(x) = eiθ(t,x) φ(x − y(t), α(t)), t θ (t, x) = v(t)x − (|v(s)|2 − α(s)) ds + γ (t), 0 t y(t) = 2 v(s) ds + D(t), 0 ∞ 2s v(s)v(∞) ˙ ds. γ (t) = (t) − (v(t) − v(∞))D(∞) +
(1.1)
t
Theorem 1.1 (Main result). There exists a local codimension-one Lipschitz manifold N in = H 1 ∩ |x|−1 L 2 , containing the 8-dimensional manifold of standing waves, such that Eq. (0.1) has a global solution ψ if we start with initial data ψ(0) on the manifold N . Furthermore, the solution depends Lipschitz continuously on the initial data and decomposes into a moving soliton and a dispersive term: ψ = W (π(t)) + R(t), with π˙ ∞ + tπ˙ (t)1 ≤ Cψ(0) − W (π(0))
(1.2)
R L ∞ L 2 ∩L 2 L 6 ∩t−1/2 L 2 L 6+∞ ≤ Cψ(0) − W (π(0)) .
(1.3)
and t
x
t
x
t
x
The dispersive term scatters: R(t) = eit f 0 + o L 2 (1), for some f 0 ∈ L 2 . Moreover, for a solution ψ of initial data ψ(0) ∈ N , one has that ψ(t) ∈ for all t and ψ(t) ∈ N for sufficiently small t. Finally, N is a centre-stable manifold for this equation in the sense of Bates, Jones [BatJon].
1.2. Background. Consider the focussing nonlinear cubic Schrödinger Eq. (0.1). It admits a particular class of solutions of the form eitα φ, where φ = φ(·, α) ∈ S, φ > 0, are solutions of (0.2). These solutions exist for all time and are periodic. Positive, smooth solutions φ to (0.2) are called ground states and solutions to (0.1) obtained from eitα φ by Galilean coordinate changes, phase changes, or scaling are called standing waves. All these transformations are symmetries of the equation, subsumed by the following formula: G(t)( f (x, t)) = ei(+vx−t|v| ) f (α 1/2 x − 2tv − D, αt). 2
(1.4)
A natural question is whether standing waves are stable under small perturbations. From a physical point of view, the NLS equation in R3 with cubic nonlinearity and the focussing sign (0.2) describes, to a first approximation, the self-focussing of optical beams due to the nonlinear increase of the refraction index. As such, the equation appeared for the first time in the physical literature in 1965, in [Kel].
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1.3. Known stability results in other cases. Concerning the general NLS problem, more results have been obtained in the defocussing case or for L 2 -subcritical and L 2 -critical power nonlinearities in the focussing case. A few negative results have been established as well. Cazenave and Lions [CazLio] and Weinstein [Wei1,Wei2] used the method of modulation to prove the orbital stability of standing waves in the focussing L 2 -subcritical case. Asymptotic stability results have been first obtained by Soffer, Weinstein [SofWei1, SofWei2], then by Pillet, Wayne, [PilWay], Buslaev, Perelman [BusPer1,BusPer2], [BusPer3], Cuccagna [Cuc], Rodnianski, Schlag, Soffer, [RoScSo1,RoScSo2], Schlag [Sch], and Krieger, Schlag [KriSch1]. Grillakis, Shatah, and Strauss [GrShSt1,GrShSt2] developed a general theory of stability of solitary waves for Hamiltonian evolution equations, which, when applied to the Schrödinger equation, shows the dichotomy between the L 2 -subcritical and critical or supercritical cases. If the nonlinearity is L 2 -critical or supercritical and focussing, negative energy x−1 H 1 initial data leads to solutions that blow up in finite time, due to the virial identity (see Glassey [Gla]). For weakening the condition on initial data and for a survey of this topic see [SulSul] and [Caz]. Berestycki, Cazenave [BerCaz] showed that blow-up can occur for arbitrarily small perturbations of ground states. Recent results concerning the blowup of the critical and supercritical equation include Merle, Raphael [MerRap] and Krieger, Schlag [KriSch2]. In 1993, Merle [Mer] showed in the L 2 -critical case the existence of a minimal blowup mass for H 1 solutions, equal to that of the standing wave solution, such that any solution with smaller mass has global existence and dispersive behavior. A comparable result was achieved in 2006 by Kenig, Merle [KenMer] for the H˙ 1 -critical equation. A similar statement is possible concerning the cubic nonlinearity studied here (which is H˙ 1/2 -critical). The present paper does not address this question, but is a first step in that direction.
1.4. The theory of Bates and Jones. In 1989, Bates, Jones [BatJon] proved that the space of solutions decomposes into an unstable and a centre-stable manifold, for a large class of semilinear equations. As far as it concerns this paper, their result is the following: consider a Banach space X and the semilinear equation u t = Au + f (u),
(1.5)
under the assumptions H1 A : X → X is a closed, densely defined linear operator that generates a C0 group. H2 The spectrum of A decomposes into σ (A) = σs (A) ∪ σc (A) ∪ σu (A) situated in the left half-plane, on the imaginary axis, and in the right half-plane respectively and σs (A) and σu (A) are bounded. H3 The nonlinearity f is locally Lipschitz, f (0) = 0, and ∀ > 0 there exists a neighborhood of 0 on which f has Lipschitz constant . Furthermore, let X u , X c , and X s be the A-invariant subspaces corresponding to σu , σc , and respectively σs , and let S c (t) be the evolution generated by A on X c . Bates and Jones further assume that C1-2 dim X u , dim X s < ∞. C3 ∀ρ > 0 ∃M > 0 such that S c (t) ≤ Meρ|t| .
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Let be the flow associated to the nonlinear equation. We call N ⊂ U t-invariant if, whenever (s)v ∈ U for s ∈ [0, t], (s)v ∈ N for s ∈ [0, t]. Let W u be the set of u for which (t)u ∈ U for all t < 0 and decays exponentially as t → −∞. Also, consider the natural direct sum projection π cs on X c ⊕ X s . Definition 1. A centre-stable manifold N ⊂ U is a Lipschitz manifold with the property that N is t-invariant relative to U , π cs (N ) contains a neighborhood of 0 in X c ⊕ X s , and N ∩ W u = {0}. The result of [BatJon] is then Theorem 1.2. Under assumptions H1-H3 and C1-C3, there exists an open neighborhood U of 0 such that W u is a Lipschitz manifold which is tangent to X u at 0 and there exists a centre-stable manifold W cs ⊂ U which is tangent to X cs . Gesztesy, Jones, Latushkin, Stanislavova [GeJoLaSt] proved in 2000 that the abstract Theorem 1.2 applies to the semilinear Schrödinger equation. More precisely, their main result was that Theorem 1.3. Given the equation iu t − u − f (x, |u|2 )u − βu = 0
(1.6)
and assuming that H1 f is C 3 and all derivatives are bounded on U × R3 , where U is a neighborhood of 0; H2 f (x, 0) → 0 exponentially as x → ∞; H3 β < 0; H4 u 0 is an exponentially decaying stationary solution to the equation (standing wave), then there exists a neighborhood of u 0 that decomposes into a centre-stable and an unstable manifold. While providing an interesting answer to the problem, the main drawback of this approach is that one cannot infer the global in time behavior of the solutions on the centre-stable manifold. Indeed, once a solution leaves the specified neighborhood of 0, one cannot say anything more about it, not even concerning its existence. 1.5. The result of Schlag. In [Sch], Schlag extended the method of modulation to the L 2 -supercritical case and proved that in the neighborhood of each ground state of Eq. (0.1) there exists a codimension-one Lipschitz submanifold of H 1 (R3 ) ∩ W 1,1 (R3 ) such that initial data on the submanifold lead to global solutions. The method used in [Sch] and applied in the current paper with some enhancements is the following: write the solution to Eq. (0.1) as = W + R, where W = eiθ φ(x − y, α) is a nonlinearly moving standing wave, determined by the parameter path π = (, D, α, v) as in (2.4), while R is an error term that needs to be controlled. One obtains the nonlinear R , with the nonselfadjoint Hamiltonian Schrödinger equation (2.8) in Z = R W (π )2 + 2|W (π )|2 HπZ = (1.7) −W (π )2 − − 2|W (π )|2
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and localized quadratic and nonlocalized cubic nonlinear terms on the right-hand side. The spectrum of the Hamiltonian determines the properties of the equation. Following an appropriate transformation, it becomes real-valued and takes the form H=
+ 2φ(·, α)2 − α −φ(·, α)2
φ(·, α)2 . − − 2φ(·, α)2 + α
(1.8)
For the rest of this paper, we make the following standard spectral assumption: Assumption 1. H has no embedded eigenvalues in the interior of its essential spectrum for any α > 0. Such assumptions are routinely made in the proof of asymptotic stability results, as for example in [BusPer1,Cuc,RoScSo2]. Even though Assumption 1 is expected to be true, it has not been proved to hold. Nevertheless, the assumption is most likely true generically, in the sense that embedded eigenvalues should, as a rule, vanish under perturbations by turning into resonances in the upper-half plane (by Fermi’s rule), see [CucPelVou]. Thus, even if Assumption 1 fails in some particular case, one should be able to reinstate it by means of perturbations. Under this assumption, we completely describe the spectrum of H following [Sch], with the proof delayed until the next section. It consists of an absolutely continuous part (−∞, −α] ∪ [α, ∞) supported on the real axis, a generalized eigenspace at 0 with 4 eigenvectors and 4 generalized eigenvectors. To each disconnected component of the spectrum there corresponds a Riesz projection (namely Pc , Pr oot , and Pim = P+ + P− respectively) given by a Cauchy integral. In the course of the proof, Schlag used the method of modulation. The necessity for it arises because the projection of the solution onto the generalized eigenspace of the Hamiltonian at zero does not disperse or satisfy Strichartz estimates. Physically, this corresponds to the fact that a nonzero displacement of the solution relative to the soliton W does not go away in time and that even a small relative velocity can lead to a large displacement in finite time. Since the right-hand side terms of the equation keep introducing small perturbations, one constantly needs to adjust the soliton path in order to eliminate them from the generalized zero eigenspace. One of the main contributions of Schlag [Sch] was adapting the modulation method to the L 2 -supercritical case. In this case, the main difficulty lies in dealing with the unstable mode of the equation, which corresponds to the imaginary eigenvalue iσ of H. To address this, [Sch] showed that the solution of the linearized equation does not grow exponentially in time if and only if the initial data Z (0) is on a certain codimensionone manifold, tangent to K er (P+ (0)). This choice eliminates the effect of the unstable eigenvalue. In this manner, Schlag [Sch] proved global existence and decay properties for the linearized equation with H 1 ∩ W 1,1 initial data on a codimension-one manifold. A fixed point argument allowed him to go back to the nonlinear equation. The main result of [Sch] states the following: Theorem 1.4. Impose the spectral Assumption 1 and fix α0 > 0. Then there exist a small δ > 0 and a Lipschitz manifold N of size δ inside W 1,2 ∩ W 1,1 , of codimension one, so that φ(·, α0 ) ∈ N , with the following property: for any choice of initial data ψ(0) ∈ N ,
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the NLS equation (0.1) has a global H 1 solution ψ(t) for t ≥ 0. Moreover, ψ(t) = W (t, ·) + R(t),
(1.9)
where W as in (2.4) is governed by a path π(t) of parameters so that |π(t)−(0, 0, 0, α0 )|≤ δ and which converges to some terminal vector π(∞) such that supt≥0 |π(t) − π(∞)| ≤ Cδ. Finally, R(t) H 1 ≤ Cδ, R(t)∞ ≤ Cδt −3/2
(1.10)
for all t > 0, and there is scattering: R(t) = eit f 0 + o L 2 (1) as t → ∞
(1.11)
for some f 0 ∈ L 2 (R3 ). The main problem here is that the H 1 ∩ W 1,1 space is not preserved by the flow. Starting with a function ψ(0) ∈ N of finite H 1 ∩ W 1,1 norm at t = 0 as initial data, there is no guarantee that ψ(t) will still have finite W 1,1 norm for any t = 0. Therefore, the question whether the manifold N is invariant under the Hamiltonian flow does not make sense in this context. One can replace the W 1,2 (R3 ) ∩ W 1,1 (R3 ) norm with the stronger invariant 5/2+ = H 5/2+ ∩ |x|−5/2− L 2 norm, but this weakens the result considerably. Another example of the same phenomenon, in the case of the wave equation, is given by Krieger, Schlag [KriSch3]. For a more general survey of this topic, see [Sch2]. 1.6. Current paper. The result of this paper represents an improvement over that of Schlag [Sch], in that it holds in the H 1/2 ∩ L 4/3− norm, which is strictly weaker than the invariant = H 1 ∩ |x|−1 L 2 space, a somewhat natural choice for Eq. (0.1). In this space, the question concerning the manifold’s invariance under the flow becomes meaningful and it turns out that the answer is affirmative. This paper follows the method of proof of [Sch] (namely the method of modulation, adapted to the L 2 -supercritical case), but some important details differ. The choice of H 1/2 for initial data is sharp and corresponds to the fact that the equation is H˙ 1/2 -critical. It is possible only due to Keel-Tao endpoint Strichartz estimates for the linearized Hamiltonian. The endpoint corresponds exactly to using half a derivative to bound the nonlocalized cubic right-hand side term of the linearized equation. The L 4/3− condition on the initial data leads to a t −1 decay in L 2 in time of the solution that compensates for the possibility of linear growth in the modulation equations. This problem arises because of the generalized eigenspace of the Hamiltonian at 0. This L 2 in time decay bound is not sharp. We expect that, due to the oscillatory nature of the integrand, further improvements are achievable by using conditionally convergent integrals, instead of absolutely converging ones as in the current paper. 1.7. Linear estimates. The first dispersive estimates concerning NLS with nonselfadjoint Hamiltonians are present in [BusPer1]. More recently, Erdogan, Schlag [ErdSch] considered Hamiltonians of the form H = H0 + V , where −U −W − + µ 0 . (1.12) , V = H0 = W U 0 −µ
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They made the following assumptions: that −σ3 V is a positive matrix, that L − = − + µ + U + W ≥ 0, that U and W have decay, and the spectral Assump polynomial 1 0 tion 1. Here σ3 denotes the Pauli matrix . 0 −1 Under these conditions, Erdogan, Schlag [ErdSch] proved the L 2 boundedness of the evolution eit H for |V | ≤ Cx−1− . In [Sch], Schlag proved the L 1 → L ∞ dispersive estimate and Strichartz nonendpoint estimates, for x−3− potential decay and under the further assumption that the edges of the spectrum are neither eigenvalues nor resonances. Erdogan, Schlag [ErdSch] obtained corresponding results for nonselfadjoint Hamiltonians in the presence of a resonance or eigenvalues at the edges of the essential spectrum, if the potential decays like x−10− . Yajima [Yaj] proved independenty the same result, assuming less decay on V . This paper establishes the following Keel-Tao endpoint Strichartz estimates for a nonselfadjoint Hamiltonian of the form (1.12): Corollary 1.5. Suppose that H = H0 + V , where − + µ 0 −U H0 = , V = 0 −µ W
−W , U
(1.13)
that −σ3 V is a positive matrix, that L − = − + µ + U + W ≥ 0, that |V | ≤ Cx−7/2− , that the spectral Assumption 1 holds, and that the edges of the spectrum ±µ are neither eigenvalues nor resonances. Then the evolution eit H Pc satisfies the following Strichartz-type estimates: eit H Pc f L qt L r x −is H Pc F(s) ds e 2 it H −is H∗ e e Pc F ds q r s
Lt Lx
≤ C f 2 , ≤ CF ≤ CF ≤ CF
q
,
q˜
,
q˜
,
L t L rx
L t L rx˜ L t L rx˜
for any sharply admissible (q, r ) (that is, such that 2 ≤ q, r ≤ ∞, (q, ˜ r˜ ). The same estimates hold after swapping H and H∗ .
(1.14)
3 3 1 + = ) and q 2r 4
Note that Corollary 1.5 is not an immediate consequence of [KeeTao], because ∗ eit H e−is H = ei(t−s)H . The exact rate of decay of V does not matter for the purpose of this paper, since we deal only with exponentially decaying potentials. However, it is important to have the endpoint Strichartz estimate for two reasons. Firstly, by linearizing the equation one obtains small localized linear terms on the right-hand side and it is useful to be able to bound their contribution using the L 2t → L 2t endpoint Strichartz estimate. Secondly, as mentioned before, the sharp estimate allows one to use exactly half a derivative in handling the nonlocalized cubic terms on the right-hand side. The difficulty in the proof lies in the fact that H is not selfadjoint, so the usual L 1 → L ∞ dispersive estimate does not imply the endpoint estimate Corollary 1.5. Therefore, we use the following strengthened version of it:
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Proposition 1.6. Under the assumptions of Corollary 1.5, ∗
eit H Pc e−is H Pc∗ 1→∞ ≤ C|t − s|−3/2 .
(1.15)
The proof of this statement is a generalization of the one given in [Sch] for the usual dispersive estimate. The argument uses the spectral representation of the evolution from that paper and the finite Born sum expansion of the resolvent for both the operator and its adjoint. Once established, the estimate (1.15), together with the L 2 theory of [ErdSch], makes possible to apply the methods of [KeeTao], leading to Corollary 1.5. Now we return to the nonlinear problem. Without loss of generality, take any standing wave W (0) and transform it, by means of a symmetry transformation G0 , into a positive ground state φ(·, α0 ) of Eq. (0.1). Then let P+ (0) and Pr oot (0) be the Riesz projections onto the eigenspace corresponding to the eigenvalue iσ of positive imaginary part and respectively onto the generalized zero eigenspace of the linearized Hamiltonian H (1.8) at time 0. Furthermore, let f + (0), f˜+ (0), and η F (0), ξ F (0) be the normalized eigenvectors of H and H∗ at iσ and the generalized eigenvectors of H and H∗ at 0, respectively. All are exponentially decaying Schwartz functions and P+ (0) = ·, f˜+ (0) f + (0), Pr oot = ·, ξ F (0)η F (0). (1.16) F
In the sequel we use the notation L p∩q = L p ∩ L q and L p+q = L p + L q . Following these preparations, we state a more technical result from which the main theorem follows almost immediately. For simplicity, we first state it in the case when the initial data is in the neighborhood of a positive ground state and its projection on the generalized zero eigenspace vanishes. Theorem 1.7. Assume that W (0) = φ(·, α0 ) is a positive ground state of Eq. (0.2). For 1 ≤ q < 4/3, let Sδ be given by R0 1/2 3 q 3 =0 . Sδ = R0 ∈ H (R ) ∩ L (R ) | R0 H 1/2 ∩L q < δ, (P+ (0)+ Pr oot (0)) R0 (1.17) Then, for some small δ, there exists a map F : Sδ → H 1/2 (R3 ) ∩ L q (R3 ), whose range is spanned by a Schwartz function, given by F1 + F(R0 ) = h(R0 ) f (0) = (1.18) F2 such that (1) F(R0 ) ≤ CR0 2H 1/2 ∩L q , (2) F(R01 ) − F(R02 ) ≤ CδR01 − R02 H 1/2 ∩L q , and, for every R0 ∈ S(δ), the Eq. (0.1) having (R0 )(0) = W (0)+ R0 +F1 (R0 ) as initial data admits a global solution (R0 ). Moreover, the solution (R0 ) has the following properties: (1) (R0 ) depends Lipschitz continuously on R0 , (R01 ) − (R02 )t1/2− L 2 xL 6+∞ ≤ CR01 − R02 2 . t
x
(1.19)
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(2) There exists a parameter path π with π(0) = (0, 0, 0, α0 ) and t π˙ (t)1 +π˙ 1∩∞ < ∞ such that (R0 ) stays close to W (π ) for all time t ≥ 0: (R0 ) = W (π ) + R, where R L ∞ H 1/2 ∩L 2 W 1/2,6 ∩t−1/2− L 2 L 6+∞ ≤ δ t
x
t
x
t
x
(1.20)
and one has scattering: for some f 0 in L 2 , R(t) = eit f 0 + o L 2 (1).
(1.21)
Note that the endpoint Strichartz estimate makes it possible to attain the critical H 1/2 norm, so Theorem 1.7 is optimal in this respect. The map R0 → (R0 )(0) = W (0) + R0 + F1 (R0 ) takes Sδ to a codimension-nine submanifold N9 of H 1/2 ∩ L q . Indeed, the map is Lipschitz bicontinuous for sufficiently small δ and Sδ is an open set in a codimension-nine linear space. Since we want to extend this result to more general standing waves instead of just ground states, we conjugate everything by means of symmetry transformations. Also note that the codimension-nine manifold N9 provided by Theorem 1.7 becomes, after applying symmetry transformations, a codimension-one submanifold. These observations are summarized in the following corollary: Corollary 1.8. Consider any standing wave W (0). Under the same assumptions as in Theorem 1.7, there exists a codimension-one Lipschitz manifold N L q ∩H 1/2 in L q ∩ H 1/2 (R3 ), 1 ≤ q < 4/3, given locally by N L q ∩H 1/2 = g(N9 ), (1.22) g
whose tangent space at W (0) is K er (P+ (0)), such that for initial data (0) on the manifold N L q ∩H 1/2 the equation has a global dispersive solution , with the same properties as in Theorem 1.7, but with respect to some more general standing wave g(W (0)), such that |g| ≤ CR0 L q ∩H 1/2 , instead of simply W (0). A straightforward consequence is that the same result holds in the strictly stronger norm of 1 = H 1 ∩ |x|−1 L 2 , which has the advantage of being locally invariant under the flow. Furthermore, in this topology one can identify N as the centre-stable manifold of [BatJon], from the previous discussion. This leads to the main Theorem 1.1, stated on the first page. We remark that 1 can be replaced in this statement with any invariant s = H s ∩ −s |x| L 2 space, for s > 3/4, this being the minimal requirement so that s ⊂ L q ∩ H 1/2 for some q < 4/3. 2. Proof of the Nonlinear Results 2.1. Formulation of the problem. We aim to prove that there exists a codimension-one submanifold of H 1/2 ∩ L 4/3− on which the focussing cubic nonlinear Schrödinger equation (0.1) has global solutions. Throughout this section we employ the Keel-Tao endpoint Strichartz estimates of Sect. 3. In the L 2 -subcritical case, Cazenave, Lions [CazLio] and Weinstein [Wei2] proved that stability occurs for any solution that starts in a sufficiently small neighborhood of the standing wave manifold. However, the presence of an unstable eigenstate of the
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linearization precludes one from achieving such a result in the L 2 -supercritical case and Berestycki, Lions [BerLio] proved that arbitrarily small perturbations of the ground state may lead to blowup in finite time. The best that one can hope for is the existence of a codimension-one manifold on which the evolution does not lead to blowup. This is indeed the result proved by Schlag [Sch] and improved here. Let φ = φ(·, α) be the radially symmetric ground state (meaning φ > 0) of the semilinear Schrödinger operator corresponding to energy α > 0, that is a solution of (0.2). The existence of such solutions to Eq. (0.2) was proved by Berestycki and Lions in [BerLio], who further showed that they are infinitely differentiable and exponentially decaying. Uniqueness was established by Coffman [Cof] for (0.2) and Kwong [Kwo] and McLeod, Serrin [McLSer] for more general nonlinearities. In the particular case of the cubic nonlinearity, Eq. (0.2) and its solutions have the scaling invariance φ(x, α) = α 1/2 φ(α 1/2 x, 1). Note that eitα φ(x, α) is a 1-parameter family of periodic solutions for Eq. (0.1). Starting from it, one can obtain more solutions by taking advantage of the symmetries of Eq. (0.1). Applying the following family of transformations G(t)( f (x, t)) = ei(+vx−t|v| ) α 1/2 f (α 1/2 x − 2tv − D, αt) 2
to
eit φ(·, 1),
(2.1)
the result is a wider 8-parameter family of solutions to (0.1),
G(t)(eit φ(x, 1)) = ei(+vx−t|v|
2 +αt)
α 1/2 φ(α 1/2 x − 2tv − D, 1)
(2.2)
or, after reparametrizing, G(t)(eit φ(x, 1)) = ei(+vx−t|v|
2 +αt)
φ(x − 2tv − D, α),
(2.3)
which we call standing waves. Here G is composed of a Galilean coordinate change, with six degrees of freedom corresponding to v and D, a phase change represented by , and a rescaling embodied by α. Henceforth we call such G as in (2.1) symmetry transformations, since they correspond to the symmetries of Eq. (0.1). In the sequel we consider the pairs made of a function and its conjugate instead of just For example, by a standing wave we will also mean the pair the itfunction alone. G(t)(e φ(x, 1)) . There is an obvious correspondence between the pair and its first G(t)(eit φ(x, 1)) component, as long as the components are conjugate to one another. All the column two-vectors that appear in this paper will have this property, related to the fact that the 01 -invariant. vector form of Eq. (0.1) is 10 The question arises whether standing waves are stable under small perturbations. We seek perturbed solutions of the form ψ = W (π ) + R with small R, where W (π(t))(x) = eiθ(t,x) φ(x − y(t), α(t)), t θ (t, x) = v(t)x − (|v(s)|2 − α(s)) ds + γ (t), 0 t y(t) = 2 v(s) ds + D(t).
(2.4)
0
W (π ) represents a moving soliton governed by the parameter path π = (, α, Di , vi ). We look for solutions ψ that remain close to the 8-dimensional manifold of solitons for all positive times t > 0, hence to a moving soliton like W (π ).
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2.2. Setting up the contraction scheme. Assume that all the parameters describing W (π ), namely γ , α, Di , and vi , have limits as t → ∞, denoted γ (∞), etc. It is more convenient in the sequel to consider an alternative to γ , namely a new parameter such that ˙ = γ˙ + v(2tv(∞) ˙ + D(∞)), more precisely (t) = γ (t) + (v(t) − v(∞))D(∞) − Henceforth, we assume that
∞ t
(2.5) 2s v(s)v(∞) ˙ ds.
π˙ ∞ + tπ˙ (t)1 < ∞,
(2.6)
where π(t) = (vk (t), Dk (t), α(t), (t)). Note that γ can be recovered from π and that γ˙ 1 < ∞ under our assumption; also γ (∞) = (∞). For F ∈ {vk , Dk , α, }, denote by ξ FZ the following family of vectors: eiθ(x,t) xk φ(x − y(t), α(t)) , = −iθ(x,t) xk φ(x − y(t), α(t)) e ieiθ(x,t) ∂k φ(x − y(t), α(t)) Z ξvk (t) = , −ie−iθ(x,t) ∂k φ(x − y(t), α(t)) ieiθ(x,t) ∂α φ(x − y(t), α(t)) ξZ (t) = , −ie−iθ(x,t) ∂α φ(x − y(t), α(t)) iθ(x,t) φ(x − y(t), α(t)) e ξαZ (t) = −iθ(x,t) . φ(x − y(t), α(t)) e
ξ DZk (t)
(2.7)
−i 0 ξ FZ . Also let = = 0 i Z Their immediate importance is that ηF (t) span the tangent space of the 8-dimensional W (π(t)) , for each individual t ≥ 0, and ξ FZ (t) form a standing wave manifold at W (π(t)) dual basis with respect to the usual dot product. From another perspective, note that if v = D = γ = 0 and W is a positive ground state of the equation, then η FZ span the generalized eigenspace of the linearized Hamiltonian H (1.8) at zero and ξ FZ fulfill the same function for its adjoint H∗ . However, the property of being an eigenvector is not preserved under symmetry transformations, so this characterization is no longer true when W is a more general standing wave instead of a positive ground state. The following lemma exhibits the equation satisfied by the error term R:
η FZ
−iσ3 ξ FZ
Lemma 2.1. = W (π ) + R is a solution of Eq. (0.1) if and only if Z =
z1 = R is z2 = R
a solution to i∂t Z + HπZ Z = −i π˙ ∂π
W (π ) W (π )
+ N Z (Z , π ),
(2.8)
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where
W (π )2 + 2|W (π )|2 , HπZ = −W (π )2 − − 2|W (π )|2 W (π ) ieiθ xφ(x − y, α) ieiθ φ(x − y, α) ω + γ ˙ ω∂ ˙ π =v˙ ω W (π ) −ie−iθ xφ(x − y, α) −ie−iθ φ(x − y, α) iθ e ∂ φ(x − y, α) −eiθ ∇φ(x − y, α) + D˙ ω + α˙ ω −iθ α e ∂α φ(x − y, α) −e−iθ ∇φ(x − y, α)
Z = α˙ ω ηZ − γ˙ ω ηαZ − D˙ kω ηvZk + v˙kω η D k
(2.9)
(2.10)
k
(with θ = θ (x, t), α = α(t), y = y(t)) and −2|z 1 |2 W (π ) − W (π )z 12 − |z 1 |2 z 1 Z N (Z , π ) = . 2|z 2 |2 W (π ) + W (π )z 22 + |z 2 |2 z 2
(2.11)
We wrote π∂ ˙ π W (π ) in a more general form that becomes convenient when linearizing the equation. In the linearized equation, the family of vectors η F depends on one path and the coefficients v˙ etc. depend on another. Proof. By direct computation. = W (π ) + R satisfies Eq. (0.1), i∂t (W + R) + (W + R) = −|W + R|2 (W + R) = −|W |2 W − 2|W |2 R − W 2 R − 2W |R|2 −2W R 2 −|R|2 R, (2.12) while W (π ) fulfills the identity i∂t W (π ) + W (π ) = −|W (π )|2 W (π ) + i π˙ ∂π W (π ).
(2.13)
Subtracting the two relations, one has that i∂t R + R = −2|W |2 R − W 2 R − 2W |R|2 − 2W R 2 − |R|2 R − i π˙ ∂π W (π ). (2.14) Joining this equation with its conjugate, one obtains exactly (2.8).
Consider the following linearized version of Eq. (2.8), in which we partly replace π and Z with the auxiliary functions π 0 and Z 0 : W (π 0 ) Z i Z t + Hπ 0 Z = −i π˙ ∂π + N Z (Z 0 , π 0 ). (2.15) W (π 0 ) We choose W (π ) such that at each time t it satisfies the orthogonality condition Z (t), ξ FZ (t) = 0,
(2.16)
which leads to a system of modulation equations for the path π . If the standing wave W (π ) is a positive ground state of the equation, this simply means that Pr oot Z (t) = 0, that is the projection of Z (t) onto the generalized eigenspace at 0 of the Hamiltonian H (1.8) is 0. Otherwise, the condition takes a more complicated meaning.
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If we try to approximate HπZ 0 by a constant Hamiltonian in order to solve Eq. (2.8), the problem is that the potential moves with nonzero velocity along the path described by π . Thus, we need to change the reference frame to one that moves with the same speed as HπZ 0 . However, since HπZ 0 does not move with constant speed, we have (in order to avoid gradient terms in the equation) to choose a reference frame moving at the constant speed that best approximates the speed of HπZ 0 . The same considerations apply to the phase of HπZ 0 . Therefore, we need to determine the uniform movement path that best approximates π 0 . Define the following limit values associated to any path π : ∞ ∞ = γ (∞) − (v 2 (∞) − v 2 (s) − α(∞) + α(s)) ds, 0
v∞ = v(∞), D∞ = D(∞) − 2
∞
(v(∞) − v(s)) ds,
(2.17)
0
α∞ = α(∞), φ∞ = φ(·, α∞ ). Given some parameter path π such that π˙ 1∩∞ + t π˙ 1 < ∞, one can distinguish a symmetry transformation (as in (2.1)) Gπ associated to π , Gπ (t)( f (x)) = e−iθ∞ (t,x+y∞ (t)) f (x + y∞ (t)),
(2.18)
where y∞ (t) = 2tv∞ + D∞ , θ∞ (t, x) = ∞ + v∞ x − t (|v∞ |2 − α∞ ). Also consider the corresponding transformation for column two-vectors, Gπ (t)u 1 u1 = . gπ (t) u2 Gπ (t)u 2
(2.19)
(2.20)
Note that Gπ and gπ only depend on the terminal values D∞ , v∞ , ∞ , and α∞ . For future reference, let ρ∞ (t, x) = θ (t, x + y∞ ) − θ∞ (t, x + y∞ ).
(2.21)
Gπ W (π ) is close to a constant ground state φ(·, α) and it turns out that the best uniformly moving approximation to π is provided by the constant path (∞ , α∞ , D∞ , v∞ ). Therefore, we apply the transformation gπ 0 to the linearized Eq. (2.15). In this context it is natural to introduce the families of functions −ieiρ∞ φ(x + y∞ − y, α) = gπ 0 (t)ηαZ (t), ηα (t) = ie−iρ∞ φ(x + y∞ − y, α) iρ e ∞ ∂α φ(x + y∞ − y, α) = gπ 0 (t)ηZ (t), η (t) = −iρ∞ e ∂α φ(x + y∞ − y, α) iρ (2.22) e ∞ ∂ φ(x + y∞ − y, α) = gπ 0 (t)ηvZk (t), ηvk (t) = −iρ∞ k e ∂k φ(x + y∞ − y, α) −i xk eiρ∞ φ(x + y∞ − y, α) Z (t) − (y ) g (t)η Z (t), = gπ 0 (t)η D η Dk (t) = ∞ k π0 α k i xk e−iρ∞ φ(x + y∞ − y, α) (where ρ∞ = ρ∞ (x, t), y∞ = y∞ (t), y = y(t), α = α(t)) and
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ξ F (t) = iσ3 η F (t) =
for F ∈ {α, , vk } gπ 0 (t)ξ FZ (t), gπ 0 (t)ξ DZk (t) − (y∞ )k gπ 0 (t)ξαZ (t), for F = Dk .
(2.23)
We made a change in the definitions of η Dk and ξ Dk so that these functions would be uniformly bounded in time, instead of linearly increasing as they would have been if we had just applied the symmetry transformation gπ 0 . The two families η F and ξ F span the generalized zero eigenspaces of H and H∗ respectively, if W (π(0 t)) is a standing wave. Furthermore,η F span the tangent space of the eight-dimensional standing wave W (π 0 (t)) manifold at gπ 0 (t) at each individual t ≥ 0. W (π 0 (t)) Lemma 2.2. (Z , π ) is a solution of Eq. (2.15) if and only if U = gπ 0 Z and π satisfy W (π 0 ) iUt + Hπ 0 U = −i π˙ ∂π gπ 0 (2.24) + N (U 0 , π 0 ) + Vπ 0 U, W (π 0 ) where U 0 = gπ 0 Z 0 and we used the notations 2 (x) − α 2 (x) + 2φ∞ φ∞ ∞ , (2.25) Hπ 0 (x) = 2 (x) 2 (x) + α −φ∞ − − 2φ∞ ∞ 2 (x)−φ 2 (x + y − y, α)) 2 (x)−e2iρ∞ φ 2 (x + y − y, α) 2(φ∞ φ∞ ∞ ∞ Vπ 0 = 2 (x)+e−2iρ∞ φ 2 (x + y − y, α) −2(φ 2 (x)−φ 2 (x + y − y, α)) , −φ∞ ∞ ∞ ∞ (2.26) ω∂ ˙ π gπ
W (π ) = α˙ ω η − γ˙ ω ηα − ( D˙ kω ηvk + v˙kω η Dk ), W (π )
(2.27)
k
and
−2|u 1 |2 eiρ∞ φ(x + y∞ − y) − u 21 e−iρ∞ φ(x + y∞ − y) − |u 1 |2 u 1 . N (U, π ) = 2|u 2 |2 e−iρ∞ φ(x + y∞ − y) + u 22 eiρ∞ φ(x + y∞ − y) + |u 2 |2 u 2 (2.28)
u1 is a C2 -valued function and π , π 0 , ω are paths. We wrote the term Here U = u2 W (π 0 ) in a more general form, in order to exhibit its dependence on two π∂ ˙ π gπ 0 W (π 0 ) paths, π and π 0 . We also recall the notations ρ∞ = ρ∞ (x, t), y∞ = y∞ (t), y = y(t), α = α(t), φ∞ = φ(·, α∞ ). Henceforth, ρ∞ , y, y∞ , φ, and φ∞ will refer to quantities derived from π 0 . Let H = Hπ 0 + Vπ 0 . H is the Hamiltonian of Eq. (2.24), but we split it into a constant part Hπ 0 and an error term Vπ 0 . Proof. Firstly, we compute the following commutators:
[∂t , gπ ] = −(|v∞ |2 + α∞ )iσ3 + 2v∞ ∇ gπ
(2.29)
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and
159
[σ3 , gπ ] = −|v∞ |2 σ3 − 2v∞ i∇ gπ .
(2.30)
U and Z 0 = g−1 U 0 into (2.15), we then have Plugging Z = g−1 π0 π0 g−1 (iUt + σ3 U ) + [i∂t + σ3 , g−1 ]U π0 π0 2 2 W W (π 0 ) 2|W | −1 g U − i π˙ ∂π U 0 , π 0 ). + N Z (g−1 = π0 −W 2 −2|W |2 π 0 W (π 0 ) Therefore, applying gπ 0 to this equation and taking into account the fact that 2|W |2 W 2 gπ 0 g−1 −W 2 −2|W |2 π 0 e2iρ∞ φ 2 (x + y∞ − y, α) 2φ 2 (x + y∞ − y, α) = −e−2iρ∞ φ 2 (x + y∞ − y, α) −2φ 2 (x + y∞ − y, α)
(2.31)
(2.32)
and gπ 0 ([i∂t + σ3 , g−1 ]) = −[i∂t + σ3 , gπ 0 ]g−1 = −α∞ σ3 , π0 π0
(2.33)
we obtain iUt + σ3 U − α∞ σ3 U e2iρ∞ φ 2 (x + y∞ − y, α) 2φ 2 (x + y∞ − y, α) U = −e−2iρ∞ φ 2 (x + y∞ − y, α) −2φ 2 (x + y∞ − y, α) −i π˙ ∂π gπ 0 W (π 0 ) + gπ 0 N Z (g−π 0 U 0 , π 0 ).
(2.34)
However, by definition 2φ 2 (x + y∞ − y, α) e2iρ∞ φ 2 (x + y∞ − y, α) U = HU. σ3 U −α∞ σ3 U − −e−2iρ∞ φ 2 (x + y∞ − y, α) −2φ 2 (x + y∞ − y, α) (2.35) Note that
π˙ ∂π gπ 0
W (π 0 ) Z = αg ˙ π 0 ηZ − γ˙ gπ 0 ηαZ − ( D˙ k gπ 0 ηvZk + v˙k gπ 0 η D ). (2.36) k 0 W (π ) k
Here the important fact is that Z v˙k gπ 0 η D + γ˙ gπ 0 ηαZ k k
−ieiρ∞ φ(x + y∞ − y, α) −i(x + y∞ )eiρ∞ φ(x + y∞ − y, α) + γ ˙ i(x + y∞ )e−iρ∞ φ(x + y∞ − y, α) ie−iρ∞ φ(x + y∞ − y, α) k −i xeiρ∞ φ(x + y − y, α) −ieiρ∞ φ(x + y∞ − y, α) ∞ ˙ + . (2.37) = v˙k i xe−iρ∞ φ(x + y∞ − y, α) ie−iρ∞ φ(x + y∞ − y, α)
=
k
v˙k
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Finally, a simple computation shows that gπ 0 N Z (g−1 U 0 , π 0 ) = N (U 0 , π 0 ) π0 and thus we have retrieved all the terms of Eq. (2.24).
(2.38)
In the next three lemmas we examine in more detail the properties of ξ F and η F . Lemma 2.3. η F and ξ F are biorthogonal, in the sense that ηα , ξG = − 21 α −1 φ22 for G = and 0 otherwise, η , ξG = 21 α −1 φ22 for G = α and 0 otherwise, ηvk , ξG = −2φ22 for G = Dk and 0 otherwise, η Dk , ξG = 2φ22 for G = vk and 0 otherwise.
(2.39)
Note that all these functions are related to π 0 , not to π . Proof. By direct computation. Most of the integrals cancel simply as the product of even and odd functions. The only nontrivial computation is that 1 F F ηα , ξ = 2φ(x, α)∂α φ(x, α) d x = ∂α φ(·, α)22 = − α −1 φ(·, α)2 . 3 2 R (2.40) Let
−|v(t) − v∞ |2 − 2i(v(t) − v∞ )∇ 0 E= 0 |v(t) − v∞ |2 − 2i(v(t) − v∞ )∇
(2.41)
and H = Hπ 0 + Vπ 0 . Lemma 2.4. H∗ ξα = Eξα , H∗ ξ = −2iξα + Eξ , H∗ ξvk = Eξvk , H∗ ξ Dk = −2iξvk + Eξ Dk .
(2.42)
Proof. By direct computation. ∗ For t = ∞ we note that ξ F actually become generalizedeigenvectors for H , because 0 W (π ) the symmetry transformation gπ 0 was chosen so that gπ 0 becomes a ground W (π 0 ) state in the limit. Otherwise, there is a small error term. Lemma 2.5. If U is a solution of (2.24), π satisfies the modulation equations 0 0 ˙ α˙ = 2αφ−2 2 (−∂t U, ξα + U, ξα − iU, Eξα − iN (U , π ), ξα ), 0 0 ˙ ˙ = 2αφ−2 2 (−∂t U, ξ +2iU, ξα +U, ξ −iU, Eξ − iN (U , π ), ξ ), v˙k = φ−2 (−∂t U, ξvk + U, ξ˙vk − iU, Eξvk − iN (U 0 , π 0 ), ξvk ), 2
0 0 ˙ D˙ k = φ−2 2 (−∂t U, ξ Dk +2iU, ξvk +U, ξ Dk −iU, Eξ Dk −iN (U , π ), ξ Dk ). (2.43)
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Proof. Consider the Schrödinger equation (2.24) and take its dot product with ξ F for each F:
iUt , ξ F + U, H∗ ξ F = −iπ˙ ∂π gπ 0
W (π 0 ) , ξ F + N (U 0 , π 0 ), ξ F . (2.44) W (π 0 )
Write Ut , ξ F = ∂t U, ξ F − U, ξ˙ F . Lemma (2.42) helps evaluate the second term and the identity (2.27) and Lemma 2.3 help evaluate the third term, leading to Eqs. (2.43) above. We restate the orthogonality condition (2.16) for U in the form U (t), ξ F (t) = 0
for F ∈ {α, , vk , Dk },
(2.45)
for every t ≥ 0, which is equivalent to condition (2.16) on Z . Indeed, they follow from one another by applying the unitary transformations gπ 0 (t). Let Lπ0U =
˙ φ−2 2 (U, ξ F − iU, Eξ F )η F
F∈{vi ,Di }
+α
˙ φ−2 2 (U, ξ F − iU, Eξ F )η F
(2.46)
F∈{α,}
and
Nπ 0 (U 0 , π 0 ) = −
0 0 φ−2 2 iN (U , π ), ξ F η F
F∈{vi ,Di }
−α
0 0 φ−2 2 iN (U , π ), ξ F η F .
(2.47)
F∈{α,}
The modulation equations can then be rewritten as − i π˙ ∂π gπ 0 W (π 0 ) = L π 0 U + Nπ 0 (U 0 , π 0 ).
(2.48)
L π 0 U represents the part that is linear in U and Nπ 0 (U 0 , π 0 ) represents the nonlinear part N (U 0 , π 0 ), ξ F . Note that the orthogonality condition for all times t is equivalent to the modulation equation (2.48) together with the orthogonality condition at time 0. Next, we estimate a few useful quantities that appear in the right-hand side terms of the equations. Let ν(T ) = tπ˙ (t) L 1 (T,∞) + π˙ L ∞ (T,∞)
(2.49)
˙ 1 ≤ ν(0) < ∞ is bounded and likewise ν 0 for π 0 . We still assume that π˙ 1∩∞ + t π and we justify this assumption later. Now we state very general estimates that are used in the proof:
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Lemma 2.6. For any path π , ∞ |v(t) − v∞ | dt ≤ t v(t) ˙ L 1 (T,∞) ≤ ν(T ), T
∞ T ∞
T
˙ T |v(T ) − v∞ | ≤ t v(t) L 1 (T,∞) ≤ ν(T ), 2 |v(t) − v∞ |2 dt ≤ v(t) ˙ ˙ L 1 (T,∞) t v(t) L 1 (T,∞) ≤ ν (T ),
(2.50)
2 t|v(t) − v∞ |2 dt ≤ t v(t) ˙ ≤ ν 2 (T ). L 1 (T,∞)
Proof. All of these estimates are straightforward.
Then, there are some more specific estimates that we need: Lemma 2.7. ˙ + |x||v| ˙ + (|v| + |v∞ |)|v − v∞ | + |α − α∞ |, |ρ˙∞ | ≤ || |∇ρ∞ | ≤ |v − v∞ |, ˙ + 2|v − v∞ |, | y˙ − y˙∞ | ≤ | D| ˙ + (1 + |x|)|v| ˙ + | D| |ξ˙ F + i Eξ F | ≤ || ˙ + |v − v∞ | + |α| ˙ + |α − α∞ |, Vπ 0 1+∞→1∩∞ ≤ (|y − y∞ | + |ρ∞ | + |α − α∞ |) ≤ ν 0 (t),
(2.51)
∇Vπ 0 1+∞→1∩∞ ≤ |v − v∞ | ≤ ν (t), L π 0 1+∞→1∩∞ ≤ C·, ξ˙ F − i·, Eξ F , 0
∇ L π 0 U (t)1∩∞
˙ ≤ ν 0 (t), ≤ C(| y˙ − y˙∞ | + |ρ˙∞ | + |α|) ≤ C|U (t), ξ˙ F − iU, Eξ F |(1 + |v − v∞ |).
Proof. ρ∞ (t) = θ (t, x + y∞ ) − θ∞ (t, x + y∞ ) = (v(t) − v(∞))(x + 2tv(∞) + D(∞)) t − (|v(s)|2 − |v(∞)|2 − α(s) + α(∞)) ds + γ (t) − ∞ 0
= (v(t) − v(∞))(x + 2tv(∞) + D(∞)) ∞ (|v(s)|2 − |v(∞)|2 − α(s) + α(∞)) ds + γ (t) − (∞). +
(2.52)
t
Therefore, ˙ ρ˙∞ (t) = v(t)x ˙ − (|v(t)|2 − |v(∞)|2 ) + α(t) − α(∞) + (t).
(2.53)
Finally, note that ξ˙ F = i ρσ ˙ 3 ξ F + ( y˙∞ − y˙ )∇ξ F + α∂ ˙ α ξF 2 2 ˙ = i(v(t)x ˙ − |v(t)| + |v(∞)| + α(t) − α(∞) + (t))σ 3ξF + 2(v(∞) − v(t))∇ξ F + α∂ ˙ α ξF ,
(2.54)
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163
so ˙ ξ˙ F + i Eξ F = i(v(t)x ˙ − |v(t)|2 + |v(∞)|2 + α(t) − α(∞) + (t))σ 3ξF 2 + i|v(t) − v∞ | σ3 ξ F + α∂ ˙ α ξF ˙ = i(v(t)x ˙ + 2v∞ (v − v∞ ) + α(t) − α(∞) + (t))σ ˙ α ξF . 3 ξ F + α∂ The other formulae follow by straightforward computations.
(2.55)
2.3. Spectrum of the Hamiltonian. Without loss of generality, we perform a symmetry transformation in the nonlinear equation (0.1) and assume that the initial data is in the neighborhood of a positive ground state of the equation W (π(0)) = φ(·, α) instead of a more general standing wave. This is possible because standing waves are, by definition, an orbit of the action of symmetry transformations. Even though symmetry transformations change the spectrum, the information gained in the manner is still useful in the general case. Consider the operators − + α − 2φ 2 (·, α) −φ 2 (·, α) H= (2.56) = H0 + V. φ 2 (·, α) − α + 2φ 2 (·, α) By rescaling, one sees that all these operators have the same spectrum up to dilation and similar spectral properties. We restate the known facts about the spectrum of H. As proved by Buslaev, Perelman [BusPer1] and also Rodnianski, Schlag, Soffer in [RoScSo2], under fairly general assumptions, σ (H) ⊂ R ∪ iR and is symmetric with respect to the coordinate axes and all eigenvalues are simple with the possible exception of 0. Furthermore, by Weyl’s criterion σess (H) = (−∞, −α] ∪ [α, +∞). Grillakis, Shatah, Strauss [GrShSt1] and Schlag [Sch] showed that there is only one pair of conjugate imaginary eigenvalues ±iσ and that the corresponding eigenvectors decay exponentially. For the decay see Hundertmark, Lee [HunLee]. The pair of conjugate imaginary eigenvalues ±iσ reflects the L 2 -supercritical nature of the problem. The generalized eigenspace at 0 arises due to the symmetries of the equation, which is invariant under Galilean coordinate changes, phase changes, and scaling. It is relatively easy to see that each of these symmetries gives rise to a generalized eigenvalue of the Hamiltonian H at 0, but proving the converse is much harder and was done by Weinstein in [Wei1], [Wei2]. Schlag [Sch] showed, using ideas of Perelman [Per], that if the operators L ± = − + α − φ 2 (·, α) ∓ 2φ 2 (·, α) (2.57) 1 i that arise by conjugating H with have no eigenvalue in (0, α] and no resonance 1 −i at α, it implies that the real discrete spectrum of H is {0} and that the edges ±α are neither eigenvalues nor resonances. A paper of Demanet, Schlag [DemSch] proved numerically that the scalar operators meet these conditions. Therefore, there are no eigenvalues in [−α, α] and ±α are neither eigenvalues nor resonances for H. Furthermore, the method of Agmon [Agm], adapted to the matrix case, enabled Erdogan, Schlag [ErdSch] and independently [CucPelVou] to prove that any resonances embedded in the interior of the essential spectrum (that is, in (−∞, −α) ∪ (α, ∞)) have to be eigenvalues, under very general assumptions.
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Under the spectral Assumption 1 we now have a complete description of the spectrum of H. It consists of a pair of conjugate purely imaginary eigenvalues, a generalized eigenspace at 0, and the essential spectrum (−∞, −α] ∪ [α, ∞). It helps in the proof to exhibit the discrete eigenspaces of H. Denote by f ± and ± ˜ f the normalized eigenfunctions of H and respectively H∗ corresponding to the ±iσ eigenvalues. Also observe that η F are the generalized eigenfunctions at zero of H and ξ F fulfill the same role for H∗ . Furthermore, now we can express the Riesz projections, following Schlag [Sch], as Pim = P+ + P− , P± = ·, f˜± f ± , Pr oot = ·, ξα η + ·, ξ ηα +
(·, ξvk η Dk + ·, ξ Dk ηvk ),
(2.58) (2.59)
k
and Pc = 1 − Pim − Pr oot .
(2.60)
Even though we do not have an explicit form of the imaginary eigenvectors, Schlag [Sch] proved that f ± , in the L 2 norm, and σ are locally Lipschitz continuous as a function of α and that f ± are exponentially decaying. Concerning the continuous spectrum, the absence of embedded eigenvalues, following the spectral Assumption 1, permitted Erdogan, Schlag [ErdSch] to state the limiting absorption principle in the following form: Lemma 2.8. Assume that the thresholds of the spectrum of H = H0 +V (1.8) are regular, meaning that the operators 1 + (H0 − (±α ± i0))−1 V are invertible from the weighted Sobolev space x1+ L 2 to itself for any > 0. Then there exists 0 < α < α such that sup
|λ|1/2 (H − (λ ± i ))−1 x−1− L 2 →x1+ L 2 < ∞
(2.61)
sup
∂λ (H − (λ ± i ))−1 x−1−− L 2 →x1++ L 2 < ∞
(2.62)
|λ|≥α , >0
and |λ|≥α , >0 =1,2
if |V | ≤ Cx−7/2− . The fact that the thresholds are neither eigenvalues nor resonances implies their regularity. 2.4. Proof of the main result. Proof. To recapitulate, we are interested in finding solutions to Eq. (0.1), starting from initial data in a neighborhood of the soliton W (0), which remain close to stationary solutions for all times. Furthermore, we take W (0) to be a positive ground state of the equation. R0 We prove that, to a first approximation, R0 | (P+ + Pr oot ) = 0 is the stable R0 submanifold. Quadratic corrections are needed, as the statement of Theorem 1.7 makes clear.
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Let, for some δ ≤ 1, 1 ≤ q < 4/3, small > 0, X δ = (U, π ) | π(0) = (0, 0, 0, α), tπ˙ (t)1 + π˙ ∞ ≤ δ,
U L 2 W 1/2,6 ∩L ∞ H 1/2 + U t1−2/q+ L 2 L 6+∞ ≤ δ . t
x
t
x
t
x
(2.63)
Let β = 2/q − 1 − . Note that, for U (x, t) = gπ 0 (t)Z (x, t), one has (U, π ) ∈ X δ if and only if (Z , π ) ∈ X δ , since gπ 0 (t) is an isometry. In the sequel we deal with both U and Z , as necessary. We define a mapping that takes the pair (Z 0 , π 0 ) to the solution (Z , π ) of the linearized equation (2.15) corresponding to the initial data R0 + h f + (0), Z (0) = R0 (2.64) π(0) =(0, 0, 0, α), where h and a F (0) will be chosen later depending on R0 , so that Z fulfills the orthogonality condition (2.45) and is globally bounded in time. The first condition can be equivalently formulated in terms of U , R0 U (0) = gπ 0 (0) + h f + (0) . (2.65) R0 Further note that, by interpolation between L 2 and H 1 , for δ < 1, U (0) L q ∩H 1/2 ≤ C(R0 L q ∩H 1/2 + |h|).
(2.66)
Now we run a fixed point argument, showing that is a contraction in X δ for small δ. This is achieved in two steps, by proving firstly that takes X δ to itself and secondly that it is distance-decreasing in a weaker metric. 2.5. Stability. Here we prove that if δ ≤ 1 is sufficiently small and (Z 0 , π 0 ) ∈ X δ , then (Z 0 , π 0 ) = (Z , π ) ∈ X δ . Since g is an isometry, this is equivalent to proving that if (U 0 , π 0 ) ∈ X δ , then the solution (U, π ) of (2.24) is in X δ , for small δ. It is more convenient to prove the statement for U than for Z . Note that, after making δ as small as needed, α(t) always belongs to a fixed compact set centered at its initial value and therefore all the Sobolev norms of φ(·, α(t)) and f ± , f˜± are uniformly bounded. Replace π˙ on the right-hand side of (2.24) by its expression given by (2.27) and assume the orthogonality condition U (t), ξ F (t) = 0 in order to obtain the system of equations iUt + Hπ 0 U = L π 0 U + Vπ 0 U + N (U 0 , π 0 ) + Nπ 0 (U 0 , π 0 ) 0 0 ˙ F˙ = 2α0 φ0 −2 2 (U, ξ F0 −iU, E 0 ξ F0 −iN (U , π ), ξ F0 ), F ∈ {α, } F˙ = φ0 −2 (U, ξ˙ F0 − iU, E 0 ξ F0 − iN (U 0 , π 0 ), ξ F0 ), F ∈ {Dk , vk }. 2
(2.67)
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W (π 0 ) = L π 0 U + Nπ 0 (U 0 , π 0 ) by W (π 0 ) virtue of the modulation equations (2.43), (2.48). L π 0 U represents the part that is linear in U and Nπ 0 (U 0 , π 0 ) represents the nonlinear part N (U 0 , π 0 ), ξ F0 . The initial data is given by condition (2.64). The orthogonality condition attime 0 is true by definition, regardless of the value R0 of R0 ∈ Sδ and h, because ⊥ ξ F (0) and, due to the spectral decomposition, R0 + f (0) ⊥ ξ F (0) too, for F ∈ {vk , Dk , α, }. Global existence of the solution U to the linearized equation (2.67) follows by a standard fixed point argument. Introduce a second auxiliary function U1 and write the equation as Here we made the replacement −i π˙ gπ 0 ∂π 0
iUt + Hπ 0 U = Vπ 0 U1 + L π 0 U1 + N (U 0 , π 0 ) + Nπ 0 (U 0 , π 0 ).
(2.68)
Note that eit Hπ 0 2→2 ≤ Ceσ |t| . For any T1 consider a small time interval [T1 , T2 ] of length at most 1. Assume that U1 L ∞ (T1 ,T2 ;L 2x ) ≤ r . One has that U L ∞ (T1 ,T2 ;L 2x ) ≤ U (T1 )2 +
T2
eσ (t−T1 ) R H S(t)2 dt
T1
≤ U (T1 )2 + C(T2 − T1 )(L π 0 + Vπ 0 L 2x →L 2x U1 L ∞ (T1 ,T2 ;L 2x ) + N (U 0 , π 0 ) + Nπ 0 (U 0 , π 0 ) L ∞ 2 ) ≤ r, t Lx
(2.69)
if r is chosen such that r ≥ C(U (T1 )2 + δ). Likewise, subtracting two copies of the equation, with two solutions U and U˜ corresponding to auxiliary functions U1 and U˜ 1 , one obtains U − U˜ L ∞ (T1 ,T2 ;L 2x ) ≤ C(T2 − T1 )L π 0 + Vπ 0 L 2x →L 2x U1 − U˜ 1 L ∞ (T1 ,T2 ;L 2x ) . (2.70) Thus, if T2 − T1 is sufficiently small, the mapping that associates U to U1 is a contraction in the set {U1 | U1 L ∞ (T1 ,T2 ;L 2x ) ≤ r }. If r is sufficiently large the set is stable under the mapping. One obtains a fixed point that is a solution to (2.67) on (T1 , T2 ), but the length T2 − T1 for which this happens does not depend on the initial data. Therefore, one can iterate and obtain a global in time solution U of (2.67). Next, we prove that the global solution U is in X δ and thus globally bounded for some unique value of the parameter h. The operator Hπ 0 induces the time-independent decomposition 1 = Pc + Pr oot + Pim on L 2 (R3 ) corresponding to the decomposition of its spectrum into the absolutely continuous part, the generalized eigenspace at zero, and the imaginary eigenvalues, respectively. Since the range and cokernel of Pr oot and Pim are spanned by finitely many Schwartz functions, they are bounded from L p to L q , for any 1 ≤ p, q ≤ ∞. Therefore Pc = 1 − Pr oot − Pim is bounded on L p , 1 ≤ p ≤ ∞, and one can write Pr oot U (t) = a F (t)η F (∞), Pim U (t) = b+ (t) f + + b− (t) f − . (2.71) F
Focussing Cubic NLS in 3D
167
We will bound each projection Pr oot U , Pim U , and Pc U , as well as ∇ Pr oot U , ∇ Pim U , and ∇ Pc U (six estimates in total). One can bound the zero generalized eigenspace component in a straightforward manner. Expanding the orthogonality condition U (t), ξ F0 (t) = 0, one has for every G that a F (t)η F (∞), ξG (t) + (Pim + Pc )U (t), ξG (t). (2.72) 0= F
Since |ξG 0 (t) − ξG 0 (∞)| ≤ C(|ρ∞ | + |y − y∞ |) ≤ Cδ and the matrix with entries η F (∞), ξG (∞) is invertible, the matrix with entries η F0 (∞), ξG 0 (t) is invertible with bounded norm for small δ. Therefore, by solving the system (2.72) one obtains that Pr oot U (t)1∩∞ ≤ C(Pc + Pim )U (t)1+∞ .
(2.73)
Since the range of Pr oot is spanned by Schwartz functions, the same holds with derivatives or weights: ∇ Pr oot U (t)1∩∞ ≤ C(Pc + Pim )U (t)1+∞ .
(2.74)
As for the other two components of U , one has that i∂t Pc U + Hπ 0 Pc U = Pc (L π 0 (U ) + Vπ 0 U + N (U 0 , π 0 ) + Nπ 0 (U 0 , π 0 )) (2.75) and i∂t Pim U + Hπ 0 Pim U = Pim (L π 0 (U ) + Vπ 0 U + N (U 0 , π 0 ) + Nπ 0 (U 0 , π 0 )). (2.76) Using the explicit form (2.71) of Pim U (t) = b− (t) f − + b+ (t) f + , the corresponding Eq. (2.76) becomes b− b σ 0 N− = , (2.77) ∂t − + 0 −σ b+ b+ N+ where |N± (t)| ≤ Pim (L π 0 (U )+Vπ 0 U + N (U 0 , π 0 )(t)+ Nπ 0 (U 0 , π 0 ))1+∞ . Here ±iσ are the imaginary eigenvalues of Hπ 0 , as in our discussion of its spectrum in Sect. 2.3. Now we state a standard elementary lemma, see [Sch]. It characterizes the unique bounded solution of the two-dimensional ODE (2.77). Lemma 2.9. Consider the equation σ 0 x˙ − x = f (t), 0 −σ
(2.78)
where f ∈ L 1∩∞ . Then x is bounded on [0, ∞) if and only if ∞ 0 = x1 (0) + e−tσ f 1 (t) dt.
(2.79)
0
In this case,
∞
x1 (t) = − t
for all t ≥ 0.
e(t−s)σ f 1 (s) ds, x2 (t) = e−tσ x2 (0) +
t 0
e−(t−s)σ f 2 (s) ds
(2.80)
168
M. Beceanu
Proof. Any solution will be a linear combination of the exponentially increasing and the exponentially decaying ones and we want to make sure that the exponentially increasing one is absent. It is always true that x1 (t) = e
tσ
x1 (0) +
t
e
−sσ
f 1 (s) ds , x2 (t) = e−tσ x2 (0) +
0
t
e−(t−s)σ f 2 (s) ds.
0
(2.81) Thus, if x1 is to remain bounded, the expression between parentheses must converge to 0, hence (2.79). Conversely, if (2.79) holds, then ∞ x1 (t) = − e(t−s)σ f 1 (s) ds (2.82) t
tends to 0.
Consequently, Eq. (2.77) has a bounded solution if and only if ∞ 0 = b+ (0) + e−tσ N+ (t) dt.
(2.83)
0
Now we establish the relation between b+ (0) and h. The initial assignment (2.65) implies that b+ (0) = U (0), f˜+ R0 + hgπ 0 (0) f + (0), f˜+ = gπ 0 (0) R0
R0 , (gπ 0 (0)−1 f˜+ ) − f˜+ (0) + h 1 + (gπ 0 f + (0)) − f + , f˜+ . (2.84) = R0 Taking into account the fact that (gπ 0 f + ) − f + (0)xL 2x ≤ (gπ 0 f + ) − f + xL 2x + f + − f + (0)xL 2x ≤ Cδ, (2.85) it follows that if (2.83) holds then, for sufficiently small δ, one can solve Eq. (2.84) for h and |h| ≤ C(|b+ (0)| + δR0 1+∞ ).
(2.86)
Clearly, condition (2.83) is then fulfilled by a suitable choice of h. U is globally bounded by the definition (2.63) of X δ , which implies the boundedness of each component, in particular Uim . Proceeding henceforth under this assumption, we get |b+ (0)| ≤ CN+ (L 1 +L ∞ )(L 1 +L ∞ ) ≤ CδU L ∞ L 2 ∩L 2 L 6 + Cδ 2 , t
t
x
t
x
x
|h| ≤ Cδ(R0 1+∞ + U L ∞ L 2 ∩L 2 L 6 ) + Cδ . 2
t
x
t
x
t
x
(2.87) (2.88)
Note that σ depends Lipschitz continuously on α. Then σ belongs to a compact subset of (0, ∞), because α belongs to a compact subset of (0, ∞). In this case, since
Focussing Cubic NLS in 3D
169
f ± are Schwartz functions, one has that 2t e(t−s)σ R H S(s)1+∞ ds P+ U (t)1∩∞∩H 1 ≤ C t ∞ (t−s)σ e R H S(s)1+∞ ds + 2t
≤C
2t
e
(t−s)
R H S(s)1+∞ ds + e
−tσ
(δU L 2 L 6 + δ ) . 2
t
t
x
(2.89) We deal with the two expressions separately. The latter poses no problem, due to the exponential decay. As for the former, we bound it in t−β L 2t L 1∩∞ ∩ Hx1 (the precise x norm in x does not matter, due to these being Schwartz functions) by means of the bilinear estimate tβ s−β et−s g(t) f (s) ds dt ≤ f 2 g2 . (2.90) t<s<2t
Indeed, note that tβ s−β is bounded from above and the conclusion follows after a dyadic decomposition. The estimate implies that 2t e(t−s) R H S(s)1+∞ ds −β 2 1∩∞ ≤ CR H St−β L 2 L 1∩∞ . (2.91) t x t
t
Lt Lx
The pointwise in time norm is easier to bound, since 2t 2 e(t−s) R H S(s)1+∞ ds ≤ C max R H S(s)1+∞ ≤ C(δU L ∞ 2 + δ ). (2.92) t Lx s∈[t,2t]
t
The same works for P− U , where we also have to take into account the contribution of the initial data. This leads to Pim U L ∞ L 2 ∩L 2 L 6 + ∇ Pim U L ∞ L 2 ∩L 2 L 6 t
t
x
t
x
t
x
x
≤ C(U (0)1+∞ + δU L ∞ L 2 ∩L 2 L 6 + δ 2 ), t
t
x
(2.93)
x
and tβ Pim U L 2 L 1∩∞ ≤ C(U (0)1+∞ + δU t−β L 2 L 6+∞ + δ 2 ). t
t
x
(2.94)
x
Now we turn to Pc U , the projection on the continuous spectrum. One has, by Lemmas 2.6 and 2.7, that N (π 0 , U ) L 2 L 6/5 ≤ C(U L ∞ 6 U L ∞ L 2 U L 2 L 6 + U L ∞ L 2 U L 2 L 6 ), t Lx t t x x t
t
x
t
x
x
Vπ 0 U L 2 L 6/5 ≤ Cν 0 (1 + ν 0 )U (t) L 2 L 6 , t
t
x
x
L π 0 U L 2 L 6/5 + ∇ L π 0 U L 2 L 6/5 ≤ Cν 0 (1 + ν 0 )U L 2 L 6 , t
x
t
t
x
x
Nπ 0 (U, π 0 ) L 2 L 6/5 + ∇ Nπ 0 (U, π 0 ) L 2 L 6/5 t
x
t
x
≤ C(U L ∞ 6 U L ∞ L 2 U L 2 L 6 + U L ∞ L 2 U L 2 L 6 ). t Lx t t x x t
t
x
We recall that, by the definition (2.63) of X δ ,
ν0
≤ δ.
x
(2.95)
170
M. Beceanu
Applying endpoint Strichartz estimates to Eq. (2.75) yields that Pc U L 2 L 6 ∩L ∞ L 2 ≤ C(U (0)2 + Pc (L π 0 (U ) + Vπ 0 U + N (U 0 , π 0 )) L 2 L 6/5 +L 1 L 2 ) t
t
x
x
x
t
t
≤ C(U (0)2 + δU L ∞ L 2 ∩L 2 L 6 ) + Cδ . 2
t
x
t
x
(2.96)
x
We now establish the bounds for Pc U . Note that the components of [∇, H] are Schwartz functions and therefore [∇, Hπ 0 ]U L 2 L 6/5 ≤ CU L 2 L 6 . By trilinear t x t x interpolation between H 1/2
Pc U L ∞ L 2 ∩L 2 L 6 ≤ C(U (0)2 + i(Pc U )t + HPc U L 1 L 2 +L 2 L 6/5 ) t
t
x
x
t
x
x
t
≤ C(U (0)2 + R H S L 2 L 6/5 +L 1 L 2 ) x
t
t
(2.97)
x
and Pc U L ∞ H 1 ∩L 2 W 1,6 ≤ C(U (0) Hx1 + i(Pc U )t + HPc U L 1 H 1 +L 2 W 1,6/5 ) t
x
t
x
t
t
x
x
≤ C(U (0) Hx1 + R H S L 2 W 1,6/5 +L 1 H 1 + U L 2 L 6 ) t
x
t
t
x
(2.98)
x
it follows that Pc U L ∞ H 1/2 ∩L 2 W 1/2,6 ≤ C(U (0) H 1/2 + R H S L 2 W 1/2,6/5 +L 1 H 1/2 + U L 2 L 6 ). x
t
t
x
x
x
t
t
t
x
x
(2.99) The fractional Sobolev spaces H s and W s, p arise naturally by interpolation and are given by H s = W s,2 , W s, p = (− + 1)−s/2 L p . In the sequel we use the Sobolev embeddings W 1/2,3 → L 6 and H 1/2 → L 3 . Now we examine each term on the right-hand side of (2.24). We use the fractional Leibniz rule, as stated, for example, in [Tay, p. 105]: f 1 f 2 f 3 W 1/2,6/5 ≤ C( f i W 1/2,3 f j 3 f k 6 + f i 3 f j 3 f k W 1/2,6 ) i, j,k
≤ C(
i, j,k
f i W 1/2,3 f j 3 f k W 1/2,3 +
i, j,k
f i 3 f j 3 f k W 1/2,6 ).
i, j,k
(2.100) Making all the f ’s equal, one gets U 3 L 2 W 1/2,6/5 ≤ C(U 2 4 t
1/2,3
L t Wx
x
≤
2 U L ∞ 3 + U 2 1/2,6 U ∞ 3 ) L L L W t Lx t
CU 3 ∞ 1/2 2 1/2,6 . L t Hx ∩L t Wx
x
t
x
(2.101)
Thus, in estimating this cubic term we had to use both the Keel-Tao endpoint Strichartz estimate and the critical half-derivative, which prevents us from doing any better (that is, from lowering the number of derivatives). The localized quadratic terms can be handled similarly with the help of (2.100), the conclusion being U 2 φ L 2 W 1/2,6/5 ≤ C(U 2 4 t
1/2,3
L t Wx
x
≤
φ L ∞ 3 + U 2 1/2,6 U L ∞ L 3 φ L ∞ L 3 ) L W t Lx t t x x
CU 2 ∞ 1/2 2 1/2,6 . L t Hx ∩L t Wx
t
x
(2.102)
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171
As for the linear terms, a satisfactory estimate is Vπ 0 U W 1/2,6/5 ≤ C(Vπ 0 3/2 U W 1/2,6 + Vπ 0 W 1/2,3/2 U 6 ),
(2.103)
x
x
which implies that Vπ 0 U L 2 W 1/2,6/5 ≤ CVπ 0 L ∞ W 1/2,3/2 U L 2 W 1/2,6 . x
t
x
t
(2.104)
x
t
Therefore Pc U L 2 W 1/2,6 ∩L ∞ H 1/2 ≤ C(U (0) H 1/2 t
x
x
t
x
+δU L 2 W 1/2,6 ∩L ∞ H 1/2 + δ 2 + U L 2 L 6 ). (2.105) x
t
t
x
t
x
Putting together estimates (2.73), (2.93), and (2.96), one has that U L ∞ L 2 ∩L 2 L 6 ≤ C(U (0)2 + δU L ∞ L 2 ∩L 2 L 6 + δ 2 ), t
x
t
t
x
t
x
(2.106)
x
which for sufficiently small δ and U (0)2 implies that U L ∞ L 2 ∩L 2 L 6 ≤ CU (0)2 + Cδ 2 ≤ δ. t
t
x
(2.107)
x
This proves, after considering (2.74), (2.93), and (2.105), that U L 2 W 1/2,6 ∩L ∞ H 1/2 ≤ C(U (0) H 1/2 + δU L 2 W 1/2,6 ∩L ∞ H 1/2 + δ 2 + U L 2 L 6 ) t
x
t
x
x
t
x
t
x
t
x
(2.108) and therefore U L 2 W 1/2,6 ∩L ∞ H 1/2 ≤ CU (0) H 1/2 + Cδ 2 ≤ δ. t
x
t
x
(2.109)
x
Lastly, for π the estimates (see Lemma 2.7) U, ξ˙ F − iU, Eξ F − iN (U 0 , π 0 ), ξ F L 1 t
0 0 2 0 2 ≤ ν 0 U L ∞ 6 + U L ∞ L 6 U 2 6 + U 2 6 t Lx t L L L L x t
t
x
(2.110)
x
and U, ξ˙ F − iU, Eξ F − iN (U 0 , π 0 ), ξ F L ∞ t 0 3 0 2 ≤ ν 0 U L ∞ 6 + U ∞ 6 + U ∞ 6 L L L L t Lx t
x
t
x
(2.111)
lead by the modulation equations (2.43) to the straightforward inequality (where we used ν 0 ≤ δ) π˙ ∞ + π˙ 1 ≤ C (U, ξ˙ F − iU, Eξ F − iN (U 0 , π 0 ), ξ F ∞ F
+U, ξ˙ F − iU, Eξ F − iN (U 0 , π 0 ), ξ F 1 ) 2 2 2 2 ≤ CδU L ∞ 6 + Cδ ≤ Cδ(U (0) W 1,2 + δ ) + Cδ ≤ Cδ , t Lx which, for small δ, proves some of the desired bounds on π .
(2.112)
172
M. Beceanu
Next we obtain decay estimates for U and π , which are necessary in order to bound the quantities ρ∞ (t) and y(t) − y∞ . The important ingredient in the proof is the evaluation of tβ U L 2 L 6+∞ by means of Lemma 3.7. Here it becomes necessary to assume t x that the initial data U (0) is in L q , q < 4/3, in addition to being in H 1/2 . We make the choice of β = 2/q − 1 − > 1/2. The reason why we need decay is ∞ tπ˙ (t)1 ≤ t|U, ξ˙ F − iU, Eξ F − iN (U 0 , π 0 ), ξ F | dt 0 ∞ t|π 0 (t) − π 0 (∞)|U (t)1+∞ + t(U 0 )2 (t)1+∞ ≤C 0
+t(U 0 )2 (t)3/2+∞ U 0 (t)3 dt ≤ C(t1/2 (π 0 (t) − π 0 (∞)) L 2 t1/2 U (t) L 2 L 1+∞ t
t
x
+t1/2 U 0 (t)2L 2 L 1+∞ (1 + δ)) t
x
1 ≤ ν 0 + CU (t)2t−1/2 L 2 L 1+∞ + Cδ 2 . t x 2
(2.113)
This inequality is the means to prove that ν(t) stays bounded. In order to apply Lemma 3.7 and bound this quantity, we evaluate the right-hand side terms of (2.24), beginning with the worst: V U t−β L 2 L 1∩6/5 ≤ CδU t−β L 2 L 6+∞ . t
x
t
(2.114)
x
Here we used the fact that ν 0 (t) is bounded. Exactly the same works for the other linear term in U , L π 0 U t−β L 2 L 1∩6/5 ≤ CδU t−β L 2 L 6+∞ . t
t
x
(2.115)
x
The remaining terms, N (U 0 , π 0 ) and Nπ 0 (U 0 , π 0 ), work out in the same fashion, provided that U is uniformly bounded in time, which it is. Special attention has to be paid to the nonlocalized term, for which note that (U 0 )3 t−β L 2 L 1∩6/5 ≤ CU 0 t−β L 2 L 6+∞ U 0 2L ∞ L 2∩3 t
t
x
x
t
x
≤ CU t−β L 2 L 6+∞ δ . 0
2
t
(2.116)
x
This is another place where the H˙ 1/2 -critical nature of the problem comes into play, since H˙ 1/2 embeds in L 3 . In conclusion, the nonlinear right-hand side terms have the same behaviour as the linear ones: N (U 0 , π 0 ) + Nπ 0 (U 0 , π 0 )t−β L 2 L 1∩6/5 ≤ CδU 0 t−β L 2 L 6+∞ . t
x
t
(2.117)
x
After applying Lemma 3.7, the result is
∞ 0
t4/q−2−2 Pc U (t)26+∞ dt ≤ C(U (0)2H 1/2 ∩L q + δ 2 U 2t1−2/q+ L 2 L 6+∞ + δ 4 ). t
x
(2.118)
Focussing Cubic NLS in 3D
173
The hyperbolic part Pim U decays just as fast by (2.94) and the projection on the generalized 0 eigenspace Pr oot U is dominated by the other two components, so we can add them both in for free. By making δ and the initial data small we obtain the desired estimates U t1−2/q+ L 2 L 6+∞ ≤ δ and tπ˙ 1 ≤ Cδ 2 ≤ δ. t
x
(2.119)
This finishes proving that the mapping takes X δ to itself, provided that δ and the initial data are sufficiently small. Next, we need to show that really is a contraction within this set.
2.6. Contraction. Here we prove that is a contraction on X δ for small δ << 1. Consider two solutions (Z 1 , π1 ) and (Z 2 , π2 ) of the linearized equation (2.67) corresponding to two different pairs of auxiliary functions, (Z 10 , π10 ) and respectively (Z 20 , π20 ). In our previous notation, we have that (Z 1 , π1 ) = (Z 10 , π10 ) and (Z 2 , π2 ) = (Z 20 , π20 ). Assume that (Z 10 , π10 ), (Z 20 , π20 ) ∈ X δ . Then, it follows from the first part of the proof that the same holds for (Z 1 , π1 ) and (Z 2 , π2 ). We perform the contraction in the following seminorm: Y = {(Z , π ) | (Z , π )Y = π˙ L 1∩∞ ∩t−β L 2 + Z t 1−β L 2 L 6+∞ < ∞}. (2.120) t
t
x
Here β = 2/q − 1 − is the same as before and −β < −1/2, 1 − β < 1/2. It is straightforward to note that this seminorm defines a metric on X δ because the elements of X δ have well-determined initial parameter values. Furthermore, since the seminorm is weaker than the one that defines X δ , it makes X δ a complete metric space. We prove that the map is a contraction on X δ in this metric, more precisely that 1 (Z 1 , π1 ) − (Z 2 , π2 )Y ≤ (Z 10 , π10 ) − (Z 20 , π20 )Y , for some sufficiently small δ. 2 Since this result will be reused later in the proof, it is convenient to state it in the form of a lemma: Lemma 2.10. Consider (Z 1 , π1 ) = (Z 10 , π10 ), (Z 2 , π2 ) = (Z 20 , π20 ) solving the linearized equation (2.15), such that (Z 1 , π1 ), (Z 10 , π10 ), (Z 2 , π2 ), (Z 20 , π20 ) ∈ X δ , π10 (0) = π20 (0) = (0, 0, 0, α0 ), and with initial data Z i (0) =
R0i + h i f + (0) R0i
(2.121)
such that R0i ⊥ f˜+ (0), R0i ⊥ ξ F (0). Further assume the orthogonality conditions Z i (t), ξ FZ i (t) = 0 for all t ≥ 0 and F ∈ {vi , Di , α, }, with ξ Z i pertaining to πi0 . Then, if δ is sufficiently small, (Z 1 , π1 ) − (Z 2 , π2 )Y ≤ C(δ(Z 10 , π10 ) − (Z 20 , π20 )Y + R01 − R02 2 ). (2.122)
174
M. Beceanu
Proof. Subtract the two copies of Eq. (2.15) corresponding to Z 1 and to Z 2 from one another and introduce the new function U = gπ 0 (Z 1 − Z 2 ). Then 1
iUt + Hπ 0 U = Vπ 0 U + gπ 0 (−i π˙ 1 ∂π W (π10 ) 1
1
1
+ i π˙ 2 ∂π W (π20 ) + (HπZ 1 − HπZ 2 )Z 1 + N Z (Z 10 , π10 ) − N Z (Z 20 , π20 )), 0
0
(2.123) with initial data
U (0) = gπ 0 (0)
2 R0 R01 + − + (h 1 − h 2 ) f (0) . R01 R02
(2.124)
It is worth noting that both paths start at the same point, so the operators HπZ 0 are the i
same and have the same eigenfunctions for i = 1, 2. Henceforth, we shall use the names ξ F , etc., in relation to π10 . The vectors f + (0) and ξ F (0) do not depend on the whole parameter paths πi0 , but only on the starting point W (0). We do not assume that R01 = R02 , in order to prove the Lipschitz continuity of at the same time. The modulation equations for Z are obtained from the orthogonality condition for Z (2.16) and have the form, similar to (2.48), − i π˙ ∂π W (π 0 ) = L πZ 0 Z + NπZ0 (Z 0 , π 0 ), where L πZ 0 Z =
Z Z Z ˙Z φ−2 2 (Z , ξ F − iZ , E ξ F )η F ,
(2.126)
Z 0 0 Z Z φ−2 2 iN (Z , π ), ξ F η F ,
(2.127)
F∈{α,,vi ,Di }
NπZ0 (Z 0 , π 0 ) = − and
(2.125)
F∈{α,,vi ,Di }
−|v(t)|2 − 2iv(t)∇ 0 E = . 0 |v(t)|2 − 2iv(t)∇ Z
(2.128)
Split Eq. (2.123) into three parts, corresponding to our decomposition of the spectrum of Hπ 0 into the absolutely continuous part, the generalized eigenspace at 0, and 1 the imaginary eigenvalues, respectively. Then, we estimate each component separately. The ranges of projections on the generalized eigenspace at 0 and on the imaginary eigenspace are spanned by finitely many Schwartz functions, Pr oot U (t) = a F (t)η F (∞), Pim U (t) = b+ (t) f + + b− (t) f − . (2.129) F
Firstly, we deal with Pr oot U . Both Z 1 and Z 2 satisfy orthogonality conditions of the form Z i (t), ξ FZ i (t) = 0. Applying the transformation g and taking the difference, one has that U, ξ F = Z 2 , ξ FZ 2 − ξ FZ 1 ,
(2.130)
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175
and therefore Z 2 (t), ξ FZ 2 (t) − ξ FZ 1 (t) =
a F (t)η F (∞), ξG (t) + (Pim + Pc )U (t), ξG (t).
F
(2.131) The matrix with entries η F (∞), ξG (t) is invertible with bounded norm. Hence, the following holds:
Pr oot U (t)1∩∞ ≤ C (Pc + Pim )U (t)1+∞ + Z 2 (t)1+∞ |π10 (t) − π20 (t)| t + |π10 (s) − π20 (s)| ds . (2.132) 0
An immediate consequence is that Pr oot U (0)1∩∞
R01 R02 ≤C 1 − +|h 1 − h 2 | + δπ1 − π2 ∞ . R0 R02 1+∞ (2.133)
The last term is not strictly necessary here, since the two parameter paths start at the same point. We still include it, in order to keep the argument general. Note that
t
(Z 2 (t)1+∞ | 0
≤
π10 (s) − π20 (s) ds|)t1−β L 2
CZ 2 t1−2/q L 2 L 2+∞ π10 t x
t
− π20 ∞
≤ Cδπ˙ 10 − π˙ 20 1 .
(2.134)
Therefore Pr oot U t1−β L 2 L 6+∞ ≤ C((Pc + Pim )U (t)t1−β L 2 L 6+∞ + δπ˙ 10 − π˙ 20 1∩∞ ). t
t
x
x
(2.135) Now we evaluate the terms on the right-hand side of Eq. (2.123), in order to bound the two remaining components, Pim U and Pc U . Since Pr oot U grows like t 2−2/q− < 1/2 in L 2 in time, we prove that the other two components have the same growth rate in L 2 in time. We evaluate the projection on the continuous spectrum. Just as in the stability part of the proof, Lemma 3.7 leads to the required weighted estimate. From Eq. (2.123) we have that Pc U t1−β L 2 L 6+∞ ≤ C(U (0)2 + R H St1−β L 2 L 1∩6/5 ). t
x
t
x
(2.136)
Thus, if we can establish the bound for the right-hand side of the equation, we then retrieve it for Pc U .
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M. Beceanu
Observe that Vπ 0 U t1−β L 2 L 1∩6/5 ≤CδU t1−β L 2 L 6+∞ ; 1
t
x
t
x
(L πZ 0 − L πZ 0 )Z 2 t1−β L 2 L 1∩6/5 ≤Cdπ 0 Y Z 2 t−β L 2 L 6+∞ 1
t
x
t
2
x
≤Cdπ 0 Y δ; (HπZ 1 0
(2.137)
gπ 0 L πZ 0 U t1−β L 2 L 1∩6/5 1 t x 1
≤δU t1−β L 2 L 6+∞ ;
− HπZ 2 )Z 1 t1−β L 2 L 1∩6/5 t x 0
≤Cdπ − dπ (∞)∞ Z 1 t−β L 2 L 6+∞
t
0
x
0
t
x
≤Cd π˙ 1∩∞ δ, 0
as well as N Z (Z 10 , π10 ) − N Z (Z 10 , π20 )t1−β L 2 L 1∩6/5 x
t
≤
Cπ10
− π20 ∞ Z 10 t−β L 2 L 1∩6/5 Z 10 L ∞ 2 t Lx t x
≤ Cπ˙ 10 − π˙ 20 1∩∞ δ 2 ; N Z (Z 10 , π20 ) − N Z (Z 20 , π20 )t1−β L 2 L 1∩6/5 x
t
≤ CZ 10
−
Z 20 t1−β L 2 L 6+∞ (δ 2 t x
NπZ0 (Z 10 , π10 ) − 1
+ δ); (2.138)
NπZ0 (Z 20 , π20 )t1−β L 2 L 1∩6/5 t x 2
≤ NπZ0 (Z 10 , π10 ) − NπZ0 (Z 10 , π10 )t1−β L 2 L 1∩6/5 1
x
t
2
+ NπZ0 (Z 10 , π10 ) − NπZ0 (Z 10 , π20 )t1−β L 2 L 1∩6/5 2
t
2
+ NπZ0 (Z 10 , π20 ) − 2
x
NπZ0 (Z 20 , π20 )t1−β L 2 L 1∩6/5 t x 2
≤ Cδ(π˙ 10 − π˙ 20 1 + Z 10 − Z 20 t1−β L 2 L 6+∞ ). t
x
We again used the H˙ 1/2 -critical nature of the equation and the Keel-Tao endpoint Strichartz estimate, in the next-to-last inequality, concerning unlocalized terms, under the following guise: (Z i0 )2 (t) L 3/2∩∞ ≤ CZ i0 (t)2
1/2
Hx
x
≤ Cδ 2 .
(2.139)
After examining each term, the overall conclusion is that R H St1−β L 2 L 1∩6/5 ≤ Cδ(U t1−β L 2 L 16+∞ + π˙ 10 − π˙ 20 1∩∞ t
t
x
+Z 10
−
x
Z 20 t1−β L 2 L 6+∞ ), t x
(2.140)
and thus 0 0 Pc U t1−β L 2 L 6+∞ ≤ C(U (0)2 + δU t 1/10 L ∞ 3+∞ + δ(dU , dπ )Y ). (2.141) t Lx t
x
Next, we bound Pim U . Note that Pim U (t)2 ≤ Z 1 L ∞ 2 + Z 2 L ∞ L 2 ≤ Cδ, t Lx t x so Pim U is a bounded solution to a hyperbolic ODE system. Therefore, by yet another application of Lemma 2.9, ∞ 0 = U (0), f˜+ + e−tσ N+ (t) dt, (2.142) 0
Focussing Cubic NLS in 3D
177
where ±iσ are the imaginary eigenvalues of Hπ 0 . Thus 1
U (0),
+ f∞
≤ CN+ t1−β L 2 L 1∩6/5 ≤ CR H St1−β L 2 L 1∩6/5 x
t
t
≤ Cδ(U t1−β L 2 L 6+∞
(2.143)
≤ C(U (0)1+∞ + δU t1−β L 2 L 6+∞ + δ(U10 − U20 , π10 − π20 )Y ).
(2.144)
x
−
Z 20 , π10
x
− π20 )Y )
t
+ (Z 10
and, by the same reasoning as in the stability proof (see 2.94), Pim U t1−β L 2 L 6+∞ t
x
t
x
Note σ belongs to a compact subset of (0, ∞), because α belongs to a compact subset of (0, ∞), so the constants are independent of σ . Now we deal with the initial data U (0): 1 2 R0 R0 U (0) p ≤ C + |h − h | . (2.145) − 1 2 p R01 R02 Furthermore, subtracting the two copies of (2.84) from one another, one has b+1 (0) − b+2 (0) = gπ 0 (0)(Z 1 (0) − Z 2 (0)), f˜+ 1 2 R0 R0 + ˜+ − + (h − h ) f (0) , f = gπ 0 1 2 R01 R02 1 2 R0 R0 −1 ˜+ + ˜ − , f − gπ 0 f (0) + (h 1 − h 2 )(1 + (gπ 0 f + (0)) − f + , f˜+ ), = R01 R02 (2.146) and thus, for small δ, by (2.143), R1 R2 1 2 0 0 − |h 1 − h 2 | ≤ C |b+ (0) − b+ (0)| + δ R01 R02 1∩∞ ≤ Cδ(U t1−β L 2 L 6+∞ + (Z 10 − Z 20 , π10 − π20 )Y + R01 − R02 1∩∞ ). t
x
(2.147) By (2.133) it follows that, for small δ,
U (0) p ≤ C R01 − R02 p + δU t1−β L 2 L 6+∞ + δ(Z 10 − Z 20 , π10 − π20 )Y . t
x
(2.148) Putting (2.135), (2.144), and (2.141) together, one has that U t1−β L 2 L 6+∞
t x ≤ C δU t1−β L 2 L 6+∞ + δ(Z 10 , π10 ) − (Z 20 , π20 )Y + R01 − R02 2 , (2.149) t
x
whence, for small δ, U t1−β L 2 L 6+∞ ≤ Cδ(Z 10 − Z 20 , π10 − π20 )Y + CR01 − R02 2 . t
x
(2.150)
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M. Beceanu
In studying the difference π1 − π2 , we switch back to using U1 and U2 instead of Z 1 and Z 2 . The reason why this is possible is that U1 − U2 t1−β L 2 xL 6+∞ = gπ 0 Z 1 − gπ 0 Z 2 t1−β L 2 xL 6+∞ t
x
1
t
2
x
≤ gπ 0 Z 1 − gπ 0 Z 1 t1−β L 2 xL 6+∞ + gπ 0 Z 1 − gπ 0 Z 2 t1−β L 2 xL 6+∞ 1
t
2
x
2
t
2
x
≤ Cπ10 − π20 ∞ tZ 1 t1−β L 2 L 6+∞ + Z 1 − Z 2 t 1−β L 2 L 6+∞ t
t
x
x
≤ max(C, Cδ)(Z 1 , π1 ) − (Z 2 , π2 )Y .
(2.151)
To put it otherwise, this norm is sufficiently weak not to see such small symmetry transformations. This helps us insofar as all the terms that appear in the modulation equations are localized. The modulation equations for π1 − π2 are of the form, derived from (2.43), − i π˙ 1
W (π10 ) W (π20 ) + i π ˙ = L π 0 U1 − L π 0 U2 + Nπ 0 (U01 , π01 ) − Nπ 0 (U02 , π02 ). 2 1 2 1 2 W (π10 ) W (π20 ) (2.152)
Denoting for convenience dπ = π1 − π2 , one has that d π˙ L 1 ≤ d π˙ t−β L 2 ≤ L π 0 U1 − L π 0 U2 t−β L 2 L 1+∞ t
t
1
t
2
x
+Nπ 0 (U10 , π10 ) − Nπ 0 (U20 , π20 )t−β L 2 L 1+∞ t x 1 2 W (π 0 ) W (π 0 ) 1 2 − +C π˙ 2 . W (π10 ) W (π20 ) t−β L 2t L 1+∞ x
(2.153)
An important fact is that L π 0 (T ) − L π 0 (T ) L 1+∞ →L 1∩∞ x x 1 2 −2 1 1 φ (ξ˙ − i Eξ 1 ) ⊗ η1 − φ 2 −2 (ξ˙ 2 − i E Z ξ 2 ) ⊗ η2 1+∞ 1∩∞ ≤ F F F F F F L 2 2 →L x
F∈{vi ,Di }
x
2α1 φ 1 −2 (ξ˙ 1 − i Eξ 1 ) ⊗ η1 + F F F 2 F∈{α,}
Z 2 2 ˙2 −2α2 φ 2 −2 2 (ξ F − i E ξ F ) ⊗ η F L 1+∞ →L 1∩∞ x x ≤ C (|π10 (T ) − π10 (∞)| + |π20 (T ) − π20 (∞)|)
T
|dπ 0 (t)| dt + |dπ 0 (T )|
0
+ |dπ 0 (T ) − dπ 0 (∞)| + |d π˙ 0 (T ) − d π˙ 0 (∞)| ≤ Cδdπ 0 ∞ .
(2.154)
Indeed, despite the fact that the parameter paths converge to different final values, the difference of these two operators decays in time.
Focussing Cubic NLS in 3D
179
Using the estimates (L π 0 − L π 0 )U1 t−β L 2 L 1∩∞ t x 1 2 t 0 0 0 0 0 0 ≤ C (|π1 (t) − π1 (∞)| + |π1 (t) − π1 (∞)|) |dπ (s)| ds + |dπ (t)| 0 + |dπ 0 (t) − dπ 0 (∞)| + |d π˙ 0 (t) − d π˙ 0 (∞)| U1 (t)t−β L 2 L 1+∞ t x
≤ Cπ˙ 10 − π˙ 20 1∩∞ t (π10 (t) − π10 (∞)) L 1∩∞ + 1 U1 t−β L 2 L 2+∞ t
t
x
≤ Cδd π˙ 1 , 0
(2.155)
L π 0 (U1 − U2 )t−β L 2 L 1∩∞ t x 2 2 2 ≤ C (|π0 (t) − π0 (∞)| + |π˙ 02 (t)|)U1 (t) − U2 L 1+∞ x ≤
t−β L 2t 2 2 CU1 − U2 t1−β L 2 xL 6+∞ t (π0 (t) − π0 (∞)) L 1∩∞ t t x
≤ CδU1 − U2 t1−β L 2 xL 6+∞ , t
(2.156)
x
Nπ 0 (U10 , π10 ) − Nπ 0 (U20 , π20 )t−β L 2 L 1∩∞ 1
t
2
x
≤ Nπ 0 (U10 , π10 ) − Nπ 0 (U10 , π10 )t−β L 2 L 1∩∞ 1
t
2
+Nπ 0 (U10 , π10 ) − 2
x
Nπ 0 (U10 , π20 )t−β L 2 L 1∩∞ t x 2
+Nπ 0 (U10 , π20 ) − Nπ 0 (U20 , π20 )t−β L 2 L 1∩∞ 2
t
2
x
≤ Cδ(d π˙ 1 + U10 − U20 t1−β L 2 xL 6+∞ ), t
(2.157)
x
and W (π 0 ) W (π 0 ) 1 2 − ≤ Cδdπ 0 ∞ ≤ Cδd π˙ 0 1 , π˙ 2 W (π10 ) W (π20 ) t−β L 2t L 1+∞ x
(2.158)
one gets the desired conclusion that d π˙ 1∩∞ ≤ Cδ((Z 10 − Z 20 , π10 − π20 )Y + U1 − U2 t1−β L 2 L 6+∞ ). (2.159) t
x
Therefore, (Z 1 − Z 2 , π1 − π2 )Y ≤ Cδ((Z 10 − Z 20 , π10 − π20 )Y + R01 − R02 2 ) for sufficiently small δ.
(2.160)
R ,π Since is a contraction in a complete metric space, it has a fixed point Z = R ∈ X δ , such that R + W (π ) is a global solution for (0.1).
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M. Beceanu
2.7. Remaining bounds. The correction term F(R0 ) = h(R0 ) f + Z satisfies the appropriate bounds since |h(R0 )| ≤ Cδ 2
(2.161)
by (2.88) and (2.107), for U (0) L 4/3− ∩H 1/2 ≤ Cδ, and |h(R01 ) − h(R02 )| ≤ CδR01 − R02 2
(2.162)
by (2.147) and (2.160). Same goes for the a F coefficients. If we did not assume that R01 = R02 in (2.160), then it implies that (R0 ) is a Lipschitz function of R0 . Indeed (Z 1 , π1 ) − (Z 2 , π2 )Y ≤ CδR01 − R02 2
(2.163)
R1 − R2 t1−β L 2 L 6+∞ ≤ CδR01 − R02 2
(2.164)
W (π1 ) − W (π2 )t1−β L 2 xL 6+∞ ≤ CδR01 − R02 3/2∩2 .
(2.165)
implies t
x
and t
x
Only the scattering is left to prove and it follows in a standard manner using the Strichartz inequalities. Indeed, observe that ∞ e−is H R H S(s) ds (2.166) U1 = 0
and the integral converges in the L 2 norm, because R H S is in L 2t L 6x . Therefore, ∞ Pc U (t) − eit H Pc (U (0) + U1 ) = −eit H Pc e−is H Pc R H S(s) ds = o L 2 (1).
is in
L2
t
(2.167) The other two components of U , Pim U and Pr oot U converge to zero in the L 2 norm. Indeed, they easily converge to zero in other norms and, being given by Schwartz functions in the space variable, all of their Lebesgue norms are comparable. Thus U behaves like eit H Pc (U (0) + U1 ) + o L 2 (1). 1 0 . We want to establish that Let H0 (α∞ ) = ( − α∞ )σ3 , where σ3 = 0 −1 L = lim eit H0 (α∞ ) e−it H Pc (U (0) + U1 ) t→∞
exists as a strong L 2 limit. But, letting M =
2 2 φ∞ 2φ∞ 2 2 −φ∞ −2φ∞
= H − H0 ,
d it H0 (α∞ ) −it H e e Pc (U (0) + U1 ) = eit H0 (α∞ ) Me−it H (U (0) + U1 ). dt In other words,
L = Pc (U (0) + U1 ) + lim
t→∞ 0
t
(2.168)
eit H0 (α∞ ) Me−it H Pc (U (0) + U1 ) dt.
(2.169)
(2.170)
Focussing Cubic NLS in 3D
181
However, we note that T2 it H0 (α∞ ) −it H e Me P (U (0) + U ) dt ≤ C c 1 2
T1
≤C
T2 T1 T2 T1
Me−it H Pc (U (0) + U1 )26/5 dt e−it H Pc (U (0) + U1 )26 dt
≤ CU (0) + U1 22 .
(2.171)
Since this last integral is absolutely convergent, the initial one also converges. Therefore L exists as a strong L 2 limit and U (t) = eit H0 (α∞ ) L + o L 2 (1).
(2.172)
Switching back to Z (t) = g−1 π (t)U (t), one has it H0 (α∞ ) L + o L 2 (1)) = eit H0 (0) g−1 Z (t) = g−1 π (t)(e π (0)L + o L 2 (1),
which finishes the proof of scattering.
(2.173)
Proof of Corollary 1.8. Firstly, apply a symmetry transformation to the whole equation in order to make W (0) = φ(·, α0 ) a positive ground state. The result that holds for the transformed problem is also valid for the original one. The previous Theorem 1.7 proves the existence of a codimension-nine Lipschitz submanifold of L q ∩ H 1/2 , 1 ≤ q < 4/3, tangent to Sδ at W (0), made of initial data (R0 )(0), R0 ∈ Sδ , for global solutions of (0.1). Since the equation is invariant under symmetry transformations, we let symmetry transformations act on this submanifold and retrieve a larger set of good initial data. More precisely, let {G((R0 )(0)) | R0 ∈ Sδ }. (2.174) N L q ∩H 1/2 = G
The action G((R0 )(0)) maps the product between an 8-dimensional manifold of symmetry transformations and the codimension-nine submanifold into L q ∩H 1/2 . The matrix ∂g W (0) , ξ F (W (0)) = ηG (W (0)), ξ F (W (0)), (2.175) ∂G W (0) where F and G stand for parameters vk , Dk , α, and is invertible by Lemma 2.3, which implies that the action of the symmetry transformations is transverse to the codimensionnine manifold. Therefore, the range of the map is locally (in a neighborhood of W (0)) a codimension-one submanifold of L q ∩ H 1/2 . Proof of Theorem 1.1. The first new claim is that if a solution obtained by the previous corollary has initial data (R0 )(0) = W (0) + R0 + F1 (R0 ) ∈ H 1 , then (R0 )(T ) is still in H 1 at any time T > 0. W (π(t)) (R0 )(t) − , we Using the change of coordinates U = gπ (t) (R0 )(t) W (π(t)) transform the equation in the same manner as in the proof of Theorem 1.7 to the form (2.24). Taking derivatives, one has that i∂t ∇U + Hπ ∇U = [Hπ , ∇]U + ∇(Vπ U ) + ∇(L π U ) + ∇ N (U, π ) + ∇(Nπ (π, U )). (2.176)
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M. Beceanu
This equation is linear in ∇U . By means of Strichartz estimates, for U ∈ X δ we obtain ∇U L 2 L 6 ∩L ∞ L 2 ≤ C(∇U (0)2 + R H S L 2 L 6/5 +L 1 L 2 ) t
x
t
x
x
t
t
x
≤ C(∇U (0)2 + δ∇U L 2 L 6 ∩L ∞ L 2 + δ) ≤ C(∇U (0)2 + δ) t
t
x
x
(2.177) for sufficiently small δ. By approximating ∇U (0) with functions of better regularity, the bound is seen to hold everywhere in time, instead of almost everywhere. This proves the statement. Secondly, we prove that if a H 1 solution has initial data (R0 )(0) = W (0) + R0 + F1 (R0 ) ∈ |x|−1 L 2 , then the property of being in |x|−1 L 2 is kept at any later time T > 0. Using the same machinery as in the previous proof, we can reduce this to showing that if |x|U (0) ∈ L 2 in the equation W (π ) + N (U, π ) + Vπ U, (2.178) iUt + Hπ U = −i π˙ ∂π gπ W (π ) then the property is preserved at time T > 0. However, xU satisfies an equation of its own, namely W (π ) + x N (U, π ) + x Vπ U. (2.179) i(xU )t + Hπ 0 (xU ) = 2∇σ3 U − i x π˙ ∂π gπ W (π ) The local terms on the right-hand side are already bounded in Strichartz norms by our knowledge about U . This leaves ∇σ3 U L 1 (0,T ;L 2x ) ≤ C T ∇U L ∞ 2 t Lx
(2.180)
2 xU 3 L 1 L 6/5 ≤ CxU L ∞ 2 U 2 6 ≤ CδxU L ∞ L 2 . t Lx t L L x
(2.181)
and t
x
t
x
As expected, the gradient term grows linearly in time and we cannot do any better. By a standard argument, for sufficiently small δ it follows that xU (T )2 ≤ C(xU (0)2 + T δ).
(2.182)
Note that ∇U (0)2 ≤ C∇ R0 2 and likewise xU (0)2 ≤ Cx R0 2 . Thus the solution stays for all times in the space, but the norm may grow linearly in time. The second claim was that the manifold of global solutions N is locally invariant under the flow. Define the manifold as N = {(0) | (0) ∈ N L q ∩H 1/2 , (0) = G(W (0) + R0 + F1 (R0 )), for R0 with R0 + F1 (R0 ) < δ0 } (2.183) with N L q ∩H 1/2 being the codimension-one submanifold from Corollary 1.8. The only new condition pertains to the size of the initial data in . Clearly N is still a codimensionone submanifold of . Note that every globally bounded solution sufficiently close to the manifold of standing waves must actually start on N L q ∩H 1/2 . We phrase this as the following lemma:
Focussing Cubic NLS in 3D
183
W (π ) , = Z + W (π ) Z (t) ⊥ ξ FZ (π(t)) for all t ≥ 0, and (Z , π ) ∈ X δ for some fixed, sufficiently small δ. Further assume that Z (0) L q ∩H 1/2 ≤ Cδ for the same δ. Then (0) ∈ N L q ∩H 1/2 .
Lemma 2.11. Consider a solution to Eq. (0.1) such that
Proof. One can perform small symmetry transformations in this situation, since N L q ∩H 1/2 is invariant under them. Therefore assume, without loss of generality, that W (π(0)) is a R0 = (Pc (0) + P− (0))Z (0); positive ground state, meaning π(0) = (0, 0, 0, α0 ). Let R0 then R0 L q ∩H 1/2 ≤ Cδ. Consider the global solution (R0 ), of initial data (R0 )(0) = W (π(0)) + R0 + F1 (R0 ).
(2.184)
It exists by Theorem 1.7 if δ is sufficiently small and it has the property that, for some path π(R0 ) with π(R0 )(0) = (0, 0, 0, α0 ), (R0 ) W (π(R0 )) = Z (R0 ) + , (Z (R0 ), π(R0 )) ∈ X δ , (2.185) (R0 ) W (π(R0 )) and Z (R0 )(t) ⊥ ξ FZ (π(R0 )(t)). Thus the conditions of Lemma 2.10 are met and one can compare the global solutions and (R0 ). The immediate result is that = (R0 ). However, (R0 ) belongs to N L q ∩H 1/2 by construction, which finishes the proof. With the help of Lemma 2.11 it is straightforward to prove that N is locally in time invariant. Indeed, consider the truncated solution τ obtained by restricting some global solution , (0) ∈ N , to the time interval [τ, ∞). Keeping the same notations, one has for every t, W (π ) , Z (t) ⊥ ξ FZ (π(t)), (Z , π ) ∈ X δ . =Z+ (2.186) W (π ) Note that Z (t) grows at most linearly in time, so if the condition Z (0) < δ0 from the definition (2.183) of N is met then it still holds for τ close to 0. Assume that δ0 is sufficiently small that, by Hölder’s inequality, Z (t) L 4/3 ∩H 1/2 ≤ CZ (t) < Cδ0 .
(2.187)
Therefore we can apply Lemma 2.11 and obtain that (τ ) = τ (0) ∈ N L q ∩H 1/2 . Since Z (τ ) < δ0 , it follows that (τ ) ∈ N . The same proof cannot yield global in time invariance, because after a strictly positive time the solution may escape the δ0 -neighborhood of the standing wave manifold in which it has to be for the argument to hold. Existence results for initial data in a weaker invariant norm, one that does not grow in time, are needed in order to approach the global result. Similar considerations apply to the proof of local in time invariance for negative time. Consider a global solution with (0) ∈ N . Instead of truncating, now we use local existence theory to get a continuation of to a small interval [−t, 0]. Let −t be the
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M. Beceanu
solution obtained by pasting this onto the original solution. By means of the modulation equations W (π(s)) (s) − , ξ F (π(s)) = 0, (2.188) (s) W (π(s)) we also extend the parameter path π to a small time interval [−t, 0]. At this point we apply Lemma 2.11 exactly as above, eventually concluding that (−t) ∈ N . There is still the issue of checking that N is a centre-stable manifold as in [BatJon]. We prove it in the neighborhood of each standing wave W , since being a centre-stable manifold is a local property. To begin with, we rewrite Eq. (0.1) to make it fit the framework of the theory of Bates, Jones [BatJon]. Consider a fixed ground state φ(·, α0 ) and the constant path π0 = (0, 0, 0, α0 ). Linearizing the equation around this constant path a and applying W (π0 ) , symmetry transformation, as in Lemma 2.2, yields, for U = gπ0 − W (π0 ) that
where
i∂t U + HU = N (U, π0 ),
(2.189)
+ 2φ 2 (·, α0 ) − α0 φ 2 (·, α0 ) H= −φ 2 (·, α0 ) − − 2φ 2 (·, α0 ) + α0
(2.190)
and N (U, π0 ) is as in (2.28). Note that here all the right-hand side terms are at least quadratic in U , due to linearizing around a constant path. The spectrum of H is known, see Sect. 2.3, namely σ (H) = (−∞, −α] ∪ [α, ∞) ∪ {0, ±iσ }. The stable spectrum is −iσ , the unstable spectrum is iσ , and everything else belongs to the centre. It is easy to check that all the conditions of [BatJon] are met, leading to the existence of a centre-stable manifold. In the sequel we prove that N is a centre-stable manifold, namely that it fulfills the three properties enumerated in Definition 1: N is t-invariant with respect to a neighborhood of φ(·, α0 ), π cs (N ) contains a neighborhood of 0 in X c ⊕ X s , and N ∩ W u = {0}. All of this is relative to a specific neighborhood of 0, V = {U | U < δ0 } for some small δ0 . The t-invariance of N relative to V follows from definition and Lemma 2.11, in the same manner in which we proved the local in time invariance of N under the Hamiltonian flow. Then, 0 π cs (N ) = {(P− + Pc + Pr oot )(U (0))} = g − φ(·, α0 ) . (2.191) 0 R0 covers a whole neighborhood of 0 in Ker(P+ ) ∩ Ker(Pr oot ) and the action Since R0 of g is transverse, with the range of its differential at 0 spanned by η F (π0 ), it follows that π cs (N ) is a neighborhood of 0 in Ker(P+ ), the second property that the centre-stable manifold must have. Finally, consider a solution U ∈ W u of (2.189), meaning that U (t) ≤ δ for all negative t and that it decays exponentially as t → −∞, U (t) ≤ CeCt (even though
Focussing Cubic NLS in 3D
185
polynomial decay is sufficient). Decompose U into its projections on the continuous, imaginary, and zero spectrum of H and let δ(T ) = U t−1 L 1 (−∞,T ]L 6/5∩6 + U L 2 (−∞,T ]L 6 ∩L ∞ (−∞,T ]L 2 . t
x
t
x
t
x
(2.192)
Observe that δ(t) → 0 as t → −∞, so we can assume it to be arbitrarily small, δ(t) < 1 to begin with. By means of Strichartz estimates one obtains that Pc U L 2 (−∞,T ]L 6 ∩L ∞ (−∞,T ]L 2 ≤ CN (U, π0 ) L 2 (−∞,T ]L 6/5 +L 1 (−∞,T ]L 2 t
x
t
x
x
t
t
≤ Cδ(T )U L ∞ 2, t (−∞,T ]L x
x
(2.193)
because now the right-hand side N contains only quadratic or higher terms. The same estimate holds for Pim U , because it is bounded at −∞, so we can use Lemma 2.9. We write it in the form t P− U (t) = − e−σ (t−s) P− N (U (s), π0 ) ds, −∞ (2.194) T P+ U (t) = e(t−T )σ P+ U (T ) −
e(t−s)σ P+ N (U (s), π0 ) ds.
t
Therefore Pim U L ∞ 2 ≤ C(P+ U (T )2 + δ(T )U L ∞ (−∞,T ]L 2 ). t (−∞,T ]L x t x
(2.195)
We assumed no orthogonality condition. The modulation equations (2.43) now give Pr oot U and contain only quadratic or higher terms. Note that lim Pr oot U (t) = 0. t→−∞
Therefore Pr oot U L ∞ 2 ≤ CN (U, π0 )t−1 L 1 (−∞,T ]L 1+∞ ≤ Cδ(t)U L ∞ (−∞,T ]L 2 . t (−∞,T ]L x t x t
x
(2.196) Putting these estimates together, one has that U L ∞ 2 ≤ C(δ(T )U L ∞ (−∞,T ]L 2 + P+ U (T )2 ). t (−∞,T ]L x t x
(2.197)
For sufficiently negative T0 , it follows that U (t)2 ≤ CP+ U (t)2 , for any t ≤ T0 . The converse is obviously true, so the two norms are comparable. Furthermore, by reiterating this argument one has that (1 − P+ )U (t)2 ≤ Cδ(t)P+ U (t)2 . Next, assume the stable manifold, meaning that, for some (0) ∈ N , that U is on W (π0 ) . has some moving soliton path π associated to it such U = gπ 0 − W(π0 ) W (π ) that − = Z , π ∈ X δ . Then, W (π ) (t) W (π (t)) W (π(t)) (t) 0 − − |π(t) − π0 (t)| ≤C + (t) W (π(t)) (t) W (π0 (t)) ≤C(δ + |π(∞) − (0, 0, 0, α0 )|),
(2.198)
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M. Beceanu
and therefore π − π0 ∞ ≤ C(δ + |π(∞) − (0, 0, 0, α0 )|). This implies that U (t) L ∞ 2∩3 ≤ C(δ + |π(∞) − (0, 0, 0, α0 )|) t Lx
(2.199)
for positive t and therefore for all t, due to the fact that U ∈ W u . If (0, 0, 0, α0 ) is not the terminal value of the path π , then, since U is a bounded solution to (2.189) at +∞, it follows by Lemma 2.9 that
∞
P+ U (t)2 ≤
e(t−s)σ P+ N (U, π )2 ds ≤ C(δ 2 + |π(∞) − (0, 0, 0, α0 )|2 ).
t
(2.200) Also, because Pr oot U (t)2 is bounded from below, (1 − P+ )U (t)2 ≥ Pr oot U (t)2 ≥ C|π(∞) − (0, 0, 0, α0 )|.
(2.201)
Therefore P+ U (t)/(1 − P+ )U (t) ≤ C(δ + |π(∞) − (0, 0, 0, α0 )|) ≤ C(δ + δ0 ).
(2.202)
If, on the other hand, π approaches (0, 0, 0, α0 ) in the limit, note that in the modulated equation (2.24) valid for (U Z = gπ Z , π ), if δ is sufficiently small, it follows that CPc U Z (0) ≥ Pc U Z (t)2 ≥ C(Pc U Z (0) − Pc U Z (0)2 ) ≥ CPc U Z (0) ≥ Cδ
(2.203)
by means of the Strichartz estimates. Such a lower bound is also implied by scattering. Furthermore, since π and π0 grow near in the +∞ limit, it follows that Pc U Z − Pc U 2 → 0. Therefore the norm of (1 − P+ )U is bounded from below at +∞ in either case, unless Pc Z (0) = 0. However, if it were so, it would imply that Z = 0 and is constant, equal to a soliton. Since − W (π0 ) decays exponentially at −∞, this would imply that they are equal and then U = 0. For P+ U , now we have that Pπ+ U Z (t)− Pπ+0 U (t)1∩∞ → 0 as t → ∞ and therefore P+ U 1∩∞ → 0. Excluding the trivial case U = 0, we obtain that either P+ U (t)/(1 − P+ )U (t) ≤ C(δ + δ0 ) or P+ U (t)/(1 − P+ )U (t) → 0 as t → ∞. This implies that one can make this ratio as small as necessary for some large T0 . Assume that δ < 1. Then Lemma 2.4 from [BatJon] states, under even more general conditions, that if the ratio P+ U (T0 )/(1 − P+ )U (T0 ) is small enough, it will stay bounded for all t ≤ T0 . The proof of this result is based on Gronwall’s inequality. However, this contradicts our previous conclusion that (1 − Pc )U (t)2 /Pc U (t)2 goes to 0 as t goes to −∞. Therefore, U can only be 0. This proves that N ∩ W u = {0}. In other words, there are no exponentially unstable solutions in N in the sense of [BatJon]. The final requirement for N to be a centre-stable manifold is thus met.
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3. Linear Estimates 3.1. The endpoint Strichartz estimate. Consider operators in R3 of the form H = H0 +V , where − + µ 0 −U −W , V = . (3.1) H0 = 0 −µ W U We assume that −σ3 V is a positive matrix, that L − = − + µ + U + W ≥ 0, that |V | ≤ Cx−7/2− , that the spectral Assumption 1 holds, and that the edges of the spectrum ±µ are neither eigenvalues nor resonances. The operator H has σ (H) ⊂ R ∪ iR and σess (H) = (−∞, −µ] ∪ [µ, ∞). We make the spectral assumption that H has no eigenvalues in the set (−∞, −µ)∪(µ, ∞) and that the thresholds ±µ are also regular, meaning that I + (H0 − µ ± i0)−1 V : x1+ L 2 → x1+ L 2 is invertible. Firstly, we need the main result of [KeeTao]: p
Theorem 3.1. Let (X, dµ) be a measure space, L x = L p (X, dµ). Suppose that for each t one has an operator U (t) such that (3.2) U (t) f 2 ≤ C f 2 , U (s)U ∗ (t) f ∞ ≤ C|t − s|−σ f 1 . σ 1 σ = , and let q be the Let σ > 1. Call (q, r ) sharp σ -admissible if q, r ≥ 2, + q r 2 exponent such that q1 + q1 = 1. Then
s
U (t) f L qt L r ≤C f 2 , x ∗ U (s)F(s) ds ≤CF L q L r , 2 t x U (t)U ∗ (s)F ds q r ≤CF q˜ r˜ , L L Lt Lx
t
(3.3) (3.4) (3.5)
x
for any sharp σ -admissible (q, r ), (q, ˜ r˜ ). The following lemma is a straightforward generalization of Theorem 3.1. Since it is important in the sequel, however, a short proof will be given. p
Lemma 3.2. Let (X, dµ) be a measure space, L µ = L p (X, dµ). Suppose that for each t one has operators U (t) and V (t) that satisfy U (t) f 2 ≤ C f H , V (t) f 2 ≤ C f H ,
(3.6)
U (s)V (t) f ∞ ≤ C|t − s|−σ f 1 , U (s)U ∗ (t) f ∞ ≤ C|t − s|−σ f 1 , V ∗ (s)V (t) f ∞ ≤ C|t − s|−σ f 1 .
(3.7)
and
σ 1 σ + = , and let q be the q r 2 = 1. Then, in addition to the Strichartz estimates (3.3-3.5)
Let σ > 1. Call (q, r ) sharp σ -admissible if q, r ≥ 2, exponent such that q1 + q1 for U and V , one has that U (t)V (s)F ds s
for any sharp σ -admissible (q, r ), (q, ˜ r˜ ).
q
L t L rx
≤ CF
q˜
L t L rx˜
(3.8)
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M. Beceanu
Proof. The proof is a rephrasing of the one given in [KeeTao]. Inequalities (3.3-3.5) are already provided, so only (3.8) is left. Consider the bilinear form T (F, G) = U ∗ (s)F(s), V (t)G(t) ds dt. (3.9) s
By interpolation between |U ∗ (s)F(s), V (t)G(t)| ≤ CF(s)2 G(t)2
(3.10)
|U ∗ (s)F(s), V (t)G(t)| ≤ C|t − s|−σ F(s)1 G(t)1
(3.11)
|U ∗ (s)F(s), V (t)G(t)| ≤ C|t − s|−1−β(r,r ) F(s)r G(t)r
(3.12)
and
we obtain σ σ where β(r, r˜ ) = σ − 1 − − . r r˜ Let T j (F, G) = U ∗ (s)F(s), V (t)G(t) ds dt. Then the estimate t−2 j+1 <s≤t−2 j
|T j (F, G)| ≤ C2− jβ(a,b) F L 2 L a G L 2 L b t
t
x
(3.13)
x
holds for all j ∈ Z and all ( a1 , b1 ) in a neighborhood of ( r1 , r1 ). The proof goes through showing (3.13) for the exponents a = b = ∞, (a, 2) with 2 ≤ a < r , and (2, b) with 2 ≤ b < r. One can now infer that |T (F, G)| ≤ F
q
L t L rx
G
q
(3.14)
L t L rx
2σ ). Since (3.14) was already true for the other endσ −1 point by (3.10), it is true for all admissible (q, r ). The general retarded estimate (3.8) follows immediately as in [KeeTao] by interpolation between (3.10), (3.14), and |T (F, G)| ≤ sup U ∗ (s)F(s) ds2 G L 1 L 2 ≤ F q r G L 1 L 2 , (3.15) for the endpoint (q, r ) = (2,
t
t
s
x
Lt Lx
and the symmetric inequality, which are both consequences of (3.4).
t
x
Applying Lemma 3.2 to the families of evolution operators U (t) = eit H Pc , V (s) = we have obtained
e−is H Pc ,
Corollary 3.3. Assume that eit H Pc 2→2 ≤ C
(3.16)
eit H Pc∗ e−is H Pc 1→∞ ≤ C|t − s|−3/2 , ∗ eit H Pc e−is H Pc∗ 1→∞ ≤ C|t − s|−3/2 .
(3.17)
and ∗
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189
Then eit H Pc f L qt L r x −is H Pc F(s) ds e 2 it H −is H ∗ ∗ e Pc e Pc F ds q r Lt Lx s
s
≤C f 2 , ≤CF ≤CF ≤CF
for any sharply admissible (q, r ) (that is, such that q, r ≥ 2, The same estimates hold after swapping H and H∗ .
q
L t L rx
q˜ L t L rx˜ q
, ,
(3.18)
L t L rx
1 3 3 + = ) and (q, ˜ r˜ ). q 2r 4
Coming back to the particular operator H given in (3.1), let Pp be the Riesz projection on the point spectrum and 1 − Pp = Pc be the projection on the continuous spectrum of H. The evolution applied to eigenfunctions may lead to exponential growth in any norm; leaving that aside, one can achieve the above bound for HPc . Indeed, the bounds eit H Pc 2→2 ≤ C
(3.19)
eit H Pc 1→∞ ≤ C|t|−3/2
(3.20)
and
were proved by Schlag in [Sch] and Schlag, Erdogan in [ErdSch]. What is left to prove is ∗
eit H Pc e−is H Pc∗ 1→∞ ≤ C|t − s|−3/2 ,
(3.21)
as well as the symmetric estimate. 3.2. Proof of the strengthened dispersive estimate. Proof. We begin with the following explicit representation derived from [Sch]: for f , g ∈ L 2,1+ , 1 it H φ, e Pc ψ = eitλ φ, R V (λ)ψ dλ, (3.22) 2πi − ∪ + where R V (λ) = (H − λ)−1 and ± are the counterclockwise contours given by ± = {z | d(z, [±µ, ±∞)) = }. The integral can be taken improper, but it is more helpful to consider instead the mollified version ∞ ∞ f (λ) d x = lim f (λ)χ (λ/R) dλ, (3.23) R→∞ 0
0
where χ is a smooth cutoff function with∗ χ (x) = 1 for |x| < 1 and χ (x) = 0 for |x| > 2. A similar formula holds for e−is H Pc∗ , with V replaced by V ∗ . The expression we need to estimate becomes ∗
∗
e−it H Pc∗ φ, e−is H Pc∗ ψ 1 =− 2 ei(tλ−sη) φ, R V (λ)R V ∗ (η)ψ dη dλ. 4π −1 ∪ + −2 ∪ + 1
2
(3.24)
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M. Beceanu
We make the arbitrary choice 1 > 2 . After splitting each contour into + and − , we obtain 4 terms to be treated separately. We begin with the + + term. Expand both R V and R V ∗ into finite Born sums consisting of 2m terms and a remainder. Let R0 (λ) = (H0 − λ)−1 . The expression becomes ∗
φ, eit H P+ e−is H P+∗ ψ 2m−1 2m−1 = (−1)k+ =0
k=0
(3.25) +
1
ei(tλ−sη) φ,
+
2
R0 (λ)(V R0 (λ)) R0 (η)(V R0 (η))k ψ dη dλ 2m−1 + (−1) ei(tλ−sη) φ,
∗
+
=0
1
(3.26)
+
2
R0 (λ)(V R0 (λ)) (R0 (η)V ∗ )m R V ∗ (η)(V ∗ R0 (η))m ψ dη dλ 2m−1 + (−1)k ei(tλ−sη) φ,
+
k=0
1
+
2
(R0 (λ)V )m R V (λ)(V R0 (λ))m R0 (η)(V ∗ R0 (η))k ψ dη dλ ei(tλ−sη) φ, + +
(3.27)
(3.28)
+
2 (R0 (λ)V )m R V (λ)(V R0 (λ))m (R0 (η)V ∗ )m R V ∗ (η)(V ∗ R0 (η))m ψ dη dλ. 1
(3.29)
In each term, the localizing potentials V or V ∗ alternate with the resolvents R0 , with the exception of exactly two resolvent operators following one another. Since this is a potentially dangerous situation, we apply the resolvent identity. For the very simplest term, this means
+
1
=
+
ei(tλ−sη) φ, R0 (λ)R0 (η)ψ dη dλ
2
+
1
ei(tλ−sη) φ, R0 (η)ψ dη dλ − λ−η
+
2
+
1
+
2
ei(tλ−sη) φ, R0 (λ)ψ dη dλ. λ−η (3.30)
Every resulting term has a kernel that can be written, by means of the resolvent identity, as a sum of two parts of the form T =
+
1
+
2
ei(tλ−sη) f (λ)g(η) dη dλ. λ−η
(3.31)
We make f and g explicit later, but for now we continue in this general setting. After making 2 = 0, we get T =
+
1
0
∞
ei(tλ−s(η+µ)) f (λ)(g(η + µ + i0) − g(η + µ − i0)) dη dλ. (3.32) λ − (η + µ)
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Let f s (λ) = 21 ( f (λ + i0) + f (λ − i0)) and f a (λ) = 21 ( f (λ + i0) − f (λ − i0)) and the same for g. We assume that g(λ ± i ) = g(λ ± i0) + O( ) uniformly on compact intervals. Letting 1 go to 0 we have ∞ ∞ T = eit (λ+µ) f s (λ + µ) lim e−is(η+µ) ga (η + µ) dη dλ →0 (λ − η)2 + 2 0 0 ∞ ∞ λ−η eit (λ+µ) f a (λ + µ) lim e−is(η+µ) ga (η + µ) dη dλ + →0 (λ − η)2 + 2 0 0 ∞
ei(t−s)λ f s (λ + µ)ga (λ + µ)+ = ei(t−s)µ 0
(3.33) +eitλ f a (λ + µ)H χ[0,∞) e−isη ga (η + µ) (λ) dλ, where H is the Hilbert transform. The limit exists because we are applying singular kernels to integrable functions of compact support. Any further error terms are in the order of and vanish. Therefore, we need to examine oscillatory integrals of the following form: s , Lemma 3.4. Assume that Fs is even and Fa and G a are odd functions and that F ∈ M (the space of finite measures) and likewise for Fa and G a . Then F s ∞ √ √ 1 + F 1 G s 1 G a 1 ), ei(t−s)λ Fs ( λ)G a ( λ) dλ ≤ C|t − s|−3/2 ( F a s 0
(3.34) as well as
∞
0
√ √ eitλ Fa ( λ)H (e−isη χ[0,∞) G a ( η))(λ) dλ
1 + a 1 ). Fa 1 G Fa 1 G ≤ C|t − s|−3/2 ( a
(3.35)
The integrals on the left-hand side are improper and computed with √ the help of a smooth cutoff. Under the assumptions, Fa ∈ C 1 (R), so χ[0,∞) Fa ( λ) ∈ C 1/2 , and likewise for G a . Proof. For (3.34) we have (3.34) =
∞
2
ei(t−s)λ λFs (λ)G a (λ) dλ −∞ ∞ 1 2 = ei(t−s)λ (Fs (λ)G a (λ)) dλ t − s −∞ 1 + F 1 G s 1 G a 1 ). ≤ C|t − s|−3/2 ( F a s
(3.36)
For terms of the form (3.35) a slight refinement is needed. First note that ∞ √ √ 1 ∞ −iτ λ2 (χ[0,∞) Fa ( ·))∧ (τ ) = e−iτ λ Fa ( λ) dλ = e λFa (λ) dλ 2 −∞ 0 ∞ ∞ 1 1 2 2 = e−iτ λ Fa (λ) dλ = √ τ −3/2 eiξ /τ Fa (ξ ) dξ. (3.37) 3/2 2τ −∞ 4 2π −∞
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M. Beceanu
The same goes for G a . Letting s − t = k we have (3.35) ∞ √ √ (eit· χ[0,∞) Fa ( ·))∧ (τ )(H (e−is· χ[0,∞) G a ( ·)))∧ (−τ ) dτ = −∞ ∞ √ √ = (χ[0,∞) Fa ( ·))∧ (τ − t)(χ[0,∞) G a ( ·))∧ (−τ + s) sgn(τ ) dτ −∞ b √ ∧ √ ∧ (χ[0,∞) Fa ( ·)) (τ )(χ[0,∞) G a ( ·)) (k − τ ) dτ ≤ 2 sup a,b
a
b ∞ ∞ 2 η2 1 −3/2 −3/2 i( λτ + k−τ ) . = sup |τ | |k − τ | e (ξ ) G (ν) dν dξ dτ F a a 16π 3 a,b a −∞ −∞ (3.38) We apply the stationary phase method to this integral (see, for example, [Ste], p. 332). We write the proof explicitly because neither the phase, nor the integrand is absolutely integrable. b ν2 ξ2 Consider + and ψ = |τ |−3/2 |k − τ |−3/2 . The aim eiφ ψ dτ , where φ = τ k−τ a is to prove that b iφ ≤ C|k|−3/2 1 + 1 . (3.39) e ψ dτ |ξ | |ν| a Without loss of generality, let ξ , ν, k > 0. First assume [a, b] ⊂ [0, k] and note that τ2 τ2 τ2 dτ iφ e ψ dτ ≤ = (τ ) (3.40) , 1 3/2 (k − τ )3/2 τ1 τ τ1 τ1 1 (τ 1/2 (k − τ )−1/2 − τ −1/2 (k − τ )1/2 ). Indeed, 2k 2 1 (τ 1/2 (k − τ )−1/2 − τ −1/2 (k − τ )1/2 ) = (τ −1/2 (k − τ )−1/2 + τ 1/2 (k − τ )−3/2 2 +τ −3/2 (k − τ )1/2 + τ −1/2 (k − τ )−1/2 ) (k − τ + τ )2 = 3/2 . (3.41) 2τ (k − τ )3/2
where the antiderivative is 1 (τ ) =
On any interval not containing a stationary point, moreover, one has τ2 iφ ≤ C sup ψ(τ ) , e ψ dτ τ1 τ ∈[τ1 ,τ2 ] φ (τ )
(3.42)
where, by the convexity of 1/x, ψ(τ ) 1 τ 1/2 (k − τ )1/2 1 1 1 τ 1/2 (k − τ )1/2 ≤ + . (3.43) = φ (τ ) ντ + ξ(k − τ ) |ντ − ξ(k − τ )| k ν ξ |ντ − ξ(k − τ )| ψ(τ ) The reason is that, after integration by parts, changes sign at most a constant φ (τ ) number of times, so one can integrate and only lose some constant.
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193
The last expression in (3.43) is 0 at the endpoints 0 and k. Note that the phase derivaξk . Surround tive φ vanishes at exactly one point in the interval [0, k], namely τ0 = ν+ξ τ0 with a small interval [τ1 , τ2 ] on which we estimate the integral in absolute value and otherwise integrate by parts as above. We obtain that b ψ(τ ) iφ ≤C + | max e ψ dτ (τ ) − (τ )| . (3.44) 1 2 1 1 τ ∈[0,τ1 ]∪[τ2 ,k] φ (τ ) a Choose τ1 and τ2 such that 1 (τ2 ) − 1 (τ0 ) = k
−3/2
1 1 + ν ξ
(3.45)
and likewise for τ1 , which takes care of the last term. This choice is possible and is unique due to the fact that 1 (τ ) → ±∞ as τ → k− and 0+, respectively, and 1 is strictly increasing. Therefore, for τ ∈ [0, τ1 ] ∪ [τ2 , k], 1 −3/2 1 + . (3.46) |1 (τ ) − 1 (τ0 )| ≥ k ν ξ It is left to prove that, under this condition, ψ(τ ) 1 1 + . ≤ Ck −3/2 φ (τ ) ν ξ
(3.47)
Rewrite condition (3.46) as 1 1/2 1/2 |ν τ − ξ 1/2 (k − τ )1/2 | · |ν −1/2 (k − τ )−1/2 − ξ −1/2 τ −1/2 | k2 1 1 + , ≥ Ck −3/2 ν ξ and further as |ντ − ξ(k − τ )| ≥ Ck 1/2 (k − τ )1/2 τ 1/2
1 1 + ν ξ
(ν 1/2 τ 1/2 + ξ 1/2 (k − τ )1/2 )ν 1/2 ξ 1/2 . |ν 1/2 (k − τ )1/2 − ξ 1/2 τ 1/2 |
Then it suffices to prove that 1 1 (ν 1/2 τ 1/2 + ξ 1/2 (k − τ )1/2 )ν 1/2 ξ 1/2 + ≥C ν ξ |ν 1/2 (k − τ )1/2 − ξ 1/2 τ 1/2 |
(3.48)
(3.49)
(3.50)
or equivalently (ν + ξ )(νξ 1/2 τ 1/2 + ξ ν 1/2 (k − τ )1/2 ) ≥ Cνξ |ν 1/2 (k − τ )1/2 − ξ 1/2 τ 1/2 |
(3.51)
which is obvious. The same goes for τ2 . Next, assume that [a, b] ⊂ [k, ∞). The proof goes along the same lines, based on (3.40) and on (3.42), with the difference that the antiderivative is now written 2 =
1 (τ 1/2 (τ − k)−1/2 + τ −1/2 (τ − k)−1/2 ) 2k 2
(3.52)
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M. Beceanu
and that ψ(τ ) τ 1/2 (τ − k)1/2 1 = . φ (τ ) ντ + ξ(τ − k) ντ − ξ(τ − k)
(3.53)
If ν > ξ the phase has no stationary points in this interval. We divide [k, ∞) into the subintervals on which 2 (τ ) ≤ ξ −1 k −3/2 and the rest. On the former the integral is trivially bounded and the number of such intervals is bounded from above. On the latter (also the union of a bounded number of intervals) one has, following integration by parts, τ ψ(τ ) 2 iφ ≤C (3.54) max e ψ dτ . τ ∈[k,∞) φ (τ ) τ1 2 (τ )≥ξ −1 k −3/2
However, the condition 2 (τ ) ≥ ξ −1 k −3/2 implies that ξ(2τ − k) ≥ Ck 1/2 , τ 1/2 (τ − k)1/2
(3.55)
τ 1/2 (τ − k)1/2 1 1 ≤ Ck −1/2 . ντ + ξ(τ − k) (ν − ξ )τ + ξ k ξk
(3.56)
and therefore
ξk , which becomes infinite if ξ −ν ξ = ν. Surround it with an interval [τ1 , τ2 ] on which we integrate the absolute value, otherwise integrate by parts. Overall, the integral is bounded by b ψ(τ ) iφ (3.57) e ψ dτ ≤ C |2 (τ2 ) − 2 (τ1 )| + max . τ ∈[k,τ1 ]∪[τ2 ,∞) φ (τ ) a A stationary point occurs if ν < ξ , namely τ0 =
Choose τ1 such that |2 (τ1 ) − 2 (τ0 )| = k −3/2
1 ν
(3.58)
and likewise for τ2 (or τ2 = ∞ if there is no such value). This takes care of the integral on [τ1 , τ2 ]. As for the remaining portion, note that for any τ ∈ [k, τ1 ] ∪ [τ2 , ∞) one has 1 |2 (τ1 ) − 2 (τ0 )| ≥ k −3/2 , ν
(3.59)
and therefore 1 1 1/2 1/2 |ν τ − ξ 1/2 (τ − k)1/2 | · |(τ − k)−1/2 ν −1/2 − τ −1/2 ξ −1/2 | ≥ Ck −3/2 . 2 k ν (3.60) Equivalently, under the assumption ν < ξ , |ντ − ξ(τ − k)| k 1/2 (τ 1/2 ν 1/2 + (τ − k)1/2 ξ 1/2 )ξ 1/2 ν 1/2 ≥C . 1/2 1/2 (τ − k) τ ν τ 1/2 ξ 1/2 − (τ − k)1/2 ν 1/2
(3.61)
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195
Note that ψ(τ ) 1 (τ − k)1/2 τ 1/2 . ≤ φ (τ ) νk |ντ − ξ(τ − k)|
(3.62)
ν(τ 1/2 ξ 1/2 − (τ − k)1/2 ν 1/2 ) ≤ C. (τ 1/2 ν 1/2 + (τ − k)1/2 ξ 1/2 )ξ 1/2 ν 1/2
(3.63)
It is left to prove that
However, the last statement is equivalent to τ 1/2 ξ 1/2 ν − (τ − k)1/2 ν 3/2 ≤ C(τ 1/2 ξ 1/2 ν + (τ − k)1/2 ξ ν 1/2 )
(3.64)
which is again obvious. The third case [a, b] ⊂ (−∞, 0] is identical to the second case [a, b] ⊂ [k, ∞). Cutting the interval [a, b] into at most three pieces according to this partition, one obtains the conclusion (3.35). The lemma relates to (3.33) in the following manner: when applying it to (3.33), we take Fs,a (λ) = 21 f ((λ + i0)2 + µ) ± f ((λ − i0)2 + µ) and likewise for G a . This leads to a bound for the conditionally converging integral T (3.31). After this general discussion of (3.31), we return to the concrete Born sum expansion (3.26-3.29), expanded again, as previously stated, by means of the resolvent identity. Now we identify Fs , Fa , and G a for this case. From the Born sum expansion (3.26-3.29) and from the known expression for the kernel of the free resolvent 1 exp(±|x − y|λ) 0 R0 (λ2 + µ ± i0)(x, y) = 0 exp(−|x − y| λ2 + 2µ) 4π |x − y| (3.65) (where factors:
√
is given by the main branch of the logarithm) we get the following types of
Fs (λ) = cos(Aλ) exp(−B λ2 + 2µ) (a) or ((R0 (λ2 + µ)V )m R V (λ2 )(V R0 (λ2 + µ))m )s (b) or ((R0 (λ2 + µ)V)m R V (λ2 + µ)(V R0 (λ2 + µ))m−1 )s (c), Fa (λ) = sin(Aλ) exp(−B λ2 + 2µ) (d) or ((R0 (λ2 + µ)V )m R V (λ2 + µ)(V R0 (λ2 + µ))m )a (e) or ((R0 (λ2 + µ)V)m R V (λ2 + µ)(V R0 (λ2 + µ))m−1 )a ( f ), G a (λ) = sin(Cλ) exp(−D λ2 + 2µ) (g) or ((R0 (λ2 + µ)V ∗ )m R V ∗ (λ2 + µ)(V ∗ R0 (λ2 + µ))m )a (h) or ((R0 (λ2 + µ)V ∗ )m−1 R V ∗ (λ2 + µ)(V R0 (λ2 + µ))m )a (i),
(3.66)
where we again used the notation f s (λ) = 21 ( f (λ + i0) + f (λ − i0)) and f a (λ) = 1 2 ( f (λ + i0) − f (λ − i0)). Factors of the form (a), (d), and (g) stem from the general terms in the Born sum expansion, as in (3.26), while the others represent the contribution of the remainders and mixed terms (3.27-3.29).
196
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We evaluate the first type of factors, coming from (3.26), in view of applying Lemma 3.4. Note that, uniformly in A and B,
∧ sin(Aλ) exp(−B λ2 + 2µ) 1 ≤ (sin(Aλ))∧ M (exp(−B λ2 + 2µ))∧ 1 ≤ C, (3.67) where M is the space of finite measures. This is true because (sin(Aλ))∧ is the sum of two point measures of mass 1/2i, while + 2µ)) 1 =
√ ∞ 2 e−B λ +2µ e−iτ λ dλ 1 (exp(−B Lτ −∞ ∞ √ √ 2 2 2 2 = e− η +2µB e−iτ η dη 1 ≤ Ce− η +2µB Hη1 ≤ Ce−|η| H 1 < ∞.
−∞
λ2
∧
Lτ
(3.68)
Similarly, d
∧ sin(Aλ) exp(−B λ2 + 2µ) 1 dλ
∧ ≤ A cos(Aλ) exp(−B λ2 + 2µ) 1 ∧ sin(Aλ) Bλ2 2 exp(−B λ + 2µ) ≤ C A, + 1 λ λ2 + 2µ
(3.69)
because sin(Aλ) ∧ 1 = χ[−A,A] 1 ≤ C A 1 λ 2
(3.70)
and ∧ Bλ2 exp(−B λ2 + 2µ) 2 1 λ + 2µ ∞ √ Bλ2 2 e−B λ +2µ e−iτ λ dλ 1 = 2 Lτ λ + 2µ −∞ ∞ √ 2 η 2 2 = e− η +2µB e−iτ η dη 1 Lτ η2 + 2µB 2 −∞ √ 2 η 2 2 ≤ C e− η +2µB 1,2 2 2 Wη η + 2µB ≤ |η|e−|η| W 1,2 < ∞. With sine replaced by cosine, the analogous estimate is ∧ d exp(−B λ2 + 2µ) ≤ C. cos(Aλ)λ 1 dλ
(3.71)
(3.72)
Focussing Cubic NLS in 3D
197
This takes care of factors of the form (a), (d), or (g). We proceed to obtain a bound for the remaining factors. Intuitively, since they represent the remainder in the Born sum expansion, the bound should be less sharp. Let −iλ|x| 0 e R0 (λ2 + µ)(x, x1 ) G x (λ2 )(x1 ) = 0 1 1 0 exp(i(|x − x1 | − |x|)λ) = 0 exp(−|x − x1 | λ2 + 2µ) 4π |x − x1 | (3.73) and
ax,y (λ2 ) = V R V (λ2 + µ)V (R0 (λ2 + µ)V )m G y (λ2 ), (R0∗ (λ2 + µ)V ∗ )m G ∗x (λ2 ) . (3.74)
The following estimates hold for G: sup ∂λ G x (λ2 )x1 3/2+ j+ L 2x ≤ Cx−1 , sup ∂λ G x (λ2 )x1 1/2+ j+ L 2x ≤ C, j
j
1
x
x
1
(3.75)
implying that j d 2 −2− a (λ ) x−1 y−1 for j = 0, 1, dλ j x,y ≤Cλ j d 2 −2− for j = 2, dλ j ax,y (λ ) ≤Cλ
(3.76)
provided that m is sufficiently large and V has sufficient decay, |V | ≤ Cx−7/2− . We used the limiting absorption principle Lemma 2.8 to bound this quantity. Incrementing m by 1 increases the decay by λ−1/2 . The decay condition on V arises as follows: if, for example, two derivatives fall on the R V factor, then V has to compensate for 3 + powers of x from its output and for another 1/2 + power coming from the pairing with R0 . Then the factors of the form (b) and (e) can be written as K V s (λ) =
1 (K V ((λ + i0)2 ) + K V ((λ − i0)2 ), 2
(3.77)
K V a (λ) =
1 (K V ((λ + i0)2 ) − K V ((λ − i0)2 ), 2
(3.78)
respectively
where K V (λ2 )(x, y) = ((R0 (λ2 + µ)V )m R V (λ2 + µ)(V R0 (λ2 + µ))m )(x, y) iλ|y| iλ|x| 0 0 e e 2 a (λ ) . (3.79) = 0 1 x,y 0 1 Clearly K V a 1 = (ax,y (λ2 ))∧ 1 ≤ ax,y (λ2 )W 1,2 (R) ≤ Cx−1 y−1 ,
(3.80)
198
M. Beceanu
√ and, taking into account the fact that K V a (λ) is of the form Fa ( λ) for antisymmetric Fa , d ax,y (λ2 )W 1,2 (R) + (|x| + |y|)(ax,y (λ2 ))∧ 1 ≤ C. K V a 1 ≤ dλ Moreover, K ∧ Va K V a (ξ )1 ≤ C K V a 1 ≤ C. ≤ Cξ 1 λ
(3.81)
(3.82)
These decay estimates carry on to factors of the form (b) or (e) and, with minimal modifications, to factors of the form (c), (f), (h), and (i), provided that m is sufficiently large. After showing that all types of terms arising from (3.66) satisfy the prerequisites for applying Lemma 3.4, we apply it to each of them, in turn. We first deal with the 4m 2 terms in (3.26). As mentioned previously, each splits into two parts after using the resolvent identity. Let us introduce the following notations: k+−2 j=1 V j (x j ) K (x0 , . . . , xk+−1 ) = k+−2 , (3.83) j=0 |x j+1 − x j | where V j are entries of V (that is, ±U or ±W ); |x j+1 − x j |, B = |x j+1 − x j |, A= j∈J c
j∈J
C=
|x j+1 − x j |, D =
j∈I
|x j+1 − x j |,
(3.84)
j∈I c
where J ⊂ {0, . . . , − 1} and J c = {0, . . . , − 1}\J ; I ⊂ {, . . . , + k − 2} and I c = {, . . . , + k − 2}\I . Also, let Fs (λ) = cos(Aλ) exp(−B λ2 + 2µ), (3.85) Fa (λ) = sin(Aλ) exp(−B λ2 + 2µ), G a (λ) = sin(Cλ) exp(−D λ2 + 2µ). After applying the resolvent identity and performing the matrix multiplication, we find that each term of (3.26) is a sum, for all possible choices of J and I , of terms with kernel of the form ∞
√ √ ei(t−s)λ Fs ( λ)G a ( λ) K (x0 , . . . , xk+−1 ) K k−1 (x0 , xk+−1 ) = R3(k+−2) 0 √ √ itλ −isη + e Fa ( λ)Hη (χ[0,∞] e G a ( η))(λ) dλ d x1 . . . d xk+−2 . (3.86) Then, in view of Lemma 3.4 and the bounds (3.67), (3.69), and (3.72), we have that ∞ √ √ √ √ ei(t−s)λ Fs ( λ)G a ( λ) + eitλ Fa ( λ)H (χ[0,∞] e−isη G a ( η))(λ) dλ 0
≤ C|t − s|−3/2 (A + C) ≤ C|t − s|−3/2
k+−2 j=0
|x j+1 − x j |.
(3.87)
Focussing Cubic NLS in 3D
199
Combining this with Lemma 2.5, p. 12 of [RodSch], which states that
R3(k+−2)
K (x0 , . . . , xk+−1 )
k+−2
|x j+1 − x j | d x1 . . . d xk+−2 ≤ (k + )V k+−1 , K
j=0
(3.88) where V K = sup
x∈R3
|V (y)| dy is the Kato norm, we obtain that |x − y|
K k−1 1→∞ ≤
sup
x0 ,xk+−1
K k−1 (x0 , xk+−1 ) ≤ C|t − s|−3/2 .
(3.89)
The same method can be applied to the remaining terms in (3.27), (3.28), and (3.29). Since (3.27) and (3.28) are similar, we look at a typical term of one of these two sums, consisting of the product between some term of the Born sum expansion, on one hand, and the remainder, on the other. The kernel of such a term is of the form K k−1 V ∗ (x0 , xk−1+2m ) = K (x0 , . . . , xk−1 )V (xk−1 ) · R3(k−2+2m)
∞
·
√ √ ei(t−s)λ Fs ( λ)eitλ K V ∗ a (xk−1 , . . . , xk−1+2m , λ)
0
√ √ + eitλ Fa ( λ)Hη (χ[0,∞] e−isη K V ∗ a (xk−1 , . . . , xk−1+2m , η))(λ) dλ d x1 . . . xk−2+2m , (3.90) where Fa , Fs are as in (3.85). The kernel K V ∗ a involves R V ∗ and is not given by an explicit formula as Fa is. However, we still have estimates (3.67), (3.69), (3.72), as well as (3.80), (3.81), (3.82), based on the limiting absorption principle. We could treat the pairing Fs K V ∗ a √ using only inequalities of the form √ (3.80), (3.81), ∗ because K (x0 , . . . , xk−1 )V (xk−1 )Fs ( λ)K V a (xk−1 , . . . , xk−1+2m , (λ)) is the sum of two antisymmetric kernels. Otherwise, keeping within the previous framework, note that we can take a factor of λ from the antisymmetric part over to the symmetric part whenever that is needed. By Lemma 3.4, ∞ √ √ ei(t−s)λ Fs ( λ)eitλ K V ∗ a (xk−1 , . . . , xk−1+2m , λ) 0
√ √ +eitλ Fa ( λ)Hη (χ[0,∞] e−isη K V ∗ a (xk−1 , xk−1+2m , η))(λ) dλ d xk . . . d xk−2+2m ⎞ ⎛ k−2 ≤ C|t − s|−3/2 ⎝ |x j+1 − x j | + 1⎠ .
(3.91)
j=0
This estimate, to which we now add back those factors of (3.90) that we omitted for convenience in (3.91), results in
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M. Beceanu
|K k−1 V ∗ (x0 , xk−1+2m )| k−2 ≤ C|t − s|−3/2 K (x0 , . . . , xk−1 )|V (xk−1 )|( |x j+1 − x j | + 1) d x1 . . . d xk−1 j=0
k−2 −3/2 ≤ C|t − s| V 1 sup K (x0 , . . . , xk ) |x j+1 − x j | d x1 . . . d xk−2 x k−1
j=0
−3/2 −3/2 +C|t − s| . K (x0 , . . . , xk−1 )V (xk−1 ) d x1 . . . d xk−1 ≤ C|t − s| (3.92)
Thus we have proved that K k−1 V ∗ 1→∞ ≤ C|t − s|−3/2 . Finally, the last term appearing in (3.29), consisting of the product of the remainders in the Born sum expansion, yields to the same approach. The final step of the computation, instead of (3.92), is |K V
V ∗ (x 0 , x 4m−1 )|
≤ C|t − s|−3/2
x0 −1 x2m −1 V (x2m ) + V (x2m )x2m −1 x4m−1 −1 d x2m
≤ C|t − s|−3/2 V 1 ≤ C|t − s|−3/2 .
(3.93) ∗
This completes the proof of the fact that eit H P+ e−is H P+∗ 1→∞ ≤ C|t − s|−3/2 . The other 3 combinations are entirely analogous. Indeed, the P− P− term can be treated by the same means. As for the mixed terms, a very similar approach works and we present the proof in brief. Again, we expand both factors into finite Born series and we obtain a sum with (2m + 1)2 terms. Each term has a kernel that can be written, by means of the resolvent identity, as a sum of 2 parts of the form T =
+
1
−2
ei(tλ−sη) f (λ)g(η) dη dλ. λ−η
(3.94)
After making 1 = 2 = 0, we get, by analogy to (3.33), T = 0
∞
eitλ f a (λ + µ)H (χ(−∞,0] e−isη ga (η − µ))(λ) dλ,
(3.95)
where H is the Hilbert transform. The other term, involving the Dirichlet kernel, cancels, because now the contours surround disjoint regoins. However, now we note upon inspection that the terms stemming from χ(−∞,0] ga (η − µ) have the same form as those we have already enumerated in (3.66). From here the proof proceeds in the same manner as in the previous case. The bound that we eventually obtain for these terms is even better, C|t + s|−3/2 instead of C|t − s|−3/2 , because the phases add instead of cancelling.
Focussing Cubic NLS in 3D
201
3.3. Other linear estimates. Proof of Corollary 1.5. The dispersive estimate (3.21) holds for H∗ as well as H, since they are conjugated by σ3 . This, together with the L 2 bound (3.19), implies the Keel-Tao endpoint Strichartz estimate. Next, by interpolating between the endpoint Strichartz estimate for L 2 initial data and the L 1 → L ∞ decay estimates, we achieve an improved decay of the solution, in norm, for L p initial data, 1 ≤ p ≤ 2. † † # $ c a a ac ad . In the sequel, denote = a b , so that = d b b bc bd Lemma 3.5. For 1 ≤ q ≤ 2, ∞ eit H Pc U1 (eit H Pc U2 )† T
3q 2(q−1)
dt ≤ C T
2− q4
U1 q U2 q .
(3.96)
Likewise, for 1 ≤ q < 4/3, β < 2/q − 1, ∞ 2− 4 +2β t2β eit H Pc U1 (eit H Pc U2 )† 3+ 3q dt ≤ CT q U1 q∩2 U2 q∩2 . 2(q−1)
T
(3.97) Proof. We obtain the first result by complex bilinear interpolation (see [BerLöf], p. 96, Theorem 4.4.1). We use it in the following form: Theorem 3.6. For i = 0, 1, let B : Ai ⊕ Bi → Ci be a bilinear mapping such that B(a, b)Ci ≤ Mi a Ai b Bi .
(3.98)
Then, for each 0 ≤ θ ≤ 1, B can be extended uniquely to a bilinear mapping from [A0 , A1 ][θ] ⊕ [B0 , B1 ][θ] to [C0 , C1 ][θ] with norm at most M0θ M11−θ . The first estimate (3.96) then follows directly if we take B(U1 , U2 ) = eit H Pc U1 (eit H Pc U2 )†
(3.99)
and interpolate between ∞ eit H Pc U1 (eit H Pc U2 )† ∞ dt ≤ C T −2 U1 1 U2 1
(3.100)
T
and
∞
eit H Pc U1 (eit H Pc U2 )† 3 dt ≤ CU1 2 U2 2 .
(3.101)
T
The second statement (3.97) follows from the first, once we write ∞ t2β eit H Pc U1 (eit H Pc U2 )† 3+ 3q dt 2(q−1) T ∞ eit H Pc U1 (eit H Pc U2 )† 3+ 3q dt ≤ T 2β 2(q−1) T ∞ ∞ t2β−1 eis H Pc U1 (eis H Pc U2 )† 3+ +C T
t
3q 2(q−1)
ds dt.
(3.102)
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M. Beceanu
From this we derive an inequality concerning the solution of the inhomogenous problem. Lemma 3.7. Consider the equation i∂t U + HPc U = R H S(t), U (0) given.
(3.103)
Then, for q < 4/3, β < 2/q − 1, ∞ t2β Pc U 26+∞ dt ≤ C(U (0)q∩2 + R H St−β L 2 L 1∩6/5 )2 . t
0
x
(3.104)
Note that for the exponent of interest, β = 1/2, it is possible to replace ∞ by 12 + and 1 by 6/5 − , if q is sufficiently close to 4/3. Proof. Firstly, we examine the source terms. Setting U1 = U2 in (3.97), one has ∞ 2 t2β eit H Pc U (0)2 3q dt ≤ CT 2−4/q+2β U (0)q∩2 , (3.105) 6+ q−1
T
6+
3q
and note that L q−1 ⊂ L 6+∞ . We then evaluate the inhomogenous terms. They have L 2 in time decay, as will follow from (we henceforth denote min(a, b) = a ∧ b) ∞ t 2 2β t ei(t−s)H Pc R H S(s) ds dt 6+∞ 0 ∞0 t 2 ≤C t2β ei(t−s)H Pc R H S(s) ds dt 6 0 t/2∧(t−1) ∞ t/2∧(t−1) 2 2β i(t−s)H t e Pc R H S(s) ds dt . (3.106) + 0
∞
0
Now we examine the two expressions separately. Concerning the first, note that what one needs to prove is equivalent to ∗ t−2β s2β e−is H Pc F1 (s), e−it H F2 (t) ds dt t/2∧(t−1)≤s≤t
≤ CF1 L 2 L 6/5 F2 L 2 L 6/5 . s
x
t
(3.107)
x
However, observe that this follows by the same means as the usual Keel-Tao endpoint Strichartz estimate, by a dyadic partition, because the extra t−2β s2β factor is bounded. The second term can be handled as follows: t/2∧(t−1) ∞ t2β ei(t−s)H Pc R H S(s) ds2∞ dt 0
0
≤
∞
t 2β−3
0
≤ 0
t/2∧(t−1) 0
∞
s2β
∞ s+1
provided that 2β < 2.
s2β R H S(s)21 ds
t/2∧(t−1)
s−2β ds
0
t 2β−3 dt R H S(s)21 ds ≤
0
∞
s2β R H S(s)21 ds, (3.108)
Focussing Cubic NLS in 3D
203
It follows that for β < 1, t 2β t ei(t−s)H Pc R H S(s) ds L 2 L 6+∞ ≤ s2β R H S(s) L 2 L 1∩6/5 . (3.109) t
0
x
s
x
This suffices to bound the product of the two inhomogenous terms. The product of a source term and an inhomogenous term can be handled in the same manner, ∞ t t2β eit H Pc U (0)6+∞ ei(t−s)H Pc R H S(s) ds dt T
0
∞
≤
t2β eit H Pc U1 (0)2
3p 2( p−1)
T ∞
T
dt
t 2 t2β ei(t−s)H Pc R H S 2 (s) ds
6+∞
0
≤ U (0)q∩2 R H St−β L 2 L 1∩6/5 . t
x
6+∞
1/2 1/2 dt
(3.110)
Acknowledgement. I would like to thank Professor Wilhelm Schlag for his suggestions and for his very careful reading of this paper.
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204
[CucPelVou] [DemSch] [ErdSch] [GeJoLaSt] [Gla] [GrShSt1] [GrShSt2] [HunLee] [KeeTao] [KenMer] [Kel] [KriSch1] [KriSch2] [KriSch3] [Kwo] [McLSer] [Mer] [MerRap] [Per] [PilWay] [RodSch] [RoScSo1] [RoScSo2] [Sch] [Sch2] [SofWei1] [SofWei2] [Ste] [SulSul] [Tay]
M. Beceanu
Cuccagna, S., Pelinovsky, D., Vougalter, V.: Spectra of positive and negative energies in the linearized NLS problem. Comm. Pure Appl. Math. 58(1), 1–29 (2005) Demanet, L., Schlag, W.: Numerical verification of a gap condition for a linearized NLS equation. Nonlinearity 19, 829–852 (2006) Erdogan, B., Schlag, W.: Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II. To appear in Journal d’Analyse Mathematique, available at http://arxiv.org/list/math/0504585, 2005 Gesztesy, F., Jones, C.K.R.T., Latushkin, Y., Stanislavova, M.: A spectral mapping theorem and invariant manifolds for nonlinear schrödinger equations. Indiana Univ. Math. J. 49(1), 221–243 (2000) Glassey, R.T.: On the blowing-up of solutions to the cauchy problem for the nonlinear schrödinger equation. J. Math. Phys. 18, 1794–1797 (1977) Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. i. J. Funct. Anal. 74(1), 160–197 (1987) Grillakis, M., Shatah, J., Strauss, W.: Stability theory of solitary waves in the presence of symmetry. ii. J. Funct. Anal. 94(1), 308–348 (1990) Hundertmark, D., Lee, Y.-R.: Exponential decay of eigenfunctions and generalized eigenfunctions of a non self-adjoint matrix schrödinger operator related to NLS. Bull. London Math. Soc. 39(5), 709–720 (2007) Keel, M., Tao, T.: Endpoint strichartz estimates. Amer. Math. J. 120, 955–980 (1998) Kenig, C., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear schrödinger equation in the radial case. Inv. Math. 166(3), 645– 675 (2006) Kelley, P.L.: Self-focusing of optical beams. Phys. Rev. Lett. 15, 1005–1008 (1965) Krieger, J., Schlag, W.: Stable manifolds for all monic supercritical NLS in one dimension. J. AMS 19(4), 815–920 (2006) Krieger, J., Schlag, W.: Non-generic blow-up solutions for the critical focusing NLS in 1-d. http://arxiv.org/list/math/0508576, 2005 Krieger, J., Schlag, W.: On the focusing critical semi-linear wave equation. Amer. J. Math. 129(3), 843–913 (2007) Kwong, M.K.: Uniqueness of positive solutions of u − u + u p = 0 in Rn . Arch. Rat. Mech. Anal. 65, 243–266 (1989) McLeod, K., Serrin, J.: Nonlinear schrödinger equation. uniqueness of positive solutions of u + f (u) = 0 in Rn . Arch. Rat. Mech. Anal. 99, 115–145 (1987) Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear schrödinger equations with critical power. Duke Math. J. 69(2), 427–454 (1993) Merle, F., Raphael, P.: On a sharp lower bound on the blow-up rate for the L 2 critical nonlinear schrödinger equation. J. Amer. Math. Soc. 19(1), 37–90 (2006) Perelman, G.: On the formation of singularities in solutions of the critical nonlinear schrödinger equation. Ann. Henri Poincaré 2(4), 605–673 (2001) Pillet, C.A., Wayne, C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Diff. Eq. 141(2), 310–326 (1997) Rodnianski, I., Schlag, W.: Time decay for solutions of schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004) Rodnianski, I., Schlag, W., Soffer, A.: Dispersive analysis of charge transfer models. Comm. Pure and Appl. Math. 58(2), 149–216 (2004) Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of N-soliton states of NLS. http:// arxiv.org/list/math/0309114, 2003 Schlag, W.: Stable Manifolds for an orbitally unstable NLS. http://arxiv.org/list/math/ 0405435, 2004 Schlag, W.: Spectral theory and nonlinear partial differential equations: a survey. Discrete Contin. Dyn. Syst. 15(3), 703–723 (2006) Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering for nonintegrable equations. Comm. Math. Phys. 133, 119–146 (1990) Soffer, A., Weinstein, M.I.: Multichannel nonlinear scattering, ii. The Case of Anisotropic Potentials and Data. J. Diff. Eq. 98, 376–390 (1992) Stein, E.: Harmonic Analysis, Princeton, NJ: Princeton University Press, 1994 Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse. Applied Mathematical Sciences, 139, New York: Springer-Verlag, 1999 Taylor, M.E.: Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials. Mathematical Surveys and Monographs, 81, Providence, RI: Amer. Math. Soc., 2000
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Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive equations. Comm. Pure Appl. Math. 39(1), 51–67 (1986) Weinstein, M.I.: Modulational stability of ground states of nonlinear schrödinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985) Yajima, K.: Dispersive estimate for schrödinger equations with threshold resonance and eigenvalue. Commun. Math. Phys. 259(2), 475–509 (2005)
Communicated by P. Constantin
Commun. Math. Phys. 280, 207–261 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0446-0
Communications in
Mathematical Physics
Open-Closed Field Algebras Liang Kong1,2, 1 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103 Leipzig, Germany 2 Institut Des Hautes Études Scientifiques, Le Bois-Marie, 35, Route De Chartres,
F-91440 Bures-sur-Yvette, France. E-mail: [email protected] Received: 26 April 2007 / Accepted: 10 August 2007 Published online: 6 March 2008 – © The Author(s) 2008
Abstract: We introduce the notions of open-closed field algebra and open-closed field algebra over a vertex operator algebra V . In the case that V satisfies certain finiteness and reductivity conditions, we show that an open-closed field algebra over V canonically gives an algebra over a C-extension of Swiss-cheese partial operad. We also give a tensor-categorical formulation and constructions of open-closed field algebras over V . 0. Introduction Open-closed (or boundary) conformal field theory was first developed in physics by Cardy [C1]-[C4, CL]. It has important applications to certain critical phenomena on surfaces with boundaries in condensed matter physics. It is also a powerful tool in the study of “D-branes” in string theory. A mathematical definition of such theory, which generalizes Segal’s definition of conformal field theory [S1, S2], was given by Huang [H7], Hu and Kriz [HKr]. However, a thorough mathematical study of this subject is still lacking. In [V], Voronov introduced the so-called “Swiss-cheese operad” to describe the interactions among open strings and closed strings. In order to incorporate the full conformal symmetry which is not included in Voronov’s notion of Swiss-cheese operad, Huang and the author introduced in [HKo1] a notion called Swiss-cheese partial operad. One of the main goals of this work is to construct examples of algebras over a certain C-extension of Swiss-cheese partial operad. Swiss-cheese partial operad consists of disks with strips and tubes, which are conformal equivalence classes of disks with oriented punctures in the interior of the disks (called tubes) and those on the boundaries of the disks (called strips), together with a local coordinate map around each puncture. On the moduli space of disks with strips and Current address: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany. E-mail: [email protected]
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tubes, which is denoted as S, there are two kinds of sewing operations. One is called a boundary sewing operation, which sews two elements in S along two oppositely oriented strips. The other one is called an interior sewing operation, which sews a disk with strips and tubes with a sphere with tubes [H4] along two oppositely oriented tubes. Disks without interior punctures are closed under boundary sewing operations. This closed structure is nothing but the operad of disks with strips, which was introduced and studied in [HKo1] and denoted as ϒ. In [HKo1], Huang and the author introduced the notion of open-string vertex operator algebra and showed that open-string vertex operator algebras of central charge c are algebras over ϒ˜ c , which is a C-extension of ϒ. The moduli space of spheres with tubes equipped with sewing operations has a structure of partial operad, called sphere partial operad and denoted as K . It is well understood by the works of Huang [H1, H2, H4]. Similar to the study of ϒ, we can study S in the framework of sphere partial operad K by embedding S into K via a doubling map δ [A, C1, HKo1]. The interior sewing operations on S correspond to a double-sewing “action” of K on the image of δ (see (2.23)). As a consequence, the closed theory in an open-closed conformal field theory must contain both chiral part and anti-chiral part. Such closed theories were studied in [HKo2, K2, HKo3] in terms of the so-called (conformal) full field algebra and variants of it. More precisely, we showed in [K2] that conformal full field algebras over V L ⊗ V R , where V L and V R are vertex operator algebras of central charge c L and c R respectively and satisfy certain finiteness and reductivity conditions, L L R R are algebras over the partial operad K˜ c ⊗ K˜ c , where K˜ c and K˜ c are line bundles over the base space K [HKo1, K2]. ˜ c , which is a C-extension of S. There is a natural We are interested in algebras over S ˜ c induced by interior sewing operations [HKo1]. Therefore, it action of K˜ c ⊗ K˜ c on S ˜ c with certain natural properties should conis natural to expect that an algebra over S tain a conformal full field algebra Vcl and an open-string vertex operator algebra Vop . ˜ c . This Moreover, Vcl should “act” on Vop according to the action of K˜ c ⊗ K˜ c on S requires certain compatibility between Vcl and Vop . One of the compatibility conditions is the so-called conformal invariant boundary condition. It roughly means that the two vertex operator algebras generated by the left and right Virasoro elements in Vcl must match in some way with that generated from the Virasoro element in Vop . In this case, we also call it a boundary condition preserving conformal symmetry. In this work, we only study those boundary conditions which preserve an enlarged symmetry given by a vertex operator algebra U (see [FS1, FS2] for general situations). Such structure is formalized by a notion called open-closed field algebra over U (see Definition 1.25). When U satisfies conditions in Theorem 0.1, we show that an open-closed field algebra ˜ c. over U canonically gives an algebra over S Open-closed field algebras over U are difficult to study for general U . The following theorem proved by Huang in [H8] is important for us to simplify the situation. Theorem 0.1. Let V be a vertex operator algebra with central charge c satisfying the following conditions: 1. Every C-graded generalized V -module is a direct sum of C-graded irreducible V -modules. 2. There are only finitely many inequivalent C-graded irreducible V -modules. 3. Every R-graded irreducible V -module satisfies the C1 -cofiniteness condition. Then the direct sum of all (in-equivalent) irreducible V -modules has a natural structure of intertwining operator algebra and the category of V -modules, denoted as CV , has
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a natural structure of vertex tensor category. In particular, CV has natural structure of braided tensor category. Assumption 0.2. In this work, we fix a vertex operator algebra V , which is assumed to satisfy the conditions in Theorem 0.1 without further announcement. The notion of open-closed field algebra over V can be described by very few data and axioms in the framework of intertwining operator algebra (see Theorem 1.28 for a precise statement). Moreover, such algebra has a very simple categorical formulation. We will introduce a tensor-categorical notion called open-closed CV |CV ⊗V -algebra (see Definition 3.13) and show that the category of open-closed field algebras over V is isomorphic to the category of open-closed CV |CV ⊗V -algebras. The categorical formulation of open-closed field algebra over V makes it very easy to construct examples. One of important developments on open-closed conformal field theories recently is a series of works [FFFS, FS3, FRS1]-[FRS4, FjFRS] by Felder, Fröhlich, Fuchs, Runkel, Schweigert, Fjelstad, on rational open-closed conformal field theories using the theories of modular tensor category and 3-dimensional topological field theories. Assuming the existence of the structure of a modular tensor category on the category of modules for a vertex operator algebra and the existence of conformal blocks with monodromies compatible with the modular tensor category and all the necessary convergence properties, they constructed conformal blocks for open-closed conformal field theories of all genus and proved their factorization properties and invariance properties under the actions of mapping class groups. Our approach [HKo1]-[HKo3, K2, K3] is somewhat complementary to theirs. We leave a detailed discussion of the relation between these two approaches to future publications. The layout of this work is as follows. In Sect. 1, we review some old notions such as open-string vertex (operator) algebra, (conformal) full field algebra and their variants. We also introduce the notions of open-closed field algebra and open-closed field algebra over U , and study their basic properties. In Sect. 2, we give an operadic formulation of open-closed field algebra over V . In particular, we show that an open-closed field ˜ c . In Sect. 3, we give a categorical algebra over V canonically gives an algebra over S formulation of the notion of open-closed field algebra over V . Convention of notations: N, Z, Z+ , R, R+ , C denote the set of natural numbers, integers, positive integers, real numbers, positive real numbers, complex numbers, respec˜ = H ∪ R. ˜ = H ∪ R, H tively. Let H = {z ∈ C|Imz > 0}, H = {z ∈ C|Imz < 0}, H ˆ to denote the one point compactification of R, H, ˜ respectively. The ˆ H, ˆ H ˜ H We use R, following notations will also be used: ∀n ∈ Z+ , n := {(r1 , . . . , rn ) ∈ Rn |r1 > · · · > rn ≥ 0}, n MH := {(z 1 , . . . , z n ) ∈ Hn |z i = z j , for i, j = 1, . . . , n and i = j}, n MC := {(z 1 , . . . , z n ) ∈ Cn |z i = z j , for i, j = 1, . . . , n and i = j}.
(0.1) (0.2) (0.3)
The ground field is always assumed to be C. 1. Open-Closed Field Algebras Let G be an abelian group. For any G-graded vector space F = ⊕n∈G F(n) and any n ∈ G, we shall use Pn to denote the projection from F or F = n∈G F(n) to F(n) .
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∗ the topology induced from the pairWe give F and its graded dual F = ⊕n∈G F(n) ing between F and F . We also give Hom(F, F) the topology induced from the linear functionals on Hom(F, F) given by f → v , f (v) for f ∈ Hom(F, F), v ∈ F and v ∈ F . For any vector space F and any set S, F ⊗0 = C and S 0 is an one-point set {∗}.
1.1. Boundary field algebras and open-string vertex algebras. Definition 1.1. A boundary field algebra is a R-graded vector space Vop with grading operator being dop , together with a correlation-function map for each n ∈ N: ⊗n n m (n) op : Vop × → V op
(u 1 ⊗ · · · ⊗ u n , (r1 , . . . , rn )) → m (n) op (u 1 , . . . , u n ; r1 , . . . , rn ) and an operator Dop ∈ End Vop , satisfying the following axioms: (n)
1. For each n ∈ Z+ , m op (u 1 , . . . , u n ; r1 , . . . , rn ) is linear in u 1 , . . . , u n and smooth in r1 , . . . , rn . (0) 2. ∀u ∈ Vop , m (1) op (u; 0) = u and 1op := m op (1) ∈ (Vop )(0) . (i) 3. Convergence property. For n ∈ Z+ , k ∈ N and i = 1, . . . , n, u 1 , . . . , u n , u 1 , . . . , (i) u k ∈ Vop , the following series1 (1) (i) (i) (i) k m (n) op (u 1 , . . . , u i−1 , Ps m op (u 1 , . . . , u k , r1 , . . . , rk ), s∈R
u i+1 , . . . , u n ; r1 , . . . , rn ),
(1.1)
(i)
converges absolutely, whenever r1 < |ri − r j | for all j = i, to (i)
(i)
(u 1 , . . . , u i−1 , u 1 , . . . , u k , u i+1 , . . . , u n ; m (n+k−1) op r1 , . . . , ri−1 , ri + r1(i) , . . . , ri + rk(i) , ri+1 , . . . , rn ).
(1.2)
4. dop -bracket property. (n) adop eadop m (n) u 1 , . . . , eadop u n ; ea r1 , . . . , ea rn ). op (u 1 , . . . , u n ; r1 , . . . , rn ) = m op (e (1.3)
for n ∈ Z+ , r1 > · · · > rn ≥ 0, r ∈ R and u 1 . . . , u n ∈ Vop . 5. Dop -property. For u 1 , . . . , u n ∈ Vop , r1 > · · · > rn ≥ 0 and rn + a ≥ 0, (n) ea Dop m (n) op (u 1 , . . . , u n ; r1 , . . . , rn ) = m op (u 1 , . . . , u n , r1 + a, . . . , rn + a).
(1.4)
We denote such a boundary field algebra as (Vop , m op , dop , Dop ). Homomorphisms, isomorphisms and subalgebras of boundary field algebra are defined in the obvious way. ⊗(2) Let the map Yop : Vop × R+ → V op be defined by Yop : (u ⊗ v, r ) → Yop (u, r )v = m 2 (u, v; r, 0). 1 That the number of nonvanishing terms in the sum is countable is automatically assumed.
(1.5)
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Then by the convergence property, we have Yop (1op , r ) = id F , lim Yop (u, r )1op = u,
(1.6) (1.7)
r →0
for u ∈ Vop . (1.7) implies that the map u → Yop (u, r ) is one-to-one. By (1.3), we have eadop Yop (u, r )e−adop = Yop (eadop u, ar )
(1.8)
for u ∈ Vop , r ∈ R+ and a ∈ R. By (1.4), we also have ea Dop Yop (u, r )e−a Dop = Yop (u, r + a)
(1.9)
for u ∈ Vop , r > 0, r + a > 0. Moreover, using (1.7) and (1.4), we obtain the identity: r Dop v. Yop (v, r )1op = m (1) op (v; r ) = e
(1.10)
(n)
If we assume some analytic properties on m op , it is possible to use Yop to generate all (n) m op . This motivate us to introduce the notion of open-string vertex algebra in [HKo1]. The definition given here is a refinement of that in [HKo1]. Definition 1.2. An open-string vertex algebra is an R-graded vector space Vop = ⊕n∈R (Vop )(n) (graded by weights) equipped with a vertex map: Yop : (Vop ⊗ Vop ) × R+ → V op , (u ⊗ v, r ) → Yop (u, r )v,
(1.11)
a vacuum 1op ∈ Vop and an operator Dop ∈ End Vop of weight 1, satisfying the following conditions: 1. Vertex map weight property. For s1 , s2 ∈ R, there exists a finite subset S(s1 , s2 ) ⊂ R such that the image of (⊕s∈s1 +Z (Vop )(s) ) ⊗ (⊕s∈s2 +Z (Vop )(s) ) under the vertex map Yop is in s∈S(s1 ,s2 )+Z (Vop )(s) . 2. Vacuum properties. a. identity property. For any r ∈ R+ , Yop (1op , r ) = idVop , b. creation property. ∀u ∈ Vop , limr →0 Yop (u, r )1op = u. 3. Convergence properties. , the series For n ∈ N, u 1 , . . . , u n , v ∈ Vop and v ∈ Vop v , Yop (u 1 , r1 ) . . . Yop (u n , rn )v v , Yop (u 1 , r1 )Pm 1 Yop (u 2 , r2 ) . . . Pm n−1 Yop (u n , rn )v = m 1 ,...,m n−1
converges absolutely when r1 > · · · > rn > 0 and is a restriction to the domain {r1 > · · · > rn > 0} of an (possibly multivalued) analytic function in (C× )n with only possible singularities at ri = r j for 1 ≤ i, j ≤ n and i = j . , the series a. For u 1 , u 2 , v ∈ Vop , v ∈ Vop v , Yop (Yop (u 1 , r0 )u 2 , r2 )v = v , Yop (Pm Yop (u 1 , r0 )u 2 , r2 )v m
converges absolutely when r2 > r0 > 0.
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, 4. Associativity. For u 1 , u 2 , v ∈ Vop and v ∈ Vop
v , Yop (u 1 , r1 )Yop (u 2 , r2 )v = v , Yop (Yop (u 1 , r1 − r2 )u 2 , r2 )v for r1 , r2 ∈ R satisfying r1 > r2 > r1 − r2 > 0. 5. dop -bracket property. Let dop be the grading operator on Vop . For u ∈ Vop and r ∈ R+ , [dop , Yop (u, r )] = Yop (dop u, r ) + r
d Yop (u, r ). dr
(1.12)
6. Dop -derivative property. We still use Dop to denote the natural extension of Dop to Hom(V op , V op ). For u ∈ Vop , Yop (u, r ) as a map from R+ to Hom(Vop , V op ) is differentiable and d Yop (u, r ) = [Dop , Yop (u, r )] = Yop (Dop u, r ). dr
(1.13)
Homomorphisms, isomorphisms, subalgebras of open-string vertex algebras are defined in the obvious way. We denote such algebra by (Vop , Yop , 1op , dop , Dop ) or simply Vop . For u ∈ Vop , it was shown in [HKo1] that there is a formal vertex operator f Yop (u, x) = u n x −n−1 , (1.14) n∈R
where u n ∈ End Vop , so that f
Yop (u, x)|x=r = Yop (u, r ).
(1.15)
We can also replace x by a complex variable z if we choose a branch cut. For any z ∈ C× and n ∈ R, we define log z := log |z| + argz, 0 ≤ argz < 2π.
(1.16)
But for power functions, we distinguish two types of complex variables, z (or z 1 , z 2 , . . . ) and z¯ 1 , ζ (or z¯ 1 , ζ1 , z¯ 2 , ζ2 , . . . ). We define z n := en log z , z¯ n := enlog z , ζ n := enlog ζ .
(1.17)
Proposition 1.3. An open-string vertex algebra canonically gives a boundary field algebra. Proof. The proof is standard. We omit it here.
Definition 1.4. An open-string vertex operator algebra is an open-string vertex algebra, together with a conformal element ωop , satisfying the following conditions: 7. Grading-restriction conditions. For all n ∈ R, dim(Vop )(n) < ∞ and (Vop )(n) = 0 when n is sufficiently negative.
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8. Virasoro relations. The vertex operator associated to ωop has the following expansion: L(n)r −n−2 Yop (ωop , r ) = n∈Z
where L(n) satisfying the following condition: ∀m, n ∈ Z, c [L(m), L(n)] = (m − n)L(m + n) − (m 3 − m)δm+n,0 , 12 for some c ∈ C. 9. Commutator formula for Virasoro operators and formal vertex operators (or comf ponent operators). For v ∈ Vop , Yop (ωop , x)v involves only finitely many negative powers of x and x1 − x0 f f f f Yop (Yop (ωop , x0 )v, x2 ). [Yop (ωop , x1 ), Yop (v, x2 )] = Resx0 x2−1 δ x2 10. L(0)-grading property and L(−1)-derivative property. L(0) = dop , L(−1) = Dop . We shall denote the open-string vertex operator algebra defined above by (Vop , Yop , 1op , ωop ) or simply Vop . The complex number c in the definition is called central charge. The meromorphic center of Vop is defined by f C0 (Vop ) = u ∈ ⊕n∈Z (Vop )(n) Yop (u, x) ∈ (End Vop )[[x, x −1 ]], f f Yop (v, x)u = e x D Yop (u, −x)v, ∀v ∈ Vop . It was shown in [HKo1] that the meromorphic center of a grading-restricted open-string vertex (operator) algebra is a grading-restricted vertex (operator) algebra. Let U be a grading-restricted vertex (operator) algebra. If there is a monomorphism ιop : U → C0 (Vop ) of grading-restricted vertex (operator) algebra, we call Vop an open-string vertex (operator) algebra over U , and denote it by (Vop , Yop , ιop ) or simply f by V . In this case, the formal vertex operator Yop is an intertwining operator of type Vopop , where Vop is a U -module. V V op op
Remark 1.5. Open-string vertex operator algebra can be viewed as a noncommutative generalization of vertex operator algebra. Other noncommutative generalizations of vertex (operator) algebra were also studied in the literature [FR, B, EK, BKa, L1, L2, L3]. 1.2. Full field algebras. Definition 1.6. A R × R-graded full field algebra is an R × R-graded vector space Vcl = ⊕m,n∈R (Vcl )(m,n) (graded by left weight wt L and right weight wt R with left and right grading operators d L and d R ), equipped with correlation-function maps (n)
m cl :
Vcl⊗n × MCn → V cl (n) (u 1 ⊗ · · · ⊗ u n , (z 1 , . . . , z n )) → m cl (u 1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n ),
for each n ∈ N, and operators D L and D R of weights (1, 0) and (0, 1) respectively, satisfying the following axioms:
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1. Single-valuedness property. e2πi(d −d ) = idVcl . (n) 2. For n ∈ Z+ , m cl (u 1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n ) is linear in u 1 , . . . , u n and smooth in the real and imaginary parts of z 1 , . . . , z n . 3. Identity properties. ∀u ∈ Vcl , m 1 (u; 0, 0) = u and 1cl := m (0) cl (1) ∈ (Vcl )(0,0) . (i) ∈ Vcl and 4. Convergence property. For k ∈ Z+ and u 1 , . . . , u n , u (i) 1 , . . . , uk i = 1, . . . , n, the series (n) (i) (i) (i) (i) m cl (u 1 , . . . , u i−1 , P( p,q) m (k) cl (v1 , . . . , vk ; z 1 , z¯ 1 , . . . , z k , z¯ k ), L
R
p,q∈R
u i+1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n )
(1.18)
converges absolutely to (n+k−1)
m cl
(u 1 , . . . , u i−1 , v1 , . . . , vk , u i+1 , . . . , u n ; z 1 , z¯ 1 , . . . , z i−1 , z¯ i−1 , (i)
(i)
(i)
(i)
z i + z 1 , z¯ i + z¯ 1 . . . , z i + z k , z¯ i + z¯ k , z i+1 , z¯ i+1 , . . . , z n , z¯ n )
(1.19)
(i)
whenever |z p | < |z i − z j | for all j = 1, . . . , n, i = j and for p = 1, . . . , k. 5. Permutation property. For n ∈ Z+ and σ ∈ Sn , we have (n)
m cl (u 1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n ) (n)
= m cl (u σ (1) , . . . , u σ (n) ; z σ (1) , z¯ σ (1) , . . . , z σ (n) , z¯ σ (n) )
(1.20)
for u 1 , . . . , u n ∈ Vcl and (z 1 , . . . , z n ) ∈ MCn . 6. d L and d R property. For u 1 , . . . , u n ∈ Vcl and a ∈ C, ¯ ead ead m (n) cl (u 1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n ) L
R
ad ad ¯ = m (n) e ¯ u 1 , . . . , ead ead u n , ea z 1 , ea¯ z¯ 1 , . . . , ea z n , ea¯ z¯ n ). cl (e L
R
L
R
(1.21)
7. D L and D R property: [D L , D R ] = 0 and for u 1 , . . . , u n ∈ Vcl and a ∈ C, (n)
ea D ea¯ D m cl (u 1 , . . . , u n ; z 1 , z¯ 1 , . . . , z n , z¯ n ) L
R
(n)
¯ . . . , z n + a, z¯ n + a). ¯ = m cl (u 1 , . . . , u n , z 1 + a, z¯ 1 + a,
(1.22)
We denote the R × R-graded full field algebra defined above by (Vcl , m cl , d L , d R , D L , D R ) or simply by Vcl . Subalgebra, homomorphism, monomorphism, epimorphism and isomorphism of a full field algebra can be naturally defined. Let Y : Vcl⊗2 × C× → V cl be so that Y : (u ⊗ v, z) → Y(u; z, z¯ )v := m 2 (u ⊗ v; z, z¯ , 0, 0).
(1.23)
Then by the convergence property, it is easy to see that Y(1cl ; z, z¯ ) = idVcl , lim Y(u; z, z¯ )1cl = u, ∀u ∈ Vcl .
z→0
Moreover, it is also not hard to show the following two properties of Y:
(1.24) (1.25)
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1. The d L - and d R -bracket properties.
∂ d L , Y(u; z, z¯ ) = z Y(u; z, z¯ ) + Y(d L u; z, z¯ ), ∂z
∂ d R , Y(u; z, z¯ ) = z¯ Y(u; z, z¯ ) + Y(d R u; z, z¯ ). ∂ z¯
(1.26)
2. The D L - and D R -derivative property.
∂ Y(u; z, z¯ ), D L , Y(u; z, z¯ ) = Y(D L u; z, z¯ ) = ∂z
∂ D R , Y(u; z, z¯ ) = Y(D R u; z, z¯ ) = Y(u; z, z¯ ). ∂ z¯
(1.28)
It was shown in [HKo2] that we have the following expansion: Y(u; z, z¯ ) = Yl,r (u)e(−l−1) log z e(−r −1) log z ,
(1.27)
(1.29)
(1.30)
r,s∈R
where Yl,r (u) ∈ End F with wt L Yl,r (u) = wt L u−l −1 and wt R Yl,r (u) = wt R u−r −1. Moreover, the expansion above is unique. Let x and x¯ be independent and commuting formal variables. We define the formal full vertex operator Y f associated to u ∈ Vcl by Y f (u; x, x) ¯ = Yl,r (u)x −l−1 x¯ −r −1 . (1.31) l,r ∈R
For nonzero complex numbers z and ζ , we can substitute er log z and es log ζ for x r and x¯ s respectively in Y f (u; x, x) ¯ to obtain an operator Yan (u; z, ζ ) : Vcl × (C× )2 → V cl called the analytic full vertex operator. Definition 1.7. An R × R-graded full field algebra (Vcl , m cl , d L , d R , D L , D R ) is called grading-restricted if it satisfies the following grading-restriction conditions: 1. There exists M ∈ R such that (Vcl )(m,n) = 0 if n < M or m < M. 2. dim(Vcl )(m,n) < ∞ for m, n ∈ R. We say that Vcl is lower-truncated if Vcl satisfies the first grading restriction condition. In this case, for u ∈ Vcl and k ∈ R, we have Yl,r (u) ∈ End Vcl l+r =k
with total weight wt u − k − 2. We denote l+r =k Yl,r (u) by Yk−1 (u). Then we have the expansion Yk (u)x −k−1 , (1.32) Y f (u; x, x) = k∈R
where wt Yk (u) = wt u − k − 1. For given u, v ∈ vcl , we have Yk (u)w = 0 for sufficiently large k.
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Let (V L , Y L , 1 L , ω L ) and (V R , Y R , 1 R , ω R ) be vertex operator algebras. It was pointed out in [HKo2] that V L ⊗ V R has a natural structure of full field algebra. Let ιcl be an injective homomorphism from the full field algebra V L ⊗ V R to Vcl . Then we have 1cl = ιcl (1 L ⊗ 1 R ), d L ◦ ιcl = ιcl ◦ (L L (0) ⊗ I V R ), d R ◦ ιcl = ιcl ◦ (I V L ⊗ L R (0))), D L ◦ ιcl = ιcl ◦ (L L (−1) ⊗ I V R ) and D R ◦ ιcl = ιcl ◦ (I V L ⊗ L R (−1)). Moreover, Vcl has a left conformal element ιcl (ω L ⊗ 1 R ) and an right conformal element ιcl (1 L ⊗ ω R ). We have the following operators on Vcl : L L (0) = Resx Resx¯ x¯ −1 Y f (ιcl (ω L ⊗ 1 R ); x, x), ¯ ¯ L R (0) = Resx Resx¯ x −1 Y f (ιcl (1 L ⊗ ω L ); x, x), ¯ L L (−1) = Resx Resx¯ x x¯ −1 Y f (ιcl (ω L ⊗ 1 R ); x, x), ¯ f (ιcl (1 L ⊗ ω L ); x, x). ¯ L R (−1) = Resx Resx¯ x −1 xY Since these operators are operators on Vcl , it should be easy to distinguish them from those operators with the same notation but acting on V L or V R . Definition 1.8. Let (V L , Y L , 1 L , ω L ) and (V R , Y R , 1 R , ω R ) be vertex operator algebras. A full field algebra over V L ⊗ V R is a grading-restricted R × R-graded full field algebra (Vcl , m cl , d L , d R , D L , D R ) equipped with an injective homomorphism ιcl from the full field algebra V L ⊗ V R to Vcl such that d L = L L (0), d R = L R (0), D L = L L (−1) and D R = L R (−1). A full field algebra over V L ⊗ V R equipped with left and right conformal elements ιcl (ω L ⊗ 1 R ) and ιcl (1 L ⊗ ω R ) is called conformal full field algebra over V L ⊗ V R . We shall denote the (conformal) full field algebra over V L ⊗ V R defined above by (Vcl , m cl , ιcl ) or simply by Vcl . From now on, we will not distinguish V L with V L ⊗ 1 R and ιcl (V L ⊗ 1 R ). Similarly for V R with 1 L ⊗ V R and ιcl (1 L ⊗ V R ). For u L ∈ V L , Yan (u L ; z, ζ ) is independent of ζ . So we simply denote it as Yan (u L , z). Similarly, we denote Yan (u R ; z, ζ ) as Yan (u R , ζ ) for u R ∈ V R . 1.3. Open-closed field algebras. Definition 1.9. An open-closed field algebra consists of a R×R-graded full field algebra (Vcl , m cl , dclL , dclR , DclL , DclR ) and a R-graded vector space Vop with the grading operator dop and an additional operator Dop ∈ End Vop , together with a map for each pair of n, l ∈ N: (l;n)
⊗n l m cl−op : Vcl⊗l ⊗ Vop × MH × n → V op
(u 1 ⊗ · · · ⊗ u l ⊗ v1 ⊗ · · · ⊗ vn , (z 1 , . . . , zl ; r1 , . . . , rn )) → (l;n)
m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 . . . , zl , z¯l ; r1 , . . . , rn ), satisfying the following axioms: (l;n)
1. m cl−op (u 1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) is linear in u 1 , . . . , vn and smooth in r1 , . . . , rn , z 1 , . . . , zl . (0;1) (0;0) 2. Identity properties. m cl−op (v; 0) = v, ∀v ∈ Vop and 1op := m cl−op (1) ∈ (Vop )(0) .
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3. Convergence properties. a. For u 1 , . . . , u l , u˜ 1 , . . . , u˜ k ∈ Vcl , v1 , . . . , vn , v˜1 , . . . , v˜m ∈ Vop and i = 1, . . . , n, the following series:
(l;n)
(k;m)
m cl−op (u 1 , . . . , u l ; v1 , . . . , vi−1 , Pn 1 m cl−op (u˜ 1 , . . . , u˜ k ;
n 1 ∈R (i)
(i)
(i)
(i)
(i)
v˜1 , . . . , v˜m ; z 1 , z 1 , . . . , z k , z k ; r1 , . . . , rm(i) ); vi+1 , . . . vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) (i)
(1.33)
(i)
converges absolutely when |z s −ri |, |rt −ri | > |z p |, rq ≥ 0 for all s = 1, . . . , l, t = 1, . . . , n, t = i, p = 1, . . . , k and q = 1, . . . , m to (l+k;n+m−1)
m cl−op
(u 1 , . . . , u l , u˜ 1 , . . . , u˜ k ; v1 , . . . , vi−1 , v˜1 , . . . , v˜m , (i)
(i)
(i)
(i)
vi+1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l , z 1 , z 1 , . . . , z k , z k ; (i)
r1 , . . . , ri−1 , ri + r1 , . . . , ri + rm(i) , ri+1 , . . . , rn ).
(1.34)
b. For u 1 , . . . , u l , u˜ 1 , . . . , u˜ k ∈ Vcl , v1 , . . . , vn ∈ Vop and i = 1, . . . , n, the following series:
(k) (i) (i) (i) (i) m (l;n) cl−op (u 1 , . . . , u i−1 , P(n 1 ,n 2 ) m cl (u˜ 1 , . . . , u˜ k ; z 1 , z 1 , . . . , z k , z k ),
n 1 ,n 2 ∈R
u i+1 , . . . , u l ; v1 , . . . vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn )
(1.35)
(i)
converges absolutely, when |z s − z i |, |rt − z i | > |z p | for all s = 1, . . . , l, s = i, t = 1, . . . , n, p = 1, . . . , k, to (l+k−1;n)
m cl−op
(u 1 , . . . , u i−1 , u˜ 1 , . . . , u˜ k , u i+1 , . . . , u l ; v1 , . . . vn ; (i)
(i)
z 1 , z¯ 1 , . . . , z i−1 , z¯ i−1 , z i + z 1 , z i + z 1 , . . . , z i + z k(i) , z i + z k(i) , z i+1 , z¯ i+1 , . . . , zl , z¯l ; r1 , . . . , rn ).
(1.36)
4. Permutation axiom. For u 1 , . . . , u l ∈ Vcl , v1 , . . . , vn ∈ Vop and σ ∈ Sl , (l;n)
m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) (l;n)
= m cl−op (u σ (1) , . . . , u σ (l) ; v1 , . . . , vn ; z σ (1) , z¯ σ (1) , . . . , z σ (l) , z¯ σ (l) ; r1 , . . . , rn ).
(1.37)
5. dop −, dclL − and dclR -property. For u 1 , . . . , u l ∈ Vcl , v1 , . . . , vn ∈ Vop and a ∈ R, (l;n)
eadop m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ; ) a(dcl +dcl ) u 1 , . . . , ea(dcl +dcl ) u l ; eadop v1 , . . . , eadop vn ; = m (l;n) cl−op (e L
R
L
R
ea z 1 , ea z¯ 1 , . . . , ea zl , ea z¯l ; ea r1 , . . . , ea rn ).
(1.38)
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6. Dop -property. For u 1 , . . . , u l ∈ Vcl , v1 , . . . , vn ∈ Vop and rn + a ≥ 0, (l;n)
ea Dop m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) (l;n)
= m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 + a, z¯ 1 + a, . . . , zl + a, z¯l + a; r1 + a, . . . , rn + a).
(1.39)
We denote such algebra by (Vcl , Vop , m cl−op ) for simplicity. Homomorphisms, isomorphisms, subalgebras of open-closed field algebras are defined in the obvious way. Remark 1.10. From the above definition, it is clear that an open-closed field algebra automatically includes a boundary field algebra as a substructure. We discuss a few results which follow immediately from the definition. By the identity properties and (1.39), we have, for a ≥ 0, a Dop (0;1) m (0;1) m cl−op (v; 0) = ea Dop v. cl−op (v; a) = e
(1.40)
By (1.40) and the convergence property, for i = 1, . . . , n and a ∈ R, we obtain (l;n)
m cl−op (u 1 , . . . , u l ; v1 , . . . , vi−1 , ea Dop vi , vi+1 , . . . , vn ; =
z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) (l;n) (0;1) m cl−op (u 1 , . . . , u l ; v1 , . . . , vi−1 , m cl−op (vi ; a),
=
vi+1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) (l;n) m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l r1 , . . . , ri−1 , ri + a, ri+1 , . . . , rn )
(1.41)
when |r j −ri |, |z k −ri | > |a| for j = i and k = 1, . . . , l. By (1.22) and the convergence property, we also have, for j = 1, . . . , l and b ∈ C, ¯
bDcl +bDcl m (l;n) u j , u j+1 , . . . , u l ; v1 , . . . , vn ; cl−op (u 1 , . . . , u j−1 , e L
R
z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) (1) ¯ = m (l;n) cl−op (u 1 , . . . , u j−1 , m cl (u j ; b, b), u j+1 , . . . , u l ;
v1 , . . . , vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ) = m (l;n) cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; z 1 , z¯ 1 , . . . , z j−1 , z¯ j−1 , z j + b, z j + b, z j+1 , z¯ j+1 , . . . , zl , z¯l ; r1 , . . . , rn )
(1.42)
when |z i − z j |, |rk − z j | > |b| for i = 1, . . . , l, i = j and k = 1, . . . , n. (0;n) Let m (n) op := m cl−op . The definition of open-closed field algebra immediately implies that (Vop , m op , dop , Dop ) is a boundary field algebra. In particular, the map Yop defined in (1.5) satisfies Eqs. (1.6), (1.7), (1.8) and (1.9). Similarly, we know that the map Y defined in (1.23) satisfies Eqs. (1.24), (1.25), (1.26), (1.27), (1.28), (1.29), (1.30) and (1.31). In an open-closed field algebra (Vcl , Vop , m cl−op ), there is an additional vertex operator map: Ycl−op : (Vcl ⊗ Vop ) × H → V op (u ⊗ v, (z, z¯ )) → Ycl−op (u; z, z¯ )v,
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defined by (1;1)
Ycl−op (u; z, z¯ )v := m cl−op (u; v; z, z¯ ; 0).
(1.43)
By the convergence property, we have the following identity property: (0) Ycl−op (1cl ; z, z¯ )v = m (1;1) cl−op (m cl (1); v; z, z¯ ; 0)
= m (0;1) cl−op (v; 0) = v.
(1.44)
Proposition 1.11. For u ∈ Vcl , we have [dop , Ycl−op (u; z, z¯ )] = Ycl−op ((dclL + dclR )u; z, z¯ ) ∂ ∂ + z + z¯ Ycl−op (u; z, z¯ ). ∂z ∂ z¯
(1.45)
∂ |a=0 to both sides of (1.38) when n = 1, l = 1 and r1 = 0, we Proof. Applying ∂a obtain (1.45) immediately.
Proposition 1.12. For u ∈ Vcl , we have [Dop , Ycl−op (u; z, z¯ )] = Ycl−op ((DclL + DclR )u; z, z¯ ), ∂ Ycl−op (u; z, z¯ ), Ycl−op (DclL u; z, z¯ ) = ∂z ∂ Ycl−op (u; z, z¯ ). Ycl−op (DclR u; z, z¯ ) = ∂ z¯
(1.46) (1.47) (1.48)
∂ |b=0 ( ∂∂b¯ |b=0 Proof. Applying ∂b ¯ ) to both sides of (1.42) when n = 1, l = 1 and r1 = 0, we obtain (1.47) and (1.48). ∂ Applying ∂a |a=0 to both sides of (1.39) when n = 1, l = 1 and r1 = 0 and using (1.47) and (1.48) we obtain the first identity in (1.46) immediately.
1.4. Analytic open-closed field algebras. The notion of open-closed field algebra introduced in the last subsection is very general. There is not much to say about open-closed field algebras in such generality. In this subsection, we study those open-closed field algebras satisfying some nice analytic properties. In this case, the whole structure can be reconstructed from some simple ingredients. Definition 1.13. An open-closed field algebra (Vcl , Vop , m cl−op ) is called analytic if it satisfies the following conditions: 1. Ycl−op can be extended to a map Vcl ⊗ Vop × H × H → V op such that for z ∈ H, ζ ∈ H, Ycl−op (u; z, z¯ ) = Ycl−op (u; z, ζ )|ζ =¯z .
(1.49)
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2. For n ∈ N, v1 , · · · , vn+1 ∈ Vop , v ∈ (Vop ) and u 1 , · · · , u n ∈ Vcl , the series v , Ycl−op (u 1 , z 1 , ζ1 )Yop (v1 , r1 ) · · · Ycl−op (u n , z n , ζn )Yop (vn , rn )vn+1 is absolutely convergent when |z 1 |, |ζ1 | > r1 > · · · > |z n |, |ζn | > rn > 0 and can be extended to a (possibly multivalued) analytic function on {(z 1 , ζ1 , r1 , . . . , z n , ζn , rn ) ∈ MC3n }. 3. For n ∈ N, v , u 1 , . . . , u n+1 ∈ Vcl , the series v , Yan (u 1 ; z 1 , ζ1 ) . . . Yan (u n ; z n , ζn )u n+1 is absolutely convergent when |z 1 | > · · · > |z n | > 0 and |ζ1 | > · · · > |ζn | > 0 and can be extended to an analytic function on MC2n . , v , v ∈ V , u ∈ V , the series 4. For v ∈ Vop 1 2 op cl v , Yop (Ycl−op (u; z, ζ )v1 , r )v2 is absolutely convergent when r > |z|, |ζ | > 0. , v ∈ V , u , u ∈ V , the series 5. For v ∈ Vop op 1 2 cl v , Ycl−op (Yan (u 1 ; z 1 , ζ1 )u 1 ; z 2 , ζ2 )v , converges absolutely when |z 2 | > |z 1 | > 0, |ζ2 | > |ζ1 | > 0 and |z 1 |+|ζ1 | < |z 2 −ζ2 |. (l;n)
By the convergence properties of open-closed field algebra, m cl−op can be expressed (l;n)
l × n . m as products of Ycl−op , Yop on a dense subdomain of MH cl−op on the complement of this dense subdomain is completely determined by analytic extension. Therefore, m (l;n) cl−op , l, n ∈ N are completely determined by 1op , Yop and Ycl−op . Similarly, (n)
m cl , n ∈ N is completely determined by 1cl and Yan . Therefore, we also denote an analytic open-closed conformal field algebra by ((Vcl , Yan , 1cl ), (Vop , Yop , 1op ), Ycl−op ), or (Vcl , Vop , Ycl−op ) for simplicity. Two immediate consequences of the definition of analytic open-closed field algebra are given in the following two lemmas. Lemma 1.14. Dop -bracket properties. u ∈ Vcl , we have [Dop , Ycl−op (u; z, ζ )] = Ycl−op ((DclL + DclR )u; z, ζ ), ∂ Ycl−op (DclL u; z, ζ ) = Ycl−op (u; z, ζ ), ∂z ∂ Ycl−op (u; z, ζ ). Ycl−op (DclR u; z, ζ ) = ∂ζ
(1.50)
Proof. By (1.46),(1.47) and (1.48), for any fixed z ∈ H, all three identities hold when ζ = z¯ . Replacing u by (DclR )k u for any k ∈ N in (1.46), (1.47) and (1.48), we obtained ∂ k that all derivatives ( ∂ζ ) |ζ =¯z , ∀k ∈ N of both sides of the above three identities are
equal respectively. By the property of analytic function and the fact that H is a simply connected domain, we obtain that both sides of above three identities, viewed as analytic functions for a fixed z ∈ H and any ζ ∈ H, must be equal respectively.
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Lemma 1.15. dop -bracket properties : u ∈ Vcl , we have [dop , Ycl−op (u; z, ζ )] = Ycl−op ((dclL + dclR )u; z, ζ ) ∂ ∂ + z +ζ Ycl−op (u; z, ζ ). ∂z ∂ζ Proof. The proof is the same as that of Lemma 1.14.
(1.51)
An analytic open-closed field algebra (Vcl , Vop , Ycl−op ) satisfies two associativities and two commutativities. These properties are very important for categorical formulation. and z ∈ H, Proposition 1.16 (Associativity I). For u ∈ Vcl , v1 , v2 ∈ Vop , v ∈ Vop ζ ∈ H, we have
v , Ycl−op (u; z, ζ )Yop (v1 , r )v2 = v , Yop (Ycl−op (u; z − r, ζ − r )v1 , r )v2 (1.52) when |z|, |ζ | > r > 0 and r > |r − z|, |r − ζ | > 0. Proof. We abbreviate the “left-(right)-hand side” as “LHS (RHS)” . For z ∈ H, (1;1)
LHS of (1.52) |ζ =¯z = m cl−op (u; m (2) op (v1 , v2 ; r, 0); z, z¯ ; 0) (1;2)
= m cl−op (u; v1 , v2 ; z, z¯ ; r, 0)
(1.53)
when |z| > r > 0, and (1;1) RHS of (1.52) |ζ =¯z = m (0;2) cl−op (m cl−op (u; v1 ; z − r, z¯ − r ; 0), v2 ; r, 0)
= m (1;2) cl−op (u; v1 , v2 ; z, z¯ ; r, 0)
(1.54)
when r > |r − z| > 0. Therefore (1.52) holds when ζ = z¯ and |z| > r > |r − z| > 0, z ∈ H. Now replace u in (1.52) by (DclR )k u, k ∈ N, we obtain: ∂ k ∂ k LHS of (1.52) = RHS of (1.52) (1.55) ∂ζ k ζ =¯z ∂ζ k ζ =¯z when z ∈ H and |z| > r > |r − z| > 0. Then by the properties of analytic function, it is clear that (1.52) holds for all z ∈ H, ζ ∈ H and |z|, |ζ | > r > 0 and r > |r − z|, |r − ζ | > 0. and Proposition 1.17 (Associativity II). For u 1 , u 2 ∈ Vcl , v1 , v2 ∈ Vop , v ∈ Vop z 1 , z 2 ∈ H, ζ1 , ζ2 ∈ H, we have
w , Ycl−op (u 1 ; z 1 , ζ1 )Ycl−op (u 2 ; z 2 , ζ2 )v2 = v , Ycl−op (Yan (u 1 ; z 1 − z 2 , ζ1 − ζ2 )u 2 ; z 2 , ζ2 )v2
(1.56)
when |z 1 |, |ζ1 | > |z 2 |, |ζ2 | and |z 2 | > |z 1 − z 2 | > 0, |ζ2 | > |ζ1 − ζ2 | > 0 and |z 2 − ζ2 | > |z 1 − z 2 | + |ζ1 − ζ2 |.
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Proof. The proof is similar to that of (1.52). So we will only sketch the difference here. First, it is easy to show that (1.56) is true for z i ∈ H, ζi = z¯ i , i = 1, 2 in a proper domain. Then replacing u 1 , u 2 by (DclR ) p u 1 , (DclR )q u 2 for p, q ∈ N and using (1.29) and (1.50), we obtain that ∂ p ∂ q ∂ p ∂ q LHS of (1.56) = RHS of (1.56) (1.57) p q p q ∂ζ1 ∂ζ2 ζi =¯zi ∂ζ1 ∂ζ2 ζi =¯zi in a proper domain. By the property of analytic function again, the analytic extension of both sides of (1.56) to the following simply connected domain D0 = {(z 1 , ζ1 , z 2 , ζ2 )|z i ∈ H, ζi ∈ H, |z 1 |, |ζ1 | > |z 2 |, |ζ2 |} must be identical. Moreover, the additional restrictions of the domain in the statement of Associativity II guarantee the absolute convergence of both sides of (1.56). For an analytic open-closed field algebra, the map Ycl−op can be uniquely extended to Vcl ⊗ Vop × R where R := {(z, ζ ) ∈ C2 |z ∈ H ∪ R+ , ζ ∈ H ∪ R+ , z = ζ }. , Proposition 1.18 (Commutativity I). For u ∈ Vcl , v1 , v2 ∈ Vop and v ∈ Vop
v , Ycl−op (u; z, ζ )Yop (v1 , r )v2 ,
(1.58)
which is absolutely convergent when z > ζ > r > 0 (recall Definition 1.13), and v , Yop (v1 , r )Ycl−op (u; z, ζ )v2 ,
(1.59)
which is absolutely convergent when r > z > ζ > 0 (recall Definition 1.13), are analytic continuations of each other along the following path. 2
(1.60) Proof. When z ∈ H, ζ = z¯ and |z| > r > 0, (1.58) is absolutely convergent (1;2) to m cl−op (u; v1 , v2 ; z, z¯ ; r, 0). Hence, if we analytically extend the analytic function (1.58) to D1 := {(z, ζ )|z ∈ H, ζ = z¯ , |z| = r }, then its value on D1 must equal to m (1;2) cl−op (u; v1 , v2 ; z, z¯ ; r, 0) when z ∈ H, |z| = r by
the continuity of m (1;2) cl−op . Similarly, the unique extension of (1.59) from r > |z|, |ζ | > 0 to D1 is also equal to (1;2) m cl−op (u; v1 , v2 ; z, z¯ ; r, 0) when z ∈ H, |z| = r . 2 The extended domain R\{z = r, or, ζ = r } is simply connected for fixed r > 0, all possible paths of analytic continuation are homotopically equivalent.
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Both (1.58) and (1.59) can be uniquely extended to two analytic functions on H × H. By the discussion above, these two extended analytic functions of (z, ζ ) take the same value on D1 which is of lower dimension. Now we replace u by (DclL )m (DclR )n u, m, n ∈ N and repeat the arguments in the proof of Proposition 1.16. We obtain that all the derivatives of the two extended analytic functions also match on D1 . By the properties of analytic functions, these two extended functions are identical on H × H. Then their extensions to (z, ζ ) ∈ R\{z = r, or, ζ = r } are also identical. Thus we have proved the first commutativity. , Proposition 1.19 (Commutativity II). For u 1 , u 2 ∈ Vcl , v ∈ Vop and v ∈ Vop
v , Ycl−op (u 1 ; z 1 , ζ1 )Ycl−op (u 2 ; z 2 , ζ2 )v ,
(1.61)
which is absolutely convergent when z 1 > ζ1 > z 2 > ζ2 > 0, and v , Ycl−op (u 2 ; z 2 , ζ2 )Ycl−op (u 1 ; z 1 , ζ1 )v ,
(1.62)
which is absolutely convergent when z 2 > ζ2 > z 1 > ζ1 > 0, are analytic continuations of each other along the following paths.
(1.63) Proof. Commutativity II follows directly from the Associativity II (1.56) and the skew-symmetry of the full field algebra Vcl [HKo2]. Here, we give a more direct proof which is similar to that of Proposition 1.18. The unique extension of (1.61) from |z 1 |, |ζ1 | > |z 2 |, |ζ2 | > 0 to 2 D2 := {(z 1 , ζ1 , z 2 , ζ2 )|z 1 , z 2 ∈ MH , |z 1 | = |z 2 |, ζi = z¯ i , i = 1, 2} (2;1)
is equal to m cl−op (u 1 , u 2 ; v; z 1 , z¯ 1 , z 2 , z¯ 2 ; 0), the unique extension of (1.62) from (2;1)
|z 2 |, |ζ2 | > |z 1 |, |ζ1 | > 0 to D2 also matches with m cl−op (u 1 , u 2 ; v; z 1 , z¯ 1 , z 2 , z¯ 2 ; 0). By the similar argument as the proof of the first commutativity, we see that (1.61) in |z 1 |, |ζ1 | > |z 2 |, |ζ2 | > 0 and (1.61) in |z 2 |, |ζ2 | > |z 1 |, |ζ1 | > 0 are analytic continuations of each other along the following paths.
(1.64) Then it is obvious to see that the unique extension of both (1.61) and (1.62) to the subdomain of {(z 1 , ζ1 , z 2 , ζ2 ) ∈ R4+ }, where z 1 , ζ1 , z 2 , ζ2 have distinct values, are analytic extensions of each other along the paths (1.63).
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1.5. Open-closed field algebras over U . We gradually add more structures on an analytic open-closed field algebra. At the end of this process, we will arrive at a notion called open-closed field algebra over a vertex operator algebra U . Let (V L , YV L , 1 L ) and (V R , YV R , 1 R ) be two vertex algebras. We now consider an analytic open-closed field algebra (Vcl , Vop , Ycl−op ) such that Vcl is a full field algebra over V L ⊗ V R . For such full field algebra Vcl , we will not distinguish V L with V L ⊗ 1 R ⊂ Vcl and V R with 1 L ⊗ V R ⊂ Vcl . Lemma 1.20. For u L ∈ V L and u R ∈ V R , Ycl−op (u L ; z, ζ ) is independent of ζ and Ycl−op (u R ; z, ζ ) is independent of z. Proof. For u L ∈ V L , w ∈ Vop , w ∈ (Vop ) , using the associativity (1.56), we have w , Ycl−op (u L ; z, ζ )w = w , Ycl−op (u L ; z, ζ )Ycl−op (1cl ; z 1 , ζ1 )w = w , Ycl−op (Yan (u L ; z − z 1 , ζ − ζ1 )1cl ; z 1 , ζ1 )w (1.65) when |z 1 | > |z − z 1 | > 0, |ζ1 | > |ζ − ζ1 | > 0 and |z − z 1 | + |ζ − ζ1 | < |z 1 − ζ1 |. The right hand side of (1.65) is independent of ζ and the left-hand side of (1.65) is analytic in ζ . Hence Ycl−op (u L ; z, ζ ) is independent of ζ for all z ∈ H. Similarly, Ycl−op (u R ; z, ζ ) is independent of z for all ζ ∈ H and u R ∈ V R . In order to emphasis these ζ - or z-independence properties, we denote them simply as Ycl−op (u L , z) and Ycl−op (u R , ζ ) for u L ∈ V L and u R ∈ V R respectively. Replace u in (1.51) by u L ∈ V L and u R ∈ V R respectively, we obtain ∂ [dop , Ycl−op (u L , z)] = Ycl−op (dclL u L , z) + z Ycl−op (u L , z), ∂z
∂ R R R dop , Ycl−op (u , ζ ) = Ycl−op (dcl u , ζ ) + ζ Ycl−op (u R , ζ ). (1.66) ∂ζ As a consequence, we have Ycl−op (u L , z) = u nL z −n−1 , n∈R
Ycl−op (u , ζ ) = R
u nR ζ −n−1 ,
(1.67)
n∈R
where u nL , u nR ∈ End Vop and wt u nL = wt L u L − n − 1 and wt u nR = wt R u R − n − 1, and z n = en log z and ζ n = enlog ζ . Moreover, we have ∂ [Dop , Ycl−op (u L , z)] = Ycl−op (DclL u L , z) = Ycl−op (u L , z), (1.68) ∂z
∂ Dop , Ycl−op (u R , ζ ) = Ycl−op (DclR u R , ζ ) = Ycl−op (u R , ζ ), ∂ζ which further implies that ea Dop Ycl−op (u L , z)e−a Dop = Ycl−op (u L , z + a) ea Dop Ycl−op (u L , ζ )e−a Dop = Ycl−op (u R , ζ + a) for |z| > |a|, z + a ∈ H and |ζ | > |a|, ζ + a ∈ H respectively.
(1.69)
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Lemma 1.21. For any u L ∈ V L ⊗ 1 R and u R ∈ 1 L ⊗ V R , the following two limits: lim Ycl−op (u L , z)1op ,
z→0
lim Ycl−op (u R , ζ )1op
ζ →0
exist in Vop . Proof. By the associativity and the creation property of open-string vertex algebra, we have Ycl−op (u L , z + r )1op = Ycl−op (u L , z + r )Yop (1op , r )1op = Yop (Ycl−op (u L , z)1op , r )1op = er Dop Ycl−op (u, z)1op
(1.70)
when |z + r | > r > |z| > 0. For fixed z ∈ H, the left-hand side of (1.70) is an analytic function valued in V op on the domain {r ∈ C|z + r = 0}. The right-hand side of (1.70), as a power series of r , is absolutely convergent when |r | > |z| > 0. By the general property of power series, the right-hand side of (1.70) is absolutely convergent for all r ∈ C to a singlevalued analytic function. Because both sides of (1.70) are analytic functions, the equality (1.70) must hold for all r ∈ C. In particular, limr →−z Ycl−op (u L , z + r )1op exists. Equivalently, lim z→0 Ycl−op (u L , z)1op exists. By the expansion (1.67), we must have u nL 1op = 0 for all n > −1. Moreover, it is also easy to see that u nL 1op = 0 for any n∈ / −Z+ by (1.68) (see the proof of Proposition 1.8 in [HKo1]). Therefore, we have L 1op ∈ Vop . lim Ycl−op (u L , z)1op = u −1
z→0
The proof of the existence of the second limit is entirely the same.
By the lemma above, we can define two maps h L : V L → Vop and h R : V R → Vop as follows: for all u L ∈ V L and u R ∈ V R , h L : u L → lim Ycl−op (u L , z)1op , z→0
h : u → lim Ycl−op (u R , ζ )1op . R
R
ζ →0
(1.71)
Notice also that h L , h R preserve the weights. Namely L 1op ) = wt L u L , wt h L (u L ) = wt (u −1 R wt h R (u R ) = wt (u −1 1op ) = wt R u R .
Therefore both h L and h R can be naturally extended to maps V L → Vop and V R → Vop . We still denote the extended maps as h L and h R respectively. Lemma 1.22. For u L ∈ V L , u R ∈ V R , we have Ycl−op (u L , z)1op = e z Dop h L (u L ), Ycl−op (u R , ζ )1op = eζ Dop h R (u R ).
(1.72)
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Proof. Since we have shown that (1.70) holds for all r ∈ C, z ∈ H and both sides of (1.70) are analytic for all r, z ∈ C, if we take the limit lim z→0 on both sides of (1.70), the equality should still hold. Thus we obtain the first identity in (1.72). The proof of the second identity is entirely the same. Proposition 1.23. h L and h R are homomorphisms from V L and V R respectively to their images, viewed as graded vertex algebras. Proof. We have shown that h L , h R preserve gradings. By the identity property of open-closed field algebra, we have h L (1cl ) = 1op . Next, for u L ∈ V L and r > 0, we define Ycl−op (u L , r )w := lim Ycl−op (u L , z)w z→r
= lim Ycl−op (u L , z)Yop (1op , r )w, z→r
(1.73)
where the limit is taken along a path from a fixed initial point in H to r > 0. Since Ycl−op (u L , z)w is analytic in C× , the limit is independent of the path we choose. So we choose a path in the domain {z ∈ H||z| > r > |z − r | > 0}. In this domain, we can apply the associativity (1.52) to the right-hand side of (1.73). We obtain Ycl−op (u L , r )w = lim Yop (Ycl−op (u L , z − r )1op , r )w z→r
= Yop (h L (u L ), r )w. For |z| > r > 0, by (1.72), we have, Ycl−op (Y(u L ; r, r )v L , z)1op = e z Dop h L (Y(u L ; r, r )v L ),
(1.74)
the right hand side of which is absolutely convergent for all z ∈ C, and both sides are analytic in z. Therefore Ycl−op (Y(u L ; r, r )v L , z)1op is absolutely convergent for all z ∈ C and (1.74) holds for all z ∈ C. By the associativity, we have Ycl−op (Y(u L ; r, r )v L , z)1op = Ycl−op (u L , r + z)Ycl−op (v L , z)1op
(1.75)
for |r + z| > |z| > r > 0. Again both sides of (1.75) are analytic in z. Hence the left hand side of (1.75) defined for all z ∈ C is the analytic extension of the right-hand side of (1.75), which is defined on {|r + z| > |z|}. Since the extension is free of singularity on entire C, the right-hand side of (1.75) must be well-defined on entire C. Therefore, we must have lim Ycl−op (Y(u L , r )v L , z)1op = lim Ycl−op (u L , r + z)Ycl−op (v L , z)1op .
z→0
z→0
Combining the above results, we have h L (Y(u L , r )v L ) = lim e z Dop h L (Y(u L , r )v L ) z→0
= lim Ycl−op (Y(u L , r )v L , z)1op z→0
= lim Ycl−op (u L , r + z)Ycl−op (v L , z)1op z→0
= Ycl−op (u L , r )h L (v L ) = Yop (h L (u L ), r )h L (v L ).
(1.76)
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227
Thus h L is a vertex algebra homomorphism. The proof for h R is entirely the same.
Let (U, Y, 1, ω) be a vertex operator algebra with central charge c. U and U ⊗ U naturally give an analytic open-closed field algebra, in which Ycl−op (·; z, ζ )· is given by Ycl−op (u ⊗ v; z, ζ )w = Y (u, z)Y (v, ζ )w, = Y (v, ζ )Y (u, z)w,
|z| > |ζ | > 0, |ζ | > |z| > 0
(1.77)
for u, v, w ∈ U . In this case, h L : u ⊗ 1 → u and h R : 1 ⊗ u → u. We denote this open-closed field algebra as (U ⊗ U, U ). In general, let ρ L , ρ R ∈ Aut(U ), where Aut(U ) is the set of automorphisms of U as vertex operator algebra. We can obtain a new action of U ⊗ U on U by composing (1.77) with the automorphism ρ L ⊗ρ R : U ⊗U → U ⊗U . Namely, there exists another open-closed field algebra structure on U and U ⊗ U , in which Ycl−op (u ⊗ v; z, ζ )w, for u, v, w ∈ U , is given by Y (ρ L (u), z)Y (ρ R (v), ζ )w, Y (ρ R (v), ζ )Y (ρ L (u), z)w,
for |z| > |ζ | > 0, for |ζ | > |z| > 0.
(1.78)
In this case, h L : u ⊗ 1 → ρ L (u) and h R : 1 ⊗ u → ρ R (u). We denote such openclosed field algebra as (U ⊗ U, U, ρ L , ρ R ). In particular, (U ⊗ U, U, idU , idU ) is just (U ⊗ U, U ). Remark 1.24. (U ⊗ U, U, ρ L , ρ R ) for general automorphisms ρ L and ρ R is very interesting in physics. But it adds some technical subtleties in later formulations. So we postpone its study to future publications. In this work, we focus on (U ⊗ U, U ). Definition 1.25. Let (U, Y, 1, ω) be a vertex operator algebra. An open-closed field algebra over U is an analytic open-closed field algebra ((Vcl , m cl , ιcl ), (Vop , Yop , ιop ), Ycl−op ), where (Vcl , m cl , ιcl ) is a conformal full field algebra over U ⊗ U and (Vop , Yop , ιop ) is an open-string vertex operator algebra over U , satisfying the following conditions: 1. U -invariant boundary condition. h L = h R = ιop . 2. Chirality splitting property. ∀u ∈ Vcl , u = u L ⊗ u R ∈ W L ⊗ W R ⊂ Vcl for some U -modules W L , W R . There exist U -modules W1 , W2 and intertwining operators V 1 Vop W2 , W R W , W L V respectively3 , such Y (1) , Y (2) , Y (3) , Y (4) of type W LopW , W W RV op op 1 2 that w , Ycl−op (u; z, ζ )w = w , Y (1) (u L , z)Y (2) (u R , ζ )w
(1.79)
when |z| > |ζ | > 0 (recall the convention (1.16) and (1.17)), and w , Ycl−op (u; z, ζ )w = w , Y (3) (u R , ζ )Y (4) (u L , z)w when |ζ | > |z| > 0 for all u ∈ Vcl , w ∈ Vop , w ∈ Vop . 3 It was proved in [HKo1] that V is a U -module, and in [HKo2] that V is a U ⊗ U -module. op cl
(1.80)
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In the case that U is generated by ω, i.e. U = ω , the ω -invariant boundary condition is simply called the conformal invariant boundary condition. We also call the open-closed field algebra over ω open-closed conformal field algebra. Remark 1.26. The U -invariant boundary condition actually says that the open-closed field algebra over U contains (U ⊗ U, U ) as a subalgebra. If we only want to construct algebras over Swiss-cheese partial operad, the U -invariant boundary condition in Definition 1.25 can be weakened to the conformal invariant boundary condition: h L | ω = h R | ω = ιop | ω .
(1.81)
These situations appear in physics in the study of the so-called symmetry breaking boundary conditions (see for example [FS1, FS2] and references therein). All examples studied in this work and [K3] satisfy the U -invariant boundary condition. We leave the study of general symmetry-broken situations to the future. Remark 1.27. The chirality splitting property is a very natural condition because the inte˜ c is defined by a double sewing operation as given in (2.25). rior sewing operation of S Unfortunately, we do not know whether this chirality splitting property is necessary for ˜ c. general constructions of algebras over S For an open-closed field algebra over V , there are three Virasoro elements, ωop := ιop (ω) and ω L := ιcl (ω ⊗ 1) and ω R := ιcl (1 ⊗ ω), and we have L(n)r −n−2 , Yop (ωop , r ) = n∈Z
Y(ω L , z) =
L L (n)z −n−2 ,
n∈Z
Y(ω , ζ ) = R
L R (n)ζ −n−2 ,
n∈Z
where L L (n) = L(n) ⊗ 1 and L R (n) = 1 ⊗ L(n) for n ∈ Z. When U = V a vertex operator algebra satisfying the conditions in Theorem 0.1, we have a very simple description of open-closed field algebra over V given in the following theorem. Theorem 1.28. An open-closed field algebra over V is equivalent to the following structure: (Vop , Yop , ιop ) an open-string vertex operator algebra over V and (Vcl , m cl , ιcl ) a conformal full field algebra over V ⊗ V , together with a vertex map Ycl−op (·; z, ζ )· given by intertwining operators Y (i) , i = 1, 2, 3, 4 as in (1.79) and (1.80), satisfying the unit property: Ycl−op (1cl ; z, z¯ )v := Y (1) (1, z)Y (2) (1, z¯ )v = v,
∀v ∈ Vop ,
(1.82)
Associativity I (1.52), Associativity II (1.56) and Commutativity I given in Proposition 1.18. Proof. It is clear that an open-closed field algebra over V gives the data and properties included in the statement of the theorem. We only need to show that such data is sufficient to reconstruct an open-closed field algebra over V . Moreover, such open-closed field algebra over V with the given data is unique.
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229
Since V satisfies the condition in Theorem 0.1, the conditions listed in Definition 1.13 are all automatically satisfied. In particular, for vi ∈ Vop , u i ∈ Vcl , i = 1, . . . , n and , the following series: v ∈ Vop v , Ycl−op (u 1 ; z 1 , ζ1 )Yop (v1 , r1 ) · · · Ycl−op (u n ; z n , ζn )Yop (vn , rn )1op
(1.83)
is absolutely convergent when z 1 > ζ1 > r1 > · · · > z n > ζn > rn > 0, and can be extended by analytic continuation to a multi-valued analytic function for variables z i , ζi ∈ C× , i = 1, . . . , n with possible singularities only when two of z i , ζ j , rk are equal. Using this property of (1.83), we can define a non-analytic but single-valued n as follows. smooth function in n × MH n , n ∈ Z . Let γ be a smooth path from Let (r1 , . . . , rn ) ∈ n and (ξ1 , . . . , ξn ) ∈ MH + 1 n . Let γ : [0, 1] → Rn (3n, 3(n − 1), . . . , 1) to (ξ1 , . . . , ξn ) such that γ1 ((0, 1)) ⊂ MH 0 + be so that γ0 (t) = ((1 − t)(3n − 1) + t3n, . . . , (1 − t)2 + t3). Clearly, γ0 is the straight line from (3n − 1, 3(n − 1) − 1, . . . , 2) to (3n, 3(n − 1), . . . , 3) in Rn+ . Then we define a path γ2 : [0, 1] ∈ (H ∪ R+ )n to be the composition γ2 = γ¯1 ◦ γ0 , where γ¯1 is the complex conjugation of γ1 . It is clear that γ2 is a path from (3n − 1, 3(n − 1) − 1, . . . , 2) to (ξ¯1 , . . . , ξ¯n ). Let γ3 : [0, 1] → Rn+ be so that γ3 (t) = ((1 − t)(3n − 2) + tr1 , . . . , (1 − t)1 + trn ). So γ3 is the straight line from (3n − 2, 3(n − 1) − 2, . . . , 1) to (r1 , . . . , rn ) ∈ n . Combining γ1 , γ2 , γ3 , we obtain a path γ in C3n from the initial point (3n, 3n − ¯ ¯ 1, 3n − 2, . . . , 3, 2, 1) ∈ R2n + to the final point (ξ1 , ξ1 , r1 , . . . , ξn , ξn , rn ) in the obvious way. Then we define (n;n) v , m cl−op (u 1 , . . . , u n , v1 , . . . , vn ; ξ1 , ξ¯1 , . . . , ξn , ξ¯n , r1 , . . . , rn )
to be the value obtained from the value of (1.83) at the initial point (3n, 3n − 1, 3n − 2, . . . , 3, 2, 1) by analytic continuation along the path γ . Following a similar argument as in the proof of Theorem 2.11 in [HKo2], it is easy to show that such defined (n;n)
m cl−op (u 1 , . . . , u n ; v1 , . . . , vn ; ξ1 , ξ¯1 , . . . , ξn , ξ¯n , r1 , . . . , rn ),
(1.84)
is independent of the choice of γ1 and its initial points. Moreover, such defined (1.84) is single-valued and smooth in H × H. For n > l ≥ 0, we define (l;n)
m cl−op (u 1 , . . . , u l ; v1 , . . . , vn ; ξ1 , ξ¯1 , . . . , ξl , ξ¯l ; r1 , . . . , rn ), (n;n) := m cl−op (u 1 , . . . , u l , 1cl , . . . , 1cl ; v1 , . . . , vn ; ξ1 , ξ¯1 , . . . , ξn , ξ¯n ; r1 , . . . , rn ),
and for l > n ≥ 0, we define (l;n) m cl−op (v1 , . . . , vl ; w1 , . . . , wn ; ξ1 , ξ¯1 , . . . , ξl , ξ¯l ; r1 , . . . , rn ),
¯ ¯ := m (l;l) cl−op (u 1 , . . . , u l ; v1 , . . . , vn , 1op , . . . , 1op ; ξ1 , ξ1 , . . . , ξl , ξl ; r1 , . . . , rl ), (1.85) (0;0)
and for n = l = 0, we define m cl−op (1) = 1op ∈ Vop .
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L. Kong (l;n)
Immediately following from the construction of m cl−op , we have, for all v ∈ Vop , (0;1)
(1;1)
m cl−op (v; 0) = m cl−op (1cl ; v; z; 0) = Ycl−op (1cl ; z, z¯ )v = v. (l;n)
Now we show the permutation axiom for m cl−op . This is enough to just consider adjacent permutations (ii + 1), i = 1, . . . , l − 1 because they generate the whole permutation group. We can just consider (12) because all the other cases are exactly the same. This amounts to show that (l;n) m cl−op (v1 , v2 , . . . , vl ; w1 , . . . , wn ; ξ1 , ξ¯1 , ξ2 , ξ¯2 , . . . , ξl , ξ¯l ; r1 , . . . , rn ), (l;n) = m cl−op (v2 , v1 , . . . , vl ; w1 , . . . , wn ; ξ2 , ξ¯2 , ξ1 , ξ¯1 , . . . , ξl , ξ¯l ; r1 , . . . , rn ).
(1.86)
By our construction, the only difference of the two sides of (1.86) is that they are obtained by analytic continuation along paths with different initial points. The initial points of the path for the left-hand side of (1.86) is z 1 > ζ1 > r1 > z 2 > ζ2 > r2 > . . . , that for the right hand side of (1.86) is z 2 > ζ2 > r1 > z 1 > ζ1 > r2 > . . . . But by the commutativity, the value of (1.83) at these two initial points are analytic continuations of each other along the paths given in the commutativity axiom of open-closed field algebra. Hence the equality (1.86) follows. Equations (1.38) and (1.39) can be proved by first proving similar properties of (1.83), which is obvious by the properties of intertwining operators. Then those properties (1.38) (l;n) and (1.39) of m cl−op follow from analytic continuations. Since Y (i) , i = 1, 2, 3, 4 are intertwining operators of V , we have, for u ∈ V , h L (u) = lim Ycl−op (u, z)1op z→0
= lim lim Y (1) (Y (u, z 1 )1, z)Y (2) (1, z¯ )ιop (1) z→0 z 1 →0
= lim lim YVop (u, z + z 1 )Y (1) (1, z)Y (2) (1, z¯ )ιop (1) z→0 z 1 →0
= lim lim YVop (u, z + z 1 )Ycl−op (1cl ; z, z¯ )ιop (1) z→0 z 1 →0
= lim lim YVop (u, z + z 1 )ιop (1) z→0 z 1 →0
= lim lim ιop (Y (u, z + z 1 )1) z→0 z 1 →0
= ιop (u).
(1.87)
Similarly, one can show that h R (u) = ιop (u), ∀u ∈ V . Thus we have proved the V -invariant boundary condition h L = h R = ιop . It remains to show the convergence properties of open-closed field algebra. For the (i) (i) first convergence property ((1.33) and (1.34)), one first considers cases when z j , r p , j = 1, . . . , k, p = 1, . . . , m in (1.33) have distinct absolute values and z j , r p , j = 1, . . . , l, p = 1, . . . , n in (1.33) have distinct absolute values. In these cases, one can (l;n) (k;m) express m cl−op and m cl−op as products of Ycl−op and Yop . Then by using the associativity (1.52) and that of open-string vertex operator algebra, it is easy to show that (1.33) converges absolutely to (1.34) in the required domain. The rest of the cases can all be reduced to the above cases by using (1.39) (see (i) (i) the proof of Theorem 2.11 in [HKo2] for reference). More precisely, for z 1 , . . . , z k ,
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(i)
(i)
z 1 , . . . , zl ∈ H; r1 > · · · > rm ≥ 0, r1 > · · · > rn > 04 , there always exists a ∈ R+ small enough so that both of the following sets: (i)
(i)
(i)
(i)
{z 1 + a, . . . , z k + a, r1 + a, . . . , rk + a}, {z 1 − a, . . . , zl − a, r1 − a, . . . , rn − a} are sets whose elements have distinct absolute values. Then (1.33) equals the following iterate series: (l;n) (m;k) m cl−op (u 1 , . . . , u l ; v1 , . . . , vi−1 , Pn 2 e−a Dop Pn 1 m cl−op (u˜ 1 , . . . , u˜ k ; n2
n1
v˜1 , . . . , v˜m ; z 1(i) + a, z 1(i) + a, . . . , z k(i) + a, z k(i) + a; r1(i) + a, . . . , rm(i) + a), vi+1 , . . . vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn )a1n 1 a2n 2
(1.88)
when a1 = a2 = 1. Hence we first switch the order of the above iterate sum. Then using (1.41) and the analytic extension properties of (1.83), we can easily show that the iterate series (1.88) with opposite summing order is absolutely convergent when 1 ≥ |a1 |, |a2 |. Then we can switch the order of iterate sum in (1.88) freely without changing the value of the double sum (1.88). By (1.41), we have reduced all the remaining cases to the previous cases. The proof of the second convergence property ((1.35) and (1.36)) is entirely the same except that one uses the associativity (1.56) and (1.42). We omit the details. It is also clear that such open-closed field algebra over V is unique because products of Ycl−op and Yop determine a dense subset of the set of all cases. The rest of the cases are uniquely determined by continuity. 2. Operadic Formulation In this section, we first review the notion of 2-colored partial operad and algebra over it. Then we recall the notion of Swiss-cheese partial operad S, its relation to sphere partial ˜ c [HKo1]. In the end, we prove that an open-closed field operad K and its C-extension S ˜ c. algebra over V canonically gives an algebra over S 2.1. 2-colored partial operads. We recall the notion of 2-colored (partial) operad [V, Kt] and algebras over it. The 2-colored operad is called relative operad in [V] and colored operad in [Kt]. We first recall some basic notions from [H4]. The notion of (partial) operad can be defined in any symmetric monoidal category [MSS]. In this work, we only work in the category of sets. We will use the definition of (partial) operad given in [H4]. We denote a (partial) operad as a triple (P, IP , γP ), where P = {P(n)}n∈N is a family of sets, IP the identity element and γP the substitution map. Note that the definition of (partial) operad in [H4] is slightly different from that in [MSS] for the appearance of P(0) which is very important for the study of conformal field theory. If a triple (P, IP , γP ) satisfies all the axioms of a (partial) operad except the associativity of γP , then it is called (partial) nonassociative operad. 4 If r = 0, we can further introduce another small real variable b, using (1.39) to move r = 0 to some n n rn > 0. We omit the details.
