Commun. Math. Phys. 245, 1–25 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1005-3
Communications in
Mathematical Physics
Two-Dimensional Gauge Theories of the Symmetric Group Sn in the Large-n Limit A. D’Adda1 , P. Provero1,2 1 2
Istituto Nazionale di Fisica Nucleare, Sezione di Torino and Dipartimento di Fisica Teorica dell’Universit`a di Torino, via P. Giuria 1, 10125 Torino, Italy. E-mail:
[email protected];
[email protected] Dipartimento di Scienze e Tecnologie Avanzate, Universit`a del Piemonte Orientale, 15100 Alessandria, Italy
Received: 19 November 2001 / Accepted: 3 November 2003 Published online: 16 January 2004 – © Springer-Verlag 2004
Abstract: We study the two-dimensional gauge theory of the symmetric group Sn describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disc and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of Sn , one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra logN compared to the standard ’t Hooft scaling. 1. Introduction There are several reasons for studying gauge theories of the symmetric group Sn in the large n limit. The first one is of course that the problem is interesting in itself, as a simple but non-trivial theory with non abelian gauge invariance. A second reason of interest is that gauge theories of Sn on a Riemann surface describe the statistics of the n-coverings of the surface, namely they address and solve the problem of counting in how many distinct ways the surface can be covered n times, without allowing folds but allowing branch points (see [1–3] and references therein). The distinct coverings of a two-dimensional surface are on the other hand the string configurations of a two-dimensional string theory, so that Sn gauge theories count the number of string configurations where the world sheet wraps n times around the target space. Another reason of interest is that gauge theories of Sn are closely related toYang-Mills theory in two dimensions, and to other gauge models, in particular the one introduced
2
A. D’Adda, P. Provero
by Kostov, Staudacher and Wynter (in brief KSW) in [1, 2]. Both YM2 [4, 5] and the KSW model with U (N ) gauge group can in fact be interpreted in the large N limit in terms of coverings of the two dimensional target space. In the present paper we consider the gauge theory of the symmetric group Sn in its own right, in the limit where n, namely the world sheet area, becomes large. This limit is different from the usual large N limit of U (N ) gauge, although a close relation between the two exists, as discussed in Sect. 5. The results obtained in this paper can be summarized as follows: – We find a non trivial phase structure in the large-n limit of the Sn gauge theory on a disc or a sphere, with a phase transition at a critical value of the “area” of the target surface1 . This is reminiscent at first sight of the Douglas-Kazakov [8] phase transition for two dimensional Yang-Mills theories, but it is in fact a completely different phenomenon, as discussed below and in full detail in Sect. 5. – We prove the equivalence between gauge theories in two dimensions and random walks on group manifolds. In the case of Sn this allows us to map our results, in particular the aforementioned phase transition, into statements about random walks on Sn . The phase transition found in the Sn gauge theory precisely corresponds to the cutoff phenomenon in random walks on Sn discovered in [15]. – We discover that the same cutoff phase transition occurs also in 2D Yang-Mills theory. This phase transition is associated to an unorthodox rescaling of the coupling constant with N , a rescaling that differs from the standard ’t Hooft rescaling by a factor log N . In this paper however we merely prove the existence of such a transition, whose details should be the object of further investigation. – Finally using a correspondence [18] between cutoff phenomena in random walks and phase transitions in random graphs we establish that the different phases in the large n limit of the Sn gauge theory can be interpreted in terms of connectivity properties of the associated world sheet. However, while these results are rigorous on a disc with free boundary conditions, their extension to a sphere may require an additional hypothesis as discussed in Sect. 4.3. The paper is organized as follows: in Sect. 2 we review the correspondence between Sn gauge theories and branched coverings, defining the models we are going to study. In Sect. 3 we study the partition function on the sphere by a saddle point analysis of the sum over the irreducible representations of Sn . This allows us to identify two lines of large-n phase transitions in the phase diagram of the model. In Sect. 4 we show the equivalence between two-dimensional gauge theories and random walks on group manifolds. This equivalence allows us to reinterpret the phase transition as a cutoff phenomenon in the corresponding random walk. The approach through the equivalent random walk is particularly suited to study the partition function on a disc with free boundary conditions, where we find a similar, if less rich, phase diagram. A mapping into the theory of random graphs allows us to determine that the order parameter in the case of the disc is the connectivity of the world sheet, and to draw some conclusions also on the connectivity of the world sheet in the case of the sphere. In Sect. 5 we establish the relation between Sn gauge theory in the large n limit and U (N ) gauge theories (Yang-Mills theory, chiral Yang-Mills theory and KSW model) in the large N limit and we prove the existence of a cutoff phenomenon also for 2D Yang-Mills. Section 6 is devoted to some concluding remarks and possible developments. Some technical details are discussed in the appendices. 1
The exact meaning of “area” in this context will be given in the next section.
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
3
2. The Model In this section we introduce the lattice gauge theory of the group Sn whose large n limit we are going to discuss in the following sections. We begin by briefly reviewing how an Sn gauge theory describes the statistics of the n-coverings of the Riemann surface, referring the reader to ref. [1, 2] for a more extensive discussion. The statistics of the n-coverings of a Riemann surface MG of genus G is given by the partition function of a Sn lattice gauge theory, defined on a cell decomposition of MG . This can be seen by the following argument (more details can be found in Ref. [3]). To construct a branched n-covering, consider n copies of each site of the cell decomposition: these will be the sites of the covering surface. For each link of the target surface, joining the sites p1 and p2 , join each of the n copies of p1 to one of the n copies of p2 ; repeat for all the links of the target surface to define a discretized covering. Each possible covering is defined by a choice of the copies to be glued for each link of the target surface, that is by assigning an element of the symmetric group Sn to each link. In general, such a covering will have branch points: consider a closed path on the target surface, and lift it to the covering surface, by starting on one of the n sheets of the covering and changing the sheet according to the element of Sn associated to the links defining the path. The lifted path is not in general closed, that is the covering is branched. Branch points are located on the plaquettes of the cell decomposition, and can be classified according to the conjugacy class of the element of Sn given by the ordered product of the elements associated to the links of the plaquette. For example, let n = 3 and consider a plaquette bordered by three links, to which the following permutations are associated: P1 = (12)(3), P2 = (13)(2), P3 = (12)(3),
(1) (2) (3)
then the permutation associated to the plaquette is P1 P2 P3 = (1)(23)
(4)
so that a quadratic branch point is associated to the plaquette. The type of branch point on each plaquette is determined by the conjugacy class of the product of the Sn elements around the plaquette, and is therefore invariant under a local Sn gauge transformation defined according to the usual rules of lattice gauge theory. Therefore a theory of n-coverings, in which the Boltzmann weight depends only on the type of branch points that are present on each plaquette, is a lattice gauge theory defined on the target surface, with gauge group Sn . The phase structure of such theories in the large n limit is the object of our study. Let us consider first the case of unbranched coverings. The Boltzmann weight associated to the plaquette s is simply δ(Ps ): the product of gauge variables around the plaquette is constrained to be the identity2 of Sn : 1 dr chr (Ps ). (5) w(Ps ) = δ(Ps ) = n! r In the l.h.s. of (5) the delta function is expressed as an expansion in the characters of Sn , with r labeling the representations of Sn , dr the dimension of the representation 2
In our notation δ(P ) is one if P is the identity in Sn and zero otherwise.
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A. D’Adda, P. Provero
r and chr (P ) the character of P in the representation r. The partition function of this model on MG is simply given by [3]: Zn,G =
dr 2−2G r
n!
.
(6)
The partition function (6) depends only on the genus G of the surface, namely the underlying theory is a topological theory. When branch points are allowed, the topological character of the theory is lost, and the partition function develops a dependence on another parameter, which we shall call “area” and denote by A, but which is not necessarily identified with the area of MG . All we require is that A is additive, namely that if we sew two surfaces (for instance two plaquettes) the total “area” is the sum of the “areas” of the constituents. In fact, by mimicking (generalized)Yang-Mills theories, one can replace [3] the Boltzmann weight (5) of the topological theory with a Boltzmann weight that allows branch points: 1 dr chr (Ps )eAs gr , n! r
w(Ps , As ) =
(7)
where gr are arbitrary coefficients and As is the area of the plaquette s. The crucial property of this Boltzmann weight is that, if s1 and s2 are two adjoining plaquettes and Q the permutation associated with their common link, we have:
w(QP1 , As1 )w(P2 Q−1 , As2 ) = w(P1 P2 , As1 + As2 ).
(8)
Q
That is by summing over Q we obtain the same Boltzmann weight that we would have if the link corresponding to the variable Q had been suppressed in the original lattice. The partition function on a Riemann surface of genus G can be easily calculated from (7) by using the orthogonality properties of the characters: Zn,G (A) =
dr 2−2G r
n!
eAgr .
(9)
As discussed in [3] the exponential factor in the partition function (9) can be thought of as due to a dense distribution of branch points, in which a branch point associated to a permutation Q appears with a probability density gQ , which is related to the coefficients gr by: gr =
Q=1
gQ
chr (Q) . dr
(10)
It is clear from (10) that gQ is a class function, namely it depends only on the conjugacy class of Q. While completely general the model defined by the Boltzmann weight (7) and the partition function (9) depends on as many independent couplings gr as the irreducible representations of Sn and is unsuitable for studying the large n limit. We are going then to restrict our considerations to models with only quadratic branch points, namely we
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
5
shall consider the case in which gQ = 0 for all Q s, except for the one consisting of a single exchange. In this case the partition function takes the form: Zn,G (A) =
dr 2−2G A chr (2) e dr , n! r
(11)
where by 2 we denote a transposition, namely a permutation with one cycle of length 2 and n − 2 cycles of length 1. In (11) the area has been redefined to absorb the factor g2 . The theory defined by the partition function (11) is the exact analogue, for the Sn gauge group, of two dimensional Yang-Mills theory. In fact the quantity chdr r(2) at the exponent coincides essentially with the quadratic Casimir C2 (r) of the U (N ) representation labeled by the same Young diagram as the representation r of Sn : C2 (r) = n(n − 1)
chr (2) + nN. dr
(12)
Instead the characters at the r.h.s. of (10) are related to higher Casimirs, and the general model defined in (7) and (9) is correspondingly the Sn analogue of the so-called generalized Yang-Mills theories [12, 13]. The number p of quadratic branch points in (11) is not fixed, and in fact the area A is the chemical potential relative to such number. The partition function at fixed p can in fact be obtained as the coefficient of Ap /p! in the expansion of (11) in a power series of A. The resulting partition function is: Zn,G,p =
dr 2−2G chr (2) p r
n!
dr
.
(13)
Here the number p of quadratic branch points is additive and plays the role of the “area” 3 . In fact the partition function (13) can be obtained from a lattice consisting of p plaquettes of unit area, each endowed with just one quadratic branch point and hence carrying a Boltzmann weight: w(Ps ) =
1 chr (2)chr (Ps ). n! r
(14)
If the p plaquettes are joined together to form a closed surface of genus G, the partition function (13) is reproduced by using again the orthogonality and fusion properties of the characters. We are ready now to introduce the most general model containing only quadratic branch points. This is obtained by assigning to each plaquette a probability x to have a single quadratic branch point and a probability 1 − x not to have any branch point at all. This amounts to consider a model with p plaquettes of Boltzmann weight w(Ps ) =
1 [(1 − x)dr + xchr (2)] chr (Ps ), n! r
(15)
which leads to the following partition function: 3 Notice that one can only have an even number of quadratic branch points on a closed Riemann surface, so that the partition function (13) vanishes for odd p.
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A. D’Adda, P. Provero
Zn,G,p (x) =
dr 2−2G r
chr (2) (1 − x) + x dr
n!
p .
(16)
In addition to the genus G this partition function depends on two independent parameters: the “area” (number of plaquettes) p and the density of branch points x. For x = 1, Zn,G,p (x) obviously coincides with Zn,G,p , but also the partition function Zn,G (A) of Eq. (11) can be obtained from Zn,G,p (x) by taking the double scaling limit x → 0 and p → ∞ keeping A = xp fixed: Zn,G,p (x) −→ e−A Zn,G (A). x→0
(17)
The partition function (16) will be the object of our investigation in the large n limit. In particular we shall study the case of the target space with the topology of a sphere (G = 0), where the model has a non-trivial phase diagram in the (x, p) plane provided p is rescaled with n in a suitable way. The existence of a phase transition at a critical value on the area p and fixed x can be argued by a simple qualitative argument. Consider a disc of area p, with holonomy at the border given by a permutation Q. The corresponding partition function (consider for simplicity the case x = 1) is then 1 chr (2) p Zn,disc,p (Q) = dr chr (Q) . n! r dr
(18)
Clearly for p < n a permutation Q consisting of a number of exchanges larger than p cannot be constructed out of p quadratic branch points, and ZN,disc,p (Q) is then necessarily zero for such Q. Instead if p n it is conceivable that beyond a critical value of p and in the large n limit all permutations have the same probability to appear on the border. Hence beyond the critical value of p, Zn,disc,p (Q) becomes independent of Q and constant in p. The existence of such a critical value will be proved in the next section, more precisely we shall prove that if we put p = αn log n a phase transition occurs in the large n limit at a critical value of α. The argument can be easily extended from the case of the disc to the one of the sphere. In fact as a sphere is a disc with the holonomy Q = 1, the same phase transition occurs on the sphere and it is seen as the critical point beyond which the partition function becomes constant in α. In the random walk approach that we will treat in Sect. 4 the same phase transition can be interpreted as a cutoff phenomenon, namely as the existence of a critical value of the number of steps after which the walker has the same probability to be in any point of the lattice. If instead of keeping x fixed we consider the more general model (16) we find, in the large n limit, a phase diagram in the (α, x) plane, with three different phases whose features we shall also study in the following sections. 3. Large n Limit – The Variational Method In this section we shall take the large n limit of the partition function (16) on a sphere and study its phase structure in the (x, p) plane. The large n limit will be obtained by determining the saddle point in the sum over the irreducible representations at the r.h.s. of (16), namely the representation that dominates the sum in the large n limit.
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
7
3.1. Representations of Sn in the large n limit. The large n limit of the symmetric group Sn is quite different from, say, the large N limit of U(N ). The difference consists mainly in the fact that the irreducible representations of SU(N ) are labeled by Young diagrams made by at most N − 1 rows and an arbitrary number of columns. Therefore in the large N limit its rows and columns can simultaneously scale like N . For instance in two dimensional Yang Mills theory on a sphere or a cylinder [11] the saddle point at large N corresponds to a representation whose Young diagram has a number of boxes of order N 2 . Instead, the irreducible representations of Sn are in one to one correspondence with the Young diagrams made of exactly n boxes. Namely the area of the Young diagrams, rather than the length of its rows and columns, scales like n. Let us first establish some notations. We shall label the lengths of the rows of the Young diagram by the positive integers r1 ≥ r2 ≥ . . . ≥ rs1 and the lengths of the columns by s1 ≥ s2 ≥ . . . ≥ sr1 , with the constraint that the total number of boxes is equal to n: s1
ri =
i=1
r1
sj = n.
(19)
j =1
In order to evaluate the partition function (16) in the large n limit we need the explicit expression of the dimension dr of the representation and of chdr r(2) . These are well known quantities in the theory of the symmetric group, and are given respectively by: dr = i≤sj
n! (r + sj − i − j + 1) , j ≤ri i
(20)
and
chr (2) 1 1 2 2 2 ri − 2iri + n . (21) = ri − sj = dr n(n − 1) n(n − 1) i
j
i
The l.h.s. of (21) coincides, up to a factor, with the quadratic Casimir for the representation of a unitary group associated to the same Young diagram. As we are interested in the evaluation of these quantities in the large n limit, we have first of all to characterize a general Young diagram consisting of n boxes in the large n limit. The most natural ansatz, in order to have a diagram of area n, would be to assume that the columns sj and the rows ri scale with n respectively as nα and n1−α with 0 ≤ α ≤ 1. However this is far from being the most general case, as different parts of the diagram may scale with different powers α. To be completely general let us introduce in place of the discrete variables i (resp. j ) labeling the rows (resp. columns) the variables, continuous in the large n limit, defined by ξ=
log i , log n
η=
log j . log n
(22)
With this definition a point (ξ0 , η0 ) in the (ξ, η) plane represents a portion of the diagram whose rows and columns scale as nξ0 and nη0 respectively. The Young diagram is represented by the functions ϕ(ξ ) =
log ri , log n
ψ(η) =
log sj , log n
(23)
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A. D’Adda, P. Provero
where in the large n limit ψ(η) = ϕ −1 (η) or, more precisely ψ[ϕ(ξ )] = ξ + o(1). We recall that the integers {ri } form a monotonic sequence, namely ri ≤ rj for i > j . It follows then from the definition (23): ϕ(ξ1 ) ≤ ϕ(ξ2 )
f or
ξ1 > ξ2 .
(24)
The same is obviously true for ψ(η). Although nothing can be said about the continuity of ϕ(ξ ) Eq. (24) ensures that the limit from above and from below of ϕ(ξ ) for ξ → ξ0 exists for any ξ0 . The discrete sum over i is replaced in the large n limit by an integral in dξ :
−→ dξ log n eξ log n , (25) i
and correspondingly the constraint (19) becomes: lim
n→∞
In = 1, n
(26)
where
ψ(0)
In =
dξ log n elog n (ξ +ϕ(ξ )) .
(27)
0
It can easily be proved, taking (24) into account, that Eq. (26) poses the following constraints on ϕ(ξ ): i. ξ + ϕ(ξ ) ≤ 1 for all ξ ’s. If this condition were not satisfied the limit at the l.h.s. of (26) would diverge. ii. There is at least one value ξ0 for which4 limε→0+ (ξ0 + ϕ(ξ0 − ε)) = 1. It can easily be checked that if this condition were not satisfied the limit in (26) would give zero. iii. If ξ + ϕ(ξ ) = 1 in a whole interval a ≤ ξ ≤ b, then In > (b − a)n log n and the limit in (26) diverges. So the points ξ0 for which limε→0+ (ξ0 + ϕ(ξ0 − ε)) = 1 form a discrete set 5 . We may conclude that in the large n limit the contributions of order n to In come from the neighborhoods of the discrete set of points αt for which αt + limε→0+ ϕ(αt − ε) = 1. If we represent the Young diagram in the (ξ, η) plane, as in Fig. 1, these points are the ones where the diagram touches the line ξ +η = 1. Each of these points corresponds to a portion of the Young diagram, which we shall denote by Yt , whose area is of order n and where rows and columns scale respectively like nαt and n1−αt . Let zt , with 0 < zt ≤ 1, be the fraction of the area n of the Young diagram which is taken by Yt . zt is given by an integral over the neighborhood of the point αt : αt + log n (ξ +ϕ(ξ )) α − dξ log n e . (28) zt = t n We do not assume the continuity of ϕ(ξ ) at ξ = ξ0 . It can actually be proved that it is a finite set. In fact an accumulation of such points would again lead to In of order n log n. This result is not essential for what follows and the proof’s is rather involved, so it will be omitted here. 4 5
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
9
η
Y1 Y2 Y3 α1
α2 α3
Y4 α4
ξ
Fig. 1. A Young diagram in the (ξ, η) plane. Only the points touching the line ξ + η = 1 contribute in the leading order in the large-n limit
In the large n limit the r.h.s. of (28) is independent of , which can be chosen arbitrarily small. The constraint (26) now reads in terms of the zt ’s: zt = 1. (29) t
In order to obtain the large n limit of the partition function (16) we need to calculate the large n behavior of dr and chdr r(2) . One way to do this would be to express these quantities in terms of the functions ϕ(ξ ) and ψ(η), to replace the discrete sums in the large n limit by integrals over the logarithmically rescaled variables ξ and η and extract from the integrals the leading contributions in the large n limit following essentially the same procedure used in the previous discussion. This procedure is completely rigorous but involves rather lengthy and involved calculations. We prefer to follow a more, euristic (but of course in the end equivalent) path which consists in calculating the contribution of each portion Yt of the Young diagram separately. The advantage of doing so is that since rows and columns in Yt scale respectively like nαt and n1−αt , the large n limit can be more conveniently done by introducing continuum variables xt and yt related to the discrete variables i and j by: xt =
i , nαt
yt =
j n1−αt
.
(30)
While Yt is represented by a single point in the plane (ξ, η) of the logarithmically rescaled variables, as shown in Fig. 1, it is “blown up” and occupies a finite area in the plane of
10
A. D’Adda, P. Provero
the new variables (xt , yt ). In this way we are able to calculate the large n limit of dr and chdr r(2) up to the next to leading order. The latter depends not just on the area zt of Yt (like the leading order) but also on its “shape” and it is depressed only by a factor log n with respect to the leading order. Although not relevant for studying the phase transitions described in the present paper, this next-to-leading order will be important for any further investigation of the model. In fact there is suggestive evidence that more phase structure (in particular the analogue of the Douglas-Kazakov phase transition) would emerge if this order was to be taken into account. The details of the calculation are given in Appendix A. Here we just give the leading term in the large n limit for the relevant quantities. Consider first the dimension dr of the representation. It is shown in Appendix A that the contribution of Yt to log dr in the large n limit is αt zt n log n for αt ≤ 1/2 and (1 − αt )zt n log n for αt ≥ 1/2. In the large n limit log dr is given by the sum of these contributions: log dr = n log n 1/2z1/2 + α t zt + (1 − αt )zt + o(n log n), (31) t:αt <1/2
t:αt >1/2
where z1/2 is short for zt with αt = 1/2. As for chdr r(2) its large n behavior is dominated by the regions of the Young diagram (which we shall denote by Y0 and Y1 ) corresponding to a scaling power α = 0 and α = 1 respectively. All other contributions, coming from Yt with αt different from 0 or 1, are depressed by a factor n−αt if αt ≤ 1/2 or nαt −1 if αt ≥ 1/2, and can be neglected in the large n limit. Y0 (resp. Y1 ) consists of a finite number of columns (resp. rows) of lengths ri (resp. sj ) proportional to n, namely: ri = nfi
sj = ngj ,
(32)
where fi and gj are finite in the large n limit. Clearly the areas z0 and z1 of Y0 and Y1 are given in terms of fi and gj by: z0 = fi , z1 = gj . (33) i
j
The large n limit of the leading term of (32) and it is the given by:
chr (2) dr
can now be easily deduced from (21) and
chr (2) 2 2 = fi − gj + o(1). dr i
(34)
j
Notice that while chdr r(2) depends in the large n limit only on z0 and z1 , log dr depends only on the zt ’s with αt different from 0 and 1 as the coefficients of z0 and z1 in (31) vanish. This means that dr and chdr r(2) are only coupled through the constraint (29). Therefore, in order to find the representation r that dominates in the large n limit of (16) we can then proceed in the following way: first find separately the extrema of chdr r(2) and log dr at fixed zˆ defined by zˆ ≡ z0 + z1 = 1 − zt (35) t=0,1
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
11
and then find the extremum in zˆ . The extrema of | chdr r(2) | and log dr can be read directly from (34) and (31) respectively. The maximum of | chdr r(2) | corresponds either to f1 = zˆ or to g1 = zˆ , with all the other coefficients fi and gj equal to zero. In other words either Y0 is a diagram consisting of a single column of length zˆ , or Y1 is a diagram with a single row of length zˆ . In either case we have |
chr (2) | = zˆ 2 . dr
(36)
For log dr it is clear from (31) that the maximum, at fixed zˆ , occurs when the whole contribution comes from Y1/2 , namely from a region of the Young diagram, of area (1 − zˆ )n, where both rows and columns scale as n1/2 . The large n limit of such diagram is: log dr =
1 n log n(1 − zˆ ). 2
(37)
In conclusion the sum over the irreducible representations in the large n limit of (16) is dominated by a representation whose Young diagram consists of two parts: a single row (or column) of length zˆ n and a region of area (1 − zˆ )n, where rows and columns scale as n1/2 . A last variation with respect to zˆ is all we need to determine the saddle point. 3.2. Phase transitions. Consider first the partition function (13) on a sphere, namely at G = 0. This can be written as: 1 2 log dr +p log chdrr(2) Zn,G=0,p = 2 e . (38) n! r In the large n limit the exponent at the r.h.s. of (38) can be approximated to the leading term by using (36) and (37). Moreover, as the number of representations only grows √ n like the number of partitions of n, namely in the large n limit as e , their entropy is negligible compared to the leading term in (38), and the sum over all representations is given by the contribution of the representation for which the exponent is maximum. As discussed in the previous section such representation is parametrized by z˜ , and we are led to the problem of finding the maximum, with respect to variations of z˜ , of n log n 1 − z˜ + 2A log z˜ , (39) where, in order to have both terms of the same order in the large n limit, we have set p = An log n.
(40)
The maximum of (39) is at z˜ = 2A. This solution however is valid only for A < 1/2, as the value of z˜ is limited to the interval (0, 1). So the model has two phases: for A < 1/2 the Young diagram of the leading representation consists of a single row (or column) of length 2An and a part of area (1 − 2A)n whose rows and columns scale like n1/2 . For A > 1/2 the sum over the representations is dominated by the representation consisting of a single row (or column) of length n. Let us consider now the more general model whose partition function is given in (16). A difference with respect to the previous case is that p does not need to be even, and the symmetry with respect to the exchange of rows and columns in the representations is broken. So for each value of z˜ there are two representations, whose contribution differ
12
A. D’Adda, P. Provero 1
III
0.8
0.6
A
II
0.4
I 0.2
0 0
0.2
0.4
0.6
0.8
1
x Fig. 2. Phase diagram in the (x, A) plane. The phase transitions between phases I/III and II/III are first order, while the II/III transition is second order
for the sign of chr (2). It easy to check from (16) that the contributions coming from representations with positive sign of chr (2) are always greater in absolute value, and give rise to the leading term in the large n limit. The leading representation is then obtained by finding the value of z˜ , constrained by 0 ≤ z˜ ≤ 1, for which
(41) n log n 1 − zˆ + A log 1 − x + x zˆ 2 is maximum. The phase structure is more complicated than in the previous case, and it is represented in the (x, A) plane in Fig. 2. The phase labeled in the figure with I corresponds to zˆ = 0, namely to a situation where the dominant representation is entirely made of rows and columns that scale as n1/2 . This phase did not exist in the previous case (x = 1) except for the trivial point A = 0. Phase II and III are the ones already studied at x = 1 and correspond respectively to 0 < z˜ < 1 and z˜ = 1. The critical line that separates I from III can be easily calculated, and is given by: AI,I I I = −
1 log(1 − x)
(42)
while the critical line separating phase II and III is simply: AI I,I I I =
1 . 2x
(43)
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
Finally the line that separates phase I and phase II is given by 1−x AI,I I = , ϕx
13
(44)
where ϕ can be expressed in terms of the coordinate xc of the triple point: ϕ = 4xc (1−xc ). The triple point is at the intersection of (42) and (43)and its coordinate xc is the solution of the transcendental equation log(1−x)+2x = 0. Its numerical value is xc = 0.796812.., from which one also obtains ϕ = 0.647611... The critical line (43) is what one expects, according to the results of the x = 1 model, from an effective number of branch points equal to the number of plaquettes An log n times the probability x for a plaquette to have a branch point. However the phase diagram discussed above shows that such naive expectation is not fulfilled everywhere. This can be understood by calculating the large n limit of (16) in a slightly different way. We first expand the binomial in (16) and write: Zn,G=0,p (x) = =
p dr 2 p chr (2) k (1 − x)p−k x k k n! dr r k=0 p k=0
p k
(1 − x)p−k x k Zn,G=0,k .
(45)
In the large n limit we parametrize p as in (40), and k as k = λp. The sum over k is replaced by an integral over λ that can be evaluated using the saddle point method. In doing this the large n solution for Zn,G=0,k must be used, keeping in mind that this consists of two phases, one for 2λA > 1 and one for 2λA < 1. The calculation reproduces the phase diagram of Fig. 2. The saddle point corresponds to λ = 0 in phase I, to λ = x in phase III and to 0 < λ < x in phase II. The free energy in the different phases can be obtained from (41) by replacing z˜ with the relevant saddle point solution. So we have: Fn (A, x) = n log n 1 − z(A, x) + A log[1 − x + xz(A, x)2 ] , (46) where z(A, x) = 0 if the point (A, x) is in I, z(A, x) = 1 if (A, x) is in III, while for (A, x) in II we have 1−x z(A, x) = A + A2 − . (47) x As the free energy is known exactly in the large n limit in any point of the (A, x) plane, the order of the phase transitions can be explicitly calculated. The (I,II) and the (I,III) phase transitions are of first order, while the (II,III) phase transition is a second order phase transition with the second derivative of Fn (A, x) with respect to A finite everywhere but with a discontinuity at the critical point A = 1/2x. The three phases are characterized by different connectivity properties of the world sheet. We have not been able to investigate these properties within the variational approach, so we have to rely on the equivalence between gauge theories of Sn and random walks on one hand, and between random walks and random graphs on the other. These will be discussed in the following sections; in particular the connectivity of the world sheet in the different phases is discussed in Sect. 4.3, as a corollary of well known results in the theory of random graphs.
14
A. D’Adda, P. Provero
T1
Tp
T2
Fig. 3. A cell decomposition of the disc. The permutations on the dashed links have been gauge-fixed to the identity, so that the ones on the boundary are forced to be transpositions. The ordered product of the p transpositions gives the holonomy
Finally let us consider the model given by the partition function (11). As already pointed out, this coincides with the small x limit of (16) provided we put A = xp = 1 xAn log n. For small x the first order phase transition occurs at A = − log(1−x) , hence for (11) at A = n log n. 4. Gauge Theories on a Disc as Random Walks on the Group Manifold A two-dimensional gauge theory on a disc is equivalent to a random walk on the gauge group manifold, the area of the disc being identified with the number of steps and the gauge theory action with the transition probability at each step. This result is completely general with respect to the choice of the gauge group and of the action, as we will prove below. However it might be useful to show first how this result emerges in the Sn gauge theory defined by Eq. (13), that is in a theory where all plaquettes variables are forced to be equal to transpositions. Suppose we want to compute such a partition function on a disc with a fixed holonomy Q ∈ Sn , Eq. (18). The latter shows that the partition function depends only on the holonomy Q and the area of the disc (that is the theory is invariant for the area-preserving diffeomorphism). It follows that we can freely choose any cell decomposition of the disc made of p plaquettes, e.g. the one shown in Fig. 3. To compute the partition function means to count the ways in which we can place permutations P on all the links in such a way that • the ordered product of the links around each plaquette is a transposition, • the ordered product of the links around the boundary of the disc is a permutation in the same conjugacy class as Q. Now we can use the gauge invariance of the theory to fix all the radial links to contain the identical permutation. At this point the links on the boundary are forced to contain
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
15
transpositions: therefore the partition function with holonomy Q is the number of ways in which one can write the permutation Q as an ordered product of p transpositions. This in turn can be seen as a random walk on Sn , in which, at each step, the permutation is multiplied by a transposition chosen at random: the gauge theory partition function for area p and holonomy Q is the probability that after p steps the walker is in Q. 4.1. The correspondence for a general gauge theory. To show that this result actually holds for all gauge groups and choice of the action, consider now a gauge theory on a disc of area p with gauge group G and holonomy g ∈ G on the disc boundary. To fix the notations, we will consider a finite group G, but the argument can be extended to Lie groups. The theory is defined by a function w(g) such that the Boltzmann weight of a configuration is given by the product of w(gpl ) over all plaquettes of the lattice, with gpl the ordered product of the links around the plaquette. For the theory to be gauge invariant, w has to bea class function; moreover we will require w(g) ≥ 0 for all g and normalize w so that g w(g) = 1. The partition function is [16, 17] 1 p dr w˜ r χr (g), (48) Zp (g) = |G| r where the sum is over all irreducible representations of G, χr (g) is the character of g in the representation r, and the w˜ r ’s are the coefficients of the character expansion of the Boltzmann weight: 1 dr w˜ r χr (g). (49) w(g) = |G| r Now consider a random walk on G with transition probability defined as follows: if the walker is in gp ∈ G at step p, then its position at step p + 1 is obtained by left multiplying gp by an element g chosen in G with a probability t (g) which is a class function, i.e. depends on the conjugacy class of g only. Suppose the random walk starts in the identity of G, and call Kp (g) the probability that the walker is in g after the pth step. Then t (g g −1 )Kp (g ). (50) Kp+1 (g) = g
Now assume Kp (g) is a class function, with character expansion Kp (g) =
1 (p) dr kr χr (g), |G| r
(51)
then it follows from Eq. (50) that also Kp+1 is a class function, and the coefficients of its character expansion are (p+1)
kr
= t˜r kr , (p)
(52)
where the t˜r ’s are the coefficients of the character expansion of the class function t: 1 dr t˜r χr (g). (53) t (g) = |G| r
16
A. D’Adda, P. Provero
Now, since K1 (g) = t (g), it follows by induction that Kp is indeed a class function, and (p)
kr
p = t˜r
(54)
so that the probability distribution after p steps of the random walk equals the gauge theory partition function Zp (g) provided the Boltzmann weight of the plaquette in the latter is identified with the transition function of the former: w(g) = t (g).
(55)
In conclusion, the partition function of a gauge theory on a disc, of area p with a certain holonomy g on the disc boundary, equals the probability that a random walk that starts in the identity of the group will reach the element g in p steps, each step consisting of left multiplication by an element chosen with a probability distribution coinciding with the plaquette Boltzmann weight of the gauge theory. 4.2. Cutoff phenomenon in random walks on Sn . The cutoff phenomenon in random walks was discovered in Ref.[15], where a random walk on Sn was studied in which at each step the permutation is multiplied by the identical permutation with probability 1/n and by a randomly chosen transposition otherwise. According to the argument of the previous section, this corresponds to our model Eq.(16) with x = 1 − 1/n. The holonomy of the gauge theory translates into constraints on the element of Sn where the random walk ends: for example the partition function on the sphere will count the walks that return to the identical permutation in p steps. The main result of Ref. [15] is that if the number of steps scales as An log n, then in the large-n limit for A > 1/2 the probability of finding the walker in any given element Q ∈ Sn is just 1/n! for all Q: complete randomization has been achieved and all memory of the initial position of the walker has been erased. In terms of the corresponding gauge theory, this can be translated into a statement about the partition function on a disc: for A > 1/2, the partition function with any given holonomy Q stops depending on A and is simply proportional to the number of permutations in the conjugacy class of Q. This is true in particular for Q = 1, corresponding to the partition function on the sphere. Therefore the phase transition found in Sect. 3 has a natural interpretation as a cutoff phenomenon in the corresponding random walk. Strictly speaking, this applies only to the specific model x = 1 − 1/n studied in Ref. [15]. However we want to argue that this is the correct interpretation of the whole line 1 of phase transitions at A = 2x found in Sect. 3. Consider first the model with x = 1, where at each step the permutation is multiplied by a random transposition. The probability distribution does not have a limit as the number of steps goes to infinity, since for an even number of steps one can only obtain an even permutation and vice versa. Therefore the probability distribution in Sn can never become uniform. However it is natural to expect that a sort of cutoff phenomenon occurs all the same at the number of steps p = 1/2 n log n, and precisely that for even p the probability distribution becomes uniform in the alternating group and for odd p in its complement. To support this conjecture, let us compute e.g. the expected number of cycles of length 1 in the permutation obtained after p steps. The calculation is described in Appendix B, and the result is n−3 p N1 (x = 1, p) = 1 + (n − 1) (56) n−1
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
so that for p = An log n we have N1 (x = 1) ∼
n1−2A for A ≤ 1/2 1 for A > 1/2.
17
(57)
The result for A > 1/2 is the one expected for a uniform probability distribution in the alternating group or its complement. Repeating the calculation for arbitrary x one finds 1−2xA n for A ≤ 1/(2x) N1 (x) ∼ (58) 1 for A > 1/(2x) so that the cutoff phenomenon occurs at A = 1/(2x), the result one intuitively expects from the fact that a fraction x of the random walk steps are “wasted” in doing nothing and do not contribute to the randomization process. 4.3. Results from random graphs theory. In the previous sections we have mainly considered the theory defined on a sphere: we have found a complex phase structure with first and second-order phase transitions. One of the transition lines can be interpreted as a cutoff phenomenon in the corresponding random walk. In this section we consider the theory with free boundary conditions: the permutations on the boundary of the disc are summed over like the internal ones. From the point of view of the random walk, this implies considering all the possible paths irrespective of the permutation they end in after p steps. We will show that for this particular choice of boundary conditions we can exploit the correspondence between cutoff phenomena in random walks and sharp thresholds in random graphs [18] to show the existence of a phase transition separating a phase in which the branched covering is connected from a phase in which it is disconnected. Consider the random walk in Sn defined by x = 1, i.e. at each step the permutation is multiplied by a random transposition (but all the arguments we will give translate trivially to the case x < 1). From the point of view of coverings, a step in which the transposition (ij ) is used corresponds to adding a simple branch point that connects the two sheets i and j of the covering surface. One can think of the process as the construction of a random graph on n sites where at each step a link between two sites is added at random. After p = An log n steps the expected number of links is equal to the number of steps (since the number of available links is O(n2 ) the fact that the same link can be added more than once can be neglected in the large n limit; see Ref. [18]). It is a classic result in the theory of random graphs [20, 21] that if the number of links p is smaller than 21 n log n then the graph is almost certainly disconnected while for p > 1/2 n log n the graph is almost certainly connected, where “almost certainly” means that the probability is one in the limit n → ∞. Since the connectedness of the random graph is exactly equivalent to the connectedness of the corresponding branched covering, we conclude that the model with free boundary conditions undergoes a phase transition at A = 1/2, where the covering surface goes from disconnected to connected. For x < 1, the same transition occurs at A = 1/(2x). Notice that the free boundary conditions are crucial for this argument to work: in the case, say, of the sphere, the corresponding random walk is forced to go back to the initial position in p steps, so that links in the graph are not added independently and the result of Ref. [20, 21] do not apply. However some consequences can be drawn from these results also for the case of the sphere. Consider first the model with x = 1 and a sphere
18
A. D’Adda, P. Provero
of area A > 1. The latter can be thought of as two discs, both with area A > 1/2, joined together. The partition function of the sphere is then obtained by multiplying together the partition functions of the two discs and by summing over the common holonomy P on the border. If the world sheets of the two discs are almost certainly connected, the same applies to the world sheet of the resulting sphere. Strictly speaking this proof holds only for A > 1, however we have shown that the leading term of order n log n of the free energy is independent of A for A > 1/2. So unless a phase transition occurs due to the next leading term of order n (which cannot be ruled out a priori) the whole phase with A > 1/2 at x = 1 (and extending the argument to x < 1 the whole phase III) will be characterized by a connected world sheet. What can be said about phase II and I? We mentioned already the result in the theory of random graphs that for a number of links p smaller than 1/2n log n the graph is almost certainly disconnected. Although this result applies to the case of free boundary conditions, it should a fortiori be true also for the sphere. In fact the sphere corresponds to a random walk which is forced to end in the identical permutation, thus favoring graphs in which less links are turned on. So for A < 1/2 at x = 1, namely in phase II, and a fortiori in phase I we expect the world sheet to be disconnected. Another result in random graphs [20, 21] states that if the number of links p grows like n log n with > 0, the size nc of the largest connected graph is n in the large n limit, namely limn→∞ nc /n = 1. Again, although the result is proved for graphs corresponding to random walks with free boundary conditions, it can be extended, at x = 1, to a sphere of area n log n which can be thought of as obtained by sewing two discs of area /2n log n. Both phase III and phase II are then characterized by the presence of a connected world sheet of size nc ∼ n in the large n limit, but phase III has completely connected world sheets while phase II has not. In phase I the number of effective branch points grows slower than any n log n, possibly like αn. If that is the case (a detailed analysis of next to leading terms would be required to check this point) then another result [20, 21] of random graphs could be applied. This states that if the number of links is αn in the large n limit, then for α > 1/2 the largest connected part has size ψ(α)n with ψ(1/2) = 0 and ψ(∞) = 1. The function ψ(α) is known and can be written as an infinite series. The point α = 1/2 is the percolation threshold, its existence may be an indication of further phase structure within phase I. 5. Relation with Two Dimensional Yang-Mills Theories Besides being linked to the theory of random walks and random graphs, Sn gauge theory is also closely related to lattice U (N ) gauge theories on a Riemann surface. This was first discovered by Gross and Taylor [4, 5] , who found that the coefficients of the large N expansion of the U (N ) partition function could be interpreted in terms of string configurations, namely of coverings of the Riemann surface. In the case of U (N ) Yang-Mills theory the maps from the string world sheet to the target space have two possible orientations and world sheets of opposite orientations can interact only through point-like singularities. As a result the theory almost exactly factorizes into two copies of a simpler, orientation preserving chiral theory. This chiral Yang-Mills theory is obtained as a truncation of the whole theory by restricting the sum over the irreducible representations of U (N) to the representations whose Young diagram contains a finite number of boxes in the large N limit. The number n of boxes in a representation coincides with the number of times the world sheet of the corresponding string configuration covers the target space. While in the gauge theory of Sn the irreducible representations are labeled by Young diagrams that contain exactly n boxes, in chiral U (N ) gauge theory
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
19
the irreducible representations are labeled byYoung diagrams which contain an arbitrary number of boxes and are only restricted by the condition that the number of rows do not exceed N − 1. The partition function of chiral Yang-Mills theory can then be written as a sum over n, and we expect each term in the sum, being the number of n-coverings, to be related to an Sn gauge theory. For chiral Yang-Mills this is true only on a torus, namely if the genus of the target space is zero. It is well known in fact that for different genuses the coefficients of the 1/N expansion of the U (N ) partition function are not directly related to the number of coverings, due to the presence of the so-called −1 points. A matrix model that gives, to all orders in the 1/N expansion, the exact statistic of branched coverings on a Riemann surface and whose restriction to a fixed value of n coincides with a Sn gauge theory was introduced by Kostov, Staudacher and Wynter (KSW in short) in [1, 2]. This model has still a U (N ) gauge invariance but realized in terms of complex rather than unitary N × N matrices. The partition function of the KSW model on a Riemann surface of genus G is [1, 2]6 : 2−2G τ n(n − 1) chr (2) (KSW ) n dr N exp − nµ , (59) ZN,G (τ, µ) = n! 2N dr n r∈U (N),|r|=n
where we denote by r both the Young diagrams and the corresponding representations of either U (N) of S|r| , with |r| the number of boxes in r. The sum is over all Young diagrams corresponding to representations of U (N ), dr and chr (2) are the same as in the previous sections. By comparing (59) with (11) we can write: (KSW )
ZN,G
(τ, µ) = Z1,G (A)N 2−2G e−µ + Z1,G (A)N 2(2−2G) e−2µ + . . . +ZN,G (A)N N(2−2G) e−Nµ +Z˜ N+1,G (A)N (N+1)(2−2G) e−(N+1)µ + . . .
(60)
The partition functions at the r.h.s. of (60), for n ≤ N , are the partition functions of the Sn gauge theory given in (11) with A = τ n(n−1) 2N . For n > N the partition func˜ are incomplete Sn partition functions, because in that case some tions, denoted by Z, irreducible representations of Sn , whose Young diagram has more than N rows, do not correspond to any representation of U (N ). The partition function of chiral Yang-Mills theory is similar to (59), but with the dimension r of the U (N ) representation replacing the factor N n dn!r at the r.h.s. The ratio between these two factors is the so called r term, whose presence prevents chiral Yang-Mills theory from having a simple interpretation in terms of coverings for G = 1. In spite of these differences two dimensional Yang-Mills theory, chiral Yang-Mills theory and the KSW model all share some common features in the large N limit. If the target space has the topology of a sphere (G = 0) all these theories exhibit a non-trivial phase structure in the large N limit. In the case of two dimensional Yang-Mills theory there is a third order phase transition, the Douglas-Kazakov phase transition [8]. This occurs at a critical value A = π 2 of the area of the sphere, measured in units of the coupling constant. This phase transition is well understood in terms of topologically non-trivial configurations [9, 10]. The phase structure of chiral Yang-Mills theory and of the KSW model has been studied in [2] and in [14] and it turns out to be richer than 6 We have restricted the KSW model to contain only quadratic branch points, for the general case, see the original papers.
20
A. D’Adda, P. Provero
pure Yang-Mills. In the KSW model four distinct phases are present in the (τ, µ) plane. In all these cases the saddle point in the large N limit corresponds to a representation of U (N) where both row and columns of the associated Young diagram scale as N , hence the total number of boxes is of order N 2 : n ∝ N 2.
(61)
This means that the saddle point at large N corresponds to a string configuration that covers the target space a number of times n proportional to N 2 , and that the associated √ Young diagram has rows and columns of order n. Besides the free energy is of order N 2 , namely of order n. This also means that the number of branch points p is of order n. In fact the free energy is the exponent at the r.h.s. of (59) calculated at the saddle point plus a term, that does not depend on the number of branch points, coming from the dimension of the representation. Let us expand the exponential in (59) in power series: p τ n(n − 1) chr (2) 1 τ n(n − 1) chr (2) exp − nµ = − nµ . (62) 2N dr p! 2N dr p Each term in the sum at the r.h.s. corresponds to a configuration with p plaquettes, each plaquette with either a single quadratic branch point or no branch point at all with proband nµ. The former grows faster with n abilities respectively proportional to τ n(n−1) 2N than the latter, so in the large n limit the probability x of having a branch point in each plaquette tends to 1. Finally we remark that the sum over p in (62) is dominated at large n by a single τ n(n−1) chr (2) 7 value of p , namely p = 2N dr − nµ . This implies that p is also of order n8 . To summarize: the standard large N limit of U (N ) gauge theories is described in terms of string configurations (coverings), which are also configurations of an Sn gauge √ theory, with n given by (61), rows and columns scaling like n and p ∝ n,
x = 1.
(63)
That means the large N limit of U (N ) gauge theories corresponds to the point at the right corner of region I in Fig. 2, where the coefficient of n log n in p is strictly zero. In order to “blow up” that point and determine whether a further phase structure is present there one would need to evaluate the terms of order n in the free energy. This will be the task of a future work. We just remark here that the presence of a non-trivial phase structure in that region is likely for at least two reasons: because that region is the section with a constant n plane of the large N limit of the KSW model, which has a non-trivial phase structure, and because we know from the random graphs theory that for p ∝ n there is at least one transition, the percolation transition. It appears from the previous discussion that the phase transition that we found in the Sn theory, and that corresponds to the well known cutoff phenomenon in the random walk, does not have a counterpart among the known phase transitions in two dimensional Yang-Mills theory. It is quite natural to ask whether a transition of this kind exists also for two dimensional Yang-Mills and other U (N ) gauge theory on a sphere. We found the answer to be affirmative. This new type of phase transition, the cutoff phenomenon, can be observed provided the coupling constant is rescaled with N with an extra log N xj j j ! in the large x limit is j = x, as shown by Stirling formula. 8 Remember that the saddle point is at N ∝ √n and chr (2) ∝ n−1/2 . dr 7
The saddle point of
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
21
with respect to the usual ’t Hooft prescription. We give here a simple, although rigorous, argument, leaving a detailed analysis of the transition to a future work. Consider the partition function of YM2 on a sphere (see for instance [10]) with gauge group U (N ): A A N 2 2 Z0 (A, N ) = e 24 (N −1)
2 (n1 , ..., nN )e− 2N i=1 ni , (64) n1 >n2 >...>nN
where the ni ’s are integers, (n1 , ..., nN ) is the Vandermonde determinant and A the area of the sphere. In the large N limit a la ’t Hooft A is kept fixed and the Douglas-Kazakov phase transition occurs at A = π 2 . Let us now allow A to rescale with N: A → A(N ). For A(N ) sufficiently large we expect the sum to be dominated in the large N limit by the configuration for which the exponential is maximum, namely N−1 N−1 {n1 , n2 , .....nN } = { N−1 2 , 2 − 1, .... − 2 }. This configuration corresponds to the trivial representation of U (N ), and it is the exact analogue of the representation of Sn consisting of a single row. Let us determine now the value of A(N ) for which this configuration ceases to be a maximum and becomes unstable with respect to a small perturbation. We can do that by comparing the contribution of the trivial representation to the sum in (64) to the contribution of a representation where n1 is increased by one unit. We find:
2 (n1 , ..., nN )e− 2N A
2 (n1 , ..., nN )e− 2N A
N
2 i=1 ni
N
2 i=1 ni
|{n1 ,n2 ,.....nN }={ N −1 , N −1 −1,....− N −1 } 2
2
2
|{n1 ,n2 ,.....nN }={ N −1 +1, N −1 −1,....− N −1 } 2
2
A(N )
=
e 2 . N2
(65)
2
For A(N ) > 4 log N the r.h.s. of (65) is greater than one and the sum over the irreducible representations in the partition function (64) is dominated by the trivial representation. In this phase the free energy is independent on A in the large N limit and the partition function on a disc is independent on the holonomy at the border, exactly as in phase III of Sect. 3.2 . According to the equivalence between gauge theories in two dimensions and random walks on the group manifold we expect the phase transition at A(N ) = 4 log N to have an interpretation as a cutoff phenomenon in the corresponding random walk on the manifold of U (N ). A better understanding of this point, as well as of the details of the phase at A < 4 log N , will require further investigation. 6. Conclusions Let us summarize the results obtained in the paper. We have studied a two-dimensional gauge theory of the symmetric group Sn that describes the statistics of branched coverings on a Riemann surface, in the large-n limit. • The theory on the sphere shows an interesting phase diagram when the number of branch points scales as n log n, with lines of first and second-order phase transitions. • All two-dimensional gauge theories on a genus-0 surface can be mapped into random walks in the corresponding group manifold. In our case, this allows us to interpret one of the transition lines as a cutoff phenomenon in the corresponding random walk. • The theory on a disc, with free boundary conditions, can be studied with methods of the theory of random graphs: this allows one to show that there is a phase transition on a disc from a disconnected to a connected covering surface. From this one can argue, and with some limitations prove, that the connectedness of the covering is what characterizes the different phases also on the sphere.
22
A. D’Adda, P. Provero
• A cutoff phenomenon is found also in 2D Yang-Mills on a sphere, if the area of the sphere scales with N like N log N . The present paper can be extended in two distinct directions. On one hand it would be desirable to understand better region I of the phase diagram of Fig. 2, and in particular its right corner which corresponds to the large N limit of U (N ) gauge theories. For this purpose the variational approach should be implemented to include the contributions of the next-to-leading order in n, which is of order n instead of n log n. This might reveal further phase structure, like for instance a percolation phase transition. Also, phase transitions in region I should be the analogue in Sn gauge theory of the Douglas-Kazakov phase transition in 2D Yang-Mills, and of the phase transitions studied in [14] and [2] for chiral Yang-Mills and KSW model respectively. It can be shown that the calculation of the correlators, which are relevant in order to determine the order parameters of the various phase transitions, also requires to know the free energy beyond the leading order. The existence of a cutoff phenomenon in two dimensional Yang-Mills, although within the framework of a non conventional scaling with N of the coupling constant, is also a fact whose meaning and physical implications, if any, should be further investigated. In particular a full description of the phase preceding the cutoff is still lacking. Acknowledgements. We are grateful to M. Bill´o and M. Caselle for many enlightening conversations.
Appendix A In this appendix we calculate in the large n limit some relevant quantities, such as log dr and chdr r(2) , in a representation r associated to a Young diagram whose rows and columns scale respectively as nα and n1−α . This is not the most general case. However we have already shown in Sect. 3 that in the large n limit the most general Young diagram can be decomposed into a discrete set of subdiagrams Yt (see Fig. 1 and the related discussion), each scaling as above with a different power αt . The calculation that we are going to present below will also apply to each Yt , and the result for the whole Young diagram is obtained by summing the contributions of the different Yt ’s. The shaded area of the Young diagram in 1 gives contributions which are subleading in the large n and can be neglected. Let us consider first a Young diagram where rows and columns scale respectively as nα and n1−α . It is convenient then to introduce the following continuous variables: x=
i , nα
y=
j
(66)
n1−α
and correspondingly f (x) =
ri , 1−α n
g(y) ≡ f −1 (y) =
sj , nα
(67)
where the derivatives f (x) and g (y) are everywhere negative or null. The variable x ranges from 0 to a maximum value xmax = f −1 (0) and similarly y ranges from 0 to ymax = f (0). Then it is easy to replace the discrete variables with the continuous ones in the expression of dr and chdr r(2) and obtain:
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
23
f −1 (0)
f (x) log dr = log n! − n dx dy 0 0 × log nα f −1 (y) − x + n1−α (f (x) − y) and
(68)
f −1 (0)
f −1 (0) chr (2) 1 = dxf (x)2 − 2nα−1 dxxf (x) n−α dr 1 − 1/n 0 0
(69)
while the constraint (19) in the large n limit becomes:
f −1 (0)
f (0)
dxf (x) =
0
dyf −1 (y) = 1.
(70)
0
Keeping only the terms of order n log n and n in Eqs (68) we have for log dr , in the large n limit:
f −1 (0) log dr = αn log n − n 1 + dxf (x) (log f (x) − 1) α < 1/2, 0
f (0)
log dr = (1 − α)n log n − n 1 +
dyf
−1
(y) log f
−1
(y) − 1
α > 1/2.
0
(71) Similarly we obtain for
chr (2) dr ,
keeping terms up to order 1: −1
f (0) chr (2) log α < 1/2, = −α log n + log dxf (x)2 dr 0 f (0) chr (2) −1 2 log = (1 − α) log n + log α > 1/2. dyf (y) dr 0
(72)
Consider now the most general case, where the Young diagram consists, in the large n limit, of a discrete set of subdiagrams Yt , each scaling with a different power αt . Each Yt contributes to log dr with a term of the form (71), with the appropriate αt in place of α and weighted with its area zt . The sum over t reproduces, for the leading n log n term, Eq.(31). Consider now chdr r(2) . It is clear from (69) that the asymptotic behavior of the contribution coming from Yt is n−αt (resp. n1−αt ) for αt < 1/2 (resp. αt > 1/2). The sum over t is then dominated by subdiagrams with scaling powers α = 0 and α = 1 which are discussed in detail in Sect. 3. Appendix B In this Appendix we derive Eqs. (57) and (58), in two different ways: first by direct combinatorial methods, then using the character expansion of the probability distribution discussed in Subsect. 4.1. Consider a random walk in Sn in which, at each step, the permutation is multiplied by a randomly chosen transposition with probability x, or by the identity with probability
24
A. D’Adda, P. Provero
1 − x. It is convenient to think of n objects and n boxes: initially the object i is in the box i. Then we start moving them around with the following rule: At each step, we exchange two randomly chosen objects with probability x, or we do nothing with probability 1−x. Choose now one of the n objects, say number 1, and compute the probability P1 (x, p) that, after p steps, it is in box number 1 (either because it never left it, or because it went back to it). The expected number of cycles of length 1 in the final permutation is then just N1 (x, p) = nP1 (x, p).
(73)
To compute P1 (x, p), write it as P1 (x, p) =
p
(74)
q(k)s(k),
k=0
where q(k) is the probability that object 1 changes box exactly k times during the walk, and s(k) is the probability that it will be back in its original box after changing box k times. We have easily q(k) =
p 2x k k
n
1−
2x n
p−k .
(75)
For s(k) we can write a recursion relation: suppose element 1 is in box 1 after being moved k times: Then it will certainly not be in box 1 after being moved k + 1 times. If it is not in box 1 after being moved k times, it will be after being moved k + 1 times with probability 1/(n − 1). Hence s(k + 1) =
1 [1 − s(k)] n−1
that together with the initial condition s(0) = 1 gives 1 1 s(k) = . 1− n (1 − n)k−1
(76)
(77)
Substituting in Eq. (74) we obtain P1 (x, p) =
1 n − 2x − 1 p 1 + (n − 1) n n−1
(this result for x = 1 − 1/n was already quoted in Ref. [15]), and n − 2x − 1 p N1 (x, p) = 1 + (n − 1) , n−1
(78)
(79)
which includes in particular Eq. (57), so that taking the limit n → ∞ with p = An log n one obtains Eq. (58). The same result can be obtained with character expansion methods by noticing that N1 (x, p) is simply the expectation value < T r Q > of the trace of the permutation obtained after p steps, where the trace is taken in the “fundamental” n-dimensional representation: Such representation is reducible to a direct sum of the trivial representation
2-D Gauge Theories of the Symmetric Group Sn in the Large-n Limit
25
(Young diagram made of one line of n boxes) and the n − 1 dimensional representation described by a diagram with two rows of length n − 1 and 1. Therefore T r Q = ch1 (Q) + chn−1 (Q).
(80)
Using the character expansion of the probability distribution as in Subsect. 4.1 we obtain N1 (x, p) = T r Q 1 chr (2) p = dr (1 − x) + x chr (Q). ch1 (Q) + chn−1 (Q) n! dr r Q∈Sn
(81) Using the orthogonality of characters and chn−1 (2) n−3 = dn−1 n−1
(82)
we find again Eq. (79). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
Kostov, I.K., Staudacher, M.: Phys. Lett. B394, 75 (1997), [hep-th/9611011] Kostov, I.K., Staudacher, M., Wynter, T.: Commun. Math. Phys. 191, 283 (1998), [hep-th/9703189] Billo, M., D’Adda, A., Provero, P.: hep-th/0103242 Gross, D.J., Taylor, W.I.: Nucl. Phys. B400, 181 (1993), [hep-th/9301068] Gross, D.J., Taylor, W.I.: Nucl. Phys. B403, 395 (1993), [hep-th/9303046] Baez, J., Taylor, W.: Nucl. Phys. B 426, 53 (1994), [arXiv:hep-th/9401041] Billo, M., D’Adda, A., Provero, P.: Unpublished Douglas, M.R., Kazakov, V.A.: Phys. Lett. B 319, 219 (1993), [hep-th/9305047] Caselle, M., D’Adda, A., Magnea, L., Panzeri, S.: arXiv:hep-th/9309107 Gross, D.J., Matytsin, A.: Nucl. Phys. B 429, 50 (1994), [arXiv:hep-th/9404004] Gross, D.J., Matytsin, A.: Nucl. Phys. B 437, 541 (1995), [arXiv:hep-th/9410054] Douglas, M.R., Li, K., Staudacher, M.: Nucl. Phys. B 420, 118 (1994), [arXiv:hep-th/9401062] Ganor, O., Sonnenschein, J., Yankielowicz, S.: Nucl. Phys. B 434, 139 (1995), [arXiv:hepth/9407114] Crescimanno, M.J., Taylor, W.: Nucl. Phys. B 437, 3 (1995), [arXiv:hep-th/9408115] Diaconis, P., Shahshahani, M.: Z. Wahrsch. Verw. Gebiete 57, 159 (1981) Migdal, A.A.: Sov. Phys. JETP 42, 413 (1975) Rusakov, B.E. Mod. Phys. Lett. A5, 693 (1990) Pak, I., Vu, V. H.: Disc. Appl. Math. 110, 251 (2001) Gross, D.J., Witten, E.: Phys. Rev. D 21, 446 (1980) Erd¨os, P., R´enyi, A.: Publ. Math. Debrecen 6, 290 (1959) Erd¨os, P., R´enyi, A.: The Art of Counting, Cambridge: MIT, 1973
Communicated by R.H. Dijkgraaf
Commun. Math. Phys. 245, 27–67 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-0995-1
Communications in
Mathematical Physics
Tensor Gauge Fields in Arbitrary Representations of GL(D, R). Duality and Poincar´e Lemma Xavier Bekaert, Nicolas Boulanger Physique Th´eorique et Math´ematique, Universit´e Libre de Bruxelles, C.P. 231, 1050 Bruxelles, Belgium Received: 28 August 2002 / Accepted: 22 August 2003 Published online: 16 January 2004 – © Springer-Verlag 2004
Abstract: Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we have studied the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D, R). We have proven a generalization of the Poincar´e lemma which enables us to solve the above-mentioned problems in a systematic and unified way. 1. Introduction The surge of interest in string field theories has refocused attention on the old problem of formulating covariant field theories of particles carrying arbitrary representation of the Lorentz group. These fields appear as massive excitations of string (for spin S > 2). It is believed that in a particular phase of M-theory, all such excitations become massless. The covariant formulation of massless gauge fields in arbitrary representations of the Lorentz group has been completed for D = 4 [1]. However, the generalization of this formulation to arbitrary values of D is a difficult problem since the case D = 4 is a very special one, as all the irreps of the little group SO(2) are totally symmetric. The covariant formulation for totally antisymmetric representations in arbitrary spacetime dimension has been easily obtained using differential forms. For mixed symmetry type gauge fields, the problem was partially solved in the late eighties [2] (for recent works, see for instance [3]). A recent approach [4] has shed new light on higher-spin gauge fields, showing how it is possible to formulate the free equations while foregoing the trace conditions of the Fang-Fronsdal formalism. In this formulation, the higher-spin gauge parameters are then not constrained to be irreducible under SO(D − 1, 1), it is sufficient for them to be irreducible under GL(D, R). Dualities are crucial in order to scrutinize non-perturbative aspects of gauge field and string theories, it is therefore of relevance to investigate the duality properties of arbitrary
“Chercheur F.R.I.A.”, Belgium
28
X. Bekaert, N. Boulanger
tensor gauge fields. It is well known that the gravity field equations in four-dimensional spacetime are formally invariant under a duality rotation (for recent papers, see for example [5, 6]). As usual the Bianchi identities get exchanged with field equations but, as for Yang-Mills theories, this duality rotation does not appear to be a true symmetry of gravity: the covariant derivative involves the gauge field which is not inert under the duality transformation. A deep analogy with the self-dual D3-brane that originates from the compactified M5-brane is expected to occur when the six-dimensional (4, 0) superconformal gravity theory is compactified over a 2-torus [7]. Thus, a SL(2, Z)-duality group for D = 4 Einstein gravity would be geometrically realized as the modular group of the torus. In any case, linearized gravity does not present the problem mentioned previously and duality is thus a true symmetry of this theory. Dualizing a free symmetric gauge field in D > 4 generates new irreps of GL(D, R). This paper provides a systematic treatment of the gauge structure and duality properties of tensor gauge fields in arbitrary representations of GL(D, R). Review and reformulation of known results ([6, 8] and the references therein) are given in a systematic unified mathematical framework and are presented together with new results and their proofs. Section 2 is a review of massless spin-two gauge field theory and its dualisation. The obtained free dual gauge fields are in representations of GL(D, R) corresponding to Young diagrams with one row of two columns and all the other rows of length one. Section 3 “N-complexes” gathers together the mathematical background needed for the following sections. Based on the works [9–12], it includes definitions and propositions together with a review of linearized gravity gauge structure in the language of N -complexes. Section 4 discusses linearized gravity field equations and their duality properties in the introduced mathematical framework. Section 5 presents our theorem, which generalizes the standard Poincar´e lemma. This theorem is then used in Sect. 6 to elucidate the gauge structure and duality properties of tensor gauge fields in arbitrary representations of GL(D, R). The proof of the theorem, contained in the Appendix, is iterative and simply proceeds by successive applications of the standard Poincar´e lemma. 2. Linearized Gravity 2.1. Pauli-Fierz action. A free symmetric tensor gauge field hµν in D dimensions has the gauge symmetry δhµν = 2∂(µ ξν) . The linearized Riemann tensor for this field is 1 Rµν σ τ ≡ (∂µ ∂σ hντ + . . . ) = −2∂[µ hν][σ,τ ] . 2 It satisfies the property Rµν σ τ = Rσ τ µν
(2.1)
(2.2)
(2.3)
together with the first Bianchi identity R[µν σ ]τ = 0
(2.4)
∂[ρ Rµν] σ τ = 0 .
(2.5)
and the second Bianchi identity
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
29
It has been shown by Pauli and Fierz [13] that there is a unique, consistent action that describes a pure massless spin-two field. This action is the Einstein action linearized around a Minkowski background1 √ 2 SEH [gµν ] = 2 d D x −g Rf ull , gµν = ηµν + κhµν , (2.6) κ where Rf ull is the scalar curvature for the metric gµν . The constant κ has mechanical dimensions LD/2−1 . The term of order 1/κ 2 in the expansion of SEH vanishes since the background is flat. The term of order 1/κ is equal to zero because it is proportional to the (sourceless) Einstein equations evaluated at the Minkowski metric. The next order term in the expansion in κ is the action for a massless spin-2 field in D-dimensional spacetime 1 SP F [hµν ] = d D x − ∂µ hνρ ∂ µ hνρ + ∂µ hµν ∂ρ hρν 2 µ 1 − ∂ν h µ ∂ρ hρν + (2.7) ∂µ hνν ∂ µ hρρ . 2 Since we linearize around a flat background, spacetime indices are raised and lowered with the flat Minkowskian metric ηµν . For D = 2 the Lagrangian is a total derivative so we will assume D ≥ 3. The (vacuum) equations of motion are the natural free field equations Rσ µ σ ν = 0
(2.8)
which are equivalent to the linearized Einstein equations. Together with (2.2) the previous equation implies that ∂ µ Rµν σ τ = 0 .
(2.9)
2.2. Minimal coupling. The Euler-Lagrange variation of the Pauli-Fierz action is δSP F 1 = R σµσ ν − ηµν R σ τ σ τ . δhµν 2
(2.10)
It can be shown that the second Bianchi identity (2.5) implies on-shell ∂ µ Rµσ νρ + ∂ρ Rνµσµ − ∂ν Rρµσµ = 0 ,
(2.11)
and taking the trace again this leads to ∂ µ R σµσ ν −
1 ∂ν R σ τ σ τ = 0 . 2
(2.12)
From another perspective, Eqs. (2.12) can be regarded as the Noether identities corresponding to the gauge transformations (2.1). 1 Notice that the way back to full gravity is quite constrained. It has been shown that there is no local consistent coupling, with two or less derivatives of the fields, that can mix various gravitons [14]. In other words, there are no Yang-Mills-like spin-2 theories. The only possible deformations are given by a sum of individual Einstein-Hilbert actions. Therefore, in the case of one graviton, [14] provides a strong proof of the uniqueness of Einstein’s theory.
30
X. Bekaert, N. Boulanger
Let us introduce a source Tµν which couples minimally to hµν through the term (2.13) Sminimal = −κ d D x hµν Tµν . We add this term to the Pauli-Fierz action (2.7), together with a kinetic term SK for the sources, to obtain the action S = SP F + SK + Sminimal .
(2.14)
The field equations for the symmetric gauge field hµν are the linearized Einstein equations R σµσ ν −
1 ηµν R σ τ σ τ = κTµν . 2
(2.15)
Consistency with (2.12) implies that the linearized energy-momentum tensor is conserved ∂ µ Tµν = 0. The simplest example of a source is that of a free particle of mass m following a worldline x µ (s) with s the proper time along the worldline. The Polyakov action for the massive particle reads dx µ dx ν . (2.16) SP olyakov [x µ (s)] = −m ds gµν ds ds It results as the sum of the two actions
dx µ dx ν , ds ds dx µ dx ν = −mκ ds hµν , ds ds
SK = −m Sminimal
ds ηµν
(2.17) (2.18)
from which it can be inferred that the (matter) source Tµν for a massive particle is equal to dx µ dx ν T µν (x) = m ds δ D (x − x(s)) . (2.19) ds ds 2
µ
This relationship is conserved if and only if ddsx2 = 0, which means that the test particle follows a straight worldline. In general, when considering a free massless spin-two theory coupled with matter, the latter has to be constrained in order to be consistent with the conservation of the linearized energy-momentum tensor2 . At first sight, it is however inconsistent with the natural expectation that matter reacts to the gravitational 2 µ field. Anyway, the constraint ddsx2 = 0 is mathematically inconsistent with the e.o.m. obtained from varying (2.17) and (2.18) with respect to the worldline x µ (s) which constrains the massive particle to follow a geodesic for gµν (and not a straight line). In fact, self for matter to respond to the gravitational field, it is necessary to add a source κTµν self for the gravitational field itself, in such a way that the sum Tµν + Tµν is conserved if the matter obeys its own equation to first order in κ and if the gravitational field obeys (2.15). This gravitational self-energy must come from a first order (in κ) deformation of 2 This should not be too surprising since it is well known that the Einstein equations simultaneously determine the gravity field and the motion of matter.
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
31
the Pauli-Fierz action. This modification was the starting point of Feynman3 and others in their derivation of the Einstein-Hilbert action by consistent deformation of the PauliFierz action with back reaction [16]. At the end of the perturbative procedure, the result obtained is that the free-falling particle must follow a geodesic for consistency with the (full) Einstein equations.
2.3. Duality in linearized gravity. Let us mention for a further purpose that Eq.(2.9) can be directly deduced from Eqs. (2.4)–(2.5)–(2.8) for the linearized Riemann tensor without using its explicit expression (2.2). To simplify the proof and initiate a discussion about duality properties, let us introduce the tensor (∗R)µ1 ...µD−2 | ρσ =
1 µ µ εµ1 ... µD R D−1 D ρσ . 2
(2.20)
The linearized second Bianchi identity and the Einstein equations can be written in terms of this new tensor respectively as (2.21) ∂ µ (∗R)µ ...ν | ρσ = 0 and (∗R)[µ ...ν | ρ]σ = 0 .
(2.22)
Taking the divergence of (2.22) with respect to the first index µ, and applying (2.21), we obtain (2.23) ∂ µ (∗R)ρ ...ν | µσ = 0 which is equivalent to ∂ µ Rαβ µσ = 0
(2.24)
as follows from the definition (2.20). Using the symmetry property (2.3) of the Riemann tensor we recover (2.9). In Corollary 1, we will prove that the equations Rσ µ σ ν = 0 ,
∂ µ Rµν σ τ = ∂ σ Rµν σ τ = 0
(2.25)
are (locally) equivalent to the following equation [6]: (∗R)µ1 ...µD−2 | ρσ = ∂[µ1 h˜ µ2 ...µD−2 ] | [ρ,σ ] ,
(2.26)
which defines the tensor field h˜ [µ1 ...µD−3 ] | ρ called the dual gauge field of hµν and which is said to have mixed symmetry because it is neither (completely) antisymmetric nor symmetric. In fact, it obeys the identity h˜ [µ1 ...µD−3 | ρ] ≡ 0 .
(2.27)
3 In 1962, Feynman presented this derivation in his sixth Caltech lecture on gravitation [15]. One of the intriguing features of this viewpoint is that the initial flat background is no longer observable in the full theory. In the same vein, the fact that the self-interacting theory has a geometric interpretation is “not readily explainable – it is just marvelous”, as Feynman expressed.
32
X. Bekaert, N. Boulanger
However, for D = 4 the dual gauge field is a symmetric tensor h˜ µν , which signals a possible duality symmetry. The curvature dual (2.26) remains unchanged by the transformations δ h˜ µ1 ...µD−3 | ρ = ∂[µ1 Sµ2 ...µD−3 ] | ρ + ∂ρ Aµ2 ...µD−3 µ1 + Aρ[µ2 ...µD−3 ,µ1 ],
(2.28)
where complete antisymmetrization of the gauge parameter S[µ1 ...µD−4 ] | µD−3 vanishes and the other gauge parameter Aµ1 ...µD−3 is completely antisymmetric. 2.4. Mixed symmetry type gauge fields. Let us consider the general case of massless gauge fields Mµ1 µ2 ...µn | µn+1 having the same symmetries as the above-mentioned dual gauge field h˜ µ1 ...µD−3 | ρ . These can be represented by the Young diagram 1 n+1 2
.. .
, (2.29)
n
which implies that the field obeys the identity M[µ1 µ2 ...µn | µn+1 ] ≡ 0 .
(2.30)
Such tensor gauge fields were studied two decades ago by the authors of [8, 17, 18] and appear in the bosonic sector of some odd-dimensional CS supergravities [19]. Here, n is used to denote the number of antisymmetric indices carried by the field Mµ1 µ2 ...µn | µn+1 , which is also the number of boxes in the first column of the corresponding Young array. n(D+1)! components in D dimensions. The tensors Mµ1 µ2 ...µn | µn+1 have (n+1)!(D−n)! The action of the free theory is (2.31) S0 [Mµ1 µ2 ...µn | µn+1 ] = d D x L, where the Lagrangian is4 [18]. µ2 ...µn | µn+1
L = Mµ1 ...µn | µn+1 ∂ 2 M µ1 ...µn | µn+1 − 2nMµ1 ...µn | µn+1 ∂ µ1 ∂ λ Mλ
µ2 ...µn−1 µ | µn µ1 2 µ1 ...µn µ1 ν µ1 + n(n − 1)Mµ1 µ2 ...µn | µ ∂ ∂ Mν µ2 ...µn | µ1 ∂ M β +n(n − 1)Mβγ µ3 ...µn ∂ γ ∂ µ Mνµµ3 ...µn | ν µ +2nM µ1 2 ...µn | µ1 ∂ µ ∂ν Mµµ2 ...µn | ν . (2.32)
−nM
The field equations derived from (2.32) are equivalent to ηµ1 ν1 Kµ1 µ2 ...µn+1 | ν1 ν2 = 0 ,
(2.33)
Kµ1 µ2 ...µn+1 | ν1 ν2 ≡ ∂[µ1 Mµ2 ...µn+1 ] | [ν1 , ν2 ]
(2.34)
where
is the curvature and obeys the algebraic identity 4
Notice that, for n = 1, the Lagrangian reproduces (2.7).
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
33
K[µ1 ...µn+1 | ν1 ]ν2 = 0 .
(2.35)
The action (2.31) and the curvature (2.34) are invariant under the following gauge transformations: δS,A Mµ1 ...µn+1 = ∂[µ1 Sµ2 ...µn ] | µn+1 + ∂[µ1 Aµ2 ...µn ]µn+1 + ∂µn+1 Aµ2 ...µn µ1 ,
(2.36)
where the gauge parameters Sµ2 ...µn | µn+1 and Aµ2 ...µn+1 have the symmetries 2 n+1 3
2 3
and
.. .
.. , respectively . .
n
n+1
These gauge transformations are accompanied by a chain of n − 1 reducibilities on the gauge parameters. These reducibilities read, with 1 ≤ i ≤ n, (i) S µ1 ...µn−i | µn−i+1
(i+1) µ2 ...µn−i ] | µn−i+1
= ∂[µ1 S
+
(n + 1) (n − i + 1)
(i+1) (i+1) × ∂[µ1 A µ2 ...µn−i ]µn−i+1 +∂µn−i+1 A µ2 ...µn−i µ1 , (2.37) (i)
(i+1) µ2 ...µn−i+1 ] ,
A µ1 ...µn−i+1 = ∂[µ1 A
(2.38)
with the conventions that (1) S µ1 ...µn−1 | µn (1) A µ1 ...µn
= Sµ1 ...µn−1 | µn ,
(n) Sµ=
0,
= Aµ1 ...µn .
The reducibility parameters at reducibility level i have the symmetry (i+1) S µ1 ...µn−i−1 | µn−i
1 2 .. .
n−i
(i+1)
and A
1 µ1 ...µn−i 2.
n−i−1
.
.. n−i
(n−1)
Note that S µν µ ν . These gauge transformations and reducibilities have already been introduced and discussed in references [17, 18, 8]. The problem of investigating all the possible consistent couplings among the fields Mµ1 µ2 |µ3 will be treated in [20]. Our theorem will provide a systematic tool for the investigation of mixed symmetry type gauge field theories. The number of physical degrees of freedom for the theory (2.31), (2.36), is equal to (D − 2)! D (D − n − 2) n . (D − n − 1)! (n + 1)!
(2.39)
This number is manifestly invariant under the exchange n ↔ D − n − 2 which corresponds to a Hodge duality transformation. This confirms that the dimension for which the theory is dual to a symmetric tensor is equal to D = n + 3, which is also the critical dimension for the theory to have local physical degrees of freedom. The theory (2.31), (2.36) is then dual to Pauli-Fierz’s action (2.7) for D = n + 3.
34
X. Bekaert, N. Boulanger
3. N -Complexes The objective of the works presented in [9, 11, 12] was to construct complexes for irreducible tensor fields of mixed Young symmetry type, thereby generalizing to some extent the calculus of differential forms. This tool provides an elegant formulation of symmetric tensor gauge fields and their Hodge duals (such as differential form notation within electrodynamics). 3.1. Young diagrams. A Young diagram Y is a diagram which consists of a finite number S > 0 of columns of identical squares (referred to as the cells) of finite non-increasing lengths l1 ≥ l2 ≥ . . . ≥ lS ≥ 0. For instance, Y ≡
.
The total number of cells of the Young diagram Y is denoted by |Y | =
S
(3.1)
li .
i=1
Order relations. For future reference, the subset Y(S) of NS is defined by Y(S) ≡ {(n1 , . . . , nS ) ∈ NS | n1 ≥ n2 ≥ ... ≥ nS ≥ 0} .
(3.2)
For two columns, the set Y(2) can be depicted as the following set of points in the plane R2 : ... (3,3)
•
(2,2) (3,2)
•
•
(1,1) (2,1) (3,1)
•
•
•
(0,0) (1,0) (2,0) (3,0)
•
•
•
•
... ...
(3.3)
... ...
Let Y be a diagram with S columns of respective lengths l1 , l2 , ..., lS . If Yp is a welldefined Young diagram, then (l1 , l2 , . . . , lS ) ∈ Y(S) . Conversely, a Young diagram Y with S columns is uniquely determined by the gift of an element of Y(S) , and can there(S) fore be labeled unambiguously as Y(l1 ,l2 ,... ,lS ) (lS = 0). We denote5 by Y (S) the set of (S)
all Young diagrams Y(l1 ,l2 ,... ,lS ) with at most S columns of respective length 0 ≤ lS ≤ lS−1 ≤ . . . ≤ l1 ≤ D − 1. This identification between Y(S) and Y (S) suggests obvious definitions of sums and differences of Young diagrams. There is a natural definition of inclusion of Young diagrams (S)
(S)
Y(m1 ,... ,mS ) ⊂ Y(n1 ,... ,nS ) ⇔ m1 ≤ n1 , m2 ≤ n2 , . . . , mS ≤ nS . 5
We will sometimes use the symbol Y(S) instead of Y (S) .
(3.4)
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
35 (S)
(S)
We can develop a stronger definition of inclusion. Let Y(m1 ,... ,mS ) and Y(n1 ,... ,nS ) be (S)
(S)
two Young diagrams of Y (S) . We say that Y(m1 ,... ,mS ) is well-included into Y(n1 ,... ,nS ) (S)
(S)
if Y(m1 ,... ,mS ) ⊂ Y(n1 ,... ,nS ) and ni − mi ≤ 1 for all i ∈ {1, . . . , S}. In other words, if no column of the Young diagrams differs by more than a single box. We denote this particular inclusion by , i.e. (S)
(S)
Y(m1 ,... ,mS ) Y(n1 ,... ,nS ) ⇔ mi ≤ ni ≤ mi + 1
∀i ∈ {1, . . . , S} .
(3.5)
This new inclusion suggests the following pictorial representation of Y(S) : .z< . . zz z zz zz (3,3) / ~·? · · > •O ~ } ~ } ~~ }} ~~ }} ~ } } ~ (2,2) / (3,2) / ·? · · ~~ }> •O }> •O ~ } } ~~ }} }} ~~ }} }} ~ } } } } ~ (1,1) / (2,1) / (3,1) / ·? · · • • • ~~ }> O }> O }> O ~ } } } ~~ }} }} }} ~~ }} }} }} ~ } } } } } ~ } (0,0) / (1,0) / (2,0) / (3,0) /· · · • • • •
(3.6)
where all the arrows represent maps . This diagram is completely commutative. The previous inclusions ⊂ and provide partial order relations for Y(S) . The order is only partial because all Young tableaux are not comparable. We now introduce a total order relation for Y(S) . If (m1 , . . . , mS ) and (n1 , . . . , nS ) belong to Y(S) , then mi = ni , ∀i ∈ {1, . . . , K} , (S) (S) Y(m1 ,... ,mS ) Y(n1 ,... ,nS ) ⇔ ∃K ∈ {1, . . . , S} : mK+1 ≤ nK+1 . (3.7) This ordering simply provides the lexicographic ordering for Y(S) . Maximal diagrams. A sequence of Y(S) which is of physical interest is the maximal sequence denoted by Y S ≡ (YpS )p∈N . This is defined as the naturally ordered sequence of maximal diagrams6 (the ordering is induced by the inclusion of Young tableaux). Maximal diagrams are diagrams with maximally filled rows, that is to say, Young diagrams YpS with p cells defined in the following manner: we add cells to a row until it contains S cells and then we proceed in the same way with the row below, and continue until all p cells have been used. Consequently all rows except the last one are of length S and, if rp is the remainder of the division of p by S (rp ≡ p mod S) then the last row 6 The subsequent notations for maximal sequences are different from the ones of [11, 12]. We have shifted the upper index by one unit.
36
X. Bekaert, N. Boulanger
of the Young diagram YpS will contain rp ≤ S cells (if rp = 0). For two columns (S = 2) the maximal sequence is represented as the following path in the plane depicting Y(2) : O (2,2)
•O
/ (3,2) •
/ (2,1) •
(1,1)
•O
/ (1,0) •
(0,0)
•
Diagrams for which all rows have exactly S cells are called rectangular diagrams. These are those represented by the leftmost diagonal of the diagram Y(S) . (S)
Duality. Let Y(l1 ,... ,lS ) be aYoung diagram in Y (S) and I a non-empty subset of {1, . . . , S}. (S)
The diagram D(1 ,... ,S ) with S columns of respective lengths i ≡
li D − li
if i ∈ I , if i ∈ I ,
(3.8)
I is, in general, not a Young diagram. We define the dual Young diagram Y (λ1 ,... ,λS ) ⊂ (S) Y associated to the set I as the Young diagram obtained by reordering the columns of (S) D(1 ,... ,S ) . In other words, its i th column has length λi = π(i) ,
λj ≤ λi for i ≤ j ,
(3.9)
where π is a permutation of the elements of {1, . . . , S}. Schur module. Multilinear applications with a definite symmetry are associated with a definite Young diagram7 . Let V be a finite-dimensional vector space of dimension D and V ∗ its dual. The dual of the nth tensor power V n of V is canonically identified with the space of multilinear forms: (V n )∗ ∼ = (V ∗ )n . Let Y be a Young diagram and let us consider that each of the |Y | copies of V ∗ in the tensor product (V ∗ )|Y | is labeled by one cell of Y . The Schur module V Y is defined as the vector space of all multilinear forms T in (V ∗ )|Y | such that: (i) T is completely antisymmetric in the entries of each column of Y , (ii) complete antisymmetrization of T in the entries of a column of Y and another entry of Y which is on the right-hand side of the column vanishes. V Y is an irreducible subspace invariant for the action of GL(D, R) on V |Y | . 7
This set of definitions essentially comes from [12].
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
37
Let Y be a Young diagram and T an arbitrary multilinear form in (V ∗ )|Y | , one defines the multilinear form Y(T ) ∈ (V ∗ )|Y | by Y(T ) = T ◦ A ◦ S with A=
(−)ε(c) c , c∈C
S=
r,
r∈R
where C is the group of permutations which permute the entries of each column, ε(c) is the parity of the permutation c, and R is the group of permutations which permute the entries of each row of Y . Any Y(T ) ∈ V Y and the application Y of V |Y | satisfies the condition Y 2 = λY for some number λ = 0. Thus Y = λ−1 Y is a projection of V |Y | onto itself, i.e. Y2 = Y, with image Im(Y) = V Y . The projection Y will be referred to as the Young symmetrizer of the Young diagram Y . 3.2. Differential N-complex. Let (Y ) = (Yp )p∈N be a given sequence of Young diagrams such that the number of cells of Yp is p, ∀p ∈ N. For each p, we assume that p there is a single shape Yp and Yp ⊂ Yq for p < q. We define (Y ) (M) as the vector space of smooth covariant tensor fields of rank p on the pseudo-Riemannian manifold M which have theYoung symmetry type Yp (i.e. their components T (x) belong to the Schur module V Yp associated to Yp ). More precisely they obey the identity Yp T (x) = T (x), ∀x ∈ M, with Yp the Young symmetrizer on tensor of rank p associated to the Young p symmetry Yp . Let (Y ) (M) be the graded vector space ⊕p (Y ) (M) of irreducible tensor fields on M. The exterior differential can then be generalized by setting [11, 12] p
p+1
d ≡ Yp+1 ◦ ∂ : (Y ) (M) → (Y ) (M) ,
(3.10) p
that is to say, first taking the partial derivative of the tensor T ∈ (Y ) (M) and applying p+1
the Young symmetrizer Yp+1 to obtain a tensor in (Y ) (M). Examples are provided in Subsect. 3.3. Let us briefly mention that there are no dx µ in this definition of the operator d. The operator d is not nilpotent in general, therefore d does not always endow (Y ) (M) with the structure of a standard differential complex. If we want to generalize the calculus of differential forms, we have to use the extension of the structure of a differential complex with higher order of nilpotency. An N-complex is defined as a graded space V = ⊕i Vi equipped with an endomorphism d of degree 1 that is nilpotent of order N ∈ N − {0, 1}: d N = 0 [21]. It is important to stress that the operator d is not necessarily a differential because, in general, d is neither nilpotent nor a derivative (for instance, even if one defines a product in (Y ) (M), the non-trivial projections affect the Leibnitz rule). A sufficient condition for d to endow (Y ) (M) with the structure of an N -complex is that the number of columns of any Young diagram be strictly smaller than N [12]: Lemma 1. Let S be a non-vanishing integer and assume that the sequence (Y ) is such that the number of columns of the Young diagram Yp is strictly smaller than S + 1 (i.e. ≤ S) for any p ∈ N. Then the space (Y ) (M), endowed with the operator d, is a (S + 1)-complex.
38
X. Bekaert, N. Boulanger
Indeed, the condition d S+1 ω = 0 for all ω ∈ (Y ) (M) is fulfilled since the indices in one column are antisymmetrized and d S+1 ω necessarily involves at least two partial derivatives ∂ in one of the columns (there are S + 1 partial derivatives involved and a maximum of S columns). Notation. The space (Y (S) ) (M) is a (S + 1)-complex that we denote (S) (M). The subcomplex Y (S)
(l1 ,l2 ,... ,lS )
(l ,l ,... ,lS )
1 2 (M) is denoted by (S)
(M).
This complex (S) (M) is the generalization of the differential form complex (M) = (1) (M) we are seeking because each proper space is invariant under the action of GL(D, R). For example, the previously mentioned mixed symmetry type gauge field (n,1) M (2.29) belongs to (2) (M). 3.3. Symmetric gauge tensors and maximal sequences. A Young diagram sequence of interest in theories of spin S ≥ 1 is the maximal sequence Y S = (YpS )p∈N [11, 12]. This sequence is defined as the sequence of diagrams with maximally filled rows naturally ordered by the number p of boxes. p
p
Notation. In order to simplify the notation, we shall denote (Y S ) (M) by S (M) and (Y S ) (M) by S (M). p If D is the dimension of the manifold M then the subcomplexes S (M) with p > SD are trivial since, for these values of p, the Young diagrams YpS have at least one column containing more than D cells. Massless spin-one gauge field. It is clear that 1 (M) with the differential d is the usual complex (M) of differential forms on M. The connection between the complex of differential forms on M and the theory of classical q-form gauge fields is well known. Indeed the subcomplex d0
d1
dq−1
dq
dq+1
0 (M) → 1 (M) → . . . → q (M) → q+1 (M) → q+2 (M)
(3.11)
with dp ≡ d : p → p+1 , has the following interpretation in terms of q-form gauge field A[q] theory. The space q+1 (M) is the space which the field strength F[q+1] lives in. The space q+2 (M) is the space of Hodge duals to magnetic sources ∗Jm (at least if we extend the space of “smooth” (q + 2)-forms to de Rham currents) since dF[q+1] = (∗Jm )[q+2] . If there is no magnetic source, the field strength belongs to the kernel of dq+1 . The Abelian gauge field A[q] belongs to q (M). The subspace Ker dq of q (M) is the space of pure gauge configurations (which are physically irrelevant). The space q−1 (M) is the space of infinitesimal gauge parameters [q−1] and q−2 (M) is the space of first reducibility parameters [q−2] , etc. If the manifold M has the topology of RD then (3.11) is an exact sequence. Massless spin-two gauge field. As another example, we demonstrate the correspondence between some Young diagrams in the maximal sequence with at most two columns and their corresponding spaces in the differential 3-complex 2 (M). The interest of 2 (M) is its direct applicability in free spin-two gauge theory. Indeed, in this case the analog of the sequence (3.11) is
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
39
Table 1. Two-column maximal sequence and its physical relevance Young tableau
Vector space
Example
Components
12 (M)
lin. diffeomorphism parameter
ξµ
22 (M)
graviton
hµν
32 (M)
mixed symmetry type field
Mµν | ρ
42 (M)
Riemann tensor
Rµν ρσ
52 (M)
Bianchi identity
∂[λ Rµν] ρσ
d
d2
d
12 (M) → 22 (M) → 42 (M) → 52 (M),
(3.12)
where 12 (M) is the space of covariant vector fields ξµ on M, 22 (M) is the space of covariant rank 2 symmetric tensor fields hµν on M, 42 (M) the space of covariant tensor fields Rµν ρσ of rank 4 having the symmetries of the Riemann curvature tensor, and 52 (M) is the space of covariant tensor fields of degree 5 having the symmetries of the left-hand side of the Bianchi II identity. The action of the operator d, whose order of nilpotency is equal to 3, is written explicitly in terms of components: 1 (∂µ ξν + ∂ν ξµ ), 2 1 = (∂λ ∂ρ hµν + ∂µ ∂ν hλρ − ∂µ ∂ρ hλν − ∂λ ∂ν hµρ ), 4 1 = (∂λ Rµν αβ + ∂µ Rνλ αβ + ∂ν Rλµ αβ ). 3
(dξ )µν = (d 2 h)λµρν (dR)λµναβ
(3.13) (3.14) (3.15)
The generalized 3-complex 2 (M) can be pictured as the commutative diagram ·: · · vv
d 2 vvv
vv vv
: 62 (M) d t O
d 2 ttt t d tt tt / 5 (M) : 42 (M) 2 d t O t 2 d tt t d tt tt / 3 (M) : 22 (M) 2 O d t t d 2 tt t d tt tt 0 1 / (M) (M) 2
d
2
/· · ·
(3.16)
40
X. Bekaert, N. Boulanger
In terms of this diagram, the higher order nilpotency d 3 = 0 translates into the fact that (i) if one takes a vertical arrow followed by a diagonal arrow, or (ii) if a diagonal arrow is followed by a horizontal arrow, it always maps to zero.
3.4. Rectangular diagrams. The generalized cohomology [21] of the N -complex N −1 (M) is the family of graded vector spaces H(k) (d) with 1 ≤ k ≤ N − 1 dep fined by H(k) (d) = Ker(d k )/Im(d N−k ). In general the cohomology groups H(k) (d) are not empty, even when M has a trivial topology. Nevertheless there exists a generalization of the Poincar´e lemma for N -complexes of interest in gauge theories. Let Y S be a maximal sequence ofYoung diagrams. The (generalized) Poincar´e lemma states that for M with the topology of RD the generalized cohomology8 of d on tensors represented by rectangular diagrams is empty in the space of maximal tensors [9, 11, 12]. Proposition 1 (Generalized Poincar´e lemma for rectangular diagrams). 0 (RD ) is the space of real polynomial functions on RD of degree strictly less – H(k) S than k (1 ≤ k ≤ N − 1) and nS (RD ) = 0 ∀n such that 1 ≤ n ≤ D − 1. – H(k) S This is the first theorem of [12], the proof of which is given therein. This theorem strengthens the analogy between the two complexes (3.11) and (3.12) since it implies that (3.12) is also an exact sequence when M has a trivial topology. 2 (RD ) = 0 and exactness at 4 (M) means Exactness at 22 (M) means H(2) 2 2 4 (RD ) = 0. These properties have a physical interpretation in terms of the H(1) 2 linearized Bianchi identity II and gauge transformations. Let Rµνρσ be a tensor that is antisymmetric in its two pairs of indices Rµνρσ = −Rνµρσ = −Rµνσρ , namely it has the symmetry of the Young diagram
.
This latter decomposes according to
=
.
(3.17)
If we impose the condition that R obeys the first Bianchi identity (2.4), we eliminate the last two terms in its decomposition (3.17) hence the tensor R has the symmetries of the Riemann tensor and belongs to 42 (M). Furthermore, from (3.15) it is apparent that the linearized second Bianchi identity (2.5) for R reads dR = 0. As the Riemann tensor has the symmetries of a rectangular diagram, we obtain R = d 2 h with h ∈ 22 (M) from the exactness of the sequence (3.12). This means that R is effectively the linearized Riemann tensor associated to the spin-two field h, as can be directly seen from (3.14). 8 Strictly speaking, the generalized Poincar´e lemma for rectangular diagrams was proved in [11, 12] with another choice of convention where one first antisymmetrizes the columns. This other choice is more convenient to prove the theorem in [12] but is inappropriate for considering Hodge dualization properties. This explains our choice of convention; still, as we will show later, the generalized Poincar´e lemma for rectangular diagrams remains true with the definition (3.10).
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
41
However, the definition of the metric fluctuation h is not unique: the gauge field h + δh is physically equivalent to h if it does not affect the physical linearized Riemann tensor, i.e. d 2 (δh) = 0. Since the sequence (3.11) is exact we find : δh = dξ with ξ ∈ 12 (M). As a result we recover the standard gauge transformations (2.1). 3.5. Multiforms, Hodge duality and trace operators. A good mathematical understanding of the gauge structure of free symmetric tensor gauge field theories is provided by the maximal sequence and the vanishing of the rectangular diagrams cohomology. However, several new mathematical ingredients are needed as well as an extension of Proposition 1 to capture their dynamics. A useful new ingredient is the obvious generalization of Hodge’s duality for S (RD ), which is obtained by contracting the columns with the epsilon tensor εµ1 ...µD of M and lowering the indices with the Minkowskian metric. For rank S symmetric tensor gauge theories there are S different Hodge operations since the corresponding diagrams may contain up to S columns. A simple but important point to note is the following: generically the Hodge duality is not an internal operation in the space S (M). For this reason, we define a new space of tensors in the next subsection. Multiforms. A key ingredient is the graded tensor product of C ∞ (M) with S copies of the exterior algebra RD∗ , where RD∗ , is the dual space of basis di x µ (1 ≤ i ≤ S, thus there are S times D of them). Elements of this space will be referred to as multiforms [12]. They are sums of products of the generators di x µ with smooth functions on M. The components of a multiform define a tensor with the symmetry properties of the product of S columns
.. .. ..
...
.. .
.. .
Notation. We shall denote this multigraded space ⊗S (RD∗ ⊗C ∞ (M) by [S] (M). 1 ,l2 ,... ,lS The subspace l[S] (M) is defined as the space of multiforms whose components S
(1) Y(li ) which represents have the symmetry properties of the diagram Dl1 ,l2 ,... ,lS := i=1
the above product of S columns with respective lengths l1 , l2 , ..., lS . 1 ,... ,lS The tensor field α[µ1 ...µ1 ]...[µS ...µS ] (x) defines a multiform α ∈ l[S] (M) which 1
explicitly reads
l1
1
lS
1
α = α[µ1 ...µ1 ]...[µS ...µS ] (x) d1 x µ1 ∧ . . . ∧ d1 x 1
l1
1
lS
µ1l
1
S
. . . dS x µ1 ∧ . . . ∧ dS x
µSl
S
. (3.18)
In the sequel, when we refer to the multiform α we will speak either of (3.18) or of its components. More accurately, we will identify [S] (M) with the space of the smooth tensor field components. We endow [S] (M) with the structure of a (multi)complex by defining S anticommuting differentials ,... ,li ,... ,lS ,... ,li +1,... ,lS di : l1[S] (M) → l1[S] (M) ,
1≤i ≤S,
(3.19)
42
X. Bekaert, N. Boulanger
defined by adding a box containing the partial derivative in the i th column. For instance, d2 acting on the previous diagrammatic example is
.. .. ..
...
.. .
.. .
∂
Summary of notations. The multicomplex [S] (M) is the subspace of S-uple multi1 ,l2 ,... ,lS (M). The space (S) (M) is the forms. It is the direct sum of subcomplexes l[S] (S + 1)-complex of tensors represented by Young tableaux with at most S columns. It is p (l1 ,l2 ,... ,lS ) (M). The space S (M) = ⊕p S (M) the direct sum of subcomplexes (S) is the space of maximal tensors. Thus we have the chain of inclusions S (M) ⊂ (S) (M) ⊂ [S] (M). Hodge and trace operators. We introduce the following notation for the S possible Hodge dual definitions: ,... ,li ,... ,lS ,... ,D−li ,... ,lS ∗i : l1[S] (M) → l1[S] (M) ,
1≤i ≤S.
(3.20)
The operator ∗i is defined as the action of the Hodge operator on the indices of the i th column. To remain in the space of covariant tensors requires the use of the flat metric to lower down indices. Using the metric, another simple operation that can be defined is the trace. The convention is that we always take the trace over indices in two different columns, say the i th and j th . We denote this operation by l ,... ,li ,... ,lj ,... ,lS
Trij : 1[S]
l ,... ,li −1,... ,lj −1,... ,lS
(M) → 1[S]
(M) ,
i = j .
(3.21)
The Schur module definition (see Subsect. 3.1) gives the necessary and sufficient set of conditions for a (covariant) tensor Tµ1 µ2 ...µp (x) of rank p to be in the irreducible representation of GL(D, R) associated with the Young diagram Y (with |Y | = p). Each index of Tµ1 µ2 ...µp (x) is placed in one box of Y . The set of conditions is the following: (i) Tµ1 µ2 ...µp (x) is completely antisymmetric in the entries of each column of Y , (ii) complete antisymmetrization of Tµ1 µ2 ...µp (x) in the entries of a column of Y and another entry of Y which is on the right of the column, vanishes. Using the previous definitions of multiforms, Hodge dual and trace operators, this set of conditions gives 1 ,... ,lS Proposition 2 (Schur module). Let α be a multiform in l[S] (M). If
lj ≤ li < D ,
∀ i, j ∈ {1, . . . , S} : i ≤ j ,
then one obtains the equivalence Trij { ∗i α } = 0 ∀ i, j : 1 ≤ i < j ≤ S
⇐⇒
(l ,... ,lS )
1 α ∈ (S)
(M) .
Indeed, condition (i) is satisfied since α is a multiform. Condition (ii) is simply rewritten in terms of tracelessness conditions.
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
43
Another useful property, which generalizes the derivation followed in the chain of Eqs. (2.20)–(2.24), is for any i, j ∈ {1, . . . , S}, Trij α = 0 • (3.22) ⇒ dj (∗j α) = 0 . di α = 0 The following property on powers of the trace operator will also be useful later on. We state it as 1 ,... ,lS (M) be a multiform. For any m ∈ N such that 0 ≤ m ≤ Proposition 3. Let α ∈ l[S] min(D − li , D − lj ), one has the equivalence
m Trij {∗i ∗j α} = 0
⇐⇒
m+li +lj −D Trij {α} = 0.
Proof. The proof of the proposition is inductive, the induction parameter being the number of traces, and is mainly based on the rule for contractions of epsilon tensors. ⇒: We start the proof of the necessity by a preliminary lemma: For any given integer p ∈ N, D−li −p l −p Trij {∗i ∗j α} = 0 (3.23) ⇒ Trij j {α} = 0 . lj −n+1 Trij {α} = 0 , ∀n ≥ p D−li −p This is true because it can be checked explicitly that Trij {∗i ∗j α} is equal to a lj −k sum of terms proportional to Trij {α} for all k ≥ p. The second hypothesis says that D−li −p l −p these last terms vanish for k ≥ p +1. As a result, Trij {∗i ∗j α} ∝ Trij j {α}. D−li −p l −p Therefore the vanishing of Trij {∗i ∗j α} implies the vanishing of Trij j {α}. Now that this preliminary lemma is given, we can turn back to our inductive proof. The induction hypothesis Im is the following: m Trij {∗i ∗j α} = 0
⇒
m+li +lj −D Trij {α} = 0.
(3.24)
The starting point of the induction is ID−li +1 [considering without loss of generality that D − li = min(D − li , D − lj )] which is obviously true since in this case where m = D − li + 1, both traces in (3.24) vanish. What we have to show now is that, if In is true ∀ n ≥ m, then Im−1 is also true. m−1 n {∗i ∗j α} = 0 implies that Trij {∗i ∗j α} = 0 for all It is obvious that Trij n+li +lj −D n ≥ m − 1. The induction hypothesis thus implies that Trij { α } = 0 for all m−1 {∗i ∗j α} = 0 and the help of the lemma (3.23), we n ≥ m. Together with Trij m−1+li +lj −D eventually obtain Trij { α } = 0, which ends the proof of the induction hypothesis. ⇐: In this case, the sufficiency is a consequence of the necessity. In other words, since we proved that the implication Im is valid from the left to the right in (3.24) we will show that then, it is also valid from the right to the left. Indeed, the relation ∗i ∗j (∗i ∗j α) = ±α allows to write m+li +lj −D m+(li −D)+(lj −D)+D { α } = ± Trij { ∗i ∗j (∗i ∗j α) } . Trij
(3.25)
44
X. Bekaert, N. Boulanger
The (proven) implication Im of Proposition 3 applied to the multiform ∗i ∗j α ∈ l ,... ,D−li ,... ,D−lj ,... ,lS
1 (S)
(M) is
m+(li −D)+(lj −D)+D m Trij { ∗i ∗j (∗i ∗j α) } = 0 ⇒ Trij { α } = 0 .
(3.26)
Combined with the relation (3.25), the previous implication is precisely the (reversed) implication in Proposition 3.
3.6. Generalized nilpotency. Let Yp be well-included9 into Yp+q , that is Yp Yp+q . Let I be the subset of {1, 2, . . . , S} containing the q elements (#I = q) corresponding to the difference between Yp+q and Yp . We “generalize” the definition (3.10) by introducing the differential operators d I as follows (see also [9, 10]) p
d I ≡ cI Yp+q ◦
p p+q ∂i : (Y ) (M) → (Y ) (M),
(3.27)
i∈I
where ∂i indicates that the index corresponding to this partial derivative is placed at the p bottom of the i th column and cI are normalization factors so that we have strict equalities 10 in the next Proposition 4 . When I contains only one element (q = 1) we recover the definition (3.10) of d. The tensor d I Yp will be represented by the Young diagram Yp+q , where we place a partial derivative symbol ∂ in the q boxes which do not belong to the subdiagram Yp Yp+q . The product of operators d I is commutative : d I ◦ d J = d J ◦ d I for all I, J ⊂ {1, . . . , S} (#I = q, #J = r) such that the product maps to a well-defined Young diagram Yp+q+r . The following proposition gathers all these properties Proposition 4. Let I and J be two subsets of {1, . . . , S}. Let α be an irreducible tensor belonging to (2) (M). The following properties are satisfied11 – If I ∩ J = φ, then (d I ◦ d J )α = d I ∪J α. Therefore, d I α = d #I α. – If I ∩ J = φ, then (d I ◦ d J )α = 0. Proposition 4 is proved in [9]. The last property states that the product of d I and d J identically vanishes if it is represented by a diagram Yp+q+r with at least one column containing two partial derivatives. Proposition 4 proves that the operator d I provides the most general non-trivial way of applying partial derivatives in (S) (M). Proposition 4 is also helpful because it makes contact with the definition (3.10) in that the operator d I can be identified, up to a constant factor, with the (non-trivial) #I th power of the operator d. Despite this identification, we frequently use the notation d I because it contains more information than the notation d #I . 9
See Subsect. 3.1. p The precise expression for the constants cI was obtained in [9]. 11 According to the terminology of [9], these properties mean that the set Y (S) is endowed with the structure of hypercomplex by means of the maps d I . 10
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
45
The space (2) (M) can be pictured analogously to the representation (3.6) of the set of Young diagrams Y (2) s. 9 . . ss s s ss ss (3,3) / ·; · · (2) (M) O 8 ww r w r w rr ww rrr ww r w r r w (2,2) / ·; · · / (3,2) (M) (2) (M) (2) 8 O 8 O ww r r w r r w rr rr ww rrr rrr ww r r w r r r r w (1,1) / ·; · · / (2,1) (M) / (3,1) (M) (2) (M) 8 O 8 (2) O 8 (2) O ww r r r w r r r r r r ww rrr rrr rrr ww r r r w r r r rr rr rr ww (0,0) / (2,0) (M) / (3,0) (M) /· · · / (1,0) (M) (2) (M) (2) (2) (2) From the previous discussions, the definitions of the arrows should be clear: → : Horizontal arrows are maps d = d {1} . ↑ : Vertical arrows are maps d = d {2} . : Diagonal arrows are maps d 2 = d {1,2} . Proposition 4 translates in terms of this diagram into the fact that – this diagram is completely commutative, and – the composition of any two arrows with at least one common direction maps to zero identically. Of course, these diagrammatic properties hold for arbitrary S (the corresponding picture would be simply a higher-dimensional generalization since Y(S) ⊂ RS ). 4. Linearized Gravity Field Equations From now on, we will restrict ourselves to the case of linearized gravity, i.e. rank-2 symmetric gauge fields. There are two possible Hodge operations, denoted by ∗, acting on the first column if it is written on the left, and on the second column if it is written on the right. Since we are no longer restricted to maximal Young diagrams the notation d is ambiguous (we do not know a priori on whichYoung symmetry type we should project in the definition (3.10)). Instead we use the above mentioned differentials di of multiform theory. There are only two of these in the case of linearized gravity: d1 called the (left) differential, denoted by dL , and d2 , the (right) differential, denoted by dR . With these differentials it is possible to rewrite (2.25) in the compact form dL ∗ R = 0 = dR R∗. The second Bianchi identity reads dL R = 0 = dR R. The convention that we use is to take the trace over indices in the first row, using the flat background metric ηµν . We denote this operation by Tr (which is Tr12 according to the definition given in the previous section). In this notation the Einstein equation (2.8) takes the form TrR = 0, while the first Bianchi identity (2.4) reads Tr∗R = 0.
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X. Bekaert, N. Boulanger
From Proposition 2, it is clear that the following property holds: let B be a “biform” in p,q [2] (M) which means B is a tensor with symmetry 1 2 .. .. .. p
1 .. . q
,
(4.1)
then, B obeys the (first) “Bianchi identity” Tr(∗B) = 0
(4.2)
(p,q)
if and only if B ∈ (2) (M). This is pictorially described by the diagram .. .. . .. ..
(4.3)
that is, the two columns of the product are attached together. With all the new artillery introduced in the previous section, it becomes easier to extend the concept of electric-magnetic duality for linearized gravity. First of all we emphasize the analogy between the Bianchi identities and the field equations by rewriting them respectively as
Tr ∗ R = 0 dL R = 0 = dR R
(4.4)
,
and
µ
TrR = 0 dL (∗R) = 0 = dR (R∗)
(4.5)
,
ρ
where Rµν ρσ ≡ ν σ . We recall that dL (∗R) = 0 = dR (R∗) was obtained in Sect. 2 by using the second Bianchi identity. As discussed in Subsect. 3.5, the first Bianchi identity implies that R effectively has µ ρ
the symmetry properties of the Riemann tensor, i.e. Rµν ρσ ≡ ν σ . Using this symmetry property the two equations dL R = 0 = dR R can now be rewritten as the single equation dR = 0. Therefore, if the manifold M is of trivial topology then, for a given multiform R ∈ 2,2 [2] (M), one obtains the equivalence
Tr ∗ R = 0 dL R = 0 = dR R
due to Proposition 1 and Proposition 2.
⇔
R = d 2h h ∈ 22 (M)
,
(4.6)
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
47
4.1. Dual linearized Riemann tensor. By Proposition 2, the (vacuum) Einstein equation TrR = 0 can then be translated into the assertion that the dual of the Riemann tensor has (D−2,2) (on-shell) the symmetries of a diagram (D − 2, 2), in other words ∗R ∈ (2) (M). As explained in Subsect. 2.3, the second Bianchi identity dR R = 0, together with the linearized Einstein equations, implies the equation dL ∗ R = 0. Furthermore the second Bianchi identity dR R = 0 is equivalent to dR ∗R = 0, therefore we have the equivalence (D−2,2) TrR = 0 (M) ∗R ∈ (2) . (4.7) ⇔ dR R = 0 dL ∗ R = 0 = dR ∗ R In addition dL ∗ R = 0 = dR ∗ R now implies ∗R = d 2 h˜ (where we denote the non(D−3,1) (M) is the dual gauge ambiguous product dL dR by d 2 ). The tensor field h˜ ∈ (2) field of h obtained in (2.26). This property (2.26), which holds for manifolds M with the topology of RD , is a direct application of Corollary 1 of the generalized Poincar´e lemma given in the following section; we anticipate this result here in order to motivate the theorem by using a specific example. We have an equivalence analogous to (4.6), ∗R = d 2 h˜ TrR = 0 . (4.8) ⇔ (D−3,1) dL ∗ R = 0 = dR ∗ R h˜ ∈ (2) (M) Therefore linearized gravity exhibits a duality symmetry similar to the electric-magnetic duality of electrodynamics, which interchanges Bianchi identities and field equations (D−3,1) (M) have mixed symmetry and were discussed [6]. Tensor gauge fields in (2) above in Sect. 2.4. The right-hand-side of (2.26) is represented by ∂ ∂ .. .. ..
.. .. ..
∂
.
∂
The appropriate symmetries are automatically implemented by the antisymmetrizations in (2.26) since the dual gauge field h˜ already has the appropriate symmetry .. .. ..
.
In other words, the two explicit antisymmetrizations in (2.26) are sufficient to ensure (2) that the dual tensor ∗R possesses the symmetries associated with Y(D−2,2) . A general explanation of this fact will be given at the end of the next section. The dual linearized Riemann tensor is invariant under the transformation (D−4,1) (D−3,0) ∼ D−3 (M) . (M) , A ∈ (M) = δ h˜ = d(S + A) with S ∈ (2)
(2)
(4.9) The right-hand side of this gauge transformation, explicitly written in (2.28), is represented by
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X. Bekaert, N. Boulanger
∂
.. .. .. ∂
.. .. ..
In this formalism, the reducibilities (2.37) and (2.38) respectively read (up to coefficient redefinitions) (i−1)
(i)
(i)
S = dL S +d A , (i)
(i−1)
(D−3−i,1)
S ∈ (2)
(i)
A = −dL A , (M) ,
(i = 2, . . . , D − 2) ,
(i)
(4.10)
A ∈ D−2−i (M) .
These reducibilities are a direct consequence of Corollary 2. 4.2. Comparison with electrodynamics. Compared to electromagnetism, linearized gravity presents several new features. First, there are now two kinds of Bianchi identities, some of which are algebraic relations (Bianchi I) while the others are differential equations (Bianchi II). In electromagnetism, only the latter are present. Second (and perhaps more importantly), the equation of motion of linearized gravity theory is an algebraic equation for the curvature (more precisely, TrR = 0). This is natural since the curvature tensor already contains two derivatives of the gauge field. Moreover, for higher (1,... ,1) (M) (S ≥ 3) the natural gauge invariant curvature spin gauge fields h ∈ S (2,... ,2) S (M) contains S derivatives of the completely symmetric gauge field, d h ∈ S hence local second order equations of motion cannot contain this curvature. Third, the current conservation in electromagnetism is a direct consequence of the field equation while for linearized gravity the Bianchi identities play a crucial role. In relation to the first remark, the introduction of sources for linearized gravity seems rather cumbersome to deal with. A natural proposal is to replace the Bianchi I identities by equations Tr ∗ R = Tˆ ,
Tˆ ∈ D−3,1 (M) . [2]
(4.11)
If one uses the terminology of electrodynamics it is natural to call Tˆ a “magnetic” source. If such a dual source is effectively present, i.e. Tˆ = 0, the tensor R is no longer irreducible under GL(D, R), that is to say R becomes a sum of tensors of different symmetry types and only one of them has the Riemann tensor symmetries. This seems a difficult starting point. The linearized Einstein equations read TrR = T ,
T ∈ 1,1 [2] (M) .
(4.12)
˜ The “electric” source The sources T and Tˆ respectively couple to the gauge fields h and h. T is a symmetric tensor (related to the energy-momentum tensor) if the dual source Tˆ vanishes, since R ∈ 42 (M) in that case. Another intriguing feature is that a violation of Bianchi II identities implies a non-conservation of the linearized energy-momentum tensor because ∂ µ Tµν =
3 µρ ∂ R , 2 [µ νρ]
according to the linearized Einstein equations (2.15).
(4.13)
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
49
Let us now stress some peculiar features of D = 4 dimensional spacetime. From our previous experience with electromagnetism and our definition of Hodge duality, we naturally expect this dimension to be privileged. In fact, the analogy between linearized gravity and electromagnetism is closer in four dimensions because less independent equations are involved : ∗R has the same symmetries as the Riemann tensor, thus the dual gauge field h˜ is a symmetric tensor in 22 (M). So the Hodge duality is a symmetry of the theory only in four-dimensional spacetime. The dual tensor ∗R is represented by a Young diagram of rectangular shape and Proposition 1 can be used to derive the existence of the dual potential as a consequence of the field equation d ∗ R = 0.
5. Generalized Poincar´e Lemma Even if we restrict our attention to completely symmetric tensor gauge field theories, the Hodge duality operation enforced the use of the space (S) (M) of tensors with at most S columns in the previous section. This unavoidable fact requires an extension of Proposition 1 to general irreducible tensors in (S) (M).
5.1. Generalized cohomology. The generalized cohomology12 of the generalized complex (S) (M) is defined to be the family of graded vector spaces H(m) (d) = (l1 ,... ,lS ) (l1 ,... ,lS ) (l1 ,... ,lS ) ⊕Y(S) H(m) (d) with 1 ≤ m ≤ S, where H(m) (d) is the set of α ∈ (S) (M) such that dI α = 0
∀I ⊂ {1, 2, . . . , S} | #I = m , d I α ∈ (S) (M)
(5.1)
with the equivalence relation α ∼ α +
d J βJ ,
βJ ∈ (S) (M) .
(5.2)
J ⊂ {1, 2, . . . , S} #J = S − m + 1 Let us stress that each βJ is a tensor in an irreducible representation of GL(D, R) such (l1 ,... ,lS ) (M). In other words, each irreducible tensor d J βJ is represented that d J βJ ∈ (S) by a specific diagram Y(lJ1 ,... ,lS ) constructed in the following way : (S)
1st. Start from the Young diagram Y(l1 ,... ,lS ) of the irreducible tensor field α. 2nd. Remove the lowest cell in S − m + 1 columns of the diagram, making sure that the remainder is still a Young diagram. 3rd. Replace all the removed cells with cells containing a partial derivative. The irreducible tensors βJ are represented by a diagram obtained at the second step. 12 This definition of generalized cohomology extends the definition of “hypercohomology” introduced in [9].
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X. Bekaert, N. Boulanger
A less explicit definition of the generalized cohomology is by the following quotient Ker d m H(m) (d) = . Im d S−m+1
(5.3)
We can now state a generalized version of the Poincar´e lemma, the proof of which will be postponed to the next subsection because it is rather lengthy and technical. (l ,... ,l )
S Theorem (Generalized Poincar´e lemma). Let Y(S)1 be a Young diagram with lS = 0 and columns of lengths strictly smaller than D : li < D, ∀i ∈ {1, 2, . . . , S}. For all m ∈ N such that 1 ≤ m ≤ S one has that (l1 ,... ,lS ) (S) (RD ) ∼ H(m) = 0.
The theorem extends Proposition 1; the latter can be recovered retrospectively by the fact that, for rectangular tensors, there exists only one d I α and one βJ .
5.2. Applications to gauge theories. In linearized gravity, one considers the action of nilpotent operators di on the tensors instead of the distinct operators d {i} . However, it is possible to show the useful Proposition 5. Let α be an irreducible tensor of (S) (M). We have the implication
di
α = 0,
∀I ⊂ {1, 2, . . . , S} | #I = m
i∈I
⇒
dI α = 0 ,
∀I ⊂ {1, 2, . . . , S} | #I = m .
Therefore, the conditions appearing in symmetric tensor gauge theories are stronger than the cocycle condition of H(m) (d) and the coboundary property also applies. Now we present the following corollary which is a specific application of the theorem together with Proposition 5. Its interest resides in its applicability in linearized gravity field equations (we anticipated the use of this corollary in the previous subsection). Corollary 1. Let κ ∈ (S) (M) be an irreducible tensor field represented by a Young diagram with at least one row of S cells and without any column of length ≥ D − 1. If the tensor κ obeys di κ = 0
∀i ∈ {1, . . . , S} ,
then κ=
S di λ, i=1
where λ belongs to (S) (M) and the tensor κ is represented by a Young diagram where all the cells of the first row are filled by partial derivatives.
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
51
Proof. The essence of the proof is that the two tensors with diagrams ... ∂ ... .. .. .. . . ...
∂ ∂
...
and
.. .
.. . ∂
.. ... ...
∂
∂
are proportional since the initial symmetrization of the partial derivatives will be cancelled out by the antisymmetrization in the columns that immediately follows for two attached columns of different length (if they have the same length the partial derivatives are in the same row and the symmetrization is automatic). By induction, starting from the left, one proves that this is true for an arbitrary number of columns. This argument remains true if we add smaller columns on the right of the Young diagram. The last subtlety in the corollary is that antisymmetrization in each column (κ = ( di )λ) automatically provides the appropriate Young symmetrization since λ has the appropriate symmetry. This can be easily checked by performing a complete antisymmetrization of the tensor κ in the entries of a column and another entry which is on its right. The result automatically vanishes because the index in the column at the right is either – attached to a partial derivative, in which case the antisymmetrization contains two partial derivatives, or – attached to the tensor λ. In this case, antisymmetrization over the indices of the column except the one in the first row (corresponding to a partial derivative) causes an antisymmetrization of the tensor λ in the entries of a column and another entry which is on the right-hand side. The result vanishes since λ has the symmetry properties corresponding to the diagram obtained after eliminating the first row of κ. This last discussion can be summarized by the operator formula S
(S)
(S)
(S)
di ◦ Y(l1 ,... ,lS ) ∝ Y(l1 +1,... ,lS +1) ◦ ∂ S ◦ Y(l1 ,... ,lS ) ,
(5.4)
i=1
where ∂ S are S partial derivatives with indices corresponding to the first row of a given Young diagram. We now present another corollary, which determines the reducibility identities for the mixed symmetry type gauge field. Corollary 2. Let λ ∈ (2) (M) be a sum of two irreducible tensors (l −1,l2 )
λ1 ∈ (2)1
(l ,l2 −1)
(M) and λ2 ∈ (2)1
(M) with l1 ≥ l2 (λ1 = 0 if l1 = l2 ).
Then, 2
i=1
where
d {i} λi = 0
⇒
λi =
2
j =1
d {j } µij
(i = 1, 2) ,
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X. Bekaert, N. Boulanger (l −2,l2 )
– µ11 ∈ (2)1 – –
(M) (which vanishes if l1 ≤ l2 + 1),
(l −1,l −1) µ12 , µ2,1 ∈ (2)1 2 (M) (l ,l −2) µ22 ∈ (2)1 2 (M).
(µ1,2 = 0 if l1 = l2 ), and
Furthermore, if l1 > l2 we can assume, without loss of generality, that µ21 = −µ12 . Proof. We apply d {1} and d {2} to the equation
2
d {i} λi = 0 and obtain 0 = d 1,2 λi ∝
i=1
d1 d2 λi in view of the remarks following Corollary 1. From the theorem, we deduce that 2 λi = d {j } µij with tensors µij in the appropriate spaces given in Corollary 2. The j =1
fact they vanish agrees with the rule given above. To finish the proof we should consider the case l1 > l2 . Assembling the results 2 together, d {i} λi = d {1,2} (µ12 + µ21 ) = 0 due to Proposition 4. Thus, d1 d2 (µ12 + i=1
µ21 ) = 0. Using Corollary 2 again, one obtains µ12 + µ21 = (l −2,l −1)
2
d {k} νk with ν1 ∈
k=1
(l −1,l −2)
(2)1 2 (M) and ν2 ∈ (2)1 2 (M) (ν1 = 0 if l1 = l2 ). Hence we can make the redefinitions µ12 → µ"12 = µ12 − d {2} ν2 and µ21 → µ"12 = µ21 − d {1} ν1 which do not affect λ, in such a way that we have µ"21 = −µ"12 . This proposition can be generalized to give a full proof of the gauge reducibility rules given in [8] and will be reviewed in Subsect. 6.2. 6. Arbitrary Young Symmetry Type Gauge Field Theories We now generalize the results of Sect. 4 to arbitrary irreducible tensor representations of GL(D, R). The discussion presented below fits into the approach followed by [18] for two columns (S = 2) and by [8, 6] for an arbitrary number of columns. The interest of this section lies in the translation of these old results in the present mathematical language and in the use of the generalized Poincar´e lemma for a more systematic mathematical foundation. 6.1. Bianchi identities. Firstly, we generalize our previous discussion on linearized gravity by introducing a tensor K, which is the future curvature. A priori, K is a multi1 ,... ,lS form of l[S] (M) (lS = 0) with 1 ≤ lj ≤ li < D for i ≤ j . Secondly, we suppose the (algebraic) Bianchi I relations to be Trij { ∗i K} = 0 ,
∀i, j : 1 ≤ i < j ≤ S ,
(6.1)
in order to obtain, from Proposition 2, that K is an irreducible tensor under GL(D, R) (l1 ,... ,lS ) (M). Thirdly, we define the (differential) Bianchi II relations as belonging to (S) di K = 0 ,
∀i : 1 ≤ i ≤ S ,
(6.2)
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
53
in such a way that, from Corollary 1, one obtains K = d1 d2 . . . dS κ .
(6.3)
In this case, the curvature is indeed a natural object for describing a theory with gauge (l1 −1,... ,lS −1) (M). The gauge invariances are then fields κ ∈ (S) κ → κ + d {i} βi ,
(6.4) (l −1,... ,l −2,... ,l −1)
1 i S where the gauge parameters βi are irreducible tensors βi in (S) (M) for any i such that li ≥ 2 (and li > li−1 ), as follows from our theorem and Proposition 5.
6.2. Reducibilities. The gauge transformations (6.4) are generally reducible, i.e. d {j } βj ≡ 0 for non-vanishing irreducible tensors βj = 0. The procedure followed in the proof of Corollary 2 can be applied to the general case. This generates the inductive rules of (i+1)
[8] to form the (i + 1)th generation reducibility parameters β
j1 j2 ...ji+1
from the i th
(i)
generation parameters β j1 j2 ...ji : – i=1 (S)
(A) Start with the Young diagram Y(l1 −1,... ,lS −1) corresponding to the tensor gauge field κ. (B) Remove a box from a row such that the result is a standard Young diagram. In other words, the gauge parameters are taken to be the first reducibility parameters: (1)
βj = β j . – i →i+1 (C) Remove a box from a row which has not previously had a box removed (in forming the lower generations of reducibility parameters) such that the result is a standard Young diagram. (D) There is one and only one reducibility parameter for each Young diagram. The Labastida-Morris rules (A)-(D) provide the complete BRST spectrum with the full tower of ghosts of ghosts. More explicitly, the chain of reducibilities is (i)
(i+1)
β j1 j2 ...ji = d {ji+1 } β
j1 j2 ...ji ji+1 =
0,
(i = 1, 2, . . . , r),
(6.5)
where r = l1 − 1 is the number of rows of κ. The chain is of length r because at each step one removes a box from a row which has not been chosen before. We can see that the order of reducibility of the gauge transformations (6.4) is equal to l1 − 2. For S = 1, we recover the fact that a p-form gauge field theory (l1 = p + 1) has its order of reducibility equal to p − 1. (i)
The subscripts of the i th reducibility parameter β j1 j2 ...ji belong to the set {1, . . . , S}. (i)
These determine the Young diagram corresponding to the irreducible tensor β j1 j2 ...ji :
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X. Bekaert, N. Boulanger
reading from the left to the right, the subscripts give the successive columns from which (i)
to remove the bottom box following the rules (A)-(C). A reducibility parameter β j1 j2 ...ji vanishes if these rules are not fulfilled. Furthermore, they are antisymmetric for any pair of different indices (i)
(i)
β j1 ...jk ...jl ...ji = − β j1 ...jl ...jk ...ji ,
∀jl = jk .
(6.6)
This property ensures the rule (D) and provides the correct signs to fulfill the reducibilities. Indeed, (i)
(i+1)
d {ji } β j1 j2 ...ji = d {ji } d {ji+1 } β
j1 j2 ...ji ji+1 =
0,
(6.7)
due to Proposition 4 and Eq. (6.6). 6.3. Field equations and dualisation properties. We make the important following assumption concerning the positive integers li (i = 1, . . . , S) associated to K ∈ 1 ,... ,lS l[S] : l i + lj ≤ D ,
∀ i, j
(6.8)
∀ i, j .
(6.9)
and take the field equations to be in that case Trij {K} = 0 ,
Indeed, if l1 + l2 > D and if Eq. (6.9) holds, then the curvature K identically vanishes, as is well known when studying irreps of O(D − 1, 1). This property is a particular instance of Proposition 3 (for m = 0). The field equations (6.9) combined with the Bianchi I identities (6.1) state that the curvature K is a tensor irreducible under O(D − 1, 1). To any non-empty subset I ⊂ {1, 2, . . . , S} (#I = m), we associate a Hodge duality operator ∗I ≡ ∗k . (6.10) k∈I 1 ,... ,S The dual ∗I K of the curvature is a multiform in [S] (M), where the lengths i are defined in Eq. (3.8). The Bianchi I identities (6.1) together with the field equations (6.9) imply the relations
Trij {∗i (∗I K )} = 0 ,
∀i, j : j ≤ i ,
(6.11)
where i is the length (3.8) of the i-th column of the dual tensor ∗I K. Indeed, let be i and j such that j ≤ i . There are essentially four possibilities: – i ∈ I and – j ∈ I : Then lj ≤ li and the Bianchi I identities (6.1) are equivalent to (6.11) since Trij and ∗k commute if i = k and j = k. – j ∈ I : Then one should have D − lj ≤ li which means that D ≤ li + lj , in contradiction with the hypothesis (6.8) except for the case where li + lj = D. From Proposition 3, we deduce that in such a case the field equations (6.9) are equivalent to (6.11).
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
55
– i ∈ I and – j ∈ I : We have lj ≤ D − li which is equivalent to li + lj ≤ D. The field equations (6.9) are of course equivalent to (6.11) since ∗2i K = ±K. – j ∈ I : We have D − lj ≤ D − li which is equivalent to li ≤ lj . The Bianchi I identities Trj i {∗j K} = 0 are therefore satisfied and equivalent to (6.11) because Trij = Trj i . (S) I Let Y (λ1 ,... ,λS ) be the Young diagram dual to Y(l1 ,... ,lS ) . We define KI to be the multi-
1 ,... ,λS (M) obtained after reordering the columns of ∗I K. The identity (6.11) form in λ[S] can then be formulated as
I } = 0 , ∀ i, j : 1 ≤ i < j ≤ S . Trij {∗i K
(6.12)
I is irreducible under Due to Proposition 2, it follows from (6.12) that the tensor K GL(D, R), I ∈ (λ1 ,... ,λS ) (M) . K (S)
(6.13)
Now we use the property (3.22) to deduce from the Bianchi II identities (6.2) and the field equations (6.9) that di (∗i K) = 0 for any i. Therefore di (∗I K) = 0 ,
∀i ∈ {1, . . . , S} ,
(6.14)
because di and ∗j commute if i = j , and either – i ∈ I so (6.14) follows from di K = 0, or – i ∈ I and then (6.14) is a consequence of di ∗i K = 0. I satisfies (on-shell) its own Bianchi II identity In other words, any dual tensor K (6.14) which, together with (6.13), implies the (local) existence of a dual gauge field κ˜ I such that the Hodge dual of the curvature is itself a curvature I = d1 d2 . . . dS κ˜ I K
(6.15)
for some gauge field (λ1 −1,... ,λS −1) (M) . (S)
κ˜ I ∈
(6.16)
The Hodge operators ∗I therefore relate different free field theories of arbitrary tensor gauge fields, extending the electric-magnetic duality property of electrodynamics. In the same way, we obtain the field equations of the dual theory m
Trij ij { ∗I K } = 0 , where
mij ≡
1 + D − l i − lj 1
∀ i, j : i < j , if i and j ∈ I , if i or j ∈ I .
(6.17)
(6.18)
Indeed, since the trace is symmetric in i and j we must consider only three distinct cases: – i ∈ I and j ∈ I : The starting field equation (6.9) is naturally equivalent to the dual field equation (6.17). – i ∈ I and
56
X. Bekaert, N. Boulanger
– j ∈ I : If i < j the Bianchi I relation (6.1) is satisfied and it implies (6.17). – j ∈ I : A direct use of Proposition 3 leads from the field equation (6.9) to (6.17). We can summarize the algebraic part of the previous discussions in terms of a remark on tensorial irreps of 0(D − 1, 1). I Remark. Let I ⊂ {1, . . . , S} be a non-empty subset. Let Y (λ1 ,... ,λS ) be the Young dia(S)
(l ,... ,lS )
1 gram dual to Y(l1 ,... ,lS ) . If α ∈ (S)
(M) is a tensor in the irreducible represen(S)
tation of O(D − 1, 1) associated to the Young diagram Y(l1 ,... ,lS ) , then the dual tensor (λ ,... ,λ )
S αI ∈ (S)1 (M) is in the irreducible representation of O(D − 1, 1) associated to I the Young diagram Y (λ1 ,... ,λS ) .
As one can see, a seemingly odd feature of some dual field theories is that their field equations (6.17) are not of the same type as (6.9). In fact, the dual field equations are of the type (6.9) for all I only in the exceptional case where D is even and li = lj = D/2. Note that this condition is satisfied for free gauge theories of completely symmetric tensors in D = 4 flat space. The point is that i + j = 2D − li − lj ≥ D for i, j ∈ I , I . therefore the property (6.8) is generally not satisfied by the dual tensor K To end up, we generalize the field equation (6.9) to the case where the hypothesis (6.8) is not satisfied. A natural idea is that when li + lj > D for a curvature tensor (l1 ,... ,lS ) (M) (lS = 0), the corresponding fields equations are [6] K ∈ (S) 1+li +lj −D
Trij
{K } = 0.
(6.19)
Acknowledgement. We are grateful to M. Henneaux for a crucial remark at the initial stage of this work. We would like to thank M. Henneaux and C. Schomblond for their careful reading and comments on intermediate versions of this paper. X.B. acknowledges D. Tonei for her help in English and the organizers of the conference “Rencontres Math´ematiques de Glanon (6`eme e´ dition)” where this work was presented [22]. This work was supported in part by the “Actions de Recherche Concert´ees” of the “Direction de la Recherche Scientifique - Communaut´e Francaise de Belgique”, IISN-Belgium (convention 4.4505.86), a “Pˆole d’Attraction Interuniversitaire” (Belgium) and the European Commission RTN programme HPRNCT-2000-00131, in which we are associated with K.U. Leuven.
Appendix A. Inductive Proof of the Generalized Poincar´e Lemma The proof of the generalized Poincar´e lemma that we give hereafter is inductive in several directions. The first induction parameter is the number S of columns; in Sect. A.1 we start from the standard Poincar´e lemma, i.e. S = 1, and compute the generalized cohomologies when a cell is added in a new, second column, i.e. S = 2. The second induction parameter is the number of cells in the new (second) column. Thus Sect. A.1, (∗,1) which gives a (pictorial) proof that the cohomological groups (2) (RD ) are trivial, also provides the starting point for the induction on , keeping S = 2 fixed. (∗,∗) The inductive proof of the vanishing of H(∗) (2) (RD ) is then given in Sect. (∗,∗) A.2. This proof of H(∗) (2) (RD ) ∼ = 0 is purely algebraic and does not contain any pictorial description, this time. However, for a better understanding of the algebraic demonstration, a pictorial translation of most of the results obtained in A.2 is furnished in Sect. A.3.
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
57
The inductive progression we have sketched above is the one used to obtain the proof (∗,... ,∗) ((∗) (RD )) ∼ of the vanishing H(∗) = 0, for diagrams obeying the assumptions of the Theorem (Sect. 5). This time, instead of progressing from S = 1 to S = 2 and then from a length-( − 1) to a length- second column, we go from S to S + 1 columns and subsequently, keeping the number of columns fixed to S + 1, we increase the length of the last (S + 1)th column. Since this progression, exposed in detail in Sect. A.1, A.2 and A.3, provides the proof of our generalized Poincar´e lemma, we only summarize those results in Sect. A.4 and cast our theorem in precise mathematical terms.
(∗,1)
A.1. Generalized cohomology in (2) (RD ). Using the standard Poincar´e lemma (S = (n,1) 1), we begin by providing a pictorial proof that the two cohomologies H(1) (2) (RD ) (n,1) and H(2) (2) (RD ) are trivial for 0 < n < D, i.e. that • (1) 1 n+1
d {i} 2 .. . implies
= 0 ,
i = 1, 2
(A.1)
n
1 n+1 2
.. .
and • (2)
1 ∂ 2
=
(A.2)
.. .
n
∂
1 n+1
d {1,2} 2 .. . implies
= 0
(A.3)
n
1 n+1 2 =
1 ∂ 2
n
n
.. .
.. .
+
1 n+1 2 .
.. .
(A.4)
∂
The numbers in the cells are irrelevant, they simply signal the length of the columns. For clarity we recall the following convention that, whenever a Young tableau Y appears with certain boxes filled in with partial derivatives ∂, one takes a field with the representation of the Young tableau obtained by removing all the ∂-boxes from Y . One differentiates this new field as many times as there are derivatives in Y and then project the result on the Young symmetry of Y .
58
X. Bekaert, N. Boulanger
First cohomology group. For the two different possible values of i in (A.1) we have the following two conditions on the field (n, 1) : 1 n+1
–
.. .
=0
for i = 1
=0
for i = 2 .
n ∂
and –
1 n+1 .. ∂
.
n
The first condition is treated now : one considers the index of the second column as a spectator, which yields 1 2
n+1
1 2
.. .
⊗ n+1
.. .
n ∂
= 0,
n ∂
where the symbol means that there is an implicit projection using Y on the right-hand side in order to agree with the left-hand side (in other words the symbol replaces the expression = Y). The Poincar´e Lemma is used for the first column to write, using branching rules for GL(D, R) : 1 n+1
.. .
1 2
.. .
n
⊗ n ∂ ⊗
1 2
.. .
n−1 ∂
⊗ n
.
n−1
In the last step, we have undone the manifest antisymmetrization with the index carrying the partial derivative; we are more interested in the symmetries of the tensor under the derivative. We first perform the product in the brackets to obtain a sum of different types of irreducible tensors. Then, we perform the product with the partial derivative to get 1 2
.. .
n−1 n
n+1
1 2
.. .
n−1 n
∂
⊕
1 2
.. .
n−1 ∂
n
⊕
1 2
.. .
.
(A.5)
n−1 n ∂
The last term in the above Eq. (A.5) does not match the symmetry of the left-hand-side, so it must vanish. Using the Poincar´e lemma, which is applicable since one is not in top form degree : n < D, one obtains
Tensor Gauge Fields in Arbitrary Representations of GL(D, R) 1 2
59
1 2
=
.. . n
(A.6)
.
.. . ∂
Substituting this result in the decomposition (A.5) yields 1 2
n+1
.. .
n−1 n
1 2
⊕
.. .
n
1 2
∂
−→
.. .
n−1 ∂
n−1 ∂
n
1 2
(A.7)
.. .
n−1 ∂
where the arrow indicates that we performed a field redefinition. Thus, without loss of generality, the right-hand-side can be assumed to contain a partial derivative in the first column. With this preliminary result, the second condition expressed in (A.1), 1 n+1
d {2} 2 .. .
≡
n
1 n+1 2 ∂ =0 ,
(A.8)
.. .
n
gives 1 2
.. .
n ∂
= 0.
(A.9)
n−1 ∂
The Poincar´e lemma on the second column leads to 1 2
.. .
n
⊗
∂
n−1 ∂
1 2
.. .
n−1 n
1 2
.. .
n−1 n ∂
⊕
1 2
.. .
∂
.
(A.10)
n−1 n
The first totally antisymmetric component vanishes since there is no component with the same symmetry on the left-hand-side, implying that 1 2
.. .
n−1 n
=
1 2
.. .
n−1 ∂
(A.11)
60
X. Bekaert, N. Boulanger
which in turn, substituted into (A.10), gives n
1 2
1 2
=
.. .
∂
n−1 ∂
(A.12)
.
.. .
n−1 ∂
Substituting this result in (A.7) proves (A.2). Second cohomology group. We now turn to the proof that (A.3) implies (A.4). The condition (A.3) reads 1 2
n+1 ∂
= 0
.. .
(A.13)
n ∂
whose type was already encountered in (A.9) above. We use our previous result (A.12) and write n+1
1 2
1 2
=
.. .
∂
(A.14)
.. .
n ∂
n ∂
or 1 2
n+1
1 2
−
.. .
∂
= 0.
.. .
n ∂
(A.15)
n ∂
This kind of equation was also found before, in (A.1), when i = 1. Then we are able to write 1 2
n+1
−
.. . n
1 2
∂
=
.. . n
1 2
n
(A.16)
.. . ∂
which is the analogue of (A.7). Equivalently, 1 2
.. . n
which is the desired result.
n+1
=
1 2
.. . n
∂
+
1 2
.. . ∂
n+1
(A.17)
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
61
(∗,∗)
A.2. Generalized cohomology in (2) (RD ). Here we proceed by induction on the number of boxes in the last (second) column. We will temporarily leave the diagrammatic exposition. For an easier understanding of the following propositions, we sketch (n,l) a pictorial translation of the proof that H(1) (2 (RD )) 0 , 0 < l < n < D in Subsect. A.3. Induction Hypothesis S . Suppose that the three following statements hold : • d {1} µ(l1 , l2 ) = 0 ⇒ µ(l1 , l2 ) = d {1} ν(l1 − 1, l2 ) , (l ,l ) • H 1 2 ((2) (RD )) ∼ = 0, •
(1) (l ,l ) H(2)1 2 ((2) (RD ))
(A.18) (A.19)
∼ = 0,
(A.20)
where 0 < l1 < D, 0 < l2 < ≤ l1 and where the notation µ(l1 , l2 ) indicates (l −1,l ) (l ,l ) that µ ∈ (2)1 2 (RD ), similarly ν ∈ (2)1 2 (RD ). The integer is fixed and is our induction parameter. The induction hypothesis S is that one knows the cohomology of d {1} and the generalized cohomology for all tensors whose second column has length strictly smaller than . What we showed in the above Sect. A.1 constitutes the “initial conditions S2 ” of our induction proof. The Poincar´e lemma actually constitutes S1 . To prove S ⇒ S+1 amounts to show that we have the three assertions (A.18), (A.19) and (A.20) with the new conditions 0 < l1 < D, 0 < l2 ≤ ≤ l1 , i.e. where the second column is now allowed to have length l2 = . 13 These three assertions (with the new conditions on the lengths of the columns) are (l ,l ) proved in the following and lead to the result that H(∗)1 2 ((2) (RD )) 0 for any (l1 , l2 ) ∈ Y(2) . Before starting these three proofs and for later purposes, we introduce a total order relation in the space (S) (RD ), naturally induced by the total order relation (3.7) for 1 Y(S) .If α(l1 , . . . , lS ) and β(l1" , . . . , lS" ) belong to (S) respectively, then
(l ,... ,lS )
(l " ,... ,lS" )
1 (RD ) and (S)
(RD ),
α(l1 , . . . , lS ) β(l1" , . . . , lS" )
(A.21)
lk = lk" , 1 ≤ k ≤ K , and " , lK+1 ≤ lK+1
(A.22)
if and only if
where K is an integer satisfying 1 ≤ K ≤ S. This lexicographic ordering induces a grading in (S) (RD ), that we call “L-grading”. This L-grading is a generalization to (S) (RD ) of the form-grading in (1) (RD ). In the sequel we will use hatted symbols to denote multiforms of [S] (RD ), while the unhatted tensors belong to (S) (RD ). Turning back to our inductive proof, we begin with the Lemma 2. If S is satisfied, then d {1} µ(l1 , ) = 0 , 0 < l1 < D ⇒ µ(l1 , ) = d {1} ν(l1 − 1, ).
(A.23)
13 The case where the fixed induction parameter satisfies = l is a little bit particular, so will have to 1 be treated separately.
62
X. Bekaert, N. Boulanger
Proof. 1. < l1 . Applying the Poincar´e lemma to the cocycle condition in Eq. (A.23), viewing the second column as a spectator, yields µ(l1 , ) d1 νˆ (l1 − 1, ), where 1 −1, νˆ (l1 − 1, ) ∈ l[2] . Decomposing the right-hand-side (expressed in terms of multiforms) into irrep. of GL(D, R) gives µ(l1 , ) d {1} ν(l1 − 1, ) + d {2} ν(l1 , − 1) + d {1} ν(l1 , − 1) +d {2} ν(l1 + 1, − 2) + (. . . ) ,
(A.24)
where (. . . ) denotes tensors of higher L-grading. Because the third and fourth terms (l ,) do not belong to (2)1 , they must cancel: d {1} ν(l1 , − 1) + d {2} ν(l1 + 1, − 2) = 0 .
(A.25)
Applying the operator d {2} to (A.25) gives d {1,2} ν(l1 , − 1) = 0. The induction hypothesis S allows us to write ν(l1 , − 1) = d {1} ν(l1 − 1, − 1) + d {2} ν(l1 , − 2). Substituting back in the decomposition of µ(l1 , ), we find, after the redefinition ν " (l1 − 1, ) = ν(l1 − 1, ) + d {2} ν(l1 − 1, − 1), the result we were looking for: µ(l1 , ) = d {1} ν " (l1 − 1, ) .