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We consider an important example of partial nonassociative operad. Let U = ⊕n∈J U(n) be a vector space graded by an index set J and EU = {EU (n)}n∈N a family of vector spaces, where EU (n) = HomC (U ⊗n , U ). For k, n 1 , . . . , n k ∈ N, f ∈ EU (k), gi ∈ EU (n i ), i = 1, . . . , k and v j ∈ U, j = 1, . . . , n 1 + · · · + n k , γ EU ( f ; g1 , . . . , gk )(v1 ⊗ · · · ⊗ vn 1 +···+n k ) f (Ps1 g1 (v1 ⊗ · · · ⊗ vn 1 ) ⊗ · · · := s1 ,...,sk ∈J
⊗Psk gk (vn 1 +···+n k−1 +1 ⊗ · · · ⊗ vn 1 +···+n k ))
(2.1)
is well-defined if the sum is countable and absolutely convergent. γ EU is not associative in general because an iterate series may converge in both order but may not converge to the same value. It is clear that (EU , idU , γ EU ) is a partial nonassociative operad. We sometimes denote it simply as EU . If J is the set of equivalence classes of irreducible modules over a group G and U(n) is a direct sum of irreducible G-modules of the equivalence class n ∈ J , we denote this partial nonassociative operad as EUG . Definition 2.1. An algebra over a partial operad (P, IP , γP ), or simply a P-algebra, is a graded vector space U , together with a partial nonassociative operad homomorphism ν : P → EU . Definition 2.2. Given a partial operad (P, IP , γP ), a subset G of P(1) is called rescaling group for P if 1. For any n ∈ N, Pi ∈ G, i = 0, . . . , n and P ∈ P(n), γP (P; P1 , . . . , Pn ) and γP (P0 ; P) are well-defined. γP 2. IP ∈ G and G together with the identity IP and multiplication map G × G −→ G is a group. Definition 2.3. A partial operad (P, IP , γP ) is called G-rescalable, if for Pi ∈ P(n i ), i = 1, . . . , k and P0 ∈ P(k), then there exists gi ∈ G, i = 1, . . . , k such that γP (γP (P0 ; g1 , . . . , gk ); P1 , . . . , Pk ) is well-defined. Definition 2.4. An algebra over a G-rescalable partial operad (P, IP , γP ), or a G-rescalable P-algebra, consists of a completely reducible G-modules U = ⊕n∈J U(n) , where J is the set of equivalence classes of irreducible G-modules and U(n) is a direct sum of irreducible G-modules of the equivalence class n, and a partial nonassociative operad homomorphism ν : P → EUG such that ν : G → End U(n) coincides with G-module structure on U(n) . We denote such algebra as (U, ν). For m ∈ Z+ , let m = m 1 + · · · + m n be an ordered partition and σ ∈ Sn . The block permutation σ(m 1 ,...,m n ) ∈ Sm is the permutation acting on {1, . . . , m} by permuting n intervals of lengths m 1 , . . . , m n in the same way that σ permute 1, . . . , n. Let σi ∈ Sm i , i = 1, . . . , n, we view the element (σ1 , . . . , σn ) ∈ Sm 1 × · · · × Sm n naturally as an element in Sm by the canonical embedding Sm 1 × · · · × Sm n → Sm .
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Definition 2.5. Let (Q, IQ , γQ ) be an operad. A right module over Q, or a right Q-module, is a family of sets P = {P(n)}n∈N with actions of permutation groups, equipped with maps: γ
P(k) × Q(n 1 ) × · · · × Q(nl ) − → P(n 1 + · · · + n k ) such that 1. For c ∈ P(k), we have γ (c; IQ , . . . , IQ ) = c.
(2.2)
2. γ is associative. Namely, for c ∈ P(k), di ∈ Q( pi ), i = 1, . . . , k, e j ∈ Q(q j ), j = 1, . . . , p1 + · · · + pk , we have γ (γ (c; d1 , . . . , dk ); e1 , . . . , e p1 +···+ pk ) = γ (c; f 1 , . . . , f k ),
(2.3)
where f s = γQ (ds ; e p1 +···+ ps−1 +1 , . . . , e p1 +···+ ps ). 3. For c ∈ P(k; l), di ∈ Q( pi ), i = 1, . . . , l, σ ∈ Sl and τ j ∈ S p j , j = 1, . . . , l, γ (σ (c); d1 , . . . , dl ) = σ( p1 ,..., pl ) (γ (c; d1 , . . . , dl )), γ (c; τ1 (d1 ), . . . , τl (dl )) = (τ1 , . . . , τl )(γ (c; d1 , . . . , dl )).
(2.4) (2.5)
Homomorphisms and isomorphisms between right Q-modules are naturally defined. The left module over a partial operad can be similarly defined. Definition 2.6. A right module P over a partial operad Q, or a right Q-module, is called G-rescalable if Q is G-rescalable and for any c ∈ P(k), di ∈ Q(n i ), i = 1, . . . , k, there exist gi ∈ G, i = 1, . . . , k such that γ (γ (c; g1 , . . . , gk ); d1 , . . . , dk ) is well-defined. Definition 2.7. A 2-colored operad consists of an operad (Q, IQ , γQ ), a family of sets P(m; n) equipped with an Sm × Sn -action for m, n ∈ N, a distinguished element IP ∈ P(1, 1) and substitution maps γ1 , γ2 given as follows: P(k; l) × P(m 1 ; n 1 ) × · · · × P(m k ; n k ) γ1
− → P(m 1 + . . . + m k ; l + n 1 + . . . + n k ), γ2
P(k; l) × Q( p1 ) × · · · × Q( pl ) − → P(k; p1 + · · · + pl ),
(2.6)
satisfying the following axioms: 1. The family of sets P := {∪n∈N P(m; n)}m∈N equipped with the natural Sm -action on P(m) = ∪n∈N P(m; n), together with identity element IP and substitution maps γ1 is an operad. 2. γ2 gives each P(k) a right Q-module structure for k ∈ N. We denote it as (P|Q, (γ1 , γ2 )) or P for simplicity.
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Remark 2.8. The substitution map γ1 , γ2 can be combined into a single substitution map γ = (γ1 , γ2 ): P(k; l) × P(m 1 ; n 1 ) × . . . × P(m k ; n k ) × Q( p1 ) × · · · × Q( pl ) γ
− → P(m 1 + . . . + m k ; n 1 + . . . + n k + p1 + · · · + pl ).
(2.7)
For this reason, we also denote (P|Q, (γ1 , γ2 )) as (P|Q, γ ). For some examples we encounter later, γ1 , γ2 are often defined all together in terms of γ . Definition 2.9. 2-colored partial operad is defined similarly as that of 2-colored operad except that γ1 , γ2 are only partially defined and Q, P are partial operads and (2.3) holds whenever both sides exist. If only the associativities of γQ , γ1 , γ2 do not hold, then it is called 2-colored nonassociative (partial) operad. We give an important example of 2-colored partial nonassociative operad. Let J1 , J2 be two index sets. U1 = ⊕n∈J1 (U1 )(n) , U2 = ⊕n∈J2 (U2 )(n) be two graded vector spaces. Consider two families of vector spaces, EU2 (n) = HomC (U2⊗n , U 2 ),
EU1 |U2 (m; n) = HomC (U1⊗m ⊗ U2⊗n , U 1 ).
(2.8)
We denote both of the projection operators U1 → (U1 )(n) , U2 → (U2 )(n) as Pn for n ∈ J1 or J2 . For any f ∈ EU1 |U2 (k; l), gi ∈ EU1 |U2 (m i ; n i ), i = 1, . . . , k, and h j ∈ EU2 ( p j ), j = 1, . . . , l, we say that ( f ; g1 , . . . , gn ; h 1 , . . . , h l ) :=
(1)
(1)
(1) f (Ps1 g1 (u 1 , . . . , u (1) m 1 , v1 , . . . , vn 1 ),
s1 ,...,sk ∈J1 ;t1 ,...,tl ∈J2 (k)
(k)
(k) . . . , Psk gk (u 1 , . . . , u (k) m k , v1 , . . . , vn k );
Pt1 h 1 (w1(1) , . . . , wn(1) ), . . . , Ptl h l (w1(l) , . . . , wn(l)l )), 1 (i) where u (i) j ∈ U1 , v j ∈ U2 , is well-defined if the multiple sum is absolutely convergent. This gives arise to a partially defined substitution map :
EU1 |U2 (k; l) × EU1 |U2 (m 1 ; n 1 ) × · · · × EU1 |U2 (m k ; n k ) × EU2 ( p1 ) × · · · × EU2 ( pl )
− → EU1 |U2 (m 1 + · · · + m k , n 1 + · · · + n k + p1 + · · · + pl ). does not satisfy the associativity in general. Let EU1 |U2 = {EU1 |U2 (m; n)}m,n∈N and EU2 = {EU2 (n)}n∈N . It is obvious that (EU1 |U2 |EU2 , ) is a 2-colored nonassociative partial operad. Let U1 be a completely reducible G 1 -module and U2 a completely reducible G 2 module. Namely, U1 = ⊕n 1 ∈J1 (U1 )(n 1 ) , U2 = ⊕n 2 ∈J2 (U2 )(n 2 ) where Ji is the set of equivalence classes of irreducible G i -modules and (Ui )(n i ) is a direct sum of irreducible G i -modules of the equivalence class n i for i = 1, 2. In this case, we denote EU1 |U2 by G |G EU11|U22 .
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Definition 2.10. A homomorphism between two 2-colored (partial) operads (Pi |Qi , γi ), i = 1, 2 consists of two (partial) operad homomorphisms: νP1 |Q1 : P1 → P2 ,
and
νQ1 : Q1 → Q2
such that νP1 |Q1 : P1 → P2 , where P2 is a right Q1 -module by νQ1 , is also a right Q1 -module homomorphism. Definition 2.11. An algebra over a 2-colored partial operad (P|Q, γ ), or a P|Qalgebra consists of two graded vector spaces U1 , U2 and a homomorphism (νP |Q , νQ ) from (P|Q, γ ) to (EU1 |U2 |EU2 , ). We denote this algebra as (U1 |U2 , νP |Q , νQ ). Definition 2.12. If a 2-colored partial operad (P|Q, γ ) is so that P is a G 1 -rescalable partial operad and a G 2 -rescalable right Q-module, then it is called G 1 |G 2 -rescalable. Definition 2.13. A G 1 |G 2 -rescalable P|Q-algebra (U1 |U2 , νP |Q , νQ ) is a P|Q-algeG |G bra so that νP |Q : P → EU11|U22 and νQ : Q → EUG22 ; moreover, νP |Q : G 1 → End U1 coincides with the G 1 -module structure on U1 and νQ : G 2 → End U2 coincides with the G 2 -module structure on U2 . 2.2. Swiss-cheese partial operad S. A disk with strips and tubes of type (m − , m + ; n − , n + ) (m − , m + , n − , n + ∈ N) is a disk S with the following additional data: B , p B , . . . , p B (called boundary B 1. m − +m + distinct ordered punctures p−m , . . . , p−1 m+ 1 − B are negatively oriented B , . . . , p punctures) on ∂ S (the boundary of S), where p−m −1 − and p1B , . . . , pmB + are positively oriented, together with local coordinates: B B B B , ϕ−m ), . . . , (U−1 , ϕ−1 ); (U1B , ϕ1B ), . . . , (UmB+ , ϕmB+ ), (U−m − −
˜ is an analytic map which where UiB is a neighborhood of piB and ϕiB : UiB → H B B vanishes at pi and maps Ui ∩ ∂ S to R, for each i = −m − , . . . , −1, 1, . . . , m + . I , p I , . . . , p I (called interior puncI 2. n − + n + distinct ordered points p−n , . . . , p−1 n+ 1 − I I are negatively oriented and tures) in the interior of S, where p−n − , . . . , p−1 p1I , . . . , pnI + are positively oriented, together with local coordinates: I I I I , ϕ−n ), . . . , (U−1 , ϕ−1 ); (U1I , ϕ1I ), . . . , (U1I , ϕnI + ), (U−n − −
where U jI is a local neighborhood of p Ij and ϕ Ij : U jI → C is an analytic map which vanishes at piI for each j = −n − , . . . , −1, 1, . . . , n + . Two disks with strips and tubes are conformal equivalent if there exists between them a biholomorphic map which maps punctures to punctures and preserves the order of punctures and the germs of local coordinate maps. We denote the moduli space of the conformal equivalence classes of disks with strips and tubes of type (m − , m + ; n − , n + ) as S(m − , m + |n − , n + ). The structure on S(m − , m + |n − , n + ) will be discussed in [K3]. In this work, we are only interested in disks with strips and tubes of types (1, m + ; 0, n + ) for n, l ∈ N. For simplicity, we denote S(1, m + |0, n + ) by ϒ(m + ; n + ). For such disks, we label the only negatively oriented boundary puncture as the 0th boundary puncture.
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We can choose a canonical representative for each conformal equivalence class in ϒ(m + ; n + ) just as we did for disks with strips [HKo1] and sphere with tubes [H4]. More precisely, for a disk with strips and tubes of type (1, m + ; 0, n + ) where m + > 0, we first ˆ Then we use an automorphism of H ˆ to use a conformal map f to map the disk to H. th move the only negatively oriented puncture (the 0 puncture) to ∞ and the smallest rk to 0, and fix the local coordinate map f 0 at ∞ to be so that limw→∞ w f 0 (w) = −1. As a consequence, the canonical representative of a generic conformal equivalent class ˆ together with a negatively of disk with strips and tubes Q ∈ ϒ(m + ; n + ) is a disk H, ˆ oriented boundary puncture at ∞ ∈ H and local coordinate map given by f 0B (w) = −e
∞
(0) − j+1 d j=1 −B j w dw
1 , w
where B (0) j ∈ R, and positively oriented boundary punctures at r1 , . . . rm + ∈ R+ ∪ {0} (rk = 0) and local coordinate maps given by ∞ (i) B j x j+1 ddx (i) x ddx B j=1 f i (w) = e (b0 ) x , i = 1, . . . , m + , x=w−ri
(i)
(i)
where B j ∈ R, b0 ∈ R+ , and positively oriented interior punctures at z 1 , . . . , z n + ∈ H with local coordinate maps given by ∞ (i) (i) x ddx A j x j+1 ddx I j=1 f i (w) = e (a0 ) x , i = 1, . . . , n + , x=w−z i
(i)
(i)
where A j ∈ C and a0 ∈ C× . Let R (C ) be the set of sequences of real numbers (complex numbers) {C j }∞ j=1
such that e define
j>0 C j x
j+1 d dx
x as a power series converges in some neighborhood of 0. We
˜ n := {(r1 , . . . , rn )| ∃σ ∈ Sn , rσ (1) > · · · > rσ (n) = 0}. Using the data on the canonical representative of Q, we denote Q ∈ ϒ(m + ; n + ) as follows (1)
(m + )
[r1 , . . . , rm + −1 ; B (0) , (b0 , B (1) ), . . . , (b0
, B (m + ) )|
z 1 , . . . , z n + ; (a0(1) , A(1) ), . . . , (a0(n + ) , A(n + ) )], (i)
( j)
(2.9) (i)
˜ m + and b ∈ R+ , a ∈ C× , and B (i) = {B }∞ ∈ R and where (r1 , . . . , rm + ) ∈ 0 0 j j=1 ( p)
∞ ∈ for all i = 1, . . . , m , p = 1, . . . , n . A( p) = {Aq }q=1 + + C Using notations (0.1) and (0.2), for m + > 0 and n + ∈ N, we can express the moduli space of disks with strips and tubes of type (1, m + ; 0, n + ) as follows:
˜ m + −1 × R × (R+ × R )m + × M n + × (C× × C )n + . ϒ(m + ; n + ) = H ˆ to fix B (0) = 0. Hence, we have For m + = 0, n + ∈ N, we used automorphism of H 1 (0)
ϒ(0; n + ) = {B (0) ∈ | B1 = 0} × (C× × C )n + .
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Note that ϒ(m + ; 0) is nothing but ϒ(m + ) introduced and studied in [HKo1]. ϒ := {ϒ(n)}n∈N is a partial operad of disks with strips [HKo1]. The identity Iϒ is an element of ϒ(1; 0). Also, for m + , n + ∈ N, Sn + acts on ϒ(m + ; n + ) in the obvious way. Let S(m) = ∪n + ∈N ϒ(m + ; n + ) for m ∈ N, and S = ∪m∈N S(m). There are two kinds of sewing operations on S, The first kind is called boundary sewing operation which sews the positively oriented boundary puncture in the first disk with a negatively oriented boundary puncture in the second disk. The second is called interior sewing operation which sews a positively oriented interior puncture in a disk with a negatively oriented puncture in a sphere with tubes. We describe these sewing operations more precisely. Let us consider P ∈ ϒ(m + ; n + ) and Q ∈ ϒ( p+ ; q+ ) (Q ∈ K ( p+ )) for boundary sewing operations (for interior sewing operations). Let B r ( B¯ r ) denote the open (closed) ball in C center at 0 with radius r , ϕi the germs of local coordinate map at i th boundary (interior) puncture p of P, and ψ0 the germs of local coordinate map at 0th puncture q of Q. Then we say that the i th strip of P can be sewn with the 0th strip (tube) of Q if there is a r ∈ R+ such that p and q are the only punctures in ϕi−1 ( B¯ r ) and ψ0−1 ( B¯ 1/r ) respectively. A new disk with strips and tubes in ϒ(m + + p+ − 1, n + + q+ ) (ϒ(m + , n + + p+ − 1)), denoted as Pi ∞0B Q (Pi ∞0I Q), is obtained by cutting out ϕi−1 (B r ) and ψ0−1 (B 1/r ) from P and Q respectively, and then identifying the boundary of ϕi−1 ( B¯ r ) and ψ0−1 ( B¯ 1/r ) via the map ψ −1 ◦ JHˆ ◦ ϕi , where JHˆ : w → − w1 . The boundary sewing operations and interior sewing operations induce the following partially defined substitution maps: ϒ(k; l) × ϒ(m 1 ; n 1 ) × . . . × ϒ(m k ; n k ) × K ( p1 ) × · · · × K ( pl ) γ
− → ϒ(m 1 + . . . + m k ; n 1 + . . . + n k + p1 + · · · + pl ).
(2.10)
The following proposition is clear. Proposition 2.14. (S|K , γ ) is a R+ |C× -rescalable 2-colored partial operad. This R+ |C× -rescalable 2-colored partial operad (S|K , γ ) is a generalization of Voronov’s Swiss-cheese operad [V]. So we will call it Swiss-cheese partial operad and sometimes denote it by S for simplicity. The relation between Swiss-cheese operad and Swiss-cheese partial operad is an analogue of that between little disk operad and sphere partial operad [H4]. 2.3. Sewing equations and the doubling map δ. We are interested in finding the canonical representative of disk with strips and tubes obtained by sewing two such disks or sewing a disk with a sphere. In the case of sphere partial operad K , such canonical representatives were obtained by Huang [H1, H2, H4] by solving the so-called sewing equation. Similarly, the canonical representatives of disks with strips and tubes obtained by two types of sewing operations can also be determined by solving two types of sewing equations. We start with the boundary sewing operations. For Q ∈ S, we denoted the canonical representative of Q as Q . Let P ∈ ϒ(m; n) and Q ∈ ϒ( p; q). Let g0 be the local coordinate map at ∞ ∈ Q and f i be that at z i ∈ P , 1 ≤ i ≤ m. We assume
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that Pi ∞0B Q exists. Then the canonical disk Pi ∞ B Q can be obtained by solving the 0
following sewing equation: −1 B B F(1) , g0−1 (w) = F(2) f i (w)
(2.11)
B is a conformal map from an open neighborhood of ∞ ∈ to an open neighwhere F(1) P B is a conformal map from an open neighborhood of borhood of ∞ ∈ Pi ∞ B Q , and F(2) 0
0 ∈ Q to an open neighborhood of 0 ∈ Pi ∞ B Q , with the following normalization 0
conditions: B F(1) (∞) = ∞,
B F(2) (0) = 0,
lim
w→∞
B (w) F(1)
w
= 1.
(2.12)
B , FB , f It is easy to see that the solution of (2.11) and (2.12) is unique. Notice that F(1) (2) 0 and f i in (2.11) are all real analytic. Similarly, let P ∈ ϒ(m; n) and Q ∈ K ( p). We denote the canonical sphere with tubes of Q as Q . Let g0 be the local coordinate map at ∞ ∈ Q and f i be that at z i ∈ P, 1 ≤ i ≤ n. We assume that Pi ∞0I Q exists. Then Pi ∞ I Q can be obtained by 0
solving the following sewing equation: −1 I I F(1) g0−1 , (w) = F(2) f i (w)
(2.13)
I is a conformal map from an open neighborhood of R ˆ in P to an open neighwhere F(1) I is a conformal map from an open neighborhood of 0 ˆ in borhood of R I , and F Pi ∞0 Q
(2)
in Q to an open subset of H ⊂ Pi ∞ I Q , with the following normalization conditions: 0
I (∞) = ∞, F(1)
I F(1) (0) = 0,
lim
w→∞
I (w) F(1)
w
= 1.
(2.14)
It is easy to see that the solution of (2.13) and (2.14) is unique as well. I is real analytic because it maps R to R. Hence, F I is also the unique Notice that F(1) (1) solution for the following equation: −1 −1 I I (2.15) F(1) (w) = F(2) g0 f i (w) with the same normalization condition (2.14). In sphere partial operad, we define a complex conjugation map Conj : K → K as follows (1)
(n)
Conj : (z 1 , . . . , z n−1 ; A(0) , (a0 , A(1) ), . . . , (a0 , A(n) ))
−→ (¯z 1 , . . . , z¯ n−1 ; A¯ (0) , (a¯ 0(1) , A¯ (1) ), . . . , (a¯ 0(n) , A¯ (n) )). For simplicity, we denote Conj(Q) as Q¯ for Q ∈ K . Proposition 2.15. Conj is a partial operad automorphism of K .
(2.16)
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Proof. It is clear that Conj((0, (1, 0))) = (0, (1, 0)) and Conj is equivariant with respect to the action of permutation group. Moreover, Conj is obviously bijective. It only remains to show that, for 1 ≤ i ≤ m, Pi ∞0 Q = P¯i ∞0 Q¯ (2.17) for any pair P ∈ K (m), Q ∈ K (n) such that Pi ∞0 Q exists. Let f i be the local coordinate map at i th puncture in P and g0 be that at ∞ in Q. Then the local coordinate map at i th puncture in P¯ is f¯i and that at ∞ in Q¯ is g¯ 0 . Also notice that the sewing equation and normalization equation for sphere partial operad (1) (2) ([H4]) are the same as Eq. (2.11) and (2.12). Let Fspher e , Fspher e be the solution of (1) (2) (2.11) and (2.12) for the sphere with tubes , then F¯spher e , F¯spher e also satisfy the same normalization condition and the following sewing equation: −1 (1) (2) −1 ¯ ¯ , Fspher e (w) = Fspher e g0 f¯i (w) ¯ Using the explicit formula ((A.6.1)–(A.6.5) which is the sewing equation for P¯i ∞0 Q. (1) (2) in [H4]) of the moduli P¯i ∞0 Q¯ in terms of F¯spher e , F¯spher e , f¯i , g¯ 0 , one can easily see that (2.17) is true. There is a canonical doubling map δ : S → K defined as follows. Let Q ∈ ϒ(n; l) with form (1)
(n)
Q = [ r1 , . . . , rn−1 ; B (0) , (b0 , B (1) ), . . . , (b0 , B (n) ) |z 1 , . . . , zl ; (a0(1) , A(1) ), . . . , (a0(l) , A(l) ) ]. Then (1)
(l)
δ(Q) = (z 1 , . . . , zl , z¯ 1 , . . . , z¯l , r1 , . . . , rn−1 ; (a0 , A(1) ), . . . , (a0 , A(l) ), (1)
(l)
(1)
(n)
(a0 , A(1) ), . . . , (a0 , A(l) ); B (0) , (b0 , B (1) ), . . . , (b0 , B (n) )). Proposition 2.16. Let P ∈ ϒ(m; n) and Q ∈ ϒ( p; q). Assume that Pi have
∞B Q
δ(Pi ∞0B Q) = δ(P)2n+i ∞0 δ(Q).
0
(2.18)
exists. We (2.19)
B , F B , g −1 and f in (2.11) are all real analytic, every solution of (2.11) Proof. Since F(1) i (2) 0 and (2.12) for disk with strips and tubes can be extended to a solution of the same sewing equation for sphere with tubes by Schwarz’s reflection principle. Then the proposition follows immediately from this fact.
Given a canonical disk with strips and tubes Q corresponding to moduli Q ∈ S, ˆ we consider its complex conjugation, denoted as . is the lower half plane H Q
Q
together with the same boundary punctures and local coordinate maps as those in Q , and interior punctures which are the complex conjugation of the interior punctures in Q with local coordinate maps being the complex conjugation of those in Q . We can denote it as [r1 , . . . , rm + −1 ; B (0) , (b0(1) , B (1) ), . . . , (b0(m + ) , B (m + ) )| (1)
(n + )
z¯ 1 , . . . , z¯ n + ; (a0 , A(1) ), . . . , (a0 where z 1 , . . . , z n + ∈ H.
, A(n + ) )],
(2.20)
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Lemma 2.17. Let P ∈ ϒ(m; n) and Q ∈ K ( p). Assume that Pi ∞0I Q exists for 1 ≤ i ≤ m. Then P i ∞0I Q¯ also exists and ¯ Pi ∞ I Q ∼ = P i ∞0I Q.
(2.21)
0
Proof. We can choose a canonical representative of P i ∞0I Q¯ as a lower half plane ˆ with two punctures at ∞ and 0 and the local coordinate map g at ∞ being so that H 0 limw→∞ wg0 (w) = −1. We denote such representative of P i ∞0I Q¯ (a lower half plane) as 1 . 1 can be obtained by solving the sewing equation −1 I I , (2.22) (w) = G (2) g0−1 G (1) f i (w) I is a conformal map from a neighborhood of R ˆ ⊂ P to a neighborhood where G (1) ˆ ⊂ 1 , and G I is a conformal map from a neighborhood of 0 ∈ Q to an open of R (2)
subset of H ⊂ 1 , with the normalization equation (2.14). By comparing (2.22) with I (real analytic) and F I of (2.15) and (2.14) (2.15), we see that the unique solution F(1) (2) I and G I of (2.22) and (2.14). Hence we have exactly gives the unique solution G (1) (2) 1 = Pi ∞ I Q . 0
Proposition 2.18. Let P ∈ ϒ(n; l), Q ∈ K (m), and Pi ∞0I Q exists for 1 ≤ i ≤ l. Then δ(Pi ∞0I Q) = (δ(P)i ∞0 Q)l+m−1+i ∞0 Q¯
(2.23)
Proof. Now we first consider the right-hand side of (2.23). We denote the canonical representative of any R ∈ K (l) as R . Then δ(P) can be viewed as a union of the closure of upper half plane, denoted as U+ , and the closure of lower half plane, denoted as U− . Let + be the Riemann surface obtained by sewing U+ with Q, and − the Riemann ¯ By identifying the real line in U+ ⊂ + with surface obtained by sewing U− with Q. the real line in U− ⊂ − using identity map, we obtain the surface + #− which is isomorphic to the canonical sphere with tubes (δ(P)i ∞0 Q)l+m−1+i ∞0 Q¯ . Both + and − are disks with strips and tubes. Since U+ = P , there is a unique biholomorphic map f from + to the canonical disk with strips and tubes Q i ∞ I P . 0
Similarly, because U− = P , by Lemma 2.17, there is a unique biholomorphic map g from − to the canonical disk with strips and tubes Pi ∞ I Q . The restriction of f on the 0
ˆ = ∂U+ is nothing but the unique real analytic map F I satisfying neighborhood of R (1) ˆ = ∂U− is noth(2.13) and (2.14). Meanwhile, the restriction of g on a neighborhood of R I satisfying (2.15) and (2.14). So we must have f | = g| , which ing but the same F(1) ˆ ˆ R R further implies that f −1 |Rˆ = g −1 |Rˆ . Hence, f −1 can be extended to a biholomorphic map from δ(Pi ∞ I Q) to + #− . 0
Therefore δ(Pi ∞ I Q) must be biholomorphic to (δ(P)i ∞0 Q)l+m−1+i ∞0 Q¯ . Since they 0
are both canonical representatives of sphere with tubes, we must have the equality (2.23).
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Remark 2.19. Proposition 2.18 is nothing but the doubling trick [A, C1] stated in our partial operad language. It also implies that the bulk theories in an open-closed conformal field theory must contain both chiral parts and anti-chiral parts. Corollary 2.20. For P ∈ δ(ϒ(n, l)) ⊂ K (n + 2l) as in (2.18) and Q ∈ K (m), the sewing operations: (Pi ∞0 Q)l+m−1+i ∞0 Q¯ for 1 ≤ i ≤ l define an action of the diagonal ¯ ∈ K × K } of K × K on δ(S). {(Q, Q) 2.4. The C-extensions of S. In order to study open-closed conformal field theories with nontrivial central charges, we need study the C-extensions of Swiss-cheese partial operad S. For c ∈ C, let K˜ c be the 2c th power of determinant line bundle over K [H4]. We denote ˜ c . δ can certainly be the pullback line bundle over S through the doubling map δ as S c ˜ extended to a map on S . We still denote it as δ. For any n ∈ N, the restrictions of the ˜ c (n) and we shall sections ψn+2l of K˜ c (n + 2l) for l ∈ N to ϒ(n; l) gives a section of S S c use ψn to denote this section. It is clear that ϒ˜ , the C-extension of the partial operad of disks with strips, is the pullback bundle of the inclusion map ϒ → S. ˜ c are naturally induced from the sewing operaThe boundary sewing operations in S ˜ c as i ∞
0B .5 More explicitly, tions of K˜ c . We denote the boundary sewing operations in S B ˜ Q˜ be elements in let P ∈ ϒ(n; l) and Q ∈ ϒ(m; k) be so that Pi ∞0 Q exists. Let P, the fiber over P and Q respectively. We define ˜ 2l+i ∞ ˜
0B Q˜ := δ −1 (δ( Q)
0 Q), P˜i ∞
(2.24)
where δ −1 is defined on the image of δ. We would also like to lift an interior sewing operation i ∞0I to a sewing operation ˜ c and an element in K˜ c ⊗ K˜ c¯ . We still call it interior sewing between an element in S
0I . Let P ∈ ϒ(n; l), Q ∈ K (m) such that Pi ∞0I Q exists. operation and denote it as i ∞ ˜ Q˜ be elements in the fibers over P and Q respectively. Let ψm ⊗ ψ¯ m be the Let P, canonical section on K˜ c ⊗ K˜ c¯ (m). Then we have Q˜ = λψm ⊗ ψ¯ m (Q) for some λ ∈ C.
0I Q˜ by Then we define P˜i ∞ ˜ i∞ ¯
0I Q˜ := δ −1 ((δ( P)
0 λψm (Q))l+m−1+i ∞
0 ψm ( Q)). P˜i ∞
(2.25)
The following lemma shows that the interior sewing operations are associative. ˜ Q˜ 1 , Q˜ 2 be elements Lemma 2.21. Let P ∈ ϒ(n; l), Q 1 ∈ K (m 1 ), Q 2 ∈ K (m 2 ) and P, c c c ¯ ˜ ˜ ˜ in fibers of line bundles S and K ⊗ K over the base points P, Q 1 , Q 2 respectively. Let 1 ≤ i ≤ l and 1 ≤ j ≤ m 1 . Then we have
0I Q˜ 1 )i+ j−1 ∞
0I Q˜ 2 = P˜i ∞
0I ( Q˜ 1 j ∞
0 Q˜ 2 ) ( P˜i ∞
(2.26)
assuming that the sewing operations appeared in (2.26) are all well-defined. 5 Since we always work with a fixed c ∈ C, it is convenient to make the dependence on c implicit in some notations.
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Proof. Let λi ∈ C, i = 1, 2 be such that Q˜ i = λi ψm i ⊗ ψ¯ m i (Q i ), i = 1, 2. Let (i) (a0 , A(i) ) be the local coordinate map at j th puncture of Q 1 and let B (0) be the local coordinate map at ∞ in Q 2 . By (2.25), the δ image of the left-hand side of (2.26) equals ˜ i∞
0 λ1 ψm 1 (Q 1 ))l+m 1 −1+i ∞
0 ψm 1 ( Q¯ 1 )) (((δ( P)
0 λ2 ψm 2 (Q 2 ))l+m 1 +m 2 +i+ j−3 ∞
0 ψm 2 ( Q¯ 2 ). i+ j−1 ∞ By the associativity of the partial operad K˜ c , the above formula equals ˜ i∞
0 (λ1 ψm 1 (Q 1 ) j ∞
0 λ2 ψm 2 (Q 2 ))) (δ( P) l+m 1 +m 2 −2+i
˜ i∞
0 (λ1 λ2 e(A = (δ( P)
0 (ψm 1 ( Q¯ 1 ) j ∞
0 ψm 2 ( Q¯ 2 )) ∞
(i) ,B (0) ,a (i) )c 0
l+m 1 +m 2 −2+i
0 e(A ∞
ψm 1 +m 2 −1 (Q 1 j ∞0 Q 2 )))
(i) ,B (0) ,a (i) )c 0
ψm 1 +m 2 −1 (Q 1 j ∞0 Q 2 )
(i) (0) (i) (i) (0) (i)
0I (λ1 λ2 e(A ,B ,a0 )c e(A ,B ,a0 )c = δ P˜i ∞ ψm 1 +m 2 −1 ⊗ ψ¯ m 1 +m 2 −1 (Q 1 j ∞0 Q 2 ))
0I (λ1 ψm 1 ⊗ ψ¯ m 1 (Q 1 ) j ∞
0 λ2 ψm 2 ⊗ ψ¯ m 2 (Q 2 )) = δ P˜i ∞
0I ( Q˜ 1 i ∞
0 Q˜ 2 )), = δ( P˜i ∞ which is nothing but the right-hand side of (2.26).
Boundary sewing operations and interior sewing operations induce the following partially defined substitution maps γ˜ : ϒ˜ c (n; l) × ϒ˜ c (m 1 ; k1 ) × · · · × ϒ˜ c (m n ; kn ) × K˜ c (n 1 ) × · · · × K˜ c (nl ) γ˜
− → ϒ˜ c (m 1 + · · · + m n ; k1 + · · · + kn + n 1 + · · · + nl ) The following proposition is clear. ˜ c | K˜ c ⊗ K˜ c¯ , γ˜ ) is a R+ |C× -rescalable 2-colored partial operad. Proposition 2.22. (S ˜ c | K˜ c ⊗ K˜ c¯ , γ˜ ) Swiss-cheese partial operad with central charge c. We will call (S ˜ c restricted on ϒ is just ϒ˜ c which was introduced in [HKo1]. Note that S ˜ c | K˜ c ⊗ K˜ c¯ -algebras. Let V O = ⊕n∈R V O , where V O has a structure of 2.5. Smooth S (n) (n) C , irreducible R+ -module given by r → r n idV O for r ∈ R+ . Let V C = ⊕(m,n)∈R×R V(m,n) (n)
C where V(m,n) has a structure of irreducible C× -module given by z → z m z¯ n idV C
C×
z∈ (recall (1.16) and (1.17)) for all m, n. This also implies that m−n ∈ / Z. Let (V O |V C , ν ˜ c
S | K˜ c ⊗ K˜ c¯
˜ c | K˜ c ⊗ K˜ c¯ -algebra. be a R+ |C× -rescalable S
,ν ˜c
K ⊗ K˜ c¯
)
C V(m,n)
(m,n)
for
= 0 for all
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243
˜ c | K˜ c ⊗ K˜ c¯ -algebra (V O |V C , ν Definition 2.23. The R+ |C× -rescalable S ˜ c | K˜ c ⊗ K˜ c¯ , S ν ˜ c ˜ c¯ ) is called smooth if it satisfies the following two conditions: K ⊗K
O < ∞ for s ∈ R, V O = 0 for n << 0 and dim V C 1. dim V(s) (n) (m,n) < ∞ for m, n ∈ Z, C V(m,n) for m << 0 or n << 0. 2. For w ∈ (V O ) , v1 , . . . , vm ∈ V O , u 1 , . . . , u n ∈ V C , and P˜ ∈ ϒ˜ c (m; n), the following map
P˜ → w, ν ˜ c
˜
( P)(v1 S | K˜ c ⊗ K˜ c¯
⊗ · · · ⊗ vm ⊗ u 1 ⊗ · · · ⊗ u n )
is linear on fiber and smooth on the base space ϒ(m; n). By [H4], a vertex operator algebra U canonically gives a K˜ c -algebra. Using the dou˜ c | K˜ c ⊗ K˜ c¯ -algebra, in which bling map δ, this K˜ c -algebra naturally gives a smooth S O C ˜ c | K˜ c ⊗ K˜ c¯ -algebra canonically V = U and V = U ⊗U . It is also easy to see that this S gives an analytic open-closed field algebra which is nothing but (U ⊗ U, U, idU , idU ) or ˜ c | K˜ c ⊗ K˜ c¯ -algebra simply (U ⊗U, U ) discussed in Sect. 1. We still denote this smooth S c c ˜ | K˜ ⊗ K˜ c¯ -algebra containing (U ⊗ U, U ) as a subalgebra as (U ⊗ U, U ). A smooth S c c ˜ ˜ is called a smooth S | K ⊗ K˜ c¯ -algebra over U . Let (Vcl , Vop , m cl−op ) be an open-closed field algebra over V . By definition, Vcl is a conformal full field algebra over V ⊗ V . By the results in [K2], Vcl has a structure of smooth K˜ c ⊗ K˜ c¯ -algebra structure, we denote it as (Vcl , ν ˜ c ˜ c¯ ). K ⊗K Let Q ∈ ϒ(n; l) of form (2.9) be such that r1 > . . . > rn−1 > rn = 0. We define a × ˜ c → E R+ |C as follows: map ν ˜ c ˜ c ˜ c¯ : S Vop |Vcl S |K ⊗K
ν˜c
S | K˜ c ⊗ K˜ c¯
(λψnS(Q))(v1 ⊗ · · · ⊗ vn ⊗ u 1 ⊗ · · · ⊗ u l )
:= λe−L − (B
(0) )
−L + (A m (l;n) cl−op (e
(l) )
(a0(l) )−L(0) ⊗ e−L + (A
. . . e−L + (A
(1) )
(a0(1) )−L(0) ⊗ e−L + (A (l) )
(1)
(1) )
(a0(1) )−L(0) u 1 ,
(a0(l) )−L(0) u l ;
(n)
e−L + (B ) (b0(1) )−L(0) v1 , . . . , e−L + (B ) (b0(n) )−L(0) vn ; z 1 , z¯ 1 , . . . , zl , z¯l ; r1 , . . . , rn ), (2.27) ∞ where L ± (A) = j=1 L(± j)A j for any A = {A1 , A2 , . . . }, A j ∈ C, for u 1 , . . . , u l ∈ Vcl , v1 , . . . , vn ∈ Vop . Let el be the identity element of Sl . ∀σ ∈ Sn , we define ν˜c
S | K˜ c ⊗ K˜ c¯
(σ, el )(λψnS(Q)) = ν ˜ c
S | K˜ c ⊗ K˜ c¯
λψnS(Q) .
(2.28)
We have finished the definition of ν ˜ c ˜ c ˜ c¯ in all cases. By results in [HKo1], the S |K ⊗K restriction of ν c c c¯ on ϒ˜ c clearly gives a morphism of partial nonassociative operad ˜ | K˜ ⊗ K˜ S
from ϒ˜ c to E VRop+ .
Theorem 2.24. (Vop |Vcl , ν ˜ c
,ν ) S | K˜ c ⊗ K˜ c¯ K˜ c ⊗ K˜ c¯
˜ c | K˜ c ⊗ K˜ c¯ -algebra. is a smooth S
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L. Kong
Proof. By the permutation property of open-closed field algebra and (2.28), it is clear that ν ˜ c ˜ c ˜ c¯ is equivariant with the actions of permutation groups. S |K ⊗K The conditions in Definition 2.23 are automatically satisfied. It remains to show that ν˜c
S | K˜ c ⊗ K˜ c¯
◦ γ˜ = ◦ (ν ˜ c
S | K˜ c ⊗ K˜ c¯
,...,ν ˜ c
S | K˜ c ⊗ K˜ c¯
,ν ˜c
K ⊗ K˜ c¯
,...,ν ˜c
K ⊗ K˜ c¯
)
(2.29)
as a map ϒ˜ c (n; l) × ϒ˜ c (m 1 ; k1 ) × · · · × ϒ˜ c (m n ; kn ) × K˜ c (n 1 ) × · · · × K˜ c (nl ) ⊗n+m 1 +···+m n → Hom(Vcl⊗k1 +···+kn +n 1 +···+nl ⊗ Vop , Vop ).
(2.30)
˜ c can be viewed as a partial suboperad of Thanks to the doubling map δ and (2.26), S c ˜ K with single-sewing operations for the punctures in R+ and double-sewing operations for mirror pairs of punctures in upper and lower half planes. By using the V -invariant boundary condition, the chirality splitting property and the convergence and extension properties of any products and iterates of intertwining operators proved by Huang for any V satisfying the condition in Theorem 0.1, it is easy to generalize the proof of Huang’s fundamental result Proposition 5.4.1 in [H4] and results in [H5, H6] to show that (2.29) holds. The arguments are standard but tedious. We omit the details. 3. Categorical Formulation In this section, we study open-closed field algebras over V from a tensor-categorical point of view. In Sect. 3.1, we recall some basic ingredients of the vertex tensor categories. In Sect. 3.2, we reformulate the notion of open-closed field algebra over V categorically by a categorical notion called open-closed CV |CV ⊗V -algebra. 3.1. Vertex tensor categories. The theory of tensor products for modules over a vertex operator algebra was developed by Huang and Lepowsky [HL2-HL5,H3]. By Theorem 0.1 and our assumption on V , the category of V -modules, denoted as CV , have a structure of vertex tensor category [HL2]. In particular, it has a structure of semisimple braided tensor category. We review some of the ingredients of vertex tensor category CV and set our notations along the way. There is a tensor product bifunctor P(z) : CV × CV → CV for each P(z), z ∈ C× in sphere partial operad K , where P(z) is the conformal equivalence class of sphere with three punctures 0, z, ∞ and standard local coordinates [H4]. We denote P(1) simply as . For any pair of V -modules W1 , W2 , the module W1 P(z) W2 is spanned by the homogeneous components of w1 P(z 1 ) w2 ∈ W1 ⊗ W2 , ∀w1 ∈ W1 , w2 ∈ W2 . For each V -module W , there is a left unit isomorphism l W : V W → W defined by l W (v w) = YW (v, 1)w,
∀v ∈ V, w ∈ W,
(3.1)
where l W is the unique extension of l W on V W and YW is the vertex operator which defines the module structure on W , and a right unit isomorphism r W : W V → W defined by r W (w v) = e L(−1) YW (v, −1)w,
∀v ∈ V, w ∈ W.
(3.2)
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245
Remark 3.1. We have used “overline” for the extensions of maps, algebraic completions of graded vector spaces and complex conjugations of complex variables. One shall not confuse them because they act on different things. Let W1 and W2 be V -modules. For a given path γ ∈ C× from a point z 1 to z 2 , there is a parallel isomorphism associated to this path Tγ : W1 P(z 1 ) W2 −→ W1 P(z 2 ) W2 . Let Y be the intertwining operator corresponding to the intertwining map P(z 2 ) and l(z 1 ) the value of the logarithm of z 1 determined by log z 2 and analytic continuation along the path γ . For w1 ∈ W1 , w2 ∈ W2 , Tγ is defined by Tγ (w1 P(z 1 ) w2 ) = Y(w1 , el(z 1 ) )w2 , where Tγ is the natural extension of Tγ . Moreover, the parallel isomorphism depends only on the homotopy class of γ . For z 1 > z 2 > z 1 − z 2 > 0 and each triple of V -modules W1 , W2 , W3 , there is an associativity isomorphism: P(z −z ),P(z 2 )
2 A P(z 11 ),P(z 2)
: W1 P(z 1 ) (W2 P(z 2 ) W3 ) → (W1 P(z 1 −z 2 ) W2 ) P(z 2 ) W3 ,
which is characterized by P(z −z ),P(z 2 )
2 A P(z 11 ),P(z 2)
(w(1) P(z 1 ) (w(2) P(z 2 ) w(3) )) = (w(1) P(z 1 −z 2 ) w(2) ) P(z 2 ) w(3) (3.3)
for w(i) ∈ Wi , i = 1, 2, 3. The associativity isomorphism A of the braided tensor category is characterized by the following commutative diagram: W1 P(z 1 ) (W2 P(z 2 ) W3 ) P(z −z ),P(z 2 ) 1 2)
2 A P(z 1 ),P(z
(W1 P(z 1 −z 2 ) W2 ) P(z 2 ) W3
Tγ1 ◦(idW1 P(z 1 ) Tγ2 )
Tγ2 ◦(Tγ3 P(z 2 ) idW3 )
/ W1 (W2 W3 )
(3.4)
A
/ (W1 W2 ) W3 ,
where γ1 , γ2 , γ3 are paths in R+ from z 1 , z 2 , z 1 − z 2 to 1, respectively. There is also a braiding isomorphism, for z > 0, R+P(z) : W1 P(z) W2 → W2 P(z) W1 for each pair of V -modules W1 , W2 , defined as P(z)
R+
(w1 P(z) w2 ) = e z L(−1) T γ+ (w2 P(−z) w1 ),
(3.5)
where γ+ is a path from −z to z given by γ+ (t) = −eiπ t z, t ∈ [0, 1]. The inverse of P(z) P(z) R+ is denoted by R− , which is characterized by P(z)
R− (w2 P(z) w1 ) = e z L(−1) T γ− (w1 P(−z) w2 ),
(3.6)
where γ− is a path given by γ− (t) = −e−iπ t z. We denote R± simply as R± . Accordingly, we define the twist by θW = e−2πi L(0) for any V -module W . P(1)
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For z 1 , z 2 ∈ R+ , the naturalness of T implies the commutativity of the following diagram: Tγ
W1 P(z 1 ) W2
/ W1 P(z 2 ) W2
P(z 1 )
R±
(3.7)
P(z 2 )
Tγ
W2 P(z 1 ) W1
R±
/ W2 P(z 2 ) W1 ,
where γ is a path in R+ from z 1 to z 2 . 3 W3 Let VW . For the chosen denote the space of intertwining operators of type WW 1 W2 1 W2
W3 is canonically isomorphic to the space of intertwining branch cut (1.16), the space VW 1 W2 3 maps denoted as M[P(z)]W W1 W2 . By the universal property of tensor product, the space
3 M[P(z)]W W1 W2 is canonically identified with the space HomV (W1 P(z) W2 , W3 ). Let
P(z)
W3 Y ∈ VW . We denote the corresponding morphism in HomV (W1 W2 , W3 ) as m Y . 1 W2 W3 W3 For r ∈ Z, r : VW → VW is an isomorphism defined as follows: 1 W2 2 W1
r (Y)(w2 , x)w1 = e x L(−1) Y(w1 , e(2r +1)πi x)w2 for all w1 ∈ W1 , w2 ∈ W2 . Then by Proposition 3.1 in [K2], where the definition of braiding is opposite to our choice here (recall (3.5), (3.6)), we obtain the following identities: P(z)
mY
P(z)
P(z)
= m 0 (Y ) ◦ R+
P(z)
P(z)
= m −1 (Y ) ◦ R− .
(3.8)
The tensor product V ⊗ V is also a vertex operator algebra [FHL]. Moreover, it was shown in [HKo2] that V ⊗ V also satisfies the conditions in Theorem 0.1. Therefore, CV ⊗V , the category of V ⊗ V -modules, also has a structure of semisimple braided tensor category. For z, ζ ∈ C× , let P(z)P(ζ ) be the tensor product bifunctor in CV ⊗V defined by (A ⊗ B) P(z)P(ζ ) (C ⊗ D) = (A P(z) B) ⊗ (C P(ζ ) D), where A, B, C, D are V -modules. We denote the bifunctor P(1)P(1) simply as . There are a few different braiding structures on CV ⊗V [K2]. We choose the one given by R+ ⊗ R− . The twist θ A : A → A, for A ∈ CV ⊗V , is given as follows: θ A = e−2πi L
L (0)
⊗ e2πi L
R (0)
.
(3.9)
An object A is said to have a trivial twist if θ A = id A . Now we recall the definition of (commutative) associative algebra in a braided tensor category C with tensor product ⊗, unit object 1C , left unit isomorphism l W and right unit isomorphism r W for any object W , the associativity A and the braiding R. Definition 3.2. An associative algebra in C (or associative C-algebra) is an object A in C together with µ A : A ⊗ A → A and a monomorphism ι A : 1C → A satisfying 1. Associativity. µ A ◦ (µ A ⊗ id A ) ◦ A = µ A ◦ (id A ⊗ µ A ). −1 2. Unit properties. µ A ◦ (ι A ⊗ id A ) ◦ l −1 A = µ A ◦ (ι A ⊗ id A ) ◦ r A = id A .
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A is called commutative if µ A = µ A ◦ R. The following theorem is proved in [HKo1]. Theorem 3.3. The category of open-string vertex operator algebras over V is isomorphic to the category of associative CV -algebras. The following theorem is proved in [K2]. Theorem 3.4. The category of conformal full field algebras over V ⊗ V is isomorphic to the category of commutative associative algebras in CV ⊗V with a trivial twist. We are interested in studying the relation between the above two algebras as ingredients of an open-closed field algebra over V . Notice that these two algebras live in different categories. So we will first discuss a functor between these two categories. Recall a functor TP(z) : CV ⊗V → CV [HL4]. In particular, for W1 , W2 being V -module, TP(z) (W1 ⊗ W2 ) = W1 P(z) W2 . We will simply write TP(1) as T . Let ϕ0 := l V−1 = r V−1 : V → V V = T (V ⊗ V ).
(3.10)
For each four V -modules WiL , WiR , i = 1, 2, notice that T (W1L ⊗ W1R ) T (W2L ⊗ W2R ) = (W1L W1R ) (W2L W2R ), T ((W1L
⊗
W2R ) (W2L
⊗
W2R ))
=
(W1L
W2L ) (W1R
W2R ).
(3.11) (3.12)
We define ϕ2 : (W1L W1R ) (W2L W2R ) → (W1L W2L ) (W1R W2R ) by ϕ2 := A ◦ (idW1 A−1 ) ◦ (idW1 R− idW4 ) ◦ (idW1 A) ◦ A−1 .
(3.13)
The above definition of ϕ2 can be naturally extended to a morphism T (A) T (B) → T (A B) for each pair of objects A and B in CV ⊗V . We still denote the extended morphism as ϕ2 . The following result is clear. Lemma 3.5. T together with ϕ0 , ϕ2 given in (3.10) (3.13) is a monoidal functor. For any four objects Wi , i = 1, 2, 3, 4 in CV , we define a morphism σ in Hom((W1 W2 ) (W3 W4 ), (W3 W4 ) (W1 W2 )) according to the following graph:
(3.14) Clearly, for any A ∈ CV ⊗V , σ can be extended to an automorphism on T (A) T (A), denoted as σ A . Proposition 3.6. Let (A, µ A , ι A ) be a commutative algebra in CV ⊗V . Let µT (A) := T (µ A ) ◦ ϕ2 and ιT (A) := T (ι A ) ◦ ϕ0 . Then (T (A), µT (A) , ιT (A) ) is an associative CV -algebra satisfying the following commutativity: µT (A) = µT (A) ◦ σ A .
(3.15)
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Proof. That (T (A), µT (A) , ιT (A) ) is an associative CV -algebra follows from Lemma 3.5. We only prove (3.15). Let us assume A = W1 ⊗ W2 with W1 , W2 being V -modules (the proof for general A is the same). Consider the following diagram: (W1 W2 ) (W1 W2 )
ϕ2
T (µcl )
R+ R−
σ
(W1 W2 ) (W1 W2 )
/ (W1 W1 ) (W2 W2 )
ϕ2
/ (W1 W1 ) (W2 W2 )
T (µcl )
/ W1 W2
(3.16)
id / W1 W2 .
The left subdiagram is commutative because two paths corresponding to the same braiding; the right subdiagram is commutative because of the commutativity of A. Hence the above diagram is commutative. Equation (3.15) follows from the commutativity of (3.16). 3.2. Open-closed CV |CV ⊗V -algebras. Let ((Vcl , m cl , ιcl ), (Vop , Yop , ιop ), Ycl−op ) be an open-closed field algebra over V throughout this subsection. By Theorem 3.4, (Vcl , m cl , ιcl ), a conformal full field algebra over V ⊗ V is equivalent to a commutative associative algebra with a trivial twist in CV ⊗V with braiding R+− . We denoted this CV ⊗V -algebra as a triple (Vcl , µcl , ιcl ), where µcl = m Y f (recall (1.31)). By Proposition 3.6, (T (Vcl ), νT (Vcl ) , ιT (Vcl ) ) is an associative CV -algebra. The property of algebra T (Vcl ) can be expressed in the following graphic equations:
(3.17)
(3.18) By Theorem 3.3, (Vop , Yop , ιop ), an open-string vertex operator algebra over V is equivalent to an associative CV -algebra, denoted as a triple (Vop , µop , ιop ), where µop = m Y f (recall (1.14) and (1.15)). The defining properties of (Vop , µop , ιop ) can be op expressed in the following graphic equations:
(3.19)
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249
It remains to study the categorical formulation of the only remaining data Ycl−op and its properties. By the chirality splitting properties and the associativity of intertwining operator algebra [H7], there exist intertwining operators Y (5) , Y (6) such that ¯ z, ζ )v = v , Y (5) (Y (6) (u L , z − ζ )u R , ζ )v v , Ycl−op (u ⊗ u,
(3.20)
, v ∈ V , u L ⊗ u R ∈ V and ζ > z − ζ > 0. For u L ⊗ u R ∈ V , we have for v ∈ Vop op cl cl L R u P(z−ζ ) u ∈ TP(z−ζ ) (Vcl ). P(z−ζ )P(ζ ) We define a map µcl−op : TP(z−ζ ) (Vcl ) P(ζ ) Vop → Vop as follows: P(z−ζ )P(ζ )
µcl−op
P(ζ )
P(z−ζ )
:= m Y (5) ◦ (m Y (6)
P(ζ ) idVop ).
(3.21)
By the convergence and analytic properties of the right-hand side of (3.20) and the fact that module maps are continuous with respect to the topology on graded vector spaces defined in Sect. 1, we obtain, for z > ζ > z − ζ > 0, u L ⊗ u R ∈ Vcl , v ∈ Vop , the following identity: P(z−ζ )P(ζ )
µcl−op
((u L P(z−ζ ) u R ) P(ζ ) v) = Y (5) (Y (6) (u L , z − ζ )u R , ζ )v, (3.22)
P(z−ζ )P(ζ )
P(z−ζ )P(ζ )
is the unique extension of µcl−op to the algebraic completion where µcl−op of TP(z−ζ ) (Vcl ) P(ζ ) Vop . Moreover, by Proposition 1.2 in [H9] (or the nondegeneracy of intertwining operator algebra [K1]), such V -module map is unique. Let γ1 be a path in R+ from 1 to z − ζ and γ2 a path in R+ from 1 to ζ . Then we define a map µcl−op : T (Vcl ) Vop → Vop as follows: P(z−ζ )P(ζ )
µcl−op := µcl−op
◦ Tγ2 ◦ (Tγ1 idVop ).
(3.23)
Since Tγ depends on path only homotopically, it is clear that the above definition of µcl−op is independent of z, ζ in R+ and paths γ1 , γ2 in R+ . In particular, one can choose γ1 and γ2 to be the straight line between 1 and z − ζ, ζ respectively. By Theorem 1.28, to obtain a categorical formulation of open-closed field algebra over V is enough to study the categorical formulations of the unit property (1.82), Associativity I (1.52), Associativity II (1.56) and Commutativity I in Proposition 1.18. We first consider the property (1.82). Proposition 3.7. The condition (1.82) is equivalent to the following condition: µcl−op ◦ ((T (ιVcl ) ◦ ϕ0 ) idVop ) ◦ l V−1 = idVop , op
(3.24)
which can also be expressed by the following graphic equation:
(3.25)
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Proof. First, (1.82) is equivalent to the following condition: Ycl−op (1cl ; z, ζ ) = idVop , for z > ζ > 0.
(3.26)
Recall that 1cl = ιVcl (1 ⊗ 1). Replacing u L ⊗ u R by 1cl in (3.20), one can see that both sides of Eq. (3.20) are independent of z and ζ . Hence (3.26) holds for all z, ζ ∈ R+ . Using (3.20) and (3.22), we obtain P(z−ζ )P(ζ )
µcl−op
(TP(z−ζ ) (1cl ) P(ζ ) v) = v
(3.27)
for v ∈ Vop and z, ζ ∈ R+ . Therefore, we have, for v ∈ Vop , P(z−ζ )P(ζ )
µcl−op =
◦ TP(z−ζ ) (ιVcl ) P(ζ ) idVop ((1 P(z−ζ ) 1) P(ζ ) v)
P(z−ζ )P(ζ ) µcl−op (TP(z−ζ ) (1cl ) P(ζ )
v)
= v,
(3.28)
which can be further expressed equivalently as ◦ (TP(z−ζ ) (ιVcl ) ◦ Tγ1 ) P(ζ ) idVop ◦ Tγ2 ◦ (l V−1 idVop ) ◦ l V−1 op −1 P(z−ζ )P(ζ ) = µcl−op ◦ (Tγ1 P(ζ ) idVop ) ◦ Tγ2 ◦ (T (ιVcl ) ◦ ϕ0 ) idVop ◦ l Vop P(z−ζ )P(ζ ) = µcl−op ◦ Tγ2 ◦ (Tγ1 idVop ) ◦ (T (ιVcl ) ◦ ϕ0 ) idVop ◦ l V−1 op −1 = µcl−op ◦ (T (ιVcl ) ◦ ϕ0 ) idVop ◦ l Vop , (3.29) P(z−ζ )P(ζ )
idVop = µcl−op
where γ1 and γ2 are paths in R+ from 1 to z − ζ and ζ respectively. Conversely, from (3.27), (3.28) and (3.29), it is clear that (3.24) or (3.25) also implies (1.82). Now we consider the associativity II (recall Proposition 1.17). Proposition 3.8. The associativity II is equivalent to the following condition: µcl−op ◦ (idT (Vcl ) µcl−op ) = µcl−op ◦ (T (µcl ) ◦ ϕ2 ) idVop ◦ A,
(3.30)
which can also be expressed by the following graphic equation
(3.31) Proof. Using the convergence property of products and iterates of intertwining operators of V , it is not hard to show that, in the domain D3 := {(z 1 , ζ1 , z 2 , ζ2 )|z 1 > ζ1 > z 2 > ζ2 > 0, 2ζ2 > z 2 , 2ζ1 > z 1 + z 2 }, (3.32)
Open-Closed Field Algebras
251
we have Ycl−op (u 1L ⊗ u 1R ; z 1 , ζ1 )Ycl−op (u 1L ⊗ u 1R ; z 1 , ζ1 )v = Y (5) (Y (6) (u 1L , z 1 − ζ1 )u 1R , ζ1 )Y (5) (Y (6) (u 2L , z 2 − ζ2 )u 2R , ζ2 )v
(3.33)
for u 1L ⊗ u 1R , u 2L ⊗ u 2R ∈ Vcl and v ∈ Vop . By (3.21)(3.22) and the fact that module maps are continuous, we obtain Ycl−op (u 1L ⊗ u 1R ; z 1 , ζ1 )Ycl−op (u 1L ⊗ u 1R ; z 1 , ζ1 )v P(z −ζ )P(ζ )
P(z −ζ )P(ζ )
1 1 1 2 2 2 = µcl−op ◦ (idTP(z1 −ζ1 ) (Vcl ) P(ζ1 ) µcl−op ) L × (u 1 P(z 1 −ζ1 ) u 1R ) P(ζ1 ) ((u 2L P(z 2 −ζ2 ) u 2R ) P(ζ2 ) v)
(3.34)
for (z 1 , ζ1 , z 2 , ζ2 ) in the domain D3 . Moreover, by the universal property of tensor product and the nondegeneracy of intertwining operator algebra [K1], P(z −ζ1 )P(ζ1 )
1 µcl−op
P(z −ζ2 )P(ζ2 )
2 ◦ (idTP(z1 −ζ1 ) (Vcl ) P(ζ1 ) µcl−op
)
(3.35)
is the unique morphism in Hom(TP(z 1 −ζ1 ) (Vcl ) P(ζ1 ) (TP(z 2 −ζ2 ) (Vcl ) P(ζ2 ) Vop ), Vop ) such that the identity (3.34) holds in D3 . On the other hand for (1.56), it is proved in [HKo2] that Y can be expanded as follows: Yan (u 1L ⊗ u 1R ; z, ζ )u 2L ⊗ u 2R =
N
YiL (u 1L , z)u 2L ⊗ YiR (u 1R , ζ )u 2R
(3.36)
i=1
for some N ∈ Z+ . There is a unique morphism P(z 1 −z 2 )P(ζ1 −ζ2 )
µcl
∈ HomCV ⊗V (Vcl P(z 1 −z 2 )P(ζ1 −ζ2 ) Vcl , Vcl )
such that, for u, v ∈ Vcl , P(z 1 −z 2 )P(ζ1 −ζ2 )
µcl
(u P(z 1 −z 2 )P(ζ1 −ζ2 ) v) = Yan (u; z 1 − z 2 , ζ1 − ζ2 )v.
(3.37)
By the convergence property of intertwining operator algebra, it not hard to see that, in the domain D4 := {(z 1 , ζ1 , z 2 , ζ2 )|z 1 > z 2 > ζ1 > ζ2 > 0, 2ζ2 > z 1 , 2z 2 > z 1 + ζ1 },
(3.38)
we have Ycl−op (Yan (u 1L ⊗ u 1R ; z 1 − z 2 , ζ1 − ζ2 )u 2L ⊗ u 2R ; z 2 , ζ2 )v =
N
Ycl−op (YiL (u 1L , z 1 − z 2 )u 2L ⊗ YiR (u 1R , ζ1 − ζ2 )u 2R ; z 2 , ζ2 )v
i=1
=
N i=1
Y (5) (Y (6) (YiL (u 1L , z 1 − z 2 )u 2L , z 2 − ζ2 )YiR (u 1R , ζ1 − ζ2 )u 2R , ζ2 )v.
(3.39)
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Combining (3.39) with (3.37), (3.20) and (3.22) and the fact that module maps are continuous, we obtain the following identity: Ycl−op (Yan (u 1L ⊗ u 1R ; z 1 − z 2 , ζ1 − ζ2 )u 2L ⊗ u 2R ; z 2 , ζ2 )v P(z −ζ )P(ζ )
P(z −z )P(ζ −ζ )
2 2 2 1 2 = µcl−op ◦ (TP(z 2 −ζ2 ) (µcl 1 2 ) P(ζ2 ) idVop ) L L R × ((u 1 P(z 1 −z 2 ) u 2 ) P(z 2 −ζ2 ) (u 1 P(ζ1 −ζ2 ) u 2R )) P(ζ2 ) v
(3.40)
for (z 1 , ζ1 , z 2 , ζ2 ) in the domain D4 . Moreover, by the universal property of tensor product and the nondegeneracy of intertwining operator algebra [K1], P(z −ζ2 )P(ζ2 )
2 µcl−op
P(z 1 −z 2 )P(ζ1 −ζ2 )
◦ (TP(z 2 −ζ2 ) (µcl
) P(ζ2 ) idVop )
(3.41)
is the unique morphism in Hom(TP(z 2 −ζ2 ) (Vcl P(z 1 −z 2 )P(ζ1 −ζ2 ) Vcl ) P(ζ2 ) Vop , Vop ) such that the identity (3.40) holds in D4 . Notice that the domains D3 and D4 are disjoint. Now we fix a point (z 1 , ζ1 , z 2 , ζ2 ) in the domain {(z 1 , ζ1 , z 2 , ζ2 )|z 1 > ζ1 > z 2 > ζ2 > 0, 2ζ2 > z 1 , 2ζ1 > z 1 + z 2 , 2z 2 > ζ1 + ζ2 }, (3.42) which is a subdomain of D3 . Let z˜ 2 = ζ1 , ζ˜1 = z 2 . Then the quadruple (z 1 , ζ˜1 , z˜ 2 , ζ2 ) is in the domain D4 . By the analytic properties guaranteed by Theorem 1.28, both sides of associativity (1.56) can be uniquely extended to the boundary (z 1 , ζ1 , z 2 , ζ2 ) ∈ R4+ ∩ MC4 × . Moreover, Ycl−op (u 1L ⊗ u 1R ; z 1 , ζ1 )Ycl−op (u 2L ⊗ u 2R ; z 2 , ζ2 )v, and Ycl−op (Yan (u 1L ⊗ u 1R ; z 1 − z˜ 2 , ζ˜1 − ζ2 )u 2L ⊗ u 2R ; z˜ 2 , ζ2 )v, can be obtained from each other by analytic continuation along the following path
(3.43) Meanwhile, if we start from the element (u 1L P(z 1 −ζ1 ) u 1R ) P(ζ1 ) ((u 2L P(z 2 −ζ2 ) u 2R ) P(ζ2 ) v)
Open-Closed Field Algebras
253
in TP(z 1 −ζ1 ) (Vcl ) P(ζ1 ) (TP(z 2 −ζ2 ) (Vcl ) P(ζ2 ) Vop ) and apply associativity isomorphisms repeatedly and braiding isomorphism on it, we obtain P(z −ζ )P(ζ1 ) −1 ) 1 1)
1 (A P(z 1 )P(ζ
−−−−−−−−−−−→ u 1L P(z 1 ) (u 1R P(ζ1 ) ((u 2L P(z 2 −ζ2 ) u 2R ) P(ζ2 ) v)), P(ζ −ζ )P(ζ )
2 2 A P(ζ1 )P(ζ 1 2) −−−−−−−−→ u 1L P(z 1 ) (u 1R P(ζ1 −ζ2 ) (u 2L P(z 2 −ζ2 ) u 2R )) P(ζ2 ) v , P(ζ −z )P(z −ζ )
1 2 2 2 −−−−−−−−−−→ u 1L P(z 1 ) ((u 1R P(ζ1 −z 2 ) u 2L ) P(z 2 −ζ2 ) u 2R ) P(ζ2 ) v , A P(ζ1 −ζ2 )P(z 2 −ζ2 )
P(ζ −z ) R− 1 2 −−−−−−→ u 1L P(z 1 ) (e(ζ1 −z 2 )L(−1) ·
Tγ− (u 2L P(−ζ1 +z 2 ) u 1R ) P(z 2 −ζ2 ) u 2R ) P(ζ2 ) v ,
(3.44)
where we have ignored the obvious identity maps and γ− is a path from −ζ1 + z 2 to ζ1 − z 2 given by γ− (t) = (−ζ1 + z 2 )e−πit , t ∈ [0, 1]. Now we analytically continue the last line of (3.44) along the path (3.43), and we obtain u 1L P(z 1 ) ((u 2L P(ζ1 −z 2 ) u 1R ) P(z 2 −ζ2 ) u 2R ) P(ζ2 ) v , (3.45) which is nothing but the third line of (3.44) except u 1R and u 2L are exchanged. Note that the last line of (3.44) and (3.45) are elements in the algebraic completion of the same V -module. We denote this V -module as W . Now we further apply associativity morphisms on (3.45) and use z 2 = ζ˜1 and ζ1 = z˜ 2 . We then obtain P(˜z −ζ˜ )P(ζ˜1 −ζ2 ) −1 ) 2 2
−−−−−−−−−−−−−→ u 1L P(z 1 ) (u 2L P(˜z 2 −ζ2 ) (u 1R P(ζ˜1 −ζ2 ) u 2R )) P(ζ2 ) v , (A P(˜z 2 −ζ2 )
P(z −ζ )P(ζ2 ) 1 2)
−−−−−−−−→ u 1L P(z 1 −ζ2 ) (u 2L P(˜z 2 −ζ2 ) (u 1R P(ζ˜1 −ζ2 ) u 2R )) P(ζ2 ) v, 2 A P(z 1 )P(ζ
P(z −˜z )P(˜z −ζ )
A P(z 1 −ζ2 )P(˜z 2 −ζ2 ) 1 2 2 2 −−−−−−−−−−→ (u 1L P(z 1 −˜z 2 ) u 2L ) P(˜z 2 −ζ2 ) (u 1R P(ζ˜1 −ζ2 ) u 2R ) P(ζ2 ) v. (3.46)
Let m be the morphism W → Vop such that P(ζ1 −z 2 )
(3.35) = m ◦ R−
P(ζ −ζ )P(ζ2 )
2 A P(ζ11 )P(ζ 2)
P(ζ −z )P(z −ζ )
◦ A P(ζ11 −ζ22 )P(z 22 −ζ22 ) ◦ P(z −ζ )P(ζ1 ) −1
1 ◦ (A P(z 11 )P(ζ 1)
)
.
(3.47)
If we apply m on both the last line of (3.44) and (3.45), the two images of m are clearly the analytic continuation of each other along the path (3.43). On the other hand, combining this fact with (3.40) and (3.46), we obtain that P(z −˜z )P(˜z −ζ )
P(z −ζ )P(ζ2 )
2 m = (3.41) ◦ A P(z 11 −ζ22 )P(˜z 22 −ζ22 ) ◦ A P(z 11 )P(ζ 2)
P(˜z −ζ˜ )P(ζ˜1 −ζ2 ) −1
◦ (A P(˜z 22 −ζ22 )
)
(3.48)
because the extensions of both sides of (3.48), applied on (3.45), give the same element in Vop . Therefore, we further obtain from (3.47) and (3.48) the following identity: P(z −˜z )P(˜z −ζ )
P(z −ζ )P(ζ2 )
2 (3.35) = (3.41) ◦ A P(z 11 −ζ22 )P(˜z 22 −ζ22 ) ◦ A P(z 11 )P(ζ 2)
P(ζ −z ) ◦R− 1 2
P(ζ −z )P(z −ζ ) ◦ A P(ζ11 −ζ22 )P(z 22 −ζ22 )
P(˜z −ζ˜ )P(ζ˜1 −ζ2 ) −1
◦ (A P(˜z 22 −ζ22 )
P(ζ −ζ2 )P(ζ2 ) ◦ A P(ζ11 )P(ζ 2)
)
P(z −ζ1 )P(ζ1 ) −1 ◦ (A P(z 11 )P(ζ ) . 1)
(3.49)
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L. Kong
Using the commutative diagram (3.4), (3.7) and the definition of ϕ2 (recall (3.13)), it is easy to see that (3.49) implies the commutativity of the following diagram: TP(z 1 −ζ1 ) (Vcl ) P(ζ1 ) (TP(z 2 −ζ2 ) (Vcl ) P(ζ2 ) Vop ) LLL LLL f1 LLL LLL (3.35) T (Vcl ) (T (Vcl ) Vop ) LLLL LLL LLL LLL A LL& *V (T (Vcl ) T (Vcl )) Vop 4 op r8 r r rr ϕ2 idVop rr rr r r rr T (Vcl Vcl ) Vop rr r rr (3.41)|z2 =˜z2 ,ζ1 =ζ˜1 rr g1 rr r r rr TP(˜z 2 −ζ2 ) (Vcl P(z 1 −˜z 2 )P(ζ˜1 −ζ2 ) Vcl ) P(ζ2 ) Vop
(3.50)
where f 1 = idT (Vcl ) (Tγ4 id Hop ) ◦ Tγ2 Tγ3 ◦ Tγ1 , g1 = TP(z 2 −ζ2 ) (Tγ7 ⊗ Tγ8 ) P(ζ2 ) id Hop ◦ (Tγ6 P(ζ2 ) idVop ) ◦ Tγ5
(3.51) (3.52)
in which γi , i = 1, . . . , 4 are paths in R+ from ζ1 , z 1 − ζ1 , ζ2 , z 2 − ζ2 to 1 respectively and γi , i = 5, . . . , 8 are paths in R+ from 1 to ζ2 , z 2 − ζ2 , z 1 − z 2 , ζ1 − ζ2 respectively. Using (3.23), it is easy to see that µcl−op ◦ (idT (Vcl ) µcl−op ) = (3.35) ◦ f 1−1 , µcl−op ◦ (T (µcl ) idVop ) = (3.41) ◦ g1 .
(3.53)
Equation (3.53) together with the commutative diagram (3.50) implies (3.30), which is nothing but the commutativity of the subdiagram in the middle of (3.50). Conversely, (3.30) implies the commutativity of the diagram (3.50). It is easy to see that the above arguments can be reversed. Therefore, we can also obtain the associativity II (1.56) from (3.30). We now study the categorical formulations of the rest of the conditions needed in Theorem 1.28. The proof is essentially the same as that of Proposition 3.8. So we will only sketch the proofs below. Proposition 3.9. The associativity I (recall Proposition 1.16) is equivalent to the following condition: µcl−op (idT (Vcl ) µop ) = µop (µcl−op idVop ) ◦ A
(3.54)
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255
which can also be expressed in the following graph:
(3.55) Proof. The left-hand side of (1.52) gives rise to a morphism P(z−ζ )P(ζ )
µcl−op
P(r ) ◦ (idTP(z−ζ ) (Vcl ) P(ζ ) µop )
(3.56)
in Hom(TP(z−ζ ) (Vcl ) P(ζ ) (Vop P(r ) Vop ), Vop ). The right-hand side of (1.52) gives rise to a morphism P(z−ζ )P(ζ −r )
P(r ) µop ◦ (µcl−op
P(ζ −r ) idVop )
(3.57)
in Hom((TP(z−ζ ) (Vcl ) P(ζ ) Vop ) P(r ) Vop , Vop ). For z, ζ, r in a proper subdomain of z > ζ > r > 0, using similar arguments as in Proposition 3.8, we obtain that the associativity (1.52), for some z, ζ, r, r˜ ∈ R+ , implies the following commutative diagram: TP(z−ζ ) (Vcl ) P(ζ ) (Vop P(r ) Vop ) TTTT TTT(3.56 TTTT) f2 TTTT TTT* /4 Vop T (Vcl ) (Vop Vop ) u: u u uu A uu u u uu (T (Vcl ) Vop ) Vop uuu u uu (3.57) uu g2 u uu uu (TP(z−ζ ) (Vcl ) P(ζ −r ) Vop ) P(r ) Vop ) ,
(3.58)
where f 2 = (Tγ2 Tγ3 ) ◦ Tγ1 , g2 = Tγ6 P(ζ ) idVop ) P(r ) idVop ◦ (Tγ5 P(r ) idVop ) ◦ Tγ4 ,
(3.59) (3.60)
where γi , i = 1, 2, 3 are paths in R+ from ζ, z − ζ and r to 1 respectively and γi , i = 4, 5, 6 are paths in R+ from 1 to r, ζ − r and z − ζ respectively. The commutativity of the outside loop in (3.58) implies immediately the commutativity of the subdiagram in the middle of (3.58), which is nothing but the identity (3.54) or (3.55). Conversely, using (3.58) and reversing the above arguments, it is clear that (3.54) or (3.55) also implies the associativity (1.52). σ1
N W L ⊗ W R . We define a map T (V ) V − Let Vcl = ⊕i=1 cl op → Vop T (Vcl ) by i i N (R+ idW R ) ◦ A ◦ (idW L R− ) ◦ A−1 . σ1 := ⊕i=1 i
i
(3.61)
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Proposition 3.10. The commutativity I (recall Proposition 1.18) is equivalent to the following identity: µcl−op (idT (Vcl ) µop ) = µop (idVop µcl−op ) ◦ A−1 ◦ σ1 ◦ A,
(3.62)
or the following graphic identities:
(3.63) Proof. There is a morphism P(z−ζ )P(ζ )
µcl−op
P(r ) ◦ (idTP(z−ζ ) (Vcl ) P(ζ ) µop )
(3.64)
in Hom(TP(z−ζ ) (Vcl ) P(ζ ) (Vop P(r ) Vop )) associated to (1.58). There is another morphism P(z−ζ )P(ζ )
P(˜r ) µop ◦ (idVop P(˜r ) µcl−op
)
(3.65)
in Hom(Vop (T (Vcl ) Vop )) associated to (1.59). For z, ζ, r, r1 in a proper subdomain of r1 > z > ζ > r > 0, using similar arguments as in Proposition 3.8, we obtain that the commutativity I, implies the following commutative diagram: TP(z−ζ ) (Vcl ) P(ζ ) (Vop P(r ) Vop ) HH HH HH f4 HH HH H(3.64) T (Vcl ) (Vop Vop ) HHH HH HH HH A HH H$ ) Vop (T (Vcl ) Vop ) Vop σ1
(Vop T (Vcl )) Vop
idVop
5 Vop v: v vv A−1 vv v v vv Vop (T (Vcl ) Vop ) vvvv v (3.65) vv vv g4 v v vv Vop P(r1 ) (TP(z−ζ ) (Vcl ) P(ζ ) Vop ),
(3.66)
Open-Closed Field Algebras
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where f 4 = (Tγ2 Tγ3 ) ◦ Tγ1 , g4 = (idVop (Tγ5 Tγ6 ) ◦ Tγ4 ,
(3.67) (3.68)
in which γi , i = 1, 2, 3 are paths in R+ from ζ , z − ζ and r respectively to 1 and γi , i = 4, 5, 6 are path in R+ from 1 to r1 , z − ζ and ζ respectively. The above commutative diagram immediately implies that the subdiagram in the middle of (3.66) is commutative. This is nothing but the commutativity (3.62) or the first formula in (3.63). Moreover, it also easy to see that the two formulas in (3.63) are actually equivalent. Conversely, using the commutative diagram (3.66) and reversing the above argu˜ c. ments, it is clear that (3.63) implies the commutativity of rational S Commutativity II (recall Proposition 1.19) is not needed in Theorem 1.28 because it automatically follows from Associativity II and skew symmetry of Vcl . It also has a very nice categorical formulation as given in the following proposition, which follows from (3.18) and (3.31) immediately. Proposition 3.11. For an open-closed field algebra over V , we have µcl−op ◦ idT (Vcl ) µcl−op = µcl−op ◦ idT (Vcl ) µcl−op ◦ A−1 ◦ σ ◦ A, or equivalently
(3.69) In summary, we have already completely reformulated all the data and conditions in Theorem 1.28 in the language of tensor category as (3.23), (3.24), (3.30), (3.54) and (3.62) or equivalently as graphic identities (3.25), (3.31), (3.55) and (3.63). We define a morphism ιcl−op : T (Vcl ) → Vop as the composition of the following maps: r T−1 id ιVop µcl−op (Vcl ) T (Vcl ) −−−→ T (Vcl ) 1CV −−−−−→ T (Vcl ) Vop −−−→ Vop ,
(3.70)
or equivalently as the following graphic formula:
(3.71) Lemma 3.12. ιcl−op is an algebra morphism from T (Vcl ) to Vop .
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Proof. That ιcl−op maps identity to identity is proved as follows:
(3.72) The homomorphism property µop ◦ (ιcl−op ιcl−op ) = ιcl−op ◦ m˜ cl is proven as follows:
(3.73) Definition 3.13. An open-closed CV |CV ⊗V -algebra, denoted as ( (Aop , µop , ιop )|(Acl , µcl , ιcl ), ιcl−op ) or simply (Aop |Acl , ιcl−op ), consists of an associative algebra (Aop , µop , ιop ) in CV , a commutative associative algebra with a trivial twist (Acl , µcl , ιcl ) in CV ⊗V and an associative algebra homomorphism ιcl−op : T (Acl ) → Aop , satisfying the following commutativity: µop ◦ (ιcl−op idVop ) = µop ◦ (idVop ιcl−op ) ◦ σ1 ,
(3.74)
or equivalently,
(3.75) Theorem 3.14. The notion of open-closed field algebra over V is equivalent to that of open-closed CV |CV ⊗V -algebra, in the sense that the categories of the above two notions are isomorphic. Proof. Given an open-closed field algebra over V , we have shown that it gives a triple (Vcl , Vop , µcl−op ), in which Vcl is a commutative associative algebra in CV ⊗V with a trivial twist, and Vop is an algebra in CV , and µcl−op satisfies (3.25), (3.31), (3.55) and (3.63). Moreover, we have shown that ιcl−op defined by (3.71) gives a morphism of the associative algebra. Now we prove (3.75) as follows:
Hence (Vop |Vcl , ιcl−op ) is an open-closed CV |CV ⊗V -algebra.
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It is easy to check that it gives a functor from the category of open-closed field algebras over V to that of open-closed CV |CV ⊗V -algebras. Conversely, given an open-closed CV |CV ⊗V -algebra, (Vop |Vcl , ιcl−op ), we define a morphism µcl−op ∈ Hom(T (Vcl ) Vop , Vop ) as
(3.76) Since ιcl−op is an algebra homomorphism, it maps unit to unit (recall (3.72)). Thus the identity property of µcl−op holds. Then the identity property of Ycl−op follows. Furthermore, we have
which gives the associativity II (3.31). The associativity I (3.55) follows from
By using (3.76), we can prove the commutativity (3.63) as follows:
Notice that the other half of (3.63) is equivalent to the first half. Also notice that the commutativity (3.69) simply follows from (3.15) and the fact that ιcl−op is an algebra homomorphism. It is easy to check that this correspondence gives a functor from the category of open-closed CV |CV ⊗V -algebras to that of open-closed field algebras over V . That the two relevant categories are isomorphic follows from (3.71) and (3.76) easily. It is very easy to construct open-closed CV |CV ⊗V -algebras. For example, let A be an associative algebra in CV ⊗V and Cl (A) the left center of A [O]. Let A0 be any subalgebra of Cl (A) and ι : A0 → A the natural embedding. Then it is clear that (T (A)|A0 , T (ι)) gives an open-closed CV |CV ⊗V -algebra, which further gives an open-closed field alge˜ c -algebra over V . We will not pursue the construction further braover V and a smooth S
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in this work. Instead, we leave it to [K3], in which we will construct explicit examples that are most relevent in physics. Moreover, a general theory of open-closed CV |CV ⊗V algebras and its variants ([K3]) will be developed and its connection to the works [FFFS, FS3, FRS1-FRS4, FjFRS] will be explained in future publications. Acknowledgement. This work grows from a chapter in the author’s thesis. I want to thank my advisor Yi-Zhi Huang for introducing this interesting field to me, for his many valuable suggestions for improvement and for his constant support. I also want to thank J. Lepowsky, J. Fuchs and C. Schweigert for many inspiring conversations, and I. Runkel for kindly sending me his thesis and one of his computer programs. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Huang, Y.-Z.: Geometric interpretation of vertex operator algebras. Proc. Natl. Acad. Sci. USA 88, 9964–9968 (1991) [H3] Huang, Y.-Z.: A theory of tensor products for module categories for a vertex operator algebra, IV. J. Pure Appl. Alg. 100, 173–216 (1995) [H4] Huang, Y.-Z.: Two-dimensional conformal geometry and vertex operator algebras. Progress in Mathematics, Vol. 148, Boston: Birkhäuser 1997 [H5] Huang, Y.-Z.: Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories. In: Operads: Proceedings of Renaissance Conferences, ed. J.-L. Loday, J. Stasheff, A. A. Voronov, Comtemporary Math., Vol. 202, Providence, RI: Amer. Math. Soc., 1997, pp. 335–355 [H6] Huang, Y.-Z.: Genus-zero modular functors and intertwining operator algebras. Internat, J. Math. 9, 845–863 (1998) [H7] Huang, Y.-Z.: Riemann surfaces with boundaries and the theory of vertex operator algebras. Fields Institute Commun. 39, 109 (2003) [H8] Huang, Y.-Z.: Differential equations and intertwining operators. Comm. Contemp. Math. 7, 375–400 (2005) [H9] Huang, Y.-Z.: Vertex operator algebras and the Verlinde conjecture. Comm. Contemp. Math., to appear; availble at http://arxiv.org/list/math.QA/0406291, 2004 [HKo1] Huang, Y.-Z., Kong, L.: Open-string vertex algebra, category and operad. Commun. Math. Phys. 250, 433–471 (2004) [HKo2] Huang, Y.-Z., Kong, L.: Full field algebra. Commun. Math. Phys. 272, 345–396 (2007) [HKo3] Huang, Y.-Z., Kong, L.: Modular invariance for full field algebras. http://arxiv.org/list/math.QA/ 0609570, 2006 [HKr] Hu, P., Kriz, I.: Closed and open conformal field theories and their anomalies. Commun. Math. Phys. 254(1), 221–253 (2005) [HL1] Huang, Y.-Z., Lepowsky, J.: Vertex operator algebras and operads. In: The Gelfand Mathematical Seminars, 1990–1992, ed. L. Corwin, I. Gelfand, J. Lepowsky, Boston: Birkhäuser 1993, pp. 145–161 [HL2] Huang, Y.-Z., Lepowsky, J.: Tensor products of modules for a vertex operator algebra and vertex tensor categories. In: Lie Theory and Geometry, in honor of Bertram Kostant, ed. R. Brylinski, J.-L. Brylinski, V. Guillemin, V. Kac, Boston: Birkhäuser, 1994, pp. 349–383 [HL3] Huang, Y.-Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra. I. Selecta Math (N.S.) 1, 699–756 (1995) [HL4] Huang, Y.-Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator, algebra II. Selecta Math. (N.S.) 1, 757–786 (1995) [HL5] Huang, Y.-Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra, III. J. Pure Appl. Alg. 100, 141–171 (1995) [K1] Kong, L.: Nondegenerate intertwining operator algebras. Unpublished [K2] Kong, L.: Full field algebras, operads and tensor categories. Adv. Math 213, 271–340 (2007) [K3] Kong, L.: Cardy condition for open-closed field algebras. http://arxiv.org/list/math/QA/061255, 2006 [Kt] Kontsevich, M.: Operads and motives in deformation quantization. Lett. Math. Phy. 48, 35–72 (1999) [L1] Li, H.S.: Axiomatic G 1 -vertex algebras. Commun. Contemp. Math. 5, 281–327 (2003) [L2] Li, H.S.: Nonlocal vertex algebras generated by formal vertex operators. Selecta Math. (N.S.) 11 (3–4), 349–397 (2005) [L3] Li, H.S.: Constructing quantum vertex algebras. Internat. J. Math. 17(4), 441–476 (2006) [MSS] Markl, M., Shnider, S., Stasheff, J.D.: Operads in Algebra, Topology and Physics. Volume 96 of Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc. 2002 [O] Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8, 177– 206 (2003) [S1] Segal, G.: The definition of conformal field theory. In: Differential geometrical methods in theoretical physics (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 250, Dordrecht: Kluwer Acad. Publ. 1988, pp. 165–171 [S2] Segal, G.: The definition of conformal field theory, preprint, 1988; also in: Topology, geometry and quantum field theory, ed. U. Tillmann, London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge: Cambridge University Press, 2004, pp. 421–577 [V] Voronov, A.A.: The Swiss-cheese operad. In: Homotopy invariant algebraic structures, in honor of J. Michael Boardman, Proc. of the AMS Special Session on Homotopy Theory, Baltimore, 1998, ed. J.-P. Meyer, J. Morava, W. S. Wilson, Contemporary Math., Vol. 239, Providence, RI: Amer. Math. Soc. 1999, pp. 365–373 Communicated by Y. Kawahigashi
Commun. Math. Phys. 280, 263–280 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0442-4
Communications in
Mathematical Physics
Gaussian Quantum Marginal Problem Jens Eisert1,2,3 , Tomáš Tyc4 , Terry Rudolph2 , Barry C. Sanders5 1 Institute for Mathematical Sciences, Imperial College London, Princes Gate, London SW7 2PE, UK. 2 3 4 5
E-mail: [email protected] Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, UK Physics Department, University of Potsdam, Am Neuen Palais, 14469 Potsdam, Germany Institute of Theoretical Physics, Masaryk University, 61137 Brno, Czech Republic Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada
Received: 30 April 2007 / Accepted: 18 July 2007 Published online: 23 February 2008 – © Springer-Verlag 2008
Abstract: The quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system. This setting, also referred to as the question of the compatibility of local spectra, has several applications in quantum information theory. Here, we introduce the analogue of this statement for Gaussian states for any number of modes, and solve it in generality, for pure and mixed states, both concerning necessary and sufficient conditions. Formally, our result can be viewed as an analogue of the Sing-Thompson Theorem (respectively Horn’s Lemma), characterizing the relationship between main diagonal elements and singular values of a complex matrix: We find necessary and sufficient conditions for vectors (d1 , . . . , dn ) and (c1 , . . . , cn ) to be the symplectic eigenvalues and symplectic main diagonal elements of a strictly positive real matrix, respectively. More physically speaking, this result determines what local temperatures or entropies are consistent with a pure or mixed Gaussian state of several modes. We find that this result implies a solution to the problem of sharing of entanglement in pure Gaussian states and allows for estimating the global entropy of non-Gaussian states based on local measurements. Implications to the actual preparation of multi-mode continuous-variable entangled states are discussed. We compare the findings with the marginal problem for qubits, the solution of which for pure states has a strikingly similar and in fact simple form. 1. Introduction What reduced states are compatible with some quantum state of a composite system having a certain spectrum? The study of this question has in fact a long tradition – as the natural quantum analogue of the marginal problem in classical probability theory. Very recently, this problem, now coined the quantum marginal problem, has seen a revival of interest, motivated by applications in the context of quantum information theory [1–6]. In fact, in the quantum information setting, notably in quantum channel capacity expressions, in assessments of quantum communication protocols, or in the
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separability problem, one often encounters questions of compatibility of reductions with global quantum states [7–11]. Since it is only natural to look at the full orbit under local unitary operations, the quantum marginal problem immediately translates to a question of the compatibility of spectra of quantum states. The mixed quantum marginal problem then amounts to the following question: Is there a state ρ of a quantum system with n subsystems, each with a reduction ρk , that is consistent with spec(ρ) = r, spec(ρk ) = rk
(1) (2)
for k = 1, . . . , n, r and rk denoting the respective vectors of spectra?. In the pure marginal problem, one assumes ρ = |ψψ| to be pure. In the condensed-matter context [12,13], related questions are also of interest: For example, once one had classified all possible two-qubit reductions of translationally invariant quantum states, then one would be able to obtain the ground state energy of any nearest-neighbor Hamiltonian of a spin chain. The quantum marginal problem was solved in several steps: Higuchi et al. [1] solved the pure quantum marginal problem for qubits. Subsequently, Bravyi was able to solve the mixed state case for two qubits, followed by Franz [6] and Higuchi [2] for a three qutrit system. The general solution of the quantum marginal problem for finite-dimensional systems was found in the celebrated work of Klyachko [5], see also Refs. [14,15]. This is indeed a closed-form solution. Yet the number of constraints grows extremely rapidly with the system size, rendering the explicit check whether the conditions are satisfied unfeasible even for relatively small systems. In this work, we introduce the Gaussian version of the quantum marginal problem. Gaussian states play a key role in a number of contexts, specifically whenever bosonic modes and quadratic Hamiltonians become relevant, which are ubiquitous in quantum optical systems, free fields, and condensed matter lattice systems. For general infinite dimensional systems the marginals problem may well be intractable. However, given that in turn these Gaussian states can be described by merely their first and second moments [16,17], one could reasonably hope that it could be possible to give a full account of the Gaussian quantum marginal problem. This gives rise, naturally, not to a condition to spectra of quantum states, but to symplectic spectra, as explained below. For the specific case of three modes, the result is known [18], see also Ref. [19]. In this work we will show that this program of characterizing the reductions of Gaussian states can be achieved in generality, even concerning both necessary and sufficient conditions. This means that one can give a complete answer to what reductions entangled Gaussian states can possible have.1 Equivalently, we can describe this Gaussian marginal problem as a problem of compatibility of temperatures of standard harmonic systems: Given a state ρ, what local temperatures – or equivalently for single modes, what local entropies – are compatible with this joint state? Of course, one can always take the temperatures to be equal. But if they are different, they constrain each other in a fairly subtle way, as we will see. In a sense, the result gives rise to the interesting situation that by looking at local temperatures, one can assess whether these reductions may possibly originate from a joint system in a pure state. Finally, it is important to note, since sufficiency of the conditions 1 We refer here to the marginal problem for Gaussian states, which are quantum states fully defined by their first and second moments of canonical coordinates. However, clearly, our result equally applies to general and hence non-Gaussian states, in that it fully answers the question what local second moments are consistent with global second moments of quantum states of several modes.
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is always proven by an explicit construction, the result also implies a recipe for preparing multi-mode continuous-variable entangled states. 2. Main Result We consider states on n modes, and consider reductions to single modes. Gaussian states are represented by the matrix of second moments, the 2n × 2n covariance matrix γ of the system, together with the vector µ of first moments. For a definition and a survey of properties, see Refs. [16,17]. In this language, the vacuum state of a standard oscillator becomes γ = 12 , as the 2 × 2 identity matrix. The canonical commutation relations are embodied in the symplectic matrix n 0 1 (3) σ = −1 0 k=1
for n modes. The covariance matrices of n modes are exactly those real matrices satisfying γ + iσ ≥ 0, (4) which is simply a statement of the Heisenberg uncertainty principle. The first moments can always be made zero locally, and are hence not interesting for our purposes here. Note also that the set of Gaussian states is closed under reductions, so reduced states of Gaussian states are always Gaussian as well. Real matrices that leave the symplectic form invariant, Sσ S T = σ , form the real symplectic group Sp(2n, R). In the same way as symmetric matrices M can be diagonalized with orthogonal matrices to a diagonal matrix O M O T = D, one can diagonalize strictly positive matrices using such S ∈ Sp(2n, R), according to S M S T = D.
(5)
The simply counted main diagonal elements of D form then the symplectic spectrum of M, and the collection of symplectic eigenvalues can be abbreviated as sspec(M) = (d1 , . . . , dn ), D = diag(d1 , d1 , . . . , dn , dn ). (6) This procedure is nothing but the familiar normal mode decomposition. In turn, by definition, the symplectic eigenvalues are given by the square roots of the eigenvalues of the matrix −Mσ Mσ . Again, for the vacuum, the symplectic eigenvalues are all given by unity. In a mild abuse of notation, we will refer to the symplectic spectrum of a Gaussian state as the symplectic spectrum of the respective covariance matrix. Finally, for a given covariance matrix γ , and in fact any strictly positive real matrix, we refer to the symplectic main diagonal elements (c1 , . . . , cn ) as the symplectic eigenvalues of the 2 × 2 main diagonal blocks. This is the natural analogue of main diagonal elements. Equivalently, the symplectic main diagonal elements are the main diagonal elements after the main diagonal 2 × 2 blocks have been brought into the form c 0 . (7) γk = k 0 ck We are now in the position to state our main result. It relates the symplectic spectrum of composite systems to the ones of the reductions. We will first state it as a mere matrix constraint, then as the actual Gaussian marginal problem, and finally for the important special case of having a pure joint state.
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Theorem 1 (Necessary and sufficient conditions). Let (d1 , . . . , dn ) and (c1 , . . . , cn ) be two vectors of positive numbers in non-decreasing order. Then there exists a strictly positive real 2n × 2n-matrix γ such that (d1 , . . . , dn ) are its symplectic eigenvalues and (c1 , . . . , cn ) the symplectic main diagonal elements if and only if the n + 1 conditions k j=1
cn −
n−1 j=1
cj ≥
k
d j , k = 1, . . . , n
(8)
j=1
c j ≤ dn −
n−1
dj
(9)
j=1
are satisfied. This set of inequalities may be conceived as a general analogue of the Sing-Thompson theorem [20–22], see below. More physically speaking, this means the following: Corollary 1 (Gaussian marginal problem). Assume that ρ is a Gaussian state of n modes satisfying sspec(ρ) = (d1 , . . . , dn ). Then the possible reduced states ρk to each of the individual modes k = 1, . . . , n are exactly those Gaussian states with sspec(ρk ) = ck
(10)
satisfying Eq. (8) and (9). These conditions hence fully characterize the possible reduced marginal states. For two modes, n = 2, for example, the given conditions read c1 + c2 ≥ d1 + d2 , c2 − c1 ≤ d2 − d1 ,
(11) (12)
for c2 ≥ c1 and d2 ≥ d1 . The constraint c1 ≥ d1 is then automatically satisfied. For pure Gaussian states, the above conditions take a specifically simple form. Quite strikingly, we will see that the resulting conditions very much resemble the situation of the marginal problem for qubits. Corollary 2 (Pure Gaussian marginal problem). Let ρ = |ψψ| be a pure Gaussian state of n modes. Then the set (b1 + 1, . . . , bn + 1) of symplectic eigenvalues sspec(ρk ) = bk + 1,
(13)
k = 1, . . . , n, of the reduced states ρk of each of the n modes is given by the set defined by bj ≤ bk (14) k= j
for all j, for b j ≥ 0.
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To reiterate, these conditions are necessary and sufficient for the local symplectic spectra being consistent with the global state being a pure Gaussian state. Equivalently, this can be put as follows: If γ is the covariance matrix of a pure Gaussian state with reductions b +1 0 γk = k , (15) 0 bk + 1 k = 1, . . . , n. Then, Eq. (14) defines the local temperatures Tk per mode consistent with the whole system being in a pure Gaussian state, according to bk = 2(exp(1/Tk ) − 1)−1 ,
(16)
for the standard harmonic oscillator (an oscillator with unit mass and frequency). The above condition hence determines the temperatures that modes can have, given that a composite system is in a pure Gaussian state. The form of Eq. (15) can always be achieved by means of local rotations and squeezings in phase space. One can hence equally think in terms of local symplectic spectra or local temperatures. It is instructive to compare the results for the pure Gaussian marginal problem with the one for qubits as solved in Ref. [1]. There, it has been found that for a system consisting of n qubits, one has λj ≤ λk (17) k= j
for the spectral values rk = (λk , 1 − λk ), λk ∈ [0, 1]. Moreover, these conditions are both necessary and sufficient. It is remarkable that this form is identical with the result for n single modes bj ≤ bk , (18) k= j
bk ≥ 0, as necessary and sufficient conditions. Again, the admissible symplectic eigenvalues are defined by a cone the base of which is formed by a simplex. Note that the methods used in Ref. [1] to arrive at the above result are entirely different. Once again, a striking formal similarity between the case of qubit systems and Gaussian states is encountered. Finally, from the perspective of matrix analysis, the above result can be seen as a general analogue of the Sing-Thompson Theorem [20–22] (or Horn’s Lemma [23] in case of Hermitian matrices), first posed in Ref. [24], where the role of singular values is taken by the symplectic eigenvalues. Sing-Thompson Theorem ([20–22]). Let (x1 , . . . , xn ) be complex numbers such that |xk | are non-increasingly ordered and let (y1 , . . . , yn ) be non-increasingly ordered positive numbers. Then an n × n matrix exists with x1 , . . . , xn as its main diagonal and y1 , . . . , yn as its singular values if and only if k
|x j | ≤
j=1 n−1 j=1
|x j | − |xn | ≤
k
y j , k = 1, . . . , n,
(19)
y j − yn .
(20)
j=1 n−1 j=1
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It is interesting to see that – although the symplectic group Sp(2n, R) is not a compact group – there is so much formal similarity concerning the implications on main diagonal elements of matrices. Note, however, that the ordering of singular values and symplectic eigenvalues, respectively, is different in Theorem 1 and in the Sing-Thompson theorem. 3. Proof As a preparation of the proof, we will identify a simple set of necessary conditions that constrains the possible reductions that are consistent with the assumption that the state is pure and Gaussian. These simple conditions derive from a connection between the symplectic trace and the trace of the covariance matrix. Quite surprisingly, we will see that they already define the full set of possible marginals consistent with a Gaussian state of n modes. We shall start by stating the condition to the reductions. Lemma 1 (Symplectic trace). Let γ be a strictly positive real 2n × 2n-matrix such that its main diagonal 2 × 2 blocks are given by Eq. (7) for ck ∈ [1, ∞). Then the symplectic eigenvalues (d1 , . . . , dn ) of the matrix γ satisfy n
dk ≤
k=1
n
ck .
(21)
k=1
Proof. Note that the right-hand side of Eq. (21) is nothing but half the trace of the covariance matrix γ , whereas the left-hand side is the symplectic trace str(γ ) of γ , so str(γ ) =
n
dj
(22)
j=1
if sspec(γ ) = (d1 , . . . , dn ), see, e.g., Ref. [25]. We arrive at this relationship by making use of a property of the trace-norm. The symplectic eigenvalues d1 , . . . , dn of γ are given by the square roots of the simply counted eigenvalues of the matrix (iσ )γ (iσ )γ [16,17]. Hence, the symplectic spectrum is just given by the spectrum of the matrix M = |γ 1/2 (iσ )γ 1/2 |,
(23)
where | · | denotes the matrix absolute value2 . So we have that 2
n
dk = tr(M) = γ 1/2 (iσ )γ 1/2 1 ,
(24)
k=1
where · 1 is the trace norm. The property we wish to prove then immediately follows from the fact that the trace-norm is a unitarily invariant norm: this implies that 2
n
dk = γ 1/2 (iσ )γ 1/2 1 ≤ (iσ )γ 1 ,
(25)
k=1
as AB1 ≤ B A1 for any matrices A, B for which AB is Hermitian. This inequality holds for any unitarily invariant norm whenever AB is a normal operator [26]. Now, 2 |A| = (A2 )1/2 for Hermitian matrices A.
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since any covariance matrix is positive, γ ≥ 0, and the largest singular value of iσ is clearly given by unity, we can finally conclude that 2
n
dk ≤ γ 1 = tr(γ ) = 2
k=1
n
ck ,
(26)
k=1
which is the statement that we intended to show.
This observation implies as an immediate consequence a necessary condition for the possible reductions, given a Gaussian state of an n-mode system: Let γ be the covariance matrix of a Gaussian pure state of n modes, with reductions as above. We can think of the state as a bi-partite state between a distinguished mode labeled k, without loss of generality being the last mode k = n, and the rest of the system. We can in fact Schmidt decompose this pure state with respect to this split using Gaussian unitary operations [29,28,34]. This means that we can find symplectic transformations S A ∈ Sp(2(n − 1), R) and S B ∈ Sp(2, R) such that A C , (S A ⊕ S B )γ (S A ⊕ S B ) = CT B
T
(27)
where A = diag(1, . . . , 1, an , an ), B = diag(an , an ),
(28) (29)
with some 2(n − 1) × 2-matrix C. The symplectic eigenvalues of modes 1, . . . , n − 1 are hence given by 1, . . . , 1, an . The above statement therefore implies the inequality n − 2 + an ≤ a1 + · · · + an−1 ,
(30)
or, by substituting bk = ak − 1 for all k = 1, . . . , n, bn ≤ b1 + b2 + · · · + bn−1 .
(31)
This must obviously hold for all distinguished modes and not only the last one, and hence, we arrive at the following simple necessary conditions: Corollary 3 (Necessary conditions for pure states). Let γ be the covariance matrix of a pure Gaussian state with thermal reductions bk + 1 0 γk = , 0 bk + 1
k = 1, . . . , n. Then, for all j, bj ≤
k= j
bk .
(32)
(33)
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That is, the largest value of b j cannot exceed the sum of all the other ones. So far, we have assumed the global state ρ to be a pure state. In the full problem, however, we may of course allow ρ to be any Gaussian state, and hence a mixed one, with symplectic spectrum sspec(ρ) = (d1 , . . . , dn ) ≥ (1, . . . , 1),
(34)
instead of being (1, . . . , 1). This is the Gaussian analogue of the mixed marginal problem. For this mixed state case, we provide necessary conditions for the main reductions, in form of n inequalities on partial sums, and one where the largest symplectic eigenvalue of a reduction plays an important role. The first set of n conditions is up to the different ordering a weak majorization relation for symplectic eigenvalues, which is in fact essentially a corollary of a result from Ref. [27] due to Hiroshima. The second statement, the n + 1th condition, as well as showing sufficiency of the general conditions, will turn out to be significantly more involved. Lemma 2 (Necessity of the first n conditions). Let (d1 , . . . , dn ), and (c1 , . . . , cn ) be defined as in Theorem 1. For any given (d1 , . . . , dn ), the admissible (c1 , . . . , cn ) satisfy k j=1
cj ≥
k
dj
(35)
j=1
for all k = 1, . . . , n. Proof. Let S ∈ Sp(2n, R) be the matrix from the symplectic group that brings γ into diagonal form, so Sγ S T = diag(d1 , d1 , . . . , dn , dn ). (36) The main diagonal elements of γ , in turn, again without loss of generality in nondecreasing order, are given by (c1 , . . . , cn ). Now according to Ref. [27], we have that min tr(T γ T T ) = 2
k
dj
(37)
j=1
for k = 1, . . . , n, where the minimum is taken over all real 2k × 2n-matrices T for which T σn T T = σk . (38) Here, σk denotes the symplectic matrix on k modes as defined in Eq. (3). Now we can actually take S ∈ Sp(2n, R) according to S = 1, we see that T , consisting of the first 2k rows of S, satisfies Eq. (38). Since this submatrix does not necessarily correspond to a minimum in Eq. (37), we find 2
k j=1
for any k = 1, . . . , n.
c j = tr(T γ T T ) ≥ 2
k
dj,
(39)
j=1
We will now prove the necessity of the n + 1th inequality constraint in Theorem 1.
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Lemma 3 (Necessity of the last condition). Let (d1 , . . . , dn ), and (c1 , . . . , cn ) be defined as in Theorem 1. For any given vector of symplectic eigenvalues (d1 , . . . , dn ), the admissible (c1 , . . . , cn ) satisfy cn −
n−1
c j ≤ dn −
j=1
n−1
dj.
(40)
j=1
Proof. We will define the function f : Sn → R, where Sn is the set of strictly positive real 2n × 2n-matrices, as follows: We define the vector c = (c1 , . . . , cn ) as c j = (γ2 j−1,2 j−1 γ2 j,2 j − γ22j−1,2 j )1/2 ,
(41)
j = 1, . . . , n, as the usual vector of symplectic spectra of each of the n modes, and then set n f (γ ) := 2 max(c) − cj. (42) j=1
For a diagonal matrix D = diag(d1 , d1 , . . . , dn , dn ) with entries in non-decreasing order, we have n−1 f (D) = dn − dj. (43) j=1
We will now investigate the orbit of this function f under the symplectic group, f˜ = sup x ∈ R : x = f (S DS T ), S ∈ Sp(2n, R) , (44) and will see that the supremum is actually attained as a maximum for S = 1. Each of the matrices γ = S DS T have by construction the same symplectic spectrum as D. This is a variation over 2n 2 + n real parameters, as any S ∈ Sp(2n, R) can be decomposed according to the Euler decomposition as S = O QV,
(45)
where O, V ∈ K (n) := Sp(2n, R) ∩ O(2n) and Q ∈ {(z 1 , 1/z 1 , . . . , z n , 1/z n ) : z k ∈ R\{0}} .
(46)
That is, O, V reflect passive operations, whereas Q stands for a squeezing operation. We will now see that the maximum of this function f – which exists, albeit the group being non-compact – is actually attained when the matrix is already diagonal. This means that in general, we have that f˜ = 2 max sspec(γ ) − str(γ ).
(47)
For any global maximum, any local variation will not increase this function further. Let us start from some γ = S DS T . For any such covariance matrix γ we can find a T ∈ Sp(2(n − 1), R) such that E F (T ⊕ 12 )γ (T ⊕ 12 )T = =: γ , (48) FT G
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where
, cn−1 ) E = diag(c1 , c1 , . . . , cn−1
(49)
is a (2n − 2) × (2n − 2) matrix and G is a 2 × 2 matrix. Using Lemma 1 again, we find that n−1 n−1 cj ≤ cj, (50) j=1
so
j=1
f (γ ) ≥ f (γ ).
(51)
In other words, it does not restrict generality to assume the final covariance matrix to be of the form as in the right hand side of Eq. (48), and we will use the notation E F (52) γ = S DS T = FT G with E = diag(c1 , c1 , . . . , cn−1 , cn−1 ) and G = diag(cn , cn ). We can now investigate submatrices of γ associated with modes m and n, 1 ≤ m < n, cm 12 Cn,m . (53) Mm,n = T Cn,m cn 12 This we can always bring to a diagonal form, using symplectic diagonalization, only affecting the main diagonal elements of modes n and m, and leaving the other main diagonal elements invariant. This brings this submatrix into the form cm 12 0 , (54) Mm,n = 0 cn 12 . From Lemma 5 we know that with cn ≥ cm ≤ cn − cm , cn − cm
(55)
so we have increased the function f , whenever Cn,m = 0. Hence, for global and hence local optimality, we have to have Cn,m = 0. However, each of the matrices Cn,m = 0 for all m = 1, . . . , n − 1 exactly if the matrix γ is already diagonal. What remains to be shown is that the function f is bounded from above, to exclude the case that the maximum does not even exist. One way to show this is to make use of the upper bound in Lemma 4 to have for every covariance matrix γ with symplectic spectrum (d1 , . . . , dn ) in non-decreasing order f (γ ) ≤
n
d j + (3 − 2n)d1 ,
(56)
j=2
which shows that f is always bounded from above. If γ is merely a strictly positive real matrix, but no covariance matrix, an upper bound follows from a rescaling with a positive number.
We now prove the upper bound required for the proof of Lemma 3.
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Lemma 4 (Upper bound). Let (d1 , . . . , dn ), and (c1 , . . . , cn ) be defined as in Theorem 1, and γ be additionally a 2n × 2n covariance matrix. For any given (d1 , . . . , dn ), the admissible (c1 , . . . , cn ) satisfy cn −
n−1
cj ≤
j=1
n
d j + (3 − 2n)d1 .
(57)
j=2
Proof. We start from a 4n × 4n-covariance matrix A C , γ = CT A
(58)
corresponding to a pure Gaussian state, where n dk 0 A= , 0 dk C=
k=1 n k=1
(59)
0 (dk2 − 1)1/2 0 −(dk2 − 1)1/2
(60)
are real 2n × 2n-matrices. Physically, this means that we start from a collection of n two mode squeezed states, with the property that the reduction to the first n modes is just a diagonal covariance matrix with symplectic eigenvalues (d1 , . . . , dn ), again in non-decreasing order. Let us first assume that d1 = 1; this assumption will be relaxed later. Let us now consider T S1 AS1T S1 C S1 0 S1 0 γ , (61) = 0 1 0 1 C T S1T A for S1 ∈ Sp(2n, R). Obviously, the set we seek to characterize is the set B of main diagonals of the upper left block U = S1 AS1T (62) of this matrix. We can always start from a diagonal matrix having the symplectic eigenvalues on the main diagonal, and consider the orbit under all symplectic transformations S ∈ Sp(4n, R). We will now relax the problem by allowing all S ∈ Sp(4n, R) instead of symplectic transformations of the form S = S1 ⊕ 1, S1 ∈ Sp(2n, R). We hence consider the full orbit under all symplectic transformations. This set C ⊃ B is characterized by the reductions to single modes of A C γ = Sγ S T = (63) CT A for some S ∈ Sp(4n, R), such that again A = diag(d1 , d1 , . . . , dn , dn ) This includes the case (61). We are now in the position to make use of the statement that we have established before: From exploiting the Schmidt decomposition on the level of second moments, and using Lemma 1 relating the trace to the symplectic trace, we find cn −
n−1 j=1
cj ≤
n j=2
d j + 3 − 2n,
(64)
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as d1 = 1 was assumed. Let us now consider the case of d1 > 1. We will apply the previous result, after appropriately rescaling the covariance matrix. Indeed, we can construct a covariance matrix γ˜ as in Eq. (58) for (d˜1 , . . . , d˜n ) = (1, d2 /d1 , . . . , dn /d1 ).
(65)
We then investigate the orbit of γ˜ under the symplectic group, and look at the main diagonal elements of S γ˜ S T . By construction, we have that γ˜ + iσ ≥ 0. We can hence apply Eq. (64) to this case. Multiplying both sides of Eq. (63) by d1 gives rise to the condition in Eq. (57).
Lemma 5 (Solution to two-mode problem). There exists a strictly positive real 4 × 4-matrix γ with main diagonal blocks diag(c1 , c1 ), diag(c2 , c2 ) and symplectic eigenvalues (d1 , d2 ) if and only if c1 + c2 ≥ d1 + d2 , c2 − c1 ≤ d2 − d1 ,
(66) (67)
assuming c2 ≥ c1 and d2 ≥ d1 . Moreover, c1 − c2 = d1 − d2 if and only if the 2 × 2 off diagonal block of γ vanishes. Proof. The necessary conditions that |c1 −c2 | ≤ |d1 −d2 | are a consequence of Lemma 4. The necessary conditions c1 + c2 ≥ d1 + d2 and c1 ≥ d1 have been previously shown in Lemma 2. Hence, we have to show that these conditions can in fact be achieved. This can be done by considering a ⎡
c1 ⎢0 γ =⎣ e 0
0 c1 0 f
e 0 c2 0
⎤ 0 f ⎥ = Sdiag(d1 , d1 , d2 , d2 )S T . 0⎦ c2
(68)
The relationship between c1 , c2 , e, f and d1 , d2 is given by
2 d1/2 = c12 + c22 + 2e f
± (c14 + c24 + 4e f c22 − 2c12 (c22 − 2e f ) + 4c1 c2 (e2 + f 2 ))1/2 /2,
(69)
as d1 , d2 are the square roots of the eigenvalues of −σ γ σ γ [16], compare also Ref. [32]. An elementary analysis shows that the above inequalities can always be achieved. Also, the extremal values are achieved if and only if e = f = 0.
What we finally need to show is that the conditions that we have derived are in fact sufficient. This will be the most involved statement. Lemma 6 (Sufficiency of the conditions). For any vectors (c1 , . . . , cn ) and (d1 , . . . , dn ) satisfying the conditions (8) and (9) there exists a 2n × 2n strictly positive real matrix with diag(c1 , . . . , cn ) as its symplectic main diagonal elements and (d1 , . . . , dn ) as its symplectic eigenvalues.
Gaussian Quantum Marginal Problem
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Proof. The argument will essentially be an argument by induction, in several ways resembling the argument put forth in Refs. [20–22]. The underlying idea of the proof is essentially as follows: On using the given constraints, one constructs an appropriate 2(n − 1) × 2(n − 1)-matrix, in a way that it can be combined to the desired 2n × 2nmatrix by means of an appropriate S ∈ Sp(4, R) acting on a 4 × 4 submatrix only. Note, however, that compared to Ref. [22], we look at variations over the non-compact symplectic group Sp(2n, R), and not the compact U (2n). For a single mode, n = 1, there is nothing to be shown. For two modes, Lemma 5 provides the sufficiency of the conditions. Let us hence assume that we are given vectors (c1 , . . . , cn ) and (d1 , . . . , dn ) as above, and that we have already shown that for 2(n − 1) × 2(n − 1)-matrices, the conditions (8) and (9) are indeed sufficient. We complete the proof by explicitly constructing an 2n × 2n-matrix with the stated property. We have that c1 ≥ d1 by assumption. We could also have c1 ≥ d j for some 2 ≤ j ≤ n, so let k ∈ {1, . . . , n} be the largest index such that c1 ≥ dk .
(70)
Let us first consider the case that k ≤ n − 2, and we will consider the cases k = n − 1 and k = n later. Then we can set x := dk + dk+1 − c1 , which means that x ≥ 0, and that all conditions c1 + x ≥ dk + dk+1 , c1 − x ≥ dk − dk+1 , −c1 + x ≥ dk − dk+1
(71) (72) (73)
are satisfied: (71) by definition, (72) because c1 ≥ dk and (73) as dk+1 ≥ c1 . This means that we can find a matrix of the form c 1 C , (74) γ := 1 T2 C x12 for some 2×2-matrix C, with symplectic eigenvalues (dk , dk+1 ), using Lemma 5. Therefore, the matrix γ = γ ⊕ diag(d1 , d1 , d2 , d2 , . . . , dk−1 , dk−1 , dk+2 , dk+2 , . . . , dn , dn )
(75)
has the symplectic spectrum (d1 , . . . , dn ). We will now show that we can construct a 2(n−1)×2(n−1) matrix γ with symplectic eigenvalues (d1 , . . . , dk−1 , x, dk+2 , . . . , dn ) and main diagonal elements (c2 , c2 , . . . , cn , cn ), by invoking the induction assumption. This matrix γ we can indeed construct, as we have c2 + · · · + cl ≥ d1 + · · · + dl−1 , l = 2, . . . , k, c2 + · · · + ck+1 ≥ d1 + · · · + dk−1 + x, c2 + · · · + cs ≥ d1 + · · · + dk−1 + x + dk+2 + · · · + ds , s = k + 2, . . . , n,
(76) (77) (78)
as one can show using dk ≤ c1 ≤ dk+1 and x = dk + dk+1 − c1 . Also, we have cn − c2 − · · · − cn−1 ≤ dn − d1 − · · · − dk−1 − x − dk+2 − · · · − dn ,
(79)
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fulfilling all of the conditions that we need invoking the induction assumption to construct γ . This matrix has the same symplectic eigenvalues as the right lower 2(n−1)×2(n−1) submatrix γ of γ . Therefore, there exists an S ∈ Sp(2(n − 1), R) such that γ = Sγ S T .
(80)
γ := (12 ⊕ S)γ (12 ⊕ S)T
(81)
So the matrix has the symplectic spectrum (d1 , . . . , dn ) and symplectic main diagonal elements (c1 , . . . , cn ). Hence, by invoking the induction assumption, we have been able to construct the desired matrix with the appropriate symplectic spectrum and main diagonal elements. Note that only two-mode operations have been needed in order to achieve this goal. We now turn to the two remaining cases, k = n and k = n − 1. In both cases this means that we have c1 ≥ dn−1 , as dn ≥ dn−1 , and both cases can be treated in actually exactly the same manner. Obviously, this implies that also cn ≥ c1 ≥ dn−1 . We can now define again an x, by means of a set of inequalities. This construction is very similar to the one in Ref. [22]. We can require on the one hand x ≥ max{dn−1 , dn−1 + dn − cn , dn−1 − dn + cn , d1 + · · · + dn−2 + cn−1 − c1 − · · · − cn−2 }.
(82)
On the other hand, we can require x ≤ min{dn − dn−1 + cn , c1 + · · · + cn−1 − d1 − · · · − dn−2 , 0}.
(83)
Both these conditions can be simultaneously satisfied, making use of cn ≥ cn−1 and cn ≥ dn−1 . This in turn means that we have cn + x ≥ dn−1 + dn , cn − x ≥ dn−1 − dn , x − cn ≥ dn−1 − dn ,
(84) (85) (86)
where the latter two inequalities mean that |x − cn | ≤ |dn−1 − dn |. Moreover, we satisfy all the inequalities c1 + · · · + cl ≥ c1 + · · · + dl , l = 1, . . . , n − 2, c1 + · · · + cn−1 ≥ d1 + · · · + dn−2 + x,
(87) (88)
cn−1 − c1 − · · · − cn−2 ≤ x − d1 − · · · − dn−2 .
(89)
and
Again, we can hence invoke the induction assumption, and construct in the same way as before the desired covariance matrix with symplectic spectrum (d1 , . . . , dn ) and symplectic main diagonal elements (c1 , . . . , cn ). This ends the proof of sufficiency of the given conditions.
Gaussian Quantum Marginal Problem
277
4. Physical Implications of the Result and Outlook The results found in this work can also be read as a full specification of what multipartite Gaussian states may be prepared. Since the argument is constructive it readily provides a recipe of how to construct multi-mode Gaussian entangled states with all possible local entropies. For pure states, starting from squeezed modes, all is needed is a network of passive operations. Applied to optical systems of several modes, notably, this gives rise to a protocol to prepare multi-mode pure-state entangled light of all possible entanglement structures from squeezed light, using passive linear optical networks, via γ = O P OT ,
(90)
with P = (z 1 , 1/z 1 , . . . , z n , 1/z n ), z k ∈ R\{0}, and O ∈ K (n). P is the covariance matrix of squeezed single modes, whereas O represents the passive optical network. The latter can readily be broken down to a network of beam splitters and phase shifters, according to Ref. [33]. Hence, our result also generalizes the preparation of Ref. [18] from the case of three modes to any number of modes. Similarly, for mixed states, the given result readily defines a preparation procedure, but now using also squeezers in general. The above statement also settles the question of the sharing of entanglement of single modes versus the rest of the system in a multi-mode system: For a pure Gaussian state with d1 = · · · = dn = 1, the entanglement entropy E j|{1,...,n}\{ j} of a mode labeled j with respect to the rest of the system is given by E j|{1,...,n}\{ j} := S(ρ j ) = s(c j ) :=
cj + 1 cj + 1 cj − 1 cj − 1 log2 − log2 , (91) 2 2 2 2
where s : [1, ∞) → [0, ∞) is a monotone increasing, concave function. Corollary 4 (Entanglement sharing in pure Gaussian states). For pure Gaussian states, the set of all possible entanglement values of a single mode with respect to the system is given by ⎧ ⎫ ⎬
⎨ E 1|{2,...,n} , . . . , E n|{1,...,n−1} ∈ (s(c1 ), . . . , s(cn )) : c j − 1 ≤ (ck − 1), c j ≥ 1 . ⎩ ⎭ k= j
(92) This result is an immediate consequence of the above pure marginal problem, Corollary 2. In fact, this is for pure Gaussian states more than a monogamy inequality: it constitutes a full characterization of the complete set of consistent degrees of entanglement. A further practically useful application of our result is the following: It tells us how pure a state must have been, based on the information available from measuring local properties like local photon numbers. This is expected to be a very desirable tool in an experimental context: In optical systems, such measurements are readily available with homodyne or photon counting measurements. Corollary 5 (Locally measuring global purity in non-Gaussian states). Let us assume that one has acquired knowledge about the local symplectic eigenvalues c1 , . . . , cn of
278
J. Eisert, T. Tyc, T. Rudolph, B. C. Sanders
a global state ρ. Then one can infer that the global von-Neumann entropy S(ρ) of ρ satisfies n (93) ck . S(ρ) ≤ s k=1
This estimate is true regardless whether the state ρ is a Gaussian state or not. Proof. Let us denote with ω the Gaussian state with the same covariance matrix γ ≥ 0 as the (unknown) state ρ. The vectors (d1 , . . . , dn ) and (c1 , . . . , cn ) are the symplectic eigenvalues and symplectic main diagonal elements of ω, respectively. From the fact that diag(d1 , d1 , . . . , dn , dn ) reflects a tensor product of Gaussian states, we can conclude that n S(ω) = s(d j ). (94) j=1
In turn, from Lemma 1 we find that n j=1
cj ≥
n
dj.
(95)
j=1
By means of an extremality property of the von-Neumann entropy (see, e.g., Ref. [34,35]) that a Gaussian state has the largest von-Neumann entropy for fixed second moments, we find that S(ρ) ≤ S(ω). Since the function s : [1, ∞) → [0, ∞) defined in Eq. (91) is concave and monotone increasing, we have that S(ρ) ≤
n
s(d j ) ≤ s(d1 + · · · + dn ) ≤ s(c1 + · · · + cn ).
(96)
j=1
This is the statement that we intended to prove. Clearly, this bound is tight, as is obvious when applying the inequality to the Gaussian state with covariance matrix diag(d1 , d1 , . . . , dn , dn ) itself.
For example, if one obtains c1 = 3/2 = c2 = 3/2 and c3 = 2 in local measurements on the local photon number, then one finds that the global state necessarily satisfies S(ρ) ≤ s(5). This is a powerful tool when local measurements in optical systems are more accessible than global ones, for example, when no phase reference is available, or bringing modes together is a difficult task. To finally turn to the role of Gaussian operations in this work: Our result highlights an observation that has been encountered already a number of times in the literature: That global Gaussian operations applied to many modes at once are often hardly more powerful than when applied to pairs of modes. This resembles to some extent the situation in the distillation of entangled Gaussian states by means of Gaussian operations [16,36–38]. In this work, we have given a complete characterization of reductions of pure or mixed Gaussian states. In this way, we have also given a general picture of the possibility of sharing quantum correlations in a continuous-variable setting. Since our proof is constructive, it also gives rise to a general recipe to generate multi-mode entangled states with all possible reductions. Formally, we established a connection to a compatibility argument of symplectic spectra, by means of new matrix inequalities fully characterizing the set in question. These matrix inequalities formally resemble the well-known
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279
Sing-Thompson Theorem relating singular values to main diagonal elements. It is the hope that this work can provide a significant insight into the achievable correlations in composite quantum systems of many modes. Acknowledgements. We would like to thank V. Buzek, P. Hyllus, and M.M. Wolf for valuable discussions on the subject of the paper, and especially K. Audenaert for many constructive and helpful comments concerning the presentation of the results, and A. Serafini and G. Adesso for further remarks on the manuscript. This work has been supported by the DFG (SPP 1116, SPP 1078), the EPSRC, the QIP-IRC, iCORE, CIFAR, Microsoft Research, the grants MSM0021622419 and MSM0021622409, and the EURYI Award.
References 1. Higuchi, A., Sudbery, A., Szulc, J.: One-qubit reduced states of a pure many-qubit state: polygon inequalities. Phys. Rev. Lett. 90, 107902 (2003) 2. Higuchi, H.: On the one-particle reduced density matrices of a pure three-qutrit quantum state. http:// arxiv.org/list/quant-ph/0309186, 2003 3. Bravyi, S.: Compatibility between local and multipartite states. Quant. Inf. Comp. 4, 12–26 (2004) 4. Han, Y.-J., Zhang, Y.-S., Guo, G.-C.: Compatibility relations between the reduced and global density matrices Phys. Rev. A 71, 052306 (2005) 5. Klyachko, A.: Quantum marginal problem and representations of the symmetric group. http://arxiv.org/ list/quant-ph/0409113, 2004 6. Franz, M.: Moment polytopes of projective G-varieties and tensor products of symmetric group representations. J. Lie Theory 12, 539–549 (2002) 7. Christandl, M., Winter, A.: Squashed entanglement: An additive entanglement measure. J. Math. Phys. 45, 829–840 (2004) 8. Terhal, B.M., Koashi, M., Imoto, N.: Unconditionally secure key distribution based on two nonorthogonal states. Phys. Rev. Lett. 90, 167904 (2003) 9. Nielsen, M.A., Kempe, J.: Separable states are more disordered globally than locally. Phys. Rev. Lett. 86, 5184–5187 (2001) 10. Eisert, J., Audenaert, K., Plenio, M.B.: Remarks on entanglement measures and non-local state distinguishability. J. Phys. A: Math. Gen. 36, 5605–5615 (2003) 11. Daftuar, S., Hayden, P.: Quantum state transformations and the Schubert calculus. Ann. Phys. 315, 80–122 (2005) 12. Hall, W.: Compatibility of subsystem states and convex geometry. Phys. Rev. A 75, 032102 (2007) 13. Liu, Y.-K., Christandl, M., Verstraete, F.: Quantum computational complexity of the N-Representability Problem: QMA Complete Phys. Rev. Lett. 98, 110503 (2007) 14. Christandl, M., Harrow, A., Mitchison, G.: On nonzero Kronecker coefficients and their consequences for spectra. Commun. Math. Phys. 270, 575–585 (2007) 15. Christandl, M.: PhD thesis, (Cambridge, October 2005) 16. Eisert, J., Plenio, M.B.: Introduction to the basics of entanglement theory in continuous-variable systems. Int. J. Quant. Inf. 1, 479–506 (2003) 17. Braunstein, S.L., van Loock, P.: Quantum information with continuous variables. Rev. Mod. Phys. 77, 513–577 (2005) 18. Adesso, G., Serafini, A., Illuminati, F.: Optical state engineering, quantum communication, and robustness of entanglement promiscuity in three-mode Gaussian states. New J. Phys. 9, 60 (2007) 19. Adesso, G., Serafini, A., Illuminati, F.: Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence. Phys. Rev. A 73, 032345 (2006) 20. Sing, F.Y.: Some results on matrices with prescribed diagonal elements and singular values. Canad. Math. Bull. 19, 89–92 (1976) 21. Thompson, R.C.: Singular values, diagonal elements, and convexity. SIAM J. Appl. Math. 32, 39–63 (1977) 22. Thompson, R.C.: Singular values and diagonal elements of complex symmetric matrices. Lin. Alg. Appl. 26, 65–106 (1979) 23. Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math. 76, 620–630 (1954) 24. Mirsky, L.: Inequalities and existence theorems in the theory of matrices. J. Math. Anal. Appl. 9, 99–118 (1964)
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25. Hyllus, P., Eisert, J.: Optimal entanglement witnesses for continuous-variable systems. New J. Phys. 8, 51 (2006) 26. Bhatia, R.: Matrix Analysis. Berlin: Springer, 1997, p. 254 27. Hiroshima, T.: Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs. Phys. Rev. A 73, 012330 (2006) 28. Botero, A., Reznik, B.: Modewise entanglement of Gaussian states. Phys. Rev. A 67, 052311 (2003) 29. Giedke, G., Eisert, J., Cirac, J.I., Plenio, M.B.: Entanglement transformations of pure Gaussian states. Quant. Inf. Comp. 3, 211–223 (2003) 30. Holevo, A.S., Werner, R.F.: Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63, 032312 (2001) 31. Arvind Dutta, B., Mukunda, N., Simon, R.: The real symplectic groups in quantum mechanics and optics. Pramana 45(6), 471–497 (1995) 32. Adesso, G., Serafini, A., Illuminati, F.: Extremal entanglement and mixedness in continuous variable systems. Phys. Rev. A 70, 022318 (2004) 33. Reck, M., Zeilinger, A., Bernstein, H.J., Bertani, P.: Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58–61 (1994) 34. Holevo, A.S., Werner, R.F.: Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63, 032312 (2001) 35. Eisert, J., Wolf, M.M.: Gaussian quantum channels. In: Quantum Information with Continuous Variables of Atoms and Light, Cerf, N.J., Leuchs, G., Polzik, E.J., eds., London: Imperial College Press, 2007, pp 23–42 36. Eisert, J., Scheel, S., Plenio, M.B.: Distilling Gaussian states with Gaussian operations is impossible. Phys. Rev. Lett. 89, 137903 (2002) 37. Fiurášek, J.: Gaussian transformations and distillation of entangled Gaussian states. Phys. Rev. Lett. 89, 137904 (2002) 38. Giedke, G., Cirac, J.I.: Characterization of Gaussian operations and distillation of Gaussian states. Phys. Rev. A 66, 032316 (2002) Communicated by M.B. Ruskai
Commun. Math. Phys. 280, 281–283 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0462-0
Communications in
Mathematical Physics
Erratum
Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrödinger Equation Driven by the Square of a Gaussian Field Philippe Mounaix1 , Pierre Collet1 , Joel L. Lebowitz2 1 Centre de Physique Théorique, UMR 7644 du CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex,
France. E-mail: [email protected]; [email protected]
2 Departments of Mathematics and Physics, Rutgers, The State University of new Jersey,
Piscataway, NJ 08854-8019, USA. E-mail: [email protected] Received: 25 October 2007 / Accepted: 5 November 2007 Published online: 7 March 2008 – © Springer-Verlag 2008
Commun. Math. Phys. 264, 741–758 (2006)
The proof of the inequality λq (x, t) ≤ (qµx,t − 0+ )−1 [p. 750, below Eq. (29)] is based on the statement that E(x, t; s) is an entire function of s ∈ C M [see below Eq. (30)]. But according to Eq. (9) and Lemma 1, all we know is that E(x, t; s) is an entire function of k(s) ∈ R N . Nevertheless, the above inequality holds, hence Proposition 1. To prove it we replace (31) with the following lemma. Lemma. If |E(x, t)|q < +∞, then lim sup e−||s|| |E(x, t; s)|q < +∞, 2
||s||→+∞
(31)
along almost every direction sˆ in C M . Proof. For given x and t, write E(x, t; s) = E(ˆs , κ), where κ = ||k(s)|| = ||s||2 . Making this change of notation in (29) it is easily seen that if |E(x, t)|q < +∞, then
+∞
|E(ˆs , κ)|q e−κ κ M−1 dκ < +∞,
(32)
0
for almost every direction sˆ in C M . Fix sˆ such that (32) is fulfilled. By Eq. (9), Lemma 1, and Lemma 2, E(ˆs , z) is an entire function of z ∈ C of finite exponential type [1]. Since q ≥ 1 and M ≥ 1 are integers, the function f (z) = E(ˆs , z)q e−z z M−1 is also an entire The online version of the original article can be found under doi:10.1007/s00220-006-1553-4.
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function of z ∈ C of finite exponential type, say γ f . Let R be a fixed positive number, let γ > max(0, γ f ), and define R ϕ± (z) = |e±iγ z f (z + u)|du. (33) 0
The functions ϕ± (z) are logarithmically subharmonic and bounded by (32) on the positive real axis. Furthermore, ϕ+ (z) and ϕ− (z) are bounded respectively on the positive and negative imaginary axis. Following then the same argument as in the proof of the Plancherel-Pólya theorem (see [1], p 51), we apply the Phragmén-Lindelöf theorem [1] to the subharmonic functions ln ϕ+ (z) in the sector 0 < argz < π/2 and ln ϕ− (z) in the sector −π/2 < argz < 0. One finds that ∃A > 0 such that, ∀z with Re(z) > 0, R | f (z + u)|du ≤ Aeγ |Im(z)| . (34) 0
Now, for all κ > R/2 one has, by the subharmonicity of | f (z)| and (34), 4 | f (κ)| ≤ | f (κ + z)| d 2 z π R2 |z| R/2, 1 8A γ R/2 e − 1 < +∞, |E(ˆs , κ)|q e−κ ≤ γ π R2 κ M−1
(35)
(36)
(hence limκ→+∞ |E(ˆs , κ)|q e−κ = 0 for M > 1). Getting back to the original notation yields the new Eq. (31), which completes the proof of the lemma. We can now proceed with the proof of λq (x, t) ≤ (qµx,t − 0+ )−1 . Since every t element of the matrix 0 γ (x(τ ), τ ) dτ is a continuous functional of x(·) ∈ B(x, t) with the uniform norm on [0, t] (see Appendix B), its largest eigenvalue, µ1 [x(·)], is also a continuous functional of x(·). Accordingly, ∀ε > 0 ∃xε (·) ∈ B(x, t) such that M µ tx,t − ε/2 ≤ µ1 [xε (·)] ≤ µx,t . Let σε ∈ C (with ||σε || = 1) be an eigenvector of 0 γ (x ε (τ ), τ ) dτ associated withthe eigenvalue µ1 [x ε (·)]. Fix x ε (·) and σε . The quat dratic form sˆ † 0 γ (xε (τ ), τ ) dτ sˆ is a continuous function of the direction sˆ . Thus, ∃δ > 0 such that ∀ˆs with ||ˆs − σε || ≤ δ, t t ε † † sˆ (37) γ (xε (τ ), τ ) dτ sˆ − σε γ (xε (τ ), τ ) dτ σε ≤ , 2 0
0
and
Hx,t (ˆs ) =
sup x(·)∈B(x,t) t †
sˆ †
t
γ (x(τ ), τ ) dτ sˆ
0
γ (xε (τ ), τ ) dτ sˆ 0 t ≥ σε† γ (xε (τ ), τ ) dτ σε − ε/2 ≥ µx,t − ε.
≥ sˆ
0
(38)
Erratum
283
From the latter inequality and Lemma 2 it follows that ∀ε > 0, ∃δ > 0 such that ∀ˆs with ||ˆs − σε || ≤ δ, ln |E(x, t; s)|q ≥ qλ(µx,t − ε), ||s||2 ||s||→+∞ lim sup
and for every λ > (qµx,t − qε)−1 , lim sup e−||s|| |E(x, t; s)|q = +∞. 2
||s||→+∞
(39)
Since δ > 0, the set of all the directions sˆ fulfilling (39) is of strictly positive measure and, according to the lemma above, |E(x, t)|q = +∞. Therefore, λq (x, t) ≤ (qµx,t −qε)−1 and taking ε arbitrarily small one obtains λq (x, t) ≤ (qµx,t − 0+ )−1 . (We use the notation qµx,t − 0+ to emphasize the fact that |E(x, t)|q may be finite at λ = 1/qµx,t . Our approach does not yield the behavior of |E(x, t)|q at λ = 1/qµx,t sharp.)
Reference 1. Levin, B.Ya.: Lectures on entire functions. Translations of mathematical monographs, No. 150.: Providence, RI: Amer. Math. Soc., 1996 Communicated by M. Aizenman
Commun. Math. Phys. 280, 285–313 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0466-9
Communications in
Mathematical Physics
Moduli Spaces of Self-Dual Connections over Asymptotically Locally Flat Gravitational Instantons Gábor Etesi1,2 , Marcos Jardim2 1 Department of Geometry, Mathematical Institute, Faculty of Science,
Budapest University of Technology and Economics, Egry J. u. 1, H ép., 1111 Budapest, Hungary. E-mail: [email protected]; [email protected]
2 Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas,
C.P. 6065, 13083-859, Campinas, SP, Brazil. E-mail: [email protected] Received: 3 November 2006 / Accepted: 22 October 2007 Published online: 28 March 2008 – © Springer-Verlag 2008
Abstract: We investigate Yang–Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel–Hunsicker–Mazzeo. First referring to the codimension 2 singularity removal theorem of Sibner–Sibner and Råde we prove that given a smooth, finite energy, self-dual SU(2) connection over a complete ALF space, its energy is congruent to a Chern–Simons invariant of the boundary three-manifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly. Then we introduce framed moduli spaces of self-dual connections over Ricci flat ALF spaces. We prove that the moduli space of smooth, irreducible, rapidly decaying self-dual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold. We calculate its dimension by a variant of the Gromov–Lawson relative index theorem. As an application, we study Yang–Mills instantons over the flat R3 × S 1 , the multi-Taub–NUT family, and the Riemannian Schwarzschild space. 1. Introduction By a gravitational instanton one usually means a connected, four dimensional complete hyper-Kähler Riemannian manifold. In particular, these spaces have SU(2) ∼ = Sp(1) holonomy; consequently, they are Ricci flat, and hence solutions of the Riemannian Einstein’s vacuum equation. The only compact, four dimensional hyper-Kähler spaces are, up to (universal) covering, diffeomorphic to the flat torus T 4 or a K 3 surface. The next natural step would be to understand non-compact gravitational instantons. Compactness in this case should be replaced by the condition that the metric be complete and decay to the flat metric at infinity somehow such that the Pontryagin number of the manifold be finite.
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Such open hyper-Kähler examples can be constructed as follows. Consider a connected, orientable compact four-manifold M with connected boundary ∂ M which is a smooth three-manifold. Then the open manifold M := M\∂ M has a decomposition M = K ∪ W , where K is a compact subset and W ∼ = ∂ M × R+ is an open annulus or neck. Parameterize the half-line R+ by r . Assume ∂ M is fibered over a base manifold B with fibers F and the complete hyper-Kähler metric g asymptotically and locally looks like g ∼ dr 2 + r 2 g B + g F . In other words the base B of the fibration blows up locally in a Euclidean way as r → ∞, while the volume of the fiber remains finite. By the curvature decay, g F must be flat, hence F is a connected, compact, orientable, flat manifold. On induction of the dimension of F, we can introduce several cases of increasing transcendentality, using the terminology of Cherkis and Kapustin [7]: (i) (M, g) is ALE (asymptotically locally Euclidean) if dim F = 0; (ii) (M, g) is ALF (asymptotically locally flat) if dim F = 1, in this case necessarily F∼ = S 1 must hold; (iii) (M, g) is ALG (this abbreviation by induction) if dim F = 2, in this case F ∼ = T 2; (iv) (M, g) is ALH if dim F = 3, in this case F is diffeomorphic to one of the six flat orientable three-manifolds. Due to their relevance in quantum gravity or recently rather in low-energy supersymmetric solutions of string theory and, last but not least, their mathematical beauty, there has been some effort to classify these spaces over the past decades. Trivial examples for any class is provided by the space R4−dim F × F with its flat product metric. The first two non-trivial, infinite families were discovered by Gibbons and Hawking in 1976 [19] in a rather explicit form. One of these families are the Ak ALE or multiEguchi–Hanson spaces. In 1989, Kronheimer gave a full classification of ALE spaces [26] constructing them as minimal resolutions of C2 / , where ⊂ SU(2) is a finite subgroup, i.e. is either a cyclic group Ak , k ≥ 0, dihedral group Dk with k > 0, or one of the exceptional groups El with l = 6, 7, 8. The other infinite family of Gibbons and Hawking is the Ak ALF or multi-Taub–NUT family. Recently another Dk ALF family has been constructed by Cherkis and Kapustin [10] and in a more explicit form by Cherkis and Hitchin [8]. Motivated by string theoretical considerations, Cherkis and Kapustin have suggested a classification scheme for ALF spaces as well as for ALG and ALH [7] although they relax the above asymptotical behaviour of the metric in these later two cases in order to obtain a natural classification. They claim that the Ak and Dk families with k ≥ 0 exhaust the ALF geometry (in this enumeration D0 is the Atiyah–Hitchin manifold). For the ALG case if we suppose that these spaces arise by deformations of elliptic fibrations with only one singular fiber, it is conjectured that the possibilities are Dk with 0 ≤ k ≤ 5 (cf. [9]) and El with l = 6, 7, 8. An example for a non-trivial ALH space is the minimal resolution of (R × T 3 )/Z2 . The trouble is that these spaces are more transcendental as dim F increases, hence their constructions, involving twistor theory, Nahm transform, etc. are less straightforward and explicit. To conclude this brief survey, we remark that the restrictive hyper-Kähler assumption on the metric, which appeared to be relevant in the more recent string theoretical investigations, excludes some examples considered as gravitational instantons in the early eighties. An important non-compact example which satisfies the ALF condition is for instance the Riemannian Schwarzschild space, which is Ricci flat but not hyperKähler [22]. For a more complete list of such “old” examples cf. [14]. From Donaldson theory we learned that the moduli spaces of SU(2) instantons over compact four-manifolds encompass a lot of information about the original manifold,
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hence understanding SU(2) instantons over gravitational instantons also might be helpful in their classification. On the compact examples T 4 and the K 3’s, Yang–Mills instantons can be studied via the usual methods, especially the celebrated Hitchin–Kobayashi correspondence. The full construction of SU(2) instantons in the hyper-Kähler ALE case was carried out in important papers by Nakajima [29] and Kronheimer–Nakajima in 1990 [27]. However, the knowledge regarding moduli spaces of instantons over nontrivial ALF spaces is rather limited, even in the hyper-Kähler ALF case, due to analytical difficulties. One has only sporadic examples of explicit solutions (cf. e.g. [17,18]). Also very little is known about instanton theory over the Riemannian Schwarzschild space [6,16]. The only well studied case is the flat R3 × S 1 space; instantons over this space, also known as calorons, have been extensively studied in the literature, cf. [4,5,30]. Close to nothing is known about instantons over non-trivial ALG and ALH geometries. Studying Yang–Mills instanton moduli spaces over ALF spaces is certainly interesting not only because understanding the reducible solutions already leads to an encouraging topological classification result in the hyper-Kähler case [15], but also due to their physical significance. In this paper we set the foundations for a general theory of Yang–Mills instantons over ALF spaces in the broad sense adopted in the eighties, i.e. including not hyper-Kähler examples, too. In Sect. 2 we exploit the existence of a natural smooth compactification X of an ALF space introduced by Hausel–Hunsicker–Mazzeo [21]. Working over this compact space, the asymptotical behaviour of any finite energy connection over an ALF space can be analyzed by the codimension 2 singularity removal theorem of Sibner–Sibner [33] and Råde [32]. This guarantees the existence of a locally flat connection ∇ with fixed constant holonomy in infinity to which the finite energy connection converges. First we prove in Sect. 2 that the energy of a smooth, self-dual SU(2) connection of finite energy which satisfies a certain holonomy condition (cf. condition (11) here) and has rapid curvature decay (in the sense of condition (16) in the paper), is congruent to the Chern–Simons invariant τ N (∞ ) of the boundary N of the ALF space (Theorem 2.2 here). If the holonomy condition holds then ∇ is in fact flat and ∞ is a fixed smooth gauge for the limiting flat connection restricted to the boundary. The relevant holonomy condition can be replaced by a simple topological criterion on the infinity of the ALF space, leading to a more explicit form of this theorem (cf. Theorem 2.3). Then in Sect. 3 we introduce framed instanton moduli spaces M(e, ) of smooth, irreducible, rapidly decaying self-dual SU(2) connections, obeying the holonomy condition, with fixed energy e < ∞ and asymptotical behaviour described by the flat connection ∇ on a fixed bundle. Referring to a variant of the Gromov–Lawson relative index theorem [20] (cf. Theorem 3.1 here) we will be able to demonstrate that a framed moduli space over a Ricci flat ALF space is either empty or forms a smooth manifold of dimension dim M(e, ) = 8 (e + τ N (∞ ) − τ N (∞ )) − 3b− (X ), where ∞ is the restriction to N of the trivial flat connection ∇ in some smooth gauge and b− (X ) is the rank of the negative definite part of the intersection form of the Hausel–Hunsicker–Mazzeo compactification (cf. Theorem 3.2). In Sect. 4 we apply our results on three classical examples, obtaining several novel facts regarding instantons over them. First, we prove in Theorem 4.1 that any smooth, finite energy caloron over R3 × S 1 automatically satisfies our holonomy condition, has integer energy e ∈ N if it decays rapidly and that the dimension of the moduli space in this case is 8e, in agreement with [4]. These moduli spaces are non-empty for all positive integer e [5].
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For the canonically oriented multi-Taub–NUT spaces, we show that the dimension of the framed moduli of smooth, irreducible, rapidly decaying anti-self-dual connections satisfying the holonomy condition is divisible by 8 (cf. Theorem 4.2). Known explicit solutions [18] show that at least a few of these moduli spaces are actually non-empty. Finally, we consider the Riemannian Schwarzschild case, and prove in Theorem 4.3 that all smooth finite energy instantons obey the holonomy condition, have integer energy e if they decay rapidly and the dimension is 8e −3. Moreover, this moduli space is surely non-empty at least for e = 1. We also enumerate the remarkably few known explicit solutions [6,16], and observe that these admit deformations. Section 5 is an Appendix containing the proof of the relative index theorem used in the paper, Theorem 3.1. 2. The Spectrum of the Yang–Mills Functional In this section we prove that the spectrum of the Yang–Mills functional evaluated on self-dual connections satisfying a certain analytical and a topological condition over a complete ALF manifold is “quantized” by the Chern–Simons invariants of the boundary. First, let us carefully define the notion of ALF space used in this paper, and describe its useful topological compactification, first used in [21]. Let (M, g) be a connected, oriented Riemannian four-manifold. This space is called an asymptotically locally flat (ALF) space if the following holds. There is a compact subset K ⊂ M such that M\K = W and W ∼ = N × R+ , with N being a connected, compact, oriented three-manifold without boundary admitting a smooth S 1 -fibration F
π : N −→ B∞
(1)
whose base space is a compact Riemann surface B∞ . For the smooth, complete Riemannian metric g there exists a diffeomorphism φ : N × R+ → W such that φ ∗ (g|W ) = dr 2 + r 2 (π ∗ g B∞ ) + h F ,
(2)
where g B∞ is a smooth metric on B∞ , h F is a symmetric 2-tensor on N which restricts to a metric along the fibers F ∼ = S 1 and (π ∗ g B∞ ) as well as h F are some finite, bounded, smooth extensions of π ∗ g B∞ and h F over W , respectively. That is, we require (π ∗ g B∞ ) (r ) ∼ O(1) and h F (r ) ∼ O(1) and the extensions for r < ∞ preserve the properties of the original fields. Furthermore, we impose that the curvature Rg of g decays like |φ ∗ (Rg |W )| ∼ O(r −3 ).
(3)
Here Rg is regarded as a map Rg : C ∞ (2 M) → C ∞ (2 M) and its pointwise norm is calculated accordingly in an orthonormal frame. Hence the Pontryagin number of our ALF spaces is finite. Examples of such metrics are the natural metric on R3 × S 1 which is in particular flat; the multi-Taub–NUT family [19] which is moreover hyper-Kähler or the Riemannian Schwarzschild space [22] which is in addition Ricci flat only. For a more detailed description of these spaces, cf. Sect. 4. We construct the compactification X of M simply by shrinking all fibers of N into points as r → ∞ like in [21]. We put an orientation onto X induced by the orientation of the original M. The space X is then a connected, oriented, smooth four-manifold without
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boundary. One clearly obtains a decomposition X ∼ = M ∪ B∞ , and consequently we can think of B∞ as a smoothly embedded submanifold of X of codimension 2. For example, for R3 × S 1 one finds X ∼ = S 4 and B∞ is an embedded S 2 [15]; for the multi-Taub–NUT space with the orientation induced by one of the complex structures, X is the connected 2 sum of s copies of CP ’s (s refers to the number of NUTs) and B∞ is homeomorphic 2 to S providing a generator of the second cohomology of X ; in case of the Riemannian Schwarzschild geometry, X ∼ = S 2 × S 2 and B∞ is again S 2 , also providing a generator for the second cohomology (cf. [15,21]). Let M R := M\(N × (R, ∞)) be the truncated manifold with boundary ∂ M R ∼ = N × {R}. Taking into account that W ∼ = N × (R, ∞), a normal neighbourhood V R of B∞ in X has a model like V R ∼ = N × (R, ∞]/ ∼, where ∼ means that N × {∞} is pinched into B∞ . We obtain W = V R \V∞ , with V∞ ∼ = B∞ . By introducing the parameter ε := R −1 we have another model Vε provided by the fibration Bε2
Vε −→ B∞ whose fibers are two-balls of radius ε. In this second picture we have the identification V0 ∼ = B∞ , so that the end W looks like Vε∗ := Vε\V0 .
(4)
Choosing a local coordinate patch U ⊂ B∞ , then locally Vε |U ∼ = U × Bε2 and Vε∗ |U ∼ = 2 U × (Bε \{0}). We introduce local coordinates (u, v) on U and polar coordinates (ρ, τ ) along the discs Bε2 with 0 ≤ ρ < ε and 0 ≤ τ < 2π . Note that in fact ρ = r −1 is a global coordinate over the whole Vε ∼ = V R . For simplicity we denote Vε |U as Uε and will call the set Uε∗ := Uε \U0
(5)
an elementary neighbourhood. Clearly, their union covers the end W . In this ε-picture we will use the notation ∂ Mε ∼ = N × {ε} for the boundary of the truncated manifold Mε = M\(N × (0, ε)), and by a slight abuse of notation we will also think of the end sometimes as W ∼ = ∂ Mε × (0, ε). We do not expect the complete ALF metric g to extend over this compactification, even conformally. However the ALF property (2) implies that we can suppose the existence of a smooth positive function f ∼ O(r −2 ) on M such that the rescaled metric g˜ := f 2 g extends smoothly as a tensor field over X (i.e., a smooth Riemannian metric degenerated along the singularity set B∞ ). In the vicinity of the singularity we find g| ˜ Vε = dρ 2 + ρ 2 (π ∗ g B∞ ) + ρ 4 h F via (2), consequently we can choose the coordinate system (u, v, ρ, τ ) on Uε such that {du, dv, dρ, dτ } forms an oriented frame on T ∗ Uε∗ and with some bounded, finite function ϕ, the metric looks like g| ˜ Uε∗ = dρ 2 + ρ 2 ϕ(u, v, ρ)(du 2 + dv 2 ) + ρ 4 (dτ 2 + . . . ). Consequently we find that Volg˜ (Vε ) ∼ O(ε5 ).
(6)
We will also need a smooth regularization of g. ˜ Taking a monotonously increasing smooth function f ε supported in V2ε and equal to 1 on Vε such that |d f ε | ∼ O(ε−1 ), as well as picking up a smooth metric h on X , we can regularize g˜ by introducing the smooth metric g˜ ε := (1 − f ε )g˜ + f ε h over X . It is clear that g˜ 0 and g˜ agree on M.
(7)
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Let F be an SU(2) vector bundle over X endowed with a fixed connection ∇ and an invariant fiberwise scalar product. Using the rescaled-degenerated metric g, ˜ define p Sobolev spaces L j, (∗ X ⊗ F) with 1 < p ≤ ∞ and j = 0, 1, . . . as the completion of C0∞ (∗ X ⊗ F), smooth sections compactly supported in M ⊂ X , with respect to the norm ⎞1 ⎛ p j p k ⎠ ⎝
ω L p (X ) := lim
∇ ω L p (Mε ,g| , (8) ˜ M ) ε→0
j,
ε
k=0
where
p
∇k ω L p (Mε ,g| ˜ M
ε)
=
|∇k ω| p ∗g˜ 1. Mε
Throughout this paper, Sobolev norms of this kind will be used unless otherwise stated. p We will write simply L p for L 0, . Notice that every 2-form with finite L 2 -norm over (M, g) will also belong to this Sobolev space, by conformal invariance and completeness. Next, we collect some useful facts regarding the Chern–Simons functional. Let E be a smooth SU(2) bundle over M. Since topological G-bundles over an open fourmanifold are classified by H 2 (M, π1 (G)), note that E is necessarily trivial. Put a smooth SU(2) connection ∇ B onto E. Consider the boundary ∂ Mε of the truncated manifold. The restricted bundle E|∂ Mε is also trivial. Therefore any restricted SU(2) connection ∇ B |∂ Mε := ∇ Bε over E|∂ Mε can be identified with a smooth su(2)-valued 1-form Bε . The Chern–Simons functional is then defined to be 1 2 τ∂ Mε (Bε ) := − 2 tr dBε ∧ Bε + Bε ∧ Bε ∧ Bε . 8π 3 ∂ Mε
This expression is gauge invariant up to an integer. Moreover, the representation space χ (∂ Mε ) := Hom(π1 (∂ Mε ), SU(2))/SU(2) is called the character variety of ∂ Mε ∼ = N and parameterizes the gauge equivalence classes of smooth flat SU(2) connections over N . Lemma 2.1. Fix an 0 < ρ < ε and let ∇ Aρ = d + Aρ and ∇ Bρ = d + Bρ be two smooth SU(2) connections in a fixed smooth gauge on the trivial SU(2) bundle E|∂ Mρ . Then there is a constant c1 = c1 (Bρ ) > 0, depending on ρ only through Bρ , such that |τ∂ Mρ (Aρ ) − τ∂ Mρ (Bρ )| ≤ c1 Aρ − Bρ L 2 (∂ Mρ ) ,
(9)
that is, the Chern–Simons functional is continuous in the L 2 norm. Moreover, for each ρ, τ∂ Mρ (Aρ ) is constant on the path connected components of the character variety χ (∂ Mρ ). Proof. The first observation follows from the identity τ∂ Mρ (Aρ ) − τ∂ Mρ (Bρ ) = 1 1 − 2 tr (FAρ + FBρ ) ∧ (Aρ − Bρ )− (Aρ − Bρ ) ∧ (Aρ − Bρ ) ∧ (Aρ−Bρ ) , 8π 3 ∂ Mρ
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which implies that there is a constant c0 = c0 (ρ, Bρ ) such that |τ∂ Mρ (Aρ ) − τ∂ Mρ (Bρ )| ≤ c0 Aρ − Bρ
3
2 L 1,B (∂ Mρ ) ρ
,
3 2 that is, the Chern–Simons functional is continuous in the L 1,B norm. Then applying ρ 3
2 ⊂ L 2 over a compact three-manifold, we find a similar the Sobolev embedding L 1,B ρ inequality with a constant c1 = c1 (ρ, Bρ ). The metric locally looks like g| ˜ ∂ Mρ ∩Uε∗ = ρ 2 ϕ(du 2 + dv 2 ) + ρ 4 (dτ 2 + 2h τ,u dτ du + 2h τ,v dτ dv + h u,v du 2 + h v,v dv 2 + 2h u,v dudv) with ϕ and h τ,u , etc. being bounded functions of (u, v, ρ) and (u, v, ρ, τ ) respectively, hence the metric coefficients are bounded functions of ρ; consequently we can suppose that c1 does not depend explicitly on ρ. Concerning the second part, assume ∇ Aρ and ∇ Bρ are two smooth, flat connections belonging to the same path connected component of χ (∂ Mρ ). Then there is a continuous path ∇ Atρ with t ∈ [0, 1] of flat connections connecting the given flat connections. Out of this we construct a connection ∇ A on ∂ Mρ × [0, 1] given by A := Atρ + 0 · dt. Clearly, this connection is flat, i.e., FA = 0. The Chern–Simons theorem [11] implies that 1 τ∂ Mρ (Aρ ) − τ∂ Mρ (Bρ ) = − 2 tr(FA ∧ FA ) = 0, 8π
∂ Mρ ×[0,1]
concluding the proof.
The last ingredient in our discussion is the fundamental theorem of Sibner–Sibner [33] and Råde [32] which allows us to study the asymptotic behaviour of finite energy connections over an ALF space. Consider a smooth (trivial) SU(2) vector bundle E over the ALF space (M, g). Let ∇ A be a Sobolev connection on E with finite energy, i.e. FA ∈ L 2 (2 M ⊗ EndE). Taking into account completeness of the ALF metric, we have |FA |(r ) → 0 almost everywhere as r → ∞. Thus we have a connection defined on X away from a smooth, codimension 2 submanifold B∞ ⊂ X and satisfies
FA L 2 (X ) < ∞. Consider a neighbourhood B∞ ⊂ Vε and write Vε∗ to describe the end W as in (4). Let ∇ be an SU(2) connection on E|Vε∗ which is locally flat and smooth. The restricted bundle E|Vε∗ is trivial, hence we can choose some global gauge such that ∇ A |Vε∗ = d+ A Vε∗ and ∇ = d + Vε∗ ; we assume with some j = 0, 1, . . . that A Vε∗ ∈ L 2j+1,,loc . Taking into account that for the elementary neighbourhood π1 (Uε∗ ) ∼ = Z, generated by a τ -circle, it is clear that locally on Uε∗ ⊂ Vε∗ we can choose a more specific gauge ∇ |Uε∗ = d + m with a constant m ∈ [0, 1) such that [33] im 0 m = dτ. (10) 0 −im Here m represents the local holonomy of the locally flat connection around the punctured discs of the space Uε∗ , see [33]. It is invariant under gauge transformations modulo an integer. For later use we impose two conditions on this local holonomy. The embedding i : Uε∗ ⊂ Vε∗ induces a group homomorphism i ∗ : π1 (Uε∗ ) → π1 (Vε∗ ). It may happen that this homomorphism has non-trivial kernel. Let l be a loop in Uε∗ such that [l] generates π1 (Uε∗ ) ∼ = Z.
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Definition 2.1. A locally flat connection ∇ on E|Vε∗ is said to satisfy the weak holonomy condition if for all Uε∗ ⊂ Vε∗ the restricted connection ∇ |Uε∗ = d + m has trivial local holonomy whenever l is contractible in Vε∗ , i.e. [l] ∈ Ker i ∗ =⇒ m = 0.
(11)
Additionally, ∇ is said to satisfy the strong holonomy condition if the local holonomy of any restriction ∇ |Uε∗ = d + m vanishes, i.e. m = 0.
(12)
Clearly, ∇ is globally a smooth, flat connection on E|Vε∗ if and only if the weak holonomy condition holds. Moreover, the strong holonomy condition implies the weak one. We are now in a position to recall the following fundamental regularity result [32,33]. Theorem 2.1. (Sibner–Sibner, 1992 and Råde, 1994) There exist a constant ε > 0 and a flat SU(2) connection ∇ |Uε∗ on E|Uε∗ with a constant holonomy m ∈ [0, 1) such that on E|Uε∗ one can find a gauge ∇ A |Uε∗ = d + AUε∗ and ∇ |Uε∗ = d + m with AUε∗ − m ∈ L 21, (Uε∗ ) such that the estimate
AUε∗ − m L 2
∗ 1, (Uε )
≤ c2 FA L 2 (Uε )
holds with a constant c2 = c2 (g| ˜ Uε ) > 0 depending only on the metric. This theorem shows that any finite energy connection is always asymptotic to a flat connection at least locally. It is therefore convenient to say that the finite energy connection ∇ A satisfies the weak or the strong holonomy condition if its associated asymptotic locally flat connection ∇ , in the sense of Theorem 2.1, satisfies the corresponding condition in the sense of Definition 2.1. We will be using this terminology. This estimate can be globalized over the whole end Vε∗ as follows. Consider a finite ∗ ⊂ V ∗. covering B∞ = ∪α Uα and denote the corresponding punctured sets as Uε,α ε ∗ These sets also give rise to a finite covering of Vε . It is clear that the weak condition (11) is independent of the index α, since by Theorem 2.1, m is constant over all Vε∗ . Imposing ∗ extend smoothly over the whole E| ∗ . That is, there is (11), the local gauges m on Uε,α Vε ∗ = γ −1 m γα + γ −1 dγα a smooth flat gauge ∇ = d + Vε∗ over E|Vε∗ such that Vε∗ |Uε,α α α ∗ → SU(2). This gauge is unique only with smooth gauge transformations γα : Uε,α up to an arbitrary smooth gauge transformation. Since this construction deals with the topology of the boundary (1) only, we can assume that these gauge transformations are independent of the (global) radial coordinate 0 < ρ < ε. Then we write this global gauge as ∇ A |Vε∗ = d + A Vε∗ ,
∇ = d + Vε∗ .
(13)
A comparison with the local gauges in Theorem 2.1 shows that −1 ∗ = γ ∗ − m )γα . (A Vε∗ − Vε∗ )|Uε,α α (AUε,α
(14)
Applying Theorem 2.1 in all coordinate patches and summing up over them we come up with
A Vε∗ − Vε∗ L 2
∗ 1, (Vε )
≤ c3 FA L 2 (Vε ) ,
(15)
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with some constant c3 = c3 (g| ˜ Vε , γα , dγα ) > 0. This is the globalized version of Theorem 2.1. Let Aε and ε be the restrictions of the connection 1-forms in the global gauge (13) to the boundary E|∂ Mε . The whole construction shows that ε is smooth and is independent of ε, consequently lim ε exists and is smooth. Note again that smoothness follows if ε→0
and only if the weak holonomy condition (11) is satisfied. In this way lim τ∂ Mε (ε ) = ε→0
τ N (0 ) also exists and gives rise to a Chern–Simons invariant of the boundary. Assume that our finite energy connection ∇ A is smooth; then in the global gauge (13) A Vε∗ is also smooth, hence τ∂ Mε (Aε ) also exists for ε > 0. Next we analyze the behaviour τ∂ Mε (Aε ) as ε tends to zero. First notice that the local flat gauge m in (10) does not have radial component, consequently A Vε∗ − Vε∗ = Aε − ε + Ar , where Ar is the radial component of A Vε∗ . Dividing the square of (9) by ε > 0 and then integrating it we obtain, making use of (15), that 1 ε
ε
c2 |τ∂ Mρ (Aρ ) − τ∂ Mρ (ρ )| dρ ≤ 1 ε
ε
Aρ − ρ 2L 2 (∂ M ) dρ ≤
2
0
c2 ≤ 1 ε =
ρ
0
ε
Aρ − ρ 2L 2 (∂ M ) + Ar 2L 2 (∂ M ) dρ ρ
ρ
0
c12 (c1 c3 )2
A Vε∗ − Vε∗ 2L 2 (V ∗ ) ≤
FA 2L 2 (V ) . ε ε ε ε
Finite energy and completeness implies that FA L 2 (Vε ) vanishes as ε tends to zero. However for our purposes we need a stronger decay assumption. Definition 2.2. The finite energy SU(2) connection ∇ A on the bundle E over M decays rapidly if its curvature satisfies √
FA L 2 (Vε ) R FA L 2 (VR ,g|V ) = 0 = lim √ R ε→0 R→∞ ε lim
(16)
along the end of the ALF space. Consequently for a rapidly decaying connection we obtain 1 lim ε→0 ε
ε |τ∂ Mρ (Aρ ) − τ∂ Mρ (ρ )|2 dρ = 0, 0
which is equivalent to lim τ∂ Mε (Aε ) = τ N (0 ).
ε→0
(17)
We are finally ready to state an energy identity for self-dual connections. Let ∇ A be a smooth, self-dual, finite energy connection on the trivial SU(2) bundle E over an ALF space (M, g): FA = ∗FA ,
FA 2L 2 (M,g) < ∞.
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Assume it satisfies the weak holonomy condition (11). In this case we can fix a gauge (13) along Vε∗ and both A Vε∗ and Vε∗ are smooth. Restrict ∇ A onto E| Mε with ε > 0. Exploiting self-duality, an application of the Chern–Simons theorem [11] along the boundary shows that
FA 2L 2 (M
ε ,g| Mε )
≡ τ∂ Mε (Aε )
mod Z.
Moreover if the connection decays rapidly in the sense of (16), then the right-hand side has a limit (17); therefore we have arrived at the following theorem: Theorem 2.2. Let (M, g) be an ALF space with an end W ∼ = N × R+ . Let E be an SU(2) vector bundle over M, necessarily trivial, with a smooth, finite energy, self-dual connection ∇ A . If it satisfies the weak holonomy condition (11) and decays rapidly in the sense of (16), then there exists a smooth flat SU(2) connection ∇ on E|W and a smooth flat gauge ∇ = d + W , unique up to a smooth gauge transformation, such that lim W | N ×{r } = ∞ exists, is smooth and
r →∞
FA 2L 2 (M,g) ≡ τ N (∞ )
mod Z.
That is, the energy is congruent to a Chern–Simons invariant of the boundary. Remark. It is clear that the above result depends only on the conformal class of the metric. One finds a similar energy identity for manifolds with conformally cylindrical ends [28] (including ALE spaces, in accordance with the energies of explicit instanton solutions of [17] and [18]) and for manifolds conformally of the form C × as in [35]. We expect that the validity of identities of this kind is more general. Taking into account the second part of Lemma 2.1 and the fact that the character variety of a compact three-manifold has finitely many connected components, we conclude that the energy spectrum of smooth, finite energy, self-dual connections over ALF spaces which satisfy the weak holonomy condition and decay rapidly is discrete. For irreducible instantons, imposing rapid decay is necessary for having discrete energies. For instance, in principle the energy formula [30, Eq. 2.32] provides a continuous energy spectrum for calorons and calorons of fractional energy are known to exist (cf. e.g. [12]). But slowly decaying reducible instantons still can have discrete spectrum; this is the case e.g. over the Schwarzschild space, cf. Sect. 4. Alternatively, instead of the rapid decay condition (16), one could also impose the possibly weaker but less natural condition that the gauge invariant limit lim |τ∂ Mε (Aε ) − τ∂ Mε (ε )| = µ
ε→0
exists. Then the identity of Theorem 2.2 would become:
FA 2L 2 (M,g) ≡ τ N (∞ ) + µ
mod Z.
By analogy with the energy formula for calorons [30, Eq. 2.32], we believe that the extra term µ is related to the overall magnetic charge of an instanton while the modified energy formula would represent the decomposition of the energy into “electric” (i.e., Chern–Simons) and “magnetic” (i.e., proportional to the µ-term) contributions. In general, proving the existence of limits for the Chern–Simons functional assuming only the finiteness of the energy of the connection ∇ A is a very hard analytical problem,
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cf. [28,35]. Therefore our rapid decay condition is a simple and natural condition which allows us to explicitly compute the limit of the Chern–Simons functional in our situation. Another example illustrates that the weak holonomy condition is also essential in Theorem 2.2. Consider R4 , equipped with the Taub–NUT metric. This geometry admits a smooth L 2 harmonic 2-form which can be identified with the curvature FB of a selfdual, rapidly decaying U(1) connection ∇ B as in [17]; hence ∇ B ⊕ ∇ B−1 is a smooth, self-dual, rapidly decaying, reducible SU(2) connection. We know that H 2 (R4 , Z) = 0, hence ∇ B lives on a trivial line bundle, consequently it can be rescaled by an arbitrary constant like B → cB without destroying its self-duality and finite energy. But the smooth, self-dual family ∇cB has continuous energy proportional to c2 . This strange phenomenon also appears over the multi-Taub–NUT spaces, although they are no more topologically trivial, cf. Sect. 4 for more details. From our holonomy viewpoint, this anomaly can be understood as follows. Let i : Uε∗ ⊂ W be an elementary neighbourhood as in (5) with the induced map i ∗ : π1 (Uε∗ ) → π1 (W ). On the one hand we have π1 (Uε∗ ) ∼ = Z as usual. On the other hand for the Taub– NUT space the asymptotical topology is W ∼ = 1, consequently = S 3 ×R+ , hence π1 (W ) ∼ i ∗ has a non-trivial kernel. However for a generic c the connection ∇cB |Uε∗ has non-trivial local holonomy m = 0, hence it does not obey the weak holonomy condition (11); therefore Theorem 2.2 fails in this case. The flat R3 × S 1 space has contrary behaviour to the multi-Taub–NUT geometries. In this case we find W ∼ = S 2 × S 1 × R+ for the end, consequently π1 (W ) ∼ = Z and the map i ∗ is an obvious isomorphism. Hence the weak holonomy condition is always obeyed. The character variety of the boundary is χ (S 2 × S 1 ) ∼ = [0, 1), hence connected. Referring to the second part of Lemma 2.1 we conclude then that the energy of any smooth, self-dual, rapidly decaying connection over the flat R3 × S 1 must be a nonnegative integer in accordance with the known explicit solutions [5]. The case of the Schwarzschild space is similar, cf. Sect. 4. These observations lead us to a more transparent form of Theorem 2.2 by replacing the weak holonomy condition with a simple, sufficient topological criterion, which amounts to a straightforward re-formulation of Definition 2.1. Theorem 2.3. Let (M, g) be an ALF space with an end W ∼ = N × R+ as before and, referring to the fibration (1), assume N is an arbitrary circle bundle over B∞ ∼ = S 2 , RP 2 , 2 2 or is a trivial circle bundle over S or RP . Then if E is the (trivial) SU(2) vector bundle over M with a smooth, finite energy connection ∇ A , then it satisfies the weak holonomy condition (11). Moreover if ∇ A is self-dual and decays rapidly as in (16), then its energy is congruent to one of the Chern–Simons invariants of the boundary N . In addition if the character variety χ (N ) is connected, then the energy of any smooth, self-dual, rapidly decaying connection must be a non-negative integer. Proof. Consider an elementary neighbourhood i : Uε∗ ⊂ W as in (5) and the induced map i ∗ : π1 (Uε∗ ) → π1 (W ). If Ker i ∗ = {0}, then ∇ A obeys (11). However one sees that π1 (Uε∗ ) ∼ = π1 (F) and π1 (W ) ∼ = π1 (N ), hence the map i ∗ fits well into the homotopy exact sequence i∗
. . . −→ π2 (B∞ ) −→ π1 (F) −→ π1 (N ) −→ π1 (B∞ ) −→ . . . of the fibration (1). This segment shows that Ker i ∗ = {0} if and only if N is either a non-trivial circle bundle over S 2 that is, N ∼ = S 3 /Zs a lens space of type L(s, 1), or a non-trivial circle bundle over RP 2 .
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G. Etesi, M. Jardim
The last part is clear via the second part of Lemma 2.1.
Finally we investigate the strong holonomy condition (12). As an important corollary of our construction we find Theorem 2.4. (Sibner–Sibner, 1992; Råde, 1994) Let (M, g) be an ALF space with an end W as before and E be the trivial SU(2) vector bundle over M with a smooth, finite energy connection ∇ A and associated locally flat connection ∇ on E|W . If and only if the strong holonomy condition (12) is satisfied then both ∇ A , as well as ∇ as a flat connection, extend smoothly over the whole X , the Hausel–Hunsicker–Mazzeo compactification of (M, g). That is, there exist bundles E˜ and E˜ 0 ∼ = X × C2 over X ∼ ˜ ˜ such that E| M = E and the connection ∇ A extends smoothly over E; in the same fashion E˜ 0 |W ∼ = E|W and ∇ extends smoothly as a flat connection over E˜ 0 . Proof. The restriction of the embedding M ⊂ X gives Uε∗ ⊂ Uε and this later space is contractible. Consequently if i : Uε∗ ⊂ X is the embedding, then for the induced map i ∗ : π1 (Uε∗ ) → π1 (X ) we always have Ker i ∗ = π1 (Uε∗ ), hence the connections ∇ A and ∇ extend smoothly over X via Theorem 2.1 if and only if the strong holonomy condition (12) holds. In particular the extension of ∇ is a flat connection. Remark. If a finite energy self-dual connection satisfies the strong holonomy condition, then its energy is integer via Theorem 2.4 regardless of its curvature decay. Consequently these instantons again have discrete energy spectrum. We may then ask ourselves about the relationship between the strong holonomy condition on the one hand and the weak holonomy condition imposed together with the rapid decay condition on the other hand. 3. The Moduli Space In this section we are going to prove that the moduli spaces of framed SU(2) instantons over ALF manifolds form smooth, finite dimensional manifolds, whenever non-empty. The argument will go along the by now familiar lines consisting of three steps: (i) Compute the dimension of the space of infinitesimal deformations of an irreducible, rapidly decaying self-dual connection, satisfying the weak holonomy condition, using a variant of the Gromov–Lawson relative index theorem [20] and a vanishing theorem; (ii) Use the Banach space inverse and implicit function theorems to integrate the infinitesimal deformations and obtain a local moduli space; (iii) Show that local moduli spaces give local coordinates on the global moduli space and that this global space is a Hausdorff manifold. We will carry out the calculations in detail for step (i) while just sketch (ii) and (iii) and refer the reader to the classical paper [2]. Let (M, g) be an ALF space with a single end W as in Sect. 2. Consider a trivial SU(2) bundle E over M with a smooth, irreducible, self-dual, finite energy connection ∇ A on it. By smoothness we mean that the connection 1-form is smooth in any smooth trivialization of E. In addition suppose ∇ A satisfies the weak holonomy condition (11) as well as decays rapidly in the sense of (16). Then by Theorem 2.2 its energy is determined by a Chern–Simons invariant. We will assume that this energy e := FA 2L 2 (M,g) is fixed. Consider the associated flat connection ∇ with holonomy m ∈ [0, 1) as in Theorem 2.2. Extend ∇ over the whole E and continue to denote it by ∇ . Take the smooth gauge (13) on the neck. Since E is trivial, we can extend this gauge smoothly over the whole M and can write ∇ A = d + A and ∇ = d + for some smooth connection 1-forms A and well defined over the whole M. We also fix this gauge once and for
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all in our forthcoming calculations. In particular the asymptotics of ∇ A is also fixed and is given by . The connection ∇ , the usual Killing form on EndE and the rescaled metric g˜ are used to construct Sobolev spaces over various subsets of X with respect to the norm (8). Both the energy e and the asymptotics are preserved under gauge transformations which tend to the identity with vanishing first derivatives everywhere in infinity. We suppose AutE ⊂ EndE and define the gauge group to be the completion G E := {γ − 1 ∈ C0∞ (EndE) | γ − 1 L 2
j+2, (M)
< ∞, γ ∈ C ∞ (AutE) a.e.},
and the gauge equivalence class of ∇ A under G E is denoted by [∇ A ]. Then we are seeking the virtual dimension of the framed moduli space M(e, ) of all such connections given up to these specified gauge transformations. Consider the usual deformation complex ∇A
∇− A
L 2j+2, (0 M ⊗ EndE) −→ L 2j+1, (1 M ⊗ EndE) −→ L 2j, (− M ⊗ EndE), where ∇ − A refers to the induced connection composed with the projection onto the antiself-dual side. Our first step is to check that the Betti numbers h 0 , h 1 , h − of this complex, given by h 0 = dim H 0 (EndE), etc., are finite. We therefore introduce an elliptic operator δ ∗A : L 2j+1, (1 M ⊗ EndE) −→ L 2j, ((0 M ⊕ − M) ⊗ EndE),
(18)
the so-called deformation operator δ ∗A := ∇ ∗A ⊕ ∇ − A , which is a conformally invariant first order elliptic operator over (M, g), ˜ hence (M, g). Here ∇ ∗A is the formal L 2 adjoint of ∇ A . We will demonstrate that δ ∗A is Fredholm, so it follows that h 1 = dim Ker δ ∗A and h 0 + h − = dim Coker δ ∗A are finite. Pick up the trivial flat SU(2) connection ∇ on E; it satisfies the strong holonomy condition (12), hence it extends smoothly over X to an operator ∇˜ by Theorem 2.4. Using the regularized metric g˜ ε of (7) it gives rise to an induced elliptic operator over the compact space (X, g˜ ε ) as ∗ ∞ 1 ∞ 0 − ˜ ˜ δε, ˜ : C 0 ( X ⊗ End E 0 ) −→ C 0 (( X ⊕ X ) ⊗ End E 0 ).
Consequently δ ∗ ˜ hence its restrictions δ ∗ ˜ | M and δ ∗ ˜ |W are Fredholm with respect to ε, ε, ε, any Sobolev completion. We construct a particular completion as follows. The smooth extended connection ∇ on E can be extended further over X to a connection ∇˜ such that it gives rise to an SU(2) Sobolev connection on the trivial bundle E˜ 0 . Consider a completion like ∗ 2 1 2 0 − ˜ ˜ δε, ˜ : L j+1,˜ ( X ⊗ End E 0 ) −→ L j,˜ (( X ⊕ X ) ⊗ End E 0 ).
(19)
The operators in (18) and (19) give rise to restrictions. The self-dual connection yields δ ∗A |W : L 2j+1,˜ (1 W ⊗ EndE|W ) −→ L 2j,˜ ((0 W ⊕ − W ) ⊗ EndE|W ), while the trivial connection gives ∗ 2 1 2 0 − ˜ ˜ δε, ˜ |W : L j+1,˜ ( W ⊗ End E 0 |W ) −→ L j,˜ (( W ⊕ W ) ⊗ End E 0 |W ).
Notice that these operators actually act on isomorphic Sobolev spaces, consequently comparing them makes sense. Working in these Sobolev spaces we claim that
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Lemma 3.1. The deformation operator δ ∗A of (18) with j = 0, 1 satisfies the operator norm inequality
∗ j 5 ∗ ≤ 3 · 2 c , (20) )|
F
+ c mε
(δ ∗A − δε, 2 V 3 A 4 L (Vε ) ˜ ε where m ∈ [0, 1) is the holonomy of ∇ A and c4 = c4 (g| ˜ Vε , γα , dγα ) > 0 is a constant depending only on the metric and the gauge transformations used in (14). Consequently δ ∗A is a Fredholom operator over (M, g). Proof. Consider the restriction of the operators constructed above to the neck W ∼ = Vε∗ 2 1 and calculate the operator norm of their difference with an a ∈ L j+1, ( M ⊗ EndE) as follows: ∗ ∗
(δ ∗A − δε, ˜ )|Vε = sup
(δ ∗A − δ ∗ ˜ )a L 2
∗ j, (Vε )
ε,
a L 2
a=0
∗ j+1, (Vε )
.
By assumption ∇ A satisfies the weak holonomy condition (11), hence the connection ∇ is flat on E|Vε∗ and it determines a deformation operator δ ∗˜ |Vε∗ over (Vε∗ , g| ˜ Vε∗ ). We use the triangle inequality: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
(δ ∗A − δε, ˜ )|Vε ≤ (δ A − δ˜ )|Vε + (δ˜ − δε, ˜ )|Vε .
Referring to the global gauge (13) and the metric g˜ we have δ ∗A a = (δ + d− )a + A∗ a + (A ∧ a + a ∧ A)− , and the same for δ ∗˜ , consequently
(δ ∗A − δ∗˜ )|Vε∗ = (A Vε∗ − Vε∗ )∗ + ((A Vε∗ − Vε∗ ) ∧ · + · ∧ (A Vε∗ − Vε∗ ))− . Taking j = 0, 1 and combining this with (15) we find an estimate for the first term like
(δ ∗A − δ∗˜ )|Vε∗ ≤ 3 · 2 j A Vε∗ − Vε∗ L 2
∗ j, (Vε )
≤ 3 · 2 j c3 FA L 2 (Vε ) .
Regarding the second term, the trivial flat connection ∇ on E|Vε∗ satisfies the strong holonomy condition (12). In the gauge (13) we use, we may suppose simply |Vε∗ = 0, consequently neither data from ∇˜ nor the perturbed metric g˜ ε influence this term. Now take a partition of the end into elementary neighbourhoods (5) and use the associated simple constant gauges (10) then ∗ j ∗ ≤ 3 · 2
(δ∗˜ − δε, )|
γα−1 m γα L 2 (U ∗ ) ≤ 3 · 2 j c4 mε5 V ˜ ε j,
α
j
ε,α
with some constant c4 = c4 (g| ˜ Vε∗ , γα , . . . , ∇ γα ) > 0 via (6). Putting these together we get (20). Taking into account that the right-hand side of (20) is arbitrarily small we conclude that δ ∗A |Vε∗ is Fredholm because so is δ ∗ ˜ |Vε∗ and Fredholmness is an open property. ε, Clearly, δ ∗A | M\Vε∗ is also Fredholm, because it is an elliptic operator over a compact manifold. Therefore glueing the parametrices of these operators together, one constructs a parametrix for δ ∗A over the whole M (see [24] for an analogous construction). This shows that δ ∗A is Fredholm over the whole (M, g) ˜ hence (M, g) as claimed.
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We assert that T[∇ A ] M(e, ) ∼ = Ker δ ∗A , consequently h 1 = dim M(e, ). Indeed, the anti-self-dual part of the curvature of a perturbed connection ∇ A+a is given by − 2 − 1 FA+a = ∇− A a + (a ∧ a) with the perturbation a ∈ L j+1, ( M ⊗ EndE). Therefore if ∗ a ∈ Ker δ A , then both self-duality and the energy of ∇ A+a are preserved infinitesimally i.e., in first order. The perturbation vanishes everywhere in infinity hence the asymptotics given by is also unchanged; in particular ∇ A+a continues to obey the weak holonomy condition. Incidentally, we note however that locally, hence also globally, some care is needed when one perturbs a connection in our moduli spaces. For a perturbation with a ∈ L 2j+1, (1 M ⊗ EndE) the asymptotics and in particular the weak holonomy condition are obeyed as we have seen. But concerning the energy, by repeating the calculation of Sect. 2 again, we obtain 1 ε
ε |τ∂ Mρ (Aρ + aρ ) − τ∂ Mρ (ρ )|2 dρ 0
1 ≤ ε ≤
ε
2 |τ∂ Mρ (Aρ + aρ ) − τ∂ Mρ (Aρ )| + |τ∂ Mρ (Aρ ) − τ∂ Mρ (ρ )| dρ
0
c1 c3 c1 √ a L 2 (Vε ) + √ FA L 2 (Vε ) ε ε
2 .
If the last line tends to zero as ε → 0, then the perturbed connection has the same limit as in (17). Consequently, we find that the energy is also unchanged. Since the original connection decays rapidly in the sense of (16) if the perturbations also decay rapidly, 1 i.e., lim ε− 2 a L 2 (Vε ) = 0, then the energy is preseved by local (i.e. small but finite) ε→0
perturbations as well. It is therefore convenient to introduce weighted Sobolev spaces with weight δ = 21 and to say that a and ∇ A decay rapidly if a ∈ L 21 (1 M ⊗ EndE) and FA ∈ L 21
2 , j,
2 , j+1,
(2 M ⊗ EndE), respectively. These are gauge invariant conditions and for ∇ A
with j = 0 it is equivalent to (16). In this framework the rough estimate
FA+a L 2
(M) 1 2 , j,
= FA + ∇ A a + a ∧ a L 2
(M) 1 2 , j,
≤ FA L 2
(M) 1 2 , j,
+ ∇ A a L 2
(M) 1 2 , j,
+ c5 a 2L 2
(M) 1 2 , j+1,
with some constant c5 = c5 (g) ˜ > 0 implies that ∇ A+a also decays rapidly. Let A E denote the affine space of rapidly decaying SU(2) connections on E as well as F E− the vector space of the anti-self-dual parts of their curvatures. Then take a complex of punctured spaces, the global version of the deformation complex above: fA
− A
(G E , 1) −→ (A E , ∇ A ) −→ (F E− , 0). We have (A E , ∇ A ) ∼ = L 21
2 , j+1,
(1 M ⊗ EndE) and (F E− , 0) ∼ = L 21
2 , j,
(− M ⊗ EndE).
The global gauge fixing map is defined as f A (γ ) := γ −1 ∇ A γ − ∇ A , while − A (a) :=
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− − ∼ FA+a = ∇− A a + (a ∧ a) . If ∇ A is irreducible then Ker f A = 1. One easily shows that if both γ − 1 and a are pointwise small, then f A (γ ) = ∇ A (log γ ) and its formal L 2 adjoint satisfying
( f A (γ ), a) L 2 (A E ,∇ A ) = (γ , f A∗ (a)) L 2 (G E ,1) looks like f A∗ (a) = exp(∇ ∗A a) if
(γ , β) L 2 (G E ,1) := −
tr(log γ log β) ∗g˜ 1, M
defined in a neighbourhood of 1 ∈ G E . Hence we have a model for O[∇ A ] ⊂ M(e, ), the vicinity of [∇ A ], as follows: O[∇ A ] ∼ = Ker( f A∗ × − A ) ⊂ (A E , ∇ A ). ∗ − The derivative of f A∗ × − A at a is δ A + (a ∧ · + · ∧ a) , which is a Fredholm operator, hence O[∇ A ] is smooth and finite dimensional. These local models match together and prove that the moduli space is indeed a smooth manifold of dimension h 1 . We return to the calculation of h 1 . We will calculate the index of δ ∗A in (18) by referring to a relative index theorem. This provides us with the alternating sum −h 0 + h 1 − h − and then we show that h 0 = h − = 0 via a vanishing theorem. First we proceed to the calculation of the index of δ ∗A . This will be carried out by a variant of the Gromov–Lawson relative index theorem [20], which we will now explain. First, let us introduce some notation. For any elliptic Fredholm operator P, let Indexa P denote its analytical index, i.e., Indexa P = dim Ker P −dim Coker P. If this P is defined over a compact manifold, Indexa P is given by a topological formula as in the Atiyah– Singer index theorem, which we denote by Indext P, the topological index of P. The following theorem will be proved in the Appendix (cf. [23]):
Theorem 3.1. Let (M, g) be a complete Riemannian manifold, and let X be some smooth compactification of M. Let also D1 : L 2j+1,1 (F1 ) → L 2j, (F1 ) and D0 : 1
L 2j+1,0 (F0 ) → L 2j, (F0 ) be two first order, elliptic Fredholm operators defined on com0
plex vector bundles F1 , F1 and F0 , F0 over M with fixed Sobolev connections ∇1 , ∇1 and ∇0 , ∇0 , respectively. Assume that given κ > 0, there is a compact subset K ⊂ M such that, for W = M\K , the following hold: (i) There are bundle isomorphisms φ : F1 |W ∼ = F0 |W and φ : F1 |W ∼ = F0 |W ; (ii) The operators asymptotically agree, that is, in some operator norm (D1 −D0 )|W < κ. If arbitrary elliptic extensions D˜ 1 and D˜ 0 of D1 and D0 to X exist, then we have Indexa D1 − Indexa D0 = Indext D˜ 1 − Indext D˜ 0 for the difference of the analytical indices. In the case at hand, F1 = F0 = 1 M ⊗ EndE ⊗ C and F1 = F0 = (0 M ⊕ − M) ⊗ EndE ⊗C; then the complexification of δ ∗A , which by Lemma 3.1 is a Fredholm operator, plays the role of D1 and via (20) it asymptotically agrees with the complexified δ ∗ ˜ | M , ε, also Fredholm by construction, which replaces D0 .
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To find the operators which correspond to D˜ 1 and D˜ 0 , we proceed as follows. Remember that in general ∇ A does not extend over X (cf. Theorem 2.4). Let E˜ be the unique vec˜ Vε ∼ ˜ X \V2ε ∼ tor bundle over X constructed as follows. Since E| = E| M\V2ε∗ and E| = Vε ×C2 , this bundle is uniquely determined by the glued connection ∇ A˜ = (1 − f ε )∇ A + f ε ∇˜ ˜ where f ε is taken from (7). We can construct the associated operator δ ∗ over on E, ε, A˜ End E˜ with respect to the metric (7) whose complexification will correspond to D˜ 1 ; the operator D˜ 0 is given by the complexification of the operator δ ∗ ˜ on the trivial bundle ε, End E˜ 0 , which we have already constructed. The right-hand side of the relative index formula in Theorem 3.1 is given by ∗ ∗ 1 − 1 − Indext (δε, ) − Indext (δε, ˜ ) = 8k − 3(1 − b (X ) + b (X )) + 3(1 − b (X ) + b (X )) = 8k, A˜
˜ Notice where k = FA˜ 2L 2 (X ) is the second Chern number of the extended bundle E. that this number might be different from the energy e = FA 2L 2 (M,g) of the original connection. We only know a priori that k ≤ e. However we claim that
Lemma 3.2. Using the notation of Theorem 2.2 and, in the same fashion, if ∇ = d+W on E|W , then letting ∞ := lim W | N ×{r } , in any smooth gauge r →∞
k = e + τ N (∞ ) − τ N (∞ )
(21)
holds. Notice that this expression is gauge invariant. Remark. Of course in any practical application it is worth taking the gauge in which simply τ N (∞ ) = 0. Proof. Using the gauge (13) for instance and applying the Chern–Simons theorem for the restricted energies to Mε , we find (k − e)| Mε = τ∂ Mε ( A˜ ε ) − τ∂ Mε (Aε ) + 2 F −˜ 2L 2 (M ) . A
ε
Therefore, since F −˜ L 2 (Mε ) = F −˜ L 2 (V2ε ) and the Chern–Simons invariants converge A A as in (17) by the rapid decay assumption, taking the limit one obtains k − e = τ N (0 ) − τ N (0 ) + 2 lim F −˜ 2L 2 (V ε→0
= τ N (∞ ) − τ N (∞ ) + 2
A
2ε )
lim F −˜ 2L 2 V ,g| . A 2R V2R R→∞
Consequently we have to demonstrate that lim F −˜ L 2 (V2ε ) = 0. There is a decompoε→0
A
sition of the glued curvature like FA˜ = + ϕ with := (1 − f ε )FA + f ε F , and a perturbation term as follows: ϕ := −d f ε ∧ (A V2ε∗ − V2ε∗ ) − f ε (1 − f ε )(A V2ε∗ − V2ε∗ ) ∧ (A V2ε∗ − V2ε∗ ). This shows that F −˜ = ϕ − , consequently it is compactly supported in V2ε \Vε . Moreover A there is a constant c6 = c6 (d f ε , g| ˜ V2ε ) > 0, independent of ε, such that d f ε L 2 (V2ε ) ≤ 1,
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c6 ; as well as | f ε (1− f ε )| ≤ we tame ϕ − like
1 4
therefore, recalling the pattern of the proof of Lemma 3.1,
c5
ϕ − L 2 (V2ε ) ≤ ϕ L 2 (V2ε ) ≤ c5 c6 A V2ε∗ − V2ε∗ L 2 (V ∗ ) + A V2ε∗ − V2ε∗ 2L 2 (V ∗ ) ≤ 1, 2ε 1, 2ε 4
c
2 5 ≤ c5 c6 c3 FA L 2 (V2ε ) + c4 m(2ε)5 + c3 FA L 2 (V2ε ) + c4 m(2ε)5 . 4 However we know that this last line can be kept as small as one likes providing the result. Regarding the left-hand side of Theorem 3.1, on the one hand we already know that Indexa δ ∗A = −h 0 + h 1 − h − = h 1 = dim M(e, ) by the promised vanishing theorem. On the other hand, since EndE ⊗ C ∼ = M × C3 , we find
− ∗ 0 1 Indexa (δε, | ) = −3 b (M, g ˜ | ) − b (M, g ˜ | ) + b (M, g ˜ | ) , M ε M ε M ε M 2 2 2 ˜ L L L (M, g˜ ε | M ) is the dimension of where biL 2 (M, g˜ ε | M ) is the i th L 2 Betti number and b− L2 the space of anti-self-dual finite energy 2-forms on the rescaled-regularized manifold (M, g˜ ε | M ), i.e. this index is the truncated L 2 Euler characteristic of (M, g˜ ε | M ). We wish to cast this subtle invariant into a more explicit form at the expense of imposing a further but natural assumption on the spaces we work with. Lemma 3.3. Let (M, g) be a Ricci flat ALF space, and let X be its compactification with induced orientation. Then one has Indexa (δ ∗ ˜ | M ) = −3b− (X ), where b− (X ) denotes ε, the rank of the negative definite part of the topological intersection form of X . Proof. Exploiting the stability of the index against small perturbations as well as the conformal invariance of the operator δ ∗ ˜ | M , without changing the index we can replace ε, the metric g˜ ε with the original ALF metric g. Consequently we can write
− ∗ 0 1 | ) = −3 b (M, g) − b (M, g) + b (M, g) Indexa (δε, M 2 2 2 ˜ L L L for the index we are seeking. Remember that this metric is complete and asymptotically looks like (2). This implies that (M, g) has infinite volume, hence a theorem of Yau [36] yields that b0L 2 (M, g) = 0. Moreover if we assume the curvature of (M, g) not only satisfies (3) but is furthermore Ricci flat, then a result of Dodziuk [13] shows that in addition b1L 2 (M, g) = 0. Concer(M, g) we use the result of [15] (based on [21, Corollary 7]) to observe that ning b− L2 any finite energy anti-self-dual 2-form over (M, g) extends smoothly as a (formally) anti-self-dual 2-form over (X, g) ˜ showing that b− (M, g) = b− (X ) as desired. L2 Finally we prove the vanishing of the numbers h 0 and h − . The proof is a combination of the standard method [2] and a Witten-type vanishing result ([31, Lemma 4.3]). In the adjoint of the elliptic complex (18) we find Indexa δ A = dim Ker δ A − dim Coker δ A = h 0 − h 1 + h − , hence proving Ker δ A = {0} is equivalent to h 0 = h − = 0.
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Lemma 3.4. Assume (M, g) is an ALF space as defined in Sect. 2. If ∇ A is irreducible and ψ ∈ L 2j, ((0 M ⊕ − M) ⊗ EndE) satisfies δ A ψ = 0, then ψ = 0. Proof. Taking into account that Ker δ A = Ker(δ ∗A δ A ) consists of smooth functions by elliptic regularity and is conformally invariant, we can use the usual Weitzenböck formula with respect to the original ALF metric g as follows: sg : C0∞ ((0 M ⊕ − M) ⊗ EndE) 3 −→ C0∞ ((0 M ⊕ − M) ⊗ EndE).
δ ∗A δ A = ∇ ∗A ∇ A + Wg− +
In this formula ∇ A : C0∞ ((0 M ⊕ − M) ⊗ EndE) → C0∞ ((1 M ⊕ 3 M) ⊗ EndE) is s the induced connection while Wg− + 3g acts only on the − M summand as a symmetric, linear, algebraic map. This implies that if δ A ψ = 0 such that ψ ∈ L 2j, (0 M ⊗ EndE) s ∼ only or Wg− + 3g = 0, then ∇ A ψ = 0. If ∇ A is irreducible, then both 0 M ⊗EndE ⊗C = 0 − − 2 − − ∼ S ⊗EndE and M ⊗EndE ⊗C = S ⊗EndE are irreducible SU(2) ×SU(2) bundles, hence ψ = 0 follows. Concerning the generic case, then as before, we find h 0 = 0 by irreducibility, conses quently we can assume ψ ∈ L 2j, (− M ⊗ EndE) only and Wg− + 3g = 0. Since Ker δ A consists of smooth functions it follows that if δ A ψ = 0, then ψ vanishes everywhere at infinity. Let · , · be a pointwise SU(2)-invariant scalar product on − M ⊗ EndE and set 1 s |ψ| := ψ, ψ 2 . Assume δ A ψ = 0 but ψ = 0. Then ∇ ∗A ∇ A ψ, ψ = −(Wg− + 3g )ψ, ψ by the Weitzenböck formula above. Combining this with the pointwise expression ∇ ∗A ∇ A ψ, ψ = |∇ A ψ|2 + 21 |ψ|2 and applying 21 |ψ|2 = |ψ||ψ| + |d|ψ||2 as well as Kato’s inequality |d|ψ|| ≤ |∇ A ψ|, valid away from the zero set of ψ, we obtain 2
|d|ψ||2 |d|ψ||2
− sg
|ψ|
− sg
= Wg + − log |ψ| − ≤ Wg + − , 2 |ψ| 3 |ψ| 3 |ψ|2
that is,
sg
3 |d log |ψ||2 + log |ψ| ≤ Wg− + . 3 ∼ N × R+ be a naturally parameterized ray running toward infinity Let λ : R+ → W = and let f (r ) := log |ψ(λ(r ))|. Observe that f is negative in the vicinity of a zero of ψ or for large r ’s. The last inequality then asymptotically cuts down along λ to 3( f (r ))2 −
2 c7 f (r ) − f (r ) ≤ 3 , r r
using the expansion of the Laplacian for a metric like (2) and referring to the curvature decay (3) providing a constant c7 = c7 (g) ≥ 0. This inequality yields −c7 r −2 ≤ (r f (r )) . Integrating it we find c7r −1 + a ≤ (r f (r )) ≤ r f (r ), showing c7 r −2 + ar −1 ≤ f (r ) with some constant a, hence there is a constant c8 ≥ 0 such that c8 := −| inf f (r )|. Integrating c7 r −1 + a ≤ (r f (r )) again we also obtain c7 (log r )r −1 + a + r ∈R+
br −1 ≤ f (r ) ≤ 0, for some real constants a, b.
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Let x0 ∈ M be such that ψ(x0 ) = 0; then by smoothness there is another point x ∈ M with this property such that |x0 | < |x|. Integrating again the inequality −c7r −2 ≤ (r f (r )) twice from x0 to x along the ray λ connecting them we finally get
|ψ(x0 )| ≤ |ψ(x)| exp (c7 |x0 |−1 + c8 |x0 |)(1 − |x0 ||x|−1 ) . Therefore either letting x be a zero of ψ along λ or, if no such point exists, taking the limit |x| → ∞, we find that ψ(x0 ) = 0 as desired. Finally, putting all of our findings together, we have arrived at the following theorem: Theorem 3.2. Let (M, g) be an ALF space with an end W ∼ = N × R+ as before. Assume furthermore that the metric is Ricci flat. Consider a rank 2 complex SU(2) vector bundle E over M, necessarily trivial, and denote by M(e, ) the framed moduli space of smooth, irreducible, self-dual SU(2) connections on E satisfying the weak holonomy condition (11) and decaying rapidly in the sense of (16) such that their energy e < ∞ is fixed and are asymptotic to a fixed smooth flat connection ∇ on E|W . Then M(e, ) is either empty or a manifold of dimension dim M(e, ) = 8 (e + τ N (∞ ) − τ N (∞ )) − 3b− (X ), where ∇ is the trivial flat connection on E|W and τ N is the Chern–Simons functional of the boundary, while X is the Hausel–Hunsicker–Mazzeo compactification of M with induced orientation. Remark. Of course, we get a dimension formula for anti-instantons by replacing b− (X ) with b+ (X ). Notice that our moduli spaces contain framings, since we have a fixed flat connection and a gauge at infinity. The virtual dimension of the moduli space of unframed instantons is given by dim M(e, ) − 3, which is is the number of effective free parameters. A dimension formula in the presence of a magnetic term µ mentioned in Sect. 2 is also easy to work out because in this case (21) is simply replaced with k = e+τ N (∞ )− τ N (∞ )−µ and then this should be inserted into the dimension formula of Theorem 3.2. Note also that our moduli spaces are naturally endowed with weighted L 2 metrics. An interesting problem is to investigate the properties of these metrics.
4. Case Studies In this section we present some applications of Theorem 3.2. We will consider rapidly decaying instantons over the flat R3 × S 1 , the multi-Taub–NUT geometries and the Riemannian Schwarzschild space. We also have the aim to enumerate the known Yang– Mills instantons over non-trivial ALF geometries. However we acknowledge that our list is surely incomplete, cf. e.g. [1]. The flat space R3 × S 1 . This is the simplest ALF space, hence instanton (or also called caloron i.e., instanton at finite temperature) theory over this space is well-known (cf. [4,5]). We claim that
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Theorem 4.1. Take M = R3 × S 1 with a fixed orientation and put the natural flat metric onto it. Let ∇ A be a smooth, self-dual, rapidly decaying SU(2) connection on a fixed rank 2 complex vector bundle E over M. Then ∇ A satisfies the weak holonomy condition and has non-negative integer energy. Let M(e, ) denote the framed moduli space of these connections which are moreover irreducible as in Theorem 3.2. Then dim M(e, ) = 8e and M(e, ) is not empty for all e ∈ N. Proof. Since the metric is flat, the conditions of Theorem 3.2 are satisfied. Furthermore, in the case at hand the asymptotical topology of the space is W ∼ = S 2 × S 1 × R+ , conse2 × S 1 , hence its character variety χ (S 2 × S 1 ) ∼ [0, 1) is connected. quently N ∼ S = = Theorem 2.3 therefore guarantees that any smooth, self-dual, rapidly decaying connection has non-negative integer energy. In the gauge in which τ S 2 ×S 1 (∞ ) = 0, we also get τ S 2 ×S 1 (∞ ) = 0. Moreover we find that X ∼ = S 4 for the Hausel–Hunsicker– − + Mazzeo compactification [15] yielding b (X ) = b (X ) = 0; putting these data into the dimension formula in Theorem 3.2 we get the dimension as stated, in agreement with [4]. The moduli spaces M(e, ) are not empty for all e ∈ N; explicit solutions with arbitrary energy were constructed via a modified ADHM construction in [4,5]. The Multi-Taub–NUT (or Ak ALF, or ALF Gibbons–Hawking) spaces. The underlying manifold MV topologically can be understood as follows. There is a circle action on MV with s distinct fixed points p1 , . . . , ps ∈ MV , called NUTs. The quotient is R3 and we denote the images of the fixed points also by p1 , . . . , ps ∈ R3 . Then UV := MV \{ p1 , . . . , ps } is fibered over Z V := R3 \{ p1 , . . . , ps } with S 1 fibers. The degree of this circle bundle around each point pi is one. The metric gV on UV looks like (cf. e.g. [14, p. 363]) ds 2 = V (dx 2 + dy 2 + dz 2 ) +
1 (dτ + α)2 , V
where τ ∈ (0, 8π m] parameterizes the circles and x = (x, y, z) ∈ R3 ; the smooth function V : Z V → R and the 1-form α ∈ C ∞ (1 Z V ) are defined as follows: V (x, τ ) = V (x) = 1 +
s i=1
2m , |x − pi |
dα = ∗3 dV.
Here m > 0 is a fixed constant and ∗3 refers to the Hodge-operation with respect to the flat metric on R3 . We can see that the metric is independent of τ , hence we have a Killing field on (MV , gV ). This Killing field provides the above mentioned U (1)-action. Furthermore it is possible to show that, despite the apparent singularities in the NUTs, these metrics extend analytically over the whole MV providing an ALF, hyper–Kähler manifold. We also notice that the Killing field makes it possible to write a particular Kähler-form in the hyper-Kähler family as ω = dβ, where β is a 1-form of linear growth. Then we can assert that
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Theorem 4.2. Let (MV , gV ) be a multi-Taub–NUT space with s NUTs and orientation induced by any of the complex structures in the hyper–Kähler family. Consider the framed moduli space M(e, ) of smooth, irreducible, rapidly decaying anti-self-dual connections satisfying the weak holonomy condition on a fixed rank 2 complex SU(2) vector bundle E as in Theorem 3.2. Then M(e, ) is either empty or a manifold of dimension dim M(e, ) = 8 e + τ L(s,1) (∞ ) − τ L(s,1) (∞ ) , where L(s, 1) is the lens space representing the boundary of MV . The moduli spaces are surely not empty for τ L(s,1) (∞ ) = τ L(s,1) (∞ ) = 0 and e = 1, . . . , s. Proof. This space is non-flat, nevertheless its curvature satisfies the cubic curvature decay (3), hence it is an ALF space in our sense. Since it is moreover hyper–Kähler, the conditions of Theorem 3.2 are satisfied. However this time the asymptotic topology is W ∼ = L(s, 1) × R+ , therefore N ∼ = L(s, 1) is a non-trivial circle bundle over S 2 ; consequently the weak holonomy condition (11) must be imposed. If the connection in addition decays rapidly as in (16) then its energy is determined by a Chern–Simons invariant via Theorem 2.2. The character variety of the boundary lens space, χ (L(s, 1)) is also non-connected if s > 1 and each connected component has a non-trivial fractional Chern–Simons invariant which is calculable (cf., e.g. [3,25]). By the result in [15] the compactified space X with its induced orientation is isomorphic to the connected sum 2 of s copies of CP ’s, therefore b+ (X ) = 0 and b− (X ) = s. Inserting these into the dimension formula of Theorem 3.2 for anti-self-dual connections we get the dimension. Concerning non-emptiness, since lacking a general ADHM-like construction, we may use a conformal rescaling method [17,18]. Take the natural orthonormal frame 1 ξ 0 = √ (dτ + α), V
ξ1 =
√
ξ2 =
V dx,
√
V dy,
ξ3 =
√
V dz
over UV and introduce the quaternion-valued 1-form ξ := ξ 0 +ξ 1 i +ξ 2 j+ξ 3 k. Moreover pick up the non-negative function f : UV → R+ defined as f (x) := λ0 +
s i=1
λi |x − pi |
with λ0 , λ1 , . . . , λs being real non-negative constants and also take the quaternion-valued 0-form d log f := −V
∂ log f ∂ log f ∂ log f ∂ log f + i+ j+ k ∂τ ∂x ∂y ∂z
f (notice that actually ∂ log = 0). Over UV ⊂ MV we have a gauge induced by the ∂τ above orthonormal frame on the positive spinor bundle + |UV . In this gauge consider ’t Hooft-like SU(2) connections ∇λ+0 ,...,λs |UV := d + A+λ0 ,...,λs ,UV on + |UV with
A+λ0 ,...,λs ,UV := Im
(d log f ) ξ . √ 2 V
It was demonstrated in [18] that these connections, parameterized by λ0 , λ1 , . . . , λs up to an overall scaling, extend over MV and provide smooth, rapidly decaying antiself-dual connections on + . They are irreducible if λ0 > 0, are non-gauge equivalent
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and have trivial holonomy at infinity; hence satisfy the weak holonomy condition (11) and in particular the corresponding Chern–Simons invariants vanish. Consequently their energies are always integers equal to e = n, where 0 ≤ n ≤ s is the number of non-zero λi ’s with i = 1, . . . , s. Therefore moduli spaces consisting of anti-instantons of these energies cannot be empty. Remark. It is reasonable to expect that the higher energy moduli spaces are also not empty. Furthermore, it is not clear whether fractionally charged, rapidly decaying irreducible instantons actually exist over multi-Taub–NUT or over more general ALF spaces at all. Consider the family ∇i+ defined by the function f i (x) = |x−λipi | . These are unital energy solutions which are reducible to U(1) (in fact these are the only reducible points in the family, cf. [18]). These provide as many as s = 1 + b2 (MV ) non-equivalent + reducible, rapidly decaying anti-self-dual solutions and if ∇i+ |UV = d + Ai,U , then the V + + connection ∇ |UV := d + Ai,UV decays rapidly, is reducible and non-topological; i
hence it admits arbitrary rescalings yielding the strange continuous energy solutions mentioned in Sect. 2. Of course these generically rescaled connections violate the weak holonomy condition. The Riemannian (or Euclidean) Schwarzschild space. The underlying space is M = S 2 × R2 . We have a particularly nice form of the metric g on a dense open subset (R2 \{0}) × S 2 ⊂ M of the Riemannian Schwarzschild manifold. It is convenient to use polar coordinates (r, τ ) on R2 \{0} in the range r ∈ (2m, ∞) and τ ∈ [0, 8π m), where m > 0 is a fixed constant. The metric then takes the form 2m 2m −1 2 ds 2 = 1 − dτ 2 + 1 − dr + r 2 d2 , r r where d2 is the line element of the round sphere. In spherical coordinates θ ∈ (0, π ) and ϕ ∈ [0, 2π ) it is d2 = dθ 2 + sin2 θ dϕ 2 on the open coordinate chart (S 2 \({S} ∪ {N })) ⊂ S 2 . Consequently the above metric takes the following form on the open, dense coordinate chart U := (R2 \{0}) × S 2 \({S} ∪ {N }) ⊂ M: 2m 2m −1 2 2 dτ + 1 − ds = 1 − dr + r 2 (dθ 2 + sin2 θ dϕ 2 ). r r 2
The metric can be extended analytically to the whole M as a complete Ricci flat metric, however this time W ± = 0. Nevertheless we obtain Theorem 4.3. Let (M, g) be the Riemannian Schwarzschild manifold with a fixed orientation. Let ∇ A be a smooth, rapidly decaying, self-dual SU(2) connection on a fixed rank 2 complex vector bundle E over M. Then ∇ A satisfies the weak holonomy condition and has non-negative integer energy. Let M(e, ) denote the framed moduli space of these connections which are moreover irreducible as in Theorem 3.2. Then it is either empty or a manifold of dimension dim M(e, ) = 8e − 3. The moduli space with e = 1 is surely non-empty.
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Proof. The metric is Ricci flat, moreover a direct calculation shows that both W ± satisfy the decay (3) hence Theorem 3.2 applies in this situation as well. Furthermore, the asymptotical topology and the character variety of the space is again W ∼ = S 2 × S 1 × R+ 2 1 ∼ and χ (S × S ) = [0, 1), consequently the energy of a rapidly decaying instanton is integer as in Theorem 4.1. We can again set the gauge in which all Chern–Simons invariants vanish and find moreover X ∼ = S 2 × S 2 yielding b− (X ) = b+ (X ) = 1; substituting these data into the dimension formula we get the desired result. Regarding non-emptiness, very little is known. The apparently different non-Abelian solutions found by Charap and Duff (cf. the 1-parameter family (I) in [6]) are in fact all gauge equivalent [34] and provide a single rapidly decaying self-dual connection which + on + | with is the positive chirality spin connection. It looks like ∇ + |U := d + AU U + AU
1 1 m
1 2m 2m dθ i + sin θ dϕj + cos θ dϕ − 2 dτ k. := 1− 1− 2 r 2 r 2 r
One can show that this connection extends smoothly over + as an SO(3) × U(1) invariant, irreducible, self-dual connection of unit energy, centered around the 2-sphere in the origin. Remark. Due to its resistance against deformations over three decades, it has been conjectured that this positive chirality spin connection is the only unit energy instanton over the Schwarzschild space (cf. e.g. [34]). However we can see now that in fact it admits a 2 parameter deformation. It would be interesting to find these solutions explicitly as well as construct higher energy irreducible solutions. In their paper Charap and Duff exhibit another family of SO(3) × U(1) invariant instantons ∇n of energies 2n 2 with n ∈ Z (cf. solutions of type (II) in [6]). However it was pointed out in [16] that these solutions are in fact reducible to U(1) and locally look like ∇n |U ± = d + An,U ± with 1 n (∓1 + cos θ )dϕ − dτ k An,U ± := 2 r over the charts U ± defined by removing the north or the south poles from S 2 respectively. They extend smoothly as slowly decaying reducible, self-dual connections over the bundles L n ⊕ L −1 n , where L n is a line bundle with c1 (L) = n. Hence they are topological in contrast to the above mentioned Abelian instantons over the multi-Taub–NUT space. This constrains them to have discrete energy spectrum despite their slow decay. It is known that these are the only reducible SU(2) instantons over the Riemannian Schwarzschild space [16,21]. 5. Appendix In this Appendix we shall prove Theorem 3.1; this proof is taken from [23], and follows closely the arguments of [20]. In the course of the proof we shall use the notation introduced in the bulk of the paper. The right-hand side of the index formula in Theorem 3.1 is called the relative topological index of the operators D0 and D1 : Indext (D1 , D0 ) := Indext D˜ 1 − Indext D˜ 0 .
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Notice that it can be computed in terms of the topology of the topological extensions of the bundles F j and F j to X , using the Atiyah–Singer index theorem. Furthermore, as we will see below, this quantity does not depend on how the operators D0 and D1 are extended to D˜ 0 and D˜ 1 (see Lemma 5.1 below). The first step in the proof of Theorem 3.1 is the construction of a new Fredholm operator D1 as follows. Let β1 and β2 be cut-off functions, respectively supported over K and W = M\K , and define D1 = β1 D1 β1 + β0 D0 β0 . Now, it is clear that D1 |W coincides with D0 |W . Furthermore, since (D1 − D1 )|W < κ with κ arbitrarily small, we know that Indexa D1 = Indexa D1 . Our strategy is to establish the index formula for the pair D1 and D0 . In order to simplify notation however, we will continue to denote by D1 and D0 a pair of elliptic Fredholm operators which coincide at infinity. Now recall that if D is any Fredholm operator over M, there is a bounded, elliptic pseudo-differential operator Q, called the parametrix of D, such that D Q = I − S and Q D = I − S , where S and S are compact smoothing operators, and I is the identity operator. Note that neither Q nor S and S are unique. In particular, there is a bounded operator G, called the Green’s operator for D, satisfying DG = I − H and G D = I − H , where H and H are finite rank projection operators; the image of H is Ker D and the image of H is Coker D. Let K H (x, y) be the Schwartzian kernel of the operator H . Its local trace function is defined by tr[H ](x) = K H (x, x); moreover, these are C ∞ functions [20]. If D is Fredholm, its (analytical) index is given by Indexa D = dim Ker D − dim Coker D = (22) tr[H ] − tr[H ] M
as it is well-known; recall that compact operators have smooth, square integrable kernels. Furthermore, if M is a closed manifold, we have [20] Indext D = tr[S] − tr[S ] . M
Let us now return to the situation set up above. Consider the parametrices and Green’s operators ( j = 0, 1), Dj Q j = I − Sj Dj G j = I − Hj (23) Q D = I − S G D = I − H . j
j
j
j
j
j
The strategy of proof is to express both sides of the index formula of Theorem 3.1 in terms of integrals, as in (22); for its left-hand side, the relative analytical index, we have Indexa (D1 , D0 ) := Indexa D1 − Indexa D0 tr[H1 ] − tr[H1 ] − tr[H0 ] − tr[H0 ] . = M
M
For the relative topological index, we have the following
(24)
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Lemma 5.1. Under the hypothesis of Theorem 3.1, we have that Indext (D1 , D0 ) = tr[S1 ] − tr[S1 ] − tr[S0 ] − tr[S0 ] . M
(25)
M
⊂ X the compactification of the end W ⊂ M. Extend D j ( j = 0, 1) Proof. Denote by W to operators D˜ j , both defined over the whole X . Then the parametrices Q˜ j of D˜ j are extensions of the parametrices Q j of D j , and the corresponding compact smoothing operators S˜ j and S˜ j are extensions of S j and S j . As explained above, we can assume that the operators D1 and D0 coincide at infinity. ˜ 0 | , and therefore This means that D1 |W D0 |W , hence also D˜ 1 |W D W S1 |W S0 |W and S1 |W S0 |W ; S˜1 |W S˜0 |W and S˜1 |W S˜0 |W . It follows that the operators S˜1 − S˜0 and S˜1 − S˜0 are supported on K = M\W = X \W furthermore, S˜1 − S˜0 = (S1 − S0 )| K and S˜1 − S˜0 = (S1 − S0 )| K . It follows that Indext (D1 , D0 ) = Indext D˜ 1 − Indext D˜ 0
˜ ˜ tr[ S1 ] − tr[ S1 ] − tr[ S˜0 ] − tr[ S˜0 ] = X
=
tr[ S˜1 ] − tr[ S˜0 ] −
X
X
(tr[S1 ] − tr[S0 ]) −
= M
=
tr[S1 ] − tr[S1 ] −
M
as desired.
X
M
tr[ S˜1 ] − tr[ S˜0 ]
tr[S1 ] − tr[S0 ] tr[S0 ] − tr[S0 ]
M
As we noted before, the proof of the lemma shows also that the definition of the relative 1 . 0 and D topological index is independent of the choice of extensions D Before we step into the proof of Theorem 3.1 itself, we must introducesome further notation. Let f : [0, 1] → [0, 1] be a smooth function such that f = 1 on 0, 13 , f = 0 on 23 , 1 and f ≈ −1 on 13 , 23 . Pick up a point x0 ∈ M and let d(x) = dist(x, x0 ). For each m ∈ Z∗ , consider the functions 1 −d(x) . e f m (x) = f m 1
3 log 4m
Note that supp d f m2 ⊂ B
3 log 2m
−B
and
d f m L 2 ≤
c9 , m
(26)
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1 where c9 = X e−d(x) 2 . Here, Br = {x ∈ M | d(x) ≤ r }, which is compact by the completeness of M. Proof of Theorem 3.1. All we have to do is to show that the right-hand sides of (24) and (25) are equal. In fact, let U ⊂ V be small neighbourhoods of the diagonal within M × M and choose ψ ∈ C ∞ (M × M) supported on V and such that ψ = 1 on U . Let Q j be the operator whose Schwartzian kernel is K Q j (x, y) = ψ(x, y)K G j (x, y), where G j is the Green’s operator for D j . Then Q j is a parametrix for D j for which the corresponding smoothing operators S j and S j , as in (23), satisfy tr[S j ] = tr[H j ] and tr[S j ] = tr[H j ],
(27)
where H j and H j are the finite rank projection operators associated with the Green’s operator G j , as in (23). But it is not necessarily the case that the two parametrices Q 0 and Q 1 so obtained must coincide at W . In order to fix that, we will glue them with the common parametrix of D0 |W and D1 |W , denoted Q (with corresponding smoothing operators S and S ), using the cut-off functions f m defined above (assume that the base points are contained in the compact set K ). More precisely, for a section s, 1
1 1 1 2 2 2 Q (1 − f m ) 2 s ; f Q (m) (s) = f Q s + (1 − f ) j m m m j (m)
(m)
clearly, for each m, the operators Q 0 and Q 1 coincide at W . For the respective smoothing operators, we get (see [20, Prop. 1.24]) ⎧ 1 1
1 1 1 1 1 ⎪ (m) 2 2 ⎪ 2 2 ⎨ S j (s) = f m S j f m s + (1 − f m ) S (1 − f m ) s + (Q j f m2 s − Q (1 − f m ) 2 s d f m2 , 1
1 1 1 ⎪ (m) ⎪ ⎩ S j (s) = f m2 S j f m2 s + (1 − f m ) 2 S (1 − f m ) 2 s .
Therefore (m)
(m)
tr[S j ] − tr[S j
1 ] = f m2 tr[S j ] − tr[S j ]
1 1 + (1 − f m ) 2 tr[S] − tr[S ] + tr (Q j − Q)d f m2
and (m)
(m)
(m)
(m)
(m)
(m)
tr[S1 ] − tr[S1 ] − tr[S0 ] + tr[S0 ] 1 1 1 = f m2 tr[S1 ] − tr[S1 ] − tr[S0 ] + tr[S0 ] + tr (Q 1 − Q)d f m2 − tr (Q 0 − Q)d f m2 , so finally we obtain (m)
(m)
tr[S1 ] − tr[S1 ] − tr[S0 ] + tr[S0 ] 1 1 = f m2 tr[S1 ] − tr[S1 ] − tr[S0 ] + tr[S0 ] + tr (Q 1 − Q 0 )d f m2 .
(28)
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We must now integrate both sides of (28) and take limits as m → ∞. For m sufficiently large, supp(1 − f m ) ⊂ W , hence the left hand side of the identity (28) equals the relative topological index Indext (D1 , D0 ), by Lemma 5.1. On the other hand, the first summand inside on the right-hand side of (28) equals Indexa (D1 , D0 ) by (27) and (24). Thus, it is enough to show that the integral of the last two terms on the right-hand side of (28) vanishes as m → ∞. Indeed, note that 1 1 2 tr (Q 1 − Q 0 )d f m = d f m2 tr[Q 1 − Q 0 ], hence, since supp(d f m ) ⊂ W for sufficiently large m and using also (26), it follows that 1 c9 tr (Q 1 − Q 0 )d f m2 ≤ tr[G 1 − G 0 ] → 0 as m → ∞ (29) m W
W
if the integral on the right-hand side of the above inequality is finite. Indeed, let D = D1 |W = D0 |W ; from the parametrix equation, we have D((G 1 − G 0 )|W ) = (H1 − H0 )|W . Observe that H = Ker((H1 − H0 )|W ) is a closed subspace of finite codimension in L 2j+1,1 (F1 |W ). Moreover H ⊆ Ker D; thus, (G 1 − G 0 )|W has finite dimensional range and hence it is of trace class, i.e. the integral on the right-hand side of inequality (29) does converge (see also [20, Lemma 4.28]). This concludes the proof. Acknowledgement. The authors thank S. Cherkis, T. Hausel and D. Nógrádi for careful reading of the manuscript and making several important comments. We also thank the organizers of the 2004 March AIM-ARCC workshop on L 2 harmonic forms for bringing the team together. G.E. is supported by the CNPq post-doctoral grant No. 150854/2005-6 (Brazil) and the OTKA grants No. T43242 and No. T046365 (Hungary). M.J. is partially supported by the CNPq grant No. 300991/2004-5 and the FAPESP grant No. 2005/04558-0.
References 1. Aliev, A.N., Saclioglu, C.: Self-dual fields harbored by a Kerr–Taub-BOLT instanton. Phys. Lett. B632, 725–727 (2006) 2. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four dimensional Riemannian geometry. Proc. Roy. Soc. London A362, 425–461 (1978) 3. Auckly, D.R.: Topological methods to compute Chern–Simons invariants. Math. Proc. Camb. Phil. Soc. 115, 229–251 (1994) 4. Bruckmann, F., van Baal, P.: Multi-caloron solutions. Nucl. Phys. B645, 105–133 (2002) 5. Bruckmann, F., Nógrádi, D., van Baal, P.: Higher charge calorons with non-trivial holonomy. Nucl. Phys. B698, 233–254 (2004) 6. Charap, J.M., Duff, M.J.: Space-time topology and a new class of Yang-Mills instanton. Phys. Lett. B71, 219–221 (1977) 7. Cherkis, S.A.: Self-dual gravitational instantons. Talk given at the AIM-ARCC workshop “L 2 cohomology in geometry and physics”, Palo Alto, USA, March 16–21, 2004 8. Cherkis, S.A., Hitchin, N.J.: Gravitational instantons of type Dk . Commun. Math. Phys. 260, 299–317 (2005) 9. Cherkis, S.A., Kapustin, A.: Hyper-Kähler metrics from periodic monopoles. Phys. Rev. D65, 084015 (2002) 10. Cherkis, S.A., Kapustin, A.: Dk gravitational instantons and Nahm equations. Adv. Theor. Math. Phys. 2, 1287–1306 (1999) 11. Chern, S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99, 48–69 (1974) 12. Derek, H.: Large scale and large period limits of symmetric calorons. J. Math. Phys. 48, 082905 (2007) 13. Dodziuk, J.: Vanishing theorems for square-integrable harmonic forms. Proc. Indian Acad. Sci. Math. Sci. 90, 21–27 (1981)
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14. Eguchi, T., Gilkey, P.B., Hanson, A.J.: Gravity, gauge theories and differential geometry. Phys. Rep. 66, 213–393 (1980) 15. Etesi, G.: The topology of asymptotically locally flat gravitational instantons. Phys. Lett. B641, 461–465 (2006) 16. Etesi, G., Hausel, T.: Geometric interpretation of Schwarzschild instantons. J. Geom. Phys. 37, 126–136 (2001) 17. Etesi, G., Hausel, T.: Geometric construction of new Yang–Mills instantons over Taub–NUT space. Phys. Lett. B514, 189–199 (2001) 18. Etesi, G., Hausel, T.: On Yang–Mills instantons over multi-centered gravitational instantons. Commun. Math. Phys. 235, 275–288 (2003) 19. Gibbons, G.W., Hawking, S.W.: Gravitational multi-instantons. Phys. Lett. B78, 430–432 (1976) 20. Gromov, M., Lawson, H.B. Jr.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. 58, 83–196 (1983) 21. Hausel, T., Hunsicker, E., Mazzeo, R.: Hodge cohomology of gravitational instantons. Duke Math. J. 122, 485–548 (2004) 22. Hawking, S.W.: Gravitational instantons. Phys. Lett. A60, 81–83 (1977) 23. Jardim, M.: Nahm transform of doubly periodic instantons. PhD Thesis, University of Oxford, 110 pp, http://arXiv.org/list/math.DG/9912028, 1999 24. Jardim, M.: Nahm transform and spectral curves for doubly-periodic instantons. Commun. Math. Phys. 225, 639–668 (2002) 25. Kirk, P., Klassen, E.: Chern–Simons invariants and representation spaces of knot groups. Math. Ann. 287, 343–367 (1990) 26. Kronheimer, P.B.: The construction of ALE spaces as hyper-Kähler quotients. J. Diff. Geom. 29, 665–683 (1989) 27. Kronheimer, P.B., Nakajima, N.: Yang–Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990) 28. Morgan, J.W., Mrowka, T., Ruberman, D.: The L 2 moduli space and a vanishing theorem for Donaldson polynomial invariants. Monographs in geometry and topology, Volume II, Cambridge, MA: Int. Press, 1994 29. Nakajima, H.: Moduli spaces of anti-self-dual connections on ALE gravitational instantons. Invent. Math. 102, 267–303 (1990) 30. Nye, T.M.W.: The geometry of calorons. PhD Thesis, University of Edinburgh, 147 pp, http://arXiv.org/ list/hep-th/0311215, 2001 31. Parker, T., Taubes, C.H.: On Witten’s proof of the positive energy theorem. Commun. Math. Phys. 84, 223–238 (1982) 32. Råde, J.: Singular Yang–Mills fields. Local theory II. J. Reine Angew. Math. 456, 197–219 (1994) 33. Sibner, L.M., Sibner, R.J.: Classification of singular Sobolev connections by their holonomy. Commun. Math. Phys. 144, 337–350 (1992) 34. Tekin, B.: Yang-Mills solutions on Euclidean Schwarzschild space. Phys. Rev. D65, 084035 (2002) 35. Wehrheim, K.: Energy identity for anti-self-dual instantons on C×. Math. Res. Lett. 13, 161–166 (2006) 36. Yau, S-T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure. Appl. Math. 28, 201–228 (1975) Communicated by G.W. Gibbons
Commun. Math. Phys. 280, 315–349 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0468-7
Communications in
Mathematical Physics
Calorons, Nahm’s Equations on S1 and Bundles over P1 × P1 Benoit Charbonneau1 , Jacques Hurtubise2 1 Mathematics Department, Duke University, Box 90320, Durham, NC 27708, USA.
E-mail: [email protected]
2 Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St. W,
Montreal, Quebec, H3A 2K6, Canada. E-mail: [email protected] Received: 8 December 2006 / Accepted: 26 October 2007 Published online: 2 April 2008 – © Springer-Verlag 2008
Abstract: The moduli space of solutions to Nahm’s equations of rank (k, k + j) on the circle, and hence, of SU (2) calorons of charge (k, j), is shown to be equivalent to the moduli of holomorphic rank 2 bundles on P1 × P1 trivialized at infinity ({∞} × P1 ∪ P1 × {∞}) with c2 = k and equipped with a flag of degree j along P1 × {0}. An explicit matrix description of these spaces is given by a monad construction. 1. Introduction The four-dimensional (anti-) self-dual Yang–Mills (ASD) equations, and their solutions (called instantons), are by now a staple of both geometry and physics, whose myriad uses and properties are too many to summarise here. The earliest base manifold on which these equations were studied was simply R4 ; various constructions in this case have been given, the most efficient of which is the well known ADHM construction [2]. Early on, solutions were produced by reducing the equations under the various symmetry groups acting on R4 (e.g. [13,14]). These reduced equations turn out to be interesting in their own right. Indeed, invariance under the action of R by translation produces monopoles, solutions to the Bogomolny equations [34]; invariance under the action of R2 produces the Hitchin equations whose analysis tells us a lot about bundles on Riemann surfaces [16]. Invariance under R3 yields Nahm’s equations [29], some important ordinary differential equations. The case (or rather two cases as we will see) that concerns us here, that of minimal (translation) invariance, under a single discrete translation (Z-invariance), with suitable boundary conditions, corresponds to the case of calorons. These gauge fields have seen a recurrence of interest recently, for a variety of reasons; see [3–5,21–27,30]. From this list of examples, it would seem that the most interesting cases were produced by considering various Abelian groups acting on R4 by translation. While this is a question of taste, these cases do possess a most interesting feature, a correspondence due to Nahm [28,8], only proven in certain cases, which postulates an isomorphism
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between the moduli of instantons on R4 invariant under the action of a closed subgroup G of translations of R4 (which can be thought of as suitable fields on R4 /G), and ˆ The boundary conditions instantons on R4 invariant under the action of a dual group G. for both sets of fields must be defined with care for the correspondence to hold. So far quite a good set of cases are known; see [18]. In the case of calorons, the heuristic suggests a correspondence between calorons (instantons on R4 /Z = S 1 × R3 , with appropriate boundary conditions), and solutions to Nahm’s equations (o.d.e.s given by reducing the ASD equations) on the circle, again with suitable boundary behavior at selected points on the circle (see below). This correspondence has been partially proved by Nye [31] and Nye–Singer [32]. We complete the proof in [7], showing the correspondence is exact. Thus the moduli space of calorons corresponds to the moduli space of solutions to Nahm’s equations on the circle, and it is this space that we examine in this paper. In deciding what this space should be, we are guided by a few basic ideas. The first of these is that calorons with gauge group K can be thought of as monopoles over R3 , with values in the Kac–Moody algebra L˜ K . This point of view has been developed by Garland and Murray [11], and is a very useful way of understanding calorons, in particular for moduli. Indeed, our second idea is that monopoles with compact simple gauge group K and maximal symmetry breaking at infinity (part of the boundary conditions) on R3 correspond to rational maps of P1 to K /T , with T the maximal torus. This is proven for K a classical group, [10,17] and then by Jarvis for all compact simple groups [19]. The space K /T can also be written as K C /B, where K C is the complexification of K , and B a Borel subgroup. Combining these two ideas, the moduli space of calorons for gauge group K , (or rather of solutions to Nahm’s equations) should be that of rational maps into the homogeneous space given by quotienting the loop group L K C of K C by the subgroup Lˆ B consisting of Fourier series with only terms of non-negative degree, and with the degree zero term lying in B. This is where a third idea, due to Atiyah [1] comes into play. One thinks of elements of L K C as transition functions for a K C -bundle on P1 ; a map of P1 into L K C then defines a bundle over P1 × P1 . Working through the quotienting by Lˆ B, as in [1], tells us the theorem that we are going to prove in this paper. We restrict to the case of SU (2) calorons. Theorem 1. The moduli space of solutions of rank (k, k + j) to Nahm’s equations on the circle (Eqs. (67) and (68)) is equivalent to the moduli space of pairs of vector bundles E of rank 2 on P1 × P1 , first Chern class zero and second Chern number k, trivialized over {∞} × P1 ∪ P1 × {∞}, and equipped with an injective map φ from the line bundle O(− j) of degree − j to the restriction of E to P1 × {0} (up to non-zero scalar multiple), such that the image of φ(∞) lies in the subspace spanned by the second vector of the trivialization at (∞, ∞). The proof of this theorem goes in several steps, each interesting in its own right: − We first show that the pairs (E, φ) are equivalent to certain matrices, satisfying some algebraic conditions. This step is a generalization of the monad construction of [9]. It is the subject of Sects. 2 and 3. − We then show that the monad is equivalent to a set of sheaves on P1 , and maps between them. We do so in Sect. 4. − Finally, in Sect. 5, we show that these sheaves are equivalent to a Nahm complex over the circle, and hence to a solution of the Nahm equations.