(A.26)
2. The case = l1 can be analyzed in the same way. The equation d {1} µ(l1 , l1 ) = 0 , 0 < l1 < D
(A.27)
gives, after applying the standard Poincar´e lemma, that µ(l1 , l1 ) d1 νˆ (l1 − 1, l1 ), 1 −1,l1 1 ,l1 −1 νˆ (l1 −1, l1 ) ∈ l[2] l[2] . In terms of irrep. of GL(D, R), we get µ(l1 , l1 ) d {2} ν(l1 , l1 − 1) + d {1} ν(l1 , l1 − 1) + d {2} ν(l1 + 1, l1 − 2) + (. . . ), where (. . . ) denote terms of higher L-grading. The sum of the second and third terms must vanish, and applying d {2} gives d {1,2} ν(l1 , l1 − 1) = 0. By virtue of our hypothesis of induction S , we obtain ν(l1 , l1 − 1) = d {1} ν(l1 − 1, l1 − 1) + d {2} ν(l1 , l1 − 2). Here, the result which emerges after substituting the above equation in the decomposition of µ(l1 , l1 ) and performing a field redefinition, is µ(l1 , l1 ) = d {1,2} ν(l1 − 1, l1 − 1) .
(A.28)
Note that the case = l1 gave us for free (,) Proposition 6. If S is satisfied, then H(1) ((2) (RD )) ∼ = 0.
Having Lemma 2 at our disposal, we now proceed to prove Proposition 7. If S is satisfied, then H(1)1 ((2) (RD )) ∼ = 0. (l ,)
It states that the cocycle conditions d {i} µ(l1 , ) = 0 , i ∈ {1, 2}, 0 < l1 < D
(A.29)
µ(l1 , ) = d {1,2} ν(l1 − 1, − 1) .
(A.30)
imply that
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
63
Proof. (1) < l1 . In the case i = 1, the conditions (A.29) give, using Lemma 2, that µ(l1 , ) = d {1} ν(l1 − 1, ).
(A.31)
Substituting this into the condition (A.29) for i = 2 yields d {1,2} ν(l1 − 1, ) = 0.
(A.32)
Using the Poincar´e lemma on the second column, we have d {1} ν(l1 − 1, ) d1 νˆ (l1 − 1, ) d {2} ν(l1 , − 1) + d {1} ν(l1 , − 1) +d {2} ν(l1 + 1, − 2) + (. . . ) ,
(A.33)
where, as before, we used the branching rules for GL(D, R) and (. . . ) corresponds to terms of higher L-grading. The sum of the second and third terms of the right-hand side must vanish, as does the action of d {2} on it. As a consequence, ν(l1 , − 1) = d {1} ν(l1 −1, −1)+d {2} ν(l1 , −2), hence d {1} ν(l1 −1, ) = d {1,2} ν(l1 −1, −1). Substituting this into (A.31) we finally have µ(l1 , ) = d {1,2} ν(l1 − 1, − 1), which proves the proposition. (2) = l1 . This case was already obtained in Proposition 6.
(A.34)
The following vanishing of cohomology still remains to be shown: (l ,) Proposition 8. If S is satisfied, then H(2)1 ((2) (RD )) ∼ = 0.
Proof. The cocycle condition with < l1 has in fact already been encountered in (A.32). We can then use results already obtained in proof of the Proposition 7 to write that the cocycle condition d {1,2} µ(l1 , ) = 0 ,
0 < l1 < D, = l1
(A.35)
leads to d {1} µ(l1 , ) = d {1,2} µ(l1 , − 1). Rewriting this equation as d {1} [µ(l1 , ) − d {2} µ(l1 , −1)] = 0 and using the results of Lemma 2 we obtain µ(l1 , )−d {2} µ(l1 , − 1) = d {1} µ(l1 − 1, ), i.e. µ(l1 , ) = d {2} µ(l1 , − 1) + d {1} µ(l1 − 1, ).
(A.36)
Had we started with the cocycle condition d {1,2} µ(l1 , = l1 ) = 0, 0 < l1 < D, we would have found d {1} µ(l1 , l1 ) = d {1,2} µ(l1 , l1 − 1), then µ(l1 , l1 ) − d {2} µ(l1 , l1 − 1) = d {1,2} µ(l1 − 1, l1 − 1), and after a field redefinition, the result µ(l1 , l1 ) = d {2} µ(l1 , l1 − 1),
(A.37)
which is the coboundary condition analogous to (A.36) in the case of maximally filled tensors in 2 (RD ). Conclusions. Our inductive proof provided us with the following results about the generalized cohomologies of d {i} , i ∈ {1, 2}, in the space (2) : (l ,l2 )
H(∗)1
((2) (RD )) ∼ = 0 , ∀(l1 , l2 ) ∈ Y(2) , l2 = 0.
(A.38)
64
X. Bekaert, N. Boulanger
A.3. Diagrammatical presentation. The pictorial translation of Lemma 2, which proved (n,l) to be crucial in proving H(1) ((2) (RD )) ∼ = 0 , reads 1 1
1 1
.. ... .
=0
1 1
.. ... .
⇒
l
.. ... .
=
l
(A.39)
.
l
.. .
.. .
.. .
n ∂
n
∂
For practical purposes and for simplicity, we fix l = 2. Using the standard Poincar´e lemma, one obtains 1 ∗ 2 ∗
.. .
1 ∗ =0 ⇒ 2 ∗
∂
.. .
n ∂
⊗
1 2
∗ ⊗ ∗
.. .
n
(A.40)
n−1
i.e. 1 2
.. . n−1 n
∗ ∗
1 2
.. .
∗ ∗ ⊕
n−1 ∂
.. .
∗
1 2
∗ ∂ ⊕
1 2
.. .
1 2
⊕ ...
n−1 ∗ ∂
n−1 ∗
n−1 ∗ ∗
∂
1 2
⊕ ...
.
(A.41)
n−1 ∗ ∗ ∂
The condition that the sum of the third and fourth terms of the right-hand side vanishes, implies, due to the induction hypothesis, that the tensor in the second term can be written as 1 2
∗
=
.. . n−1 ∗
1 2
∗
+
.. . n−1 ∂
1 2
.. .
∂
(A.42)
n−1 ∗
which, substituted into (A.42), gives 1 2
.. . n−1 n
∗ ∗
=
1 2
.. . n−1 ∂
∗ ∗ .
(A.43)
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
65
Once Lemma 2 is obtained, we turn to the pictorial description of the proof for (n,l) H(1) ((2) (RD )) ∼ = 0 . The first cocycle condition d {1} α(n, l) = 0
(A.44)
is represented by 1 1
.. ... . l
= 0.
.. .
(A.45)
n ∂
Using Lemma 2, its solution is (as we showed pictorially in the case where l = 2), 1 1
1 1
.. ... .
.. ... .
=
l
.. .
.. .
n
∂
(A.46)
.
l
Substituting in the second cocycle condition d {2} α(n, l) = 0
(A.47)
yields 1
1
.. .
.. .
= 0.
l ∂
.. .
(A.48)
n−1 ∂
Applying the Poincar´e lemma on the second column, viewing the first one as a spectator, gives 1
.. .
1
.. . l
∂
⊗
.. .
1
1
.. .
⊗ .. . l−1
n
n−1 ∂
1
.. . .. . n
1
.. . l−1 ∂
1
. ⊕ ..
1
.. . l−1
1
⊕
.. .
.. .
.. .
n ∂
n n+1
1
..
l−2 ⊕ . . . , ∂
(A.49)
66
X. Bekaert, N. Boulanger
where the dots in the above equation correspond to tensors of higher order L-grading (whose first column has a length greater or equal to n + 2). The second and third terms must cancel because they do not have the symmetry of the left-hand side. Applying d {2} on the sum of the second and third term and using our hypothesis of induction, we obtain 1
.. .
1
1
.. .
l−1 l
n−1 ∂
.. .
1
.. .
l−1 ∂
(A.50)
.
n−1 ∂ (n,l)
This, substituted back into (A.46), gives us the vanishing of H(1) ((2) (RD )) for n = D, l = D and l = 0. (∗,... ,∗)
(RD ). Here we present the final result A.4. Generalized Poincar´e lemma in (∗) concerning our generalized Poincar´e lemma: (∗,... ,∗)
H(∗)
((∗) (RD )) ∼ = 0,
(A.51)
for diagrams obeying the assumption of the Theorem (cfr. Sect. 5). Equation (A.51) is really proved if one has the following inductive progression: Under the assumption that (l ,... ,lS−1 ,lS )
H(∗)1
((S) (RD )) ∼ =0
(A.52)
∀(l1 , . . . , lS−1 ) ∈ Y(S−1) and lS fixed such that 0 < lS < lS−1 , the following holds: (l ,... ,lS−1 ,lS +1)
H(k)1
((S) (RD )) 0,
(A.53)
((S+1) (RD )) 0 ,
(A.54)
and (l ,... ,lS−1 ,lS ,1)
H(l)1
0 < l < S + 2. More explicitly, if the following statements are satisfied: d I µ(l1 , . . . , lS−1 , lS ) = 0 ∀I ⊂ {1, 2, . . . , S} | # I = m ⇒ µ(l1 , . . . , lS−1 , lS ) = J d J νJ (l ,... ,lS−1 ,lS )
1 ∀J ⊂ {1, 2, . . . , S} | # J = S +1−m and d J νJ ∈ (S)
(RD ) ,
it can be shown that these statements are also true: • d I µ(l1 , . . . , lS−1 , lS + 1) = 0 ∀I ⊂ {1, 2, . . . , S} | # I = m ⇒ µ(l1 , . . . , lS−1 , lS + 1) = J d J νJ (l ,... ,lS−1 ,lS +1)
1 ∀J ⊂ {1, 2, . . . , S} | # J = S +1−m and d J νJ ∈ (S)
(RD )
Tensor Gauge Fields in Arbitrary Representations of GL(D, R)
67
and • d I µ(l1 , . . . , lS−1 , lS , 1) = 0 ∀I ⊂ {1, 2, . . . , S, S + 1} | # I = m ⇒ µ(l1 , . . . , lS−1 , lS , 1) = J d J νJ 1 ∀J ⊂ {1, . . . , S, S + 1} | # J = S +2−m and d {J } νJ ∈ (S)
(l ,... ,lS ,1)
(RD ) .
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Commun. Math. Phys. 245, 69–103 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1003-5
Communications in
Mathematical Physics
On the Singularity of the Free Energy at a First Order Phase Transition Sacha Friedli1 , Charles-E. Pfister2 1
Institut de Th´eorie des Ph´enom`enes Physiques, EPF-L, 1015 Lausanne, Switzerland. E-mail:
[email protected] 2 Institut d’Analyse et de Calcul Scientifique, EPF-L, 1015 Lausanne, Switzerland. E-mail:
[email protected] Received: 30 November 2002 / Accepted: 23 June 2003 Published online: 12 December 2003 – © Springer-Verlag 2003
Abstract: At first order phase transition the free energy does not have an analytic continuation in the thermodynamical variable, which is conjugate to an order parameter for the transition. This result is proved at low temperature for lattice models with finite range interaction and two periodic ground-states, under the only condition that they satisfy the Peierls condition.
1. Introduction We study a lattice model with finite state space on Zd , d ≥ 2. The Hamiltonian Hµ = H0 + µH1 is the sum of two Hamiltonians, which have finite-range and periodic interactions. We assume that H0 has two periodic ground-states ψ1 and ψ2 , and so that the Peierls condition is satisfied, and that H1 splits the degeneracy of the ground-states of H0 : if µ < 0, then Hµ has a unique ground-state ψ2 , and if µ > 0, then Hµ has a unique ground-state ψ1 . The free energy of the model, at inverse temperature β, is denoted by f (µ, β). Our main result is Theorem 1.1. Under the above setting, there exist an open interval U0 0, β ∗ ∈ R+ and, for all β ≥ β ∗ , µ∗ (β) ∈ U0 with the following properties: 1. There is a first-order phase transition at µ∗ (β). 2. The free energy f (µ, β) is real-analytic in µ in {µ ∈ U0 : µ < µ∗ (β)}; it has a C ∞ continuation in {µ ∈ U0 : µ ≤ µ∗ (β)}. 3. The free energy f (µ, β) is real-analytic in µ in {µ ∈ U0 : µ > µ∗ (β)}; it has a C ∞ continuation in {µ ∈ U0 : µ ≥ µ∗ (β)}. 4. There is no analytic continuation of f along a real path from µ < µ∗ (β) to µ > µ∗ (β) crossing µ∗ (β), or vice-versa.
Supported by Fonds National Suisse de la Recherche Scientifique.
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This theorem answers a fundamental theoretical question: does the free energy, which is analytic in the region of a single phase, have an analytic continuation beyond a firstorder phase transition point? The answer is yes for the theory of a simple fluid of van der Waals or for mean-field theories. The analytic continuation of the free energy beyond the transition point was interpreted as the free energy of a metastable phase. The answer is no for models with finite range interaction, under very general conditions, as Theorem 1.1 shows. This contrasted behavior has its origin in the fact that for models with finite range interaction there is spatial phase separation at first order phase transition, contrary to what happens in a mean-field model. Theorem 1.1 and its proof confirm the prediction of the droplet model [1]. Theorem 1.1 generalizes the works of Isakov [2] for the Ising model and [3], where a similar theorem is proven under additional assumptions, which are not easy to verify in a concrete model. Our version of Theorem 1.1, which relies uniquely on Peierls condition, is therefore a genuine improvement of [3]. The first result of this kind was proven by Kunz and Souillard [4]; it concerns the non-analytic behavior of the generating function of the cluster size distribution in percolation, which plays the role of a free energy in that model. The first statement of Theorem 1.1 is a particular case of the theory of Pirogov and Sinai (see [6]). We give a proof of this result, as far as it concerns the free energy, since we need detailed information about the phase diagram in the complex plane of the parameter µ. The obstruction to an analytic continuation of the free energy in the variable µ is due to the stability of the droplets of both phases in a neighborhood of µ∗ . Our proof follows essentially that of Isakov in [2]. We give a detailed proof of Theorem 1.1, and do not assume any familiarity with [2] or [3]. On the other hand we assume that the reader is familiar with the cluster expansion technique. The results presented here are true for a much larger class of systems, but for the sake of simplicity we restrict our discussion in that paper to the above setting, which is already quite general. For example, Theorem 1.1 is true for the Potts model with a high number q of components at the first order phase transition point βc , where the q ordered phases coexist with the disordered phase. Here µ = β, the inverse temperature, and the statement is that the free energy, which is analytic for β > βc , or for β < βc , does not have an analytic continuation across βc . Theorem 1.1 is also true when the model has more than two ground-states. For example, for the Blume-Capel model, whose Hamiltonian is (si − sj )2 − h si − λ si2 with si ∈ {−1, 0, 1} , i,j
i
i
the free energy is an analytic function of h and λ in the single phase regions. At low temperature, at the triple point occurring at h = 0 and λ = λ∗ (β) there is no analytic continuation of the free energy in λ, along the path h = 0, or in the variable h, along the path λ = λ∗ . The case of coexistence of more than two phases will be treated in a separate paper. In the rest of the section we fix the main notations following chapter two of Sinai’s book [6], so that the reader may easily find more information if necessary. We also state Lemma 1.1 which contains all estimates on partition functions or free energies. The model is defined on the lattice Zd , d ≥ 2. The spin variables ϕ(x), x ∈ Zd , take values in a finite state space. If ϕ, ψ are two spin configurations, then ϕ = ψ (a.s.) means that ϕ(x) = ψ(x) holds only on a finite subset of Zd . The restriction of ϕ to a subset A ⊂ Zd is denoted by ϕ(A). The cardinality of a subset S is denoted by |S|. If x, y ∈ Zd , then |x−y| := maxdi=1 |xi −yi |; if W ⊂ Zd and x ∈ Zd , then d(x, W ) := miny∈W |x−y|
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and if W, W are subsets of Zd , then d(W, W ) = minx∈W d(x, W ). We define for W ⊂ Zd , ∂W := {x ∈ W : d(x, Zd \W ) = 1} . A subset W ⊂ Zd is connected if any two points x, y ∈ W are connected by a path {x0 , x1 , . . . , xn } ⊂ W , with x0 = x, xn = y and |xi − xi+1 | = 1, i = 0, 1, . . . , n − 1. A component is a maximally connected subset. Let H be a Hamiltonian with finite-range and periodic bounded interaction. By introducing an equivalent model on a sublattice, with a larger state space, we can assume that the model is translation invariant with interaction between neighboring spins ϕ(x) and ϕ(y), |x − y| = 1, only. Therefore, without restricting the generality, we assume that this is the case and that the interaction is Zd -invariant. The Hamiltonian is written Hµ = H0 + µH1 ,
µ ∈ R.
H0 has two Zd -invariant ground-states ψ1 and ψ2 , and the perturbation H1 splits the degeneracy of the ground-states of H0 . We assume that the energy (per unit spin) of the µ ground-states of H0 is 0. Ux (ϕ) ≡ U0,x + µ U1,x is the interaction energy of the spin located at x for the configuration ϕ, so that by definition Hµ (ϕ) =
Uxµ (ϕ) (formal sum) .
x∈Zd
U1,x is an order parameter for the phase transition. If ϕ and ψ are two configurations and ϕ = ψ (a.s.), then Hµ (ϕ|ψ) :=
Uxµ (ϕ) − Uxµ (ψ) .
x∈Zd
This last sum is finite since only finitely many terms are non-zero. The main condition, which we impose on H0 , is Peierls condition for the ground-states ψ1 and ψ2 . Let x ∈ Zd and W1 (x) := {y ∈ Zd : |y − x| ≤ 1} . The boundary ∂ϕ of the configuration ϕ is the subset of Zd defined by ∂ϕ :=
W1 (x) : ϕ(W1 (x)) = ψm (W1 (x)) , m = 1, 2 .
x∈Zd
Peierls condition means that there exists a positive constant ρ such that for m = 1, 2, H0 (ϕ|ψm ) ≥ ρ|∂ϕ| ∀ ϕ such that ϕ = ψm (a.s.) . We shall not usually write the µ-dependence of some quantity; we write for example H µ or Ux instead of Hµ or Ux . Definition 1.1. Let M denote a finite connected subset of Zd , and let ϕ be a configuration. Then a couple = (M, ϕ(M)) is called a contour of ϕ if M is a component of the boundary ∂ϕ of ϕ. A couple = (M, ϕ(M)) of this type is called a contour if there exists at least one configuration ϕ such that is a contour of ϕ.
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If = (M, ϕ(M)) is a contour, then M is the support of , which we also denote by supp . Suppose that = (M, ϕ(M)) is a contour and consider the components Aα of Zd \M. Then for each component Aα there exists a unique ground-state ψq(α) , such that for each x ∈ ∂Aα one has ϕ(W1 (x)) = ψq(α) (W1 (x)). The index q(α) is the label of the component Aα . For any contour there exists a unique infinite component of Zd \supp , Ext , called the exterior of ; all other components are called internal components of . The ground-state corresponding to the label of Ext is the boundary condition of ; the superscript q in q indicates that is a contour with boundary condition ψq . Int m is the union of all internal components of with label m; Int := m=1,2 Intm is the interior of . We use the abbreviations || := |supp | and Vm () := |Intm |. We define1 V ( q ) := Vm ( q ) m = q . For x ∈ Zd , let
(1.1)
d c(x) := y ∈ Rd : max |xi − yi | ≤ 1/2 i=1
be the unit cube of center x in
Rd .
If ⊂ Zd , then | | is equal to the d-volume of c(x) ⊂ Rd . (1.2) x∈
The (d − 1)-volume of the boundary of the set (1.2) is denoted by ∂| |. We have 2d | |
d−1 d
≤ ∂| | .
(1.3)
The equality in (1.3) is true for cubes only. When = Int m q , m = q, V ( q ) ≡ | | and ∂V ( q ) ≡ ∂| |; there exists a positive constant C0 such that ∂V ( q ) ≤ C0 | q | q = 1, 2 .
(1.4)
For each contour = (M, ϕ(M)) there corresponds a unique configuration ϕ with the properties: ϕ = ψq on Ext , where q is the label of Ext , ϕ (M) = ϕ(M), ϕ = ψm on Int m , m = 1, 2. is the only contour of ϕ . Let ⊂ Zd ; the notation ⊂ means that supp ⊂ , Int ⊂ and d(supp , c ) > 1. A contour of a configuration ϕ is an external contour of ϕ if and only if supp ⊂ Ext for any contour of ϕ. Definition 1.2. Let ( q ) be the set of configurations ϕ = ψq (a.s.) such that q is the only external contour of ϕ. Then exp − βH(ϕ|ψq ) . ( q ) := ϕ∈ ( q )
Let ⊂ Zd be a finite subset; let q ( ) be the set of configurations ϕ = ψq (a.s.) such that ⊂ whenever is a contour of ϕ. Then exp − βH(ϕ|ψq ) . q ( ) := ϕ∈ q ( ) 1
Here our convention differs from [6].
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Two fundamental identities relate the partition functions ( q ) and q ( ), q ( ) =
n
q
(i ) ,
i=1 q where the sum is over the set of all families {1 , . . .
(1.5)
q
, n } of external contours in , and
2
m (Int m q ) . ( q ) = exp − βH(ϕ q |ψq )
(1.6)
m=1
We define (limit in the sense of van Hove) gq := lim −
↑Zd
1 log q ( ) . β| |
The energy (per unit volume) of ψm for the Hamiltonian H1 is h(ψm ) := U1,x (ψm ) . By definition of H1 , h(ψ2 ) − h(ψ1 ) = 0, and we assume that
:= h(ψ2 ) − h(ψ1 ) > 0 . The free energy in the thermodynamical limit is 1 1 f = lim − log q ( ) + lim Ux (ψq ) = gq + µ h(ψq ) . β| |
↑Zd
↑Zd | |
(1.7)
x∈
It is independent of the boundary condition ψq . Definition 1.3. Let q be a contour with boundary condition ψq . The weight ω( q ) of q is m (Int m q ) ω( q ) := exp − βH(ϕ q |ψq ) (m = q) . q (Int m q ) The (bare) surface energy of a contour q is
q := H0 (ϕ q |ψq ) . For a contour q we set a(ϕ q ) :=
U1,x (ϕ q ) − U1,x (ψq ) .
x∈supp q
Since the interaction is bounded, there exists a constant C1 so that |a(ϕ q )| ≤ C1 | q | .
(1.8)
Using these notations we have H(ϕ q |ψq ) = Ux (ϕ q ) − Ux (ψq ) + Ux (ϕ q ) − Ux (ψq ) x∈supp q
x∈Int q
= H0 (ϕ q |ψq ) + µa(ϕ q ) + µ(h(ψm ) − h(ψq ))V ( q ) = q + µa(ϕ q ) + µ(h(ψm ) − h(ψq ))V ( q ) (m = q) .
(1.9)
The surface energy q is always strictly positive since Peierls condition holds, and there exists a constant C2 , independent of q = 1, 2, such that ρ| q | ≤ q ≤ C2 | q | .
(1.10)
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Definition 1.4. The weight ω( q ) is τ -stable for q if |ω( q )| ≤ exp(−τ | q |) . For a finite subset ⊂ Zd , using (1.5) and (1.6), one obtains easily the following identity for the partition function q ( ), q ( ) = 1 +
n
q
(1.11)
ω(i ) ,
i=1 q
q
where the sum is over all families of compatible contours {1 , . . . , n } with boundary q q q condition ψq , that is, i ⊂ and d(supp i , supp j ) > 1 for all i = j , i, j = 1, . . . , n, n ≥ 1. We also introduce restricted partition functions and free energies. For each n = 0, 1, . . . , we define new weights ωn ( q ), ω( q ) if V ( q ) ≤ n, q (1.12) ωn ( ) := 0 otherwise. For q = 1, 2, we define nq by Eq. (1.11), using ωn ( q ) instead of ω( q ). It is essential later on to replace the real parameter µ by a complex parameter z; we set (provided that nq ( )(z) = 0 for all ) gqn (z) := − lim
↑Zd
1 log nq ( )(z) and β| |
fqn (z) := gqn (z) + z h(ψq ) .
(1.13)
fqn is the restricted free energy of order n and boundary condition ψq . Let d−1 l(n) := C0−1 2dn d n ≥ 1.
(1.14)
Notice that nq ( ) = q ( ) if | | ≤ n, and that V ( q ) ≥ n implies that | q | ≥ l(n) since (1.3) and (1.4) hold. Lemma 1.1 gives basic, but essential, estimates for the rest of the paper. The only hypothesis for this lemma is that the weights of the contours are τ -stable. Lemma 1.1. Let ω( q ) be any complex weights, and define ωn ( q ) by (1.12). Suppose that the weights ωn ( q ) are τ -stable for all q . Then there exists K0 < ∞ and τ0∗ < ∞ independent of n, so that for all τ ≥ τ0∗ , β|gqn | ≤ K0 e−τ .
(1.15)
For all finite subsets ⊂ Zd , log n ( ) + βg n | | ≤ K0 e−τ ∂| | . q q
(1.16)
If ωn ( q ) = 0 for all q such that | q | ≤ k, then k β|gqn | ≤ K0 e−τ .
(1.17)
For n ≥ k and k ≥ 1, l(k) β|gqn − gqk−1 | ≤ K0 e−τ .
(1.18)
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Furthermore, if ωn ( q ) depends on a parameter t and d 2 ωn ( q ) ≤ D1 e−τ | q | and d ωn ( q ) ≤ D2 e−τ | q | , dt dt 2
(1.19)
then there exists Kj < ∞ and τj∗ < ∞ independent of n, j = 1, 2, so that for all dj τ ≥ τj∗ , j gqn exists and dt d β gqn ≤ D1 K1 e−τ dt
d2 and β 2 gqn ≤ max{D2 , D12 }K2 e−τ . dt
(1.20)
For all finite subsets ⊂ Zd , d log n ( ) + β d g n | | ≤ D1 K1 e−τ ∂| | q q dt dt
(1.21)
d2 d2 n n log ( ) + β g | | ≤ max{D2 , D12 }K2 e−τ ∂| | . q dt 2 dt 2 q
(1.22)
and
If the weights ωn ( q ) are τ -stable for all q and all n ≥ 1, then all these estimates hold for gq and q instead of gqn and nq . Moreover, gqn and its first two derivatives converge to gq and its first two derivatives. Proof. Let ω( q ) be an arbitrary weight, satisfying the only condition that it is τ -stable for any q . The partition function q ( ) is defined in (1.11) by q ( ) = 1 +
n
q
ω(i ) ,
i=1 q
q
where the sum is over all families of compatible contours {1 , . . . , n } with boundary q q q condition ψq , that is, i ⊂ and d(supp i , supp j ) > 1 for all i = j , i, j = 1, . . . , n, n ≥ 1. We set, following reference [5] Sect. 3 2 , q := {x ∈ Zd : d(x, supp 2 ) ≤ 1} .
(1.23)
There exists a constant C5 such that | q | ≤ C5 | q |, and q
q
q
q
(i and j not compatible) ⇒ (supp i ∩ j = ∅) . We introduce q q ϕ2 (i , j ) 2
:=
q
q
0 if i and j compatible q q −1 if i and j not compatible .
In [5] q is denoted by i( q ), which has another meaning here (see Subsect. 2.3).
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If the weights of all contours with boundary condition ψq are τ -stable and if τ is large enough, then one can express the logarithm of q ( ) as log q ( ) =
m 1
q q q T ··· ϕm (1 , . . . , m ) ω(i ) m! q q 1 ⊂
m≥1
1 = m!
x∈
m≥1
q
m ⊂
i=1
···
q
T ( q , . . . ϕm 1
q
|supp 1 |
q
1 ⊂
q x∈supp 1
m q , m )
m ⊂
q
ω(i ) .
(1.24)
i=1
q
T ( , . . . , ) is a purely combinatorial factor (see [5], Formulas (3.20) and In (1.24) ϕm m 1 (3.42)). This is the basic identity which is used for controlling q ( ). An important T ( q , . . . , q ) is that ϕ T ( q , . . . , q ) = 0 if the following graph is not property of ϕm m m m 1 1 q connected (Lemma 3.3 in [5]): to each i we associate a vertex vi , and to each pair q q {vi , vj } we associate an edge if and only if ϕ2 (i , j ) = 0.
Lemma 1.2. Assume that
C :=
|ω( q )| exp(| q |) < ∞ .
q :supp q 0
Then
q 1 :
q 2
q
···
q
q
T |ϕm (1 , . . . , m )|
q m
m
q
|ω(i )| ≤ (m − 1)!C m .
i=1
0∈supp 1
If, furthermore C < 1, then (1.24) is true, and the right-hand side of (1.24) is an absolutely convergent sum. Lemma 1.2 is Lemma 3.5 in [5], where a proof is given. There exists a constant, KP , called the Peierls constant, such that |{ q : supp q 0 and |supp q | = n}| ≤ KPn . ∗
If ω( q ) is τ -stable, then there exist Kˆ 0 < ∞ and τ0∗ < ∞ so that Kˆ 0 e−τ0 < 1, and for all τ ≥ τ0∗ , C=
|ω( q )| exp(| q |) ≤
q :supp q 0
KP e−(τ −C5 )j ≤ Kˆ 0 e−τ . j
j ≥1
If this is true, (1.24) implies3 that −β gq =
1 m!
m≥1
3
q
1 q 0∈supp 1
···
q
m
m
1 q q q T ( , . . . , ) ω(i ) . ϕ m q m 1 |supp 1 | i=1
The corresponding formula (3.58) in [5] is incorrect; a factor |γ1 ∩ Z2∗ |−1 is missing.
(1.25)
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Therefore, there exists K0 < ∞ so that for all τ ≥ τ0∗ , β|gq | ≤
Kˆ 0 C ≤ e−τ ≡ K0 e−τ . 1−C 1 − Kˆ 0
We have for all finite subsets ⊂ Zd , 1 log q ( ) + βgq | | ≤ m! x∈∂ m≥1
q
q
T |ϕm (1 , . . . , m )|
q q 1 ,... ,m q ∃i i x
m
q
|ω(i )|
i=1
≤ K0 e−τ ∂| | . If ω( q ) = 0 for all q such that | q | ≤ m, then C ≤ Kˆ 0m e−τ m , and m β|gq | ≤ K0 e−τ . If n ≥ k and k ≥ 1, then β|gqn − gqk−1 | ≤
1 j! j ≥1
q
q
q
|ϕjT (1 , . . . , j )|
q
q
1 0,2 ,... ,j
j
q
|ωn (i )|
i=1
q ∃i V (i )≥k
≤
j 1 j! j ≥1
i=1 q 0, q ,... , q 1 2 j
q
q
|ϕkT (1 , . . . , j )|
j
q
|ωn (i )|
i=1
q
l(k) ≤ K0 e−τ .
V (i ) ≥ k
The last inequality is proved by a straightforward generalization of the proof of Lemma 3.5 in [5]. The last statements of Lemma 1.1 are proven in the same way, by deriving (1.24) term by term. 2. Proof of Theorem 1.1 The proof of Theorem 1.1 is given in the next five subsections. In Subsect. 2.1 we construct the phase diagram and in Subsect. 2.2 we study the analytic continuation of the weights of contours in a neighborhood of the point of phase coexistence µ∗ . These results about the analytic continuation are crucial for the rest of the analysis and cannot be found in the literature. We need stronger results than those of Isakov [3] in order to prove Theorem 1.1 under the only assumption that Peierls condition is true. For the construction of the phase diagram in the complex plane we follow Isakov [3] and Zahradnik [7]. In Subsect. 2.3 we derive an expression of the derivatives of the free energy at finite volume. We prove a lower bound for a restricted class of terms of this expression. This is an improved version of a similar analysis of Isakov [2]. From these results we obtain a lower bound for the derivatives of the free energy f in a finite box . We show in Subsect. 2.4 that for large β, there exists an increasing diverging sequence {kn }, so that d the knth -derivative of f with respect to µ, evaluated at µ∗ , behaves like kn ! d−1 (provided that is large enough). In the last subsection we end the proof of the impossibility of an analytic continuation of the free energy across µ∗ , by showing that the results of Subsect. 2.4 remain true in the thermodynamical limit.
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2.1. Construction of the phase diagram in the complex plane. We construct the phase diagram for complex values of the parameter µ, by constructing iteratively the phase diagram for the restricted free energies fqn (see (1.13)). We set z := µ + iν. The method consists in finding a sequence of intervals for each ν ∈ R, Un (ν; β) := (µ∗n (ν; β) − bn1 , µ∗n (ν; β) + bn2 ) , with the properties 1 2 (µ∗n (ν; β) − bn1 , µ∗n (ν; β) + bn2 ) ⊂ (µ∗n−1 (ν; β) − bn−1 , µ∗n−1 (ν; β) + bn−1 ) (2.1) q
and limn bn = 0, q = 1, 2. By construction of the intervals Un−1 (ν; β) the restricted free energies fqn−1 of order n − 1, q = 1, 2, are well-defined and analytic on Un−1 := {z ∈ C : Rez ∈ Un−1 (Imz; β)} . The point µ∗n (ν; β), n ≥ 1, is the solution of the equation Re f2n−1 (µ∗n (ν; β) + iν) − f1n−1 (µ∗n (ν; β) + iν) = 0 . µ∗n (0; β) is the point of phase coexistence for the restricted free energies of order n − 1, and the point of phase coexistence of the model is given by µ∗ (0; β) = limn µ∗n (0; β). This iterative construction is as important as the statement of Proposition 2.1, which is the main result of Subsect. 2.1. Proposition 2.1. Let 0 < ε < ρ and 0 < δ < 1 so that − 2δ > 0. Set U0 := (−C1−1 ε, C1−1 ε) and U0 := {z ∈ C : Rez ∈ U0 } and
τ (β) := β(ρ − ε) − 3C0 δ .
There exists β0 ∈ R+ such that for all β ≥ β0 the following holds: 1. There exists a continuous real-valued function on R, ν → µ∗ (ν; β) ∈ U0 , so that µ∗ (ν; β) + iν ∈ U0 . 2. If µ + iν ∈ U0 and µ ≤ µ∗ (ν; β), then the weight ω( 2 ) is τ (β)-stable for all contours 2 with boundary condition ψ2 , and analytic in z = µ + iν if µ < µ∗ (ν; β). 3. If µ + iν ∈ U0 and µ ≥ µ∗ (ν; β), then the weight ω( 1 ) is τ (β)-stable for all contours 1 with boundary condition ψ1 , and analytic in z = µ + iν if µ > µ∗ (ν; β). It is useful to put into evidence here some points of the proof of Proposition 2.1, before giving it in detail. Remark 2.1. The iterative method depends on a free parameter θ , 0 < θ < 1, which is fixed at the end of the proof of Theorem 1.1. Let 0 < θ < 1 be given, as well as ε and δ as in the proposition. We list here all major constants which appear in the proof, since these constants are used at different places in the paper. We use the isoperimetric constant χ , which is defined as the best constant in (2.2), V ( q )
d−1 d
≤ χ −1 q ∀ q , q = 1, 2 .
Existence of χ in (2.2) follows from (1.3), (1.4) and (1.10). We set τ1 (β; θ ) := β ρ(1 − θ ) − ε − 2δC0 ;
(2.2)
(2.3)
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τ2 (β; θ ) := τ1 (β; θ ) −
d ; d −1
(2.4) d
C3 := C1 + 2δC0 + ( + 2δ)(χ −1 C2 ) d−1 .
(2.5)
We choose β0 so that, for all β ≥ β0 , τ2 (β) > max{τ0∗ , τ1∗ , τ2∗ }, (2.18) holds4 , Ke−τ1 (β) ≤ δ
and
C3 Ke−τ2 (β) ≤ δ .
(2.6)
K is a constant, which is greater than max{K0 , K1 }, and K0 , K1 are the constants of Lemma 1.1; ρ is the constant of Peierls condition and = h(ψ2 ) − h(ψ1 ) > 0. We also require for Proposition 2.3 that τ (β) − max
d , p ≥ τ2 (β; θ ) d −1
∀ β ≥ β0 .
Here p ∈ N is fixed in the proof of Proposition 2.2. Remark 2.2. In the above formulas we may choose δ in such a way that δ = δ(β) and limβ→∞ δ(β) = 0. Indeed, the only condition which we need to satisfy is (2.6). So, whenever we need it, we consider δ as a function of β, so that by taking β large enough, we have δ as small as we wish. Remark 2.3. The main technical part of the proof of Proposition 2.1 is the proof of point D below. If we want to prove only the first statement of Theorem 1.1, then it is sufficient to prove points A, B and C below. This gives a constructive definition of the point of phase coexistence µ∗ (β), as well as the main estimates necessary to construct the different phases at this point, since we get that all contours are τ1 (β)-stable at µ∗ (β). For example, existence of two phases follows from a straightforward Peierls argument. Remark 2.4. We emphasize here a key step of the iterative proof of Proposition 2.1. Assume that β ≥ β0 , and for q = 1, 2, that the weights ωn−1 ( q ) are τ1 (β; θ )-stable and d ωn−1 () ≤ βC3 e−τ2 (β;θ )|| . dz From (1.20) and (1.16), d n−1 f2 − f1n−1 − ) ≤ 2δ , dz and (m = q)
(2.7)
log n−1 (Int m q ) + βg n−1 V ( q )| ≤ δ C0 | q | q q log n−1 (Int m q ) + βg n−1 V ( q )| ≤ δ C0 | q | . m m
Let q be a contour with V ( q ) = n. Then m (Int m q ) (m = q) |ω( q )| = exp − βReH(ϕ q |ψq ) q (Int m q ) ≤ exp − β q + βε + 2C0 δ | q | − βRe fmn−1 − fqn−1 V ( q ) ,
(2.8)
4 τ ∗ , k = 0, 1, 2, are defined in Lemma 1.1. Condition τ (β) > τ ∗ is needed only in Lemma 2.1. We 2 2 k have stated Lemma 2.1 separately in order to simplify the proof of Proposition 2.1.
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since all contours inside Int m q have a volume smaller than n − 1, and (see (1.8)) |Rez a(ϕ q )| ≤ ε
∀ z ∈ U0 .
To prove the stability of ω( q ) we must control the volume term in the right-hand side of inequality (2.8). If (2.9) −Re f1n−1 − f2n−1 V ( 2 ) ≤ θ 2 , n−1 n−1 1 1 −Re f2 − f1 (2.10) V ( ) ≤ θ , then ω( 2 ) and ω( 1 ) are τ1 (β; θ )-stable. Indeed, these inequalities imply |ω( q )| ≤ exp − β(1 − θ ) q + βε + 2C0 δ | q | ≤ exp − β(1 − θ )ρ − βε − 2C0 δ | q | . Verification of the inequalities (2.9) and (2.10) is possible because (2.7) provides a sharp estimate of the derivative of f2n−1 −f1n−1 . We also use the isoperimetric inequality (2.2). Proof. Let θ , 0 < θ < 1. On the interval U0 (ν; β) := (−b0 , b0 ) with b0 = εC1−1 , q fq0 (µ + iν) is defined and we set µ∗0 (ν; β) := 0. The two decreasing sequences {bn }, q = 1, 2 and n ≥ 1, are defined by bn1 ≡ bn2 :=
χθ 1
( + 2δ)n d
, n ≥ 1.
(2.11)
Then it is immediate to verify, when β is large enough or δ small enough, that q
q
bn − bn+1 >
2δ l(n) , β( − 2δ)
∀n ≥ 1 .
(2.12)
On U0 all contours with empty interior are β(ρ − ε)-stable, and d ω() ≤ βC1 ||e−β(ρ−ε)|| ≤ βC1 e−[β(ρ−ε)−1]|| ≤ βC3 e−τ2 (β)|| . dz We prove iteratively the following statements. A. There exists a continuous solution ν → µ∗n (ν; β) of the equation Re f2n−1 (µ∗n (ν; β) + iν) − f1n−1 (µ∗n (ν; β) + iν) = 0 , so that (2.1) holds. B. ωn ( q ) is well-defined and analytic on Un , for any contour q , q = 1, 2, and ωn ( q ) is τ1 (β)-stable. Moreover, nq ( ) = 0 for any finite , and fqn (z; β) is analytic on Un . d q C. On Un , ωn ( q ) ≤ βC3 e−τ2 (β)| | . dz D. If z = µ + iν ∈ U0 and µ ≤ µ∗n (ν; β) − bn1 , then ω( 2 ) is τ (β)-stable for any 2 with boundary condition ψ2 . If z = µ + iν ∈ U0 and µ ≥ µ∗n (ν; β) + bn2 , then ω( 1 ) is τ (β)-stable for any 1 with boundary condition ψ1 .
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81
From these results the proposition follows with µ∗ (ν; β) = lim µ∗n (ν; β) . n→∞
We assume that the construction has been done for all k ≤ n − 1. A. We prove the existence of µ∗n (ν; β) ∈ Un−1 . µ∗n (ν; β) is a solution of the equation Re f2n−1 (µ∗n (ν; β) + iν) − f1n−1 (µ∗n (ν; β) + iν) = 0 . Let F k (z) := f2k (z) − f1k (z). Then, for µ + iν ∈ Un−1 , F n−1 (µ + iν) = F n−1 (µ + iν) − F n−2 (µ∗n−1 + iν) = F n−1 (µ + iν) − F n−1 (µ∗n−1 + iν) + F n−1 (µ∗n−1 + iν) − F n−2 (µ∗n−1 + iν) µ d n−1 = F (µ + iν) dµ + g2n−1 − g2n−2 (µ∗n−1 + iν) µ∗n−1 dµ − g1n−1 − g1n−2 (µ∗n−1 + iν) . (2.13) If V () = n − 1, then || ≥ l(n − 1). Therefore, (1.18) gives | gqn−1 − gqn−2 (µ∗n−1 + iν)| ≤ β −1 δ l(n−1) .
(2.14)
If z = µ + iν ∈ Un−1 , then (2.13), (2.7) and (2.14) imply
(µ − µ∗n−1 ) + 2δ|µ − µ∗n−1 | + 2β −1 δ l(n−1) ≥ ReF n−1 (z ) ≥ (µ − µ∗n−1 ) − 2δ|µ − µ∗n−1 | − 2β −1 δ l(n−1) . Since (2.12) holds, q
q
q
bn−1 > bn−1 − bn >
2δ l(n−1) , β( − 2δ)
1 2 so that ReF n−1 (µ∗n−1 − bn−1 + iν) < 0 and ReF n−1 (µ∗n−1 + bn−1 + iν) > 0. This proves the existence of µ∗n and its uniqueness, since µ → ReF n−1 (µ + iν) is strictly increasing (see (2.7)). Moreover, by putting µ = µ∗n (ν; β) in (2.13), we get
|µ∗n (ν; β) − µ∗n−1 (ν; β)| ≤
2δ l(n−1) . β( − 2δ)
Therefore Un ⊂ Un−1 . The implicit function theorem implies that ν → µ∗n (ν; β) is continuous (even C ∞ ). B. By the induction hypothesis the weights ωn ( q ) are analytic in Un−1 . We prove that on Un ωn ( q ) is τ1 -stable for all contours q , q = 1, 2. This implies that fqn is analytic on Un . The proof of the stability of the contours is the content of Remark 2.4. Let q be a contour with V ( q ) = n. We verify (2.9) if µ ≤ µ∗n + bn2 , and (2.10) q if µ ≥ µ∗n − bn1 . The choice of {bn } and the isoperimetric inequality (2.2) imply µ d q n−1 V ( q ) n−1 V ( ) n−1 n−1 Re f − fq − f dµ = f Re m m q
q
q µ∗n dµ ≤ |µ − µ∗n |( + 2δ)
V ( q ) ≤ θ .
q
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S. Friedli, C.-E. Pfister
C. We prove that on Un
d ωn () ≤ βC3 e−τ2 (β)|| . dz q Let V ( ) = n; from (1.9) d ωn ( q ) = ωn ( q ) − βa(ϕ q ) − β h(ψm ) − h(ψq ) V ( q ) dz d + log m (Int m q ) − log q (Int m q ) . dz
Equations (1.20), (1.21), (1.4), (1.8), (2.2) and (1.10) imply d ωn ( q ) ≤ β|ωn ( q )| | q |(C1 + 2δC0 ) + V ( q )( + 2δ) dz d
≤ βC3 |ωn ( q )|| q | d−1 ≤ βC3 e−τ2 (β)| | . q
D. We prove that ω( 2 )(z) is τ (β)-stable for any contour 2 with boundary condition ψ2 , if µ ≤ µ∗n (ν; β) − bn1 . Using the induction hypothesis it is sufficient to prove this statement for z = µ + iν ∈ Un−1 and µ ≤ µ∗n (ν; β) − bn1 . The next observation, leading to (2.15) and (2.17), is the key point of the proof of D. If z = µ + iν ∈ Un−1 , then all contours with volume V () ≤ n − 1 are τ1 (β)-stable; (2.7) and µ ≤ µ∗n imply that µ → Re(f1n−1 − f2n−1 )(µ + iν) is strictly decreasing. If µ ≤ µ∗n (ν; β) − bn1 , then (see (2.11) and (2.12)) µ∗n d βRe(f1n−1 − f2n−1 )(µ + iν) = −β Re(f1n−1 − f2n−1 )(µ + iν) dµ dµ µ µ∗n d Re(f1n−1 − f2n−1 )(µ + iν) dµ ≥ −β ∗ 1 dµ µn −bn ≥ βbn1 ( − 2δ) ≥ 2δ l(n) .
(2.15)
First suppose that V ( 2 ) ≤ n. From (2.15) and (2.8) it follows that ω( 2 ) is β(ρ − ε − 2β −1 C0 δ)-stable, in particular τ (β)-stable. Moreover, if | | ≤ n, then 1 ( ) (2.16) exp − βz(h(ψ1 ) − h(ψ2 ))| | ≤ e3δ∂| | . 2 ( ) Indeed, all contours inside are τ1 (β)-stable. By (1.16) and (2.15), −βz(h(ψ1 )−h(ψ2 ))| | 1 ( ) −β(zh(ψ1 )−zh(ψ2 )+g1n−1 −g2n−1 )| | 2δ∂| | e e ≤ e 2 ( ) n−1
= e−βRe(f1 ≤e
2δ∂| |
(z)−f2n−1 (z))| | 2δ∂| |
e
.
To prove point D, we prove by induction on | | that (2.16) holds for any . Indeed, if (2.16) is true and if we set := Int1 2 , then it follows easily from the definition of ω( 2 ) and from (1.9) that ω( 2 ) is τ (β)-stable.
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83
The argument to prove (2.16) is due to Zahradnik [7]. The statement is true for | | ≤ n. Suppose that it is true for | | ≤ k, k > n, and let | | = k + 1. The induction hypothesis implies that ω( 2 )(z) is τ (β)-stable if V ( 2 ) ≤ k. Therefore (1.16) gives −βz(h(ψ1 )−h(ψ2 ))| | 1 ( ) −β(zh(ψ1 )−zh(ψ2 )−g2k )| | 1 ( )eδ∂| | . e ≤ e 2 ( ) From (1.5) 1 ( ) =
r
(j1 ) ,
j =1
where the sum is over all families {11 , . . . , r1 } of compatible external contours in . We say that an external contour j1 is large if V (j1 ) ≥ n. Suppose that the contours 1 , . . . 1 not large. We set 11 , . . . p1 are large and all other contours p+1 r p
p Ext1 ( )
:=
Extj1 ∩ .
j =1
Summing over all contours which are not large, we get from (1.6) and (1.9), 1 ( ) = =
p n−1 exp − βH(ϕ 1 |ψ1 ) 1 (Int 1 j1 ) 2 (Int 2 j1 ) Ext 1 ( ) 1 p
p n−1 Ext 1 ( ) 1
j =1 p
j
e
−β j1 −βza(ϕ 1 )+βz(h(ψ1 )−h(ψ2 ))|Int2 j1 | j
j =1
·
1 (Int 1 j1 ) 2 (Int 1 j1 )
2 (Int 1 j1 ) 2 (Int 2 j1 ) ;
the sums are over all families {11 , . . . , p1 } of compatible external large contours in
. All contourswhich are not large are τ1 (β)-stable, and we use Lemma 1.1 to control p n−1 Ext 1 ( ) , 2 (Int 1 j1 ) and 2 (Int 2 j1 ). We have 1 p
∂|Ext 1 ( )| ≤ ∂| | +
p
C0 |j1 | .
j =1
Hence, (1.16), Intj1 = Int1 j1 ∪ Int2 j1 and the induction hypothesis imply that (Rez = µ) p
p 1 n−1 1 ( ) ≤ eδ∂| | e−βReg1 |Ext1 ( )| e−(β(ρ−ε)−C0 δ)|j | j =1
1 1 1 1 (Int 1 j ) · eβµ(h(ψ1 )−h(ψ2 ))(|Intj |−|Int1 j |) 2 (Int 1 j1 ) 2 (Int 2 j1 ) 1 2 (Int 1 j )
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S. Friedli, C.-E. Pfister
≤ eδ∂| |
n−1
e−βReg1
p
p
|Ext 1 ( )|
e−(β(ρ−ε)−4C0 δ)|j | 1
j =1
·e
βµ(h(ψ1 )−h(ψ2 ))|Intj1 |
≤ eδ∂| |
n−1
e−βReg1
2 (Int 1 1 ) 2 (Int 2 1 ) j
p
p
|Ext 1 ( )|
j
e−(βρ−βε−5C0 δ)|j | 1
j =1
·e
β(µh(ψ1 )−µh(ψ2 )−g2k )|Intj1 |
.
We have
p
p
| | = |Ext1 ( )| +
|j1 | +
j =1
p
|Intj1 | .
j =1
Writing zh(ψ2 ) = f2n−1 − g2n−1 , and adding and subtracting
j
g1n−1 |j1 |, we get
−βz(h(ψ1 )−h(ψ2 ))| | 1 ( ) e 2 ( ) p n−1 n−1 n−1 k e−βRe(f1 −f2 +g2 −g2 )|Ext1 ( )| ≤ e2δ∂| | ·
p
n−1
e−(βρ−βε−6C0 δ)|j | e−βRe(f1 1
−f2n−1 +g2n−1 −g2k )|j1 |
.
j =1
We define τˆ (β) := β(ρ − ε) − 6C0 δ . From (2.15) and (1.18) we have βRe(f1n−1 − f2n−1 − g2k + g2n−1 ) ≥ δ l(n) .
(2.17)
Hence, p
p 1 l(n) l(n) −βz(h(ψ1 )−h(ψ2 ))| | 1 ( ) e−δ |Ext1 ( )| e−(δ +τˆ (β))|j | . ≤ e2δ∂| | e 2 ( ) j =1
We define (C0 δ is introduced for controlling boundary terms later on) e−(τˆ (β)−C0 δ)|| if || ≥ l(n); ω() ˆ := 0 otherwise. ˆ Let ( ) be defined by (1.11), replacing ω( q ) by ω(), ˆ and let gˆ := lim −
↑Zd
1 ˆ log ( ) . β| |
Our definition of β0 is such that for all β ≥ β0 , Ke−τˆ (β) ≤ δ .
(2.18)
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85
ˆ Since β|g| ˆ ≤ δ l(n) , putting into evidence a factor eβ g| | , we get p
1 −βz(h(ψ1 )−h(ψ2 ))| | 1 ( ) ˆ j1 | ˆ e−τˆ (β)|j | e−β g|Int e ≤ e2δ∂| |+β g| | 2 ( ) ˆ ≤ e2δ∂| |+β g| |
j =1 p
ˆ e−(τˆ (β)−C0 δ)|j | (Int j1 ) . 1
j =1 ˆ | as a partition function (up to a boundary term), since by We have interpreted e−β g|Int (1.16) 1 ˆ 1 | ˆ ≤ (Int 1 ) eC0 δ| | . e−β g|Int 1
We sum over external contours and get −βz(h(ψ1 )−h(ψ2 ))| | 1 ( ) ˆ ˆ ( ) ≤ e3δ∂| | . ≤ e2δ∂| |+β g| | e 2 ( )
It is not difficult to prove more regularity for the curve ν → µ∗ (ν; β). We need below only the following result. Lemma 2.1. Let 0 < δ < 1. If β is sufficiently large, then for all n ≥ 1 and
d ∗ µ (0; β) = 0, dν n
2δ 2 d2 ∗ 2δ ≤ 2δ µ (ν; β) + + 1 . dν 2 n
− 2δ − 2δ
− 2δ
Proof. Let δ be as in the proof of Proposition 2.1. Because the free energies f1n−1 and f2n−1 are real on the real axis, it follows that they satisfy fqn−1 (z) = fqn−1 (z), and thered ∗ fore ν → µ∗n (ν; β) is even, and µ (0; β) = 0. By definition µ∗n (ν; β) is solution dν n of Re f2n−1 (µ∗n (ν; β) + iν) − f1n−1 (µ∗n (ν; β) + iν) = 0 , which implies that
dµ∗n d dµ∗n d = Re g1n−1 − g2n−1 + Re g1n−1 − g2n−1 dν dµ dν dν
and
2 ∗ ∗ n−1 n−1 d 2 µ∗n d d2 n−1 d µn n−1 dµn 2 = − g + Re g − g Re g 1 2 1 2 dν 2 dµ dν 2 dµ2 dν ∗ 2 n−1 d2 dµ d n + Re g1 − g2n−1 + 2 Re g1n−1 − g2n−1 . dµdν dν dν
From the Proof of Proposition 2.1, Step C, we have on Uk , d ωk () ≤ βC3 e−τ2 (β)|| . dz
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S. Friedli, C.-E. Pfister
d Let τ3 (β) := τ1 (β) − 2 d−1 . A similar proof shows for β sufficiently large, that there exists C4 with the property
d2 ωk () ≤ β 2 C4 e−τ3 (β)|| . 2 dz Assume that β is large enough so that
Let Gn−1
β max{C4 , C32 }K2 e−τ3 (β)|| ≤ δ . := Re g1n−1 − g2n−1 ; by Lemma 1.1 d n−1 d G ≤ 2δ , Gn−1 ≤ 2δ , dµ dν d 2 n−1 2 2 ≤ 2δ , d Gn−1 ≤ 2δ , d Gn−1 ≤ 2δ . G 2 2 dµ dν dµdν
Hence 2δ 2 2 ∗ dµ∗n 2δ ≤ 2δ , d µn ≤ 2δ +1 . + 2 dν
− 2δ dν
− 2δ − 2δ
− 2δ
Proposition 2.2. Under the conditions of Proposition 2.1, there exist β0 ∈ R+ and p ∈ N so that the following holds for all β ≥ β0 . Let τ (β) := τ (β) − max
d ,p . d −1
1. If µ + iν ∈ U0 and µ ≤ µ∗ (ν; β), then d ω( 2 )(z) ≤ βC3 e−τ (β)| 2 | . dz 2. If µ + iν ∈ U0 and µ ≥ µ∗ (ν; β), then d ω( 1 )(z) ≤ βC3 e−τ (β)| 1 | . dz q
Proof. Let Un and bn be as in Proposition 2.1. Suppose that z = µ + iν ∈ Un−1 \Un and µ ≤ µ∗ (ν; β). We distinguish two cases, V ( 2 ) ≤ n and V ( 2 ) > n. If V ( 2 ) ≤ n, then Step C of the iteration method of Proposition 2.1 implies that d ω( 2 ) ≤ β|ω( 2 )| | 2 |(C1 + 2δC0 ) + V ( 2 )( + 2δ) dz d
≤ βC3 | q | d−1 |ω( 2 )| . Since by Proposition 2.1 ω( 2 ) is τ (β)-stable, we get for all 2 such that V ( 2 ) ≤ n, d d 2 2 ω( 2 ) ≤ βC3 | 2 | d−1 e−τ (β)| | ≤ βC3 e−τ (β)| | . dz
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87
Suppose that V ( 2 ) ≥ n + 1. We estimate the derivative at z of ω( 2 ) using Cauchy’s formula with a circle of center z contained in {µ + iν : µ ≤ µ∗ (ν; β)}. We estimate from below |Rez − µ∗ (ν; β)| when z ∈ Un−1 \Un , uniformly in ν. |Rez − µ∗ | ≥ |Rez − µ∗n | − |µ∗n − µ∗ | ≥ bn2 − |µ∗n − µ∗ | . We estimate |µ∗n − µ∗ | by first estimating |µ∗k − µ∗n |. Let k > n; then, since µ∗k ∈ Un , 0 = Re f2k−1 (µ∗k ) − f1k−1 (µ∗k ) − Re f2n−1 (µ∗n ) − f1n−1 (µ∗n ) = Re f2k−1 (µ∗k ) − f2n−1 (µ∗k ) − Re f1k−1 (µ∗k ) − f1n−1 (µ∗k ) +Re f2n−1 (µ∗k ) − f2n−1 (µ∗n ) − Re f1n−1 (µ∗k ) − f1n−1 (µ∗n ) . From (2.14) we get |µ∗k (ν; β) − µ∗n (ν; β)| ≤
2δ l(n) β( − 2δ)
∀ k > n,
so that |µ∗ (ν; β) − µ∗n (ν; β)| ≤
2δ l(n) . β( − 2δ)
(2.19)
If V ( 2 ) ≥ n + 1, then | 2 | ≥ l(n + 1). Choose p ∈ N so that for all n ≥ 1, 1 p 1 χθ 2δ l(n) ≤ ≤ − 1 d−1 2 p | | β( − 2δ) ( + 2δ)n d 2dn d ≤ bn2 − |µ∗ − µ∗n | ≤ |Rez − µ∗ | . We use Cauchy’s formula, with a circle of center z and radius | 2 |−p , to estimate d 2 dz ω( ), d ω( 2 ) ≤ | 2 |p e−τ (β)| 2 | ≤ e−τ (β)| 2 | . dz 2.2. Analytic continuation of the weights of contours at µ∗ . In this subsection we consider how the weight ω( 2 ) for a contour with boundary condition ψ2 behaves as a function of z = µ + iν in the vicinity of z∗ := µ∗ (ν; β) + iν. We obtain new domains of analyticity of the weights of contours, by introducing the isoperimetric constant χ2 (n) (see (2.20)), which differs from that used in [3]. This is one very important point of our analysis. The main result of this subsection is Proposition 2.3. At z∗ the (complex) free energies fq , q = 1, 2, are well-defined and can be computed by the cluster expansion method. Moreover, Ref2 (z∗ ) = Ref1 (z∗ ) . Therefore Reg1 (z∗ ) + µ∗ (ν; β)h(ψ1 ) = Reg2 (z∗ ) + µ∗ (ν; β)h(ψ2 ) . With δ as in the proof of Proposition 2.1, we get |µ∗ (ν; β)| ≤
2δ , β
88
and
S. Friedli, C.-E. Pfister
2C1 δ q |ω( q )(z∗ )| ≤ exp − β q + | | + δC0 | q | , ∀ q .
We set
µ∗ := µ∗ (0; β) ,
and adopt the following convention: if a quantity, say H or fq , is evaluated at the transition point µ∗ , we simply write H∗ or fq∗ . The analyticity properties of ω( 2 ) near µ∗ are controlled by isoperimetric inequalities V ( 2 )
d−1 d
≤ χ2 (n)−1 2
∀ 2 , V ( 2 ) ≥ n .
(2.20)
The difference with (2.2) is that only contours with boundary condition ψ2 and V ( 2 ) ≥ n are considered for a given n. By definition the isoperimetric constants χ2 (n) satisfy χ2 (n)
−1
d−1 V ( 2 ) d 2 2 ≤ C , ∀ := inf C : such that V ( ) ≥ n .
2
χ2 (n) is a bounded increasing sequence; we set χ2 (∞) := limn χ2 (n), and define χ2 (m)
R2 (n) := inf
1
m:m≤n
.
md
There are similar definitions for χ1 (n) and R1 (n). The corresponding isoperimetric inequalities control the analyticity properties of ω( 1 ) around µ∗ . Lemma 2.2. For any χq < χq (∞), there exists N (χq ) such that for all n ≥ N (χq ), χq n
≤ Rq (n) ≤
1 d
χq (∞) 1
.
nd
For q = 1, 2, n → na Rq (n) is increasing in n, provided that a ≥ d1 . Proof. Let q = 2 and suppose that χ2 (m)
R2 (n) =
1
for m < n.
md
Then R2 (m ) = R2 (n) for all m ≤ m ≤ n. Let n be the largest n ≥ m such that R2 (n) =
χ2 (m) 1
.
md
We have n < ∞, otherwise 0 < R2 (m) = R2 (n) ≤
χ2 (∞) 1
nd
∀ n ≥ m,
which is impossible. Therefore, either R2 (n ) =
χ2 (n ) n
1 d
or
R2 (n + 1) =
χ2 (n + 1) 1
(n + 1) d
,
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89
and for all k ≥ n + 1, since χ2 (m) is increasing, R2 (k) = inf
m≤k
χ2 (m) m
1 d
=
inf
n ≤m≤k
χ2 (m) m
1 d
≥
inf
n ≤m≤k
χ2 (n ) m
1 d
=
χ2 (n ) 1
(2.21)
.
kd
Inequality (2.21) is true for infinitely many n ; since there exists m such that χ2 ≤ χ2 (m), the first statement is proved. On an interval of constancy of R2 (n), n → na R2 (n) is increasing. On the other hand, if on [m1 , m2 ] χ2 (n) R2 (n) = , 1 nd 1
then n → na R2 (n) is increasing on [m1 , m2 ] since n → χ2 (n) and n → na− d are increasing. The next proposition gives the domains of analyticity and the stability properties of the weights ω() needed for estimating the derivatives of the free energy. Proposition 2.3. Let 0 < θ < 1 and 0 < ε < 1 so that ρ(1 − θ) − ε > 0. There exist 0 < δ < 1, 0 < θ < 1 and β0 ≥ β0 , such that for all β ≥ β0 ω( 2 ) is analytic and τ1 (β; θ )-stable in a complex neighborhood of z ∈ C : Rez ≤ µ∗ (Imz; β) + θ −1 R2 (V ( 2 )) ∩ U0 . Moreover
d ω( 2 ) ≤ βC3 e−τ2 (β;θ )| 2 | . dz
Similar properties hold for ω( 1 ) in a complex neighborhood of z ∈ C : µ∗ (Imz; β) − θ −1 R1 (V ( 1 )) ≤ Rez ∩ U0 . τ1 (β; θ ) and τ2 (β; θ ) are defined at (2.3) and (2.4). Proof. We prove the proposition for ω( 2 ). By Proposition 2.1 ω( 2 ) is τ (β)-stable d if Rez ≤ µ∗ (ν; β) ∩ U0 , and by Proposition 2.2 dz ω( 2 ) is τ (β)-stable on the same region. Let (2.22) In (ν; β) := µ∗ (ν; β) − θ −1 R1 (n), µ∗ (ν; β) + θ −1 R2 (n) . We prove by iteration, that on the intervals In (ν; β) ω( q ), q = 1, 2, is τ1 (β; θ )-stable, d and dz ω( q ) is τ2 (β; θ )-stable. To prove the stability of ω( q ) it is sufficient by Remark 2.4 to verify (2.9) and (2.10). Suppose that the statement is correct for V ( q ) ≤ n − 1. Let V ( 2 ) = n, z = µ + iν, and µ ≥ µ∗ (ν; β). Then µ V ( 2 ) V ( 2 ) d n−1 n−1 = −Re f (z) − f (z) −Re f1n−1 (z) − f2n−1 (z) 1 2
2
2 µ∗n dµ 1
nd ≤ ( + 2δ) |µ − µ∗ | + |µ∗ − µ∗n | χ2 (n) 1
+ 2δ 2( + 2δ)δ l(n) n d ≤ θ+
β( − 2δ) χ2 (n) ≤ θ .
90
S. Friedli, C.-E. Pfister
We used (2.19) to control |µ∗ − µ∗n |. If β is large enough and δ small enough, then there d exists θ < 1. The stability of dz ω( 2 ) is a consequence of d ω( 2 ) ≤ β|ω( 2 )| | 2 |(C1 + 2δC0 ) + V ( 2 )( + 2δ) dz d
≤ βC3 | q | d−1 |ω( 2 )| .
2.3. Derivatives of the free energy at finite volume. Although non-analytic behavior of the free energy occurs only in the thermodynamical limit, most of the analysis is done at finite volume. We write dk (k) [g] t := k g(t) dt t=t for the k th order derivative at t of the function g. The method of Isakov [2] allows to get estimates of the derivatives of the free energy at µ∗ , which are uniform in the volume. We consider the case of the boundary condition ψ2 . The other case is similar. We tacitly assume that β is large enough so that Lemma 1.1 and all results of Subsects. 2.1 and 2.2 are valid. The main tool for estimating the derivatives of the free energy is Cauchy’s formula. However, we need to establish several results before we can obtain the desired estimates on the derivatives of the free energy. The preparatory work is done in this subsection, which is divided into three subsections. In 2.3.1 we give an expression of the derivatives of the free energy in terms of the derivatives of a free energy of a contour u( 2 ) = − log(1 + φ ( 2 )) ≈ −φ ( 2 ) (see (2.24)). The main work is to estimate φ ( 2 )n (z) k! dz . 2πi ∂Dr (z − µ∗ )k+1 g
The boundary of the disc Dr is decomposed naturally into two parts, ∂Dr and ∂Drd , g d ( 2 ) (see (2.26) and (2.27)). In 2.3.2 and the integral into two integrals Ik,n ( 2 ) and Ik,n g d ( 2 ) by we prove the upper bound (2.28) for Ik,n ( 2 ), and in 2.3.3 we evaluate Ik,n the stationary phase method, see (2.34) and (2.35). This is a key point in the proof of d ( 2 ). Theorem 1.1, since we obtain lower and upper bounds for Ik,n 2.3.1. An expression for the derivatives of the free energy Let = (L) be the cubic box
(L) := {x ∈ Zd : |x| ≤ L} . We introduce a linear order, denoted by ≤, among all contours q ⊂ with boundary condition ψq . We assume that the linear order is such that V ( q ) ≤ V ( q ) if q ≤ q . There exists a natural enumeration of the contours by the positive integers. The predecessor of q in that enumeration (if q is not the smallest contour) is denoted by i( q ). We introduce the restricted partition function q ( ), which is computed with the contours of C ( q ) := { q ⊂ : q ≤ q } , that is q ( ) := 1 +
n
i=1
q
ω(i ) ,
(2.23)
Singularity of the Free Energy
91 q
q
where the sum is over all families of compatible contours {1 , . . . , n } which belong to C ( q ). The partition function q ( ) is written as a finite product q ( ) =
q ⊂
q ( ) . i( q ) ( )
By convention i( q ) ( ) := 1 when q is the smallest contour. We set u ( q ) := − log
q ( ) . i( q ) ( )
u ( q ) is the free energy cost for introducing the new contour q in the restricted model, where all contours satisfy q ≤ q . We have the identity q ( ) = i( q ) ( ) + ω( q ) i( q ) ( ( q )) q q i( q ) ( ( )) q . = i( ) ( ) 1 + ω( ) i( q ) ( ) In this last expression i( q ) ( ( q )) denotes the restricted partition function i( q ) ( ( q )) := 1 +
n
q
ω(i ) ,
i=1 q
q
where the sum is over all families of compatible contours {1 , . . . , n } which belong q q to C (i( q )), and such that { q , 1 , . . . , n } is a compatible family. We also set φ ( q ) := ω( q )
i( q ) ( ( q )) . i( q ) ( )
With these notations (−1)n u ( q ) = − log 1 + φ ( q ) = φ ( q )n , n
(2.24)
n≥1
and for k ≥ 2, q (k)
| |β[f ] µ∗ =
(k)
[u ( q )] µ∗ .
q ⊂
(k)
We consider the case of the boundary condition ψ2 . [φ ( 2 )n ]µ∗ is computed using Cauchy’s formula, φ ( 2 )n (z) k! (k) dz , [φ ( 2 )n ] µ∗ = 2πi ∂Dr (z − µ∗ )k+1 where ∂Dr is the boundary of a disc Dr of radius r and center µ∗ inside the analyticity region of Proposition 2.3, U0 ∩ z ∈ C : Rez ≤ µ∗ (Im(z); β) + θ −1 R2 (V ( 2 )) .
92
S. Friedli, C.-E. Pfister φ ( 2 )n (z) (z−µ∗ )k+1
The function z →
is real on the real axis, so that
φ ( 2 )n (z) φ ( 2 )n (z)
= , (z − µ∗ )k+1 (z − µ∗ )k+1 and consequently k! 2π i
∂Dr
k! φ ( 2 )n (z) dz = Re (z − µ∗ )k+1 2πi
∂Dr
φ ( 2 )n (z) dz . (z − µ∗ )k+1
(2.25)
Remark 2.5. From Lemma 2.1, there exists C independent of ν and n, so that µ∗n (ν; β) ≥ µ∗n (0; β) − C ν 2 . This implies that the region {Rez ≤ µ∗ − C (Imz)2 + θ −1 R2 (V ( 2 ))} is always in the analyticity region of ω( 2 ), which is given in Proposition 2.3. Therefore, if 1 C ≤ 2 , −1 2 θ R2 (V ( 2 )) then the disc Dr of center µ∗ and radius r = θ −1 R2 (V ( 2 )) is inside the analyticity region of ω( 2 ). This happens as soon as V ( 2 ) is large enough. Assuming that the disc Dr is inside the analyticity region of ω( 2 ), we decompose ∂Dr into ∂Dr := ∂Dr ∩ {z : Rez ≤ µ∗ (Im(z); β) − θ −1 R1 (V ( 2 ))} , g
and ∂Drd := ∂Dr ∩ {z : Rez ≥ µ∗ (Im(z); β) − θ −1 R1 (V ( 2 ))} , g
d ( 2 ), and write (2.25) as a sum of two integrals Ik,n ( 2 ) and Ik,n
g Ik,n ( 2 )
k! φ ( 2 )n (z) := Re dz 2πi ∂Drg (z − µ∗ )k+1
(2.26)
d ( 2 ) Ik,n
k! φ ( 2 )n (z) := Re dz . 2πi ∂Drd (z − µ∗ )k+1
(2.27)
and
Singularity of the Free Energy
93 g
g
2.3.2. An upper bound for Ik,n ( 2 ) Ik,n ( 2 ) is not the main contribution to (2.25), so that it is sufficient to get an upper bound for this integral. Let z ∈ U0 and Rez ≤ µ∗ Im(z); β . From (2.16) we get |ω( 2 )| ≤ exp − β 2 + β|Rez|C1 | 2 | + 3C0 δ| 2 | . Using formula (1.24), we get after cancellation and the use of Lemma 1.2 and Proposition 2.3 (see also (1.23)), 2 ( ( 2 )) 2 2 i( ) ≤ eδ| | ≤ eδC5 | | . i( 2 ) ( ) We set
ζ := z − µ∗ .
Therefore, there exists a constant C6 so that |φ ( 2 )| ≤ e−β
2 (1−C δ−|Reζ |C ρ −1 ) 6 1
if
Reζ ≤ µ∗ Im(ζ ); β − µ∗ .
This upper bound implies g
Ik,n ( 2 ) ≤
k! −nβ 2 (1−C6 δ−rC1 ρ −1 ) e . rk
(2.28)
d ( 2 ). In order to apply the stationary phase 2.3.3. Lower and upper bounds for Ik,n d 2 method to evaluate Ik,n ( ), we first rewrite φ ( 2 ) in the following form, ∗ φ ( 2 )(z) = φ
( 2 ) eβ V (
2 )(ζ +g( 2 )(ζ ))
,
(2.29)
where g( 2 ) is an analytic function of ζ in a neighborhood of ζ = 0 and g( 2 )(0) = 0. Let µ∗ Im(z); β − θ −1 R1 (V ( 2 )) ≤ Rez ≤ µ∗ Im(z); β + θ −1 R2 (V ( 2 )) . In this region (see Fig. 2.1) we control the weights of contours with boundary conditions ψ2 and ψ1 . Therefore, by the cluster expansion method, we control log 1 (Int 1 2 ), and we can write i( 2 ) ( ( 2 )) 1 (Int 1 2 ) φ ( 2 ) = exp − βH(ϕ 2 |ψ2 ) + log . + log i( 2 ) ( ) 2 (Int 1 2 ) :=G( 2 )
By definition z = ζ + µ∗ , so that we have (see (1.9)) −βH(ϕ 2 |ψ2 )(z) + G( 2 )(z) = −βH(ϕ 2 |ψ2 )(µ∗ ) + β V ( 2 )ζ µ∗ +ζ d G( 2 )(z )dz + G( 2 )(µ∗ ) −βa(ϕ 2 )ζ + dz µ∗ = −βH(ϕ 2 |ψ2 )(µ∗ ) + G( 2 )(µ∗ ) + β V ( 2 )ζ µ∗ +ζ d 2 G( )(z ) − βa(ϕ + 2 ) dz . ∗ dz µ :=β V ( 2 )g( 2 )(ζ )
94
S. Friedli, C.-E. Pfister
µ∗ (ν; β) ν ∂Drd ∂Drg µ∗ (0; β)
µ
r
θ −1 R2 (V ( 2 )) θ −1 R1 (V ( 2 )) g
d ( 2 ) Fig. 2.1. The decomposition of the integral into Ik,n ( 2 ) and Ik,n
This proves (2.29). For large enough β, τ (β) ≥ τ1 (β, θ ) ≥ τ2 (β, θ ), d d 1 d g( 2 )(ζ ) = log 1 (Int 1 2 ) − log 2 (Int 1 2 ) 2 dζ β V ( ) dζ dζ i( 2 ) ( ( 2 )) d (2.30) + log − βa(ϕ 2 ) . dζ i( 2 ) ( ) The last term of the right-hand side of (2.30) is estimated using (1.8). The first two terms are estimated using Proposition 2.3, (1.21) and (1.20). The third term is estimated by writing explicitly the logarithm of the quotient, using (1.24). After cancellation the resulting series is derived term by term and is estimated as in Lemma 1.2 using the basic estimates of Proposition 2.3. There exists K ≥ max{K1 , K0 }, such that 2 2 d C1 | 2 | g( 2 )(ζ ) ≤ 2C3 Ke−τ2 (β;θ ) 1 + C0 | | + | | + dζ
V ( 2 ) V ( 2 )
V ( 2 ) 2 | | ≤ C7 e−τ2 (β;θ ) + C8 , (2.31) V ( 2 ) for suitable constants C7 and C8 . Moreover, there exists a constant C9 so that ∗ exp − β 2 (1 + C9 δ)] ≤ φ
( 2 ) ≤ exp[−β 2 (1 − C9 δ)] . Let
c(n) := nβ V ( 2 ) .
We parametrize ∂Drd by z := µ∗ + reiα , −α1 ≤ α ≤ α2 , 0 < αi ≤ π , φ ∗ ( 2 )n α2 c(n)r cos α+c(n)Re g( 2 )(ζ ) d (α)) dα , Ik,n cos(ψ ( 2 ) = k! k e 2πr −α1
(2.32)
Singularity of the Free Energy
95
where (α) := c(n)r sin α + c(n) Im g( 2 )(ζ ) − kα . ψ We search for a stationary phase point ζk,n = rk,n eiαk,n defined by the equations d c(n)r cos α + c(n)Re g( 2 ) reiα = 0 dα
d (α) = 0 . ψ dα
and
These equations are equivalent to the equations ( denotes the derivative with respect to ζ ) c(n) sin α 1 + Re g( 2 ) (ζ ) + cos αIm g( 2 ) (ζ ) = 0 ; c(n)r cos α 1 + Re g( 2 ) (ζ ) − r sin αIm g( 2 ) (ζ ) = k . Since g( 2 ) is real on the real axis, αk,n = 0 and rk,n is a solution of c(n)r 1 + g( 2 ) (r) = k .
(2.33)
Lemma 2.3. Let αi ≥ π/4, i = 1, 2, A ≤ 1/25 and c(n) ≥ 1. If g(ζ ) is analytic in ζ in the disc {ζ : |ζ | ≤ R}, real on the real axis, and for all ζ in that disc d g( 2 )(ζ ) ≤ A , dζ then there exists k0 (A) ∈ N, such that for all integers k, √ k ∈ k0 (A), c(n)(1 − 2 A)R , there is a unique solution 0 < rk,n < R of (2.33). Moreover, ecrk,n +c(n) g(
2 )(r ) k,n
10 c(n)rk,n
1 ≤ 2π ≤
α2
−α1
ec(n)r cos α+c(n)Re g(
2 ec(n)rk,n +c(n) g( )(rk,n )
c(n)rk,n
2)
(α)) dα cos(ψ
.
Proof. Existence and uniqueness of rk,n is a consequence of the monotonicity of r → c(n)r 1 + g( 2 ) (r) . The last part of Lemma 2.3 is proven in the Appendix of [2]. The computation is relatively long, but standard. Setting c(n) = nβ V ( 2 ) and R = θ −1 R2 (V ( 2 )) in Lemma 2.3 we get sufficient conditions for the existence of a stationary phase point and the following evaluation d ( 2 ) by that method. Since r of the integral Ik,n k,n is solution of (2.33), we have k− and
k kA kA k ≤ c(n)rk,n ≤ =k+ , = (1 + A) (1 − A) (1 − A) (1 + A)
c(n)|g( 2 )(rk,n )| = c(n)
0
rk,n
g( 2 ) (ζ )dζ ≤ Ac(n)rk,n ≤ k
A . 1−A
96
S. Friedli, C.-E. Pfister
Therefore Lemma 2.3 implies √ 1−A k k! ek ∗ 2 n d ( ) ≤ Ik,n ( 2 ) √ c− c(n)k k φ
k 10 k √ 1+A k k! ek ∗ 2 n ≤ √ ( ) , c+ c(n)k k φ
k k with 2A c+ (A) : = (1 + A) exp , 1−A 2A . c− (A) : = (1 − A) exp − 1 − A2 If A converges to 0, then c± converges to 1. We assume that (see (2.31))
C7 e−τ2 (β;θ ) ≤
A 2
and C8
A | 2 | ≤ . V ( 2 ) 2
A can be chosen as small as we wish, provided that β is large enough and enough.
(2.34)
(2.35)
(2.36) | 2 | V ( 2 )
small
2.4. Lower bounds on the derivatives of the free energy at finite volume. We estimate the (k) derivative of [f 2 ]µ∗ for large enough k. The main result of this subsection is Proposition 2.4. Subsection 2.4.2 is a very important point of our analysis. Let 0 < θ < 1, A ≤ 1/25, and set √ θˆ := θ (1 − 2 A) . Let ε > 0 and χ2 so that (1 + ε )χ2 > χ2 (∞) .
(2.37)
and ε . We fix the values of θ, and ε
The whole analysis depends on the parameters θ by the following conditions, which are needed for the proof of Proposition 2.4. We choose 0 < A0 < 1/25, θ and ε so that √ d−1 1 1 1 − 2 A0 d d c− (A0 ) d ed > 1. (2.38) and < √ d − 1 1 + ε 1 + ε d − 1 θ (1 − 2 A0 ) This is possible, since d 1
(d − 1) e d Indeed,
> 1.
1 1 1 1 1 n−1 1 d ed − 1 = d ed − 1 − + = +1 d d n! d n≥2 1 n 1 = 1+ (n + 1)! d n≥1 1 1 n 1 1 1 < 1− = ed − + . 2d n! d 2d n≥1
Singularity of the Free Energy
97
Notice that conditions (2.38) are still satisfied with the same values of θ and ε if we replace A0 by 0 < A < A0 . Given θ , the value of θ is fixed in Proposition 2.3. From now we assume that β is so large that all results of Subsects. 2.1 and 2.2 are valid. The value of 0 < A < A0 is fixed in the proof of Lemma 2.5. Given k large enough, there is a natural distinction between contours 2 such that ˆ ( 2 )R2 (V ( 2 )) ≤ k and those such that θˆ βV ( 2 )R2 (V ( 2 )) > k. For the latter θβV d ( 2 ) by the stationary phase method. We need as a matter of fact a we can estimate Ik,n finer distinction between contours. We distinguish three classes of contours: 1. k-small contours: θˆ βV ( 2 )R2 (V ( 2 )) ≤ k; d−1 2. fat contours: for η ≥ 0, fixed later by (2.41), V ( 2 ) d ≤ η 2 ; d−1 3. k-large and thin contours: θˆ βV ( 2 )R2 (V ( 2 )) > k, V ( 2 ) d > η 2 . We make precise the meaning of k large enough. By Lemma 2.2 V → V R2 (V ) is increasing in V , and there exists N (χ2 ) such that R2 (V ) ≥
χ2 V
if
1 d
V ≥ N (χ2 ) .
We assume that there is a k-small contour 2 such that V ( 2 ) ≥ N (χ2 ), and that the maximal volume of the k-small contours is so large that Remark 2.5 is valid. We also assume (see Lemma 2.3) that k > k0 (A) and that for a k-large and thin contour (see (2.31) and (2.36)) C8
| 2 | A C8 ≤ , ≤ 1 V ( 2 ) 2 2 ρηV ( ) d
so that |g( 2 ) | ≤ A, and C1 k ρ (1 − A0
≤
1
)ηV ( 2 ) d
k 10
(2.39)
are verified. There exists K(A, η, β) such that if k ≥ K(A, η, β), then k is large enough. From now on k ≥ K(A, η, β). q (k)
2.4.1. Contribution to [f ]µ∗ from the k-small and fat contours. Let 2 be a k-small contour. Since V → R2 (V ) is decreasing in V , u ( 2 ) is analytic in the region {z : Rez ≤ µ∗ (Imz; β) + θ −1 R2 (V ∗ )} ∩ U0 , where V ∗ is the maximal volume of k-small contours. V ∗ satisfies V∗ Hence
d−1 d
≤
k θˆ βχ2
.
d 1 1 1 ˆ 2 d−1 β d−1 k − d−1 . θ −1 R2 (V ∗ ) ≥ θˆ −1 χ2 V ∗ − d ≥ −1 θχ
98
S. Friedli, C.-E. Pfister
Since Remark 2.5 is valid, we estimate the derivative of u ( 2 ) by Cauchy’s formula d 1 1 with a disc centered at µ∗ with radius −1 θˆ χ2 d−1 β d−1 k − d−1 . There exists a constant C10 such that
(k) [u ( 2 )] µ∗ ≤ C10
2 :Int 2 0 V ( 2 )
d−1 d ≤
k
β
1 d−1
ˆ ) (θχ 2
k
k! k d−1 .
d d−1
(2.40)
k ˆ θβχ 2
Let 2 be a fat contour, which is not k-small. We use in Cauchy’s formula a disc centered at µ∗ with radius θˆ −1 χ2 (1)V ( 2 )− d ≤ θ −1 R2 (V ( 2 )) . 1
We get (see (1.10)) [φ ( 2 )n ](k)∗ ≤ k! µ
! !
≤ k!
"k
1
V ( 2 ) d χ2 (1)θˆ
e−n[τ1 (β;θ )−C5 δ]| 1
(C2 η) d−1 χ2 (1)θˆ
2|
"k
k
| 2 | d−1 e−n[τ1 (β;θ )−C5 δ]| | . 2
We sum over n and over 2 using the inequality
mp e−qm ≤
m≥1
1 (p + 1) qp
(p ≥ 2 , q ≥ 2) .
There exist C11 and C12 (θ ) > 0 so that
[u ( 2 )](k)∗ ≤ C11 µ
!
"k
1
(C2 η) d−1 1
(C12 β) d−1 χ2 (1)θˆ
2 :Int d 0
k!
k +1 d −1
d−1
V ( 2 ) d ≤η 2 2 not k-small
!
"k
1
(C2 η) d−1
≤ C11
(C12 β)
1 d−1
k
χ2 (1)θˆ
k! k d−1 .
We choose η so small that (see (2.40)) 1
(C2 η) d−1 (C12 β)
1 d−1
χ2 (1)θˆ
< β
1 d−1
(θˆ χ2 (∞))
d d−1
< β
1 d−1
d
ˆ ) d−1 (θχ 2
.
(2.41)
Singularity of the Free Energy
99
q (k)
2.4.2. Contribution to [f ]µ∗ from the k-large and thin contours. For k-large and thin (k)
contours we get lower and upper bounds for [φ ( 2 )n ] µ∗ . There are two cases. A. Assume that R1 (V ( 2 )) ≥ R2 (V ( 2 )), or that V ( 2 ) is so large that θˆ βV ( 2 )R1 (V ( 2 )) > k . For each n ≥ 1 let c(n) = nβ V ( 2 ). Under these conditions we can apply Lemma 2.3 with a disc Drk,n so that ∂Drk,n = ∂Drdk,n . Indeed, if R1 (V ( 2 )) ≥ R2 (V ( 2 )), then we apply Lemma 2.3 with R = θ −1 R2 (V ( 2 )), and in the other case we set R = θ −1 R1 (V ( 2 )). In both cases rk,n < R, which implies ∂Drk,n = ∂Drdk,n . Thered ( 2 ) the lower and upper bounds (2.34). fore we get for Ik,n Lemma 2.4. There exists a function D(k), limk→∞ D(k) = 0, such that for β sufficiently large and A sufficiently small the following holds. If k ≥ K(A, η, β) and R1 (V ( 2 )) ≥ R2 (V ( 2 )) or θˆ βV ( 2 )R1 (V ( 2 )) > k, then (k)
(k)
(k)
(1 − D(k)) [φ ( 2 )] µ∗ ≤ −[u ( 2 )] µ∗ ≤ (1 + D(k)) [φ ( 2 )] µ∗ . Proof. We have (k) −[u ( 2 )] µ∗
=
(k) [φ ( 2 )] µ∗
(k) + [φ ( 2 )] µ∗
(−1)(n−1) [φ ( 2 )n ](k) µ∗ (k)
n
n≥2
[φ ( 2 )] µ∗
.
From (2.34) there exists a constant C13 , (k)
[φ ( 2 )n ] µ∗ (k) [φ ( 2 )] µ∗
∗ ≤ C13 φ
( 2 )(n−1)
c k +
c−
nk .
1
The isoperimetric inequality (2.20), R2 (n) ≤ χ2 (n)n− d and the definition of k-large volume contour imply β 2 ≥ βχ2 (V ( 2 ))V ( 2 )
≥ θˆ βR2 (V ( 2 ))V ( 2 ) ≥ k .
d−1 d
Let b := C9 δ (see (2.32)); we may assume k c+ k c−
nk−1 e−(n−1)(1−b)k ≤
n≥2
≤ ≤
9 10
k c+ k c−
4 5
by taking β large enough. Then
e− 10 (n−1)k e−k 1
n≥2
k c+ k c− c
−b ≥
e− 10 (n−1)k e−k 1
n≥2
+ −1 k 10
c−
e
1
1
9 ( 10 −b)(n−1)−ln n
4
5 (n−1)−ln n
e− 10 nk .
n≥1
We choose A so small that c+ (A)c− (A)−1 e− 10 ≤ 1.
100
S. Friedli, C.-E. Pfister
B. The second case is when θˆ βV ( 2 )R1 (V ( 2 )) ≤ k ≤ θˆ βV ( 2 )R2 (V ( 2 )) . Since the contours are also thin, β 2 ≤ η−1 θˆ −1 χ1 (1)−1 β θˆ χ1 (1)V ( 2 ) d ≤ η−1 θˆ −1 χ1 (1)−1 β θˆ V ( 2 )R1 (V ( 2 )) d−1
≤ η−1 θˆ −1 χ1 (1)−1 k ≡ λk . We choose R = β −1 R2 (V ( 2 )) in Lemma 2.3. The integration in (2.25) is decomposed into two parts (see Fig. 2.1). We show that the contribution from the integration g over ∂Drk,n is negligible for large enough β. Since k ≥ K(A, η, β) and the contours satisfy V ( 2 )
d−1 d
> η 2 , we have
nβ 2 rk,n ≤
k 1
(1 − A)ηV ( 2 ) d
≤
k
.
1
(1 − A0 )ηV ( 2 ) d
By definition of K(A, η, β) (see (2.39)) nβ 2 ρ −1 C1 rk,n ≤
k . 10 (k)
From (2.28) with r = rk,n we obtain that the contribution to |[u ( q )] µ∗ | is at most k k k! k −nβ 2 (1−C δ) 6 n e . (1 + A)k β V ( 2 ) exp 10 k k n≥1
As in the proof of Lemma 2.4, we choose β large enough so that we can assume that 9 4 10 − C6 δ ≥ 5 . Then 4 1 2 2 nk e−nβ (1−C6 δ) ≤ e−β (1−C6 δ) 1 + e− 10 (n−1)k e−k 5 (n−1)−ln n n≥1
≤e
−β 2 (1−C6 δ)
=e
−β 2 (1−C6 δ)
n≥2
1+
1
e− 10 nk
n≥1
1 + D(k) .
Since β 2 ≤ λk, by choosing A small enough and β large enough, so that δ is small enough, we have k k −β e e (1 − D(k))c−
2k
2 C δ 9
k k −kλC9 δ ≥ (1 − D(k))c− e e >e3
2 C δ 6
≤ (1 + D(k))(1 + A)k e 10 eλkC6 δ < e 3 .
and k
(1 + D(k))(1 + A)k e 10 eβ
k
k
(k)
If these inequalities are satisfied, then the contribution to −[u ( q )] µ∗ coming from the g integrations over ∂Drk,n is negligible with respect to that coming from the integrations d over ∂Drk,n . Taking into account (2.34) we get Lemma 2.5.
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Lemma 2.5. There exists 0 < A ≤ A0 so that for all β sufficiently large, the following holds. If k ≥ K(A , η, β) and 2 is a k-large and thin contour, then (k)
−[u ( 2 )] µ∗ ≥
k k ∗ 2 1 φ ( ) . (1 − D(k)) β V ( 2 ) c− 20
Proposition 2.4. There exists β so that for all β > β , the following holds. There exists an increasing diverging sequence {kn } such that for each kn there exists (Ln ) such that for all ⊃ (Ln ), kn
kn −[f 2 ] µn∗ ≥ C14 kn ! d−1 kn β − d−1 χ2 d
(k )
dkn − d−1
.
C14 > 0 is a constant independent of β, kn and . Proof. We compare the contribution of the small and fat contours with that of the large (k) and thin contours for k ≥ K(A , η, β). The contribution of the small contours to |[f 2 ] µ∗ | is at most C10 k β − d−1 (θˆ χ2 )− d−1 k! k d−1 ≤ C10 k β − d−1 k
kd
k
k
e d1 k d d d−1 k! d−1 . ˆ θχ 2
The contribution of the fat contours is much smaller by our choice of η (see (2.41)). The (k) contribution to −[f 2 ] µ∗ of each large and thin contour is nonnegative. By assumption (2.37) and the definition of the isoperimetric constant χ2 , there exists a sequence n2 , n ≥ 1, such that lim n2 → ∞
n→∞
and V (n2 )
d−1 d
≥
n2 . (1 + ε )χ2
d
d Since x k d−1 e−x has its maximum at x = k d−1 , we set
# kn :=
$ d −1 β n2 . d
For any n, n2 is a thin and kn -large volume contour, since by (2.38) √ √ d−1 β (1 − 2 A )V ( 2 )R2 (V ( 2 )) ≥ β (1 − 2 A )V ( 2 ) d χ2 √ (1 − 2 A ) ≥ β n2 ≥ kn . 1 + ε If ⊃ n2 , then (k )
k ∗ 2 1 − D(k) (n ) β c− V (n2 ) n φ
20 d−1 dkn kn d d c−d 1 − D(k) kn − kn d−1 d−1 ∗ ≥ kn φ (n2 )
β d−1 20 (d − 1)(1 + ε )χ2
−[u (n2 )] µn∗ ≥
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and (see (2.32)) kn d kn d ∗ (n2 ) ≥ knd−1 exp − kn knd−1 φ
d + 1 (1 + C9 δ) d −1 d d e−1−C9 δ . ∼ kn ! d−1 e−C9 δ d−1 kn d (2πkn ) 2(d−1)
By the choice (2.38) of the parameters θ and ε , if δ is small enough, i.e. β large enough, then d−1
1
c−d −C9 δ d < e . √ d − 1 1 + ε θ (1 − 2 A ) ed
Hence the contributions of the small and fat contours are negligible for large kn (see (2.40) and (2.41)). Let (Ln ) be a box which contains at least | (Ln )|/4 translates of n2 . For any ⊃ (Ln ), if kn and β are large enough, then there exists a constant C14 > 0, independent of β, kn and ⊃ (Ln ), such that kn
kn −[f 2 ] µn∗ ≥ C14 kn ! d−1 kn β − d−1 χ2 d
(k )
dkn − d−1
.
2.5. Lower bounds of the derivatives of the free energy at infinite volume. We show that it is possible to interchange the thermodynamical limit and the operation of taking the derivatives, and that the Taylor series, which exists, has a radius of convergence equal to 0. These statements are the consequence of Lemmas 2.6 and 2.7. Lemma 2.6. If β is sufficiently large, and ε > 0 sufficiently small, then for any k ∈ N there exists Mk = Mk (β) < ∞, such that for all t ∈ (µ∗ − ε, µ∗ ] and for all finite , 2 (k) [f ] t ≤ Mk .
Proof. For sufficiently large contours, ω( 2 ) is analytic and τ1 (β, θ )-stable on a disc of radius θ −1 R2 (V ( 2 )). From the Cauchy formula k 2 ≤ k! C k | 2 | d−1 [u ( 2 )](k) e−βκ| | , t 15 for some constants C15 and κ > 0. Therefore, for sufficiently large contours, k 2 [u ( 2 )](k) ≤ k! C k | 2 | d−1 e−βκ| | ≡ | |βMk < ∞ . t 15 2 ⊂
2 ⊂
(k) This implies the existence of Mk such that [f 2 ] t ≤ Mk . Lemma 2.7. (k)
(k)
2 ] µ∗ = lim∗ [f ] t . lim [f (L)
L→∞
t↑µ
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Proof. We compute the first derivative at the origin. Let η > 0, f (µ∗ ) − f (µ∗ − η) η 2 2 f (L) (µ∗ ) − f (L) (µ∗ − η) = lim L→∞ η
A(η) : =
(1)
= lim
L→∞
= lim
L→∞
2 [f (L) ] µ∗ η +
(1)
2 [f (L) ] µ∗
(2) 1 2 2 2! [f (L) ]µ∗ −xL (η) η
η 1 2 (2) + [f (L) ]µ∗ −xL (η) η . 2!
(2)
2 By Lemma 2.6, |[f (L) ]µ∗ −xL (η) | ≤ M2 . Therefore {A(η)}η is a Cauchy sequence. Hence the following limits exist, (1)
[f ] µ∗ = lim η↓0
f (µ∗ ) − f (µ∗ − η) (1) (1) 2 = lim∗ [f ] t = lim [f (L) ] µ∗ . t↑µ L→∞ η
The proof is the same for the derivatives of any order.
References 1. Fisher, M.E.: The Theory of Condensation and the Critical Point. Physics 3, 255–283 (1967) 2. Isakov, S.N.: Nonanalytic Features of the First Order Phase Transition in the Ising Model. Commun. Math. Phys. 95, 427–443 (1984) 3. Isakov, S.N.: Phase Diagrams and Singularity at the Point of a Phase Transition of the First Kind in Lattice Gas Models. Teor. Mat. Fiz. 71, 426–440 (1987) 4. Kunz, H., Souillard, B.: Essential Singularity and Asymptotic Behavior of Cluster Size Distribution in Percolation Problems. J. Stat. Phys. 19, 77–106 (1978) 5. Pfister, C.-E.: Large Deviations and Phase Separation in the Two Dimensional Ising Model. Helv. Phys. Acta 64, 953–1054 (1991) 6. Sinai, Ya.G.: Theory of Phase Transitions: Rigorous Results. Oxford: Pergamon Press, 1982 7. Zahradnik, M.: An Alternate Version of Pirogov-Sinai Theory. Commun. Math. Phys. 93, 559–581 (1984) Communicated by H. Spohn
Commun. Math. Phys. 245, 105–121 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1004-4
Communications in
Mathematical Physics
Uniqueness and Inviscid Limits of Solutions for the Complex Ginzburg-Landau Equation in a Two-Dimensional Domain Takayoshi Ogawa1 , Tomomi Yokota2 1
Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan. E-mail:
[email protected] 2 Department of Mathematics, Tokyo University of Science, Tokyo 162-0827, Japan. E-mail:
[email protected] Received: 30 December 2002 / Accepted: 18 September 2003 Published online: 12 December 2003 – © Springer-Verlag 2003
1. Introduction and Results The complex Ginzburg-Landau equation is, as is well known, an important model equation appearing in the theory of superconductivity, and it also describes spatial pattern formation or the onset of instabilities in non-equilibrium fluid dynamical systems (see Cross-Hohenberg [5]). Successful studies on the equation are increasing recently in both theoretical physics and mathematics. On the other hand, the equation meets the nonlinear Schr¨odinger equation in its special cases and from this fact, the inviscid limiting problem is attracting a great deal of attention. In this paper we discuss the uniqueness and inviscid limit of weak solutions to the initial-boundary value problem for the complex Ginzburg-Landau equation:
∂t u − (λ + iα)u + (κ + iβ)|u|p−1 u − γ u = 0, u(t, x) = 0, t > 0, x ∈ ∂, u(0, x) = u (x), t = 0, x ∈ , 0
t > 0,
x ∈ , (1.1)
where is a domain in the two-dimensional Euclidean space R2 (not necessary having compact smooth boundaries) and p > 1, λ ≥ 0, κ, α, β and γ are real constants. Though it is usually assumed that λ > 0, we consider (1.1) including the case where λ = 0. When p = 3 and ⊂ R2 is a bounded domain with smooth boundary, the initial-boundary value problem (1.1) with λ > 0 and κ > 0 was studied by Temam [20]. The case on the n-dimensional torus Tn = (R/2πZ)n with periodic boundary condition is also considered by Bartuccelli-Constantin-Doering-Gibbon-Gisself¨alt [2] and Levermore-Oliver [11]. Concerning the Cauchy problem with H 1 initial data, Ginibre-Velo [7, 8] established the time global well-posedness for (1.1) with λ > 0 and κ > 0 in the whole space Rn with almost no restriction on the coefficients under the condition that p < ∞ for n = 1, 2 and p ≤ (n+2)/(n−2) for n ≥ 3. They also proved the uniqueness
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of the weak solution for L2 initial data under the condition that the coefficient κ + iβ of the nonlinear term satisfies √ 2 p |β| ≤ κ. (1.2) p−1 This uniqueness result also holds for the case of the initial-boundary value problem (1.1) in general domains in Rn as shown by Okazawa-Yokota [17–19] by using monotonicity methods (for the simplest case where λ = β = γ = 0 and α = κ = 1, see Lions [12]). The condition (1.2) assures that the nonlinear term F (u) := (κ + iβ)|u|p−1 u maintains the monotone structure: Re (F (u1 ) − F (u2 ), u1 − u2 ) ≥ 0,
(1.3)
because the argument of (F (u1 ) − F (u2 ), u1 − u2 ) is in [−π/2, π/2] so that the lefthand side of (1.3) is nonnegative (cf. [18, Lemma 3.1]). In [17 and 19], not only the uniqueness has been proved, but also the smoothing effect on the solutions to (1.1) with λ > 0. However, without restriction (1.2), there is essentially no result on the uniqueness of weak solutions to the initial-boundary value problem (1.1) in general domains in R2 . The first purpose of this paper is to prove the uniqueness of weak solutions to (1.1) in a general domain ⊂ R2 without assuming (1.2), provided p ≤ 3 which includes the physically important situation under n = 2. The uniqueness problem for the weak solution is not quite well understood for the initial-boundary value problem (1.1) compared to the Cauchy problem (cf. [6, 17, 22]). We give two kinds of uniqueness theorem for different classes of weak solutions. One is for a class with L2 initial data (Theorem 1.1) and the other is for a class with H 1 initial data (Theorem 1.2). The first statement covers the second one except for the case λ = 0. This second theorem generalize the known result for the nonlinear Schr¨odinger equation [14]. The second purpose here is to discuss the inviscid limit for the initial-boundary value problem (1.1) in a general domain ⊂ R2 . By letting λ = 0 in (1.1), we have the initial-boundary value problem for the nonlinear Schr¨odinger equation: p−1 v + iγ v, t > 0, x ∈ , i∂t v + αv = (β − iκ)|v| (1.4) v(t, x) = 0, t > 0, x ∈ ∂, v(0, x) = u (x), t = 0, x ∈ . 0 In addition, we consider the simpler case κ = 0: p−1 w + iγ w, t > 0, i∂t w + αw = β|w| w(t, x) = 0, t > 0, x ∈ ∂, w(0, x) = u (x), t = 0, x ∈ . 0
x ∈ , (1.5)
Analogous in fluid mechanics, the inviscid limiting problem is to show how the solution of (1.1) converges to the one of (1.4) or (1.5) as λ → 0 (or both λ and κ → 0). Due to the physical interest, the inviscid limit of the complex Ginzburg-Landau equation has been studied by several authors. When = Rn , Wu [24] established the convergence rate for the difference of the solutions to (1.1) and (1.5) as λ → 0 and κ → 0 under condition (1.2). Bechouche-J¨ungel [3] dealt with the same problem in the torus Tn with the periodic boundary condition or in the whole space Rn (for some stability result see
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Weinstein-Xin [23]). In [3], the inviscid limit was proved rigorously with H 1 initial data. Namely the solution u of (1.1) converges to the solution w of (1.5) as u − wL∞ (0,T ;Lq (Rn )) → 0, loc where q < 2n/(n − 2) (see [3, Theorem 1.1]), because their proof is based on compactness methods. To obtain the convergence in L∞ (0, T ; Lq (Rn )) and its rate, in [3, Theorem 1.2], it is assumed that u0 ∈ H m () with m ∈ N and m > n/2 so that the Sobolev embedding H m () → L∞ () is effectively used. We should emphasize that most of those results were restricted only for the Cauchy problem in the whole space or the periodic initial-boundary value problem on the torus. In this paper we deal with a much more general case of the coefficient (namely, without restriction like (1.2)) and we show the strong inviscid limit in the global space L∞ (0, T ; Lq ()), where 2 ≤ q < ∞, for the initial-boundary value problem (1.1) in arbitrary domains in R2 (see Theorem 1.3 below). Here our theorem does not require that u0 ∈ L∞ (); nevertheless, we can estimate directly the difference of the solutions to (1.1) and (1.4) or (1.5). We note that the case we consider here includes a typical situation appearing as the important model of the theory of superconductivity. Before stating our results, we define a weak solution to (1.1). Given u0 ∈ L2 (), we say that u is a weak solution to (1.1) if u ∈ C([0, T ); L2 ()) ∩ L2 (0, T ; H01 ()) ∩ Lp+1 (0, T ; Lp+1 ()) and u satisfies (1.1) in the sense of distributions. If we assume that λ > 0, then the weak solution u would be a strong solution to (1.1) under a suitable condition in the sense that ∂t u, u ∈ L2 (0, T ; L2 ()) (parabolic regularity). Analogously for (1.4) (similarly for (1.5)), we say that for u0 ∈ H01 (), u is a weak solution to (1.4) if u ∈ C([0, T ); H −1 ()) ∩ L∞ (0, T ; H01 ()) and u satisfies (1.4). Let u0 ∈ H01 () and 1 < p ≤ 3. If κ ≥ 0, then the global existence of weak solutions to (1.1) has essentially been proved by [20 and 7] without any restriction on the coefficients when λ > 0. For the case where λ = 0 see e.g., [12, 14 and 4]; note that u0 2 is assumed to be sufficiently small only if αβ < 0 and p = 3. On the other hand, if κ < 0, then the local existence of weak solutions to (1.1) can be proved. Now we state our results on the uniqueness of weak solutions to (1.1). We start with the result for L2 initial data, excluding the case where λ = 0, because we can prove the uniqueness of weak solutions to (1.1) under a weaker regularity assumption when λ > 0 and the proof is also useful when λ = 0. We establish the uniqueness part of the following theorem; note that the existence part was proved by [7 and 19]. Theorem 1.1. Let ⊂ R2 be a (not necessary bounded) domain and λ > 0, κ ≥ 0, α, β, γ ∈ R and 1 < p ≤ 3. Then for any u0 ∈ L2 () there exists a unique global weak solution u to (1.1) such that u ∈ C([0, ∞); L2 ()) ∩ L2 (0, T ; H01 ()) ∩ Lp+1 (0, T ; Lp+1 ()) for all T > 0. Remark. Theorem 1.1 can be generalized to the n-dimensional case. Namely, we can prove the uniqueness of weak solutions to the initial-boundary value problem for the generalized complex Ginzburg-Landau equation which is the equation in (1.1) with replaced with the n-Laplacian. We shall discuss the details in Sect. 6. Next we state the result on the uniqueness of weak solutions to (1.1) for H 1 initial data, including the case where λ = 0 or κ < 0. We establish the uniqueness part of the following theorem; note that the existence part was essentially proved by [4, 7, 12, 14 and 20].
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Theorem 1.2. Let be as in Theorem 1.1 and λ ≥ 0, κ, α, β, γ ∈ R and 1 < p ≤ 3. (i-a) If κ ≥ 0 and αβ ≥ 0(α = 0), then for any u0 ∈ H01 () there exists a unique global weak solution u ∈ C([0, ∞); H01 ()) to (1.1). (i-b) If κ ≥ 0 and αβ < 0, then for any u0 ∈ H01 () there exist T ∗ > 0 and a unique local weak solution u ∈ C([0, T ∗ ); H01 ()) to (1.1). In addition, if λ > 0 or if 1 < p < 3 or if u0 2 is sufficiently small, then T ∗ = ∞. (ii) If κ < 0, then for any u0 ∈ H01 () there exist T ∗ > 0 and a unique local weak solution u ∈ C([0, T ∗ ); H01 ()) to (1.1). Remark. We should note that the above results do not essentially require any restriction on the coefficients except λ ≥ 0, while the exponent of the nonlinear term is restricted to 1 < p ≤ 3. The physically important case where p = 3 and κ ≥ 0 is still covered here. The uniqueness for (1.1) with β = γ = 0 and α = κ = 1, that is, the special case of Theorem 1.2 (i-a) was proved by Lions [12]. The smallness assumption for the case p = 3, αβ < 0 appears as a natural restriction since if u0 2 is large enough, the weak solution may blow up in a finite time when λ = κ = γ = 0 (cf. [9, 10 and 16]. Once the uniqueness is established, we turn our attention to the inviscid limit of the complex Ginzburg-Landau equation. The following theorem shows that the inviscid limit also holds for the general domain in R2 . Theorem 1.3. Let ⊂ R2 be as in Theorem 1.1 and α, β, γ ∈ R. Assume that 1 < p ≤ 3. Let u, v and w be the unique local or global weak solutions to (1.1), (1.4) and (1.5) with the same initial data u0 ∈ H01 (), respectively. (i-a) If κ ≥ 0 and αβ ≥ 0(α = 0), then for all T > 0 and for all 2 ≤ q < ∞, sup u(t) − v(t)q → 0 as λ → 0,
(1.6)
sup u(t) − w(t)q → 0 as λ → 0, κ → 0.
(1.7)
t∈[0,T ] t∈[0,T ]
(i-b) If κ ≥ 0 and αβ < 0(u0 2 is sufficiently small only if p = 3), and κ/λ(p−1)/2 is bounded as λ → 0, κ → 0, then (1.7) holds for all T > 0 and for all 2 ≤ q < ∞. (ii) If κ < 0 (u0 2 is sufficiently small when p = 3), and |κ|/λ(p+1)/4 is bounded as λ → 0, κ → 0, then (1.7) holds for some T > 0 and for all 2 ≤ q < ∞. Remark. For a higher dimensional case than n ≥ 2, it is also possible to show the uniqueness and inviscid limit for the complex Ginzburg-Landau equation provided the weak or strong solution belongs to L∞ (0, T ; H n/2 () ∩ H01 ()). This is the critical space where the solution has uniqueness in a general context (see e.g., [15]). For example, when n = 3, it is possible to show the local well-posedness for the solution to (1.1) in C([0, T ]; H 2 () ∩ H01 ()) according to the theory of evolution equations. Since the proof is very much similar to the above, we do not go further in this direction. We should remark that the solution does not necessarily belong to L∞ () when λ = 0 so that the limiting case q = ∞ is eliminated in Theorem 1.3. One of the difficulties to show the result is that in general, f ∈ W01,n () does not yield f ∈ L∞ (). Relating to this failure of the embedding, there is a limiting Sobolev
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type inequality initially established by Trudinger [21] and Moser [13], which is our key tool for proving the uniqueness and inviscid result: |f (x)| n/(n−1) dx ≤ Cn ||, exp νn ∇f n for f ∈ W01,n (), where νn is the best constant for the isoperimetric inequality. It is also known that the generalized version of the above inequality also holds in the whole space. We generalize this inequality in the following section. From this inequality, one can estimate an asymptotically sharp upper bound of the best possible constant of the Gagliardo-Nirenberg inequality in W01,n (). The uniqueness and inviscid limit follow from this sharp upper bound and some argument found in [14 and 15]. In this paper we use the following notation. We denote by ·p the Lp norm. W k,p () is the Sobolev space with a k th order weak derivative belonging to Lp () with norm 1,p ·W k,p . In particular, H 1 () := W 1,2 (). W0 () and H01 () stand for the W 1,p () and H 1 () completions of C0∞ (). Also, ωn stands for the surface volume of the n − 1 dimensional unit sphere. 2. The Gagliardo-Nirenberg Inequality The basic tool to show the uniqueness is the Gagliardo-Nirenberg inequality (cf. [14]). Since we consider the higher dimensional case in Sect. 6, we prepare the inequality for a general domain ⊂ Rn (n ∈ N). The following inequality is the limiting form of the critical Sobolev inequality initially driven by Trudinger [21] and it provides the crucial inequality. Lemma 2.1. Let ⊂ Rn (n ∈ N) be a (not necessary bounded) domain and f ∈ 1/n−1 W01,n (). Then for any 0 < ν < nωn there exists a constant C0 > 0 which is independent of such that n−2
ν k |f (x)| nk/(n−1) |f (x)| n/(n−1) f nn exp ν − dx ≤ C0 . ∇f n k! ∇f n ∇f nn k=0
(2.1) Here ωn is the surface volume of the n − 1 dimensional unit sphere. The above type of inequality was proved by Trudinger [21] and its sharp form by Moser [13] in the bounded domain . In this case the inequality appears as |f (x)| n/(n−1) dx ≤ C0 ||, exp νn (2.2) ∇f n 1/(n−1)
and the sharp constant is given by νn = nωn which is the sharp constant of the isoperimetric inequality. In the unbounded domain or whole space case, Ogawa [14] extended (2.2) to an arbitrary domain in R2 and it is generalized to the higher dimensional cases [15]. The sharp form of the inequality was obtained by Adachi-Tanaka [1] for = Rn . For the case f ∈ W01,n (), the inequality (2.1) follows immediately from
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the inequality on Rn and density argument. Note that for the general domain, the sharp 1/(n−1) exponent nωn can not be attained (see [1]). The following lemma is used in Sects. 4 and 5 for the case n = 2 and in Sect. 6 for the case n ≥ 3. Lemma 2.2. Let ⊂ Rn be a (not necessary bounded) domain and f ∈ W01,n (). Then 1/n−1 , there exists a constant C0 > 0 which is independent for any q ≥ n and ν < nωn of q such that
√ 1 f q ≤ C0 a(q) q q
(n − 1)q neν
n−1 n
n
1− qn
f nq ∇f n
,
(2.3)
q >> n,
where a(q) is a bounded L1 function of q ∈ [n, ∞) depending only on n and ν. As proved in [14], for the n = 2 case, we have for f ∈ H01 (), f q ≤ (4π)(2−q)/2q
q 1/2 2
2/q
1−(2/q)
f 2 ∇f 2
(2.4)
which is also used in [24]. The asymptotic upper estimate for the constant of the Gagliardo-Nirenberg inequality in the above Lemma 2.2 is better than this estimate, although this is not quite required for the proof of the main results. Proof of Lemma 2.2. It follows from the Trudinger-Moser inequality that ∞ k=n−1
k 1 f nn |f (x)| n/(n−1) dx ≤ C0 . ν k! ∇f n ∇f nn
Namely we have for some {ak } ∈ l 1 which depends only on ν such that
νk nk
k!∇f nn−1
kn
|f (x)| n−1 dx ≤ C0 ak
Noticing the Stirling formula for k! f
kn n−1
≤
f nn , ∇f nn
k = n − 1, n, . . . .
√ 2πke−k k k as k → ∞,
n−1 √ n−1 C0 ak 2πkk k nk 1− n−1 f n k ∇f n k , k (eν)
k >> n − 1.
(2.5)
By interpolating (2.5) between each k, we have for q > n, f q ≤
√ n−1 1 n 1− n C0 (n − 1)aq 2π(n − 1)q q (n − 1)q n f nq ∇f n q , √ neν n
This proves the desired inequality (2.3).
q >> n.
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3. Energy Bounds The following proposition plays an essential role in the proof of Theorem 1.3. Proposition 3.1. Let ⊂ R2 be a (not necessary bounded) domain and u0 ∈ H01 (). Assume that 0 < λ ≤ 1, |κ| ≤ 1 and 1 < p ≤ 3. Then the local or global (strong) solution u to (1.1) is bounded in H01 () uniformly in λ, κ in the following sense: (i-a) If κ ≥ 0 and αβ ≥ 0(α = 0), then for all T > 0, sup u(t)H 1 () ≤ E1 .
t∈[0,T ]
0
(i-b) If κ ≥ 0 and αβ < 0(u0 2 is sufficiently small only if p = 3), and κ/λ(p−1)/2 is bounded as λ → 0, κ → 0, then for all T > 0, sup u(t)H 1 () ≤ E2 .
t∈[0,T ]
0
(ii) If κ < 0 (u0 2 is sufficiently small when p = 3), and |κ|/λ(p+1)/4 is bounded as λ → 0, κ → 0, then for some T > 0, sup u(t)H 1 () ≤ E3 .
t∈[0,T ]
0
Here E1 , E2 and E3 are positive constants depending only on p, α, β, γ , T and u0 H 1 () 0 but not on λ and κ. The case (i-a) was proved in [7 and 19] while the cases (i-b) and (ii) can be proved the same way as in [3, Lemma 3.4] for the Cauchy problem in the whole space case and the initial-boundary value problem for a periodic boundary condition. Proof of Proposition 3.1. Let u be a solution to (1.1) with u0 ∈ H01 (); note that u is a strong solution such that ∂t u, u ∈ L2 (0, T ; L2 ()), since it is assumed that λ > 0. Taking the real part of the inner product of the equation in (1.1) with u, −u and |u|p−1 u, we have 1 d p+1 u22 + λ∇u22 + κup+1 = γ u22 , 2 dt
(3.1)
1 d ∇u22 + λu22 + κ Re (−u, |u|p−1 u) + β Im (−u, |u|p−1 u) = γ ∇u22 , 2 dt (3.2) 1 d p+1 2p up+1 + κu2p + λ Re (−u, |u|p−1 u) − α Im (−u, |u|p−1 u) p + 1 dt p+1
= γ up+1 . Adding (3.1), (3.2) and (3.3) multiplied by β/α (α = 0) gives β β d 1 p+1 p+1 2 2 uH 1 + up+1 = γ uH 1 + up+1 + F (u), dt 2 α(p + 1) α
(3.3)
(3.4)
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where we set βκ p+1 2p F (u) := −λ∇u22 − κup+1 − λu22 − u2p α βλ p−1 − κ+ u). Re (−u, |u| α
(3.5)
Noting that Re (−u, |u|p−1 u) ≥ 0, we proceed with the proof as follows. Case (i-a). κ ≥ 0, αβ ≥ 0 (α = 0). Since F (u) ≤ 0, it follows from (3.4) that 1 β β p+1 p+1 2 (p+1)|γ |t 1 2 u(t)H 1 + u(t)p+1 ≤ e u0 H 1 + u0 p+1 . 2 α(p + 1) 2 α(p + 1) Therefore we can obtain the assertion. Case (i-b). κ ≥ 0, αβ < 0(u0 2 is sufficiently small only if p = 3). In this case we see from (3.1) and the definition of F (see (3.5)) that u(t)2 ≤ eγ t u0 2 , |β|κ |β|λ 2p p u2p + u2 u2p . F (u) ≤ −λu22 + |α| |α|
(3.6) (3.7)
Using the Gagliardo-Nirenberg inequality (2.4) with q = 2p and the moment inequality ∇u22 ≤ u2 u2 , we have 2p
2(p−1)
u2p ≤ cp u22 ∇u2
p+1
≤ cp u2
p−1
u2
.
Applying this inequality to the right-hand side of (3.7) yields F (u) ≤ −λu22 +
cp |β|κ cp |β|λ p+1 p−1 (p+1)/2 (p+1)/2 u2 u2 + u2 u2 . |α| |α| (3.8)
Hence, if 1 < p < 3, then the Young inequality and (3.6) give 2/(3−p) cp |β| κ F (u) ≤ k1 + λ (eγ t u0 2 )2(p+1)/(3−p) , |α| λ(p−1)/2
(3.9)
where k1 is a positive constant depending only on α, β and p. Therefore, applying (3.9) to (3.4) and using the Gagliardo-Nirenberg inequality (2.3), we can obtain the assertion when 1 < p < 3. If p = 3, then we assume that u0 2 is sufficiently small and so is u(t)2 by virtue of (3.6). Consequently, we see from (3.8) that F (u) ≤ 0 and hence the assertion follows. Case (ii). κ < 0(u0 2 is sufficiently small when p = 3). First suppose that p = 3. Using the Gagliardo-Nirenberg inequality, we see by (3.1) that 1 d p−1 u22 + λ∇u22 − c|κ|·u22 ∇u2 ≤ γ u22 , 2 dt and hence the Young inequality gives d 4/(3−p) + (p − 1), u22 ≤ (3 − p)A2/(3−p) u2 dt
(3.10)
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where A := |γ | + cp |κ|/λ(p−1)/2 . Solving this differential inequality, we obtain u(t)22 ≤ u0 22 +
p−1 − 3−p
p−1 3−p
3−p 2
3−p 2
− p−1 1 A
3−p
− 3−p p−1 2 − (p − 1)A 3−p t
1 , A
(3.11)
for all t ∈ [0, T ∗ ), where −1 2 p − 1 3−p 1 p−1 3−p T ∗ := (p − 1)A 3−p u0 22 + ( . ) 2 3−p A Now suppose that αβ > 0. Then it follows from the definition (3.5) of F that p+1
F (u) ≤ −λ∇u22 + |κ|·up+1 − λu22 +
β|κ| 2p p u2p + |κ|·u2 u2p . α
Using the Gagliardo-Nirenberg inequality as above, we obtain p−1
F (u) ≤ −λ∇u22 + c|κ|·u22 ∇u2 cp β|κ| p+1 p−1 (p+1)/2 (p+1)/2 − λu22 + u2 . u2 u2 + cp |κ|·u2 α Hence the Young inequality yields F (u) ≤ k2
|κ|
2/(3−p)
4/(3−p)
u2 λ(p−1)/2 2/(3−p) 4/(3−p) |κ| |κ| 2(p+1)/(3−p) u2 + k3 + , λ(p−1)/2 λ(p+1)/4
(3.12)
where k2 and k3 are positive constants depending only on α, β and p. Therefore, applying (3.11) to (3.12), we can obtain the assertion by integrating (3.4) on [0, T ] for some T < T ∗ ; note that λ(p−1)/2 ≥ λ(p+1)/4 if λ ≤ 1. Next suppose that αβ < 0. Then we have |β|λ p+1 p F (u) ≤ −λ∇u22 + |κ|·up+1 − λu22 + |κ| + u2 u2p . |α| Using the Gagliardo-Nirenberg inequality and Young’s inequality gives 2/(3−p) |κ| 4/(3−p) u2 F (u) ≤ k2 λ(p−1)/2 4/(3−p) |κ| 2(p+1)/(3−p) +k4 + λ u2 , λ(p+1)/4 where k4 is a positive constant depending only on α, β and p. Therefore we can obtain the assertion in the same way as above.
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It remains to prove the case where κ < 0 and p = 3. In this case we can control the L2 norm of the solution as follows. In view of (3.10) with p = 3 we have 1 d u22 + (λ − c|κ|·u22 )∇u22 ≤ γ u22 . 2 dt
(3.13)
Let |κ| ≤ Kλ for some constant K > 0 and let u0 22 ≤ (2cK)−1 . Then by continuity we see that T0 := sup{t; u(s)22 < (cK)−1 for all s ∈ [0, t]} > 0. Therefore the second term on the left-hand side of (3.13) is nonnegative for t ∈ [0, T0 ] and hence u(t)22 ≤ e2γ t u0 22 ≤ e2γ t (2cK)−1
for t ∈ [0, T0 ].
(3.14)
2 2 Now suppose that γ > 0. Then it follow from (3.14) that if T0 < log 2γ , then u(T0 )2 ≤ e2γ T0 (2cK)−1 < (cK)−1 which contradicts the definition of T0 . Thus we see that 2 T0 ≥ log 2γ so that T0 is independent of λ and κ. Therefore, if u0 2 is sufficiently small, then so is u(t)2 by virtue of (3.14). This is also true when γ ≤ 0. Consequently, estimating F (u) in analogous way, we can obtain the desired assertion.
4. Proof for the Uniqueness Proof of Theorem 1.1. Since the existence part is already known as mentioned in Sect. 1, we prove the uniqueness part. Let u1 and u2 be two weak solutions to (1.1) with the same initial data u0 ∈ L2 () such that u1 , u2 ∈ C([0, ∞); L2 ())∩L2 (0, T ; H01 ())∩ Lp+1 (0, T ; Lp+1 ()). Put w ≡ u1 − u2 . Then w satisfies ∂t w − (λ + iα)w − γ w = −(κ + iβ)(|u1 |p−1 u1 − |u2 |p−1 u2 )
in H −1 ,
and hence H −1 ∂t w − (λ + iα)w − γ w, wH01
= −(κ + iβ)H −1 |u1 |p−1 u1 − |u2 |p−1 u2 , wH 1 . 0
Taking the real parts in the both sides gives 1 d 2 2 2 max{|u1 (t)|p−1 , |u2 (t)|p−1 }|w(t)|2 dx, w(t)2 − γ w(t)2 + λ∇w2 ≤ C1 2 dt where C1 := p κ 2 + β 2 . Setting z(t, x) := max{|u1 (t, x)|p−1 , |u2 (t, x)|p−1 }, we have 1 d z(t)|w(t)|2 dx. (4.1) w(t)22 − γ w(t)22 ≤ C1 2 dt Applying the H¨older inequality to the right-hand side of (4.1) yields ε 1 d 2(1−ε) z(t)1/ε |w(t)|2 dx w(t)2 . w(t)22 − γ w(t)22 ≤ C1 2 dt
(4.2)
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Hence it follows that ε d −2εγ t 2ε −2εγ t e w(t)2 ≤ 2εC1 · e z(t)1/ε |w(t)|2 dx . dt
(4.3)
Integrating (4.3) over [0, t] and using that w(0)2 = 0, we obtain t ε 2εγ (t−s) 1/ε 2 w(t)2ε ≤2εC e z(s) |w(s)| dx ds 1 2 0 t ε/2 ≤2εC1 · e2εγ t z(s)2/ε dx w(s)2ε 4 ds. 0
(4.4)
The above procedure can be justified by taking the test function as the regularized ρn ∗ w(t) by the mollifier or the Yosida approximation of the Laplacian in . On the other hand, we note that 2(p−1) 2(p−1) 2/ε ε z dx ≤ |u1 | dx + |u2 | ε dx. (4.5)
Here it follows from Lemma 2.2 with n = 2 that p − 1 p−1 2(p−1) 2(p−1) −2 ε |uj | ε dx ≤ C0 uj 22 ∇uj 2 ε ε
(4.6)
for j = 1, 2 and any small ε > 0. Noting that C2 := supt∈[0,T ] uj (t)2 < ∞ by assumption, we see from (4.5) and (4.6) that z
2/ε
ε/2 dx
≤
ε/2 C0 C2ε
p−1 ε
p−1 2
p−1−ε
(∇u1 2
p−1−ε
+ ∇u2 2 1/2
).
(4.7)
1/2
Moreover, the Gagliardo-Nirenberg inequality (f 4 ≤ C3 f 2 ∇f 2 ) yields 2ε ε ε ε 2ε ε ε w(t)2ε 4 ≤ C3 w(t)2 ∇w(t)2 ≤ C2 C3 (∇u1 2 + ∇u2 2 ).
(4.8)
Applying (4.7) and (4.8) to the right-hand side of (4.4), we obtain p−1
ε/2
2ε 2εγ t 2 ε (3−p)/2 C w(t)2ε 2 ≤ 4(p − 1) 0 C1 C 2 · e t p−1 p−1 × u1 (s)H 1 () + u2 (s)H 1 () ds. 0
Hence, if 0 < ε < 1 and 1 < p ≤ 3, then t 1/ε 1/2 p−1 p−1 w(t)22 ≤ C0 C22 · e2γ t 8C1 u1 (s)H 1 () + u2 (s)H 1 () ds .
(4.9)
0
Since u1 , u2 ∈ L2 (0, T ; H01 ()) and p − 1 ≤ 2, we can take T1 > 0 such that T1 p−1 p−1 u1 (s)H 1 () + u2 (s)H 1 () ds < 1. 8C1 0
Therefore, letting ε → 0 in (4.9) for each t ∈ [0, T1 ], we see that w(t) = 0 for t ∈ [0, T1 ]. Repeating this procedure, we can conclude that w(t) = 0 for all t ∈ [0, T ]. This completes the proof of Theorem 1.1.
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Proof of Theorem 1.2. As in the proof of Theorem 1.1, we can obtain (4.1) even in the case where λ = 0 or κ < 0. Therefore the proof of Theorem 1.2 is parallel to that of Theorem 1.1; note that the regularity of the solution in Theorem 1.2 is stronger than that in Theorem 1.1. 5. Proof for the Inviscid Limit Proof of Theorem 1.3. Let u be the unique weak solution to (1.1) with u0 ∈ H01 (). Following the statement of the result, we divide the proof into three cases. Case (i-a). Let v be the unique weak solution to (1.4) with the same initial data u0 ∈ H01 (). Similarly to the above, we set w := u − v. Then w satisfies ∂t w − iαw − γ w = λu − (κ + iβ)(|u|p−1 u − |v|p−1 v)
in H −1 .
In an analogous way, multiplying the above equation by w and integrating its real part over , we have 1 d 2 2 max{|u|p−1 , |v|p−1 }|w|2 dx w2 − γ w2 ≤ −λ Re(∇u, ∇w) + C1 2 dt ≤ λ Re(∇u, ∇v) + C1
max{|u|p−1 , |v|p−1 }|w|2 dx,
where C1 = p κ 2 + β 2 again. Setting z := max{|u|p−1 , |v|p−1 }, we see by the same way as in (4.2) that ε 1 d 2(1−ε) z1/ε |w|2 dx w2 . w22 − γ w22 ≤ λ∇u2 ∇v2 + C1 2 dt Now let δ ∈ C 1 ([0, T ]), δ(0) = 0 and δ(t) > 0 for t ∈ (0, T ]. Denoting again by γ the positive part of γ , we have ε d −2εγ t w22 + δ(t) e dt 1 d 1 ≤ 2ε · e−2εγ t w22 − γ w22 + δ (t) − γ δ(t) (w22 + δ(t))ε−1 2 dt 2 ε 2(1−ε) ≤ 2ε · e−2εγ t C1 z1/ε |w|2 dx w2 1 + λ∇u2 ∇v2 + δ (t) − γ δ(t) (w22 + δ(t))ε−1 2 ε 1 −2εγ t 1/ε 2 ε−1 ε−1 C1 . ≤ 2ε · e z |w| dx + δ(t) λ∇u2 ∇v2 + δ (t)δ(t) 2 (5.1) Now we set δ(t) := λ(1 − e−t ) for small λ > 0. Integrating (5.1) over [0, t] gives (w(t)22 + λ(1 − e−t ))ε t ε/2 2εγ t C1 ≤ 2ε · e z(τ )2/ε dx w(τ )2ε 4 0 + λε (1 − e−τ )ε−1 ∇u(τ )2 ∇v(τ )2 dτ + e2εγ t λε (1 − e−t )ε .
(5.2)
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From Lemma 2.2 and Proposition 3.1 it follows that for any small ε > 0, z2/ε dx
ε/2
ε/2
≤ 2ε/2 C0
p−1 ε
(p−1)/2
p−1
(5.3)
E1
and 1/2
1/2
sup w(t)4 ≤ C3 sup w(t)2 ∇w(t)2
t∈[0,T ]
˜ ≤ 2C3 E1 ≤ E,
(5.4)
t∈[0,T ]
where E1 and E˜ are determined by the H 1 norm of the solutions and independent of λ. Hence, applying (5.3) and (5.4) to the right-hand side of (5.2), we see that (w(t)22 + λ(1 − e−t ))ε (p−1)/2 ε/2 p − 1 p−1 ≤ 2e2εγ t εtC1 2ε/2 C0 E1 E˜ 2ε + λε et E12 + λε ε p−1 ε/2 2εγ t (3−p)/2 ≤ 2e tC1 E1 (p − 1)(p−1)/2 2ε/2 C0 E˜ 2ε + (et E12 + 1)λε ε ε 1/2 ≤ 2e2εγ t tC(p, E1 ) 2C0 E˜ 2 + (et E12 + 1)λε for t ∈ [0, T ], (5.5) where we used ε(3−p)/2 ≤ 1 under the assumption 1 < p ≤ 3 with small ε and p−1 C(p, E1 ) := C1 E1 (p − 1)(p−1)/2 . Therefore we obtain ε 1/ε 1/2 , w(t)22 ≤ e2γ t 2tC(p, E1 ) 2C0 E˜ 2 + 2(et E12 + 1)λε and hence 1/ε 1/2 . lim sup sup w(t)22 ≤ 2e2γ T1 C0 E˜ 2 2T1 C(p, E1 ) λ→0
(5.6)
t∈[0,T1 ]
Here we take T1 > 0 as follows: 2T1 C(p, E1 ) < 1. Letting ε → 0 in (5.6), we see that lim sup w(t)22 = 0.
λ→0 t∈[0,T1 ]
Repeating this procedure yields that lim supt∈[0,T ] w(t)22 = 0. Therefore, the Gagliardo-Nirenberg inequality
λ→0
2/q
1−2/q
w(t)q ≤ C4 w(t)2 ∇w(t)2
yields the conclusion (1.6) for the case (i-a). Next we prove (1.7) for the case (i-a). Namely, we consider the case where v(t) is the solution to (1.5). Let v be the unique weak solution to (1.5) with the same initial data u0 ∈ H01 (). Similarly to above, we set w := u − v. Then w satisfies ∂t w − iαw − γ w = λu − κ|u|p−1 u − iβ(|u|p−1 u − |v|p−1 v)
in H −1 .
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In an analogous way, we see that 1 d w22 ≤ γ w22 + λ Re(∇u, ∇v) + κ Re(|u|p−1 u, v) 2 dt + C1
max{|u|p−1 , |v|p−1 }|w|2 dx,
where C1 = p κ 2 + β 2 again. Now we set δ(t) := δ(1 − e−t ) for some small δ > 0 which will be determined later. As in the proof of (5.2), we have (w(t)22 + δ(1 − e−t ))ε t ε/2 2εγ t C1 ≤ 2ε · e z(τ )2/ε dx w(τ )2ε 4 0
+δ
ε−1
(1 − e
−τ ε−1
)
p (λ∇u(τ )2 ∇v(τ )2 + κu(τ )p+1 v(τ )p+1 )
+ e2εγ t δ ε (1 − e−t )ε .
dτ (5.7)
By virtue of Lemma 2.2 and Proposition 3.1, we obtain (5.3), (5.4) and 2
p−1
sup u(t)p+1 ≤ C5 sup u(t)2p+1 ∇u(t)2p+1 ≤ C5 E1 ,
t∈[0,T ]
(5.8)
t∈[0,T ]
where E1 is determined by the H 1 norm of the solutions and independent of λ and κ. Thus applying (5.3), (5.4) and (5.8) to the right-hand side of (5.7), we see that for t ∈ [0, T ], (w(t)22 + δ(1 − e−t ))ε 1/2 ˜ 2 ε 2εγ t t 2 ε−1 t p+1 p+1 ε−1 ε tC(p, E1 ) 2C0 E ≤ 2e + e E1 δ λ + e C 5 E 1 δ κ + δ , (5.9) where we used ε(3−p)/2 ≤ 1 under the assumption 1 < p ≤√ 3 with small ε and p−1 C(p, E1 ) := C1 E1 (p − 1)(p−1)/2 . Therefore, choosing δ := λ2 + κ 2 , we obtain ε 1/2 2 2γ t 2tC(p, E1 ) 2C0 E˜ 2 + 2et E12 λε w(t)2 ≤ e 1/ε p+1
+ 2et C5 and hence
p+1 ε
E1
κ + 2(λ2 + κ 2 )ε/2
,
1/ε 1/2 , lim sup sup w(t)22 ≤ 2e2γ T1 C0 E˜ 2 2T1 C(p, E1 ) λ,κ→0 t∈[0,T1 ]
where T1 > 0 is taken as follows: 2T1 C(p, E1 ) < 1. Therefore we can obtain the conclusion (1.7) for the case (i-a) in the same way as in the proof of (1.6). Case (i-b) and Case (ii). The proofs are almost the same as in the former case except the energy bound E1 is exchanged into E2 and E3 given by Proposition 3.1.
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6. Generalized Complex Ginzburg-Landau Equations In this section we generalize the uniqueness theorem for (1.1) (Theorem 1.1) to the initial-boundary value problem for the generalized complex Ginzburg-Landau equation with the n-Laplacian: n−2 p−1 u − γ u = 0, t > 0, x ∈ , ∂t u − (λ + iα) div (|∇u| ∇u) + (κ + iβ)|u| u(t, x) = 0, t > 0, x ∈ ∂, u(0, x) = u (x), t = 0, x ∈ , 0 (6.1) where ⊂ Rn (n ≥ 2) is a bounded domain and p > 1, λ > 0, κ ≥ 0, α, β and γ are real constants. This problem is the special case of the p-Laplace case [19] in which the existence of global weak solutions is established for all p > 1 under the following condition: √ |α| 2 n−1 ≤ . (6.2) λ n−2 However, in [19], the uniqueness of solutions is not obtained except the restricted condition (1.2). The following theorem gives new information on the uniqueness in the case n where 1 < p ≤ 1 + n−1 . Theorem 6.1. Let ⊂ Rn (n ≥ 2) be a bounded domain and λ > 0, κ ≥ 0, α, β, γ ∈ R n and 1 < p ≤ 1 + n−1 . Assume that condition (6.2) is satisfied. Then for any u0 ∈ L2 () there exists a unique global weak solution u to (6.1) such that u ∈ C([0, ∞); L2 ()) ∩ Ln (0, T ; W01,n ()) ∩ Lp+1 (0, T ; Lp+1 ()) for all T > 0. Proof of Theorem 6.1. Since the two-dimensional case is proved above, we prove the case where n ≥ 3. Let u1 and u2 be two weak solutions to (6.1) with the same initial data u0 ∈ L2 () such that u1 , u2 ∈ C([0, ∞); L2 ()) ∩ Ln (0, T ; W01,n ()) ∩ Lp+1 (0, T ; Lp+1 ()). Put w := u1 − u2 . Then w satisfies ∂t w − (λ + iα) div (|∇u1 |n−2 ∇u1 − |∇u2 |n−2 ∇u2 ) − γ w = −(κ + iβ)(|u1 |p−1 u1 − |u2 |p−1 u2 ). Multiplying both sides by w and then integrating its real part over , we have 1 d max{|u1 |p−1 , |u2 |p−1 }|w|2 dx, (6.3) w22 + Re (λ + iα)Z − γ w22 ≤ C1 2 dt where Z := Ln |∇u1 |n−2 ∇u1 − |∇u2 |n−2 ∇u2 , ∇wLn and C1 = p κ 2 + β 2 . It follows from [19, Lemma 2.1] that n−2 |Im Z| ≤ √ Re Z. 2 n−1
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Hence, condition (6.2) implies that the second term on the left-hand side of (6.3) is nonnegative: √ 2 n−1 λ − |α| |Im Z| ≥ 0. Re (λ + iα)Z ≥ n−2 So we see from (6.3) that 1 d w(t)22 − γ w(t)22 ≤ C1 2 dt
z(t)|w(t)|2 dx,
where z := max{|u1 |p−1 , |u2 |p−1 }. As in the proof of (4.4), we have w(t)2ε 2
≤ 2εC1 · e
2εγ t
t 0
z(s)2/ε dx
ε/2
w(s)2ε 4 ds.
(6.4)
Using Lemma 2.2, we obtain z
2/ε
ε/2 dx
≤
ε/2 C0
2(n − 1)(p − 1) nνε
(n−1)(p−1) n
p−1 p−1 u1 W 1,n () + u2 W 1,n () .
Applying this inequality to (6.4) gives w(t)2ε 2
≤ε
1− (n−1)(p−1) n
ε/2 C0 C62ε e2εγ t C7
t 0
p−1+2ε p−1+2ε u1 (s)W 1,n () + u2 (s)W 1,n () ds,
where C6 > 0 is the Sobolev imbedding constant for W 1,n () → L4 () and C7 :=
(n−1)(p−1) n n . Since it is assumed that 1 < p ≤ 1 + n−1 , it follows that for 4C1 2(n−1)(p−1) nν any small ε > 0, t 1/ε 1/2 p−1+2ε p−1+2ε w(t)22 ≤ C0 C62 e2γ t C7 u1 (s)W 1,n () + u2 (s)W 1,n () ds .
(6.5)
0
Now let 0 < δ < 1. Noting that u1 , u2 ∈ Ln (0, T ; W01,n ()) and p − 1 + 2ε ≤ n if ε ≤ n(n−2) 2(n−1) (n ≥ 3), we can take Tδ > 0 such that
Tδ
C2 0
p−1+2ε p−1+2ε u1 W 1,n () + u2 W 1,n () ds ≤ δ < 1.
Therefore, letting ε → 0 in (6.5) for each t ∈ [0, Tδ ], we see that w(t) = 0 for t ∈ [0, Tδ ]. Repeating this procedure, we can conclude that w(t) = 0 for all t ∈ [0, T ]. This completes the proof of Theorem 6.1.
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References 1. Adachi, S., Tanaka, K.: Trudinger type inequalities in Rn and their best exponents. Proc. Am. Math. Soc. 128, 2051–2057 (2000) 2. Bartuccelli, M., Constantin, P., Doering, C.R., Gibbon, J.D., Gisself¨alt, M.: On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation. Physica D 44, 421–444 (1990) 3. Bechouche, P., J¨ungel, A.: Inviscid limits of the complex Ginzburg-Landau equation. Commun. Math. Phys. 214, 201–226 (2000) 4. Cazenave, T.: Introduction to Nonlinear Schr¨odinger equations. Textos de M´etodos Matem´aticos 22, Riode Janeiro: Univ. Federal do Rio de Janeiro, 1989 5. Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851– 1089 (1993) 6. Ginibre, J., Velo, G.: On a class of nonlinear Schr¨odinger equation I, the Cauchy problem, general case. J. Funct. Anal. 32, 1–32 (1979) 7. Ginibre, J., Velo, G.: The Cauchy problem in local spaces for the complex Ginzburg-Landau equation I, Compactness methods. Physica D 95, 191–228 (1996) 8. Ginibre, J., Velo, G.: The Cauchy problem in local spaces for the complex Ginzburg-Landau equation II, Contraction methods. Commun. Math. Phys. 187, 45–79 (1997) 9. Kavian, O.: A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schr¨odinger equations. Trans. Am. Math. Soc. 299, 113–129 (1987) 10. Kurata, K., Ogawa, T.: Remarks on blowing-up of solutions for some nonlinear Schr¨odinger equations. Tokyo J. Math. 13, 339–419 (1990) 11. Levermore, C.D., Oliver, M.: The complex Ginzburg-Landau equation as a model problem. Lect. Appl. Math. 31, 141–190 (1996) 12. Lions, J.L.: Quelques m´ethodes de r´esolution des probl´emes aux limites non lin´eaires. Paris: Dunod, 1969 13. Moser, J.: A sharp form of an inequality by N.Trudinger. Indiana Univ. Math. J. 11, 1077–1092 (1971) 14. Ogawa, T.: A proof of Trudinger’s inequality and its application to nonlinear Schr¨odinger equations. Nonlinear Anal. T.M.A. 14, 765–769 (1990) 15. Ogawa, T., Ozawa, T.: Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schr¨odinger mixed problem. J. Math. Anal. Appl. 155, 531–540 (1991) 16. Ogawa, T., Tsutsumi, Y.: Blow-up of H 1 solution for the nonlinear Schr¨odinger equation. J. Diff. Eqs. 92, 317–330 (1991) 17. Okazawa, N., Yokota, T.: Monotonicity method for the complex Ginzburg-Landau equation, including smoothing effect. Nonlinear Anal. T.M.A. 47, 49–88 (2001) 18. Okazawa, N.,Yokota, T.: Monotonicity method applied to the complex Ginzburg-Landau and related equations. J. Math. Anal. Appl. 267, 247–263 (2002) 19. Okazawa, N., Yokota, T.: Global existence and smoothing effect for the complex Ginzburg-Landau equation with p-Laplacian. J. Diff. Eqs. 182, 541–576 (2002) 20. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Math. Sci. Vol. 68, Berlin and New York: Springer-Verlag, 1988; 2nd ed., 1997 21. Trudinger, N.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–484 (1967) 22. Wang, B.: The limit behavior of solutions for the Cauchy problem of the complex Ginzburg-Landau equation. Comm. Pure Appl. Math. 55, 481–508 (2002) 23. Weinstein, M., Xin, J.: Dynamic stability of vortex solution of Ginzburg-Landau and nonlinear Schr¨odinger equations. Commun. Math. Phys. 180, 389–428 (1996) 24. Wu, J.: The inviscid limit of the complex Ginzburg-Landau equation. J. Diff. Eqs. 142, 413–433 (1998) Communicated by P. Constantin
Commun. Math. Phys. 245, 123–147 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1006-2
Communications in
Mathematical Physics
The Interaction Energy of Well-Separated Skyrme Solitons N.S. Manton1 , B.J. Schroers2 , M.A. Singer3 1
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK. E-mail:
[email protected] 2 Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK. E-mail:
[email protected] 3 School of Mathematics, University of Edinburgh, Kings Buildings, Edinburgh EH9 3JZ, UK. E-mail:
[email protected] Received: 8 January 2003 / Accepted: 31 January 2003 Published online: 17 December 2003 – © Springer-Verlag 2003
Abstract: We prove that the asymptotic field of a Skyrme soliton of any degree has a non-trivial multipole expansion. It follows that every Skyrme soliton has a well-defined leading multipole moment. We derive an expression for the linear interaction energy of well-separated Skyrme solitons in terms of their leading multipole moments. This expression can always be made negative by suitable rotations of one of the Skyrme solitons in space and iso-space. We show that the linear interaction energy dominates for large separation if the orders of the Skyrme solitons’ multipole moments differ by at most two. In that case there are therefore always attractive forces between the Skyrme solitons.
1. Skyrme Solitons The fundamental field of Skyrme’s theory [1] is a map U : R3 → SU (2).
(1.1)
We denote points in R3 by x with coordinates xi , i = 1, 2, 3, and Euclidean length r = |x| = x12 + x22 + x32 . Sometimes we write xˆ for the unit vector x/r. It is often useful to parametrise U in terms of the Pauli matrices τ1 , τ2 and τ3 and the triplet of pion fields π1 , π2 and π2 as U (x) = σ (x) + iπa (x)τa ,
(1.2)
where summation over the repeated index a is implied and the field σ is determined by the constraint σ 2 + π12 + π22 + π32 = 1. In this introductory section we do not specify the class of functions to which U belongs. It is assumed to be sufficiently smooth for all the following operations to make sense.
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The Skyrme energy functional is best written in terms of the Lie-algebra valued currents L i = U † ∂i U
(1.3)
Ri = U ∂i U † ,
(1.4)
or
where ∂i = ∂/∂xi . It is E[U ] = −
3
d x
1 1 tr(Li Li ) + tr([Lj , Li ][Lj , Li ]) . 2 16
(1.5)
The Euler-Lagrange equation for stationary points of this functional is conveniently expressed in terms of the modified currents 1 L˜ i = Li − [Lj , [Lj , Li ]] 4
(1.6)
1 R˜ i = Ri − [Rj , [Rj , Ri ]], 4
(1.7)
and
where we again use the convention that repeated indices are summed over. It reads ∂i L˜ i = 0
(1.8)
∂i R˜ i = 0.
(1.9)
or, equivalently,
Here we are interested in finite-energy solutions of the Euler-Lagrange equation. It is shown in [2] that the finite energy requirement implies that the map U tends to a constant value at infinity in a weak sense. We choose that constant to be the identity element 1 ∈ SU (2) and demand lim U (x) = 1.
(1.10)
r→∞
The boundary condition (1.10) means that the domain of U is effectively compactified to a three-sphere. Since the target space is also a three-sphere, maps satisfying (1.10) have an associated integer degree. The first rigorous proof that for a finite-energy Skyrme configuration in a very general class of functions the degree 1 d 3 x ij k tr Li Lj Lk (1.11) deg[U ] = − 2 24π is an integer was given in [3]. This result means that the space C = {U : R3 → SU (2) | E[U ] < ∞}
(1.12)
of finite-energy configurations is a disjoint union of sectors Ck = {U : R3 → SU (2) | E[U ] < ∞ labelled by the integers k ∈ Z.
and deg[U ] = k}
(1.13)
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The symmetry group of Skyrme’s theory will play an important role in our discussion. The energy functional (1.5), the boundary condition (1.10) and the degree (1.11) are invariant under the action of the Euclidean group of translations and rotations in R3 and under rotations of the pion fields πa → Gab πb ,
G ∈ SO(3),
(1.14)
which we call iso-rotations. Reflections in space S : x → −x and in iso-space πa → −πa both leave the energy invariant but each changes the sign of the degree. The pullback of Skyrme configurations U via S provides a map S˜ : Ck → C−k ,
U → U ◦ S
(1.15)
which preserves the energy. It was shown in [4] that the energy in each topological sector is bounded below by a multiple of the degree. It follows from the results of [5] that the bound cannot be attained for the standard version of the Skyrme model described here, so that we have the strict inequality E[U ] > 12π 2 |k|.
(1.16)
The bound ensures that the infima Ik = inf{E[U ] | U ∈ Ck }
(1.17)
are well-defined. The question of whether the infima are attained was first addressed by Esteban in the paper [2]. Amongst other things Esteban proved that, for a suitable class of functions, Ik ≤ Il + Ik−l
(1.18)
for all k, l ∈ Z. She also showed that infima are attained provided one assumes the strict inequality Ik < Il + Ik−l
(1.19) √ for all k ∈ Z − {0, ±1} and l ∈ Z − {0, k} in the range |l| + |k − l| < 2|k|. In [3] it was shown that the result still holds if one widens the class of allowed functions but the inequality (1.19) remains a necessary assumption in the proof. The strict inequality is also of interest in physics. As we shall see it is related to the question of attractive forces in the Skyrme model. For low values of k the existence and nature of minima of the Skyrme energy functional is understood in more detail. For fields of degree one, the highly symmetric hedgehog ansatz UH (x) = exp(if (r)xˆa τa ),
(1.20)
introduced already by Skyrme, leads to an ordinary differential equation for the profile function f . With the boundary conditions f (0) = π and f (∞) = 0 the resulting Skyrme configuration has degree 1 and is called the Skyrmion. It was shown in [6] that the Skyrmion minimises the Skyrme energy functional amongst all degree one configurations of the hedgehog form. In [2] the existence of the minimum of the Skyrme energy functional in C1 was proved, but it has not been established rigorously that minimising
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configurations have the symmetry of the hedgehog field (1.20). Note that if U1 is a minimal energy configuration in C1 then the reflected configuration S˜ (U1 ) has the same energy and minimises the energy in C−1 . In the following we use the term Skyrme solitons for minimal energy solutions of the Skyrme equation with non-zero degree k. There is overwhelming numerical evidence that the minimum in C2 is attained by a configuration of toroidal symmetry [7]. For higher degree, too, much is known numerically about the minima of Skyrme’s energy functional in the sector Ck . Numerical searches, assisted by analytical ans¨atze and investigations of the possible symmetries of Skyrme solitons, suggest the existence of Skyrme solitons in all sectors Ck up to k = 22. The energies are sufficiently accurately computed that it appears that the inequality (1.19) is satisfied for all k in the range 2 ≤ k ≤ 20 and all l in the range 0 < l < k, see [8–10]. The existence of attractive forces between two Skyrmions was already shown by Skyrme, using the product ansatz. In this paper we investigate the existence of attractive forces between general Skyrme solitons. It is perhaps worth stressing that our arguments in fact apply to any finite-energy solution of the Skyrme equation, not just minimal ones. An earlier attempt at proving the existence of attractive forces between Skyrme solitons was made in the unpublished paper [11]. Our approach is partly inspired by ideas in [11] but also fills important gaps left there. Our main tool is an asymptotic expansion of Skyrme solitons. In Sect. 2 we show that Skyrme fields have an asymptotic expansion in powers of 1/r and that for non-trivial Skyrme solitons that expansion always has a non-trivial leading multipole. In Sect. 3 we study the interaction energy of two Skyrme solitons and show that it is dominated by a certain linear interaction energy provided the orders of the leading multipoles of the Skyrme solitons do not differ by more than 2. In Sect. 4 we derive an expression for the linear interaction energy of two multipoles, and show that it can always be made negative by suitable relative rotations in space and iso-space. At the end of this paper we briefly comment on the relationship between our results and Esteban’s work, and on the implications for the existence of Skyrme solitons of arbitrary degree. 2. The Asymptotics of Skyrme Solitons The aim of this section is to show that if U and the currents Lj are a little better than continuous, then U is smooth in R3 , and has a non-trivial asymptotic expansion in powers of 1/r and log r as r → ∞. By non-trivial we mean here that if U is non-constant, then it cannot happen that all terms in the asymptotic expansion vanish. Put another way, it is not possible for U to approach 1 at infinity faster than every power of 1/r unless U is identically equal to 1 in R3 . To make a precise statement, say that the (possibly matrix-valued) function f is in Cb0,α (R3 ), with 0 < α < 1, if f satisfies sup |f | +
x∈R3
sup x,x ∈R3 ,x =x
(1 + r + r )α |f (x) − f (x )| < ∞. |r − r |α + (1 + r + r )α d(ω, ω )α
(2.1)
Here x = rω, x = r ω , where ω is regarded as an angular variable (unit vector) living on the unit 2-sphere, and d(ω, ω ) is the 2-sphere distance between ω and ω . If we consider the space Cb0,α (K), where K is a bounded subset of R3 , then this is precisely the same as the usual space of (bounded) H¨older-continuous functions, with H¨older
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exponent α. However, Cb0,α (R3 ) is slightly different from the usual space of bounded H¨older-continuous functions on R3 . The main result of this section can be summarized as follows: Theorem 2.1. Suppose that for some δ > 0, (1 + r)δ (U − 1) ∈ Cb0,α , (1 + r)δ Lj ∈ Cb0,α .
(2.2)
Suppose further that U satisfies the Skyrme differential Eqs. (1.8) in the sense of distributions. Then U is smooth in R3 and has a complete asymptotic expansion in powers of 1/r and log r, for large r. If U is non-constant, then this expansion has a leading term which is harmonic, hence a multipole. Note that the hypotheses (2.2) force U to approach 1 and the Lj to approach 0 like r −δ as r → ∞. As a technical remark, we point out that the assumption of H¨older-continuous currents implies that U will have a H¨older-continuous first derivative. The second derivatives ∂j ∂k U are then defined only in the sense of distributions, but in the Skyrme equation ∂j ∂k U enters linearly, and is multiplied by continuous functions (smooth functions of the currents) see Sect. 2.1 below. In particular, the left-hand side of the Skyrme equation ∂j L˜ j = 0 makes sense as a distribution if (2.2) holds. The assumptions (2.2) of Theorem (2.1) do not follow immediately from the variational analysis used by Esteban in [2]. Her methods only give that the derivatives ∂j U are locally square-integrable (and that the components of U are locally bounded). On physical grounds, one expects minimizers of the Skyrme energy functional to satisfy (2.2), but it would be desirable to bridge the gap between the analysis given here and what was proved rigorously in [2 and 3]. This issue will not be pursued further here. The proof of Theorem (2.1) proceeds in four steps, each of which takes up one of the following subsections. In the first subsection we rewrite the Skyrme equation in order to make explicit the form of the non-linearities. The equation is quasilinear, in the sense that the derivatives of highest order (2) enter linearly. The Skyrme equation can therefore be regarded as a second-order linear elliptic PDE, with H¨older-continuous coefficients. Then standard regularity theorems (Schauder estimates) yield that U is smooth. In Sect. 2.2 we study the behaviour of the Skyrme equation at spatial infinity, by introducing coordinates (s, ω), where s = r −1 , so that the 2-sphere at infinity becomes a genuine boundary at s = 0. After a rescaling, the Laplacian of R3 is replaced by b = (s∂s )2 − s∂s + ω ,
(2.3)
where ω is the Laplacian of the unit 2-sphere. The analysis is now guided by the corresponding analysis of a system of ordinary differential equations with regular singular point at s = 0. In particular, one does not expect the solutions to be smooth near s = 0, but one does expect a non-trivial expansion in powers of s (and possibly log s). The b-calculus of [12] enables us to make this kind of argument precise. Thus in Sect. 2.2 we show that if U = 1 + u, then u is conormal at s = 0, which is to say that u and all derivatives of the form (s∂s )m ∇ωn u are continuous as s → 0. (This condition is strictly weaker than u being smooth near s = 0.) In Sect. 2.3, we show that it is not possible for U to approach 1 faster than r −µ for every µ > 0 unless U = 1 in R3 . Finally in Sect. 2.4 we combine this fact with another application of the b-calculus to show the existence of a non-trivial asymptotic expansion in powers of r −1 and log r (or equivalently in powers of s and log s).
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2.1. Rewriting the Skyrme equation. To begin with we work near a fixed point of R3 , which we may as well take to be the origin 0. By replacing U (x) by U (0)−1 U (x) we can assume that U (0) = 1. Write U (x) = 1 + u(x),
(2.4)
so that u(0) = 0. Because U is unitary, the 2 × 2 complex matrix u will satisfy the algebraic constraints u + u† + uu† = 0, tr(u) + det u = 0.
(2.5)
In particular, u is neither exactly skew-hermitian nor trace-free. Then Li = ∂i u + u† ∂i u
(2.6)
∂i Li = (1 + u† ) u + ∂i u† ∂i u.
(2.7)
and
The cubic term in the currents can be written 1 1 Lj [Lj , Li ] = (∂j u + u† ∂j u)[∂j u + u† ∂j u, ∂i u + u† ∂i u]. 2 2
(2.8)
Taking the divergence, and using the notation vij = ∂i u† ∂j u,
(2.9)
1 Lj [Lj , Li ] = (1 + u† )(T + F ), 2
(2.10)
we obtain
where T = T (u, ∂u, ∂ 2 u) 1+u (Lj [Lj , (1 + u† ) u] + Lj [(1 + u† )∂i u∂j u, Li ]) = 2
(2.11)
and F = F (u, ∂u) =
1+u (vij [Li , Lj ] + Lj [Lj , vii ] + Lj [(1 + u† )vij , Li ]). 2
(2.12)
The nonlinear terms have been divided here so that T is polynomial in the first derivatives of u and linear in its second derivatives, while F only contains u and its first derivatives. With this notation, the full Skyrme equation can be written P (u, ∂u, ∂ 2 u) = Q(u, ∂u) + F (u, ∂u),
(2.13)
where P (u, ∂u, ∂ 2 u) = u + T (u, ∂u, ∂ 2 u) and Q(u, ∂u) = −(1 + u)vii .
(2.14)
Notice that Q is (approximately) quadratic in u, T is of degree three and F is of degree four. Note also that T is linear in ∂i ∂j u (and quadratic in ∂i u).
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Because of the assumed H¨older continuity of the currents, the coefficients of the differential operator f −→ P (u, ∂u, ∂ 2 f )
(2.15)
are H¨older continuous, and this operator is linear and elliptic in a small neighbourhood K of 0. Moreover, the RHS Q + F of (2.13) is also in Cb0,α (K), so by elliptic regularity, it follows that u ∈ Cb2,α (K). Then the currents are in Cb1,α (K) and the process continues to show that u is in Cbk,α (K) for every k. Thus u is smooth near 0. Since the point 0 was arbitrary, the argument shows that u is smooth in R3 . 2.2. Boundary regularity. In order to analyze the asymptotic behaviour of the field U , we shall make a transformation of the problem which involves passing from R3 to a “compactification” R3 in which the sphere at infinity becomes a genuine boundary. This is easily achieved by introducing the coordinate s = 1/r along with angular coordinates θ and ϕ in R3 . Then in the space [0, ∞) × S 2 with coordinates (s, θ, ϕ), the boundary s = 0 corresponds to r = ∞ and θ and ϕ give coordinates on the boundary, which is the 2-sphere “at infinity”. R3 is defined to be the union of R3 with the 2-sphere at infinity attached in this way. It can be cumbersome to work with explicit coordinates on the 2-sphere, so we again use ω for points on S 2 and represent any point other than the origin of R3 in the form (s, ω). Next, introduce rescaled derivatives, Di = r∂i =
1 ∂i . s
(2.16)
These vector fields have the property that they are linear combinations, with coefficients that are smooth, down to s = 0, of the basic vector fields s∂s , ∂θ and ∂ϕ . The Euclidean Laplacian takes the form = s 2 b
(2.17)
with b defined in (2.3). Now we write U = 1 + u for large r (that is, for small positive s) and make the rescalings of (2.13) suggested by (2.16) and (2.17). The result is a ‘b’ version of the Skyrme equation, Pb (u, Du, D 2 u) = Qb (u, Du) + s 2 Fb (u, Du),
(2.18)
Pb (u, Du, D 2 u) = b u + s 2 Tb (u, Du, D 2 u), Qb (u, Du) = −(1 + u)Di u† Di u,
(2.19) (2.20)
where
these terms being obtained by replacing ∂i by the rescaled derivative Di wherever they occur. The reason for reformulating the equation in this way is that there is a well-established theory, called the b-calculus, which can be used to analyze equations of this kind
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[12]. The b-calculus is concerned with b-differential operators. In the present situation, a b-differential operator is just a differential operator of the form
P =
Cabc (s, ω)(s∂s )a ∂θb ∂ϕc ,
(2.21)
a+b+c≤m
where the coefficients Cabc are smooth up to the boundary s = 0. From (2.3) it is clear that the rescaled Laplacian b is an example of such an operator. The set of all such operators will be denoted by Diffb , those of order k by Diffkb . The first aspect of this theory that is needed is the counterpart of the elliptic regularity for H¨older spaces that we used in the previous subsection. For this we need b-H¨older spaces, already introduced in (2.1). With the new variable s, we have f (s, ω) ∈ Cb0,α if sup |f | +
x∈R3
(s + s )α |f (s, ω) − f (s , ω )| < ∞. α α α (s,ω) =(s ,ω ) |s − s | + (s + s ) d(ω, ω ) sup
(2.22)
Now put Cbk,α = {f : Lf ∈ Cb0,α for all b-differential operators L ∈ Diffkb } and A = {u : Lu ∈ Cb0,α for all L ∈ Diffb }. In order to force functions to decay as s → 0, we introduce weighted versions of these spaces, s δ Cbk,α = {u = s δ v : v ∈ Cbk,α }, s δ A = {u = s δ v : v ∈ A}. The following is a very special case of elliptic regularity for b-differential operators [13]. Theorem 2.2. Consider the differential operator P = b + s 2 E, where E is a secondorder differential operator with coefficients smooth up to the boundary s = 0. Suppose that P u = f near s = 0, with u and f in s δ Cb0,α . Then if δ is not an integer, it follows that u ∈ s δ Cb2,α . We want to apply this to the Skyrme equation, written in the form (2.19). After the last section, we know that the coefficients of the perturbing term E are smooth for s > 0, but all we know at s = 0 is the original assumption that the currents are in s δ Cb0,α . The elliptic regularity statement still holds in this case (though this statement does not seem to be available in the literature). Applying this result, for δ a small positive number, we obtain first that u ∈ s δ Cb2,α , which gives that the coefficients are in s δ Cb1,α , so u ∈ s δ Cb3,α and so on. Iterating, we find u ∈ s δ A. This is a major step forward: in particular, the angular variation of u is now very well controlled. However, u may still be very far from being smooth up to the boundary or having an asymptotic expansion there. Indeed, horrors like s δ sin log s lie in s δ A.
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2.3. Power-law decay of u. In this section and the next we establish that a topologically non-trivial solution of the Skyrme equation must have a non-trivial asymptotic expansion in powers of 1/r. We show first that it is not possible for a topologically non-trivial solution to approach 1 faster than every power of 1/r. This result is then fed into an iterative analysis of the equation in the next section. These two sections can, however, be read in either order. The main result of this section is as follows. Theorem 2.3. If the topological charge of the Skyrme field U is non-zero and if U satisfies the hypotheses of Theorem 2.1, then there exists some µ ∈ R+ such that r µ u(r, ω) does not tend to zero as r → ∞. To set this result in a more general context, recall that a partial differential equation Lu = 0 is said to have the unique continuation property at a point 0, say, if the following is true: If all derivatives of u vanish at 0, then u = 0 in some neighbourhood of 0.
(2.23)
The methods used to establish that a second-order PDE has the unique continuation property establish analogous statements with somewhat weaker hypotheses. For example, a simplified version of Theorem 17.2.6 of [16] is as follows. Theorem 2.4. Let aj k (x) be smooth and positive-definite in a neighbourhood X of 0, and suppose that aj k (0) = δj k . Suppose that for x ∈ X, the smooth function u satisfies |aj k (x)∂j ∂k u| ≤ A (|u(x)| + |∇u(x)|)
(2.24)
for some constant A, and |x|−µ |u(x)| → 0 as |x| → 0 for every µ ∈ R+ .
(2.25)
Then u = 0 identically in a neighbourhood of 0. Using this result we can prove the following: Theorem 2.5. Let U satisfy the hypotheses of Theorem (2.1). Then, if the topological charge of U is non-zero, U cannot be constant in any open subset of R3 . Proof. Define W = {x ∈ R3 : U (y) = U (x) for all y in some neighbourhood of x}.
(2.26)
Then W is open by definition. By Theorem 2.4, W is also closed. To see this, suppose that xn ∈ W , xn → x0 ∈ R3 . Then all derivatives of U are zero at x0 , so (2.25) holds (with 0 replaced by x0 ). The Skyrme equation, in the form (2.13), implies a differential inequality of the form (2.24) in a neighbourhood of x0 . It follows1 that U is identically constant in a neighbourhood of x0 , so that x0 ∈ W . Since R3 is connected, W = ∅ or W = R3 .
1 Unique continuation theorems do not extend wholesale to systems. However, in our case, the leading term is the Laplacian and the other second-order terms C(u, ∂u, ∂ 2 u) are non-scalar but small near x0 . The proof of Theorem 2.4 goes through in this case.
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We will use this theorem to give an indirect proof of Theorem 2.3. Suppose r µ u(x) → 0 as r → ∞ for every µ. Adapting Theorem 2.4 we shall show that then u = 0 for all sufficiently large r. Thus U is constant in an open set, hence by the previous result constant everywhere. So consider the b-differential operator Pb = b + s 2 Tb
(2.27)
on R3 , where the coefficients of Tb are in Cb0,α . By rescaling Theorem 17.2.6 in [16], we obtain Theorem 2.6. Suppose that |Pb u(s, ω)| ≤ As δ (|u(s, ω)| + |Du(s, ω)|) for all 0 < s < s0 , ω ∈ S 2
(2.28)
and that s −µ |u(s, ω)| → 0 as s → 0 for all µ.
(2.29)
Then u(s, ω) = 0 for 0 ≤ s < s1 , where s1 is some small positive number. The “b” version (2.18) of the Skyrme equation implies a differential inequality of the form (2.28), just as before. It follows that if u decays faster than any power of r, then u = 0 for all sufficiently large r. By the remarks before the statement of Theorem 2.6, the proof of Theorem 2.3 is now complete.
2.4. Refined regularity, asymptotic expansions. The main result of this section can be stated as follows: Theorem 2.7. Let U = 1 + u satisfy the hypotheses of Theorem 2.1. Then there is some integer M ≥ 1 and an asymptotic expansion u∼
2M j =M
Yj (ω)r
−(j +1)
+
∞
r −(j +1) wj (ω, log r) for large r.
(2.30)
j =2M+1
Here the Yj are Lie-algebra valued spherical harmonics, ω Yj = −j (j + 1)Yj ,
(2.31)
−(j +1) is a non-zero harmonic function. and YM = 0, so that the piece 2M j =M Yj (ω)r The functions wj are smooth in ω and polynomial in log r. It will follow from the proof that the terms w2M+1 to w3M+1 are of at most degree 1 in log r, the terms w3M+2 to w4M+2 are of at most degree 2 in log r and so on. We remark also that the asymptotic expansion can safely be differentiated term by term to give asymptotic expansions of all derivatives of u. In order to motivate the proof, consider the equation b u = f,
(2.32)
Interaction Energy of Well-Separated Skyrme Solitons
133
where u and f are defined for small s. If f has the form f = s λ g(ω), then a solution can be found as follows. Expand g as a sum of spherical harmonics, g=
∞
gj , where ω gj = −j (j + 1)gj
j =0
and seek a solution u =
uj (s)gj . Then uj must satisfy
[(s∂s )2 − (s∂s ) − j (j + 1)]uj (s) = s λ . This is solved by uj (s) =
sλ λ(λ − 1) − j (j + 1)
provided there is no resonance, that is to say λ = −j, j + 1. In the resonant case, uj (s) has the form s λ (A+B log s). The general solution is obtained by combining this with an arbitrary solution of the homogeneous equation b v = 0. If we require u → 0 as s → 0, then v must itself go to zero and hence will be a sum of j +1 , where Y satisfies (2.31). multipoles, v = ∞ j j =0 Yj (ω)s The b-calculus extends results of this kind to functions in s δ A (which, as we have seen, can be far from having expansions in powers of s). In order to summarize the needed results, write f = O(s δ ) instead of f ∈ s δ A.
(2.33)
This conforms to the use of the “O”-notation in the rest of the paper, but has the additional property If f = O(s δ ), then Lf = O(s δ ) for all L ∈ Diffb .
(2.34)
Lemma 2.8. Suppose that u and f defined near s = 0 satisfy (2.32). If u = O(s δ ) and f = O(s n+δ ), where δ > 0 and n is a positive integer, then u=
n−1
Yj (ω)s j +1 + O(s n+δ ),
(2.35)
j =0
where Yj satisfies (2.31). In particular the sum on the RHS is harmonic b
n−1
Yj (ω)s j +1 = 0.
(2.36)
j =0
Lemma 2.9. Suppose that u and f defined near s = 0 satisfy (2.32). If u = O(s δ ) and f =
n−1 j =0
fj (ω, log s)s j +1 + O(s n+δ ),
(2.37)
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N.S. Manton, B.J. Schroers, M.A. Singer
where δ > 0, n is a positive integer and fj is polynomial of degree mj in log s, then u=
n−1
wj (ω, log s)s j +1 + O(s n+δ ),
(2.38)
j =0
where wj is a polynomial of degree mj + 1 in log s. These results will be applied to the Skyrme equation, now rewritten as b u = Zb (u) := Qb (u) − s 2 Tb (u) + s 2 Fb (u).
(2.39)
Lemma 2.10. Suppose that U = 1 + u satisfies the Skyrme equation and u = O(s δ ). Then u = O(s 2 ). Proof. On the RHS of (2.39), Qb is quadratic in Du and the other terms in Zb are of even higher degree. Hence Zb = O(s 2δ ). Applying Lemma 2.8 we obtain that u = O(s 2δ ) + O(s). If 2δ < 1, we can iterate this argument to obtain eventually u ∈ O(s). In [14], it is shown that a solution of the Skyrme equation cannot have leading term 1/r = s in its asymptotic expansion. It follows that u = O(s 1+δ ) (for a possibly smaller δ > 0). Hence Zb (u) = O(s 2+2δ ) and so, applying Lemma 2.8 again, u = Y1 s 2 + O(s 2+δ ).
Combining Theorem 2.3 with Lemma 2.8, we see that there exists an integer M ≥ 1 with the property that u = YM (ω)s M+1 + O(s M+1+δ ),
(2.40)
where YM is a non-vanishing spherical harmonic. We can now complete the proof of Theorem 2.7 in the following iterative fashion. From the structure of Zb (u) it follows from (2.40) that Zb (u) = f2M+1 s 2M+2 + O(s 2M+2+δ ).
(2.41)
Applying Lemma 2.9, u=
2M
Yj (ω)s j +1 + w2M+1 (ω, log s)s 2M+2 + O(s 2M+2+δ ),
(2.42)
j =M
where w2M+1 (ω, log s) is of degree at most 1 in log s. Now this expression for u is substituted into Zb , to give Zb =
3M+1
fj (ω)s j +1 + f3M+2 (ω, log s)s 3M+3 + +O(s 3M+3+δ ),
(2.43)
2M+1
where f3M+2 is of degree at most 1 in log s. Now apply Lemma 2.9 to get u=
2M j =M
Yj (ω)s j +1 +
3M+1
wj (ω, log s)s j +1
j =2M+1
+w3M+2 (ω, log s)s 3M+3 + O(s 3M+3+δ ),
(2.44)
Interaction Energy of Well-Separated Skyrme Solitons
135
where the functions w2M+1 , ..., w3M+1 are of degree at most 1 in log s and w3M+2 is of degree at most 2 in log s. Carrying on in this way we obtain a complete asymptotic expansion. To complete the proof of Theorem 2.7 note finally that the first M + 1 terms in the expansion are actually the pion field. Indeed, if the expansion (2.30) is substituted into the j +1 is skew-adjoint algebraic constraints (2.5) we see that the harmonic piece 2M M Yj s 2M+2 and trace-free – the quadratic corrections enter at order s .
The upshot of this section is that every Skyrme soliton has a leading Lie-algebra valued multipole field (called a 2M -pole) uM (x) = iτa
M m=−M
4π YMm (θ, ϕ) , Qa 2M + 1 Mm r M+1
(2.45)
where YMm are the usual spherical harmonics on S 2 , see Appendix A. The leading multipole moment QaMm is independent of the location of the Skyrme soliton, and is acted on naturally by rotations and iso-rotations. It is a key ingredient in the calculations of the following sections. As already mentioned one can show that Skyrme solitons never have asymptotic monopole fields [14]. The leading multipole field of the B = 1 hedgehog (1.20) is a triplet of dipoles, and dipoles are known to occur as leading multipoles in a number of Skyrme solitons. The highest leading multipole known from numerical work is an octupole, which occurs in a B = 7 configuration with icosahedral symmetry [9]. 3. The Interaction Energy of Two Skyrme Solitons Suppose we have Skyrme solitons U (1) and U (2) of degrees k and l which minimise the energy in the sectors Ck and Cl . Since the total energies of U (1) and U (2) are finite there must be balls B1 and B2 in R3 so that most of the energy of U (1) and U (2) is concentrated in, respectively, B1 and B2 . Outside the balls B1 and B2 the asymptotic analysis of the previous section applies. Suppose that the leading multipole of U (1) is a 2M -pole and the leading multipole of U (2) is a 2N -pole. Denoting the radii of B1 and B2 by D1 and D2 , and with the abbreviation (2.45) for a generic Lie-algebra valued multipole field we have U (1) (x) ∼ 1 + uM (x) for
x ∈ B1
(3.1)
U (2) (x) ∼ 1 + vN (x) for
x ∈ B2 .
(3.2)
and
Using the translational invariance of the Skyrme energy functional we can assume without loss of generality that B1 is centred at X+ = (0, 0, R/2) and that B2 is centred at X− = (0, 0, −R/2), where R is so large that B1 and B2 do not overlap, i.e. R > D1 + D2 . The parameter R will be interpreted as the separation of the Skyrme solitons. There is clearly an ambiguity in the definition of such a separation parameter, but this does not affect our calculation of leading terms in the limit where R becomes large. Then we define the following product configuration: UR (x) = U (1) (x)U (2) (x).
(3.3)
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N.S. Manton, B.J. Schroers, M.A. Singer
This configuration has degree k + l and we shall see shortly that its energy is finite, so that UR ∈ Ck+l . Our goal is to study the energy of the product configuration UR perturbatively in the limit of large R and to compute the leading terms in powers of 1/R. A similar calculation for moving and spinning Skyrmions was performed in [15], where some further details (1) (1) are given. Let Li and L˜ i be the currents (1.3) and (1.6) constructed out of U (1) , and (2) (2) Ri and R˜ i be the currents (1.4) and (1.7) constructed out of U (2) . Then one computes E[UR ] = E[U (1) ] + E[U (2) ] + W2 + W4 .
(3.4)
The energies E[U (1) ] and E[U (2) ] are simply the energies of the Skyrme solitons U (1) and U (2) and therefore independent of R. The interaction terms W2 and W4 are given by integrals over R3 , W2 = d 3 x w2 and W4 = d 3 x w4 , (3.5) with integrands
(1) (2) (1) (2) (1) (2) w2 = tr Li R˜ i + L˜ i Ri − Li Ri
(3.6)
1 (1) (2) (1) (2) w4 = − tr [Li , Rj ][Li , Rj ] 8
(1) (2) (2) (1) (1) (1) (2) (2) +[Li , Rj ][Ri , Lj ] + [Li , Lj ][Ri , Rj ] .
(3.7)
and
We shall see shortly that the term W2 contains the leading contribution to the interaction energy. However, for the presentation of our method of computation it is more convenient to begin with the quartic term W4 . We split the integration region R3 into the balls B1 and B2 and the complement C = R3 − (B1 ∪ B2 ). To illustrate our method, consider the integral
1 (1) (2) (1) (2) I =− d 3 x tr [Li , Rj ][Li , Rj ] 8 B1
1 (1) (1) (2) (2) ≤ d 3 x tr Li Li tr Rj Rj . (3.8) 4 B1 (1)
The currents Li Therefore
are smooth functions and hence bounded on the compact domain B1 . |I | ≤ −K B1
(2) (2) d 3 x tr Rj Rj
(3.9)
for some positive constant K. Since B1 is far away from the centre of soliton U (2) , the (2) leading contribution to the integral (3.9) is obtained by replacing Rj by the leading multipole component −i∂j vN . Using |∂j vN |(x) ≤
K |x − X− |N+2
(3.10)
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137
for a further positive constant K we conclude that 1 . I =O R 2N+4 A similar calculation for the other terms in W4 shows that 1 3 . d x w4 = O R 2N+4 B1 Considering the contribution from B2 we find by the same argument 1 3 . d x w4 = O R 2M+4 B2 Finally, the remaining integral over C can be estimated as follows. Define 1 1 d 3x , F (R) = 2M+4 |x − X |2N+4 |x − X | + − C
(3.11)
(3.12)
(3.13)
(3.14)
noting that the integral converges for all values of R > D1 + D2 . Then there is a positive constant K such that d 3 x w4 < K F (R). (3.15) C
The large R behaviour of F can be estimated by a scaling analysis. Fix R0 > D1 + D2 and consider R > R0 . Changing integration variables x → (R0 /R)x one computes 2M+2N+5 R0 F (R) = F (R0 ) R 1 1 + , (3.16) d 3x |x − X+ |2M+4 |x − X− |2N+4 S1 (R)∪S2 (R) where S1 (R) and S2 (R) are the thick shells, S1 (R) = {x ∈ R3 |(R0 /R)D1 ≤ |(x1 , x2 , x3 − (R0 /2)| < D1 }, S2 (R) = {x ∈ R3 |(R0 /R)D2 ≤ |(x1 , x2 , x3 + (R0 /2)| < D2 },
(3.17)
which converge to punctured balls B10 = {x ∈ R3 − {(0, 0, R0 /2)}| |(x1 , x2 , x3 − (R0 /2)| < D1 }, B20 = {x ∈ R3 − {(0, 0, −R0 /2)} | |(x1 , x2 , x3 + (R0 /2)| < D2 }
(3.18)
in the limit R → ∞. In that limit the integral over S1 (R) diverges like R 2N+1 and that over S2 (R) like R 2M+1 . Combining this with the factor R −(2M+2N+5) we deduce that F (R) decays for large R like R −(2N+4) and R −(2M+4) , just like the contributions (3.12) and (3.13). Combining all the terms, we conclude that the leading terms in W4 decay for large R according to R −(2N+4) and R −(2M+4) . In order to evaluate W2 we first divide the region of integration into the half-spaces H + = {x ∈ R3 | x3 > 0}
and
H − = {x ∈ R3 | x3 < 0}.
(3.19)
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N.S. Manton, B.J. Schroers, M.A. Singer
In H + we replace U (2) by the leading term 1 + vN and in H − we replace U (1) by the leading contribution 1 + uM . The result is
1 (1) (1) W2 ≈ d 3 x tr Li [∂j vN , [∂j vN , ∂i vN ]] − tr ∂i vN L˜ i 4 H+
1 (2) (2) − . (3.20) d 3 x tr Ri [∂j uM , [∂j uM , ∂i uM ]] + tr ∂i uM R˜ i 4 H− (1) (2) Now integrating by parts and using the Euler-Lagrange equations ∂i L˜ i = ∂i R˜ i = 0 for the individual Skyrme solitons we convert two of the terms into an area integral
(1) (2) − + d 3 x tr ∂i vN L˜ d 3 xtr ∂i uM R˜ i
H+
=
x3 =0
i
H−
(1) (2) dx1 dx2 tr vN L˜ 3 + uM R˜ 3 .
(3.21)
Since the x1 x2 -plane is far away from both Skyrme solitons the leading contribution to this area integral can be expressed entirely in terms of the asymptotic fields: E = 2
3 a=1 x3 =0
a a dx1 dx2 (uaM ∂3 vN − vN ∂3 uaM ).
(3.22)
A simple scaling analysis shows that E falls off like R −(N+M+1) for large R. We skip the details here because we shall show how to evaluate E exactly in the next section. The remaining terms in (3.20) can be estimated with the techniques used in estimating W2 . The result is 1 1 W2 = E + O + O . (3.23) R 3N+6 R 3M+6 Combining all terms in (3.4) we conclude that E[UR ] = E[U (1) ] + E[U (2) ] + E + O
1 R 2N+4
+O
1
R 2M+4
.
(3.24)
Note that E is the leading contribution to the interaction energy if |N − M| ≤ 2, i.e. if the orders of the leading multipoles of the two Skyrme solitons differ by at most two. We will comment on the validity of this assumption at the end of this paper. 4. Harmonic Functions and Their Interaction Energy In order to compute the interaction energy E we need to derive some general results about harmonic functions. We define the regions Hδ− = {x ∈ R3 | x3 < δ}
and
Hδ+ = {x ∈ R3 | x3 > −δ},
(4.1)
where the positive parameter δ is introduced for technical reasons. Then we introduce the spaces H− = {f : Hδ− → R | f = 0,
lim f (x) = 0}
r→∞
(4.2)
Interaction Energy of Well-Separated Skyrme Solitons
H+ = {g : Hδ+ → R | g = 0,
139
lim g(x) = 0}.
r→∞
(4.3)
Elements of H− tend to zero at the boundary “at infinity” of Hδ− , elements of H+ tend to zero at the boundary “at infinity” of Hδ+ . No additional restriction is placed on the behaviour at the boundaries x3 = ±δ. For the calculations in this section it is convenient to split R3 into R2 × R and denote vectors in R2 by bold letters, e.g. x = (x1 , x2 ). We then write x = (x, x3 ). The most general element of H− can be written as f (x) =
d 2k p(k) exp(ik · x + kx3 ), (2π)2 2k
(4.4)
where k = k 2 and the volume element d 2 k/((2π )2 2k) arises from the combination of d 3 k with the delta-function δ(k 2 − k 2 ) which ensures that f satisfies the Laplace equation. Since f is real the Fourier transform p satisfies p(k) ¯ = p(−k).
(4.5)
Similarly, the most general element of H+ can be written as g(x) = with l =
d 2l q(l) exp(il · x − lx3 ), (2π)2 2l
(4.6)
l 2 and q(k) ¯ = q(−k).
(4.7)
There is a natural pairing between elements of H− and those of H+ , f, g =
x3 =0
dx1 dx2 (g∂3 f − f ∂3 g)
for
f ∈ H− , g ∈ H+ .
(4.8)
Using the expansion (4.4) and (4.6) we find, in terms of the Fourier modes, f, g =
d 2k p(k)q(−k). (2π)2 2k
(4.9)
This pairing is of interest to us since the interaction energy (3.22) is proportional to the a . Therefore, we also refer to the expressions (4.8) and sum over the pairings uaM , vN (4.9) as the interaction energy of the harmonic functions f and g. It is clear that the pairing (4.8) may vanish for some pairs of harmonic functions f ∈ H− , g ∈ H+ . This happens for example if the support of the Fourier transform p is complementary to that of the Fourier transform q. However, we shall now show that the interaction energy of multipoles can always be made non-zero by rotating one of the functions.
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4.1. Multipole fields. For f ∈ H− we have an alternative expansion in spherical harmonics YMm , f (x) =
M M≥0 m=−M
4π YMm (θ+ , ϕ+ ) QMm , M+1 (2M + 1) r+
(4.10)
where r+ = |x−X+ | and (θ+ , ϕ+ ) are spherical coordinates centred at X+ = (0, 0, R/2). In this section we only need to assume R > 0, but in our applications we will be interested in the large R limit. The coefficients QMm are called the multipole moments of the function f . Assume that f has non-vanishing multipole moments and suppose M is the smallest integer such that QMm = 0 for some m = −M, ..., M. The function fM (x) =
M YMm (θ+ , ϕ+ ) 4π QMm M+1 (2M + 1) r+
(4.11)
m=−M
is a 2M -pole field and QMm are the leading multipole moments of f . It is often convenient to write multipole fields in terms of partial derivatives of the Coulomb potential centred at X+ : φ+ (x) =
1 1 = , 2 r+ ρ + (x3 − R/2)2
(4.12)
where ρ 2 = x 2 . The function ∂3m3 ∂2m2 ∂1m1 φ+ (x) is a 2M -pole field if m1 +m2 +m3 = M. However, not all of the fields obtained in this way are independent. We introduce the complex derivatives ∂=
1 (∂1 − i∂2 ) and 2
1 ∂¯ = (∂1 + i∂2 ) 2
(4.13)
¯ Then, since φ+ (x) = 0 we have and note that = ∂32 + 4∂ ∂. 1 ∂ ∂¯ φ+ (x) = − ∂32 φ+ (x). 4
(4.14)
Thus a basis for 2M -pole fields is given by ∂3m3 ∂ n ∂¯ n¯ φ+ (x), where M = m3 + n + n¯ and either n or n¯ can be taken to be zero. In Appendix A we derive the exact relation between the functions ∂3M+m ∂ −m φ+ (x), −M ≤ m < 0, and ∂3M−m ∂¯ m φ+ (x), 0 ≤ m ≤ M, on the one hand and the spherical harmonics YMm centred at X+ on the other. The result is that we have the alternative expansion of an 2M -pole field fM (x) = AMm ∂3M+m ∂ −m φ+ (x) + AMm ∂3M−m ∂¯ m φ+ (x), (4.15) −M≤m≤0
1≤m≤M
where the coefficients AMm , m = −M, ..., M are directly proportional to the multipole moments QMm . It follows from the results in Appendix A that 4π (−1)M+m 2m AMm = (4.16) QMm √ 2M + 1 (M − m)!(M + m)!
Interaction Energy of Well-Separated Skyrme Solitons
for m ≥ 0 and
AMm =
141
4π (−1)M 2|m| QMm √ 2M + 1 (M − m)!(M + m)!
(4.17)
for m < 0. Note in particular that the reality of fM is equivalent to AM(−m) = A¯ Mm .
(4.18)
Multipole fields have a remarkably simple Fourier transform, which will be important for us. We use the representation 2 d k −k|x3 − R | 2 exp(ik · x), φ+ (x) = (4.19) e 2πk which can be verified as follows. Exploiting the invariance of φ+ under rotations in the x1 x2 -plane we may assume that x = (ρ, 0). Using polar coordinates (k, ψ) for k we first carry out the dk integration and then the angular integration: 2π R dψ ∞ dk e−k|x3 − 2 | exp(ik · x) 2π 0 0 2π dψ −1 (4.20) = 2π (iρ cos ψ − |x3 − R/2|) 0 dw −2 = , (4.21) 2 S 1 2πi (iρw − 2|x3 − R/2|w + iρ) where we changed variables to w = eiψ in the last line. Expanding the integrand in partial fractions and using the residue theorem then yields the expression (4.12). We compute the Fourier transform of the multipole field (4.15) by differentiating (4.19) under the integral sign. Note that i −iψ exp(ik · x) ke 2 i ∂¯ exp(ik · x) = keiψ exp(ik · x), 2 ∂ exp(ik · x) =
so we find fM (x) =
1 2π
d 2 k e−k( 2 −x3 ) k M−1 R
−M≤m≤M
and (4.22)
|m| i AMm eimψ exp(ik · x). 2 (4.23) R 2 . Thus with the normalisation
Here we have used that x3 < δ so that in particular x3 < (4.4) we arrive at the following simple expression for the Fourier transform i |m| −k R2 M k AMm eimψ . pM (k) = 4πe 2
(4.24)
−M≤m≤M
This function factorises into a k-dependent part and the function i |m| AMm eimψ (ψ) = 2 −M≤m≤M
(4.25)
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N.S. Manton, B.J. Schroers, M.A. Singer
of the angle ψ. The k-dependent part e−k 2 k M is non-zero for k = 0 and the function only vanishes identically if AMm = 0 for all m = −M, ..., M, i.e. if the 2M -pole field is trivial. R
4.2. The interaction energy of two scalar multipoles. The interaction energy of two multipoles can be expressed in a remarkably compact way. Let φ− (x) =
1 |x − X− |
(4.26)
be the Coulomb potential centred at X− = (0, 0, −R/2) and consider the multipole field gN (x) = BNn ∂3N+n ∂ −n φ− (x) + BNn ∂3N−n ∂¯ n φ− (x) (4.27) −N≤n≤0
1≤n≤N
with BN(−n) = B¯ Nn . By the same calculation as for fM above we find the Fourier transform of gN in the x1 x2 -plane: i |n| R qN (k) = 4πe−k 2 k M BNn einψ . (4.28) 2 −N ≤n≤N
The interaction energy of the two multipole fields fM and gN VMN = fM , gN
(4.29)
can now be computed using the formula (4.9). Using the factorisation property of the Fourier transforms pN and qN , it is easy to perform the integration over k. Assuming without loss of generality that M ≤ N we first carry out the integration over the angle ψ to find VMN = 4π
∞
0
dke−kR k N+M
M
2−2|m| A¯ Mm BNm ,
(4.30)
m=−M
where we have used the reality condition for the coefficients AMm and BNm . Computing the remaining integral we obtain the final result VMN = 4π
M (M + N )! −2|m| ¯ AMm BNm . 2 R M+N+1
(4.31)
m=−M
This formula has a number of interesting features. The interaction energy depends only −2|m| A ¯ Mm BNm of on the separation of the multipoles and on the combination M m=−M 2 the multipole components. As explained in Appendix A, the multipole moments QNn of a 2N -pole can be thought of as elements of the (2N + 1)-dimensional unitary irreducible representation WN of SO(3). The vector B with 2N +1 components BNn , −N ≤ n ≤ N is naturally an element of WN . Rotations G ∈ SO(3) about the centre X− of the mul N tipole field gN act on the multipole components via BNn → N n =−N Unn (G)BNn , where U N is a (2N + 1)-dimensional irreducible representation of SO(3) (because of the rescaling (4.16) and (4.17) this is not the standard unitary representation). With our
Interaction Energy of Well-Separated Skyrme Solitons
143
assumption that M ≤ N we can use the multipole components AMm , −M ≤ m ≤ M, to define the linear form FA : WN → R,
M
B →
2−2|m| A¯ Mm BNm .
(4.32)
m=−M
By assumption, the components AMm are not all zero, and therefore the map FA is non-degenerate. Writing the formula (4.31) in terms of this map as VMN = 4π
(M + N )! FA (B), R M+N+1
(4.33)
we immediately deduce the following result. Theorem 4.1. The interaction energy of an 2M -pole and a 2N -pole separated by a distance R is always non-vanishing for some relative orientation of the two multipoles. When such an orientation is chosen, the modulus of the interaction energy decreases with the separation as R −(M+N+1) . Proof. Assuming without loss of generality that the multipoles are separated along the x3 -axis and that M ≤ N , we have the formula (4.33) for the interaction energy. Since the map (4.32) defined in terms of the (non-vanishing) multipole components AMm of the 2M pole at X+ is non-degenerate it has a 2N -dimensional kernel. It then follows from the irreducibility of the (2N + 1)-dimensional representation WN that there exists a rotation G ∈ SO(3) such that U N (G)B is not in the kernel of FA for some G. For that G we thus have FA (U N (G)B) = κ = 0 and VMN = 4π κ(M + N )!R −(M+N+1) .
5. Attractive Forces and Existence of Minima The arguments of the previous section apply to the asymptotic pion fields of the Skyrme solitons U (1) and U (2) discussed in Sect. 3. In particular we note that the interaction energy E (3.22) for the leading multipole fields uM and vN is just the sum over iso-components of pairings of the form (4.29) a a a = 2 dx1 dx2 (uaM ∂3 vN − vN ∂3 uaM ). (5.1) Ea = −2uaM , vN x3 =0
Now pick one of the iso-indices, say a = 1, and use iso-rotations of the Skyrme 1 are non-vanishing. It solitons to make sure that the first iso-components u1M and vN then follows from Theorem 4.1 that we can make the multipole interaction energy E1 non-zero by spatial rotations of one of the Skyrme solitons. This fact is the main upshot of the calculations in the previous section and a crucial input for the following argument which was missing in [11]. Now consider the sum E = E1 + E2 + E3 .
(5.2)
We would like to show that we can always arrange for E to be negative by a suitable iso-rotation of one of the Skyrme solitons. We may assume that, possibly after re-labelling the pion fields, E1 ≥ E2 ≥ E3 .
(5.3)
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If E < 0 we are done, so suppose that E ≥ 0. Since we know that not all Ea vanish we can conclude that E1 > 0. Now perform an iso-rotation of Skyrme soliton (1) (1) 1 by 180 degrees around the third iso-spin axis. This reverses the sign of π1 and π2 and hence also the sign of E1 and E2 . The new value of E is E = − E1 − E2 + E3 = − E1 − ( E2 − E3 ) < 0
(5.4)
since − E1 < 0 and, with our ordering, −( E2 − E3 ) ≤ 0. Thus, the contribution E to the interaction energy of two Skyrme solitons U (1) and (2) U can always be made less than zero by suitable rotations and iso-rotations of Skyrme soliton 1. It follows from the discussion at the end of Sect. 3 that for |N − M| ≤ 2 and sufficiently large separation parameter R, E[UR ] < E[U (1) ] + E[U (1) ].
(5.5)
We conclude with a few comments on the implications of our result for the question of existence of general Skyrme solitons. As explained in Sect. 1, Esteban proved the existence of Skyrme solitons of arbitrary degree provided the strict inequality (1.19) holds. Our result (5.5) implies the inequality in those cases where minima exist in the sectors l and k − l, and where the associated multipoles have orders which do not differ by more than two. Since monopole fields do not arise in Skyrme solitons, the interaction energy E dominates at large separation if the leading multipole moments in Skyrme solitons are at most octupoles. As explained at the end of Sect. 2, the B = 7 Skyrme soliton is believed to have octupoles as leading multipoles, but there is no numerical evidence for leading multipoles of higher order. Unfortunately, it seems very difficult to rule out this possibility in general. Even if one could prove (or circumvent) the assumption concerning multipoles, the existence of attractive forces between Skyrme solitons is not sufficient to establish the inequality (1.19) for infima. Physically it seems reasonable that the existence of attractive forces should imply the existence of minima in every sector. However, we have not been able to develop this observation into a mathematical proof. Further thoughts and speculations in this direction can be found in [17]. Acknowledgement. BJS acknowledges financial support through an Advanced Research Fellowship of the Engineering and Physical Sciences Research Council. MAS thanks Rafe Mazzeo for directing him to the unique continuation theorem used in Sect. 2. NSM is grateful to Walter Kohn for drawing his attention to the formula (4.4).
A. Spherical Harmonics In this appendix we derive the relation between the standard spherical harmonics and the following functions on R3 − {0} used in the multipole expansion in Sect. 4.1: n N−n 1 ∂¯ ∂3 if n ≥ 0 r FNn (x) = . (A.1) N+n 1 −n ∂ ∂3 r if n < 0 Here N ≥ 0 and −N ≤ n ≤ N and ∂ = 21 (∂1 − i∂2 ). These functions are harmonic in their domain: FNn = 0.
(A.2)
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They are also homogeneous of degree −(N + 1) so that they can be written as FNn =
1 Nn (θ, ϕ), r N+1
(A.3)
where (θ, ϕ) are the usual spherical coordinates on the two-sphere centred at the origin. Since the Laplace operator takes the following form in spherical coordinates: 1 ∂2 1 r + 2 ω , r ∂r 2 r
=
(A.4)
where ω is the Laplace operator on the 2-sphere of unit radius, it follows from (A.2) that ω Nn = −N (N + 1)Nn .
(A.5)
Define the generator of rotations about the 3-axis J3 = −i
∂ ∂ϕ
(A.6)
and express it in terms of complex coordinates z = x1 + ix2 and complex derivatives in the x1 x2 plane: ¯ J3 = z∂ − z¯ ∂.
(A.7)
The operator ∂ acts as a raising operator and the operator ∂¯ acts as a lowering operator for J3 : ¯ = ∂¯ [J3 , ∂]
and
[J3 , ∂] = −∂.
(A.8)
Thus if φn is a function on R3 which is an eigenfunction of J3 with eigenvalue n then ¯ n is an eigenfunction with eigenvalue n + 1 provided it is not zero. Similarly ∂φn is ∂φ an eigenfunction of J3 with eigenvalue n − 1 provided it is not zero. It follows from (A.1) that n ∂¯ FN−n,0 if n ≥ 0 (A.9) FNn = ∂ −n FN+n,0 if n < 0. Noting that, by rotational symmetry about the 3-axis, J3 N0 = 0
for all N,
(A.10)
we conclude that J3 FNn = nFNn .
(A.11)
Nn = r N+1 FNn
(A.12)
J3 Nn = nNn .
(A.13)
It follows that
also satisfies
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Thus, to sum up, the Nn are functions on S 2 , which are eigenfunctions of both the Laplace operator (A.5) and the operator J3 with eigenvalues respectively −N (N + 1) and n. It follows from standard harmonic analysis on S 2 that they must be proportional to the spherical harmonics YNn . In the case n = 0 we can determine the proportionality constant by evaluating both N 0 and YN0 on the positive 3-axis. With the usual normalisation [18] we find 2N + 1 (−1)N YN0 = (A.14) N0 . 4π N! The relation between YNn and Nn for n = 0 is harder to compute. Let us assume initially that n > 0. Starting with the standard expression for the associated Legendre function in terms of Legendre polynomials n d PN (cos θ) (A.15) PNn (cos θ ) = (−1)n sinn θ d cos θ and the expression of the spherical harmonic in terms of the associated Legendre function (see e.g. [18] p. 99) we have the relation
n ∂ n n (N − n)! iϕ YNn (θ, ϕ) = (−1) YN0 (θ, ϕ). (A.16) sin θ e (N + n)! ∂ cos θ Then using (A.14) and the definition of N0 we deduce N+n (N − n)! 2N + 1 (−1) N+1 n n N 1 YNn (θ, ϕ) = r z Dθ ∂3 , 4π N! (N + n)! r
(A.17)
where Dθ =
1 ∂ x3 ∂ =− + ∂3 r ∂ cos θ ρ ∂ρ
(A.18)
and z = ρeiϕ with ρ = r sin θ as in the main text of the paper. Then we use the commutation relation [Dθ , ∂3 ] =
1 ∂ ρ ∂ρ
(A.19)
to move Dθ past ∂3 in (A.17). Noting that Dθ r = 0 we find 1 ∂ n N−n 1 (−1)N+n N+1 2N + 1 YNn (θ, ϕ) = r . (A.20) ∂3 zn √ 4π ρ ∂ρ r (N − n)!(N + n)! Now exploit that on any function f which only depends on ρ and x3 , 1 ∂f 2¯ (ρ, x3 ) = ∂f (ρ, x3 ) ρ ∂ρ z to conclude YNn (θ, ϕ) = r
N+1
2N + 1 (−1)N+n 2n ∂¯ n ∂3N−n √ 4π (N − n)!(N + n)!
(A.21) 1 . r
(A.22)
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Thus we finally arrive at the promised relationship between YNn and Nn , valid for n ≥ 0: 2N + 1 (−1)N+n 2n (A.23) YNn = Nn . √ 4π (N − n)!(N + n)! ¯ N(−n) and YNn = To deduce the corresponding result for n < 0 we note that Nn = n ¯ (−1) YN(−n) for n < 0. Thus for n < 0: 2N + 1 (−1)N 2|n| YNn = (A.24) Nn . √ 4π (N − n)!(N + n)! References 1. Skyrme, T.H.: A non-linear field theory. Proc. Roy. Soc. London A 260, 127–138 (1961) 2. Esteban, M.J.: A direct variational approach to Skyrme’s model for meson fields. Commun. Math. Phys. 105, 571–591 (1986) 3. Esteban, M.J., M¨uller, S.: Sobolev maps with integer degree and applications to Skyrme’s problem. Proc. Roy. Soc. London A 436, 197–201 (1992) 4. Faddeev, L.D.: Some comments on the many-dimensional solitons. Lett. Math. Phys. 1, 289–293 (1976) 5. Manton, N.S.: Geometry of Skyrmions. Commun. Math. Phys. 111, 469–478 (1987) 6. Kapitanski, L.B., Ladyzenskaia, O.A.: On the Coleman’s principle concerning the stationary points of invariant functionals. Zap. Nauchn. Semin, LOMI 127, 84–102 (1983) 7. Kopeliovich, V.B., Stern, B.E.: Exotic Skyrmions. JETP Lett. 45, 203–207 (1987); Verbaarschot, J.J.M.: Axial symmetry of bound baryon number two solution of the Skyrme model. Phys. Lett. 195 B, 235–239 (1987) 8. Braaten, E., Carson, L., Townsend, S.: Novel structure of static multisoliton solutions in the Skyrme model. Phys. Lett. 235 B, 147–152 (1990) 9. Battye, R.A., Sutcliffe, P.M.: Symmetric Skyrmions. Phys. Rev. Lett. 79, 363–366 (1997) 10. Battye, R.A., Sutcliffe, P.M.: Skyrmions, fullerenes and rational maps. Rev. Math. Phys. 14, 29–86 (2002) 11. Castillejo, L., Kugler, M.: The interaction of Skyrmions. Unpublished preprint, 1987 12. Melrose, R.B.: The Atiyah-Patodi-Singer Index Theorem. Research Notes in Mathematics 4, Wellesley: A.K. Peters, 1993 13. Mazzeo, R.R.: Elliptic theory of differential edge operators I. Commun. Partial Diff. Eqs. 16, 1615– 1664 (1991) 14. Manton, N.S.: Skyrmions and their multipole moments. Acta Phys. Polonica B 25, 1757–1764 (1994) 15. Schroers, B.J.: Dynamics of moving and spinning Skyrmions. Zeitschrift f¨ur Physik C 61, 479–494 (1994) 16. H¨ormander, L.: The analysis of linear partial differential operators III. Grundlehren der mathematischen Wissenschaften 274, Berlin: Springer Verlag, 1985 17. Schroers, B.J.: On the existence of minima in the Skyrme model. JHEP Proceedings PRHEPunesp2002/034 Integrable Theories, Solitons and Duality, Sao Paulo, 2002 18. Jackson, J.D.: Classical Electrodynamics. New York: Wiley, 1972 Communicated by P. Constantin
Commun. Math. Phys. 245, 149–176 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1007-1
Communications in
Mathematical Physics
The Selberg Zeta Function for Convex Co-Compact Schottky Groups Laurent Guillop´e1 , Kevin K. Lin2 , Maciej Zworski2 1
Laboratoire Jean Leray (UMR CNRS-UN 6629), D´epartement de Math´ematiques, Facult´e des Sciences et des Techniques, 2, rue de la Houssini`ere. 44322 Nantes Cedex 3, France. E-mail:
[email protected] 2 Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, USA. E-mail:
[email protected];
[email protected] Received: 27 January 2003 / Accepted: 23 June 2003 Published online: 20 January 2004 – © Springer-Verlag 2004
Abstract: We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on the hyperbolic space Hn+1 : in strips parallel to the imaginary axis the zeta function is bounded by exp(C|s|δ ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp(C|s|n+1 ), and it gives new bounds on the number of resonances (scattering poles) of \Hn+1 . The proof of this result is based on the application of holomorphic L2 techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider \Hn+1 as the simplest model of quantum chaotic scattering. 1. Introduction In this paper we give an upper bound for the Selberg zeta function of a convex co-compact Schottky group in terms of the dimension of its limit set. This leads to a Weyl-type upper bound for the number of zeros of the zeta function in a strip with the number of degrees of freedom given by the dimension of the limit set plus one. We also report on preliminary numerical computations which indicate that our upper bound may be sharp, and close to a possible lower bound. Our motivation comes from the study of the distribution of quantum resonances – see [39] for a general introduction. Since the work of Sj¨ostrand [33] on geometric upper bounds for the number of resonances, it has been expected that for chaotic scattering systems the density of resonances near the real axis can be approximately given by a power law with the power equal to half of the dimension of the trapped set (see (1.1) below). Upper bounds in geometric situations have been obtained in [36] and [38]. Recent numerical studies in the semi-classical and several convex obstacles settings, [12, 13 and 14] respectively, have provided evidence that the density of resonances satisfies a lower bound related to the dimension of the trapped set. In complicated situations
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which were studied numerically, the dimension is a delicate concept and it may be that different notions of dimension have to be used for upper and lower bounds – this point has been emphasized in [14]. Generally, the zeros of dynamical zeta functions are interpreted as the classical correlation spectrum [32]. In the case of convex co-compact hyperbolic quotients, X = \Hn+1 quantum resonances also coincide with the singularities of the zeta function – see [22]. The notion of the dimension of the trapped set is also clear as it is given by 2(1 + δ). Here δ = dim () is the dimension of the limit set of , that is the set of accumulation points of any -orbit in Hn+1 , () ⊂ ∂Hn+1 . Hence we may expect that m (s) ∼ r 1+δ , (1.1) | Im s|≤r, Re s>−C
where m (s) is the multiplicity of the zero of the zeta function of at s. Referring for definitions of Schottky groups and zeta functions to Sects. 2 and 3 respectively we have Theorem. Suppose that is a convex co-compact Schottky group and that Z (s) is its Selberg zeta function. Then for any C0 > 0 there exists C1 such that for | Re s| < C0 , |Z (s)| ≤ C1 exp(C1 |s|δ ),
δ = dim ().
(1.2)
The proof of this result is based on the quasi-self-similarity of limit sets of convex co-compact Schottky groups and on the application of holomorphic L2 -techniques to the study of the determinants of the Ruelle transfer operators. If we use the convergence of the product representation (3.1) of the zeta function for Re s large and apply Jensen’s theorem we obtain the following Corollary 1. Let m (s) be the multiplicity of a zero of Z at s. Then, for any C0 , there exists some constant C1 such that for r > 1, {m (s) : r ≤ | Im s| ≤ r + 1, Re s > −C0 } ≤ C1 r δ , (1.3) where δ = dim (). We can apply the preceding results to Schottky manifolds: a hyperbolic manifold is called Schottky if its fundamental group is Schottky. The case of surfaces is of special interest: any convex co-compact hyperbolic surface is Schottky. With the description of the divisor of the zeta function through spectral data established by Patterson and Perry [22] (using the results by Bunke and Olbrich [2] in odd dimension), we can reformulate the preceding corollary nicely in the resonance setting. We do it only for surfaces (see below for short comments on higher dimensions). Corollary 2. Let X be a convex co-compact hyperbolic surface, SX be the set of the scattering resonances of the Laplace-Beltrami operator on X and mX (s) be the multiplicity of resonance s. Then, for any C0 , there exists some constant C1 such that for r > 1, {mX (s) : s ∈ R, r ≤ | Im s| ≤ r + 1, Re s > −C0 } ≤ C1 r δ , (1.4) where 2(1 + δ) is the Hausdorff dimension of the recurrent set for the geodesic flow on T ∗ X.
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This corollary is stronger than the result obtained in [38] where the upper bound of the type (1.1) was given. In fact, the upper bound (1.4) is what we would obtain had we had a Weyl law of the form r 1+δ with a remainder O(r δ ). That local upper bounds of this type are expected despite the absence of a Weyl law has been known since [25]. F. Naud [21] has proved the existence of ε > 0 such that the domain {Re s > δ − ε} \ {δ} is resonance free. Section 7 deals with numerical computations of the density of zeros. They show that (1.1) may be true. In fact, in the range of Im s used in the computation we see that the number of zeros grows fast. If the range of Re s is large (and fixed) we need very large Im s to see the upper bound of Corollary 2. The computations also show that our bound on the zeta function is optimal. For values of Z (s) with Re s negative we see that we need very large Im s to see the onset of the upper bound. That is not surprising since we recall in Proposition 3.2 that log |Z (s)| = O(|s|n+1 ), and that this bound is optimal (and of course δ < n). We refer to Sect. 7 for the details and present here two pictures only. We take for a group generated by compositions of reflections in three symmetrically spaced circles perpendicular to the unit circle, and cutting it at the angles 30◦ (see Fig.3 for the 110◦ angle). Figure 1 shows the density of zeros of Z in that case and Fig. 2 plots the values of log | log |Z ||.
0.25
(log(#(zeros))−const)/log(y)−1
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 200
400
600
800
1000
y Fig. 1. The plot of (log N(y) − C)/ log y − 1, where N(y) is the number of zeros with | Im s| ≤ y, for a Schottky reflection group with δ 0.184. Different lines represent different strips | Re s| ≤ C1 , and the thick blue line gives δ. The constant C is determined by least squares regression; see Sect. 7 for details
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1 0.9
log(log(|Z(s)|))/log(|s|)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
200
400
600
800
1000
|s| Fig. 2. Density of values of log | log |Z ||/ log |s| for a Schottky reflection group with δ 0.184
Finally, we stress that our main theorem is most likely to be a special case of a more general statement relating the growth (and hence the density of zeros) of zeta functions to the dimensions of natural measures appearing in the underlying dynamics. Finding this general statement is an interesting problem. In that direction the methods of this paper have been applied in [34] to give bounds on zeta functions associated to the dynamics of z → z2 + c, c < −2. Unlike in this paper the numerical study in [34] was based on the proof of the upper bound (1.2). 2. Schottky Groups The hyperbolic geometry on the simply connected curvature −1 space Hn+1 and the conformal geometry on its boundary at infinity ∂Hn+1 = Sn share the same automorphism group: the isometry group Isom(Hn+1 ) and the conformal group Conf(Sn ) (with the conformal structure given by the standard metric on Sn of curvature +1) are isomorphic. In particular any isometry g of Hn+1 induces on Sn a conformal map γ , whose conformal distortion at the point w ∈ Sn will be denoted by Dγ (w) . There is also a correspondence between balls D and spheres C on Sn (for n = 2, the original setting for Kleinian groups, these are discs and circles) and half-spaces P and geodesic hyperplanes H in Hn+1 : D = P ∩ ∂Hn+1 and C = H ∩ ∂Hn+1 . Given a hyperplane H (a sphere C resp.), its interior hyperplane (ball) will be given by the choice of a component of Hn+1 \ H (Sn \ C resp.)
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Let us review the definition of a Schottky group (see [16, 18, 30] and references given there). Let k, be integers with 0 ≤ k ≤ , k + ≥ 3 and Di , i = 1, . . . , k + be a collection of mutually disjoint topological balls on the sphere Sn . We suppose that, for each i = 1, . . . , k, there exists a conformal map γi such that γi (Sn \ Di ) = Di+ , and, for i = k + 1, . . . , , there exists a conformal symmetry γi such that γi (Sn \ Di ) = Di . The Schottky marked group = (D1 , . . . , D+k , γ1 , . . . , γ ), is the group of conformal transformations generated by the γ1 , . . . , γ . We take k+ ≥ 3 to exclude elementary groups. If in addition the closures Di are mutually disjoint, which will be assumed here, the Schottky group is convex co-compact. If, for i = 1, . . . , , i denotes the cyclic group generated by γi , the group is the free product 1 ∗ . . . ∗ , with fundamental domain Sn \ ∪+k i=1 Di for its action on the sphere Sn . If we introduce, for j = + 1, . . . , + k, the transformation γj = γj−1 − , every non-trivial element γ ∈ is uniquely written as γ = γ (1)γ (2) . . . γ (N ) with each γ (I ) in {γ1 , . . . , γ+k } and γ (I )γ (I + 1) = 1, I = 1, . . . , N − 1. The uniquely defined integer N is the word length |γ | of γ (with respect to the generators set {γ1 , . . . , γ }). Let us discuss some particular cases. We suppose that each Di is a geometric ball, boundary at infinity of an hyperbolic half-space Pi : the marked Schottky group is said to be classical and can be described as an isometry group of the interior hyperbolic space Hn+1 . If k = 0, the group is called a Schottky reflection group. For i = 1, . . . , , the symmetry γi is the conformal symmetry with respect to the sphere ∂Di and is omitted in the marking : = (D1 , . . . , D ). The corresponding hyperbolic isometry group is the Schottky marked reflection group (P1 , . . . , P ) , generated by the hyperbolic symmetries si , i = 1, . . . , with respect to the hyperplane Hi = ∂Pi (with infinite boundary ∂Di ). Figure 3 shows the fundamental domain of a reflection group in H2 with = 3. If k = , the Schottky group contains only orientation preserving transformations. If gi is the hyperbolic isometry of Hn+1 with action at infinity given by γi , then the classical Schottky group has a hyperbolic marking = (P1 , . . . , P2 , g1 , . . . , g ). The Schottky domain Hn+1 \ ∪2 i=1 Pi is a fundamental domain for the action of on n+1 H . A group is said to be a Schottky group if it admits a presentation induced by a configuration of balls as described above. The subgroup + of orientation preserving transformations of a Schottky group is Schottky: for the reflection Schottky group = (D1 , . . . , D ), we have + = (D1 , . . . D−1 , γ D1 , . . . , γ D−1 , γ γ1 , . . . , γ γ−1 ), where γi denotes the conformal symmetry with respect to the sphere ∂Di . Such a group was called symmetrical by Poincar´e [26]. An oriented hyperbolic manifold M is said to be (classical) Schottky if its fundamental group π1 (M) (realized as a discrete subgroup of Isom+ (Hn+1 )) admits a (classical) Schottky marking.
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Fig. 3. Tessellation in H2 by the group, θ , θ = 110◦ , generated by symmetries in three symmetrically placed lines each cutting the unit circle in an 110◦ angle, with the fundamental domain of its Schottky subgroup of direct isometries, θ+ , and the associated Riemann surface θ+ \H2 . The dimension of the limit set is δ = 0.70055063 . . .
Non-trivial elements of are either symmetries or hyperbolic. For a hyperbolic element γ ∈ , there exists α ∈ Isom(Hn+1 ) such that, in the Poincar´e model Hn+1 n+1 R+ = R+ × R n , n+1 , O(γ ) ∈ O(n), (γ ) > 0. (2.1) α −1 γ α(x, y) = e(γ ) (x, O(γ )y), (x, y) ∈ R+
If ⊂ Isom+ (Hn+1 ), the conjugacy classes of hyperbolic elements, [γ1 ] = [γ2 ] ⇐⇒ ∃ β ∈ βγ1 β −1 = γ2 , are in one-to-one correspondence with closed geodesics of X = \Hn+1 . The primitive geodesics correspond to conjugacy classes of primitive elements of (that is, elements which are not non-trivial powers). The magnification factor exp (γ ) in (2.1) gives the length (γ ) of the closed geodesic. The limit set, () of a discrete subgroup, , of Isom(Hn+1 ), is defined as the set in Hn+1 = Hn+1 ∪ ∂Hn+1 of accumulation points of any -orbit in Hn+1 : the limit set () is included in the boundary ∂Hn+1 . In the convex co-compact case it has a particularly nice structure; furthermore, for Schottky groups, it is totally disconnected and included in D = ∪+k i=1 Di . The aspects relevant to us come from the work of Patterson and Sullivan – see [35] and references given there. As will be discussed in more detail in Sect. 4, the limit set has a quasi-self-similar structure and a finite Hausdorff measure at dimension δ = δ(). The limit set is related to the trapped set, K, of the usual scattering [33, 36], that is the set of points in phase space such that the trajectory through that point does not escape to infinity in either direction: if π ∗ is the projection from T ∗ Hn+1 on T ∗ (\Hn+1 ), the trapped set K is the union of the projections π ∗ (Cξ η ), where Cξ η is the geodesic line (ξ, η) with extremities ξ and η, distinct points of the limit set (). In particular, we have dim K = 2(δ + 1), see [38]. To stress the connection to closed geodesics on \Hn+1 let us also mention that, generalizing earlier results of Guillop´e [7] and Lalley [10], Perry [23] showed that {[γ ] : γ primitive, (γ ) < r} ∼
eδr . δr
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Fig. 4. A typical limit set for a convex co-compact Schottky group, ⊂ Isom(H3 ), taken from [18]
3. Properties of the Selberg Zeta Function For , a discrete subgroup of Conf(Hn+1 ), the Selberg zeta function is defined as follows: Z (s) = 1 − e−iθ(γ ),α e−(s+|α|)(γ ) . (3.1) {[γ ]} α∈Nn0
Here, γ ∈ are hyperbolic, exp((γ ) + iθj (γ )) are the eigenvalues of the derivative of the action of γ on Sn at the repelling fixed point f+ (γ ) (exp(iθj (γ )) are the eigenvalues of the isometry O(γ ) in the normal form (2.1)) and [γ ] its conjugacy class in . The first product in (3.1) goes over the primitive conjugacy classes. The real exp (γ ) is called the dilation factor of γ (we have always (γ ) > 0 because we consider the fixed repelling point). An element is called primitive if it is not a non-trivial power of another element. In terms of hyperbolic geometry, the isometry γ keeps invariant the geodesic line (f+ (γ ), f+ (γ −1 )) in its action on the hyperbolic space Hn+1 , whose projection on \Hn+1 is a closed geodesic of length (γ ) and holonomy spectrum θj (γ ), j = 1, . . . , n. The induced correspondence between conjugacy classes of and closed geodesics of \Hn+1 is one to one. The word length |[γ ]| of the conjugacy class [γ ] is the minimum of the word length of the elements in this conjugacy class. For the Schottky group = (D1 , . . . , D+k , γ1 , . . . , γ ), we define the following map T = T on D = +k i=1 Di : T : D −→ Sn , T (x) = γi (x), x ∈ Di .
(3.2)
We need to find an open neighbourhood of the limit set where T is strictly expanding in the following sense : F defined on V is said to be (strictly) expanding on V ⊂ Sn with respect to the metric if there exists θ ≥ 1 (θ > 1) such that
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DF (v)ξ ≥ θ ξ ,
v ∈ V , ξ ∈ Tv V .
In the case when is a symmetrical Schottky group, we can suppose that up to a conformal identification ∂D is a great circle of the sphere Sn . For the metric on Sn we can take the metric induced by its embedding in Rn+1 . The inversion, σ , is the restriction to Sn of the symmetry on Rn+1 with respect to the euclidean hyperplane containing ∂D , hence an isometry. Each inversion σi , i = 1, . . . , − 1 is expanding on the ball Di . Hence, the map T is expanding on D = −1 i=1 Di ∪ σ Di , strictly expanding on any open set precompact in D. However, the map T is not expanding on D in general. To circumvent that we need to consider refinements, DN , defined by recurrence: D1 = D,
DN = T −1 (DN−1 ) ∩ DN−1 ,
N > 1.
N DiN . The collection of sets {DiN }i coincides with Each set DN is a disjoint union ∪di=1 the collection
{Dγ }γ ∈N , N = {γ ∈ : |γ | = N }, where Dγ = γ (N)−1 . . . γ (2)−1 Dγ (1)
γ = γ (1) . . . γ (N ),
if
with Dγi = Di , i = 1, . . . , 2. The iterated map, T N , is defined on DN , and T |γ | |Dγ = γ . The map T is strictly expanding on DN for N big enough as explained in the following lemma (see Lemma 9.2 in Lalley [10] for a similar result). Lemma 3.1. Let be a Schottky group and D, T defined as in (3.2). There exist an integer N ≥ 1, a metric , defined on DN , and a real β > 1 such that DT (w) ≥ β, w ∈ DN . The metric can be taken analytic on DN . Proof. Let us recall that any M¨obius transformation γ of Rk which does not fix the point at infinity, ∞, has an isometric sphere Sγ , and that γ is strictly contracting on any compact subset of its exterior (the unbounded component of Rk \ Sγ ). The sphere Sγ is centered at γ −1 ∞ and if rγ is its radius, we have (see [30]) ||γ x − γ y|| =
rγ2 ||x − y|| ||x − γ −1 ∞|| ||y − γ −1 ∞||
,
x, y ∈ Rk \ {γ −1 ∞}.
(3.3)
Up to a conformal transformation, we can suppose that is a subset of Conf(Rn ), with the point at infinity in its ordinary set. No non-trivial element in fixes ∞ and, taking in (3.3) as x, y points in the upper half-plane Hn+1 (and the Poincar´e extension of γ to Rn+1 ), we deduce that the set of radii {rγ , γ ∈ } accumulates only at 0. For γ = γ (1) . . . γ (|γ |), we have γ −1 (∞) ∈ Dγ ⊂ Dγ (2)...γ (|γ |) ⊂ . . . ⊂ Dγ (|γ |−1) γ (|γ |) ⊂ Dγ (|γ |) , hence there exists N0 such that the isometric sphere Sγ is included in Dγ (|γ |) if |γ | ≥ N0 . For such a γ , the interior of the isometric sphere Sγ −1 is included in Dγ (1)−1 : its exterior
The Selberg Zeta Function for Convex Co-Compact Schottky Groups
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contains all the Dγi , γi = γ (1)−1 , hence γ −1 is strictly contracting on ∪γi =γ (1)−1 Di . As γ (Dγ ) ⊂ ∪γi =γ (1)−1 Di , the map γ = T |γ | |Dγ is expanding on Dγ ⊂ D|γ | . We have just proved the existence of η0 > 1 such that DT N0 (w) ≥ η0 , w ∈ DN0 , hence there exist constants C > 0, θ > 1 such that DT p (w) ≥ Cθ p , w ∈ Dp , p ≥ 1. Taking an integer N such that Cθ N > 1, we define on DN the metric (introduced by Mather [17])
V =
N−1
DT p (w)V ,
V ∈ Tw D N ,
p=0
which concludes the proof.
Let us fix now an integer N as in Lemma 3.1. Let Sn be a Grauert tube of Sn , that is a complex n-manifold containing Sn as a totally real submanifold (that is all we need). Sn . By further Let us then choose open neighbourhoods1 Di , i = 1, . . . , dN of DiN in shrinking, we can suppose that the open sets Di are mutually disjoint, and that the real N analytic maps T and DT extend holomorphically to D = ∪di=1 Di , with DT ≥ β δi for some β > 1. The open sets Di can be chosen to be a union Di = ∪k=1 Dik of open sets, each one biholomorphic to the ball BCn (0, 1) in Cn . With this formalism in place we define the Ruelle transfer operator L(s)u(z) = [DT (w)]−s u(w), z ∈ D, T w=z
u ∈ H (D), H (D) = {u holomorphic in D : 2
|u(z)|2 dm(z) < ∞},
2
(3.4)
D 1
[DT (w)] is holomorphic in D, [DT (w)]Sn =| det DT | n . The only difference from the standard definition lies in choosing L2 spaces of holomorphic functions instead of Banach spaces. However we still obtain the analogue of a (special case of a) result of Ruelle [31] and Fried [5]: Proposition 3.2. Suppose that L(s) : H 2 (D) → H 2 (D) is defined by (3.4). Then for all s ∈ C L(s) is a trace class operator and | det(I − L(s))| ≤ exp(C|s|n+1 ).
(3.5)
Proof. The proof is based on estimates of the characteristic values, µ (L(s)). We will show that there exists C > 0 such that µ (L(s)) ≤ CeC|s|−
1 n /C
.
(3.6)
To see how that is obtained and how it implies (3.5) let us first recall some basic properties of characteristic values of a compact operator A : H1 → H2 , where Hj ’s are Hilbert spaces. We define A = µ0 (A) ≥ µ1 (A) ≥ · · · ≥ µ (A) → 0, to be 1 1 the eigenvalues of (A∗ A) 2 : H1 → H1 , or equivalently of (AA∗ ) 2 : H2 → H2 . The min-max principle shows that µ (A) = 1
min
max Av H2 .
V ⊂H1 v∈V codim V = v H1 =1
We drop the index N in the open sets DiN for the purpose of notational simplicity.
(3.7)
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The following rough estimate will be enough for us here: suppose that {xj }∞ j =0 is an orthonormal basis of H1 , then µ (A) ≤
∞
Axj H2 .
(3.8)
j =
To see this we will use V = span {xj }∞ j = in (3.7): for v ∈ V we have, ∞ ∞ ∞ Axj ≤ v
Av H2 = v, xj H1 Axj ≤ |v, x |
Axj H2 , j H 1 H2 j = j = j = H2
from which (3.7) gives (3.8). We will also need some real results about characteristic values. The first is the Weyl inequality (see [6], and also [33, Appendix A]). It says that if H1 = H2 and λj (A) are the eigenvalues of A, |λ0 (A)| ≥ |λ1 (A)| ≥ · · · ≥ |λ (A)| → 0, then for any N , N
(1 + |λ (A)|) ≤
=0
N
(1 + µ (A)).
=0
In particular if the operator A is of trace class, that is if, determinant def
det(I + A) =
∞
µ (A)
< ∞, then the
(1 + λ (A)),
=0
is well defined and | det(I + A)| ≤
∞
(1 + µ (A)).
(3.9)
=0
We also need to recall the following standard inequality about characteristic values (see [6]): µ1 +2 (A + B) ≤ µ1 (A) + µ2 (B).
(3.10)
We finish the review, as we started, with an obvious equality: suppose that Aj : H1j →
H2j and we form Jj=1 Aj : Jj=1 H1j → Jj=1 H2j , as usual, Jj=1 Aj (v1 ⊕ · · · ⊕ vJ ) = A1 v1 ⊕ · · · ⊕ AJ vJ . Then ∞ ∞ J J
µ Aj = µ (Aj ). (3.11) =0
j =1
j =1 =0
With these preliminary facts taken care of, we see that (3.6) implies (3.5). In fact, (3.9) shows that | det(I − L(s))| ≤
∞ =0
(1 + eC|s|−
1 n /C
) ≤ eC1 |s|
n+1
.
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Hence it remains to establish (3.6). For that we will write H 2 (D) =
dN
H 2 (Di ),
i=1
and introduce, for i, j = 1, . . . , d N , the operator Lij (s) : H 2 (Di ) → H 2 (Dj ), nonzero only when T (Di ) and Dj are not disjoint, where Lij (s)u(z) = [Dfij (z)]s u(fij (z)), z ∈ Dj , fij = (T Di )−1Dj , def
(3.12)
where [Df ] is defined as in (3.4). From (3.10) and a version of (3.11) we then have µ (L(s)) ≤
max 2µ[/C] (Lij (s)).
1≤i,j ≤dN
To estimate µk (Lij (s)), let us recall that Di was taken as a union of open sets Dik , k = 1, . . . , δi biholomorphic to BCn (0, 1): as fij (Dj ) is relatively compact in Di , ρ we can find ρ ∈ (0, 1) (independent of i, j = 1, . . . , dN ) such that fij (Dj ) ⊂ Di , ρ ρ ρ δi where Di = ∪k=1 Dik with Dik ⊂ Dik the pullback of the ball BCn (0, ρ) through the biholomorphism of Dik onto BCn (0, 1). The map Lij (s) is the composition H 2 (D
i)
R
/ ⊕δi H 2 (Dik ) k=1
ρ
⊕Rik
/ ⊕δi H 2 (D ρ ) ik k=1
πρ
/ H 2 (D ρ ) i
Lρij (s)
/ H 2 (Dj ) ,
ρ
where R and Rik are the natural restrictions, π ρ is the orthogonal projection on the space ρ ρ ρ i H 2 (Di ) immersed in ⊕δk=1 H 2 (Dik ) by the natural restrictions and Lij (s) is defined by ρ the same formula (3.12) as Lij (s). The maps R and π are bounded, while the norm of ρ ρ Lij (s) is bounded by CeC|s| . The bounds on the singular values of Rik , given up to a bounded factor by the following lemma, give the bound 1/n /C
µ (Lij (s)) ≤ CeC|s|−
,
for some C, which completes the proof of (3.6).
2 Lemma 3.3. Let ρ ∈ (0, 1) and : Cn (0, 1)) → H (BCn (0, ρ)) induced by the restriction map of BCn (0, 1) to BCn (0, ρ). Then, for any ρ ∈ (ρ, 1) there exits a constant C such that
Rρ
H 2 (B
1/n
ρ µ (R ρ ) ≤ C
.
Proof. We use (3.8) with the standard basis (xα )α∈Nn of H 2 (BCn (0, 1)): α1 αn |xα (z)|2 dm(z) = 1, α ∈ N0n , xα (z) = cα z1 · · · zn ,
(3.13)
BCn (0,1)
for which we have
R ρ (xα ) 2 =
|xα (w)|2 dm(w) = ρ 2(|α|+n) . BCn (0,ρ)
The number of α’s with |α| ≤ m is approximately mn and hence by (3.8) we have 1/n k n−1 ρ k ≤ C ρ . µ (R ρ ) ≤ C Ck≥1/n
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The next proposition is a modification of standard zeta function arguments – see [27 and 28] for the discussion of the hyperbolic case. Proposition 3.4. Let L(s) be defined by (3.4). Then, if Z is the zeta function (3.1) corresponding to the group , Z (s) = det(I − L(s)). Proof. For s fixed and z ∈ C, def
h(z) = det(I − zL(s)) is, in view of (3.6) and (3.9), an entire function of order 0. For |z| sufficiently small log(I − zL(s)) is well defined and we have ∞ zm m (3.14) tr(L(s) ) . det(I − zL(s)) = exp − m m=1
The correspondence between the conjugacy classes of hyperbolic elements and the periodic orbits of T is particularly simple for Schottky groups and we recall it in the form given in [28] (where it is given in the more complicated setting of co-compact groups): Conjugacy classes of with contraction factor exp (γ ) and word length |[γ ]| are in one to one correspondence with periodic orbits {x, T x, · · · , T m−1 x} such that [DT m (x)] = exp (γ ), and m = |[γ ]|. For prime closed geodesics we have the same correspondence with primitive periodic orbits of T . It is not needed for us to recall the precise definition of the word length. Roughly speaking it is the number of generators of needed to write down γ . To evaluate tr(L(s)m ) we write tr Li1 i2 (s) ◦ · · · ◦ Lim i1 (s) , tr L(s)m = (i1 ,··· ,im )
where in the notation of (3.12) we have Li1 i2 (s) ◦ · · · ◦ Lim i1 (s)u(z) = [D(fi1 i2 ◦ · · · ◦ fim i1 )(z)]s u(fi1 i2 ◦ · · · ◦ fim i1 (z)), fi1 i2 ◦ · · · ◦ fim i1 : Di1 −→ Di1 . The trace of this operator is non-zero only if fi1 i2 ◦ · · · ◦ fim i1 has a fixed point in Di1 . Since this transformation corresponds to an element of that fixed point is unique. Let us call this element γ −1 . Since it corresponds to a given periodic point, x, of T n (corresponding to a fixed point of fi1 i2 ◦ · · · ◦ fim i1 ), γ is determined uniquely by x and n: γ = γ (x, n), T n x = x. By conjugation and a choice of coordinates z = (z1 , . . . , zn ) it can be put into the form γ (z) = e(γ ) (eiθ1 (γ ) z1 , · · · , eiθn (γ ) zn ),
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and the trace can be evaluated on the Hilbert space H 2 (BCn (0, 1)). Using the basis (3.13) we can write the kernel of Li1 i2 ◦ · · · ◦ Lim−1 im as Li1 i2 ◦ · · · ◦ Lim−1 im (z, w) = |γ (0)|−s cα (γ −1 (z))α w α =
α∈Nn0
cα e−(s+|α|)(γ )−iθ(γ ),α zα w α .
α∈Nn0
The evaluation of the trace is now clear. Returning to (3.14), we obtain for Re s sufficiently large (using {[γ ]}’s to denote the conjugacy classes of primitive elements of ), ∞ n z det(I − zL(s)) = exp − e−(s+|α|)(γ (x,n))−iθ(γ (x,n)),α n n n n=1 T x=x α∈N0 ∞ nk ∞ z = exp − e−k((s+|α|)(γ )−iθ(γ ),α) k n n=1 {[γ ]} k=1 |[γ ]|=n
=
α∈N0
1 − z|γ | e−iθ(γ ),α e−(s+|α|)(γ )
{[γ ]} α∈Nn0
which in view of (3.1) proves the proposition once we put z = 1.
Remark. The proof above is inspired by the work on the distribution of resonances in Euclidean scattering - see [37, Prop. 2]. The Fredholm determinant method and the use of Weyl inequalities in the study of resonances were introduced by Melrose [20] and developed further by many authors – see [33, 39], and references given there. That was done at about the same time as David Fried (across the Charles River from Melrose) was applying the Grothendieck-Fredholm theory to multidimensional zeta-functions [5]. In both situation the enemy is the exponential growth for complex energies s, which is eliminated thanks to analyticity properties of the kernel of the operator. Finally, we remark that in view of the lower bounds on the number of zeros of Z obtained in [8] in dimension two, and in [24] in general, we see from Proposition 3.4 that the upper bound (3.5) is optimal for any . 4. Applications of Quasi-Self-Similarity of () In this section we will review the results on convex co-compact Schottky groups (coming essentially from [35]) and apply them to refine the domain D used in the definition of the transfer operator (3.4). We start with a more general definition of convex co-compact subgroups of Isom(Hn+1 ). A discrete subgroup is called convex co-compact if def
\C(()) is compact, C(()) = convex hull(()).
(4.1)
Here, the convex hull is meant in the sense of the hyperbolic metric on Hn+1 : () ⊂ ∂Hn+1 , and acts on it in the usual way. In particular this implies that \C(()) has a compact fundamental domain in Hn+1 . The first result gives a quasi-self-similarity for arbitrary convex co-compact groups:
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Proposition 4.1. Suppose that ⊂ Isom(Hn+1 ) is convex co-compact in the sense of (4.1). Then there exist c > 0 and r0 > 0 such that for any x0 ∈ () and r < r0 there exists a map g : BSn (x0 , r) → Sn with the properties g(() ∩ BSn (x0 , r)) ⊂ (), cr −1 d
Sn (x, y)
≤ dSn (g(x), g(y)) ≤ c−1 r −1 dSn (x, y), x, y ∈ BSn (x0 , r).
(4.2)
Proof. We proceed following the argument in [35, Sect.3]. Let us fix z0 ∈ C(()). If L is the geodesic ray through z0 and x0 , then ∃ C > 0 ∀z ∈ L ∃ γ ∈ d(γ −1 z0 , z) < C. This follows from the compactness of \C(()): for any point on the ray, z, there exists an element of the orbit of z0 within a finite distance from z. We can now choose z = z(r) on the ray L so that d(z, z0 ) = log(1/r), and then γ such that d(γ −1 z0 , z0 ) = log(1/r) + O(1). If xγ is the end point of the geodesic ray through z0 and γ −1 z0 , then for a fixed C1 , the ball BSn (xγ , C1 r) covers BSn (x0 , r). The action of γ on BSn (xγ , C1 r) satisfies (4.2): () is -invariant, and the other property follows by putting γ into the normal form (2.1). Since z0 was fixed and we have no dependence on γ , the proof is completed. Using the self-similarity we will obtain neighbourhoods of (), D = D(h), which can be used in place of the fixed domain D of Sect. 3. We will use the upper half space model for Hn+1 , and assume (as we may) that () Rn ⊂ Cn . When talking about the expanding property of T near () we will use the metric obtained in Lemma 3.1. The distance we use below is given in terms of that (analytic) metric. Proposition 4.2. For any h > 0, sufficiently small we can find D(h) = ∪j Dj (h), an open neighbourhood of () in Cn , with Dj (h) its connected components, and such that T (Di (h)) ∩ Dj (h) = ∅ ⇒ dCn ((T Di (h) )−1 (Dj (h)), ∂Di (h)) > h/C.
(4.3)
In addition there exists K independent of h such that Dj (h) is a union of at most K balls of radius h.
(4.4)
Proof. We start by considering D(h) = {z : dCn ((), z) < (1 − η)h}, where 0 < η < 1 will be chosen later. j (h) be the Let h be small enough, so that D(h) ⊂ D, where D is as in (3.4). Let D connected components of D(h). Then (4.3) holds due to the expanding property of T j (h) = ∅. on D and the fact that T preserves (), () ∩ D The self-similarity provided in Proposition 4.1 shows that each connected component j (h)’s is contained in a ball of radius bounded by K1 h, and that the distances between D are bounded from below by h/K2 , with K1 , K2 fixed. In fact, () is totally disconnected and for a sufficiently small , the -neighbourhood of () has more than one connected components, each contained in a ball, of radius at most K0 , and separated from the others by the distance at least 1/K0 . By applying the self-similarity transformation with r = h(c)−1 , we see that the diameter of each
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connected component of D(h) is bounded by 2K1 h, K1 = K0 (c2 )−1 . Similarly we obtain a separation condition. j (h). We now modify D(h) so that (4.4) holds. That is done by modifying each D Observe that def j (h) = j (h). D BCn (z, (1 − η)h), j (h) = () ∩ D z∈j (h)
Choose z0 ∈ j (h). Since for z in BCn (z0 , ηh), BCn (z, (1 − η)h) ⊂ BCn (z0 , h), we have j (h). BCn (z, (1 − η)h) ⊃ D BCn (z0 , h) ∪ z∈j (h)\BCn (z0 ,ηh)
If z0 , · · · , zk , are chosen, we then find zk+1 ∈ j (h) \
k
BCn (zm , hη),
m=0
so that now k+1 m=0
BCn (zm , h) ∪
k+1
z∈j (h)\
m=0
j (h). BCn (z, (1 − η)h) ⊃ D BCn (zm ,ηh)
This process has to terminate in a uniformly bounded number of steps, as the number of points separated by hη in a set of diameter K1 h is uniformly bounded (independently of h, by C n (K/η)2n , where C depends on the metric; this can be seen, for instance, by volume comparisons). Hence def
Dj (h) =
K
j (h). BCn (zm , h) ⊃ D
m=0
We now choose η small enough depending on the expansion constant of T and the separation constant, so that (4.3) holds and that Dj (h)’s are mutually disjoint. 5. Estimates in Terms of the Dimension of () In the definition (3.4) and Proposition 3.4 we used the neighbourhood D of () given by Lemma 3.1. It is clear from the proof that we can, in place of D use any neighbourhood for which (4.3) holds. For the proof of the Theorem stated in Sect. 1 we will modify Dj ’s in the definition of L(s) in a way dependent on the size of s: we will use Proposition 4.2 with h = 1/|s|. The self-similarity structure of () will show that we can choose Dj = Dj (h) to be a union of O(h−δ ) disjoint balls of radii ∼ h. A modification of the argument used in the proof of Proposition 3.2 will then give (1.2). We start with the following lemma which is a more precise version of the argument already used in the proof of Proposition 3.2:
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Lemma 5.1. Suppose that j ⊂ Cn , j = 1, 2, are open sets, and 1 = K k=1 BCn (zk , rk ). 1 of 1 with values in Let g be a holomorphic mapping defined on a neighbourhood, 2 , satisfying dCn (g(1 ), ∂2 ) > 1/C0 > 0,
0 < Dg(z) < 1, z ∈ 1 .
If def
A : H 2 (2 ) −→ H 2 (1 ), Au(z) = u(g(z)), z ∈ 1 , then for some C1 depending only on rk ’s, K, dCn (g(1 ), ∂2 ), and min 1 Dg Cn →Cn , we have µ (A) ≤ C1 e−
1/n /C 1
,
where µ (A)’s are the characteristic values of A. Proof. We define a new Hilbert space def
H =
K
H 2 (Bk ), Bk = BCn (zk , rk ),
k=1
and a natural operator J : H 2 (1 ) −→ H, (J u)k = uBk . We easily check that J ∗ J : H 2 (1 ) → H 2 (1 ) is invertible with constants depending only on K. Hence µ (A) = µ ((J ∗ J )−1 J ∗ J A) ≤ (J ∗ J )−1 J ∗ µ (J A). We then notice that µk (J A) ≤ k max µ (Ak ), 1≤k≤K
where Ak : H 2 (2 ) −→ H 2 (Bk ), Ak u(z) = u(gk (z)), gk = gBk . To estimate the characteristic values of Ak we observe that we can extend gk to a larger k (contained in 1 ) and such that the image of its closure still lies in 2 (since ball, B we know that min 1 Dg Cn →Cn is strictly less than 1). That gives us the operators k ) → H 2 (Bk ), Rk u = uBk , and A k defined as Ak but with Bk replaced by Rk : H 2 (B Bk . We now have Ak = Rk Ak and consequently, k µ (Rk ). µ (Ak ) ≤ A Lemma 3.3 gives µ (Rk ) ≤ C2 exp(−1/n /C2 ) completing the proof.
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Proof of Theorem. As outlined in the beginning of the section we put h = 1/|s|, where |s| is large but | Re s| is uniformly bounded. In Proposition 4.2 each Dj (h) is given as a union of (a fixed number of) balls with respect to some fixed metric for which T is uniformly expanding. For h small each ball in the family can be replaced by a linearly distorted ball with all the constants uniform. Hence we can apply Lemma 5.1 with g given by rescaled fij (defined in (3.12)). The now classical results of Patterson and Sullivan [35] on the dimension of the limit set show that the total number of the balls is O(h−δ ): what we are using here is the fact that the Hausdorff measure of () is finite. We can now apply the same procedure as in the proof of Proposition 3.2 using Lemma 5.1. What we have gained is a bound on the weight: since | Re s| ≤ C and [Dfij ] is real on the real Sn , |[Dfij (z)]s | ≤ C exp(|s|| arg[Dfij (z)]|) ≤ C exp(C1 |s|| Im z|) ≤ C2 , z ∈ Dj (h). We write L(s) as a sum of fixed number of operators Lij (s) each of which is a direct sum of O(h−δ ) operators. The balls and contractions are uniform after rescaling by h and hence the characteristic values of each of these operators satisfy the bound µ ≤ Cγ l , 0 < γ < 1. Using (3.9) and (3.11) we obtain the bound log | det(I − L(s))| ≤ CP (h) = O(h−δ ), and this is (1.2).
Proof of Corollary 1. The definition of Z (s) (3.1) shows that for Re s > C1 we have |Z (s)| > 1/2. The Jensen formula then shows that the left-hand side of (1.3) is bounded by {m (s) : |s − ir − C1 | ≤ C2 } ≤ 2 max log |Z (s)| + C4 , |s|≤r+C3 | Re s|≤C0
and (1.3) follows from (1.2).
6. Schottky Manifolds and Resonances We recall that a complete Riemannian manifold of constant curvature −1 is said to be Schottky if its fundamental group is Schottky. In low dimensions Schottky manifolds can be described geometrically. Proposition 6.1. Any convex co-compact hyperbolic surface is Schottky. This result is proved by Button [3] and for the reader’s convenience we sketch the proof. Proof. Any convex co-compact, non-elementary surface X is topologically described by two integers (g, f ) : its numbers g of holes and f of funnels, with the conditions g ≥ 0, f ≥ 1 and f ≥ 3 if g = 0. For any such pair (g, f ), there does exist a Schottky surface of this type and we choose for each type (g, f ) such a surface Xg,f . The projection onto Xg,f of the boundary of the Schottky domain is a collection L1 , . . . , L , of mutually disjoint geodesic lines. Let X be any hyperbolic convex co-compact surface . The surface X is homeomorphic to some Xg,f . Pushing back on X the geodesic lines Li , i = 1, . . . , of Xg,f and cutting
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X along these curves, we obtain in the hyperbolic plane a domain whose boundary is the union of paired mutually disjoint curves Ci , C+i , i = 1, . . . , , each one with a pair of points at infinity. These point pairs determine intervals, which are mutually disjoint (the curves Cj , j = 1, . . . , 2 don’t intersect). The intervals are paired with an hyperbolic transformation, so give a Schottky group, which coincide with the fundamental group of the surface X. Proof of Corollary 2. For a Schottky manifold, the fundamental group is Schottky, and hence, X = \Hn+1 , ⊂ Isom+ (Hn+1 ). We then introduce its zeta function ZX as the zeta function Z of the group . Following Patterson and Perry [22] we introduce the spectral sets PX and SX defined by the Laplace-Beltrami operator X on X: PX = {s : Re s > n/2, s(n − s) is a L2 eigenvalue of X }, SX = {s : Re s < n/2, s is a singularity of the scattering matrix SX }. Moreover, each complex s in PX has a multiplicity denoted by mX (s), each s in SX a pole multiplicity denoted by m− X (s). In the case of surfaces, the divisor of the Selberg zeta function ZX is given by the following formula: −χX
∞ k=0
(2k + 1)[−k] + mX
n n 2
2
+
s∈PX
mX (s)[s] +
m− X (s)[s],
s∈SX
where χX is the Euler characteristic of X, see [22, Theorem 1.2]. The zeta function, ZX , is entire and in any half-plane {Re s > −C0 }, the formula above shows that the bounds on the number of its zeros provide bounds on the number of resonances. The dimension of the limit set, δ depends only on and, as shown in [35], it gives the Hausdorff dimension of the recurrent set for the geodesic flow on T ∗ X by the formula 2(1 + δ). For a convex co-compact hyperbolic manifold X, Patterson and Perry give a formula for the divisor of the zeta function ZX in any (even) dimension, but it does not imply (in the non-Schottky case) that the zeta function is entire. In the case of Schottky groups, the zeta function ZX is entire, as it was shown in Proposition 3.4. Hence we concluded that Corollary 2 holds also for Schottky manifolds. We conclude with some remarks about Kleinian groups in dimension n + 1 = 3. Schottky 3-manifolds are geometrically described by Maskit [15]: Proposition 6.2. A hyperbolic convex co-compact, non compact 3-manifold is Schottky if and only if its fundamental group is a free group of finite type. While non-compact surfaces of finite geometric type always have a free fundamental group, that is not true for the 3-manifold. For instance, if is a co-compact surface group, the 3-manifold H3 / is convex co-compact with a non-free fundamental group. Quasi-fuchsian groups (that is, deformation of such a in Isom(H3 )) give similar examples. Finally, we note that the bound on the number of zeros of Z established here for Schottky groups is valid for any group , for which an expanding Markov partition can be built. Anderson and Rocha [1] construct such a Markov partition for any function group. This class of groups does not exhaust all convex co-compact groups (the complement in the 3-sphere of a regular neighbourhood of a graph is not in this class) and it is not known if all convex co-compact Kleinian groups admit an expanding Markov partition.
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7. Numerical Results 7.1. Discussion. In this section, we present empirical numerical results on the distribution of zeros of the zeta function Z(s) for the simple case of the group θ . As a hyperbolic geometry group, θ is generated by reflections s0 , s1 , s2 in three symmetrically placed geodesics in the Poincar´e disc, which intersect its boundary, the unit circle, at angles θ – see Fig. 3 where θ = 110◦ . The corresponding conformal symmetries are denoted by φ0 , φ1 , φ2 . Numerical computations of the zeta function in that case have been already performed by Jenkinson-Pollicott [9]. Their goal was to find an efficient way of computing the dimension of limit sets (see also the earlier work of McMullen [19]). Table 1 gives Table 1. Dimensions of the limit set for relevant values of θ θ 10◦ 20◦ 30◦ 40◦
δ = dim (θ ) 0.116009447786 0.151183682038 0.183983061248 0.217765810254
0.25
(log(#(zeros))−const)/log(y)−1
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 200
400
600
800
1000
y 0 ,y]:Z(s)=0}|)−C − 1 as a function of y, for different values of Fig. 5. This plot shows log(|{s∈[x0 ,x1 ]×[y log(y) x0 : The thin blue line is for x0 = −0.2, the red line for x0 = −0.1, and the black line for x0 = +0.1. The thick horizontal line indicates the dimension of the corresponding limit set. In this plot, θ = 10◦ . The constant C is determined by least squares regression, as explained in this section
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0.25
(log(#(zeros))−const)/log(y)−1
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 200
400
600
800
1000
y 0 ,y]:Z(s)=0}|)−C − 1 as a function of y, for different values of Fig. 6. This plot shows log(|{s∈[x0 ,x1 ]×[y log(y) x0 : The thin blue line is for x0 = −0.2, the red line for x0 = −0.1, and the black line for x0 = +0.1. The thick horizontal line indicates the dimension of the corresponding limit set. In this plot, θ = 20◦ . The constant C is determined by least squares regression, as explained in this section
the (approximate) dimensions of the limit sets for the relevant angles, calculated as the largest real zero of Z(s) [9] using Newton’s method. Figures 1 and 5–7 show log(|{s ∈ [x0 , x1 ] × [y0 , y] : Z(s) = 0}|) − C −1 log(y)
(7.1)
as a function of y, where the constant C is chosen to minimize the usual mean square error err(C , C) =
N
|{s ∈ [x0 , x1 ] × [y0 , yk ] : Z(s) = 0}| − C log(yk ) − C
2
,
k=1
defined using the numerically computed data {(yk , |{s ∈ [x0 , x1 ] × [y0 , yk ] : Z(s) = 0}|) : k = 1, ..., N}. In each plot, the value of x0 is varied to test the dependence of the distribution on the region in which we count: The blue line corresponds to x0 = −0.2, the red line x0 = −0.1, and the black line x0 = +0.1. The data show that most of the zeros
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0.25
(log(#(zeros))−const)/log(y)−1
0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 −0.25 200
400
600
800
1000
y 0 ,y]:Z(s)=0}|)−C − 1 as a function of y, for different values of Fig. 7. This plot shows log(|{s∈[x0 ,x1 ]×[y log(y) x0 : The thin blue line is for x0 = −0.2, the red line for x0 = −0.1, and the black line for x0 = +0.1. The thick horizontal line indicates the dimension of the corresponding limit set. In this plot, θ = 40◦ . The constant C is determined by least squares regression, as explained in this section
lie in the left half plane. Based on the theorems proved in earlier sections, we expect the curves to be bounded above by the dimension (the thick blue line) asymptotically. This is not the case, except for the black line, which represents zeros with Re(s) > x0 = +0.1. Note that the value of x1 is not very important because Z(s) − 1 decays very rapidly for large Re(s). Thus, we set x1 = 10 throughout. The value of y0 is fixed at −0.1, to avoid integrating over any zeros. Similarly, Fig. 2 and 8–10 show log(log(|Z(s)|)) as a function of |s|, for a large number log(|s|) of points in the rectangle [−0.2, 1.0] × [0, 103 ]. In this case, we also expect the curves to be asymptotically bounded by the dimension. This is also not the case. The only reasonable explanation, barring errors in the numerical calculations, is that the asymptotic upper bound is accurate only for very large values of Im(s), and we were not able to calculate Z(s) reliably for such values. These results also show that Z(s) has plenty of zeros in regions of interest.
7.2. Implementation notes. To count the number of zeros of Z(s) in a given region in the complex plane, we rely on the Argument Principle:
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1 0.9
log(log(|Z(s)|))/log(|s|)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
200
400
600
800
1000
|s| Fig. 8. This plot shows log(log(|Z(s)|))/ log(|s|) for a large number of points in the rectangle [−0.2, 1]× [0, 103 ]. The horizontal line indicates dimension. Here, θ = 10◦
1 |{s ∈ : Z(s) = 0}| = 2πi
∂
Z (s) ds. Z(s)
(7.2)
To evaluate Z(s), our main technical tool comes from Jenkinson and Pollicott [9], though we note that the essential ideas were used in Eckhardt, et. al. [4] and date back to Ruelle [31]. First, some notation: Let us denote symbolic sequences on the three characters 0, 1, 2 of length |σ | = n by σ . That is, σ = (σ (0), σ (1), ..., σ (n)), σ (k) ∈ {0, 1, 2}, and σ (0) = σ (n). Symbolic sequences represent periodic orbits : to each sequence σ we associate a composition of reflections φσ = φσ (n) ◦ ... ◦ φσ (1) : Dσ (0) → Dσ (0) . As φσ is a contraction of Dσ (0) into itself, it has a unique fixed point zσ . It is shown in Jenkinson and Pollicott [9] that Z(s) = limM→∞ ZM (s), where ZM (s) = 1 +
N M (−1)r N=1 r=1
r!
r 1 |φσ (zσ )|s , nk 1 − φσ (zσ )
n∈P (N,r) k=1
(7.3)
|σ |=nk
where P (N, r) is the set of all r-tuples of positive integers (n1 , ..., nr ) such that n1 + ... + nr = N. The series (in N ) converges absolutely in {s : Re(s) > −a} for some positive a.
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1 0.9
log(log(|Z(s)|))/log(|s|)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
200
400
600
800
1000
|s| Fig. 9. This plot shows log(log(|Z(s)|))/ log(|s|) for a large number of points in the rectangle [−0.2, 1]× [0, 103 ]. The horizontal line indicates dimension. Here, θ = 20◦
Equation (7.3) lets us evaluate Z(s) for reasonable values of s in a straightforward manner. In addition, we found two simple and useful observations during the course of this calculation: (1) Define an (s) = −
1 |φσ (zσ )|s n 1 − φσ (zσ )
(7.4)
|σ |=n
and BN,r (s) =
1 r!
r
ank (s).
(7.5)
n∈P (N,r) k=1
Then the recursion relation BN,r (s) =
N−r+1 1 BN−n,r−1 (s) · an (s), r
(7.6)
n=1
with initial conditions BN,1 (s) = aN (s), provides an efficient way to evaluate the sum in (7.3). A similar relation can be derived for Z (s) by differentiation.
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1 0.9
log(log(|Z(s)|))/log(|s|)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
200
400
600
800
1000
|s| Fig. 10. This plot shows log(log(|Z(s)|))/ log(|s|) for a large number of points in the rectangle [−0.2, 1] × [0, 103 ]. The horizontal line indicates dimension. Here, θ = 40◦
(2) Recall that the maps φσ are compositions of linear fractional transformations. Identifying these transformations with elements of GL(2, R) in the usual way, we can compute the numbers φσ (zσ ) via matrix multiplications. However, long matrix products can become numerically unstable for larger values of |σ |.An alternative involves the observation that the matrices Aσ = Aσ (n) · ... · Aσ (1) corresponding to the maps φσ have distinct nonzero real eigenvalues. Let us denote these eigenvalues by λ+ and λ− so that |λ+ | > |λ− |. Then a simple calculation shows that φσ (zσ ) = λ− /λ+ . This becomes simply (−1)|σ | /λ2+ if we normalize the determinants of the generators A0 , A1 , and A2 . The larger eigenvalue λ+ can be easily computed using a naive power method: (a) Choose a random v0 . (k) (b) For each k ≥ 0, set vk+1 = Aσ vk /||Aσ vk || and λ+ = Aσ vk , vk . (k) (c) Iterate until the sequence (λ+ ) converges, up to some prespecified error tolerance. The resulting algorithm is slightly less efficient than direct matrix multiplication, but it is much less susceptible to the effects of round-off error. Note that it is certainly possible, even desirable, to apply to this problem modern linear algebraic techniques, such as those implemented in ARPACK [11]. But, we found that the power method suffices in these calculations.
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0 −2 −4 −6 −8 −10 −12 −14 −16
0
200
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|s| |R12 (s)−R13 (s)| Fig. 11. Logarithmic plot (base 10) of the modified relative error 1+|R along the line 12 (s)|+|R13 (s)| Re(s) = −0.2, where RN (s) = ZN (s)/ZN (s). The blue curve is θ = 10◦ , the red curve θ = 20◦ , the green curve θ = 30◦ , and the black curve θ = 40◦
These two simple observations let us calculate the values of Z(s) for a wide range of values in an efficient manner. When combined with adaptive gaussian quadrature, Eq. (7.3) allows us to evaluate the relevant contour integrals. Note that: (1) To calculate the Selberg zeta function Z2 (s) for closed geodesics on the quotient space \H2 , we simply sum over periodic orbits of even length, and additionally use a2 (n, s) = 2a(n, s) instead of a(n, s) in the recursion relations above. This counts the number of equivalence classes of orbits correctly. (2) The work of Pollicott and Rocha [29] revolves around a closely-related trace formula: Z(s) = 1 +
N ∞ N=1 r=1
(−1)
r
r |φσ k (zσk )|s
{[σ1 ], ..., [σr ]} ∈ P (N, r) k=1
1 − φσ k (zσk )
,
(7.7)
where P (N, r) = {{[σ1 ], ..., [σr ]} : |σ1 | + ... + |σr | = N, σk primitive}, and [σ ] is the equivalence class of σ under shifts. The primary difference between (7.3) and
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0 −2 −4 −6 −8 −10 −12 −14 −16
0
200
400
600
800
1000
|s| |R12 (s)−R13 (s)| Fig. 12. Logarithmic plot (base 10) of the modified relative error 1+|R along the line 12 (s)|+|R13 (s)| Re(s) = −0.1, where RN (s) = ZN (s)/ZN (s). The blue curve is θ = 10◦ , the red curve θ = 20◦ , the green curve θ = 30◦ , and the black curve θ = 40◦
(7.7) is that the latter sums over sets of equivalence classes of primitive periodic orbits (equivalent up to shifts), whereas the former sums over all periodic orbits. While it is possible to enumerate primitive periodic orbits efficiently, for example by a simple sieve method, Eq. (7.3) still provides a better numerical algorithm, as it is easier to implement and results in faster and more stable code. 7.3. Error analysis. Figures 11–13 show the logarithms (base 10) of the modified relative errors |R12 (s) − R13 (s)| , (7.8) 1 + |R12 (s)| + |R13 (s)| (s)/Z (s). on the lines x0 +i[0, 103 ], for x0 ∈ {−0.2, −0.1, 0.1} and where RN (s) = ZN N This formula interpolates between the absolute and the relative errors, and measures the convergence of the integrand in (7.2). These results lend some weight to the reliability (s)/Z (s) used in the calculations above. (i.e. convergence) of the values of ZN N
Note added in proof. Using some of the techniques of this paper, H. Christianson has recently generalized the theoretical results of [34] (where only quadradic functions with real Julia sets were treated) to the case any hyperbolic rational function.
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0
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600
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|s| |R12 (s)−R13 (s)| Fig. 13. Logarithmic plot (base 10) of the modified relative error 1+|R along the line 12 (s)|+|R13 (s)| Re(s) = +0.1, where RN (s) = ZN (s)/ZN (s). The blue curve is θ = 10◦ , the red curve θ = 20◦ , the green curve θ = 30◦ , and the black curve θ = 40◦
Acknowledgements. We would like to thank Curt McMullen for explaining the proof of Proposition 4.1 to us and for providing some of the figures. The first author would like to thank J. Anderson for helpful mail discussion. The third author would like to thank John Lott for introducing him to dynamical zeta functions, and to Mike Christ for helpful discussions. KL was supported by a Fannie and John Hertz Foundation Fellowship, and partially by the Office of Science, Office of Advanced Scientific Computing Research, Mathematical, Information, and Computational Sciences Division, Applied Mathematical Sciences Subprogram, of the U.S. Department of Energy, under Contract No. DE-AC03-76SF00098. MZ gratefully acknowledges partial support by the National Science Foundation under the grant DMS-0200732.
References 1. Anderson, J.W., Rocha, A.C.: Analyticity of Hausdorff dimension of limit sets of Kleinian groups. Ann. Acad. Sci. Fennicae 22, 349–364 (1997) 2. Bunke, U., Olbrich, M.: Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group. Ann. Math. 149, 627–689 (1999) 3. Button, J.: All Fuchsian Schottky groups are classical Schottky groups. Geometry and Topology Monographs, 1, 1998, The Epstein Birthday Schrift, pp. 117–125 4. Eckhardt, B., Russberg, G., Cvitanovi´c, P., Rosenqvist, P.E., Scherer, P.: Pinball Scattering. In: Quantum Chaos, Cambridge: Cambridge Univ. Press, 1995, pp. 405–433 ´ Norm. Sup 19, 491–517 (1986) 5. Fried, D.: The Zeta functions of Ruelle and Selberg. Ann. Sc. Ec. 6. Gohberg, I.C., Krein, M.G.: An introduction to the theory of linear non-self-adjoint operators. Translation of Mathematical Monographs, 18, Providence, RI: A.M.S., 1969
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7. Guillop´e, L.: Sur la distribution des longueurs des g´eod´esiques ferm´ees d’une surface compacte a` bord totalement g´eod´esique. Duke Math. J. 53, 827–848 (1986) 8. Guillop´e, L., Zworski, M.: Scattering asymptotics for Riemann surfaces. Ann. Math. 145, 597–660 (1997) 9. Jenkinson, O., Pollicott, M.: Calculating Hausdorff dimension of Julia sets and Kleinian limit sets. Am. J. Math. 124, 495–545 (2002) 10. Lalley, S.: Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits. Acta Math. 163, 1–55 (1989) 11. Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK User’s Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM, 1998 12. Lin, K.: Numerical study of quantum resonances in chaotic scattering. J. Comp. Phys. 176, 295–329 (2002) 13. Lin, K., Zworski, M.: Quantum resonances in chaotic scattering. Chem. Phys. Lett. 355, 201–205 (2002) 14. Lu, W., Sridhar, S., Zworski, M.: Fractal Weyl laws for chaotic open systems. Phys. Rev. Lett. 91, 154101 (2003) 15. Maskit, B.: A characterization of Schottky groups. J. Analyse Math. 19, 227–230 (1967) 16. Maskit, B.: Kleinian groups. Grundlehren Math. Wiss. 287, Berlin: Springer-Verlag, 1988 17. Mather, J.N.: Characterization of Anosov diffeomorphisms. Nederl. Akad. Wet. Proc., Ser. A 71, 479–483 (1968) 18. McMullen, C.: The classification of conformal dynamical systems. In: Current Developments in Mathematics, 1995, Cambridge: International Press, 1995, pp. 323–360 19. McMullen, C.: Hausdorff dimension and conformal dynamics III: Computation of dimension. Am. J. Math. 120, 691–721 (1998) 20. Melrose, R.B.: Polynomial bounds on the number of scattering poles. J. Funct. Anal. 53, 287–303 (1983) 21. Naud, F.: Expanding maps on Cantor sets, analytic continuation of zeta functions with applications to convex co-compact surfaces. Preprint, 2003 22. Patterson, S.J., Perry, P.: Divisor of the Selberg zeta function for Kleinian groups in even dimensions, with an appendix by C. Epstein. Duke. Math. J. 326, 321–390 (2001) 23. Perry, P.: Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume. Geom. Funct. Anal. 11, 132–141 (2001) 24. Perry, P.: A Poisson formula and lower bounds on the number of resonances for hyperbolic manifolds. Int. Math. Res. Notices 34, 1837–1851 (2003) 25. Petkov, V., Zworski, M.: Breit-Wigner approximation and the distribution of resonances. Commun. Math. Phys. 204, 329–351 (1999); Erratum, Commun. Math. Phys. 214, 733–735 (2000) 26. Poincar´e, H.: M´emoire sur les groupes Klein´eens. Acta Math. 3, 49–92 (1883) 27. Pollicott, M.: Kleinian groups, Laplacian on forms and currents at infinity. Proc. Am. Math. Soc. 110, 269–279 (1990) 28. Pollicott, M.: Some applications of thermodynamic formalism to manifolds with constant negative curvature. Adv. Math. 85, 161–192 (1991) 29. Pollicott, M., Rocha, A.C.: A remarkable formula for the determinant of the Laplacian. Inv. Math. 130, 399–414 (1997) 30. Ratcliffe, J.G.: Foundations of hyperbolic manifolds. Graduate Texts in Mathematics. 149, Berlin: Springer-Verlag, 1994 31. Ruelle, D.: Zeta-functions for expanding Anosov maps and flows. Inv. Math. 34, 231–242 (1976) 32. Rugh, H.H.: The correlation spectrum for hyperbolic analytic maps. Nonlinearity 5, 1237–1263 (1992) 33. Sj¨ostrand, J.: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60, 1–57 (1990) 34. Strain, J., Zworski, M.: Growth of the zeta function for a quadratic map and the dimension of the Julia set. Preprint, 2003 ´ 50, 35. Sullivan, D.: The density at infinity of a discrete group of hyperbolic motions. Publ. Math. IHES 259–277 (1979) 36. Wunsch, J., Zworski, M.: Distribution of resonances for asymptotically euclidean manifolds. J. Diff. Geom. 55, 43–82 (2000) 37. Zworski, M.: Sharp polynomial bounds on the number of scattering poles. Duke Math. J. 59, 311–323 (1989) 38. Zworski, M.: Dimension of the limit set and the distribution of resonances for convex co-compact hyperbolic surfaces. Inv. Math. 136, 353–409 (1999) 39. Zworski, M.: Resonances in Physics and Geometry. Notices Am. Math. Soc. 46, 319–328 (1999) Communicated by P. Sarnak
Commun. Math. Phys. 245, 177–200 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1009-z
Communications in
Mathematical Physics
On Algebraic Supergroups, Coadjoint Orbits and Their Deformations R. Fioresi1, , M.A. Lled´o2 1
Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy. E-mail:
[email protected] 2 INFN, Sezione di Torino, Italy and Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. E-mail:
[email protected] Received: 10 February 2003 / Accepted: 3 September 2003 Published online: 16 December 2003 – © Springer-Verlag 2003
Abstract: In this paper we study algebraic supergroups and their coadjoint orbits as affine algebraic supervarieties. We find an algebraic deformation quantization of them that can be related to the fuzzy spaces of non-commutative geometry. 1. Introduction The use of “odd variables” in physics is very old and quite natural. Since then the concept of supermanifold in mathematics has evolved to a very precise formulation, starting with the superanalysis of Berezin [1], and together with other works by Kostant [2], Leites [3], Manin [4], Bernstein, Deligne and Morgan [5] to mention some of the most representative. Supermanifolds are seen roughly as ordinary manifolds together with a sheaf of superalgebras. The odd coordinates appear in the stalks of the sheaf. This approach allows considerable freedom. Following Manin [4], one can define different kinds of superspaces, supermanifolds or algebraic supervarieties by choosing a base space with the appropriate topology. The sheaves considered are superalgebra valued, so that one can generalize the concepts of complex and algebraic geometry to this new setting. In this paper we will deal strictly with algebraic supervarieties, but the concept of supermanifold has been treated extensively in the literature mentioned above. It is interesting to note that there is an alternative definition of algebraic variety in terms of its functor of points. Essentially, an algebraic variety can be defined as a certain functor from some category of commutative algebras to the category of sets. It is then very natural to substitute the category of algebras by an appropriate category of commutative superalgebras and to call this a supervariety. The same can be done for supergroups, super Lie algebras, coadjoint orbits of supergroups and other “super” objects. The elegance of this approach cannot hide the many non-trivial steps involved in the generalization. As an example, it should be enough to remember the profound
Investigation supported by the University of Bologna, funds for selected research topics.
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differences between the classifications of semisimple Lie algebras and semisimple Lie superalgebras [1, 6, 7]. The purpose of this paper is to study the coadjoint orbits of certain supergroups, to establish their structure as algebraic supermanifolds and to obtain a quantum deformation of the superalgebras that represent them. Recently there has been a growing interest in the physics literature on noncommutative spaces (also called “fuzzy spaces”). The idea that spacetime may have non-commuting coordinates which then cannot be determined simultaneously has been proposed at different times for diverse motives [10–13]. In particular, these spaces have been considered as possible compactification manifolds of string theory [13]. In general, we can say that a “fuzzy” space is an algebra of operators on a Hilbert space obtained by some quantization procedure. This means that it is possible to define a classical limit for such an algebra, which will be a commutative algebra with a Poisson structure, that is, a phase space. As it is well known, the coadjoint orbits of Lie groups are symplectic manifolds and hamiltonian spaces with the Kirillov Poisson structure of the corresponding Lie group. There is extensive literature on the quantization of coadjoint orbits, most of the works based on the Kirillov-Kostant orbit principle which associates to every orbit (under some conditions) a unitary representation of the group. Let us consider as an example the sphere S 2 , a regular coadjoint orbit of the group SU(2). In physical terms, the quantization of S 2 is the quantization of the spin; it is perhaps the most classical example of quantization, other than flat space R2n . In the approach of geometric quantization (see Ref. [8] for a review) the sphere must have half integer radius j and to this orbit it corresponds the unitary representation of SU(2) of spin j (which is finite dimensional). Let us take the classical algebra of observables to be the polynomials on the algebraic variety S 2 . Given that the Hilbert space of the quantum system is finite dimensional, the algebra of observables is also finite dimensional. It is in fact of dimension (2j +1)2 and isomorphic to the algebra of (2j +1)×(2j +1)-matrices [9, 10]. After a rescaling of the coordinates, we can take the limit j → ∞ maintaining the radius of the sphere constant. In this way the algebra becomes infinite dimensional and all polynomials are quantized. This procedure is most appropriately described by Madore [10], being the algebras with finite j approximations to the non commutative or fuzzy sphere. The deformation quantization approach [14] is inspired in the correspondence principle: to each classical observable there must correspond a quantum observable (operator on a Hilbert space). This quantization map will induce a non-commutative, associative star product on the algebra of classical observables which will be expressed as a power series in , having as a zero order term the ordinary, commutative product and as first order term the Poisson bracket. One can ask the reverse question, and explore the possible deformations of the commutative product independently of any quantization map or Hilbert space. It is somehow a semiclassical approach. Very often, the star product is only a formal star product, in the sense that the series in does not converge. It however has the advantage that many calculations and questions may be posed in a simpler manner (see [22] for the string theory application of the star product). The deformation quantization of the sphere and other coadjoint orbits has been treated in the literature [15–19]. The immediate question is if deformation theory can give some information on representation theory and the Kirillov-Kostant orbit principle [20, 21]. In Ref. [18] a family of algebraic star products is defined on regular orbits of semisimple groups. The star product algebra is isomorphic to the quotient of the enveloping algebra by some ideal. It is in fact possible to select this ideal in such a way that it is in the kernel of some unitary representation. The image of this algebra by the representation
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map will give a finite dimensional algebra. In this way we can obtain the fuzzy sphere and other fuzzy coadjoint orbits from the star product. The star product is then seen as a structure underlying and unifying all the finite dimensional fuzzy algebras. The classical limit and the correspondence principle are seen naturally in this approach. In this paper we extend the approach of Ref. [18] to some coadjoint orbits of some semisimple supergroups. The extension to the super category is not straightforward and this is reflected in the fact that we have to restrict ourselves to the supergroups SLm|n and OSpm|n , and orbits of elements with distinct eigenvalues (see Theorem 4.1) to obtain the extension. This is by no means an affirmation that the procedure could not in principle be applied to more general supergroups and orbits, but it is a consequence of the technical difficulties involved. Other treatments of the “fuzzy supershere” can be found in Refs. [25, 24, 23]. A comparison with these approaches as well as a relation of our star product with representation theory of supergroups will be discussed elsewhere. The paper is organized as follows: Sect. 2 is dedicated to the definition of algebraic supervariety and its functor of points. The definitions and results that we mention are somehow scattered in the literature and we want to give a comprehensive account here [26, 10]. Section 3 is dedicated to algebraic supergroups and their associated Lie superalgebras in terms of their functors. The correspondence between the algebraic Lie supergroup and its Lie superalgebra is treated in detail. Most of the definitions and theorems extend easily from the classical case, but we are not aware of any reference where this account has been done explicitly for the algebraic case (instead, the differential case is better known). We show proofs when we think it will be helpful to read the paper. We also illustrate the abstract definitions with the examples of GLm|n and SLm|n . In Sect. 4 we retrieve the coadjoint orbits of the supergroups mentioned above as certain representable functors, and then as affine algebraic supervarieties. Finally, in Sect. 5 we present a deformation of the superalgebra that represents the functor associated to the coadjoint orbit. 2. Algebraic Supervarieties and Superschemes In this section we want to give a definition of algebraic supervarieties and superschemes. Our description is self-contained and very much inspired in the approach of Refs. [4, 5]. We assume some knowledge of basic algebraic geometry and of super vector calculus. We are especially interested in the description of a supervariety in terms of its functor of points. Let k be a commutative ring. All algebras and superalgebras will be intended to be over k and are assumed to be commutative unless otherwise stated. If A is a superalgebra we will denote by A0 the even part and by A1 the odd part, so A = A0 + A1 . Let IAodd be the ideal in A generated by the odd part. The quotient A◦ = A/IAodd is an ordinary algebra. Notice that A is both an A0 -module and an A◦ -module. ◦ are If OX is a sheaf of superalgebras over a topological space X, then OX,0 and OX sheaves of algebras with OX,0 (U ) = OX (U )0 ,
◦ OX (U ) = OX (U )◦
U ⊂open X.
We denote by (algrf ) the category of commutative k-algebras which are reduced1 and finitely generated (often called affine algebras), and by (salgrf ) the category of 1
A reduced algebra is an algebra that has no nilpotent elements.
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commutative k-superalgebras finitely generated, and such that, modulo the ideal generated by the odd elements, they are reduced. We denote also by (alg) and (salg) the categories of commutative k-algebras and commutative k-superalgebras respectively.
2.1. Ringed superspaces. Definition 2.1. A superspace (X, OX ) is a topological space X together with a sheaf of superalgebras OX , such that: a. (X, OX,0 ) is a noetherian scheme, b. OX is a coherent sheaf of OX,0 -modules. Definition 2.2. A morphism of superspaces (X, OX ) and (Y, OY ) is given by a pair (f, ψ) where f : X → Y is a continuous map, ψ : OY → f∗ OX is a map of sheaves of superalgebras on Y and (f, ψ|OY ) is a morphism of schemes. f∗ OX denotes the push-forward of the sheaf OX under f . Example 2.1. Let A be an object of (salg). We consider the topological space XA := Spec(A0 ) = Spec(A◦ ) (they are isomorphic since the algebras differ only by nilpotent elements) with the Zariski topology. On XA we have the structural sheaf, OA0 . The stalk of the sheaf at the prime p ∈ Spec(A0 ) is the localization of A0 at p. As for any superalgebra, A is a module over A0 , so we have indeed a sheaf of OA0 -modules over XA with stalk the localization of the A0 -module A over each prime p ∈ Spec(A0 ). It is a sheaf of superalgebras that we will ˜ (XA , A) ˜ is a superspace. (For more details on the construction of the sheaf denote by A. ˜ M, for a generic A0 -module M, see [27] II §5). ˜ Definition 2.3. An affine algebraic supervariety is a superspace isomorphic to (XA , A) for some superalgebra A in (salgrf ). A is often called the coordinate superalgebra of the supervariety. The affine algebraic supervarieties form a category denoted by (svar aff ). Given an affine algebraic supervariety (V , OV ) (or just V for short) we have an ordinary affine algebraic variety associated to it, (V , OV◦ ). It is called the reduced variety of V and denoted also by V ◦ . It is straightforward to generalize the construction to an arbitrary commutative superalgebra A. We then have the following ˜ for some Definition 2.4. An affine superscheme is a superspace isomorphic to (XA , A) superalgebra A in (salg). The affine superschemes over k form a category denoted by (sschemesaff ). As in the classical case, general supervarieties and superschemes are defined as superspaces that are locally affine supervarieties and affine superschemes. To any superscheme (S, OS ) one can associate an ordinary scheme (S, OS◦ ), called the reduced scheme. Example 2.2. Supervariety over the sphere S 2 . (See Ref.[1], p. 8). This is an example of a supervariety explicitly given by the superalgebra representing the functor. We will see how the sheaf is constructed in the closed points (maximal ideals) of the topological
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space. Consider the free commutative superalgebra generated by three even variables x1 , x2 , x3 and three odd variables ξ1 , ξ2 , ξ3 , k[x1 , x2 , x3 , ξ1 , ξ2 , ξ3 ], and the ideal I = (x12 + x22 + x32 − 1, x1 · ξ1 + x2 · ξ2 + x3 · ξ3 ). The superalgebra k[V ] = k[x1 , x2 , x3 , ξ1 , ξ2 , ξ3 ]/I represents a supervariety whose reduced variety V ◦ is the sphere S 2 . At each maximal ideal in k[V ]0 , m = (xi − ai , ξi ξj )
with i, j = 1, 2, 3, ai ∈ k and a12 + a22 + a32 = 1,
the local ring of k[V ]0 is the ring of fractions f / f, g ∈ k[V ]0 , g ∈ /m . (k[V ]0 )m = g The stalk of the structural sheaf at m is the localization of k[V ] as a k[V ]0 -module, that is m k[V ]m = /m . / m ∈ k[V ], g ∈ k[V ]0 , g ∈ g Notice that if a1 = 0 (not all ai are zero simultaneously), then x1 is invertible in the localization and we have 1 ξ1 = (x2 ξ2 + x3 ξ3 ), x1 so {ξ2 , ξ3 } generate k[V ]m as an Ok[V ]0 -module. 2.2. The functor of points. We recall first the definition of the functor of points in the classical (non super) case. Let X be an affine variety. The representable functor hX : (var aff )opp → (sets) 2 from the category of affine varieties to the category of sets, hX (Y ) = hom(varaff ) (Y, X) is the functor of points of X. An element of hom(varaff ) (Y, X) is an Y -point of X. Given the equivalence of categories F : (algrf ) −→ (var aff ) R −→ Spec(R), we can equivalently define the functor of points of an affine variety X = Spec(R) as the representable functor hR (T ) = hom(algrf ) (R, T ),
T ∈ (algrf ).
This leads to an alternative definition of affine variety as a representable functor between the categories (algrf ) and (sets). These definitions and observations can be extended immediately to the super case. Definition 2.5. The functor of points of a supervariety X is a representable functor hX : (svar aff )opp → (sets). 2 By the label opp we denote the category with the direction of morphisms inverted. We could equally speak about contravariant functors.
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Observation 2.1. The construction of Example 2.1 defines an equivalence of categories between (svar aff ) and (salgrf ) and between (sschemesaff ) and (salg). So the functor of points can be equivalently given by a representable functor hA : (salgrf ) → (sets). For example, as we will see in the next section, algebraic supergroups can easily be defined as certain representable functors. In the general case, for a general superscheme X, its functor of points can be defined either as a representable functor from (sschemes)opp → (sets) or as the functor hX : (salg)opp −→ (sets),
hX (A) = hom(sschemes) (XA , X),
where XA is the affine superscheme associated to the superalgebra A (see Example 2.1). In fact, as it happens in the non-super case (see Ref. [31], p. 253), the functor of points of a superscheme is determined by looking at its restriction to affine superschemes. Since we are mainly interested in affine supervarieties and affine superschemes we will not pursue further this subject. 3. Algebraic Supergroups and Their Lie Superalgebras In this section we generalize the basic notions of algebraic groups to the super case. 3.1. Supergroups and supergroup functors. Definition 3.1. An affine algebraic supergroup is a representable, group valued functor G : (salgrf ) → (sets). The superalgebra k[G] representing a supergroup has the additional structure of a commutative super Hopf algebra. The coproduct : k[G] −→ k[G] ⊗ k[G] f −→ f is such that for any affine superalgebra A and two morphisms (A-points) x, y ∈ G(A) = homk−superalg (k[G], A) the following relation holds: mA x ⊗ y(f ) = x · y(f ) ∀f ∈ k[G], where mA denotes the multiplication in A and “·” denotes the multiplication in the group G(A). The tensor product is understood in the tensor category of super vector spaces [5]. The counit is given by E : k[G] −→ k, f −→ e(f ), where e is the identity in G(k). The antipode is defined as S : k[G] −→ k[G], f −→ S(f ), with x(f ) = x −1 (S(f )) ∀x ∈ G(A), and where x −1 denotes the group inverse of x in G(A).
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Example 3.1. The supergroups GLm|n and SLm|n . Let A be a commutative superalgebra. We denote by Am|n the free module over A generated by m even generators and n odd generators. The endomorphisms of this super vector space (linear maps that preserve the grading) are given by matrices (we use the conventions of Ref. [5]) pm×m qm×n , (1) rn×m sn×n where p and s have even entries and q and r have odd entries in A. We denote the set of these matrices as glm|n (A). We can define a functor glm|n : (salgrf ) −→ (sets) A −→ glm|n (A). glm|n is a representable functor. It is represented by the superalgebra k[glm|n ] := k[xij , yαβ , ξiβ , γαj ],
i, j = 1, . . . m, α, β = 1, . . . n,
where x and y are even generators and ξ and γ are odd generators. In fact, writing the generators in matrix form, {xij } {ξiβ } , {γαj } {yαβ } any matrix as in (1) assigns to a generator of k[glm|n ] an element of A, and the assignment has the right parity. Hence, it defines a superalgebra morphism k[glm|n ] → A. GLm|n (A) is defined as the set of all morphisms g : Am|n → Am|n which are invertible. In terms of the matrix (1), this means that the Berezinian [1] or superdeterminant Ber(g) = det(p − qs −1 r) det(s −1 ) is invertible in A. A necessary and sufficient condition for g to be invertible is that p and s are invertible. The group valued functor GLm|n : (salgrf ) −→ (sets), A −→ GLm|n (A), is an affine supergroup represented by the algebra [30] k[GLm|n ] := k[xij , yαβ , ξiβ , γαj , z, w]/ (w det(x) − 1, z det(y) − 1 , i, j = 1, . . . m, α, β = 1, . . . n. Requiring that the berezinian is equal to 1 gives the supergroup SLm|n , represented by k[GLm|n ]/ det(x − ξy −1 γ )z − 1 , where y −1 is the matrix of indeterminates, inverse of the matrix y, whose determinant is invertible and has inverse z. In Ref.[30] the Hopf superalgebra structure of this affine supergroup was explicitly computed. In the classical case, the concept of group functor is a generalization of the concept of algebraic group. This is treated extensively in [32] II, §1. It will be useful to introduce this notion for the super case, which can be done easily with suitable changes.
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Definition 3.2. Let G be a functor from (salg) to (sets). We say that G is a supergroup functor if: 1. There exists a natural transformation called the composition law m : G × G −→ G satisfying the associativity condition: m · (m × ?) = m · (? × m). 2. There exists a natural transformation, the unit section, u : ek −→ G, where ek : (salg) −→ (sets), ek (A) = 1A , satisfying the commutative diagram: u×id
id×u
G × ek −→ G ×G −→ ek × G m G 3. There exists a natural transformation σ : G −→ G satisfying: (id,σ )
G −−−−→ G × G m u
ek −−−−→
G
A morphism of supergroup functors is defined as a natural transformation preserving the composition law. Because of their representability property, affine algebraic supergroups are supergroup functors. 3.2. Lie superalgebras. Let Ok : (salgrf ) −→ (sets) be the functor represented by k[x]. k[x] corresponds in fact to an ordinary algebraic variety, Spec(k[x]), the affine line. For a superalgebra A we have that Ok (A) = A0 . Definition 3.3. A Lie superalgebra is a representable, group valued functor g : (salgrf ) −→ (sets) with the following properties: 1. g has the structure of Ok -module, that is, there is a natural transformation Ok ×g−→g. For each superalgebra A we have an A0 -module structure on g(A). 2. There is a natural transformation [ , ] : g × g −→ g which is Ok -linear and that satisfies commutative diagrams corresponding to the antisymmetric property and the Jacobi identity. For each superalgebra A, [ , ] defines a Lie algebra structure on g(A), hence the functor g is Lie algebra valued. For any algebraic supergroup there is a Lie superalgebra which is naturally associated. It is our purpose to construct it explicitly. Again, the construction is a generalization (not completely straightforward) of what happens in the non-super case. Our treatment is very similar to the one in Ref.[5] II, 4, no. 1. Let A be a commutative superalgebra and let A( ) =def A[ ]/( 2 ) be the algebra of dual numbers ( here is taken as an even indeterminate). We have that A[ ] = A ⊕ A and there are two homorphisms: i : A → A( ) defined by i(1) = 1 and p : A( ) → A defined by p(1) = 1, p( ) = 0.
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Definition 3.4. Let G be a supergroup functor. Consider the homomorphism G(p) : G(A( )) −→ G(A). For each G there is a supergroup functor, Lie(G), defined as Lie(G)(A) =def ker(G(p)). We will see that when G is an affine algebraic supergroup (so it is a representable group functor) Lie(G) is indeed a Lie superalgebra. It is instructive to see first an example. Example 3.2. Lie(GLm|n ), Lie(SLm|n ). We want to determine the functor Lie(GLm|n ). Consider the map: GLm|n (p) :
GLm|n (A( ))
p + p q + q r + r s + s
−→ GLm|n (A) pq → r s
with p, p , s, s having entries in A0 and q, q , r, r having entries in A1 . p and s are invertible matrices. One can see immediately that 1 + p q . Lie(GLm|n )(A) = ker(GLm|n (p)) = r 1 + s The functor Lie(GLm|n ) is clearly group valued and can be identified with the (additive) group functor glm|n (see Example 3.1). For Lie(SLm|n ) one gets the extra condition Ber = det(1 + p ) det(1 − s ) = 1, which implies the zero supertrace condition, that is, tr(p ) − tr(s ) = 0. The functor Lie(SLm|n ) is then identified with the (additive) group functor slm|n (A) = {x ∈ glm|n (A) / str(x) = 0}. The functors glm|n and slm|n are representable and Lie algebra valued. We have already seen that glm|n is representable (see Example 3.1). slm|n is represented by the superalgebra: k[slm|n ] = k[glm|n ]/str(m), where m is the matrix of indeterminates generating the algebra k[glm|n ].
We want to show that when G is an affine algebraic group then Lie(G) is a superalgebra. We will show first that the functor satisfies both properties in Definition 3.3. Then we will show that it is representable. We start by seeing that Lie(G) has a structure of Ok -module. Let ua : A( ) −→ A( ) be the endomorphism, ua (1) = 1, ua ( ) = a , for a ∈ A0 . Lie(G) admits a Ok -module structure, i.e. there is a natural transformation Ok × Lie(G) −−−−→ Lie(G) such that for any superalgebra A, a ∈ Ok (A), x ∈ Lie(G)(A), (a, x) → ax = Lie(G)(ua )x. Notice that for subgroups of GLm|n (A), ax corresponds to the multiplication of the matrix x by the even scalar a.
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Let us consider the group of linear automorphisms of Lie(G)(A). Because of the natural Ok -module structure of Lie(G) we have a group functor GL(Lie(G)) : (salgrf ) −→ (sets). Alternatively one can also denote GL(Lie(G)) = Aut(Lie(G)). We now want to introduce the Lie algebra structure (Property 2 in Definition 3.3). We will do it through the adjoint actions, seen as natural transformations. Definition 3.5. Let G be a supergroup functor. The adjoint action of G on Lie(G) is defined as the natural transformation Ad
G −−−−→ GL(Lie(G)) which for any superalgebra A and g ∈ G(A), x ∈ Lie(G)(A), AdA (g)x =def G(i)(g)xG(i)(g)−1 , AdA (g)x ∈ G(A( )) but since G(p) is a group homomorphism, AdA (g)x ∈ Lie(G)(A). Ad is a morphism in the category of group functors. So we can make the following definition: Definition 3.6. Let G be a supergroup functor. The adjoint action of Lie(G) on Lie(G) is defined as ad =def Lie(Ad) : Lie(G) −→ Lie(GL(Lie(G))) = End(Lie(G)). Finally, the bracket on Lie(G)(A) is defined as [x, y] =def ad(x)y,
x, y ∈ Lie(G)(A).