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Of the various steps in the chain of equivalences, perhaps the one expressing the caloron, or bundle plus map, as a diagram of sheaves over P1 is most deserving of comment. The twistor construction of calorons, at least from a Kac–Moody point of view, is given by some algebro-geometric data over T P1 , and the diagram of sheaves is given by restricting this data to a fiber C of the projection T P1 → P1 , and extending it to P1 . This is quite similar to what happens for monopoles. One sees again the theme of Kac–Moody groups. Our moduli space, let us not forget, is supposed to correspond to that of maps from P1 to a homogeneous space corresponding to this group. In the finite dimensional case of homogeneous spaces for Gl(n, C), one can describe maps into a flag manifold in terms of similar diagrams of sheaves [17]; it is not surprising that this pattern reoccurs here, and that it is a case in which the “finite-dimensional” aspects of the Kac–Moody group (root spaces, etc.) predominate.
2. Monad Construction Let us use standard affine coordinates (x, y) on P1 × P1 , denote π the projection on the first factor, and i y0 : P1 → P1 × P1 the injection x → (x, y0 ). Set H1 := {∞} × P1 and H2 := P1 × {∞}. For any sheaf F on P1 × P1 , denote F y0 := i y0 ∗ i y∗0 F the extension by zero of the restriction of F at level y0 and F( p, q) := F ⊗ O( p H1 + q H2 ). Let E be a Sl(2, C)-bundle over P1 × P1 , with c2 (E) = k, trivial over the fiber {∞} × P1 , trivialized over the section P1 × {∞} (thus equipped with a standard degree zero flag E ∞− ⊂ E ∞ defined by the first basis vector) and with given flag E 0+ ⊂ E 0 of degree j over P1 × {0}, and such that identifying the fiber of E at (∞, 0) and (∞, ∞), the flags E 0+ and E ∞− are transverse. We define three locally free sheaves K 0 , K ∞ and K 0∞ by the exact sequences - E 0 - K0 0 - K∞ - E 0 - K 0∞ - E
- E 0 /E 0,+ ∼ = O( j, 0)0 - 0, - E ∞ /E ∞,− ∼ = O∞ - 0, - E 0 /E 0,+ ⊕ E ∞ /E ∞,− - 0.
(1) (2) (3)
As a consequence we have supplementary sequences 0 - K0 0 - K∞
- K 0∞ (0, 1) - O∞ - 0, - K 0∞ (0, 1) - O(− j, 0)0 - 0.
(4) (5)
In Sect. 5, we define the Nahm complex using the diagram of sheaves E(0, −1) K ∞ (0, −1) ? 6 - K 0∞ , K 0 (0, −1) taking direct images onto P1 . We have the easily proven lemma.
(6)
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Lemma 1. 1) H 0 (P1 , R 1 π∗ (E(0, −1)) = Ck , and R 1 π∗ (E(0, −1)) is supported on k points, counted with multiplicity. 2) H 0 (P1 , R 1 π∗ (K 0∞ )) = Ck+ j , and R 1 π∗ (K 0∞ ) is supported on k + j points, counted with multiplicity. 3) R 1 π∗ (K ∞ (0, −1)) and R 1 π∗ (K 0 (0, −1)) are supported over the whole line; generically, R 1 π∗ (K ∞ (0, −1)) = O(k), R 1 π∗ (K 0 (0, −1)) = O(k + j). 4) These sheaves fit in the exact sequences 0 → O( j) 0→ O 0→ O 0 → O(− j)
→ → → →
R 1 π∗ (K 0 (0, −1)) R 1 π∗ (K ∞ (0, −1)) R 1 π∗ (K 0 (0, −1)) R 1 π∗ (K ∞ (0, −1))
→ R 1 π∗ (E(0, −1)) → R 1 π∗ (E(0, −1)) → R 1 π∗ (K 0∞ ) → R 1 π∗ (K 0∞ )
→ → → →
0, 0, 0, 0.
(7)
In particular, the natural maps from R 1 π∗ (K ∞ (0, −1)) and R 1 π∗ (K 0 (0, −1)) to R 1 π∗ (E(0, −1)) and R 1 π∗ (K 0∞ ) are all surjections, and the kernels are all torsion free. Proof. On a generic fiber of π , we have E = O ⊕ O, K 0 = O ⊕ O(−1), K ∞ = O ⊕ O(−1), and K 0∞ = O(−1) ⊕ O(−1). Hence for F = E(0, −1), K 0 (0, −1), K ∞ (0, −1), or K 0∞ , and for w generic, F|π −1 (w) has no global sections. Thus π∗ F = 0, and R 1 π∗ E(0, −1), R 1 π∗ K 0∞ are torsion, while the sheaves R 1 π∗ K 0 (0, −1), and R 1 π∗ K ∞ (0, −1) are line bundles over the generic set of w for which K 0 = O ⊕ O(−1), and K ∞ = O ⊕ O(−1); for the generic bundle, this is all of P1 . Then Statement 4 follows from taking the direct image of the Sequences (1)–(3). Statements 1, 2, and 3 follow from the Grothendieck–Riemann–Roch theorem, and Sequences (1)–(3). The proof is now complete.
Let F be a vector bundle on P1 × P1 of rank r with first and second Chern classes c1 and c2 . Using the Riemann–Roch theorem, we find 1 2 c − c2 + (H1 + H2 ).c1 + r, 2 1 χ (F( p, q)) = χ (F) + r ( p + q) + p H1 .c1 + q H2 .c1 + r pq. χ (F) =
(8) (9)
Lemma 2. Let F be a vector bundle on P1 × P1 , trivial on a section P1 × {y}. Then H2 .c1 = 0, and as functions of q, h 0 (F( p, q)) is constant for fixed p ≤ −1, h 1 (F(−1, q)) is constant, h 2 (F( p, q)) is constant for fixed p ≥ −1. Similarly, for F trivial on a fiber {x} × P1 , we have H1 .c1 = 0, and h 0 (F( p, q)) is constant for fixed q ≤ −1, h 1 (F( p, −1)) is constant as functions of p, h 2 (F( p, q)) is constant for fixed q ≥ −1. Proof. Suppose F is a rank r vector bundle on P1 × P1 which is trivial on P1 × {y}. We have an exact sequence of sheaves 0 → F(0, −1) → F → Fy → 0. Tensored by O( p, q), this sequence gives us a long exact sequence in cohomology, which ensures an isomorphism between H i (F( p, q)) and H i (F( p, q − 1)) whenever both H i (Fy ( p, q))
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and H i−1 (Fy ( p, q)) are 0. Because of the equality H i (Fy ( p, q)) = H i (P1 , O( p)r ), this condition happens precisely to ensure the h i are constant as specified. Since F is trivial along a section, we just proved that χ (F(−1, q)) = χ (F) − r − H1 .c1 + q H2 .c1 is constant as a function of q. Hence H2 .c1 = 0. With this remark, the first half of this lemma is proved. The second half is obtained by symmetry.
Our bundle E, being trivial over the fiber and section at ∞, has thus c1 = 0. Lemma 3. For K 0 , K ∞ and K 0∞ defined by Sequences (1), (2), and (3), c(K 0 ) = 1 − H2 + (k + j)H1 H2 , c(K ∞ ) = 1 − H2 + k H1 H2 , c(K 0∞ ) = 1 + 2H2 + (k + j)H1 H2 , and χ (K 0 ( p, q)) = −(k + j) + (1 + p)(1 + 2q), χ (K ∞ ( p, q)) = −k + (1 + p)(1 + 2q), χ (K 0∞ ( p, q)) = −(k + j) + 2q(1 + p). Proof. On the following exact sequences, use that c(F3 ) = c(F1 )c(F2 )−1 when 0 → F1 → F2 → F3 → 0 is exact: 0 → E(0, −1) → E → E 0 → 0, 0 → O(− j, −1) → O(− j, 0) → O(− j, 0)0 → 0, 0 → O(− j, 0)0 → E 0 → E 0 /E 0+ → 0, 0 → K 0 → E → E 0 /E 0+ → 0, 0 → K 0∞ → K 0 → E ∞ /E ∞− → 0. Using Eqs. (8) and (9), we obtain from those Chern classes the Euler characteristics. Setting j = 0 gives the answers for K ∞ .
Using this result, we can compute many of the cohomology groups of the bundles E, K 0 , K ∞ , K 0∞ . Theorem 2 (Vanishing). The cohomology groups of the bundles E, K 0 , K ∞ , and K 0∞ defined by Sequences (1), (2), and (3) vanish as follows: p p p p
≤ −1 or q ≤ −1 or q ≥ −1 or q ≥ −1 or q
≤ −1 =⇒ h 0 (E( p, q)) = h 0 (K 0 ( p, q)) = h 0 (K ∞ ( p, q)) = 0, ≤ 0 =⇒ h 0 (K 0∞ ( p, q)) = 0, ≥ −1 =⇒ h 2 (E( p, q)) = 0, ≥ 0 =⇒ h 2 (K 0 ( p, q)) = h 2 (K ∞ ( p, q)) = h 2 (K ∞ ( p, q)) = 0.
(10) (11) (12) (13)
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When h 0 = h 2 = 0, we get an exact formula for h 1 : ( p ≤ −1 and q ≥ −1) =⇒ h 1 (E( p, q)) = k − 2(1 + q)(1 + p), (14) or ( p ≥ −1 and q ≤ −1) ⎫ ( p ≤ −1 and q ≥ 0) ⎬ or =⇒ h 1 (K 0 ( p, q)) = (k + j) − (1 + 2q)(1 + p), (15) ⎭ ( p ≥ −1 and q ≤ −1) ( p ≤ −1 and q ≥ 0) =⇒ h 1 (K 0∞ ( p, q)) = (k + j) − 2q(1 + p). (16) or ( p ≥ −1 and q ≤ 0) Equation (15) is valid for K ∞ by setting j = 0. When j ≥ 1, we get extra information for K 0 : for p ≤ j − 1, we have h 0 (K 0 ( p, 0)) = 0, h 1 (K 0 ( p, 0)) = k + j − 1 − p.
(17) (18)
Proof. Lemma 2 tell us that h 2 (E( p, q)) is constant in the region {( p, q) | p ≥ −1 or q ≥ −1}. For any i > 0, and N big enough, Theorem B of Serre (see [12, p.700]) says h i (F(N , N )) = 0. Thus h 2 (E( p, q)) = 0 throughout this region, proving (12). By Serre duality, we have the corresponding result for h 0 . Restricted to a generic section, K 0 , K ∞ and K 0∞ are trivial. Hence the first part of Lemma 2 applies. Restricted to the fiber above ∞ however, K 0 and K ∞ are O ⊕ O(−1) while K 0∞ is O(−1) ⊕ O(−1). Working as in the proof of Lemma 2, we have for F ∈ {K 0 , K ∞ } an exact sequence · · · → H i (F( p − 1, q)) → H i (F( p, q)) → H i (P1 , O(q) ⊕ O(q − 1)) · · · ensuring that h 0 (F( p, q)) is constant for fixed q ≤ −1, h 2 (F( p, q)) is constant for fixed q ≥ 0. Similarly, we obtain h 0 (K 0∞ ( p, q)) is constant for fixed q ≤ 0, h 2 (K 0∞ ( p, q)) is constant for fixed q ≥ 0. Again, we deduct from Theorem B the wanted vanishing for the h 0 and h 2 . Whenever h 0 = h 2 = 0, we have h 1 = −χ . The exact formula for h 1 thus follow from the Riemann–Roch Equation (9) used with Lemma 3.
Because of all this vanishing, the following theorem guarantees a monad description of those bundles.
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Theorem 3 (Buchdahl’s Beilinson’s theorem). For any holomorphic vector bundle F on P1 × P1 , there is a spectral sequence with E 1 -term O(−1, −1)h (F(−1,−1)) O(−1, 0)h (F(−1,0)) ⊕ O(0, −1)h (F(0,−1)) O h (F) , 1 1 1 1 O(−1, −1)h (F(−1,−1)) O(−1, 0)h (F(−1,0)) ⊕ O(0, −1)h (F(0,−1)) O h (F) , 0 0 0 0 O(−1, −1)h (F(−1,−1)) O(−1, 0)h (F(−1,0)) ⊕ O(0, −1)h (F(0,−1)) O h (F) , 2
2
and
p,q
E1
p+q
⇒ E∞ =
2
F, 0,
2
if p + q = 0, otherwise.
Proof. See [6, p. 144]. Use [15, Ex. 6.1, p. 237] and [15, Prop. 6.3, p. 234] to see that the only extension 0 → O(1, 0)(1, 0) → R → O(0, 1)(0, 1) → 0 on H0 = P1 × P1 −1,q is the trivial one. Hence we replace the term E 1 in the Hn -analogue of Beilinson’s Theorem by the direct sum as in the statement above.
Now let us exploit the vanishing theorem together with the machinery of monads. First, let us check existence. Theorem 4 (Existence of monad). Let E be one of the bundles we are considering, that is trivial on {∞} × P1 and on P1 × {∞}. The bundles E, K 0 ,K ∞ and K 0∞ (0, 1) are the cohomology of monads of respective type O(−1, 0)k
→ O(−1, 1)k
⊕ Ok+2
→ O(0, 1)k ,
O(−1, 0)k+ j → O(−1, 1)k+ j ⊕ Ok+ j+3 → O(0, 1)k+ j+1 , O(−1, 0)k
→ O(−1, 1)k
⊕ Ok+3
→ O(0, 1)k+1 ,
(19)
O(−1, 0)k+ j → O(−1, 1)k+ j ⊕ Ok+ j+2 → O(0, 1)k+ j . Before starting the proof, set V1 (F) := H 1 (F(−1, −2)),
V2 (F) := H 1 (F(−1, −1)),
V3 (F) := H 1 (F(0, −2)),
V4 (F) := H 1 (F(0, −1)).
(20)
Proof. We use extensively the Vanishing Theorem 2. Suppose F ∈ {E, K 0 , K ∞ }. The cohomology groups appearing in the E 1 -term for F(0, −1) are H i (F( p, q)) with p ∈ {−1, 0} and q ∈ {−2, −1}. Then q ≤ −1 implies those groups with i = 0 are trivial, and p ≥ −1 implies those groups with i = 2 are also trivial. For the K 0∞ ( p, q) appearing, with p, q ∈ {−1, 0}, the fact that q ≤ 0 ensures h 0 = 0 while p ≥ −1 ensures h 2 = 0. The E 1 -term for F(0, −1) with F ∈ {E, K 0 , K ∞ , K 0∞ (0, 1)} thus reduces to the middle row, so that F(0, −1) is the cohomology of a monad of the form O(−1, −1) ⊗ V1 (F) → O(−1, 0) ⊗ V2 (F) ⊕ O(0, −1) ⊗ V3 (F) → O ⊗ V4 (F). The exact formula for the dimensions of those spaces are given in Theorem 2, and correspond to the values given in Eq. (19). Tensoring by O(0, 1) concludes the proof.
The following lemma can be used to double check the result.
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Lemma 4. For the cohomology F of a monad O(−1, 0)k1 - O(−1, 1)k2 ⊕ Ok3 - O(0, 1)k4 ,
(21)
we have rk F = k2 + k3 − k1 − k4 , c(F) = 1 + (k1 − k2 )H1 + (k2 − k4 )H2 + (k2 (k1 + k4 + 1 − k2 ) − k1 k4 ) H1 H2 . Proof. From [33, Lemma 3.1.2, p. 240], we have −1 −1 c O(0, 1)k4 . c(F) = c O(−1, 1)k2 ⊕ Ok3 c O(−1, 0)k1 Since c(O( p, q)r ) = 1 + r ( p H1 + q H2 ) + r (r − 1) pq H1 H2 , we have the proof.
Lemma 5. For the monads of Theorem 4, written as M(F) :
V1 V2 ⊗ O(−1, 0) V4 -
⊕ ⊗ ⊗ , µ1 x + µ0 ν1 y + ν0 O(0, 1) V3 ⊗ O O(−1, 0) α1 y + α0 β1 x + β0
the map α1 is an isomorphism, β1 and µ1 are injective, ν1 is surjective, and ker(ν1 ) ∩ Im(β1 ) = 0. Proof. Note first that for F ∈ {E, K 0 , K ∞ , K 0∞ (0, 1)}, we have F|P1 ×{∞} trivial and F|{∞}×P1 isomorphic to O2 , O ⊕ O(−1), or O(−1)2 . Let us first study the restriction to P1 × {∞}. Consider the monad M(F|) : V1 ⊗ O(−1)
α1 β1 x + β0
V2 ⊗ O(−1) - V4 ⊗ O. ⊕
µ x + µ ν 1 0 1 V3 ⊗ O
Since 0 → V1 ⊗ O(−1) → ker → F| → 0 is exact, we have H 1 (ker) = H 1 (F|) = 0. Since 0 → ker → V2 ⊗ O(−1) ⊕ V3 ⊗ O → V4 ⊗ O → 0 is exact, we obtain ν V3 ⊗ H 0 (O) -1 V4 ⊗ H 0 (O) - H 1 (ker) = 0, whence ν1 is surjective. Consider the monad ∗
M(F| (−1)) :
V4∗
⊗ O(−1) -
V2∗ ⊗ O - V ∗ ⊗ O. ⊕ 1 ∗ V3 ⊗ O(−1)
Note that as before, H 1 (ker) = H 1 (F|∗ (−1)) = 0, but here H 0 = 0 as well. Thus α1∗ ∗ V1 ⊗ H 0 (O) - H 1 (ker) = 0, 0 = H 0 (ker) - V2∗ ⊗ H 0 (O) being exact, we have that α1 is an isomorphism.
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Now let us study the restriction to {∞} × P1 . Consider the monad V2 ⊗ O - V4 ⊗ O. ⊕
M(F|(−1)) : V1 ⊗ O(−1) α1 y + α0 V3 ⊗ O(−1) µ1 ν1 y + ν0 β1 As before, we have an exact sequence µ 0 = H 0 (ker) - V2 ⊗ H 0 (O) -1 V4 ⊗ H 0 (O), whence µ1 is injective. Consider now ∗
M(F| ) :
V4∗
V2∗ ⊗ O(−1) - V ∗ ⊗ O. ⊕ ⊗ O(−1) 1 ∗ V3 ⊗ O
From this exact sequence we get in cohomology the exact sequence β1∗ ∗ V3∗ ⊗ H 0 (O) V1 ⊗ H 0 (O) - H 1 (ker) = 0, whence β1 is injective. The monad equation µ1 α1 + ν1 β1 = 0 and the injectivity of µ1 α1 imposes ker(ν1 ) ∩ Im(β1 ) = 0. The proof of Lemma 5 is now complete.
Since the construction of Buchdahl is natural, a map φ : F → F induces maps in cohomology φ∗ : Vi (F) → Vi (F ), which we denote Φi,F,F , or Φi when there is no risk of confusion. Using Sequences (1), (2), and (4), twisted by O( p, q), we have that all the Φi,K 0 ,E , Φi,K ∞ ,E and Φi,K 0 ,K 0∞ (0,1) induced by the injections are surjective. Moreover, because k1 = k2 sometimes, four of those maps are isomorphisms. Sequence (5), tensored by O( p, q), yields the exact sequence H 0 (P1 , O( p − j)) → H 1 (K ∞ ( p, q)) → H 1 (K 0∞ ( p, q + 1)) → H 1 (P1 , O( p − j)). Depending on p and j, we obtain surjectivity and/or injectivity. We can summarize this information with the diagrams of the following lemma. Lemma 6. The four maps of Diagram (6) induce monad maps with the following surjectivity/injectivity properties: ⊂ - V1 (K 0∞ (0, 1)) V3 (K 0 ) - V3 (K 0∞ (0, 1)) V1 (K 0 ) 6 ? ? V1 (E)
∪
⊃
V1 (K ∞ )
⊂ - V2 (K 0∞ (0, 1)) V2 (K 0 ) 6
? ? V2 (E)
∪
⊃
V2 (K ∞ )
6 surj. if j ≤ 1, ? ? V3 (E)
inj. if j ≥ 1
V3 (K ∞ )
- V4 (K 0∞ (0, 1)) V4 (K 0 ) 6 surj. if j ≤ 1, ? ? V4 (E)
inj. if j ≥ 1
V4 (K ∞ ).
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Remark 1. The monads given by Theorem 4, of the type given by Eq. (21), are uniquely determined up to the action of Aut O(−1, 0)k1 × Aut O(−1, 1)k2 ⊕ Ok3 × Aut O(0, 1)k4 ; see [33, Lemma 4.1.3 on p. 276]. This group is exactly Gl(k1 , C) × Gl(k2 , C) × Gl(k3 , C) × Gl(k4 , C). Let us exploit those symmetries to Before doing so, set ⎡ 0 0 ··· 0 ⎢1 0 · · · 0 ⎢ ⎢0 1 · · · 0 s := ⎢ ⎢ .. .. ⎢ .. .. ⎣. . . . 0 0 ··· 1
give normal forms for the monads of Theorem 4. ⎤ 0 0⎥ ⎥ 0⎥ ⎥ ⎥ .. ⎥ .⎦ 0
and
e+ := 0
···
0
1.
(22)
Theorem 5. For j > 0, the bundle E, K 0 , K ∞ and K 0∞ (0, 1) are respectively the cohomology of the monads A(E)⎤ A−y ⎢ ⎥ ⎣B − x⎦ D
O(−1, 1)k
⊕
O(−1, 0)k ⎡
Ok+2
B(E) - O(0, 1)k , x−B A−y C
(23)
O(−1, 1)k+ j O(−1, 0)k+ j A(K 0 ) B(K 0 ) k+ j+1 , ⎡ ⎤ ⊕ ⎡ ⎤ O(0, 1) A−y 0 x−B 0 A−y 0 C 0 k+ j+3 O ⎢ A ⎢ ⎥ −y ⎥ ⎢ ⎥ ⎢ −B ⎥ ⎢ ⎥ ⎢ 0 ⎥ x − s A −y C (24) ⎢B − x 0 ⎥ ⎣ ⎦ 0 ⎢ ⎥
⎢ ⎥ 0 −e+ 0 0 1 0 −y ⎢ B ⎥ ⎢ ⎥ ⎢ 0 s − x⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎦ ⎣ D 0 e+
A(K ∞) O(−1, 0)k ⎡ ⎤ A−y ⎢B − x⎥ ⎥ ⎢ ⎥ ⎢ ⎣ D ⎦ D2 A
O(−1, 1)k ⊕ Ok+3
B(K ∞ ) O(0, 1)k+1 , x−B A−y C 0
0 −y 1 −D2 0
(25)
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O(−1, 1)k+ j A(K 0∞ (0, 1)) B(K 0∞ (0, 1)) O(0, 1)k+ j , O(−1, 0) ⎡ ⎤ ⊕ A−y 0 k+ j+2 O −y ⎢ A ⎥ ⎢ ⎥ −C1 e+ ⎥ ⎢ B − x ⎢ ⎥ ⎢ B ⎥ ⎢ 0 s − x − C1 e+ ⎥ (26) ⎢ ⎥ ⎣ D ⎦ 0 2 0 e+ ⎡ ⎤ C1 e+ A − y 0 C2 AC1 x − B ⎦, B(K 0∞ (0, 1)) = ⎣ −B x − s + C1 e+ A −y C2 A C1 0 k+ j
with A, B, C, D, A , B and C being matrices of respective size k × k, k × k, k × 2, 2 × k, j × k, 1 × k, j × 2, (Ci , Ci being the i th column of C, C , and Di the i th row of D), and satisfying the monad equations [A, B] + C D = 0, B A + s A − A B − C D = 0, 0
−e+ A + 1 0 D = 0,
(27) (28) (29)
and the genericity conditions ⎡
⎤ A−y ⎣ B − x ⎦ injective for all x, y ∈ C, D
x − B A − y C surjective for all x, y ∈ C, ⎤ ⎡ A C 0 x − B ⎥ ⎢ −B ⎢ A C x − s ⎥ ⎦ surjective for all x ∈ C, ⎣ 0
0 0 1 0 −e+ A C2 C2 j−1 C 2 N= M · · · M is an isomorphism, A C2 C2 C2
(30) (31) (32)
(33)
where ⎡ ⎤ B −C1 e+ ⎦, M =⎣ B s − C1 e+ 0
(34)
modulo the action of Gl(k, C) (A, B, C, D, A , B , C ) → (g Ag −1 , g Bg −1 , gC, Dg −1 , A g −1 , B g −1 , C ).
(35)
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The maps between the bundles are mediated by the following maps of monads: - O(−1, 1)k+ j ⊕ Ok+ j+3 - O(0, 1)k+ j+1 A(K 0 ) B(K 0 ) ⎡ ⎤ 100000 0 ⎢0 1 0 0 0 0 0 ⎥ ⎢ ⎥ 10 1 0 −C1 ⎢0 0 1 0 0 0 −C1 ⎥ ⎢0 0 0 1 0 0 −C ⎥ 0 1 −C1 01 ⎢ ⎥ ⎣0 0 0 0 0 1 0 1 ⎦ 000000 1 ? ? ? A(K 0∞ (0, 1)) - O(−1, 1)k+ j ⊕ Ok+ j+2 B(K 0∞ (0, 1))- O(0, 1)k+ j O(−1, 0)k+ j 6 6 6 ⎤ ⎡ A 0 0 0 ⎢A 0 0 0 ⎥ ⎥ ⎢0 A 0C A A C2 ⎢ 2 0 ⎥
⎢ 0 A 0 C 0 ⎥ A C2 A ⎢ 2 ⎥ ⎦ ⎣ 1 00 0 0 0 10 O(−1, 0)k+ j
O(−1, 0)k
1 ? O(−1, 0)k 6
10 O(−1, 0)k+ j
- O(−1, 1)k ⊕ Ok+3 ⎡ ⎤ 1000 ⎣0 1 0 0⎦ 0010 ? A(E) O(−1, 1)k ⊕ Ok+2 ⎡ 6 10000 ⎣0 0 1 0 0 00001
A(K ∞ )
A(K 0 )-
O(−1, 1)k+ j ⊕ Ok+ j+3
B(K ∞ )
- O(0, 1)k+1
10
B(E) ⎤ 0 0⎦ 0 B(K 0 )
? - O(0, 1)k 6
100 - O(0, 1)k+ j+1 . (36)
When j = 0, the bundle E and K ∞ are the cohomology of the same monads given by Eq. (23) and (25), with the matrices A, B, C, D satisfying Eq. (27), (30), and (31), the matrix A is invertible, and the matrices for K 0 and K 0∞ (0, 1) are ⎤ ⎡ A−y ⎢B−x⎥ A(K 0 ) = ⎣ D ⎦, D1 A−1 x−B A−y C 0
B(K 0 ) = , 1 0 −y −D1 A−1 0 ⎡ ⎤ A−y ⎢ B − C1 D1 A−1 − x ⎥ ⎥, A(K 0∞ (0, 1)) = ⎢ ⎣ ⎦ D2 D1 A−1
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B(K 0∞ (0, 1)) = x − B + C1 D1 A−1 A − y C2 AC1 . The maps between the bundle are mediated by the j = 0 version of Diagram (36). The proof of this theorem is postponed to Sect. 3. Remark 2. A flag of degree j in E|P1 ×{0} is given by the projective equivalence class of a (pointwise) injective map of bundle O(− j) → E|P1 ×{0} . The bundle E|P1 ×{0} on P1 splits as a sum O(n) ⊕ O(−n) for some n ∈ N. The injection O(− j) → O(n) ⊕ O(−n) is equivalent to a nowhere vanishing section of O(n + j) ⊕ O(−n + j). Thus the existence of a flag O(− j) → E|P1 ×{0} guarantees that E|P1 ×{0} splits as O(− j) ⊕ O( j) when j ≤ 0, and as O(n) ⊕ O(−n) for some 0 ≤ n ≤ j when j ≥ 0. Obviously, only j ≥ 0 matters for studying E, but it turns out the result for j ≤ 0 is useful for studying K 0 . The splitting of E 0 imposed by the existence of the flag forces dim ker(A) = n. Indeed, restrict Monad (23) to P1 × {0}, and tensor with O(−1) throughout to obtain E|(−1) as the cohomology of the monad β α O(−2)k - O(−2)k ⊕ O(−1)k+2 - O(−1)k . One then has h 1 (P1 × {0}, E(−1)) = dim coker(A), hence dim ker(A) = n (or | j|, if j ≤ 0).
(37)
In particular, A is invertible when j = 0. Let us now consider the converse to Theorem 5. Given matrices satisfying the monad equations (27), (28) and (29), we can construct cohomology sheaves E, K 0 , K ∞ , K 0∞ (0, 1) with maps K 0 → E, K ∞ → E, K 0 → K 0∞ (0, 1), K ∞ → K 0∞ (0, 1). The genericity conditions (30), (31), and (32) ensure those sheaves are bundles. It is routine, but lengthy, work to verify that those maps are injective sheaf maps. Similarly, it is not hard to verify that the cokernels of K 0 → K 0∞ (0, 1) and K ∞ → E are O∞ , as wanted. There remains the cokernels of the maps K ∞ → K 0∞ (0, 1) and K 0 → E. Suppose f ∈ ker(B(E)), with components f 1 , f 2 , f 3 in the decomposition Ck ⊕ C j ⊕ C2 used in Theorem 5. Away from y = 0, f is the image of ⎡
f1 0 f2
⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − B f + A f + C f /y ⎥ ∈ ker(B(K 0 )), ⎢ ⎥ 1 2 3 0 ⎢ ⎥ ⎣ ⎦ f 3
1 0 f 3 /y hence the cokernel Q 0 of the map K 0 → E is supported on y = 0. To verify that Q 0 = O( j, 0)0 , as in Eq. (1), we verify that π∗ Q 0 = O( j), using the following lemma.
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Lemma 7. Suppose that a bundle F is defined by a monad O(−1, 0)
k1
A1 y + A0 B1 x + B0
O(−1, 1)k2 - O(0, 1)k4 ,
⊕ C x + C D y + D k 1 0 1 0 3 O
(38)
and that π∗ F(0, −1) = 0. Then R 1 π∗ F(0, −1) has a resolution - O(−1)k2
- Ok4 - R 1 π∗ F(0, −1) C1 x + C0
0
- 0.
(39)
Proof. Use the Künneth Theorem for sheaves [20, Prop 9.2.4, p. 116] to see that R i π∗ O( p, q) = O( p) ⊗ H i P1 , O(q) . Alternatively, use the projection formula [15, Ex. 8.3, p. 253], and prove that R i π∗ O(0, q) = i O h (q) using the exact sequence O(0, q) → O(0, q + 1) → O|x=∞ . Tensor Monad (38) by O(0, −1), and let K = ker C1 x + C0 D1 x + D0 . The short exact sequence 0 → O(−1, −1)k1 → K → F(0, −1) → 0 on P1 × P1 gives on P1 the isomorphisms R i π∗ K = R i π∗ F(0, −1), for i ≥ 0. The short exact sequence 0 → K → O(−1, 0)k2 ⊕O(0, −1)k3 → Ok4 → 0 induces the exact sequence 0
- π∗ K - O(−1)k2
- O k4
- R 1 π∗ K
- 0
on P1 , and R i π∗ K = 0 for i ≥ 2. The proof is now complete.
This lemma gives resolutions for R 1 π∗ K 0 (0, −1) and R 1 π∗ E(0, −1), with kernels the zero sheaves π∗ K 0 (0, −1) and π∗ E(0, −1). We then have a resolution diagram 0
0
0
- O(−1)k 6
- O(−1)k+ j 6 - O(−1) j
x−B ⎡
- Ok 6 ⎤
- R 1 π∗ E(0, −1) 6
x − B 0 ⎢ −B ⎥ x − s⎦ ⎣ 0 0 −e+ - Ok+ j+1 - R 1 π∗ K 0 (0, −1) 6 6 x −s −e+ - O j+1 - K
- 0
- 0
- 0.
Diagram chasing gives us a kernel K , with its resolution; however, the last row is a fairly standard resolution of O( j), and so K = O( j). It remains to verify that the cokernel Q ∞ of the map K ∞ → K 0∞ (0, 1) is O(− j, 0)0 . We first check that the map is surjective away from y = 0. Suppose f ∈ ker(B(K 0∞ (0, 1))), with components f 1 , . . . , f 6 in the decomposition Ck ⊕ C j ⊕ Ck ⊕ C j ⊕ C ⊕ C
Calorons, Nahm’s Equations on S 1 and Bundles over P1 × P1
used in Theorem 5. Since Set
A A
329
is injective, there exists P Q such that P A + Q A = 1.
g1 := P f 1 + Q f 2 , g3 := f 6 , g5 := f 5 , g5 − D2 g1 g4 := , y g2 := P f 3 + Q f 4 − (PC2 + QC2 )g4 ,
and
⎡ ⎤ g1 ⎢ .. ⎥ g := ⎣ . ⎦. g5
Away from y = 0, we have g is mapped to f , and g ∈ ker(B(K ∞ )). To verify this last statement, the only difficulty is proving that (x − B)g1 + (A − y)g2 + C1 g3 + C2 g4 = 0. The trick is to prove that
A A and A times the left-hand-side of the equation is 0 and then use the injectivity of A . Thus the sheaf Q ∞ is supported over the line y = 0. We now only need to work in a neighborhood of y = 0. In the special case j = 0, the components f 2 and f 4 are automatically zero since they belong to C j = {0}. Equation (37) guarantees that A is invertible, and P is its inverse. We can then choose a different representative of f for which f 1 = 0. But then g4 = f 5 /y. When f 5 = 0, the g defined above that maps to f is a genuine element in ker(B(K ∞ )). The only problem is when f 5 = 0. Hence Q ∞ = O0 , as desired. Suppose then that j = 0. Applying Lemma 7, we have a resolution ⎡ ⎤ C1 e+ x − B ⎣ −B ⎦ (x − s) + C1 e+ 0 - Ok+ j - R 1 π∗ (K 0∞ ) O(−1)k+ j 6 A C2 6 A 6 C (40) A A 2 x−B −D2 - Ok+1 - R 1 π∗ (K ∞ (0, −1)). O(−1)k
Note that R 1 π∗ (K 0∞ ) is supported on points, away from ∞. We want to build up a resolution of the twists R 1 π∗ (K ∞ (0, −1))( ), and then show that R 1 π∗ (K ∞ (0, −1))( j) maps to R 1 π∗ (K 0∞ ) with kernel O. Suppose for the moment that we have a sheaf F with resolution x +α β - Oa+1 - F - 0. 0 - O(−1)a The injection O → Oa+1 in the last coordinate induces a map s1 : O → F which is non-zero at ∞. Let s( ) and e+ ( ) be × and 1 × versions of the matrices defined by Eq. (22). The sheaf O( ) has resolution x − s( ) −e+ ( ) - O +1 - O( ) - 0, 0 - O(−1)
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and the injection O → O +1 in the first coordinate induces a map s2 : O → O( ). The maps s1 and s2 induce naturally s˜1 : O( ) → F( ) and s˜2 : F → F( ). We then have a short exact sequence s1
−s2 s˜2 s˜1 - 0. F( ) F ⊕ O( ) O 0 Using the snake lemma on the resolutions - 0
0
- O
- O
? ? ? - Oa+ - Oa+1+ +1 - F ⊕ O( )
0
- 0
- 0,
we get the resolution 0
- Oa+ ⎡
a+ +1 F( ) ⎤ O x+ α 0 ⎢ β ⎥ ⎣ 0 x − s( )⎦ 0 −e+ ( )
- 0.
Setting, as above, ⎡ ⎤ B −C1 e+ ⎦, M := ⎣ B s − C1 e+ 0 and
A A A N := A
N :=
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C2 C2 j−2 C 2 M · · · M , C2 C2 C2 C2 C2 j−1 C 2 M · · · M C2 C2 C2
(42) (43)
(so that M, N are (k + j) × (k + j) matrices, and N is (k + j) × (k + j − 1)), we build up a twist by O( j − 1) of Diagram (40), O(−1)k+ j 6 N
x−M
- Ok+ j 6 N
- R 1 π∗ (K 0∞ ) 6
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k+ j R 1 π∗ (K ∞ (0, −1))( j − 1). O(−1)k+ j−1 ⎡ ⎤ O x − B 0 ⎢ −D2 ⎥ x − s( j − 1)⎦ ⎣ 0 0 −e+ ( j − 1)
The kernel π∗ Q( j − 1) of the map R 1 π∗ (K ∞ (0, −1))( j − 1) → R 1 π∗ (K 0∞ ) is the desired O(−1) if and only if the induced map N on sections is an isomorphism, hence the genericity condition we have imposed on the matrix N in our theorem above.
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While Theorem 5 gives us the matrices starting from the bundles created from the knowledge of the flag and trivialisations, we just proved that the matrices give us the bundles E, K 0 , K ∞ , and K 0∞ (0, 1), from which we can extract the flag and trivialisations. We can then end this section with a theorem, summarising the passage from a bundle and flag to their associated monad. Theorem 6. There is an equivalence between 1) Vector bundles E of rank two on P1 × P1 , with c1 (E) = 0, c2 (E) = k, trivialized along P1 × {∞} ∪ {∞} × P1 , and with a based flag φ : O(− j) → E of degree j along P1 × {0} (up to non-zero scalar multiple), with the basing condition φ(∞)(O(− j)) = span(0, 1). 2) Matrices A, B (k × k), C (k × 2), D (2 × k), A ( j × k), B (1 × k), C ( j × 2), satisfying the monad equations (27), (28), (29) and the genericity conditions (30), (31), (32), (33) modulo the action of Gl(k, C) given by Eq. (35). 3. Normal Forms: the Proof of Theorem 5 We now prove Theorem 5. To do so, we normalize the monads given by Theorem 4 so that in the end they are defined only up to a Gl(k, C) action. To simplify the notation given by Eq. (20), set Vi := Vi (E), W i := Vi (K ∞ ),
Wi := Vi (K 0 ), and
W i := Vi (K 0∞ (0, 1)).
We first normalize the monads for E. From Lemma 5, we know that if
α1 y + α0 and B(E) = µ1 x + µ0 ν1 y + ν0 , A(E) = β1 x + β0 then α1 is an isomorphism, β1 is injective and ν1 is surjective. We also know that µ1 is injective, and since k2 = k = k4 , it must be an isomorphism. From Lemma 5, we also have that V3 = β1 (V1 ) ⊕ ker(ν1 ). Given any basis of V1 and ker(ν1 ), we can pick bases of V2 = α1 (V1 ), β1 (V1 ) and V4 = µ1 α1 (V1 ) so that α1 = −1k×k , µ1 = 1k×k ,
−1k×k , ν1 = −1k×k 0k×r . β1 = 0r ×k So we get ⎡
⎤ A−y A(E) = ⎣ B − x ⎦, D
and
B(E) = x + µ0 ν00 − y C .
Since 0 = B(E)A(E) = (µ0 A + ν00 B + C D) + (A − ν00 )x − (µ0 + B)y, we must have µ0 = −B, ν00 = A, and [A, B] + C D = 0. Notice that we did not use all the freedom we were given by Remark 1: the basis of Ck = V1 is still totally arbitrary. However, the basis of C2 = ker(ν1 ) is induced by the trivialization of E along P1 × {∞}. The residual freedom is Gl(k).
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Let us continue with K 0 , and normalize its monad. Some of the normalization is inherited from that of E. We know the inclusion K 0 → E gives surjections Φi : Wi → Vi . Consider the diagram α10 y + α00
W1 W4 β10 x + β00 W2 ⊗ O(−1, 1) µ01 x + µ00 ν10 y + ν00 ⊕ ⊗ ⊗ W3 ⊗ O O(−1, 0) O(0, 1) Φ1 Φ4 Φ23 = Φ2 ⊕ Φ3 ⎡ ⎤ A−y (45) ? ? ? ? ? ? ⎣B − x⎦ V2 ⊗ O(−1, 1) V4 V1 D ⊕ ⊗ ⊗
x−B A−y C V3 ⊗ O O(−1, 0) O(0, 1). From the coefficient of x y in the monad equations for K 0 and E, we get the commutative diagram
W1
α =∼ 1 -
? ? V1 ⊂
β10? ? µ V2 ∼1 =
β1
- W4
− ν0 1 --
Φ2
⊂
Φ1
µ01
⊂
Φ4
W3
Φ3
? ? - V3 .
? ? - V4
− ν1 --
α0 =∼ 1 -
W2
Let V¯1 be a copy of V1 in W1 so that Φ1 |V¯1 is an isomorphism. Set Z i := ker(Φi ). We have W1 = V¯1 ⊕ Z 1 , W2 = α10 (V¯1 ) ⊕ Z 2 .
(46) (47)
From Lemma 5, α10 is an isomorphism, β10 and µ01 are injective, and ν10 is surjective. While ν1 β1 is an isomorphism, hence V3 = Im(β1 ) ⊕ ker(ν1 ), at the level of K 0 , we still have ker(ν10 ) ∩ Im(β10 ) = {0} but the direct sum doesn’t fill all of W3 . Note that β10 (ker Φ1 ) ⊂ ker(Φ3 ). But more important is that β10 (V¯1 ) ∩ ker(Φ3 ) = {0} because β1 is injective. Restrict the monads in Diagram (45) to P1 × {∞}. From the display of those monads, we get information about the various Φi . First, from the exact sequence 0 → W1 ⊗ O(−1) → ker(B(K 0 |)) → K 0 | → 0 and its equivalent for E, we find in cohomology
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333
that the map H 0 (ker(B(K 0 |))) → H 0 (ker(B(E|))) is injective because it is exactly the map H 0 (K 0 |) → H 0 (E|). But from the sequence 0 → ker(B(K 0 |)) → W2 ⊗ O(−1) ⊕ W3 ⊗ O → W4 ⊗ O → 0 and its equivalent for E, we get the identifications H 0 (ker(B(K 0 |))) = ker(ν10 ) and H 0 (ker(B(E|))) = ker(ν1 ) compatible with the Φi , it must be that the restriction of Φ3 gives an isomorphism ker(ν10 ) → ker(ν1 ). Let L ⊂ Z 3 be a one-dimensional complement to β10 (Z 1 ) in Z 3 . Then W3 = β10 (V¯1 ) ⊕ β10 (Z 1 ) ⊕ ker(ν10 ) ⊕ L .
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Note that again, µ01 (Z 2 ) ⊂ Z 4 . Thus W4 = µ01 α10 (V¯1 ) ⊕ µ01 (Z 2 ) ⊕ ν10 (L).
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The basis we have for the Vi can be lifted to induce basis of V¯1 , α10 (V¯1 ), β10 (V¯1 ), and ker(ν10 ). We can then write
Φ1 = 1k×k 0k× j = Φ2 ,
1k×k 0k× j 0k×2 0k×1 Φ4 = 1k×k 0k× j 0k×1 , Φ3 = 02×k 02× j 12×2 02×1
µ01 α10 (V¯1 )
as in Diagram (36). Given any bases for Z 1 and L, we can pick basis of Z 2 , β10 (Z 1 ), and µ01 (Z 1 ) so that ⎡ ⎤ ⎡ ⎤ −1 0 10 −1 0 ⎢ 0 −1⎥ α10 = , β10 = ⎣ , µ01 = ⎣0 1⎦, ⎦ 0 0 0 −1 00 0 0 ⎡ ⎤ −1 0 0 0 ν10 = ⎣ 0 −1 0 0⎦. 0 0 0 −1 The coefficients of x and y in the monad equation for K 0 and the commutativity of diagram (45) force ⎤ ⎡ −B 0 A 0 ⎢ 0 0 ⎥ 0 α00 = 0 , µ0 = ⎣µ01 µ02 ⎦, A α01 µ003 µ004 ⎤ ⎡ ⎤ ⎡ B 0 A 0 C 0 0 0 ⎢−µ −µ ⎥ 0 C ν0 ⎥ 01 02 ⎥, ν 0 = ⎢ β00 = ⎢ ⎣ A α01 0 01 ⎦. ⎣ D 0 ⎦ 0 ν0 0 0 0 0 ν −µ03 −µ04 02 03
Restrict K 0 and E to y = = 0, take duals and tensor by O(−1). We have W ∗ ⊗ O(−1) - W ∗ ⊗ O(−1) ⊕ W ∗ ⊗ O - W ∗ ⊗ O 4
3
6
2
6
1
6
V4∗ ⊗ O(−1) - V3∗ ⊗ O(−1) ⊕ V2∗ ⊗ O - V1∗ ⊗ O.
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The isomorphism E ∗ |(−1) → K 0∗ |(−1) is mediated by the Φi∗ . From the display of the monads, we have H i (ker B ∗ (K 0 |)) = H i (K 0∗ |(−1)), and similarly for E. Hence we have the commutative diagram of exact sequences W1∗ - H 1 (K 0∗ |(−1)) → 0 A ∗ A∗ −
6 6 6∗ 0 ∗−
0 α01 ∼ Φ2∗ Φ1 =
0 →H 0 (K 0∗ |(−1)) - W2∗ 6 ∼ = 0 → H 0 (E ∗ |(−1))
- V∗ 2
hence
ker
- V∗ 1
A∗ −
- H 1 (E ∗ |(−1)) → 0,
A−
0 0 − → ker(A − ) A α01
0 − for all = 0. is an isomorphism through Φ1 . There can therefore be no kernel for α01 0 = 0. Hence the eigenvalues of α01 Now, restricted to y = 0, the sequence 0 → K 0 → E → E 0 /E 0+ → 0 becomes, for some T,
0 → T → K 0 | → E| → O( j) → 0. Since c1 (K 0 |) = 0, we have K 0 | = O(m)⊕O(−m). The exact sequence forces T = O( j). Then the injection O( j) → O(m) ⊕ O(−m) forces m = j, as seen in Remark 2. From the monad of K 0 |∗ (−1), A∗ O(−1) ⊗ W4∗ - O ⊗ W2∗ ⊕ O(−1) ⊗ W3∗ - O ⊗ W1∗ , we find that A∗ A ∗ C = H (K 0 | (−1))) = H (ker(A )) = coker 0 ∗ . 0 α01 j
1
∗
1
Thus it must be that
dim ker
∗
A 0 = j, 0 A α01
whence, since the map K 0 | → E| is zero on sections, A 0 is injective, and α01 = 0. A
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Since A(K 0 ) is injective for all (x, y) = (x, 0), it must be that 0 µ02 + x : Z 1 → α10 (Z 1 ) ⊕ L µ004 is injective for all x. Owing to Lemma 8 below, we can then choose a basis of Z 1 and α10 (Z 1 ) ⊕ L such that µ002 = −s and µ004 = −e+ ; recall Eq. (22).
Calorons, Nahm’s Equations on S 1 and Bundles over P1 × P1
Lemma 8 (Cyclicity). Suppose
335
T −λ is an injective map Cd → Cd+1 for all λ. Then v
(vT d−1 , . . . , vT, v) is a basis for Cd . Proof. The result is invariant under conjugation (T, v) → (P T P −1 , v P −1 ) and translation (T, v) → (T − λ, v), hence we only need to prove it for a matrix T in Jordan normal form, with one eigenvalue zero. In fact, the injectivity hypothesis forces all the blocks to have different eigenvalues. We can then finish the proof by induction on the number of Jordan blocks.
Now that µ002 = −s and µ004 = −e+ , we can perform elementary column operations on β00 to kill all but the first line B of −µ001 and all of −µ003 . Such column operations correspond to right multiplying by a matrix of the type 10 : V¯1 ⊕ Z 1 → V¯1 ⊕ Z 1 , ∗1 hence a repositioning of V¯1 in W1 , while keeping Z 1 fixed. Consider now the constant term of B(K 0 )A(K 0 ) = 0. Due to the splitting of W1 and W4 , we find six equations, three being Eqs. (27), (28), and (29), one being tautologically 0, and the remaining two being 0 0 = ν02 e+ : Z 1 → L → µ00 (Z 2 ), 0 0 = ν04 e+ : Z 1 → L → ν10 (L). 0 = 0 and ν 0 = 0. Once we consider that the flag at (∞, 0) lives in the second Hence ν02 04
0 = 1 0 in the appropriate basis. vector of the trivialization, we have ν03 We thus reduced the symmetries enough to establish the validity of Eq. (24), and of the fourth row of vertical maps in Diagram (36). Thus the monad for K 0 is as advertised, and the residual symmetry is Gl(k), isomorphic to the symmetry of the monad for E. Let us now continue with K ∞ , with an obvious translate in the notation. The problem is simplified as Z¯ 1 = Z¯ 2 = {0}, and dim Z¯ 3 = dim Z¯ 4 = 1. We thus have, for some lift F of ker(ν1 ),
W 2 = α1∞ (W 1 ),
W 3 = β1∞ (W 1 ) ⊕ F ⊕ Z¯ 3 , ∞ ¯ W 4 = µ∞ 1 α1 (W 1 ) ⊕ Z 4 .
Note that contrary to what happened for K 0 , we cannot choose F to be ker(ν1∞ ), as it contains Z¯ 3 . Indeed, the exact sequence (2) for K ∞ , restricted to y = ∞, becomes 0 → O → K ∞ | → E| → O → 0, whence the map H 0 (K 0 |) → H 0 (E|) has a one-dimensional kernel. Going through the same analysis as before, where H 0 (K 0 |) = ker(ν1∞ ) and H 0 (E|) = ker(ν1 ), we see that Φ¯ 3 restricted to ker(ν1∞ ) has a one-dimensional kernel, Z¯ 3 itself, as claimed. We lift the basis of the Vi to induce basis on all those pieces of the W i . We can then write
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B. Charbonneau, J. Hurtubise
Φ¯ 2 = 1k ,
Φ¯ 4 = 1k 0 , ⎡ ⎤ −1k 1 ∞ ∞ ∞ ⎣ ⎦ 0 , α1 = −1k , β1 = µ1 = k , 0 0 −1k 0 0 . ν1∞ = 0 0 −1 0 Φ¯ 1 = 1k , 1 0 0 , Φ¯ 3 = k 0 12 0
The commutativity of a diagram for K ∞ analogous to Diagram (45) and the coefficients of x and y in the monad equation B(K ∞ )A(K ∞ ) = 0 ensure ⎡ ⎤ B −B ∞ ⎣ ⎦, D , β = = α0∞ = A, µ∞ 0 0 −D2 ∞ β01 A C 0 ∞ ν0 = ∞ ν∞ . 0 ν01 02 Restricting the map K ∞ → E at y = 0, where it is an isomorphism, we have at the level of the cohomology of the monads that projection on the first two factors must be an isomorphism
A C 0 ker → ker A C . ∞ ∞ 0 ν01 ν02 ∞ = 0, and by choosing the For the projection to be an isomorphism, it must be that ν02 ∞ ∞, basis of Z¯ 3 and Z¯ 4 , we can have ν02 = 1. With column operations, we can kill ν01 hence repositioning F inside W 3 . The constant term of the monad equation then implies β01 = D2 A. We thus established the validity of Eq. (25), and the third row of vertical maps in Diagram (36).Thus the monad for K ∞ is as announced, with the same residual Gl(k) symmetry. Let us continue with K 0∞ (0, 1). Notice that K 0∞ (0, 1) is to K 0 what E is to K ∞ . Indeed, in a small neighborhood U intersecting y = ∞,
E(U ) =O(U ) ⊕ O(U ), K 0∞ (0, 1)(U ) =yO(U ) ⊕ O(U ),
and and
O(U ) ⊕ O(U ), y yO(U ) ⊕ O(U ). K 0 (U ) = y
K ∞ (U ) =
while
Also, K 0∞ (0, 1) is trivial on P1 × {∞} ∪ {∞} × P1 , and has a choice of a flag in the section at ∞. We can then use the monad of K 0 to extract the monad of K 0∞ (0, 1), once however we normalize it correctly. Staring at the monad given by Eq. (25), we see we want an expression for K 0 of the type ⎡ ⎤ A˜ − y ⎢ B˜ − x ⎥ x − B˜ A˜ − y C˜ 0 ⎢ ⎥ A2 (K 0 ) = ⎣ ˜ ⎦, B2 (K 0 ) = . 0 −y 1 − D˜ 2 0 D D˜ 2 A˜
Calorons, Nahm’s Equations on S 1 and Bundles over P1 × P1
337
To get an expression of this type, we set ⎡
⎡
⎤ 1 0 −C1 P := ⎣0 1 −C1 ⎦, 00 1
1 ⎢0 ⎢ ⎢0 ⎢ Q := ⎢0 ⎢0 ⎢ ⎣0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 0 C1 0 C1 0 0 1 0 0 1
⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0 ⎥, 1⎥ ⎥ 0⎦ 0
and A2 (K 0 ) := Q −1 A(K 0 ), B2 (K 0 ) := PB(K 0 )Q. Now deleting the last row and column of B2 (K 0 ) and the last row of A2 (K 0 ), we obtain the monad of Eq. (26), and establish the validity of the first row of vertical maps in Diagram (36). The only part of Theorem 5 that remains to be proved is the validity of the second row of vertical maps in Diagram (36). Notice the map from sections of K ∞ to sections K 0∞ (0, 1) is multiplication by y, as K 0∞ (0, 1) equals K 0∞ (P1 × {∞}), not K 0∞ (P1 × {0}). On the dense set {(x, y) ∈ P1 × P1 | y = 0 and (A − y) is invertible}, we can trivialize the bundles by sending γ ∈ C2 to ⎡
⎤ 0 ⎡ ⎤ 0 0 ⎢ ⎥ 0 ⎢ ⎥ −1 −(A − y)−1 Cγ ⎥ −(A − y) Cγ ⎥ ⎢ ⎣−(A − y)−1 Cγ ⎦, ⎢ , ⎢ −1 ⎥, ⎣ ⎦ γ −y A (A − y)−1 Cγ + y −1 C γ ⎥ ⎢ γ ⎣ ⎦ yγ2 γ −1 y γ1 ⎡ ⎤ 0 0 ⎢ ⎥ ⎢ ⎥ −(A − y)−1 (y −1 AC1 γ1 + C2 γ2 ) ⎢ ⎥ ⎢ −1 −1 ⎥ −1 −1 −1 ⎢ y (y A C1 γ1 + C2 γ2 ) − y A (A − y) (AC1 γ1 y + C2 γ2 )⎥ ⎣ ⎦ γ2 −1 y γ1 ⎡
⎤
respectively for E, K ∞ , K 0 , and K 0∞ (0, 1). This choice of trivialization is preserved by the various Φ23 of Diagram (36) whose validity we already established. For the proposed Φ23 of K ∞ → K 0∞ (0, 1), we have, using an obvious notation, Φ23 (γ K ∞ ) = yγ K 0∞ (0,1) . Since the candidate Φ23 is globally defined and agrees with the actual Φ23 on a dense subset of P1 × P1 , its validity is established. The commutativity of the diagram forces Φ1 and Φ4 to be as claimed in the second row of Diagram (36). The genericity conditions are simply those for monads implied by Buchdahl’s Beilinson’s theorem, along with the constraint on the matrix N which was proven in the previous section. The proof of Theorem 5 is now complete.
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4. From Monads to Sequences of Sheaves on P1 , and Back Again We show in this section how Diagram (6), encoding the bundle E and the flag, with the trivialization over y = ∞, leads one to a Nahm complex, and inversely how the Nahm complex encodes the diagram. We already have that the diagram gives the monads of Theorem 5 and morphisms between them, and conversely, monads of our normalized form give back the diagram of bundles. It thus suffices to show how our monad matrices encode, and are encoded by, a Nahm complex. The intermediary step we introduce are the exact sequences (7) of Lemma 1. More specifically, set P0 := R 1 π∗ (K 0 (0, −1)), P∞ := R 1 π∗ (K ∞ (0, −1)),
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Q ∞0 := R 1 π∗ (E(0, −1)), Q 0∞ := R 1 π∗ (K 0∞ ). Those exact sequences can now be read 0 0 0 0
→ O( j) → O → O → O(− j)
→ → → →
P0 P∞ P0 P∞
→ → → →
Q ∞0 Q ∞0 Q 0∞ Q 0∞
→ → → →
0, 0, 0, 0,
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with Q 0∞ , Q ∞0 torsion sheaves of length k + j, k respectively, supported away from infinity, while P0 , P∞ are trivialized over infinity on the line. We note that the sheaves come with resolutions, as given by Lemma 7. From Theorem 5, in the case j ≥ 1, we get a diagram of resolutions - Ok+ j+1 - P0 O(−1)k+ j 10 1 0 −C1 ⎡ ⎤ 0 1 −C1 01 C1 e+ x − B ⎣ −B ⎦ (x − s) + C1 e+ ? ? ? 0 - Ok+ j - Q 0∞ O(−1)k+ j 6 6 A 6 A C2 C A A 2 x−B −D 2 - Ok+1 - P∞ O(−1)k
1 10
? ? ? x−B - Ok - Q ∞0 O(−1)k 6
6
6 10 100 ⎡ ⎤ x − B 0 ⎢ −B ⎥ x − s⎦ ⎣ 0
O(−1)k+ j
0
−e+
- Ok+ j+1
- P0
- 0
- 0
- 0
- 0
- 0.
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The next step is to prove that these resolutions always exist in this form, given the sheaves fitting into Sequences (52). Lemma 9. Let j ≥ 1 and let P0 , P∞ , Q 0∞ , Q ∞0 be sheaves over P1 fitting into exact sequences (52), with Q 0∞ , Q ∞0 torsion of length k + j, k respectively, supported away from infinity in P1 . Then one has a commuting diagram of resolutions as Diagram (53). Furthermore, the matrix N defined by Eq. (43) using the matrices of the diagram is an isomorphism. Proof. If one takes the resolution 0
y - O(−1, −1) x −O
- O∆
- 0,
of the diagonal in P1 × P1 , lifts a sheaf F from P1 , tensors it with the resolution, and pushes down to the other factor, one obtains a resolution 0
-
H 0 (P1 , F(−1)) H 0 (P1 , F) α0 (F) + α1 (F)x ⊗ ⊗ O(−1) O [ f ] ⊗ g → [ f ] ⊗ xg − [y f ] ⊗ g
- F
- 0,
as long as H 1 (P1 , F(−1)) vanishes, which is the case for our sheaves. Taking the sheaves and the maps P0 → Q 0∞ ← P∞ → Q 0∞ ← P0 and applying this process to them, we obtain a diagram akin to Diagram (53), with the right sheaves. One must show that the maps can be normalized as advertised. We first note that as Q ∞0 is supported away from infinity, one can identify H 0 (P1 , Q ∞0 (−1)) = H 0 (P1 , Q ∞0 ) and normalize xα1 (Q ∞0 ) + α0 (Q ∞0 ) to x − B, for some B. The fact that P0 , P∞ are line bundles at infinity allow us to filter their sections by order of vanishing at infinity. We can split H 0 (P1 , P0 (−1)), and H 0 (P1 , P0 ) as sums H 0 (P1 , Q ∞0 ) ⊕ H 0 (P1 , O( j − 1)),
and
H 0 (P1 , Q ∞0 ) ⊕ H 0 (P1 , O( j)),
with the second summands being the kernels of projection to Q ∞0 , and the first identifying H 0 (P1 , Q ∞0 ) with the subspace of sections of P0 (−1), P0 vanishing at least to order j, j + 1 at infinity. The spaces H 0 (P1 , O( )) have natural bases of sections 1, . . . , y in terms of which the resolution naturally gets expressed in terms of the shift matrix s. Finally, for a class [ f ] ∈ H 0 (P1 , Q∞0 ) ⊂ H 0 (P1 , P0 (−1)), we have y [y f ] ∈ Im H 0 (P1 , O( j − 1)) - H 0 (P1 , O( j)) , hence the need for the matrix B . The last two lines of the diagram, and the maps between them, are then as advertised. Similarly for P∞ , we can write H 0 (P1 , P∞ (−1)), and H 0 (P1 , P∞ ) as H 0 (P1 , Q ∞0 ), and H 0 (P1 , Q ∞0 ) ⊕ H 0 (P1 , O), and show that the maps of the third row and between the third and fourth row of the diagram is of the form given for some row vector D2 . We then have isomorphisms H 0 (P1 , P0 (−1)) H 0 (P1 , Q 0∞ (−1)) showing us that we can take the map between the first elements of the first and second rows to be the identity. The isomorphism H 0 (P1 , Q 0∞ (−1)) H 0 (P1 , Q 0∞ ) shows us that we can ˜ for some B. ˜ The commutativity of the normalize xα1 (Q 0∞ ) + α0 (Q 0∞ ) to x − B,
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diagrams then tells us that B˜ is of the form given, and that the remaining maps are also of the form given, for suitable C1 , C1 , C2 , C2 , A, A . Use the monad equation (27) to set D1 = e+ A . The commutativity of Diagram (53) implies the monad equations (27) and (28). Finally, the fact that N is an isomorphism follows from the fact that the map P∞ ( j) → Q 0∞ must induce an isomorphism on sections, as in Diagram (44). The proof is now complete.
The genericity conditions on the matrices are equivalent to some non-degeneracy conditions on the sheaves of Eq. (51). We first note that one of the genericity conditions on the matrices is automatic if they come from our sheaves. Indeed, if the condition (32) is not satisfied (here j > 0), we have that for some x, ⎡ ⎤ A C 0 x − B ⎢ −B ⎥ A C x − s ⎥ Im⎢ (54) ⎣ 0 ⎦⊂V
0 0 1 0 −e+ with V a proper codimension one subspace of Ck+ j+1 . Then ⎤ ⎡ 0 x − B A C ⎥ ⎢ −B x − s ⎦ ⊂ V, Im 2 ⊂ (V ∩ Ck+ j ), Im⎣ 0 A C2 0 −e+ ⎛⎛ ⎡ ⎤ ⎡ ⎤⎞ ⎡ x − B 0 ⎤⎞ x − B C1 e+ 0 0 C1 ⎥⎟ ⎜⎝ ⎢ −B ⎣ ⎦ ⎣ ⎦ ⎠ B x − s ⎦⎠ Im = Im ⎝ 1 − 0 0 C1 ⎣ 0 x − s + C1 e+ − 0 00 1 0 −e+ ⊂ V ∩ Ck+ j . Hence by replacing the spaces in the second column by V, V ∩ Ck+ j , Ck+1 , Ck , and V , we can “reduce” the diagram at x: there are subsheaves P˜0 , Q˜ 0∞ which, together with P∞ and Q ∞0 , fit in a variation of Sequences (52) for which j is replaced by j − 1, and giving as quotients of P0 , Q 0∞ , P∞ , Q ∞0 the sheaves Cx , Cx , 0, 0. In particular, the map P∞ → Q 0∞ is not surjective and we have left our class of sheaves. The remaining conditions on our matrices do both translate into irreducibility for our diagram. Let us say that the “complex” of sheaves is reducible at x if either − Case 1. There are skyscraper subsheaves Cx , Cx , Cx , Cx of P0 , Q 0∞ , P∞ , Q ∞0 , localized at x, mapping to each other by Sequences (52). − Case 2. There are subsheaves P˜0 , Q˜ 0∞ , P˜∞ , Q˜ ∞0 of P0 , Q 0∞ , P∞ , Q ∞0 , fitting in Sequences (52), and giving as quotients the sheaves Cx , Cx , Cx , Cx . Translated to the world of resolutions, we can say that Diagram (53) is reducible at x if either − Case 1. There are, restricting at x so that we are dealing with vector spaces, one dimensional subspaces V1 , V2 , V3 , V4 , V5 = V1 of the spaces in the first column, (the subscript indicates the row) that lie in the kernel of the maps from the first column to the second and that are mapped to each other under the vertical maps; the spaces in the first columns can then be replaced by quotients, giving a “smaller” diagram;
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− Case 2. There are, restricting at x, codimension one subspaces V1 , V2 , V3 , V4 , V5 = V1 of the spaces in the second column, containing the images of the maps between the first and second column, and mapped to each other under the vertical maps, so that the diagram can then be “reduced” to a smaller one. We remark that it suffices to take dimension one or codimension one subspaces; other cases are reducible to this one, as can easily be checked. We can then see that the genericity conditions (30), (31) on the matrices are equivalent to Diagram (53) being irreducible at all x. − Case 1. Suppose there exists a one-dimensional subspace L ⊂ Ck such that ⎡ ⎤ A−y L ⊂ ker ⎣ B − x ⎦ D
(55)
for some x and y. The monad equations then imply B A + (s − x)A (L) = 0, (B − x)(L) = 0, 0 D2 (L) = 0,
A(L) ⊂ L ,
D1 (L) = e+ A (L) = 0.
Hence there are subspaces AA L, AA L, L, L, AA L of the kernels in all the exact sequences in Diagram (53), which are mapped to each other under the vertical maps. We can reduce the diagram at x. − Case 2. Suppose that for some x and y,
Im x − B A − y C ⊂ V, (56) with V a proper codimension one subspace of Ck , hence Im(x − B) ⊂ V, A(V ) ⊂ V,
Im(C1 ) ⊂ V, Im(C2 ) ⊂ V.
Hence by replacing the spaces in the second column by V ⊕C j+1 , V ⊕C j , V ⊕C, V , and V ⊕ C j+1 , we can again reduce the diagram at x Conversely, if the diagram is reducible at x, with a common kernel through the diagram, one finds an L as in Eq. (55). If the diagram is reducible at x, not with a common kernel, but with a common one-dimensional cokernel, we have at x codimension one subspaces V1 , V2 , V3 , V4 of the spaces in the second column, mapped to each other under the vertical maps and containing the images of the horizontal maps. Because codimV4 = 1, there must be a line W ⊂ Ck ⊂ Ck+1 so that V3 ⊕ W = Ck+1 and V4 ⊕ W = Ck . Since 1 0 0 V1 ⊂ V4 , we must have V1 = V4 ⊕C j+1 . To have codimV2 = 1, we must then have V2 = V4 ⊕C j . Since A C2 (V3 ) ⊂ V2 , A C2
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we have A C2 (V3 ) ⊂ V4 . In the decomposition V4 ⊕ W → V3 ⊕ W we can write
∗∗ A C2 = 0y
for some y ∈ C. Then Im A − y C2 ⊂ V 4 . Since V 2 contains the image of the second horizontal map, we must have as well Im x − B C1 ⊂ V4 . But then
Im A − y x − B C ⊂ V4 = C k , hence the non-degeneracy condition (31) is not satisfied. Theorem 7. There is an equivalence between 1) Matrices A, B (k × k), C (k × 2), D (2 × k), A ( j × k), B (1 × k), C ( j × 2), satisfying the monad equations (27), (28), (29), and the genericity conditions (30), (31), (32), (33), modulo the action of Gl(k, C) given by Eq. (35); 2) Exact sequences of sheaves 0 0 0 0
→ O( j) → → O → → O → → O(− j) →
P0 P∞ P0 P∞
→ → → →
Q ∞0 Q ∞0 Q 0∞ Q 0∞
→ → → →
0, 0, 0, 0,
(57)
on P1 , with Q 0∞ , Q ∞0 torsion sheaves of length k + j, k respectively, supported away from infinity, and with P0 , P∞ trivialized over infinity on the line, and irreducible. 5. To Nahm Complexes, and Back Again We next show that the sheaves fitting in the exact sequences (57) define, and are defined by, a Nahm complex on the circle. We define these for the integers k > 0, j ≥ 0, If j > 0, the Nahm complexes that we consider over the circle are defined by − A bundle V∞0 of rank k over the interval [θ0 , θ∞ ], equipped with a smooth connection α∞0 , and a covariant constant smooth section β∞0 of End(V∞0 ); − A bundle V0∞ of rank k + j over the interval [θ∞ , 2π + θ0 ], equipped with an smooth connection α0∞ on the interior, analytic near the boundary points, and a covariant constant section β0∞ of End(V0∞ ) smooth on the interior, analytic near the boundary points; − At the boundary point θ0 , an injection i 0 : V∞0 → V0∞ and a surjection π0 : V0∞ → V∞0 , such that π0 i 0 = I d, so that one can decompose V0∞ as ker(π0 )⊕Im(i 0 ). One asks that there be an extension of this decomposition to a trivialization on the interior of the interval such that one can write the connection α0∞ and the endomorphism β0∞ in block form as j−1 j−1 U (t) t 2 W (t) P(t) t 2 Q(t) α0∞ (t) = j−1 , β0∞ (t) = j−1 , t 2 V (t) X (t) t 2 R(t) S(t) where t is a local parameter with the point θ0 corresponding to t = 0. The top left blocks are k × k, the bottom right block is j × j; U, W, V, P, Q, R are analytic at
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t = 0, and X, S are meromorphic with simple poles at t = 0, and residues conjugate to ( j − 1) −( j − 1) 2 − ( j − 1) X −1 = diag , ,..., , (58) 4 4 4 ⎤ ⎡ 0 0 0 ... 0 0 ⎢ 1 0 0 ... 0 0 ⎥ ⎥ ⎢ (59) S−1 = ⎢ 0 1 0 . . . 0 0 ⎥. ⎣. . . . . . . . . . . . . . .⎦ 0 0 0 ... 1 0 Furthermore, U (0) = α∞0 (0) P(0) = β∞0 − At the boundary point θ∞ , the boundary conditions are the same as at θ0 . − At both boundary points, some extra data, consisting of a trivialization (choice of vectors v0 , v∞ ) of the −( j−1) eigenspace of X −1 . 4 For j = 0, the constraints are simpler: the Nahm complexes over the circle that we consider are defined by − A bundle V∞0 of rank k over the interval [θ0 , θ∞ ], equipped with a smooth connection α∞0 , and a covariant constant smooth section β∞0 of End(V∞0 ); − A bundle V0∞ of rank k over the interval [θ∞ , 2π + θ0 ], equipped with an smooth connection α0∞ and a covariant constant smooth section β0∞ of End(V0∞ ). − At the boundary point θ0 , isomorphisms i 0 : V∞0 → V0∞ , π0 = i 0−1 with the gluing condition that β∞0 − π0 β0∞ i 0 has rank one. −1 with the − At the boundary point θ∞ , isomorphisms i ∞ : V∞0 → V0∞ , π∞ = i ∞ gluing condition that β∞0 − π∞ β0∞ i ∞ has rank one. − At both boundary points, extra data consisting of decompositions v0 = (u 0 , w0 ), v∞ = (u ∞ , w∞ ) of the rank one boundary difference matrices β∞0 − π0 β0∞ i 0 , β∞0 − π∞ β0∞ i ∞ into products of a column and a row vector: β∞0 − π0 β0∞ i 0 = u 0 · w0 , β∞0 − π∞ β0∞ i ∞ = u ∞ · w∞ .
(60)
There is a group G of gauge transformations which acts on the complex and that can be used to normalize the complex as in the lemma below. This group is constructed as follows: one takes smooth g∞0 (z) ∈ Aut (V∞0 ) on [θ0 , θ∞ ], g0∞ (z) ∈ Aut (V0∞ ) on [θ∞ , 2π + θ0 ] with, on the “large” side of the boundary points, in the trivialisations used above, the constraint that g0∞ (z) be analytic, with a decomposition j+1 K (t) t 2 L(t) g0∞ (t) = j+1 t 2 M(t) N (t) with K , L , M, N analytic at t = 0, and K (0) = g∞0 (0). The group G acts as 1 −1 ˙ , gβg −1 . g · (α, β) = gαg −1 − gg 2 Lemma 10 (Prop 1.15 of [17]).
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− Away from the boundary points, or even at the boundary points, if one is on V∞0 , one can gauge to α = 0, β = constant. − At the boundary, over V0∞ , one can gauge transform to the block form 0 1 0 α0∞ = , (61) −( j−1) 2−( j−1) , , . . . , ( j−1) t 0 diag 4 4 4 ⎡ ⎤ j−1 t 2 q0 e+ P0 ⎦. (62) β0∞ = ⎣ j−1 r0 −t −1 s + t j−1 s˜0 e+ t 2 0 Here P0 is k × k, r0 is 1 × k, q0 is k × 1, s˜0 is j × 1 and P0 , r0 , q0 are constant in t, and, setting (˜s0 )i = t i−1 (s0 )i , then s0 is also constant in t. − Using the gauge transformation G(t) := diag(1, . . . , 1, t (− j+1)/2 , t (− j+3)/2 , . . . , t ( j−1)/2 ), (which does not lie in our gauge group), we transform further to α0∞ = 0, and ⎡ ⎤ P0 q0 e+ ⎦. β0∞ = ⎣ r0 (63) −s + s0 e+ 0 These normal forms are unique up to the action of Gl(k, C), if in addition one asks that the “extra data” vector v be mapped to the (k + 1)th basis vector in the normal form. One now must create a Nahm complex from the data of the sheaves and the exact sequences. On the interior of the first interval, we use the matrix B coming from Diagram (53) and set V∞0 = (θ0 , θ∞ ) × H 0 (P1 , Q ∞0 ), α∞0 = 0, β∞0 = −B.
(64)
In the same vein, we set V0∞ = (θ∞ , 2π + θ0 ) × H 0 (P1 , Q 0∞ ), α0∞ = 0, ⎡ ⎤ −B C1 e+ ⎦ = −M. β0∞ = ⎣ −B −s + C1 e+ 0
(65)
Note that the endomorphism is already in the normal form (63) given by Lemma 10. There remains the gluing on the ends of the interval, which involves introducing some form of monodromy, as we are on a circle. The gluing will be mediated by the sheaves P0 , P∞ . The basic trick is that, for n ≤ m, there are inclusions 0 f m,n : H 0 (P1 , P0 (n)) → H 0 (P1 , P0 (m)), ∞ f m,n : H 0 (P1 , P∞ (n)) → H 0 (P1 , P∞ (m)),
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as sections vanishing to an appropriate order at infinity. In addition, there are maps arising from the exact sequences (57), n 0 : H 0 (P1 , P0 ( )) → H 0 (P1 , Q ∞0 ),
0 1 0 1 n∞ : H (P , P∞ ( )) → H (P , Q ∞0 ),
m 0 : H 0 (P1 , P0 ( )) → H 0 (P1 , Q 0∞ ),
0 1 0 1 m∞ : H (P , P∞ ( )) → H (P , Q 0∞ ).
∞ The maps m 0−1 , n 0− j−1 , m ∞ j−1 , n −1 are isomorphisms. Schematically,
1 0 j− n−
-
∼=
H 0 (Q ∞0 ) = Ck n∞ −1 =∼
H 0 (P0 (− j − 1))
H 0 (P∞ (−1)) ∩
∩
∞ f j−1,−1 ? H 0 (P∞ ( j − 1))
0 f −1,− j−1
? H 0 (P0 (−1))
m0
−1
=∼
∼= ∞ −1 j m 0 k+ j H (Q 0∞ ) = C .
At θ0 , we define maps π0 : V0∞ (θ0 ) = H 0 (P1 , Q 0∞ ) → V∞0 (θ0 ) = H 0 (P1 , Q ∞0 ), i 0 : V∞0 (θ0 ) = H 0 (P1 , Q ∞0 ) → V0∞ (θ0 ) = H 0 (P1 , Q 0∞ ) 0 0 0 0 −1 −1 by i 0 = m 0−1 ◦ f −1,− j−1 ◦ (n − j−1 ) , π0 = n −1 ◦ (m −1 ) . The composition π0 ◦ i 0 is the identity. Similarly, at θ∞ , we define maps
π∞ : V0∞ (θ∞ ) = H 0 (P1 , Q 0∞ ) → V∞0 (θ∞ ) = H 0 (P1 , Q ∞0 ), i ∞ : V∞0 (θ∞ ) = H 0 (P1 , Q ∞0 ) → V0∞ (θ∞ ) = H 0 (P1 , Q 0∞ ) ∞ ∞ −1 ∞ ∞ −1 by i ∞ = m ∞ j−1 ◦ f j−1,−1 ◦ (n −1 ) , π∞ = n j−1 ◦ (m j−1 ) . Again, π∞ ◦ i ∞ is the identity. In the bases used in Diagram (53), one has the block decompositions
1 , π0 = 1 0 , i0 = 0
1 A = N · , π∞ = 1 0 · N −1 . i∞ = 0 A
We set C˜ 1 −1 j C2 := N M C2 C˜ 1
⎤ ˜ −B C1 e+ ⎦, := ⎣ −D2 −s + C˜ 1 e+ 0 ⎡
and
β˜0∞
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and then the monad equations imply β0∞ N = N β˜0∞ . We can then introduce N as the parallel transport from θ∞ to θ0 +2π over the big side, as well as introducing the necessary poles. Indeed over the interval (θ∞ , 2π + θ0 ), we begin with the complex α0∞ , β0∞ of Eq. (65), and then gauge it with a transformation g, given by choosing a smooth path h(θ ) in Gl(k + j, C) equal to N at θ∞ + 2 and the identity at 2π + θ0 − 2 , and setting ⎧ ⎨ G(θ − θ∞ ) ◦ N −1 , t ∈ (θ∞ , θ∞ + ), t ∈ (θ∞ + 2 , 2π + θ0 − 2 ), g(θ ) = h(θ ), ⎩ G(2π + θ0 − θ )−1 , t ∈ (2π + θ0 − , 2π + θ0 ). We then smooth g over the remaining small intervals, so that the result is C ∞ . Applying g to our Nahm complex over the interval, we obtain ⎛ ⎡ ⎤ ⎞ −B C1 e+ 1 ⎦g −1 ⎠ . (γ0∞ , δ0∞ ) = ⎝− gg ˙ −1 , g ⎣ B −s + C1 e+ − 2 0 Under the gauge transformation g, the gluing maps become
1 , and π0 = π∞ = 1 0 . i0 = i∞ = 0 The Nahm complex over the circle associated to the complex of sheaves is then given by (α∞0 , β∞0 ) = (0, B), (α0∞ , β0∞ ) = (γ0∞ , δ0∞ ), 1 1 , i∞ = , i0 = 0 0
π0 = 1 0 , π∞ = 1 0 . The “extra data” vectors are obtained from the trivializations of P0 , P∞ at infinity. Conversely, given a Nahm complex, we can recover the matrix data, and hence the sheaves, by first gauging α∞0 to zero, and setting B = β∞0 . Next gauging (α0∞ , β0∞ ) to their normal form near the poles, as in the lemma above, so that α0∞ = 0 near the poles, gives the matrix data B , C1 , C1 from the normal form near θ0 , and D2 from the normal form near θ∞ . The gauge transformation relating the two normal forms (that is the integration of the connection α0∞ between θ0 and θ∞ ) is the matrix N defined above; from it, one can recuperate A, A , C2 , C2 . Finally, setting D1 = e+ A gives us the remaining data. The fact that the matrix N conjugates one normal form to the other then yields back the monad equations. For j = 0, the correspondence is much simpler. We build our bundles, and the maps i 0 , π0 , i ∞ , π∞ in the same way. The resolutions for Q ∞0 , Q 0∞ then give us matrices ˜ 0 , B − i ∞ Bπ ˜ 0 are of rank one. In ˜ and it is straightforward to see that B − i 0 Bπ B, B, the trivialization given in the previous sections, i 0 = 1, i ∞ = A.
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We then have, as above, the rank one jumps B − B˜ = C1 D1 A−1 ,
and
B − A−1 B˜ A = −A−1 C2 D2 .
One defines the Nahm complex by choosing a path g(t) from the identity to A, and setting 1 −1 ˙ , β∞0 = g Bg −1 . α∞0 = 0, β∞0 = B, α0∞ = − gg 2
(66)
Again, the trivializations of P0 , P∞ at infinity give us the normalizations of the decompositions of the jumps as a product of a column and a row. Conversely, from the Nahm complex, it is straightforward to extract the matrix information, and so the sheaves. The final correspondence which must be checked is the irreducibility conditions. A Nahm complex over the circle is reducible if there exists a subbundle of each V∗ , parallel for the α∗ and invariant under the β∗ , mapping to each other by the gluing maps at the boundary point, and proper on at least one interval. Let us consider the three cases of reducibility for the complex of sheaves. Because of the way the gluing maps, the connections α0∞ , α∞0 and the endomorphisms β0∞ , β∞0 are built for the cohomology of the P∗ and Q ∗ , we can see that − Case 1 corresponds to the existence of a sub-line bundle of the V∗ , invariant and parallel, − Case 2 corresponds to the existence of a co-rank 1 subbundle of the V∗ , invariant and parallel, Summarizing: Theorem 8. Let k ≥ 1, j ≥ 0 be integers. There is an equivalence between 1) Exact sequences of sheaves 0 0 0 0
→ O( j) → O → O → O(− j)
→ → → →
P0 P∞ P0 P∞
→ → → →
Q ∞0 Q ∞0 Q 0∞ Q 0∞
→ 0, → 0, → 0, → 0
on P1 , with Q 0∞ , Q ∞0 torsion sheaves of length k + j, k respectively, supported away from infinity, and with P0 , P∞ trivialized over infinity on the line, and irreducible. 2) Irreducible Nahm complexes α, β on the circle, with rank k over (θ0 , θ∞ ), rank k + j over (θ∞ , 2π + θ0 ), with the boundary conditions defined above, modulo the action of the complex gauge group. The last major step is to pass from Nahm complexes to solutions of Nahm’s equations. These equations are obtained by adding to the covariant constancy condition dβ + [α, β] = 0, dt
(67)
d(α + α ∗ ) + [α, α ∗ ] + [β, β ∗ ] = 0. dt
(68)
the additional “real” equation
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B. Charbonneau, J. Hurtubise
These equations are invariant under unitary gauge transformations. One then has the theorem that orbits of irreducible Nahm complexes under the action of the complex gauge group contain a unique solution to Nahm’s equations, up to the action of the unitary gauge group. The idea of the proof, due to Donaldson [10], is to give a variational formulation to the equations, and to show that each orbit contains a unique critical point. The proof given in [17, Sect. 2] in the context of SU (N ) monopoles extends verbatim to the case we consider here, with one main difference, that of irreducibility. In [17, Sect. 2], the irreducibility is automatic, because of the pole structure. Here, as we have seen for the Nahm complexes, the irreducibility must be put in as a supplementary condition. We note that, with the addition of a metric structure, the notions of codimension one invariant subbundle and dimension one invariant subbundle fuse as one solves the variational problem, into a dimension one subbundle, invariant under the Ti ; in other words, minimizing the energy takes one from a block upper triangular form or block lower triangular form to a block diagonal form. This last equivalence, combined with Theorems 6, 7, and 8, provides a proof of our main result, Theorem 1, which we now rewrite in the language we absorbed throughout our journey. Theorem 9. Let k ≥ 0, j ≥ 0 be integers. There is an equivalence between 1) Vector bundles E of rank two on P1 × P1 , with c1 (E) = 0, c2 (E) = k, trivialized along P1 × {∞} ∪ {∞} × P1 , and with a based flag φ : O(− j) → E of degree j along P1 × {0} (up to non-zero scalar multiple), with the basing condition φ(∞)(O(− j)) = span(0, 1), and 2) Irreducible solutions to Nahm’s equations α, β on the circle, with rank k over (θ0 , θ∞ ), rank k + j over (θ∞ , 2π + θ0 ), with the boundary conditions defined above, modulo the action of the unitary gauge group. The case k = 0 is a version of a theorem of Donaldson [10], and the other cases have been dealt with above. Acknowledgements. This research was conducted while the first author was at McGill University and was supported by an NSERC PDF. He wishes to thank Alexandru Ghitza for useful discussions. The second author is supported by NSERC and FQRNT grants. The diagrams in this paper were created using Paul Taylor’s Commutative Diagram package.
References 1. Atiyah, M.F.: Instantons in two and four dimensions. Commun. Math. Phys. 93(4), 437–451 (1984) 2. Atiyah, M.F., Drinfel’d, V.G., Hitchin, N.J., Manin, Yu.I.: Construction of instantons. Phys. Lett. A 65(3), 185–187 (1978) 3. Bruckmann, F., Nógrádi, D., van Baal, P.: Constituent monopoles through the eyes of fermion zero-modes. Nucl. Phys. B 666(1–2), 197–229 (2003) 4. Bruckmann, F., Nógrádi, D., van Baal, P.: Higher charge calorons with non-trivial holonomy. Nucl. Phys. B 698(1–2), 233–254 (2004) 5. Bruckmann, F., van Baal, P.: Multi-caloron solutions. Nucl. Phys. B 645(1–2), 105–133 (2002) 6. Buchdahl, N.P.: Stable 2-bundles on Hirzebruch surfaces. Math. Z. 194(1), 143–152 (1987) 7. Charbonneau, B., Hurtubise, J.: The Nahm transform for calorons. to appear in Proceedings of Hitchin Birthday Conference, 25 pages, 2007. http://arxiv.org/abs/0705.2412, 2007 8. Corrigan, E., Goddard, P.: Construction of instanton and monopole solutions and reciprocity. Ann. Phys. 154(1), 253–279 (1984) 9. Donaldson, S.K.: Instantons and geometric invariant theory. Commun. Math. Phys. 93(4), 453–460 (1984) 10. Donaldson, S.K.: Nahm’s equations and the classification of monopoles. Commun. Math. Phys. 96(3), 387–407 (1984)
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11. Garland, H., Murray, M.K.: Kac–Moody monopoles and periodic instantons. Commun. Math. Phys. 120(2), 335–351 (1988) 12. Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley-Interscience [John Wiley & Sons], 1978 13. Harnad, J., Shnider, S., Vinet, L.: The Yang–Mills system in compactified Minkowski space; invariance conditions and SU(2) invariant solutions. J. Math. Phys. 20(5), 931–942 (1979) 14. Harnad, J., Vinet, L.: On the U(2) invariant solutions to Yang–Mills equations in compactified Minkowski space. Phys. Lett. B 76(5), 589–592 (1978) 15. Hartshorne, R.: Algebraic geometry. New York: Springer-Verlag, 1977 16. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55(1), 59–126 (1987) 17. Hurtubise, J.: The classification of monopoles for the classical groups. Commun. Math. Phys. 120(4), 613–641 (1989) 18. Jardim, M.: A survey on Nahm transform. J. Geom. Phys. 52(3), 313–327 (2004) 19. Jarvis, S.: Euclidean monopoles and rational maps. Proc. London Math. Soc. 77(1), 170–192 (1998) 20. Kempf, G.R.: Algebraic varieties. Volume 172 of London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1993 21. Kraan, T.C.: Instantons, monopoles and toric hyperKähler manifolds. Commun. Math. Phys. 212(3), 503–533 (2000) 22. Kraan, T.C., van Baal, P.: Exact T -duality between calorons and Taub-NUT spaces. Phys. Lett. B 428(3– 4), 268–276 (1998) 23. Kraan, T.C., van Baal, P.: Monopole constituents inside SU (n) calorons. Phys. Lett. B 435, 389–395 (1998) 24. Kraan, T.C., van Baal, P.: Periodic instantons with non-trivial holonomy. Nucl. Phys. B 533(1–3), 627–659 (1998) 25. Lee, K.: Instantons and magnetic monopoles on R3 × S 1 with arbitrary simple gauge groups. Phys. Lett. B 426(3–4), 323–328 (1998) 26. Lee, K., Lu., C.: SU(2) calorons and magnetic monopoles. Phys. Rev. D 58, 025011 (1998) 27. Lee, K., Yi, S.-H.: 1/4 BPS dyonic calorons. Phys. Rev. D 67(2), 025012 (2003) 28. Nahm, W.: Self-dual monopoles and calorons. In: Group theoretical methods in physics (Trieste, 1983), Volume 201 of Lecture Notes in Phys., Berlin: Springer, 1984, pp. 189–200 29. Nahm, W.: All self-dual multimonopoles for arbitrary gauge groups. In: Structural elements in particle physics and statistical mechanics (Freiburg, 1981), Volume 82 of NATO Adv. Study Inst. Ser. B: Physics, New York: Plenum 1983, pp. 301–310 30. Norbury, P.: Periodic instantons and the loop group. Commun. Math. Phys. 212(3), 557–569 (2000) 31. Nye, T.M.W.: The geometry of calorons. PhD thesis, University of Edinburgh, 2001, available at http:// arxiv.org/abs/hep-th/0311215, 2003 32. Nye, T.M.W., Singer, M.A.: An L 2 -index theorem for Dirac operators on S 1 × R3 . J. Funct. Anal. 177(1), 203–218 (2000) 33. Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces, Volume 3 of Progress in Mathematics. Boston, Mass: Birkhäuser, 1980 34. Ward, R.S.: A Yang–Mills–Higgs monopole of charge 2. Commun. Math. Phys. 79(3), 317–325 (1981) Communicated by G.W. Gibbons
Commun. Math. Phys. 280, 351–388 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0470-0
Communications in
Mathematical Physics
A Probabilistic Approach to Zhang’s Sandpile Model Anne Fey-den Boer1 , Ronald Meester1 , Corrie Quant1 , Frank Redig2 1 Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands.
E-mail: [email protected]; [email protected]; [email protected]
2 Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden,
The Netherlands. E-mail: [email protected] Received: 12 January 2007 / Accepted: 6 November 2007 Published online: 28 March 2008 – © The Author(s) 2008
Abstract: The current literature on sandpile models mainly deals with the abelian sandpile model (ASM) and its variants. We treat a less known - but equally interesting - model, namely Zhang’s sandpile. This model differs in two aspects from the ASM. First, additions are not discrete, but random amounts with a uniform distribution on an interval [a, b]. Second, if a site topples - which happens if the amount at that site is larger than a threshold value E c (which is a model parameter), then it divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the infinite volume limit, this model tends to behave like the ASM in the sense that the stationary measure for the system in large volumes tends to be peaked narrowly around a finite set. This belief is supported by simulations, but so far not by analytical investigations. We study the stationary distribution of this model in one dimension, for several values of a and b. When there is only one site, exact computations are possible. Our main result concerns the limit as the number of sites tends to infinity. We find that the stationary distribution, in the case a ≥ E c /2, indeed tends to that of the ASM (up to a scaling factor), in agreement with Zhang’s conjecture. For the √ case a = 0, b = 1 we provide strong evidence that the stationary expectation tends to 1/2. 1. Introduction and Main Results With the introduction of the sandpile model by Bak, Tang and Wiesenfeld (BTW), the notion of self-organized criticality was introduced, and subsequently applied to several other models such as forest-fire models, and the Bak-Sneppen model for evolution. In turn, these models serve as a paradigm for a variety of natural phenomena in which, empirically, power laws of avalanche characteristics and/or correlations are found, such as the Gutenberg-Richter law for earthquakes. See [12] for a extended overview. After the work of Dhar [2], the BTW model was later renamed ‘abelian sandpile model’ (ASM), referring to the abelian group structure of addition operators. This abelianness has since served as the main tool of analysis for this model.
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A less known variant of the BTW-model has been introduced by Zhang [13], where instead of discrete sand grains, continuous height variables are used. This lattice model is described informally as follows. Consider a finite subset ⊂ Zd . Initially, every lattice site i ∈ is given an energy 0 ≤ E i < E c , where E c is the so called critical threshold, and often chosen to be equal to 1. Then, at each discrete time step, one adds a random amount of energy, uniformly distributed on some interval [a, b] ⊂ [0, E c ], at a randomly chosen lattice site. If the resulting energy at this site is still below the critical value then we have arrived at the new configuration. If not, an avalanche is started, in which all unstable sites (that is, sites with energy at least E c ) ‘topple’ in parallel, i.e., give a fraction 1/2d of their energy to each neighbor in . As usual in sandpile models, upon toppling of boundary sites, energy is lost. As in the BTW-model, the stabilization of an unstable configuration is performed instantaneously, i.e., one only looks at the final stable result of the random addition. In his original paper, Zhang observes, based on results of numerical simulation (see also [6]), that for large lattices, the energy variables in the stationary state tend to concentrate around discrete values of energy; he calls this the emergence of energy ‘quasi-units’. Therefore, he argues that in the thermodynamic limit, the stationary dynamics should behave as in the discrete ASM. However, Zhang’s model is not abelian (the next configuration depends on the order of topplings in each avalanche; see below), and thus represents a challenge from the analytical point of view. There is no mentioning of this fact in [6,13], see however [8]; probably they chose the usual parallel order of topplings in simulations. After its introduction, a model of Zhang’s type (the toppling rule is the same as Zhang’s, but the addition is a deterministic amount larger than the critical energy) has been studied further in the language of dynamical systems theory in [1]. The stationary distributions found for this model concentrate on fractal sets. Furthermore, in these studies, emergence of self-organized criticality is linked to the behavior of the smallest Lyapounov exponents for large system sizes. From the dynamical systems point of view, Zhang’s model is a non-trivial example of an iterated function system, or of a coupled map lattice with strong coupling. In this paper we rigorously study Zhang’s model in dimension d = 1 with probabilistic techniques, investigating uniqueness and deriving certain properties of the stationary distribution. Without loss of generality, we take E c = 1 throughout the paper. In Sect. 2 we rigorously define the model for d = 1. We show that in the particular case of d = 1 and stabilizing after every addition, the topplings are in fact abelian, so that the model can be defined without specifying the order of topplings. In that section, we also include a number of general properties of stationary distributions. For instance, we prove that if the number of sites is finite, then every stationary distribution is absolutely continuous with respect to Lebesgue measure on (0, 1), in contrast with the fractal distributions for the model defined in [1] (where the additions are deterministic). We then study several specific cases of Zhang’s model. For each case, we prove by coupling that the stationary distribution is unique. In Sect. 4, we explicitly compute the stationary distribution for the model on one site, with a = 0, by reducing it to the solution of a delay equation [3]. Our main result is in Sect. 5, for the model with a ≥ 1/2. We show that in the infinite volume limit, every one-site marginal of the stationary distribution concentrates on a non-random value, which is the expectation of the addition distribution (Theorem 5.5). This supports Zhang’s conjecture that in the infinite volume limit, his model tends to behave like the abelian sandpile. Section 5 contains a number of technical results necessary for proving Theorem 5.5, but which are also of independent interest. For instance,
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we construct a coupling of the so-called reduction of Zhang’s model to the abelian sandpile model, and we prove that any initial distribution converges exponentially fast to the stationary distribution. In Sect. 6, we treat the model for [a, b] = [0, 1]. We present simulations that indicate the emergence of quasi-units also for this case. However, since in this case there is less correspondence with the abelian sandpile model, we cannot fully prove this. We can prove that the stationary distribution is unique, and we show that if every one-site marginal of the stationary distribution tends to the same value in the infinite volume limit, and √in addition if there is a certain amount of asymptotic independence, then this value is 1/2. This value is consistent with our own simulations. 2. Model Definition We define Zhang’s model in dimension one as a discrete-time Markov process with state space N := [0, 1){1,2,...,N } ⊂ [0, ∞){1,2,...,N } := N , endowed with the usual sigma-algebra. We write η, ξ ∈ N , configurations of Zhang’s model and η j for the j th coordinate of η. We interpret η j as the amount of energy at site j. By Pη , we denote the probability measure on (the usual sigma-algebra on) the path space N N for the process started in η. Likewise we use Pν when the process is started from a probability measure ν on N , that is, with initial configuration chosen according to ν. The configuration at time t is denoted as η(t) and its j th component as η j (t). We next describe the evolution of the process. Let 0 ≤ a < b ≤ 1. At time 0 the process starts in some configuration η ∈ N . For every t = 1, 2, . . ., the configuration η(t) is obtained from η(t − 1) as follows. At time t, a random amount of energy Ut , uniformly distributed on [a, b], is added to a uniformly chosen site X t ∈ {1, . . . , N }, hence P(X t = j) = 1/N for all j = 1, . . . , N . We assume that Ut and X t are independent of each other and of the past of the process. If, after the addition, the energies at all sites are still smaller than 1, then the resulting configuration is in N and this is the new configuration of the process. If however after the addition the energy of site X t is at least 1 - such a site is called unstable - then this site will topple, i.e., transfer half of its energy to its left neighbor and the other half to its right neighbor. In case of a toppling of a boundary site, this means that half of the energy disappears. The resulting configuration after one toppling may still not be in N , because a toppling may give rise to other unstable sites. Toppling continues until all sites have energy smaller than 1 (i.e., until all sites are stable). This final result of the addition is the new configuration of the process in N . The entire sequence of topplings after one addition is called an avalanche. We call the above model the (N , [a, b])-model. We use the symbol Tx (ξ ) for the result of toppling of site x in configuration ξ ∈ N . We write Au,x (η) for the result of adding an amount u at site x of η, and stabilizing through topplings. It is not a priori clear that the process described above is well defined. By this we mean that it is not a priori clear that every order in which we perform the various topplings leads to the same final configuration η(t). In fact, unlike in the abelian sandpile, topplings according to Zhang’s toppling rule are not abelian in general. To give an example of nonabelian behavior, let N = 2 and ξ = (1.2, 1.6). Then T1 (T2 (ξ )) = T1 ((2, 0)) = (0, 1), whereas T2 (T1 (ξ )) = T2 ((0, 2.2)) = (1.1, 0).
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Despite this non-abelianness of certain topplings, we will now show that in the process defined above, we only encounter avalanches that consist of topplings with the abelian property. When restricted to a certain subset of N , topplings are abelian, and it turns out that this subset is all we use. (In particular, the example that we just gave cannot occur in our process.) Proposition 2.1. The (N , [a, b])-model is well defined. To prove this, we will need the following lemma. First we introduce some notation. We call a site j of a configuration η: empty nonempty unstable
if η j = 0, if η j ∈ (0, 1), if η j ≥ 1.
˜ N ⊂ N be the set of all (possibly unstable) configurations such that Lemma 2.2. Let between every pair of unstable sites there is at least one empty site, and such that the energy of any unstable site is strictly smaller than 2. During stabilization after an addition to a configuration in N , only configurations ˜ N are encountered. in ˜ N , the resulting configuration Proof. We first prove that for every configuration η˜ ∈ ˜ ˜ N can after toppling of one of the unstable sites is still in N . An unstable site i of η˜ in have either two empty neighbors (first case), two nonempty neighbors (second case) or one nonempty and one empty (third case). In the first case, toppling of site i cannot create a new unstable site, since 21 η˜ i < 1, but i itself becomes empty. Thus, if there were unstable sites to the left and to the right of i, after the toppling there still is an empty site between them. In the second and third case, the nonempty neighbor(s) of i can become unstable. Suppose the left neighbor i − 1 becomes unstable. Directly to its right, at i, an empty site is created. To its left, there was either no unstable site, or first an empty site and then somewhere an unstable site. The empty site can not have been site i − 1 itself, because to have become unstable it must have been nonempty. For the right neighbor the same ˜ N. argument applies. Therefore, the new configuration is still in ˜ Since N ⊂ N , and since by making an addition to a stable configuration, we arrive ˜ N , the above argument shows that in the process of stabilization after addition to a in ˜ N are encountered. stable configuration, only configurations in Proof of Proposition 2.1. By Lemma 2.2, we only need to consider configurations in ˜ N and i and j are unstable sites in η, then ˜ N . Now we show that, if η ∈ Ti (T j (η)) = T j (Ti (η)).
(1)
To prove this, we consider all different possibilities for x. If x is not a neighbor of either i or j, then toppling of i or j does not change ηx , so that (1) is obvious. If x is equal to i or j, or neighbor to only one of them, then only one of the topplings changes ηx , so ˜ N, that again (1) is obvious. Finally, if x is a neighbor of both i and j, then, since η ∈ x must be empty before the topplings at i and j. We then have T j (η)x =
1 ηj, 2
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so that Ti (T j (η))x =
1 1 η j + ηi = T j (Ti (η))x . 2 2
Therefore, also in this last case (1) is true. Having established that the topplings of two unstable sites commute, it follows that the final stable result after an addition is independent of the order in which we topple, and hence Au,x (η(t)) is well-defined; see [7], Sect. 2.3 for a proof of this latter fact. Remark 2.3. It will be convenient to order the topplings in so-called waves [9]. Suppose the addition of energy at time t takes place at site k and makes this site unstable. In the first wave, we topple site k and then all other sites that become unstable, but we do not topple site k again. After this wave only site k can possibly be unstable. If site k is unstable after this first wave, the second wave starts with toppling site k (for the second time) and then all other sites that become unstable, leaving site k alone, until we reach a configuration in which all sites are stable. This is the state of the process at time t. It is easy to see that in each wave, every site can topple at most once. 3. Preliminaries and Technicalities In this section, we discuss a number of technical results which are needed in the sequel, and which are also interesting in their own right. The section is subdivided into three subsections, dealing with connections to the abelian sandpile, avalanches, and nonsingularity of the marginals of stationary distributions, respectively.
3.1. Comparison with the abelian sandpile model. We start by giving some background on the abelian sandpile model in one dimension. In the abelian sandpile model on a finite set ⊂ Z, the amount of energy added is a nonrandom quantity: each time step one grain of sand is added to a random site. When a site is unstable, i.e., it contains at least two grains, it topples by transferring one grain of sand to each of its two neighbors (at the boundary grains are lost). The abelian addition operator is as follows: add a particle at site x and stabilize by toppling unstable sites, in any order. We denote this operator by ax : {0, 1} → {0, 1} . For toppling of site x in the abelian sandpile model, we use the symbol Tx . Abelian sandpiles have some convenient properties [2]: topplings on different sites commute, addition operators commute, and the stationary measure on finitely many sites is the uniform measure on the set of so-called recurrent configurations. Recurrent (or allowed) configurations are characterized by the fact that they do not contain a forbidden subconfiguration (FSC). A FSC is defined as a restriction of η to a subset W of , such that ηx is strictly less than the number of neighbors of x in W , for all x. In [7], a proof can be found that a FSC cannot be created by an addition or by a toppling. In the one-dimensional case on N sites, the abelian sandpile model behaves as follows. Sites are either empty, containing no grains, or full, containing one grain. When an empty site receives a grain, it becomes full, and when a full site receives a grain, it becomes unstable. In the latter case, the configuration changes in the following manner. Suppose the addition site was x. We call the distance to the first site that is empty to the left i. If there is no empty site to the left, then i − 1 is the distance to the boundary. j is defined similarly, but now to the right. After stabilization, the sites in
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{x − i, . . . , x + j} ∩ {1, . . . , N } are full, except for a new empty site at x − i + j. Only sites in {x − i, . . . , x + j} ∩ {1, . . . , N } have toppled. The number of topplings of each site is at most equal to min{i, j}, but is equal to the minimum of its distances to the endsites of the avalanche, if this is less than min{i, j}. For example, boundary sites can never topple more than once in an avalanche. These results follow straightforwardly from working out the avalanche. The recurrent configurations are those with at most one empty site; in the one-dimensional case, a connected subset of of more than one site, with empty sites at its boundary, is a FSC. If we have a configuration with exactly one empty site x say, then after the next addition, there is either no empty site (if the addition was at x) or exactly one empty site whose distribution is uniform over all sites except x. Here is an example of how a non-recurrent state on 11 sites relaxes through topplings. An addition was made to the 7th site; underlined sites are the sites that topple. The topplings are ordered into waves (see Remark 2.3). In the example, the second wave starts on the 5th configuration: 11011121101 → 11011202101 → 11012020201 → 11020121011 → 11101121011 → 11101202011 → 11102020111 → 11110120111 → 11110201111 → 11111011111. To compare Zhang’s model to the abelian sandpile, we label the different states of a ˜ N as follows: site j ∈ {1, . . . , N } in η ∈ ˜ N . For every j ∈ {1, . . . , N }, we say that η j is Definition 3.1. Let η ∈ empty(0) full(1) unstable(2) anomalous(a)
if η j if η j if η j if η j
= 0, ∈ [ 21 , 1), ≥ 1, ∈ (0, 21 ).
(2)
˜ N is the configuration denoted Definition 3.2. The reduction of a configuration η ∈ by R(η) ∈ {0, 1, 2, a}{1,...,N } corresponding to η by Definition (2). We denote with R(ηi ) the reduced value of site i, that is, R(ηi ) = R(η)i . For general 0 ≤ a < b ≤ 1, we have the following result. Proposition 3.3. For any starting configuration η ∈ N , there exists a random variable T ≥ 0 with P(T < ∞) = 1 such that for all t ≥ T , η(t) contains at most one empty or anomalous site. To prove this proposition, we first introduce FSC’s for Zhang’s model. We define a FSC in Zhang’s model in one dimension as the restriction of η to a subset W of {1, . . . , N }, in such a way that 2η j is strictly less than the number of neighbors of j in W , for all j ∈ W . From here on, we will denote the number of neighbors of j in W by degW ( j). To distinguish between the two models, we will from now on call the above Zhang-FSC, and the definition given in Sect. 3.1 abelian-FSC. From the definition, it follows that in a stable configuration, any restriction to a connected subset of more than one site, with the boundary sites either empty or anomalous, is a Zhang-FSC. Note that according to this definition, a stable configuration without Zhang-FSC’s can be equivalently described as a configuration with at most one empty or anomalous site.
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Lemma 3.4. A Zhang-FSC cannot be created by an addition in Zhang’s model. Proof. The proof is similar to the proof of the corresponding fact for abelian-FSC’s, which can be found for instance in [7], Sect. 5. We suppose that η(t) does not contain an FSC, and an addition was made at site x. If the addition caused no toppling, then it cannot create a Zhang-FSC, because no site decreased its energy. Suppose therefore that the addition caused a toppling in x. Then for each neighbor y of x, 1 Tx (η) y ≥ η y + , 2 so that 2Tx (η) y ≥ 2η y + 1. Also Tx (η)x = 0, and all other sites are unchanged by the toppling. We will now derive a contradiction. Suppose the toppling created a Zhang-FSC, on a subset which we call W . It is clear that this means that x should be in W , because it is the only site that decreased its energy by the toppling. For all j ∈ W , we should have that 2Tx (η) j < degW ( j). This means that for all neighbors y of x in W , we have 2η y < degW (y) − 1, and for all other j ∈ W we have 2η j < degW ( j). From these inequalities it follows that W \ {x} was already a Zhang-FSC before the toppling, which is not possible, because we supposed that η(t) contained no Zhang-FSC. By the same argument, further topplings cannot create a Zhang-FSC either, and the proof is complete. Remark 3.5. We have not defined Zhang’s model in dimension d > 1, because in that case the resulting configuration of stabilization through topplings is not independent of the order of topplings. But since the proof above only discusses the result of one toppling, Lemma 3.4 remains valid for any choice of order of topplings. The proof is extended simply by replacing the factor 2 by 2d. Proof of Proposition 3.3. If η already contains at most one non-full, i.e., empty or anomalous site, then it contains no Zhang-FSC’s, and the first part of the proposition follows. Suppose therefore that at some time t, η(t) contains M(t) non-full sites, with 1 < M(t) ≤ N . We denote the positions of the non-full sites of η(t) by Yi (t), i = 1, . . . , M(t), and we will show that M(t) is nonincreasing in t, and decreases to 1 in finite time. Note that for all 1 ≤ i < j ≤ M(t), the restriction of η(t) to {Yi (t), Yi (t) + 1, . . . , Y j (t)} is a Zhang-FSC. At time t + 1, we have the following two possibilities. Either the addition causes no avalanche, in that case M(t + 1) ≤ M(t), or it causes an avalanche. We will call the set of sites that change in an avalanche (that is, all sites that topple at least once, together with their neighbors) the range of the avalanche. We first show that if the range at time t + 1 contains a site y ∈ {Yi (t), . . . , Yi+1 (t)} for some i, then M(t + 1) < M(t). Suppose there is such a site. Then, since {Yi (t) + 1, . . . , Yi+1 (t) − 1} contains only full sites, all sites in this subset will topple and after stabilization of this subset, it will not contain a Zhang-FSC by Lemma 3.4. In other words, in this subset at most one empty site is created. But since Yi (t) and Yi+1 (t) received energy from a toppling neighbor, they are no longer empty or anomalous. Therefore, M(t + 1) < M(t). If there is no such site, then the range is either {1, . . . , Y1 (t)}, or {Y M(t) (t), . . . , N }, and M(t + 1) = M(t). With the same reasoning as above, we can conclude that in these cases, Y1 (t + 1) < Y1 (t), resp. Y M(t) (t + 1) > Y M(t) (t). Thus, M(t) strictly decreases at every time step where an avalanche contains topplings between two non-full sites. As long as there are at least two non-full sites, such
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an avalanche must occur eventually. We cannot make infinitely many additions without causing topplings, and we cannot infinitely many times cause an avalanche at x < Y1 (t) or x > Y M(t) (t) without decreasing M(t), since after each such an avalanche, these non-full sites ‘move’ closer to the boundary. Remark 3.6. This proof also shows that a.s. within finite time, each site topples at least once. In the case that a ≥ 1/2, we can further specify some characteristics of the model by comparing it to the one-dimensional abelian sandpile discussed above. We define regular configurations as follows: Definition 3.7. We call a configuration η ∈ N regular if η contains no anomalous sites, and at most one empty site. Proposition 3.8. Suppose a ≥ 21 . Then 1. For any initial configuration η, for all t ≥ N (N − 1), η(t) is regular, 2. If η(t) is regular with no empty site, then η(t + 1) contains one empty site whose position is uniform over all sites. If η(t) is regular with one empty site at x, say, then η(t + 1) either contains no empty site (if the addition is at x) or one empty site whose distribution is uniform over all sites except x (if the addition is not at x). 3. For every stationary distribution µ, and for all i ∈ {1, . . . , N }, µ(ηi = 0) =
1 . N +1
In words, this proposition states that if a ≥ 1/2, then every stationary distribution concentrates on regular configurations. Moreover, the stationary probability that a certain site i is empty, does not depend on i. Note that as a consequence, the stationary probability that all sites are full, is also N1+1 . The property mentioned in part 2 of the proposition will be referred to as the empty site being almost uniform on 1, . . . , N , this notion will be used in Sect. 5. To prove this proposition, we need the following lemma. In words, it states that if a ≥ 1/2 and η contains no anomalous sites, then the reduction of Zhang’s model (according to Definition 3.2) behaves just as the abelian sandpile model. Lemma 3.9. For all u ∈ [ 21 , 1), for all η ∈ N which do not contain anomalous sites, and for all x ∈ {1, . . . , N }, R(Au,x (η)) = ax (R(η)),
(3)
where ax is the addition operator of the abelian sandpile model. In both avalanches, corresponding sites topple the same number of times. Proof. Under the conditions of the lemma, site x can be either full or empty. If x is empty, then upon the addition of u ≥ 21 it becomes full. No topplings follow, so that in that case we directly have R(Au,x (η)) = ax (R(η)). If η is such that site x is full, then upon addition it becomes unstable. We call the ˜ N (see configuration after addition, but before any topplings η, ˜ and use that it is in Lemma 2.2). To check if in that case R(Au,x (η)) = ax (R(η)), we only need to prove R(Tx (η)) ˜ = Tx (R(η)), ˜ with R(η˜ x ) = 2, since we already know that in both models, the final configuration after one addition is independent of the order of topplings.
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In Tx (η), ˜ site x will be empty. This corresponds to the abelian toppling, because site x contained two grains after the addition, and by toppling it gave one to each neighbor. In Tx (η), ˜ the energy of the neighbors of x is their energy in η, plus at least 21 . Thus the neighbors of site x will in Tx (η) ˜ be full if they were empty, or unstable if they were full. Both correspond to the abelian toppling, where the neighbors of x received one grain. Proof of Proposition 3.8. To prove part (1), we note that any amount of energy that a site can receive during the process, i.e., either an addition or half the content of an unstable neighbor, is at least 1/2. Thus, anomalous sites can not be created in the process. Anomalous sites can however disappear, either by receiving an addition, or, as we have seen in the proof of Proposition 3.3, when they are in the range of an avalanche. When we make an addition of at least 1/2 to a configuration with more than one non-full site, then either the number of non-full sites strictly decreases, or one of the outer non-full sites moves at least one step closer to the boundary. We note that η contains at most N non-full sites, and the distance to the boundary is at most N − 1. When finally there is only one non-full site, then in the next time step it must either become full or be in the range of an avalanche. Thus, there is a random time T ≤ N (N − 1) such that η(T ) is regular for the first time, and as anomalous sites cannot be created, by Proposition 3.3, η(t) is regular for all t ≥ T . For t ≥ T , η(t) satisfies the condition of Lemma 3.9. This means that the evolution of the reduction of Zhang’s model coincide s with that of the abelian sandpile model. Parts (2) and (3) then follow from the corresponding properties of the abelian sandpile. 3.2. Avalanches in Zhang’s model. We next describe in full detail the effect of an avalanche, started by an addition to a configuration η(t) in Zhang’s model. Let C(t + 1) be the range of this avalanche. Recall that we defined the range of an avalanche as the set of sites that change their energy at least once in the course of the avalanche (that is, all sites that topple at least once, together with their neighbors). We denote by T (t + 1) the collection of sites that topple at least once in the avalanche. Finally, C (t + 1) ⊂ C(t + 1) denotes the collection of anomalous sites that change, but do not topple in the avalanche. During the avalanche, the energies of sites in the range, as well as Ut+1 , get redistributed through topplings in a rather complicated manner. By decomposing the avalanche into waves (see Remark 2.3), we prove the following properties of this redistribution. Proposition 3.10. Suppose an avalanche is started by an addition at site x to configuration η(t). For all sites j in C(t + 1), there exist Fi j = Fi j (η(t), x, Ut+1 ) such that we can write Fi j ηi (t) + Fx j Ut+1 + η j (t)1 j∈C (t+1) , (4) η j (t + 1) = i∈T (t+1)
with 1.
Fx j +
Fi j = R(η(t + 1) j );
i∈T (t+1)
2. for all j ∈ C(t + 1) such that η j (t + 1) = 0, Fx j ≥ 2− 3N /2 ;
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3. for all j ∈ C(t + 1) such that η j (t + 1) = 0, j ≥ x, we have Fx, j+1 ≤ Fx j ; and similarly, Fx, j−1 ≤ Fx j for j ≤ x. In words, we can write the new energy of each site in the range of the avalanche at time t + 1 as a linear combination of energies at time t and the addition Ut+1 , in such a way that the prefactors sum up to 1 or 0. Furthermore, every site in the range receives a positive fraction of at least 2− 3N /2 of the addition. These received fractions are such that larger fractions are found closer to the addition site. We will need this last property in the proof of Theorem 5.5. Proof of Proposition 3.10. We start with part 1. First, we decompose the avalanche started at site x into waves. We index the waves with k = 1, . . . , K , and write out explicitly the configuration after wave k, in terms of the configuration after wave k − 1. The energy of site i after wave k is denoted by η˜ i,k ; we use the tilde to emphasize that these energies are not really encountered in the process. We define η˜ i,0 = ηi (t) + Ut+1 1i=x ; note that η˜ i,K = ηi (t + 1). In each wave, all participating sites topple only once by Remark 2.3. We call the outermost sites that toppled in wave k, the endsites of this wave, and we denote them by Mk and Mk , with Mk > Mk . For the first wave, this is either a boundary site, or the site next to the empty or anomalous site that stops the wave. Thus, M1 and M1 depend on η(t), x and Ut+1 . For K > 1, all further waves are stopped by the empty sites that were created when the endsites of the previous wave toppled, so that for each k < K , Mk+1 = Mk − 1 and Mk+1 = Mk + 1. In every wave but the last, site x becomes again unstable. Only in the last wave, x is an endsite, so that at most one of its neighbors topples. In wave k, first site x topples, transferring half its energy, that is, 21 η˜ x,k−1 , to each neighbor. Then, if x is not an endsite (that is, k < K ), both its neighbors topple, transferring half of their current energy, that is, 21 η˜ x±1,k−1 + 41 η˜ x,k−1 , to their respective neighbors. Site x is then again unstable, but it does not topple again in this wave. Thus, the topplings propagate away from x in both directions, until the endsites are reached. Every toppling site in its turn transfers half its current energy, including the energy received from its toppling neighbor, to both its neighbors. Writing out all topplings leads to the following expression, for all sites i ≥ x. A similar expression gives the updated energies for the sites with i < x. Note that, when K > 1, for every k > 1, η˜ Mk +1,k−1 = 0. Only when k = 1, it can be the case that site M1 + 1 was anomalous, so that η˜ M1 +1,0 > 0: η˜ x,k = η˜ i,k =
1 1 1 1 η˜ x+1,k−1 + η˜ x,k−1 1 Mk >x + η˜ x−1,k−1 + η˜ x,k−1 1 Mk <x , 2 4 2 4
i+1 n=x
1 η˜ n,k−1 , 2i+2−n
for i = x + 1, . . . , Mk − 1,
η˜ Mk ,k = 0, if Mk = x, ⎧ + η ˜ if Mk ≥ x + 2, η ˜ ⎨ Mk −1,k Mk +1,k−1 1 1 η ˜ + η ˜ + η ˜ η˜ Mk +1,k = 2 x+1,k−1 4 x,k−1 x+2,k−1 if Mk = x + 1, ⎩1 if Mk = x. 2 η˜ x,k−1 + η˜ x+1,k−1
(6)
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We write for all j ∈ C(t + 1), with f i j (k) implicitly defined by the coefficients in (6), η˜ j,k = f i j (k)η˜ i,k−1 + 1 j∈C (t+1) η˜ j,k−1 . (7) i∈T (t+1)
Since we made an addition to a stable configuration, by Lemma 2.2, we only encoun˜ N . From a case by case analysis of (6), we claim that for all ter configurations in j ∈ C(t + 1) we have R(η˜ j,k ) = f i j (k)R(η˜ i,k−1 ); (8) i∈T (t+1)
the reader can verify this for all cases. Note that for all j ∈ C(t + 1), R(η˜ j,k ) = a. If j ∈ C (t + 1), then in the first wave j becomes full. To prove the proposition, we start with the case K = 1, for which we have f i j (1)η˜ i,0 + η j (t)1 j∈C (t+1) η j (t + 1) = η˜ j,1 = i∈T (t+1)
=
f i j (1)ηi (t) + f x j (1)Ut+1 + η j (t)1 j∈C (t+1) .
i∈T (t+1)
We also have, according to (8), R(η j (t + 1)) =
i∈T (t+1)
= f x j (1) +
f i j (1)R(η˜ i,0 )
f i j (1).
i∈T (t+1)
Hence if K = 1 we choose Fi j = f i j (1) and part 1 of the proposition is proved for this case. For K > 1, we use induction in k. Here we only consider sites that are not in C (t +1); we already treated these above in the case K = 1. For wave k −1, we make the induction hypothesis that η˜ j,k−1 = Fm j (k − 1)ηm (t) + Fx j (k − 1)Ut+1 , (9) m∈T (t+1)
with
Fm j (k − 1) + Fx j (k − 1) = R(η˜ j,k−1 ).
(10)
m∈T (t+1)
Inserting this in (7), we get f i j (k)η˜ i,k−1 η˜ j,k = i∈T (t+1)
=
m∈T (t+1) i∈T (t+1)
Fmi (k − 1) f i j (k)ηm (t) +
i∈T (t+1)
f i j (k)Fxi (k − 1)Ut+1 ,
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and inserting this in (8), we get f i j (k)R(η˜ i,k−1 ) R(η˜ j,k ) = i∈T (t+1)
=
⎡
f i j (k) ⎣
i∈T (t+1)
⎤ Fmi (k − 1) + Fxi (k − 1)⎦ .
m∈T (t+1)
Hence, if we define Fm j (k) =
f i j (k)Fmi (k − 1),
i∈T (t+1)
then (9) and (10) are also true for wave k. For k − 1 = 0, the hypothesis is also true, with Fmi (0) = 1m=i . We define Fi j := Fi j (K ), and then the first part of the proposition is proved for all K . To prove part 2 of the proposition, we derive a lower bound for Fx j . The number K of waves in an avalanche is equal to the minimum of the distance to the end sites, leading to the upper bound K ≤ N /2. After the first wave, (6) gives for all nonempty j = x, Fx j (1) ≥ ( 21 ) N +1 . At the start of the next wave, if there is one, the fraction of Ut+1 present at x is equal to Fx x (1) = 21 . Hence, if after the second wave there is a third one, even if we ignore all fractions of Ut on sites other than x, then we still have, again by (6), Fx j (2) > 21 ( 21 ) N +1 . So if before each wave we always ignore all fractions of Ut on sites other than x, and if we assume the maximum number of waves, then we arrive at a lower bound for nonempty sites j: Fx j
N /2−1 N +1 1 1 ≥ ≥ 2− 3N /2 . 2 2
We now prove part 3. For K = 1, part 3 of the proposition follows directly from (6), so we discuss the case K > 1. Moreover, we only discuss the case j ≥ x, since by symmetry, the case j ≤ x is similar. We will show that for every k ∈ {1, . . . , K − 1}, 1 Fx x (k) > Fx,x+1 (k) > · · · > Fx,Mk −1 (k) = Fx,Mk +1 (k), 2
(11)
Fx,Mk +1 (k) ≥ Fx,Mk−1 +1 (k − 1),
(12)
and
where we define Fx,M0 +1 (0) = 0. In the final wave, the sites j > M K + 1 do not change, so the inequality in part 3 of the proposition for these sites follows from (12). Therefore, after proving (11) and (12) for all k ∈ {1, . . . , K − 1}, we will show that the required result follows for the sites x ≤ j ≤ M K . We will now prove (11) and (12) for all k ∈ {1, . . . , K − 1}, using induction in k. After the first wave, we have from (6) that 1 Fx x (1) > Fx,x+1 (1) > · · · > Fx,Mk −1 (1) = Fx,Mk +1 (1), 2 so that (11) and (12) are satisfied after the first wave.
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Now assume as induction hypothesis that (11) is true after wave k, with k < K − 1. We have seen that this is true after the first wave. We rewrite (6), for every k < K − 1, so that Mk+1 > x. In the first line, we use that for all k < K − 1, Fx,x+1 (k) = Fx,x−1 (k). First, we explain why this is true. By (6), we have in general, for positive and negative y, that Fx,x+y (k) is a function of Fx z (k − 1), with z ∈ Z1 (y) = {x + y − 1, . . . , x + y + 1}, that is symmetric in y as long as all sites in Z1 (y) ∪ Z1 (−y) toppled in wave k − 1. Continuing this reasoning, we have that Fx,x+y (k) is a function of Fx z (0), with z ∈ Zk (y) = {x + y − k, . . . , x + y + k}, that is symmetric in y as long as all sites in Zk (y) ∪ Zk (−y) topple in the first wave. If k < K − 1, this requirement is satisfied for all x − 1 − k, . . . , x + 1 + k. Moreover, we have that Fx z (0) = 1x=z , so that we obtain that Fx,x+1 (k) = Fx,x−1 (k). We now write, for every k < K − 1, 1 Fx x (k + 1) = Fx,x+1 (k) + Fx x (k), 2 1 F (k) + 41 Fx x (k + 1) if Mk+1 > x + 1, x,x+2 Fx,x+1 (k + 1) = 2 0 if Mk+1 = x + 1, 1 1 for i = x + 2, . . . , Mk+1 − 1, Fx,x+i (k + 1) = Fx,x+i+1 (k) + Fx,x+i−1 (k + 1) 2 2 Fx Mk (k + 1) = 0, if Mk+1 > x + 1, Fx,M −1 (k + 1) (13) Fx,Mk +1 (k + 1) = 1 k F (k + 1) if Mk+1 = x + 1. 2 x,Mk −1 If Mk+1 = x + 1, then (11) and (12) follow directly for k + 1 from this expression. However, when Mk+1 > x + 1, we need the following derivation. From (13) and the induction hypothesis, we find the following inequalities, each one following from the previous one: 1 1 1 Fx x (k) < Fx x (k) + Fx x (k) = Fx x (k), 2 2 2 1 1 1 Fx,x+1 (k + 1) = Fx,x+2 (k) + Fx x (k + 1) < Fx,x+1 (k) 2 4 2 1 1 + Fx x (k) = Fx x (k + 1), 4 2 Fx x (k + 1) = Fx,x+1 (k) +
If Mk = x + 2, then Fx,x+2 = 0, and (11) and (12) are satisfied. For Mk > x + 2, we have 1 1 1 1 Fx,x+3 (k) + Fx,x+1 (k + 1) < Fx,x+2 (k) + Fx x (k + 1) 2 2 2 4 = Fx,x+1 (k + 1).
Fx,x+2 (k + 1) =
For all i = 2, . . . , Mk+1 − x − 1, if Fx,x+i (k + 1) < Fx,x+i−1 (k + 1) then Fx,x+i+1 (k + 1) =
1 1 1 Fx,x+i+2 (k) + Fx,x+i (k + 1) < Fx,x+i+1 (k) 2 2 2 1 + Fx,x+i−1 (k + 1) = Fx,x+i (k + 1). 2
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Since Fx,x+i (k + 1) < Fx,x+i−1 (k + 1) is true for i = 2, (11) follows for wave k + 1, and is thus proved for every k < K . Moreover, we have Fx,Mk+1 +1 (k + 1) = Fx,Mk+1 −1 (k + 1) =
1 1 Fx,Mk+1 (k) + Fx,Mk+1 −2 (k + 1). 2 2
With the above derived Fx,Mk+1 −1 (k +1) < Fx,Mk+1 −2 (k +1), it follows that Fx,Mk+1 (k) < Fx,Mk+1 −2 (k + 1), so that Fx,Mk+1 −1 (k + 1) > Fx,Mk+1 (k) = Fx,Mk +1 (k), which is (12). Finally we discuss the last wave. For the last wave, we need to discuss several cases. If M K = x, then either Mk < x or M K = x, but if M K > x, then M K = x, because at least one of the end sites of the last wave is x. In case M K = M K = x we have Fx x (K ) = 0, if Mk = x and M K < x then Fx x (K ) =
1 1 1 1 Fx,x−1 (K − 1) + Fx x (K − 1) = Fx,x+1 (K − 1) + Fx x (K − 1). 2 4 2 4
In both cases, we have Fx,x+1 (K ) =
1 Fx x (K − 1) = Fx,x+2 (K − 1), 2
so that Fx x (K ) > Fx,x+1 (K ). Part 3 follows for M K = x. In case M K = x + 1 we have 1 1 Fx,x+1 (K − 1) + Fx x (K − 1), 2 4 Fx,x+1 (K ) = 0, 1 1 Fx,x+2 (K ) = Fx x (K ) = Fx,x+1 (K − 1) + Fx x (K − 1) 2 4 1 1 > Fx,x+1 (K − 1) + Fx,x+1 (K − 1) > Fx,x+1 (K − 1). 2 2 Fx x (K ) =
For all M K > x + 1 we have 1 Fx,x+1 (K − 1) + 2 1 Fx,x+1 (K ) = Fx,x+2 (K − 1) + 2 Fx x (K ) =
1 Fx x (K − 1), 4 1 Fx x (K − 1) < Fx x (K ). 4
In this case, verifying (11) and (12) proceeds as in the previous derivation for the case k < K , Mk ≥ x + 2.
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3.3. Absolute continuity of one-site marginals of stationary distributions. Consider a one-site marginal ν j of any stationary distribution ν of Zhang’s sandpile model. It is easy to see that ν j will have an atom at 0, because after each avalanche there remains at least one empty site. It is intuitively clear that there can be no other atoms: by only making uniformly distributed additions, it seems impossible to create further atoms. Here we prove the stronger statement that the one-site marginals of any stationary distribution are absolutely continuous with respect to Lebesgue measure on (0, 1). Theorem 3.11. Let ν be a stationary distribution for Zhang’s model on N sites. Every one-site marginal of ν is on (0, 1) absolutely continuous with respect to Lebesgue measure. Proof. Let A ⊂ (0, 1) be so that λ(A) = 0, where λ denotes Lebesgue measure. We pick a starting configuration η according to ν. We define a stopping time τ as the first time t such that all non-zero energies ηi (t) contain a nonzero contribution of at least one of the added amounts U1 , U2 , . . . , Ut . We then write, for an arbitrary nonzero site j, Pν (η j (t) ∈ A) ≤ Pν (η j (t) ∈ A, τ < t) + Pν (t ≤ τ ).
(15)
The second term at the right-hand side tends to 0 as t → ∞ because by Remark 3.6, a.s. within finite time each site has participated in an avalanche at least once, and by Proposition 3.10, part 2, each site contains a nonzero contribution of the addition that started the last avalanche it participated in. We claim that the first term at the right-hand side is equal to zero. To this end, we first observe that η j (t) is built up of fractions of ηi (0), i = 1, . . . , N , and the additions U1 , U2 , . . . , Ut . These fractions are random variables themselves, and we can bound this term by
N t t Pν Z i ηi (0) + Ys Us ∈ A, Ys > 0 , (16) s=1
i=1
s=1
where Z i represents the (random) fraction of ηi (0) in η j (t), and Ys represents the (random) fraction of Us in η j (t). We clearly have that the Us are all independent of each other and of ηi (0) for all i. However, the Us are not necessarily independent of the Z i and the Ys , since the numerical value of the Us affects the relevant fractions. Also, we know from the analysis in the previous subsection that the Z i and Ys can only take non-negative values in a countable set. Summing over all elements in this set, we rewrite (16) as N
t Pν z i ηi (0) + ys Us ∈ A, Z i = z i , Ys = ys z i ,ys ; s ys >0
s=1
i=1
which is at most z i ,ys ; s ys >0
Pν
N
z i ηi (0) +
t
ys Us ∈ A ,
s=1
i=1
which, by the independence of the Us and the ηi (0), is equal to
N t z i xi + ys Us ∈ A dν(x1 , . . . , x N ). Pν z i ,ys ; s ys >0
i=1
s=1
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Since ts=1 ys > 0, Us are independent uniforms, and by assumption λ(A) = 0, the probabilities inside the integral are clearly zero. Since the left hand side of (15) is equal to ν j (A) for all t, we now take the limit t → ∞ on both sides, and we conclude that ν j (A) = 0. Remark 3.12. The same proof shows that for every stationary measure ν, and for every i 1 , . . . , i k ∈ {1, . . . , N }, conditional on all sites i 1 , . . . , i k being nonempty, the joint distribution of ηi1 , . . . , ηik under ν is absolutely continuous with respect to Lebesgue measure on (0, 1)k . 4. The (1, [a, b])-Model In this section we consider the simplest version of Zhang’s model: the (1, [a, b])-model. In words: there is only one site and we add amounts of energy that are uniformly distributed on the interval [a, b], with 0 ≤ a < b ≤ 1. 4.1. Uniqueness of the stationary distribution. Before turning to the particular case a = 0, we prove uniqueness of the stationary distribution for all [a, b] ⊆ [0, 1]. We also prove that every initial distribution on 1 converges to this stationary measure. We find two different kinds of convergence; convergence in total variation is the strongest, but we cannot obtain this for all values of a and b. Theorem 4.1. (a) The (1, [a, b]) model has a unique stationary distribution ρ = ρ ab . For every initial distribution Pη on 1 , we have time-average total variation convergence to ρ, i.e., t 1 lim sup Pη (η(s) ∈ A) − ρ(A) = 0. t→∞ A⊂ t 1 s=0
1 (b) In addition, if there exists no integer m > 1 such that [a, b] ⊆ [ m1 , m−1 ], (hence in particular if a = 0), then we have convergence in total variation to ρ for every initial distribution Pη on 1 , i.e., lim sup Pη (η(t) ∈ A) − ρ(A) = 0. t→∞ A⊂ 1
Proof. We prove this theorem by constructing a coupling. The two processes to be coupled have initial configurations η1 and η2 , with η1 ,η2 ∈ 1 . We denote by η1 (t), η2 (t) two independent copies of the process starting from η1 and η2 respectively. The corresponding independent additions at each time step are denoted by Ut1 and Ut2 , respectively. Let T1 = min{t : η1 (t) = 0} and T2 = min{t : η2 (t) = 0}. Suppose (without loss of generality) that T2 ≥ T1 . We define a shift-coupling ([11], Chap. 5) as follows: ηˆ 1 (t) = η1 (t) η2 (t) ηˆ 2 (t) = η1 (t − (T2 − T1 ))
for all t, for t < T2 , for t ≥ T2 .
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Defining T = min{t : η1 (t) = η2 (t) = 0}, we also define the exact coupling ηˆ 1 (t) = η1 (t) η2 (t) ηˆ 3 (t) = η1 (t)
for all t, for t < T, for t ≥ T.
Since the process is Markov, both couplings have the correct distribution. We write ˜ 2 < ∞) = P(T ˜ 1 < ∞) = 1, the shift-coupling is always P˜ = Pη1 × Pη2 . Since P(T successful, and (a) follows. ˜ To investigate whether η1 (t) = η2 (t) = 0 occurs infinitely often P-a.s., we define N = {n : (n − 1)a < 1, nb > 1}; this is the set of possible numbers of time steps between successive events η1 (t) = 0. In words, an n ∈ N is such that, starting from η1 = 0, it is possible that in n − 1 steps we do not yet reach energy 1, but in n steps we do. To give an example, if a ≥ 1/2, then N = {2}. If the gcd of N is 1 (this is in particular the case if a = 0), then the processes {t : η1 (t) = 0} and {t : η2 (t) = 0} are independent aperiodic renewal processes, and it ˜ follows that η1 (t) = η2 (t) = 0 happens infinitely often P-a.s. As we have seen, for a > 0, the gcd of N need not be 1. In fact, we can see from the definition of N that this is the case if (and only if) there is an integer m > 1 such 1 that [a, b] ⊆ [ m1 , m−1 ]. Then N = {m}. For such values of a and b, the processes 1 {t : η (t) = 0} and {t : η2 (t) = 0} are periodic, so that we do not have a successful exact coupling. 4.2. The stationary distribution of the (1, [0, b])-model. We write ρ b for the stationary measure ρ 0b of the (1, [0, b])-model and F b for the distribution function of the amount of energy at stationarity, that is, F b (h) = ρ b (η : 0 ≤ η ≤ h). We prove the following explicit solution for F b (h). Theorem 4.2. (a) The distribution function of the energy in the (1, [0, b])-model at stationarity is given by ⎧ 0 ⎪ ⎪ ⎨ F b (0) > 0 h F b (h) = ⎪ F b (0) m ⎪ κ=0 ⎩ 1
(−1)κ bκ κ! (h
− κb)κ e
h−κb b
for h < 0, for h = 0, for 0 < h ≤ 1, for h > 1,
where m h = hb − 1 and where F b (0) = m1
1
1−κb (−1)κ κ b κ=0 bκ κ! (1 − κb) e
follows from the identity F b (1) = 1. (b) For h ∈ [0, 1] we have lim F b (h) = h.
b→0
(17)
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We remark that although in (a) we have a more or less explicit expression for F b (h), the convergence in (b) is not proved analytically, but rather probabilistically. Proof of Theorem 4.2, part (a). Observe that the process for one site is defined as η(t + 1) = (η(t) + Ut+1 ) 1η(t)+Ut+1 <1 .
(18)
b (h) in terms of We define Ftb (h) = P(η(t) ≤ h), and derive an expression for Ft+1 Ftb (h). In the stationary situation, these two functions should be equal. We deduce from (18) that for 0 ≤ h ≤ 1, b Ft+1 (h) = P(η(t) + Ut+1 < h) + P(η(t) + Ut+1 ≥ 1).
(19)
We compute for 0 ≤ h ≤ b, P(η(t) + Ut+1 ≤ h) = P(η(t) ≤ h − Ut+1 ) h 1 P(η(t) ≤ h − u) du = b 0 h 1 b = Ft (h − u) du, b 0
(20)
and likewise for b ≤ h ≤ 1 we find P(η(t) + Ut+1 ≤ h) = 0
b
1 b (F (h − u)) du. b t
Finally, using that F b (1) = 1 (this follows from (18)), b 1 b (Ft (1) − Ftb (1 − u)) du P(η(t) + Ut+1 ≥ 1) = b 0 b (1 − Ftb (1 − u)) b = (0). du = Ft+1 b 0
(21)
(22)
Putting (19), (20), (21) and (22) together leads to the conclusion that the stationary distribution F b (h) satisfies h F b (h−u) du + F b (0) if 0 ≤ h ≤ b, b 0 b (23) F (h) = b F b (h−u) du + F b (0) if b ≤ h ≤ 1. 0 b Furthermore, since F b (h) is a distribution function, F b (h) = 0 for h < 0 and F b (h) = 1 for h > 1. We can rewrite Eq. (23) as a differential delay equation. Let f b (h) be a density corresponding to F b for 0 < h < 1; this density exists according to Theorem 3.11. We differentiate (23) twice on both sides, to get in the case 0 < h < b, d f b (h) 1 = f b (h), dh b and in the case b < h ≤ 1, f b (h) =
1 b (F (h) − F b (h − b)). b
(24)
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0.2
0.4
369
0.6
0.8
1
Fig. 1. f b (h) for b = 21 . Note the discontinuity at h = 21
At this point, we can conclude that the solution is unique and could in principle be found using the method of steps. However, since we already have the candidate solution given in Theorem 4.2, we only need to check that it indeed satisfies Eq. (23). h In the case 0 < h < b, in which case m h = 0, we have F b (h) = F b (0)e b , which is consistent with Theorem 24. We check that for the derivative f b of F b as defined in (17), for b ≤ h ≤ 1, mh h−κb 1 κ−1 1 F b (0) f (h) = − (h − κb)κ−1 e b − b b (κ − 1)! κ=1 m h h−κb 1 κ 1 F b (0) − + (h − κb)κ e b , b b κ! b
κ=0
whereas mh h−κb 1 κ 1 F b (0) F b (h) − = (h − κb)κ e b b b b κ! κ=0
and −
m h −1 h−(κ+1)b 1 κ 1 F b (0) F b (h − b) − =− κ(h − (κ + 1)b)κ e b b b b κ! κ=0 mh h−κb 1 κ−1 1 F b (0) − (h − κb)κ−1 e b , =− b b (κ − 1)! κ=1
which leads to (24) as required. We remark that the probability density function f b (h) has an essential point of discontinuity at h = b. Figures 1 and 2 show two examples of f b (h). Proof of Theorem 4.2, part (b). For a while, we fix b > 0 and write ηb (t) for the state of our process when we start with ηb (0) = 0. For fixed h, define a process X b (t) by X b (t) = 1 if ηb (t) ≤ h and X b (t) = 0 if ηb (t) > h. The process X b is a delayed renewal
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0.2
0.4
0.6
0.8
1
1 . This figure illustrates that for small b, f b (h) tends to the uniform distribution Fig. 2. f b (h) for b = 10
process; a renewal occurs at t if X b (t − 1) = 1 and X b (t) = 0. Let K (t) be the number of renewals up to (and including) time t, where we take K (0) = 0. The k th renewal takes place at time Tk , where we define T0 = 0. The number of indices t ∈ [Tk−1 , Tk ) with X b (t) = 0 is denoted by Z b (k); the number of indices t in that interval with X b (t) = 1 by W b (k) . Typical random variables with these distributions are denoted by Z b and W b respectively. By the identity K (t−1) k=1
W b (k) ≤
t−1
X b (i) ≤
i=0
K (t−1)
W b ( j) + W b (K (t − 1) + 1),
j=1
and the well known fact that K (t)/t → 1/E(Z b + W b ) as t → ∞, we find that F b (h) = lim P(ηb (t) ≤ h) = lim P(X b (t) = 1) t→∞
= lim
t→∞
t−1 1
t→∞
t
X b (i)
i=0
K (t) E(W b ) = lim t→∞ t E(W b ) . = E(Z b + W b ) To compute these expectations in the limit b → 0, we use another renewal process. Let U1 , U2 , . . . be independent uniform random variables on [0, 1] and write Sn1 = U1 + · · · + Un . We define N (s) = max{n ∈ N : Sn1 ≤ s}. The process {N (s) : s ≥ 0} is a renewal process, and lim
s→∞
E(N (s)) = E(U1 )−1 = 2. s
(25)
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371
Now observe that W b and N (h/b) − 1 have the same distribution, and that Z b + W b has the same distribution as N (1/b) − 1. Hence lim
b→0
E(W b ) E(N (h/b) − 1) = lim , b b E(Z + W ) b→0 E(N (1/b) − 1)
and this last limit is equal to h, according to (25). 5. The (N, [a, b])-Model with N ≥ 2 and a ≥
1 2
5.1. Uniqueness of stationary distribution. In the course of the process of Zhang’s model, the energies of all sites can be randomly augmented through additions, and randomly redistributed among other sites through avalanches. Thus at time t, every site contains some linear combination of all additions up to time t, and the energies at time 0. In a series of lemmas we derive very detailed properties of these combinations in the case a ≥ 1/2. These properties are crucial to prove the following result. Theorem 5.1. The (N , [a, b]) model with a ≥ 21 , has a unique stationary distribution µ = µab . For every initial distribution ν on N , Pν converges exponentially fast in total variation to µ. We have demonstrated for the case a ≥ 21 that after a finite (random) time, we only encounter regular configurations (Proposition 3.8). By Lemma 3.9, if η(t − 1) is regular, then the knowledge of R(η(t − 1)) and X t suffices to know the number of topplings of each site at time t. From these numbers, we can infer the endsites of all waves, and then by repeatedly applying (6), we can find the factors Fi j in Proposition 3.10. Thus, also these factors are functions of R(η(t − 1)) and X t only. Using this observation, we prove the following. Lemma 5.2. Let a ≥ 21 . Suppose at some (possibly random) time τ we have a configuration ξ(τ ) with no anomalous sites. Then for all j = 1, . . . , N and for t ≥ τ , we can write ξ j (t) =
t
Aθ j (t)Uθ +
θ=τ +1
N
Bm j (t)ξm (τ )
m=1
in such a way that the coefficients in (26) satisfy Aθ j (t) = Aθ j (R(ξ(τ )), X τ +1 , . . . , X t ) and Bm j (t) = Bm j (R(ξ(τ )), X τ +1 , . . . , X t ), and such that for every j and every t ≥ τ , t θ=τ +1
Aθ j (t) +
N m=1
Bm j (t) = R(ξ j (t)) = 1ξ j (t) =0 .
(26)
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Remark 5.3. Notice that, in the special case that τ is a stopping time, Aθ j (t) is independent of the amounts added after time τ , i.e., Aθ j (t) and {Uθ , θ ≥ τ + 1} are independent. We will make use of this observation in Sect. 5.2. Proof of Lemma 5.2. We use induction. We start at t = τ , where we choose Bm j (τ ) = N R(ξ(τ ) j ) 1m= j , and Aτ j (t) = 0 for all t ≥ τ and for all j. We then have m=1 Bm j (τ ) = R(ξ(τ ) j ), so that at t = τ the statement in the lemma is true. We next show that if the statement in the lemma is true at time t ≥ τ , then it is also true at time t + 1. At time t we have for every j = 1, . . . , N , t
ξ j (t) =
Aθ j (t)Uθ +
θ=τ +1
N
Bm j (t)ξm (τ ),
m=1
with t
N
Aθ j (t) +
θ=τ +1
Bm j (t) = R(ξ(t) j ),
m=1
where all Aθ j (t) and Bm j (t) are determined by R(ξ(τ )), X τ +1 , . . . , X t , so that R(ξ(t)) is also determined by R(ξ(τ )), X τ +1 , . . . , X t . We first discuss the case where we added to a full site, so that an avalanche is started. In that case, the knowledge of R(ξ(τ )), X τ +1 , . . . , X t+1 determines the sets C(t + 1), T (t + 1) and the factors Fi j from Proposition 3.10 (since a ≥ 21 ). We write, denoting X t+1 = x, ξ j (t + 1) = Fi j ξi (t) + Fx j Ut+1 i∈T (t+1)
=
Fi j
i∈T (t+1)
t
Aθi (t)Uθ +
θ=1
N
Bmi (t)ξm (τ ) + Fx j Ut+1 .
m=1
(27) Thus we can identify Aθ j (t + 1) =
Fi j Aθi (t),
(28)
i∈T (t+1)
Bm j (t + 1) =
Fi j Bmi (t),
i∈T (t+1)
and At+1, j (t + 1) = Fx j , so that indeed all Aθ j (t + 1) and Bm j (t + 1) are functions of R(ξ(τ )), X τ +1 , . . . , X t+1 only. Furthermore, t t+1 N N Aθ j (t + 1) + Bm j (t + 1) = Fi j Aθi (t) + Bmi (t) + Fx j θ=1
i∈T (t+1)
m=1
=
i∈T (t+1)
θ=1
m=1
Fi j + Fx j = R(η(t + 1) j ),
(29)
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373
where we used that by Lemma 3.9 all sites that toppled must have been full, therefore had reduced value 1. If no avalanche was started, then the only site that changed is the addition site x, and it must have been empty at time t. Therefore, we have for all τ < θ < t + 1, Aθ x (t + 1) = Aθ x (t) = 0, for all m, Bmx (t + 1) = Bmx (t) = 0 and At+1,x (t + 1) = 1, so that the above conclusion is the same. For every θ , we have i∈T (t+1) Aθi (t) ≤ 1, because the addition Uθ gets redistributed by avalanches, but some part disappears through topplings of boundary sites. One might expect that, as an addition gets redistributed multiple times and many times some parts disappear at the boundary, the entire addition eventually disappears, and similarly for the energies ξ j (τ ). Indeed, we have the following results about the behavior of Aθi (t) for fixed θ , and about the behavior of Bm j (t) for fixed m. Lemma 5.4. For every θ , and for t > θ , 1. max1≤i≤N Aθi (t) and max1≤i≤N Bmi (t) are both non-increasing in t. 2. For all θ, m and i, limt→∞ Aθi (t) = 0, and limt→∞ Bmi (t) = 0. Proof. We can assume that t > θ . The proofs for Aθ j (t) and for Bm j (t) proceed along the same line, so we will only discuss Aθ j (t). We will show that for every j, Aθ j (t + 1) ≤ maxi Aθi (t), by considering one fixed j. If the energy of site j did not change in an avalanche at time t + 1, then Aθ j (t + 1) = Aθ j (t) ≤ max Aθi (t). i
If site j became empty in the avalanche, then Aθ j (t + 1) = 0 < max Aθi (t). i
For the third possibility-the energy of site j changed to a nonzero value in an avalanche at time t + 1-we use (28), and estimate Aθ j (t + 1) = Fi j Aθi (t) ≤ max Aθi (t) Fi j . i
i∈T (t+1)
i∈T (t+1)
By Proposition 3.10 part (1) and (2), i∈T (t+1) Fi j ≤ 1 − 2− 3N /2 , so that in this third case, Aθ j (t + 1) ≤ 1 − 2− 3N /2 max Aθi (t) < max Aθi (t). i
i
Thus, it follows that maxi Aθi (t +1) can never be larger than maxi Aθi (t). This proves part (1). It also follows that when between t and t + t all sites have changed at least once, we are sure that maxi Aθi (t + t ) ≤ (1 − 2− 3N /2 ) maxi Aθi (t). We next derive an upper bound for the time that one of the sites can remain unchanged. Suppose at some finite time t when η(t) is regular (see Proposition 3.8), we try never to change some sites again. If all sites are full, then this is impossible: the next avalanche will change all sites. If there is an empty site x, then the next addition (if it is not at site x) changes either the sites 1, . . . , x, or the sites x, . . . , N . In the first case, after the avalanche we have a new empty site x < x. If we keep trying not to change the sites x, . . . , N , we have to keep making additions that move the empty site closer to the
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boundary (site 1). It will therefore reach the boundary in at most N − 1 time steps. Then we have no choice but to change all sites: we can either add to the empty site and obtain the full configuration, so that with the next addition all sites will change, or add to any other site, which immediately changes all sites. This argument shows that the largest possible number of time steps between changing all sites is N + 1. We therefore have t−θ N +1 max Aθi (t) < 1 − 2− 3N /2 , i
(30)
so that lim max Aθi (t) < lim
t→∞
t→∞
i
t−θ N +1 1 − 2− 3N /2 = 0.
With the above results, we can now prove uniqueness of the stationary distribution. Proof of Theorem 5.1. By compactness, there is at least one stationary measure µ. To prove the theorem, we will show that there is a coupling (ηˆ 1 (t), ηˆ 2 (t))∞ 0 with probability law Pˆ (η1 ,η2 ) for two realizations of the (N , [a, b]) model with a ≥ 21 , such that for all > 0, and for all starting configurations η1 and η2 , for t → ∞ we have Pˆ (η1 ,η2 ) max |ηˆ 1j (t) − ηˆ 2j (t)| > = O(e−α N t )), j
(31)
with α N > 0. From (31), it follows that the Wasserstein distance ([4], Chap. 11.8) between any two measures µ1 and µ2 on N vanishes exponentially fast as t → ∞. If we choose η1 distributed according to µ stationary, then it is clear that every other measure on N converges exponentially fast to µ. In particular, it follows that µ is unique. As in the proof of Theorem 4.1, the two processes to be coupled have initial configurations η1 and η2 , with η1 , η2 ∈ N . The independent additions at each time step are denoted by Ut1 and Ut2 , the addition sites X t1 and X t2 . We define the coupling as follows: ηˆ 1 (t) = η1 (t) η2 (t) ηˆ 2 (t) = AU 1 ,X 1 (ηˆ 2 (t − 1)) t
t
for all t, for t ≤ T, for t > T,
where T = min{t > T : R(η1 (t)) = R(η2 (t))}, and T the first time that both η1 (t) and η2 (t) are regular. In Proposition 3.8 it was proven that T ≤ N (N − 1), uniformly in η. In words, this coupling is such that from the first time on where the reductions of ηˆ 1 (t) and ηˆ 2 (t) are the same, we make additions to both copies in the same manner, i.e., we add the same amounts at the same location to both copies. Then, by Lemma 3.9, in both copies the same avalanches will occur. We will then use Lemma 5.4 to show that, from time T on, the difference between ηˆ 1 (t) and ηˆ 2 (t) vanishes exponentially fast. First we show that Pˆ (η1 ,η2 ) (T > t) is exponentially decreasing in t. There are N + 1 1 possible reduced regular configurations. Once ηˆ 1 (t) is regular, the addition site X t+1 1 uniquely determines the new reduced regular configuration R(η (t + 1)). This new
Probabilistic Approach to Zhang’s Sandpile Model
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reduced configuration cannot be the same as R(η1 (t)). Thus, there are N equally likely possibilities for R(η1 (t + 1)), and likewise for R(η2 (t + 1)). If R(η1 (t)) = R(η2 (t)), then one of the possibilities for R(η1 (t + 1)) is the same as R(η2 (t)), so that there are N − 1 possible reduced configurations that can be reached both from η1 (t) and η2 (t). The probability that R(η1 (t + 1)) is one of these is NN−1 , and the probability that R(η2 (t + 1)) is the same is N1 . Therefore, T is geometrically distributed, with parameter p N = NN−1 2 . We now use Lemma 5.2 with τ = T . For t > T , we have in this case that A1jθ (t) = A2jθ (t) and B 1jm (t) = B 2jm (t), because from time T on, in both processes the same avalanches occur. Also, for t > T , we have chosen U 1 (t) = U 2 (t). Therefore, for t > T, ηˆ 1j (t) − ηˆ 2j (t) =
N
1 2 B 1jm (t) ηˆ m (T ) − ηˆ m (T ) .
m=1
From (30) in the proof of Lemma 5.4 we know that t−T N +1 B 1jm (t) ≤ 1 − 2− 3N /2 , so that N
t−T N +1 1 B 1jm (t)ηˆ m (T ) ≤ N 1 − 2− 3N /2 ,
m=1
so that for t > T , we arrive at t−T −1 N +1 max |ηˆ 1j (t) − ηˆ 2j (t)| ≤ 2N 1 − 2− 3N /2 . j
We now split Pˆ (η1 ,η2 ) (max j |ηˆ 1j (t) − ηˆ 2j (t)| > ) into two terms, by conditioning on t < 2T and t ≥ 2T respectively. Both terms decrease exponentially in t: the first term because the probability of t < 2T is exponentially decreasing in t, and the second term because for t ≥ 2T , max j |ηˆ 1j (t) − ηˆ 2j (t)| itself is exponentially decreasing in t. A comparison of the two terms P(t < 2T ) and max j |ηˆ 1j (t) − ηˆ 2j (t)| yields that for N large, the second term dominates. We find that α N depends, for large N , on N as 1 α N = − 21 ln(1 − 2− 3N /2 ) N +1 . We see that as N increases, our bound on the speed of convergence decreases exponentially fast to zero. 5.2. Emergence of quasi-units in the infinite volume limit. In Proposition 3.3, we already noticed a close similarity between the stationary distribution of Zhang’s model with a ≥ 1/2, and the abelian sandpile model. We found that the stationary distribution of the reduced Zhang’s model, in which we label full sites as 1 and empty sites as 0, is equal to that of the abelian sandpile model (Proposition 3.8). In this section, we find that in the limit N → ∞, the similarity is even stronger. We find emergence of Zhang’s quasi-units in the following sense: as N → ∞, all one-site
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marginals of the stationary distribution concentrate on a single, nonrandom value. We believe that the same is true for a < 1/2 also (see Sect. 6.3 for a related result), but our proof is not applicable in this case, since it heavily depends on Proposition 3.8. To state and prove our result, we introduce the notation µ N for the stationary distribution for the model on N sites, with expectation and variance E N and Var N , respectively. Theorem 5.5. In the (N , [a, b]) model with a ≥ 21 , for the unique stationary measure µ N we have lim µ N = δE(U ) ,
(32)
N →∞
where δE(U ) denotes the Dirac measure concentrating on the (infinite-volume) constant configuration ηi = E(U ) for all i ∈ N, and where the limit is in the sense of weak convergence of probability measures. We will prove this theorem by showing that for η distributed according to µ N , in the limit N → ∞, for every sequence 1 ≤ j N ≤ N , 1. lim N →∞ E N η j N = EU , 2. lim N →∞ Var N (η j N ) = 0. The proof of the first item is not difficult. However, the proof of the second part is complicated, and is split up into several lemmas. Proof of Theorem 5.5, part (1). We choose as initial configuration η ≡ 0, the configuration with all N sites empty, so that according to Lemma 5.2, we can write η j N (t) =
t
Aθ j N (t)Uθ .
(33)
θ=1
Denoting expectation for this process as E0N , we find, using Remark 5.3, that E0N η j N (t) = EU E0N R(η(t) j N ). First, we take the limit t → ∞. By Theorem 5.1, E0N η j N (t) converges to E N η j N . From Proposition 3.8, it likewise follows that limt→∞ E0N R(η(t) j ) = NN+1 . Inserting these and subsequently taking the limit N → ∞ proves the first part. For the proof of the second, more complicated part, we need a number of lemmas. First, we rewrite Var N (η j N ) in the following manner. Lemma 5.6. Var (η j N ) = N
Var(U ) lim E0N t→∞
t θ=1
(Aθ j N (t))
2
+ (EU )2
N . (N + 1)2
Proof. We start from expression (33), and use that the corresponding variance Var 0N converges to the stationary Var N as t → ∞ by Theorem 5.1. We rewrite, for fixed N and j N = j,
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377
⎡
2 ⎤ t 2 Var 0N (η j (t)) = E0N (η j (t))2 − E0N η j (t) = E0N ⎣ Aθ j (t)Uθ ⎦ − E0N
t
θ=1
2 Aθ j (t)Uθ
θ=1
⎡
⎤ t = E0N ⎣ (Aθ j (t))2 Uθ2 + Aθ j (t)Uθ Aθ j (t)Uθ ⎦
θ=1
− E0N
t
2
θ =θ
Aθ j (t)Uθ
θ=1
= E(U 2 )E0N
t (Aθ j (t))2
θ=1
⎡
+ (EU )
2
E0N
⎣
⎤
Aθ j (t)Aθ j (t)⎦ − (EU )
θ =θ
t 2 2 N 2 = E(U ) − (EU ) E0 (Aθ j (t)) ⎡ + (EU )2 ⎣E0N
t
θ=1
2
Aθ j (t)
θ=1
= Var(U ) E0N
t
− E0N
t
2
E0N
t
2 Aθ j (t)
θ=1
2 ⎤ Aθ j (t) ⎦
θ=1
(Aθ j (t))2 + (EU )2 Var 0N (R(η(t) j )),
θ=1
where in the third equality we used the independence of the A-coefficients of the added amounts Uθ . We now insert j = j N , take the limit t → ∞, and insert limt→∞ Var 0N (R(η(t) j N )) = Var N (R(η j N )) = (N N+1)2 . Arriving at this point, in order to prove Theorem 5.5, it suffices to show that lim
N →∞
lim E N t→∞ 0
t 2 (Aθ j N (t)) = 0.
(34)
θ=1
The next lemmas are needed to obtain an estimate for this expectation. We will adopt the strategy of showing that the factors Aθ j (t) are typically small, so that the energy of a typical site consists of many tiny fractions of additions. To make this precise, we start with considering one fixed θ > N (N − 1), a time t > θ , and we fix > 0. Definition 5.7. We say that the event G t (α) occurs, if max j Aθ j (t) ≤ α. We say that the event Ht ( ) occurs, if max j Aθ j (t) ≥ , and if in addition there is a lattice interval of size at most M = 1 + 1, containing X θ , such that for all sites j outside this interval, Aθ j (t) ≤ . (We call the mentioned interval the θ -heavy interval.)
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Note that since we have j Aθ j (t) ≤ 1 for every θ , the number of sites where Aθ j (t) ≥ , cannot exceed 1 . In Lemma 5.4, we proved that max j Aθ j (t) is nonincreasing in t, for t ≥ θ . Therefore, also G t (α) is increasing in t. This is not true for Ht ( ), because after an avalanche, the sites where Aθ j (t) > might not form an appropriate interval around X θ . In view of what we want to prove, the events G t ( ) and Ht ( ) are good events, because they imply that (if we think of N as being much larger than M) Aθ j (t) ≤ ‘with large probability’. In the case that G t ( ) occurs, Aθi (t) ≤ for all i, and in the case that Ht ( ) occurs, there can be sites that contain a large Aθi (t), but these sites are in the θ -heavy interval containing X θ . This latter random variable is uniformly distributed on {1, . . . , N }, so that there is a large probability that a particular j does not happen to be among them. If we only know that G t (α) occurs for some α > , then we cannot draw such a conclusion. However, we will see that this is rarely the case. Lemma 5.8. For every N , for every θ > N (N − 1), for every > 0, for every K and j, 1. there exists a constant c = c( ), such that for θ ≤ t ≤ θ + K , P0N (Aθ j (t) > ) ≤
cK ; N
2. for every N large enough, there exist constants w = w( ) and 0 < γ = γ (N , ) < 1, such that for t > θ , P0N (Aθ j (t) > ) ≤ (1 − γ )t−θ−3w . In the proof of Lemma 5.8, we need the following technical lemma. Lemma 5.9. Consider a collection of real numbers yi ≥ 0, indexed by N, with i yi ≤ 1 and such that for some x ∈ N, maxi =x yi ≤ α. Then, for j ≥ x + α1 , we have j−x+2 i=1
1 1 α. y j−i+2 ≤ f (α) := 1 − 1 2i 2 α
Proof. We write j−x+2 i=1
1
α j−x+2 1 1 1 y = y + y j−i+2 j−i+2 j−i+2 i i 2 2 2i 1 i=1
i= α +1
α1
≤
1 1 y j−i+2 + 1
α +1 2i 2 i=1
j−x+2
y j−i+2 .
i= α1 +1
Note that index x is in the second sum. For i = 1, . . . , α1 , write y j−i+2 = α − z i ,
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with 0 ≤ z i ≤ α. Then, since α α1 ≥ 1 and α1 i=1 z i , so that 1
α 1 1 y j−i+2 + 1 i 2 2 α +1 i=1
j−x+2
yi ≤ 1, we have
i
j−x+2
1
y j−i+2
i= α1 +1
i= α1 +1
y j−i+2 ≤
1
α
α 1 1 ≤ (α − z ) + zi i 1 2i 2 α +1 i=1 i=1
α
α
α 1 1 1 1 zi ≤ = α− − 1 α, i
α +1 2i 2i 2 2 i=1 i=1 i=1 1
1
1
where in the last step we used z i ≥ 0. Thus j−x+2 i=1
α1 1 1 1 y j−i+2 ≤ α = 1− 1 α. 2i 2i 2 α i=1
Proof of Lemma 5.8, part (1). We first discuss the case t = θ . We show that Hθ ( ) occurs, for arbitrary . At t = θ , the addition is made at X θ . From Proposition 3.10, part (3), it follows, for every , that if after an avalanche there are sites j with Aθ j (θ ) > (we will call such sites ‘θ -heavy’ sites), then these sites form a set of adjacent sites including X θ , except for a possible empty site among them. Since we have Nj=1 Aθ j (θ ) ≤ 1, there can be at most
1 θ -heavy sites. If the addition was made to an empty site, then Aθ X θ (θ ) = 1. Thus, the θ -heavy interval has length at most 1 + 1 , and we conclude that Hθ ( ) occurs. To estimate the probability that Aθ j (θ ) > , or in other words, the probability that a given site j is in the θ -heavy interval, we use that X θ is uniformly distributed on {1, . . . , N }. Site j can be in the θ -heavy interval if the distance between X θ and j is at most 1 + 1 , 1+ 1
so that P0N (Aθ j (θ ) > ) ≤ 2 N =: c2N( ) . We next discuss θ < t ≤ θ + K . We introduce the following constants. We choose a number w such that f w (1) ≤ , with f as in Lemma 5.9, and where f w denotes f composed with itself w times. Note that this is possible because limk→∞ f k (1) = 0. We choose a combination of ˜0 and d such that ˜w ≤ , with ˜k+1 defined as ˜k + 1 ( f k (1) − ˜k ). Finally, we define M˜ k = ˜10 + 1 + k(1 + d). 2d+1 For fixed θ and a time t > θ , we define three types of avalanches at time t: ‘good’, ‘neutral’ and ‘bad’. Definition 5.10. For a fixed θ , the avalanche at time t is • a good avalanche if the following conditions are satisfied: 1. X t and X θ are on the same side of the empty site (if present) at t − 1, 2. X θ is at distance at least M˜ w from the boundary, 3. X t is at distance at least w from the boundary, and from the empty site (if present) at t − 1, 4. X t is at distance at least M˜ w + 1 from X θ , 5. X θ is at distance at least M˜ w from the empty site (if present) at t − 1,
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• a neutral avalanche if condition (5) is satisfied, but (1) is not, • a bad avalanche in all other cases. Having defined the three kinds of avalanches, we now claim the following: • If Ht−1 ( ) occurs, then after a neutral avalanche, Ht ( ) occurs. • If Ht−1 ( ) occurs, then after a good avalanche, G t ( ) occurs. The first claim (about the neutral avalanche) holds because if condition (5) is satisfied, but (1) is not, then not only is X θ not in the range of the avalanche, but the distance of X θ to the empty site is large enough to guarantee that the entire θ -heavy interval is not in the range. Thus, the θ -heavy interval does not topple in the avalanche. It then automatically follows that Ht ( ) occurs. To show the second claim (about the good avalanche), more work is required. We break the avalanche up into waves. Using a similar notation as in the proof of Proposition 3.10, we will denote by A˜ θ j (k) the fraction of Uθ at site j after wave k. We also define another event: we say that H˜ k ( M˜ k , α˜ k , ˜k ) occurs, if max j A˜ θ j (k) ≤ α˜ k , and all sites where A˜ θ j (k) > are in an interval of length at most M˜ k containing X θ (we will call this the (k, θ )-heavy interval), with the exception of site X t when it is unstable, in which case we require that A˜ θ X t (k) ≤ 2 ˜k . We define a ‘good’ wave, as a wave in which all sites of the (k, θ )-heavy interval topple, and the starting site X t is at a distance of at least α1k from the (k, θ )-heavy interval. It might become clear now that Definition 5.10 has been designed precisely so that a good avalanche is an avalanche that starts with at least w good waves. We will now show by induction in the number of waves that after an avalanche that starts with w good waves, G t ( ) occurs. For k = 0, we choose α˜ 0 = 1, so that at k = 0, H˜ 0 ( M˜ 0 , α˜ 0 , ˜0 ) occurs. We will 1 choose α˜ k+1 = f (α˜ k ) and ˜k+1 = ˜k + 2d+1 (α˜ k − ˜k ), so that once H˜ w ( M˜ w , α˜ w , ˜w ) occurred, we are sure that after the avalanche G t ( ) occurs, because both α˜ w and ˜w are smaller than . Now all we need to show is that, if H˜ k ( M˜ k , α˜ k , ˜k ) occurs, then after a good wave H˜ k+1 ( M˜ k+1 , α˜ k+1 , ˜k+1 ) occurs. From (6), we see that, for all j > X t that topple (and do not become empty), 1 1 1 1 A˜ θ j (k + 1) = A˜ θ, j+1 (k) + A˜ θ j (k) + A˜ θ, j−1 (k) + · · · + j−X +2 A˜ θ,X t (k), (36) t 2 4 8 2 and similarly for j < X t . For j = X t , we have 1 ˜ 1 1 ˜ A˜ θ j (k + 1) = Aθ, j+1 (k) + A˜ j,θ (k) 1 A˜ θ, j+1 (k) =0 + Aθ, j−1 (k) 2 4 2 1 (37) + A˜ j,θ (k) 1 A˜ θ, j−1 (k) =0 . 4 First we use that in a good wave, all sites in the θ -heavy interval topple, and X t is not in this interval. We denote by m the leftmost site of the θ -heavy interval, so that ˜ the rightmost site is m + M(k). Suppose, without loss of generality, that X t < m. We substitute A˜ θ j (k) ≤ α˜ k for all j in the θ -heavy interval, A˜ X t θ (k) ≤ 2 ˜k , and A˜ θ j (k) ≤ ˜k otherwise into (36), to derive for all j that topple: j < m − 1, j = x, ˜ j = m − 1, . . . , m + M(k), ˜ j = m + M(k) +d ,
A˜ θ j (k + 1) ≤ ˜k , A˜ θ j (k + 1) < α˜ k , A˜ θ j (k + 1) < ˜k +
1
2
d +1
(α˜ k − ˜k ), d = 1, 2, . . . .
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Additionally, by (37), we have A˜ θ,X t (k + 1) ≤ 2 ˜k . The factor 2 is only there as long as site X t is unstable. From (38) we have that α˜ k+1 < α˜ k , but moreover, in a good wave, the variables A˜ θ j (k) satisfy the conditions of Lemma 5.9, so that in fact α˜ k+1 ≤ f (α˜ k ). If we insert our choice of d for d , then we can see from (38) that indeed after the good wave H˜ k+1 ( M˜ k+1 , α˜ k+1 , ˜k+1 ) occurs. Now we are ready to evaluate P0N (Aθ j (t) > ), for t ∈ {θ + 1, . . . , θ + K }. As is clear by now, there are three possibilities: G t ( ) occurs, so that max j Aθ j (t) ≤ , or Ht ( ) occurs, in which case Aθ j (t) can be larger than if j is in the θ -heavy interval. We derived in the case t = θ that the probability for this is bounded above by cN2 . Finally, it is possible that neither occurs, in which case we do not have an estimate for the probability that Aθ j (t) > . But for this last case, we must have had at least one bad avalanche between θ + 1 and t. We will now show that the probability of this event is bounded above by KNc1 , where c1 depends only on . As stated in Definition 5.10, a bad avalanche can occur at time t if at least one of the conditions (2) through (5) is not satisfied. Thus, we can bound the total probability of a bad avalanche at time t, by summing the probabilities that the various conditions are not satisfied. We discuss the conditions one by one. ˜
w • The probability that condition (2) is not satisfied, is bounded above by 2 M N , since X θ is distributed uniformly on {1, . . . , N }. • The probability that condition (3) is not satisfied, is bounded above by 4w N , since X t is distributed uniformly on {1, . . . , N }, and independent of the position of the empty site at t − 1, if present.
2( M˜ + 1 )
w , • The probability that condition (4) is not satisfied, is bounded above by N since X t and X θ are independent. ˜w , since • The probability that condition (5) is not satisfied is bounded above by 2NM−1 the position of the empty site at t − 1 is almost uniform on {1, . . . , N }. (Recall the notion of almost uniformity, mentioned after Proposition 3.8.)
Thus, the total probability of a bad avalanche at time t is bounded by 2( M˜ w + 1 ) N
2 M˜ w N
2 M˜ w N
+
4w N
+
c1 N,
+ ≡ so that the probability of at least one bad avalanche between θ + 1 and t is bounded by KNc1 . We conclude that for t ∈ {θ + 1, . . . , θ + K }, P0N (Aθ j (t) > ) ≤ K cN1 +c2 ≤ cK N , for some c > 0. Proof of Lemma 5.8, part (2). From Lemma 5.9, it follows that if G t (α) occurs, and after s time steps all θ -heavy sites have toppled at least once, in avalanches that all start at least a distance α1 from all current θ -heavy sites, then G t+s ( f (α)) occurs; we will exploit this fact. Suppose that G t (α) occurs and that in addition, at time t the empty set is almost uniform. We claim that this implies that G t+2 ( f (α)) occurs with a probability that is bounded below, uniformly in N and . To see this, observe that if there is no empty site, then all N sites topple in one time step. If there is an empty site, this (meaning all sites topple) also happens in two steps if in the first step, we add to one side of the empty site, and in the second step to the other side. Denote by e1 the position of the empty site before the first addition (at X t+1 ), and e2 before the second addition (at X t+2 ). If X t+1 < e1 , then e2 < e1 . Therefore, all sites topple if X t+1 < e1 and X t+2 > e1 . With the distribution of e1 uniform on {1, . . . , N }, the probability that this happens is bounded below by some constant γ independent of N and . However we have the extra demand
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that both additions should start at least a distance α1 ≤ 1 from all current θ -heavy sites, of which there are at most 1 . Thus, both additions should avoid at most 1 2 sites. The probability that this happens is therefore less than some γ > 0, but it is easy to see that the difference decreases with N . We can then conclude that there is an N large enough so that the probability that this happens is at least γ > 0 for all N ≥ N , with 0 < γ < 1 independent of N and . In view of this, the probability that G θ+2 ( f (1)) occurs, for N large enough, is at least γ . We wish to iterate this argument w times. However, the lower bound γ is only valid when the empty site is almost uniform on {1, . . . , N }. We do not have this for η(θ + 2) since we have information about what happened in the time interval (θ, θ + 2). However, after one more addition at an unknown position, the empty site in η(θ +3) is again almost uniform on {1, . . . , N }. Since f w (1) ≤ , iterating this argument gives P0N (max Aθ j (t) > ) ≤ (1 − γ )t−θ−3w . j
Proof of Theorem 5.5, part (2). By Lemma 34, it suffices to prove (34). We estimate, using that θ Aθ j N (t) ≤ 1, t t N 2 N (Aθ j N (t)) ≤ E0 max Aθ j N (t) Aθ j N (t) ≤ E0N max Aθ j N (t) E0 1≤θ≤t
θ=1
≤ E0N
max
t−K ≤θ≤t
Aθ j N (t) + E0N
1≤θ≤t
θ=1
max
N (N +1)<θ
Aθ j N (t) + E0N
max
1≤θ≤N (N +1)
Aθ j N (t) .
We then for the first two terms estimate, using that maxθ Aθ j N (t) ≤ 1, P0N (Aθ j N (t) > ). E0N [max Aθ j N (t)] ≤ + P0N (max Aθ j N (t) > ) ≤ + θ
θ
θ
We finally use Lemma 5.8, and choose K = K N increasing with N . For θ ∈ [t, t − K N ], 2 t KN N we straightforwardly obtain θ=t−K N P0 Aθ j N (t) > = O N , uniformly in t, as N → ∞. For θ < t − K N we calculate P0N Aθ j N (t) > ≤ (1 − γ )t−θ−3w = O((1 − γ ) K N ), θ
N → ∞,
t−θ>K N
so that E0N
t
(Aθ j N (t))2 ≤ 2 + O
θ=1
+E0N
max
1≤θ≤N (N +1)
Aθ j N (t) ,
K N2 N
+ O((1 − γ ) K N )
N → ∞.
In the limit t → ∞, by Lemma 5.4, part (2), the last term vanishes. We now choose K N = N 1/3 , to obtain t N 2 lim sup lim E0 (Aθ j N (t)) ≤ 2 . N →∞ t→∞
θ=1
Probabilistic Approach to Zhang’s Sandpile Model
383
Since > 0 is arbitrary, we finally conclude that
lim
N →∞
lim E N t→∞ 0
t
(Aθ j N (t))
2
= 0.
θ=1
6. The (N, [0,1])-Model 6.1. Uniqueness of the stationary distribution. Theorem 6.1. The (N , [0, 1]) model has a unique stationary distribution υ N . For every initial distribution ν on N , Pν converges in total variation to υ N . Proof. We prove this theorem again by constructing a successful coupling. For clarity, we first treat the case N = 2, and then generalize to N > 2. The coupling is best described in words. Using the same notation as in previous couplings, we call two independent copies of the process η1 (t) and η2 (t), and call the coupled processes ηˆ 1 (t) and ηˆ 2 (t). Initially, we choose ηˆ 1 (t) = η1 (t), and ηˆ 2 (t) = η2 (t). It is easy, but tedious, to show that η11 (t) = η12 (t) = 0, while R(η21 (t)) = R(η22 (t)) = 1, occurs infinitely often. At the first such time T1 that this occurs, we choose the next addition as follows. Call (t) = η21 (t) − η22 (t). We choose Xˆ T21 +1 = X T11 +1 , and Uˆ T21 +1 = (UT11 +1 + (T1 ))mod 1. Observe that the distribution of Uˆ T21 +1 is uniform on [0, 1]. This addition is such that with positive probability the full sites are chosen for the addition, and the difference (T1 ) is canceled. More precisely, this occurs if X T11 +1 = 2, which has probability 1/2, and (UT11 +1 + (T1 ))mod 1 = UT11 +1 + (T1 ), which has probability at least 1/2, since η21 (T1 ) and η22 (T1 ) are both full, therefore (T1 ) ≤ 1/2. If this occurs, then we achieve success, i.e., ηˆ 1 (T1 + 1) = ηˆ 2 (T1 + 1), and from that time on we can let the two coupled processes evolve together. If ηˆ 1 (T1 + 1) = ηˆ 2 (T1 + 1), then we evolve the two coupled processes independently, and repeat the above procedure at the next instant that ηˆ 11 (t) = ηˆ 12 (t) = 0. Since at every such instant, the probability of success is positive, we only need a finite number of attempts. Therefore, the above constructed coupling is successful, and this proves the claim for N = 2. We now describe the coupling in the case N > 2. We will again evolve two processes independently, until a time where η11 (t) = η12 (t) = 0, while all other sites are full. At this time we will attempt to cancel the differences on the other N − 1 sites one by one. We define j (t) = η1j (t) − η2j (t), and as before we would be successful if we could cancel all these differences. However, now that N > 2, we do not want an avalanche to occur during this equalizing procedure, because we need η11 (t) = η12 (t) = 0 during the entire procedure. Therefore, we specify T1 further: T1 is the first time where not only η11 (t) = η12 (t) = 0 and all other sites are full, but also η1j (t) < 1 − and η2j (t) < 1 − , for all j = 2, . . . , N , with = 2 N1+1 . At such a time, a positive amount can be added to each site without starting an avalanche. We will first show that this occurs infinitely often, which also settles the case N = 2.
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By Proposition 3.3, after a finite time η1 (t) and η2 (t) contain at most one non-full site. It now suffices to show that for any ξ(t) ∈ N with at most one non-full site, with positive probability the event that ξ1 (t + 4) = 0, while ξ j (t + 4) ≤ 1 − 2 N1+1 for every 2 ≤ j ≤ N , occurs. One explicit possibility is as follows. The first addition should cause an avalanche. This will ensure that ξ(t + 1) contains one empty site. This occurs if the addition site is a full site, and the addition is at least 1/2. The probability of this is at least 21 (1 − N1 ). The second addition should change the empty site into full. For this to occur, the addition 1 should be at least 1/2, and the empty site should be chosen. This has probability 2N . The third addition should be at least 1/2 to site 1, so that an avalanche is started that will 1 result in ξ N (t + 3) = 0. This has again probability 2N . Finally, the last addition should 1 3 be an amount in [ 2 , 4 ], to site N − 1. Then by (6), every site but site N will topple once, and after this avalanche, site 1 will be empty, while every other site contains at most 1 1 − 2 N1+1 . This last addition has probability 4N . Now we show that at time T1 defined as above, there is a positive probability of success. To choose all full sites one by one, we require, first, for all j = 2, . . . , N that X T11 + j−1 = j. This has probability ( N1 ) N −1 . Second, we need (UT11 + j−1 + j (T1 )) mod1 = UT11 + j−1 + j (T1 ) for all j = 2, . . . , N . This event is independent of the previous event and has probability at least ( 21 ) N −1 . If this second condition is met, then third, we need to avoid avalanches, so for all j = 2, . . . , N , η1j (T1 + j − 1) + UT11 + j−1 = η2j (T1 + j − 1) + Uˆ T21 + j−1 < 1. It is not hard to see that this has positive conditional probability, given the previous events. We conclude that the probability of success at time T1 + N − 1 is positive, so that we only need a finite number of such attempts. Therefore, the coupling is successful, and we are done. 6.2. Simulations. We performed Monte Carlo simulations of the (N , [0, 1])-model, for various values of N . Figure 3 shows histograms of the energies that a site assumes during all the iterations. We started from the empty configuration, but omitted the first 10% of the observations to avoid recording transient behavior. Further increasing this percentage, or the number of iterations, had no visible influence on the results. The presented results show that, as the number of sites of the model increases, the energy becomes more and more concentrated around√a value close to 0.7. In the next section, we present an argument for this value to be 1/2. We further observe that it seems to make a difference where the site is located: at the boundary the variance seems to be larger than in the middle. 6.3. The expected stationary energy per site as N → ∞. From the simulations it appears that for large values of N , the energy per site concentrates at a value close to 0.7, for every site. Below we argue, under √ some assumptions that are consistent with our simulations, that this value should be 1/2. First, we assume that every site has the same expected stationary energy. Moreover, we assume that pairs of sites are asymptotically independent, i.e., ηx becomes independent of η y as |x − y| → ∞. (If the stationary measure is indeed such that the energy of every site is a.s. equal to a constant, then this second assumption is clearly true.) With Eυ N denoting expectation with respect to the stationary distribution υ N , we say that (υ N ) N is asymptotically independent if for any 1 ≤ x N , y N ≤
Probabilistic Approach to Zhang’s Sandpile Model
2.5
x 10
385
4
2.5
a
x 10
2
2
1.5
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1
1
0.5
0.5
0 0
0.1
0.2
0.3
0.4
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0.8
0.9
1
0
x 10
3.5
c
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x 10
d
3
3
2.5
2.5
2
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1.5
1
1
0.5
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0
0 0
0.1
0.2
0.3
0.4
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0.6
0.7
0.8
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0
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1
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0
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b
x 10
4.5
e
4
4
3.5
3.5
3
3
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2
2
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1
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0
x 10
f
0 0
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0.3
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1
0
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1
Fig. 3. Simulation results for the (N , [0, 1])-model. The histograms represent observed energies during 100,000 (a,b) and 200,000 iterations (c-f). The system size was 3 sites (a,b), 30 sites (c,d) and 100 sites (e,f). (a),(c) and (e) are boundary sites, (b), (d) and (f) are central sites
N with |x N − y N | → ∞, and for any A, B subsets of R with positive Lebesgue measure, we have (39) lim Eυ N (1ηx N ∈B |η y N ∈ A) − Eυ N (1ηx N ∈B = 0. N →∞
Theorem 6.2. Suppose that in the (N , [0, 1]) model, for any sequence j N ∈ {1, . . . , N }, lim Eυ N (η j N ) = ρ,
N →∞
(40)
for some constant ρ. Suppose in addition that (υ N ) N is asymptotically independent. Then we have ρ = 21 .
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Proof. The proof is based on a conservation argument. If we pick a configuration according to υ N and we make an addition U , we denote the random amount that leaves the system by E out,N . By stationarity, the expectation of U must be the same as the expectation of E out,N . The amount of energy that leaves the system in case of an avalanche, depends on whether or not one of the sites is empty (or behaves as empty). Remember (Proposition 3.3) that when we pick a configuration according to the stationary distribution, then there can be at most one empty or anomalous site. If there is one empty site, then the avalanche reaches one boundary. If there are only full sites, then the avalanche reaches both boundaries, and in case of one anomalous site, both can happen. However, configurations with no empty site have vanishing probability as N → ∞: we claim that the stationary probability for a configuration to have no empty site, is bounded above by p N , with lim N →∞ p N = 0. To see this, we divide the support of the stationary distribution into two sets: E, the set of configurations with one empty site, and N , the set of configurations with no empty site. The only way to reach N from E, is to make an addition precisely at the empty site. As X is uniformly distributed on {1, . . . , N }, this has probability N1 , irrespective of the details of the configuration. The only way to reach E from N , is to cause an avalanche; this certainly happens if an addition of at least 1/2 is made to a full site. Again, since X is uniformly distributed on {1, . . . , N }, and since there is at most one non-full site, this has probability at least 1 N −1 2 N . Now let X be the (random) addition site at a given time, and denote by A x the event that X = x and that this addition causes the start of an avalanche. Since E out,N = 0 when no avalanche is started, we can write Eυ N (E out,N ) =
N
Eυ N (E out,N |A x )Pυ N (A x ).
(41)
x=1
We calculate Pυ N (A x ) as follows, writing U for the value of the addition: 1 1 Pυ N (ηx + U ≥ 1) = Pυ N (U ≥ 1 − ηx ) N N 1 1 = Pυ N (U ≥ 1 − ηx )dυ N (η) = ηx dυ N (η) N N 1 = Eυ N (ηx ). N
Pυ N (A x ) =
(42)
Let L N = log N . Even if the avalanche reaches both boundary sites, the amount of energy that leaves the system can never exceed 2, which implies that N N −2L N E (E |A ) − E (E |A ) (43) υN out,N x υN out,N x ≤ 8L N . x=1 x=2L N It follows from (41), (42) and (43) that Eυ N (E out,N ) =
1 N
N −2L N x=2L N
Eυ N (E out,N |A x )Eυ N (ηx ) + O(L N/N ).
(44)
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If the avalanche, started at site x, reaches the boundary at site 1, then the amount of energy that leaves the system is given by 21 η1 + 41 η2 + · · · + ( 21 )x (ηx + U ). For all x ∈ {2L N , . . . , N − 2L N }, this can be written as L N L N +1 x 1 1 1 1 1 η1 + η2 + · · · + ηL N + η L N +1 + · · · + (ηx + U ), 2 4 2 2 2 where for the last part of this expression, we have the bound L N +1 x L N 1 1 1 η L N +1 + · · · + (ηx + U ) ≤ . 2 2 2 Since the occurrence of A x depends only on ηx (and on X and U ), for 2L N ≤ x ≤ N − 2L N , by asymptotic independence there is an α N , with lim N →∞ α N = 0, such that for all 1 ≤ i ≤ L N and 2L N ≤ x ≤ N − 2L N , we have |Eυ N (ηi |A x ) − Eυ N (ηi )| ≤ α N , so that
1 1 1 LN 1 LN Eυ N 2 η1 + · · · + ( 2 ) η L N |A x − Eυ N 2 η1 + · · · + ( 2 ) η L N 1 1 ≤ + + · · · αN , 2 4
which is bounded above by α N . By symmetry, we have a similar result in the case that the other boundary is reached. In case both boundaries are reached, we simply use that the amount of energy that leaves the system is bounded above by 2. In view of this, we continue the bound in (44) as follows:
L N N −2L 1 LN 1 N 1 1 1 η1 + η2 + · · · + Eυ N (E out,N ) = η L N |A x + O Eυ N N 2 4 2 2 x=2L N
Eυ N (ηx ) + +O(L N /N )
L N N −2L 1 1 N 1 1 η1 + η2 + · · · + = Eυ N η L N Eυ N (ηx ) + N 2 4 2 x=2L N 1 LN LN +O +O + O(α N ) + O( p N ), N 2 as N → ∞. Letting N → ∞ and inserting (40) now gives lim Eυ N (E out,N ) = ρ 2 .
N →∞
As the expectation of U is 21 , we conclude that ρ =
1 2.
Acknowledgement. We thank an anonymous referee for his/her very careful reading of the manuscript, which lead to many improvements and corrections. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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References 1. Blanchard, P., Cessac, B., Krüger, T.: A Dynamical System Approach to SOC Models of Zhang’s Type. J. Stat. Phys. 88(1/2), 307–318 (1997) 2. Dhar, D.: Self-Organized Critical State of Sandpile Automaton Models. Phys. Rev. Lett. 64(14), 1613–1616 (1990) 3. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.-O.: Delay equations. Applied Mathematical Sciences, Vol. 110. New York: Springer-Verlag, 1995 4. Dudley, R.M.: Real Analysis and Probability. Pacificrove, CA: Wadsworth & Brooks/Cole, 1989 5. Feller, W.: An Introduction to Probability Theory and its Applications, Vol. II. New York: John Wiley and Sons, Inc., 1966 6. Janosi, I.M.: Effect of anisotropy on the self-organized critical state. Phys. Rev. A 42(2), 769–774 (1989) 7. Meester, R., Redig, F., Znamenski, D.: The abelian sandpile model; a mathematical introduction. Markov Proc. Rel. Fields 7, 509–523 (2001) 8. Pastor-Satorras, R., Vespignani, A.: Anomalous scaling in the Zhang model. Eur. Phys. J. B 18, 197–200 (2000) 9. Ivashkevich, E.V., Priezzhev, V.B.: Introduction to the sandpile model.. Physica A 254, 97–116 (1998) 10. Maes, C., Redig, F., Saada, E., van Moffaert, A.: On the thermodynamic limit for a one-dimensional sandpile process. Markov Proc. and Rel. Fields 6(1), 1–21 (2000) 11. Thorisson, H.: Coupling, Stationarity, and Regeneration. New York: Springer Verlag, 2000 12. Turcotte, D.L.: Self-organized criticality. Rep. Prog. Phys. 62(10), 1377–1429 (1999) 13. Zhang, Y.-C.: Scaling theory of Self-Organized Criticality. Phys. Rev. Lett. 63(5), 470–473 (1989) Communicated by H. Spohn
Commun. Math. Phys. 280, 389–401 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0469-6
Communications in
Mathematical Physics
A Replica-Coupling Approach to Disordered Pinning Models Fabio Lucio Toninelli Université de Lyon, Laboratoire de Physique de l’Ecole Normale Supérieure de Lyon, CNRS UMR 5672, 46 Allée d’Italie, 69364 Lyon, France. E-mail: [email protected] Received: 5 February 2007 / Accepted: 11 October 2007 Published online: 28 March 2008 – © Springer-Verlag 2008
Abstract: We consider a renewal process τ = {τ0 , τ1 , . . .} on the integers, where the law of τi − τi−1 has a power-like tail P(τi − τi−1 = n) = n −(α+1) L(n) with α ≥ 0 and L(·) slowly varying. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to τ . In such generality this class of problems includes, among others, (1 + d)-dimensional models of pinning of directed polymers on a one-dimensional random defect, (1 + 1)-dimensional models of wetting of disordered substrates, and the Poland-Scheraga model of DNA denaturation. By varying the average of the reward, the system undergoes a transition from a localized phase, where τ occupies a finite fraction of N to a delocalized phase, where the density of τ vanishes. In absence of disorder (i.e., if the reward is independent of n), the transition is of first order for α > 1 and of higher order for α < 1. Moreover, for α ranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small (but extensive) amount of disorder is known to modify the order of transition as soon as α > 1/2 [11]. In physical terms, disorder is relevant in this situation, in agreement with the heuristic Harris criterion. On the other hand, for 0 < α < 1/2 it has been proven recently by K. Alexander [2] that, if disorder is sufficiently weak, critical exponents are not modified by randomness: disorder is irrelevant. In this work, generalizing techniques which in the framework of spin glasses are known as replica coupling and interpolation, we give a new, simpler proof of the main results of [2]. Moreover, we (partially) justify a small-disorder expansion worked out in [9] for α < 1/2, showing that it provides a free energy upper bound which improves the annealed one. 1. Introduction Consider a (recurrent or transient) Markov chain {Sn }n≥0 started from a particular point, call it 0 by convention, of the state space . Assume that the distribution of the interarrival times to the state 0 has a power-like tail: if τ := {n ≥ 0 : Sn = 0}, we require P(τi − τi−1 = n) n −α−1 for n large (see Eq. (2.1) below for precise definitions and
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conditions). This is true, for instance, if S is the simple random walk (SRW) in = Zd , in which case α = 1/2 for d = 1 and α = d/2 − 1 for d ≥ 2. One may naturally think of {(n, Sn )}n≥0 as a directed polymer configuration in × N. We want to model the situation where the polymer interacts with the one-dimensional defect line {0} × N. To this purpose, we introduce the Hamiltonian H N (S) = −
N
εn 1 Sn =0 ,
(1.1)
n=1
which gives a reward (if εn > 0) or a penalty (if εn < 0) to the occurrence of a polymer-line contact at step n. Typically, we have in mind the situation where {εn }n∈N is a sequence of IID (possibly degenerate) random variables. Let h and β 2 be the average and variance of εn , respectively. Varying h at β fixed, the system undergoes a phase transition: for h > h c (β) the Boltzmann average of the contact fraction N := |{1 ≤ n ≤ N : Sn = 0}|/N converges almost surely to a positive constant, call it (β, h), for N → ∞ (localized phase), while for h < h c (β) it converges to zero (delocalized phase). Models of this kind are employed in the physics literature to describe, for instance, the interaction of (1 + 1)-dimensional interfaces with disordered walls [6], of flux lines with columnar defects in type-II superconductors [17], and the DNA denaturation transition in the Poland-Scheraga approximation [5]. In absence of disorder (β = 0) it is known that the transition is of first order ((0, h) has a discontinuity at h c (0)) if α > 1, while for 0 ≤ α < 1 the transition is continuous: in particular, (0, h) vanishes like (h − h c (0))(1−α)/α for h h c (0) if 0 < α < 1 and faster than any power of (h − h c (0)) if α = 0. For α = 1, finally, transition can be either continuous or discontinuous, depending on the slowly varying function L(·) in (2.1). An interesting question concerns the effect of disorder on the nature of the transition. In terms of the non-rigorous Harris criterion, disorder is believed to be irrelevant for α < 1/2 and relevant for α > 1/2, where “relevance” refers to the property of changing the critical exponents. The question of disorder relevance in the (so called “marginal”) case α = 1/2 is not settled yet, even on heuristic grounds. Recently, rigorous methods have allowed to put this belief on firmer ground. In particular, in Refs. [11,12] it was proved that, for every β > 0, α ≥ 0 and h sufficiently close to (but larger than) h c (β), one has (β, h) ≤ (1 + α)c(β)(h − h c (β)), for some c(β) < ∞. This result, compared with the critical behavior mentioned above of the non-random model, proves relevance of disorder for α > 1/2, since 1 > (1 − α)/α. On the other hand, in a recent remarkable work K. Alexander showed [2] that the opposite is true for 0 < α < 1/2: if disorder is sufficiently weak, (β, h) vanishes like (h−h c (β))(1−α)/α as in the homogeneous model. Moreover, the critical point h c (β) coincides, always for β small and 0 < α < 1/2, with the critical point h ac (β) of the corresponding annealed model (cf. Sect. 2). Always in [2], for 1/2 ≤ α < 1 it was proven that the ratio h c (β)/ h ac (β) converges to 1 for β 0. This, on the other hand, is expected to be false for α > 1. The purpose of this work is twofold. Firstly, we present a method which allows to re-obtain the main results of [2] in a simpler way. Secondly, we show that the well known inequality between quenched and annealed free energies is strict as soon as the annealed model is localized and β > 0. Moreover, we prove that a small-disorder expansion for the quenched free energy, worked out in [9] for 0 ≤ α < 1/2, provides at least a free energy upper bound. As far as the first point is concerned, our strategy is a generalization of techniques which in the domain of mean field spin glasses are known as replica coupling [18,15]
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and interpolation. These methods had a remarkable impact on the understanding of spin glasses in recent years (see, e.g., [16,14,1,19]). In particular the “quadratic replica coupling” method, introduced in [15], gives a very efficient control of the SherringtonKirkpatrick model at high temperature (β small), i.e., for weak disorder, which is the same situation we are after here. Our method is not unrelated to that of [2]: the two share the idea that the basic object to look at is the law of the intersection of two independent copies of the renewal τ . However, our strategy allows to bypass the need of performing refined second-moment computations on a suitably truncated partition function as in [2] and gives, in the case of Gaussian disorder, particularly neat proofs. In the rest of the paper, we will forget the polymer-like interpretation and the Markov chain structure, and define the model directly starting from the process τ of the “returns to 0”. 2. Model and Results Consider a recurrent renewal sequence 0 = τ0 < τ1 < . . ., where {τi − τi−1 }i≥0 are integer-valued IID random variables with law K (n) := P(τ1 = n) =
L(n) n 1+α
∀n ∈ N.
(2.1)
We assume that the function L(·) is slowly varying at infinity [4], α ≥ 0 and n∈N K (n) = 1. Recall that a slowly varying function L(·) is a positive function (0, ∞) x → L(x) ∈ (0, ∞) such that, for every r > 0, lim
x→∞
L(r x) = 1. L(x)
(2.2)
We denote by E the expectation on τ := {τi }i≥0 and we put for notational simplicity δn := 1n∈τ , where 1 A is the indicator of the event A. We define, for β ≥ 0 and h ∈ R, the quenched free energy as F(β, h) = lim FN (β, h) := lim N →∞
N →∞
N 1 E log E e n=1 (βωn +h)δn δ N , N
(2.3)
where {ωn }n∈N are IID centered random variables with finite second moment, law denoted by P and corresponding expectation E, and normalized so that E ω12 = 1. In this d
work, we restrict to the case where disorder has a Gaussian distribution: ω1 = N (0, 1). Some degree of generalization is possible: for instance, results and proofs can be extended to the situation where ωn are IID bounded random variables. The existence of the N → ∞ limit in (2.3) is well known, see for instance [9, Sect. 4.2]. The limit actually exists, and is almost-surely equal to F(β, h), even omitting the disorder average E in (2.3). We point out that, by superadditivity, for every N ∈ N, FN (β, h) ≤ F(β, h)
(2.4)
and that, from Jensen’s inequality, FN (β, h) ≤ FNa (β, h) :=
N 1 log E E e n=1 (βωn +h)δn δ N = FN (0, h + β 2 /2). (2.5) N
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(If disorder is not Gaussian, β 2 /2 is replaced by log E exp(βω1 ).) FNa (β, h) is known as the (finite-volume) annealed free energy which, as (2.5) shows, coincides with the free energy of the homogeneous model (β = 0) for a shifted value of h. The limit free energy F(β, h) would not change (see, e.g., [11, Remark 1.1]) if the factor δ N were omitted in (2.3), i.e., if the boundary condition {N ∈ τ } were replaced by a free boundary condition at N . However, in that case exact superadditivity, and (2.4), would not hold. Another well-established fact is that F(β, h) ≥ 0 (cf. for instance [11]), which allows the definition of the critical point, for a given β ≥ 0, as h c (β) := sup{h ∈ R : F(β, h) = 0}. Note that Eq. (2.5) implies h c (β) ≥ h ac (β) := sup{h ∈ R : F a (β, h) = 0} = h c (0) − β 2 /2. For obvious reasons, h ac (β) is referred to as the annealed critical point. Concerning upper bounds for h c (β), already before [2] it was known (see [3] and [9, Theorem 5.2]) that h c (β) < h c (0) for every β > 0. To make a link with the discussion in the introduction, note that the contact fraction (β, h) is just ∂h F(β, h). With the exception of Theorem 2.6, we will consider from now on only the values 0 < α < 1, as in [2], in which case τ is null-recurrent under P. For the homogeneous system it is known [9, Theorem 2.1] that F(0, h) = 0 if h ≤ 0, while for h > 0, F(0, h) = h 1/α L(1/ h). L(·) is a slowly varying function satisfying 1/α α h −1/α Rα (h), L(1/ h) =
(1 − α)
(2.6)
(2.7)
and Rα (·) is asymptotically equivalent to the inverse of the map x → x α L(1/x). The fact that L(·) is slowly varying follows from [4, Theorem 1.5.12]. In particular, notice that h c (0) = 0, so that h ac (β) = −β 2 /2. We want to prove first of all that, if 0 < α < 1/2 and β is sufficiently small (i.e., if disorder is sufficiently weak), h c (β) = h ac (β). Keeping in mind that F a (β, h ac (β)+ ) = F(0, ), this is an immediate consequence of Theorem 2.1. Assume that either 0 < α < 1/2 or that α = 1/2 and n∈N n −1 L(n)−2 < ∞. Then, for every > 0 there exist β0 ( ) > 0 and 0 ( ) > 0 such that, for every β ≤ β0 ( ) and 0 < < 0 ( ), one has (1 − )F(0, ) ≤ F(β, h ac (β) + ) ≤ F(0, ).
(2.8)
In view of [11, Theorem 2.1], the same cannot hold for α > 1/2. However, one has: ˇ Theorem 2.2. Assume that 1/2 < α < 1. There exists a slowly varying function L(·) and, for every > 0, constants a1 ( ) < ∞ and 0 ( ) > 0 such that, if ˇ a1 ( )β 2α/(2α−1) L(1/β) ≤ ≤ 0 ( ),
(2.9)
the inequalities (2.8) hold. As already pointed out in [2], since 2α/(2α −1) > 2 Theorem 2.2 shows in particular that h c (β) lim = 1. (2.10) β0 h ac (β) On the other hand, it is unknown whether there exist non-zero values of β such that the equality h c (β) = h ac (β) holds, for 1/2 < α < 1.
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Remark 2.3. The lower bound we obtain for F(β, h) (and, as a consequence, for h ac (β)− h c (β)) in Theorem 2.2 differs from the analogous one of [2, Theorem 3] only in the form ˇ (an explicit choice of L(·) ˇ can be extracted from of the slowly varying function L(·) ˇ Eq. (3.28) below). In general, our L(·) is larger due to the logarithmic denominator in (3.28). However, the important point is that the exponent of β in (2.9) agrees with that in the analogous condition of [2, Theorem 3]. Indeed, with the conventions of [2] (which amount to replacing everywhere h by βh and α by c − 1), the exponent 2α/(2α − 1) = 1 + 1/(2α − 1) would be instead 1/(2α − 1) = 1/(2c − 3), as in [2, Theorem 3]. Finally, for the “marginal case” we have Theorem 2.4. Assume that α = 1/2 and n∈N n −1 L(n)−2 = ∞. Let (·) be the slowly varying function (diverging at infinity) defined by N n=1
1 n L(n)2
N →∞
∼
(N ).
(2.11)
For every > 0 there exist constants a2 ( ) < ∞ and 0 ( ) > 0 such that, if 0 < ≤ 0 ( ) and if the condition a2 ( )| log F(0, )| 1 ≥ a2 ( ) (2.12) β2 F(0, ) is verified, then Eq. (2.8) holds. Remark 2.5. Note that, thanks to Theorem 2.4 and the property of slow variation of (·), the difference h c (β) − h ac (β) vanishes faster than any power of β, for β 0. Again, it is unknown whether h c (β) = h ac (β) for some β > 0. In general, our condition (2.12) is different from the one in the analogous Theorem 4 of [2], due to the presence of the factor | log F(0, )| in the argument of (·). However, for many “reasonable” and physically interesting choices of L(·) in (2.1), the two results are equivalent. In particular, if P is the law of the returns to zero of the SRW {Sn }n≥0 in one dimension, i.e. τ = {n ≥ 0 : S2n = 0}, in which case L(·) and L(·) are asymptotically constant and (N ) ∼ a3 log N , one sees easily that (2.12) is verified as soon as ≥ a4 ( )e
−
a5 ( ) β2
,
(2.13)
which is the same condition which was found in [2]. Another case where Theorem 2.4 and [2, Theorem 4] are equivalent is when L(n) ∼ a6 (log n)(1−γ )/2 for γ > 0, in which case (N ) ∼ a7 (log N )γ . While we focused up to now on free energy lower bounds, one may wonder whether it is possible to improve the Jensen upper bound (2.5). For α > 1/2 it follows from [11] that F(β, h) < F a (β, h) as soon as β is positive and h − h ac (β) is positive and small. We conclude this section with a theorem which generalizes this result to arbitrary α and h > h ac (β), and which justifies (as an upper bound) a small-β expansion worked out in [9, Sect. 5.5].
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Theorem 2.6. For every β > 0, α ≥ 0 and > 0, 2 2 β q a 2 + F(0, − β q) < F(0, ). inf F(β, h c (β) + ) ≤ 2 0≤q≤ /β 2
(2.14)
As a consequence, for 0 ≤ α < 1/2 there exist β0 > 0 and 0 > 0 such that F(β, h ac (β) + ) ≤ F(0, ) −
β2 (∂ F(0, ))2 (1 + O(β 2 )) 2
(2.15)
for β ≤ β0 and ≤ 0 , where O(β 2 ) is independent of . The first inequality in (2.14) is somewhat reminiscent of the “replica-symmetric” free energy upper bound [13] for the Sherrington-Kirkpatrick model. The reader who wonders why we stopped at order β 2 in the “expansion” (2.15) should look at Remark 3.1 below. Note that, in view of Eqs. (2.6) and (3.47), (∂ F(0, ))2 F(0, ) if is small and α < 1/2. Observe also that, for α > 1/2 and β, small, taking the infimum in (2.14) gives nothing substantially better than just choosing q = /β 2 , from which F(β, h ac (β) + ) ≤ 2 /(2β 2 ); essentially the same bound (with an extra factor (1 + α) in the right-hand side) is however already implied by [11, Theorem 2.1] (see also [9, Remark 5.7]). Remark 2.7. As a general remark, we emphasize that the assumption of recurrence for τ , i.e., n∈N K (n) = 1 is by no means a restriction. Indeed, as has been observed several times in the literature (including [11] and [2]), if K := n∈N K (n) < 1 one can (n) is (n) := K (n)/ K , and of course the renewal τ with law P(τ1 = n) = K define K recurrent. Then, it is immediate to realize from definition (2.3) that h + log K ), F(β, h) = F(β,
(2.16)
being the free energy of the model defined as in (2.3) but with P replaced by F P. In particular, h ac (β) = − log K − β 2 /2. Theorems 2.1-2.6 are therefore transferred with obvious changes to this situation. This observation allows to apply the results, for instance, to the case where we consider the SRW {Sn }n≥0 in Z3 , and we let P be the law of τ := {n ≥ 0 : S2n = 0}, i.e., the law of its returns to the origin. In this case, assumption (2.1) holds with α = 1/2, L(·) asymptotically constant and, due to transience, K < 1. The same is true if {Sn }n≥0 is the SRW on Z, conditioned to be non-negative. 3. Proofs Proof of Theorem 2.1. In view of Eq. (2.5), we have to prove only the first inequality in (2.8). This is based on an adaptation of the quadratic replica coupling method of [15], plus ideas suggested by [2]. Let > 0 and start from the identity F(β, −β 2 /2 + ) = F(0, ) + lim R N , (β), N →∞
where R N , (β) :=
N 1 2 E log e n=1 (βωn −β /2)δn , N , N
(3.1)
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and, for a P-measurable function f (τ ),
f N ,
N E e n=1 δn f δ N . N := E e n=1 δn δ N
(3.2)
Via the Gaussian integration by parts formula E (ω f (ω)) = E f (ω),
(3.3)
valid (if ω is a standard Gaussian random variable N (0, 1)) for every differentiable function f (·) such that lim|x|→∞ exp(−x 2 /2) f (x) = 0 , one finds for 0 < t < 1: ⎧⎛ ⎞2 ⎫ √ N (β tωn −tβ 2 /2)δn ⎪ ⎪ n=1 N δ e ⎨ ⎬ m √ d β2 ⎜ ⎟ N , ⎠ . (3.4) R N , ( tβ) = − E ⎝ N √ 2 ⎪ ⎪ dt 2N ⎭ e n=1 (β tωn −tβ /2)δn m=1 ⎩ N ,
Define also, for λ ≥ 0,
1 (1) (2) ⊗2 E log e HN (t,λ,β;τ ,τ ) N , 2N N √ 1 2 /2)(δ (1) +δ (2) )+λβ 2 N δ (1) δ (2) ⊗2 (β tω −tβ n n n n=1 n n := , E log e n=1 N , 2N
ψ N , (t, λ, β) :=
(3.5)
(i)
(1) (2) where the product measure ·⊗2 N , acts on the pair (τ , τ ), while δn := 1n∈τ (i) . Note that ψ N , (t, λ, β) actually depends on (t, λ, β) only through the two combinations β 2 t and β 2 λ. We add also that the introduction of the parameter t, which could in principle be avoided, allows for more natural expressions in the formulas which follow. One has immediately 2 N (1) (2) ⊗2 1 log eλβ n=1 δn δn ψ N , (0, λ, β) = (3.6) N , 2N and √ (3.7) ψ N , (t, 0, β) = R N , ( tβ).
Again via integration by parts, (1) (2) ⊗2 (1) (2) δm δm e HN (t,λ,β;τ ,τ ) d N , ψ N , (t, λ, β) = E ⊗2 dt 2N (1) (2) m=1 e HN (t,λ,β;τ ,τ ) N , ⎧⎛ ⎞2 ⎫ (1) ,τ (2) ) ⊗2 ⎪ ⎪ (1) (2) H (t,λ,β;τ ⎪ ⎪ N N (δ + δm )e ⎬ β 2 ⎨⎜ m N , ⎟ − E ⎝ ⎠ ⊗2 ⎪ ⎪ 4N (1) (2) ⎪ m=1 ⎪ ⎭ ⎩ e HN (t,λ,β;τ ,τ )
β2
N
≤
(1) (2) ⊗2 (1) (2) N δm δm e HN (t,λ,β;τ ,τ ) 2 β N , E ⊗2 2N (1) (2) m=1 e HN (t,λ,β;τ ,τ ) N ,
N ,
=
d ψ N , (t, λ, β), dλ (3.8)
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so that, for every 0 ≤ t ≤ 1 and λ, ψ N , (t, λ, β) ≤ ψ N , (0, λ + t, β).
(3.9)
Going back to Eq. (3.4), using convexity and monotonicity of ψ N , (t, λ, β) with respect to λ and (3.7), one finds √ d d −R N , ( tβ) = ψ N , (t, λ, β)λ=0 dt dλ √ ψ N , (t, 2 − t, β) − R N , ( tβ) ≤ 2−t √ ≤ ψ N , (0, 2, β) − R N , ( tβ), (3.10) where in the last inequality we used (3.9) and the fact that 2 − t ≥ 1. Integrating this differential inequality between 0 and 1 and observing that R N , (0) = 0, one has 0 ≤ −R N , (β) ≤ (e − 1)ψ N , (0, 2, β).
(3.11)
Now we estimate
2 N (1) (2) N (1) (2) 1 (1) (2) log E⊗2 e2β n=1 δn δn + n=1 (δn +δn ) δ N δ N 2N (1) (2) 1 FN (0, q ) 2 N + log E⊗2 e2 pβ n=1 δn δn , ≤ −FN (0, ) + q 2N p (3.12)
ψ N , (0, 2, β) = −FN (0, ) +
where we used Hölder’s inequality and p, q (satisfying 1/ p + 1/q = 1) are to be determined. One finds then (1) (2) 1 2 N log E⊗2 e2 pβ n=1 δn δn lim sup ψ N , (0, 2, β) ≤ lim sup N →∞ N →∞ 2N p 1 F(0, q ) −1 . (3.13) +F(0, ) q F(0, ) We know from (2.6) and the property (2.2) of slow variation that, for every q > 0, lim
0
F(0, q ) = q 1/α . F(0, )
(3.14)
Therefore, choosing q = q( ) sufficiently close to (but not equal to) 1 and 0 ( ) > 0 sufficiently small one has, uniformly on β ≥ 0 and on 0 < ≤ 0 ( ), lim sup ψ N , (0, 2, β) ≤ N →∞
1 F(0, ) + lim sup log E⊗2 e−1 2N p( ) N →∞ N (1) (2) 2 p( )β 2 n=1 δn δn . × e
(3.15)
Of course, p( ) = q( )/(q( ) − 1) < ∞ as long as > 0. Finally, we observe that (1) (2) under the assumptions of the theorem, the renewal transient under the law τ ∩−1τ is−2 ⊗2 P . Indeed, if 0 < α < 1/2 or if α = 1/2 and n∈N n L(n) < ∞ one has ⎛ ⎞ 1n∈τ (1) ∩τ (2) ⎠ = P(n ∈ τ )2 < ∞ (3.16) E⊗2 ⎝ n≥1
n≥1
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since, as proven in [7], n→∞
P(n ∈ τ ) ∼
1 Cα α sin(π α) := . L(n)n 1−α π L(n)n 1−α
Actually, Eq. (3.17) holds more generally for 0 < α < 1. Therefore, there exists β1 > 0 such that (1) (2) 2 N sup E⊗2 e2 p( )β n=1 δn δn < ∞
(3.17)
(3.18)
N
for every β 2 p( ) ≤ β12 . Together with (3.15) and (3.1), this implies F(β, −β 2 /2 + ) ≥ (1 − )F(0, ) as soon as β 2 ≤ β02 ( ) := β12 / p( ).
(3.19)
Proof of Theorem 2.2. In what follows we assume that is sufficiently small so that F(0, ) < 1. Let N = N ( ) := c| log F(0, )|/F(0, ) with c > 0. By Eq. (2.4) we have, in analogy with (3.1), F(β, −β 2 /2 + ) ≥ FN ( ) (0, ) + R N ( ), (β).
(3.20)
As follows from Proposition 2.7 of [10], there exists a8 ∈ (0, ∞) (depending only on the law K (·) of the renewal) such that FN (0, ) ≥ F(0, ) − a8
log N N
(3.21)
for every N . Choosing c = c( ) large enough, Eq. (3.21) implies that FN ( ) (0, ) ≥ (1 − )F(0, ). As for R N ( ), (β), we have from (3.11) and (3.12), 1 F(0, q ) − 1 + F(0, ) (1 − e)−1 R N ( ), (β) ≤ F(0, ) q F(0, ) 1 2 N ( ) (1) (2) + log E⊗2 e2 pβ n=1 δn δn , 2N ( ) p
(3.22)
(3.23)
where we used Eqs. (3.22) and (2.4) to bound (1/q)FN ( ) (0, q ) − FN ( ) (0, ) from above. Choosing again q = q( ) we obtain, for ≤ 0 ( ), 1 2 N ( ) (1) (2) (1 − e)−1 R N ( ), (β) ≤ 2 F(0, ) + log E⊗2 e2 p( )β n=1 δn δn . 2N ( ) p( ) (3.24) Now observe that, if 1/2 < α < 1, there exists a9 = a9 (α) ∈ (0, ∞) such that for every integers N and k, N k L(N )2 ⊗2 (1) (2) P δn δn ≥ k ≤ 1 − a9 2α−1 . (3.25) N n=1
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This geometric bound is proven in [2, Lemma 3], but in Subsect. 3.1 we give another simple proof. Thanks to (3.25) we have E
⊗2
−1 N ( ) (1) (2) L(N ( ))2 2 p( )β 2 n=1 δn δn 2β 2 p( ) e ≤ 1−e , (3.26) 1 − a9 N ( )2α−1
whenever the right-hand side is positive, and this is of course the case under the stronger requirement e
2β 2 p( )
a9 L(N ( ))2 L(N ( ))2 1 − a9 ≤ 1− . N ( )2α−1 2 N ( )2α−1
(3.27)
At this point, using the definition of N ( ), it is not difficult to see that there exists a positive constant a10 ( ) such that (3.27) holds if ˆ β 2 p( ) ≤ a10 ( ) (2α−1)/α L(1/ ) 2α−1 | log F(0, )| 2 L(1/ ) (2α−1)/α L . := a10 ( ) |log F(0, )| F(0, )
(3.28)
ˆ The fact that L(·) is slowly varying follows from [4, Prop. 1.5.7] and Eq. (2.6). For ˆ instance, if L(·) is asymptotically constant one has L(x) ∼ a11 | log x|1−2α . Condition ˇ . (3.28) is equivalent to the first inequality in (2.9), for suitably chosen a1 ( ) and L(·) As a consequence, F(0, ) 1 2 N ( ) (1) (2) log E⊗2 e2 p( )β n=1 δn δn ≤ 2N ( ) p( ) 2c( ) p( )| log F(0, )| 2N ( )2α−1 . (3.29) × log a9 L(N ( ))2 Recalling Eq. (2.6) one sees that, if c( ) is chosen large enough, 1 2 N ( ) (1) (2) log E⊗2 e2 p( )β n=1 δn δn ≤ F(0, ). 2N ( ) p( )
(3.30)
Together with Eqs. (3.20), (3.22) and (3.24), this concludes the proof of the theorem. Proof of Theorem 2.4. The proof is almost identical to that of Theorem 2.2 and up to Eq. (3.24) no changes are needed. The estimate (3.25) is then replaced by P
⊗2
N n=1
δn(1) δn(2)
≥k
a12 k ≤ 1− (N )
(3.31)
for every N , for some a12 > 0 (see [2, Lemma 3], or the alternative argument given in Subsect. 3.1). In analogy with Eq. (3.26) one obtains then 2 N ( ) (1) (2) 2 E⊗2 e2 p( )β n=1 δn δn ≤ 1 − e2β p( ) 1 −
a12 (N ( ))
−1 (3.32)
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whenever the right-hand side is positive. Choosing a2 ( ) large enough one sees that if condition (2.12) is fulfilled then a12 a12 2 e2β p( ) 1 − ≤ 1− (3.33) (N ( )) 2(N ( )) and, in analogy with (3.29), 1 F(0, ) 2 N ( ) (1) (2) log E⊗2 e2( )β n=1 δn δn ≤ 2N ( ) p( ) 2c( ) p( )| log F(0, )| 2(N ( )) . (3.34) × log a12 From this estimate, for c( ) sufficiently large one obtains again (3.30) and as a consequence the statement of Theorem 2.4. 3.1. Proof of (3.25) and (3.31). For what concerns (3.25), start from the obvious bound N k ⊗2 (1) (2) P δn δn ≥ k ≤ 1 − P⊗2 (inf{n > 0 : n ∈ τ (1) ∩ τ (2) } > N ) . (3.35) n=1
Next note that, by Eq. (3.17), n→∞
u n := P⊗2 (n ∈ τ (1) ∩ τ (2) ) ∼
Cα2 L(n)2 n 2(1−α)
(3.36)
and that u n satisfies the renewal equation u n = δn,0 +
n−1
u k Q(n − k),
(3.37)
k=0
where Q(k) := P⊗2 (inf{n > 0 : n ∈ τ (1) ∩ τ (2) } = k) is the probability we need to estimate in (3.35). Q(·) is a probability on N since the renewal τ (1) ∩ τ (2) is recurrent for 1/2 < α < 1, as can be seen from the fact that, due to Eq. (3.17), the expectation in (3.16) diverges in this case. After a Laplace transform, one finds for s > 0, 1 ˆ (3.38) e−ns Q(n) = 1 − Q(s) := u(s) ˆ n≥0
and, by [4, Theorem 1.7.1] and the asymptotic behavior (3.36), one finds s→0+
u(s) ˆ ∼
1 Cα2 (2α) . 2α − 1 s 2α−1 (L(1/s))2
(3.39)
Note that 0 < 2α−1 < 1. By the classical Tauberian theorem (in particular, [4, Corollary 8.1.7] is enough in this case), one obtains then n≥N
Qn
N →∞
∼
L(N )2 2α − 1 Cα2 (2α) (2(1 − α)) N 2α−1
which, together with (3.35), completes the proof of (3.25).
(3.40)
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We turn now to the proof of (3.31). From Eq. (3.36) with α = 1/2 and [4, Theorem 1.7.1], one finds, in analogy with (3.39), s→0+
2 u(s) ˆ ∼ C1/2 (1/s).
(3.41)
Then, Eq. (3.38) and [4, Corollary 8.1.7] imply 1 N →∞ Qn ∼ , 2 (N ) C 1/2 n≥N and therefore Eq. (3.31).
(3.42)
3.2. Proof of Theorem 2.6. Start again from (3.1) and define, for q ∈ R, N √ 1 2 2 φ N , (t, β) := E log e n=1 [β tωn −tβ /2+β q(t−1)]δn N , N so that φ N , (0, β) = FN (0, − β 2 q) − FN (0, )
(3.43)
(3.44)
and φ N , (1, β) = R N , (β). In analogy with Eq. (3.4) one has ⎧⎛ ⎞2 ⎫ √ N [β tωn −tβ 2 /2+β 2 q(t−1)]δn ⎪ ⎪ n=1 N δ e ⎨ ⎬ m d β2 ⎜ ⎟ N , − q ⎠ φ N , (t, β) = − E ⎝ N √ 2 2 ⎪ ⎪ dt 2N ⎭ e n=1 [β tωn −tβ /2+β q(t−1)]δn m=1 ⎩ N ,
β 2q 2 + 2 β 2q 2 , (3.45) ≤ 2 from which statement (2.14) follows after an integration on t (it is clear that taking the infimum over q ∈ R or over 0 ≤ q ≤ /β 2 gives the same result.) The strict inequality in (2.14) holds since the quantity to be minimized in (2.14) has negative derivative at q = 0. To prove (2.15) recall that F(0, ) satisfies for > 0 the identity [11, Appendix A] e−F(0, )n K (n) = e− , (3.46) n∈N
(so that, in particular, F(0, ) is real analytic for > 0). An application of [4, Theorem 1.7.1] gives therefore, for α < 1/2, ∂ F(0, ) = (1−α)/α L (1) (1/ ),
2 ∂ F(0, ) = (1−2α)/α L (2) (1/ ),
(3.47)
where the slowly varying functions L (i) (·) can be expressed through L(·) (cf., for instance, [9, Sect. 2.4] for the first equality). For α = 0, (3.47) is understood to mean 2 F(0, ) is that the two derivatives vanish faster than any power of . This shows that ∂ bounded above by a constant for, say, ≤ 1 if α < 1/2. Then, choosing q = ∂ F(0, ) in (2.14) (which is the minimizer of β 2 q 2 /2 + F(0, − β 2 q) at lowest order in β) yields (2.15). It is important to note that, thanks to the first equality in (3.47) and the assumption α < 1/2, this choice is compatible with the constraint q ≤ /β 2 , for and β sufficiently small.
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Remark 3.1. The reason why we stopped at order β 2 in (2.15) is that at next order the 3 F(0, ), which diverges for 0 if α > 1/3. In analogy error term O(β 6 ) involves ∂ with (2.15), one can however prove that, if α < 1/k with 2 < k ∈ N, the expansion (2.15) can be pushed to order β 2(k−1) with a uniform control in of the error term O(β 2k ). We do not detail this point, the computations involved being straightforward. Acknowledgments. I am extremely grateful to Giambattista Giacomin for many motivating discussions and for constructive comments on this manuscript. This research has been conducted in the framework of the GIP-ANR project JC05_42461 (POLINTBIO).
References 1. Aizenman, M., Sims, R., Starr, S.L.: Extended variational principle for the Sherrington-Kirkpatrick spinglass model. Phys. Rev. B 68, 214403 (2003) 2. Alexander, K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279(1), 117–146 (2008) 3. Alexander, K.S., Sidoravicius, V.: Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16, 636–669 (2006) 4. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge: Cambridge University Press, 1987 5. Cule, D., Hwa, T.: Denaturation of Heterogeneous DNA. Phys. Rev. Lett. 79, 2375–2378 (1997) 6. Derrida, B., Hakim, V., Vannimenus, J.: Effect of disorder on two-dimensional wetting. J. Stati. Phys. 66, 1189–1213 (1992) 7. Doney, R.A.: One-sided large deviation and renewal theorems in the case of infinite mean. Probab. Theory Rel. Fields 107, 451–465 (1997) 8. Forgacs, G., Luck, J.M., Nieuwenhuizen, Th.M., Orland, H.: Wetting of a Disordered Substrate: Exact Critical behavior in Two Dimensions. Phys. Rev. Lett. 57, 2184–2187 (1986) 9. Giacomin, G.: Random Polymer Models. London-Singapore: Imperial College Press/World Scientific, 2007 10. Giacomin, G., Toninelli, F.L.: The localized phase of disordered copolymers with adsorption. ALEA 1, 149–180 (2006) 11. Giacomin, G., Toninelli, F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006) 12. Giacomin, G., Toninelli, F.L.: Smoothing of Depinning Transitions for Directed Polymers with Quenched Disorder. Phys. Rev. Lett. 96, 060702 (2006) 13. Guerra, F.: Sum rules for the free energy in the mean field spin glass model. In: Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects. Fields Inst. Commun. 30, Providence, RI: Amer. Math. Soc., 2001 14. Guerra, F.: Replica Broken Bounds in the Mean Field Spin Glass Model. Commun. Math. Phys. 233, 1–12 (2003) 15. Guerra, F., Toninelli, F.L.: Quadratic replica coupling for the Sherrington-Kirkpatrick mean field spin glass model. J. Math. Phys. 43, 3704–3716 (2002) 16. Guerra, F., Toninelli, F.L.: The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230, 71–79 (2002) 17. Nelson, D.R., Vinokur, V.M.: Boson localization and correlated pinning of superconducting vortex arrays. Phys. Rev. B 48, 13060–13097 (1993) 18. Talagrand, M.: On the high temperature region of the Sherrigton-Kirkpatrick model. Ann. Probab. 30, 364–381 (2002) 19. Talagrand, M.: The Parisi Formula. Ann. Math. 163, 221–263 (2006) Communicated by M. Aizenman
Commun. Math. Phys. 280, 403–425 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0467-8
Communications in
Mathematical Physics
P(φ)2 Quantum Field Theories and Segal’s Axioms Doug Pickrell Mathematics Department, University of Arizona, Tucson, AZ 85721, USA. E-mail: [email protected] Received: 4 March 2007 / Accepted: 11 October 2007 Published online: 28 March 2008 – © Springer-Verlag 2008
Abstract: The purpose of this paper is to show that P(φ)2 Euclidean quantum field theories satisfy axioms of the type advocated by Graeme Segal. 1. Introduction Throughout this paper, we fix (a bare mass) m 0 > 0, and a polynomial P : R → R which is bounded from below. If Σˆ is a closed Riemannian surface, the classical P(φ)2 -action is the local functional 1 2 2 2 ˆ A : F(Σ) → R : φ → |dφ| + m 0 φ + P(φ) d A, (1) 2 Σˆ ˆ is the appropriate domain of R-valued fields on Σˆ for A. A heuristic where F(Σ) expression for the P(φ)2 -Feynmann-Kac measure is ex p(−A(φ)) dλ(φ(x)), (2) x∈Σˆ
where dλ(φ(x)) denotes Lebesgue measure for φ(x) ∈ R. It is notoriously difficult to understand the meaning of a generic heuristic FeynmannKac expression. Such an expression may not be usefully represented by a measure at all. However, for the P(φ)2 action (1), there is a well-known interpretation of (2), as a finite measure on generalized functions, e−
Σˆ :P(φ):C0
detζ (m 20 + ∆)−1/2 dφC ,
(3)
1 where C0 = − 2π ln(m 0 d(x, y)), C = (m 20 + ∆)−1 , dφC is the Gaussian probability measure with covariance C, : P(φ) :C0 denotes a regularization of the nonlinear interaction, and detζ denotes the zeta function determinant.
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Our main purpose is to show that these Feynmann-Kac measures lead naturally to a theory satisfying a primitive form of Segal’s axioms for a quantum field theory: to a circle S 1R of radius R, there is an associated Hilbert space, to a compact Riemannian surface with geodesic boundary components there is an associated operator, and these assignments have functorial properties consistent with heuristic manipulations of path integrals. The plan of the paper is the following. In Sect. 2 we introduce some notation used throughout the paper (we largely follow the conventions in [9]). We also recall the primitive form of Segal’s axioms, roughly expressed above. In Sect. 3, and Appendix A, we discuss the P(φ)2 -Hilbert spaces. The main point is that for P(φ)2 theories, in Segal’s framework, the Hilbert space is independent of P, m 0 , and the metric on space (a union of circles). Moreover, we can focus on the real part of the Hilbert space, which simplifies matters somewhat. This real Hilbert space is defined in terms of the notion of the space of half-densities associated to a measure class (Appendix A). To define the vector that corresponds to a Riemannian surface with geodesic boundary, in Sect. 4 we consider the Feynmann-Kac measure which is associated to the double of the surface (following [9] or [15]). The fundamental result, established by constructive field theorists in the 70’s, is that (3) is indeed a well-defined finite measure. In Sect. 5, we show that the Feynman-Kac measures naturally lead to a representation of Segal’s category of compact Riemannian surfaces with geodesic boundaries. The free case (P = 0) has been considered previously, and more deeply, by Segal ([13,14]), and, from a different point of view, by Dimock ([5]). The main technical tool is the work of Burghelea, Friedlander, and Kappeler on locality properties of zeta function determinants ([3]). 2. Preliminaries Throughout this paper all function spaces are real, and all manifolds are oriented. Suppose that X is a closed Riemannian manifold. The test function space is D(X ) = C ∞ (X ; R), with the Frechet topology of uniform convergence of all derivatives. We will write f, g, h, .. for test functions. The space of distributions is D (X ), with the weak topology relative to D(X ). The Riemannian volume induces a map with dense image D(X ) → D (X ) : f → f d V.
(4)
We will write φ, ψ, .. for distributions. The pairing of a test function and distribution will be denoted by ( f, φ). The positive Laplacian on functions will be denoted by ∆ = ∆ X , and C(m, X ) will denote the operator (m 2 + ∆)−d/2 , where d = dim(X ). In this paper we will only consider d = 1, 2. We will often abbreviate C(m, X ) to C, when there is minimal risk of confusion. The Gaussian probability measure on D (X ) with Cameron-Martin Hilbert space W d/2 (X, m) = {φ : C(m, X )−1/2 φ ∈ L 2 (X, d V )}, with inner product
φ, ψW d/2 =
C −1/2 φC −1/2 ψd V, X
(5)
(6)
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405
will be denoted by dφC(m,X ) . Heuristically, dφC = dφC(m,X ) =
1 − 1 φ(m 2 +∆ X )d/2 φd V e 2 X dλ(φ), Z
(7)
where dλ(φ) denotes the heuristic Riemannian volume on fields induced by d V ; rigorously, the Fourier transform is given by 1 (8) e−i( f,φ) dφC = e− 2 ( f,C f ) . Remark 1. (a) An f ∈ D(X ) defines a linear function ( f, ·) on D (X ). One has |( f, φ)|2 dφC = | f |2W −d/2 (X,m) .
(9)
Therefore there is an isometric injection W −d/2 (X, m) → L 2 (dφC )
(10)
(and this can be extended to an isomorphism of Hilbert spaces ˆ −d/2 (X, m)) → L 2 (dφC ), S(W
(11)
ˆ denotes a Hilbert space completion of the symmetric using normal ordering, where S(·) algebra). Whereas we prefer to parameterize the Gaussian dφC using the Cameron-Martin Hilbert space W d/2 (X, m), others prefer to think in terms of a random process indexed by the dual Hilbert space W −d/2 (X, m) (see Chap. 1 of [15] for a lucid discussion). (b) Given x ∈ X , δx lies just outside of W −d/2 , and hence does not quite define an L 2 random variable. This is one point of view on the main technical difficulty of quantum field theory. Lemma 1. If ρ is a positive constant, and ρ X denotes the space obtained by dilating all distances by ρ, then dφC(m,ρ X ) = dφC(ρm,X ) . (12) Proof. Let d = dim(X ) and d V X the Riemannian volume for X . Then d Vρ X = ρ d d V X , ∆ρ X = ρ −2 ∆ X , and the Cameron-Martin norm for dφC(m,ρ X ) equals ∂ φ(m 2 + ρ −2 ∆ X )d/2 φρ d d V X = φ((ρm)2 + | |2 )d/2 φd V X , (13) ∂θ X the Cameron-Martin norm for dφC(ρm,X ) .
We will write S 1R , rather than RS 1 , to denote S 1 with the metric ds = Rdθ . Suppose that Σ is a compact Riemannian surface with boundary, S. We are assuming that S has an intrinsic orientation which, at a given point, may or may not agree with the orientation induced by Σ. We define W 1 (Σ, m) to consist of L 2 functions with locally L 2 -integrable partial derivatives such that the norm squared (dφ ∧ ∗dφ + ∗m 2 φ 2 ) = (|dφ|2 + m 2 φ 2 )d A < ∞, (14) Σ
Σ
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where ∗ = ∗Σ denotes the star operator. This is consistent with (5)-(6), when S is empty. As a topological space, W 1 (Σ, m) is independent of m. When the specific metric is not needed, we will simply write W 1 (Σ). Because S is smooth, smooth functions are dense in W 1 (Σ). The restriction map C ∞ (Σ) → C ∞ (S)
(15)
extends continuously to a map, the trace, W 1 (Σ) → W 1/2 (S).
(16)
The trace induces a short exact sequence of topological spaces, 0 → W01 (Σ, m) → W 1 (Σ, m) → W 1/2 (S) → 0.
(17)
The orthogonal complement of the kernel is W01 (Σ, m)⊥ = {φ ∈ W 1 (Σ) : (m 2 + ∆)φ = 0 in Σ \ S},
(18)
the solution space of the Helmholtz equation. The quotient Hilbert space structure on W 1/2 (S) is defined by a positive first order pseudodifferential operator DΣ on S. The expression for this operator can be derived from the isomorphism induced by the trace, W01 (Σ, m)⊥ → W 1/2 (S) : Φ → φ = Φ| S . For a smooth solution Φ of the Helmholtz equation, using Stokes’s theorem, 2 2 (dΦ ∧ ∗Σ dΦ + m ∗Σ Φ ) = d(Φ ∧ ∗Σ dΦ) = Φ ∧ ∗Σ dΦ Σ
Σ
∂Σ
(19)
(20)
(here ∂Σ denotes the boundary with induced orientation). Consequently DΣ φ = ± ∗ S (∗Σ dΦ)| S ,
(21)
where the sign is positive if the intrinsic and induced orientations agree. When S is totally geodesic, this is simply the unit outward normal derivative of Φ along S. The operator DΣ is often referred to as the Dirichlet to Neumann operator. The principal 2 is the induced metric on T ∗ S (see Subsect. 4.4 of [3]). symbol of the operator DΣ 2.1. Segal’s definition (a primitive version). As in Sect. 4 of [13], let Cmetric denote the category for which the objects are oriented closed Riemannian 1-manifolds, and the morphisms are oriented compact Riemannian 2-manifolds with totally geodesic boundaries. Definition 1. A primitive 2-dimensional unitary quantum field theory is a representation of Cmetric by separable Hilbert spaces and Hilbert-Schmidt operators such that disjoint union corresponds to tensor product, orientation reversal corresponds to adjoint, Cmetric -isomorphisms correspond to natural Hilbert space isomorphisms.