The arguments in Ref.[32] II §4 4.2, 4.3 apply with small changes to our case and prove that [ , ] is a Lie bracket. Example 3.3. GLm|n . We want to see that in the case of GLm|n the Lie bracket [ , ] is the commutator. We have AdA : GL(A) −→ GL(Lie(GLm|n ))(A) = GL(glm|n (A)) g → Ad(A)(g), Ad(g)x = gxg −1 ,
x ∈ glm|n (A).
By definition we have: Lie(GL(glm|n ))(A) = {1 + α / α ∈ GL(glm|n )(A)} So we have, for a, b ∈ glm|n (A) ∼ = Lie(GLm|n )(A) = {1 + a | a ∈ glm|n (A)}: ad(1 + a)b = (1 + a)b(1 − a) = b + (ab − ba) = b + [a, b]. Hence ad(1 + a) = ? + α(a), with α(a) = [a, ] as we wanted to show. It is also clear that the same computation will hold for any closed subgroup of GLm|n .
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Finally we have to address the issue of the representability of the functor Lie(G). From the classical (non-super) case, we know that it is not, in general, representable (see Ref.[32] II, 4, 4.8). The most useful example is when G is a linear group, that is, a closed subgroup of GLn . Then Lie(G) is representable, and we will see that the same is true for the super case. More generally, for any affine algebraic supergroup G, the associated functor Lie(G) can be shown to be representable. Theorem 3.1 characterizes Lie(G) geometrically as the tangent space at the identity. In the classical case, this result is a particular case of a more general one involving schemes, whose proof can be found in Ref. [32]. Since we are only concerned with affine algebraic supergroups, all we need is an extension to the super category of the (simpler) proof for affine algebraic groups. We find it then convenient to write explicitly the proof of the following theorem, although the super extension in this particular case presented no difficulty. Theorem 3.1. Let k be a field and G be an affine algebraic group, with k[G] its coordinate superalgebra. As in Sect. 3.1, let E denote the counit in the superHopf algebra k[G]. We denote by mE = ker(E) and by ω the super vector space mE /mE2 . Then, Lie(G) is a representable functor and it is represented by k[ω] where k[ω] = k[x1 , . . . xp , ξ1 , . . . ξq ] with x1 . . . xp , and ξ1 . . . ξq being even and odd indeterminates respectively, and p|q is the superdimension of ω. Proof. We have to prove that Lie(G)(A) = homk−superalg (k[ω], A). It is immediate to verify that homk−superalg (k[ω], A) ∼ = homk−supermod (ω, A) ∼ = (ω∗ ⊗ A)0 , hence it is enough to show that Lie(G)(A) ∼ = homk−supermod (ω, A). We will define a map ρ : homk−supermod (ω, A) → Lie(G)(A) and then show that it is a bijection. Let d ∈ homk−supermod (ω, A), d : ω −→ A. We first consider the following maps: φ : k[G] −→ k ⊕ ω = k ⊕ mE /mE2 , f → (E(f ), f − E(f ) + mE2 ), and for each d, d : k ⊕ ω −→ A( ), (s, t) → s + d(t) . We define ρ(d) as the composition ρ(d) = d ◦ φ. Then we have that G(p)(ρ(d))(f ) = E(f ), so ρ(d) ∈ ker G(p) = Lie(G)(A). We want now to give the inverse z : Lie(G)(A) → homk−supermod (ω, A). Assume ψ ∈ Lie(G)(A). We can write: ψ(f ) = E(f ) + f . Now consider ψ|mE : mE −→ A( ), f → f . Observe that ψ|m2 = 0, hence going to the quotient we have a supermodule map E ˜ ) = f . Now define z(ψ)(f ) = f . This is the inverse of a. ψ˜ : ω −→ A( ), ψ(f The fact z · ρ = id is straightforward. For ρ · z observe that given ψ : k[G] −→ A( ), this can always be written as: ψ(g) = E(g) + g . Observe that g − E(g) ∈ mE , hence ψ|mE (g − E(g)) = g , now the result follows easily.
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Corollary 3.1. Let k be a field and G be an affine algebraic supergroup. Then Lie(G) is a Lie superalgebra. We want to remark that ω = mE /mE2 can be regarded as the dual of the tangent space at the identity of the supergroup G. Theorem 3.1 hence states that such tangent space is the same as Lie(G), as it happens in the non-super case. Observation 3.1. Lie superalgebras as super vector spaces with a graded bracket. Lie superalgebras were first introduced in physics with a different definition. A Lie superalgebra is a super (Z2 -graded) vector space g = g0 + g1 with a bilinear, graded operation [ , ] : g ⊗ g −→ g, X ⊗ Y −→ [X, Y ], such that 1. [X, Y ] = −(−1)pX pY [Y, X], 2. [X, [Y, Z]] + (−1)pX pY +pX pZ [Y, [Z, X]] + (−1)pX pZ +pY pZ [Z, [X, Y ]] = 0, where X, Y, Z are homogeneous elements of g with parities pX , pY , pZ . This definition is used mostly when Lie superalgebras are treated independently of Lie supergroups [1, 6]. This definition of Lie superalgebra can be shown to be equivalent to Definition 3.3 in functorial terms. This is proven in Ref. [5], Corollary 1.7.3, p. 57, using the even rules principle. We will show how it works for the specific example glm|n (Example 3.4). We have then that in the super vector space ω∗ there is a graded bracket [ , ] and a Lie superalgebra structure in the sense mentioned above. Example 3.4. Let us consider the set of (m + n) × (m + n) matrices with arbitrary entries in A denoted as glm|n (A). We also denote I = 1, . . . m + n,
I = i for I = 1, . . . m, I = n + α for I = n + 1, . . . m + n.
Let {EI J } be the standard basis of matrices with 1 in the place I J and 0 everywhere else. An element X ∈ glm|n (A) can be written in terms of the standard basis X = XI J EI J = pij Eij + qiβ Eiβ + rαj Eαj + sαβ Eαβ , where the sum over repeated indices is understood and the parities of p, q, r and s are arbitrary. We assign even degree to Eij and Eαβ (block diagonal matrices) and odd degree to Eiβ and Eαj (block off diagonal matrices). This corresponds to even and odd linear maps. With these assignments glm|n (A) is a non-commutative, associative superalgebra, and glm|n (A) is its even part. It corresponds to the even linear maps or super vector space morphisms. We can give it a super Lie algebra structure with the ordinary commutator of matrices among even elements or among an even and an odd one and the ordinary anticommutator of matrices among odd elements. Then, the even part of this Lie superalgebra is the Lie algebra glm|n (A). In general, giving a representable, Lie algebra valued functor is equivalent to give a super Lie algebra through the (anti)commutation rules of the generators.
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4. Coadjoint Orbits of Supergroups Let G ⊂ GL(m|n) be an algebraic Lie supergroup and g = Lie(G) the associated Lie superalgebra. Let g∗ be the functor g∗ (A) = g(A)∗ . We want to define a coadjoint orbit of the supergroup G. Let X0 be a geometric point of g∗ , that is X0 ∈ g∗ (k). g(k) is an ordinary Lie algebra over k, and g(k) ⊂ g(A) for any A through the unit map of A. We consider the following functor, CX0 : (salg) −→ (sets), A −→ {Ad∗g X0 , ∀g ∈ G(A)} = G(A)/H (A),
(2)
where H (A) is the stability group of X0 . Notice that it is necessary to choose a geometric point in order to have a functor. The functor CX0 is not, in general, representable. The problem arises already at the classical level. We can see it with an example. Let us consider the algebraic group SLn over the complex numbers and its Lie algebra sln . Their functors of points are represented respectively by C[SLn ] = C[xij ]/ det(x) − 1 , i, j = 1, . . . n. C[sln ] = C[xij ]/ tr(x) , Let X0 = diag(l1 . . . ln ) ∈ sln (C), li = lj , for i = j . The coadjoint orbit of X0 is an algebraic variety represented by C[CX0 ] = C[xij ]/(p1 − c1 , · · · pl − cl ),
(3)
where pi = tr(X i ) and ci = pi (X0 ). An A-point of CX0 is a morphism C[CX0 ] → A and it is given by a matrix (aij ) ∈ gln (A), such that pk (aij ) = ck , k = 1 . . . m + n. r , defined as C If the functor CX X0 but restricted to the category of commutative alge0 bras were representable, any such matrix would be of the form X = gX0 g −1 , that is, conjugate to a diagonal one through an element g ∈ SLn (A). But this is not necessarily true in an arbitrary algebra A. In the classical case the functor of points of the coadjoint orbit is obtained as the r ([32], p. 341). We will see that the same is true sheafification (see Definition 4.1) of CX 0 for the super case. We will start by considering a functor F : (salg) −→ (sets). This is a generalization of the concept of Z-functor defined in [32], p. 9. CX0 is an example of such a functor. Due to the equivalence of categories between (salg) and (sschemesaff ) we can give equivalently the functor: F : (sschemesaff )opp −→ (sets). (Abusing the notation, we use the same letter for both functors.) For each affine superscheme X the restriction FX of F to Top(X) defines a presheaf on X (Top(X) denotes the category of open sets in X). A functor F : (salg) −→ (sets) is said to be local if FX is a sheaf for each X ∈ (sschemes).
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Equivalently, F is local if for any A ∈ (salgrf ) and elements f1 . . . fs ∈ A such that (f1 . . . fs ) = (1) we have the exact sequence: F (αi )
F (A) → F (Afi )
F (αij )
⇒
F (Afi fj ).
In particular, any representable functor is local. Definition 4.1. Let us consider a functor F : (salg) −→ (sets). We will denote by F˜ the ˜ fi ) = F(Afi ). unique local functor such that for f1 . . . fs ∈ A, (f1 . . . fs ) = (1), F(A ˜ F is called the sheafification of F. It is perhaps more natural to introduce the sheafification in terms of the presheaves FX . To any presheaf one can associate a sheaf that is the “closest” sheaf to the given presheaf; it is called its sheafification. Then F˜ can be equivalently defined as the local functor obtained by doing the sheafification of the presheaves FX for all X ∈ (sschemes). Observation 4.1. If P is a subfunctor of a local functor F and for all R ∈ (salg) there ˜ F. (R exist f1 . . . fr ∈ R ◦ such that (f1 . . . fr ) = (1) and P(Rfi ) = F(Rfi ), then P= is viewed as an R0 module.) Our strategy to determine the functor of the coadjoint orbits of supergroups will be to find a representable (hence local) functor and to prove that it is the sheafification of (2). We need first some notions about the invariant polynomials of a superalgebra. 4.1. Invariant polynomials. For the rest of the section we take k = C. The Cartan-Killing form of a super Lie algebra g is a natural transformation: B : g × g −→ g, BA (X, Y ) = str(adX adY )
X, Y ∈ g(A).
BA is an invariant, supersymmetric bilinear form. In the following, we will consider simple Lie superalgebras whose Cartan-Killing form is non-degenerate, namely slm|n with m = n, ospm|n with m2 − n2 = 1 (m, n even), f4 and g3 [6]. There is a natural isomorphism between the functors g and g∗ such that to each object A in (salgrf ) assigns a morphism ϕA : g(A) −→ g∗ (A) X −→ ϕA (X) with ϕA (X)(Y ) = BA (X, Y ). This isomorphism gives also an isomorphism between the superalgebras representing both functors, C[g] C[g∗ ] and intertwines the adjoint and coadjoint representation, so the adjoint orbits are the same as the coadjoint orbits. From now on we will use the algebra C[g]. Let G be an affine algebraic supergroup, subfunctor of GL(m|n), with super Lie algebra g. We say that p ∈ C[g] is an invariant polynomial if for any A-point x : C[g] → A of g and g : C[G] → A of G we have that x(p) = AdA (g)x(p).
(4)
The invariant polynomials are a subalgebra of C[g]. Contrary to what happens in the classical case, this algebra may be not finitely generated [1, 34]. This is the case for the algebra of invariant polynomials on glm|n . The generators can be taken to be the supertraces of arbitrary order, str(Xk ), which are independent. The invariant polynomials in the reduced Lie algebra, glm × gln are generated by traces tr(X k ) with k = 1, . . . m + n, since higher order traces can be expressed in terms of the first m + n ones.
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4.2. The coadjoint orbits of a supergroup as algebraic supervarieties. For a regular, semisimple element X0 of g(C), its orbit under the adjoint action of the group G(C) is an algebraic variety defined by the values of the homogeneous Chevalley polynomials p1 , . . . pl , where l is the rank of the group. We will see that the supersymmetric extensions of these polynomials define the adjoint orbit of the supergroup. More specifically, we have the following Theorem 4.1. Let pˆ 1 , . . . pˆ l be polynomials on a simple Lie superalgebra of the type slm|n or ospm|n with the following properties: 1. They are invariant polynomials under the adjoint action (4). 2. Let p1 , . . . pl be the projections of pˆ 1 , . . . pˆ l onto the reduced algebra C[g◦ ] = C[g]/I odd . The ideal of the orbit of a regular, semisimple element X0 ∈ g(C) with distinct eigenvalues, is J = (p1 − c1 , . . . pl − cl ), ci = pi (X0 ). So the orbit of X0 is an algebraic variety whose functor is represented by C[g◦ ]/J . Then the sheafification of the functor CX0 (2) is representable and is represented by AX0 = C[g]/I,
(5)
with I = (pˆ 1 − c1 , . . . pˆ l − cl ). Proof. Let us denote by F : (salgrf ) → (sets) the functor represented by AX0 in (5). It is clear that CX0 is a subfunctor of F , since for any superalgebra R, an element of CX0 (R) = {gX0 g −1 , g ∈ G(R)} is given by a matrix in g(R), pij qiβ M: rαj sαβ whose entries satisfy p1 − c1 = 0, . . . pl − cl = 0, and then defines a homomorphism M : AX0 −→ R. In view of Observation (4.1) we just have to show that for f1 . . . fs ∈ R ◦ , (f1 . . . fs ) = (1), F (Rfi ) = CX0 (Rfi ). Let f ∈ R ◦ . By the previous observation there is an obvious injective map CX0 (Rf ) → F (Rf ). We have to show that the fi can be chosen in such a way that the map is also surjective. This means that we need to prove that given W ∈ CX0 (R), there exist f ∈ R ◦ and g ∈ G(Rf ) such that gWg −1 = D,
or
gW = Dg,
(6)
where D is a diagonal matrix diagonal. Later on we will prove that D = X0 . We consider first a superalgebra R is a free superalgebra in the odd generators. We decompose the matrices in (6) as sums of matrices whose elements are homogeneous in the odd variables (g0 + g1 + · · · )(W0 + W1 + · · · ) = (D0 + D1 + · · · )(g0 + g1 + · · · ). Then we can compare elements of the same degree obtaining g0 Wn + g1 Wn−1 + · · · gn W0 = D0 gn + D1 gn−1 + · · · + Dn g0 .
(7)
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We will prove the result by induction. For n = 0 we have g0 W0 = D0 g0 .
(8)
But this is the classical result, with D0 = X0 . By Hypothesis 2 of the theorem, (p1 − c1 , . . . pl − cl ) is the ideal of the reduced orbit, so when we restrict to the category of commutative algebras we know that F is the sheafification of CX0 and so it is represented by C[g◦ ]/(p1 − c1 . . . pl − cl ). This means that there exist f ∈ R ◦ , g0 ∈ G(R ◦ ), such that g0 W0 g0−1 = D0 . Moreover, the classical results guarantee that one can choose f1 . . . fr among all possible f ’s in such a way that the ideal that they generate in R ◦ , (f1 . . . fr ) = (1) = R ◦ . The induction proof is based on an argument given in Ref. [1], p. 117. We assume that the result is true up to order n − 1. Then we multiply (7) by g0−1 to the right. Using (8) we obtain X0 gn g0−1 − gn g0−1 X0 + Dn = Kn ,
(9)
where Kn is a known matrix Kn = −D1 gn−1 g0−1 − D2 gn−2 g0−1 − · · · + g0 Wn−1 g0−1 + · · · + gn−1 W1 g0−1 and Dn is a diagonal matrix. The matrix X0 gn g0−1 − gn g0−1 X0 has only elements outside the diagonal, and if (λi ) are the eigenvalues of X0 we have that the entry (ij ) is given by (X0 gn g0−1 − gn g0−1 X0 )ij = (λi − λj )(gn g0−1 )ij . Then, gn and Dn can be computed from (9) provided λi = λj . To finish the proof we will have to show that Dn = 0 for n ≥ 0. Let us consider the case of sl(m|n). Then the invariant polynomials pi ∈ C[sl(m|n)] that we have to consider are pi (M) = strM i , i = 1, . . . m + n. We want to prove that for a diagonal matrix D = D0 + D1 + · · · , if pi (D) = pi (D0 ),
(10)
then D = D0 . Let D = diag(λ1 + λ 1 , λ2 + λ 2 , . . . λm+n + λ m+n ), with λ i contain all the terms in the odd variables. Then (10) implies the following homogeneous system: 1 ··· 1 −1 · · · −1 λ1 λ1 · · · λm −λm+1 · · · −λm+n λ2 ··· ··· m+n m+n m+n m+n λm+n · · · λm −λm+1 · · · −λm+n λ1 which can have a non-trivial solution if the determinant 1 ··· 1 −1 · · · −1 λ1 · · · λm −λm+1 · · · −λm+n nm det · · · (λi − λj ) = (−1) i>j m+n m+n m+n · · · λm+n −λm+1 · · · −λm+n λ1 m
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is different from zero (Vandermonde determinant). So we have our result if all the eigenvalues are different. In the case of ospm|n , the relevant polynomials are of the form strM 2i , so the result can be reproduced without difficulty. In the case we consider a superalgebra R/J with J an ideal, it is enough to look at the images of the (f1 , · · · fr ) under the projection F /J . 5. Deformation Quantization of Coadjoint Orbits of Supergroups Let k = C. We start with the definitions of Poisson superalgebra and its deformation. Definition 5.1. Let A be a commutative superalgebra. We say that A is a Poisson superalgebra if there exists a linear map (Poisson superbracket) { , } : A ⊗ A −→ A, f ⊗ g −→ {f, g}, such that 1. {a, b} = −(−1)pa pb {b, a}, 2. {a, {b, c}} + (−1)pa pb +pa pc {b, {c, a}} + (−1)pa pc +pb pc {c, {a, b}} = 0, 3. {a, bc} = {a, b}c + (−1)pa pb b{a, c}, where a, b, c are homogeneous elements of A with parities pa , pb , pc 3 . Let g be a Lie superalgebra with Lie superbracket [XI , XJ ] = cIKJ XK for a certain homogeneous basis {XI }s+r I =1 with the first s vectors even and the last r odd. Then C[g∗ ] C[x1 , . . . xs , xs+1 , · · · xs+r ] has a Poisson superalgebra structure with superbracket given by {xI , xJ } = xK ([XI , XJ ])xK = cIKJ xK K
K
and extended to the whole algebra by Property 3 of Definition 5.1. Let g be one of the Lie algebras considered in Theorem 4.1, which have a CartanKilling form, so g g∗ and C[g] C[g∗ ]. Let A = C[g]/I be the algebra associated to the superorbit of a geometric element X0 with the conditions of Theorem 4.1. Then A is also a Poisson superalgebra with the Poisson superbracket induced by the one in C[g]. This follows from the derivation Property 3 in Definition 5.1 and the fact that pˆ i are invariant polynomials. In this section we want to construct a deformation quantization of the superalgebra representing the orbit of a supergroup. We will extend the method used in Refs. [18, 19] for the classical (non-super) case. Definition 5.2. Given a Poisson superalgebra A, a formal deformation (or deformation quantization) of A is an associative non-commutative superalgebra algebra Ah over C[[h]], where h is a formal parameter, with the following properties: 3
We want to remark that with this definition we have only even Poisson brackets.
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1. Ah is isomorphic to A[[h]] as a C[[h]]-module. 2. The multiplication ∗h in Ah reduces mod(h) to the one in A. 3. a˜ ∗h b˜ − b˜ ∗h a˜ = h{a, b} mod (h2 ), where a, ˜ b˜ ∈ Ah reduce to a, b ∈ A mod(h) and { , } is the Poisson superbracket in A. Let g be a Lie superalgebra and gh the Lie superalgebra over C[[h]] obtained by multiplying the Lie bracket of g by the formal parameter h. Let us denote by Uh the universal enveloping algebra of gh ([1], p. 279). As in the classical case, it is easy to prove that the associative, non-commutative superalgebra Uh is a deformation quantization of C[g∗ ]. Let g be a Lie superalgebra over C of the type considered in Theorem 4.1. By Property 1 in Definition 5.2, there exists an isomorphism of C[[h]]-modules ψ : C[g∗ ][[h]]] → Uh . We want to prove that ψ can be chosen in such a way that there exists an ideal Ih ⊂ Uh such that ψ(I) = Ih 4 , and the map induced on the quotients ψh : C[g∗ ]/I[[h]] → Uh /Ih is an isomorphism of C[[h]]-modules. We have then that the following diagram C[g∗ ][[h]] π
ψ
−−−−→
Uh π h
ψh
C[g∗ ]/I[[h]] −−−−→ Uh /Ih with π and πh the canonical projections, commutes. We list first some known results about the enveloping algebra of g, U (g) [1]. We denote by S(g) the algebra of super symmetric tensors on g. We have that S(g) C[g∗ ]. As before, {XI }s+r I =1 denotes a homogeneous basis of g as vector superspace. Let τ : C[g∗ ] → U (g) be the supersymmetrizer map τ (xI1 · · · xIp ) =
1 (−1)σ¯ (s) Xs(I1 ) ⊗ · · · ⊗ Xs(Ip ) , p! s∈Sp
where Sp is the group of permutations of order p and σ¯ (s) is the sign arising when s performing the permutation XI1 ⊗ · · · ⊗ XIp −−−−→ Xs(I1 ) ⊗ · · · ⊗ Xs(Ip ) as if the homogeneous elements XIj were supercommuting. τ is an isomorphism of g-modules. Let C[g]g denote the set of polynomials invariant under the adjoint action of g (in particular pˆ i ∈ C[g]g). Then τ induces an isomorphism of g-modules: C[g]g Z(U (g)), where Z(U (g)) is the center of U (g). Let I = (p1 − c1 , . . . pl − cl ) as in Theorem 4.1. We set Ih = (P1 − c1 (h) . . . Pl − cl (h)) ⊂ Uh ,
Pi = τ (pˆ i ),
ck (0) = ck ,
with ci (h) ∈ C[[h]]. Theorem 5.1. In the settings of Theorem 4.1, Uh /Ih is a deformation quantization of C[g∗ ]/I. 4
The ideal I is understood here as an ideal of C[g∗ ][[h]]
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Proof. The only property in Definition 5.2 which is not immediate is property a, that is, the fact of Uh /Ih C[g∗ ]/I[[h]] as C[[h]]-modules. The proof can be almost translated from the classical case in Ref. [18]. Let {xI1 , . . . , xIk }(I1 ,...,Ik )∈S be a basis of C[g∗ ]/I as a C-module, where S is a fixed set of multiindices appropriate to describe the basis. In particular, we can take them such that I1 ≤ · · · ≤ Ik . Proving that the monomials B = {XI1 · · · XIk }(I1 ,...,Ik )∈S are a basis for Uh /Ih will be enough. The proof that B is a generating set for Uh /Ih is identical to the proof of Proposition 3.13 in Ref. [18] and we will not repeat it here. For the linear independence, we have to show that there is no relation among them. Suppose that G ∈ Ih is such relation, G = G0 + G1 h + · · · ,
Gi ∈ spanC {XI1 · · · XIk }(I1 ...Ik )∈S .
Assume Gi = 0, i < r, Gr = 0 so we can write G = hr F , with F = F0 + hF1 + · · · ,
F0 = 0.
(11)
We need to prove the non trivial fact that if hF ∈ Ih then F ∈ Ih . We will denote by a capital letter, say P , an element in Uh , by pˆ its projection onto C[g∗ ] and by p its further projection onto C[g◦∗ ]. Assume that Ai (Pi − ci (h)) (12) hF = i
ci + ci1 h + . . . cn hn .
with ci (h) = Then, reducing mod h (taking h = 0) we obtain aˆ i (pˆ i − ci ). 0=
(13)
i
Setting all the odd variables to 0, we have that (13) implies that there exist bij , antisymmetric in i and j such that ai = bij (pj − cj ) j
(see for example Ref. [35], p. 81, or Lemma 3.8 of Ref. [18]), provided that the differentials dpi are independent on the orbit (a condition which is satisfied [36] in our case). The generalization of this property to the supersymmetric case deserves special treatment, but the proof is rather technical and we will do it separately in Lemma 5.1. Assuming that this is true, then there exist bˆij , antisymmetric in i and j such that aˆ i = bˆij (pˆ j − cj ). j
It is easy to convince oneself that this equation can be lifted to Uh , so there exists Ai and Bij antisymmetric in i and j such that Ai = Bij (Pj − cj (h)), j
with Ai = Ai mod h, since they both project to ai , i.e. Ai = Ai + hCi . If one substitutes in (12): hF = h Ci (Pi − ci (h)), i
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since j Bij (Pj − cj (h))(Pi − ci (h)) = 0 (the Bij ’s are antisymmetric). Hence we get the fact: hF ∈ Ih then F ∈ Ih . So being F ∈ Ih , we can reduce (11) mod h, f = ai (pˆ i − ci ). But f would represent a non trivial relation among the monomials {xI1 · · · xIk }(I1 ...Ik )∈S in C[g∗ ]/I, which is a contradiction, so the linear independence is proven. Lemma 5.1. Let A be the free commutative superalgebra over C generated by M even variables x1 . . . xM and N odd variables ξ1 . . . ξN . Let q1 . . . qn even polynomials in A and denote by q1◦ , . . . qn◦ their projections onto A◦ . Assume that the qi◦ ’s satisfy the following property: If i qi◦ fi◦ = 0 for some fi◦ ∈ A◦ , then there exist Fij◦ ∈ A◦ such that fi◦ = j Fij◦ qj◦ , with Fij◦ = −Fj◦i , i, j = 1 . . . n. Then, if i qi fi = 0 for some fi ∈ A, there exist Fij ∈ A such that fi = j Fij qj , with Fij = −Fj i , i, j = 1 . . . n. Proof. We write fi =
N
fiα1 ...αk ξα1 · · · ξαk ,
k=0 1≤α1 <...αk ≤N
qi =
N
β ...βl
qi 1
ξβ1 · · · ξβl .
l=0 1≤β1 <...βl ≤N
Since qi are even functions we have that l is always an even number (that is, the components with l odd are zero). The terms for k = 0 or l = 0 correspond to fi◦ and qi◦ respectively. The condition ni=1 fi qi = 0 reads in components p n
ˆ
ˆ
αk1 ···αks α1 ···αˆ k1 ···αˆ ks ···αp qi
(−1)(k1 ,...ks ,1,...k1 ,...ks ,...p) fi
= 0,
(14)
i=1 s=0 1≤k1 <···
for p = 1, . . . N and where (k1 , . . . ks , 1, . . . kˆ1 , . . . kˆs , . . . p) is the length of the permutation (k1 , . . . ks , 1, . . . kˆ1 , . . . kˆs , . . . p). For p = 0 we have n
fi◦ qi◦ = 0.
(15)
i=1 odd be the ideal in A generated by the We will show the lemma by induction. Let Im products odd Im = (ξα1 · · · ξαm+1 , . . . , ξ1 · · · ξN ), odd . Assume that there exists and consider the projections πm : A → A/Im
Fijm =
m r=0
Fijδ1 ···δr ξδ1 · · · ξδr ,
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with Fijm = −Fjmi , such that πs (fi ) = πs (Fijm pj )
(16)
for all s ≤ m. (To simplify the notation the sum over repeated indices i and j will be understood.) For m = 0 this is the classical condition, that is guaranteed by (15) and the δ ···δ hypothesis of the theorem. We must show that there exist Fij1 m+1 such that Fijm+1
=
m+1
Fijδ1 ···δr ξδ1 · · · ξδr
r=0
satisfies πs (fi ) = πs (Fijm+1 pj ) for all s ≤ m + 1. Since πN is the identity, we will have our result. Let us write the induction hypothesis (16) in components, fiα1 ···αs =
s
ˆ
ˆ
αl ...αlr
(−1)(l1 ...lr ,1,...l1 ,...lr ,...s) Fij 1
α1 ...αˆ l1 ...αˆ lr ,...αs
qj
,
(17)
r=0 1≤l1 <···lr ≤s
with s ≤ m. We substitute this expression for s ≤ m in (14). Taking p = m + 1 we have α1 ...αm+1 0 qi
fi
+
m
s
s=0 1≤k1 <···
ˆ
ˆ
ˆ
× (−1)(k1 ,...ks ,1,...k1 ,...ks ,...m+1) (−1)(kl1 ...klr k1 ...kl1 ...klr ...ks ) αkl ...αkl
× Fij
r
1
αk1 ...αˆ kl ...αˆ kl ...αks α1 ,...αˆ k ...αˆ k ...αm+1 r s 1 1 qi
qj
= 0.
(18)
We distinguish two kinds of terms in (18), the ones that are multiplied by qi◦ and the ones that are not. A generic term that does not contain qi◦ is of the form ˆ
ˆ
ˆ
ˆ
(−1)(k1 ,...ks ,1,...k1 ,...ks ,...m+1) (−1)(kl1 ...klr k1 ...kl1 ...klr ...ks ) αkl ...αkl
× Fij
1
r
αk1 ...αˆ kl ...αˆ kl ...αks α1 ,...αˆ k ...αˆ k ...αm+1 r s 1 1 qi
qj
(19)
with r = s and s = m + 1. We are going to see that these terms cancel. The reason is that for each term as (19) there exists an identical term where qi is interchanged with qj . Then, they cancel because of the antisymmetry of Fij . We first consider the following set of indices: {k1 , . . . ks } = {1, . . . m + 1} − {k1 , . . . ks } ∪{kl1 , . . . klr }. Then {k1 , . . . ks } − {kl1 , . . . klr } = {1, . . . m + 1} − {k1 , . . . ks },
{k1 , . . . ks } − {kl1 , . . . klr } = {1, . . . m + 1} − {k1 , . . . ks },
(20)
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and s = m + 1 − s + r. By construction {kl1 , . . . klr } ⊂ {k1 , . . . ks }, so we can consider the term
ˆ
ˆ
ˆ
ˆ
(−1)(k1 ,...ks ,1,...k1 ,...ks ,...m+1) (−1)(kl1 ...klr k1 ...kl1 ...klr ...ks ) αkl ...αkl
× Fij
r
1
αk ...αˆ kl ...αˆ kl ...αk
qj
1
1
s
r
α1 ,...αˆ k ...αˆ k ...αm+1
qi
1
s
(21)
.
Equation (21) is equal to (19) with qi in the place of qj , except possibly by a sign. We are going to see that the sign is the same. We start by observing that (k1 . . . ks , 1 . . . kˆ1 . . . kˆs . . . m + 1) = (kl1 . . . klr , 1 . . . kˆ1 . . . kˆs . . . m + 1) +(k1 . . . kˆl1 . . . kˆlr . . . ks , 1 . . . kˆ1 . . . kˆs . . . m + 1), (k1 . . . ks , 1 . . . kˆ1 . . . kˆs . . . m + 1) = (kl1 . . . klr , 1 . . . kˆ1 . . . kˆs . . . m + 1) +(k1 . . . kˆl1 . . . kˆlr . . . k , 1 . . . kˆ1 . . . kˆ . . . m + 1). s
s s
It is now a crucial fact that s − r, m + 1 − s, − r, m + 1 − s are all even numbers. The last two terms in the above equalities are easily seen equal modulo 2 by using (20). Then we have (k1 . . . ks , 1 . . . kˆ1 . . . kˆs . . . m + 1) − (k1 . . . ks , 1 . . . kˆ1 . . . kˆs . . . m + 1) = (kl1 . . . klr , 1 . . . kˆ1 . . . kˆs . . . m + 1) −(kl1 . . . klr , 1 . . . kˆ1 . . . kˆs . . . m + 1) mod2 = (kl1 . . . klr , k1 , . . . kˆl1 . . . kˆlr , . . . k ) s
−(kl1 . . . klr , k1 , . . . kˆl1 . . . kˆlr , . . . ks ) mod2, where we have used (20) again. We have proven our claim and all the terms in (18) that do not contain qi0 cancel. Then from (18) we deduce
α1 ...αm+1
fi
−
m
ˆ
ˆ
(−1)(k1 ,...ks ,1,...k1 ,...ks ,...m+1)
s=0 1≤k1 <···
αk ...αk α1 ,...αˆ k1 ...αˆ ks ...αm+1 ◦ × Fij 1 s qj qi
= 0. α ...αm+1
From the hypothesis of the theorem, this means that there exists Fij 1 metric in i and j such that α1 ...αm+1
fi
m
−
ˆ
ˆ
(−1)(k1 ,...ks ,1,...k1 ,...ks ,...m+1)
s=0 1≤k1 <···
× Fij
α ...αm+1 ◦ qj ,
= Fij 1
from which one can compute α1 ...αm+1
fi
=
m+1
s=0 1≤k1 <···
× Fij terminating our induction.
ˆ
ˆ
(−1)(k1 ,...ks ,1,...k1 ,...ks ,...m+1)
, antisym-
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Acknowledgements. We would like to thank Prof. V. S. Varadarajan and Prof. A. Vistoli for many helpful comments. Work supported in part by the European Community’s Human Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time in which M. A. Ll. is associated to Torino University. The work of M. A. Ll. has also been supported by the research grant BFM 2002-03681 from the Ministerio de Ciencia y Tecnolog´ıa (Spain) and from EU FEDER funds.
References 1. Berezin, F.A.: Introduction to superanalysis. A.A. Kirillov, (ed.), Dordrecht, (Holland): D. Reidel Publishing Company, 1987. With an appendix by V. I. Ogievetsky. Translated from the Russian by J. Niederle and R. Koteck´y. Translation edited by Dimitri Le˘ıtes 2. Kostant, B.: Graded manifolds, Graded Lie theory and prequantization. Lect. Notes Math. 570, 1977 3. Leites, D.A.: Introduction to the theory of supermanifolds. Russian Math. Surv. 35(1), 1–64 (1980) 4. Manin,Y.: Gauge field theory and complex geometry. Berlin-Heidelberg-NewYork: Springer Verlag, 1988 5. Deligne, P., Morgan, J.: Notes on supersymmetry (following J. Bernstein). In: Quantum fields and strings. A course for mathematicians, Vol 1, Evidence, RI: AMS, 1999 6. Kac, V.G.: Lie superalgebras. Adv. Math. 26, 8–26 (1977) 7. Rittenberg, V.: A guide to Lie superalgebras. Invited Talk at Int. Colloquium on Group Theoretical Methods in Physics, Tubingen, Germany, Jul 18-22, 1977 8. Vogan, D.: The Orbit Method and Unitary Representations for Reductive Lie groups. In: Algebraic and Analytic Methods in Representation Theory. Perspectives in Mathematics, Vol 17. London-New York-San Diago: Academic Press, 1996 9. V´arilly, J.C., Gracia-Bond´ıa, J.M.: The Moyal representation for spin. Ann. Phys. 190, 107–148 (1989) 10. Madore, J.: The fuzzy sphere. Class. Quantum Grav. 9, 69–87 (1992) 11. Doplicher, S., Fredenhagen, K., Roberts, J.E.: The quantum structure of space-time at the Planck scale and quantum fields. Commun. Math. Phys. 172, 187–220 (1995) 12. Connes, A.: Non Commutative Geometry. London-New York: Academic press, 1994 13. Connes, A., Douglas, M.R., Schwarz, A.: Non commutative geometry and Matrix theory: Compactification on tori. JHEP 9802, 003 (1998) 14. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, D.: Sternheimer, Deformation theory and quantization, I and II. Ann. Phys. 111, 1–151 (1978) 15. Cahen, B.: Deformation program for principal series representations. Lett. Math. Phys. 36, 65–75 (1996) 16. Arnal, D., Cahen, M., Gutt, S.: Deformations on coadjoint orbits. J. Geom. Phys. 3, 327–351 (1986) 17. Cahen, M., Gutt, S.: Produits ∗ sur les orbites des groupes semi-simples de rang 1. C.R. Acad. Sci. Paris 296, s´erie I, 821–823 (1983); An algebraic construction of ∗ product on the regular orbits of semisimple Lie groups. In: Gravitation and Cosmology. Monographs and Textbooks in Physical Sciences. A volume in honor of Ivor Robinson, W. Rundler and A. Trautman, (eds.), Bibliopolis, 1987; Non localit´e d’une d´eformation symplectique sur la sph`ere S 2 . Bull. Soc. Math. Belg. 36B, 207–221 (1987) 18. Fioresi, R., Lled´o, M.A.: On the deformation quantization of coadjoint orbits of semisimple Lie Groups. Pacific J. Math 198(2), 411–436 (2001) 19. Fioresi, R., Lled´o, M.A.: A Comparison between Star Products on Regular Orbits of Compact Lie Groups. To appear in the Journal of Physics A: Mathematical and General. math.QA/0106129; Fioresi, R., Levrero, A., Lled´o, M.A.: Algebraic and Differential Star Products on Regular Orbits of Compact Lie Groups. To appear in the Pacific J. of Math. math.QA/0011172; Lled´o, M.A.: Deformation Quantization of Non-Regular Orbits of Compact Lie Groups. Lett. Math. Phys. 58, 57–62 (2001) 20. Fronsdal, C.: Some ideas about Quantization. Rep. Math. Phys. 15, 111–145 (1978) 21. Moreno, C.: Invariant star products and representations of compact semi-simple Lie groups. Lett. Math. Phys. 12, 217–229 (1986) 22. Seiberg, N., Witten, E.: String Theory and Noncommutative Geometry. JHEP 09, 032 (1999) 23. Klimcik, C.: An extended fuzzy supersphere and twisted chiral superfields. Commun. Math. Phys. 206, 587 (1999) 24. Grosse, H., Reiter, G.: The fuzzy (super)sphere and field theory. Prepared for Conference on New Insights in Quantum Mechanics, Goslar, Germany, 31 Aug - 3 Sep 1998 25. Balachandran, A.P., Kurkcuoglu, S., Rojas, E.: The star product on the fuzzy supersphere. JHEP 0207, 056 (2002)
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26. Leites, D.A.: Spectra of graded commutative rings. Uspehi Matem. Nauk 30(3), 209 (1974) (In Russian) 27. Hartshorne, R.: Algebraic geometry. New York: Springer Verlag, 1977 28. Milne, J.S.: Algebraic geometry. http://www.jmilne.org, 1996 29. Shafarevich, I.R.: Basic algebraic geometry. Berlin-Heidelberg: Springer-Verlag, 1977 30. Fioresi, R.: On algebraic supergroups and quantum deformations. math.QA/0111113; Supergroups, quantum supergroups and their homogeneous spaces. Euroconference on Brane New World and Noncommutative Geometry (Torino, 2000). Modern Phys. Lett. A 16, 269–274 (2001) 31. Eisenbud, D., Harris, J.: The geometry of schemes. New York: Springer Verlag, 2000 32. Demazure, M., Gabriel, P.: Groupes Alg´ebriques, Tome 1. Mason&Cie, e´ diteur. The Netherlands: North-Holland Publishing Company, 1970, p. 341 33. Borel, A.: Linear algebraic groups. New York: Springer Verlag, 1991 34. Kac, V.G.: Characters of typical representations of classical Lie superalgebras. Commun. Alg. 5, 889–897 (1977); Sergeev, A.: The invariant polynomials on simple Lie superalgebras, Represent. Theory 3, 250–280 (1999); Leites, D., Sergeev, A.: Casimir operators for Lie superalgebras. arXiv:math.RT/0202180 35. Olver, P.J.: Applications of Lie groups to differential equations. Berlin-Heidelberg-New York: Springer-Verlag, 1986 36. Varadarajan, V.S.: On the ring of invariant polynomials on a semisimple Lie algebra. Am. J. Math 90, 1968 37. Manin, Y.: Topics in non-commutative geometry. Princeton, NJ: Princeton University Press, 1991 Communicated by A. Connes
Commun. Math. Phys. 245, 201–214 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1019-x
Communications in
Mathematical Physics
On the Maximal Scarring for Quantum Cat Map Eigenstates Fr´ed´eric Faure1 , St´ephane Nonnenmacher2,3 1
Laboratoire de Physique et Mod´elisation des Milieux Condens´es (LPM2C), BP 166, 38042 Grenoble C´edex 9, France. E-mail:
[email protected] 2 Service de Physique Th´eorique, CEA/DSM/PhT Unit´e de Recherche Associ´ee au CNRS, CEA/Saclay, 91191 Gif-sur-Yvette C´edex, France. E-mail:
[email protected] 3 Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070, USA Received: 23 April 2003 / Accepted: 29 July 2003 Published online: 16 January 2004 – © Springer-Verlag 2004
Abstract: We consider the quantized hyperbolic automorphisms on the 2-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the semiclassical measure corresponding to a sequence of normalized eigenstates has a pure point component (phenomenon of “strong scarring”), then the weight of this component cannot be larger than the weight of the Lebesgue component, and therefore admits the sharp upper bound 1/2.
1. Introduction We are interested in the semiclassical properties of quantum maps on the twodimensional torus T2 , that is the unitary transformations Mˆ h which quantize a symplectic map M on T2 (h = 2π is Planck’s constant). More precisely, we focus on maps M having a chaotic dynamics (that is, at least ergodic w.r. to the Lebesgue measure), and investigate the phase space properties of the eigenstates of Mˆ h in the semiclassical limit h → 0. To any sequence of eigenstates {|ψh }h→0 corresponds a sequence of probability measures on the torus {µh }h→0 . In the weak-∗ topology, the set of Borel probability measures on the torus is compact so the sequence {µh }h→0 admits at least one accumulation point µ; such a µ is called a “semiclassical measure” (also a “quantum limit”) of the map M, related with the sequence {|ψh }h→0 . From Egorov’s theorem, the measure µ is invariant through the classical map M. A natural question is the following: «For any M-invariant probability measure ν, does there exist a sequence of eigenstates of the quantized map Mˆ h admitting ν as a semiclassical measure ?» If the answer is negative, then one wants to determine the set of semiclassical measures generated by all possible sequences of eigenstates.
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If the map M is ergodic (w.r. to the Lebesgue measure on T2 ), it admits a unique absolutely continuous invariant measure, namely the Lebesgue measure itself (which is also the Liouville measure on the symplectic manifold T2 ). On the other hand, each periodic orbit of M supports an invariant Dirac (atomic, pure point) measure. As a result, if M is an Anosov map (i.e. uniformly hyperbolic on T2 ), the space of invariant pure point measures is infinite-dimensional (its closure yields the full set of invariant probability measures MM [19]). We now review some results obtained so far on this issue. Schnirelman’s theorem provides a partial answer to the above question, in the case of an ergodic map: “almost all sequences” of eigenstates admit for semiclassical measure the Lebesgue measure [18, 5, 21, 3]. This phenomenon is called “quantum ergodicity” in the mathematics literature. Still, this theorem does not exclude “exceptional sequences” of eigenstates converging to a different semiclassical measure. Extensive numerical studies have shown that many eigenstates of quantum hyperbolic systems show an enhanced concentration on one or several (unstable) periodic orbits [9]. Still, it is commonly believed that this enhancement (called “scarring”) is weak enough to allow the concerned eigenstates to converge (in the weak-∗ sense) to the Liouville measure. Thus, “scarring” must not be mistaken with “strong scarring”, that is the existence of a sequence of eigenstates, the semiclassical measure of which contains a pure point component on some periodic orbit. “Quantum unique ergodicity”, that is, the absence of any exceptional sequence of eigenstates, was proven for some families of ergodic linear parabolic maps on T2 [3, 15], using the fact that for these maps the Lebesgue measure is the unique invariant measure. On the other hand, a special class of ergodic piecewise affine transformations on T2 have been studied and quantized in [4], for which every classical invariant measure is a semiclassical measure. In both these cases, the maps are only “weakly chaotic”, in particular they are not mixing and have no periodic point. In the continuous-time framework, precise results have been obtained for the eigenstates of the Laplace-Beltrami operator on arithmetic manifolds of constant negative curvature in dimension 2 or 3 [17, 13, 16, 10]; the corresponding classical dynamics, namely the geodesic flows on the manifolds, are known to be of Anosov type. E. Lindenstrauss [12] recently proved quantum unique ergodicity for sequences of joint eigenstates of the Laplacian and the Hecke operators on arithmetic surfaces (all eigenstates of the Laplacian are conjectured to be of this type). In this paper, we restrict ourselves to a very special family of Anosov maps on the torus, namely the linear hyperbolic automorphisms of the 2-torus, also called generalized Arnold’s cat maps. For any automorphism A of the form A ≡ Id mod 4 and any value of h, Rudnick and Kurlberg have defined a family of “Hecke operators” commuting with the quantum map Aˆ h , and proven unique quantum ergodicity for the sequences of joint eigenstates [11]. However, as opposed to the case of arithmetic surfaces, many eigenstates of Aˆ h are not Hecke eigenstates, leaving open the possibility of exceptional sequences. In [2], Bonechi and De Bièvre have shown that for any automorphism A, a semiclassical measure of A cannot be completely localized (or completely √ “scarred”), in that the weight of its pure point component is bounded above by 5 − 1 /2 0.62 (their proof also applies to ergodic automorphisms on higher-dimensional symplectic tori). In [7], sequences of eigenstates of Aˆ h were explicitly constructed, for which the semiclassical measure has a pure point component of weight 1/2 localized on a finite set of periodic orbits, the remaining part of the measure being Lebesgue. In this paper we improve the results of [2] as follows:
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Theorem 1.1. Let A be a hyperbolic automorphism of T2 , and µ be a normalized semiclassical measure of A. Splitting µ into its pure point, Lebesgue and singular continuous components, µ = µpp +µLeb +µsc , the following inequalities hold between the weights of these components: µLeb (T2 ) ≥ µpp (T2 ), which implies µpp (T2 ) ≤ 1/2. The states constructed in [7] saturate this upper bound: they are “maximally scarred”. After recalling the definition of the quantized automorphisms (Sect. 2), we prove in Sect. 3 two “dynamical” propositions (the first one was proven in [2, Sect. 5] in a more general context). They both deal with the evolution through Aˆ h up to the “Ehrenfest time” T ∼ | log h|/λ, of quantum states with prescribed initial localization properties. Using these propositions, we then show in Sect. 4 that for any finite union S of periodic orbits, any semiclassical measure µ of A satisfies µ(S) ≤ µLeb (T2 ) (Theorem 4.1), from where the above theorem is a straightforward corollary. In final remarks, we draw consequences of the above theorem, concerning the determination of the set MA,SC of semiclassical measures for the automorphism A. We also discuss possible extensions of these results to a broader class of Anosov systems. Let us mention that our methods are quite similar to the one used in [2]. The reason why we obtain a sharper bound lies in a cautious use of the Cauchy-Schwarz inequality. In the proof of [2, Thm. 1.2], this inequality is directly used to estimate the localization properties of the eigenstate, thereby introducing a loss of information. On the other hand, we apply this inequality in Sect. 3 to obtain sharp estimates on the localization of “partial eigenstates”, while Eqs. 4.5–4.7 dealing with the full eigenstate remain equalities. 2. Quantum Hyperbolic Automorphisms on T2 2.1. Quantum mechanics on T2 . We briefly describe the quantum mechanics on the 2-torus phase space as defined in [8, 6]. The Hilbert space of quantum states corresponding to Planck’s constant h will be called Hh : quantum states can be identified with distributions ψ(q) ∈ S (R) such that ψ(q +1) = ψ(q) and (Fψ)(k+h−1 ) = (Fψ)(k), where F is the Fourier transform. Hh is a nontrivial vector space iff h−1 ∈ N∗ , and then −1 Hh C(h ) . In what follows, h will always be taken of that form; the semiclassical limit is therefore defined as the limit h → 0, h−1 ∈ N. For any classical observable f ∈ C ∞ (T2 ), we note respectively fˆ = Oph (f ) its Weyl quantization and fˆAW = OphAW (f ) its anti-Wick quantization on Hh [3]. The anti-Wick quantized operator satisfies the property fˆAW ≤ f ∞ = supT2 (|f |). To any state |ψh ∈ Hh correspond the Wigner and Husimi measures µ˜ h , µh defined as [3] dx Wψh (x) f (x) = ψh |fˆ|ψh , (2.1) µ˜ h (f ) = 2 T dx dx Hψh (x) f (x) = µh (f ) = |ψh |x|2 f (x) = ψh |fˆAW |ψh . (2.2) T2 T2 h The ket |x denotes the (asymptotically normalized) coherent state in Hh localized at the point x = (q, p), with widths q ∼ p ∼ /2. While the Husimi density Hψh (x) is a non-negative smooth function on T2 , the “Wigner function” Wψh (x) is a finite combination of delta peaks with real (possibly negative) coefficients. We now consider an infinite sequence of states {|ψh ∈ Hh }h→0 . By definition, the corresponding sequence of Husimi measures {µh } weak-∗ converges to µ iff for any
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smooth observable f , one has µh (f ) −−→ µ(f ). The sequence of signed measures {µ˜ h } then admits the √ same weak-∗ limit (µh is the convolution of µ˜ h by a Gaussian kernel of width ∼ ). The limit (Borel) measure µ is then called the semiclassical measure of the sequence {|ψh }h→0 (by a slight abuse of language, we also say that the sequence of states {|ψh }h→0 converges to µ). If the states |ψh are (normalized) eigenstates of a quantized map Mˆ h , then µ is called a semiclassical measure for M. In that case, Egorov’s property, that is the semiclassical commutation of time evolution and quantization: ∀t ∈ Z,
Mˆ h−t Oph (f )Mˆ ht − Oph (f ◦ M t ) −−→ 0, h→0
(2.3)
implies that µ is an invariant measure for the classical map M. A sequence of states {|ψh }h→0 is said to be equidistributed if it converges semiclassically to (a multiple of) the Lebesgue measure on T2 (noted dx), i.e. ifffor a certain C > 0 and for any observable f , µh (f ) (equivalently µ˜ h (f )) converges to C T2 dx f (x). On the other hand, a sequence {|ψh }h→0 is called localized iff it converges to a pure point measure on T2 , that is if there exists a countable set of points {xi } and weights h→0 αi > 0, i αi < ∞ such that for any observable f , µh (f ) −−→ i αi f (xi ). 2.2. Quantum yperbolic automorphisms. An automorphism of T2 , or generalized cat map, is thediffeomorphism on T2 defined by the action of a matrix A ∈ SL(2, Z) on the q 2 point x = p ∈ T . The map itself will also be denoted by A. It is uniformly hyperbolic on T2 (therefore of Anosov type) iff |tr(A)| > 2. Depending on the sign of the trace, the eigenvalues of A are of the form {±eλ , ±e−λ }, where λ > 0 is the Lyapounov exponent. The corresponding eigenaxes define the unstable/stable directions at any point x ∈ T2 , and their projections on the torus make up the unstable and stable manifolds of the origin (which is an obvious fixed point). We will use the property that the slopes of these axes 2 are irrational, so that both manifolds are dense
on T . 10 01 Under the condition A ≡ or mod 2, the linear automorphism A can be 01 10 quantized on any Hh , yielding a unitary matrix Aˆ h of dimension h−1 [8, 6]. For simplicity, we will assume in the following sections that A is of that form. Yet, this restriction can easily be lifted: any matrix A ∈ SL(2, Z) can be quantized on Hh if we restrict h−1 to take even values, or extend the definition of Hh to allow nontrivial “Bloch angles” [6, 3]. All our results can be straightforwardly generalized to those cases. Let us call Tv the classical translation by the vector v ∈ R2 . It can be naturally quantized as a unitary matrix Tˆv on Hh iff v belongs to the square lattice (hZ)2 , that is iff v = hk for a certain k ∈ Z2 . The operator Tˆhk can also be interpreted as the Weyl quanti- zation Oph (Fk ) of the complex-valued observable Fk (q, p) = exp − 2π i(k1 p − k2 q) on T2 . For any h, the following intertwining relation holds between the quantum automorphism Aˆ h and the quantized translations: ∀k ∈ Z2 ,
ˆ ˆ ˆ ˆ −1 ˆ Aˆ −1 h Thk Ah = ThA−1 k ⇐⇒ Ah Oph (Fk ) Ah = Oph (Fk ◦ A). (2.4)
Comparing with Eq. (2.3), we see that the above identity realizes an exact Egorov property: quantization exactly commutes with time evolution, for arbitrary times t and
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arbitrary h [8]. This exact equality is characteristic of linear maps, and will be crucial in the next sections. 3. Localized States Evolve into Equidistributed States We define the Ehrenfest time or log-time corresponding to the quantum map Aˆ h by T =
|log h| , λ
where [.] denotes the integer part (we will always omit indicating the h-dependence of T ). Consider a set of n points S = {xi }i=1→n on T2 , such that for a certain integer d > 0, all vectors (xi − xj ) belong to the lattice d1 Z2 . Assign to each point xi a weight αi > 0, with i αi = 1, and define the Dirac probability measure δS ,α = ni=1 αi δxi on the torus. Proposition 3.1. Suppose that a sequence of states {|ϕh ∈ Hh }h→0 converges to the measure δS ,α . Then the sequence of states {|ϕh = Aˆ Th |ϕh } is semiclassically equidistributed. This property has been proven in a more general setting (higher dimensional automorphisms, time T replaced by a time interval) in [2, Theorem 5.1]. To be self-contained, we give below a “fast” proof of this proposition for our case. h→0
Proof. As a first step, we draw consequences from the assumption µh −→ δS ,α , where µh is the Husimi measure of |ϕh . Denoting by D(xi , a) the disk of radius a > 0 cendef
tered at xi , and by D(S, a) = ∪ni=1 D(xi , a) the corresponding neighbourhood of S, h→0
the weak-∗ convergence of µh implies that µh (T2 ) −→ 1 and that for any a > 0, h→0 µh T2 \D(S, a) −→ 0. From a standard diagonal argument, one can then construct a h→0
decreasing sequence ah > 0, ah −→ 0 such that |µh T2 − 1| ≤ ah
and µh T2 \ D(S, ah ) ≤ ah .
(3.1)
In a second step, we remark that proving the equidistribution of the sequence |ϕh amounts to prove that for any fixed k ∈ Z2 , h→0 ˆ Fk (x) dx = δk,0 . (3.2) µ˜ h (Fk ) = ϕh |Thk |ϕh −−→ T2
h→0
The case k = 0 is obvious since µ˜ h (T2 ) = µ˜ h (T2 ) = µh (T2 ) −→ 1 by assumption. We now select a fixed wave vector 0 = k ∈ Z2 , and study the above LHS. The intertwining relation (2.4) allows us to rewrite it as ˆ ˆ ˆT ϕh |Tˆhk |ϕh = ϕh |Aˆ −T h Thk Ah |ϕh = ϕh |Thkh |ϕh ,
(3.3)
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with the vector kh = A−T k. From the definition of the Ehrenfest time, the “large” eigenvalue of A−T is (±eλ )T = ±C(h) h1 , where the prefactor satisfies e−λ ≤ C(h) ≤ 1. The decomposition of that vector in the eigenbasis of A then reads: hkh = ±eλT hkstable ± e−λT hkunstable = ±C(h)kstable + O(h2 ).
(3.4)
The vector hkh thus has a finite length (bounded from above and from below uniformly in h), and is asymptotically parallel to the stable axis. Because the slope of that axis is irrational, the distance between hkh and the lattice d1 Z2 is bounded from below by a constant c(k) > 0 uniformly w.r. to h. To finish the proof, we write the above overlap as a coherent state integral: ˆ |ϕh |Thkh |ϕh | =
dx dx ˆ ϕh |xx|Thkh |ϕh ≤ |x|ϕh | |x − hkh |ϕh |. 2 2 T h T h (3.5)
We then split the integral on the RHS between D(S, ah ) and its complement. The integral on D(S, ah ) is estimated through a Cauchy-Schwarz inequality: D(S ,ah )
dx |x|ϕh | |x − hkh |ϕh | ≤ h
µh D(S, ah ) µh D(S, ah ) − hkh .
Due to above-mentioned property of hkh and the fact that xi − xj ∈ d1 Z2 , for small enough ah the set D(S, ah ) − hkh has no intersection with D(S, ah ). As a consequence, using (3.1), the second √ factor under the square-root on the RHS is bounded above by ah , and the full RHS by ah (1 + ah ). The remaining integral over T2 \D(S, ah ) admits the same upper bound for similar reasons. Proposition 3.2. Let (S, α) be a finite weighted set of A-periodic points, and ν an A-invariant probability measure satisfying ν(S) = 0. Suppose that the sequence {|ϕS ,h }h→0 converges semiclassically to the measure δS ,α , and that a second sequence = A ˆ T |ϕν,h {|ϕν,h }h→0 converges to ν. Then the states |ϕS ,h = Aˆ Th |ϕS ,h , |ϕν,h h satisfy: ∀k ∈ Z2 ,
h→0 ϕS ,h |Tˆhk |ϕν,h −−→ 0.
Proof. We use similar methods as for the previous proposition. Namely, as in Eq. (3.3) we rewrite the overlap as ϕS ,h |Tˆhk |ϕν,h = ϕS ,h |Tˆhkh |ϕν,h =
T2
dx ϕS ,h |xx|Tˆhkh |ϕν,h h
(3.6)
with hkh given by Eq. (3.4). We want to cut this integral into two parts. Using the same notations as above, the assumptions of the proposition imply the existence of a h→0
decreasing sequence ah −−→ 0 such that the Husimi measures of |ϕS ,h and |ϕν,h satisfy µS ,h T2 \ D(S, ah ) ≤ ah ,
µS ,h T2 ≤ 1 + ah
and
µν,h T2 ≤ 1 + ah .
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These inequalities also hold if we replace ah by any decreasing sequence bh ≥ ah , bh → 0. We can now bound the part of the integral (3.6) over the set T2 \D(S, bh ): dx ˆ |xx| T |ϕ ≤ µS ,h T2 \D(S, bh ) µν,h T2 ϕ S ,h hkh ν,h 2 T \D(S ,bh ) h h→0 ≤ bh (1 + bh ) −−→ 0. The second part of the integral is also estimated by Cauchy-Schwarz: dx ˆ ϕS ,h |xx|Thkh |ϕν,h ≤ µS ,h T2 µν,h D(S, bh ) − hkh . D(S ,bh ) h We now show that the sequence bh can be chosen such that the second factor under the square root vanishes in the semiclassical limit, so that the full RHS vanishes as well. Indeed, using Eq. (3.4) and the bound on C(h), one realizes that for h small enough, the set D(S, bh ) − hkh is contained in the thin “tube” def Tbh = D(S, 2bh ) + [−1, −e−λ ] ∪ [e−λ , 1] kstable . As bh → 0, this tube decreases to T0 = S + [−1, −e−λ ] ∪ [e−λ , 1] kstable , that is a union of segments of the stable manifold (T0 = S if k = 0). The following simple lemma is proven in Appendix B. Lemma 3.3. For any invariant probability measure ν, any finite set of periodic points S and any 0 < a < b < ∞, one has ν(S + [a, b]estable ) = 0, where estable is a unit vector in the stable direction. This lemma implies that ν(T0 ) = 0 in the case k = 0, the case k = 0 being trivial. Since bh 0, the regularity of the Borel measure ν entails: lim ν(Tbh ) = ν(T0 ) = 0.
bh →0
Since by assumption the Husimi measures µν,h converge towards ν, one can by a diagonal argument choose the sequence bh 0 (making sure that bh ≥ ah ) such that: h→0 µν,h Tbh −−→ 0. h→0 This finally implies that µν,h D(S, bh ) − hkh −−→ 0.
4. Maximal Localization of the Semiclassical Measures We call τ a periodic orbit of A of period Tτ , and denote its associated measure by 1 δτ = δx . Tτ x∈τ We consider a finite set F of periodic orbits, associate a weight wτ > 0 to each orbit ( τ ∈F wτ = 1), and construct the pure point invariant measure δF ,w = w τ δτ . τ ∈F
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All periodic points of A have rational coordinates, so this measure is a particular instance of the measure δS ,α considered in Proposition 3.1. Indeed, grouping the periodic orbits of F together yields the set of rational points S = SF = ∪τ ∈F τ = {x1 , . . . , xn }, and the weight allocated to each point xi reads αi = wTττ . From now on, the two notations δF ,w and δS ,α will refer to the same invariant measure. We now state the central result of this article. Theorem 4.1. Let µ be a normalized A-invariant Borel measure of A, and (F, w) a finite weighted set of periodic orbits. µ may be decomposed into µ = βδF ,w + (1 − β)ν, where ν is a normalized invariant measure satisfying ν(SF ) = 0. If µ is a semiclassical measure of A, then its Lebesgue component has a weight ≥ β, which in turn implies β ≤ 1/2. Any invariant Borel measure µ ∈ MA can obviously be decomposed in the above way, with 0 ≤ β ≤ 1. We do not assume the measure ν to be continuous, but allow it to contain a pure point component localized on a (possibly countable) set of periodic orbits disjoint with F. The statement of the theorem is of course stronger if we include in (F, w) as many orbits as possible. By a simple limit argument, one may eventually take for ν the continuous component of µ, allowing (F, w) to be a countable weighted set of orbits (still taking τ ∈F wτ = 1): one then obtains Theorem 1.1 as a simple corollary of the one above. Proof of Theorem 4.1. There are two steps in the proof. Let {|ψh }h→0 be a sequence of eigenstates of Aˆ h admitting µ as a semiclassical measure. Our first objective is to decompose the state |ψh into |ψS ,h + |ψν,h , such that the sequence {|ψS ,h } (resp. {|ψν,h }) converges to the measure βδS ,α (resp. the measure (1−β)ν). This decomposition will be obtained by “projecting” |ψh on appropriate (h-dependent) small neighborhoods of S. In the second part we will be guided by the following simple idea. Because |ψh is an eigenstate of Aˆ h , |ψh ∝ Aˆ th |ψh = Aˆ th |ψS ,h + Aˆ th |ψν,h for any t ∈ Z, in particular for the Ehrenfest time t = T . From Proposition 3.1, the sequence of states {Aˆ Th |ψS ,h } is equidistributed; together with Proposition 3.2, that implies that the semiclassical measure of Aˆ Th |ψh (that is, µ) contains a Lebesgue part of weight ≥ β. First part. Splitting the eigenstates. To “project” the eigenstates in a small neighbourhood of S, we will use a function ϑ ∈ C ∞ (R2 ) satisfying 0 ≤ ϑ(x) ≤ 1 on R2 , ϑ(x) = 0 outside the disk D(0, 2) and ϑ(x) = 1 inside the disk D(0, 1). For any (small) r > 0, and any point xi ∈ S, let ϑi,r (x) = n∈Z2 ϑ x−xri −n , which can be seen as a smooth function on T2 , localized around xi . We will also need the function ϑr = ni=1 ϑi,r to take all points of S into account: this function is supported in the neighborhood D(S, 2r) of S. There is an r0 > 0 such that the disks D(xi , 2r0 ) do not overlap each other, which implies ϑi,r ϑj,r ≡ 0 if i = j . AW Fixing 0 < r ≤ r0 , we consider the anti-Wick quantization of these functions, ϑˆ i,r and use them to decompose |ψh . We first observe that for any xi ∈ S, h→0 AW ψh |ϑˆ i,r |ψh −−→ βδS ,α (ϑi,r ) + (1 − β)ν(ϑi,r ) = βαi + (1 − β)ν(ϑi,r ).
Furthermore, one has for any pair xi , xj ∈ S, h→0 AW ˆ AW ψh |ϑˆ i,r ϑj,r |ψh −−→ βδS ,α (ϑi,r ϑj,r ) + (1 − β)ν(ϑi,r ϑj,r ) 2 = δij βαi + (1 − β)ν(ϑi,r ) .
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On the other hand, the regularity of the (Borel) measure ν entails: r→0
∀xi ∈ S,
ν(ϑi,r ) −−→ ν({xi }) = 0
and similarly
r→0
2 ν(ϑi,r ) −−→ 0.
From the above limits, one can construct by a diagonal argument a decreasing sequence h→0
of radii r(h) −→ 0 such that ∀xi , xj ∈ S,
AW ψh |ϑˆ i,r(h) |ψh −−→ βαi h→0
AW ˆ AW and ψh |ϑˆ i,r(h) ϑj,r(h) |ψh −−→ δij βαi . (4.1) h→0
We now show that the two sequences of vectors def AW |ψh |ψν,h = Id − ϑˆ r(h)
def AW |ψh , |ψS ,h = ϑˆ r(h)
(4.2)
provide the desired decomposition of |ψh , that is the corresponding sequences respech→0
tively converge to the measures βδS ,α and (1−β)ν. The first statement µS ,h −→ βδS ,α AW acts as a “microlocal projector” onto the set S. seems quite natural: the operator ϑˆ r(h) This property is precisely expressed in the following lemma, which we prove in Appendix A. h→0
Lemma 4.2. For any f ∈ C ∞ (T2 ), xi ∈ S and any decreasing sequence r(h) −→ 0, one has h→0 AW AW
fˆAW ϑˆ i,r(h) − f (xi ) ϑˆ i,r(h)
−−→ 0, where . is the operator norm on Hh . This lemma together with the properties (4.1) immediately yield the required limits: h→0 ∀f ∈ C ∞ (T2 ), ψS ,h |fˆAW |ψS ,h −−→ βαi f (xi ) = βδS ,α (f ), (4.3) i
ˆAW
ψν,h |f
h→0
|ψν,h −−→ µ(f ) − βδS ,α (f ) = (1−β)ν(f ). (4.4)
Second part. Playing with time evolution. We will follow a strategy close to the one used to prove Proposition 3.1. We consider a fixed k ∈ Z2 , and focus on the overlap ψh |Tˆhk |ψh . From the semiclassical assumption on {|ψh }, one has h→0 ψh |Tˆhk |ψh −−→ µ(Fk ) = βδS ,α (Fk ) + (1 − β)ν(Fk ).
(4.5)
On the other hand, since |ψh = |ψS ,h + |ψν,h is an eigenstate of Aˆ h , one may rewrite ˆ ˆT ψh |Tˆhk |ψh = ψh |Aˆ −T h Thk Ah |ψh
(4.6)
= ψS ,h |Tˆhk |ψS ,h + 2 ψν,h |Tˆhk |ψS ,h + ψν,h |Tˆhk |ψν,h , (4.7) =A ˆ T |ψν,h and T is the Ehrenfest time. with the notation |ψS ,h = Aˆ Th |ψS ,h , |ψν,h h We are now in position to collect the dynamical results of Sect. 3:
• From the asymptotic localization (4.3) of |ψS ,h and Proposition 3.1, the first term in (4.7) converges to βdx(Fk ) = βδk,0 as h → 0.
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• From the properties (4.3–4.4) and Proposition 3.2, the cross-terms in (4.7) vanish in h→0 the semiclassical limit: ψ |Tˆhk |ψ −−→ 0. ν,h
S ,h
Using Eq. (4.5), the last term in (4.7) has therefore the semiclassical behaviour h→0
ψν,h |Oph (Fk )|ψν,h −−→ µ(Fk ) − βdx(Fk ). } Since this limit holds for any k ∈ Z2 , it means that the sequence {|ψν,h h→0 admits the semiclassical measure µ − βdx. Because semiclassical measures are limits of Husimi measures, they cannot contain any negative part. Therefore, µ − βdx must be a positive measure, which implies that the Lebesgue component of µ has a weight ≥ β. This component being contained in (1 − β)ν, one has finally (1 − β) ≥ β ⇔ β ≤ 1/2.
5. Remarks and Comments 5.1. On the set of semiclassical measures. Theorem 1.1 constrains the set of semiclassical measures MA,SC to be a proper subset of the set MA of invariant Borel measures. One can easily show that MA,SC is a closed set of MA . Indeed, if for any n > 0 the sequence Sn = {|ψh,n }h→0 converges towards a normalized semiclassical measure µn , and that in turn the measures µn weak-∗ converge to a measure µ, then one can extract h→0
a function n(h) −→ ∞ such that {|ψh,n(h) }h→0 converges to µ. Every open neighbourhood of MA contains a pure point measure of type δτ (τ a periodic orbit) [19, 14], therefore Theorem 1.1 implies that the set MA,SC is nowhere dense in MA (i.e. its interior is empty). On the other hand, the results of [7] show that MA,SC contains all measures of the type δτ +dx 2 . Since the measures {δτ } are dense in MA and the set MA,SC is closed, this implies ν + dx ∀ν ∈ MA , ∈ MA,SC . 2 This inclusion together with the constraint imposed by Thm. 1.1 do not suffice to fully identify the set MA,SC . We do not know if a singular continuous invariant measure can be a semiclassical measure. The set of invariant continuous measures is dense in MA [19], so in any case MA,SC cannot contain all continuous invariant measures. 5.2. About the Ehrenfest time. Proposition 3.1 means that any sequence of localized states {|ϕh }h→0 evolves towards a sequence of equidistributed states at the Ehrenfest time T = | logλ h| +O(1). To achieve this goal, the prefactor 1/λ defining T is crucial, and cannot be modified without stronger assumptions on the localization of |ϕh . Indeed, for any > 0, one can construct a sequence of states |ϕh semiclassically localized at the origin, such that the evolved states |ψh = Aˆ (1− )T |ϕh are still localized at the same point. Explicitly, consider the coherent state at the origin |0 = |0h , and take the sequences def −(1− )T /2 |0, |ϕh = Aˆ h
|ψh = Aˆ h
(1− )T
|ϕh = Aˆ h
(1− )T /2
|0.
At the “microscopic scale”, the states |ϕh and |ψh are very different: the former is stretched along the stable axis, the latter along the unstable axis. However, the “length”
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of both states is of the order h /2 , so this difference of shape is invisible at the measuretheoretic level, and both sequences admit the semiclassical measure δ0 . On the other hand, there exist an infinite sequence of Planck’s constants h−1 k ∈ N, hk → 0 such that starting from the states {|ψhk } defined above (localized at the origin), the evolved states {Aˆ (1+ )T |ψhk } are localized at the origin as well. These special values of h correspond to “short quantum periods” of the quantized cat map [1]. Let us remind that for any h−1 ∈ N, the quantum cat map Aˆ h is periodic, meaning that there exists Ph ∈ N and θh ∈ [0, 2π] such that Aˆ Ph h = eiθh Idh [8]. Besides, there exists an infinite subsequence hk → 0 for which these periods are as short as Phk = 2Thk + O(1). As a result, one has 3(1+ )T /2 3(1+ )T /2−Pk (−1+ )T /2+O (1) (1+ )T |ψhk = Aˆ hk |0 = eiθhk Aˆ hk |0 = eiθhk Aˆ hk |0. Aˆ hk
The state on the RHS is close to |ϕhk at the microscopic level, and therefore admits δ0 for semiclassical measure. In conclusion, the time (1 − )T can be too short, and (1 + )T too long to produce the transition localized → equidistributed described in Prop. 3.1. 5.3 More general maps? Proposition 3.1 can be extended to a broad class of quantized automorphisms on tori of dimension 2d with d > 1 [2, Thm. 5.1]. Let eλ be the maximal modulus of the eigenvalues of A, Eλ the direct sum of the generalized eigenspaces corresponding to eigenvalues of modulus eλ , and E<λ the direct sum of the remaining eigenspaces. The precise conditions on the automorphism A for the proposition to be satisfied are the following: (1) A is ergodic, meaning that λ > 0 and that none of its eigenvalues is a root of unity. A is then automatically mixing, but need not be hyperbolic (it may have eigenvalues on the unit circle). (2) the subspace E<λ has a trivial intersection with Z2d . (3) A restricted to Eλ is diagonalizable. The two last conditions are always satisfied for ergodic automorphisms in the case d = 1, but not necessarily in higher dimension. They are needed to obtain the crucial estimate (5.37) of [2]. One easily checks that Proposition 3.2 (and Lemma 3.3 which it depends on) also holds for higher-dimensional automorphisms satisfying the above conditions. As opposed to the proof of [2, Thm. 1.2], our Thm. 4.1 crucially relies on the two dynamical propositions of Sect. 3, while the remaining ingredient in the proof of the theorem (namely, the splitting of eigenstates into two parts) can be straightforwardly transposed to higher dimension. Therefore, Theorems 4.1 and 1.1 also hold in higher dimension for the class of ergodic automorphisms described above. We do not know if the upper bound 1/2 is sharp in dimension d > 1; in fact, the periods Ph of the quantized h→0
Ph automorphisms then satisfy | log h| −→ ∞ ([5 ]), which makes the construction of [7] impossible to generalize. Back to the 2-dimensional torus, a natural extension of the above results would concern the perturbations of the linear map A of the form M = e−tXH ◦ A, where XH is the vector field generated by a Hamiltonian H ∈ C ∞ (T2 ). For t sufficiently small, this map is still Anosov. The challenge consists in generalizing Propositions 3.1 and 3.2 to those maps, with an appropriate definition of the Ehrenfest time. The trick (3.3) used in the linear case to prove these propositions cannot be used for a nonlinear perturbation, the problem starting from the poor control of Egorov’s property (2.3) for times of order T .
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Finally, one may also try to prove a similar property for chaotic flows, e.g. the geodesic flow on a compact Riemann surface of negative curvature. In such a setting, some interesting results have been recently obtained by R. Schubert, pertaining to the long-time evolution of Lagrangian states, which is a first step towards the proof of Proposition 3.1 in this setting.
Appendix A. Proof of Lemma 4.2 AW |x is We start by showing that for x outside a small disk around xi , the state ϑˆ i,r(h) asymptotically small (|x is a torus coherent state at the point x). More precisely, there exists a sequence of radii R(h) 0 such that, for any xi ∈ S and any sequence of points AW |x ≤ h2 for sufficiently small h. {xh ∈ T2 } satisfying xh ∈ D(xi , R(h)), then ϑˆ i,r(h) h First of all, we recall a couple of estimates on torus coherent states, valid for small enough h.
• For any x ∈ T2 , |x ≤ 2. • For any x, y ∈ T2 , one has |x|y| ≤ 5 exp{π|x − y|2 /2h}, where |x − y| denotes the torus distance between the points x, y. AW is a combination of projectors |xx| for points x in the disk Now, the operator ϑˆ i,r(h) √ D(xi , 2r(h)). Therefore, taking R(h) = 2r(h) + 2h|log h| will √ do the job: any xh ∈ D(xi , R(h)) and any x ∈ D(xi , 2r(h)) satisfy |x − xh | ≥ 2h|log h|, which implies for h small enough |x|xh | ≤ 5 exp(−2πh|log h|/2h) ≤ h3 . One finally gets AW
ϑˆ i,r(h) |xh ≤
D(xi ,2r(h))
dx
|x |x|xh | ≤ 2h2 V ol(D(xi , 2r(h))) ≤ h2 . h
We are now in position to estimate the operator product
AW fˆAW ϑˆ i,r(h) =
T2
dy AW |yf (y)y| ϑˆ i,r(h) h
by separating the integral into two parts. On the one hand, the integral over T2 \ D(xi , R(h)) is bounded above by 2h f ∞ from the above results. On the other hand, on def
D(xi , R(h)) the function f (y) is equal to the function f (xi ) + gh (y), where gh (y) = (f (y) − f (xi ))ϑi,R(h) (y). Since gh (y) is uniformly bounded on T2 , the same arguments as above yield AW AW AW fˆAW ϑˆ i,r(h) = f (xi )ϑˆ i,r(h) + gˆ hAW ϑˆ i,r(h) + O(h).
The function gh actually decreases uniformly with h due to the smoothness of f :
gh ∞ ≤
sup
|y−xi |≤2R(h)
|f (y) − f (xi )| ≤ 2 df ∞ R(h).
This upper bound also applies to the anti-Wick quantization of gh , so that h→0
gˆ AW ϑˆ AW ≤ 2 df ∞ R(h) −→ 0. h
i,r(h)
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Appendix B. Proof of Lemma 3.3 We first replace S by the finite invariant set it generates, S = ∪n∈Z An (S). We then def want to prove that if I0 = S + [a, b]estable with 0 < a < b, then ν(I0 ) = 0 if ν is an invariant probability measure. Let n0 be an integer such that aeλn0 > b. Then, the sets Ij = Aj n0 (I0 ) = S + [aeλj n0 , beλj n0 ]estable , def
j ∈Z
are pairwise disjoint. The invariant measure ν will satisfy for any J ≥ 0: ν
J j =−J
Ij
=
J
ν(Ij ) = (2J + 1)ν(I0 ).
j =−J
Since ν(T2 ) = 1, one must therefore have ν(I0 ) = 0. This lemma can be easily generalized to the case of the higher-dimensional ergodic toral automorphisms satisfying the conditions stated in Sect. 5.3. It can also be extended to any Anosov map M on T2 , the straight segments making up I0 being replaced by segments of the stable manifolds of a set of periodic points. Acknowledgements. We have benefitted from insightful discussions with R. Schubert, and Y. Colin de Verdière, whose remarks also motivated this work. We are grateful to M. Dimassi for pointing out to us a gap in the proof of Prop. 3.2. Both authors acknowledge the support of the Mathematical Sciences Research Institute (Berkeley) where this work was completed.
References 1. Bonechi, F., De Bi`evre, S.: Exponential mixing and ln timescales in quantized hyperbolic maps on the torus. Commun. Math. Phys. 211, 659–686 (2000) 2. Bonechi, F., De Bi`evre, S.:Controlling strong scarring for quantized ergodic toral automorphisms. Duke Math J. 117, 571–587 (2003); Section 5 appears only in the preprint mp-arc/02-81 (2002) 3. Bouzouina, A., De Bi`evre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178, 83–105 (1996) 4. Chang, C.-H., Kr¨uger, T., Schubert, R.: Quantizations of piecewise affine maps on the torus and their quantum limits. In preparation 5. Colin de Verdi`ere, Y.: Ergodicit´e et fonctions propres du Laplacien. Commun. Math. Phys. 102, 497–502 (1985) 5 . Corvaja, P., Rudnick, Z., Zannier, U.: A lower bound for periods of matrices. Preprint math. NT/0309215 (2003) 6. Degli Esposti, M.: Quantization of the orientation preserving automorphisms of the torus. Ann. Inst. Henri Poincar´e 58, 323–341 (1993) 7. Faure, F., Nonnenmacher, S., De Bi`evre, S.: Scarred eigenstates for quantum cat maps of minimal periods, Commun. Math. Phys. 239, 449–492 (2003) 8. Hannay, J.H., Berry, M.V.: Quantization of linear maps-Fresnel diffraction by a periodic grating. Physica D 1, 267–290 (1980) 9. Heller, E.J.: Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits. Phys. Rev. Lett. 53, 1515–1518 (1984) 10. Koyama, S.: Quantum ergodicity of Eisenstein series for arithmetic 3-manifolds. Commun. Math. Phys. 215, 477–486 (2000) 11. Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103, 47–77 (2000) 12. Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Preprint (2003) 13. Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on P SL2 (Z) \ H 2 . Publ. Math. IHES 81, 207–237 (1995) 14. Marcus, B.: A note on periodic points of toral automorphisms. Monatsh. Math. 89, 121–129 (1980)
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15. Marklof, J., Rudnick, Z.: Quantum unique ergodicity for parabolic maps. Geom. Funct. Anal. 10, 1554–1578 (2000) 16. Petridis, Y., Sarnak, P.: Quantum unique ergodicity for SL2 (O) \ H 3 and estimates for L-functions. J. Evol. Equ. 1, 277–290 (2001) 17. Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161, 195–231 (1994) 18. Schnirelman, A.: Ergodic properties of eigenfunctions. Usp. Math. Nauk 29, 181–182 (1974) 19. Sigmund, K.: Generic Properties of Invariant Measures for Axiom A-Diffeormorphisms. Invent. Math. 11, 99–109 (1970) 20. Wolpert, S. A.: The modulus of continuity for 0 (m)\H semi-classical limits. Commun. Math. Phys. 216, 313–323 (2001) 21. Zelditch, S.: Uniform distribution of the eigenfunctions on compact hyperbolic surfaces. Duke Math. J 55, 919–941 (1987); Index and dynamics of quantized contact transformations, Ann. Inst. Fourier 47, 673–682 (1996) Communicated by P. Sarnak
Commun. Math. Phys. 245, 215–247 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1001-7
Communications in
Mathematical Physics
Existence and Asymptotic Behavior of Multi-Dimensional Quantum Hydrodynamic Model for Semiconductors Hailiang Li1,2, , Pierangelo Marcati3 1 2 3
Department of Mathematics, Capital Normal University, Beijing 100037, P.R. China Institute of Mathematics, University of Vienna, Austria Dipartimento di Matematica, Universit`a dell’Aquila, 67100 L’Aquila, Italy. E-mail:
[email protected]
Received: 26 November 2002 / Accepted: 21 August 2003 Published online: 29 January 2004 – © Springer-Verlag 2004
Abstract: This paper is devoted to the study of the existence and the time-asymptotic of multi-dimensional quantum hydrodynamic equations for the electron particle density, the current density and the electrostatic potential in spatial periodic domain. The equations are formally analogous to classical hydrodynamics but differ in the momentum equation, which is forced by an additional nonlinear dispersion term, (due to the quantum Bohm potential) and are used in the modelling of quantum effects on semiconductor devices. We prove the local-in-time existence of the solutions, in the case of the general, nonconvex pressure-density relation and large and regular initial data. Furthermore we propose a “subsonic” type stability condition related to one of the classical hydrodynamical equations. When this condition is satisfied, the local-in-time solutions exist globally in-time and converge time exponentially toward the corresponding steady-state. Since for this problem classical methods like, for instance, the Friedrichs theory for symmetric hyperbolic systems cannot be used, we investigate via an iterative procedure an extended system, which incorporates the one under investigation as a special case. In particular the dispersive terms appear in the form of a fourth-order wave type equation.
1. Introduction and Main Results Quantum hydrodynamic models become important and necessary to model and simulate electron transport, affected by extremely high electric fields, in ultra-small sub-micron semiconductor devices, such as resonant tunnelling diodes, where quantum effects (like particle tunnelling through potential barriers and build-up in quantum wells [10, 21]) take place and dominate the process. Such kinds of quantum mechanical phenomena cannot Current address: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560–0043, Japan. E-mail:
[email protected]. osaka-u.ac.jp
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be simulated by classical hydrodynamical models. The advantage of the macroscopic quantum hydrodynamical models relies on the facts that they are not only able to describe directly the dynamics of the physical observable and simulate the main characters of quantum effects, but are also numerically less expensive than those microscopic models like the Schr¨odinger and Wigner-Boltzmann equations. Moreover, even in the process of the semiclassical (or zero dispersion) limit, the macroscopic quantum quantities like density, momentum, and temperature converge in some sense to those of Newtonian fluid-dynamical quantities [13]. Similar macroscopic quantum models are also used in other physical area such as superfluid [26] and superconductivity [5]. The idea to derive quantum fluid-type equations goes back to Madelung in 1927 [27, 24], where the relation between the (linear) Schr¨odinger equation and quantum fluid equation was described in view of the nonlinear geometric optic (WKB)–ansatz of the wave function for irrotational flow away from vacuum. This in fact gives a way to derive quantum fluid type equations, i.e., to make use of the WKB–expansion and derive the equations for (macroscopic) density and momentum from the single-state Schr¨odinger equation, or those with temperature involved from the mixed-state Schr¨odinger equation [14, 18, 13]. Another practicable way to derive quantum hydrodynamic equations is to take advantage of the kinetic structure behind the Schr¨odinger Hamiltonian through Wigner transformation [37]. In fact, the action of the Wigner transformation on the wave function describes the equivalence between the (linear) Schr¨odinger equation and Wigner-Boltzmann equation [31], the quantum kinetic transport equation. The application of the moment method to the Wigner-Boltzmann (or Wigner-Poisson) equation, yields the macroscopic quantities density, momentum and temperature, whose timeevolutions obey the quantum hydrodynamic equations [10, 11]. This is done in analogy with derivation of the first three moment equations, in the moment expansion for the Wigner (distribution) function of the Wigner-Boltzmann equation, under appropriate closure conditions [15] near the “quantum Maxwellian”. For further references on the quantum modelling of semiconductor devices, we refer to [32, 10, 14, 18, 11] and the references quoted therein. We are interested in the mathematical analysis of the quantum hydrodynamic model for semiconductors. In the present paper we consider the initial value problem (IVP) of the quantum hydrodynamic model for semiconductors where an additional relaxation term is involved in the linear momentum equation to model the interaction between the electron and crystal lattice. The re-scaled multi-dimensional quantum hydrodynamic models for semiconductors (QHD) then is given by ∂t ρ + ∇ · (ρu) = 0, ∂t (ρu) + ∇ · (ρu ⊗ u) + ∇P = ρ∇V + 41 ε 2 ∇ · ρ∇ 2 log ρ − λ2 V = ρ − C, ρ(x, 0) = ρ1 (x), u(x, 0) = u1 (x),
ρu τ ,
(1.1) (1.2) (1.3) (1.4)
where ρ > 0, u, J = ρu denote the density, velocity and momentum respectively. ε > 0 the scaled Planck constant, τ > 0 is the (scaled) momentum relaxation time, λ > 0 the (re-scaled) Debye length, and C = C(x) > 0 the doping profile simulating the semiconductor device under consideration [18, 32]. The pressure P = P (ρ), like in classical fluid dynamics, often satisfies the γ -law expression P (ρ) =
T γ ρ , γ
ρ ≥ 0,
γ ≥1
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with the temperature T > 0 [10, 17]. Notice that the particle temperature is T (ρ) = T ρ γ −1 . Moreover, the nonlinear dispersive term √ 1 ρ 1 2 2 2 ε ∇ · ρ∇ log ρ = ε ρ∇ √ 4 2 ρ is produced by the gradient of the quantum Bohm potential √ 1 2 ρ Q(ρ) = ε √ , 2 ρ which requires the strict positivity of density for the classical solution. Recently, many efforts have been made on the existence of (steady-state or timedependent) solutions of QHD (1.1)–(1.3). The existence and uniqueness of (classical) steady-state solutions to the QHD (1.1)–(1.3) for current density J = 0 (thermal equilibrium) has been studied in one dimensional and multi-dimensional bounded domains for density and electrostatic potential boundary conditions [1, 12]. The thermal equilibrium state of the bipolar isothermic model in a bounded domain was considered in [36]. The stationary QHD (1.1)–(1.3) for J > 0 (non-thermal equilibrium) has been considered in [9, 17, 38] for general monotone pressure functions, but, with different boundary conditions, i.e., Dirichlet data for the velocity potential S [17] or by using nonlinear boundary conditions [9, 38]. The existence of the one-dimensional steady-state solutions to (1.1)– (1.3) subject to boundary conditions on the density and the electrostatic potential has been proved in [16], for the case of a linear pressure function P (ρ) = ρ, and in [19] for general pressure functions P (ρ). The local in-time existence of the classical solution was obtained in the one-dimensional bounded domain [20] (subject to boundary conditions on the density and the electrostatic potential). In this case additional boundedness restrictions on initial velocity were required to keep the strict positivity of density. The case of large initial data and the strictly convex pressure function in Rn has been investigated by [25] . In both of these cases, the classical solutions exist globally in time for initial data which are small perturbations of stationary states [20, 25] (which are time exponentially stable). In the present paper we consider the initial value problem (1.1)–(1.4) for a general, nonconvex pressure function in multi-dimension, and we focus on the local existence of the classical solutions (ρ, u, V ) of IVP (1.1)–(1.4) for regular large initial data, and their time-asymptotic convergence to an asymptotic state under small perturbation. We give a general framework to show the local in-time existence of classical solutions for a general (nonconvex) pressure density function and for regular large initial data. Then, we propose a (generic) “subsonic” condition to prove the global existence of the classical solutions in the “subsonic” region and investigate their large time behavior. It is convenient to make use of the variable transformation ρ = ψ 2 in (1.1)–(1.4). Then, we derive the corresponding IVP for (ψ, u, V ): 2ψ · ∂t ψ + ∇ · (ψ 2 u) = 0, ∂t u + (u·∇)u + ∇h(ψ 2 ) +
ε2
ψ u = ∇V + ∇ τ 2 ψ
V = ψ 2 − C, ψ(x, 0) = ψ1 (x) := ρ1 (x), u(x, 0) = u1 (x),
(1.5)
,
(1.6) (1.7) (1.8)
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with ρh (ρ) = P (ρ). Note here the two problems (1.1)–(1.4) and (1.5)–(1.8) are equivalent for classical solutions. For simplicity in this paper we consider the initial value problem (1.5)–(1.8) on the multi-dimensional torus Tn , with T = [0, L] and L > 0 representing the period length. The doping profile C is therefore assumed to be a periodic function and in the present paper is set to be a positive constant for mathematical simplicity. Because of the periodicity in the space variables, the solution of the Poisson equation is not unique since each combination of one solution and a constant is another solution. It is natural to consider the Poisson equation (1.7) in homogeneous Sobolev space. And by choosing an appropriate reference value of voltage, we can consider the Poisson equation (1.7) for V satisfying V (x, t)dx = 0, t ≥ 0. Tn
In analogy, the right hand side term of Eq. (1.7) is required to belong to the homogeneous Sobolev space, i.e., (ψ 2 − C)(x, t)dx = 0, t ≥ 0. Tn
This can be guaranteed due to the conservation (neutrality) of density (1.5) and neutrality assumption on the initial datum (ψ12 − C)(x)dx = 0. (1.9) Tn
In the present paper we consider the problem (1.5)–(1.8) for irrotational (quantum) flow. We describe some ideas to prove both the local and the global existence and we investigate the large time behavior in the “subsonic” regime. The general situation for rotational flow is more complicated and it is expected to be investigated in a forthcoming paper. The first result is the following local existence theorem: Theorem 1.1. Suppose P (ρ) ∈ C 5 (0, +∞). Assume (ψ1 , u1 ) ∈ H 6 (Tn ) × H 5 (Tn ) (n = 2, 3) satisfying (1.9), ∇ × u1 = 0, and minx∈[0,1] ψ1 (x) > 0. Then, there exists T∗∗ > 0, such that there exists a unique solution (ψ, u, V ) to the IVP (1.5)–(1.8), with ψ > 0, which satisfies C 3 ([0, T∗∗ ]; L2 (Tn )), i = 0, 1, 2; ψ ∈ C i ([0, T∗∗ ]; H 6−2i (Tn )) u ∈ C i ([0, T∗∗ ]; H 5−2i (Tn )), i = 0, 1, 2;
V ∈ C 1 ([0, T∗∗ ]; H˙ 4 (Tn )).
Remark 1.2. The irrotationality assumption on the velocity vector fields u is consistent with Eq. (1.6), namely it keeps this property as long as it is true initially. This can be justified via standard arguments as used in the case of ideal fluids in classical hydrodynamics based on Kelvin’s theorem and Stokes’s theorem, see for instance [23] for details. The proof of the above local-in-time existence is based on the construction of approximate solutions and the application of compactness arguments. The main difficulties are given by the following facts. The former arises since the general pressure P (ρ) can be non-convex (even zero), then the left part of (1.5)–(1.7) (or (1.1)–(1.3) resp.) may not
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be hyperbolic anymore and we cannot apply the theory of quasilinear symmetric hyperbolic systems like [25] to obtain the local existence. The latter is given by the nonlinear dispersion term in (1.6), which requires the density ψ (or ρ resp.) to be strictly positive, for regular solutions. Hence we have to establish the local-in-time existence of solutions in a less traditional way. Indeed we are going to construct approximate solutions and to prove the local in-time existence of classical solutions (v, ϕ, ψ, u, V ) for an extended system, which incorporates our problem, constructed in a suitable way based on (1.5)–(1.8). Note that in this new system, there are two additional equations for the variable v, the artificial “velocity” (a sort of Lagrangian type velocity), and the artificial “density” ϕ > 0. The key point is that the local in-time existence of classical solutions for this extended system for the unknown (v, ϕ, ψ, u, V ) will be equivalent to the original one given by (1.5)–(1.8), when v = u and ψ = ϕ (see Sect. 3 for a proof in detail). In order to extend the local-in-time solution globally in time, we will need uniform a-priori estimates, that can be proved by assuming the initial data close to the time¯ u, ¯ V¯ ). Actually it will be possible to extend globally, asymptotic (stationary) state (ψ, the local-in-time solutions, in the “subsonic” region (in the sense defined by (1.10) or (1.12) below); namely we will prove the global existence of the local-in-time solution ¯ u, ¯ V¯ ) located in the so called “subsonic” region when it starts near a stationary state (ψ, (this notion to be provided later in a more precise fashion). ¯ u, ¯ V¯ ) of the boundary value probThe well-posedness of the stationary state (ψ, lem (1.5)–(1.7) subject to density and electrostatic potential boundary conditions was established in one dimension [19] for a general (nonconvex) pressure function P (ρ), and was obtained for multi-dimensional irrotational flow [17] for a monotone enthalpy function where an additional boundary condition was imposed for the Fermi potential. The argument [17, 19] could be applied also here to obtain the existence of the stationary solution with periodic boundary conditions. However, since here we are focusing our attention only on the global existence, for simplicity we √ will bound ourselves to ¯ u, ¯ V¯ ) = ( C, 0, 0) and study the consider only the very special stationary state (ψ, situation when the initial data are assumed in a small neighborhood of the stationary √ solution ( C, 0, 0) to (1.5)–(1.7). Here note that the same argument can be applied to treat the more general case, see item (1) of Remark 1.4 and Theorem 1.5 below for an explanation in detail. Theorem 1.3. Let P (ρ) ∈ C 5 (0, +∞) satisfying A0 =:
π2 2 ε + P (C) > 0, L2
(1.10)
√ where L > 0 is the space period length. Let us assume (ψ1 − C, u1 ) ∈ H 6 (Tn )×H 5 (Tn ) (n = 2, 3), √ the condition (1.9) and moreover ∇× u1 = 0. There exists η > 0 such that, if ψ1 − C H 6 (Tn ) + u1 H 5 (Tn ) ≤ η, the solution (ψ, u, V ) of the IVP (1.5)–(1.8) exists globally in time and moreover one has √ (ψ − C)(t) 2H 6 (Tn ) + u(t) 2H 5 (Tn ) + V (t) 2H 4 (Tn ) ≤ Cδ0 e−0 t , for all t ≥ 0, where C > 0, 0 > 0 are suitable constants, and √ δ0 = ψ1 − C 2H 6 (Tn ) + u1 2H 5 (Tn ) .
(1.11)
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Remark √ 1.4. (1) Although in Theorem 1.3 we choose the special stationary state ( C, 0, 0), we claim that the method used here can be applied to prove the timeasymptotic convergence toward any stationary state of (1.5)–(1.7) on the multi¯ u, ¯ V¯ ), with ∇×u¯ = 0. Their well-posedness can dimensional torus Tn , say (ψ, be obtained by applying the arguments of [17], with suitable modifications. In this case, the corresponding “subsonic” condition has to be changed in the following way: π2 2 ¯ 2. ε + P (ψ¯ 2 ) > |u| (1.12) L2 (2) It is known that classical solutions of the hydrodynamical model for semiconductors (without dispersion term) for large initial data may blow up in finite time to form singularities [3]. Analogous results on the existence of the L∞ solution and one or two dimensional transonic solutions for the hydrodynamical model for semiconductors was proven [6, 7]. However when dispersive regularity is involved in (1.10) or (1.12), it may prevent the formation of singularities, and classical solutions exist globally in time even in the transonic or supersonic region, in the classical sense [2, 4]. (3) Note here that the conditions (1.10) and (1.12) are exactly the subsonic conditions in the classical sense [2], when the re-scaled Planck constant ε goes to zero. If ε > 0 2 2 and P (ρ) > 0, the √ “sound” speed c( ˜ ρ) ¯ = π ε /L2 + P (ρ) ¯ is bigger than the sound speed c(ρ) = P (ρ) ¯ for the classical hydrodynamic equations. Theorems 1.1–1.3 can be extended to the multi-dimensional torus Tn , n ≥ 2, for the IVP (1.5)–(1.8) with smooth initial data. Indeed, we have
Theorem 1.5. Let P ∈ C m (0, ∞), with m ≥ s − 1 and s > n2 + 5. Let us assume that (ψ1 , u1 ) ∈ H s (Tn ) × H s−1 (Tn ), ∇ × u1 = 0, and minx∈[0,1] ψ1 (x) > 0, then, there exists T > 0 such that a solution (ψ, u, V )(t) ∈ H s (Tn )×H s−1 (Tn )×H s−2 (Tn ) of the IVP (1.5)–(1.8), with ψ > 0, exists on [0, T ]. ¯ u, ¯ V¯ ), with ∇×u¯ = 0 and ψ¯ > 0, is a classical stationary Moreover, assume that (ψ, ¯ H s (Tn ) + state of (1.5)–(1.7) with small oscillation and satisfies (1.12). Then, if ψ1 − ψ ¯ H s−1 (Tn ) is sufficiently small, the solution (ψ, u, V )(t) of IVP (1.5)–(1.8) exists u1 − u globally in time and satisfies 2 2 ¯ ¯ (ψ − ψ)(t) + (V − V¯ )(t) 2H s−2 (Tn ) ≤ Cδ1 e−2 t , H s (Tn ) + (u − u)(t) H s−1 (Tn )
with 2 > 0 and ¯ 2 s n + (u1 − u) ¯ 2H s−1 (Tn ) . δ1 = (ψ1 − ψ) H (T ) Remark 1.6. Once we prove the local existence (resp. global existence) of solutions (ψ, u, V ) of IVP (1.5)–(1.8), we can obtain the local existence (resp. global existence) of solutions (ρ, u, V ) of IVP (1.1)–(1.4) by setting ρ = ψ 2 . This paper is organized in the following way. In Sect. 2, we present preliminary results on the divergence equation, Poisson equation, and a fourth order semilinear wave type equation on Tn , then we list some known calculus inequalities. We prove Theorem 1.1 in Sect. 3. After the construction of our extended system in Sect. 3.1, we show the construction of the approximate solutions, we derive the uniform estimates, and we prove Theorem 1.1 in Sect. 3.2. Section 4 is concerned with the proof of Theorem 1.3. After the reformulation of original problem in Sect. 4.1, we establish the a-priori estimates on the local solutions in Sect. 4.2, and prove the global existence and the large time behavior in the remaining part.
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Notation. C always denotes the generic positive constant. L2 (Tn ) is the space of square integral functions on Tn with the norm · . H k (Tn ) with integer k ≥ 1 denotes the usual Sobolev space of function f , satisfying ∂xi f ∈ L2 (0 ≤ i ≤ k), with norm D l f 2 , f k = 0≤|l|≤m
here and after D α = ∂1α1 ∂2α2 · · · ∂nαn for |α| = α1 + α2 + · · · + αn and ∂j = ∂xj , j = 1, 2, ..., n, for abbreviation. In particular, · 0 = · . H˙ k (Tn ) denotes the subspace of function in H k () satisfying u(x) dx = 0.
Let T > 0 and let B be a Banach space. C k (0, T ; B) (C k ([0, T ]; B) resp.) denotes the space of B-valued k-times continuously differentiable functions on (0, T ) (or [0, T ] resp.), L2 ([0, T ]; B) the space of B-valued L2 -functions on [0, T ], and H k ([0, T ]; B) the spaces of functions f , such that ∂ti f ∈ L2 ([0, T ]; B), 1 ≤ i ≤ k, 1 ≤ p ≤ ∞. 2. Preliminaries In this section, we prove the existence and uniqueness of solutions of the divergence equation on Tn and we recall a known result on the multi-dimensional Poisson equation with periodic boundary conditions. Then, we turn to prove the well-posedness for an abstract second order semi-linear evolution equation. Finally, some calculus inequalities are listed without proof. First, we have the following theorem on the divergence operator and Laplace operator on Tn : n Theorem 2.1. Let f ∈ H˙ s (Tn ), s ≥ 0. There exists a unique solution u ∈ H s+1 (Tn ) satisfying ∇ · u = f, ∇×u = 0, (u − u) ˆ dx = 0, (2.1) Tn
and (u − u) ˆ H s+1 (Tn ) ≤ c1 f H˙ s (Tn ) ,
(2.2)
where c1 > 0 is a suitable constant and uˆ a vector in Rn . Theorem 2.2. Let f ∈ H˙ s (Tn ), s ≥ 0. There exists a unique solution u ∈ H˙ s+2 (Tn ) to the Poisson equation u = f satisfying u H˙ s+2 (Tn ) ≤ c2 f H˙ s (Tn ) with c2 > 0.
(2.3)
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The proofs of Theorems 2.1–2.2 can be completed with the help of the Fourier series expansion of the functions u, u and f . Here we omit the details. Based on Theorem 2.2, we obtain the initial potential V1 through (1.7) in view of the initial density: 2 V1 = ψ1 − C, V1 (x)dx = 0. (2.4) Tn
√
C ∈ H 3 , we obtain that V1 ∈ H˙ 5 and satisfies √ V1 H˙ 5 (Tn ) ≤ c3 ψ12 − C H˙ 3 (Tn ) ≤ c4 ψ1 − C H 3 (Tn ) ,
By (1.9) and ψ1 −
(2.5)
with c3 , c4 > 0 constants. Finally, let us consider the abstract initial value problem in the periodic Hilbert space L2 (Tn ): 1 u + Au + Lu = F (t), τ u(0) = u0 , u (0) = u1 .
u + Hereafter u denotes
du dt .
(2.6) (2.7)
The operator A is defined by Au = ν0 2 u + ν1 u,
(2.8)
Rn ,
where is the Laplacian operator on and ν0 , ν1 > 0 are given constants. The domain of the linear operator A is D(A) = H 4 (Tn ). Related to the operator A, we define a continuous and symmetric bilinear form a(u, v) on H 2 (Tn ), a(u, v) = (ν0 uv + ν1 uv)dx, ∀ u, v ∈ H 2 (Tn ), (2.9) Tn
which is coercive, i.e., ∃ ν > 0,
a(u, u) ≥ ν u H 2 (Tn ) ,
∀ u ∈ H 2 (Tn ).
(2.10)
This means that there is a complete orthogonal family {rl }l∈N of L2 (Tn ) and a family {µl }l∈N consisting of the eigenvectors and eigenvalues of operator A Arl = µl rl , l = 1, 2, · · · , 0 < µ1 ≤ µ2 , · · · , µl → ∞ as l → ∞. The family {rl }l∈N is also orthogonal for a(u, v) on
H 2 (Tn ),
a(rl , rj ) =< Arl , rj >= µl (rl , rj ) = µl δlj ,
i.e., ∀ l, j,
where δlj denotes the Kronecker symbol. Related to Lu and F (t), we have < Lu, v >= (b(x, t)·∇u)vdx, u, v ∈ H 2 (Tn ), n T < F (t), v >= f (x, t)vdx, v ∈ H 2 (Tn ), Tn
(2.11)
(2.12) (2.13)
where b : T × [0, T ] → Rn and f : T × [0, T ] → R are measurable functions. By applying the Faedo-Galerkin method [35, 39], we can obtain the existence of solutions to (2.6)–(2.7) in a standard way.
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Theorem 2.3. Let T > 0, n = 2, 3, and assume that F ∈ H 1 ([0, T ]; L2 (Tn )),
b ∈ L2 ([0, T ]; H 3 (Tn ))
H 1 ([0, T ]; H 2 (Tn )). (2.14)
Then, if u0 ∈ H 4 (Tn ) and u1 ∈ H 2 (Tn ), the solution to (2.6)–(2.7) exists and satisfies u ∈ C i ([0, T ]; H 4−2j (Tn )), j = 0, 1,
u ∈ L∞ ([0, T ]; L2 (Tn )).
(2.15)
Moreover, assume that F , F ∈ L2 ([0, T ]; H 2 (Tn )),
(2.16)
then, if u0 ∈ H 6 (Tn ) and u1 ∈ H 4 (Tn ), it follows u ∈ C i ([0, T ]; H 6−2j (Tn )), j = 0, 1, 2,
u ∈ L∞ ([0, T ]; L2 (Tn )).
(2.17)
Proof. The statement (2.17) follows from (2.15), if we consider the same type of problem for new unknown v = D2 u. The statement (2.15) can be proved by applying the Faedo-Galerkin method. We omit the details here since everything is quite standard. For general stability theory of abstract second order equations, the reader can refer to [29, 30]. Remark 2.4. Note that if (2.14) is replaced by F ∈ C 1 ([0, T ]; L2 (Tn )),
b ∈ C i [0, T ]; H 3−i (Tn )), i = 0, 1,
(2.18)
then in (2.15) it follows u ∈ C([0, T ]; L2 (Tn )). Furthermore, when (2.16) is replaced by F ∈ C 1 ([0, T ]; H 2 (Tn )),
(2.19)
it also holds in (2.17) that u ∈ C([0, T ]; L2 (Tn )). Finally, we list below the Moser-type calculus inequalities [22, 28, 34]: Lemma 2.5. Let f, g ∈ L∞ (Tn ) H s (Tn ). Then, it follows D α (f g) ≤ C g L∞ D α f + C f L∞ D α g , D (f g) − f D g ≤ C g L∞ D f + C f L∞ D α
α
α
α−1
g ,
(2.20) (2.21)
for 1 ≤ |α| ≤ s. 3. Local Existence This section is concerned with the proof of Theorem 1.1. We construct the new extended system based on (1.5)–(1.8) in Sect. 3.1, then we build up the approximate solutions, derive the uniform estimates, and prove Theorem 1.1 in Sect. 3.2. For simplicity, we set τ = 1.
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3.1. Construction of the extended system. We construct the extended system in this subsection. For irrotational flow, the velocity field can be represented as the gradient field of a phase function S: u = ∇S.
(3.1)
In analogy, the continuous equation (1.6) for the irrotational velocity vector field u is changed into 1 ε2 ψ 2 2 ∂t u + ∇(|u| ) + ∇h(ψ ) + u = ∇V + ∇ , (3.2) 2 2 ψ which, together with the initial data u(x, 0) = u1 (x), provides the time-decay of mean velocity on Tn : −t ¯ u(x, t)dx = u(t) =: e u1 (x)dx, t ≥ 0. (3.3) Tn
Tn
For ψ > 0 Eq. (1.5) becomes 2∂t ψ + 2u·∇ψ + ψ∇ · u = 0.
(3.4)
We want to explain the main steps that we will use in the next subsection to implement an iterative procedure. Once we know u and ψ based on Eq. (3.4) and the previous observation, we introduce two new equations for the artificial “velocity” v and artificial “density” ϕ > 0, 2(∂t ψ + u·∇ψ) ¯ ∇· v = − v(x, t)dx = u(t), (3.5) , ∇×v = 0, ϕ Tn 1 ∂t ϕ + ϕ∇ · v + u·∇ψ = 0, 2
ϕ(x, 0) = ψ1 (x) > 0.
(3.6)
Clearly to re-initialize the procedure, we have to determine ψ and u as long as we know ϕ and v (we will propose the corresponding equations, used in the next subsection, for ψ and u below). By a simple combination of Eqs. (3.5)–(3.6), we obtain ∂t [ϕ − ψ](x, t) = 0,
∀ x ∈ Tn ,
which implies [ϕ − ψ](x, t) = 0 for (x, t) ∈ Tn × (0, ∞),
if [ϕ − ψ](x, 0) = 0.
(3.7)
By applying to (3.6) a standard argument in the theory of O.D.E. namely by multiplying t Eq. (3.6) by the function exp{ 21 0 ∇·v(x, s)ds} and by integrating the resulting equation with respect to time, we can represent ϕ for (x, t) ∈ Tn × [0, +∞) by the identity t t 1 t − 21 0 ∇·v(x,s)ds ϕ(x, t) = ψ1 (x)e − u·∇ψ(x, s)e− 2 s ∇·v(x,ξ )dξ ds. (3.8) 0
This means that for short time (smooth) solutions (if they exist) satisfy ϕ(x, t) > 0,
if
ψ1 (x) > 0,
x ∈ Tn .
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Based on Eq. (3.4) and Eq. (3.2), we show how to reconstruct the density ψ. Here we use the following second order evolutional problem: 1 1 |ψ|2 1 1 ψtt + ψt + ε 2 2 ψ − ε 2 − P (ψ 2 ) + ψV + ∇ψ · ∇V 4 4 ϕ 2ϕ 2 1 1 + (u + v)·∇ψt − ∇ψ · ∇(|v|2 ) − ψ∇v : ∇v + v·∇(u·∇ψ) 2 2 1 ψt − (ψt + u·∇ψ)(v·∇ψ) − (ψt + u·∇ψ) = 0, (3.9) ϕ ϕ with initial data 1 ψ(x, 0) = ψ1 , ψt (x, 0) = ψ0 =: − ψ1 ∇ · u1 − u·∇ψ1 , 2 where v = (v 1 , v 2 , ..., v n ) and ∇v : ∇v =
(3.10)
|∂j v i |2 .
i,j
Indeed, let us multiply (3.2) by ψ 2 , take divergence of the resulting equation, then use (3.4), the irrotationality assumption of velocity vector fields plus the relation ψ |ψ|2 ∇ · ψ 2∇ = ψ 2 ψ − , ψ ψ replace the nonlinear term
1 2 2 4ψ ∇ · (ψ ∇(|u| ))
by
1 1 1 ∇ψ · ∇(|v|2 ) + ψ∇v : ∇v − v·∇ψt − (v·∇)(u·∇ψ) + (ψt + u·∇ψ)(v·∇ψ), 2 2 ψ and finally replace ψ1 in the resulting equation by ϕ1 ; we get Eq. (3.9). Similarly we can construct from (3.2) the equation for reconstructing the velocity u, 1 ε 2 ∇ψ ψ∇ψ 2 2 ∂t u + u + ∇(|v| ) + ∇h(ψ ) = ∇V + , u(x, 0) = u1 (x). − 2 2 ϕ ϕ2 (3.11) Here we have used the identity ∇ψ ψ∇ψ ψ , = − ∇ ψ ψ ψ2
(3.12)
and we replaced ψ1 and |u|2 by ϕ1 and |v|2 respectively in (3.2). Finally, from (1.7) the reconstruction of V is done directly by using the Poisson equation on Tn and involves only ψ: 1 (ψ 2 − C)(x, t)dx, V (x, t)dx = 0. (3.13) V = ψ 2 − C − n L Tn Tn So far, we have constructed an extended coupled and closed system for the new unknown U = (v, ϕ, ψ, u, V ), which consists of two O.D.E.s (3.6) for ϕ and (3.11) for
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u, a second order evolutional equation (3.9) for ψ, a divergence equation (3.5) for v, and an elliptic equation (3.13) for V . The most important fact (which we will be able to show later on) is to note that this extended system for U = (v, ϕ, ψ, u, V ) is equivalent to the original equations (1.5)–(1.7) of (ψ, u, V ), as far as we look for classical solutions, when u = v and ψ = ϕ > 0. 3.2. Iteration scheme and local existence. Now, we consider the corresponding probp lem for an approximate solution {U i }∞ i=1 with U = (vp , ϕp , ψp , up , Vp ) based on the extended system constructed in Subsect. (3.1). The iteration scheme for the approximate solution U p+1 = (vp+1 , ϕp+1 , ψp+1 , up+1 , Vp+1 ), p ≥ 1, is defined by solving the following problems on Tn : ¯ vp+1 (x, t) dx = u(t), (3.14) ∇ · vp+1 = rp (t), ∇×vp+1 = 0,
Tn
ϕp+1 + 21 (∇ · vp )ϕp+1 + up ·∇ψp = 0, ϕp+1 (x, 0) = ψ1 (x),
(3.15)
+ ψp+1 + ν2 ψp+1 + νψp+1 + kp (t) · ∇ψp+1 = hp (t), ψp+1
ψp+1 (x, 0) = ψ1 (x),
t > 0,
ψp+1 (x, 0)
= ψ0 =:
− 21 ψ1 ∇ · u1
t > 0,
− u·∇ψ1 ,
up+1 + up+1 = gp (t), t > 0, up+1 (0) = u1 , ∇×u1 = 0, Vp+1 (x, t)dx = 0, Vp+1 = qp (t),
(3.17) (3.18)
Tn
where ν = 41 ε 2 , and rp (t) = rp (x, t) = −
2(ψp + up ·∇ψp ) ϕp
+
(3.16)
2(ψp + up ·∇ψp ) 1 (x, t)dx, Ln Tn ϕp (3.19)
kp (t) = kp (x, t) =up (x, t) + vp (x, t), |ψp |2
ψp
(3.20) |ψp 4 ϕp
ε2
|2
1 − ψp Vp − ∇ψp · ∇Vp ϕp ϕp 2 2
1 P (ψp ) 1 1 + + νψp + ∇ψp ·∇(|vp |2 ) + ψp |∂j vpi |2 2 ϕp 2 2 j,i 1 ψp + up ·∇ψp vp ·∇ψp , − vp ·∇ up ·∇ψp + (3.21) ϕp (ψp )∇ψp 1 2 1 2 ∇ψp 2 gp (t) = gp (x, t) =∇Vp − ∇ vp , − − ∇h(ψp ) + ε 2 2 ϕp ϕp2 (3.22) 1 qp (t) = qp (x, t) =ψp2 − C − n (ψ 2 − C)(x, t)dx, (3.23) L Tn p
hp (t) = hp (x, t) =
+
up ·∇ψp +
where up = (u1p , u2p , · · · , unp ) and vp = (vp1 , vp2 , · · · , vpn ).
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Let us emphasize that here the functions rp (0), kp (0), hp (0), gp (0), qp (0) depend only upon the initial data (ψ1 , u1 ) and moreover they are periodic in the space variables. The main result in this section is the following concerning “a-priori estimates”. Lemma 3.1. Let us assume that P ∈ C 5 (0, ∞) and (ψ1 , u1 ) ∈ H 6 × H 5 , ∇×u1 = 0, such that ψ ∗ = max ψ1 (x), ψ∗ =: min ψ1 (x) > 0. x∈Tn
(3.24)
x∈Tn
Then, there exist a positive time T∗ and a sequence {U p }∞ p=1 of approximate solutions, which solve the system (3.14)–(3.18) for t ∈ [0, T∗ ] and satisfy vp ∈ C j ([0, T∗ ]; H 4−j (Tn )) C 2 ([0, T∗ ]; H 1 (Tn )), j = 0, 1, 2 1 3 n ([0, T∗ ]; H 2 (Tn )) C 3 ([0, T∗ ]; L2 (Tn )), ϕp ∈ C ([0, T∗ ]; H (T )) C ψp ∈ C l ([0, T∗ ]; H 6−2l (Tn )) C 3 ([0, T∗ ]; L2 (Tn )), l = 0, 1, 2, u ∈ C 1 ([0, T∗ ]; H 3 (Tn )) C 2 ([0, T∗ ]; H 1 (Tn )), p Vp ∈ C([0, T∗ ]; H˙ 4 (Tn )) C 1 ([0, T∗ ]; H˙ 4 (Tn )). (3.25) Moreover, there is a positive constant M∗ so that for all t ∈ [0, T∗ ], we have 2 2 2 2 2 (up , up )(t) 3 + (up , vp )(t) 1 + vp (t) 4 + vp (t) 3 + (Vp , Vp )(t) 4 ≤ M∗ , (ψp , ψp , ψp , ψp )(t) 2
H 6 ×H 4 ×H 2 ×L2
+ (ϕp , ϕp , ϕp , ϕp )(t) 2
H 3 ×H 3 ×H 2 ×L2
≤ M∗ ,
(3.26)
uniformly with respect to p ≥ 1. Proof. Step 1: Estimates for p = 1. Obviously, U 1 = (u1 (x), ψ1 (x), ψ1 (x),u1 (x),V1 (x)) satisfies (3.25)–(3.26) for the time interval [0, 1] with M∗ replaced by some constant B1 > 0 and V1 determined by (2.4). We start the iterative process with U 1 = (u1 , ψ1 , ψ1 , u1 , V1 ); then by solving the problems (3.14)–(3.18) for p = 1, we can prove the (local in time) existence of a solution U 2 = (v2 , ψ2 , ϕ2 , u2 , V2 ) which also satisfies (3.25)–(3.26) for a time interval (which without loss of generality is chosen to be [0, 1] since we focus on local in-time existence of solutions) and with M∗ replaced by another constant B2 > 0. In fact, for U 1 = (u1 , ψ1 , ψ1 , u1 , V1 ) the functions r1 , k1 , h1 , g1 , q1 depend only on the initial data (ψ1 , u1 ), i.e., r1 (x, t) = r˜1 (x), k1 (x, t) = k˜1 (x), h1 (x, t) = h˜ 1 (x), g1 (x, t) = g˜ 1 (x), q1 (x, t) = q˜1 (x), and ˜r1 22 + k˜1 23 + h˜ 1 23 + g˜ 1 23 + q˜1 22 ≤ N a0 I04 eN u1 3 .
(3.27)
From now on, N > 0 denotes a generic constant independent of U p , p ≥ 1, a0 =
(1 + ψ ∗ )m , ψ∗m
for a integer m ≥ 10,
(3.28)
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H.-L. Li, P. Marcati
and I0 = (ψ1 −
√
C) 2 + ∇ψ1 25 + u1 25 .
(3.29)
The system (3.14)–(3.18) with p = 1 is linear on the unknown U 2 = (v2 , ψ2 , ϕ2 , u2 , V2 ), therefore it can be solved based on the estimates (3.27) for the corresponding right-hand side terms as follows. Namely, by Theorem 2.1, we obtain the existence of the solution v2 to the divergence equation (3.14), with r1 (x, t) replaced by r˜1 (x), satisfying C 2 ([0, 1]; H 1 (Tn )), j = 0, 1. v2 ∈ C j ([0, 1]; H 4−j (Tn )) Then by making use of the theory of the linear O.D.E. system, we prove the existence of the solution u2 of (3.17) for g1 (x, t) = g˜ 1 (x) and then ϕ2 of (3.15): C 2 ([0, 1]; H 1 (Tn )), u2 ∈ C 1 ([0, 1]; H 3 (Tn )) ϕ2 ∈ C 1 ([0, 1]; H 3 (Tn )) C 2 ([0, 1]; H 2 (Tn )) C 3 ([0, 1]; L2 (Tn )). By applying Theorem 2.3 to (3.16), with b(x, t) = 2u1 (x) in (2.12) and f (x, t) = h˜ 1 (x) in (2.13), we obtain the existence of a solution ψ2 satisfying C 3 ([0, 1]; L2 (Tn )), j = 0, 1, 2. ψ2 ∈ C j ([0, 1]; H 6−2j (Tn )) Finally, the existence of a solution V2 satisfying V2 ∈ C 1 ([0, 1]; H˙ 4 (Tn )) follows from the application of Theorem 2.2 to Eq. (3.18) on Tn , with q1 (x, t) replaced by q˜1 (x). Moreover, based on the estimates (3.27), we conclude there is a constant B2 > 0, such that U 2 satisfies (u2 , u2 )(t) 23 + (u2 , v2 )(t) 21 + v2 (t) 24 + v2 (t) 23 + (V2 , V2 )(t) 24 ≤ B2 , (ϕ2 , ϕ2 , ϕ2 , ϕ2 )(t) 2H 6 ×H 3 ×H 2 ×L2 + (ψ2 , ψ2 , ψ2 , ψ2 )(t) 2H 6 ×H 4 ×H 2 ×L2 ≤ B2 , for all t ∈ [0, 1]. p
Step 2: Estimates for p ≥ 2. Now, assume that {U i }i=1 (p ≥ 2) exist in the time interval [0, 1], solve the system (3.14)–(3.18), and satisfy (3.25)–(3.26), with M∗ replaced by the max Bp ( ≥ max1≤j ≤p−1 {Bj }). For given U p , the system (3.17)–(3.18) is linear in U p+1 = (vp+1 , ϕp+1 , ψp+1 , up+1 , Vp+1 ). As before, we apply Theorem 2.1 to Eq. (3.14) for vp+1 , the theory of linear O.D.E. systems to Eq. (3.15) for ϕp+1 and Eq. (3.17) for up+1 , Theorem 2.3 to wave type equation (3.16) for ψp+1 with f (x, t) = hp (t) and b(x, t) = kp (t), and Theorem 2.2 to the Poisson equation (3.18) for Vp+1 . Therefore we obtain the existence of U p+1 = (vp+1 , ψp+1 , ϕp+1 , up+1 , Vp+1 ) on the time interval [0, 1] and moreover it follows: )) C 2 ([0, 1]; H 1 (Tn )), j = 0, 1, vp+1 ∈ C j ([0, 1]; H 4−j (Tn 2 ([0, 1]; H 2 (Tn )) C 3 ([0, 1]; L2 (Tn )), ϕp+1 ∈ C 1 ([0, 1]; H 3 (Tn )) C j 6−2j n 3 2 n (T)) C ([0, 1]; L (T )), j = 0, 1, 2, ψp+1 ∈ C ([0, 1]; H 1 ([0, 1]; H 3 (Tn )) 2 1 n u ∈ C p+1 C1 ([0, 1]; H4 (Tn )), 4 n ˙ ˙ Vp+1 ∈ C([0, 1]; H (T )) C ([0, 1]; H (T )).
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Now our goal is to deduce uniform bounds for U j +1 , 1 ≤ j ≤ p, for some time interval. , where the Let us first estimate the L2 norms of the initial value of ψp+1 , ψp+1 , ψp+1 initial value 0 of ψp+1 is obtained through (3.16)1 at t = 0, where ψp+1 and ψp+1 are replaced by the initial data ψ1 , ψ0 : ˜ 0 = −ψ0 − ν2 ψ0 − νψ1 − 2u1 ·∇ψ0 + h(0),
(3.30)
˜ and h(0) = hp (0) depending only on (ψ1 , u1 ). Hence these initial values will depend only on (ψ1 , u1 ) and are periodic functions of the space variables. Obviously, there is a constant M2 > 0, such that the initial values of ψp+1 , ψp+1 , ψp+1 for p ≥ 1 are bounded by M2 I0 ≥ max ψ1 22 , ψ0 22 , 0 22 , u1 23 . (3.31) Here, we recall that I0 is defined by means of (3.29). Denote by M0 =40M2 I0 · max{1, ν −1 }, M1 =3N a02 (I0 and choose
T∗ = min 1,
+ 1 + M0 ) · max{1, ν 7
(3.32) −2
},
ψ∗ M2 I0 ln 2 2M2 I0 2M2 I0 , , , , , 4M0 N M3 N M4 N M5 N M6
(3.33)
(3.34)
where M3 = 5a02 (I0 + 1 + M0 + M1 )6 , M4 = 2a03 (I0 + 1 + M0 + M1 )8 , M5 = a02 (I0 + 1 + M0 + M1 )7 , M6 = a05 (I0 + 1 + M0 + M1 )14 .
(3.35)
As before N ≥ M2 denotes a generic constant independent of U p , p ≥ 1, and a0 is defined by (3.28). p
Step 2.1: We claim that if the solution {U j }j =1 , (p ≥ 2), to the problems (3.14)–(3.18) satisfies uj (t) 23 + (ψj , ψj )(t) 24 + ψj (t) 22 ≤ M0 , (3.36) vj (t) 24 + Dψj (t) 21 ≤ a0 M1 , for all 1 ≤ j ≤ p and t ∈ [0, T∗ ], then this is also true for U p+1 , namely (t) 2 ≤ M , )(t) 24 + ψp+1 up+1 (t) 23 + (ψp+1 , ψp+1 0 2 vp+1 (t) 24 + Dψp+1 (t) 21 ≤ a0 M1 ,
(3.37)
for all t ∈ [0, T∗ ]. Here M0 and M1 are given by (3.32) and (3.33). We prove (3.37) in the following Steps 2.2–2.4, namely, we first obtain the uniform bounds for Vj +1 (1 ≤ j ≤ p) based on (3.36), then we estimate uniform bounds of ϕj +1 , vj +1 , uj +1 (1 ≤ j ≤ p) and their time derivatives in Sobolev space and prove that vp+1 , up+1 satisfy (3.37), and finally we estimate ψj +1 (1 ≤ j ≤ p). Meanwhile, related to this, we can get uniform estimates on the time derivatives of up+1 , vp+1 and . on ψp+1
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Step 2.2: Estimate on Vj +1 . Based on (3.36) we derive the estimates on Vj +1 (1 ≤ j ≤ p) by solving the Poisson equation (3.18) on Tn for Vj +1 , 1 ≤ j ≤ p. Since it always holds Tn
qj (x, t)dx = 0,
1 ≤ j ≤ p,
t ∈ [0, T∗ ],
by using Theorem 2.2 there exists a unique solution Vj +1 of Eq. (3.18) satisfying Vj +1 (t) 24 ≤N qj (t) 22 ≤ N ψj (t) 42 ≤ N M02 ,
t ∈ [0, T∗ ], 1 ≤ j ≤ p, (3.38)
Vj+1 (t) 24
≤N qj (t) 22
≤
N (ψj , ψj )(t) 42
≤
N M02 ,
t ∈ [0, T∗ ], 1 ≤ j ≤ p. (3.39)
Thus, we conclude that Vp+1 ∈ C 1 ([0, T∗ ]; H˙ 4 (Tn )) is uniformly bounded so long as (3.36) is true. Step 2.3: Estimates on ϕj , vj , uj . We estimate ϕj , vj , uj , 1 ≤ j ≤ p for (x, t) ∈ Tn × [0, T∗ ] based on (3.36). For (x, t) ∈ Tn × [0, T∗ ] by using the same ideas as in deriving (3.8) it follows for ϕj +1 from (3.15) that ϕj +1 (x, t) = ψ1 (x) − t e 21 0s ∇·vj (x,ξ )dξ uj ·∇ψj (x, s)ds e− 21 0t ∇·vj (x,s)ds , 0
ϕj +1 ∈
C 1 ([0, 1]; H 3 (Tn ))
C 2 ([0, 1]; H 2 (Tn ))
C 3 ([0, 1]; L2 (Tn )), (3.40)
which satisfies for all (x, t) ∈ Tn × [0, T∗ ], 1 1 ψ∗ ≤ ψ∗ e−N(1+M1 )T∗ ≤ ϕj +1 (x, t) ≤ (ψ ∗ + ψ∗ )eN(1+M1 )T∗ ≤ 2(ψ ∗ + ψ∗ ). 4 2 (3.41) Moreover the L2 norm of ϕj +1 , with 1 ≤ j ≤ p, and its derivatives are bounded for all t ∈ [0, T∗ ], through those of vj , uj and through the initial data by ϕj +1 (t) 2 ≤ N eN(1+M1 )T∗ ( ψ1 2 + T∗ ( uj (t) 2 · ψj (t) 2 )) ≤ N I0 , 3 3 4 3
(3.42)
and 2 2 ϕj +1 (t) ≤N I0 + vj 24 + uj (t) 23 + ψj (t) 24 3
≤N a0 (I0 + 1 + M0 + M1 )2 ,
(3.43)
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2 ϕj +1 (t) ≤N M0 ϕj +1 (t) 22 + uj (t) 22 + M0 + N ϕj +1 (t) 23 · vj (t) 23 2
≤N a02 (I0 + 1 + M0 + M1 )3 + N (I0 + M0 ) (uj , vj )(t) 23 ,
(3.44)
2 ϕj +1 (t) ≤N (I0 + M0 ) ϕj+1 (t) 2 + (uj , uj )(t) 2 + vj (t) 21 + M0 + N ϕj +1 (t) 43 + vj (t) 43 ≤N vj (t) 43 + (I0 + M0 )2 (uj , vj )(t) 23 + N (I0 + M0 ) uj (t) 2 + vj (t) 21 + N a02 (I0 + 1 + M0 + M1 )4 .
(3.45)
Let us consider the divergence equation (3.14) for vj +1 , with 1 ≤ j ≤ p. Since one has rj (x, t) dx = 0, 1 ≤ j ≤ p, t ∈ [0, T∗ ], Tn
the application of Theorem 2.1 yields the existence of a unique solution vj +1 of Eq. (3.14) for t ∈ [0, T∗ ], which, in view of (3.40)–(3.44) and (3.36), satisfies the following bounds: vj +1 (t) 24 ≤ N rj (t) 23 ≤N a0 ϕj (t) 23 ψj (t) 23 + ψj (t) 24 + uj (t) 23 ≤N a0 (I0 + 1 + M0 )3 1 ≤ M1 , t ∈ [0, T∗ ], 1 ≤ j ≤ p, 3
(3.46) (3.47)
vj +1 (t) 23 ≤N rj (t) 22
≤N a0 ϕj 22 ψj (t) 22 + M0 (ψj , uj )(t) 22 + N a0 I0 ϕj (t) 23 M0 uj 22 + ψj (t) 22 ≤Na02 (I0 + 1 + M0 + M1 )5 + N a0 (I0 + M0 )2 uj (t) 22 ,
t ∈ [0, T∗ ], 1 ≤ j ≤ p,
(3.48)
and vj+1 (t) 21 ≤N rj (t) 2 ≤Na0 ψj (t) 2 + M0 (uj , uj )(t) 2 + M0 ψj (t) 22 + N a0 ϕj (t) 2 ψj (t) 22 + M0 uj (t) 22 + N a0 (1 + M0 )2 ϕj (t) 43 + N a0 ϕj (t) 23 ψj (t) 2 + ψj (t) 24 + (uj , uj )(t) 2
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≤N a0 ψj (t) 23 + M0 uj (t) 2 + N a03 (I0 + 1 + M0 + M1 )6 + N a0 (I0 + 1 + M0 + M1 )3 uj (t) 23 + vj (t) 23 ≤Na0 ψj (t) 23 + M0 uj (t) 2 + N a03 (I0 + 1 + M0 + M1 )8 + N a02 (I0 + 1 + M0 + M1 )3 uj (t) 23 + N a02 (I0 + 1 + M0 )5 uj −1 (t) 22 ,
t ∈ [0, T∗ ], 2 ≤ j ≤ p,
(3.49)
where we have already used (3.48) for vj .
For the functions U i (1 ≤ j ≤ p) satisfying (3.36), it is easy to verify that gj , gj (1 ≤ j ≤ p) belong to H 3 (Tn ) and H 1 (Tn ). By (3.36), (3.38)–(3.43) and (3.47)–(3.48), we can obtain the L2 norm of gj , uj , vj +1 (1 ≤ j ≤ p) and those of their derivatives as follows. We observe that 5 gj (t) 23 ≤Na0 ψj (t) 26 + ϕj (t) 23 + N ∇Vj (t) 23 + vj (t) 44 ≤Na02 (I0 + 1 + M0 + M1 )6 ,
t ∈ [0, T∗ ], 1 ≤ j ≤ p.
(3.50)
Then from (3.17) and (3.36) one has uj (t) 23 ≤N uj 23 + gj −1 (t) 23 ≤N a02 (I0 + 1 + M0 + M1 )6 ,
t ∈ [0, T∗ ], 1 ≤ j ≤ p.
(3.51)
And we can estimate vj +1 in view of (3.48) as follows: vj +1 (t) 23 ≤Na02 (I0 + 1 + M0 + M1 )5 + N a0 (I0 + M0 )2 uj (t) 22 ≤N a03 (I0 + 1 + M0 + M1 )8 ,
t ∈ [0, T∗ ], 1 ≤ j ≤ p.
(3.52)
By differentiating (3.22) with respect to t, and using (3.36), (3.39), (3.42)–(3.43), (3.47) and (3.52), we obtain 3 gj (t) 21 ≤Na0 (ψj , ψj )(t) 24 + ϕj (t) 23 4 + N a0 ϕj (t) 23 ψj (t) 24 + ϕj (t) 23 + N ∇Vj (t) 21 + vj (t) 23 · vj (t) 23 ≤Na04 (I0 + 1 + M0 + M1 )11 ,
t ∈ [0, T∗ ], 1 ≤ j ≤ p.
(3.53)
Hence, we obtain, after differentiating (3.17) with respect to time, that uj (t) 21 ≤N ( uj 21 + gj −1 (t) 21 ) ≤N a04 (I0 + 1 + M0 + M1 )11 ,
t ∈ [0, T∗ ], 2 ≤ j ≤ p,
(3.54)
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233
and from (3.49) that
vj+1 (t) 21 ≤N a0 ψj (t) 23 + M0 uj (t) 2 + N a03 (I0 + 1 + M0 + M1 )8 + N a02 (I0 + 1 + M0 + M1 )3 uj (t) 23 + N a02 (I0 + 1 + M0 )5 uj −1 (t) 22 ≤N a05 (I0 + 1 + M0 + M1 )12 + N a0 ψj (t) 23 ,
t ∈ [0, T∗ ], 1 ≤ j ≤ p.
(3.55)
By the previous estimates, it is easy to obtain the estimates for up+1 . In fact, by taking the inner product between D α (3.17)1 (0 ≤ |α| ≤ 3) and 2D α up+1 over Tn , we obtain d (3.56) D α up+1 2 + D α up+1 2 ≤ D α gp (t) 2 . dt Hence by summing (3.56) with respect to |α| = 0, 1, 2, 3, and integrating it over [0, t], and by the Gronwall lemma, we have t gp (s) 23 e−(t−s) ds up+1 (t) 23 ≤ u1 23 + 0
2 M0 , t ∈ [0, T∗ ], (3.57) 5 with T∗ defined by (3.34). With the help of (3.50), (3.53) and (3.57), the corresponding H 3 and H 1 norms of up+1 and up+1 are bounded, similarly to (3.51) and (3.54), by up+1 (t) 23 ≤N up (t) 23 + gp (t) 23 ≤ N a02 (I0 + 1 + M0 + M1 )6 , (3.58) ≤M2 I0 + T∗ N a02 (I0 + 1 + M0 + M1 )6 ≤
up+1 (t) 21 ≤N ( up (t) 21 + gp (t) 21 ) ≤ N a04 (I0 + 1 + M0 + M1 )11 ,
(3.59)
for t ∈ [0, T∗ ]. In addition, with the help of previous estimates on v, u (i.e., (3.51), (3.52), (3.54), and (3.55)), we obtain from (3.44)–(3.45) that 2 ϕj +1 (t) ≤N a02 (I0 + 1 + M0 + M1 )3 + N (I0 + M0 ) (uj , vj )(t) 23 2
≤N a03 (I0 + 1 + M0 + M1 )9 , and
(3.60)
2 ϕj +1 (t) ≤N vj (t) 43 + (I0 + M0 )2 (uj , vj )(t) 23 + N (I0 + M0 ) uj (t) 2 + vj (t) 21 + N a02 (I0 + 1 + M0 + M1 )4 ≤N a06 (I0 + 1 + M0 + M1 )16 + N a0 (I0 + M0 ) ψj (t) 23 .
(3.61)
So far, we have proved that vp+1 and up+1 satisfy (3.37) (i.e., (3.47) and (3.57)) as long as (3.36) holds, and the time derivatives of them (i.e., (3.52), (3.58), and (3.59)) are also bounded uniformly in Sobolev space, with the exception of (3.55) for vj+1 relative to ψj+1 (1 ≤ j ≤ p). Furthermore, from (3.42)–(3.43) and (3.60)–(3.61) we conclude that ϕp+1 and its time derivatives are uniformly bounded in Sobolev space, with the exception of ϕj+1 , i.e., (3.61), relative to ψj+1 (1 ≤ j ≤ p).
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Step 2.4: Estimates on ψj +1 , vj+1 , ϕj+1 . We estimate ψj +1 and then vj+1 and ϕj+1 , 1 ≤ j ≤ p, for (x, t) ∈ Tn × [0, T∗ ]. By (3.36), (3.46), (3.51), and (3.52), it is easy to verify the upper bounds of kj , kj in H 3 (Tn ), for all t ∈ [0, T∗ ], namely kj (t) 23 ≤ N( uj (t) 23 + vj (t) 23 ) ≤ N a0 (I0 + 1 + M0 )3 ,
1 ≤ j ≤ p,
(3.62)
and kj (t) 23 ≤ N( uj (t) 23 + vj (t) 23 ) ≤ N a03 (I0 + 1 + M0 + M1 )8 , 1 ≤ j ≤ p. (3.63) With the help of (3.36), (3.38)–(3.39), (3.42), (3.43), (3.46), (3.51) and (3.52), we obtain, from (3.21), the following bounds on hp (t), hp (t): 4 hj (t) 22 ≤Na0 ϕj (t) 23 + ψj (t) 24 + ψj (t) 22 + uj (t) 23 3 + N a0 vj (t) 24 ψj (t) 24 + (ψj , ϕj )(t) 22 + uj (t) 23 + N ψj (t) 24 Vj (t) 24 + vj (t) 44 ≤Na02 (I0 + 1 + M0 )7 ,
1 ≤ j ≤ p, t ∈ [0, T∗ ],
(3.64)
and 5 hj (t) 22 ≤Na0 (ϕj , ϕj )(t) 23 + (ψj , ψj )(t) 24 + ψj (t) 22 + uj (t) 23 4 + Na0 uj (t) 23 1 + vj (t) 24 ϕj (t) 22 + (ψj , ψj )(t) 24 4 + Na0 vj (t) 24 (ψj , ψj )(t) 24 + (ψj , ϕj , ϕj )(t) 22 + uj (t) 23 5 + Na0 vj (t) 24 ψj (t) 24 + (ψj , ϕj )(t) 22 + uj (t) 24 + N ψj (t) 24 Vj (t) 24 + vj (t) 23 · vj (t) 24 + N ψj (t) 24 Vj (t) 24 + vj (t) 44 ≤Na05 (I0 + 1 + M0 + M1 )14 ,
2 ≤ j ≤ p, t ∈ [0, T∗ ].
(3.65)
To obtain the bounds on the L2 norm of ψp+1 and its derivatives, we first take the inner product between Eq. (3.16)1 and 2ψp+1 and then we integrate by parts. By using Lemma 2.5, we have d ( ψp+1 (t) 2 + ν ψp+1 (t) 2 + ν ψp+1 (t) 2 ) dt ≤ |∇ · kp (t)|L∞ ψp+1 2 + hp (t) 2 ≤ N (1 + kp (t) 23 ) ψp+1 (t) 2 + hp (t) 2 .
(3.66)
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235
Take the inner product between Eq. D α (3.16)1 and 2D α ψp+1 with 1 ≤ |α| ≤ 2 and n integrate it by parts over T . It follows
d ( D α ψp+1 (t) 2 + ν D α ψp+1 (t) 2 + ν D α ψp+1 (t) 2 ) dt ≤ |∇ · kp (t)|L∞ D α ψp+1 (t) 2 + D α hp (t) 2 + N
Tn
≤ N(1 + kp (t) 23 ) Dψp+1 (t) 2 + D α hp (t) 2 + N
|Hα (ψp+1 , kp )|2 dx
Tn
|Hα (ψp+1 , kp )|2 dx,
(3.67)
where Hα (ψ, k) = D α (k·∇ψ) − k·∇(D α ψ). By Lemma 2.5, (3.62), we get N (1 + k(t) 23 ) Dψ 2 , |α| = 1, |Hα (ψ, k)|2 dx ≤ 2 2 α 2 n N (1 + k(t) 3 )( Dψ + D ψ ), |α| = 2. T
(3.68)
By substituting (3.68) into (3.67) and taking summation of these differential inequalities with respect to |α| = 0, 1, 2, we have d ψp+1 (t) 22 + ν ψp+1 (t) 22 + ν ψp+1 (t) 22 dt
≤ N(1 + kp (t) 23 ) ψp+1 (t) 22 + ν ψp+1 (t) 22 + ν ψp+1 (t) 22
+ hp (t) 22 .
(3.69)
By applying the Gronwall inequality and by using (3.62), (3.64), we obtain (t) 22 + ψp+1 (t) 22 + ψp+1 (t) 22 ψp+1
≤ max{1, ν −1 } · ( ψ0 22 + ψ1 24 + T∗ N M5 )eT∗ Na0 (1+M0 +M1 )
3
≤ 2(2M2 I0 + T∗ M5 ) · max{1, ν −1 } 1 ≤ 8M2 I0 = M0 , t ∈ [0, T∗ ], p ≥ 1, 5
(3.70)
where we recall that M0 , T∗ and M5 are defined by (3.32), (3.34), and (3.35) respectively. , with 0 ≤ Let us take the inner product between Eq. D α ∂t (3.16)1 and 2D α ψp+1 |α| ≤ 2, and integrate by parts over Tn , then by summing the resulting differential inequality with respect to α, by (3.68) and by the following estimates: (t) 2 ψ 2 + ψ 2 , α = 0, (t) (t) N ν k p 2 p+1 p+1 |D α kp ·∇ψp+1 21 + ψp+1 (t) 2 , |α| = 1, |2 ≤ N ν kp (t) 22 Dψp+1 Tn N ν k (t) 2 Dψ 2 + ψ 2 , |α| = 2, p 2 p+1 p+1 1
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we obtain, in analogy to (3.66), (3.69), that d ψp+1 (t) 22 + ν ψp+1 (t) 22 + ν ψp+1 (t) 22 dt ≤ N(1 + kp (t) 23 ) Dψp+1 (t) 21 + ν Dψp+1 (t) 21 + ν Dψp+1 (t) 21
+ hp (t) 22 + N (|D α kp ·∇ψp+1 , kp )|2 ) dx |2 + |Hα (ψp+1
n 0≤|α|≤2 T
≤ NB1 ψp+1 22 + ν ψp+1 (t) 22 + ν ψp+1 (t) 22 + hp (t) 22 ,
(3.71)
where B1 = a03 (I0 + 1 + M0 + M1 )8 . By applying the Gronwall inequality to (3.71), it follows ψp+1 (t) 22 + ψp+1 (t) 22 + ψp+1 (t) 22
≤ max{1, ν −1 } · ( 0 22 + ψ0 24 + T∗ N M6 )eT∗ Na0 (I0 +1+M0 +M1 ) 3
8
≤ 2(2M2 I0 + T∗ N M6 ) · max{1, ν −1 } 1 ≤ 8M2 I0 = M0 , t ∈ [0, T∗ ], p ≥ 1, (3.72) 5 where we recall that M0 , T∗ and M6 are defined by (3.32), (3.34), and (3.35) respectively. To estimate the L2 bounds of D 5 ψp+1 and D 6 ψp+1 , it is sufficient to estimate those of 2 Dψp+1 and 2 D 2 ψp+1 . By differentiating Eq. (3.16)1 twice with respect to x and by taking the inner product with 2 Dψp+1 and 2 D 2 ψp+1 over Tn , and using the estimates (3.62), (3.64), (3.70), and (3.72), one has N 2 Dψp+1 (t) 2 ≤ 2 ψp+1 (t) 21 + ψp+1 (t) 21 + ψp+1 (t) 21 ν N N + 2 D(kp ·∇ψp+1 )(t) 2 + 2 hp (t) 21 ν ν N 1 ≤ 2 a02 (I0 + 1 + M0 )7 ≤ M1 , t ∈ [0, T∗ ], p ≥ 1, (3.73) ν 3 N 2 D 2 ψp+1 (t) 2 ≤ 2 ψp+1 (t) 22 + ψp+1 (t) 22 + ψp+1 (t) 22 ν N N + 2 D 2 (kp ·∇ψp+1 )(t) 2 + 2 hp (t) 22 ν ν N 2 1 7 ≤ 2 a0 (I0 + 1 + M0 ) ≤ M1 , t ∈ [0, T∗ ], p ≥ 1, (3.74) ν 3 where we recall M1 and T∗ are defined by (3.33) and (3.34) respectively. We now need to show the L2 norm of ψj+1 and vj+1 for 1 ≤ j ≤ p. By taking the and using the above estimates, we obtain inner product between ∂t (3.16)1 and ψp+1 ψp+1 (t) 2 ≤N ψp+1 (t) 2 [1 + kp (t) 22 ] + hp (t) 2 + N ψp+1 (t) 24 1 + kp (t) 22 ≤N a05 (I0 + 1 + M0 + M1 )14 , t ∈ [0, T∗ ], 1 ≤ j ≤ p,
(3.75)
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which gives from (3.55) that vj+1 (t) 21 ≤N a05 (I0 + 1 + M0 + M1 )12 + N a0 ψj (t) 23 ≤N a06 (I0 + 1 + M0 + M1 )14 , t ∈ [0, T∗ ], 1 ≤ j ≤ p,
(3.76)
and from (3.61) that 2 ϕj +1 (t) ≤N a06 (I0 + 1 + M0 + M1 )16 + N a0 (I0 + M0 ) ψj (t) 23 ≤N a06 (I0 + 1 + M0 + M1 )16 , t ∈ [0, T∗ ], 1 ≤ j ≤ p.
(3.77)
Step 3: End of proof. By the previous estimates (3.38)–(3.39) on Vp+1 , (3.47), (3.52), and (3.76) on vp+1 , (3.42)–(3.43), (3.60), and (3.77) on ϕp+1 , (3.57)–(3.59) on up+1 , and (3.70) and (3.72)–(3.75) on ψp+1 , we conclude that the approximate solution U p+1 = (vp+1 , ϕp+1 , ψp+1 , up+1 , Vp+1 ) is uniformly bounded in the time interval [0, T∗ ] and it satisfies (3.37) for each p ≥ 1 as long as U p satisfies (3.36) with M0 , M1 , and T∗ defined by (3.32), (3.33), and (3.34) respectively, which are independent of U p+1 , p ≥ 1. By repeating the procedure used above, we can construct the approximate solution {U i }∞ i=1 , which solves (3.25)–(3.26) on [0, T∗ ], with T∗ defined by (3.34) and the constant M∗ > 0 chosen by M∗ = max M0 , M1 , Na06 (I0 + 1 + M0 + M1 )16 . (3.78) Let us recall here that M0 , M1 and a0 are defined by (3.32), (3.33) and (3.28) respectively and N > 0 is a generic constant independent of U p+1 , p ≥ 1. Therefore, the proof of Lemma 3.1 is completed. Proof of Theorem 1.1. By means of Lemma 3.1, we obtain an approximate solution sequence {U p }∞ p=1 satisfying (3.25)–(3.26). Therefore, the proof of Theorem 1.1 is completed if we show that the whole sequence converges. Indeed, based on Lemma 3.1, we can obtain the estimates of the difference Y p+1 =: U p+1 −U p , p ≥ 1, of the approxip+1 = (¯ mate solution sequence {U p }∞ vp+1 , ϕ¯p+1 , ψ¯ p+1 , u¯ p+1 , V¯p+1) p=1 . Let us denote Y by v¯ p+1 = vp+1 − vp , ψ¯ p+1 = ψp+1 − ψp ,
u¯ p+1
ϕ¯p+1 = ϕp+1 − ϕp , = up+1 − up , V¯p+1 = Vp+1 − Vp .
We can obtain for p ≥ 4, ¯vp+1 (t) 24 + (V¯p+1 , V¯p+1 )(t) 23 ≤ N∗ (ψ¯ p , ψ¯ p )(t) 24 + (ϕ¯p , u¯ p )(t) 23 , (¯vp+1 , u¯ p+1 , ϕ¯p+1 )(t) 23
≤ N∗
2
(ψ¯ p−j , ψ¯ p−j )(t) 24
j =0
+N∗
2
ψ¯ p−j (t) 22 + (ϕ¯p−j , u¯ p−j )(t) 23 ,
j =0
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H.-L. Li, P. Marcati
D α ψ¯ p+1 (t) 2 ≤ N∗ ψ¯ p−j (t) 22 + N∗
2
(ψ¯ p+1−j , ψ¯ p+1−j )(t) 24
j =0
5≤|α|≤6
+N∗
1
(t) 22 + (ϕ¯p−j , u¯ p−j )(t) 23 . ψ¯ p−j
j =0
Here N∗ denotes a constant dependent on M∗ . By using the previous estimates, Lemma 3.1, and an argument similar to the one used to get (3.42), (3.57), (3.70), and (3.72), we show, after a tedious computation, that there exists 0 < T∗∗ ≤ T∗ , such that the difference Y p+1 = U p+1 − U p , p ≥ 1, of the approximate solution sequence satisfies the following estimates ∞
(u¯ p+1 , ϕ¯p+1 ) 2C 1 ([0,T
3 ∗∗ ];H )
+ V¯p+1 2C 1 ([0,T
p=1 ∞
ψ¯ p+1 2C i ([0,T
p=1
∗∗
];H 6−2i )
4 ∗∗ ];H˙ )
+ ¯vp+1 2C([0,T
∗∗
];H 4 )
≤ C∗ ,
+ ¯vp+1 2C([0,T
∗∗
(3.79)
];H 3 )
≤ C∗ , (3.80)
where i = 0, 1, 2, and C∗ = C∗ (N, M∗ ) denote a positive constant depending on N and M∗ . Then by applying the Ascoli-Arzela Theorem (to the time variable) and the Rellich-Kondrachev Theorem (to the spatial variables) [33], we prove, in a standard way (see for instance [28]), that there exists a (unique) U = (v, ϕ, ψ, u, V ), such that as p → ∞, vp → v strongly in C i ([0, T∗∗ ]; H 4−i−σ (Tn )), 2 T∗∗ ]; H 2−σ (Tn )), ϕp → ϕ strongly in C 1 ([0, T∗∗ ]; H 3−σ (Tn )) C ([0, i 6−2i−σ n 2 (T )) C ([0, T∗∗ ]; H 2−σ (Tn )), ψp → ψ strongly in C ([0, T∗∗ ]; H i 3−σ n (T )), u → u strongly in C ([0, T∗∗ ]; H p Vp → V strongly in C i ([0, T∗∗ ]; H˙ 4−σ (Tn )), (3.81) holds with i = 0, 1, and σ > 0. Moreover, by (3.41) one has ϕ(x, t) ≥
1 ψ∗ > 0, 4
(x, t) ∈ Tn × [0, T∗∗ ].
(3.82)
If we take σ 1 in (3.81) and we pass into the limit as p → ∞ in (3.14)–(3.18), we obtain the (short time) existence and uniqueness of the classical solution of the system (3.5), (3.6), (3.9), (3.11), and (3.13) constructed in Sect. 3.1. Next, we claim the local in-time classical solution (v, ϕ, ψ, u, V ), with initial data (v, ϕ, ψ, u, V )(x, 0) = (u1 , ψ1 , ψ1 , u1 , V1 )(x) also satisfies ψ = ϕ,
u = v,
(3.83)
and then solves the IVP (1.5)–(1.8). Indeed by passing into the limit in (3.18)1 , we have 1 ϕt + u·∇ψ + ϕ∇ · v = 0, 2
(3.84)
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which yields 2(ϕt + u·∇ψ) = −∇ · v, (3.85) ϕ 2(ϕt + u·∇ψ) (x, t)dx = − ∇ · v(x, t)dx = 0. (3.86) n ϕ T Tn Let us note here that ϕ > 0. Then by taking the limiting equation of v (passing into the limit in (3.14) and (3.19)) 2(ψt + u·∇ψ) 1 2(ψt + u·∇ψ) ∇· v = − + n (x, t)dx, (3.87) ϕ L Tn ϕ and using (3.85), (3.86), one has (ϕ − ψ)t (x, t) (ϕ − ψ)t 1 − n (x, t)dx = 0, ϕ L Tn ϕ
∀ x ∈ Tn , t ≥ 0.
(3.88)
Since by a straightforward computation we obtain (ϕ − ψ)t (x, 0) = 0 from (3.85) and (3.87) with t = 0, then, from (3.88), we conclude that (ϕ − ψ)t (x, t) = ϕ(x, t)f (t),
t ≥ 0,
for any f ∈ C 2 ([0, T∗∗ ]), with f (0) = 0. In particular we can choose f (t) = 0,
t ≥ 0,
hence by (3.82) and the fact ϕ(x, 0) = ψ(x, 0) = ψ1 (x)
⇒
(ϕ − ψ)(x, 0) = 0,
we obtain ψ(x, t) = ϕ(x, t) ≥
1 ψ∗ > 0, 4
1 ψt + u·∇ψ + ψ∇ · v = 0, 2
t ∈ [0, T∗∗ ], x ∈ Tn , t ∈ [0, T∗∗ ], x ∈ Tn .
(3.89) (3.90)
By passing into the limit p → ∞ in (3.17) we recover the equation for u, i.e., (3.11). By using (3.89) and (3.12), from (3.11), one has ψ 1 ε2 2 2 ∂t u + ∇(|v| ) + ∇h(ψ ) + u = ∇V + ∇ . (3.91) 2 2 ψ This equation, together with the fact ∇×u1 (x) = 0, implies ∇×u = 0,
∀ x ∈ Tn , t ≥ 0.
(3.92)
Similarly, by passing into limit in (3.16) we recover Eq. (3.9) for ψ, hence recombining the various terms, with the help of (3.89) and (3.90), we get 1 1 1 ψtt + ψt + u·∇ψt + ψt (∇ · v) − ∇ · (ψ 2 ∇(|v|2 )) − P (ψ 2 ) 2 4ψ 2ψ 1 1 2 ψ 2 2 + ∇ · (ψ ∇V ) + ε ∇· ψ ∇ = 0. (3.93) 2ψ 4ψ ψ
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H.-L. Li, P. Marcati
From (3.90) we have ψt = −u·∇ψ − 21 ψ∇· v, then by substituting it into (3.93) and by representing ut by (3.91), it follows ∇ · (u − v)t + ∇ · (u − v) = 0. By integrating previously the above equation with respect to time on [0, T∗∗ ], since ∇ · (u − v)(x, 0) = 0 and ¯ u(x, t)dx = v(x, t)dx = u(t), Tn
Tn
we get the conclusion, by applying Theorem 2.1 where we choose f = 0, namely we have uˆ = 0, that u(x, t) = v(x, t),
t ∈ [0, T∗∗ ], x ∈ Tn ,
(3.94)
for the irrotational flow. Thus, by (3.91) and (3.94), we recover the equation for u which is exactly Eq. (3.2) (and then Eq. (1.6) for the irrotational flow). Multiplying (3.90) by ψ and by using (3.94) we recover the equation for ψ (which is exactly Eq. (1.5)) ∂t (ψ 2 ) + ∇ · (ψ 2 u) = 0. From (3.95) the conservation (neutrality) of the density (ψ 2 − C)(x, t)dx = (ψ12 − C)(x)dx = 0, Tn
Tn
(3.95)
t >0
(3.96)
follows. Therefore passing into the limit as p → ∞, by (3.18) and by Theorem 2.2 one has that V ∈ C 1 ([0, T∗∗ ]; H˙ 4 ) is the unique solution of the periodic boundary problem of the Poisson equation: 2 V = ψ − C, V dx = 0. Tn
Therefore (ψ, u, V ) with ψ ≥ 21 ψ∗ > 0 is the unique local (in time) solution of IVP (1.5)–(1.8). By a straightforward computation once more, we get ψ ∈ C i ([0, T∗∗ ]; H 6−2i (Tn ))
C 3 ([0, T∗∗ ]; L2 (Tn )), i = 0, 1, 2;
u ∈ C i ([0, T∗∗ ]; H 5−2i (Tn )), i = 0, 1, 2; The proof of Theorem 1.1 is completed.
V ∈ C 1 ([0, T∗∗ ]; H˙ 4 (Tn )).
4. Global Existence and Large Time Behavior We prove here uniform a-priori estimates for the local classical solutions (ψ, u, V ) of√IVP (1.5)–(1.8) for any fixed T > 0, when (ψ, u, V ) is close to the steady state ( C, 0, 0).
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4.1. Reformulation of original problem. In this subsection, we reformulate the original problem (1.5)–(1.8) into an equivalent one for classical solutions. For simplicity, we still set τ = 1. Set √ w = ψ − C. By using (1.5), (1.7) and (3.9), we have the following systems for (w, u, V ): ut + (u · ∇)u + u = f1 (x, t), 1 wtt + wt + ε 2 2 w + Cw = f2 (x, t) + f3 (x, t), 4 √ V = (2 C + w)w,
(4.1) (4.2) (4.3)
and the corresponding initial values are w(x, 0) = w1 (x), wt (x, 0) = w2 (x), u(x, 0) = u1 (x),
(4.4)
with w1 (x) =: ψ1 −
√
√ 1 √ C, w2 (x) =: u1 ·∇( C + w1 ) − ( C + w1 )∇ · u1 . 2
Here √
1 f1 (x, t) =∇V − ∇(h(( C + w) ) − h(C)) + ε 2 ∇ 2 f2 (x, t) = − 2u · ∇wt + P (C)w, 2
w √ w+ C
(4.5)
,
(4.6) (4.7)
√ wt2 ε 2 |w|2 1 √ − w 2 (3 C + w) − ∇w · ∇V + √ 2 4 ( C + w) w+ C √ √ √ (P (( C + w)2 )( C + w)) 2 + (P (( C + w) ) − P (C))w + |∇w|2 √ C+w √ 1 + √ (4.8) ∇ 2 · [ C + w]2 u ⊗ u + 2u · ∇wt . 2( C + w)
f3 (x, t) = −
The derivatives of w and u satisfy: √ √ 2wt + 2u·∇( C + w) + ( C + w)∇ · u = 0.
(4.9)
4.2. The a-priori estimates. For all T > 0, define a suitable function space for the unknown (w, u, V ) of the IVP (4.2)–(4.4) in the following way: X(T ) = {(w, u, V ) ∈ H 6 (Tn ) × H 5 (Tn ) × H˙ 4 (Tn ),
0 ≤ t ≤ T}
with norm M(0, T ) = max { w(t) H 6 (Tn ) + u(t) H 5 (Tn ) + V (t) H˙ 4 (Tn ) }, 0≤t≤T
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H.-L. Li, P. Marcati
and assume that δT = max ( w(t) H 6 (Tn ) + u(t) H 5 (Tn ) ) 1.
(4.10)
0≤t≤T
Under the assumption (4.10), it follows immediately −
1√ 1√ C≤w≤ C. 2 2
(4.11)
Lemma 4.1. Let (w, u, V ) ∈ X(T ), let the multi-index α satisfy 0 ≤ |α| ≤ 4, then the following inequality holds |∇V |2 + V 25 ≤ C w 23 ,
|Vt |2 + |∇Vt |2 + Vt 24 ≤ C D α wt 22 , 2 −t
D u ≤ C D u1 e α
2
α
2 −t
D f3 ≤ C D u1 e α
2
α
+ C D (∇V , wt , w, ∇w, w) ,
(4.12)
2
(4.13)
+ CδT D (∇V , wt , w, ∇w, w) ,
(4.14)
α
α
2
provided that δT 1. Proof. The estimates (4.12) follows from Theorem 2.3, since the integral of the righthand side term of (4.3) equals zero due to the conservation of density and (1.9). By (4.12) and by (4.10), we have |∇V | + |Vt | + |∇Vt | + ||(∇V , ∇Vt )|| ≤ CδT .
(4.15)
In order to estimate (4.13) we take the inner product between (4.1) and u on Tn , then 1 d 1 u · ∇(|u|2 )dx + f1 · u dx u 2 + u 2 = − 2 dt 2 Tn Tn 1 ≤ + CδT u 2 + C ∇ · u 2 + C (w, ∇V , w) 2 . (4.16) 4 By replacing ∇ · u in (4.16) by (4.9) and by (4.12), one has d u 2 + (3/2 − CδT ) u 2 ≤ C (∇w, wt , w) 2 dx. dt
(4.17)
By applying the Gronwall Lemma, by taking δT small enough such that 1 − CδT ≤ 1/2, we get (4.13) for α = 0. In order to get higher order estimates, we set uˆ = D α u. It satisfies the equation1 uˆ t + (u · ∇)uˆ + uˆ = f5 + ∇f6 ,
(4.18)
where
√ f5 (x, t) =∇(D α V ) − D α ∇h( C + w) − [D α ((u·∇)u) − (u·∇)D α u], w 1 2 α . f6 (x, t) = ε D √ 2 C+w
(4.19) (4.20)
1 For the proof of the case |α| = 4, we can assume that the solutions (w, u, V ) have high order regularity to have enough smooth derivatives, since the a-priori estimates (4.24) and (4.31) below are still valid for these solutions when smoothed by Friedrich’s mollifier under assumptions similar to (4.10). We omit all the details here.
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Let us take the inner product between (4.18) and uˆ and integrate by parts over Tn . Then, it follows 1 d 3 1 2 u ˆ + − ∇ · u u ˆ 2 2 dt 4 2 √ 1 ≤ − C f5 2 + ε 2 |D α ∇ · u| D α (( C + w)−1 w) dx n 2 T 1 α ≤C D (∇V , w, ∇w, w) 2 + CδT u ˆ 2 + ∇ · (D α u) 2 . (4.21) 4 By Lemma 2.5 and by (4.9), one has √ √ ∇ · (D α u) 2 ≤C D α (( C + w)−1 wt ) 2 + C D α (( C + w)−1 (u·∇)w) 2 ≤C D α (wt , w, ∇w) 2 + CδT D α u 2 .
(4.22)
By substituting (4.22) into (4.21) and by using the Gronwall inequality, one obtains (4.13) for 1 ≤ |α| ≤ 4, provided that δT is small enough. Finally, we estimate (4.14), with the help of Lemma 2.5, (4.10)–(4.13), (4.9), as D α f3 2 ≤CδT D α (∇V , w, wt , ∇w, w, u, D α ∇ 2 w) 2 + CδT
D α ∂l uj 2
l,j
≤CδT D (∇V , w, wt , ∇w, w, u, ∇ · u)
2
α
≤C D α u1 2 e−t + CδT D α (∇V , w, wt , ∇w, w) 2 . Thus, the proof of Lemma 4.1 is complete.
(4.23)
We have the following basic estimates: Lemma 4.2. Let (w, u, V ) ∈ X(T ), then there exists β1 > 0, such that (w, ∇w, w, wt )(t) 2 + u(t) 21 + V (t) 22 ≤ C(||w1 ||22 + ||u1 ||21 )e−β1 t ,
(4.24)
provided that δT is small enough. Proof. Take the inner product between (4.2) and w + 2wt and integrate by parts over Tn . Therefore one has d 1 2 1 w + wwt + wt2 + Cw 2 + ε 2 |w|2 dx dt Tn 2 4 1 2 + ε w 2 + C w 2 + wt 2 4 =
Tn
(f2 + f3 )(w + 2wt )dx
≤ CδT (wt , w, ∇w, w) 2 + C u1 2 e−t 1 1 + C w 2 + wt 2 + f2 (w + 2wt )dx. 4 4 Tn
(4.25)
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H.-L. Li, P. Marcati
By integration by parts and (4.9), the last term on the right hand side of (4.25) can be estimated by f2 (w + 2wt )dx = (2wwt ∇ · u + 2wt u·∇w + wt2 ∇ · u)dx Tn
Tn
d ∇w 2 − P (C) ∇w 2 dt d ≤CδT (w, wt , ∇w) 2 − P (C) ∇w 2 − P (C) ∇w 2 . dt (4.26) − P (C)
Since ∇w 2 ≤
L2 w 2 , 4π 2
(4.27)
it follows
1 2 1 d w + wwt + wt2 + Cw 2 + ε2 |w|2 + P (C)|∇w|2 dx dt Tn 2 4 1 3 3 + A0 − CδT w 2 + C − CδT w 2 + − CδT wt 2 4 4 4 ≤ C u1 2 e−t ,
(4.28)
where A0 is defined by the “subsonic” condition (1.10) A0 =
π2 2 ε + P (C) > 0. L2
Note that there are positive constants κ1 , β0 such that ||(w, wt , ∇w, w)||2 1 2 1 2 2 2 2 2 ≤ κ1 w + wwt + wt + Cw + ε |w| + P (C)|∇w| dx 4 Tn 2 ≤ κ1 β0−1 (wt , w, w) 2 .
(4.29)
Hence, by applying the Gronwall lemma to (4.28) and using (4.29), we get ||(w, wt , ∇w, w)||2 ≤ C(||w1 ||22 + ||u1 ||21 )e−β1 t
(4.30)
with 0 < β1 < min{1, κ2 β0 }, provided that δT is sufficiently small to have 3 3 1 A0 − CδT , C − CδT , − CδT =: κ2 > 0. min 4 4 4 The combination of (4.30) and (4.12)–(4.13) with α = 0 yields (4.24).
In order to obtain higher order estimates, we differentiate (4.1)–(4.2) with respect to x; therefore by repeating the previous steps and by using Lemmas 4.1–4.2, we have
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Lemma 4.3. Let(w, u, V ) ∈ X(T ), then there exists β4 > 0, such that the following inequality holds: (w, w, wt )(t) 2|α| + u(t) 21+|α| + V (t) 24 ≤ C(||w1 ||22+|α| + ||u1 ||21+|α| )e−β4 t (4.31) for 1 ≤ |α| ≤ 4, provided that δT 1. Proof. Let w˜ = D α w, with 1 ≤ |α| ≤ 4. Then w˜ satisfies the equation 1 w˜ tt + w˜ t + ε 2 2 w˜ + C w˜ = D α f2 (x, t) + D α f3 (x, t). 4
(4.32)
Let us take the inner product between (4.32) by w˜ + 2w˜ t and integrate it by parts over Tn . By using (4.10), (4.11), and (4.14), we obtain 1 2 1 2 d 2 2 2 dx ˜ w˜ + w˜ w˜ t + w˜ t + C w˜ + ε |w| dt Tn 2 4 1 + ε 2 w ˜ 2 + C w ˜ 2 + w˜ t 2 4 1 1 ≤ CδT (w˜ t , w, ˜ ∇ w, ˜ w, ˜ ∇V ) 2 + C w ˜ 2 + w˜ t 2 8 8 + C D α u1 2 exp{−t} +
Tn
D α f2 (w˜ + 2w˜ t )dx.
(4.33)
By integrating by parts and by using (4.9), (4.13), the last term on the right hand side of (4.33) can be estimated as follows: D α f2 (w˜ + 2w˜ t )dx = − 2 [D α (u·∇wt ) − u·∇ w˜ t ](w˜ + 2w˜ t )dx n Tn T + (2w˜ w˜ t ∇ · u + 2w˜ t u·∇ w˜ + w˜ t2 ∇ · u)dx Tn
d ˜ 2 ∇ w ˜ 2 − P (C) ∇ w dt 1 1 ˜ 2 + w˜ t 2 ≤ CδT (D α u, ∇wt , w, ˜ ∇ w) ˜ 2 + C w 8 8 d − P (C) ∇ w ˜ 2 ˜ 2 − P (C) ∇ w dt ≤ CδT (wt , w, w, D α ∇V ) 2 + C D α u1 2 e−t 1 1 ˜ 2 + w˜ t 2 + C w 8 8 d ˜ 2, (4.34) ˜ 2 − P (C) ∇ w − P (C) ∇ w dt − P (C)
where we used the Nirenberg type inequality ˜ 2 ). ∇ w ˜ ≤ C( w ˜ 2 + w
(4.35)
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By substituting (4.34) into (4.33), by using the Gronwall inequality, (4.24), (4.35), and an argument similar to the one for (4.29), we have, for 1 ≤ |α| ≤ 4, that ˜ w) ˜ 2 ≤ C( w1 22+|α| + u1 21+|α| )e−β2 t , (w, ˜ w˜ t , w,
(4.36)
where β2 is a suitable positive constant. Finally we have D α+1 u 2 ≤C ∇ · (D α u) 2 ≤ C D α (wt , w, ∇w, u) 2 ≤C D α (wt , w, ∇w, w) 2 + C D α u1 2 e−t ≤C( w1 22+|α| + u1 21+|α| )e−β3 t with β3 = min{β2 , 1}. The estimate (4.31) follows from (4.36)–(4.37) and Lemma 4.1.
(4.37)
Hence by Lemmas 4.1–4.3, (4.35) and by the Sobolev embedding theorem, we get the following result. Theorem 4.4. Let (w, u, V ) ∈ X(T ), then the following inequality holds: w(t) 2H 6 (Tn ) + u(t) 2H 5 (Tn ) + V (t) 2H˙ 4 (Tn ) ≤ Cδ0 e−β5 t ,
(4.38)
provided that δT 1. Here β5 = min{β4 , β1 } and δ0 is given by (1.11). Proof of the Theorem 1.3. Based on Theorem 4.4, we can prove that (4.10) √ is true for the classical solution existing locally in time, as long as δ0 = ||ψ1 − C||26 + ||u1 ||25 is small enough (e.g. Cδ0 1). Then via the classical continuity argument and the uniform a-priori bounds (4.38) we have the global existence, and the time-asymptotic behavior of our solutions. Acknowledgement. The authors thank the referees for useful comments on the presentation of this paper. The authors thank Professor C. Dafermos for his interest and discussion. H.L. is supported by JSPS post-doctor fellowship and by the Wittgenstein Award 2000 of Peter A. Markowich, funded by the Austrian FWF. Part of the research was made when H.L. visited the Departmento di Matematica Pura e Applicata, University of L’Aquila; he is grateful for the hospitality of the department. P.M. is partially supported by RTN Grant HPRN-CT-2002-00282 (HYKE European Network) and MIUR-COFIN-2002.
References 1. Brezzi, F., Gasser, I., Markowich, P., Schmeiser, C.: Thermal equilibrium state of the quantum hydrodynamic model for semiconductor in one dimension. Appl. Math. Lett. 8, 47–52 (1995) 2. Courant, R., Friedrichs, K.O.: Supersonic flow and shock waves. AMS, Vol. 21, New YorkHeidelberg: Springer-Verlag, 1976 3. Chen, G., Wang, D.: Formation of singularities in compressible Euler-Poisson fluids with heat diffusion and damping relaxation. J. Diff. Eqs. 144, 44–65 (1998) 4. Dafermos, C.: Hyperbolic conservation law in continous mechanics. Grundlehren der mathematischen Wissenschaften Vol. 325, Berlin: Springer, 2000 5. Feynman, R.: Statistical mechanics, A set of lectures. New York: W.A. Benjamin, 1972 6. Gamba, I.: Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor. Commun. PDEs. 17, 553–577 (1992) 7. Gamba, I., Morawetz, C.S., A viscous approximation for a 2-D steady semiconductor or transonic gas dynamic flow. Existence theorem for potential flow. Comm. Pure Appl. Math. 49, 999–1049 (1996) 8. Gamba, I., J¨ungel, A.: Asymptotic limits in quantum trajectory models. To appear in Commun. PDEs., 2001
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9. Gamba, I., J¨ungel, A.: Positive solutions to singular second and third order differential equations for quantum fluids. Arch. Rat. Mech. Anal. 156, 183–203 (2001) 10. Gardner, C.: The quantum hydrodynamic model for semiconductors devices. SIAM J. Appl. Math. 54, 409–427 (1994) 11. Gardner, C., Ringhofer, C.: Dispersive/hyperbolic models for transport in semiconductor devices. Accepted for publication in IMA Volumes in Mathematics and its Applications 12. Gasser, I., J¨ungel, A.: The quantum hydrodynamic model for semiconductors in thermal equilibrium. Z. Angew. Math. Phys. 48, 45–59 (1997) 13. Gasser, I., Lin, C.-K., Markowich, P.: A review of dispersive limits of the (non)linear Schr¨odingertype equation. Taiwanese J. Math. 4, 501–529 (2000) 14. Gasser, I., Markowich, P.: Quantum hydrodynamics, Wigner transforms and the classical limit. Asymptotic Anal. 14, 97–116 (1997) 15. Gasser, I., Markowich, P.A., Ringhofer, C.: Closure conditions for classical and quantum moment hierarchies in the small temperature limit. Transp. Theory Stat. Phys. 25, 409–423 (1996) 16. Gyi, M.T., J¨ungel, A.: A quantum regularization of the one-dimensional hydrodynamic model for semiconductors. Adv. Diff. Eqs. 5, 773–800 (2000) 17. J¨ungel, A.: A steady-state potential flow Euler-Poisson system for charged quantum fluids. Comm. Math. Phys. 194, 463–479 (1998) 18. J¨ungel, A.: Quasi-hydrodynamic semiconductor equations. Progress in Nonlinear Differential Equations, Basel: Birkh¨auser, 2001 19. J¨ungel, A., Li,H.-L.: Quantum Euler-Poisson system: Existence of stationary states. Preprint 2001 20. J¨ungel, A., Li, H.-L.: Quantum Euler-Poisson system: Global existence and exponential decay. Preprint 2002 21. Klusdahl, N., Kriman, A., Ferry, D., Ringhofer, C.: Self-consistent study of the resonant-tunneling diode. Phys. Rev. B. 39, 7720–7735 (1989) 22. Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. Lecture Notes in Math. 448. Berlin: Springer, 1975, pp. 25–70 23. Landau, L.D., Lifshitz, E.M.: Fluid dynamics. Oxford: Pergamon Press, 1959 24. Landau, L.D., Lifshitz, E.M.: Quantum mechanics: Non-relativistic theory. New York: Pergamon Press, 1965 25. Li, H.-L., Lin, C.-K.: Semiclassical limit and well-posedness of nonlinear Schr¨odinger-Poisson. Preprint 2001 26. Loffredo, M., Morato, L.: On the creation of quantum vortex lines in rotating HeII. Il nouvo cimento 108B, 205–215 (1993) 27. Madelung, E.: Quantentheorie in hydrodynamischer Form. Z. Physik 40, 322 (1927) 28. Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Berlin Heidelberg-New York: Springer-Verlag, 1984 29. Marcati, P.: Stability for second order abstract evolution equations. Nonl. Anal. TMA. 8, 237–252 (1984) 30. Marcati, P.: Decay and stability for nonlinear hyperbolic equations. J. Diff. Eqs. 55, 30–58 (1984) 31. Markowich, P.: On the equivalentce of the Schr¨odinger and the quantum Liouville equations. Math. Meth. Appl. Sci. 11, 459–469 (1989) 32. Markowich, P.A., Ringhofer, C., Schmeiser, C.: Semiconductor Equations. Wien: Springer, 1990 33. Simon, J.: Compact sets in the space Lp (0, T ; B). Ann. Math. Pura. Appl. 146, 65–96 (1987) 34. Taylor, M.E.: Pseudodifferential operators and nonlinear PDE. Progress in Mathematics Vol. 100, Boston: Birkhauser 1991 35. Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Appl. Math. Sci. 68. Berlin-Heidelberg-New York: Springer-Verlag, 1988 36. Unterreiter, A.: The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model. Commun. Math. Phys. 188, 69–88 (1997) 37. Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932) 38. Zhang, B., Jerome, W.: On a steady-state quantum hydrodynamic model for semiconductors. Nonlinear Anal. TMA 26, 845–856 (1996) 39. Zeidler, E.: Nonlinear functional analysis and its applications. Vol. II: Nonlinear monotone operators. Berlin-Heidelberg-New York: Springer-Verlag, 1990 Communicated by P. Constantin
Commun. Math. Phys. 245, 249–278 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1012-4
Communications in
Mathematical Physics
Deformed Quantum Calogero-Moser Problems and Lie Superalgebras A.N. Sergeev1,2 , A.P. Veselov2,3 1 2
Balakovo Institute of Technology and Control, 413800 Balakovo, Russia Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK. E-mail:
[email protected];
[email protected] 3 Landau Institute for Theoretical Physics, Kosygina 2, 117940 Moscow, Russia
Received: 7 March 2003 / Accepted: 12 September 2003 Published online: 23 December 2003 – © Springer-Verlag 2003
Abstract: The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova. For the classical series a recurrent formula for the quantum integrals is found, which implies the integrability of these problems. The corresponding algebras of the quantum integrals are investigated, the explicit formulas for their Poincare series for generic values of the deformation parameter are presented. 1. Introduction The Calogero-Moser-Sutherland (CMS) problem in its original form [1, 2] describes the particles on the line pairwise interacting with the potential U (x1 , . . . , xn ) =
g 2 ω2 . sin2 ω(xi − xj ) 1≤i<j ≤n
In the limit ω → 0 one has the rational potential U (x1 , . . . , xn ) =
1≤i<j ≤n
g2 . (xi − xj )2
Olshanetsky and Perelomov [3] proposed a generalization of these problems related to any root system. The corresponding quantum Hamiltonian has the form L = − +
mα (mα + 2m2α + 1)(α, α) , 2 (α, x) sin α∈R +
(1)
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where R+ is a positive part of a root system R (which could be non-reduced), and m(α) = mα is a function on R which is invariant under the corresponding Weyl group W . If R is a root system of a compact symmetric space X and mα = µ2α − 1, where µα are the multiplicities of the roots then the operator (1) is conjugated to the radial part of the Laplace-Beltrami operator on X, L = − + 2 mα cot(α, x)∂α , (2) α∈R+
namely L = ψˆ 0−1 ◦ L ◦ ψˆ 0 , where ψˆ 0 is the multiplication operator by the function ψˆ 0 = sin−mα (α, x).
(3)
α∈R+
In this general form this was shown by Olshanetsky and Perelomov [4] but in the special case X = SU (n)/SO(n) it was known long before that (see e.g. [5]). So integrability of the generalized CMS quantum problem (1) in this case follows from the general theory of the invariant operators on the symmetric spaces (see e.g. [6]). In the general case the proof of integrability was found later by Heckman and Opdam [7, 8] who used very different arguments. It turned out however that there are other, non-symmetric integrable generalizations of the quantum CMS problems. The first series An (m) of such “deformed” CMS problems have been found by Chalykh, Feigin and one of the authors in [9, 10], later another series Cn (m, l) was discovered [11]. Although these deformations since then appeared in a different context (see e.g. [12] about relations to WDVV equations) their algebraic nature until recently was totally unclear. An important step was made by one of the authors in [13, 14] who observed that the deformed CMS operator of the type An (m) for a special value of the parameter can be interpreted as the radial part of the Laplace-Beltrami operator on a certain symmetric superspace. This indicated that the clue to this mystery could be found in this direction. The goal of the present paper is to develop a systematic theory of the deformed CMS quantum systems related to Lie superalgebras and symmetric superspaces. A crucial role in our approach is played by the notion of the generalized root systems introduced by V. Serganova in the paper [15]. Serganova suggested a version of the standard geometric description of the root systems [16] in the presence of the isotropic roots. She classified all such irreducible systems and showed that they are essentially the root systems of the contragredient Lie superalgebras classified by V. Kac in [17]. All the generalized root systems have partial symmetry described by the Weyl group W0 generated by reflections corresponding to the non-isotropic (“real”) roots. For any such generalized root system R we construct a family of the deformed quantum CMS systems in the following way. The system R stays the same, we change the scalar product and the multiplicities of the roots (which are not integers anymore) in such a way that the following conditions are satisfied: 1) the new bilinear form B and the multiplicity function stay W0 -invariant, 2) multiplicities of all the imaginary roots are fixed to be 1,
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3) the corresponding Schr¨odinger operator (1) can be conjugated to a form (2) or equivalently, has an eigenfunction of the form (3). The last condition leads to certain relations between the multiplicities and parameters of the form. Our analysis shows that the admissible forms for all generalized root systems depend on one deformation parameter (in the case D(2, 1, λ) we have three parameters but two of them are already in the Lie superalbegra itself), although multiplicities have sometimes more freedom (see Sect. 2). We call the corresponding operators the deformed CMS operators related to the generalized root system R. According to Serganova’s classification we have two infinite series of such operators related to the classical systems R of the type A(n, m) and BC(n, m), and three exceptional cases corresponding to the exceptional systems G(1, 2), AB(1, 3) and D(2, 1, λ). The deformed operators corresponding to the classical series have the form: 2 2 ∂ ∂ ∂2 ∂2 LA(n−1,m−1) = − − k + · · · + + · · · + ∂x1 2 ∂xn 2 ∂y1 2 ∂ym 2 n m 2k(k + 1) 2(k −1 + 1) + + sin2 (xi − xj ) i<j sin2 (yi − yj ) i<j +
m n i=1 j =1
2(k + 1) sin2 (xi − yj )
(4)
for A(n − 1, m − 1), where k is an arbitrary parameter, and 2 2 ∂ ∂ ∂2 ∂2 −k + ··· + + ··· + LBC(n,m) = − ∂x1 2 ∂xn 2 ∂y1 2 ∂ym 2 n 2k(k + 1) 2k(k + 1) + + 2 (x − x ) 2 (x + x ) sin sin i j i j i<j m 2(k −1 + 1) 2(k −1 + 1) + + sin2 (yi − yj ) sin2 (yi + yj ) i<j m n n 2(k + 1) 2(k + 1) p(p + 2q + 1) + + + 2 2 sin (xi − yj ) sin (xi + yj ) sin2 xi i=1 j =1 i=1 +
n 4q(q + 1) i=1
sin2 2xi
m m kr(r + 2s + 1) 4ks(s + 1) + + sin2 yj sin2 2yj j =1 j =1
(5)
for BC(n, m), where the parameters k, p, q, r, s satisfy the following relations: p = kr,
2q + 1 = k(2s + 1).
(6)
The system (4) can be considered as the interaction of two groups of particles of masses 1 and k1 respectively with the special parameters of interaction depending on k. When m = 1 ( i.e. when the second group consists only of one particle) this system was first proposed in [9]. For the general n and m the operator (4) was first introduced by one of the authors in [13] but its rational limit was discovered earlier by Berest and Yakimov, who were looking for Darboux-type transformations for Calogero-Moser systems [18].
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The system (5) can be interpreted in a similar way under the assumption that the configuration of particles is symmetric with respect to the origin. Although it depends on 5 parameters only 3 of them are independent due to relations (6) (say k, p and q). The system BC(n, m) with m = 1 and p = 0 was first considered in [11]. The case m = 1 is special as the only one when all the parameters could be integer. As far as we know the operator (5) for the general m, n as well as the deformed CMS systems related to the exceptional root systems G(1, 2), AB(1, 3) and D(2, 1, λ) (see the next section) were not considered before. The conjecture is that all these quantum systems are integrable in the sense that they have enough commuting integrals. In this paper we prove this conjecture for the classical series ( i.e. for the operators (4) and (5)) explicitly constructing the integrals. The corresponding algebra of integrals seems to be interesting by itself. To describe it we introduce the following algebra R,B related to a deformed generalized root system (R, B). It consists of the polynomials p(x) on V which are invariant under reflections sα corresponding to the real roots (i.e. are W0 -invariant) and satisfy the conditions 1 1 p x+ α ≡p x− α (7) 2 2 on the hyperplane B(α, x) = 0 for each imaginary root α. Our main result can be formulated as follows. Theorem. For the classical generalized root systems R and generic values of the deformation parameter k in the form B there exists a monomorphism χ from the commutative algebra R,B into the algebra of differential operators on V such that χ (x 2 ) is the corresponding deformed CMS operator related to R. In the rational limit a similar result holds for a closely related algebra 0R,B with the condition (7) in the definition of R,B replaced by its differential version: ∂α p(x) = 0 when B(α, x) = 0. The structure of the paper follows. We start with the precise definition of the generalized root systems R and formulate Serganova’s classification theorem. Then we define the deformed CMS operators related to such a system R and classify all of them for each generalized root system. In Sect. 3 we prove the integrability of the deformed quantum CMS problems for the classical series A(n, m) and BC(n, m). The proof is effective: the quantum integrals are given explicitly by some recurrent formula. Our formula can be considered as a deformed version of the Matsuo’s formula (2.3.6) from [19]. In section 4 we introduce and investigate the algebras R,B and 0R,B . In particular we show that for the classical series and generic values of the deformation parameter k these algebras are finitely generated and compute the Poincare series for them. We show that the image of the Harish-Chandra homomorphism from the rings of quantum integrals of the deformed CMS problems described in section 2 for generic k is exactly the algebra R,B (in the rational case - 0R,B ). In the last section 5 we discuss the elliptic and difference generalizations of our operators. 2. Generalized Root Systems and Deformed Quantum CMS Problems We start with the definition of the generalized root systems due to V. Serganova [15]. We should mention that there are three slightly different definitions of generalized root systems in [15], we choose one of them which suits best for our purpose.
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Let V be a finite dimensional complex vector space with a non-degenerate bilinear form <, >. Definition. The finite set R ⊂ V \ {0} is called a generalized root system if the following conditions are fulfilled : 1) R spans V and R = −R ; 2<α,β> 2) if α, β ∈ R and < α, α >= 0 then 2<α,β> <α,α> ∈ Z and sα (β) = β − <α,α> α ∈ R; 3) if α ∈ R and < α, α >= 0 then for any β ∈ R such that < α, β >= 0 at least one of the vectors β + α or β − α belongs to R. The non-isotropic roots are called real, the isotropic roots are called imaginary: Rre = {α ∈ R :< α, α >= 0} Rim = {α ∈ R :< α, α >= 0}. A generalized root system R is called reducible if it can be represented as a direct sum of two non-empty generalized root systems R1 and R2 , i.e. V = V1 ⊕ V2 , R1 ⊂ V1 , R2 ⊂ V2 , and R = R1 ∪ R2 . Otherwise the system is called irreducible. Any generalized root system has a partial symmetry described by the finite group W0 generated by the reflections with respect to the real roots. The main result of a remarkable Serganova’s paper [15] is the classification theorem for the irreducible generalized root systems which says that they all are contained in the following list. List of the irreducible generalized root systems Classical series 1. A(n − 1, m − 1), n = m. Let Vn,m = V1 ⊕ V2 be a vector space with the basis {e1 , . . . , en+m }, such that {e1 , . . . , en } is a basis of V1 and {en+1 , . . . , en+m } is a basis ∗ . of V2 . Let ei , i = 1, ..., n + m denote the corresponding basis in the dual space Vn,m Consider the following bilinear (indefinite) symmetric form on Vn,m : B(u, v) =
n i=1
ui v i −
n+m
uj v j ,
(8)
j =n+1
where ui , v i are the coordinates of vectors u, v in the basis ei . Let us split the set of the indices I = {1, . . . , n + m} into two groups: I = I1 ∪ I2 , where I1 = {1, . . . , n}, I2 = {n + 1, . . . , n + m} and rewrite the last formula as B= ei ⊗ ei − ej ⊗ ej , (9) i∈I1
j ∈I2
where B is now considered as an element of V ∗ ⊗ V ∗ . The generalized root system of type A(n − 1, m − 1), n = m is defined as the set R = {ei − ej , i = j, i, j ∈ I } and the corresponding space V is the hyperplane in Vn,m generated by this set with the induced bilinear form. It is easy to see that in this case Rre = An−1 ⊕ Am−1 , Rim = {±(ei − ej ), i ∈ I1 , j ∈ I2 }. The corresponding Lie superalgebra is sl(n|m).
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2. A(n − 1, n − 1). If we would do the same when m = n then we will have a problem: the restriction of the form B on the corresponding hyperplane V is degenerate. Indeed the vector v = i∈I1 ei − j ∈I2 ej belongs to V and orthogonal to all the roots (and thus to all V ). In order to have a proper generalized root system in this case we should consider the quotient V = V / < v > and the corresponding set R which is the image of R after such a projection. This is the system of the type A(n − 1, n − 1). The corresponding Lie superalgebra is psl(n|n). 3. B(n, m). In this case V = Vn,m which is defined above with the same bilinear form B and R is the set {±ei ± ej , i = j, i, j ∈ I, ±ei , i ∈ I, ±2ei , i ∈ I2 }. The real and imaginary roots are Rre = Bn ⊕ BCm , Rim = {±ei ± ej , i ∈ I1 , j ∈ I2 }. This corresponds to the orthosymplectic Lie superalgebra osp(2n + 1|2m). 4. D(n, m), n ≥ 2. V = Vn,m is the same as in the previous case, but R is the set {±ei ± ej , i = j, i, j ∈ I, ±2ei , i ∈ I2 }. We have Rre = Dn ⊕ Cm , Rim = {±ei ± ej , i ∈ I1 , j ∈ I2 }. The corresponding Lie superalgebra is osp(2n|2m). 5. C(0, m). Here V = V1,m and R is the set {±ei ± ej , i = j, i, j ∈ I, ±2ei , i ∈ I2 }. In this case Rre = Cm , Rim = {±e1 ±ej , j ∈ I2 , }. The corresponding Lie superalgebra is osp(2|2m). 6. C(n, m). Here V = Vn,m and R is the set {±ei ± ej , i = j, i, j ∈ I, ±2ei , i ∈ I }, so that Rre = Cn ⊕ Cm , Rim = {±ei ± ej , i ∈ I1 , j ∈ I2 , }. In this case and in the next one there are no related Lie superalgebras but there are symmetric superspaces with such root systems. 7. BC(n, m). V = Vn,m and R consists of {±ei ±ej , i = j, i, j ∈ I, ±ei , ±2ei , i ∈ I }. In this case Rre = BCn ⊕ BCm , Rim = {±ei ± ej , i ∈ I1 , j ∈ I2 }.
Exceptional cases 8. AB(1, 3) (also known as F (4)). Here V = V1 ⊕ V2 , where V1 is a three dimensional space with the basis {e1 , e2 , e3 } and V2 is a one-dimensional space generated by e4 . The bilinear form B is B(u, v) = u1 v 1 + u2 v 2 + u3 v 3 − 3u4 v 4 . The root system R is the set ±ei ± ej ,
i = j,
±ei ,
i, j = 1, 2, 3,
±e4 ,
1 (±e1 ± e2 ± e3 ± e4 ), 2
Rre = B3 ⊕ A1 , Rim = { 21 (±e1 ± e2 ± e3 ± e4 )}. 9. G(1, 2) (also known as G(3)). Here V = V1 ⊕ V2 , where V1 is a two-dimensional space, generated by three vectors e1 , e2 , e3 with the condition that e1 + e2 + e3 = 0 and V2 is a one-dimensional space generated by e4 . The form B is determined by the following conditions: B(ei , ej ) = −1
if i = j,
B(ei , ei ) = 2,
B(ei , e4 ) = 0,
where i, j = 1, 2, 3. R consists of the vectors ±ei , (ei − ej ), ±e4 , ±2e4 , ±ei ± e4 , Rre = G2 ⊕ BC1 , Rim = {±ei ± e4 , i = 1, 2, 3}.
B(e4 , e4 ) = −2, i = j, i, j ≤ 3,
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10. D(2, 1, λ). Here λ = (λ1 , λ2 , λ3 ) are the parameters, satisfying the relation λ1 + λ2 + λ3 = 0. The space V is V1 ⊕ V2 ⊕ V3 the direct sum of three one-dimensional spaces generated by e1 , e2 , e3 respectively. The form B is B(u, v) = λ1 u1 v 1 + λ2 u2 v 2 + λ3 u3 v 3 . R is the set {±2e1 , ±2e2 , ±2e3 , ±e1 ± e2 ± e3 }, Rre = A1 ⊕ A1 ⊕ A1 , Rim = {±e1 ± e2 ± e3 }. Now we are going to explain how one can construct a family of the Schr¨odinger operators related to a generalized root system R ⊂ V . These operators are defined on the same space V and have the form L = − +
mα (mα + 2m2α + 1)(α, α) , sin2 (α, x) α∈R
(10)
+
but now the brackets ( , ) and the Laplacian correspond to the new (“deformed”) bilinear form B on V . Sometime we would consider this operator on V ∗ (which can be identified with V using B); in that case the brackets (α, x) should be understood as a natural pairing between vectors and covectors. The multiplicities mα are related to B in such a way that the following conditions are satisfied: 1) the new form B and the multiplicities are W0 -invariant; 2) all imaginary roots have
the multiplicity 1; 3) the function ψ0 = α∈R+ sin−mα (α, x) is a (formal) eigenfunction of the corresponding Schr¨odinger operator (10). We will call such forms B and multiplicities admissible and the corresponding operators (10) the deformed CMS operators related to the generalized root system R. If we replace x in (10) by ωx and let ω tend to 0 (in other words if we replace sin z by z) we will have the rational limits of these operators. Let us comment on the conditions 1)–3). The first one is very natural: we would like to preserve the (partial) symmetry of the system. The third condition is responsible for the existence of the “radial gauge” of the operator L: L = − + 2 mα cot(α, x)∂α (11) α∈R +
(see [3, 20]), and thus is motivated by the theory of symmetric spaces. The second condition (related to condition 3) in Serganova’s axiomatics of the generalized root systems) looks very simple but actually is the most difficult to justify. The motivation comes from the theory of the locus configurations, where the first examples of such deformations have been found [10, 11]. In that theory all the multiplicities are integers and 1 is the smallest possible option. A straightforward check shows that condition 3) is equivalent to the following main identity: mα mβ (α, β)(cot(α, x) cot(β, x) + 1) ≡ 0, (12) α∼β,α,β∈R+
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where α ∼ β means that α is not proportional to β (note that in BC(n, m) there are 2 proportional roots). In that case Lψ0 = λψ0 with the eigenvalue λ = |ρ(m)| where, ρ(m) = mα α (cf. [3, 20]). α∈R+
To describe all possible deformations for a given generalized root system one can use the fact that the condition (12) can be checked separately for all two-dimensional subsystems (cf. [11, 20]). It is enough to consider only the following classical root systems A(n, m), BC(n, m) (from the deformation point of view others are simply special cases) and the exceptional root systems G(1, 2), AB(1, 3), D(2, 1, λ). The forms B are obviously defined up to a multiple so we will choose some normalization to avoid unnecessary constants. A straightforward analysis leads to the following List of admissible deformations of generalized root systems. Here B is considered as an element of V ∗ ⊗ V ∗ which is non-degenerate and thus determines an isomorphism between V and its dual V ∗ . The formulas for the operators will be written on V ∗ : this is more convenient for several reasons. In all the cases the admissible forms depend on one parameter. We denote this parameter k and choose it in such a way that the value k = −1 corresponds to the Lie superalgebra case. Classical series A(n, m). Form B is B=
ei ⊗ ei + k
ej ⊗ ej ,
(13)
j ∈I2
i∈I1
where k is an arbitrary non-zero parameter. Multiplicities mα = m(α) of the real roots are m(ei − ej ) = k, i, j ∈ I1 , m(ei − ej ) = k −1 , i, j ∈ I2 (recall that all imaginary roots have multiplicity 1). The corresponding one-parameter family of the deformed CMS operators has the form (4). We should mention that in the case m = n the vector v = i∈I1 ei − i∈I2 ei is not isotropic for the deformed form, so strictly speaking we deform not the generalized system of type A(n − 1, n − 1) but its (degenerate) extension. BC(n, m). B is the same as above, multiplicities are m(ei ± ej ) = k, m(ei ± ej ) = k
−1
m(ei ) = p, ,
m(ej ) = r,
m(2ei ) = q, i, j ∈ I1 , m(2ej ) = s, i, j ∈ I2 ,
where p, q, r, s are satisfying the relations p = kr,
2q + 1 = k(2s + 1).
The corresponding deformed CMS operators depend on three free parameters and are given by (5).
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257
Exceptional cases AB(1, 3).
B = e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3 + 3ke4 ⊗ e4 ,
multiplicities are 3k + 1 1−k , m(e4 ) = b = , 2 2k 3k − 1 m(ei ± ej ) = c = , i, j = 1, 2, 3. 4 The deformed CMS operator in this case has the form 2 3 ∂2 ∂ a(a + 1) 3kb(b + 1) ∂2 ∂2 LAB(1,3) = − − 3k + + + + 2 2 2 2 sin2 xi sin2 y ∂x1 ∂x2 ∂x3 ∂y i=1 4c(c + 1) 4c(c + 1) + + 2 sin (xi − xj ) sin2 (xi + xj ) 1≤i<j ≤3 m(ei ) = a =
+
1 (3k + 3) , 1 2 4 ± sin 2 (y ± x1 ± x2 ± x3 )
(14)
where the parameters a, b, c are given in terms of the deformation parameter k above and the last sum is over all 8 possible combinations of the signs. In this case we have only one free parameter k. G(1, 2). In the basis e1 , e2 , e4 the form B has the form 1 B = e1 ⊗ e1 + e2 ⊗ e2 − (e1 ⊗ e2 + e2 ⊗ e1 ) + ke4 ⊗ e4 . 2 Multiplicities are m(ei ) = a = 1 + 2k, m(e4 ) = c =
1 + 2, k
2k − 1 , 3 1 1 m(2e4 ) = d = − , 2k 2 m(ei − ej ) = b =
where i, j = 1, 2, 3. The corresponding deformation of the CMS operator has the form 2 3 a(a + 1) ∂2 ∂ ∂2 ∂2 LG(1,2) = − − k − + + ∂x1 ∂x2 ∂x1 2 ∂x2 2 ∂y 2 i=1 sin2 xi kc(c + 2d + 1) 4kd(d + 1) 3b(b + 1) + + + 2 sin (xi − xj ) sin2 y sin2 2y 1≤i<j ≤3 3 2(k + 1) 2(k + 1) + + , sin2 (xi − y) sin2 (xi + y) i=1
(15)
where the parameters a, b, c, d are given in terms of k above. Again we have the oneparameter family.
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D(2, 1, λ). The form B is B = λ 1 e 1 ⊗ e 1 + λ2 e 2 ⊗ e 2 + λ3 e 3 ⊗ e 3 , where λi ,
i = 1, 2, 3 are arbitrary non-zero parameters. Let us introduce the parameter k = λ1 + λ2 + λ3 − 1
so that when k = −1 we have the Lie superalgebra case. The multiplicities have the form k+1 m(2ei ) = mi = − 1, i = 1, 2, 3. 2λi The corresponding deformed CMS operators are 4λi mi (mi + 1) ∂2 ∂2 ∂2 + λ + λ + 2 3 2 2 2 ∂x2 sin2 2xi ∂x1 ∂x3 i=1 2(k + 1) , + 2 sin (x1 ± x2 ± x3 ) ± 3
LD(2,1,λ) = λ1
(16)
where the last sum is again over all possible combinations of signs (4 in this case). This family is living on the projective plane with projective coordinates λ1 : λ2 : λ3 . The Lie superalgebra case corresponds to the line λ1 + λ2 + λ3 = 0, so we have again only one deformation parameter (say k). It is interesting to note that the potentials U of all operators listed above satisfy the so-called locus conditions (see [11]): the first coefficient in the Laurent expansion of U in the direction of α is identically zero on the hyperplanes sin(α, x) = 0. For the real roots this is obvious by symmetry reasons, but for imaginary roots this is not and follows only from a direct case by case check. If we replace the third condition for admissible deformations by these locus relations we will have essentially the same list of the potentials (some additional possibilities are due simply to the symmetry of the potential under the change mα → (−1 − mα )). We do not have satisfactory explanation of this phenomenon, which seems to be important (cf. [20, 21]). Since the locus configurations (in the rational case) are known to be related to the Huygens’ Principle (see [11]) it is natural to ask about the deformed CMS operators with all multiplicities being integers. A simple check of the cases listed above shows that the only non-trivial cases correspond to the systems A(n, 1) and BC(n, 1) which has already been discovered in [10, 11]. This indicates that the list of known locus configurations [21] could be in fact complete and therefore could give the answer to the famous Hadamard problem in the theory of Huygens’ principle. 3. Construction of Quantum Integrals for Classical Series In this section we present a recursive formula for the quantum integrals of the deformed CMS problems (4), (5) (more precisely, for its hyperbolic versions). Our formula is a properly deformed version of the formula used by A. Matsuo in his paper [19] on the relations between the CMS quantum problem and the KZ equation. We would like to mention also that for m = 1 an equivalent set of integrals was found earlier in [10] using very different ideas.
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259
Let V be an n + m-dimensional vector space with the deformed inner product (u, v) =
n
ui v i + k
n+m
uj v j
j =n+1
i=1
corresponding to the form B given by (13). When k = 1 we have the standard Euclidean scalar product which we denote by <, >: < u, v >= u1 v 1 + · · · + un+m v n+m . We will be using the fact that the generalized root systems R of type A(n − 1, m − 1) and BC(n, m) with respect to the scalar product < u, v > coincide with the usual root systems A(n + m − 1) and BC(n + m) (see the previous section). The corresponding Weyl groups W are generated by reflections with respect to the Euclidean form which we denote as s<α> . Let us introduce the operator B by the relation B(u, v) =< Bu, v >: Bei = ei , i = 1, ..., n, Bej = kej , j = n + 1, ..., n + m. We will call a vector v ∈ V homogeneous if it is an eigenvector of B (and thus for any pair of forms in our family). For any homogeneous vector v the following relation holds: (x, v) (v, v) = < x, v > < v, v > for any vector x ∈ V . Obviously in our case the set of homogeneous vectors is V1 ∪ V2 , where V1 and V2 are generated by the first n and last m basic vectors respectively. It will be important for us that for the classical systems there exists an orbit O of the Weyl group W , which consists of homogeneous vectors. Indeed, for the A(n − 1, m − 1) root system one can take O = {ei } and for BC(n, m) such an orbit is O = {±ei }, i = 1, . . . , n + m. In this section we will assume that x ∈ V and the brackets (α, x) denote the deformed product given by B. Let us define for α ∈ R the following functions on V : fα (x) =
1 e(α,x) + 1 (α, x) 1 , = coth (α,x) 2e 2 2 −1
ϕα (x) =
1 1 − fα (x)2 = − 4 4 sinh2
(α,x) 2
.
They satisfy the following relations: ∂v fα = (v, α)ϕα ,
∂v ϕα = −2(v, α)fα ϕα
for any v ∈ V . (p) Let us define now for any natural number p the operator ∂v by the following recurrent procedure: ∂v(1) = ∂v , ∂v(p) = ∂v ∂v(p−1) −
α∈R +
(p−1)
mα (α, v)fα (∂v(p−1) − ∂s<α> v ),
(17)
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where R + is a positive part of R and s<α> is the reflection corresponding to the root α with respect to the Euclidean form <, >. The formula (17) is a deformed version of the formula (2.3.6) from A. Matsuo’s paper [19]. Take now an orbit O of the corresponding Weyl group W consisting of homogeneous elements (see above) and define Lp =
∂v(p) . (v, v)
(18)
v∈O
One can easily check using the relation ∂v − ∂s<α> v = 2
< α, v > ∂α < α, α >
that L2 =
v∈O
∂v2 −2 m α f α ∂α (v, v) + α∈R
is up to a coefficient the deformed CMS operator (11) (in the hyperbolic version and radial gauge) after a scaling x → 2x. Theorem 1. The operators Lp given by the formula (18) commute with each other: [Lp , Lq ] = 0 and thus are the quantum integrals of the corresponding deformed CMS problem (11) related to classical generalized root systems. The proof is based on the following (p)
Proposition 1. The operators ∂v deformed CMS operator:
satisfy the following commutation relation with the
< α, α > (p) ϕα (∂v(p) − ∂s<α> v ). mα L2 , ∂v(p) = (v, v) < v, v > +
(19)
α∈R
The proof is by induction in p. For p = 1 we have the relation [L2 , ∂v ] = (v, v)
α∈R +
mα
< α, α > ϕα (∂v − ∂s<α> v ), < v, v >
(20)
which is easy to check. The proof of the induction step is a long but straightforward calculation. We reproduce the main steps to show the role of the properties of admissible deformations here.
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261
Let us assume that the statement is true for all natural numbers less than p. We have L2 , ∂v(p) = [L2 , ∂v ] ∂v(p−1) + ∂v L2 , ∂v(p−1) (p−1) − mα (α, v) [L2 , fα ] (∂v(p−1) − ∂s<α> v ) α∈R +
−
(p−1) mα (α, v)fα L2 , ∂v(p−1) − ∂s<α> v
α∈R +
= (v, v)
mα
α∈R +
+(v, v)∂v
−2
< α, α > ϕα (∂v − ∂s<α> v )∂v(p−1) < v, v >
α∈R +
mα
< α, α > (p−1) ϕα (∂v(p−1) − ∂s<α> v ) < v, v > (p−1)
mα (α, v)ϕα ∂α (∂v(p−1) − ∂s<α> v )
α∈R +
+2
(p−1)
mα (α, v)fα ∂α (fα )(∂v(p−1) − ∂s<α> v )
α∈R +
−
(p−1) mα (α, v)fα L2 , ∂v(p−1) − ∂s<α> v
α∈R +
+2
(p−1)
mα mβ (α, v)fβ ∂β (fα )(∂v(p−1) − ∂s<α> v ),
α,β∈R +
where we have used the induction assumption and the relation mβ fβ ∂β (fα ). [L2 , fα ] = 2ϕα ∂α − 2fα ∂α (fα ) − 2 β∈R +
Let us denote the sum of the last two sums in the previous expression as B: (p−1) mα mβ (α, v)fβ ∂β (fα )(∂v(p−1) − ∂s<α> v ) B=2 α,β∈R +
−
(p−1) mα (α, v)fα L2 , ∂v(p−1) − ∂s<α> v
α∈R +
and the rest of the previous expression as A. Using the homogeneity of v we can rewrite A as < α, α > (p−1) ϕα (∂v − ∂s<α> v )∂v(p−1) + ∂v (∂v(p−1) − ∂s<α> v ) mα A = (v, v) < v, v > α∈R + < α, α > (p−1) ∂v (ϕα )(∂v(p−1) − ∂s<α> v ) mα +(v, v) < v, v > + α∈R (p−1) mα (α, v)ϕα ∂α (∂v(p−1) − ∂s<α> v ) −2 +2
+ α∈R
α∈R +
(p−1)
mα (α, v)fα ∂α (fα )(∂v(p−1) − ∂s<α> v )
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= (v, v)
mα
α∈R +
+2
< α, α > (p−1) ϕα (∂v ∂v(p−1) − ∂sα v ∂s<α> v ) < v, v >
mα (α, v)fα (∂α (fα ) −
α∈R +
Now let us use that ∂v ∂v(p−1) = ∂v(p) +
(v, v) (p−1) < α, α > ϕα )(∂v(p−1) − ∂s<α> v ). < v, v > (p−1)
mβ (β, v)fβ (∂v(p−1) − ∂s<β> v ),
β∈R + (p−1)
(p)
∂s<α> v ∂s<α> v = ∂s<α> v +
(p−1)
(p−1)
mβ (β, s<α> v)fβ (∂s<α> v − ∂s<β> s<α> v ),
β∈R +
to rewrite the last expression as < α, α > (p) A = (v, v) ϕα (∂v(p) − ∂s<α> v ) mα < v, v > α∈R + < α, α > (p−1) ϕα mα mβ (β, v)fβ (∂v(p−1) − ∂s<β> v ) +(v, v) < v, v > α∈R + β∈R + < α, α > (p−1) (p−1) mα mβ (β, s<α> v)fβ (∂s<α> v − ∂s<β> s<α> v ) ϕα −(v, v) < v, v > + + +2
α∈R
β∈R
mα (α, v)fα (∂α (fα ) −
α∈R +
= (v, v)
mα
α∈R +
(v, v) (p−1) < α, α > ϕα )(∂v(p−1) − ∂s<α> v ) < v, v >
< α, α > (p) ϕα (∂v(p) − ∂s<α> v ) < v, v >
< α, α > (p−1) ϕα mα fα ((α, v) + (α, s<α> v)) (∂v(p−1) − ∂s<α> v ) < v, v > α∈R + (v, v) (p−1) < α, α > (∂v(p−1) − ∂s<α> v ) mα (α, v)fα ϕα (α, α) − +2 < v, v > α∈R + < α, α > (p−1) ϕα mα mβ (β, v)fβ (∂v(p−1) − ∂s<β> v ) +(v, v) < v, v > α∈R + β∈R + ,β=α < α, α > (p−1) (p−1) −(v, v) mα ϕα mβ (β, s<α> v)fβ (∂s<α> v − ∂s<β> s<α> v ). < v, v > + + +(v, v)
α∈R
mα
β∈R ,β=α
Combining the second and third sums we come to the following expression: (v, v) (α, α) mα {(α, v) + (α, s<α> v)} + 2 < α, v > { − } , < α, α > < v, v > which can be rewritten in the form (v, v) (α, α) −2 < α, v > (mα − 1) − . < α, α > < v, v > We claim that this is 0 for any root α. Indeed if α is imaginary then mα = 1 by our assumption (property 2 of admissible deformations). If α is real and < α, v > is not (α,α) (v,v) zero, then <α,α> −
= 0 for any v from our orbit.
Deformed Quantum Calogero-Moser Problems and Lie Superalgebras
263
Thus we come to the following expression for A :
< α, α > (p) ϕα (∂v(p) − ∂s<α> v ) < v, v > α∈R + < α, α > (p−1) mα mβ (β, v)fβ ∂v(p−1) − ∂s<β> v ) ϕα +(v, v) < v, v > α∈R + β∈R + β=α < α, α > (p−1) (p−1) −(v, v) mα mβ (β, s<α> v)fβ (∂s<α> v − ∂s<β> s<α> v ). ϕα < v, v > + +
A = (v, v)
mα
β∈R β=α
α∈R
Now let us look at term B. We have (p−1) mα (α, v)fα L2 , ∂v(p−1) − ∂s<α> v α∈R +
=
mα (α, v)fα mβ < β, β > ϕβ
α,β∈R +
−
mα (α, v)fα mβ < β, β > ϕβ
α,β∈R +
=
(v, v) (p−1) (∂ (p−1) − ∂s<β> v ) < v, v > v
mα (α, v)fα mβ < β, β > ϕβ
α,β∈R + β=α (p−1)
(s<α> v, s<α> v) (p−1) (p−1) (∂s v − ∂s<β> s<α> v ) < s<α> v, s<α> v > <α>
mα (α, v)fα mβ < β, β > ϕβ
α,β∈R + β=α
−
(v, v) (p−1) (∂ (p−1) − ∂s<β> v ) < v, v > v
(s<α> v, s<α> v) < s<α> v, s<α> v >
(p−1)
×(∂s<α> v − ∂s<β> s<α> v ) (v, v) (s<α> v, s<α> v) 2 + + mα (α, v)fα < α, α > ϕα < v, v > < s<α> v, s<α> v > + α∈R
(p−1)
×(∂v(p−1) − ∂s<α> v ). Combining the last term with the term in the second sum of B corresponding to β = α and using the relation (α, α) (v, v) < α, v >2 (s<α> v, s<α> v) = (v, v) + 4 − < α, α > < α, α > < v, v > we have
(v, v) (s<α> v, s<α> v) m2α (α, v)fα 2∂α (fα )− < α, α > ϕα + < v, v > < s<α> v, s<α> v > (p−1)
×(∂v(p−1) − ∂s<α> v ), which is zero since (α, α) (v, v) < α, v >2 (p−1) − (∂v(p−1) − ∂s<α> v ) = 0 2 1−2 < α, α >< v, v > < α, α > < v, v >
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for any root: for the real roots the product of the last two brackets is zero while for imaginary roots the first bracket is zero. Thus we arrive at the following expression for
< α, α > (p) L2 , ∂v(p) = A + B = (v, v) mα ϕα (∂v(p) − ∂s<α> v ) < v, v > α∈R + < α, α > mα m β ϕα fβ (β, v)(∂v(p−1) +(v, v) < v, v > β=α (p−1) (p−1) (p−1) −∂s<β> v ) − (β, s<α> v)(∂s<α> v − ∂s<β> s<α> v ) −
β=α
< β, β > mα mβ (α, v) fα ϕβ (v, v)(∂v(p−1) < v, v >
(p−1)
(p−1)
(p−1)
−∂s<β> v ) − (s<α> v, s<α> v)(∂s<α> v − ∂s<β> s<α> v )
+2
(p−1)
mα mβ (α, v)fβ ∂β (fα )(∂v(p−1) − ∂s<α> v ).
α=β
Let us denote the sum of the last three sums in the previous expression as C. We must show that C is identically zero. Let us notice that
mα mβ (α, v)fβ ∂β (fα ) =
α=β
mα mβ (α, v)fβ (β, α)ϕα
α=β
= ∂v
mα mβ (α, β)fβ fα .
α=β
But according to our assumption (third property of admissible deformations)
mα mβ (α, β)fβ fα = const + 4
α=β
mα m2α (α, α)f2α fα
α
(see the formula (12) above). Thus C can be rewritten as
(v, v)mα mβ
β=α
< α, α > ϕα fβ (β, v)(∂v(p−1) < v, v >
(p−1) (p−1) −∂s<β> v ) − (β, s<α> v)(∂s<α> v
−
β=α
mα mβ (α, v)
(p−1) − ∂s<β> s<α> v )
< β, β > fα ϕβ (v, v)(∂v(p−1) < v, v >
(p−1) (p−1) −∂s<β> v ) − (s<α> v, s<α> v)(∂s<α> v
(p−1) − ∂s<β> s<α> v )
Deformed Quantum Calogero-Moser Problems and Lie Superalgebras
−2
(p−1)
mα mβ (α, v)fβ ∂β (fα )∂s<α> v + 4∂v
α=β
=
S1 (α, β, v) −
α=β
S2 (α, β, v) −
α=β
α=β
265
mα m2α (α, α)f2α fα ∂v(p−1)
α
S3 (α, β, v) +
S4 (α, v),
α
where S1 , S2 , S3 , S4 denote the terms in the first, second, third and fourth sums respectively. Choose in the first three sums the terms with β = 2α. We have α S1 (α, 2α, v) = S (2α, α, v) = 0 and 1 α −
(S2 (α, 2α, v)+S2 (2α, α, v))− (S3 (α, 2α, v)+S3 (2α, α, v))+ S4 (α, v) = 0. α
α
α
Thus we must show only that
C=
(S1 (α, β, v) − S2 (α, β, v) − S3 (α, β, v)) = 0.
α∼β,α,β∈R+
This can be done separately for each two-dimensional subsystem (cf. [11]). Proposition 1 is proven. Now we are ready to prove Theorem 1. When p = 2 (i.e. when Lp = L2 = L is the deformed CMS operator) this follows immediately from the lemma. Indeed [L2 , Lp ] =
L2 ,
v∈O
(p) < α, α > ∂v (p) = ϕα (∂v(p) − ∂s<α> v ), mα (v, v) < v, v > + v∈O α∈R
which obviously is identically zero. To prove that these operators commute for any p, q we borrow the idea from T. Oshima’s paper [25]. Consider an involution σ on the space of all differential operators on V corresponding to the change x → −x and the standard anti-involution ∗: operator L∗ is a formal adjoint to L. We have [Lσ1 , Lσ2 ] = [L1 , L2 ]σ and [L∗1 , L∗2 ] = −[L1 , L2 ]∗ . Our operators Lp have the following properties with respect to these involutions: L∗p = Lσp = (−1)p Lp . Now let us consider the commutator C = [Lp , Lq ]. By Jacobi identity [C, L2 ] = 0, so we can use Berezin’s lemma [26] which says that in such a case the highest symbol of C must be polynomial in x. Since in our case it must also be periodic this implies that the highest symbol is constant. We claim that it is actually zero. Indeed C ∗ = [Lp , Lq ]∗ = −[L∗p , L∗q ] = −[Lσp , Lσq ] = −[Lp , Lq ]σ = −C σ , so C ∗ = −C σ . Notice that since the highest symbol P (ξ ) of C does not depend on x the highest symbols of C ∗ and C σ are the same and equal to (−1)N P (ξ ), where N is the order of C. On the other hand we see that they must be different by a sign. This means that P and hence C are zero. This completes the proof of Theorem 1. Notice that as follows from the formula for the integrals in the BCn,m case, all the integrals Lp with odd p actually vanish, so in that case we will consider only even p. Recall now that the quantum system in Rn is called integrable if it has at least n commuting independent quantum integrals. Corollary. Deformed CMS problems (4), (5) related to the classical generalized root systems are integrable. The same is true for their rational limits.
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To have the integrals in the rational limit one should replace in the formulas of this section sinh z by z and coth z by z−1 , so that fα (x) = (α, x)−1 , ϕα (x) = −(α, x)−2 . For non-classical generalized root systems there are no analogues of the form <, > which is related to the fact that their affine structure is different from any of the usual root systems. This means that our proof can not be extended to this case, at least in a straightforward way. How to prove integrability of the deformed CMS problems in the exceptional cases is still an open question at the moment. 4. Algebra R,B and Harish-Chandra Homomorphism Let R ⊂ V be a classical generalized root system and (R, m, B) be its admissible deformation described in Sect. 1. Let us introduce the corresponding algebra ωR,B as the algebra of polynomial functions p(x) on V , which satisfy the following properties (cf. [22]): 1) p(x) are invariant with respect to the Weyl group W0 , corresponding to the real roots of the system; 2) p(x + ωα) ≡ p(x − ωα) on the hyperplane (α, x) = 0 for any imaginary root α, where (α, x) is the deformed scalar product determined by B. In the limit ω → 0 we have the algebra 0R,B of W0 -invariant polynomials with the properties 1) and 2)0 ∂α p(x) ≡ 0 on the hyperplane (α, x) = 0 for any imaginary root α ∈ R. One can consider this algebra also as a subalgebra of polynomial functions on V ∗ satisfying the same relation 2)0 , where (α, x) is understood as pairing between vector and covector and ∂α is defined using the deformed form B. Below we will be using this realization. Since the algebras ωR,B are obviously isomorphic for all ω = 0 we will assume 1/2
later on that ω = 1/2 considering only algebras R,B = R,B and 0R,B . We are going to show that for generic k these two algebras are actually isomorphic to the algebras generated by the quantum integrals of the deformed CMS problems from the previous section in the trigonometric and rational case respectively. Remark. We should mention that in the case when all the multiplicities are integer the algebra of quantum integrals is actually much bigger and is called the algebra of quasiinvariants, see [11, 23]). For example when k = 1 the quasi-invariants are polynomials satisfying the property 2)0 for all roots, but no symmetry is imposed (see [32] for the latest results in this direction). It is obvious that the highest order component of any polynomial P ∈ R,B belongs to the algebra 0R,B . A more subtle question is whether for any homogeneous Q ∈ 0R,B there exists P ∈ R,B such that Q is the highest term of P . We will show that at least for generic values of the deformation parameter k this is true, which means that 0R,B is the associated graded algebra for R,B . We are now going to describe the algebras 0R,B more explicitly. Let us start with the type A(n − 1, m − 1). The corresponding algebra can be realized as the following algebra 0n,m;k ⊂ C[V ∗ ] = C[x1 , . . . , xn , y1 , . . . , ym ] consisting of the polynomials f (x1 , . . . , xn , y1 , . . . , ym ) which are symmetric in x1 , . . . , xn and y1 , . . . , ym separately and satisfy the conditions ∂ ∂ −k f ≡0 ∂xi ∂yj on each hyperplane xi − yj = 0 for i = 1, ..., n and j = 1, ..., m.
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It is very easy to check that the deformed Newton sums pr (x, y, k) =
n
1 r yj k m
xir +
i=1
(21)
j =1
belong to 0n,m;k for all nonnegative integers r. Theorem 2. If k is not a positive rational number then the algebra 0n,m;k is generated by the deformed Newton polynomials pr (x, y, k), r ∈ Z+ . Notice that for special values of k this is not true. For example, if k = 1 the deformed Newton sums generate the algebra of symmetric polynomials
in n + m variables, while 0n,m;k is a much bigger algebra containing for example p = (xi − yj )3 . To prove the theorem let us recall that the partition λ of a natural number N is a decreasing sequence of non-negative integers λ1 ≥ λ2 ≥ λ3 ≥ . . . such that only a finite number of them are non-zero and their sum is equal to N . This sum λ1 + λ2 + λ3 + · · · is usually denoted as |λ|. To each partition one can relate a Young diagram with N squares in a natural way (see e.g. [24]). Proposition 2. If k is not a positive rational then the dimension of the homogeneous component 0n,m;k of degree N is less than or equal to the number of partitions λ of N such that λn+1 ≤ m. Notice that the corresponding Young diagrams are precisely the ones contained in the fat (n, m)-hook (see Fig. 1). Denote by DN (n, m) the number of such partitions (or diagrams in this fat hook). Let I = (i1 , i2 , . . . , in ) and J = (j1 , j2 , . . . , jm ) be some sequences (at the beginning, unordered) of nonnegative integers such that nr=1 ir + m s=1 js = N . Let N (J ) be the
Fig. 1. Fat hook
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number of nonzero elements of J , M(I, J ) be the number of elements of I which are greater or equal to N(J ). Define the sets Ereg = {(I, J ) | M(I, J ) = n} ,
Enreg = {(I, J ) | M(I, J ) < n} .
The pairs from Ereg will be called regular, otherwise - irregular. Let us prescribe to each pair (I, J ) a variable C(I, J ) in such a way that C(I, J ) = C(σ (I ), τ (J )), σ ∈ Sn , τ ∈ Sm is the same for all orderings of I and J. It is easy to see that the number of different C(I, J ), (I, J ) ∈ Ereg is equal to DN (n, m). Indeed for given I one can construct a Young diagram in the usual way (ordered ik are the horizontal rows) and then attach at the bottom the transposed Young diagram related to J. The condition M(I, J ) = n guarantees that altogether we will have a Young diagram, which obviously is contained in the (n, m)-hook. For any sequence I let us rewrite the elements of I in non-decreasing order and denote this sequence I + . Choose some integers 1 ≤ r ≤ n and 1 ≤ s ≤ m and 1 ≤ p ≤ N and consider the equations (i − kj )C(I, J ) = 0, (22) i+j =p
where i occupies the r th place in I , j occupies the s th place in J and all other elements of I, J are fixed. All such equations form the system of linear equations on C(I, J ), which has the following meaning. Let f = C(I, J )x I y J be a homogeneous polynomial of degree N symmetric in x and y separately, then the system (22) is nothing else but the condition 2)0 for the polynomials f from 0n,m;k . Thus to prove the proposition it is enough to show that every irregular C(I, J ), (I, J ) ∈ Enreg can be expressed from the system (22) as a linear combination of C(I, J ), (I, J ) ∈ Ereg . Let us first prove this statement for n = 1. We will use the induction with respect to the following total order on Enreg . Let (I, J ), (K, L) ∈ Enreg . We will say that +) (I, J ) < (K, L) if for the corresponding ordered sets (i, j1+ . . . jm+ and (k, l1+ . . . lm + + either N (J ) < N (L) or N (J ) = N (L) and for q = min{i, k} (jm , . . . , jm−q , i) < + , . . . , l + , k) in lexicographic order. (lm m−q If N(J ) = 1 then we have kj C(0, j ) + (−1 + (j − 1)k)C(1, j − 1) + (−2 + (j − 2)k)C(2, j − 2) + · · · = 0. So we have for k = 0 that C(0, j ) ∈ Span{C(i, j ), (i, j ) ∈ Ereg }. For general N(J ) > 1 take any (I, J ) ∈ Enreg with an ordered J = J + : (i, j1 , j2 , . . . ). Consider Eq. (22) corresponding to r = 1, s = N (J ) − i, p = i + js . One can check that (I, J ) is the largest pair with respect to our order among all irregular pairs (K, L) corresponding to C(K, L) entering the equation with non-zero coefficients. Since the coefficient at C(I, J ) is −i + kjs and thus (because k is not a positive rational) is not zero we can express C(I, J ) as a linear combination of lower variables. This proves the proposition for n = 1. ˜ ˜ For general n we can use the induction in M(I, J ) = n − M(I, J ). If M(I, J) = 1 ˜ we can use the previous arguments. Assume now that M(I, J ) > 1. Consider one index r, for which ir < N (J ) and apply previous arguments to express C(I, J ) as a linear combination of C(I ∗ , J ∗ ) with (I ∗ , J ∗ ) such that ir∗ ≥ N (J ∗ ). According to the inductive hypothesis we can express C(I ∗ , J ∗ ) as a linear combination C(I ∗∗ , J ∗∗ ), where (I ∗∗ , J ∗∗ ) ∈ Ereg . Proposition 2 is proved.
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Now let us prove the theorem. Let us denote by Nn,m;k the algebra generated by the deformed Newton sums (21). As we have already mentioned Nn,m;k ⊂ 0n,m;k . To show that Nn,m;k = 0n,m;k it is enough to prove that the dimension of the homogeneous component of degree N of Nn,m;k is not less than DN (n, m). To produce enough independent polynomials we will use the theory of Jack polynomials (see e.g. [24]). Let be the algebra of symmetric functions in an infinite number of variables z1 , z2 , . . . and pr (z) = z1r + z2r + · · · be the power sum, Pλ (z, θ ) be the Jack polynomial depending on the partition λ (see [24] ). Consider a homomorphism φ from to 0n,m;k such that φ(pr (z)) = pr (x, y, k). Such a homomorphism was first used by Kerov, Okounkov and Olshanski in [27]. The image of the Jack polynomials under this homomorphism sometimes is called super-Jack polynomials (see e.g. [28]). One can show using some results from [24] (see formulas (7.9’) and (10.19) from Chapter 6) that for θ = −k, φ(Pλ (z, θ )) =
bλ/µ (θ )Pµ (x, θ )Pλ /µ (y, θ −1 ),
(23)
µ⊂λ
where bλ/µ is some rational function of θ with poles in non-positive rational numbers. Since θ = −k is not such a number, by assumption these super-Jack polynomials are well-defined. From the formula (23) it follows that the leading term in lexicographic order of φ(Pλ (z, θ )) has a form <λ 1 −n>
x1λ1 . . . xnλn y1
<λ m −n>
. . . ym
,
= max(0, x). From the where λ is the partition conjugate to λ and < x >= x+|x| 2 definition φ(Pλ (z, θ )) ∈ Nn,m;k . It is clear that all these polynomials corresponding to the diagrams contained in the fat hook are linearly independent. This completes the proof of Theorem 2. Remark. The relation with the theory of Jack polynomials is actually much deeper. We discuss this in detail in our paper [38] (see also [13, 14]). As a corollary we can give a formula for the Poincar´e series Pn,m (t) = ⊕i dim(0n,m;k )(i) t i of the algebra 0n,m;k for generic k. Theorem 3. The Poincare´ series of the algebra 0n,m;k for generic k has the following form m t i(n+1) 1 Pn,m (t) = 1+ . (24) (1 − t)(1 − t 2 ) . . . (1 − t n ) (1 − t)(1 − t 2 ) . . . (1 − t i ) i=1
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Proof. From Theorem 2 it follows that the corresponding Poincar´e series is the sum of t |λ| over all partitions λ which fit into the fat (n, m) hook: Pn,m (t) = t |λ| = t |λ| + t |λ| + · · · + t |λ| , λn+1 ≤m
λn+1 =0
λn+1 =m
λn+1 =1
where |λ| = λ1 + λ2 + · · · + λN . It is easy to see that t |λ| = t i(n+1)+|µ|+|ν| = t i(n+1) t |µ| t |ν| . λn+1 =i
µn+1 =0 νi+1 =0
µn+1 =0
νi+1 =0
Since µn+1 =0
t |µ| =
1 , (1 − t)(1 − t 2 ) . . . (1 − t n )
we arrive at the formula (24).
νi+1 =0
t |ν| =
1 , (1 − t)(1 − t 2 ) . . . (1 − t i )
Remark. Another (recurrent) formula for the generating function of the Young diagrams which fit into the fat hook was found recently by Orellana and Zabrocki [29] in relation with the theory of the Schur functions and characters of Lie superalgebras. Notice that the symmetry between n and m is not obvious from our formula (24) and leads to some identities which might be interesting. For R = BC(n, m) the algebra 0R,B is related to 0n,m;k in a very simple way: it is easy to check from the definition that it consists of the polynomials p(x12 , x22 , . . . , 2 ), where p belong to 0 y12 , y22 , . . . , ym n,m;k . BC (t) of the algebra 0 Corollary. The Poincare´ series Pn,m R,B for the generalized system R of type BC(n, m) and generic values of the deformation parameter has the following form: BC Pn,m (t) = Pn,m (t 2 ),
where Pn,m (t) is given by the formula (24). Let us discuss now the Harish-Chandra homomorphism. Let R ∈ V be a generalized root system and R + be a set of positive roots. Let us denote by D[R − ] the algebra of differential operators on V ∗ with coefficients in C[e−α , (e−α − 1)−1 ], where α ∈ R + . The Harish-Chandra homomorphism ϕ : D[R − ] −→ D, where D is the algebra of differential operators on V ∗ with constant coefficients, is uniquely determined by the condition ϕ(e−α ) = 0. The algebra D is isomorphic to the algebra of polynomial functions on the space V ∗ . Let now R be a classical generalized root system and consider the algebra QR,m,B generated by the quantum integrals Ls of the corresponding deformed CMS problem (10): Ls = ψˆ 0 ◦ Ls ◦ ψˆ 0−1 ,
where ψˆ 0 is the multiplication operator by the function ψˆ 0 = α∈R+ sin−mα (α, x) and Ls given by (18). It is easy to check that all the operators Ls belong to the algebra D[R − ], so QR,m,B is a subalgebra in D[R − ].
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Theorem 4. For generic values of the deformation parameter the Harish-Chandra homomorphism maps the algebra of quantum integrals of the deformed CMS problems QR,m,B onto the algebra R,B . In the rational limit the same is true for the algebra 0R,B . Proof. We will identify V and V ∗ using the form B. Take λ ∈ V and define xv (λ) = (p) e−(λ,x) ϕ(∂v )e(λ,x) . From (17) we have the following recurrent relations p
xv(p) (λ) = (λ, v)xv(p−1) (λ) −
1 (p−1) mα (α, v) xv(p−1) (λ) − xs<α> (v) (λ) , 2 + α∈R
which can be rewritten as xv(p) (λ) = (λ − ρ, v)xv(p−1) (λ) +
1 (p−1) mα (α, v)xs<α> (v) (λ). 2 + α∈R
(p)
(p)
The shifted functions yv (λ) = xv (λ + ρ) satisfy the relations yv(p) (λ) = (λ, v)yv(p−1) (λ) +
1 (p−1) mα (α, v)ys<α> (v) (λ). 2 + α∈R
It is easy to see that the image Zp = ϕ(Lp ) of the quantum integrals Lp under HarishChandra homomorphism has the form Zp =
yv(p) . (v, v)
v∈O
Let (λ, γ ) = 0, where γ ∈ Rim is an imaginary root. We should prove that γ γ Zp λ − = Zp λ + . 2 2
(25)
We will prove this for the root system of type An,m , the case of BCn,m root system is very similar. Without loss of generality we can assume that γ = en − en+1 . Let us introduce (p)− (p) (p)+ (p) yv (λ) = yv (λ − γ2 ), yv (λ) = yv (λ + γ2 ). We have the following recurrent relations 1 1 (p−1)− yv(p)− (λ) = λ − γ , v yv(p−1)− (λ) + mα (α, v)ys<α> (v) (λ), 2 2 α∈R + 1 1 (p−1)+ mα (α, v)ys<α> (v) (λ). yv(p)+ (λ) = λ + γ , v yv(p−1)+ (λ) + 2 2 + α∈R
(p)±
Let us denote by v = en , u = s<γ > (v) = en+1 and introduce yv,u (λ) as yv,u (λ) = ((u, u)−1 + (v, v)−1 )−1 [(u, u)−1 yu (p)±
(p)±
+ (v, v)−1 yv(p)± ].
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Lemma. On the hyperplane (λ, γ ) = 0 the following relations hold: (p)±
(p−1)±
1)yv,u (λ) = (λ, v)yv,u (p)−
2) yv
(p)−
(λ) = yu
(λ) +
1 2
(p−1)±
s<α> (v)=u,v
(26)
mα (α, v)ys<α> (v) (λ),
(λ).
Proof is by induction and based on the following fact, which can be easily checked directly: if s<α> (v) = s<β> (u) then mβ (β, u) −1 −1 −1 mα (α, v) ((u, u) + (v, v) ) + = mα (α, v) = mβ (β, v). (v, v) (u, u) Now let w = es , w = u, v, then one can check that (p−1)+
mα (α, w)yv(p−1)+ + mβ (β, w)yu
(p−1)+
= (mα (α, w) + mβ (β, w))yv,u
,
where s<α> (v) = s<β> (u) = w. Notice that the last relation determines α and β uniquely in our case. Using this one can show that 1 (p−1)± yw(p)± = (λ, w)yw(p−1)± + (mα (α, w) + mβ (β, w))yv,u 2 1 (p−1)± + mδ (δ, w)ys<δ>(w) 2
(27)
s<δ >(w)=u,v
provided (λ, γ ) = 0. (p)+ (p)− (p)− (p)+ From the relations (26), (27) it follows that yw = yw , yv,u = yv,u on the hyperplane (λ, γ ) = 0, which imply the relation (25). Note that the highest term of Zr (λ) is λr1 + · · · + λrn + k r−1 (λrn+1 + · · · + λrn+m ).
(28)
Now from Theorem 2 it follows that for generic k, the homomorphism ϕ is surjective. The fact that it is injective is obvious. This completes the proof in the trigonometric case, the rational case easily follows. In fact ω in the definition of ωR,B and ω in the limiting procedure from the trigonometric to rational case could be identified. Remark. Notice that we have proved that the image of the Harish-Chandra homomorphism belong to the algebra R,B for all values of the parameter k. The condition that k is generic is used only to claim that the image coincides with this algebra. As a corollary we have the following statement which is probably true for all generalized root systems and all values of deformation parameter. Proposition 3. For the classical generalized root systems and generic values of the deformation parameter the algebra 0R,B is the associated graded algebra for R,B . In the A(n, m) case we can give an explicit formula for the generators of the algebra R,B : Yr (λ) =
n i=1
Br (λi + 1/2) + k
r−1
m j =1
Br (λj +n + 1/2),
(29)
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273
where Br (x) are the classical Bernoulli polynomials. One can easily check using the relation Br (x +1)−Br (x) = rx r−1 that Yr satisfy the relations (25) and have the highest term (28). We finish this section with the following Theorem 5. Algebra 0n,m;k is finitely generated if and only if k is not a negative rational number of the form − rs , where 1 ≤ r ≤ n, 1 ≤ s ≤ m. Consider the subalgebra P (k) = C[p1 , ..., pn+m ] generated by the first n + m deformed Newton sums (21). We need the following result about common zeros of these polynomials (cf. Proposition 1 in [32]). Proposition 4. Consider the following system of algebraic equations: x1 + x2 + · · · + xn + k −1 (xn+1 + xn+2 + · · · + xn+m ) = 0 2 2 2 x12 + x22 + · · · + xn2 + k −1 (xn+1 + xn+2 + · · · + xn+m )=0 ··· n+m n+m n+m n+m + x2n+m + · · · + xnn+m + k −1 (xn+1 + xn+2 + · · · + xn+m ) = 0. x1 If parameter k is not a negative rational number of the form − rs , where 1 ≤ r ≤ n, 1 ≤ s ≤ m, then the system has only the trivial (zero) solution in Cn+m . Converse statement is also true. To prove this suppose that the system has a nontrivial solution x1 , . . . , xn+m . We can assume that xi = 0 for all i = 1, . . . , n + m. Let us re-group the set X = {x1 , x2 , . . . , xn+m } ⊂ C identifying equal xi ’s as {z1 , . . . , zp }, p ≤ n + m, where all zj are different. Multiplicity of zj is a pair (rj , sj ), where rj shows how many times zj enters the set {x1 , x2 , . . . , xn } and sj is the same for the rest of the set X. For the numbers zj , 1 ≤ j ≤ p we have the system p
aj zji = 0,
i = 1, . . . , n + m,
j =1
where aj = rj + k −1 sj . Consider the first p of these equations as the linear system on aj . Its determinant is of Vandermonde type and is not zero since all zj are different and non-zero. Hence all aj must be zero which may happen only if k = − rs for some 1 ≤ r ≤ n, 1 ≤ s ≤ m. The converse statement is obvious: if k = − rs then we can take x1 = x2 = · · · = xr = z = xn+1 = · · · = xn+s and other xi being zero to have the non-trivial solutions of the system with arbitrary z. From Proposition 4 it follows that for k = − rs the algebra of all polynomials on V is a finitely generated module over the subalgebra P (k). By a general result from the commutative algebra (see e.g. Proposition 7.8 from [30]) this implies that 0n,m;k is finitely generated. Now suppose that k = − rs for some 1 ≤ r ≤ n, 1 ≤ s ≤ m. Consider the following homomorphism: φr,s : 0n,m;k → 01,1;−1 by sending a polynomial f (x1 , . . . , xn , y1 , . . . , ym ) into fˆ(x, y) = f (x, x, . . . , x, 0, . . . , 0, y, y, . . . , y, 0, . . . , 0),
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where x is repeated r times and y is repeated s times. One can easily check that if f is in 0n,m;k with this particular k then fˆ(x, y) satisfies the condition (∂x + ∂y )fˆ = 0 when x = y, i.e. fˆ belongs to 01,1;−1 . The last algebra is actually very simple: it consists of the polynomials which are constant on the line x = y and thus have the form c + q(x, y)(x − y) with arbitrary polynomial q. It is easy to see that this algebra is not finitely generated. Since it is a homomorphic image of the algebra 0n,m;k this implies that the algebra 0n,m;k with k = − rs is also not finitely generated. Theorem 5 is proved. Remark. The case k = −1 is actually very special: in that case the algebra 0n,m;k is known as the algebra of supersymmetric polynomials and plays an important role in geometry (see e.g. [31], Chapter 3). Notice also that the special values of k in Theorem 2 are positive rationals while in Theorem 5 they are negative. Corollary. For generic values of the deformation parameters the algebras 0R,B and R,B for the classical generalized root systems are finitely generated. Indeed, from Proposition 3 it follows that it is enough to show that 0R,B is finitely generated. For the A(n, m) type this follows directly from Theorem 5, to prove this for BC(n, m) one should replace in Theorem 5 all the coordinates by their squares. An interesting question is whether the algebra 0n,m;k is free as a module over its polynomial subalgebra P (k), i.e. has the Cohen-Macaulay property. If this is true (which we believe to be so) then our formula (24) gives the degrees of its generators for generic values of k. For example, when m = 1 we have Pn,1 (t) = =
t n+1 1 1 + (1 − t)(1 − t 2 ) . . . (1 − t n ) (1 − t) 1 + t n+2 + t n+3 + · · · + t 2n+1 , (1 − t)(1 − t 2 ) . . . (1 − t n+1 )
which shows that the generators should have the degrees 0, n + 2, n + 3, ..., 2n + 1. The conjecture is that one can take the corresponding deformed Newton sums as such generators. For n = 2 this is in a good agreement with the results from [32], where the Cohen-Macaulay property for the rings of quasi-invariants related to A(n, 1) and BC(n, 1) is established (for any n) and the corresponding Poincare series are found (for n = 2).
5. Generalizations: Elliptic and Difference Versions The deformed quantum CMS systems we discussed have some natural generalizations. First of all if we replace in all the formulas for these operators the function sin12 z by the Weierstrass’ elliptic function ℘ (z) we will have the deformed elliptic CMS operators.
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For the generalized root system of type A(n − 1, m − 1) we have 2 ∂ ∂2 ∂2 ∂2 − k + · · · + + · · · + ∂x1 2 ∂xn 2 ∂y1 2 ∂ym 2 n m 2k(k + 1)℘ (xi − xj ) + 2(k −1 + 1)℘ (yi − yj ) +
Lell A(n−1,m−1) = −
+
i<j n m
i<j
2(k + 1)℘ (xi − yj ).
(30)
i=1 j =1
For BC(n, m) we can write a more general deformed Inozemtsev operator: 2 ∂ ∂2 ∂2 ∂2 −k =− + ··· + + ··· + ∂x1 2 ∂xn 2 ∂y1 2 ∂ym 2 n 2k(k + 1)(℘ (xi − xj ) + ℘ (xi + xj )) +
Lell BC(n,m)
i<j
+
m
2(k −1 + 1)(℘ (yi − yj ) + ℘ (yi + yj ))
i<j
+
n m
2(k + 1)(℘ (xi − yj ) + ℘ (xi + yj ))
i=1 j =1
+
3 n
ql (ql + 1)℘ (xi + ωl ) +
3 m
sl (sl + 1)℘ (yj + ωl ),
(31)
j =1 l=0
i=1 l=0
where ω0 = 0, ωl , l = 1, 2, 3 are the half-periods of the corresponding elliptic curve and the 9 parameters k, ql , sl , l = 0, 1, 2, 3 satisfy the following 4 relations 2ql + 1 = k(2sl + 1)
(32)
for all l. We conjecture that the elliptic versions of the deformed CMS problems are integrable as well. For m = 1 some results in this direction are found in [33, 34] (see also the recent paper [35]). Consider now the difference case. Let us introduce the following deformed Macdonald-Ruijsenaars operator. It depends on two parameters t and q and has the form D n,m =
n m 1 1 Ai Tq,xi + Bj Tt,yj , 1−q 1−t j =1
i=1
where Ai =
n m (xi − txk ) (xi − qyj ) , (xi − xk ) (xi − yj ) k=i
j =1
Bj =
n m (yj − txi ) (yj − qyl ) (yj − xi ) (yj − yl ) i=1
l=j
(33)
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and Tq,xi , Tt,yj are the “shift operators”: (Tq,xi f )(x1 , . . . , xi , . . . , xn , y1 , . . . , ym ) = f (x1 , . . . , qxi , . . . , xn , y1 , . . . , ym ), (Tt,yj f )(x1 , . . . , xn , y1 , . . . , yj , . . . , ym ) = f (x1 , . . . , xn , y1 , . . . , tyj , . . . , ym ). For m = 1 a similar operator was considered by O. Chalykh in [37]. We should mention that there is a small discrepancy between his and our formulas because of the misprint in formula (7.3) in [37]. It is interesting that the form of the operator (33) is invariant under the simultaneous interchange of q ↔ t and x ↔ y. In a way the deformed operator (33) is more symmetric than the original Macdonald-Ruijsenaars operator [24, 36]. In the differential case this duality corresponds to the invariance of the family of the deformed CM operators (4) under the interchange x ↔ y and k ↔ k1 . One can verify that the operator (33) can be rewritten in terms of the root system R of type A(n, m) as follows (notations are as in Sect. 3) (α,v) −α 1 1 − t q α Tv , DR = (34) 1 − q −α 1 − q (v,v) v∈O
α∈R,<α,v>>0
where tα = q mα ,
q α (u) = q (α,u) ,
(Tv f )(u) = f (u + v)
and the relation with the previous formula is given by t = qk,
xi = q ei ,
yj = q en+j .
In this form the operator can be immediately generalized for any reduced generalized root system Bn,m , Cn,m , Dn,m (but not for the general BC(n, m) case). We intend to discuss the properties of these deformed Macdonald operators in a separate paper. Concluding Remarks We have constructed for each generalized root system a family of the deformed quantum CMS problems and proved their integrability for the classical series. The proof is effective but not conceptual. We present more conceptual proof for the A(n, m) series in our paper [38], where the algebraic varieties corresponding to the rings 0R,B and relations with the theory of Jack polynomials are also discussed in more detail. It is clear that these relations between the theory of Lie superalgebras and quantum integrable systems should be understood better. In particular, the spectral theory for the deformed CMS operators should be related to the representation theory of Lie superalgebras and spherical functions on the symmetric superspaces. We hope to come back to this problem soon. Acknowledgements. We are grateful to O. Chalykh, A. Okounkov, G. Olshanski and especially to M. Feigin for useful discussions and helpful remarks. We would like also to thank the referee for a wonderful job which helped us to improve the paper. This work was supported by EPSRC (grants GR/R70194/01 and GR/M69548). The second author (A.P.V.) is grateful to IHES (Bures-sur-Yvette, France) for the hospitality in February 2003 when the first version of this paper was prepared.
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References 1. Calogero, F.: Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436 (1971) 2. Sutherland, B.: Exact results for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019–2021 (1971) 3. Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep. 94, 313–404 (1983) 4. Olshanetsky, M.A., Perelomov, A.M.: Quantum systems related to root systems and radial parts of Laplace operators. Funct. Anal. Appl. 12, 121–128 (1978) 5. Berezin, F.A., Pokhil, G.P., Finkelberg, V.M.: Schr¨odinger equation for a system of one-dimensional particles with point interaction. Vestnik MGU 1, 21–28 (1964) 6. Helgason, S.: Groups and Geometric Analysis. London-New York: Academic Press, 1984 7. Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions I. Comp. Math. 64, 329–352 (1987) 8. Heckman, G.J.: A remark on the Dunkl differential-difference operators. Progress in Math. 101, 181–191 (1991) 9. Veselov, A.P., Feigin, M.V., Chalykh, O.A.: New integrable deformations of quantum CalogeroMoser problem. Russ. Math. Surv. 51(3), 185–186 (1996) 10. Chalykh, O.A., Feigin, M.V., Veselov, A.P.: New integrable generalizations of Calogero-Moser quantum problem. J. Math. Phys. 39(2), 695–703 (1998) 11. Chalykh, O.A., Feigin, M.V., Veselov, A.P.: Multidimensional Baker-Akhiezer Functions and Huygens’ Principle. Commun. Math. Phys. 206, 533–566 (1999) 12. Veselov, A.P.: Deformations of the root systems and new solutions to generalized WDVV equations. Phys. Lett. A 261, 297–302 (1999) 13. Sergeev, A.N.: Superanalogs of the Calogero operators and Jack polynomials. J. Nonlin. Math. Phys. 8(1), 59–64 (2001) 14. Sergeev, A.N.: Calogero operator and Lie superalgebras. Theor. Math. Phys. 131(3), 747–764 (2002) 15. Serganova, V.: On generalization of root system. Commun. in Algebra 24(13), 4281–4299 (1996) 16. Bourbaki, N.: Groupes et alg`ebres de Lie. Chap. VI, Paris: Masson, 1981 17. Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977) 18. Berest, Yu.: Private communication to A.P. Veselov. October 1997 19. Matsuo, A.: Integrable connections related to zonal spherical functions. Invent. Math. 110, 95–121 (1992) 20. Veselov, A.P.: On generalisations of the Calogero-Moser-Sutherland quantum problem and WDVV equations. J. Math. Phys. 43(11), 5675–5682 (2002) 21. Chalykh, O.A., Veselov, A.P.: Locus configurations and ∨-systems. Phys. Lett. A 285, 339–349 (2001) 22. Chalykh, O.A., Veselov, A.P.: Commutative rings of partial differential operators and Lie algebras. Commun. Math. Phys. 126, 597–611 (1990) 23. Feigin, M., Veselov, A.P.: Quasi-invariants of Coxeter groups and m-harmonic polynomials. Int. Math. Res. Notices 10, 521–545 (2002) 24. Macdonald, I.: Symmetric functions and Hall polynomials. 2nd edition, Oxford: Oxford Univ. Press, 1995 25. Oshima, T.: Completely integrable systems with a symmmetry in coordinates. Asian J. Math. 2, 935–955 (1998) 26. Berezin, F.A.: Laplace operators on semisimple Lie groups. Proc. Moscow Math. Soc. 6, 371–463 (1971) (Russian) 27. Kerov, S., Okounkov, A., Olshanski, G.: The boundary of theYoung graph with Jack edge multipliers. Intern. Math. Res. Notices 4, 173–199 (1998) 28. Okounkov, A.: On N-point correlations in the log-gas at rational temperature. hep-th/9702001 29. Orellana, R.C., Zabrocki, M.: Some remarks on the characters of the general Lie superalgebra. math.CO/0008152 30. Atiyah, M., Macdonald, I.G.: Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969 31. Fulton, W., Pragacz, P.: Schubert Varieties and Degeneracy Loci. Lect. Notes in Math. 1689, Berlin Heidelberg-New York: Springer, 1998 32. Feigin, M., Veselov, A.P.: Quasi-invariants and quantum integrals of the deformed Calogero-Moser systems. Intern. Math. Research Notices 46, 2487–2511 (2003) 33. Khodarinova, L.A.: On quantum elliptic Calogero-Moser problem. Vestnik MGU, Ser.I Math. Mech. 5, 16–19 (1998) 34. Khodarinova, L.A., Prikhodsky, I.A.: On algebraic integrability of the deformed elliptic CalogeroMoser problem. J. Nonlin. Math. Phys. 8(1), 50–53 (2001)
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35. Chalykh, O., Etingof, P., Oblomkov, A.: Generalized Lam´e operators. Commun. Math. Phys. 239, 115–153 (2003) 36. Ruijsenaars, S.N.M.: Complete integrability of relativistic Calogero-Moser systems and elliptic functions identities. Commun. Math. Phys. 110, 191–213 (1987) 37. Chalykh, O.: Macdonald polynomials and algebraic integrability. Adv. Math. 166, 193–259 (2002) 38. Sergeev, A.N., Veselov, A.P.: Generalised discriminants, deformed quantum Calogero-Moser system and Jack polynomials. math-ph/ 0307036 Communicated by L. Takhtajan
Commun. Math. Phys. 245, 279–296 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1013-3
Communications in
Mathematical Physics
The Chiral Space of Local Operators in SU (2)-Invariant Thirring Model Atsushi Nakayashiki Faculty of Mathematics, Kyushu University, Ropponmatsu 4-2-1, Fukuoka 810-8560, Japan. E-mail: [email protected] Received: 24 March 2003 / Accepted: 21 August 2003 Published online: 23 December 2003 – © Springer-Verlag 2003
Abstract: The space of local operators in the SU (2) invariant Thirring model (SU (2) ITM) is studied by the form factor bootstrap method. By constructing sets of form factors explicitly we define a susbspace of operators which has the same character as the 2 . This makes a correspondence level one integrable highest weight representation of sl between this subspace and the chiral space of local operators in the underlying conformal field theory, the su(2) Wess-Zumino-Witten model at level one.
1. Introduction The SU (2) invariant Thirring model (SU (2) ITM) is a massive integrable model in two dimensional quantum field theory which can be considered as a deformation of the level one su(2) Wess-Zumino-Witten (WZW) model in conformal field theory. It is expected that there is a one to one correspondence between local operators in a CFT model and those in a massive deformation of it [20]. In particular there should be a subspace of operators in SU (2) ITM which is isomorphic to the level one integrable highest weight 2 , the chiral space of the level one su(2) WZW model. In this paper representation of sl we single out such a space by constructing sets of form factors explicitly. Let V = Cv+ ⊕ Cv− be the two dimensional vector space on which sl2 acts. The S-matrix of SU (2) ITM is the operator S(β) : V ⊗2 −→ V ⊗2 , the explicit form of which ⊗2n -valued meromorphic is given in (6). The set of form factors (F2n )∞ n=0 , F2n being a V function, of a local operator in SU (2) ITM satisfies the following system of equations: Pi,i+1 Si,i+1 (βi − βi+1 )F2n (β1 , · · · , β2n ) = F2n (· · · , βi+1 , βi , · · · ), P2n−1,2n P2n−1,2n−2 · · · P1,2 F2n (β1 − 2πi, · · · , β2n ) = (−1)n F2n (β2 , · · · , β2n , β1 ),
(1) (2)
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2π ires β2n =β2n−1 +πi F2n (β1 , · · · , β2n )
= I − (−1)n−1 S2n−1,2n−2 (β2n−1 − β2n−2 ) · · · S2n−1,1 (β2n−1 − β1 ) × F2n−2 (β1 , · · · , β2n−2 ) ⊗ e0 ,
(3)
where e0 = v+ ⊗ v− − v− ⊗ v+ , P : V ⊗2 −→ V ⊗2 is the permutation operator and Sij (β) acts on the i th and j th components of V ⊗2n as S(β), etc. Conversely any solution (F2n )∞ n=0 of (1), (2), (3) determines a local operator such that its 2n-particle form factor is F2n [17]. Thus the space of local operators can be identified with the space of solutions of (1), (2), (3). The complete description of the solution space of (1) and (2) is known (see [11] and references therein). Let Rn be the ring of symmetric polynomials of x1 , · · · , xn with xj = exp(βj ) and Cn the field of symmetric, 2πi-periodic and meromorphic functions of β1 , · · · , βn . For ≤ n consider a polynomial P = P (X1 , · · · , X |x1 , · · · , x2n ) such that it is anti-symmetric in Xa ’s with the coefficients in R2n satisfying degXa P ≤ 2n−1 for any a. To each P a solution P of (1), (2) is constructed through multi-dimensional q-hypergeometric integrals. It takes the value in the space of sl2 highest weight vectors with the weight 2n − 2. Since S(β) and P (the permutation operator) commute with sl2 , other solutions are created from highest weight solutions by the actions of sl2 and C2n . It is convenient to describe the space of polynomials as above by some exterior space. (2n) 2n−1 j j (2n) Let H (2n) = ⊕2n−1 j =0 R2n X and HC2n = ⊕j =0 C2n X . Then ∧ HC2n can be identified with the space of polynomials satisfying the conditions above and ∧ H (2n) becomes a subspace of it. For each n ≥ 0 we classify solutions of (1)-(3) into the solutions of the form (0, · · · , 0, F2n , F2n+2 , · · · ). Then Eq. (3) implies resβ2n =β2n−1 +πi F2n (β1 , · · · , β2n ) = 0.
(4)
If we write F2n = P , Eq. (4) should be rewritten as equations for a polynomial P . In fact the following equations imply (4) P |(x2n−1 ,x2n )=(x,−x),X =±x −1 = 0.
(5)
The converse is conjectured to be true. Let U2n, = {P ∈ ∧ H (2n) | P is a solution to (5)}. In [11] U2n, is proved to be a free R2n -module and its basis has been constructed explicitly. To obtain the solution space of (1), (2), (4) one has to consider U2n, modulo polynomials P such that P vanishes identically. Such polynomials have been determined (2n) (2n) completely [19]. Namely there are polynomials 1 ∈ U2n,1 and 2 ∈ U2n,2 such (2n) (2n) (2n) (2n) that P is identically zero if and only if P ∈ 1 ∧−1 HC2n + 2 ∧−2 HC2n . We define U2n, , M2n, = (2n) (2n) 1 ∧ U2n,−1 + 2 ∧ U2n,−2 (0)
(0)
and, for λ ≥ 0, Mλ = ⊕2n−2=λ M2n, . Then Mλ becomes isomorphic to the space of sl2 highest weight vectors in the level one integrable highest weight representation 2 [11]. V (0 ) of sl
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In this paper, to each P2m ∈ U2m, , we construct polynomials P2n , n > m such that (P2n )∞ n=0 satisfies (3) (Theorem 3), where we set P2n = 0 for n < m . This set of functions specifies a local operator which is a highest weight vector of sl2 . The operators corresponding to non-highest weight vectors are obtained from the ones above by the action of sl2 . Let Vλ be the finite dimensional irreducible representation of sl2 with the (0) highest weight λ. Then ⊕λ Mλ ⊗ Vλ specifies the subspace of local operators in SU (2) ITM which is isomorphic to V (0 ). Finally we briefly remark on the previous results on the determination of the space of local operators in massive integrable models. For several models with diagonal S-matrices the characters of the space of initial form factors, which are the first non-vanishing form factors, have been calculated [6]. They are shown to take fermionic forms [5] of the corresponding CFT characters. For the Sine-Gordon model and SU(2) ITM, which are two of the simple models with non-diagonal S-matrices, important results toward determining all local operators have been obtained in [18] and [9]. However the construction of 2n-particle form factors for every n to each element of the underlying CFT characters has never been done even for models with diagonal S-matrices other than the Ising model [2, 3] and the model considered in [1]. We solve this problem for SU(2) ITM under certain assumptions. The plan of the present paper is as follows. In Sect. 2 we review the results of [11], in particular, the description of the basis of U2n, . The formulae for polynomials P2n , n ≥ m and consequently for form factors are given in Sect. 3. In Sect. 4 some theorems on symmetric polynomials which is used in Sect. 3 are proved. They are interesting by itself. In Appendix A the definitions and properties of the function ζ (β) which appeared in the formulae of form factors is briefly reviewed for the sake of the readers’ convenience. 2. Space of Minimal Form Factors In this section we recall the results of [11]. Let V = Cv+ ⊕ Cv− be the vector representation of sl2 = Ce ⊕ Ch ⊕ Cf with the correspondence v+ = t (1, 0), v− = t (0, 1) and P : V ⊗2 −→ V ⊗2 the permutation operator P (v 1 ⊗ v 2 ) = v 2 ⊗ v 1 . The S-matrix of the SU (2) invariant Thirring model is the linear operator in End(V ⊗2 ) given as β − πiP ˆ , S(β) = β − πi
ˆ S(β) = S0 (β)S(β),
S0 (β) =
−β
( πi+β 2πi ) ( 2πi ) β
( πi−β 2πi ) ( 2πi )
.
(6)
For a linear operator A in End(V ⊗2 ) we define Aij ∈ End(V ⊗n ) as follows. Write A= Ba ⊗ Ca , Ba , Ca ∈ End(V ). a
Then Aij =
1 ⊗ · · · ⊗ Ba ⊗ · · · ⊗ Ca ⊗ · · · ⊗ 1,
a
where Ba and Ca are in ith and jth components respectively. Consider the following system of equations for a V ⊗n -valued function F : Pi,i+1 Si,i+1 (βi − βi+1 )F (β1 , · · · , βn ) = F (· · · , βi+1 , βi , · · · ), n 2
(7)
Pn−1,n Pn−1,n−2 · · · P1,2 F (β1 − 2πi, · · · , βn ) = (−1) F (β2 , · · · , βn , β1 ). (8)
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All meromorphic solutions of those equations are given by the multi-dimensional integrals of q-hypergeometric type in the following manner. Notice that S(β) commutes with the action of sl2 . Therefore any solution can be obtained, by the action of sl2 , from the solutions taking values in the space of highest sing weight vectors (V ⊗n )λ : (V ⊗n )λ
= {v ∈ V ⊗n | ev = 0, hv = λv}.
sing
We remark that (V ⊗n )λ = {0} if and only if λ is written as λ = n − 2 for some 0 ≤ ≤ n/2. For a subset M ⊂ {1, 2, ..., n} we set sing
vM = v 1 ⊗ · · · ⊗ v n ∈ V ⊗n ,
M = {j | j = − }.
Let Cn be the field of symmetric, 2πi-periodic and meromorphic functions of β1 , ..., βn and (n)
k H (n) := ⊕n−1 k=0 Rn X ,
k HCn := ⊕n−1 k=0 Cn X
the space of polynomials of X of degree at most n − 1 with the coefficients in Rn and (n) Cn respectively. We identify the th exterior product space ∧ HCn with the space of anti-symmetric polynomials of X1 , ..., X with the coefficients in Cn by Xi1 ∧ · · · ∧ X i → Asym(X1i1 · · · Xi ) = sgn σ Xσi1(1) · · · Xσi() . σ ∈S
We use the variables Xa , xj and αa , βj which are related by Xa = exp(−αa ), xj = exp(βj ). (n) Now for each P ∈ ∧ HCn define the V ⊗n -valued function ψP by ψP =
IM (P )vM ,
M=
where IM (P ) :=
C a=1
dαa
P (X1 , · · · , X |x1 , · · · , xn ) , φn (αa )wM n a=1 j =1 (1 − Xa xj ) a=1
(9)
where, wM = Asym(gM ) and, for M = (m1 , · · · , m ) with m1 < · · · < m , φn (α) =
α−βj +πi n
( −2πi ) j =1
α−β
( −2πij )
,
gM =
a=1
×
m a −1 αa − β j + π i 1 αa − βma αa − β j j =1 (αa − αb + π i).
1≤a
The integration contour C is defined as follows. It goes from −∞ to +∞, separating two sets {βj − 2π ik | 1 ≤ j ≤ n, k ≥ 0} and {βj + (2k − 1)π i |1 ≤ j ≤ n, k ≥ 0}.
Chiral Space of Local Operators in SU (2)-Invariant Thirring Model
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We set P = exp
n n
4
j =1
βj
ζ (βj − βk )ψP ,
(10)
j
where ζ (β) is a certain meromorphic function described in Appendix A. Let Solλn be the space of meromorphic solutions of (7) and (8) taking values in sing n (V ⊗n )λ . Then P ∈ Soln−2 . Remark. The integral IM (P ) is well defined for any polynomial P (X1 , ..., X |x1 , ..., xn ) such that degXa P ≤ n for any a. If P is symmetric in xj ’s, the function P becomes a solution of (7) and (8). Due to the anti-symmetry of wM the relation !P = Asym(P ) holds. The highest degree Xan can be eliminated by Ker below. Thus for any P we (n) have P = P for some P ∈ ∧ HCn . We later use this property in the description of polynomials corresponding to form factors. We have the linear map: (n)
n : ∧ HCn −→ Soln−2 ,
P → P . (n)
This map is surjective. The kernel of is described as follows. Let ± (X) = Xxj ) and (n)
(n)
n
j =1 (1±
(n)
21 (X) = + (X) + (−1)n−1 − (X), X − X 1 2 (n) (n) (n) (n) (n) 22 (X1 , X2 ) = + (X1 )+ (X2 ) − − (X1 )− (X2 ) X1 + X 2 (n) (n) (n) (n) +(−1)n + (X1 )− (X2 ) − + (X2 )− (X1 ) . Then (n)
(n)
(n)
(n)
Ker = ∧−1 HCn ∧ 1 + ∧−2 HCn ∧ 2 . Consider the restriction of to ∧ H (n) . Then the function P has at most a simple pole at βn = βn−1 + πi as a function of βn . In [11] the subspace of ∧ H (n) such that the function P is holomorphic at βn = βn−1 + πi has been studied. Let us recall the results. For a polynomial P (X1 , ..., X |x1 , · · · , xn ) in Xa ’s with the coefficients in Laurent polynomials of xj ’s define ρ± (P ) = P (X1 , · · · , X−1 , ±x −1 |x1 , · · · , xn−2 , x, −x), and set Un, = { P ∈ ∧ H (n) | ρ+ (P ) = ρ− (P ) = 0 }. (n)
(n)
Then 1 ∈ Un,1 , 2 βn−1 + π i. Let
∈ Un,2 and if P is in Un, then P becomes regular at βn =
Mn, =
Un, (n) Un,−1 ∧ 1
and, for i = 0, 1, λ ∈ Z≥0 , (i)
(n)
+ Un,−2 ∧ 2
Mλ = ⊕n≡i mod. 2, n−2=λ Mn, .
,
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Remark. We conjecture the following equation: (n) (n) (n) (n) (n) (n) Un, ∩ ∧−1 HCn ∧ 1 + ∧−2 HCn ∧ 2 = Un,−1 ∧ 1 + Un,−2 ∧ 2 . (11) For n = 1, 2 (11) holds. If (11) holds, then the space Mn, becomes isomorphic to n the subspace of Soln−2 . We define a degree of an element in Un, , denoted by deg1 , assigning deg1 Xa = −1, deg1 xj = 1. Then we set deg2 P = deg1 P + n2 /4 for a homogeneous element P in (n) (n) Un, . Then deg1 1 = deg1 2 = 0. We introduce a grading on Mn, by deg2 . In general for a graded vector space V = ⊕n Vn we define its character by ch V = q n dim Vn . n
2 , V (i ) the integrable highest weight repLet i be the fundamental weights of sl 2 with the highest weight i , V (i |λ), λ ∈ Z≥0 , the subspace of V (i ) resentation of sl consisting of sl2 -highest weight vectors with the weight λ and d the scaling element of 2 [7]. sl Theorem 1 ([11]). (i) ch Mλ
= trV (i |λ) (q
−d+i/4
n2
)=
n−2=λ, n≡i mod. 2
q4 [n]q !
n n , − −1 q q (12)
where [n]q = 1 − q n ,
[n]q ! =
n
[j ]q ,
j =1
[n]q ! n = . q []q ![n − ]q !
The Rn -module Un, becomes a free module and a basis is given explicitly. Since we consider the case of n even in this paper, we recall the basis only in the case of n even here. (n) Let ek be the elementary symmetric polynomial defined by n
(1 + xj t) =
j =1
n
(n)
ek t k .
k=0
(2n)
The symmetric polynomial Pr,s , 1 ≤ r ≤ n, s ∈ Z is defined by the following recursion relations: (2n)
(2n)
P1,s = e2s−1 , (2n)
(2n)
(2n)
(2n) Pr,s = Pr−1,s+1 − e2s Pr−1,1
for r ≥ 2.
For a function f (x1 , · · · , xn ) of xj ’s we denote f¯ the function obtained from f by specializing the last two variables to x, −x: f¯ = f (x1 , · · · , xn−2 , x, −x),
x = xn−1 ,
Chiral Space of Local Operators in SU (2)-Invariant Thirring Model
285
(2n)
if it has a sense. Then Pr,s ’s satisfy (2n)
Pr,s
(2n−2)
(2n−2) = Pr,s − x 2 Pr,s−1 .
(13)
We set (2n)
v0
=
n
(2n)
ej
X 2j ,
vr(2n) =
j =0
n
wr(2n) = Xvr(2n) ,
(2n) 2(s−1) Pr,s X ,
1 ≤ r ≤ n,
s=1
(14) and, for 1 ≤ j ≤ n, (2n)
2ξj
(X1 , X2 ) =
X1 − X2 (2n) (2n) (2n) (2n) v0 (X1 )wj (X2 ) + v0 (X2 )wj (X1 ) X1 + X 2 (2n)
−v0
(2n)
(2n)
(X1 )wj
(2n)
(X2 ) + v0
(2n)
(X2 )wj
(2n)
(15)
(X1 ).
(2n)
Equation (13) implies that vr , wr ∈ U2n,1 , 1 ≤ r ≤ n and ξj ∈ U2n,2 . We use the multi-index notations like vI = vi1 ∧ · · · ∧ vi for I = (i1 , · · · , i ). Then, in general, we have the following theorem. Theorem 2. As an R2n -module, U2n, is a free module with the following set of elements as a basis: (2n)
(2n)
(2n)
vI ∧ wJ ∧ ξK , I = (i1 , · · · , i1 ), 1 ≤ i1 < · · · < i1 ≤ n, J = (j1 , · · · , j2 ), 1 ≤ j1 < · · · < j2 ≤ n − 1 − 3 , K = (k1 , · · · , k3 ), 1 ≤ k1 ≤ · · · ≤ k3 ≤ n − 1 − 3 + 1, 1 + 2 + 23 = . 3. Chiral Subspace Let m and r be non-negative integers such that 0 ≤ r ≤ m. We fix them once and for all in this section. We set λ = 2m − 2r and 2n = r + n − m. Define P2n = 0 for n < m. To each P2m ∈ U2m,r we shall construct polynomials P2n (X1 , ..., X2n |x1 , ..., x2n ), n ≥ m such that (P2n )∞ n=0 satisfy (1), (2), (3). (n) i Let us first discuss the symmetry of Eq. (1), (2), (3). Let pi = ∞ j =1 xj and pi = n i ∞ j =1 xj for i = 0. Suppose that (F2n )n=0 is a solution of (1), (2), (3). Then, for any s ∈ Z, (p2s−1 F2n )∞ n=0 is again a solution of the same equations, since p2s−1 = p2s−1 . Thus the space of solutions of (1), (2), (3) becomes a module over the ring Z[p±1 , p±3 , · · · ]. The action of p2s−1 is that of the local integral of motion with spin 2s − 1. We introduce (n) the generating function of p2s−1 : (n) (n) t2j −1 p2j −1 , Eodd (t) = exp (2n)
(2n)
j ∈Z
which satisfies the equation (n)
(n−2)
Eodd (t) = Eodd (t).
(2n−2)
286
A. Nakayashiki
Next we consider the functions 2n (±) Q2n (z)
∓2 2 a=r+1 (1 − Xa z ) , 2n ±1 j =1 (1 − xj z)
=
for n ≥ m. For n = m the empty product in the numerator is understood as 1, that is, 1
(±)
Q2m (z) = 2m
±1 j =1 (1 − xj z)
(2m)
where hj tion
(2m)+
:= hj
=
∞
(2m)± j
hj
z ,
j =0
is the complete symmetric function [10]. They satisfy the equa ( ) ( ) ρ± Q2n (z) = Q2n−2 (z),
(16)
for = ±. These functions were extracted from the integral formulae for the form factors of local operators defined by Lukyanov [13, 12, 9]. Take I, J, K such that they satisfy conditions in Theorem 2 with 1 + 2 + 23 = r. We set (2m)
P2m = Eodd (t)
m
(+)
(2m)
Q2m (zi )vI
(2m)
∧ wJ
(2m)
∧ ξK
(17)
,
i=1
P2n =
(2n) c2n Eodd (t)
m
(+) (2n) Q2n (zi )vI
(2n) ∧ wJ
(2n) ∧ ξK
2n
Xa2n+1+2r−2a , for n > m,
a=r+1
i=1
(18) where the constant c2n is given by 1
c2n = (−1) 2 (n−m)(n+m−2r−1) (iζ (−πi))m−n (2π i)(m−n)(n+r) . (2n)
In (17), (18), vI expand P2n as
(2n)
∧ wJ
(2n)
∧ ξK
(19)
is understood as the polynomial of X1 , · · · , Xr . We
P2n =
P2n,α,γ t α zγ ,
α,γ
where α = (..., α−1 , α1 , α3 , ...), γ = (γ1 , ..., γm ) and tα =
tiαi , zγ =
γ
zi i .
Theorem 3. (1) The set of functions (P2n )∞ n=0 satisfies (1), (2), (3) and each P2n takes sing the value in (V ⊗2n )λ , λ = 2m − 2r. (2) The following property holds for all n: P2n,α,γ (β1 + θ, · · · , β2n + θ ) = exp(θ deg2 P2m,α,γ )P2n,α,γ (β1 , · · · , β2n ). The property (2) in Theorem 3 was proved in [11, 13]. It means that the Lorentz spin of the local operator specified by the set of form factors (P2n )∞ n=0 is deg2 P2m,α,γ . Explicitly deg2 P2m,α,γ is given by
Chiral Space of Local Operators in SU (2)-Invariant Thirring Model
deg2 P2m,α,γ = m2 +
i αi +
287 (2m)
γi + deg1 (vI
(2m)
∧ wJ (2m)
(2m)
As shown in Theorem 5 in the next section R2m is a free C[p1 , p3 module with the nth ordered products of h2j ’s as a basis, that is, (2m)
R2m = ⊕0≤r1 ≤···≤rm C[p1
(2m)
, p3
(2m)
(2m)
∧ ξK
(2m)
). (2m)
, · · · , p2m−1 ]
(2m)
, · · · , p2m−1 ]h2r1 · · · h2rm .
Thus any element of U2m,r can be expressed as a linear combination of P2m,α,γ ’s, that is, U2m,r = CP2m,α,γ , α,γ
where the summation is taken for all γ and α with αi = 0, i < 0 or i > 2m. sing Let Oλ (2m) be the space of meromorphic solutions to (1), (2), (3) such that F2n = sing sing 0, n < m and F2n ∈ Solλ2n for all n. We set Oλ = Oλ (0). Then we have sing
Oλ
sing
= Oλ
sing
(0) ⊃ Oλ
sing
(2) ⊃ Oλ
(4) ⊃ · · · .
For each m there is a map sing
: U2m,r −→ O2m−2r (2m),
P2m,α,γ → (P2n,α,γ )∞ n=0 ,
where α is such that αi = 0 for i < 0 or i > 2m. It induces the map sing
: M2m,r −→ O2m−2r (2m). Finally we obtain the map sing
(0)
: Mλ = ⊕λ=2m−2r M2m,r −→ Oλ
.
(20)
Let O be the space of meromorphic solutions of (1), (2), (3) and Vλ Cλ+1 the (0) irreducible resresentation of sl2 with the highest weight λ. We consider Mλ as the trivial sl2 -module. Then (20) induces the map of sl2 -modules (0)
⊕λ Mλ ⊗ Vλ −→ O.
(21)
We call the image of (21) the chiral subspace of the space of local operators in SU(2) ITM. If we assume the conjecture (11), the map (21) is injective and the chiral subspace is isomorphic to V (0 ) due to Theorem 1. Now let us prove Theorem 3. The following theorem has been proved in [13]. Theorem 4 ([13]). Let P2n (X1 , ..., X2n |x1 , ..., x2n ), n ≥ m be polynomials in X1 , ..., X2n with the coefficients in the ring of symmetric Laurent polynomials of xj ’s such that degXa P ≤ 2n for 1 ≤ a ≤ 2n . Set P2n = 0 for n < m. Suppose that (P2n )∞ n=0 satisfy ) such that the following conditions: there exists a set of polynomials (P2n −1 2n Asym P2n = Asym (1 − x 2 Xa2 )P2n , P2n |X
d2n =
2n =±x
−1
= ±x
a=1 −(2n−1)
d2n P2n−2 ,
2π (−2πi)−r+m−2n , ζ (−πi)
(22) (23) (24)
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A. Nakayashiki
where Asym is the anti-symmetrization with respect to Xr+1 , ..., X2n . Then (P2n )∞ n=0 satisfies (1), (2), (3). Let us show that (17), (18) satisfy (22) and (23). Lemma 1. For 0 ≤ i ≤ n, 1 ≤ j ≤ n we have (2n)
vi (2n)
ξj
(2n−2)
(X) = (1 − x 2 X 2 )vi 2
(X1 , X2 ) =
(2n)
(X),
(2n−2)
(1 − x 2 Xa2 )ξj
wj
(2n−2)
(X) = (1 − x 2 X 2 )wj
(X),
(X1 , X2 ).
a=1 (2n)
Proof. Using (13) we easily have the results for vi (2n) for v0 is similarly verified using (2n)
ek (2n)
The equation for ξj
(2n−2)
= ek
(2n)
and wi
(2n−2)
− x 2 ek−2 (2n)
follows from those of vi
for i = 0. The equation
.
(2n)
and wi
.
(2n−2)
∧ wJ
By this lemma we have (2n)
vI
(2n)
∧ wJ
(2n)
∧ ξK
=
r
(1 − x 2 Xa2 ) vI
(2n−2)
(2n−2)
∧ ξK
.
(25)
a=1
The following lemma is proved in [13]. Lemma 2. Let Asym denote the anti-symmetrization with respect to Xr+1 , ..., X2n . We have 2n
Xa2n+1+2r−2a Asym a=r+1 2n −1
= (−1)2n −r−1 Asym X2n−1 (1 − x 2 Xa2 )Xa2n−1+2r−2a . 2n
(26)
a=r+1
It follows from (25) that P2n =
r
(1 − x 2 Xa2 )P2n ,
a=1 P2n = c2n Eodd
(2n−2)
(t)
m
(+)
(2n−2)
Q2n (zi )vI
(2n−2)
∧ wJ
2n
(2n−2)
∧ ξK
Xa2n+1+2r−2a .
a=r+1
i=1
m
(2n−2)
Notice that i=1 Q2n (zi ) is symmetric with respect to Xr+1 , ..., X2n and vI (2n−2) (2n−2) wJ ∧ ξK does not contain Xr+1 , ..., X2n . Then using Lemma 2 we have Asym(P2n ) =
r
(1 − x 2 Xa2 )Asym(P2n )
a=1 −1
2n
= Asym
a=1
, (1 − x 2 Xa2 )P2n
∧
Chiral Space of Local Operators in SU (2)-Invariant Thirring Model
289
where P2n
= (−1)
2n −r−1
(2n−2) c2n Eodd (t)
m
(+)
(2n−2)
Q2n (zi )vI
(2n−2)
∧ wJ
(2n−2)
∧ ξK
i=1
×X2n−1 2n
2n −1
Xa2n−1+2r−2a .
a=r+1
Due to (16) we have |X P2n
2n
=±x −1
(2n−2)
= ±(−1)n−m−1 c2n Eodd
(t)
m
(+)
(2n−2)
Q2n−2 (zi )vI
(2n−2)
∧wJ
(2n−2)
∧ξK
i=1 2(n−1)
× x −(2n−1)
Xa2n−1+2r−2a
a=r+1
= ±(−1)
n−m−1
c2n c2n−1
x −(2n−1) P2n−2 .
Finally the recursion relation c2n = (−1)n−m−1 d2n , c2n−1
c2m = 1,
uniquely determines c2n , n ≥ m and the explicit form of which is given by (19).
4. Some Properties of Symmetric Functions (n)
(n)
(n)
In this section we omit (n) of ej , pj , hj if there is no fear of confusion on the number of the variables. Let Rnodd = C[p2j +1 |0 ≤ j ≤ (n − 1)/2] ⊂ Rn and n = [n/2] be the largest integer not exceeding n/2. Theorem 5. The module Rn is a free Rnodd -module with the elements {h2i1 · · · h2in |0 ≤ i1 ≤ · · · ≤ in } as a basis: Rn = ⊕0≤i1 ≤···≤in Rnodd h2i1 · · · h2in . For the proof of this theorem we need to prepare some propositions and lemmas. We denote Rn the subring of Rn generated by ej with j odd, j ≤ n: n − 1 . Rn = C e2j +1 |0 ≤ j ≤ 2 Proposition 1. If Rn is generated by {hi1 · · · hin |i1 , · · · , in ≥ 0} over Rn , it is generated by the same set over Rnodd . To prove this proposition we notice the following lemma. Lemma 3. In Rn the following conditions are equivalent: (1) e2j +1 = 0 for 0 ≤ j ≤ (n − 1)/2. (2) p2j +1 = 0 for 0 ≤ j ≤ (n − 1)/2.
290
A. Nakayashiki
Proof. By the formula (cf. [10]) kek =
k
(−1)r−1 pr ek−r ,
(27)
r=1
each ek is expressed as a homogeneous polynomial of p1 ,...,pk and vice versa where the degrees are defined by deg pk = k, deg ek = k. Suppose that ej = 0 for all odd j . Then, for odd k, if pk is written as a polynomial of ei ’s, each monomial contains at least one el with l being odd and pk = 0. Thus (1)⇒(2) is proved. The converse is similarly proved. Proof of Proposition 1. To avoid notational complexity we give a proof in the case of n even and use 2n instead of n. The case of n odd is similarly proved. rn Let Er1 ···rn = e2r1 e4r2 · · · e2n . Since R2n = C[e1 , e3 , · · · , e2n−1 ], R2n is a free R2n -module with the elements {Er1 ···rn |rj ∈ Z≥0 , ∀j } as a basis. By the assumption one can write ···sn Er1 ···rn = frs11···r (e1 , e3 , · · · , e2n−1 )hs1 · · · hsn , n s1 ,...,sn ≥0
···sn for some frs11···r n ∈ R2n . Set
Fr1 ···rn =
s1 ,...,sn ≥0
···sn frs11···r (p1 , p3 , · · · , p2n−1 )hs1 · · · hsn . n
Lemma 4. The set of polynomials {Fr1 ···rn |rj ∈ Z≥0 ∀j } is linearly independent over odd . R2n Proof. It is sufficient to prove that {Fr1 ···rn } is linearly independent over C at p1 = p3 = · · · = p2n−1 = 0. By Lemma 3, ···sn Fr1 ···rn |p2j −1 =0,j ≤n = frs11···r (0, · · · , 0)(hs1 · · · hsn )|p2j −1 =0,j ≤n n s1 ,...,sn ≥0
= Er1 ···rn |p2j −1 =0,j ≤n . Thus we have to prove that E˜ r1 ···rn := Er1 ···rn |p2j −1 =0,j ≤n are linearly independent. Lemma 5. At e2j −1 = 0, 1 ≤ j ≤ n we have e2k = −
1 p2k + · · · , 2k
1 ≤ k ≤ n,
where · · · part is a polynomial of p2 , p4 , ..., p2k−2 . Proof. The lemma easily follows from (27).
On the set {(r1 , · · · , rn )} of non-negative integers we introduce the lexicographical order from the right. Suppose that cr1 ···rn E˜ r1 ···rn = 0,
Chiral Space of Local Operators in SU (2)-Invariant Thirring Model
291
and (r10 , · · · , rn0 ) is the largest index such that cr1 ···rn = 0. If one expresses each E˜ r1 ···rn as a polynomial of p2 , p4 , ..., p2n , one has, by Lemma 5, r0
r0
cr 0 ···r 0 p21 · · · p2nn + 1
rn cr 1 ···rn p2r1 · · · p2n = 0,
n
(r1 ,··· ,rn )<(r10 ,··· ,rn0 )
for some constants cr 1 ···rn . Thus cr 0 ···r 0 = 0 which contradicts the assumption. Therefore n 1 Lemma 4 is proved. odd E Since R2n = ⊕R2n r1 ···rn , deg Fr1 ···rn = deg Er1 ···rn and ch R2n = ch R2n , we have odd Fr1 ···rn = ch R2n . ch ⊕ R2n odd F Thus R2n = ⊕R2n r1 ···rn which proves Proposition 1.
Proposition 2. The module Rn is generated by {hi1 · · · hin |ij ∈ Z≥0 ∀j } over Rn . Proof. As in the previous proposition we give a proof for n being even. The proof of the case n odd is similar. In this proof we again use 2n instead of n. Then (2n) = n. For a partition λ = (λ1 , · · · , λm ) we denote sλ the Schur function and (λ) the length of λ [10]. . Due to the Let L2n be the submodule of R2n generated by {sλ |(λ) ≤ n} over R2n formula sλ = det(hλi −i+j )1≤i,j ≤(λ) , it is sufficient to prove R2n = L2n . Since {sλ |(λ) ≤ 2n} is a C-linear basis of R2n , we have to prove sλ ∈ L2n for any partition λ = (λ1 , · · · , λ2n ). We prove this by induction on the degree |λ| = λ1 + · · · + λ2n of sλ . If |λ| ≤ n for λ = (λ1 , · · · , λ2n ), λj = 0 for j ≥ n + 1 and sλ ∈ L2n . Let d > n. Suppose that sµ ∈ L2n for any partition µ = (µ1 , · · · , µ2n ) with |µ| < d. We prove sλ ∈ L2n for any λ = (λ1 , · · · , λ2n ) with |λ| = d. To this end we introduce the lexicographical order from the left on the set of partitions {(µ1 , · · · , µ2n )}. We prove sλ ∈ L2n by descending induction on this order of λ. For λ = (d) we have sλ ∈ L2n since (λ) = 1 ≤ n. Let λ be such that λ < (d). Suppose that sµ ∈ L2n for any µ > λ with |µ| = d. We assume that λ is of the form λ = (λ1 , · · · , λm ), λm = 0. Since sλ ∈ L2n for m ≤ n, we assume that n + 1 ≤ m ≤ 2n. ˜ = |λ| − 1 and First consider the case λm = 1. Let λ˜ = (λ1 , · · · , λm−1 ). Then |λ| sλ˜ ∈ L2n by the assumption of induction on |λ|. Recall that s(1k ) = ek . Then sλ˜ s(1) ∈ L2n . By Pieri’s formula (cf. [10]) c µ sµ , (28) sλ˜ s(1) = sλ + µ>λ
for some constants cµ . By the hypothesis of induction on the order of λ the second term of the right-hand side of (28) is in L2n . Thus sλ ∈ L2n . Next consider the case m = 2n. We set λ˜ = (λ1 , · · · , λ2n−1 , λ2n − 1). Then λ˜ is a partition and sλ˜ ∈ L2n by the induction hypothesis on |λ|. Then sλ˜ s(1) ∈ L2n and Eq. (28) holds. Thus sλ ∈ L2n . Let us assume λm ≥ 2 and m < 2n.
292
A. Nakayashiki
Lemma 6. Let m ≤ m ≤ 2n and ij ∈ Z≥0 , m + m − 2n + 1 ≤ j ≤ m be such that ij = 2n − m and
λ(m ) := (λ1 , · · · , λm+m −2n , λm+m −2n+1 − im+m −2n+1 , · · · , λm − im , 12n−m ) is a partition. Then sλ(m ) ∈ L2n . Proof. In this proof we use the following notations. Let i = (0, · · · , 1, · · · , 0) where 1 is on the i-th position. For N ≥ 1 and a = (ai ), b = (bi ) ∈ ZN we define a + b = (ai + bi ). For N1 ≤ N2 , a = (ai ) ∈ ZN1 is considered as an element of ZN2 by setting a = (ai ), ai = 0 for N1 < i. We prove the lemma by induction on m . Suppose that m = m. Set λ˜ (m) := (λ1 , · · · , λ2m−2n−1 , λ2m−2n − 1, λ2m−2n+1 − i2m−2n+1 − 1, · · · , λm − im − 1). Since m < 2n and λ(m) is a partition, λm − im ≥ 1. Then λ˜ (m) becomes a partition and |λ˜ (m) | = |λ| − (2(2n − m) + 1) < |λ|. Thus sλ˜ (m) ∈ L2n by the assumption of induction on |λ|. Then sλ˜ (m) s(12(2n−m)+1 ) = sλ(m) + c µ sµ , µ>λ
for some constant cµ ’s. Thus sλ(m) ∈ L2n by induction on the order of λ. Assume that m < m ≤ 2n and that the lemma holds for any m satisfying m ≤ m < m . Let λ˜ (m ) = (λ1 , · · · , λm+m −2n−1 , λm+m −2n − 1, λm+m −2n+1 − im+m −2n+1 − 1, · · · , λm − im − 1).