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Remark 2. (a) The naturality of the isomorphisms has to be spelled out in terms of various commuting diagrams, which we will leave to the reader’s imagination (see Sect. 4 of [13] for some additional details). (b) It is interesting to ask to what extent this definition captures the notion of locality for a qft. Segal has recently advocated additional axioms, which address the following two (apparent) shortcomings: (1) a generic surface does not have many closed geodesics, and in particular a morphism may not be divisible (i.e. expressible as a composition); and (2) a circle can be cut into intervals, and the Hilbert space should be recoverable from data associated to the intervals (see pp. 424–425 of [13]). (c) For a divisible morphism Σ : S → S, the definition implies that the corresponding operator is trace class. In this case it also follows that the trace equals the partition function of the closed surface obtained by sewing along S. To show that P(φ)2 satisfies this primitive form of Segal’s axioms, we will do the following. To S 1R we will associate a real Hilbert space, which we will ultimately denote by H(S 1 ), because this space will not depend on R, P, or m 0 . This space will carry a natural Rot (S 1 ) action. Since disjoint union of circles corresponds to a tensor product of Hilbert spaces, and a connected oriented Riemannian 1-manifold is isomorphic to S 1R , for a uniquely determined R, where the isomorphism is determined up to a rotation, this determines the Hilbert space for more general 1-manifolds. Since we will work with real Hilbert spaces, we will not have to explicitly keep track of duals. Let Σ denote an oriented compact Riemannian surface with geodesic and arclength parameterized boundary components. A component of ∂Σ is said to be outgoing if the parameterization agrees with the induced orientation, and ingoing otherwise. The union of outgoing boundary components will be denoted by (∂Σ)out , and the union of ingoing boundary components will be denoted by (∂Σ)in . To this surface we will associate a trace class operator Z(Σ) : H((∂Σ)in ) → H((∂Σ)out ). (22) Let |Σ| denote the morphism obtained from Σ by reversing the orientation of all incoming circles. Because the Hilbert spaces we consider are real, so that we can identify such a space with its dual, there are equalities Z(Σ) = Z(|Σ|) ∈ H(∂Σ) = H(∂|Σ|).
(23)
Suppose that Σ1 and Σ2 are two such surfaces, and the number of outgoing boundary components of Σ1 is the same as the number of ingoing boundary components of Σ2 . We can glue these Riemannian manifolds along (∂Σ1 )out and (∂Σ2 )in to obtain another such surface Σ2 ◦ Σ1 . We will show Z(Σ2 ◦ Σ1 ) = Z(Σ2 ) ◦ Z(Σ1 ).
(24)
3. The Hilbert Space H(S1 ) To define the Hilbert space, we will use the notion of the space of half-densities of a measure class. This is described in Appendix A. Suppose that M > 0. For all of the P(φ)2 theories, H(S 1R ) = H(C(M, S 1R )),
(25)
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where C(M, S 1R ) is the measure class on D (S 1 ) represented by the probability measure dφC(M,S 1 ) on D (S 1 ). R We also want to allow the possibility that M = 0. This is the nonfinite measure √ 2π dφC(M,R) . dφC(0,S 1 ) = lim (26) R M↓0 M A real generalized function on S 1 has a Fourier series φ = φ0 +
∞
(φn einθ + φ¯ n e−inθ ).
(27)
1
In these coordinates, if M > 0, dφC(M,S 1 ) is the infinite product of probability measures R
∞ 1 2 2 M R) dφC(M,S 1 ) = √ e− 2 M φ0 dλ(φ0 ) dµ(M , n R 2π n=1
(28)
(M 2 + n 2 )1/2 − 1 (M 2 +n 2 )1/2 |φn |2 e 2 dλ(φn ). 2π
(29)
where dµ(M) = n If M = 0, then
dφC(0,S 1 ) = dλ(φ0 )
∞
dµ(0) n .
(30)
n=1
Note there is no dependence on R when M = 0. Lemma 2. The measure class C(M, S 1R ) is independent of M ≥ 0 and R. Proof. In addressing this question, we can ignore the φ0 factor. Kakutani’s theorem (Theorem 2.12.7, p. 92, of [1]), asserts that the two infinite product measures R) dµn(mr ) and dµ(M (31) n are either equivalent or disjoint, and they are equivalent if and only if the inner product between the corresponding positive half-densities is positive, i.e. ∞ (mr ) (M R) dµn dµn > 0. (32) n=1
In doing this calculation, we can clearly assume r = R = 1. The n th factor of (32) equals 1 (M 2 + n 2 )1/4 (m 2 + n 2 )1/4 2 2 1/2 2 2 1/2 2 e− 4 ((M +n ) +(m +n ) )|xn | dλ(xn ) 2π C 1/2 1/2 m2 2 M2 1+ 2 =n 1+ 2 2 2 1/2 n n (M + n ) + (m 2 + n 2 )1/2 1/2 1/2 m2 2 M2 1+ 2 = 1+ 2 1/2 1/2 2 n n m2 1+ M + 1 + n2 n2 This has a positive infinite product over n.
(33) (34) (35)
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We will need a more sophisticated result along these same lines. Suppose that D is a positive classical pseudodifferential operator of order 1 on S, a compact connected one-manifold (e.g. D = (M 2 + ∆ S 1 )1/2 ). The principal symbol of the operator D 2 determines a Riemannian metric on S, hence a radius R. By choosing an arclength coordinate Rθ , we can suppose S = S 1 and the metric is Rdθ . Proposition 1. Let D1 and D2 denote two operators as above such that D1 and D2 have the same principal symbols. Let Rdθ denote the corresponding metric. Then the Gaussian measures µi with Cameron-Martin inner products (36) φ, ψi = φ Di ψ Rdθ S
are equivalent. Proof. Obviously
φ, ψ2 = D1−1 D2 φ, ψ1 .
(37)
Because D1 and D2 are classical pseudodifferential operators, and they have the same principal symbols, D1−1 D2 = 1 + A, (38) where A is a pseudodifferential operator of order −1. Because S is one dimensional, A is Hilbert-Schmidt. This implies that the µi are equivalent (see Theorem 6.3.2, p. 286, of [1], or Theorem I.23, p. 41, of [15]). Since the Hilbert space corresponding to a circle is independent of R, M, and P, we will denote it simply by H(S 1 ). More generally, given a closed 1-manifold S, there is a measure class associated to W 1/2 (S), and we will denote the associated real Hilbert space of half-densities by H(S). This space is intrinsic to S, and it is naturally isomorphic to the tensor product of the H(Si ), where the Si (ordered in some way) denote the connected components of S; see (5) of Appendix A. 4. Feynmann-Kac Measures To define the trace class operators corresponding to surfaces, we will need a number of technical results about Feynmann-Kac measures for closed Riemannian surfaces. Suppose that Σˆ is a closed oriented Riemannian surface. Let { f k } denote an orthonormal basis of real eigenfunctions for the positive Laplace operator, ∆, where ∆f k = λk f k , 0 = λ0 < λ1 ≤ λ2 ≤ ... . A generalized function on Σˆ has an expansion φ= φk f k = φ0 + ψ. (39) In the coordinates φk ∈ R, dφC(M,Σ) ˆ is the infinite product measure dφC(M,Σ) ˆ =
where dµ(M) = n
dµ(M) n ,
M 2 + λk − (M 2 +λk ) φ 2 k dφ . 2 e k 2π
(40)
(41)
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D. Pickrell
We also define dφC(0,Σ) ˆ = dλ(φ0 ) × dψC(0,Σ) ˆ = dλ(φ0 )
∞ k=1
λk − λk φ 2 e 2 k dφk . 2π
(42)
ˆ Remark 3. (a) The measure dψC(0,Σ) ˆ is a Gaussian measure on D (Σ)0 , where ψ ∈ D0 means ψ(1) = 0 (D0 is the dual of D/R), and the Cameron-Martin inner product is dψ1 ∧ ∗dψ2 . (43) Σˆ
(b) The space D0 depends on the C ∞ structure of Σˆ (diffeomorphisms act naturally on D/R, and hence its dual). The Cameron-Martin inner product depends on the conformal structure of Σˆ (because it involves the star operator on one-forms). The decomposition of distributions ˆ = R ⊕ D (Σ) ˆ 0 : φ = φ0 + ψ, D (Σ) (44) as in (39), depends on the volume element of Σˆ (so that φ0 can be interpreted as a distribution). Consequently the measure dφC(0,Σ) ˆ depends on the Riemannian structure ˆ of Σ; the measure dψ ˆ is conformally invariant. C(0,Σ)
ˆ denote the measure class of dφ Let C(M, Σ) ˆ . C(M,Σ) ˆ is independent of M ≥ 0 Lemma 3. For constant ρ > 0, the measure class C(M, ρ Σ) and ρ. Proof. The independence of M is essentially the same as for Lemma 2. The point is that λk is asymptotically k. We again apply Kakutani’s criterion for equivalence, as in (32). We can also ignore the zero mode, φ0 . (M) (m) The n th factor, dµn dµn equals 1 (M 2 + λn )1/4 (m 2 + λn )1/4 2 2 2 e− 4 (M +λn +m +λn )|φn | dλ(φn ) (2π )1/2 R 1/4 1/4 1/2 1/2 M2 m2 4π λn 1+ 1+ = 2π λn λn 2λn + M 2 + m 2 −1/2 1/4 1/4 m2 M 2 + m2 M2 1+ 1+ = 1+ λn λn 2λn 1 . =1+O λ2n
(45) (46) (47) (48)
Thus the inner product is positive, and the measures are equivalent. This proves the independence of M. The independence of ρ now follows from Lemma 1. Remark 4. If Σˆ is replaced by a manifold of dimension d, and we consider an action defined by a second order operator, then independence of mass holds if and only if d < 4, because λn is asymptotic to n 2/d .
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In the formulation of the following lemma, we will use a basic fact, due to Colella and Lanford, about the free field dφC(M,Σ) ˆ . This will be used frequently in the remainder of the paper. A typical configuration φ for the free field is not an ordinary function (or even a signed measure). However, given a nice foliation of Σˆ by 1-submanifolds, a typical configuration can be thought of as a continuous function (of a transverse parameter) with values in distributions along the leaves. A precise formulation of this, in the case of R2 , can be found in [4] (Theorem 1.1, part (b), p. 45, and see the paragraph following the theorem, for further comment). Lemma 4. Suppose that c : S 1R → Σˆ is an isometric embedding. Then the projection 1 1 of dφC(M,Σ) ˆ to a measure on D (S ) belongs to the measure class C(M, S R ). Proof. Since dφC(M,Σ) ˆ is a Gaussian measure, its projection must be a Gaussian measure. One way to calculate the image of a Gaussian is to consider the map of CameronMartin spaces, which in this case is the trace map ˆ M) → W 1/2 (S 1 ), W 1 (Σ,
(49)
where the inner product on the target is determined by a positive first order pseudodifferential operator D, obtained by considering the W 1 inner product on Helmholtz solutions on Σˆ \ c(S 1 ), as in (17)-(21). To relate this directly to (17)–(21), cut the closed surface Σˆ along c to obtain a compact surface Σ with two boundary components, one of which is positively parameterized by c, and one of which is negatively parameterized by c. This induces a pseudodifferential operator DΣ on ∂Σ, as in (17)-(21). This yields two pseudodifferential operators D± on S 1 , corresponding to the positive and negative c- parameterizations. The operator D = D+ + D− , by (21). Thus D 2 has principal symbol which is proportional to the induced metric on T ∗ S 1 , and the lemma follows from Proposition 1 (recall also that the measure class C(M, S 1R ) is independent of M and R, by Lemma 2). ˆ the covariance for the Alternatively, if C(x, y) denotes the kernel for C(M, Σ), projection is given by C (θ, θ ) = C(c(θ ), c(θ )). (50) One can read off the principal symbols for C and its inverse D from the fact that C is 1 asymptotically − 2π ln(Md(x, y)), as d(x, y) → 0. 4.1. Normal Ordering. From now on we will use our fixed bare mass m 0 > 0. Let ˆ If f ∈ D(Σ), ˆ then ( f, ·) ∈ L 2 (dφC ). By definition C = C(m 0 , Σ). ( f,C f )
: ( f, ·)n :C = Hn
(( f, ·)) ∈ L 2 (dφC ),
(51) 1
where Hnα denotes the n th Hermite polynomial for the Gaussian (2π α)−1/2 e− 2α x dλ(x) (there are a number of different ways to motivate this definition; see either Sect. 6.3 of [9] or Chap. 1 of [15]). For example : ( f, ·)4 :C = ( f, ·)4 − 6( f, C f )( f, ·)2 + 3( f, C f )2 .
2
(52)
ˆ because of (10). UnfortuOne can define : ( f, ·)n :C equally well for f ∈ W −1 (Σ), ˆ δx is not in W −1 , and in fact it is impossible to define (δx , ·) nately, given a point x ∈ Σ,
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D. Pickrell
as a random variable with respect to dφC(m 0 ,Σ) ˆ (the support of this measure consists of genuine distributions). However, for n ≥ 0, it is possible to define a regularization ˆ : (δx , ·)n :C , as a distribution; that is, given ρ ∈ D(Σ), : (δx , ·)n :C ρ(x)d A(x) (53) Σˆ
is a well-defined integrable random variable with respect to dφC . For example (see Sect. 8.5, p. 152, of [9]), : (δx , ·)4 :C = lim[(δt,x , ·)4 − 6(δt,x , Cδt,x )(δt,x , ·)2 + 3(δt,x , Cδt,x )2 ], t↓0
(54)
ˆ satisfies δt,x → δx as t ↓ 0. We will always choose the functions δt,x where δt,x ∈ D(Σ) to have compact support which shrinks to x, and for these functions to depend smoothly on x. Now suppose that we think of C as a kernel function (which we can do because we have a Riemannian background, and in particular an area form). A fundamental fact is that, near the diagonal, ˆ C(m 0 , Σ)(x, y) = C0 (m 0 , x, y) + C f (m 0 , x, y),
(55)
1 where C0 (m 0 , x, y) = − 2π ln(m 0 d(x, y)) and C f is smooth. We will often suppress the argument m 0 . ˆ we define For ρ ∈ D(Σ), (δ ,C δ ) : (δx , ·)n :C0 ρ(x)d A(x) = lim Hn t,x 0 t,x ((δt,x , ·))ρ(x)d A(x). (56) t↓0
For example : (δx , ·)4 :C0 = lim[(δt,x , ·)4 − 6(δt,x , C0 δt,x )(δt,x , ·)2 + 3(δt,x , C0 δt,x )2 ]. t↓0
(57)
Remark 5. This is local: the calculation of (δt,x , C0 δt,x ) depends on arbitrarily small neighborhoods of x as t ↓ 0. In a first version of this paper, I claimed that one could just as well use C. But in general this is false, because for fixed x, there is a constant in the asymptotic expansion of C (the value C f (x, x)), which is not zero, and which is not locally determined. One can also express (56) in terms of regularization by C: by a standard formula for ‘finite change of Wick order’ (see (8.6.1) of [9]), (56) equals [n/2] j=0
n! (n − 2 j)! j!2 j
C f (x, x) j : (δx , ·)n−2 j :C ρ(x)d A(x).
(58)
For example : (δx , ·)4 :C0 =: (δx , ·)4 :C +6C f (x, x) : (δx , ·)2 :C +3C f (x, x)2 .
(59)
The important point is that these regularizations agree up to lower order terms. In general we define : P((δx , ·)) :C0 by linear extension. We will occasionally abbreviate this simply to : P :C0 , or, if we need to display the argument, to : P(φ) :C0 [rather than the more cumbersome : P((δx , ·)) :C0 (φ)]. The following is one of the fundamental results of constructive quantum field theory.
P(φ)2 Quantum Field Theories and Segal’s Axioms
413
Theorem 1. Suppose that P(φ) is bounded below. Then ex p(− (dφC(m 0 ,Σ) ˆ ).
Σˆ
: P :C0 ) ∈ L 1
This follows, with relatively minor modifications, from the arguments in Sect. 8.6 of [9], or V.2 of [15] (Note that a closed Riemannian surface is conformally equivalent to a constant curvature surface, and hence by uniformization can be presented as a nice bounded region with generalized periodic boundary conditions, and conformally Euclidean metric - with the exception of the sphere.) ˆ Definition 2. The Feynman-Kac measure for Σˆ is the finite measure on D (Σ), e−
Σˆ :P:C0 d A(x)
detζ (∆ + m 20 )−1/2 dφC .
(60)
At a heuristic level, we can say that the ζ -determinant is essential because we have (for no good reason) normalized the free background dφC to have unit mass; we have to add back in the Gaussian volume of the Cameron-Martin space. 5. Surfaces, Operators, and Sewing Suppose that Σ is a compact oriented Riemannian surface, with geodesic and geodesically parameterized boundary components. We also initially assume that all of the boundary components are outgoing, i.e Σ = |Σ|. We consider the closed Riemannian surface (61) Σˆ = Σ ∗ ◦ Σ, where Σ ∗ is the surface obtained by reversing the orientation of everything. Of fundamental importance is the existence of a reflection symmetry through ∂Σ. ˆ We will write Let S denote ∂Σ, and C = C(m 0 , Σ). S∗ (e−
Σˆ :P:C0 d A
dφC )
(62)
for the projection of this measure to a finite measure on D (S), which exists by Lemma 4. Definition 3. For Σ as above, we define Z1 (Σ) = Z1 (|Σ|) = (S∗ (e− and
Σˆ :P:C0 d A
dφC ))1/2 ∈ H(S),
Z(Σ) = detζ (∆Σˆ + m 20 )−1/4 Z1 (Σ) ∈ H(S).
(63) (64)
ˆ we define Z(Σ) ˆ to be the integral of its Feynman-Kac measure. For a closed surface Σ, Note that for a morphism Σ : S1 → S2 , it follows immediately from this definition that Z(Σ) represents a Hilbert-Schmidt operator. Theorem 2. Suppose that Σ1 and Σ2 are two morphisms which can be composed. Then Z(Σ3 ) = Z(Σ2 ) ◦ Z(Σ1 ),
(65)
where Σ3 = Σ2 ◦ Σ1 . (b) Suppose Σ : S → S is divisible. Then Z(Σ) is trace class, and ˆ trace(Z(Σ)) = Z(Σ), where Σˆ is the closed surface obtained by gluing Σ to itself along S.
(66)
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D. Pickrell
The rest of this section is devoted to the proof of this theorem. For (a) there are three possibilities: both of (∂Σ1 )in and (∂Σ2 )out are empty, one is empty, and neither is empty. The line of argument for each of these cases is exactly the same, but the notational details vary. We will carry out all the details for the second possibility. There are basically four parts to the argument. In the first part, we study the disintegration of the free Feynmann-Kac measure with respect to its projection to a measure on generalized functions on the boundary. The second part involves the local character of the nonlinear interaction. The third and fourth parts are tightly intertwined: these parts concern the sewing properties for the normalized background Gaussian measures, and the ζ -regularized Gaussian volumes, respectively.
5.1. Part 1. Decomposition of free backgrounds relative to traces. As above, we initially suppose that Σ = |Σ|, with outgoing boundary S. The trace map ˆ → W 1/2 (S) : φˆ → φˆ S W 1 (Σ)
(67)
corresponds to a Hilbert space decomposition ˆ m 0 ) = W01 (Σ, ˆ m 0 ) ⊕ W01 (Σ, ˆ m 0 )⊥ . W 1 (Σ,
(68)
ˆ m 0 ) = W01 (Σ, m 0 ) ⊕ W01 (Σ ∗ , m 0 ) W01 (Σ,
(69)
ˆ m 0 )⊥ = W 1 (Σ, ˆ m 0 ) ∩ ker (∆ + m 20 )| ˆ . W01 (Σ, Σ\S
(70)
In turn, and The latter space has two other realizations. On the one hand it is essentially isomorphic to W 1 (Σ, m 0 ) ∩ ker (∆ + m 20 )|Σ\S , (71) because a W 1 -solution φˆ s of the Helmholtz equation on Σˆ \ S is necessarily even, i.e. invariant with respect to the mirror symmetry of Σˆ through S (the even and odd parts of a Helmholtz solution would also be solutions; the odd part vanishes on S, hence it must be identically zero); hence φˆ s is determined by its restriction to Σ, which we denote by φs . On the other hand it is also isomorphic to W 1/2 (S), with the inner product determined by 2DΣ , as in (21). We now want to apply these Hilbert space decompositions to obtain decompositions of the corresponding Gaussian measures, in particular our background Gaussian measures. ˆ m 0 ), In the following we will have to distinguish, for example, between φˆ ∈ W 1 (Σ, ˆ ˆ and a typical φ in the support of d φC ; we will refer to the latter as a random field (rather than introducing some additional notation). We will also implicitly invoke the theorem of Collella-Lansford, which, for example, allows us to make sense of the restriction of a random φˆ to Σ or S. The Gaussian measure d φˆ C(m 0 ,Σ) ˆ has a disintegration relative to its projection to fields on S: ˆ ˆ d φˆ C(m 0 ,Σ) = [d φˆ C(m 0 ,Σ) (72) ˆ ˆ |φ S = φ1 ]d(S∗ (d φC(m 0 ,Σ) ˆ ))(φ1 ).
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The existence of this disintegration is a general fact (Prop. 13, Sect. 2, No. 7, of [2]). But as we will explain in the following paragraphs, the ‘normalized conditional measure’ [d φˆ C |φˆ S = φ1 ] is a Gaussian probability measure centered at (a classical solution corresponding to) φ1 . The Hilbert space decompositions (68) and (69), and the isomorphism (70), imply that a sample field for the Gaussian d φˆ C(m 0 ,Σ) ˆ can be uniquely decomposed as a sum of independent terms: φˆ = φˆ 0 + φˆ s = φ0 + φˆ s + φ0∗ , (73) where φ0 (φ0∗ , respectively) is a generalized function which is supported on Σ (Σ ∗ , respectively) and vanishing on S, and φˆ s is a solution of the Helmholtz equation in Σˆ \ S, and determined by its (distributional) boundary value φ1 on S. We will write φ = φ0 + φs (φ ∗ = φ0∗ + φs∗ , respectively) for the restriction of a random φˆ to Σ (Σ ∗ , respectively). In particular for a.e. φ1 , the φ1 (normalized) conditioned measure in (72) is a direct product ˆ [d φˆ C(m 0 ,Σ) ˆ |φ| S = φ1 ] =
(74)
∗ ∗ [dφC(m 0 ,Σ) |φ S = φ1 ] × [dφC(m ∗ |φ S = φ1 ]. 0 ,Σ )
(75)
Remark 6. (a) A random φ for [dφC(m 0 ,Σ) |φ S = φ1 ] is of the form φ = φ0 + φs , φ0 is a normalized Gaussian with Cameron-Martin space W01 (m 0 , Σ), and (φs ) S = φ1 . Thus we could also write [dφC(m 0 ,Σ) |φ S = φ1 ] = dφC(m 0 ,Σ,D) (φ − φs ),
(76)
where C(m 0 , Σ, D) is the inverse of m 20 + ∆Σ with Dirichlet boundary condition. The right hand side is defined for all solutions φs of the Helmholtz equation in Σ \ S. If one considers a collar {0 ≤ t ≤ δ} × S 1 ⊂ Σ for a boundary component ({t = 0}), then the Collela-Lanford theorem says that for any > 0, with probability one, φ0 is a continuous function of t with values in W − (S 1 ), which vanishes when t = 0. (b) To this point we have not given an independent meaning to C(m 0 , Σ) or dφC(m 0 ,Σ) . The measure dφC(m 0 ,Σ) can be understood as the Gaussian with Cameron-Martin space W 1 (Σ, m 0 ) (see (14)); a random φ is a restriction of a random φˆ to Σ. However ‘C(m 0 , Σ) = (m 20 + ∆)−1 ’ does not have an independent meaning, in reference to Σ alone (because we are interested in a free boundary condition, which is why we introduce the double of Σ). In terms of this notation, and using reflection symmetry through S, we obtain the following Lemma 5. The pushforward measure Z1 (Σ)2 (see Definition 3), is given by ˆ ˆ e− Σˆ :P(φ):C0 [d φˆ C(m 0 ,Σ) | φ| = φ ] d(S∗ d φˆ C(m 0 ,Σ) S 1 ˆ ˆ )(φ1 ) φˆ 0
=
e
−
Σ :P(φ):C0
2 [dφC(m 0 ,Σ) |φ| S = φ1 ]
d(S∗ d φˆ C(m 0 ,Σ) ˆ )(φ1 ).
(77) (78)
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D. Pickrell
We now turn to the setup of the theorem. Suppose that we are given Σ1 and Σ2 . We first suppose that Σ1 has empty incoming boundary, and Σ2 has nonempty outgoing boundary. Thus Σ3 = Σ2 ◦Σ1 also has empty incoming boundary. Let S1 denote the outgoing boundary of Σ1 (which is the same as the incoming boundary for Σ2 ), and let S2 denote the outgoing boundary for Σ2 . We will write φ for a field on Σ1 . This field has a decomposition φ = φ0 + φs , where φ0 is Gaussian and φs is a solution of the Helmholtz equation and determined by the boundary value φ S1 . We will similarly write ψ for a field on Σ2 , with decomposition ψ = ψ0 + ψs . We will also write φi for a field on Si , and Ci will denote the covariance (m 20 + ∆)−1 associated to |Σˆ i |. For Σ3 = Σ2 ◦ Σ1 , there is a finer decomposition, corresponding to the trace map W 1 (Σ3 ) → W 1/2 (S1 ) ⊕ W 1/2 (S2 )
(79)
W01 (Σ3 , m 0 ) = W01 (Σ1 , m 0 ) ⊕ W01 (Σ2 , m 0 ).
(80)
and the isomorphism
A random field Φ on Σ3 with distribution dΦC3 can be written as a sum of independent Gaussians (81) Φ = φ0 + ψ0 + Φ s , Φ s = φs ψs , where φs and ψs (are random Helmholtz solutions, as before, and) have common boundary value φ1 on S1 , ψs has boundary value φ2 on S2 , and the dΦC3 -distribution for Φ s , in the coordinates (φ1 , φ2 ), is a Gaussian measure with covariance (m 20 + ∆Σˆ 3 )−1 restricted to S1 ∪ S2 . We will write the dΦC3 distribution for Φ s as dΦCs 3 . 5.2. Part 2. Locality of nonlinear interactions. We now want to calculate Z1 (Σ2 )Z1 (Σ1 ), φ1 ∈D (S1 )
(82)
where the integral is over the common boundary value φ1 = φ S1 = ψ S1 . By Lemma 5 this − :P(φ):C0 = [dφC1 |φ S1 = φ1 ]) (S1∗ (d φˆ C1 ))1/2 (φ1 ) (83) e Σ1 φ1 − Σ :P(ψ):C0 2 [dψC2 |ψ Si = φi ) ((S1 S2 )∗ (d ψˆ C2 ))1/2 (φ1 , φ2 ). (84) e Proposition 2. For a random field Φ as in (81), : P(φ) :C0 + : P(ψ) :C0 = : P(Φ) :C0 , a.e. [dΦC3 ]. Σ1
Σ2
Σ2 ◦Σ1
(85)
1 log Proof. We first remark that we have not indicated the dependence of C0 = − 2π (m 0 d(x, y)) on the underlying surface, because when there is an ambiguity, the metrics are the same. The proposition follows from the definition (56) for C0 -regularization, and Remark 5.
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417
Corollary 1. =
φ1
φ1
Z1 (Σ2 )(φ2 , φ1 )Z1 (Σ1 )(φ1 )
FP (φ2 , φ1 )((S1 S2 )∗ (d ψˆ C2 ))1/2 (φ1 , φ2 )(S1∗ (d φˆ C1 ))1/2 (φ1 ),
where
FP (φ2 , φ1 ) =
e
−
Σ3 :P(Φ):C0
[dΦC3 |Φ S1 = φ1 , Φ S2 = φ2 ].
Proof. Proposition 2, applied to (83), implies that (86) equals − :P(Φ):C0 [dφC1 |φ S1 = φ1 ] × [dψC2 |ψ S1 = φ1 , ψ S2 = φ2 ] , = e Σ3 φ1
((S1 S2 )∗ (d ψˆ C2 ))1/2 (φ2 , φ1 )(S1∗ (d φˆ C1 ))1/2 (φ1 ).
(86) (87)
(88)
(89) (90)
By (81) dΦC3 is obtained by (normalized) conditioning dφC1 × dψC2 so that (φs ) S1 = (ψs ) S1 . Thus [dφC1 |φ S1 = φ1 ] × [dψC2 |ψ S1 = φ1 , ψ S2 = φ2 ] = [dΦC3 |Φ S1 = φ1 , Φ S2 = φ2 ].
(91) (92)
Inserting this into (90) yields the proposition. 5.3. Parts 3 and 4. Sewing of normalized background measures and ζ -regularized volumes. Now we need to compare the expression in Corollary 1 with Z1 (Σ3 ). By Lemma 5 − :P(Φ):C0 Z1 (Σ3 ) = e Σ3 [dΦC3 |Φ S2 = φ2 ]((S2 )∗ (d Φˆ C3 ))1/2 (φ2 ) (93) = FP (φ2 , φ1 )([dΦCs 3 : Φ Ss 2 = φ2 ])((S2 )∗ (d Φˆ C3 ))1/2 (φ2 ), (94) where FP is as in Corollary 1. To complete the proof of the theorem, in comparing (86) and (94), it is clear that we need to compare the measures (with values in half-densities) in the two integrals. These measures do not depend upon P (all the P-dependence is in FP ). Proposition 3. Suppose that P = 0. For a.e. φ2 , the following equality of measures on fields φ1 holds: Z(Σ2 )(φ2 , φ1 )Z(Σ1 )(φ1 ) = [dΦCs 3 |Φ Ss 2 = φ2 ](φ1 )Z(Σ3 )(φ2 ).
(95) (96)
Remark 7. (a) The measures involved in this statement are Gaussian, hence eminently computable. The nontrivial content of the statement involves understanding the way in which ζ -determinants mesh with the determinants which arise in calculating compositions of half-densities.
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D. Pickrell
(b) Since [dΦCs 3 |Φ Ss 2 = φ2 ] is a probability measure, the free version of the Theorem follows from this proposition by integrating 1 on both sides: Z(Σ2 )(φ2 , φ1 )Z(Σ1 )(φ1 ) = Z(Σ3 )(φ2 ). (97) φ1
At the projective level, this equality has an important interpretation in terms of the composition of Lagrangian subspaces (see p. 147 of [10] or [16] for the general definitions). Given a 1-manifold S, let Q(S) denote ‘position space’ W 1/2 (S). Then as Lagrangian subspaces, the composition of φ1 1 ⊥ ⊂ T ∗ Q(S1 ) = Q(S1 ) ⊕ Q(S1 )∗ (98) W0 (Σ1 , m) = DΣ1 φ1 (the Cameron-Martin space of Z1 (Σ1 )2 , and Helmholtz solution space on Σ1 ) with φ1 φ2 (99) , W01 (Σ2 , m)⊥ = Aφ2 + Bφ1 −(B t φ2 + Dφ1 ) ⊂ T ∗ Q(S2 ) × T ∗ Q(S1 )
(100)
(the Cameron-Martin space of Z1 (Σ2 )2 , and Helmholtz solution space on Σ2 , where DΣ2 has been written as a 2 × 2 matrix, as in (108) below, and the minus sign has been inserted because the intrinsic orientation of S1 is opposite the Σ2 -induced orientation (see (21)), is φ2 1 ⊥ ⊂ T ∗ Q(S2 ) W0 (Σ2 ◦ Σ1 , m) = (101) DΣ3 φ2 (the Cameron-Martin space of Z1 (Σ3 )2 , and Helmholtz solution space on Σ3 ). Proof. In the course of the proof, we will apply Theorem B of [3] a number of times. In applying this theorem, when we consider the Laplacian ∆Σi , it will be understood that we are imposing a Dirichlet boundary condition. Reflecting the decomposition (73), Theorem B of [3] implies that detζ (m 20 + ∆Σˆ i ) = detζ (m 20 + ∆Σi )detζ (2DΣi )detζ (m 20 + ∆Σi∗ ) =
detζ (m 20
+ ∆Σi ) detζ (2DΣi ). 2
(102) (103)
Reflecting the decomposition (81), a slightly extended version of Theorem B implies that detζ (m 20 + ∆Σ3 ) = detζ (m 20 + ∆Σ1 )detζ (m 20 + ∆Σ2 )detζ (DΣ1 ,Σ2 ),
(104)
where DΣ1 ,Σ2 is the pseudodifferential operator on S1 which has an inverse with kernel (m 20 + ∆Σ3 )−1 . The statement of the proposition involves half-densities, in the variable φ2 . To prove the proposition, it suffices to show that Z(Σ1∗ )(φ1∗ )Z(Σ2∗ )(φ1∗ , φ2 )Z(Σ2 )(φ2 , φ1 )Z(Σ1 )(φ1 ) = [dΦCs 3 |Φ Ss 2
= φ2 ](φ1 )Z(Σ3 )
2
(φ2 )[dΦC∗s3 |Φ S∗s∗ 2
=
φ2 ](φ1∗ ),
(105) (106)
P(φ)2 Quantum Field Theories and Segal’s Axioms
419
as measured on random fields φ1 , φ2 , and φ1∗ . To clarify the notation involved in the statement, there is an underlying factorization Σˆ 3 = Σ1∗ ◦ Σ2∗ ◦ Σ2 ◦ Σ1 ,
(107)
and φ1 is a random field on S1 , the outgoing boundary of Σ1 , φ2 is a random field on S2 , the outgoing boundary of Σ2 , and φ1∗ is a random field on S1∗ , the outgoing boundary of Σ2∗ . To prove (106), we will compute the Fourier transforms of both sides. Our strategy of proof will involve first doing some intermediate calculations heuristically (which should serve the dual purpose of illuminating the meaning of the statements), and then justifying the answers (by noting that the calculations are valid in finite dimensions, and taking limits). We will write DΣ2 as a 2 × 2 matrix, DΣ2 =
A B Bt D
(108)
relative to the coordinates (φ2 , φ1 ). Thus for example D has the following meaning: given φ1 , calculate the Helmholtz solution on int (Σ2 ) which has boundary value φ1 on S1 and vanishing boundary value on S2 ; then Dφ1 is the inward (from the perspective of Σ2 ) normal derivative along S1 . We will use the two identities: DΣ1 + D = DΣ1 ,Σ2 , A − B(DΣ1 +
D)−1 B t
(109)
= DΣ3 .
(110)
The first is straightforward. The second is a coordinate expression of (b) of Remark 7, because (110) is equivalent to DΣ3 φ2 = Aφ2 + Bφ1 , −(B t φ2 + Dφ1 ) = DΣ1 φ1 .
(111)
We will similarly write DΣ2∗ , in terms of A∗ , B ∗ , and D ∗ . In the calculations which follow, we will, in intermediate heuristic steps, use matrix notation for various pairings. For example the probability measure Z1 (Σ1 )2 will be represented by the heuristic expression 1 t
det (2DΣ1 )1/2 e− 2 φ1 (2DΣ1 )φ1 dφ1 .
(112)
We will also use the identity (valid in finite dimensions) det (2DΣ2 ) = det (2D)det (2(A − B D −1 B t )),
(113)
which follows from the factorization 2DΣ2 =
1 B D −1 0 1
2(A − B D −1 B t ) 0 0 2D
1
D −1 B t
0 . 1
(114)
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We first calculate the Gaussian integral e−i( f1 ,φ1 ) Z1 (Σ2 )(φ2 , φ1 )Z1 (Σ1 )(φ1 ) = φ1
φ1
e
−i( f 1 ,φ1 )
det (2DΣ2 )
1/4
e
− 12 (φ2t ,φ1t )
A B Bt D
φ2 φ1
(115)
(dφ1 dφ2 )1/2 ,
(116)
1 t
det (2DΣ1 )1/4 e− 2 φ1 DΣ1 φ1 (dφ1 )1/2
(117)
= det (2DΣ2 ) det (2DΣ1 ) , t 1 −1 ex p − ( DΣ1 + Dφ1 + DΣ1 + D (B t φ2 + i f 1 ) 2 φ 1 −1 DΣ1 + Dφ1 + DΣ1 + D (B t φ2 + i f 1 ) dφ1 × 1 t 1 t t −1 t ex p (B φ2 + i f 1 ) (DΣ1 + D) (B φ2 + i f 1 ) ex p − φ2 Aφ2 (dφ2 )1/2 2 2 1/4
1/4
= det (2D2DΣ1 (DΣ1 + D)−2 )1/4 det (2(A − B D −1 B t ))1/4
(118) (119) (120) (121) (122)
e
− 21 f 1t (DΣ1 +D)−1 f 1 iφ2t B(DΣ1 +D)−1 f 1
(123)
e
− 21 φ2t {A−B(DΣ1 +D)−1 B t }φ2
(124)
e
(dφ2 )
1/2
(we also used (113) in the last step). We now calculate, in terms of the identities (109)-(110) (and using reflection symmetry), that ∗ ∗ (125) e−i(( f1 ,φ1 )+( f 2 ,φ2 )+( f 1 ,φ1 )) Z1 (Σ1∗ )(φ1∗ )Z1 (Σ2∗ )(φ1∗ , φ2 )Z1 (Σ2 )(φ2 , φ1 )Z1 (Σ1 )(φ1 ) −1
−1 1/2
(126) −1
= det (4D(DΣ1 + D) DΣ1 (DΣ1 + D) ) det (2(A − B D B )) −1 −1 −1 t ∗ ∗ ∗ −1 − 1 ( f 1t DΣ f 1 + f 1∗t DΣ ∗ ∗ f 1 ) iφ2 (B DΣ1 ,Σ2 f 1 +B DΣ ∗ ,Σ ∗ f 1 − f 2 ) 1 ,Σ2 1 ,Σ2 1 2 e e 2 t
1/2
(127) (128)
1 t
e− 2 φ2 2DΣ3 φ2 dφ2 = det (4(1 +
−1 DΣ D)−1 (1 + 1
D
−1
DΣ1 )
(129) −1 1/2
)
det (2(A − B D
−1
B )(2DΣ3 ) t
−1 1/2
)
(130) e e
− 12 ( f 1 ,(m 20 +∆Σ3 )−1 f 1 )+( f 1∗ ,(m 20 +∆Σ ∗ )−1 f 1∗ ) 3
(131)
− 21 (B(m 20 +∆Σ3 )−1 f 1 +B ∗ (m 20 +∆Σ ∗ )−1 f 1∗ − f 2 ),(m 20 +∆Σˆ )−1 (B(m 20 +∆Σ3 )−1 f 1 +B ∗ (m 20 +∆Σ ∗ )−1 f 1∗ − f 2 ) 3
3
3
. (132)
The expression we have obtained for the Fourier transform is correct for the following reasons. Our intermediate calculations are valid provided that all the objects involved are understood to be finite dimensional. In particular we can consider compatible compressions of the positive operators DΣ1 , DΣ2 , DΣ2∗ , and DΣ1∗ . For example we can con p 0 p 0 p 0 p 0 DΣ2∗ , , and DΣ2 sider the positive operators p DΣ1 p, 0 p 0 p 0 p 0 p
P(φ)2 Quantum Field Theories and Segal’s Axioms
421
p DΣ1∗ p, where p is the projection corresponding to a bounded portion of the spectrum of DΣ1 (where DΣ2 is written as in (108)). As the cutoff p is removed, the Gaussian measure corresponding to p DΣ1 p will converge weakly to Z1 (Σ1 )2 (the Gaussian corresponding to DΣ1 ), and so on. We also observe that 1 −1 D) (2 + D −1 DΣ1 + DΣ 1 4
(133)
and (A − B D −1 B t )−1 (A − B(DΣ1 + D)−1 B t ) = (1 −
A−1 B D −1 B t )−1 (1 −
A−1 B(1 +
D −1 D
Σ1
(134)
)−1 D −1 B t )
(135)
are of the form 1 + T , where T is trace class. This is true of (133), because D −1 DΣ1 = 1 + H , where H is Hilbert-Schmidt, hence D −1 DΣ1 + (D −1 DΣ1 )−1 = 2 + T, T = (1 + H )−1 H 2 ,
(136)
and T is trace class (the fact is that D − DΣ1 is a smoothing operator, so that H itself is trace class; this follows from use of (55)). This is true for (135), because A−1 B and D −1 B t are smoothing operators. These considerations imply that the determinants in the last line of (132) are well-defined. Furthermore, if we insert the cutoff p, the corresponding determinants will converge, as p → 1. This implies that we can take a limit of finite dimensional approximations to justify our formula for the Fourier transform (126). We now claim that the Fourier transform of the left-hand side of (106) = det (m 20 + ∆Σˆ 3 )−1/2 e e
− 21 ( f 1 ,(m 20 +∆Σ3 )−1 f 1 )+( f 1∗ ,(m 20 +∆Σ ∗ )−1 f 1∗ )
(137)
3
− 12 (B(m 20 +∆Σ3 )−1 f 1 +B ∗ (m 20 +∆Σ ∗ )−1 f 1∗ − f 2 ),(m 20 +∆Σˆ )−1 (B(m 20 +∆Σ3 )−1 f 1 +B ∗ (m 20 +∆Σ ∗ )−1 f 1∗ − f 2 ) 3
3
3
. (138)
To justify this claim, we need to show det (2DΣ1 2D(DΣ1 + D)−2 )1/2 det (2(A − B D −1 B t )(2DΣ3 )−1 )1/2 detζ (m 20
+ ∆Σˆ 2
)−1/2 det
2 ζ (m 0
+ ∆Σˆ 1
)−1/2
=
detζ (m 20
+ ∆Σˆ 3
)−1/2 .
(139) (140)
Using (4.32) and (4.33), this is equivalent to −2 det (2DΣ1 2D DΣ )1/2 det (2(A − B D −1 B t )(2DΣ3 )−1 )1/2 1 ,Σ2
(141)
detζ (2DΣ1 )−1/2 detζ (2DΣ2 )−1/2 detζ (DΣ1 ,Σ2 )detζ (2DΣ3 )1/2 = 1.
(142)
To simplify this, we will use the well-known fact that detζ (AB) = detζ (A)det (B), when B = 1 + T , T trace class (see [8] or [12]). This implies that (142) is equivalent to detζ (2DΣ1 2D)1/2 detζ (2(A − B D −1 B t ))1/2 detζ (2DΣ1
)−1/2 det
ζ (2DΣ2
)−1/2
= 1.
(143) (144)
Together with the factorization following (113), this also implies that detζ (2DΣ2 ) = detζ (2D)detζ (2(A − B D −1 B t )).
(145)
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Thus (146) is equivalent to showing that the multiplicative anomaly F(2DΣ1 , 2D) = detζ (2DΣ1 2D)1/2 detζ (2DΣ1 )−1/2 detζ (2D)−1/2 = 1.
(146)
It is well-known that this vanishes, because D − DΣ1 is a smoothing operator (see [8] or [12]). We have now established that (138) is an expression for the Fourier transform of the left-hand side of (106). We will now calculate the Fourier transform of the right-hand side of (106), along the same lines. As we did for DΣ2 , we will write DΣ1 ,Σ2 as a 2 × 2 matrix α β (147) DΣ1 ,Σ2 = βt δ relative to the coordinates (φ2 , φ1 ). The crucial fact is that B = β. We first calculate (heuristically) e−i( f1 ,φ1 ) [dΦCs 3 |Φ S2 = φ2 ]Z1 (Σ3 )(φ2 ) φ1 1 t −1 t e−i( f1 ,φ1 ) det (δ)1/2 e 2 φ2 (α−βδ β )φ2 , =
(148) (149)
φ1
e
− 12 (φ2t ,φ1t )
α β βt δ
= det (2DΣ3 ) This implies
φ2 φ1 dφ det (2D )1/4 e− 12 φ2t DΣ3 φ2 (dφ )1/2 1 Σ3 2
1/4 − 12 f 1t δ −1 f 1 iφ2t βδ −1 f 1 − 12 φ2t DΣ3 φ2
e
e
∗
e
(dφ2 )
1/2
.
∗
e−i(( f1 ,φ1 )+( f 2 ,φ2 )+( f 1 ,φ1 ))
[dΦCs 3 |Φ Ss 2 = φ2 ](φ1 )Z1 (Σ3 )2 (φ2 )[dΦC∗s3 |Φ S∗s∗ = φ2 ](φ1∗ ) = e e
2 − 21 ( f 1 ,(m 20 +∆Σ3 )−1 f 1 )+( f 1∗ ,(m 20 +∆Σ ∗ )−1 f 1∗ ) 3
(150) (151)
(152) (153) (154)
− 12 (β(m 20 +∆Σ3 )−1 f 1 −β ∗ (m 20 +∆Σ ∗ )−1 f 1∗ − f 2 ,(m 20 +∆Σˆ )−1 (β(m 20 +∆Σ3 )−1 f 1 −β ∗ (m 20 +∆Σ ∗ )−1 f 1∗ − f 2 ) 3 3 3
. (155)
This last equation is justified, by noting that the calculations leading to it are valid in finite dimensions and taking limits. Using B = β, it is now clear that the Fourier transform of the right-hand side of (106) equals (138). This proves (106), and completes the proof of the proposition. As we remarked above, this proves part (a) of the theorem, assuming that the incoming boundary of Σ1 is empty and the outgoing boundary of Σ2 is nonempty. The proofs in the other two cases for (a) involve straightforward modifications. To prove (b), suppose that Σ = Σ1 ◦ Σ2 . Then ˆ trace(Z(Σ)) = Z(|Σ1 |∗ ) ◦ Z(|Σ2 |) = Z(|Σ1 |∗ ◦ |Σ2 |) = Z(Σ). This completes the proof of the theorem.
(156)
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6. Appendix A: Half-Densities Suppose that X is a standard Borel space, and C is a measure class on X . Let C¯ denote the union of all measure classes which are absolutely continuous with respect to C, and let C¯ f denote the subset of finite measures. There is a real separable Hilbert space, H(C), the space of half-densities relative to C, and a bilinear map H(C) × H(C) → C¯ f ,
(157)
which are canonically associated to C. We will define the space of half densities in terms of its representations. Fix a positive representative ν for C. There is an isomorphism of Hilbert spaces L 2 (X, ν; R) → H(C) : f → f (dν)1/2 ,
(158)
and in terms of this isomorphism, the map (157) is given by f (dν)1/2 ⊗ g(dν)1/2 → f gdν. If one chooses another positive representative for C, say µ, then dµ 1/2 , f (dν)1/2 = h(dµ)1/2 ⇔ f = h dν
(159)
(160)
1/2 denotes the positive square root of this where f ∈ L 2 (dν), h ∈ L 2 (dµ), and ( dµ dν ) positive function. In an obvious way, these identifications can be used to give a formal definition of H(C). We now list a number of elementary facts about spaces of half-densities. (1) If C1 << C2 , then there is a canonical isometric embedding dν1 1/2 1/2 (dν2 )1/2 , (161) HC1 ) → H(C2 ) : f (dν1 ) → f dν2
where νi is a positive representative for Ci . (2) The isomorphism (158), and the coordinate transformation (161), show that there is a distinguished positive cone inside H(C), corresponding to nonnegative functions in (158). We will denote this cone by H(C)+ . Given a positive finite measure, ν, ν 1/2 will denote the positive square root in the space of half densities. (3) The natural representation of L ∞ (C) by multiplication operators on L 2 (X, ν) corresponds to a well-defined natural action L ∞ (C) × H(C) → H(C) : F ⊗ δ → F · δ.
(162)
Conversely given a faithful multiplicity free representation of a commutative Von Neumann algebra A × H → H, (163) there is a measure class C, unique up to isomorphism, such that (163) is realized as (162). This is a special case of the spectral theorem (see [6], p. 210, Theorem 2). (4) Given disjoint measure spaces Ci , there is a canonical isomorphism H(Ci ) ⊕ H(C2 ) → H(C1 C2 ).
(164)
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(5) Given a pair of spaces and measure classes (X i , Ci ), there is a measure class C1 ⊗ C2 on X 1 × X 2 generated by C1 × C2 . There is a canonical isomorphism H(C1 ) ⊗ H(C2 ) → H(C1 ⊗ C2 ).
(165)
(6) Suppose that ν is a finite positive measure belonging to the measure class C ⊗ C on X × X . Then ν 1/2 ∈ H(C) ⊗ H(C), (166) and hence ν 1/2 can be interpreted as a Hilbert-Schmidt operator on H(C). This operator is positivity-preserving: ν 1/2 : H(C)+ → H(C)+ (167) (in [15], p. 30, the phrase ‘doubly Markovian map’ is used for this property). Given ν1 and ν2 , 1/2 1/2 1/2 ν1 ◦ ν2 = ν3 ,
(168)
where ν3 is another finite positive measure, 2 ν3 (φ, ψ) = ν1 (φ, η)1/2 ν2 (η, ψ)1/2 .
(169)
η
This can be summarized as follows. Proposition 4. The finite positive measures in C ⊗ C form a semigroup, with multiplication (168), and this semigroup is represented by positivity-preserving Hilbert-Schmidt operators on H(C). (7) Now suppose that the Borel structure on X is derived from a locally convex linear structure, and C is the measure class on X of a Gaussian measure (Chap. 2 of [1]). Proposition 5. Suppose that ν1 and ν2 are Gaussian measures belonging to C ⊗ C. Then ν3 , as in (168)–(169), is a multiple of a Gaussian measure. If X is finite dimensional, this is a consequence of the theorem in Sect. 3, p. 65 of [11]. The proposition follows in a routine way, after rewriting Howe’s formulas to account for normalizations of measures, by taking limits. Acknowledgement. I thank Lennie Friedlander and John Palmer for useful conversations. I also thank a referee for pointing out a serious error in a first version of this paper; see Remark 5.
References 1. Bogachev, V.: Gaussian Measures. Mathematical Surveys and Monographs 62, Providence, RI: Amer. Math. Soc. 1998 2. Bourbaki, N.: Elements of Mathematique. Fasc. XXXV, Livre VI: Integration, Ch IX, Actualites Sci. Indust., No. 1343, Paris: Hermann, 1969 3. Burghelea, D., Friedlander, L., Kappeler, T.: Mayer-Vietoris type formulas for determinants of elliptic operators. J. Funct. Anal. 107, 34–65 (1992) 4. Colella, P., Lanford, O.E.: Sample field behavior for the free Markov random field. In: Constructive Quantum Field Theory, edited by G. Velo and A. Wightman, Lecture Notes in physics, 25, Berlin:SpringerVerlag, 1973, pp. 44–70 5. Dimock, J.: Transition amplitudes and sewing properties for bosons on the Riemann sphere. J. Math. Phys. 48 (5) (2007)
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6. Dixmier, J.: Les algebres d’operateurs dans l’espace Hilbertien. Paris:Gauthier Villars, 1969 7. Folland, G.: Introduction to Partial Differential Equations. Mathematical Notes, Princeton, NJ: Princeton University Press, 1976 8. Friedlander, L.: PhD thesis, Dept. Math., MIT, 1989 9. Glimm, J., Jaffe, A.: Quantum Physics, a Functional Integral Point of View. Berlin-HeidelbergNew York:Springer-Verlag, 1981 10. Guillemin, V., Sternberg, S.: Geometric Asymptotics. Math. Surveys. no. 14, Providence, RI: Amer. Math. Soc., 1977 11. Howe, R.: The oscillator semigroup. In: The Mathematical Heritage of Hermann Weyl. edited by R. Wells, Proceedings of Symposia in Pure Mathematics, Vol. 48, Providence, RI: Amer. Math. Soc., 1989, pp. 61–132 12. Kontsevich, M., Vishik, S.: Geometry of determinants of elliptic operators. In: Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswich, NJ, 1993) Progr. Math. 131, Boston, MA: Birkhäuser, 1995, pp. 173–197 13. Segal, G.: The definition of conformal field theory, In: Geometry and Quantum Field Theory, edited by U. Tillmann, Oxford: Oxford Univ. Press, 2004, pp. 423–577 14. Segal, G.: Lectures at Stanford University, 1996 and 2006, unpublished 15. Simon, B.: The P(φ)2 Euclidean (Quantum) Field Theory. Princeton Series in Physics, Princeton, NJ: Princeton Univ. Press, 1974 16. Weinstein, A.: Symplectic geometry. Bull. Amer. Math. Soc. 5(1), 1–13 (1981) Communicated by M. Aizenman
Commun. Math. Phys. 280, 427–444 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0478-5
Communications in
Mathematical Physics
The Automorphism Group of a Simple Tracially AI Algebra Ping Wong Ng1 , Efren Ruiz2 1 University of Louisiana at Lafayette, 217 Maxim D. Doucet Hall, P.O. Box 41010,
Lafayette, LA 70504-1010, USA. E-mail: [email protected]
2 Department of Mathematics, University of Hawaii Hilo, 200 W. Kawili St.,
Hilo, Hawaii 96720, USA. E-mail: [email protected] Received: 27 March 2007 / Accepted: 11 October 2007 Published online: 8 April 2008 – © Springer-Verlag 2008
Abstract: The structure of the automorphism group of a simple TAI algebra is studied. Inn(A) In particular, we show that Inn is isomorphic (as a topological group) to an inverse 0 (A) limit of discrete abelian groups for a unital, simple, AH algebra with bounded dimension Inn(A) is totally disconnected. growth. Consequently, Inn 0 (A) Another consequence of our results is the following: Suppose A is the transformation group C ∗ -algebra of a minimal Furstenberg transformation with a unique invariant probability measure. Then the automorphism group of A is an extension of a simple topological group by the discrete group Aut(K (A))+,1 . 1. Introduction Denote the group of automorphisms of A equipped with the topology of pointwise convergence by Aut(A), denote the closure of the group of inner automorphisms of A by Inn(A), and denote the closure of the group of inner automorphisms of A whose implementing unitaries are connected to 1 A via a norm continuous path of unitaries by Inn0 (A). We then have that Aut(A) decomposes into the following series of closed normal subgroups: Inn0 (A) Inn(A) Aut(A). In [6], Elliott and Rørdam showed that for a simple unital C ∗ -algebra A that is an AT algebra with real rank zero, Inn0 (A) is a simple topological group (no non-trivial Inn(A) is totally disconnected. Hence, the result of Elliott closed normal subgroup) and Inn 0 (A) and Rørdam gives a structure theorem for Aut(A) since Aut(A) fits into the following exact sequence: {1} → Inn(A) → Aut(A) → Aut(K ∗ (A))+,1 → {1}.
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Inn(A) In their paper, they asked if Inn0 (A) is a simple topological group and if Inn is totally 0 (A) ∗ disconnected for every unital simple C -algebra A. Inn(A) is totally disconnected for A belonging to a class of In this paper, we show that Inn 0 (A) ∗ C -algebras that are classified by K -theory and traces. The class includes all unital, separable, nuclear, purely infinite, simple C ∗ -algebras satisfying the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet [24] and all unital, simple, AH algebras Inn(A) with bounded dimension growth. We show that Inn is isomorphic to an inverse limit 0 (A) of discrete abelian groups (similar to the one used by Elliott and Rørdam in [6]). Recently in [15] we have shown that Inn0 (A) is a simple topological group whenever A is a unital simple AH algebra with bounded dimension growth. This result and the result of this paper give a structure theorem for Aut(A). The paper is organized as follows. In Sect. 2, we define Property (C) and show that Inn(A) if A is a unital, separable C ∗ -algebra satisfying Property (C), then Inn is a totally 0 (A) disconnected group (see Theorem 2). Section 3 is devoted to showing that A satisfies Property (C) whenever A is a unital, separable, nuclear, simple C ∗ -algebra satisfying the UCT, and A is either a purely infinite C ∗ -algebra or a tracially AI algebra (Theorem 3). In the last section, we apply our results to study the automorphism group of A, where A is the transformation group C ∗ -algebra associated to a minimal Furstenberg transformation with a unique invariant probability measure. We also give an explicit description of Aut(K (A))+,1 when n is equal to 2.
2. The Closure of the Group of Inner Automorphisms We will use the following conventions throughout the paper. 1. Let A be a unital C ∗ -algebra. Set T (A) = {τ ∈ A∗ : τ tracial state}. The space of all affine continuous maps from T (A) to R will be denoted by Aff(T (A)). For every projection p ∈ Mn (A), define f p ∈ Aff(T (A)) by f p (τ ) = τ ( p). Define ρ A : K 0 (A) → Aff(T (A)) by ρ A ([ p]) = f p . 2. Denote the class of all separable, nuclear, C ∗ -algebras satisfying the UCT by C. 3. Let G and H be groups and g ∈ G. Set Hg (G, H ) = {h ∈ H : ∃ α ∈ Hom(G, H ) such that α(g) = h} . Lin in [13] used H[1 A ] (K 0 (A), K 0 (B)) to study unitary equivalence between two extensions. We will now explain the notation used in Lemma 1 below and in the process prove Lemma 1. Let A be a separable unital C ∗ -algebra and let {G n }∞ n=1 be an increasing sequence of finitely generated subgroups of K 0 (A) with [1 A ] ∈ G n . Note that there exists a sequence of surjections K 1 (A) K 1 (A) ← ← ··· , H[1 A ] (G 1 , K 1 (A)) H[1 A ] (G 2 , K 1 (A)) K 1 (A) K 1 (A) with . Equip H[1 ] (G n ,K 1 (A)) A H[1 A ] (G n , K 1 (A)) the discrete topology and give the inverse limit group the natural inverse limit topology. which gives an inverse limit group lim ←−
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Then the inverse limit is a complete topological space. Denote the natural projection lim ←−
K 1 (A) K 1 (A) → H[1 A ] (G n , K 1 (A)) H[1 A ] (G n , K 1 (A))
by πn . Then xk → x in the inverse limit if and only if for all n ∈ N, πn (xk ) → πn (x) K 1 (A) in H[1 ] (G , if and only if for all n ∈ N, there exists k(n) ∈ N such that πn (xk ) = n ,K 1 (A)) A
K 1 (A) H[1 A ] (G n ,K 1 (A)) commute K 1 (A) . with the surjections in the inverse limit, they lift to a map K 1 (A) → lim H[1 ] (G n ,K 1 (A)) ←− A K 1 (A) ˇ From the above For g ∈ K 1 (A), denote the image of g in lim H[1 ] (G n ,K 1 (A)) by g. ←− A
πn (x) for all k ≥ k(n). Since the quotient maps from K 1 (A) to
paragraphs, we have the follow result.
∞ Lemma 1. Let A and {G n }∞ n=1 be as in the above paragraphs. If {gk }k=1 is a sequence in K 1 (A), then gˇ k → 0 if and only if for each n ∈ N, there exists k(n) ∈ N such that for all K 1 (A) k ≥ k(n), gk is in H[1 A ] (G n , K 1 (A)). Also, the image of K 1 (A) in lim ←− H[1 ] (G n , K 1 (A)) A is dense. Inn(A) Note that there exists a homomorphism U (A)/U0 (A) → Inn such that the dia0 (A) gram / U (A) / U (A)/U0 (A) / U0 (A) / {1} {1} Ad
{1}
/ Inn (A) 0
Ad
/ Inn(A) /
Inn(A) Inn0 (A)
(1) / {1}
Inn(A) Inn(A) is commutative. If Inn is given the quotient topology, then Inn is a complete 0 (A) 0 (A) topological group. Inn(A) For g ∈ U (A)/U0 (A), denote the image of g in Inn by g. ˆ The proof of the 0 (A) following lemma is similar to the proof of Lemma 4.2 of [6]. In fact, the proof carries over to our case.
Lemma 2. Let A be a unital, separable C ∗ -algebra. The image of U (A)/U0 (A) (via the Inn(A) . Moreover, if {gk }∞ map g → g) ˆ is dense in Inn k=1 is a sequence in U (A)/U0 (A), 0 (A) then gk → 0 if and only if there exists a sequence {u k }∞ k=1 ⊂ U (A) such that 1. for all k, gk = [u k ] in U (A)/U0 (A) and 2. limk→∞ u k a − au k = 0 for all a in A. We will also need the following fact from group theory. We leave the proof to the reader. Lemma 3. Let {G n }∞ n=1 be a sequence of abelian groups and G be an abelian group. ∞
G
n is a group homomorphism. Denote the quotient homomorSuppose α : G → n=1 ∞ n=1 G n G n phism from G n to ∞ G by . Then for every finitely generated subgroup H of G, n=1 n there exists a group homomorphism β H : H → ∞ n=1 G n such that α| H = ◦ β H .
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Let A be a unital C ∗ -algebra. For notational convenience, for g ∈ U (A)/U0 (A) we will again denote the image of g under the canonical map from U (A)/U0 (A) to K 1 (A) by g. Theorem 1. Let A be a separable, unital C ∗ -algebra. If {G n }∞ n=1 is an increasing sequence of finitely generated subgroups of K 0 (A) with [1 A ] ∈ G n and ∪∞ n=1 G n = K 0 (A), then µ : gˆ → gˇ extends uniquely to a continuous group homomorphism K 1 (A) Inn(A) → lim H[1 ] (G . µ : Inn n ,K 1 (A)) (A) ←−
0
A
Inn(A) is the completion of the image of U (A)/U0 (A) and by Proof. By Lemma 2, Inn 0 (A) K 1 (A) is the completion of the image of K 1 (A). Hence, to Lemma 1, lim ←− H[1 ] (G n , K 1 (A)) A prove the theorem it is enough to show that for every sequence {gk }∞ k=1 in U (A)/U0 (A), gk → 0 implies that gˇ k → 0. Suppose gk → 0. By Lemma 2, there exists a sequence {u k }∞ k=1 in U (A) such that
1. for all k ∈ N, we have that [u k ] = gk in U (A)/U0 (A) and 2. limk→∞ u k a − au k = 0 for all a ∈ A. Set ∞ (A) = {{xn } : xn ∈ A and sup xn < ∞} and c0 (A) = {{xn } ∈ ∞ (A) : limn→∞ xn = 0}. Set q∞ (A) = ∞ (A)/c0 (A) and denote the quotient map from ∞ (A) to q∞ (A) by π∞ . Define ϕ : A ⊗ C(S 1 ) → q∞ (A) by ϕ(a ⊗ 1C(S 1 ) ) = π∞ ({a}) and ϕ(1 A ⊗ z) = π∞ ({u k }). Note that ϕ is a unital ∗-homomorphism. Note that in the commutative diagram 0
/ K 1 (c0 (A))
/ K 1 (∞ (A))
/ K 1 (q∞ (A))
/0
0
/ ∞ K 1 (A) k=1
/ ∞ K 1 (A) k=1 /
∞ K 1 (A) k=1 ∞ k=1 K 1 (A)
/0
the rows are exact and the middle vertical map is defined by [{wk }] → {[wk ]}. By composing K 1 (ϕ) with the vertical map in thethird column of the above diagram, we get a ∞ K 1 (A) homomorphism α : K 1 (A ⊗ C(S 1 )) → k=1 . ∞ K (A) k=1
1
Note that there exists a homomorphism β : K 0 (A) → K 1 (A ⊗ C(S 1 )) which maps [1 A ] to [1 A ⊗z]. By Lemma 3, for any finitely generated subgroup G of K 0 (A) containing [1 A ], α ◦ β|G lifts to a homomorphism from G to ∞ k=1 K 1 (A). Note that β ◦ α maps [1 A ] to the image of {[u k ]}∞ k=1 in
∞
k=1 ∞
K 1 (A) . K 1 (A)
So, for each
∞ ∞ ∞ n, {γk ([1 A ]) − [u k ]}∞ k=1 K 1 (A) is k=1 is in ⊕k=1 K 1 (A) if γ = {γk }k=1 : G n → a lifting of α ◦ β|G n . Hence, there exists k(n) such that for all k ≥ k(n), we have that γk ([1 A ]) = [u k ] = gk in K 1 (A). This implies that for all k ≥ k(n), gk is in H[1 A ] (G n , K 1 (A)). By Lemma 1, gˇ k → 0. The uniqueness of µ is clear. k=1
Definition 1. A unital C ∗ -algebra A is said to satisfy Property (C) if for every > 0 and finite subset F of A, there exists a finitely generated subgroup G 0 of K 0 (A) containing [1 A ] such that the following holds: for every u ∈ U (A) with [u] ∈ H[1 A ] (G 0 , K 1 (A)), there exists w ∈ U (A) such that 1. wa − aw < for all a ∈ F and
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2. [w] = [u] in U (A)/U0 (A). Theorem 2. Let A be a separable, unital C ∗ -algebra that satisfies Property (C). Then for any increasing sequence {G n }∞ n=1 of finitely generated subgroups of K 0 (A) with [1 A ] ∈ G n and ∪∞ G = K (A), the continuous map µ defined in Theorem 1 is 0 n=1 n Inn(A) injective. Consequently, Inn is totally disconnected. 0 (A) If, in addition, the natural map from U (A)/U0 (A) to K 1 (A) is an isomorphism, then µ is an isomorphism.
Proof. Note that to prove the first part of the theorem it is enough to show that for every sequence {gk }∞ k → 0 if and only if gˇ k → 0. By Theorem 1, the k=1 in U (A)/U0 (A), g only if direction holds since µ is continuous. Suppose that gˇ k → 0. By Lemma 1, for each n ∈ N, there exists k(n) ∈ N such that gk ∈ H[1 A ] (G n , K 1 (A)) for all k ≥ k(n). Let {Fm }∞ m=1 be an increasing sequence of finite subsets of A whose union is dense in A. For m ∈ N, let Hm be the finitely generated subgroup of K 0 (A) given in Definition 1 corresponding to Fm and 21m . Since ∪∞ n=1 G n = K 0 (A) and G n ⊂ G n+1 , there exists a strictly increasing sequence {n(m)}∞ m=1 ⊂ N such that Hm is a subgroup of G n(m) . Since gˇ k → 0, there exists a strictly increasing sequence {k(m)}∞ m=1 ⊂ N such that gk ∈ H[1 A ] (G n(m) , K 1 (A)) ⊂ H[1 A ] (Hm , K 1 (A)) for all k ≥ k(m). By the choice Hm , for m, s ∈ N with k(m) ≤ s < k(m + 1), there exists ws ∈ U (A) such that 1. ws a − aws < 21m for all a ∈ Fm and 2. [ws ] = gs in U (A)/U0 (A).
For k ≥ k(1), set u k = wk . For k < k(1), let u k be any lifting of gk . Since ∞ m=1 Fm is dense in A, it is easy to check that limk→∞ u k a − au k = 0 for all a ∈ A and [u k ] = gk in U (A)/U0 (A) for all k. Therefore, by Lemma 2, gk → 0. If the natural map from U (A)/U0 (A) to K 1 (A) is an isomorphism, then K 1 (A) is Inn(A) dense in the complete topological group Inn . Hence, we have that µ is surjective. 0 (A) Therefore, µ is an isomorphism.
Corollary 1. Let A be a separable, unital C ∗ -algebra that satisfies Property (C). Suppose that the natural map from U (A)/U0 (A) to K 1 (A) is an isomorphism. Then 1.
Inn(A) Inn0 (A)
is compact if and only if for every finitely generated subgroup G of K 0 (A) containing [1 A ], the group K 1 (A) H[1 A ] (G, K 1 (A))
is a finite group. Inn(A) is locally compact if and only if there exists a finitely generated subgroup 2. Inn 0 (A) G 0 of K 0 (A) containing [1 A ] such that for every finitely generated subgroup G of K 0 (A) containing G 0 , the kernel of the surjective group homomorphism K 1 (A) K 1 (A) −→ H[1 A ] (G 0 , K 1 (A)) H[1 A ] (G, K 1 (A)) is finite.
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3.
P. W. Ng, E. Ruiz Inn(A) Inn0 (A) K 0 (A)
is discrete if and only if there exists a finitely generated subgroup G 0 for containing [1 A ] such that for every finitely generated subgroup G of K 0 (A) containing G 0 , we have that H[1 A ] (G 0 , K 1 (A)) = H[1 A ] (G, K 1 (A)). Inn(A) Consequently, Inn is isomorphic to 0 (A) then it has no isolated points.
K 1 (A) Inn(A) H[1 A ] (G 0 ,K 1 (A)) . Also, if Inn0 (A)
is not discrete,
Proof. It is easy to see that each statement is true for the inverse limit since the collection K 1 (A) is a basis of closed open sets for of subsets πn−1 ({x}) for n ∈ N and x ∈ H[1 ] (G n ,K 1 (A)) A its topology. The corollary now follows from Theorem 2. 3. C ∗-algebras Satisfying Property (C) The purpose of this section is to prove the following theorem: Theorem 3. Let A be a unital, simple C ∗ -algebra in C. If A is either a purely infinite C ∗ -algebra or a tracially AI algebra, then A satisfies Property (C). In order to prove Theorem 3, we first develop some notation that will be used throughout the rest of this section. Let A be a unital C ∗ -algebra and let Cn be the mapping cone of the degree n map on C0 (R). The set of all projections and unitaries in ∞ m,n=1 Mm ( A ⊗ C n ) will be denoted by P(A). Denote the total K -theory of B defined in [4] by K (B). For any finite subset P of P(A), there exist a finite subset G(P) of A and δ(P) > 0 such that if B is any C ∗ -algebra and L : A → B is a contractive, completely positive, linear map which is G(P)-δ(P)-multiplicative, then L defines a map K (L)|P : P → K (B), where P is the image of P in K (A). By enlarging G(P) and choosing a smaller δ(P), if necessary, K (L)|P is defined on the subgroup generated by P. Suppose A satisfies the UCT. Then, by Theorem 1.4 of [5], KL(A, B) (see [22] for the definition of KL(A, B)) is naturally isomorphic to HomΛ (K (A), K (B)) and under this isomorphism KL(ϕ) is precisely K (ϕ) for any ∗-homomorphism ϕ : A → B. In the sequel, we will make this identification without further mention. Throughout the rest of this section ι will denote the embedding of A to A ⊗ C(S 1 ) defined by ι(a) = a⊗1C(S 1 ) . Theorem 3 will be a consequence of the following theorem: Theorem 4. Let A be a unital, simple C ∗ -algebra in C. Suppose A is a tracially AI algebra or A is a purely infinite C ∗ -algebra. Then for every > 0 and finite subset F of A, there exists a finitely generated subgroup G of K 0 (A) containing [1 A ] such that the following holds: for every u ∈ U (A) with [u] ∈ H[1 A ] (G, K 1 (A)), there exist a unital, contractive, completely positive, linear map Φ : A ⊗ C(S 1 ) → A and w ∈ U (A) such that Φ is ι(F) ∪ {1 A ⊗ z, 1 A ⊗ z ∗ }--multiplicative, K 1 (Φ)([1 A ⊗ z]) = [u] in K 1 (A), and w(Φ ◦ ι)(a)w∗ − a < for all a ∈ F. Note that if A is a purely infinite simple C ∗ -algebra, then by Theorem 1.9 of [3] the natural map from U (A)/U0 (A) to K 1 (A) is an isomorphism. Also, in [20, Theorem 10.12] and [2, Corollary 7.14], it was shown that if A is a simple unital C ∗ -algebra of stable rank one, then the natural map U (A)/U (A)0 to K 1 (A) is an isomorphism. See also [19,1], and [21].
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Proof of Theorem 3. Let 0 < < 21 and F be a finite subset of A. By Lemma 4.1.1 of [9], there exists 0 < δ < 6 such that if x ∈ A satisfies ∗ x x − 1 A < δ and x x ∗ − 1 A < δ, then there exists u ∈ U (A) such that u − x < 6 . Let G be the finitely generated subgroup of K 0 (A) containing [1 A ] given by Theorem 4 which corresponds to A, the finite set F, and δ > 0. Hence, by Theorem 4 there exist a unital, completely positive, linear map Φ : A ⊗ C(S 1 ) → A and w ∈ U (A) such that 1. Φ is ι(F) ∪ {1 A ⊗ z, 1 A ⊗ z ∗ }-δ-multiplicative; 2. K 1 (Φ)([1 A ⊗ z]) = [u] in K 1 (A); 3. w(Φ ◦ ι)(a)w∗ − a < δ for all a in F, whenever u is in U (A) with u ∈ H[1 A ] (G, K 1 (A)). By the choice of δ, there exists v0 ∈ U (A) such that Φ(1 A ⊗ z) − v0 < 6 . Set v = wv0 w ∗ . Then [v] = [v0 ] = K 1 (Φ)([1 A ⊗ z]) = [u] in K 1 (A) and hence [v] = [u] in U (A)/U0 (A) since A is a simple tracially AI algebra or A is a purely infinite simple C ∗ -algebra. A computation shows that va − av < for all a ∈ F. The rest of the section is devoted to proving Theorem 4. We will need the following lemma. Lemma 4. Let A and B be unital C ∗ -algebras with A in C. Suppose ϕ ∈ HomΛ (K (A), K (B)) satisfies ϕ([1 A ]) = [1 B ] and ϕ| K 0 (A) is positive. If ζ is in H[1 A ] (K 0 (A), K 1 (B)), then there exists α ∈ HomΛ (K (A ⊗ C(S 1 )), K (B)) such that 1. α| K 0 (A⊗C(S 1 )) is positive; 2. α ◦ K (ι) = K (ϕ); and 3. α([1 A ⊗ z]) = ζ . Proof. Throughout the proof a × b will denote the Kasparov product of a and b. Note that the sequence 0
/ C0 ((0, 1), A)
/ A ⊗ C(S 1 )
π
/ A
/0
(2)
is a split exact sequence, where the splitting map is given by ι(a) = a ⊗ 1C(S 1 ) . Hence, K 0 (A ⊗ C(S 1 )) ∼ = K 0 (A) ⊕ K 1 (A) and K 1 (A ⊗ C(S 1 )) ∼ = K 1 (A) ⊕ K 0 (A). Moreover, under these isomorphisms, K 0 (ι) : K 0 (A) → K 0 (A) ⊕ K 1 (A) may be written as K 0 (ι)(x) = (x, 0) and K 1 (ι) : K 1 (A) → K 1 (A) ⊕ K 0 (A) may be written as K 1 (ι)(x) = (x, 0). Define γ0 : K 0 (A⊗C(S 1 )) → K 0 (B) by γ0 = ϕ◦ K 0 (π ). Since ζ is in H[1 A ] (K 0 (A), K 1 (B)), there exists α ∈ Hom(K 0 (A), K 1 (B)) such that α([1 A ]) = ζ . Define γ1 : K 1 (A ⊗ C(S 1 )) → K 1 (B) by γ1 ((x, y)) = ϕ(x) + α(y). Since A satisfies the UCT, A ⊗ C(S 1 ) satisfies the UCT. Hence, there exists ξ ∈ KL(A ⊗ C(S 1 ), B) such that Γ (ξ )i = γi . Therefore, ϕ − KL(ι) × ξ ∈ ext 1Z (K ∗ (A), K ∗+1 (B)), where ext 1Z (K ∗ (A), K ∗+1 (B)) is the quotient of Ext1Z (K ∗ (A), K ∗+1 (B)) by all locally trivial extensions. Since ι is a splitting map for the sequence in (2), there exists δ ∈ ext 1Z (K ∗ (A ⊗ C(S 1 )), K ∗+1 (B)) such that KL(ι) × δ = ϕ − KL(ι) × ξ . A computation shows that α = ξ + δ satisfies the desired properties.
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3.1. Proof of Theorem 4: The Purely Infinite Case. Let > 0 and F be a finite subset of A. By Proposition 4.3.3 and Theorem 8.4.1 of [23], there exists a sub-C ∗ -algebra B of A and a finite subset F B of B such that (i) (ii) (iii) (iv)
B is a unital, purely infinite, simple C ∗ -algebra in C; 1B = 1 A; K ∗ (B) is finitely generated; and every element of F is within 7 of an element of F B .
Denote the inclusion of B into A by ψ. Set G 0 = K 0 (ψ)(K 0 (B)). Suppose [u] is in H[1 A ] (G 0 , K 1 (A)) for some u ∈ U (A). Since ψ(1 B ) = 1 B = 1 A , [u] ∈ H[1 B ] (K 0 (B), K 1 (A)). By Lemma 4, there exists α ∈ HomΛ (K (B ⊗ C(S 1 )), K (A)) such that 1. α ◦ K (ι) = K (ψ) and 2. α([1 B ⊗ z]) = [u]. By Theorem 6.7 of [11], there exists a unital injective ∗-homomorphism β : B ⊗ C(S 1 ) → A such that K (β) = α. By Theorem 6.7 of [11], there exists w ∈ U (A) such that w(β ◦ ι)(b)w ∗ − ψ(b) < 4 7 for all b ∈ F B . Since ψ(b) = b for all b ∈ B, we ∗ have that w(β ◦ ι)(b)w − b < 4 7 for all b ∈ F B . Since B ⊗ C(S 1 ) is a nuclear sub-C ∗ -algebra of A ⊗ C(S 1 ), there exists a unital, contractive, completely positive, linear map Φ : A ⊗ C(S 1 ) → A such that
Φ(x) − β(x) < 7 for all x in the set ι(F B ) ∪ {1 A ⊗ z, 1 A ⊗ z ∗ } ∪ x y : x, y ∈ ι(F B ) ∪ {1 A ⊗ z, 1 A ⊗ z ∗ } . A computation shows that Φ and w satisfy the desired properties. 3.2. Proof of Theorem 4: The tracially AI case. Before proving Theorem 4 in the tracially AI case, we would like to give an outline of the proof. The proof is similar to the proof in the purely infinite case. The most difficult part of the proof is that total K -theory no longer classifies (up to approximate unitary equivalence) unital ∗-homomorphisms between tracially AI algebras. One needs to add the tracial states space as well as control the exponential length of certain unitaries. Another thing that adds to the complexity is that ∗-homomorphisms are not readily available, hence one works with “almost” ∗-homomorphism. We use Lemma 6 and Lemma 7 to lift α as in Lemma 4 to an almost ∗-homomorphism Φ with the right K -theory information and Φ ◦ ι and id A agree on total K -theory and traces. The rest of the proof involves perturbing Φ to get Φ such that Φ ◦ ι and id A “agree” on total K -theory, traces, and U (A)/CU (A). This requires similar techniques used in Lin’s classification of tracially AI algebras (see the proof of Theorem 10.4 of [12]). To do this we will use the results of Lin in Sects. 6 and 7 of [12]. These results allow us to perturb Φ on a large corner of A to get a new map that agrees with Φ on total K -theory and is close to id A in traces and U (A)/CU (A). Using Lin’s stable uniqueness theorem (Theorem 8.6 of [12]), we get that the map Φ gives the right K -theory information and Φ ◦ ι and id A are approximately unitarily equivalent. Lemma 5. Let A be a unital C ∗ -algebra, F be a finite subset of A containing the identity of A, P be a finite subset of P(A). For every > 0, there exists δ > 0 such that the following holds: If B is a C ∗ -algebra and φ : A → B is a F-δ-multiplicative, contractive, completely positive, linear map with K (φ)|P well-defined, then there exists a F--multiplicative, contractive, completely positive, linear map ψ : A → B such that ψ(1 A ) is a projection in B, φ(a) − ψ(a) < for all a ∈ F, and K (φ)|P = K (ψ)|P .
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Lemma 6. Let A be a unital, simple C ∗ -algebra in C. Suppose A is a tracially AI algebra. Then for every > 0, finite subset P of P(A), and finite subset F of A⊗C(S 1 ), there exists a finitely generated subgroup G of K 0 (A) containing [1 A ] such that the following holds: for every u ∈ U (A) with [u] ∈ H[1 A ] (G, K 1 (A)), there exists a contractive, completely positive, linear map ψ : A ⊗ C(S 1 ) → A such that 1. ψ is F--multiplicative; 2. K (ψ ◦ ι)|P = K (id A )|P ; and 3. K (ψ)([1 A ⊗ z]) = [u]. Proof. Note that without loss of generality, we may assume that F is contained in the unit ball of A ⊗ C(S 1 ). We also may assume that 1 A ⊗ z and 1 A ⊗ z ∗ are in F and < 1. Set Q = ι(P) ∪ {1 A ⊗ z, 1 A ⊗ 1C(S 1 ) }. Suppose A is finite dimensional. Then A = Mn (C). Take G = K 0 (A). Note that K 1 (A) = 0. Thus, if [u] is in H[1 A ] (K 0 (A), K 1 (A)) = 0, then [u] = 0. Therefore, we can take ψ to be the natural projection from A ⊗ C(S 1 ) to A. Suppose A is infinite dimensional. By Theorem 10.9 and Theorem 10.1 of [12], A is k(n) isomorphic to lim(An , φn,n+1 ), where An = i=1 P[n,i] M[n,i] (C(X [n,i] ))P[n,i] , where −→ each X [n,i] is a connected finite CW-complex and φn,n+1 is a unital, injective ∗-homomorphism. So we may assume that A is this direct limit decomposition. By [10] and [12], there exists a unital, separable, simple AH algebra B such that B is a tracially AF algebra satisfying the UCT and (K 0 (A), K 0 (A)+ , [1 A ], K 1 (A)) ∼ = (K 0 (B), K 0 (B)+ , [1 B ], K 1 (B)). Denote the above isomorphism by β. Since A and B satisfy the UCT, β lifts to an isomorphism from K (A) to K (B). By an abuse of notation, we denote this lifting by β. Denote the natural embedding of An into An ⊗ C(S 1 ) by ι An . Since A ⊗ C(S 1 ) = lim(An ⊗ C(S 1 ), φn,n+1 ⊗ idC(S 1 ) ) and since K (·) is a continuous functor with respect −→
to direct limits, there exist n 0 ∈ N, a finite subset Pn 0 of P(An 0 ), a finite subset Qn 0 of P(An 0 ⊗ C(S 1 )) containing ι An0 (Pn 0 ), and a finite subset Fn 0 of An 0 ⊗ C(S 1 ) such that 1. for p ∈ P, there exists e p ∈ Pn 0 such that [ p] = K (φn 0 ,∞ )([e p ]); 2. for p ∈ Q, there exists q p ∈ Qn 0 such that [ p] = K (φn 0 ,∞ ⊗ idC(S 1 ) )([q p ]); 3. every element of F is within 20 to an element of (φn 0 ,∞ ⊗ idC(S 1 ) )(Fn 0 ). Set G = K 0 (φn 0 ,∞ )(K 0 (An 0 )). Suppose [u] is in H[1 A ] (G, K 1 (A)). Then β([u]) is an element of H[1 An ] (K 0 (An 0 ), K 1 (B)). Therefore, by Lemma 4 there exists α in 0
HomΛ (K (An 0 ⊗ C(S 1 )), K (B)) such that 1. α| K 0 (An ⊗C(S 1 )) is positive; 0 2. α ◦ K (ι An0 ) = β ◦ K (φn 0 ,∞ ); and 3. α([1 An0 ⊗ z]) = β([u]) in K 1 (B). By Theorem 6.2.9 of [9], there exists a sequence of contractive, completely positive, 1 linear maps {L n 0 ,k }∞ k=1 from An 0 ⊗ C(S ) to B such that 1. limk→∞ L n 0 ,k (x y) − L n 0 ,k (x)L n 0 ,k (y) = 0 for all x, y ∈ An 0 ⊗ C(S 1 ), and 2. K (L n 0 ,k )|Qn0 = α|Qn0 for all k.
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Since K (L n 0 ,k )([1 An0 ⊗ 1C(S 1 ) ]) = α([1 An0 ⊗ 1C(S 1 ) ]) = [1 B ], by a small perturbation we may assume that L n 0 ,k is unital for all k. Since An 0 ⊗ C(S 1 ) is nuclear, there exists 1 a sequence of contractive, completely positive, linear maps {ψn 0 ,k }∞ k=1 from A ⊗ C(S ) 1 to An 0 ⊗ C(S ) such that lim ψn 0 ,k ◦ (φn 0 ,∞ ⊗ idC(S 1 ) ) (x) − x = 0 k→∞
for all x ∈ An 0 ⊗ C(S 1 ). Set a and b are in F. Choose xa and xb in Fn 0 such βn 0 ,k = L n 0 ,k ◦ ψn 0 ,k . Suppose that a − (φn 0 ,∞ ⊗ idC(S 1 ) )(xa ) < 10 and b − (φn 0 ,∞ ⊗ idC(S 1 ) )(xb ) < 10 . Then an easy computation shows that βn ,k (ab) − βn ,k (a)βn ,k (b) < + βn ,k (x0 y0 ) − βn ,k (x0 )βn ,k (y0 ), 0 0 0 0 0 0 4 where x0 = (φn 0 ,∞ ⊗ idC(S 1 ) )(xa ) and y0 = (φn 0 ,∞ ⊗ idC(S 1 ) )(xb ). Since lim βn 0 ,k (x y) − βn 0 ,k (x)βn 0 ,k (y) = 0 k→∞
for all x, y ∈ (φn 0 ,∞ ⊗ idC(S 1 ) )(An 0 ⊗ C(S 1 )), we have that lim βn 0 ,k (ab) − βn 0 ,k (a)βn 0 ,k (b) ≤ 4
k→∞
for all a, b ∈ F. It is easy to check that K (βn 0 ,k ◦ ι)|P = β|P and K (βn 0 ,k )([1 A ⊗ z]) = β([u]) for k sufficiently large. Choose k0 ∈ N such that βn 0 ,k0 (ab) − βn 0 ,k0 (a)βn 0 ,k0 (b) < 2 for all a, b ∈ F. Choose a finite subset H of P(B) such that for q ∈ Q, there exists pq ∈ H such that [ pq ] = β([q]). Since B is a unital, separable, nuclear, simple, tracially AF algebra, by Proposition 9.10 of [12], there exists a sequence of contractive, completely positive, linear maps {γk }∞ k=1 from B to A such that lim γk (x y) − γk (x)γk (y) = 0
k→∞
for all x, y ∈ B and K (γk )|H = (β −1 )|H for all k. Choose k1 such that γk1 (x y) −γk1 (x)γk1 (y) < 2 for all x, y ∈ βn 0 ,k0 (F). Set φ = γk1 ◦ βn 0 ,k0 . Then K (φ)([1 A ⊗ z]) = [u], K (φ ◦ ι)|P = K (id A )|P , and
φ(x y) − φ(x)φ(y) < for all x, y ∈ F. We next show that the contractive, completely positive, linear map obtained in the above lemma can be perturbed in such a way that the contractive, completely positive, linear map obtained by this small perturbation is “close” to id A on the tracial state space of A. Lemma 7. Let A be as in Lemma 6. Then for every > 0, finite subset P of P(A), finite subset F1 of As.a. , and finite subset F2 of A ⊗ C(S 1 ), there exists a finitely generated subgroup G of K 0 (A) containing [1 A ] such that the following holds: for every u ∈ U (A) with [u] ∈ H[1 A ] (G, K 1 (A)), there exists a unital, completely positive, linear map ψ : A ⊗ C(S 1 ) → A such that 1. ψ is F2 --multiplicative;
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2. K (ψ ◦ ι)|P = K (id A )|P ; 3. K (ψ)([1 A ⊗ z]) = [u]; and 4. sup {|(τ ◦ ψ ◦ ι)(a) − τ (a)| : τ ∈ T (A)} < for all a ∈ F1 . Proof. Let δ and G be the quantities given in Lemma 5 corresponding to A ⊗ C(S 1 ), ι(F1 ) ∪ F2 ∪ {1 A⊗C(S 1 ) }, ι(P) ∪ {1 A ⊗ z, 1 A⊗C(S 1 ) }, and 2 . Let {Hn }∞ n=1 be an increasing sequence of finite subsets of A whose union is dense in A. Note that we may assume that δ < 2 . Now, for n ∈ N, there exist a projection pn ∈ A, a sub-C ∗ -algebra k(n) Dn = i=1 Mm(i,n) (C(X [i,n] )) of A, where X [i,n] is either C or [0, 1] with 1 Dn = pn , and a sequence of contractive, completely positive, linear maps {L n }∞ n=1 from A to Dn such that 1. 2. 3. 4.
pn x − x pn < 21n for all x ∈ Hn ;
pn x pn − L n (x) < 21n for all x ∈ Hn ;
x − (1 A − pn )x(1 A − pn ) − ψn (x) < τ (1 A − pn ) < 21n for all τ ∈ T (A).
1 2n
for all x ∈ Hn with x ≤ 1; and
Note that limn→∞ L n (x y) − L n (x)L n (y) = 0 for all x, y ∈ A. Denote the i th summand of Dn by D[n,i] . Let d[n,i] = 1 D[n,i] . Choose n large enough such that 21n < 3δ . Let P1 be a finite subset of P(A ⊗ C(S 1 )) such that P1 contains ι(P), d[n,i] ⊗ 1C(S 1 ) , pn ⊗ 1C(S 1 ) . Choose a finite subset X2 of A ⊗ C(S 1 ) such that X2 contains ι(F1 ) ∪ F2 and the set (1 A − pn ) ⊗ 1C(S 1 ) x (1 A − pn ) ⊗ 1C(S 1 ) : x ∈ F . Let G be the finitely generated subgroup of K 0 (A) in Lemma 6 which corresponds to 21n , P, and X2 . Suppose [u] is in H[1 A ] (G, K 1 (A)). Then by Lemma 6, there exists a contractive, completely positive, linear map L : A ⊗ C(S 1 ) → A such that 1. 2. 3. 4.
L is X2 - 21n -multiplicative; K (L)|P1 is well-defined; K (L ◦ ι)|P = K (id A )|P ; and K (L)([1 A ⊗ z]) = [u] in K 1 (A).
k(n) Choose a projection qn ∈ A such that [qn ] = K (L)( i=1 [d[n,i] ⊗ 1C(S 1 ) ]). Let Gn be a finite subset of Dn such that Gn contains the generators of Dn . Define γ : T (A) → T (Dn ) by γ (τ ) = τ (1pn ) τ | Dn . Using the same argument as in Proposition 9.7 of [12], we get a ∗-homomorphism h : Dn → qn Aqn such that sup {|(τ ◦ h)(g) − γ (τ )(g)| : τ ∈ T (A)} <
1 2n
for all g ∈ Gn . Define ψ : A ⊗ C(S 1 ) → A by ψ0 (x) = L([(1 A − pn ) ⊗ 1C(S 1 ) ]x[(1 A − pn ) ⊗ 1C(S 1 ) ]) + (h ◦ L n ◦ π )(x). Hence, ψ0 is X2 -δ-multiplicative. By construction, we have that 1. K (ψ0 )([1 A ⊗ z]) = K (L)([1 A ⊗ z]) = [u]; 2. K (ψ0 ◦ ι)|P = K (id A )|P ; and 3. for all a ∈ F1 , sup {|(τ ◦ ψ0 ◦ ι)(a) − τ (a)| : τ ∈ T (A)} < δ.
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By Lemma 5, there exists a contractive, completely positive, linear map β : A ⊗ C(S 1 ) → A such that β is X2 - 2 -multiplicative, β(1 A⊗C(S 1 ) ) is a projection in A, and
ψ0 (x) − β(x) < 2 for all x ∈ X2 . Also, we have that K (β ◦ ι)|P = K (ψ0 ◦ ι)|P and K (β)([1 A ⊗ z]) = K (ψ0 )([1 A ⊗ z]). Hence, K (β)([1 A⊗C(S 1 ) ]) = [1 A ]. Since A has stable rank one, there exists w ∈ U (A) such that wβ(1 A⊗C(S 1 ) )w ∗ = 1 A . Set ψ = Ad(w) ◦ β. It is easy to check that ψ is the desired unital, completely positive, linear map from A ⊗ C(S 1 ) to A. Suppose that ψ : A → B is a G-δ-multiplicative, contractive, completely positive, linear map, u is a normal partial isometry, and a projection p in B is given so that 1
ψ(u ∗ u) − p < 32 . We define ψ˜ as follows. With δ chosen to be sufficiently small ˜ and G chosen to be sufficiently large, it is well-known that there exists ψ(u) a normal 6 ˜ . This notation will partial isometry (unitary in a corner) such that ψ(u) − ψ(u) < 13 be used later. Note also, if u is in U0 (A), then with sufficiently large G and sufficiently ˜ small δ, we may assume that ψ(u) is in U0 (B). Let φ : [0, a] → X be a continuous map, where X is a normed
nspace. Let P = {0 =
φ(ti ) − φ(ti−1 ) . t0 < t1 < · · · < tn = a} be a partition of [0, a]. Set L(φ)(P) = i=1 The length of φ, denoted as L(φ), is defined to be the supremum of L(φ)(P) over all partitions P of [0, a]. If A is a unital C ∗ -algebra and u is in U (A)0 , then cel(u) is defined to be cel(u) = inf {L(φ) : φ : [0, 1] → U (A)0 , φ(0) = 1 A , φ(1) = u} . Definition 2. Let A be a unital C ∗ -algebra. Let CU (A) be the closure of the commutator subgroup of U (A). Clearly, the commutator subgroup forms a normal subgroup of U (A). Also note that U (A)/CU (A) is commutative. If the natural map from U (A)/U0 (A) to K 1 (A) is injective, then CU (A) is a normal subgroup of U0 (A). If u is in U (A), we will denote the image of u in U (A)/CU (A) by u, ¯ and if F is a subgroup of U (A), then F¯ will denote the image of F in U (A)/CU (A). If u¯ and v¯ are in U (A)/CU (A), define dist(u, ¯ v) ¯ = inf { x − y : x, y ∈ U (A) such that x¯ = u, ¯ y¯ = v} ¯ . If u and v are in U (A), then dist(u, ¯ v) ¯ = inf { uv ∗ − x : x ∈ CU (A)}. Let g = n −1 −1 i=1 ai bi ai bi , where ai and bi are in U (A). Let G be a finite subset of A, δ > 0, and ψ : A → B be a G-δ-multiplicative, contractive, completely positive, linear map, where B is a unital C ∗ -algebra. From the paragraphs before the definition, for > 0 if G is sufficiently large and δ is sufficiently small, n −1 −1 xi yi (xi ) (yi ) < , ψ(g) − 2 i=1
where xi and yi are in U (B). Thus, for g ∈ CU (A), with sufficiently large G and sufficiently small δ, L(g) − u < for some u ∈ CU (B). Moreover, for any finite subset U of U (B) and subgroup F of U (B) generated by U, and > 0, there exist a finite subset G and δ > 0 such that, for any G-δ-multiplicative, contractive, completely positive, linear map ψ : A → B, we have that ψ induces a homomorphism, ψ ‡ : F¯ → U (B)/CU (B) ˜ ψ ‡ (u)) ¯ < for all u ∈ U. such that dist(ψ(u), Note that for any ∗-homomorphism from A to B, φ induces a continuous homomorphism φ ‡ : U (A)/CU (A) → U (B)/CU (B).
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We are now ready to prove Theorem 4 in the tracially AI case. Proof of Theorem 4: The tracially AI case. Let > 0 and F be a finite subset of A. Suppose A is a finite dimensional simple tracially AI algebra. Then A = Mn . Let G = K 0 (A) = Z. Since K 1 (A) = 0, we can take Φ to be the canonical projection of A ⊗ C(S 1 ) to A and w = 1 A . Suppose A is an infinite dimensional C ∗ -algebra. By Lemma 10.9 and Theorem 10.10 of [12], A is isomorphic to lim(An , φn,n+1 ), where An = m(n) j=1 A[n, j] is as described −→ in Theorem 10.1 of [12] such that φn,m , φn,∞ and K ∗ (φn,m ) are injective maps for all m > n. For notational convenience, we may identify An with φn,n+1 (An ). We will use this identification without further warning. π ; Define L : U (A) → R≥0 as follows: for u ∈ U0 (A), define L(u) = 2cel(u)+8π + 16 u ∈ U (A) but not in U0 (A) with finite order in U (A)/U0 (A), define L(u) = 16π + cel(u k(u) ) π + 16 , where k(u) is the order of [u] in U (A)/U0 (A); and for u ∈ U (A) but not 16 π in U0 (A) with infinite order in U (A)/U0 (A), define L(u) = 16π + 16 . π Note that we may assume that < 128 . Given A, L, , and F, Theorem 8.6 of [12] provides us with δ1 > 0, n ∈ N, a finite subset P of P(A), and a finite subset S of A. Theorem 8.6 of [12] also provides us with mutually orthogonal projections q, p1 , . . . , pn such that for each i, we have that q is a Murray–von Neumann equivalent to a subprojection of pi and pi is a Murray–von Neumann equivalent to p1 and there exist a sub-C ∗ -algebra C1 that is a finite direct sum of C ∗ -algebras of the form Mn (C(X )) (X is either [0, 1] or C) with 1C1 = p1 and unital, contractive, completely positive, linear maps h 0 : A → q Aq and h 1 : A → C1 such that 1. h 0 and h 1 are S- δ41 -multiplicative; 2. h 0 (x) = q xq; and < x − (h (x) ⊕ h (x) ⊕ · · · ⊕ h (x) ) 3. 0 1 1
δ1 16
for all x ∈ S.
n
Set C = Mn (C1 ) which is identified as a sub-C ∗ -algebra of (1 A − q)A(1 A − q). With this choice of C1 , Theorem 8.6 of [12] provides us with a finite subset G0 of A, a finite subset P0 of projections in MN (C), a finite subset H of As.a. , δ0 > 0 and σ > 0. Set δ equal to the minimum of δ0 and δ1 . Note that we may assume that P0 contains 1 A − q and contains at least one minimal projection of each summand of C. We also may assume that for u ∈ U (A) ∩ P, u has the form quq ⊕ (1 A − q)u(1 A − q), where quq is in U (q Aq) and (1 A − q)u(1 A − q) is in U (C). Note that there exists n 0 ∈ N and a projection q0 ∈ An 0 such that φn 0 ,∞ (q0 ) is unitarily equivalent to q via a unitary v ∈ U0 (A). Hence, by conjugating φn,∞ by Ad(v), we may assume that φn 0 ,∞ (q0 ) = q. Also, we may assume that quq is in φn 0 ,∞ (An 0 ). Set U = {quq : u ∈ U (A) ∩ P}. Denote by F the subgroup of U (q Aq) generated by U and denote the image of F in U (q Aq)/CU (q Aq) by F. By Theorem 6.6(3) of [12], F = F ∩ (U0 (q Aq)/CU (q Aq)) ⊕ F 0 ⊕ F 1 , where F 0 is a torsion group and F 1 is a free group. If κ denotes the homomorphism from U (q Aq)/CU (q Aq) to K 1 (q Aq), then κ(F 1 ) is isomorphic to F 1 . Suppose U has the following decomposition: U = U0 ∪ U1 ,
(3)
such that U0 generates F ∩ (U0 (q Aq)/CU (q Aq)) ⊕ F 0 and U1 generates F 1 . It turns out that we can reduce the general case to this case since by Lemma 6.9 of [12], modulo
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unitaries in CU (q Aq), this decomposition can be made with the cost of no more than 8π in the estimation of the exponential length. Also, choosing a larger n 0 if necessary, we may assume that U0 and U1 are subsets of qφn 0 ,∞ (An 0 )q. , For the quantities ι(S ∪ G0 ∪ F) ∪ {1 A ⊗ z, 1 A ⊗ z ∗ }, H, P ∪ P0 , and δ2 = min{δ,σ,} 100 Lemma 7 provides us with a finitely generated subgroup G of K 0 (A) containing [1 A ]. Suppose [u] is in H[1 A ] (G, K 1 (A)) for some u ∈ U (A). By Lemma 7, there exists a unital, completely positive, linear map ψ : A ⊗ C(S 1 ) → A such that ψ is ι(S ∪ G0 ∪ F) ∪ {1 A ⊗ z, 1 A ⊗ z ∗ }-δ2 -multiplicative and 1. K (ψ ◦ ι)|P ∪P0 = K (id A )|P ∪P0 ; 2. supτ ∈T (A) {|(τ ◦ ψ ◦ ι)(x) − τ (x)| : τ ∈ T (A)} < δ2 for all x ∈ H; and 3. K (ψ)([1 A ⊗ z]) = [u] in K 1 (A). Since An and A are separable, nuclear C ∗ -algebras and A = lim(An , φn,n+1 ), there −→
exists a sequence of unital, completely positive, linear maps {µn : A → An }∞ n=1 such that lim µn (x y) − µn (x)µn (y) = 0 n→∞ for all x, y ∈ A and limn→∞ (φn,∞ ◦ µn )(a) − a = 0 for all a ∈ A. Therefore we may choose n 1 ≥ n 0 such that 1. 2. 3. 4. 5.
∗ µ n 1 ◦ ψ is ι(S ∪ G0 ∪ F) ∪ {1 A⊗ z, 1 A ⊗ z }-2δ2 -multiplicative; (φn ,∞ ◦ µn ◦ ψ)(x) − ψ(x) < 2δ2 for all x ∈ ι(S ∪G0 ∪F)∪{1 A ⊗z, 1 A ⊗z ∗ }; 1 1 K (φn 1 ,∞ ◦ µn 1 ◦ ψ ◦ ι)|P ∪P0 = K (ψ ◦ ι)|P ∪P0 ; K (φn 1 ,∞ ◦ µn 1 ◦ ψ)([1 A ⊗ z]) = K (ψ)([1 A ⊗ z]) = [u] in K 1 (A); and sup |(τ ◦ φn 1 ,∞ ◦ µn 1 ◦ ψ ◦ ι)(a) − τ (a)| : τ ∈ T (A) < 2δ2 for all a ∈ H.
Let B1 = q An 1 q. Since A is simple, it is known and easy to see that, by choosing possibly a larger n 1 , we may assume that the rank of q at each point is at least 6 (in An 1 ). Note that we have assumed that q is in An 1 . So B1 is a corner of An 1 . By construction and the fact that each K ∗ (φn,∞ ) is injective, we have that K (µn 1 ◦ ψ ◦ ι)([q]) = [q]. By conjugating µn 1 ◦ ψ ◦ ι by some unitary w if necessary, we may assume that (µn ◦ ψ ◦ ι)(q) − q < δ . Define Λ(b) = aq[(µn ◦ ψ ◦ ι)(qbq)]qa (where a = 1 4 1 [q(µn 1 ◦ ψ ◦ ι)(q)q]−1/2 ) for b ∈ q Aq. Note that Λ − (µn 1 ◦ ψ ◦ ι)|q Aq < 2δ . (k) as described Write An 1 = ⊕m k=1 A[n(k),n 1 ] , where each A[n(k),n 1 ] has the form C in Definition 7.1 of [12]. According to this direct sum decomposition, we may write q = q1 ⊕ q2 ⊕ · · · ⊕ ql with 0 ≤ l ≤ m and qk = 0, for 1 ≤ k ≤ l. Choose N1 ∈ N such that N1 [qk ] ≥ 3[1 A[n(k),n1 ] ] for k ≤ l. Note that we may assume that qk has rank at least 6. By applying an inner automorphism, we may assume that ⊕lk=1 A[n(k),n 1 ] is a hereditary sub-C ∗ -algebra of M N1 (B1 ). Since F1 is finitely generated, with sufficiently large n 1 , we obtain (see Definition 6.2 of [12]) a homomorphism j : F¯1 → U (B1 )/CU (B1 ) such that φn‡1 ◦ j = idF1 , where φn 1 = φn 1 ,∞ . Then (since the canonical map from K 1 (An 1 ) to K 1 (A) is injective), κ1 ◦ φn‡1 ◦ (µn 1 ◦ ψ ◦ ι)‡ | F 1 = κ1 ◦ (φn 1 )‡ ◦ j = (κ1 ) F 1 , where κ1 : U (q Aq)/CU (q Aq) → K 1 (q Aq) is the quotient map. Note that K 1 (q Aq) = K 1 (A). Lemma 7.5 of [12] provides us with ∆ > 0 corresponding to A and 4 . We may assume that ∆ < σ4 . To simplify the notation, without loss of generality, we may assume that φn 1 (q) = q. By the assumption on A, we have that φn 1 | B1 = (φn 1 )0 ⊕ (φn 1 )1 , where
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1. τ ((φn 1 )0 (1 B1 )) < 2(N∆+1)2 for all τ ∈ T (A) and 1 2. (φn 1 )0 is homotopically trivial (but nonzero) (see Theorem 10.1 of [12]). It follows from Lemma 7.5 of [12] and the choice of ∆ that there exists a ∗-homomorphism h : B1 → e0 Ae0 such that (i) K (h) = K ((φn 1 )0 ) in HomΛ (K (B1 ), K (A)) and ¯ −1 (h ⊕ (φn 1 )1 )‡ (Λ‡ (w)) ¯ = gw , where gw is in U0 (q Aq) and (ii) (φn‡1 ◦ j (w)) cel(gw ) < 4 (in U (q Aq)) for all w ∈ U1 . Recall that we have assumed that An 1 is a sub-C ∗ -algebra of M N1 (B1 ). Set t = (h ⊕ (φn 1 )1 ) ⊗ idMN1 |⊕l
k=1 A[n( j),n 1 ]
and Ψ = t ⊕ (φn 1 )|⊕mk=l+1 A[n( j),n ] . Let Φ = Ψ ◦ µn 1 ◦ ψ. It is clear that (since ∆ < 1
σ 4)
1. K (Φ ◦ ι)|P ∪P0 = K (φn 1 ,∞ ◦ µn 1 ◦ ψ ◦ ι)|P ∪P0 = K (ψ ◦ ι)|P ∪P0 ; 2. K (Φ ◦ ι)([1 A ⊗ z]) = K (φn 1 ,∞ ◦ µn 1 ◦ ψ)([1 A ⊗ z]) = K (ψ)([1 A ⊗ z]) = [u] in K 1 (A); and 3. sup |(τ ◦ Φ ◦ ι)(a) − (τ ◦ φn 1 ,∞ ◦ µn 1 ◦ ψ)(a)| : τ ∈ T (A) < σ2 for all a ∈ A. 4. For all w ∈ U1 , by (ii) we have that cel(w ∗ (Φ ◦ ι)(w)) < 8π +
in U (q Aq); and 2
5. for w ∈ U0 , by Lemma 6.8, Theorem 6.10, and Lemma 6.9 of [12] we have that ◦ ι)(w)) cel(w ∗ (Φ π 2cel(w) + 64 , < 2cel(wk(w) ) 8π + k(w) +
in U (q Aq) if [w] = 0 in K 1 (A) π 16 ,
in U (q Aq) if the order of [w] is k(w).
Therefore, even after we add 8π for the decomposition of F as in (3), we get ◦ ι)(h 0 (u))) < L(u) in U (q Aq) cel(id A (h 0 (u)∗ )(Φ for all u ∈ U (A) ∩ P. Since we also have that K (Φ ◦ ι)|P ∪P0 = K (id A )|P ∪P0 and sup {|(τ ◦ Φ ◦ ι)(a) − τ (a)| : τ ∈ T (A)} < σ for all a ∈ H, we can apply Theorem 8.6 of [12] to Φ ◦ ι and id A to get w ∈ U (A) such that w(Φ ◦ ι)(a)w∗ − a < for all a ∈ F. Note that Φ is a unital, contractive, completely positive, linear map such that Φ is ι(F) ∪ {1 A ⊗ z, 1 A ⊗ z ∗ }--multiplicative and K (Φ)([1 A ⊗ z]) = [u] in K 1 (A). Remark 1. If, in the above theorem, we assumed that A is tracially AF or A is tracially AI with K 1 (A) a torsion group, then the proof of the above theorem would be much easier. In these two cases, total K -theory and traces are enough to determine when two almost multiplicative linear maps are approximately unitarily equivalent (see Theorem 6.3.3 of [9] and Theorem 8.7 of [12]). The proof becomes much easier and shorter.
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4. The Automorphism Group of a Simple C ∗-Algebra For any C ∗ -algebra A, we will denote the subgroup of HomΛ (K (A), K (A)) consisting of all elements α such that α is an isomorphism which sends [1 A ] to [1 A ] and α| K 0 (A) and α −1 | K 0 (A) are positive homomorphisms by AutΛ (K (A))+,1 . Definition 3. Let θ be an irrational number, f k : Tk → R be a continuous function, and di j be nonnegative integers. Define h n : Tn → Tn to be the inverse of the homeomorphism (ζ1 , ζ2 , . . . , ζn ) → (e2πiθ ζ1 , e2πi f1 (ζ1 ) ζ1d12 ζ2 , e2πi f2 (ζ1 ,ζ2 ) ζ1d13 ζ2d23 ζ3 , . . . ). The homeomorphism h n will be called a n-dimensional Furstenberg transformation. If di,i+1 is nonzero for all i, then by Theorem 2.1 of [8], the dynamical system (Tn , h n ) is minimal. This implies that the transformation group C ∗ -algebra associated to (Tn , h n ) is simple. Also, if each f i satisfies a certain Lipschitz property, then Furstenberg in [8] proved that (Tn , h n ) has a unique h n -invariant probability measure on Tn . This implies that the transformation group C ∗ -algebra associated to (Tn , h n ) is simple with a unique tracial state. Proposition 1. Let An be the transformation group C ∗ -algebra associated to a n-dimensional Furstenberg transformation (Tn , h n ). If (Tn , h n ) is minimal with a unique h n -invariant probability measure on Tn , then ρ An (K 0 (An )) is dense in Aff(T (An )). Proof. We follow the computation given in Example 4.9 of [16]. Define α : C(Tn ) → ∗ n ∗ n C(Tn ) by α( f ) = f ◦ h −1 n . Set An = C (Z, T , h n ) = C (Z, C(T , α)). By the Pimsner–Voiculescu exact sequence [17], the following sequence n )) K 0 (C(T O
id −α∗−1
/ K 0 (C(Tn ))
/ K 0 (An ) ∂
exp
K 1 (An ) o
K 1 (C(Tn )) o
id −α∗−1
K 1 (C(Tn ))
is exact. Let z ∈ U (C(T)) be the element which sends ζ to ζ . Then K 1 (C(Tn )) is a finitely generated free abelian group with [z ⊗ 1 ⊗ · · · ⊗ 1] as one of the generators. Note that α is homotopic to the ∗-homomorphism given by f → f ◦ h −1 with h −1 (ζ1 , ζ2 , . . . , ζn ) = (ζ1 , ζ1d12 ζ2 , ζ1d13 ζ2d23 ζ3 , . . . ). Hence, α([z ⊗ 1 ⊗ · · · ⊗ 1]) = [z ⊗ 1 ⊗ · · · ⊗ 1]. Therefore, by Proposition 6.1 of [18], we have that K 0 (An ) ∼ = Zm ⊕ G, where G is a finitely generated torsion group and one of the generators of K 0 (An ) is an element η0 such that ∂(η0 ) = [z −1 ⊗ 1 ⊗ · · · ⊗ 1]. Let τ be the unique tracial state of An . Then τ is induced by a unique h n -invariant probability measure µ on Tn . We compute the image of η0 under the map τ∗ : K 0 (An ) → R. Combining Definition VI.8 and Theorem V.12 and VI.11 of [7] to get (notation explained afterwards) exp(2πiτ∗ (η0 )) = Rαµ ([z −1 ⊗ 1 ⊗ · · · ⊗ 1]). Here [z −1 ⊗ 1 ⊗ · · · ⊗ 1] now represents the homotopy class of the function (ζ1 , ζ2 , . . . , ζn ) → ζ1−1 .
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µ
Following Definitions VI.3 and VI.5 of [7], Rα ([v]) is computed by finding a contin∗ i f (x) for all x ∈ Tn . With uous function f : Tn → R such that v(h −1 n (x)) v(x) = e −1 v = z ⊗ 1 ⊗ · · · ⊗ 1 one can easily check that we may choose the function f to be the constant function 2π θ . Hence, exp(2πiτ∗ (η0 )) = exp(2πiθ ). Therefore, there exists k ∈ Z such that τ∗ (η0 ) = θ + k. Since τ∗ ([1 An ]) = 1, we have that ρ An (K 0 (An )) contains Z + θ Z ⊂ R = Aff(T (An )). Therefore, the image of ρ An (K 0 (An )) is dense in Aff(T (An )). Theorem 5. Let (Tn , h n ) and An be as in Proposition 1. Then Inn(An ) is a simple topological group and Aut(An ) fits into the following exact sequence: {1} → Inn(An ) → Aut(An ) → Aut Λ (K (A))+,1 → {1}. Proof. By Proposition 1 and Theorem 4.6 of [14], An is a separable, simple, unital, tracially AF algebra satisfying the UCT. Hence, by the results of Lin in [10] the above sequence is exact. By the results of [18], we have that K 0 (An ) is isomorphic to Z ⊕ G in which the isomorphism sends [1 An ] to (1, 0). Hence, H[1 An ] (K 0 (An ), K 1 (An )) = K 1 (An ). Therefore, Inn(An ) = Inn0 (An ) by Corollary 1. By Corollary 2.5 of [6], Inn0 (An ) is simple. In the following example, we will consider a 2-dimensional Furstenberg transformation. In particular, we would like to give an explicit description of the group Aut(K (A2 ))+,1 . Example 1. Let A be the transformation group C ∗ -algebra associated to a minimal Furstenberg transformation (T2 , h 2 ) that has a unique h 2 -invariant probability measure. Set d12 = d in Definition 3. Then Phillips in [16] (see Example 4.9) showed that K 0 (A) ∼ = Z3 , K 1 (A) ∼ = Z3 ⊕ Z/dZ, and K 0 (A)+ can be identified with (m 1 , m 2 , m 3 ) ∈ Z3 : m 1 + m 2 θ > 0 or m 1 = m 2 = m 3 = 0 . Let G to be the subgroup of GL3 (Z) generated by ⎛ ⎞ ⎛ ⎞ 100 10 0 ⎝ 0 1 0 ⎠ and ⎝ 0 1 0 ⎠ . 011 0 1 −1 By a simple computation, one can show that Aut(K 0 (A))+,1 is isomorphic to G. Since A satisfies the UCT, as noted in [5] pp. 375, Aut Λ (K (A)) can be identified with the groups of units in the ring KL(A, A). Note the KL(A, A) is naturally isomorphic to KK(A, A) since the K -groups of A are finitely generated. Using the fact that the UCT of Rosenberg and Schochet splits, we have that KK(A, A) ∼ = Hom(K ∗ (A), K ∗ (A)) ⊕ Ext 1Z (K ∗ (A), K ∗+1 (A)). The multiplication is induced by the usual action of Hom on Ext. The product of any two elements of Ext is zero. Hence, Aut Λ (K (A)) ∼ = Aut(K ∗ (A)) × Ext 1Z (K 1 (A), K 0 (A)). Therefore, we have a bijection Aut Λ (K (A))+,1 ∼ = Aut(K 0 (A))+,1 × Aut(K 1 (A)) × Ext 1Z (K 1 (A), K 0 (A)) ∼ = G × Aut(Z3 ⊕ Z/dZ) × (Z/dZ)3 . The product of two elements is given by (α 0 , α 1 , x) ◦ (β 0 , β 1 , y) = (α 0 β 0 , α 1 β 1 , α 0 (y) + kβ 1 x), where kβ 1 is in Z/dZ such that the projection of β 1 (0, 1) onto the Z/dZ is kβ 1 .
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Acknowledgements. The first author would like to thank George Elliott and the Fields Institute for their hospitality while he was a post doctoral fellow at the Fields Institute (December 2005 to July 2006), where part of this research was completed. The second author would like to thank Frédéric Latrémolière for many stimulating conversations. The authors are grateful to the referee for a careful reading of the paper and useful suggestions.
References 1. Blackadar, B.: A stable cancellation theorem for simple C ∗ -algebras. Proc. London Math. Soc. (3), 47, 303–305 (1983) 2. Blackadar, B.: Comparison theory for simple C ∗ -algebras. In: Operator algebras and applications, Vol. 1. Vol. 135 of London Math. Soc. Lecture Note Ser., Cambridge: Cambridge Univ. Press, 1988, pp. 21–54 3. Cuntz, J.: K -theory for certain C ∗ -algebras. Ann. Math. (2), 113, 181–197 (1981) 4. Dadarlat, M., Gong, G.: A classification result for approximately homogeneous C ∗ -algebras of real rank zero. Geom. Funct. Anal. 7, 646–711 (1997) 5. Dadarlat, M., Loring, T.A.: A universal multicoefficient theorem for the Kasparov groups. Duke Math. J. 84, 355–377 (1996) 6. Elliott, G.A., Rørdam, M.: The automorphism group of the irrational rotation C ∗ -algebra. Commun. Math. Phys. 155, 3–26 (1993) 7. Exel, R.: Rotation numbers for automorphisms of C ∗ algebras. Pacific J. Math. 127, 31–89 (1987) 8. Furstenberg, H.: Strict ergodicity and transformation of the torus. Amer. J. Math. 83, 573–601 (1961) 9. Lin, H.: An introduction to the classification of amenable C ∗ -algebras. River Edge, NJ: World Scientific Publishing Co. Inc., 2001 10. Lin, H.: Classification of simple C ∗ -algebras of tracial topological rank zero. Duke Math. J. 125, 91–119 (2004) 11. Lin, H.: A separable Brown-Douglas-Fillmore theorem and weak stability. Trans. Amer. Math. Soc. 356, 2889–2925 (2004) (electronic) 12. Lin, H.: Simple nuclear C ∗ -algebras of tracial topological rank one. http://arxiv.org/list/math.OA/ 0401240, 2004 13. Lin, H.: Unitary equivalences for essential extensions of C ∗ -algebras. http://arxiv.org/list/math.OA/ 0403236, 2004 14. Lin, H., Phillips, N.C.: Crossed products by minimal homeomorphisms. http://arxiv.org/list/math.OA/ 0408291, 2004 15. Ng, P.W., Ruiz, E.: The structure of the unitary groups of certain simple C ∗ -algebras. Preprint, 2007. To appear in Houston J. Math. 16. Phillips, N.C.: Cancellation and stable rank for direct limits of recursive subhomogeneous algebras. Trans. Amer. Math. Soc. 359, 4625–4652 (2007) (electronic) 17. Pimsner, M., Voiculescu, D.: Exact sequences for K -groups and Ext-groups of certain cross-product C ∗ -algebras. J. Operator Theory 4, 93–118 (1980) 18. Reihani, K., Milnes, P.: C ∗ -algebras from anzai flows and their K -groups. http://arxiv.org/list/math.OA/ 0311425, 2003 19. Rieffel, M.A.: The cancellation theorem for projective modules over irrational rotation C ∗ -algebras. Proc. London Math. Soc. (3), 47, 285–302 (1983) 20. Rieffel, M.A.: Dimension and stable rank in k-theory of C ∗ -algebras. Proc. London Math. Soc. (3), 46, 301–333 (1983) 21. Rieffel, M.A.: The homotopy groups of the unitary groups of non-commutative tori. J. Operator Theory 17, 237–254 (1987) 22. Rørdam, M.: Classification of certain infinite simple C ∗ -algebras. J. Funct. Anal. 131, 415–458 (1995) 23. Rørdam, M., Størmer, E.: Classification of nuclear C ∗ -algebras. Entropy in operator algebras. In: Vol. 126 of Encyclopaedia of Mathematical Sciences, Berlin: Springer-Verlag, 2002 24. Rosenberg, J., Schochet, C.: The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K -functor. Duke Math. J. 55, 431–474 (1987) Communicated by Y. Kawahigashi
Commun. Math. Phys. 280, 445–462 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0476-7
Communications in
Mathematical Physics
Submean Variance Bound for Effective Resistance of Random Electric Networks Itai Benjamini1 , Raphaël Rossignol2, 1 The Weizmann Institute, Rehovot 76100, Israël 2 Université de Neuchâtel, Institut de Mathématiques, 11 rue Emile Argand, Case postale 158,
2009 Neuchâtel, Suisse. E-mail: [email protected] Received: 13 April 2007 / Accepted: 12 October 2007 Published online: 4 April 2008 – © Springer-Verlag 2008
Abstract: We study a model of random electric networks with Bernoulli resistances. In the case of the lattice Z2 , we show that the point-to-point effective resistance between 0 2 and a vertex v has a variance of order at most (log |v|) 3 , whereas its expected value is of order log |v|, when v goes to infinity. When d = 2, expectation and variance are of the same order. Similar results are obtained in the context of p- resistance. The proofs rely on a modified Poincaré inequality due to Falik and Samorodnitsky [7]. 1. Introduction The main goal of this paper is to study the effective resistance between two finite sets of vertices in a random electric network with i.i.d resistances. The infinite grid Zd will be the essential graph that we will focus on. Let us first briefly describe our notation for a deterministic electrical network (for more background, see Doyle and Snell [6], Peres [15], Lyons and Peres [12] and Soardi [17]). Let G = (V, E) be an unoriented locally finite graph with an at most countable set of vertices V and a set of edges E (we allow multiple edges between two vertices). Let r = (re )e∈E be a collection of positive real numbers, which are called resistances. To each edge e, one may associate two oriented − → edges, and we shall denote by E the set of all these oriented edges. Let A and Z be two finite, disjoint, non-empty sets of vertices of G: A will denote the source of the network, − → and Z the sink. A function θ on E is called a flow from A to Z with strength θ if it is → → antisymmetric, i.e θ− x y = −θ− yx , if it satisfies the node law at each vertex x of V \(A ∪ Z ):
→ θ− x y = 0,
y∼x Raphaël Rossignol was supported by the Swiss National Science Foundation grants 200021-1036251/1 and 200020-112316/1.
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and if the “flow in” at A and the “flow out” at Z equal θ : − − θ (→ ay) = θ (→ yz). θ = a∈A y∼a y∈ A
z∈Z y∼z y∈ A
In this definition, it is assumed that all the vertices in A are considered as a single one, as if they were linked by a wire with null resistance (and the same is true for Z ). The effective resistance Rr (A ↔ Z ) may be defined in different ways, the following is the most appropriate for us: re θ (e)2 , (1) Rr (A ↔ Z ) = inf θ=1
e∈E
where the infimum is taken over all flows θ from A to Z with strength 1. This infimum is always attained at what is called the unit minimal (or wired) current (see [17] Theorem 3.25, p. 40). A current is a flow which satisfies, in addition to the node law, Kirchhoff’s loop law (see [17], p. 12). In finite graphs, currents are unique whereas in infinite graphs, there may exist more than one current. But in Zd , for instance, between two finite sets A and Z , it is known to be attained uniquely when the resistances are bounded away from 0 and infinity (see Lyons and Peres [12], p. 82 or Soardi [17], p. 39–43). Electric networks have been thoroughly studied by probabilists since there is a correspondence between electrical networks on a given graph and reversible Markov chains on the same graph. Let us introduce randomness on the electrical network itself by choosing the resistances independently and identically distributed. That is to say, let ν be a probability measure on R+ , and equip R+E with the tensor product ν ⊗E . When the resistances are bounded away from 0 and ∞, it is easy to see that the mean of the effective resistance is of the same order of that in the network where all resistances are equal to 1. In fact, different realizations of this network are “roughly equivalent” (see Lyons and Peres [12], p. 42), and for example, the associated random walks are of the same type. Related results are those of Berger [5], p. 550 and Pemantle and Peres [14], which give respectively sufficient conditions for almost sure recurrence of the network and a necessary and sufficient condition for almost sure transience. In this paper, we are mainly concerned with the typical fluctuations of the function r → Rr (A ↔ Z ) around its mean when A and Z are “far apart”. For simplicity, we choose to focus on the variance of the effective resistance. Typically, we will take A and Z reduced to two vertices far apart: A = {a} and Z = {z}, and we shall note Rr (a ↔ z) instead of Rr ({a} ↔ {z}). Following the terminology used in First Passage Percolation (see [10]), we shall call this the point-to-point effective resistance from a to z. In this paper, we prove that the type of fluctuations of the point-to- point effective resistance on Zd is qualitatively different when d = 2 and d = 2. Indeed, when d = 2, we will see, quite easily, that these fluctuations are of the same order as its mean. On the other hand, when G = Z2 , and the resistances are bounded away from 0 and ∞, it is easy to show that the mean of Rr (0 ↔ v) is of order log |v|, where |v| stands for the l 1 -norm of the vertex v (see Sect. 3). The main result of this paper is the following variance bound on Z2 when the resistances are distributed according to a Bernoulli distribution bounded away from 0. Theorem 1.1. Suppose that ν = 21 δa + 21 δb , with 0 < a ≤ b < +∞. Let E be the set of edges in Z2 , and define µ = ν E . Then, as v goes to infinity: 2 V arµ (Rr (0 ↔ v)) = O (log |v|) 3 .
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Here as in the rest of the article, when f and g are two functions on Zd , we use the notation “ f (v) = O(g(v)) as v goes to infinity” to mean there is a positive constant C such that, for v large enough, f (v) ≤ Cg(v). We shall also use the notation “ f (v) = (g(v))” to mean “ f (v) = O(g(v)) and g(v) = O( f (v))”. The paper is organised as follows. In Sect. 2 we introduce the main tool of this paper, which is a modified Poincaré inequality due to Falik and Samorodnitsky [7]. A first result is given in Proposition 2.2, which announces our main result, on Z2 . Section 3 is devoted to the analysis of Z2 : we prove our main result, Theorem 1.1 and compare it to the simpler case of Zd , for d = 2. The choice of the Bernoulli setting has been done for the sake of simplicity, but it is possible to extend our variance bound to other distributions, and even to obtain the corresponding exponential concentration inequalities. This is developped in Sect. 4. In Sect. 5, we make some remarks and conjectures on the a priori simpler case of the left-right resitance on the n × n grid. Finally, Sect. 6 is devoted to an extension of Theorem 1.1 to the non-linear setting of p-networks.
2. A General Result in a Bernoulli Setting In this section, we suppose that ν is the Bernoulli probability measure ν = 21 δa + 21 δb , with 0 < a ≤ b < +∞ and that for each collection of resistances r in = {a, b} E , there exists a unique current flow between two finite sets of vertices of the graph G. We want to bound from above the variance of the effective resistance. Let us denote: f (r ) = Rr (A ↔ Z ). A first idea is to use the Poincaré inequality, which in this setting is equivalent to the Efron-Stein inequality (see e.g. Steele [19] or Ané et al. [1]): V ar ( f ) ≤
e f 22 ,
e∈E
where e is the following discrete gradient: 1 e f (r ) = ( f (r ) − f (σe r )), 2 r if e = e . (σe r )e = e b + a − re if e = e Let θr be a flow attaining the minimum in the definition of Rr (a ↔ z). Using the definition of the effective resistance (1), f (σe r ) − f (r ) ≤
(σe r )e θr (e )2 −
e ∈E
re θr (e )2 ,
e ∈E
f (σe r ) − f (r ) ≤ (b − a)θr (e) . 2
(2)
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For any real number h, we denote by h + the number max{h, 0},
e f 22 =
e∈E
1 E(( f (σe r ) − f (r ))2+ ), 2 e∈E
e f 22 ≤
e∈E
(b − a)2 E( θr (e)4 ). 2
(3)
e∈E
It is quite possible that this last bound is sharp in numerous settings of interest, including Z2 (see Sect. 3), but in general we do not know how to evaluate the right-hand side of inequality (3). We are just able to bound it from above using the fact that when θ is a unit current flow, |θ (e)| ≤ 1 for every edge e. This last fact is intuitive, but for a formal proof, one can see Lyons and Peres [12], p. 49–50. Therefore
e f 22 ≤
e∈E
(b − a)2 re θr (e)2 ), E( 2a e∈E
e f 22 ≤
(b
− a)2 2a
e∈E
E( f ).
(4)
We have shown that the variance of f is at most of the order of its mean. It is possible to improve on this, under some suitable assumption, by using the following inequality, due to Falik and Samorodnitsky [7]: V ar ( f ) log
V ar ( f ) e f 22 . ≤2 2 f e e∈E 1 e∈E
(5)
In order to state it as a bound on the variance of f , and to avoid repetitions, we present it in a slightly different way: Lemma 2.1. Falik and Samorodnitsky. Let f belong to L 1 ({a, b} E ). Suppose that E1 ( f ) and E2 ( f ) are two real numbers such that: E2 ( f ) ≥
e f 22 ,
e∈E
E1 ( f ) ≥
e f 21 ,
e∈E
and E2 ( f ) ≥ e. E1 ( f ) Then, V ar ( f ) ≤ 2
E2 ( f ) log
E2 ( f ) E (f) E1 ( f ) log E2 ( f ) 1
.
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Proof. Inequality (5) is proved by Falik and Samorodnitsky only for a finite set E, but it extends straightforwardly to a countable set E, for functions in L 1 ({a, b} E ). Therefore, we have: V ar ( f ) ≤ 2E2 ( f ). E1 ( f )
V ar ( f ) log
(6)
Now, consider the following disjunction: • either V ar ( f ) ≤ E2E(2f()f ) , • or V ar ( f ) ≥
log E ( f ) 1 E2 ( f ) E2 ( f ) , and log E ( f )
plugging this inequality into (6) gives us:
1
V ar ( f ) ≤ 2
E2 ( f ) log
E2 ( f ) E (f) E1 ( f ) log E2 ( f )
.
1
In any case, since E2 ( f )/E1 ( f ) ≥ e, the second possibility is weaker than the first one, and we get: V ar ( f ) ≤ 2
E2 ( f ) log
E2 ( f ) E (f) E1 ( f ) log E2 ( f )
.
1
This inequality is very much in the spirit of an inequality by Talagrand [20] and could be called a modified Poincaré inequality (see also [2,3] and [16] for more information on such inequalities). The idea to use such a type of inequalities in order to improve variance bounds is due to Benjamini, Kalai and Schramm [4] in the context of First Passage Percolation. In our setting of random electric networks, it allows us to show that as soon as the expected resistance is large but the minimal energy flow via all but few resistances is small, then the variance of the resistance is small compared to the expected resistance. This statement is reflected in the following proposition, which is an introduction to the case of Z2 in Sect. 3. Proposition 2.2. Let G = (V, E) be an unoriented graph with an at most countable set of vertices V , a set of edges E. Let A and Z be two disjoint non-empty subsets of V . Let a and b be two positive real numbers, and (re )e∈E be i.i.d resistances with common law 1 1 c 2 δa + 2 δb . Let E m be any subset of E such that E m is finite. Define: αm = sup E(re θr2 ), e∈E m
βm =
|E mc | , E(Rr (A ↔ Z ))
and εm =
b−a a
2 αm + (b − a)2 βm .
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Suppose that εm < 1. Then, V ar (Rr (A ↔ Z )) ≤ 2K
where K = sup
(b−a)2 2a , e
E( f ) log
K εm log
, K εm
.
Proof. We want to use Lemma 2.1. Define:
Let us evaluate the terms inequality (4) that:
f (r ) = Rr (a ↔ z). 2 2 e∈E e f 1 and e∈E e f 2 . We have already seen in
e f 22 ≤
e∈E
(b − a)2 E( f ), 2a
and so, define:
where K = sup
(b−a)2 2a
e f 21 =
e∈E
E2 ( f ) = K E( f ),
, e . Besides,
e f 21 +
e f 21 ,
c e∈E m
e∈E m
b−a e f 1 sup E(re θr2 ) + |E mc | sup e f 21 , a e∈E m e∈E e∈E b−a e f 1 + sup e f 21 βm E( f ). αm = a e∈E ≤
e∈E
Recall that when θ is a unit current flow, |θ (e)| ≤ 1 for every edge e. Therefore, sup e f 1 ≤ (b − a).
e∈E
Also, using inequality (2), e f 1 = E(( f (σe r ) − f (r ))+ ), e∈E
e∈E
≤ (b − a)E(
θr (e)2 ),
e∈E
b−a E( f ). ≤ a Therefore, e∈E
e f 21 ≤ E( f )εm .
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Define E1 ( f ) = E( f )εm . The assumption εm < 1 ensures that E2 ( f )/E1 ( f ) ≥ e. We conclude by applying Lemma 2.1. Unfortunately, it is not very easy to bound the terms αm and βm in an efficient way, essentially because in a random setting, we have no good bound on the amount of current through a particular edge. For example, in Sect. 3, we will have to resort to an averaging trick, and we shall not be able to use directly Proposition 2.2. Nevertheless, for some interesting graphs such as trees, but also the lattices Zd , exact calculations are available when all resistances are equal. Therefore, Proposition 2.2 would become more helpful if one could prove the following stability result, which we deliberately state in an informal way. Question 2.3. Assume the flow on the fixed resistance 1 environment on a graph satisfies the condition that, except for a small set of edges, only o(1) flow goes via all the other edges, then is the same true for a perturbed environment? 3. The Zd Case It is natural to inspect the Bernoulli setting on the most studied electrical networks, which are Zd , d ≥ 1. Here, we focus on the point- to-point resistance between the origin and a vertex v when v goes to infinity (+∞ when d = 1). Let us denote it as f v : f v (r ) = Rr (0 ↔ v). 3.1. The Zd case for d = 2. When d = 2, one can see easily that the variance of f v is of the same order as its mean value (when b > a). Indeed, when d = 1, f n is just na + (b − a)Bn , where Bn is a random variable of binomial distribution with parameters n and 1/2. When d ≥ 3, remark first that, denoting a = (a, a, . . .), aR1 (0 ↔ v) ≤ Rr (0 ↔ v) ≤ bR1 (0 ↔ v).
(7)
Therefore, the mean of f v is of the same order (up to a multiplicative constant) as in the network where all resistances equal 1. Thus, when d ≥ 3, the mean of f v is of order (1) (see Lyons and Peres [12], p. 39–40). The variance of f v is also of order (1) when b > a, as follows from the following simple lemma. Lemma 3.1. Let G = (V, E) be a unoriented connected graph, s ∈ V a vertex with finite degree D and Z be a finite subset of V such that s ∈ Z . If the resistances on E are independently distributed according to a symmetric Bernoulli law on {a, b}, with b ≥ a > 0, V ar (Rr (s ↔ Z )) ≥ C(b − a)2 , where C is a positive constant depending only on D.
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Proof. Denote by D = e1 , . . . , e D the D edges incident to s. For any r in R+E , denote f (r ) = Rr (s ↔ Z ), and let (b(D) , r −D ) be the the set of resistances obtained from r by switching all resistances on D to b. One has:
V ar ( f ) ≥ E(( f − f dre1 . . . dre D )2 ),
1 (D) −D ≥ D E(( f (b , r ) − f dre1 . . . dre D )2 ), 2 2 1 ≥ E(( f (b(D) , r −D ) − ( f (a (D) , r −D ))2 ), 2D 2 1 2 ≥ (b − a) E(( θ(b(D) ,r −D ) (e)2 )2 ), 2D e∈D 2 1 1 ≥ (b − a)2 2 . 2D D 3.2. The case of Z2 : some heuristics. Now, let us examine the case of Z2 . When the resistances are bounded away from 0 and infinity, f v , and therefore its expectation, is of order (log |v|). Indeed, Eq. (7) implies that it is of the same order as the resistance on the network where all resistances equal 1. This more simple resistance can be explicitly computed using Fourier transform on the lattice Z2 (see Soardi [17], p. 104–107). In a more simple way, it can be easily bounded from below by using Nash-Williams inequality, and from above by embedding a suitable tree in Z2 (see Doyle and Snell [6], p. 85, or alternatively Lyons and Peres [12], p. 39–40). A more complicated question to address is the existence of a precise limit of the ratio E( f v )/ log |v|. This would lead to an analog of the “time constant” arising in the context of First Passage Percolation (see Kesten [10]). Closely related questions are the existence of an asymptotic shape and, if it exists, whether it is an euclidean ball or not. We believe that the time constant and the asymptotic shape exist. Conjecture 3.2. Define Bt = {v ∈ Z2 s.t. Rr (0 ↔ v) ≤ t}. There exists a non-empty, compact subset of R2 , B0 such that, for every positive number ε, (1 − ε)B0 ⊂
1 Bt ⊂ (1 + ε)B0 . log t
What about the order of the variance of f v , when v goes to infinity? Reasonably, it should be of order (1). Since we did not manage to prove this, we state it as a conjecture:
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Conjecture 3.3. Suppose that ν = 21 δa + 21 δb , with 0 < a ≤ b < +∞. Let E be the set of edges in Z2 , and define µ = ν E . Then, as v tends to infinity: V arµ (Rr (0 ↔ v)) = (1). A first intuitive support to this conjecture comes from inequality (3). It is quite possible that it gives a bound of order O(1). Indeed, this would be the case if the current in the perturbed environment remained “close” (for example at a l 4 -distance of order 1) to the current in the uniform network (with all resistances equal to 1). A second support to this conjecture comes from the analysis of the graph G n = (Vn , E n ). This one arises when one applies in a classical way the Nash-Williams inequality to get a lower bound on the resistance between the origin and the border of the box {−n, . . . , n} × {−n, . . . , n}. The set of vertices Vn is just {0, . . . , n}. For i in {0, . . . , n − 1}, draw 2i + 1 parallel edges between i and i + 1, and call them ei,1 , . . . , ei,2i+1 . This is a Parallel-Series electric network, and the effective resistance is easy to compute: Rr (0 ↔ n) =
n−1 i=0
1 2i+1
1 k=1 ri,k
.
One can show the following result. Proposition 3.4. If the resistances on G n are independently distributed according to a symmetric Bernoulli law on {a, b}, with b ≥ a > 0, E(Rr (0 ↔ n)) = (log n), and V ar (Rr (0 ↔ n)) = (1). Proof. To shorten the notations, we treat the case a = 1/2 and b = 1. For any r in {a, b} E , denote f (r ) = Rr (0 ↔ n). The estimate on the mean is obvious. The estimate on the variance is easy too. First note that ⎛ ⎞ n−1 1 V ar ⎝ 2i+1 1 ⎠ . V ar ( f ) = k=1 ri,k
i=0
Denote by Yi the random variable: 1 Yi = 2i+1
1 k=1 ri,k
.
Remark that 2i+1
1
r k=1 i,k
= 2i + 1 + B2i+1 ,
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where B2i+1 has a binomial distribution with parameters 2i + 1 and 1/2. Therefore, denoting Ni =
B2i+1 −
2i+1 2
2i+1 4
,
when i tends to infinity, Ni converges weakly to a standard Gaussian variable, and Yi =
3 (2i+1) 2
1 . + 2i+1 N i 4
Therefore, ⎛ ⎡ √ 1 1 ⎣3 2i + 1 ⎝ Yi = 3/2 9(i + 1/2) 1 + √Ni
⎞⎤ − 1⎠⎦ +
3 2i+1
1 . 3(i + 1/2)
Define ⎛ √ Z i = 3 2i + 1 ⎝
⎞ 1 1+
√Ni 3 2i+1
− 1⎠ .
Since the sequence (Ni ) is weakly convergent, it is bounded in probability. Hence, using that: 1 − 1 + x = O(x 2 ), 1+x √ as x goes to zero, we deduce that 2i + 1(Z i + Ni ) is bounded in probability, and therefore Z i converges in distribution to a standard Gaussian random variable. Using the concentration properties of the binomial distribution, it is easy to show that Z i and Z i2 are asymptotically uniformly integrable. This implies that the variance of Z i tends to 1 as i tends to infinity. Thus, V ar (Yi ) =
1 1 V ar (Z i ) = (1 + o(1)), 92 (i + 1/2)3 9(i + 1/2)3
and consequently, V ar ( f ) = (1). Of course, on Z2 , things are more difficult to compute.
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3.3. The case of Z2 : proof of Theorem 1.1. We shall prove below that the variance of f v 2 is of order O((log |v|) 3 ). We shall proceed very much as in [4], resorting to an averaging trick to trade the study of f v against the study of a randomized version of it. Proof of Theorem 1.1. Let m be a positive integer to be fixed later, and z a random variable, independent from the edge-resistances, and distributed according to µm , the uniform distribution on the box Bm = [0, m − 1]2 ∩ Z2 . Define f˜(z, r ) := Rr (z, v + z). ˜ := Bm × {a, b} E which is endowed with the We think of f˜ as a function on the space ⊗E probability measure µm ⊗ ν . The first thing to note is that f v and f˜ are not too far apart. To see this, we can use the following triangle inequality (see [12], Exercise 2.65, p. 67), which holds for every three vertices x, y, z in Z2 , and any r ∈ , Rr (x, z) ≤ Rr (x, y) + Rr (y, z).
(8)
Therefore, taking L 2 -norms in L 2 (µm ⊗ ν ⊗E ), and noting that |z| ≤ 2m, f v − f˜2 ≤ Rr (0, z)2 + Rr (z, z + v)2 , ≤ C log m, where C is a universal constant. Noting that f˜ has the same expectation as f v , we get thus: f v − E( f v )2 ≤ f v − f˜2 + f˜ − E( f˜)2 , ≤ C log m + f˜ − E( f˜)2 . Therefore:
V ar ( f v ) ≤ (C log m)2 + 2C log m V ar ( f˜) + V ar ( f˜).
(9)
Now, we want to bound the variance of f˜ from above. Define:
˜ Eµ ( f ) = f (z, r ) dµm (z),
f (z, r ) dν ⊗E (r ), Eν ( f˜) = V arµ ( f˜) = Eµ ( f (z, r )2 ) − Eµ ( f (z, r ))2 , V arµ ( f˜) = Eν ( f (z, r )2 ) − Eν ( f (z, r ))2 . Then, we split the variance of f˜ into two parts: the one due to z and the other due to r , V ar ( f˜) = Eν (V arµ ( f˜)) + V arν (Eµ ( f˜)) .
(10)
Thanks to the triangle inequality (8), Eν (V arµ ( f˜)) ≤ (C log m)2 .
(11)
To bound the last term of the sum in (10), we apply Lemma 2.1 to Eµ ( f˜). Remark that, thanks to Jensen’s inequality, e Eµ ( f˜) ≤ e f˜ , 1
1
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where the first L 1 -norm integrates against ν ⊗E , and the second one integrates against µm ⊗ ν ⊗E . Also, e Eµ ( f˜) ≤ e f˜ . 2
2
Let us denote by θrz the unit current flow from z to z + v, when the resistances are r . Using inequality (2), and the translation invariance of this setting, we get, for every edge e: e f˜ ≤ (b − a)E(θrz (e)2 ), 1
= (b − a)E(θr0 (e − z)2 ), b−a = E(θr (e − z 0 )2 ). (m + 1)2 z 0 ∈Bm
Now we claim that:
∀e ∈ E,
E(θr (e − z 0 )2 ) ≤
z 0 ∈Bm
5b(m + 1) . a
(12)
Assuming this claim, we have: 1 . sup e f˜ = O 1 m e∈E
(13)
To see that claim (12) is true, let e− be the left-most or lower end-point of e and remark that the set of edges described by e − z is included in the set of edges of the box Bme := e− + Bm+1 . Therefore,
E(θr (e − z 0 )2 ) ≤
z 0 ∈Bm
E(θr (e )2 ).
e ⊂Bme
Let ∂ Bme be the (inner) border of the box Bme : ∂ Bm = {0} × [0, m] ∪ {m} × [0, m] ∪ [0, m] × {0} ∪ [0, m] × {m}, ∂ Bme = e− + ∂ Bm . First, suppose that neither 0 nor v belongs to Bme . We define a flow η from 0 to v such that: η(e ) = θr (e ) if e ⊂ Bme , η(e ) = 0 if e ⊂ Bme and e ⊂ ∂ Bme . These conditions do not suffice to determine uniquely the flow η, but one can then choose the flow on ∂ Bme that minimizes the energy e ∈∂ Bme re η(e )2 . This is the current flow on ∂ Bme when the flow entering and going outside ∂ Bme is fixed by θr . For a formal proof
Variance Bounds and Random Networks
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of the existence of such a flow, see Soardi [17] Theorem 2.2, p. 22. This flow has a strength less than 1. Therefore, the flow through each edge of ∂ Bme is less than 1, and: re θr (e)2 = re θr (e)2 − re θr (e)2 , e ⊂Bme
e⊂ Bme
e∈E
≤
re θr (e)2 ,
e⊂ Bme
e∈E
=
re η(e)2 − re η(e)2 ,
e⊂∂ Bme
≤ 4b(m + 1). Therefore,
θr (e)2 ≤
e ⊂Bme
4b(m + 1) . a
Now, suppose that a or v belongs to Bme . Let us say v belongs to Bme , the other situation being symmetrical. We define a flow η from 0 to v as before, except that instead of assigning 0 to each value inside Bme , we keep a path from ∂ Bme to v on which the flow is assigned to 1, directed towards v. The same considerations as before lead to
θr (e)2 ≤
e ⊂Bme
5b(m + 1) , a
and claim (12) is proved. Finally notice that, using inequality (2): b−a E( f˜), e f˜ ≤ 1 a e∈E 2 (b − a)2 E( f˜). e f˜ ≤ 2 2a e∈E
Recall that E( f˜) = E( f v ) = (log |v|). Therefore, there exist constants K and K such that: 2 e f˜ ≤ e f˜ . sup e f˜ , e∈E
1
1 e∈E
e∈E
≤K
log |v| , m
and: 2 e f˜ ≤ K log |v|. e∈E
2
1
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Denoting E1 ( f˜) = K logm|v| and E2 ( f˜) = K log |v|, the hypotheses of Lemma 2.1 are fulfilled, at least for m larger than e KK . Assume that m is a function of |v| which goes to infinity when |v| goes to infinity. Lemma 2.1 together with inequalities (10) and (11) gives us: log |v| + (log m)2 . V ar ( f˜) ≤ O log m 1
Thus, choose m the greatest integer such that log m ≤ (log |v|) 3 and the result follows from inequality (9). 4. Other Distributions and Exponential Concentration Inequalities Using a forthcoming paper of Benaim and Rossignol [2], one can derive an exponential concentration inequality on the effective resistance for various distributions, Bernoulli or continuous ones. The only estimates needed to use the results in [2] are the main estimates in the proof of Theorem 1.1, and they can be obtained easily when the resistances are bounded away from 0 and infinity. For example, suppose that ν is bounded away from 0 and infinity, and that it is either a Bernoulli distribution or absolutely continuous with respect to the Lebesgue measure with a density which is bounded away from 0 on its support, then, there exist two positive constants C1 and C2 such that: 1 ∀t > 0, P |Rr (0 ↔ v) − E(Rr (0 ↔ v))| > t (log |v|) 3 ≤ C1 e−C2 t . Whether this result may be extended to distributions which are not bounded away from 0 is still uncertain. 5. Left-Right Resistance on the n × n-Grid This section is purely prospective, and focuses on another interesting case of study: the left-right resistance on the n × n-grid on Z2 . The graph is Z2 ∩ [0, n] × [0, n], the source is An = {0} × [0, n] and the sink is Z n = {n} × [0, n]. When all resistances are equal to 1, one may easily see that R1 (An ↔ Z n ) equals n/(n + 1) and therefore tends to 1, as n tends to infinity. In a random setting, where all resistances are independently and identically distributed with respect to ν, this implies that:
lim sup E(Rr (An ↔ Z n )) ≤ x dν(x), n→∞
where the inequality follows from: E(inf f i ) ≤ inf E( f i ). i∈I
i∈I
Recall the dual characterisation of the resistance: 1 1 = inf (F(e+ ) − F(e− ))2 , F Rr (An ↔ Z n ) r e e
Variance Bounds and Random Networks
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where the infimum is taken over all functions F on the vertices which equal 1 on Z n and 0 on An . This implies:
1 1 lim sup E ≤ dν(x). R (A ↔ Z ) x n→∞ r n n Using E(1/R) ≥ 1/E(R), we finally obtain:
1 1 x
dν(x)
≤ lim inf E(Rr (An ↔ Z n )) ≤ lim sup E(Rr (An ↔ Z n )) ≤ n→∞
x dν(x).
n→∞
See also Theorem 2 in Hammersley [8]. In fact, it is natural to conjecture that the limit of Rr (An ↔ Z n ) as n tends to infinity exists almost surely and is constant (see Hammersley [8], p. 350). This is indeed the case, at least under an ellipticity condition, as follows from the work by Künnemann [11] (see Theorem 7.4, p. 230 in the book by Jikov et al. [9]). This work relies on homogenization techniques introduced by Papanicolaou and Varadhan [13]. Notice that in the book by Jikov et al., the law of large numbers is even stated for conductances which are allowed to take the value 0 (see [9] Chapters 8 and 9, notably Eq. (9.16), p. 303 and Theorem 9.6, p. 314). Returning to resistances with finite mean, the variance in this setting is obviously less than 1, but inequality (3) suggests that it is much lower. Conjecture 5.1. If the resistances are bounded away from 0 and infinity, 1 V ar (Rr (An ↔ Z n )) = O . n2 A lower bound of this order has been proven by Wehr [21] for some absolutely continuous distributions ν under the assumption that the convergence of the effective resistance holds almost surely. Actually, Wehr’s result is stated for effective conductivity, that is the inverse of effective resistance, but in this context, they both are of order (1), and the lower bound of Wehr implies a lower bound of the same order on the variance of the resistance. A very appealing question is therefore: Question 5.2. Defining Rn = Rr (An ↔ Z n ), does n(Rn − E(Rn )) converge in distribution as n tends to infinity? What is the limit law? 6. Submean Variance Bound for p-Resistance In the analysis presented so far, the probabilistic interpretation of electrical networks has played no role. It is therefore tempting to extend our work to the setting of p-networks (see for instance Soardi [17], p.176-178). As before, consider an unoriented, at most countable and locally finite graph G = (V, E). Let r = (re )e∈E be a collection of resistances. For any p > 1, we define the p-resistance between two vertices x and y as p re |θ (e)| p , (14) Rr (x ↔ y) = inf θ=1
e∈E
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where the infimum is taken over all flows θ from x to y with strength 1. It is known that the p-resistance from 0 to infinity on Zd , when all resistances equal 1, is finite if and only if p > d/(d − 1) (see Soardi and Yamasaki [18]). More precisely, the flow described in [18] and the usual shorting argument to lowerbound resistance from 0 to the border of the box Bn allow to obtain the following estimate of the p-resistance on Zd : ⎞ ⎛ |v| 1 p ⎠. E(Rr (0 ↔ v)) = ⎝ (2n + 1)(d−1)( p−1) k=0
Whenever this expectation tends to infinity as |v| tends to infinity, and when d ≥ 2, one may hope to obtain a similar result as in Theorem 1.1. This is indeed the case: we obtain a weaker result, but still, the variance is small compared to the mean. The proof is essentially the same as in the case where p = 2. There are two main important points to take care of. First, it remains true that for a unit flow which minimizes the p-energy e∈E re |θ (e)| p , the flow on each edge is less than 1. This follows from the same argument as in the linear case (see Lyons and Peres [12], p. 49-50). Second, it is not clear whether inequality (8) remains true or not. Nevertheless, we can easily obtain the following weaker inequality. For every three vertices x, y, z in Zd , and any r ∈ , p
p
Rr (x, z) ≤ Rr (x, y) + 2 p b|z − y|.
(15)
θ x,y
To see this, let be the unit current flow (for p-energy) from x to y and π = (u 0 = y, u 1 , . . . , u |z−y| = z) be a deterministic oriented path from y to z. Define a flow θ y,z from y to z as follows: θ y,z (e) = 0, = 1,
−→ u− θ y,z (− i u i+1 )
if e ∈ π, . if i ∈ {0, . . . , |z − y| − 1}
Now, let η x,z be the unit flow θ x,y + θ y,z , which goes from x to z. p Rr (x, z) ≤ re |η x,z (e)| p , e
= ≤
re |θ x,y (e)| p +
e∈π p Rr (x,
re |θ x,y (e) + 1| p ,
e∈π
y) + 2 b|z − y|, p
x,y
where the last inequality follows from the fact that |θe | ≤ 1 for every edge e. Inequality (15) is proved. This allows to adapt the proof of Theorem 1.1 as follows. Inequalities (9) and (11) become respectively: V ar ( f v ) ≤ (Cm)2 + 2Cm V ar ( f˜) + V ar ( f˜), and Eν (V arµ ( f˜)) = O(m 2 ). The rest of the proof is the same, and leads to: E( f ) V ar ( f˜) = O + m2 . log m We can choose, for instance m = (E( f ))1/3 , to get the following weaker analog of Theorem 1.1.
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Proposition 6.1. Suppose that ν = 21 δa + 21 δb , with 0 < a ≤ b < +∞. Let d ≥ 1 be an integer, E be the set of edges in Zd , and define µ = ν E . Then, for any real number p in ]1, +∞[: p
E(Rr (0 ↔ v)) = (ad (|v|, p)), and if d ≥ 2,
p
V arµ (Rr (0 ↔ v)) = O where
ad (|v|, p) , log ad (|v|, p)
⎧ 1−(d−1)( p−1) ⎨n ad (n, p) = log(n) ⎩ 1
if p < if p = if p >
d d−1 , d d−1 , d d−1 .
Remark also that Lemma 3.1 is easily extended to this setting, and we get therefore that, d for any p > d−1 , p
V arµ (Rr (0 ↔ v)) = (1). Acknowledgements. Itai Benjamini would like to thank Gady Kozma, Noam Berger and Oded Schramm for useful discussions. R. Rossignol would like to thank Michel Benaim for useful discussions, and Yuval Peres for useful remarks on a first version of this paper.
References 1. Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques. Paris: Société Mathématique de France, 2000 2. Benaim, M., Rossignol, R.: Exponential concentration for First Passage Percolation through modified Poincaré inequalities. http://arxiv.org/abs/math.PR/0609730; 2006, to appear in Annales de l’IHP 3. Benaim, M., Rossignol, R.: A modified Poincaré inequality and its application to first passage percolation. http://arxiv.org/abs/math.PR/0602496, 2006 4. Benjamini, I., Kalai, G., Schramm, O.: First passage percolation has sublinear distance variance. Ann. Probab. 31(4), 1970–1978 (2003) 5. Berger, N.: Transience, recurrence and critical behavior for long-range percolation. Commun. Math. Phys. 226, 531–558 (2002) 6. Doyle, P., Snell, J.: Random walks and electric networks. Washington, DC: Mathematical Association of America, 1984 7. Falik, D., Samorodnitsky, A.: Edge-isoperimetric inequalities and influences. http://arxiv.org/abs/math. CO/0512636, 2005 8. Hammersley, J.M.: Mesoadditive processes and the specific conductivity of lattices. J. Appl. Probab., Special 25A, 347–358 (1988) 9. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of differential operators and integral functionals. Berlin-Heidelberg-New York: Springer-Verlag, 1994 10. Kesten, H.: Aspects of first passage percolation. In: Ecole d’été de probabilité de Saint-Flour XIV–1984, Volume 1180 of Lecture Notes in Math., Berlin: Springer, 1986, pp. 125–264 11. Künnemann, R.: The diffusion limit for reversible jump processes on Zd with ergodic random bond conductivities. Commun. Math. Phys. 90(1), 27–68 (1983) 12. Lyons, R., Peres, Y.: Probability on trees and networks. Book in progress. Available at http://mypage.iu. edu/~rdlyons/prbtree/prbtree.html, 1997–2006 13. Papanicolaou, G., Varadhan, S.: Boundary value problems with rapidly oscillating random coefficients. In: Random fields, Vol. I, II (Esztergom, 1979), Volume 27 of Colloq. Math. Soc. János Bolyai, Amsterdam: North- Holland, 1981, pp. 835–873
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14. Pemantle, R., Peres, Y.: On which graphs are all random walks in random environments transient? Random Disc. Struct. 76, 207–211 (1996) 15. Peres, Y.: Probability on trees: an introductory climb. In: Lectures on probability theory and statistics (Saint-Flour, 1997), Volume 1717, Berlin: Springer, 1999, pp. 193–280 16. Rossignol, R.: Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Ann. Probab. 34(5), 1707–1725 (2006) 17. Soardi, P.: Potential theory on infinite networks. Number 1590 in Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1994 18. Soardi, P., Yamasaki, M.: Parabolic index and rough isometries. Hiroshima Math. J. 23, 333–342 (1993) 19. Steele, J.: An Efron-Stein inequality for nonsymmetric statistics. Ann. Stat. 14, 753–758 (1986) 20. Talagrand, M.: On Russo’s approximate zero-one law. Ann. Probab. 22, 1576–1587 (1994) 21. Wehr, J.: A lower bound on the variance of conductance in random resistor networks. J. Stat. Phys. 86(5–6), 1359–1365 (1997) Communicated by M. Aizenman
Commun. Math. Phys. 280, 463–497 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0433-5
Communications in
Mathematical Physics
Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function Jinho Baik1 , Robert Buckingham1 , Jeffery DiFranco2 1 Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA.
E-mail: [email protected]; [email protected]
2 Department of Mathematics, Seattle University, Seattle, WA, 98122, USA.
E-mail: [email protected] Received: 27 April 2007 / Accepted: 21 June 2007 Published online: 26 February 2008 – © Springer-Verlag 2008
Abstract: The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE TracyWidom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new. 1. Introduction Let F1 (x), F2 (x), and F4 (x) denote the GOE, GUE, and GSE Tracy-Widom distribution functions, respectively. They are defined as [24,25] 1 1 2 E(x) + F(x), (1) F1 (x) = F(x)E(x), F2 (x) = F(x) , F4 (x) = 2 E(x) where
F(x) = exp
−
1 2
∞
R(s)ds ,
E(x) = exp
x
−
1 2
∞
q(s)ds .
(2)
x
Here the (real) function q(x) is the solution to the Painlevé II equation q = 2q 3 + xq,
(3)
that satisfies the boundary condition q(x) ∼ Ai(x),
x → +∞.
(4)
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J. Baik, R. Buckingham, J. DiFranco
Recall [1] that the Airy function Ai(x) satisfies Ai (x) = xAi(x) and 2 3/2 1 Ai(x) ∼ √ 1/4 e− 3 x , 2 πx
x → +∞.
(5)
There is a unique global solution q(x) to Eq. (3) with the condition (4) (the HastingsMcLeod solution) [19]. The function R(s) is defined as ∞ R(x) = (q(s))2 ds. (6) x
By taking derivatives and using (3) and (4) (see, for example, (1.15) of [24] and (2.6) of [3]), the function R(x) can also be written as R(x) = (q (x))2 − x(q(x))2 − (q(x))4 . Integrating by parts, F(x) can be written as 1 ∞ 2 F(x) = exp − (s − x)(q(s)) ds , 2 x
(7)
(8)
which is commonly used in the literature. Notice that (2) involves integrals from x to positive infinity. The main results of this paper are the following representations of F(x) and E(x), which involve integrals from minus infinity to x. Theorem 1.1. For x < 0, 1 (−1)
F(x) = 21/48 e 2 ζ
x 1 3 1 2 1 e− 24 |x| 1 R(y) − y dy , exp + |x|1/16 2 −∞ 4 8y
where ζ (z) is the Riemann-zeta function, and x 1 − √1 |x|3/2 |y| 1 3 2 E(x) = 1/4 e q(y) − dy . exp 2 2 −∞ 2
(9)
(10)
Remark 1. The formula (9) also follows from the recent work [8] of Deift, Its, and Krasovsky. See Subsect. 1.2 below for further discussion. The integrals in (9) and (10) converge. Indeed, it is known that [19,12] 1 −x 73 10219 −12 q(x) = 1+ 3 − + + O(|x| ) , x → −∞. 2 8x 128x 6 1024x 9
(11)
This asymptotic behavior of q was obtained using the integrable structure of the Painlevé II equation (see, for example, [16]). The coefficients of the higher terms in the above asymptotic expansion can also be computed recursively (see for example, Theorem 1.28 of [12]). For R(x), (7) and (11) imply that x2 1 9 128 −12 R(x) = 1− 3 + − + O(x ) , x → −∞. (12) 4 2x 16x 6 62x 9 We now discuss two consequences of Theorem 1.1.
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465
1.1. Total integrals of q(x) and R(x). Comparing with (2), Theorem 1.1 is equivalent to the following. Corollary 1.2. For c < 0, ∞
1 2 1 R(y) − y + dy R(y)dy + 4 8y c −∞ 1 1 1 = − log 2 − ζ (−1) + |c|3 + log |c| 24 12 8
c
(13)
and
∞ c
√ |y| 2 3/2 1 |c| . q(y)dy + q(y) − dy = log 2 + 2 2 3 −∞
c
(14)
These formulas should be compared with the evaluation of the total integral of the Airy function [1]: ∞ Ai(y)dy = 1. (15) −∞
Recall that the Airy differential equation is the small amplitude limit of the Painlevé II equation. Unlike the Airy function, R(x) and q(x) do not decay as x → −∞, and hence we need to subtract out the diverging terms in order to make the integrals finite. 1.2. Asymptotics of Tracy-Widom distribution functions as x → −∞. Using formulas (2) and (12), Tracy and Widom computed that (see Sect. 1.D of [24]) as x → −∞, 1
e− 12 |x| F2 (x) = τ2 |x|1/8
3
1+
3 + O(|x|−6 ) 6 2 |x|3
(16)
for some undetermined constant τ2 . The constant τ2 was conjectured in the same paper [24] to be τ2 = 21/24 eζ
(−1)
.
(17)
This conjecture (17) was recently proved by Deift, Its, and Krasovsky [8]. In this paper, we present an alternative proof of (17). Moreover, we also compute the similar constants τ1 and τ4 for the GOE and GSE Tracy-Widom distribution functions. The asymptotics similar to (16) follow from (1) and (11): as x → −∞, F1 (x) = τ1
e
F4 (x) = τ4
e
1 − 24 |x|3 −
1 √ |x|3/2 3 2
1−
|x|1/16 1 − 24 |x|3 +
1 √ |x|3/2 3 2
|x|1/16
1
+ O(|x| √ 24 2|x|3/2
1+
√
1
24 2|x|3/2
Using (11) and (12), Theorem 1.1 implies the following.
−3
) ,
+ O(|x|−3 ) .
(18)
(19)
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Corollary 1.3. As x → −∞, 1
1 (−1)
F(x) = 21/48 e 2 ζ
e− 24 |x| |x|1/16
and E(x) =
1 21/4
e
−
1 √ |x|3/2 3 2
3
1−
1+
3 + O(|x|−6 ) 27 |x|3 1
+ O(|x| √ 24 2|x|3/2
−3
(20)
) .
(21)
Hence 1 (−1)
τ1 = 2−11/48 e 2 ζ
τ2 = 21/24 eζ
,
(−1)
,
1 (−1)
τ4 = 2−35/48 e 2 ζ
.
(22)
Conversely, using (11) and (12), this corollary together with (2) implies Corollary 1.2, and hence Theorem 1.1. This is one example of so-called constant problems in random matrix theory. One can ask the same question of evaluating the constant term in the asymptotic expansion in other distribution functions such as the limiting gap distribution in the bulk or in the hard edge. For the gap probability distribution in the bulk scaling limit which is given by the Fredholm determinant of the sine-kernel, Dyson [13] first conjectured the constant term for β = 2 in terms of ζ (−1) using a formula in an earlier work [27] of Widom. This conjecture was proved by Ehrhardt [14] and Krasovsky [20], independently and simultaneously. A third proof was given in [9]. The constant problem for β = 1 and β = 4 ensembles in the bulk scaling limit was recently obtained by Ehrhardt [15]. For the hard edge of the β-Laguerre ensemble associated with the weight x m e−x , the constant was obtained by Forrester [18] (Eq. (2.26a)) when m is a non-negative integer and 2/β is a positive integer. The above limiting distribution functions in random matrix theory are expressed in terms of a Fredholm determinant or an integral involving a Painlevé function. For example, the proof of [8] used the Fredholm determinant formula of the GUE TracyWidom distribution: F2 (x) = det(1 − Ax ),
(23)
where Ax is the operator on L 2 ((x, ∞)) whose kernel is A(u, v) =
Ai(u)Ai (v) − Ai (u)Ai(v) . u−v
(24)
In terms of the Fredholm determinant formula, the difficulty comes from the fact that even if we know all the eigenvalues λ j (x) of Ax , we still need to evaluate the product ∞ j=1 (1 − λ j (x)). When one uses the Painlevé function, one faces a similar difficulty of evaluating the total integral of the Painlevé function. We remark that the asymptotics as x → +∞ of F(x) and E(x) (and hence Fβ (x)) are, using (4), 4 3/2 e− 3 x 35 −3 1 − + O(x ) , 32π x 3/2 24x 3/2 2 3/2 41 e− 3 x −3 E(x) = 1 − √ 3/2 1 − + O(x ) . 48x 3/2 4 πx
F(x) = 1 −
(25) (26)
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
467
1.3. Outline of the proof. The Tracy-Widom distribution functions are the limits of a variety of objects such as the largest eigenvalue of certain ensembles of random matrices, the length of the longest increasing subsequence of a random permutation, the last passage time of a certain last passage percolation model, and the height of a certain random growth model (see, for example, the survey [21]). Dyson [13] exploited this notion of universality to solve the constant problem for the sine-kernel determinant. Namely, among the many different quantities whose limit is the sine-kernel determinant, he chose one for which the associated constant term is explicitly computable (specifically, a certain Toeplitz determinant on an arc for which the constant term had been obtained by Widom [27]), and then took the appropriate limit while checking the limit of the constant term. However, the rigorous proof of this idea was only obtained in the subsequent work of Ehrhardt [14] and Krasovsky [20]. In order to apply this idea for F2 (x), the key step is to choose the appropriate approximate ensemble. In the work of Deift, Its, and Krasovsky [8], the authors started with the Laguerre unitary ensemble and took the appropriate limit while controlling the error terms. In this paper, we use the fact that Fβ (x) is a (double-scaling) limit of a Toeplitz/Hankel determinant. Let Dn (t) denote the n × n Toeplitz determinant with symbol f (eiθ ) = e2t cos(θ) on the unit circle: π 1 2t cos θ i( j−k)θ Dn (t) = det e e dθ 2π −π 0≤ j,k≤n−1 n
2
1 2t nj=1 cos θ j iθk iθ
= e − e dθ j . (27)
e
(2π )n n! [−π,π ]n j=1
1≤k<≤n
Note that some references (e.g. [7]) define Dn as an (n + 1) × (n + 1) determinant, whereas others (e.g. [5]) use our convention. In studying the asymptotics of the length of the longest increasing subsequence in random permutations, in [2], the authors proved that when n = [2t + xt 1/3 ],
(28)
e−t Dn (t) → F2 (x).
(29)
as t → ∞, 2
The idea of the proof of (29) in [2] is as follows. The Toeplitz determinants are intimately related to orthogonal polynomials on the unit circle. Let p j (z) = κ j z j + · · · be the dθ orthonormal polynomial of degree j with respect to the weight e2t cos θ 2π :
π −π
p j (eiθ ) pk (eiθ )e2t cos θ
dθ = δ jk 2π
for j, k ≥ 0.
(30)
If κ j > 0 then p j is unique. We denote by π j (z; t) = π j (z) the monic orthogonal polynomial: pk (z) = κk π j (z). Then (see, for example, [23]) the leading coefficient κ j = κ j (t) is given by κ j (t) =
D j (t) . D j+1 (t)
(31)
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J. Baik, R. Buckingham, J. DiFranco 2
As the strong Szegö limit theorem implies that Dn (t) → et as n → ∞ for fixed t, the left-hand-side of (29) can be written as e−t Dn (t) = 2
∞ ∞ Dq (t) κq2 (t). = D (t) q+1 q=n q=n
(32)
The basic result of [2] is that κq2 (t) ∼ 1 −
R(y) , t 1/3
t → ∞, q = [2t + yt 1/3 ]
(33)
for y in a compact subset of R. (In [2], the notations v(x) = −R(x) and u(x) = −q(x) are used.) Hence formally, as t → ∞ with n = 2t + xt 1/3 , ∞ ∞
2 −t 2 R(y) 1/3 log κq (t) ∼ t log 1 − 1/3 dy ∼ log e Dn (t) = t x q=n ∞ R(y)dy = log F2 (x). (34) − x
The first step of this paper is to write, instead of (32), e−t Dn (t) = e−t 2
2
n n Dq (t) 1 2 = e−t . 2 D (t) κ (t) q=1 q−1 q=1 q−1
(35)
Here D0 (t) := 1. Then formally, we expect that as t → ∞ with n = 2t + xt 1/3 , (35) converges to an integral from −∞ to x. For this to work, we need the asymptotics of κq (t) for the whole range of q and t such that 1 ≤ q ≤ 2t + xt 1/3 as t → ∞. It turns out it is more convenient to write, for an arbitrary fixed L, n
e−t Dn (t) = e−t D L (t) 2
2
q=L+1
n Dq (t) 1 2 = e−t D L (t) . 2 Dq−1 (t) κ (t) q=L+1 q−1
(36)
We introduce another fixed large number M > 0 and write log(e
−t 2
Dn ) = −t + log(D L ) + 2
exact part
1/3 [2t−Mt −1]
q=L+1
1/3 [2t+xt ]
−2 log(κq−1 )+
Airy part
−2 log(κq−1 ) . (37)
q=[2t−Mt 1/3 ]
Painlevé part
Since L and M are arbitrary, we can compute the desired limit by computing lim
lim log(e−t Dn (t)),
L ,M→∞ t→∞
2
n = [2t + xt 1/3 ].
(38)
From (33), the Painlevé part converges to a finite integral of R(y) from y = −M to y = x as t → ∞. For the Airy part, we need the asymptotics of κq (t) for L +1 ≤ q ≤ 2t −Mt 1/3 as t → ∞ for fixed L , M > 0. The paper [2] obtains a weak one-sided bound of κq (t) for t ≤ q ≤ 2t − Mt 1/3 as t → ∞, where > 0 is small but fixed. The technical part of this paper is to compute the leading asymptotics of κq (t) in L +1 ≤ q ≤ 2t − Mt 1/3 with proper control of the errors so that the Airy part converges. The advantage of introducing
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
469
L is that we do not need small values of q, which simplifies the analysis. The calculation is carried out in Sect. 3. Finally, for the exact part, the asymptotics of D L (t) as t → ∞ are straightforward using a steepest-descent method since the size of the determinant is fixed and only the weight varies. The limit is given in terms of the Selberg integral for the L × L Gaussian unitary ensemble, which is given by a product of Gamma functions, the Barnes G-function. The asymptotics of the Barnes G-function as L → ∞ are related to the term ζ (−1) (see (48) below). The computation is carried out in Sect. 2. Now we outline the proof of the formula (10) for E(x). In the study of symmetrized random permutations it was proven in [4,5] that, in a similar double scaling limit, certain other determinants converge to F1 (x) and F4 (x). But it was observed in [4,5] that these determinants can be expressed in terms of κq (t) and πq (0; t) for the same orthonormal polynomials (30) above. Hence by using the same idea for Dn (t), we only need to keep track of πq (0; t) in the asymptotic analysis of the orthogonal polynomials. See Sect. 5 below for more details. This paper is organized as follows. In Sect. 2, the asymptotics of the exact part of (37) are computed. We compute the asymptotics of the Airy part in Sect. 3. The proof of (9) for F(x) in Theorem 1.1 is then given in Sect. 4. The proof of (10) for E(x) in Theorem 1.1 is given in Sect. 5. While we were writing up this paper, Alexander Its told us that there is another way to compute the constant term for E(x) using a formula in [5]. This idea will be explored in a later publication together with Its to compute the total integrals of other Painlevé solutions, such as the Ablowitz-Segur solution. 2. The Exact Part We compute the exact part of (37). From Eq. (27), D L (t) =
1 (2π ) L L!
[−π,π ] L
e2t
L
j=1 cos θ j
L
2
iθk
dθ j .
e − eiθ
(39)
j=1
1≤k<≤L
Following the standard stationary phase method of restricting each integral to a small interval − ≤ θ ≤ and expanding eiθ and e2t cos θ in Taylor series, D L (t) is approximately 1 (2π ) L L!
[− , ] L
e2t L−t
L
2 j=1 θ j
|θk − θ |2
L
dθ j
(40)
j=1
1≤k<≤L
as t → ∞. By extending the range of integration to R L , we obtain lim D L (t) ·
t→∞
e2t L Herm D (t) (2π ) L L
−1 = 1,
(41)
where D Herm (t) L
1 = L!
[−∞,∞] L
e−t
L
2 j=1 θ j
1≤k<≤n
|θk − θ |2
L j=1
dθ j .
(42)
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J. Baik, R. Buckingham, J. DiFranco
This integral is known as a Selberg integral and is computed explicitly as (see for example, [22], Eq. (17.6.7)) (t) = D Herm L
L−1
π L/2 2 L(L−1)/2 t
L 2 /2
π L/2
q! =
2 L(L−1)/2 t L
q=0
2 /2
G(L + 1),
(43)
where G(z) denotes the Barnes G-function, or double gamma function. Some properties of the Barnes G-functions are (see, for example, [26,6]) G(z + 1) = (z)G(z),
(44)
G(1) = G(2) = G(3) = 1.
(45)
1 = 2 3 = log G 2
log G
1 1 3 log 2 − log π + ζ (−1), 24 4 2 1 1 3 log 2 + log π + ζ (−1), 24 4 2
G(n) = 1!2! · · · (n − 2)!, log G(z + 1) =
(46)
n = 2, 3, 4, . . . ,
z2 3 1 z log z − z 2 + log(2π ) − log z + ζ (−1) + O 2 4 2 12 as z → ∞.
(47)
1 z2
(48)
Therefore, L2 log(2t) lim lim log(D L ) − 2Lt − L→∞ t→∞ 2 2 L 1 3 2 + − log L − L + ζ (−1) = 0. 2 12 4
(49)
3. The Airy Part The main result of this section is Lemma 3.10 which computes lim
lim
L ,M→∞ t→∞
1/3 [2t−Mt −1]
−2 log(κq−1 ),
(50)
q=L+1
the Airy part of (37). We use the notation γ =
2t . q
(51)
It is well known that the leading coefficients κq−2 of orthonormal polynomials can be expressed in terms of the solution of a matrix Riemann-Hilbert Problem (RHP) [17]. We start with the RHP for m (5) defined in Sect. 6 (p.1156) of [2], which is obtained through
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
(a)
(b)
471
(c)
Fig. 1. Contours used in the definition of the RHP for m (5) and m (R)
a series of explicit transformations of the original RHP for orthogonal polynomials. For notational ease, we drop the tildes. Let θc be defined such that 0 < θc < π and sin2 θ2c = γ1 . For q in the regime L + 1 ≤ q ≤ 2t − Mt 1/3 − 1, we have γ > 1. Define the contours C1 = {eiθ : θc < |θ | ≤ π } and C2 = {eiθ : 0 ≤ |θ | ≤ θc } with the orientations given as in Fig. 1(a). Also define the contours Cin and Cout as in Fig. 1(a). Let (5) = C1 ∪ C2 ∪ Cin ∪ Cout . Now let m (5) (z) = m (5) (z; t, q) be the solution to the following RHP: ⎧ (5) ⎨ m is analytic in z ∈ C\ 5 , (5) (5) (52) m (z) = m − (z)v (5) (z) for z ∈ 5 , ⎩ +(5) m = I as z → ∞, where the jump matrix v (5) (z) = v (5) (z; t, q) is given by ⎧ 0 1 ⎪ ⎪ z ∈ C2 ⎪ −1 0 ⎪ ⎪ ⎪ ⎪ ⎨ 1 e−2qα z ∈ C1 v (5) (z) = 0 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 12qα 0 z ∈ Cin ∪ Cout . e 1 Here α(z) = −
γ 4
ξ
z
(53)
s+1 (s − ξ )(s − ξ )ds, s2
where ξ = eiθc and the branch is chosen to be analytic in C\C2 and +s for s → ∞. Then (see (6.40) of [2]) (5)
2 κq−1 = −eq(−γ +log γ +1) m 21 (0)
(54)
(s − ξ )(s − ξ ) ∼ (55)
and (5)
πq (0) = (−1)q m 11 (0).
(56)
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J. Baik, R. Buckingham, J. DiFranco
We analyze the solution m (5) to this RHP for the regime L + 1 ≤ q ≤ 2t − Mt 1/3 − 1 as t → ∞. Our analysis builds on the work of [2] and makes two main technical improvements. The first is that the paper [2] only considered the regime when t ≤ q. Hence q necessarily grows to infinity. In this work, we allow q to be finite. The second is that we compute a higher order correction explicitly to the asymptotics obtained in [2]. This higher-order correction contributes to the sum (50). In [2], only a one-sided bound of a similar sum was obtained. We merely outline the analysis for the parts that overlap with the analysis of [2].
From the construction of α in [2] we have that e−α(z) < 1 for z ∈ C1 and eα(z) < 1 for z ∈ Cin ∪ Cout . If we formally take the limit of our jump matrix v (5) as q → ∞ the jumps on the contours Cin and Cout approach the identity matrix and the jumps on C1 and C2 approach constant jumps. This limiting RHP is solved explicitly by ⎛ 1
β + β −1 m (5,∞) (z) = ⎝
− 2i1 β − β −1 2
where β(z) = that
z−ξ z−ξ −1
1/4
⎞ β − β −1
⎠, 1 −1 β + β 2 1 2i
, which is analytic for z ∈ C\C2 and β → 1 as z → ∞. Note
(5,∞) m 21 (0)
(57)
q 1 = −√ = − , γ 2t
(5,∞) m 11 (0)
=
γ −1 = γ
2t − q . 2t
(58)
However, the convergence of the jump matrix v (5) (z) is not uniform near the points ξ and ξ , since α(ξ ) = α(ξ ) = 0. Therefore a parametrix is introduced around these 1 points. For fixed δ < 100 , define Oξ = {z : |z − ξ | ≤ δ|ξ − ξ |}, Oξ = {z : |z − ξ | ≤ δ|ξ − ξ |}.
(59)
√
Note that the diameter of Oξ is of order γγ−1 and varies as t and q vary. The diameter approaches 0 as γ → 1 or γ → ∞, which happens when q is close to 2t − Mt 1/3 or L + 1, respectively. However, the point is that in the regime L + 1 ≤ q ≤ 2t − Mt 1/3 − 1, the diameter of Oξ cannot shrink “too fast.” Therefore, the usual Airy parametrix for a domain of fixed size still yields a good parametrix for the RHP in the regime under consideration. The case when γ → 1 “slowly” was analyzed in [2] for the leading asymptotics of m (5) . In this section, we also analyze the case when γ → ∞ “slowly,” and also improve the work in [2] to obtain a higher-order correction term. Orient the boundary of both Oξ and Oξ in the counterclockwise direction. Now as in [2] (see also [12]) for z ∈ Oξ \ (5) define the matrix-valued function m p as 2 3 3 z−ξ 1 −1 √ iπ/6 σ63 m p (z) = πe q α(z) −i −i 2 z−ξ 2/3 3 qα(z) × eqα(z)σ3 , 2
σ3 4
(60)
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
where ω = e2πi/3 and ⎧ ⎪ Ai(s) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ai (s) ⎪ ⎪ ⎪ ⎪ ⎪ Ai(s) ⎪ ⎪ ⎪ ⎨ Ai (s) (s) = ⎪ ⎪ Ai(s) ⎪ ⎪ ⎪ (s) ⎪ Ai ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ai(s) ⎪ ⎪ ⎩ Ai (s)
πi Ai(ω2 s) e− 6 σ3 , 2 2 ω Ai (ω s) Ai(ω2 s) 1 0 − πi σ e 6 3 , ω2 Ai (ω2 s) −1 1 −ω2 Ai(ωs) − πi σ3 1 0 e 6 , −Ai (ωs) 1 1
−ω2 Ai(ωs) − πi σ3 e 6 , −Ai (ωs)
473
0 < arg(s) < 2π 3
2π 3 ,
< arg(s) < π, (61)
π < arg(s) < 4π 3
4π 3 ,
< arg(s) < 2π.
We can define m p (z) = m p (z) for z ∈ Oξ by (60). For z ∈ / (5) ∪ (Oξ ∪ Oξ ), let (5,∞) (z). It is shown in [2] that m p then solves a RHP that has the same m p (z) = m jump conditions as m (5) on the contour C2 as well as on (5) ∩ O, where we define O = Oξ ∪ Oξ . Define R(z) = m (5) (z)m −1 p (z). Then R solves a RHP on = ∂O ∪ ((C 1 ∪ C in ∪ −1 Cout ) ∩ Oc ) with jump v R = m p− v (5) v −1 p m p− . Explicitly, the jump matrix v R is given by ⎧ 0, z ∈ ( (5) ∩ O) ∪ C2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 e−2qα ⎪ (5,∞) ⎪ (m (5,∞) )−1 , z ∈ C1 ∩ Oc , ⎨m 0 0 vR = I + (62) ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ m (5,∞) 2qα z ∈ (Cin ∪ Cout ) ∩ Oc , (m (5,∞) )−1 , ⎪ ⎪ 0 e ⎪ ⎪ ⎪ ⎪ ⎩ 1 qα E z ∈ ∂O, qα v R + v R , qα
where v R is given explicitly in Lemma 3.1 below, and the matrix v RE is defined as 1 qα v R for z ∈ ∂O. Since m (5) (0) = R(0)m p (0) = R(0)m (5,∞) (0), v RE = v R − I − qα √ (5) (5) we have m 21 (0) = − √1γ (R22 (0) − γ − 1R21 (0)) and m 11 (0) = − √1γ (R12 (0) − √ γ − 1R11 (0)). Therefore
1 2 κq−1 = √ eq(−γ +log γ +1) R22 (0) − γ − 1R21 (0) γ
(63)
(−1)q R12 (0) − γ − 1R11 (0) . πq (0) = − √ γ
(64)
and
In [2], for z ∈ ∂O, the jump matrix v R is approximated by the identity matrix I and 1 qα the terms qα v R + v RE are treated as an error (for the case when et ≤ q). For our purpose, we need to compute the contribution from the next order term
1 qα qα v R
explicitly.
474
J. Baik, R. Buckingham, J. DiFranco
It will be shown in the following subsections that for any > 0, there are L 0 and M0 such that for fixed L ≥ L 0 and M ≥ M0 , there is t0 = t0 (L , M) such that ||v R − I || L ∞ ( ) < for all t ≥ t0 and L + 1 ≤ q ≤ 2t − Mt 1/3 − 1. This was shown in [2] for
1 t ≤ q ≤ 2t − Mt 1/3 − 1. Then we proceed via the standard Riemann-Hilbert analysis ! f (s) 1 as, for example, in [2]. Let C( f )(z) := 2πi
s−z ds be the Cauchy operator defined for z ∈ / . For z ∈ , C− ( f )(z) is defined as the nontangential limit of C( f )(z ) as z approaches z from the right-hand side of . Define the operator C R ( f ) = C− ( f (v R − I )) for f ∈ L 2 ( ) and the function µ = I + (1 − C R )−1 C R I . A simple scaling argument shows that C R is a uniformly bounded operator for L + 1 ≤ q ≤ 2t − Mt 1/3 − 1. Since the supremum norm of v R − I can be made as small as necessary, we find that for L and M fixed but chosen large enough, (1 − C R )−1 is a bounded L 2 operator with norm uniformly bounded for t sufficiently large for all q such that L +1 ≤ q ≤ 2t − Mt 1/3 −1. By the theory of Riemann-Hilbert problems, µ(s)(v R − I ) 1 ds. (65) R(z) − I = 2πi s−z Define the contours ± as the part of in the upper-half and lower-half planes, respectively. That is,
± = ∩ (± (z) > 0).
(66)
± ± C1± = C1 ∩ ± , Cin = Cin ∩ ± , Cout = Cout ∩ ±
(67)
Also define
as shown in Fig. 1(c). Now, by the Schwartz-reflexivity of v R (see [2], p. 1159) and µ, 1 µ(s)(v R (s) − I ) 1 µ(s)(v R (s) − I ) ds = ds, (68) 2πi − s 2πi + s and therefore
"
1 R(0) − I = πi
+
# µ(s)(v R − I ) ds . s
(69)
We write this as R(0) − I = R (1) + R (2) + R (3) + R (4) + R (5) , where
$
R (1) $
1 R (2) = πi
1 = πi
(70)
% 1 qα ds v R (s) , s ∂ Oξ qα
$ % % 1 ds ds µ(s) · v RE (s) µ(s)(v R (s) − I ) , R (3) = , s πi C1+ s ∂ Oξ
(71)
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
475
$
R (4) R (5)
% 1 ds = µ(s)(v R (s) − I ) , + πi Cin+ ∪Cout s $ % 1 1 qα ds = (µ(s) − I ) v R (s) . πi ∂ Oξ qα s
Hence using (63), 1/3 [2t−Mt −1]
−2 log(κq−1 )
q=L+1
=
1/3 [2t−Mt −1]
q=L+1
1 − q(−γ + log γ + 1)+ log γ −log R22 (0) − γ − 1R21 (0) 2
1/3 [2t−Mt −1]
2t 2t 1 2t − q − + log + 1 + log = q q 2 q q=L+1
(1) (1) − log 1 + R22 (0) − γ − 1R21 (0) √ 5 (i) (i) R22 (0) − γ − 1R21 (0) (72) . − log 1 + √ (1) (1) 1 + R (0) − γ − 1R (0) i=2 22 21 ' & ! qα 1 1 qα ds v (s) 3.1. Calculation of R (1) = πi ∂ Oξ qα R s . First, we compute v R explicitly. qα
1 Lemma 3.1. For z ∈ ∂Oξ , the jump matrix v R (z) can be written as v R = I + qα v R +v RE , where 1 5 7 d1 β 2 + c1 β −2 i(d1 β 2 − c1 β −2 ) qα vR = , d1 = − (73) , c1 = 2 i(d1 β 2 − c1 β −2 ) −(d1 β 2 + c1 β −2 ) 72 72
and
v RE = O
1 |qα|2
.
(74)
(5,∞) and m Proof. On ∂Oξ , v R = m p− m −1 p+ is given by p+ . On this contour m p− = m (60). Thus for z ∈ ∂Oξ ,
For 0 < arg(s) <
σ3 6
2/3
3 qα(z) 2 −1 1 1 −1 . √ π −i −i
v R = m (5,∞) e−qα(z)σ3 −1 ×e−iπ/6 q −
3 α(z) 2
2 3
z−ξ z−ξ
−
σ3 4
(75)
2π 3 ,
consider (see (61)) Ai(s) Ai(ω2 s) e−(iπ/6)σ3 (s) = Ai (s) ω2 Ai (ω2 s)
(76)
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J. Baik, R. Buckingham, J. DiFranco
with ω = e2iπ/3 . From Abramowitz and Stegun [1] (10.4.59) and (10.4.61), for | arg(s)| < π, 3/2 e−(2/3)s 1 c1 Ai(s) = √ 1/4 1 − 2 +O , (77) 3/2 |s|3 2 πs 3s s 1/4 e−(2/3)s Ai (s) = − √ 2 π
3/2
wherein c1 =
d1
1−
2 3/2 3s
+O
1 |s|3
,
(78)
7 and d1 = − 72 . Also note the identity
5 72
Ai(s) + ωAi(ωs) + ω2 Ai(ω2 s) = 0
(79)
and, from Abramowitz and Stegun (10.4.11.13), W [Ai(s), Ai(ωs)] =
1 −iπ/6 e , 2π
(80)
1 iπ/6 e , 2π
(81)
W [Ai(s), Ai(ω2 s)] =
W [Ai(ωs), Ai(ω2 s)] = Using det (s) = W [Ai(s), Ai(ω2 s)] =
−1
1 iπ/6 , 2π e
ω2 Ai (ω2 s) (s) = 2π −Ai (s)e−iπ/3
1 iπ/2 . e 2π
(82)
we have −Ai(ω2 s) . Ai(s)e−iπ/3
(83)
Using 23 λ(z)3/2 = α(z), Eqs. (77) and (78) yield Ai(q
Ai (q
2/3
2/3
e−qα λ(z)) = √ 2 π (q 2/3 λ)1/4
(84)
(q 2/3 λ)1/4 e−qα λ(z)) = − √ 2 π
1 d1 , +O 1− qα |qα|2
(85)
eiπ/6 eqα λ(z)) = √ 2 π (q 2/3 λ)1/4
c1 1 1+ , +O qα |qα|2
(86)
2 2/3
Ai(ω q
c1 1 1− , +O qα |qα|2
ω2 Ai (ω2 q 2/3 λ(z)) = −
ω2 (q 2/3 λ)1/4 eqα √ 2 πeiπ/6
1+
d1 +O qα
1 |qα|2
.
(87)
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
477
We insert the asymptotics (84)-(87) into (83) resulting in the asymptotic formulas for 0 < arg(q 2/3 λ(z)) < π and qα large, √ iπ/6 qασ3 1 d1 −c1 1 −1 −1 2/3 + e (q λ(z)) = π e −i −i qα id1 ic1 σ3 /6 3 1 qα +O . (88) 2 |qα| 2 It is straightforward to compute an analogous expansion for −1 (s) for the other values of arg(q 2/3 λ(z)) in Eq. (76). To do this one must use the asymptotic formulas (77) and (78) as well as the additional expansions (10.4.60) and (10.4.62) from Abramowitz and Stegun [1]. Namely, for | arg(s)| < 2π/3, # " 3c1 1 s −1/4 2 3/2 π 2 3/2 π , (89) s + − 3/2 cos s + +O Ai(−s) = √ sin 3 4 2s 3 4 |s|3 π # " s 1/4 3d1 1 2 3/2 π 2 3/2 π Ai(−s) = − √ . s + − 3/2 sin s + +O cos 3 4 2s 3 4 |s|3 π
(90)
After carrying out this computation, the first two terms in the expansion are the same in all four regions. In other words, (88) is valid not only for 0 < arg(q 2/3 λ(z)) < 2π/3 but for all regions in the definition of in (76). Inserting the expansion in (88), Eq. (75) reduces to vR = I + for all z ∈ ∂Oξ .
1 1 d1 β 2 + c1 β −2 i(d1 β 2 − c1 β −2 ) , + O 2qα i(d1 β 2 − c1 β −2 ) −(d1 β 2 + c1 β −2 ) |qα|2
(91)
Now we explicitly evaluate R (1) . Lemma 3.2. We have R (1) =
⎛
⎞
(γ −1)1/2 1 1 1 − 24qγ 1/2 − 8q(γ −1) 24qγ ⎠ 8q(γ −1) ⎝ . (γ −1)1/2 1 1 1 − − + 24qγ 8q(γ −1) 24qγ 8q(γ −1)1/2
Proof. From Lemma 3.1, it is sufficient to compute the integrals β(s)2 1 1 1 ds and I2 = ds. I1 = 2 2πi ∂ Oξ α(s)s 2πi ∂ Oξ β(s) α(s)s
(92)
(93)
We will use the relations ξ = eiθc and 1 2(γ − 1)1/2 γ −2 θc = 1/2 , sin(θc ) = , cos(θc ) = , sin γ γ 2 γ γ − 1 1/2 θc = . cos 2 γ
(94)
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J. Baik, R. Buckingham, J. DiFranco
Note that α(z) = 23 (z − ξ )3/2 G(z) for an analytic function G(z) in Oξ (see the bottom line at p.1157 of [2]). Hence by residue calculations, I1 =
1 2πi
3
1/2 G(z)z ∂ Oξ 2(z − ξ )(z − ξ )
dz =
1 3 2 (ξ − ξ )1/2 G(ξ )ξ
(95)
and $ %
3 d (z − ξ )1/2
3(z − ξ )1/2 1 dz = I2 =
2πi ∂ Oξ 2(z − ξ )2 G(z)z 2 dz G(z)z z=ξ % $ 1/2 1/2 1 (ξ − ξ ) G (ξ ) (ξ − ξ ) 3 − − = . 1/2 2 2(ξ − ξ ) G(ξ )ξ G(ξ )2 ξ G(ξ )ξ 2 But since α(z) = 23 (z − ξ )3/2 G(z) and α (z) = − γ4 ward computation yields that G(ξ ) = lim
z→ξ
z+1 z2
(96)
(z − ξ )(z − ξ ), a straightfor-
α (z) γ ξ +1 =− (ξ − ξ )1/2 , 1/2 (z − ξ ) 4 ξ2
(97)
and 3γ G (ξ ) = 20
ξ +1 (ξ + 2)(ξ − ξ )1/2 − 3 2 ξ 2ξ (ξ − ξ )1/2
.
(98)
Using (94), we obtain I1 =
−3 3 i, + 4(γ − 1) 4(γ − 1)1/2
(99)
and I2 =
3 3 − + 4(γ − 1) 5γ
3(γ − 1)1/2 3 − 1/2 4(γ − 1) 5γ
i.
(100)
Therefore, (1)
# " 1 7 1 1 5 − I1 + I2 = − q 72 72 8q(γ − 1) 24qγ
(101)
" # 7 1 1 5 (γ − 1)1/2 I1 + I2 = . − q 72 72 8q(γ − 1)1/2 24qγ
(102)
(1)
R11 = −R22 = and (1) (1) R12 = R21 =
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
479
' & ! 1 1 ds 3.2. Bound on R (2) = πi ∂ Oξ µ(s) · O |qα|2 s . We begin by establishing the leading term of α(z) for z near ξ . Lemma 3.3. For 1 ≤ q < 2t and for z such that |z − ξ | ≤ min{ 21 , |ξ − ξ |},
3/4 5/2
α(z) − 2 (γ − 1)3/4 (z − ξ )3/2 e−3iπ/4 e−3iθc /2 ≤ 50(γ − 1) |z − ξ | .
3 |ξ − ξ |
(103)
Proof. Write z = ξ(1 + ). Then | | = |z − ξ | ≤ min{ 21 , |ξ − ξ |}. Under the change of variables s = ξ(1 + u), Eq. (54) for α(z) becomes 1/2 1 + ξ u γ 3/2 (1 + ξ ) ξ − ξ 1 √ ξ
1+ξ α(z( )) = − u u 1+ du. (104) √ (1 + u)2 4 ξ ξ −ξ 0 Using ξ = eiθc and (94), we have γ 3/2 (1 + ξ ) ξ − ξ − = (γ − 1)3/4 e−3iπ/4 3/2 √ 4 ξ = (γ − 1)3/4 (z − ξ )3/2 e−3iπ/4 e−3iθc /2 .
(105)
For the integrand in (104), using the inequalities |(1 + w)1/2 − 1| ≤ |w| for |w| ≤ 1 1 2 and |(1 + w)−2 − 1| ≤ 10|w| for |w| ≤ 21 , and using the fact that |1+ξ | ≤ |ξ −ξ | and 1≤
2 , |ξ −ξ |
we obtain
1/2 1 + ξ u
1+ξ
1 + ξ u
≤ 50| | = 50|z − ξ | . − 1
|ξ − ξ | 2 (1 + u) ξ −ξ |ξ − ξ |
Therefore, we obtain (103).
(106)
Lemma 3.4. For L + 1 ≤ q ≤ 2t − Mt 1/3 − 1, there is a constant c > 0 such that |R (2) | ≤ Proof. On ∂Oξ , |z − ξ | = δ|ξ − ξ | ≤
c(2t)2 . q 3/2 (2t − q)5/2 1 40 |ξ
(107)
− ξ |. Hence from Lemma 3.3, we have
δ 3/2 4 3/2 γ − 1 3/2 1 3/4 3/2 3/4 3/2 (γ − 1) |ξ − ξ | = δ |α(z)| ≥ (γ − 1) |z − ξ | = 6 6 3 γ (108) for z ∈ ∂Oξ . Therefore, as µ and |R
(2)
|≤c
1 s
are bounded on ∂Oξ ,
γ3 2π c γ 3 δ|ξ − ξ | cγ 2 |ds| = = 2 3 q 2 (γ − 1)3 q 2 (γ − 1)5/2 ∂ Oξ q (γ − 1)
for some constants c , c > 0, as |ξ − ξ | =
4(γ −1)1/2 . γ
(109)
480
J. Baik, R. Buckingham, J. DiFranco
&
3.3. Bound on R (3) =
!
1 πi
C1+
' µ(s)(v R (s) − I ) ds s . Since µ(s) and 1/s are bounded
on C1+ , |R (3) | ≤ c ||v R − I || L 1 (C + ) for some constant c > 0. But on C1+ , v R (z) − I = 1 O(e−2qα(z) ). Hence |R (3) | ≤ c||e−2qα(z) || L 1 (C + ) ,
(110)
1
for some constant c > 0. For z = eiθ ∈ C+ (hence θc < θ ≤ π ), using (eiφ − ξ )(eiφ − ξ ) = |(eiφ − eiφc )(eiφ − e−iφc )|1/2 ei(π +φ)/2 , we have
(111)
1 + eiφ iφ (e − eiθc )(eiφ − e−iθc ) · ieiφ dφ 2iφ e θc θ φ 1/2 φ + θc 1/2 φ − θc sin sin dφ. cos =γ (112) 2 2 2 θc (Recall that γ = 2tq .) Note that α(ξ ) = 0, α(s) is real and positive on C1+ , and α(eiθ ) increases as θ increases. α(eiθ ) = −
γ 4
θ
Lemma 3.5. For 1 ≤ q ≤ 2t, α(eiθ ) ≥
1 γ (γ − 1)(θ − θc )2 12π
(113)
for θc ≤ θ ≤ π . Proof. We consider two cases separately: θc ≤ π3 and θc ≥ π3 . 2π Start with the case when θc ≤ π3 . We consider two sub-cases: θ ≤ 2π 3 and θ ≥ 3 . φ φ+θc φ−θc π π π When θ ≤ 2π 3 , 0 ≤ 2 ≤ 3 , 0 ≤ 2 ≤ 2 and 0 ≤ 2 ≤ 3 . Hence using the basic 1 π 2 inequalities cos(x) ≥ 2 for 0 ≤ x ≤ 3 and sin(x) ≥ π x for 0 ≤ x ≤ π2 , we find that γ α(eiθ ) ≥ 2π
!θ
θc (φ +θc )
1/2 (φ
γ − θc )1/2 dφ ≥ 2π
!θ
θc (φ
γ − θc )dφ = 4π (θ − θc )2 .
(114) When θ ≥
2π 3 ,
from the monotonicity of α(eiθ ) and using (114), α(eiθ ) ≥ α(e
For 0 ≤ θc ≤
π 3
and
2π 3
2π 3 i
)≥
≤ θ ≤ π , we have α(eiθ ) ≥
γ 4π 2π 3
2π − θc 3
− θc ≥
π 3
2 γ θ − θc . 12π
2 .
(115)
≥ 13 (θ − θc ). Therefore, (116)
For the second case, when θc ≥ π3 , using the change of variables φ → π − φ, π −θc φ + (π − θc ) (π − θc ) − φ φ sin1/2 sin1/2 dφ. sin α(eiθ ) = γ 2 2 2 π −θ (117)
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
Note that 0 ≤
φ 2
≤
π 3,0
481
φ+(π −θc ) (π −θc )−φ ≤ π − θc ≤ 2π ≤ π3 . Using the 3 , and 0 ≤ √2 √2 ≥ 32π3 x for 0 ≤ x ≤ π3 and sin(x) ≥ 34π3 x for 0 ≤ x ≤ 2π 3 ,
≤
basic inequalities sin(x) we find that π −θc 27γ φ(φ + (π − θc ))1/2 ((π − θc ) − φ)1/2 dφ α(eiθ ) ≥ √ 8 2π 2 π −θ π −θc 27γ ≥ √ φ((π − θc ) − φ)dφ 8 2π 2 π −θ
9γ 9γ (π − θc )(θ − θc )2 . (118) = √ π − θc + 2(π − θ ) (θ − θc )2 ≥ √ 16 2 16 2π 2 θc π −θc c = sin ≤ π −θ Since 1−γ = cos γ 2 2 2 , we have 9 α(eiθ ) ≥ √ γ (γ − 1)(θ − θc )2 . 8 2π 2 Combining (114), (116), and (119) completes the proof.
(119)
Lemma 3.6. For L + 1 ≤ q ≤ 2t − Mt 1/3 − 1, there is a constant c > 0 such that |R (3) | ≤
c(2t)2 . − q)5/2
q 3/2 (2t
(120)
Proof. Let eiθ∗√ be the endpoint of C1+ on ∂Oξ . Note that since the radius of ∂Oξ is δ|ξ − ξ | = 4δ
γ −1 γ ,
√ θ∗ − θc ≥ 4δ
γ −1 . γ
(121)
Using Lemma 3.5 and changing variables, e−2qα L 1 (C + )≤ 1
π
θ∗
e−
√ q γ (γ −1) (θ−θc )2 6π
1/2 ∞ 1 2 3π e− 2 x d x, q γ (γ − 1) x∗
dθ ≤
√
(122)
where √ q γ (γ − 1) 1/2 4δ √ γ − 1 3/4 (θ∗ − θc ) ≥ √ q . x∗ = 3π γ 3π Using the inequality
!∞ a
e−2qα L 1 (C + ) ≤ 1
1 2
e− 2 x d x ≤
1 a3
for a > 0,
3π √ q γ (γ − 1)
Hence from (110) we obtain (120).
(123)
1/2
1 (2t)2 9π 2 ≤ . x∗3 128δ 3 q 3/2 (2t − q)5/2
(124)
482
J. Baik, R. Buckingham, J. DiFranco
3.4. Bound on R (4) =
&
+ the functions µ(s) and Cout
' + µ(s)(v R (s) − I ) ds s . As before, since on C in ∪
!
1 πi
+ ∪C + Cin out
are uniformly bounded and v R (z) − I = O(e−2qα(z) ),
1 s
2qα + ) ≤ ce + ) |R (4) | ≤ c ||v R − I || L 1 (C + ∪Cout L 1 (C + ∪Cout in
in
(125)
for some constants c , c > 0. Lemma 3.7. For L + 1 ≤ q ≤ 2t − Mt 1/3 − 1, there is a constant c > 0 such that |R (4) | ≤
c(2t)2 . q 3/2 (2t − q)5/2
(126)
π ). Let δ4 > 0 be a small positive number defined on p. 1152 of Proof. Let γ0 = csc2 ( 24 [2]. We estimate e−2qα in the following three cases separately: (i) 2t (1 + δ4 )−1 ≤ q ≤ 2t − Mt 1/3 + 1, (ii) γ2t0 ≤ q ≤ 2t (1 + δ4 )−1 , and (iii) L + 1 ≤ q ≤ γ2t0 . (i) For 2t (1 + δ4 )−1 ≤ q ≤ 2t − Mt 1/3 − 1, from (6.37) of [2], there are constants c1 , c2 , c3 , c4 , c5 > 0 such that ∞ 3 + ) ≤ c1 ||e2qα || L 1 (C + ∪Cout e−c3 q x d x + c4 e−c5 q . (127) √ in
c2 γ −1
Using the change of variables y = c3
c1
+ ) ≤ ||e2qα || L 1 (C + ∪Cout in
q x 3,
1/3
∞
e−y dy + c4 e−c5 q y 2/3
3c3 q 1/3 c2 q(γ −1)3/2 c1 3 3/2 ≤ 2 e−c2 c3 q(γ −1) + c4 e−c5 q 3c2 c3 q(γ − 1) c1 c4 ≤ 5 2 + 2 2. 2 5/2 c5 q 3c2 c3 q (γ − 1)
(128)
Since γ ≥ 1 and γ − 1 ≤ 1 + δ4 , we find that + ) ≤ ||e2qα || L 1 (C + ∪Cout in
cγ 2 c(2t)2 = q 2 (γ − 1)5/2 q 3/2 (2t − q)5/2
(129)
for a constant c > 0. (ii) For γ2t0 ≤ q ≤ 2t (1+δ4 )−1 , note that the radius of Oξ is of O(1). Hence a standard + ∩ C+ , calculation in Riemann-Hilbert steepest-descent analysis shows that for z ∈ Cin out + + 1 (α(z)) ≤ −c for some constant c > 0. Since the length of L (Cin ∪ Cout ) is bounded,
−2c q + ) ≤c e ||e2qα || L 1 (C + ∪Cout ≤ in
c c γ 2 c (2t)2 ≤ = (2c )2 q 2 q 2 (γ − 1)5/2 q 3/2 (2t − q)5/2
(130)
for some constants c , c > 0. π (iii) Consider the case when L + 1 ≤ q ≤ γ2t0 . Then 0 ≤ θc ≤ 12 . In this case, we + + make a specific choice of Cin and Cout : 7 1 + , = ξ + ρ sin θc ei(θc + 6 π ) : 2δ ≤ ρ ≤ Cin − sin(θc + 76 π ) (131) 1 + i(θc − 16 π ) Cout = ξ + ρ sinc e . : 2δ ≤ ρ ≤ − sin(θc − 16 π )
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
483
+ are straight line segments from ξ to a point on the positive real axis. The contours Cin (Recall that Oξ has the radius δ|ξ − ξ | = 2δ sin θc .) Now we estimate (α(z)) for + ∪ C + . For z ∈ C + , take the contour in (54) to be the straight line from ξ to z. z ∈ Cin out in Then one can check from the geometry that
0 ≤ arg(1 + s) ≤ θc ,
0 ≤ arg(s) ≤ θc ,
π 5π ≤ arg(s − ξ ) ≤ − θc , 2 6
arg(s − ξ ) = θc +
arg(ds) = θc +
7π , 6
2π . 6
(132)
Therefore the argument of the integrand in (54) is in [2π, 2π + π6 +2θc ] ⊂ [2π, 2π + π3 ] π since 0 ≤ θc ≤ 12 . Thus, the cosine of the argument is greater than or equal to cos( π3 ) = 1 2.
7
+ , by the change of variables s = Therefore, for z = ξ + ρ sin θc ei(θc + 6 π ) ∈ Cin 7
ξ + y sin θc ei(θc + 6 π ) ,
(α(z)) ≤ −
γ 8
ξ
z
1 + s
|ds| (s − ξ )(s − ξ )
s2
γ sin2 θc cos( θ2c ) ≤− √ 2 2
ρ
√
0
|1 + y
y sin θc i( θ2c + 7π 6 )| θ e
2 cos( 22 )
|1 + y sin θc ei
7π 6
|2
(133)
1−i 1 yei(θc+7π6 ) 1/2 dy. 2
Using the inequality |1 + xeiφ | ≥ | sin φ| for all x ∈ R, and using 0 ≤ θc ≤ 1 |y| ≤ 7π ≤ 2, we have − sin(θc +
6
π 12
and
)
γ sin2 θc cos( θ2c ) ρ √ ydy √ 36 2 0 √ 2 γ − 1 3/2 3/2 ρ =− 27 γ √ 2 γ − 1 3/2 ≤− (2δ)3/2 27 γ
(α(z)) ≤ −
(134)
+ . For z ∈ C + , taking the contour in (54) to be the straight line from ξ to z, for z ∈ Cin out we can check that
0 ≤ arg(1 + s) ≤ θc ,
0 ≤ arg(s) ≤ θc ,
π π − θc ≤ arg(s − ξ ) ≤ , 6 2
arg(s − ξ ) = θc −
arg(ds) = θc −
π , 6
π , 6 (135)
484
J. Baik, R. Buckingham, J. DiFranco
Hence the argument of the integrand in (54) is in [−θc − π6 , 2θc ] ⊂ [− π4 , π6 ]. Therefore, 1
+ , for z = ξ + ρ sin θc ei(θc − 6 π ) ∈ Cout
z
1 + s
γ
(α(z)) ≤ − √ (s − ξ )(s − ξ )
|ds|
2 s 4 2 ξ
≤−
γ
sin2 θc
cos( θ2c )
ρ
√
|1 + y
y sin θc i( θ2c − π6 ) |
θ e 2 cos( 22 ) π 6
2 |1 + y sin θc e−i |2 0 3/2 γ −1 1 ≤− √ ρ 3/2 γ 27 2 γ − 1 3/2 1 ≤− √ (2δ)3/2 . γ 27 2
1 − i 1 yei(θc − π6 ) 1/2 dy 2
(136)
From (134) and (136), arguing as in (130), we obtain + ) ≤ ||e2qα || L 1 (C + ∪Cout in
c (2t)2 − q)5/2
for a constant c > 0. Hence we obtain the estimate for |R (4) |. 3.5. Bound on R (5) =
&
1 πi
!
(137)
q 3/2 (2t
qα ds s
1 ∂ Oξ (µ(s) − I ) qα v R
' .
Lemma 3.8. For L + 1 ≤ q ≤ 2t − Mt 1/3 − 1, there is a constant c > 0 such that |R (5) | ≤
c(2t)2 . q 3/2 (2t − q)5/2
(138)
Proof. As µ − I = (1 − C R )−1 C R I , and as (1 − C)−1 and C− are uniformly bounded, ||µ − I || L 2 ( ) ≤ c0 ||C R I || L 2 ( ) = c0 ||C− (v R − I )|| L 2 ( ) ≤ c1 ||v R − I || L 2 ( ) (139) for some constants c0 , c1 > 0 when t is large enough. Below, we assume that t is large enough so that the above estimate holds. Now
qα
v R
|ds|
(5) |R | ≤
(µ(s) − I )
qα |s| ∂ Oξ
qα
v
≤ 2||µ − I || L 2 (∂ Oξ )
R
qα
2 L (∂ Oξ )
qα
v
≤ 2c1 ||v R − I || L 2 ( )
R
. (140)
qα
2 L (∂ Oξ )
Since β(z) =
z−ξ 1/4 z−ξ
qα
is bounded above and below for z ∈ Oξ , v R (z) in Lemma 3.1
is bounded. Using (108) and the fact that the radius of Oξ is δ|ξ − ξ | =
√ 4δ γ −1 , γ
we
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
485
have
qα
2
v
R
qα
2
≤
L (∂ Oξ )
c2 γ 2 c2 (2t)2 = q 2 (γ − 1)5/2 q 3/2 (2t − q)5/2
for a constant c2 > 0. Write ||v R − I ||2L 2 ( ) = ||v R − I ||2L 2 (∂ O) + ||v R − I ||2L 2 ( \∂ O) . As v R − I = c3 in Lemma 3.1 is bounded by |qα| for a constant c3 > 0, we have, as in (141), ||v R − I ||2L 2 (∂ O) = 2||v R − I ||2L 2 (∂ O) ≤ ξ
c4 (2t)2 q 3/2 (2t − q)5/2
(141)
qα
vR qα
+ v RE
(142)
for a constant c4 > 0. On the other hand, ||v R − I ||2L 2 ( \∂ O) = 2||v R − I ||2L 2 (C + ) + 2||v R − I ||2L 2 (C + ∪C + ) out 1 in
+ ) ≤ c3 ||e−2qα || L 2 (C + ) + ||e2qα || L 2 (C + ∪Cout 1 in
+ ) ≤ c3 ||e−2qα || L 1 (C + ) + ||e2qα || L 1 (C + ∪Cout 1
in
(143)
+ ∪ C+ . for a constant c3 > 0, since e−2qα ≤ 1 for z ∈ C1+ and e2qα ≤ 1 for z ∈ Cin out Hence from (124) and (130), we have
||v R − I ||2L 2 ( \∂ O) ≤
c4 (2t)2 q 3/2 (2t − q)5/2
for a constant c4 > 0. By combining (141), (142), and (144), we obtain (138).
(144)
3.6. The Airy part. From Lemmas 3.4, 3.6, 3.7, and 3.8, we find that, for L + 1 ≤ q ≤ [2t − Mt 1/3 − 1], there is a constant c > 0 such that
5 c(2t)2 c(2t)2 (i) (i)
≤
(R (0) − γ − 1R (0)) 22 21
q 3/2 (2t − q)5/2 + q 2 (2t − q)2 .
(145)
i=2
We need the following result. Lemma 3.9. We have lim lim sup
1/3 [2t−Mt −1]
M→∞ t→∞
q=L+1
(2t)2 =0 q 3/2 (2t − q)5/2
(146)
and lim lim sup
L→∞ t→∞
1/3 [2t−Mt −1]
q=L+1
(2t)2 = 0. q 2 (2t − q)2
(147)
486
J. Baik, R. Buckingham, J. DiFranco
Proof. We use the following basic inequality. Let a, b be integers. Let s(x) be a positive differentiable function in an interval [a − 1, b + 1] and there is c ∈ (a, b) such that s (x) < 0 for x ∈ [a − 1, c) and s (x) > 0 for x ∈ (c, b + 1]. Then b
s(q) ≤
As a function of 0 < q < 2t, increases for
< q < 2t. Hence lim lim sup
M→∞ t→∞
s(x)d x.
(148)
a−1
q=a
3 4t
b+1
(2t)2 q 3/2 (2t−q)5/2
1/3 [2t−Mt −1]
q=L+1
decreases for 0 < q <
3 4t
(2t)2 q 3/2 (2t − q)5/2
2t−Mt 1/3
(2t)2 dq M→∞ t→∞ q 3/2 (2t − q)5/2 L " #2t−Mt 1/3 4(−3t 2 + 6tq − 2q 2 ) = lim lim sup = 0. M→∞ t→∞ 3t (2t − q)3/2 q 1/2 L
≤ lim lim sup
As a function of 0 < q < 2t, t < q < 2t. Hence lim lim sup
(2t)2 q 2 (2t−q)2
1/3 [2t−Mt −1]
L→∞ t→∞
q=L+1
(149)
decreases for 0 < q < t and then increases for (2t)2 − q)2
q 2 (2t
2t−Mt 1/3
(2t)2 dq L→∞ t→∞ L − q)2 1/3 " # −2t + 2q 1 2t − q 2t−Mt = lim lim − log = L→∞ t→∞ q(2t − q) t q L
≤ lim lim
and then
q 2 (2t
0.
(150)
Now we prove the main result of Sect. 3. Lemma 3.10 (Airy part). We have
[2t−Mt 1/3 −1]
1 1 −2
log(κq−1 ) − t 2 − 2t L + L 2 log(2t) − ( L 2 lim lim
t→∞ L ,M→∞ 2 2 q=L+1
3 1 1 1 3 1 M − log M + log 2
= 0. (151) − ) log L + L 2 − 12 4 12 8 24 Proof. We first prove that 1/3 1/3 [2t−Mt [2t−Mt −1] −1] 2t
2t −2 lim lim + log +1 −q − log(κq−1 ) − L ,M→∞ t→∞ q q q=L+1 q=L+1 2t 1 1 1 + log + + = 0. (152) 2 q 8(2t − q) 12q
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
487
√ (1) (1) 1 1 Using 1 + R22 (0) − γ − 1R21 (0) = 1 − 8(2t−q) − 12q ≥ mas 3.4, 3.6, 3.7, and 3.8, we find that √
5
(i) (i)
i=2 (R22 (0) − γ − 1R21 (0))
1
√
≤ 2 (1) (1) 1 + R22 (0) − γ − 1R21 (0)
1 2
and using Lem-
(153)
for L+1 ≤ q ≤ [2t −Mt 1/3 −1], when we take t large enough. Hence using | log(1+x)| ≤ 2|x| for |x| ≥ 21 , 1/3 5 √
[2t−Mt
(i) (i) −1]
i=2 (R22 (0) − γ − 1R21 (0))
log 1 + lim lim
√
(1) (1) t→∞ L ,M→∞ 1 + R22 (0) − γ − 1R21 (0) q=L+1
≤
≤
lim
lim 4
1/3 [2t−Mt 5 −1]
L ,M→∞ t→∞
lim
q=L+1
lim 4c
L ,M→∞ t→∞
(i) R22 (0) −
(i) γ − 1R21 (0)
i=2
1/3 [2t−Mt −1]
q=L+1
(2t)2 (2t)2 + =0 q 3/2 (2t − q)5/2 q 2 (2t − q)2
(154)
using (145) and Lemma 3.9. On the other hand, from Lemma 3.2, 1/3 [2t−Mt −1]
q=L+1
log(1 +
× log 1 −
(1) R22 (0) −
γ
(1) − 1R21 (0))
=
1/3 [2t−Mt −1]
q=L+1
1 1 . − 8(2t − q) 12q
(155)
Using −x 2 ≤ log(1 − x) + x ≤ 0 for 0 ≤ x ≤ 21 , 1/3 1/3
[2t−Mt [2t−Mt
−1] −1]
1 1
1 1
− + + log 1 −
8(2t − q) 12q 8(2t − q) 12q
q=L+1
q=L+1
[2t−Mt 1/3 −1]
≤
q=L+1
2t−Mt 1/3
≤
L
1 1 1 + + 2 64(2t − q) 48(2t − q)q 144q 2
1 1 + 2 64(2t − q) 96t
1 1 + 2t − q q
1 dq → 0 + 144q 2
(156)
as t → ∞ by evaluating the integral explicitly. Using (154) and (156) in (72), we obtain (152). Now we compute each term of the sum in (152). Note that [2t − Mt 1/3 − 1] = 2t − Mt 1/3 − 1 − for 0 ≤ < 1. We have 1/3 [2t−Mt −1]
q=L+1
1/3 [2t−Mt −1] 2t 1 −q − + 1 (2t − q) = (2t − L − 1)(2t − L) = q 2
q=L+1
1 − (Mt 1/3 + )(Mt 1/3 + 1 + ) 2
(157)
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J. Baik, R. Buckingham, J. DiFranco
and 1/3 [2t−Mt −1]
q=L+1
1 1 (−q + ) log(2t) = − ((2t − Mt 1/3 − 1 − )2 − L 2 ) log(2t). 2 2
(158)
Since for positive integer m, m−1
k log k = m log((m − 1)!)−
k=1
m−1
log(k!) = m log( (m)) − log G(m + 1),
(159)
k=1
we find that 1/3 [2t−Mt −1]
1 1 1/3 q− log q = 2t − Mt − − log (2t − Mt 1/3 − ) 2 2 q=L+1 (160) 1 log (L + 1) + log G(L + 2). − log G(2t − Mt 1/3 − + 1) − L + 2
Note that from Stirling’s formula for the Gamma function and the asymptotics (48) for the Barnes G-function, as z → ∞, z+ −
2 1 z 1 log (z + 1) − log G(z + 2) = − log z 2 2 6
z2 z 1 1 + − log(2π ) + − ζ (−1) + o(1). 4 2 4 12
(161)
Now using the fact that K
1 − log K − γ = 0, lim
K →∞ q
(162)
q=1
where γ is Euler’s constant, we obtain
[2t−Mt 1/3 −1]
1 1 1 lim
− log(2t) + log(Mt 1/3 )
= 0 t→∞ 8(2t − q) 8 8
(163)
[2t−Mt 1/3 −1] L
1 1 1 1 1
= 0. lim − log(2t) − γ + t→∞
12q 12 12 12 q
(164)
q=L+1
and
q=L+1
q=1
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
489
Therefore, using (157), (158), (160), (163), (164), and (161), we obtain 1/3
[2t−Mt −1] 2t
1 2t 2t 1 1 lim
−q − + 1 + log + + log + t→∞ q q 2 q 8(2t − q) 12q q=L+1
L2 log(2t)
− t 2 − 2t L + 2 1 3 1 1 1 1 = − M − log M + log 2 − log(2π ) + − ζ (−1) 12 8 24 4 12 L 1 1 1 1 log (L + 1) + log G(L + 2) + γ− . (165) − (L + 1) L + 2 2 12 q
q=1
Hence using (161) and (162) again, we obtain (151).
4. Proof of (9) in Theorem 1.1: Computation of F(x) Recall Eq. (37) which we rewrite here: log(e
−t 2
1/3 [2t−Mt −1]
Dn ) = −t + log(D L )+ exact part 2
q=L+1
1/3 [2t+xt ] −2 −2 log(κq−1 )+ log(κq−1 ). q=[2t−Mt 1/3 ]
Airy part
(166)
Painlevé part
For the Painlevé part, from [2], 1/3 [2t+xt ]
lim
t→∞
−2 log(κq−1 )=−
q=[2t−Mt 1/3 ]
x −M
R(y)dy
1 2 1 1 1 R(y) − y + dy + x 3 − log |x| =− 4 8y 12 8 −M 1 3 1 (167) + M + log M. 12 8
x
By combining (167), (151), and (49), we obtain lim
lim log(e−t Dn (t)) 2
M,L→∞ t→∞ x
=−
−∞
R(y) −
1 2 1 1 1 1 y + dy + x 3 − log |x| + log 2 + ζ (−1). (168) 4 8y 12 8 24
5. Proof of (10) in Theorem 1.1: Computation of E(x) Let I j (2t) =
1 2π
π
−π
e2t cos θ ei jθ dθ.
(169)
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J. Baik, R. Buckingham, J. DiFranco
Set (see [4]) D++ (t) = det(I j−k (2t) + I j+k+2 (2t))0≤ j,k≤−1 ,
(170)
D−+ (t) = det(I j−k (2t) + I j+k+1 (2t))0≤ j,k≤−1 .
(171)
It is shown in Corollary 7.2 of [5] that for = [t +
x 1/3 t ], 2
(172)
where x lies in a compact subset of R, we have lim |e−t
2 /2
t→∞
lim |e−t
++ D−1 (t) − F(x)E(x)| = 0,
2 /2−t
t→∞
(173)
D−+ (t) − F(x)E(x)| = 0.
(174)
In [5], the above results are shown (with ( x in place of x) for the alternate scaling t = − (x2 1/3 . With the above scaling = [t + x2 t 1/3 ], we find ( x = x(1 + x2 t −2/3 )−1/3 . Since x is in a compact set, and E(x) and F(x) are continuous, the above results follow. From Eqs. (173) and (174), for a fixed x ∈ R, F(x)2 E(x)2 = lim e−t t→∞
2 −t
++ D−1 D−+ ,
= [t +
x 1/3 t ]. 2
(175)
Let π j (z; t) be the monic orthogonal polynomial of degree k with respect to the 1 2t cos θ e dθ on the unit circle, as introduced in Sect. 1. It is shown in Corollary weight 2π 2.7 of [4] that (cf. (32) above) e−t
2 /2
D++ (t) =
∞
κ22 j+2 (t)(1 − π2 j+2 (0; t)) =
j=
e−t
2 /2−t
D−+ (t) =
∞
κ22 j+1 (t)(1 + π2 j+1 (0; t)) =
j=
∞
κ22 j+1
j=
1 + π2 j+2 (0; t)
,
∞
κ22 j
j=
1 − π2 j+1 (0; t)
(176)
,
(177)
where the last equalities in (176) and (177) use the basic identity (see e.g [23]) 1 − πk2 (0; t) =
2 κk−1
κk2
.
(178)
Using Eqs. (176) and (177), we can write ++ D−1 = D ++ L−1
D−+
=
D −+ L
−1
D ++ j
j=L
D ++ j−1
D −+ j
j=L+1
D −+ j−1
= D ++ L−1
= D −+ L
−1
κ2−2 j−1 (t)(1 + π2 j (0; t)),
(179)
κ2−2 j−2 (t)(1 − π2 j−1 (0; t)),
(180)
j=L j=L+1
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
491
and hence we have 2−2
−+ ++ D−1 D−+ = D ++ L−1 D L
−1
κk−2 (t) ·
k=2L−1
) * (1 + π2 j (0; t))(1 − π2 j+1 (0; t)) .
j=L
(181) Using (recall that D = det(I j−k (2t))0≤ j,k≤−1 is the × Toeplitz determinant given in (27)) a−1 Da = κ −2 j (t), Db
(182)
j=b
we find that −+ ++ D−1 D−+ = D ++ L−1 D L
−1 D2−1 [(1 + π2 j (0; t))(1 − π2 j+1 (0; t))]. D2L−1
(183)
j=L
Inserting this equation into (175), and using (29), F(x)2 E(x)2 = F2 (x) lim e−t
−+ −1 D ++ L−1 D L
D2L−1
t→∞
[(1 + π2 j (0; t))(1 − π2 j+1 (0; t))].
j=L
(184) Hence we find (cf. (36)) E(x)2 = lim e−t
−+ −1 D ++ L−1 D L
D2L−1 & ' x = t + t 1/3 . 2
t→∞
[(1 + π2 j (0; t))(1 − π2 j+1 (0; t))],
j=L
(185)
In analogy to Eq. (37), we write 2 log E(x) =
lim
lim
L ,M→∞ t→∞
− t + log
−+ D ++ L−1 D L
D2L−1 Exact part
1/3 ] [t− M 2 t
+
log[(1 + π2 j (0; t))(1 − π2 j+1 (0; t))]
j=L
Airy part [t+ x2 t 1/3 −1]
+
1/3 +1] j=[t− M 2 t
log[(1 + π2 j (0; t))(1 − π2 j+1 (0; t))] . Painlevé part
We compute each term as in the case of log F(x).
(186)
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J. Baik, R. Buckingham, J. DiFranco
Lemma 5.1. (The Painlevé part). We have for x < 0 and M > 0, [t+ x2 t 1/3 −1]
lim
t→∞
log[(1 + π2 j (0; t))(1 − π2 j+1 (0; t))]
1/3 +1] j=[t− M 2 t
=
x −M
q(y) −
√ √ |y| 2 3/2 2(−x)3/2 M − . dy + 2 3 3
(187)
Proof. This follows from the following results in Corollaries 7.1 of [5]: for x in a compact subset of R,
∞
(188) lim
log(1 + π2 j+2 (0; t)) + log E(x)
= 0, t→∞
j=[t+ x t 1/3 ] 2
∞
lim
log(1 − π2 j+1 (0; t)) + log E(x)
= 0. t→∞
j=[t+ x t 1/3 ]
(189)
2
We remark that in [5], the notations v(x) = −R(x) and u(x) = −q(x) are used. Lemma 5.2. (The Airy part).
[t− M t 1/3 ] 2
lim lim
log[(1 + π2 j (0; t))(1 − π2 j+1 (0; t))] L ,M→∞ t→∞
j=L
√
2 3/2 1
M − 2L − log 2 = 0. − t−
3 2
(190)
Proof. Using (64), 1/3 ] [t− M 2 t
log[(1 + π2 j (0))(1 − π2 j+1 (0))]
j=L 1/3 ]+1 2[t− M 2 t
=
q=2L
γ −1 1 R11 (0; q) − √ R12 (0; q) . log 1 + γ γ
Using (70) and the fact that Lemma 3.2, this equals
log 1 +
q=2L 1/3 ]+1 2[t− M 2 t
q=2L
√ (1) (1) γ − 1R11 (0) − R12 (0) = 0, which follows from
1/3 ]+1 2[t− M 2 t
(191)
γ −1 γ
+
5 √ (i) (i) i=2 ( γ − 1R11 (0; q) − R12 (0; q)) . log 1 + √ √ γ + γ −1
(192)
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
493
From Lemmas 3.4, 3.6, 3.7, and 3.8, the same estimate as in (145) holds for 5 √ (i) (i) 1/3 − 1]. Therei=2 ( γ − 1R11 (0; q) − R12 (0; q)) for L + 1 ≤ q ≤ [2t − Mt fore the same argument as in (154) implies that the second sum in (192) vanishes in the limit. Therefore,
[t− M t 1/3 ] 2
log[(1 + π2 j (0; t))(1 − π2 j+1 (0; t))] lim lim
L ,M→∞ t→∞
j=L
1/3 ]+1 2[t− M
2 t 2t − q
(193) − log 1 +
= 0.
2t q=2L
We use the Euler-Maclaurin summation formula b b f (a) + f (b) f (b) − b (a) + + Err, f (k) = f (x)d x + 2 6 · 4! a k=a b 2 |Err| ≤ | f (3) (x)|d x. (194) (2π )2 a 1/3 ] = t − M t 1/3 − , where 0 ≤ < 1. and write [t − M Set f (q) = log 1 + 2t−q 2t 2 t 2 Then lim
t→∞
f (2t − Mt 1/3 − 2 + 1) + f (2L) 1 = log 2, 2 2
(195)
f (2t − Mt 1/3 − 2 + 1) + f (2L) = 0. 6 · 4!
(196)
and lim
t→∞
Also using f (3) (q) ≥ 0, 2 |Err| ≤ (2π )2
2t−Mt 1/3 −2 +1
2L
| f (3) (q)|dq =
2 (2π )2
( f (2t − Mt 1/3 − 2 + 1) − f (2L)) → 0, (197) t−L as t → ∞. Finally, changing variables to w = 1 + 2t−q 2t and setting δ1 = 1 + t Mt 1/3 +2 −1 and δ2 = , 2t 2t−Mt 1/3 −2 +1 1+δ2 2t − q log 1 + (1 − w) log(w)dw dq = 4t 2t δ1 2L #1+δ2 " 1 1 = 4t w − w 2 log w + w 2 − w 2 4 δ1 √ 2 3/2 1 M − 2L log 2 + O 1/3 . =t− 3 t (198)
494
J. Baik, R. Buckingham, J. DiFranco
Hence the result follows.
Lemma 5.3. (The exact part) We have % $ −+ D ++ L−1 D L lim lim log − (2L − 1) log 2 = 0. L→∞ t→∞ D2L−1
(199)
Proof. Note (see [4]) that " # π 1 i( j−k)θ i( j+k+2)θ 2t cos θ D ++ = det [e − e ]e dθ L−1 2π −π 0≤ j,k≤L−2 " π # 1 = det ei( j+1)θ sin((k + 1)θ )e2t cos θ dθ iπ −π 0≤ j,k≤L−2 " π # sin(( j + 1)θ ) sin((k + 1)θ ) 2 2t cos θ 2 = det dθ . · sin θ e π 0 sin θ sin θ 0≤ j,k≤L−2 (200) Changing variables to x = cos θ and noting that sin((sinj+1)θ) is a polynomial in x of the θ j j form 2 x + · · · (a constant multiple of the Chebyshev polynomial of the second kind), we have " j+k+1 1 # 2 ++ j+k 2t x 2 D L−1 = det x 1 − x e dx π −1 0≤ j,k≤L−2 L−1 L−1 2 2(L−1)(L−2) = |xk − x |2 1 − x 2j e2t x j d x j . π (L − 1)! [−1,1] L−1 j=1
1≤k<≤L−1
(201) A steepest descent analysis yields that lim D ++ L−1 ·
t→∞
×
t (L−1)
e2t (L−1)
2 +(L−1)/2
[0,∞] L−1 1≤k<≤L−1
π (L−1) (L − 1)!
|yk − y |2 ·
L−1
√
⎞−1
y j e−y j dy j ⎠
= 1.
(202)
j=1
The multiple integral is another Selberg integral (corresponding to a Laguerre ensemble) which is evaluated explicitly as (see (17.6.5) of [22])
[0,∞] L−1 1≤k<≤L−1
|yk − y |2 ·
L−1
√
y j e−y j dy j = (L − 1)!
G(L + 21 )G(L)
j=1
G( 23 )
.
(203)
Hence lim D ++ t→∞ L−1
·
e2t (L−1) t (L−1)
2 +(L−1)/2
π (L−1)
·
G(L + 21 )G(L) G( 23 )
−1
= 1.
(204)
Asymptotics of Tracy-Widom Distributions and Painlevé II Function Integral
495
Similarly, " # π 1 i( j−k)θ i( j+k+1)θ 2t cos θ = det (e + e )e dθ D −+ L 2π −π 0≤ j,k≤L−1 " π # 1 1 1 i( j+ 2 )θ 2t cos θ = det e cos k+ dθ θ e π −π 2 0≤ j,k≤L−1 " π # 1 1 1 θ cos k+ θ e2t cos θ dθ = det cos j+ π −π 2 2 0≤ j,k≤L−1 $ %
1 1 π cos j + 2 θ cos k + 2 θ θ 1 = det cos2 . e2t cos θ dθ π −π 2 cos θ2 cos θ2 0≤ j,k≤L−1 (205) Setting x = cos θ and noting that (− 12 , 21 )
Pj
cos(( j+ 21 )θ) cos θ2
is a constant multiple of a Jacobi polynomial
(x) of the form 2 j x j + · · · , we have $ % 1 1 + x 2t x 2 L(L−1) −+ j+k e dx det x DL = πL 1−x −1 2 L(L−1) = π L L!