Here λm − im ≥ 1. In fact if m < 2n then it is obvious that λm − im ≥ 1. If m = 2n then ij = 0 for all j and λm − im = λm ≥ 2 by assumption. Then λ˜ (m ) is a partition. Since |λ˜ (m ) | < |λ|, sλ˜ (m ) ∈ L2n by the hypothesis of induction on |λ|. Then sλ[k] ({jl }) + cµ s µ . sλ˜ (m ) s(12(2n−m )+1 ) = sλ(m ) + k≥1 {jl }
µ>λ
Here {jl } and λ[k] ({jl }) are as follows. For a given k, {jl } is a set of numbers satisfying m + m − 2n ≤ j1 < · · · < j2n−m −k+1 ≤ m. Then λ[k] ({jl }) is defined by
λ[k] ({jl }) = λ˜ (m ) +
jl + (0m , 12n−m +k ).
l
If λ[k] ({jl }) is not a partition we define sλ[k] ({jl }) = 0. If λ[k] ({jl }) is a partition and
λ[k] ({jl }) > 2n, then sλ[k] ({jl }) = 0. Consider k such that λ[k] ({jl }) is a partition and
λ[k] ({jl }) ≤ 2n. Since λ = λ˜ (m ) +
m p=m+m −2n
p +
m j =m+m −2n+1
ij j ,
Chiral Space of Local Operators in SU (2)-Invariant Thirring Model
293
we have m
λ[k] ({jl }) − λ = −
p +
−k+1 2n−m
p=m+m −2n
ij j
j =m+m −2n+1
l=1
2n−m +k
+(0 , 1 m
m
jl −
(29)
).
Let us write m
p −
−k+1 2n−m
p=m+m −2n m+m −2n−k
m
jl +
ij j l=1 j =m+m −2n+1 , im+m −k−2n+1 , · · · , im ) =: i .
= (0 Then ij ≥ 0 for any j , ij = 2n − (m − k) and
λ − i + (0m , 12n−m +k ) = λ[k] ({jl }) is a partition. Thus (ij ) satisfies the condition of (ij ) for m −k. Since k ≥ 1, sλ[k] ({jl }) ∈ L2n by the assumption of induction on m . Thus sλ(m ) ∈ L2n . Consider the case m = 2n in Lemma 6. Then λ(2m) = λ. Thus sλ ∈ L2n . This completes the proof of Proposition 2. Lemma 7. If e2j +1 = 0 for all j ≤ (n − 1)/2, h2r−1 = 0 for any r ≥ 1. Proof. The lemma is easily proved by induction on r using the relation m
(−1)i ei hm−i = 0,
m ≥ 1.
i=0
Proof of Theorem 5. Consider the specialization p2j +1 = 0 for all j ≤ (n − 1)/2. Then Rn = C[p2 , p4 , · · · , p2n ] and
ch Rn =
n j =1
1 . 1 − q 2j
(30)
By Lemma 3 we have e2j +1 = 0 for j ≤ (n − 1)/2. Then h2r−1 = 0 for any r ≥ 1 by Lemma 7. Thus Ch2r1 · · · h2rn , (31) Rn = 0≤r1 ≤···≤rn
by Proposition 1 and Proposition 2. For two formal power series f (q), g(q) of q with the coefficients in R we denote g(q) ≤ f (q) if all coefficients of f (q) − g(q) is non-negative. Then ch(RHS of (31)) ≤
0≤r1 ≤···≤rn
q
2(r1 +···+rn )
=
n j =1
1 = ch Rn . 1 − q 2j
(32)
294
A. Nakayashiki
Thus the inequality in (32) is in fact the equality. This means that {h2r1 · · · h2rn |0 ≤ r1 ≤ · · · ≤ rn } are linearly independent over C at p2j +1 = 0, j ≤ (n − 1)/2. Thus {h2r1 · · · h2rn |0 ≤ r1 ≤ · · · ≤ rn } are linearly independent over Rnodd . Then n ch ⊕0≤r1 ≤···≤rn Rnodd h2r1 · · · h2rn = j =1
This completes the proof.
1 = ch Rn . 1 − qj
5. Concluding Remarks In [18] Smirnov has given a systematic construction of form factors of a large number of chargeless local operators in the Sine-Gordon model. This construction is extended to the charged operators in SU(2) ITM in [13]. In these descriptions of local operators, however, finding the subspace which has the same character, with respect to spins, as the chiral subspace of local operators in the corresponding conformal field theory has not been successful up to now. In this paper we have given an alternative construction of local operators in SU(2) ITM based on the results of our previous paper [11] and the ideas of [18, 13]. With this description of local operators we have specified the subspace of operators isomorphic 2 which is the chiral space to the level one integrable highest weight representation of sl of the level one su(2) WZW model. Let us give a brief comment here on the differences of the formulae (17), (18) and those of Smirnov in [18]. In [18] the initial condition is taken as P2m = (e2m )−N f + 2m (2m)
r a=1
Xara ,
0 ≤ ra ≤ 2m − 1,
+ 2m =
(xi + xj ),
i<j
(2m) N for some N ≥ 0 and f ∈ R2m . Since + = 0, Asym ) P (e ∈ U2m, . X ,··· ,X 2m r 1 2m 2m The important point is that, in general, U2m,r is not generated by such functions only. (n) (n) (n) That is why we have introduced the polynomials vi , wj and ξk . This difference is crucial in the calculation of the character of the chiral subspace of local operators. Due to this difference of the initial condition the degrees of freedom corresponding to the parameter t2s , s ∈ Z in [18] could not be introduced in our formula. This drawback (±) is compensated by the introduction of the functions Q2n (z). In this respect the structure of symmetric polynomials given in Theorem 5 is important. A. Function ζ (β) The function ζ (β) was introduced in [8, 15, 17]. It is conveniently described using the double gamma function of Barnes [13, 4]. We review it here. We follow [14] about the notations of the double gamma function. For a set of positive real numbers ω = (ω1 , ω2 ) define z z2
2 (z|ω)−1 = z exp γ22 (ω) + γ21 (ω) 1 + 2 mω1 + nω2 2 z z × exp − , + 2 mω1 + nω2 2(mω1 + nω2 )
Chiral Space of Local Operators in SU (2)-Invariant Thirring Model
295
where the product is over all sets of non-negative integers except (m, n) = (0, 0) and γ22 (ω), γ21 (ω) are given by the formulae ∞ nω 1 nω2 1 ω1 1 1 2 + ψ − log + + log ω1 ω1 ω1 ω1 2nω2 2 ω1 ω2 n=1 1 ω 1 − ω2 ω2 γ 1 1 − (γ − log 2π) + log − + , (33) 2ω1 2ω1 ω2 ω1 2 ω1 ω2 ω1 π2 1 nω2 1 γ −γ21 (ω) = 2 ψ − + − log ω2 + . (34) 2 ω1 nω2 ω1 ω2 ω1 ω2 ω1 6ω1
−γ22 (ω) =
Here ψ is the logarithmic derivative of the gamma function and γ is the Euler’s constant. Then 2 (z|ω)−1 becomes an entire function of z, is symmetric with respect to ω1 and ω2 and satisfies the following difference equation: 1 z 1
2 (z + ω1 |ω) √ log ω2 + = 2π exp − . ω2 2
( ωz2 )
2 (z|ω)
(35)
Set 2 (z) = 2 (z|2π, 2π) and define ζ (β) =
2 (−iβ + π) 2 (iβ + 3π ) .
2 (−iβ) 2 (iβ + 2π )
(36)
Then it satisfies the following equations ζ (β − 2πi) = ζ (−β), ζ (β)ζ (β − πi) =
(37) (2π )3/2
iβ
( −iβ+π 2π ) ( 2π )
−β
( πi+β ζ (−β) 2πi ) ( 2πi ) = . β ζ (β)
( πi−β ) ( ) 2πi 2πi
,
(38) (39)
The right-hand side of (39) is S0 (β) in the S-matrix. References 1. Babelon, O., Bernard, D., Smirnov, F.: Null-vectors in integrable field theory. Commun. Math. Phys. 186, 601–648 (1997) 2. Cardy, J., Mussardo, G.: Form factors of descendent operators in perturbed conformal field theories. Nucl. Phys. B340, 387–402 (1990) 3. Christe, P.: Factorized characters and form factors of descendant operators in perturbed conformal systems. Int. J. Mod. Phys. A6, 5271–5286 (1991) 4. Jimbo, M., Miwa, T.: Quantum KZ equation with |q| = 1 and correlation functions of the XXZ model in the gapless regime. J. Phys. A: Math. Gen. 29, 2923–2958 (1996) 5. Kedem, R., McCoy, B., Melzer, E.: The sums of Rogers, Schur and Ramanujan and the BoseFermi correspondence in 1 + 1-dimensional quantum field theory. In: Recent Progress in Statistical Mechanics and Quantum Field Theory P. Bouwknegt, et al., (eds.), Singapore: World Scientific, 1995, pp. 195–219 6. Koubek, A.: The space of local operators in perturbed conformal field theories. Nucl. Phys. B. 435, 703–734 (1995) 7. Kac, V.: Infinite dimensional Lie algebras. Third edition. Cambridge: Cambridge University Press, 1992
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8. Kirillov, A.N., Smirnov, F.: A representation of the current algebra connected with the su(2)-invariant Thirring model. Phys. Lett. B 198, 506–510 (1987) 9. Lukyanov, S.: Free field representation for massive integrable models. Commun. Math. Phys. 167, 183–226 (1995) 10. Macdonald, I.G.: Symmetric functions and Hall polynomials. Second edition, Oxford: Oxford University Press, 1995 11. Nakayashiki, A.: Residues of q-hypergeometric integrals and characters of affine Lie algebras. Commun. Math. Phys. 240, 197–241 (2003) 12. Nakayashiki, A., Pakuliak, S., Tarasov, V.: On solutions of the KZ and qKZ equations at level zero. Ann. Inst. Henri Poincar´e 71, 459–496 (1999) 13. Nakayashiki, A., Takeyama, Y.: On form factors of SU(2) invariant Thirring model. In: MathPhys Odyssey 2001, Integrable Models and Beyond- in honor of Barry M. McCoy, Kashiwara, M. and Miwa, T., (eds.), Progr. Math. Phys., Basel-Boston: Birkh¨auser, 2002, pp. 357–390 14. Shintani, T.: On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo 24, 167–199 (1977) 15. Smirnov, F.: Lectures on integrable massive models of quantum field theory, In: Nankai Lectures on Mathem. Physics Ge, M-L., Zhao, B-H., (eds.), Singapore: World Scientific, 1990, pp. 1–68 16. Smirnov, F.: Dynamical symmetries of massive integrable models. Int. J. Mod. Phys. A 7 Suppl. 1B, 813–837 (1992) 17. Smirnov, F.: Form factors in completely integrable models of quantum field theories. Singapore: World Scientific, 1992 18. Smirnov, F.: Counting the local fields in SG theory. Nucl. Phys. B 453, 807–824 (1995) 19. Tarasov, V.: Completeness of the hypergeometric solutions of the qKZ equations at level zero. Am. Math. Soc. Translations Ser. 2 201, 309–321 (2000) 20. Zamolodchikov, A.B.: Integrable field theory from conformal field theory. Adv. Stud. Pure Math. 19, 641–674 (1989) Communicated by L. Takhtajan
Commun. Math. Phys. 245, 297–354 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1017-z
Communications in
Mathematical Physics
Dynamics Near an Unstable Kirchhoff Ellipse Yan Guo1 , Chris Hallstrom2 , Daniel Spirn3 1
Lefshetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: [email protected] 2 Department of Mathematics, University of Portland, Portland, OR 97203, USA. E-mail: [email protected] 3 Department of Mathematics, Brown University, 151 Thayer St., Providence, RI 02912, USA. E-mail: [email protected] Received: 14 April 2003 / Accepted: 18 September 2003 Published online: 20 January 2004 – © Springer-Verlag 2004
Abstract: We describe the dynamics of the Kirchhoff ellipse by formulating a nonlinear equation for the boundary of a perturbed vortex patch in elliptical coordinates. We demonstrate that in the regime for which the linearized equation of motion is unstable, the nonlinear dynamics of a rather general initial perturbation of the Kirchhoff ellipse are determined by the fastest growing mode for the corresponding linearized equation, on a time scale when the nonlinear instability occurs. In particular, we resolve a question suggested by Love’s results and prove that such elliptical patches are indeed unstable in the full nonlinear sense. 1. Introduction The Kirchhoff elliptical vortex patch (KVP) is one of the few exact solutions to the two-dimensional Euler equations. As such, it is important to understand the stability of pertubations of these solutions. The classical question of KVP stability dates back to the linear stability analysis of A.E. Love in 1893 [16]. By relating the perturbation of the boundary to a corresponding perturbation of the stream function, Love established that linear stability exists when the ratio of major to minor axes is not too large. Nearly a century later, Wan and Pulvirenti [23] used an energy method to address the nonlinear stability of circular patches, and subsequently, Wan [22] and Tang [21] adapted this technique to elliptical patches in the stable regime. The question of whether a linearly unstable KVP is unstable in the full nonlinear sense has remained open and is the subject of our paper.
1.1. Formulation and Equations of Motion. Consider the motion of an incompressible and inviscid fluid in two dimensions. The evolution of a divergence-free velocity field u = (u, v) can be described by the usual Euler equations. Equivalently, the flow can be described by the vorticity equation
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∂t ω + u∂x ω + v∂y ω = 0,
(1.1)
where ω = ∂x v − ∂y u. One important feature of this equation is that in contrast to the three dimensional case, vorticity is conserved along particle trajectories, a fact that has important consequences for the existence of solutions to the 2D Euler equations [3] (see also [17]). Another key aspect to this equation is that we can eliminate the velocity from the vorticity equation (1.1) by recovering the velocity from the vorticity. Because of the divergence-free condition, the velocity field u = (u, v) can be expressed in terms of a stream function ψ via the relation u = (−∂y ψ, ∂x ψ) ≡ ∇ ⊥ ψ. This stream function is unique up to an additive constant. In terms of the two-dimensional vorticity, the stream function satisfies the Poisson equation ψ = ω for which we can write down an explicit solution in terms of a convolution of ω. This then allows us to express the velocity in terms of this solution via the so-called Biot-Savart law 1 ⊥ ⊥ log |x − y|ω(y) dy. (1.2) ∇ u(x) = ∇ ψ(x) = 2π R2 Since vorticity is conserved along particle trajectories, it follows that if we consider a vorticity distribution defined by the characteristic function of some bounded, simply connected region E ⊂ R2 (so that ω = ω0 χE ), then the evolution of the vorticity is equivalent to the evolution of the set E under the flow. In other words, given a vorticity field ω0 x ∈ E ω(x, t) = 0 ∈ E we are interested in the evolution of the boundary (t) = ∂E(t). Such a vorticity distribution is known as a vortex patch and is an example of a L∞ ∩ L1 weak solution to the vorticity equation (1.1), which has been well studied by numerous authors in recent years [24, 17, 4]. See especially the monographs of Saffman [18] and of Majda and Bertozzi [17] for good overviews of recent work. Also of considerable interest is the numerical solution of the vortex patch equations [9, 15]. A particular example of an exact vortex patch solution is defined on an elliptical region x2 y2 E0 = (x, y) | 2 + 2 ≤ 1 , (1.3) a b where we assume that 0 < b < a, and the corresponding a vortex patch ωe = χE0 . This is the so-called Kirchhoff ellipse and is well known that ωe evolves by rotation at a uniform angular velocity = ω0 ab/(a + b)2 (see [14] for example). By adding a uniform rotation to the velocity field, we can consider the Kirchhoff ellipse to be a stationary solution. The stream function is then decomposed ψK = ψE + ψR ,
(1.4)
where ψE is given by solving the Poisson equation over the elliptical patch and ψR corresponds to a uniform rotation with angular velocity . A natural choice of variables
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for describing our elliptical vortex patch are the so-called elliptical coordinates defined by x = c cosh(ξ ) cos(η), y = c sinh(ξ ) sin(η),
(1.5) (1.6)
√ where c = a 2 − b2 . In this coordinate system, an ellipse is described by an equation of the form ξ = ξ0 .
(1.7)
In our case, where the boundary of E0 has major and minor axes of length a and b respectively, then ξ0 satisfies the equations cosh(ξ0 ) = a/c, sinh(ξ0 ) = b/c.
(1.8)
Throughout this paper, we will make use of the Jacobian for the change of variables from Cartesian coordinates to elliptical coordinates J (ξ, η) ≡
1 ∂(x, y) = c2 cosh(2ξ ) − cos(2η) . ∂(ξ, η) 2
(1.9)
It is well known (e.g. see [16, 14]) that the stream function for the exterior of the Kirchhoff ellipse (ξ > ξ0 ) in a nonrotating frame is given by ψE =
abω0 −2ξ abω0 ξ+ e cos(2η) 2 4
and for uniform rotation with angular velocity the stream function is ψR = −
2 abω0 −2ξ0 cosh(2ξ ) + cos(2η) . (x + y 2 ) = − e 2 4
The stream function for the Kirchhoff ellipse in a rotating reference frame is therefore the sum ψK = ψE + ψR . Since this part of the stream function is known explicitly, we directly compute derivatives of ψK giving abω0 −2ξ0 abω0 abω0 −2ξ ∂ψK cos(2η) − sinh(2ξ ) = − e e ∂ξ 2 2 2 and abω0 −2ξ0 abω0 −2ξ ∂ψK sin(2η) + sin(2η). =− e e ∂η 2 2 We are interested in the dynamics of perturbations about these elliptical patch solutions. To describe the dynamics of KVP perturbations, we use a level set method to derive an evolution equation for the patch boundary. By introducing a scalar function φ that is positive inside the patch, negative outside, and zero on the boundary, the boundary is defined as a level set of this function. By advecting this function, we can then describe the evolution of the boundary. Since we are interested in patches that are perturbations
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from the elliptical patch, it is natural to choose a function φ(ξ, η) that is defined in terms of elliptical coordinates. Specifically, we let φ = ξ − ξ0 + ξ (x, y, t) , (1.10) ξ is the where ξ0 is the value of the elliptical coordinate that defines the base ellipse and perturbation. To derive an equation for the perturbation we observe that the boundary of the patch is advected by the fluid flow so that the material derivative of φ vanishes ∂ φ + u · ∇φ = 0. ∂t
(1.11)
Here, u is the velocity generated by the vortex patch viewed in a rotating reference frame so that the base elliptical patch is a steady solution. The velocity can be represented in terms of the stream function = ψK + ψ , ψ
(1.12)
where ψK corresponds to the steady Kirchhoff ellipse and is known explicitly. The other is due to the perturbed patch boundary. The advection equation (1.11) then piece ψ becomes ∂ ∂ ∂ ∂ ∂ φ+ φ=0 φ− ψ ψ ∂t ∂y ∂x ∂x ∂y or with a change of coordinates , φ) 1 ∂(ψ ∂ φ+ = 0, ∂t J ∂(ξ, η) where J is the Jacobian given by Eq. (1.9). Under the assumption that the perturbation is a function only of η and t, computing derivatives of φ yields , φ ∂ ψ ∂ ∂ ∂ . =− ψ ξ− ψ ∂ (ξ, η) ∂ξ ∂η ∂η Together with the advection equation this gives the following nonlinear equation for the perturbation ∂ ξ 1 d = 0. + ψ ∂t J dη
(1.13)
Given the assumption that the perturbation depends only on η and t, we will simplify the notation throughout the remainder of the paper by dropping the tilde and denoting the perturbation as simply ξ(η, t). Equation (1.13) can be thought of as a balance of the rate of motion of the boundary in the normal direction with the normal velocity. Since the integral that appears in the explicit calculation of the stream function can be expressed in terms of a contour integral around the boundary (see [17] for example), we can reformulate (1.13) as an integro-differential equation. It is convenient to express the points on the boundary using the complex notation z(η, t) = c cosh(ξ0 + ξ(η, t) + iη)
(1.14)
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which then allows us to express, using a complex inner product, the derivative of ψ in (1.13) as
2π d 1 ∂z log z(η) − z(η ) ∂η z(η ) dη , ψ= , ∇ψ = − i∂η z, dη ∂η 2π 0 where we make use of the symmetry of the kernal and Stoke’s theorem, ω0 ω0 ∇y⊥ log |x − y|dy = log |x − y|dl. ∇ ⊥ψ = 2π E 2π ∂E into a rotational piece and that due to the Now, splitting the stream function ψ perturbed vortex patch, we have the following integro-differential equation for the perturbation: 1 d ψR (ξ, η) J dη
2π 1 1 log z(η, t) − z(η , t) ∂η z(η , t) dη . (1.15) i∂η z, − J 2π 0
∂t ξ(η, t) = −
The derivation of this equation is based upon arguments used by Love [16], and so we will refer to this as the Love equation. We remark that although we have computed the stream function in terms of a contour integral around the boundary of the patch, our formulation is different from that of the contour dynamics equation, see [7] and [17]. Smooth solutions to the contour dynamics equation maintain global regularity, [5, 4]. 1.2. Linear stability analysis. If we linearize (1.15) and set q(η, t) = J0 (ξ0 , η)ξ(η, t) ≡ J0 (η)ξ(η, t), then the linearized Love equation satisfies abω0 ∂q(η, t) ≡ ∂η q(η, t) ∂t (a + b)2 2π ω0 log c cosh(ξ0 + iη) − cosh(ξ0 + iη ) q(η , t) dη . (1.16) + ∂η 2π 0 Although this equation does not appear explicitly in Love’s paper [16], it is equivalent and yields the same linear stability result. This equation is also equivalent to the linearized equations found in the work of Wan and Pulverenti [23], Wan [22], and Tang [21]. The remainder of this subsection is devoted to characterizing the dynamics of equation (1.16). Given initially q(η, 0) =
∞
[Am (cos mη) + Bm sin(mη)],
(1.17)
m=1
in order to write down its explicit solution formula, we introduce the notation γ = a/b, then by (1.8) and the analysis of Subsect. 2.1,
γ −1 m ω0 2mγ µ+ = + − 1 , (1.18) m 2 γ +1 (γ + 1)2
ω0 γ −1 m 2mγ µ− = − − 1 , (1.19) m 2 γ +1 (γ + 1)2
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and
2 γ − 1 2m 2mγ ω 0 − λm = µ+ − − 1 . m µm = 2 γ +1 (γ + 1)2
The solutions to (1.16) are given by
µ+ q(η, t) = Am cosh(λm t) − m Bm sinh(λm t) cos(mη) λm m,µ− m >0
λm + Bm cosh(λm t) − + Am sinh(λm t) sin(mη) µm
µ+ + Am cos(λm t) − m Bm sin(λm t) cos(mη) λm m,µ− m <0
λm + Bm cos(λm t) + + Am sin(λm t) sin(mη). µm Clearly there exist unstable growing modes if the quantity
2 γ − 1 2m 2mγ L= − −1 γ +1 (γ + 1)2
(1.20)
(1.21)
(1.22)
is positive. It is straightforward to check that for m = 1, the quantity L is equal to −γ 2 /(γ + 1)2 and is therefore negative for all values of γ . Also, for m = 2, L is identically equal to 0. At m = 3, however, we find that L is equal to 4
γ 2 (γ − 3)(3γ − 1) , (γ + 1)6
which is positive for all γ > 3. This is the linear stability result obtained by Love. L has a sharp upper bound which implies an upper bound on λ. By computing the gradient of L and looking for critical points, we find that at any local maximum, γ must satisfy the condition
γ +1 2 (γ − 1) log = 2. (1.23) γ −1 It can be shown that this has no solutions for γ > 3 and therefore we need to calculate the supremum of L over arbitrarily large m and γ . This is facilitated by making the change of variables γ = αm, where α is an arbitrary positive constant, so that the quantity L can be written as
2m
2 4/α 2αm2 L= 1− − −1 . 2m + 2/α (αm + 1)2 2 The limit of large m is easily evaluated as e−4/α − α2 − 1 . Optimizing in α shows that L ≤ Lmax ≈ 0.162 and we conclude that if λ denotes the maximum growth rate at a fixed value of γ then λm ≤ λ < ω0 Lmax /4. (1.24)
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m=3 m=4 m=8 m=12 L_max
L
3
10
20
30 γ
40
50
60
Fig. 1. The growth rates for various modes is determined by the magnitude of λ2m . The first excited mode is m = 3 which has a nonzero growth rate for γ > 3. Higher modes are excited at larger values of γ . For a fixed value of γ , only a finite number of modes are excited. Note that λ2m is bounded above uniformly by Lmax , where L is defined by (1.22)
We note that this bound holds for arbitrary values of γ > 3 and m > 0 but that for a fixed value of γ , there are only a finite number of growing modes and that there is one mode (or possibly two) with a maximal growth rate. This maximal growth rate depends upon γ , although it will always be less than λ found above. 1.3. Main Results. As Love himself showed, if γ = a/b denotes the ratio of major to minor axes of our base ellipse, then for γ > 3, the linearization of (1.15) has positive eigenvalues and therefore unstable growing modes. We now fix γ > 3. As we show in Sect. 2, for a fixed value of γ , only finitely many modes are excited and the growth rates for those modes are bounded. Let λ > 0 denote the largest eigenvalue for the linearized Love equation (1.16) and let m(λ) be the corresponding wave number. Note that at certain values of γ there can be two modes m1 (λ) < m2 (λ) with the same maximal growth rate, see Fig. 1. Now let θ0 > 0 be a sufficiently small fixed constant. Then for any δ > 0 that is δ arbitrarily small we define the escape time T δ by the relation δeλT = θ0 or equivalently Tδ =
1 θ0 ln . λ δ
(1.25)
We also define an acceptable class of perturbations of the Kirchhoff vortex patch. Definition 1. Let γ and ξ0 be the parameters for our base ellipse E0 . ξ (η) ∈ C ∞ [0, 2π ] is called a generic profile if
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ξ (η) =
J0−1 (η)
∞
Am cos (mη) + Bm sin (mη) ,
m=1
and if λ is the dominant eigenvalue from the (γ , m) curve (1.20), then there exists m(λ) such that
µ+ m(λ) Am(λ) =
µ− m(λ) Bm(λ) .
We point out that there can be two distinct eigenmodes with respect to λ. Such a generic profile ensures the excitation of the dominant linear growing mode(s). We feel this may be an unnecessary condition in general since higher-order approximations from Sect. 5 have forcings that couple all modes. For notational simplicity, we also define the generator, eLt , of the linear Love equation with initial data q(η, 0) = J0 (η)ξ (η), ξ1 (η, t) = J0−1 (η)q(η, t) ≡ eLt ξ ,
(1.26)
where q(η, t) is the solution given by (1.21). Theorem 1. Let γ > 3 and let ξ (η) be a generic profile. Then there exist constants δ0 > 0, 0 > 0, C > 0 and a sufficiently small θ0 > 0, depending only on ξ0 such that for all 0 < δ ≤ δ0 , let ξ δ (0, η) = δξ (η) with Ep , the associated perturbed ellipse: • If ξ δ (η, t) is a solution of the Love equation (1.15) with ξ(η, 0) = δξ (η), then for all t ∈ [0, Tδ ], δ (1.27) ξ (t) − eLt δξ 2 ≤ C δ 2 e2λt . L
• For t ∈ [Tδ /2, Tδ ] then + k µ δ (λ) m −1 i Ami (λ) − Bmi (λ) cos (mi (λ)η) ξ (t) − δeλt J0 (η) λ i=1 λ + Bmi (λ) − + Ami (λ) sin (mi (λ)η) ≤ C δ 2 e2λt , (1.28) 2 µmi (λ) L
where k = 1 unless there are two dominant eigenmodes, in which case k = 2. • If t is the flow operator for the vortex patch then (1.29) χTδ (E0 ) − χTδ (Ep ) 1 ≥ 0 . L
Remark 1. We note that in fact (1.27) implies nonlinear orbital instability of elliptical patches for γ > 3. In other words, (1.29) can be strengthened so that for all t ≥ 0, χt (E0 ) − χTδ (Ep ) 1 ≥ 0 > 0. L
This follows from a consideration of the exponential growth rate associated to the maximal growing modes as opposed to the linear growth of the rotational displacement. This proves the converse of the orbital stability results of Wan [22] and Tang [21].
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Normally, the difference between characterizing growing modes of a linearized partial differential equation and fully understanding nonlinear dynamic instability is significant, due to the presence of unbounded higher order derivatives as well as the expected presence of continuous spectra. Indeed, nonlinearities present challenges for most conservative PDEs that arise in mathematical physics. Although no general theory exists to date for proving nonlinear instability, there have been recent successes in specific areas. For example, Guo and Strauss [12] proved nonlinear instability for BGK equilibria of the Vlasov-Maxwell system in plasma physics using careful linear analysis as well as a delicate bootstrap argument. In addition, Almeida, Bethuel, and Guo showed that non-unit valenced Ginzburg-Landau vortices are dynamically unstable for large coupling constants [1]. There have also been several nonlinear instability results regarding incompressible and compressible Euler equations by Friedlander, Strauss, and Vishik [10], Grenier [11], Bardos, Guo, and Strauss [2], and Hwang and Guo [13]. Most notably, Grenier introduced a novel method of higher order approximation to prove the instability for several types of ideal fluids [11] for much larger classes of perturbations. Beyond the proof of the nonlinear instability, the novelty of our paper here is the characterization of the unstable dynamics of a rather general perturbation, not just along the maximal growing mode. This is useful for both theoretical understanding as well as numerical simulations. Our success is based on two important ingredients. First of all, we have obtained the exact solution formula (1.21), which gives a complete and precise description of the linearized equation. We then use this precise information to estimate general higher order approximations for generic perturbations, following Grenier’s method, as well as the full dynamics of the boundary. We note that there are many important physical problems with a similarly precise structure for linear equations that have been investigated intensely by physicists, and our method should also apply. The second ingredient is a new type of Sobolev estimates for a class of logarithmically singular integral operators for a smooth, non self-intersecting contour w(η) on the plane: 2π log |w(η) − w(η )|φ(η )dη ≤ Cw ||φ||H s−1 , (1.30) 0
Hs
for s ≥ 3. Notice that even the expression of high derivatives for such a singular integral operator is delicate. We introduce an integration multiplier ∂η z(η), z(η) − z(η ) M= ∂η z(η ), z(η) − z(η ) which will allow us to integrate by parts repeatedly in the singular integral to express its high derivatives. Moreover, the particular form of M is crucial for certain exact cancellations to obtain (1.30). In fact, an additional multiplier, Mδ , is also used. Due to its technical nature, our self-contained proof of Theorem 1 is divided into a series of steps in the following sections. In Sect. 2, we obtain the linear estimate. Section 3 is devoted to the crucial estimate (1.30). Section 4 is devoted to obtaining an a priori estimate needed for a local existence argument. Section 5 describes the construction of an approximate equation of motion that generalizes to higher orders the linearization of Sect. 2. Finally, in Sect. 6, the difference between the solution to the Love equation and the n-th order approximate solution is estimated in H s using standard energy techniques,
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and finally we establish our instability theorem in Subsect. 6.4. In order to make our paper more readable, we have left some background for Sect. 3 in an Appendix. Finally, we also remark that in addition to the particular type of perturbation described herein, other types of perturbations are also possible. In particular, Constantin [6] has considered the perturbation in the Yudovich norm obtained by introducing a second vortex patch whose boundary is widely separated from the first and whose vortex strength is small. In this case, the far-field approximation leads to a decoupling of the contour equations and an interesting instability analysis for the Kirchhoff ellipse which is independent of the shape of the ellipse. 2. Linearization Since our argument relies heavily on precise information on the linearized Love equation (1.16), we first establish the crucial formula (1.21). In the second subsection we prove a generic H s upper bound and a lower bound for solutions with generic profile initial data. The latter result is Theorem 2. 2.1. Linearized Love Equation. To show (1.21), we postulate the Fourier series representation for a solution q(η, t) =
∞
Am (t) cos(mη) + Bm (t) sin(mη),
(2.1)
m=1
and insert into the linear equation (1.16). It is useful to observe that cosh(ξ0 + iη) − cosh(ξ0 + iη ) = 1 e−ξ0 1 − e−iη+iη 1 − e2ξ0 +iη+iη 2 so that the integral in (1.16) can be split into a sum of three integrals, the first of which will vanish upon integrating sine or cosine against a constant. For the others, we need to evaluate four integrals in which a sine or cosine is integrated against the logarithmic terms that appear above. Using the following notation introduced by Tang [21], 1 2π Im (α + iβ) ≡ log 1 − eα+iβ+iη eimη dη , (2.2) π 0 and evaluating (see Lemma 1 below, also [21, 22]), we have integrals such as 2π 1 1 log 1 − e−iη+iη cos(mη ) dη = (Im (−iη) + I−m (−iη)) 2π 0 4 1 =− cos(mη) 2m
(2.3)
and 1 2π
0
1 log 1 − e−2ξ0 +iη+iη cos(mη ) dη = (Im (−2ξ0 + iη) + I−m (−2ξ0 + iη)) 4 1 −2ξ0 =− cos(mη). (2.4) e 2m
2π
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Combining (2.3) and (2.4) together yields
1 1 −2ξ0 ∂ cos(mη) − Am (t)ω0 ∂η cos(mη) cos(mη) + e Am (t) 2 η 2m 2m (a + b)
abmω0 + ω0 e−mξ0 cosh(mξ0 ) sin(mη). = Am (t) − (a + b)2 abω0
A similar computation holds for Bm (t). From (1.16) we find the following expansion: ∞
ω0 ab −mξ0 ˙ Am (t) + Bm (t)m − Bm (t)ω0 e sinh(mξ0 ) cos(mη) (a + b)2 m=1
ω0 ab −mξ0 ˙ + Bm (t) − Am (t)m + Am (t)ω0 e cosh(mξ0 ) sin(mη) = 0. (a + b)2 Notice that from definitions of µ± m and λm , we have equivalently A˙ m (t) + µ+ m Bm (t) = 0,
B˙ m (t) + µ− m Am (t) = 0.
Using (1.17) we set Am (0) = Am ,
Bm (0) = Bm ,
then (1.21) follows from elementary ODE theory. For the sake of completeness, we state and prove the following lemma allowing us to calculate the integrals Im (α + iβ) that appear in (2.3), (2.4), and other terms of (1.16). See also Tang [21] and Wan [22]. Lemma 1. For α, β ∈ R and m = 0 the integral 1 2π Im (α + iβ) ≡ log 1 − eα+iβ+iη eimη dη π 0
(2.5)
is equal to Im (α + iβ) = −
1 −|mα|−imβ e . |m|
(2.6)
Proof. First assume that m > 0. If we let w = eiη then the integral in (2.5) becomes the contour integral 1 log 1 − eα+iβ w w m−1 dw, (2.7) iπ |w|=1 which we split into two pieces by writing 2 1 − eα+iβ w = 1 − eα+iβ w 1 − eα−iβ /w . Each of the resulting integrals is then handled by integrating by parts. For instance wm −1 1 α+iβ m−1 log 1 − e w w dw = dw. 2iπ |w|=1 2miπ |w|=1 w − e−α−iβ
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For α > 0, the singularity at w = e−α−iβ lies inside the contour and we evaluate by residues so that Im = −
1 −mα −imβ e . e m
Similarly, the second piece of Eq. (2.7) can be written as 1 2iπ
|w|=1
log 1 − e
α−iβ
/w w
m−1
−1 dw = 2miπ
|w|=1
eα−iβ w m−1 dw. w − eα−iβ
For α > 0 the singularity at eα−iβ is outside the contour and the integral vanishes. On the other hand, if α < 0 then the second integral is nonzero and we have Im = −
1 mα −imβ . e e m
Finally, if α = 0 then both singularities lie on the unit circle and we must take the limit of a small detour around the singularities. The integral is given by π i times the residues: Im = −
1 1 (π i) e−iβm + e−iβ e−iβ(m−1) = − e−imβ . 2mπ i m
A similar argument works for the case m < 0, reversing the orientation of the contour |w| = 1. 2.2. Bounds on growth rate. We have the following result which we will need later on to prove a similar statement about the growth rates of higher order approximate solutions. Theorem 2. Suppose that q(η, t) = J0 (η)ξ1 (η, t) is a solution to the linearized Love equation (1.16). Then there exists a positive constant C such that ξ1 (·, t)H s ≤ Ceλt ξ1 (·, 0)H s
(2.8)
for all s > 0 and all t ≥ 0. Furthermore, if ξ1 (η, 0) = ξ¯ is a generic profile then there exists a positive constant C and a time tξ both of which depend on ξ1 (η, 0) such that ξ1 (·, t)L2 ≥ Ceλt ξ1 (·, 0)L2
(2.9)
for all time t ≥ tξ . Proof. Recall (1.21) and ξ(η, t) = J0−1 q(η, t). Since |λm | ≤ λ for all real valued λm , we can conclude that for all terms in this sum, and therefore for q itself, the growth is bounded in terms of eλt . In the case where λm = 0, the growth of the coefficients Am and Bm is linear and the bound in terms of eλt still applies. Inequality (2.8) of the theorem then follows upon taking derivatives and using the boundedness of J0 .
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309
To finish proving the lower bound, we observe that ξ1 (·, t)2L2 =
∞
2 A2m (t) + Bm (t)
≥
m=1
k
2 A2mi (λ) (t) + Bm (t), i (λ)
i=1
where λ is the dominant eigenvalue and k = 1 unless there are two dominant eigenvalues (in which case k = 2). From (1.21) there exists tξ that depends on Am (0) and Bm (0) such that k
2 A2mi (λ) (t) + Bm (t) i (λ)
i=1
≥
k
1 +
i=1
λmi (λ) µ+ mi (λ)
2 2 µ+ (λ) m i (Ami (λ) (0) − Bmi (λ) (0) e2λt λmi (λ)
≥ ce2λt for all t ≥ tξ , where c > 0 so long as Ami (λ) (0) =
µ+ mi (λ) λmi (λ)
Bmi (λ) (0)
for some i. This condition is equivalent to µ+ A = µ− m(λ) m(λ) m(λ) Bm(λ) , our condition on a generic profile, as described by Definition 1.
(2.10)
3. Sobolev Estimates of the Singular Integral Operator Since the Love equation (1.15) consists of a logarithmically singular integral operator, in this section we establish H k estimates on such operators of the form (3.1) log w(η) − w(η ) ϕ(η )dη , where w ∈ C and ϕ ∈ H s−1 and 0 ≤ k ≤ s. In particular we prove the following theorem. We prove the following theorem, in which PsK is defined below. Theorem 3. Suppose ϕ(η) ∈ H s−1 and suppose w ∈ PsK for s ≥ 3, then there exists a constant C such that if k ≤ s − 2 then k ∂ k w ϕH s−1 . log w(η) − w(η ) ϕ(η )dη ≤ C + 1 (3.2) s H ∞ ∂ηk L On the other hand, if s − 1 ≤ k ≤ s then k ∂ k w ϕH s−1 . log w(η) − w(η ) ϕ(η )dη ≤ C + 1 s H 2 ∂ηk L In either case C depends only on k and K.
(3.3)
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For the convenience of the reader we leave some of the technical preparation for the proof of Theorem 3 to the Appendix. We first introduce a function space PsK which prevents our contour from having self-intersections and allows us to establish necessary lower and upper bounds. Let ∂ α denote the multi-index ∂ α u = ∂ηα1 ∂ηα 2 u, then we recall the following norms: φ(η, η ) φ(η, η )
L2 ⊗L2
L∞ ⊗L∞
=
φ(η, η ) 2 dη dη
1 2
,
= ess sup φ(η, η ) ,
k α φ(η, η ) k k = ∂ φ(η, η ) 2 2 . H ⊗H L ⊗L α=0
As in [4, 17] we define w = inf
η1 =η2
|w(η1 ) − w(η2 )| . |η1 − η2 |
(3.4)
If w ≥ c then w will have no self-intersections, and we also note that w ≥ c implies ∂η w ≥ c. We now define a function space PsK . Definition 2. We say w(η) ∈ PsK if w(η) ∈ H s for some s ≥ 3 and ∂η w(η) 1 ≤ K, C 1 w(η) ≥ . K
(3.5) (3.6)
It turns out that z = c cosh (ξ0 + ξ(η) + iη) will lie in PsK if ξ(η) is smooth and small enough, see Proposition 9. We remark that this space has many similarities to the space OM found in [4, 17], but we ask for even stronger restrictions. It is within this unusual space that we will derive estimates on our singular integral operator. In order to differentiate (3.1), we need to be able to move derivatives off of the logarithm term and onto the integrand. We can do this by expressing derivatives with respect to the η variable in terms of derivatives with respect to η and then integrating by parts. We begin by differentiating the logarithm term that appears in the singular integral, ∂η w(η), w(η) − w(η ) ∂η log w(η) − w(η ) = , |w(η) − w(η )|2 where the numerator is the complex inner product ∂η w(η), w(η) − w(η ) = ∂η w(η)(w(η) − w(η )) . We can express the derivative of the log with respect to η as a derivative with respect to η as follows: ∂η w(η), w(η) − w(η ) . ∂η log w(η) − w(η ) = −∂η log w(η) − w(η ) ∂η w(η ), w(η) − w(η ) Therefore, if we are integrating over η , we can integrate by parts and move the derivative onto the integrand. From this formal calculation, we introduce the following notation.
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Definition 3. Let M = M(η, η ) denote the multiplier ∂η w(η), w(η) − w(η ) . M(η, η ) = ∂η w(η ), w(η) − w(η )
(3.7)
Furthermore, we define DM as the operation DM φ = ∂η φ + ∂η (M φ) .
(3.8)
In order to establish Theorem 3 we assume w ∈ PsK and divide up the estimate into special cases. We would like to make use of Lemma 7 of the Appendix and perform successive integration by parts to move the derivatives off the kernal; however this lemma requires an integrand with support near the singularity. Therefore, let 1 on |η − η | < ρ2 µ(η, η ) = (3.9) 0 on |η − η | > ρ (η, η ) = 1 − µ(η, η ) ϕ(η ). with µ ∈ C0∞ (Bρ ) and set φ(η, η ) = µ(η, η )ϕ(η ) and φ Then (7.1) becomes ∂k log w(η) − w(η ) ϕ(η )dη k ∂η ∂k (η, η )dη log w(η) − w(η ) φ = k ∂η ∂k log w(η) − w(η ) φ(η, η )dη . + k (3.10) ∂η We can now consider the proof of Theorem 3 as an estimate on a non-singular integral and an estimate on a singular integral with support so small that we may use Proposition 13 of the Appendix. Since the first term of (3.10) is isolated away from the singularity, the estimates are simple and direct, and they are found in Proposition 1 and Proposition 2. The estimates of the second term of (3.10) are much more involved. When k ≤ s − 2 we establish L∞ estimates on the operator, and this is proved in Proposition 3. When k = s − 1 or s then we establish L2 estimates on the operator, and their proofs are found in Proposition 4 and Proposition 5, respectively. We remark that the k = s case is very delicate, and the majority of Sect. 3 is devoted to establishing such a bound. In the following we let ρ=
8 , K2
(3.11)
as in the proof of Proposition 13 in the Appendix. We first consider estimates of the singular integral away from the singularity. (η, η ) = 1 − µ(η, η ) ϕ(η ), where (1 − µ) ∈ C ∞ B Cρ . Proposition 1. Suppose φ 0 If w ∈ PsK then for k ≤ s − 2, k ∂ w(η) − w(η ) φ(η, η )dη ≤ C ϕH s−1 wH s , log ∂ηk 2 L where C depends only on K.
2
(3.12)
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Proof. Since we are away from the singularity, we can differentiate inside the integral to get ∂k (η, η )dη log w(η) − w(η ) φ k ∂η k 2 k−l 1 k (η, η ) dη . ∂ηl log w(η) − w(η ) = ∂η φ l 2 l=0
From Lemma 5 in the Appendix, the first term can be bounded 2 l ∂η log w(η) − w(η ) ∞ ∞ L ⊗L 1 2(γ −1) w(η) − w(η ) ∞ ∞ ≤ C max L ⊗L 1≤γ ≤l |w(η) − w(η )|2γ L∞ ⊗L∞ 2 2 × ∂ηl x(η) − x(η ) + y(η) − y(η ) ∞ ∞ , L ⊗L
where w = x + iy. From (3.6), Lemma 4, and Lemma 5 in the Appendix, we get 2 2 l ∂η x(η) − x(η ) + y(η) − y(η ) ∞ ∞ L ⊗L l ≤ C x(η) − x(η )L∞ ⊗L∞ ∂η x(η) ∞ ∞ L ⊗L l +C y(η) − y(η )L∞ ⊗L∞ ∂η y(η) ∞ ∞ L ⊗L
≤ C wH s . Furthermore, since w ∈ PsK , 1 |w(η) − w(η )|2γ
=
η − η 2γ
1
|w(η) − w(η )|2γ
|η − η |2γ
≤C
so long as |η − η | > ρ2 . Therefore, we get the uniform bound 2 l ∂η log w(η) − w(η ) ∞ ∞ ≤ C wH s . L ⊗L
Our desired estimate follows from k ∂ w(η) − w(η ) φ (η, η )dη log ∂ηk ∞ L k k−l s−1 s−1 wH s , ≤ C wH s ∂η φ ∞ ∞ ≤ C φ H ⊗H l=0
L ⊗L
s−1 s−1 ≤ C ϕH s−1 . and the bound φ H ⊗H
To establish the bounds for the case k = s − 1 or k = s in L2 , we follow the same method as in Proposition 1 and get the following result.
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(η, η ) = 1 − µ(η, η ) ϕ(η ), where (1 − µ) ∈ C ∞ B Cρ . Proposition 2. Suppose φ 0 2
If w ∈ PsK then for s − 1 ≤ k ≤ s, k ∂ log w(η) − w(η ) φ (η, η )dη ≤ C ϕH s−1 wH s , ∂ηk L2
(3.13)
where C depends only on K. We now consider the estimates about the second singular term of (3.10). Since we will be using multipliers to successively integrate by parts, we will need to bound a varik φ. Set α = ety of terms that arise from DM (α0 , α1 , . . . , αl ) and β = (β0 , β1 , . . . , βl ). k,l We define mα,β by the following expansion: k DM φ
=
∂ηk φ
+
k
β ∂ηα0 ∂η 0 φ
l=1
l "
β
∂ηαi ∂η i M =
mk,l α,β ,
(3.14)
i=1
where α0 + α1 + · · · + αl = k − l and β0 + β1 + · · · + βl = l in order to simplify notation. We make extensive use of Proposition 13 and Lemma 6 in the Appendix to prove Proposition 3–Proposition 5. We consider the k ≤ s − 2 case first. Proposition 3. Suppose φ(η, η ) = µ(η, η )ϕ(η ), where µ ∈ C0∞ Bρ . If w ∈ PsK then for 0 ≤ k ≤ s − 2, k ∂ k w(η) − w(η ) φ(η, η )dη log ∞ ≤ C ϕH s−1 wH s + 1 , (3.15) ∂ηk L where C depends only on K. Proof. Since supp φ ⊂ {|η − η | < ρ} then by Lemma 7 in the Appendix, ∂k w(η) − w(η ) φ(η, η )dη = log w(η) − w(η ) D k φ dη log M k ∂η for multiplier M. Then k ∂ log w(η) − w(η ) φ(η, η )dη ∂ηk L∞ 1 1 2 2 2 k w(η) − w(η ) 2 dη D ≤ log φ dη M ∞ ∞. Since w ∈
PsK
then log w(η) − w(η )
L
L
L2 (supp φ)
k ∂ log w(η) − w(η ) φ(η, η )dη ∂ηk
≤ C. Thus
L∞
≤
k,l mα,β
L∞ ⊗L∞
,
where βi ≤ l, αi ≤ k − l, and k ≤ s − 2. We now bound these mk,l α,β terms. From Proposition 13 αi βi ∂η ∂η M ∞ ∞ ≤ C wH s ; L ⊗L
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therefore, k DM φ
L∞ ⊗L∞
≤ ∂ηk φ
L∞ ⊗L∞
+C
k l ∂η φ
l "
l=1
i=1
≤ C φH s−1 ⊗H s−1 + C
L∞ ⊗L∞
k
wlH s φH s−1 ⊗H s−1
l=1
≤ C φH s−1 ⊗H s−1 wkH s + 1 for a large constant C = C(K).
wH s
Next we consider k = s − 1. The proof of the bound is very similar to the proof of Proposition 3 since log w(η) − w(η ) ∈ L2 ⊗ L2 . Proposition 4. Suppose φ(η, η ) = µ(η, η )ϕ(η ), where µ ∈ C0∞ Bρ . If w ∈ PsK then s−1 ∂ s−1 ∂ηs−1 log w(η) − w(η ) φ(η, η )dη 2 ≤ C ϕH s−1 wH s + 1 , (3.16) L where C depends only on K. Proof. Note that s−1 ∂ s−1 ∂ηs−1 log w(η) − w(η ) φ(η, η )dη 2 ≤ log w(η) − w(η ) DM φ dη 2 L L s−1 ≤ C DM φ 2 2 . L ⊗L
s−1 φ. From (3.14) It is therefore sufficient to establish an L2 ⊗ L2 estimate on DM s−1,l s−1 mα,β 2 2 , DM φ 2 2 ≤ L ⊗L
L ⊗L
and we divide up the estimates of ms−1,l α,β into a few cases. (i) If αi + βi ≤ s − 2 and α0 , β0 ≤ s − 2, then we can estimate each term using Proposition 13. From (3.14), s−1,l mα,β
L2 ⊗L2
≤ ∂ηα0 φ
L2 ⊗L2
β + C ∂ηα0 ∂η 0 φ
l " αi βi ∂ ∂ M η η 2
L2 ⊗L
i=1
≤ C φH s−1 ⊗H s−1 wlH s + 1 for 1 ≤ l ≤ s − 2. (ii) If α0 = s − 1 then by Proposition 13 s−1,l mα,β 2 2 = ∂ηs−1 φ L ⊗L
L2 ⊗L2
≤ φH s−1 ⊗H s−1 .
L∞ ⊗L∞
Dynamics Near an Unstable Kirchhoff Ellipse
315
(iii) If β0 = s − 1 then s−1,l mα,β
L2 ⊗L2
≤ ∂ηs−1 φ
L2 ⊗L2
s−1 "
ML∞ ⊗L∞ ≤ C φH s−1 ⊗H s−1 .
i=1
(iv) Finally, we consider αj + βj = s − 1 for some j ≥ 1. Then s−1,l mα,β
L2 ⊗L2
α β ≤ C φL∞ ⊗L∞ ∂η j ∂η j M
L2 ⊗L2
βj −1
"
ML∞ ⊗L∞
i=j
≤ C φH s−1 ⊗H s−1 wH s . Combining (i) − (iv) yields estimate (3.16) after a simple interpolation.
s−1 φ We now study the end case k = s which is much more delicate. This is because DM s−1 depends on both η, η , so that extra regularity for DM φ is needed by Theorem 8 in the Appendix. It is important to make use of the fact that φ(η, η ) = µ(η, η )ϕ(η ) as well as the special structure of M in order to obtain such regularity. Proposition 5. Suppose φ(η, η ) = µ(η, η )ϕ(η ), where µ ∈ C0∞ Bρ . If w ∈ PsK then s ∂ w(η) − w(η ) φ(η, η )dη ≤ C ϕH s−1 ws s + 1 , log (3.17) H 2 ∂ηs L
where C depends only on K. Proof. We have to be much more careful about the bounds. In the following proof we let C denote any constant that depends only on K: s ∂ w(η) − w(η ) φ(η, η )dη log 2 ∂ηs L ∂ s−1 = log w(η) − w(η ) DM φ dη 2 ∂η L
∂η w(η) s−1 ≤ w(η) − w(η ) DM φ dη 2 L s−1 + log w(η) − w(η ) ∂η DM φ dη ≡ (I ) + (I I ).
L2
(3.18)
We first concern ourselves with (I ) and divide our estimates into four cases (i) − (iv) of increasing difficulty. The last case (iv) is further subdivided into several cases. Recall # s−1,l s−1 DM φ = ms−1,l α,β , and we organize the mα,β into several cases. (i) Consider αi$ + βi ≤ s − 2%for all i ∈ {0, . . . , l}. In order to use Theorem 8, we ∈ L2 , which lets will show ∂η ∂η w(η)ms−1,l ∈ L2 ⊗ L2 and ∂η w(η)ms−1,l α,β α,β η=η
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us use (7.16), & ' l " β β α α 0 i ∂η ∂η i M ∂η ∂η w(η) ∂η 0 ∂η φ i=1 L2 ⊗L2 l " β +1 β ≤ ∂η w(η)∂ηα0 ∂η 0 φ ∂ηαi ∂η i M i=1 L2 ⊗L2 l l " βi αi ∂η w(η) ∂ α0 ∂ β 0 φ ∂ αi ∂ β i +1 M + ∂ ∂ M η η η η η η j =1 i=j
.
L2 ⊗L2
For the first line we use (3.6) and (7.5), (7.6) from Proposition 13 in the Appendix and find $ % s−1,l ∂η ∂η w(η) mα,β 2 2 ≤ C φH s−1 ⊗H s−1 wlH s , L ⊗L
where 1 ≤ l ≤ s − 2. It is straightforward to establish the trace estimate & ' l " αi βi ∂η w(η)∂ α0 ∂ β 0 φ ≤ C φ(η, η)H s−1 wl s , ∂η ∂ η M η η H 2 i=1 η=η L
and combining the two cases together, along with Theorem 8 in the Appendix, yields the correct bound. (ii) Similarly, if α0 = s − 1 then from (3.6) we find $ % ∂η ∂η w(η) ∂ηs−1 φ 2 2 ≤ C φH s−1 ⊗H s−1 L ⊗L
and the trace estimate $ % ∂η w(η)∂ s−1 φ η
η=η
L2
≤ C φH s−1 ⊗H s−1 ,
which is sufficient to use Theorem 8. (iii) If β0 = s − 1 then we use (7.17) of Theorem 8 in the Appendix, $ % 2 1/2 ∂η ∂η w(η) M s−1 ∂ s−1 φ dη ∞ η L (η ) s−2 ≤ C ϕH s−1 M ∂η M ∞ ∞ ≤ C ϕH s−1 wH s
L ⊗L
and $ % w(η)M s ∂ s−1 η φ η=η
L2
≤ C φ(η, η)H s−1 .
Dynamics Near an Unstable Kirchhoff Ellipse
317
(iv) If there exists a j ≥ 1 such that αj + βj = s − 1, then we have terms of the form ∂η w(η) mlα,β = ∂η w(η)φ M βj −1 ∂η j ∂η j M α
β
with 0 ≤ αj ≤ s − 2 and 1 ≤ βj ≤ s − 1. These terms must be further subdivided, as there is a special case that cannot be estimated by Theorem 8. We note α β α β ∂η j ∂η j M = ∂η j ∂η j (M − 1) where ) ∂η w(η) − ∂η w(η ), w(η)−w(η η−η . M −1= w(η)−w(η ) ∂η w(η ), η−η This extra difference in the numerator allows us to control the special case. Let
w(η) − w(η ) , (A, D) = ∂η w(η) − ∂η w(η ), η − η
w(η) − w(η ) . (B, D) = ∂η w(η ), η − η
(3.19) (3.20)
Our primary concern is when all s − 1 derivatives land on D since k j ∂η ∂η D
L2 ⊗L2
≤ ∂ηk+j +1 w
L2
.
In this case, we cannot differentiate again in either η or η –a sufficient condition to use Theorem 8. Instead we will use A to control the estimate. We list a few simple estimates (A, D)L∞ ⊗L∞ ≥ K 2 ,
(3.21)
(B, D)L∞ ⊗L∞ ≥ K , k j ∂η ∂η (A, D) 2 2 ≤ C ∂ηk+j +1 w 2 , L ⊗L L k+j +1 k j w 2 , ∂η ∂η (B, D) 2 2 ≤ C ∂η 2
L ⊗L
L
(3.22) (3.23) (3.24)
which follow from the proof of Proposition 13. To simplify notation, let α = αj and β = βj , then β
∂ηα ∂η (M − 1) =
$ % β β β c(α, β) ∂ηα1 ∂η 1 A, ∂ηα2 ∂η 2 D ∂ηα3 ∂η 3 (B, D)−1
(3.25)
with α1 + α2 + α3 = α, β1 + β2 + β3 = β, and α + β = s − 1. We study several cases of αj ’s and βj ’s. (1) First, we consider α1 = β1 = 0. The integral operator is no longer singular, so we do not need to use Theorem 8. From H¨older’s inequality
318
Y. Guo, C. Hallstrom, D. Spirn
$ % ∂η w(η) β−1 α2 β2 α3 β3 −1 A, ∂η ∂η D ∂η ∂η (B, D) dη φM w(η) − w(η ) L2
η − η ≤ ∂η w(η) φM βj −1 w(η) − w(η )
2 21
$ % ∂η w(η) − ∂η w(η ) α2 β2 α3 β3 −1 × , ∂η ∂η D ∂η ∂η (B, D) dη dη η − η η − η Mβ−1 ≤ ∂η w L∞ sup L∞ ⊗L∞ φL∞ ⊗L∞ w(η) − w(η ) η=η $ % ∂η w(η) − ∂η w(η ) α2 β2 α3 β3 −1 D) × ∂ D ∂ (B, ∂ 2 2 , ∂ η η η η ∞ ∞ L ⊗L η − η L ⊗L
where α2 + β2 + α3 + β3 = s − 1. It suffices to estimate the last term in L2 ⊗ L2 , and we further separate it into two cases. If α2 +β2 ≤ s −2 then we use (7.2)-(7.3) in the Appendix and (3.21)-(3.24) to get $ % α2 β2 α3 β3 ∂η ∂η D ∂η ∂η (B, D)−1 2 2 L ⊗L γ
d 1 α2 β2 ◦ (B, D) ≤ C ∂η ∂η D ∞ ∞ max ∞ ∞ γ L ⊗L 1≤γ ≤α3 +β3 dx x L ⊗L γ −1 α3 +β3 2 × (B, D)L∞ ⊗L∞ ∂ (B, D) L2 ⊗L2 ≤ C wH s . On the other hand if α2 + β2 = s − 1, then we find a similar estimate, $ % α2 β2 α3 β3 ∂η ∂η D ∂η ∂η (B, D)−1 2 2 L ⊗L γ
d 1 α2 β2 ≤ C ∂η ∂η D 2 2 max ◦ D) (B, ∞ ∞ γ L ⊗L 1≤γ ≤α3 +β3 dx x L ⊗L γ −1 α +β 3 3 × (B, D) ∞ ∞ ∂ (B, D) ∞ ∞ L ⊗L
≤
C w2H s
L ⊗L
.
Combining both estimates we get
$ % ∂η w(η) β−1 α2 β2 α3 β3 −1 A, ∂η ∂η D ∂η ∂η (B, D) dη φM w(η) − w(η ) L2 3 4 ≤ C φH s−1 ⊗H s−1 wH s + wH s . (2) Second, we consider 1 ≤ α1 ≤ s −1 or 1 ≤ β1 ≤ s −2. Then we can differentiate in η and use (7.16). We find $ $ %% β β β ∂η ∂η w(η) φ M β−1 ∂ηα1 ∂η 1 A, ∂ηα2 ∂η 2 D ∂ηα3 ∂η 3 (B, D)−1 2 2 L ⊗L ≤ C φH s−1 ⊗H s−1 wsH s + 1 ,
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and the trace estimate $ % ∂η w(η) φ M β−1 ∂ α1 ∂ β 1 A, ∂ α2 ∂ β 2 D ∂ α3 ∂ β 3 (B, D)−1 η η η η η η η=η L2 ≤ C φ(η , η ) s−1 ws s + 1 . H
H
(3) Finally, we consider β1 = s −1, then we can differentiate in η and use Theorem 8 in a similar way as in (2), $ %% $ ∂η ∂η w(η) φ M β−1 ∂ β 1 A, D (B, D)−1 2 η L (η) L∞ (η ) ≤ C φH s−1 ⊗H s−1 wsH s + 1 with the trace estimate the same as in (2). This completes the bound for (I ) of (3.18). Next we bound (I I ) of (3.18), log w(η) − w(η ) ∂η D s−1 φ(η, η )dη M
,
L2
and we divide the integrand terms ∂η ms−1,l α,β into three cases (v) − (vii). (v) If αi + βi ≤ s − 2 for all i then we show that each term is L2 ⊗ L2 , which is sufficient since log is L2 integrable. The estimates are the same as in the proof of Proposition 4 part (i). (vi) If α0 + β0 = s − 1 then we have terms of the form
% $ ( β log w(η) − w(η ) ∂η ∂ηα0 ∂η 0 φM β0 dη
for 0 ≤ α0 ≤ s − 1. This is again estimated as in part (v), but we have the bound % $ ( log w(η) − w(η ) ∂η ∂ α0 ∂ β 0 φM β0 dη η η
L2
≤ C ∂η φ H s−1 ⊗H s−1 wH s
since there can be s derivatives in η on φ. (vii) If αj + βj = s − 1 for some j ≥ 1 then we have a term of the form
α +1 β log w(η) − w(η ) φ M βj −1 ∂η j ∂η j Mdη .
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To bound the last term, we note that βj ≥ 1; therefore, we can use Lemma 6 in the Appendix and integrate by parts,
α +1 β log w(η) − w(η ) φ M βj −1 ∂η j ∂η j Mdη $ % α +1 β −1 = −P .V . ∂η log w(η) − w(η ) φ M βj −1 ∂η j ∂η j Mdη α +1 β −1 = − log w(η) − w(η ) ∂η φM βj −1 ∂η j ∂η j Mdη
∂η w(η ) α +1 β −1 φM βj −1 ∂η j ∂η j Mdη . −P .V . (3.26) w(η) − w(η )
α +1 β −1 We control the first line of (3.26) directly since ∂η φM βj −1 ∂η j ∂η j M∈L2 ⊗L2 . In particular α +1 β −1 j ∂η j M 2 2 ∂η φ M βj −1 ∂η L ⊗L αj +1 βj −1 α +1 β −1 β −1 j ≤ ∂η φM ∂η ∂η M 2 2 + C φM βj −2 ∂η M ∂η j ∂η j M 2 2 L ⊗L L ⊗L βj −1 αj +1 βj −1 ≤ ∂η φ L∞ ⊗L∞ ML∞ ⊗L∞ ∂η ∂η M 2 2 L ⊗L βj −2 α +1 β −1 +C φL∞ ⊗L∞ ML∞ ⊗L∞ ∂η M L∞ ⊗L∞ ∂η j ∂η j M 2 2 L ⊗L
≤ C φH s−1 ⊗H s−1
w2H s
.
Therefore, log w(η) − w(η ) ∂η φ M βj −1 ∂ηαj +1 ∂ β j −1 Mdη η
L2
≤ C φH s−1 ⊗H s−1
w4H s
.
This finishes the bound on the first term of (3.26). Finally, we bound the second term of (3.26), where αj + 1 + βj − 1 = s − 1. This integral operator is bounded in L2 by copying the method in step (iv), replacing ∂η w(η ) with ∂η w(η). This finishes the proof of Proposition 5.
4. Energy Estimates and Local Existence The goal of this section is to establish the local existence of solutions to the nonlinear Love equation (1.15) for sufficiently small initial data. The key to this argument is an energy inequality from which an a priori estimate can be derived. The result is the following local existence theorem.
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Theorem 4. Let s ≥ 3. There exists a constant ω0 > 0 and a time T0 > 0, both depending only on ξ0 , such that if ξ(·, 0)H s ≤ ω0 then there exists a unique solution ξ(η, t) to the Love equation (1.15) for all t ∈ [0, T0 ]. Furthermore, ξ(η, t) satisfies ∂ ξ H s ≤ C(ξ0 ) 1 + ξ s+3 (4.1) Hs ∂t and ξ(·, t)H s ≤ C(ξ0 , ω0 ) ξ(·, 0)H s
(4.2)
for all t ∈ [0, T0 ]. In the next subsection, we prove a number of preliminary estimates that are needed to derive the key differential inequality (4.1) which itself is obtained in the second subsection. Once we have established the a priori estimate, the proof of this theorem is standard; an outline of the argument is presented in the final subsection.
4.1. Preliminary Estimates. It will prove useful to state and prove some estimates needed to control various quantities that appear in the energy argument. Recall from Eq. (1.9) 2 that the Jacobian is defined as J = c2 (cosh(2ξ0 + 2ξ ) − cos(2η)), where c and ξ0 are constants that define the base ellipse. Proposition 6. Suppose |ξ | ≤ ξ0 /4, then there exists a constant C, depending upon r, k, and ξ0 , such that for 1 ≤ r ≤ ∞ the Jacobian satisfies the estimates k (4.3) ∂η J r ≤ C 1 + ∂ηk ξ r L
and
L
k −1 ∂η J
Lr
≤ C 1 + ∂ηk ξ r . L
(4.4)
Proof. Applying the Moser inequality (7.3) to derivatives of cosh(2ξ0 + 2ξ ) and using the given bound on ξ gives
ξ0 k γ −1 k γ max cosh + 2ξ ≤ C 2 cosh(2ξ + )ξ (2ξ ) ∂η ∂η ξ r . (4.5) r ∞ 0 m 0 L 1≤γ ≤k L L 2 Inequality (4.3) then follows by taking C to be the larger of c2 2k−1 and the coefficient in (4.5). To establish the bound (4.4) we first note that the condition on ξ implies the simple upper and lower bounds,
ξ0 c2 cosh 2ξ0 − −1 >0 J ≥ 2 2 and c2 J ≤ 2
ξ0 cosh 2ξ0 + 2
+1 .
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Then by applying the Moser inequality (7.3) we have k −1 ∂η J
Lr
γ −1 ≤ Cm max γ ! J −γ −1 ∞ J L∞ ∂ηk J r 1≤γ ≤k L L k ≤ C ∂η J r . L
The previously established bound (4.3) finishes the proposition.
We also need a similar bound for derivatives of the rotational part of the stream function. Proposition 7. Suppose |ξ | ≤ ξ0 /4, then there exists a constant C depending only on k, r, and ξ0 such that for 1 ≤ r ≤ ∞ the rotational part of the stream function ψR satisfies k ∂η ψR
Lr
≤ C 1 + ∂ηk ξ r .
(4.6)
L
2 Proof. Since ψR = 2 x 2 + y 2 = c4 (cosh(2ξ0 + 2ξ ) + cos(2η)) the argument is the same as that for ∂ηk J in the previous proposition. Finally, we will need control on derivatives of z(η). Proposition 8. Suppose z(η) = c cosh (ξ0 + ξ(η) + iη) and that |ξ | ≤ ξ0 /4. Then there exists a constant C depending on k, r, and ξ0 such that for 1 ≤ r ≤ ∞, k ∂η z
Lr
≤ C 1 + ∂ηk ξ r .
(4.7)
L
Proof. We write z = x + iy so that k ∂η z
Lr
≤ ∂ηk x
Lr
+ ∂ηk y
Lr
.
Each of the terms on the right can bounded in the same way. For example, since x = c cosh (ξ0 + ξ ) cos(η), the Leibniz rule and bounds on cos(η) and its derivatives gives k ∂η x
Lr
≤ C ∂ηk cosh (ξ0 + ξ )
Lr
+ cosh (ξ0 + ξ )L∞ .
From inequality (4.5) in the proof of Proposition 6 and the straightforward bound on cosh(ξ0 + ξ ) we have k ∂η x
Lr
≤ C 1 + ∂ηk ξ r L
and the bound on y = c sinh(ξ0 + ξ ) sin(η) is exactly the same.
Proposition 9. Suppose z(η) = c cosh (ξ0 + ξ(η) + iη) for ξ ∈ H s and |ξ |C 2 ≤ ξ0 /4. Then z(η) ∈ PsK for K = K(ξ0 ) > 0.
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Proof. From the definition of z(η) it is clear that z ∈ H s as long as ξ ∈ H s . It also follows from the definition and the bounds on ξ that |∂η z| is bounded below and above by constants that depend on ξ0 which we denote k1 and k2 respectively. For instance,
c2 3ξ0 |∂η z| ≥ cosh − 1 ≡ k1 . 2 2 Now by writing z(η1 ) − z(η2 ) = η1 − η 2 it follows that
1
∂η z(η2 + δ(η1 − η2 )) dδ,
0
z(η1 ) − z(η2 ) ≥ k1 (ξ0 ), inf η1 =η2 η −η 1
and by taking K(ξ0 ) = max
)
(4.8)
2
*
1 k1 (ξ0 ) , k2 (ξ0 )
the proposition holds.
4.2. A priori bounds. To establish the needed differential inequality we use a standard energy argument, using the Love equation written in integro-differential form. For the most part, the bounds will follow from the propositions stated in the previous subsection plus the estimates on the singular integral established in Theorem 3. There are two cases, however, that are more delicate. When s derivatives fall on either ∂η ψR or ∂η z we must control those terms via an integration by parts argument. Upon differentiating the Love equation s times in η, we have
∂ s ∂ξ 1 ∂ ∂s = − ψ R ∂ηs ∂t ∂ηs J ∂η
2π 1 1 ∂s . (4.9) log z(η) − z(η ) ∂η z(η )dη i∂η z, − s ∂η J 2π 0 We first consider the term on the right hand side of (4.9) which includes the rotational piece of the stream function. Separating the one term in which all s derivatives hit on ψR we have ∂ηs
J
−1
∂η ψR
s−1
s s−k −1 k+1 ∂η = J ∂η ψR + J −1 ∂ηs+1 ψR . k
(4.10)
k=0
Multiplying by ∂ηs ξ and integrating, we can bound both ∂ηs ξ and ∂ηs−k J −1 in L2 as long as k ≤ s − 2. For the case k = s − 1, we bound ∂ηs ξ and ∂ηs ψR in L2 . In either case, Propositions 6 and 7 imply s s−k −1 k+1 J ∂η ψR 1 ≤ C ξ H s + ξ 3H s . (4.11) ∂η ξ ∂η L
We have also used Young’s inequality and the Sobolev embedding H s → W s−1,∞ . In addition, all constants have been absorbed into C.
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The second term in Eq. (4.10) is more delicate as now the stream function ψR is differentiated s + 1 times. However, we can control this term by integrating by parts. From the definition of ψR , 2π 2π 1 s s+1 1 s s+1 ∂η ξ ∂η ψR dη = C ∂η ξ ∂η (cosh (2ξ0 + 2ξ ) − cos(2η)) dη J J 0 0 2π 1 s s+1 ∂η ξ ∂η (cosh (2ξ0 + 2ξ )) dη . ≤ C ∂ηs ξ 2 + C L J 0 Due to the chain rule and the Leibniz rule, the last term is a linear combination of terms of the following form:
s+1 γ ∂ s+1 d cosh(2ξ0 + 2x) ◦ ξ (cosh (2ξ0 + 2ξ )) = ∂ηs+1 dx γ γ =1 α α1 i1 α2 i2 αγ iγ ∂ ∂ ∂ × ξ ξ ··· ξ , ∂ηα1 ∂ηα2 ∂ηαγ where α1 + α2 + · · · + αγ = s + 1. There are three situations to consider, namely when all αi ≤ s − 1, when there is an αi = s, and when there is an αi = s + 1. For the case when all αi ≤ s − 1 we can bound the terms in the same way that one proves the Moser lemma: 2π
" γ s+1 γ i j 1 s d αj ξ cosh(2ξ + 2x) ◦ ξ ξ dη ∂ ∂ 0 η η γ dx 0 J γ =2 αi ≤s−1 j =0 ≤ C ξ H s 1 + ξ s+1 . (4.12) Hs For the case when αi = s, we bound 2
1 s d s cosh(2ξ0 + 2x) ◦ ξ ∂η ξ ∂η ξ dη J ∂η ξ dx 2 2 ≤ C ∂ηs ξ 2 ∂η ξ L∞ cosh(2ξ0 + 2ξ )L∞ L
≤ C ξ 3H s .
(4.13)
Finally, when αi = s + 1 we have only the term
d cosh(2ξ0 + 2x) ◦ ξ ∂ηs+1 ξ . dx We can bound the contribution of this last term to the energy integral by first observing 2π 2π 2 2 s 1 ∂η ξ sinh(2ξ0 + 2ξ ) ∂ηs+1 ξ dη = sinh(2ξ0 + 2ξ )∂η ∂ηs ξ dη, J J 0 0 which allows us to integrate by parts and use (4.4) to bound the result by C ξ 2H s + ξ 3H s .
(4.14)
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Combining the bounds (4.11)–(4.14) gives
s s 1 s+2 ∂ ξ ∂ ψ ∂ η η J η R dη 1 ≤ C ξ H s + ξ H s . L
(4.15)
We now want to establish a priori bounds on the last term in (4.9), which includes the singular integral operator. Multiplying by ∂ηs ξ and integrating over η yields s k 2π s k ∂ηs ξ ∂ηs−k J −1 k j 0 k=0 j =0
×
i∂ηk−j +1 z,
1 j ∂ 2π η
2π
log z(η) − z(η ) ∂η z(η )dη dη.
(4.16)
0
With the exception of the case k = s, j = 0, which we must handle separately, the terms in (4.16) can be directly estimated using Propositions 6 and 8 as well as the estimates for the singular integral established by Theorem 3. We note that the constants introduced by the singular integral estimates depend on K, but K is in turn determined by ξ0 by Proposition 9. As in (4.10), the arrangements of derivatives in (4.16) will determine what can be bounded in L∞ and what must be bounded in L2 . For instance, if k ≤ s − 2, then j ≤ s − 2 as well, and so the expressions in (4.16) are bounded by C ∂ηs ξ 2 ∂ηs−k J −1 2 ∂ηk−j +1 z ∞ L L L 2π j × log z(η) − z(η ) ∂η z(η )dη (4.17) . ∂η 0
L∞
By (4.4), (4.7) as well as (3.2), we then have the bound j +1 . C(K) ξ H s (1 + ξ H s )2 1 + ξ H s ≤ C(K) ξ H s + ξ s+2 s H
(4.18)
If k = s − 1 and j ≤ s − 2 then we obtain the same estimate by bounding ∂ s−j z in L2 . In the case where j = s − 1 then we must bound the singular integral in L2 and then the bound is . (4.19) C(K) ξ H s (1 + ξ H s )2 1 + ξ sH s ≤ C(K) ξ H s + ξ s+3 Hs If j = s then the L2 bound on the singular integral gives a bound of ≤ C(K) ξ H s + ξ s+4 . C(K) ξ H s (1 + ξ H s )2 1 + ξ s+1 Hs Hs
(4.20)
If k = s and j = 0 then we must integrate by parts. Namely, if we write the singular integral in terms of real and imaginary parts 1 2π
0
2π
log z(η) − z(η ) ∂η z(η )dη = A + iB,
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then upon computing the complex inner product we have
2π
0
∂ηs ξ
J −1
i∂ηs+1 z, A + iB dη =
J −1 ∂ηs ξ B∂ηs+1 x − A∂ηs+1 y dη.
2π 0
Then, for instance, ∂ηs+1 x
s+1 s+1 k ∂η cosh (ξ0 + ξ ) ∂ηs+1−k cos (η) , = k k=0
with the end term k = s + 1 producing s + 1 derivatives of ξ . We can then integrate by parts to obtain bounds such as
2π 0
2 J −1 B∂η ∂ηs ξ dη ≤ CK ξ 2H s + ξ 4H s .
(4.21)
Combining estimates (4.18)–(4.21) then gives the inequality (4.1). At this point, we have a differential inequality which we can solve to find a bound on solutions ξ on some finite time interval.
4.3. Local Existence. Proof of Theorem 4. In the following steps, we outline a method for constructing local solutions in H s , given the a priori bound implied by (4.1). Let ξ ∈ H s and suppose ξ satisfies ξ C 2 ≤ ξ40 and the differential inequality. We let x = 1 + ξ H s , then (4.1) implies x˙ ≤ Cx s+3 from which we conclude that x s+2 ≤
x0s+2 1 − C(s + 2)x0s+2 t
as long as t < 1/(C(s + 2)x0s+2 ). This shows that there exists T0 > 0 depending on ξ0 such that ξ(t)H s ≤ C (ξ(·, 0)H s ) for all t ∈ [0, T0 ], which is to say (4.2) is satisfied. We define ξ to be a solution to a de-singularized Love equation, ∂t ξ =
% 1 d $ R ψ + ψ , J dη
where z = c cosh (ξ0 + ξ (η) + iη) , c2 J = [cosh (2ξ0 + 2ξ (η)) − cos(2η)] , 2 c2 ψR = [cosh (2ξ0 + 2ξ (η)) + cos(2η)] ,
4 1 |z (η) − z (η )|2 + 2 ∂η z (η )dη . ψ = i∂η z (η), log 2π
(4.22)
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Rewriting this equation ∂t ξ = X(ξ , ∂η ξ )∂η ξ + Y (ξ , ∂η ξ ),
(4.23)
we get a nonlinear transport equation. It is standard to use the characteristic method to show that there exists a local-in-time solution to (4.23) in H s by a fixed-point argument. It follows that there exists a uniform bound on ξ , , ∂t ξ H s ≤ C() 1 + ξ s+3 Hs where C() ≤ C(ξ0 ). Therefore, the lifespan of the approximate equation is at least up to T0 . By a continuation argument, we get existence of a solution for t ∈ [0, T0 ] of the de-singularized Love equation (4.22). Letting → 0 we obtain an H s solution ξ(η, t) to the Love equation (1.15). Since there is a uniform bound on {ξ } in H s , we use Rellich’s Theorem to take a Cauchy sequence ξ → ξ in H s−1 . Finally, we check that ξ is a strong solution to the Love equation. We need to make sure that our solution retains the uniform bound ξ C 2 ≤ ξ40 . Finally, we check that ξ C 2 ≤ ξ0 /4. As we’ve shown, there exists ω0 > 0 and T0 > 0 such that if ξ(η, 0)H s ≤ ω0 then ξ(t)H s ≤ α for all t ∈ [0, T0 ]. By Morξ0 rey’s inequality, ξ C 2 ≤ C ξ H s ≤ Cα ≤ ξ40 by choosing α ≤ 4C . This ω0 ensures local existence for times up to T0 . 5. Approximate Equations of Motion From Sect. 2, we learned that the linearized solutions to the Love equation satisfy strict growth rates depending on the dominant eigenvalue, see Fig. 1. In this section we aim to improve these growth rates by constructing higher-order approximations to the Love equation. This method originated in [11] and was used in [13], and it allows for much more flexibility in the proof of instability results. In the following let λ = λmax , where λmax is the dominant eigenvalue from a given γ , see Fig. 1. Definition 4. Let ξ1 (η, t), . . . , ξn (η, t) be n, C ∞ [0, 2π ] functions such that ξk (·, t)H s ≤ c(s, k)ekλt
(5.1)
for all s ≥ 0. Then ξ a (η, t) =
n
δ k ξk (η, t)
(5.2)
k=1
is called an n-th order approximation. Let N (ξ ) ≡
1 d 1 d (ξ ) = ψ (ψR + ψ) , J (ξ ) dη J (ξ ) dη
then the Love equation can be written ∂t ξ = −N (ξ ). We can state our result for high-order approximations to the Love equation.
(5.3)
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Theorem 5. Fix n ≥ 3 and θ0 > 0, sufficiently small.+Then there exists δ0 > 0 such that for all δ ∈ (0, δ0 ) and all ξ (η) ∈ C ∞ [0, 2π] with ξ (η)dη = 0, there exists an nth order approximation ξ a such that for all t ∈ [0, Tδ ], ∂t ξ a + N (ξ a ) = Ran , ξ a (η, 0) = δξ (η) with
(5.4)
a R s ≤ c(s, n)δ n+1 e(n+1)λt n H
for constant c(s, n) and Tδ =
1 λ
log
(5.5)
θ0 δ .
In order to prove this theorem, we start with an expansion of N (ξ a ) in δ. The following lemma will be crucial to the proof. Lemma 2. Let ξ a be an nth order approximation. If ξ a C 2 ≤
ξ0 4
then
∂k
N (ξ a ) = (I ) + (I I ) + (I I I ) ∂δ k for k ≤ n. There exists Pα,β,γ (η) ∈ C ∞ and Pα,β,γ (η, η ) ∈ C ∞ , such that (I ) =
= k. Next,
∂δαi ∂ηβi ξ a (η)
|η−η |≥ ρ2
i=1 αi
Pα,β,γ1 (η)
∂δαi ∂ηβi ξ a (η )dη ,
(5.8)
i=γ1 +1
#γ2
γ1 "
= k. Finally, we have
∂δαi ∂ηβi ξ a (η)
i=1
×
γ2 "
Pα,β,γ2 (η, η )
where 0 ≤ γi ≤ k, 0 ≤ βi ≤ 1, and (I I I ) =
(5.7)
i=1
×
i=1 αi
γ1 "
Pα,β,γ1 (η)
∂δαi ∂ηβi ξ a (η),
i=1
#γ
where 0 ≤ γ ≤ k, 0 ≤ βi ≤ 1, and (I I ) =
γ "
Pα,β,γ (η)
(5.6)
|η−η |≤ρ
log za (η) − za (η ) G(η, η )dη ,
(5.9)
where G(η, η ) = Pα,β,γ2 (η )Pα,β,γ3 (η, η ) 0 ≤ γi ≤ k, 0 ≤ βi ≤ k, and
∂δαi ∂η i ξ a (η ) β
i=γ1 +1
#γ3
ξαi ,βi (η, η )
γ2 "
i=1 αi
i=γ2 +1
ξαi ,βi (η, η ), (5.10)
= k. Here
1
=
γ3 "
Pα,β,γ (η ) 0
# where η = η + η − η and αj = αi .
γ " j =1
α
β
∂δ j ∂η j ξ a (η )d,
(5.11)
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Remark 2. We intentionally leave Pα,β,γ ambiguous since we will be able to bound ∂k a (5.7)–(5.9) so long as 1 ≤ γj ≤ k. However, we emphasize that each term of ∂δ k N (ξ ) satisfies (5.12) αi = k. We will use this crucial fact to prove the correct bounds on the approximate solution. Proof. By (5.3) k k−j 1 ∂j d ∂k k ∂ a (ξ a ) N (ξ ) = ψ k k−j j ∂δ ∂δ J (ξ a ) ∂δ j dη
(5.13)
j =0
and γ j " 1 ∂ αi a 1 ∂j c(k, γ ) J ξ , = ∂δ j J (ξ a ) ∂δ αi (J (ξ a ))γ γ =1
(5.14)
i=1
where α1 + · · · + αγ = k. Furthermore,
" γ
γ j ∂ j a c2 ∂ αi a d a J ξ c(j, γ ) cosh(2ξ + 2x) ◦ ξ ξ . = 0 ∂δ j 2 dx γ ∂δ αi γ =1
(5.15)
i=1
Similarly we find
" γ
γ j a ∂j d d a c(j, γ ) cosh(2ξ0 + 2x) ◦ ξ ∂η αi !ξαi . ψR ξ = ∂δ j dη dx γ γ =1
i=1
(5.16) Combining (5.13)–(5.16) we establish (5.7). ∂k 1 d The terms (II) and (III) come from ∂δ k J dη ψ , which includes the singular integral operator. In particular, ∂j d a ψ ξ ∂δ j dη
j ∂ 1 = j i∂η za , log za (η) − za (η ) ∂η za (η )dη ∂δ 2π
j 1 j ∂ j −l ∂l log za (η) − za (η ) ∂η za (η )dη , = i j −l ∂η za , l l 2π ∂δ ∂δ l=0
where za (η) = c cosh (ξ0 + ξ a (η) + iη). Like in Sect. 3 we isolate the integral about the singularity using (3.9),
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log za (η) − za (η ) ∂η za (η )dη ∂j = j log za (η) − za (η ) (1 − µ) ∂η za (η )dη ∂δ |η−η |≥ ρ2 ∂j + j log za (η) − za (η ) µ ∂η za (η )dη , ∂δ |η−η |≤ρ
∂j ∂δ j
and investigate the two cases separately. (II) Away from the singularity, ∂j log za (η) − za (η ) (1 − µ) ∂η za (η )dη j ρ ∂δ |η−η |≥ 2 j j = ∂δl log za (η) − za (η ) l |η−η |≥ ρ2 l=0 j −l × (1 − µ) ∂δ ∂η za (η ) dη
(5.17)
and j ∂δ
j a a log z (η) − z (η ) = γ =1
c(k, α) |za (η) − za (η )|2γ
γ " 2 d αi a z (η) − za (η ) . dδ αi i=1
(5.18) If
za (η)
=
∂j ∂δ j
x a (η) + iy a (η),
a z (η) − za (η ) 2
2 ∂j a ∂j a a 2 x y (η) − y a (η ) (η) − x (η ) + ∂δ j ∂δ j
γ 2 γ a " αi a d 2 a = x ◦ x (η) − x (η ) ∂δ x (η) − ∂δαi x a (η ) γ dx =
γ =1
+ where
#
2
γ =1
i=1
γ " αi a dγ 2 a a ∂δ y (η) − ∂δαi y a (η ) , (5.19) x ◦ y (η) − y (η ) dx γ i=1
αi = j . Finally,
" γ j γ d ∂j a a x (η) = cos(η) cosh + x) ◦ ξ (η) ∂δαi ξ a (η), (ξ 0 ∂δ j dx γ γ =1 i=1
" γ j γ j d ∂ a a y (η) = sin(η) sinh (ξ0 + x) ◦ ξ (η) ∂δαi ξ a (η). (5.20) ∂δ j dx γ γ =1
i=1
Combining (5.17)–(5.20) together with (5.13)–(5.16) yields (5.8). (III) In order to study the integral about the singularity, we define a new multiplier that allows us to integrate by parts.
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# Definition 5. Let ξ a = nk=1 δ k ξk (η) and za (η) = c cosh(ξ0 + ξ a (η) + iη), then we define Mδ to be the multiplier a a (η ) a ∂δ za (η) − ∂δ za (η ), z (η)−z ∂δ z (η) − ∂δ za (η ), za (η) − za (η ) η−η Mδ = − =− a a (η ) ∂η za (η ), za (η) − za (η ) ∂η za (η ), z (η)−z η−η for the contour of the approximation za . We also let DMδ φ = ∂δ φ + ∂η (Mδ φ). As in Sect. 3 this multiplier allows us to lower the degree of the singularity. Since the approximate contours are C ∞ , it is not necessary to close the singular integral in H s , and we will be content to close the estimate in some finite Sobolev norm. This is in sharp contrast to the energy estimates of Sect. 4 for solutions to the Love equation. Therefore, we can write j ∂δ log za (η) − za (η ) µ∂η za (η )dη j = log za (η) − za (η ) DMδ µ∂η za (η ) dη for j = 1, . . . , s so long as za ∈ PsK . +1 a a (η ) Since z (η)−z = 0 ∂η za (η )d, where η = η + η − η , then η−η
α β α2 β2 1 α β a a a 1 1 ∂δ ∂η Mδ = ∂δ ∂η ∂δ z (η) − ∂δ z (η ) , ∂δ ∂η ∂η z (η )d 0 & '
−1 1
∂η za (η ),
β
×∂δα3 ∂η 3
∂η za (η )d
,
0
where α1 + α2 + α3 = α and β1 + β2 + β3 = β. We find
" 1 1 γ α γ d β β a ∂δα ∂η ∂η za (η )d = ∂η cos(η ) cosh(ξ + x) ◦ ξ ∂δαi ξ a (η ) 0 dx γ 0 0 γ =1
β
+i∂η
1
sin(η ) 0
α
γ =1
i=1
dγ sinh(ξ0 + x) ◦ ξ a dx γ
" γ
∂δαi ξ a (η ).
i=1
Setting ξα,β (η, η )
with
#
αi = α and β
#
=
1
Pγ (η ) 0
γ "
β
∂η i ξαi (η )d
(5.21)
i=1
βi ≤ β + 1, we find
∂δα ∂η Mδ =
×
γ1 " c(α, β) β ∂η i ξαi (η) γ4 +1 a a ∂η z (η ), 0 ∂η z (η )d i=1
γ2 " i=γ1 +1
ξαi ,βi (η, η )
γ3 " i=γ2 +1
∂η i ξαi (η ) β
(5.22)
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with
#γ3 i
α i = α + 1, γ4 ≤ β + α, and β i ≤ β + 1. Therefore, l α0 β0 " j β Dδ µ ∂η za (η ) = ∂δ ∂η µ∂η za (η ) ∂δαi ∂η i Mδ ,
where
#
αi = j − l and
#
i=1
βi = l. Using (5.22) we get
l α0 β0 " j β ∂δ ∂η µ∂η za (η ) ∂δαi ∂η i Mδ Dδ µ ∂η za (η ) = i=1
=
× with
#
αi = j and
#
γ1 " Pγ6 (η ) β ∂δαi ∂η i ξ a (η) γ7 +1 a a ∂η z (η ), 0 ∂η z (η )d i=1
c(α, β) γ2 "
i=γ1 +1
ξαi ,βi (η, η )
k "
∂δαi ∂η i ξ a (η ) β
(5.23)
i=γ2 +1
βi ≤ j . This finishes the proof of Lemma 2.
Theorem 5 Proof. this theorem we need to first construct our nth order approx#n To prove a j imation ξ = j =1 δ ξj and check that the ξj satisfy the appropriate decay estimate (5.2). Second, we check that this particular nth order approximation satisfies (5.4). The construction of our nth order approximation is by induction. We define ξ1 (t, η) to be the linearization from Sect. 2. Namely, 1 ∂η Lξ1 , J0 ξ1 (η, 0) = ξ (η). ∂t ξ1 = −
By Theorem 2 we recover the estimate
ξ1 (·, t)H s ≤ Ceλt ξ H s .
To finish the induction we assume that {ξ1 , . . . , ξk−1 } are defined and satisfy the growth rate (5.2). We constructed the k th C ∞ function ξk of the nth order approximation and check that it also satisfies (5.2). We define ξk (t, η) to be a solution to the PDE 1 ∂k 1 ∂k a ∂ t ξk =− N (ξ ) , k k k! ∂δ k! ∂δ (5.24) δ=0 δ=0 ξk (η, 0) = 0. We now show that ξk satisfies the correct growth rate using our induction hypothesis. Using Lemma 2 we separate 1 ∂k 1 a − N (ξ ) = − Lξk + f (ξ1 , . . . , ξk−1 ), (5.25) k k! ∂δ J 0 δ=0 where L is the linearized operator of Sect. 2 and f (ξ1 , . . . , ξk−1 ) is a product ξk ’s. We let f (ξj ) = f1 (ξj ) + f2 (ξj ) + f3 (ξj ),
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depending on if it comes from (I ), (I I ), or (I I I ) in Lemma 2. We claim that the induction assumption and the form of f (ξ1 , . . . , ξk−1 ) yields an estimate f (ξ1 , . . . , ξk−1 )H s ≤ c(s, k)ekλt .
(5.26)
In particular # when δ = 0, each (I ) − (I I I ) is comprised of a sum of a product of ξαi ’s such that αi = k for each term. This follows from ∂ j a ξ = j ! ξj (t, η) ∂δ j δ=0 and a careful examination of (I ), (I I ), and (I I I ). By the induction hypothesis, each ξαi has a simple H s bound on the growth rate ξα (·, t) s ≤ c(s, k)eαj λt . j H This implies f1 H s ≤ c(s, n)eλt
#
αj
= c(s, k)ekλt .
A similar argument controls f2 by the same bound. For f3 , we use j ∂ log z0 (η) − z0 (η ) G(η, η )dη ∂ηj L2 j = log z0 (η) − z0 (η ) DM0 G(η, η )dη L2 j ≤ C DM0 G(η, η )dη 2 , L
(5.27)
where M0 = M is the multiplier from Sect. 3 with w = c cosh(ξ0 + iη). Combining (5.27) and Theorem 3, we find after a long calculation that f3 L2 ≤ c(s, k)ekλt . Combining the bounds on the fj ’s, we finish the proof of (5.26). Finally, we use (1.16), (5.26), and Duhamel’s Principle to get t ξk (η, t)H s ≤ c(s, k) eλ(t−τ ) f (ξ1 , . . . , ξk−1 )H s (τ )dτ 0
≤ c(s, k)ekλt .
(5.28)
This finishes the proof of (5.2). In order to estimate the error of using ξ a , we use a Taylor expansion ∂t ξ a + N (ξ a ) =
n
δ j ∂t ξj + N (ξ a )
j =1
n δj ∂ j a N (ξ ) j j ! ∂δ δ=0 j =1 n+1 n+1 ∂ δ = N (ξ a ) n+1 (n + 1)! ∂δ δ=δ = N (ξ a ) −
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for some δ ∈ (0, δ) and define the remainder as Ran =
δ n+1 ∂ n+1 a N (ξ ) . n+1 (n + 1)! ∂δ δ=δ
(5.29)
We bound Ran in H s by following the same method as above except we have terms of the form n i−j ∂j a ξ (η, t) = c(i, j, n) δ ξi (η, t). ∂δ j δ=δ i=j
However, for all t ∈ [0, Tδ ], δ eλt ≤ δeλTδ = θ0 , which implies j n ∂ a i−j ≤ ξi (η, t)H s δ ξ (η, t) ∂δ j s δ=δ H i=j
≤ c(s, i)ej λt
n−j
θ0i = c(s, n, θ0 )ej λt .
(5.30)
i=0
Given estimate (5.30), the calculation is very similar to the δ = 0 case for θ0 sufficiently small. 6. Difference Inequality and Instability The goal of this section is to derive a differential inequality for the difference of solutions to the Love equation and the approximate Love equation as stated in the following theorem. Theorem 6. Suppose that ξ is a solution to the Love equation (1.13) and ξ a is a solution to the nth order approximate Love equation (5.4), then the difference ξ d = ξ a − ξ satisfies the differential inequality
2 s 2 2 ∂t ξ d s ≤ C(n, s, ξ0 ) ξ d s + ξ d s + Rna H s , (6.1) H
H
H
Rna
is the remainder to the approximate solution as defined by Theorem 5. The where constant C depends on n, s and ξ0 and ξ a s+2 . H s+1 Proof. From Eqs. (1.13) and (5.4) the difference ξ d = ξ a − ξ satisfies the equation
1 d d a 1 1 1 d d + − a ψ ψR − ψRa + ψ − ψ a + Rna . (6.2) ∂t ξ = J J dη J dη J dη We now use a standard energy technique. Upon differentiating both sides of this equation s times in η and multiplying the result by ∂ηs ξ , we have the following inequality:
1 s d 2 s d s 1 d a s d s 1 d a 1 ∂t ∂η ξ ≤ ∂η ξ ∂η + ∂ − a ψ ψ ξ ∂ − ψ R η η 2 J J dη J dη R
1 d a + ∂ηs ξ d ∂ηs ψ − ψ + ∂ηs ξ d ∂ηs Rna J dη = (I ) + (I I ) + (I I I ) + (I V ).
(6.3)
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Integrating each of these terms in η and bounding the results in terms of ξ d H s then yields the theorem. The first term follows from some straightforward estimates and the bounds on the singular integral given by Theorem 3. Differentiating the stream function s + 1 times is not a problem since it is defined by the approximate solution which is C ∞ . The second term is also relatively straightforward, although now we must be careful about the case in which the stream functions are differentiated s + 1 times. Here we make use of an integration by parts argument. The third term, I I I , is much more delicate, and we will need to rewrite that term to allow us to extract bounds in terms of the difference of solutions. We then follow the general strategy used to bound the singular integral operator in higher Sobolev spaces as described in Sect. 3. In particular, upon each differentiation of the singular integral, a multiplier is used to move derivatives off of the logarithm terms. In this case, however, we will have two multipliers corresponding to the solutions z and za respectively. It is from this difference in multipliers that we will extract bounds in terms of ξ d . As before, we can only do this s − 1 times, leaving a final derivative which must be considered carefully. As many of the details are similar to those used in Sect. 3, we provide a sketch of the argument, highlighting the unique features of having the difference of multipliers. For the sake of clarity, we consider each of the terms I − I I I in the following subsections. Estimates on the remainder term Rna are given in Theorem 5. 6.1. Estimates for I . For the first term of (6.3), we need only to establish an elementary a is defined in terms of the approxiH s bound for the difference J −1 − (J a )−1 . Since ψ mate solution which is C ∞ , we’re free to differentiate s + 1 times. We begin with a few preliminary propositions. Proposition 10. Suppose |ξ |, |ξ a | ≤ ξ0 , then there exists a constant C, depending only on s, ξ0 , and c such that for k ≤ s − 1, k (6.4) ∂η J − J a ∞ ≤ C ξ d s L
H
and s ∂η J − J a
L2
≤ C ξ d
Hs
.
(6.5)
Proof. Let D c denote the difference between cosh evaluated at 2ξ0 +2ξ and at 2ξ0 +2ξ a , and let D s denote a similarly defined difference for sinh. Then ∂η D c = 2 ∂η ξ a D s − sinh(2ξ0 + 2ξ )∂η ξ d and ∂η D s = 2 ∂η ξ a D c − cosh(2ξ0 + 2ξ )∂η ξ d . Then in terms of these differences, derivatives of J − J a can be computed as ∂ηk J − J a = c2 ∂ηk−1 ∂η ξ a D s − sinh(2ξ0 + 2ξ )∂η ξ d ,
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and so by the Leibniz formula (7.2) we have k a ∂η J − J r ≤ C ∂ηk ξ a r D s L∞ + ∂η ξ a L∞ ∂ηk−1 D s r L L L + ∂ηk−1 sinh(2ξ0 + 2ξ ) r ∂η ξ d ∞ L L
k d + sinh(2ξ0 + 2ξ )L∞ ∂η ξ r . L
The first term can be bounded in terms of ξ d while the second term will generate four more terms upon each differentiation, each of which will have similar terms. The end result is a sum k k j d (6.6) ∂η J − J a r ≤ C ∂η ξ r + ξ d ∞ . L
The proposition then follows.
j =0
L
L
Proposition 11. Suppose |ξ |, |ξ a | ≤ ξ0 , then there exists a constant C depending only on s, ξ0 , and c such that for k ≤ s − 1,
k 1 1 d ∂ ≤ C (6.7) − ξ s , η J H J a L∞ and for k = s
Proof. First note that
k 1 1 ∂ ≤C − ξ d s . η J H J a L2
1 1 − a J J
=
(6.8)
−1 Ja − J JJa
so that by the Leibniz formula (7.2)
k 1 a −1 1 k a ∂ ∞ η J − J a r ≤ C ∂η J − J Lr J J L L −1 + J − J a L∞ ∂ηk J J a r . L
(6.9)
By Proposition 10, the terms involving the difference J − J a are bounded by ξ d H s . We also note that by the Moser lemma (7.3) and the fact established in Lemma 6 that J and J a are both bounded above and away from zero we have k d a −1 k a a −k−1 a k−1 d max γ ! J J ∞ J J L∞ k J J dηk J J r ≤ C 1≤γ r ≤k L dη L L k d a ≤ C dηk J J r . L By the results of Lemma 6, the terms in Eq. (6.9) involving J J a are bounded in L∞ . The proposition then follows.
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Proposition 11 now allows us to integrate the first term (I ) in (6.3) and conclude that
2 s d s 1 d a 1 ≤C ∂ ξ ∂ a s . − ψ (6.10) ξ d s ∂η ψ η η H a H J J dη 1 L
ψa
is defined in terms of the approximate solution, we can bound any number of Since a in L2 . In this case, by the same arguments presented in Sect. 4.2 to derivatives of ψ bound derivatives of ψ we have s+2 s+1 ,a ∂η ψ 2 ≤ C 1 + ξ a s+1 . L
6.2. Estimates for I I . For the second term in (6.3) we have the following
s ds 1 ∂ a d s−k −1 d k+1 a J = ψR − ψ R . ψ − ψ R R s−k k+1 s dη J ∂η dη dη k=0
The only difficulty in establishing bounds comes when there are s + 1 derivatives on the difference ψRa − ψR in which case we need an integration by parts argument. For the other cases we can use previous propositions as follows. If k ≤ s − 2, then making use of the observation that ψRa − ψR = − J a − J we have upon integrating (II), 2 d −1 k+1 a (6.11) ψR − ψR ∞ ≤ C ξ d s , ξ s J s ∂η H
H
L
H
where the constant C depends on the H s norm of ξ and ξ a . For the case k = s − 1 then we have a similar bound as the previous case, namely 2 d (6.12) ξ s ∂η J −1 ∞ ψRa − ψR H s ≤ C(ξ, ξ a ) ξ d s . H
H
L
Finally, when k = s we recall the notation from Proposition 10 and note that ψRa −ψR = D c . Differentiating then gives ∂ηs ∂η D c = 2∂ηs ∂η ξ a + 2D s ∂ηs+1 ξ a − ∂ηs sinh(2ξ0 + 2ξ )∂η ξ d − sinh(2ξ0 + 2ξ )∂ηs+1 ξ d . Upon integrating in η, the first two terms yield a bound of Cξ d 2H s from Proposition 10 while for the last term we have 2π 2π 2 1 1 1 s d s+1 d sinh(2ξ0 + 2ξ )∂η ξ ∂η ξ dη = sinh(2ξ0 + 2ξ ) ∂η ∂ηs ξ d dη. J J 2 0 0 (6.13) We can now integrate by parts so that s ∂η ∂η D c
L1
2 ≤ C ξ d s . H
From the inequalities (6.11)–(6.14) we then conclude that
2 s d s 1 a ≤C ∂ ξ ∂ ψ ∂ − ψ ξ d s , η R η R η H J 1 L where the constant C depends on ξ , ξ a , and K.
(6.14)
(6.15)
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6.3. Estimates for I I I . For the third term in (6.3) we make use of the fact that the derivative of the stream function ψ can be represented in terms of a complex inner product of ∂η z with the singular integral, as in Eq. (1.15). In particular, we compute the derivative of the difference ψ a − ψ, where ψ a is the stream function defined by za (η), the contour defined by the approximate equation (5.4). If we introduce zd (η) = za (η) − z(η), then it is possible to write, after rearranging terms,
2π d a a 1 d log z(η) − z(η ) ∂η z (η ) dη ψ − ψ = i∂η z , dη 2π 0
2π d 1 log z(η) − z(η ) ∂η z(η ) dη + i∂η z , 2π 0 2π 1 log za (η) − za (η ) + i∂η za , 2π 0
a − log z(η) − z(η ) ∂η z (η ) dη .
(6.16)
We now consider each of these three terms separately. The first two will follow from previous propositions, while the third will be much more delicate. For the first term, we note that by a straightforward application of the singular integral estimate given in Theorem 3 we have
2π
d log z(η) − z(η ) ∂η z (η )dη
Hs
0
≤ C(ξ, s) zd
Hs
,
(6.17)
where the constant depends on K, s, and ξ . We therefore need only an H s bound on zd . Note that k d ∂η z r ≤ ∂ηk x d r + ∂ηk y d r , L
L
L
where x d = c cosh(ξ0 + ξ a ) − cosh(ξ0 + ξ ) cos(η), y d = c sinh(ξ0 + ξ a ) − sinh(ξ0 + ξ ) sin(η), and so by (7.2) k d ∂η x
Lr
≤ C ∂ηk cosh(ξ + ξ a ) − cosh(ξ0 + ξ ) r cos(η)L∞ L k a + cosh(ξ + ξ ) − cosh(ξ0 + ξ ) L∞ ∂η cos(η) r . L
Bounds on the derivatives of the differences of the cosh terms in terms of ξ d are obtained exactly like the difference J a − J obtained in Proposition 10, as is the L∞ norm of the difference, and so for k ≤ s we have k d (6.18) ∂η z r ≤ C ξ d s . L
H
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The inequalities (6.17) and (6.18) then imply that the first term in (6.16) is bounded by
2π 2 s d s 1 a 1 d ≤C ∂ ξ ∂ z , log z(η) − z(η ) ∂ z (η )dη i∂ ξ d s , η η η 1 η H J 2π 0 L (6.19) where the constant C depends on K and ξ a H s . For the second term in (6.16) we need to consider the following combination of derivatives:
s−k s 1 j 2π ∂η ∂ηk J −1 i∂ηs−k−j +1 zd , log z(η) − z(η ) ∂η z(η )dη 2π 0 k=0 j =0
which can be handled exactly as the previous term except when k = j = 0. In this case, there are s + 1 derivatives on zd and we need to integrate by parts as we did to derive the inequality (6.14). In all cases, we have
2π 2 s d s 1 d 1 ≤C ∂ ξ ∂ i∂ z , log z(η) − z(η ) ∂ z(η )dη ξ d s , η η η η H J 2π 0 1 L (6.20) where the constant depends on ξ , ξ a , K and s. To obtain bounds on the third term in (6.16), it is sufficient to prove the following lemma. Lemma 3. Let ξ, ξ a < ξ0 /4, then there exists a constant C such that 2π . - a d a a z (η ) dη (η) − z (η ) − log z(η) − z(η ) ∂ ≤ C log z ξ η Hs
0
Hs
,
(6.21) where the constant C depends on s, ξ0 , and ξ a sH s+1 . Before proving this lemma, recall from Sect. 3 that we can isolate the singularity by defining φ(η, η ) = µ(η, η )∂η za (η ), (η, η ) = (1 − µ(η, η ))∂η za (η ), φ
(6.22) (6.23)
where µ(η, η ) is the smooth cutoff function defined by (3.9). Now the singular integral can be separated into two pieces, the first of which is concentrated around the singularity allowing us to use our multiplier technique to move derivatives off of the logarithm terms. For the second integral, we are free to differentiate the logarithm terms and estimate directly. We also need to introduce an additional multiplier analogous to the one introduced in Definition 2. Let M a = M a (η, η ) denote the multiplier a ∂η z (η), za (η) − za (η ) . M a (η, η ) = (6.24) ∂η za (η ), za (η) − za (η ) The following proposition bounds derivatives of the multiplier difference.
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Proposition 12. Suppose that z, za ∈ PsK and |η − η | ≤ ρ. If j + k ≤ s − 2 then j k a ∂η ∂η M − M
L∞ ⊗L∞
≤ C ξ d
Hs
(6.25)
,
while for j + k = s − 1, j k a ∂η ∂η M − M
≤ C ξ d
,
(6.26)
F d G − F Gd Fa F = − , a G G GGa
(6.27)
L2 ⊗L2
Hs
where the constant C depends on ξ , ξ a , s and K. Proof. We first write the multipliers as quotients Ma − M = where F d = F a − F = ∂η zd (η), za (η) − za (η ) + ∂η z(η), zd (η) − zd (η ) and Gd = Ga − G = ∂η zd (η ), za (η) − za (η ) + ∂η z(η ), zd (η) − zd (η ) . Then just as in the proof of the bounds on M, the Leibniz rule gives α β F Gd ∂ ∂ η η GGa
Lr
−1 β ≤ C ∂ηα ∂η F Gd r GGa ∞ L L −1 β + F Gd ∞ ∂ηα ∂η GGa r , L
L
and we need only to observe that d G
L∞
≤ C(z, za ) zd H s ,
and for α + β ≤ s − 1, α β ∂ ∂ d a d G ≤ C(z, z ) z s . ∂η∂η 2 H L The second term in (6.27) is handled in the same way. The fact that d z
H2
finishes the proof of the proposition.
≤ C(z, za ) ξ d
Hs
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341
Proof of Lemma 3. We first consider the more difficult case in which the support of φ is concentrated around the singularity. If we differentiate such an integral s times, then all but one of the derivatives may be moved off of the logarithm through the use of multipliers. By adding and subtracting terms to introduce differences, we have ∂ηs
2π 0
log za (η) − za (η ) − log |z(η) − z(η )| φ dη
s−1 log za (η) − za (η ) − log |z(η) − z(η )| DM a φ dη
2π
= ∂η 0
2π
+ ∂η 0
s−1 s−1 log |z(η) − z(η )| DM a φ − DM φ dη .
(6.28) (6.29)
Recall from Definition 2 the operator DM φ = ∂η φ + ∂η (Mφ) and define similarly DM a φ = ∂η φ + ∂η (M a φ). We now consider these two integrals separately. The first can be expressed as s−1 s−1 log |za (η) − za (η )|DM a DM a φ − log |z(η) − z(η )|DM DM a φ dη s−1 = log |za (η) − za (η )| − log |z(η) − z(η )| ∂η DM dη (6.30) a φ s−1 + log |z(η) − z(η )|∂η MDM a φ s−1 dη . (6.31) − log |za (η) − za (η )|∂η M a DM a φ Now consider the integral (6.30). If we introduce the variable z = za + (z − za ). Then
log |z (η) − z (η )| − log |z(η) − z(η )| = a
a
1
0
d log |z (η) − z (η )| d d
and (6.30) = = 0
1
d log |z (η) − z (η )| d d
∂ s−1 D a φ dη ∂η M
0 1 (z (η) − z (η ), zd (η) − zd (η ))
∂ s−1 D a φ d dη . (z (η) − z (η ), z (η) − z (η )) ∂η M
We claim that this last quantity is bounded in L2 in terms of ξ d . This follows by writing
z (η) − z (η ), zd (η) − zd (η ) z (η) − z (η ) zd (η) − zd (η ) , = η − η η − η (z (η) − z (η ), z (η) − z (η ))
z (η) − z (η ) z (η) − z (η ) −1 × , , (6.32) η − η η − η and making use of the facts that z and za are in PsK and as was shown in a previous proposition, d ∂η z 2 ≤ Cξ d H s . L
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In addition, we observe that since za ∈ C ∞ we can follow the proof of Proposition 4 to obtain the bound s−1 φ (6.33) ∂η DM 2 ≤ C ξ H s+1 + ξ sH s+1 . a L
We absorb this quantity into the constant in (6.21). Now consider the integral (6.31). By adding and subtracting terms, we can organize this term as s−1 a a a s−1 log |z(η) − z(η )|∂η MDM dη a φ − log |z (η) − z (η )|∂η M DM a φ s−1 = log |z(η) − z(η )|∂η M − M a DM (6.34) a φ dη s−1 + log |z(η) − z(η )| − log |za (η) − za (η )| ∂η M a DM (6.35) a φ dη s−1 + log |z(η) − z(η )| M − M a ∂η DM (6.36) a φ dη s−1 + log |z(η) − z(η )| − log |za (η) − za (η )| M a ∂η DM (6.37) a φ dη . Each of these four terms contains a difference which we can bound in terms of ξ d . In particular, the terms (6.35) and (6.37) both have a difference of logarithms which can be handled exactly as in (6.32) above. The other two terms (6.34) and (6.36) contain a difference in multipliers M − M a which is handled via Proposition 12. Finally, we consider the integral (6.29). We cannot introduce any more factors of DM and so we simply carry through the final derivative, giving two terms ∂ s−1 s−1 dη log |z(η) − z(η )| DM a φ − DM φ ∂η
∂η z(η) s−1 s−1 = φ − D φ dη (6.38) D a M M z(η) − z(η ) s−1 s−1 + log |z(η) − z(η )|∂η DM dη . (6.39) a φ − DM φ The bounds on these two terms are handled in the same way as in the end case for the singular integral operator in H s taking into account the fact that here we have the difference of two multiplier derivatives. In particular, we have s−1 s−1 φ − DM DM a φ =
s−1
mlα,β ,
(6.40)
l=1
where mlα,β
≡
β ∂ηα0 ∂η 0 φ
l " i=1
β ∂ηαi ∂η i M a
−
l "
β ∂ηαi ∂η i M
.
(6.41)
i=1
The multi-indices satisfy α0 +α1 +. . .+αl = s −1−l and β0 +β1 +. . . βl = l. Note that because of the difference in Eq. (6.40), the case α0 = s − 1 vanishes. Through judicious
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rearrangement of terms, this can be expressed as a sum of derivatives on M a − M times products of derivatives of M a and of M. Indeed, the key observation is that β
mlα,β = ∂ηα0 ∂η 0 φ
l
" αj βj a " αj βj β ∂ηαi ∂η i M a − M · ∂ η ∂η M · ∂η ∂η M.
i=1
j
(6.42)
j >i
We now bound (6.38). This is done using the same argument as in Proposition 5, namely by considering the different possible arrangements of derivatives in mlα,β . We refer the reader to that proposition for additional details. If αi + βi ≤ s − 2 for all i and β0 ≤ s − 2, then the estimate (6.43) ∂η ∂η z(η)mlα,β 2 2 ≤ Cξ d H s L ⊗L
follows immediately from Propositions 12 and 13, and the analogous bounds for M a . Theorem 8 then applies. If β0 = s − 1, then all the derivatives are applied to φ and we are left only with the difference M a − M. Again, we can use Theorem 8 obtain the same bound as in the previous case. Finally we consider the case where αi + βi = s − 1 for some i so that we have terms like q β φ∂ηαi ∂η i M a − M M p M a , where p + q = s − 1. Now we need to consider the explicit structure of M a − M as a fraction of complex inner products and how the s − 1 derivatives are then distributed over the various terms in that fraction. For most of those terms, there are fewer than s − 1 derivatives on any one term and we can differentiate once more and use Theorem 8 as in the previous cases. The delicate end case is treated as in Proposition 5, making use of β β the observation that ∂ηα ∂η M = ∂ηα ∂η (M − 1). Instead of M a − M we write
F Fa Ga G − − − Ga Ga G G d d F − G G − (F − G) Gd = , GGa
(M a − 1) − (M − 1) =
where for example, F d − Gd = ∂η zd (η) − ∂η zd (η ), za (η) − za (η ) + ∂η z(η) − ∂η z(η ), zd (η) − zd (η ) . This introduces an additional difference which can be used to estimate the singular s−1 integral directly. Again, the L2 bound on this term will be ξ d H s . For the second term (6.39), we need L2 bounds on s−1 s−1 dη . log |z − z |∂η DM a φ − DM φ
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Just as in the proof of Proposition 5, we look at ∂η mlα,β and consider the possible ways to distribute the derivatives. The only difficulty comes when αi + βi = s − 1 for some i > 0. But in this case, we can integrate by parts in η as in Eq. (3.26). Finally, we note that the case where we are isolated away from the singularity in (6.21) is straightforward since we are free to differentiate the logarithm terms s times. The details are similar to the proof of Proposition 1, and the difference ξ d can be extracted from the difference in the derivatives of the logarithms by techniques similar to those used for the difference M a − M. 6.4. Proof of Instability Theorem. Proof of Theorem 1. The method of proof is similar to that used by Grenier [11]. Begin by fixing the generic perturbation ξ ∈ C ∞ . We will fix our parameters δ0 , θ0 , and n as a function of ξ0 through a series of steps. From Theorem 4 there is an ω0 such that if ξ(·, 0)H s ≤ ω0 then there exists a solution ξ(t) for all t ∈ [0, T0 ]. We fix δ0 such that δ0 ξ s = ξ(·, 0)H s = ω0 (6.44) H 4 −1 or δ0 = ξ H s ω0 . Therefore, for all δ ∈ (0, δ0 ) there exists a solution ξ δ (t, η) to (1.15) with ξ δ (η, 0) = δξ for a time span at least up to T0 . We further shrink δ0 so that T0 ≥ tξ , where tξ comes from Theorem 2. Using Theorem 5, we also have the existence of an nth order approximation, ξ a (η, t) such that ξ a (η, 0) = δξ (η). We let ) ω0 ω0 * and ξ δ H s ≤ (6.45) T = sup t : ξ a H s+1 ≤ 2 2 which in general will be much larger than T0 . Our difference ξ d = ξ δ − ξ a is defined for an approximate solution ξ a of order n, and we determine n = n(ω0 ) below. By Theorem 5 and Theorem 6,
2 δ a d d 2 2(n+1) 2(n+1)λt . (6.46) e ∂t ξ s ≤ C ξ H s+1 , ξ H s ξ s + δ H
H
For all t ∈ [0, T ], (6.45) implies C ξ a H s+1 , ξ δ H s ≤ C ω20 . We choose n so that C ω20 ≤ 2 (n + 1) λ, or
n≥
1 ω0 C − 1. 2λ 2
(6.47)
We combine (6.46), (6.47), Gronwall’s inequality, along with ξ d (η, 0) = 0 to get d (6.48) ξ s ≤ C1 δ n+1 e(n+1)λt , H
where C1 = C1 (ω0 ). We now choose θ0 = θ0 (ω0 ) > 0. Define Tδ by Tδ =
θ0 1 log . λ δ
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Tδ is the time frame on which instability will start to manifest. We choose θ0 so that Tδ ≤ T . In other words we want to extend the time scale for the local solution to extend past Tδ . If Tδ > T then by Theorem 5 for all t ≤ T , a ξ
Hs
≤
n j =1
n δ j ξj H s ≤ c(s, j ) δ j eλj t j =1
≤ c(s, j )
n
δ j eλj Tδ ≤ c(s, j )
j =1
≤ C 2 θ0
1 − θ0n 1 − θ0
n
j
θ0
j =1
C2 ≤ θ0 , 2
(6.49)
where C2 = max{c(s, j )}, so long as, say, θ0 < 41 . By (6.48) d ξ
Hs
≤ C1 δ n+1 eλ(n+1)t ≤ C1 δ n+1 eλ(n+1)Tδ = C1 θ0n+1 .
(6.50)
Choosing
1
1 1 C3 n ω0 n+1 ω0 C3 θ0 = min > 0, , , , , 4 2C1 4C2 C2 4C1
(6.51)
where C3 is defined below, then both ξ a H s+1 ≤ ω20 and ξ δ H s ≤ ω20 for all t ∈ T , which contradicts T < Tδ . Therefore Tδ ≤ T , and we have finished defining δ0 , n, and θ0 . Theorem 1 directly follows from (6.48) and (6.51). If we set t = Tδ , then by (2.9) and (6.48), a ξ
L2
≥ δ ξ1 L2 −
n j =2
≥ C 3 θ0 −
n
n δ j ξj L2 ≥ C3 δeλTδ − c(s, j )δ j eλj Tδ
j
c(s, j )θ0 ≥ C3 θ0 −
j =2
since ξ1 L2 ≥ C3 eλt and θ0 ≤
j =2
C3 C2 .
C2 2 C3 θ ≥ θ0 , 2 0 2
(6.52)
Therefore, (6.48) and (6.52) implies
δ ξ (·, Tδ ) 2 ≥ ξ a 2 − ξ − ξ a 2 ≥ C3 θ0 − ξ d s L L L H 2 C3 C3 C3 n+1 λ(n+1)Tδ θ0 − C 1 δ e θ0 − C1 θ0n+1 ≥ θ0 , ≥ ≥ 2 2 4 since θ0 ≤
C3 4C1
1
n
, and this yields (1.27) and (1.28). Since cosh(2x) − 1 ≥ 2x 2 then
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2π 0
ξ0 + ξ δ ξ0
c2 J (ξ , η )dξ dη ≥ 2
≥c
2π
0 2π
2 0
ξ0 + ξ δ
cosh 2ξ − 1 dξ dη
ξ0
ξ0 + ξ δ
ξ
2
dξ dη
ξ0
2 3 c2 2π 2 δ ≥ 3ξ0 ξ + 3ξ0 ξ δ + ξ δ dη 3 0 c 2 C 3 ξ0 ≥ θ0 , 4 and we set 0 =
c2 C3 ξ0 4 θ0 .
Therefore,
χTδ (E0 ) − χTδ (Ep )
L1
This proves (1.29).
≥ 0 > 0.
7. Appendix: Background Our method relies on establishing H s bounds on the Love equation, for s ≥ 3. Since (1.15) is defined in terms of a line integral, similar to the contour dynamics equation, see [17], we need to consider Sobolev estimates for a class of singular integral operators of the form log w(η) − w(η ) ϕ(η )dη , (7.1) where w ∈ H s ([0, 2π ]; C) and ϕ ∈ H s−1 ([0, 2π ]; C). If our contour w(η) is smooth enough and has no self-intersections, then we will be able to bound the operator in H s . Lemma 4 (Leibniz’ rule). Let u, v ∈ L∞ (Rn ) be such that ∂ α u, ∂ α v ∈ Lr (Rn ) with 1 ≤ r ≤ ∞. Then ∂ α (uv) ∈ Lr (Rn ) and 2 α ∂ (uv) r n ≤ 4 α2 ∂ α u r n vL∞ (Rn ) + uL∞ (Rn ) ∂ α v r n . (7.2) L (R ) L (R ) L (R ) Lemma 5 (Moser). Let u ∈ L∞ (Rn ) be such that ∂ α u ∈ Lr (Rn ) with 1 ≤ r ≤ ∞. If g ∈ C r (R), then ∂ α (g ◦ u) ∈ Lr (Rn ) and there is a constant C that depends on r and α such that
α γ −1 (γ ) ∂ (g ◦ u) r n ≤ C max ◦ u ∞ n uL∞ (Rn ) ∂ α uLr (Rn ) . g L (R ) 1≤γ ≤α
L (R )
(7.3) In our case n = {1, 2}. We need estimates on M and its derivatives in order to close the H s bound on (7.1). Since it is possible that the denominator of M vanishes for arbitrary values of η and η , we need to restrict η to be within a small distance of η .
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Proposition 13. Let w ∈ PsK and suppose Bρ = |η − η | ≤ ρ with ρ ≤
1 . 8K 2
Then
ML∞ ⊗L∞ (Bρ ) ≤ C. If j + k ≤ s − 2 then
k j ∂η ∂η M
L∞ ⊗L∞ (Bρ )
while if j + k = s − 1 then k j ∂η ∂η M
(7.4)
≤ C wH s ,
(7.5)
≤ C wH s ,
L2 ⊗L2 (Bρ )
(7.6)
where C depends on K and s. Proof. Dividing both numerator and denominator by the quantity η − η , we proceed by first observing that because w ∈ PsK , ∂η w(η), w(η) − w(η ) ≤ w2 ∞,1 . W η − η
We now bound the denominator from below. Note that
w(η) − w(η ) w(η) − w(η ) ∂η w(η ), = ∂η w(η ), ∂η w(η ) + − ∂η w(η ) η − η η − η 2 w(η) − w(η ) ≥ ∂η w(η ) − ∂η w(η ) · − ∂ w(η ) η . η − η (7.7) Since
w(η) − w(η ) − ∂ w(η ) η ∞ ≤ Kρ η−η L
and ∂η w(η )L∞ is bounded from below by the constant 1/(2K), it follows that for ρ small enough, the right-hand side of (7.7) is bounded from below by 1/(8K 2 ). The L∞ bound follows. For higher derivatives, we express M as the product F G−1 , where
w(η) − w(η ) , F = ∂η w(η), η − η
w(η) − w(η ) G = ∂η w(η ), , η − η and then using (7.2), k j j ∂η ∂η M r 2 ≤ C ∂ηk ∂η F L (R )
Lr
−1 G
L∞
j + F L∞ ∂ηk ∂η G−1 r .
Note F L∞ ≤ 2∂η w2L∞ ≤ 2K 2 , and by (7.7), G−1 L∞ ≤ C.
L
(7.8)
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Y. Guo, C. Hallstrom, D. Spirn j
Now consider ∂ηk ∂η F . Again, using the Leibniz formula, the defintion of F , and the observation that 1 w(η) − w(η ) = wη (η + (η − η )) d, η − η 0 then we find
k j ∂η ∂η F
Lr
≤ C wW ∞,1 wW r,j +k+1 ≤ C wW r,j +k+1 ,
and a similar bound holds for GLr . Now by (7.3), derivatives on G−1 are bounded as follows: j k −1 γ −1 max ∂ α GLr ∂η ∂η G r ≤ C max γ !G−1−γ ∞ GL∞ 1≤γ ≤j +k |α|=j +k L L ≤ C 1 + K 2 wW ∞,1 wW r,j +k+1 ≤ C K + K 3 wW r,j +k+1 , where we’ve made use of the bound G−γ ≤ K 2γ . Bound (7.8) follows.
Given the multiplier M, we move derivatives off the kernal and onto the integrand by integrating by parts, so long as it is well-defined. We mimic [17] with the following lemma that allow us to perform such an integration by parts by changing the derivative in η on the singular integral operator (7.1) into a covariant-like derivative DM on ϕ. We include the proof for the interest of completeness. Lemma 6. Let w ∈ PsK , ∂η φ(η, η ) ∈ L2 ⊗ L2 , and φ(η , η ) ∈ L2 , then P .V . ∂η log w(η) − w(η ) φ(η, η )dη 2π =− log w(η) − w(η ) ∂η φ(η, η )dη
(7.9)
0
and ∂η
2π
log w(η) − w(η ) φ(η, η )dη =
0
log w(η) − w(η ) ∂η φ(η, η )dη 0
∂η w(η) +P .V . φ(η, η )dη w(η) − w(η ) (7.10) 2π
holds in the sense of distribution. Proof. Recall the definition of principle value integration, P .V . ∂η log w(η) − w(η ) φ(η, η )dη = lim ∂η log w(η) − w(η ) φ(η, η )dη . →0 |η −η|≥
Dynamics Near an Unstable Kirchhoff Ellipse
For a fixed |η −η|≥
∂η log w(η) − w(η ) φ(η, η )dη
=−
349
|η −η|≥
/ log w(η) − w(η ) ∂η φ(η, η ) dη
(
+ log |w(η) − w(η + )| (φ(η, η + ) − φ(η, η − )) w(η) − w(η + ) φ(η, η − ). + log w(η) − w(η − ) We would like the last two lines to go to zero as → 0, and this follows from the regularity of φ and w. Since ∂η φ(η, η ) ∈ L2 ⊗ L2 , then 1 φ (η, η + ) − φ (η, η − )L2 = 2 ∂η φ(η, η − + 2t)dt 2 L 0 ≤ C ∂η φ(η, η )L2 ⊗L2 . Since w ∈ PsK , |log |w(η) − w(η + )|| ≤ C |log | and log |w(η) − w(η + )| (φ(η, η + ) − φ(η, η − ))L2 ≤ C |log | .
(7.11)
For the last line we use an argument of [17], w(η) − w(η + ) w(η) − w(η + ) log = log − log w(η) − w(η − ) w(η) − w(η − ) wη (η1 ) − wη (η2 ) wη (η1 ) ≤ log 1 + = log wη (η2 ) wη (η2 ) wη (η1 ) − wη (η2 ) −1 ≤ ≤ C inf{wη L∞ }, wη (η2 ) since w ∈ PsK . Here η1 ∈ (η, η + ) and η2 ∈ (η − , η), and w(η) − w(η + ) log φ(η, η − ) ≤ C φ(η , η − ) 2 , w(η) − w(η − ) 2 L L since φ(η , η ) ∈ L2 . Finally, we need to bound log w(η) − w(η ) ∂η φ(η, η )dη |η−η |≤
(7.12)
.
L2
Since w ∈ PsK then C −1 η − η ≤ w(η) − w(η ) ≤ C η − η . Therefore, log w(η) − w(η ) ∂η φ(η, η )dη 2 |η−η |≤ L log η − η ∂η φ(η, η ) dη + C ∂η φ(η, η ) dη ≤ |η−η |≤
L2
|η−η |≤
L2
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Y. Guo, C. Hallstrom, D. Spirn
log η ∂η φ(η, η − η ) dη + C 21 ∂η φ(η, η ) 2 2 2 L ⊗L 0 L * ) 1 ≤ C min |log |2 , 2 ∂η φ(η, η )L2 ⊗L2 (7.13) ≤
for all . Combining (7.11)–(7.13), for all small enough,
∂η log w(η) − w(η ) φ(η, η )dη
|η−η |≥ 2π
log |w(η) − w(η )|∂η φ(η, η )dη
+
0
1 = o 2 ,
L2
which completes the proof of (7.9). Next we note
log w(η) − w(η ) φ(η, η )dη
2π
∂η 0
2π
=
log w(η) − w(η ) ∂η φ(η, η )dη
0
+
|η −η|≥
+
|η −η|≤
∂η log w(η) − w(η ) φ(η, η )dη ∂η log w(η) − w(η ) φ(η, η )dη .
We want to check that the last line is bounded as o (1). To control this third term, we use the multiplier M. From (7.5), if w ∈ PsK then ∂η M(η, η ) ∈ L2 ⊗ L2 . Therefore, |η −η|≤
∂η log w(η) − w(η ) φ(η, η )dη
=−
|η −η|≤
∂η log w(η) − w(η ) M(η, η )φ(η, η )dη
η+ = − log w(η) − w(η ) M(η, η )φ(η, η ) η− + log w(η) − w(η ) ∂η M(η, η )φ(η, η ) dη . |η −η|≤
We can directly show ∂η M(η, η )φ(η, η ) ∈ L2 ⊗ L2 and M(η , η )φ(η , η ) = φ(η , η ) ∈ L2 . Following the proof of (7.9) we find 1 2 |log | ∂ log w(η) − w(η ) φ(η, η )dη = o η |η −η|≤
for all small enough.
L2
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351
Lemma 7. Under the assumptions of Lemma 6 and Proposition 13, and ∂η φ(η, η ) ∈ L2 ⊗ L2 , then ∂ log w(η) − w(η ) φ(η, η )dη ∂η = log w(η) − w(η ) DM φ dη . (7.14) Proof. This result directly follows from (7.9) and (7.10), since ∂ log w(η) − w(η ) φ(η, η )dη ∂η = log w(η) − w(η ) ∂η φ dη + P .V . ∂η log w(η) − w(η ) φ dη = log w(η) − w(η ) ∂η φ dη − P .V . ∂η log w(η) − w(η ) Mφ dη = log w(η) − w(η ) ∂η φ dη + log w(η) − w(η ) ∂η (Mφ) dη = log w(η) − w(η ) DM φ dη holds so long as DM φ ∈ L2 ⊗ L2 . k + + k ϕ(η )dη In fact we find ∂η log w(η) − w(η ) ϕ(η )dη = log w(η) − w(η ) DM k so long as DM ϕ satisfies the assumptions of Lemma 6. However, since we will need such s ϕ, we are left with terms in the integrand an estimate when k = s, if try to control DM of the form ∂ηs+1 w. To deal with this case we move only s − 1 derivatives onto φ, and we are left with an operator that resembles the classical Hilbert transform on a contour. Therefore, we need to establish an L2 estimate on such a singular integral operator. Recall the following classical result on Hilbert transforms on smooth contours. Theorem 7. Let g be L2 () and ∈ C 1,α , then the following estimate holds: g(ω) z − ω dω 2 ≤ c() gL2 () , L () 1
where c() ≤ c + cC2 1,α for α ∈ (0, 1). Proof. See [19], p. 165, for example. We generalize such L2
estimates for a Hilbert transform-like operator to functions of
two variables. Theorem 8. Let w(η) ∈ PsK and φ(η) ∈ L2 , then for a.e.η we have 2π φ(η ) P .V . dη ≤ C φ(η )L2 . w(η) − w(η ) 0 L2
(7.15)
If φ(η, η ) is a function of two variables, then the following two estimates hold: 2π % $ φ(η, η ) P .V . ∂η φ(η, η ) 2 2 + φ(η, η)L2 (7.16) dη ≤ C L ⊗L w(η) − w(η ) 2 0
L
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Y. Guo, C. Hallstrom, D. Spirn
and 2π P .V . &0 ≤C
φ(η, η ) dη w(η) − w(η ) L2 1 2 ∂η φ(·, η ) 2 dη ∞
L (η )
' + φ(η , η ) L2 .
(7.17)
The constant C depends only on K. 1
Proof. Since our curve w ∈ PsK , then it is C 2, 2 and has no self-intersections. Therefore, w−1 is well-defined. Let ζ = w(η) then P .V .
2 φ(η ) dη w(η) − w(η ) L2 0 2 φ w−1 (ζ ) dζ dζ . = P .V . −1 −1 ∂η w w (ζ ) ζ − ζ ∂η w w (ζ )
Let
2π
φ w−1 (ζ ) . (ζ ) = ∂η w w−1 (ζ )
Then by Theorem 7 P .V .
2π
0
2 φ(η ) (ζ ) 2 dζ P .V . dη = dζ w(η) − w(η ) ∂η w w−1 (ζ ) ζ −ζ L2
−1 (ζ ) 2 dζ ≤ C() inf ∂η w w−1 (ζ ) ζ ∈
−1
≤ C() inf ∂η w(η) η
× ∂η w(η ) dη ≤ C
2 2π φ(η ) ) ∂ w(η 0 η
2π
φ(η ) 2 dη
0
since |∂η w| ≥ K −1 , which finishes (7.15). In general our integrand will depend on both η and η . We improve our result to functions of two variables. Let φ(η, η ) dη , F (η) = P .V . w(η) − w(η ) then F (η)L2
φ(η, η ) − φ(η, η) φ(η, η) dη + P .V . dη , ≤ P .V . w(η) − w(η ) w(η) − w(η ) L2 L2
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353
and we bound these terms separately. We first consider the first term. By Morrey’s inequality
φ(η, η ) − φ(η, η) 2 P .V . dη dη w(η) − w(η ) 2 η − η φ(η, η ) − φ(η, η) dη dη ≤ |η − η | |w(η) − w(η )| 2 + 1/2 ∂η φ(η, ·) 2 dη ≤ K2 dη dη |η − η |1/2
∂η φ(η, ·) 2 dη
≤C ≤ C ∂η φ
L2 ⊗L2
2
dη |η − η |1/2
dη
.
For the second term we note φ(η, η) 1 P .V . = φ(η, η)P .V . dη dη w(η) − w(η ) w(η) − w(η ) L2 L2 ≤ C φ(η, η)L2 . This finishes (7.16). The proof of (7.17) follows in a similar fashion. Let φ(η, η ) dη , F (η) = P .V . w(η) − w(η ) then φ(η, η ) − φ(η , η ) φ(η , η ) F (η)L2 ≤ P .V . dη + P .V . dη , w(η) − w(η ) w(η) − w(η ) L2 L2 and we bound these terms separately. We first consider the first term. By Morrey’s inequality
φ(η, η ) − φ(η , η ) 2 P .V . dη dη w(η) − w(η ) 2 η − η φ(η, η ) − φ(η , η ) dη dη ≤ |η − η | |w(η) − w(η )| + 2 1/2 ∂η φ(·, η ) 2 dη ≤ K2 dη dη |η − η |1/2
2 1 2 dη dη ≤ K ∂η φ(·, η ) dη |η − η |1/2 L∞ (η ) ∂η φ(·, η ) 2 dη ≤C ∞ . 2
L (η )
For the second term we use (7.15) and this finishes (7.17).
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Acknowledgement. The authors wish to thank Jill Pipher for many interesting and helpful discussions. They would also like to thank Peter Constantin for his interest in the work. The authors also thank the anonymous referee for the comments that helped improve the presentation of the work. Y. G. was supported in part by NSF grants DMS-0305161, INT-9815432 and a Salomon award of Brown University. C.H. was supported in part by the NSF. D. S. was supported in part by NSF grant DMS-0306398.
References 1. Almeida, L., Bethuel, F., Guo, Y.: A remark on the instability of symmetric vortices with large coupling constant. Commun. Pure Appl. Math. 50, 1295–1300 (1997) 2. Bardos, C., Guo, Y., Strauss, W.: Stable and unstable ideal plane flows. Dedicated to the memory of Jacques-Louis Lions. Chinese Ann. Math. Ser. B 23, 149–164 (2002) 3. Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984) 4. Bertozzi, A., Constantin, P.: Global regularity for vortex patches. Commun. Math. Phys. 152, 19–28 (1993) 5. Chemin, J.-Y.: Persistance de structures geometriques dans les fluides incompressibles bidimensionals. Ann. Ec. Norm. Super. 26, 1–16 (1993) 6. Constantin, P.: Unpublished material 7. Constantin, P., Titi, E.: On the evolution of nearly circular vortex patches. Commun. Math. Phys. 119, 177–198 (1988) 8. Cordier, S., Grenier, E., Guo, Y.: Two-stream instabilities in plasmas. Cathleen Morawetz: a great mathematician. Methods Appl. Anal. 7, 391–405 (2000) 9. Dritschel, D., Legras, B.: The elliptical model of two-dimensional vortex dynamics, II. Disturbance equations. Phys. Fluids A 5, 855–869 (1991) 10. Friedlander, S., Strauss, W., Vishik, M.: Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincar´e Anal. Non Lineaire 14, 187–209 (1997) 11. Grenier, E.: On the nonlinear instability of Euler and Prandtl equations. Commun. Pure Appl. Math. 53, 1067–1091 (2000) 12. Guo,Y., Strauss, W.: Instability of periodic BGK equilibria. Commun. Pure Appl. Math. 48, 861–894 (1995) 13. Hwang, H.J., Guo, Y.: On the dynamical Rayleigh-Taylor instability. Arch. Rat. Mech. Anal. 167, 235–253 (2003) 14. Lamb, H.: Hydrodynamics. Cambridge: Cambridge University Press, 1993 15. Legras, B., Dritschel, D.: The elliptical model of two-dimensional vortex dynamics, I. The basic state. Phys. Fluids A 5, 845–854 (1991) 16. Love, A.E.: On the stability of certain vortex motion. Proc. Soc. Lond. 23, 18–42 (1893) 17. Majda,A., Bertozzi,A.: Vorticity and Incompressible Flow. Cambridge: Cambridge University Press, 2002 18. Saffman, P.: Vortex dynamics, Cambridge Monographs on Mechanics and Applied Mathematics. New York: Cambridge University Press, 1992 19. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton, NJ: Princeton University Press, 1993 20. Su, C.H.: Motion of fluid with constant vorticity in a singly-connected region. Phys. Fluids 22(10), 2032–2033 (1979) 21. Tang, Yun: Nonlinear stability of vortex patches. Trans. Am. Math. Soc. 304, 617–638 (1987) 22. Wan, Y.H.: The stability of rotating vortex patches. Commun. Math. Phys. 107, 1–20 (1986) 23. Wan, Y.H., Pulvirenti, M.: Nonlinear stability of circular vortex patches. Commun. Math. Phys. 99, 435–450 (1985) 24. Yudovitch, V.I.: Non-stationary flow of an incompressible liquid. Zh. Vychisl. mat. Math. Fiz. 3, 1032–1066 (1963) Communicated by P. Constantin
Commun. Math. Phys. 245, 355–382 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1021-3
Communications in
Mathematical Physics
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model Haru Pinson Mathematics Department, University of Arizona, Tucson, AZ 85721, USA Received: 1 May 2003 / Accepted: 9 August 2003 Published online: 20 January 2004 – © Springer-Verlag 2004
Abstract: We exploit the discrete analytic structure present in the 2d dimer model to rigorously compute the exponents of a class of two point functions in all directions. This is a dimer analogue of the critical 2d Ising spin spin correlation function.
1. Introduction The exponent of the spin spin correlation functions of the 2d critical Ising model is still known rigorously only along certain directions. In one formulation, the computation boils down to an asymptotic analysis of large matrices whose entries can be derived from lattice Green’s functions for a certain operator. Unfortunately the lack of pliable formula for the lattice Green’s function in all directions has hindered the computation. More precisely, it appears that the short distance information contained in the lattice Green’s function appears to influence or renormalize the actual exponent, and this short distance information is not so easily accessible. However, as it is well known that along certain directions the lattice Green’s function takes on a particularly simple form, and the resulting simple formulas can be utilized through, for example, the Cauchy determinant formula to yield the exponent. As of yet no simple formula for the Green’s function has appeared in all directions. Instead for a model similar to the Ising namely the 2d dimer model, we rely on the discrete analytic structure of the Green’s functions to extract enough short distance information to precisely compute the exponent for a large class of two point functions. Our hope is to ultimately obtain the exponent for the spin spin correlation function in the critical 2d Ising model in all directions. However, we first analyze an analogous two point function in the 2d dimer model. Similar to the Ising model, the partition function of this model can be written in terms of the Pfaffian. In the continuum, the spin spin correlation is described in terms of bosonic variables (see [ID])
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which arises from a certain identity between fermionic and bosonic functional integrals. Unfortunately this appears to be pliable only in the continuum. Similarly the dimer two point function we consider is defined in terms of a bosonic variable or a height function. The class of two point functions when formulated in the Pfaffian language becomes nonlocal (as the Ising spin spin correlator). There is a certain parameter which appears in these two point functions which can be chosen to be small and thus helpful since we use a perturbative analysis. Dotsenko and Dotsenko [DD] use a standard formula to expand the determinant. They use the explicit formula for the Green’s function to sum every other index in this expansion; something that resembles a first renormalization group step. Then they compute the leading order contribution exactly. We use their strategy of integrating out every other index and their change of variables. The importance of this first integration step is that it localizes the short distance information to a certain sum over an unbounded interval. It is interesting that the short distance information can be “localized” in such a simple way so that in the end precise exponents can be computed. We use the discrete analyticity of the Green’s function to evaluate this sum precisely. In the case of Dotsenko and Dotsenko, the exact formula for the Green’s function permits them the exact computation; this exact formula does not exist in all directions. The long distance asymptotics of the Green’s functions suffice for almost all other computations involving the Green’s function. The importance of the Dotsenko and Dotsenko change of variables stems from at least two reasons. First it decouples a huge chain of interconnected factors so that basically independent sums occur at the end. Second, it highlights the distinct growth behavior of the numerator and the denominator of a certain ratio that appears in the computations. Past Works. We cannot cite all the references in regard to the dimer model; in fact, when we typed in “dimer” at the physics archive, xxx.lanl.gov, we got 680 entries! Let us be content with some references on some initial works and recent works which have influenced us. Kasteleyn [Ka] and Temperley and Fisher [TF, F], first solved the 2d dimer model. There were subsequent developments by Fisher and Stephenson [FS]. At least on the mathematical side, Kenyon [Ke, Ke2] has proven numerous results about the dimer model in recent years. For example, he [Ke] has proven conformal invariance for a certain correlation function in the dimer model. Although we do not use any theorems from the theory of discrete analytic functions, we point out that this theory has been developed by Ferrand [F], Duffin [D] and others (there are references to other works in [D]). One example from this theory is an analogue of the standard Liouville theorem in complex analysis. In some ways, the thrust of this paper is an asymptotic analysis of the determinant of certain large matrices. There has been a large body of work on the asymptotic analysis of the determinants of a certain class of matrices although they appear not to be directly relevant for the class of matrices considered in this paper. However, we should mention the work of Deift, Its and Zhou [DIZ] where the Riemann-Hilbert methods are applied to the asymptotic analysis of certain determinants. Their methods seem to work not only for the leading order terms but also the higher order corrections. Also Basor [B] and Widom [W] have obtained asymptotic formulas for the determinant of Toeplitz operators generated from functions with singularities. Finally we mention the work of Palmer and Tracy [PT] on the scaling limit of the 2d Ising model from above and below the critical temperature.