[−1,1] L 1≤k<≤L
|xk − x |
2
0≤ j,k≤L−1
L j=1
1 + x j 2t x j e dx j. 1 − xj
(206)
A steepest-descent analysis implies that ⎛ ⎞−1 L 2t L e −1/2 lim D −+ · ⎝ 2 |yk − y |2 y j e−y j dy j⎠ = 1. t→∞ L t L −L/2 π L L! [0,∞] L 1≤k<≤L j=1 (207) The multiple integral is also a Selberg integral (see (17.6.5) of [22]), and we obtain −1 G(L + 1)G(L + 21 ) e2t L −+ · = 1. (208) lim D · 2 t→∞ L t L −L/2 π L G( 21 ) Using (204), (208), and lim D2L−1 ·
t→∞
−1
e4t L−2t (2t)2L
2 −2L+ 1 2
1
(2π ) L− 2
G(2L)
= 1,
(209)
which follows from Sect. 2 (the exact part for F2 (x)), we obtain $ % 2 −+ D ++ 22L −L [G(L + 21 )]2 G(L)G(L + 1) L−1 D L lim log − log · = 0. (210) 1 t→∞ D2L−1 G( 21 )G( 23 )G(2L) π L− 2 The result is now proved using the properties (46) and (48) for the Barnes G-function.
496
J. Baik, R. Buckingham, J. DiFranco
Combining Eq. (186) and Lemmas 5.1, 5.2, and 5.3, we obtain √ x |y| 2(−x)3/2 1 2 log E(x) = q(y) − dy − − log 2. 2 3 2 −∞
(211)
Acknowledgements. The authors would like to thank P. Deift and A. Its for useful communications. The work of the first author was supported in part by NSF Grant # DMS-0457335 and the Sloan Fellowship. The second and third authors were partially supported by NSF Focused Research Group grant # DMS-0354373.
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25. Tracy, C., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996) 26. Voros, A.: Spectral functions, special functions and the Selberg zeta function. Commun. Math. Phys. 110, 439–465 (1987) 27. Widom, H.: The strong Szegö limit theorem for circular arcs. Indiana Univ. Math. J. 21, 277–283 (1971) Communicated by P. Sarnak
Commun. Math. Phys. 280, 499–516 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0451-3
Communications in
Mathematical Physics
The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian David Damanik1, , Mark Embree2, , Anton Gorodetski3 , Serguei Tcheremchantsev4 1 Department of Mathematics, Rice University, Houston, TX 77005, USA.
E-mail: [email protected]
2 Department of Computational and Applied Mathematics, Rice University, Houston,
TX 77005, USA. E-mail: [email protected]
3 Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA.
E-mail: [email protected]
4 Université d’Orléans, Laboratoire MAPMO, CNRS-UMR 6628, B.P. 6759,
F-45067 Orléans Cedex, France. E-mail: [email protected] Received: 2 May 2007 / Accepted: 20 August 2007 Published online: 4 March 2008 – © Springer-Verlag 2008
Abstract: We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds √show that as λ → ∞, dim(σ (Hλ )) · log λ converges to an explicit constant, log(1 + 2) ≈ 0.88137. We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian. 1. Introduction The Fibonacci Hamiltonian is a discrete one-dimensional Schrödinger operator [H u](n) = u(n + 1) + u(n − 1) + V (n)u(n) in 2 (Z). The potential V : Z → R is given by V (n) = λχ[1−φ −1 ,1) (nφ −1 + θ mod 1), √ where λ > 0 is the coupling constant, φ = 21 ( 5 + 1) is the golden mean, and θ ∈ [0, 1) is the phase. This operator is important for both physical and mathematical reasons. On the one hand, it is the most popular quantum model of a one-dimensional quasicrystal, that is, a structure that shares many features with one displaying global order, but which in fact lacks global translation invariance. On the other hand, this operator has zeromeasure Cantor spectrum and all spectral measures are purely singular continuous. These properties had been regarded as “exotic” in the context of general Schrödinger operators up until the 1980s, but for this operator family, they occur persistently for all parameter values. The Fibonacci Hamiltonian has been heavily studied since the early 1980s; see [6] for a recent review of the results obtained for it and related models. D. D. was supported in part by NSF grant DMS–0653720.
M. E. was supported by NSF grant DMS–CAREER–0449973.
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Let us recall some specific results and references that will be important for what follows. It is known that the spectrum of the Fibonacci Hamiltonian is independent of θ (see, e.g., [4]). This follows quickly from strong convergence once one realizes that for each pair θ, θ˜ , there is a sequence n k → ∞ such that θ + n k φ −1 converges to θ˜ in R/Z “from the right.” We denote this common spectrum by λ , that is, λ = σ (H ) for every θ ∈ [0, 1). It is natural to study the spectrum as a set. As we shall see, such a study is also motivated by the consequences one can draw for the long-time behavior of the solution of the time-dependent Schrödinger equation. Süt˝o has shown that the spectrum always has zero Lebesgue measure [29], Leb(λ ) = 0
for every λ > 0.
(1)
This immediately implies the absence of absolutely continuous spectrum for all parameter values.1 It was later seen that one also has the absence of point spectrum for all parameter values; see Süt˝o [28], Hof-Knill-Simon [13], and Kaminaga [16] for partial results and Damanik-Lenz [7] for the full result. Thus, the Fibonacci model exhibits purely singular continuous spectrum that is very rigid in the sense that it is not affected by a change of the defining parameters. The result on zero measure spectrum, (1), naturally leads one to ask about the dimension of this set. There are several popular ways to measure the fractal dimension of a nowhere dense subset of the real line. Let us recall the definition of two of these dimensions. Suppose we are given a bounded and infinite set S ⊆ R. A δ-cover of S is a countable union of real intervals, {Im }m≥1 , such that each of these intervals has length bounded by δ > 0. For α ∈ [0, 1], let h α (S) = lim inf |Im |α . δ→0 δ-covers
m≥1
It is clear that the limit exists in [0, ∞]. Moreover, if h α (S) = 0 for some α, then h α (S) = 0 for every α > α. Similarly, if h α (S) = ∞ for some α, then h α (S) = ∞ for every α < α. Thus, the following quantity is well-defined: dim H (S) = inf α : h α (S) < ∞ = sup α : h α (S) = ∞ . The number dim H (S) ∈ [0, 1] is called the Hausdorff dimension of the set S. A different way to measure the fractal dimension of S is via the box counting dimension. The lower box counting dimension is defined as follows: dim− B (S) = lim inf ε→0
log N S (ε) , log 1/ε
where N S (ε) = #{ j ∈ Z : [ jε, ( j + 1)ε) ∩ S = ∅}. The upper box counting dimension, dim+B (S), is defined similarly, with the lim inf replaced by a lim sup. When dim+B (S) and dim− B (S) are equal, we denote their common value by dim B (S) and call this number the box counting dimension of S. These dimensions are related by the inequalities 1 Historically, these two properties were established in the reverse order. The methods used by Kotani [19] in the proof of absence of absolutely continuous spectrum (for almost every θ ) were the key to proving zero measure spectrum.
Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
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+ dim H (S) ≤ dim− B (S) ≤ dim B (S). In general, both inequalities may be strict; see, for example, [21, pp. 76–77]. The main goal of this paper is to study the fractal dimension of the spectrum of the Fibonacci Hamiltonian. The following result shows that for sufficiently large coupling, the dimensions just introduced coincide.
Theorem 1. Suppose that λ ≥ 16. Then the box counting dimension of λ exists and obeys dim B (λ ) = dim H (λ ). While this theorem has not appeared in print explicitly before, it follows quickly from a combination of known results. We present the relevant facts in the Appendix. Theorem 1 will allow us to obtain precise asymptotics for the fractal dimension of λ as λ → ∞. The reason for this is the following. The box counting dimension is easier to bound from below, while the Hausdorff dimension is easier to bound from above. Consequently, we will prove a lower bound for the box counting dimension in Sect. 3 and an upper bound for the Hausdorff dimension in Sect. 4. It is known how to describe the spectrum of the Fibonacci Hamiltonian in terms of the spectra of canonical periodic approximants. We will recall this in Sect. 2. The general theory of periodic discrete one-dimensional Schrödinger operators shows that the spectrum of a such a periodic operator is always given by a finite union of compact intervals. Our crucial new insight is a way to describe the asymptotic distribution of bandwidths in these periodic spectra. In this description, the following function plays an important role. Define f (x) =
1 (2 − 3x) log 2 + (1 − x) log(1 − x) − (2x − 1) log(2x − 1) x −(2 − 3x) log(2 − 3x)
on the interval 21 , 23 . Setting f 21 = log 2 and f 23 = 0, it is not hard to see that f extends to a continuous function on [ 21 , 23 ], and with the aid of symbolic computation one can confirm that it takes its maximum at the unique point √ 12 − 2 2 ∗ = 0.5395042867796 . . . , x = 17 with √ f ∗ = f (x ∗ ) = log(1 + 2) = 0.8813735870195 . . . .
Write Su (λ) = 2λ+22 and Sl (λ) = 21 (λ − 4) + (λ − 4)2 − 12 . With these functions of (sufficiently large) λ we can now state the bounds on the fractal dimension of λ that we will prove in Sects. 3 and 4, respectively. ∗ Theorem 2. (a) Suppose λ > 4. Then, dim− B (λ ) ≥ f / log Su (λ).
(b) Suppose λ ≥ 8. Then, dim H (λ ) ≤ f ∗ / log Sl (λ). Since both Su (λ) and Sl (λ) behave asymptotically like log λ, we obtain the following result as an immediate consequence. We write dim for either dim H or dim B , which is justified by Theorem 1.
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Corollary 1. We have lim dim(λ ) · log λ = f ∗ . λ→∞
In particular, the constant f ∗ is the best possible in both bounds in Theorem 2. Our interest in obtaining the optimal constant does not only stem from natural curiosity. An interesting and mathematically challenging problem is to study the spreading of a wavepacket in a quantum system in the case where the initial state has a purely singular continuous spectral measure. This is the case for every initial state from 2 (Z) for a system governed by the Fibonacci Hamiltonian. One is often especially interested in the spreading of a wavepacket that is initially localized on just one site. That is, with H as above, we consider ψ(t) = e−it H δ1 and study its spreading via the time-averaged outside probabilities; namely, 2 2 ∞ Pr (N , T ) = e−2t/T e−it H δ1 , δn dt T 0 n>N
and
2 2 ∞ Pl (N , T ) = e−2t/T e−it H δ1 , δn dt. T 0 n<−N
Let P(N , T ) = Pl (N , T ) + Pr (N , T ) and define S − (α) = − lim inf T →∞
log P(T α − 2, T ) log P(T α − 2, T ) , S + (α) = − lim sup . log T log T T →∞
For every α, 0 ≤ S + (α) ≤ S − (α) ≤ ∞. These numbers control the power decaying tails of the wavepacket. In particular, the following critical exponents are of interest: αu± = sup{α ≥ 0 : S ± (α) < ∞}. One can interpret αu± as the rates of propagation of the fastest (polynomially small) part of the wavepacket; compare [11]. In particular, if α > αu+ , then P(T α , T ) goes to 0 faster than any inverse power of T , and if α > αu− , then there is a sequence of times Tk → ∞ such that P(Tkα , Tk ) goes to 0 faster than any inverse power of Tk . In Sect. 5 we prove a result for general Schrödinger operators on 2 (Z) that will imply the following consequence for the Fibonacci Hamiltonian. Theorem 3. For every λ > 0 and every θ ∈ [0, 1), we have that αu± ≥ dim± B (λ ). ± ∗ Consequently, for λ > 4 and every θ , we have αu ≥ f / log Su (λ). 2. The Band-Gap Structure of the Approximating Periodic Spectra In this section we describe the canonical coverings of λ by (unions of) periodic spectra and the hierarchical structure of these sets. We use the combinatorics of this description to derive detailed results about the distribution of bandwidths in these spectra. For E ∈ R and λ > 0, we define a sequence of numbers xk = xk (E, λ) as follows: x−1 = 2, x0 = E, x1 = E − λ, xk+1 = xk xk−1 − xk−2
for k ≥ 1.
(2)
Using this recurrence and the initial values, one can quickly check (Süt˝o [28]) that 2 2 xk+1 + xk2 + xk−1 − xk+1 xk xk−1 = 4 + λ2 for every k ≥ 0.
(3)
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For fixed λ > 0, define σk = {E ∈ R : |xk (E, λ)| ≤ 2}. The set σk is actually equal to the spectrum of the Schrödinger operator H whose potential Vk results from V (with θ = 0) by replacing φ −1 with Fk−1 /Fk (cf. [28]). Here, {Fk }k≥0 denotes the sequence of Fibonacci numbers, that is, F0 = 1, F1 = 1, and Fk+1 = Fk + Fk−1 for k ≥ 1. Hence, Vk is periodic and σk consists of Fk bands (closed intervals). Süt˝o also proved that (σk−1 ∪ σk ) ⊃ (σk ∪ σk+1 ) and λ =
(σk ∪ σk+1 ) = {E : {xk } is a bounded sequence} .
(4) (5)
k≥1
From now on, we assume λ > 4, since we will make critical use of the fact that in this case, it follows from the invariant (3) that three consecutive xk ’s cannot all be bounded in absolute value by 2: (6) σk ∩ σk+1 ∩ σk+2 = ∅. The identity (6) is the basis for work done by Raymond [27]; see also [10,17], which describe the band structure on the various levels in an inductive way. Let us recall this result. Following [17], we call a band Ik ⊂ σk a “type A band” if Ik ⊂ σk−1 (and hence Ik ∩ (σk+1 ∪ σk−2 ) = ∅). We call a band Ik ⊂ σk a “type B band” if Ik ⊂ σk−2 (and therefore Ik ∩ σk−1 = ∅). Then we have the following result (Lemma 5.3 of [17], essentially Lemma 6.1 of [27]). Lemma 1. For every λ > 4 and every k ≥ 1, (a) Every type A band Ik ⊂ σk contains exactly one type B band Ik+2 ⊂ σk+2 , and no other bands from σk+1 , σk+2 . (b) Every type B band Ik ⊂ σk contains exactly one type A band Ik+1 ⊂ σk+1 and two type B bands from σk+2 , positioned around Ik+1 . Lemma 2. For every band I of σk , we have that I ∩ λ = ∅. Proof. Let I be a band of σk . Choose a band I (1) in σk+1 ∪ σk+2 with I ⊃ I (1) , as is possible by Lemma 1. Iterating this procedure, we obtain a nested sequence of intervals and hence a point E ∈ I for which the corresponding trace map orbit is bounded. Thus, by (5), E ∈ λ . We denote by ak the number of bands of type A in σk and by bk the number of bands of type B in σk . By Raymond’s work, it follows immediately that ak + bk = Fk for every k. In fact, we have the following result. Lemma 3. The constants {ak } and {bk } obey the relations ak = bk−1 , bk = ak−2 + 2bk−2
(7)
with initial values a0 = 1, a1 = 0, b0 = 0, and b1 = 1. Consequently, for k ≥ 2, ak = bk−1 = Fk−2 .
(8)
Proof. The recursions (7) hold by definition. The explicit expressions in (8) then follow quickly by induction.
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We are interested in the size of the bands of a given type on a given level. To capture the distribution of these lengths, we denote by ak,m the number of bands b of type A in σk with #{0 ≤ j < k : b ∩ σ j = ∅} = m and by bk,m the number of bands b of type B in σk with #{0 ≤ j < k : b ∩ σ j = ∅} = m. The motivation for this definition is the following: If we consider a given band and its location relative to the bands on the previous levels, each time the given band is contained in a band on a previous level, we essentially pick up a factor roughly of size λ−1 (for λ large). Thus, for example, there are ak,m bands of size ≈ λ−m of type A in σk . Since the combinatorics of the situation is λ-independent, we choose to separate the two aspects. This will allow us to get much more precise information in the regime of large λ, which is extremely hard to study numerically. Lemma 4. We have ak,m = bk−1,m−1 , bk,m = ak−2,m−1 + 2bk−2,m−1
(9)
with initial values a0,m = 0 for m > 0, a0,0 = 1, a1,m = 0 for m ≥ 0, b0,m = 0 for m ≥ 0, b1,m = 0 for m > 0, and b1,0 = 1. Consequently, k−m m 22k−3m−1 k−m when k2 ≤ m ≤ 2k 2m−k 3 ; (10) ak,m = bk−1,m−1 = 0 otherwise. Remark. The fact that ak,m is zero when k2 ≤ m ≤ 2k 3 fails is an immediate consequence of the properties (4), (5), and (6) established by Süt˝o. As a slight variation of the proof of Theorem 4 below shows, this fact alone is sufficient for a quick proof of log φ dim± B (λ ) ≥ 1.5 log Su (λ) . Our more detailed description of the numbers ak,m will then enable us to prove the stronger lower bound with f # = f ∗ / log φ ≈ 1.83157 in place of 1.5, which is optimal, as discussed in the Introduction. Proof. It is immediate from the definition and (7) that the recursions (9) hold. Recall that the Chebyshev polynomials of the first kind are defined by the recurrence relation T0 (x) = 1, T1 (x) = x, Tm+1 (x) = 2x Tm (x) − Tm−1 (x). Write Tm (x) as Tm (x) = m/2 r m−2r . On the one hand, it is known (see, e.g., [1]) that for m ≥ 1 and r =0 (−1) cr,m x 0 ≤ r ≤ m/2, m −r m . (11) cr,m = 2m−2r −1 m −r r On the other hand, the recursion for the polynomials gives c0,0 = 1, c0,1 = 1, and m+1 2
m
(−1) cr,m+1 x r
m−2r +1
m−1
2 2 = 2x (−1)r cr,m x m−2r − (−1)r cr,m−1 x m−2r −1
r =0
r =0
=
r =0
m2
m−1 2
r =0
r =0
m2
m+1
r =0
r =1
(−1)r 2cr,m x m−2r +1 + (−1)r +1 cr,m−1 x m−2r −1
2 = (−1)r 2cr,m x m−2r +1 + (−1)r cr −1,m−1 x m−2r +1 .
Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
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It follows that cr,m+1 = 2cr,m + cr −1,m−1 .
(12)
Denote a˜ k,m = c2m−k,m . Then, using (12), we find that a˜ k+1,m+1 = 2a˜ k−1,m + a˜ k−2,m−1 . Comparing this with the recursion ak+1,m+1 = 2ak−1,m +ak−2,m−1 we established above, along with the initial values, we see that ak,m = a˜ k,m = c2m−k,m . Combining (11) and (13), we obtain (10).
(13)
Recall that we introduced the function f (x) =
1 (2 − 3x) log 2 + (1 − x) log(1 − x) − (2x − 1) log(2x − 1) x −(2 − 3x) log(2 − 3x)
√ 2 = on the interval 21 , 23 . We set f 21 = log 2 and f 23 = 0 and write x ∗ = 12−2 17 1 2 0.5395042867796 . . . for the unique point in 2 , 3 where f takes its maximum value, √ f ∗ = f (x ∗ ) = log(1 + 2) = 0.8813735870195 . . . .
Proposition 1. For
k 2
≤m≤
2k 3 ,
we have
m
m
k −1/2 exp m f ak,m k 1/2 exp m f . k k
(14)
Consequently, lim max
k→∞ m
Proof. As we saw above, when k
k 2
1 log am,k = f ∗ . m
≤m≤
2k 3 ,
(15)
m ak,m = 22k−3m−1 k−m
k−m 2m−k . Thus, we
see that ak, k = 2 2 −1 and ak, 2k = 1, so the estimate (14) holds when m = 2
Let us now consider
k 2
3
<m<
2k 3 ,
k 2
or m =
2k 3 .
in which case
k k k k < k − m < , 1 ≤ 2m − k < , 1 ≤ 2k − 3m < . (16) 3 2 3 2 √ Stirling’s approximation gives n! n (n/e)n for every n ≥ 1, where we write a b if a b and a b. Thus, ak,m 2
2k−3m
m k−m
k−m (2m − k)(2k − 3m)
k−m k−m
1/2
e
2m−k 2m−k 2k−3m 2k−3m . e
e
Due to the estimates (16), (14) will follow once we show that k−m k−m a˜ k,m := 2
2k−3m
e 2m−k 2m−k e
2k−3m 2k−3m e
m
. = exp m f k
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Writing x = m/k, we find that (k − m)k−m
a˜ k,m = 22k−3m =2 =2
(2m − k)2m−k (2k − 3m)2k−3m (k − kx)k−kx 2k−3kx (2kx − k)2kx−k (2k − 3kx)2k−3kx (1 − x)k(1−x) 2k−3kx
(2x − 1)k(2x−1) (2 − 3x)k(2−3x) = exp (kx f (x)) = exp (m f (m/k)) . This completes the proof of (14), which in turn immediately implies that lim supk→∞ maxm m −1 log am,k ≤ f ∗ . On the other hand, as k gets large, we can choose m so that m/k gets arbitrarily close to x ∗ , so that by (14) again, lim inf k→∞ maxm m −1 log am,k ≥ f ∗ . Thus, we have established (15). 3. A Lower Bound for the Box Counting Dimension of the Spectrum In this section we will prove a lower bound for the (lower) box counting dimension of the spectrum of the Fibonacci Hamiltonian. Recall that the lower box counting dimension of a bounded set S ⊂ R is defined as dim− B (S) = lim inf ε→0
log N S (ε) , log 1/ε
where N S (ε) = #{ j ∈ Z : [ jε, ( j + 1)ε) ∩ S = ∅}. The following was shown in [9]. Lemma 5. For every λ > 4, there exists Su (λ) such that the following holds: (a) Given any (type A) band Ik+1 ⊂ σk+1 lying in the band Ik ⊂ σk , we have for every x (E) E ∈ Ik+1 , xk+1 ≤ Su (λ). k (E) (b) Given any (type B) band Ik+2 ⊂ σk+2 lying in the band Ik ⊂ σk , we have for every x (E) E ∈ Ik+2 , xk+2 (E) ≤ Su (λ). k
For example, one can choose Su (λ) = 2λ + 22. We use the symbol Su (λ) instead of the explicit 2λ + 22 to make the dependence of everything that follows on this quantity explicit. Inductively, Lemma 5 gives an upper bound for |xk | on each band of σk and this in turn will give a lower bound on the length of a band, which will be crucial in the proof of the following result. Theorem 4. For all λ > 4, dim− B (λ ) ≥
f∗ . log Su (λ)
(17)
Proof. Let m k = 3kx ∗ . Since x ∗ > 0.4, the sequence {m k } is strictly increasing and limk→∞ m3kk = x ∗ . Write f k = m −1 k log a3k,m k . From Proposition 1, we know lim f k = f ∗
k→∞
(18)
Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
507
and it is obvious that
m k+1 = 1. (19) k→∞ m k For a given k, let us consider the type A bands in σ3k that lie in m k bands on previous levels. By definition, there are Nk := a3k,m k such bands. By Lemma 5, each of them has length at least εk := 4Su (λ)−m k , since on each band of σ3k , the function x3k is k be these bands, enumerated strictly monotone and runs from ±2 to ∓2. Let {A3k, j } Nj=1 so that A3k, j is to the left of A3k, j+1 for every j. Each band has non-empty intersection with λ by Lemma 2. Thus, for every j, there exists E 3k, j ∈ A3k, j ∩ λ . Clearly, the energies {E 3k, j } are increasing in j. Consider {E 3k, j } with j odd, that is, j = 2s + 1, 0 ≤ s ≤ Nk /2. Since any two bands A3k,2s−1 , A3k,2s+1 are separated by the band A3k,2s , which has length at least εk , we get |x3k,2s−1 − x3k,2s+1 | ≥ εk for every s. Thus, the E 3k,2s+1 belong to different ε-boxes if ε < εk . We conclude that Nλ (ε) ≥ Nk /2 for every ε < εk . Given any ε > 0, choose k with εk+1 ≤ ε < εk . Then, lim
f k − m1k log 2 log(a3k,m k ) − log 2 log Nλ (ε) ≥ . ≥ m k+1 log 1/ε log 1/εk+1 m k log Su (λ) Since, as ε → 0, we have k, m k → ∞ and lim
m k→∞ mk+1 k
f∗ fk = log Su (λ) log Su (λ)
by (18) and (19), the result follows. 4. An Upper Bound for the Hausdorff Dimension of the Spectrum In this section we prove an upper bound for the Hausdorff dimension of λ . We will use the canonical coverings σk ∪ σk+1 of λ and estimate the lengths of the intervals in these periodic spectra from above using our combinatorial result from Sect. 2 together with a scaling result that is analogous to Lemma 5, but which gives bounds from the other side. Namely, the following was shown by Killip, Kiselev, and Last [17, Lemma 5.5]. Lemma 6. For every λ ≥ 8, there exists Sl (λ) such that the following holds: (a) Given any (type A) band Ik+1 ⊂ σk+1 lying in the band Ik ⊂ σk , we have for every x (E) E ∈ Ik+1 , xk+1 ≥ Sl (λ). k (E) (b) Given any (type B) band Ik+2 ⊂ σk+2 lying in the band Ik ⊂ σk , we have for every x (E) E ∈ Ik+2 , xk+2 (E) ≥ Sl (λ). k
For example, one can choose Sl (λ) =
1 2
(λ − 4) + (λ − 4)2 − 12 .
As before, we use the symbol Sl (λ) instead of the explicit possible choice to make the dependence of everything that follows on this quantity explicit. Inductively, Lemma 6 gives a lower bound for |xk | on each band of σk and hence an upper bound for the length of a band. Theorem 5. Suppose λ ≥ 8. Then dim H (λ ) ≤ f ∗ / log Sl (λ).
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2 log φ Remark. Raymond [27] proved an upper bound of the form dim H (λ ) ≤ log Sl (λ) . As above, to get this weaker result, all one needs is information about the support of ak,m , which in turn follows quickly from the properties (4), (5), and (6). Using our more detailed information about the values of ak,m on the support, we can improve the constant 2 to the value f # = f ∗ / log φ ≈ 1.83157, which is optimal by our discussion in the introduction. Our proof is inspired by Raymond’s proof, and the improvement stems from our more detailed analysis of the distributions of bandwidths in the approximating periodic spectra.
Proof. As recalled above, the set σk ∪ σk+1 is a finite union of compact real intervals that covers λ . There are ak,m + bk,m bands in σk that lie in exactly m intervals on previous levels. We know that the length of each of these bands is bounded from above by 4Sl (λ)−m . Thus, it suffices to show that, given any s>
f∗ , log Sl (λ)
(20)
we have lim
k→∞
2k 3
2(k+1) 3
m= k2
m= k+1 2
ak,m + bk,m (4Sl (λ))−sm +
ak+1,m + bk+1,m (4Sl (λ))−sm = 0.
2k/3 For simplicity, we will only consider Ck = m=k/2 ak,m Sl (λ)−sm ; the other terms can be dealt with in an analogous way. By our upper bound for ak,m established earlier and the assumption (20), we have Ck k 1/2
2k/3 m=k/2
k ∗ ( f − s log Sl (λ)) . exp m( f ( mk ) − s log Sl (λ)) k 3/2 exp 2
Using (20) again, we see that the right-hand side goes to zero as k → ∞. Theorem 4 and Theorem 5 together establish Theorem 2 from the Introduction. 5. Transfer Matrices, Box Counting Dimension, and Wavepacket Spreading In this section we will first consider general half-line Schrödinger operators and derive a number of consequences from polynomially bounded transfer matrices. We then turn to general Schrödinger operators on the line and use our half-line results to derive a lower bound for the exponent governing the spreading of the fastest part of the wavepacket in terms of the box counting dimension of the spectrum. These results are relevant in our context since for the Fibonacci potential, it is known that the transfer matrices are polynomially bounded for all parameter values and all energies in the spectrum. Consider a discrete half-line Schrödinger operator on 2 (Z+ ), [H+ u](n) = u(n + 1) + u(n − 1) + V (n)u(n),
(21)
with Dirichlet boundary condition u(0) = 0. Here, V is a bounded real-valued function. H+ is a bounded self-adjoint operator in 2 (Z+ ). The vector δ1 is cyclic and we denote its spectral measure by µ.
Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
509
For z ∈ C and m, n ∈ Z+ , we denote by T (n, m; z) the transfer matrix associated with the difference equation u(n + 1) + u(n − 1) + V (n)u(n) = zu(n).
(22)
That is, T (n, m; z) is the unique unimodular 2 × 2 matrix for which we have u(n + 1) u(m + 1) = T (n, m; z) u(n) u(m) for every solution of (22). To study the spreading of a wavepacket with initial state δ1 under the dynamics generated by H , one usually considers for p > 0, 2 ∞ −2t/T p −it H+ p |X |δ1 (T ) = e n |e δ1 , δn |2 dt. T 0 n∈Z+
One is interested in the power-law growth of this quantity and therefore studies the lower and upper transport exponents p
βδ−1 ( p)
= lim inf T →∞
log|X |δ1 (T ) p log T
p
,
βδ+1 ( p)
= lim sup T →∞
log|X |δ1 (T ) p log T
.
Both functions βδ±1 ( p) are nondecreasing in p and hence the following limits exist: αu± = lim p→∞ βδ±1 ( p). Alternatively (see [11, Theorem 4.1]), we also have αu± = sup{α ≥ 0 : S ± (α) < ∞}, where S − (α) = − lim inf T →∞
and
log P+ (T α − 2, T ) log P+ (T α − 2, T ) , S + (α) = − lim sup , log T log T T →∞
2 ∞ P+ (N , T ) = e−2t/T |e−it H+ δ1 , δn |2 dt. T 0 n>N
The following result derives several consequences from the fact that the transfer matrices are polynomially bounded for energies from some subset of the real line. We denote by µ the spectral measure associated with the vector δ1 . Theorem 6. Consider a half-line Schrödinger operator H+ with bounded potential as above. Let A ⊂ [−B, B], B > 0, and assume that there exist positive constants C, α such that for every E ∈ A and every N ≥ 1, T (n, m; E) ≤ C N α
for all m, n ∈ [1, N ].
(23)
(a) For every σ > 0, there exists ε0 (σ ) > 0 such that µ([E − ε, E + ε]) ≥ Dε L
(24)
for ε ∈ (0, ε0 ) and E ∈ A. Here, L = (1 + σ )(1 + 3α) and D depends only on σ .
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(b) For any p > 0, we have that βδ±1 ( p)
1 1 + 3α ≥ 1+ dim± . B (A) − p p
In particular,
αu± ≥ dim± B (A).
(25)
(26)
Proof. Germinet, Kiselev and Tcheremchantsev [11, Prop. 2.1] have shown that for any M > 0, µ ([E − ε, E + ε]) ≥ C1
E+ε/2
T (N , 0; x)−2 d x − C2 ε M .
(27)
E−ε/2
Here ε ∈ (0, 1), E ∈ [−B, B], N = ε−1−σ , σ > 0, and the constants C1 , C2 depend only on B, M, σ . Let E ∈ A. To bound the integral in (27) from below, we need an upper bound for T (N , 1; x) for x close to E. This can be accomplished using a perturbative argument of Simon; compare [9, Lemma 2.1]. Namely, if D(N , E) =
sup
T (n, m; E),
1≤n,m≤N
then for any δ ∈ C and 1 ≤ n ≤ N we have T (n, 1; E + δ) ≤ D(N , E) exp(D(N , E)|n|δ). By assumption, D(N , E) ≤ C N α for E ∈ A and N ≥ 1. Thus,
T (N , 0; E + δ) ≤ C N α exp C N 1+α δ , and for any δ with |δ| ≤ γ = 1/2N −1−α , we get T (N , 1; E + δ) ≤ C N α
(28)
with a uniform constant. Since N = ε−1−σ , ε ∈ (0, 1), we see that γ < ε/2 for ε small enough and therefore [E − γ , E + γ ] ⊂ [E − ε/2, E + ε/2]. Using (27) and (28), we find µ([E − ε, E + ε]) ≥ C1 ε(1+σ )(1+3α) − C2 ε M with appropriate constants. Taking M = (1 + σ )(1 + 3α) + 1, we obtain (24). To prove the dynamical lower bound (25), we will apply [31, Coro. 4.1]; see also [2]. In our notation, this result reads
βδ±1 ( p) ≥ Dµ± (q), q ∈ (1 + p)−1 , 1 , (29) where Dµ (q), 0 < q < 1 are the generalized fractal dimensions of the spectral measure; compare [2,3]. One of the equivalent ways of defining these dimensions is the following (cf. [31, Theorem 4.3]): Dµ+ (q) = lim sup ε→0
log Sµ (q, ε) , (q − 1) log ε
(30)
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511
and similarly for Dµ− (q) with lim sup replaced by lim inf, where Sµ (q, ε) = j∈Z (µ([ jε, ( j + 1)ε))q . To bound Sµ (q, ε) from below, we can follow the proof of [31, Theorem 4.5]. Let I j = [ jε, ( j + 1)ε). Denote by J A the set of j ∈ Z for which I j ∩ A = ∅. Thus, for every j ∈ J A , there exists E j ∈ [ jε, ( j + 1)ε) such that E j ∈ A. The bound (24) implies for ε small enough, µ(I j−1 ) + µ(I j ) + µ(I j+1 ) ≥ µ([E j − ε, E j + ε]) ≥ Dε L for every j ∈ J A , where L = (1 + σ )(1 + 3α). Thus, q B≡ µ(I j−1 ) + µ(I j ) + µ(I j+1 ) ≥ D q εq L 1. j∈Z
j∈J A
On the other hand, since (a+b)q
≤
a q +bq ,
Therefore, Sµ (q, ε) ≥ C(q)εq L
a, b > 0, q ∈ (0, 1), one has B ≤ 3Sµ (q, ε). 1 ≡ C(q)εq L N (ε) (31) j∈J A
with N (ε) = #{ j ∈ Z : I j ∩ A = ∅}. It follows from (30)–(31) that Dµ± (q) ≥
dim± B (A) − q L , q ∈ (0, 1). 1−q
(32)
The bounds (29) and (32) imply, for any q ∈ ((1 + p)−1 , 1), βδ±1 ( p) ≥
dim± B (A) − q L . 1−q
As this holds for L = (1+σ )(1+3α) with any σ > 0, letting σ → 0 and q → (1+ p)−1 , we obtain (25). Since αu± = lim p→+∞ βδ±1 ( p), the estimate (26) follows. Remark. One can improve the bounds (24), (25) if one has a nontrivial (i.e., better than ballistic) upper bound on αu+ . In this case [32], (27) holds with N = ε−ρ , where ρ = αu+ + σ , σ > 0. Thus, (24) holds with a smaller value of L. Let us now turn to the whole-line case and prove a result analogous to (26). Consider the operator H acting on 2 (Z) as [H u](n) = u(n + 1) + u(n − 1) + V (n)u(n) with bounded V : Z → R, and choose K so that σ (H ) ⊂ [−K + 1, K − 1]. The transfer matrices T (n, m; z) are defined in an analogous way. For N ≥ 1, recall that the time-averaged right outside probabilities are given by ∞ 2 Pr (N , T ) = T2 0 e−2t/T n>N e−it H δ1 , δn dt. It was proved in [10] that (33) C T −3 Ir (N , T ) ≤ Pr (N , T ) ≤ Ce−cN + C T 3 Ir (N , T ), K where Ir (N , T ) = −K T (N , 1; E + i/T )−2 d E. Although not stated explicitly, this result from [10] can be extended easily to the half-line case (21). Thus, for the timeaveraged outside probabilities P+ (N , T ), we have C T −3 I+ (N , T ) ≤ P+ (N , T ) ≤ Ce−cN + C T 3 I+ (N , T ), K where I+ (N , T ) = −K T (N , 1; E + i/T )−2 d E.
(34)
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Corollary 2. Suppose that H is a discrete Schrödinger operator on the line with bounded potential. Assume A ⊂ [−B, B] ⊂ R is such that for E ∈ A and N ≥ 1, T (n, m; E) ≤ C N α
for all m, n ∈ [1, N ]
(35)
with uniform constants C, α. Then, for the initial state δ1 , we have αu± ≥ dim± B (A). Proof. Let H+ be the half-line operator of the form (21) with potential V+ given by V+ (n) = V (n) for n ≥ 1. The transfer matrices are the same for both operators since they are defined locally, and thus we have Ir (N , T ) = I+ (N , T ). The condition (23) ± holds for the operator H+ and Theorem 6 therefore yields αu,+ ≥ dim± B (A) for the corresponding dynamics. − Let us assume that dim− B (A) > 0 (if dim B (A) = 0, the result is trivially true) and − − − show that αu ≥ dim B (A). Take any α with 0 < α < dim− B (A) ≤ αu,+ . Due to the ± definition of αu,+ , we have that P+ (T α − 2, T ) ≥ T −M
(36)
with some finite M > 0 for T sufficiently large. Since Ir (N , T) = I+ (N , T ), it follows from (33) and (34) that Pr (N , T ) ≥ C T −3 I+ (N , T ) ≥ C T −6 P+ (N , T ) − Ce−cN . It follows from (36) that for T sufficiently large, Pr (T α −2, T ) ≥ C T −M−6 . Consequently, for the full time-averaged outside probabilities, we find 2 2 ∞ −2t/T −it H e δ1 , δn dt ≥ C T −M−6 . P(T α − 2, T ) = e T 0 α |n|>T
It now follows directly from the definition of αu− that αu− ≥ α. Since this holds for every + α < dim− B (A), the result follows. For αu , the proof is completely analogous. (The bound (36) then holds for some sequence of times.) Remarks. (a) One can prove a similar result under the condition T (n, m; E) ≤ C N α for all m, n ∈ [−N , −1], where E ∈ A, N ≥ 1. (b) This establishes Theorem 3 as stated in the Introduction since it has been shown that in the Fibonacci case, the transfer matrices are polynomially bounded for all energies in the spectrum with uniform constants C, α. See Iochum-Testard [14] for the case θ = 0 and Damanik-Lenz [8] for the case of general θ . Appendix A. The Hyperbolicity of the Trace Map at Large Coupling and Some of its Consequences A.1. Description of the trace map. The main tool that we are using here is the trace map. It was originally introduced in [15,18]; see also [28] for proofs of the results described below. Let us recall that the numbers xk = xk (E, λ) introduced in Sect. 2 satisfy the recursion relation (37) xk+1 = xk xk−1 − xk−2 , with initial conditions x−1 = 2, x0 = E, x1 = E − λ, and the invariance relation 2 2 + xk2 + xk−1 − xk+1 xk xk−1 = 4 + λ2 xk+1
(38)
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for every k ∈ Z+ . Because of (37), it is natural to consider the so-called trace map, T : R3 → R3 , T (x, y, z) = (x y − z, x, y). The sequence {x1 , x2 , x3 , . . .} can be considered as the sequence of first coordinates of points in the trace map orbit having initial condition (x1 , x0 , x−1 ). By (38), the following function is invariant under the action of T : I = x 2 + y 2 + 2 z − x yz − 4. That is, T preserves the family of cubic surfaces S I = {(x, y, z) ∈ R3 : x 2 + y 2 + z 2 − x yz − 4 = I }. The surface S0 is called the Cayley cubic, and we define the line lλ = {(E − λ, E, 2) : E ∈ R}. It is easy to check that lλ ⊂ Sλ2 . The following result, proved in [28], characterizes the spectrum of the Fibonacci Hamiltonian in terms of trace map dynamics and therefore establishes an important and fruitful connection between spectral and dynamical issues in this context. Theorem 7. The energy E belongs to the spectrum λ of the Fibonacci Hamiltonian if and only if the positive semiorbit of the point (E − λ, E, 2) under iterates of the trace map T is bounded. A.2. Hyperbolicity of the trace map for large λ. Denote by f λ the restriction of the trace map to the invariant surface Sλ2 . That is, f λ : Sλ2 → Sλ2 and f λ = T |Sλ2 . Denote by λ the set of points in Sλ2 whose full orbits under f λ are bounded. Let us recall that an invariant set of a diffeomorphism f : M → M is locally maximal if there exists a neighborhood U () such that = n∈Z f n (U ). An invariant closed set of a diffeomorphism f : M → M is hyperbolic if there exists a splitting of the tangent space Tx M = E xs ⊕ E xu at every point x ∈ such that it is invariant under D f , and D f exponentially contracts vectors from the stable subspaces {E xs } and exponentially expands vectors from the unstable subspaces {E xu }. See [12] for a detailed survey of hyperbolic dynamics and an extensive list of references. Casdagli proved the following result [5]. Theorem 8. For every λ ≥ 16, the set λ is a locally maximal invariant hyperbolic set of f λ : Sλ2 → Sλ2 . A.3. Some properties of locally maximal hyperbolic invariant sets of surface diffeomorphisms. Consider a locally maximal invariant transitive hyperbolic set ⊂ M, dim M = 2, of a diffeomorphism f ∈ Diff r (M), r ≥ 1. We have = ∩n∈Z f n (U ()) for some neighborhood U (). Assume also that dim E u = dim E s = 1. Let us gather several results in this general context that we will eventually specialize to the case where is given by λ for f λ : Sλ2 → Sλ2 , λ ≥ 16. I. Stability. There is a neighborhood U ⊂ Diff 1 (M) of the map f such that for every g ∈ U, the set g = ∩n∈Z g(U ()) is a locally maximal invariant hyperbolic set of g. Moreover, there is a homeomorphism h : → g that conjugates f | and g|g , that is, the following diagram commutes: f |
−−−−→ ⏐ ⏐ h g|g
⏐ ⏐ h
g −−−−→ g
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II. Invariant manifolds. For x ∈ and small ε > 0, consider the local stable and unstable sets, Wεs (x) = {w ∈ M : d( f n (x), f n (w)) ≤ ε for all n ≥ 0} and Wεu (x) = {w ∈ M : d( f n (x), f n (w)) ≤ ε for all n ≤ 0}. If ε > 0 is small enough, then these are embedded C r -disks with Tx Wεs (x) = E xs and Tx Wεu (x) = E xu . Define the (global) stable and unstable sets as W s (x) = ∪n∈Z+ f −n (Wεs (x)) and W u (x) = ∪n∈Z+ f n (Wεu (x)). Define also W s () = ∪x∈ W s (x) and W u () = ∪x∈ W u (x). III. Invariant foliations. A stable foliation for is a foliation F s of a neighborhood of such that (a) for each x ∈ , F(x), the leaf containing x, is tangent to E xs ; (b) for each x ∈ , sufficiently near , f (F s (x)) ⊂ F s ( f (x)). An unstable foliation F u can be defined in a similar way. For a locally maximal hyperbolic set ⊂ M of a C 1 -diffeomorphism f : M → M, dim M = 2, stable and unstable C 0 foliations with C 1 -leaves can be constructed [20]. In the case of C 2 -diffeomorphisms, C 1 invariant foliations exist (see [24], Theorem 8 in Appendix 1). IV. Local Hausdorff dimension and box counting dimension. Consider, for x ∈ and small ε > 0, the set Wεu (x) ∩ . The Hausdorff dimension of this set does not depend on x ∈ and ε > 0, and coincides with its box counting dimension (see [22,30]): dim H Wεu (x) ∩ = dim B Wεu (x) ∩ . In a similar way, dim H Wεs (x) ∩ = dim B Wεs (x) ∩ . Denote h s = dim H Wεs (x) ∩ and h u = dim H Wεu (x) ∩ . We will call h s and h u the local stable and unstable Hausdorff dimensions of , respectively. V. Global Hausdorff dimension. Moreover, the Hausdorff dimension of is equal to its box counting dimension and dim H = dim B = h s + h u ; see [22,25]. VI. Continuity of the Hausdorff dimension. The local Hausdorff dimensions h s () and h u () depend continuously on f : M → M in the C 1 -topology; see [22,25]. Therefore, dim H f = dim B f = h s ( f ) + h u ( f ) also depends continuously on f in the C 1 -topology. Moreover, for r ≥ 2 and C r -diffeomorphisms f : M → M, the Hausdorff dimension of a hyperbolic set f is a C r −1 function of f ; see [20]. Remark. For hyperbolic sets in dimension greater than two, most of these properties do not hold in general; see [26] for more details. A.4. Implications for the trace map and the spectrum. Due to Theorem 8, Properties I–VI can all be applied to the hyperbolic set λ of the trace map f λ : Sλ2 → Sλ2 for every λ ≥ 16. One can extract the following statement from the material in [5, Sect. 2]. Lemma 7. For λ ≥ 16 and every x ∈ λ , the stable manifold W s (x) intersects the line lλ transversally. The existence of a C 1 -foliation F s allows us to locally consider the set W s (λ ) ∩ lλ as a C 1 -image of a set Wεu (x) ∩ λ . Therefore, we obtain the following consequences for the spectrum of the Fibonacci Hamiltonian. Theorem 9. For λ ≥ 16, the following statements hold:
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(a) The spectrum λ depends continuously on λ in the Hausdorff metric. (b) We have dim H (λ ) = dim B (λ ). (c) For every small ε > 0 and every E ∈ λ , we have dim H ((E − ε, E + ε) ∩ λ ) = dim H (λ ) and dim B ((E − ε, E + ε) ∩ λ ) = dim B (λ ). (d) The Hausdorff dimension dim H (λ ) is a C ∞ -function of λ. In particular, part (b) establishes Theorem 1 as formulated in the Introduction. Acknowledgements. We are grateful for the On-Line Encylopedia of Integer Sequences and the Inverse Symbolic Calculator, both of which assisted our hunt for f ∗ .
References 1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1965 2. Barbaroux, J.-M., Germinet, F., Tcheremchantsev, S.: Fractal dimensions and the phenomenon of intermittency in quantum dynamics. Duke Math. J. 110, 161–193 (2001) 3. Barbaroux, J.-M., Germinet, F., Tcheremchantsev, S.: Generalized fractal dimensions: equivalences and basic properties. J. Math. Pures Appl. 80, 977–1012 (2001) 4. Bellissard, J., Iochum, B., Scoppola, E., Testard, D.: Spectral properties of one-dimensional quasicrystals. Commun. Math. Phys. 125, 527–543 (1989) 5. Casdagli, M.: Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Commun. Math. Phys. 107, 295–318 (1986) 6. Damanik, D.: Strictly ergodic subshifts and nassociated operators. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, Proceedings of Symposia in Pure Mathematics 74, Providence, RI: Amer. Math. Soc. 2006, pp. 505–538 7. Damanik, D., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals. I. Absence of eigenvalues. Commun. Math. Phys. 207, 687–696 (1999) 8. Damanik, D., Lenz, D.: Uniform spectral properties of one-dimensional quasicrystals. II. The Lyapunov exponent. Lett. Math. Phys. 50, 245–257 (1999) 9. Damanik, D., Tcheremchantsev, S.: Power-law bounds on transfer matrices and quantum dynamics in one dimension. Commun. Math. Phys. 236, 513–534 (2003) 10. Damanik, D., Tcheremchantsev, S.: Upper bounds in quantum dynamics. J. Amer. Math. Soc. 20, 799–827 (2007) 11. Germinet, F., Kiselev, A., Tcheremchantsev, S.: Transfer matrices and transport for Schrödinger operators. Ann. Inst. Fourier (Grenoble) 54, 787–830 (2004) 12. Hasselblatt, B.: Hyperbolic dynamical systems. In: Handbook of Dynamical Systems,Vol. 1A. Amsterdam: North-Holland, 2002, pp. 239–319 13. Hof, A., Knill, O., Simon, B.: Singular continuous spectrum for palindromic Schrödinger operators. Commun. Math. Phys. 174, 149–159 (1995) 14. Iochum, B., Testard, D.: Power law growth for the resistance in the Fibonacci model. J. Stat. Phys. 65, 715–723 (1991) 15. Kadanoff, L.: Analysis of cycles for a volume preserving map. Unpublished 16. Kaminaga, M.: Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential. Forum Math. 8, 63–69 (1996) 17. Killip, R., Kiselev, A., Last, Y.: Dynamical upper bounds on wavepacket spreading. Amer. J. Math. 125, 1165–1198 (2003) 18. Kohmoto, M., Kadanoff, L.P., Tang, C.: Localization problem in one dimension: Mapping and escape. Phys. Rev. Lett. 50, 1870–1872 (1983) 19. Kotani, S.: Jacobi matrices with random potentials taking finitely many values. Rev. Math. Phys. 1, 129–133 (1989)
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20. Mañé, R.: The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces. Bol. Soc. Brasil. Mat. (N.S.) 20, 1–24 (1990) 21. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability. Cambridge: Cambridge University Press, 1995 22. McCluskey, H., Manning, A.: Hausdorff dimension for horseshoes. Ergodic Theory Dynam. Systems 3, 251–261 (1983); Erratum. Ergodic Theory Dynam. Systems 5, 319 (1985) 23. Ostlund, S., Pandit, R., Rand, D., Schellnhuber, H.J., Siggia, E.D.: One-dimensional Schrödinger equation with an almost periodic potential. Phys. Rev. Lett. 50, 1873–1877 (1983) 24. Palis, J., Takens, F.: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors. Cambridge: Cambridge University Press, 1993 25. Palis, J., Viana, M.: On the continuity of the Hausdorff dimension and limit capacity for horseshoes, In: Dynamical Systems, Lecture Notes in Mathematics 1331, Berlin: Springer, 1988, pp. 150–160 26. Pesin, Ya.: Dimension Theory in Dynamical Systems. Chicago: University of Chicago Press, 1997 27. Raymond, L.: A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain. Preprint (1997) 28. Süt˝o, A.: The spectrum of a quasiperiodic Schrödinger operator. Commun. Math. Phys. 111, 409–415 (1987) 29. Süt˝o, A.: Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian. J. Stat. Phys. 56, 525–531 (1989) 30. Takens, F.: Limit capacity and Hausdorff dimension of dynamically defined Cantor sets. In: Dynamical Systems, Lecture Notes in Mathematics 1331, Springer, Berlin, 1988, pp. 196–212 31. Tcheremchantsev, S.: Mixed lower bounds for quantum transport. J. Funct. Anal. 197, 247–282 (2003) 32. Tcheremchantsev, S.: In preparation Communicated by B. Simon
Commun. Math. Phys. 280, 517–544 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0455-z
Communications in
Mathematical Physics
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps J.-B. Bru1 , M. Correggi2 , P. Pickl1 , J. Yngvason1,3 1 Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090 Vienna, Austria 2 Scuola Normale Superiore SNS, Piazza dei Cavalieri 7, 56126 Pisa, Italy 3 Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna, Austria.
E-mail: [email protected] Received: 10 May 2007 / Accepted: 20 August 2007 Published online: 14 March 2008 – © Springer-Verlag 2008
Abstract Starting from the full many body Hamiltonian we derive the leading order energy and density asymptotics for the ground state of a dilute, rotating Bose gas in an anharmonic trap in the ‘Thomas Fermi’ (TF) limit when the Gross-Pitaevskii coupling parameter and/or the rotation velocity tend to infinity. Although the many-body wave function is expected to have a complicated phase, the leading order contribution to the energy can be computed by minimizing a simple functional of the density alone. 1. Introduction Rotating Bose-Einstein condensates exhibit fascinating quantum phenomena like superfluidity and quantization of vorticity and their study is currently an active area of both experimental and theoretical research. Much of the theoretical work (see, e.g., the monograph [A] where an extensive list of references can be found) is based on an effective description of the ground state in terms of the Gross-Pitaeveskii (GP) equation that has recently been proved [LSe,S1] to be exact in a suitable limit. The corresponding result for the non-rotating case was obtained in [LSY1], but the rotating case was a longstanding problem whose solution required different techniques from those of [LSY1]. In fact, the rotating case differs markedly from the non-rotating one since the absolute many-body ground state is in general not the same as the bosonic ground state [S1]. The limit considered in [LSe,S1] is the GP limit of the many-body ground state which means that the particle number N tends to infinity while the GP parameter N a, with a the scattering length of the interaction potential, as well as the rotational velocity, Ω, are kept fixed. In several experiments the GP parameter can be quite large, however, and also the rotational velocity can be so large that the effect of the rotation becomes comparable with that of the interactions. Some of the interesting phenomena expected under such conditions are discussed in [B,BP,ECHSC,F,FB,FZ,KTU,KB,Lu,WGBP]. To describe such cases theoretically it is natural to consider the limit when N a and/or Ω tend to infinity. Since the error estimates in [LSe,S1] are not uniform in these parameters
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the results of [LSe,S1] do not apply to this situation and a complementary investigation is called for. In order to allow arbitrarily large rotational velocities we shall require that the trap potential increases more rapidly at infinity than quadratically (anharmonic traps). In a recent paper [CRY1] the limit of GP theory (but not the many-body problem) for large N a and large Ω was studied for a rotating 2D gas in a ‘flat’ trap with walls that confine the gas to a fixed bounded region. (See also [CRY2] for an extension to more general 2D traps.) It was proved that the leading order energy and density asymptotics is correctly described by an energy functional of the density without a gradient term. This functional for rapidly rotating gases was first introduced and studied in [FB]. Because of its formal analogies with the Thomas-Fermi (TF) theory for fermions it is also referred to as a TF functional although the physical situation is quite different. In the present paper we derive the TF description from the many-body problem in 3D. This might at first sight appear to be a simple combination of the results of [LSe,S1] with those of [CRY1] (extended to 3D and more general traps): first take the GP limit of the many-body theory and then the TF limit of the GP theory. But such an argument is not valid because the error estimates in [LSe] blow up if N a and/or Ω tend to infinity. In fact, it is clear that it will at least be necessary to require explicitly that the gas is dilute in the sense that the average particle distance is much larger than the scattering length. (This condition is automatically fulfilled if N a is kept fixed.) It is also instructive to compare with the proof of dimensional reduction of Bose gases in tightly confining traps [LSY4,SY] where one also has the option of taking a limit in two steps. Here the 1D or 2D limits of the 3D GP theory do not, in fact, cover all cases that can occur when one starts with the 3D many-body theory and takes its tight confinement limit directly. In the present situation, however, the TF limit of the GP theory exhausts the leading order asymptotics for the many-body ground state, provided only the gas remains dilute as the limit is taken. Because the many-body wave function will have a complicated phase it is not self-evident that the ground state energy can be computed accurately by a functional of the density alone. That this is indeed possible is a basic message of the present paper. The main techniques applied in our derivation of the TF limit are on the one hand an extension of [CRY1] to 3D and anharmonic, homogeneous potentials, and on the other hand the techniques described in [LSY5] for treating the many-body problem in the non-rotating case. Additional tools are the diamagnetic inequality as well as the method used in [S1] to bound the many-body energy from above by the GP energy in the rotating case. Our results concern the ground state energy and density to leading order in small parameters (the reciprocal of the coupling constant or the rotational velocity as well as the ratio of a to the mean particle spacing). In contrast to [LSe] Bose-Einstein condensation (BEC) is not proved. In fact, even in the non-rotating case a proof of BEC in the TF limit is still an open problem. The variational wave functions we employ for deriving upper bounds to the energy reflect the expected short scale structure due to interactions as well as the quantized vortices generated by the rotation. In 3D the exact vortex structure is presumably rather complicated, e.g., due to bending of vortex lines [A1]. Our trial functions do not model such details but they are sufficient for leading order calculations and leave room for improvements. The paper is organized as follows. In the next section we describe the general setting and state our main results about the energy and density asymptotics. Section 3 is concerned with the TF limit of GP theory, generalizing the results of [CRY1] to 3D and homogeneous, anharmonic trapping potentials, as well as some basic properties of the TF
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theory needed for the proofs of the quantum mechanical (QM) limit theorems. In Sect. 4 we prove the upper and lower bounds on the many-body quantum-mechanical energy and obtain limit theorems for the density as corollaries. Some additional properties of the TF density are discussed in an Appendix. 2. General Setting and Main Results We consider a system of N spinless bosons in R3 with mass m, trapped in an external potential V and interacting with a nonnegative, radially symmetric pair potential v of finite range. The basic quantum mechanical Hamiltonian in a reference frame that rotates with angular velocity Ω is N HN ≡ − j − L j · Ω + V (x j ) + j=1
v(|x i − x j |),
(2.1)
1≤i< j≤N
where units have been chosen so that 2m = = 1. Also1 x j ∈ R3 and L j ≡ −i x j ∧ ∇ j for j = 1, . . . , N are respectively the positions and the angular momentum operators of the operates on symmetric (bosonic) wave functions in particles. The Hamiltonian L 2 R3N , dx 1 . . . dx N . We keep the external potential fixed and choose the associated length scale (the extension of the ground state of − + V ) as a length unit. In order to be able to vary the scattering length2 of the interaction potential v with N we write v(|x|) = a −2 v1 (|x|/a), where v1 is a potential with scattering length 1. Then v has scattering length a and for fixed v1 the Hamiltonian is parametrized by N and a besides Ω. For convenience, we include a factor 4π in the definition of the GP parameter g ≡ 4π N a.
(2.2)
The ground state energy of (2.1), i.e., the infimum of its spectrum, will be denoted by QM E g,Ω (N ). Introducing the vector potential AΩ ≡ 21 Ω(ez ∧ x) associated with a rotation Ω = Ω ez , where ez is the unit vector in z-direction, the Hamiltonian H N can be rewritten in the form N 2 HN = −i∇ j − AΩ (x j ) + V (x j ) − 41 Ω 2 r 2j + j=1
v(|x i − x j |),
(2.3)
1≤i< j≤N
where r ≡ |ez ∧ x| is the distance from the axis of rotation. The splitting of the term −L · Ω into the contribution of the vector potential and the term −Ω 2 r 2 /4 corresponds respectively to the Coriolis and the centrifugal force in the rotating frame. The vector potential is primarily responsible for the formation of vortices while the centrifugal potential affects the overall density profile if the rotational velocity is high enough. 1 We use the following notation: x ≡ (x, y, z) will always denote a point in R3 , while r and z are its projections on the x, y−plane and the z−axis respectively (cylindrical coordinates), namely x ≡ (r, z). Moreover |x| and r ≡ |r| will denote the moduli of the corresponding vectors, whereas ζ ≡ x + i y will be the complex number associated with the x, y−coordinates of x. 2 Alternatively, one could keep a fixed and vary the external potential and hence the length scale. What matters is only the ratio of a to this scale.
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In order that (2.3) is bounded from below and trapping for all Ω we require the external potential V to be bounded from below and moreover that V (x) − 41 Ω 2 r 2 → ∞
for |x| → ∞.
(2.4)
Due to the particle repulsion and the centrifugal force the gas cloud expands as g or Ω tend to infinity so essentially only the behavior of V for large |x| matters. For simplicity we shall assume that V is a homogeneous function3 of order s > 2, i.e. V (λx) = λs V (λx) for any λ > 0 and x ∈ R3 , but we need not assume that V is symmetric w.r.t. rotations about the z-axis. In order to obtain explicit error estimates in Sect. 3 we shall assume that V is twice continuously differentiable but Hölder continuity would in fact be sufficient for the main results. The case of a ‘flat’ trapping potential, i.e., V = 0 within a bounded, open set B with a smooth boundary and ∞ outside, can also be treated by our methods. This case corresponds formally to s = ∞ and the formulas for the energy and density asymptotics can be obtained as the s → ∞ limit of the formulas for finite s. At the end of Sect. 4 we shall comment on the results for flat traps and the minor modifications required of the proofs. The GP functional in the rotating frame is defined as
GP Eg,Ω (2.5) dx |[∇ − i AΩ ] φ|2 + V |φ|2 − 41 Ω 2 r 2 |φ|2 + g |φ|4 [φ] ≡ R3
on the domain
DGP ≡ φ ∈ L 4 R3 : V |φ|2 ∈ L 1 R3 and [∇ − i AΩ ] φ ∈ L 2 R3 .
(2.6)
The corresponding energy is GP E g,Ω ≡
inf
φ∈D GP ,φ2 =1
GP Eg,Ω [φ] .
(2.7)
GP . The infimum is, in fact, a minimum and we denote any normalized minimizer by φg,Ω GP ≡ |φ GP |2 . The minimizer may not be unique because The corresponding density is ρg,Ω g,Ω vortices can break rotational symmetry, but any minimizer satisfies the variational (GP) equation GP 2 GP 2 GP 1 2 2 −(∇ − i AΩ ) + V − 4 Ω r + 2g φg,Ω φg,Ω = µGP (2.8) g,Ω φg,Ω , GP where µGP g,Ω is the GP chemical potential. Multiplying (2.8) with φg,Ω and integrating gives GP GP 2 µGP g,Ω = E g,Ω + 2gρg,Ω 2 . QM
(2.9)
GP → 1 as N → ∞ if g and Ω are In [LSe,S1] it is proved that E g,Ω (N )/N E g,Ω fixed. In the present paper we are concerned with the situation where g and/or Ω tend to 3 Asymptotic homogeneity in the sense of [LSY2] would also suffice.
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infinity together with N . As we shall show, the first term in (2.5) is negligible in this limit and the ground state energy and density can be described in terms of the TF functional
TF (2.10) dx Vρ − 41 Ω 2 r 2 ρ + gρ 2 Eg,Ω [ρ] ≡ R3
defined on the domain
DTF ≡ ρ ∈ L 2 R3 : ρ ≥ 0, Vρ ∈ L 1 R3
(2.11)
with the energy TF E g,Ω ≡
inf
ρ∈D TF ,ρ1 =1
TF Eg,Ω [ρ] .
The minimization problem (2.12) has a unique solution given by 1 TF TF µg,Ω + 41 Ω 2 r 2 − V (x) , ρg,Ω (x) = + 2g
(2.12)
(2.13)
where [·]+ denotes the positive part and µTF g,Ω is the TF chemical potential determined TF TF and integrating gives by the normalization ||ρg,Ω ||1 = 1. Multiplying (2.13) by ρg,Ω TF TF 2 µTF g,Ω = E g,Ω + 2g||ρg,Ω ||2 .
(2.14)
By simple rescaling (explained at the beginning of Sect. 3) we obtain the relations TF TF TF = E 1,ω and g −s/(s+3) µTF g −s/(s+3) E g,Ω g,Ω = µ1,ω
(2.15)
ω ≡ g −(s−2)/(2s+6) Ω
(2.16)
TF TF g 1/(s+3) x = ρ1,ω g 3/(s+3) ρg,Ω (x) .
(2.17)
with
and likewise
The case ω = 0 corresponds to the standard TF functional without rotation whose relation to the many-body problem was already discussed in [LSY5]. Note also that TF is a decreasing function of ω ≥ 0 with range (−∞, E TF ] with E TF > 0. In E 1,ω 1,0 1,0 TF = 0. particular, there is an ω > 0 such that E 1,ω In the case that ω tends to infinity the rotational term completely dominates the interaction term. In this case it is appropriate to scale lengths with Ω 2/(s−2) rather than g 1/(s+3) (cf. Sect. 3), obtaining TF TF = E γTF,1 and Ω −2s/(s−2) µTF Ω −2s/(s−2) E g,Ω g,Ω = µγ ,1
(2.18)
γ ≡ Ω −2(s+3)/(s−2) g = ω−2(s+3)/(s−2)
(2.19)
TF Ω 2/(s−2) x = ργTF,1 (x) . Ω 6/(s−2) ρg,Ω
(2.20)
with
and
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Moreover, as ω → ∞, i.e., γ → 0, we have TF = inf lim E γTF,1 = E 0,1
γ →0
x ∈R3
V (x) − 41 r 2 < 0,
(2.21)
while ργTF,1 converges to a measure supported on the set M of minima of the function W (x) ≡ V (x) − 41 r 2 . This together with the other facts mentioned about the TF theory is discussed further in Sect. 3.2. The scaling properties of the TF theory already suggest that one should distinguish between the following three cases when the N → ∞ limit of the many-body ground state with g and/or Ω also tending to infinity is considered: – Slow or moderate rotation, ω 1: The effect of the rotation is negligible to leading order.4 – Rapid rotation, ω ∼ 1: Rotational effects are comparable to those of the interactions. – Ultrarapid rotation, ω 1: Rotational effects dominate. Moreover, a description in terms of the simple density functional (2.10) can only be expected in a dilute limit. A convenient measure for diluteness turns out to be smallness of the parameter TF TF a 3 N ρg,Ω ∞ ∼ N −2 g 3 ρg,Ω ∞ .
(2.22)
Our main result about the energy asymptotics is the following Theorem 2.1 (QM energy asymptotics). Let V be a homogenous potential of order s > TF → 0 2, define g = 4πa N and ω = g −(s−2)/(2s+6) Ω, and assume that N −2 g 3 ρg,Ω ∞ as N → ∞. (i) If g → ∞ and ω → 0 as N → ∞, then
QM TF lim g −s/(s+3) N −1 E g,Ω (N ) = E 1,0 ; N →∞
(ii) If g → ∞ and ω > 0 is fixed as N → ∞, then
QM TF ; lim g −s/(s+3) N −1 E g,Ω (N ) = E 1,ω N →∞
(iii) If Ω → ∞ and ω → ∞ as N → ∞, then
QM TF . lim Ω −2s/(s−2) N −1 E g,Ω (N ) = E 0,1 N →∞
Although stated separately, cases (i) and (ii) can, in fact, be treated together because the convergence of the scaled energy is uniform in ω as long as ω stays bounded. Besides the energy asymptotics we shall also consider the convergence of the quantum mechanical particle density of ground states or approximate ground states. We say that a sequence of bosonic, normalized wave functions Ψ N ∈ L 2 (R3N ) (depending also on the parameters g and Ω) is an approximate ground state if QM
Ψ N , H N Ψ N /E g,Ω (N ) → 1.
(2.23)
4 This regime could be subdivided further into ‘slow’ rotations, where vortices do not yet form, and ‘moderate’ rotations where vortices are present. This finer division goes, however, beyond the leading order considerations of the present paper.
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QM
(Recall that E g,Ω (N ) is, by definition, the infimum of the spectrum of (2.1).) The particle density, normalized so that its integral is 1, is defined by QM |Ψ N (x, x 2 , . . . , x N )|2 dx 2 ....dx N . ρ N ,g,Ω (x) ≡ (2.24) R3(N −1)
QM
To ensure convergence of ρ N ,g,Ω (x) we have to rescale it in accord with (2.17). For bounded ω, convergence of the density follows from the convergence of the energy using standard arguments [G,LSi]. Theorem 2.2 (QM density asymptotics for ω < ∞). Under the conditions of Theorem 2.1 (i) or (ii) we have QM TF g 3/(s+3) ρ N ,g,Ω g 1/(s+3) x → ρ1,ω (2.25) (x) in weak L 1 sense. The case of ultrarapid rotations, ω → ∞, is a little more delicate. Recall that M ⊂ R3 was defined as the set where the function V (x)− 41 r 2 attains its minimum. In Lemma 3.1 it is shown that this set is a subset of a cylinder and that the scaled TF density (2.20) is eventually concentrated on M. In physical terms, the ultrastrong centrifugal forces outweigh the repulsion between particles and constrain them to the boundary of the available region (in the scaled variables). We show that the same holds for the scaled QM density. Theorem 2.3 (QM density asymptotics for ω → ∞). Under the conditions of Theorem 2.1 (iii) the scaled particle density QM Ω 6/(s−2) ρ N ,g,Ω Ω 2/(s−2) x (2.26) becomes concentrated on the set M as N → ∞ in the sense that the integral of (2.26) over any measurable set that has a strictly positive distance from M tends to zero. The proofs of Theorems 2.1–2.3 will be given in several steps. In the next section, we analyze the asymptotic properties of the GP and TF theories when g and/or Ω tend to infinity. Section 4 contains the upper and lower bounds to the quantum mechanical energy that complete the proof of Theorem 2.1. Theorem 2.2 is derived from the convergence of the energy by the arguments of [G,LSi] and Theorem 2.3 is proved by showing that any mass outside the set M would be in conflict with Theorem 2.1 (iii). 3. From GP to TF In this section we consider the TF limit of GP theory. Since for s < ∞ the GP and TF minimizers spread out over an increasingly large region as g and/or Ω tend to infinity, we write x = λx and φ(x) = λ−3/2 φ (x ) with a suitable length scale λ depending on g or Ω, so that the relevant x remain essentially bounded. We have φ 2 = φ2 = 1, and since V (x) = λs V (x ) by assumption, the GP functional can be written 2 2 GP −2 Eg,Ω [φ] = λ dx ∇ − i Aλ2 Ω (x ) φ + λs+2 V (x ) φ 3 R 4
2 2 1 2 (3.1) − 4 (λ Ω)2 r φ + gλ−1 φ .
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We now distinguish two cases. When the rotational contribution to the energy is smaller than or at most comparable to the interaction energy we equate λs+2 with gλ−1 , i.e., choose λ = g 1/(s+3) .
(3.2)
For convenience and comparison with [CRY1,CRY2] and [A] we define a small parameter ε by 1/ε2 ≡ λs+2 = λ−1 g = g (s+2)/(s+3) .
(3.3)
g −(s−2)/(2s+6) Ω,
The parameter ω = that measures the relative strength of the rotation with respect to the interactions can then be written ω = ε(s−2)/(s+2) Ω.
(3.4) GP [φ] = λ−2 E˜ GP φ , or by writing Eg,Ω ε,ω
GP We now define a rescaled GP functional E˜ε,ω explicitly, dropping the primes, 2 1 ω2 r 2 GP |φ|2 + |φ|4 . (3.5) E˜ε,ω dx ∇ − i Aω/ε φ + 2 V |φ|2 − [φ] ≡ ε 4 R3
In the next subsection we study the ε → 0 limit of this functional with ω < ∞ fixed or tending to zero. When the rotation dominates the interaction we take λs+2 to be equal to (λ2 Ω)2 in (3.1), i.e, we take λ = Ω 2/(s−2) .
GP [φ] = λ−2 Eˆ GP φ with Proceeding as before we now write Eg,Ω Ω,ω
2 GP EˆΩ,ω V − 41 r 2 |φ|2 + γ |φ|4 , dx |[∇ − i AΩ ] φ|2 + Ω [φ] ≡ R3
(3.6)
(3.7)
where Ω ≡ Ω (s+2)/(s−2) and γ = ω−2(s+3)/(s−2) . The limit Ω → ∞, ω →
∞ (i.e., Ω
(3.8)
→ ∞, γ → 0) will be considered in Subsect. 3.2.
3.1. The Regime ω < ∞. For ω < ∞ fixed or tending to zero, we study the rescaled GP by GP functional introduced in (3.5) and define E˜ ε,ω GP ≡ E˜ ε,ω
inf
φ∈D GP ,φ2 =1
GP GP E˜ε,ω [φ] = g −2/(s+3) E g,Ω .
(3.9)
GP as g → ∞ is then given by the limit of E˜ GP The asymptotic behavior of g −2/(s+3) E g,Ω ε,ω GP in the following theorem. as ε → 0, see (3.3). We state this result about E g,Ω
Theorem 3.1 (GP energy asymptotics for ω < ∞). Under the conditions of Theorem 2.1 (i) or (ii) one has, as g → ∞, GP TF = E 1,ω + O g −(s+2)/(2s+6) log g . (3.10) g −s/(s+3) E g,Ω
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Proof. The proof is obtained by a comparison between suitable lower and upper bounds GP , following closely the proof of Theorem 2.1 in [CRY1]. for E˜ ε,ω The two regimes of slow (i) and rapid (ii) rotations can be treated together. Actually, the lower and upper bounds in the case (ii) are sufficient to get the result in the case (i) because the estimates are uniform on bounded intervals of ω and the TF ground state energy is a continuous function of ω. To simplify our notation, we denote by Cω any constant independent of ε. In particular, Cω needs not be the same from one equation to another, but Cω is always uniformly bounded for ω bounded. By simply neglecting the positive contribution of the kinetic energy in (3.5) one obtains the lower bound GP TF TF ε2 E˜ε,ω [φ] ≥ E1,ω [|φ|2 ] ≥ E 1,ω .
(3.11)
For an upper bound we test the functional (3.5) with a trial function of the form ˜ φ(x) = cε ˜ ε (x) χε (r)gε (r),
(3.12)
where gε is a phase factor, χε (r) a function that vanishes at the singularities of gε and TF . Note that both g and χ depend only on the 2D ˜ ε a suitable regularization of ρ1,ω ε ε coordinate r. For simplicity of notation we have suppressed the dependence on ω that is regarded as fixed. TF is analogous to the one used in [LSY2], Lemma 2.3, and The regularization of ρ1,ω TF , with is explicitly given by ˜ ε ≡ jε ρ1,ω 1 |x| . exp − jε (x) ≡ 4π ε3 ε
(3.13)
Since jε 1 = 1, ˜ ε is L 2 −normalized. It is also clear that ˜ ε converges uniformly TF as ε → 0 and it is uniformly bounded in ε, i.e., there exists a constant C such to ρ1,ω ω that ˜ ε ≤ Cω . Furthermore, although ˜ ε is not compactly supported, it is exponentially TF . More precisely, denoting5 small in ε for x sufficiently far from the support of ρ1,ω TF Rω ≡ sup{|x| : ρ1,ω (x) > 0} < ∞,
(3.14)
one has for any x ∈ R3 , |x| > Rω , x − x TF (x ) ρ1,ω dx exp − TF ε supp ρ1,ω 1 |x| − Rω . ≤ exp − 4π ε3 ε
1 ˜ ε (x) = 4π ε3
(3.15)
We also observe that the gradient of ˜ ε can be bounded in two ways. By using the fact that |∇ jε | = ε−1 jε , one can easily prove that |∇ ˜ ε | ≤ ε−1 | ˜ ε | , 5 By (2.13) the support of ρ TF is a compact set, uniformly bounded in ω, if ω remains bounded. 1,ω
(3.16)
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TF , i.e., TF ≤ C , one has whereas, by exploiting the regularity of ρ1,ω ∇ρ1,ω ω 1 TF TF dx |∇ ˜ ε | ≤ dx dx ∇ρ1,ω (x − x ) jε (x ) = ∇ρ1,ω ≤ Cω . R3
R3
R3
1
(3.17)
Writing points r = (x, y) ∈ R2 as complex numbers ζ ≡ x + i y, the phase gε is defined by ζ − ζj , (3.18) gε (ζ ) ≡ |ζ − ζ j | ζ j ∈L
where L is a square lattice of spacing ε defined in the following way: (3.19) L ≡ {r = (mε , nε ), m, n ∈ Z : r < 2Rω − ε } , √ √ with Rω defined by (3.14). We also assume that the spacing is of order ε, i.e., ε = δ ε, for some δ > 0 independent of ε. Note that the phase gε carries lines of vortices of degree 1 passing through the lattice points and it coincides with the trial function chosen in [CRY1]. Moreover the number, Nε , of lines of vortices included in the support of φ˜ is bounded by Cω /ε, because this support is contained in the finite region x ≤ Rω . The function χε is given by ⎧ if |r − r j | ≥ εη for all r j ∈ L, ⎪ ⎨1 χε (r) ≡ |r − r | (3.20) j ⎪ η, ⎩ if |r − r | ≤ ε j εη for some6 η > 5/2. ˜ 2 = 1, and, since Finally the constant cε is fixed by the normalization condition, φ χε ≤ 1 and Nε ≤ C/ε, 1 ≤ cε2 ≤ 1 + O(ε2η−1 ) = 1 + o(ε4 ).
(3.21)
GP evaluated on the trial function (3.12) is given by The rescaled functional E˜ε,ω 2 2 GP ˜ E˜ε,ω [φ] = cε2 dx ∇ χε ˜ ε + cε2 d x ˜ ε χε2 ∇ − i Aω/ε gε
R3
+
TF [|φ| ˜ 2] E1,ω
ε2
R3
.
(3.22)
We estimate the three energy contributions separately. Using that (a + b)2 ≤ 2a 2 + 2b2 the first term can be bounded by7 2 2 cε2 dx ∇ χε ˜ ε ≤ 2cε2 dx ∇ ˜ ε + cε2 Cω dr |∇χε |2 , R3
R3
B Rω
where we have used the trivial bound dz ˜ ε (x) ≤ Cω . R
6 This requirement on η is needed in order to apply Theorem 3.1 in [CRY1] (see below in the proof). 7 In this proof B , R > 0, will always denote a two-dimensional disc of radius R, centered at the origin. R
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Denoting by Bε a two-dimensional disc of radius εη centered at r j ∈ L ∩ B Rω , one has j j∈L Bε Cω , (3.23) dr |∇χε |2 ≤ ≤ Nε ≤ 2η ε ε B Rω j
while, by using both (3.16) and (3.17), we get R3
2 dx ∇ ˜ ε =
R3
dx
|∇ ˜ ε |2 Cω 1 . ≤ dx |∇ ˜ ε | ≤ 4 ˜ ε 4ε R3 ε
Hence (3.21) implies the bound 2 C ω 2 cε . dx ∇ χε ˜ ε ≤ 3 ε R
(3.24)
(3.25)
In order to estimate the second term in (3.22), we first need to restrict the integration to a suitable two-dimensional compact set, by exploiting the exponential smallness of ˜ ε given by (3.15): 2 dx ˜ ε χε2 ∇ − i Aω/ε gε 3 R 2 2 = dx ˜ ε χ 2 ∇ − i Aω/ε gε + dx ˜ ε χ 2 ∇ − i Aω/ε gε ε
| x |≤2Rω
| x |≥2Rω
ε
2 C ω |x| − Rω ∇ − i Aω/ε gε + 6 ≤ Cω dr dx exp − ε x≥2Rω ε B 2Rω 2 2 ≤ Cω dr ∇ − i Aω/ε gε + Cω dr χε2 ∇ − i Aω/ε gε j ∪i∈L Bε Cω Rω , + 9 exp − ε ε
χε2
where ≡ B2Rω \
j∈L
Bεj ,
and we have used the pointwise estimate (see also (3.10) in [CRY1]) ∇ − i Aω/ε gε ≤ |∇gε | + Cω ≤ Cω ε ε3/2 which holds for any x ∈ R3 such that |x| ≥ 2Rω . The second term of the right hand side of the above expression can be bounded in the following way (see also the proof of Theorem 2.1 in [CRY1]): 2 2 2 2 dr χε ∇ − i Aω/ε gε ≤ 2 dr |∇gε | + 2 dr Aω/ε j j j ∪ j∈L Bε
∪ j∈L Bε
≤
∪ j∈L Bε
Cω Cω Cω Cω + 4−2η + 3−4η ≤ . ε ε ε ε
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On the other hand we can apply8 Theorem 3.1 in [CRY1] to get 2 π R 2 ω π 2 Cω | log ε| − 2 + dr ∇ − i Aω/ε gε ≤ 2ω ε 2 δ ε √ and, by choosing δ = 2π/ω, 2 Cω | log ε| Cω Cω Rω 2 2 dx ˜ ε χε ∇ − i Aω/ε gε ≤ cε + + 9 exp − ε ε ε ε R3 Cω | log ε| . (3.26) ≤ ε ˜ 2 . We It remains then to estimate the last term in (3.22), namely the TF energy of |φ| have TF TF ˜ 2 ≤ E1,ω E1,ω |φ| ˜ ε + o(ε) and, defining Wω (x) ≡ V (x) − ω2 r 2 /4, TF TF TF TF TF E1,ω ρ1,ω = ˜ ε − E1,ω dx Wω jε ρ1,ω − ρ1,ω 3 R TF = dx ρ1,ω ( jε Wω − Wω ) . TF supp ρ1,ω
Differentiabilty of V implies 1 dx Wω (x − εx ) − Wω (x) e−| x | 4π R3 ≤ Cω ε dx r e−| x | ≤ Cω ε,
|( jε Wω ) (x) − Wω (x)| ≤
R3
so that
TF TF TF TF ˜ 2 ≤ E1,ω |φ| ρ1,ω + Cω ε = E 1,ω E1,ω + Cω ε.
(3.27)
Putting all the three estimates (3.25), (3.26) and (3.27) together we obtain the upper bound GP E˜ ε,ω ≤
TF E 1,ω
ε2
+
Cω | log ε| . ε
(3.28)
The upper bound and the lower bound together give the desired result in the rapid rotation regime (ii). Indeed, writing ε2 as g −(s−2)/(s+3) and using (3.9), we get, as g → ∞, TF GP TF E 1,ω ≤ g −s/(s+3) E g,Ω ≤ E 1,ω + Cω g −(s+2)/(2s+6) log g.
The same estimates prove the convergence in the slow rotation regime (i), by uniformity TF . of the bounds and continuity of E 1,ω 8 By scaling the expression can be reduced to an integral over a set contained in a ball of radius 1. The only 2. change with respect to Theorem 3.1 in [CRY1] is then the multiplying factor Rω
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
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As in the two-dimensional case (see [CRY1,CRY2]), a simple corollary of this result GP 2 TF . In is the L 1 -convergence of the (scaled) GP density φg,Ω to the TF minimizer ρ1,ω particular, in the slow rotation regime (i) this gives the convergence of the GP density to TF . In that case the proof of the upper bound could have been simplified a lot by taking ρ1,0 as trial function a suitable regularization of the (real) TF minimizer without rotation (ω = 0). Also in the rapid rotation regime (ii) other trial functions could be chosen with the same leading order contribution to the energy as (3.12). One possibility is to cover TF with Dirichlet boxes and choose a phase factor in each box so that the support of ρ1,ω the contribution from Aω/ε to the kinetic energy is gauged away to leading order. The reason we have chosen a trial function of the form (3.12) is that the error term in (3.10) has the expected dependence on the parameters of a next to leading order term, albeit with an unspecified constant in front. In fact, based on considerations of special cases (see, e.g., [A]) the true GP minimizer is expected to contain, like (3.12), a large number Nε ∼ ω/ε of vortex lines, each giving a kinetic contribution of order | log ε| beyond the leading order term. We note that the optimal vortex lattice is expected to be triangular rather than rectangular and bending of lattice lines [A1] will also occur but such details only affect the constant factor and higher order contributions in the error term in (3.10). 3.2. The regime ω → ∞. We now consider the case of ultrarapid rotations. In the proofs of the quantum mechanical limit theorems in Sect. 4 we shall not make direct use of the GP minimizers to construct trial functions for an upper bound to the energy, but the asymptotics of the TF energy and minimizer as ω → ∞ will be important. In this subsection we first derive these properties. The limit theorem for the GP energy for ultrarapid rotations can be proved by a simple extension of the corresponding TF result and is included for completeness. We start with a lemma on the structure of the set M of minimizing points for the sum of the external potential V and the centrifugal potential −r 2 /4. As always, V is assumed to be homogeneous of order s > 2. Lemma 3.1 (M is a subset of a cylinder). The set M of minimizing points for W (x) = V (x) − r 2 /4 is a compact subset of a cylindrical surface with a fixed radial coordinate sm , (3.29) r0 = 2 s−2 where m ≡ − min W ≥ 0. By scaling it follows that all points of the set MΩ of minimizing points for WΩ (x) ≡ V (x) − Ω 2 r 2 /4 have the same radial coordinate rΩ = Ω 2/(s−2) r0 . Proof. That M is compact follows from continuity of V and the assumption that W tends to ∞ as |x| → ∞. On M we have ∇W = 0 and W = −m, and thus ∂r V = r/2, ∂z V = 0 and V = r 2 /4 − m. On the other hand, since V is homogeneous of order s, Euler’s relation gives r ∂r V + z∂z V = sV and hence (3.29). If V is rotationally symmetric, i.e., V (x) = V (r, z), and strictly monotonously increasing in |z| (examples: V (x) = a|x|s , or V (x) = ar s + b|z|s ), then M is clearly a circle in the z = 0 plane. It is, however also possible that M consists of a twodimensional subset of the cylinder r = r0 (example: V (x) = r s f (|z|/r ) with f = 1 on
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J.-B. Bru, M. Correggi, P. Pickl, J. Yngvason
some interval but > 1 and increasing outside the interval), or of discrete points (example: V (x) = a|x|s + b|y|s + c|z|s with a = b). TF = inf W as Next we consider the convergence of the TF energy E γTF,1 to E 0,1 −2(s+3)/(s−2) → 0. γ =ω Theorem 3.2 (TF energy and density for ω → ∞). For γ → 0 TF + O(γ 2/5 ). E γTF,1 = E 0,1
(3.30)
Moreover, the TF minimizer ργTF,1 satisfies the bound ργTF,1 ∞ ≤ const.γ −3/5
(3.31)
and for any > 0 there is a γ such that for γ < γ the support of ργTF,1 is contained in M ≡ {x : |x − x | ≤ for some x ∈ M}. TF . For an upper bound we provide a trial function with Proof. It is clear that E γTF,1 ≥ E 0,1 TF 2/5 energy at most E 0,1 + O(γ ). Let h be any continuous, nonnegative function with support in the unit ball in R3 and h = 1. For δ > 0 and a point x 0 ∈ M define ρδ (x) = δ −3 h((x − x 0 )/δ). Then, using that W is C 2 and that ρδ is supported in a ball of radius δ around x 0 ∈ M, we have
TF [ρ ] = dx Wρδ + γρδ2 ≤ E 0,1 + Cδ 2 + γ δ −3 h22 . (3.32) E γTF,1 ≤ EγTF ,1 δ
R3
Choosing δ = γ 1/5 now proves (3.30). The TF minimizer is explicitly given by ργTF,1 (x) =
1 TF µγ ,1 − W (x) . + 2γ
(3.33)
Since ργTF,1 remains normalized as γ → 0, it is clear by continuity of W that µTF γ ,1 must converge to the minimum of W and the support of ργTF,1 shrinks to the set M as stated in the TF + γ ||ρ TF ||2 and, by (2.14), µTF = E TF + g||ρ TF ||2 . lemma. Moreover, E γTF,1 ≥ E 0,1 g,Ω g,Ω g,Ω 2 γ ,1 2 This together with (3.30) implies TF 2/5 0 ≤ µTF ≤ µTF ), (3.34) γ ,1 − W (x) γ ,1 − E 0,1 ≤ O(γ +
and hence ργTF,1 ∞ ≤ const.γ −3/5 .
Remark 3.1. In the case that M does not consist of discrete points but is one- or twodimensional the power of γ in the optimal error terms is of higher order than γ 2/5 . For instance if V is radially symmetric and M is a circle, the trial function can be taken to be radially symmetric and the error is O(γ 1/2 ). As a complementary statement to Theorem 3.1 we now prove the convergence of the TF as Ω and ω → ∞. scaled GP ground state energy to E 0,1
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
531
Theorem 3.3 (GP energy asymptotics for ω → ∞). As Ω → ∞ and ω → ∞, GP TF (3.35) Ω −2s/(s−2) E g,Ω = E 0,1 + O Ω −1 + γ 2/5 , with Ω = Ω (s+2)/(s−2) and γ = ω−2(s+3)/(s−2) . GP = Ω −2 Eˆ GP where Eˆ GP is the ground state Proof. Note first that Ω −2s/(s−2) E g,Ω Ω,ω Ω,ω GP , cf. (3.7). Dropping the positive kinetic term as energy of the scaled GP functional EˆΩ,ω in the proof of Theorem 3.1 immediately gives the lower bound GP TF Ω −2 Eˆ Ω,ω ≥ E 0,1 .
(3.36)
For an upper bound we make use of a nonnegative function h with support in the unit ball√and h = 1 as in the proof of Theorem 3.2, this time requiring h to be C ∞ so that ∇ h2 < ∞. For δ > 0 we put h δ (x) = δ −3 h(x/δ). Let x 0 ∈ M and define (3.37) φ(x) = h δ (x − x 0 ) exp{iΩ x · (ez ∧ x 0 )/2}. GP with this function we obtain Testing EˆΩ,ω
GP ≤ 2Ω −2 ∇ h δ 22 + γ h δ 22 Ω −2 Eˆ Ω,ω + dx 41 |ez ∧ (x 0 − x)|2 + W (x) h δ (x − x 0 ) . R3
(3.38)
Since h δ (x) ≡ δ −3 h(x/δ), the first term is O Ω −2 δ −2 . Since h δ 22 ≤ h δ ∞ = −3 −3 O δ , the second term is O γ δ . In the last integral we use that W ∈ C 2 , that TF , and ||h || = 1 supp h δ is contained in a ball of radius δ around x 0 with W (x 0 ) = E 0,1 δ 1 to get TF dx 41 |ez ∧ (x 0 − x)|2 + W (x) h δ (x − x 0 ) ≤ E 0,1 + O(δ 2 ). (3.39) R3
We thus have
GP TF Ω −2 Eˆ Ω,ω ≤ E 0,1 + O Ω −2 δ −2 + δ 2 + γ δ −3 .
(3.40)
Equating the second and the last error term leads to the choice δ = γ 1/5 and an error O(Ω −2 γ −2/5 + γ 2/5 ) = O(γ 2/5 ) provided Ω −1 ≤ γ 2/5 . For Ω −1 > γ 2/5 we choose δ = Ω −1/2 (this corresponds to equating the first and the second error term in (3.40)). Then the errors are O(Ω −1 ). Altogether we obtain (3.35). Remark 3.2. The true GP density is in general not concentrated around a single point in M and the trial function (3.37) is not designed to give optimal error bounds. Another obvious possibility is to replace h δ by a regularization of the TF density and choose a phase factor of the ‘giant vortex’ type like in [CRY1]. In fact, in the proof of (4.3) we use a function of this form as an ingredient of the many-body trial function. Since Theorem 3.3 is not directly used for the proof of the corresponding many-body result we do not elaborate on this point further here.
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4. Proofs of the QM Limit Theorems QM
In this section we derive the bounds on the quantum mechanical ground state energy E g,Ω that lead to the proofs of Theorems 2.1–2.3. The lower bound in the case of ultrarapid rotations is simply obtained by dropping positive terms from the Hamiltonian. In the case ω < ∞ one uses first the diamagnetic inequality [LL] to eliminate the vector potential from the Hamiltonian (2.3) and then proceeds with the techniques described in [LSY5] for the non-rotating case. The upper bound for ω < ∞ is obtained by first bounding the QM energy by the GP energy. The method, that is a generalization of [LSY1], is described briefly in [S1] for fixed g and Ω, but in order to keep track of the error terms as g and/or Ω tend to ∞ and for completeness we carry it out in more detail. Once a bound in terms of the GP energy has been obtained, we can use Theorem 3.1 of the previous section to relate it to the TF energy. In the regime of ultrarapid rotation, ω → ∞, we use a slightly different method that gives an estimate in terms of the TF energy and error terms involving directly the TF density whose relevant properties were described in Theorem 3.2. The limit Theorems 2.2-2.3 for the density are simple consequences of the energy bounds and are discussed in Subsect. 4.2.
4.1. Bounds on the QM energy. Proposition 4.1 (Lower bound for the QM energy). Let the potential V be homogenous of order s > 2. Then TF Ω −2s/(s−2) N −1 E g,Ω (N ) ≥ E 0,1 . QM
(4.1)
TF Furthermore, if ω = g −(s−2)/(2s+6) Ω is fixed and N −2 g 3 ρg,Ω ∞ → 0 as N → ∞ then
s QM TF lim inf g − s+3 N −1 E g,Ω (N ) ≥ E 1,ω (4.2) N →∞
uniformly in ω on any bounded interval. Proof. To prove (4.1) consider a normalized N -particle wave function Ψ N and let ρˆ N (x) = λ3 ρ N (λx) with λ = Ω 2/s−2 be the corresponding scaled density. Then we can write Ω −2s/(s−2) N −1 Ψ N , H N Ψ N = CΨ N + inf W,
(4.3)
R3
with W (x) = V (x) − r 2 /4 and CΨ N ≡ Ω −2s/(s−2) [∇ − i AΩ ]Ψ N 22 + + Ω −2s/(s−2) N −1
!
R3
ρˆ N (x) W (x)− inf W dx
" Ψ N , v(|x i − x j |)Ψ N .
R3
(4.4)
1≤i< j≤N
Since the interaction potential v is by assumption nonnegative the same holds for CΨ N TF . so the left hand side of (4.1) is ≥ inf W = E 0,1
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
533
Now, let us consider the case when ω < ∞ is fixed as N → ∞. By the diamagnetic inequality, |(∇ − i A(x)) f (x)| ≥ |∇| f (x)||, cf. [LL], and the bound TF 1 2 2 V (x) ≥ µTF g,Ω + 4 Ω r − 2gρg,Ω (x)
(4.5)
that follows from Eq. (2.13) we obtain QM
E g,Ω (N ) ≥ N µTF g,Ω + with Q (Ψ ) ≡
N
∇i Ψ 2 +
i=1
−2g
inf
Ψ,Ψ =1
Q (Ψ ) ,
(4.6)
v(|x i − x j |) |Ψ |2 d3N x
1≤i< j≤N
N
TF ρg,Ω (x i ) |Ψ |2 d3N x.
(4.7)
i=1 TF (x) tends to zero for every x as g → ∞, cf. (2.20), it is convenient at this Since ρg,Ω point to carry out a rescaling by writing x = λx with λ = g 1/(s+3) , cf. (3.2). The scaled interaction potential v (x ) = λ2 v(λx ) has scattering length a = g −1/(s+3) a and the corresponding coupling parameter is
g = 4π N a = g (s+2)/(s+3) . Using (2.14) and (2.15) we obtain g
−s/(s+3)
N
−1
QM E g,Ω
−
TF E 1,ω
≥
TF 2 (ρ1,ω ) + (N g )−1
(4.8)
inf
Ψ,Ψ =1
Q (Ψ ),
(4.9)
where, dropping the primes on the integration variables, N ∇i Ψ 2 + Q (Ψ ) ≡ v (|x i − x j |) |Ψ |2 d3N x i=1
−2g
1≤i< j≤N N
TF ρ1,ω (x i ) |Ψ |2 d3N x .
(4.10)
i=1
We are now exactly in the situation discussed in [LSY5] for the nonrotating case, cf. Eq. (6.61) in [LSY5]. Like there, the next step is to divide space into boxes, labeled by α and of side length l, with Neumann boundary conditions and use the lower bound of [LY] for the homogeneous gas in each box. The result is (cf. Eq. (6.62) in [LSY5]) s 1/17 QM TF TF 2 g − s+3 N −1 E g,Ω (N ) − E 1,ω ≥ (ρ1,ω ) − dα2 l 3 (1 − CY ). (4.11) α
Here dα is the maximum value of Since
TF ρ1,ω
in the box α and Y = a 3 N /l 3 ∼ g N −2 /l 3 .
TF TF g N −2 ρg,Ω ∞ = g N −2 ρ1,ω ∞
(4.12)
the diluteness condition implies that Y → 0 for fixed l. If we now first take N → ∞ TF )2 implies that the right hand side and then l → 0, the Riemann approximation of (ρ1,ω of (4.11) tends to zero, proving (4.2) for fixed ω. It is also clear that all estimates are uniform in ω on bounded sets, so in particular one can take ω to 0.
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Proposition 4.2 (Upper bound on the QM energy for ω < ∞). Let the potential V be TF → 0 homogenous of order s > 2 and suppose the diluteness condition, N −2 g 3 ρg,Ω ∞ as N → ∞, is fulfilled. If ω is fixed, then
s QM TF lim sup g − s+3 N −1 E g,Ω (N ) ≤ E 1,ω (4.13) N →∞
uniformly in ω on bounded intervals. Proof. The proof is a combination of a variational bound on the QM energy in terms of the GP energy and the bounds of the GP energy in terms of the TF energy that were discussed in Sect. 3. For the former we can use the same method as in [LSY1] and [S2] and not all details will be repeated here, but we shall keep track of the error terms and their dependence on the various parameters. The main step is to show that under the stated assumptions QM
GP E g,Ω (N ) − N E g,Ω GP N gρg,Ω ∞
≤ o (1),
(4.14)
by exhibiting a sequence of N particle trial functions Ψ N , N = 1, 2, . . . , such that GP
Ψ N , H, Ψ N Ψ N , Ψ N −1 − N E g,Ω GP N gρg,Ω ∞
≤ o (1) .
(4.15)
We write the trial functions in the form Ψ N = F(x 1 , . . . , x N )G(x 1 , . . . , x N )
(4.16)
with G(x 1 , . . . , x N ) ≡
#N
GP i=1 φg,Ω (x i )
(4.17)
and a real function F. Partial integration, using the variational equation (2.8) and the reality of F, leads to GP
Ψ N , H, Ψ N = N µg,Ω Ψ N , Ψ N + |∇i F|2 |G|2 +
1≤i≤N
3N 1≤i< j≤N R
R3N
v(|x i − x j |)|F| |G| − 2g 2
2
3N 1≤i≤N R
GP ρg,Ω (x i )|F|2 |G|2 .
(4.18) The second line of (4.18) is a real quadratic form in F and we shall make use of the fact that for an upper bound on the bosonic ground state energy it is not necessary to require that the trial function F is symmetric under permutations of the variables. This can be seen by a simple adaption of an argument of Lieb [L1] which implies that the infimum over all functions F is the same as the infimum over all nonnegative, symmetric functions. Like in [LSY1] we shall take a trial function of the Dyson form [D] #N F(x 1 , . . . , x N ) = i=1 Fi (x 1 , . . . , x i ), (4.19)
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
where
535
Fi (x 1 , . . . , x i ) = f (ti ), ti = min |x i − x j |, j = 1, . . . , i − 1 ,
(4.20)
with a function f satisfying f ≥ 0.
0 ≤ f ≤ 1,
The function f will be specified shortly. Our estimates involve the quantities 2 2 2 1 f + 2v f , K ≡ (1 − f ), J ≡ f f . I ≡ R3
R3
R3
(4.21)
By exactly the same computation as the one leading to Eq. (3.29) in [LSY1] we obtain, GP I < 1, provided N ρg,Ω ∞ F G−2 2
⎧ ⎨ ⎩ R3N
|∇ F|2 |G|2 +
i< j
1 ≤ GP I )2 (1 − N ρg,Ω ∞
R3N
v(|x i − x j |)F 2 |G|2
2
N J
R3
GP ρg,Ω (x)2
⎫ ⎬ ⎭
2 3 2 GP 2 + N K ρg,Ω ∞ , 3
(4.22)
and the same technique gives also a bound on the last term in (4.18), GP −2gF G−2 ρg,Ω (x i )|F|2 |G|2 2 ≤ −2g N
0
R3
R3N
GP GP 2 dxρg,Ω (x)2 + 2g N 2 I ρg,Ω ∞ .
(4.23)
We now choose the function f . For a parameter b > a that will soon be fixed we define (1 + 1 )u(r )/r for r ≤ b f (r ) = , (4.24) 1 for r > b where u(r ) is the solution of the scattering equation −u (r ) + 21 v(r )u(r ) = 0
with u(0) = 0, lim u (r ) = 1 r →∞
and 1 is determined by requiring f to be continuous. Convexity of u gives 0 for r ≤ a 0 for r ≤ a , 1 ≥ u (r ) ≥ r ≥ u(r ) ≥ r − a for r > a 1 − ar for r > a . These estimates imply J ≤ (1 + 1 )2 4πa, 3 a + ab(b − a) , I ≤ 4π 3 a , K ≤ 4π(1 + 1 )a b − 2 a . 0 ≤ 1 ≤ b−a
(4.25) (4.26) (4.27) (4.28)
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J.-B. Bru, M. Correggi, P. Pickl, J. Yngvason
Before proceeding further, we need to relate the supremum of the GP density, GP || , to ||ρ TF || ||ρg,Ω ∞ g,Ω ∞ since the diluteness condition is stated in terms of the latter. GP = Rei S with real S and the nonnegative amplitude R. For this purpose we write φg,Ω A straightforward computation, using ∇ · A = 0, gives GP GP −(∇ − i A)2 φg,Ω = (− + 2i A · ∇ + A2 )φg,Ω
= −(R)ei S − 2i(∇ R) · (∇ S)ei S + (∇ S)2 Rei S − i(S)Rei S + 2i A · (∇ R)ei S − 2 A · (∇ S)Rei S + A2 Rei S and from the GP equation (2.8) one obtains GP + V − 41 Ω 2 r 2 R = µGP − + (∇ S)2 − 2 A · (∇ S) + A2 + 2gρg,Ω g,Ω R. GP (x) = ρ GP and R(x) ≤ 0. Thus, At any point x ∈ R3 where R is maximal ρg,Ω g,Ω ∞ GP ∞ ≤ −(∇ S(x))2 + 2 A(x) · ∇ S(x) − A2 (x) − V (x) + 41 Ω 2 r 2 + µGP 2gρg,Ω g,Ω ,
and since −(∇ S)2 + 2 A · ∇ S ≤ A2 we obtain GP 2gρg,Ω ∞ ≤ µGP g,Ω − inf
x ∈R3
V (x) − 41 Ω 2 r 2
.
On the other hand, by (2.13) we have TF 2g||ρg,Ω ||∞ = µTF g,Ω − inf
x ∈R3
V (x) − 41 Ω 2 r 2 ,
(4.29)
and therefore, using (2.9) and (2.14), GP TF TF 2g||ρg,Ω ||∞ ≤ µGP g,Ω − µg,Ω + 2g||ρg,Ω ||∞ GP TF GP TF ≤ E g,Ω − E g,Ω + g||ρg,Ω ||∞ + 2g||ρg,Ω ||∞ ,
i.e.,
GP ||∞ ||ρg,Ω
≤
TF 2||ρg,Ω ||∞
1+
GP − E TF E g,Ω g,Ω TF || g||ρg,Ω ∞
.
(4.30)
By using (2.15) and Theorem 3.1 one sees that GP − E TF E g,Ω g,Ω TF || g||ρg,Ω ∞
TF −1 GP TF = o(1), = ||ρ1,ω ||∞ g −s/(s+3) E g,Ω − E 1,ω
(4.31)
so GP TF ||ρg,Ω ||∞ ≤ 2||ρg,Ω ||∞ {1 + o(1)} .
(4.32)
TF || The diluteness condition N −2 g 3 ||ρg,Ω ∞ → 0 thus implies the corresponding condiGP −2 3 tion for the GP density, i.e., N g ||ρg,Ω ||∞ → 0. Therefore, by choosing 1
GP b = (N ρg,Ω ∞ )− 3 ,
(4.33)
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
537
GP I → 0 and it follows from (4.25)-(4.28) that 1 → 0, 2 ≡ N ρg,Ω ∞ GP −1 2 2 3 ≡ g N K ρg,Ω ∞ → 0 for N → ∞. Altogether one gets from (4.18), using (2.9), (4.22) and (4.23), that GP GP 2 GP N −1 E g,Ω ≤ E g,Ω + gρg,Ω 2 {O(1 ) + O(2 )} + gρg,Ω ∞ O(3 ) QM
GP GP ≤ E g,Ω + o(1) gρg,Ω ∞ ,
i.e. (4.15). By (2.17) we have
TF TF TF g −s/(s+3) g||ρg,Ω ||∞ = g 3/(s+3) ||ρg,Ω ||∞ = ||ρ1,ω ||∞ ,
(4.34)
and by (4.32) and Theorem 3.1 we can conclude that GP TF TF g −s/(s+3) N −1 E g,Ω ≤ g −s/(s+3) E g,Ω + o(1)||ρ1,ω ||∞ = E 1,ω + o (1) . QM
(4.35)
For ultrarapid rotations the proof of (4.32) given above is not valid because the error term may blow up as ω → ∞. We shall therefore treat this case separately, using a GP || ≤ (const.)||ρ TF || trial function different from (4.17). If a general proof of ||ρg,Ω ∞ g,Ω ∞ can be found, then Eq. (4.15) is verified also for ω → ∞ and the proof of the next proposition would follow in the same way as the previous one. Proposition 4.3 (Upper bound on the QM energy for ω → ∞). Let the potential V be TF → 0 homogenous of order s > 2 and suppose the diluteness condition, N −2 g 3 ρg,Ω ∞ as N → ∞, is fulfilled. If Ω → ∞ and ω → ∞ as N → ∞, then
QM TF lim sup Ω −2s/(s−2) N −1 E N ,g,Ω (N ) ≤ E 0,1 . N →∞
Proof. The first step is to choose a suitable phase factor for the trial function to compensate the vector potential in the kinetic term as far as possible. As shown in Lemma 3.1 the set of minimizers of WΩ (x) = V (x) − 41 Ω 2 r 2
(4.36)
is a subset of a cylinder with radius rΩ > 0. We define the phase factor as follows: (x 1 , . . . , x N ) ≡
N
θ (x j ),
(4.37)
j=1
with
1 2 rΩ Ω ϑ for x = (r, ϑ, z), θ (x) ≡ exp i 2
(4.38)
and where [·] stands for the integer part. Any function Ψ ∈ L 2 can be written as Ψ = with ∈ L 2 and a straightforward computation gives ' ( N Ω 2 r 2j 2 QM QM 2 2 || + −i∇ j − AΩ (x j ) || E N ,g,Ω (Ψ ) = E N ,g,0 () + − 4 R3N j=1
−2
N j=1
Re
R3N
∗
∗ −i∇ j − AΩ (x j ) ∇ j .
(4.39)
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J.-B. Bru, M. Correggi, P. Pickl, J. Yngvason
Since
rΩ 1 1 2 θ eϑ , r Ω − (−i∇ − AΩ ) θ = r 2 Ω 2
(4.40)
we get, using the Cauchy-Schwartz inequality combined with 2ab ≤ a 2 + b2 , QM
QM
E N ,g,Ω (Ψ ) ≤ E N ,g,0 () ' (
N Ω 2 r 2j 1 1 2 2 r j Ω 2 2 2 || + 2 r Ω − + || + ∇ j − . 4 rj 2 Ω 2 R3N
(4.41)
j=1
In particular, since
1
2 2 rΩ Ω
2 Ω + κ with |κ| < 1, we then have = 21 rΩ
E N ,g,Ω (Ψ ) − N inf WΩ ≤ E)N ,g,Ω (), QM
QM
(4.42)
with QM E)N ,g,Ω () ≡
N 3N j=1 R
+
2 )Ω − inf WΩ ||2 2 ∇ j + W
3N 1≤i< j≤N R
v(|x i − x j |)||2
(4.43)
and ⎛
⎞ 2 r2 )Ω (x) ≡ V (x) + Ω 2 ⎝− + Ω − r + 4 ⎠. W 4 r Ω 2r 2 r2
(4.44)
The functional (4.43) describes the QM energy of a non-rotating system of particles )Ω − inf WΩ ) and with a two-body with mass 1/2 in the positive external potential (W interaction potential v ≥ 0. We need to choose a trial function for (4.43). As as in the proof of the previous proposition we are dealing with a real quadratic form so the infimum over all ∈ L 2 (R3 ) is the same as the infimum over all symmetric ∈ L 2 (R3 ). Thus, by (4.42), QM
E N ,g,Ω − N inf WΩ ≤inf
QM E)N ,g,Ω ()
||||22
.
(4.45)
TF the regularized TF density with j defined by (3.13) for We denote by ρ ≡ j ρg,Ω √ 9 −2 any > 0. Observe that ||ρ ||1 = 1 and ||∇ ρ ||2 = since |∇ j | = −1 j and || j ||1 = 1 for any > 0. Our trial function is defined as
˜ 1 , . . . , x N ) ≡ F(x 1 , . . . , x N )G(x 1 , . . . , x N ), (x 9 This is unrelated to the ε defined in (3.3).
(4.46)
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
539
with G(x 1 , . . . , x N ) ≡
N .
ρ (x j ),
(4.47)
j=1
while the function F is of the Dyson form, cf. (4.19)-(4.20) and (4.24). QM ˜ ˜ 2 follows closely the computations in [LSY1], The estimation of E)N ,g,Ω ()/|| || 2 Eqs. (3.11)–(3.29), but with the regularized TF density ρ instead of the GP density. TF || → 0 implies that the same condition is also The diluteness condition N −2 g 3 ||ρg,Ω ∞ fulfilled for ρ because TF TF ||∞ || j ||1 = ||ρg,Ω ||∞ . ||ρ ||∞ ≤ ||ρg,Ω
(4.48)
The GP equation, that was used in the computations in [LSY1] to obtain Eq. (3.28) in that paper, is not at our disposal for ρ but we can instead use the Cauchy-Schwarz inequality combined with 2ab ≤ a 2 + b2 . In this way, using also the positivity of v, we obtain QM
E N ,g,Ω − N inf WΩ ≤
N
F G−2 2
4
j=1
+4
N j=1
F G−2 2
∇ j G 2 F 2 +
R3N
⎧ ⎨
⎩ R3N
R3N
∇ j F 2 G 2 + i< j
)Ω x j − inf WΩ F 2 G 2 W
R3N
v(|x i − x j |)F 2 G 2
⎫ ⎬ ⎭
. (4.49)
Since |∇ j | = −1 j we have for the first term of (4.49) the bound 4 F G−2 2
R3N
= 4 F G−2 2
∇ j G 2 F 2
R3N
2 ∇ ρ 1 1 j xj . F 2 G 2 ≤ 2 . ρ x j 2 ρ x j
(4.50)
)Ω −inf WΩ ) ≥ 0, we obtain Using Eqs. (3.15)–(3.16) and (3.21) in [LSY1] as well as (W the bound −2 )Ω x j − inf WΩ F 2 G 2 F G2 W 3N R 1 )Ω − inf WΩ ρ ≤ d3 x W (4.51) 1 − N ||ρ ||∞ I R3 with I defined by (4.21), provided N ||ρ ||∞ I < 1 that is guaranteed by the diluteness condition. The last two terms in (4.49) are bounded in exactly the same way GP . We omit the details. Altogether we which leads to (4.22) with ρ in the place of ρg,Ω
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have the upper bound N
−1
QM E N ,g,Ω
− inf WΩ
1 )Ω − inf WΩ ρ + 4gρ2 ≤ 2 + (1 + o (1)) dx W R3 +o (1) g||ρ ||∞ . (4.52)
Since ρ is normalized, TF ||ρ ||2 ≤ ||ρ ||∞ ≤ ||ρg,Ω ||∞
(4.53)
TF Ω −2s/(s−2) inf WΩ = E 0,1 ,
(4.54)
by (4.48). Also,
2
and by Lemma 3.1 all minimizing points have the same radial coordinate rΩ = r0 Ω s−2 with r0 the radius of the set M. The inequality (4.52) now implies TF Ω −2s/(s−2) N −1 E N ,g,Ω − E 0,1 )Ω − inf WΩ ρ (1 + o (1)) ≤ Ω −2s/(s−2) dx W QM
R3
TF +5Ω −2s/(s−2) g||ρg,Ω ||∞ (1 + o (1)) + Ω −2s/(s−2) −2 .
(4.55)
We now have to bound each term of the right hand side. For the second term we use that TF Ω −2s/(s−2) g||ρg,Ω ||∞ = γ ||ργTF,1 ||∞ ≤ o (1)
(4.56)
by (2.20) and because ||ργTF,1 ||∞ ≤ O(γ −3/5 ) by Theorem 3.2. The last term in (4.55) is o(1) as long as Ω −s/(s−2) . It remains to consider the first term in (4.55). By scaling we have −2s/(s−2) )Ω − inf WΩ ρ Ω dx W R3
⎧ ⎨
⎫ 2 r2 4 ⎬ TF = dx V (x) − 41 r 2 + 0 − r + 2 2 ρ , ) (x) − E 0,1 ⎩ ⎭ 3 r Ω r R
with
2 6 ρ ) (x) = Ω s−2 ρ Ω s−2 x = j) ργTF,1 (x),
2
) ≡ Ω − s−2 .
(4.57)
(4.58)
As γ → 0 the support of ργTF,1 becomes concentrated on the set M where V (x) − 14 r 2 is minimized and all points in M have the same radial coordinate, r0 . Moreover, outside of the support of ργTF,1 the regularized density ρ ) decreases exponentially if d(x)/) → ∞, where d(x) is the distance of x from the support of ργTF,1 (see also the proof of Theorem 3.1). This implies that the right hand side of (4.57) tends to zero as γ → 0, Ω → ∞ and → 0 with −1 Ω −s/(s−2) → 0.
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
541
4.2. Convergence of the QM particle density. 4.2.1. The case ω < ∞ Proof of Theorem 2.2. We use Griffiths’ argument [G] in the same way as for an analogous problem in [LSi]. Take any bounded function f : R3 → R and for any σ ∈ [−δ, δ] (δ > 0) perturb the Hamiltonian H N with the external potential σ g s/(s+3) f (g −1/(s+3) x). Because f is bounded, the statements (i)-(ii) of Theorem 2.1 can also be proven for the perturbed external potential {V (x) + σ g s/(s+3) f (g −1/(s+3) x)}. Namely, if g → ∞ and QM ω ≥ 0 is fixed then the corresponding ground state E g,Ω,σ (N ) converges to
QM TF g −s/(s+3) N −1 E g,Ω,σ (N ) = E 1,ω,σ for any σ ∈ [−δ, δ] , δ > 0, (4.59) lim N →+∞
TF where E 1,ω,σ is the TF energy with V replaced by V + σ f . Consequently, for any approximated ground states Ψ N of H N we have " ! (4.60) )N , Ψ N , H N ,σ − H N Ψ N = N g s/(s+3) σ f, ρ
with ρ )N (x) = g 3/(s+3) ρ N (g 1/(s+3) x). Hence, the Rayleigh-Ritz principle for any σ ∈ [−δ, δ] leads to / 0 QM QM QM )N + Ψ N , H N − E g,Ω (N ) Ψ N . E g,Ω,σ (N ) − E g,Ω,0 (N ) ≤ N g s/(s+3) σ f, ρ (4.61) Because (2.15) and (2.17) are still valid with V (x) replaced by V (x)+ +σ g s/(s+3) f (g −1/(s+3) x), the previous inequality combined with (4.59) implies in the limit N → ∞ that TF TF )N for any σ ∈ (0, δ] , (4.62) σ −1 E 1,ω,σ ≤lim inf f, ρ − E 1,ω,0 N →+∞
whereas a negative parameter σ ∈ [−δ, 0) reverses the inequality. The proof of the TF differentiability of E 1,ω,σ at σ = 0 is deduced from similar estimations as from ( 4.60) 0 / TF TF , where ρ1,ω,σ to (4.62) combined with the continuity of the function σ → f, ρ1,ω,σ TF . We omit the details. In other words, is the minimizer of the variational problem E 1,ω,σ 0 / TF TF . Consequently, by (4.62) and its reversed inequality, we = f, ρ1,ω we have ∂σ E 1,ω,σ obtain Theorem 2.2.
4.2.2. The case ω → ∞. In the case of ultrarapid rotations Griffiths’ argument is not as easily applicable as in the previous situation. There are two complications. First, perturbing V with a scaled additional term σ f leads to a potential that is not homogeneous and the proof of the upper bound for the QM energy has to be modified in order to get TF TF . Secondly, if M consists of more than one point (4.62) with E 0,1,σ replacing E 1,ω,σ TF TF the variational problem for E0,1,σ does not have a unique minimizer and E 0,1,σ is not differentiable at σ = 0 in general. To get around this complication, we would need to perturb the Hamiltonian with an additional term to avoid the degeneracy of the variational problem. An example of such an argument is given in [LSe]. But since we are content with proving that the density is concentrated on M in the limit and not striving to obtain the exact limiting measure on M we can ignore all these problems and use the following simpler argument.
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Proof of Theorem 2.3. If K is any set with a positive distance from M, then W (x) − TF ≥ c > 0 on K with some strictly positive number c. Hence, by Eqs. (4.3)-(4.4), E 0,1 −2s/(s−2) −1 TF Ω N Ψ N , H N Ψ N − E 0,1 ≥ c ρˆ N . (4.63) K
On the other hand, by Proposition 4.3 we know that the left hand side of (4.63) tends to zero as N → ∞, so K ρˆ N → 0. 4.3. Remarks on ‘flat’ trapping potentials. As mentioned in Sect. 2 the case of a ‘flat’ trap, that corresponds formally to s = ∞, can be treated in essentially the same way as we have done for s < ∞. By a flat potential we mean that V is 0 inside some open, bounded set B with a smooth boundary and ∞ outside. More precisely, the kinetic term in the many-body Hamiltonian (2.3) and the GP functional (2.5) are defined with Dirichlet conditions on the boundary of B, but Neumann conditions lead, in fact, to the same results in the large g, large Ω limit. For s = ∞, Eq. (2.16) resp. (2.19) reduces to ω = g −1/2 Ω resp. γ = Ω −2 g and the TF = E TF , resp. Ω −2 E TF = E TF . Theorem 2.1 becomes TF energy scales as g −1 E g,Ω g,Ω 1,ω γ ,1 TF , in case (ii) g −1 N −1 E TF in case (i) g −1 N −1 E g,Ω (N ) → E 1,0 g,Ω (N ) → E 1,ω , and in QM
QM
TF . case (iii) Ω −2 N −1 E g,Ω (N ) → E 0,1 The proofs require only some minor modifications. For instance, in the proofs of Therorem 3.1 and Propositions 4.2-4.3 the trial functions have to be modified in order to take the boundary condition into account. In the case of ultrarapid rotations it has to be noted that the set M is now a subset of the boundary ∂B of B, consisting of the points on ∂B where the centrifugal potential −r 2 /4 is minimal, i.e., where r is maximal. If the boundary is smooth one can still use a Taylor expansion for the proof of an analogue of Theorem 3.3 (with different error terms). QM
Appendix A. The TF Density at Large Rotational Velocities The TF density is explicitly given by (2.13), i.e., 1 TF TF ρg,Ω µg,Ω + 14 Ω 2 r 2 − V (x) . (x) = + 2g
(A.1)
To get a picture how it changes with the parameters, in particular as ω = g −(s−2)/(2s+6) → ∞, it is convenient to use the scaling TF Ω 2/(s−2) x = ργTF,1 (x) (A.2) Ω 6/(s−2) ρg,Ω to eliminate the dependence of the potential on Ω and consider 1 TF 1 2 ργTF,1 (x) = µγ ,1 + 4 r − V (x) + 2γ
(A.3)
with γ = ω−2(s+3)/(s−2) . As ω increases from 0 to ∞, γ decreases from +∞ to zero and, due to the normalization, the chemical potential µTF γ ,1 decreases monotonically from 2 1 TF = min V (x) − 1 r 2 . Since V is homogeneous of order s > 2 it is clear +∞ to E 0,1 4
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543
TF < 0. By continuity there is a γ (and a corresponding ω ) such that µTF = 0. that E 0,1 c c γc ,1 Explicitly, 1 1 2 γc = 2 r − V dx. (A.4) (x) 4 +
Since V (0) = 0 and V is continuous, we have ργTF,1 (x) > 0 in a neighborhood of 0 for γ > γc (i.e., ω < ωc ). For γ < γc , on the other hand, ργTF,1 (x) = 0 in a cylinder around the z axis, because µγ ,1 < 0 and V ≥ 0. In other √ words, for γ < γc (i.e., ω > ωc ), the centrifugal force creates a ‘hole’ of radius ≥ 2 −µγ ,1 in the density. To describe this in a little more detail we introduce cylindrical coordinates (r, z, ϑ) and consider the density as a function of r at fixed (z, ϑ), first for z = 0. Using homogeneity of V we see that the boundary of the support of ργTF,1 is determined by the solutions to the equation a(ϑ)r s − r 2 /4 = µTF γ ,1
(A.5)
with a(ϑ) = V (1, 0, ϑ) > 0. If γ > γc (i.e., ω < ωc ) so µTF γ ,1 > 0, the equation has TF + one solution, r (ϑ)γ > 0, and ργ ,1 (r, 0, ϑ) > 0 for r < r (ϑ)+γ but ργTF,1 = 0 outside. Equation (A.5) has two solutions, r (ϑ)± γ , if γc > γ > γϑ (i.e., ωc < ω < ωϑ ) where γϑ ≥ 0 is determined by solving (A.5) together with sa(ϑ)r s−1 − r 2 /2 = 0.
(A.6)
+ The density ργTF,1 (r, 0, ϑ) is nonzero for r inside the interval (r (ϑ)− γ , r (ϑ)γ ) but vanishes outside. For γ = γϑ the interval shrinks to a point with radial coordinate + r (ϑ)lim = (2sa(ϑ))−1/(s−2) = sup r (ϑ)− γ = inf r (ϑ)γ . γ ≥γϑ
γ ≥γϑ
(A.7)
The chemical potential µTF γϑ ,1 and hence γϑ is determined by inserting (A.7) into (A.5). If γϑ > 0, then ργTF,1 is identically zero in the ϑ direction for γϑ ≥ γ ≥ 0. If γϑ = 0, on the other hand, then (A.5) implies that (r (ϑ)lim , 0, ϑ) ∈ M and thus r (ϑ)lim = r0 by Lemma 3.1. For z = 0 we can use the homogeneity of V to write V (x) = r s V (1, z/r, ϑ) and apply the considerations above to a ray with fixed z/r and ϑ. It is, however, more interesting to consider what happens at fixed z = 0 in the case that V is monotonically increasing in |z|, for instance if V (x) = V0 (r) + c|z|s . At fixed z the term c|z|s acts as a TF s shift of the chemical potential, µTF γ ,1 → µγ ,1 −c|z| . Hence for a ‘hole’ to appear at z > 0 TF s it is not necessary that µγ ,1 becomes negative, it appears already when µTF γ ,1 − c|z| < 0. For any γ < ∞ the density has therefore a ‘hole’ for sufficiently large |z|, the width increasing with |z|. As γ decreases from an initial value > γc the ‘hole’ moves down to lower values of z, reaching z = 0 at γ = γc . As an example, consider V (r, z, ϑ) = r s (1 − ε sin2 ϑ) + |z|s with 0 < ε < 1. Here γϑ > 0 for all 1 ϑ ∈ [0, 2π ) except for 2 ϑ = 0 and ϑ = π . As γ → 0 the density converges to 21 δ(x − 1) + 21 δ(x + 1) δ(y)δ(z). If ε = 0, i.e., V = r s + c|z s |, then γϑ = 0 for all ϑ and the limiting density for γ → 0 is (2π )−1 δ(r − 1)δ(z). Acknowledgement. This work was supported by an Austrian Science Fund (FWF) grant P17176-N02 and the ESF Programme INSTANS. JY would like to thank the Niels Bohr International Academy and Nordita, Copenhagen, for hospitality and Chris Pethick, Gentaro Watanabe and Gordon Baym for discussions.
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References [A] [A1] [B] [BP] [CRY1] [CRY2] [D] [ECHSC] [F] [FB] [FZ] [G] [KTU] [KB] [L1] [L2] [LL] [LSe] [LSi] [LSY1] [LSY2] [LSY3] [LSY4] [LSY5] [LY] [Lu] [SY] [S1] [S2] [WGBP]
Aftalion, A.: Vortices in Bose-Einstein Condensates. Progress in Nonlinear Differential Equations and their Applications 67, Boston: Birkhäuser Boston, 2006 Aftalion, A., Riviere, T.: Vortex energy and vortex bending for a rotating Bose-Einstein condensate. Phys. Rev. A 64, 043611 (2001) Baym, G.: Rapidly rotating Bose-Einstein condensates. J. Low Temp. Phys. 138, 601–610 (2005) Baym, G., Pethick, C.J.: Vortex core structure and global properties of rapidly rotating BoseEinstein condensates. Phys. Rev. A 69, 043619 (2004) Correggi, M., Rindler-Daller, T., Yngvason, J.: Rapidly rotating Bose-Einstein condensates in strongly anharmonic traps. J. Math. Phys. 48, 042104–30 (2007) Correggi, M., Rindler-Daller, T., Yngvason, J.: Rapidly rotating Bose-Einstein condensates in homogeneous traps. J. Math. Phys. 78, 102103 (2007) Dyson, F.J.: Ground-state energy of a hard sphere gas. Phys. Rev. 106, 20–26 (1957) Engels, P., Coddington, I., Haljan, P.C., Schweikhardt, V., Cornell, E.A.: Observation of long-lived vortex aggregates in rapidly rotating Bose-Einstein condensates. Phys. Rev. Lett. 90, 170405 (2003) Fetter, A.L.: Rotating vortex lattice in a Bose-Einstein condensate trapped in combined quadratic and quartic radial potentials. Phys. Rev. A 64, 063608 (2001) Fischer, U.R., Baym, G.: Vortex states of rapidly rotating dilute Bose-Einstein condensates. Phys. Rev. Lett. 90, 140402 (2003) Fu, H., Zaremba, E.: Transition to the giant vortex state in a harmonic-plus-quartic trap. Phys. Rev. A 73, 013614 (2006) Griffiths, R.: A proof that the free energy of a spin system is extensive. J. Math. Phys. 5, 1215 (1964) Kasamatsu, K., Tsubota, M., Ueda, M.: Giant hole and circular superflow in a fast rotating BoseEinstein condensate. Phys. Rev. A 66, 053606 (2002) Kavoulakis, G.M., Baym, G.: Rapidly rotating Bose-Einstein condensates in anharmonic potentials. New J. Phys. 5, 51 (2003) Lieb, E.H.: The stability of matter: from atoms to stars. Bull. Am. Math. Soc. 22, 1–49 (1990) Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981) Lieb, E.H., Loss, M.: Analysis (2nd. ed.), Graduate Studies in Mathematics 14, Providence, RI: Amer, Math. Soc. (2001) Lieb, E.H., Seiringer, R.: Derivation of the Gross-Pitaevskii equation for rotating Bose gases. Commun. Math. Phys. 264, 505–537 (2006) Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977) Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a trap: a rigorous derivation of the GrossPitaevskii energy functional. Phys. Rev. A 61, 0436021-13 (2000) Lieb, E.H., Seiringer, R., Yngvason, J.: A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Commun. Math. Phys. 224, 17–31 (2001) Lieb, E.H., Seiringer, R., Yngvason, J.: One-dimensional bosons in three-dimensional traps. Phys. Rev. Lett. 91, 1504011–4 (2003) Lieb, E.H., Seiringer, R., Yngvason, J.: One-dimensional behavior of dilute, trapped Bose gases. Commun. Math. Phys. 244, 347–393 (2004) Lieb, E.H., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation, Oberwolfach Seminars 34, Birkhäuser Verlag, Basel, 2005 Lieb, E.H., Yngvason, J.: Ground state energy of the low density Bose gas. Phys. Rev. Lett. 80, 2504–2507 (1998) Lundh, E.: Multiply quantized vortices in trapped Bose-Einstein condensates. Phys. Rev. A 65, 043604 (2002) Schnee, K., Yngvason, J.: Bosons in disc-shaped traps: from 3d to 2d. Commun. Math. Phys. 269, 659–691 (2007) Seiringer, R.: Ground state asymptotics of a dilute, rotating gas. J. Phys. A: Math. Gen. 36, 9755–9778 (2003) Seiringer, R.: Dilute, Trapped Bose Gases and Bose-Einstein Condensation. In: Large Coulomb Systems, Lect. Notes Phys. 695, J. Derezinski, H. Siedentop, eds., Berlin: Springer-Verlag, 2006 Watanabe, G., Baym, G., Gifford, S.A., Pethick, C.J.: Global structure of vortices in rotating Bose-Einstein condensates. Phys. Rev. A 74, 063621 (2006)
Communicated by I.M. Sigal
Commun. Math. Phys. 280, 545–562 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0450-4
Communications in
Mathematical Physics
Orthosymplectic Lie Superalgebras in Superspace Analogues of Quantum Kepler Problems R. B. Zhang School of Mathematics and Statistics, University of Sydney, Sydney, Australia. E-mail: [email protected] Received: 21 May 2007 / Accepted: 4 August 2007 Published online: 8 March 2008 – © Springer-Verlag 2008
Abstract: A Schrödinger type equation on the superspace R D|2n is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant “distance” away from the origin. An osp(2, D + 1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well as their respective dimensions. 1. Introduction The quantum Kepler problem and its analogues in higher dimensions are a series of soluble quantum mechanical systems with − r1 potentials. The simplest of such systems is the hydrogen atom. It was discovered in the 60s by McIntosh and Cisneros [MC] and Zwanziger [Z] independently that the quantum system describing the hydrogen atom remained soluble when coupled to a magnetic charge. Since then the quantum Kepler problem has been generalised to include couplings to nonabelian magnetic monopoles in 5-dimensions in [I] and in arbitrary dimensions in [M1]. Such generalisations are referred to as the generalised MICZ-Kepler problem in the literature. It has long been known that the original quantum Kepler problem had an so(2, 4) dynamical symmetry. In [BB], Barut and Bornzin showed that the dynamical symmetry survived when the system was under the influence of a magnetic charge. They used the dynamical symmetry to give a beautiful solution of the problem. In a recent joint publication [MZ] with Meng, we demonstrated that the generalised MICZ-Kepler problem in odd dimension D has an so(2, D + 1) dynamical symmetry, and gave a solution to the problem by algebraic means using a particular irreducible unitary highest weight representation of the dynamical symmetry. This work has also been extended to even dimensions (but for restricted classes of magnetic monopoles) in [M2].
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A natural problem is to introduce supersymmetries into the Kepler problem and its generalisations and to study the resulting supersymmetric quantum mechanical systems. The N = 2 supersymmetric case was studied in [KLPW] in arbitrary dimensions but without magnetic monopoles. An so(D + 1) dynamical symmetry remained in this case, which helped to obtain the bound states spectrum and the multiplicities of the eigenvalues. In this paper we investigate the Schrödinger equation on the superspace R D|2n involving a potential inversely proportional to the negative of the osp(D|2n) invariant “distance” away from the origin (see Eq. (2.3)). We shall refer to the study of this eigenvalue problem as the quantum Kepler problem on the superspace R D|2n without magnetic monopoles. As we shall see, the system is integrable, and will be solved by algebraic means. It will be very interesting to extend the study to include couplings to magnetic monopoles. Schrödinger equations on superspaces were studied by Delbourgo in the late 80s [D1] as a method to incorporate spin. This developed into a fruitful programme (see, e.g., [DJW] and references therein) on using supermanifolds to describe gauge symmetries and also to explain internal degrees of freedom of elementary particles (see [D2,D3] for recent developments). We hope that the present work and further studies will provide useful mathematical information for the programme of Delbourgo and co-workers. Our primary interest in this paper is the integrability of the quantum Kepler problem on superspace without magnetic monopoles, and also the relevant representations of its dynamical supersymmetry. We shall give the precise definition of the quantum Kepler problem on the superspace R D|2n in Sect. 2, and reformulate the problem algebraically following ideas of Barut and Bornzin [BB]. We show in Theorem 3.1 that the problem has an osp(2, D + 1|2n) dynamical supersymmetry, where the generators of the orthosymplectic Lie superalgebra are constructed explicitly. The bound states are shown to form an infinite dimensional irreducible highest weight module of this Lie superalgebra (Theorem 4.1), and a thorough understanding of the structure of this module is also given. Using this information we obtain the bound state spectrum and also the corresponding eigenspaces in Theorem 4.2. It is quite remarkable that the Kepler problem remains integrable when generalised to superspaces (see Remarks 2.1 and 4.2). Furthermore, the appearance of the dynamical supersymmetry and the way in which its representation theory enables us to solve the problem all appear to be quite fascinating from the point of view of the theory of Lie superalgebras. It is well known that even the finite dimensional representations of orthosymplectic Lie superalgebras are extremely hard to study and very little is known about them. Thus it is a nice surprise that the infinite dimensional irreducible representation of osp(2, D + 1|2n) appearing in the problem can be understood for all D > 2n + 1. Therefore, results in this paper should be of interest to the representation theory of Lie superalgebras as well. 2. Quantum Kepler Problem on Superspace In this section we introduce the quantum Kepler problems on superspaces, and also give an algebraic formulation for the problems following the strategy of Barut and Bornzin [BB]. 2.1. Generalities. Let R D|2n denote the superspace with D even dimensions and 2n odd dimensions. Denote by X a with a = 1, 2, . . . , D +2n the coordinate of the superspace,
Quantum Kepler Problems on Superspaces
547
D|2n the mewhere X a is even ⎛ if a ≤ D and ⎞ odd if a > D. We shall assign to R ID 0 0 tric η = (ηab ) = ⎝ 0 0 −In ⎠, where I D and In are the identity matrices of sizes 0 In 0 D × D and n × n respectively. By a function on R D|2n we shall mean a map from R D to the complex Grassmann algebra 2n generated by the odd coordinates X ν with ν = D + 1, D + 2, . . . , D + 2n. Denote ∂a = ∂ X∂ a which acts on functions from the ab b a ab −1 left. Let X aa = b ηab X and ∂ = b η ∂b , where η are the entries of η . Set = a ∂ ∂a . Given a function V(X ) on R D|2n which is assumed to be even in the Grassmann variables, we introduce the operator
1 H = − + V(X ), 2 which will be referred to as a quantum Hamiltonian operator. Our broad aim is to investigate the eigenvalue problem for the quantum Hamiltonian operator, that is, to solve the Schrödinger equation H = E,
(2.1)
where the eigenvalue E is required to belong to R (thus V(X ) has to be even). We shall be particularly interested in systems of the form of (2.1) which are integrable. For each given V(X ), we shall need to specify the class of functions on R D|2n , to which the solutions of the Schrödinger equation belong. For the potential corresponding to the Kepler problem, this will be discussed in some detail in the next subsection. Here we merely point out that the eigenfunction is a polynomial in the odd coordinates with coefficients being complex valued functions on R D , which will be referred to as coefficient functions. Remark 2.1. The Schrödinger equation on R D|2n is equivalent to a system of partial differential equations on R D for the coefficient functions. Note that since the Hamiltonian operator is even, the Schrödinger equation separates into two independent equations for the even and odd parts of respectively. 2.2. Quantum Kepler problem on superspace. Let us now introduce the quantum a 1 2 Hamiltonian operator which we shall study in this paper. Let R = a X Xa , D which is only defined away from the origin of R , and should be interpreted as a D i j polynomial in the odd coordinates. More precisely, let r 2 = i, j = 1 X ηi j X and D+2n 2 2 X µ ηµν X ν . Then R = r 1 + r 2 , where 1 + r 2 should be unders2 = µ,ν=D+1 tood as a Taylor expansion in expansion terminates at order n We have ∂a (R) =
2 . Since the odd coordinates are Grassmannian, the r2 2 in r 2 , and we have a polynomial in the odd coordinates.
Xa , (R 2 ) = 2d, where d = D − 2n. R
(2.2)
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We shall take the following quantum Hamiltonian operator: 1 1 H =− − . 2 R
(2.3)
Our purpose is to determine the spectrum of H and the corresponding eigenvectors. In the remainder of this paper, we shall only consider the Schrödinger equation (2.1) with this quantum Hamiltonian. Let us now specify the class of functions on R D|2n to which the eigenfunctions belong. We should mention that the so-called superanalysis (analysis of functions on superspace) is yet to develop into a coherent theory. It will take us too far astray to investigate superanalysis in any depth here, as we shall adopt an algebraic approach to the quantum Kepler problem on superspace, which by-passes many analytic issues. The rather superfluous discussion below on functions on R D|2n suffices for us to get by. The Grassmann algebra 2n generated by the odd coordinates is Z+ -graded with X µ (1 + D ≤ µ ≤ 2n + D) having degree 1. Let ζs (0 ≤ s ≤ 22n − 1) be a homogeneous basis of 2n consisting of products of the odd coordinates, and denote by deg(ζs ) the degree of ζs . We order the basis elements in such a way that deg(ζs ) ≤ deg(ζs+1 ), and thus ζ0 = 1. Introduce the conjugate linear algebra automorphism ¯ : 2n −→ 2n defined by X µ = X µ , µ = D + 1, D + 2, . . . , D + 2n. As usual, conjugate linear means that for any λ = c1 λ1 + c2 λ2 with λ1 , λ2 ∈ 2n and c1 , c2 ∈ C, λ¯ = c¯1 λ¯ 1 + c¯2 λ¯ 2 , where c¯1 and c¯2 are the complex conjugates of c1 and c1 . Also, being an algebra automorphism, the map ¯ obeys the rule λ1 λ2 = λ¯ 1 λ¯ 2 . Note that ζ¯s ζs = 0 if deg(ζs ) ≤ n, but ζ¯t ζt = 0 if deg(ζt ) > n. The conjugate linear automorphism on 2n extends to the superalgebra of functions on R D|2n in a natural way, and we shall still denote the resulting map by ¯. More explicitly, write a function as = s ζs ψs where the ψs are complex valued functions on R D . ¯ ¯ ¯ = ¯ Then conjugate of ψs . For any two s ζs ψs , where ψs is the usual complex functions and on R D|2n , we let | = R D if the integral over R D exists (thus lies in 2n ). Let F denote the set of functions on R D|2n such that for every ∈ F, (1) the integral | = R D exists; and (2) the coefficient functions of are twice differentiable on R D \{0}. Then for any = s ζs ψs ∈ F, the coefficient function ψ0 , which will be called the body of , must be square integrable in the usual sense. However note that all functions satisfying ψs = 0 for all s ≤ 2n − 1 belong to F. Remark 2.2. Even though our definition of F imposes conditions on all the coefficient functions ψs with s ≤ 2n − 1, it has in essence adopted the type of thinking common in physics that appropriate conditions only need to be imposed on the body (if the body is nonzero) of a function and then the supersymmetry of the physical problem will determine properties of the function as a whole. Remark 2.3. When defining F, one may be tempted to use the more “natural” valuation R D|2n instead, where the integration over the odd coordinates is given by the Berezin integral. However, this does not make the body of square integrable.
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We require that a solution of the Schrödinger equation (2.1) with the quantum Hamiltonian operator (2.3) belongs to the set F. The complex vector space spanned by all the solutions is evidently stable under the action of the quantum Hamiltonian operator, and a major aim is to understand this vector space. It is also this vector space on which the dynamical supersymmetry algebra acts. As we shall see later, the space in fact forms an irreducible module over the dynamical supersymmetry algebra. It is worth pointing out that functions ∈ F satisfying the condition ψs = 0 for all s ≤ 2n − 1 are irrelevant for us here as the Schrödinger equation does not admit any nontrivial solution of this kind. Note that 2 = 2 , and hence R¯ = R. Thus for any function , we have H = H . If is a solution of the Schödinger equation satisfying = , then the corresponding eigenvalue will necessarily be real. Thus we may regard the quantum Hamiltonian operator as Hermitian in a generalised sense. We shall further restrict ourselves to considering only the bound states, that is, the eigenvectors of H associated with negative eigenvalues. In order to have bound state solutions in F, we need to impose the following condition on the superspace: D > 2n + 1. In this case, the ground state eigenvalue E0 and the associated eigenvector 0 of the system are given by
2
2 1 2 E0 = − R . , 0 = exp − 2 d −1 d −1 ¯ 0 0 exists thus 0 ∈ F. It is also easy to check that 0 indeed Note that R D satisfies the Schrödinger equation with the energy eigenvalue E0 (also see Eq. (4.3) and discussions after). 2.3. Algebraic formulation. We shall solve the quantum Kepler problem algebraically by using the representation theory of a dynamical supersymmetry algebra, which will be constructed later. For this purpose we need to reformulate the Schrödinger equation (2.1) algebraically for the quantum Hamiltonian (2.3). It is well known that the following differential operators: Jab = X a ∂b − (−1)[a][b] X b ∂a
(2.4)
form the orthosymplectic Lie superalgebra osp(D|2n) with the commutation relations (see, e.g., [JG, (40)]) [Jab , Jcd ] = ηcb Jad + (−1)[a]([b]+[c]) ηda Jbc − (−1)[c][d] ηdb Jac − (−1)[a][b] ηca Jbd . (2.5) Here [a] = 0 if a ≤ D and [a] = 1 otherwise. As customary, [A, B] represents the usual commutator unless both A and B are odd and in that case [A, B] = AB + B A. One can easily check that [Jab , ] = 0, [Jab , R] = 0.
(2.6)
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Therefore, [Jab , H ] = 0 for all a, b, and the system has an osp(D|2n) symmetry. Let E=
D+2n
X a ∂a , T = E +
a=1
d −1 , 2
where E is the Euler operator. The following lemma is a generalisation of a result in [BB, Appendix] to the superspace setting. Lemma 2.1. Define the following operators: i i R (− − 1) , J0 = R (− + 1) . (2.7) 2 2 (1) The operators satisfy the commutation relations of the Lie algebra so(2, 1): J−2 = T, J−1 =
[J−1 , J0 ] = J−2 , [J−2 , J−1 ] = −J0 , [J0 , J−2 ] = J−1 .
(2.8)
(2) This Lie algebra commutes with the Lie superalgebra osp(D|2n) spanned by the operators Jab : [Jab , J0 ] = [Jab , J−1 ] = [Jab , J−2 ] = 0, ∀a, b. Proof. Note that the Euler operator satisfies [E, X a ] = X a and [E, ∂a ] = −∂a . This immediately leads to [E, Jab ] = 0. Now part (2) easily follows from this commutation relation and also the commutation relations (2.6). To prove the first relation in (2.8), note that [J−1 , J0 ] = 21 R[, R]. Using
2 d −1 [, R] = +E , (2.9) R 2 we obtain the desired result. The other two relations are easily proven by using properties of the Euler operator E. Theorem 2.1. For the bound states (with E < 0), let
√ = g, where g = exp −T ln −2E . Then the Schrödinger equation (2.1) is equivalent to 1 where h 0 = i J0 . h0 = − √ −2E
(2.10)
Proof. Let = R(H − E). The Schrödinger equation can be re-written as g = 0. Using the following relations 1 g Rg −1 = √ R, gg −1 = −2E −2E in the equation we obtain (2.10). In the remainder of this paper we shall consider bound states only. The algebraic formulation of the quantum Kepler problem allows us to obtain the spectrum of the Hamiltonian for the bound states by using the representation theory of the algebras described in Lemma 2.1. However, in order to determine the multiplicities of the eigenvalues and also to construct the corresponding eigenfunctions, we need to explore a larger dynamical symmetry of the problem. This is done in the next section.
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3. Dynamical Supersymmetry In this section we shall show that the quantum Kepler problem in the superspace R D|2n has a dynamical supersymmetry described by the orthosymplectic Lie superalgebra osp(2, D + 1|2n). The generators and commutation relations of the superalgebra will be given explicitly. A parabolic subalgebra and some nilpotent subalgebra of the dynamical supersymmetry algebra will be studied in detail, which play crucial roles in solving the quantum Kepler problem. 3.1. Dynamical supersymmetry. We have the following result: Lemma 3.1. Let a = R∂a for all a = 1, 2, . . . , D + 2n. (1) We have [J−2 , a ] = 0, and Aa := [J−1 , a ] =
i X a ( + 1) − i T ∂a , 2
i X a ( − 1) − i T ∂a . 2 (2) The operators a , Aa and Ma satisfy the following commutation relations: Ma := [J0 , a ] =
[ a , b ] = Jab , [ a , Ab ] = −ηba J−1 , [ a , Mb ] = −ηba J0 , [Aa , Ab ] = −Jab , [Ma , Mb ] = Jab , [Aa , Mb ] = −ηba J−2 .
(3.1)
(3.2)
Proof. The lemma can be proved by straightforward computations. To prove part (1), note that i i Aa = − R[, R]∂a + R[∂a , R]( + 1). 2 2 Using the first relation in (2.2) and also (2.9), we easily arrive at the result. The formula for Ma can be proved in exactly the same way. The proof for the first formula in (3.2) is very simple, thus we omit the details. Now consider [ a , Ab ]. We have [ a , Ab ] =
i i ηab R − i[R∂a , T ∂b ] + [R∂a , X b ]. 2 2
Using the following relations [R∂a , T ∂b ] = −(−1)[a][b]
Xb T ∂a , R
2X b T ∂a , (3.3) R and also (2.9) we easily arrive at the desired formula. The commutator [ a , Mb ] can be similarly proved by using relations (3.3) and (2.9). Let us consider [Aa , Ab ]. We have [R∂a , X b ] = ηab R − (−1)[a][b]
1 −[Aa , Ab ] = [T ∂a , T ∂b ] − [T ∂a , X b ( + 1)] 2 1 +(−1)[a][b] [T ∂b , X a ( + 1)] 2 1 + X a [, X b ] − (−1)[a][b] X b [, X a ] ( + 1). 4
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To simplify the right hand side of the above equation, we use the following relations: [, X a ] = 2∂a , [T ∂a , T ∂b ] = 0, [T ∂a , X b ] = T ηba + (−1)[a][b] X b ∂a ,
[T ∂a , X b ] = T ηba − (−1)[a][b] X b ∂a .
(3.4)
Carefully combining similar terms we arrive at [Aa , Ab ] = −Jab . The commutators [Ma , Mb ] and [Aa , Mb ] can be calculated in the same way by using the relations in (3.4).
I 0 , where Let us introduce a (D + 2n + 3) × (D + 2n + 3) matrix (η K L ) = 2,1 0 η ⎛ ⎞ −1 0 0 I2,1 = ⎝ 0 −1 0⎠ and η is the metric of R D|2n . The indices K and L take values 0 0 1 −2, −1, 0, 1, 2, . . . , D ordered as shown. We set [K ] = 0 if K = i ≤ 0, and [K ] = [a] if K = a ≥ 1. Theorem 3.1. Introduce the operators JK L such that JK L = −(−1)[K ][L] JL K with Ji j = i jk Jk , for i, j, k ∈ {−2, −1, 0}, J−2,a = a , J0,a = −Aa , J−1,a = Ma , Jab is defined by (2.4), for a, b ∈ {1, 2, . . . , D + 2n},
(3.5)
where i jk is totally skew symmetric in the three indices and −2,−1,0 = 1. These operators form a basis for the orthosymplectic Lie superalgebra osp(2, D + 1|2n) with the commutation relations [JK L , J P Q ] = η P L JK Q + (−1)[K ]([L]+[P]) η Q K JL P −(−1)[P][Q] η Q L JK P − (−1)[K ][L] η P K JL Q .
(3.6)
Proof. Lemma 2.1 and Lemma 3.1 have verified all the commutation relations except [Jab , Jic ] for a, b, c ≥ 1 and i ≤ 0, and some cases of [Ji j , Jkc ] with i, j, k ≤ 0 and c ≥ 1. The relations for [Ji j , Jkc ] easily follow from Lemma 3.1(1) and Lemma 2.1(1). A direct calculation gives [Jab , J−2,c ] = ηcb J−2,a − (−1)[b][c] ηca J−2,b . Since Jab commutes with all Ji j by Lemma 2.1(2), we may replace the index −2 in this equation by any i ∈ {−2, −1, 0}. This completes the proof of the theorem. 3.2. Root system. Denote by g the complexification of the real Lie superalgebra osp(2, D + 1|2n) spanned by the operators in (3.5), then g ∼ = osp(D + 3|2n). Let us now specify the root system for g that will be used for characterising highest weight representations. We adopt notations and conventions from [K] (see also [S]), but take a nonstandard root system for the Lie superalgebra g. Dynkin diagrams corresponding to the root systems of g for even and odd D are respectively given by
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...
...
if D is even,
...
...
if D is odd.
Here the black and grey nodes correspond to odd simple roots. When D = 2m, there are m + 1 + n nodes in the Dynkin diagram. Order the nodes from left to right. They respectively correspond to the simple roots
0 − 1 , 1 − 2 , . . . , m−1 − m , m − δ1 , δ1 − δ2 , . . . , δn−1 − δn , δn . The element h 0 belongs to the Cartan subalgebra with 0 (h 0 ) = 1 and i (h 0 ) = 0 = δ j (h 0 ) for all the other i and all the δ j . When D = 2m + 1, there are m + 2 + n nodes, respectively corresponding to the simple roots
−1 − 0 , 0 − 1 , 1 − 2 , . . . , m−1 − m , m − δ1 , δ1 − δ2 , . . . , δn−1 − δn , 2δn . In this case, −1 (h 0 ) = 1 while the evaluations of all the other i and all the δ j on h 0 are zero. + n tuple, which is An element in the weight space of g will be written as an D+3 2 the coordinate in the basis consisting of the i and δ j . The basis is ordered in such a way that the δ j are positioned after all i , and the i appear in their natural order, and so do also the δ j . Since osp(D + 1|2n) is a regular subalgebra of g, we shall take for it the root system compatible with that of g. This is specified by the Dynkin diagram obtained by removing the left most node from the Dynkin diagram of g. 3.3. Parabolic subalgebra. In this subsection we discuss subalgebras of the Lie superalgebra g. The main result established is Proposition 3.1, which is of crucial importance for solving the Kepler problem on the superspace. Unfortunately the material presented here is quite technical, so we have relegated some of it to the Appendix. Define the linear operator adh 0 on g by adh 0 (Y ) = [h 0 , Y ] for all Y ∈ g (recall that h 0 = i J0 ). Now g decomposes into a direct sum g = g−1 ⊕ g0 ⊕ g+1 of eigenspaces of adh 0 with eigenvalues +1, 0 and −1 respectively. Here (1) (2) (3)
g+1 is spanned by i(J−1 − i J−2 ) and Ma − i a (a ≥ 1), g−1 is spanned by i(J−1 + i J−2 ) and Ma + i a (a ≥ 1), g0 is spanned by h 0 , Aa and Jab (a, b ≥ 1).
The subspaces g+1 , g−1 and g0 all form subalgebras of g. In particular, g0 is the subalgebra gl1 ⊕ osp(D + 1|2n), and [g+1 , g+1 ] = [g−1 , g−1 ] = 0. Note that if D is even, g+1 is spanned by the root vectors associated with the positive roots
0 , 0 ± i , ∀i > 0; 0 ± δ j , ∀ j. If D is odd, the subalgebra g+1 is spanned by the root vectors associated with the positive roots
−1 ± i , ∀i ≥ 0; −1 ± δ j , ∀ j. Both g+1 and g−1 are stable under the adjoint action of g0 in the sense that [g0 , g±1 ] = g±1 , thus form g0 -modules. When restricted to modules for the osp(D+1|2n) subalgebra of g0 , both g+1 and g−1 are isomorphic to the natural osp(D + 1|2n)-module.
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We have the following parabolic subalgebra of g: p := g0 ⊕ g+1
(3.7)
with the nilpotent radical g+1 . Furthermore, g = p ⊕ g−1 . Denote by Ug the complex linear span of products of the operators in g. Let Up and Ug−1 be the sub superalgebras of Ug generated by the elements of p and g−1 respectively. Then the underlying vector space of Ug is isomorphic to Ug−1 Up. Note that Ug−1 is Z2 -graded commutative. Remark 3.1. The associative superalgebra Ug is a quotient of the universal enveloping algebra U(g) of g = osp(D + 3|2n). Similar comments apply to Up and U(p), etc.. ∞ U The subalgebra Ug−1 has a Z-grading Ug−1 = ⊕l=0 g−1 (−l). Let K 0 = i(J−1 +i J−2 ) and K a = Ma + i a (a ≥ 1), which form a basis of the subalgebra g−1 . Then Ug−1 (−l) is spanned by all the homogeneous polynomials of order l in the elements K A with A = 0, 1, . . . , D + 2n. Here we should note that K A are odd in the Z2 grading for all A > D, and in this case (K A )2 = 0. However, K 0 is even and (K 0 )l = 0 for all l. Let η0 = (η AB ) with A, B = 0, 1, . . . , D + 2n be the matrix obtained from (η K L ) by removing the first two rows and first two columns. This is the metric relative to which the osp(D + 1|2n) subalgebra of g0 is defined. Now the (adjoint) action of g0 on g−1 naturally extends to an action on Ug−1 (−l). Denote by η0 (g−1 , g−1 ) the subspace of osp(D + 1|2n)-invariants in Ug−1 (−2). Then η0 (g−1 , g−1 ) is spanned by
(K )2 :=
D+2n
ηB A K A K B,
A,B=0
where η AB are the matrix elements of η0−1 . We have the following result. Lemma 3.2. The operators K A satisfy (K )2 = 0. Proof. The proof of this claim involves a considerable amount of calculations. Thus we shall present only the main formulae needed. Let us first calculate K 02 . We have 2
1 2 R( + 1) − T K0 = 2 1 1 = (R( + 1))2 − T R( + 1) − [R( + 1), T ] + T 2 . 4 2 Using [R( + 1), T ] = R( − 1), we obtain 1 1 (R( + 1))2 − T R( + 1) − R( − 1) + T 2 . 4 2 K a K a = a (M a Ma + i(M a a + a Ma ) − a a ). Note that
K 02 = Now
D+2n a=1
D+2n a=1
1 a X ( − 1)X a ( − 1) 4 a 1 a + (X ( − 1)T ∂a + T ∂ a X a ( − 1)) 2 a T ∂ a T ∂a . −
M a Ma = −
a
(3.8)
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The first term can be expressed as − 41 R 2 ( − 1)2 − 41 a X a [( − 1), X a ]( − 1). Using [( − 1), X a ] = 2∂a we obtain − 41 R 2 ( − 1)2 − 21 E( − 1). The second term can be simplified to T (T + 1/2)( − 1) + 21 E( + 1), while the third yields −T (T + 1). Combining these results together we arrive at D+2n a=1
1 1 M a Ma = − R 2 ( − 1)2 − T ( + 1) − T 2 + E. 4 2
(3.9)
It is easy to show that D+2n
a a = E + R 2 .
(3.10)
a=1
D+2n a (M a + a Ma ) can be expressed as Now 2i a=1 − X a ( − 1)R∂a − R∂ a X a ( − 1) + 2 (T ∂ a R∂a + R∂ a T ∂a ). a
a
(3.11)
a
The first sum can be rewritten as a X a [( − 1), R]∂a + R E( − 1). Using Eq. (2.9) a a we obtain a X [( − 1), R]∂a = a 2XR T ∂a = 2T R1 E. This leads to 1 X a ( − 1)R∂a = 2T E + R E( − 1). R a We also have
R∂ a X a ( − 1) = R(d + E)( − 1).
a
Finally the third sum in (3.11) gives 1 2 (T ∂ a R∂a + R∂ a T ∂a ) = 2T E + 4T R. R a Combining the results together we obtain i
D+2n
(M a a + a Ma ) =
a=1
1 R( − 1) + T R( + 1). 2
(3.12)
Combining Eqs. (3.9), (3.10) and (3.12) we readily obtain D+2n a=1
1 1 K a K a = − R 2 ( − 1)2 − R 2 − T ( + 1) 4 2 +T R( + 1) +
1 R( − 1) − T 2 . 2
Using R 2 ( + 1)2 = (R( + 1))2 − 2T ( + 1), we immediately arrive at D+2n a=1
1 1 K a K a = − (R( + 1))2 + T R( + 1) + R( − 1) − T 2 . 4 2
Recalling Eq. (3.8), we see that K 02 +
D+2n a=1
(3.13)
K a K a = 0. This completes the proof.
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With the help of Lemma 3.2, we can prove the following result, which will play a crucial role in understanding the osp(2, D + 1|2n) representation appearing in the quantum Kepler problem. Proposition 3.1. As a module for the subalgebra osp(D + 1|2n) of g0 , the subspace Ug−1 (−l) of Ug−1 is irreducible and isomorphic to the irreducible rank l symmetric (in the Z2 -graded sense) tensor of the natural module for osp(D +1|2n). The osp(D +1|2n) highest weight of Ug−1 (−l) is given by (l, 0, . . . , 0). Remark 3.2. The analogous statement for the subalgebra U(g−1 ) of the universal enveloping algebra U(g) of g = osp(D + 3|2n) is completely false. Proof for Proposition 3.1. We need the general facts about symmetric tensors of the natural module V for osp(D +1|2n) discussed in Appendix A. Here it is more convenient to choose a basis {v A |A = 0, 1, . . . , D +2n} for V such that the non-degenerate invariant bilinear form , : V × V −→ C gives the metric (η AB ). Now we use this matrix to define and ∗ . Under the condition that D > 2n + 1, every symmetric tensor power S(V )l of V is semisimple as a osp(D + 1|2n)-module by Lemma A.1. Furthermore, S(V )l = S(V )l0 ⊕ S(V )l−2 ∗ , and the harmonic space S(V )l0 defined by S(V )l0 = {w ∈ S(V )l | w = 0} is irreducible with highest weight (l, 0, . . . , 0). Now we turn to Ug−1 . The Z-graded superalgebra homomorphism S(V ) → Ug−1 is also an osp(D + 1|2n) map. The restriction of this map to any homogeneous component is nonzero. Since (K )2 = 0 by Lemma 3.2, the map sends S(V )l−2 ∗ to zero. Thus Ug−1 (−l) is the image of S(V )l0 . Since S(V )l0 is irreducible, we must have Ug−1 (−l) ∼ = S(V )l0 as osp(D + 1|2n)-modules. 4. Solution of Quantum Kepler Problem on Superspace
√ 4.1. Induced representations. Let 0 = g0 0 with g0 = exp −T ln −2E0 , where 0 is the ground state wave function. Then 0 = (−2E0 )
d−1 4
e−R .
Lemma 4.1. The function 0 spans a 1-dimensional module for the parabolic subalgebra p defined by Eq. (3.7). Proof. It is evident that Jab ( 0 ) = 0 for all a, b ≥ 1. Also the definition of 0 implies that it is an eigenvector of h 0 . Therefore the lemma will be valid if 0 is annihilated by i(J−1 − i J−2 ), Aa and Ma − i a (a ≥ 1). To proceed further, we need to use the following formulae:
d − 1 −R e , (e−R ) = 1 − R
d −1 − R e−R , T (e−R ) = 2
d − 1 −R e . (4.1) T ∂a (e−R ) = X a 1 − 2R Using the first and third formulae in the following equation: Aa (e−R ) =
i X a ( + 1)(e−R ) − i T ∂a (e−R ), 2
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we immediately arrive at Aa (e−R ) = 0. Note that Ma (e−R ) = Aa (e−R ) − i X a e−R = −i X a e−R . This easily leads to (Ma − i a )(e−R ) = 0, ∀a ≥ 1. Finally by using the first and second formulae of (4.1), we can show that i(J−1 − i J−2 )(e−R ) = 0. Let H = Ug 0 , which has a natural g-module structure. Then ∞ Ug−1 (−l) 0 . H = Ug−1 0 = ⊕l=0
It follows that the operator h 0 is diagonalisable in H and its eigenvalues must be of the form − d−1 2 − k (k ∈ Z+ ). 0 If L λ denotes the irreducible p-module with highest weight λ, we have the generalised Verma module Vλ = U(g) ⊗U(p) L 0λ . The p-module L 0λ is 1-dimensional if
d −1 λ= − , 0, . . . , 0 , (4.2) 2 and it is evident that H is a quotient module of the corresponding generalised Verma module Vλ . The highest weight vector 0 of H generatesa module for the so(2, D +1) subalgebra D+3 of osp(2, D +1|2n) with a highest weight λ0 = − d−1 entries , 0, . . . , 0 (the first 2 2 of (4.2)). This module is necessarily infinite dimensional as the highest weight is not dominant. This in particular implies that Ug−1 (−l) 0 = 0, for all l. Another fact which can be deduced is that if H contains any nontrivial submodule H1 , then H/H1 has to be infinite dimensional as well, as the image of 0 generates an infinite dimensional so(2, D + 1)-submodule in the quotient. Since 0 has trivial g0 -action and Ug−1 (−l) is irreducible under the action of g0 by Proposition 3.1, we immediately see that Ug−1 (−l) 0 is irreducible as a g0 -module. Now let H1 be a non-trivial g-submodule of H, then there exists some integer l0 such that all Ug−1 (−l) 0 with l ≥ l0 belong to H1 . This contradicts the fact that H/H1 is infinite dimensional. Thus H must be irreducible as a g-module. D+1|2n Let L µ denote the irreducible osp(D + 1|2n)-module with highest weight µ. We have proved the following theorem. Theorem 4.1. (1) As an osp(2, D + 1|2n)-module H is irreducible. ∞ L D+1|2n as an osp(D + 1|2n)-module. (2) The restriction of H is isomorphic to ⊕l=0 (l,0,...,0) Remark 4.1. It is known [J,EHW] that λ0 does not give rise to a unitarisable highest weight so(2, D + 1)-module. Thus the so(2, D + 1)-submodule of H generated by 0 is not unitarisable, and this in turn implies that H as an osp(2, D + 1|2n)-module is not unitarisable.
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4.2. Solution of quantum Kepler problem on superspace. In this subsection we use the results on the irreducible osp(D + 3|2n)-module H obtained to determine the bound states of the quantum Kepler problem on the superspace R D|2n . Let Hk = Ug−1 (−k) 0 , and for k ∈ Z+ , 2 1 1 . Ek = − 2 d−1 2 +k √ Define gk = exp −T ln −2Ek , and set H˜ =
∞
H˜ k , where H˜ k = {gk−1 v | v ∈ Hk }.
k=0
We have the following result. Theorem 4.2. (1) The quantum Hamiltonian operator H is diagonalisable when acting on H˜ and has the eigenvalues El for l ∈ Z+ . (2) The dimension of the subspace H˜ l is given by the formula ⎛ ⎞ ⎛⎛ ⎞ ⎛ ⎞⎞ l D+k 2n 2n ⎝ ⎠ ⎝⎝ ⎠−⎝ ⎠⎠ , k l −k l −2−k k=0 ⎛ ⎞ a where the binomial coefficient ⎝ ⎠ is assumed to be zero if b < 0 or b > a. b (3) The subspace H˜ I (I = 0, 1, . . . ) is the entire E I -eigenspace of the quantum Hamiltonian operator H . Proof. It follows from Theorem 2.1 that every nonzero vector in H˜ l is indeed an eigenvector of the quantum Hamiltonian operator H with eigenvalue El . To prove the formula for the dimension of H˜ l , we note that dim H˜ l = dim Hl , and by Theorem 4.1, D+1|2n dim H˜ l = dim L (l,0,...,0) . We have D+1|2n
dim L (k,0,...,0) = dim S(V )l − dim S(V )l−2 . It is well known that dim S(V )l =
l k=0
⎛ ⎝
D+k k
⎞⎛
2n
⎞
⎠⎝ ⎠. l −k
Hence the formula for dim H˜ l follows. To prove the third claim, we need some input from the Schrödinger equation. Let us return to the original form (2.1) of the equation. Because of the osp(D|2n) symmetry of the equation, the wave functions can be written as C-linear combinations of functions of the form Rωl χl by the first part of Lemma A.1, where ω is a harmonic polynomial (that is, ω = 0) in the coordinates X a , which is homogeneous of degree l. The function χl
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depends on R only, and the factor R −l is introduced for convenience. Then Eq. (2.1) reduces to the following equation for χl :
1 1 d 2 χl d − 1 dχl l(d − 2 + l) − − + χl − χl = Eχl . (4.3) 2 2 2 dR R dR R R This has the same form as the radial part of the Schrödinger equation for the Kepler problem on Rd , and can be solved in terms of generalised Laguerre polynomials (see, e.g., [Al, (4), (14)]). Such a solution, when its argument R is replaced by r , is square integrable over the positive half line R+ with respect to the measure r d−1 dr . For a fixed l, we denote by χl, j ( j = 0, 1, 2, . . . ) the independent solutions of (4.3).
2 χl, j ¯ , the integrant of which vanishes Consider Rωl χl, j | Rωl χl, j = R D ωω Rl exponentially fast as r goes to infinity. Thus we only need to analyse the r → 0 end of the integral to see whether the integral converges. By inspecting the form of the generalised Laguerre polynomials we can see that at the worst, the integral behaves ∞ 2 l 2 n−l like R D (χl, j (R))2 ∝ (2 )n 0 (χl, j (r ))2 r d−1 dr . It follows from the R2 R2 square integrability of χl, j over R+ with respect to the measure r d−1 dr that all Rωl χl, j belong to the set F defined in Subsect. 2.2. For a fixed l, the solutions χl, j of (4.3) respectively correspond, in a one-to-one manner, to the eigenvalues El+ j of the quantum Hamiltonian operator (2.3). In particular, the ground state corresponds to the energy eigenvalue E0 . Thus for a given non-negative integer I , the eigenspace of H corresponding to the energy E I is spanned by Rωl χl, j for all homogeneous harmonic polynomials ω of degree l, and for all l, j ≥ 0 with l + j = I . I D|2n L (l,0,0,...,0) . By As a module over osp(D|2n), the E I -eigenspace is isomorphic to l=0 using the branching rule (A.3), we have I
D|2n D+1|2n L (l,0,0,...,0) ∼ = L (I,0,0,...,0) ∼ = H˜ I .
l=0
Remark 4.2. In view of Remark 2.1, the generalised Kepler problem on the superspace is equivalent to an eigenvalue problem involving a system of partial differential equations. It is a rather non-trivial matter that the problem is still soluble. Appendix A. Symmetric Superalgebra For proving Theorem 3.1, we need some general facts about symmetric tensors of the natural module V for osp(M|2n), which we briefly discuss in this Appendix. In general the symmetric tensors are not completely reducible, and this complicates the matter of decomposing these representations enormously. Fortunately we only need to consider the case M − 2n > 1 for the purpose of this paper. In this case, the symmetric tensors are semi-simple, as we shall show. We work over the complex field and use the root system described in Sect. 3 for osp(M|2n). For the sake of being concrete, we label the basis elements i and δ j of the a weight space by i = 1, 2, . . . , M 2 and j = 1, 2, . . . , n. Let v with a = 1, 2, . . . , M + 2n be a weight basis for the natural module V , where the weight wt (v a ) of v a is greater
560
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than that of v b if a < b. Then the highest weight is 1 , and the lowest weight is − 1 . We assume that the highest weight vector v 1 is even. There exists a non-degenerate osp(M|2n)-invariant bilinear form , : V × V −→ C, which is unique up to scalar multiples. Note that v a , v b = 0 if and only if wt (v a ) = −wt (v b ). Let ηab = v a , v b and form the matrix η−1 = (ηab ). We also write η = (ηab ) for the inverse matrix. Let S(V ) be the Z2 -graded symmetric superalgebra of V , that is, the superalgebra generated by the v a subject to the relations that v a v b = −v b v a if both v a and v b are ∞ S(V ) odd, and v a v b = v b v a otherwise. Then this is a Z-graded algebra S(V ) = ⊕l=0 l with elements of V having degree 1. Define operators ∗ =
M+2n a=1
v a ηab v b , =
M+2n a=1
ηba
M+2n ∂ ∂ M + 2n − 1 a ∂ + , T = v , ∂v a ∂v b 2 ∂v a a=1
which all commute with osp(M|2n). The operators satisfy the commutation relations [T, ∗ ] = 2∗ , [T, ] = −2, [∗ , ] = −T, thus their real spanned is isomorphic to the Lie algebra su(1, 1). We consider S(V ) as a complex module for su(1, 1). The operator T is diagonalisable with eigenvalues M+2n−1 + l for l ∈ Z+ . Now we need to assume that 2 M − 2n > 1.
(A.1)
Evidently every submodule of S(V ) is of highest weight type. Since the eigenvalues of −T are all strictly negative, the sub-representations are necessarily infinite dimensional. The eigenvalues of the quadratic Casimir of su(1, 1) corresponding to distinct highest weights λl := − M+2n−1 + l with l ∈ Z+ are different, thus S(V ) decomposes into 2 a direct sum S(V ) = ⊕l C (l) , where each submodule C (l) has the property that all irreducible subquotients are isomorphic with the same highest weight λl . An irreducible su(1, 1)-module with highest weight λl for l ∈ Z+ is unitarisable. It follows that every isotypical component is unitarisable and hence completely reducible. Thus S(V ) is completely reducible with respect to su(1, 1). This in particular implies that every weight vector in S(V )l can be uniquely expressed as v + w with v ∈ S(V )l ∩ ker being a highest weight vector, and w ∈ ∗ S(V )l . We can also deduce that ∗ S(V )l = S(V )l ∩ im∗ for each S(V )l by noting the obvious fact that if v belongs to a weight space in an irreducible su(1, 1)-module, then ∗ v is a nonzero scalar multiple of v unless v is the highest weight vector. Thus we have the following vector space decomposition S(V ) = ker ⊕ im∗ .
(A.2)
Since the su(1, 1) algebra commutes with osp(M|2n), Eq. (A.2) is a decomposition of osp(M|2n)-modules. Denote S(V )l ∩ ker by S(V )l0 and call it the harmonic space of S(V )l . It is easy to see that for all l ≤ 2, S(V )l0 is isomorphic to the irreducible osp(M|2n)-module with highest weight (l, 0, . . . , 0). Assume that this is also true for l > 2, then the 0 must highest weight vector for S(V )l0 is (v 1 )l . Each highest weight vector in S(V )l+1 1 l b contain a term (v ) v for some b. In order for the corresponding weight to be dominant,
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561
v b is either v 1 , v 2 or the lowest weight vector of V , which respectively have weights (l + 1, 0, . . . , 0), (l, 1, 0, . . . , 0) and (l − 1, 0, . . . , 0). We can write down all the vectors of the same weights in S(V )l+1 , and try to make linear combinations of them to obtain highest weight vectors. Simple calculations show that there can not be any highest weight vector with weight (l, 1, 0, . . . , 0) as S(V ) is the symmetric superalgebra. The highest weight vector corresponding to the weight (l − 1, 0, . . . , 0) is (v 1 )l−1 ∗ , which belongs 0 . This proves that S(V )0 is isomorphic to the irreducible to im∗ but not S(V )l+1 l+1 osp(M|2n)-module with highest weight (l + 1, 0, . . . , 0). To summarise, we have the following result. Lemma A.1. Keep notations as above. (1) Under condition (A.1), S(V )l = S(V )l0 ⊕ S(V )l−2 ∗ as osp(M|2n)-module, M|2n and S(V )l0 is isomorphic to the irreducible module L (l,0,...,0) with highest weight (l, 0, . . . , 0). ∞ M|2n (2) As an su(1, 1) × osp(M|2n)-module, S(V ) ∼ = L=0 L (l) ⊗ L (l,0,...,0) , where L (l) is the irreducible su(1, 1)-module with highest weight − M+2n−1 − l. 2 A further result which can be deduced from the above lemma is the branching rule M|2n of the irreducible symmetric tensor module L (l,0,...,0) to osp(M − 1|2n)-modules. We assume that the condition M −1−2n > 1 is satisfied. Then the symmetric tensor powers of the natural module V for osp(M − 1|2n) is completely reducible by the lemma. We l have the following osp(M − 1|2n)-module isomorphism S(V )l ∼ = k=0 S(V )l−k . Then by the first part of Lemma A.1,
0 S(V )l ∼ ⊕l−2 = ⊕lk=0 S(V )l−k k=0 S(V )l−k , where S(V )i0 is the harmonic subspace of S(V )i . Using the first part of Lemma A.1 to the left-hand side, and also noting that the second term on the right hand side can be ∼ re-written as ⊕l−2 k=0 S(V )l−k = S(V )l−2 , we obtain
0 S(V )l−2 ∼ S(V )l−2 . S(V )l0 = ⊕lk=0 S(V )l−k That is, the following branching rule holds: M|2n L (l,0,...,0) ∼ =
l
M−1|2n
L (l−k,0,...,0) as osp(M − 1|2n)-module.
(A.3)
k=0
Acknowledgements. I thank Guowu Meng for helpful correspondence. Financial support from the Australian Research Council is gratefully acknowledged.
References [Al] [BB] [D1] [D2]
Al-Jaber, S.M.: Hydrogen atom in n dimensions. Intl. J. Theor. Phys. 37, 1289–1298 (1998) Barut, A.O., Bornzin, G.L.: SO(4, 2)-formulation of the symmetry breaking in relativistic kepler problems with or without magnetic charges. J. Math. Physics 12, 841–846 (1971) Delbourgo, R.: Grassmann wave functions and intrinsic spin. Intl. J. Mod. Phys. A 3(3), 591–602 (1988) Delbourgo, R.: The flavour of gravity. J. Phys. A 39(18), 5175–5187 (2006)
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Delbourgo, R.: Flavour mixing and mass matrices via anticommuting properties. J Phys. A39, 14735–14744 (2006) Delbourgo, R., Jarvis, P.D., Warner, R.C.: Schizosymmetry: a new paradigm for superfield expansions. Mod. Phys. Lett. A 9(25), 2305–2313 (1994) Enright, T.J., Howe, R., Wallach, N.R.: A classification of unitary highest weight modules. In: Representation theory of reductive groups (Park City, Utah, 1982), Progr. Math. 40, 97–143, (1983) Iwai, T.: The geometry of the SU(2) kepler problem. J. Geom. Phys. 7, 507–535 (1990) Jackobsen, H.P.: Hermitian symmetric spaces and their unitary highest weight modules. J. Funct. Anal. 52(3), 385–412 (1983) Jarvis, P.D., Green, H.S.: Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded lie algebras. J. Math. Phys. 20(10), 2115–2122 (1979) Kac, V.G.: Lie superalgebras. Adv. in Math. 26(1), 8–96 (1977) Kirchberg, A., Länge, J.D., Pisani, P.A.G., Wipf, A.: Algebraic solution of the supersymmetric hydrogen atom in d dimensions. Ann. Phys. 303, 359–388 (2003) Meng, G.: MICZ-Kepler problems in all dimensions. J. Math. Physics, 48(3), 032105, 14p. (2007) Meng, G.: To appear Meng, G., Zhang, R.B.: Generalised MICZ-Kepler problems and unitary highest weight modules. http://arxiv.org/list/math-ph/0702086, 2007 McIntosh, H.V., Cisneros, A.: Degeneracy in the presence of a magnetic monopole. J. Math. Physics 11, 896–916 (1970) Scheunert, M.: The theory of Lie superalgebras. An introduction. Lecture Notes in Mathematics 716, Berlin: Springer, 1979 Zwanziger, D.: Exactly soluble nonrelativistic model of particles with both electric and magnetic charges. Phys. Rev. 176, 1480–1488 (1968)
Communicated by Y. Kawahigashi
Commun. Math. Phys. 280, 563–573 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0458-9
Communications in
Mathematical Physics
Erratum
Decay of Solutions of the Wave Equation in the Kerr Geometry F. Finster1, , N. Kamran2, , J. Smoller3, , S.-T. Yau4, 1 NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany.
E-mail: [email protected]
2 Department of Math. and Statistics, McGill University, Montréal, Québec, Canada H3A 2K6.
E-mail: [email protected]
3 Mathematics Department, The University of Michigan, Ann Arbor, MI 48109, USA.
E-mail: [email protected]
4 Mathematics Department, Harvard University, Cambridge, MA 02138, USA.
E-mail: [email protected] Received: 4 April 2006 / Accepted: 30 November 2007 Published online: 18 March 2008 – © Springer-Verlag 2008 Commun. Math. Phys. 264, 465–503 (2006)
As was recently pointed out to us by Thierry Daudé, there is an error on the last page of our paper [2]. Namely, the inequality (8.3) cannot be applied to the function (t) because it does not satisfy the correct boundary conditions. This invalidates the last two inequalities of the paper, and thus the proof of decay is incomplete. We here fill the gap using a different method. At the same time, we will clarify in which sense the sum over the angular momentum modes converges in [1, Theorem 1.1] and [2, Theorem 7.1], an issue which in these papers was not treated in sufficient detail. The arguments in these papers certainly yield weak convergence in L 2loc ; here we will prove strong convergence. Our method here is to split the wave function into the high and low energy components. For the high energy component, we show that the L 2 -norm of the wave function can be bounded by the energy integral, even though the energy density need not be everywhere positive (Sect. 1). For the low energy component we refine our ODE techniques (Sect. 2). Combining these arguments with a Sobolev estimate and the Riemann-Lebesgue lemma completes the proof (Sect. 3). We begin by considering the integral representation of [2, Theorem 7.1], for fixed k and a finite number n 0 of angular momentum modes, n 0 (t, r, ϑ) :=
∞ −∞
ˆ n 0 (ω, r, ϑ), dω e−iωt
The online version of the original article can be found under doi:10.1007/s00220-006-1525-8. Research supported in part by the Deutsche Forschungsgemeinschaft.
Research supported by NSERC grant # RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998.
Research supported in part by the NSF, Grant No. 33-585-7510-2-30.
(1)
564
F. Finster, N. Kamran, J. Smoller, S.-T. Yau
where for notational convenience we have omitted the ϕ-dependence (i.e. the factor ˆ n 0 (ω) is defined by e−ikϕ ), and ˆ n 0 (ω, r, ϑ) =
n0 2 1 1 ωn a b tab ωn (r, ϑ) <ωn , 0 > 2π ω n=1 a,b=1
(as in [1], we always denote the scalar wave function by , whereas = (, ∂t ) is a two-component vector). We recall that for large ω, the WKB-estimates of [1, Sect. 6] ensure that the fundamental solutions bkωn go over to plane waves, and thus, since the ˆ n 0 (ω, r, ϑ) decays initial data 0 is smooth and compactly supported, the function rapidly in ω (for details on this method see [3, proof of Theorem 6.5]). As a consequence, n 0 and its derivatives are, for r and ϑ in any compact set, uniformly bounded in time. Our goal is to obtain estimates uniform in n 0 , justifying that, as n 0 → ∞, n 0 converges in L 2loc to the solution of the wave equation. To arrange the energy splitting we choose for a given parameter J > 0, a positive smooth function χH+ supported on (J, ∞) with χH+ |[2J,∞) ≡ 1. We define χH− by χH− (ω) = χH+ (−ω) and set χL = 1 − χH+ − χH− . We introduce the high-energy ˆ nH±0 and the low-energy contribution ˆ nL 0 by contributions ˆ nH±0 (ω, r, ϑ) = χH± (ω) ˆ n 0 (ω, r, ϑ), ˆ nL 0 (ω, r, ϑ) = χL (ω) ˆ n 0 (ω, r, ϑ). 1. L 2 -Estimates of the High-Energy Contribution We recall from [1, (2.5)] that the energy density of a wave function in the Kerr geometry is given by 2 (r + a 2 )2 − a 2 sin2 ϑ |∂t |2 + |∂r |2 E() = 1 a2 k 2 ||2 . + sin2 ϑ |∂cos ϑ |2 + (2) − sin2 ϑ Note that the energy density need not be positive due to the last term. However, the next theorem shows that the energy integral ∞ 1 E() := dr d cos ϑ E((t)) r1
−1
(which is independent of time due to energy conservation), in the high-energy region is both positive and can be bounded from below by the L 2 -norm. In what follows, we only consider nH+0 because nH−0 can be treated similarly. Theorem 1.1. There exists a positive constant J0 (depending only on k, but independent of n 0 and 0 ), such that for all J ≥ J0 the following inequality holds for every t: 1 J 2 ∞ (r 2 + a 2 )2 n0 dr d cos ϑ |nH+0 (t)|2 . E(H+ ) ≥ 2 r1 −1 The remainder of this section is devoted to the proof of Theorem 1.1. We begin with the following lemma.
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565
Lemma 1.2. Let g, be measurable functions, g real and complex, such that and g are in L 1 (R). Then 2 dω dω min g(ω), g(ω ) (ω) (ω ) ≥ inf g . R
R
R
Proof. Using a standard approximation argument, it suffices to consider the case that g and are simple functions of the form g(ω) =
A
ga χ (K a ), (ω) =
a=1
A
a χ (K a ),
a=1
where χ (K a ) is the characteristic function of the set K a , and (K a )a=1,...,A forms a partition of R. Then the above inequality reduces to A
min(ga , gb ) a |K a | b |K b | ≥ min g
a,b=1
A
a |K a | b |K b |.
a,b=1
In the case A = 2 and g1 ≤ g2 , this inequality follows immediately from the calculation g1 |c1 |2 + g1 (c1 c2 + c1 c2 ) + g2 |c2 |2 ≥ g1 |c1 |2 + g1 (c1 c2 + c1 c2 ) +g1 |c2 |2 = g1 |c1 + c2 |2 , where ca := a |K a |. In the case A = 3 and g1 ≤ g2 ≤ g3 , we get g1 |c1 |2 + 2Re(c1 c2 + c1 c3 ) + g2 |c2 |2 + 2Re(c2 c3 ) + g3 |c3 |2 ≥ g1 |c1 |2 + 2Re(c1 c2 + c1 c3 ) + g2 |c2 + c3 |2 ≥ g1 |c1 + c2 + c3 |2 . The general case is similar.
The next lemma bounds the L 2 -norm of nH+0 and its partial derivatives by a constant depending on n 0 and 0 . Lemma 1.3. There is a constant C = C(n 0 , 0 ) such that for every t,
∞
r1
(r 2 + a 2 )2 dr
1
−1
3 d cos ϑ |nH+0 (t)|2 + |∂r nH+0 (t)|2 + |∂tk nH+0 (t)|2 ≤ C. k=1
Proof. It suffices to consider one angular momentum mode. For notational simplicity we omit the angular dependence. Since nH+0 and its derivatives are locally pointwise bounded uniformly in time, it follows that their L 2 -norms on any compact set are bounded in time. Near the event horizon, we work with the fundamental solution φ´ in the ReggeWheeler variable u (see [1, (2.18, 5.2)] and [2, Sect. 3]). Then for any sufficiently small u 0 , the integral of ||2 over the region u < u 0 can be written as u0 du |φ|2 , −∞
566
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where it is now convenient to write our integral representation (1) in the form ∞ dω h + (ω) φ´ ω (u) + h − (ω) φ´ ω (u) e−iωt . φ(t, u) = −∞
Here the functions h ± have rapid decay and, as they are supported away from the set {0, ω0 }, they are also smooth (see [2, Sect. 3.1]). Using the Jost representation [2, (3.7, 3.10)], the function φ(t, u) can be decomposed as φ(t, u) = φ+ (t, u) + φ− (t, u) + ρ(t, u), where
φ± (t, u) :=
∞
dω h ± (ω) e±i(ω−ω0 )u−iωt ,
−∞ γu
|ρ(t, u)| ≤ C e
for all t, where C, γ > 0.
Note that the smoothness of h ± implies that φ± decay rapidly in u. From the exponential decay of the factor eγ u it is obvious that the L 2 -norm of ρ is bounded uniformly in t. The L 2 -norms of φ± can be estimated as ∞ ∞ u0 |φ± (t, u)|2 du ≤ |φ± (t, u)|2 du = |φ± (0, u )|2 du =: c, −∞
−∞
−∞
where u = u ∓ t. Near infinity, we work similarly with the fundamental solutions φ` [2, (3.2)]. Again using the Jost representation [2, (3.15) and Lemma 3.3], we get terms depending only on t ± u as well as error terms which decay like 1/u and are thus in L 2 . The time derivatives can be treated in the same way, since a time derivative merely gives a factor of ω which can be absorbed into h ± . For the spatial derivatives we use similarly the estimates for the first derivatives of the Jost functions. Our next step is to decompose the energy integral into a convenient form. To do this, we introduce a positive mollifier α ∈ C0∞ ([−1, 1]) with the properties α(−ω) = α(ω) and α(ω)dω = 1. We define the function (ω − ω ) by mollifying the Heaviside function , (ω − ω ) = ( ∗ α)(ω − ω ).
(3)
We now substitute the Fourier representation of nH+0 into the formula for the energy density (2). For simplicity we omit the indices n 0 and H + in what follows. Omitting the first positive summand in (2), we get the inequality ∞ ∞ dω dω e−i(ω−ω )t E()(t, r, ϑ) ≥ −∞
−∞
1 a2 ˆ ˆ ) (ω k 2 (ω) × (4) − sin2 ϑ ˆ ˆ ) + sin2 ϑ ∂cos ϑ (ω) ˆ ˆ ) + (ω − ω ) ∂r (ω) ∂r (ω ∂cos ϑ (ω (5)
ˆ ˆ ) + sin2 ϑ ∂cos ϑ (ω) ˆ ˆ ) . (6) + (1 − )(ω − ω ) ∂r (ω) ∂r (ω ∂cos ϑ (ω
Erratum: Decay of Solutions of the Wave Equation in the Kerr Geometry
567
We multiply by a positive test function η(u) ∈ C0∞ (R) and integrate over r and cos ϑ. Integrating by parts in (5) and (6) to the right and left, respectively, we can use the wave equation 2 ∂ ∂ 1 2 − − (r + a 2 )ω + ak ∂r ∂r 1 ∂ ∂ 2 2 2 ˆ sin ϑ + − (aω sin ϑ + k) (ω) = 0 ∂ cos ϑ ∂ cos ϑ sin2 ϑ to obtain ∞ 1 ∞ ∞ ∞ 1 dr d cos ϑ η(u) (5) + (6) = dr d cos ϑ dω dω e−i(ω−ω )t r1 −1 r1 −1 −∞ −∞ ˆ ) + (1 − ) ∂r (ω) ˆ ˆ ) ˆ (ω × −η (u) (r 2 + a 2 ) (ω) ∂r (ω (7)
ˆ ) , ˆ (ω (8) + η(u) g(ω ) + (1 − ) g(ω) (ω) where g(ω, r, ϑ) =
2 1 1 2 (r + a 2 )ω + ak − (aω sin2 ϑ + k)2 , sin2 ϑ
and we used that r 2 + a2 d η(u) = η (u) . dr We interchange the orders of integration of the spatial and frequency integrals and let η tend to the constant function one. In the term corresponding to (4), Lemma 1.3 allows us to pass to the limit. In (7, 8) the situation is a bit more involved due to the factors of . However, since multiplication by (ω − ω ) corresponds to convolution with its Fourier ˇ transform (t), we can again apply Lemma 1.3 and pass to the limit using Lebesgue’s dominated convergence theorem. To make this method more precise, let us show in detail that the expression ∞ 1 ∞ ∞ ˆ ) ˆ φ(ω dr d cos ϑ dω dω e−i(ω−ω )t 1−η(u) (ω − ω ) g(ω ) (ω) r1
−1
−∞
−∞
(9) tends to zero as η converges to the constant function one. Rewriting the factor with a time convolution, we obtain the expression ∞ ˇ ) F(τ ), dτ (τ −∞
where F and ˇ are defined by ∞ 1 F(τ ) = dr d cos ϑ (1 − η(u)) (t − τ, r, ϑ) (gˇ ∗ )(t − τ, r, ϑ), r1 −1 ∞ 1 ˇ )= (b) e−ibτ db. (τ 2π −∞
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Writing g as a polynomial in ω, g(ω) = g0 + g1 ω + g2 ω2 ,
(10)
where g0 =
k2 a2k2 − , g1 = 2ak sin2 ϑ
r 2 + a2 (r 2 + a 2 )2 − 1 , g2 = − a 2 sin2 ϑ, (11)
the function gˇ ∗ can be expressed explicitly in terms of and its time derivatives of ˇ we first note that the Fourier transform of the order at most two. In order to compute , Heaviside function is 1 PP ˇ ) = (τ −i + π δ(τ ) , (12) 2π τ where “PP” denotes the principal part. Using (3) together with the fact that convolution in momentum space corresponds to multiplication in position space, we find that PP ˇ ) = −i + π δ(τ ) α(τ ˇ ), (13) (τ τ where αˇ is a Schwartz function with α(0) ˇ = (2π )−1 . According to Lemma 1.3, the function F is uniformly bounded, |F(τ )| ≤ sup |η| for all τ ∈ R. ˇ we can for any given ε choose a parameter L > 0 such that Using the rapid decay of , ˇ )||F(τ )| ≤ ε sup |η|. |(τ R\[−L ,L]
On the interval [−L , L], on the other hand, the singularity of ˆ at τ = 0 can be controlled by at most first derivatives of F, and thus for a suitable constant C = C(L), L ˇ (τ ) F(τ ) ≤ 2C sup (|F| + |F |). −L
[−L ,L]
Using the rapid decay of and its time derivatives in u, locally uniformly in τ , we can make sup[−L ,L] (|F| + |F |) as small as we like. This shows that (9) really tends to zero as η goes to the constant function one. We conclude that ∞ 1 ∞ ∞ n0 dr d cos ϑ dω dω e−i(ω−ω )t E(H+ ) ≥ r1
−1
× min(g(ω), g(ω )) +
−∞
−∞
k2 a2k2 − sin2 ϑ
ˆ ) ˆ (ω (ω)
(14)
ˆ ) . (15) ˆ (ω + g(ω ) + (1 − ) g(ω) − min(g(ω), g(ω )) (ω)
Erratum: Decay of Solutions of the Wave Equation in the Kerr Geometry
We apply Lemma 1.2 to obtain ∞ 1 dr d cos ϑ inf g(ω) + (14) ≥ ω≥J
−1
r1
569
k2 a2k2 − sin2 ϑ
|(t, r, ϑ)|2 .
Using the explicit form of g, (10), we find that for sufficiently large J , 1 J 2 ∞ (r 2 + a 2 )2 d cos ϑ |(t, r, ϑ)|2 . (14) ≥ dr 2 r1 −1
(16)
It remains to control the term (15). We write the ω, ω -integral of (15) in the form ∞ ∞ ˆ B := dω dω e−i(ω−ω )t h(ω, ω ) (ω) (ω ), −∞
−∞
where h(ω, ω ) = g(ω ) + (1 − ) g(ω) − min(g(ω), g(ω )). Introducing the variables a = 21 (ω + ω ) and b = 21 (ω − ω ), and using that g(ω) is a polynomial in ω, a short calculation yields h(a + b, a − b) = (g1 + 2g2 a) S(2b) where S(b) := b (b) − (b) . Using (12, 13) together with the fact that the factor b corresponds to a derivative in position space, we obtain 1 ˇ ) d ˇ ) = 1 d 1 − 2π α(τ = − S(τ α (sτ ) ds. 2π dτ τ dτ 0 This is a smooth function which decays quadratically at infinity; in particular, it is integrable. We thus obtain for the Fourier transform of h the explicit formula ∞ ∞ ˇ τ ) = 1 h(τ, dω dω h(ω, ω ) e−i(ωt−ω t ) 2 (4π ) −∞ −∞ τ + τ . = g1 δ(τ − τ ) + 2ig2 δ (τ − τ ) Sˇ 2 Using Plancherel for distributions, we obtain ∞ ∞ ˇ τ ) (t − τ ) (t − τ ) B= dτ dτ h(τ, −∞ −∞ ∞
ˇ ) g1 (t − τ ) (t − τ ) = dτ S(τ −∞ + ig2 ∂t (t − τ ) (t + τ ) − (t − τ ) ∂t (t + τ ) . Integrating over space, we can use the explicit formulas for g1 and g2 and apply Lemma 1.3 to obtain ∞ ∞ 1 ˇ )| dτ. dr d cos ϑ B ≥ − C(n 0 , 0 ) | S(τ (15) = r1
−1
−∞
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We now let α tend to the Dirac delta; then αˇ tends to the constant function (2π )−1 . As a consequence, the L 1 -norm of Sˇ tends to zero, and thus (15) becomes positive in this limit. Hence the energy is bounded from below by (16). This concludes the proof of Theorem 1.1. 2. Pointwise Estimates for the Low-Energy Contribution The low-energy contribution can be written as n0 ∞ 2 dω −iωt 1 ωn a b L (t, r, ϑ) = e χL (ω) tab ωn (r, ϑ) <ωn , 0 >. 2π −∞ ω n0
n=1
a,b=1
We now derive pointwise estimates for the large angular momentum modes. Theorem 2.1. For any u 0 < u 1 there is a constant C > 0 such that for all ω ∈ (−2J, 2J ) \ {ω0 , 0} and for all u, u ∈ (u 0 , u 1 ), ∞ 2 1 a b (17) t φ (u) φ (u ) ab < C. n=1
a,b=1
Proof. From [2, Sect. 5] the coefficients tab have the explicit form ⎛ α ⎞ α −Im 1 + Re β β ⎟ ⎜ T := (tab ) = ⎝ α⎠, α 1 − Re −Im β β where the transmission coefficients α and β are defined by ´ φ` = α φ´ + β φ, ´ φ2 = Im φ. ´ The estimates of [2, Sect. 4.3] are obviously valid for ω in and φ1 = Re φ, any bounded set; in particular for ω ∈ (−2J, 2J ) \ {ω0 , 0}. We use these estimates in what follows, also using the same notation. We choose n 1 so large that u + < u 0 , and thus on the whole interval (u 0 , 2u 1 ) the invariant disk estimates of Lemmas 4.2 and 4.8 in [2] hold. Rewriting the expression tab φ a φ b with the Green’s function (see the proof of Lemma 5.1 in [2]), this expression is clearly invariant under the phase transforma´ Thus we can arrange that φ(2u ´ tion φ´ → eiϑ φ. 1 ) is real. Then the transmission coefficients are computed at u = 2u 1 by 1 1 φ´ φ´ α α φ` ´ = φ . = β β y ´ y ´ φ` φ´ φ´ We thus obtain α = −
φ` 2 φ´ Im y´
y´ − y`
u=2u 1
β =
φ` 2 φ´ Im y´
y´ − y`
u=2u 1
.
Erratum: Decay of Solutions of the Wave Equation in the Kerr Geometry
Hence
571
4 || 1 + α = 2 |Im y´ | ≤ , β | y´ − y` | ρ´ 2 Re ( y´ − y` )
where all functions are evaluated at u = 2u 1 , and where we used the relation || ρ´ 2 = , Im y´
(18)
´ = 2i. From [2, Lemma 4.6 ´ φ) which is an immediate consequence of [2, (4.9)] and w(φ, and (4.42)] we know that for λ sufficiently large, 1 . C Using the above formulas for tab , we conclude that ρ(u) ´ ρ(u ´ ) 1 a b ≤ 12 C t . φ (u) φ (u ) ab ρ(2u ´ 1 ) ρ(2u ´ 1) (a,b) =(2,2) Re ( y´ − y` ) ≥
The argument after [2, (4.43)] shows that the two factors on the right decay like √ exp(− λ/c). It remains to consider the case a = b = 2. Taking the imaginary part of the identity u ´ ´ ) exp y´ φ(u) = φ(2u 1 2u 1
´ and using that φ(2u 1 ) is real, we find that
ρ(u) ´ = ρ(2u ´ 1 ) exp
u
Re y´
2u 1
and thus
sin ´ |φ2 (u)| = ρ(u)
u 2u 1
´ Im y´ ≤ ρ(u)
2u 1
Im y´ .
u
From [2, (4.40)] we see that Re y is positive on [u, 2u 1 ]. Using the relation ρ /ρ = Re y, we conclude that ρ´ is monotone increasing, and (18) yields that Im y´ is decreasing. Hence, again using (18), |φ2 (u)| ≤ ρ(u) ´ Im y´ (u) (2u 1 − u 0 ) ≤ || Im y´ (u 0 ) (2u 1 − u 0 ). Using the above estimates for α/β, we conclude that 1 t22 φ 2 (u) φ 2 (u ) ≤ 3 Im y´ (u 0 ) (2u 1 − u 0 )2 .
(19)
The invariant region estimate of Lemmas 4.2 and 4.8 in [2] yield that 7 u0 √ Im y´ (u 0 ) ≤ c|| exp − V , 8 u+ √ and we conclude that (19) again decays like exp(− λ/c). Since the eigenvalues λn scale quadratically in n [2, (2.11)], the summands in (17) decay exponentially in n.
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This theorem gives us pointwise control of the low-energy contribution, uniformly in time and in n 0 , locally uniformly in space. To see this, we estimate the integral representation (1) by n0 ∞ 2 1 ωn a 1 b tab ωn (r, ϑ) <ωn , 0 > . |L (t, r, ϑ)| ≤ dω |χ L (ω)| 2π ω −∞ n0
n=1
a,b=1
In order to control the factor ω−1 , we write the energy scalar product on the right in the form [1, (2.15)], which involves an overall factor ω. Now we can apply Theorem 2.1. 3. Decay in L ∞ loc In this section we complete the proof of Theorem 1.1 in [2]. Let K ⊂ (r1 , ∞) × S 2 be a compact set. The L 2 -norm of n 0 can be estimated by n 0 (t) L 2 (K ) ≤ nH+0 (t) L 2 (K ) + nH−0 (t) L 2 (K ) + nL 0 (t) L 2 (K ) . According to Theorems 1.1 and 2.1, these norms are bounded uniformly in n 0 and t. Furthermore, our estimates imply that the sequence n 0 (t) forms a Cauchy sequence in L 2 (K ). To see this, we note that for any n 1 , n 2 , n 1 − n 2 L 2 (K ) ≤ E nH+1 − nH+2 + E nH−1 − nH−2 + nL 1 − nL 2 L 2 (K ) , uniformly in t. The energy terms on the right are the sums of the energies of the individual angular momentum modes. All the summands are positive due to Theorem 1.1, and thus the energy terms become small as n 1 , n 2 → ∞. The same is true for the last summand due to our ODE estimates of Theorem 2.1. We conclude that n 0 (t) converges in L 2loc as n 0 → ∞, and the limit coincides with the weak limit, which in [1,2] is shown to be the solution (t) of the Cauchy problem. To prove decay, given any ε > 0 we choose n 0 such that (t) − n 0 (t) L 2 (K ) < ˆ n 0 is continuous in ω and has rapid decay, uniformly on K , the ε for all t. Since Riemann-Lebesgue lemma yields that n 0 (t) decays in L ∞ (K ) ⊂ L 2 (K ). Since ε is arbitrary, we conclude that (t) decays in L 2 (K ). Applying the same argument to the initial data H n 0 , we conclude that the partial derivatives of (t) also decay in L 2 (K ). The Sobolev embedding H 2,2 (K ) → L ∞ (K ) completes the proof. We wish to take the opportunity to correct a few other typos in the article: On page 476, line 12, the text “Proof of Theorem 3.1” should be replaced by “Proof of Theorem 1 3.2.” On page 477, line 10, the two factors ωµ should be replaced by ωµ− 2 . On page 477, line 26, the two factors π u should be replaced by π uω. On page 477, last line, the 1 two factors ωµ should be replaced by ωµ 2 . On page 490, equation (4.37), the summand − T20 should be replaced by − T20 ||. On page 499, line 12, the function η should be replaced by η L . Acknowledgements. We would like to thank Johann Kronthaler and Thierry Daudé for helpful discussions.
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References 1. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: An integral spectral representation of the propagator for the wave equation in the Kerr geometry. Commun. Math. Phys. 260, 257–298 (2005) 2. Finster, F., Kamran, N., Smoller, J., Yau, S.-T.: Decay of Solutions of the Wave Equation in the Kerr Geometry. Commun. Math. Phys. 264, 465–503 (2006) 3. Kronthaler, J.: The Cauchy problem for the wave equation in the Schwarzschild geometry. J. Math. Phys. 47, 042501 (2006) Communicated by G.W. Gibbons
Commun. Math. Phys. 280, 575–610 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0465-x
Communications in
Mathematical Physics
Quantum Stochastic Convolution Cocycles II J. Martin Lindsay, Adam G. Skalski Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. E-mail: [email protected]; [email protected] Received: 29 August 2006 / Accepted: 27 September 2007 Published online: 22 April 2008 – © Springer-Verlag 2008
Abstract: Schürmann’s theory of quantum Lévy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C ∗ -hyperbialgebra, which are Markov-regular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic generators is obtained. Automatic complete boundedness of a class of derivations is established, leading to a characterisation of the stochastic generators of *-homomorphic convolution cocycles on a C ∗ -bialgebra. Two tentative definitions of quantum Lévy process on a compact quantum group are given and, with respect to both of these, it is shown that an equivalent process on Fock space may be reconstructed from the generator of the quantum Lévy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups. Introduction In this paper we investigate quantum stochastic evolutions with independent identically distributed increments on compact quantum groups, in other words quantum Lévy processes. The natural setting for this analysis is the somewhat wider one of quantum stochastic convolution cocycles. For a compact quantum group B, a quantum stochastic convolution cocycle on B is a family of linear maps (lt )t≥0 from B to operators on the symmetric Fock space F, over a Hilbert space of the form L 2 (R+ ; k), satisfying ls+t = ls (σs ◦ lt ),
s, t ≥ 0
and some regularity and natural adaptedness conditions. Here (σs )s≥0 is the semigroup of time-shifts on B(F) and the convolution is induced by the quantum group structure; Permanent address of AGS: Department of Mathematics, University of Łód´z, ul. Banacha 22, 90-238 Łód´z, Poland.
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the initial condition is specified by the counit: l0 = (·)IF . Thus the increment of the process over the interval [0, s + t] coincides with the increment over [0, s] convolved with the (shifted) increment over [0, t]. We show that such families may be obtained as solutions of quantum stochastic differential equations with completely bounded coefficients, we analyse their positivity and multiplicativity properties, and we establish natural conditions under which all sufficiently regular cocycles arise in this way. Motivated by these results (and the purely algebraic theory), we propose two abstract definitions of quantum Lévy process on a compact quantum group and show that any process which has bounded ‘generator’ has an equivalent Fock space realisation. Precise definitions are given below. Stochastic cocycles on operator algebras were introduced by Accardi (under the name quantum Markovian cocycles) for Feynman-Kac type perturbation of quantum dynamical semigroups ([Acc]). Earlier work on a cocycle approach to classical Markov processes and their Itô integral representation may be found in [Pin]. Quantum stochastic differential equations ([HuP]) were quickly seen to provide examples of stochastic cocycles and in fact to characterise large classes of them in the Fock space context (see [L1 ] and references therein). The theory of quantum Lévy processes, developed by Schürmann and others, generalises the classical theory of Lévy processes on groups ([Hey]), and Skorohod’s theory of stochastic semigroups ([Sko]), to the context of quantum groups or, more generally, *-bialgebras (see [Sch,FrScho,Glo] and references therein). A quantum Lévy process on a quantum group B is a time-indexed family of unital *-homomorphisms from B to some noncommutative probability space, with identically distributed and (tensor-)independent increments, satisfying the convolution increment relation given by the coproduct of B, and with initial condition given by the counit of B. Schürmann showed that each quantum Lévy process may be equivalently realised in a symmetric Fock space as a solution of a quantum stochastic differential equation. This led us to introduce and investigate, in this algebraic context, quantum stochastic convolution cocycles ([LS1 ]). These are linear (but not necessarily unital or *-homomorphic) maps from a coalgebra to a space of Fock space operator processes, satisfying the convolution increment relation and counital initial condition. In the last twenty years there has been a growing interest in the theory of topological quantum groups. Starting from the fundamental paper of Woronowicz ([Wor1 ]), where the concept of compact quantum groups was first introduced (under the name of compact matrix pseudogroups), it has led to a rich and well-developed theory, with a satisfactory notion of locally compact quantum group eventually emerging in the work of Kustermans and Vaes ([KuV]). The main object becomes a C ∗ -algebra, equipped with a coproduct and counit satisfying a corresponding form of coassociativity and counit relations. In this paper we go beyond the purely algebraic context treated in [LS1 ] and initiate the study of quantum Lévy processes on a compact quantum group, or more generally on a C ∗ -bialgebra. Heeding P.-A. Meyer’s dictum once more, we again set our work in the wider context of quantum stochastic convolution cocycles on a coalgebra. The coalgebras here though are operator-space-theoretic rather than being purely algebraic. Nevertheless the stochastic cocycles in question may be obtained by solving coalgebraic quantum stochastic differential equations. In turn, every sufficiently regular completely positive and contractive quantum stochastic convolution cocycle on a C ∗ -hyperbialgebra is shown to satisfy a quantum stochastic differential equation of the above type. These results are obtained by, on the one hand applying techniques of operator space theory ([EfR,Pis2 ]), and on the other hand using known facts about standard quantum stochastic
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cocycles (see [LW2 ,L1 ] and references therein). Here it is natural to work with processes on abstract operator spaces and C ∗ -bialgebras. For this we use a theory developed in [L2 ] and summarised in the first section. When the spaces are concrete this reduces to the existing theory. A key tool of our analysis is a convolution operation which we call the R-map. This transforms coalgebraic objects such as convolution cocycles and coalgebraic quantum stochastic differential equations to standard objects of quantum stochastic analysis ([L1 ]), setting up a traffic of properties and relationships which we systematically exploit. The R-map gives rise to a noncommutative avatar of the transformation between convolution semigroups of measures and Markov semigroups (of operators), familiar from classical probability theory. The structure of the stochastic generators of Markov-regular, *-homomorphic convolution cocycles on a C ∗ -bialgebra may be characterised in terms of -structure maps, where is the counit, or topological Schürmann triples (cf. their purely algebraic counterparts). The complete boundedness of such generators, indeed their implementability, follows from their algebraic properties alone. We prove this by first extending wellknown results of Sakai, Ringrose and Christensen, on automatic continuity and innerness properties of derivations, to the case of (π , π )-derivations. The fact that every -structure map defined on the whole C ∗ -bialgebra must be implemented may be viewed as a noncommutative counterpart to the fact that every classical Lévy process on a topological group which has a bounded generator must be a compound Poisson process. In this connection we note the definition of quantum Poisson process on a *-bialgebra proposed in [Fra]. The axiomatisation of quantum Lévy processes on a C ∗ -bialgebra raises several problems connected with the fact that the product on a C ∗ -algebra A usually fails to extend to a continuous map from the spatial tensor product A ⊗ A to the algebra. We offer two different ways of overcoming this obstacle, for both of which a topological version of Schürmann’s reconstruction theorem remains valid. Our choice of examples is designed to expose the variety of connections of this work with the classical and quantum probabilistic literature. The analysis of quantum stochastic convolution cocycles in the topological context requires different methods and techniques to that of the purely algebraic and poses new nontrivial problems. However, according to our philosophy (explicitly described in the expository paper [LS2 ]), purely algebraic and topological convolution cocycles may nevertheless usefully be viewed from a common vantage point. This perspective is particularly well illustrated in the last class of examples discussed here, namely that of *-homomorphic quantum stochastic convolution cocycles on a full compact quantum group. We would also like to point out that conversely, due to the Fundamental Theorem on Coalgebras, one can view the purely algebraic situation as a finite dimensional version of the topological theory. An example of reasoning along these lines may be found in the final section of [LS3 ]. The plan of the paper is as follows. In the first section we review the basic facts needed from operator space theory and quantum stochastic analysis. We work with processes on abstract operator spaces. The transition from concrete to abstract exploits a number of natural identifications and inclusions, the key ones being (1.2), (1.3) and (1.7). In Sect. 2 the notion of operator space coalgebra is introduced and basic properties of the R-map are established, facilitating a correspondence between mapping composition structures and convolution-type structures. Section 3 contains proofs of the existence, uniqueness and regularity of solutions of coalgebraic quantum stochastic differential equations with completely bounded coefficients. There also the ground is prepared for a traffic between standard quantum stochastic cocycles and quantum stochastic convolution cocycles.
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The latter are defined in Sect. 4 where the solutions of coalgebraic quantum stochastic differential equations are shown to lie in this class. The section concludes with a brief discussion of opposite convolution cocycles. In Sect. 5 the converse result is established for Markov-regular, completely positive, contractive quantum stochastic convolution cocycles on a C ∗ -hyperbialgebra: they are characterised as solutions of the coalgebraic quantum stochastic differential equation with completely bounded coefficient of a particular form. Section 6 deals with *-homomorphic convolution cocycles on a C ∗ -bialgebra. As in the purely algebraic case, their stochastic generators are characterised by structure relations involving the counit; in the topological case these amount to the generator being an -structure map where is the counit of the bialgebra. In Sect. 7 two candidates for the axiomatisation of quantum Lévy processes on a C ∗ -bialgebra are proposed; firstly, in a weak sense of distributions, and secondly, as processes whose values are operators from a product system, in the sense of Arveson. Basic consequences of the proposed axioms are discussed, and reconstruction theorems established. Section 8 is devoted to examples, first the commutative case of classical compact groups, then the cocommutative case of the universal C ∗ -algebra of a discrete group, and finally the case of full compact quantum groups. In the latter case a link is established with the purely algebraic quantum stochastic convolution cocycles investigated in [LS1 ]. In an appendix some results on derivations are established; these are applied to yield the automatic implementedness of -structure maps used in Sect. 6. Some of the results proved here have been announced in [LS2 ]. Note added in proof. It is now clear that our results extend to the context of locally compact quantum groups in the sense of Kustermans and Vaes ([LS4 ]). Notation. All vector spaces arising in this paper are complex; inner products (and all sesquilinear maps) are linear in their second argument. For a dense subspace E of a Hilbert space h, O(E) denotes the space of operators h → h with domain E and O‡ (E) := {T ∈ O(E) : Dom T ∗ ⊃ E}. Thus O‡ (E) has the natural conjugation T → T † := T ∗ | E . We view B(h) as a subspace of O‡ (E) (via restriction/continuous linear extension). For vectors ζ ∈ E and ζ ∈ h, ωζ ,ζ denotes the linear functional on O(E) given by T → ζ , T ζ . We use the Dirac-inspired notations |E := {|ζ : ζ ∈ E} and E| := { ζ | : ζ ∈ E}, where |ζ ∈ |h := B(C; h) and ζ | ∈ h| := B(h; C) are defined by λ → λζ and η → ζ, η respectively. A class of ampliations frequently met here is denoted as follows: ιh : V → V ⊗ B(h), x → x ⊗ Ih ,
(0.1)
where this time the operator space V is determined by context (and ⊗ denotes spatial tensor product). For a vector-valued function f on R+ and subinterval I of R+ f I denotes the function on R+ which agrees with f on I and vanishes outside I . Similarly, for a vector ξ , ξ I is defined by viewing ξ as a constant function. This extends the standard indicator function notation. The symmetric measure spaceover the Lebesgue measure space R+ ([Gui]) is denoted , with integration denoted · · · dσ , thus = {σ ⊂ R+ : #σ < ∞} = (n) where (n) = {σ ⊂ R : #σ = n} and ∅ is an atom having unit measure. If + n≥0 (n)
(n)
R+ is replaced by a subinterval I then we write I and I , thus the measure of I is |I |n /n! where |I | denotes the length of I .
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For a linear map ψ : U → V the corresponding linear map between conjugate vector spaces U † → V † , x † → ψ(x)† (0.2) is denoted ψ † ; L(U † ; V † ) is thereby the natural conjugate space of L(U ; V ). The collection of sesqilinear maps φ : U × V → W is denoted S L(U, V ; W ); when W is a space of maps we denote values of φ by φ u,v (u ∈ U, v ∈ V ). The collection of bilinear maps U × V → W is denoted L(U, V ; W ). If A is an involutive algebra and E is a dense subspace of a Hilbert space h then weak multiplicativity for a map φ : A → O‡ (E), is the property φ(a ∗ b) = φ † (a)∗ φ(b) (a, b ∈ A), (0.3) where φ † : a → φ(a ∗ )∗ | E . Remark. If φ : A → O‡ (E) is a linear map, defined on a C ∗ -algebra, which is real (that is φ = φ † ) and weakly multiplicative then φ is necessarily bounded-operator-valued and thus may be viewed as a *-homomorphism A → B(h). 1. Operator Space and Quantum Stochastic Preliminaries In this section we collect some relevant facts from operator space theory, recall the matrixspace construction and describe the basic properties of tensor-extended compositions. We also recall relevant results from quantum stochastic (QS) analysis. Operator spaces ([EfR,Pis2 ]). For operator spaces V and W the Banach space of completely bounded maps from V to W is endowed with operator space structure via the linear identifications Mn (C B(V; W)) = C B (V; Mn (W)) (n ∈ N), where Mn (W) denotes the linear space Mn (W) with its natural operator space structure. When viewed as a C ∗ -algebra or operator space, Mn (C) is denoted Mn . The operator space spatial/minimal tensor product of V and W is here denoted simply V ⊗ W. For example Mn (W) may be identified with the spatial tensor product W ⊗ Mn . When V and W are realised in B(H) and B(K) respectively, V ⊗ W is realised concretely in B(H ⊗ K) = B(H)⊗B(K) as the norm closure of the algebraic tensor product V W. In fact V ⊗ W does not depend on concrete realisation of V and W; an abstract model arises from the natural linear embedding V W → C B(W∗ ; V)
(1.1)
(where W∗ is defined below). For any completely bounded maps φ : V → V and ψ : W → W into further operator spaces, the linear map φ ψ extends uniquely to a completely bounded map V ⊗ W → V ⊗ W ; the extension is denoted φ ⊗ ψ and satisfies φ ⊗ ψcb = φcb ψcb . Each bounded operator φ : V → Mn is automatically completely bounded and satisfies φcb = φ (n) in the notation φ (n) : [xi j ] → [φ(xi j )], in other words φ (n) = φ ⊗ idMn . In particular, the operator space C B(V; C) coincides with the Banach space dual B(V; C) and has the same norm; it is therefore denoted V∗ . Note the natural completely isometric isomorphisms C B(U, V; W) = C B (U; C B(V; W))
(1.2)
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for operator spaces U, V and W. We shall also exploit the natural completely isometric inclusions V ⊗ B(H; H ) → C B H |, |H ; V (1.3) for operator space V and Hilbert spaces H and H . (See below for the tensor product which delivers isomorphism here.) The following short-hand notation for tensor-extended composition is useful. Let U, V, W and X be operator spaces, and let V be a vector space. If φ ∈ L(V ; U ⊗ V ⊗ W) and ψ ∈ C B(V; X) then we compose in the obvious way: ψ • φ := (idU ⊗ψ ⊗ idW ) ◦ φ ∈ L(V ; U ⊗ X ⊗ W).
(1.4)
Ambiguity is avoided provided that the context dictates which tensor component the second-to-be-applied map also applies to the case where ψ should act on. This φ ∈ S L H , H; L(V ; V) as follows:ψ • φ ∈ S L H , H; L(V ; X) is given by
(1.5) (ψ • φ)ξ ,ξ = ψ ◦ φ ξ ,ξ . The natural inclusion L V ; C B H |, |H ; V ⊂ S L H , H; L(V ; V) is relevant here. Matrix spaces ([LW4 ]). For an operator space Y in B(H; H ) and Hilbert spaces h and h define Y ⊗M B(h; h ) := {T ∈ B(H ⊗ h; H ⊗ h ) = B(H; H )⊗B(h; h ) : ζ ,ζ (T ) ∈ Y}, (1.6) where ζ ,ζ denotes the slice map id B(H;H ) ⊗ωζ ,ζ (allowing the context to dictate the Hilbert spaces H and H ). For us the relevant cases are Y⊗M B(h) and Y⊗M |h , referred to respectively as the h-matrix space over Y and the h-column space over Y. Matrix spaces are operator spaces which lie between the spatial tensor product Y ⊗ B(h; h ) and the ultraweak tensor product Y⊗B(h; h ), coinciding with the latter when Y is ultraweakly closed (Y here denotes the ultraweak closure of Y). They arise naturally in quantum stochastic analysis where a topological state space is to be coupled with the measuretheoretic noise — if Y is a C ∗ -algebra then typically the inclusion Y⊗ B(h) ⊂ Y⊗M B(h) is proper and Y ⊗M B(h) is not a C ∗ -algebra. Completely bounded maps between concrete operator spaces lift to completely bounded maps between corresponding matrix spaces: if Y is another concrete operator space, for φ ∈ C B(Y; Y ) there is a unique map : Y ⊗M B(h; h ) → Y ⊗M B(h; h ) satisfying ζ ,ζ ◦ = φ ◦ ζ ,ζ (ζ ∈ h, ζ ∈ h ); it is denoted φ ⊗M id B(h;h ) . Using these matrix liftings, tensor-extended compositions work in the same way for matrix spaces as for spatial tensor products. There are natural completely isometric isomorphisms Y ⊗M B(h; h ) = C B( h |, |h ; Y)
(1.7)
(cf. (1.3)) under which φ ⊗M id B(h;h ) corresponds to φ ◦ , composition with φ ([L2 ]). The two tensor-extended compositions are consistent.
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Quantum stochastics ([Par,Mey]; we follow [L1 ,LS3 ], modified for abstract spaces). Fix now, and for the rest of the paper, a complex Hilbert space k which we refer to as the noise 1 dimension space, and let k denote the orthogonal sum C ⊕ k. Whenever c ∈ k, := Lin{ k; for E ⊂ k, E c : c ∈ E} and when g is a function with values c := c ∈ in k, g denotes the corresponding function with values in k, defined by g (s) := g(s). 2 Let F I denote the symmetric Fock space over L (I ; k), dropping the subscript when the interval I is all of R+ . For any dense subspace D of k let S D denote the linear span of {d[0,t[ : d ∈ D, t ∈ R+ } in L 2 (R+ ; k) (we always take these right-continuous versions) and let E D denote span of {ε(g) : g ∈ S D } in F, where ε(g) denotes the1 linear − 2 ⊗n . The subscript D is dropped when D = k. We the exponential vector (n!) g n≥0
usually drop the tensor symbol and denote simple tensors such as v ⊗ ε( f ) by vε( f ). Also define
1 0 ∈ B( k). (1.8) e0 := ∈ k and Q S := P{0}⊕k = Ik 0 The basic objects we consider in this paper are completely bounded quantum stochastic mapping processes on operator spaces. These are time-indexed families of completely bounded maps {kt : t ≥ 0} from an operator space to the algebra of bounded operators on h⊗F, for a Hilbert space h, satisfying standard adaptedness and measurability conditions. For technical reasons we also need to consider mapping processes whose values are (at least, a priori) unbounded operators. The crucial point here is that the naturally arising operators have ‘bounded slices’: for any vectors ε, ε ∈ E the maps v → Ih ⊗ ε | kt (v) Ih ⊗ |ε (t ∈ R+ ) have values in B(h), and are (completely) bounded, even though the global maps kt may not be – more precisely they have (completely) bounded columns (see Property 2, following Theorem 1.1). This point of view, where each kt is taken to be a family of maps indexed by pairs of exponential vectors, allows the replacement of B(h) by an abstract operator space and, once the somewhat technical definitions below are accepted, leads to a development of the theory which is straight-forward and effective with more transparent proofs. This said, to follow the arguments it is safe to keep in mind sesquilinear maps induced by mapping processes in the familiar sense. Let V and W be operator spaces. In this paper we denote by P(V → W) the collection of families k = (kt )t≥0 of maps in L (V; L (E; C B( F|; W))) ⊂ S L (E, E; L(V; W)) satisfying the following measurability and adaptedness conditions:
s → ksε ,ε is pointwise weakly measurable, and
ε ,ε1
ktε ,ε = ε2 , ε2 kt 1
,
for ε = ε( f ), ε = ε( f ) ∈ E and t ∈ R+ , where ε1 = ε( f [0,t[ ) and ε2 = ε( f [t,∞[ ) with ε1 and ε2 defined in the same way for f . When W = C (as is the case for quantum stochastic convolution cocycles) we write P (V) instead of P(V → C). Then kt ∈ L (V; O(E)) for each t ≥ 0 and, in terms of the exponential property of Fock space: F = F[0,t] ⊗ F[t,∞[ , adaptedness reads kt (x)ε( f ) = u t ⊗ ε( f [t,∞[ ) where u t = kt (x)ε( f [0,t[ ) ∈ F[0,t] .
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Here the following Banach space identifications are used: C B ( F|; C) = B ( F|; C) = F. When kt is viewed as a map in L (V, E; C B( F|; W)) we use the notation kt,|ε (x). Note that if k ∈ P(V → Y) for a concrete operator space Y then, invoking the complete isometry (1.7), kt,|ε ∈ L (V; Y ⊗M |F ). Quantum stochastic processes k and j are identified if, for all ε , ε ∈ E, x ∈ V and ϕ ∈ W∗ , the scalar-valued functions t → ϕ ◦ ktε ,ε (x) and t → ϕ ◦ jtε ,ε (x) agree almost everywhere. We also denote by P‡ (V → W) the subspace of processes k for which
each map ε | → ktε,ε (x) is completely bounded E| → W (ε ∈ E, x ∈ V, t ∈ R+ ). Then, for k ∈ P‡ (V → W),
(k † )εt ,ε := (ktε,ε )† defines a process k † ∈ P‡ (V† → W† ), where † denotes conjugate operator space. When the operator space W is concrete this amounts to the usual notion of adjoint(able) process. Complete boundedness for a process k ∈ P(V → W) means kt ∈ C B ( F|, |F ; C B(V; W)) ⊂ L (V, E; C B( F|; W)) for each t ∈ R+ ; similarly we write P cb (V) for the corresponding subspace of P (V). Thus Pcb (V → W), the class of such processes, is a subspace of P‡ (V → W). The natural inclusion C B (V; W ⊗ B(F)) ⊂ C B ( F|, |F ; C B(V; W)) (1.9) and, for a concrete operator space Y, the natural identification C B ( F|, |F ; C B(V; Y)) = C B (V; Y ⊗M B(F))
(1.10)
([L2 ]) are both worth noting here (cf. (1.3)); they explain the terminology. We need two further properties for processes: k ∈ P(V → W) is weakly initial space bounded if
ktε ,ε : V → W is bounded (ε, ε ∈ E, t ∈ R+ ) and is weakly regular if further
sup ksε ,ε : 0 ≤ s ≤ t < ∞, for all t ≥ 0. We shall be dealing with QS differential equations of the form dkt = kt • dφ (t), k0 = ιF ◦ κ, where φ ∈ C B V; V ⊗ B( k) and κ ∈ C B(V; W). Here the natural inclusion C B V; V ⊗ B( k) ⊂ C B k|, | k ; C B(V)
(1.12)
and, for a concrete operator space Y in B(h), the natural complete isometries C B k|, | k ; C B(Y) = C B Y; Y ⊗M B( k) ⊂ C B V; B(h ⊗ k)
(1.13)
(1.11)
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are relevant (cf. (1.9) and (1.10)). A process k ∈ P(V → W) is a weak solution of the QS differential equation (1.11) if
s → ksε ,ε ◦ φ ζ ε ,ε
kt
,ζ
(x) is weakly continuous, and t (x) = ε , ε κ(x) + w- ksε ,ε ◦ φ f (s), f (s) (x) ds 0
k, ε = ε( f ), ε = ε( f ) ∈ E, x ∈ V, t ∈ R+ ); it is so-called due to the (ζ, ζ ∈ First Fundamental Formula of quantum stochastic calculus. The theorem we need is the following special case of Proposition 3.5 in [LS3 ] which generalises [LW4 ] to allow nontrivial initial conditions and abstract spaces. Theorem 1.1. Let φ ∈ C B V; V ⊗ B( k) and κ ∈ C B(V; W). Then there is a unique weakly regular weak solution of the quantum stochastic differential equation (1.11). Notation. k κ,φ , simplifying to k φ for the case where V = W and κ = idW . We list key properties of the solution processes needed in this paper next (see [LS3 ]). Let k = k κ,φ for κ and φ as above. 1. 2.
k ∈ P‡ (V → W) and k † = k κ ,φ . For all ε ∈ E and t ∈ R+ , kt,|ε ∈ C B (V; C B( F|; W)) = C B ( F|; C B(V; W)) (the process has completely bounded columns) and the map s → ks,|ε is locally Hölder-continuous with exponent 1/2. Moreover if φ(V) ⊂ V ⊗ B( k), then k satisfies †
†
kt,|ε (V) ⊂ W ⊗ |F 3.
(in terms of the inclusion (1.1)). If κ = κ2 ◦ κ1 , where κ1 ∈ C B(V; U) and κ2 ∈ C B(U; W) then kt,|ε = κ2 • kt,|ε : x → κ2 ◦ kt,|ε (x) where k = k κ1 ,φ (t ∈ R+ , ε ∈ E). If the process k is completely bounded then so is k and we have the identity κ ,φ
k t = κ2 • k t 1 (t ∈ R+ ). In particular, φ
kt,|ε = κ • kt,|ε 4.
φ resp. kt = κ • kt when k φ ∈ Pcb (V → W) .
The following useful ‘form representation’ holds ktε ,ε (x) = ε , ε wdσ σ ◦ τ#σ ◦ κ • φ •#σ (x) [0,t]
(1.14)
(weak integral) where σ := ωξ ,ξ for ξ = π f (σ ) and ξ = π f (σ ), when ε = ε( f ) and ε = ε( f ) and, for n ∈ N, τn is the permutation of tensor components which reverses the order of the k’s.
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Let ε, ε ∈ E and t ∈ R+ . If each kt,|ε is W ⊗ |F -valued then
ktε ,ε = (idW ⊗λε ) ◦ kt,|ε , and if further k ∈ Pcb (V → W) and each kt maps V into the spatial tensor product W ⊗ B(F), then kt,|ε = (idW ⊗ρε ) ◦ kt .
6.
Here λε : |F → C denotes left multiplication by ε |, and ρε : B(F) → |F right multiplication by |ε . In fact k is a strong solution of the QS differential equation, meaning that the integral equation t kt = ιF ◦ κ + ks • dφ (s) 0
7.
is valid in a strong sense ([L1 ]). The process k is expressible in terms of the multiple QS integral operation: kt = t ◦ υ, where υ = (υn )n≥0 and υn = τn ◦ κ • φ •n
8.
(cf. Property 4). When V = W and κ = idW , k0 = ιF and k enjoys the following weak cocycle property: for s, t ∈ R+ , ε = ε( f ) and ε = ε( f ) in E, ε ,ε1
ε ,ε = ε3 , ε3 ks 1 ks+t
where ε1 ,
ε2
ε ,ε2
◦ kt 2
,
ε1 = ε( f [0,s[ ), ε2 = ε(Ss∗ f [s,s+t[ ) and ε3 = ε( f [s+t,∞[ ),
ε3
(1.15)
f
and being defined similarly with in place of f , (St )t≥0 is the oneparameter semigroup of right shifts on L 2 (R+ ; k) and (σt )t≥0 is the induced endomorphism semigroup on B(F), ampliated to S L (E, E; W). The cocycle property simplifies to ks+t = ks • σs ◦ kt
when k is completely bounded. Property 8, namely the fact that processes of the form k φ are weak QS cocycles (also called Markovian cocycles), has a converse — subject to certain constraints, weak QS cocycles are necessarily of this form. The main results in this direction concern completely positive, contractive QS cocycles on a unital C ∗ -algebra and are collected next — they originate in [LPa]; a direct proof is given in [LW5 ]. Theorem 1.2. ([LW1−3 ]) Let A be a unital C ∗ -algebra and let k ∈ P(A → A). Then the following are equivalent: (i) k is a Markov-regular, completely positiveand contractive QS cocycle on A; k|, | k ; C B(A) = C B A; C B( k|, | k ; A) satisfies (ii) k = k φ , where φ ∈ C B φ(1) ≤ 0 and, in any faithful, nondegenerate representation of A, φ may be decomposed as follows: φ(x) = (x) − x ⊗ Q S − (x ⊗ |e0 ) J − J ∗ (x ⊗ e0 |) (1.16) (x ∈ A), for some map ∈ C P A; A ⊗B( k) and operator J ∈ A ⊗ k|.
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Here, with respect to the representation, A denotes the double commutant of A, ⊗ denotes the ultraweak tensor product and φ(x) ∈ A ⊗M B( k). The form (1.16) taken by the stochastic generator (see also [Bel]) generalises the Christensen-Evans Theorem on the generators of norm continuous completely positive contractive semigroups ([ChE]). 2. Operator Space Coalgebras In this section we adapt the basic notions of coalgebra to the category of operator spaces, and consider convolution semigroups of functionals in this context. Three structures are considered: operator space coalgebras, operator system coalgebras and C ∗ -bialgebras, corresponding to the hierarchy of properties investigated in this paper: linearity, positivity and multiplicativity. In fact a hybrid structure, called C ∗ -hyperbialgebra, plays a more prominent role than operator system coalgebras do. Definition. An operator space coalgebra is an operator space C equipped with complete contractions : C → C and : C → C ⊗ C, called the counit and coproduct respectively, satisfying (OSC1) (OSC2)
( ⊗ idC ) ◦ = (idC ⊗) ◦ (coassociativity), ( ⊗ idC ) ◦ = idC = (idC ⊗) ◦ (counit property);
it is an operator system coalgebra if C is an operator system and (OSyC) and are both unital and completely positive; a C ∗ -hyperbialgebra if C is a unital C ∗ -algebra and (C ∗ -Hy) is a character (i.e. it is nonzero and multiplicative) and is unital and completely positive; and finally it is a C ∗ -bialgebra if C is a (unital) C ∗ -algebra and (C ∗ -Bi)
and are both unital and *-homomorphic.
By (OSC1), •2 : C → C⊗3 is defined unambiguously, as is •2 : C → C⊗(n+1) for all n ∈ N, and we define •0 := idC . Similarly (OSC2) gives • = idC , with unambiguous meaning. An operator space coalgebra is cocommutative if = op , where op := τ ◦ , τ being the tensor flip on C ⊗ C. The opposite operator space coalgebra results from replacing by op . An operator space coalgebra is thus typically not a coalgebra in the algebraic sense ([Swe]) since the coproduct is not required to map C into C C. A (unital) C ∗ -bialgebra is a C ∗ -hyperbialgebra B whose coproduct is also multiplicative (and thus a unital *-homomorphism). Some authors (for example [Kus2 ]) drop the unital condition on C ∗ -bialgebras, and require instead the counit to be a nondegenerate *-homomorphism into the multiplier algebra M(C ⊗ C). The asymmetry in the definition of C ∗ -hyperbialgebra—whereby is required to be multiplicative but only to be completely positive — is motivated by the example of compact quantum hypergroups ([ChV]). Multiplicativity of the counit is used extensively in characterising generators of completely positive convolution cocycles (Sect. 5 and [S]). Finally note that the conjugate operator space of an operator space coalgebra has natural operator space coalgebra structure.
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Convolution. For an operator space coalgebra C and operators spaces V1 and V2 , the convolution of ϕ1 ∈ C B(C; V1 ) and ϕ2 ∈ C B(C; V2 ) is defined by ϕ1 ϕ2 := (ϕ1 ⊗ ϕ2 ) ◦ ∈ C B(C; V1 ⊗ V2 ). It is easily seen that convolution is associative (in the same sense as the spatial tensor product is) and enjoys submultiplicativity and unital properties: (ϕ1 ϕ2 ) ϕ3 = ϕ1 (ϕ2 ϕ3 ), ϕ1 ϕ2 cb ≤ ϕ1 cb ϕ2 cb , and ϕ = ϕ = ϕ .
(2.1) (2.2)
In particular, (C∗ , ) is a unital Banach algebra. For n ∈ N and ϕ1 , . . . , ϕn ∈ C B(C; V), n-fold convolution is defined via n-fold tensor products: ϕ1 · · · ϕn = (ϕ1 ⊗ · · · ⊗ ϕn ) ◦ •(n−1) .
(2.3)
We also define ϕ 0 := , which is consistent with (2.2). Given an operator space coalgebra C, each operator space V determines maps RV : C B(C; V) → C B(C; C ⊗ V), ϕ → (idC ⊗ϕ) ◦ ; E V : C B(C; C ⊗ V) → C B(C; V), φ → ( ⊗ idV ) ◦ φ. Thus the action of RV is given by convolution with the identity map on C, putting the argument on the right, and that of E V is given by composition in the tensor-extended sense with the counit: RV ϕ = idC ϕ, and E V φ = • φ. In the noncocommutative case we are therefore making a choice here. We abbreviate RC to R . The basic properties of these maps are collected below. They are all easily proved from the definitions, noting that under the completely isometric identification Mn (C ⊗ V) = C ⊗ Mn (V), (RV )(n) = RMn (V) . Proposition 2.1. Let C be an operator space coalgebra, and let V1 , V2 and V be operator spaces. (a)
RV and E V are complete isometries satisfying E V ◦ RV = idC B(C;V) .
(b) If ϕ1 ∈ C B(C; V1 ) and ϕ2 ∈ C B(C; V2 ), then RV1 ⊗V2 (ϕ1 ϕ2 ) = RV1 ϕ1 • RV2 ϕ2 . (c) If ϕ ∈ C B(C; V), then RV† (ϕ † ) = (RV ϕ)† . Remark. Noting that = E C (idC ) and = RC (idC ), it is clear that operator space coalgebras could be axiomatised in terms of R- and E-maps in lieu of and . Write C B (C; C ⊗ V) for Ran RV .
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Corollary 2.2. For each operator space V, RV determines a complete isometry of operator spaces C B(C; V) ∼ = C B (C; C ⊗ V), by corestriction. In case V = C this gives an isometric isomorphism of unital Banach algebras (C∗ , ) ∼ = C B (C), ◦ . A further noteworthy consequence is the following identity. Corollary 2.3. In C B (C; C ⊗ Mn ), φcb = φ (n) . Proof. Let φ ∈ C B (C; C ⊗ Mn ), say φ = RMn ϕ. Then ϕ ∈ C B(C; Mn ) so φcb = ϕcb = ϕ (n) = • φ (n) ≤ φ (n) . The result follows.
In particular, in C B (C) the completely bounded norm coincides with the bounded operator norm. As a result C B (C) is a closed subspace of B(C). The next proposition collects the structure-preserving properties of RV and its inverse, under a number of pertinent assumptions on C and V. Proposition 2.4. Let C be an operator space coalgebra and V an operator space, let ϕ ∈ C B(C; V) and φ = RV ϕ ∈ C B (C; C ⊗ V). (a) The map φ is completely contractive if and only if ϕ is. (b) If C is an operator system coalgebra and V is an operator system then φ is real (respectively, completely positive, or unital) if and only if ϕ is. (c) If C is a C ∗ -bialgebra and V is a C ∗ -algebra then φ is multiplicative if and only if ϕ is. A convolution semigroup of functionals on an operator space coalgebra C is a one-parameter family λ = (λt )t≥0 in C∗ satisfying λ0 = and λs+t = λs λt . In other words a convolution semigroup of functionals on C is a one-parameter semigroup in the unital algebra (C∗ , ). Proposition 2.5. Let C be an operator space coalgebra. The map λ → P := (R λt )t≥0 is a bijection from the set of convolution semigroups of functionals on C to the set of one-parameter semigroups in C B (C). Moreover, the conditions in (a) below are equivalent, and so are the conditions in (b): (a) (b)
(i) λt → pointwise as t → 0; (ii) P is a C0 -semigroup on C. (i) λ is norm continuous in t; (ii) P is norm continuous in t; (iii) P is cb-norm continuous in t; (iv) P has a completely bounded generator.
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Proof. The first part follows from Corollary 2.2. Since ◦ Pt = λt , (aii) implies (ai). Suppose therefore that (ai) holds. Then, for any ϕ ∈ C∗ , ϕ ◦ Pt = λt ◦ (ϕ ⊗ idC ) ◦ and ◦ (ϕ ⊗ idC ) ◦ = ϕ, so Pt x → x weakly as t 0, for all x ∈ C. But this implies that P is strongly continuous ([Dav], Prop. 1.23) and thus a C0 -semigroup, so (aii) holds. By Corollary 2.3, Pt − idC cb = λt − = Pt − idC , and so (b) follows.
Thus each norm-continuous convolution semigroup of functionals λ on C has a generator: γ := lim t −1 (λt − ) t0
from which the convolution semigroup of functionals may be recovered tn γ n . λt = exp tγ := n! n≥0
The corresponding one-parameter semigroup on C has completely bounded generator: R λt = etτ , where τ = R γ ∈ C B (C). 3. Operator Space Coalgebraic Quantum Stochastic Differential Equations In this section we consider operator space coalgebraic quantum stochastic differential equations with completely bounded coefficients, and relate their solutions to those of standard QS differential equations by means of R-maps. In particular we show that the complete boundedness property is preserved when moving between these two kinds of solutions. For this section C is a fixed operator space coalgebra. Let ϕ ∈ C B C; B( k) . A weakly initial space bounded process k ∈ P (C) is a weak solution of the operator space coalgebraic QS differential equation dkt = kt dϕ (t), k0 = ιF ◦ , if
s → ksε ,ε ϕ ζ ,ζ (x) is continuous, and t ktε ,ε (x) = ε , ε κ(x) + (ksε ,ε ϕ f (s), f (s) )(x) d x
(3.1) (3.2) (3.3)
0
k, ε = ε( f ), ε = ε( f ) ∈ E, x ∈ C and t ∈ R+ . for ζ, ζ ∈ Remark. By the Banach-Steinhaus Theorem x ∈ C, x ≤ 1, s ∈ [0, t] < ∞ sup (ωε( f ),ε( f ) ◦ ks ) (ω f (s), ◦ ϕ)(x) f (s) for each f, f ∈ S and t ∈ R+ . It follows therefore that weak solutions of the operator space coalgebraic QS differential equation in the above sense are automatically weakly regular.
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Theorem 3.1. Let ϕ ∈ C B C; B( k) . Then the operator space coalgebraic quantum stochastic differential equation (3.1) has a unique weak solution. Proof. Let k ∈ P (C) be weakly regular. Then
ktε ,ε ϕ ζ
,ζ
= ktε ,ε ◦ φ ζ
,ζ
(ε, ε ∈ E, ζ, ζ ∈ k, t ∈ R+ ), where φ ∈ R B(k) ϕ. It follows that k weakly satisfies the operator space coalgebraic QS differential equation (3.1) if and only if k weakly satisfies the operator space QS differential equation dkt = kt • dφ (t), k0 = ιF ◦ . Since φ ∈ C B C; C ⊗ B( k) and ∈ C∗ = C B(C; C) the result therefore follows from Theorem 1.1, and the automatic weak regularity of weak solutions of (3.1). Notation. We denote the unique weak solution of (3.1), for completely bounded ϕ, by l ϕ . From the above proof we see that l ϕ = k ,φ , where k) . φ = R B(k) ϕ ∈ C B C; C ⊗ B( Note that Proposition 2.1 implies that • φ •n = • R B(k)⊗n ϕ n = ϕ n , n ≥ 0. The properties of solutions of operator spaceQS differential equations listed in Sect. 1 entail the following for l = l ϕ , where ϕ ∈ C B C; B( k) : k) . 1 . l ∈ P‡ (C) and l † = l ψ where ψ = ϕ † ∈ C B C† ; B( 2 . lt,|ε ∈ C B (C; |F ) and the map s → ls,|ε is locally Hölder continuous with φ exponent 21 , moreover kt,|ε (C) ⊂ C ⊗ |F for all ε ∈ E and t ∈ R+ . 3 . Since l = k ,φ , φ (3.4) lt,|ε = • kt,|ε , (ε ∈ E, t ∈ R+ ); also if k φ is completely bounded then l is too and φ
lt = • kt , t ∈ R+ . 4 .
In the notation of Property 4,
ltε ,ε = ε , ε
5 .
[0,t]
dσ σ ◦ τ#σ ◦ ϕ #σ
for ε = ε( f ), ε ∈ ε( f ) ∈ E and t ∈ R+ . In the notation of Property 5,
ltε ,ε = λε ◦ lt,|ε and, if l is completely bounded so that lt ∈ C B (C; B(F)), then lt,|ε = ρε ◦ lt (ε, ε ∈ E, t ∈ R+ ).
(3.5)
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l is a strong solution of the operator space coalgebraic QS differential equation: t ls dϕ (s) lt = ιF ◦ + 0
7 .
is valid in a strong sense. l is given explicitly by lt = t ◦ υ where υ = (υ)n≥0 and υn = τn ◦ ϕ n , n ∈ Z+ .
(3.6)
Remark. In view of the injectivity of the quantum stochastic operation ([LW4 ], Proposition 2.3), Property 7 implies that the map ϕ → l ϕ is injective. The next two results strengthen Property
(3.7)
3 .
Proposition 3.2. Let l = l ϕ and k = k φ , where ϕ ∈ C B C; B( k) and φ = R B(k) ϕ. Then lt,|ε ∈ C B (C; |F ) and kt,|ε = R|F lt,|ε , t ∈ R+ , ε ∈ E. In particular, k satisfies kt,|ε ∈ C B (C; C ⊗ |F ) , ε ∈ E, t ∈ R+ . Proof. The first (and last) part has already been noted in Property 2 . Write k ∈ P (C) for the process defined by kt,|ε = R|F lt,|ε ∈ C B (C; C ⊗ |F ) (ε ∈ E, t ∈ R+ ). Let ε = ε( f ), ε = ε( f ) ∈ E and t ∈ R+ , and consider the ‘form representation’ of l given in Property 7 and the corresponding representation of k. Writing Rσ for RV , where V = B( k)⊗#σ (and using the notation σ adopted in Properties 4 and 4), Proposition 2.1 yields the identity R σ ◦ τ#σ ◦ ϕ #σ = σ • Rσ τ#σ ◦ ϕ #σ = σ • τ#σ ◦ φ •#σ (σ ∈ ). Thus, integrating over [0,t] , R ltε ,ε = ktε ,ε . Therefore, using Property 5 ,
ktε ,ε = R (λε ◦ lt,|ε ) = (idC ⊗λε ) ◦ R|F lt,|ε = ktε ,ε . The result follows.
Proposition 3.3. Let l = l ϕ and k = k φ , where ϕ ∈ C B C; B( k) and φ = R B(k) ϕ. Then the process l is completely bounded if and only if k is, and in this case kt = R B(F )lt , t ∈ R+ ,
(3.8)
in particular k is C ⊗ B(F)-valued. Proof. Suppose that l is completely bounded and define the process k by kt = R B(F )lt . By Proposition 3.2 and Properties 5 and 5, kt,|ε = R|F (ρε ◦ lt ) = (idC ⊗ρε ) ◦ kt = kt,|ε (ε ∈ E, t ∈ R+ ), and so k is the completely bounded C ⊗ B(F)-valued process (R B(F )lt )t≥0 . Conversely if k is completely bounded then l is too, by Property 3 .
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4. Quantum Stochastic Convolution Cocycles In this section we study quantum stochastic convolution cocycles on an operator space coalgebra by applying the R-map to the theory of quantum stochastic cocycles on an operator space ([LW2 ]). For this section an operator space coalgebra C is fixed. Definition. A completely bounded process l ∈ P (C) is called a quantum stochastic convolution cocycle if it satisfies l0 = ιF ◦ and ls+t = ls (σs ◦ lt ) for s, t ∈ R+ .
(4.1)
QS convolution cocycles therefore satisfy ε ,ε1
ε ,ε = ε3 , ε3 ls 1 ls+t
ε ,ε2
lt 2
(4.2)
for ε = ε( f ), ε = ε( f ) and s, t ∈ R+ , where ε1 , . . . , ε3 are defined by (1.15). More generally, if l is a weakly initial space bounded process C → C satisfying (4.2) then it is called a weak quantum stochastic convolution cocycle. Compare this with the cocycle property for a weakly initial space bounded process on an operator space (see Property 8 in the list of properties of solutions of QS differential equations). For a weak QS convolution cocycle l on C define
λct ,c := e−t c ,c ltε ,ε where ε = ε(c[0,t[ ) and ε = ε(c[0,t[ )
(c, c ∈ k, t ∈ R+ ). Then λc ,c := (λct ,c )t≥0 is a convolution semigroup and we refer to {λc ,c : c, c ∈ k} as the cocycle’s associated convolution semigroups of functionals and call l Markov-regular if λ0,0 is norm continuous, in analogy to Markov-regular QS cocycles ([LW2 ]). As for standard QS cocycles, if the cocycle is contractive then Markov-regularity implies that all of its associated convolution semigroups of functionals are norm continuous. Repeated application of the defining property (4.2) shows that, for each ε = ε( f ), ε = ε( f ) ∈ E and t ∈ R+ , ε , ε −1 ltε ,ε is the convolute of a finite number of associated convolution semigroups of functionals of l. In particular two weak QS convolution cocycles are the same if each of their corresponding associated convolution semigroups of functionals coincide. Lemma 4.1. Let l ∈ P (C) and k ∈ P(C → C) be weakly initial space bounded processes related by ktε ,ε = Rltε ,ε , (4.3) for ε, ε ∈ E, t ∈ R+ . Then l is a weak QS convolution cocycle if and only if k is a weak QS cocycle, and in this case l is Markov-regular if and only if k is. Proof. In view of the identity ε ,ε ε ,ε2 ε ,ε1 ε ,ε2 1 R l s 1 l t 2 = ks 1 ◦ kt 2 (in the notation (1.15)) the result follows from the complete isometry of R . Proposition 4.2. Let ϕ ∈ C B C; B( k) . Then l ϕ is a Markov-regular weak QS convolution cocycle, each of whose convolution semigroups of functionals is norm continuous.
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Proof. Let k = k φ , where φ = R B(k) ϕ. Then k is a Markov-regular quantum stochastic cocycle all of whose associated semigroups are norm continuous ([LW2 ]). Since, by Proposition 3.2, l and k are related by (4.3) the result therefore follows from Lemma 4.1. In the next section we obtain a converse by restricting to completely positive, contractive QS convolution cocycles on a C ∗ -hyperbialgebra. In view of the identity tn #σ dσ π ( · )πc (σ ) = (ω ◦ ϕ)n , c (σ ), ϕ n! c ,c [0,t] n≥0
the convolution semigroup of functionals λc ,c cocycle l ϕ has generator
associated with the weak QS convolution
ω (4.4) c , c ◦ ϕ. This corresponds to the fact that the semigroups associated with a Markov-regular QS cocycle k φ on an operator space have generators ω c , c • φ.
(4.5)
Below we initiate a traffic between properties of a QS convolution cocycle and those of its stochastic generator. Recall Property 1 for processes l ϕ . The following is easily proved either using the R-map, or directly. Proposition 4.3. Let l = l ϕ , where ϕ ∈ C B C; B( k) and C is an operator system coalgebra. Then (a) l is unital if and only if ϕ(1) = 0, (b) l is real if and only if ϕ is real. Opposite QS convolution cocycles. The opposite QS convolution cocycle relation, for processes in P cb (C), is l0 = ιF ◦ and ls+t = (σs ◦ lt ) ls , which involves the natural identifications B(F[s,s+t[ )⊗B(F[0,s[ ) = B(F[0,s+t[ ), for s, t ∈ R+ . Completely bounded processes which satisfy a QS differential equation of the form dlt = dϕ (t) lt , l0 = ιF ◦ , for ϕ ∈ C B(C; B( k)), are opposite QS convolution cocycles; they are given explicitly by lt = t ◦ υ where υn = ϕ n , n ∈ Z+ (cf. Properties 6 and 7 in Sect. 3), with ϕ l being an appropriate notation. There is a bijective correspondence between the set of QS convolution cocycles treated in this paper and the set of opposite QS convolution cocycles. This is effected by time-reversal, as in [LW2 ]. Opposite QS convolution cocycles have convolution semigroup representation as QS convolution cocycles do, but with the semigroups appearing in the reverse order. In particular time-reversal exchanges l ϕ and ϕ l. One may also view the correspondence in terms of the opposite coproduct op . In [LS1 ] we actually worked with opposite cocycles (thus the convolvands in (5.1) and the a(i) ’s in (5.2), on p. 595 of that paper, should both have appeared in the reverse order, with the notation ϕ l being more appropriate for the opposite QS convolution cocycles generated there). The results of that paper are equally valid for QS convolution cocycles on coalgebras defined as here through the relations (4.1) and (4.2).
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5. Completely Positive Quantum Stochastic Convolution Cocycles In this section we characterise the Markov-regular QS convolution cocycles amongst the completely positive and contractive processes on a C ∗ -hyperbialgebra E, as those which satisfy a coalgebraic QS differential equation with completely bounded coefficient of a particular form. We also give the general form of the coefficient of such QS differential equation. Recall Theorem 1.2 and the notations (1.8). Theorem 5.1. Let E be a C ∗ -hyperbialgebra and let l ∈ P (E). Then the following are equivalent: (i) l is a Markov-regular, completely positive, contractive QS convolution cocycle; k) satisfies ϕ(1) ≤ 0 and may be decomposed as (ii) l = l ϕ , where ϕ ∈ C B E; B( follows: ϕ = ψ − (·) Q S + |e0 χ | + |χ e0 | (5.1) for some completely positive map ψ : E → B( k) and vector χ ∈ k; (iii) there is a *-representation (ρ, K) of E, a contraction D ∈ B(k; K) and a vector ξ ∈ K, such that l = l ϕ , where
ξ | ϕ(x) = (5.2) (ρ(x) − (x)IK ) |ξ D + (x)ϕ(1), ∗ D (x ∈ E), and ϕ(1) is nonpositive with block matrix of the form
∗ ∗ . ∗ D ∗ D − Ik Proof. For the proof of the equivalence of (i) and (ii) we may suppose that E is faithfully and nondegenerately represented in B(h), say, in such a way that the counit extends to a normal state on E . (This may be achieved by taking the direct sum of an arbitrary faithful nondegenerate representation and the GNS representation (h , π , ξ ), so that is extended by the vector state ω(0,ξ ) . Alternatively, take the universal representation and bidual map ∗∗ .) Note that is necessarily *-homomorphic. Suppose first that (i) holds and let {γc ,c : c , c ∈ k} be the generators of the associated convolution semigroups of functionals of l. Let k ∈ P‡ (E → E) be the process (R B(F )lt )t≥0 . By Proposition 2.4, k is completely positive and contractive. Moreover (4.3) holds so that k is a Markov-regular QS cocycle on E. In view of Theorem 1.2, it that k = k φ for some map φ of the form (1.16) with constituents follows ∈ C P E; E ⊗ B( k) and J ∈ E ⊗ | k , say. Thus, from the definition of k, ω c , c • φ = (idE ⊗γc ,c ) ◦ (c, c ∈ k). Now define ϕ, ψ ∈ C B E; B( k) and χ ∈ k by
(5.3)
ϕ = • φ, ψ = ( ⊗ id B(k) ) ◦ and χ | = ( ⊗ id k| )(J ), noting that, by the complete positivity of , ϕ(1) ≤ 0 and ψ is completely positive. We claim that l = l ϕ and that ϕ has the decomposition (5.1). By (5.3), ω c , c ◦ ϕ = γc ,c and so, by (4.4), the QS convolution cocycles l ϕ and l have the same associated convolution semigroups and are therefore equal. In view of the multiplicativity of , ( ⊗ id B(k) ) ((x ⊗ |e0 )J ) = (x)|e0 χ |,
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for x ∈ E. Now, using the fact that is real to obtain the adjoint identity, collecting terms yields the decomposition (5.1), so (ii) holds. φ Suppose conversely that (ii) holds. As in the proof of Proposition 4.2, let k = k where φ = R B(k) ϕ ∈ C B E; E ⊗ B(k) . Then φ has the form (1.16), with = R B(k) ψ and J = Ih ⊗ χ |, moreover it follows from Proposition 2.4 that is completely positive and φ(1) ≤ 0. Thus, by Theorem 1.2, the Markov-regular weak QS cocycle k is completely positive and contractive. Therefore, by Proposition 3.3, Proposition 2.4 and Lemma 4.1, (i) holds. Again suppose that (ii) holds. Let
ξ | ρ(·) |ξ D (5.4) D∗ be a minimal Stinespring decomposition of ψ. Thus (ρ, K) is a unital C ∗ -representation of E, ξ is a vector in K, D is an operator in B(k; K) (and (5.5) below holds). Identity (5.2) follows, with
ξ 2 − 2 Re α D ∗ ξ − c| ϕ(1) = , |D ∗ ξ − c D ∗ D − Ik where αc = χ , so (iii) holds. Conversely, suppose that (iii) holds. Then, writing
t
d| |d D ∗ D − Ik for the block matrix form of ϕ(1), ϕ has the form (5.1), where ψ is given by (5.4) and 1 (ξ 2 − t) χ= 2 ∗ D ξ −d so (ii) holds. This completes the proof.
Remarks. An alternative proof of the above theorem, which directly establishes the equivalence of (i) and (iii) without appeal to Theorem 1.2 on standard QS cocycles (whose proof depends on the Christensen-Evans Theorem), is given in [S]. In (iii) the following minimality condition on the quadruple (ρ, K, D, ξ ) may be assumed: ρ(E) (Cξ + Ran D) is dense in K. (5.5) Under minimality there is uniqueness too: if (ρ , K , D , ξ ) is another quadruple as in (iii) then there is a unique isometry V ∈ B(K; K ) (unitary if the second quadruple is also minimal) satisfying V D = D , V ξ = ξ and Vρ(x) = ρ (x)V for x ∈ E. By a characterisation of nonnegative block matrix operators (see, for example, Lemma 2.2 in [GLSW]), if ϕ is the stochastic generator of a Markov-regular, completely positive, contractive QS convolution cocycle, then ϕ(1) has the form
t
C 1/2 e| |C 1/2 e −C
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for a nonnegative contraction C, a unique vector e ∈ Ran C and a real number t satisfying t ≤ −e2 . Moreover, with respect to any decomposition (5.2), C = Ik − D ∗ D. Unitality for the cocycle is equivalent to its stochastic generator being expressible in the form
ξ | (ρ − ιK ◦ ) (·) |ξ D , ∗ D where D is isometric and ρ(1) coincides with the identity operator on Cξ + Ran D. It also follows from the above proof that if k is the QS cocycle on a C ∗ -hyperbialgebra E, given by kt = R B(F )lt , where l is a Markov-regular, completely positive, contractive QS convolution cocycle on E, then k = k φ , where the stochastic generator φ is expressible in the form x → ψ(x) − x ⊗ Q S + |χ e0 | + |e0 χ | for some completely positive map ψ : E → E ⊗ B( k) and vector χ ∈ k. Note that no appeal to a concrete realisation of E is needed in this decomposition. 6. Homomorphic Quantum Stochastic Convolution Cocycles In this section we characterise the stochastic generators of Markov-regular *-homomorphic QS convolution cocycles on a C ∗ -bialgebra, by applying the R-map to the characterisation of the generators of Markov-regular multiplicative cocycles obtained in [LW4 ]. Thus let B be a C ∗ -bialgebra. Weak multiplicativity for a process l ∈ P‡ (B) is the following property:
† ∗ ltε ,ε (x ∗ y) = lt,|ε (x) l t,|ε (y)
(ε, ε ∈ E, x, y ∈ B, t ∈ R+ ). If the C ∗ -bialgebra is concretely realised on a Hilbert space then weak multiplicativity for a process k ∈ P‡ (B → B) reads kt (x ∗ y) = kt† (x)∗ kt (y) (x, y ∈ B, t ∈ R+ ), an identity in Hilbert space operators. In view of the remark at the end of the introduction, if k ∈ P‡ (B → B) is both weakly multiplicative and real then it is bounded, and so *-homomorphic — in particular it is completely bounded. Theorem 6.1. Let l = l ϕ , where ϕ ∈ C B B; B( k) . Then the following are equivalent (a) l is weakly multiplicative; (b) ϕ satisfies ϕ(x y) = ϕ(x)(y) + (x)ϕ(y) + ϕ(x)Q S ϕ(y)
(6.1)
(x, y ∈ B). Proof. For the proof we may suppose without loss of generality that the C ∗ -bialgebra B is concretely realised, in B(h) say. Let φ = R B(k) ϕ and set k = k φ . Since Ran φ ⊂ B ⊗ B( k), Theorem 3.4 and Corollary 4.2 of [LW4 ] imply that k is weakly multiplicative if and only if φ satisfies φ(x y) = φ(x)ιk (y) + ιk (x)φ(y) + φ(x) Ih ⊗ Q S φ(y). (6.2)
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If (6.2) holds then applying the homomorphism ⊗ id B(k) to both sides yields (6.1). = idB ⊗(ιk ◦ ), so that Conversely, suppose that (6.1) holds and set ϕ = idB ⊗ϕ and φ= ϕ ◦ and ◦ = ιk .
(6.3)
Then, for simple tensors X = x1 ⊗ x2 and Y = y1 ⊗ y2 in B ⊗ B,
x1 y1 ⊗ ϕ(x2 y2 ) = x1 y1 ⊗ ϕ(x2 )(y2 ) + (x2 )ϕ(y2 ) + ϕ(x2 )Q S ϕ(y2 ) , or
ϕ (X Y ) = ϕ (X ) (Y ) + (X ) ϕ (Y ) + ϕ (X ) Ih ⊗ Q S ϕ (Y ).
By linearity and continuity this holds for all X, Y ∈ B ⊗ B, in particular, for X = x and Y = y. Therefore, by (6.3) and the multiplicativity of , (6.2) holds. It therefore remains only to show that l is weakly multiplicative if and only if k is. Recall that l ∈ P‡ (B) and l † = l ψ , where ψ = ϕ† . Let u, u ∈ h, t ∈ R+ and ε, ε ∈ E. If l is weakly multiplicative then ε , lt (x y)ε = lt† (x ∗ )ε , lt (y)ε , so u ε , (idB lt )(X Y )uε = (idB lt† )(X ∗ )u ε , (idB lt )(Y )uε (6.4) holds for all X, Y ∈ B B. Now it follows, from the identity # (idB lt# )(X )uε = idB ⊗lt,|ε (X )u (where l # stands for l or l † ) and Property 2 , that both sides of (6.4) are continuous in both X and Y , giving an identity for all X, Y ∈ B⊗B. Setting X = x and Y = y and using the multiplicativity of , this identity becomes a statement of the weak multiplicativity of k. Suppose conversely that k is weakly multiplicative. Set k † = k ψ for ψ = φ † , and let x, y ∈ B. First note that the identity ( ⊗ id F | )(X ∗ )( ⊗ id|F )(Y ) = (X ∗ Y ) is obvious for X, Y ∈ B |F and so holds for X, Y ∈ B ⊗ |F by continuity. Set † X = kt,|ε (x ∗ ) and Y = kt,|ε (y). Then X, Y ∈ B ⊗ |F and so † ∗ lt (x )ε, lt (y)ε = ( ⊗ id|F )(X )∗ ( ⊗ id|F )(Y ) = (X ∗ Y ) = ( ◦ ωε,ε • kt )(x y) = (ωε,ε ◦ • kt )(x y) = ε, lt (x y)ε . Thus l is weakly multiplicative. This completes the proof.
Combining this result with Proposition 4.3, Theorem 5.1 and Theorem A.6 we obtain the advertised characterisation of the stochastic generators of Markov-regular *-homomorphic convolution cocycles on a C ∗ -bialgebra. Theorem 6.2. Let B be a C ∗ -bialgebra and let l ∈ P (B). Then the following are equivalent: (i) l is a Markov-regular, *-homomorphic (and unital) QS convolution cocycle on B;
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(ii) l = l ϕ , where ϕ ∈ C B B; B( k) satisfies ϕ(x ∗ y) = ϕ(x)∗ (y) + (x)∗ ϕ(y) + ϕ(x)∗ Q S ϕ(y) (and ϕ(1) = 0) ; (6.5) (iii) there is a vector c ∈ k and (unital) *-homomorphism π : A → B(k) such that l = l ϕ , where
c| (6.6) ϕ(x) = (π(x) − (x)Ik ) |c Ik , x ∈ B. Ik Remark. In fact, as is shown in the Appendix, the relation (6.5) for a linear map ϕ (an -structure map in the terminology used there) entails the implemented form (6.6), in particular the complete boundedness of ϕ. The characterisations of stochastic generators of completely positive, contractive QS convolution cocycles and *-homomorphic QS convolution cocycles in Theorems 5.1 and 6.2 may be used to derive dilation theorems for QS convolution cocycles (see [S]), of the type obtained for standard QS cocycles in [GLW] and [GLSW]. These characterisations are also used to establish the main result in [FrS], that every Markov-regular Fock space quantum Lévy process can be realised as a limit of (suitably scaled) random walks. 7. Axiomatisation of Topological Quantum Lévy Processes Defining quantum Lévy process on a C ∗ -bialgebra requires certain modifications of the original, purely algebraic, definition of Accardi, Schürmann and von Waldenfels ([ASW,Sch]). The problem is how to build convolution increments of the process given that, in general, multiplication B B → B need not extend continuously to B ⊗ B. (This is a commonly met difficulty in the theory of topological quantum groups, see [Kus1 ].) Below we outline two ways of overcoming this obstacle. The simplest idea is to define a quantum Lévy process using only the concept of distributions. Definition. A weak quantum Lévy process on a C ∗ -bialgebra B over a unital *algebra-with-state (A, ω) is a family js,t : B → A 0≤s≤t of unital *-homomorphisms such that the functional λs,t := ω ◦ js,t is continuous and satisfies the following conditions, for 0 ≤ r ≤ s ≤ t: (wQLP i) (wQLP ii) (wQLP iii) (wQLP iv)
λr,t = λr,s λs,t ; λt,t = ; λs,t = λ0,t−s ;
ω
n i=1
(wQLP v)
jsi ,ti (xi ) =
n
λsi ,ti (xi )
i=1
whenever n ∈ N, x1 , . . . , xn ∈ B and the intervals [s1 , t1 [, . . . , [sn , tn [ are disjoint; λ0,t → pointwise as t → 0.
A weak quantum Lévy process on a C ∗ -bialgebra B is called Markov-regular if λ0,t → in norm, as t → 0.
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The family λ := λ0,t t≥0 is a pointwise continuous convolution semigroup of functionals on B, called the one-dimensional distribution of the process; if the process is Markov-regular then λ has a convolution generator which is also referred to as the generator of the weak quantum Lévy process. Two weak quantum Lévy processes on B, j 1 over (A1 , ω1 ) and j 2 over (A2 , ω2 ), are said to be equivalent if they satisfy 1 2 ω1 ◦ js,t = ω2 ◦ js,t
for all 0 ≤ s ≤ t, in other words if their one-dimensional distributions coincide; if they are Markov-regular then this is equivalent to equality of their generators. Remarks. Note that the above definition of a weak quantum Lévy process, in contrast to the definition of a quantum Lévy process on an algebraic *-bialgebra, does not yield a recipe for expressing the joint moments of the process increments corresponding to overlapping time intervals, such as ω( jr,t (x) js,t (y)), where r, s < t. To achieve the latter, one would have to formulate the weak convolution increment property (wQLPi) in greater generality and assume certain commutation relations between the increments corresponding to disjoint time intervals. For other investigations of the notion of independence in noncommutative probability, in the absence of commutation relations being imposed, we refer to the recent paper [HKK]. As in the algebraic case, the generator of a Markov-regular weak quantum Lévy process vanishes on 1B , is real and is conditionally positive, that is positive on the kernel of the counit. Observe that if l ∈ P (B) is a unital *-homomorphic QS convolution cocycle then, defining A := B(F), ω := ωε(0) , and js,t := σs ◦ lt−s for all 0 ≤ s ≤ t, we obtain a weak quantum Lévy process on B, called a Fock space quantum Lévy process, Markov-regular if l is. The proof of the following theorem closely mirrors the proof of Schürmann’s reconstruction theorem for the purely algebraic case ([Sch], see also [LS1 ]); all the necessary continuity properties follow from the results in the Appendix. Theorem 7.1. Let γ be a real, conditionally positive linear functional on B vanishing at 1B . Then there is a (Markov-regular) Fock space quantum Lévy process with generator γ. Proof. The proof uses a GNS-style construction. Let D = Ker N , where N is the following subspace of Ker :
x ∈ Ker γ (x ∗ x) = 0 . Then ([x], [y]) → γ (x ∗ y) defines an inner product on D. Let k be the Hilbert space completion of D. The prescription π(x) : [z] → [x z] defines bounded operators on D, whose extensions make up a unital representation of B on k satisfying π(x)[y], [z] = [y], π(x ∗ )[z] . Furthermore the linear map δ : x → |d(x) , where d(x) = [x − (x)Ik ], is easily seen to be a (π , )-derivation B → |k satisfying δ(x)∗ δ(y) = γ (x ∗ y) − γ (x)∗ (y) − (x)∗ γ (y). Theorem A.6 therefore implies that the map ϕ : B → B( k), with block matrix form given by the prescription (A.8) (with λ = γ and χ = ), is completely bounded. Setting l = l ϕ , Theorem 6.2 implies that the Markov-regular weak QS convolution cocycle l is unital and *-homomorphic. Since ϕ00 = γ the result follows.
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Corollary 7.2. Every Markov-regular weak quantum Lévy process is equivalent to a Fock space quantum Lévy process. Another notion, in a sense intermediate between weak quantum Lévy processes and Fock space quantum Lévy processes, can be formulated in terms of product systems—a similar idea is mentioned in a recent paper of Skeide ([Ske]). Recall that a product system of Hilbert spaces is a ‘measurable’ family of Hilbert spaces E = {E t : t ≥ 0}, together with unitaries Us,t : E s ⊗ E t → E s+t (s, t ≥ 0) satisfying associativity relations: Ur +s,t (Ur,s ⊗ It ) = Ur,s+t (Ir ⊗ Us,t )
(7.1)
(r, s, t ∈ R+ ), where Is denotes the identity operator on E s . A unit for the product system E is a ‘measurable’ family {u(t) : t ≥ 0} of vectors with u(t) ∈ E t and u(s + t) = Us,t (u(s) ⊗ u(t)) for all s, t ≥ 0 (the unit is normalised if, for all t ≥ 0, u(t) = 1). For the precise definition we refer to [Arv]. The unitaries Us,t implement isomorphisms σs,t : B(E s ⊗ E t ) → B(E s+t ). Definition. A product system quantum Lévy process on B over a product-systemwith-normalised-unit (E, u) is a family ( jt : B → B(E t ))t≥0 of unital *-homomorphisms satisfying the following conditions: (psQLPi) (psQLPii) (psQLPiii)
jr +s = σr,s ◦ ( jr js ), j0 = ι0 ◦ , ωu(t) ◦ jt → pointwise as t → 0,
for r, s ≥ 0, where ι0 denotes the ampliation C → B(E 0 ). The family {ωu(t) ◦ jt : t ≥ 0} is a continuous convolution semigroup of functionals on B called the one-dimensional distribution of the product system quantum Lévy process. The ‘exponential’ product system is given by E t = F[0,t[ and Us,t = Is ⊗ Ss,t , where Ss,t denotes the natural shift F[0,t[ → F[s,s+t] and the exponential property of symmetric Fock space is invoked. Clearly every Fock space quantum Lévy process may be viewed as a product system quantum Lévy process over (E, ), where is the normalised unit given by (t) = ε(0) ∈ F[0,t[ , t ≥ 0. Proposition 7.3. Each product system quantum Lévy process on B naturally determines a weak quantum Lévy process on B with the same one-dimensional distribution. Proof. Let j be a quantum Lévy process on B over a product-system-with-normalised := unit (E, u). We use an inductive limit construction. Define A t≥0 (B(E t ), t) and the relation: (T, r ) ≡ (S, s) if there is t ≥ max{r, s} such that σr,t−r (T ⊗ introduce on A It−r ) = σs,t−s (S⊗It−s ), in other words we identify operators with common ampliations. The associativity relations (7.1) imply that ≡ is an equivalence relation. Define A = ≡ and introduce the structure of a unital *-algebra on A, consistent with the pointwise A/ operations: (T, t) + (S, t) = (T + S, t), (S, t) · (T, t) = (ST, t), (T, t)∗ = (T ∗ , t) → C defined by ω:A ω(T, t) = ωu(t) (T ) induces a (t ≥ 0, S, T ∈ B(E t )). The map state ω on A. For s, t ∈ R+ define js,t : B → A by x → σs,t−s (Is ⊗ jt−s (x)) ≡ . It is easy to see that the family js,t 0≤s≤t is a weak quantum Lévy process on B over (A, ω).
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The construction in the above proof, informed by the case of QS convolution cocycles, is a special case of the familiar construction of C ∗ -algebraic inductive limits. The com is a unital C ∗ -algebra that may pletion of A with respect to the norm induced from A ∗ be called the C -algebra of finite range operators on the product system E. Remark. A form of reconstruction theorem also holds for completely positive QS convolution cocycles. It is easily seen that if l ∈ P (E) is a Markov-regular, unital, completely positive QS convolution cocycle on a C ∗ -hyperbialgebra E, then the generator of its Markov convolution semigroup is real, vanishes at 1E and is conditionally positive. The GNS-type construction from the proof of Theorem 7.1 yields a completely bounded map ϕ : A → B( k) for which the cocycle l ϕ is unital and completely positive according to Proposition 4.3 and Theorem 5.1 (of course there is no reason why it should be *homomorphic, if E is not a C ∗ -bialgebra). Clearly the Markov convolution semigroup of l ϕ coincides with that of l. 8. Examples In this section we consider *-homomorphic QS convolution cocycles on three types of C ∗ -bialgebra, namely algebras of continuous functions on compact semigroups, universal C ∗ -algebras of discrete groups, and full compact quantum groups. We focus on connections between the results obtained in this paper and the case of purely algebraic convolution cocycles analysed in its predecessor, [LS1 ]. Recall that in [LS1 ] the basic object is a purely algebraic *-bialgebra (or even coalgebra) B, and coalgebraic where D is some QS differential equations are driven by coefficients in L(B; O‡ ( D)), dense subspace of the noise dimension space k. Processes V → C, now for a vector space V , are families k = (kt )t≥0 of maps V → O(E D ); we denote the space of these by P (V : E D ), and write P‡ (B : E D ) for the subspace of O‡ (E D )-valued processes. Pointwise Hölder-continuity for such a process k means that each of the vector-valued functions t → kt (x)ε should be locally Hölder-continuous with exponent 1/2. Note that it is a weaker form of continuity than the one that arises when V is an operator space (cf. Properties 2 and 2 after Theorems 1.1 and 3.1). The notation introduced after Theorem 3.1 extends as follows: for ϕ ∈ L(B; O( D)), (3.6) still defines a process P (B : E D ) (again written l ϕ ) which (uniquely) satisfies the QS differential equation (3.1), now understood in the sense of [LS1 ], and is a QS convolution cocycle with respect to the purely algebraic coalgebra structure. If the coefficient then l ϕ ∈ P‡ (B : E D ). ϕ lies in L(B; O‡ ( D)) Commutative case: Continuous functions on a semigroup. Let H be a compact semigroup with identity e and let B denote C(H ), the algebra of continuous complex-valued functions on H . Then B has the structure of a C ∗ -bialgebra with comultiplication and counit given by (F)(h, h ) = F(hh ) and (F) = F(e) (h, h ∈ H, F ∈ B), courtesy of the natural identification B ⊗ B ∼ = C(H × H ). Following standard practice in quantum probability (going back to [AFL] and beyond), any H -valued stochastic process X = (X t )t≥0 on the probability space (, F, P), may be described by a family of unital *-homomorphisms (lt )t≥0 given by lt : B → L ∞ (, F, P),
F → F ◦ X t ,
in turn these homomorphisms uniquely determine the original process.
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Recall that a process X on a semigroup with identity is called a Lévy process if it has identically distributed, independent increments, P({X 0 = e}) = 1 and the distribution of X t converges weakly to the Dirac measure δ{e} (the distribution of X 0 ) as t tends to 0. In general every Lévy process on a semigroup may be equivalently realised, in the sense of equal finite-dimensional distributions (see [Sch,LS1 ]), as a quantum Lévy process on a *-bialgebra ([Sch,FrScho]). As is well known, not all Lévy processes have stochastic generators defined on the whole of B. In our language, this corresponds to the fact that not all *-homomorphic processes on B are Markov-regular. Now Markov-regularity of the process corresponds to norm continuity of the convolution semigroup given by λt (F) = F ◦ Xt d P
(F ∈ B, t ≥ 0). Note that the usual notion of weak continuity for this semigroup corresponds, in the algebraic formulation, to pointwise continuity of the Markov semigroup. We therefore obtain the following result. Proposition 8.1. Let X be Lévy processes on a compact semigroup with identity H . Suppose that as a topological space H is normal. Then X is equivalent to a Markov-regular *-homomorphic QS convolution cocycle on B if and only if it satisfies the following condition: P ({X t = e}) → 1 as t → 0. (8.1) Proof. It is easily seen that condition (8.1) implies the existence of a bounded generator γ : B → C from which the process can be reconstructed. The other direction can be seen by considering the Markov semigroup of a given QS convolution cocycle and judiciously choosing continuous functions on H with values in [0, 1], which are equal to 1 at e and vanish outside of some neighbourhood of the identity element e. Processes satisfying (8.1) were investigated for example in [Gre]. They are called homogenous processes of discontinuous type and their laws are compound Poisson distributions ([Gre], Theorem 2.3.5). Cocommutative case: Group algebras. Let be a discrete group. Denote by B = C ∗ ( ) the enveloping C ∗ -algebra of the Banach algebra l 1 ( ) ([Ped]), called the universal (or full) C ∗ -algebra of . By construction (the algebra of functions on with finite support being dense in B), there is a universal unitary representation L : → B such that B := Lin{L g : g ∈ } is dense in B. Due to universality the mappings and defined on the image of L by (L g ) = L g ⊗ L g and (L g ) = 1, extend to *-homomorphisms on B. It is easily checked that B, equipped with the resulting comultiplication and counit, becomes a cocommutative C ∗ -bialgebra. Theorem 8.2. Let B = C ∗ ( ) for a discrete group . Then W (t, g) = lt (L g ) (g ∈ , t ≥ 0)
(8.2)
defines a bijective correspondence between unital *-homomorphic QS convolution cocycles on the C ∗ -bialgebra B and maps W : R+ × → B(F) satisfying the following conditions:
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(i) for each g ∈ the family {W (t, g) : t ≥ 0} is a left QS operator cocycle; (ii) for each t ≥ 0 the family {W (t, g) : g ∈ } is a unitary representation of on F. Proof. Let l ∈ P (B) be a *-homomorphic QS convolution cocycle and define a map W : R+ × → B(F) by (8.2). Then, for all g, h ∈ and s, t ≥ 0, ls+t (L g ) = (ls ⊗ (σt ◦ ls )) (L g ) = ls (L g ) ⊗ σs (lt (L g )) = W (s, g) ⊗ σs (W (t, g)), lt (L g )lt (L h ) = lt (L g L h ) = lt (L gh ) = W (t, gh), lt (L g )∗ = lt (L ∗g ) = lt (L g−1 ) = W (t, g −1 ), lt (L e ) = lt (1B ) = IF , and l0 (L g ) = IF , so W satisfies (i) and (ii). Conversely, suppose that W : R+ × → B(F) is a map satisfying conditions (i) and (ii). Due to universality there are maps lt : B → B(F), t ≥ 0, satisfying (8.2). The properties of W imply that they are unital *-homomorphisms and that they satisfy l0 (x) = (x)IF and ls+t (x) = (ls ⊗ (σs ◦ lt )) (x) for s, t ≥ 0 and x ∈ B. Continuity ensures that these remain valid for x ∈ B and so the result follows. On the level of stochastic generators the above correspondence takes the following form. Theorem 8.3. Let B := Lin{L g : g ∈ } for a discrete group . Then ψg = ϕ(L g ), g ∈ , determines a bijective correspondence between maps ϕ ∈ L(B; B( k)) satisfying ϕ(ab) = ϕ(a)(b) + (a)ϕ(b) + ϕ(a)Q S ϕ(b), ϕ(a)∗ = ϕ(a ∗ ), ϕ(1) = 0, (8.3) and maps ψ : → B( k) satisfying ψgh = ψg + ψh + ψg Q S ψh , (ψg )∗ = ψg−1 , ψe = 0; Proof. Elementary calculation.
(8.4)
Remarks. Identities (8.4) may be considered as a special (time-independent) case of formulae (4.2-4) in [HLP]. They are equivalent to ψ having the block matrix form
iλg − 21 ξg 2 − ξg |Ug ψg = , (8.5) |ξg U g − Ik for a unitary representation U of on k and maps λ : → R and ξ : → k satisfying ξgh = ξg + Ug ξh and λgh = λg + λh − Im ξg , Ug ξh .
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Note that, according to Theorem 6.3 of [LS1 ], each map ϕ ∈ L(B; B( k)) satisfying (8.3) generates a unital, real and weakly multiplicative QS convolution cocycle l ϕ on B. The process l ϕ continuously extends to a *-homomorphic QS convolution cocycle on B (see Lemma 8.7 below). On the other hand, given a map ψ : → B( k) satisfying (8.4), for each fixed g ∈ the unique (weakly regular, weak) solution of the operator QS differential equation X 0 = IF , d X t = X t d L (t), where L = ψg , is a unitary left QS cocycle W g ([LW2 ]). The map W : R+ × given g by W (t, g) = Wt satisfies the conditions of Theorem 8.2. One can easily see that the correspondences described in Theorems 8.2 and 8.3 are consistent with this construction. Proposition 8.4. A unital *-homomorphic QS convolution cocycle l on B is equal to l ϕ for some ϕ ∈ L(B; B( k)) if and only if it is pointwise weakly measurable. Proof. One direction is trivial. For the other consider the unitary cocycles {W (·, g) : g ∈ } associated with l by Theorem 8.2. Theorem 6.7 of [LW2 ] implies that each of these cocycles is stochastically generated (as it is weakly measurable). Denoting the respective generators by ψg one can see that the map ψ : → B(k) so obtained satisfies the conditions (8.4). The desired conclusion therefore follows from Theorem 8.3 and the subsequent discussion. If a *-homomorphic QS convolution cocycle l on B is Markov-regular, the automatic implementedness of its stochastic generator ϕ (Theorem 6.2) implies in particular that the triple (λ, ξ, U ) corresponding to ϕ by (8.5) and Theorem 8.3 must also be implemented, in the following sense: there is a vector η ∈ k such that ξg = Ug η − η and λg = Im η, Ug η , g ∈ G. In the language of group cohomology, the first order cocycle ξ is a coboundary. In this connection, see [PaS]. Elements of a C ∗ -bialgebra B are called group-like when they satisfy b = b ⊗ b, as the L g ’s do. On such elements the solution (kt (b))t≥0 , of the mapping QS differential equation (3.1), is given by the solution of the operator QS differential equation d X t = X t d L (t),
X 0 = IF ,
where L = ϕ(b) ∈ B( k). For more on this we refer to Sect. 4.1 of [Sch]. Full compact quantum groups. A concept of compact quantum groups was introduced by Woronowicz, in [Wor1 ]. For our purposes it is most convenient to adopt the following definition: Definition. ([Wor2 ]) A compact quantum group is a pair (B, ), where B is a unital C ∗ -algebra, and : B → B⊗B is a unital, *-homomorphic map which is coassociative and satisfies the quantum cancellation properties: Lin((1 ⊗ B)(B)) = Lin((B ⊗ 1)(B)) = B ⊗ B.
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For the concept of Hopf *-algebras and their unitary corepresentations, as well as unitary corepresentations of compact quantum groups, we refer the reader to [KlS]. For our purposes it is sufficient to note the facts contained in the following theorem. Theorem 8.5. ([Wor2 ]) Let B be a compact quantum group and let B denote the linear span of the matrix coefficients of irreducible unitary corepresentations of B. Then B is a dense *-subalgebra of B, the coproduct of B restricts to an algebraic coproduct 0 on B and there is a natural counit and coinverse S on B which makes it a Hopf *-algebra. Remark ([BMT]). In the above theorem (B, 0 , , S) is the unique dense Hopf *-subalgebra of B, in the following sense: if (B , 0 , , S ) is a Hopf *-algebra, in which B is a dense *-subalgebra of B and the coproduct of B restricts to the algebraic coproduct 0 on B , then (B , 0 , , S ) equals (B, 0 , , S). The Hopf *-algebra arising here is called the associated Hopf *-algebra of (B, ). When B = C(G) for a compact group G, B is the algebra of all matrix coefficients of unitary representations of G; when B is the universal C ∗ -algebra of a discrete group , B = Lin{L g : g ∈ } (see the beginning of the previous subsection). Dijkhuizen and Koornwinder observed that the Hopf *-algebras arising in this way have intrinsic algebraic structure. Definition. A Hopf *-algebra B is called a CQG algebra if it is the linear span of all matrix elements of its finite dimensional unitary corepresentations. Theorem 8.6. ([DiK]) Each Hopf *-algebra associated with a compact quantum group is a CQG algebra. Conversely, if B is a CQG algebra then
x := sup π(x) : π is a *-representation of B on a Hilbert space (8.6) defines a C ∗ -norm on B and the completion of B with respect to this norm is a compact quantum group whose comultiplication extends that of B. The compact quantum group obtained from a Hopf *-algebra B in this theorem is called its universal compact quantum group and is denoted Bu . For later use note the following extension of Lemma 11.31 in [KlS]: Lemma 8.7. Let E be a dense subspace of a Hilbert space H and let B be a CQG algebra. Suppose that π : B → O‡ (E) is real, unital and weakly multiplicative. Then π is bounded-operator-valued and admits a continuous extension to a unital *-homomorphism from Bu to B(H). Proof. Let [xi, j ]i,n j=1 be any finite dimensional unitary corepresentation of B. Then, ∗ x since nk=1 xk, j k, j = 1B for j ∈ {1, . . . , n}, π(xi, j )ξ 2 ≤
n k=1
π(xk, j )ξ 2 =
n
π(xk, j )ξ, π(xk, j )ξ k=1
= ξ, π
n
∗ xk, j x k, j
ξ = ξ 2
k=1
for i, j ∈ {1, . . . , n} and ξ ∈ E. This implies that, for each x ∈ B, π(x) is bounded—let π1 (x) denote its continuous extension to a bounded operator on H. The resulting map π1 : B → B(H) is then a unital *-homomorphism, moreover it is clearly contractive with respect to the canonical norm on B, given by (8.6); the result follows.
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Definition. A compact quantum group (B, ) is called full if the C ∗ -norm it induces on its associated CQG algebra B coincides with its canonical norm defined in (8.6)— equivalently, if B is *-isomorphic to Bu . The notion of full compact quantum groups was introduced in [BMT] and in [BaS] (in the first paper they were called universal compact quantum groups). It is very relevant for our context, as the above facts imply the following Proposition 8.8. Let B be a full compact quantum group with associated Hopf *-algebra B. Then B is a C ∗ -bialgebra whose counit is the continuous extension of the counit on B. Moreover restriction induces a bijective correspondence between unital, *-homomorphic QS convolution cocycles on B and unital, real, weakly multiplicative QS convolution cocycles (in the sense of [LS1 ]) on B. Both families of examples described in the previous two subsections, namely algebras of continuous functions on compact groups and full C ∗ -algebras of discrete groups, are full compact quantum groups. Moreover most of the genuinely quantum (i.e. neither commutative nor cocommutative) compact quantum groups considered in the literature also fall into this category, including the queen of examples, SUq (2). Reconnecting further with our previous work, we obtain the following result. Theorem 8.9. Let k ∈ P cb (B), where B is a full compact quantum group with associated Hopf *-algebra B. Then the following are equivalent: (i) k and k † are pointwise Hölder-continuous QS convolution cocycles; (ii) k|B = l ϕ for some map ϕ ∈ L(B; B( k)). Proof. One direction follows from the fact that B is an (algebraic) coalgebra and Theorem 5.8 of [LS1 ]. The other is trivial. Specialising to *-homomorphic cocycles yields the following much stronger result. Theorem 8.10. Let k ∈ P (B : E D ), where B is a full compact quantum group with associated Hopf *-algebra B and D is a dense subspace of k. Then the following are equivalent: (i) k is pointwise Hölder-continuous, unital and *-homomorphic (thus bounded) and a −→ kt (a) defines a QS convolution cocycle; satisfying the structure (ii) k is bounded and k|B = l ϕ for some ϕ ∈ L(B; O‡ ( D)) relations (6.5). Proof. The implication (i)⇒(ii) follows from the previous theorem and implication (i)⇒(ii) of Theorem 6.3 of [LS1 ] (note that it even yields ϕ ∈ L(B; O‡ ( k)) = L(B; B( k)). Suppose conversely that (ii) holds. Theorem 6.3 of [LS1 ] guarantees that l = k|B is real, unital, and weakly multiplicative. Lemma 8.7 shows that l admits a continuous extension to a *-homomorphic unital process B → C defined on E D , which must coincide with k. Application of the previous theorem therefore completes the proof. The above theorem may be equivalently formulated in the following way. Theorem 8.11. Let k ∈ P (B : E D ), where B is the Hopf *-algebra associated with a full compact quantum group B and D is a dense subspace of k. Then the following are equivalent:
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(i) k extends to a pointwise Hölder-continuous, unital, *-homomorphic QS convolution cocycle on B; satisfying the structure relations (6.5). (ii) k = l ϕ for some ϕ ∈ L B; O‡ ( D) Remark. In the course of the proof of the previous theorem it was established that each satisfying the conditions (6.5) map ϕ defined on a CQG algebra B with values in O‡ ( D) must be bounded-operator-valued. We stress however, that ϕ need not extend continuously to B (for examples see [SchS]). On the other hand if ϕ is continuous, then it is necessarily completely bounded. Appendix: (π , π )-Derivations and χ -Structure Maps In this Appendix we give an extension of the innerness theorem of Christensen, for completely bounded derivations on a C ∗ -algebra, to (π , π )-derivations, and prove automatic complete boundedness for (π, χ )-derivations, when χ is a character. These are then applied to prove the innerness of what we call χ -structure maps. We first recall the relevant theorems on derivations. Theorem A.1. ([Sak1 ,Rin]) Let δ : A → X be a derivation from a C ∗ -algebra A into a Banach A-bimodule. Then δ is bounded. Theorem A.2. ([Chr]) Let A be a C ∗ -algebra in B(h) and let δ : A → B(h) be a derivation. If δ is completely bounded then it is inner: there is R ∈ B(h) such that δ(a) = a R − Ra, a ∈ A. A simple proof of the first theorem in the case X = A (Sakai’s Theorem), due to Kishimoto, may be found in [Sak2 ], and a good reference for the second, along with connections to not-necessarily-involutive homomorphisms between C ∗ -algebras, is [Pis1 ]. We are interested in the particular class of Banach A bimodule-valued derivations captured by the following definition. Definition. Let A be a C ∗ -algebra with representations (π, h) and (π , h ). A map δ : A → B(h; h ) is called a (π , π)-derivation if it satisfies δ(ab) = δ(a)π(b) + π (a)δ(b); it is inner if it is implemented by an operator T ∈ B(h; h ) in the sense that δ : a → π (a)T − T π(a). Theorem A.3. Let A be a C ∗ -algebra with representations (π, h) and (π , h ), and let δ : A → B(h; h ) be a completely bounded (π , π )-derivation. Then δ is inner. A= π (A), Proof. Let (ρ, K) be a faithful representation of A and set H = h⊕h ⊕K and where π is the faithful representation π ⊕ π ⊕ ρ. Then A is a C ∗ -subalgebra of B(H) and it is easily verified that ⎡ ⎤ 0 ⎦ π (a) → ⎣δ(a) 0 0 defines a derivation δ : A → B(H). It is also clear that δ is completely bounded if and only if δ is. Moreover, if δ is inner then the (π , π )-derivation δ is implemented by S21 ∈ B(h; h ) for any operator S = [Si j ] ∈ B(H) implementing the derivation δ . The result therefore follows from Theorem A.2.
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Theorem A.4. Let A be a C ∗ -algebra with representation (π, h) and character χ , and let δ : A → |h be a (π, χ )-derivation. Then δ is inner. Proof. Without loss of generality we may suppose that the C ∗ -algebra A and representation π are both unital; if necessary by extending π , χ and δ to the unitisation of A in the following natural way: (a, z) → π(a) + z Ih , (a, z) → χ (a) + z and (a, z) → δ(a). By Theorem A.1, δ is bounded. Let A0 = Ker χ and let ψ : A → A0 be the projection a → a − χ (a)1. Then A0 is a C ∗ -subalgebra of A, ψ is completely bounded and δ = δ ◦ ψ, where δ = δ|A0 . Therefore, by the previous theorem, it suffices to show that δ is completely bounded. Now δ (ab) = π(a)δ(b) for all a, b ∈ A0 . Since δ is bounded this implies that δ (n) (A) = lim π (n) (A) (δ(eλ ) ⊗ In ) λ
# (n) # δ # ≤ δ. (n ∈ N, A ∈ Mn (A0 )), for any C ∗ -approximate identity (eλ ) for A0 , and so # The result follows. We note two consequences; the first is used in [S]. Corollary A.5. Let A be a C ∗ -algebra with characters (i.e. nonzero multiplicative linear functionals) χ and χ . Then every (χ , χ ) derivation on A vanishes. For the second the following definitions are convenient. If A is a C ∗ -algebra with character χ , then a χ -structure map on A is a linear map ϕ : A → B(C ⊕ h), for some Hilbert space h, satisfying ϕ(a ∗ b) = ϕ(a)∗ χ (b) + χ (a)∗ ϕ(b) + ϕ(a)∗ ϕ(b), where := 0 Ih . For any C ∗ -representation (π, h) and vector ξ ∈ h,
ξ | a → (π(a) − χ (a)Ih ) |ξ Ih Ih
(A.7)
defines a χ -structure map. Such χ -structure maps are said to be implemented. Thus implementation involves a pair (π, ξ ). Note that implemented χ -structure maps are completely bounded. Theorem A.6. Let A be a C ∗ -algebra with character χ and let ϕ be a χ -structure map on A. Then ϕ is implemented. Proof. Without loss of generality we may suppose that A is unital, since otherwise (invoking the reality of ϕ) the prescriptions (a, z) → χ (a) + z, respectively (a, z) → ϕ(a), extend χ and ϕ to the unitisation of A, maintaining the χ -structure relation (A.7). Now the χ -structure relation is equivalent to ϕ having block matrix form
† λδ , (A.8) δ ν
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where ν = π − ιk ◦ χ for a *-homomorphism π : A → B(h), δ is a (π, χ )-derivation and the linear functional λ satisfies λ(a ∗ b) = λ(a)∗ χ (b) + χ (a)∗ λ(b) + δ(a)∗ δ(b) (a, b ∈ A)—in particular, λ is real and satisfies λ(1) = −δ(1)∗ δ(1) and λ(a ∗ b) = δ(a)∗ δ(b) for a, b ∈ A0 ,
(A.9)
where A0 = Ker χ . By Theorem A.4, there is a vector ξ ∈ k such that δ(a) = ν(a)|ξ . Now define a bounded linear functional λ on A by λ(a) = ξ, ν(a)ξ . It is easily checked that λ also satisfies (A.9), thus λ agrees with λ on A00 + C1A , where A00 = Lin{a ∗ b : a, b ∈ A0 }. But A00 is dense in A0 and A = A0 ⊕ C1A , so λ equals λ. The result follows. References [Acc]
Accardi, L.: On the quantum Feynman-Kac formula. Rend. Sem. Mat. Fis. Milano 48, 135–180 (1980) [AFL] Accardi, L., Frigerio, A., Lewis, J.: Quantum stochastic processes. Publ. Res. Inst. Math. Sci. 18(1), 97–133 (1982) [ABKL] Applebaum, D., Bhat, B.V.R., Kustermans, J., Lindsay, J.M.: Quantum Independent Increment Processes. Vol. I: From Classical Probability to Quantum Stochastics, eds. U. Franz, M. Schürmann, Lecture Notes in Mathematics 1865, Heidelberg: Springer, 2005 [Arv] Arveson, W.: Noncommutative dynamics and E-semigroups. New York: Springer, 2003 [ASW] Accardi, L., Schürmann, M., von Waldenfels, W.: Quantum independent increment processes on superalgebras. Math. Z. 198(4), 451–477 (1988) [BaS] Baaj, S., Skandalis, G.: Unitaires multiplicatifs et dualité pour les produits croisés de C ∗ -algèbres. Ann. Sci. École Norm. Sup. 26(4), 425–488 (1993) [BFGKT] Barndorff-Nielsen, O.E., Franz, U., Gohm, R., Kümmerer, B., Thorbjørnsen, S.: Quantum Independent Increment Processes, Vol. II: Structure of Quantum Lévy Processes, Classical Probability and Physics. eds. U. Franz, M. Schürmann, Lecture Notes in Mathematics 1866, Heidelberg: Springer, 2006 [Bel] Belavkin, V.P.: Quantum stochastic positive evolutions: characterization, construction, dilation. Comm. Math. Phys. 184(3), 533–566 (1997) [BMT] Bedos, E., Murphy, G., Tuset, L.: Co-amenability for compact quantum groups. J. Geom. Phys. 40(2), 130–153 (2001) [ChV] Chapovsky, Yu., Vainerman, L.: Compact quantum hypergroups. J. Operator Theory 41(2), 261–289 (1999) [Chr] Christensen, E.: Extensions of derivations II. Math. Scand. 50, 111–122 (1982) [ChE] Christensen, E., Evans, D.E.: Cohomology of operator algebras and quantum dynamical semigroups. J. London Math. Soc. 20, 358–368 (1979) [Dav] Davies, E.B.: One-parameter Semigroups London:Academic Press, 1980 [DiK] Dijkhuizen, M., Koornwinder, T.: CQG algebras—a direct algebraic approach to compact quantum groups. Lett. Math. Phys. 32(4), 315–330 (1994) [EfR] Effros, E.G., Ruan, Z.-J.: Operator Spaces. London Mathematical Society Monographs, New Series 23,Oxford: Oxford University Press, 2000 [Fra] Franz, U.: Lévy processes on quantum groups and dual groups. In: [BFGKT] [FrScho] Franz U., Schott R., Stochastic Processes and Operator Calculus on Quantum Groups. Mathematics and its Applications 490 Dordrecht: Kluwer, 1999 [FrSch] Franz, U., Schürmann, M.: Lévy processes on quantum hypergroups. In: Infinite Dimensional Harmonic Analysis, eds. H. Heyer, T. Hirai & N. Obata, Altendorff: Gräbner, 2000, pp. 93–114 [FrS] Franz, U., Skalski, A.G.: Approximation of quantum Lévy processes by quantum random walks. Proc. Ind. Acad. Sci., Math. Sci. (to appear in May 2008) [Glo] Glockner, P.: Quantum stochastic differential equations on *-bialgebras. Math. Proc. Camb. Phil. Soc. 109(3), 571–595 (1991) [GLW] Goswami, D., Lindsay, J.M., Wills, S.J.: A stochastic Stinespring theorem. Math. Ann. 319(4), 647–673 (2001)
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[GLSW] [Gre] [Gui] [HKK] [Hey] [HLP] [HuP] [KlS] [Kus1 ] [Kus2 ] [KuV] [L1 ] [L2 ] [LPa] [LS1 ] [LS2 ] [LS3 ] [LS4 ] [LW1 ] [LW2 ] [LW3 ] [LW4 ] [LW5 ] [Mey] [Par] [PaS] [Ped] [Pin] [Pis1 ] [Pis2 ] [Rin] [Sak1 ] [Sak2 ]
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Goswami, D., Lindsay, J.M., Sinha, K.B., Wills, S.J.: Dilation of markovian cocycles on a von Neumann algebra. Pacific J. Math. 211(2), 221–247 (2003) Grenander, U.: Probabilities on Algebraic Structures. New York-London: John Wiley & Sons, 1963 Guichardet, A.: Symmetric Hilbert Spaces and Related Topics. Lecture Notes in Mathematics 267, Heidelberg: Springer, 1970 Hellmich, J., Köstler, C., Kümmerer, B.: Noncommutative continuous Bernoulli shifts. http://arxiv. org/list/math.OA/0411565, 2004 Heyer, H.: Probability Measures on Locally Compact Groups. Berlin: Springer, 1977 Hudson, R.L., Lindsay, J.M., Parthasarathy, K.R.: Flows of quantum noise. J. Appl. Anal. 4(2), 143–160 (1998) Hudson, R.L., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys. 93(3), 301–323 (1984) Klimyk, A., Schmüdgen, K.: Quantum Groups and their Representations, Texts and Monographs in Physics, Berlin: Springer, 1997 Kustermans, J.: Locally compact quantum groups. In: [ABKL] Kustermans, J.: Locally compact quantum groups in the universal setting. Internat. J. Math. 12(3), 289–338 (2001) Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Sci. École Norm. Sup. (4) 33(6), 837–934 (2000) Lindsay, J.M.: Quantum stochastic analysis—an introduction, In: [ABKL]. Lindsay, J.M.: A note on matrix spaces. In preparation Lindsay, J.M., Parthasarathy, K.R.: On the generators of quantum stochastic flows. J. Funct. Anal. 158(2), 521–549 (1998) Lindsay, J.M., Skalski, A.G.: Quantum stochastic convolution cocycles. Ann. Inst. H. Poincaré, Probab. Statist. 41(3) (En hommage à Paul-André Meyer), 581–604 (2005) Lindsay, J.M., Skalski, A.G.: Quantum stochastic convolution cocycles—algebraic and C ∗ -algebraic. Banach Center Publ. 73, 313–324 (2006) Lindsay, J.M., Skalski, A.G.: On quantum stochastic differential equations. J. Math. Anal. Appl. 330(2), 1093–1114 (2007) Lindsay, J.M., Skalski, A.G.: Quantum stochastic convolution cocycles III. In preparation. Lindsay, J.M., Wills, S.J.: Existence, positivity, and contractivity for quantum stochastic flows with infinite dimensional noise. Probab. Theory Related Fields 116(4), 505–543 (2000) Lindsay, J.M., Wills, S.J.: Markovian cocycles on operator algebras, adapted to a fock filtration. J. Funct. Anal. 178(2), 269–305 (2000) Lindsay, J.M., Wills, S.J.: Existence of Feller cocycles on a C ∗ -algebra. Bull. London Math. Soc. 33(5), 613–621 (2001) Lindsay, J.M., Wills, S.J.: Homomorphic Feller cocycles on a C ∗ -algebra. J.London Math. Soc. (2) 68(1), 255–272 (2003) Lindsay, J.M., Wills, S.J.: Quantum stochastic cocycles and completely bounded semigroups on operator spaces II. In preparation [Preprint(2005): On the generators of completely positive Markovian cocycles] Meyer, P.-A.: Quantum Probability for Probabilists. 2nd Edition, Lecture Notes in Mathematics 1538, Berlin: Springer, 1995 Parthasarathy, K.R.: Introduction to Quantum Stochastic Calculus. Basel: Birkhäuser, 1992 Parthasarathy, K.R., Schmidt, K.: Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory. Lecture Notes in Mathematics 272, Berlin: Springer 1972 Pedersen, G.K.: C ∗ -algebras and their automorphism groups. London Mathematical Society Monographs 14, London-New York: Academic Press Inc., 1979 Pinsky, M.: Stochastic integral representation of multiplicative functionals of a Wiener process. Trans. Amer. Math. Soc. 167, 89–113 (1972) Pisier, G.: Similarity problems and completely bounded maps. Lecture Notes in Mathematics 1618, Berlin: Springer, 2001 Pisier, G.: Introduction to Operator Space Theory. London Mathematical Society Lecture Note Series 294, Cambridge: Cambridge University Press, 2003 Ringrose, J.R.: Automatic continuity of derivations of operator algebras. J. London Math. Soc. 5, 432–438 (1972) Sakai, S.: On a conjecture of Kaplansky. Tôhoku Math. J. 12, 31–33 (1960) Sakai, S.: Operator algebras in dynamical systems. The theory of unbounded derivations in C ∗ -algebras. Encyclopedia of Mathematics and its Applications 41. Cambridge: Cambridge University Press, 1991
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Schürmann, M.: White Noise on Bialgebras, Lecture Notes in Mathematics 1544. Heidelberg: Springer, 1993 Schürmann, M., Skeide, M.: Infinitesimal generators on the quantum group SUq (2). Infin. Dimens. Anal. Quantum Prob. Relat. Top. 1(4), 573–598 (1998) Skalski, A.: Completely positive quantum stochastic convolution cocycles and their dilations. Math. Proc. Camb. Phil. Soc. 143, 201–219 (2007) Skeide, M.: Lévy processes and tensor product systems of Hilbert modules, in “Quantum Probability and Infinite Dimensional Analysis, From Foundations to Applications” eds. M. Schürmann & U. Franz, Singapore: World Scientific Publishing, 2004, pp. 492–503 Skorohod, A.V.: Operator stochastic differential equations and stochastic semigroups (Russian). Usp. Mat. Nauk 37(6), 228, 157–183; (1982). [Transln. Russi. Math. Surv. 37(6), 177–204 (1982)] Sweedler, M.E.: Hopf Algebras. New York: Benjamin, 1969 Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys. 111(4), 613–665 (1987) Woronowicz, S.L.: Compact quantum groups. In: Symétries Quantiques, Proceedings, Les Houches 1995, eds. A. Connes, K. Gawedzki, J. Zinn-Justin, Amsterdam: North-Holland, 1998, pp. 845–884
Communicated by A. Connes
Commun. Math. Phys. 280, 611–673 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0479-4
Communications in
Mathematical Physics
Sasaki–Einstein Manifolds and Volume Minimisation Dario Martelli1 , James Sparks2,3 , Shing-Tung Yau2 1 Department of Physics, CERN Theory Division, 1211 Geneva 23, Switzerland.
E-mail: [email protected]
2 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA.
E-mail: [email protected]; [email protected]
3 Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Received: 28 September 2006 / Accepted: 3 December 2007 Published online: 9 April 2008 – © Springer-Verlag 2008
Abstract: We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein– Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X , is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat–Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki– Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kähler–Einstein metric. Contents 1. 2.
Introduction and Summary . . . . . . . 1.1 Background . . . . . . . . . . . . 1.2 Outline . . . . . . . . . . . . . . . Sasakian Geometry . . . . . . . . . . 2.1 Kähler cones . . . . . . . . . . . . 2.2 The Calabi–Yau condition . . . . . 2.3 The Reeb foliation . . . . . . . . . 2.4 Transverse Kähler deformations . 2.5 Moment maps . . . . . . . . . . . 2.6 Killing spinors and the (n, 0)–form 2.7 The homogeneous gauge for . .
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The Variational Problem . . . . . . . . . . . . . . . . . 3.1 The Einstein–Hilbert action . . . . . . . . . . . . . 3.2 Varying the Reeb vector field . . . . . . . . . . . . 3.3 Uniqueness of critical points . . . . . . . . . . . . 4. The Futaki Invariant . . . . . . . . . . . . . . . . . . . 4.1 Brief review of the Futaki invariant . . . . . . . . . 4.2 Relation to the volume . . . . . . . . . . . . . . . 4.3 Isometries of Sasaki–Einstein manifolds . . . . . . 5. A Localisation Formula for the Volume . . . . . . . . . 5.1 The volume and the Duistermaat–Heckman formula 5.2 The Duistermaat–Heckman Theorem . . . . . . . . 5.3 Application to Sasakian geometry . . . . . . . . . 5.4 Sasakian 5–manifolds and an example . . . . . . . 6. The Index–Character . . . . . . . . . . . . . . . . . . . 6.1 The character . . . . . . . . . . . . . . . . . . . . 6.2 Relation to the ordinary index . . . . . . . . . . . . 6.3 Localisation and relation to the volume . . . . . . . 7. Toric Sasakian Manifolds . . . . . . . . . . . . . . . . 7.1 Affine toric varieties . . . . . . . . . . . . . . . . . 7.2 Relation of the character to the volume . . . . . . . 7.3 Localisation formula . . . . . . . . . . . . . . . . 7.4 Examples . . . . . . . . . . . . . . . . . . . . . . 7.4.1 The conifold . . . . . . . . . . . . . . . . . . 7.4.2 The first del Pezzo surface . . . . . . . . . . . 7.4.3 The second del Pezzo surface. . . . . . . . . . 7.4.4 An orbifold resolution: Y p,q singularities. . . . A. The Reeb Vector Field is Holomorphic and Killing . . . B. More on the Holomorphic (n, 0)–Form . . . . . . . . . C. Variation Formulae . . . . . . . . . . . . . . . . . . . . C.1 First variation . . . . . . . . . . . . . . . . . . . . C.2 Second variation . . . . . . . . . . . . . . . . . . .
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1. Introduction and Summary 1.1. Background. The AdS/CFT correspondence [1] is one of the most important advancements in string theory. It provides a detailed correspondence between certain conformal field theories and geometries, and has led to remarkable new results on both sides. A large class of examples consists of type IIB string theory on the background AdS5 × L, where L is a Sasaki–Einstein five–manifold and the dual theory is a four–dimensional N = 1 superconformal field theory [2–5]. This has recently led to considerable interest in Sasaki–Einstein geometry. Geometry. Recall that a Sasakian manifold (L , g L ) is a Riemannian manifold of dimension (2n − 1) whose metric cone gC(L) = dr 2 + r 2 g L
(1.1)
is Kähler. (L , g L ) is Sasaki–Einstein if the cone (1.1) is also Ricci–flat. It follows that a Sasaki–Einstein manifold is a positively curved Einstein manifold. The canonical
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example is an odd–dimensional round sphere S 2n−1 ; the metric cone (1.1) is then Cn with its flat metric. All Sasakian manifolds have a canonically defined Killing vector field ξ , called the Reeb vector field. This vector field will play a central role in this paper. To define ξ note that, since the cone is Kähler, it comes equipped with a covariantly constant complex structure tensor J . The Reeb vector field is then defined to be ∂ ξ=J r . (1.2) ∂r As a vector field on the link1 L = {r = 1} this has norm one, and hence its orbits define a foliation of L. Either these orbits all close, or they don’t. If they all close, the flow generated by ξ induces a U (1) action on L which, since the vector field is nowhere– vanishing, is locally free. The orbit, or quotient, space is then a Kähler orbifold, which is a manifold if the U (1) action is actually free. These Sasakian metrics are referred to as quasi–regular and regular, respectively. More generally, the generic orbits of ξ might not close. In this case there is no quotient space, and the Kähler structure exists only as a transverse structure. The closure of the orbits of ξ is at least a two–torus, and thus these so–called irregular Sasakian metrics admit at least a two–torus of isometries. This will also be crucial in what follows. Note that Sasakian geometries are then sandwiched between two Kähler geometries: one of complex dimension n on the cone, and one, which is generally only a transverse structure, of dimension n − 1. For Sasaki–Einstein manifolds, the transverse metric is in fact Kähler–Einstein. Until very recently, the only explicit examples of simply–connected Sasaki–Einstein manifolds in dimension five (equivalently complex dimension n = 3) were the round sphere S 5 and the homogeneous metric on S 2 × S 3 , known as T 1,1 in the physics literature. These are both regular Sasakian structures, the orbit spaces being CP 2 and CP 1 × CP 1 with their Kähler–Einstein metrics. All other Sasaki–Einstein metrics, in dimension five, were known only through existence arguments. The remaining regular metrics are based on circle bundles over del Pezzo surfaces d Pk , 3 ≤ k ≤ 8, equipped with their Kähler Einstein metrics – these are known to exist through the work of Tian and Yau [6,7]. On the other hand, Boyer and Galicki have produced many examples of quasi–regular Sasaki–Einstein metrics using existence results of Kollár and collaborators for Kähler–Einstein metrics on orbifolds. For a review of their work, see [8]. In references [9–11] infinite families of explicit inhomogeneous Sasaki–Einstein metrics in all dimensions have been constructed. In particular, when n = 3 there is a family of cohomogeneity one five–metrics, denoted Y p,q , where q < p with p, q positive integers [10]. This has subsequently been generalised to a three–parameter cohomogeneity two family L a,b,c [12–14]. Provided the integers a, b, c are chosen such that L a,b,c is smooth and simply–connected, these manifolds are all diffeomorphic to S 2 × S 3 . Further generalisations in complex dimension n ≥ 4 have appeared in [15–17]. The metrics Y p,q are quasi–regular when 4 p 2 − 3q 2 is a square. However, for general q < p, they are irregular. These were the first examples of irregular Sasaki–Einstein manifolds, which in particular disproved the conjecture of Cheeger and Tian [18] that irregular Sasaki–Einstein manifolds do not exist. Field theory. In general, the field theory duals of Sasaki–Einstein five–manifolds may be thought of as arising from a stack of D3–branes sitting at the apex r = 0 of the Ricci–flat 1 We prefer this choice of terminology to “base of the cone”, or “horizon”.
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Kähler cone (1.1). Alternatively, the Calabi–Yau geometry may be thought of as arising from the moduli space of the Higgs branch of the gauge theory on the D3–branes. Through simple AdS/CFT arguments, one can show that the symmetry generated by the Reeb vector field and the volume of the Sasaki–Einstein manifold correspond to the R–symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. All N = 1 superconformal field theories in four dimensions possess a global R–symmetry which is part of the superconformal algebra. The a central charge appears as a coefficient in the one–point function of the trace of the energy–momentum tensor on a general background2 , and its value may be computed exactly, once the R–symmetry is correctly identified. A general procedure that determines this symmetry is a–maximisation [19]. One defines a function atrial on an appropriate space of potential (or “trial”) R–symmetries. The local maximum of this function determines the R–symmetry of the theory at its superconformal point. Moreover, the critical value of atrial is precisely the central charge a of the superconformal theory. Since atrial is a cubic function with rational coefficients, it follows3 that the R–charges of fields are algebraic numbers [19]. The AdS/CFT correspondence relates these R–charges to the volume of the dual Sasaki–Einstein manifold, as well as the volumes of certain supersymmetric three– dimensional submanifolds of L. In particular, we have the relation [20,21] aL vol[S 5 ] . = aS5 vol[L]
(1.3)
Since the left-hand side is determined by a–maximisation, we thus learn that the volume of a Sasaki–Einstein five–manifold vol[L], relative to that of the round sphere, is an algebraic number. Moreover, in the field theory, this number has been determined by a finite dimensional extremal problem. Our aim in [22], of which this paper is a continuation, was to try to understand, from a purely geometrical viewpoint, where these statements are coming from. Toric geometries and their duals. Given a Sasaki–Einstein manifold, it is in general a difficult problem to determine the dual field theory. However, in the case that the local Calabi–Yau singularity is toric there exist techniques that allow one to determine a dual gauge theory, starting from the combinatorial data that defines the toric variety. Using these methods it has been possible to construct gauge theory duals for the infinite family of Sasaki–Einstein manifolds Y p,q [23,24], thus furnishing a countably infinite set of AdS/CFT duals where both sides of the duality are known explicitly. Indeed, remarkable agreement was found between the geometrical computation in the case of the Y p,q metrics [10,23] and the a–maximisation calculation [24,25] for the corresponding quiver gauge theories. Thus the relation (1.3) was confirmed for a non–trivial infinite family of examples. Further developments [26–31] have resulted in the determination 2 The other coefficient is usually called c. However, superconformal field theories with a Sasaki–Einstein dual have a = c. 3 Since we make a similar claim in this paper, we recall here a proof of this fact: suppose we have a vector v ∈ Rs which is an isolated zero of a set of polynomials in the components of v with rational coefficients. Consider the Galois group Gal(C/Q). This group fixes the set of polynomials, and thus in particular the Galois orbit of the zeros is finite. An algebraic number may be defined as an element of C with finite Galois orbit, and thus we see that the components of the vector v are algebraic numbers. We thank Dorian Goldfeld for this argument. Recall also that the set of algebraic numbers form a field. Thus, in the present example, the R–charges of fields being algebraic implies that the a–central charge, which is a polynomial function of the R–charges with rational coefficients, is also an algebraic number.
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of families of gauge theories that are dual to the wider class of toric Sasaki–Einstein manifolds L a,b,c [28,32] (see also [33]). Of course, given this success, it is natural to try to obtain a general understanding of the geometry underlying a–maximisation and the AdS/CFT correspondence. To this end, in [22] we studied a variational problem on a space of toric Sasakian metrics. Let us recall the essential points of [22]. Let (X, ω) be a toric Kähler cone of complex dimension n. This means that X is an affine toric variety, equipped with a conical Kähler metric that is invariant under a holomorphic action of the n–torus Tn . X has an isolated singular point at the tip of the cone, the complement of which is X 0 = C(L) ∼ = R+ × L. A conical metric on X which is Kähler (but in general not Ricci–flat) then gives a Sasakian metric on the link L. The moment map for the torus action exhibits X as a Lagrangian Tn fibration over a strictly convex rational polyhedral cone4 C ∗ ⊂ t∗n ∼ = Rn . This is a n subset of R of the form C ∗ = {y ∈ Rn | (y, va ) ≥ 0, a = 1, . . . , D}.
(1.4)
Thus C ∗ is made by intersecting D hyperplanes through the origin in order to make a convex polyhedral cone. Here y ∈ Rn are coordinates on Rn and va are the inward pointing normal vectors to the D hyperplanes, or facets, that bound the polyhedral cone. The condition that X is Calabi–Yau implies that the vectors va may, by an appropriate S L(n; Z) transformation of the torus, be all written as va = (1, wa ). In particular, in complex dimension n = 3 we may therefore represent any toric Calabi–Yau cone X by a convex lattice polytope in Z2 , where the vertices are simply the vectors wa . This is usually called the toric diagram. Note that the cone C ∗ may also be defined in terms of its generating edge vectors {u α } giving the directions of the lines going through the origin. When n = 3 the projection of these lines onto the plane with normal (1, 0, 0) are the external legs of the so–called pq–web appearing in the physics literature. These are also weight vectors for the torus action and generalise to the non–toric case. For a toric Kähler cone (X, ω), one can introduce symplectic coordinates (yi , φi ), where φi ∼ φi + 2π are angular coordinates along the orbits of the torus action, and yi are the associated moment map coordinates. These may be considered as coordinates on t∗n ∼ = Rn . The symplectic (Kähler) form is then ω=
n
dyi ∧ dφi .
(1.5)
i=1
In this coordinate system, the metric degrees of freedom are therefore entirely encoded in the complex structure tensor J – see [22] for further details. In [22] we considered the space of all smooth toric Kähler cone metrics on such an affine toric variety X . The space of metrics naturally factors into the space of Reeb vector fields, which live in the interior C0 of the dual cone C ⊂ tn ∼ = Rn to C ∗ , and then an infinite dimensional space of transverse Kähler metrics. A general Reeb vector field may be written ξ=
n i=1
bi
∂ , ∂φi
(1.6)
4 We make a change of notation from our previous paper [22]: specifically, we exchange the roles of cone and dual cone. This is more in line with algebro–geometric terminology, and is more natural in the sense that the moment cone C ∗ lives in the dual Lie algebra t∗n of the torus.
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where b ∈ C0 ⊂ Rn . In this toric setting, the remaining degrees of freedom in the metric are described by the space of all homogeneous degree 1 functions on C ∗ which are smooth up to the boundary (together with a convexity requirement). The main result of [22] is that the Einstein–Hilbert action on L, restricted to this space of toric Sasakian metrics on L, reduces to the volume function vol[L] : C0 → R, which depends only on the Reeb vector field in C0 . Moreover, this is essentially just the Euclidean volume of the polytope formed by C ∗ and the hyperplane 2(b, y) = 1. This depends only on b and the toric data {va }. In particular, for n = 3, we have the formula V (b) ≡
D D 1 (va−1 , va , va+1 ) vol[L](b) (va−1 , va , va+1 ) 1 = = b1 (b, va−1 , va )(b, va , va+1 ) b1 (b, u a )(b, u a+1 ) vol[S 5 ] a=1
a=1
(1.7) for the normalised volume of L. The symbol (·, ·, ·) denotes a 3 × 3 determinant, while (·, ·) is the usual scalar product on Rn (or dual pairing between tn and t∗n , whichever the reader prefers). The function V (b) diverges to +∞ at the boundary ∂C of C – this is because the Reeb vector field develops a fixed point set in this limit, as will be explained in Sect. 2. Once the critical Reeb vector field b = b∗ is obtained one can compute the volume of the Sasaki–Einstein manifold, as well as the volumes of certain toric submanifolds, without explicit knowledge of the metric5 . From the explicit form of the Einstein–Hilbert action, it follows that the ratios of these volumes to those of round spheres are in general algebraic numbers. This method of determining the critical Reeb vector field, and the corresponding volume, has been referred to as “Z –minimisation”, where Z is just the restriction of the function V (b) to the hyperplane b1 = n. Indeed, the results of [22], for n = 3, were interpreted as the geometric “dual” of a–maximisation for the case that the Sasaki–Einstein manifolds, and hence the superconformal gauge theories, were toric. An analytic proof of the equivalence of these two optimisation problems was given in the work of [29], modulo certain assumptions on the matter content of the field theory. The key point is that, following on from results in [28], the trial a–function atrial for the gauge theory may be defined in closed form in terms of the toric data, i.e. the normal vectors {va }, independently of the precise details of the gauge theory – for example the form of the superpotential. This is a priori a function of D − 1 variables, the trial R–charges, where D is the number of facets of the cone. The global baryonic symmetries are U (1) D−3 [28]. Once one maximises atrial over this space, one is left with B a function of two variables which geometrically are the components of the Reeb vector field. Rather surprisingly, the functions atrial and 1/Z are then identically equal6 . This of course explains why maximising a in the field theory is the same as minimising Z in the geometry. Further work on the relation between a–maximisation and Z –minimisation has appeared in [35–37]. 1.2. Outline. The main result of the present work is to extend to general Sasaki–Einstein manifolds the toric results obtained in [22]. This was initially a technical problem – some of the methods described above simply do not extend when X is not toric. However, in the process of solving this problem, we will also gain further insight into the results of [22]. 5 In [22] the issue of existence of this metric was not addressed. However, the real Monge–Ampère equation derived in [22] has recently been shown to always admit a solution [34], thus solving the existence problem for toric Sasaki–Einstein manifolds. 6 This fact was also observed by two of us (D.M. and J.F.S.) in unpublished work.
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We begin by fixing a complex manifold X , which is topologically a real cone over a compact manifold L. Thus X 0 = R+ × L, where r > 0 is a coordinate on R+ , and r = 0 is always an isolated singular point of X , unless L is a sphere. For most of the paper it will be irrelevant whether we are referring to X 0 or the singular cone X . This is because we shall mainly be interested in the Sasakian geometry of the link L – the embedding into X is then purely for convenience, since it is generally easier to work with the Kähler geometry of the cone. Since we are interested in Ricci–flat Kähler metrics, we certainly require the canonical bundle of X 0 to be trivial. We implement this by assuming7 we have a nowhere vanishing holomorphic (n, 0)–form on X 0 . By definition the singularity X is therefore Gorenstein. We also require there to be a space of Kähler cone metrics on X . The space of orbits of every homothetic vector field r ∂/∂r in this space is required to be diffeomorphic to L. The closure of the orbits of the corresponding Reeb vector field ξ = J (r ∂/∂r ) defines some torus Tm ⊂ Aut(X ) since ξ is holomorphic. Thus, as for the toric geometries above, we fix a (maximal) torus Ts ⊂ Aut(X ) and assume that it acts isometrically on our space of Kähler cone metrics on X . The Reeb vector fields in our space of metrics are all required to lie in the Lie algebra of this torus. Note that there is no loss of generality in making these assumptions: the Reeb vector field for a Sasaki–Einstein metric defines some torus that acts isometrically: by going “off–shell” and studying a space of Sasakian metrics on which this torus (or a larger torus containing this) also acts isometrically, we shall learn rather a lot about Sasaki–Einstein manifolds, realised as critical points of the Einstein–Hilbert action on this space of metrics. The first result is that the Einstein–Hilbert action S on L, restricted to the space of Sasakian metrics, is essentially just the volume functional vol[L] of L. More precisely, we prove that S = 4(n − 1)(1 + γ − n)vol[L],
(1.8)
where one can show that, for any Kähler cone metric, there exists a gauge in which is homogeneous degree γ under r ∂/∂r , where γ is unique. Given any homothetic vector field r ∂/∂r , cr ∂/∂r is another homothetic vector field for a Kähler cone metric on X , where c is any positive constant8 . Setting to zero the variation of (1.8) in this direction gives γ = n, since vol[L] is homogeneous degree −n under this scaling. Thus we may think of S as the volume functional: S = 4(n − 1)vol[L],
(1.9)
provided we consider only metrics for which is homogeneous degree n under r ∂/∂r . This condition is the generalisation of the constraint b1 = n in the context of toric geometries [22]. The next result is that the volume of the link L depends only on the Reeb vector field ξ , and not on the remaining degrees of freedom in the metric. The first and second derivatives of this volume function are computed in Sect. 3: dvol[L](Y ) = −n η(Y )dµ, L 2 d vol[L](Y, Z ) = n(n + 1) η(Y )η(Z )dµ . (1.10) L 7 so defined is far from unique – one is always free to multiply by a nowhere vanishing holomorphic
function. This is an important difference to the case of compact Calabi–Yau manifolds. This degree of freedom can be fixed by imposing a “homogeneous gauge” for , as we discuss later. 8 This is a transverse homothety, in the language of Boyer and Galicki [38].
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Here Y, Z are holomorphic Killing vector fields in ts , the Lie algebra of the torus Ts , and η(Y ) denotes the contraction of Y with the one–form η, the latter being dual to the Reeb vector field. In particular, note that the second equation shows that vol[L] is strictly convex – one may use this to argue uniqueness of critical points. We shall return to discuss the first equation in detail later. Note that, when the background (L , g L ) is Sasaki–Einstein, the right hand sides of these formulae essentially appeared in [36]. In this context these formulae arose from Kaluza–Klein reduction on AdS5 × L. In particular, we see that the first derivative dvol[L](Y ) is proportional to the coefficient τ RY of a two–point function in the CFT, via the AdS/CFT correspondence. This relates the geometric problem considered here to τ –minimisation [39] in the field theory. Since the torus Ts acts isometrically on each metric, there is again a moment map and a fixed convex rational polyhedral cone C ∗ ⊂ t∗s . Any Reeb vector field must then lie in the interior of the dual cone C to C ∗ . The space of Reeb vector fields under which has charge n form a convex polytope in C0 – this is formed by the hyperplane b1 = n in the toric case [22]. The boundary of C is a singular limit, since ξ develops a fixed point set there. We again write the Reeb vector field as ξ=
s i=1
bi
∂ , ∂φi
(1.11)
where ∂/∂φi generate the torus action. The volume of the link is then a function vol[L] : C0 → R .
(1.12)
At this point, the current set–up is not dissimilar to that in our previous paper [22] – essentially the only difference is that the torus Ts no longer has maximal possible dimension s = n. The crucial point is that, for s < n, the volume function (1.12) is no longer given just in terms of the combinatorial data specifying C ∗ . It should be clear, for example by simply restricting the toric case to a subtorus, that this data is insufficient to determine the volume as a function of b. The key step to making progress in general is to write the volume functional of L in the form 1 2 vol[L] = n−1 e−r /2 eω . (1.13) 2 (n − 1)! X The integrand in (1.13) may be interpreted as an equivariantly closed form, since r 2 /2 is precisely the Hamiltonian function for the Reeb vector field ξ . The right-hand side of (1.13) takes the form of the Duistermaat–Heckman formula [40,41]. This may then be localised with respect to the Reeb vector field ξ . Our general formula is: 1 vol[L](b) V (b) ≡ = vol[S 2n−1 ] dF {F}
R F λ=1
⎡ ⎤−1 ca (Eλ ) 1 ⎣ ⎦ . (b, u λ )n λ (b, u λ )a
(1.14)
a≥0
Since ξ vanishes only at the tip of the cone r = 0, the right hand side of (1.14) requires one to resolve the singular cone X – the left-hand side is of course independent of the choice of resolution. This resolution can always be made, and any equivariant (orbifold)
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resolution will suffice9 . The first sum in (1.14) is over connected components of the fixed point set of the Ts action on the resolved space W . The u λ are weights of the Ts action on the normal bundle to each connected component F of fixed point set. These essentially enter into defining the moment cone C ∗ . The ca (Eλ ) are Chern classes of the normal bundle to F. The term d F denotes the order of an orbifold structure group – these terms are all equal to 1 when the resolved space W is completely smooth. Precise definitions will appear later in Sect. 5. We also note that the right-hand side of (1.14) is homogeneous degree −n in b, precisely as in the toric case, and is manifestly a rational function of b with rational coefficients, since the weight vectors u λ and Chern classes ca (Eλ ) (and 1/d F ) are generally rational. From this formula for the volume, which recall is essentially the Einstein–Hilbert action, it follows immediately that the volume of a Sasaki–Einstein manifold, relative to that of the round sphere, is an algebraic number. When the complex dimension n = 3, this result is AdS/CFT “dual” to the fact that the central charges of four dimensional superconformal field theories are indeed algebraic numbers. When X is toric, that is the torus action is n–dimensional, the formula (1.14) simplifies and reduces to a sum over the fixed points of the torus action over any toric resolution of the Kähler cone: V (b) =
n pA
1 . A) (b, u i ∈P i=1
(1.15)
Here p A are the vertices of the polytope P of the resolved toric variety – these are the very same vertices that enter into the topological vertex in topological string theory. The u iA ∈ Zn , i = 1, . . . , n, are the n primitive edge vectors that describe the A−th vertex. The right-hand side of formula (1.15) is of course necessarily independent of the choice of resolving polytope P, in order that this formula makes sense. It is a non– trivial fact that (1.15) is equivalent to the previous toric formula for the volume (1.7). For instance, the number of terms in the sum in (1.15) is given by the Euler number of any crepant resolution – that is, the number of gauge groups of the dual gauge theory; while the number of terms in (1.7) is D – that is, the number of facets of the polyhedral cone. However, it can be shown that (1.15) is finite everywhere in the interior C0 of the polyhedral cone C, and has simple poles at the facets of C, precisely as the expression10 (1.7). These formulae show that the volumes of general Sasakian manifolds, as a function of the Reeb vector field, are topological. For toric geometries, this topological data is captured by the normal vectors va that define C ∗ . For non–toric geometries, there are additional Chern classes that enter. In fact, we will also show that these formulae may be recovered from a particular limit of an equivariant index on X , which roughly counts holomorphic functions according to their charges under Ts . Specifically, we define C(q, X ) = Tr{q | H0 (X )} .
(1.16)
9 We may in general resolve X , in an equivariant manner, by blowing up the Fano orbifold V associated to any quasi–regular Kähler cone structure on X , as we shall explain later. It is interesting to note that, when constructing the gauge theory that lives on D3–branes probing the conical singularity, one also makes such a resolution. Specifically, an exceptional collection of sheaves on V may then, in principle, be used to derive the gauge theory (see e.g. [42,43]). 10 This follows from the fact that the Duistermaat–Heckman formula reduces to the characteristic function [44] of the cone (see also [45]), as we will show later.
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Here q ∈ (C∗ )s lives in (a subspace of) the algebraic torus associated to Ts , and the notation denotes a trace of the induced action of this torus on the space of holomorphic functions on X 11 . This equivariant index is clearly a holomorphic invariant, and the volume of the corresponding Sasakian link will turn out to appear as the coefficient of the leading divergent term of this index in a certain expansion: V (b) = lim t n C(q = exp(−tb), X ). t→0
(1.17)
The character C(q, X ) has a pole of order n at qi = 1, i = 1, . . . , n, and this limit picks out the leading behaviour near this pole. For regular Sasaki–Einstein manifolds this relation, with critical b = b∗ , was noted already in [46] – the essential difference here is that we interpret this relation as a function of b, by using the equivariant index rather than just the index. This result is again perhaps most easily described in the toric setting. In this case, the equivariant index counts holomorphic functions on X weighted by their U (1)n charges. It is known that these are in one–to–one correspondence with integral points inside the polyhedral cone, the U (1)n charges being precisely the location of the lattice points in SC ∗ = Zn ∩ C ∗ . In a limit in which the lattice spacing tends to zero, the distribution of points gives an increasingly better approximation to a volume measure on the cone. The slightly non–obvious point is that this measure in fact reduces to the measure on the Sasakian link L, giving (1.17). From a physical viewpoint, the equivariant index12 is counting BPS mesonic operators of the dual gauge theory, weighted by their U (1)s charges. This is because the set of holomorphic functions on the Calabi–Yau cone correspond to elements of the chiral ring in the dual gauge theory. In [47,48] some indices counting BPS operators of superconformal field theories have been introduced and studied. In contrast to the equivariant index defined here, those indices take into account states with arbitrary spin. On the other hand, the fact that the index considered here is equivariant means that it is twisted with respect to the global flavour symmetries of the gauge theory. Moreover, the equivariant index is a holomorphic invariant, and may be computed without knowledge of the Kaluza–Klein spectrum. Rather interestingly, our results in Sect. 6 may then be interpreted as saying that the trial central charge of the dual gauge theory emerges as an asymptotic coefficient of the generating function of (scalar) BPS operators. It would be very interesting to study in more detail the relation between these results and the work of [47,48]. Let us now return to the expression for the first derivative of vol[L] in (1.10). This is zero for a Sasaki–Einstein manifold, since Sasaki–Einstein metrics are critical points of the Einstein–Hilbert action. Moreover, for fixed Reeb vector field ξ , this derivative is independent of the metric. Thus, dvol[L] is a linear map on a space of holomorphic vector fields which is also a holomorphic invariant and vanishes identically when ξ is the critical Reeb vector field for a Sasaki–Einstein metric. Those readers that are familiar with Kähler geometry will recognise these as properties of the Futaki invariant in Kähler geometry [49]. Indeed, if ξ is quasi–regular, we show that
dvol[L](Y ) = − · F[JV (YV )] . (1.18) 2 Here YV is the push down of Y to the Fano orbifold V , JV is its complex structure tensor, and is the length of the circle fibre. We will define the Futaki invariant F, and review 11 We will not worry too much about where this trace converges, as we are mainly interested in its behaviour near a certain pole. 12 We would like to thank S. Benvenuti and A. Hanany for discussions on this.
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some of its properties, in Sect. 4. Thus the dynamical problem of finding the critical Reeb vector field can be understood as varying ξ such that the transverse Kähler orbifold V , when ξ is quasi–regular, has zero Futaki invariant. This is a well–known obstruction to the existence of a Kähler–Einstein metric on V [49]. This new interpretation of the Futaki invariant places the problem of finding Kähler–Einstein metrics on Fanos into a more general context. For example, the Futaki invariant of the first del Pezzo surface is well– known to be non–zero. It therefore cannot admit a Kähler–Einstein metric. However, the canonical circle bundle over this del Pezzo surface does admit a Sasaki–Einstein metric [10,23] – the Kähler–Einstein metric is only a transverse metric13 . From the point of view of our variational problem, there is nothing mysterious about this: the vector field that rotates the S 1 fibre of the circle bundle is simply not a critical point of the Einstein–Hilbert action. This also leads to a result concerning the isometry group of Sasaki–Einstein manifolds. In particular, we will argue14 that the isometry group of a Sasaki–Einstein manifold is a maximal compact subgroup K ⊃ Ts of the holomorphic automorphism group Aut(X ) of the Kähler cone. The Reeb vector field then lies in the centre of the Lie algebra of K . This gives a rigorous account of the expectation that flavour symmetries of the field theory must be realised as isometries of the dual geometry, and that the R–symmetry does not mix with non–abelian flavour symmetries. Let us conclude this outline with an observation on the types of algebraic numbers that arise from the volume minimisation problem studied here. We have shown that all Sasaki–Einstein manifolds have a normalised volume, relative to the round sphere, which is an algebraic number, that is the (real) root of a polynomial over the rationals. Let us say that the degree, denoted deg(L), of a Sasaki–Einstein manifold L is the degree of this algebraic number. Thus, for instance, if deg(L) = 2 the normalised volume is quadratic irrational. Recall that the rank of a Sasaki–Einstein manifold is the dimension of the closure of the orbits of the Reeb vector field. We write this as rank(L). We then make the following conjecture: for a Sasaki–Einstein manifold L of dimension 2n − 1, the degree and rank are related as follows: deg(L) = (n − 1)rank(L)−1 .
(1.19)
For example, all quasi–regular Sasaki–Einstein manifolds have degree one since the normalised volume is a rational number. By definition they also have rank one. In all the irregular cases that we have examined, which by now include a number of infinite families, this relation holds. Although we have obtained explicit expressions for the volume function V (b), it seems to be a non–trivial fact that one obtains algebraic numbers of such low degree obeying (1.19) from extremising this function – a priori, the degree would seem to be much larger. It would be interesting to investigate this further, and prove or disprove the conjecture. 2. Sasakian Geometry In this section we present a formulation of Sasakian geometry in terms of the geometry of Kähler cones. This way of formulating Sasakian geometry, although equivalent to the original description in terms of metric contact geometry, turns out to be more natural for 13 In general, we may define the Futaki invariant for transverse metrics by (1.18). 14 We will give a rigorous proof only for quasi–regular structures.
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describing the problems of interest here. A review of Sasakian geometry, where further details may be found, is contained in [8]. A central fact we use here, that will be useful for later computations, is that the radial coordinate r determines not only the link L in X , but also that r 2 may be interpreted as the Kähler potential for the Kähler cone [22]. A choice of Sasakian metric on L, for fixed complex structure J on X , requires a choice of Reeb vector field ξ = J (r ∂/∂r ) and a choice of transverse Kähler metric. We also discuss moment maps for torus actions on the cone. By a result of [50], the image of the moment map is a convex rational polyhedral cone C ∗ in the dual Lie algebra t∗s of the torus, provided ξ ∈ ts . Here we identify elements of the Lie algebra with the corresponding vector fields on X (or L). We show that the space of Reeb vector fields in ts lies in the interior C0 of the dual cone to C ∗ . Finally, we discuss the existence of certain Killing spinors on Sasakian manifolds and their relation to the Sasakian structure. We will make use of some of these formulae in later sections. 2.1. Kähler cones. A Sasakian manifold is a compact Riemannian manifold (L , g L ) whose metric cone (X, g X ) is Kähler. Specifically, g X = dr 2 + r 2 g L ,
(2.1)
where X 0 = {r > 0} is diffeomorphic to R+ × L = C(L). We also typically take L to be simply–connected. The Kähler condition on (X, g X ) means that, by definition, the holonomy group of (X, g X ) reduces to a subgroup of U (n), where n = dimC X . In particular, this means that there is a parallel complex structure J , ∇ X J = 0,
(2.2)
where ∇ X is the Levi–Civita connection of (X, g X ). We refer to the vector field r ∂/∂r as the homothetic vector field. The Reeb vector field is defined15 to be ∂ . (2.3) ξ=J r ∂r A straightforward calculation shows that r ∂/∂r and ξ are both holomorphic. Moreover, ξ is Killing. A proof of these statements may be found in Appendix A. Provided L is not locally isometric to the round sphere16 , Killing vector fields on the cone X are in one–to–one correspondence with Killing vector fields on the link L. Since L is compact, the group of isometries of (L , g L ) is a compact Lie group. Since all holomorphic Killing vector fields on (X, g X ) arise from Killing vector fields on the link L, they therefore commute with r ∂/∂r and thus also commute with ξ = J (r ∂/∂r ). Since ξ is itself Killing, it follows that ξ lies in the centre of the Lie algebra of the isometry group. 15 An alternative definition using spinors, perhaps more familiar to physicists, will be given later in Subsect. 2.6. 16 If L is locally isometric to the round sphere then Killing vector fields on the cone (X, g ) may be X constructed from solutions to Obata’s equation [51], which in turn relates to conformal Killing vector fields on the link L.
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We now define the 1–form on X , 1 dr = 2 g X (ξ, ·). η=J r r
(2.4)
This is the contact form of the Sasakian structure when pulled back to the link L via the embedding i : L → X
(2.5)
that embeds L in X at r = 1. Note that η is homogeneous degree zero under r ∂/∂r . In terms of the dc and ∂ operator on X we have η = J d log r = dc log r = i(∂¯ − ∂) log r,
(2.6)
dη = 2i∂ ∂¯ log r.
(2.7)
and thus
We may now write the metric g X on X as g X = dr 2 + r 2 (η ⊗ η + gT ),
(2.8)
where one can show that gT is a Kähler metric on the distribution orthogonal to the span of r ∂/∂r and ξ . The corresponding transverse Kähler form is easily computed to be ωT =
1 dη. 2
(2.9)
The Kähler form on X is thus ω = ωX =
1 2 d(r η). 2
(2.10)
In particular ω is exact due to the homothetic symmetry generated by r ∂/∂r . We may rewrite (2.10) as ω=
1 c 2 1 ¯ 2 dd r = i∂ ∂r . 4 2
(2.11)
The function r 2 thus serves a dual purpose: it defines the link L = X |r =1 and is also the Kähler potential that defines the metric. Note from (2.11) that any holomorphic vector field χ which is tangent to L, dr (χ ) = 0, is automatically Killing. Here we have used the notation α(χ ) for the pairing between a 1–form α and vector field χ . Conversely, recall that all Killing vector fields on (X, g X ) are tangent to L. Thus for any holomorphic Killing vector field Y we have17 LY η = 0.
(2.12)
A Riemannian manifold (X, g X ) is a cone if and only if the metric takes the form (2.1). We end this subsection by reformulating this in terms of the Kähler form ω on X when (X, J, g X ) is a Kähler cone. Thus, let (X 0 , J ) be a complex manifold, with a 17 In this paper we will use extensively the Lie derivative L along vector fields Y . It is useful to recall Y the standard formula for Lie derivatives acting on a differential form α: LY α = d(Y α) + Y dα. Note in µ components we have (Y α)µ1 ...µ p−1 ≡ Y p αµ p µ1 ...µ p−1 .
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diffeomorphism onto R+ × L with r > 0 a coordinate on R+ . We require that r ∂/∂r be holomorphic with respect to J . We may then simply define ω in terms of r by Eq. (2.11). One must also ensure that the corresponding metric g = ω(·, J ·) is positive definite – typically we shall use this formulation only for infinitesimal deformations around some fixed background Sasakian metric and thus this will not be an issue. Since r ∂/∂r is holomorphic, and ω is clearly homogeneous of degree two under r ∂/∂r , the metric g will be also homogeneous degree two under r ∂/∂r . However, this is not sufficient for g to be a cone – one also requires g(r ∂/∂r, Y ) = 0 for all vector fields Y tangent to L, dr (Y ) = 0. Equivalently, ω(ξ, Y ) = 0, where we define ξ = J (r ∂/∂r ). It is simple to check that the necessary condition Lξ r = dr (ξ ) = 0
(2.13)
that ξ is tangent to L is also sufficient for g = ω(·, J ·) to be a cone. It follows now that ∂/∂r has unit norm and that the metric g is a Kähler metric which is a cone of the form (2.1).
2.2. The Calabi–Yau condition. So far we have not fixed any Calabi–Yau condition on (X, J ). We are interested in finding a Ricci–flat Kähler metric on X , and thus we certainly require c1 (X 0 ) = 0. We may impose this by assuming that there is a nowhere vanishing holomorphic section of n,0 X 0 . In particular, is then closed d = 0.
(2.14)
One can regard the (n, 0)–form as defining a reduction of the structure group of the tangent bundle of X 0 from G L(2n; R) to S L(n; C). The corresponding almost complex structure is then integrable if (2.14) holds. The conical singularity X , including the isolated singular point r = 0, is then by definition a Gorenstein singularity. On a compact manifold, such an is always unique up to a constant multiple. However, when X is non–compact, and in particular a Kähler cone, so defined is certainly not unique – one is free to multiply by any nowhere zero holomorphic function on X , and this will also satisfy (2.14). However, for a Ricci–flat Kähler cone, with homothetic vector field r ∂/∂r , we may always choose18 such that Lr ∂/∂r = n.
(2.15)
In fact, one may construct this as a bilinear in the covariantly constant spinor on X , as we recall in Sect. 2.6. In Sect. 2.7 we show that, for any fixed Kähler cone metric – not necessarily Ricci–flat – with homothetic vector field r ∂/∂r , one can always choose a gauge for in which it is homogeneous degree γ under r ∂/∂r , with γ a unique constant. Then (2.15) will follow from varying the Einstein–Hilbert action on the link L, as we show in Sect. 3.1. However, until Sect. 2.7, we fix (X, ) together with a space of Kähler cone metrics on X such that satisfies (2.15) for every metric. This is then unique up to a constant multiple19 . Given such an , for any Kähler form ω on X there exists 18 For the toric geometries studied in [22], this condition is equivalent to b = n, as can be seen by writing 1
n ∂ ) and using the explicit form of given in [22]. r ∂/∂r = − i=1 bi J ( ∂φ i 19 To see this, pick a quasi–regular r ∂/∂r . Any other such holomorphic (n, 0)–form is α, where α is a
nowhere zero holomorphic function on X . Since α is degree zero under r ∂/∂r , it descends to a holomorphic function on V , where V is the space of orbits of ξ on L. Since V is compact, α is constant.
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a real function f on X such that in 1 ¯ = exp( f ) ωn . (−1)n(n−1)/2 ∧ n 2 n!
(2.16)
A Ricci–flat Kähler metric with Kähler form ω of course has f constant. 2.3. The Reeb foliation. The vector field ξ restricts to a unit norm Killing vector field on L, which by an abuse of notation we also denote by ξ . Since ξ is nowhere–vanishing, its orbits define a foliation of L. There is then a classification of Sasakian structures according to the global properties of this foliation: • If all the orbits close, ξ generates a circle action on L. If, moreover, the action is free the Sasakian manifold is said to be regular. All the orbits have the same length, and L is the total space of a principal circle bundle π : L → V over a Kähler manifold V . This inherits a metric gV and Kähler form ωV , where gV is the push–down to V of the transverse metric gT . • More generally, if ξ generates a U (1) action on L, this action will be locally free, but not free. The Sasakian manifold is then said to be quasi–regular. Suppose that x ∈ L is a point which has some non–trivial isotropy subgroup x ⊂ U (1). Thus x ∼ = Zm for some positive integer m. The length of the orbit through x is then 1/m times the length of the generic orbit. The orbit space is naturally an orbifold, with L being the total space of an orbifold circle bundle π : L → V over a Kähler orbifold V . Moreover, the point x descends to a singular point of the orbifold with local orbifold structure group Zm . • If the generic orbit of ξ does not close, the Sasakian manifold is said to be irregular. In this case the generic orbits are diffeomorphic to the real line R. Recall that the isometry group of (L , g L ) is a compact Lie group. The orbits of a Killing vector field define a one–parameter subgroup, the closure of which will always be an abelian subgroup and thus a torus. The dimension of the closure of the generic orbit is called the rank of the Sasakian metric, denoted rank(L , g L ). Thus irregular Sasakian metrics have rank > 1. A straightforward calculation gives Ric(g X ) = Ric(g L ) − (2n − 2)g L = Ric(gT ) − 2ngT
(2.17)
and thus in particular ρ = ρT − 2nωT ,
(2.18)
where ρT denotes the transverse Ricci–form. We also have20 ρ = i∂ ∂¯ log 2g X ,
(2.19)
1 ¯ The Ricci–potential for the Kähler cone is where we have defined 2g X = n! . 2 thus log g X . Since we assume that is homogeneous degree n under r ∂/∂r , this is 20 Note that multiplying by a nowhere zero holomorphic function α on X leaves the right hand side of (2.19) invariant.
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homogeneous degree zero, i.e. it is independent of r , and hence is the pull–back under p ∗ of a global function on L. Here p:X→L
(2.20)
projects points (r, x) ∈ R+ × L onto x ∈ L. Moreover, since Lξ = ni and ξ is Killing it follows that the Ricci–potential is basic with respect to the foliation defined by ξ . Recall that a p–form α on L is said to be basic with respect to the foliation induced by ξ if and only if Lξ α = 0,
ξ α = 0.
(2.21)
Thus α has no component along g(ξ, ·) and is independent of ξ . It is straightforward to check that the transverse Kähler form ωT , and its Ricci–form, are also basic. Suppose now that (L , g L ) is quasi–regular21 . Thus the space of orbits of ξ is a compact complex orbifold V . The transverse Kähler and Ricci forms push down to ωV , and ρV on V , respectively. Thus (2.18) may be interpreted as an equation on V . The left-hand side is i∂ ∂¯ exact on the orbifold V , and hence [ρV ] − 2n[ωV ] = 0 ∈ H 2 (V ; R).
(2.22)
In particular this shows that V is Fano, since c1 (V ) = [ρV /2π ] is positive. Note that η satisfies dη = 2π ∗ (ωV ). The Kähler class of V is then proportional to the first Chern class of the orbifold circle bundle π : L → V . By definition, the orbifold is thus Hodge. We denote the associated orbifold complex line bundle over V by L. Note from (2.22) that, since [ωV ] is proportional to the anti–canonical class c1 (V ) of V , the orbifold line bundle L is closely related to the canonical bundle K → V over V . To see this more clearly, let U ⊂ V denote a smooth open subset of V over which L trivialises. We may then introduce a coordinate ψ on the circle fibre of π : L |U → U such that on L |U = π −1 (U ) we have η = dψ + π ∗ (σ ),
(2.23)
where σ is a one–form on U with dσ = 2ωV . Note that, although this cannot be extended to a one–form on all of V , η is globally defined on L – one may cover V by open sets, and the σ and ψ are related on overlaps by opposite gauge transformations. From Eq. (2.22), we see that if ψ ∼ ψ + 2π/n, then X is the total space of the canonical complex cone over V – that is, the associated line bundle L to π is L = K. More generally, we may set ψ ∼ψ+
2πβ ; n
−c1 (L) =
c1 (V ) 2 (V ; Z). ∈ Horb β
(2.24)
∗ (V ; Z) Then Lβ ∼ = K. Here we have introduced the integral orbifold cohomology Horb of Haefliger, such that orbifold line bundles are classified up to isomorphism by 2 c1 (L) ∈ Horb (V ; Z).
(2.25)
This reduces to the usual integral cohomology when V is a manifold. The maximal integer β in (2.24) is called the Fano index of V . If L is simply–connected, then c1 (L) 21 We include the regular case when V is a manifold in this terminology.
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2 (V ; Z) and β is then equal to the index of V . For example, if V = is primitive22 in Horb n−1 2 CP , then H (V ; Z) ∼ = Z. The line bundle L with c1 (L) = −1 ∈ H 2 (V ; Z) ∼ =Z 2n−1 gives L = S , the complex cone being isomorphic to L = O(−1) → CP n−1 with the zero section contracted. Note that the total space of L is the blow–up of Cn at the origin. On the other hand, the canonical bundle is K = O(−n) → CP n−1 which gives the link S 2n−1 /Zn . Thus the index is equal to n. From the Kodaira–Bailey embedding theorem, (V, gV ), with the induced complex structure, is necessarily a normal projective algebraic variety [8]. Let T → V be the orbifold holomorphic line bundle over V with first Chern class c1 (V )/Ind(V ), with Ind(V ) being the index of V . In particular, T is ample and has a primitive first Chern class. By the Kodaira–Bailey embedding theorem, for k ∈ N sufficiently large, T k defines an embedding of V into CP N −1 via its space of global holomorphic sections. Thus, a basis sα , α = 1, . . . , N , of H 0 (V ; T k ) may be regarded as homogeneous coordinates on CP N −1 , with V p → [s1 ( p), . . . , s N ( p)] being an embedding. The image is a projective algebraic variety, and thus the zero locus of a set of homogeneous polynomials { f A = 0} in the homogeneous coordinates. If H = O(1) denotes the hyperplane bundle on CP N −1 then its pull–back to V is of course isomorphic to T k . On the other hand, as described above, L∗ ∼ = T Ind(V )/β for some positive β, where recall Ind(V ) −1 ∼ that T = K is the anti–canonical bundle. By taking the period β = Ind(V )/k in the above it follows that the corresponding cone X is the affine algebraic variety defined by { f A = 0} ⊂ C N . The maximal value of β, given by β = Ind(V ), is then a k–fold cover of this X . Moreover, by our earlier assumptions, X constitutes an isolated Gorenstein singularity. We note that there is therefore a natural (orbifold) resolution of any (X, J ) equipped with a quasi–regular Kähler cone metric: one simply takes the total space of the orbifold complex line bundle L → V . The resulting space W has at worst orbifold singularities, and W \ V ∼ = X 0 is a biholomorphism. Thus W is birational to X . One might be able to resolve the cone X completely, but the existence of a resolution with at worst orbifold singularities will be sufficient for our needs later.
2.4. Transverse Kähler deformations. In order to specify a Sasakian structure on L, one clearly needs to give the Reeb vector field ξ . By embedding L as a link in a fixed non–compact X , this is equivalent to choosing a homothetic vector field r ∂/∂r on X . Having determined this vector field, the remaining freedom in the choice of Sasakian metric consists of transverse Kähler deformations. Suppose that we have two Kähler potentials r 2 , r˜ 2 on X such that their respective homothetic vector fields coincide: ∂ ∂ = r˜ . (2.26) r ∂r ∂ r˜ This equation may be read as saying that r˜ is a homogeneous degree one function under r ∂/∂r , and thus r˜ 2 = r 2 exp φ
(2.27)
for some homogeneous degree zero function φ. Thus φ is a pull–back of a function, that we also call φ, from the link L under p ∗ . We compute 1 1 (2.28) η˜ = dc log r 2 + φ = η + dc φ. 2 2 22 That is, there is no γ ∈ H 2 (V ; Z) and integer m ∈ Z, |m| > 1, such that mγ = c (L). 1 orb
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In order that r˜ defines a metric cone, recall from the end of Sect. 2.1 that we require Lξ r˜ = 0,
(2.29)
which implies that φ is basic with respect to the foliation induced by ξ , Lξ φ = 0. Introducing local transverse CR coordinates, i.e. local complex coordinates on the transverse space, (z i , z¯ i ) on L, one thus has φ = φ(z i , z¯ i ). Note also that ¯ dη˜ = dη + i∂ ∂φ,
(2.30)
˜ ωT = (1/2)dη differ precisely by a so that the transverse Kähler forms ω˜T = (1/2)dη, transverse Kähler deformation. When the Sasakian structure is quasi–regular, φ pushes down to a global function on the orbifold V , and transformations of the metric of the form (2.30) are precisely those preserving the Kähler class [ωV ] ∈ H 2 (V ; R). 2.5. Moment maps. In this subsection we consider a space of Kähler cone metrics on X such that each metric has isometry group containing a torus Ts . Moreover, the flow of the Reeb vector field is assumed to lie in this torus. For each metric, there is an associated moment map whose image is a convex rational polyhedral cone C ∗ ⊂ t∗s ∼ = Rs . Moreover, these cones are all isomorphic. We show that the space of Reeb vector fields on L is (contained in) the interior C0 ⊂ ts of the dual cone to C ∗ . Suppose then that Ts acts holomorphically on the cone X , preserving a fixed choice of Kähler form (2.11) on X . Let ts denote the Lie algebra of Ts . We suppose that ξ ∈ ts – the torus action is then said to be of Reeb type [52]. Let us introduce a basis ∂/∂φi of vector fields generating the torus action, with φi ∼ φi + 2π . Then we may write ξ=
s i=1
bi
∂ . ∂φi
(2.31)
Since L∂/∂φi ω = 0
(2.32)
and X has b1 (X 0 ) = 0, it follows that for each i = 1, . . . , s there exists a function yi on X such that dyi = − In fact, it is simple to verify that
∂ ω. ∂φi
1 2 ∂ yi = r η 2 ∂φi
(2.33)
(2.34)
is the homogeneous solution. The functions yi may be considered as coordinates on the dual Lie algebra t∗s . This is often referred to as the moment map µ : X → t∗s ,
(2.35)
where for Y ∈ ts we have (Y, µ) =
1 2 r η(Y ). 2
(2.36)
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Under these conditions, [50] proved that the image of X under µ is a convex rational polyhedral cone C ∗ ⊂ t∗s . This is a convex polyhedral cone whose generators are all vectors whose components are rational numbers. The image of the link L = X |r =1 is given by 2(b, y) = 1
(2.37)
as follows by setting Y = ξ in (2.36). The hyperplane (2.37) intersects the cone C ∗ to form a compact polytope (b) = C ∗ ∩ {2(b, y) ≤ 1}
(2.38)
if and only if b lies in the interior of the dual cone to C ∗ , which we denote C ⊂ ts . Note that this analysis is essentially the same as that appearing in the toric context in our previous paper [22]. The only difference is that the torus no longer has maximal possible rank s = n, and thus the cone need not be toric. However, the Euclidean volume of (b) is no longer the volume of the Sasakian metric on L. The cone C is a convex rational polyhedral cone by Farkas’ Theorem. Geometrically, the limit in which the Reeb vector field ξ approaches the boundary ∂C of this cone is precisely the limit in which ξ develops a non–trivial fixed point set on X . Recall that ξ has square norm r 2 on X and thus in particular is nowhere vanishing on X 0 = {r > 0}. Thus the boundary of the cone C is a singular limit of the space of Sasakian metrics on L. To see this, let Fα denote the facets of C ⊂ ts , and let the associated primitive inward pointing normal vectors be u α ∈ t∗s . The u α are precisely the generating rays of the dual cone C ∗ to C. Thus we may exhibit C ∗ ⊂ t∗s ∼ = Rs as
∗ ∗ tα u α ∈ ts | tα ≥ 0 . (2.39) C = α
If ξ ∈ Fα , for some α, then (ξ, u α ) = 0. We may reinterpret this equation in terms of the moment cone C ∗ . Let Rα denote the 1–dimensional face, or ray, of C ∗ generated by the vector u α . The inverse image X α = µ−1 (Rα ) is a Ts –invariant conic symplectic subspace of X [50] and is a vanishing set for the vector field ξ ∈ Fα ⊂ ts . It may help to give a simple example. Thus, let X = Cn . Taking the flat metric gives ω=
n 1 i=1
2
d(ρi2 ) ∧ dφi ,
(2.40)
where (ρi , φi ) are polar coordinates on the i th complex plane of Cn . We have C ∗ = (R+ )n , with coordinates yi = ρi2 /2 ≥ 0, which happens to be isomorphic to its dual cone, C = (R+ )n . This orthant is bounded by n hyperplanes, with primitive inward–pointing j j normals u i = ei , where ei is the i th standard orthonormal basis vector: ei = δi . The ∗ {u i } indeed also generate the moment cone C . The inverse image under the moment map of the ray Ri generated by u i is the subspace X i = {z j = 0 | j = i} ∼ (2.41) = C ⊂ Cn . Any vector field ξ of the form ξ=
j=i
clearly vanishes on X i .
cj
∂ ∂φ j
(2.42)
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2.6. Killing spinors and the (n, 0)–form. We now discuss the existence of certain Killing spinors on Sasakian manifolds, and their relation to the differential forms defining the Sasakian structure which we have already introduced. The use of spinors often provides a quick and elegant method for obtaining various results, as we shall see in Sect. 4. In fact, these methods are perhaps more familiar to physicists. Recall that all Kähler manifolds admit a gauge covariantly constant spinor. More precisely, the spinor in question is in fact a section of a specific spinc bundle that is intrinsically defined on any Kähler manifold. We will not give a complete account of this in the following, but simply make a note of the results that we will need in this paper, especially in Sect. 4. For further details, the reader might consult a number of standard references [53,54]. The spinors on a Sasakian manifold are induced from those on the Kähler cone, again in a rather standard way. This is treated, for Sasaki– Einstein manifolds, in the paper of Bär [55]. The extension to Sasakian manifolds is straightforward. Let (X, J, g) be a Kähler manifold. The bundle of complex spinors S does not necessarily exist globally on X , the canonical example being CP 2 . However, the spinc bundle −1/2
V = S ⊗ KX
(2.43)
always exists. Here K X denotes the canonical bundle of X , which is the (complex line) bundle n,0 (X ) of forms of Hodge type (n, 0) with respect to J . The idea in (2.43) is that, although neither bundle may exist separately due to w2 (X ) ∈ H 2 (X ; Z2 ) being non–zero, the obstructing cocycle cancels out in the tensor product and V exists as a genuine complex vector bundle. The metric on X induces the usual spin connection on S, and the canonical bundle K X inherits a connection one–form with curvature −ρ, where ρ is the Ricci–form on X . Thus the bundle V has defined on it a standard connection form. A key result is that, as a complex vector bundle, V∼ = 0,∗ (X ).
(2.44)
In fact, since X is even dimensional, the spinc bundle V decomposes into spinors of positive and negative chirality, V = V + ⊕ V − and V+ ∼ = 0,even (X ),
V− ∼ = 0,odd (X ).
(2.45)
The connection on V referred to above is then equal to the standard metric–induced connection on 0,∗ (X ). In particular, there is always a covariantly constant section of 0,∗ (X ) – it is just the constant function on X . Via the isomorphism23 (2.44) this gets interpreted as a gauge covariantly constant spinor on X , the gauge connection being −1/2 the standard metric–induced one on K X . We conclude then that there is always a spinor field24 on X satisfying (in a local coordinate patch over which V trivialises) ∇YX −
i A(Y ) = 0, 2
(2.46)
23 This is essentially the same twisting of spinors that occurs on the worldvolumes of D–branes wrapping calibrated submanifolds. 24 By an abuse of terminology we refer to sections of V as “spinors”.
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where Y is any vector field on X and ∇ X denotes the spin connection on (X, g). The −1/2 connection one–form A/2 on K X satisfies, as mentioned above, d A = ρ.
(2.47)
For our Calabi–Yau cone25 (X, ), K X is of course topologically trivial by definition. Thus, topologically, is a genuine spinor field. However, unless the metric is Ricci– flat, the connection form A will be non–zero. The reduction of to a spinor on the link L = X |r =1 is again rather standard [55]. Either spinor bundle V + |r =1 or V − |r =1 is isomorphic to the spinor (or rather spinc ) bundle on L. By writing out the spin connection on the cone in terms of that on the link L, one easily shows that there is a spinor θ = |r =1 on L satisfying i i ∇YL θ − Y · θ − A(Y )θ = 0. 2 2
(2.48)
Here Y · θ denotes Clifford multiplication: Y · θ = Y µ γµ θ , and γµ generate the Clifford algebra Cliff(2n − 1, 0). Thus {γµ , γν } = 2g L µν . It is now simple to check from (2.48) that θ has constant norm, and we normalise it so that θ¯ θ = 1. Then the contact one–form η on L is given by the bilinear ¯ (1) θ. η = θγ
(2.49)
Note that one may define the Reeb vector field as the dual of the contact one–form η. µν Thus, in components, ξ µ = g L ην . It is then straightforward to check, using (2.48), that L ∇(µ ην) = 0, so that ξ is indeed a Killing vector field. One also easily verifies, again using (2.48), that dη = −2i θ¯ γ(2) θ = 2ωT .
(2.50)
We may define an (n, 0)–form K on X as a bilinear in the spinor , namely ¯ c γ(n) . K =
(2.51)
It is important to note that this is different from the holomorphic (n, 0)–form on X we introduced earlier. The two are of course necessarily proportional, and in fact are related by = exp ( f /2)K .
(2.52)
Here f is the same function as that appearing in (2.16). Indeed, we can write the Ricci– form on X as ρ = i∂ ∂¯ f , so that we may take A=
1 c d f. 2
(2.53)
Equivalently, we have f = log 2g X .
(2.54)
From (2.46) we have, as usual on a Kähler manifold, dK = i A ∧ K =
i c d f ∧ K. 2
25 Strictly, one should write X in most of what follows. 0
(2.55)
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D. Martelli, J. Sparks, S.-T. Yau
Then d =
1 (d f + idc f ) ∧ = ∂ f ∧ = 0. 2
(2.56)
In an orthonormal frame eµ , µ = 1, . . . , 2n, for (X, g) note that ω = e1 ∧ e2 + · · · + e2n−1 ∧ e2n , K = (e1 + ie2 ) ∧ · · · ∧ (e2n−1 + ie2n ),
(2.57)
so that in 1 (−1)n(n−1)/2 K ∧ K¯ = ωn . n 2 n!
(2.58)
The relation (2.52) is then consistent with the normalisation in (2.16). Finally, we introduce the space of Reeb vector fields under which has charge n. Thus, we define = ξ ∈ C0 ⊂ ts | Lξ = in . (2.59) Clearly, if ξ ∈ is fixed, any other ξ ∈ is given by ξ = ξ + Y , where is uncharged with respect to Y : LY = 0.
(2.60)
The space of all Y satisfying (2.60) forms a vector subspace of ts . Moreover, this subspace has codimension one, so that the corresponding plane through ξ forms a finite polytope with C. Thus is an (r − 1)–dimensional polytope. In the toric language of [22], this is just the intersection of the plane b1 = n with Reeb polytope C. 2.7. The homogeneous gauge for . Suppose that X 0 = R+ × L is a complex manifold admitting a nowhere vanishing holomorphic section of n,0 (X 0 ). Recall that, in contrast to the case of compact X , is unique only up to multiplying by a nowhere vanishing holomorphic function. Suppose moreover that g X is a quasi–regular Kähler cone metric on X , with homothetic vector field r ∂/∂r . The aim in this section is to prove that there always exists a “gauge” in which the holomorphic (n, 0)–form is homogeneous of constant degree γ ∈ R under r ∂/∂r , where γ is unique. We shall use this result in Sect. 3.1 to argue that γ = n arises by varying the Einstein–Hilbert action of the link. Since r ∂/∂r is a holomorphic vector field Lr ∂/∂r = κ ,
(2.61)
where κ is a holomorphic function. We must then find a nowhere zero holomorphic function α and a constant γ such that Lr ∂/∂r log α = γ − κ, since then ≡ α is homogeneous degree γ . Let us expand γ −κ = ak , k≥0
(2.62)
(2.63)
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where ak are holomorphic functions of weight k under the U (1) action generated by ξ = J (r ∂/∂r ). Roughly, we are doing a Taylor expansion in the fibre of L∗ → V , where V is the Kähler orbifold base defined by the quasi–regular Reeb vector field ξ . Here L is the associated complex line orbifold bundle to U (1) → L → V . The ak , in their V dependence, may be considered as sections of (L∗ )k → V . If ∂/∂ν, ν ∼ ν + 2π rotates the fibre of L with weight one, then we have ∂ = hξ ∂ν
(2.64)
k Lξ ak = i ak . h
(2.65)
for some positive constant h. Thus
Eq. (2.62) is then straightforward to solve: log α =
h k≥1
k
ak + log δ.
(2.66)
Here we have used the fact that each ak is holomorphic. Moreover, δ is holomorphic of homogeneous degree a0 , where a 0 = γ − κ0
(2.67)
and the constant κ0 is the degree zero part of κ. In order that α be nowhere vanishing, we now require a0 = 0. This is because δ is homogeneous of fixed degree, and thus corresponds to a section of (L∗ )m → V , where a0 = m/ h. However, since (L∗ )m is a non–trivial bundle for m = 0, any section must vanish somewhere, unless m = 0. Thus a0 = 0, which in fact fixes γ uniquely because of the latter argument. Finally, the resulting expression (2.66) for α is clearly nowhere vanishing, holomorphic, and satisfies (2.62). This completes the proof. 3. The Variational Problem In this section we show that the Einstein–Hilbert action on L, restricted to the space of Sasakian metrics on L, is essentially the volume functional of L. Moreover, the volume depends only on the choice of Reeb vector field, and not on the remaining degrees of freedom in the metric. We give general formulae for the first and second variations, in particular showing that the volume of L is a strictly convex function. The derivations of these formulae, which are straightforward but rather technical, are relegated to Appendix C. The first variation will be related to the Futaki invariant for quasi–regular Sasakian metrics in Sect. 4. From the second variation formula it follows that there is a unique critical point of the Einstein–Hilbert action in a given Reeb cone ⊂ C0 ⊂ ts . 3.1. The Einstein–Hilbert action. As is well–known, a metric g L on L satisfying the Einstein equation Ric(g L ) = (2n − 2)g L
(3.1)
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D. Martelli, J. Sparks, S.-T. Yau
is a critical point of the Einstein–Hilbert action S : Met(L) → R given by
(3.2)
S[g L ] =
[s(g L ) + 2(n − 1)(3 − 2n)] dµ,
(3.3)
L
where s(g L ) is the scalar curvature of g L , and dµ is the associated Riemannian measure. We would like to restrict S to a space of Sasakian metrics on L. Recall that we require X 0 = R+ × L to be Calabi–Yau, meaning that there is a nowhere vanishing holomorphic (n, 0)–form . At this stage, we are not assuming that obeys any additional property. The Ricci–form on X is then ρ = i∂ ∂¯ f , where f = log 2g X .
(3.4)
Note that of course the Ricci–form is independent of multiplying by any nowhere zero holomorphic function on X . The scalar curvature of (X, g X ) is (3.5) s(g X ) = Tr g −1 X Ric(g X ) = − X f, where X is the Laplacian on (X, g X ). Using the relation s(g X ) =
1 [s(g L ) − 2(n − 1)(2n − 1)] , r2
(3.6)
one easily sees that S[g L ] = 2(n − 1) (R + 2vol[L]) ,
(3.7)
where R is defined by
ωn =− s(g X ) R= n! r ≤1
r ≤1
X f
ωn . n!
(3.8)
Note that this is independent of the gauge choice for , i.e. it is independent of the choice of nowhere zero holomorphic multiple. However, in order to relate R to an expression on the link, it is useful to impose a homogeneity property on . This can always be done, as we showed in Sect. 2.7. Strictly speaking, we only proved this for quasi–regular Sasakian metrics. However, since the rationals are dense in the reals and S is continuous, this will in fact be sufficient. Thus, let be homogeneous degree γ and r ∂/∂r be quasi–regular. It follows that f satisfies Lr ∂/∂r f = 2(γ − n). Recall that on a cone X =
1 1 ∂ L − 2n−1 r2 r ∂r
∂ r 2n−1 , ∂r
where L is the Laplacian on (L , g L ). We therefore have L f r 2n−3 dr ∧ dµ. R = 2(γ − n)vol[L] − r ≤1
(3.9)
(3.10)
(3.11)
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The integrand in the second term is now homogeneous under r ∂/∂r , so we may perform the r integration trivially. Using Stokes’ theorem on L we conclude that this term is identically zero. Thus we find that S[g L ] = 4(n − 1)(1 + γ − n)vol[L].
(3.12)
Thus the Einstein–Hilbert action on L is related simply to the volume functional on L. Given any homothetic vector field r ∂/∂r , cr ∂/∂r is always26 another homothetic vector field for a Kähler cone metric on X , where c is a positive constant. Since vol[L] is homogeneous degree −n under this scaling, one may immediately extremise (3.12) in this direction, obtaining γ = n.
(3.13)
Note that this is precisely analogous to the argument in our previous paper [22], which sets b1 = n, in the notation there. Thus, provided we restrict to Kähler cone metrics for which γ = n, i.e. there is a nowhere vanishing holomorphic (n, 0)–form of homogeneous degree n, the Einstein–Hilbert action is just the volume functional for the link: S[g L ] = 4(n − 1)vol[L].
(3.14)
This reduces our problem to studying the volume of the link in the remainder of the paper. We next show that S is independent of the choice of transverse Kähler metric. Hence S is a function on the space of Reeb vector fields, or equivalently, of homothetic vector fields. Thus, consider vol[L] : Sas(L) → R.
(3.15)
We may write the volume as
vol[L] =
dµ = 2n vol[X 1 ] = 2n L
r ≤1
ωn , n!
(3.16)
where we define X 1 = X |r ≤1 . Here we have simply written the measure on X in polar coordinates. Note that, since we are regarding r 2 as the Kähler potential, changing the metric also changes the definition of X 1 . Let us now fix a background with Kähler potential r 2 and set r 2 (t) = r 2 exp(tφ),
(3.17)
where t is a (small) parameter and φ is a basic function on L = X |r =1 . Thus dω 1 (t = 0) = ddc (r 2 φ). dt 4
(3.18)
To first order in t, the hypersurface r (t) = 1 is given by 1 r = 1 − tφ. 2 26 That is, the space of homothetic vector fields is itself a cone.
(3.19)
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We now write
vol[L] =
r ≤1
d(r 2n ) ∧ dµ.
(3.20)
The first order variation in vol[L](t) from that of t = 0 contains a contribution from the domain of integration, as well as from the integrand. The former is slightly more subtle. Consider the expression d(r 2n ) ∧ dµ (3.21) 1 r ≤1− 2 tφ
which is vol[L] together with the first order variation due to the change of integration domain. By performing the r integration pointwise over the link L we obtain, to first order in t, (3.22) vol[L] − t n φdµ. L
The total derivative of vol[L] at t = 0 is thus dvol[L] n ωn−1 (t = 0) = −n φdµ + , ddc (r 2 φ) ∧ dt 2 r ≤1 (n − 1)! L
(3.23)
where the second term arises by varying the Liouville measure ωn /n!. We may now apply Stokes’ theorem to the second term on the right-hand side of (3.23). Using i ∗ ω = ωT together with the equation dc r 2 = 2r 2 η (see Eq. (2.6)), we obtain ωTn−1 dvol[L] (t = 0) = −n φdµ + n φη ∧ . dt (n − 1)! L L
(3.24)
(3.25)
Notice that the term involving dc φ does not contribute, since ξ contracted into the integrand is identically zero. Indeed, we have ξ dc φ = Lr ∂/∂r φ = 0
(3.26)
and ωT = (1/2)dη is basic. Noting that ωTn−1 , (n − 1)!
(3.27)
dvol[L] (t = 0) = 0 dt
(3.28)
dµ = η ∧ we have thus shown that
identically for all transverse Kähler deformations27 . It follows that vol[L] may be interpreted as a function vol[L] : C0 → R.
(3.29)
Our task in the remainder of this paper is to understand the properties of this function. 27 Note that we didn’t use the equation L φ = 0 anywhere. Any deformation φ of the metric not satisfying ξ this equation will preserve the homothetic scaling of the metric on X , but it will no longer be a cone.
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3.2. Varying the Reeb vector field. In the previous subsection we saw that vol[L] may be regarded as a function on the space of Reeb vector fields, since the volume is independent of transverse Kähler deformations. We now fix a maximal torus Ts ⊂ Aut(X ), acting by isometries on each metric in a space of Sasakian metrics, and consider the properties of the functional vol[L] on this space as we vary the Reeb vector field. In particular, in the remainder of this section we give formulae for the first and second variations. We would like to differentiate the function vol[L]. Thus, we fix an arbitrary background Kähler cone metric, with Kähler potential r 2 , and linearise the deformation equations around this. We set ξ(t) = ξ + tY, r 2 (t) = r 2 (1 + tφ),
(3.30) (3.31)
where t is a (small) parameter and Y ∈ ts is holomorphic and Killing. A priori, φ is any function on X . Working to first order in t, the calculation goes much as in the last subsection. The details may be found in Appendix C. We obtain dvol[L](Y ) = −n η(Y )dµ. (3.32) L
This is our general result for the first derivative of vol[L]. As one can see, it manifestly depends only on the direction Y in which we deform the Reeb vector field, and not on the function φ, in accord with the previous section. Note that the integrand in (3.32) is twice the Hamiltonian function for Y . Indeed, in this case, dr (Y ) = 0 and hence LY ω = 0. Since X necessarily has b1 (X 0 ) = 0, there is therefore a function yY such that 1 dyY = − Y ddc r 2 . (3.33) 4 Using 1 Y ω = − η(Y )dr 2 − r 2 Y dη (3.34) 2 and (2.12), one finds that the homogeneous solution to this equation is yY = 21 r 2 η(Y ).
(3.35)
The Hamiltonian yY is then homogeneous degree two under r ∂/∂r . Substituting this into (3.32) we recover the toric formula (3.18) in [22], obtained in a completely different way using convex polytopes. In order to compute the second variation, we note that, since Y is Killing, it commutes with r ∂/∂r : [Y, r ∂/∂r ] = 0. As a result, η(Y ) is independent of r . Using this property, we may write ωn dvol[L](Y ) = −n(n + 1) (3.36) (dc r 2 )(Y ) , n! r ≤1 where recall that dc r 2 = 2r 2 η. We now differentiate again to obtain 2 d vol[L](Y, Z ) = n(n + 1) η(Y )η(Z )dµ.
(3.37)
L
This is a general form for the second variation of the volume of a Sasakian manifold. The derivation, which is a little lengthy, is contained in Appendix C. Note that (3.37) is manifestly positive definite, and hence the volume is a strictly convex function. Again, for toric geometries, this reduces to a formula in [22].
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3.3. Uniqueness of critical points. Using this last result, we may prove uniqueness of a critical point rather simply. We regard the volume as a function vol[L] : C0 → R.
(3.38)
The Reeb vector field for a Sasaki–Einstein metric is a critical point of vol[L], restricted to the subspace for which the holomorphic (n, 0)–form has charge n. This defines a compact convex polytope ⊂ ts , and we may hence regard the Einstein–Hilbert action as a function S : → R.
(3.39)
Since we have just shown that S is strictly convex on this space, and is itself convex, standard convexity arguments show that S has a unique critical point. Thus, assuming a Sasaki–Einstein metric exists in our space of Sasakian metrics on L, its Reeb vector field is unique in ⊂ ts . 4. The Futaki Invariant In this section, we consider a fixed background Sasakian metric which is quasi–regular. We show that the first derivative dvol[L], as a linear function on the Lie algebra ts , is closely related to the Futaki invariant of V . This is a well–known [49] obstruction to the existence of a Kähler–Einstein metric on V . Using this relation, together with Matsushima’s theorem [56], we show that the group of holomorphic isometries of a quasi–regular Sasaki–Einstein metric on L is a maximal compact subgroup of Aut(X ). We conjecture this to be true also for the more generic irregular case. 4.1. Brief review of the Futaki invariant. Let (V, JV , gV ) be a Kähler orbifold28 with Kähler form ωV and corresponding Ricci–form ρV such that [ρV ] = λ[ωV ] ∈ H 2 (V ; R),
(4.1)
where λ is a real positive constant. The value of λ is irrelevant since one can always ¯ rescale the metric, leaving ρV invariant29 . By the global i∂ ∂–lemma, there is a globally defined smooth function f = f gV such that ρV − λωV = i∂ ∂¯ f
(4.2)
where, throughout this section, the ∂¯ operator is that defined on V . Note that this is the same f as that appearing in Sect. 2. We now define a linear map F : aut R (V ) → R
(4.3)
from the Lie algebra of real holomorphic vector fields aut R (V ) on V to the real numbers by assigning to each holomorphic vector field ζ on V the number ωn−1 V . (4.4) Lζ f F[ζ ] = (n − 1)! V 28 One usually works with manifolds. Passing to the larger orbifold category involves no essential differences. 29 Nevertheless, the value λ = 2n is that relevant for Kähler cones in complex dimension n.
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It is more natural, and more standard, to define F on the space of complex holomorphic vector fields: FC : aut(V ) → C
(4.5)
by simply complexifying (4.4). The functional F, introduced by Futaki in 1983 [49], has the following rather striking properties: • F is independent of the choice of Kähler metric representing [ωV ]. That is, it is invariant under Kähler deformations. In this sense it is a topological invariant. • FC is a Lie algebra homomorphism. • If V admits a Kähler–Einstein metric, so that for some gV the function f is constant, the Futaki invariant F clearly vanishes identically. Because of the second item, Calabi named FC the Futaki character [57] – it is a character because FC is a homomorphism onto the complex numbers. Let G = Aut(V ) denote the group of holomorphic automorphisms of V , and g its Lie algebra. Thus FC : g → C. Mabuchi [58] proved that the nilpotent radical of g lies in ker FC . Thus if g = h ⊕ Lie(Rad(G))
(4.6)
denotes the Levi decomposition of g, it follows that FC is completely determined by its restriction to the maximal reductive algebra h. Since FC is a Lie algebra character of h, it vanishes on the derived algebra [h, h], and is therefore determined by its restriction to the Lie algebra of the centre of G = Aut(V ). The upshot then is that FC is non–zero only on the centre of g. 4.2. Relation to the volume. Note that our derivative dvol[L] : ts → R is a linear map which is also independent of transverse Kähler deformations. This follows since vol[L] itself has this property for all Kähler cones. Moreover, if L admits a Sasaki–Einstein metric with Reeb vector field ξ , then dvol[L](Y ) = 0 for all vector fields Y ∈ ts with LY = 0. This is true simply because Sasaki–Einstein metrics are critical points of the Einstein–Hilbert action, which is equal to vol[L] on the subspace . Thus, given the exposition in the previous subsection, it is not surprising that dvol[L] is related to Futaki’s invariant. We now investigate this in more detail. We regard dvol[L](Y ) = −n η(Y )dµ (4.7) L
as a linear map dvol[L] : ts → R
(4.8)
with Y ∈ ts . Since all such Y commute with r ∂/∂r and ξ , Y descends to a holomorphic vector field YV = π∗ Y on V , where recall that π : L → V is the orbifold circle fibration. In fact, when interpreting vol[L] as the Einstein–Hilbert action, we should consider only those Y ∈ ts such that LY = 0. These form a linear subspace in ts .
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In order to relate dvol[L] to the Futaki invariant, we shall use the existence of a certain spinor field θ on the Sasakian link L, as discussed in Subsect. 2.6. We shall also need the Lie derivative, acting30 on spinor fields, along a Killing vector field31 Y: 1 LY θ = ∇Y θ + dY · θ. 4
(4.9)
Suppose now that Y is Killing and satisfies LY θ = iαθ.
(4.10)
Of course, we are interested in those Y with α = 0 since then LY = 0 also, as follows by writing the holomorphic (n, 0)–form as a bilinear in the spinor field. The equivalence of these two conditions is not quite obvious – a detailed argument proving this is given in Appendix B. We now compute 1 1 ¯ ¯ −i θ ∇Y θ − A(Y )θ θ dµ η(Y )dµ = 2 L 2 L i 1 ¯ θ αθ + dY · θ − A(Y )θ dµ (4.11) = 4 2 L 1 1 = αvol[L] − (dη, dY )dµ − A(Y )dµ, 8 L 2 L where recall that the spinor is normalised so that θ¯ θ = 1. Here we have denoted the pointwise inner product between two two–forms A, B as (A, B) ≡
1 Aµν B µν . 2
(4.12)
Next, we obtain 1 − 8
1 (dη, dY )dµ = − (d∗ dη, Y )dµ 8 L L 1 =− ( L η, Y )dµ 8 L n−1 =− η(Y )dµ. 2 L
(4.13)
Here we have used the fact that d∗ η = 0 as η is a Killing one–form. Moreover, we have L η = 2Ric(g L )(η) = 4(n − 1)η + 2Ric(g X )(η) = 4(n − 1)η.
(4.14)
The first equality is true for any Killing one–form. In the second we have used Eq. (2.17) to relate the Ricci curvature of the link L to that of the cone X . Note that from the same 30 Recall that the differential forms act on spinors via Clifford multiplication; that is for a p–form A, we 1 A µ ...µ p θ . have A · θ ≡ p! µ1 ...µ p γ 1 31 We denote the one–form dual to Y by Y = g (Y, ·). L
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equation it also follows that ξ contracted into Ric(g X ) is zero, which gives the last equality. Finally, substituting (4.14) into (4.11) gives us
η(Y )dµ = −2αvol[L] +
dvol[L](Y ) = −n L
A(Y )dµ.
(4.15)
L
Clearly, to relate to the Futaki invariant (4.4), we must now integrate over the circle fibre of π : L → V . Indeed, recall that ρ = ρV − 2nωV =
1 c dd f 2
(4.16)
as an equation on the Fano orbifold V . We thus have A=
1 ∗ c π (d f ). 2
(4.17)
Since f is basic by definition, and Y commutes with ξ , it follows that A(Y ) is also a basic function, i.e. it is independent of ξ . Thus we may trivially integrate over the circle fibre. Denote the length of this by
=
2πβ . n
(4.18)
Then
ωn−1 V (n − 1)! V ωn−1
V . = −2αvol[L] − L JV (YV ) f 2 V (n − 1)!
dvol[L](Y ) = −2αvol[L] +
2
dc f (YV )
(4.19)
Thus we have shown that dvol[L](Y ) = −2αvol[L] −
· F[JV (YV )]. 2
(4.20)
Of course, for Y preserving we have α = 0 and we are done. Note also that, when Y = ξ the spinor has charge α = n/2, as follows from a simple calculation. In this case YV = 0 and (4.20) shows that vol[L] is homogeneous degree −n under deformations along the Reeb vector field. Later we will see this rather directly from our explicit formula for the volume. Thus, as expected, the derivative of the volume is directly related to the Futaki invariant of V . The dynamical problem of finding a critical point of the Sasakian volume V may be interpreted as choosing the Reeb vector field in such a way so that the corresponding Kähler orbifold V has zero Futaki invariant. This is certainly a necessary condition for existence of a Kähler–Einstein metric on V , which clearly is also necessary in order for a Sasaki–Einstein metric to exist on L. Of course, this interpretation requires us to stick with quasi–regular stuctures, which is unnatural. Nevertheless, it gives a very interesting new interpretation of the Futaki invariant.
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4.3. Isometries of Sasaki–Einstein manifolds. Using the result of the last subsection, together with known properties of the Futaki invariant described above, we may now deduce some additional properties of Sasaki–Einstein manifolds. In particular, we first show that the critical Reeb vector field is in the centre of a maximal compact subgroup K ⊂ Aut(X ), where Ts ⊂ K . Then we use this fact to argue that the group of (holomorphic) isometries of a Sasaki–Einstein metric on L is a maximal compact subgroup of Aut(X ). Let K ⊂ Aut(X ) be a maximal compact subgroup of the holomorphic automorphism group of X , containing the maximal torus Ts ⊂ K . Let z ⊂ k denote the centre of k, and write ts = z ⊕ t . (4.21) m ∼ Pick a basis for z so that we may identify z = R . We may consider the space of Reeb (m)
vector fields in this subspace – by the same reasoning as in Sect. 2, these form a cone C0 where we now keep track of the dimension of the cone. There will be a unique critical (m) point of the Einstein–Hilbert action on this space. Let us suppose that b∗ ∈ Qm , so that this critical point is quasi–regular. Clearly we have considered only a subspace z ⊂ ts , or Rm ⊂ Rs , and the minimum we have found in Rm might not be a minimum on the larger space. However, using the relation between dvol[L] and the Futaki invariant we (m) may in fact argue that the critical point b∗ ∈ Rm is necessarily a critical point of the s minimisation problem on R . To see this, note that t ∼ = Rs−m descends to a subalgebra t ⊂ aut R (V ). Since (m) m s−m b = (b∗ , 0) ∈ R ⊕ R is in the centre of k by construction, in fact the whole of k descends to a subalgebra of aut R (V ). Recall now that FC : aut(V ) → C vanishes on the complexification of t . This is because FC is non–zero only on the centre of aut(V ). Thus the derivative of vol[L] in the directions t is automatically zero. Since the critical point is unique, this proves that b∗ = (b∗(m) , 0) ∈ Rm ⊕ Rs−m is the critical point also for the larger extremal problem on Rs . This argument may of course be made for any K ⊃ Ts , but in particular applies to a maximal such K . Hence we learn that the critical Reeb vector field, for a Sasaki–Einstein metric, necessarily lies in the centre32 of the Lie algebra of a maximal compact subgroup of Aut(X ). Using this last fact, we may now prove that, for fixed Ts ⊂ K , the isometry group of a Sasaki–Einstein metric on L with Reeb vector field in ts is a maximal compact subgroup of Aut(X ). To see this, note that since the critical Reeb vector field ξ∗ ⊂ z lies in the centre of k, the whole of K , modulo the U (1) generated by the Reeb vector field, descends to a compact subgroup of the complex automorphisms G = Aut(V ) of the orbifold V . The latter is Kähler–Einstein, and by Matsushima’s theorem33 [56], we learn that K in fact acts isometrically on V . Thus K acts isometrically on L, and we are done. Of course, this is also likely to be true for irregular Sasaki–Einstein metrics. Thus we make a more general conjecture • The group of holomorphic isometries of a Sasaki–Einstein metric on L is a maximal compact subgroup of Aut(X ). 32 Note this statement is different from the statement that the Reeb vector field for any Sasakian metric lies in the centre of the isometry group: the group of automorphisms of (L , g L ) i.e. the isometry group, depends on the choice of Reeb vector field, whereas K depends only on (X, J ). 33 Recall that Matsushima’s theorem states that on a Kähler–Einstein manifold V (or, more generally, orbifold) the Lie algebra of holomorphic vector fields aut(V ) is the complexification of the Lie algebra generated by Killing vector fields.
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5. A Localisation Formula for the Volume In this section we explain that the volume of a Sasakian manifold may be interpreted in terms of the Duistermaat–Heckman formula. The essential point is that the Kähler potential r 2 /2 may also be interpreted as the Hamiltonian function for the Reeb vector field. By taking any equivariant orbifold resolution of the cone X , we obtain an explicit formula for the volume, as a function of the Reeb vector ξ ∈ C0 , in terms of topological fixed point data. If we write ξ ∈ C0 ⊂ ts as ξ=
s i=1
bi
∂ , ∂φi
(5.1)
then as a result the volume vol[L] : C0 → R, relative to the volume of the round sphere, is a rational function of b ∈ Rs with rational coefficients. The Reeb vector field b∗ for a Sasaki–Einstein metric is a critical point of vol[L] on the convex subspace , formed by intersecting C0 with the rational hyperplane of Reeb vectors under which has charge n. It follows that b∗ is an algebraic vector – that is, a vector whose components are all algebraic numbers. It thus also follows that the volume vol[L](b∗ ) of a Sasaki–Einstein metric, relative to the round sphere, is an algebraic number. 5.1. The volume and the Duistermaat–Heckman formula. There is an alternative way of writing the volume of (L , g L ). In the previous subsections we used the fact that ωn . (5.2) dµ = 2n vol[L] = L r ≤1 n! This follows simply by writing out the measure on the cone X in polar coordinates and cutting off the r integral at r = 1. However, we may also write 1 ωn 2 . (5.3) vol[L] = n−1 e−r /2 2 (n − 1)! X n! This is now an integral over the whole cone. Note that the term in the exponent is the Kähler potential, which here acts as a convergence factor. So far, the function r 2 has played a dual role: it determines the link L = X |r =1 , and is also the Kähler potential. However, the function r 2 /2 is also the Hamiltonian function for the Reeb vector field. To see this, recall from (3.35) that the Hamiltonian function associated to the vector field Y is yY = 21 r 2 η(Y ). Setting Y = ξ , we thus see that r 2 /2 is precisely the Hamiltonian associated to the Reeb vector field. Thus we may suggestively write the volume (5.3) as 1 vol[L] = n−1 e−H eω , (5.4) 2 (n − 1)! X where H = r 2 /2 is the Hamiltonian. The integrand on the right-hand side of (5.4) is precisely that appearing in the Duistermaat–Heckman formula [40,41] for a (non–compact) symplectic manifold X with symplectic form ω. H is the Hamiltonian function for a Hamiltonian vector field. The Duistermaat–Heckman theorem expresses this integral as an integral of local data over the fixed point set of the vector field. Of course, for a Kähler cone, the action generated by the flow of the Reeb vector field is locally free on r > 0,
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since ξ has square norm r 2 . The fixed point contribution is therefore, formally, entirely from the isolated singular point r = 0. Thus in order to apply the theorem, one must first resolve the singularity. Taking a limit in which the resolved space approaches the cone, we will obtain a well–defined expression for the volume in terms of purely topological fixed point data. The volume is of course independent of the choice of resolution. 5.2. The Duistermaat–Heckman Theorem. In this subsection we give a review of the Duistermaat–Heckman theorem [40,41] for compact Kähler manifolds. The non–compact case of interest will follow straightforwardly from this, as we shall explain. Since the proof of the Duistermaat–Heckman theorem [41] is entirely differentio–geometric, the result is also valid for orbifolds, with a simple modification that we describe. Let W be a compact Kähler manifold with Kähler form ω, dimC W = n, and let Ts ⊂ Aut(W ) act on W in a Hamiltonian fashion. Let ξ ∈ ts with Hamiltonian H . Thus dH = −ξ ω.
(5.5)
The flow on W generated by ξ will have some fixed point set, which is also the zero set of the vector field ξ . In general, the fixed point set {F} will consist of a number of disconnected components F of different dimensions; each component is a Kähler submanifold of W . Let f : F → W denote the embedding, so that f ∗ ω is a Kähler form on F. From now on, we focus on a particular connected component F, of complex codimension k. The normal bundle E of F in W is a rank k complex vector bundle over F. The flow generated by ξ induces a linear action on E. Let u 1 , . . . , u R ∈ Zs ⊂ t∗s be the set of distinct weights of this linear action on E. This splits E into a direct sum of complex vector bundles E=
R
Eλ .
(5.6)
λ=1
Here Eλ is a complex vector bundle over F of rank n λ , and hence R
n λ = k.
(5.7)
λ=1
Thus, for example, if E splits into a sum of complex line bundles then each n λ = 1 and R = k. We denote the linear action of ξ on E by Lξ . This acts on the λth factor in (5.6) by multiplication by i(b, u λ ), where recall that b ∈ Rs are the components of the vector field ξ as in (5.1). With respect to the decomposition (5.6), we thus have Lξ = i diag 1n 1 (b, u 1 ), 1n 2 (b, u 2 ), . . . , 1n R (b, u R ) , (5.8) where 1n λ denotes the unit n λ × n λ matrix. Hence the determinant of this transformation is R Lξ = det (b, u λ )n λ . (5.9) i λ=1
Note this is homogeneous degree k in b.
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Finally, choose any Ts –invariant Hermitian connection on E, with curvature E . The Duistermaat–Heckman theorem then states that e− f ∗ H e f ∗ ω ωn . e−H (5.10) = Lξ −E n! W F det {F} 2πi The sum is over each connected component F of the fixed point set. The determinant is a k × k determinant and should be expanded formally into a differential form of mixed degree. Moreover, the inverse is understood to mean one should expand this formally in a Taylor series. This notation is standard in index theory. We will now examine the right-hand side of (5.10) in more detail. If we let λ be the curvature of Eλ , we may write det
Lξ − E 2πi
1 = det (2π )k
Lξ i
R
det(1 − (Lξ )−1 λ ).
(5.11)
λ=1
Fix one of the vector bundles Eλ . Then det(1 − (Lξ )
−1
iλ , λ ) = det 1 + w 2π
(5.12)
where w=
2π . (b, u λ )
(5.13)
The right-hand side of (5.12) is precisely the Chern polynomial of Eλ . As a cohomology relation, we have iλ = det 1 + w ca (Eλ )wa ∈ H ∗ (F; R). (5.14) 2π a≥0
Recall that each Eλ has Chern classes ca (Eλ ) ∈ H 2a (F; R),
(5.15)
where 0 ≤ a ≤ n λ and c0 = 1. Thus we may write −1 Lξ − E −1 Lξ k det = (2π ) det βa (b) 2πi i a≥0
= R
(2π )k
λ=1 (b, u λ )
nλ
βa (b).
(5.16)
a≥0
The βa (b) are closed differential forms on F of degree 2a, with β0 (b) = 1. The cohomology class of βa (b) in H 2a (F; R) is then a polynomial in the Chern classes of the Eλ . Moreover, the coefficients are rational functions of b, of degree −a. Since f ∗ H is constant on each connected component of F, it follows that, for each connected component, the integrand on the right-hand side of (5.10) is a polynomial in the Chern classes of Eλ and the pulled–back Kähler form f ∗ ω on F. The coefficients
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are memomorphic34 in b, and analytic provided (b, u λ ) = 0 for all λ = 1, . . . , R. Of course, the left-hand side of (5.10) is certainly analytic. As a special case of this result, suppose that F is an isolated fixed point. Thus k = n, and E is a trivial bundle. We may then write the n, possibly indistinct, weights as u λ , λ = 1, . . . , n. Neither the Chern classes nor the Liouville measure exp( f ∗ ω) contribute non–trivially, and we are left with (2π )n e− f
∗H
n λ=1
1 . (b, u λ )
(5.17)
This is the general formula for the contribution of an isolated fixed point to the Duistermaat–Heckman formula (5.10). The proof of the Duistermaat–Heckman theorem [41] is entirely differentio– geometric, and thus the proof also goes through easily for non–compact manifolds, and orbifolds. The proof goes roughly as follows. The integrand on the left-hand side of the Duistermaat–Heckman formula (5.10) is exact on W minus the set of fixed points {F}. One then applies Stokes’ theorem to obtain a sum of integrals over boundaries around each connected component F. This boundary is diffeomorphic, via the exponential map, to the total space of the normal sphere bundle to F in W , of radius . One should eventually take the limit → 0. By introducing an Hermitian connection on the normal bundle E with curvature E , one can perform the integral over the normal sphere explicitly, resulting in the formula (5.10). Thus we may extend this proof straightforwardly to our non–compact case, and to orbifolds, as follows: • For non–compact manifolds, provided the fixed point sets are in the interior, and that the measure tends to zero at infinity (so that the boundary at infinity makes no contribution in Stokes’ theorem), the formula (5.10) is still valid. • For orbifolds, we must modify the formula (5.10) slightly. The normal space to a generic point in a connected component F is not a sphere, but rather a quotient S 2k−1 / , where is a finite group of order d. Thus, when we integrate over this normal space, we pick up a factor 1/d. For each connected component F we denote this integer, called the order of F, by d F . The general formula is then almost identical
e−H W
ωn = n! {F}
∗
F
∗
1 e− f H e f ω . d F det Lξ −E 2πi
(5.18)
In the orbifold case, the Chern classes, defined in terms of a curvature form E on the vector orbibundle E, are in general rational, i.e. images under the natural map H ∗ (F; Q) → H ∗ (F; R).
(5.19)
These slight generalisations will be crucial for the application to Sasakian geometry, to which we now turn. 34 We may analytically continue the right-hand side to b ∈ Cs .
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5.3. Application to Sasakian geometry. Let (X, ω) be a Kähler cone, with Kähler potential r 2 . The volume of the link L = X |r =1 may be written as in (5.4). In order to apply the Duistermaat–Heckman theorem, we must first resolve the cone X . In fact in order to compute the volume vol[L](b) as a function of b, we pick, topologically, a fixed (orbifold) resolution W of X . Thus we have a map :W →X
(5.20)
∼ X 0 is a biholomorphism. Moreover, the map and an exceptional set E such that W \ E = should be equivariant with respect to the action of Ts . Thus all the fixed points of Ts on W necessarily lie on the exceptional set. There is a natural way to do this: we simply choose a (any) quasi–regular Reeb vector field ξ0 and blow up the Kähler orbifold V0 to obtain the total space W of the bundle L → V0 . Note that then W \ V0 ∼ = X 0 , and that Ts acts on W . Thus this resolution is obviously equivariant. We then assume35 that, for every Reeb vector field ξ ∈ C0 , there is a 1–parameter family of Ts –invariant Kähler metrics g(T ), 0 < T < δ for some δ > 0, on W such that g(T ) smoothly approaches a Kähler cone metric with Reeb vector field ξ as T → 0. We may then apply the Duistermaat–Heckman theorem (5.18) to the Kähler metric g(T ) on the orbifold W . In the limit that T → 0, the exceptional set collapses to zero volume and we recover the cone X . Since all fixed points of Ts lie on the exceptional set, the pull–back of the Hamiltonian f ∗ H tends to zero in this limit, as H → r 2 /2 which is zero at r = 0. Moreover, the pull–back of the Kähler form f ∗ ω is also zero in this limit. Hence the exponential terms on the right-hand side of the Duistermaat–Heckman formula are equal to 1 in the conical limit T → 0. This leaves us with the formula 1 1 (2π )k vol[L](b) = n−1 βa (b). (5.21) R nλ 2 (n − 1)! d λ=1 (b, u λ ) a≥0 {F} F F This formula is valid for b a generic element of ts . Then the vanishing set of the Reeb vector field is the fixed point set of Ts . For certain special values of b the set of fixed points changes. For example, when W is obtained by taking a quasi–regular Reeb vector field ξ0 and blowing up V0 , the fixed point set of ξ0 is the whole of V0 . Note, however, that vol[L](b) is still a smooth function of b. The integral over F picks out the term in the sum of degree a = (n − k). Recall that the a th term is homogeneous in b of degree −a. In particular, we may extract the factor (2π )n−k and write 2π n 1 β(b) . (n − 1)! (b, u λ )n λ R
vol[L](b) =
{F}
(5.22)
λ=1
Here β(b) is a sum of Chern numbers of the normal bundle E of F in W , with coefficients which are homogeneous degree −(n − k) in b. Specifically, ⎡ ⎤−1 R 1 ⎣ ca (Eλ ) ⎦ β(b) = . (5.23) (b, u λ )a F dF λ=1
a≥0
35 We assume there is no obstruction to doing this. In any case, we shall also prove the localisation formula (5.25) in the next section using a different relation to an equivariant index on the cone X . This doesn’t assume the existence of any metric on W .
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The Chern polynomials in (5.23) should be expanded in a Taylor series, and the integral over F picks out the differential form of degree 2(n − k). Recall that Chern numbers are defined as integrals over F of wedge products of Chern classes. Thus when W is a manifold the coefficients in β(b) are integers. For the more general case of orbifolds, the Chern numbers are rational numbers. Noting that vol[S 2n−1 ] =
2π n (n − 1)!
(5.24)
is precisely the volume of the round (2n − 1)–sphere, we thus have ⎡ ⎤−1 R vol[L](b) 1 1 ca (Eλ ) ⎦ ⎣ V (b) ≡ = . vol[S 2n−1 ] dF F (b, u λ )n λ (b, u λ )a {F}
λ=1
(5.25)
a≥0
Here we have defined the normalised volume V (b). The right-hand side of (5.25) is homogeneous degree −n in b, and is manifestly a rational function of b with rational coefficients, since the weights and Chern numbers are generally rational numbers. This is our general formula for the volume of a Sasakian metric on L with Reeb vector field b. Using this result, we may now prove: • The volume of a Sasaki–Einstein manifold, relative to that of the round sphere, is an algebraic number. This follows trivially, since the Reeb vector b∗ for a Sasaki–Einstein metric is a critical point of (5.25) on the subspace of vector fields under which the holomorphic (n, 0)– form has charge n. Thus b∗ is an isolated zero of a system of algebraic equations with rational coefficients, and hence is algebraic. The normalised volume V (b∗ ) is thus also an algebraic number. We conclude this section by relating the formula (5.25) to the volume of quasi–regular Sasakian metrics. Let ξ be the Reeb vector field for a quasi–regular Sasakian structure, and choose the resolution W above so that V is the Kähler orbifold of the Sasakian structure. ξ generates an action of U (1) on W which is locally free outside the zero section, and fixes the zero section V . Thus in this case E = L and the formula (5.25) simplifies considerably. The weight u = 1, the codimension k = 1, d = 1, and hence (5.25) gives −1 1 V (b) = 1 + b−1 c1 (L) b V 1 = n c1 (L∗ )n−1 , (5.26) b V where c1 (L) = −c1 (L∗ ). Recall from Sect. 2 that L → V is always given by some root of the canonical bundle over V . The first Chern class of L∗ is then c1 (L∗ ) =
c1 (V ) 2 (V ; Z), ∈ Horb β
(5.27)
2 (V ; Z) is the first Chern class of the complex orbifold V . The where c1 = c1 (V ) ∈ Horb total space L of the associated circle bundle to L will be simply connected if and only
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if β is the maximal positive integer such that (5.27) is an integer class, which recall is called the index of V . The general volume formula (5.25) gives 1 c1n−1 . (5.28) V (b) = n n−1 b β V We now compute the volume V (b) directly. Recall that, in general, the volume of a quasi–regular Sasakian manifold is given by the formula 2π n β vol[L] = n cn−1 . (5.29) n (n − 1)! V 1 To see this, recall from (2.22) that the Kähler class of V is [ωV ] = [ρV /2n], where in the latter equation we have used the fact that the holomorphic (n, 0)–form is homogeneous degree n under r ∂/∂r . We also have c1 = [ρV /2π ]. Thus the volume of V is given by vol[V ] = V
ωn−1 π n−1 V = n−1 (n − 1)! n (n − 1)!
V
c1n−1 .
(5.30)
The length of the circle fibre is 2πβ/n, and hence (5.29) follows. For example, when V = CP n−1 and β = n this leads to the formula vol[S 2n−1 ] =
2π n (n − 1)!
for the volume of the round sphere. Thus from (5.29) we have β V (b) = n cn−1 n V 1
(5.31)
(5.32)
which is precisely formula (5.28) with b = n/β.
(5.33)
This equation for b must be imposed to compare (5.32) to (5.28), since in the former equation we have assumed that has charge n under the Reeb vector field. More precisely, in (5.28) we have written ξ =b
∂ , ∂ν
(5.34)
where ν ∼ ν + 2π , and ∂/∂ν rotates the fibre of the line bundle L with weight one. Thus ν=
nψ , β
(5.35)
where ψ was defined in Sect. 2.3. Recalling that has charge n under ∂/∂ψ, we may thus also impose this on the vector field ξ in (5.34): β in = Lξ = bL∂/∂ν = b L∂/∂ψ = ibβ n from which (5.33) follows.
(5.36)
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5.4. Sasakian 5–manifolds and an example. The most physically interesting dimension is n = 3. Then the AdS/CFT correspondence conjectures that string theory on Ad S5 × L is dual to an N = 1 superconformal field theory in four dimensions. In this section we therefore specialise the formula (5.25) to complex dimension n = 3. Suppose that X 0 ∼ = R+ × L is a Calabi–Yau 3–fold, and suppose we have a space of Kähler cone metrics on X which are invariant under a T2 action. We choose this case since a T3 action would mean that L were toric, which we treat in Sect. 7. Resolving the cone X to W , as before, the fixed point sets on V0 will consist of isolated fixed points p ∈ V0 and curves C ⊂ V0 . We have already treated the isolated fixed points in (5.17). Let the normal bundle of C in V0 be M. The total normal bundle of C in W is then E = h ∗ L ⊕ M, where h : C → V0 is the inclusion. Thus the normal bundle E to C in W splits into a sum of two line bundles. We denote the weights as u λ ∈ Z2 , λ = 1, 2 for L and M, respectively. We then get the following general formula for the volume, in terms of topological fixed point data: 2 3 1 1 1 c1 (L) c1 (M) 1 − + . V (b) = dp (b, u µ ) dC (b, u λ ) C (b, u 1 ) (b, u 2 ) { p}
µ=1
λ=1
{C}
(5.37) Here u µ ∈ Z2 , µ = 1, 2, 3 are the weights on the tangent space at p. Note that V (b) is clearly homogeneous degree −3 in b, as it should be. Example. As an example of formula (5.37), let us calculate the volume of Sasakian metrics on the complex cone over the first del Pezzo surface. Of course, this is toric, so one can use the toric methods developed in [22]. However, the point here is that we will rederive the result using non–toric methods. Specifically, we’ll use the formula (5.37). We think of the del Pezzo as the first Hirzebruch surface, F1 . This is a CP 1 bundle over CP 1 , which may be realised as the projectivisation F1 = P(O(0) ⊕ O(−1)) → CP 1 .
(5.38)
We take W to be the total space of the canonical bundle K → F1 , and the T2 to act by rotating the fibre of K and the fibre CP 1 of F1 . The fixed point set of this T2 action consists of two curves on V0 = F1 , which are the north and south pole sections of F1 . We denote these by H and E, which are two copies of CP 1 . In fact, [H ] is the hyperplane class on d P1 , and [E] is the exceptional divisor. The normal bundles over H and E are H : E = O(−3) ⊕ O(1), E : E = O(−1) ⊕ O(−1).
(5.39)
Note that the Chern numbers sum to −2 in each case, as is necessary to cancel the Chern number +2 of CP 1 . Write T2 = U (1)1 × U (1)2 . Let U (1)1 rotate the fibre of L = K with weight one, and let U (1)2 rotate the fibre CP 1 of F1 with weight one, where we take the canonical lifting of this action to the canonical bundle K. We write ξ=
2 i=1
bi
∂ . ∂φi
(5.40)
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We need to compute the weights u λ ∈ Z2 ⊂ t∗2 of the T2 action on the normal bundles (5.39). Here λ = 1, 2 denote the two line bundles in the splitting (5.39). We compute the weight of each U (1) ⊂ T2 in turn. U (1)1 has weights [1, 0] with respect to the splitting (5.39) for both H and E since this simply rotates the fibre of K with weight one. As for U (1)2 , note that K restricted to any point x on the base CP 1 of F1 is a copy of T ∗ CP 1 = O(−2) → CP 1 . By definition, U (1)2 rotates this fibre CP 1 with weight one, and thus fixes its north and south poles. Thus as we vary x on the base CP 1 , we sweep out H and E, respectively. The weights of U (1)2 on the tangent space to x in T ∗ CP 1 are thus [−1, 1] and [1, −1], respectively – the opposite signs appear because we have the cotangent bundle, rather than the tangent bundle. These also give the weights for U (1)2 acting on H and E, with respect to the decomposition (5.39), respectively. Thus, to summarise, the weights u λ ∈ Z2 ⊂ t∗2 are H : u 1 = (1, −1), E : u 1 = (1, 1),
u 2 = (0, 1), u 2 = (0, −1).
(5.41)
The formula (5.37) thus gives 1 3 1 1 −1 1 V (b) = + + + (b1 − b2 )b2 b1 − b2 b2 (b1 + b2 )(−b2 ) b1 + b2 −b2 8b1 + 4b2 = 2 . (5.42) (b1 − b22 )2 One can verify that, on setting b1 = 3, corresponding to the holomorphic (3, 0)–form having charge√ 3, the remaining function of b2 has a critical point, inside the Reeb cone, at b2 = −4 + 13. The volume at the critical point is then √ 43 + 13 13 , (5.43) V∗ = 324 which is indeed the correct result [10,23]. The reason that we get the correct result here is that any circle that rotates the base CP 1 of F1 is not in the centre of the Lie algebra of the compact group K = U (1)2 × SU (2) acting on the cone X . From the results on the Futaki invariant in Sect. 4, it follows that b3 = 0 necessarily at any critical point. Indeed, the Lie algebra k = t2 ⊕ su(2). The reason that we do not need to extremise over t3 is that k = z ⊕ t , where t ⊂ su(2). Thus the derivative of vol[L] is automatically zero in the direction t , provided ξ ∈ z = t2 . The isometry algebra of the metric is then, according to our conjecture, the maximal k = t2 ⊕ su(2), which indeed it is [10,23]. The formula (5.42) may indeed be recovered from the toric result on setting b3 = 0, as we show in Sect. 7. 6. The Index–Character In this section we show that the volume of a Sasakian manifold, as a function of b, is also related to a limit of the equivariant index of the Cauchy–Riemann operator ∂¯ on the cone X . This equivariant index essentially counts holomorphic functions on X according to their charges under the Ts action. The key to this relation is the Lefschetz fixed point theorem for the ∂¯ operator. In fact, by taking a limit of this formula, we will recover our general formula (5.25) for the volume in terms of fixed point data.
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6.1. The character. Recall that ∂¯ is the Cauchy–Riemann operator on X . We may consider the elliptic complex ∂¯
∂¯
∂¯
0 −→ 0,0 (X ) −→ 0,1 (X ) −→ · · · −→ 0,n (X ) −→ 0
(6.1)
on X . Here 0, p (X ) denotes the differential forms of Hodge type (0, p) with respect to the complex structure of X . We denote the cohomology groups of this sequence as H p (X ) ∼ = H 0, p (X ; C). In fact, the groups H p (X ), for p > 0, are all zero. This follows since X may be realised as the total space of a negative complex line bundle L over a compact Fano orbifold V . The property is then inherited from the Fano V . On the other hand, H0 (X ) is clearly infinite dimensional, in contrast to the compact case. ¯ Hence there is The action of Ts on X , since it is holomorphic, commutes with ∂. s ¯ an induced action of T on the cohomology groups of ∂. The equivariant index, or holomorphic Lefschetz number, for an element q ∈ Ts , is defined to be ¯ X) = L(q, ∂,
n
(−1) p Tr{q | H p (X )}.
(6.2)
p=0
Here the notation Tr{q | H p (X )} means one should take the trace of the induced action of q on H p (X ). The index itself, given by setting q = 1, is clearly infinite: the action of q is trivial and the trace simply counts holomorphic functions on X . However, the equivariant index is well–defined, provided the trace converges. In fact, we may analytically continue (6.2) to q ∈ TsC . The singular behaviour at q = 1 will then show up as a pole. Note we ¯ We shall henceforth write the have not imposed any type of boundary conditions on ∂. equivariant index as ¯ X ) = Tr{q | H0 (X )} C(q, X ) = L(q, ∂,
(6.3)
and refer to it simply as the character.
6.2. Relation to the ordinary index. Suppose that we have a regular Sasakian structure on L, and consider the corresponding circle action on X . Holomorphic functions on the cone X of charge k under this circle action may be identified with holomorphic sections of the bundle (L∗ )k → V,
(6.4)
where recall that L∗ is an ample line bundle over V whose dual is the associated complex line bundle to the projection π : L → V . Canonically, we may take L = K, the canonical bundle over V . The trace of q ∈ C∗ on the space of holomorphic functions of charge k is given by (6.5) Tr{q | H0 (X )k } = q k dim H 0 V ; (L∗ )k . The right-hand side is given by the Riemann–Roch theorem. Indeed, we have e−kc1 (L) · Todd(V ). dim H 0 V ; (L∗ )k = χ V, (L∗ )k = V
(6.6)
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In the first equality we have used the fact that dim H i V ; (L∗ )k = 0 for i > 0 since V is Fano and hence L∗ is ample. The second equality is the Riemann–Roch theorem36 . The Todd class is a certain polynomial in the Chern classes of V , whose precise form we won’t need. It follows then that the character, for a regular U (1) action, is simply given by C(q, X ) = qk e−kc1 (L) · Todd(V ). (6.7) V
k≥0
More generally, we can interpret the character C(q, X ) in terms of the equivariant index theorem for Ts−1 on V . However, it is easier to keep things defined on the cone. The relation between C(q, X ) defined in (6.7) and the volume of a regular Sasakian (–Einstein) manifold has been noted before in [46]. The key in this section is to extend this to the equivariant case. Then the relation of the volume to the equivariant index becomes a function of the Reeb vector field.
6.3. Localisation and relation to the volume. In the above discussion we have been slightly cavalier in defining C(q, X ) as a trace over holomorphic functions on X , since X is singular. Recall from Sect. 2 that X is an affine algebraic variety with an isolated Gorenstein singularity at r = 0, that is defined by polynomial equations { f 1 = 0, . . . , f S = 0} ⊂ C N . The space of holomorphic functions on X that we want is then given by elements of the coordinate ring of X , C[X ] = C[z 1 , . . . , z N ]/ f 1 , . . . , f S ,
(6.8)
where C[z 1 , . . . , z N ] is simply the polynomial ring on C N and f 1 , . . . , f S is the ideal generated by the functions { f A , A = 1, . . . , S}. Also as discussed in Sect. 2, we may always resolve X to a space W with at worst orbifold singularities: for example, one can take any quasi–regular Sasakian structure and take W to be the total space of L → V0 . It is important that the resolution is equivariant with respect to the torus action. Thus, in general, we have a Ts –equivariant birational map f :W →X
(6.9)
which maps some exceptional set E ⊂ W to the singular point p = {r = 0} ∈ X . In particular, f : W \ E → X 0 = X \ { p} is a biholomorphism. The fixed point set of Ts on W is then necessarily supported on E. The character is conveniently computed on W , as in the previous section, and is independent of the choice of resolution. In fact this set–up is identical to that in Sect. 5 – the resulting formula is rather similar to (5.10), although for orbifolds the equivariant index theorem is rather more involved than for manifolds37 . We claim that the volume V (b) is given in terms of the character C(q, X ) by the simple formula V (b) = lim t n C(exp(−tb), X ). t→0
(6.10)
36 This does not generalise as straightforwardly to orbifolds as one might have hoped. We shall make some comments on the (equivariant) Riemann–Roch theorem for orbifolds at the end of this section. 37 The additional technicalities for orbifolds drop out on taking the limit to obtain the volume.
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Recall that C(q, X ) is singular at q = 1. By defining qi = exp(−tbi ), and sending t → 0, we are essentially picking out the leading singular behaviour. As we shall explain, the leading term in t is always a pole of order n. Let W be a completely smooth resolution of the cone X . For example, if X admits any regular Sasakian structure, as in the previous section, we take W to be the total space of L → V0 with V0 a manifold. With definitions as in the last section, the equivariant index theorem in [59] gives Todd(F) C(q, W ) = (6.11) . R u λ −x j λ=1 j 1−q e {F} F Here the x j are the basic characters38 for the bundle Eλ → F. These are defined via the splitting principle. This says that, for practical calculations, we may assume that Eλ splits as a direct sum of complex line bundles Eλ =
nλ
Lj.
(6.12)
j=1
Then x j = c1 (L j ) ∈ H 2 (F; Z).
(6.13)
For a justification of this, the reader might consult reference [60]. The Chern classes of Eλ may then be written straightforwardly in terms of the basic characters. For example, (1 + x j ) (6.14) c(Eλ ) = j
so that c1 = j x j . The Chern character is given by ch(Eλ ) = To illustrate this, we note that one defines the Todd class as Todd =
a
j
exp(x j ).
1 xa 1 = 1 + c1 + (c12 + c2 ) + · · · . 1 − exp(−xa ) 2 12
(6.15)
Here xa are the basic characters for the complex tangent bundle of F. We have also expanded the Todd class in terms of Chern classes of F. It will turn out that the Todd class does not contribute to the volume formula in the limit (6.10). The denominator in (6.11), and (6.15), are again understood to be expanded in a formal Taylor expansion. Before proceeding to the limit (6.10), we note that one can recover (6.7) from (6.11) rather straightforwardly. The fixed point set of the free U (1) action is the zero section, or exceptional divisor, V . The normal bundle is then L, and the weight u = 1. Hence (6.11) gives Todd(V ) C(q, X ) = C(q, W ) = 1 − qe−c1 (L) V = q k e−kc1 (L) · Todd(V ). (6.16) V k≥0 38 We suppress the λ–dependence for simplicity of notation.
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We now turn to proving (6.10). Set q = exp(−tb) with t a (small) real positive number. The denominator in (6.11) is then, to leading order in t, given by R R n nλ R λ t (b, u λ ) + x j = t k 1 + zx j , (6.17) (b, u λ )n λ λ=1 j=1
λ=1
λ=1 j=1
where z=
1 . t (b, u λ )
(6.18)
The higher order terms in x j will not contribute at leading order in t, once one integrates over the fixed point set, which is why they do not appear in (6.17). Recalling the definition of the Chern polynomial, we thus see that, to leading order in t, (6.11) is given by C(e
−tb
, W) ∼
t
−k
λ=1
{F}
=
{F}
R
t
−k
R λ=1
1 (b, u λ )n λ 1 (b, u λ )n λ
F
Todd(F) λ λ=1 det 1 + zi 2π
R
F
Todd(F) .
a λ=1 a≥0 ca (Eλ )z
R
(6.19)
Again, it is simple to verify that the Todd class (6.15) contributes only the constant term 1 to leading order in t. Moreover, the integral over F picks out the differential form of degree 2(n − k), the coefficient of which is homogeneous degree −(n − k) in t. Thus we have C(e−tb , W ) ∼ t −n
R {F} λ=1
1 β(b), (b, u λ )n λ
(6.20)
where ⎡ ⎤−1 R ca (Eλ ) ⎣ ⎦ . β(b) = (b, u λ )a F λ=1
(6.21)
a≥0
Thus we have shown that the expression lim t n C(e−tb , W )
t→0
(6.22)
gives precisely the earlier volume formula (5.25). We have recovered it here by taking a limit of the equivariant index theorem for the ∂¯ operator. One can argue that, for a fixed quasi–regular Sasakian metric, this limit of the index gives the volume – such an argument was given in [46] and is similar to that at the end of Sect. 5.3. Since the rationals are dense in the reals, this proves (6.10) in general, as a function of the Reeb vector field. We finish this subsection with some comments on the extension of this result to orbifold resolutions of X – that is, resolutions with at worst orbifold singularities. Unfortunately, the Lefschetz formula (6.11) is not true for orbifolds. Recall that, for the Duistermaat–Heckman theorem, the only essential difference was the order d of the
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fixed point set which enters into the formula. For the character C(q, W ) the difference is more substantial. The order again appears, but the integral over a connected component of the fixed point set F is replaced by an integral over the associated orbifold Fˆ to F. Moreover, there are additional terms in the integrand. For a complete account, we refer the reader to the (fairly recent) original paper [61]. We shall not enter into the details of the general equivariant index theorem for orbifolds, since we do not need it. Instead, we simply note that the orbifold version of Duistermaat–Heckman may be recovered from an expression for C(q, W ) using the general results of [61] in much the same way as the smooth manifold case treated here.
7. Toric Sasakian Manifolds In this section we turn our attention to toric Sasakian manifolds. In this case the Kähler cone X is an affine toric variety. The equivariant index, which is a character on the space of holomorphic functions, may be computed as a sum over integral points inside the polyhedral cone C ∗ . The toric setting also allows us to obtain a “hands–on” derivation of the volume function from the index–character. For the purpose of being self–contained, we begin by recalling the well–known correspondence between the combinatorial data of an affine toric variety and the set of holomorphic functions defined on it.
7.1. Affine toric varieties. When X is toric, that is the torus has maximal possible rank s = n, it is specified by a convex rational polyhedral cone C ∗ ⊂ Rn . Let SC ∗ = C ∗ ∩ Zn . As is well–known, SC ∗ ⊂ Zn is an abelian semi–group, which by Gordan’s lemma is finitely generated. This means that there are a finite number of generators m 1 , . . . , m N ∈ SC ∗ , such that every element of SC ∗ is of the form a1 m 1 + · · · + a N m N ,
a A ∈ N.
(7.1)
To SC ∗ there is an associated semi–group algebra, denoted C[SC ∗ ], given by the characters w m : (C∗ )n → C∗ defined as wm =
n
wim
i
(7.2)
i=1
with multiplication rule
w m · w m = w m+m .
(7.3)
Notice that C[SC ∗ ] is generated by the elements {w m A |m A generate SC ∗ }. In algebraic geometry, the toric variety X C ∗ associated to a strictly convex rational polyhedral cone C ∗ is defined as the maximal spectrum39 X C ∗ = Specm C[SC ∗ ]
(7.4)
39 The maximal spectrum of an algebra A is defined to be the set Spec A = {maximal ideals in A} equipped m with the Zariski topology. An ideal I in A is said to be maximal if I = A and the only proper ideal in A containing I is I itself.
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of the semi–group algebra C[C ∗ ∩ Zn ]. In general, one can show that there exist suitable binomial40 functions f A ⊂ C N , where N is the number of generators of SC ∗ , such that C[SC ∗ ] = C[z 1 , . . . , z N ]/ f 1 , . . . , f S .
(7.5)
Then, more concretely, X C ∗ = { f 1 = 0, . . . , f S = 0} ⊂ C N
(7.6)
presents X C ∗ as an affine variety, with ring of holomorphic functions given precisely by (7.5). To exemplify this, let us describe briefly the conifold singularity. A set of generators of SC ∗ is given, in this case, by the four outward primitive edge vectors that generate the polyhedral cone C ∗ , which we will present shortly, see (7.27). Denoting w = (x, y, z), the corresponding generators of C[SC ∗ ] are given by Y = y, W = x z −1 ,
X = x y −1 ,
Z=z
(7.7)
respectively. It then follows, as is well–known, that the conifold singularity can be represented as the single equation f = X Y − Z W = 0 ⊂ C4
(7.8)
and the coordinate ring is simply C[X, Y, Z , W ]/X Y − Z W . The vanishing of X Y − Z W follows from the relation m 1 + m 3 = m 2 + m 4 between the generators of C ∗ ∩ Z3 , and the ideal X Y − Z W is determined by the integer linear relation among these generators. While this is a rather trivial example, it is important to note that in general to construct the monomial ideal one has to include all the generators of SC ∗ (otherwise the resulting variety is not normal) and these are generally many more than the generating edge vectors of C ∗ . For instance, for the complex cone over the first del Pezzo surface, whose link is the Sasaki–Einstein manifold Y 2,1 , there are 9 generators of SC ∗ , so that N = 9, while there are 20 relations among them, so that S = 20. As a result this is not a complete intersection. Some discussion illustrating these points in the physics literature can be found in [63,64]. 7.2. Relation of the character to the volume. As we have explained, when X is toric, by construction a basis of holomorphic functions on X is given by the wm above. Thus counting holomorphic functions on X according to their charges under Tn is equivalent to counting the elements of the semi–group SC ∗ . The character41 C(q, X ) is thus given by C(q, X ) = qm. (7.9) m∈SC ∗
We are again tacitly assuming here that the series defining C(q, X ) converges. 40 A binomial is a difference of two monomials. Then the affine toric variety is defined by equations of the type ‘monomial equals monomial’. 41 In fact, the character is also very closely related to the Ehrhart polynomial when L admits a regular Sasakian structure, with Fano V . In this case V is also toric and is thus associated to a convex lattice polytope in Rn−1 . The Ehrhart polynomial E(, k) is defined to be the number of lattice points inside the dilated polytope k. One can then show that E(, k) is a polynomial in k of degree n − 1. The coefficient of the leading term is precisely the volume of the polytope . This is analogous to the relation we discussed in Sect. 6. For a nice account of this, and the relation to toric geometry, see David Cox’s notes [62].
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As we proved in the last section, in general the normalised volume V (b) is related to the character C(q, X ) by the simple formula V (b) = lim t n C(e−tb , X ).
(7.10)
t→0
We again remind the reader that the notation q = e−tb is shorthand for defining the components qi = e−tbi .
(7.11)
In this section we shall prove this relation more directly, using the formula (7.9). The limit t → 0 may be understood as a Riemann integral, with the limit giving the volume formula (5.3). We first discuss a toy example – the generalisation will be straightforward. Consider the following limit t 1 = . −tb t→0 1 − e b
(7.12)
lim
Now, let us expand the fraction in a Taylor series. The radius of convergence of this series is precisely 1, so that for b > 0, we require that t also be positive. We will be particularly interested in isolating the singular behaviour as t → 0. We claim that one can deduce the above limit via lim
t→0
∞
te−tmb =
∞
e−yb dy =
0
m=0
1 . b
(7.13)
The integral arises simply from the definition of the Riemann integral. In particular, we subdivide the interval [0, ∞] into intervals of length t, and sum the contributions of the function e−yb evaluated at the end–points of each interval ym = mt. The limit t → 0 is then precisely a definition of the Riemann integral. This easily generalises, and we obtain lim t n C(e−tb , X ) = lim t n e−t (b,m) = e−(b,y) dy1 . . . dyn . (7.14) t→0
t→0
m∈SC ∗
C∗
The term appearing on the right-hand side is called the characteristic function of the cone, and was introduced in [44]. Of course, the interpretation of this function as the volume of a toric Sasakian manifold is new. Specialising (5.3) to the toric case, we can relate this to the volume of the Reeb polytope. Thus, from (5.3), we have (n − 1)! e−(b,y) dy1 . . . dyn = n!2n vol((b)) = vol[L](b). (7.15) 2π n C∗ Putting (7.15) together with (7.14), we have thus shown that the volume of L follows from a simple limit of the index–character lim t n C(e−tb , X ) =
t→0
(n − 1)! vol[L](b) 2π n
(7.16)
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in a very direct manner. Note that here we did not use any fixed point theorems. One can verify (7.16) directly in some simple cases. Consider, for instance, X = C3 , with the canonical basis vi = ei for the toric data. We then have t3 e−(b,y) dy1 . . . dyn lim = t→0 (1 − e−tb1 )(1 − e−tb2 )(1 − e−tb3 ) C∗ 1 = = 48 vol((b)), (7.17) b1 b2 b3 where the last equality is computed using the formulae in [22]. Finally, we can also give an independent proof of (7.15) by induction, which uses the particular structure of polyhedral cones. First, we note that (7.15) can be proved by direct calculation for n = 2: without loss of generality, we can take the primitive normals to the cone to be (v1 , v2 ) and (0, −1) respectively. Then the evaluation of the integral yields C∗
e
−(b,y)
dy1 dy2 =
∞
dy1
dy2 e−(b1 y1 +b2 y2 )
0
0
=
v 2
− v1 y1
v1 = 8 vol((b)), b1 (v1 b2 − v2 b1 )
(7.18)
where the last equality follows by calculating the area of the triangular region (b). A result in [65] shows that the integral of an exponential of a linear function on a polyhedral cone (more generally on a polytope) can be reduced to integrals over its facets Ca∗ , namely b
C∗
e−(b,y) dy1 . . . dyn =
d va e−(b,y) dσ |va | Ca∗
(7.19)
a=1
for any b ∈ Rn . Now we proceed by induction, where the hypothesis is that (7.15) holds at the (n − 1)th step. Using the first component of (7.19), one obtains C∗
e−(b,y) dy1 . . . dyn =
2n−1 (n − 1)! 1 vol(Fa ), b1 |va | d
(7.20)
a=1
which upon using the relation (2.91) of [22] gives 2n n!vol((b)),
(7.21)
concluding the proof. Notice that this expression for the volume allows one to compute derivatives in b straightforwardly: ∂k V (b) = (−1)k yi1 . . . yik e−(b,y) dy1 . . . dyn , ∗ ∂bi1 · · · ∂bik C
(7.22)
thus generalising in a natural way the formulae in [22]. In particular, convexity of V (b) is now immediate from this form of the volume.
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7.3. Localisation formula. In the case of toric cones X , the general fixed point formula (6.11) for the character has a very simple presentation. Recall that every toric X may be completely resolved by intersecting the cone C ∗ with enough hyperplanes in generic position. Specifically, the primitive normal vectors va ∈ Zn that define the cone C ∗ may, by an S L(n; Z) transformtion, be put in the form va = (1, wa ), where each wa ∈ Zn−1 . The convex hull of {wa } in Rn−1 defines a convex lattice polytope. Each interior point in this polytope defines a normal vector to a hyperplane in Rn . If all such hyperplanes are included, in generic position42 , it is well–known that the corresponding toric manifold is in fact completely smooth. Let W = X P be the resolved toric Calabi–Yau manifold43 corresponding to the resulting non–compact polytope P. Thus P ⊂ Rn is the image of X P under the moment map for the Tn action. The vertices of P are precisely the images of the fixed points under the Tn action on X P . Denote these as p A . Since each p A corresponds to a smooth point, it follows that there are n primitive edge vectors u iA ∈ Zn ⊂ t∗n , i = 1, . . . , n, meeting at p A , which moreover span Zn over Z. In particular, this ensures that a small neighbourhood of p A is equivariantly biholomorphic to Cn . The action of q ∈ (C∗ )n on complex coordinates (z 1 , . . . , z n ) in this neighbourhood is given by A
A
q : (z 1 , . . . , z n ) → (q u 1 z 1 , . . . , q u n z n ).
(7.23)
We may then define the character: C(q, Cn ; {u iA }) =
n
1
i=1
(1 − q u i )
A
.
(7.24)
The fixed point theorem for the equivariant index of ∂¯ on X P is now very simple to state. It says that C(q, X P ) = C(q, Cn ; {u iA }), (7.25) p A ∈P
where the {u iA } are the outward–pointing primitive edge vectors at each vertex p A ∈ P. As explained earlier, C(q, X ) = C(q, X P ). Thus we may in fact choose any toric resolution X P . One can prove invariance directly in dimension n = 3 as follows. For dimension n = 3, any toric resolution of the cone X may be reached from any other by a sequence of local toric flop transitions. Since each flop is a local modification of the formula (7.25), one needs to only focus on the relevant vertices p near the flop at each step. One can show that the formula for the conifold is itself invariant under the flop transition, as we discuss in the next subsection. This proves rather directly that the fixed point formula is invariant under toric flops, in complex dimension n = 3. It is now simple to take the limit (7.10), giving V (b) =
n pA
1 , (b, u iA ) ∈P i=1
(7.26)
42 The positions are the Fayet–Iliopoulos parameters, in the language of gauged linear sigma models. 43 The construction described above ensures that the resolution is Calabi–Yau. However, more generally
there is no need to impose this condition in order to compute the character. For example, one can compute the character for the canonical action of T2 on C2 by blowing up origin of the latter to give O(−1) → CP 1 . This is not a Calabi–Yau manifold. Similarly, the character for the conifold may be computed by resolving to O(−1, −1) → CP 1 × CP 1 .
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Fig. 1. Toric diagram for the conifold
Fig. 2. A small resolution of the conifold
where again the u iA are the outward–pointing primitive normals at the vertex p A . Clearly, this is a special case of our general result (5.25). The number of fixed points in the sum (7.26) is given by the Euler number χ (X P ) of the resolution X P . One can deduce this simply from the Lefschetz fixed point formula: one applies the equivariant index theorem to the de Rham complex. This expresses the Euler number of the resolved space X P as a sum of the Euler numbers of the fixed point sets. The Euler number of each fixed point contributes 1 to the total Euler number, and the result follows.
7.4. Examples. In this subsection, we compute explicitly the character C(q, X ) in a number of examples, and verify that we correctly reproduce the volume V (b) of the Sasakian metric as a limit. In particular, we consider three smooth resolutions of simple toric Gorenstein singularities. In order to demonstrate that in general we only need to consider orbifold resolutions, we recover the Sasakian volume V (b) for the Y p,q singularities by applying our more general orbifold localisation formula to a partial resolution of the singularity obtained by blowing up a Fano. 7.4.1. The conifold We take the toric data w1 = (0, 0), w2 = (0, 1), w3 = (1, 1), w4 = (1, 0). The outward primitive edge vectors for the polyhedral cone are then easily determined to be (0, 1, 0), (1, 0, −1), (1, −1, 0), (0, 0, 1).
(7.27)
Each vector has zero dot products with precisely two of the va = (1, wa ), and positive dot products with the remaining two. We must now choose a resolution of the cone. There are two choices, related by the flop transition. There are two vertices in each case. First resolution. We choose the following resolution: (1)
(1)
(1)
p1 : u 1 = (0, 1, 0), u 2 = (0, 0, 1), u 3 = (1, −1, −1), (2)
(2)
(2)
p2 : u 1 = (1, −1, 0), u 2 = (1, 0, −1), u 3 = (−1, 1, 1).
(7.28)
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D. Martelli, J. Sparks, S.-T. Yau
Fig. 3. The other small resolution of the conifold
The fixed point formula thus gives C(q, X ) =
3
(α)
pα i=1
=
1 − q ui
1 (1 − q2 )(1 − q3 )(1 − q1 q2−1 q3−1 ) +
=
1
(7.29)
1 (1 − q1 q2−1 )(1 − q1 q3−1 )(1 − q1−1 q2 q3 ) 1 − q1
(1 − q2 )(1 − q3 )(1 − q1 q2−1 )(1 − q1 q3−1 )
.
This is the general result for the character44 . We may now set q = exp(−tb) and take the limit V (b) = lim t 3 C(e−tb , X ) = t→0
b1 . b2 b3 (b1 − b2 )(b1 − b3 )
(7.30)
This indeed correctly reproduces the result of [22] for the volume. Second resolution. The other small resolution of the conifold has fixed points (1)
(1)
(1)
p1 : u 1 = (1, −1, 0), u 2 = (0, 0, 1), u 3 = (0, 1, −1), (2)
(2)
(2)
p2 : u 1 = (0, 1, 0), u 2 = (1, 0, −1), u 3 = (0, −1, 1).
(7.31)
The fixed point formula thus gives C(q, X ) =
3
(α)
pα i=1
= =
1 1 − q ui
1 (1 − q1 q2−1 )(1 − q3 )(1 − q2 q3−1 )
+
1 − q1
1 (1 − q2 )(1 − q1 q3−1 )(1 − q2−1 q3 )
(1 − q2 )(1 − q3 )(1 − q1 q2−1 )(1 − q1 q3−1 )
.
Of course, as expected, this is the same as (7.29). 44 Similar computations in related contexts have appeared before in [66–68].
(7.32)
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Fig. 4. Toric diagram for the complex cone over d P1
Fig. 5. Canonical bundle over d P1
7.4.2. The first del Pezzo surface Recall that this singularity is the lowest member of the Y p,q family of toric singularities [23]. Here we take the toric data w1 = (−1, −1), w2 = (−1, 0), w3 = (0, 1), w4 = (1, 0). The outward primitive edge vectors for the polyhedral cone are then easily determined to be (1, 1, 0), (1, 1, −1), (1, −1, −1), (1, −1, 2).
(7.33)
We resolve the cone by simply blowing up the del Pezzo surface, corresponding to including the interior point w = (0, 0). This leads to four vertices, with edges: (1)
(1)
(1)
p1 : u 1 = (1, 1, 0), u 2 = (0, 0, −1), u 3 = (0, −1, 1), (2) (2) p2 : u (2) 1 = (1, 1, −1), u 2 = (0, 0, 1), u 3 = (0, −1, 0), (3)
(3)
(7.34)
(3)
p3 : u 1 = (1, −1, −1), u 2 = (0, 0, 1), u 3 = (0, 1, 0), (4)
(4)
(4)
p4 : u 1 = (1, −1, 2), u 2 = (0, 0, −1), u 3 = (0, 1, −1). The fixed point formula gives, after some algebra: C(q, X ) =
N (q) (1 − q1 q2 )(1 − q1 q2 q3−1 )(1 − q1 q2−1 q3−1 )(1 − q1 q2−1 q32 )
,
(7.35)
where the numerator is given by N (q) = 1 + q1 + q1 q3 + q1 q3−1 + q1 q2−1 + q1 q2−1 q3 −q12 (1 + q1 + q2 + q3 + q3−1 + q2 q3−1 ).
(7.36)
Either by taking a limit of this expression, or else using (7.26) directly, one obtains V (b) = lim t 3 C(e−tb , X ) t→0
=
2(4b1 + 2b2 − b3 ) . (b1 + b2 )(b1 − b2 + 2b3 )(b1 − b2 − b3 )(b1 + b2 − b3 )
(7.37)
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Fig. 6. Toric diagram for the complex cone over d P2
Fig. 7. Canonical bundle over d P2
This is indeed correct, although we have chosen a different basis from that of [22]. Note also that, setting b3 = 0, we recover the formula (5.42) derived earlier without using toric geometry. 7.4.3. The second del Pezzo surface. We take the toric data w1 = (−1, −1), w2 = (−1, 0), w3 = (0, 1), w4 = (1, 0), w5 = (0, −1). This is the blow–up of the first del Pezzo surface, introducing an exceptional divisor corresponding to w5 . The outward primitive edge vectors for the polyhedral cone are then easily determined to be (1, 1, 0), (1, 1, −1), (1, −1, −1), (1, −1, 1), (1, 0, 1).
(7.38)
We resolve the cone by simply blowing up the del Pezzo surface, corresponding to including the interior point w5 = (0, 0). This leads to five vertices, with edges: (1) (1) p1 : u (1) 1 = (1, 1, 0), u 2 = (0, 0, −1), u 3 = (0, −1, 1), (2)
(2)
(2)
p2 : u 1 = (1, 1, −1), u 2 = (0, 0, 1), u 3 = (0, −1, 0), (3)
(3)
(3)
p3 : u 1 = (1, −1, −1), u 2 = (0, 0, 1), u 3 = (0, 1, 0), p4 : p5 :
(4) u1 u (5) 1
= =
(7.39)
(4) (4) (1, −1, 1), u 2 = (0, 0, −1), u 3 = (0, 1, 0), (5) (1, 0, 1), u (5) 2 = (0, −1, 0), u 3 = (0, 1, −1).
Rather than give the full character, we simply state the result for the volume: V (b) =
7b12 + 2b1 b2 + 2b1 b3 − b22 − b32 + 2b2 b3 . (7.40) (b1 + b2 )(b1 + b2 − b3 )(b1 − b2 − b3 )(b1 − b2 + b3 )(b1 + b3 )
Setting b1 = 3, it is straightforward to determine that the critical point, inside the Reeb cone, lies at √ −57 + 9 33 , (7.41) b∗2 = b∗3 = 16
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Fig. 8. Toric diagram for Y 5,3
and that the volume at the critical point is √ 59 + 11 33 . V (b∗ ) = 486
(7.42)
7.4.4. An orbifold resolution: Y p,q singularities. Recall that the Y p,q singularities are affine toric Gorenstein singularities generated by four rays, with toric data w1 = (0, 0), w2 = ( p − q − 1, p − q), w3 = ( p, p), w4 = (1, 0) [23]. This includes our earlier example of the complex cone over the first del Pezzo surface, although here we use a different basis for convenience. The outward edge vectors for the polyhedral cone are then easily determined to be (0, p − q, − p + q + 1), ( p, q, −1 − q), ( p, − p, p − 1), (0, 0, 1). (7.43) We now partially resolve the cone by blowing up the Fano corresponding to the interior point45 w = (1, 1). This leads to a non–compact polytope P ⊂ R3 with four vertices, with outward–pointing edge vectors: (1) p1 : u (1) 1 = (0, p − q, − p + q + 1), u 2 = (0, −1, 1), (1)
u 3 = (1, − p + q + 1, p − q − 2), (2)
(2)
p2 : ( p − 1)u 1 = ( p, q, −1 − q), ( p − 1)u 2 = (−1, p − q − 1, − p + q + 2), (2)
( p − 1)u 3 = (0, − p + 1, p − 1), (3) p3 : ( p − 1)u (3) 1 = ( p, − p, p − 1), ( p − 1)u 2 = (0, p − 1, − p + 1), (3)
( p − 1)u 3 = (−1, 1, 0), (4)
(4)
(4)
p4 : u 1 = (0, 0, 1), u 2 = (1, −1, 0), u 3 = (0, 1, −1).
(7.44)
The normalisations here ensure that we correctly get the corresponding weights for the torus action that enter the orbifold localisation formula. Note that they are generally rational vectors, for vertices p2 and p3 . Indeed, it is straightforward to show that the three primitive outward–pointing edge vectors at these vertices span Z3 over Q, but not over Z. In both cases, Z3 modulo this span is isomorphic to Z p−1 . This immediately implies that these vertices are Z p−1 orbifold singularities, and thus the orders of these fixed points are d p2 = d p3 = p − 1. On the other hand, the vertices p1 and p4 are smooth, and thus d p1 = d p4 = 1. 45 Note that any interior point would do.
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D. Martelli, J. Sparks, S.-T. Yau
Fig. 9. Partially resolved polytope P for Y 5,3
We must now use the localisation formula for orbifolds, which includes the inverse orders 1/d p at each vertex p as a multiplicative factor. This is straightforward to compute: V (b) =
p A ,A=1,...,4
=
3 1
d pA
1 (A)
i=1
(b, u i )
p [ p( p − q)b1 + q( p − q)b2 + q(2 − p + q)b3 ] , b3 ( pb1 − pb2 + ( p−1)b3 )(( p−q)b2 + (1− p + q)b3 )( pb1 + qb2 −(1 + q)b3 )
which is indeed the correct expression [22]. Acknowledgement. We would like to thank S. Benvenuti, D. Goldfeld, P. A. Grassi, A. Hanany, J. F. Morales and R. Thomas for interesting discussions. D. M. and J. F. S would also like to thank the Institute for Mathematical Sciences, Imperial College London, for hospitality during the final stages of this work. J. F. S. is supported by NSF grants DMS–0244464, DMS–0074329 and DMS–9803347. S.–T. Y. is supported in part by NSF grants DMS–0306600 and DMS–0074329.
A. The Reeb Vector Field is Holomorphic and Killing In this appendix we give a proof that ξ = J (r ∂/∂r ) is both Killing and holomorphic. This fact is well–known in the literature, although it seems the derivation is not (however, see [69]). Thus, for completeness, we give one here. We begin with the following simple formulae for covariant derivatives on X : ∂ ∂ ∂ = r , ∇r ∂/∂r Y = ∇Y r = Y, ∇r ∂/∂r r ∂r ∂r ∂r ∂ (A.1) ∇Y Z = ∇YL Z − g X (Y, Z )r , ∂r which may easily be checked by computing the Christoffel symbols of the metric g X = dr 2 + r 2 g L . Here ∇ denotes the Levi–Civita connection on (X, g X ), ∇ L is that on (L , g L ), and Y, Z are vector fields on L, viewed as vector fields on X 0 = R+ × L. A straightforward calculation, using ∇ J = 0, then shows that ξ is in fact a Killing vector field on X : ∂ ∂ g X (∇Y ξ, Z ) = g X (∇Y J (r ∂r ) , Z ) = g X (J (∇Y r ∂r ), Z ) = g X (J Y, Z ), (A.2)
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where Z and Y are any two vector fields on L. The last term is skew in Z and Y . Similarly, g X (∇r ∂/∂r ξ, Y ) = g X (ξ, Y ),
(A.3)
g X (∇Y ξ, r ∂/∂r ) = −g X (J Y, r ∂/∂r ) = −g X (Y, ξ ),
(A.4)
and
so that (A.3) is also skew; while the diagonal element g X (∇r ∂/∂r ξ, r ∂/∂r ) = 0
(A.5)
clearly vanishes. Thus we conclude that g X (∇U ξ, V ) is skew in U, V for any vector fields U and V on X , and hence ξ is Killing. One can similarly check that ξ pushes forward to a unit Killing vector on L, where we identify L = X |r =1 . In fact r ∂/∂r and ξ are both holomorphic vector fields. Indeed, for any vector fields U and V , we have the general formula (LU J )V = (∇U J )V + J ∇V U − ∇ J V U,
(A.6)
relating the Lie derivative to the covariant derivative. Using this and the fact that J is covariantly constant, ∇U J = 0, one now easily sees that Lr ∂/∂r J = 0,
Lξ J = 0.
(A.7)
B. More on the Holomorphic (n, 0)–Form In the main text we used the fact that LY = 0 is equivalent to LY = 0, where Y is holomorphic, Killing, and commutes with ξ . Although intuitively clear, one has to do a little work to prove this. We include the details here for completeness. Suppose that LY = 0, where is the canonically defined spinor on the Kähler cone X and Y is a holomorphic Killing vector field that commutes with ξ . This is true of all Y ∈ ts in the main text. The restriction of to L is the spinor θ . Writing = exp( f /2)K , we of course have LY K = 0. Thus we must prove that LY f = 0. Since i∂ ∂¯ f = ρ = J Ric(g X ), and Y is both holomorphic and Killing, we immediately have ¯ Y f = 0. i∂ ∂L
(B.1)
Recall that f = log 2g is degree zero under r ∂/∂r and basic with respect to ξ . Thus the same is true of LY f . We may then interpret (B.1) as a transverse equation, or, when L is quasi–regular, as an equation on the Fano V . In the latter case it follows immediately that LY f = c, a constant. In the irregular case, the paper [70] also claims that a transverse i∂ ∂¯ lemma holds in general. We now compute in in ωn n(n−1)/2 n(n−1)/2 ¯ = ¯ = d e f Y (−1) c ∧ (−1) L ( ∧ ) Y 2n 2n n! n−1 ω = −d e f ∧ dyY ∧ . (B.2) (n − 1)!
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D. Martelli, J. Sparks, S.-T. Yau
Here we have used the fact that is closed, and recall that dyY = −Y ω, where yY is the Hamiltonian function for Y . The right hand side of (B.2) is clearly exact. Hence we may integrate this equation over r ≤ 1 and use Stokes’ Theorem to deduce that in (−1)n(n−1)/2 c 2n
r ≤1
¯ =− ∧
e f dyY ∧ L
ωTn−1 . (n − 1)!
(B.3)
Now, every term in the integrand on the right-hand side of this equation is basic with respect to ξ . In particular, ξ contracted into the integrand is zero46 . However, this means ¯ is certainly non–zero, we that the integral is itself zero. Since the integral of ∧ conclude that c = 0 and hence that LY f = 0, as desired. Conversely, suppose that LY = 0. From Eq. (2.16) we immediately deduce now that LY f = 0 since Y is holomorphic and Killing by assumption. Thus LY K = 0. Hence ¯ c )γ(n) + ¯ c γ(n) LY = 2 ¯ c γ(n) LY . 0 = LY K = (LY
(B.4)
Consider now LY . In fact this must be proportional to . An easy way to see this is to go back to the isomorphism (2.44). The splitting of 0,∗ (X ) into differential forms of different degrees is realised on the space of spinors V via the Clifford action of the Kähler form ω. The latter splits the bundle V into eigenspaces V=
n
Va ,
(B.5)
a=0
where Va is an eigenspace of ω· with eigenvalue i(n − 2a). Moreover, dim Va = and Va ∼ = 0,a (X ).
n a
(B.6)
Recall now that corresponds to a section of 0,0 (X ) under the isomorphism (2.44); hence has eigenvalue in under the Clifford action of ω. Indeed, one can check this eigenvalue rather straightforwardly, without appealing to the isomorphism (2.44). We may now consider LY . Since Y is holomorphic and Killing, it preserves ω. Thus the Clifford action commutes past the Lie derivative, and we learn that LY has the same eigenvalue as . But since this eigenbundle is one–dimensional, they must in fact be proportional: LY = F for some function F. Thus (B.4) says that ¯ c γ(n) = 2F K . 0 = 2F
(B.7)
Since K is certainly non–zero, we conclude that F = 0, and we are done. C. Variation Formulae In this appendix we derive the first and second variation formulae (3.32), (3.37). 46 To see that dy is basic, simply notice that dy (ξ ) = L y = 1 L (r 2 η(Y )) = 0, the last equality ξ Y Y Y 2 ξ following from the fact that ξ preserves η, Y and r , as discussed at various points in the text.
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C.1. First variation. Recall that we linearise the equations for deforming the Reeb vector field around a given background Kähler cone with Kähler potential r 2 . We set ξ(t) = ξ + tY, r 2 (t) = r 2 (1 + tφ).
(C.1) (C.2)
We work to first order in t. Note that contracting (C.1) with J we have r (t)
∂ ∂ =r − t J (Y ). ∂r (t) ∂r
(C.3)
Expanding Lr (t)∂/∂r (t) r 2 (t) = 2r 2 (t)
(C.4)
Lr ∂/∂r φ = 2L J (Y ) log r = −2η(Y ).
(C.5)
Lξ(t)r 2 (t) = 0
(C.6)
Lξ φ = −2d log r (Y ) = 0.
(C.7)
to first order in t gives Recall we also require
which gives, again to first order, In particular, note that when Y = 0 we recover that φ should be homogeneous degree zero and basic, and thus gives a transverse Kähler deformation. Note also that the righthand side of (C.7) is zero if and only if Y is a holomorphic Killing vector field of the background. We may use these equations to compute the derivative of the volume vol[L], which we think of as vol[ξ ], in the direction Y . We write this as dvol[L](Y ). Arguing much as in Sect. 3.1, we obtain n ωn−1 dvol[L](Y ) = −n φdµ + . (C.8) ddc (r 2 φ) ∧ 2 r ≤1 (n − 1)! L The first term arises from the variation of the domain, as in Eqs. (3.21), (3.22). However, one must be careful to note that here φ = φ(r, x) is a function of both r and the point x ∈ L, where L is the unperturbed link L = X |r =1 . We must integrate up to the hypersurface r (t) = 1, which one can check is, to first order in t, given by r = 1 − (1/2)tφ(r = 1, x). Thus one should replace φ by φ(r = 1) in (3.21). Using Stokes’ theorem on the second term on the right-hand side of (C.8), the first term is cancelled, as before, leaving ωTn−1 n dvol[L](Y ) = . (C.9) dc φ ∧ 2 L (n − 1)! Now ξ dc φ = Lr ∂/∂r φ = −2η(Y ), where in the last equality we have used the linearised equation. We thus have dvol[L](Y ) = −n η(Y )dµ, L
which is formula (3.32) in the main text.
(C.10)
(C.11)
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C.2. Second variation. We now take the second variation of the volume. We write ωn (C.12) dvol[L](Y ) = −n(n + 1) (dc r 2 )(Y ) n! r ≤1 as described in the main text. We now again deform r 2 (t) = r 2 (1 + tψ), ∂ ∂ =r − t J (Z ), r (t) ∂r (t) ∂r
(C.13)
giving linearised equations Lr ∂/∂r ψ = −2η(Z ), Lξ ψ = −2d log r (Z ) = 0.
(C.14)
In fact the second equation again will not be used. The derivative of (C.12) gives ωn 1 ψη(Y )dµ − (dc (r 2 ψ))(Y ) d2 vol[L](Y, Z ) = n(n + 1) n! L r ≤1 n−1 1 ω − . (C.15) 2r 2 η(Y ) ddc (r 2 ψ) ∧ 4 (n − 1)! r ≤1 The three terms occur from the variation in domain, the variation of dc (r 2 ), and the variation of the measure, respectively. The last term in (C.15) may be integrated by parts, with respect to d, giving the two terms ωTn−1 1 ωn−1 − , (C.16) − η(Y )dc (r 2 ψ) ∧ (Y ω) ∧ dc (r 2 ψ) ∧ 2 L (n − 1)! (n − 1)! r ≤1 where we have used 2Y ω = −d(r 2 η(Y )). Expanding the first term in (C.16), with respect to dc , gives − ψη(Y )dµ + η(Y )η(Z )dµ. (C.17) L
L
To produce the form of the second term in (C.17), we have used the trick (C.10) of writing the linearised Eq. (C.14) as (dc ψ)(ξ ) = −2η(Z ). Now, the first term in (C.17) precisely cancels the first term in (C.15). Hence we are left with 1 d2 vol[L](Y, Z ) = η(Y )η(Z )dµ n(n + 1) L ωn ωn−1 + Y ω ∧ dc (r 2 ψ) ∧ . (C.18) − (dc (r 2 ψ))(Y ) n! (n − 1)! r ≤1 Finally, we note the identity ωn 0 = Y dc (r 2 ψ) ∧ n! = (dc (r 2 ψ))(Y )
ωn ωn−1 − dc (r 2 ψ) ∧ Y ω ∧ . n! (n − 1)!
(C.19)
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Thus we have shown that d vol[L](Y, Z ) = n(n + 1)
η(Y )η(Z )dµ,
2
(C.20)
L
which is Eq. (3.37) in the main text.
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Commun. Math. Phys. 280, 675–725 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0473-x
Communications in
Mathematical Physics
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries Charles Doran1 , Brian Greene2,3 , Simon Judes2 1 Department of Mathematics, University of Washington, Seattle, WA 98195, USA 2 Institute for Strings, Cosmology and Astroparticle Physics, Department of Physics,
Columbia University, New York, NY 10027, USA. E-mail: [email protected]
3 Department of Mathematics, Columbia University, New York, NY 10027, USA
Received: 25 January 2007 / Accepted: 27 November 2007 Published online: 15 April 2008 – © Springer-Verlag 2008
Abstract: At special loci in their moduli spaces, Calabi–Yau manifolds are endowed with discrete symmetries. Over the years, such spaces have been intensely studied and have found a variety of important applications. As string compactifications they are phenomenologically favored, and considerably simplify many important calculations. Mathematically, they provided the framework for the first construction of mirror manifolds, and the resulting rational curve counts. Thus, it is of significant interest to investigate such manifolds further. In this paper, we consider several unexplored loci within familiar families of Calabi–Yau hypersurfaces that have large but unexpected discrete symmetry groups. By deriving, correcting, and generalizing a technique similar to that of Candelas, de la Ossa and Rodriguez–Villegas, we find a calculationally tractable means of finding the Picard–Fuchs equations satisfied by the periods of all 3–forms in these families. To provide a modest point of comparison, we then briefly investigate the relation between the size of the symmetry group along these loci and the number of nonzero Yukawa couplings. We include an introductory exposition of the mathematics involved, intended to be accessible to physicists, in order to make the discussion self–contained.
Contents 1. 2. 3. 4. 5. A. B. C. D.
Introduction — Mirror Manifolds far from the Fermat Point . . . . . . Cohomology of Hypersurfaces . . . . . . . . . . . . . . . . . . . . . Quintic Calabi–Yau 3–Folds along Enhanced Discrete Symmetry Loci Yukawa couplings of (2, 1)–forms . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Picard–Fuchs Equations for the Z51 Quintic . . . . . . . . . . . . . . Symmetric Hypersurfaces in Weighted Projective Space . . . . . . . . Examples of the Griffiths–Dwork Technique . . . . . . . . . . . . . . Geometric Mondromy . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
676 681 690 713 715 716 718 720 722
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1. Introduction — Mirror Manifolds far from the Fermat Point Although a global description of the complex structure moduli space of many Calabi– Yau manifolds is available, it is often very useful to consider special loci with discrete symmetries. For example, in the context of the E 8 × E 8 heterotic string compactified to 4 dimensions, a simple but powerful way to break the gauge symmetry to SU (3)×SU (2)×U (1)n is to allow Wilson lines, which require that the compact 6d manifold is not simply connected[1–3]. Suitable manifolds with nontrivial fundamental group are most easily constructed by starting out with a simply connected space X˜ with a freely acting discrete symmetry group G, and taking the quotient X = X˜ /G, which then has π1 = G. A more technical but equally important reason for focusing on models with a discrete symmetry group G (or after quotienting with π1 = G), is that many interesting calculations are considerably simpler than in the general case with trivial G. For example, in heterotic compactifications that pass through an E 6 GUT phase, [1,2] phenomenological information is contained in the 27 ⊗ 27 ⊗ 27 Yukawa couplings: √ ¯ ¯¯ j κ(α, β, γ ) = d6 x gΩi jk Ω i j k α ii¯ β j¯ γ kk¯ . (1) CY
Here Ω is the holomorphic 3–form, and α, β, γ ∈ H 1 (T ) correspond to four-dimensional fields that lie in the 27 of E 6 . In general, there are a large number of such integrals, and each is burdensome to calculate. If discrete symmetries are present, then there are relations among the couplings, and many vanish [1,4,5]. Other simplifications have been found in calculations of the stabilized values of moduli in type IIB string backgrounds with nontrivial flux [6]. The vacua are determined by the Gukov–Vafa–Witten superpotential: [7,8]1 W = G ∧ Ω(t) ∝ gi i (t) (2) CY
i
where gi are integers specifying the number of units of flux around the i th 3–cycle of the Calabi–Yau, i (t) is the integral of Ω over the i th 3–cycle (the i th period of Ω), and t denotes the coordinates on the complex structure moduli space. One usually finds the periods by solving certain differential equations that they satisfy, the so–called Picard– Fuchs equations. Unfortunately the order of the equations is in general b3 (CY), which can be very large (∼ 100). It was noted in [9] that if the manifold has discrete symmetries, then the order of the Picard–Fuchs equations is vastly reduced, greatly facilitating their solution. From a mathematical point of view as well, Calabi–Yau manifolds with discrete symmetries have been instrumental to key developments. For example, the first construction of a pair of mirror manifolds [10] involved the prototypical family of 3–folds with discrete symmetry: the Fermat family of quintic hypersurfaces in P4 , defined in homogeneous coordinates [x0 , . . . , x4 ] by: Q(t) = (x0 )5 + (x1 )5 + (x2 )5 + (x3 )5 + (x4 )5 − 5t x0 x1 x2 x3 x4 = 0.
(3)
1 G is a combination of the 3–form fluxes and the dilaton. So in type IIB string theory, generic nonzero flux creates a potential for the complex structure moduli and the dilaton, but not for the Kähler moduli.
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
Quintic Moduli Space Complex Structure
677
Quintic Mirror Moduli Space Complex Structure
(dim = 101)
V (t)
(dim = 1)
V (0) V (0) = (Z5 )3
V (t) (Z5 )3
V (0)
V (t)
Kähler structure
Kähler structure
(dim = 1)
(dim = 101)
Fig. 1. A schematic diagram of the moduli spaces of the quintic in P4 and its mirror. The line shown in the quintic moduli space is mirror to the horizontal line on the mirror side. But the quotient by (Z5 )3 gives the vertical line
This family (denoted V (t)) is invariant under the S5 group of permutations of the xi ’s, as well as 4 Z5 scalings2 generated by: g1 = (1, 0, 0, 0, 4), g2 = (1, 0, 0, 4, 0),
g3 = (1, 0, 4, 0, 0), g4 = (1, 4, 0, 0, 0),
(4)
where (a, b, c, d, e) means (x0 , x1 , x2 , x3 , x4 ) → (γ a x0 , γ b x1 , γ c x2 , γ d x3 , γ e x4 ) and γ 5 = 1 = γ . Because we are working in projective space, g1 , g2 , g3 , g4 are not independent symmetries of V (t), since g1 g2 g3 g4 = (4, 4, 4, 4, 4) = I , implying that the symmetry group is [S5 (Z5 )4 ]/Z5 . Quotienting V (0) by (Z5 )3 (generated by particular combinations of the 4 gi ’s) and resolving the orbifold singularities appropriately (0), the mirror to the Fermat quintic[10]. The single complex structure paraproduces V (t) (the meter of the mirror is then t, but somewhat confusingly, V (t)/(Z5 )3 is not V 3 mirror to V (t)) except at t = 0. This is because V (t) and V (t)/(Z5 ) differ from V (0) and V (0)/(Z5 )3 respectively only in their complex structure, but moving in the complex structure moduli space of the mirror corresponds to moving in the Kähler moduli space of the original manifold and vice versa. See Fig. 1. One can see that a mirror must exist for any nonsingular quintic in P4 by considering deformations of the conformal field theory away from the Fermat point by truly marginal operators. These are interpreted differently on the original and mirror manifolds (complex structure and Kähler deformations switch roles), but they exist on both sides. It is clear that it is the symmetry of the Fermat quintic that makes this construction possible, by allowing an explicit realization of the mirror at one point in moduli space. The goal of [14] was to find other points in moduli space where the mirror can be presented as a resolution of a quotient by a discrete symmetry. One might guess from the simplicity of the form of (3) that moving away from the Fermat family will reduce the symmetry to a subgroup of [S5 (Z5 )4 ]/Z5 . Indeed this seems to be the case for local deformations, but it turns out to be spectacularly false if one searches the moduli 2 We use the notation: Z = Z/nZ. n
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space sufficiently. For example, in [14] the hypersurface V41 (t) defined by: Q 41 (t) = (y0 )4 y1 + (y1 )4 y2 + (y2 )4 y3 + (y3 )4 y4 + (y4 )4 y0 − 5t y0 y1 y2 y3 y4 = 0
(5)
was found to possess a Z41 scaling symmetry generated by (1, 37, 16, 18, 10) where the entries now indicate nontrivial 41st roots of unity. It is worthwhile to recall the reasoning that leads to the form of (5). The idea is to implement the (Z5 )3 quotient of the Fermat quintic by making an unusual, apparently ill-defined, change of variables: 4/5 1/5
4/5 1/5
4/5 1/5
4/5 1/5
4/5 1/5
(x0 , x1 , x2 , x3 , x4 ) → (y0 y1 , y1 y2 , y2 y3 , y3 y4 , y4 y0 ).
(6)
Generally, the fractional powers would require, at the very least, choices of branch cuts to make the map and its inverse well defined. However, it is easy to see that away from coordinate hyperplanes, appropriate coordinate identifications make this unnecessary. From (6) it is immediate that imposing a (Z5 )3 group of identifications on the (x0 , x1 , x2 , x3 , x4 )–the very same group, in fact, that yields the mirror Calabi–Yau family–the map from the y’s to the x’s becomes well defined. However, this is not yet sufficient for (6) to be one–to–one. By solving for the inverse, we find, for example: 256
−64
16
−4
1
y0 = x0205 x1205 x2205 x3205 x4205 and cyclic permutations
(7)
which requires further identifications be made on the yi ’s. One can check that (7) is well defined if one identifies yi with its image under the Z41 scaling symmetry indicated above. This suggests a relationship: VFermat (t) V41 (t) ∼ . (Z5 )3 Z41
(8)
But since the fractional change of variables is only invertible away from coordinate hyperplanes, the relationship in (8) is not a biholomorphism. It was argued in [14] using the methods of toric geometry, that the two quotients are nevertheless topologically identical, representing two parametrizations of the complex structure moduli space of the quintic mirror, at different points in the Kähler moduli space. The relationship (8) suggests that by focusing on points in the quintic moduli space that have a maximal discrete symmetry group in their local neighborhood, and by then quotienting by this maximal group, we generate the mirror partners to these manifolds. Since the initial manifolds differ by deformations of their complex structures, their mirrors would then differ by deformations of their Kahler structures. In particular, this would mean that the Picard–Fuchs equations for the periods of the holomorphic 3–form of the Z41 quotient, or equivalently for those periods of V41 (t) invariant under Z41 , should agree with the standard Picard-Fuchs equation on the mirror quintic. In [14] a tedious calculation using the Griffiths–Dwork technique on V41 (t)/Z41 was shown to yield: 4 3 2 5 d 4 d 3 d 2 d 1−t − 10t − 25t − 15t −t Ω=0 (9) dt 4 dt 3 dt 2 dt which is precisely the Picard–Fuchs equation satisfied by the periods of the holomorphic (t) [12]. 3–form of the mirror family V
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Table 1. Six 1–parameter families of quintic hypersurfaces with discrete symmetries
1
2 3 4 5 6
Q(t) 1 a 5 + b5 + c5 + d 5 + e5 − tabcde 5 1 5 1 5 1 5 1 5 1 5
a 4 b + b4 c + c4 d + d 4 e + e4 a − tabcde a 4 b + b4 c + c4 d + d 4 a + e5 − tabcde a 4 b + b4 c + c4 a + d 5 + e5 − tabcde a 4 b + b4 c + c4 a + d 4 e + e4 d − tabcde a 4 b + b4 a + c5 + d 5 + e5 − tabcde
Scaling Symmetries
Action
(Z5 )3
(4, 1, 0, 0, 0), (4, 0, 1, 0, 0), (4, 0, 0, 1, 0)
Z41
(1, 37, 16, 18, 10)
Z51
(1, 47, 16, 38, 0)
Z5 × Z13
(0, 0, 0, 4, 1), (1, 9, 3)
Z3 × Z13
(0, 0, 0, 1, 2), (1, 9, 3)
(Z5 )2 × Z3
(0, 0, 4, 1, 0), (0, 0, 4, 0, 1), (1, 2, 0, 0, 0)
It is easily seen that the technique illustrated by (6) offers numerous variations, providing a rich set of new enhanced symmetry loci. For example, one can consider: 4/5 1/5
4/5 1/5
4/5 1/5
4/5 1/5
(x0 , x1 , x2 , x3 , x4 ) → (y0 , y1 y2 , y2 y3 , y3 y4 , y4 y1 ),
(10)
which leads to a family with another unfamiliar symmetry group, Z51 . Several other examples were tabulated in [14,19], which we repeat in Table 1. Given how useful loci with discrete symmetry have been in the development of our understanding of Calabi–Yau moduli spaces, new symmetric families are of great interest. They expand the range of examples to which analytic methods can be applied, and provide new testing grounds for mirror symmetry, rational curve counts, moduli stabilization, and phenomenology. To orient our analysis, it is interesting to ask where in moduli space these new loci reside; for example, where is the V41 (t) family in relation to the Fermat locus? We might attempt a linear transformation x(y) on Q 41 (t) to bring it into the form: Q 41 = (y0 )5 + (y1 )5 + (y2 )5 + (y3 )5 + (y4 )5 + · · · = 0,
(11)
where the ellipsis indicates a specific combination of quintic monomials at most cubic in any of the yi ’s. But this brings a more pressing issue into sharp relief. Notice that such a linear transformation would obscure the presence of the Z41 symmetry. Similarly, it is clear that an arbitrary linear transformation of the Fermat quintic would make the (Z5 )3 symmetry significantly less obvious because the symmetry would no longer act diagonally on the homogeneous coordinates. This may lead one to wonder whether the Z41 symmetric family and the Fermat family are even distinct loci. Perhaps Z41 acts nondiagonally on (3), and (Z5 )3 nondiagonally on (5). To show that this is not the case, we write the polynomial defining the Fermat family as: 15 Q 0 − t Q ∞ = 0, where Q 0 = (x0 )5 + (x1 )5 + (x2 )5 + (x3 )5 + (x4 )5 is the Fermat polynomial and Q ∞ = x0 x1 x2 x3 x4 . If the symmetry groups of Q 0 and Q ∞ are denoted G 0 and G ∞ respectively, then the automorphism group of a generic member of the Fermat family is just G 0 ∩ G ∞ . In [15] it was shown that G 0 = S5 (Z5 )4 , i.e. the automorphisms of the Fermat quintic are permutations of the homogeneous coordinates, and scalings by 5th roots of unity, excluding an overall scaling which is
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trivial in projective space.3 On the other hand G ∞ = S5 (C∗ )4 , i.e. we can permute the coordinates, and scale them. The full automorphism group of a generic member of the pencil is then G = S5 (Z5 )3 , i.e. the subgroup of G 0 ∩ G ∞ that scales 15 Q 0 − t Q ∞ by an overall factor. In other words there are no more symmetries than those found above by inspection. G is a finite group with 53 × 5! elements, so it has no subgroup of order 41 or 51. It follows that the families with Z41 and Z51 symmetries cannot be isomorphic to the Fermat family. The new loci are thus distinct from the Fermat family, and therefore constitute a new probe for enriching our understanding of Calabi–Yau manifolds and their moduli spaces. Utilizing this probe requires that we’re able to perform the basic calculations of periods, familiar from studies of Fermat families, which contain essential information about the complex structure and geometric monodromy of the families. The purpose of this paper is to set up the formalism for doing so. Plan of the paper. In Sect. 2, we review the aspects of the cohomology of families of hypersurfaces required to understand the Picard–Fuchs equations and the Griffiths– Dwork procedure for finding them. The expert reader will find much in this section that is already familiar. However, because our results and, in particular, the way they differ from [9], depend critically on this background, a self-contained summary is essential. We emphasize those aspects which play key roles in the sections that follow. Some examples and further aspects of the formalism are developed in Appendices C and D. In Sect. 3 we derive an alternative to the Griffiths–Dwork method for symmetric hypersurfaces in Pn which greatly reduces the labor of computation. The technique is similar in its details to that applied to the quintic 3–fold by Candelas, de la Ossa and Rodriguez–Villegas [9], but our results improve on [9] in three key respects: – Here (in Sect. 3.1) the technique is derived in a rigorous fashion from fundamental results of Griffiths [16]. – In [9] it is claimed that these methods compute differential equations whose solutions are periods of the holomorphic 3–form. By carefully relating the method to the work of Griffiths [16] (reviewed in Section 2), we show that this is generally not true. Instead, we show that most of the resulting equations are satisfied by other elements of the period matrix, i.e. integrals of elements of H 3 (X, C) not contained in H 3,0 (X ). – The technique involves constructing diagrams which display the relevant relations among periods in a useful way. We find an algorithm for sytematically constructing these diagrams, summarized in Sect. 3.1.2. We then apply the procedure to calculate the Picard–Fuchs equations for the Z41 – symmetric family of 3–folds. Other families can be treated in the same way, and the results for the Z51 case are tabulated in Appendix A. To give a feel for the new loci and to see another way in which they differ from the Fermat family, Sect. 4 examines the effect of the discrete symmetries on the Yukawa couplings of the 6 quintics in Table 1. We find a somewhat surprising relation between the number of nonzero couplings and the size of the symmetry group. Finally, in Appendix B we indicate how to extend the technique to symmetric Calabi– Yau hypersurfaces in weighted projective spaces, and we compute the Picard–Fuchs equation for the example of a Fermat–type hypersurface in WP[41,48,51,52,64] [256]. 3 The result: Lemma 3.2 of [15] is rather more general. For d ≥ 3 and n ≥ 2 and (d, n) = (3, 2) or (4, 3), the symmetries of the degree d Fermat hypersurface in Pn are Sn+1 (Zd )n . In words, a semi–direct product of permutations and scalings by d th roots of unity.