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
357
Fig. 1.
1
2
0
0
3
3
1
2
3
0
2
2
1
3
1
0
Fig. 2.
2. Determinant Formulation This paper focuses on the analytical issues of analyzing the asymptotics of large determinants. There are numerous combinatoric lemmas which one must prove in order to derive the desired determinant formulation. In this section we basically quote the needed results to state the desired determinant formulas. Consider a finite square lattice Z2 ∩ ([−t, t] × [−t, t]) for a large positive number t. The partition function of the dimer model is the total number of perfect matchings on the lattice. See Fig. 1. The number of perfect matchings can be expressed as a determinant; it is often formulated in terms of the Pfaffian. See [P, Ka, Ke, Ke2]. To each dimer configuration we can assign an integer-valued height configuration. The height is defined on the dual lattice or the face of each square. This height function was introduced by Thurston [T]. As in Fig. 1, there is a dimer emanating from each open site. We can locally assign integer values by increasing the value by one as we go anti-clockwise around an open site. See Fig. 2. Using this assignment, a global height function can be assigned to each dimer configuration (see Fig. 3). The value of the height is fixed at some point.
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1
0
1
0
2
3
2
3
1
4
5
4
2
3
2
3
Fig. 3.
To each dimer configuration there is a height function. We denote this height function as h. Consider the correlation function αi(h(q )−h(p )) dimer configurations e αi(h(q )−h(p )) >= . (1) <e dimer configurations 1 The summation as indicated runs over all dimer configurations or perfect matchings. Now we take the infinite volume limit (this can be done rigorously; see [Ke2]). We can work with the finite lattice; however the form of the Green’s functions becomes slightly more complicated and some work must be done to deal with that. Anyway we will deal with the infinite volume limit. The correlation (1) then take the following simple form. We will write the correlation explicitly for a simple case, but the general form should be clear from this special case. We have
< eαi(h(q )−h(p )) >= det (I + (e−4iα − 1)A)e2iα5+iα , where the 5 by 5 matrix entries are given by Az1 ,zj Az2 ,zj Az3 ,zj Az4 ,zj Az5 ,zj
= = = = =
C(z1 − zj ) − iC(z0 − zj ), C(z2 − zj ) − iC(z1 − zj ), iC(z4 − zj ) + C(z3 − zj ) − iC(z2 − zj ), iC(z5 − zj ) + C(z4 − zj ), iC(z6 − zj ) + C(z5 − zj ),
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
359
z’6 q’ i 1
z5
z’5 i 1
z4
z’4 i
1
z3 z2 i 1
z1
1
z’3
i z’2
z’1
p’ i z’0 Fig. 4.
for j = 1, . . . , 5. The Green’s function1 for the discrete ∂¯ operator satisfies the discrete ∂¯ relations C(x + 1 − y) + iC(x + i − y) − C(x − 1 − y) − iC(x − i − y) = δx,y .
(2)
See Fig. 5. From now on, we will just refer to this function as Green’s function.
Sketch of the derivation of the determinant form of the correlation. Take a lattice such as the one in Fig. 1 (except much bigger). Let O = {open sites}, B = {blackened sites}. Let K : CB → CO be a matrix given by 1 There appears to be some disagreement on the usage of the word “Green’s function” in the mathematical community. There are many instances in which the word “Green’s function” is restricted to mean the inverse of the Laplacian. However, there are instances in which the word “Green’s function” means the inverse of more general operators. In this paper, we will take the later approach.
360
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x+i
i 1
x1
1 x+1
x i
xi Fig. 5.
Kx,y
1 i = −1 −i 0
if y = x + 1 if y = x + i if y = x − 1 if y = x − i otherwise .
See Fig. 5. Then, we know [Ke, Ke2, P] that |det (K)| = number of perfect matchings. : CB → CO where We modify the matrix K to K if x = zi or y = zj for i = 1, . . . , 5, j = 0, . . . , 6 Kx,y x,y = K −4iα Kx,y if x = zi and y = zj for some i = 1, . . . , 5, j = 0, . . . , 6. e Then, by considering the local height assignment (Fig. 2), we deduce that
< eαi(h(q )−h(p )) >= (e2iα )5 eiα as We rewrite K where Kx,y =
0 Kx,y
det (K) . det (K)
x,y = Kx,y + (e−4iα − 1)Kx,y K ,
if x = zi or y = zj for i = 1, . . . , 5, j = 0, . . . , 6 if x = zi and y = zj for some i = 1, . . . , 5, j = 0, . . . , 6.
The determinant representation of the correlation function, thus, follows. The Green’s function takes the form 2π 2π ei(uθ−vφ) 1 dθdφ. C(u + iv) = 2 8π 0 isin(θ ) + sin(φ) 0 See [Ke].
(3)
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
361
We proceed to state the desired theorem. Let r ∈ (0, +∞). Define reiθ to be “the” dual lattice point “nearest” the complex number reiθ . There is ambiguity here, but any reasonable choices will do. For example, we can take any dual lattice point within distance 1. Theorem. Let α be sufficiently small. Then < eiα(h(re
iθ )−h( 1 +i 1 )) 2 2
2
>= r
− 8α2 π
eC(θ,r,α) ,
where |C(θ, r, α)| ≤ C for some constant C. Remark. The proof actually shows that the term C(θ, r, α) is analytic in α, and moreover, n ∂ ≤ Cn C(θ, r, α) ∂α n for some constant Cn . We will actually prove the case when θ ∈ ( π4 , 3π 4 ). The other cases can be easily done in a similar way. We require numerous lemmas. Let us first proceed with those. 3. Contour Deformation Assume f, g are discrete analytic except possibly at one point fx+1 + ifx+i − fx−1 − ifx−i = −cf δx,p0 , gx+1 + igx+i − gx−1 − igx−i = cg δx,p0 ,
(4)
where cf , cg = 0, 1. Define SumH = (−fq1 + ifq2 )igq2 +
n
(−fqj + ifqj +1 )(−gqj + igqj +1 )
j =2
SumH
+(−fqn+1 + ifqn+2 )(−gqn+1 ), n
= (−fq1 + ifq2 )igq2 + (−fqj + ifqj +1 )(−gqj + igqj +1 ) j =2
+(−fqn+1 + ifqn+2 )(−gqn+1 ).
See Fig. 6 for the meaning of the points. In Fig. 6, the lattice has been rotated 45 degrees clockwise. We can deform the summation by the following analogue of the complex contour deformation which appears in standard complex analysis.
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q’1
q’2 1
i
i q1
q’3 1
i 1
1
1
q
q’n+2
1
i
i
1
q’n+1
i
2
1
i
q n+1 1
3
P0
i
....
1
q
i 1
i
q n+2
Fig. 6.
Lemma 1. SumH − (−fq1 + ifq2 )(−gq1 − igq1 ) − (−fqn+1 + ifqn+2 )(igqn+2 + gqn+2 ) = SumH + (−fqn+1 + ifqn+2 )(−cg δx,p0 ) + (igqn+1 )(cf δx,p0 ).
Proof. SumH = (−fq1 + ifq2 )igq2 +
n
(−fqj + ifqj +1 )(−gqj + igqj +1 )
j =2
+(−fqn+1 + ifqn+2 )(−gqn+1 ) = (−ifq1 + fq2 )gq2 + (−ifq2 + fq3 )(−igq2 + gq3 ) + . . . +(−ifqn + fqn+1 )(−igqn + gqn+1 ) + (−ifqn+1 + fqn+2 )(−igqn+1 ) (5) = (−fq1 + ifq2 )(−igq1 − gq1 + igq2 ) + (−fq2 + ifq3 )(−gq2 + igq3 ) + . . . +(−fqn + ifqn+1 )(−gqn + igqn+1 ) −(−fqn+1 + ifqn+2 + cf δx,p0 )(−igqn+1 ) (6) = (−fq1 + ifq2 )(−igq1 − gq1 + igq2 ) + (−fq2 + ifq3 )(−gq2 + igq3 ) + . . . +(−fqn + ifqn+1 )(−gqn + igqn+1 ) + (−fqn+1 + ifqn+2 )(gqn+2 − gqn+1 +igqn+2 − cg δx,p0 ) − (cf δx,p0 )(−igqn+1 ). (7) To get (5), there are neighboring cancellations. To get (6), we used the relations (4). Using (7), SumH − (−fq1 + ifq2 )(−gq1 − igq1 ) − (−fqn+1 + ifqn+2 )(igqn+2 + gqn+2 )
= (−fq1 + ifq2 )(igq2 ) + (−fq2 + ifq3 )(−gq2 + igq3 ) + . . . +(−fqn + ifqn+1 )(−gqn + igqn+1 ) +(−fqn+1 + ifqn+2 )(−gqn+1 − cg δx,p0 ) − (cf δx,p0 )(−igqn+1 ) = SumH + (−fqn+1 + ifqn+2 )(−cg δx,p0 ) − (cf δx,p0 )(−igqn+1 ).
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
363
L’ j
p
k L Fig. 7.
Let
Am,k
m∈L −iC(m − i − k) + C(m + 1 − k) = iC(m + i − k) + C(m + 1 − k) m ∈ L iC(m + i − k) + C(m + 1 − k) − iC(m − i − k) m = p
and U = L ∪ L ∪ {p}. See Fig. 7 for the definitions of L, p, L . In Fig. 7, k ∈ L and all open points in this direction also belong to L. A similar convention exists for L . Lemma 2.
Am,k Ak,l = Am,l .
k∈U
Proof. The basic idea is to deform the left side of the equality to infinity. What remains is the right side. If the Green’s function C(z) is purely discrete analytic (meaning that the relation (2) is satisfied with the delta function being replaced by zero), then the sum k∈U Am,k Ak,l would be identically zero. However due to the presence of the delta function we actually get Am,l .
364
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There are three possibilities m ∈ L, m ∈ L and m = p. The first two cases are similar, and first we do the case m ∈ L. Let f (k) = C(m − k), g(k) = C(k − l) and e = −1 + i. Then,
Am,k Ak,l + Am,p Ap,l + Am,p+e Ap+e,l
k∈L
=
(−if (k + i) + f (k − 1))(−ig(k − i) + g(k + 1)) k∈L
+(−if (p + i) + f (p − 1))(ig(p + i) + g(p + 1) − ig(p − i)) +(−if
(p + e + i) + f (p + e − 1))(ig(p + e + i) + g(p + e + 1)) = (if (k + i) − f (k − 1))(ig(k + i) − g(k − 1) − δk,l ) k∈L
+(if (p + i) − f (p − 1))(−g(p − 1) − δp,l ) − (if (p + e + i) −f (p + e − 1))(ig(p + e + i) + g(p + e + 1)).
(8)
The relation g(k + 1) + ig(k + i) − g(k − 1) − ig(k − i) = δk,l is used to get (8). The non-delta function terms in (8)
(if (k + i) − f (k − 1))(ig(k + i) − g(k − 1)) + (if (p + i) − f (p − 1))
k∈L
× (−g(p − 1)) − (if (p + e + i) − f (p + e − 1))(ig(p + e + i) + g(p + e + 1)) can be identified as SumH (and the term −(−fqn+1 + ifqn+2 )(igqn+2 + gqn+2 ) in Lemma 1), and this contour can be successively deformed by using Lemma 1. One of the end point terms is not present but that term does not contribute since it can be chosen to be infinitely far away. What remains in (8) are the delta function terms which give either Am,l or zero. In the case of zero, the contour deformation procedure generates Am,l through the cg δ, cf δ function in Lemma 1. Note that the function f is strictly discrete analytic on the region where the contour deformation is taking place. We consider the case m = p:
Am,k Ak,l + Am,p Ap,l + Am,p+e Ap+e,l
k∈L
=
(−if (k + i) + f (k − 1) + if (k − i))(−ig(k − i) + g(k + 1)) k∈L
+(−if (p + i) + f (p − 1) + if (p − i))(−ig(p − i) + g(p + 1) + ig(p + i)) +(−if (p + e + i) + f (p + e − 1) +if (p + e − i))(ig(p + e + i) + g(p + e + 1))
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
=
365
(if (k + i) − f (k − 1) − if (k − i))(ig(k + i) − g(k − 1) − δk,l ) k∈L
+(if (p + i) − f (p − 1) − if (p − i))(−g(p − 1) − δp,l ) + (−if (p + e + i) +f (p + e − 1) + if (p + e − i))(ig(p + e + i) + g(p + e + 1)) = Sum + del, where Sum =
(if (k + i) − f (k − 1) − if (k − i))(ig(k + i) − g(k − 1))
k∈L
+(if (p + i) − f (p − 1) − if (p − i))(−g(p − 1)) + (−if (p + e + i) +f (p + e − 1) + if (p + e − i))(ig(p + e + i) + g(p + e + 1))
del = (if1 (k + i) − f1 (k − 1) − if1 (k − i))(−δk,l ) k∈L
+(if (p + i) − f (p − 1) − if (p − i))(−δp,l ). If l ∈ L or l = p, then
del = Am,l . Otherwise, the term Am,l as in the statement of this lemma will arise later. We want to use the analysis for the case m ∈ L to analyze Sum. Consider if (k + i) − f (k − 1) − if (k − i) 1 1 = (if (k + i) − f (k − 1) − if (k − i)) + (−f (k + 1) − δk,p ) 2 2 1 1 1 = (if (k + i) − f (k − 1)) + (−if (k − i) − f (k + 1)) − δk,p 2 2 2 i 1 1 = (if (k + i) − f (k − 1)) + (if2 (k + i) − f2 (k − 1)) − δk,p , 2 2 2 where
f2 (k) = f (1 − i + k).
This satisfies the relation f2 (k + 1) + if2 (k + i) − f2 (k − 1) − if2 (k − i) = −δk,p+e . Then, 1 Sum = Sum1 + Sum2 + (g(p − 1)), 2 where Sum1 =
1
{ (if (k + i) − f (k − 1))(ig(k + i) − g(k − 1)) 2 k∈L
+(if (p + i) − f (p − 1))(−g(p − 1)) − (if (p + e + i) −f (p + e − 1))(ig(p + e + i) + g(p + e + 1))} i
Sum2 = { (if2 (k + i) − f2 (k − 1))(ig(k + i) − g(k − 1)) 2 k∈L
+(if2 (p + i) − f2 (p − 1))(−g(p − 1)) − (if2 (p + e + i) −f2 (p + e − 1))(ig(p + e + i) + g(p + e + 1))}.
(9)
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Both sums Sum1 , Sum2 can be analyzed just in the case m ∈ L by deforming the contour. There is an extra term 21 (−g(p − 1)) in (9). In using Lemma 1 for the sum Sum2 , the delta function −δk,p+e here appears as −cf δk,p0 in Lemma 1. This cancels the term in (9). Again deforming the sums Sum1 , Sum2 using Lemma 1 generates Am,l . 4. Expansion Clearly,
Am,k Ak,l =
k∈S
Am,k Ak,l − Fm,l ,
k∈U
where Fm,l =
Am,k Ak,l ,
k∈U
S = S1 ∪ S2 ∪ {p}, U = U1 ∪ U 2 . (See Fig. 8 in regard to the last two equations.) From Lemma 1, we have
Am,k Ak,l = Am,l − Fm,l .
(10)
k∈S
By using (10), we can contract expressions as A2 = A − F, A3 = (A − F )A = A − F − F A, A4 = (A − F − F A)A = A − F − F A − F (A − F ) = A − F − 2F A + F 2 . (11) This notation A2 , for example, is the usual matrix multiplication. The index of the matrices runs over the set S. We then have As =
s−1
n=1
ans F n +
s−1
bns F n A
n=0
for some coefficients ans , bns . By some recursive relations, they can easily be evaluated. Lemma 3. For n ≥ 2, s ≥ 3, (−1)n (s − n)(s − (n + 1)) . . . (s − (2n − 2)), (n − 1)! (−1)n+1 = (s − n)(s − (n + 1)) . . . (s − (2n − 2)), (n − 1)!
ans+1 = s bn−1
and for 2 ≤ s, a1s = −1, b0s = 1.
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
U
367
R’0 2
S2
..
2’ 1’
0’ p n ..
S1 2 1
U1 0
Fig. 8.
Proof. As+1 = As A =
s−1
ans F n A +
n=1
=
s−1
n=1
(ans + bns )F n A −
s−1
bns F n (A − F )
n=0 s−1
bns F n+1 + b0s A.
n=0
Thus, we have s+1 an+1 = −bns , bns+1 = ans + bns b0s+1 = b0s .
(12)
3 , n = 1, 2 and m = 0, 1, 2. Use relations We can verify that this lemma holds for an3 , bm (12) to prove this inductively in s.
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Letting j
Fx,y =
Ax,k Ak,y
k∈Uj
for j = 1, 2 (see Fig. 8 for the definition of the sets Uj ) we have 1 2 + Fx,y . Fx,y = Fx,y
Formulas. 1 s1 (x, y) + O( |x||y| ) if x, y ∈ S1 x log( ) y 1 1 1 Fx,y = M2 ( e2 π 2 x−y ) + O( |x||y| ) if x, y ∈ S2 log( x ) 1 M3 ( e 1π 2 x−yy ) + O( |x||y| ) if y ∈ S1 , x ∈ S2 or x ∈ S1 , y ∈ S2 , 2 R−x 1 log( R−y ) 1 M1 ( e1 π 2 x−y ) + O( |R−x||R−y| ) if x, y ∈ S1 1 2 if x, y ∈ S2 Fx,y = s2 (x, y) + O( |R−x||R−y| ) R−x log( ) R−y 1 M3 ( e 1π 2 x−y ) + O( |R−x||R−y| ) if y ∈ S1 , x ∈ S2 or x ∈ S1 , y ∈ S2 , 1
and |Fp,x | ≤ C where
s1 (x, y) = s2 (x, y) =
|log( px )| + |log( R−p R−x )| |p − x| u 1 log( v ) 2π 2 u−v
0
,
u = v mod 2 , otherwise
R −v
0 1 log( R0 −u ) 2 u−v 2π
0
u = v mod 2 , otherwise
If x, y ∈ S1 , then x = u(1 + i), y = v(1 + i), If x, y ∈ S2 , then x = u(−1 + i) + p, y = v(−1 + i) + p, e1 = 2(−1 + i), e2 = −2(1 + i), M1 = −2iRe or 2I m, M2 = 2iRe or − 2I m, M3 = 2Re or 2iI m, and the logarithmic function log is the complex analytic one which has very small angles near the positive real axis. Also, (1 + i), 2(1 + i), . . . , n(1 + i) ∈ S1 , 1(−1 + i) + p, 2(−1 + i) + p, . . . , (R0 − 1)(−1 + i) + p ∈ S2 . See Fig. 8. The notation M1 (z) for a complex number z indicates one of two possible values −2iRe(z) or 2I m(z). Also, Mj is symmetric in x ↔ y, i.e., if M2 ( e12 log( y )
log( x )
log( xy ) x−y ) appears
y 1 1 , then M ( 1 1 x in Fx,y 2 e2 y−x )(=M2 ( e2 x−y )) will appear in Fy,x . A precise rule for Mj (z) can be given, but we do not need it here. It will turn out that we only need the precise formulas for s1 , s2 to compute the exponents. The complex number R0 (−1 + i) + p equals the complex number R.
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
369
1 with x, y ∈ S , but the Sketch of the proof. We analyze the simplest case for Fx,y 1 elementary method works for all the cases. Let us consider the case x = u(1 + i), y = v(1 + i), where u, v are even positive integers. The asymptotics ([Ke], Theorem 11) of the Green’s function indicates that there is staggering, meaning that the behavior of the Green’s function is qualitatively different on the two sublattices. In the sum for 1 , we combine every two terms. Using this combination and the asymptotics of the Fx,y Green’s function, we arrive at 1 = Fx,y
∞ 1 2
1 + O . I m π2 (x − 2n(−1 − i))(2n(−1 − i) − y) |x||y| n=0
Note that we are summing over the even integers 2n; this stems from the combination x of two terms. Factoring out the complex number 2(−1 − i) and letting x0 = 2(−1−i) , y y0 = 2(−1−i) , we are led to the analysis of ∞
n=0
1 = (x0 − n)(n − y0 )
∞ 0
1 dn + (x0 − n)(n − y0 )
∞
dn{g(n)− 0
1 }, (x0 − n)(n − y0 )
where g(n) is a piecewise constant function which will reproduce the left side of this 1 equation. The second term on the right side of this equation is clearly O( |x||y| ). The first term on the right side of this equation is exactly computable and leads to the leading order asymptotics in our formulas. As a side note, some accurate asymptotics of the Green’s function for the lattice Laplacian appears in [DS]. Estimates. Consider x, y, x , y ∈ [1, R ∗ − 21 ] and x ∈ [x−1, x+1], y ∈ [y−1, y+1]; R ∗ is some large positive number. Then there exists a positive number C so that log( y ) x 1 1 y−x − 1 ≤ C + , log( yx ) |x| |y| y −x log( RR∗∗ −x ) −y 1 1 y−x − 1 ≤ C + . log( RR∗∗ −x |R ∗ − x| |R ∗ − y| ) −y y −x
From these two inequalities, we have R ∗ −x y log( y ) log( R ∗∗ −x ) log( ) log( ) R −y R ∗ −y x . + ≤ C x + y−x y − x y−x y −x Sketch of the proof. Let
y x
= eζ . Then, y x ζ y = e x y x = eδ1 eδ2 eζ ,
370
H. Pinson
where δ1 = O( y1 ) and δ2 = O( x1 ). Thus, log( yx ) y−x
log( yx ) y −x
=
ζ x(eζ −1) δ1 +δ2 +ζ x (eδ1 +δ2 +ζ −1)
.
The right side of this equation becomes x f (ζ + δ1 + δ2 ) , x f (ζ ) where f (ζ ) =
(eζ − 1) . ζ
Simple (Taylor expansion) analysis reveals that f (ζ + δ1 + δ2 ) = 1 + O(1)(δ1 + δ2 ). f (ζ ) Thus, the sketch is complete. Lemma 4. There exists a positive constant C so that 1 |T r(F n A) − T r(F n )| ≤ C n . 2 Proof. In the expression T r(F n A), the summation with the presence of the diagonal term Ax,x (= 21 , basically) (see a few lines below) leads to no cancellation properties. However, if we take this term out (as in the left side of the inequality in the statement of this lemma), there are cancellations due to the anti-symmetry property (13). We exploit this cancellation to prove this lemma. First if xi , yi ∈ Si for i = 1, 2 and xi = yi , then Axi ,yi = −Ayi ,xi and Axi ,xi =
(13)
1 . 2
The first equality follows from the symmetry C(−z) = −C(z).
(14)
The second equality follows from the relation (2). Thus, Fxii ,yi = Fyii ,xi for i = 1, 2. Combining this with the Formulas (and the subsequent comments), C , |R − x1 ||R − y1 | C − Fy2 ,x2 | ≤ |x2 ||y2 |
|Fx1 ,y1 − Fy1 ,x1 | ≤ |Fx2 ,y2
(15)
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
371
for xi , yi ∈ Si , i = 1, 2. For all other cases not involving the special point p, we have 1 1 |Fx,y − Fy,x | ≤ C + . (16) |R − x||R − y| |x||y| Consider 1 T r(F n A) − T r(F n ) 2
= Ax1 ,x2 Fx2 ,x3 . . . Fxn+1 ,x1 + x1 ,x2 ∈S1 ,x1 =x2
+
Ax1 ,x2 Fx2 ,x3 . . . Fxn+1 ,x1
x1 ,x2 ∈S2 ,x1 =x2
Ax1 ,x2 Fx2 ,x3 . . . Fxn+1 ,x1 −
others
1
Fp,x3 . . . Fxn+1 ,p . 2
(17)
In the first sum of the expression (17), we use the symmetry (13) to group the terms x1 , x2 and x2 , x1 . We then use the trivial identity Fxi ,xi+1 = (Fxi ,xi+1 − Fxi+1 ,xi ) + Fxi+1 ,xi . Inserting this identity into the first sum of (17) and expanding, we obtain a finite sum of terms of the form
Ax1 ,x2 Fx2 ,x3 . . . (Fxi ,xi+1 − Fxi+1 ,xi ) . . . Fxn+1 ,x1 . x1 ,x2 ∈S1 ,|x1 |<|x2 |
There will be at least one difference factor. We have grouped the terms in pairs (x1 , x2 ), (x2 , x3 ), . . . , (xn+1 , x1 ) and (x2 , x1 ), (x1 , xn+1 ), (xn , xn−1 ), . . . , (x3 , x2 ). If xi or xi+1 equals p, then the difference does not help, and we will comment about this case in the very last paragraph of this lemma. Thus, let us assume that we can use either (15) or (16). We are summing over the set of complex number S1 ∪ S2 ∪ {p}; it is more convenient to sum over a straight line. Thus, we can project the point x2 to x2∗ as in Fig. 9. We can then project to the real part of x2∗ . An elementary argument can be invoked to prove S2 x2
2
x*
d d
x1 S 1
Fig. 9.
372
H. Pinson
Re(x1 ) log( x1 ) log( Re(x ∗ ) x2 2) . ≤C x1 − x 2 Re(x1 ) − Re(x2∗ )
(18)
If both x1 and x2 belong to S2 , then both points are projected as x2 in the previous example. Then those are projected to the real parts. We take the same convention for the logarithm as specified in the text following the Formulas. ∗ Let us denote the corresponding real parts yi , i = 1, . . . , n + 1 of x1∗ , . . . , xn+1 (after this projection business). Let us assume that C |R − x||R − y| appears as in (15) or (16). The subsequent formulas actually depend on the cases (15) or (16) (since we have to keep track of the location of the variables). The two cases are 1 similar, so we will just do (15). First note that this term is very small ≈ |R| . Using the bounds in the Estimate, we arrive at the bound
A F . . . (F − F ) . . . F x1 ,x2 x2 ,x3 xi ,xi+1 xi+1 ,xi xn+1 ,x1 x1 ,x2 ∈S1 ,|x1 |<|x2 |
≤C
cR ∗
n
dy1 dy2 dyi dyi+1 1
log(
yi (R ∗ −yi−1 ) yi−1 (R ∗ −yi )
j
R ∗ − 21 1
y3 (R ∗ −y2 )
log( y2 (R ∗ −y3 ) ) 1 dyj |y1 − y2 | y3 − y 2
)
1 yi − yi−1 − yi )(R ∗ − yi+1 ) yi+2 (R ∗ −yi+1 ) (R ∗ −yn+1 ) log( yi+1 (R ∗ −yi+2 ) ) log( yy1n+1 (R ∗ −y1 ) ) ... × y3 − y2 y1 − yn+1
×
(R ∗
(19)
for some constant 0 < c < 1. The constant c arises due to the restriction x1 , x2 ∈ S1 . R ∗ is the real part of the projection of R. Here we have used the assumption θ ∈ ( π4 , 3π 4 ) R ∗ − 21 so that such c exists. We have used the notation j 1 dyj to denote the measure over all variables different from y1 , y2 , yi , yi+1 . In this case (19), we have
j
R ∗ − 21 1
R ∗ − 21
dyj =
R ∗ − 21
dy3 · · ·
1
1
R ∗ − 21
×
dyi+2
1
dyi+3 · · ·
1
R ∗ − 21
dyi−1 R ∗ − 21
dyn+1 .
1
Notation. We will subsequently use this short hand notation to denote the measure over all other unspecified variables. Hopefully this will be clear from the context. Since there are no restrictions on the variables yj , the limits of integration range from 1 to R ∗ − 21 . We also implicitly assume the condition |y1 − y2 | ≥ 1 since x1 = x2 . We will often implicitly assume this without writing this condition in the subsequent discussions. Otherwise we would have manifestly divergent integrals! To evaluate this integral, we perform two successive change of variables
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
373
R∗ + ui , 2∗ R + ui . wi = R2∗ 2 − ui yi =
(20)
Then, the right side of (19) becomes
Cn R∗
c 1−c
dw1 dw2 dwi dwi+1
1 R ∗ −1
wi+2 − wi+1
...
2R ∗ −1 1 R ∗ −1
j
i+2 log( w wi+1 )
×
1 ) log( wwn+1
w1 − wn+1
w3 i ) log( wwi−1 ) log( w 1 2 dwj ... |w1 − w2 | w3 − w2 wi − wi−1
(21)
.
In order to evaluate this integral, we use the change of variables of Dotsenko and Dotsenko [DD]. The change of variables is given by wi+2 wi+3 wn+1 = eζi+2 , = eζi+3 . . . , = eζn+1 , wi+1 wi+2 wn w2 wi w1 = eζ1 = eζ2 . . . = eζi . wn+1 w1 wi−1
(22)
The integral now becomes bounded above by Cn R∗
c 1−c 1 R ∗ −1
×
dwi+1
( R1∗ )2 ≤|ζ2 |
ζi
ζi+2
2sinh( ζ2i )
2sinh( ζi+2 2 )
Since
dζ2
j
...
+∞
dζj
−∞
ζn+1
1
ζ3
2sinh( ζ21 )
2|sinh( ζ22 )|
2sinh( ζ23 )
...
1
2sinh( ζn+1 2 )
eζ1 +···+ζi +ζi+2 +···+ζn+1 =
ζ1
e 2 (ζ1 +···+ζi +ζi+2 +···+ζn+1 ) .
wi c )(R ∗ − 1), ≤( wi+1 1−c
the above integral is bounded by Cn
logR ∗ (R ∗ )
1 2
(
c 3 )2 . 1−c
We will comment about these contributions at the end of the proof of the theorem. In particular the role of c will be discussed. Let us comment on the case C . |x||y| At least one variable x or y belongs to S2 . For definiteness let us assume one belongs to S1 , and the other is in S2 . After the change of variables (20) and (22), we arrive at ∗
+∞ 1 ζ1 C n 2R −1 1 dw dζ dζj ... i+1 2 2 ζ ζ2 ∗ 1 c 1 R 1−c wi+1 2sinh( 2|sinh( ) )| ≤|ζ2 | −∞ ∗ 2 2 2 j (R ) ×
ζi 2sinh( ζ2i )
1
e− 2 (ζ1 +···+ζi +ζi+2 +···+ζn+1 )
ζi+2 2sinh( ζi+2 2 )
...
ζn+1 2sinh( ζn+1 2 )
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H. Pinson
with 1
e− 2 (ζ1 +···+ζi +ζi+2 +···+ζn+1 ) =
√ wi+1 √ ≤ wi+1 R ∗ − 1. wi
c ]. Thus, the above integral is bounded The last inequality follows from wi ∈ [ R ∗1−1 , 1−c above by 1 2 logR ∗ ( 1−c c ) Cn . 1 (R ∗ ) 2
Again we will comment about these types of terms at the very end of the proof of the theorem. Let us now consider the summation over the “others” in (17). Now there are several cases x1 ∈ S1 , x2 ∈ S2 , x2 ∈ S1 , x1 ∈ S2 or x1 = p or x2 = p. The latter two cases are not hard, and we will comment later about those two cases. Let us consider the first case (the second is basically the same). The first point is that no symmetry (13) is needed. We just use the location x1 ∈ S1 , x2 ∈ S2 ,
A F . . . F x1 ,x2 x2 ,x3 xn+1 ,x1 x1 ∈S1 ,x2 ∈S2
≤C
n
cR ∗
dy1 1
×
R ∗ − 21
dy2
cR ∗
y1 log( yn+1 ) + log(
R ∗ −y2
y3
log( y2 ) + log( R ∗ −y3 ) 1 dyj ... |y1 − y2 | y3 − y 2
1
j R ∗ −yn+1 R ∗ −y1
R ∗ − 21
) .
yn+1 − y1
We perform the identical change of variables (20) to obtain Cn
c 1−c 1 R ∗ −1
dw1
2R ∗ −1
dw2
c 1−c
2R ∗ −1
dwj
1 R ∗ −1
j
w3 1 ) log( wwn+1 ) log( w 1 2 ... . (23) |w1 − w2 | w3 − w2 w1 − wn+1
We perform a similar change of variables as above (22). The indices have changed w2 w3 wn+1 = eζ2 , = eζ3 , . . . , = eζn+1 . w1 w2 wn We need to keep careful track of the lower bound for ζ2 . We can easily follow the restrictions y1 ∈ [1, cR ∗ ], y2 ∈ [cR ∗ , R ∗ − 21 ] to the restrictions on ζ2 . This results in 1−
w1
≤ ζ2 .
c 1−c
Thus, Cn ×
c 1−c 1 R ∗ −1
dw1
1 w1
+∞ 1−
w1 c 1−c
dζ2
j
+∞ −∞
ζn+1
ζ2 + · · · + ζn+1
2sinh( ζn+1 2 )
n+1 2sinh( ζ2 +···+ζ ) 2
dζj
1
ζ3
2|sinh( ζ22 )|
2sinh( ζ23 )
...
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
375
which is bounded by C
n
c 1−c
−log(1 −
w1 c 1−c
w1
1 R ∗ −1
) (24)
.
The later integral is finite. We sketch the other case x1 = p or in other words the part with the factor 21 in (17). We proceed as in the previous case except that now the integral over x1 is replaced by 1 a fixed x1 = p. We do all the change of variables as before. Instead of we ζ2 2|sinh(
will have
ζ2 ζ . 2sinh( 22 )
2
)|
Finally at the end, there will be no integral over w1 . There will be
an overall bound by
Cn . R ∗ (1 − c)c
1 may appear strange, but it arises due to the fact Perhaps the presence of the factor (1−c)c c w1 = 1−c and the appearance of the factor (w1 + 1)2 if we wrote the full expression; a similar factor arises in the next paragraph. As promised earlier, we make a comment on the case when one of the summation variables take the value p in
Ax1 ,x2 Fx2 ,x3 . . . Fxn+1 ,x1 others
of (17). Let us say p = x3 ; it really does not make any difference where the point p appears. We make the usual change of variables yi → ui , ui → wi ; we have assumed that this sum has been bounded by an integral with variables yi as usual. Also since the point x3 = p is fixed, no integration is performed with respect to this variable although we can still perform the change of variable. It is just a different way to write that number. We continue with the similar change of variables w4 wn+1 w1 w2 = eζ4 . . . = eζn+1 , = eζ1 , = eζ2 . w3 wn wn+1 w1 Then this sum is bounded above by ∗ w3 1 ) log( wwn+1 ) log( w C n 2R −1 1 2 2 dw dw dw dw . . . dw (w + 1) . . . 1 2 4 n+1 3 5 ∗ 1 R |w1 − w2 | w3 − w2 w1 − wn+1 ∗ R −1
(as usual there are restrictions on |w1 − w2 |) which in turn is bounded by Cn
logR ∗ . R ∗ c(1 − c)
We will comment about the dependence of the constant c at the end of the proof of the theorem. Lemma 5. 1 4 T r(F ) = log(R ) 2π (2π)2n n
∗
+∞ −∞
dp(
π2 )n + O(C n ). cosh2 (πp)
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H. Pinson
Proof. T r(F n ) =
log( xx21 )
xi ∈S1
...
2π 2 (x2 − x1 ) R0 −x1 log( R ) 0 −xn
2π 2 (xn − x1 )
...
log( xxn1 )
0 −x2
log( R R0 −x1 )
+
2π 2 (x1 − xn )
xi ∈S2
2π 2 (x1 − x2 )
...
+ other.
(25)
Let us first consider the other terms. These will produce O(C n ). Two such terms are ∗
∗
−xn −x2
1 log( xx3 ) + log( R log( xxn1 ) + log( R R ∗ −x1 ) R ∗ −x3 ) 2 ... x1 x2 x1 − x n x3 − x 2
(26)
and ∗
∗
R −x1 log( xx21 ) + log( R ∗ −x ) 2
...
x2 − x1
x1 ≤cR ∗ ≤x2
−xn log( xxn1 ) + log( R R ∗ −x1 )
x1 − x n
(27)
.
We have invoked the projection argument in arriving at (26) and (27). For the sake of clarification, the number R ∗ is the real part of the projection of the complex number R. R0 is the label corresponding to R. This is measured from the point p. See Fig. 8. The expressions (26) and (27) are bounded by C
n
R ∗ − 21 1
j
R ∗ −y2
y3
R ∗ −yn
y1
log( yn ) + log( R ∗ −y1 ) 1 log( y2 ) + log( R ∗ −y3 ) dyj ... y1 y2 y3 − y 2 y1 − y n
and C
n
cR ∗ dy1
1
R∗ − 1 2 cR ∗
dy2
R ∗ − 21 j
1
∗
dyj
R −y1 log( yy2 ) + log( R ∗ −y ) 1
2
y2 − y1
∗
...
−yn log( yyn1 ) + log( R R ∗ −y ) 1
y1 − yn
.
Again perform the usual change of variables, and we get ∗ w1 w3 ) log( w C n 2R −1 1 log( w2 ) n dw . . . j ∗ 1 w1 − w n R w 1 w 2 w3 − w 2 R∗−1
(28)
j
and C
n
c 1−c 1 R ∗ −1
dw1
2R ∗ −1 c 1−c
dw2
j
2R ∗ −1 1 R ∗ −1
dwj
2 log( w w1 )
w2 − w 1
...
w1 log( w ) n
w1 − w n
A calculus argument can be invoked to bound (28) above by ∗ w1 w3 2 log( w ) log( w C n 2R −1 1 1 w1 ) log( w2 ) n dwj + ... ∗ 1 R w1 w2 w2 − w 1 w3 − w 2 w1 − w n R∗−1 j
.
(29)
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
377
which can be bounded above by
Cn R∗
2R ∗ −1
dw1
1 R∗−1
×
1 +∞ ζ2 ζn dζj ... ζ 2 2 w1 j −∞ 2sinh( 2 ) 2sinh( ζ2n )
ζ2 + · · · + ζn n) 2sinh( ζ2 +···+ζ ) 2
.
This is bounded by C n . Again invoking the same change of variables w2 w3 wn = eζ2 , = eζ3 , . . . , = e ζn , w2 w1 w1 we arrive at a bound for (27) (together with an elementary argument to figure out the limits of integrations) Cn
c 1−c 1 R ∗ −1
×
1 dw1 w1
+∞ 1−
w1 c 1−c
dζ2
j
+∞ −∞
dζj
ζ2 2sinh( ζ22 )
...
ζn 2sinh( ζ2n )
ζ2 + · · · + ζn . 2sinh(ζ2 + · · · + ζn )
This can be bounded above by C
c 1−c
n
−log(1 −
w1 c 1−c
w1
1 R ∗ −1
) dw1
which is finite. Our last term takes the form (for example) 2
log( R−x log( xxn1 ) log( xx23 ) R−x1 ) | | ... . 2π(x1 − xn ) (x1 − x2 ) 2π(x3 − x2 )
xi ∈S1
Again using the projection argument and the Estimates, this is bounded by
C
n
j
R ∗ − 21
∗
dyj
1
−y2 y3 log( R R ∗ −y1 ) log( y2 )
(y1 − y2 ) (y3 − y2 )
...
log( yyn1 ) (y1 − yn )
.
This expression arises as a term in
C
n
j
1
R ∗ − 21
∗
dyj
R −y1 log( yy21 ) + log( R ∗ −y ) 2
(y2 − y1 )
when this is expanded. In fact it arises as a term in
∗
...
−yn log( yyn1 ) + log( R R ∗ −y1 )
(y1 − yn )
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H. Pinson
C
n
R ∗ − 21
dyj
(y2 − y1 )
1
j
−C n
n
R ∗ − 21
dyj
1
j
−C
∗
R −y1 log( yy21 ) + log R ∗ −y ) 2
R ∗ − 21
1
j
dyj
log( yy21 ) (y2 − y1 )
...
R ∗ −y1 log( R ∗ −y ) 2
(y2 − y1 )
∗
...
−yn log( yyn1 ) + log R R ∗ −y1 )
(y1 − yn )
log( yyn1 ) (y1 − yn ) ∗
...
−yn log( R R ∗ −y1 )
(y1 − yn )
(30)
.
We can explicitly compute the three terms to show that these three terms can be bounded by C n . To do the first integral, we again use the change of variables yi → ui , ui → wi , wi → ζi . We can directly proceed to the ζ variables immediately in regard to the second. The third integral is basically the same as the second integral after a change of variables yi → R ∗ − yi . The first two integrals become 2R ∗ −1 1 +∞ ζ2 ζn ζ2 + · · · + ζn dw1 dζj ... + O(C n ) ζ2 ζn ζ2 +···+ζn 1 w 1 j 2sinh( 2 ) 2sinh( 2 ) 2sinh( ) −∞ R ∗ −1 2 and R∗ − 1 2
1
1 +∞ ζ2 ζn ζ2 + · · · + ζn dw1 dζj ... + O(C n ). ζ2 ζn w1 2sinh( ) 2sinh( ) 2sinh( ζ2 +···+ζn ) −∞ 2
j
2
2
(31) The third integral also takes the form (31). Thus, after a subtraction, (30) becomes O(C n ). C C Let us comment on the other possibilities. If (R ∗ −x1 )(R ∗ −x ) appears instead of x x 1 2 2 in (26) (after the usual “projection” argument), then we can do the change of variable xi → R ∗ −xi , which brings us back to the case (26). If a mixture, say, (R ∗ −x1 )(RC∗ −x2 )x3 x4 occurs, then we can use the inequality log( xx43 ) 1 ≤ x4 − x 3 x3 x4 log(
x4
)
to replace x31x4 by x4 −xx33 , and we are back to the previous case. Finally we deal with the first two terms in (25),
log( xx21 )
...
log( xxn1 )
2π 2 (x1 − xn ) 2π 2 (x2 − x1 ) ∗ log( yy21 ) log( yyn1 ) 1 cR 1 =2 dy . . . +2 j (2π )2n (y − y ) (y − y ) (2π )2n 1 n 2 1 1 j y ∗
cR log( y21 ) log( yyn1 ) log( xxn1 ) log( xx21 ) × dyj − ... ... . (x2 − x1 ) (x1 − xn ) (y2 − y1 ) (y1 − yn ) 1
xi ∈S1
(32)
j
The factor 2 next to the equal sign comes from the fact that we sum over odd integers and even integers separately. See the definitions of s1 , s2 in the Formulas. Since the
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model
379
coordinates xi run over even integers (or odd integers), we pick up an extra factor of 21n . The variables xi are assumed to be piecewise constant ( and depend on the variables yj ) so that the original sum can be recovered. First, we show Lemma 6.
cR ∗
dyi (
1
j
log( xx21 )
...
(x2 − x1 )
log( xxn1 ) (x1 − xn )
−
log( yy21 ) (y2 − y1 )
...
log( yyn1 ) (y1 − yn )
) = O(C n ).
Comment. To repeat, the variables xi are piecewise constant, so that the original sum can be recovered. Proof. The left side of the equality in this lemma is x x log( x2 ) log( xn1 ) y2 1
cR ∗ log( yyn1 ) (x −x ) log( y1 ) (x −x )) dyj 2 y21 . . . 1 y1n − 1 ... . log( y ) log( yn ) (y2 − y1 ) 2π(y1 − yn ) 1 1 j
(y1 −yn )
(y2 −y1 )
Using an inequality from the Estimate x
log( i+1 xi ) (xi+1 −xi ) y
log( i+1 yi ) (yi+1 −yi )
=1+O
1 1 + |yi | |yi+1 |
over the intervals as stated in the Estimate. Thus, it suffices to bound such terms as
j
cR ∗
dyj 1
y2 log( yyn1 ) 1 log( y1 ) ... . y1 (y2 − y1 ) (y1 − yn )
But as in the usual integration of this form, the presence of the extra factor bound by C n . The other terms can be handled easily.
1 y1
leads to a
The first integral on the right side of the equality in (32) becomes
j
cR ∗
dyj
log( yy21 )
...
log( yyn1 )
(y2 − y1 ) (y1 − yn ) +∞ 1 ζ2 ζn ζ2 + · · · + ζn = dy1 dζj ... + O(C n ) ζ ζ ζ2 +···+ζn n 2 y1 ) ) ) 2sinh( 2sinh( 2sinh( 1 −∞ 2 2 2 j +∞ 2 1 π = logR ∗ )n + O(C n ). dp( (33) 2π −∞ cosh2 (πp) 1
cR ∗
There is a factor log(c) contained in O(C n ) here. We have used the formula +∞ ζ π2 iζp . dζ = e cosh2 (pπ ) 2sinh( ζ2 ) −∞ A similar argument can be invoked to deal with the second sum in (25).
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H. Pinson
Remark. To repeat, let us comment on the origin of the prefactor 1 4 . 2π (2π)2n The factor 2n π1 2n comes from the Green’s functions. The extra factor 21n comes from the fact that we sum over a subset of either the even integers or the odd integers. The factor 1 2π comes from the Fourier inversion theorem. The factor 2 comes from the fact that we sum over the subset of the odd integers and even integers separately. Finally the last factor 2 comes from the summation over the set S1 and S2 . Proof of the Theorem. Consider det (I + (e−4iα − 1)A) = eT rlog(I +xA) , where x = e−4iα − 1. Let us expand this logarithm in a power series
1
dtT r( 0
∞
(−1)s+1 t s−1 x s As ).
s=1
Substituting As =
s−1
ans F n +
s−1
n=1
n=0
1 1 bns F n ((A − I ) + I ) 2 2
into the sum (putting the part A− 21 I as an O(1) contribution since, according to Lemma 4, that part does not give a logarithmic contribution), we arrive at
1
dt 0
∞
n=1
1 1 An T r(F n ) + Bn T r(F n ) + B0 T r(I ) + O(1), 2 2
where An = Bn = B0 =
∞
s=n+1 ∞
(−1)s+1 t s−1 x s ans , (−1)s+1 t s−1 x s bns ,
s=n+1 ∞
(−1)s+1 t s−1 x s b0s .
s=1
In getting the O(1) term, we have used the smallness of |x| and Lemma 4. Using Lemma 3, we arrive at (−xt)2n 1 An = − (−1)n t (1 + xt)n (−xt)2n+1 1 Bn = − (−1)n t (1 + xt)n+1
Rotational Invariance and Discrete Analyticity in the 2d Dimer Model 4 for n = 1, . . . . By Lemma 5, we can replace T r(F n ) by logR ∗ 2π where 1 . y= 2 2 cosh2 (πp)
381
∞
−∞ dpy
n + O(C n ),
Then ∞
n=1 ∞
4 An T r(F ) = logR 2π ∗
n
Bn T r(F n ) = logR ∗
n=1
−4 2π
∞
dp
−∞
(xt)2 y + O(1), t (1 + xt + x 2 t 2 y)
∞
−∞
dp x(
1 1 − ) + O(1). 1 + xt 1 + xt + x 2 t 2 y
Integrating over the variable t, we arrive at
1
∞
1 An T r(F n ) + Bn T r(F n ) 2 0 n=1 4 1 ∞ dp(log(1 + x + x 2 y) − log(1 + x)) + O(1) = logR ∗ 2π 2 −∞ ∞ sin2 (2α) ∗ 4 1 = logR dp log(1 − ) + O(1) 2π 2 −∞ cosh2 (πp) dt
= −logR ∗
8α 2 + O(1). π2
Finally we have
1 0
1 dt B0 T r(I ) + 2iα(|S1 | + |S2 | + 1) = O(1). 2
Now in regard to the contribution C(θ, r, α) or the O(1) contribution to the two point function, it appears that this quantity satisfies only supr |C(θ, r, α)| < +∞, since we have only obtained bounds of the form, for example, Cn
logR ∗ , R ∗ c(1 − c)
See, for example, the paragraph just above QED near Lemma 5. Also there is a factor log(c) in O(C n ) in (33). Thus, it appears that there is a dangerous dependence on the angle θ near c = 0, 1. However, it is clear that we could have started this whole proof 3π 5π from a perturbation around the angles θ = −π 4 , 4 or 4 . In doing that this defect is removed, and we obtain the theorem. Acknowledgement. I would like to thank Professor Thomas Spencer for initial consultations. I would also like to thank Professor Robert Langlands for pointing out these types of problems. I would also like to thank Professor Robert Maier for some help on the use of computer graphics. I would like to thank the referee for the extensive comments which lead to the improvement of this paper.
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References [B]
Basor, E.: Asymptotic formulas for Toeplitz determinants. Trans. Am. Math. Soc. 239, 33–65 (1978) [DIZ] Deift, P., Its, A., Zhou, X.: A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. (2) 146(1), 149–235 (1997) [DD] Dotsenko, V., Dotsenko, S., Vladimir, S.: Critical behaviour of the phase transition in the 2D Ising model with impurities. Adv. Phys. 32(2), 129–172 (1983) [D] Duffin, R.J.: Basic properties of discrete analytic functions. Duke Math. J. 23, 335–363 (1956) [DS] Duffin, R.J., Shaffer, D.H.: Asymptotic expansion of double Fourier transforms. Duke Math. J 27, 581–596 (1960) [F] Ferrand, J.: Fonctions prharmoniques et fonctions prholomorphes. Bull. Sci. Math. (2) 68, 152– 180 (1944) [FS] Fisher, M., Stephenson, J.: Statistical mechanics of dimers on a plane lattice. II. Dimer correlations and monomers. Phys. Rev. (2) 132, 1411–1431 (1963) [ID] Itzykson, C., Drouffe, J.: Statistical field theory. Vol. 1. From Brownian motion to renormalization and lattice gauge theory. Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge University Press, 1991 [Ka] Kasteleyn, P.: The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961) [Ke] Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28(2), 759–795 (2000) [Ke2] Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincar´e Probab. Statist. 33(5), 591– 618 (1997) [PT] Palmer, J., Tracy, C.: Two-dimensional Ising correlations: Convergence of the scaling limit. Adv. Appl. Math. 2(3), 329–388 (1981) [P] Propp, J.: Dimers and Dominoes. Preprint [TF] Temperley, H., Fisher, M.: Dimer problem in statistical mechanics-an exact result. Phil. Mag. 6, 1061 (1961) [T] Thurston, W.P.: Conway’s tiling groups. Am. Math. Monthly 97, 757–773 (1990) [W] Widom, H.: Toeplitz determinants with singular generating functions. Am. J. Math. 95, 333–383 (1973) Communicated by A. Kupiainen
Commun. Math. Phys. 245, 383–405 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1022-2
Communications in
Mathematical Physics
Existence of Baryons, Baryon Spectrum and Mass Splitting in Strong Coupling Lattice QCD Paulo A. Faria da Veiga1 , M. O’Carroll1 , Ricardo Schor2 1 2
Departamento de Matem´atica, ICMC-USP, C.P. 668, 13560-970 S˜ao Carlos, SP, Brazil. E-mail: [email protected]; [email protected] Departamento de F´ısica-ICEx, UFMG, C.P. 702, 30161-970 Belo Horizonte MG, Brazil. E-mail: [email protected]
Received: 12 May 2003 / Accepted: 24 September 2003 Published online: 23 January 2004 – © Springer-Verlag 2004
Abstract: We consider a functional integral formulation for one-flavor lattice Quantum Chromodynamics in d = 2, 3 space dimensions and imaginary time, and work in the regime of the small hopping parameter 0 < κ ≪ 1, and zero plaquette coupling. Following the standard construction, this model exhibits positivity which is used to obtain the underlying physical Hilbert space H. Then, using a Feynman-Kac formalism, we write the correlation functions for the model as functional integrals over the space of Grassmannian (fermionic) fields for one quark specie and the SU(3) gauge fields. We determine the energy-momentum spectrum associated with gauge invariant local baryon (anti-baryon) fields which are composites of three quark (anti-quark) fields. With the associated correlation functions, we establish a Feynman-Kac formula, and a spectral representation for the Fourier transform of the two-point functions. This representation allows us to show that baryons and anti-baryons arise as tightly bound, bound states of three (anti-)quarks. Labelling the components of the baryon fields by s = 3/2, 1/2, −1/2, −3/2, we show that the baryon and anti-baryon mass spectrum only depends on |s|, and the associated masses are given by Ms = −3 ln κ + rs (κ), where rs (κ) is real analytic in κ, for each d = 2, 3. The mass splitting is M3/2 − M1/2 = 18κ 6 , for d = 2 and, if any, is at least of O(κ 7 ), for d = 3. In the subspace Ho ⊂ H generated by an odd number of fermions, the baryon and anti-baryon energy-momentum dispersion curves are isolated up to near the baryon-meson threshold −5 ln κ (upper gap property), identical and are determined up to O(κ 5 ). The symmetries of coordinate reflections, spatial lattice rotations, parity and charge conjugation are established for the correlation functions, and are shown to be implemented on H by unitary (anti-unitary, for time reversal) operators.
1. Introduction The existence and spectral properties of quantum field theories arise as fundamental problems in physics. In quantum chromodynamics (QCD), it is fundamental to establish
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P.A. Faria da Veiga, M. O’Carroll, R. Schor
on a rigorous basis the low energy-momentum (e-m) spectrum of particles and their bound states, in particular, to prove the existence of mesons and baryons and their bound states. Spectral properties of this sort involve low-energies and one way to study them is to use a lattice regularization of the continuum. Strong coupling lattice models can provide valuable insight into the low energy behavior of QCD. Our partial understanding of confinement comes in this way, and in fact was a reason for the introduction of lattice QCD by Wilson (see [1–3] and the recent reviews [4, 5]). As one of our main results, in the regime of the small hopping parameter 0 < κ ≪ 1, and zero plaquette coupling (such as strong coupling), we obtain an understanding of baryons from first principles by showing that they arise as tightly bound, bound states of three quarks and their occurrence is manifested by the appearance of an isolated dispersion curve in the e-m spectrum. We work with a lattice regularization and obtain a qualitative picture of the spectrum from first principles in the strong coupling regime, corresponding to quark confinement. No approximations are made so our results are exact in this context. More specifically, we consider lattice QCD as in [3, 6, 7]. We work in space dimension d = 2 and d = 3, 4 × 4 Dirac spin matrices and imaginary time, with small hopping parameter 0 < κ ≪ 1 (strong coupling) and large glueball mass (see [8]) in the baryon Fermi sector. For the Hamiltonian approach to baryon masses we refer to [9, 10]. For the determination of masses for the meson sector, in the Euclidean approach, see [11]. However, we point out that determining exponential decay rates of cf’s, as in Ref. [11], is not sufficient to identify these rates with the mass spectrum, since a connection with the energy-momentum spectrum is not established. This is especially true in the case where there are multiplicities involved and mass splitting, as here. A treatment for mesons, including a spectral representation, can be found in Ref. [12]. Taking a simpler version of the QCD model, similar results were recently reported by us in [7], using 2 × 2 Pauli spin matrices and for d = 2. After establishing positivity and constructing the underlying physical Hilbert space H, we consider the two-point functions for the baryon (anti-baryon) fields composed of three quarks (anti-quarks). Analogous to the K¨allen-Lehman representation in quantum field theory, we obtain a spectral representation for the Fourier transform of the two-point function. This is the fundamental tool which allows us to relate the singularities of their Fourier transforms to the e-m spectrum via a Feynman-Kac (F-K) formula. We stress that a knowledge of decay rates of correlation functions does not mean that they can be identified with masses, especially in the case of nearly degenerate masses. For this reason, it is essential to establish a spectral representation. Let Ho ⊂ H denote the subspace generated by an odd number of fermions. In the even space, the mesons, which are quark, anti-quark composite fields, have asymptotic mass −2 ln κ. We show that there is a baryon particle and an anti-particle, with asymptotic mass −3 ln κ, manifested by isolated dispersions curves in the e-m spectrum (upper mass gap property) and more: it is the only spectrum in the subspace Ho ⊂ H, up to near the meson-baryon threshold −5 ln κ. Thus, the upper gap persists in Ho . With s = 3/2, 1/2, −1/2, −3/2 labelling (spin-like) components of the baryon composite fields, we show that the mass spectrum only depends on |s| and the masses are given by Ms = −3 ln κ + rs (κ), with rs (κ) real analytic in κ. The mass splitting is M3/2 − M1/2 = 18κ 6 , for d = 2, and zero at least up to and including O(κ 6 ), for d = 3. The symmetries of parity, charge conjugation, lattice spatial rotations and coordinate reflections are also established for the correlation functions, and are shown to be implemented on H by unitary operators, or an anti-unitary operator in the case of time reversal.
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385
It is important to emphasize that the upper mass gap property, has not been established in the Hamiltonian formulation treatments, and that the above mentioned spectral representation does not appear in the literature, as far as we know. Besides its intrinsic importance, the determination and control of the baryonic mass spectrum is an essential step towards the understanding of e.g. baryon-baryon bound states (such as for the nuclear force) from first principles, and thus bridging the gap between QCD and the effective baryon, effective meson picture of nuclear forces through a single and multiple boson exchange arising from a Yukawa interaction (see [13, 14]). This work is being continued and the analysis of the existence of two-particle bound states is performed using our techniques from previous works on the determination of bound state spectrum in spin and stochastic Ginzburg-Landau lattice systems. (See [15 and 16]). 2. The Model and Results Let us define the gauge-matter model and give our main results. The quantum mechanical Hilbert space and e-m operators are obtained by a standard construction from the thermodynamic limit of gauge invariant correlation functions of a statistical mechanical model where the Fermionic degrees of freedom are described by Grassmann variables (see [3, 6, 17] for details). The partition function of the model is given formally by ¯ Z= e−S(ψ,ψ,g) dψ d ψ¯ dµ(g) , (2.1) ¯ ψ, g), the normalized expectations are denoted by and, for a function F (ψ, 1 ¯ ¯ ¯ ψ, g) e−S(ψ,ψ,g) F (ψ, ψ, g) = F (ψ, dψ d ψ¯ dµ(g) . Z We consider the cases where the space dimension is d = 2, 3. Labelling by 0 the time 0 0 1 d direction, the unit lattice is taken as Zd+1 o , where u = (u , u) = (u , u , . . . , u ) ∈ d+1 d Zo ≡ Z1/2 × Z , where Z1/2 = {±1/2, ±3/2, . . . }. The choice of the shifted lattice for the time direction, avoiding the zero-time coordinate, is so that, in the continuum limit, two-sided equal time limits of Fermi field correlations can be accommodated. At each site u ∈ Zd+1 o , there are Fermion Grassmann fields ψaα (u), associated with a quark, ¯ and fields ψaα (u), associated with an anti-quark, which carry a Dirac spin-like index α = 1, 2, 3, 4 and an SU(3) color index a = 1, 2, 3. We refer to α = 1, 2 as upper spin indices and α = 3, 4 as lower ones, denoted respectively by u and . Also, letting eµ , µ = 0, 1, 2, 3, denote the unit lattice vector for the µ-direction, for each nearest neighbor oriented lattice bond < u, u ± eµ > there is an SU(3) matrix U (gu,u±eµ ) parametrized by the gauge group element gu,u±eµ and satisfying U (gu,u+eµ )−1 = U (gu+eµ ,u ). Associated with each lattice oriented plaquette p there is a plaquette variable χ (U (gp )) where U (gp ) is the orientation-ordered product of matrices of SU(3) of the plaquette oriented bonds, and χ is the trace. The model action is given by κ eµ ¯ g) = S(ψ, ψ, (gu,u+eµ )ab ψbβ (u + eµ ) ψ¯ aα (u) αβ 2 1 + χ (gp ) , (2.2) ψ¯ aα (u)Mαβ ψaβ (u) − 2 g0 p d+1 u∈Zo
386
P.A. Faria da Veiga, M. O’Carroll, R. Schor
where the first sum runs over u ∈ Zd+1 o , = ±1 and µ = 0, 1, . . . , d. For notational simplicity, we sometimes drop U from U (g). Concerning the parameters, we take m > 0, and the quark-gauge coupling or hopping parameter κ > 0. Also, g0 > 0 describes the pure gauge strength and M ≡ M(m, κ) = (m + 2κ)I4 , I4µbeing the 4 × 4 identity matrix. Also, within the family of actions of [3], we take ±e = −1 ± γ µ . For d = 3, 1 0 0 γµ are the 4 × 4 hermitian traceless anti-commuting Dirac matrices γ = 0 −1 j 0 iσ ; and σ j , j = 1, 2, 3, denotes the hermitian traceless antiand γ j = −iσ j 0 commuting 2 × 2 Pauli spin matrices. For d = 2, all but the γ 3 matrix appear in the action. The measure dµ(g) is the product measure over non-oriented bonds of normalized SU(3) Haar measures (see [18]) and the integrals over Grassmann fields are defined according to [17]. For a polynomial in the Grassmann variables with coefficients depending on the gauge variables, the fermionic integral is defined as the coefficient of the monomial of maximum degree, i.e. of u,α,a ψ¯ aα (u)ψaα (u). In Eq. (2.1), dψ d ψ¯ means u,α,a dψaα (u) d ψ¯ aα (u) such that, with a normalization N1 = 1, we ¯ have ψaα (x) ψ¯ bβ (y) = (1/N1 ) ψaα (x) ψ¯ bβ (y) e− u,a ,α ,β ψa α (u)Mα β ψa β (u) × −1 dψ d ψ¯ = Mα,β δab δ(x −y), with a Kronecker delta for space-time coordinates. With our restrictions on the parameters, there is a quantum mechanical Hilbert space of physical states (see below), for κ > 0; and the condition m > 0 guarantees that the one-particle free Fermion dispersion curve increases in each positive momentum component. Throughout the paper we take the hopping parameter sufficiently small 0 < κ ≪ 1, and without loss of generality we set M = 1 in the action (2.2). An important feature of the action (2.2) is that it is invariant by the gauge transformations given by, for x ∈ Zd+1 and h(x) ∈ SU(3), o ψ(x) → h(x) ψ(x), ¯ ¯ ψ(x) → ψ(x) [h(x)]−1 , U (g
x+eµ ,x
) → [h(x
+ eµ )]−1 U (g
(2.3)
x+eµ ,x
) h(x) .
Other symmetries of the action (2.2), such as charge conjugation, parity, coordinate reflections and spatial rotations, will be considered below. For small enough couplings κ and g0−2 , by polymer expansion methods (see [3, 19]), the thermodynamic limit of correlations exists and truncated correlations have exponential tree decay. The limiting correlation functions are lattice translational invariant. Furthermore, the correlation functions extend to analytic functions in the coupling parameters. For the formal hopping parameter expansion, or strong coupling expansion, see [20, 2]. Next, we recall the definition of the quantum mechanical Hilbert space H and the e-m operators starting from gauge invariant correlation functions, with support restricted 0 i to u0 = 1/2. Letting T0x , Tix , i = 1, 2, 3, denote translation of the functions of Grassmann and gauge variables by x 0 ≥ 0, x ∈ Zd+1 ; and for F and G only depending on coordinates with u0 = 1/2, we have the F-K formula 0
1
d
0
1
d
(G, T0x T1x . . . Tdx F )H = [T0x T1x . . . Tdx F ] G ,
(2.4)
Existence of Baryons and Baryon Spectrum in Lattice QCD
387
where is an anti-linear operator which involves time reflection. Following [3], with the usual sum convention, the action of on single fields is given by ψ¯ aα (u) = (γ0 )αβ ψaβ (tu) , ψaα (u) = ψ¯ aβ (tu)(γ0 )βα ; where t (u0 , u) = (−u0 , u), for A and B monomials, (AB) = (B) (A); and for a function of the gauge fields f ({guv }) = f ∗ ({g(tu)(tv) }), u, v ∈ Zd+1 o , where ∗ means complex conjugate. extends anti-linearly to the algebra. We do not distinguish between Grassmann, gauge variables and their associated Hilbert space vectors in our notation. As linear operators in H, Tµ , µ = 0, 1, . . . , d, are mutually commuting; T0 is self-adjoint, with −1 ≤ T0 ≤ 1, and Tj =1,... ,d are unitary, so that we write j Tj = eiP and P = (P 1 , . . . , P d ) is the self-adjoint momentum operator, with spectral points p ∈ Td ≡ (−π, π ]d . Since T02 ≥ 0, we define the energy operator H ≥ 0 by T02 = e−2H . We refer to a point in the e-m spectrum associated with spatial momentum zero as a mass. More precisely, the positivity condition F F ≥ 0 is established in [3] but there may be nonzero F ’s such that F F = 0. The collection of such F ’s is denoted by N . Thus, a pre-Hilbert space H can be constructed from the inner product G F . The physical Hilbert space H is defined as the completion of the quotient space H /N . For the construction of the physical Hilbert space and a nonnegative self-adjoint transfer matrix in a finite space lattice model with a not too large hopping parameter, we refer to [21], and we mention that this construction gives reflection positivity for reflections in lattice planes, leading to nonnegativity of the transfer matrix T0 . For the absence of a nonzero kernel of the transfer matrix, for the pure gauge model, we refer to [22]. We now turn to our results on the existence of particles, their masses and dispersion ˜ We restrict curves. Recall Ho is the subspace of H generated by an odd number of ψ. −2 our attention to the subspace Ho ⊂ H, for g0 << κ. For the pure gauge case and small g0−2 , the low-lying glueball spectrum is found in [8]. For large g0 , the glueball mass is ≈ 8 ln g0 . For the even subspace, quark-anti-quark mesons are treated in [11], and have asymptotic mass −2 ln κ. We define the Grassmann gauge invariant baryon composite fields by, with abc denoting the Levi-Civita symbol, 1 ¯ ¯ ¯ , s = 3/2 6 1 abc ψa1 ψb1 ψc1 √ abc ψ¯ a1 ψ¯ b1 ψ¯ c2 , s = 1/2 φ¯ su u0 = 1/2, u = 2 1 3 (2.5) ¯ a1 ψ¯ b2 ψ¯ c2 , s = −1/2 √ abc ψ 2 3 1 ¯ ¯ ¯ , s = −3/2 ; 6 abc ψa2 ψb2 ψc2 and
φ¯ s u0 = 1/2, u =
1 ¯ ¯ ¯ 6 abc ψa3 ψb3 ψc3 1 √ ψ¯ ψ¯ ψ¯ 2 3 abc
a3 b3 c4
, s = 3/2 , s = 1/2
1 √ abc ψ¯ a3 ψ¯ b4 ψ¯ c4 , s = −1/2 21 3 ¯ ¯ ¯ , s = −3/2 ; 6 abc ψa4 ψb4 ψc4
(2.6)
where we recall the superscript u selects upper indices 1, 2 for the selects the spin and lower indices 3, 4. Similarly, composite fields φsu u0 = 1/2, u and φs u0 = 1/2, u are defined by dropping the bars from the ψ’s.
388
P.A. Faria da Veiga, M. O’Carroll, R. Schor
Note that the linear combinations that occur in Eqs. (2.5) and (2.6), as in simpler quantum mechanical systems, are those which are present in the coupling of three 1/2 spins in the continuum, associated with total spin 3/2. We remark that the spin labels on φ˜ are analogous to the total z spin component and are not to be taken as an exact correspondence. On the internal spinor indices, we only have the lattice (discrete) rotation symmetries (see Theorem 4 below). In the context of our lattice model, we can classify ¯ s ≡ ¯ the improper zero-momentum states Zd−1 φs (1/2, x) under the irreducible representations of the group Z4 generated by rotations of π/2 in the x 1 x 2 −plane and the group Z2 generated by x 3 coordinate reflections. Of course, the commuting groups Z2 and Z4 are subgroups of the lattice group of rotations and reflections. Using the results ¯ of Theorem 4 below, on symmetries, we find that s=3/2,1/2,−1/2,−3/2 is a basis for the irreducible representation of Z4 generated by 1 (trivial), i, −1 and −i, respectively; and is a basis for the irreducible representation of Z2 generated by 1, −1, −1, −1, respectively. The normalizations are chosen so that the associated two-point free functions, obtained by setting κ = 0 in the first term in the action of Eq. (2.2), the hopping term, but keeping κ = 0 in M, are equal to −1 at coincident points. For some considerations, it is convenient to use the following reduced equivalent expressions for e.g. φ¯ s : 1 φ¯ 3/2 = ψ¯ 13 ψ¯ 23 ψ¯ 33 , φ¯ 1/2 = √ ψ¯ 13 ψ¯ 23 ψ¯ 34 + ψ¯ 13 ψ¯ 24 ψ¯ 33 + ψ¯ 14 ψ¯ 23 ψ¯ 33 , 3 1 φ¯ −3/2 = ψ¯ 14 ψ¯ 24 ψ¯ 34 , φ¯ −1/2 = √ ψ¯ 13 ψ¯ 24 ψ¯ 34 + ψ14 ψ¯ 23 ψ¯ 34 + ψ14 ψ¯ 24 ψ¯ 33 , 3 (2.7) which uses the Grassmannian character of ψ. Similar expressions hold for the composite fields φs , φ¯ su and φsu . In the sequel, our analysis deals with the subspace Hb ⊂ H generated by the vectors φ¯ su , ..., φs , s = 3/2, 1/2, −1/2, −3/2. We will show that the φ¯ s generate particles, i.e. are associated with isolated dispersion curves in the low-energy spectrum, with asymptotic mass −3 ln κ, and which correspond to baryons. Similarly, φsu also generate particles associated with isolated dispersion curves. To see the connection with the e-m spectrum, we define two-point correlation functions, establish F-K formulas and spectral representations. The normalized two-point correlation functions are defined by (χ here denotes the characteristic function) Guss (u, v) = φ¯ su (u)φsu (v) χu0 ≤v 0 − φsu (u)φ¯ su (v)∗ χu0 >v 0 = Guss (u − v), G (u, v) = φs (u)φ¯ (v) χu0 ≤v 0 − φ¯ s (u)φ (v)∗ χu0 >v 0 = G (u − v) , (2.8) ss
s
s
ss
where ∗ denotes complex conjugation. These seemingly awkward definitions allow us to show the existence of particles (upper gap property), and we observe that the u0 > v 0 definition, extended to u0 = v 0 , agrees with the u0 ≤ v 0 definition, using time reversal and parity (see Theorem 4 below). It is enough to restrict our attention to the φ¯ s as later on, using a charge conjugation symmetry, identical spectral results hold for φsu , and their identification as anti-baryons is made. Within this context, from now on, we suppress the index from φ¯ s in order to simplify the notation. We set E(λ0 , λ) = E0 (λ0 )E(λ), where E0 (λ0 ) is the spectral family for the operator T0 , and E(λ) = di=1 Ei (λi ) is the product of the spectral families for P i . The spectral representation of the next proposition is an important tool.
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Proposition 1. For u0 = v 0 , φ¯ s ≡ φ¯ s (1/2, 0), the following F-K formula holds, 1 0 0 Gss (u, v) = − (λ0 )|v −u |−1 eiλ.(v−u) d(φ¯ s , E(λ0 , λ)φ¯ s )H , (2.9) −1
Td
and is an even function of v − u. We now obtain a spectral representation for the Fourier transform of the two=point function Gss (u, v). For x ∈ Zd+1 , with an abuse of notation, we define Gss (x = u − v) ≡ Gss (u, v). Then, the Fourier transform ˜ ss (p) = G Gss (x)e−ipx , p ∈ Td+1 , x∈Zd+1
admits the spectral representation ˜ ss (p) = G ˜ ss (p) − (2π)d G ×
1
δ(p − λ)
1
0 eip − λ0 −1 Td dλ0 dλ (φ¯ s (1/2, 0), E0 (λ0 )E(λ)φ¯ s (1/2, 0))H ,
+
1
e−ip − λ0 0
(2.10)
˜ where G(p) = x∈Zd e−ip.x G(x 0 = 0, x). From the above spectral representation, we see that points of non-analyticity in p0 , on the imaginary axis, are points in the e-m spectrum. It is possible that points of nonanalyticity of the form p0 = ±π + iq 0 can occur but this is shown not to be the case in our analysis. We determine the spectrum and show the existence of isolated dispersion curves, up to the meson-baryon threshold −5 ln κ, by showing that ss (u, v), the convolution inverse of the two-point function Gss (u, v) decays faster than Gss (u, v), and hence the Fourier transform ˜ ss (p) of ss (x = u − v) ≡ ss (u, v) has a larger region −1 ˜ s s (p)/det[(p)] ˜ ≡ ˜ ss of analyticity in p0 . Thus, the cofactor [cof ] (p) provides a ˜ meromorphic extension of Gss (p), and the e-m spectrum occurs, for each p, as points ˜ given by the p0 imaginary axis zeroes of det[(p)]. ˜ To analyze det (p), it suffices to obtain a long range bound for (x), but we need its precise short distance behavior, for |x| ≤ 2, to determine the masses and the mass splitting up to and including order κ 6 . However, to control the error, bounds on (x) (which improve those obtained by the hyperplane decoupling method [23, 24, 19, 15]) are needed for some x’s with |x 0 | ≤ 4. These bounds are obtained by showing explicit cancellations in the Neumann series. The two-point function convolution inverse (x) is defined by −1 −1 Gd , = (1 + G−1 d Gn )
(2.11)
using a Neumann series, where Gd is the diagonal part of G given by Gd,ss (u, v) = Gss (u, u)δss δuv ,
(2.12)
Gn,ss (u, v) = Gss (u, v) − Gd,ss (u, v) .
(2.13)
and Gn is the remainder
By the bounds in Theorems 1 and 2 below, G, G−1 d , Gn and are bounded, as matrix operators on 2 (C4 × Zd+1 ). o
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Moreover, ss (x) is analytic in κ as Gss (x) is, and its short distance behavior is determined by expanding in κ. Long-range bounds on the decay of Gss (x) and ss (x) are obtained by the decoupling of the hyperplane method (see [23, 19, 24, 15]). Our results hold for all sufficiently small hopping parameter κ. The bounds on G and are given in the theorem below. d i Theorem 1. Let |x| ≡ i=1 |x |. For 0 < κ ≪ 1, and for c denoting a positive constant, independent of κ, the following bounds hold: |Gss (x)| ≤ cκ 3|x |ss (x)| ≤
0 |+3|x|
;
cκ 3+5(|x |−1)+3|x| , |x 0 | ≥ 2 0 cκ 3|x |+3|x| , |x 0 | ≤ 1 .
(2.14)
0
(2.15)
The short-distance behaviors of G(x) and (x) are given by: Theorem 2. Let 0 < κ ≪ 1, µ, ν = 0, 1, . . . , d and e0 , ei and ej , i, j = 1, . . . , d, denote the time and space unit vectors, and , = ±1. The following short-distance behaviors hold: −δss + O(κ 8 ) , x = 0; 3 δ + O(κ 9 ) c κ , x = e0 ; 3,0 ss |Gss (x)| = c3 κ 3 δss + O(κ 7 ) (2.16) , x = ej ; 6 + O(κ 10 ) , x = 2eµ ; c δ + c δ κ δ 6,0 µ 0 6 µ j ss cs,µν κ 6 δss + O(κ 10 ) , x = eµ + eν , µ < ν ; and
2 )κ 6 δ + O(κ 8 ) , x = 0 ; −δss − (2dc32 + 2c3,0 ss 3 11 κ δ + O(κ ) , x = e0 −c 3,03 ss 11 −c3 κ δss + O(κ ) , x = ej O(κ 12 ) , x = 2e0 ; −(c32 + c6 ) κ 6 δss + O(κ 10 ) , x = 2ej |ss (x)| = −[cs,µ,ν + 2c32 ]κ 6 (1 − δ0µ )(1 − δ0ν )δss + O(κ 10 ) , x = eµ + eν , µ < ν ; 10 0 O(κ ) , |x | = 1, 0 < |x| ≤ 2 ; O(κ 13 ) , |x 0 | = 2, |x| ≤ 1 ; O(κ 16 ) , |x 0 | = 3, |x| = 0 ; O(κ 19 ) , |x 0 | = 4, |x| = 0 ,
(2.17)
where the κ independent constants are given by 3 c3 0 = −6 , c3 = − , 4 9 c60 = −c32 0 , c6 = − , 2
(2.18)
and cs µν = Ae s
µ ,x
ν ,x
+ As e
, µ < ν , x = eµ + eν ,
(2.19)
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with Ae s
µ ,x
=−
9 µ eν eµ eν eµ eν + δ + 2 , δ|s| 3/2 e |s| 1/2 33,33,33 33,33,44 33,34,43 16
(2.20)
where we set µ ν
e e α1 β1 ,α2 β2 ,α3 β3 =
3
µ
ν
e e
k=1
αk βk
.
(2.21)
Remark 1. The absence of lower order terms in ss , as compared to Gss , for |x 0 | = 1, 2, 3, 4, is due to explicit cancellations in the Neumann series and improves the hyperplane method bounds obtained in Theorem 1. Concerning the mass spectrum, i.e. the e-m spectrum at zero-space momentum, it turns out that the mass is determined up to O(κ 6 ) by the values of ss (x) up to distance |x| ≤ 2, and s = s . The κ 6 contribution to (x), for these values of x, comes from the first and second order terms in Gn [see Eq. (2.13)] in the Neumann series to Eq. (2.11). The second order terms are independent of s since they are products of two κ 3 terms of Gn,ss (x), for points x of distance one, which are diagonal and independent of s. For the first order term in Gn (x), |x| = 2, the κ 6 contributions come from straight contributions and angle contributions. Straight contributions have two subsequent sets of three identical bonds, with the same orientation, connecting e.g. the point 0 to eµ and then to 2eµ ); and the property for the matrices of Eq. (3.31) guarantees that these contributions behave like the κ 3 , diagonal and s-independent contributions Gn,ss (x), |x| = 1, and do not give rise to mass splitting as well. Angles are contributions to Gn,ss (x) for points of the form x = ei + ej , i, j = 1, 2, . . . , d, i < j , , = ±1. These are L-type contributions associated with two sets of three lattice bonds, with the same orientation; one set connecting the points 0 → ei and the other connecting ei → ei + ej , or one set connecting the points 0 → ej and the other connecting ej → ei + ej . They contribute to mass splitting in O(κ 6 ), for d = 2. For d = 3, they add up to zero and no mass splitting appears (see Theorem 3). Before stating our results on the mass spectrum and dispersion curves, we give an intuitive picture for the asymptotic behavior of the mass. Retaining only the diagonal part of ss (x), x = 0 and |x 0 | = 1, the equation for the mass is ˜ 0 = iM, p = 0) (1 + c3,0 κ 3 eM )4 = 0 , M ∈ R , det (p so we have a mass M with magnitude of order −3 ln κ, with a four-fold degeneracy. To rigorously determine the e-m spectrum, we first consider the subspace Hb ⊂ Ho ⊂ H, generated by three quarks or anti-quarks. Our analysis begins by exploiting symmetries for p = 0, namely the reflection on the coordinates x 1 or x 2 , which has the effect of changing the sign of the index s on the composite fields φ˜ s,u (analogous to a ˜ 0 = iM, p = 0) reversal of the third spin component, spin flip, in the continuum). (p is shown to be diagonal and only depends on |s|, s = ±1/2, ± 3/2. The determination of the non-singular part can be cast into an analytic implicit func˜ 0 = iM, p) is diagonal, but the tion problem. For p = 0, we have not shown that (p asymptotic form of the dispersion curve can be obtained by a Rouch´e theorem argument ˜ 0 = iw(p), p). Our results in Hb are extended to the whole for the zeroes of det (p odd Hilbert space Ho by adapting the Euclidean subtraction method of [8].
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The results for the e-m spectrum are given in the theorem below. Theorem 3. The spectral multiplicity of the spectrum described below is doubled due to identical contributions of baryonic particles and anti-particles. For small κ, we have: 1. The mass spectrum in Ho and for the energy interval (0, −(5 − ) ln κ), > 0, contains two-masses (not necessarily distinct) M3/2 = M−3/2 and M1/2 = M−1/2 , each of multiplicity two and Ms = −3 ln κ + rs (κ) ,
(2.22)
and rs (κ) ≡ rs (κ, d) is real analytic in κ, for each d. We obtain rs (κ) = − ln |c3 0 | + 2dc3 κ 3 + [4dc32 + 2dc6 1 +c32 0 − (2dc3 )2 + 4(cs 12 + cs 13 + cs 23 )]κ 6 + O(κ 7 ) , 2 where cs 13 and cs 23 are to be omitted for d = 2. For d = 3, cs 13 + cs 23 + cs 12 = 0, so there is no mass splitting up to and including O(κ 6 ). For d = 2, the mass splitting is given by M3/2 − M1/2 = 4(c3/2 12 − c1/2 12 )κ 6 + O(κ 7 ) = 18κ 6 + O(κ 7 ). 2. The e-m spectrum in Ho and for the energy interval (0, −(5−) ln κ), > 0, consists of four dispersion curves, two by two identical and not necessarily distinct from each other, each of which has the form w(p) = −3 ln κ − ln |c3 0 | + 2dc3 κ − c3 κ 3
3
d
2(1 − cos p j ) + O(κ 6 ) . (2.23)
j =1
The curves w(p) are increasing functions of each component pj of p, and are convex for small |p|. Remark 2. In contrast to Eq. (2.23), for free fermions and small |p|, the coefficient of the |p|2 term in w(p) is proportional to κ. Remark 3. The nonzero mass splitting in d = 2, in contrast to the case d = 3, may have its origin in the fact the spatial rotation group, even on the lattice, is abelian for d = 2 whereas it is non-abelian for d = 3. (As we will see below, the definition of a charge conjugation transformation is also d dependent.) As remarked before, symmetries are only used to show that the mass depends on |s|. Whether or not this continuum spin analogy can be carried over into a more precise correspondence is an interesting question to be investigated. However, if we take this continuum spin analogy seriously, then the physical significance of the mass splitting, for d = 2, is that the lower energy state for three quarks occurs when there is less spin alignment. For d = 3, if present, the mass splitting is a higher order effect. Let us now turn to a discussion of symmetries, other than gauge, and give our symmetry results. Symmetry operations are defined on the field (gauge and Grassmann) algebra. A symmetry operation Y maps single Grassmannian fields by Y ψa,α (x) = Aαβ ψ˜ aβ (Yx), Y ψ¯ (x) = ψ˜¯ (Yx)B , a,α
aβ
βα
(2.24)
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393
and acts on functions f (gxz ) of the gauge group SU(3) by Y f (gxz ) = (Yf )(gY x Y z ) ,
(2.25)
where Y is a transformation acting on the coordinates, ∼ means a bar or removal of a bar, if one is already present. On Grassmannian monomials M1 , M2 , Y is taken to have the property Y(M1 M2 ) = Y(M1 )Y(M2 ) , which we call order preserving (homomorphism) or Y(M1 M2 ) = Y(M2 )Y(M1 ) , which we call order reversing (anti-homomorphism). The symmetry operation is extended to the whole algebra of the fields by linearity or anti-linearity. A symmetry of the model is defined to be a symmetry operation which leaves invariant the action S of Eq. (2.2) and which has the following property involving the normalized expectations of gauge invariant functions F of the fields: YF = F ,
(2.26)
where F means either F or F ∗ . Furthermore, the symmetry can be implemented on the quantum mechanical Hilbert space H by a linear or anti-linear operator stabilizing the null space N , i.e. such that if F ∈ N then YF ∈ N . Our results on rotation, coordinate reflection, parity and charge conjugation (see [25]) are given in the next theorem. Theorem 4. Each of the symmetry operations specified below is a symmetry of the model. The Case d = 3: 1. Spatial rotations r by π/2 and π about: eiπ/4 (a) The direction e1 : r1 (e1 , e2 , e3 ) = (e1 , e3 , −e2 ), with B = √ (1 − γ 2 γ 3 ) = 2 A−1 , and f (gxy ) → f (gr1 x r1 y ). For rotations of angle π , B = −iγ 2 γ 3 . eiπ/4 (b) The direction e2 : r2 (e1 , e2 , e3 ) = (−e3 , e2 , e1 ), with B = √ (1 − γ 3 γ 1 ) = 2 A−1 , and f (gxy ) → f (gr2 x r2 y ). For rotations of angle π , B = −iγ 3 γ 1 . eiπ/4 (c) The direction e3 : r1 (e1 , e2 , e3 ) = (e2 , −e1 , e3 ), with B = √ (1 − γ 1 γ 2 ) 2 = A−1 , and f (gxy ) → f (gr3 x r3 y ). For rotations of angle π , B = −iγ 1 γ 2 . 2. Spatial coordinate reflections R in: (a) The coordinate e1 : R1 (e1 , e2 , e3 ) = (−e1 , e2 , e3 ), with B = iγ 0 γ 2 γ 3 = A−1 , and such that f (gxy ) → f (gR1 x R1 y ). (b) The coordinate e2 : R2 (e1 , e2 , e3 ) = (e1 , −e2 , e3 ), with B = iγ 0 γ 1 γ 3 = A−1 , and such that f (gxy ) → f (gR2 x R2 y ). (c) The coordinate e3 : R3 (e1 , e2 , e3 ) = (e1 , e2 , −e3 ), with B = −iγ 0 γ 1 γ 2 = A−1 , and such that f (gxy ) → f (gR3 x R3 y ).
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P.A. Faria da Veiga, M. O’Carroll, R. Schor
3. Parity P: P(e1 , e2 , e3 ) = (−e1 , −e2 , −e3 ), with B = γ 0 = A−1 , and such that f (gxy ) → f (gP x Py ). ∗ ). 4. Charge Conjugation C: ∼ means bar, B = γ 2 γ 0 , and such that f (gxy ) → f (gxy 5. Time Reversal T : T (e0 , e1 , e2 , e3 ) = (−e0 , e1 , e2 , e3 ), B = γ 0 , ∼ means bar, means complex conjugation, f (gxy ) → f (gT x T y )∗ . The symmetry operations 1–3 are order preserving, 4–5 are order reversing, 1–4 (respectively, 5) are linear (respectively, anti-linear) and are implemented on H by unitary linear (respectively, anti-linear) operators. The Case d = 2. Of course, we do not have rotations r1 and r2 , and also the e3 axis reflection. The matrices A and B are the same as the above, except for charge conjugation, where A = γ 1 = B. Remark 4. We refer the reader to [25], for comparison with the continuum. Before we close the Introduction, we make some remarks on the consequences of some of the symmetries of Theorem 4. Charge conjugation implies Gss (u, v) = Sss1 Gus1 s2 (u, v)Sst2 s , where S t S = I4 = SS t , S 2 = −I4 , leading to det ˜ (p) = det ˜ u (p) so that the mass spectrum and dispersion curves for the baryon particles and anti-particles are identical. Spatial rotations, coordinate reflections and parity are used to obtain symmetry prop˜ ss (p), which in turn follow from symmetry properties erties of ˜ ss (p) via those of G of Gss (x). Recall that the masses and dispersion curves are determined to O(κ 6 ) by the short distance behavior of Gss (x) for distances |x| ≤ 2. It is enough to determine Gss (x), e.g. for x = 0, e0 , e1 , 2e1 , e1 + e3 , and the other contributions are determined by the use of symmetries, and indeed to all orders in κ. The rest of the paper is devoted to prove Proposition 1 and our four main theorems and is organized as follows: in Sect. 3 we first prove Proposition 1. Next, Theorem 1 is proved, using a decoupling of hyperplane method, and last we prove Theorem 2; in Sect. 4, we prove Theorem 3; in Sect. 5, we prove Theorem 4; and leave Sect. 6 for concluding remarks. We point out that many of the proofs involve long and tedious computations. In these cases, we only give prototype detailed calculations, as brief as possible, but try to keep this work as self-contained as possible. 3. Decay Bounds and Short-Distance Behavior of G and Γ We begin this section by giving the proof of Proposition 1. Proof of Proposition 1. For u0 < v 0 , Gss (u, v) = φs (u)φ¯ s (v) = −[T0v T v−u φ¯ s (1/2, 0)]φs (−1/2, 0). But φs (−1/2, 0) = φs (1/2, 0) so that Gss (u, v) = [T0v
0 −u0 −1
0 −u0 −1
×
T v−u φ¯ s (1/2, 0)][ φ¯ s (1/2, 0)] .
Note that by the parity symmetry of Theorem 4, since φ˜ η (w 0 , 0) has parity −1, v − u can be replaced by u − v here and for u0 > v 0 below. The F-K formula of Eq. (2.4) 0 0 gives the result. For u0 > v 0 , write Gss (u, v) = −φ¯ s (u)φs (v)∗ = −[T0u −v −1 T u−v φ¯ s (1/2, 0)] φs (1/2, 0)∗ , and the result follows by the F-K formula of Eq. (2.4).
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We now use the decoupling of the hyperplane method to obtain bounds on G and , as given in Theorem 1. We assume that the reader has some familiarity with this method and refer to [23, 24, 19, 15] for more details. We will encounter gauge group integrals of monomials in the group matrix elements gij (i, j = 1, 2, 3 denote SU(3) matrix elements, and we suppress the lattice points from the notation) and the inverses gij−1 . Following the techniques developed in Chapter 8 of [20] and in [26, 27], for the general SU(N ) case, if gij−1 occurs, it is substituted by the cofactor expression 1 gij−1 = j i2 i3 ij2 j3 gi2 j2 gi3 j3 . 2 Next, the following generating function formula will be the basis for computing gauge group integrals (see [27]), eJ g dµ(g) =
∞
an (detJ )n ,
(3.27)
n=0
where J g = 3i,j =1 Jij gij , and the coefficients an > 0 are determined recursively and 2! given by an = n!(n+1)!(n+2)! . Integrals of monomials in the g’s are given by the deriva 1 tives with respect to the “sources” J at J = 0. Since detJ = 3i1 ,... , j3 =1 3! i1 i2 i3 j1 j2 j3 Ji1 j1 Ji2 j2 Ji3 j3 , the integral of a monomial with only g’s vanishes unless its degree is a multiple of three. In particular, we obtain 1 ga1 b1 ga−1 dµ(g) = δa1 b2 δa2 b1 , (3.28) 2 b2 3 1 ga1 b1 ga2 b2 ga3 b3 dµ(g) = a1 a2 a3 b1 b2 b3 . (3.29) 6 These results can also be obtained using the decomposition of tensor product representations and the Peter-Weyl orthogonality relations (see [18]). Also, we use the following properties involving matrices (µ, ν = 0, 1, 2, 3, and , = ±1) e −e = 0 , µ µ µ e e = −2 e , µ
µ
ν
ν
(3.30) (3.31)
e e = 2I4 − − e −e . µ
µ
(3.32)
We will obtain decay properties for the more general gauge invariant correlation functions defined by 0 −1/2
F (u)L(v) = [(T0 )u
T u F ][(T 0 )v
0 −1/2
T v L] ,
(3.33)
where F, L ∈ Ho , and are supported in w 0 = 1/2, T0 is time translation by e0 and 1 d T u = T 1,u . . . T d,u is space translation by u = (u1 , . . . , ud ) ∈ Zd . We discuss explicitly the decoupling procedure for the time (vertical) direction. The space directions are treated similarly, and together with the time direction. The arguments are carried out for correlation functions in a finite volume ∈ Zd+1 o , with bounds uniform in the volume ||, and extend to the infinite volume using standard consequences of the polymer expansion (see e.g. [25, 19]).
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P.A. Faria da Veiga, M. O’Carroll, R. Schor
For u0 < v 0 , p ∈ Z, u0 + 1/2 ≤ p ≤ v 0 − 1/2 (or, if u0 > v 0 , v 0 + 1/2 ≤ p ≤ − 1/2), replace the hopping parameter κ > 0 multiplying the nonlocal fermionic part of the action (2.2) (not the κ in M) by κp ∈ C. With α = (α1 , α2 , α3 ) denoting a multiple spin index, letting u0
b˜α (u) = a1 a2 a3 ψ˜ a1 α1 (u)ψ˜ a2 α2 (u)ψ˜ a3 α3 (u) , and denoting ∂ r /∂κpr by ∂ r and by ∂0r its κp = 0 value, the following properties hold. Lemma 1. Concerning the derivatives of G, we have: 1. If u0 = v 0 , ∂0r F (u)L(v) = 0 , r = 0, 1, 2, 4. 2. If u0 < v 0 , and r = 3, 1 F (u)b¯σ (w) bσ (w + e0 )L(v) ∂03 F (u)L(v) = − 6 σ w|w0 =−1/2+p − F (u)bσu (w) b¯σu (w + e0 )L(v) . (3.34) κp =0
3. If u0 > v 0 , and r = 3, ∂03 F (u)L(v) = −
1 6
w|w0 =−1/2+p
σ
L(v)b¯σ (w) bσ (w + e0 )F (u)
− L(v)bσ (w) b¯σ (w + e0 )F (u)
κp =0
.
(3.35)
Proof. Consider u0 < v 0 ; the argument for u0 > v 0 is similar. Expanding F (u)L(v) in powers of κp brings powers of the inter-hyperplane part of the action (2.2) into the integrands of the numerator and denominator (partition function Z of Eq. 2.1) of the expression for this correlation function. For the denominator, the κp coefficient is a sum of single bond terms, and each term is a product of expectations containing a single Grassmann field (fermion) ψ or ψ¯ which integrates to zero. The coefficient of κp2 is a sum of terms with two bonds which must be coincident and of opposite orientation in order that the gauge group integral be nonzero. The gauge group integral can then be performed using (3.28) and is nonzero. For the numerator, we consider the coefficients of κp0 , κp1 , ..., κp4 . For κp0 , the expectation factorizes and each of the two factors has an odd number of fermions and gives zero upon integration. The same argument holds for the coefficients of κp2 and κp4 . For κp1 , the integral over the single inter-hyperplane gauge field gives zero. Finally, for the coefficient of κp3 , we have terms with three bonds. For a nonzero contribution, these bonds must be coincident and have the same orientation and the gauge group integral is performed using Eq. (3.29). However, by applying the property of matrices given in Eq. (3.30), for the time direction µ = 0, gives the result. To calculate the κp derivatives of , it is convenient to consider instead ≡ −, minus the convolution inverse of G, and use the formula r−1 r ∂ r = ∂ r−s G ∂ s . (3.36) s s=0
The first four κ = 0 derivatives of are given in the next lemma.
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Lemma 2. For the derivatives of , we have: 1. If u0 = v 0 , ∂0r (u, v) = 0, r = 0, 1, 2. 2. If u0 < v 0 , ∂03 (u, v) = − 16 w|w0 =−1/2+p δuw δw+e0 ,v . 3. If u0 > v 0 , ∂03 (u, v) = − 16 w|w0 =−1/2+p δvw δw+e0 ,u . 4. If u0 + 1 < v 0 or u0 − 1 > v 0 , ∂04 (u, v) = 0. Proof. For the first statement, consider first u0 < v 0 . Using Eq. (3.36) and Lemma 1, the result follows directly; and similarly for u0 > v 0 . For the second statement, using the first one and the second statement of Lemma 1, we have (u, w)∂ 3 G(w, z)(z, v) ∂03 (u, v) = κp =0
w,z|w0 +1/2≤p≤z0 −1/2
=−
1 6
(u, w)
w,z|w0 +1/2≤p≤z0 −1/2
×G(r + e0 , z) (z, v)κ
p =0
G(w, r)
r|r 0 +1/2=p
.
(3.37)
Using the first item of Lemma 1, for r = 0, the w and z sums can be taken over all values to give the result. The third statement follows from a similar argument. The fourth one follows from the first three and using Eq. (3.36) again. We are now ready to prove Theorem 1. Proof of Theorem 1. Using a Cauchy integral representation for each κp and for analogous spatial complex hyperplane decoupling parameters, taking into account the number of vanishing derivatives as given in Lemmas 1 and 2, and using Cauchy estimates on the multiple integral gives the result. Let us now turn to the proof of Theorem 2. The corrections to the asymptotic mass value of −3 ln κ which we need for the determination of mass splitting require precise values of ss (x) for small |x|. The results go beyond those obtained by the hyperplane method, and rely on explicit cancellations in the Neumann series for . The results below are obtained expanding in powers of κ and controlling the remainders using the analyticity of G and , and the decay bounds of Theorem 1. Proof of Theorem 2. Consider the expansion of the denominator of G in powers of κ. Here, the contributions are associated with bond cycles. For a point where the bonds arrive and leave in opposite directions, the fermi integration gives products of matrices in which the property of Eq. (3.30) is used to give zero. The first nonvanishing contribution occurs for κ 8 , corresponding to two sets of four bonds going around an elementary square in opposite directions. For the numerators in G(u, v), we obtain bond contributions connecting u to v. Bonds are always oriented in such a way they have a ψ¯ field attached at the source and a ψ at the end. For the lowest orders in κ, the bond connecting u to v must be made of coinciding bonds, two by two (with opposite directions) or three by three (with the same direction); and the gauge group integrals can be done using Eqs. (3.28) and (3.29). Otherwise, the gauge group integral gives zero. An exception here is the κ 5 contribution to G(x), with |x| = 1, with four coinciding bonds with the same direction and one bond with the opposite one. This contribution gives zero, after integrating over the fermions and gauge fields, because of the structure of the
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matrices . We explicitly carry out two typical calculations: one for the κ 3 contribution to G(x), associated with x = eµ (|x| = 1), involving one distance unit, and another for the case of an angle contribution to G(x) for x = eµ + eν , µ < ν, so that |x| = 2. The straight |x| = 2 contribution case µ = ν and = is simpler as the property (3.31) ensures its behavior is like the case |x| = 1. The other contributions are obtained similarly. The κ 3 contribution to the case |x = u − v| = 1, involves three identical bonds connecting u to v, with the same orientation. After performing the gauge group integral, using Eq. (3.29), we obtain, for Gss (x), 1 κ 3 a1 a2 a3 b1 b2 b3 φs (u)ψ¯ a1 α1 (u)ψ¯ a2 α2 (u)ψ¯ a3 α3 (u)0 6 2 µ µ µ × αe1 β1 αe2 β2 αe3 β3 ψb1 β1 (v)ψb2 β2 (v)ψb3 β3 φ¯ s (v)0 . The presence of the lower field φs (φ¯ s ) forces the α’s (β’s) to be lower indices in the µ first (second) expectation. In the lower index subspace, e is diagonal with values −2 (time direction) and −1 (space directions). With this, we obtain 1 κ 3 − [8δµ0 + δµj ] a1 a2 a3 b1 b2 b3 φs (u)ψ¯ a1 α1 (u)ψ¯ a2 α2 (u)ψ¯ a3 α3 (u)0 6 2 × ψb1 α1 (v)ψb2 α2 (v)ψb3 α3 (v)φ¯ s (v)0 . Now, in order to have a nonzero contribution, the spin indices for φs and φ¯ s must be equal to the fermions in each · 0 factor, which gives s = s . The computation continues using Wick’s theorem, paying attention to the normalization factors for φs and φ¯ s given in Eq. (2.7) and, next, performing the sum over the color indices, which gives us a multiplicative factor 36. The κ 6 contribution of the angle 0 → eµ → eµ + eν ≡ x to Gss (x = u − v) is µ ,x denoted by κ 6 Ae and has three g0 eµ emanating from zero in the eµ direction and ss three geµ x emanating from eµ and going to x. After the gauge integrations of (g0 eµ )3 and (geµ x )3 , using Eq. (3.29), and fermi integration of the fields at eµ using Wick’s theorem, with 0 denoting the expectation with the hopping parameter set to zero in the action of Eq. (2.2), we get 6 µ ,x 1 Ae = − φs (u)a1 a2 a3 ψ¯ a1 α1 (u)ψ¯ a2 α2 (u)ψ¯ a3 α3 (u)0 ss 2 1 µ µ µ ν ν ν × δβ1 σ (α4 ) δβ2 σ (α5 ) δβ3 σ (α6 ) αe1 β1 αe2 β2 αe3 β3 α 4eβ4 α 5eβ5 α 6eβ6 6 σ × b4 b5 b6 ψ¯ b4 β4 (v)ψ¯ b5 β5 (v)ψ¯ b6 β6 (v)φ¯ s (v)0 ,
where σ means sum over permutations of 4, 5, 6. Carrying out the β1 , β2 and β3 sums, we get 6 1 eµ ,x Ass = − φs (u)a1 a2 a3 ψ¯ a1 α1 (u)ψ¯ a2 α2 (u)ψ¯ a3 α3 (u)0 2 µ ν µ ν 1 µ ν × e e e e e e α1 σ (β4 ) α2 σ (β5 ) α3 σ (β6 ) 6 σ × b4 b5 b6 ψ¯ b4 β4 (v)ψ¯ b5 β5 (v)ψ¯ b6 β6 (v)φ¯ s (v)0 .
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Since the last factor is unchanged by permutations of β4 , β5 and β6 , we can perform σ to get µ ,x
Ae ss
6 1 µ eν φs (u)a1 a2 a3 ψ¯ a1 α1 (u)ψ¯ a2 α2 (u)ψ¯ a3 α3 (u)0 e α1 β4 ,α2 β5 ,α3 β6 2 × b4 b5 b6 ψ¯ b4 β4 (v)ψ¯ b5 β5 (v)ψ¯ b6 β6 (v)φ¯ s (v)0 ,
= −
where we have used the definition given in Eq. (2.21). The presence of φs and φ¯ s constrains the indices α1 , . . . , α3 , β4 , . . . , β6 to be lower indices (i.e. to assume values 3 or 4). Calculating the 0 factors, we finally get, for s = s , Eq. (2.20), with the µ ,x µ ,x ≡ Ae . identification Ae ss s These results lead to Eq. (2.16). To obtain the results for given in the second part of Theorem 2, we recall the discussion given in Sect. 1 and note that ss is obtained from Gss from the Neumann series [see Eq. (2.11)]. We show how cancellations occur, which improves the hyperplane method bound. We explicitly consider the case x = e0 + ej , j = 1, . . . , d; the other cases where there are one, two, three or four time units are treated similarly. Recall that [see Eq. (2.12)] Gd,ss (u, v) = Gss (u, v)δss δuv and from Theorem 2 we obtain Gss (0, 0) = −1 + O(κ 8 ). With Gn given in Eq. (2.12) and using −1 i i −1 = ∞ i=0 (−1) [Gd Gn ] Gd , for x = u − v = 0, we write −1 ss (u, v) = −G−1 G−1 d,ss (0)Gn,ss (u, v)Gd,s s (0) + d,ss (0)Gn,ss1 (u, w) w,s1
×
−1 G−1 d,s1 s1 (0)Gn,s1 s (w, v)Gd,s s (0)
+ O(G3n ) .
(3.38)
For s = s , there are two κ 6 angle contributions to Gn,ss (u, v) in the first term of Eq. (3.38), and these are cancelled by the product of two O(κ 3 ) contributions for w−v = e0 and w − v = ej in the second term of Eq. (3.38), using Eq. (2.16). 4. Spectral Results We now prove Theorem 3. The proof is done in two steps. First, we consider the restriction to the baryonic space Hb . Then, we extend the results to the whole odd space Ho , up to −(5−) ln κ, i.e. up to near the meson-baryon threshold. To determine the baryon masses ˜ 0 , p) = 0. For and dispersion curves, we find the p0 imaginary axis solutions of det (p 0 ˜ the mass spectrum, we find, by the use of symmetries, that s,s (p , p = 0) is diagonal and only depends on |s|. Furthermore, we show that Ms + 3 ln κ is real analytic in κ. For p = 0, as we have not found symmetries which simplify the matrix structure, and ˜ 0 = iw(p), p) = 0, by an we determine the dispersion curves w(p), where det (p application of Rouch´e’s Theorem. We state some symmetry properties in the next lemma. Lemma 3. The following symmetry properties hold for G and : 1. Gss (x) = [Gs s (x)]∗ and ss (x) = [s s (x)]∗ ; ˜ ss (iχ , p) = [G ˜ s s (iχ , p)]∗ and ˜ ss (iχ , p) = 2. For χ ∈ R, let p 0 = iχ . We have G ∗ [˜ s s (iχ , p)] ; ˜ ss (p 0 , p = 0) = G ˜ −s −s (p 0 , p = 0) and ˜ ss (p 0 , p = 0) = 3. At p = 0, G 0 ˜ −s −s (p , p = 0);
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˜ ss (p 0 , p = 0) = δss G ˜ ss (p 0 , p = 0) and ˜ ss (p 0 , p = 0) = 4. At p = 0, G 0 δss ˜ ss (p , p = 0). Proof. Here we use several symmetries given in Theorem 4. For x 0 = 0, the first item follows directly from the spectral representation for Gss [see Eq. (2.9)]. For x 0 = 0, the result follows from time reversal and parity. Thus, Item 1 holds for all x. The proof of Item 2 follows from parity invariance Gss (x 0 , x) = Gss (x 0 , −x). To prove Item 3, we use reflection symmetry in the coordinate x 1 . Here x = (x 1 , . . . , x d ) → x = (−x 1 , . . . , x d ). Item 4 is a consequence of π/2 rotations about the e3 axis. Here, for d = 3, x = (x 0 , x 1 , . . . , x d ) → x = (x 0 , −x 2 , x 1 , x 3 ) [x = (x 0 , x 1 , x 2 ) → x = (x 0 , −x 2 , x 1 ), for d = 2] and ψ˜ a 1 (x) → −ψ˜ a 2 (x ), ψ˜ a 2 (x) → −ψ˜ a 2 (x ), ψ˜ a 3 (x) → ψ˜ a 3 (x ), ψa 4 (x) → −iψa 4 (x ) and ψ¯ a 4 (x) → i ψ¯ a 4 (x ). After using Theorem 2 and taking the Fourier transform, we have ˜ ss (p 0 , p = 0) = −1 − 2dc3 κ 3 − 2[c32 0 + 2dc32 + dc6 ]κ 6 − 4(cs 12 + cs 13 + cs 23 )κ 6 0 0 + O(κ 8 ) − (c3 0 κ 3 + O(κ 11 )) (eip + e−ip ) + · · · . (4.39) We introduce an auxiliary function Hs (w, κ), jointly analytic in w and κ, for small 0 |κ| and |w|, such that Hs (w = −1 − c3 0 κ 3 e−ip , κ) = ˜ ss (p 0 , p = 0). The function Hs (w, κ) is defined by Hs (w, κ) = w − 2dc3 κ 3 − 2[2dc32 + c32 0 + dc6 ]κ 6 − 4(cs 12 + cs 13 + cs 23 )κ 6 + c32 0 6 0 1+w c3 0 κ 3 κ + + ss (x = 0, x) − ss (x 0 = 1, x) 1+w 1+w c3 0 κ 3 x x n −c3 0 κ 3 1+w n 0 + , (4.40) ss (x = n, x) + 1+w −c3 0 κ 3 n≥1,x|(n,x)=(1,0)
(x 0 = 0, x) (respectively, (x 0 = 1, x)) contains the contributions of O(κ 8 ) where ss ss s (respectively, O(κ 11 )) or higher. Hs (0, 0) = 0 and ∂H ∂w (0, 0) = 1 and hence the analytic implicit function theorem applies and gives us an analytic function w(κ) such that H (w(κ), κ) = 0. Thus, for κ real positive, the mass is given by
Ms = − ln −c3 0 κ 3 + ln (1 + w) . By an analysis of the formulas for the implicit function derivatives, with dw = −(∂H )−1 ∂k H ≡ −(H w)−1 , we find d0r w = 0, r = 0, 1, 2, 4, 5 and d03 w = −∂κ3 H (0, 0) = 12c3 d , d06 w = −∂κ6 H (0, 0) = −6! −4dc32 − 2dc6 − c32 0 − 4(cs 12 + cs 13 + cs 23 ) .
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Hence, for d = 3, Ms = −3 ln κ − ln |c3 0 | + 2dc3 κ 3 1 + 4dc32 + 2dc6 + c32 0 − (2dc3 )2 + 4(cs 12 + cs 13 + cs 23 ) κ 6 + O(κ 7 ) , 2 and similarly, for d = 2, with cs 13 = 0 = cs 23 . ij ij ij For an explicit calculation, for i < j , we use the compact notation 0 , 1 and 2 i j i j i j for e33e33 33 , e33e33 44 and e33e34 43 , respectively. Thus, from Eq. (2.20), 6 1 ij ij ij ei ej Ass = −36 δ|s| 3/2 0 + δ|s| 1/2 1 + 22 , 2 and
Gss (x = ei + ej ) = Aess e +e + Aess e +e ≡ cs ij κ 6 . By a direct computation, using the definition of Eq. (2.21), we find, i j
21 ∗ 3 12 0 = (1 − i) = [0 ]
,
i
j j
i
31 13 0 = 1 = 0
,
32 23 0 = 1 = 0 , 32 23 1 = 1 = 1 ,
21 ∗ 2 12 1 = (1 − i) (1 + i) = [1 ]
31 , 13 1 = −1 = 1
,
21 12 2 = 0 = 2
31 , 13 2 = −1 = 2
32 , 23 2 = −1 = 2 ;
so that (recall cs 13 = cs 23 = 0, if d = 2) 9 −4δ|s| 3/2 + 4δ|s| 1/2 , cs 12 = − 16 9 cs 13 = cs 23 = − 2δ|s| 3/2 − 2δ|s| 1/2 , d = 3 . 16 Thus, if d = 3, we have cs 12 + cs 13 + cs 23 = 0. This relation says that no mass splitting appears in d = 3 up to O(κ 6 ). On the contrary, cs 12 = 0 gives rise to a mass splitting for d = 2. Thus, up to and including O(κ 6 ), we obtain 18κ 6 , d = 2 (4.41) M3/2 − M1/2 = 0 , d = 3. ˜ 0 = iw(p), p) = 0. Let us now turn to the dispersion curves. They satisfy det (ip d j To determine them, with c3 (p) ≡ c3 j =1 2 cos p , we write the Fourier transform of ss (x) as 0 0 ˜ ss (p 0 , p) = −1 − c3 (p)κ 3 − c3 0 κ 3 (e−ip + eip ) δss 0 0 + ss (n, x) e−ip.x (e−ip n + eip n ) , (4.42)
n,x
where n,x means that all terms of order κ 6 or higher in ss (x) are included. Introduce the auxiliary matrix function Hss (w, κ) ≡ Hss (w, k, p) such that Hss (w = 0 −1 − c3 (p)κ 3 − c3 0 κ 3 e−ip , κ) = ˜ ss (p 0 , p). Hss (w, κ) is defined by Hss (w, κ) = wδss + ss (n, x) e−ip.x ×
n,x
1 + w + c3 (p)κ 3 −c3 0 κ 3
n
+
−c3 0 κ 3 1 + w + c3 (p)κ 3
n ,
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where n,x means that only O(κ 6 ) terms or higher order terms are to be included. H (w, κ) is jointly analytic in κ and w at (w, κ) = (0, 0). Letting f (w) ≡ det H (w, κ) = det wI + [det H (w, κ) − det wI ] ≡ g(w) + h(w) , we can apply Rouch´e’s Theorem to f (w), with the circle |w| = c|κ|6 , c ≫ 1. On the circle, |g(w)| = c4 |κ|24 and |h(w)| ≤ c1 |κ|24 < c4 |κ 24 | = |g(w)|, so that f (w) has four zeroes inside |w| = c|κ|6 as g(w) = w 4 has a fourth order zero. Notice that the upper bound for |h(w)| comes from an upper bound on the remaining twenty three terms in the difference between the two determinants. Now, for p 0 = iw(p), and κ real ˜ 0 = iw(p), p) = 0 has the form positive, each of the four zeroes satisfying det (p w(p) = −3 ln κ − ln |c3 0 | + 2dc3 (p)κ 3 − 2c3 κ 3
d
j =1 (1 − cos p
j ) + O(κ 6 ) .
To extend the spectral results from Hb to Ho , we adapt the Euclidean subtraction method established in [8]. Omitting the spin indices, and reinstating now the upper and lower indices introduced in Eqs. (2.5) and (2.6), we consider the subtracted two-point correlation function u u u F(u1 , u2 ) = GLL (u1 , u2 ) − GL, u (u1 , v1 ) (v1 , v2 )Gχ,L (v2 , u2 ) −
v1 ,v2 ∈Zd GL, (u1 , v1 ) (v1 , v2 )Gχ ,L (v2 , u2 ) ,
v1 ,v2 ∈Zd
where u (s) = (φ1u (s), φ¯ 2u (s)), (s) = (φ3 (s), φ¯ 4 (s)); χ u (v) = (φ¯ 1u (v), −φ2u (v)) and χ (v) = (φ¯ 3 (v), −φ4 (v)) have two components; u, (s, v) is a given by the convolution inverse of the two-point functions Gu, of Eq. (2.8), and we suppress the sum over spin components. Finally, SF,H (u, v) , u0 ≤ v 0 GF H (u, v) = S ∗ , u0 > v 0 (T 0 )−tF −1/2 F,− (T 0 )−tH −1/2 H with SF,H (u, v) = F (u)H (v), for F supported on 1/2 ≤ u0 ≤ tF and H on 1/2 ≤ v 0 ≤ tH . In this way, the extension to Ho is a consequence of the lemma below. Lemma 4. For u0 < v 0 , p ∈ Z, u0 + 1/2 ≤ p ≤ v 0 − 1/2 (or, if u0 > v 0 , v 0 + 1/2 ≤ p ≤ u0 − 1/2), and again denoting by ∂0 the κp derivative at κp = 0, we have ∂0r F(u, v) = 0, for r = 0, 1, 2, 3, 4. Proof. The proof is standard and follows from direct computation, paying attention to the support restrictions. It is trivial for r = 0. As before, it relies on the product structure for ∂03 , as the one given in Lemma 1, for the two-point function G. For r = 0, 3, the result follows from imbalance of fermion fields appearing in the Fermi expectations. This ends the proof of Theorem 3.
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5. Symmetries Here, we turn to the proof of the symmetries given by Theorem 4. We first obtain the symmetries at the level of correlation functions and then we show that they can be implemented on the Hilbert space H. To show F = YF , as defined in Eq. (2.26), it is enough to show that Y leaves the action invariant and that F = YF , where denotes the expectation with the action set to zero. In the course of the proof, we utilize a change of variables formula for linear transformations of ψ and separately for ψ¯ or, as in the case of charge conjugation and time ¯ More precisely, reversal, a formula for linear transformations that interchange ψ and ψ. we use variations of the formula we now give. If ψi = Uij−1 ψj , ψ¯ i = Vij−1 ψ¯ j , i.e., ψi = Uij ψj , ψ¯ i = Vij ψ¯ j , we have
¯ F (ψ, ψ)
n
dψk d ψ¯ k = detU detV
F (U −1 ψ , V −1 ψ¯ )
k=1
n
dψk d ψ¯ k . (5.43)
k=1
The above equation is established by noting that the integral on the l.h.s. is the coefficient of the maximal degree monomial nj=1 ψ¯ j ψj of F , written in this order; writing this monomial in terms of the ψ˜ , and identifying the coefficients of maximal degree yields formula (5.43). For all symmetries except for rotations, the Grassmann variables transformation matrices are diagonal or anti-diagonal (i.e. diagonal along the secondary diagonal) so that the change of variables formula is simple. We explicitly give the proof of Theorem 4 for the charge conjugation symmetry C, as the other proofs are simpler variations of this case. We show the invariance of the action under C. We have eµ C ψ¯ aα (u)αβ g(u, u + eµ )ab ψbβ (u + eµ ) = ψ¯ bγ (u + eµ )Aγβ µ
e ×αβ Bαρ g(u + eµ , u)ba ψaρ (u) , −e g(u + eµ , u)ba ψaβ (u) which corresponds to the term of the action ψ¯ bα (u + eµ )αβ imposing the condition (t denotes the transpose) µ −eµ A( e )t B = γρ . µ
γρ
For d = 3, the above condition is satisfied if AB = I4 and A (γ µ )t = −γ µ A. Since = γ 0,2 and (γ 1,3 )t = −γ 1,3 , the above is satisfied taking A = γ 0 γ 2 ; so that −1 B = A = γ 2 γ 0 and A2 = −I4 . For d = 2, the condition Aγ 3 = γ 3 A is dropped and the above is satisfied taking A = γ 1 = B. Next, we show C F = F . Using the change of variable of Eq. (5.43) formula ∗ }) dµ(g). Fixand performing the Fermi integrals, we are left with the integral f ({guv ing a gauge bond variable and, for a monomial, the gauge bond integral is real by Eq. (3.27). The result extends to polynomials and, by a limiting argument, to continuous functions. Using this property, we obtain f ({guv }) dµ(g) and we are done. We now show that C can be implemented on the whole Hilbert space H. We see that ˜ ˜ C ψ(u) = − ψ(u) and, for a function f ({guv }), we have C f ({guv }) = Cf ({guv }). ˜ then C F = − CF and C F = CF if F is even. If If F is an odd monomial in ψ, (γ 0,2 )t
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F is supported on u0 ≥ 1/2, and F is even or odd, then (σ = 1 if F is odd and zero for even F ’s) (C F )( C F ) e−S = (−1)σ ( F ) F e−S = F F e−S .
For general F , we decompose it into its even and odd components, i.e. F = Fe + Fo . Since (C Fe )( C Fo ) e−S = Fe Fo ) e−S = 0, we obtain that (C F )( C F ) e−S = F F ) e−S , for an arbitrary F . Thus, if F ∈ N then CF ∈ N and C can be lifted to H as a unitary operator obeying C 2 = −I . In the next step, we obtain the relation between the particle and anti-particle two-point correlation functions φs (u)φ¯ s (v) and φ¯ su (u)φsu (v), respectively. We take d = 3. The analysis is similar for d = 2. From the definition given in the statement of Theorem 4, ˜ in the following way (retaining only spin indices): ψ1 → −ψ¯ 1 , C transforms the ψ’s ψ2 → ψ¯ 3 , ψ3 → −ψ¯ 2 , ψ4 → ψ¯ 1 , and the same upon interchanging bar and unbarred ˜ Thus, Cφ ¯u ¯u ψ’s. ±3/2 = ±φ∓3/2 , Cφ±1/2 = ∓φ∓3/2 , and the same interchanging bar and u (u)φ u (v), with the minus sign unbarred φ’s. Thus, we obtain φs (u)φ¯ s (v) = ∓φ¯ −s −s for (s, s ) = (±3/2, ±3/2), (±1/2, ±1/2), (±3/2, ∓1/2) and (±1/2, ∓3/2), and the plus sign for the other (s, s ). As the masses are determined by analyzing the diagonal two-point correlation functions, with s = s , we see that the particle and anti-particle masses are the same. In general, let S denote the transformation that takes the φsu ’s into φs ’s. Explicitly, S has nonzero entries only on the secondary diagonal and Sss = 1, for (s, s ) = (3/2, −3/2) and (−1/2, 1/2), and −1 for (s, s ) = (−3/2, 3/2) and (1/2, −1/2). It is easy to check that S satisfies S t S = I4 = SS t , S 2 = −I4 and S t = −S. Then t Gss (u, v) = Ssj Gujk (u, v)Sks ,
˜ u and ˜ u , i.e. ˜ = S ˜ u S t . Hence, det ˜ = and we have the same transformation for G u ˜ det , which implies that the particle and anti-particle dispersion curves are identical. This concludes the proof of Theorem 4. 6. Concluding Remarks With more work, our methods can be used for the case of more than one flavor. It would be interesting to determine in what order in the hopping parameter mass splitting occurs (if at all) for the case of space dimension d = 3. As we have shown the existence of baryons, and we have good control of their dispersion curves, the question of the existence of baryon-baryon bound states is a most interesting one. Acknowledgements. We thank the anonymous referee for suggestions that improved the paper. This work was partially supported by Pronex and CNPq.
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3. Seiler, E.: Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Lecture Notes in Physics 159, New York: Springer, 1982 4. Montvay, I.: Numerical Calculation of Hadron Masses in Quantum Chromodynamics. Rev. Mod. Phys. 59, 263–285 (1987) 5. Creutz, M.: Lattice Gauge Theory: A Retrospective. Nucl. Phys. B, (Proc. Suppl.) 94, 219–226 (2001) 6. Osterwalder, K., Seiler, E.: Gauge Field Theories on a Lattice. Ann. Phys. (N.Y.) 110, 440–471 (1978) 7. Faria da Veiga, P.A., O’Carroll, M., Schor, R.: Understanding Baryons From First Principles. Phys. Rev. D67, 017501, 1–4 (2003) 8. Schor, R.: Nucl. Phys. B 222, 71–82 (1983); B 231, 321 (1984) 9. Banks, T., Raby, S., Susskind, L., Kogut, J., Jones, D.R.T., Scharbach, P.N., Sinclair, D.K.: t StrongCoupling Calculations of Hadron Spectrum of Quantum Chromodynamics. Phys. Rev. D 15, 1111– 1127 (1977) 10. Schreiber, D.: T-Expansion of QCD Baryons. Phys. Rev. D 48, 5393–5402 (1993) 11. Fr¨ohlich, J., King, C.: Meson Masses and the U(1) Problem in Lattice QCD. Nucl. Phys B 290, 157–187 (1987) 12. Neto, A.F., Faria da Veiga, P.A., O’Carroll, M.: Existence of Mesons and Mass Splitting in Strong Coupling Lattice QCD. J. Math. Phys., to appear. Available from http://www.icmc.usp.br/∼veiga/ short mesons.pdf 13. Machleidt, R.: The Nuclear Force in the Third Millennium. Nucl. Phys. A 689, 11c–22c (2001) 14. Machleidt, R., Holinde, K., Elster, Ch.: The Bonn Meson-Exchange Model for the Nucleon-Nucleon Interaction. Phys. Rep. 149, 1–89 (1987) 15. Faria da Veiga, P.A., O’Carroll, M., Pereira, E., Schor, R.: Spectral Analysis of Weakly Coupled Stochastic Lattice Ginzburg-Landau Models. Commun. Math. Phys. 220, 377–402 (2001) 16. Schor, R., O’Carroll, M.: J. Stat. Phys. 99, 1265–1279 (2000); J. Stat. Phys. 109, 279–288 (2002) 17. Berezin, F.A.: The Method of Second Quantization. NY: Academic Press, 1966 18. Simon, B.: Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996 19. Simon, B.: Statistical Mechanics of Lattice Models. Princeton, NJ: Princeton University Press, 1994 20. Creutz, M.: Quarks, Gluons and Lattices. Cambridge: Cambridge University Press, 1983 21. Luscher, M.: Construction of a self-adjoint, strictly positive transfer matrix for Euclidean lattice gauge theories. Commun. Math. Phys. 54, 283–292 (1977) 22. Fredenhagen, K.: On the existence of the real time evolution in Euclidean lattice gauge theories. Commun. Math. Phys. 101, 579–587 (1985) 23. Spencer, T.: The Decay of the Bethe-Salpeter Kernel in P (ϕ)2 Quantum Field Models. Commun. Math. Phys. 44, 143–164 (1975). Spencer, T., Zirilli, F.: Scattering States and Bound States in λP (φ)2 Models. Commun. Math. Phys. 49, 1–16 (1976) 24. Dimock, J., Eckmann, J.-P.: On the Bound State in Weakly Coupled λ φ 6 −φ 4 2 Models. Commun. Math. Phys. 51, 41–54 (1976); Ann. of Phys. (NY) 103, 289–314 (1977). Dimock, J.: Cluster Expansion for Stochastic Lattice Fields. J. Stat. Phys. 58, 1181–1207 (1990) 25. Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View. New York: Springer Verlag, 1986 26. Creutz, M.: Feynman Rules for Lattice Gauge Theories. Rev. Mod. Phys. 50, 561–571 (1978) 27. Creutz, M.: On Invariant Integration Over SU(N). J. Math. Phys. 19, 2043–2046 (1978) Communicated by J.Z. Imbrie
Commun. Math. Phys. 245, 407–424 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1023-1
Communications in
Mathematical Physics
Baryon Charges in 4d Superconformal Field Theories and Their AdS Duals Ken Intriligator, Brian Wecht Department of Physics, University of California, San Diego, La Jolla, CA 92093-0354, USA Received: 22 May 2003 / Accepted: 1 August 2003 Published online: 20 January 2004 – © Springer-Verlag 2004
Abstract: We consider general aspects of the realization of R and non-R flavor symmetries in the AdS5 × H5 dual of 4d N = 1 superconformal field theories. We find a general prescription for computing the charges under these symmetries for baryonic operators, which uses only topological information (intersection numbers) on H5 . We find and discuss a new correspondence between the nodes of the SCFT quiver diagrams and certain divisors in the associated geometry. We also discuss connections between the non-R flavor symmetries and the enhanced gauge symmetries in non-conformal theories obtained by adding wrapped branes. 1. Introduction Interesting gauge theories arise in string theory from D-branes at geometric singularities. This was first studied for orbifold singularities [1, 2] and, more recently, for more general singularities. In this paper, we will be especially interested in 4d N = 1 superconformal field theories (SCFTs) which can be string engineered by placing N D3 branes at the conical singularity of a general (local) Calabi-Yau 3-fold X6 . For large N , it is useful to consider the AdS dual [3] description of these SCFTs, which is IIB string theory on AdS5 × H5 with H5 the horizon manifold [4] of X6 . A mirror IIA construction is to 6 . wrap N D6 branes on the SYZ [5] T 3 of the mirror geometry X String theory and AdS/CFT provide useful insights into the SCFTs which can be so constructed. For example, AdS/CFT implies that the D3 brane world-volume gauge theory actually does flow to an interacting RG fixed point in the IR. It is thus interesting to study generally which quiver gauge theories can be string engineered, and what sorts of general predictions string theory makes about these SCFTs. Many techniques have been developed over the past several years to help determine, given a general singularity X6 , precisely what is the associated world-volume quiver gauge theory (or rather theories, since there can be many Seiberg dual descriptions of the same superconformal field theory). As of yet, however, there is no completely general
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method to systematically answer this question. A partial answer can be found via partial resolution of orbifold singularities [6], but there is no systematic method for following the RG flow from the orbifold SCFT to that of the partially resolved singularity. Another method, which is useful for toric singularities, is to go to the mirror IIA description, where the gauge group and matter content can be determined in terms of intersections of 3-cycles [7–9]. Still another method, which seems to give correct results [10] even outside of its expected regime of validity, is to analyze the IIB D3-brane gauge theories (“bundles on cycles”) in the large volume limit, and then just extrapolate to the opposite limit of singular X6 . For a selection of additional relevant references, see [11–17]. We will here study some general aspects of the world-volume SCFTs which can be constructed via IIB D3-branes at singularities and aspects of their AdS duals, focusing particularly on the geometric realization of the flavor symmetries of SCFTs. We will here only consider cases where the world-volume gauge theory has already been determined by the above mentioned methods, but we hope that some of the methods we discuss could also be helpful in determining the world-volume gauge theories for more general singularities. The world-volume gauge theories thus obtained are of the general quiver form U (N dα ), (1.1) α
where α run over the nodes of the quiver. The quiver and coefficients dα depend on the particular singularity. The theories are generally chiral, with nαβ > 0 chiral superfield bifundamentals Qiαβ , i = 1 . . . nαβ , in the (N dα , N dβ ) of U (N dα ) × U (N dβ ). We take nβα = −nαβ , and draw the arrows on the quiver so that nαβ > 0 means that the arrow(s) point from node α to node β. Absence of gauge anomalies requires each node to have the same number of incoming and outgoing arrows, so β nαβ dβ = 0 for every α. The SCFT is specified by the quiver diagram and the superpotential, which is also determined from the geometry and is a sum of terms of the form W = ai1 ...ik TrQiα11 α2 . . . Qiαl l αl+1 . . . Qiαkk α1
(1.2)
with the bifundamental gauge indices contracted in a closed loop to form a gauge invariant meson of the quiver theory. There are also gauge invariant baryons, but these generally do not enter into the superpotential. We will be interested in studying the bifundamentals Qαβ and their charges under the flavor symmetries. In the AdS dual, we only see the gauge invariant operators; in particular, we see the baryons formed from the bifundamentals rather than the bifundamentals themselves. To simplify our discussion, we will here only consider cases where all dα = 1 in (1.1), so the baryons are simply Bαβ = detN×N (Qαβ ). Thus, the charges of the Bαβ are just a factor of N times those of the Qαβ . In AdS/CFT the baryonic operators map to particles in the AdS5 bulk, which arise as D3 branes wrapping 3-cycles of H5 . The 4d N = 1 SCFT has a global symmetry group SU (2, 2|1)⊗F, where SU (2, 2|1) contains the superconformal U (1)R symmetry whose existence is necessary for a SCFT and where F are non-R flavor symmetries. We will be especially interested in a U (1)n subgroup of F; these are the flavor symmetries under which all nαβ bifundamentals Qαβ carry the same charge, so the baryons are charged under these. In the AdS dual, the continuous global symmetries are all gauge symmetries in the AdS5 bulk. In particular, U (1)R arises as a Kaluza-Klein gauge field coming from the metric; it is associated with a geometric isometry of the horizon manifold H5 . The U (1)n ⊂ F gauge fields in AdS5
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arise via reduction of the IIB RR gauge field C4 on n independent 3-cycles of H5 . Since the baryons are wrapped D3s, they are charged under these gauge fields. Supposing that H5 is a regular Einstein-Sasaki manifold (this assumption might not actually be necessary for our discussion to apply), it can be written as a U (1) fibration over a four dimensional surface V4 [18]. The U (1) fiber is the isometry associated with the U (1)R symmetry, and the baryons Bαβ must wrap this fiber since they are charged under the superconformal U (1)R [12, 14]. In addition, the baryons wrap certain holomorphic 2-cycles Lαβ ⊂ V4 . The holomorphic condition on the 2-cycles is necessary for the 3-cycle obtained via including the U (1)R fiber to be supersymmetric. Therefore the baryons, and thus also the bifundamentals Qαβ in our quiver gauge theory, are associated with divisors Lαβ on V4 . All nαβ bifundamentals connecting nodes α and β are associated with the same divisor Lαβ , and we take Lαβ = Lβα . As far as we know, a general method for determining the correct Lαβ has not appeared in the literature, though they were discussed in detail for a particular example, V4 = dP3 , in [12]. As we discuss, the U (1)R and flavor charges of the Qαβ are determined via topological intersections with the corresponding Lαβ . For example, the U (1)R charge of the baryons is related to their dimension via R[Bαβ ] = 23 [Bαβ ], which is proportional to the volume of the 3-cycles which the baryon wraps [12, 14]. This yields (when all dα = 1 in (1.1)) 2c1 · Lαβ R[Qαβ ] = , (1.3) c1 · c 1 measured by the intersection of the divisor with the first Chern class of V4 . The U (1)n non-R flavor charges in F are given by all possible independent divisors Ji of V4 which are orthogonal to the first Chern class of V4 : Ji · c1 = 0,
i = 1 . . . n.
(1.4)
This condition, via (1.3), is required for the flavor current to be U (1)R neutral. We can pick an arbitrary basis of such Ji , satisfying Ji ·c1 = 0. The charges of the bifundamentals under these flavor symmetries is Fi [Qαβ ] = Ji · Lαβ .
(1.5)
While the overall normalization of the R-symmetry is fixed, that of the other flavor symmetries is irrelevant. It is interesting that string theory “knows” which is the correct superconformal U (1)R symmetry, i.e. precisely which U (1)R is the one which is in the same supermultiplet as the stress tensor. In the geometry, this preferred U (1)R is precisely that which is measured by c1 , rather than some linear combination of c1 and the Ji . Finding the correct superconformal U (1)R directly via field theory methods was, until recently, an open problem. Inspired by our geometric results discussed here, we have very recently found [19] a field theory method to determine the superconformal U (1)R . We will verify in examples that our field theory condition [19] agrees with the result (1.3). We find some interesting properties which the divisors Lαβ , which describe the bifundamentals in the quiver, must satisfy. We now summarize these results for the simplifying case where all dα = 1 in (1.1). First, our U (1)R and flavor symmetries (1.3) and (1.5) must not have any ABJ anomalies. This is equivalent to the requirement that, for every node α, we must have |nαβ |Lαβ = (Nf (α) − 1)c1 . (1.6) β
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Nf (α) is the total number of flavors at node α: Nf (α) = 21 β |nαβ |. In addition, the superpotential must respect these charges. This implies that every term in the superpotential must have net divisor equal to c1 , since then (1.3) and (1.5) properly assign the superpotential R-charge 2 and flavor charge 0. Hence, for every non-zero superpotential m term, αβ Qαβαβ , we must have
mαβ Lαβ = c1 .
(1.7)
αβ
Furthermore, we find that the Lαβ can be written as differences of divisors Lα , which are associated with the nodes of the quiver: n (Lβ − Lα ) ≥ 0 0 if |nαβ nαβ αβ | (Lβ − Lα ) + c1 θαβ where θαβ ≡ Lαβ = n (Lβ − Lα ) < 0. 1 if |nαβ |nαβ | αβ | (1.8) By taking β to be the endpoint of the |nαβ | arrows, and α the start, the factors nαβ /|nαβ | become +1. The sign of L in (1.8) refers to the sign of c1 · L, and we’ll always choose the Lα > 0 in this sense. We require that all Lαβ ≥ 0 because the expression (1.3) must assign non-negative R-charge to all chiral superfields. For most Lαβ , the θαβ term in (1.8) vanishes. In fact, every term in the superpotential (1.2) has precisely one Qαβ for which the associated θαβ = 1, with the others having θαβ = 0, and this ensures that (1.7) is satisfied: every superpotential term has net divisor c1 . The anomaly free condition (1.6) implies that, for every node α of the quiver, nαβ Lβ = |nαβ |Lβ − |nαβ |Lβ = 0 mod c1 ; (1.9) β
outgoing β
incoming β
we could write the specific coefficient of c1 on the RHS in terms of Nf (α) and the θαβ , but (1.9) suffices for a later application. Outgoing β means those nodes where the arrow goes out from α, toward β. We used the fact that, mod c1 , 0 = β |nαβ |Lαβ = β nαβ (Lβ − Lα ), and β nαβ = 0. Using (1.8), the superconformal U (1)R charges (1.3) and other flavor charges (1.5) can be expressed as differences of charges associated with the nodes of the quiver: R[Qαβ ] = R(β) − R(α) + 2θαβ , Fi [Qαβ ] = Fi (β) − Fi (α),
2c1 · Lα , c1 · c 1 Fi (α) ≡ Ji · Lα .
R(α) ≡
(1.10)
As we discuss in Sect. 2, the Lα in (1.8) are expected to have some natural mathematical meaning, in some way related to a dual version of the collection of bundles on divisors O(Dα ). However, we were not able to make this precise here, and did not find a fully general method to independently obtain the Lα from first principles. As we also discuss, Seiberg duality [20] has a simple action on the Lα . To simplify the discussion, we consider the case where the dualized gauge group at node α has Nf = 2Nc , so that the rank of the dualized gauge group is the same as it was originally; then all dα = 1, for both the original and also the dualized quivers. We write the bifundamentals associated with node α as Qαβ , Qαγ , Qρα , and Qσ α with the arrows going out from node α out to β and γ (which could be the same node) and into node α from
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ρ and σ (which could also be the same). The dualized quiver has dual quark bifundamentals, with reversed arrows, and also bifundamentals corresponding to the mesons of the original U (N)α theory. We show that the duality correspondences and R-charge and flavor charge assignments imply that the Seiberg dualizing node α only changes the Lα of that node, as Lα = Lβ + Lγ − Lα ,
(1.11)
with the L’s associated with the other nodes remaining unchanged after Seiberg duality. One can also construct non-conformal theories, e.g. by wrapping D5 branes on cycles i ⊂ X6 , with the other directions filling the uncompactified 4d space transverse to X6 . As discussed in [13], there is a flux condition which requires that the two-cycles i of X6 not intersect any compact 4-cycles (this condition also rules out wrapping D7s on 4-cycles). The cycles i which the D5’s wrap correspond to divisors in V4 , and the flux condition implies that they must have zero intersection with c1 (V4 ). Thus the total 5-brane charge must be that of Ni D5s wrapped on divisors Ji of V4 , i.e. i Ni Ji , where every Ji satisfies c1 · Ji = 0. These are the same Ji in (1.4), corresponding to the non-R flavor symmetries of the SCFT theory without wrapped D5s. Indeed, the flavor symmetries of the SCFT without wrapped D5s become part of the gauge symmetry in the theory with wrapped D5’s: U (N + Mα ) with Mα = N i Ji · L α . (1.12) α
i
Because the flavor charges (1.5) of the bifundamentals have become part of their gauge charge in the theory (1.12) with added wrapped D5s, consistency of the theory (1.12) requires that the flavor symmetries Fi have vanishing ’t Hooft anomalies, TrFi = 0
and
TrFi Fj Fk = 0
for all
i, j, k.
(1.13)
This can be seen to be the case from the origin of these symmetries in the AdS5 × H5 dual, as the reduction of C4 on 3-cycles of H5 : the C4 gauge field does have the particular Chern-Simons type terms which would be needed to yield non-zero ’t Hooft anomalies upon reduction on H5 . The outline of this paper is as follows: In Sect. 2, we discuss how our main results, reviewed above, are obtained. In Sect. 3, we discuss aspects of ’t Hooft anomalies and our field theory result [19] for determining the superconformal U (1)R . In Sect. 4, we illustrate our ideas for the examples of certain toric and non-toric del Pezzo surfaces. We expect that the methods apply more generally. While this paper was in preparation, Chris Herzog and James McKernan alerted us to their related work [21]. Several of the loose ends raised in this paper were subsequently analyzed and clarified in a nice paper by Herzog and Walcher [22]. Among other things, they presented a precise notion of the “dual” to the exceptional collection, which is related to the Lα that we introduced in (1.8). 2. Some String Predictions One way to find the quiver gauge theory associated with a singularity is in terms of a collection of sheaves. These often can be written as O(Dα ), where Dα is some set
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of divisors of V4 . Given such an collection, the number of bifundamentals can then be computed by the formula (2.1) nαβ = χ (O(Dα ), O(Dβ )) − χ (O(Dβ ), O(Dα )), 3 where χ (O(Dα ), O(Dβ )) = i=0 (−1)i dimExt i (O(Dα ), O(Dβ )) is the relative Euler characteristic of the two sheaves. (For an exceptional collection one of the two terms in (2.1) vanishes.) A small modification of (2.1), though, is generally needed: at certain nodes, the directions of the bifundamentals need to be flipped. This is seen in the gauge theory because otherwise some gauge groups would be anomalous. The flip is a continuation of N → −N for the corresponding node. Precisely which nodes require such a flip can be determined by the methods of [7, 8]. We consider the situation where all gauge groups at nodes α are U (N ), to simplify the analysis of the baryons Bαβ = detN×N (Qαβ ). If there is a multiplicity nαβ > 1 of bifundamentals, there will be a corresponding multiplicity of baryons; we do not consider such baryon multiplicities further. In AdS/CFT, the baryons arise as D3 branes wrapped on 3-cycles of H5 , with the dimension of the corresponding operator directly proportional to the volume of the corresponding 3-cycle, see [12, 14]. This yields R(Bαβ ) = 2 2 3 3 4 3 (Bαβ ) = 3 µ3 L Vol( αβ ), where µ3 is the tension of the brane. The αβ corre1 sponds to a holomorphic divisor Lαβ of V , combined with the S fiber. As discussed in [12, 14], we have 3 2πq π q Vol(H5 ) = (2.2) Vol(V ) = c1 · c1 , 3 27 where 2π q/3 is the length of the U (1) fiber and c1 is the first Chern class of V , with c1 · c1 ≡ V c1 ∧ c1 . Here q is defined by c1 (V ) = qc1 (U (1)), with c1 (V ) the first Chern class of the 2 complex dimensional K¨ahler-Einstein manifold V , which satisfies ω = π3 c1 (V ) with ω the K¨ahler form of V , and c1 (U (1)) is the first Chern class of the 3 ) yields U (1) line bundle. Then R(Bαβ ) = 23 µ3 L4 Vol( αβ c1 · Lαβ 2 Nπ 2πq π R(Bαβ ) = . (2.3) c1 · Lαβ = 2N 3 2Vol(H5 ) 3 3 c1 · c 1 Since R[Bαβ ] = NR[Qαβ ], this yields (1.3). The non-R flavor symmetries F under which the baryons are charged come from reducing the IIB gauge field C4 on 3-cycles of H5 . These 3-cycles must include the U (1) fiber direction, along with some divisors Ji of V4 . Since these flavor symmetries must be R-neutral, (1.3) implies that the Ji must satisfy ci · Ji = 0. A similar consideration as in (2.3) then leads to the flavor charge assignments of the baryons and hence the bifundamentals, as in (1.5). Again, the overall normalization of these non-R U (1) flavor charges is irrelevant, so we drop the normalization factor of 2/(c1 · c1 ) for these. The condition that the U (1)R symmetry be anomaly free is that, at every node α, |nαβ |(R[Qαβ ] − 1)dβ = 0. (2.4) 2dα + β
In our situation, where all dα = 1, this together with (1.3) requires |nαβ |Lαβ = (Nf (α) − 1)c1 · c1 , c1 · β
(2.5)
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where Nf (α) = 21 β |nαβ | is the number of flavors at node α. Likewise, the condition that the U (1)i flavor symmetries associated with Ji have vanishing ABJ anomaly at every node α is |nαβ |Fi [Qαβ ] = 0. (2.6) β
This, together with (1.5), implies that |nαβ |Lαβ = 0 Ji ·
for all
Ji · c1 = 0.