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2. Cohomology of Hypersurfaces The Calabi–Yau manifolds we will consider are all hypersurfaces in Pn , i.e. submanifolds of Pn defined by the vanishing locus of a single homogeneous polynomial.4 This is a rather special choice, which can be generalized in many ways, but is particularly convenient to analyze. The goal of this section is to review a useful way to get our hands on elements of the Dolbeault cohomology groups of such a hypersurface. We consider a smooth hypersurface V ⊂ Pn defined by the zero locus of an irreducible degree l polynomial Q(x) in the homogeneous coordinates [x0 , . . . , xn ]. As might be expected, it is hard to set up coordinates on V and describe differential forms on the hypersurface directly. Instead we make use of a beautiful generalization of the Cauchy integral formula due to Griffiths [16]. Starting from Cauchy’s theorem 1 P(z) wi (i th Residue), (12) dz = 2πi C Q(z) i
where the sum is over the poles enclosed by the contour C that winds around the i th pole wi times, and the residue is the coefficient of the 1/z term in a Laurent expansion of P(z) f (z) = Q(z) about the pole. Griffiths interpreted the right-hand side as the integral of a 0–form over a 0–cycle on the Riemann sphere P1 . The 0–cycle (which we suggestively P(z) denote by V ) is the set of poles of Q(z) dz, each weighted by the number of times the contour C winds around, and the 0–form is the value of the residue at each pole. Notice that adding an exact rational 1–form (whose poles are contained in V ) to the left-hand side integrand makes no difference, so we can think of the residue as a map: Res : H(V ) → H 0 (V, C),
(13)
where H(V ) is like the de Rham cohomology group H 1 (P1 − V, C), but using only rational forms. The purpose of generalizing this story to higher dimensions is to represent (n − 1)–forms on V (which becomes a hypersurface), as residues of rational n–forms on the complement Pn − V . The latter are considerably easier to work with. 2.1. Some results of Griffiths. Let An (V ) be the space of rational n–forms on Pn with polar locus V . Then we define H(V ) = An (V )/d An−1 (V ), i.e. the de Rham cohomology of rational n–forms on Pn − V . 5 The residue map Res : H(V ) → H n−1 (V, C) is then defined by the property: 1 ϕ= Res(ϕ). (14) 2πi T (γ ) γ Here ϕ ∈ H(V ), γ is an (n − 1)–cycle in V and T (γ ) is a tubular neighborhood of γ in Pn − V . More precisely, T (γ ) is a circle bundle over γ with an embedding into Pn − V such that it encloses γ . For small enough radii, any two such bundles are homologous in Hn (Pn − V, Z), so the construction is unique. A rather abstract way to 4 In Appendix B we generalize this slightly to hypersurfaces in weighted projective space. 5 Note that ∂ An (V ) = 0 since n is the maximal holomorphic degree of a form on P , and ∂¯ An (V ) = 0 n
because rational forms are by definition holomorphic. In defining H(V ) there is therefore no need to restrict the numerator of the quotient to closed forms only — they are all closed.
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state the definition (14) is that Res is the dual of the so–called Leray coboundary map: Hn−1 (V, Z) → Hn (Pn − V, Z) which sends [γ ] to [T (γ )]. A more concrete description of the residue map can be given as follows. Let ϕ be a smooth differential form on Pn , except for possible singularities on V . To indicate the order of the singularities, suppose that for some positive integer k, f k ϕ and f k−1 d f ∧ ϕ are smooth everywhere if f = 0 is a local defining equation for V . In terms of local (affine) coordinates z 1 , . . . , z n , we have ϕ = ϕ(z)dz 1 ∧ . . . ∧ dz n , but close to the hypersurface (i.e. near f = 0), we can choose coordinates (z 1 , . . . , z n−1 , f ) and write: df ∧α β + k−1 fk f
1 dα β + k−1 α 1 d =− , + k−1 k−1 k−1 f f
ϕ=
(15)
where α and β are smooth forms and do not contain d f . This expression is only valid in a single patch, but by making use of a partition of unity, one can show that for k = 1, ϕ = dψ + η, where ψ and η are globally defined smooth forms with poles of order k − 1 along V . It follows that up to an exact form, ϕ can be reduced to a form with a pole of order 1 along V : ϕ − d(ψ1 + · · · + ψl−1 ) =
α ∧ d f + β , f
(16)
where ψa has a pole of order k − a along V . The residue is then the coefficient of d f / f restricted to the hypersurface: Res(ϕ) = α |V .
(17)
This is precisely analogous to the usual definition of the residue as the coefficient of the 1/z term in the Laurent expansion. Note that the residue of a rational n–form with k = 1 is necessarily holomorphic, but since the construction above uses a partition of unity, residues of forms with k > 1 are only smooth in general. Having defined the residue map, we now need to explore its properties. In particular, we have the following question: which rational n–forms on Pn − V map to which cohomology classes on V ? The answer is provided by another beautiful theorem of Griffiths, in preparation for which we must introduce some further formalism. Let Ank (V ) ⊂ An (V ) denote the rational n–forms on Pn − V with poles of order k along V . By analogy with H(V ), we can define the cohomology groups: Hk (V ) =
Ank (V ) d An−1 k−1 (V )
.
(18)
It is important to note that two such groups Hk (V ) and Hk (V ) generally have a nonzero intersection. If for example we take k = 1 and k = 2, the statement is just that there are rational forms with double poles that differ from rational forms with simple poles only by exact rational forms with simple poles. We will explicitly delineate such intersections shortly, but notice that what we’re speaking of here is different from the reduction of pole order in the residue construction. There we were interested in lowering the pole order by adding smooth forms. Here we are only allowed to add rational forms. We will return shortly to the question of when such a reduction is possible.
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For the moment, let us follow Griffiths and write the groups Hk (V ) as a sequence of inclusions: H1 (V ) ⊂ H2 (V ) ⊂ · · · · · · ⊂ Hn (V ) = H(V ).
(19)
The nontrivial claim here is the final equality: Hn (V ) = H(V ), the proof of which can be found in [16]. A decomposition like (19) with inclusions (as opposed to a direct sum decomposition) is called a filtration, and we will refer to (19) as the filtration of H(V ) by order of pole. There is another filtration we are interested in, the so–called Hodge filtration of H n−1 (V, C), given by: Fn−1,n−1 (V ) ⊂ Fn−1,n−2 (V ) ⊂ · · · · · · ⊂ Fn−1,0 (V ) = H n−1 (V, C),
(20)
where Fa,b (V ) = H a,0 (V ) ⊕ H a−1,1 (V ) ⊕ . . . ⊕ H b,a−b . This time the equality on the right-hand side is just the Hodge decomposition of cohomology, which holds for all Kähler manifolds: H n−1 (V, C) = H n−1,0 (V ) ⊕ H n−2,1 (V ) ⊕ . . . ⊕ H 0,n−1 (V ).
(21)
Any algebraic submanifold of Pn (a hypersurface for example) is necessarily Kähler, so we are not imposing any new restriction. The essence of Griffiths’ theorem is that the residue map acts in a very nice way between the order of pole filtration and the Hodge filtration: H1 (V ) ⊂ H2 (V ) ⊂ · · · · · · ⊂ Hn (V ) = H(V ) ↓ Res ↓ Res ↓ Res ↓ Res Fn−1,n−1 (V ) ⊂ Fn−1,n−2 (V ) ⊂ · · · · · · ⊂ Fn−1,0 (V ) = H n−1 (V, C)
(22)
We have already confirmed the far left part of the diagram: rational n–forms on Pn with poles of order 1 on V map to holomorphic (n − 1)–forms on V , i.e. elements of H n−1,0 (V ) = Fn−1,n−1 (V ). The remainder of the proof can be found in [16]. This is the answer we were looking for to the question: which (n − 1)–forms on V are the residues of which n–forms on Pn − V ? The order of the pole of the form on Pn − V (and hence the degree of the numerator) determines which of the Hodge filtrants the residue lies in. We can say more. It is clear when a form in Fn−1,n−2 is also in Fn−1,n−1 , but not so clear yet when a form in H2 (V ) is also in H1 (V ). In words: we know from (16) that the order of the pole of a form can be lowered arbitrarily by adding an appropriate smooth form, but when can this decrease be accomplished using only rational forms? The key to finding out is to look more closely at rational n–forms on Pn − V . Working in homogeneous coordinates [x 0 , . . . , x n ], one can show that any such n–form ϕ can be written:6 ϕ=
P(x) Ω0 , Q(x)k
Ω0 =
n i . . . ∧ dx n , (−1)i x i dx 0 ∧ . . . dx
(23)
i=0
where Q(x) = 0 defines the hypersurface V , and P(x) is a homogeneous polynomial obeying deg P = k deg Q − (n + 1) in order that ϕ is well defined on projective space. It follows that a rational n–form in Pn − V can be specified by an element of i indicates that it should be left out of the wedge product. 6 This is Corollary 2.1 of [16]. The hat on the dx
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C[x 0 , . . . , x n ]kl−(n+1) , i.e. a polynomial of degree kl −(n +1) with complex coefficients, where l = deg Q. The information we need is contained in the formula:7 n n Ω0 ∂ Q(x) 1 Ω0 ∂ Pi (x) P (x) = + exact rational forms. i Q(x)k+1 ∂ xi k Q(x)k ∂ xi i=0
(24)
i=0
P(x) Ω Q(x)k 0
Thus the order of pole of a form ϕ = ∈ Hk (V ) can be lowered using a rational n ∂ Q(x) form iff P(x) = i=0 Pi (x) ∂ x i for some polynomials Pi (x). The ideal generated by [∂0 Q(x), . . . , ∂n Q(x)] is called the Jacobian ideal of Q(x), and denoted J (Q). So the order of pole can be lowered iff the numerator is in J (Q). This relates to the Hodge filtration as follows. If we quotient a filtrant (a single group in the filtration) by the filtrant to its left, we find Fn−1,k (V )/Fn−1,k+1 (V ) = H k,n−1−k (V ). We have just seen that for the order of pole filtration we have:8 C[x 0 , . . . , x n ]kl−(n+1) Hk (V ) = . Hk−1 (V ) J (Q) So the residue map induces a homomorphism:
(25)
C[x 0 , . . . , x n ]kl−(n+1) → H n−k,k−1 (V ). (26) J (Q) One further note of importance is the following. The image of the map (26) in H n−k,k−1 (V ) is called the primitive cohomology of V , and denoted P H n−k,k−1 (V ). If in (22) we replace the Hodge filtrants Fa,b (V ) with their analogs constructed from primitive cohomology groups (denoted Fa,b 0 (V )), then the residue maps become isomorphisms. This is the most we will say about the general properties of the residue map. Later we will consider the simplifications that arise if V has discrete symmetries. 2.2. Example: Quintic Calabi–Yau 3–folds. Consider the case n = 4, l = 5. Since l = n + 1 is precisely the condition c1 = 0, V is a Calabi–Yau 3–fold. The Hodge filtrants are: 3,0 (V ), F3,3 0 (V ) = P H 3,0 F3,2 (V ) ⊕ P H 2,1 (V ), 0 (V ) = P H 3,0 F3,1 (V ) ⊕ P H 2,1 (V ) ⊕ P H 1,2 (V ), 0 (V ) = P H 3,0 F3,0 (V ) ⊕ P H 2,1 (V ) ⊕ P H 1,2 (V ) ⊕ P H 0,3 (V ). 0 (V ) = P H
The isomorphism with the filtration given by order of pole is:9 H1 (V ) ⊂ H2 (V ) ⊂ H3 (V ) ⊂ H4 (V ) = H(V ) ↓ Res ↓ Res ↓ Res ↓ Res ↓ Res 3,3 3,2 3,1 3,0 3 F0 (V ) ⊂ F0 (V ) ⊂ F0 (V ) ⊂ F0 (V ) = H (V, C)
(27)
7 Formula 4.5 of [16] 8 A common alternative notation for the Jacobian ring C[x 0 ,...,x n ]m is R m . Q J (Q) 9 For odd dimensional hypersurfaces we have P H p,q (V ) H p,q (V ). In the case of Calabi–Yau 3–folds this can be seen as follows: One can define P H p,3− p (V ) alternatively as the kernel of the Lefschetz map
L : H p,3− p (V ) → H 5 (V ) defined by L([φ]) = [J ∧ φ], where J is a Kähler form on V . For a Calabi–Yau 3–fold with SU (3) holonomy, b5 = 0, so the kernel of L is the whole of H p,3− p (V ).
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Now look for example at Res: H2 (V ) → F3,2 0 (V ). Notice that [α2 ] ∈ H2 (V ) maps to F3,3 (V ) ⊂ F3,2 (V ) iff [α2 ] = [α1 ], where α1 has a pole of order 1, i.e. if α2 = α1 + dη. From [16], this is equivalent to P ∈ J (Q), where α2 = QP2 Ω0 . We therefore have: F3,2 (V ) C[x0 , . . . , x4 ]5 H2 (V ) 03,3 P H 2,1 (V ). J (Q) H1 (V ) F0 (V )
(28)
Similarly, we have: C[x0 , . . . , x4 ]0 H1 (V ) 3,0 (V ), H F3,3 0 (V ) P H 0 (V ) J (Q) C[x0 , . . . , x4 ]10 F3,1 H3 (V ) 0 (V ) P H 1,2 (V ), H 3,2 (V ) 2 F J (Q) 0 (V ) C[x0 , . . . , x4 ]15 F3,0 4 (V ) 0 (V ) H P H 0,3 (V ). H (V ) 3 F3,1 J (Q) 0 (V )
(29) (30) (31)
The maps are given explicitly by: C[x0 , . . . , x4 ]5n [P] ←→ (3 − n, n) piece of Res J (Q)
P Ω0 Qk
∈ P H 3 (V, C), (32)
where k = deg5 P + 1. This is just the standard and often–used association between 5n th order monomials and elements of H 3−n,n (V ). For instance, the isomorphism (28) (n = 1) has a familiar interpretation as two different ways of looking at deformations of complex structure: on the one hand as an element of H 2,1 (V ) and on the other as an additional monomial term in the defining equation of the hypersurface V . As an example of the use of the above formalism, we derive a common expression for the unique holomorphic 3–form Ω = Res ΩQ0 . Working in the patch x0 = 0, we can scale the homogeneous coordinates so that x0 = 1, and therefore Ω0 = dx1 ∧dx2 ∧dx3 ∧ dx4 = dz 1 ∧ dz 2 ∧ dz 3 ∧ dz 4 , where z i = xi /x0 . Writing 0 ,x 1 , x 2 , x 3 , x 4 ), we 1 Q2 as Q(x dz ∧dz ∧dz 3 ∧dz 4 1 2 3 4 . Next we replace define f = Q(1, z , z , z , z ), and so Ω = Res f the coordinate z 4 with f , and using d f =
Ω = Res
∂f ∂z i
dz i , find:
dz 1 ∧ dz 2 ∧ dz 3 ∧ d f f
∂f ∂z 4
dz 1 ∧ dz 2 ∧ dz 3 = ∂f 4 ∂z
(33) V
which can be found in [1,4,11] for example. It is more difficult to identify forms with monomials of higher order, but explicit expressions are derived in [4]. Aside from the case of the holomorphic 3–form, the residues are generally not of pure Hodge type, i.e. they are not elements of any single group H p,3− p (V ). This conclusion requires modification if V possesses discrete symmetries.
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Hodge type of the non–holomorphic forms. We saw that for a generic quintic in P4 , the 3–forms corresponding to monomials of order 5n live in the (3 − n)th Hodge filtrant: F3,n = H 3,0 ⊕ . . . ⊕ H 3−n,n . In the presence of a discrete scaling symmetry, which is also a symmetry of the holomorphic 3–form,10 we can make a stronger statement. For example, let [ω] ∈ H 3,0 ⊕ H 2,1 be a class corresponding to a 5th order monomial. ¯ If Ω is the holomorphic 3–form, and ω2,1 a ∂–closed (2, 1)–form, we can write [ω] = c[Ω] + [ω2,1 ] for some constant c ∈ C. Now consider the integral: ω ∧ Ω = c Ω ∧ Ω. (34) The right-hand side is manifestly invariant, so if [ω] transforms by a nontrivial scaling, then c = 0. This however is just the same as saying that [ω] ∈ H 2,1 . A similar argument shows that the noninvariant 10th order monomials correspond to classes in H 2,1 ⊕ H 1,2 rather than the full F3,1 . 2.3. Families of hypersurfaces and the period matrix. Say we allow Q(x) and hence V to depend on a parameter t which takes values in a space T . If we vary t smoothly from t1 to t2 and do not allow V to become singular along the way, then V (t1 ) and V (t2 ) are diffeomorphic, but not in general biholomorphic to one another.11 Our interest is in the cohomology of the hypersurface V , and in particular in the Hodge decomposition, H n−1 (V, C) = H p,q (V ). (35) p+q=n−1
As we move around in complex structure moduli space, the Hodge decomposition changes because what we mean by a ( p, q)–form changes. However, one can always define a so–called topological basis of H n−1 (V, C) that does not change (at least under local deformations). This basis consists of the duals of a basis of topological cycles in Hn−1 (V, Z).12 The purpose of this section is to study in concrete terms how the Hodge basis varies with respect to the ‘anchor’ of the topological basis.13 We can phrase the discussion in terms of a fixed real differentiable manifold X diffeomorphic to V (t), and a basis of t–dependent (n − 1)–forms Ω Xi (t) ∈ Λn−1 (X, C), where i = 0, . . . , b3 − 1. We can choose the forms Ω Xi (t) to have fixed bidegree ( p, q) in the complex structure at point t, so that their cohomology classes provide the Hodge decomposition for each t ∈ T . But because residues are generally not of pure Hodge type (see (22)), it makes more sense to work in terms of the Hodge filtration. In other words, we choose forms Ω Xi (t) to be elements of Fn−1, p (V ) rather than H n−1− p, p (V ). Next we define the period matrix of V (t) as the integrals of Ω Xi (t) over a topological basis of Hn−1 (X, Z). Denoting such a basis γi , i = 1, . . . , 2(n − 1), the periods are: j Ω X (t). (36) γi
10 In the mathematics literature, symmetries that preserve the holomorphic 3–form are called symplectic automorphisms. 11 The complex structure we are talking about on V (t) is the one inherited from the embedding in P . n 12 We make use of the inclusion: H n−1 (V, Z) → H n−1 (V, C). 13 This subject has been formulated in a more abstract fashion, under the name variations of Hodge structure. An excellent and readable introduction can be found in [17].
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It is these integrals that contain the information about how the Hodge structure varies with t. We would therefore like to differentiate them with respect to t. It might seem naive to just differentiate under the integral sign, but in fact that is the correct thing to do. Technically we are looking for a connection on the bundle over T whose fibers are H n−1 (V (t), C) (the so–called Hodge bundle), but this bundle admits a flat connection ∇t known as the Gauss–Manin connection that can be defined by the property: d j j ∇t Ω X (t) = (37) Ω X (t). dt γi γi j
Notice that by repeatedly differentiating Ω X (t) we generate a sequence of representatives of classes in H n−1 (X, C). Since dim H n−1 (X, C) is finite (for X compact) we must j eventually find that some derivative of Ω X (t) can be related to lower derivatives up to exact forms, which disappear upon integration. It follows that each column of the j period matrix, i.e. the periods γi Ω X (t) for fixed j, obeys a differential equation in the variables t, called a Picard–Fuchs equation. By comparing these equations, we will be able to distinguish between different families of Calabi–Yau manifolds. First though, despite the simplification of (37), we still have a hurdle to overcome. The problem is that in Sect. 2 we worked with t–dependent hypersurfaces in Pn rather than an underlying differentiable manifold X with t–dependent (n − 1)–forms. Given a (n − 1)–cycle (and hence a period) in V (t1 ), what is the corresponding (n − 1)–cycle in V (t2 )? The conclusion we will find is that we should use the tools of Sect. 2 to rewrite the period matrix as integrals of rational forms on the complement of V ; the reader interested more in the final result than the technical details may want to skip directly to the result, (41). We would like to have a precise notion of cycles varying smoothly with t. To this end, consider π : X → T a differentiable proper mapping of differentiable manifolds, with rank = dim T , so that X t = π −1 (t) is diffeomorphic to a compact manifold X for any t. We are interested in the case where X t inherits a complex structure from X , and is biholomorphic to V (t). The Ehresmann theorem implies that the fibration X → T is locally trivial, so we can think of X as a fiber bundle with base T . We can then specify an Ehresmann connection on X → T , i.e. a decomposition of the tangent space T X into vertical and horizontal subspaces T h X and T v X respectively. This defines a notion of parallel transport along a path in T . So given a cycle γ (t1 ) ∈ X t1 and a smooth path linking t1 and t2 , an Ehresmann connection defines a cycle γ (t2 ) ∈ X t2 . Although γ (t2 ) depends on the path taken in T as well as the choice of connection, [γ (t2 )] ∈ Hn−1 (X t2 , Z) (locally) does not. Moreover, if α(t) : X t → X is a diffeomorphism, then in Hn−1 (X, Z) we have: [α(t1 )γ (t1 )] = [α(t2 )γ (t2 )]. To summarize, one can choose a basis of Hn−1 (V (t0 ), Z), and parallel transport it using an Ehresmann connection to obtain a (locally) unique horizontal family of homology classes. How does this help us? It means we can rewrite the period integrals: Ω Xi (t) = ΩVi (t) (t), (38) γi
γi (t)
where for example ΩV0 (t) (t) is the holomorphic (n − 1)–form on V (t), found using the techniques that lead to (33). We have transformed the integrand to something we know how to work with, but at the expense of introducing t dependence into the cycle over
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which we are integrating. How do we then differentiate the periods with respect to t? The answer is to represent ΩVi (t) (t) as a meromorphic form on Pn :
γi (t)
ΩVi (t) (t) =
which is just to say that ΩVi (t) (t) = Res holomorphic (n − 1)–form:
T (γi (t))
PΩ0 Q(t)k
P Ω0 , Q(t)k
. For example, the case i = 0 is the
γi (t)
ΩV0 (t) (t)
=
(39)
T (γi (t))
Ω0 . Q(t)
(40)
For i = 0, there will be a nontrivial P and a higher power of Q(t) in the denominator. Now for t in a sufficiently small neighborhood of t0 , T (γi (t)) is homologous to T (γi (t0 )) in Hn (Pn − V, C). For the case of a single parameter, we can then differentiate as follows: d PΩ0 PΩ0 PΩ0 dQ d = = −k . (41) k+1 dt dt T (γi (t)) Q(t)k dt T (γi (t0 )) Q(t)k Q(t) T (γi (t0 )) If r = dimC (Hn (V )) = dimC H n−1 (V, C) , only the first r − 1 derivatives can be linearly independent. Therefore the periods must satisfy a linear ordinary differential equation of order at most r — this is a Picard–Fuchs equation.
2.4. Picard–Fuchs equations à la Griffiths–Dwork. The tools introduced above provide a systematic, but usually tedious, technique for calculating Picard–Fuchs equations, outlined for example in [18]: 1. Differentiate the period r times. If Q(t) is linear in t, then one finds: dr dt r
T (γi (t))
PΩ0 = Q(t)k
T (γi (t))
(k + r − 1)! Ω0 ∂Q r P − . (k − 1)! Q(t)k+r ∂t
(42)
r 2. Write P − ∂∂tQ explicitly as an element of J (Q), i.e. as i Ai (x) ∂∂ xQi , where Ai (x) are polynomials of degree (n + 1)(r + k − 1) − n. 3. Use Formula 4.5 from [16] to reduce the order of the pole: n n Ω0 ∂ Q 1 Ω0 ∂ Ai A = + d(· · · ). i Q k+1 ∂ xi k Qk ∂ xi i=0
(43)
i=0
4. Repeat the above steps until the rth derivative has been expressed in terms of lower derivatives. This is the required equation. Appendix C contains two worked out applications of the above steps: the Hesse family of elliptic curves and the Fermat family of quintics in P4 .
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Parameterizing the moduli space. Conventionally, the Picard–Fuchs equation satisfied by periods of the holomorphic 3–form on the Fermat quintic or its mirror, is written as in Sect. 2.4:
2 3 4 1 θ4 − x θ + θ+ θ+ θ+ tP1 = 0. (204) 5 5 5 5 d , while t is the parameter of the Fermat family: Recall that x = t −5 and θ = x dx
Q(t) =
1 5 a + b5 + c5 + d 5 + e5 − tabcde. 5
(200)
For some purposes this is rather convenient; for example, since (204) is in generalized hypergeometric form, one can make use of standard results about the monodromy of its solutions [22]. However, the coordinate x is not particularly useful for analysing the equations satisfied by the other periods. To see why, we review the usual argument for the change of variables. It is noted14 that the transformation t → e2πi/5 t can be undone by a simple change of coordinates: xi → e−2πi/5 xi , where xi is any of the homogeneous coordinates [x0 , x1 , x2 , x3 , x4 ] = [a, b, c, d, e]. Since this transformation is holomorphic, it follows that the hypersurfaces specified by t and by e2πi/5 t are biholomorphic. The natural coordinate on the complex structure moduli space therefore seems to be t 5 or t −5 . Let us then take x = t 5 , and consider the following period of a (2, 1)–form:
a 3 b2 Ω0 = Q(x)2
a 3 b2 1 Ω0 . 5 + b5 + c5 + d 5 + e5 − x 1/5 abcde 2 a 5
(44)
Suppose we want to interpret its monodromy around x = t = 0. We note that circling around x = 0 corresponds to t → e2πi/5 t, which can then be undone by a → e−2πi/5 a. 4 The form as a whole receives a scaling by e2πi/5 = e− 5 2πi/5 , coming from the a 3 in the numerator, as well as the factor of da in Ω0 . If on the other hand we decided to absorb the change into a scaling of b instead of a, then the form would scale by a different factor. Working in terms of x, it is therefore tricky to determine what part of the monodromy of the periods comes from geometric monodromy of the cycles, and what part comes from the scalings of the forms. This is not an issue for the holomorphic 3–form (or its derivatives), and hence does not arise in discussions of the quintic mirror. In that case, the numerator of the form is some power of abcde, so the overall scaling is the same no matter which of the coordinates one chooses to absorb the scaling of t. In the case of the holomorphic 3–form, the numerator is 1, so the only scaling comes from Ω0 which behaves the same as abcde. One can then make the form as a whole invariant under the monodromy by including an additional factor of t in the numerator. This explains the (sometimes obscure) appearance of the factor of t in Eq. (204). The monodromy of the periods is then entirely geometric in origin. For the non–invariant forms it is unclear how to achieve the same outcome, so we work in terms of the original variable t, thus ensuring that the monodromy of the Picard–Fuchs equations comes only from the cycles. 14 See for example Sect. 2.2 of [18].
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3. Quintic Calabi–Yau 3–Folds along Enhanced Discrete Symmetry Loci As shown by the examples in Appendix C, the Griffiths–Dwork method involves some algebraic tedium, particularly at Step 2 of the procedure outlined in 2.4. In this section we exploit a much simpler technique, similar to that found in [9] but one in which our derivation, utilizing the results reviewed in 2, establishes a different interpretation than that suggested in [9]. We then apply it to two families of Calabi–Yau 3–folds. 3.1. The diagram technique for the Fermat quintic. We start with Formula 4.5 from [16], which in our notation is: n n Ω0 ∂ Q(t) 1 Ω0 ∂ Ai A = + exact forms. i Q(t)k+1 ∂ xi k Q(t)k ∂ xi i=0
(45)
i=0
Recall that Q(t) is the defining equation of the hypersurface, Ai are homogeneous i . . .∧dx n . We now specialize polynomials in the [x i ], and Ω0 = i (−1)i x i dx 0 ∧. . . dx to the Fermat family of quintics, i.e. n = 4, and: 1 5 a + b5 + c5 + d 5 + e5 − tabcde, (46) Q(t) = 5 where [a, b, c, d, e] = [x 0 , x 1 , x 2 , x 3 , x 4 ] is an alternative notation for the homogeneous coordinates. For this case, we have: ∂ Q(t) = (x i )4 − t x 0 . . . xi . . . x 4 . ∂ xi
(47)
Next we choose Ai = δi j xi A, where A = (x 0 )v0 (x 1 )v1 (x 2 )v2 (x 3 )v3 (x 4 )v4 = a v0 bv1 cv2 d v3 ev4 . Griffiths’ formula then becomes: 1 Ω Ω0 0 i 5 0 1 2 3 4 = A (x ) − t x x x x x (1 + vi )A + exact forms. (48) Q(t)k+1 k Q(t)k In order that these forms are well defined on P4 , we must have: k=
1 1 deg A + 1 = (v0 + v1 + v2 + v3 + v4 ) + 1, 5 5
(49)
so we will write k(v) from now on. Integrating over a cycle in P4 − V gives: Ω0 (1 + vi ) Ω0 Ω0 i 5 0 1 2 3 4 A(x ) = t Ax x x x x + A. (50) k(v) Q(t)k(v)+1 Q(t)k(v)+1 Q(t)k(v) We can write this relation in the form: (v0 , . . . , vi + 5, . . . , v4 ) =
(1 + vi ) (v0 , v1 , v2 , v3 , v4 ) k(v) + t (v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1)
which uses the following shorthand for the periods: Ω0 (x 0 )v0 (x 1 )v1 (x 2 )v2 (x 3 )v3 (x 4 )v4 . (v0 , v1 , v2 , v3 , v4 ) = Q(t)k(v)
(51)
(52)
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
The opportunity to write a differential equation arises because: d d Ω0 (v0 , v1 , v2 , v3 , v4 ) = a v0 b v1 c v2 d v3 e v4 dt dt Q(t)k(v) Ω0 = k(v) a v0 +1 bv1 +1 cv2 +1 d v3 +1 ev4 +1 Q(t)k(v)+1 = k(v)(v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1).
691
(53)
To organize these relations in a useful way, the authors of [9] presented (51) as a diagram: (v0 , v1 , v2 , v3 , v4 ) −→ (v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1) ↓ Di (v0 , . . . , vi + 5, . . . , v4 )
(54)
One should read this simply as saying that these three periods are linearly related, the subscript on Di indicating which linear relation is being used. It’s also useful to keep in mind that the period on the top right is proportional to the derivative of the period on the top left. One can then build up larger diagrams, for example: (0, 0, 0, 0, 0) ↓ D0 (5, 0, 0, 0, 0) ↓ D1 (5, 5, 0, 0, 0) ↓ D2 (5, 5, 5, 0, 0) ↓ D3 (4, 4, 4, 4, −1) → (5, 5, 5, 5, 0) ↓ D4 (4, 4, 4, 4, 4)
→ (1, 1, 1, 1, 1) → (2, 2, 2, 2, 2) → (3, 3, 3, 3, 3) → (4, 4, 4, 4, 4) ↓ D0 ↓ D0 ↓ D0 → (6, 1, 1, 1, 1) → (7, 2, 2, 2, 2) → (8, 3, 3, 3, 3) ↓ D1 ↓ D1 → (6, 6, 1, 1, 1) → (7, 7, 2, 2, 2) ↓ D2 → (6, 6, 1, 1, 1)
Several comments are in order: 1. The entry (4, 4, 4, 4, −1) does not correspond to a period. As one can see from (51), this part of the diagram just says that (4, 4, 4, 4, 4) is proportional to (5, 5, 5, 5, 0). 2. Only one of the Di is used in each row, and each Di is used once. 3. Working up the diagram using (51), one can write (4, 4, 4, 4, 4) at the bottom in terms of the top row of periods. This is then a 4th order differential equation for the period (0, 0, 0, 0, 0), i.e. the periods of the holomorphic 3–form. 2 2 d4 5 4 d 3 d 2 d (t − 1) 4 + 10t + t (0, 0, 0, 0) = 0. (55) + 25t + 15t dt dt 2 dt 2 dt d Or, in terms of η = t dt : 1 (η + 1)4 − 5 η(η − 1)(η − 2)(η − 3) (0, 0, 0, 0, 0) = 0. t
(56)
One can put this in generalized hypergeometric form with a change of variables: d λ = t 5 , θ = λ dλ :
2 3 1 4 1 θ− θ− −λ θ + θ θ− (0, 0, 0, 0, 0) = 0. (57) 5 5 5 5
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5 t . 4. This procedure is rather more convenient than the Griffiths–Dwork approach [16,20].
This is the equation satisfied by 4 F3
1 1 1 1 5, 5, 5, 5 4 3 2 5, 5, 5
The reason that a 4th order equation appears as opposed to order 204 = dimC P H 3 (V (t), C) = b3 can be traced back to the discrete symmetries of V (t) mentioned in the introduction. Recall that the symmetry group is (S5 (Z5 )4 )/Z5 . For the subgroup of scaling symmetries we can take the following for generators: g1 = (1, 0, 0, 0, 4),
g2 = (1, 0, 0, 4, 0),
g3 = (1, 0, 4, 0, 0),
(58)
where as before the entries indicate powers of a nontrivial 5th root of unity. Notice that the rule (54) ensures that all periods in a given diagram transform in the same representation of (Z5 )3 . It follows that as one moves around the base of the family (i.e. as t varies), each 3–form only samples a subspace of H 3 (X, C) spanned by those 3–forms transforming in the same representation of (S5 (Z5 )4 )/Z5 . 3.1.1. Interpretation of equations from the diagram technique. Equation (51) and its diagramatic representation (54) can be found in Sect. 3.1 of [9]. However, in this paper a different meaning is attached to the components of the diagram: (v0 , v1 , v2 , v3 , v4 ). In particular the authors of [9] define: 1 (x 0 )v0 (x 1 )v1 (x 2 )v2 (x 3 )v3 (x 4 )v4 (v0 , v1 , v2 , v3 , v4 ) = d5 x , (59) 2πi Γ Q(t)k(v)+1 where Γ is a 5–torus in C5 whose factors are loops winding around the 5 varieties ∂i Q = 0. Equation 3.2 of [9] then claims that this is in fact a period of the holomorphic 3–form: Ω, (60) (v0 , v1 , v2 , v3 , v4 ) = γv
where γv is a 3–cycle whose homology class corresponds to the element of the Jacobian ideal represented by the monomial (x 0 )v0 (x 1 )v1 (x 2 )v2 (x 3 )v3 (x 4 )v4 . The purpose of the extended introduction in Sect. 2 (and in particular the statement of Griffith’s theorems) is to show that this interpretation cannot be correct. It is clear from the definition of the Gauss–Manin connection in Eq. (37) that all 204 periods of the holomorphic 3–form obey the 4th order Eq. (55). And it is clear from (22) and the derivation above that the equations corresponding to other monomials are not satisfied by different periods of Ω, but rather by all 204 periods of other, non–holomorphic forms.15 Periods of the forms corresponding to 5th order monomials. In general, the quintic monomials map to classes in F3,2 = H 3,0 ⊕ H 2,1 , but we saw in Sect. 2.4 that the symmetries can entail further restrictions. We therefore classify the elements of C[a, b, c, d, e]5 by their transformation properties under the (Z5 )3 symmetries of the Fermat quintic. Fortunately there is no need to look at each of the 126 monomials separately, because they fall into 5 sets which transform among themselves under permutations of the homogeneous coordinates. See Table 2. 15 There is a trivial sense in which our interpretation of the diagrams is consistent with that of [9]. One can choose a basis of cycles so that all but four periods of the holomorphic 3–form Ω vanish identically over the entire moduli space. These 200 vanishing periods of Ω then satisfy the ODEs assigned to them in [9], simply because zero is a solution of any linear homogeneous differential equation.
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Table 2. The 126 quintic monomials split into 5 sets under the permutation and scaling symmetries. The number in the right column is the number of example monomials listed supplemented by their permutations Representation Class
Example Monomials
Number of Monomials in Class
A B C D E
a 5 , b5 , c5 , d 5 , e5 , abcde a 4 b, b2 cde a 3 b2 a 3 bc a 2 b2 c
6 40 20 30 30
Table 3. After the quotient by the Jacobian ideal J (Q) = [∂i Q(t)], there are 101 independent monomials Representation Class
Independent Example Monomials
Number of Independent Monomials in Class
A B C D E
abcde b2 cde a 3 b2 a 3 bc a 2 b2 c
1 20 20 30 30
As is well known, finding a set of 101 of these 126 that are independent as elements of C[a, b, c, d, e]5 /J (Q) is immediate. From (47), we see that a 5 , b5 , c5 , d 5 , e5 and abcde are all equivalent, and from b∂a Q(t) = a 4 b − tb2 cde,
(61)
we see that we can get rid of half of the monomials in representation class B. The result is summarized in Table 3. 5 By the argument at the end of Sect. 2.4, if [m 5 ] ∈ C[a,b,c,d,e] is in representations J (Q) m 5 Ω0 2,1 B, C, D or E, then the classes Res Q 2 are elements of H rather than H 3,0 ⊕ H 2,1 . There is no such restriction for the single class in representation A: the derivative of the holomorphic 3–form with respect to t. 3.1.2. Algorithm for diagram construction. In the previous section as well as in [9], the diagram for the holomorphic 3–form was constructed and utilized in an ad hoc manner. We now present a general algorithm which can be applied straightforwardly to all the forms on several families of hypersurfaces. Representation class A: {abcde}. 1. In this case, we are looking for an equation satisfied by (1, 1, 1, 1, 1). First, differentiate with respect to t as many times as is necessary to create the following ‘staircase diagram’: (3, 3, 3, 3, 3) → (4, 4, 4, 4, 4) ↓ D0 (7, 2, 2, 2, 2) → (8, 3, 3, 3, 3) ↓ D1 (6, 6, 1, 1, 1) → (7, 7, 2, 2, 2) ↓ D2 (62) (5, 5, 5, 0, 0) → (6, 6, 6, 1, 1) ↓ D3 (4, 4, 4, 4, −1) → (5, 5, 5, 5, 0) ↓ D4 (4, 4, 4, 4, 4)
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As before, one uses each of the Di ’s once, so that the bottom left and top right periods match. 2. Next extend the diagram to the left as far as possible by adding extra mini–diagrams:
(5, 0, 0, 0, 0) B ↓ D1 (5, 5, 0, 0, 0)C ↓ D2 (5, 5, 5, 0, 0) D ↓ D3 (4, 4, 4, 4, −1) → (5, 5, 5, 5, 0) E ↓ D4 (4, 4, 4, 4, 4)
(1, 1, 1, 1, 1) A → (2, 2, 2, 2, 2) → (3, 3, 3, 3, 3) → (4, 4, 4, 4, 4) ↓ D0 ↓ D0 ↓ D0 → (6, 1, 1, 1, 1) → (7, 2, 2, 2, 2) → (8, 3, 3, 3, 3) ↓ D1 ↓ D1 → (6, 6, 1, 1, 1) → (7, 7, 2, 2, 2) (63) ↓ D2 → (6, 6, 6, 1, 1)
The one exception is that we do not add a piece to the left of the period we are interested in: (1, 1, 1, 1, 1). The subscripts on the leftmost periods in each row are the symbols we will use to denote them in equations. 3. We now write the relations corresponding to the leftmost mini–diagrams, i.e. a coupled system in (A, B, C, D, E): 1 1 1 η(η − 1)(η − 2)A = t 4 E, E = (η + 1)D, D = (η + 1)C, 24 4 3 1 C = (η + 1)B, ηB = t (η + 2)A, 2
(64) (65)
d d where as before η is the logarithmic derivative d(log t) = t dt . 4. Finally we manipulate the coupled system to find an equation containing A alone, η(η − 1)(η − 2)(η − 4) − t 5 (η + 2)4 A = 0. (66)
For this last step relations like ηt = (t + 1)η are particularly helpful. Representation class B: b2 cde . (0, 2, 1, 1, 1) A → (1, 3, 2, 2, 2) ↓ D4 ↓ D4 (0, 2, 1, 1, 6) B → (1, 3, 2, 2, 7) ↓ D3 (−1, 1, 0, 5, 5) → (0, 2, 1, 6, 6)C ↓ D0 (3, 0, −1, 4, 4) → (4, 1, 0, 5, 5) D ↓ D2 (2, −1, 3, 3, 3) → (3, 0, 4, 4, 4) E ↓ D1 (2, 4, 3, 3, 3)
The coupled system is: 1 η(η − 1)A = t 3 E, E = t D, D = tC, 6 1 1 C = (η + 2)B, B = (η + 2)A. 3 2
(67) (68)
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
The equation for A is:
η(η − 1) − t 5 (η + 2)2 A = 0.
695
(69)
Representation class C: a 3 b2 .
(0, −1, 2, 2, 2) → (1, 0, 3, 3, 3) D ↓ D1 (−1, 3, 1, 1, 1) → (0, 4, 2, 2, 2) E ↓ D0 (4, 3, 1, 1, 1)
(3, 2, 0, 0, 0) A → (4, 3, 1, 1, 1) ↓ D4 ↓ D4 (3, 2, 0, 0, 5) B → (4, 3, 1, 1, 6) ↓ D3 (2, 1, −1, 4, 4) → (3, 2, 0, 5, 5)C (70) ↓ D2 → (2, 1, 4, 4, 4)
The coupled system is: 1 1 η A = t 2 E, E = t D, ηD = t 2 C, 2 3 1 1 C = (η + 1)B, B = (η + 1)A. 3 2 The equation for A is:
η(η − 3) − t 5 (η + 1)2 A = 0.
(71) (72)
(73)
Representation class D: a 3 bc .
(−1, 2, 2, 1, 1) → (0, 3, 3, 2, 2) E ↓ D0 (4, 2, 2, 1, 1)
(3, 1, 1, 0, 0) A → (4, 2, 2, 1, 1) ↓ D4 ↓ D4 (2, 0, 0, −1, 4) → (3, 1, 1, 0, 5) B → (4, 2, 2, 1, 6) ↓ D3 ↓ D3 (2, 0, 0, 4, 4)C → (3, 1, 1, 5, 5) ↓ D2 (1, −1, 4, 3, 3) → (2, 0, 5, 4, 4) D ↓ D1 → (1, 4, 4, 3, 3)
The coupled system is: 1 η A = t 2 E, 2
1 ηE = t 2 D, 3 C = t B,
The equation for A is:
1 (η + 1)C, 3 1 B = (η + 1)A. 2
D=
η(η − 2) − t 5 (η + 1)(η + 2) A = 0.
(74) (75)
(76)
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Representation class E: a 2 b2 c . (2, 2, 1, 0, 0) A → (3, 3, 2, 1, 1) ↓ D4 ↓ D4 (1, 1, 0, −1, 4) → (2, 2, 1, 0, 5) B → (3, 3, 2, 1, 6) ↓ D3 ↓ D3 (0, 0, −1, 3, 3) → (1, 1, 0, 4, 4)C → (2, 2, 1, 5, 5) ↓ D2 ↓ D2 (0, 0, 4, 3, 3) D → (1, 1, 5, 4, 4) ↓ D1 (−1, 4, 3, 2, 2) → (0, 5, 4, 3, 3) E ↓ D0 (4, 4, 3, 2, 2)
The coupled system is: 1 η(η − 1)A = t 3 E, 6
E=
1 (η + 1)D, 3
C = t B,
B=
D = tC,
(77)
1 (η + 1)A. 2
(78)
The equation for A is: η(η − 1) − t 5 (η + 3)(η + 1) A = 0.
(79)
Summary of Columns of the Period Matrix Corresponding to 5 t h Order Monomials Number of Classes A B C D E
1 20 20 30 30
Operator Annihilating Periods
Hodge Type
η(η − 1)(η − 2)(η − 4) − t 5 (η + 2)4 H 3,0 ⊕ H 2,1 η(η − 1) − t 5 (η + 2)2 H 2,1 5 2 η(η − 3) − t (η + 1) H 2,1 5 η(η − 2) − t (η + 1)(η + 2) H 2,1 5 η(η − 1) − t (η + 3)(η + 1) H 2,1
Periods of the forms corresponding to 10th order monomials It looks like an unpleasant 14 task to sift through the 4 = 1001 10th order monomials, classifying them by their transformations under (Z5 )3 , and checking for relations in J (Q). So we take a different route. Our aim is to find a convenient basis of C[a, b, c, d, e]10 /J (Q). To this end, notice that we can choose a basis of C[a, b, c, d, e]5 /J (Q) not containing any homogeneous coordinate raised to the 4th or 5th power. If the basis elements are restricted to being monomials, then the basis is unique: [m i ] = [abcde], [b2 cde], [a 3 b2 ], [a 3 bc], [a 2 b2 c] + permutations. (80)
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
697
Here i = 1, . . . , 101. In other words, any element of C[a, b, c, d, e]5 /J (Q) can be written: i αi [m i ] with αi ∈ C. We now define a map denoted : C[a, b, c, d, e]5 C[a, b, c, d, e]10 → J (Q) J (Q) 3 3 3 3 3 a b c d e . such that [m] = m
:
(81) (82)
Note that it is crucial that the basis [m i ] contains no elements like [a 4 b] or [a 5 ] in order that the map is well defined. We define the action of to be linear: (83) α1 [m 1 ] + α2 [m 2 ] = α1 [m 1 ] + α2 [m 2 ]. 5 The claim is that if [m i ] is the above basis of C[a,b,c,d,e] , then [m i ] is a basis of J (Q)
C[a,b,c,d,e]10 . J (Q)
To prove it, consider the pairing: F:
given by
C[a, b, c, d, e]10 C[a, b, c, d, e]15 C[a, b, c, d, e]5 × → C (84) J (Q) J (Q) J (Q) ⎡ ⎛ ⎞ ⎤ F⎣ αi [m i ] , ⎝ β j [m˜ j ]⎠⎦ = i, j αi β j m i m˜ j , (85) i
j
where [m˜ i ] is a basis of C[a, b, c, d, e]10 /J (Q). The isomorphism with C is realized ˆ by taking the coefficient of [a 3 b3 c3 d 3 e3 ] in the sum. We’ll denote this coefficient F. Now we know that: C[a, b, c, d, e]5 C[a, b, c, d, e]10 dim = dim = 101. (86) J (Q) J (Q) So, [m i ] is a basis of C[a, b, c, d, e]10 /J (Q) if the 101×101 matrix Fˆ [m i ], [m j ] is nondegenerate. But it’s not hard to see that Fˆ [m i ], [m j ] = δi j , the 101 × 101 identity matrix. So [m i ] is the dual basis to [m i ]. We have therefore found the basis we were looking for. Before constructing diagrams for the periods corresponding to [m i ], it is worth looking at how the operator interacts with the discrete symmetries of the Fermat quintic. First some notation: for the (Z5 )3 generated by g1 , g2 and g3 , we say that a monomial [m] is in the (n 1 , n 2 , n 3 ) representation if gi [m] = γ n i [m] for i = 1, 2, 3, where γ is a nontrivial 5th root of unity. It is easy to see that if [m] transforms in the representation (n 1 , n 2 , n 3 ), then [m] transforms in the representation (5 − n, 5 − n, 5 − n). We can go a step further, and think of as acting on the classes of irreps of (Z5 )3 that transform into each other under permutations. (We labeled these A, B, C, D and E.) One finds a simple action: A = A,
B = B,
C = C,
D = D,
E = E. (87)
For example, representation class C includes the monomial with (n 1 , n 2 , n 3 ) = (3, 3, 3). We find [a 3 b2 ] = [bc3 d 3 e3 ], which has (n 1 , n 2 , n 3 ) = (2, 2, 2). One can check that this representation is also in class C. Even better, a permutation of [bc3 d 3 e3 ] already appears in diagram (70) for representation class C, so there is no need to construct a new diagram. We will see that this happens for the other representations as well. a 3 b2 ,
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Representation class A. The 5th order monomial was abcde, with period (1, 1, 1, 1, 1) = ˜ All we need to do is write A. Acting with the map gives (2, 2, 2, 2, 2) = 21 ddtA = A. ˜ the relations for A in terms of A, to get the coupled system: 1 1 1 η(η − 1) A˜ = t 3 E, E = (η + 1)D, D = (η + 1)C, 12 4 3 1 1 ˜ C = (η + 1)B, η(η − 1)B = t 2 (η + 3) A, 2 2
(88) (89)
˜ which leads to the following equation for A:
η(η − 1)(η − 3)(η − 4) − t 5 (η + 3)4 A˜ = 0.
(90)
Representation class B. The (2, 1) period was (0, 2, 1, 1, 1) = A. The corresponding ˜ Again we write the (1, 2) period16 is (3, 1, 2, 2, 2) = (1, 3, 2, 2, 2) = 21 ddtA = A. ˜ relations for A in terms of A, to get the coupled system: 1 ˜ η A = t 2 E, E = t D, D = tC, 3 1 ˜ C = (η + 2)B, ηB = t (η + 3) A, 3
(91) (92)
˜ which leads to the following equation for A: η(η − 4) − t 5 (η + 3)2 A˜ = 0.
(93)
Representation class C. The (2, 1) period was (3, 2, 0, 0, 0). The corresponding (1, 2) period is (0, 1, 3, 3, 3) = (1, 0, 3, 3, 3) which appears in the diagram denoted D. We can therefore just use the same coupled system as before to solve for D: η(η − 2) − t 5 (η + 4)2 D = 0.
(94)
Representation class D. The (2, 1) period was (3, 1, 1, 0, 0). The corresponding (1, 2) period is (0, 2, 2, 3, 3) = (0, 3, 3, 2, 2) which appears in the diagram as E. So as with representation class C, we just use the same coupled system as found for the (2, 1)–forms to solve for E: η(η − 3) − t 5 (η + 3)(η + 4) E = 0.
(95)
16 We refer to the periods of forms corresponding to 10th order monomials as (1, 2)–periods, but one should keep in mind that generally these are integrals of classes contained in H 3,0 ⊕ H 2,1 ⊕ H 1,2 , not just H 1,2 .
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
699
Representation class E. The (2, 1) period was (2, 2, 1, 0, 0) = A. The corresponding ˜ As with representation (1, 2) period is (1, 1, 2, 3, 3) = (3, 3, 2, 1, 1) = 21 ddtA = A. ˜ classes A and B, we write the relations for A in terms of A, to get the coupled system: 1 ˜ η A = t 2 E, 3
1 (η + 1)D, D = tC, 3 ˜ C = t B, ηB = t (η + 2) A,
E=
˜ which leads to the following equation for A: η(η − 4) − t 5 (η + 2)(η + 3) A˜ = 0.
(96) (97)
(98)
Summary of Columns of the Period Matrix Corresponding to 10 t h Order Monomials # Classes A B C D E
1 20 20 30 30
Operator Annihilating Periods
Hodge Type
η(η − 1)(η − 3)(η − 4) − t 5 (η + 3)4 H 3,0 ⊕ H 2,1 ⊕ H 1,2 η(η − 4) − t 5 (η + 3)2 H 2,1 ⊕ H 1,2 5 2 η(η − 2) − t (η + 4) H 2,1 ⊕ H 1,2 5 η(η − 3) − t (η + 3)(η + 4) H 2,1 ⊕ H 1,2 5 η(η − 4) − t (η + 2)(η + 3) H 2,1 ⊕ H 1,2
Periods of the class corresponding to 15th order monomials The space C[a, b, c, d, e]15 / J (Q) is 1 dimensional, and we can take the single nonzero basis vector to be the monomial a 3 b3 c3 d 3 e3 . This choice allows us to reuse the diagram for (1, 1, 1, 1, 1), now 3 defining A = (3, 3, 3, 3, 3) = ddtΩ3 ∈ H 3,0 ⊕ H 2,1 ⊕ H 1,2 ⊕ H 0,3 . The coupled system becomes: 1 1 1 η A = t 2 E, E = (η + 1)D, D = (η + 1)C, 4 4 3 1 1 C = (η + 1)B, η(η − 1)(η − 2)B = t 3 (η + 4)A. 2 6 The resulting equation is: η(η − 2)(η − 3)(η − 4) − t 5 (η + 4)4 A = 0.
(99) (100)
(101)
The algorithm we’ve given for the diagrammatic method is both systematic and powerful. As our discussion of the Fermat pencil has made clear, the key prerequisite is focusing on a family of varieties each of whose members respects a large discrete symmetry group. Earlier, we emphasized that the quintic moduli space has other, less familiar, loci that respect other, less familiar, discrete symmetries. We now extend the diagrammatic technique to these families, focussing for definiteness on the Z41 case. The results for the Z51 family are summarized in Appendix A.
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3.2. The Z41 quintic: Q(t) = 15 a 4 b + b4 c + c4 d + d 4 e + e4 a −tabcde = 0. The symmetries of this quintic family are the Z41 scalings generated by g = (1, 37, 16, 18, 10) where the entries now indicate the powers of a nontrivial 41st root of unity multiplying each homogeneous coordinate. There is also a Z5 group of cyclic permutations of the homogeneous coordinates generated by α : (a, b, c, d, e) → (b, c, d, e, a), which is intertwined with the scalings by the relation: αgα −1 = g 10 . As for the Fermat family we have: j Ω (t) = γi
T (γi )
a v0 b v1 c v2 d v3 e v4 Ω0 . Q(t)k(v)
(102)
(103)
But the relation that previously was interpreted diagramatically is now: ∂ Q(t) Ω0 Ax i k(v)+1 ∂ xi Q(t) ∂ Q(t) a ∂a ∂ Q(t) with b ∂b ∂ Q(t) c ∂c ∂ Q(t) d ∂d ∂ Q(t) e ∂e
= = = = = =
1 Ω0 A(1 + vi ) + exact forms, k Q(t)k(v) 1 4 4 a b + e4 a − tabcde, 5 5 4 4 1 4 b c + a b − tabcde, 5 5 1 4 4 c d + b4 c − tabcde, 5 5 1 4 4 d e + c4 d − tabcde. 5 5 4 4 1 e a + d 4 e − tabcde. 5 5
(104) (105) (106) (107) (108) (109)
Equations (105–109) now have three terms on the right hand side, in contrast with their counterparts in the (Z5 )3 case, so these relations cannot be used to construct diagrams as before. But one can rectify the problem by taking particular linear combinations: ⎛ 4 ⎞ ⎞⎛ ⎛ ⎞ a b − tabcde a∂a Q(t) 256 1 −4 16 −64 ⎜ b4 c − tabcde ⎟ ⎜ −64 256 1 −4 16 ⎟ ⎜ b∂b Q(t) ⎟ ⎜ ⎟ ⎟⎜ ⎜ ⎟ 4 ⎟ ⎜ 16 −64 256 1 −4 ⎟ ⎜ c∂c Q(t) ⎟ = 205 ⎜ ⎜ c d − tabcde ⎟ . (110) ⎝ −4 16 −64 256 1 ⎠ ⎝ d∂ Q(t) ⎠ 4 ⎝ d e − tabcde ⎠ d 1 −4 16 −64 256 e∂e Q(t) e4 a − tabcde Performing the same manipulations on (104) and integrating gives: (v0 , . . . , vi + 4, vi+1 + 1, . . . , v4 ) =
f (i, v) (v0 , v1 , v2 , v3 , v4 ) 205k(v) +t (v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1), (111)
where f (i, v) is the i th component of the column vector: ⎞⎛ ⎞ ⎛ v0 + 1 256 1 −4 16 −64 ⎜ −64 256 1 −4 16 ⎟ ⎜ v1 + 1 ⎟ ⎟⎜ ⎟ ⎜ ⎜ 16 −64 256 1 −4 ⎟ ⎜ v2 + 1 ⎟ , ⎝ −4 16 −64 256 0 ⎠ ⎝ v + 1 ⎠ 3 1 −4 16 −64 256 v4 + 1
(112)
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
and again k(v) = 1 +
i
701
vi . The above is encoded in the following diagrams:
(v0 , v1 , v2 , v3 , v4 ) −→ (v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1), ↓ D0 (v0 + 4, v1 + 1, v2 , v3 , v4 )
(113)
(v0 , v1 , v2 , v3 , v4 ) −→ (v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1), ↓ D1 (v0 , v1 + 4, v2 + 1, v3 , v4 )
(114)
(v0 , v1 , v2 , v3 , v4 ) −→ (v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1), ↓ D2 (v0 , v1 , v2 + 4, v3 + 1, v4 )
(115)
(v0 , v1 , v2 , v3 , v4 ) −→ (v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1), ↓ D3 (v0 , v1 , v2 , v3 + 4, v4 + 1)
(116)
(v0 , v1 , v2 , v3 , v4 ) −→ (v0 + 1, v1 + 1, v2 + 1, v3 + 1, v4 + 1). ↓ D4 (v0 + 1, v1 , v2 , v3 , v4 + 4)
(117)
The algorithm for finding Picard–Fuchs equations is the same as in the Fermat case except that we can no longer use diagrams with −1 appearing in any of the entries of (v0 , v1 , v2 , v3 , v4 ). Periods of the holomorphic 3–form. (0, 0, 0, 0, 0) A ↓ D0 (4, 1, 0, 0, 0) B ↓ D1 (4, 5, 1, 0, 0)C ↓ D2 (4, 5, 5, 1, 0) D ↓ D3 (3, 4, 4, 4, 0) E → (4, 5, 5, 5, 1) ↓ D4 (4, 4, 4, 4, 4)
→ (1, 1, 1, 1, 1) → (2, 2, 2, 2, 2) → (3, 3, 3, 3, 3) → (4, 4, 4, 4, 4) ↓ D0 ↓ D0 ↓ D0 → (5, 2, 1, 1, 1) → (6, 3, 2, 2, 2) → (7, 4, 3, 3, 3) ↓ D1 ↓ D1 → (5, 6, 2, 1, 1) → (6, 7, 3, 2, 2) ↓ D2 → (5, 6, 6, 2, 1)
The coupled system is: 1 η(η − 1)(η − 2)(η − 3)A = t 4 ηE, ηE = t (η + 1)D, 6 1 1 D = (η + 1)C, C = (η + 1)B, B = (η + 1)A, 3 2 which leads to the equation: η(η − 1)(η − 2)(η − 3) − t 5 (η + 1)4 A = 0.
(118) (119)
(120)
Notice that the equation for the periods of the holomorphic 3–form is the same as for the Fermat family. As indicated in the introduction, this fact can be interpreted as a consequence of mirror symmetry. The Greene–Plesser mirror construction [10,14]
702
C. Doran, B. Greene, S. Judes Table 4. The 126 quintic monomials according to their transformation under Z41
0
1
2
3
4
5
6
7
8
9
10
a4b b4 c c4 d d4e e4 a abcd 11 a 2 c2 d d 2 e2 b
e3 bc b2 c2 d
b4 d c3 d 2
e3 bd a 2 c2 e b2 d 2 c
d 3 c2 c3 ab a 2 b2 e
a5 d 3 b2 b3 ac a 2 cde
d4c e3 ac c2 abd
b3 ad a2 d 2 e c2 e2 b
d5 b3 e2 e3 ad d 2 abc
e5 c3 a 2 a 3 be e2 bcd
d 3 ab a 2 b2 c
12 a 2 b2 d c2 e2 a
13 c3 be a2 d 2 c b2 e2 a
14 a4e b3 ce e2 acd
16 b3 de d 2 e2 a
17 e4 d a 3 bd d 2 bce
18 c3 ae a 2 e2 b
19 c4 b d 3 be b2 ace
20 a4 c b3 c2 c2 ade
23 a 3 e2 e3 cd c2 d 2 b
24 c4 a b3 d 2 d 3 ae a 2 bce 35 b4 e a 3 c2 c3 de e2 abd
25 e3 d 2 d 3 bc b2 c2 a
15 e4 c d 3a2 a 3 bc c2 bde 26 b4 a c3 ad a 2 bde
27 d4b c3 e2 e3 ab b2 acd 38 a 2 e2 c b2 d 2 e
28 b2 e2 c c2 d 2 a
29 a 3 ce b2 d 2 a c2 e2 d
30 d 3 ac a 2 c2 b b2 e2 d
31 b3 a 2 a 3 de d 2 e2 c
21 b5 e3 c2 c3 bd b2 ade 32 d4a e2 a 2 a 2 bcd
39 c5 a3d 2 d 3 ce c2 abe
40 c3 b2 b3 ac a 2 e2 d
22 a4d e3 b2 b3 cd d 2 ace 33 c4 e d 3 e2 e2 abc
34 a2 d 2 b b2 c2 e
36 e4 b a 3 b2 b2 cde
37 a 3 cd c2 d 2 e
involves quotienting the manifold at t = 0 by the group of scaling symmetries that preserve the holomorphic 3–form, so the differential forms that descend to the mirror are precisely those that transform trivially. Now recall from Fig. 1 that the quotient of the Fermat family is a 1–parameter familiy in mirror moduli space, varying in complex structure with t. The same is true for the quotient of the Z41 family. But the mirror family of quintics in P4 has h 2,1 = 1 and h 1,1 = 101, so the complex structure moduli space is one dimensional. It follows that the quotients of the Fermat and Z41 families are isomorphic in terms of their complex structure, differing only in their Kähler structure. We therefore expect the invariant periods of both loci to obey the same Picard–Fuchs equations. Periods of the forms corresponding to 5th order monomials. As before, elements of C[a, b, c, d, e]5 /J (Q) correspond to cohomology classes in F3,2 = H 3,0 ⊕ H 2,1 . In this case it helps to classify the 126 quintic monomials by their transformation under the Z41 scaling symmetry generated by g = (1, 37, 16, 18, 10). We say a monomial m is in representation n if g(m) = γ n m, where γ is a nontrivial 41st root of unity. The monomials are listed by representation in Table 4. The decomposition into representations of Z41 must be compatible with the permutation symmetry in the sense that cyclic permutation of all the monomials in a given representation should give the monomials in some other representation. We therefore group the representations by their behavior under cyclic permutations in Table 5. We now need to take account of the quotient of C[a, b, c, d, e]5 by J (Q) to find out how many forms are associated with each representation class. The results are summarized in Table 6. In total there are 1 + 5(3 + 2 + 2 + 3 + 3 + 3 + 2 + 2) = 101 independent monomials as expected.
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
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Table 5. The representations of a given row of the right column transform into each other under cyclic permutations of the homogeneous coordinates Representation Class A
Z41 Representations Contained Therein 0
B1 B2 B3 B4 C1 C2 C3 C4
5,8,9,21,39 1,10,16,18,37 2,20,32,33,36 4,23,25,31,40 15,22,24,27,35 3,7,13,29,30 6,14,17,19,26 11,12,28,34,38
Table 6. The 25 relations among the 126 quintic monomials, considered as elements of C[a, b, c, d, e]5 /J (Q) Representation Class
Relations
# Forms per Representation
Hodge Type
A B1 B2 B3 B4 C1 C2 C3 C4
a 4 b ∼ b4 c ∼ c4 d ∼ d 4 e ∼ e4 a ∼ abcd a∂b Q = 45 b3 ac + 15 a 5 − ta 2 cde No relations d∂c Q = 45 c3 d 2 + 15 b4 d − td 2 abe No relations c∂a Q = 45 a 3 be + 15 e4 c − tc2 bde No relations c∂e Q = 45 e3 ac+ 15 d 4 c − tc2 abd No relations
1 3 2 2 3 3 3 2 2
H 3,0 ⊕ H 2,1 H 2,1 H 2,1 H 2,1 H 2,1 H 2,1 H 2,1 H 2,1 H 2,1
Representation class A: {abcde}. (1, 1, 1, 1, 1) A → (2, 2, 2, 2, 2) → (3, 3, 3, 3, 3) → (4, 4, 4, 4, 4) ↓ D0 ↓ D0 ↓ D0 (4, 1, 0, 0, 0) B → (5, 2, 1, 1, 1) → (6, 3, 2, 2, 2) → (7, 4, 3, 3, 3) ↓ D1 ↓ D1 ↓ D1 (4, 5, 1, 0, 0)C → (5, 6, 2, 1, 1) → (6, 7, 3, 2, 2) ↓ D2 ↓ D2 (4, 5, 5, 1, 0) D → (5, 6, 6, 2, 1) ↓ D3 (3, 4, 4, 4, 0) E → (4, 5, 5, 5, 1) ↓ D4 (4, 4, 4, 4, 4)
The coupled system is: 1 1 η(η − 1)(η − 2)A = t 3 ηE, ηE = t (η + 1)D, D = (η + 1)C, 6 3 1 C = (η + 1)B, ηB = t (η + 2)A, 2 from which one can find the following equation for the period A: η(η − 1)(η − 2)(η − 4) − t 5 (η + 2)4 A = 0.
(121) (122)
(123)
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Representation class B1: 3 of a 5 , d 3 b2 , b3 ac, a 2 cde . In anticipation of using the map to find the (1, 2)–periods, it will be helpful to find a diagram without a 5 :
(1, 3, 1, 0, 0) E ↓ D3 (1, 3, 1, 4, 1)
(2, 0, 1, 1, 1) D ↓ D1 → (2, 4, 2, 1, 1)
(0, 1, 3, 3, 3)C ↓ D0 → (3, 1, 2, 2, 2) → (4, 2, 3, 3, 3)
The coupled system is:
18 37 E, ηE = t η + D, ηA = t η + 41 41
16 ηC = t η + B, 41
1 1 B= η+ A, 2 41
(0, 2, 0, 3, 0) A → (1, 3, 1, 4, 1) ↓ D4 ↓ D4 (1, 2, 0, 3, 4) B → (2, 3, 1, 4, 5) ↓ D2 → (1, 2, 4, 4, 4)
10 1 2 η(η − 1)D = t η + C, 2 41 (124) (125)
which results in the following equations for the 3 independent periods:
201 141 51 16 1 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ A = 0, 41 41 41 41 41
133 98 83 78 18 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ D = 0, 41 41 41 41 41
182 92 57 42 37 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ E = 0. 41 41 41 41 41 (126)
Representation class B2: e3 bc, b2 c2 d .
(0, 2, 2, 1, 0) E ↓ D4 (1, 2, 2, 1, 4)
(2, 0, 3, 3, 2) D ↓ D1 → (1, 3, 3, 2, 1) → (2, 4, 4, 3, 2)
(3, 1, 0, 3, 3)C ↓ D2 → (3, 1, 4, 4, 3)
(0, 1, 1, 0, 3) A → (1, 2, 2, 1, 4) ↓ D3 ↓ D3 (0, 1, 1, 4, 4) B → (1, 2, 2, 5, 5) ↓ D0 → (4, 2, 1, 4, 4)
The coupled system is:
20 1 36 33 E, η(η − 1)E = t 2 η + D, ηD = t η + C, ηA = t η + 41 2 41 41 (127)
2 1 32 ηC = t η + B, B = η+ A. (128) 41 2 41
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
705
The equations for the 2 independent periods are:
197 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ 41
143 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ 41
102 η+ 41
118 η+ 41
77 η+ 41
73 η+ 41
32 η+ 41
43 η+ 41
2 A = 0, 41
33 E = 0. 41 (129)
Representation class B3: 2 of b4 d, c3 d 2 , d 2 abe .
(1, 1, 0, 2, 1) E ↓ D1 (1, 5, 1, 2, 1)
(3, 3, 2, 0, 2) D ↓ D3 → (2, 2, 1, 3, 2) → (3, 3, 2, 4, 3)
(0, 3, 3, 1, 3)C ↓ D0 → (4, 4, 3, 1, 3)
(0, 4, 0, 1, 0) A → (1, 5, 1, 2, 1) ↓ D4 ↓ D4 (1, 4, 0, 1, 4) B → (2, 5, 1, 2, 5) ↓ D2 → (1, 4, 4, 2, 4)
The coupled system is:
81 23 1 4 η(η − 1)E = t 2 η + ηA = t η + E, D, ηD = t η + C, 41 2 41 41 (130)
1 25 10 B, B = η+ A. ηC = t η − 41 2 41 (131) The equations for the 2 independent periods are:
245 105 45 25 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η− 41 41 41 41
146 86 81 66 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
10 A = 0, 41
31 E = 0. 41 (132)
Representation class B4: d 3 c2 , c3 ab, a 2 b2 e .
(1, 1, 3, 0, 0) E ↓ D3 (1, 1, 3, 4, 1)
(2, 2, 0, 0, 1) D ↓ D2 → (2, 2, 4, 1, 1)
(0, 3, 2, 2, 3)C ↓ D0 → (3, 3, 1, 1, 2) → (4, 4, 2, 2, 3)
(0, 0, 2, 3, 0) A → (1, 1, 3, 4, 1) ↓ D4 ↓ D4 (1, 0, 2, 3, 4) B → (2, 1, 3, 4, 5) ↓ D1 → (1, 4, 3, 3, 4)
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The coupled system is:
21 5 E, ηE = t η + D, ηA = t η + 41 41
39 ηC = t η + B, 41
B=
1 9 η+ A, 2 41
1 8 η(η − 1)D = t 2 η + C, 2 41 (133) (134)
which results in the following equations for the 3 independent periods:
169 144 49 39 9 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ A = 0, 41 41 41 41 41
131 121 91 46 21 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ D = 0, 41 41 41 41 41
185 90 80 50 5 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ E = 0. 41 41 41 41 41 (135)
Representation class C1: 3 of e4 c, d 3 a 2 , a 3 bc, c2 bde . Here it will be convenient to leave out the monomial e4 c.
(0, 1, 2, 1, 1) E ↓ D0 (4, 2, 2, 1, 1)
(2, 3, 0, 2, 3) D ↓ D2 → (1, 2, 3, 2, 2) → (2, 3, 4, 3, 3)
(2, 0, 0, 3, 0) B ↓ D4 (3, 0, 0, 3, 4)C ↓ D1 → (3, 4, 1, 3, 4)
(3, 1, 1, 0, 0) A → (4, 2, 2, 1, 1) ↓ D3 ↓ D3 → (3, 1, 1, 4, 1) → (4, 2, 2, 5, 2) ↓ D4 → (4, 1, 1, 4, 5)
The coupled system is:
7 1 13 30 2 E, η(η − 1)E = t η + D, ηD = t η + C, ηA = t η + 41 2 41 41 (136)
3 29 1 η+ B, ηB = t η + A. C= 2 41 41
(137)
The equations for the 3 independent periods are:
194 89 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ 41 41
193 153 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ 41 41
95 130 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ 41 41
54 η+ 41
48 η+ 41
85 η+ 41
44 η+ 41
13 η+ 41
70 η+ 41
29 A = 0, 41
3 B = 0, 41
30 E = 0. 41 (138)
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
707
Representation class C2: e3 bd, a 2 c2 e, b2 d 2 c .
(0, 2, 1, 2, 0) E ↓ D4 (1, 2, 1, 2, 4)
(2, 0, 2, 0, 1)C ↓ D1 (2, 4, 3, 0, 1) D ↓ D3 → (1, 3, 2, 3, 1) → (2, 4, 3, 4, 2)
(0, 1, 0, 1, 3) A → (1, 2, 1, 2, 4) ↓ D0 ↓ D0 (4, 2, 0, 1, 3) B → (5, 3, 1, 2, 4) ↓ D2 → (3, 1, 3, 1, 2) → (4, 2, 4, 2, 3) ↓ D1 → (3, 5, 4, 1, 2)
The coupled system is:
14 1 1 19 17 E, η(η − 1)E = t 2 η + D, D = η+ C, (139) ηA = t η + 41 2 41 2 41
1 26 1 6 η(η − 1)C = t 2 η + B, B = η+ A. (140) 2 41 2 41 The equations for the 3 independent periods are:
181 101 96 26 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η− 41 41 41 41
149 129 99 19 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
142 137 67 47 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
6 A = 0, 41
14 C = 0, 41
17 E = 0. 41 (141)
Representation class C3: 2 of d 4 c, e3 ac, c2 abd .
(1, 1, 2, 1, 0) E ↓ D4 (2, 1, 2, 1, 4)
(3, 3, 0, 2, 2) D ↓ D2 → (2, 2, 3, 2, 1) → (3, 3, 4, 3, 2)
(0, 3, 1, 3, 3)C ↓ D0 → (4, 4, 1, 3, 3)
(1, 0, 1, 0, 3) A → (2, 1, 2, 1, 4) ↓ D1 ↓ D1 (1, 4, 2, 0, 3) B → (2, 5, 3, 1, 4) ↓ D3 → (1, 4, 2, 4, 4)
The coupled system is:
11 1 12 34 E, η(η − 1)E = t 2 η + D, ηD = t η + C, (142) ηA = t η + 41 2 41 41
1 38 28 B, B = η+ A. (143) ηC = t η + 41 2 41
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The equations for the 2 independent periods are:
198 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ 41
134 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ 41
93 η+ 41
94 η+ 41
53 η+ 41
79 η+ 41
38 η+ 41
69 η+ 41
28 A = 0, 41
34 E = 0. 41 (144)
Representation class C4: a 2 c2 d, d 2 e2 b . (2, 0, 2, 1, 0) A → (3, 1, 3, 2, 1) ↓ D1 ↓ D1 (2, 4, 3, 1, 0) B → (3, 5, 4, 2, 1) ↓ D4 (2, 3, 2, 0, 3)C → (3, 4, 3, 1, 4) ↓ D3 (0, 1, 0, 2, 2) D → (1, 2, 1, 3, 3) → (2, 3, 2, 4, 4) ↓ D2 ↓ D2 (0, 1, 4, 3, 2) E → (1, 2, 5, 4, 3) ↓ D0 (4, 2, 4, 3, 2)
The coupled system is:
1 22 24 1 27 1 2 2 η(η − 1)A = t η + E, E = η+ D, η(η − 1)D = t η + C, 2 41 2 41 2 41 (145)
1 15 35 B, B = η+ A. (146) ηC = t η + 41 2 41
The equations for the 2 independent periods are:
150 145 65 35 15 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ A = 0, 41 41 41 41 41
147 117 97 27 22 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ D = 0. 41 41 41 41 41 (147)
In summary, the classes corresponding to representations in B1, . . . , B4, C1, . . . , C4 are of pure Hodge type H 2,1 , and their periods all obey 5th order generalized hypergeometric equations. The periods of the single class in representation A (the trivial representation) are of mixed Hodge type, and obey a 4th order generalized hypergeometric equation — the same as that obeyed by the corresponding periods in the Fermat family. Periods of the forms corresponding to 10th order monomials. As for the Fermat quintic, we would like to use the map to generate a basis of C[a, b, c, d, e]10 /J (Q). This requires that we find a basis of C[a, b, c, d, e]5 /J (Q) with no monomials containing 4th or 5th powers of any coordinate. One can see by examining the relations in Table 6 as well as the monomial content of the representation classes, that such a basis can be found for the Z41 quintic.
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709
As with the Fermat quintic, acts nicely on the representations of the discrete symmetry group. If [m] is in representation n of Z41 , then [m] is in representation 41 − n. Here though the map acts in a nontrivial way on the representation classes: A = A,
B1 = B3, B2 = B4, B3 = B1, B4 = B2, C1 = C3, C2 = C4, C3 = C1, C4 = C2.
(148) (149)
For example, the 10th order monomials found by acting with on the 5th order monomials of B1 appear in the diagram for B3 up to cyclic permutations. Representation class A. The 5th order monomial in class A is abcde, which maps to a 2 b2 c2 d 2 e2 under . We can therefore reuse the diagram (63), but solve for ddtA ∝ (2, 2, 2, 2, 2) rather than A = (1, 1, 1, 1, 1). The result is: dA = 0. η(η − 1)(η − 3)(η − 4) − t 5 (η + 3)4 dt
(150)
Representation (148) we see thatwe should look at the 5th order mono class B1. From mials in B3: c3 d 2 , d 2 abe . Acting with gives a 3 b3 de3 , a 2 b2 c3 de2 , corresponding to periods (3, 3, 0, 1, 3) and (2, 2, 3, 1, 2). These are cyclic permutations of (and hence equal to) (3, 1, 2, 2, 2) and (0, 1, 3, 3, 3) which appear in the B1 diagram as C and 21 dD dt respectively. So all we need to do is write the coupled system in terms of dD dt rather than D and then solve for C and dD dt . The coupled system is:
37 59 dD ηA = t η + E, η(η − 1)E = t 2 η + , 41 41 dt
16 1 1 ηC = t η + B, B = η+ A. 41 2 41
1 dD 10 η =t η+ C, 2 dt 41
(151) (152)
The equations for the 2 independent periods are:
174 η+ η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + 41
180 η+ η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + 41
139 η+ 41
165 η+ 41
124 η+ 41
160 η+ 41
119 η+ 41
100 η+ 41
59 dD = 0, dt 41
10 C = 0. 41
Representation class B2. The 5th order monomials in B4 are d 3 c2 , c3 ab, a 2 b2 e . Acting with gives {a 3 b3 ce3 , a 2 b2 d 3 e3 , abc3 d 3 e2 }, corresponding to periods (3, 3, 1, 0, 3), (2, 2, 0, 3, 3) and (1, 1, 3, 3, 2). These are cyclic permutations of (3, 1, 0, 3, 3), (2, 0, 3, 3, 2) and (1, 3, 3, 2, 1) which appear in the B2 diagram as C, D and 21 dE dt respectively. rather than E and then solve for We therefore write the coupled system in terms of dE dt dE C, D and dt . The coupled system is:
1 dE 20 36 74 dE η(η − 1)A = t 2 η + , η =t η+ D, ηD = t η + C, (153) 41 dt 2 dt 41 41
2 1 32 ηC = t η + B, B = η+ A. (154) 41 2 41
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The equations for the 3 independent periods are:
159 114 84 184 η+ η+ η+ η+ η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + 41 41 41 41
166 156 61 196 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
200 155 125 115 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
74 41
36 41
20 41
dE = 0, dt
C = 0, D = 0.
Representation class B3. The independent 5th order monomials in B1 can be chosen to be d 3 b2 , b3 ac, a 2 cde . Acting with gives a 3 bc3 e3 , a 2 c2 d 3 e3 , ab3 c2 d 2 e2 , corresponding to periods (3, 1, 3, 0, 3), (2, 0, 2, 3, 3) and (1, 3, 2, 2, 2). These are cyclic permutations of (0, 3, 3, 1, 3), (3, 3, 2, 0, 2) and (2, 2, 1, 3, 2) which appear in the B3 diagram as C, D and 21 dE dt respectively. We therefore write the coupled system in terms rather than E and then solve for C, D and dE of dE dt dt . The coupled system is:
81 dE 23 1 dE 4 ηA = t η + , η(η − 1) = t2 η + D, ηD = t η + C, (155) 41 dt 2 dt 41 41
10 1 25 ηC = t η − B, B = η+ A. (156) 41 2 41
The equations for the 2 independent periods are:
204 189 154 64 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
168 163 148 113 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
187 127 122 107 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
4 41
23 41
72 41
C = 0, D = 0, dE = 0. dt
Representation class B4. The 5th order monomials in B2 are e3 bc, b2 c2 d . Acting with gives {a 3 b2 c2 d 3 , a 3 bcd 2 e3 }, corresponding to periods (3, 2, 2, 3, 0), and (3, 1, 1, 2, 3). These are cyclic permutations of (0, 3, 2, 2, 3) and (3, 3, 1, 1, 2) which appear in the B4 dD diagram as C, and 21 dD dt respectively. We write the coupled system in terms of dt rather than D and then solve for C and dD dt . The coupled system is:
5 62 dD ηA = t η + E, η(η − 1)E = t 2 η + , 41 41 dt
39 1 9 ηC = t η + B, B = η+ A, 41 2 41
8 1 dD η =t η+ C, (157) 2 dt 41 (158)
which results in the following equations for the 3 independent periods:
203 173 128 108 8 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ C = 0, 41 41 41 41 41
dD 172 162 132 87 62 = 0. η+ η+ η+ η+ η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + 41 41 41 41 41 dt
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Representation class C1. The independent 5th order monomials in C3 can be chosen 3 2 to be e ac, c abd . Acting with gives {a 2 b3 c2 d 3 , a 2 b2 cd 2 e3 }, corresponding to periods (2, 3, 2, 3, 0), and (2, 2, 1, 2, 3). These are cyclic permutations of (2, 3, 0, 2, 3) and (1, 2, 3, 2, 2) which appear in the C1 diagram as D, and 21 dE dt respectively. We write rather than E and then solve for D and dE the coupled system in terms of dE dt dt . The coupled system is:
71 dE 1 dE 7 η(η − 1)A = t 2 η + , η =t η+ D, (159) 41 dt 2 dt 41
1 3 29 13 C, C = η+ B, ηB = t η + A, (160) ηD = t η + 41 2 41 41 which results in the following equations for the 2 independent periods:
177 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ 41
171 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ 41
167 η+ 41
136 η+ 41
152 η+ 41
126 η+ 41
112 η+ 41
111 η+ 41
7 D = 0, 41
71 dE = 0. dt 41
Representation class C2. The 5th order monomials in C4 are a 2 c2 d, d 2 e2 b . Acting with gives {ab3 cd 2 e3 , a 3 b2 c3 de}, corresponding to periods (1, 3, 1, 2, 3), and (3, 2, 3, 1, 1). These are cyclic permutations of (3, 1, 3, 1, 2) and (1, 3, 2, 3, 1) which 1 dE appear in the C2 diagram as 21 dC dt , and 2 dt respectively. Rewriting the coupled system in terms of these variables:
58 dE 1 dE 14 2 , η =t η+ D, (161) η(η − 1)A = t η + 41 dt 2 dt 41
60 dC 1 dC 26 1 6 1 , η =t η+ B, B = η+ A, (162) ηD = t η + 2 41 dt 2 dt 41 2 41 which results in the following equations for the periods:
190 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ 41
183 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ 41
170 η+ 41
178 η+ 41
140 η+ 41
108 η+ 41
60 η+ 41
88 η+ 41
55 dC = 0, 41 dt
58 dE = 0. 41 dt
Representation class C3. The independent 5th order monomials in C1 can be chosen to be 3 2 3 d a , a bc, c2 bde . Acting with gives ab3 c3 e3 , b2 c2 d 3 e3 , a 3 b2 cd 2 e2 , corresponding to periods (1, 3, 3, 0, 3), (0, 2, 2, 3, 3) and (3, 2, 1, 2, 2). These are cyclic permutations of (0, 3, 1, 3, 3), (3, 3, 0, 2, 2) and (2, 2, 3, 2, 1) which appear in the C3 diagram as C, D and 21 dE dt respectively. The coupled system is:
1 dE 11 12 75 dE , η =t η+ D, ηD = t η + C, η(η − 1)A = t η + 41 dt 2 dt 41 41
1 38 28 B, B = η+ A. ηC = t η + 41 2 41
(163) (164)
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The equations for the 3 independent periods are:
202 192 157 52 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
161 151 116 176 η+ η+ η+ η+ η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + 41 41 41 41
175 135 120 110 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
12 41
11 41
75 41
C = 0, D = 0, dE = 0. dt
Representation class C4. The 5th order monomials in C2 are e3 bd, a 2 c2 e, b2 d 2 c . Acting with gives {a 3 b2 c3 d 2 , ab3 cd 3 e2 , a 3 bc2 de3 }, corresponding to periods (3, 2, 3, 2, 0), (1, 3, 1, 3, 2) and (3, 1, 2, 1, 3). These are cyclic permutations of (2, 3, 2, 0, 3), (3, 1, 3, 2, 1) and (1, 2, 1, 3, 3) which appear in the C4 diagram as C, 1 dA 1 dD 2 dt and 2 dt respectively. The coupled system is:
1 dA 22 1 68 dD 1 dD 24 η =t η+ E, ηE = t η + , η =t η+ C, 2 dt 41 2 41 dt 2 dt 41 (165)
1 35 56 d A B, ηB = t η + ηC = t η + . (166) 41 2 41 dt The equations for the 3 independent periods are:
186 106 76 191 η+ η+ η+ η+ η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + 41 41 41 41
199 179 109 104 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
188 158 138 68 η(η − 1)(η − 2)(η − 3)(η − 4) − t 5 η + η+ η+ η+ η+ 41 41 41 41
56 41
24 41
63 41
d A = 0, dt
C = 0, dD = 0. dt
Periods of the class corresponding to 15th order monomials. As for the Fermat quintic, we choose the monomial a 3 b3 c3 d 3 e3 to represent the single independent class in C[a, b, c, d, e]15 /J (Q), so we can reuse the diagram for (1, 1, 1, 1, 1) and solve for the 2 period (3, 3, 3, 3, 3) = 16 ddt 2A . The resulting equation is: d2 A η(η − 2)(η − 3)(η − 4) − t 5 (η + 4)4 = 0. dt 2
(167)
3.3. Decomposition of the monodromy representations. With the Picard–Fuchs data in hand, we now see what we can learn about the corresponding monodromy representations. Recall from Sect. 2 that the forms we integrate to get periods are single–valued as functions of t, i.e. as sections of the Hodge bundle over P1 − F, where the set F consists of 5th roots of unity and ∞. The only source of the monodromy of the solutions to the Picard–Fuchs equations is therefore the geometric monodromy of the cycles. However, in general the Picard–Fuchs equations contain less information than the monodromy of the cycles. For example, the holomorphic 3–form ψ obeys a 4th order equation. This means that as t varies, ψ moves around in a 4 dimensional space Ψ ⊂ H 3 (X, C). If [γ ] ∈ H3 (X, C) is a class whose dual is in Ψ , then ψ will pick up
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
713
monodromy [γ ] → [γ ] + [δ] only if [δ] ψ = 0, i.e. if [δ] has a component in the dual of Ψ as well. Another way to say this is that the Picard–Fuchs equations tell us only about particular block diagonal pieces of the monodromy matrices. In particular, we get: ⎞ ⎞ ⎛ ⎛ 4×4 4×4 5×5 2×2 ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ 5×5 2×2 Fermat: ⎜ ⎟. ⎟, Z41 : ⎜ ⎠ ⎠ ⎝ ⎝ ... ... 5×5 2×2 (168) In both cases, the single 4 × 4 block is the monodromy representation of Eq. (57). For the Fermat family, the 2 × 2 blocks correspond to the 2nd order equations satisfied by the 200 other forms. There is a block for each of the 100 other representations of (Z5 )3 instantiated by the degree 5 monomials. For the Z41 family, there is a 5 × 5 block for each of the 40 representations of Z41 . In general, the non–block diagonal pieces of the monodromy matrices will be nonzero, but the extra symmetry in the Fermat and Z41 examples provides nongeneric constraints. We now construct a basis in which the action is purely block–diagonal. Start with the 204 forms, at a point t0 such that t05 = 1, ∞:
Pi φi = Res Ω0 , (169) 1 Q(t0 ) 5 degPi +1 where as before Pi are monomials, Q is the polynomial defining the family of hypersurfaces, and Ω0 is as in Eq. (23). Let [φ¯i ] ∈ H3 (V (t0 ), C) denote the dual classes to the φi , and let φ¯i be representative cycles of these classes. We choose the Pi to transform in a representation gi of the symmetry group, so that the classes transform in the representations −gi . One can then use an Ehresmann connection to generate a family of cycles φ¯i (t) in some neighborhood of t0 , such that for each t, φ¯i (t) transforms in the representation −gi .17 Since the connection by definition respects the (Z5 )3 or Z41 symmetry, cycles can only mix under monodromy with cycles in the same representation of the symmetry group. This is equivalent to the monodromy representation being block diagonal as above. 4. Yukawa couplings of (2, 1)–forms We stressed in the introduction that the families in Table 1 are distinct from the more familiar Fermat locus (3). As a first step to seeing how these differences play out in the more detailed properties of the loci, we work out the number of Yukawa couplings constrained to vanish by the discrete symmetry group. This information is also useful for applications to string compactification, since the Yukawa couplings are intimately related to physically measurable constants in the 4d low energy effective theory. Suppose Ω(t) is the holomorphic 3–form on a family of Calabi–Yau 3–folds para3,0 (X ), but is instead meterized by t. The derivative dΩ(t) dt is no longer restricted to H dΩ(t) 3,0 2,1 contained in the second Hodge filtrant: dt ∈ H (X ) ⊕ H (X ).18 Similarly for the
17 Note that in general φ¯ (t) is only dual to φ (t) when t = t . 0 i i 18 In the context of abstract variations of Hodge structure, this property is known as Griffiths transversality,
and is a useful necessary condition for the variation of Hodge structure to be geometrical in origin. For hypersurfaces in projective space, Griffiths transversality follows from the results of Sect. 2.
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second derivative, we have: d dtΩ(t) ∈ H 3,0 (X ) ⊕ H 2,1 (X ) ⊕ H 1,2 (X ). The following 2 integrals therefore vanish identically: d2 Ω(t) dΩ(t) = Ω(t) ∧ Ω(t) ∧ = 0. (170) dt dt 2 2
But including third derivatives gives a nonzero result: d3 Ω(t) Ω(t) ∧ = 0 for general t. dt 3
(171)
This is the prototype Yukawa coupling. More generally we can look at the dependence of Ω over the whole complex structure moduli space (as opposed to just a 1–parameter family). Ω will then depend on h 2,1 parameters ti , and the Yukawa couplings are: d d d Yi jk (ti ) = Ω(ti ) ∧ Ω(ti ) . (172) dti dt j dtk ti =ti
Alternatively, with a given normalization for Ω(ti ), we can interpret the ti ’s as different directions in Tti M H 2,1 (X (ti )), the tangent space to the complex structure moduli space. The Yukawa couplings are then a map: H 2,1 (X (ti )) × H 2,1 (X (ti )) × H 2,1 (X (ti )) → C.
(173)
Yi jk is clearly symmetric in its 3 indices, each of which takes h 2,1 different values. The number of independent Yukawa couplings is therefore: NYukawas =
1 h 2,1 h 2,1 + 1 h 2,1 + 2 . 6
(174)
For example, quintic hypersurfaces in P4 have h 2,1 = 101, so NYukawas = 176851. The technique for performing detailed calculations of Yukawa couplings was presented in [4]. Here we find the number of Yi jk ’s that are potentially nonzero in the presence of various discrete symmetries. For symmetries that preserve Ω(t) (i.e. projective linear transformations that act trivially on abcde), the only Yukawa couplings that are allowed to be finite are those corresponding to 3 monomial deformations of X whose product is invariant. A computer search for such triples (summarized in Table 7) yields numbers that approximately satisfy: # nonzero Yukawas
Total # of Yukawas . Ord G
(175)
A relation of this form is somewhat surprising for the following reason. The transformations of the 101 monomials do not exhaust the 125 irreducible representations of (Z5 )3 , whereas in the Z41 case there is some monomial transforming in each of the 41 representations. Therefore one might not expect the numbers of nonzero Yukawa couplings for G = (Z5 )3 to fit the line defined by the cases with smaller G. It would be interesting to more fully examine the dependence of the number of (potentially) nonzero cubic invariants on the size of the manifold’s discrete symmetry group.
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
715
Table 7. Numbers of potentially nonvanishing Yukawa couplings for the six families of quintics in P4 listed at the end of Sect. 3.1 Symmetry Group G (Z5 )3 Z41 Z51 Z5 × Z13 Z3 × Z13 (Z5 )2 × Z3
# Nonzero Yukawas 1431 4321 3477 2736 4554 2391
(Total # Yukawas)/(Ord G) 1414.8 4313.4 3467.7 2720.8 4534.6 2358.0
5. Conclusion We have investigated some well–known Calabi–Yau 3–folds but focused on unfamiliar loci in their complex structure moduli that give rise to unexpected discrete symmetry groups. With the important role that Calabi–Yau manifolds with enhanced symmetries have played in both the physics and mathematics literatures, there is strong motivation to study these new families. By carefully deriving a technique apparently similar to that of [9] but differing significantly in interpretation, we succeeded in developing a systematic method for computing the Picard–Fuchs equations satisfied by each entry in the full period matrix of along these loci. To illustrate the method, we applied it to the Fermat family (3) as well as the Z41 quintic hypersurface family (the Z51 family and a weighted projective space example are handled the appendix). We then saw how discrete symmetries are reflected in the detailed structure of the geometric monodromy representations. In particular, aside from the 4×4 invariant part the monodromy matrices decompose into block diagonal pieces of different sizes in the different families. Finally we found the number of Yukawa couplings constrained to vanish by the symmetries and noted an intriguing approximate relation between the number of nonzero couplings and the size of the symmetry group. The Z41 and Z51 families and their cousins in Table 1 are thus a new testing ground for many calculations. For example, as with the Fermat family, computations of periods and Yukawa couplings are more tractable than for a general hypersurface. Such calculations are of interest because in heterotic string compactifications, the Yukawa couplings are eventually nothing but the parameters of the standard model, as well as because of the role periods play in various moduli stabilization schemes. For instance, in any model that purports phenomenological realism, the Yukawas must be able to incorporate the range of observed particle masses, spanning at least 14 orders of magnitude.19 It would be interesting to know if the moduli space of quintics in P4 (which is only a toy example in this context) admits regions with a large enough range of Yukawa couplings, and if so how many flux–stabilized vacua they contain. This could potentially amount to a very severe phenomenological restriction on the ‘landscape’ of vacua which currently plagues attempts to extract TeV scale predictions from string theory. A mathematical direction for future work is to use the non–Fermat families to test some of the claims of mirror symmetry. In particular Morrison has constructed [23,18] a variation of Hodge structure on the even dimensional cohomology of the mirror (the so– called A–model variation) in analogy with that coming from the middle–dimensional cohomology of the original manifold (the B–model variation). Corresponding to the B–model monodromy action on H3 (X, C), there are conjectured automorphisms of 19 Neutrinos are now known to have a mass approximately 10−3 eV, whereas the Z boson has a mass of 9.1 × 1010 eV.
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the topological K–theory of the mirror.20 [24]. Making this correspondence explicit for the special families considered here should provide new insights into the mathematical structure (quantum cohomology and Gromov–Witten theory) of the A–model on Calabi– Yau threefold hypersurfaces in toric orbifolds[26]. Our derivation of the corrected version of the technique outlined by Candelas, de la Ossa and Rodriguez–Villegas greatly reduces the computation required to find the Picard–Fuchs equations for a variety of families of Calabi–Yau manifolds with discrete symmetries. The method summarized in Sect. (3) readily extends to the other examples of 3–folds with discrete symmetries, as well as symmetric Calabi–Yau hypersurfaces of other dimensions.21 In [27] we examine 1–parameter families of K3 surfaces. Though the Picard–Fuchs equations can be derived in the same way, the interpretation of the results is more complicated than for 3–folds. The reason is essentially that H n−1,1 which controls the deformations of complex structure coincides with H 1,1 which contains the Kähler form, as well as information about algebraic cycles. For example, there is an important sublattice of H 1,1 ∩ H 2 (V (t), Z) known as the Picard group, whose classes consist of algebraic cycles. The rank of this group (the Picard rank) can jump discontinuously as one deforms the hypersurface, even without passing through singular configurations. Moreover, it has been shown that loci endowed with discrete symmetries are some of the places where such jumps take place [28]. K3 surfaces also display some extraordinary phenomena that are apparently unrelated to enhancements of Picard rank. An example is the theorem of Oguiso [29], that nontrivial projective families (such as quartic hypersurfaces in P3 ) contain dense subsets where the automorphism group is of infinite order. Nothing analogous to this occurs in families of 3–folds. Acknowledgement. The authors thank Johan de Jong, Brent Doran, and John Morgan for helpful conversations during the course of this work. BG and SJ gratefully acknowledge the support of DOE grant DE-FG0292ER40699. SJ acknowledges support from Columbia University ISE and the Pfister Foundation. C.F.D. is supported in part by a Royalty Research Fund Scholar Award from the Office of Research, University of Washington.
A. Picard–Fuchs Equations for the Z51 Quintic The calculation of the Picard–Fuchs equations for the Z51 quintic: Q(t) =
1 4 a b + b4 c + c4 d + d 4 a + e5 − tabcde = 0 5
(176)
differs only in detail from that of the Z41 case. The final results are as follows. The 51 representations of Z51 group into 14 permutation classes as follows: 20 One often imagines mirror symmetry exchanging middle cohomology H 3 with even cohomology H even = H 0 ⊕ H 2 ⊕ H 4 ⊕ H 6 . But the Chern map, which sends an element (E, F) ∈ K 0 to c(E)/c(F) (the quotient of the total Chern classes) is in fact an isomorphism when (as for quintics in P4 ) H even (Z) contains
no torsion classes. 21 One might hope to generalize the technique further to hypersurfaces and complete intersections in toric varieties, perhaps with a view to bridge the gap between GKZ systems (which can be derived algorithmically) and true Picard–Fuchs differential equations [25].
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
Rep. Class
Reps.
A B C D E F G
0 1, 16, 38, 47 2, 25, 32, 43 3, 12, 39, 48 4, 13, 35, 50 5, 29, 31, 37 6, 24, 27, 45
#(2,1)–Forms Rep. Class 1 1 2 2 3 2 2
H I J K L M N
717
Reps.
#(2,1)–Forms
7, 10, 11, 23 8, 19, 26, 49 9, 15, 36, 42 14, 20, 22, 46 17, 34 18, 21, 30, 33 28, 40, 41, 44
1 2 2 2 2 2 3
The operators annihilating the periods of the (2,1)–forms are then: A η(η − 1)(η − 2)(η − 4) − t 5 (η + 2)4 B η(η − 1)(η − 2)(η − 3) − t 5 η + 86 51 η + C η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + D η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + E η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + F η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + G η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + H η(η − 1)(η − 2)(η − 3) − t 5 η + I η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + J η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + K η(η − 1)(η − 2)(η − 3) − t 5 η +
η + 106 η + 166 51 51 26 121 151 η + 161 η + η + 51 51 51 51 19 49 59 η + 179 η + η + 51 51 51 51 39 114 η + 99 η + 54 51 51 51 η + 51 3 48 63 243 51 η + 51 η + 51 η + 51 1 16 191 251 η + η + η + 51 51 51 51 38 98 103 η + 118 51 η + 51 η + 51 51 47 242 η + 67 η + 52 51 51 51 η + 51 46 71 116 η + 226 51 η + 51 η + 51 51 22 97 122 η + 167 51 η + 51 η + 51 51 6 96 126 η + 231 51 η + 51 η + 51 51 27 η + 147 η + 57 η + 177 51 51 51 51 41 91 146 η + 181 51 η + 51 η + 51 51 2 32 127 η + 247 51 η + 51 η + 51 51 43 178 η + 83 η + 53 51 51 51 η + 51 42 87 117 η + 162 51 η + 51 η + 51 51 36 66 111 η + 246 51 η + 51 η + 51 51 37 82 107 η + 182 51 η + 51 η + 51 51 101 51
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η + 131 η + 56 η + 241 51 51 51 2 17 2 η + 187 51 51 2 34 2 η + 119 51 51 21 81 171 η + 186 51 η + 51 η + 51 51 18 33 123 η + 183 51 η + 51 η + 51 51 11 61 176 η + 211 51 η + 51 η + 51 51 58 η + 113 η + η + 163 η(η − 1)(η − 2)(η − 3) − t 5 η + 23 51 51 51 51 7 112 η + 62 η + 227 η(η − 1)(η − 2)(η − 3) − t 5 η + 51 51 η + 51 51
η(η − 1)(η − 2)(η − 3) − t 5 η + L η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + M η(η − 1)(η − 2)(η − 3) − t 5 η + η(η − 1)(η − 2)(η − 3) − t 5 η + N η(η − 1)(η − 2)(η − 3) − t 5 η +
31 51
B. Symmetric Hypersurfaces in Weighted Projective Space It is known that any Calabi–Yau 3–fold can be embedded in Pn for some sufficiently large n, but the case where the embedding is a hypersurface is the exception rather than the rule. More often the embedding can only be realized as an intersection of a large number of hypersurfaces. It is therefore useful to consider other constructions of Calabi–Yau 3–folds. One of the simplest generalizations of a hypersurface in Pn is a hypersurface in a weighted projective space: (Cn − {0}) / ∼ where the equivalence relation ∼ is given by: [x0 , . . . , xn ] ∼ [λk0 x0 , . . . , λkn ].
(177)
Here λ is any nonzero complex number, and {k0 , . . . , kn } are a collection of integers called the weights. This space is denoted WP[k0 ,...,kn ] , and one easily sees that ordinary projective space is a special case: Pn = WP[1,...,1] . The formulas relating to hypersurfaces in Pn generalize straightforwardly to the case of nontrivial weights [30]. As before we have: n n Ω0 ∂ Q(t) 1 Ω0 ∂ Ai Ai = + exact forms. Q(t)k+1 ∂ xi k Q(t)k ∂ xi i=0
(45)
i=0
But the n–form Ω0 is now given by: i . . . ∧ dx n . Ω0 = (−1)i ki x i dx 0 ∧ . . . dx
(178)
i
As an example, consider the collection of weights [k0 , k1 , k2 , k3 , k4 ] = [41, 51, 52, 48, 64]. The condition for a hypersurface to have zero first Chern class is: deg Q = i ki = 256, so the Fermat–like Calabi–Yau hypersurface is: Q(t) =
1 5 a b + b4 c + c4 d + d 4 e + e4 − tabcde = 0. 5
(179)
One can check that this hypersurface has h 2,1 = 1, and so the only periods to consider are those of the holomorphic 3–form and its derivatives.
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
719
As in ordinary projective space, one can find linear combinations of derivatives of Q suitable for constructing diagrams involving 3 periods: ⎛
256 ⎜ −64 ⎜ ⎜ 16 ⎝ −4 1
0 320 −80 20 −5
0 0 320 −80 20
0 0 0 320 −80
⎛ 5 ⎞ ⎞⎛ ⎞ a b − tabcde a∂a Q(t) 0 ⎜ b4 c − tabcde ⎟ 0 ⎟ ⎜ b∂b Q(t) ⎟ ⎜ ⎟ ⎟⎜ ⎟ 4 ⎟ 0 ⎟ ⎜ c∂c Q(t) ⎟ = 256 ⎜ ⎜ c d − tabcde ⎟ , ⎠ ⎝ ⎠ ⎝ d 4 e − tabcde ⎠ 0 d∂d Q(t) 320 e∂e Q(t) e4 − tabcde
(180)
and the relations corresponding to diagrams are therefore: (v0 + 5, v1 + 1, v2 , v3 , v4 ) = (v0 , v1 + 4, v2 + 1, v3 , v4 ) = (v0 , v1 , v2 + 4, v3 + 1, v4 ) = (v0 , v1 , v2 , v3 + 4, v4 + 1) = (v0 , v1 , v2 , v3 , v4 + 4) =
f (0,v ) 256k(v) (v) + t (v + 1), f (1,v ) 256k(v) (v) + t (v + 1), f (2,v ) 256k(v) (v) + t (v + 1), f (3,v ) 256k(v) (v) + t (v + 1), f (4,v ) 256k(v) (v) + t (v + 1),
(181) (182) (183) (184) (185)
with the coefficients f (i, v) given by: ⎛ ⎜ ⎜ ⎜ ⎝
⎞ ⎛ 256 f (0, v) f (1, v) ⎟ ⎜ −64 ⎟ ⎜ f (2, v) ⎟ = ⎜ 16 f (3, v) ⎠ ⎝ −4 1 f (4, v)
0 320 −80 20 −5
0 0 320 −80 20
0 0 0 320 −80
⎞⎛ ⎞ v0 + 1 0 0 ⎟ ⎜ v1 + 1 ⎟ ⎟⎜ ⎟ 0 ⎟ ⎜ v2 + 1 ⎟ . ⎠ ⎝ 0 v3 + 1 ⎠ 320 v4 + 1
(186)
The diagram for the periods of the holomorphic 3–form is then: (0, 0, 0, 0, 0) A ↓ D4 (0, 0, 0, 0, 4) B ↓ D3 (0, 0, 0, 4, 5)C ↓ D2 (0, 0, 4, 5, 5) D ↓ D1 (−1, 3, 4, 4, 4) → (0, 4, 5, 5, 5) E ↓ D0 (4, 4, 4, 4, 4)
→ (1, 1, 1, 1, 1) → (2, 2, 2, 2, 2) → (3, 3, 3, 3, 3) → (4, 4, 4, 4, 4) ↓ D4 ↓ D4 ↓ D4 → (1, 1, 1, 1, 5) → (2, 2, 2, 2, 6) → (3, 3, 3, 3, 7) ↓ D3 ↓ D3 → (1, 1, 1, 5, 6) → (2, 2, 2, 6, 7) ↓ D2 → (1, 1, 5, 6, 6)
The encoded coupled system is:
D
1 4! η(η − 1)(η − 2)(η = 13 (η + 1) C,
− 3)A = t 5 E, C=
1 2
(η + 1) B,
E=
1 4
(η + 1) D,
B = (η + 1) A,
(187) (188)
which results in the following Picard–Fuchs equation:
η(η − 1)(η − 2)(η − 3) − t 5 (η + 1)4 A = 0.
(189)
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C. Examples of the Griffiths–Dwork Technique As examples of the formalism outlined in Sect. 2 we compute some of the Picard–Fuchs equations for a family of elliptic curves and the Fermat family of quintics in P4 . Hesse form cubics in P2 . Cubic hypersurfaces in P2 are elliptic curves, and hence admit a single holomorphic 1–form. We have the following correspondence in general:22 Ω ∈ H 1,0 = F1,1 → [1] ∈ C[a, b, c]0 /[∂i Q] → P1 = Q1 Ω0 , abc dΩ 1,0 ⊕ H 0,1 = F1,0 → [abc] ∈ C[a, b, c] /[∂ Q] → P = 3 i 2 dt ∈ H Q Ω0 . Here [a, b, c] are homogeneous coordinates, P is generic notation for a column of the period matrix, and the polynomial Q(t) defining the hypersurface is: 1 3 a + b3 + c3 − tabc = 0. (190) Q(t) = 3 Holomorphic Differentiating a period of the holomorphic 1–form twice gives: 2Ω0 1–Form 2 P1 = (abc) , where P1 is a period, and the prime denotes differentiation with Q3 respect to t. As an element of the Jacobian ideal, we find: (1 − t 3 )(abc)2 = t 2 a 2 bc Applying (43) results in: d2 P1 (1 − t ) 2 = 2 dt 3
∂Q ∂Q ∂Q + ta 3 b + a 2 b2 . ∂a ∂b ∂c Ω0 2 t abc + Q2
Ω0 3 ta . Q2
Now writing a 3 = tabc + a ∂∂aQ , we find: dP1 Ω0 2 Ω0 ∂ Q Ω0 3 ta = t abc + t a = t2 + tP1 . Q2 Q2 Q 2 ∂a dt
(191)
(192)
(193)
Substituting back into (192) gives the Picard–Fuchs equation: (t 3 − 1)
dP1 d2 P1 + tP1 = 0. + 3t 2 2 dt dt
In terms of the logarithmic derivative: η = t dtd , this becomes: 1 (η + 1)2 − 3 η(η − 1) P1 = 0. t
(194)
(195)
d Making the substitution x = t −3 and θ = x dx results in an equation in standard hypergeometric form:
2 1 θ2 − x θ + θ+ tP1 = 0. (196) 3 3
Since nothing in the reasoning above depends on the choice of cycle integrated over, we conclude that both of the b1 = 2 integrals of the holomorphic 1–form (one column of j the period matrix γi Ω X ) obey Eq. (196). 22 For the rest of this section we will abbreviate H p,q (V (t)) and F p,q (V (t)) with H p,q and F p,q .
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
721
Mixed 1–form The Griffiths–Dwork technique doesn’t work so well for the periods of the derivative of the holomorphic 1–form: P2 =
abc d Ω0 = Q(t)2 dt
Ω0 . Q(t)
(197)
The problem is that we only have to differentiate once in order that the numerator of the integrand is in the ideal [∂i Q(t)], so one might guess that the P2 obeys a 1st order equation. Indeed a 1st order equation can be derived, but it contains P1 as well, so it is not a Picard–Fuchs equation. To eliminate P1 one must differentiate P2 a second time. In the method introduced in Sect. 3.1, this happens automatically. The end result is: (t 3 − 1)
d2 P2 1 dP2 2 + 4tP2 = 0 + 5t + dt 2 t dt
(198)
d Or, with x = t 3 and θ = x dx , in hypergeometric form:
2 2 2 −x θ+ θ θ− P2 = 0. 3 3
(199)
Again, since nothing depends on the cycle integrated over, both periods obey the above equation. The Hesse form cubic is the simplest possible example; in general computational techniques are required to do the algebra.
Fermat form quintics in P4 . Smooth quintic hypersurfaces in P4 are Calabi–Yau 3–folds with b3 = 204. The results of Sect. 2 give the following correspondence: C[a,b,c,d,e]0 [∂i Q] C[a,b,c,d,e]5 → [Mα ] ∈ ωα ∈ H 3,0 ⊕ H 2,1 [∂i Q] C[a,b,c,d,e]10 ωζ ∈ H 3,0 ⊕ H 2,1 ⊕ H 1,2 → [Mζ ] ∈ [∂i Q] d 3 Ω ∈ H 3,0 ⊕ H 2,1 ⊕ H 1,2 ⊕ H 0,3 → [a 3 b3 c3 d 3 e3 ] ∈ C[a,b,c,d,e]15 3 [∂i Q] dt
Ω ∈ H 3,0
→
[1] ∈
1 Q Ω0 , Mα → Pα = Q Ω0 , Mζ → Pζ = Q Ω0 , a 3 b3 c3 d 3 e3 → P204 = Ω0 . Q →
P1 =
Again [a, b, c, d, e] are homogeneous coordinates, and Q(t) is the polynomial defining the hypersurface. The indices α and ζ have the ranges {2, . . . , 102} and {103, . . . , 203} respectively. If we now specialize to the Fermat family of quintic hypersurfaces: Q(t) =
1 5 a + b5 + c5 + d 5 + e5 − tabcde, 5
(200)
then there are two simplifications. For a general quintic, each period satisfies a 204th order differential equation. In computational terms, one expects to have to differentiate periods 204 times before the numerator in the integrand lies in the Jacobian ideal J (Q). For (200) the order of the equations is reduced to 4 or less. The other simplification is that we can say more about the Hodge type of the forms than merely which Hodge filtrant they are in.
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Periods of the holomorphic 3–form. One finds (by Gröbner basis techniques for example) that it is sufficient to differentiate the periods of the holomorphic 3–form just 4 times, d4 P1 (4!)Ω0 = (abcde)4 , (201) dt 4 Q5 (1 − t 5 )(abcde)4 = t 4 a 4 (bcde)3 ∂a Q + t 3 a 7 b3 (cde)2 ∂b Q + t 2 (ab)6 c2 de ∂c Q + t (abc)5 d ∂d Q + (abcd)4 ∂e Q. (202) Then proceeding in the same way as with the Hesse cubic, one finds the equation: (t 5 − 1)
d3 P1 d2 P1 dP1 d4 P1 + tP1 = 0. + 10t 4 3 + 25t 3 2 + 15t 2 4 dt dt dt dt
(203)
d gives an equation in generalized hypergeometric Substituting x = t −5 and θ = x dx form:
2 3 4 1 θ+ θ+ θ+ tP1 = 0. θ4 − x θ + (204) 5 5 5 5
Indeed this is how the Picard–Fuchs equation for the invariant periods is most often presented. From hereon though we will not make such changes of variables, but rather work directly in terms of the variable t, and the logarithmic derivative η = dtd . The reasons for this choice are summarized at the end of Sect. 2.4. As before, nothing depends on which cycle is integrated over, so all 204 integrals of Ω X0 obey the 4th order Eq. (203). This fact is related to the symmetries of the Fermat locus in Sect. 3.1. Periods of the other forms. In a similar way, one can pick a basis of the rest of C[a, b, c, d, e]/J (Q) and work out the equations satisfied by each of the 203 other forms. It is easier to do this with the techniques introduced in Sect. 3.1. In particular we take advantage of the symmetries of the Fermat–form quintic with greater ease. D. Geometric Mondromy In Sect. 2 we explored the connections between two descriptions of hypersurfaces; on the one hand as objects embedded in projective space (via the order of pole filtration), and on the other as complex manifolds (via the Hodge filtration). As already alluded to, a great deal of information about the relation between these two points of view is contained in the period matrix: Hn−1 V (t), Z × H n−1 V (t), C −→ C, (205) [γi (t)], [Ω j (t)] −→ Πi j (t) = Ω j (t). (206) γi (t)
The t dependence in [γi (t)] is locally trivial, but if one follows the homology classes [γi (t)] around a path enclosing a singular hypersurface, one finds that the class at the finish is not the same as at the beginning.
Families of Quintic Calabi–Yau 3–Folds with Discrete Symmetries
723
B
0
t
1
∞
0
t
1
∞
B
A
Fig. 2. The two sheets of the Riemann surface of y 2 = x(x − 1)(x − t), which topologically glue together to make a torus. The A and B cycles are also shown
Though the reader may be familiar with monodromy of in general, for the purpose of interpreting Picard–Fuchs equations, it is useful to have in mind a simple example of geometric monodromy acting on the homology groups of a manifold. As is often the case, elliptic curves provide a beautifully concrete case study.23 Consider for example the Riemann surface of the function:24 y 2 = x(x − 1)(x − t),
(207)
which is singular at t = 0, 1 and ∞. There are 2 sheets, and we choose the branch cuts to be as in Fig. 2. As defined pictorially, the intersection matrix is given by:
0 −1 A∩ A A∩B . (208) = 1 0 B∩A B∩B But now imagine that t executes a small circle around 0. The consequences for the cycles A and B are shown in Fig. 3. In particular, the bottom diagram shows both the old B cycle (in red) and the new B (in blue). One can then read off the intersections: A ∩ A = 0, B ∩ A = 1,
A ∩ B = −1, B ∩ B = 2.
(209) (210)
From this it follows that the homology classes of the primed cycles are related to those of the unprimed cycles in the following way:
[A] A = 1 0 . (211) −2 1 [B] B In general, it is easy to see that moving cycles around 0, 1, ∞ gives a map: π1 P1 − {0, 1, ∞} → Sp(2, Z) 23 The following argument and diagrams are adapted from [21]. 24 This set of curves parametrized by t is called the Legendre family.
(212)
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t
B
B
t
0
0
1
∞
∞
1 A
A
B
B 0
t
1
∞
A
Fig. 3. The A and B cycles after t rotates by π/2, π and 2π . The dashed lines are on the 2nd sheet
called the monodromy representation, or the monodromy action. The image is Sp(2, Z) rather than a more general matrix because the intersection form (208) must be preserved. An important consequence of nontrivial monodromy is that the period matrix is a multivalued function of t. To distinguish the monodromy of cycles from the monodromy of anything else (hypergeometric functions say), we call the former geometric monodromy. The families of Calabi–Yau 3–folds we consider in Sect. 3 have a somewhat different singularity structure from the elliptic curves (207). They have the singularities of the Fermat form quintic (200).25 Rather than at t = 0, 1, ∞, the singular hypersurfaces are at t 5 = 1 and t = ∞. References 1. Witten, E.: Symmetry Breaking Patterns In Superstring Models. Nucl. Phys. B 258, 75 (1985) 2. Greene, B.R., Kirklin, K.H., Miron, P.J., Ross, G.G.: A Three Generation Superstring Model. 1. Compactification And Discrete Symmetries. Nucl. Phys. B 278, 667 (1986) low energy theory. Greene, B.R., Kirklin, K.H., Miron, P.J., Ross, G.G.: A Three Generation Superstring Model. 2. Symmetry Breaking And The Nucl. Phys. B 292, 606 (1987) 3. Braun, V., He, Y.H., Ovrut, B.A., Pantev, T.: A heterotic standard model. Phys. Lett. B 618, 252 (2005) Braun, V., Ovrut, B.A., Pantev, T., Reinbacher, R.: Elliptic Calabi–Yau threefolds with Z(3) x Z(3) Wilson lines. JHEP 0412, 062 (2004) 25 One of them is the Fermat form quintic. The other is the Z –symmetric example mentioned in the 41 introduction.
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4. Candelas, P.: Yukawa Couplings Between (2,1) Forms. Nucl. Phys. B 298, 458 (1988) 5. Greene, B.R., Kirklin, K.H., Miron, P.J., Ross, G.G.: 273 Yukawa Couplings For A Three Generation Superstring Model. Phys. Lett. B 192, 111 (1987) 6. Giryavets, A., Kachru, S., Tripathy, P.K., Trivedi, S.P.: Flux compactifications on Calabi–Yau threefolds. JHEP 0404, 003 (2004) 7. Gukov, S., Vafa, C., Witten, E.: CFT’s from Calabi–Yau four-folds. Nucl. Phys. B 584, 69 (2000) [Erratumibid. B 608, 477 (2001)] 8. Giddings, S.B., Kachru, S., Polchinski, J.: Hierarchies from fluxes in string compactifications. Phys. Rev. D 66, 106006 (2002) [arXiv:hep-th/0105097] 9. Candelas, P., de la Ossa, X., Rodriguez-Villegas, F.: Calabi–Yau manifolds over finite fields. I. http:// arXiv.org/list/hep-th/0012233,2000 10. Greene, B.R., Plesser, M.R.: Duality In Calabi–Yau Moduli Space. Nucl. Phys. B 338, 15 (1990) 11. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology. Cambridge: Camb. Univ. Press. 1987 12. Candelas, P., De La Ossa, X.C., Green, P.S., Parkes, L.: A Pair Of Calabi–Yau Manifolds As An Exactly Soluble Superconformal Theory. Nucl. Phys. B 359, 21 (1991) 13. Headrick, M., Wiseman, T.: Numerical Ricci-flat metrics on K3. Class. Quant. Orav. 22, 4931–4960 (2005) 14. Greene, B.R., Plesser, M.R., Roan, S.S.: In: ‘Mirror Symmetry I’, New constructions of mirror manifolds: Probing moduli space far from Fermat points, Somerville, MA: International Press, 1998 15. Szendröi, B.: On an example of Aspinwall and Morrison. Proc. Amer. Math. Soc. 132(3), 621–632 (2004) 16. Griffiths, P.A.: On the Periods of Certain Rational Integrals. Ann. of Math. (2) 90, 460–495 (1969) 17. Griffiths, P.A. (Ed.): Topics in transcendental algebraic geometry Annals of Mathematics Studies, 106. Princeton, NJ: Princeton University Press, 1984 18. Cox, D.A., Katz, S.: Mirror Symmetry and Algebraic Geometry. Providence, RI: Amer. Math. Sec., 1999 19. Greene, B.R., Plesser, M.R.: Mirror manifolds: A Brief review and progress report. http://arXiv:hep-th/ 9110014 20. Dwork, B.: On The Zeta function of a hypersurface. II. Ann. of Math. (2) 80, 227–299 (1964) 21. Carlson, J., Müller–Stach, S., Peters, C.: Period Mappings and Period Domains. Cambridge: Cambridge University Press, 2003 22. Ohara, K.: Computation of the Monodromy of the Generalized Hypergeometric Function. Kyushu J. Math. 51, 101–124 (1997) 23. Morrison, D.: Mathematical aspects of mirror symmetry Complex. In: Algebraic Geometry, J. Kollár, ed., IAS/Park City Math. Series, Vol. 3, 1997, pp. 265–340 24. Doran, C.F., Morgan, J.W.: Mirror symmetry and integral variations of Hodge structure underlying one parameter families of Calabi–Yau threefolds. In: Mirror Symmetry v., AMS/IP Vol. 38, Providence, RI: Amer. Math. Sec., 2006 25. Hosono, S., Klemm, A., Theisen, S., Yau, S.-T.: Mirror symmetry, mirror map and applications to Calabi– Yau hypersurfaces. Commun. Math. Phys. 167, 301–350 (1995) 26. Coates, T., Corti, A., Iritani, H., Tseng, H.: Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples. http://arxiv.org/list/math.AG/0611550, 2006 27. Doran, C.F., Greene, B.R., Judes, S., Whitcher, U.: In preparation 28. Nikulin, V.V.: Finite groups of automorphisms of Kählerian K 3 surfaces. (Russian) Trudy Moskov. Mat. Obshch. 38, 75–137 (1979) 29. Oguiso, K.: Automorphism groups in a family of K3 surfaces. http://arxiv.org/list/0104049, 2001 30. Dolgachev, I.: Weighted projective varieties. In: Group Actions and Vector Fields, J. B. Carell, ed., Lecture Notes in Math., Vol. 956, Springer–Verlag, Berlin, Heidelberg, New York: 1982, pp. 34–37; J. Steenbrink, Intersection form for quasi–homogeneous singularities. Comp. Math. 34 (1977), 211–223 D. Morrison, Picard–Fuchs equations and mirror maps for hypersurfaces. http://arxiv.org/list/9111025, 1991 Communicated by N.A. Nekrasov
Commun. Math. Phys. 280, 727–735 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0471-z
Communications in
Mathematical Physics
Hyperbolic Calorons, Monopoles, and Instantons Derek Harland Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, UK. E-mail: [email protected] Received: 1 March 2007 / Accepted: 25 September 2007 Published online: 12 April 2008 – © Springer-Verlag 2008
Abstract: We construct families of SO(3)-symmetric charge 1 instantons and calorons on the space H3 × R. We show how the calorons include instantons and hyperbolic monopoles as limiting cases. We show how Euclidean calorons are the flat space limit of this family. 1. Introduction Calorons are instantons on R3 × S 1 . It has been known for some time that they are related both to instantons on R4 and BPS monopoles on R3 . For example, the large period limits of calorons are normally instantons [6], while the large scale limit of a charge 1 caloron is a charge 1 monopole [15]. It is interesting to study monopoles on hyperbolic space H3 , [1,12,2]. These “hyperbolic monopoles” are related to their Euclidean counterparts, because one can recover Euclidean monopoles in the limit where the curvature of hyperbolic space tends to zero [14]. Hyperbolic monopoles were first constructed by means of a conformal equivalence [1,2]: since R4 is conformally equivalent H3 × S 1 , hyperbolic monopoles can be obtained from instantons invariant under an action of U (1). Hyperbolic monopoles constructed in this way have the property that the asymptotic norm of their Higgs field is always an integer, and are sometimes called “integral” for this reason. Non-integral hyperbolic monopoles were constructed later by different means [12]. The conformal equivalence used to construct integral hyperbolic monopoles has also been used to construct hyperbolic calorons, that is, instantons on H3 × S 1 [4]. Hyperbolic calorons constructed in this way have the property that their period (the circumference of S 1 ) is proportional to the radius of curvature of the hyperbolic space. This property is the analogue of the integral property which arises when hyperbolic monopoles are constructed from Euclidean instantons. The purpose of this article is to show the Supported by PPARC.
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existence of hyperbolic calorons which do not have this integral property. We will give explicit examples of charge 1 calorons with arbitrary period on hyperbolic spaces with arbitrary curvature. We will show that these are related to Euclidean calorons and hyperbolic monopoles in the ways that one might expect; we will also show that they have a well-defined large period limit, which is an instanton on H3 × R. We will now briefly outline the content of the remainder of this article. In Sect. 2, we will give precise definitions for hyperbolic instantons and calorons. In Sect. 3, we will outline two simple methods that can be used to construct them. In Sect. 4 we will give explicit examples of hyperbolic calorons and instantons, and in Sect. 5 we will explore some of their properties. We will conclude with some brief remarks in Sect. 6. 2. Hyperbolic Calorons and Instantons Let (x 1 , x 2 , x 3 ) be coordinates on the hyperbolic ball H3 and let R = (x 1 )2 + (x 2 )2 + (x 3 )2 be the radial coordinate, with 0 ≤ R < S for some fixed S (S will determine the scalar curvature of H3 ). Let τ = x 0 be a coordinate on S 1 , with period β. The metric on H3 × S 1 is 2 = dτ 2 + Λ2 (d R 2 + R 2 dΩ 2 ), ds H
where Λ := (1 − (R/S)2 )−1 and dΩ 2 represents the metric on the 2-sphere. It will also be convenient to introduce a coordinate µ = (S/2)arctanh(R/S) and a complex coordinate z = µ + iτ . In terms of µ and τ , the metric is 2 ds H = dτ 2 + dµ2 + Ξ 2 dΩ 2 ,
with Ξ := (S/2) sinh (2µ/S). Let (y 0 , y 1 , y 2 , y 3 ) be standard coordinates on R4 , and let t = y 0 and r = (y 1 )2 + (y 2 )2 + (y 3 )2 . The metric on R4 is ds E2 = dt 2 + dr 2 + r 2 dΩ 2 . We let Z = r + it be a complex coordinate. When β = Sπ , H3 × S 1 is conformally equivalent to R4 \R2 . We define a map from H3 × S 1 to R4 by the equation Z = tanh(z/S). Then it is simple to show that 2 ds H = ξ 2 ds E2 ,
with ξ = (S/2)(cosh(2µ/S) + cos(2τ/S)). Gauge fields on M = H3 × S 1 or H3 × R will be denoted A = Aα d x α , with Aα traceless anti-hermitian matrix-valued functions. The field strength tensor of a gauge field is F = (1/2)Fαν d x α ∧ d x ν , where Fαν = ∂α Aν − ∂ν Aα + [Aα , Aν ]. The action
Hyperbolic Calorons, Monopoles, and Instantons
of a gauge field on M is S=−
1 8
M
729
Tr(Fαν F αν ) Ξ 2 dτ dµ dΩ
(dΩ represents the volume form on the 2-sphere). Hyperbolic calorons are defined to be gauge fields on H3 × S 1 , whose curvature satisfies the self-dual equations, 1 i jk F jk , (1) 2Λ and whose action is finite, with Tr(Fαν F αν ) → 0 as µ → ∞. Similarly, hyperbolic instantons are self-dual gauge fields on H3 ×R with finite action and with Tr(Fαν F αν ) → 0 as µ2 + τ 2 → ∞. Finite action gauge fields on M = H3 × S 1 or H3 × R have a charge, 1 W =− 2 Tr(F ∧ F). 8π M F0i =
The action is bounded from below by the charge, S ≥ 2π 2 |W |. Instantons and calorons attain this bound and have W > 0, since they solve (1). For instantons, W is an integer. This follows from the usual argument: the gauge field extends to the manifold S 3 at infinity, and since the curvature is zero there, the gauge field is zero up to a gauge transformation g : S 3 → SU (2). The map g has integer degree, which is computed by W . The situation for calorons is a bit more complicated. A caloron is characterised by two integer topological charges, called the instanton charge (or Pontryagin index) and the magnetic charge, and a boundary condition, called the holonomy [5]. The integral W is equal to a combination of these three quantities, and is not necessarily an integer. However, the calorons we consider here will have zero magnetic charge and trivial holonomy, and in this case W will be an integer. 3. Construction 2 on H3 × S 1 is locally conformally equivalent to the As we have seen, the metric ds H 2 4 metric ds E on R . It is well known that the action of the Hodge star operator on 2-forms on R4 is invariant under conformal rescalings of the metric. Therefore the self-dual equations on H3 × S 1 are locally the same as the self-dual equations on R4 . There are a number of methods available for constructing self-dual gauge fields on R4 , and all of these can be applied directly on H3 × S 1 . In this section we will show how two well-known constructions for Euclidean instantons can be applied to give locally self-dual gauge fields on H3 × S 1 . We will show later that these methods can in fact be used to generate finite action gauge fields, that is, hyperbolic calorons (and hyperbolic instantons).
3.1. Harmonic function ansatz. The first ansatz we shall consider is the harmonic function ansatz, due to Corrigan, Fairlie, and ’t Hooft [3]. On Euclidean space, one supposes that the gauge field is written in the form, A0 = ∂ j ln ϕ(σ j /2i),
(2)
A j = (−∂0 ln ϕ δ jl + jlk ∂k ln ϕ)(σ /2i) l
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for some function ϕ. Then the gauge field will be self dual when ϕ satisfies the Laplace equation, E ϕ = 0, where E = (∂/∂ y α )2 is the (Euclidean) Laplacian. 2 , we let ρ = ϕ/ξ and change coordinates in the To use this ansatz on the metric ds H above expressions. The resulting ansatz for the gauge field is j 1 σ , (3) A0 = ∂ j ln ρ Λ 2i l σ 2Λ k A j = −Λ∂0 ln ρ δ jl + jlk ∂k ln ρ + 2 x . (4) S 2i This will be self-dual if ρ satisfies the equation, H ρ = −
4 ρ, S2
(5)
2, where H is the Laplace-Beltrami operator for the metric ds H
H =
∂ ∂τ
2 +
1 Λ2
∂ ∂ xi
2 +
2 i ∂ x . ΛS 2 ∂ x i
3.2. S O(3)-symmetric ansatz. The second ansatz we shall consider is due to Witten [16]. Witten simplified the problem of finding Euclidean instantons by restricting attention to gauge fields invariant under an action of S O(3); he was able to solve the reduced self-duality equations exactly, yielding a large family of instantons. Interestingly, the S O(3)-invariant instantons turned out to be related to twodimensional vortices. The components of the S O(3)-invariant gauge field were interpreted as a U (1) gauge field and a 2-component Higgs field, and the action of the 4-dimensional gauge field was identical to that of a vortex model on 2-dimensional hyperbolic space. Configurations in the vortex model had a topological charge, and this was equal to the topological charge of the corresponding 4-dimensional gauge theory. Witten’s method is also applicable to self-dual gauge fields on H3 × S 1 and 3 H × R, because the SO(3) action pulls back to these manifolds. We will see that the dimensionally-reduced equations of motion are solvable using an adaptation of Witten’s method. As in the Euclidean case, hyperbolic instantons will be related to a 2-dimensional vortex model, but this time the underlying 2-manifold will not be hyperbolic space. In this section, our notation will closely follow that of [9] We make the following S O(3)-symmetric ansatz for a gauge field on H3 × S 1 or 3 H × R: 1 A = − (Qa + φ1 d Q + (φ2 + 1)Qd Q), 2
(6)
where Q = x a σ a /R, a = aµ dµ + aτ dτ , and φ1 , φ2 , aµ , and aτ are real functions of µ and τ . We let φ = φ1 − iφ2 be a Higgs field, and let Dµ φ = ∂µ φ + iaµ φ and Dτ φ = ∂τ φ + iaτ φ denote the components of its covariant derivative with respect to the gauge
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field a. We will also write Dφ = Dµ φdµ + Dτ φdτ and Dz¯ = (Dµ + i Dτ )/2. Then the self-dual equations (1) for A can be succinctly written, Dz¯ φ = 0,
(7)
Ξ (∂τ aµ − ∂µ aτ ) = 1 − |φ| . 2
2
The action of the gauge field A is equal to 2 π 1 − |φ|2 2 2 2 2 S= + 2|Dµ φ| + 2|Dτ φ| dτ dµ. Ξ (∂µ aτ − ∂τ aµ ) + 2 Ξ
(8)
(9)
This is the action of a vortex model on the 2-manifold M coordinatised by µ and τ , with metric dsh2 = Ξ −2 (dµ2 + dτ 2 ). In the case of hyperbolic instantons (β = ∞), the manifold M is a universal cover of the hyperbolic disc with a point removed. Vortex configurations on M possess a topological charge, k, given by 1 k=− da. 2π A standard Bogomolny argument shows that 2π 2 k forms a lower bound for the action (9) within the set of vortex configurations with charge k > 0; Eqs. (7), (8) are precisely the equations which guarantee the bound is attained. It follows that k = W for self-dual gauge fields. We note that the ansatz (6) does not fix the gauge; one is free to make gauge transformations of the form g = exp(λQ/2) for some real function λ(µ, τ ). These gauge transformations correspond to U (1) gauge transformations in the vortex model. The solution of the self-dual equations (7), (8) is relatively simple. One first needs to choose a meromorphic function g(z) satisfying |g|2 ≤ 1, with equality when µ = 0, and a non-zero holomorphic function h(z). Then a self-dual gauge field is obtained from φ = eψ h∂z g, aµ = −∂τ ψ, aτ = ∂µ ψ,
(10) (11) (12)
ψ = ln(2Ξ ) − ln(1 − |g|2 ) − ln |h|.
(13)
The function h can be removed by gauge transformation (h is included here for later convenience). The derivation of these solutions is similar to that given by Witten in the Euclidean case. 3.3. Equivalence. After the discoveries of the S O(3)-symmetric ansatz and the harmonic function ansatz for Euclidean instantons, Manton showed that the latter encompasses the former [11]. More specifically, all self-dual gauge fields obtained via Witten’s ansatz can also be expressed in the harmonic function form (2). A similar result holds for the two hyperbolic ansätze discussed above.
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Suppose that a hyperbolic gauge field is given by (6) and (10)-(13) in terms of a meromorphic function g. Then the gauge h can be chosen so that the gauge field is in the form (3), (4), with 1 1 − |g|2 ρ= (14) 2Ξ |1 − g|2 satisfying (5). This result can be derived by following the method of Manton. 4. Examples We will now give examples of hyperbolic calorons and instantons, using the SO(3)-symmetric ansatz (6). 4.1. Charge 1 instanton. A charge 1 instanton is obtained from (10)-(13) when λ − 2z/S 2z g(z) = g1 (z) := exp − , S λ + 2z/S where λ > 0 is a real parameter. To verify that this gauge field is an instanton, one must check that the action is finite. This has been done directly, by computing all of the terms in the action (9) and verifying that their integrals converge. The charge of the instanton was computed using methods from Chap. II of [7]. 4.2. Charge k instanton. Consider the function, k 2z λi − 2z/S gk (z) := exp − S λ¯ + 2z/S i=1 i with λi complex parameters with positive real part. It is believed that when g(z) = gk (z) the gauge field is a charge k instanton. Certainly, gk (z) agrees with g1 when k = 1, and the Higgs field φ has k zeros when g = gk . Normally the charge of a vortex configuration is equal to the number of zeros of the Higgs field. Numerical results suggest that the action of the gauge field is finite and proportional to k, however, we have not yet proved this analytically. 4.3. Charge 1 caloron. The following function is a special case of gk with k = ∞: k λ + 2 jβi/S − 2z/S 2z lim , g∞ (z) := exp − S k→∞ λ − 2 jβi/S + 2z/S j=−k
where λ and β are positive real parameters. The infinite product in g∞ (z) converges uniformly on any compact set, after singular terms have been removed, and the limit has the following closed form: lim
k→∞
k sinh((z − Sλ/2)π/β) λ + 2 jβi/S − 2z/S = . λ − 2 jβi/S + 2z/S sinh((z + Sλ/2)π/β)
j=−k
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The gauge choice h = exp(2z/S) makes the gauge field obtained from g∞ explicitly periodic, with period β. This gauge field is in fact a hyperbolic caloron with finite action and charge 1, as has been verified analytically. We mentioned in the introduction that integral hyperbolic calorons (with β = Sπ ) can be obtained directly from Euclidean instantons. For example, a charge 1 instanton in the form (2), with ϕ =1+
α2 + t2
r2
is also a charge 1 integral hyperbolic caloron. One can also obtain charge 1 integral hyperbolic calorons from g∞ simply by setting β = Sπ . In fact these two sets of integral hyperbolic calorons are identical; if one takes α 2 = tanh(λ/2) then the two calorons are related by a gauge transformation. Therefore, the hyperbolic calorons obtained from g∞ are a sensible generalisation of the known integral charge 1 calorons. 5. Properties The non-integral charge 1 hyperbolic calorons we have constructed are determined by three parameters, S, β and λ, which determine the curvature of the hyperbolic space, the period of the caloron, and the shape of the caloron. By taking limits of these parameters we are able to relate the hyperbolic calorons to Euclidean calorons, hyperbolic instantons, and non-integral hyperbolic monopoles. 2 converges to the 5.1. Zero curvature limit. When S → ∞, the hyperbolic metric ds H 2 Euclidean metric ds E if one makes the identifications τ = t, µ = r . Charge 1 calorons were first constructed on Euclidean space by Harrington and Shepard [6] using the ansatz (2), with
ϕ =1+
sinh(2πr/β) π ν2 . βr cosh(2πr/β) − cos(2π t/β)
Here ν > 0 is a real parameter and β determines the period of the caloron. We will show here that Euclidean 1-calorons are obtained from hyperbolic 1-calorons by taking a limit where S → ∞. The zeros of the Higgs field φ obtained from (10) when g = g∞ are located at the points (µ, τ ) determined by τ = nβ, n ∈ Z, Sπ λ Sπ Sπ λ 2π µ = cosh + sinh . cosh β β β β
In order that these points remain finite as S → ∞, we must also let λ → 0 such that λS 2 remains finite. In the gauge h = (S/2) exp(2z/S), the functions φ, aµ , aτ converge pointwise. In fact, it can be shown that their limit describes a Euclidean 1-caloron, with ν2 =
λS 2 . S→∞,λ→0 2 lim
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5.2. Instanton limit. In the limit β → ∞, the function g∞ (z) converges pointwise to g1 (z), so one might expect the hyperbolic caloron gauge field to converge to a hyperbolic instanton gauge field in this limit. This can be verified directly, using expressions derived from (10)–(13). 5.3. Monopole limit. Non-integral hyperbolic monopoles were constructed in [12]. In the formalism of this report, the monopoles are obtained from (10)–(13), with 2Bz g(z) = g M (z) := exp − . S B > 1 is a positive real parameter and the asymptotic norm of the Higgs field of the monopole is equal to B − 1. Hyperbolic calorons have a well-defined limit as λ → ∞. One can show that this limit is gauge equivalent to a hyperbolic monopole, with B =1+
Sπ . β
6. Conclusion We have investigated the notion of non-integral hyperbolic calorons by constructing explicit examples. These generalise integral hyperbolic calorons in the same way that non-integral hyperbolic monopoles generalise integral hyperbolic monopoles. Like Euclidean 1-calorons, hyperbolic 1-calorons have a monopole limit. The non-integral property also makes it possible to obtain a large period limit, which we have called a hyperbolic instanton. The zero curvature limits of our hyperbolic calorons are Euclidean calorons. In the case of hyperbolic monopoles, the zero curvature limit is always a Euclidean monopole [14], but it is not yet known whether a hyperbolic caloron is guaranteed to have a zero curvature limit. The examples we have given are all rotationally symmetric, but this need not be the case; probably more general examples can be found using the harmonic function ansatz (3), (4). It is not clear whether hyperbolic calorons exist with non-trivial asymptotic holonomy, and a more complete account of hyperbolic calorons would address this issue. Examples of Euclidean calorons with non-trivial holonomy are known [10,8], but their construction depends on more sophisticated techniques than those presented here. The study of hyperbolic monopoles, both integral and non-integral, has yielded many interesting results. For example, hyperbolic monopoles are determined completely by their behaviour on the 2-sphere at infinity [13]. It would be interesting to see whether this, or other properties of hyperbolic monopoles, have analogues for hyperbolic calorons. Acknowledgement. I am grateful to my supervisor, Prof. R. S. Ward, for many useful discussions.
References 1. Atiyah, M.F.: Magnetic monopoles in hyperbolic spaces. In: Vector Bundles on Algebraic Varieties, Oxford: Oxford University Press, 1087, pp. 1–34 2. Chakrabarti, A.: Construction of hyeprbolic monopoles. J. Math. Phys. 27, 340–348 (1985) 3. Corrigan, E., Fairlie, D.B.: Scalar field theory and exact solutions to a classical su(2)-gauge theory. Phys. Lett. 67B, 69–71 (1977)
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4. Garland, H., Murray, M.K.: Why instantons are monopoles. Commun. Math. Phys. 121, 85–90 (1989) 5. Gross, D.J., Pisarski, R.D., Yaffe, L.G.: Qcd and instantons at finite temperature. Rev. Mod. Phys. 53, 43 (1978) 6. Harrington, B.J., Shepard, H.K.: Periodic euclidean solutions and the finite-temperature yang-mills gas. Phys. Rev. D17(8), 2122–2125 (1978) 7. Jaffe, A., Taubes, C.: Vortices and Monopoles. Boston: Birkhäuser, 1980 8. Kraan, T.C., van Baal, P.: Periodic instantons with non-trivial holonomy. Nucl. Phys. B533, 627–659 (1998) 9. Landweber, G.D.: Singular instantons with so(3) symmetry. http://arxiv.org/math.dg/0503611, 2005 10. Lee, K., Lu, C.: su(2) calorons and magnetic monopoles. Phys. Rev. D15, 025011 (1998) 11. Manton, N.S.: Instantons on a line. Phys. Lett. 76B, 111–112 (1978) 12. Nash, C.: Geometry of hyperbolic monopoles. J. Math. Phys. 27, 2160–2164 (1986) 13. Norbury, P.: Asymptotic values of hyperbolic monopoles. J. Geom. Phys. 51, 13–33 (2004) 14. Norbury, P., Jarvis, S.: Zero and infinite curvature limits of hyperbolic monopoles. Bull. of the L.M.S. 29, 737–744 (1997) 15. Rossi, P.: Propagation functions in the field of a monopole. Nucl. Phys. B149, 170–188 (1979) 16. Witten, E.: Some exact multipseudoparticle solutions of classical yang-mills theory. Phys. Rev. Lett 38(3), 121–124 (1977) Communicated by G.W. Gibbons
Commun. Math. Phys. 280, 737–749 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0477-6
Communications in
Mathematical Physics
The Geometry of Recursion Operators G. Bande1 , D. Kotschick2 1 Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari,
Via Ospedale 72, 09124 Cagliari, Italy. E-mail: [email protected]
2 Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39,
80333 München, Germany. E-mail: [email protected] Received: 12 March 2007 / Accepted: 25 October 2007 Published online: 17 April 2008 – © G. Bande, D. Kotschick 2008
Abstract: We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser’s theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squares of the intertwining endomorphisms are plus or minus the identity. For pairs of forms we recover the notions of symplectic pairs and of holomorphic symplectic structures. For triples we recover the notion of a hypersymplectic structure, and we also find three new structures that have not been considered before. One of these is the symplectic formulation of hyper-Kähler geometry, which turns out to be a strict generalization of the usual definition in terms of differential or Kähler geometry. 1. Introduction It is now well known that many geometric structures, particularly on four-manifolds, can be defined in terms of pairs of two-forms; see for example Donaldson [10]. In this paper we study the fields of endomorphisms intertwining such pairs of forms. This leads to a natural generalization of Moser’s theorem [22] on isotopies of symplectic forms and to a generalization of known geometric structures on four-manifolds to arbitrary dimensions. Suppose we are given two non-degenerate 2-forms ω and η on the same manifold M. Then there exists a unique field of invertible endomorphisms A of the tangent bundle T M defined by the equation i X ω = i AX η.
(1)
The important special case when the two 2-forms involved are closed, and therefore symplectic, is very interesting both from the point of view of physics, where it arises in © 2008 The authors. Faithful reproduction for non-commercial purpose is permitted.
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the context of bi-Hamiltonian systems, and from a purely mathematical viewpoint. In physics the field of endomorphisms A is called a recursion operator, and we shall adopt this terminology here. We shall study the global geometry and topology of a manifold endowed with two (or more) symplectic forms, which we discuss using the associated recursion operator A. For local considerations in the case when the Nijenhuis tensor of A vanishes see Turiel [24]. In Sect. 2 we shall show that the recursion operator neatly encapsulates the necessary and sufficient condition for the existence of a simultaneous isotopy of two families of symplectic forms. In Sect. 3 we consider the simplest examples, where the recursion operator A satisfies A2 = ±1. We shall find that these most basic cases correspond precisely to the symplectic pairs studied in [4], and to holomorphic symplectic forms respectively. Our discussion of holomorphic symplectic structures in terms of recursion operators generalizes the work of Geiges [13] on conformal symplectic couples from dimension four to arbitrary dimensions. In Sect. 4 we introduce the four geometries defined by triples of symplectic forms whose pairwise recursion operators all satisfy A2 = ±1. Throughout our point of view is that of symplectic geometry, taking as our geometric data only the symplectic forms and the recursion operators they define. Nevertheless, we shall see that in two of the four cases the data encoded by the triple of symplectic forms define a pseudo-Riemannian metric leading to the kind of geometry that is used in supersymmetric string theory; see for example [5,17]. One of these cases is that of hypersymplectic structures in the sense of Hitchin [16], the other one is a symplectic analogue of hyper-Kähler structures. We will show that this symplectic formulation of hyper-Kähler geometry is not equivalent to the usual one, because the symplectic data does not force the associated pseudo-Riemannian metric to be definite. Hyper-Kähler geometry corresponds precisely to the special case in which the natural metric is definite. We shall also discuss briefly the geometries defined by triples of symplectic forms with recursion operators of square ±1 which do not have natural pseudo-Riemannian metrics attached to them. These structures have more to do with foliations than with differential geometry. 2. Simultaneous Isotopies of Symplectic Forms Given a family ωt of smoothly varying symplectic forms on a compact manifold M, with t ∈ [0, 1], Moser [22] showed that there is an isotopy ϕt with ϕt∗ ωt = ω0 if and only if the cohomology class of ωt is independent of t. This condition ensures that ω˙ t is exact, and every choice of a primitive αt depending smoothly on t defines a time-dependent vector field X t by the equation i X t ωt = −αt . The isotopy ϕt is obtained by integrating X t . Conversely, every isotopy with the property ϕt∗ ωt = ω0 is generated by a vector field of this form, as one sees by differentiation. Suppose now that we have two smoothly varying families of symplectic forms ωt and ηt on a compact manifold M. (We do not make any assumption on the orientations they induce.) When is there an isotopy ϕt with ϕt∗ ωt = ω0 and ϕt∗ ηt = η0 ? As the vector fields generating isotopies for a single family are very special, one can not in general expect that there is a vector field which works for both families simultaneously. Let A be the time-dependent recursion operator defined by i X ωt = i AX ηt . If an isotopy ϕt makes both ωt and ηt constant, then it makes A constant in t. Therefore, an isotopy can only exist, if the diffeomorphism type of the recursion operator is constant
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in t. If this is the case, we may as well assume that A is independent of t. Then we have the following isotopy result à la Moser: Theorem 1. Let ωt and ηt with t ∈ [0, 1] be smoothly varying families of symplectic forms on a compact manifold M, and assume that the associated recursion operator A is independent of t. Then there exists an isotopy ϕt with ϕt∗ ωt = ω0 and ϕt∗ ηt = η0 if and only if ω˙ t and η˙ t are exact, and their primitives can be chosen in such a way that ω˙ t = dαt and η˙ t = dβt with αt = βt ◦ A. Proof. Suppose that the desired isotopy exists. Then 0=
d ∗ ϕ ωt = ϕt∗ (ω˙ t + L X t ωt ) = ϕt∗ (ω˙ t + di X t ωt ), dt t
and thus we may take αt = −i X t ωt as a primitive of ω˙ t . Similarly we may take βt = −i X t ηt as a primitive of η˙ t . With these choices we have for any Y ∈ T M: αt (Y ) = −i X t ωt (Y ) = −i AX t ηt (Y ) = −i X t ηt (AY ) = (βt ◦ A)(Y ) because ηt (AX, Y ) = ηt (X, AY ). Thus the chosen primitives satisfy αt = βt ◦ A. Conversely, assume that ω˙ t = dαt and η˙ t = dβt with αt = βt ◦ A. Define two vector fields X t and Yt by i X t ωt = −αt and i Yt ηt = −βt . We claim that X t = Yt . For the proof we calculate for an arbitrary Z ∈ T M: i X t ωt (Z ) = −αt (Z ) = −βt (AZ ) = i Yt ηt (AZ ) = i AYt ηt (Z ) = i Yt ωt (Z ). The non-degeneracy of ωt now implies that X t and Yt agree. Denote by ϕt the isotopy they generate. Then ϕt∗ ωt = ω0 and ϕt∗ ηt = η0 . Theorem 1 may look rather ad hoc at first sight, as the geometric meaning of the conditions that A be independent of t and that it intertwine the primitives αt and βt is not at all obvious. It may also not be clear that there are families in which the diffeomorphism type of the recursion operator does change. Nevertheless, we maintain that this is the natural formulation of the criterion for the existence of simultaneous isotopies. In Sect. 3 we will specialize this result by making assumptions on A, and thereby clarify the geometric content of Theorem 1. For example, when A2 = I dT M but A = ±I dT M , we shall see that ω and η are equivalent to a symplectic pair in the sense of [4], and A contains the information about the pair of foliations induced by the symplectic pair. In this case it is possible that the foliations could vary in a non-diffeomorphic way, see [3], so the assumption about the diffeomorphism type of A is not vacuous. Moreover, Theorem 1 for this case is equivalent to the stability theorem for symplectic pairs formulated and proved in [3] using the basic cohomology of foliations.
3. Symplectic Pairs and Holomorphic Symplectic Forms The recursion operator A is the identity if and only if ω and η agree. It is minus the identity if and only if ω = −η. From now on we exclude these trivial cases, so we always assume A = ±I d.
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3.1. Symplectic pairs. Consider first the case A2 = I d, but A = ±I d. Then the eigenvalues of A are ±1, and X =
1 1 (X + AX ) + (X − AX ) 2 2
is the unique decomposition of an arbitrary tangent vector X into a sum of eigenvectors of A. Thus the eigenspaces of A give a splitting T M = D+ ⊕ D− . Lemma 2. The eigenspaces D± for the eigenvalues ±1 are precisely the kernels of ∓ = ω ∓ η. Proof. Let X be an arbitrary tangent vector. Then i X ∓ = i X ω ∓ i X η = i AX η ∓ i X η = i AX ∓X η. As η is non-degenerate, the condition i X ∓ = 0 is equivalent to AX = ±X .
The dimensions of the kernels of ∓ are semi-continuous, in that each can only increase on a closed subvariety. However, the lemma shows that if the dimension of the kernel of one of the two forms ∓ jumps up, then the dimension of the kernel of the other one has to decrease. Therefore, the dimensions of the kernels are actually constant on a connected manifold M, so that the forms ∓ have constant ranks. Moreover, as the ∓ are closed, their kernel distributions are integrable. Thus, the forms ∓ are a symplectic pair in the sense of [4]. Conversely, suppose that we have a symplectic pair ± on M, that is a pair of closed 2-forms of constant ranks, whose kernel foliations F and G are complementary. Then ω = 21 (+ + − ) and η = 21 (+ − − ) are symplectic forms, and the corresponding recursion operator is A = I dT G − I dT F . Thus A2 = I dT M . We have proved: Theorem 3. Two symplectic forms ω and η on a connected manifold M whose recursion operator A satisfies A2 = I d and A = ±I d give rise to a symplectic pair ± , and every symplectic pair ± arises in this way. Remark 4. The condition A2 = I d implies that the Nijenhuis tensor of A vanishes identically. Therefore, in this case, ω and η are compatible in the sense of Poisson geometry. The following stability result was proved in [3]: Theorem 5. Let ± t be a smooth family of symplectic pairs on a closed smooth manifold M, such that the kernel foliations F = Ker(+t ) and G = Ker(− t ) are independent ± of t ∈ [0, 1]. Then there exists an isotopy ϕt with ϕt∗ ± t = 0 if and only if the basic 2 cohomology classes [+t ] ∈ Hb2 (F) and [− t ] ∈ Hb (G) are constant. We now want to explain the equivalence between this result and Theorem 1 in the case when A2 = I d. Consider the symplectic forms ω = 21 (+ + − ) and η = 21 (+ − − ), and the corresponding recursion operator A = I dT G − I dT F . First of all, assuming that A is independent of t is the same thing as assuming that its eigenfoliations F and G are independent of t. Example 3.1 of [3] shows that there are smooth families of symplectic pairs, for which the diffeomorphism type of the foliations is not constant. In such examples, the diffeomorphism type of the recursion operator is not constant. In the two theorems we assume that A, equivalently F and G, are independent of t. The
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constancy of the cohomology classes of ω and η is obviously equivalent to the constancy of the cohomology classes of ± t , as long as we use de Rham cohomology in both cases. Now the conditions ω˙ t = dαt and η˙ t = dβt with αt = βt ◦ A in Theorem 1 are equivalent ± ± ± ˙± to the conditions t = dγt with γt in the ideal of the kernel foliation of t . This ± means that the cohomology class [t ] is in fact constant in the cohomology of the ideal of the kernel foliation. As explained in [3], this in turn is equivalent to the constancy of [± t ] in the basic cohomology of the kernel foliation. 3.2. Holomorphic symplectic structures. Throughout this subsection we assume that we have two symplectic forms ω and η on a manifold M of dimension 2n, such that the recursion operator defined by i X ω = i AX η satisfies A2 = −I dT M . This implies i AX ω = −i X η. We shall prove the following: Theorem 6. If the recursion operator A satisfies A2 = −I dT M , then it defines an integrable complex structure with a holomorphic symplectic form whose real and imaginary parts are ω and η. Every holomorphic symplectic form arises in this way. Proof. In this case A defines an almost complex structure on M. We extend A complex linearly to the complexified tangent bundle TC M = T M ⊗R C. The eigenvalues of A are ±i, and X =
1 1 (X − i AX ) + (X + i AX ) 2 2
is the unique decomposition of a complex tangent vector X into a sum of eigenvectors of A. As usual, the eigenspaces of A give a splitting TC M = T 1,0 ⊕ T 0,1 , where T 1,0 is the +i eigenspace, and T 0,1 is the −i eigenspace. The two are complex conjugates of each other. Lemma 7. The eigenspaces T 0,1 and T 1,0 are precisely the kernels of = ω + iη and ¯ = ω − iη. of its complex conjugate Proof. It suffices to prove the statement for the −i eigenspace T 0,1 . The other case then follows by complex conjugation. Let X = u + iv be a complex tangent vector. Then i X = i u ω − i v η + i(i u η + i v ω). The real part of the equation i X = 0 is equivalent to its imaginary part, and each is equivalent to Au = v, which is obviously equivalent to X ∈ T 0,1 . Now we want to see that the almost complex structure A is in fact integrable. By the Newlander–Nirenberg theorem it suffices to check that one, and hence both, eigendistributions of A are closed under commutation. To do this, suppose X and Y are complex vector fields in T 1,0 , so that AX = i X , AY = iY . Then, extending L X = i X ◦ d + d ◦ i X complex linearly to complex tangent vectors, and using that ω and η are closed, we find i A[X,Y ] η = i [X,Y ] ω = L X i Y ω − i Y L X ω = L X i AY η − i Y L AX η = i(L X i Y η − i Y L X η) = i i[X,Y ] η.
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The non-degeneracy of η now implies that A[X, Y ] = i[X, Y ], so that T 1,0 is closed under commutation. Thus, we have seen that two symplectic forms ω and η whose recursion operator satisfies A2 = −I d give rise to an integrable complex structure, for which T 0,1 is precisely the kernel of = ω + iη. Thus is a closed form of type (2, 0) and rank n, where n is the complex dimension of M. Conversely, if a manifold is complex and carries a holomorphic symplectic form, then the real and imaginary parts of this form are real symplectic forms whose recursion operator is just the complex structure. This completes the proof of Theorem 6. Remark 8. The above proof of the integrability of the almost complex structure defined by the recursion operator is the same as that in Lemma (6.8) of Hitchin’s paper [15], or in Lemma (4.1) of [2]. However, unlike those references, we do not assume the symplectic forms to be compatible with any metric. We shall return to a discussion of this in Subsect. 4.5. The interpretation of Theorem 1 in the case of holomorphic symplectic structures is clear. It says that a family of holomorphic symplectic structures can be made constant by an isotopy if and only if the complex structure A is, up to diffeomorphism, independent of t, the holomorphic symplectic form t has constant cohomology class, and the primitive ˙ t can be taken to be a holomorphic form of type (1, 0) with respect to the fixed of complex structure. Note that for a manifold with a holomorphic symplectic form the complex dimension n is even, that n/2 is nowhere zero, and that (n/2)+1 is identically zero. If n = 2, the latter condition becomes 2 = 0, whose real and imaginary parts lead to ω ∧ ω = η ∧ η and ω∧η = 0. Thus ω and η precisely form a conformal symplectic couple in the sense of Geiges [13]. A possibly non-conformal couple is a pair of symplectic forms inducing the same orientation and satisfying the condition ω ∧ η = 0. For such a couple the recursion operator may be more complicated, and does not necessarily define a complex structure. However, no closed manifold admitting a non-conformal couple is known, other than the holomorphic symplectic four-manifolds, which are the K 3 surfaces, the four-torus, and the primary Kodaira surfaces. Donaldson [10] has outlined a strategy that might be applied to prove that every four-manifold with a symplectic couple also has a conformal one, and is therefore holomorphic symplectic. Note that it follows from recent work of Li [21] that any four-manifold that has a symplectic structure with vanishing first Chern class (as is clearly the case for the symplectic couples, conformal or not), must have the Betti numbers and intersection form of a holomorphic symplectic four-manifold. 4. Triples of Symplectic Forms We now want to discuss the geometries defined by a triple of symplectic forms ω1 , ω2 , ω3 whose recursion operators Ai defined by i X ωi = i Ai+2 X ωi+1 ,
(2)
satisfy Ai2 = ±I d and Ai = ±I d. Here and in the sequel all indices are taken modulo 3. Note that by the definition all cyclic compositions Ai+2 ◦ Ai+1 ◦ Ai = I d. Depending on how many of the squares of the Ai are −I d and how many are +I d, there are four different cases to consider. We shall see that in the two cases when there is an
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odd number of Ai with square −I d there are natural pseudo-Riemannian metrics defined by the triple of two-forms. When exactly one Ai has square −I d, we recover the known concept of a hypersymplectic structure. When all three Ai have square −I d, we find a new geometry consisting of a hypercomplex structure for which all complex structures admit holomorphic symplectic forms. Examples for this new geometric structure, which we call a hyperholomorphic symplectic structure, are provided by hyper-Kähler structures. The latter are precisely those hyperholomorphic symplectic structures for which the natural pseudo-Riemannian metric is in fact Riemannian. We shall see that there are non-Riemannian examples as well. In the cases where the number of Ai with square −I d is even there are no natural metrics, and those structures are rather more flexible than the metric ones. 4.1. Hyperholomorphic symplectic structures. Recall that a hypercomplex structure on a manifold is a triple of integrable complex structures satisfying the quaternion relations; see for example [18,23]. Our first structure given by a triple of symplectic forms is: Definition 9. A triple of symplectic forms ωi whose pairwise recursion operators satisfy Ai2 = −I d for all i = 1, 2, 3 is called a hyperholomorphic symplectic structure. In this case Ai+2 ◦ Ai+1 ◦ Ai = I d implies that the Ai anti-commute and satisfy the quaternion relations. By Theorem 6 each Ai is an integrable complex structure, and so the Ai together form a hypercomplex structure. Furthermore, each Ai admits a holomorphic symplectic form, justifying the name hyperholomorphic symplectic structure for such a triple1 . There are now many examples of hypercomplex structures, including many on compact manifolds that are not even cohomologically symplectic; see Sects. 7.5 and 7.6 of [18], and the references given there. Therefore, hyperholomorphic symplectic structures are much more restrictive than hypercomplex ones, but, as the following example shows, every hypercomplex structure on M does give rise to a natural hyperholomorphic symplectic structure on T ∗ M. Example 10. Let M be a manifold with an integrable complex structure J . Then lifting J to T ∗ M, the total space of the cotangent bundle is also a complex manifold. It is also holomorphic symplectic, because if ω is the exact symplectic form given by the exterior derivative of the Liouville one-form, then (X, Y ) = ω(X, Y ) + iω(J X, Y ) is holomorphic symplectic for the lifted J . If M has a hypercomplex structure, then the lifts of the three complex structures to T ∗ M still satisfy the quaternion relations, and are the recursion operators for the triple of symplectic forms given by the imaginary parts of the three holomorphic symplectic forms. Now we show that hyperholomorphic symplectic structures have natural metrics associated with them. Proposition 11. Let M be a manifold with a hyperholomorphic symplectic structure. Then the bilinear form on T M defined by g(X, Y ) = ωi (X, Ai Y ) is independent of i = 1, 2, 3. It is non-degenerate and symmetric, and invariant under all Ai . 1 This is different from the hypersymplectic structures discussed in 4.2 below.
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Proof. We first prove independence of i as follows: ωi (X, Ai Y ) = ωi (X, Ai+1 Ai+2 Y ) = ωi+2 (X, Ai+2 Y ) = . . . = ωi+1 (X, Ai+1 Y ). Note that g is non-degenerate because Ai is invertible and ωi is non-degenerate. We prove invariance under the Ai using independence of i: g(Ai X, Ai Y ) = ωi+1 (Ai X, Ai+1 Ai Y ) = −ωi+1 (Ai X, Ai+2 Y ) = ωi+2 (X, Ai+2 Y ) = g(X, Y ). Finally we prove symmetry using the invariance under Ai : g(Y, X ) = ωi (Y, Ai X ) = −ωi (Ai X, Y ) = ωi (Ai X, Ai2 Y ) = g(Ai X, Ai Y ) = g(X, Y ). The proposition shows that g is a pseudo-Riemannian metric compatible with the symplectic forms ωi . As it is symmetric and non-degenerate, there must be tangent vectors X with g(X, X ) = 0. Take such a vector X and consider also A1 X , A2 X and A3 X . By invariance of g we have g(Ai X, Ai X ) = g(X, X ), and by the definition of g and the skew-symmetry of ωi , the Ai X are g-orthogonal to each other and to X . Replacing g by its negative if necessary, we find Corollary 12. Every hyperholomorphic symplectic structure in complex dimension two is hyper-Kähler. Proof. Indeed, the pseudo-Riemannian metric g is a definite Kähler metric compatible with the underlying hypercomplex structure, whose Kähler forms with respect to Ai are the ωi (up to sign). Remark 13. Interpreting a hyperholomorphic symplectic structure in complex dimension two as a conformal symplectic triple in the sense of Geiges [13], Corollary 12 is equivalent to Theorem 2.8 of [13]. Remark 14. For any hyperholomorphic symplectic structure the symplectic forms ωi and the complex structures Ai are parallel with respect to the Levi-Cività connection of the pseudo-Kähler metric g. In particular the Obata connection of the underlying hypercomplex structure, which is the unique torsion-free connection for which the Ai are parallel, must equal the Levi-Cività connection of g. In higher dimensions hyper-Kähler structures provide examples of hyperholomorphic symplectic structures for which the natural pseudo-Riemannian metric g is definite. However, there are many other examples, even on manifolds that do not support any Kähler structure, so that Corollary 12 does not generalize to higher dimensions, as shown by the following result. Theorem 15. In every even complex dimension ≥ 4 there exist hyperholomorphic symplectic structures on closed manifolds that do not support any Kähler structure.
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Proof. Our examples will be nilmanifolds. Among such manifolds, it is known that only tori admit Kähler structures; see Benson and Gordon [7]. Therefore it is enough to produce a nilmanifold of real dimension 8 that is not a torus but admits a hyperholomorphic symplectic structure. Then we can take products with T 4 to prove the result in all dimensions. Our eight-dimensional example comes from the work of Dotti and Fino [11], who found a non-Abelian two-step nilpotent Lie algebra which is both hypercomplex and symplectic. What is new here, is that we write down three invariant symplectic forms such that the recursion operators are complex structures forming a hyperholomorphic symplectic structure. Consider the real Lie algebra g spanned by 8 vectors e1 , . . . , e8 such that [e1 , e3 ] = −[e2 , e4 ] = e7 ,
[e1 , e4 ] = [e2 , e3 ] = e8 ,
and all other commutators vanish. Clearly this is nilpotent. As the structure constants are rational, the corresponding simply connected nilpotent Lie group G admits cocompact discrete subgroups , and our example will be M = G/ . This is a nilmanifold, and is not a torus because g is not Abelian. We can take a framing of G by left-invariant vector fields corresponding to the ei . Let ei be the dual left-invariant one-forms. Then e1 , . . . , e6 are closed and, by the above formulae, we have de7 = −e1 ∧ e3 + e2 ∧ e4 ,
de8 = −e1 ∧ e4 − e2 ∧ e3 .
Now we consider the following left-invariant two-forms: ω1 = e8 ∧ e1 + e7 ∧ e2 − e6 ∧ e3 + e5 ∧ e4 , ω2 = e8 ∧ e2 − e7 ∧ e1 + e6 ∧ e4 + e5 ∧ e3 , ω3 = e8 ∧ e3 + e7 ∧ e4 + e6 ∧ e1 − e5 ∧ e2 . These forms are clearly non-degenerate, and by substituting from the formulae for de7 and de8 we see that they are closed. A direct calculation shows that the recursion operators are almost complex structures. Thus, by our previous discussion, we have a left-invariant hyperholomorphic symplectic structure, which descends to G/ . In the 8-dimensional example used in the proof the pseudo-Riemannian metric g has signature (4, 4). By taking products of this and of hyper-Kähler examples, we can realize all possible signatures of the form (4k, 4l) with k + l ≥ 2 as signatures of hyperholomorphic symplectic structures. Other examples can be constructed using the nilpotent Lie algebras also used in [12].
4.2. Hypersymplectic structures. Next we consider a triple of symplectic forms such that two recursion operators have square the identity, and one has square minus the identity. After renumbering we may assume A21 = −I d and A22 = A23 = I d. Then the cyclic relations Ai+2 ◦ Ai+1 ◦ Ai = I d show that the Ai anti-commute and A2 A1 = A3 . It follows that Ai = ±I d, so the trivial cases are excluded automatically. We have the following result analogous to Proposition 11:
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Proposition 16. Let M be a manifold with three symplectic forms whose recursion operators satisfy A21 = −I d and A22 = A23 = I d. Then ω1 (X, A1 Y ) = −ω2 (X, A2 Y ) = −ω3 (X, A3 Y ), and these expressions define a bilinear form g(X, Y ) on T M. It is non-degenerate and symmetric, invariant under A1 , and satisfies g(Ai X, Ai Y ) = −g(X, Y ) for i = 2, 3. We omit the proof as it is literally the same as for Proposition 11. Now in this case if we take a vector X with g(X, X ) = 0, then g(A1 X, A1 X ) = g(X, X ), and g(A2 X, A2 X ) = g(A3 X, A3 X ) = −g(X, X ), and the Ai X are g-orthogonal to each other and to X . Thus, we have a 4-dimensional subspace on which g is non-degenerate and has signature (2, 2). Looking at the orthogonal complement of this subspace and proceeding inductively, we see that the metric g has neutral signature. We can compare this data with the following definition due to Hitchin [16]; see also [9,12]. Definition 17. A hypersymplectic structure on a manifold is a pseudo-Riemannian metric g of neutral signature, together with three endomorphisms I , S and T of the tangent bundle satisfying I 2 = −I d, S 2 = T 2 = I d, I S = −S I = T, g(I X, I Y ) = g(X, Y ), g(S X, SY ) = −g(X, Y ), g(T X, T Y ) = −g(X, Y ), and such that the following three two-forms are closed: ω I (X, Y ) = g(I X, Y ), ω S (X, Y ) = g(S X, Y ), ωT (X, Y ) = g(T X, Y ). Given a hypersymplectic structure in this sense, the recursion operators intertwining the three symplectic forms are, up to sign, precisely the endomorphisms I , S and T . Conversely, given three symplectic forms for which one of the pairwise recursion operators has square −I d and the other two have square the identity, Proposition 16 shows that we can recover a uniquely defined hypersymplectic structure. Thus we have proved: Corollary 18. A hypersymplectic structure is equivalent to a unique triple of symplectic forms for which two of the recursion operators have square the identity, and one has square minus the identity. In real dimension 4 we have the following classification of closed hypersymplectic manifolds, which one can think of as a hypersymplectic analogue of Corollary 12. Proposition 19 (cf. [19]). A closed oriented four-manifold admits a hypersymplectic structure if and only it is T 4 or a nilmanifold for N il 3 × R. Proof. A closed oriented four-manifold with a hypersymplectic structure is holomorphic symplectic, and so by a result of Kodaira is T 4 , a primary Kodaira surface, or a K 3 surface; see [6]. Clearly T 4 inherits the standard hypersymplectic structure of R4 . By a result of Wall [25], primary Kodaira surfaces are precisely the nilmanifolds of N il 3 ×R. Recall from [25] or [4] that N il 3 ×R has a framing by left-invariant one-forms α1 , . . . , α4 with dα3 = α1 ∧ α2 , and αi closed for i = 3. The left-invariant two-forms
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ω1 = α3 ∧ α1 + α2 ∧ α4 , ω2 = α3 ∧ α2 − α1 ∧ α4 , ω3 = α3 ∧ α2 + α1 ∧ α4 , define an invariant hypersymplectic structure that descends to all compact quotients. A hypersymplectic structure also defines a symplectic pair, and therefore a fourmanifold with such a structure is symplectic for both choices of orientation. But a K 3 surface endowed with the non-complex orientation can not be symplectic, because it has vanishing Seiberg–Witten invariants. This follows from the existence of smoothly embedded spheres whose selfintersection number is positive for the non-complex orientation; cf. [20]. High-dimensional examples of hypersymplectic structures on closed manifolds have recently appeared in [12,1]. 4.3. Holomorphic symplectic pairs. Next we consider the following: Definition 20. A triple of symplectic forms ωi is called a holomorphic symplectic pair if two of the pairwise recursion operators have square −I dT M and one has square I dT M , but is not itself ±I dT M . After renumbering we may assume A21 = A22 = −I d and A23 = I d. Then the cyclic relations Ai+2 ◦ Ai+1 ◦ Ai = I d show that the Ai commute and A2 A1 = A3 . Now A3 has square the identity, but is not itself plus or minus the identity, and so defines a symplectic pair. The other two recursion operators, A1 and A2 define integrable complex structures. As they commute with A3 , they preserve its eigenfoliations and restrict as complex structures to the leaves. On the +1 eigenfoliation of A3 the two complex structures are complex conjugates of each other, and on the −1 eigenfoliation they agree. The two complex structures also have holomorphic symplectic forms which restrict to the leaves of the eigenfoliations of A3 . Thus a holomorphic symplectic pair is a symplectic pair whose leaves are not just symplectic, but are holomorphic symplectic submanifolds. It follows in particular that the real dimensions of the leaves are multiples of 4, and the smallest dimension in which this structure can occur is 8. Here are some examples. Example 21. Consider the eight-dimensional nilpotent Lie group G from the proof of Theorem 15. The forms ω1 = e8 ∧ e1 + e7 ∧ e2 − e6 ∧ e3 + e5 ∧ e4 , ω2 = e8 ∧ e2 − e7 ∧ e1 + e6 ∧ e4 + e5 ∧ e3 , ω3 = e8 ∧ e2 − e7 ∧ e1 − e6 ∧ e4 − e5 ∧ e3 , are a left-invariant holomorphic symplectic pair, that descends to all compact quotients. Example 22. Let M1 and M2 be holomorphic symplectic manifolds, and let (ωi , ηi ) be the two symplectic forms defining the structure on Mi . On the product M1 × M2 the three symplectic forms ω1 + ω2 , η1 + η2 and −ω1 + ω2 are a holomorphic symplectic pair. In this case the foliations are given by the factors of the product. For the factors in this construction we can use any holomorphic symplectic manifold. This could be hyper-Kähler, or a nilmanifold, or one of the simply connected non-Kähler examples constructed by Guan [14], compare also [8].
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4.4. Symplectic triples. We use the name symplectic triple2 for a triple of symplectic forms such that Ai2 = I d and Ai = ±I d for all three i. By the discussion in Subsect. 3.1 above this is equivalent to requiring that for i = j, the forms ωi and ω j define a symplectic pair. In other words, ωi ± ω j are forms of constant (non-maximal and non-zero) rank, and each one restricts as a symplectic form to the leaves of the kernel foliation of the other one. In dimension four, the two symplectic forms making up a symplectic pair induce opposite orientations. This means in particular that there can not be any symplectic triples. Starting in dimension 6, however, symplectic triples exist in abundance; see [4], particularly Remark 7. Using the examples of symplectic pairs on four-manifolds constructed in [4], one immediately obtains many examples of symplectic triples in higher dimensions by taking products with other symplectic manifolds. In this way many different topological types can be realized. Example 23. The simplest example is given by considering three closed two-forms ηi of constant rank = 2 on a 6-manifold, with the property that η1 ∧ η2 ∧ η3 is nowhere zero. Then we can take ω1 = η1 + η2 + η3 , ω2 = η1 + η2 − η3 and ω3 = η1 − η2 − η3 . This works for example by taking a product of three surfaces, or, more interestingly, by taking a quotient of the polydisk H2 ×H2 ×H2 by an irreducible lattice as in Subsect. 5.2 of [4].
4.5. Final comments and remarks. We have seen in Corollary 18 that the metric definition of hypersymplectic structures given in [16] is in fact equivalent to the symplectic definition we have given in terms of symplectic forms and their recursion operators. In particular, the fact that the signature of the natural pseudo-Kähler metric is always neutral follows purely from the algebraic relations between the recursion operators. The analogous result is false for hyper-Kähler structures. If one retains from a hyper-Kähler structure only the triple of symplectic forms together with the property that the intertwining recursion operators be almost complex structures, then one finds hyperholomorphic symplectic structures. As we have seen in Subsect. 4.1, the ensuing algebraic identities define a pseudo-Riemannian metric of a priori unknown signature. Therefore, when viewing hyper-Kähler structures as symplectic objects, as is done for example in [16], one either has to allow pseudo-hyper-Kähler structures, or one has to build the definiteness of the metric into the definition. It is not clear to us how one would do this in terms of symplectic geometry alone. In this direction, the discussion on p. 172 of [16] is really based on Lemma (6.8) of [15], where definiteness of the compatible metric is part of the definition. Finally, for holomorphic symplectic pairs and for symplectic triples there are no natural metrics, definite or otherwise. We leave it to the interested reader to work out why Proposition 11 has no analogue in these cases. Acknowledgement. This work was begun while the second author was a Visiting Professor at the Università degli Studi di Cagliari, supported by I. N. d. A. M. and was completed during a visit of the first author to Ludwig-Maximilians-Universität München supported in part by G. N. S. A. G. A. and P. R. I. N. We are grateful to N. Hitchin for some useful comments and for sending us a copy of [16].
2 This terminology clashes with that of Geiges [13].
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References 1. Andrada, A., Dotti, I.G.: Double products and hypersymplectic structures on R4n. Commun. Math. Phys. 262, 1–16 (2006) 2. Atiyah, M.F., Hitchin, N.J.: The Geometry and Dynamics of Magnetic Monopoles. Princeton, NJ: Princeton University Press, 1988 3. Bande, G., Ghiggini, P., Kotschick, D.: Stability theorems for symplectic and contact pairs. Int. Math. Res. Not. 68, 3673–3688 (2004) 4. Bande, G., Kotschick, D.: The geometry of symplectic pairs. Trans. Amer. Math. Soc. 358, 1643–1655 (2006) 5. Barrett, J., Gibbons, G.W., Perry, M.J., Pope, C.N., Ruback, P.: Kleinian geometry and the N = 2 superstring. Int. J. Mod. Phys. A4, 1457–1494 (1994) 6. Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Berlin: Springer-Verlag, 1984 7. Benson, C., Gordon, C.: Kähler and symplectic structures on nilmanifolds. Topology 27, 513–518 (1988) 8. Bogomolov, F.A.: On Guan’s examples of simply connected non-Kähler compact complex manifolds. Amer. J. Math. 118, 1037–1046 (1996) 9. Dancer, A.S., Jørgensen, H.R., Swann, A.F.: Metric geometries over the split quaternions. Rend. Sem. Mat. Univ. Pol. Torino 63, 119–139 (2005) 10. Donaldson, S.K.: Two-forms on four-manifolds and elliptic equations. In: Chern, S.S. (ed.) Inspired. Nankai Tracts Math., vol. 11. pp. 153–172, World Scientific, Hackensack (2006) 11. Dotti, I.G., Fino, A.: Abelian hypercomplex 8-dimensional nilmanifolds. Ann. of Global Anal. and Geom. 18, 47–59 (2000) 12. Fino, A., Pedersen, H., Poon, Y.-S., Sørensen, M.W.: Neutral Calabi–Yau structures on Kodaira manifolds. Commun. Math. Phys. 248, 255–268 (2004) 13. Geiges, H.: Symplectic couples on 4-manifolds. Duke Math. J. 85, 701–711 (1996) 14. Guan, D.: Examples of compact holomorphic symplectic manifolds which are not Kählerian, II. Invent. Math. 121, 135–145 (1995) 15. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc. 55, 59–126 (1987) 16. Hitchin, N.J.: Hypersymplectic quotients. In: La “Mécanique analytique” de Lagrange et son héritage, Supplemento al numero 124 degli Atti della Accademia delle Scienze di Torino, Classe di Scienze Fisiche, Matematiche e Naturali, 1990, pp. 169–180 17. Hull, C.M.: Actions for (2, 1) sigma models and strings. Nucl. Phys. B 509, 252–272 (1998) 18. Joyce, D.D.: Compact manifolds with special holonomy. Oxford Math. Monographs, Oxford: Oxford University Press, 2000 19. Kamada, H.: Self-dual Kähler metrics of neutral signature on complex surfaces. Ph. D. thesis, Tohoku University, Sendai 2002; published as Tohoku Math. Publ. 24, Sendai: Tohoku Univ., 2002 20. Kotschick, D.: Orientations and geometrisations of compact complex surfaces. Bull. London Math. Soc. 29, 145–149 (1997) 21. Li, T.-J.: Quaternionic bundles and Betti numbers of symplectic four-manifolds with Kodaira dimension zero. Int. Math. Res. Not. 2006, Article ID 37385, 1–28, (2006) 22. Moser, J.: On the volume elements on a manifold. Trans. Amer. Math. Soc. 120, 286–294 (1965) 23. Salamon, S.: Riemannian geometry and holonomy groups. Pitman Res. Notes, Harlow: Longman, 1989 24. Turiel, F.J.: Classification locale simultanée de deux formes symplectiques compatibles. Manu. Math. 82, 349–362 (1994) 25. Wall, C.T.C.: Geometric structures on compact complex analytic surfaces. Topology 25, 119–153 (1986) Communicated by G.W. Gibbons
Commun. Math. Phys. 280, 751–805 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0474-9
Communications in
Mathematical Physics
Effective Dynamics for Particles Coupled to a Quantized Scalar Field L. Tenuta, S. Teufel Mathematisches Institut, Eberhard-Karls-Universität, Auf der Morgenstelle 10, 72076 Tübingen, Germany. E-mail: [email protected]; [email protected] Received: 20 March 2007 / Accepted: 26 September 2007 Published online: 9 April 2008 – © Springer-Verlag 2008
Abstract: We consider a system of N non-relativistic spinless quantum particles (“electrons”) interacting with a quantized scalar Bose field (whose excitations we call “photons”). We examine the case when the velocity v of the electrons is small with respect to the one of the photons, denoted by c (v/c = ε 1). We show that dressed particle states exist (particles surrounded by “virtual photons”), which, up to terms of order (v/c)3 , follow Hamiltonian dynamics. The effective N -particle Hamiltonian contains the kinetic energies of the particles and Coulomb-like pair potentials at order (v/c)0 and the velocity dependent Darwin interaction and a mass renormalization at order (v/c)2 . Beyond that order the effective dynamics are expected to be dissipative. The main mathematical tool we use is adiabatic perturbation theory. However, in the present case there is no eigenvalue which is separated by a gap from the rest of the spectrum, but its role is taken by the bottom of the absolutely continuous spectrum, which is not an eigenvalue. Nevertheless we construct approximate dressed electron subspaces, which are adiabatically invariant for the dynamics up to order (v/c) ln[(v/c)−1 ]. We also give an explicit expression for the non-adiabatic transitions corresponding to emission of free photons. For the radiated energy we obtain the quantum analogue of the Larmor formula of classical electrodynamics. 1. Introduction In a system of classical charges the interactions are mediated through the electromagnetic field. In the case when the velocities of the particles are small with respect to the speed of light, it is possible, loosely speaking, to expand their dynamics in powers of v/c. The qualitative picture which emerges has three main features. Up to terms of order (v/c)3 , the dynamics of the particles are still of Hamiltonian form. Moreover, – at leading order, (v/c)0 , the retardation effects can be neglected and the particles interact through an instantaneous pair potential;
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– at order (v/c)2 , the particles acquire an effective mass, due to the contribution of the electromagnetic mass and, to take into account the retardation effects, one has to add to the potential a velocity-dependent term (the so-called Darwin term). Including the terms of order (v/c)3 , the dynamics are not Hamiltonian anymore, instead – there is dissipation of energy through radiation. In the dipole approximation, the rate of emitted energy is proportional to the acceleration of a particle squared. A formal derivation of this picture, which does not consider the problem of mass renormalization, can be found in [LaLi]. A mathematical analysis in the framework of the Abraham model, i. e., for charges which have a rigid charge distribution, is given in [KuSp1 ,KuSp2 ]. The above description is expected to remain true also for nonrelativistic quantum electrodynamics, where, neglecting the possibility of pair creation, one considers a system of N nonrelativistic particles interacting with the quantized Maxwell field. In physical terms, the particles carry now a cloud of virtual photons, which makes them heavier, and interact exchanging them or dissipate energy through photons travelling freely to infinity. However, in quantum mechanics one describes the interaction of charged particles in good approximation by introducing instantaneous pair potentials and without treating the field as dynamical variable. If the particles move sufficiently slowly this is known to be a very good approximation. One goal of our paper is the mathematical derivation of quantum mechanics from a model of particles that are coupled to a quantized field, but do not interact directly. Instead of nonrelativistic QED we consider the massless Nelson model, which describes N spinless particles (which will be called “electrons”) coupled to a scalar Bose field of zero mass (whose excitations will be called “photons”). In spite of the simplifications introduced, this model is expected to retain the main physical features of the original one. Therefore, since its introduction by Nelson [Ne], who analyzed its ultraviolet behavior, it has been extensively studied to get information about the spectral and scattering features of QED, mostly concerning its infrared behavior (not pretending to be exhaustive, some papers related to this aspect are [Ar,Fr,LMS,Pi] and references therein). The recent monograph by Spohn [Sp] contains detailed descriptions of the classical and the quantum mechanical models and results. In this paper we define and analyze the dynamics of dressed electron states in the Nelson model in the limit of small particle velocities. Loosely speaking a dressed electron is a bare electron dragging with it a cloud of “virtual” photons. We show that the dynamics of dressed electrons have the features discussed above: an instantaneous pair interaction at leading order (v/c)0 and a renormalized mass together with the velocity-dependent Darwin interaction at order (v/c)2 . We also provide an analogue to the classical Larmor formula for the radiated energy, i.e. for the energy carried away by “real” photons travelling freely to infinity (the heuristic concept of “real” and “virtual” photons will be made precise below). It is important to note that we consider the massless Nelson model with an ultraviolet cutoff but no infrared cutoff. Indeed, the leading order effective dynamics were already analyzed in [Te2 ] assuming an infrared regularization. One expects, and we will show it in this paper, that the dynamics of the dressed electrons even at higher orders are essentially independent of an infrared regularization, but the radiation is instead very sensitive to it, which makes the mathematical analysis much more delicate. To explain in more detail the kind of scaling we are interested in it is convenient to look first at the classical case. The classical equations of motion for N particles with
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positions q j , mass m j and a rigid “charge” distribution j coupled to a scalar field φ(x, t) 1 with propagation speed c are 1 2 ∂t φ(x, t) = ∆ x φ(x, t) − j (x − q j (t)), 2 c j=1 m j q¨ j (t) = − d x (∇ x φ)(x, t) j (x − q j (t)), N
(1) 1 ≤ j ≤ N.
(2)
We assume for simplicity that j (x) = e j ϕ(x), where the form factor ϕ gives rise to a sharp ultraviolet cutoff, (2π )−3/2 |k| < Λ, ϕ(k) ˆ = 0 otherwise.
(3)
Taking formally the limit c → ∞ in (1), we get the Poisson equation for the field, and so, eliminating the field from (2), we obtain equations of motion describing N particles interacting through smeared Coulomb potentials. Mass renormalization does not appear at the leading order. Instead of taking c → ∞, we regard as more natural to explore the regime of particle properties which gives rise to effective equations. Indeed, if we look at heavy particles for long times, i. e., if we substitute t = εt and m j = ε2 m j in (1) and (2), we find that, up to rescaling, the limit ε → 0 is equivalent to the limit c → ∞. After quantization however, the two limiting procedures are not equivalent anymore. The case c → ∞ was analyzed by Davies [Da] and by Hiroshima [Hi], who at the same time removed the ultraviolet cutoff, applying methods from the theory of the weak coupling limit. In this paper we analyze the limit ε → 0. We define now briefly the massless Nelson model (whose presentation will be completed in Sect. 2), we state our main results and the principal ideas of the proof and then compare them to the above mentioned weak coupling limit. The model is obtained through canonical quantization of the classical system described by (1) and (2). The state space for N spinless particles is L 2 (R3N ) and the Hamiltonian, assuming for simplicity that all the particles have equal mass, is given by (we switch to natural units, fixing = 1 and c = 1) Hp := −
N 1 ∆x j . 2m j=1
As explained above, we consider the case of heavy particles, i. e., m = ε−2 ,
0 < ε 1,
(4)
therefore the Hamiltonian becomes Hpε = −
ε2 1 ∆x =: pˆ 2 , 2 2
x ∈ R3N ,
pˆ := −iε∇x .
(5)
1 We use bold italic font, x, to denote vectors in R3 . The only exception to this, since there is no possibility of misunderstanding, is the three-dimensional momentum of the photons, denoted by k. The lightface font, x, will be used to denote vectors in R3N .
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The particles are coupled to a scalar field, whose states are elements of the bosonic Fock space over L 2 (R3 ), defined by M 2 3 F := ⊕∞ M=0 ⊗(s) L (R ),
(6)
M denotes the M-times symmetric tensor product and ⊗0 L 2 (R3 ) := C. We where ⊗(s) (s) denote by Q M the projector on the M-particles subspace of F and by ΩF the vector (1, 0, . . .) ∈ F , called the Fock vacuum. The Hamiltonian for the free bosonic field is
Hf := d (|k|),
(7)
where k is the momentum of the photons (the reader who is not familiar with the notation can find more details in Sect. 2). The particle j is linearly coupled to the field through the interaction Hamiltonian HI, j := d y φ( y) j ( y − x j ), (8) R3
where φ is the field operator in position representation. The state space of the combined system particles + field is H := L 2 (R3N ) ⊗ F L 2 (R3N , F )
(9)
and its time evolution is generated by the Hamiltonian H ε := H0ε +
N
HI, j ,
H0ε := Hpε ⊗ 1 + 1 ⊗ d (|k|),
(10)
j=1
with domain H0 := H 2 (R3N , F ) ∩ L 2 (R3N , D(Hf )),
H 2 (R3N , F ) := D( pˆ 2 ⊗ 1),
(11)
which is a Hilbert space with the graph norm associated to H0ε . Note that there are no direct forces acting between the particles, all the interactions are mediated through the field. Our goal is to understand the dynamics of the particles for times of order ε−1 . It is necessary to look at long times in order to see nontrivial dynamics of the particles, because, since their mass is of order O(ε−2 ) and we consider states of finite kinetic energy, their velocity is of order O(ε). However, since the coupling between the electrons and the field is not small, standard perturbation theory is of no use initially. Indeed, since the charge of the particles is of order one, the local deformation of the field, i. e. the “cloud of virtual photons”, is of order one. However, the influence of real photons with finite energy and momentum on the dynamics of the heavy electrons, whose mass is of order ε−2 , is small. Hence one expects that the coupling between properly defined dressed electron states and real photons is small. To make this precise, we construct a dressing transformation Uε : H → H , which allows us to define the dressed particle states as follows. In the new representation the vacuum sector L 2 (R3N ) ⊗ ΩF of H corresponds to states of dressed electrons without real photons, while in the original Hilbert space a state of dressed electrons with M real photons would be a linear combination of states of the form Uε−1 (ψ ⊗ a( f 1 )∗ · · · a( f M )∗ ΩF ) with ψ ∈ L 2 (R3N ), f 1 , . . . , f M ∈ L 2 (R3 ).
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Recall that Q M denotes the projector on the M-particles subspace of Fock space, then the projector on the subspace corresponding to dressed electrons with M real photons is ε PM := Uε∗ (1 ⊗ Q M )Uε . ε H are In a nutshell, the main results we prove are the following: the subspaces PM ε approximately invariant under the dynamics generated by H for times of order ε−1 . For states inside such a subspace the dynamics of the particles can be described by an effective Hamiltonian for the particles alone on the above time scale and with errors of order O ε2 log(ε−1 ) . Finally we can compute the leading order part of the state which leaves the subspace P0ε H under the time evolution, which corresponds to emission of real photons. The formula for the energy of the real photons traveling to infinity, i.e. the radiated energy, yields a quantum mechanical analogue of the classical Larmor formula for the radiation of accelerated charges. Before we can state our results in detail, we need to explain the adiabatic structure of the problem in some detail. The Hamiltonian H ε is the perturbation of a fibered Hamiltonian, because, since HI, j depends only on the configuration x j of the j th particle, the operator
Hfib (x) := d (|k|) +
N
HI, j (x j )
(12)
j=1
acts on F for every fixed x ∈ R3N . This means that ⊕ ε2 Hfib (x). H ε = − ∆x ⊗ 1 + 2 R3N Note the structural similarity with the Born-Oppenheimer approximation. There the Hamiltonian describes the interaction between nuclei and electrons in a molecule and the former have a mass of order O(ε−2 ) with respect to the latter (the typical spectrum of Hfib (x) for a diatomic molecule is shown in Fig. 1). In the present case, the particles take the role of the nuclei, and the bosons the one of the electrons. In contrast to the molecular case however, in the Nelson model Hfib (x) has typically (see Lemma 2 and Corollaries 1 and 2) purely absolutely continuous spectrum, which does not display a structure with pointwise separated bands (absence of both eigenvalues and spectral gap) (see Fig. 2). The bottom of the spectrum, E(x), can be explicitly calculated, E(x) =
N 1 Vi j (x i − x j ) + e0 , 2
(13)
i, j=1 i = j
where
Vi j (z) := −
R 3 ×R 3
dvdw
i (v − z) j (w) , 4π |v − w|
(14)
and N j (v) j (w) 1 e0 := − . dvdw 3 3 2 4π |v − w| j=1 R ×R
(15)
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Fig. 1. Schematic spectrum of the fibered Hamiltonian in the case of a diatomic molecule for energies below the dissociation threshold (r = |x 1 − x 2 |). The different eigenvalues are pointwise separated by a gap
Fig. 2. Spectrum of Hfib (x) for N = 2 (r = |x 1 − x 2 |). The spectrum is absolutely continuous, there is no eigenvalue at the bottom. The oscillations in r caused by the sharp ultraviolet cutoff are irrelevant, and therefore we do not show them
The effective pair-potential Vi j (z) coincides, up to the sign, with the electrostatic interaction energy of the charge distributions i and j at distance z, while e0 is the sum of all the self-energies. E(x) becomes an eigenvalue of Hfib (x) if the total charge of the N particles system is equal to zero, as it happens for example in the presence of an infrared cutoff. In this case, it is possible to build for every x a unitary operator, V (x), called the dressing operator, which diagonalizes Hfib (x) in the sense that Hfib (x) = V (x)Hf V (x)∗ + E(x).
(16) Hε
in two Exploiting this remark, we approximate the time evolution generated by steps. First we define an infrared regular Hamiltonian, H ε,σ , where the form factor ϕˆ (Eq. (3)) is replaced by (2π )−3/2 σ < |k| < Λ, (17) ϕˆσ (k) := 0 otherwise.
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
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1/2
Proposition (see Proposition 1). Let L(H0 , H ) be the space of bounded linear 1/2 operators from H0 to H equipped with the operator norm. It holds then that −it H ε /ε ε,σ e − e−it H /ε
1/2 L(H0 ,H
1/2
where H0
)
≤ C|t|
σ 1/2 , ε
(18)
:= D (H0ε )1/2 with the corresponding graph norm.
Choosing σ as a power of ε we can then replace the original dynamics with infrared regular ones. The latter contain however two parameters, ε and σ , therefore it is necessary to control the behavior of all the quantities that appear with respect to both. The advantage is that for H ε,σ we can build an approximate dressing operator Uε,σ , acting on the whole Hilbert space H , which is defined for every positive σ , but whose limit when σ → 0+ does not exist if the system has a total charge different from zero. Uε,σ is unitary and can be expanded in a series of powers of ε, with coefficients which are at most logarithmically divergent in σ . Moreover, the zero order coefficient is given by Vσ , the dressing operator associated to the infrared regular fibered Hamiltonian, which has therefore the property that H ε,σ =
1 2 pˆ ⊗ 1 + Vσ (x)Hf Vσ (x)∗ + E σ (x). 2
Using Uε,σ , we define the transformed Hamiltonian ε,σ ∗ Hdres := Uε,σ H ε,σ Uε,σ
(19)
which can be expanded in a series of powers of ε in L(H0 , H ), with coefficients which are also at most logarithmically divergent in σ . Thus the gain in switching to the representation defined by Uε,σ is twofold. First we can easily separate dressed electrons from real photons and second we can expand the Hamiltonian in powers of the small parameter ε and thereby separate different physical effects according to their order of magnitude. The first result we find is that, even though the fibered Hamiltonian has no eigenvalues and no spectral gap, there are approximate M-photons dressed subspaces which are almost invariant for the dynamics. Theorem (Adiabatic invariance of M-photons dressed particle subspaces). (see Theorem 3 and Remark 6). Given any χ ∈ C0∞ (R) and any function σ (ε) such that σ (ε)1/2 ε−3 → 0,
ε log(σ (ε)−1 ) → 0,
ε → 0+ ,
(20)
then √ −it H ε /ε ε ε [e , PM ]χ (H ) L(H ) ≤ C M + 1|t|ε log(σ (ε)−1 ),
(21)
where ε ∗ PM := Uε,σ (ε) (1 ⊗ Q M )Uε,σ (ε) .
(22)
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The physical mechanism which leads to the almost invariance of the subspaces is adiabatic decoupling, i. e., the separation of scales for the motion of the different parts of the system, which lets the fast degrees of freedom, in our case the photons, instantaneously adjust to the motion of the slow degrees of freedom, the electrons. It is a well-known fact however, that the decoupling becomes poorer and poorer when the kinetic energies and thus the velocities of the heavy particles grow. The quadratic dispersion relation for the electrons allows them to become arbitrarily fast, therefore the decoupling holds uniformly just on states of bounded kinetic energy. This is the reason why we introduce a uniform bound on the total energy of the system through the function χ , which obviously implies a bound on the kinetic energy of the electrons. For the following we fix the function σ (ε) in some way compatible with (20), say σ (ε) = ε8 . Then we can approximate the dynamics of the particles for states in the range ε in the following sense. of PM Theorem (Effective dynamics of the particles). (see Theorem 5). Let ω be a (mixed) dressed electrons state with finite energy and a fixed number of real photons, i. e. ε χ (H ε )H ) and ω ∈ I1 (PM (23) ωd := tr F Uε ω Uε∗ , the partial trace over the real photons, i. e. the reduced dressed electron density matrix. Let the time evolution of ω be the full time evolution ω(t) := e−it H
ε /ε
ω eit H
ε /ε
,
(24)
and define the effective time evolution of ωd by ε
ε
ωd (t) := e−it Heff /ε ωd eit Heff /ε , ε given below. with effective dressed electrons Hamiltonian Heff ∞ 3N 3N Let a ∈ Cb (R × R ) be a “macroscopic” observable on the classical phase space of the electrons and OpεW (a) its Weyl quantization acting on L 2 (R3N ). Then
tr H OpεW (a) ⊗ 1F ω(t) = tr L 2 (R3N ) OpεW (a) ωd (t) + O(ε3/2 |t|)(1 − δ M0 )
+ O ε2 log(ε−1 )(|t| + |t|2 ) ,
(25)
where δ M0 = 1, when M = 0, 0 otherwise. The effective dressed electrons Hamiltonian is infrared regular and given by ε := Heff
N 1 2 pˆ + E(x) 2m εj j j=1
−
N ˆ l (k)∗ ˆ j (k) ik·( x j − x l ) ε2 e dk (κ· pˆ l )(κ· pˆ j ) 3 4 |k|2 l, j=1 R (l = j)
+ (κ· pˆ l )(κ· pˆ j )eik·( x j − x l ) ,
(26)
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
with m εj := 1/(1 +
ε2 2 e˜ j )
759
and
e˜ j :=
j (x) j ( y) 1 . dx d y 4π R3 ×R3 |x − y|
(27)
ε is equal to the Weyl quantization of the Darwin Remark 1. The Hamiltonian Heff Hamiltonian, which, as we mentioned above, appears in classical electrodynamics when one expands the dynamics for the particles including a term of order ε2 ∼ = (v/c)2 . At leading order the dressed electrons interact through instantaneous pair potentials E(x) given in (13). At order ε2 the mass of the electrons is modified (renormalized) and a velocity dependent interaction, the so called Darwin interaction, appears.
Remark 2. The statement (25) remains true if one replaces in (23) the dressing operator Uε by its leading order approximation Vε . ε H are only approximately invariant and transiThe dressed electrons subspaces PM tions between them correspond to emission or absorption of real photons. The following theorem describes at leading order the radiated part of the wave function for an initial state in the dressed vacuum.
Theorem (Radiation). (see Corollary 6 and the subsequent remark). For a system starting in the approximate dressed vacuum (M = 0) we have, t
ε
Ψrad (t) := (1 − P0ε )e−i ε H P0ε χ (H ε )Ψ
(28) N t t t s e j ϕˆε (k) ε 2 κ · ds ei ε |k| OpεW ( x¨ cj (s; x, p)) ψd = − √ P1ε Vε e−i ε ( pˆ /2+E(x)) e−i ε |k| 3/2 |k| 2 0 j=1
+ R(t, ε), where R(t, ε)H ≤ Cε2 log(ε−1 )(|t| + |t|2 )(ψd H + |x|ψd H + | p|ψ ˆ d H ), x cj is the solution to the classical equations of motion x¨ cj (s; x, p) = −∇ x j E(x c (s; x, p)), x cj (0; x, p) = x j ,
x˙ cj (0; x, p) = p j ,
j = 1, . . . , N ,
(29)
κ := k/|k| and ψd := Vε ΩF , χ (H ε )Ψ F
(30)
is the projection of the initial state on the dressed vacuum in the field component. Remark 3. Generically (for the precise meaning of this we refer to Corollary 6) the norm of the radiated piece is bounded below by O ε log(ε−1 ) , which means that the subspace P0ε is near optimal, in the sense that the transitions are at least of order O ε log(ε−1 ) .
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Remark 4. The radiated wave function Ψrad (t) as given in (28) has no limit as ε → 0 because limε→0 ϕˆε /|k|3/2 = ϕˆ 0 /|k|3/2 ∈ / L 2 (R3 ). However, the radiated energy, i. e. the energy of the real photons in (28) has a limit. Indeed, if we compute the time derivative of the radiated energy in (28) we obtain the quantum analogue of the Larmor formula from classical electrodynamics. Let E rad (t) := Ψrad (t), Vε Hf Vε∗ Ψrad (t),
(31)
then Prad (t) :=
N d ε3 E rad (t) ∼ ei e j ψd , OpεW x¨ ic (t; x, p) · x¨ cj (t; x, p) ψd = dt 12π i, j=1
=
ε3 12π
¨ x, p)|2 ψd , ψd , OpεW | d(t;
(32)
¨ x, p) is the second time derivative of the dipole moment where d(t; d(t; x, p) :=
N
ei x ic (t; x, p),
i=1
and the symbol ∼ = means that we keep just the leading order in the expansion in powers of ε. A formal derivation of Eq. (32) is given in Remark 8. The technique used for proving these theorems is based on space-adiabatic perturbation theory, a general scheme designed to expand the dynamics generated by a pseudodifferential operator with a semiclassical symbol [NeSo,PST1 ,Te3 ]. However, this method exploits the assumption that the fibered Hamiltonian has a spectral gap, of the kind showed in Fig. 1, therefore is not directly applicable and needs some modifications. In particular, the infrared cutoff σ plays in our context the role of an effective gap. In the more usual framework of time-dependent Hamiltonians, adiabatic theorems without gap condition were first proven by Bornemann [Bo] and Avron and Elgart [AvEl1 ] (their proof was simplified and the result somewhat strengthened in [Te1 ]). Using similar techniques, a space-adiabatic theorem without gap for the Nelson model was proven in [Te2 ]. In the context of quantum statistical mechanics, an adiabatic theorem without gap for the Liouvillian (the generator of the dynamics of the states in a suitable representation of the von Neumann algebra associated to the system) is discussed in [AbFr1 ], while in [AbFr2 ] a general time-adiabatic theorem for resonances is proved. All these results are of the first order type, i. e., they describe the leading order adiabatic evolution and give upper bounds for the transitions. In our case instead, we build adiabatic dynamics including terms of the second order, and give an explicit expression for the non-adiabatic transitions, which allows us to give also a lower bound for them. In the time-dependent case, lower bounds for the transitions were calculated by Avron and Elgart [AvEl2 ] for the Friedrichs model, which describes a “small system”, whose Hilbert space is one-dimensional, interacting with a “reservoir”, whose Hilbert space is L 2 (R+ , dµ(k)), with a suitable spectral density µ. Finally let us compare our results with those obtained by Davies [Da]. He considers the limit c → ∞ for the Hamiltonian √ H c := Hp ⊗ 1 + 1 ⊗ d (c|k|) + cH I
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which is equal to H ε=1 not setting c = 1 as we did before. Davies proves then that for all t ∈ R, lim e−iH t (ψ ⊗ ΩF ) = (e−i(−∆/2+E(x))t ψ) ⊗ ΩF . c
c→∞
This shows that, while at the classical level the limit ε → 0 and c → ∞ are equivalent, they differ at the quantum one. Indeed, the limit ε → 0 is singular, because no limiting dynamics for ε = 0 exist. Moreover, the effective dynamics we get refer to dressed states, which contain a non zero number of photons, while the c → ∞ limit is taken on states which contain no photons. We summarize the structure of our paper. After some basic facts about the model recalled in Sect. 2, we explain how to construct the approximate dressing operator U in Sect. 3. In Sect. 4, we analyze the expansion of the transformed Hamiltonian and then apply these results to the study of the effective dynamics and the radiation in Sect. 5. 2. Description of the Model and Preliminary Facts In this section we complete the presentation of the Nelson model, discussing also the spectrum of the fibered Hamiltonian, and collect some basic facts we will use in the following.
2.1. Fock space and field operator. (The proofs of the statements we claim can be found in ([ReSi2 ], Sect. X.7)). We denote by Ffin the subspace of the Fock space, defined in (6), for which Ψ (M) = 0 for all but finitely many M. Given f ∈ L 2 (R3 ), one defines on Ffin the annihilation operator by √ (M) dk f¯(k)Ψ (M+1) (k, k1 , . . . , k M ). (33) (a( f )Ψ ) (k1 , . . . , k M ) := M + 1 R3
The adjoint of a( f ) is called the creation operator, and its domain contains Ffin . On this subspace they satisfy the canonical commutation relations [a( f ), a(g)∗ ] = f, g L 2 (R3 ) ,
[a( f ), a(g)] = 0, [a( f )∗ , a(g)∗ ] = 0.
(34)
Since the commutator between a( f ) and a( f )∗ is bounded, it follows that a( f ) can be extended to a closed operator on the same domain of a( f )∗ . On this domain one defines the Segal field operator 1 Φ( f ) := √ (a( f ) + a( f )∗ ) 2
(35)
which is essentially self-adjoint on Ffin . Moreover, Ffin is a set of analytic vectors for Φ( f ). From the canonical commutation relations it follows that [Φ( f ), Φ(g)] = i f, g L 2 (R3 ) .
(36)
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Given a self-adjoint multiplication operator by the function ω on the domain D(ω) ⊂ L 2 (R3 ), we define Fω,fin := L{ΩF , a( f 1 )∗ · · · a( f M )∗ ΩF : M ∈ N, f j ∈ D(ω), j = 1, . . . , M},
(37)
where L means “finite linear combinations of ”. On Fω,fin we define the second quantization of ω, d (ω), by (d (ω)Ψ )
(M)
(k1 , . . . , k M ) :=
M
ω(k j )Ψ (M) (k1 , . . . , k M ),
(38)
j=1
which is essentially self-adjoint. In particular, the free field Hamiltonian Hf , Eq. (7), acts as (Hf Ψ )(M) (k1 , . . . , k M ) =
M
|k j |Ψ (M) (k1 , . . . , k M )
j=1
and is self-adjoint on its maximal domain. From the previous definitions, given f ∈ D(ω), one gets the commutation properties [d (ω), a( f )∗ ] = a(ω f )∗ , [d (ω), a( f )] = −a(ω f ), [d (ω), iΦ( f )] = Φ(iω f ).
(39)
2.2. The Nelson model. Using the Segal field operator (which involves taking a Fourier transform) the interaction Hamiltonian, Eqs. (8) and (10), can be written as N
HI, j (x j ) = Φ(|k|v(x, k)),
(40)
j=1
where v(x, k) :=
N j=1
ˆ j (k) ik· x j ϕ(k) ˆ = e e j 3/2 . 3/2 |k| |k| N
eik· x j
(41)
j=1
This form is useful to prove some standard properties of H ε and Hfib (x). Lemma 1. 1. H ε is self-adjoint on H0 (see Eq. (11)). 2. For every x ∈ R3N , Hfib (x) is self-adjoint on D(Hf ). Proof. The claims are based on the standard estimates (see, e. g., [Be], Proposition 1.3.8) (42) Φ( f )ψ2F ≤ 2 f / |k|2L 2 (R3 ) ψ, Hf ψF + 2 f 2L 2 (R3 ) ψ2F , for ψ ∈ D(Hf ), and
Φ( f )Ψ 2H ≤ 2 f / |k|2L 2 (R3 ) Ψ, (1 ⊗ Hf )Ψ H + 2 f 2L 2 (R3 ) Ψ 2H ,
(43)
for Ψ ∈ L 2 (R3N , D(Hf )), which imply respectively that, for fixed x, Φ(|k|v(x, k)) is infinitesimally small with respect to Hf and, as an operator on H , is infinitesimally small with respect to H0ε , uniformly in ε.
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The fibered Hamiltonian has the form of a quadratic part, which corresponds to the free field, plus a term linear in the annihilation and creation operators. Hamiltonians of this form are usually called in the literature “van Hove Hamiltonians” (a review on the subject is given in [De]). Their simple form allows to analyze in detail their spectral and dynamical features. In the finite dimensional case, if one considers an harmonic oscillator with a linear perturbation, like an external electric field for example, the natural way to recover the spectrum of the Hamiltonian is to translate the x variable, transforming the initial Hamiltonian into a purely quadratic one. In quantum field theory, the analogous strategy would be to find a unitary operator which translates the annihilation and creation operators. Such an operator would implement what is called a Bogoliubov transformation. While for a finite number of degrees of freedom the Bogoliubov transformation is always implementable, this may not be the case if the phase space is infinite dimensional. In particular this is not possible if the van Hove Hamiltonian is not sufficiently regular in the infrared region. After these preliminary remarks, we can state Lemma 2. Given the fibered Hamiltonian K fib (x) := Hf + Φ(z(x, k)), where z(x, k) satisfies 1 |z(x, k)|2 <∞ α(x) := − dk 2 R3 |k|
∀x ∈ R3N
one has: 1. The spectrum of K fib (x) is given by [α(x), +∞), and σac (K fib (x)) = (α(x), +∞).
(44)
2. The infimum of the spectrum, α(x), is an eigenvalue if and only if |z(x, k)|2 dk < ∞. |k|2 R3 In this case, moreover, the unitary operator V (x) := eiΦ(iz(x,k)|k|
−1 )
(45)
is well-defined on the Fock space and K fib (x) = V (x)Hf V (x)∗ + α(x).
(46)
If z(x, k)|k|−1 ∈ / L 2 (R3 , dk), then the spectrum is absolutely continuous. Proof. Point 1 is Proposition 3.10 of [De] and point 2 is Proposition 3.13 of the same paper. Corollary 1. σ (Hfib (x)) = σac (Hfib (x)) = [E(x), +∞), where 1 E(x) = − dk |k||v(x, k)|2 . 2 R3
(47)
A more explicit expression for E(x) is given in (13). We note also for later use that E is a smooth function, bounded with all its derivatives.
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L. Tenuta, S. Teufel
Corollary 2. The infrared regularized fibered Hamiltonian Hfib,σ (x) (see Eq. (17)) can be written as Hfib,σ (x) = Vσ (x)Hf Vσ (x)∗ + E σ (x), where Vσ (x) := eiΦ(ivσ (x,k)) .
(48)
E σ (x) is the only eigenvalue, with eigenvector Ωσ (x) := Vσ (x)ΩF . Moreover, for every α ∈ N3N , ∂xα E σ = ∂xα E + O(σ |α|+1 )L(H ) .
(49)
Proof. The only thing left to prove is Eq. (49), which follows immediately from the fact that 1 α α ∂x E(x) − ∂x E σ (x) = − dk |k| ∂xα |v(x, k)|2 2 |k|<σ and |k| ∂xα |v(x, k)|2 ≤ C|k||α|−2 . As last preliminaries, we show that we can approximate the true dynamics with infrared regular ones and that the same holds for the cutoffs in the energy. Proposition 1.
1/2
where H0
−it H ε /ε σ 1/2 ε,σ e , − e−it H /ε L(H 1/2 ,H ) ≤ C|t| 0 ε = D (H0ε )1/2 with the corresponding graph norm.
(50)
Proof. Both Hamiltonians are self-adjoint on H0 , therefore, given Ψ ∈ H0 we can apply the Duhamel formula to get i t ε,σ ε −it H ε /ε −it H ε,σ /ε −e )Ψ = ds ei(s−t)H /ε (H ε − H ε,σ )e−is H /ε Ψ (e ε 0 i t ε,σ ε = ds ei(s−t)H /ε Φ |k|v(x, k)1(0,σ ) (k) e−is H /ε Ψ ε 0 1 t ε −it H ε /ε −it H ε,σ /ε ⇒ (e −e )Ψ H ≤ ds Φ |k|v(x, k)1(0,σ ) (k) e−is H /ε Ψ H , ε 0 where 1(0,σ ) (k) is the characteristic function of the interval indicated. Using Eq. (43) we find that ε Φ |k|v(x, k)1(0,σ ) (k) e−is H /ε Ψ H √ ε ≤ 2|k|1/2 v(x, k)1(0,σ ) (k) L 2 (R3 ) (H0ε )1/2 e−is H /ε Ψ H √ + 2|k|v(x, k)1(0,σ ) (k) L 2 (R3 ) Ψ H .
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765
Moreover, for β > 0, |k|β v(x, k)1(0,σ ) (k) L 2 (R3 ) ≤
N
|e j | · |k|β−3/2 ϕ(k)1 ˆ (0,σ ) (k) L 2 (R3 ) ,
j=1
(51)
4π σ 2β 2 . ˆ (k) = |k|β−3/2 ϕ(k)1 (0,σ ) L 2 (R 3 ) (2π )3 2β Since Φ(|k|v(x, k)) is infinitesimally small with respect both to 1 ⊗ Hf and H0ε (as it follows again from (43)), the graph norm defined by (H0ε )1/2 and the one defined by (H ε )1/2 are equivalent, uniformly in ε (Theorem X.18, [ReSi2 ]). This implies that, for σ sufficiently small, ε ˜ 1/2 (H0ε )1/2 Ψ 2 + Ψ 2 1/2 , Φ |k|v(x, k)1(0,σ ) (k) e−is H /ε Ψ H ≤ Cσ H H which proves the statement.
Lemma 3. Given a function χ ∈ C0∞ (R), then χ (H ε ) − χ (H ε,σ )L(H ) ≤ Cσ 1/2 .
(52)
Proof. Using the Hellfer-Sjöstrand formula (see, e. g., [DiSj] Chap. 8), given a selfadjoint operator A, we can write 1 ¯ a (z)(A − z)−1 , z := x + iy, χ (A) = d xd y ∂χ (53) π R2 where χ a ∈ C0∞ (C) is an almost analytic extension of χ , which satisfies the properties ¯ a | ≤ D ¯ |z| N¯ , ∀ N¯ ∈ N ∃ D N¯ : |∂χ N χ|aR = χ . (For the explicit construction of such a χ a see [DiSj].) Applied to our case (53) yields
1 ¯ a (z) (H ε − z)−1 − (H ε,σ − z)−1 . χ (H ε ) − χ (H ε,σ ) = d xd y ∂χ π R2 Since both Hamiltonians are self-adjoint on H0 we have (H ε − z)−1 − (H ε,σ − z)−1 = (H ε,σ − z)−1 (H ε,σ − H ε )(H ε − z)−1 = −(H ε,σ − z)−1 Φ(|k|v1(0,σ ) )(H ε − z)−1 , and hence χ (H ε ) − χ (H ε,σ )L(H ) 1 ¯ a (z)|(H ε,σ − z)−1 L(H ) Φ(|k|v1(0,σ ) )(H ε − z)−1 L(H ) . ≤ d xd y |∂χ π R2 In addition we have that (H ε,σ − z)−1 L(H ) ≤
C . |z|
(54)
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L. Tenuta, S. Teufel
This follows because H0 is dense in the domain of H ε=0,σ =0 = L 2 (R3N , D(Hf )), and for every Ψ ∈ H0 , H ε,σ Ψ → H ε=0,σ =0 Ψ as (ε, σ ) → (0, 0). According to Theorem VIII.25 [ReSi1 ], this implies that (H ε,σ − z)−1 Ψ → (H ε=0,σ =0 − z)−1 Ψ, therefore |z|(H ε,σ − z)−1 Ψ is bounded for every Ψ and the uniform boundedness principle gives (54). For the second norm we find that for z ∈ supp χ a , Φ(|k|v1(0,σ ) )(H ε − z)−1 L(H ) ≤ Φ(|k|v1(0,σ ) )L(H0 ,H ) · (H ε − z)−1 L(H ,H0 ) C Φ(|k|v1(0,σ ) )L(H0 ,H ) , ≤ |z| because H ε and H0ε define the same graph norm uniformly in ε, as it follows from (43). From the same equation and from (51) it follows also that √ Φ(|k|v1(0,σ ) )L(H0 ,H ) ≤ |k|1/2 v1(0,σ ) L 2 (R3 ) + 2|k|v1(0,σ ) L 2 (R3 ) ≤ Dσ 1/2 , which proves the statement.
3. The Approximate Dressing Operator In this section we construct the approximate dressing operator Uε,σ for the infrared regularized Hamiltonian H ε,σ . We recall at the beginning the formal procedure to build the so called superadiabatic projectors, introduced first by Berry [Ber] for time-dependent Hamiltonians and generalized by Nenciu and Sordoni [NeSo] to the space-adiabatic setting. The basic idea is to construct a sequence of projectors P jε , j ∈ N, whose commutator with the Hamiltonian is of order O(ε j+1 ). Then the range of P jε is almost invariant for the corresponding time evolution in the sense that [e−it H
ε /ε
, P jε ] = O(ε j |t|).
As we have briefly mentioned in the introduction, this procedure, which allows to build adiabatic dynamics beyond the leading order, has been implemented till now only for fibered Hamiltonians whose spectrum is made up of eigenvalues separated by a gap. Exploiting the fact that the infrared cutoff σ gives rise effectively to a gap, we can however carry out the calculations also in our case. We proceed first in a formal way, using the superadiabatic projectors just as a formal tool to deduce the expression of the unitary intertwiner U .2 We will then show that the expression we find for U actually defines a unitary operator on H and we will prove some of its useful properties, which allow us to fully characterize the domain of the 2 To simplify the notation, we drop from now on the dependence of the unitary on σ and ε, denoting it simply by U instead that by Uε,σ , as we did in the introduction.
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
767
effective Hamiltonian, and to expand it in a series of powers of ε which is convergent in L(H0 , H ). The projectors PM on the almost invariant subspaces are then recovered from U via PM = U ∗ Q M U . It is important to stress that, to characterize the transitions between the different almost invariant subspaces, we need a unitary which maps all of them simultaneously to the corresponding reference subspaces. For this reason, we cannot use the procedure based on the Nagy formula proposed in [MaSo,NeSo] and we employ instead a simpler one, based on the construction of the projectors. In what follows, σ is considered a parameter independent of ε. Only at the end, we will choose it as a suitable function of ε. 3.1. Formal definition. Using the dressing operator Vσ (x), whose expression is given in (48), we can define dressed M-particles projections: (0)
Π M (x) := Vσ (x)Q M Vσ (x)∗ .
(55)
Π0(0) (x)
= |Ωσ (x)Ωσ (x)|, the projection onto the Note that, for M = 0, we have (0) ground state of Hfib,σ (x), but, for M = 0, Π M (x) is not a spectral projection of the fibered Hamiltonian. The standard construction we briefly mentioned above, which is used in the case with gap, is based on an iterative procedure. Starting from a zero order projection Π (0) = π0 , which corresponds to a spectral subspace of thefibered Hamiltonian, it allows to build an approximate orthogonal projection Π (n) := nj=1 ε j π j which satisfies (Π (n) )∗ = Π (n) ,
(Π (n) )2 − Π (n) = O(εn+1 ),
[H ε , Π (n) ] = O(εn+1 ).
(56)
As we already stressed, we avoid on purpose to be more precise on the sense in which Eq. (56) holds and on which norm one should use on the right-hand side, because we will use it just as a formal tool to deduce the expression of U . To clarify the method we employ, we recall briefly the derivation of the first order almost projection Π (1) in the case with gap, i. e., we assume for the moment that Hfib,σ (x) has a nondegenerate eigenvalue E σ (x) which is isolated from the rest of the spectrum. Our discussion is taken from [PST2 ]. Therefore, we start from a projection π0 (x) which projects onto the eigenspace associated to E σ (x), and we proceed formally, without worrying about specifying any regularity assumption. It holds then that π0∗ = π0 ,
(π0 )2 = π0 ,
(57)
1 2 ε2 pˆ ] = iε(∇π0 ) · pˆ + ∆π0 , (58) 2 2 where the right-hand side of the commutator is of order O(ε) when applied to functions of bounded kinetic energy. Applying the inductive scheme, we determine now the coefficient π1 in Π (1) = π0 + επ1 by requiring that [π0 , H ε,σ ] = [π0 ,
(Π (1) )∗ = Π (1) ,
(Π (1) )2 − Π (1) = O(ε2 ),
[Π (1) , H ε ] = O(ε2 ).
(59)
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L. Tenuta, S. Teufel
The first condition gives !
(π0 + επ1 )2 − (π0 + επ1 ) = ε(π0 π1 + π1 π0 − π1 ) + O(ε2 ) = O(ε2 ), so that we must have π1 = π0 π1 + π1 π0 + O(ε).
(60)
Concerning the commutator, we have !
[Π (1) , H ε,σ ] = iε(∇π0 ) · pˆ + ε[π1 , Hfib,σ ] + O(ε2 ) = O(ε2 ), so π1 has to satisfy [π1 , Hfib,σ ] = −i(∇π0 ) · pˆ + O(ε).
(61)
It can be shown ([Te3 ], Lemma 3.8) that the unique solution, up to terms of order O(ε), of (61) is given by π1 = −iπ0 (∇π0 )(Hfib,σ − E)−1 π0⊥ · pˆ + adj.,
(62)
where π0⊥ := 1 − π0 , and + adj. means that the adjoint of everything which lies to the left is added. This ensures also that π1 is formally self-adjoint. If there is a gap, the reduced resolvent is certainly bounded in norm:
−1 −1 ⊥ (Hfib,σ − E) π0 L(H ) = inf dist E σ (x), σ (Hfib,σ (x))\{E σ (x)} , (63) x∈R3N
but in general the construction breaks down in the case without gap, because the reduced resolvent can be unbounded. A possible way out of this difficulty was proposed in [Te1 ] and [Te2 ] to prove the adiabatic decoupling for the zero order subspaces. It consists in introducing a second parameter δ > 0, by which the reduced resolvent is shifted into the complex plane: instead of (Hfib − E)−1 π0⊥ , one considers (Hfib − E − iδ)−1 π0⊥ . In our case, the infrared cutoff σ plays an analogous role. Actually, at the end of our formal calculations, it will result that the reduced resolvent of Hfib,σ is bounded. We carry out another generalization of the standard construction, extending it to the situation when M = 0. To understand the way we proceed it is useful to analyze more explicitly the case M = 0. According to the above discussion we define (1)
(0)
Π0 (ε, σ ) := Π0 + επ10 (σ ), (0)
⊥ π10 (σ ) := −i(∇Π0 )Rfib (E σ ) · pˆ + adj., ⊥ (E ) := (H −1 (0) ⊥ . We have omitted the Π (0) because of the where Rfib σ fib,σ − E σ ) Π0 0 (0) well-known fact that ∇Π0 is off-diagonal with respect to the block decomposition (0) induced by Π0 . Using the fact that ∇Π0(0) = iVσ (x)[Φ(i∇x vσ ), Q 0 ]Vσ (x)∗ , and introducing the notation Rf⊥ (0) := Hf−1 Q ⊥ 0 , we get then
π10 = Vσ (x)[Φ(i∇x vσ ), Q 0 ]Rf⊥ (0)Vσ (x)∗ · p− ˆ pˆ · Vσ (x)Rf⊥ (0)[Φ(i∇x vσ ), Q 0 ]Vσ (x)∗
= −Vσ (x) d (|k|−1 ), [Φ(i∇x vσ ), Q 0 ] · pV ˆ σ (x)∗ + O(ε),
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
769
where we have used that the commutator between pˆ and a function of x is of order O(ε) (we stress again that we don’t worry here about smoothness assumptions, the calculations should be considered as formal) and Rf⊥ (0)Q ≤1 = d (|k|−1 )Q ≤1 , together with Q 0 Φ(i∇x vσ ) = Q 0 Φ(i∇x vσ )Q ≤1 , Φ(i∇x vσ )Q 0 = Q ≤1 Φ(i∇x vσ )Q 0 . Calculating the commutator using Eq. (39), we get π10 = iVσ [Q 0 , Φσ ] · pV ˆ σ∗ , where we define
∇x vσ (x, k) . Φσ (x) := Φ |k|
(64)
Since ∇x vσ (x, k)|k|−1 ∈ L 2 (R3 , dk), π10 is well-defined, and one can symmetrize it to get a symmetric operator (the fact that π10 is not bounded due to the presence of pˆ will be dealt with below). It is now fairly clear how to proceed. For any M, we define (1) (0) ΠM := Π M + επ1M , i π1M := Vσ [Q M , Φσ ] · pˆ + pˆ · [Q M , Φσ ] Vσ∗ . 2
(65)
(1) Formally Π M satisfies (59) for any M. In fact, Φσ is self-adjoint, therefore i[Q M , Φσ ] (1) (1) is self-adjoint, and (Π M )∗ = Π M . Moreover, since π1M is off-diagonal with respect to π0M ,
π0M π1M π0M = (1 − π0M )π1M (1 − π0M ) = 0, Eq. (60) holds exactly, without O(ε) corrections. Concerning the commutator Eq. (61), we get
[π1M , Hfib,σ ] = Vσ [Q M , iΦσ ] · p, ˆ Hf + E(x) Vσ∗ + O(ε)
ˆ σ∗ + O(ε). ˆ σ∗ + O(ε) = Vσ [Hf , iΦσ ], Q M · pV = Vσ [Q M , iΦσ ], Hf ] · pV Applying again Eq. (39) we get, on F|k|,fin , [Hf , Φσ ] = Φ(i∇x vσ ),
(66)
[π1M , Hfib,σ ] = −i∇π0M · pˆ + O(ε).
(67)
so that
The next step is to find a unitary operator which intertwines the almost projections (1) Π M with the reference projections given by the Q M up to terms of order O(ε2 ).
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L. Tenuta, S. Teufel
Employing a procedure analogous to the one we used for the projections, we assume that we can write an expansion U
(n)
:=
n
ε jUj,
(68)
j=1
starting from a known U0 , and imposing that U (n) U (n)∗ − 1 = O(εn+1 ),
U (n)∗ U (n) − 1 = O(εn+1 ),
(1)
U (n) Π M U (n)∗ = Q M + O(εn+1 ),
(69)
to deduce the coefficients Un iteratively. An obvious choice for the zero order unitary is given by U0 (x) := Vσ (x)∗ ,
(70)
(0)
which satisfies U0 (x)Π M (x)U0 (x)∗ = Q M . Without loss of generality, we assume that U1 = T U0 , for some operator T on H . The requirements (69) then lead to (U0 + εU1 )(U0∗ + εU1∗ ) = 1 + ε(U0 U1∗ + U1 U0∗ ) + O(ε2 ) !
= 1 + ε(T ∗ + T ) + O(ε2 ) = O(ε2 )
⇒
T ∗ + T = O(ε),
and (0)
(U0 + εU1 )(Π M + επ1M )(U0∗ + εU1∗ ) = (1 + εT )(Q M + ε[Q M , iΦσ ] · p)(1 ˆ − εT ) + O(ε2 ) = Q M + ε([Q M , iΦσ ] · pˆ − [Q M , T ]) + O(ε2 ). Therefore, by choosing T = iΦσ · p, ˆ we satisfy both requirements for every M. The first order almost unitary is then given by U (1) = (1 + iεΦσ · p)V ˆ σ∗ .
(71)
3.2. Rigorous definition and properties. To give a meaning to the till now formal expression for U (1) we have to deal with two problems. The first is due to the fact that the operator Φσ is unbounded. We introduce therefore in its definition a cutoff in the number of particles, replacing it by ΦσJ := Q ≤J Φσ Q ≤J . The operator (1)
UJ
:= (1 + iεΦσJ · p)V ˆ σ∗
(72)
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
771
is again formally almost unitary up to order O(ε2 ), and intertwines the superadiabatic almost projectors for M < J, (1)
(1)
(1)∗
UJ Π M UJ
= Q M + O(ε2 ),
M < J.
(73)
(1)
This means that we can use UJ to study the transitions among superadiabatic subspaces up to an arbitrary, but fixed J. The second problem, already mentioned in the introduction, is related to the presence of the unbounded momentum operator p. ˆ To make the whole expression bounded we introduce a cutoff in the total energy. More precisely, given a function χ ∈ C0∞ (R), we define (1) := Vσ∗ [1 + iεVσ ΦσJ · pV ˆ σ∗ − iε(1 − χ (H ε,σ ))Vσ ΦσJ · pV ˆ σ∗ (1 − χ (H ε,σ ))] UJ,χ
ˆ σ∗ + iε(1 − χ (H ε,σ ))Vσ ΦσJ · pV ˆ σ∗ χ (H ε,σ )]. = Vσ∗ [1 + iεχ (H ε,σ )Vσ ΦσJ · pV (74) (1)
(1)
Note that UJ,χ χ˜ (H ε,σ ) = UJ χ˜ (H ε,σ ), for every χ˜ ∈ C0∞ (R) such that χ χ˜ = χ˜ . In the context of Born-Oppenheimer approximation, Sordoni [So] applied a similar strategy in order to define bounded superadiabatic projections. (1) Our aim in this section is first to prove that UJ,χ ∈ L(H ) ∩ L(H0 ). Once we have shown this, we can define a true unitary U through the formula (1)
(1) ∗
(1)
U := UJ,χ [UJ,χ UJ,χ ]−1/2 . We will then prove that both U and U −1 = U ∗ belong to L(H ) ∩ L(H0 ), i. e., that U is a bijection on H0 . We will in addition show that it can be expanded in a convergent power series both in L(H ) and in L(H0 ). Lemma 4. Vσ and Vσ∗ belong to L(H0 ). Moreover Vσ L(H0 ) ≤ C,
(75)
with a constant C < ∞ independent of σ . An analogous estimate holds for
Vσ∗ .
Proof. (We give the proof for Vσ . The one for Vσ∗ is the same up to some changes in the signs.) We can calculate the norm on a dense subset. It is known ([ReSi1 ], Theorem VIII.33) that H0 is essentially self-adjoint on a set of the form Dp ⊗ Df := L{ψ ⊗ ϕ : ψ ∈ Dp , ϕ ∈ Df }, where L means “finite linear combinations of ”, Dp is a core for pˆ 2 and Df is a core for Hf . We can choose then Dp = C0∞ (R3N ) and ([ReSi2 ], Sect. X.7) Df = {ϕ ∈ Ffin : ϕ (N) ∈ ⊗Nj=1 C0∞ (R3 ) ∩ L 2s (R3N )},
(76)
where ⊗Nj=1 C0∞ (R3 ) := L{ϕ1 ⊗ . . . ⊗ ϕN : ϕ j ∈ C0∞ (R3 ), j = 1, . . . , N}. The vectors in Ffin are analytic vectors for Vσ (x) ([ReSi2 ], Theorem X.41), so, if ψ ∈ Dp , ϕ ∈ Df , we have Vσ (ψ ⊗ ϕ)(x) = ψ(x)Vσ (x)ϕ =
∞ j=0
ij
Φ(ivσ (x, ·)) j ψ(x)ϕ. j!
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L. Tenuta, S. Teufel
Moreover ( pˆ 2 ⊗ 1)Φ(ivσ (x, ·)) j ψ(x)ϕ = −ε2 jΦ(ivσ ) j−1 Φ(i∆vσ )ψ(x)ϕ − ε2 j ( j − 1)Φ(ivσ ) j−2 Φ(i∇x vσ ) · Φ(i∇x vσ )ψ(x)ϕ − 2iεjΦ(ivσ ) j−1 Φ(i∇x vσ ) · pψ(x)ϕ ˆ + Φ(ivσ ) j pˆ 2 ψ(x)ϕ, where we have used (36) and the fact that vσ (x, ·), ∇x vσ (x, ·) L 2 (R3 ) = 0. From the previous equations it follows that J Φ(ivσ (x, ·)) j ψ(x)ϕ ( pˆ 2 ⊗ 1)i j j! j=0
is convergent, so that Vσ (ψ ⊗ ϕ) ⊂ D( pˆ 2 ⊗ 1) and (77) ( pˆ 2 ⊗ 1)Vσ (ψ ⊗ ϕ) = −iε2 Vσ Φ(i∆vσ )(ψ ⊗ ϕ) + ε2 Vσ Φ(i∇x vσ )· ·Φ(i∇x vσ )(ψ ⊗ ϕ) + 2εVσ Φ(i∇x vσ ) · p(ψ ˆ ⊗ ϕ) + Vσ ( pˆ 2 ⊗ 1)(ψ ⊗ ϕ). This implies ( pˆ 2 ⊗ 1)Vσ (ψ ⊗ ϕ) ≤ ε2 Φ(i∆vσ )(ψ ⊗ ϕ) + ε2 Φ(i∇x vσ ) · Φ(i∇x vσ )(ψ ⊗ ϕ) + 2εΦ(i∇x vσ ) · p(ψ ˆ ⊗ ϕ) 2 + ( pˆ ⊗ 1)(ψ ⊗ ϕ).
(78)
Using Eq. (43) we can bound the first term with some constant independent of σ and ε times ψ ⊗ ϕH0 . This √ happens because the constants involved in (43) contain terms of the form f / |k| L 2 (R3 ) , but when one differentiates vσ with respect to x, one gets an addi√ tional |k|, therefore all the terms of the form ∇x vσ / |k| and so on are uniformly bounded in σ . The same reasoning applies also to all the estimates which follow. For the second term in (77), using again (43), and the notation Ψ = ψ ⊗ ϕ and f = i∂ j vσ , we get Φ( f )2 Ψ 2H ≤ 2 f / | · |2L 2 (R3 ) Φ( f )Ψ, (1 ⊗ Hf )Φ( f )Ψ H + 2 f 2L 2 (R3 ) Φ( f )Ψ 2H √
≤ 2 f / | · |2L 2 (R3 ) Φ( f )Ψ, a(| · | f )∗ − a(| · | f ) Ψ H + 2 f / | · |2L 2 (R3 ) Φ( f )2 Ψ, Hf Ψ H + 2 f 2L 2 (R3 ) Φ( f )Ψ 2H √
≤ 2 f / | · |2L 2 (R3 ) · Φ( f )Ψ H · a(| · | f )∗ − a(| · | f ) Ψ H + 2 f / | · |2L 2 (R3 ) · Φ( f )2 Ψ H Hf Ψ H + 2 f 2L 2 (R3 ) Φ( f )Ψ 2H
a2 1 ≤ √ f / | · |2L 2 (R3 ) Φ( f )Ψ 2H + √ a(| · | f )∗ − a(| · | f ) Ψ 2H 2 a2 2 1 + b2 f / | · |2L 2 (R3 ) Φ( f )2 Ψ 2H + 2 f / | · |2L 2 (R3 ) Hf Ψ 2H b + 2 f 2L 2 (R3 ) Φ( f )Ψ 2H , ∀a, b > 0.
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
773
Hence Φ( f )2 Ψ 2H 1 − b2 f / | · |2 )
1 a2 ≤ √ f / | · |2L 2 (R3 ) Φ( f )Ψ 2H + √ a(| · | f )∗ − a(| · | f ) Ψ 2H 2 2 a 2 1 + 2 f / | · |2L 2 (R3 ) Hf Ψ 2H + 2 f 2L 2 (R3 ) Φ( f )Ψ 2H . b Each term on the right-hand side can be bounded by a constant independent of σ times Ψ H0 . For the third term in (78) we get Φ( f ) pˆ j Ψ 2H ≤ 2 f / | · |2L 2 (R3 ) pˆ j Ψ, Hf pˆ j Ψ + 2 f 2L 2 (R3 ) pˆ j Ψ 2H ≤ 2 f / | · |2L 2 (R3 ) Ψ, Hf pˆ 2j Ψ + 2 f 2L 2 (R3 ) pˆ j Ψ 2H ≤ f / | · |2L 2 (R3 ) ( pˆ 2j + Hf )Ψ 2H + 2 f 2L 2 (R3 ) pˆ j Ψ 2H , and we can again bound the right-hand side by a constant times Ψ H0 . We have now to prove similar estimates for (1 ⊗ Hf )Vσ (ψ ⊗ ϕ), and we are done. For this we need that, on Dp ⊗ Df , [Hf , Φ(ivσ )] = iΦ(|k|vσ ), and that [Φ(|k|vσ ), Φ(ivσ )] = i|k|vσ , ivσ = i
dk |k||vσ (x, k)|2 = −2iE σ (x)
⇒ [Φ(|k|vσ ), Φ(ivσ ) j ] = −2i j E σ (x)Φ(ivσ ) j−1 . We have then (1 ⊗ Hf )Φ(ivσ (x, ·)) j (ψ ⊗ ϕ) =i
j−1
Φ(ivσ )l Φ(|k|vσ )Φ(ivσ ) j−l−1 (ψ ⊗ ϕ) + Φ(ivσ ) j (1 ⊗ Hf )(ψ ⊗ ϕ)
l=0
= i jΦ(ivσ ) j−1 Φ(|k|vσ )(ψ ⊗ ϕ) + E σ (x) j ( j − 1)Φ(ivσ ) j−2 ×(ψ ⊗ ϕ) + Φ(ivσ ) j (1 ⊗ Hf )(ψ ⊗ ϕ).
(79)
This means, as for pˆ 2 ⊗ 1, that Vσ (ψ ⊗ ϕ) ⊂ D(1 ⊗ Hf ), and that (1 ⊗ Hf )Vσ (ψ ⊗ ϕ) = −Vσ Φ(|k|vσ )(ψ ⊗ ϕ) − E σ (x)Vσ (ψ ⊗ ϕ) +Vσ (1 ⊗ Hf )(ψ ⊗ ϕ).
(80)
Using Eq. (43), we can again bound each term by a constant independent of σ times ψ ⊗ ϕH0 .
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J of the Lemma 5. 1. For each x ∈ R3N and l = 1, . . . , 3N , the components Φσ,l J (x) ∈ L(F ) and Φ J (x)∗ = Φ J (x). operator ΦσJ satisfy Φσ,l σ,l σ,l Moreover, J Φσ,l : R3N → L(F ),
J x → Φσ,l (x), ∈ Cb∞ (R3N , L(F )),
(81)
and, for σ small enough, √ J Φσ,l L(H ) ≤ C J + 1 log(σ −1 ).
(82)
Given α ∈ N3N with |α| > 0, it holds instead √ J J = ∂xα Φ0,l + O(σ |α| J + 1)L(H ) , ∂xα Φσ,l
(83)
J J ∂xα Φ0,l := (∂xα Φσ,l )|σ =0
(84)
where
is a well-defined bounded operator on H . J (x)) remain true if 2. The statements of point 1 (except for the self-adjointness of Φσ,l F is replaced by D(Hf ). Proof. The proof follows applying the standard inequality √ Q ≤J Φ( f (x, k))Q ≤J L(H ) ≤ 21/2 J + 1 · sup f (x, k) L 2 (R3 ,dk) , x∈R3N
to the case f = ∂xα ∂l vσ
J belong to L(H ) ∩ L(H ) ∀α ∈ N3N . Corollary 3. The fibered operators ∂xα Φσ,l 0 (1)
Lemma 6. UJ,χ belongs to L(H ) ∩ L(H0 ) and (1) UJ,χ L(K ) ≤ C(1 + ε log(σ −1 )),
(85)
(1) ∗ where K = H or H0 . The same holds for UJ,χ . (1)
Proof. We start considering UJ,χ . We have already shown that Vσ and Vσ∗ belong to L(H ) ∩ L(H0 ) with uniformly bounded norms, so we have to examine εχ (H ε,σ )Vσ ΦσJ · pV ˆ σ∗ , ε(1 − χ (H ε,σ ))V ΦσJ · pV ˆ σ∗ χ (H ε,σ ). Given Ψ ∈ Dp ⊗ Df (for the definitions, see the proof of Lemma 4) we have χ (H ε,σ )Vσ ΦσJ · pV ˆ σ∗ Ψ = −iεχ (H ε,σ ){∇x · (Vσ ΦσJ )}Vσ∗ Ψ + χ (H ε,σ ) pˆ · Vσ ΦσJ Vσ∗ Ψ. Since ∇x · (Vσ ΦσJ ) and χ (H ε,σ ) pˆ are bounded operators (see Lemmas 4 and 5) the lefthand side belongs to L(H ) ∩ L(H0 ). Moreover, from the previous lemma it follows that
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ΦσJ L(K ) ≤ C log(σ −1 ), while all the other terms have uniformly bounded norms. For the second one, Vσ ΦσJ · pV ˆ σ∗ χ (H ε,σ )Ψ = Vσ ΦσJ · [ p, ˆ Vσ∗ ]χ (H ε,σ )Ψ + Vσ ΦσJ · Vσ∗ pχ ˆ (H ε,σ )Ψ, pχ ˆ (H ε ) ∈ L(H ) ∩ L(H0 ), and so do all the other terms, as it follows from Lemmas 4 and Lemma 5. (1) ∗ Concerning UJ,χ , from Eq. (74) it follows that (1) ∗
UJ,χ = [1 − iεχ (H ε,σ )Vσ pˆ · ΦσJ Vσ∗ − iε(1 − χ (H ε,σ ))Vσ pˆ · ΦσJ Vσ∗ χ (H ε,σ )]Vσ , (86) so one can apply the same reasoning as above.
Theorem 1. Assume that σ = σ (ε) and that conditions (20) hold, then the operator (1)
(1) ∗
(1)
U := UJ,χ [UJ,χ UJ,χ ]−1/2
(87)
is well-defined and unitary, for ε small enough. Both U and U ∗ belong to L(H ) ∩ L(H0 ), with the property that U L(H0 ) , U ∗ L(H0 ) ≤ C,
(88)
where C is independent of ε and σ . Moreover we can expand them in a power series which converges both in L(H ) and in L(H0 ). Proof. Combining Eqs. (74) and (86), and using the fact that pˆ ·ΦσJ = ΦσJ · pˆ −iε∇ ·ΦσJ , we have that (1) ∗ (1) UJ,χ UJ,χ = 1 + ε2 Bε ,
where Bε := B0 + ε B1 , B0 := −χ (H ε,σ )Vσ ∇ · ΦσJ Vσ∗ − (1 − χ (H ε,σ ))Vσ ∇ · ΦσJ Vσ∗ χ (H ε,σ )
+ [χ (H ε,σ )Vσ (ΦσJ · p)V ˆ σ∗ + (1 − χ (H ε,σ ))Vσ (ΦσJ · p)V ˆ σ∗ χ (H ε,σ )]2 ,
(89)
B1 := −i[χ (H ε,σ )Vσ ∇ · ΦσJ Vσ∗ + (1 − χ (H ε,σ ))Vσ ∇ · ΦσJ Vσ∗ χ (H ε,σ )]
·[χ (H ε,σ )Vσ (ΦσJ · p)V ˆ σ∗ + (1 − χ (H ε,σ ))Vσ (ΦσJ · p)V ˆ σ∗ χ (H ε,σ )].
From Lemma 6 it follows that Bε is a bounded operator both on H and H0 , self-adjoint on H . From equation (85) we have also that Bε L(K ) = O log(σ −1 ) , K = H , H0 , (1) ∗ (1) so, under conditions (20), we can assume that ε2 Bε L(K ) < 1. UJ,χ UJ,χ is therefore strictly positive, and the square root in (87) is well-defined.
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Moreover, we can express it through a convergent power series, both in L(H ) and in L(H0 ): (1) ∗
(1)
[UJ,χ UJ,χ ]−1/2 = (1 + ε2 Bε )−1/2 =
∞ (2 j − 1)!! 2 j j 3 (−1) j ε Bε , . (2 j)!!
(90)
j=0
We can also explicitly calculate that (1)
(1) ∗
(1)
(1) ∗
U U ∗ = UJ,χ [UJ,χ UJ,χ ]−1 UJ,χ = 1; (1) ∗
(1)
(1) ∗
(1)
(1) ∗
(1)
U ∗ U = [UJ,χ UJ,χ ]−1/2 UJ,χ UJ,χ [UJ,χ UJ,χ ]−1/2 = 1, (1)
(1) ∗
where we have used in the first equation the fact that both UJ,χ and UJ,χ are invertible, since they differ by a term of order O(ε log(σ −1 )) from a unitary operator. Putting together (87) and (90) we get in the end the expansion for U . 4. The Dressed Hamiltonian Using the results of the last section, we can define the dressed Hamiltonian just as the unitary transform of H ε,σ , ε := U H ε,σ U ∗ . Hdres
(91)
ε is self-adjoint onH . It follows from Theorem 1 that U is a bijection on H0 , so Hdres 0 ε in Moreover, since U can be expanded in a power series in L(H0 ), we can expand Hdres L(H0 , H ). However, putting directly the expansion for U in (91) we get ε−dependent coefficients, because of the ε−dependence in H ε,σ . To get the correct expansion we
need then to rearrange some terms, but the remainder we get will in the end be bounded in L(H0 , H ). Theorem 2. The expansion up to the second order of the dressed Hamiltonian is given by 3/2 ε Hdres = h 0 + εh 1,χ + ε2 h 2,χ + O(σ )L(H0 ,H ) + O(ε3 log(σ −1 ) )L(H0 ,H ) , (92) where h i,χ ∈ L(H0 , H ), h0 =
1 2 pˆ + Hf + E(x), 2
(93)
h 1,χ is given in Eq. (95) and h 2,χ is given in Eq. (97). ε Remark 5. The coefficients in the expansion of Hdres depend explicitly on the cutoff function χ (and also on the cutoff J in the number of particles, even though we have not stressed it in the notation). However, as we have already mentioned, we expect the adiabatic decoupling to be meaningful only on states in the range of χ (H ε,σ ) (or χ (H ε ), according to Lemma (3) ε ). we can interchange the two), which becomes in the representation space χ (Hdres With this in mind, we will prove later that the effective dynamics on the range of ε ) (M < J + 1) is generated by an Hamiltonian independent of χ and J. Q M χ (Hdres 3 (2 j − 1)!! = 1 · 3 · 5 · · · (2 j − 1); (2 j)!! = 2 · 4 · 6 · · · 2 j = 2 j j!
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Proof. To simplify the following reasoning we write (1) UJ,χ = Vσ∗ (1 + εT ),
ˆ σ∗ + i(1 − χ (H ε,σ ))Vσ ΦσJ · pV ˆ σ∗ χ (H ε,σ ), T := iχ (H ε,σ )Vσ ΦσJ · pV
(1) ∗
UJ,χ
= (1 − εT + iε2 S)Vσ ,
S := iχ (H ε,σ )Vσ ∇ · ΦσJ Vσ∗ + i(1 − χ (H ε,σ ))Vσ ∇ · ΦσJ Vσ∗ χ (H ε,σ ). We omit the dependence on σ and ε in T and S to streamline the presentation. We can then write Bε = iS − T 2 + iεST. Expanding the square root according to formula (90) we get ε Hdres
ε2 2 =UH U = 1 − (iS − T ) H ε,σ 2 ε2 · 1 − (iS − T 2 ) (1 − εT + iε2 S)Vσ + O ε3 (log(σ −1 ))3/2 L(H ,H ) 0 2 ε,σ
∗
Vσ∗ (1 + εT )
= Vσ∗ H ε,σ Vσ + εVσ∗ T H ε,σ Vσ − εVσ∗ H ε,σ T Vσ + iε2 Vσ∗ H ε,σ SVσ ε2 ∗ ε,σ ε2 Vσ H (iS − T 2 )Vσ − ε2 Vσ∗ T H ε,σ T Vσ − Vσ∗ (iS − T 2 )H ε,σ Vσ 2 2 + O ε3 (log(σ −1 ))3/2 L(H ,H ) = Vσ∗ H ε,σ Vσ + εVσ∗ [T, H ε,σ ]Vσ −
0
iε2 ∗ ε,σ ε2 Vσ [H , S]Vσ + Vσ∗ [H ε,σ , T ], T Vσ + O ε3 (log(σ −1 ))3/2 L(H ,H ) . + 0 2 2 We examine now separately the terms coming from different powers of ε: 1 Vσ (x)∗ pˆ 2 Vσ (x) + Hf + E σ (x) 2 1 1 = pˆ 2 + Hf + E σ (x) + Vσ (x)∗ [ pˆ 2 , Vσ (x)] 2 2 (77) 1 2 pˆ + Hf + E σ (x) + εΦ(i∇x vσ ) · pˆ = 2
(0) = Vσ∗ H ε,σ Vσ =
ε2 ε2 Φ(i∇x vσ ) · Φ(i∇x vσ ) − i Φ(i∆vσ ) 2 2 1 ε ε (49) = pˆ 2 + Hf + E(x) + Φ(i∇x vσ ) · pˆ + pˆ · Φ(i∇x vσ ) 2 2 2 +
+
ε2 Φ(i∇x vσ ) · Φ(i∇x vσ ) + O(σ )L(H ) . 2
This implies that h0 =
1 2 pˆ + Hf + E(x). 2
(94)
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Summing the term of order O(ε) coming from (0) to the ones coming from the expansion of U we get ε ε (1) = Φ(i∇x vσ ) · pˆ + pˆ · Φ(i∇x vσ ) + εVσ∗ [T, H ε,σ ]Vσ . 2 2 Here [T, H ε,σ ] = i χ (H ε,σ )[Vσ ΦσJ · pV ˆ σ∗ , H ε,σ ] + i(1 − χ (H ε,σ ))[Vσ ΦσJ · pV ˆ σ∗ , H ε,σ ]χ (H ε,σ ) i i ˆ σ∗ , pˆ 2 ] + (1 − χ (H ε,σ ))[Vσ ΦσJ · pV ˆ σ∗ , pˆ 2 ]χ (H ε,σ ) = χ (H ε,σ )[Vσ ΦσJ · pV 2 2 ˆ Hf + E σ (x)]Vσ∗ + i χ (H ε,σ )Vσ [ΦσJ · p, + i(1 − χ (H ε,σ ))Vσ [ΦσJ · p, ˆ Hf + E σ (x)]Vσ∗ χ (H ε,σ ) .
The commutator appearing in the last two lines gives [ΦσJ · p, ˆ Hf + E σ (x)] = [ΦσJ , Hf ] · pˆ + ΦσJ · [ p, ˆ E σ (x)] = iQ ≤J Φ(i∇x vσ )Q ≤J · pˆ − iεΦσJ · ∇ E σ (x), where we have used Eq. (66) to calculate the first commutator. Putting together all the terms we have therefore ε ε (1) = Φ(i∇x vσ ) · pˆ + pˆ · Φ(i∇x vσ ) − εχ (Vσ∗ H ε,σ Vσ )[Φ J (i∇x vσ ) · pˆ 2 2 − εΦσJ · ∇ E σ (x)] − ε(1 − χ (Vσ∗ H ε,σ Vσ ))[Φ J (i∇x vσ ) · pˆ − εΦσJ · ∇ E σ (x)]χ (Vσ∗ H ε,σ Vσ ) + O(ε2 log(σ −1 ))L(H0 ,H ) , where the terms containing the commutator with pˆ 2 give rise to higher order contributions and we abbreviated Φ J ( f ) := Q ≤J Φ( f )Q ≤J . Analyzing the terms of order O(ε2 ) we will see that they yield, as it happens for the zero order one, terms of the form pˆ ·Φ, which make the previous expression a symmetric operator. We have therefore in the end that h 1,χ =
1 1 Φ(i∇x vσ ) · pˆ − χ (Vσ∗ H ε,σ Vσ )Φ J (i∇x vσ ) · pˆ 2 2 1 ∗ ε,σ − (1 − χ (Vσ H Vσ ))Φ J (i∇x vσ ) · pχ ˆ (Vσ∗ H ε,σ Vσ ) + “ pˆ · Φ”, 2
(95)
where the symbol “ pˆ · Φ” means that, associated to each term of the form Φ · p, ˆ there is another one of the form pˆ · Φ which makes the sum a symmetric operator. To calculate h 2 we follow the same route, (2) =
ε2 Φ(i∇x vσ ) · Φ(i∇x vσ ) + ε2 χ (Vσ∗ H ε,σ Vσ )ΦσJ · ∇ E σ (x) 2 iε + ε2 (1 − χ (Vσ∗ H ε,σ Vσ ))ΦσJ · ∇ E σ (x)χ (Vσ∗ H ε,σ Vσ ) + χ (Vσ∗ H ε,σ Vσ ) 2 iε J ∗ 2 ∗ ε,σ J ∗ 2 ·[ Φσ · p, ˆ Vσ pˆ Vσ ] + (1 − χ (Vσ H Vσ ))[Φσ · p, ˆ Vσ pˆ Vσ ]χ (Vσ∗ H ε,σ Vσ ) 2 iε2 ∗ ε,σ ε2 Vσ [H , S]Vσ + Vσ∗ [H ε,σ , T ], T Vσ . + 2 2
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
779
We examine separately the different terms. •
ε2 ε2 Φ(i∇x vσ ) · Φ(i∇x vσ ) = a(i∂ j vσ ) + a(i∂ j vσ )∗ a(i∂ j vσ ) + a(i∂ j vσ )∗ 2 4 3N 2 ε a(i∂ j vσ )2 + a(i∂ j vσ )∗ 2 + a(i∂ j vσ )∗ a(i∂ j vσ ) + ∇vσ 2L 2 (R3 ,dk)⊗C3N . = 4 j=1
We remark that the last term, being x-independent, is a c-number. • [ΦσJ · p, ˆ Vσ∗ pˆ 2 Vσ ] = [ΦσJ · p, ˆ pˆ 2 ] + [ΦσJ · p, ˆ Vσ∗ [ pˆ 2 , Vσ ]] J = 2iε∂ j Φσ,l pˆl pˆ j + ε[ΦσJ · p, ˆ Φ(i∇x vσ ) · pˆ + pˆ · Φ(i∇x vσ )] + O(ε2 )L(H0 ,H ) J J J = 2iε∂ j Φσ,l pˆl pˆ j +ε[Φσ,l , Φ(i∂ j v)] pˆl pˆ j +ε pˆl pˆ j [Φσ,l , Φ(i∂ j vσ )]+O(ε2 )L(H0 ,H ) J J J = 2iε∂ j Φ0,l pˆl pˆ j +ε[Φσ,l , Φ(i∂ j vσ )] pˆl pˆ j +ε pˆl pˆ j [Φσ,l , Φ(i∂ j vσ )]+O(ε2 )L(H0 ,H ) + O(εσ )L(H0 ,H ) , (96)
where we have used Lemma 5. J • [Φσ,l , Φ(i∂ j vσ )] = [Q ≤J Φσ,l Q ≤J , Φ(i∂ j v)] = [Q ≤J , Φ(i∂ j vσ )]ΦσJ Q ≤J
+ Q ≤J ΦσJ [Q ≤J , Φ(i∂ j vσ )] + Q ≤J [ΦσJ , Φ(i∂ j vσ )]Q ≤J 1 1 = √ Q J a(i∂ j vσ )ΦσJ Q ≤J − √ a(i∂ j vσ )∗ Q J ΦσJ Q ≤J 2 2 1 1 + √ Q ≤J ΦσJ Q J a(i∂ j vσ ) − √ Q ≤J ΦσJ a(i∂ j vσ )∗ Q J 2 2 ∂l vσ (x, ·) + iQ ≤J ∂ j vσ (x, ·), L 2 (R3 ,dk) |·| ∂ v (x, ·) ˜ J + iQ ≤J ∂ vσ (x, ·), l σ =: R j L 2 (R3 ,dk) |·| ∂ v(x, ·) ∂l v(x, ·) ˜ J + iQ ≤J j =R , + O(σ 1/2 )L(H ) , 2 3 |·|1/2 |·|1/2 L (R ,dk) where we have bounded the scalar product using the same procedure applied in Lemma 5. ˜ J vanishes when applied to states in the range of Q M with The remainder term R M < J − 1. The scalar product can be written in a clearer way using the explicit expression of the function v (l1 , j1 = 1, . . . N , l2 , j2 = 1, . . . 3): •
∂(l1 ,l2 ) v(x, ·) ∂( j1 , j2 ) v(x, ·) , L 2 (R3 ,dk) |·|1/2 |·|1/2 eik·( x j1 − x l1 ) k . = dk ˆ l1 (k)∗ ˆ j1 (k) κl2 κ j2 , κ := 2 |k| |k| R3
When l1 = j1 , since the charge densities are spherically symmetric, we get |ˆ l (k)|2 1 |ˆ l (k)|2 dk 1 2 κl2 κ j2 = dk 1 2 δl2 , j2 |k| 3 R3 |k| R3 1 l1 (x)l1 ( y) 1 = dx d y δl2 , j2 =: el1 δl2 , j2 . 12π R3 ×R3 |x − y| 3
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Putting all the terms together, we have in the end ε J J ˆ V ∗ pˆ 2 V ] = iε2 ∂l Φ0,l pˆ j pˆl + iε2 Q ≤J el1 pˆ l21 [Φσ · p, 2 N
+i
ε2 Q ≤J 2
N
l1 =1
3 l1 , j1 =1 R (l1 = j1 )
dk
ˆ l1 (k)∗ ˆ j1 (k) ik·( x j − x l ) 1 1 (κ · p ˆ l1 )(κ · pˆ j1 ) e |k|2
+ (κ · pˆ l1 )(κ · pˆ j1 )eik·( x j1 − x l1 ) + ε2 RM + O(ε3 )L(H0 ,H ) + O(ε2 σ )L(H0 ,H ) , where RJ is a remainder term that vanishes when applied to states in the range of Q M , with M < J − 1, and pˆ l1 denotes the three-dimensional momentum operator associated to each particle. •
iε2 ∗ ε,σ ε2 Vσ [H , S]Vσ = − Vσ∗ χ (H ε,σ )[H ε,σ , Vσ ∇ · ΦσJ Vσ∗ ]Vσ 2 2 ε2 ∗ − Vσ (1 − χ (H ε,σ ))[H ε,σ , Vσ ∇ · ΦσJ Vσ∗ ]χ (H ε,σ )Vσ 2 ε2 ε2 = − χ (Vσ∗ H ε,σ Vσ )[Vσ∗ H ε,σ Vσ , ∇ · ΦσJ ] − (1 − χ (Vσ∗ H ε,σ Vσ )) 2 2 ∗ ε,σ J ∗ ε,σ ·[Vσ H Vσ , ∇ · Φσ ]χ (Vσ H Vσ ) ε2 ε2 χ (Vσ∗ H ε,σ Vσ )[Hf , ∇ · ΦσJ ] − (1 − χ (Vσ∗ H ε,σ Vσ )) 2 2 ·[Hf , ∇ · ΦσJ ]χ (Vσ∗ H ε,σ Vσ ) + O(ε3 )L(H0 ,H )
=−
=
iε2 iε2 χ (Vσ∗ H ε,σ Vσ )Φ J (i∆vσ ) + (1 − χ (Vσ∗ H ε,σ Vσ ))Φ J (i∆vσ )χ (Vσ∗ H ε,σ Vσ ) 2 2 + O(ε3 )L(H0 ,H ) .
This gives exactly the terms needed to make h 1 a symmetric operator. •
ε2 ∗ ε,σ V [H , T ], T Vσ . 2 σ
Using the calculations for the first order part h 1 , we get that ˆ σ∗ [H ε,σ , T ] = χ (H ε,σ )Vσ Φ J (i∇x vσ ) · pV
+ (1−χ (H ε,σ ))Vσ Φ J (i∇x vσ ) · pV ˆ σ∗ χ (H ε,σ )+O(ε log(σ −1 )))L(H0 ,H ) .
We can keep therefore just the first two terms, omitting the remainder. •
ε2 ∗ ε,σ ε2 Vσ [H , T ], T Vσ = Vσ∗ χ (H ε )Vσ Φ J (i∇x vσ ) · pV ˆ σ∗ 2 2 + (1 − χ (H ε,σ ))Vσ Φ J (i∇x vσ ) · pV ˆ σ∗ χ (H ε,σ ), T Vσ + O(ε3 log(σ −1 ))
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
=
781
iε2 ∗ ˆ σ∗ , χ (H ε,σ )Vσ ΦσJ · pV ˆ σ∗ Vσ V χ (H ε,σ )Vσ Φ J (i∇x vσ ) · pV 2 σ iε2 ∗ + ˆ σ∗ χ (H ε,σ ), χ (H ε,σ )Vσ ΦσJ · pV ˆ σ∗ Vσ Vσ (1 − χ (H ε,σ ))Vσ Φ J (i∇x vσ ) · pV 2 iε2 ∗ Vσ χ (H ε,σ )Vσ Φ J (i∇x vσ ) · pV + ˆ σ∗ , (1 − χ (H ε,σ ))Vσ ΦσJ · pV ˆ σ∗ χ (H ε,σ ) Vσ 2 iε2 ∗ + ˆ σ∗ χ (H ε,σ ), V (1 − χ (H ε,σ ))Vσ Φ J (i∇x vσ ) · pV 2 σ ε,σ J ∗ ε,σ 3 (1 − χ (H ))Vσ Φσ · pV ˆ σ χ (H ) Vσ + O(ε log(σ −1 ))L(H0 ,H ) .
Summing up, we get in the end that h 2,χ =
3N 1 a(i∂ j vσ )2 + a(i∂ j vσ )∗ 2 + a(i∂ j vσ )∗ a(i∂ j vσ ) 4 j=1 2 ∗ ε,σ + ∇vσ L 2 (R3 ,dk)⊗C3n + χ (Vσ H Vσ ) ΦσJ · ∇ E(x)
(97)
1 el1 pˆ l21 − ∂ j Φ0,l pˆl pˆ j + pˆl pˆ j ∂ j Φ0,l − Q ≤J 2 N
1 − Q ≤J 2
N
l1 =1
3 l1 , j1 =1 R (l1 = j1 )
dk
+ (κ · pˆ l1 )(κ · pˆ j1 )e
ˆ l1
(k)∗ ˆ |k|2
ik·( x j1 − x l1 )
j1 (k) ik·( x j − x l ) 1 1
e
(κ · pˆ l1 )(κ · pˆ j1 )
+ RJ + (1 − χ (Vσ∗ H ε,σ Vσ )) · · · χ (Vσ∗ H ε,σ Vσ )
i + Vσ∗ χ (H ε,σ )Vσ Φ J (i∇x vσ ) · pV ˆ σ∗ , χ (H ε,σ )Vσ ΦσJ · pV ˆ σ∗ Vσ 2 i + Vσ∗ (1−χ (H ε,σ ))Vσ Φ J (i∇x vσ ) · pV ˆ σ∗ χ (H ε,σ ), χ (H ε,σ )Vσ ΦσJ · pV ˆ σ∗ Vσ 2 1 + Vσ∗ χ (H ε,σ )Vσ Φ J (i∇x vσ ) · pV ˆ σ∗ , (1−χ (H ε,σ ))Vσ ΦσJ · pV ˆ σ∗ χ (H ε,σ ) Vσ 2 1 + Vσ∗ (1 − χ (H ε,σ ))Vσ Φ J (i∇x vσ ) · pV ˆ σ∗ χ (H ε,σ ) 2 ˆ σ∗ χ (H ε,σ ) Vσ , ·(1 − χ (H ε,σ ))Vσ ΦσJ · pV where we have symmetrized ∂ j Φ0,l pˆl pˆ j , which is possible up to terms of order O(ε). The expression is fairly lengthy, but we will show below that many terms vanish when ε ) (M < J − 1). applied to a state in the range of Q M χ (Hdres 5. The Effective Dynamics We start with a number of lemmas we need to analyze the effective time evolution.
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Lemma 7. Assume that σ satisfies conditions (20), then 1. Given a function χ˜ ∈ C0∞ (R), we have χ(V ˜ σ∗ H ε,σ Vσ ) − χ˜ (h 0 ) = εRεχ ,
(98)
where Rεχ ∈ L(H , H0 ), Rεχ L(H ,H0 ) = O(1) and Rεχ Q M = (Q M+1 + Q M−1 )Rεχ Q M + O(ε2 )L(H ,H0 ) .
(99)
2. Moreover, we have that
and that
ε χ˜ (Hdres ) − χ(h ˜ 0 ) = O(ε)L(H ,H0 ) ,
(100)
ε ε Q M χ(H ˜ dres ) = Q M χ˜ (h 0 )χ˜˜ (Hdres ) + O(ε2 log(σ −1 ))L(H ,H0 ) ,
(101)
where χ˜˜ is any C0∞ (R) function such that χ˜ χ˜˜ = χ˜ and χ˜˜ χ = χ˜˜ , M < J − 1. Proof. Applying the Hellfer-Sjöstrand formula, Eq. (53), we get
1 χ(V ˜ σ∗ H ε,σ Vσ ) − χ˜ (h 0 ) = d xd y ∂¯ χ˜ a (z) (Vσ∗ H ε,σ Vσ − z)−1 − (h 0 − z)−1 . π R2 Since both Hamiltonians are self-adjoint on H0 , we get (Vσ∗ H ε,σ Vσ − z)−1 − (h 0 − z)−1
= (h 0 − z)−1 (h 0 − Vσ∗ H ε,σ Vσ )(Vσ∗ H ε,σ Vσ − z)−1 ε ε = −(h 0 − z)−1 Φ(i∇x vσ ) · pˆ + pˆ · Φ(i∇x vσ ) 2 2 2 ε + Φ(i∇x vσ ) · Φ(i∇x vσ ) (Vσ∗ H ε,σ Vσ − z)−1 + O(σ |z|−2 )L(H ,H0 ) , 2
where we have used Eq. (94) to calculate the difference of the two Hamiltonians and estimated the integrand proceeding in the same way as in Lemma 3. Moreover, iterating the formula we get (Vσ∗ H ε,σ Vσ − z)−1 − (h 0 − z)−1 = (h 0 − z)
−1
Vσ∗ H ε,σ Vσ )(Vσ∗ H ε,σ Vσ
(102) −1
(h 0 − − z) ε ε = −(h 0 − z)−1 Φ(i∇x v) · pˆ + pˆ · Φ(i∇x v) (h 0 − z)−1 2 2 + O(ε2 |z|−3 )L(H ,H0 ) , so
∗ ε,σ (Vσ H Vσ − z)−1 − (h 0 − z)−1 Q M
= (Q M+1 + Q M−1 ) (Vσ∗ H ε,σ Vσ − z)−1 − (h 0 − z)−1 Q M + O(ε2 |z|−3 )L(H ,H0 ) .
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
783
Concerning point 2, Eq. (100) follows immediately from the fact that ε ε ε Hdres − z)−1 − (h 0 − z)−1 = (h 0 − z)−1 (h 0 − Hdres )(Hdres − z)−1 ε = ε(h 0 − z)−1 h 1,χ (Hdres − z)−1 + O(ε2 log(σ −1 ))L(H0 ,H ) ,
while, for Eq. (101), we have by definition that ε ε ε Q M χ(H ˜ dres ) = Q M χ˜ (Hdres )χ˜˜ (Hdres ) ε ε ε = Q M χ˜ (h 0 )χ˜˜ (Hdres ) + Q M [χ˜ (Hdres ) − χ(h ˜ 0 )]χ˜˜ (Hdres ).
Proceeding as above we find ε ε ˜ dres ) − χ˜ (h 0 )]χ˜˜ (Hdres ) Q M [χ(H
ε 1 ε = d xd y ∂¯ χ˜ a (z)Q M (Hdres − z)−1 − (h 0 − z)−1 χ˜˜ (Hdres ), π R2
(103)
so, if we show that ε Q M h 1,χ χ˜˜ (Hdres ) = O(ε log(σ −1 ))L(H0 ,H )
(104)
we are done. From Eq. (95) (omitting the “ pˆ · Φ” part, which can be treated in the same way) we get 1 1 ε ε Q M h 1,χ χ˜˜ (Hdres ) = Q M Φ(i∇x vσ ) · pˆ χ˜˜ (Hdres ) − Q M χ (Vσ∗ H ε,σ Vσ ) 2 2 1 J ε ∗ ε,σ · Φ (i∇x vσ ) · pˆ χ˜˜ (Hdres ) − Q M (1 − χ (Vσ H Vσ ))Φ J (i∇x vσ ) · pχ ˆ (Vσ∗ H ε,σ Vσ ) 2 ε · χ˜˜ (Hdres ), but it follows from point 1 that we can replace χ (V ∗ H ε V ) with χ (h 0 ) up to terms of ε ) with χ˜˜ (h ) up to terms of order O(ε), and from Eq. (103) that we can replace χ˜˜ (Hdres 0 order O(ε), therefore we have ε )= Q M h 1,χ χ˜˜ (Hdres
1 1 Q M Φ(i∇x vσ ) · pˆ χ˜˜ (h 0 ) − Q M χ (Vσ∗ H ε,σ Vσ )Φ J (i∇x vσ ) 2 2
1 Q M (1 − χ (Vσ∗ H ε,σ Vσ ))Φ J (i∇x vσ ) · pˆ χ˜˜ (h 0 ) + O(ε) 2 1 1 = Q M Φ(i∇x vσ ) · pˆ χ˜˜ (h 0 ) − Q M Φ J (i∇x vσ ) · pˆ χ˜˜ (h 0 ) + O(ε) = O(ε), 2 2 · pˆ χ˜˜ (h 0 ) −
so point 2 is proved.
The following lemma was proved in ([FGS], Appendix B) and we will use it to characterize the range of χ(−∞,c) (h 0 ), where χ(−∞,c) is the characteristic function of the indicated interval.
784
L. Tenuta, S. Teufel
Lemma 8. Let H˜ be a Hamiltonian of the form H˜ := 1 ⊗ dΓ (|k|) + H ⊗ 1,
(105)
acting on the Hilbert space H˜ = H ⊗ F , where F is the bosonic Fock space over L 2 (Rd ) and H is a generic Hilbert space. Then the set of all linear combinations of vectors of the form ϕ ⊗ a(g1 )∗ · · · a(gN )∗ ΩF , λ +
N
M j < c, (c > 0),
(106)
j=1
where ϕ = χ(−∞,λ) (H )ϕ for some λ < c, N ∈ N and M j := sup{|k| : k ∈ supp g j } is dense in χ(−∞,c) ( H˜ )H˜ . Lemma 9. Let χ ∈ C0∞ (R) and h˜ 0 := 1 ⊗ Hf + pˆ 2 /2 ⊗ 1. Then, ∃ξ ∈ C0∞ (R) such that χ ξ = χ and ξ c (h˜ 0 )χ (h 0 ) = O(ε∞ )L(H ) ,
(107)
where ξ c := 1 − ξ . Moreover, denoting by cχ := sup{|k| : k ∈ supp χ }, and defining cξ := 2cχ + E ∞ ,
(108)
where E ∞ := supx∈R3n |E(x)|, we can choose sup{|k| : k ∈ supp ξ } arbitrarily close to cξ . The statement remains true also if we invert the roles of h 0 and h˜ 0 . Proof. It follows immediately from the spectral theorem and the fact that Hf is a non(s) negative operator that, if χ(−E ∞ ,cχ ) denotes a smoothed version of the characteristic function of the interval indicated, then (s)
χ(−E ∞ ,cχ ) (h p )χ (h 0 ) = χ (h 0 ), where h p :=
1 2 pˆ + E(x). 2
Our aim is now to use the functional calculus for pseudodifferential operators developed in [HeRo] (see also [DiSj], Chap. 8) to show that, if ξ ∈ C0∞ (R), ξ = 1 on a neighborhood of [0, cχ + E ∞ ], then (s)
ξ c (h˜ p )χ(−E ∞ ,cχ ) (h p ) = O(ε∞ )L(H ) , where h˜ p := pˆ 2 /2.
(109)
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
785
Once we have shown this, we will have χ (h 0 ) = ξ(h˜ p )χ (h 0 ) + O(ε∞ )L(H ) , but, applying Lemma 8, we can write ξ(h˜ p )χ (h 0 )Ψ = lim ξ(h˜ p )Ψn , n→∞
where the Ψn are finite linear combinations of vectors of the form (106), with ϕ = χ(−∞,λ) (h p )ϕ, 0 < λ < cχ , so that (s) n→∞ n→∞ (s) χ(0,2cχ +E ∞ ) (h˜ 0 )ξ(h˜ p )χ (h 0 )Ψ
ξ(h˜ p )χ (h 0 )Ψ = lim ξ(h˜ p )Ψn = lim χ(0,2cχ +E ∞ ) (h˜ 0 )ξ(h˜ p )Ψn =
(s) ⇒ χ (h 0 ) = χ(0,2cχ +E ∞ ) (h˜ 0 )χ (h 0 ) + O(ε∞ )L(H ) .
We proceed now to prove (109). We first recall some facts we need from [HeRo] and [DiSj]. Given a function χ ∈ C0∞ (R) and a pseudodifferential operator P with symbol in a suitable symbol class (for our aims it is enough to say that this holds for both h p and h˜ p ), then also χ (P) is a pseudodifferential operator, with symbol χ (P) = OpεW (a), a ∼
∞
εjaj,
j=0
a0 = χ ( p0 ), a j =
2 j−1 k=1
d j,k (k) χ ( p0 ), k!
where OpεW denotes the Weyl quantization, p0 is the principal symbol of P and the coefficients d j,k depend on the higher order terms in the expansion of the symbol of P (their precise form is given in [HeRo]). We remark that the previous expressions are local in p0 , and that for h p and h˜ p the symbol is just the principal symbol, and is given by h0 (x, p) :=
1 2 1 p + E(x), h˜ 0 (x, p) := p 2 . 2 2
If we multiply two pseudodifferential operators, the symbol of the product is given by the twisted product of the symbols of the two operators involved: OpεW (a1 ) · OpεW (a2 ) = OpεW (a1 ε a2 ),
j
∞ iε 1 (∇ p · ∇x − ∇ξ · ∇q ) a1 (q, p)a2 (x, ξ ) a 1 ε a 2 ∼ j! 2 j=0
.
(110)
x=q,ξ = p
(s) Applying these formulas to calculate the product ξ c (h˜ p )χ(−E ∞ ,cχ ) (h p ) and using the locality in the principal symbol, we get that all the terms in the expansion of the product vanish, i.e., Eq. (109).
786
L. Tenuta, S. Teufel
Corollary 4. Given a function χ ∈ C0∞ (R) and a σ > 0, we have a
i∂ j vσ (x, ·)
Q M χ (h 0 ) = Q M−1 ξ(h 0 )a
|k|
i∂ j vσ (x, ·) |k|
Q M χ (h 0 ) + O0 (ε∞ ), (111)
where ξ ∈ C0∞ (R) and cξ = 2dχ + E ∞ , where dχ := 2cχ + E ∞ + min{cχ , Λ}, cχ := sup{|k| : k ∈ supp χ }, E ∞ := sup |E(x)|, x∈Rn
and we can choose sup{|k| : k ∈ supp ξ } arbitrarily close to cξ . An analogous statement holds for the creation operator. Proof. Applying twice Lemma 9, the thesis will follow if we prove (111) replacing h 0 with h˜ 0 . Applying Lemma 8, we have χ(−∞,cχ ) (h˜ 0 )Q M χ (h˜ 0 )Ψ = Q M χ (h˜ 0 )Ψ = lim Ψn , n→∞
where Ψn is a linear combination of vectors of the form described in (106), with ϕ = χ(−∞,λ) ( pˆ 2 )ϕ. We have therefore a
i∂ j vσ (x, ·) |k|
Q M χ (h 0 )Ψ = lim a
i∂ j vσ (x, ·) |k|
n→∞
Ψn .
Applying the operator to a vector of the form (106) we get a
i∂ j vσ (x, ·) |k|
N i∂ v (x, k) j σ ϕ ⊗ a(g1 )∗ · · · a(g M )∗ ΩF = ϕ(x) , gl (k) L 2 (R3 ,dk) |k| l=1
∨
·a(g1 )∗ · · · a(gl )∗ · · · a(g M )∗ ΩF , with i∂ j vσ (x, k) |k|
, gl (k)
L 2 (R3 ,dk)
=−
dk
1(σ,∞) (k)ˆ j1 (k)∗ −ik·x j 1 gl (k)κ j . e 2 |k|3/2
The functions gl have by hypothesis compact support in k, with radius uniformly bounded by cχ . For the part depending on x, calculating the Fourier transform with the convolution theorem, we have
i∂ j vσ (x, k) 1(σ,∞) (k)ˆ j1 (k)∗ , gl (k) ( p) = − dk F ϕ(x) gl (k)κ j2 ϕ( ˆ p + k˜ j1 ), |k| |k|3/2
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
787
where k˜ j1 = (0, . . . , k, 0, . . . 0) ∈ R3N , with the k in the entry j1 . Now, since ˆ j1 , gl and ϕˆ have compact support, also the Fourier transform will have compact support in p, with radius bounded by cχ +min{cχ , K } =: dξ . This means that
i∂ j vσ (x, ·) ϕ ⊗ a(g1 )∗ · · · a(g M )∗ ΩF a |k|
i∂ j vσ (x, ·) = χ(−∞,dξ ) (h˜ 0 )a ϕ ⊗ a(g1 )∗ · · · a(g M )∗ ΩF , |k| so, remarking that h˜ 0 is a positive operator and that we can smooth χ[0,dξ ) by a C0∞ function with arbitrarily close support, we prove the thesis. Theorem 3 (Zero order approximation to the time evolution). The following two estimates hold: √ ε t −iHdres −ih 0 εt ε ε (e −e )Q M χ˜ (Hdres )L(H ) = O M+1|t|ε log(σ (ε)−1 ) , (112) ε
t
t
ε )L(H ) = O Q M (e−iHdres ε − e−ih 0 ε )χ˜ (Hdres
√
M+1|t|ε log(σ (ε)−1 ) ,
(113)
˜ = χ˜ . for every χ˜ ∈ C0∞ (R) such that χχ Corollary 5. The subspaces associated to the Q M s are almost invariant with respect to ε ∀M ∈ N, i.e., the dynamics generated by Hdres √ ε ε [e−it Hdres /ε , Q M ]χ(H ˜ dres )L(H ) = O M + 1|t|ε log(σ (ε)−1 ) . (114) Remark 6 (Adiabatic invariance of M-photons dressed particles subspaces). Using the unitary U we can translate all the previous results from the representation space to the original dynamics. This means that if we define the perturbed dressed projectors ε PM := U ∗ Q M U ,
(115)
which satisfy by construction (0) ε − ΠM L(H ) = O(ε log(σ (ε)−1 )), PM we get that [e
−it H ε,σ /ε
,
ε PM ]χ(H ˜ ε,σ )L(H )
=O
√
M + 1|t|ε log(σ (ε))−1 ,
and [e−it H
ε,σ /ε
(0) , Π M ]χ˜ (H ε,σ )L(H ) = O (1 + |t|)ε log(σ (ε)−1 ) , ∀χ˜ ∈ C0∞ (R).
788
L. Tenuta, S. Teufel
For the original dynamics we have then [e−it H
ε /ε
ε , PM ]χ˜ (H ε ) = [(e−it H
ε /ε
−e−it H
ε,σ /ε
ε ), PM ]χ (H ε )+[e−it H
×(χ (H ε ) − χ (H ε,σ )) + [e−it H
ε,σ /ε
ε,σ /ε
ε , PM ]
ε , PM ]χ(H ˜ ε,σ ).
The first term can be bounded by O(σ (ε)1/2 ε−1 ) = O(ε2 ) using Proposition 1 and Eq. (88). The second one by O(σ (ε)1/2 ) using Lemma 3, and the third one has been just estimated above. Putting together these facts, Eq. (21) is proved. For the particular case M = 0, this result, together with the expression of the zero order Hamiltonian h 0 , was already shown in [Te2 ], assuming an infrared regularized interaction and a relativistic dispersion relation for the particles, which automatically implies that they have a bounded maximal velocity, and avoids therefore the introduction of cutoff functions. ε ) with Proof. First of all we remark that, employing Lemma 7, we can replace χ(H ˜ dres χ(h ˜ 0 ) up to terms of order O(ε)L(H ) , which are smaller than the error we want to prove. ε and h are both self-adjoint on H , we can apply the Duhamel formula, Since Hdres 0 0 and obtain ε
(e−it Hdres /ε − e−ith 0 /ε )Q M χ˜ (h 0 ) i t ε ε =− ds ei(s−t)Hdres /ε (Hdres − h 0 )e−ish 0 /ε Q M χ˜ (h 0 ) ε 0 t √ ε = −i ds ei(s−t)Hdres /ε h 1,χ e−ish 0 /ε Q M χ˜ (h 0 ) + O(ε M + 1 log(σ −1 )) 0 t √ ε = −i ds ei(s−t)Hdres /ε h 1,χ Q M χ(h ˜ 0 )e−ish 0 /ε + O(ε M + 1 log(σ −1 )), 0
which implies ε
˜ 0 )L(H ) (e−it Hdres /ε − e−ith 0 /ε )Q M χ(h
≤ |t| · h 1,χ Q M χ˜ (h 0 )L(H ) + O(ε|t| M + 1 log(σ −1 )). √
We proceed now as in the proof of Lemma 7. From Eq. (95) it follows (omitting the terms of the form “ pˆ · Φ”, which can be treated in the same way) 1 1 Φ(i∇x vσ ) · pˆ − χ (Vσ∗ H ε,σ Vσ )Φ J (i∇x vσ ) · pˆ h 1,χ Q M χ˜ (h 0 ) = 2 2 1 − 1 − χ (Vσ∗ H ε,σ Vσ ) Φ J (i∇x vσ ) · pˆ χ (Vσ∗ H ε,σ Vσ ) Q M χ˜ (h 0 ) 2 1 1 = Φ(i∇x vσ ) · pˆ Q M χ(h ˜ 0 ) − χ (Vσ∗ H ε,σ Vσ )Φ J (i∇x vσ ) · pˆ Q M χ(h ˜ 0) 2 2 1 − 1 − χ (Vσ∗ H ε,σ Vσ ) Φ J (i∇x vσ ) · pˆ Q M χ(h ˜ 0) 2 1 − 1 − χ (Vσ∗ H ε,σ Vσ ) Φ J (i∇x vσ ) · p[χ ˆ (Vσ∗ H ε,σ Vσ ) − χ (h 0 )]Q M χ˜ (h 0 ) 2 1 = − [1 − χ (h 0 )]Φ J (i∇x vσ ) · p[χ ˆ (Vσ∗ H ε,σ Vσ ) − χ (h 0 )]Q M χ(h ˜ 0) 2 1 + [χ (Vσ∗ H ε,σ Vσ ) − χ (h 0 )]Φ J (i∇x vσ ) · p[χ ˆ (Vσ∗ H ε,σ Vσ ) − χ (h 0 )]Q M χ(h ˜ 0 ). 2
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
789
Applying Lemma 7, we have immediately that the last term is of order O(ε2 ). We examine therefore more closely the second one. Using Eq. (102) and the corollary to Lemma 9, we get that [χ (Vσ∗ H ε,σ Vσ ) − χ (h 0 )]Q M χ˜ (h 0 )
= (Q M+1 + Q M−1 )ξ(h 0 )[χ (Vσ∗ H ε,σ Vσ ) − χ (h 0 )] · Q M χ˜ (h 0 ) + O(ε2 )L(H ) ,
where the function ξ has a support slightly larger than that of the function χ . Applying again the corollary, we get that, if we choose the support of the function χ˜ sufficiently smaller than the support of the function χ , then h 1,χ Q M χ(h ˜ 0 ) = O(ε2 ),
(116)
so we get the first estimate. For the second one, we apply again the Duhamel formula, but inverting the position of the two unitaries: ε
ε Q M (e−it Hdres /ε − e−ith 0 /ε )χ˜ (Hdres ) t i ε ε ε =− ds Q M ei(s−t)h 0 /ε (Hdres − h 0 )e−is Hdres /ε χ˜ (Hdres ) ε 0 t √ ε ε = −i ds ei(s−t)h 0 /ε Q M h 1,χ χ˜ (Hdres )e−is Hdres /ε + O(ε M + 1 log(σ −1 )). 0
It follows from the proof of Lemma 7 (in particular Eq. (104) and what follows) that √ ε ) = O(ε M + 1 log(σ −1 ))L(H ) , Q M h 1,χ χ˜ (Hdres
so also the second estimate is proved.
Lemma 10. The truncated dressed Hamiltonian (2)
Hdres,χ := h 0 + εh 1,χ + ε2 h 2,χ
(117)
is self-adjoint on H0 for ε small enough. Proof. By construction, the coefficients h i,χ belong to L(H0 , H ), and define symmetric operators on H0 . Moreover h 0 is self-adjoint on H0 , therefore (2) (Hdres,χ − h 0 )Ψ H ≤ Cε(h 0 Ψ H + ψH ), ∀Ψ ∈ H0 .
By a symmetric version of the Kato Theorem ([ReSi2 ], Theorem X.13) the claim follows. Theorem 4 (First order approximation to the time evolution). Given a function χ˜ ∈ C0∞ (R), ε
ε e−it Hdres /ε Q M χ˜ (Hdres ) (2)
ε Q M χ(H ˜ dres ) − iε
t
ε ds ei(s−t)h 0 /ε h 2,OD e−ish 0 /ε Q M χ(H ˜ dres ) + O(ε3/2 |t|)L(H ) (1 − δ M0 ) + O ε2 (|t| + |t|2 ) log(σ −1 ) L(H ) , (118)
=e
−it HD /ε
0
790
L. Tenuta, S. Teufel (2)
where δ M0 = 1 when M = 0, 0 otherwise, [HD , Q M ] = 0 ∀ M, (2)
HD :=
N 1 2 pˆ + E(x) + Hf 2m lε l l=1
−
N ˆ l (k)∗ ˆ j1 (k) ik·(x j −xl ) ε2 1 1 (κ · p ˆ l1 )(κ · pˆ j1 ) e dk 1 3 4 |k|2 l1 , j1 =1 R (l1 = j1 )
+ (κ · pˆ l1 )(κ · pˆ j1 )eik·(x j1 −xl1 ) , with m lε := 1/(1 +
(119)
ε2 2 el ),
κ := k/|k| and 1 l (x)l (y) el := dx dy . 4π R3 ×R3 |x − y|
(120)
The off-diagonal Hamiltonian is defined by h 2,OD := Φσ · ∇ E(x).
(121)
Proof. We split the proof into three parts. In the first one, we show that Eq. (118) is true (2) with a diagonal Hamiltonian H˜ D given by (2) H˜ D :=
N 3N 1 2 ε2 ˆ p + E(x) + H + a(i∂ j vσ )∗ a(i∂ j vσ ) f 2m lε l 4 l=1
−
j=1
N ε2
4
3 l1 , j1 =1 R (l1 = j1 )
dk
ˆ l1 (k)∗ ˆ j1 (k) ik·(x j −xl ) 1 1 (κ · p ˆ l1 )(κ · pˆ j1 ) e |k|2
+ (κ · pˆ l1 )(κ · pˆ j1 )eik·(x j1 −xl1 ) ,
(122)
and an off-diagonal one h˜ 2,OD defined by 3N 1 h˜ 2,OD := Φσ · ∇ E(x) − (∂ j Φ0,l pˆl pˆ j + pˆl pˆ j ∂ j Φ0,l ) 2 j,l=1
+
1 4
3N
a(i∂ j vσ )2 + a(i∂ j vσ )∗ 2 ].
(123)
j=1
In the second part we prove that if one neglects the term 3N ε2 a(i∂ j vσ )∗ a(i∂ j vσ ) 4 j=1
in H˜ D(2) , one gets an error of order O(ε3/2 |t|) in the time evolution. Note that this term is exactly zero if the initial state for the field is the Fock vacuum. In the third part, we prove analogously that we can replace h˜ 2,OD with h 2,OD .
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
791
(2) More specifically, the terms which we neglect in H˜ D and h˜ 2,OD give rise to higher order contributions to the time evolution, although their norm in L(H0 , H ) is not small. This is caused by the fact that they are strongly oscillating in |k|, so that their behavior is determined by the value of the density of states in a neighborhood of k = 0. For all these terms, the density however vanishes for k = 0, uniformly in σ , and this implies that they are of lower order with respect to the leading piece Φσ · ∇ E(x), whose density instead diverges logarithmically in σ (we elaborate on this last observation in a corollary to this theorem). ε by H (2) . By We start showing that we can, up to the desired error, replace Hdres dres,χ (2)
ε , therefore we can apply the Duhamel Lemma 10, Hdres,χ is self-adjoint on H0 like Hdres formula and use Theorem 2 to get (2) (2) ε /ε ε i t (2) −it Hdres,χ /ε −it Hdres ε e −e =− ds ei(s−t)Hdres /ε (Hdres − Hdres,χ )e−is Hdres,χ /ε ε 0 3/2 = O(ε2 log(σ −1 ) ). ε ) by Q χ˜ (h )χ˜˜ (H ε ). Moreover, using Lemma 7, we can replace Q M χ˜ (Hdres M 0 dres (2) Since the diagonal Hamiltonian H˜ D is also self-adjoint on H0 for ε sufficiently small (the proof can be given along the same lines of Lemma 10), we apply again the Duhamel formula, (2)
(2)
˜ ε (e−it Hdres,χ /ε − e−it HD /ε )Q M χ(h ˜ 0 )χ˜˜ (Hdres ) t (2) i ˜ (2) (2) (2) ε =− ds ei(s−t)Hdres,χ /ε (Hdres,χ − H˜ D )e−is HD /ε Q M χ˜ (h 0 )χ˜˜ (Hdres ) ε 0 t (2) ˜ (2) ε = −i ds ei(s−t)Hdres,χ /ε h 1,χ e−is HD /ε Q M χ(h ˜ 0 )χ˜˜ (Hdres ) 0 t (2) ˜ (2) ε − iε ds ei(s−t)Hdres,χ /ε (h 2,χ − h˜ 2,D )e−is HD /ε Q M χ(h ˜ 0 )χ˜˜ (Hdres ), 0
where 1 h˜ 2,D := 4
3N
a(i∂ j vσ )∗ a(i∂ j vσ )
j=1
N ˆ l (k)∗ ˆ j1 (k) ik·(x j −xl ) ε2 1 1 (κ · p ˆ l1 )(κ · pˆ j1 ) − e dk 1 3 4 |k|2 R l1 , j1 =1 (l1 = j1 )
+ (κ · pˆ l1 )(κ · pˆ j1 )eik·(x j1 −xl1 ) . To analyze the first term, we remark that, proceeding as in Lemma 7, one can prove that (2) χ(h ˜ 0 ) − χ˜ ( H˜ D ) = O(ε2 log(σ −1 ))L(H ) , so ˜ (2) /ε
[e−it HD
, χ˜ (h 0 )] = O(ε2 log(σ −1 ))L(H ) ,
792
L. Tenuta, S. Teufel
therefore, with Eq. (116), −i
t
(2)
˜ (2) /ε
ds ei(s−t)Hdres,χ /ε h 1,χ e−is HD
0
t
= −i
ε Q M χ˜ (h 0 )χ˜˜ (Hdres )
(2)
˜ (2) /ε
ds ei(s−t)Hdres,χ /ε h 1,χ Q M χ˜ (h 0 )e−is HD
0
ε ) χ˜˜ (Hdres
+ O(ε2 |t| log(σ −1 ))L(H ) = O(ε2 |t| log(σ −1 ))L(H ) . Concerning the second one, applying once again the Duhamel formula, we have
t
−iε
(2)
˜ (2) /ε
ds ei(s−t)Hdres,χ /ε (h 2,χ − h˜ 2,D )e−is HD
0
t
= − iε 0
ε Q M χ(h ˜ 0 )χ˜˜ (Hdres )
(2)
ε ds ei(s−t)Hdres,χ /ε (h 2,χ − h˜ 2,D )e−ish 0 /ε Q M χ(h ˜ 0 )χ˜˜ (Hdres )
+ O(ε2 |t|2 log(σ −1 ))L(H ) , (2)
so we have to look at ei(s−t)Hdres,χ /ε (h 2,χ − h˜ 2,D )Q M χ(h ˜ 0 ). Following a procedure already employed several times, we first observe that, in the expression for h 2,χ , Eq. (97), we can replace, making an error of order O(ε), χ (Vσ∗ H ε,σ Vσ ) with χ (h 0 ). Using the corollary to Lemma 9, we can then eliminate from h 2,χ Q M χ˜ (h 0 ) all the terms containing (1 − χ (h 0 )). What remains is then (2)
ei(s−t)Hdres,χ /ε (h 2,χ − h˜ 2,D )Q M χ(h ˜ 0) =e
(2)
i(s−t)Hdres,χ /ε
2 i=−2
J − ∂ j Φ0,l pˆl pˆ j −
3N
1 a(i∂ j vσ )2 + a(i∂ j vσ )∗ 2 ] + ΦσJ · ∇ E(x) Q M+i ξ(h 0 ) 4 j=1
N N ˆ l (k)∗ ˆ j1 (k) 1 1 el1 pˆ l21 − dk 1 2 4 |k|2 R3 l1 =1
l1 , j1 =1 (l1 = j1 )
· eik·(x j1 −xl1 ) (κ · pˆ l1 )(κ · pˆ j1 ) + (κ · pˆ l1 )(κ · pˆ j1 )eik·(x j1 −xl1 ) Q M χ˜ (h 0 ) 2 (2)
i + ei(s−t)Hdres,χ /ε Q M+i ξ(h 0 ) χ (h 0 )Φ J (i∇x vσ ) · p, ˆ χ (h 0 )ΦσJ · pˆ Q M χ(h ˜ 0) 2 i=−2
+ O(ε log(σ −1 ))L(H ) . (2)
i(s−t)Hdres,χ /ε Applying Theorem with ei(s−t)h 0 /ε making an error 3, we can now replace e of order O(|t|ε log(σ −1 )).
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
793
Concerning the last term, we get, expanding the commutator and applying again the corollary to Lemma 9,
χ (h 0 )Φ J (i∇x vσ ) · p, ˆ χ (h 0 )ΦσJ · pˆ Q M χ˜ (h 0 ) 1
=
Q M+i ξ(h 0 ) Φ J (i∇x vσ ) · p, ˆ ΦσJ · pˆ Q M χ(h ˜ 0)
i=−1
J pˆl pˆ j Q M χ˜ (h 0 ) + O(ε log(σ −1 ))L(H ) . Q M+i ξ(h 0 ) Φ J (i∂l vσ ), Φσ,l
1
=
i=−1
Using Eq. (96), we can check that this cancels exactly the diagonal terms in h 2,χ−h˜ 2,D , so that (2)
ei(s−t)Hdres,χ /ε (h 2,χ − h˜ 2,D )Q M χ˜ (h 0 ) =e
h 2,OD Q M χ˜ (h 0 ) + O(ε|t| log(σ −1 )).
i(s−t)h 0 /ε ˜
(2)
(2)
We proceed now to show that we can replace H˜ D with HD , up to an error of order O(ε3/2 |t|)L(H ) . ε )Ψ , we Applying repeatedly the Duhamel formula, and putting Ψ := Q M χ(H ˜ dres 0 get 3N (2) (2) iε t −it H˜ D /ε −it HD /ε (e −e )Ψ = − ds ei(s−t)h 0 /ε a(i∂ j vσ )∗ a(i∂ j vσ )e−ish 0 /ε Ψ 4 0 j=1
+ O(ε2 |t|2 )L(H ) . To streamline the presentation, we assume that M = 1, the calculations for M > 1 are basically the same, but more cumbersome. The time integral gives N ϕˆ σ (k) −ith 0 ϕˆ σ (k1 ) 2 e ej dk (κ1 · κ) 1/2 3 |k1 |1/2 |k| R j=1 t · ds eis(|k1 |−|k|)/ε eish p /ε eix j ·(k1 −k) e−ish p /ε Ψ (x, k) 0
= e−ith 0
t
·
N ϕˆσ (k1 ) 2 (κ1 · κ)ϕˆ σ (k) [1 + i(|k1 | − |k|)ε−1 ] e dk 1/2 j 1/2 3 |k1 | |k| [1 + i(|k1 | − |k|)ε−1 ] R j=1
ds eis(|k1 |−|k|)/ε eish p /ε eix j ·(k1 −k) e−ish p /ε Ψ (x, k).
0
Integrating by parts we get t i(|k1 | − |k|)ε−1 ds eis(|k1 |−|k|)/ε eish p /ε eix j ·(k−k1 ) e−ish p /ε Ψ (x, k) =e
0 it (|k1 |−|k|)/ε ith p /ε ix j ·(k−k1 ) −ith p /ε
−
i ε
0
e
t
e
e
Ψ − eix j ·(k−k1 ) Ψ
ds eis(|k1 |−|k|)/ε eish p [h p , eix j ·(k−k1 ) ]e−ish p Ψ,
(124)
794
L. Tenuta, S. Teufel
where the commutator is of order O(ε) when applied to functions of bounded kinetic energy, so that the right-hand side is uniformly bounded in ε. We have now to put this expression back in (124) and estimate the single terms. We show how to do this for the first one, the others being entirely analogous. We ignore the unitary on the left, which does not change the norm, so we have to consider ϕˆσ (k1 ) 2 (κ1 · κ)ϕˆ σ (k) e dk 1/2 j 1/2 |k1 | |k| [1 + i(|k1 | − |k|)ε−1 ] R3 t ds eis(|k1 |−|k|)/ε eish p /ε eix j ·(k1 −k) e−ish p /ε Ψ (x, k) · 0
for one fixed j. Using twice the Cauchy-Schwarz inequality we get (we put for brevity f (k) := e j ϕ(k)|k|−1/2 ) · · ·H =
2
·
R3N t
dk
0
dx
dk1 |· · ·| ≤ |t| 2
R3
dx
dk1 dk | f (k1 )|2
2 ds eish p eix j ·(k−k1 ) e−ish p Ψ (x, k)
| f (k)|2 1 + (|k1 | − |k|)2 ε−2
| f (k, λ) f (k1 , λ1 )|2 1 + (|k1 | − |k|)2 ε−2 Λ/ε Λ/ε k1 k 4 2 2 ≤ Cε |t| Ψ H dk1 dk = O(ε|t|2 Ψ 2H ). 2 1 + (k 1 − k) 0 0 = |t|2 Ψ 2H
dk1 dk
We examine now separately the last two terms in h˜ 2,OD . For the first we get t ε iε ds ei(s−t)h 0 /ε ∂ j Φ0,l pˆl pˆ j e−ish 0 /ε Q M χ˜ (Hdres )Ψ0 0
t ∂l ∂ j v(x, k) ∗ ∂l ∂ j v(x, k) ε pˆ j pˆl e−ish 0 /ε Ψ = ie−ith 0 /ε √ ds eish 0 /ε a +a |k| |k| 2 0
t ˆ j (k) ∗ ε = − ie−ith 0 /ε √ ds eish 0 /ε a κ j2 κl2 eik·x j1 1 1/2 |k| 2 0
ˆ (k) j pˆ ( j1 , j2 ) pˆ ( j1 ,l2 ) e−ish 0 /ε Ψ. + a κ j2 κl2 eik·x j1 1 1/2 |k| The part with the annihilation operator gives
t ˆ j1 (k) ik·x j1 −ith 0 /ε ε ish 0 /ε pˆ ( j1 , j2 ) pˆ ( j1 ,l2 ) e−ish 0 /ε Ψ e ds e a κ j2 κl2 e √ |k|1/2 2 0 t M−1 ˆ j (k)∗ ε √ = e−ith 0 /ε √ M ds eis µ=1 |kµ |/ε eish p /ε dk e−ik·x j1 1 1/2 (κ · pˆ j1 )2 |k| 2 R3 0 M−1
·e−ish p /ε e−is|k|/ε e−is µ=1 |kµ |/ε Ψ (x; k, k1 , . . . , k M−1 ) √ ˆ j1 (k)∗ t −ith 0 /ε ε M dk ds eish p /ε e−ik·x j1 (κ · pˆ j1 )2 e−ish p /ε =e √ |k|1/2 0 2 R3 ·e−is|k|/ε Ψ (x; k, k1 , . . . , k M−1 )
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
t ˆ j (k)∗ ε √ −1 (1 − i|k|ε = e−ith 0 /ε √ M dk 1/2 1 ) ds eish p /ε −1 ) 3 |k| (1 − i|k|ε 2 R 0 ·e−ik·x j1 (κ · pˆ j1 )2 e−ish p /ε e−is|k|/ε Ψ (x; k, k1 , . . . , k M−1 ).
795
(125)
Integrating by parts we have t ds eish p /ε e−ik·x j1 (κ · pˆ j1 )2 e−ish p /ε e−is|k|/ε Ψ (x; k, k1 , . . . , k M−1 ) −i|k|ε−1 0 t = ds eish p /ε e−ik·x j1 (κ · pˆ j1 )2 e−ish p /ε ∂s e−is|k|/ε Ψ (x; k, k1 , . . . , k M−1 ) =e
0 ith p /ε −ik·x j1
e
(κ · pˆ j1 )2 e−ith p /ε e−it|k|/ε Ψ (x; k, k1 , . . . , k M−1 )
− e−ik·x j1 (κ · pˆ j1 )2 Ψ (x; k, k1 , . . . , k M−1 ) i t − ds eish p /ε [h p , e−ik·x j1 (κ · pˆ j1 )2 ]e−ish p /ε e−is|k|/ε Ψ (x; k, k1 , . . . , k M−1 ) ε 0 = eith p /ε e−ik·x j1 (κ · pˆ j1 )2 e−ith p /ε e−it|k|/ε Ψ (x; k, k1 , . . . , k M−1 ) − e−ik·x j1 (κ · pˆ j1 )2 Ψ (x; k, k1 , . . . , k M−1 ) t −i ds eish p /ε − i∇x j1 (e−ik·x j1 ) · pˆ j1 − ε∆x (e−ik·x j1 ) (κ · pˆ j1 )2 0 + e−ik·x j1 [E(x), (κ · pˆ j1 )2 ] e−ish p /ε e−is|k|/ε Ψ (x; k, k1 , . . . , k M−1 ) = eith p /ε e−ik·x j1 (κ · pˆ j1 )2 e−ith p /ε e−it|k|/ε Ψ (x; k, k1 , . . . , k M−1 ) − e−ik·x j1 (κ · pˆ j1 )2 Ψ (x; k, k1 , . . . , k M−1 ) t −ik·x j1 −ik·x j1 ish p /ε e (κ · pˆ j1 )2 +i ds e κ · pˆ j1 − ε|k|e 0 −ik·x j1 2 +e [E(x), (κ · pˆ j1 ) ] e−ish p /ε e−is|k|/ε |k|Ψ (x; k, k1 , . . . , k M−1 ).
(126)
We have to put now this expression back into (125) and estimate the result. We show how to proceed for the most singular term, i. e. the one containing i 0
t
ds eish p /ε e−ik·x j1 (κ · pˆ j1 )3 e−ish p /ε e−is|k|/ε |k|Ψ (x; k, k1 , . . . , k M−1 ),
the others can be treated in the same way. Putting this term back in (125), and ignoring the unitary e−ith 0 /ε , which does not change the norm, we have to estimate in the end t ˆ j1 (k)∗ iε √ M dk 1/2 ds eish p /ε e−ik·x j1 (κ · pˆ j1 )3 e−ish p /ε √ |k| (1 − i|k|ε−1 ) 0 2 R3 ·e−is|k|/ε |k|Ψ (x; k, k1 , . . . , k M−1 ).
796
L. Tenuta, S. Teufel
Applying the Cauchy-Schwarz inequality we get |ˆ j1 (k)|2 ε2 |t|2 M dk |· · ·|2 ≤ 2 |k|(1 + |k|2 ε−2 ) R3 t dk ds eish p /ε e−ik·x j1 (κ · pˆ j1 )3 e−ish p /ε e−is|k|/ε |k| · R3
0
2 ·Ψ (x; k, k1 , . . . , k M−1 ) . The first integral gives Λ Λ/ε |ˆ j1 (k)|2 k k 2 = C dk dk = Cε 2 ε −2 ) 2 ε −2 3 |k|(1 + |k| 1 + k 1 + k2 R 0 0 = O(ε2 log(1/ε)), so, calculating the norm we get · · ·2H ≤ Cε4 log(1/ε)|t|2 M
t
ds eish p /ε e−ik·x j1 ·
0
2 ·(κ · pˆ j1 )3 e−ish p /ε e−is|k|/ε |k|Ψ (x; k, k1 , . . . , k M−1 )H t 2 = Cε4 log(1/ε)|t|2 M ds (κ · pˆ j1 )3 e−ish p /ε |k|Ψ (x; k, k1 , . . . , k M−1 )H 0 t 2 ε ≤ Cε4 log(1/ε)|t|2 M ds (κ · pˆ j1 )3 e−ish p /ε |k|Q M χ(H ˜ dres )L(H ) Ψ0 2H , 0
which shows that the norm of this term is of order O(ε2 |t| log(1/ε)) in L(H ). The part with the creation operator gives
t ˆ j1 (k) ∗ ik·x j1 −ith 0 /ε ε ish 0 /ε e ds e a κ j2 κl2 e pˆ ( j1 , j2 ) pˆ ( j1 ,l2 ) e−ish 0 /ε Ψ √ |k|1/2 2 0 t M+1 M+1 ˆ j (kµ ) 1 −ith 0 /ε ε =e ds eis µ=1 |kµ |/ε eish p /ε √ κµ, j2 κµ,l2 eiκµ ·x j1 1 1/2 √ |kµ | 2 0 M + 1 µ=1 M+1
· pˆ ( j1 , j2 ) pˆ ( j1 ,l2 ) e−is ν=1(ν =µ) |kν |/ε e−ish p /ε Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ) ˆ j1 (kµ ) t ε = e−ith 0 /ε √ ds eish p /ε eiκµ ·x j1 (κµ · pˆ j1 )2 e−ish p /ε e−is|kµ |/ε 2(M + 1) |kµ |1/2 0 ·Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ) t ˆ j1 (kµ ) ε −1 (1 − i|k |ε ) ds eish p /ε eiκµ ·x j1 = e−ith 0 /ε √ µ 2(M + 1) |kµ |1/2 (1 − i|kµ |ε−1 ) 0 ·(κµ · pˆ j )2 e−ish p /ε e−is|kµ |/ε Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ). 1
We can now integrate by parts using Eq. (126). As for the annihilation part, we examine just one term, ˆ j1 (kµ )|kµ |1/2 t ε ds eish p /ε eikµ ·x j1 (κµ · pˆ j1 )3 e−ish p /ε √ 2(M + 1) (1 − i|kµ |ε−1 ) 0 ·e−is|kµ |/ε Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ).
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
797
The norm squared is given by |ˆ j1 (kµ )|2 |kµ | t ds e−is|kµ |/ε eish p /ε eikµ ·x j1 d x dk1 . . . dk M+1 1 + |kµ |2 ε−2 0 2 ·(κµ · pˆ j1 )3 e−ish p /ε Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ) |ˆ j1 (εkµ )|2 |kµ | t ε6 ds e−is|kµ | eish p /ε eiεkµ ·x j1 d x dk1 . . . dk M+1 2(M + 1) 1 + |kµ |2 0 2 ·(κµ · pˆ j1 )3 e−ish p /ε Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ) |ˆ j1 (εkµ )|2 |kµ | t ε6 |t| ds e−is|kµ | eish p /ε eiεkµ ·x j1 dk1 . . . dk M+1 2(M + 1) 1 + |kµ |2 0 2 ·(κµ · pˆ j1 )3 e−ish p /ε Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ) |ˆ j1 (εkµ )|2 |kµ | t ε6 |t| ds e−is|kµ | eish p /ε eiεkµ ·x j1 dk1 . . . dk M+1 2(M + 1) 1 + |kµ |2 0 2 ·(κµ · pˆ j1 )3 e−ish p /ε Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ) L 2 (R3N ,d x) |ˆ j (εkµ )|2 |kµ | t ε6 |t| ds (κµ · pˆ j1 )3 e−ish p /ε dk1 . . . dk M+1 1 2(M + 1) 1 + |kµ |2 0 2 ·Ψ (x; k1 , . . . , kˆµ , . . . , k M+1 ) L 2 (R3N ,d x) t 0 /ε |kµ |3 Cε6 |t| dΩ d|kµ | ds (κµ · pˆ j1 )3 e−ish p /ε µ 2 2(M + 1) 0 1 + |kµ | 0 2 ˆ ·Ψ (x; k1 , . . . , kµ , . . . , k M+1 ) H t Cε6 |t|02 −2 [ε − log(ε−2 + 1)] dΩµ ds (κµ · pˆ j1 )3 e−ish p /ε 4(M + 1) 0 2 ·Q M χ˜ (H ε )Ψ0 ,
ε2 2(M + 1)
=
≤
=
=
≤
=
dres
H
so we get that this term gives a contribution of order O(ε2 |t|) in the norm of L(H ). We separate also in the second term of h˜ 2,OD the annihilation and the creation part. For the annihilation part we get
−
iε 4
t
ds ei(s−t)h 0 /ε
0
3N j=1
a(i∂ j vσ )2 e−ish 0 /ε Ψ
ˆ σj (ξ )∗ ˆ σj (ζ )∗ iε −ith 0 /ε =− M(M − 1)e dξ dζ ξˆ j2 ζˆ j2 1 1/2 11/2 4 |ξ | |ζ | j1 , j2 t · ds e−is(|ξ |+|ζ |)/ε eish p /ε e−i(ξ +ζ )·x j1 e−ish p /ε Ψ. 0
798
L. Tenuta, S. Teufel
We proceed now in the same way as we did for the first term. Integrating by parts we get −i(|ξ | + |ζ |) t ds e−is(|ξ |+|ζ |)/ε eish p /ε e−i(ξ +ζ )·x j1 e−ish p /ε Ψ ε 0 = e−it (|ξ |+|ζ |)/ε eith p /ε e−i(ξ +ζ )·x j1 e−ith p /ε Ψ − e−i(ξ +ζ )·x j1 Ψ t ds e−is(|ξ |+|ζ |)/ε eish p /ε e−i(ξ +ζ )·x j1 [(ξ + ζ ) · pˆ j1 − ε|ξ + ζ |2 ]e−ish p /ε Ψ. − 0
As in the previous case, we examine just one term, ˆ σj (ξ )∗ ˆ σj1 (ζ )∗ iε M(M − 1) dξ dζ ξˆ j2 ζˆ j2 1/2 1/21 4 |ξ | |ζ | [1 − i(|ξ | + |ζ |)ε−1 ] t ds e−is(|ξ |+|ζ |)/ε eish p /ε e−i(ξ +ζ )·x j1 (ξ + ζ ) · pˆ j1 e−ish p /ε · · 0
·Ψ (x; ξ, ζ, k1 , . . . , k M−2 ), the others can be treated in the same way. Using the Cauchy-Schwarz inequality we get that, independently of σ , Λ Λ |ξ ||ζ |(|ξ |2 + |ζ |2 ) · · ·2H ≤ Cε2 |t|M(M − 1) d|ξ | d|ζ | 1 + (|ξ | + |ζ |)2 ε−2 σ σ 2 t ξ +ζ −ish p /ε ε · ds Q N χ˜ (Hdres ) Ψ0 2H = O(ε4 |t|2 )H . |ξ + ζ | · pˆ j1 e L(H )
0
The creation part can be estimated in a way entirely analogous to the one already employed for the first term of h˜ 2,OD . Corollary 6. The radiated piece (i. e. the piece of the wave function which makes a transition between the almost invariant subspaces) for a system starting in the Fock vacuum is given by ε
Ψ˜ rad (t) := (1 − Q 0 )e−i ε Hdres ψ(x)ΩF t
σ (ε) t N ε −ith 0 /ε ˆ j (k) −√ e κ· ds eis|k|/ε OpεW ( x¨ cj (s; x, p))ψ(x) |k|3/2 2 0 j=1
˜ ε), + R(t,
(127)
where ˜ ε)H ≤ Cε2 log(ε−1 )(|t| + |t|2 )(ψH + |x|ψH + | p|ψ R(t, ˆ H ), and x cj is the solution to the classical equations of motion x¨ cj (s; x, p) = −∇ x j E(x c (s; x, p)), x cj (0; x, p) = x j ,
x˙ cj (0; x, p) = p j ,
j = 1, . . . , n.
(128)
We get the leading order of the radiated piece corresponding to the original Hamiltonian H ε , for a system starting in the approximate dressed vacuum Ωσ (ε) (x), applying to this wave function the dressing operator Vσ (ε) (x).
Effective Dynamics for Particles Coupled to a Quantized Scalar Field
799
Proof. Applying Eq. (118) for the case M = 0 we get at the leading order t ε /ε ⊥ −it Hdres Q0 e ψ(x)ΩF = −iε ds ei(s−t)h 0 /ε h 2,OD e−ish 0 /ε ψ(x)ΩF 0 t ε i∇x vσ (x, k) −ish p /ε e = −√ ds ei(s−t)h p /ε ei(s−t)|k|/ε ∇ E(x) · ψ(x) |k| 2 0 t N σ ε ˆ j (k) =√ κ· ds ei(s−t)|k|/ε ei(s−t)h p /ε eik· x j ∇ x j E(x)e−ish p /ε ψ(x) 2 j=1 |k|3/2 0 t N σ ε ˆ j (k) =√ κ · ds ei(s−t)|k|/ε ei(s−t)h p /ε (eik· x j − 1)∇ x j E(x)e−ish p /ε ψ(x) 2 j=1 |k|3/2 0 t N σ ε ˆ j (k) +√ κ · ds ei(s−t)|k|/ε ei(s−t)h p /ε ∇ x j E(x)e−ish p /ε ψ(x) 3/2 |k| 2 j=1 0 σ
ε ˆ j (k) =√ κ· 2 j=1 |k|3/2 N
0
t
ds ei(s−t)|k|/ε ei(s−t)h p /ε (eik· x j − 1)∇ x j E(x)e−ish p /ε ψ(x)
t N σ ε ˆ j (k) − e−ith 0 √ κ · ds eis|k|/ε OpεW ( x¨ j (s; x, p))ψ(x) 2 j=1 |k|3/2 0 + O(ε2 |t|)L(H ) ψ L 2 (R3n ) , where we have used Egorov’s theorem to approximate eish p /ε ∇x j E(x)e−ish p /ε (see, e. g., [Ro]). To end the proof we have to show that the norm of the first term is small. The procedure to employ is identical to the one applied several times in the proof of Theorem 4: First integrate by parts with respect to s and then estimate the resulting terms. For the sake of completeness we show how to estimate one of these terms t εˆ σj (k)κ ds ei(s−t)|k|/ε ei(s−t)h p /ε (eik· x j − 1)∇ x j E(x)e−ish p /ε ψ(x), · √ 2|k|3/2 (1 + i|k|ε−1 ) 0 because the others have an analogous structure. Its norm satisfies · · ·2H ≤
ε2 |t| 2
dk
|ˆ σj (εk)|2 |k|3 (1 + |k|2 )
0
t
2 iεk· x −ish p /ε j − 1)∇ (e ds E e ψ xj
L 2 (R3N )
2 Λ/ε t |k| |x j |∇ x E e−ish p /ε ψ ˜ 4 |t| ≤ Cε d|k| ds j 2 3N 2 1 + |k| 0 σ L (R ) 2 −2 t 1 + Λε |x j |e−ish p /ε ψ ˜ 4 |t| sup |∇ x j E(x)| log = Cε ds 2 3N 2 1 + σ (ε) 0 L (R ) x∈R3N
−2 1 + Λε (| pˆ j |ψ2 + |x j |ψ2 + ψ2 ), ≤ Cε4 |t|(|t| + |t|2 ) log 1 + σ (ε)2
where in the last inequality we have applied Theorem 2.1 from [RaSi].
800
L. Tenuta, S. Teufel
Remark 7. The norm squared of the leading part of the radiated piece is 2ε2 3π ≥ ≥
2 t
N |ϕˆ σ (k)|2 −isk/ε c OpW ¨ x ds e (s; x, p) ψ(x) ε j 3 k 0 C
∞
dk
dx 0
ε2 12π 4 ε2 12π 4
j=1
2 t
N Λ 1 W −isk/ε c x¨ j (s; x, p) ψ(x) dk Opε ds e dx 3 k σ 0 C
2 t
N Λ 1 W −isk c . ¨ Op x dk ds e (s; x, p) ψ(x) dx ε j 3 k σ (ε)ε−1 0 C
j=1
j=1
The symbol which appears in the Weyl quantization
t
ds e−isk
0
N
x¨ cj (s; x, p)
j=1
is independent of ε and for k = 0 is a non-null function, N N [ x˙ cj (t; x, p) − x˙ cj (0, x, p)] = [ x˙ cj (t; x, p) − p j ], j=1
j=1
therefore the corresponding operator will be also non-null. Therefore, we expect for a generic state ψ that t
n W −isk c ¨ Op inf x ds e (s; x, p) ψ(x) ε j 0≤k≤Λ 0
j=1
L 2 (R3N )⊗C3
> 0.
(129)
Then the norm of the radiated piece will be bigger than t
N W −isk c inf Opε x¨ j (s; x, p) ψ(x) ds e 0≤k≤Λ 0
·
ε2 12π 4
j=1
Λ
1 dk k σ (ε)ε−1
1/2
L 2 (R3n )⊗C3
·
= O ε log(εσ (ε)−1 ) ,
which gives a lower bound on the transition almost of the same order of the upper bound. Remark 8. The radiated energy, defined in Eq. (31), can be written at the leading order as ε
ε
E rad (t) = e−it Hdres /ε ψd (x)ΩF , (1 ⊗ Hf )e−it Hdres /ε ψd (x)ΩF ,
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where ψd is defined in (30). Using the expression for the radiated piece we get ε
ε
−it Hdres /ε −it Hdres /ε E rad (t) = Q ⊥ ψd ΩF , (1 ⊗ Hf )Q ⊥ ψd ΩF 0e 0e σ (ε)∗ σ (ε) t N t 2 ˆ j ˆ i ε ∼ dk ds ds ei(s−s )|k|/ε = 2 2 |k| R3 0 0 i, j=1
·κ · OpεW ( x¨ ic (s ))ψd , κ · OpεW ( x¨ cj (s))ψd L 2 (R3N ) t t N ε ε2 = ei e j ds ds (ei(s−s )Λ/ε − 1) 2 12π i(s − s ) 0 0 i, j=1
·ψd , OpεW ( x¨ ic (s ) · x¨ cj (s))ψd L 2 (R3N ) , where we have inserted the explicit expression of the form factor given in (3) and used the product formula (110) at leading order. The radiated power is then Prad (t) =
N d ε3 t sin[(t − s)Λ/ε] E rad (t) = · ds dt 6π 2 t −s 0 i, j=1
· ψd , OpεW ( x¨ ic (t) · x¨ cj (s))ψd L 2 (R3N ) , which converges formally to the expression given in (32) when ε → 0+ . Corollary 7. Let ε
ε
ω(t) := e−it Hdres /ε ω0 eit Hdres /ε , ε )H ), the Banach space of trace class operators on where ω0 ∈ I1 (Q M χ˜ (Hdres ε Q M χ˜ (Hdres )H , and let ωp be the partial trace over the field states
ωp (t) := tr F ω(t), then ωp (t) = e
(2)
−it HD,p /ε
it H
(2)
/ε
ωp (0)e D,p + O(ε3/2 |t|)I1 (L 2 (R3N )) (1 − δ M0 ) + O(ε2 |t| log(σ (ε)−1 ))I1 (L 2 (R3N )) + O(ε2 |t|2 log(σ (ε)−1 ))I1 (L 2 (R3N )) ,
where (2) HD,p
N 1 2 pˆ + E(x) := 2m lε l l=1
N ˆ l (k)∗ ˆ j1 (k) ik·( x j − x l ) ε2 1 1 (κ · p ˆ l1 )(κ · pˆ j1 ) − e dk 1 4 |k|2 R3 l1 , j1 =1 (l1 = j1 )
+ (κ · pˆ l1 )(κ · pˆ j1 )eik·( x j1 − x l1 ) , and I1 (L 2 (R3N )) denotes the space of trace class operators on L 2 (R3N ).
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Proof. The proof follows from the following three facts: 1. the term of order ε in Eq. (118) is off-diagonal with respect to the Q M ; (2) (2) 2. the diagonal Hamiltonian HD , defined in (119), is equal to HD,p ⊗ 1 + 1 ⊗ Hf , so we have that (2) (2) (2) (2) −it HD,p /ε it H /ε tr F (ω0 )e D,p ; tr F e−it HD /ε ω0 eit HD /ε = e 3. the following well known inequality, which holds for any Hilbert space H : ABI1 (H ) ≤ AI1 (H ) · BL(H ) . Theorem 5. Given a macroscopic observable for the particles, OpεW (a), where a is a smooth function bounded with all its derivatives, and a density matrix ε χ (H ε )H ) whose time evolution is defined by ω ∈ I1 (PM ω(t) := e−it H then
ε /ε
ωeit H
ε /ε
,
(130)
(2) (2) W −it HD,p it HD,p W tr F ω(0)e ˜ tr H Opε (a) ⊗ 1F ω(t) = tr L 2 (R3N ) Opε (a)e +O(ε3/2 |t|)(1 − δ M0 ) + O ε2 (|t| + |t|2 ) log(σ (ε)−1 ) , (131)
where ω(0) ˜ := Vσ∗(ε) ωVσ (ε) .
(132)
Proof. First of all we observe that, using Proposition 1 and Lemma 3, we have
ε,σ (ε) /ε ωσ (ε) tr H OpεW (a) ⊗ 1F ω(t) = tr H OpεW (a) ⊗ 1F e−it H
ε,σ (ε) /ε + O(σ (ε)1/2 ) + O(σ (ε)ε−1 ), ·eit H ε χ (H ε,σ (ε) )H ). By the definition of the dressed Hamiltonian and where ωσ (ε) ∈ I1 (PM the cyclicity of the trace we have then at the leading order
tr H OpεW (a) ⊗ 1F ω(t) = O(σ (ε)1/2 ) + O(σ (ε)ε−1 )
ε ε +tr H U OpεW (a) ⊗ 1F U ∗ e−it Hdres /ε U ωσ (ε) U ∗ eit Hdres /ε .
The transformed observable, using the definition of U and Lemma 7, is given by U OpεW (a) ⊗ 1F U ∗ = Vσ∗ OpεW (a) ⊗ 1F Vσ + iεχ (h 0 )ΦσJ · pV ˆ σ∗ W J · Opε (a) ⊗ 1F Vσ + iε 1 − χ (h 0 ) Φσ · pχ ˆ (h 0 )Vσ∗ OpεW (a) ⊗ 1F Vσ −iεVσ∗ OpεW (a) ⊗ 1F Vσ χ (h 0 ) pˆ · ΦσJ − iεVσ∗ OpεW (a) ⊗ 1F Vσ 1 − χ (h 0 ) pˆ · ΦσJ χ (h 0 ) + O(ε2 log(σ −1 )).
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The operator Vσ can be considered as the Weyl quantization of the operator valued symbol (x, p) → e−iΦ(ivσ (x,·)) ∈ L(F ). It is not in the standard symbol classes because the derivative is an unbounded operator, but if we multiply it by Q M we get a smooth bounded symbol. In the calculation of the trace Vσ is always multiplied by Q M or Q M±1 , therefore we can proceed as if it were in a standard symbol class. A simple application of the product rule for pseudodifferential operators (Eq. (A.11) [Te3 ]) gives then Vσ∗ OpεW (a) ⊗ 1F Vσ = OpεW (a) ⊗ 1F + Vσ∗ [ OpεW (a) ⊗ 1F , Vσ ] = OpεW (a) ⊗ 1F − iεVσ∗ OpεW {a, Vσ } + O(ε3 )L(H ) , where {·, ·} denotes the Poisson bracket. We get therefore {a, Vσ } = (∇ p a) · iΦ(i∇x vσ )Vσ and Vσ∗ OpεW (a) ⊗ 1F Vσ = OpεW (a) ⊗ 1F + εΦ(i∇x vσ ) OpεW (∇ p a) ⊗ 1F + O(ε2 log(σ −1 )).
Using this expression we have U OpεW (a) ⊗ 1F U ∗ = OpεW (a) ⊗ 1F + εΦ(i∇x vσ ) OpεW (∇ p a) ⊗ 1F ˆ (h 0 ) OpεW (a) ⊗ 1F + iεχ (h 0 )ΦσJ · pˆ OpεW (a) ⊗ 1F + iε 1 − χ (h 0 ) ΦσJ · pχ − iε OpεW (a) ⊗ 1F χ (h 0 ) pˆ · ΦσJ − iε OpεW (a) ⊗ 1F 1 − χ (h 0 ) pˆ · ΦσJ χ (h 0 ) + O(ε2 log(σ −1 )). All the terms of order ε in the previous expression are off-diagonal with respect to the Q M s, and the same holds for the term of order ε in (118). Therefore, they all vanish when we calculate the trace. Using point 2 and 3 of the last corollary we get then (131) with ω(0) ˜ = U ωσ (ε) U ∗ . Using again Lemma 3 and the fact that the terms of order ε in the expansion of U are off-diagonal we get in the end also (132). Acknowledgement. We thank Alessandro Pizzo for showing us the strategy to prove Proposition 1 and the German Research Foundation DFG for financial support.
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References [AbFr1 ] [AbFr2 ] [Ar] [AvEl1 ] [AvEl2 ] [Ber] [Be] [Bo] [Da] [De] [DiSj] [Fr] [FGS] [HeRo] [Hi] [KuSp1 ] [KuSp2 ] [LaLi] [LMS] [MaSo] [Ne] [NeSo] [Pi] [PST1 ] [PST2 ] [RaSi] [ReSi1 ] [ReSi2 ] [Ro] [So]
Abou-Salem, W.K., Fröhlich, J.: Adiabatic theorems and reversible isothermal processes. Lett. Math. Phys. 72, 153–163 (2005) Abou-Salem, W.K., Fröhlich, J.: Adiabatic theorems for quantum resonances. Commun. Math. Phys. 273(3), 651–675 (2007) Arai, A.: Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation. Rev. Math. Phys. 13, 1075–1094 (2001) Avron, J.E., Elgart, A.: Adiabatic theorem without a gap condition. Commun. Math. Phys. 203, 445–463 (1999) Avron, J.E., Elgart, A.: Smooth adiabatic evolutions with leaky power tails. J. Phys. A: Math. Gen. 32, L537–L546 (1999) Berry, M.V.: Histories of adiabatic quantum transitions. Proc. R. Soc. Lond. A 429, 61–72 (1990) Betz, V.: Gibbs measures relative to Brownian motion and Nelson’s model. Dissertation, TU München, 2002 Bornemann, F.: Homogenization in time of singularly perturbed mechanical systems. Lecture Notes in Mathematics 1687, Berlin-Heidelberg-New York: Springer, 1998 Davies, E.B.: Particle-boson interactions and the weak coupling limit. J. Math. Phys. 20, 345–351 (1978) Derezi´nski, J.: Van Hove hamiltonians—exactly solvable models of the infrared and ultraviolet problem. Ann. Henri Poincaré 4, 713–738 (2003) Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical Limit. London Mathematical Society Lecture Note Series 268, Cambridge: Cambridge University Press, 1999 Fröhlich, J.: On the infrared problem in a model of scalar electrons and massless scalar bosons. Ann. Ist. H. Poincaré 19, 1–103 (1973) Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré 3, 107–170 (2002) Helffer, B., Robert, D.: Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles. J. Funct. Anal. 53, 246–268 (1983) Hiroshima, F.: Weak coupling limit and removing an ultraviolet cutoff for a hamiltonian of particles interacting with a quantized scalar field. J. Math. Phys. 40, 1215–1236 (1999) Kunze, M., Spohn, H.: Slow motion of charges interacting through the Maxwell field. Commun. Math. Phys. 212, 437–467 (2000) Kunze, M., Spohn, H.: Post-coulombian dynamics at order c−3. J. Nonlinear Sci. 11, 321–396 (2001) Landau, L.D., Lifshitz, E.M.: The classical theory of fields. 4th rev. English ed., London: Pergamon Press, 1975 Lörinczi, J., Minlos, R.A., Spohn, H.: Infrared regular representation of the three-dimensional massless Nelson model. Lett. Math. Phys. 59, 189–198 (2002) Martinez, A., Sordoni, V.: On the time-dependent Born-Oppenheimer approximation with smooth potential. Comptes Rendu Math. 334, 185–188 (2002) Nelson, E.: Interaction of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964) Nenciu, G., Sordoni, V.: Semiclassical limit for multistate Klein-Gordon systems: almost invariant subspaces, and scattering theory. J. Math. Phys. 45, 3676–3696 (2004) Pizzo, A.: Scattering of an infraparticle: the one particle sector in Nelson’s massless model. Ann. Henri Poincaré 6, 553–606 (2005) Panati, G., Spohn, H., Teufel, S.: Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7, 145–204 (2003) Panati, G., Spohn, H., Teufel, S.: The Time-Dependent Born-Oppenheimer Approximation, Mathematical Modelling and Numerical Analysis, special issue on molecular modelling, 2007 Radin, C., Simon, B.: Invariant domains for the time-dependent Schrödinger equation. J. Diff. Eq. 29, 289–296 (1978) Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I:Functional Analysis. New York: Academic Press, 1972 Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness. New York: Academic Press, 1975 Robert, D.: Autour de l’Approximation Semi-Classique. Progress in Mathematics 68, BaselBoston: Birkhäuser, 1987 Sordoni, V.: Reduction scheme for semiclassical operator-valued Schrödinger type equation and application to scattering. Commun. Part. Diff. Eq. 28, 1221–1236 (2003)
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Spohn, H.: Dynamics of charged particles and their radiation field. Cambridge: Cambridge University Press, 2004 Teufel, S.: A note on the adiabatic theorem without gap condition. Lett. Math. Phys. 58, 261–266 (2001) Teufel, S.: Effective n-body dynamics for the massless Nelson model and adiabatic decoupling without spectral gap. Ann. H. Poincaré 3, 939–965 (2002) Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics, Lecture Notes in Mathematics 1821, Berlin-Heidelberg-New York: Springer, 2003
Communicated by H. Spohn
Commun. Math. Phys. 280, 807–829 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0449-x
Communications in
Mathematical Physics
Fredholm Determinants and the Statistics of Charge Transport J. E. Avron1 , S. Bachmann2 , G. M. Graf2 , I. Klich3 1 Department of Physics, Technion, 32000 Haifa, Israel 2 Theoretische Physik, ETH-Hönggerberg, 8093 Zürich, Switzerland. E-mail: [email protected] 3 Condensed Matter Department, Caltech, MC 114-36, Pasadena, CA 91125, USA
Received: 1 May 2007 / Accepted: 20 August 2007 Published online: 13 March 2008 – © Springer-Verlag 2008
Abstract: Using operator algebraic methods we show that the moment generating function of charge transport in a system with infinitely many non-interacting Fermions is given by a determinant of a certain operator in the one-particle Hilbert space. The formula is equivalent to a formula of Levitov and Lesovik in the finite dimensional case and may be viewed as its regularized form in general. Our result embodies two tenets often realized in mesoscopic physics, namely, that the transport properties are essentially independent of the length of the leads and of the depth of the Fermi sea. 1. Introduction Models of physical systems are often formulated with the help of one or a few parameters which guarantee that whatever one computes is well defined and finite while, at the same time, are believed not to affect properties of physical interest. Examples are: the number of particles in a macroscopic system, and the lattice spacing (ultraviolet cutoffs) in the study of critical phenomena. The theory of transport in mesoscopic systems has two such parameters: the length of the incoming leads that connect to the system and the depth of the Fermi sea. The independence of the length of the leads is the statement that well designed experiments measure the transport properties of the mesoscopic system and are independent of the measuring circuit. The independence of the depth of the Fermi sea expresses the irrelevance for transport of electrons that are buried deep in the Fermi sea, since in most situations they can not be excited above it. In this sense there is freedom from both the volume and the ultraviolet scale. See [21] for a numerical investigation of these properties. One strategy to address this type of behavior is to consider idealized systems where the parameters are taken to be infinitely large. The limiting idealized system comes with the price tag that expressions for physical quantities that are otherwise guaranteed to be finite, may become ambiguous, formal and even infinite. The value in worrying about
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J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
this idealized, possibly un-physical system, is precisely in that once the ambiguities and infinities are resolved, they teach us something important about the finite physical model, namely, that the parameters used in its formulation, do indeed effectively disappear from the physical properties. Their role is effectively reduced to the control of the small differences between the idealized model and the physical one. We shall consider a problem of this kind that arises in the context of modeling the statistics of charge transport from one reservoir to another. Levitov and Lesovik [13] wrote a formula for the appropriate generating function in terms of a certain infinite dimensional determinant. The formula has found a number of applications to shot and thermal noise in devices like transmission barriers, cavities, and interfaces. When one wants to apply this formula to the idealized cases one finds ambiguities and, as emphasized by Levitov et al. [8,11,12], the determinant requires proper definition through regularization. We intend to further the understanding of these points by providing an alternative, mathematically consistent, form for the determinant. As we shall see, the “regularized form” of the determinant naturally emerges once the quantum dynamics is formulated on the state space of the idealized system. In the next section we introduce the statistics of charge transport, review the LevitovLesovik determinant, and propose a regularization. In Sect. 3 we state the main results. Section 4 is devoted to proofs and begins with a short overview thereof. Finally, Sect. 5 exemplifies the assumptions made in this work. 2. The Levitov-Lesovik Formula and Its Regularization We consider a lead, where independent electrons are evolved over some time interval and ask about the statistics of the charge transferred from the left to the right portion of the lead. To begin, we recall the result obtained in [13] and further elaborated in [12,8]. We present its derivation and generalization to finite times along the heuristic lines given in [10], in the sense that we do as if the one-particle Hilbert space H were finite-dimensional. The fermionic Fock space F over H contains a distinguished state, the vacuum, with the physical interpretation of a no particle state. Let Tr, resp. tr, denote the trace on F, resp. H. Let U be the unitary on H representing the time evolution, and Q the projection corresponding to the right portion of the lead. Their second quantizations, k U , resp. d(Q) = k (U ) = ∧i=1 i i=1 Q i on k-particle states, then stand for the evolution on F, resp. for the charge in that portion. We suppose that the initial many particle (mixed) state is of the form P = Z −1 (M) for some operator M ≥ 0, where Z = Tr (M) = det(1 + M) ensures that Tr P = 1. The reduced one-particle density matrix N is defined by the property that tr(AN ) = Tr (d(A)P) for any one-particle operator A on H. In our case, N = M(1 + M)−1 . This follows from det(1 + eiλA M) Tr eiλd(A) P = Tr (eiλA )P = Z −1 Tr (eiλA M) = det(1 + M) iλA (1) = det(1 − N + e N ) by taking the derivative at λ = 0.
Fredholm Determinants and the Statistics of Charge Transport
809
In the following, we assume that M and Q, and hence P and d(Q), commute, which physically means that in the state defined by P, charge in the lead measured by d(Q) is a good quantum number. Hence P|α = ρα |α,
d(Q)|α = n α |α,
for some basis {|α} of F. The moment generating function for the charge transfer statistics is pn eiλn , χ (λ) = n∈Z
where pn is the probability for n electrons being deposited into the right portion of the lead by the end of the time interval. It may be computed as a sum over initial and final states, α resp. β, with the former weighted according to their probabilities ρα : χ (λ) = |β|(U )|α|2 ρα eiλ(n β −n α ) = Tr (U )∗ eiλd(Q) (U )e−iλd(Q) P α,β
∗ = Z −1 Tr (U ∗ eiλQ U e−iλQ M) = det 1 − N + eiλU QU N e−iλQ ,
(2)
where the trace has been computed in the basis |α, with an identity |ββ| = 1 absorbed at the left of (U ); the last equality is by (1). This is the Levitov-Lesovik formula: χ (λ) = det D(λ),
D(λ) = N + eiλQ U N e−iλQ ,
(3)
with N = 1 − N and Q U = U ∗ QU . Since Q is a projection, e2π iQ = e2π iQ U = 1 and D(λ) is a periodic function with period 2π . This expresses the integrality of charge transport. An example of a state of interest is that of a system at inverse temperature β having one-particle Hamiltonian H ; it is P = Z −1 (M) with M = exp(−β H ) and N = [1 + exp(β H )]−1 . In the limit β → ∞, P describes the Fermi sea, whence N is the projection onto the occupied one-particle states. The above derivation would be rigorous if the one-particle Hilbert space were finite dimensional. The question we want to address here is what is the correct replacement for D(λ) when P describes infinitely many particles, both because the lead may be infinitely extended spatially (as appropriate for an open system) and because the Fermi sea may be very or even infinitely deep. The first concern appears to affect only the derivation, but not the result, Eq. (3). However, by the second, D(λ) differs from the identity by more than a trace class operator, as would be required by the definition of a Fredholm determinant. A manifestation thereof (and in a sense the only one) is that the expected charge transport d n = −iχ (0) = −i det D(λ) = tr ((Q U − Q)N ) (4) dλ λ=0 involves an operator which is not trace class in the stated situation. These statements are illustrated (in the β = ∞ case) in Fig. 1 representing the phase space of a single particle moving freely. The Fermi sea N corresponds to | p| < p F , p F being the Fermi momentum, and similarly the right half of the lead Q to x > 0. The free evolution, which we take
810
J. E. Avron, S. Bachmann, G. M. Graf, I. Klich p E −pF EF
pF
p
N
+
x
− Q
QU
Fig. 1. Left: Dispersion relation E( p) of free particles, and its linearization. Right: Phase space (coordinates x, p) with regions selected by N , Q and Q U , and hatched along their boundaries with slanted, horizontal, and vertical dashes, respectively
as a simple example for U , is a horizontal shear, so that Q U − Q is associated with two sectors, labelled + and −. Their intersection with the horizontal strip associated with N delineates the phase space support of (Q U − Q)N . Its area, which is a rough estimate of the trace class norm of the operator, is proportional to the depth of the sea. If the dispersion relation is conveniently linearized at ± p F , the depth becomes infinite, implying that the operator is not trace class. As a remedy, we note that the expression tr (Q N − Q U NU ) = 0, U ∗ NU , vanishes by splitting the trace, though only suggestively so, because
with NU = the traces fail to exist separately due to the infinite spatial extent of the leads. Adding nevertheless that expression to (4) yields n = tr (Q U (N − NU )),
(5)
which vanishes in the special case of the free evolution, NU = N , and is expected to be finite in others. This way of renormalizing the expression is actually declaring that the Fermi sea does not contribute to the current, instead of relying on a compensation between left and right movers, as indicated by + and − in the figure. This heuristic manipulation motivates the following regularization of the LevitovLesovik determinant. Replacing D(λ) by D(λ) = e−iλNU Q U D(λ)eiλN Q
(6)
should not change the value of the determinant, since informally det(e−iλNU Q U ) · det(eiλN Q ) = eiλ tr(Q N −Q U NU ) = 1.
(7)
Incidentally, this regularization affects only the first cumulant of the statistics, i.e. the average charge transfer, since the full set of cumulants is generated by log det D(λ). We are thus led to recast Eq. (3) as χ (λ) = det D(λ), −iλN U Q U N eiλN Q + eiλNU Q U N e−iλN Q . D(λ) =e
(8) (9)
Fredholm Determinants and the Statistics of Charge Transport
811
It is to be noted that this representation of χ (λ) is manifestly particle-hole symmetric: χ N (λ) = χ N (−λ).
(10)
It is also 2π -periodic in λ, though manifestly so only at T = 0 since N Q, N Q etc. are all projections. In that case, Eq. (9) reduces to D(λ) = 1 + Q U (N − NU ) (eiλ − 1)N − (e−iλ − 1)N , which shows that the generating function χ (λ) is well-defined whenever its first cumulant (5) is. As we shall see, a slightly weaker result holds at positive temperature. Let us mention a few connections to other works. A related regularization of the Levitov-Lesovik determinant at zero temperature was used in [15], where the relation of counting statistics to a Riemann-Hilbert problem was studied. Another one, exhibiting the symmetry (10), was proposed in [16]. On the more mathematical side, regularizations of determinants have been related to renormalization in [22], though by means of a somewhat different regularization known as detn (1 + A) = det(1 + n−1 j j A) exp tr j=1 (−1) A /j . The role of C*-algebras in the theory of open systems has recently been advocated by Jakši´c and Pillet, see e.g. [9], in general, but also to fluctuations in particular. A generating function for fluctuations of energy in bosonic systems has been proposed by [19]. The purpose of this work is to show that, under reasonable assumptions, Eq. (8) is obtained without recourse to regularizations, if the second quantization is built upon the Fermi sea rather than on the vacuum N = 0. 3. Results Let H be a separable Hilbert space with the following operators acting on it: An orthogonal projection Q, a unitary U , and a selfadjoint N , with 0 ≤ N ≤ 1,
(11)
whose physical interpretations have been described in the previous section. Let N = 1 − N . We denote by I p , ( p ≥ 1) the Schatten trace ideals, i.e. the space of all bounded p operators A on H such that A p := tr|A| p < ∞. The algebra of canonical anticommutation relations (CAR) over H is the C*-algebra A(H) generated by 1, and the elements a( f ) and a ∗ ( f ), ( f ∈ H), such that i. ii. iii.
the map f −→ a( f ) is antilinear, a ∗ ( f ) = a( f )∗ , these elements satisfy the following anticommutation relations: {a( f ), a ∗ (g)} = ( f, g)1, all other anticommutators vanishing.
A (global) gauge transformation is expressed by the automorphism αλ : a( f ) → a(eiλ f ). A state ω on A(H) is gauge-invariant if ω(αλ (A)) = ω(A) for all A ∈ A(H). The operator N defines a gauge-invariant quasi-free state ω N through ω N (a ∗ ( f n ) . . . a ∗ ( f 1 ) a(g1 ) . . . a(gm )) = δnm det(gi , N f j ),
(12)
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or equivalently by ω N (a ∗ ( f )a(g)) = (g, N f ) and Wick’s lemma. Let (H N , π N , N ) be the cyclic (or GNS) representation of ω N : ω N (A) = ( N , π N (A) N ),
(A ∈ A(H)).
(13)
The algebra of observables is the (strong) closure of the range of π N , which is equal to its double commutant π N (A(H)) = π N (A(H)) . We also recall that a state is pure if and only if π N (A(H)) is irreducible, i.e. π N (A(H)) = {c · 1 | c ∈ C}, see e.g. [5], Thm. 2.3.19. This is equivalent to N being a projection operator. These concepts briefly reviewed, we are now ready to state our main theorem. Its significance is discussed below in a series of remarks. The key result, which is part (v) together with Corollary 2, states that the moment generating function is given by the regularized determinant, as described in the previous section. Theorem 1. Assume that [Q, N ] = 0, √ √ N − NU ∗ , N − NU ∗ ∈ I1 ,
(14) (15)
where NU ∗ = U NU ∗ . Pure state. Suppose N = N 2 . Then we have − 1 ∈ I1 , where D(λ) is given in Eq. (9). i. D(λ) ii. The Bogoliubov automorphisms induced on A(H) by the unitary operators U and and a exp(iλQ) are implementable on H N : There exist a unitary operator U on H N such that selfadjoint Q π N a # ( f ) U ∗ = π N a # (U f ) , (16) U eiλ Q π N a # ( f ) e−iλ Q = π N a # (eiλQ f ) , (17) for all f ∈ H. ∈ π N (A(H)) for any bounded iii. eiλ Q ∈ π N (A(H)) . More generally, f ( Q) function f . uniquely up to left multiplication with an element iv. The above properties define U ∗ eiλ Q U e−iλ Q from π N (A(H)) , and Q up to an additive constant. In particular, U is unaffected by the ambiguities. v. ∗ eiλ Q U e−iλ Q N ) = det D(λ). ( N , U Mixed state. The above conclusions hold also for 0 < N < 1 if, in addition, √ Q N N ∈ I1 .
(18)
(19)
Remark. 1) Equation (15) demands that the evolution U preserves N , except for creating excitations within an essentially finite region in space and energy, as can be seen from the phase space picture given in the introduction. This assumption is appropriate for the evolution induced by a compact device operating smoothly during a finite time interval. , Q in (ii) are replacements for the non-existent (U ) and d(Q) 2) The operators U mentioned in the introduction. Equations (16, 17) state that any additional particle in the system evolves by U , resp. contributes to the charge as described by Q.
Fredholm Determinants and the Statistics of Charge Transport
813
3) If the state is pure, the pair of Eqs. (15) reduce to the first one with square roots dropped, and property (iii) holds trivially, since B(H N ) = π N (A(H)) , the bounded is unique up to a phase. Incidentally, condition (19) operators on H N . Moreover, U would be trivial in this case. is an observable, and the same is true for U ∗ Q U , 4) Property (iii) states that Q ⊂ π N (A(H)), see (16). Thus, the total charges before and ∗ π N (A(H))U because of U after the evolution are separately bestowed with physical meaning. 5) The physical origin of the extra assumption (19) needed in the mixed state case is as follows. In both cases, the expected charge contained in a portion of the lead is of order of its length L, or zero if renormalized by subtraction of a background charge. In the pure case however, the Fermi sea is an eigenvector of the charge operator, while for the mixed state, the variance of the charge must itself be of order L, because the occupation of the one-particle states is fluctuating, since N N = 0. Hence, in this latter situation, the measurement of the renormalized charge yields finite values only as long as L is finite, of which Eq. (19) is a mathematical abstraction. This condition, while unnecessary for property (ii), is essential for (iii). Without the latter, the l.h.s. of Eq. (18) appears to be ambiguous. On the other hand, the weaker condition √ (Q U − Q) N N ∈ I1 ,
(20)
is sufficient for property (i) and to ensure that the difference Q U − Q is an observable. 6) The theorem does not apply to the general case (11). The two cases considered suffice for thermal states with β = ∞ and 0 < β < ∞.
= n d P(n) be the spectral representation of Q. According to quantum Let Q with outcome n in dn collapses N mechanical principles an ideal measurement of Q to the state d P(n) N , normalized to the probability (d P(n) N , d P(n) N ) of that outcome. Effectively, this means that d P(n) N is the state relevant for a second mea surement. The charge transfer is inferred from two measurements [15] of the charge Q, one before and one after the evolution of the system by U . The joint probability for mea d P(n) N , d P(m)U d P(n) N ) and the generating function surements n and m is (U appropriately defined as χ N (λ) =
∗ d P(m)U d P(n) N )eiλ(m−n) . (d P(n) N , U
consists of integers, up to an additive constant. The Corollary 2. The spectrum of Q generating function is
∗ eiλ Q U e−iλ Q N ) χ N (λ) = ( N , U and describes the transport of integer charges n with non-negative probabilities: χ N (λ) =
pn eiλn ,
pn ≥ 0,
n∈Z
Moreover, the particle-hole symmetry (10) holds true.
n∈Z
pn = 1.
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J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
4. Proofs We begin by giving the proof of the corollary and continue with that of part (i) of the theorem. Then we give some preliminaries, including details such as inner Bogoliubov automorphisms and the Shale-Stinespring criterion for general ones. Thereafter we prove parts (ii-iv) readily if the state is pure, and using its purification, if it is mixed. Finally, the main formula (v) is obtained using an approximation procedure in terms of inner automorphisms and finite dimensional determinants. 4.1. Proof of Corollary 2. We begin by recalling that every gauge-invariant state is a factor state (see [17], Thm. 5.1), i.e. π N (A(H)) ∩ π N (A(H)) = {c · 1| c ∈ C}.
(21)
From Eq. (17) and e2π iQ = 1, we see that e2π i Q ∈ π N (A(H)) , while by (iii) we have e2π i Q ∈ π N (A(H)) . Thus e2π i Q = c, (|c| = 1) and we may assume c = 1 by adding see (iv). The spectral representation of Q is then of the form an additive constant to Q, = Q n Pn . (22) n∈Z
We note that
( N , eiλ Q Ae−iλ Q N ) = ( N , A N )
(23)
for A ∈ π N (A(H)) . Indeed, for A = π N (a ∗ ( f ) a(g)), we have eiλ Q Ae−iλ Q = π N (a ∗ (eiλQ f ) a(eiλQ g)) by (17). The expectations (23) agree because of (eiλQ g, N eiλQ f ) = (g, N f ) by [Q, N ] = 0. The same holds true by (12) for arbitrary products of a ∗ ( f i ), a(gi ), and by density, for A ∈ π N (A(H)) . By (iii) we may apply (23) to Aeiλ Q instead of A, and obtain ( N , Pn A N ) = ( N , A Pn N ); then this to A Pn ∈ π N (A(H)) instead of A, and get ( N , A Pn N ) = ( N , Pn A Pn N ). More ∗ eiλ Q U ∈U ∗ π N (A(H)) U ⊂ π N (A(H)) by (16). Hence, using (22), over, we have U we see that ∗ eiλ Q U e−iλ Q N ) = ∗ eiλ Q U Pn N )e−iλn ( N , U ( N , Pn U n∈Z
=
∗ Pm U Pn N )eiλ(m−n) ( N , Pn U
n,m∈Z
is of the stated form. 4.2. Part (i). Since the projection Q commutes with N , see (14), we have eiλN Q = 1 + (eiλN − 1)Q, e−iλNU Q U = 1 + Q U (e−iλNU − 1). We insert these equations in the definition (9) of D(λ). Moreover, 1/2
1/2
1/2
NU − N = N 1/2 (NU − N 1/2 ) + (NU − N 1/2 )NU
∈ I1 ,
(24)
Fredholm Determinants and the Statistics of Charge Transport
so that e−iλN − e−iλNU = i
λ
815
e−i(λ−s)NU (NU − N )e−is N ds
0
we also belongs to the trace class ideal. Rather than proving D(λ) ∈ 1 + I1 for D(λ) may thus do so for the expression
[1+ Q U (e−iλN −1)]N [1+(eiλN − 1)Q] + [1 + Q U (eiλN − 1)]N [1 + (e−iλN − 1)Q]
= N + N + Q U [(e−iλN − 1)N + (eiλN − 1)N ] + [N (eiλN − 1) + N (e−iλN − 1)]Q
+Q U [(e−iλN − 1)N (eiλN − 1) + (eiλN − 1)N (e−iλN − 1)]Q
= 1 + Q U (cos(λN ) − 1)N + (cos(λN ) − 1)N − i sin(λN )N + i sin(λN )N
+Q (cos(λN ) − 1)N + (cos(λN ) − 1)N + i sin(λN )N − i sin(λN )N
+2Q U Q (1 − cos(λN ))N + (1 − cos(λN ))N 2 = 1 + (Q U + Q 2 − 2Q U Q)[(cos(λN ) − 1)N + (cos(λN ) − 1)N ] +i(Q − Q U )[sin(λN )N − sin(λN )N ].
With the help of the functions f (x) = (cos x − 1)/x and g(x) = (sin x)/x, which are bounded also at x = 0, the expression is rewritten as 1 + [(Q − Q U )Q − Q U (Q − Q U )]N N λ( f (λN ) + f (λN )) +i(Q − Q U )N N λ(g(λN ) − g(λN )). √ √ Besides Q N N ∈ I1 , see Eq. (19), we have Q U N N = U ∗ Q NU ∗ NU ∗ U ∈ I1 by √ Eq. (15), and hence (Q − Q U ) N N ∈ I1 , cf. (20). This makes the claim manifest. In the zero temperature case, where N is a projection, the above proof simplifies considerably due to N N = 0. 4.3. Preliminaries. We recall a few results about Bogoliubov transformations, first inner and then others. Given a bounded operator A on H, operators (A) and d(A) are usually defined on the Fock space over H. Following [2] we define them instead as elements of the CAR-algebra A(H), when A is of finite rank. •
For rank one operators Ai = | f i gi |, (i = 1, . . . , n), we set d(A1 , . . . , An ) = a ∗ ( f n ) · · · a ∗ ( f 1 ) a(g1 ) · · · a(gn ).
•
(25)
The definition is extended by multilinearity to operators Ai of finite rank. The result is independent of the particular decomposition into rank one operators. For U − 1 of finite rank, we set ∞ 1 (U ) = d(U − 1, . . . , U − 1), n! n=0
n
where the term n = 0, in which no arguments are present, is read as d = 1. The sum is finite, because the terms with n > rank(U − 1) vanish.
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J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
The elements of A(H) just defined share the properties of the operators on Fock space known by the same notation. Lemma 3. Let U − 1 be of finite rank. Then (U )a ∗ ( f ) = a ∗ (U f )(U ), (U1 U2 ) = (U1 )(U2 ).
(26) (27)
In particular, (U ) is unitary if U is. Proof. We have d(A1 , . . . , An )a ∗ ( f ) = a ∗ ( f )d(A1 , . . . , An ) n i , . . . , An ), a ∗ (Ai f )d(A1 , . . . , A +
(28)
i=1
where the hat indicates omission. In the rank one case, Ai = | f i gi |, this follows from (25) and from (gi , f )a ∗ ( f i ) = a ∗ (Ai f ). In the general case, by multilinearity. Thus, (U )a ∗ ( f ) = a ∗ ( f )(U ) + a ∗ ((U − 1) f )
∞ n=1
1 d(U − 1, . . . , U − 1) (n − 1)!
= a ∗ ( f )(U ) + a ∗ ((U − 1) f )(U ) = a ∗ (U f )(U ), since we applied (28) with n equal entries Ai = U − 1. We have d(A1 , . . . , An )d(B1 , . . . , Bm ) =
min(n,m) l=0
is , . . . , An , B1 , . . . , d(Ai1 B j1 , . . . , Ail B jl , A1 , . . . , A B js , . . . , Bm ),
Cl
where the second sum runs over all l-contractions (i 1 , j1 ), . . . , (il , jl ) with i 1 < . . . < il , jis = jir . In the rank one case, which implies the general one, this is just Wick’s lemma for normal ordered products. Thus (U1 )(U2 ) =
∞ ∞ 1 1 d(U1 − 1, . . . , U1 − 1) · d(U2 − 1, . . . , U2 − 1) n! m! n=0
=
m=0
∞
min(n,m)
n,m=0
l=0
1 d((U1 − 1)(U2 − 1), . . . , l!(n − l)!(m − l)!
×U1 − 1, . . . , U2 − 1, . . .) with entries repeated l, n − l, m − l times. In fact, the number of l-contractions is m! 1 n! . l! (n − l)! (m − l)!
Fredholm Determinants and the Statistics of Charge Transport
817
Setting n − l =: s, m − l =: t, l + s + t =: r , we have (U1 )(U2 ) =
∞ r =0
=
l,s,t l+s+t=r
1 d((U1 − 1)(U2 − 1), . . . , U1 −1, . . . , U2 −1, . . .) l! s! t!
∞ 1 d((U1 − 1)(U2 − 1) + (U1 − 1) + (U2 − 1), . . .), r! r =0
since there are r !/l! s! t! ways to pick terms from each entry of the last line. Since (U1 − 1)(U2 − 1) + (U1 − 1) + (U2 − 1) = U1 U2 − 1, the proof is complete. If O is an operator on H such that O −1 is in the trace class, its Fredholm determinant is defined by det O =
∞
tr ∧k (O − 1).
(29)
k=0
This extends the usual definition of the determinant in the finite dimensional case. Lemma 4. Let A be a finite rank operator, and 0 ≤ N ≤ 1. Then ω N (d(A, . . . , A)) = tr ∧k (AN ). k
Moreover, if U is such that U − 1 is of finite rank, then ω N ((U )) = det((1 − N ) + U N ). (30) m m Proof. The trace of a finite rank operator A = i=1 f i (gi , ·) is tr A = i=1 (gi , f i ). By the same token, that of ∧k A =
m i 1 ,...i k =1
is tr ∧k A =
1 (−1)σ ⊗kα=1 f iσ (α) (giα , ·) k! σ ∈Sk
1≤i 1 <...
det(giα , f iβ )kα,β=1 .
Since the a # ( f ) anticommute, we have d(A, . . . , A) = a ∗ ( f i1 ) · · · a ∗ ( f ik ) a(gik ) · · · a(gi1 ), 1≤i 1 <...
whose expectation value is computed by (12) as det(N giα , f iβ )kα,β=1 = tr ∧k (AN ), ω N (d(A, . . . , A)) = 1≤i 1 <...
because the decomposition of AN differs from that of A by N gi in place of gi . This proves the first part of the lemma. The second part is now an application of the definition k of the determinant, Eq. (29). Indeed, ω N ((U )) = 1 + ∞ k=1 tr ∧ ((U − 1)N ) = det(1 + (U − 1)N ).
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J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
We recall a few results on Bogoliubov transformations. Their proofs can be found e.g. in [1], where however CAR-algebras are introduced in the self-dual guise. A remark at the end of this subsection is intended as an aid to translation. The first result is the Shale-Stinespring criterion [24] about unitary implementability (see e.g. [14,1], Thm. 6.3 (1)). Proposition 5. Let P be a projection and V a unitary operator on H. The Bogoliubov automorphism induced by V on H, i.e. a( f ) → a(V f ), is unitarily implementable in the representation π P if and only if P V (1 − P) and (1 − P)V P are in the Hilbert-Schmidt class, I2 . In particular, an equivalent condition is [P, V ] = P V (1 − P) − (1 − P)V P ∈ I2 . There is a version of this proposition for groups ([1], Thm. 6.10 (2, 3)). Proposition 6. Let V in the previous proposition be replaced by a 1-parameter unitary group Vλ , (λ ∈ R), such that Vλ is norm continuous and P Vλ (1 − P) is continuous λ = exp(iλ in the I2 -norm. Then Vλ has an implementer of the form V v ), where v is a self-adjoint operator on H P . The requirement ( P , v P ) = 0
(31)
may be imposed, in which case v is unique. An equivalent condition, is Vλ = exp(iλv) with v a bounded, selfadjoint operator on H, and Pv(1 − P) ∈ I2 . The next result is about the continuity of the implementation, see [1], Thm. 6.10 (7). Proposition 7. Let v and vn , (n = 0, 1, . . .), satisfy the hypotheses of Prop. 6, and let v, v n satisfy the normalization (31). If (vn ) converge strongly to v and if P(vn − v)(1 − P) 2 → 0 as n → ∞, then v s− lim eiλv n = eiλ . n
The last preliminary is concerned with the twisted duality of CAR-algebras. Let P be an orthogonal projection on H, K ⊂ H a closed subspace, and K⊥ its orthogonal ⊥ ) be the von Neumann algebra generated by π P (a( f )), ( f ∈ complement. Let A(K is the parity. Then [7,4] K⊥ ), where ⊥ ). π P (A(K)) = A(K
(32)
from Prop. 5 commute with parity: Let : H P → H P be The implementers V the unitary implementation of the *-automorphism a( f ) → a(− f ) which is uniquely P = P . Then determined by , ] = 0, [V
(33)
=V λ as see [1], Thm. 6.3 (3), Thm. 6.7 (2). We will actually apply this fact only to V in Prop. 6, in which case it can be verified as follows. Since [Vλ , −1] = 0, the operators λ = cλ V λ and V λ λ implement the same Bogoliubov automorphism, whence V V with |cλ | = 1. From ( P , Vλ P ) = ( P , Vλ P ) = cλ ( P , Vλ P ) we find cλ = 1 λ P ) → 1, (λ → 0). The conclusion extends to all λ by for small λ, because ( P , V the group property.
Fredholm Determinants and the Statistics of Charge Transport
819
Remark. In order to make contact with the repeatedly cited article [1] we recall that ) is given in terms of a separable Hilbert space H a self-dual CAR-algebra A(H, equipped with a conjugation . Its generators B(h) are linear in h ∈ H and the relations satisfying on H are B(h)∗ = B(h) and {B(h), B(h )∗ } = (h , h)1. A projection P + P = 1 defines a pure state ω on the algebra through P ω(B(h)∗ B(h)) = 0,
= 0). ( Ph
The algebra A(H) is connected to the above by picking a conjugation C on H and by setting = H ⊕ H, H
( f ⊕ g) = Cg ⊕ C f,
B( f ⊕ g) = a ∗ ( f ) + a(Cg).
then agree if P( f ⊕ g) = (1 − P) f ⊕ C PCg. States defined by P and P 4.4. Parts (ii-iv). Pure state, N = N 2 . Existence: In the case Vλ = exp(iλQ) we have λ = exp(iλ Q) is [N , Vλ ] = 0 by Eq. (14), so that existence of a unitary implementer V trivial by Prop. 6. Similarly, in the case V = U we have [N , U ] = (N − NU ∗ )U ∈ I2 by Eq. (24). Hence it is also implementable by Prop. 5. denote either exp(iλ Q) or U . Suppose V 1 and V 2 both implement Uniqueness: Let V 1 V 1 V 1 = ∗ π N (a(V f )) = π N (a(V f ))V ∗ . Thus V the same transformation. Then V 2 2 ∗ ∗ (V1 V2 )V2 and V2 differ by left multiplication with V1 V2 ∈ π N (A(H)) . In the pure is unique up to a phase and Q up case the cyclic representation is irreducible, whence U to an additive constant. As mentioned in Remark 3, property (iii) is empty in this case. Mixed state, 0 < N < 1. Given 0 ≤ N ≤ 1 on H, we consider its purification √ N N N √ PN = = PN2 N N N on H ⊕ H, together with the cyclic representation (H PN , π PN , PN ) of the state defined by PN on A(H ⊕ H). We can identify A(H) ∼ = A(H ⊕ 0) via a( f ) = a( f ⊕ 0), and H N ≡ π PN (A(H ⊕ 0)) PN ⊂ H PN ,
N ≡ PN ,
π N (a) ≡ π PN (a) H N , (34)
since these objects satisfy ( PN , π PN (a ∗ ( f ⊕ 0)a(g ⊕ 0)) PN ) = (g ⊕ 0, PN ( f ⊕ 0)) = (g, N f ),
(35)
as required by (13). We can not handle the most general mixed case 0 ≤ N ≤ 1. The reason comes from the following lemma, whose proof is postponed till the end of the section. Lemma 8. Assume 0 < N < 1 (strict inequality). Then PN is cyclic in H PN for π PN (A(H ⊕ 0)). In particular, we have equality in (34), H N = H PN .
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J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
A unitary V on H induces two automorphisms on A(H ⊕ H): (a) a( f ⊕ g) → a(V f ⊕ g), and (b) a( f ⊕ g) → a(V f ⊕ V g), whose implementation may be envisaged: (a) (b)
π PN (a( f ⊕ g))V ∗ = π PN (a(V f ⊕ g)), V ∗ = π PN (a(V f ⊕ V g)). π PN (a( f ⊕ g))V V
(36)
would provide an implementation for V in the representation π N Both choices for V ∈ π N (A(H)) . Indeed, by (32) applied to on H N . Only in the first case we have V ⊕ H) . This however ∈ A(0 H ⊕ 0 ⊂ H ⊕ H instead of K ⊂ H, we need to check V ] = 0, see (33). follows from [V , π PN (a(0 ⊕ g))] = 0, see (36), and from [V , In order to determine the existence of these implementations we compute √ √ [N , U1 ] √ U1 0 N N U2 − U1 N N PN , = √ , 0 U2 N N U1 − U2 N N [N , U2 ] and see that that of (a) (i.e. U1 = V and U2 = 1) is granted if √ [N , V ] ∈ I2 , (1 − V ) N N ∈ I2 ; and that of (b) (U1 = V = U2 ) if [N , V ] ∈ I2 ,
√ [ N N , V ] ∈ I2 .
(37)
(38)
We can now complete the proof of parts (ii) and (iii) of the theorem. √ For V = iλQ ) N N = exp(iλQ), we use (a). Then Eqs. (37) hold true by Eqs. (14, 19): (1 − e √ (1 − eiλ )Q N N . This also proves (iii). For V = U , we use (b), with conditions (38) holding by Eqs. (15, 24). Part (iv) is readily proven as follows. Like in the pure case, the implementers are unique up to left multiplication by an element of π N (A(H)) (which is larger than ∈ the multiples of the identity since the representation is reducible). Thus exp (iλ Q) π N (A(H)) still implies its uniqueness up to a phase because of (21). Remark. If it were for property (ii) only, one could adopt method (b) also for V = exp(iλQ). Proof of Lemma 8. The space H PN is spanned by the vectors n
π PN (a # ( f i ⊕ gi )) PN .
(39)
i=1
It suffices to show that they can be approximated arbitrarily well by a sum of such vectors where, however, gi = 0. To this end, we first note that π PN (a ∗ (PN ( f ⊕ g))) PN = 0, π PN (a(PN ( f ⊕ g))) PN = 0, where PN = 1 − PN . This follows from (12) for PN instead of N , and implies in turn π PN (a ∗ ( f ⊕ g)) PN ≤ PN ( f ⊕ g) , π PN (a( f ⊕ g)) PN ≤ PN ( f ⊕ g) .
Fredholm Determinants and the Statistics of Charge Transport
821
Let us first consider the case where the last factor in (39) is an annihilation operator and set f n ⊕ gn =: f ⊕ g. We have √ √ √ N PN ( f ⊕ g − f ⊕ 0) = √ N( f − f ) + N g . N A vector f ∈ H is well-defined by √ √ √ N f := N f + N F(N ≥ )g, where F(N ≥ ) is the spectral projection for N on [ , ∞) and > 0. Thus, f ⊕ 0) ≤ F(N < )g PN ( f ⊕ g − can be made arbitrarily small because of Ker N = {0}. If the last factor is a creation operator, the arguments proceed similarly using Ker N = {0}. Hence the announced replacement can be performed in the last factor. After anticommuting it to the left, the claim is reduced to products with fewer factors, for which it holds by induction. 4.5. Part (v). The idea of the proof is to approximate the Bogoliubov automorphism induced by eiλQ by means of inner automorphisms, as introduced in Subsect. 4.3. The generating function on the l.h.s. of (18) then becomes computable by Lemma 4. We present separate proofs in the pure and the mixed case. The second proof, while applying to both cases, is longer than the one we give for pure states. Both depend on Prop. 7. Pure state. Let F be a finite rank operator on H with [F, N ] = 0. As such, it has an implementation in the cyclic representation π N ; its non-uniqueness does not affect the l.h.s. of ∗ eiλ F U ∗ π N ((eiλF ))U π N ((e−iλF )) N ) e−iλ F N ) = ( N , U ( N , U = ( N , π N ((U ∗ eiλF U e−iλF )) N );
(40)
on the r.h.s. we used that π N ((eiλF )) is one possible implementation of eiλF by (26) with ∗ π N ((eiλF ))U = eiλF in place of U ; the second line follows by (16), which implies U ∗ iλF is fixed by π N ((U e U )), and by (27). Another choice for F N ) = 0, ( N , F
(41)
and we may ask the same normalization for Q. Lemma 9. There is a sequence of finite dimensional orthogonal projections Fn such that [Fn , N ] = 0,
s− lim Fn = Q. n
(42)
Proof. We note that (N Q)2 = N Q, so that Q = N Q + N Q is an orthogonal split(1) (2) (1) (2) ting of Q. Let Fn = Fn + Fn , where Fn , resp. Fn , is a subprojection of N Q s (1) (1) (1) (2) s (i.e. Fn N Q = Fn ), resp. of N Q, with Fn → N Q, and Fn → N Q. Clearly, s Fn → Q and [Fn(1) , N ] = [N , Fn(1) ] = N Fn(1) − Fn(1) N = N N Q Fn(1) − Fn(1) Q N N = 0, (2)
since N N = 0. The same holds for Fn , and thus for Fn .
822
J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
By (42, 41) the assumptions of Prop. 7 are satisfied for the sequence (Fn ) and its limit Q. Therefore, Fn −iλ ∗ eiλ Q U ∗ eiλ e−iλ Q N ) = lim ( N , U ( N , U U e Fn N ).
(43)
n→∞
By Eqs. (40, 30, 42) the inner product on the r.h.s. equals det(N + eiλU
∗F U n
e−iλFn N ) = det(e−iλNU FnU N eiλN Fn + eiλNU FnU N e−iλN
F n
),
where we multiplied the determinant by 1 = det(e−iλNU FnU ) · det(eiλN Fn ),
(44)
like in the heuristic derivation (7); but unlike there, this step is now correct, since Fn is of finite rank. We also used [Fn , N ] = 0. Finally, we claim that the operator under the last determinant converges to
e−iλNU Q U N eiλN Q + eiλNU Q U N e−iλN
Q
= e−iλNU Q U N + eiλNU Q U N
(45)
in trace class norm, i.e. the same expression with Q in place of Fn . The r.h.s. is obtained using exp(iλN Q) = 1 + N Q(exp(iλ) − 1) and N N = 0. The convergence implies that of the determinants: Indeed, for A − 1, B − 1 ∈ I1 , we have ([18], Lemma XIII.17.1 (d)) | det A − det B| ≤ A − B 1 e( A−1 1 + B−1 1 +1) . Upon conjugating with U , it is enough to show (e−iλN Fn − e−iλN Q )NU ∗ 1 −→ 0, and similarly with N and N interchanged. This operator equals e−iλ − 1 times N (Fn − Q)NU ∗ = (Fn − Q)N N + (Fn − Q)N (NU ∗ − N ). The first term vanishes, and the second tends to 0 in the trace class norm as n → ∞, because of s
X n −→ 0, Y ∈ I1
=⇒
X n Y 1 −→ 0 .
(46)
Mixed state. Let us start by proving a result analogous to Lemma 9: Lemma 10. Let P, Q be orthogonal projections in a separable Hilbert space H with [Q, P] ∈ I1 .
(47)
Then there are finite dimensional subprojections Fn of Q with [Fn − Q, P] 1 −→ 0, (n → ∞).
(48)
Fredholm Determinants and the Statistics of Charge Transport
823
Proof. We split Q as Q = Q P Q + Q(1 − P)Q ≡ L 1 + L 0 ,
(49)
and observe that [Q, L 1 ]=0 and (P − 1)L 1 ∈ I1 , L 21 − L 1 = Q P[Q, P]Q ∈ I1 .
(50)
By the last property, the only possible accumulation points in the spectrum of L 1 are 0 and 1. In particular, there is an x ∈ (0, 1) which is not in the spectrum. Let Q 1 be the spectral projection of L 1 associated with (x, ∞). It may be represented as 1 Q1 = (z − L 1 )−1 dz, 2π i C where C ⊂ C is a contour encircling that part of the spectrum only. Using C z −1 dz = 0, due to x > 0, we have 1 (P − 1)Q 1 = (P − 1) (z − L 1 )−1 − z −1 dz 2π i C 1 = (P − 1)L 1 (z − L 1 )−1 z −1 dz ∈ I1 (51) 2π i C by (50). On the subspace Ran Q, the projection Q 1 , defined in terms of L 1 and x is complementary to the one, Q 0 , similarly defined by L 0 and 1 − x, see (49). Since 1 − x > 0, we have P Q 0 ∈ I1
(52)
(i)
by analogy to (51). Let now Fn , (i = 0, 1), be a sequence of finite dimensional s subprojections of Q i with Fn(i) → Q i . Then [Fn(0) − Q 0 , P] = (Fn(0) − Q 0 )P − P(Fn(0) − Q 0 ) = (Fn(0) − Q 0 )Q 0 P −P Q 0 (Fn(0) − Q 0 ), [Fn(1) − Q 1 , P] = [Fn(1) − Q 1 , P − 1] = (Fn(1) − Q 1 )Q 1 (P − 1) −(P − 1)Q 1 (Fn(1) − Q 1 ), are trace class by (51, 52), and converge to zero in the corresponding norm by (46) and (0) (1) T ∗ 1 = T 1 . Thus Fn = Fn + Fn is seen to have the stated properties. = Q ⊕ 0 instead of H, P and Q; in this We apply the lemma to H ⊕ H, PN and Q are of the form F ⊕ 0, with F a subprojection of Q. Since case, subprojections of Q √ [N , Q] −Q N N √ [PN , Q] = , N NQ 0 the hypothesis (47) of Lemma 10 is fulfilled. The claim yields n→∞
[Fn − Q, N ] 1 −→ 0,
(53)
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J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
√ as well as N N (Fn − Q) 1 → 0, which however is already known by (19) and Fn = Fn Q. We thus have a sequence (Fn ) of unitarily implementable transformations: the conditions (37) are both fulfilled, the first one because √ [N , exp(−iλFn )] = [N , F√n ](exp(−iλ)−1) and the second because (1−exp(−iλFn )) N N = (exp(−iλ)−1) Fn Q N N . Moreover, the assumptions of Prop. 7 are satisfied, so that Eqs. (43,40) are true again. To complete the proof, it remains to show that
det(N + eiλFnU e−iλFn N ) −→ det(e−iλNU Q U N eiλN Q + eiλNU Q U N e−iλN Q ).
(54)
To this end, we multiply the determinant by det(1 + FnU (e−iλNU − 1)), det(1 + (eiλN − 1)Fn ),
(55)
from the left, resp. from the right. These factors would be identical to those in (44) if Fn and N commuted, which is however no longer the case. Also, their product is not 1, but rather equals det(1 + FnU (e−iλNU − 1)) · det(1 + (eiλN − 1)Fn ) = det(1 + (eiλN − 1)Fn ) · det(1 + Fn (e−iλN − 1)) = det(1 − Fn + eiλN Fn e−iλN ),
(56)
where eiλN Fn e−iλN − Fn = i
λ
eis N [N , Fn ]e−is N ds ∈ I1
0
and n→∞
eiλN Fn e−iλN − Fn 1 −→ 0
(57)
by [N , Q] = 0 and (53). Therefore, (56) converges to 1 and it suffices to prove (54) with the l.h.s. multiplied by (55). The determinant becomes that of (1 + FnU (e−iλNU − 1))(N + eiλFnU e−iλFn N )(1 + (eiλN − 1)Fn ).
(58)
By means of (1 + FnU (e−iλNU − 1))N − e−iλNU Q U N 1 −→ 0, e−iλFn N eiλFn − N 1 −→ 0,
(59) (60)
which we shall prove momentarily, we may replace (58) by e−iλNU Q U N (1 + (eiλN − 1)Fn ) + (1 + FnU (e−iλNU − 1))eiλFnU N e−iλFn ×(1 + (eiλN − 1)Fn ). The claim then follows from N (1 + (eiλN − 1)Fn ) − N eiλN Q 1 −→ 0, N e
−iλFn
(1 + (e
(1 + FnU (e
iλN
−iλNU
− 1)Fn ) − N e
− 1))e
iλFnU
−iλN Q
N −e
1 iλNU Q U
(61) −→ 0,
(62)
N 1 −→ 0.
(63)
Fredholm Determinants and the Statistics of Charge Transport
825
Fig. 2. A simple model with two infinite chiral leads
It remains to prove (59–63). The limit (60) follows like (57). The expression in (61) is N (exp(iλN )−1)(Fn − Q) = f (N ) N N Q(Fn−Q), where f (N ) = N −1 (exp(iλN )−1) is a bounded operator; its convergence to zero follows from (19). As for (62) we have e−iλFn (1 + (eiλN − 1)Fn ) = (1 + (e−iλ − 1)Fn )(1 + (eiλN − 1)Fn ) = 1 + (e−iλ eiλN − 1)Fn + (e−iλ − 1)[Fn , eiλN ]Fn , so that by using (57) it remains to show
N (1 + (e−iλN − 1)Fn ) − N e−iλN Q 1 −→ 0. This, however, is just (61) with N and N interchanged. Finally, in (59, 63) we may, by (15), replace N and N by NU and NU in those places where the subscript is not already present. By passing to a unitary conjugate and adjoint, they reduce to (61, 62). 5. Examples We illustrate the hypotheses (14, 15) by presenting a model in which they can be verified. The left and right portions of the single lead mentioned in Sect. 1 are replaced by two infinite leads, which are however chiral. The interaction between them occurs in a finite interval and allows particles to scatter between the leads. Let H = L 2 (R) ⊕ L 2 (R) be the one-particle space with operators (− p) 0 1 0 . (64) , N= Q= 0 ( p) 0 0 Here, x ∈ R is the position variable, p = −id/d x the conjugate momentum, and the Heaviside function. The projection N describes the Fermi sea of the free Hamiltonian p 0 H0 = (65) 0 −p for vanishing Fermi energy. Clearly, [Q, N ] = 0. 5.1. Example 1. The example is conveniently stated by passing to another pair of conjugate variables, E and t: The energy E := ± p yields the spectral representation of H0 in multiplication form, E 0 , H0 = 0 E
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J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
while the operator T =
−x 0 0 x
=:
t 0 , 0 t
represents, due to i[H0 , T ] = −1, the time t of passage at x = 0 of a freely moving particle, which is presently elsewhere. In this example only, the meaning of t is therefore that of a dynamical variable, and not that of the parameter governing evolution. Rather than specifying an interacting Hamiltonian, we model the scattering process by directly giving the propagator U for the time interval under consideration. We assume it to be given by a unitary multiplication operator U (t) with U (t)−1 of compact support, see Remark 1 in Sect. 3. Such a simple kind of evolution should be seen as an effective description in the adiabatic limit and in the interaction picture. The passage across the interaction region maps the incoming state to the outgoing one by means of a scattering matrix which, in the limit of low frequencies ω, is that of the static scatterer in effect at time t, S(t). In the same limit, only electrons within an interval ∼ ω of the Fermi energy ought to matter for transport. Thus, U (t) = S(t, 0), where the 2 × 2-matrix S(t, E) is the fiber of S(t) at energy E. For a more thorough justification, see [6,3]. Proposition 11. Suppose U − 1 ∈ M2 (C0∞ (Rt )). Then [N , U ] ∈ I1 . Here M2 (X ) are the 2 × 2 matrices with entries in X . Proof. We may rename U (t) − 1 by U (t) without loss. By the assumption we may write U = f U , where f = f (t) satisfies f ∈ C0∞ (Rt ), too. Then [N , U ] = f [N , U ] + [N , f ]U , with f [N , U ] = f (E + i)−1 · (E + i)[N , U ], and similarly for the second term. We claim that both factors are Hilbert-Schmidt and hence their product trace class. The first one is, because the functions f and g(E) ≡ (E + i)−1 are in L 2 (Rt ), resp. L 2 (R E ). As for the second one, we note that N = (−E)⊗12 , whence [N , U ] has matrix entries [(−E), Ui j ]. That leads to integral operators K acting merely on L 2 (R E ) with kernels i j (E − E) (−E) − (−E ) . K (E, E ) = (E + i)U They are supported where sgn E = −sgn E and satisfy ∞ ∞ i j (E + E )|2 + |U i j (−E − E )|2 |K (E, E )|2 d E d E = |E + i|2 |U 0
0
×d E d E , which is finite. Thus, the corresponding operator is in I2 . By contrast, but under the same assumption as in the proposition, the operator (Q U − Q)N may fail to be trace class. By (4), this shows the need for regularizing (3). Indeed, we may arrange for a ψ ∈ H and U such that (Q U − Q)ψ = 0. The sequence ψn = exp(inT )ψ tends to zero weakly. Using (−E) exp(int) = exp(int)(n − E) and s (n − E) → 1, we have N ψn − ψn → 0 and, since Q, U are multiplication operators in t, (Q U − Q)N ψn → (Q U − Q)ψ = 0. As a result, (Q U − Q)N
Fredholm Determinants and the Statistics of Charge Transport
827
is not even compact. The argument just given may be summarized in physical terms as follows: Whatever contribution to transport, as signified by (Q U − Q)N , comes from one energy in the Fermi sea, it is repeated at all such energies, because the evolution U proceeds with the same velocity ±1 at all energies. It should be remarked that [N , U ] may fail to be in I2 if, unlike in Prop. 11, U (t) attains different limits at t → ±∞. This fact has been pointed out in [12] in slightly different terms as a manifestation of the orthogonality catastrophe. Consider for instance a potential drop V (t) of finite duration being applied between the leads, with
∞ / 2π Z. That situation can be modeled in the context of the present example −∞ V (t)dt ∈ by means of a vector potential, where it gives rise to the catastrophe. The same physical situation is however tame in the context of the next example. 5.2. Example 2. Here we specify a time-dependent perturbation of (65), H (t) = H0 + V (t), where V (t) is multiplication by a 2 × 2 matrix V (t, x). Let U = U (t2 , t1 ) be the propagator for H (t) between times t1 and t2 . Proposition 12. Suppose V (t, ·), ∂t V (t, ·) ∈ M2 (C0∞ (Rx )). Then [N , U ] ∈ I2 . Note that the commutator is claimed to be Hilbert-Schmidt only, which covers only the statements (ii-iv) of Theorem 1. Proof. By [23], Lemma 4 or [20], Thm. 2.8 it suffices to show that the statement holds true for the first term in the Dyson expansion of U , i.e. for s2 U˜ (s2 , s1 ) = −i eiH0 t V (t)e−iH0 t dt, (66) s1
with estimates uniform in the sub-interval [s1 , s2 ] ⊂ [t1 , t2 ]. By writing V++ (t) V+− (t) , V (t) = V−+ (t) V−− (t) the kernel of [N , V (t)] in momentum space becomes [N , V (t)]( p, p ) Vˆ++ (t, p − p )((− p) − (− p )) Vˆ+− (t, p − p )((− p) − ( p )) . = Vˆ−+ (t, p − p )(( p) − (− p )) Vˆ−− (t, p − p )(( p) − ( p )) The diagonal contributions are in I2 without recourse to the integration (66). For instance, ∞ du |u||Vˆ−− (t, u)|2 < ∞. dpdp |Vˆ−− (t, p − p )|2 |( p) − ( p )| = −∞
The off-diagonal contributions improve once the time integral is performed. We compute it by parts and obtain, for instance, the kernel s2 −i Vˆ+− (t, p − p )ei( p+ p )t dt s1 s2 s2 ei( p+ p )t − 1 ei( p+ p )t − 1 ˆ =− (t, p − p ) + ∂t Vˆ+− (t, p − p )dt, (67) V +− p + p p + p s1 s1
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J. E. Avron, S. Bachmann, G. M. Graf, I. Klich
times (− p)−( p ). The boundary terms are separately in I2 , since their corresponding square norm is sin2 (( p + p )si /2) ˆ |V+− (si , p − p )|2 |(− p) − ( p )| 4 dpdp ( p + p )2 ∞ sin2 (usi /2) |u|/2 du dv |Vˆ+− (si , v)|2 ≤ π |si | V+− (si ) 22 . =4 u2 −∞ −|u|/2 By the same estimate, but with ∂t V+− (t) in place of V+− (si ), also the integrand in (67) is in I2 . We recall that in [23,20] the implementation of the propagator of a time-dependent Dirac Hamiltonian was studied, of which the above H (t) is the 1-dimensional version. In larger dimensions, as considered there, the implementability is ensured only in some cases. We remark that by the method used in Example 1 one can show that diagonal perturbations lead to [N , U ] ∈ I1 , but not for off-diagonal ones. Acknowledgements. We would like to thank H. Araki, P. Deift, G. Dell’Antonio, G. Kottanattu, G. Lesovik and W. de Roeck for discussions. We also thank the Erwin Schrödinger Institute (Vienna) and the Lewiner Institute for Theoretical Physics at the Technion (Haifa) for hospitality.
References 1. Araki, H.: Bogoliubov automorphisms and Fock representations of the canonical anticommutation relations. Contemp. Math. 62, 23–141 (1987) 2. Araki, H., Wyss, W.: Representations of canonical anticommutation relations. Helv. Phys. Acta 37, 136– 159 (1964) 3. Avron, J.E., Elgart, A., Graf, G.M., Sadun, L., Schnee, K.: Adiabatic charge pumping in open quantum systems. Comm. Pure Appl. Math. 57, 528–561 (2004) 4. Baumgärtel, H., Jurke, M., Lledó, F.: Twisted duality of the CAR-algebra. J. Math. Phys. 43, 4158–4179 (2002) 5. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Berlin-HeidelbergNew York: Springer-Verlag, 1979 6. Büttiker, M., Thomas, H., Prêtre, A.: Current partition in multi-probe conductors in the presence of slowly oscillating external potentials. Z. Phys. B 94, 133–137 (1994) 7. Doplicher, S., Haag, R., Roberts, J.E.: Fields, observables and gauge transformations I. Commun. Math. Phys. 13, 1–23 (1969) 8. Ivanov, D.A., Lee, H.W., Levitov, L.S.: Coherent states of alternating current. Phys. Rev. B 56, 6839– 6850 (1997) 9. Jakši´c, V., Pillet, C.-A.: Mathematical theory of non-equilibrium quantum statistical mechanics. J. Stat. Phys. 108, 787–829 (2002) 10. Klich, I.: Full counting statistics: An elementary derivation of Levitov’s formula. In: Yu. V. Nazarov, Ya. M. Blanter, eds, Quantum Noise. Dordrecht: Kluwer, 2003 11. Levitov, L.S.: Counting statistics of charge pumping in open systems. http://arxiv.org/list/cond-mat/ 0103617, 2001 12. Levitov, L.S., Lee, H.W., Lesovik, G.B.: Electron counting statistics and coherent states of electric current. J. Math. Phys. 37, 4845–4866 (1996) 13. Levitov, L.S., Lesovik, G.B.: Charge distribution in quantum shot noise. JETP Lett. 58, 230–235 (1993) 14. Lundberg, L.-E.: Quasi-free ‘second quantization’. Commun. Math. Phys. 50, 103–112 (1976) 15. Muzykanskii, B.A., Adamov, Y.: Scattering approach to counting statistics in quantum pumps. Phys. Rev. B 68, 155304–155313 (2003) 16. Pilgram, S., Büttiker, M.: Statistics of charge fluctuations in chaotic cavities. Phys. Rev. B 67, 235308 (2003)
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17. Powers, R.T., Størmer, E.: Free states of the canonical anticommutation relations. Commun. Math. Phys. 16, 1–33 (1970) 18. Reed, M., Simon, B.: Methods of Modern Mathematical Physics: IV, Analysis of Operators. New York: Academic Press, 1978 19. De Roeck, W.: Large deviation generating function for energy transport in the Pauli-Fierz model. http:// arxiv.org/abs/0704.3400v3, 2007 20. Ruijsenaars, S.N.M.: Charged particles in external fields. I. Classical theory. J. Math. Phys. 18, 720–737 (1977) 21. Schönhammer, K.: Full counting statistics for noninteracting fermions: Exact results and the LevitovLesovik formula. Phys. Rev. B 75, 205329 (2007) 22. Seiler, E.: Schwinger functions for the Yukawa model in two dimensions with space-time cutoff. Commun. Math. Phys. 42, 163–182 (1975) 23. Seiler, R.: Quantum theory of particles with spin zero and one half in external fields. Commun. Math. Phys. 25, 127–151 (1972) 24. Shale, D., Stinespring, W.F.: Spinor representations of infinite orthogonal groups. J. Math. Mech. 14, 315–322 (1965) Communicated by M. Aizenman
Commun. Math. Phys. 280, 831–841 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0452-2
Communications in
Mathematical Physics
Zygmund Spaces, Inviscid Limit and Uniqueness of Euler Flows Piotr Bogusław Mucha1 , Walter M. Rusin2 1 Institute of Applied Mathematics and Mechanics, Warsaw University, ul. Banacha 2,
02-097 Warszawa, Poland. E-mail: [email protected]
2 School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis,
55455 MN, USA. E-mail: [email protected] Received: 24 May 2007 / Accepted: 7 August 2007 Published online: 13 March 2008 – © Springer-Verlag 2008
Abstract: The paper improves the classical uniqueness result for the incompressible Euler system in the n dimensional case assuming that ∇u E ∈ L 1 (0, T ; BMO()), only. Moreover the rate of the convergence for the inviscid limit of solutions to the NavierStokes equations is obtained, under the same regularity of the limit Eulerian flow. A key element of the proof is a logarithmic inequality between the Hardy and L 1 spaces which is a consequence of the basic properties of the Zygmund space L ln L. 1. Introduction The analysis of the evolutionary Euler system modeling the motion of incompressible flows in n dimensional bounded domains is the subject of this paper. We want to study the issue of uniqueness and the problem of the inviscid limit for the Navier-Stokes equations treated as an approximation of the system of inviscid flows. The classical results [6] and [16] require that solutions to the Euler system belong at least to the class of regularity which guarantees that the velocity is in the space 1 ()). Due to that fact we obtain the following estimate: u E ∈ L 1 (0, T ; W∞ T E ≤ C∇u E L (0,T ;L ()) v2 v · ∇u v d xdt (1.1) ∞ 1 L ∞ (0,T ;L 2 ()) , 0
which is the core of methods in [6,16]. Given inequality (1.1), uniqueness of solutions to the Euler system follows from elementary energy estimates. The goal of our paper is to improve the classical methods to the Euler system replacing L ∞ by the BMO-space. Because of relatively low regularity in the studied problem we cannot apply the properties of the BMO-space directly. A key element of our technique will be an application of properties of Zygmund spaces L ln L (see [20]). This analysis allows us to prove the following bound: wH1 () ≤ Cw L 1 () | ln w L 1 () | + ln(1 + w L ∞ () ) , (1.2) which measures the difference between the L 1 and Hardy space H1 .
832
P. B. Mucha, W. M. Rusin
We will study the uniqueness criteria which play an important role in analysis based on weak solutions, where – by definition – high regularity is not admitted. Weak solutions allow considerations of a larger class of external/initial data to obtain information almost the same as for smooth data. Thanks to (1.2) we will be able to prove that the criteria for the incompressible Euler system should guarantee that ∇u E ∈ L 1 (0, T ; B M O()), only, replacing the stronger condition from (1.1). Moreover the analysis will enable to consider the approximation of solutions to the Euler system by solutions to the NavierStokes equation with a small viscosity coefficient. The B M O-space is specially distinguished in two dimensions, because it is the target space for the imbedding H 1 (R2 ) ⊂ B M O(R2 ) (we cannot obtain L ∞ here). This case is of special interest to us, since we are able to point out good examples for which the L ∞ -regularity with respect to spatial coordinates is too strong. Our first result, being the fundamental tool of analysis of the Euler system, is the following Theorem 1.1. Let f ∈ B M O(Rn ), supp f – compact in Rn and g ∈ L 1 (Rn )∩ L ∞ (Rn ), then (1.3) n f g d x ≤ C f B M O(Rn ) g L 1 (Rn ) | ln g L 1 (Rn ) | + ln(1+g L ∞ (Rn ) ) . R
Theorem 1.1 is a version of the logarithmic Sobolev inequality for the Hardy and L 1 -spaces. The structure of (1.3) and its proof is essentially based on the properties of the Zygmund spaces L ln L – see [15,20]. Inequality (1.2) (or (1.3)) can be compared with a similar estimate between the L ∞ - and B M O-spaces from [7]. The authors have shown that n (1.4) f L ∞ () ≤ C 1 + f B M O() (1 + ln+ f W ps () ) for s > . p Inequality (1.4) helped to improve the classical result for the Euler system for the blowup criteria replacing L ∞ by the B M O-space. In our problems we cannot apply estimate (1.4), however it remains a motivation for Theorem 1.1. We want to apply Theorem 1.1 to analyze the Euler system u tE + u E · ∇u E + ∇ p E = 0 in × (0, T ), div u E = 0 in × (0, T ), n · u E = 0 on ∂ × (0, T ), u E |t=0 = u 0 in ,
(1.5)
where u E : × (0, T ) → Rn is the velocity field, p E : × (0, T ) → R is the pressure, n - unit, outward normal to ∂, u 0 : → Rn - a divergence-free initial velocity field. We exclude the external forces (the r.h.s. of (1.5)1 ), since it would not provide any new analytical difficulties, but only a few technical elementary estimates. We prove the following result concerning the issue of uniqueness of solutions to system (1.5) Theorem 1.2. Let be a bounded domain in Rn with smooth boundary. Let u 1E and u 2E be two solutions to the Euler system (1.5) with initial data u 0 such that ∇ u 1E , ∇ u 2E ∈ L 1 (0, T ; B M O()) and u 1E , u 2E ∈ L ∞ (0, T ; L 2+σ ()) for given σ > 0, then u 1E ≡ u 2E .
(1.6)
Zygmund Spaces, Inviscid Limit and Uniqueness of Euler Flows
833
The above result generalizes the classical theory. The proof of Theorem 1.2 is essentially based on Theorem 1.1 and an application of the Osgood theorem (see [5]) giving uniqueness in the ODE theory. Additionally, the properties of the B M O-space allow to exchange the gradient ∇u by the vorticity r ot u in the condition (1.6) which may simplify the application of Theorem 1.1. An improvement of the classical results as [6] and [16] can be found in [18] and [16]. However the relaxation of the L ∞ -regularity in the space (in [18] we even have a bit weaker space than B M O) forces that the regularity with respect to time is required to belong to the L ∞ -class. Hence the regularity with respect to time has to be much stronger than in (1.1). Our approach enables to keep the weak condition in the L 1 -norm in (0, T ). Moreover thanks to Theorem 1.1 we omit numerous technical estimates which often appear in results of that type. Our last result concerns the inviscid limit for solutions to the Navier-Stokes equations under the slip boundary conditions u νt + u ν · ∇u ν − div T(u ν , p ν ) = 0 in × (0, T ), div u ν = 0 in × (0, T ), n · T(u ν , p) · τ + α u ν · τ = 0 on ∂ × (0, T ), n · uν = 0 on ∂ × (0, T ), u ν |t=0 = u 0 in ,
(1.7)
where u ν : × (0, T ) → Rn is the velocity field, p ν : × (0, T ) → R is the pressure, n - unit, outward normal to ∂, τ - unit, tangent to ∂, T(u ν , p ν ) = νD(u) − p ν Id - stress tensor, α - describes the friction coefficient of the boundary, u 0 : → R - a divergence-free initial velocity field. At this point we could consider boundary conditions different than (1.7)3,4 , however by the results from [1,8,9], the form of (1.7) seems to be the most suitable for the issue. A comprehensive description of the properties of this type of boundary relations can be found in [4,10,11,19]. We prove the following Theorem 1.3. Let be a bounded domain in Rn with smooth boundary, let u ν be a solution to the Navier-Stokes system (1.7) and u E be the solution to the Euler system (1.5) both with initial divergence-free data u 0 . Fix T > 0, σ > 0 and consider ν such that 0 < ν ≤ 1. Assume that ∇u E (·, t) B M O() ≤ f 0 (t) and f 0 ∈ L 1 (0, T ), E L 2 () + ∇u (·, t) L 2 () ≤ g0 (t) and g0 ∈ L 2 (0, T ),
∇u ν (·, t) u ν (·, t)
L 2+σ ()
+ u E (·, t)
L 2+σ ()
(1.8)
≤ h 0 (t) and h 0 ∈ L ∞ (0, T ).
Then considering the inviscid limit of solutions to (1.7) we obtain sup (u ν − u E )(·, t) L 2 () → 0
as
ν → 0+ ,
(1.9)
0≤t≤T
where the precise rate can be expressed by the properties of functions f 0 , g0 and h 0 . If we assume extra that sup | f 0 (t) + g02 (t)| ≤ M,
(1.10)
0≤t≤T
then we obtain the following explicit rate of convergence sup (u ν − u E )(·, t) L 2 () ≤ Cν e
0≤t≤T
−2M T
.
(1.11)
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P. B. Mucha, W. M. Rusin
Theorem 1.3 gives general conditions for the inviscid limit to solutions of the Navier-Stokes equations, provided very low (lowest known) conditions on the regularity of solutions to (1.7) with respect to the viscosity coefficient ν. The main disadvantage is that in the general case we are not able to construct solutions fulfilling (1.8). However in a special case of two dimensions (see [12] and [9] for the case with homogeneous boundary data) we find a class of solutions to (1.7) which fulfill assumptions (1.10). Then by Theorem 1.3 we obtain an explicit rate of convergence to solution of the Euler system given by (1.11). A similar result has been known only for the two dimensional case [2] in whole space under the classical assumption ∇u E ∈ L 1 (0,√ T ; L ∞ (R2 )). The ν E rate of convergence of sup0≤t≤T u − u L 2 (R2 ) is estimated by ∼ νT , however the initial data considered in [2] correspond to a vortex patch – vorticity is localized to a bounded domain with smooth boundary. Maybe there is hope to find a realization of (1.8) by some class of solutions to the Navier-Stokes equations, however the problem seems to be challenging. In the proceeding, will be always understood as a bounded subset of Rn with smooth boundary ∂. Spaces (L p (), · L p () ), (L p (Rn ), · L p (Rn ) ) for p ∈ [1, ∞] denote the common Lebesgue spaces. Spaces B M O(Rn ) and B M O() are understood as space of locally integrable functions with corresponding semi-norms 1 || f || B M O(Rn ) = sup | f (s) − { f } B(x,r ) | ds x∈Rn ,r >0 |B(x, r )| B(x,r ) and
1 || f || B M O() = sup | f (s) − { f } B(x,r )∩ | ds, x∈Rn ,r >0 |B(x, r ) ∩ | B(x,r )∩ 1 where { f } A = |A| A f (s) ds, are finite. By C we denote a generic constant that is independent from ν.
2. Proofs of Theorems We start with the proof of the estimate (1.3) which will be used throughout the proceeding Proof of Theorem 1.1. Consider g ∈ H1 (Rn ). By characterization of H1 (Rn ) we have gH1 (Rn ) = g L 1 (Rn ) +
n
Rk g L 1 (Rn ) ,
k=1
where the Riesz transform is given in the usual way as F[Rk f k ] = |ξξk| F[ f k ]. Using the fact that B M O(Rn ) = (H1 (Rn ))∗ we get ≤ f B M O(Rn ) gH1 (Rn ) f g d x n R
n ≤ f B M O(Rn ) g L 1 (Rn ) + Rk g L 1 (Rn ) . (2.1) k=1
For the characterization of the Hardy space H1 (Rn ) the reader may refer to [3,13,14]. Hence it suffices to obtain an estimate on the L 1 -norm of Rk g. We use the classical Zygmund result that can be found in [15].
Zygmund Spaces, Inviscid Limit and Uniqueness of Euler Flows
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Proposition 2.1. Let h be a sufficiently smooth non-negative function with bounded support. Then n Rk h L 1 (R ) ≤ C + C h ln+ h d x, (2.2) Rn
where ln+ a = max{ln a, 0} and the constant C depends on the measure of support of h. To apply Proposition 2.1 we split g into its positive and negative parts and consider only the positive part which for simplicity is denoted by g. Estimates for the negativepart are analogous. By elementary scaling we change inequality (2.2) to get Rk g L 1 (Rn ) ≤ λ + C g ln+ g/λ d x Rn
for any λ ∈ R+ . Consider ln+ g/λ = ln g − ln λ for g ≥ λ, then g ln(g|{g≥λ} ) ≤ ln(1 + g L (Rn ) ) + ln |{g≥λ} . ∞ g n + 1 L ∞ (R ) Since
1 1+g L ∞ (Rn )
≤ 1 by properties of the logarithm we obtain
g n |{g≥λ} ≤ ln(1 + g L ∞ (Rn ) ) ∞ (R ) ) + ln g L ∞ (Rn ) + 1 λ . + ln g L ∞ (Rn ) + 1
ln(1 + g L
Since it suffices to consider λ ≤ g L ∞ (Rn ) , we get | ln(g|{g≥λ} )| ≤ 2 ln(g L ∞ (Rn ) + 1) + | ln λ|. Choose λ = g L 1 (Rn ) . We then have g ln(g L ∞ (Rn ) + 1)+| ln g L 1 (Rn ) | d x Rk g L 1 (Rn ) ≤ cg L 1 (Rn ) +2 n R ≤ cg L 1 (Rn ) 1 + 2 ln(g L ∞ (Rn ) + 1) + | ln g L 1 (Rn ) | . (2.3) Inequality (1.3) follows from inequalities (2.1),(2.3). Remark 2.1. Let be a bounded subset of Rn , f ∈ B M O(), g ∈ L 1 () ∩ L ∞ (). Then f g d x ≤ C f B M O() g L 1 () | ln g L 1 () | + ln(1 + g L ∞ () ) . (2.4)
Proof. Extending f, g by 0 outside we can apply Theorem 1.1. Now notice that for such extension g L 1 (Rn ) = g L 1 () , g L ∞ (Rn ) = g L ∞ () and f B M O(Rn ) ≤ C f B M O() . Such an extension of f is possible due to its local integrability. Inequality (1.3) is a key estimate in the proof of Theorem 1.2.
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Proof of Theorem 1.2. Let {u 1E , p1E } and {u 2E , p2E } be two different solutions of (1.5). Subtracting we get the momentum equation in the form (u 1E − u 2E ) + u 1E · ∇(u 1E − u 2E ) + (u 1E − u 2E )∇u 2E = −∇( p1E − p2E ).
(2.5)
Multiplying both sides by (u 1E − u 2E ) and integrating over we obtain 1 d (u 1E − u 2E )2 d x + u 1E ∇(u 1E − u 2E )(u 1E − u 2E ) d x (2.6) 2 dt + (u 1E − u 2E )∇u 2E (u 1E − u 2E ) d x = − (u 1E − u 2E )∇( p1E − p2E ) d x.
Integrating by parts, using boundary conditions and incompressibility of flow we reduce (2.6) to 1 d E E 2 (u − u 2 ) d x + (u 1E − u 2E )∇u 2E (u 1E − u 2E ) d x = 0. (2.7) 2 dt 1 Let α = (u 1E − u 2E )2 and β = ∇u 2E . Notice that from the assumptions on u 1E , u 2E and (2.7) it follows that u 1E − u 2E L 2 () ∈ C([0, T ]). We split α = αm + αr , where |αm | = min(|α|, m) for some m > 1. Upon Theorem 1.1 and Remark 2.1 we get |αβ| d x ≤ Cβ B M O() αm L () (1+| ln αm L () |+ln(1 + m))+ |αr β| d x. 1 1
(2.8) Denote
f (t) = 2Cβ(t) B M O , g(t) =
|αr β| d x, x(t) = u 1E − u 2E 2L 2 () .
Consider x(t) small enough so that the function |x(t) ln x(t)| is increasing, which by the continuity of x(t) is equivalent to restricting our attention to sufficiently small T0 . Then (2.8) can be restated as follows: |αβ| d x ≤ Cβ B M O() α L () (1 + | ln α L () | + ln(1 + m))+ |αr β| d x. 1 1
(2.9) Thus from (2.9) we obtain the following differential inequality: x˙ ≤ f (t)x(t)(| ln x(t)| + 1 + ln(1 + m)) + g(t), x(0) = 0.
(2.10)
To find a good estimate on x(t) we introduce the following equation: y˙ = f (t)y(t)(| ln y(t)| + 1 + ln(1 + m)) + g(t), y(0) = 1/m,
(2.11)
for some m large enough. From the Osgood existence theorem we know that there exists a unique local solution to (2.11). Additionally the r.h.s. of (2.11)1 guarantees that y(·) is increasing. This implies that the solution of (2.11) majorizes x(t), i.e.: 0 ≤ x(t) ≤ y(t) ≤ 1
for t ∈ [0, T0 ].
Zygmund Spaces, Inviscid Limit and Uniqueness of Euler Flows
837
Hence we investigate the behavior of solutions to (2.11). By Gronwall’s inequality we get
t 1 y(t) ≤ exp f (s) (| ln y(s)| + 1 + ln(1 + m)) ds m 0
t t g(s) exp f (τ ) (| ln y(τ )| + 1 + ln(1 + m)) dτ ds + 0 s (2.12)
t
t 1 ≤ exp (1 + ln(1 + m)) f (s) ds exp f (s)| ln y(s)| ds m 0 0
t
t t f (τ )| ln y(τ )| dτ exp (1 + ln(1 + m)) f (τ ) dτ g(s) ds. +exp 0
0
0
Since 1 ≥ y(t) ≥ 1/m implies | ln y(t)| ≤ ln m we can estimate the right-hand-side of (2.12) (up to multiplication by some constant) by t t t t t 1 (1 + m) 0 f (s) ds m 0 f (s) ds + m 0 f (s) ds (1+m) 0 f (s) ds g(s) ds m 0
t t t t 1 1 + g(s) ds ≤ (2m 2 ) 0 f (s) ds + = (m(1+m)) 0 f (s) ds |αr β| d xds . m 0 m 0 (2.13) This shows that it suffices to control the part αr . In this case estimates on the measure of the support come handy. Since α ∈ L ∞ (0, T ; L 1+σ/2 ()) and β ∈ L 1 (0, T ; B M O()) we have β ∈ L 1 (0, T ; L p ()) for any p < ∞, hence by elementary Hölder’s inequality |αr β| d x ≤ αr L 1+σ/4 () β L (1+σ/4) () ,
thus we obtain a bound y(t) ≤ (2m 2 ) ≤ (2m 2 )
t 0
t 0
t 1 + |αr β| d xds 0
m (2.14) 1 f (s) ds + αr L ∞ (0,T ;L 1+σ/4 ()) β L 1 (0,T ;L (1+σ/4) ()) . m f (s) ds
From the Chebyschev inequality we have
α L ∞ (0,T ;L 1+σ/2 ()) 1+σ/2 |supp αr | ≤ m
(2.15)
uniformly in time. Notice that by Hölder’s inequality
αr L 1+σ/4 () =
supp αr
|αr |
1+σ/4
1 1+σ/4
dx
2σ
≤ |supp αr | (4+σ )(2+σ ) · αr L 1+σ/2 () . (2.16)
Inequalities (2.16) and (2.15) imply σ
1+
σ
(4+σ ) αr L 1+σ/4 () ≤ m − 4+σ · α L 1+σ/2 () ,
(2.17)
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P. B. Mucha, W. M. Rusin
hence σ
1+
σ
) αr L ∞ (0,T ;L 1+σ/4 ()) ≤ m − 4+σ · α L ∞(4+σ (0,T ;L 1+σ/2 ()) .
The above estimate leads to the following: t σ y(t) ≤ C m 2 0 f (s) ds−1 + m − 4+σ 1+ σ 4+σ E E . × u 1 L ∞ (0,T ;L 2+σ ()) + u 2 L ∞ (0,T ;L 2+σ ()) Choose 0 < t1 ≤ T0 small enough so that 2
t1 0 σ
y(t) ≤ C(D AT A)m − 4+σ
(2.18)
(2.19)
σ f (s) ds − 1 < − 4+σ , then we have
for 0 ≤ t ≤ t1 .
(2.20)
Letting m → ∞ we get y(t) = 0 for 0 ≤ t ≤ t1 which implies x(t) = 0 for 0 ≤ t ≤ t1 hence u 1E = u 2E for 0 ≤ t ≤ t1 . We can continue this procedure starting at t = t1 and get uniqueness for all t ∈ [0, T ]. Estimate (1.3) can also be used to give insight into the rate of convergence in the inviscid limit of the system (1.7). Proof of Theorem 1.3. Let {u ν , p ν } and {u E , p E } be solutions to problems (1.7) and (1.5) respectively. Subtracting the momentum equations we get (u ν − u E )t − ν u ν + u ν · ∇(u ν − u E ) + (u ν − u E )∇u E = −∇( p ν − p E ). (u ν
(2.21)
and integrating over we obtain Multiplying both sides by 1 d (u ν − u E )2 d x − ν (u ν − u E ) u ν d x + (u ν − u E )∇(u ν − u E )u ν d x 2 dt + (u ν − u E )∇u E (u ν − u E ) d x = − (u ν − u E )∇( p ν − p E ) d x. (2.22)
− uE)
Integrating by parts, using boundary conditions with zero friction coefficient and incompressibility of flow we reduce (2.22) to 1 d (u ν − u E )2 d x + (u ν − u E )∇u E (u ν − u E ) d x 2 dt +ν Du ν D(u ν − u E ) d x = 0. (2.23)
(u ν
β = ∇u E . By Theorem 1.1 and Remark 2.1 we have Let α = |αβ| d x ≤ Cβ B M O α L () (1+| ln α L () | + ln(1 + α L ())). ∞ 1 1
− u E )2 ,
(2.24)
Let α = αν + αr so that |αν | = min( ν1 , |α|). Proceeding as in the proof of Theorem 1.2 we denote f (t) = 2Cβ(t) B M O() , g(t) = |αr β| d x, x(t) = u ν − u E 2L 2 () . From (2.24) we get the following inequality:
1 + g(t) + νg0 (t)2 , x˙ ≤ f (t)x(t) | ln x(t)| + 1 + ln 1 + (2.25) ν x(0) = 0.
Zygmund Spaces, Inviscid Limit and Uniqueness of Euler Flows
To find a good estimate on x(t) we introduce the following equation:
1 + g(t) + νg0 (t)2 , y˙ = f (t)y(t) | ln y(t)| + 1 + ln 1 + ν y(0) = ν
839
(2.26)
for some ν sufficiently small. From the Osgood existence theorem we know that there exists a unique local solution to (2.26). The solution of (2.26) majorizes x(t), i.e.: 0 ≤ x(t) ≤ y(t)
for t ∈ [0, T0 ],
where T0 is chosen by similar rules as in the proof of Theorem 1.2. From (2.26) by Gronwall’s inequality we have
t
1 y(t) ≤ ν exp ds f (s) | ln y(s)| + 1 + ln 1 + ν 0
t
t 1 2 dτ ds + g(s) + g0 (s) ν exp f (τ ) | ln y(τ )| + 1 + ln 1 + ν 0 s
t t 1 (2.27) ≤ ν exp 1 + ln 1 + f (s) ds exp f (s)| ln y(s)| ds ν 0 0
t
t 1 +exp 1 + ln 1 + f (τ ) dτ exp f (s)| ln y(s)| ds ν 0 0 t · (g(s) + g0 (s)2 ν) ds. 0
The condition y(t) ≥ ν for sufficiently small ν gives | ln y(t)| ≤ − ln ν = ln ν1 . Also let ν be small enough so that ν1 1 + ν1 ≤ ν22 , thus we can estimate the right-hand-side of (2.27) (up to a constant factor) by t
t f (s) ds t 2 0 2 0 f (s) ds + (g(s) + νg0 (s)2 ) ds ν 2 2 ν ν 0
t f (s) ds t t 2 0 2 ν + = |α β| d xds + ν g (s) ds dt . r 0 ν2 0 0
(2.28)
Since α ∈ L ∞ (0, T ; L 1+σ/2 ()) and β ∈ L 1 (0, T ; B M O()), hence β ∈ L 1 (0, T ; L p ()) for any p < ∞, thus we have |αr β| d x ≤ αr L 1+σ/4 () β L (1+σ/4) () ,
thus t ≤
t 2 ν2
0
2 ν2
f (s) ds
0
f (s) ds
t t ν + 0 |αr β| d xds + ν 0 |g0 (s)|2 ds
ν + αr L ∞ (0,T ;L 1+σ/4 ()) β L 1 (0,T ;L (1+σ/4) ()) + νg0 2L 2 (0,T ) . (2.29)
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P. B. Mucha, W. M. Rusin
From the Chebyshev inequality we notice that 1+σ/2 |supp αr | ≤ να L ∞ (0,T ;L 1+σ/2 ()) ,
(2.30)
uniformly in time. By Hölder’s inequality
1 1+σ/4 2σ 1+σ/4 αr L 1+σ/4 () = |αr | dx ≤ |supp αr | (4+σ )(2+σ ) · αr L 1+σ/2 () . supp αr
(2.31) Inequalities (2.31) and (2.30) imply σ
1+
σ
σ
1+
σ
4+σ αr L 1+σ/4 () ≤ ν 4+σ · α L 1+σ/2 () ,
(2.32)
hence αr L ∞ (0,T ;L 1+σ/4 ()) ≤ ν 4+σ · α L ∞4+σ (0,T ;L 1+σ/2 ()) .
(2.33)
As in the proof of Theorem 1.2 the above implies the following estimate
1+ σ t 4 4+σ + y(t) ≤ C ν 1−2 0 f (s) ds +ν 4+σ u ν L ∞ (0,T ;L 2+σ ()) +u E L ∞ (0,T ;L 2+σ ()) t Cν 1−2 0 f (s) ds ∇u ν 2L 1 (0,T ;L 2 ()) + ∇u ν 2L 1 (0,T ;L 2 ()) . (2.34) Choose 0 < t1 ≤ T0 small enough so that 1 − 2 there is
t1 0
f (s) ds <
σ 4+σ ,
σ
y(t) ≤ C(D AT A)ν 4+σ .
then for 0 ≤ t ≤ t1 (2.35)
Consider now, t1 ≤ t ≤ T and a problem analogous to (2.26) but with the initial conσ dition y(t1 ) = ν 4+σ . Repeating all above estimates we pick t1 < t2 ≤ T such that σ supt1 ≤t≤t2 u ν − u E L 2 () ≤ Cν 8+2σ . Due to integrability of f (t) iterating the procedure we eventually cover the whole interval [0, T ]. This way we obtain the explicit rate of the convergence which depends mainly on the structure of integrability of function f 0 . Thus we proved (1.9). An additional condition (1.10) improves the result and gives an explicit uniform rate of convergence. The basic estimate presented above gives
Mt 2 x(t) ≤ Cν (2.36) ν2 with M as in (1.10). Fix some n ∈ N and consider 0 ≤ t ≤ T /(2Mn). We have sup0≤t≤T /(2Mn) u ν − u E L 2 () ≤ Cν 1−T /n . The time interval has been divided into 2Mn parts and repeating the estimate we get sup u ν − u E L 2 () ≤ Cν (1−T /n)
2Mn
,
(2.37)
0≤t≤T
and taking limit n → ∞ we obtain sup u ν − u E L 2 () ≤ Cν e
0≤t≤T
Theorem 1.3 is proved.
−2M T
.
(2.38)
Zygmund Spaces, Inviscid Limit and Uniqueness of Euler Flows
841
Acknowledgements. The authors wish to express gratitude to Professor Przemysław Wojtaszczyk and Professor Vladimir Šverák for useful discussions. The first author (PBM) has been partly supported by Polish KBN grant No. 1 P03A 021 30.
References 1. Clopeau, T., Mikeli´c, A., Robert, R.: On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with friction type boundary conditions. Nonlinearity 11, 1625–1636 (1998) 2. Constantin, P., Wu, J.: Inviscid limit for vortex patches. Nonlinearity 8, 735–742 (1995) 3. Fefferman, C., Stein, E.M.: H p spaces of several variables. Acta Math. 129(3-4), 137–193 (1972) 4. Fujita, H.: Remarks on the Stokes flow under slip and leak boundary conditions of friction type, Topics in mathematical fluid mechanics. Quad. Mat. 10, 73–94 (2002) 5. Hartman, F.: Ordinary differential equations. NY-London-Sydney: John Wiley & Sons, 1964 6. Kato, T.: On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Rat. Mech. Anal. 25, 188–200 (1967) 7. Kozono, H., Taniuchi, Y.: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Commun. Math. Phys. 214(1), 191–200 (2000) 8. Masmoudi, N.: Remarks about the inviscid limit of the Navier-Stokes system. Commun. Math. Phys. 270(3), 777–788 (2007) 9. Mucha, P.B.: On the inviscid limit of the Navier-Stokes equations for flows with large flux. Nonlinearity 16(5), 1715–1732 (2003) 10. Mucha, P.B.: The Navier-Stokes equations and the maximum principle. Int. Math. Res. Not. 2004(67), 3585–3605 (2004) 11. Rencławowicz, J., Zaj¸aczkowski, W.M.: Weak solutions to the Navier-Stokes equations in a Y-shaped domain. Appl. Math. (Warsaw) 33(1), 111–127 (2006) 12. Rusin, W.M.: On the inviscid limit for the solutions of two-dimensional incompressible Navier-Stokes equations with slip-type boundary conditions. Nonlinearity 19(6), 1349–1363 (2006) 13. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, Princeton, NJ: Princeton University Press, 1970 14. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton, NJ: Princeton University Press, 1993 15. Torchinsky, A.: Real-variable methods in harmonic analysis. Pure and Applied Mathematics, 123. Orlando, FL: Academic Press, Inc., 1986 16. Yudovich, V.: Nonstationary flow of an ideal incompressible liquid. Zhurn. Vych. Mat. 3, 1032–1066 (1963) 17. Yudovich, V.: Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2, 27–38 (1995) 18. Vishik, M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. Cole Norm. Sup. (4) 32(6), 769–812 (1999) 19. Xiao, Y., Xin, Z.: On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Comm. Pure Appl. Math. 60(7), 1027–1055 (2007) 20. Zygmund, A.: Trygonometric Series. London-NY: Cambridge Univ. Press, 1959 Communicated by P. Constantin
Commun. Math. Phys. 280, 843–858 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0485-6
Communications in
Mathematical Physics
On Uniqueness in the General Inverse Transmisson Problem Victor Isakov Department of Mathematics and Statistics, Wichita State University, Wichita, KS 67206, USA. E-mail: [email protected] Received: 8 June 2007 / Accepted: 21 September 2007 Published online: 22 April 2008 – © Springer-Verlag 2008
Abstract: In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic elliptic equation with discontinuous conductivity coefficient. Uniqueness results are shown to be optimal. Hence the considered form can be viewed as a canonical form of isotropic elliptic transmission problems. Proofs use singular solutions of elliptic equations and complex geometrical optics. Determining an obstacle and boundary conditions (i.e. reflecting and transmitting properties of its boundary and interior) is of interest for acoustical and electromagnetic inverse scattering, for modeling fluid/structure interaction, and for defects detection.
1. Introduction We consider the problem of identifying a bounded Lipschitz open set D entering an elliptic transmission problem and some boundary and interior medium coefficients from a (local) Dirichlet-to Neumann map or from scattering data at fixed frequency. We assume that is a bounded Lipschitz domain or the whole space Rn , D is a bounded Lipschitz domain, D¯ ⊂ , and D + = \ D¯ is connected. Let u solve the following transmission problem: − u + + c+ u + = 0 in D + , −u − + c− u − = 0 in D, u + = a0 u − , ∂ν u + = a1 ∂ν u − + bu − on = ∂ D,
(1.1) (1.2)
with the Dirichlet data u + = g0 on ∂.
(1.3)
844
V. Isakov
We will assume that a0 , a1 ∈ C 2 (), b ∈ C(), c+ , c− ∈ L ∞ (). Under the condition a0 = −a1 on ,
(1.4)
the boundary value problem (1.1), (1.2), (1.3) is elliptic. In particular, for any g0 ∈ 1 H () where H () is a closed subspace of H 2 () of finite codimension, there is a + − 1 1 ¯ × H (D). We will give more detail after formulating solution (u , u ) ∈ H ( \ D) our main results. When 0 < a0 , 0 < a1 , 0 ≤ b, 0 ≤ c+ , 0 ≤ c− , then by using classical maximum principles one can guarantee uniqueness (and therefore solvability in the mentioned spaces) for the transmission problem (1.1), (1.2), (1.3). The conductivity ¯ can be reduced to equation div(a∇u) = 0 with a = a + ∈ C 2 ( D¯ + ), a = a − ∈ C 2 ( D) the problem (1.1), (1.2) by letting u + = (a + )1/2 u on D + , u − = (a − )1/2 u on D, a0 =
a+ a−
1 2
1
, a1 =
a− a+
21
,b =
1
1 2
a+ a−
1
1 2
(∂ν a + − ∂ν a − ),
1
c+ = (a + )− 2 (a + ) 2 , c− = (a − )− 2 (a − ) 2 on D. Other isotropic problems with interface also can be transformed into (1.1), (1.2), (1.3). So this problem can be viewed as a canonical form of isotropic elliptic problem with interface. We will show that this canonical form can be completely identified by all possible sets of boundary data. Since nonuniqueness in inverse problems is sometimes due to a form of a boundary value problem (which can be written in many different ways), our results suggest that (1.1), (1.2) is a standard (canonical) form for an isotropic inverse transmission problem. Let 0 be an open nonempty subset of ∂. Let us define the (local) Cauchy set C(0 ) = {(u + , ∂ν u + ) on 0 , u + = 0 on ∂ \ 0 }, where (u + , u − ) are all possible solutions to the transmission problem. We assume that c+ is a known function. Theorem 1.1. Assume that (1.4) holds, that a0 = a1 on ,
(1.5)
and that c+ is a known compactly supported function. Then C(0 ) uniquely determines D, αa0 , αa1 , αb on for some nonzero α which is constant on any connected component of ∂ D, and c− on D. We observe that the uniqueness result of Theorem 1.1 is optimal. Indeed, letting u − = αU − we obtain the new transmission problem for (u + , U − ) with transmission boundary coefficients αa0 , αa1 , αb and the same D and c− . Obviously, the Cauchy sets for the original and new transmission problems are the same. So one can only identify boundary coefficients modulo the multiplicative factor α. We emphasize also that we use data from a part of the boundary. When there is no obstacle, but c+ is not known, there is a recent progress in the problem with partial boundary data ([5] and more recently [8]). Now we consider the inverse scattering problem when = R3 and when in addition to Eqs. (1.1), (1.2) one requires that c+ = −k 2 outside some bounded set and u + (x) = eikξ ·x + u +0 (x),
(1.6)
Uniqueness in the General Inverse Transmisson Problem
845
where u +0 satisfies the Sommerfeld radiation condition lim r (∂r u + (x) − iku +0 (x)) = 0, as r = |x| → ∞.
(1.7)
As known [26], under the condition a0 a¯1 = C there is an unique solution to the scattering problem (1.1), (1.2), (1.6) and 1 A(σ, ξ, k) + u 0 (x) = +o . r r
(1.8)
(1.9)
Theorem 1.2. Under conditions (1.4), (1.5) and (1.8) the scattering pattern A(σ, ξ, k) given at all σ, ξ ∈ S 2 and some fixed k uniquely determines D, αa0 , αa1 , αb on for some nonzero α which is constant on any connected component of ∂ D, and c− on D. We briefly describe available results. The first uniqueness theorems for Lipschitz transparent obstacles from all possible boundary measuremenst (the Dirichletto-Neumann map) or from scattering data at fixed frequency was obtained by the author [10,11] where the method of singular solutions for domains identification was introduced. Solutions of elliptic equations with higher order singularities were constructed in [1]. This method and Schiffer’s method [19] are the only available techniques to prove uniqueness of general obstacles in inverse scattering at fixed frequency. Uniqueness of some analytic transparent obstacles follows from the methods and the results of Kohn and Vogelius [16]. Kirsch and Kress [14] applied the method of singular solutions to get the first uniqueness results for hard obstacles. In the anisotropic case most general results on identification of inclusions of different condictivity are obtained in [18]. Logarithmic stability estimates for inisotropic inclusions have been proven in [2]. These estimates are shown to be optimal. Partial uniqueness results for transmisson conditions (constant a0 , a1 = 1, b = 0) are obtained by Kirsch and Päivärinta [15] and more general results (uniqueness of ratios aa01 , ab0 for scattering problems) by a longer and more involved argument were achieved in [26]. In [3] there are uniqueness results and constructive methods for identification of obstacle D and of the special second order transmission boundary condition with applications to optical tomography. Various range type algorithms based on use of the method of singular solutions have been developed starting with the linear sampling method of Colton and Kirsch. The sampling method proved to be a valuable computational tool, however it does not produce new results about uniqueness or stability in the inverse obstacle problem and it is not yet justified in complete generality. A review of related results is given in [6,7]. When the scattering data are given at all frequencies there is a different method going back to J. Keller and developed by Majda and Taylor [21]. In [21] they recover the convex hull of an obstacle from high-frequency behavior of the scattering pattern by using the Minkowsky problem. Complete dynamic or scattering data (at all frequencies) uniquely determine the interior coefficients and obstacles in several interesting cases as follows from the Boundary Control Method [4,17], however the general problem (1.1), (1.2) with complete boundary data was not considered. The paper is organized as follows. In Sect. 2 we give two Runge type results on approximation of solutions of transmission problems in a subdomain by solutions in a larger domain and by scattering solutions. These results enable to remove several technical difficulties in proofs of Theorems 1.1, 1.2. In particular we can avoid study of structure of fundamental solution of
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the transmission problem with pole close to replacing it by fundamenal solutions of the Schrödinger equation in R3 with well-known singular structure. Since fundamental solutions of genetral transmission problems are not completely understood (at least for Lipschitz interfaces) Lemma 2.1 is indeed a major tool in proofs of our main results. This lemma also enables us to generate a complete Dirichlet-to-Neumann map from a partial one and might be useful in other situations. Approximation of solutions to (1.1), (1.2) by scattering solutions implies that the Dirichlet-to-Neumann map is uniquely determined by the scattering data and hence Theorem 1.1 implies Theorem 1.2. In Sect. 3 we prove the main Theorem 1.1. First we interpret the definition of a weak solution as orthogonality relations for solutions to (1.1), (1.2) and to the adjoint problem in . These relations remind of us energy integrals. Then by using approximation lemmas of Sect. 2 we extend these orthogonality relations onto solutions to (1.1), (1.2) with singularities outside obstacles. Assuming that there are two obstacles generating the same data we let singularity approach one of obstacles, then in one of the integrals in the orthogonality relations singularities of solutions multiply, this integral becomes unbounded, while the remaining integrals are bounded, and we arrive at a contradiction. After showing that two obstacles coincide we again use singular solutions in the orthogonality relations to show that ratios of the boundary coefficients are unique. At this stage by using approximation results of Sect. 2 we can extend orthogonality relations onto solutions to the Schrödinger equation inside an obstacle without transmission boundary conditions. This extension substantially simplifies proofs since structure of fundamental solutions of such equations is known and is easy to use. The concluding step of the proof is use of complex geometrical optics solutions to the Schrödinger equation inside an obstacle in the orthogonality relations to demonstrate that a leading coefficient a1 of the transmission boundary condition is unique (up to a multiplicative constant). To achieve our goal we also use some results on the vector Radon (so-called ray) transform. In the remaining part of then introduction we collect some results on the (direct) transmission problem. We denote by C generic constants depending only on , D, a0 , a1 , b, c. Any additional dependence will be indicated in parenthesis. Our first claim is that a solution (u + , u − ) to the transmission problem − u + + c+ u + = f 0+ + div(f + ) in D + ,
−u − + c− u − = f 0− + div(f − ) in D, −
u = a0 u + +
g0− ,
(1.10) −
−
∂ν u = a1 ∂ν u + bu + +
g1−
on ,
(1.11)
with the Dirichlet data (1.3) satisfies the following Schauder type estimate: u + (1) (D + ) + u − (1) (D) ≤ C( f 0+ (0) (D + ) +f + (0) (D + ) + f 0− (0) (D) + f − (0) (D)
+g0 ( 1 ) (∂) + g0− ( 1 ) () + g1− (− 1 ) () + u + (0) (D + ) + u − (0) (D)). 2
2
2
(1.12) We will show that the bound (1.12) follows from standard results about Sobolev spaces in Lipschitz domains and known elliptic estimates for equations with discontinuous coefficients.
Uniqueness in the General Inverse Transmisson Problem
847
¯ Letting a0 u − = w − we Indeed, we can assume that a0 is a positive function on . transform Eqs. (1.10), (1.11) into −u + + c+ u + = f 0+ + div(f + ) in D + , −div(a1 a0−1 ∇w − ) = (a0−1 a1 − c− a1 a0−1 )w − + div(w − (2a1 ∇a0−1 − ∇(a1 a0−1 )) + a1 f 0− + div(a1 f − ) − ∇a1 · f − in D,
u + = w − + g0− , ∂ν u + = a1 a0−1 ∂ν w − + (a1 (∂ν a0−1 ) + ba0−1 )w − + g1− on . By trace and extension theorems for Sobolev spaces (e.g., [22], p. 101, Lemma 3.36 1 1 with j = 0, 1) for g0 ∈ H 2 (), g1 ∈ H − 2 () there is w0+ ∈ H 1 () such that w0+ = g0− , ∂ν w0+ = g1 on , w0+ = 0 on ∂ and w0+ (1) () ≤ C(g0− ( 1 ) () + g1 (− 1 ) ()). 2
(a1 (∂ν a0−1 )
Letting g1 = following equations:
+
ba0−1 )w −
+
g1−
2
and w + = u + − w0+ on D + we obtain the
−w + = −c+ (w + + w0+ ) + f 0+ + div(f + + ∇w0+ ) on D + −div(a1 a0−1 ∇w − ) = (a0−1 a1 − c− a1 a0−1 )w − + a1 f 0− − ∇a1 · f − + div(a1 f − + w − (2a1 ∇a0−1 − ∇(a1 a0−1 )) in D, w + = w − , ∂ν w + = a1 a0−1 ∂ν w − on . Introducing w = w+ on D + , w = w − on D, a = 1 on D + , a = a1 a0−1 on D, f 0 to be −c+ (w + + w0+ ) + f 0+ on D + and (a0−1 (a1 ) − c− a1 a0−1 )w − + a1 f 0− − ∇a1 · f − on D, and f to be f + + ∇w0+ on D + , a1 f − + w − (2a1 ∇a0−1 − ∇(a1 a0−1 )) on D, we conclude that w satisfies the following elliptic equation in the divergent form: −div(a∇w) = f 0 + div(f) in . Well known energy estimates for this equation [20] imply that w(1) () ≤ C(g0 ( 1 ) (∂) + f 0 (0) () + f(0) () + w(0) ()) 2
≤ C(g0 ( 1 ) (∂) + w + (0) (D + ) + w0+ (1) (D + ) + w − (0) (D) 2
+ f 0− (0) (D) + f − (0) (D) + f 0+ (0) (D + ) + f + (0) (D + )).
(1.13)
According to the bounds on w0+ and the definition of g1 , w0+ (1) (D + ) ≤ C(u − ( 1 ) () + . . .), 4
where . . . denote the terms in the right side of (1.12). By known trace theorems for Sobolev spaces (e.g. [22], p. 100) u − ( 1 ) () ≤ Cu − ( 3 ) (D) ≤ εu − (1) (D) + C(ε)u − (0) (D) 4
4
848
V. Isakov 3
due to compactness of embedding of H 1 into H 4 . Using these bounds and the definition of w + , w − from (1.13) we obtain u + (1) (D + ) + u − (1) (D) ≤ C(εu − (1) (D) + u + (0) (D + ) + C(ε)u − (0) (D) + . . .). 1 Choosing ε = 2C we absorb the first term in the right side by the left side and complete the proof of (1.12). The energy bound (1.12) guarantees that the transmission problem (1.10), (1.11), (1.3) is Fredholm (in energy spaces) and hence uniqueness of solution implies its existence. If the solution is unique then the terms u + (0) , u − (0) on the right side of (1.12) can be dropped by a standard compactness-uniqueness argument. ¯ so that the transmisWe can choose a bounded smooth domain 0 containing sion problem with the Dirichlet data on ∂0 is uniquely solvable. Let + (x, y) be a fundamental solution to the equation −u + c+ u = 0 in . As known [1,23],
1 + 0+ (x, y), 4π |x − y|
+ (x, y) =
(1.14)
where | 0+ (x, y)| + |x − y||∇x 0+ (x, y)| ≤ C. The function (, y) = + (, y) + ¯ solves the homogeneous transmission problem (1.1), (1.2) if
0 (, y), y ∈ \ D,
0 (, y) solves the transmission problem (1.10),(1.11) with f 0+ = 0, f + = 0, f 0− = −c− (, y), f − = 0 and the transmission boundary data g0 = (a0 − 1) + (, y), g1 = (a1 − 1)∂ν + (, y) + (b − 1) + (, y). Observe that 1
+ (, y)( 1 ) () + ∂ν + (, y)(− 1 ) () ≤ C + (, y)(1) (D) ≤ Cdist (y, D)− 2 . 2
2
Hence from the bound (1.12) and remarks on Dirichlet uniqueness we have −2
(, y)= + (, y) + 0 (, y), +0 (, y)(1) (D + )+ − . 0 (, y)(1) (D)≤C(dist (y, D)) (1.15) 1
2. Approximation Results In the proofs of the main results we use some auxilliary statements. ¯ and Lemma 2.1. Let V be either D or an open set with V¯ ⊂ , D¯ ⊂ V , connected \ D, ¯ u − ∈ H 1 (D) satisfy the following differential with C 2 -boundary. Let u +0 ∈ H 1 (V \ D), 0 equations and transmission conditions: ¯ −u − + c− u − = 0 in D, − u +0 + c+ u +0 = 0 in V \ D, 0 0
− − + u +0 = a0 u − 0 , ∂ν u 0 = a1 ∂ν u 0 + bu 0 on = ∂ D.
(2.1)
If V = D then we only assume the second equation (2.1) for u − 0. 1 (V \ D) ¯ × H 1 (D)-approximated by solutions to the transThen (u +0 , u − ) can be H 0 mission problem (1.1), (1.2) which are zero on ∂ \ 0 .
Uniqueness in the General Inverse Transmisson Problem
849
Proof. As in standard proof of the Runge property we assume the opposite. Then by the ¯ H 1 (D) Hahn-Banach Theorem there are f ∗+ , f ∗− in the L 2 -dual space of H 1 (V \ D), such that f ∗+ (u + ) + f ∗− (u − ) = 0, for all solutions to the transmission problem, but f ∗+ (u +0 ) + f ∗− (u 0 ) = 0. (2.2) We recall that f ∗− can be defined by some functions f 0− , . . . , f n− ∈ L 2 (D) as ⎛ ⎞ n − − ⎝ f u− + f ∗− (u − ) = f j ∂ j u − ⎠ , u − ∈ H 1 (D). 0 D
j=1
can be defined similarly. We will extend f ∗+ onto the complement of V¯ as zero. It is known that by a small perturbation of we can achieve unique solvability of the transmission problem with the Dirichlet data on the outer boundary. So we can assume that for some Lipschitz domain (∗) containing and with (∂ \ 0 ) ⊂ ∂(∗) there 1 1 ¯ is an unique solution (u +∗ , u − ∗ ) ∈ H ((∗) \ D) × H (D) to the following adjoint transmission problem: − − − ¯ −u − −u +∗ + c+ u +∗ = f ∗+ in (∗) \ D, in D, (2.3) ∗ + c u ∗ = f∗ f ∗+
−1 + − u +∗ = a1−1 u − ∗ , ∂ν u ∗ = a 0 ∂ν u ∗ +
b − u on , a0 a1 ∗
(2.4)
with the Dirichlet data u +∗ = 0 on ∂(∗).
(2.5)
We re-call that weak solutions u +∗ , u − ∗ to the Eqs. (2.3) are defined by the integral identities ∂ν u +∗ u + + ∂ν u +∗ u + + (∇u +∗ · ∇u + + c+ u +∗ u + ) = f ∗+ (u + ), − ∂(∗) (∗)\D − − − − − − − − ∂ν u − u + (∇u − ∗ ∗ · ∇u + c u ∗ u ) = f ∗ (u ) (2.6)
H 1 ((∗) \
¯ u− D),
D
for all ∈ ∈ ¯ From (2.2) and (2.6) we have Let (∗) = ∂(∗) \ . + + + + 0=− ∂ν u ∗ u + ∂ν u ∗ u + (∇u +∗ · ∇u + + c+ u +∗ u + ) (∗) (∗)\D − − − − − − − ∂ν u ∗ u + (∇u − ∗ · ∇u + c u ∗ u ) D − − = (−u + ∂ν u +∗ + u +∗ ∂ν u + ) + (u + ∂ν u +∗ − u +∗ ∂ν u + − u − ∂ν u − ∗ + u ∗ ∂ν u ) u+
(∗)
H 1 (D).
due to the definition of weak solutions to −u + + c+ u + = 0 in (∗) \ D¯ and of −u − + c− u − = 0 in D. Using (2.5), the transmission boundary conditions (2.4) and continuing the previous equalities we yield b − − − 0=− u + ∂ν u +∗ + (a0 u − (a0−1 ∂ν u − u ) − a1−1 u − ∗ + ∗ (a1 ∂ν u ∗ + bu ) a0 a1 ∗ (∗) − − − u − ∂ν u − + u ∂ u ) = − u + ∂ν u +∗ . (2.7) ν ∗ ∗ (∗)
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V. Isakov
The equality (2.7) holds for all u + solving the transmission problem (1.1), (1.2), (1.3) in (∗) instead of with Dirichlet data which are zero on ∂(∗) \ (∗). Due to our choice of (∗) the Dirichlet data u + on (∗) can be an arbitrary compactly supported Lipschitz function on (∗), so from (2.7) we have ∂ν u +∗ = 0 on (∗). Since u +∗ solves the homogeneous second order elliptic equation on \ V¯ and has zero Cauchy data on (∗) by uniqueness in the Cauchy problem u +∗ = 0 on \ V . Let V be not D. Now due to the definition of weak solutions to (2.3) with test functions u +0 , u − 0, f ∗+ (u +0 ) + f ∗− (u − 0)
=−
∂V
u +0 ∂ν u +∗
+
(u +0 ∂ν u +∗ − u − ∂ν u − ∗)
− − − − (∇u +∗ · ∇u +0 + c+ u +∗ u +0 ) + (∇u − + ∗ · ∇u 0 + c u ∗ u 0 ) V \D D − − + + − + + = (u 0 ∂ν u ∗ − u 0 ∂ν u ∗ − u ∗ ∂ν u 0 − u − ∗ ∂ν u 0 ) = 0
− − due to definitions of weak solutions to −u +0 +c+ u +0 = 0 on V \ D¯ and −u − 0 +c u 0 = 0 on D and to the transmission conditions on , as in (2.7). If V = D, then as above − − − − − − f ∗− (u − ) = − u ∂ u + (∇u − ∗ · ∇u 0 + c u ∗ u 0 ) 0 0 ν ∗ D − − − = (−u 0 ∂ν u ∗ + u − ∗ ∂ν u 0 ) = 0,
− because u +∗ = 0 on (∗) \ D and hence u − ∗ = ∂ν u ∗ = 0 on . In both cases we have a contradiction with (2.2).
¯ c+ = −k 2 outside B and the transmission proLemma 2.2. Let B be a ball containing , blem (1.1), (1.2), (1.3) be uniquely solvable in H 1 . Then the scattering pattern A(σ, ξ, k) given for all σ, ξ and some fixed k uniquely determines the Dirichlet-to-Neumann map (∂ B). Proof. By the Rellich Theorem the scattering pattern uniquely determines the scattering solution u(; ξ, k) outside B. Hence Lemma 2.2 follows if we show that elements of ¯ span{u(; ξ, k)} can H 1 (B \ D)-approximate any solution to the transmission problem (1.1), (1.2), (1.3). As in the proof of Lemma 2.1, assuming the opposite we conclude that there is f ∗ ¯ and a solution (u + , u − ) to the transmission problem in the L 2 dual space of H 1 (B \ D) (1.1), (1.2), (1.3) such that f ∗ (u + (; ξ, k)) = 0 for all scattering solutions to the transmission problem, but f ∗ (u + ) = 0. (2.8) We extend the distribution f ∗ outside B¯ \ D as zero. Let (w + , w − ) be the solution of the transmission problem ¯ −w − + c− w − = 0 in D − , − w + + c+ w + = f ∗ in R3 \ D, b − u on , w + = a1−1 w − , ∂ν w + = a0−1 ∂ν w − + a0 a1 ∗
(2.9) (2.10)
Uniqueness in the General Inverse Transmisson Problem
851
satisfying the Sommerfeld radiation condition. From (2.8) as in the proof of Lemma 2.1 (derivation of (2.7)) we yield 0 = f ∗ (u + (; ξ, k)) = (−u + (; ξ, k)∂ν w + + w + ∂ν u + (; ξ, k)). (2.11) ∂B
From the radiation condition we have |u +0 (x)| ≤ C|x|−1 , |∇u +0 (x)| ≤ C|x|−2 and similar bounds hold for w+ . Since u +0 , w + solve the Helmholtz equation outside B, + + + + (u 0 (; ξ, k)∂ν w − w ∂ν u 0 (; ξ, k)) = (u +0 (; ξ, k)∂ν w + − w + ∂ν u +0 (; ξ, k)) ∂B
|x|=R
u +0 , w +
and letting R → ∞ we conclude that the for large R. Using the bounds on integral in the left side is zero and using (1.9) we can replace u + in (2.11) by eikξ ·. . To show that w + = 0 outside B we introduce the solution w0 to the Helmholtz equation in B with the Dirichlet data w0 = w + on ∂ B. Since both w0 and eikξ ·. solve the Helmholtz equation in B we have the equality (eikξ ·. ∂ν w0 − w + ∂ν eikξ ·. ) = 0, ∂B
and using (2.11) we yield
∂B
(∂ν w0 − ∂ν w + )eikξ ·. = 0
for all ξ ∈ S 2 . Since k 2 is not a Dirichlet eigenvalue for the Laplace operator in B and span{eikξ ·. } is (H 1 (B)-) dense in the space of all (H 1 -) solutions to the Helmholtz equation in B (see details in [11], Lemmas 3.2, 3.3) we can conclude from the last integral identity that ∂ν w0+ = ∂ν w + on ∂ B. Let us define w ∗ as w0 on B and as w + outside B. Since w0 , w + solve the Helmholtz equation in B and outside B¯ and have the same Cauchy data on ∂ B the function w ∗ solves the Helmholtz equation in R3 and it satisfies the Sommerfeld radiation condition (1.7), because w + does. Hence w ∗ = 0 in R3 which implies that w + = 0 on R3 \ B. Then as at the end of the proof of Lemma 2.1 + − + − + − + (replacing V, u +0 , u − 0 , u ∗ , u ∗ by B, u , u , w , w ) we conclude that f ∗ (u ) = 0, and we have a contradiction with (2.8).
3. Proofs of Main Results Proof of Theorem 1.1. Let two domains D1 , D2 and coefficients a01 , a11 , b1 , c1− , a02 , + − a12 , b2 , c2− produce solutions (u +1 , u − 1 ), (u 2 , u 2 ) with the same Cauchy sets. Slightly increasing we can assume that the Dirichlet transmission problems for two domains and two sets of coefficients are uniquely solvable. First we will show that D1 = D2 . Let D1 = D2 . Let D(e) be the connected component of \ ( D¯ 1 ∪ D¯ 2 ) whose boundary intersects ∂. Due to connectedness of \ D¯ 1 , \ D¯ 2 we can assume that there is a point x(0) ∈ ∂ D2 \ D¯ 1 which is contained ¯ in D(e). Since ∂ D2 is Lipschitz we can also assume that at x(0) there is the normal ν(x(0)) to ∂ D2 . Let us choose a ball B centered at x(0), B¯ ⊂ \ D¯ 1 . We will use x(δ) = x(0) + δν(x(0)) with small positive δ so that x(δ) ∈ B \ D¯ 2 . ¯ ¯ Let V be a neighborhood of D(i), where D(i) = \ D(e), with ∂ V ∈ C 2 . Slightly varying V we can assume that the transmission problem in V with the Dirichlet data on
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V. Isakov
∂ V is uniquely solvable for both sets of the coefficients. Let v be a H 1 (V )-solution to the equation − v + c+ v = 0 in V.
(3.1)
Since u +1 = u +2 , ∂ν u +1 = ∂ν u +2 on 0 and u +1 , u +2 solve (3.1) in D(e) by uniqueness in the Cauchy problem u +1 = u +2 on D(e) and therefore near ∂ V . Since u +j , v solve the differential equation (3.1) in V \ D¯ j , by the definition of weak solutions we have (v∂ν u +1 − u +1 ∂ν v) = (v∂ν u +1 − u +1 ∂ν v) = (v∂ν u +2 − u +2 ∂ν v). (3.2) ∂ D1
∂V
∂ D2
From the transmission boundary conditions on (B) = ∂ D2 ∩ B we have − − + + (v∂ν u 2 − u 2 ∂ν v) = (a12 v∂ν u − 2 + b2 u 2 v − a02 u 2 ∂ν v) (B)
(B)
(3.3)
for all solutions u 1 , u 2 to the transmission problems in with same Dirichlet data on ∂. Now we will extend the equalities (3.2), (3.3) to solutions in V . Let (u +1 , u − 1 ) be any H 1 (D1 ) × H 1 (D1 )- solution to the transmission problem. By Lemma 2.1 one can H 1 -approximate this solution by a sequence (u +1k , u − 1k ) of solutions in (which are zero on ∂ \ 0 ). Let (u +2k , u − ) be a solution of the transmission problem in with the 2k second set of the coefficients and the Dirichlet data u +2k = u +1k on ∂. Since both sets of the coefficients produce the same Cauchy sets, as above, u +2k = u +1k on D(e). Since u 1k converge (in energy norm), so do u +2k on V ∩ D(e). Again by energy estimates (1.12) in + − V it follows that (u +2k , u − 2k ) converge in energy norm on V to the solution (u 2 , u 2 ) of + + the transmission problem on V with Dirichlet data u 2 = u 1 on ∂ V . Since (3.2), (3.3) hold for u +1k , u +2k passing to limits we conclude that these equalities also hold for all H 1 -solutions u 1 , u 2 to the transmission problem in V with u +2 = u +1 on ∂ V . In addition, for such solutions u +1 = u +2 on V \ D(e). − − Using the definition of a weak solution to u − 2 = c2 u 2 we yield − − − a12 ∇u − · ∇v = a v∂ u − (∇a12 · ∇u − 12 ν 2 2 2 v + a12 c2 u 2 v), B∩D2 ∂(B∩D2 ) B∩D2 − − − a02 ∇u − · ∇v = a u ∂ v − (∇a02 · ∇vu − 02 2 ν 2 2 + a02 c u 2 v). B∩D2
∂(B∩D2 )
B∩D2
Using these relations, from (3.2), (3.3) we derive that − (a12 − a02 )∇u 2 · ∇v = (∂ν u +1 v − u +1 ∂ν v) B∩D2 ∂ D1 − − + + (∂ν u 2 v − u 2 ∂ν v) − b2 u 2 v + (a12 ∂ν u − − 2 v − a02 u 2 ∂ν v) ∂ D2 \(B) (B) ∂ B∩D2 − − − + − − (∇a12 · ∇u − (3.4) 2 v + a12 c2 u 2 v − ∇a02 · ∇vu 2 − a02 c u 2 v) D2 ∩B
+ − + + for all solutions (u +1 , u − 1 ), (u 2 , u 2 ) in V with u 2 = u 1 on ∂ V . We will use (3.4) for fundamental solutions u 1 = 1 (, y), v = + (, y) to these equations with poles at y = x(δ). As 1 (, y) we will use a fundamental solution to the
Uniqueness in the General Inverse Transmisson Problem
853
transmission problem with the first set of coefficients and obstacle D1 constructed at the end of the introductory section. Hence we have bound (1.15) with dist (y, D1 ) replaced 1 by δ. Observe that all terms on the right side of (3.4) are bounded by Cδ − 2 . Indeed, it is obvious for the first integral due to (1.14) and bound (1.15) since dist (y, D1 ) > C1 . The absolute value of the second integral is not greater than 1
(u +2 ( 1 ) (∂ D2 ) + ∂ν u +2 (− 1 ) (∂ D2 ))∇v(0) (∂ D2 \ (B)) ≤ Cu +2 (1) (D2 ) ≤ Cδ − 2 2
2
due to trace theorems, bounds (1.14), (1.15) and because dist (y, ∂ D2 \(B)) > C1 . The −1 + −1 + third integral is bounded by C|logδ| due to equality u − 2 = a02 u 2 = a02 u 1 on (B) and (1.14). The fourth and fifth integrals are bounded similarly to the second integral. We only give details for − − ∇a02 · ∇vu − = (∂ a )vu − (v∇a02 · ∇u − ν 02 2 2 2 + (a02 )vu 2 ), D2 ∩B
∂(D2 ∩B)
D2 ∩B
where we integrated by parts. Using again trace theorems the boundary integral is bounded by 1
3
Cv(0) (∂(D2 ∩ B))u 2 (1) (D2 ) ≤ C|logδ|δ − 2 ≤ Cδ − 4 . The integral over D2 ∩ B is bounded by 1
Cu 2 (1) (D2 ∩ B)v(0) (B) ≤ Cδ − 2 due to (1.14), (1.15).
As in the derivation of (3.4), by using the above bound on v, u − 2 we obtain (a12 − a02 )∇u − (a12 − a02 )u − 2 · ∇v = 2 ∂ν v B∩D2 ∂(D2 ∩B) − u− 2 div((a12 − a02 )∇v) B∩D2 a01 a01 (a12 − a02 )u − ∂ v − . . . = (a12 − a02 )∇u − = 1 ν 1 · ∇v + . . . , a a (B) 02 B∩D2 02 3
where . . . denotes terms bounded by Cδ − 4 . Hence from (3.4) we conclude that a01 (a12 − a02 )∇u − 1 · ∇v B∩D2 a02 3
is bounded by Cδ − 4 . On the other hand, using (1.14) we conclude that 1 |x − x(δ)|−4 − C|x − x(δ)|−3 ≤ (a12 − a02 )∇u − 1 · ∇v, C and hence as in [12], pp. 131–133, the integral is not less than a contradiction which shows that D1 = D2 .
1 −1 C δ . As δ
→ 0 we have
854
V. Isakov − − − Letting u − = u − 2 − u 1 and subtracting transmission conditions for u 2 , u 1 we yield
−u − + c2− u − = (c1− − c2− )u − 1 in D, a a11 1 01 − u− = − 1 u− , ∂ u = − 1 ∂ν u − (b1 a02 − b2 a01 )u − ν 1 1 + 1 on . a02 a12 a12 a02 (3.5) If −v2 + c2− v2 = 0 in D, v ∈ H 1 (D), then using the definition of a weak solution, we yield − − − (c2 − c1 )u 1 v2 = (v2 ∂ν u − − u − ∂ν v2 ) D a11 a01 1 − − − − 1 v2 ∂ν u 1 − − 1 u 1 ∂ν v2 + (b1 a02 − b2 a01 )u 1 v2 . = a12 a02 a12 a02 (3.6) Now we will use a special linear continuous extension operator of a function a ∈ Li p() ¯ To define such an operator we cover by balls B( j), j = to a function a ∗ ∈ Li p( D). 1, . . . , J, centered at such that ∩ B( j) after appropriate relabeling of coordinates is the graph of a Lipschitz function xn = γ (x1 , . . . , xn−1 ). Let χ j ∈ C0∞ (B( j)), be nonnegative, zero outside B( j) and Jj=1 χ 2j = 1 on . In B( j) we define a ∗j (x , xn ) = χ j (x)(aχ j )(x , γ (x )) and a ∗j = 0 outside B( j). Finally, we let a ∗ (x) =
J
a ∗j (x).
j=1
Since for Lipschitz A1 , A2 , − − − − − − − − A1 ∂ν u 1 v2 = A1 ∇u 1 · ∇v2 + (u − 1 ∇ A1 · ∇v2 + A1 c1 u 1 v2 ), D D − − − − − A2 ∂ν v2− u − = A ∇u · ∇v + (v2− ∇ A2 · ∇u − 2 1 1 2 1 + A2 c2 u 1 v2 ),
D
D
we have from (3.6) a11 ∗ 1 a01 ∗ − − − ∇u 1 · ∇v2 ) + − (b1 a02 − b2 a01 )u − 1 v2 . a a a 12 02 D 12 a02 a11 ∗ a01 ∗ − − − − ∇ = · ∇u 1 v2 − ∇ · ∇v2 u 1 a12 a02 D ∗ ∗ a11 a01 − u− (3.7) + c1− − c2− 1 v2 . a12 a02 So far this equality is valid for all solutions u − 1 to the transmission problem in . By − using Lemma 2.1 as above (3.7) can be extended to all solutions u − 1 , v2 to the equations
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− − ¯− (− + c1− )u − 1 = 0, (− + c2 )v2 = 0 near D . We will use (3.7) for fundamenal ¯ solutions to these equations with poles outside D. 11 01 11 ∗ 01 ∗ If aa12 − aa02 = 0 on , then we may assume that ε0 ≤ aa12 ) − ( aa02 ) on a ball B centered at a point x(0) ∈ . Let x(δ) = x(0) + δν(x(0)). Using the structure (1.14) of fundamental solutions as in [12], Sect. 5.2, we conclude that the left side in (3.7) is greater than (Cδ)−1 while the right side is bounded by C|logδ|. So we have a 11 01 contradiction which shows that aa12 − aa02 = 0 on . Now we will show that b1 a02 = b2 a01 on . If this equality does not hold, then as 11 01 above we can assume that ε0 < b1 a02 − b2 a01 on some ball B. Using that aa12 − aa02 =0 on , and hence a01 ∗ a11 ∗ − = 0 on D, a12 a02
and that |∇
a11 a12
∗
− − − −2 · (∇u − 1 v2 − ∇v2 u 1 )| ≤ C|x − x(δ)| , x ∈ D2 ∩ B,
we conclude from (3.7) that
1 − (b1 a02 − b2 a01 )u − 1 v2 = . . . , a12 a02
(3.8)
where . . . denotes a bounded function of δ. On the other hand, due to our assumption about b1 , b2 , as in [12], pp. 131–133, the left side in (3.8) is greater than C1 |lnδ|. So again we have a contradiction which shows that b1 a02 = b2 a01 on . 11 is constant on each connected component of ∂ D. Using Now we will show that aa12 already proved statements about boundary coefficients, from (3.6) we obtain a11 − (c2− − c1− )u − v = − 1 (∂ν u − (3.9) 1 2 1 v2 − u 1 ∂ν v2 ) a 12 D − − for all H 1 (D)-solutions u 1 , v2 to the equations −u 1 + c1− u − 1 = 0, −v2 + c2 v2 = 0 ¯ integrating by parts we yield in D. For A ∈ Li p( D), − − A(∂ν u − v − u ∂ v ) = (div(A∇u − 1 2 1 ν 2 1 )v2 − u 1 div(A∇v2 )) D − − − = (A(c1− − c2− )u − 1 v2 + (∇ A · ∇u 1 v2 − u 1 ∇ A · ∇v2 u 1 )). D
11 ∗ Hence from (3.6) letting A = ( aa12 ) − 1 we have
D
(c2−
− c1− )(1 +
A)u − 1 v2
=
− − (∇ A · ∇u − 1 v2 − u 1 ∇ A · ∇v2 u 1 )
(3.10)
− − − for all H 1 (D)-solutions u − 1 , v2 to the Schrödinger equations −u 1 +c1 u 1 = 0, −v2 + − c2 v2 = 0.
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Now we will make use of special almost complex-exponential solutions u − 1 , v2 . Let e3 be any direction in R3 . Let ξ be any nonzero vector orthogonal to e3 and let e1 be its direction. Let a direction e2 be orthogonal to both e1 , e3 . We define 1 1 |ξ |2 2 ξ |ξ |2 2 ξ 2 2 e2 + τ e3 , ζ (2) = − i τ + e2 − τ e3 . ζ (1) = + i τ + 2 4 2 4 As known [25,12], Sect. 5.3, for τ > C there are almost complex exponential soluiζ (1)·x (1 + w ), v (x) = eiζ (2)·x (1 + w ) with w (D) ≤ Cτ −1 , tions u − 1 2 2 j 2 1 (x) = e ∇w j 2 (D) ≤ C. Observing that ζ (1) + ζ (2) = ξ and using these solutions in (3.10) we yield − − iξ ·x (c2 − c1 )(1 + A)e (1 + w1 )(1 + w2 ) = (∇ A · i(ζ (1) − ζ (2))eiξ ·x D D + (∇ A · i(ζ (1) − ζ (2)(w1 + w2 + w1 w2 ) + (1 + w2 )∇w1 − (1 + w1 )∇w2 ))eiξ ·x . D
(3.11) From the definition ζ (1) =
ξ ξ + iτ e2 + τ e3 + O(τ −1 ), ζ (2) = − iτ e2 − τ e3 + O(τ −1 ), 2 2
and hence ζ (1) − ζ (2) = 2τ (ie2 + e3 ) + O(τ −1 ). Observing that the left side in (3.11) is bounded, and the second integral in the right side is bounded due to bounds on w j (D) and Hölder’s inequality, dividing all terms in (3.11) by τ and letting τ → ∞ we conclude that ∇ A · (e2 − ie3 )eiξ · = 0. D
Since we can change e2 to −e2 we conclude that ∇ A · e3 eiξ · = 0 D
for any ξ orthogonal to e3 . By using a partial Fourier transform, we conclude that ∇ A · e3 = 0 (3.12) l∩D
for any line l parallel to e3 . Let us extend the vector field ∇ A as zero onto R3 \D and denote the extension by F. Since (3.12) holds for any direction e3 , it follows that the ray ¯ By known results transform of the vector field F is zero. Let B be a ball containing D. [24], Theorem 5.1, there is a distribution f supported in a ball B containing D¯ such that F = ∇ f in B. By standard elliptic regularity f ∈ H 1 (B). We have ∇ f = F = 0 on ¯ hence f = 0 on B\ D. ¯ Since ∇(A − f ) = F − ∇ f = 0 on D we the domain B\ D, conclude that A − f = α, where α is constant on any connected component of D. Since D is Lipschitz, f ∈ H 1 (B) and f = 0 on B\D, we have f = 0 on ∂ D. Finally, A = α on ∂ D. 11 If aa12 is constant on any connected component of then we extend it as constant onto any connected component of D. Now using (3.9) and the formulas after it we yield
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11 (3.10) with A = aa12 − 1, and from the known [25,12], Sect. 5.3, completeness of {u − 1 v2 } − − we conclude that c1 = c2 in D. The proof is complete.
Proof of Theorem 1.2. By Lemma 2.2 the scattering pattern A(σ, ξ, k) given for all σ, ξ and some fixed k uniquely determines the Dirichlet-to-Neumann map (∂ B). Hence Theorem 1.2 follows from Theorem 1.1 with = B and 0 = ∂ B.
4. Conclusion We review some open problems. It should be not hard to obtain similar results for identification of cracks, when at the crack one assumes the general boundary condition. General boundary conditions for cracks make physical sense since there is no universally accepted mathematical model for a crack and probably one should find such a model by matching results of various experiments and looking for boundary conditions which best fit these experiments. With slight modifications we expect the results of this paper to hold in the two-dimensional case provided we assume that the interior potential is generated by a real-valued conductivity coefficient (as in the reduction after (1.4) and in [13]). It is interesting to obtain similar results for elliptic equations with first order terms modeling magnetic potential and Yang-Mills equations. In the general case we can not claim uniqueness of such equations [12], Sect. 5.5, but it is worth trying to find a canonical unique representative. Most advanced results on identification of magnetic potential can be found in [9] and [8]. We believe that their methods and the technique we used in this paper will be sufficient to obtain complete results for first order perturbations. The problem of great difficulty, see [12], Problem 5.8, is to find in addition the potential c+ outside an obstacle. We assumed c+ to be known. So far all attempts to solve this problem for nonanalytic coefficients failed, old ideas seem to be exhausted, and a new fruitful idea is needed. This new idea probably should be about spatial localization in the inverse problem for the Schrödinger equation, which is a quite delicate issue for elliptic equations. General anisotropic elliptic equations present a formidable challenge for inverse problems theory due to the substantial nonuniqueness of their identification from boundary measurements. Finding a canonical unique form of this equation among all equations producing the same boundary data is certainly of great interest for theory and applications. Acknowledgements. The author is grateful to Alexander Bukhgeim and Vladimir Scharafutdinov for fruitful discussions of the results and advice on vector tomography which helped to eliminate some geometrical restrictions of Theorems 1.1, 1.2 in the first version of this paper. This work was supported in part by the NSF grants DMS 04-05976 and DMS 07-07734 and Emylou Keith and Betty Dutcher Distinguished professorship at Wichita State University.
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4. Belishev, M.: Boundary Control in Reconstruction of Manifolds and Metrics. Inverse Problems 13, R1–R45 (1997) 5. Bukhgeim, A., Uhlmann, G.: Recovering a Potential from Partial Cauchy Data. Comm. Part. Diff. Eq. 27, 653–668 (2002) 6. Colton, D., Kress, R.: Inverse acoustic and electromagnetic scattering theory. New York: Springer-Verlag, 1998 7. Colton, D., Kress, R.: Using fundamental solutions in inverse scattering. Inverse Problems 22, R49–R66 (2006) ¨ 8. Dos Santos Ferreira, D., Kenig, C., Sjöstrand, J., Uhlmann, G.: Determining a magnetic Schrdinger operator from partial Cauchy data. Commun. Math. Phys. 271, 467–488 (2007) ¨ 9. Eskin, G., Ralston, J.: Inverse scattering problem for the Schrdinger equation with magnetic potential at a fixed energy. Commun. Math. Phys. 173, 199–224 (1995) 10. Isakov, V.: On uniqueness of recovery of a discontinuous conductivity coefficient. Commun. Pure Appl. Math. 41, 865–877 (1988) 11. Isakov, V.: On uniqueness in the inverse transmission scattering problem. Comm. Part. Diff. Eq. 15, 1565–1587 (1990) 12. Isakov, V.: Inverse Problems for Partial Differential Equations. New York: Springer-Verlag, 2006 13. Isakov, V., Nachman, A.: Global uniqueness for a two-dimensional elliptic inverse problem. Trans. AMS 347, 3375–3391 (1995) 14. Kirsch, A., Kress, R.: Uniqueness in Inverse Obstacle Problems. Inverse Problems 9, 285–299 (1993) 15. Kirsch, A., Päivärinta, L.: On recovering obstacles inside inhomogeneites. Math. Meth. Appl. Sci. 21, 619–651 (1998) 16. Kohn, R., Vogelius, M.: Determining Conductivity by Boundary Measurements, Interior Results, II. Comm. Pure Appl. Math. 38, 643–667 (1985) 17. Katchalov, A., Kurylev, Y., Lassas, M.: Inverse Boundary Spectral Problems. London: Chapman and Hall-CRC, 2000 18. Kwon, K.: Identification of anisotropic anomalous region in inverse problems. Inverse Problems 20, 1117–1136 (2004) 19. Lax, P., Phillips, R.: Scattering Theory. New York-London: Academic Press, 1989 20. Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. New York-London: Academic Press, 1969 21. Majda, A., Taylor, M.: Inverse Scattering Problems for transparent obstacles, electromagnetic waves, and hyperbolic systems. Comm. Part. Diff. Eq. 2, 395–438 (1977) 22. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge: Cambridge Univ. Press, 2000 23. Miranda, C.: Partial Differential Equations of Elliptic Type. New-York-Berlin: Springer-Verlag, 1970 24. Scharafutdinov, V.: Integral Geometry of Tensor Fields. Utrecht: VSP, 1994 25. Sylvester, J., Uhlmann, G.: Global Uniqueness Theorem for an Inverse Boundary Problem. Ann. Math. 125, 153–169 (1987) 26. Valdivia, N.: Uniqueness in Inverse Obstacle Scattering with Conductive Boundary Conditions. Applic. Anal. 83, 825–853 (2004) Communicated by B. Simon
Commun. Math. Phys. 280, 859 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0486-5
Communications in
Mathematical Physics
Publisher’s Erratum
Unstable and Stable Galaxy Models Yan Guo1 , Zhiwu Lin2 1 Lefschetz Center for Dynamical Systems, Division of Applied Mathematics,
Brown University, Providence, RI 02912, USA. E-mail: [email protected] 2 Mathematics Department, University of Missouri, Columbia, MO 65211, USA. E-mail: [email protected] Published online: 15 April 2008 – © Springer-Verlag 2008 Commun. Math. Phys. 279, 789–813 (2008)
The original version of this article unfortunately contained a mistake. The spelling of Zhiwu Lin’s name was incorrect in the HTML version.
The online version of the original article can be found under doi: 10.1007/s00220-008-0439-z