(2.7)
β
Taken together, (2.5) and (2.7) imply (1.6), which is a very restrictive condition on the Lαβ . In addition, the Lαβ must satisfy another condition in order that the superpotential respect the U (1)R and flavor symmetries. Since every term in the superpotential must have R-charge 2 and non-R flavor charge zero, the total divisor associated with any superpotential term must be precisely c1 . Thus a necessary (but generally not sufficient) m condition for a non-zero superpotential term αβ Qαβαβ is mαβ Lαβ = c1 . (2.8) αβ
We have found that, furthermore, the Lαβ can be written as a difference of divisors Lα which are associated with the nodes of the quiver, as in (1.8). The condition (1.8) is sufficiently restrictive so that, given the Lαβ , the Lα can be determined – up to the addition of an overall constant divisor to all Lα , which would cancel on the RHS of (1.8). We expect that the Lα must have a natural mathematical interpretation, which could be used to independently determine them. To get some insight into what this direct interpretation of the Lα might be, consider the process of partially resolving the geometric singularity. This corresponds to turning on a FI term at some node, which forces some bifundamentals to then have a non-zero expectation value, Higgsing the world-volume gauge theory down to that of the resolved singularity. Thus the bifundamental divisors Lαβ are naturally associated with differences of FI terms at the nodes α and β: roughly, Lαβ ∼ ζβ −ζα . We interpret this as in (1.8). Note that the FI terms are dual to the bundles at the nodes, since we have a corresponding coupling d 4 xd 4 θζα Vα , so this suggests that the Lα are dual 1 to the O(Dα ). We do not presently, however, have a prescription for how to make this precise. So, for the present work, we found the Lα on a case-by-case basis by first obtaining the Lαβ , and then using (1.8). We have found that, at least in all those cases which we have considered, it is possible to take Lα = Dα for most, but not all, of the nodes. In particular, in the examples that we considered, we can take Lα = Dα for all of those nodes for which no N → −N flip is required to get the bifundamentals via (2.1). For those nodes which do require such a flip, the Lα = Dα and, since we do not yet know the general procedure for determining these Lα , we determined them on a case-by-case basis in the examples by imposing the very restrictive consistency conditions, discussed above, which the Lαβ must satisfy. As mentioned in the introduction, we can also consider non-conformal theories, obtained by wrapping Ni D5 branes on 2-cycles of X6 . Doing so leads to a quiver of the 1
We thank M. Douglas for suggesting this to us.
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same form as in the conformal case, but with the gauge group modified, as in (1.12); as indicated, the Lα determine the gauge group modification: U (N + Mα ) with Mα = Ni Ji · L α . (2.9) α
i
Absence of gauge anomalies at every node α requires nαβ (N + Mβ ) = 0,
(2.10)
β
which is indeed satisfied thanks to (1.9), since Ji · c1 = 0. These theories with the wrapped D5s effectively gauge our previously global U (1)n flavor symmetries. Consider, as an example, the case where N1 = 1 and all other Ni = 0. The gauge group is then α U (N +J1 ·Lα ), which has as a subgroup U (1)× α U (N ), with bifundamentals Qαβ having charge J1 · (Lβ − Lα ) under the U (1). The additional gauged U (1) here is just the flavor U (1) associated with J1 . Likewise, the general gauge theory (2.9) has as a subgroup U (1)n × α U (N )α , where the U (1)n correspond to what were flavor symmetries before we added the wrapped D5s. There is another condition on the Mα appearing above, which is required by the flux condition discussed e.g. in [13]: (N + Mα )α Dα = 0, (2.11) c1 · α
where α = +1 for all nodes except for those which require a N → −N sign flip in (2.1); for these flipped nodes, α = −1. Considering the case of no wrapped D5s, the Dα must satisfy i Ji . α D α = 0 thus α D α = (2.12) N c1 · α
α
i
If we took for the α Dα the “fractional brane” charges of the sort discussed e.g. in [9], i = 0 in (2.12), but generally we will not make that choice, instead we would have N takingthe Dα to satisfy the weaker condition (2.12). Including wrapped D5s, we must have α Mα α c1 · Dα = 0, implying that α α (Ji · Lα )Dα is in the span of the Ji . Finally, consider the action of Seiberg duality [20] on a node α, which we suppose has Nf = 2Nc in order for the gauge group to be self-dual. We also suppose that all dα = 1 in (1.1). We write the bifundamentals associated with node α as Qαβ , Qαγ , Qρα , and Qσ α with the arrows going out from node α out to β and γ (which could be the same node) and into node α from ρ and σ (which could also be the same). The dualized quiver has dual quark bifundamentals, with reversed arrows, and also bifundamentals corresponding to the mesons of the original U (N )α theory. We write the dual quarks as Qβα , Qγ α , Qαρ , and Qασ . The bifundamentals coming from the mesons are Qρβ = Qρα Qαβ , Qσβ = Qσ α Qαβ , Qργ = Qρα Qαγ , and Qσ γ = Qσ α Qαγ . Since the R-charge and flavor charges, given by (1.3) and (1.5), must respect this map, the divisors associated with the meson legs of the dual quiver must satisfy Lρβ = Lρα + Lαβ , Lργ = Lρα + Lαγ ,
Lσβ = Lσ α + Lαβ , Lσ γ = Lσ α + Lαγ .
(2.13)
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Duality maps the baryons of the original theory to those of the dual, as Qf1 . . . QfNc =
f1 ...fNf q fNc +1 . . . q Nf [20]. For our theory, this implies a mapping det Qαβ = det Qγ α , and hence the map Qαβ ↔ Qγ α . We reversed the direction of the arrows, because the dual quarks transform in the conjugate flavor representation (and we then need to apply charge conjugation on node α to get back bifundamentals). We exchanged the β and γ because of the f1 ...fNf in the baryon map, which maps e.g. det Qαβ to det qγ α . Likewise, the other bifundamentals map as Qαγ ↔ Qβα , Qρα ↔ Qασ , and Qσ α ↔ Qαρ . Since the R-charge and flavor charge assignments, given by (1.3) and (1.5), must respect this map, the divisors associated with the dual quark legs of the dual quiver must satisfy f
Lγ α = Lαβ ,
Lβα = Lαγ ,
Lαρ = Lσ α ,
Lασ = Lρα .
(2.14)
The dual theory has superpotential terms such as W = µ1 Qρβ Qβα Qαρ + . . . , which must respect the U (1)R and flavor symmetries, and hence must have total divisor c1 : Lρβ + Lβα + Lαρ = c1 .
(2.15)
Finally, all of the other nodes and legs of the original quiver are otherwise untouched by the Seiberg duality on node α, so their charges, and hence leg divisor assignments, are the same in the dual as in the original theory. All of these conditions can be satisfied very simply in terms of our relation (1.8) for writing the divisors of the quiver’s legs in terms of divisors associated with the nodes. Seiberg duality only acts on the Lα of the dualized node α, with the L’s of all other nodes unchanged. The conditions (2.13) are then almost immediately satisfied, though there is are apparently non-trivial conditions coming from the terms proportional to c1 : θρβ = θρα + θαβ etc.; we verified that these conditions are indeed satisfied in all of our examples. The conditions in (2.14) are also satisfied by Lα as in (1.11), with the other node L s untouched. For example, (1.8) gives for the first relation in (2.14): Lα − Lγ + c1 θ (Lα − Lγ ) = Lβ − Lα + c1 θ (Lβ − Lα ), which is indeed satisfied when Lα = Lβ + Lγ − Lα . Note also that using (2.13), (2.14), and Nf (α) = 2, (2.15) is equivalent to (1.6). 3. ’t Hooft Anomalies It is useful to consider the ’t Hooft anomalies of the global flavor symmetries U (1)R and U (1)Fi of the SCFTs. The conditions (2.4) and (2.6), that U (1)R and U (1)Fi have vanishing ABJ anomalies at each node, ends up implying that they also have vanishing linear ’t Hooft anomalies (relevant for coupling to gravity): TrR = TrFi = 0.
(3.1)
This is a consequence of the quiver gauge group form (1.1), with only purely chiral, bifundamental matter. The various cubic ’t Hooft anomalies (again, taking all dα = 1 in (1.1) to simplify) are 1 + 1 TrR 3 = N 2 |nαβ |(R[Qαβ ] − 1)3 , 2 α
β
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TrRFi Fj = 21 N 2
|nαβ |(R[Qαβ ] − 1)Fi [Qαβ ]Fj [Qαβ ],
αβ
TrR 2 Fi = 21 N 2 TrFi Fj Fk = 21 N 2
αβ
|nαβ |(R[Qαβ ] − 1)2 Fi [Qαβ ], |nαβ |Fi [Qαβ ]Fj [Qαβ ]Fk [Qαβ ].
(3.2)
αβ
We can evaluate these in terms of the geometry via (1.3) and (1.5). Interestingly, for each of the ’t Hooft anomalies (3.2), we can also make an independent prediction. For example, using the AdS/CFT prediction for the central charges a and c [23] and their relation with the TrR 3 ’t Hooft anomaly [24, 25] leads to the prediction TrR 3 =
8 2 Vol(S 5 ) N , 9 Vol(H5 )
(3.3)
which we can write in terms of q and c1 ·c1 using (2.2). (And also TrR = 0, which we’ve already seen to indeed be the case.) So both (3.2) and (3.3) compute TrR 3 via geometric data; hence, some mathematical identity must ensure that the two, apparently different, geometric computations always agree. We do not yet have a general understanding of this expected identity, but check that the computations indeed agree in all of our examples. As discussed in [19], the superconformal U (1)R charge has the property that, among all possibilities, it maximizes 3TrR 3 − TrR. If we write the most general U (1)R sym metry as R = R0 + i si Fi , where R0 is an arbitrary initial R-symmetry and si are real parameters, maximizing 3TrR 3 − TrR with respect to the si yields: 9TrR 2 Fi = TrFi and TrRFi Fj < 0 [19]. In the present context, where all U (1)Fi have TrFi = 0, we thus must have TrR 2 Fi = 0
and
TrRFi Fj < 0,
(3.4)
specifically the latter matrix in i and j must have all negative eigenvalues. We check in all cases that (3.2), using (1.3) and (1.5), indeed satisfies (3.4). Again, we suspect that some mathematical identity ensures that this is indeed always the case, e.g. the first identity in (3.4) requires αβ
|nαβ |(2
c1 · Lαβ − 1)2 (Ji · Lαβ ) = 0, c1 · c 1
(3.5)
for all Ji satisfying c1 · Ji = 0. Finally, the flavor symmetries Fi are expected to have all vanishing cubic ’t Hooft anomalies Tr Fi Fj Fk = 0
for all
i, j, k.
(3.6)
Any non-zero such cubic ’t Hooft anomalies would require the presence of Chern-Simons 5-forms ∼ A ∧ F ∧ F in the AdS5 bulk [26, 27], but such a term does not have a candidate 10d origin, in terms of the 10d gauge fields C4 ∼ A ∧ η and F5 = F ∧ η, with η a 3-form on H5 . Further, as we mentioned above, the Fi flavor symmetries become part
Baryon Charges in 4d Superconformal Field Theories
417
of the gauge symmetry upon including wrapped D5s. Hence absence of gauge anomalies of those theories requires (3.6). Again we can check in all examples that, indeed,
|nαβ |(Ji · Lαβ )(Ji · Lαβ )(Jk · Lαβ ) = 0,
(3.7)
αβ
for all i, j, k, with Ji,j,k · c1 = 0. 4. del Pezzo Examples Consider the case where X6 is a local Calabi-Yau which is a complex cone over the del Pezzo surface dPn . Recall that dPn is a copy of P2 blown up at n points, where 0 ≤ n ≤ 8. Each blown-up point corresponds to an exceptional divisor Ei , and there is also a divisor D which is the pullback of a hyperplane on P2 . The intersection numbers of these divisors are D · D = 1,
Ei · Ej = −δij ,
D · Ei = 0,
(4.1)
and the first Chern class (anti-canonical class) is c1 = 3D −
n
Ei .
(4.2)
i=1
There are n linearly independent divisors Ji satisfying Ji · c1 = 0, so there will be a nonR U (1)n flavor symmetry under which the baryons are charged. These Ji correspond to the root lattice of the exceptional group En , with E1 = A1 , E2 = A1 + A1 , E3 = A3 , E4 = A4 , E5 = D5 , and E6,7,8 as expected. In particular, if we take for our basis Ji = Ei − Ei+1
for i = 1 . . . n − 1, and Jn = D − E1 − E2 − E3 ,
(4.3)
their intersections Ji · Jj are given by the En Cartan matrix. The dPn automorphisms correspond to the En Weyl reflections on the Ji . Rewriting (2.3) in this language, the R-charge of a baryon Bαβ corresponding to a holomorphic 2-cycle Lαβ is R(Bαβ ) = 2N
c1 · Lαβ c1 · Lαβ = 2N . c1 · c 1 9−n
(4.4)
Since the numerator is an integer, this implies that the R-charge of any baryon in the 2N dPn theory is an integer multiple of 9−n . Also, using (3.3) and (2.2), we get that the cubic ’t Hooft anomaly must be TrR 3 =
24N 2 . 9−n
(4.5)
In the following sections, we will work out our prescription in detail for the case of the toric del Pezzos dPn≤3 and the non-toric del Pezzo dP4 . The dP3 case was studied extensively in [12], so we start with that case first.
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K. Intriligator, B. Wecht 2
6
4
1
5
3
Fig. 1. The Model III dP3 quiver
4.1. Cone over dP3 . There are four known field theories that arise from the cone over dP3 which are related to each other via Seiberg duality. One of these (usually called Model III) is described by the U (N )6 theory given by the quiver in Fig. 1. The correspondence between divisors and bifundamentals has already been worked out in [12] and is given in the following table: Qαβ Lαβ E1 X51 X24 E2 X53 E3 X43 D − E 1 − E2 X25 D − E 1 − E3 . X41 D − E 2 − E3 X16 D − E1 X62 D − E2 X36 D − E3 X64 D X65 2D − E1 − E2 − E3
(4.6)
Note that, because the assignment of divisors and charges is the same for any of the |nαβ | bifundamentals connecting the same nodes, we have not explicitly written the fields Y16 , Y36 , and Y62 in this table. It is of course important to include these multiplicities in all computations, e.g. when computing traces. It is easily verified that the Lαβ (4.6) indeed satisfy our vanishing ABJ anomaly condition (1.6). Furthermore, the superpotential (found in [12, 15]) respects the symmetries, because every term in the superpotential has exactly one field for which θαβ = 1. We now write these Lαβ as in (1.8). As seen in the table below, we can take the Lα to equal the Dα which define the collection of bundles, except at nodes 2,4,5. These are precisely the nodes where a flip is required [13] to obtain the quiver diagram; this seems to be a general connection. We also include in the table below the Mα = i Ni Ji · Lα , which give the modification of the gauge groups in the quiver diagram with added wrapped D5’s, as in (1.12). The Ji are as in (4.3) : J1 = E1 − E2 , J2 = E2 − E3 , J3 = D − E1 − E2 − E3 .
Baryon Charges in 4d Superconformal Field Theories
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2
6
4
1
5
3
Fig. 2. The Model IV dP3 quiver
Node Lα Dα Mα 1 E1 E1 N3 − N 1 2 2D − E2 E2 N3 − N1 + N2 3 E3 E3 N2 + N 3 4 2D D − E2 2N3 5 0 D − E3 0 6 D D N3 .
(4.7)
One can readily check that these Lα and (1.8) reproduce the required Lαβ in (4.6). Since dP3 has c1 = 3D − E1 − E2 − E3 , we write the three flavor currents as J1 = E2 − E1 , J2 = E2 − E3 , and J3 = D − E1 − E2 − E3 . Then, we can use (1.3) and (1.5) to read off the charges: Qαβ X51 X24 X53 X43 X25 X41 X16 X62 X36 X64 X65
J1 −1 1 0 0 1 −1 1 −1 0 0 0
J 2 J3 R 0 1 1/3 −1 1 1/3 1 1 1/3 1 −1 1/3 −1 −1 1/3 0 −1 1/3 0 0 2/3 1 0 2/3 −1 0 2/3 0 1 1 0 −1 1
(4.8)
These are exactly the −U (1)C , U (1)D , −U (1)E , and R charges found in [12]. Let’s now examine a Seiberg dual theory, known as Model IV. The quiver for this theory is obtained by Seiberg dualizing node 2; see Fig. 2. The bifundamental/divisor correspondence has been worked out already in [12], so we only need to check that our prescription for Seiberg dualizing the Lα agrees. The only difference is in L2 , which becomes L2 = L4 + L6 − L2 = E2 , which checks with the results of [12].
4.2. Cone over dP2 . Since blowing down a divisor is equivalent to Higgsing an appropriate bifundamental, one can easily obtain the dP2 quiver by Higgsing any bifundamental
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4
2
5
3
Fig. 3. The dP2 quiver resulting from Higgsing X24
field in the dP3 quiver that corresponds to an exceptional divisor. Depending on which divisor gets Higgsed, the resulting quiver will be one of two possible Seiberg dual theories. We choose to blow down E2 or, equivalently, Higgs X24 . The resulting quiver appears in Fig. 3 (for simplicity, we have relabeled nodes 6 → 4 and 4/2 → 2). It is easy to figure out the appropriate assignment of divisors here: We simply take the divisors from our dP3 model and remove any E2 ’s. In the case where bifundamental fields combine and point in the same direction, the divisors never differ by more than an E2 , and thus there is no ambiguity. In the case where bifundamentals combine that point in opposite directions, the resulting divisor is the one corresponding to the bifundamental that did not change direction, i.e. the one that had more flavors. The Lα can be easily derived from our dP3 example by simply blowing down the divisor E2 . Since the Lα for the nodes from the dP3 quiver that get combined are the same up to an E2 , there is no ambiguity in how to assign the Lα for the dP2 theory. We note that this also is true for any of the Ei we could have chosen to blow down. This yields (relabeling E3 → E2 ) Node 1 2 3 4 5
Lα Mα E1 2N2 − N1 2D 2N2 E2 N1 + N 2 D N2 0 0.
(4.9)
The Mα in (4.9) give the gauge groups in the theory with added wrapped D5’s. We take J1 = E1 −E2 and J2 = D −2E1 −E2 , which satisfy Ji ·c1 = 0 with c1 = 3D −E1 −E2 for dP2 . For the theory without wrapped branes, we get the following assignment of divisors and flavor charges: Qαβ Lαβ X51 E1 E2 X53 X25 D − E1 − E2 D − E1 X23 X14 D − E1 D − E2 X21 X34 D − E2 X42 D X45 2D − E1 − E2
J1 J2 −1 2 1 1 0 −2 1 −1 1 −1 −1 0 −1 0 0 1 0 −1
R 2/7 2/7 2/7 4/7 4/7 4/7 4/7 6/7 8/7
(4.10)
Baryon Charges in 4d Superconformal Field Theories
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1
2
4
5
3
Fig. 4. The other phase of dP2
Notice that the non-R flavor charges here are given by linear combinations of the U (1)’s from dP3 under which X42 is neutral, J1dP2 = J1dP3 + J3dP3 and J2dP2 = J2dP3 − J1dP3 . This is also consistent with the divisors assigned to these flavor charges, as one sees by taking the appropriate linear combinations and removing any instances of the blowndown divisor. The superpotential for this theory is [15] WdP2 = X51 X14 X45 + X53 X34 X45 + X51 Y14 X42 X25 + X53 Y34 Y42 X25 +X21 X14 Y42 + X23 X34 X42 + X21 Y14 Z42 + X23 Y34 Z42 , (4.11) where we don’t bother recording the exact coefficients. This indeed obeys the condition that every term has precisely one field with θαβ = 1. The reader can easily verify that our ’t Hooft anomaly conditions are also satisfied: TrR 3 is indeed given by (4.5) for n = 2, as required by (3.3). The condition (3.4) of [19] is indeed satisfied, showing that the geometry knows how to pick out the correct superconformal U (1)R , via c1 . Finally, the flavor ’t Hooft anomalies (3.6) vanish, as generally happens for these string-constructed theories. We also check that our prescription for Seiberg duality works. Dualizing on node 3 yields the other phase of dP2 , given in Fig. 4: The only L that changes is L3 , which becomes L3 = L2 + L5 − L3 = 2D − E2 . It is easy to check that this is consistent with the divisors one gets by appropriately Higgsing dP3 .
4.3. Cone over dP1 . It is useful here to Higgs the dP2 theory to the dP1 theory, since this is an especially simple example. To obtain this theory, Higgs the field X51 from dP2 , which corresponds to blowing down the exceptional curve E1 . This yields the quiver in Fig. 5. The Lα , Dα , and Mα for wrapped branes are given by Node Lα Dα 1 2D − E E 2 D D 3 2D D − E 4 0 0
Mα −N1 N1 2N1 0.
(4.12)
422
K. Intriligator, B. Wecht 1 1 0 0 1 0 1
3 1 0 0 1 0 1
1 0 0 1
1 0 0 1
2
4
Fig. 5. The dP1 quiver
Note that here the flipped nodes are 1 and 3, where the Lα and Dα differ. Here, c1 = 3D − E, so we take J = D − 3E. This yields the following fields and charges. Qαβ Lαβ X13 E X21 D − E X34 D − E X14 D X23 D X42 D
J 3 −2 −2 1 1 1
R 1/4 1/2 1/2 3/4 3/4 3/4
(4.13)
The superpotential here is W = X42 X21 X14 + X42 X23 X34 + X42 X21 X13 X34
(4.14)
which obeys the required conditions. Seiberg dualizing on either node 1 or node 3 yields the same theory; one can check that the new Lα are identical to the original after relabeling nodes. 4.4. Cone over dP4 . As with the other del Pezzo surfaces, there are many different Seiberg dual quiver theories possible for dP4 . Here, we will use the one given in Fig. 6 [17, 9]. It is straightforward to check that by Higgsing X67 , one returns to the Model III dP3 quiver. There is a unique assignment of Lα which reproduces the divisors on the above dP3 Model III theory. These were found by enforcing that the field X67 corresponds to the exceptional curve we’re blowing down, L67 = E4 , and that the remaining divisors can only differ from their dP3 counterparts by this same exceptional curve. We also list the Mα = i Ni Ji · Lα , relevant for the theory with added wrapped D5s. Node Lα Mα 1 E1 N4 − N 1 2 2D − E2 − E4 N4 − N1 + N2 − N3 3 E3 N2 − N 3 + N 4 4 2D − E4 2N4 − N3 5 0 0 6 D − E4 N4 − N 3 7 D N4 .
(4.15)
Baryon Charges in 4d Superconformal Field Theories
423
7
11 00 00 11
6
1 0 0 1
112 00 00 11
1 0 0 1
11 00 1 00 11
1 0 0 1
4
3
1 0 05 1
Fig. 6. One possible quiver for dP4
On dP4 , the first Chern class is c1 = 3D − E1 − E2 − E3 − E4 . Thus, we can take as our Ji to be J1 = E1 − E2 , J2 = E2 − E3 , J3 = E3 − E4 , and J4 = D − E1 − E2 − E3 . We thus find the divisors and charges to be Qαβ X51 X24 X53 X67 X43 X25 X16 X41 X72 X36 X17 X62 X37 X74 X75
Lαβ E1 E2 E3 E4 D − E 1 − E2 D − E 1 − E3 D − E 1 − E4 D − E 2 − E3 D − E 2 − E4 D − E 3 − E4 D − E1 D − E2 D − E3 D − E4 2D − i Ei
J1 −1 1 0 0 0 1 1 −1 −1 0 1 −1 0 0 0
J2 J3 J4 R 0 0 1 2/5 −1 0 1 2/5 1 −1 1 2/5 0 1 0 2/5 1 0 −1 2/5 −1 1 −1 2/5 0 −1 0 2/5 0 1 −1 2/5 1 −1 0 2/5 −1 0 0 2/5 0 0 0 4/5 1 0 0 4/5 −1 1 0 4/5 0 −1 1 4/5 0 0 −1 4/5
(4.16)
The superpotential for this theory [10] indeed obeys the condition that each term has precisely one field with nonzero θαβ . (These charge and divisor assignments also apply for the P dP4 case considered in [17], which has a slightly different superpotential.) We can also check that our ’t Hooft anomaly conditions (4.5), (3.4) and (3.6) are also satisfied. It is also worth checking that one can Higgs this theory to dP3 and watch the divisor E4 collapse in the same manner we observed in the Higgsing of dP3 down to dP2 . This indeed works; we note that Higgsing X67 and relabeling the node 6/7 → 7 produces exactly the results found above. Finally, we can immediately construct the quivers and Lα for Seiberg dual theories. For example, dualizing on node 2 yields a quiver with L2 = L7 + L6 − L2 = L4 + L5 − L2 = E2 and all other Lα unchanged. Acknowledgements. We would like to thank M. Douglas, J. Kumar, J. Roberts, R.P. Thomas, C. Vafa, and especially Mark Gross for discussions. We would also like to thank Chris Herzog and James McKernan for alerting us to their related work. This work was supported by DOE-FG03-97ER40546.
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References 1. Douglas, M.R., Moore, G.W.: D-branes, Quivers, and ALE Instantons. arXiv:hep-th/9603167 2. Douglas, M.R., Greene, B.R., Morrison, D.R.: Nucl. Phys. B 506, 84 (1997) [arXiv:hep-th/9704151] 3. Aharony, O., Gubser, S.S., Maldacena, J.M., Ooguri, H., Oz, Y.: Large N field theories, string theory and gravity. Phys. Rept. 323, 183 (2000) [arXiv:hep-th/9905111] 4. Morrison, D.R., Plesser, M.R.: Non-spherical horizons. I. Adv. Theor. Math. Phys. 3, 1 (1999) [arXiv:hep-th/9810201] 5. Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243 (1996) [arXiv:hep-th/9606040] 6. Beasley, C., Greene, B.R., Lazaroiu, C.I., Plesser, M.R.: D3-branes on partial resolutions of abelian quotient singularities of Calabi-Yau threefolds. Nucl. Phys. B 566, 599 (2000) [arXiv:hepth/9907186] 7. Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222 8. Hori, K., Iqbal, A., Vafa, C.: D-branes and mirror symmetry. arXiv:hep-th/0005247 9. Hanany, A., Iqbal, A.: Quiver theories from D6-branes via mirror symmetry. JHEP 0204, 009 (2002) [arXiv:hep-th/0108137] 10. Wijnholt, M.: Large volume perspective on branes at singularities. arXiv:hep-th/0212021 11. Bergman, A., Herzog, C.P.: The volume of some non-spherical horizons and the AdS/CFT correspondence. JHEP 0201, 030 (2002) [arXiv:hep-th/0108020] 12. Beasley, C.E., Plesser, M.R.: Toric duality is Seiberg duality. JHEP 0112, 001 (2001) [arXiv:hepth/0109053] 13. Cachazo, F., Fiol, B., Intriligator, K.A., Katz, S., Vafa, C.: A geometric unification of dualities. Nucl. Phys. B 628, 3 (2002) [arXiv:hep-th/0110028] 14. Berenstein, D., Herzog, C.P., Klebanov, I.R.: Baryon spectra and AdS/CFT correspondence. JHEP 0206, 047 (2002) [arXiv:hep-th/0202150] 15. Feng, B., Franco, S., Hanany, A., He, Y.H.: Symmetries of toric duality. JHEP 0212, 076 (2002) [arXiv:hep-th/0205144] 16. Franco, S., Hanany, A.: Geometric dualities in 4d field theories and their 5d interpretation. JHEP 0304, 043 (2003) [arXiv:hep-th/0207006] 17. Feng, B., Franco, S., Hanany, A., He, Y.H.: Unhiggsing the del Pezzo. arXiv:hep-th/0209228 18. Boyer, C.P., Galicki, K.: On Sasakian-Einstein Geometry. Int. J. Math. 11, 873 (2000) [arXiv:math.dg/9811098] 19. Intriligator, K., Wecht, B.: The exact superconformal R-symmetry maximizes a. Nucl. Phys, B 667, 183–200 (2003) [arXiv:hep-th/0304128] 20. Seiberg, N.: Electric - magnetic duality in supersymmetric nonAbelian gauge theories. Nucl. Phys. B 435, 129 (1995) [arXiv:hep-th/9411149] 21. Herzog, C., McKernan, J.: Dibaryon Spectroscopy. JHEP 0308, 054 (2003) [hep-th/0305048] 22. Herzog,C.P. Walcher, J.: Dibaryons from Exceptional Collections. JHEP 0309, 060 (2003) [hep-th/0306298] 23. Gubser, S.S.: Einstein manifolds and conformal field theories. Phys. Rev. D 59, 025006 (1999) [arXiv:hep-th/9807164] 24. Anselmi, D., Freedman, D.Z., Grisaru, M.T., Johansen, A.A.: Nonperturbative formulas for central functions of supersymmetric gauge theories. Nucl. Phys. B 526, 543 (1998) [arXiv:hep-th/9708042] 25. Anselmi, D., Erlich, J., Freedman, D.Z., Johansen, A.A.: Positivity constraints on anomalies in supersymmetric gauge theories. Phys. Rev. D 57, 7570 (1998) [arXiv:hep-th/9711035] 26. Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hepth/9802150] 27. Freedman, D.Z., Mathur, S.D., Matusis, A., Rastelli, L.: Correlation functions in the CFT(d)/AdS(d + 1) correspondence. Nucl. Phys. B 546, 96 (1999) [arXiv:hep-th/9804058] 28. Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hepth/9802150] 29. Lawrence, A.E., Nekrasov, N., Vafa, C.: On conformal field theories in four dimensions. Nucl. Phys. B 533, 199 (1998) [arXiv:hep-th/9803015] 30. Feng, B., Hanany, A., He, Y.H., Iqbal, A.: Quiver theories, soliton spectra and Picard-Lefschetz transformations. JHEP 0302, 056 (2003) [arXiv:hep-th/0206152] 31. Witten, E.: Baryons and branes in anti de Sitter space. JHEP 9807, 006 (1998) [arXiv:hepth/9805112] Communicated by M.R. Douglas
Commun. Math. Phys. 245, 425–428 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1025-z
Communications in
Mathematical Physics
Erratum
Boundary Scattering, Symmetric Spaces and the Principal Chiral Model on the Half-Line N.J. MacKay, B.J. Short Department of Mathematics, University of York, York YO10 5DD, U.K. Received: 29 July 2003 / Accepted: 3 November 2003 Erratum published online: 29 January 2004 – © Springer-Verlag 2004 Commun. Math. Phys. 233, 313–354 (2003)
In the above article, two points require correction. First, the discussion regarding charge conjugation in Sect. 3.2.2 is unnecessary and may be replaced by an appeal only to crossing unitarity. Consequently (3.16) should now read E†E = I ρ(θ ) = ρ(−θ ∗ )∗
det E = 1, ρ(θ )ρ(−θ) = 1.
and and
(3.16)
Second, the crossing unitarity relation ij iπ il lk iπ − θ = Sj¯k¯ (2θ )K +θ K 2 2
(3.17)
was incorrectly stated, and should be replaced by iπ ¯ ¯ ¯ iπ − θ = Sji¯lk (2θ )K l k +θ . K ij 2 2
(3.17)
These corrections result in some changes to the allowed K-matrices. We give here only the list of the final boundary S-matrices, which replaces the existing Sect. 4.2: 4.2. The boundary S-matrices. First we give a scalar factor which has altered slightly from the original version and also an additional scalar factor: 1 4π 2 c2 −θ 2iπ +
µ(θ ) =
× θ 2iπ +
1 2
+
−θ
1 1 θ 1 θ 2iπ + 2 − 2iπc 2iπ + 2iπc 2iπ 1 θ 1 1 −θ 1 −θ 2iπc 2iπ + 2 − 2iπc 2iπ + 1 + 2iπc 2iπ
1 2
+
1 2iπc
−
1 2iπc
+1−
1 2iπc
,
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N.J. MacKay, B.J. Short
1 λ(θ) = 4π 2 c2 −θ −θ θ θ 1 M 1 1 1 2iπ + M N + 2iπc 2iπ + 1 − N − 2iπc 2iπ + 2iπc 2iπ − 2iπc θ −θ −θ , θ 1 M 1 1 1 2iπ +M N + 2iπc 2iπ + 1 − N − 2iπc 2iπ + 1 + 2iπc 2iπ + 1 − 2iπc in both of which c is a freely varying parameter unless otherwise stated below. The PCM S-matrices may then be written as follows. 4.2.1. SU (N ). We have found two types of boundary S-matrix for SU (N ), KP CM (θ ) = −(1 − h)θ µ(θ) ν(θ )(I + cθ EL ) ⊗ ν(θ )(I + cθ ER )
(4.7a)
(whose c → ∞ limit:
KP CM (θ ) = −(1 − h)θ η(θ )EL ⊗ η(θ )ER
(4.7)
is a valid PCM boundary scattering matrix) where EL/R ∈ and
SU (N ) , S(U (N/2) × U (N/2))
KP CM (θ ) = −(1 − h)θ λ(θ ) ν(θ )(I + cθ EL ) ⊗ ν(θ )(I + cθ ER ) ,
where EL/R ∈
(4.8)
SU (N ) . S(U (N − M) × U (M))
4.2.2. SO(N ). For SO(N ) three types have been found, h h +2 + 1 (1 − h)θ µ(θ) KP CM (θ ) = − 2 θ 2 θ × ν(θ )3,3 (θ )(I + cθ EL ) ⊗ ν(θ)3,3 (θ )(I + cθER )
(4.9a)
(whose c → ∞ limit: h h +2 + 1 (1 − h)θ η(θ )3,3 (θ )EL ⊗ η(θ )3,3 (θ )ER KP CM (θ ) = − 2 θ 2 θ (4.9) is a valid PCM boundary scattering matrix) where EL/R ∈
h KP CM (θ ) = − +2 2
θ
SO(N ) × {+1, −1} , U (N/2)
h +1 2
θ
(1 − h)θ η(θ )3,1 (θ )EL ⊗ η(θ )3,1 (θ )ER , (4.10)
Boundary Scattering, Symmetric Spaces and the Principal Chiral Model on the Half-Line
where
SO(N ) , S(O(N/2) × O(N/2))
EL/R ∈ and
KP CM (θ ) = −
h +2 2
427
θ
h +1 2
θ
(1 − h)θ µ(θ)
× ν(θ )3,1 (θ )(I + cθ EL ) ⊗ ν(θ)3,1 (θ )(I + cθER ) ,
(4.11)
where EL/R ∈
SO(N ) S(O(N − M) × O(M))
and c =
2ih in µ(θ). π(2M − N )
4.2.3. Sp(N ). Three types of KP CM (θ ) have also been found for Sp(N ), h h +2 + 1 (1 − h)θ µ(θ) KP CM (θ ) = − 2 θ 2 θ × ν(θ )3,1 (θ )(I + cθ EL ) ⊗ ν(θ )3,1 (θ )(I + cθ ER ) (4.12a) (whose c → ∞ limit: h h +2 + 1 (1 − h)θ η(θ )3,1 (θ )EL ⊗ η(θ )3,1 (θ )ER KP CM (θ ) = − 2 θ 2 θ (4.12) is a valid PCM boundary scattering matrix) where EL/R ∈ KP CM (θ ) = −
h +2 2
θ
h +1 2
where EL/R ∈ and
KP CM (θ ) = −
h +2 2
(1 − h)θ η(θ )3,3 (θ )EL ⊗ η(θ )3,3 (θ )ER , (4.13)
Sp(N ) , Sp(N/2) × Sp(N/2))
θ
θ
Sp(N ) , U (N/2)
h +1 2
θ
(1 − h)θ µ(θ)
× ν(θ )3,3 (θ )(I + cθ EL ) ⊗ ν(θ)3,3 (θ )(I + cθER ) , where EL/R ∈
Sp(N ) Sp(N − M) × Sp(M))
and c =
2ih in µ(θ). π(2M − N )
(4.14)
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4.2.4. SU (N )-conjugating. Lastly, we have found two types of representationconjugating boundary S-matrix for SU (N ) h h 1,1 (θ )EL ⊗ 1,1 (θ )ER , (4.15) +2 +1 KP CM (θ ) = 2 θ 2 θ where EL/R ∈ and
KP CM (θ ) =
h +2 2
θ
h +1 2
where EL/R ∈ {1, ω2 } ×
SU (N ) , SO(N )
θ
1,3 (θ )EL ⊗ 1,3 (θ )ER ,
SU (N ) Sp(N )
(ωN = −1) .
The discussion and conclusions are not significantly affected. Communicated by M. Aizenman
(4.16)
Commun. Math. Phys. 245, 429–448 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1034-y
Communications in
Mathematical Physics
Topologizations of Chiral Representations Florian Conrady1,2 , Christoph Schweigert3,4 1
Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany 2 Max-Planck-Institut f¨ ur Gravitationsphysik, Albert-Einstein-Institut, Am M¨uhlenberg 1, 14476 Golm, Germany. E-mail: [email protected] 3 LPTHE, Universit´e Paris VI, 4 place Jussieu, 75252 Paris Cedex 05, France 4 Institut f¨ ur Theoretische Physik, RWTH Aachen, Sommerfeldstrasse 28, 52074 Aachen, Germany. E-mail: [email protected] Received: 29 October 2002 / Accepted: 24 October 2003 Published online: 6 February 2004 – © Springer-Verlag 2004
Abstract: We analyze and compare two families of topologies that have been proposed for representation spaces of chiral algebras by Huang and Gaberdiel & Goddard respectively. We show, in particular, that for suitable pairs the topology of Gaberdiel & Goddard is coarser. We also give a new proof that the chiral two-point blocks are continuous in the topology of Huang. 1. Introduction and Summary Two-dimensional conformal field theories (CFTs) play a key role in the worldsheet formulation of string theory and in the description of universality classes of critical phenomena. In the attempt to gain a better understanding of their mathematical structure, several axiomatic approaches have been developed. When using an operator calculus, a space of states V has to be specified that consists of representations of the symmetry algebra. The representation spaces form the basic structure of V , but it is not fully determined by physical requirements what topology should be given to this space, and hence how it should be completed. In unitary CFTs, one has, by definition, a positive √ definite inner product , ; the conventional approach is then to use the norm . = , for completing V so that one obtains a Hilbert space H. Problems usually arise from the fact that domains and ranges for different operators do not coincide. Special care has to be taken when considering operator products. Often this aspect is left aside and one works on the assumption that domain issues can be settled. To define convergence in terms of an inner product is by no means the only possibility, nor is it clear that it provides the best starting point for dealing satisfactorily with domain questions. The theory of distributions shows that it can prove extremely useful to introduce topologies different from that of a (pre-) Hilbert space. It was especially B¨ohm who argued for the application of such topologies to quantum theory (see [Ma], Sect. 1 for references). More recently this idea has reappeared in the context of CFT when Gaberdiel & Goddard [GG] and Huang [Hu] proposed new topologies for representation
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spaces of chiral symmetry algebras. A central role is played by chiral (or conformal) blocks whose properties lead to the definition of locally convex state spaces that are not Hilbert spaces. In both cases, one deals with a whole family of topologies which are parametrized by suitable subsets of the complex plane. In this article we investigate and compare the two approaches. Questions of topology naturally occur when it comes to constructing a mathematically rigorous operator formalism. Gaberdiel, Goddard and Huang have achieved that for chiral CFT on the Riemann sphere. One hopes that these results can be further generalized, and that the topological properties of the spaces help in deriving statements that could not be proven so far. Let us mention a few possibilities: • String theory makes it necessary to deal with CFTs on surfaces of arbitrary genus g. For g > 1 and interacting theories, an operator formalism has yet to be developed. In a different approach one defines conformal blocks as linear functionals on tensor products of representation spaces [FB]. The nuclear mapping theorem for nuclear spaces could provide a way to construct vertex operators from these functionals. • The space of physical superstring states Vphys is obtained by taking the BRST cohomology of the combined matter-ghost system. The choice of topology can influence the content of Vphys , and should play a role in the construction of picture-changing operators [BZ]. • A cohomological approach to the Verlinde formula has been advocated in the literature [Te, FS]. The correct dimension of chiral blocks is obtained provided a certain sequence of coinvariants is exact. This may be easier to show if one chooses a suitable, possibly nuclear, topology on the state space. The results of this paper contribute to a better understanding of the topologies given in [GG] and [Hu]. As they were defined in rather different axiomatic settings, we have translated them into a common framework that allows for the construction of both types of topologies. This framework is specified in the next section. Section 3 and 4 explain O and T D and parametrized by the definition of the topologies: they are denoted by TGG Hu open sets O ⊂ C and open disks D ⊂ C centered at 0 respectively. In Sect. 5, we derive O and T D : it is shown that T D is nuclear and that T O is some simple properties of TGG Hu Hu GG nuclear if it is Hausdorff. We also take a look at the O- and D-dependence:
O is coarser than T O , if O ⊂ O, TGG GG D behaves in the opposite way: whereas THu
D is finer than T D for D ⊂ D. THu Hu
Section 6 deals with the comparison of the topologies. We prove that D is finer than T O if inf |ζ | > r, THu GG ζ ∈O
where r is the radius of the disk D. The techniques employed allow us to show in Sect. 7 D if the radius that conformal two-point blocks of the vacuum sector are continuous in THu of D is less than half the distance of the two points. This result generalizes to an arbitrary number of points. (One can give another proof of continuity which is based on Theorem 2.5 in [Hu].) Sections 6 and 7 can be read independently of Sect. 5. We assume that the reader is familiar with the basics of the theory of vertex algebras and the theory of topological vector spaces. The necessary background material can be found in [Ka, FB] and [Na, Tr].
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2. Mathematical Framework The state space of a CFT is built from representation spaces of a chiral symmetry algebra. The work of Huang and Gaberdiel & Goddard provides us with methods to topologize such spaces. This section fixes the definition of the chiral representations and specifies the additional assumptions needed for their topologization. 2.1. Vertex algebras and vertex algebra modules. In this article, we use the concept of vertex algebras to formally define the chiral symmetry algebra [FLM, Ka, FB]. The topologies will be defined on certain vertex algebras and modules of them. Let V be a Z+ -graded vertex algebra consisting of finite-dimensional graded components, that is V = Vh h∈Z+
and
dim Vh < ∞ .
The vacuum vector is denoted by ( ∈ V0 ). The map φ : V → End V [[z, z−1 ]] , v → φ(v, z) = (v)n z−n−1 , n∈Z
establishes the state-field correspondence. In physical jargon, the endomorphisms (v)n are called mode operators of the field φ(v, z) associated to the state v. (The use of the letter φ is conventional in quantum field theory; Y is the standard symbol used in the theory of vertex algebras.) In a conformal vertex algebra, the grading corresponds to the assignment of conformal weights. The choice of an integer grading means that we only consider bosonic fields; the restriction to positive values follows from unitarity requirements (see below). Take W to be a R+ -graded V -module [FB] satisfying Wh W = h∈R+
and
dim Wh < ∞ .
The fields of the chiral algebra V are represented on W by φW : V → End W [[z, z−1 ]] , v → φW (v, z) = (v)n z−n−1 . n∈Z
In the remainder of the text the index W is omitted: it will be clear from the context whether one deals with fields and operators of the vertex algebra or those of its module. Let X ⊂ V and Y ⊂ W be subspaces which generate V and W respectively: V = span {(x1 )n1 · · · (xk )nk xk+1 | xi ∈ X, ni ∈ Z+ , k ∈ N} , W = span {(v1 )n1 · · · (vk )nk y | vi ∈ V , y ∈ Y, ni ∈ Z+ , k ∈ N} .
(1) (2)
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We assume that X contains . Line (1) implies that every mode operator (v)n (v ∈ V , n ∈ Z+ ) is a linear combination of products (x1 )n1 · · · (xk )nk . This can be shown by induction and a suitable integration of the operator product expansion (see Sec. 1.6 of [FB]). Hence, as a consequence of (1) and (2), W is spanned by vectors of the form (x1 )n1 · · · (xk )nk y
(k ∈ N),
where x1 , . . . xk ∈ X, n1 , . . . , nk ∈ Z+ and y ∈ Y . 2.2. Unitarity, correlation functions and finiteness. Gaberdiel’s and Goddard’s axioms lead to state spaces of a chiral symmetry algebra that contains at least the M¨obius algebra. An additional condition on the amplitudes implies the existence of an inner product and that M¨obius transformations are unitary w.r.t. it (see Sect. 3.5, [G]). In this paper, we assume that this condition is fulfilled, so effectively one deals with chiral representations that carry a (pseudo-)unitary structure. We implement these requirements as follows: V and W are equipped with inner products , (antilinear in the first variable), and the Lie algebra sl(2, C) is unitarily represented on them; the associated operators L0 and L−1 can be identified with the grading and shift operators respectively1 . It follows that the inner products are compatible with the grading of V and W . Note also that the inner products can be indefinite; for sake of simplicity, we restrict ourselves to unitary CFTs and thus assume that , is positive definite. As a result, the grading of V and W has to be real and positive. Matrix elements w, ˜ φ(v1 , z1 ) · · · φ(vk , zk )w
(3)
of field products are obtained by inserting the formal sum φ(v1 , z1 ) · · · φ(vk , zk ) between states w, w˜ ∈ W and replacing the formal variables by complex numbers z1 , . . . , zk . In other words, we consider k + 2 point blocks on the sphere with in- and out-state taken from the module W and k insertions that are descendants of the vacuum. It follows from the axioms of the vertex algebra module that (3) converges absolutely in the region |z1 | > . . . > |zk | > 0 , and can be analytically extended to a meromorphic function on the domain M k = {(z1 , . . . , zk ) ∈ (C× )k | zi = zj for i = j } . M k is the moduli space of n different ordered points on C× . In the theory of vertex algebras, one frequently considers “matrix elements” of the form w (φ(v1 , z1 ) · · · φ(vk , zk )w) , 1
Cf. the definition of a M¨obius-conformal vertex algebra (see, e.g. [Ka]).
(4)
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where w is an element of the graded dual W = (Wh )∗ . h∈R+
Since the graded components of W are finite-dimensional, every bra-vector w˜ | can be represented by some dual vector w ∈ W , and all theorems for matrix elements (4) apply as well to (3). For later use, we note here that if W = V and w = w˜ = , the amplitude (3) is translation-invariant: , φ(v1 , z1 ) · · · φ(vk , zk ) = , φ(v1 , z1 + z) · · · φ(vk , zk + z) ,
z ∈ C.
Given an open set D ⊂ C we define the space of “correlation functions”2 F˜kD (k ∈ N) to be the vector space of all functions w, ˜ φ(v1 , .) · · · φ(vk , .)w , with v1 , . . . vk ∈ V , w, w˜ ∈ W and arguments (z1 , . . . , zk ) in the domain k MD = {(z1 , . . . , zk ) ∈ Dk | zi = zj for i = j ; zi = 0} .
F˜kD is endowed with the topology of compact convergence, i.e. the topology of uniform k . We denote by F D the completion of F˜ D . The convergence on compact subsets of MD k k topological dual FkD∗ receives the strong topology — the topology of uniform convergence on all weakly bounded subsets of FkD . For the construction of Huang’s topology it is necessary to impose two additional conditions on the vertex algebra V and the V -module W . Both should be finitely generated: the spaces X and Y are assumed to be finite-dimensional; we write d = dim X and n = dim Y . (Gaberdiel and Goddard only require that X has a countable basis. Various other finiteness conditions have been studied in the literature, see e.g. [NT].) 3. Gaberdiel’s and Goddard’s Topology A set of meromorphic and M¨obius covariant amplitudes provides the starting point for Gaberdiel’s and Goddard’s definition of chiral CFT. It allows for a direct construction of vertex operators as continuous maps between topological spaces. In Sects. 4 and 8 of [GG], it is explained how this leads to the more common description in terms of chiral algebras and their representations. We will not discuss this relation and define the topologies directly using the vertex algebra V and the module W . Below we give the construction for the module W ; it applies in particular to V , since V is a finitely generated module over itself; in this case, X plays the role of Y . Let us first sketch the idea: we seek to define seminorms on W ; to this end, we fix vectors w˜ ∈ W and v1 , . . . , vl ∈ V , and consider, for each vector w, the correlator w, ˜ φ(v1 , .) · · · φ(vk , .)w as a function of k arguments in the complex plane. After choosing a suitable domain D for these functions, a seminorm is provided by the supremum norm on compact subsets 2 Note that these functions are objects of the chiral CFT. They are the chiral (or conformal) blocks from which the physical correlators of the full CFT are constructed.
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K of D. In other words, one uses the topology of compact convergence on function spaces in order to topologize the vector space W . To keep the number of seminorms countable, we restrict the choice of out-states and insertions to Y and X respectively. That is, we only consider seminorms of the type w =
sup
|y, φ(x1 , ζ1 ) · · · φ(xl , ζl )w| .
(ζ1 ,... ,ζl )∈K
The proofs in the following sections require us to put this scheme into more formal l and F O language: Let O be an arbitrary open subset of C and l ∈ N. (The spaces MO l were defined in Sect. 2.2.) There is a linear map glO : Y ⊗ X ⊗l ⊗ W → FlO defined by glO (y ⊗ x1 ⊗ · · · ⊗ xl ⊗ w) := y, φ(x1 , .) · · · φ(xl , .)w for y ∈ Y , x1 , . . . , xl ∈ X and w ∈ W . Here, we use the complex conjugate space Y of Y and thereby avoid the antilinearity of the inner product3 . For fixed l ∈ N and x ∈ Y ⊗ X ⊗l , we obtain a linear map glO (x ⊗ .) : W → FlO .
(5)
The family of mappings glO (x ⊗ .), l ∈ N, x ∈ Y ⊗ X ⊗l , determines an initial topology O on W , i.e. the weakest topology with respect to which all g O (x ⊗ .) are continuous. TGG l It is locally convex, but not necessarily Hausdorff. At this point, Goddard and Gaberdiel divide out the subspace of vectors that have zero length with regard to all seminorms and obtain a Hausdorff space. In this paper, we will not do so, as we want to compare topologies on W (and not on some quotient space whose content depends on one of the O on W as Gaberdiel’s and Goddard’s topology, but it should topologies). We refer to TGG be kept in mind that their space of states arises only after division by the “null states”. Given bases x1 , . . . , xd
(6)
y1 , . . . , yn
(7)
for X and
for Y , multi-indices I = (i0 , i1 , . . . , il ) ∈ {1, . . . , n} × {1, . . . , d}l can be used to label a basis xI = yi0 ⊗ xi1 ⊗ · · · ⊗ xil for Y ⊗ X ⊗l . By linearity, glO (x ⊗ .) is continuous for every x ∈ Y ⊗ X ⊗l iff it is O is therefore the weakest topology on W for which each continuous for every xI . TGG 3 Y and Y are identical as sets and additive groups, only the scalar multiplications ¯· and · differ: they are related by complex conjugation, a ¯· y = a · y ≡ a y for a ∈ C.
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glO (xI ⊗ .), l ∈ N, I ∈ {1, . . . , n} × {1, . . . , d}l , is continuous. It is characterized by the family of seminorms wI,K := glO (xI ⊗ .)K = yi0 , φ(xi1 , .) · · · φ(xil , .)wK ,
(8)
where the multiindex I specifies the basis element xI and .K is the supremum norm l . This family of seminorms is equivalent to a countable set on compact subsets K ⊂ MO of seminorms, since we may restrict our choice of K to a sequence {Kn }n∈N of compacta l . Note that in the definition we are free to replace the completion F O which exhaust MO l O. by F˜lO itself without affecting the topology TGG 4. Huang’s Topology Huang constructs a topology for finitely generated conformal vertex algebras and for finitely generated modules associated to them [Hu]. His formalism does not rely on the existence of an inner product: the graded dual is employed for defining matrix elements. We use the inner product instead and adapt Huang’s scheme accordingly. The differences are pointed out at the end of this section. For the complete proofs, we refer the reader to Huang’s paper. Again, we describe the topology for a finitely generated module W ; this includes the O was obtained by mapping W into specific case W = V , where Y is given by X. TGG spaces whose topology was already known. Huang takes the reverse approach: he maps a sequence of topological spaces into a vector space containing W , and equips it with the strict inductive limit topology (for a definition, see e.g. [Na], Chap. 12). We take D to be an open disk of arbitrary radius r > 0 around 0. Let Hom(W, C) be the space of antilinear functionals on W . Again, the conformal blocks are used as a key input; we specify a map ekD : X⊗k ⊗ Y ⊗ FkD∗ → Hom(W, C)
(9)
ekD (x1 ⊗ · · · ⊗ xk ⊗ y ⊗ µ)(w) ˜ := µ(w, ˜ φ(x1 , .) · · · φ(xk , .)y)
(10)
by
for x1 , . . . , xk ∈ X, y ∈ Y , µ ∈ FkD∗ and w˜ ∈ W . Here, w, ˜ φ(x1 , .) · · · φ(xk , .)y k. is to be understood as a function on the domain MD Consider the image
D ⊗k GD ⊗ Y ⊗ FkD∗ ) k := ek (X
and the union over all k
GD :=
GD k .
k∈N
The construction of the topology proceeds in two steps: first we show that W can be embedded into GD ; then a topology is given to GD and thus also to W .
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The space W can be embedded into GD as follows: for any k-tuple n1 , . . . , nk ∈ Z one defines functionals µn1 ,... ,nk ∈ FkD∗ by µn1 ,... ,nk (w, ˜ φ(v1 , .) · · · φ(vk , .)w) 1 1 = ··· zn1 · · · zknk v , φ(v1 , z1 ) · · · φ(vk , zk )w dz1 · · · dzk 2π i |z1 |=r1 2πi |zk |=rk 1 = w, ˜ (v1 )n1 · · · (vk )nk w , where r > r1 > · · · > rk > 0. Note that the inner product provides an isomorphism between W and a subspace of Hom(W, C). As explained in Sect. 1, W is spanned by vectors of the form w = (x1 )n1 · · · (xk )nk y (k ∈ N), where x1 , . . . xk ∈ X, n1 , . . . , nk ∈ Z+ and y ∈ Y . The value of the inner product , w coincides with the value of the functional ekD (x1 ⊗ · · · ⊗ xk ⊗ y ⊗ µn1 ,... ,nk ) on any vector w˜ ∈ W . Indeed, ekD (x1 ⊗ · · · ⊗ xk ⊗ y ⊗ µn1 ,... ,nk )(w) ˜ = µn1 ,... ,nk (w, ˜ φ(x1 , .) · · · φ(xk , .)y) = w, ˜ (x1 )n1 · · · (xk )nk y = w, ˜ w . D Thus, w can be identified with the vector ekD (x1 ⊗· · ·⊗xk ⊗y ⊗µn1 ,... ,nk ) in GD k ⊂G . D This defines our embedding of W into G . D Next one constructs a canonical embedding of GD k into Gk+1 : define the linear map D → FkD γk : Fk+1
by γk (w, ˜ φ(v1 , .) · · · φ(vk+1 , .)w) = (v1 )∗−1 w, ˜ φ(v2 , .) · · · φ(vk+1 , .)w . (v1 )∗−1 is the adjoint of the −1st mode of φ(v1 , z1 ). For arbitrary w˜ ∈ W , we have ˜ = µ(w, ˜ φ(x1 , .) · · · φ(xk , .)y) ekD (x1 ⊗ · · · ⊗ xk ⊗ y ⊗ µ)(w) = µ(γk (w, ˜ φ(, .)φ(x1 , .) · · · φ(xk , .)y)) ∗ = (γk (µ))(w, ˜ φ(, .)φ(x1 , .) · · · φ(xk , .)y) D ( ⊗ x1 ⊗ · · · ⊗ xk ⊗ y ⊗ γk∗ (µ))(w) ˜ , = ek+1
where the adjoint
D∗ γk∗ : FkD∗ → Fk+1
D D has been used. This shows that GD k ⊂ Gk+1 , and that the union G of all such spaces is a vector space. How can GD be made topological? Both X and Y are finite-dimensional and carry a unique Banach space structure. So does the tensor product X ⊗k ⊗ Y . FkD∗ has the strong topology, and we equip X⊗k ⊗ Y ⊗ FkD∗ with the projective tensor product topology. ⊗k ⊗ Y ⊗ F D ∗ under the linear and surjective map eD , and is GD k is the image of X k k
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given the final (identification) topology. It can be shown then that for any k ∈ N, GD k is a topological subspace of GD k+1 . We have an increasing sequence of locally convex spaces whose union yields the vector space GD . The topology on GD is defined as the strict inductive limit determined by this sequence. As a subspace, W inherits a locally convex and Hausdorff topology from GD ; we D. denote it by THu The proofs are analogous to those in Sect. 1 and 3 of [Hu] except for the following ˜ becomes W , i.e. the value λ, w of a functional λ ∈ G ˜ on a vector replacements: G w ∈ W is replaced by the inner product w, ˜ w between a vector w˜ ∈ W and w. Instead ˜ ∗ we use the space Hom(W, C) of antilinear functionals on W , of the dual space G equipped with the weak topology. The function spaces in [Hu] correspond to FkD with D the open unit disk in C. 5. Properties of the Topologies The proofs in Sect. 5 and 6 are again formulated for a general finitely-generated V -module W . D is nuclear, and that T O is nuclear if it is 5.1. Nuclearity. We show below that THu GG Hausdorff. Gaberdiel’s and Goddard’s space of states results from dividing W by the O , which renders it Hausdorff and nuclear. “null states” with respect to TGG In the proof the following properties of nuclear spaces are used:
1. 2. 3. 4. 5. 6.
A linear subspace of a nuclear space is nuclear. The quotient of a nuclear space modulo a closed linear subspace is nuclear. A projective limit of nuclear spaces is nuclear if it is Hausdorff. A countable inductive limit of nuclear spaces is nuclear. The projective tensor product of two nuclear spaces is nuclear. A Fr´echet space is nuclear if and only if its strong dual is nuclear. (A topological vector space is called a Fr´echet space if it is complete, metrizable, locally convex and Hausdorff.)
For detailed definitions and proofs see, for instance, [Tr], Chap. 50. The following theorem provides an alternative characterization for locally convex metrizable spaces: 7. A locally convex space is metrizable iff its topology can be described by a countable family of seminorms. Given some open subset D of Cn , n ∈ N, the space H (D) of holomorphic functions l ) and H (M k ) are nuclear spaces, on it is nuclear ([Tr], Chap. 51). Accordingly, H (MO D O is a projective limit and the same holds true for the subspaces FlO , FkD and F˜kD . TGG of the spaces FlO , and hence nuclear if it is Hausdorff (3 above). If W is divided by all null states, the projective limit becomes Hausdorff and therefore nuclear (3 above). Consider now Huang’s topology: Clearly, FkD is an example of a Fr´echet space. By 6 above the strong dual FkD∗ of FkD is nuclear. Note that this conclusion cannot be made for F˜kD∗ , since F˜kD may not be complete and hence not a Fr´echet space. It follows from the definition that ∼ ⊗k ⊗ Y ⊗ F D∗ )/(eD )−1 (0) , GD k = (X k k
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where ∼ = denotes a linear and topological isomorphism. The finite-dimensional space X ⊗k ⊗ Y is nuclear and according to 5 above the tensor product with FkD∗ is nuclear as well. ekD is continuous (Proposition 1.5 in [Hu]) and (ekD )−1 (0) closed, so 2 above tells us that GD k has the nuclear property. By 4 above, the latter is preserved under the inductive limit GD = GD k , k∈N
and W , as a subspace of
GD ,
must again be nuclear.
5.2. Dependence on O and D. In the case of Gaberdiel’s and Goddard’s topologies, it is immediate from the definition that
O is coarser than T O when O ⊂ O. TGG GG
Due to their definition by functionals, Huang’s topologies behave in the opposite way: C is finer than T D for C ⊂ D. THu Hu
This can be seen as follows: Proof. Suppose that C ⊂ D ⊂ C, where C and D are open disks centered at 0. Consider the map from FkD to FkC given by restriction to MCk : it is linear, surjective, and injective, since both pre-image and image are restrictions of a single meromorphic function on the domain MCk = M k . FkC and FkD can be identified as vector spaces, but the topology on FkC is weaker. Therefore, its dual space FkC ∗ is a subspace of FkD∗ . Both dual spaces carry the strong topology: the topology of uniform convergence on weakly bounded4 subsets. Neighbourhood bases at 0 are given by the polar sets ◦ BD = {µ ∈ FkD∗ | sup |µ(f )| ≤ 1} for B bounded in FkD , f ∈B
and
BC◦ = {µ ∈ FkC ∗ | sup |µ(f )| ≤ 1} f ∈B
for B bounded in FkC
respectively. The topology induced on FkC ∗ by FkD∗ has the base B˜ C◦ = {µ ∈ FkC ∗ | sup |µ(f )| ≤ 1} for B bounded in FkD . f ∈B
A set B is bounded in FkD iff it is bounded w.r.t. each seminorm in FkD . Hence it is also bounded in FkC , and B˜ C◦ = BC◦ . Thus we see that the topology on FkC ∗ is finer than that induced by FkD∗ . Furthermore, as topological vector spaces, ∼ ⊗k ⊗ Y ⊗ F D∗ )/(eD )−1 (0) , GD k = (X k k and
4
GCk ∼ = (X⊗k ⊗ Y ⊗ FkC ∗ )/(ekC )−1 (0). Bounded = weakly bounded in locally convex Hausdorff spaces (see [Na], (9.7.6)).
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Since ekC is simply the restriction of ekD to X ⊗k ⊗ Y ⊗ FkC ∗ , GCk is a subspace of GD k . By D ∗ ⊗k ⊗k definition, the topology on X ⊗ Y ⊗ Fk is the projective tensor product of X ⊗ Y and FkD∗ . A neighbourhood base at 0 of the space X ⊗k ⊗ Y ⊗ FkD∗ is constituted by the sets conv (U ⊗ N ), where U and N are neighbourhoods in X⊗k ⊗ Y and FkD∗ respectively. The set U ⊗ N consists of all u ⊗ µ, u ∈ U , µ ∈ N , and conv stands for the convex hull. We have conv (U ⊗ N ) ∩ (X ⊗k ⊗ Y ⊗ FkC ∗ )
⊃ conv ((U ⊗ N ) ∩ (X ⊗k ⊗ Y ⊗ FkC ∗ )) ⊃ conv (U ⊗ (N ∩ F C ∗ )) , k
and N ∩ FkC ∗ is a neighbourhood of 0 in FkC ∗ . This implies that the topology of X ⊗k ⊗ Y ⊗ FkC ∗ is finer than that induced on it by X ⊗k ⊗ Y ⊗ FkD∗ . Therefore, the topology C of GCk is finer than that induced by GD k . The same holds true for the inductive limits G C D D and G , and we conclude that THu is finer than THu . 6. Comparison of the Topologies D is finer than T O for suitable choices of D and O. For that We would like to show that THu GG purpose it suffices to prove that each seminorm of Gaberdiel’s & Goddard’s topology is continuous in Huang’s topology. In the notation of Sect. 3, this means that for each l , the seminorm l ∈ N, I ∈ {1, . . . , n} × {1, . . . , d}l and compact subset K ⊂ MO
.I,K := glO (xI ⊗ .)K D . Let us therefore consider I and K to be fixed. We have to show is continuous in THu D , the net that for any net {ws }s∈S (S an index set) that converges to 0 in THu
ws I,K = yi0 , φ(xi1 , .) · · · φ(xil , .)ws K = sup |yi0 , φ(xi1 , ζ1 ) · · · φ(xil , ζl )ws | ζ ∈K
(11)
goes to 0 as well. To simplify notation we drop the index i and write y, x1 , . . . , xl from now on. D and The proof proceeds in three steps: We specify a neighbourhood base at 0 for THu express the convergence of {ws }s∈S in terms of it. To apply this convergence property, we need to cast the correlator yi0 , φ(xi1 , ζ1 ) · · · φ(xil , ζl )ws into a different form. Equation (21) below provides the desired reordering, and is proved by using the Laurent expansion of correlation functions. This equality is also essential D for the proof in Sect. 7. The third step consists in choosing a neighbourhood at 0 of THu such that (11) becomes smaller than a given .
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6.1. Convergence in Huang’s topology. Let us recall what spaces were involved in the construction of Huang’s topology: X ⊗k ⊗ Y is of finite dimension nd k and has a norm topology. All norms on X⊗k ⊗ Y are equivalent, so we can take it to be the 1-norm w.r.t. some basis (i.e. the sum of the absolute values of the coefficients in this basis). Let Uδ (0) denote the associated ball of radius δ > 0 around 0. FkD∗ carries the strong topology, and a base for the neighbourhoods of 0 in FkD∗ is given by the polars B ◦ = {µ ∈ FkD∗ | sup |µ(f )| ≤ 1}, f ∈B
where B is bounded in FkD . As already mentioned in Sec. 5.2, a neighbourhood base at 0 for X ⊗k ⊗ Y ⊗ FkD∗ is provided by the sets conv (U ⊗ N ), where U and N are neighbourhoods in X ⊗k ⊗ Y and FkD∗ respectively. Clearly, the sets conv (Uδ (0) ⊗ B ◦ )
(δ > 0, B bounded in FkD )
⊗k ⊗ Y ⊗ F D ∗ under the map form an equivalent base. The space GD k is the image of X k D ek and carries the associated final or identification topology. Therefore, the sets
ekD (conv (Uδ (0) ⊗ B ◦ ))
(12)
provide us with a neighbourhood base at 0 for GD k . The space GD = GD k k∈N
D is the strict inductive limit of the spaces GD k , and induces the topology THu on its (embedded) subspace W . A base at 0 for GD is constituted by the sets of the form conv (13) Uk , k∈N
where each Uk is a neighbourhood of 0 in GD k (see [Na], p.287, Sect. 12.1). Combining (12) and (13), we see that the sets W ∩ conv ekD (conv (Uδk (0) ⊗ Bk◦ )) k∈N
give a base at 0 for Huang’s topology5 . Since ekD is linear, the latter simplifies to D ◦ W ∩ conv ek (Uδk (0) ⊗ Bk ) . (14) k∈N
Note that in writing so we have identified W with its image under the embedding in GD ⊂ Hom(W, C). 5
It is understood that Uδk (0) and Bk◦ belong to the spaces X⊗k ⊗ Y and FkD∗ respectively.
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D on W . Given Consider now the net {ws }s∈S which converges to 0 in the topology THu a sequence of pairs (δk , Bk ), k ∈ N, there is an index s0 such that for each s ≥ s0 , ws can be expressed as a finite sum ws = ai ekDi (ui ⊗ µi ) (15) i
with
µi ∈ Bk◦i ,
ui ∈ Uδki (0) ,
and coefficients obeying
ki ∈ N ,
|ai | ≤ 1 .
i
To simplify notation we write the right-hand side of (15) without index s. Laurent expansion. We want to give an upper estimate for expression (11) when s ≥ s0 . Let us first consider the case when the sum (15) consists of only one term, i.e. ws = ekD (u ⊗ µ) ,
u ∈ Uδk (0), µ ∈ Bk◦
for some s ≥ s0 and k ∈ N. In the following the index k is fixed, so we will omit it from Uδ (0) and B. The correlator in (11) can now be written as y0 , φ(x1 , ζ1 ) · · · φ(xl , ζl )ws = y0 , φ(x1 , ζ1 ) · · · φ(xl , ζl )ekD (u ⊗ µ) .
(16)
In (16) we would like to apply the definition of ekD and make the functional µ appear explicitly (see Eq. (10)). The operators φ(x1 , ζ1 ) · · · φ(xl , ζl ) prevent us from doing so l such that and should be removed somehow. Given a ζ = (ζ1 , . . . , ζl ) ∈ MO |ζ1 | > . . . > |ζl | > 0 , one can expand the correlation function in its natural power series y, (x1 )m1 · · · (xl )ml ekD (u ⊗ µ) ζ1−m1 −1 · · · ζl−ml −1 m∈Zl
=
(xl )∗ml · · · (x1 )∗m1 y, ekD (u ⊗ µ) ζ1−m1 −1 · · · ζl−ml −1
m∈Zl
=
ekD (u ⊗ µ)((xl )∗ml · · · (x1 )∗m1 y) ζ1−m1 −1 · · · ζl−ml −1 .
(17)
m∈Zl
Next we specify a basis j
j
j
uj = u1 ⊗ · · · ⊗ uk ⊗ uk+1 , j
j
j
u1 , . . . , uk ∈ X, uk+1 ∈ Y , j = 1, . . . , nd k , for X⊗k ⊗ Y , and choose the associated 1-norm to be the norm on X ⊗k ⊗ Y . Then, each u ∈ Uδ (0) ⊂ X⊗k ⊗ Y is a linear combination k
u=
nd j =1
bj uj ,
|bj | ≤ δ ,
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and after applying the definition of ekD , the power series (17) becomes k
nd
bj
j =1
ekD (uj ⊗ µ)((xl )∗ml · · · (x1 )∗m1 y) ζ1−m1 −1 · · · ζl−ml −1
m∈Zl k
=
nd
bj
j =1
µ((xl )∗ml · · · (x1 )∗m1 y, φ(u1 , .) · · · φ(uk , .)uk+1 )ζ1−m1 −1 · · · ζl−ml −1 . j
j
j
m∈Zl
(18) Each term in the sum over j looks like the functional µ applied to j j j (xl )∗ml · · · (x1 )∗m1 y, φ(u1 , .) · · · φ(uk , .)uk+1 ζ1−m1 −1 · · · ζl−ml −1 m∈Zl
=
y, (x1 )m1 · · · (xl )ml φ(u1 , .) · · · φ(uk , .)uk+1 ζ1−m1 −1 · · · ζl−ml −1 . (19) j
j
j
m∈Zl
Note that (19) is a Hartogs expansion in ζ of 6 j
j
j
fj = y, φ(x1 , ζ1 ) · · · φ(xl , ζl )φ(u1 , .) · · · φ(uk , .) uk+1 ,
(20)
provided that sup |z| < |ζl | .
z∈D
The partial sums of (19) take their values in the dense subspace F˜kD of FkD . It is a sequence of functions in F˜kD , but in general not convergent to a function of F˜kD . At this D we have used the completion point it becomes important that in the construction of THu FkD instead of F˜kD . A theorem of complex analysis states that the Hartogs series (19) converges compactly to fj . As a result, fj is contained in the completion FkD , and with µ being an element of FkD∗ the infinite sums in (18) can be written as
µ(y, (x1 )m1 · · · (xl )ml φ(u1 , .) · · · φ(uk , .)uk+1 ) ζ1−m1 −1 · · · ζl−ml −1 j
j
j
m∈Zl j
j
j
= µ(y, φ(x1 , ζ1 ) · · · φ(xl , ζl )φ(u1 , .) · · · φ(uk , .) uk+1 ) . Recalling our starting point (Eq. (16)) we get y, φ(x1 , ζ1 ) · · · φ(xl , ζl ) ekD (u ⊗ µ) k
=
nd
j
j
j
bj µ(y, φ(x1 , ζ1 ) · · · φ(xl , ζl )φ(u1 , .) · · · φ(uk , .) uk+1 ) .
(21)
j =1
Let us recollect what assumptions were needed in order to arrive at the relation (21): The values of ζ1 , . . . , ζl are taken from an open subset O of C and (20) only makes sense as 6
Be reminded that dots represent variables of the function, whereas ζ1 to ζl are fixed.
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k if O and D do not overlap. Furthermore, the Hartogs expansion (19) a function on MD requires that
|ζ1 | > · · · > |ζl | > sup |z| .
(22)
z∈D
Equation (21) continues to hold for arbitrary orderings of ζ1 , . . . , ζl ∈ O provided inf |ζ | > sup |z| = r .
ζ ∈O
z∈D
For i = j , ζi = ζj , but what about values where |ζi | = |ζj |? Let ζ 0 = (ζ1,0 , . . . ζl,0 ) be a point where at least two radii coincide. Clearly, there is a sequence ζ n in a region of the type (22) such that lim ζ n = ζ 0 . n→∞
The corresponding sequence of functions j
j
j
y, φ(x1 , ζ1,n ) · · · φ(xl , ζl,n )φ(u1 , .) · · · φ(uk , .) uk+1 converges compactly to j
j
j
y, φ(x1 , ζ1,0 ) · · · φ(xl , ζl,0 )φ(u1 , .) · · · φ(uk , .) uk+1 . By continuity of µ it follows that Eq. (21) is valid for ζ 0 and thus for arbitrary values l . of ζ ∈ MO Choice of neighbourhood. Given > 0, we seek neighbourhoods Uδ (0) and B ◦ such that sup |y, φ(x1 , ζ1 ) · · · φ(xl , ζl ) ekD (u ⊗ µ)| ≤ ζ ∈K
if u ∈ Uδ (0) and µ ∈ B ◦ . For each point ζ = (ζ1 , . . . , ζl ) ∈ K there is a correlation function j
j
j
fj = y, φ(x1 , ζ1 ) · · · φ(xl , ζl )φ(u1 , .) · · · φ(uk , .) uk+1 . Let Bj denote the set of these functions. When regarded as a function of l + k variables, j j j f˜j = y, φ(x1 , .) · · · φ(xl , .)φ(u1 , .) · · · φ(uk , .) uk+1 l × M k , and for any compact subset K of M k , is holomorphic on MO D D
sup
(ζ ,z)∈K×K
|f˜j (ζ , z)| < ∞ .
This means that Bj is bounded in FkD , and the same holds true for the union over j , k
B=
nd j =1
Bj .
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Then, if µ ∈ B ◦ and u ∈ Uδ (0), δ = /(nd k ), Eq. (21) implies that |y, φ(xi1 , ζ1 ) · · · φ(xil , ζl ) ekD (u ⊗ µ)| k
≤
nd
j
j
j
|bj | |µ(y, φ(x1 , ζ1 ) · · · φ(xl , ζl )φ(u1 , .) · · · φ(uk , .) uk+1 )|
j =1
≤ nd k δ ≤
∀ζ ∈ K ,
as required. In general, ws is a finite sum of the type (15) for s ≥ s0 , and one has to consider the expression ai ekDi (ui ⊗ µi ) | . | y, φ(x1 , ζ1 ) · · · φ(xl , ζl ) i
Using linearity and
|ai | ≤ 1,
i
we can repeat the same arguments to obtain sup |y, φ(x1 , ζ1 ) · · · φ(xl , ζl )ws | ≤
ζ ∈K
D . The for s ≥ s0 . Therefore the seminorms .I,K are continuous in the topology THu proof was based on the validity of (21), i.e. we need that
inf |ζ | > sup |z| = r ,
ζ ∈O
D is finer than T O . and in that case THu GG
z∈D
7. Continuity of Conformal Blocks Associated to a choice of m points z1 , . . . , zn ∈ C, we define a conformal m-point block as the linear functional Cm (z1 , . . . , zm ) : V ⊗m
→ C,
v1 ⊗ · · · ⊗ vm → , φ(v1 , z1 ) · · · φ(vm , zm ) .
(23)
Note that we are now dealing with matrix elements of the vertex algebra V . Physically speaking, these are conformal blocks whose insertions are in the bosonic vacuum sector V ; we do not consider conformal blocks of other sectors, since we have not introduced general intertwining operators. In an arbitrary topology on V , the functionals (23) need not be continuous. We present D , provided D is a proof that two-point blocks are continuous in Huang’s topology THu small enough. The method can be generalized to arbitrary m-point blocks in principle, though it becomes rather unwieldy for m > 2. For fixed points z, z˜ ∈ C, the two-point block C2 (z, z˜ ) : V ⊗ V → C , v ⊗ v˜ → , φ(v, ˜ z˜ )φ(v, z)
Topologizations of Chiral Representations
445
is continuous on V ⊗ V iff it is continuous as a bilinear map from V × V into the complex numbers. A net {(v˜s , vs )}s∈S converges to 0 in V × V iff both {v˜s }s∈S and {vs }s∈S converge to 0 in V . Given such nets we want to demonstrate that |, φ(v˜s , z˜ )φ(vs , z)|
(24)
goes to zero. The proof is similar to the one of Sect. 6: We express the convergence of vs and v˜s in terms of neighbourhoods at 0, and manipulate expression (24) such that Eq. (21) can be applied. Then we choose suitable neighbourhoods to make the value of (24) smaller than . Repeating arguments of Sect. 6.1 we see that given a sequence of δk > 0 and bounded sets Bk , B˜ k in FkD , there is an index s0 such that for each s ≥ s0 , vs and v˜s are finite sums of the type ai ekDi (ui ⊗ µi ) , v˜s = a˜ i ekDi (u˜ i ⊗ µ˜ i ) , (25) vs = i
i
with ui , u˜ i ∈ Uδki (0) ,
µi ∈ B ◦ , µ˜ i ∈ B˜ ◦ ,
ki ∈ N ,
and coefficients obeying
|ai | ≤ 1 ,
i
|a˜ i | ≤ 1 .
i
Suppose for the moment that for an s ≥ s0 each of the two sums contains only one term, that is ˜ vs = ekD (u ⊗ µ) , v˜s = ekD (u˜ ⊗ µ), and
µ ∈ Bk◦ , µ˜ ∈ B˜ k◦
u, u˜ ∈ Uδk (0) ,
˜ Again, we write u for some k ∈ N. Below the index k is omitted from Uδ (0), B and B. and u˜ as linear combinations of orthonormal basis vectors: k
u=
nd
k
u˜ =
j
bj u ,
where j
b˜j u˜ j ,
j =1
j =1
|bj | ≤ δ ,
nd
j
j
j
uj = u1 ⊗ · · · ⊗ uk ⊗ uk+1 ,
j
j
u1 , . . . , uk ∈ X, uk+1 ∈ Y , j = 1, . . . , nd k , and
|b˜j | ≤ δ , j
j
j
j
j
u˜ j = u˜ 1 ⊗ · · · ⊗ u˜ k ⊗ u˜ k+1 , j
u˜ 1 , . . . , u˜ k ∈ X, u˜ k+1 ∈ Y ,
j = 1, . . . , nd k .
We are now ready to express the two-point correlator , φ(v˜s , z˜ )φ(vs , z) in terms of the defining maps of Huang’s topology. The calculation employs translation invariance
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(t), locality (l), Eq. (21) and the state-operator correspondence (s). Within correlation functions φ(v, 0) stands for the zero limit in the complex variable. , φ(v˜s , z˜ )φ(vs , z) t
= , φ(v˜s , z˜ − z)φ(vs , 0) = , φ(v˜s , z˜ − z)ekD (u ⊗ µ) k
(21)
=
nd
j
j
j
j
j
bj µ(, φ(v˜s , z˜ − z)φ(u1 , .) · · · φ(uk , .) uk+1 )
j =1 k
s
=
nd
j
bj µ(, φ(v˜s , z˜ − z)φ(u1 , .) · · · φ(uk , .)φ(uk+1 , 0))
j =1 k
l,t
=
nd
j
j
j
bj µ(, φ(u1 , . − z˜ + z) · · · φ(uk , . − z˜ + z)φ(uk+1 , −˜z + z)φ(v˜s , 0))
j =1 k
(21)
=
nd
k
bj
nd j =1
j =1
j j j ˜ φ(u1 , . − z˜ + z) · · · φ(uk , . − z˜ + z)φ(uk+1 , −˜z + z) b˜j µ(µ(, j
j
j
× φ(u˜ 1 , ˜.) · · · φ(u˜ k , ˜.)u˜ k+1 )) k
t
=
nd
k
bj
nd j =1
j =1
j j j b˜j µ(µ(, ˜ φ(u1 , z + .) · · · φ(uk , z + .)φ(uk+1 , z) j
j
j
× φ(u˜ 1 , z˜ + ˜.) · · · φ(u˜ k , z˜ + ˜.)φ(u˜ k+1 , z˜ ))) .
(26)
The notation should be understood as follows: µ˜ acts on j
j
j
j
j
j
, φ(u1 , z + .) · · · φ(uk , z + .)φ(uk+1 , z)φ(u˜ 1 , z˜ + ˜.) · · · φ(u˜ k , z˜ + ˜.)φ(u˜ k+1 , z˜ ) as a function of the variables marked by ˜. while the remaining points are fixed parameters. The expression j
j
j
j
j
j
µ(, ˜ φ(u1 , z+.) · · · φ(uk , z+.)φ(uk+1 , z)φ(u˜ 1 , z˜ +˜.) · · · φ(u˜ k , z˜ +˜.)φ(u˜ k+1 , z˜ )) is a function of the variables marked by a dot (without tilde) and serves, in turn, as an argument for the functional µ. Note that for Eq. (21) to be applicable in the third and sixth equality, it is necessary that |˜z − z| > sup |ζ | ζ ∈D
and
inf |ζ − z˜ + z| > sup |ζ | .
ζ ∈D
ζ ∈D
This is ensured if the radius r of the disk D is less than half the distance |˜z − z|. Following the same approach as in the previous section we try to make (26) arbi˜ Take a sequence of compact sets trarily small by a suitable choice of the sets B and B. k Km ⊂ MD such that ∞ k MD = Km . m=1
Topologizations of Chiral Representations
447
For each m we define Bm to be the set of functions j
j
j
fζ jj = , φ(u1 , z + ζ1 ) · · · φ(uk , z + ζk )φ(uk+1 , z) j
j
j
× φ(u˜ 1 , z˜ + ˜.) · · · φ(u˜ k , z˜ + ˜.)φ(u˜ k+1 , z˜ ) , with ζ = (ζ1 , . . . , ζl ) running through Km and j, j = 1, . . . , nd k . Bm is bounded in FkD . For a sequence of bounded sets Bm one can find ρm > 0 such that the union ∞
B˜ =
ρm B m
m=1
is again bounded. This is true in any space described by a countable family of seminorms (see [Ko], p.397). If µ˜ is taken from B˜ ◦ , max
sup |µ(f ˜ ζ jj )| ≤
max
1≤j ≤nd k 1≤j ≤nd k ζ ∈Kn
1 ρn
for each n ∈ N. The set B of functions µ˜ ∈ B˜ ◦ , j, j = 1, . . . , nd k
k hµjj ˜ ζ jj ) , ˜ : MD → C, ζ → µ(f
is therefore bounded in FkD . For µ ∈ B ◦ , µ˜ ∈ B˜ ◦ and δ = 1/2 /(nd k ), we obtain k
|, φ(v˜s , z˜ )φ(vs , z)| ≤
nd
k
|bj |
≤
|b˜j | |µ(µ(f ˜ .jj ))|
j =1
j =1 k
nd
k
nd nd
δ 2 |µ(B)|
j =1 j =1
≤ (nd k δ)2 = . The inequality remains valid for vs and v˜s of the form (25). Thus, we arrive at the result D if the open disk D has radius that two-point blocks C2 (z, z˜ ) are continuous in THu r<
|˜z − z| . 2
Acknowledgements. This work is part of F. Conrady’s diploma thesis and was supported by the “Schwerpunktprogramm String-Theorie” of the Deutsche Forschungsgemeinschaft. We are grateful to M.G. Schmidt and J. Fuchs for their continuous interest in this work. We thank Y.-Zh. Huang for helpful correspondence.
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Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves. In: Mathematical Surveys and Monographs, Vol. 88, Providence RI: American Mathematical Society, 2001 [FLM] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. NewYork: Academic Press, 1988 [FS] Fuchs, J., Schweigert, C.: A representation theoretic approach to the WZW Verlinde formula. hep-th/9707069 [G] Gaberdiel, M.R.: An Introduction to Conformal Field Theory. Rept. Prog. Phys. 63, 607–667 (2000), hep-th/9910156 [GG] Gaberdiel, M.R., Goddard, P.: Axiomatic Conformal Field Theory. Commun. Math. Phys. 209, 549–594 (2000), hep-th/9810019 [Hu] Huang,Y.-Z.: A functional-analytic theory of vertex (operator) algebras, I. Commun. Math. Phys. 204, 61–84 (1999), math.QA/9808022 [Ka] Kac, V.G.: Vertex Algebras for Beginners. University Lecture Series, Providence, RI: American Mathematical Society, 1997 [Ko] K¨othe, G.: Topologische Lineare R¨aume I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Berlin-Heidelberg-New York: Springer-Verlag, 1966 [Ma] de la Madrid, R.: Rigged Hilbert space approach to the Schr¨odinger equation. J. Phys. A: Math. Gen. 35, 319–342 (2002) [Na] Narici, L., Beckenstein, E.: Topological Vector Spaces. Pure and Applied Mathematics, New York: Marcel Dekker, Inc., 1985 [NT] Nagatomo, K., Tsuchiya, A.: Conformal field theories associated to regular vertex operator algebras I: Theories over the projective line. math.QA/0206223 [Scha] Sch¨afer, H.H.: Topological Vector Spaces. Macmillan Series in Advanced Mathematics and Theoretical Physics, New York: The Macmillan Company, 1967 [Te] Teleman, C.: Lie algebra cohomology and the fusion rules. Commun. Math. Phys. 173, 265 (1995) [Tr] Treves, F.: Topological Vector Spaces, Distributions and Kernels. London-New York-San Diego: Academic Press, 1967 Communicated by R.H. Dijkgraaf
Commun. Math. Phys. 245, 449–489 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1035-x
Communications in
Mathematical Physics
Gerbes over Orbifolds and Twisted K-Theory Ernesto Lupercio1, , Bernardo Uribe2 1 2
Departamento de Matem´aticas, CINVESTAV, Apartado Postal 14-740 07000 M´exico, D.F. M´exico. E-mail: [email protected];[email protected] Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA. E-mail: [email protected]
Received: 20 December 2002 / Accepted: 23 October 2003 Published online: 6 February 2004 – © Springer-Verlag 2004
Abstract: In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via Chern-Weil theory. For an arbitrary gerbe L, a twisting L Korb (X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal [2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. A Review of Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Orbifolds, good maps and orbibundles . . . . . . . . . . . . . 2.2 Orbifold cohomology . . . . . . . . . . . . . . . . . . . . . . 2.3 Orbifold K-theory . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Twisted K-theory on smooth manifolds . . . . . . . . . . . . 3. Gerbes over Smooth Manifolds . . . . . . . . . . . . . . . . . . . . 3.1 Gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Connections over gerbes . . . . . . . . . . . . . . . . . . . . 4. Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Orbifolds and Groupoids . . . . . . . . . . . . . . . . . . . . . . . 5.1 The groupoid associated to an orbifold . . . . . . . . . . . . . 5.2 The category associated to a groupoid and its classifying space 5.3 Sheaf cohomology and Deligne cohomology . . . . . . . . . . 6. Gerbes over Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Gerbes and inner local systems . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
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450 451 451 453 456 457 458 458 459 460 460 463 463 467 469 470 470
The first author was partially supported by the National Science Foundation and Conacyt-M´exico
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6.2 The characteristic class of a gerbe . . . . . . . . . . . . . . . 6.3 Differential geometry of gerbes over orbifolds and the B-field. 7. Twisted L Kgpd -Theory . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The twisted theory. . . . . . . . . . . . . . . . . . . . . . . . 7.3 Murray’s bundle gerbes . . . . . . . . . . . . . . . . . . . . . 8. Appendix: Stacks, Gerbes and Groupoids . . . . . . . . . . . . . . 8.1 Categories fibered by groupoids . . . . . . . . . . . . . . . . 8.2 Sheaves of categories . . . . . . . . . . . . . . . . . . . . . . 8.3 Gerbes as stacks . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Orbifolds as stacks . . . . . . . . . . . . . . . . . . . . . . . 8.5 Stacks as groupoids . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction An orbifold is a very natural generalization of a manifold. Locally it looks like the quotient of an open set of a vector space divided by the action of a group, in such a way that the stabilizer of the action at every point is a finite group. Many moduli spaces, for example, appear with canonical orbifold structures. Recently Chen and Ruan [8] motivated by their ideas in quantum cohomology and by orbifold string theory models discovered a remarkable cohomology theory of orbifolds that they have coined orbifold cohomology. Adem and Ruan [1] went on to define the corresponding orbifold K-theory and to study the resulting Chern isomorphism. One of the remarkable properties of the theory is that both theories can be twisted by what Ruan has called an inner local system coming from a third group-cohomology class called discrete torsion. Independently of that, Witten [35], while studying K-theory as the natural recipient of the charge of a D-brane in type IIA superstring theories was motivated to define a twisting of K(M) for M smooth by a third cohomology class in H 3 (M) coming from a codimension 3-cycle in M and Poincar´e duality. This twisting appeared previously in the literature in different forms [11, 14, 28]. In this paper we show that if an orbifold is interpreted as a stack then we can define a twisting of the natural K-theory of the stack that generalizes both Witten’s and AdemRuan’s twistings. We also show how we can interpret the theory of bundle gerbes over a smooth manifold and their K-theory [25, 26, 4] in terms of the theory developed here. Since the approach to the theory of stacks that we will follow is not yet published [3], we try very hard to work in very concrete terms and so our study includes a very simple definition of a gerbe over a stack motivated by that of Chaterjee and Hitchin [15] on a smooth manifold. This definition is easy to understand from the point of view of differential geometry, and of algebraic geometry. Using results of Segal [31, 33] on the topology of classifying spaces of categories and of Crainic, Moerdijk and Pronk on sheaf cohomology over orbifolds [10, 23, 24, 20] we show that the usual theory for the characteristic class of a gerbe over a smooth manifold [5] extends to the orbifold case. Then we explain how Witten’s arguments relating the charge of a D-brane generalize. A lot of what we will show is valid for foliation groupoids and also for a category of Artin stacks - roughly speaking spaces that are like orbifolds except that we allow the stabilizers of the local actions to be Lie groups. In particular we will explain how the
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twisting proposed here can be used to realize the Freed-Hopkins-Teleman twisting used in their topological interpretation of the Verlinde algebra [12]. 2. A Review of Orbifolds In this section we will review the classical construction of the category of orbifolds. This category of orbifolds is essentially that introduced by Satake [30] under the name of V-manifolds, but with a fundamental difference introduced by Chen and Ruan. They have restricted the morphisms of the category from orbifold maps to good maps, in fact Moerdijk and Pronk have found this category previously [23] where good maps go by the name of strict maps. This is the correct class of morphisms from the point of view of stack theory as we will see later. 2.1. Orbifolds, good maps and orbibundles. Following Ruan [29, 8] we will use the following definition for an orbifold. Definition 2.1.1. An n-dimensional uniformizing system for a connected topological space U is a triple (V , G, π) where • V is a connected n-dimensional smooth manifold • G is a finite group acting on V smoothly (C ∞ automorphisms) • π : V −→ U is a continuous map inducing a homeomorphism π˜ : V /G → U. Two uniformizing systems, (V1 , G,1 π1 ) and (V2 , G2 , π2 ) are isomorphic if there exists a pair of functions (φ, λ) such that: • φ : V1 −→ V2 a diffeomorphism, • λ : G1 −→ G2 an isomorphism with φ being λ-equivariant and π2 ◦ φ = π1 . Let i : U → U be a connected open subset of U and (V , G , π ) a uniformizing system of U . Definition 2.1.2. (V , G , π ) is induced from (V , G, π ) if there exist: ∼ =
• a monomorphism λ : G → G inducing an isomorphism λ : ker G → ker G, where ker G and ker G are the subgroups of G and G respectively that act trivially on V and V , and • a λ-equivariant open embedding φ : V → V with i ◦ π = π ◦ φ. We call (φ, λ) : (V , G , π ) → (V , G, π ) an injection. Two injections (φi , λi ) : (Vi , Gi , πi ) → (V , G, π ), i = 1, 2, are isomorphic if there exist: • an isomorphism (ψ, τ ) : (V1 , G1 , π1 ) → (V2 , G2 , π2 ) and ˜ τ˜ ) : (V , G, π) → (V , G, π ) • an automorphism (ψ, ˜ τ˜ ) ◦ (φ1 , λ1 ) = (φ2 , λ2 ) ◦ (ψ, τ ) such that (ψ, Remark 2.1.3. Since for a given uniformizing system (V , G, π ) of U , and any connected open set U of U , (V , G, π) induces a unique isomorphism class of uniformizing systems of U we can define the germ of a uniformizing system localized at a point.
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Let U be a connected and locally connected topological space, p ∈ U a point, and (V1 , G1 , π1 ) and (V2 , G2 , π2 ) uniformizing systems of the neighborhoods U1 and U2 of p respectively, then Definition 2.1.4. (V1 , G1 , π1 ) and (V2 , G2 , π2 ) are equivalent at p if they induce uniformizing systems for a neighborhood U3 ⊂ U1 ∩ U2 of p The germ of (V , G, π) at p is defined as the set of uniformizing systems of neighborhoods of p which are equivalent at p with (V , G, π). Definition 2.1.5. Let X be a Hausdorff, second countable topological space. An n-dimensional orbifold structure on X is a set {(Vp , Gp , πp )|p ∈ X} such that • (Vp , Gp , πp ) is a uniformizing system of Up , neighborhood of p in X, • for any point q ∈ Up , (Vp , Gp , πp ) and (Vq , Gq , πq ) are equivalent at q. We say that two orbifold structures on X, {(Vp , Gp , πp )}p∈X and {(Vp , Gp , πp )}p∈X , are equivalent if for any q ∈ X (Vq , Gq , πq ) and (Vq , Gq , πq ) are equivalent at q. Definition 2.1.6. With a given orbifold structure, X is called an orbifold. Sometimes we will simply denote by X the pair (X, {(Vp , Gp , πp )}p∈X ). When we want to make the distinction between the underlying topological space X and the orbifold (X, {(Vp , Gp , πp )}p∈X ) we will write X for the latter. For any p ∈ X let (V , G, π) be a uniformizing of a neighborhood around p and p¯ ∈ π −1 (x). Let Gp be the stabilizer of G at p. Up to conjugation the group Gp is independent of the choice of p¯ and is called the isotropy group or local group at p. Definition 2.1.7. An orbifold X is called reduced if the isotropy groups Gp act effectively for all p ∈ X. In particular this implies that an orbifold is reduced if and only if the groups ker G of Definition 2.1.2 are all trivial. Example 2.1.8. Let X = Y /G be the orbifold which is the global quotient of the finite group G acting on a connected space Y via automorphisms. Then {(X, G, π )} is trivially an orbifold structure for X. We can also define another equivalent orbifold structure in the following way: for p ∈ X, let Up ⊂ X be a sufficiently small neighborhood of p such that π −1 (Up ) = Vpα , α α the disjoint union of neighborhoods Vp , where G acts as permutations on the connected components of π −1 (Up ). Let Vp be one of these connected components, and let Gp be the subgroup of G which fixes this component Vp (we could have taken Up so that Gp is the isotropy group of the ∼ = point y ∈ π −1 (p) ∩ Vp ) and take πp = π|Vp , then Vp /Gp → Up and (Vp , Gp , πp ) is a uniformizing system for Up . This is a direct application of the previous remark.
Now we can define the notion of an orbifold vector bundle or orbibundle of rank k. Given a uniformized topological space U and a topological space E with a surjective continuous map pr : E → U , a uniformizing system of a rank k vector bundle E over U is given by the following information:
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• A uniformizing system (V , G, π) of U , • A uniformizing system (V × Rk , G, π) ˜ for E such that the action of G on V × Rk is an extension of the action of G on V given by g(x, v) = (gx, ρ(x, g)v), where ρ : V × G → Aut (Rk ) is a smooth map which satisfies ρ(gx, v) ◦ ρ(x, g) = ρ(x, h ◦ g), g, h ∈ G, x ∈ V , • The natural projection map pr ˜ : V × Rk → V satisfies π ◦ pr ˜ = pr ◦ π˜ . In the same way the orbifolds were defined, once we have the uniformizing systems of rank k we can define the germ of orbibundle structures. Definition 2.1.9. The topological space E provided with a given germ of vector bundle structures over the orbifold structure of X, is an orbibundle over X. Let’s consider now orbifolds X and X and a continuous map f : X → X . A lifting of f is the following: for any point p ∈ X there are charts (Vp , Gp , πp ) at p and (Vf (p) , Gf (p) , πf (p) ) at f (p), and a lifting f˜p of fπp (Vp ) : πp (Vp ) → πf (p) (Vf (p) ) such that for any q ∈ πp (Vp ), f˜q and f˜p define the same germ of liftings of f at q. Definition 2.1.10. A C ∞ map between orbifolds X and X (orbifold-map) is a germ of C ∞ liftings of a continuous map between X and X . We would like to be able to pull-back bundles using maps between orbifolds, but it turns out that with general orbifold-maps they cannot be defined. We need to restrict ourselves to a more specific kind of maps between orbifolds; they were named good maps by Chen and Ruan (see [8]). These good maps will precisely match the definition of a morphism in the category of groupoids (see Proposition 5.1.7). Let f˜ : X → X be a C ∞ orbifold-map whose underlying continuous function is f . Suppose there is a compatible cover U of X and a collection of open subsets U of X defining the same germs, such that there is a 1−1 correspondence between elements of U and U , say U ↔ U , with f (U ) ⊂ U and U1 ⊂ U2 implies U1 ⊂ U2 . Moreover, there is a collection of local C ∞ liftings of f where f˜U U : (V , G, π ) → (V , G , π, ) satisfies that for each injection (i, φ) : (V1 , G1 , π1 ) → (V2 , G2 , π2 ) there is another injection associated to it (ν(i), ν(φ)) : (V1 , G1 , π1 ) → (V2 , G2 , π2 ) with f˜U1 U1 ◦i = ν(i)◦f˜U2 U2 ; and for any composition of injections j ◦ i, ν(j ◦ i) = ν(j ) ◦ ν(i) should hold. The collection of maps {f˜U U , ν} defines a C ∞ lifting of f . If it is in the same germ as f˜ it is called a compatible system of f˜. Definition 2.1.11. A C ∞ map is called good if it admits a compatible system. Lemma 2.1.12 [29, Lemma 2.3.2]. Let pr : E → X be an orbifold vector bundle over X . For any compatible system ξ = {f˜U U , ν} of a good C ∞ map f : X → X , there is a canonically constructed pull-back bundle of E via f˜ (a bundle pr : Eξ → X together with a C ∞ map f˜ξ : Eξ → E covering f˜.) 2.2. Orbifold cohomology. Motivated by index theory and by string theory Chen and Ruan have defined a remarkable cohomology theory for orbifolds. One must point out that while as a group it had appeared before in the literature in several forms, its product is completely new and has very beautiful properties.
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For X an orbifold, and p a point in Up ⊂ X with (Vp , Gp , πp ), π(Vp ) = Up a local chart around it, the multi-sector k X is defined as the set of pairs (p, (g)), where (g) stands for the conjugacy class of g = (g1 , . . . , gk ) in Gp . For the point (p, (g)) ∈ k X the multisector can be seen locally as g
Vp /C(g), g
g
g
g
g
where Vp = Vp 1 ∩ Vp 2 ∩ · · · ∩ Vp k and C(g)) = C(g1 ) ∩ C(g2 ) ∩ · · · ∩ C(gk ). Vp stands for the fixed-point set of g ∈ Gp in Vp , and C(g) for the centralizer of g in Gp . Its connected components are described in the following way. For q ∈ Up , up to conjugation, there is an injective homomorphism Gq → Gp , so for g in Gq the conjugacy class (g)Gp is well defined. In this way we can define an equivalence relation (g)Gq ∼ = (g)Gp and we call Tk the set of such equivalence classes. We will abuse notation and will write (g) to denote the equivalence class to which (g)Gp belongs. Let Tk0 ⊂ Tk be the set of equivalence classes (g) such that g1 g2 . . . gk = 1. X is decomposed as k (g)∈Tk X(g) , where X(g) = (p, (g )Gp )|g ∈ Gp & (g )Gp ∈ (g) . X(g) for g = 1 is called a twisted sector and X(1) the non-twisted one. Example 2.2.1. Let’s consider the global quotient X = Y /G, G a finite group. Then X(g) ∼ = Y g /C(g) where Y g is the fixed-point set of g ∈ G and C(g) is its centralizer, hence ∼ Y g /C(g) X = {(g)|g∈G}
An important concept in the theory is that of an inner local system as defined by Y. Ruan [29]. We will show below that the gerbes are models of these systems. Definition 2.2.2. Let X be an orbifold. An inner local system L = {L(g) }(g)∈T1 is an assignment of a flat complex line orbibundle L(g) → X(g) to each twisted sector X(g) satisfying the compatibility conditions: (1) L(1) = 1 is trivial. (2) I ∗ L(g −1 ) = L(g) (3) Over each X(g) with (g) ∈ T30 (g1 g2 g3 = 1), e1∗ L(g1 ) ⊗ e2∗ L(g2 ) ⊗ e3∗ L(g3 ) = 1. One way to introduce inner local systems is by discrete torsion. Let Y be the universal orbifold cover of the orbifold Z, and let π1orb (Z) be the group of deck transformations (see [34]). For X = Z/G, Y is an orbifold universal cover of X and we have the following short exact sequence: 1 −→ π1 (Z) −→ π1orb (X) −→ G −→ 1. We call an element in H 2 (π1orb (X), U(1)) a discrete torsion of X. Using the previous short exact sequence H 2 (G, U(1)) → H 2 (π1orb (X), U(1)), therefore elements α ∈ H 2 (G, U(1)) induce discrete torsions.
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We can see α : G × G → U(1) as a two-cocycle satisfying α1,g = αg,1 = 1 and αg,hk αh,k = αg,h αgh,k for any g, h, k ∈ G. We can define its phase as γ (α)g,h := −1 αg,h αh,g which induces a representation of C(g), Lαg := γ (α)g, : C(g) → U(1). Example 2.2.3. In the case that Y → X is the orbifold universal cover and G is the orbifold fundamental group such that X = Y /G, we can construct a complex line bundle Lg = Y g ×Lαg C over X(g) . We get that Ltgt −1 is naturally isomorphic to Lg so we can denote the latter one by L(g) , and restricting to X(g1 ,...,gk ) , L(g1 ,...,gk ) = L(g1 ) · · · L(gk ) ; then L = {L(g) }(g)∈T1 is an inner local system for X. To define the orbifold cohomology group we need to add a shifting to the cohomology of the twisted sectors, and for that we are going to assume that the orbifold X is almost complex with complex structure J ; recall that J will be a smooth section of End(T X) such that J 2 = −I d. For p ∈ X the almost complex structure gives rise to an effective representation ρp : Gp → GLn (C) (n = dimC X) that could be diagonalized as m1,g m 2π mg 2π mn,g g diag e , ,...,e →Q where mg is the order of g in Gp and 0 ≤ mj,g < mg . We define a function ι : X by n mj,g ι(p, (g)Gp ) = . mg j =1
It is easy to see that it is locally constant, hence we call it ι(g) ; it is an integer if and only if ρp (g) ∈ SLn (C) and ι(g) + ι(g −1 ) = rank(ρp (g) − I ), which is the complex codimension dimC X −dimC X(g) . ι(g) is called the degree shifting number. Definition 2.2.4. Let L be an inner local system; the orbifold cohomology groups are defined as d (X; L) = Horb H d−2ι(g) (X(g) ; L(g) ). (g)∈T1
If L = Lα for some discrete torsion α we define ∗ ∗ (X, C) = Horb (X, Lα ). Horb,α
Example 2.2.5. For the global quotient X = Y /G and α ∈ H 2 (G, U(1)), Lαg induces a twisted action of C(g) on the cohomology αof the fixed point set H ∗ (Y g , C) by β → Lαg (h)h∗ β for h ∈ C(g). Let H ∗ (Y g , C)C (g) be the invariant subspace under this twisted action. Then α d Horb,α (X; C) = H d−2ι(g) (Y g ; C)C (g) . (g)∈T1
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2.3. Orbifold K-theory. In this section we will briefly describe a construction by Adem and Ruan of the so-called twisted orbifold K-theory. The following construction will generate a twisting of the orbifold K-theory by a certain class in a group cohomology group. We will recover this twisting later, as a particular case of a twisting of K-theory on a groupoid by an arbitrary gerbe. The following constructions are based on projective representations. A function ρ : G → GL(V ), for V a finite dimensional complex vector space, is a projective representation of G if there exists a function α : G×G → C∗ such that ρ(x)ρ(y) = α(x, y)ρ(xy). Such α defines a two-cocycle on G, and ρ is said to be α-representation on the space V . We can take the sum of any two α-representations, hence we can define the Grothendieck group, Rα (G), associated to the monoid of linear isomorphism classes of such α-representations. Let’s assume that is a semi-direct product of a compact Lie group H and a discrete group G. Let α ∈ H 2 (G, U(1)) so we have a group extension ˜ → G → 1, 1 → U(1) → G ˜ and ˜ is the semi-direct product of H and G. Suppose that acts on a smooth manifold X such that X/ is compact and the action has only finite isotropy, then Y = X/ is an orbifold. Definition 2.3.1. An α twisted -vector bundle on X is a complex vector bundle E → X such that U(1) acts on the fibers through complex multiplication extending the action of in X by an action of ˜ in E. We define α K (X) the α-twisted -equivariant K-theory of X as the Grothendieck group of isomorphism classes of α twisted -bundles over X. For an α-twisted bundle E → X and a β-twisted bundle F → X consider the tensor product bundle E ⊗ F → X; it becomes an α + β-twisted bundle. So we have a product α
K (X) ⊗ β K (X) → α+β K (X).
And so we call the total twisted equivariant K-theory of a space as: α T K (X) = K (X) α∈H 2 (G,U(1))
When is a finite group, there is the following decomposition theorem, Theorem 2.3.2 [29, Th. 4.2.6.]. Let be a finite group that acts on X, then for any α ∈ H 2 (G, U(1)), α ∗ ∗ K (X) ⊗ C ∼ (X/ ; C). = Horb,α The decomposition is as follows: α α ∗ ∗ K (X) ⊗ C ∼ (K(X g ) ⊗ Lαg )C(g) ∼ H ∗ (X g ; C)C (g) ∼ (X/ ; C). = = = Horb,α (g)
(g)
Definition 2.3.3. In the case that Y → X is the orbifold universal cover and α ∈ H 2 (π1orb (X), U(1)), the α twisted orbifold K-theory, α Korb (X), is the Grothendieck group of isomorphism classes of α-twisted π1orb (X)-orbifold bundles over Y and the total orbifold K-theory is: α T Korb (X) = Korb (X). α∈H 2 (π1orb (X);U(1))
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2.4. Twisted K-theory on smooth manifolds. In [35] Witten shows that the D-brane charge for Type IIB superstring theories (in the case of 9-branes) should lie on a twisted K-theory group that he denotes as K[H ] (M), where a 3-form H ∈ 3 (M; R) models the Neveu-Schwarz B-field and [H ] ∈ H 3 (M; Z) is an integer cohomology class. The manifold M is supposed smooth and it is where the D-branes can be wrapped. The class [H ] is not torsion, but in any case when [H ] is a torsion class Witten gives a very elementary definition of K[H ] (M). This will also be a particular class of the twisting of K-theory on a stack by a gerbe defined below. The construction of K[H ] (M) is as follows. Consider the long exact sequence in simplicial cohomology with constant coefficients i
· · · → H 2 (M; R) → H 2 (M; U(1)) → H 3 (M; Z) → H 3 (M; R) → · · · i
(2.4.1)
exp
induced by the exponential sequence 0 → Z → R → U(1) → 1. Since [H ] is torsion, it can be lifted to a class H ∗ ∈ H 2 (M; U(1)), and if n is its order, then for a ˇ fine covering U = {Ui }i of M the class H ∗ will be represented by a Cech cocycle hij k ∈ Cˇ 3 (M)(Q(ζn )) valued on nth roots of unity. Now we can consider a vector bundle as a collection of functions gij : Uij → GLm (C) such that gij gj k gki = idGLm (C) . Definition 2.4.1. We say that a collection of functions gij : Uij → GLm (C) is an [H ]twisted vector bundle E if gij gj k gki = hij k · idGLm (C) . The Grothendieck group of such twisted bundles is K[H ] (M). This definition does not depend on the choice of cover, for it can be written in terms of a Grothendieck group of modules over the algebra of sections END(E) of the endomorphism bundle E ⊗ E ∗ , that in particular is an ordinary vector bundle [11]. In the case in which the class α = [H ] is not a torsion class one can still define a twisting and interpret it in terms of Fredholm operators on a Hilbert space. The following description is due to Atiyah and Segal [2]. Let H be a fixed Hilbert space. We let B(H) be the Banach algebra of bounded operators on H and F(H) ⊂ B(H) be the space of Fredholm operators on H, namely, those operators in B(H) that are invertible in B(H)/K(H), where K(H) is the ideal in B(H) consisting of compact operators. Then we have the following classical theorem of Atiyah, for X a topological space: K(X) = [X, F], where the right-hand side means all the homotopy classes of maps X → F. In particular F BU . For a cohomology class α ∈ H 3 (X, Z) Atiyah and Segal construct a bundle Fα over X with fiber F(H), and then define the twisted Kα -theory as Kα (X) = [(Fα )],
(2.4.2)
namely the homotopy classes of sections of the bundle Fα . To construct Fα we will use Kuiper’s theorem that states that the group U (H) of unitary operators in H is contractible and therefore one has P(C∞ ) K(Z, 2) BU (1) U (H)/U (1) = PU (H).
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This fact immediately implies K(Z, 3) BPU (H). Hence the class α ∈ H 3 (X, Z) = [X, K(Z, 3)] = [X, BPU (H)] produces a Hilbert projective bundle Pα . The class α is called the Dixmier-Douady class of the projective bundle. We define Fα := Pα ×PU (H) F(H), where PU (H) acts on F(H) by conjugation1 . It is worthwhile to mention that J. Rosenberg has previously defined Kα (X) in [28]. His definition is clearly equivalent to the one explained above. 3. Gerbes over Smooth Manifolds 3.1. Gerbes. As a way of motivation for what follows, later we will summarize the facts about gerbes over smooth manifolds; we recommend to see [5, 15] for a more detailed description of the subject. Just as a line bundle can be given by transition functions, a gerbe can be given by transition data, namely line bundles. But the “total space” of a gerbe is a stack, as explained in the appendix. The same gerbe can be given as transition data in several ways. Let’s suppose M is a smooth manifold and {Uα }α an open cover. Let’s consider the functions gαβγ : Uα ∩ Uβ ∩ Uγ −→ U(1) defined on the threefold intersections satisfying −1 −1 −1 gαβγ = gαγβ = gβαγ = gγβα
and the cocycle condition −1 −1 (δg)αβγ η = gβγ η gαγ η gαβη gαβγ = 1
on the four-fold intersection Uα ∩ Uβ ∩ Uγ ∩ Uη . All these data define a gerbe. We could ˇ think of g as a Cech cocycle of H 2 (M, C ∞ (U(1))) and therefore we can tensor them using the product of cocycles. It also defines a class in H 3 (M; Z); taking the long exact sequence of cohomology · · · → H i (M, C ∞ (R)) → H i (M, C ∞ (U(1))) → H i+1 (M, Z) → · · · given from the exact sequence of sheaves 0 → Z → C ∞ (R) → C ∞ (U(1)) → 1 and using that C ∞ (R) is a fine sheaf, we get H 2 (M, C ∞ (U(1))) ∼ = H 3 (M, Z). We might say that a gerbe is determined topologically by its characteristic class. A trivialization of a gerbe is defined by functions fαβ = fβα : Uα ∩ Uβ → U(1) on the twofold intersections such that gαβγ = fαβ fβγ fγ α . In other words g is represented as a coboundary δf = g. 1
The ordinary action of U (H) on F (H) by conjugation clearly descends to PU (H).
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given by h The difference of two trivializations fαβ and fαβ αβ becomes a line bundle (hαβ hβγ hγ α = 1). Over a particular open subset U0 we can define a trivialization, for β, γ = 0 we take fβγ := g0βγ and because of the cocycle condition we have gαβγ = fαβ fβγ fγ α . Adding f0β = 1 we get a trivialization localized at U0 and we could do the same over each Uα . Then on the intersections Uα ∩ Uβ we get two trivializations that differ by a line bundle Lαβ . Thus a gerbe can also be seen as the following data:
• A line bundle Lαβ over each Uα ∩ Uβ • Lαβ ∼ = L−1 βα • A trivialization θαβγ of Lαβ Lβγ Lγ α ∼ = 1, where θαβγ : Uαβγ → U(1) is a 2-cocycle. Example 3.1.1 [15, Ex. 1.3]. Let M n−3 ⊂ X n be an oriented codimension 3 submanifold of a compact oriented one X. Take coordinate neighborhoods Uα of X along M; we could think of them as Uα ∼ = (Uα ∩ M) × R3 , and let U0 = X\N (M), where N (M) is the closure of a small neighborhood of M, diffeomorphic to the disc bundle in the normal bundle. We have U0 ∩ U α ∼ = Uα ∩ M × {x ∈ R3 : ||x|| > } and let’s is define the bundle Lα0 as the pullback by x → x/||x|| of the line bundle of degree 1 over S 2 . 3 The line bundles Lαβ = Lα0 L−1 0β are defined on (Uα ∩Uβ ∩M)×{x ∈ R : ||x|| > } and by construction c1 (Lαβ ) = 0 over S 2 , then they can be extended to trivial ones on the whole Uα ∩ Uβ . This information provides us with a gerbe and the characteristic class of it in H 3 (X, Z) is precisely the Poincar´e dual to the homology class of the submanifold M. This is the gerbe that we will use to recover Witten’s twisting of K-theory.
3.2. Connections over gerbes. We can also do differential geometry over gerbes [15] and we will describe what is a connection over a gerbe. For {Uα } a cover such that all finite non-empty intersection are contractible (a Leray cover), a connection will consist of 1-forms over the double intersections Aαβ , such that −1 iAαβ + iAβγ + iAγ α = gαβγ dgαβγ ,
where gαβγ : Uα ∩ Uβ ∩ Uγ → U(1) is the cocycle defined by the gerbe. −1 dgαβγ ) = 0 there are 2-forms Fα defined over Uα such that Fα − Because d(gαβγ Fβ = dAαβ ; as dFα = dFβ then we define a global 3-form G such that G|Uα = dFα . This 3-form G is called the curvature of the gerbe connection. As the Aαβ are 1-forms over the double intersections, we could reinterpret them as connection forms over the line bundles. So, using the line bundle definition of gerbe, a connection in that formalism is: • A connection αβ on Lαβ such that • αβγ θαβγ = 0 where αβγ is the connection over Lαβ Lβγ Lγ α induced by the αβ • A 2-form Fα ∈ 2 (Uα ) such that on Uα ∩ Uβ , Fβ − Fα equals the curvature of αβ . When the curvature G vanishes we say that the connection on the gerbe is flat.
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4. Groupoids The underlying idea of everything we do here is that an orbifold is best understood as a stack. A stack X is a “space” in which we can’t talk of a point in X but rather only of functions S → X , where S is any space, much in the same manner in which it makes no sense to talk of the value of the Dirac delta δ(x) at a particular point, but it makes perfect sense to write R δ(x)f (x)dx. To be fair there are points in a stack, but they carry automorphism groups in a completely analogous way to an orbifold. We refer the reader to the Appendix for more on this. In any case, just as a smooth manifold is completely determined by an open cover and the corresponding gluing maps, in the same manner a stack will be completely determined by a groupoid representing it. Of course there may be more than one such groupoid, so we use the notion of Morita equivalence to deal with this issue. A groupoid can be thought of as a generalization of a group, a manifold and an equivalence relation. First an equivalence relation. A groupoid has a set of relations R that we will think of as arrows. These arrows relate elements in a set U. Given an arrow r r r →∈ R it has a source x = s(→) ∈ U and a target y = t (→) ∈ U. Then we say that r x → y, namely x is related to y. We want to have an equivalence relation, for example r s we want transitivity and then we will need a way to compose arrows x → y → z. We also require R and U to be more than mere sets. Sometimes we want them to be locally Hausdorff, paracompact, locally compact topological spaces, sometimes schemes. Consider an example. Let X = S 2 be the smooth 2-dimensional sphere. Let p, q be the north and the south poles of S 2 , and define U1 = S 2 − {p} and U2 = S 2 − {q}. Let U12 = U1 ∩ U2 and U21 = U2 ∩ U1 be two disjoint annuli. Similarly take two disjoint disks U11 = U1 ∩ U1 and U22 = U2 ∩ U2 . Consider a category where the objects are U = U1 U2 , where means disjoint union. The set of arrows will be R = U11 U12 U21 U22 . For example the point x ∈ U12 ⊂ R is thought of as an x arrow from x ∈ U1 ⊂ U to x ∈ U2 ⊂ U, namely x → x. This is a groupoid associated to the sphere. In this example we can write the disjoint union of all possible triple intersections as R t ×s R. 4.1. Definitions. A groupoid is a pair of objects in a category R, U and morphisms s, t : R ⇒ U called respectively source and target, provided with an identity e : U −→ R, a multiplication
m : R t ×s R −→ R,
and an inverse
i : R −→ R
satisfying the following properties: (1) The identity inverts both s and t: /R U@ @@ @@ s idU @@ U e
/R U@ @@ @@ t idU @@ U e
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(2) Multiplication is compatible with both s and t: R t ×s R
m
/R
s
/U
π1
R
R t ×s R
m
/R
t
/U
π2
s
R
t
(3) Associativity: R t × s R t ×s R
idR ×m
/ R t ×s R m
m×idR
R t ×s R
/U
m
(4) Unit condition: (es,idR ) / R t ×s R R HH HH HH m H idR HHH $ R
(5) Inverse:
(idR ,et) / R t ×s R R HH HH HH m H idR HHH $ R
i ◦ i = idR , s ◦ i = t, t ◦ i = s,
with R
(idR ,i)
s
/ R t ×s R m
U
e
R t
/R
U
(i,idR )
/ R t ×s R m
e
/R
We denote the groupoid by R ⇒ U := (R, U, s, t, e, m, i), and the groupoid is called e´ tale if the base category is that of locally Hausdorff, paracompact, locally compact topological spaces and the maps s, t : R → U are local homeomorphisms (diffeomorphisms). We will say that a groupoid is proper if s × t : R → U × U is a proper (separated) map. We can of course work in the category of schemes or of differentiable manifolds as well. Remark 4.1.1. From now on we will assume that our groupoids are differentiable, e´ tale and proper. The spaces R and U will be manifolds and the structure maps (s, t, e, m, i) will be smooth. And the maps s, t will be submersions in order for the space R t ×s R to be also a manifold. Example 4.1.2. For M a manifold and {Uα } and open cover, let U= Uα R = Uα ∩ Uβ (α, β) = (β, α) α
(α,β)
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s|Uαβ : Uαβ → Uα , t|Uαβ : Uαβ → Uβ e|Uα : Uα → Uα i|Uαβ : Uαβ → Uβα & m|Uαβγ : Uαβγ → Uαγ the natural maps. Note that in this example R t ×s R coincides with the subset of R t ×s R of pairs (u, v) so that t (u) = s(v), namely the disjoint union of all possible triple intersections Uαβγ of open sets in the open cover {Uα }. We will denote this groupoid R ⇒ U by M(M, Uα ). Example 4.1.3. Let G be a group and U a set provided with a left G action G × U −→ U, (g, u) → gu, we put U = U and R = G × U with s(g, u) = u and t (g, u) = gu. The domain of m is the same as G × G × U, where m(g, h, u) = (gh, u), i(g, u) = (g −1 , gu) and e(u) = (idG , u). We will write G × U ⇒ U (or sometimes X = [U/G],) to denote this groupoid. Definition 4.1.4. A morphism of groupoids (, ψ) : (R ⇒ U ) −→ (R ⇒ U) are the following commutative diagrams:
R s
U
t
R t ×s R
/R s
ψ
/U
R
/R O
e
t
e ψ
U
/ R t ×s R
/R
m
RO
R
/U
/R
ψ
/R
i
m
R
i
Now we need to say when two groupoids are “equivalent”. Definition 4.1.5. A morphism of e´ tale groupoids (, ψ) is called an e´ tale Morita morphism whenever: • The map s ◦ π2 : U ψ ×t R → U is an e´ tale surjection, • The following square is a fibered product R
/R
ψ×ψ
/ U ×U
(s ,t )
U × U
(s,t)
where only the second condition is the required for a morphism of general groupoids to be Morita. When working on e´ tale groupoids, the Morita morphisms are understood to be e´ tale.
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Two groupoids R1 ⇒ U1 , R2 ⇒ U2 are called Morita equivalent if there are Morita morphisms (i , ψi ) : R ⇒ U −→ Ri ⇒ Ui for i = 1, 2. This is an equivalence relation and in general we will consider the category of e´ tale groupoids obtained by formally inverting the Morita equivalences (see [20] for details). It is not hard to define principal bundles over groupoids where the fibers are groupoids (cf. [10]), but here we will restrict ourselves to the construction of principal G bundles over groupoids, where G is a Lie group (or an algebraic group). This will facilitate the construction of the desired twistings in K-theory. We give ourselves a groupoid s, t : R ⇒ U. Definition 4.1.6. A principal G-bundle over the groupoid R ⇒ U is the groupoid s˜ , t˜ : R × G ⇒ U × G given by the following structure: s˜ (r, h) := (s(r), h)), e(u, ˜ h) := (e(u), h),
˜ h) := (i(r), ρ(r)h), t˜(r, h) := (t (r), ρ(r)h), i(r, and m ˜ (r, h), (r , ρ(r)h) := m(r, r ), ρ(m(r, r ))h ,
where ρ : R → G is a map satisfying: i ∗ ρ = ρ −1
(π1∗ ρ) · (π2∗ ρ) = m∗ ρ.
¯ to denote the groupoid × G ⇒ . Definition 4.1.7. For a group G we write G Proposition 4.1.8. To have a principal G-bundle over G = (R ⇒ U) is the same thing ¯ as to have a morphism of groupoids G → G. This definition coincides with the one of orbibundle given previously in Sect. 2.1 when we work with the groupoid associated to the orbifold, this will be discussed in detail in the next section. 5. Orbifolds and Groupoids 5.1. The groupoid associated to an orbifold. The underlying idea behind what follows is that an orbifold is best understood when it is interpreted as a stack. We will expand this idea in the Appendix. There we explain separately the procedures to go first from an orbifold to a stack, in such a way that the category of orbifolds constructed above turns out to be a full subcategory of the category of stacks; and then, from a stack to a groupoid, producing again an embedding of categories. But there is a more direct way to pass directly from the orbifold to the groupoid and we explain it now. We recommend to see [10, 22, 24] for a detailed exposition of this issue. Then we complete the dictionary between the orbifold approach of [8] and the groupoid approach. Let X be an orbifold and {(Vp , Gp , πp )}p∈X its orbifold structure, the groupoid R ⇒ U associated to X will be defined as follows: U := p∈X Vp and an element g : (v1 , V1 ) → (v2 , V2 ) (an arrow) in R with vi ∈ Vi , i = 1, 2, will be a equivalence class of triples g = [λ1 , w, λ2 ], where w ∈ W for another uniformizing system (W, H, ρ), and the λi ’s are injections (λi , φi ) : (W, H, ρ) → (Vi , Gi , πi ) with λi (w) = vi , i = 1, 2 as in Definition 2.1.2. For another injection (γ , ψ) : (W , H , ρ ) → (W, H, ρ) and w ∈ W with γ (w ) = w then [λ1 , w, λ2 ] = [λ1 ◦ γ , w , λ2 ◦ γ ].
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Now the maps s, t, e, i, m are naturally described: s([λ1 , w, λ2 ]) = (λ1 (w), V1 ), t ([λ1 , w, λ2 ]) = (λ2 (w), V2 ), e(x, V ) = [idV , x, idV ], i([λ1 , w, λ2 ]) = [λ2 , w, λ1 ],
m([[λ1 , w, λ2 ], [µ1 , z, µ2 ]) = [λ1 ◦ ν1 , y, µ2 ◦ ν2 ],
where h = [ν1 , y, ν2 ] is an arrow joining w and z (i.e. ν1 (y) = w & ν2 (y) = z). It can be given a topology to R so that s, t will be e´ tale maps, making it into a proper, e´ tale, differentiable groupoid, and it is not hard to check that all the properties of groupoid are satisfied. A detailed proof of this fact can be found in [24, Thm 4.1.1]. Remark 5.1.1. Two equivalent orbifold structures (as in Def. 2.1.5) will induce Morita equivalent groupoids and vice versa. Thus, the choice of groupoid in the Morita equivalence class that we will use for a specific orbifold will depend on the setting, it may change once we take finer covers, but it will be clear that it represents the same orbifold. This fact is proven by Moerdijk and Pronk in [23]. This is a good place to note that an orbifold X given by a groupoid R ⇒ U will be a smooth manifold if and only if the map (s, t) : R → U × U is one-to-one. Now we can construct principal -bundles on the groupoid R ⇒ U associated to the orbifold X getting, Proposition 5.1.2. Principal bundles over the groupoid R ⇒ U are in 1–1 correspondence with -orbibundles over X. Proof. Let’s suppose the bundles are complex, in other words = GLn (C). The proof for general is exactly the same. For an n-dimensional complex bundle over R ⇒ U we have a map ρ : R → GLn (C) and a groupoid structure R×Cn ⇒ U ×Cn as in Definition 4.1.6. Let U be an open set of X uniformized by (V , G, π) which belongs to its orbifold structure; for g ∈ G and x ∈ V , ξ = [idG , x, g] is an element of R ( via the identity on V , and the action of g in V and the conjugation by g on G thought of as an automorphism of V ) and we can define ρV ,G : V ×G → GLn (C) by ρV ,G (x, g) → ρ([idG , x, g]).As m([idG , x, g], [idG , gx, hg]) = [idG , x, hg], we have ρ([idG , gx, h])◦ρ([idG , x, g]) = ρ([idG , x, hg]), which implies ρV ,G (gx, h) ◦ ρV ,G (x, g) = ρV ,G (x, hg). So (V × Cn , G, π˜ ) with ρV ,G extending the action of G in Cn is a uniformizing system for the orbibundle we are constructing, we need to prove now that they define the same germs and then we would get an orbibundle E → X using its bundle orbifold structure. Let (λi , φi ) : (W, H, µ) → (Vi , Gi , πi ) be injections of uniformizing systems of X, with corresponding bundle uniformizing systems (W ×Cn , H, µ) ˜ and (Vi ×Cn , Gi , π˜i ). n n For x ∈ W , ξ ∈ C and h ∈ H , ([idH , x, h], ξ ) ∈ R × C and t˜([idH , x, h], ξ ) = (hx, ρ([idH , x, h])ξ ) = ρW,H (x, h)ξ . As [λi , x, φi (h)◦λi ] = [idH , x, h] for i ∈ {1, 2} then ρV1 ,G1 (λ1 (x), φ1 (h)) = ρV2 ,G2 (λ2 (x), φ2 (h)); so the bundle uniformizing systems (Vi × Cn , Gi , π˜i ) define the same germs, thus they form a bundle orbifold structure over X. Conversely, if we have the orbibundle structure for E → X we need to define the function ρ : R → GLn (C). So, for injections (λ˜i , φi ) : (W ×Cn , H, µ) → (Vi ×Cn , Gi , πi ) (where λ˜i extends the λi ’s previously defined), ρ([λ1 , x, λ2 ]) will be the element in GLn (C) such that maps pr2 (λ˜1 (x, ξ )) → pr2 (λ˜2 (x, ξ )); here pr2 stands for the projection on the second coordinate; in other words ρ([λ1 , x, λ2 ])pr2 (λ˜1 (x, ξ )) = pr2 (λ˜2 (x, ξ )).
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Because this bundle uniformizing systems define the same germs, ρ satisfies the product formula; the inverse formula is clearly satisfied. Proposition 5.1.3. Isomorphic -bundles over R ⇒ U correspond to isomorphic -orbibundles over X, and vice versa. Proof. We will focus again on complex bundles. To understand what relevant information we have from isomorphic bundles, let’s see the following lemmas: Lemma 5.1.4. An isomorphism of bundles over R ⇒ U (with maps ρi : R → GLn (C) for i = 1, 2) is determined by a map δ : R → GLn (C) such that
R × Cn −→ R × Cn , (r, ξ ) → (r, δ(r)ξ ),
ψ
U × Cn −→ U × Cn , (u, ξ ) → (u, δ(e(u))ξ ),
satisfying δ(i(r))ρ1 (r) = ρ2 (r)δ(r) and δ(r) = δ(es(r)). Proof. It is easy to check that (, ψ) defined in this way is a morphism between the bundles; the equality δ(r) = δ(es(r)) comes from the diagram of the source map and δ(i(r))ρ1 (r) = ρ2 (r)δ(r) from the one of the target map, the rest of the diagrams follow from those two. In the same way we could do this procedure for complex orbibundles: Lemma 5.1.5. An isomorphism of complex orbibundles over X (with maps ρVi ,G : V × G → GLn (C) for i = 1, 2 and {(V , G, π)} orbifold structure of X) is determined by the maps δ˜V : V → GLn (C) such that V × C n → V × Cn (r, ξ ) → (r, δ˜V (r)ξ ) 1 2 ˜ ˜ ˜ satisfying δ(gr)ρ V ,G (r, g) = ρV .G (r, g)δ(r). The δV ’s form a good map.
Proof. Because the underlying orbifold structure needs to be mapped to itself, we obtain 1 2 ˜ ˜ the δV ’s. The equality δ(gr)ρ V ,G (r, g) = ρV .G (r, g)δ(r) holds because of the good map condition. The proof of the proposition is straightforward from these lemmas. The map δ that comes from the isomorphism of the complex bundles determines uniquely the δ˜V ’s, and vice versa. Example 5.1.6. The tangent bundle T X of an orbifold X is an orbibundle over X. If U = V /G is a local uniformizing system, then a corresponding local uniformizing system for T X will be T U/G with the action g · (x, v) = (gx, dgx (v)). Similarly the frame bundle P (X) is a principal orbibundle over X. The local uniformizing system is U ×GLn (C)/G with local action g ·(x, A) = (gx, dg ◦A). Notice that P (X) is always a smooth manifold for the local action, is free, and (s, t) : R → U × U is one-to-one. We want the morphism between orbifolds to be morphisms of groupoids, and this is precisely the case for the good maps given in Definition 2.1.11.
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Proposition 5.1.7. A morphism of groupoids induces a good map between the underlying orbifolds, and conversely, every good map arises in this way. Proof. For f : X → X a good map between orbifolds, we have a correspondence U ↔ U between open subsets of a compatible cover of X and open subsets of X , such that f (U ) ⊂ U , and U1 ⊂ U2 implies U1 ⊂ U2 . Moreover, we are provided with local liftings fU U : (V , G, π) → (V , G , π ) as in Definition 2.1.11. Let R ⇒ U and R ⇒ U be the groupoids constructed from the orbifold structures of X and X respectively, determined by the compatible cover {Ui }i of X and a cover of X that uniformizes {Uj }j . Define ψ : U → U such that ψ|U = fU U and : R → R by ([λ1 , w, λ2 ]) = ([ν(λ1 ), ψ(w), ν(λ2 )], where the λi ’s are injections between W and Vi and the ν(λi )’s are the corresponding injections between W and Vi given by the definition of good map; because ν(λi ) ◦ fW W = fVi Vi ◦ λi the function is well defined and together with ψ, satisfy all the conditions for a morphism of groupoids. It is clear that the groupoids just used could differ from the groupoids one obtain after performing the construction defined at the beginning of this chapter, but they are respectively Morita equivalent. On the other hand, if we are given : R → R and ψ : U → U , we can take a sufficiently small open compatible cover for X such that for U in its cover there is an open set U of X with the desired properties. For (V , G, π ) and (V , G , π ) uniformizing systems of U and U respectively, we need to define fU U . The map between V and V is given by ψ|V , and the injection between G and G is given as follows. Let’s take x ∈ V and g ∈ G. We have an automorphism of (V , G, π ) given by the action by g on V and by conjugation on G, call this automorphism λg ; then [idG , x, g] is an element of R. Using the properties of and ψ we get that ([I dG , x, g]) = [I dG , ψ(x), g ], where g ∈ G . This because every automorphism of (V , G , π ) comes from the action of an element in G (see [29, Lemma 2.1.1]). Moreover, we have that g ◦ ψ(x) = ψ ◦ g(x). This will give us an homomorphism ρU U : G → G sending g → g , that together with fU U form the compatible system we required. Reduced orbifolds have the property that they can be seen as the quotient of a manifold by a Lie group. We just construct the frame bundle P (X) of X, which is a manifold, together with the natural action of O(n) as in Example 5.1.6 (cf. [1]). Example 5.1.8. Let X be a n-dimensional orbifold, Y its orbifold universal cover and H = π1orb (X) its fundamental orbifold group and f : Y → X the cover good map. Let P (Y ) be the frame bundle of X. By 5.1.6 we know that P (X) is a smooth manifold and it is endowed with a smooth and effective O(n) action with finite isotropy subgroups such that X [P (X)/O(n)] in the category of orbifolds (cf. [1] Prop. 2.3). The frame bundle P (Y ) is isomorphic to f ∗ P (X) and lifting the action of H in Y to a free action of H in P (Y ) with P (Y )/H P (X) we obtain the following diagram: P (Y )
f /H
/O(n)
Y
/ P (X) /O(n)
f /H
/X
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sX ,tX
Let’s consider now the groupoids RY ⇒ UY and RX ⇒ UX associated to the orbifolds Y and X by using their frame bundles (i.e. RY = P (Y )×O(n) and UY = P (Y ) with sY and tY as in Example 4.1.3). Every h ∈ H induces a morphism of groupoids RY UY
h
/ RX
h
/ UX
since the action of h in P (Y ) × O(n) commutes with the action of O(n), for P (Y ) is simply f ∗ P (X). As we are working in the reduced case, the orbifold structures of Y and X can be obtained using the frame bundles P (Y ) and P (X) so we can choose a sufficiently small orbifold cover {U } of Y , such that for (V , G, π ) a uniformizing system of U , and h ∈ H , we have an isomorphism ηh : (V , G, π) ∼ = (V , G, π ), where (V , G , π ) is a uniformizing system for U = hU . In other words, the map ηh induces a groupoid automorphism of the orbifold (a good map). s,t
Let RY × H ⇒ UY be the groupoid defined by the following maps: s(r, h) = sY (r),
t (r, h) = h(tY (r)),
i(r, h) = (h(iY (r)), h−1 ),
e(x) = (eY (x), idH ),
m((r1 , h1 ), (r2 , h2 )) = (m(r1 , h−1 (r2 )), h2 h1 ),
then the following holds. Proposition 5.1.9. The groupoids RY ×H ⇒ UY and RX ⇒ UX are Morita equivalent. Proof. Noting that the map P (Y ) → P (X) is a surjection and recalling that RX = P (X) × O(n), we can see that s ◦ π2 : UY f ×t RX → UX is an e´ tale surjection. Finally because the action of H in RY and UY is free and RY /H RX , UY /H UX , it is immediate to verify that RY × H
f π1
(s,t)
UY × UY is a fibered square.
/ RX (sX ,tY )
h
/ UX × UX
5.2. The category associated to a groupoid and its classifying space. To every groupoid R ⇒ U we can associate a category C whose objects are the objects in U and whose morphisms are the objects in R that we have called arrows before. We can see R(n) := R t ×s · · · t ×s R
n
as the composition of n morphisms. In the case in which R is a set then R(n) is the set of sequences (γ1 , γ2 , . . . , γn ) so that we can form the composition γ1 ◦ γ2 ◦ · · · ◦ γn . With this data we can form a simplicial set [19, 31].
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Definition 5.2.1. A semi-simplicial set (resp. group, space, scheme) X• is a sequence of sets {Xn }n∈N (resp. groups, spaces, schemes) together with maps X0 X1 X2 · · · Xm · · · , ∂i : Xm → Xm−1 ,
sj : Xm → Xm+1 ,
0 ≤ i, j ≤ m.
(5.2.1)
called boundary and degeneracy maps, satisfying ∂i ∂j = ∂j −1 ∂i if i < j, si sj = sj +1 si if i < j, sj −1 ∂i if i < j if i = j, j + 1. ∂ i sj = 1 s ∂ j i−1 if i > j + 1 The nerve of a category (see [31]) is a semi-simplicial set N C, where the objects of C are the vertices, the morphisms the 1-simplices, the triangular commutative diagrams the 2-simplices, and so on. For a category coming from a groupoid then the corresponding simplicial object will satisfy N Cn = Xn = R(n) . We can define the boundary maps ∂i : R(n) → R(n−1) by: if i = 0 (γ2 , . . . , γn ) ∂i (γ1 , . . . , γn ) = (γ1 , . . . , m(γi , γi+1 ), . . . , γn ) if 1 ≤ i ≤ n − 1 (γ , . . . , γ ) if i = n 1 n−1 and the degeneracy maps by (e(s(γ1 )), γ1 , . . . , γn ) for j = 0 . sj (γ1 , . . . , γn ) = (γ1 , . . . , γj , e(t (γj )), γj +1 , . . . , γn ) for j ≥ 1 We will write n to denote the standard n-simplex in Rn . Let δi : n−1 → n be the linear embedding of n−1 into n as the i th face, and let σj : n+1 → n be the linear projection of n+1 onto its j th face. Definition 5.2.2. The geometric realization |X• | of the simplicial object X• is the space (z, ∂i (x)) ∼ (δi (z), x) n × Xn |X• | = . (z, sj (x)) ∼ (σj (z), x) n∈N
Notice that the topologies of Xn are relevant to this definition. The semi-simplicial object N C determines C and its topological realization is called BC, the classifying space of the category. Again in our case C is a topological category in Segal’s sense [31]. For a groupoid R ⇒ U we will call B(R ⇒ U) = BC = |N C| the classifying space of the groupoid. The following proposition establishes that B is a functor from the category of groupoids to that of topological spaces. Recall that we say that two morphisms of groupoids are Morita related if the corresponding functors for the associated categories are connected by a morphism of functors.
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Proposition 5.2.3 [21] (cf. [31, Prop. 2.1]). A morphism of groupoids X1 → X2 induces a continuous map BX1 → BX2 . Two morphisms that are Morita related will produce homotopic maps. In particular a Morita equivalence X1 ∼ X2 will induce a homotopy equivalence BX1 BX2 . This assignment is functorial. ¯ = ( × G ⇒ ) the space B G ¯ coincides with the Example 5.2.4. For the groupoid G classifying space BG of G. Consider now the groupoid X = (G×G ⇒ G), where s(g1 , g2 ) = g1 , t (g1 , g2 ) = g2 and m((g1 , g2 ); (g2 , g3 )) = (g1 , g3 ), then it is easy to see that BX is contractible and has a G action. Usually BX is written EG. ¯ is the same thing as a principal G bundle over X and A morphism of groupoids X → G therefore can be written by means of a map G×G → G. If we choose (g2 , g2 ) → g1−1 g2 the induced map of classifying spaces EG −→ BG is the universal principal G-bundle fibration over BG. Example 5.2.5. Consider a smooth manifold X and a good open cover U = {Uα }α . Consider the groupoid G = (R ⇒ U), where R consists of the disjoint union of the double intersections Uαβ . Segal [31, Prop. 4.1] calls XU the corresponding topological category. There he proves that BG = BXU X. ¯ of If we are given a principal G bundle over G then we have a morphism G → G groupoids, that in turn induces a map X → BG. Suppose that in the previous example we take G = GLn (C). Then we get a map X → BGLn (C) → BU and since K(X) = [X, BU ] this is an element in K-theory. Example 5.2.6. Consider a groupoid X of the form M × G ⇒ M, where G is acting on M continuously. Then BX EG ×G M is the Borel construction for the action M × G → M. 5.3. Sheaf cohomology and Deligne cohomology. On a smooth manifold X a sheaf S can be defined as a functor from the category whose objects are open sets of X, and whose morphisms are inclusions to the category (for example) of abelian groups, and a gluing condition of the type described in the Appendix. So for every open set Uα in X we have an abelian group SUα = S (Uα ) called the sections of S in Uα . In the representation of a smooth manifold as a groupoid R ⇒ U, where U= Uα R = Uα ∩ Uβ (α, β) = (β, α). α
(α,β)
A sheaf can be encoded by giving a sheaf over U with additional gluing conditions given by R. Definition 5.3.1 [10]. A sheaf S on a groupoid R ⇒ U consists of (1) A sheaf S on U. (2) A continuous (right) action of R on the total space of S. π
An action of R on S → U is a map S π×t R → S satisfying the obvious identities.
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The theory of sheaves over groupoids and their cohomology has been developed by Crainic and Moerdijk [10]. There is a canonical notion of morphism of sheaves. So we can define the category Sh(X) of sheaves over the groupoid X. Morita equivalent groupoids have equivalent categories of sheaves. There is a notion of sheaf cohomology ˇ of sheaves over groupoids defined in terms of resolutions. There is also a Cech version of this cohomology developed by Moerdijk and Pronk [24]. Definition 5.3.2. A groupoid R ⇒ U is called Leray if the spaces R(n) (see below) are diffeomorphic to a disjoint union of contractible open sets. Remark 5.3.3. The existence of such a Leray groupoid for orbifolds was proven by Moerdijk and Pronk in [24, Cor. 1.2.5]. From now on we will always take a representative of the Morita class of the groupoid that is of Leray type. The basic idea is just as in the case of a smooth manifold, an S ˇ valued n Cech cocycle is an element c ∈ S ( Uα1 ···αn ), and in a similar fashion we can define cocycles in the groupoid R ⇒ U in terms of the sheaf and the products R(n) := R t ×s · · · t ×s R. Then using alternating sums of the natural collection of
maps
n
−R(2) ← −R(3) · · · , R(0) ⇔ R(1) ← ⇔ ⇔ ← − we can produce boundary homomorphisms and define the cohomology theory. The resulting groupoid sheaf cohomology satisfies the usual long exact sequences and spectral sequences. In particular we can use the exponential sequence induced by i
exp
the sequence of sheaves 0 → Z → R → U(1) → 1. In [24, 20, 10] we find a theorem that implies the following Theorem 5.3.4. For an orbifold with groupoid X and a locally constant system A of coefficients (for example A = Z) we have H ∗ (X, A) ∼ = H ∗ (BX, A), where the left-hand side is orbifold sheaf cohomology and the right hand side is ordinary simplicial cohomology. Moerdijk has proved that the previous theorem is true for arbitrary coefficients A [21]. Crainic and Moerdijk have also defined hypercohomology for a bounded complex of sheaves in a groupoid, and they obtained the basic spectral sequence. In [16, 17] we define Deligne cohomology for groupoids associated to orbifolds and also Cheeger-Simons cohomology. 6. Gerbes over Orbifolds 6.1. Gerbes and inner local systems. From this section on we are going to work over the groupoid associated to an orbifold. For R ⇒ U the groupoid associated to an orbifold X defined in 5.1 we will consider the following Definition 6.1.1. A gerbe over an orbifold R ⇒ U, is a complex line bundle L over R satisfying the following conditions:
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• i∗L ∼ = L−1
471
θ
∼1 • π1∗ L ⊗ π2∗ L ⊗ m∗ i ∗ L = • θ : R t ×s R → U(1) is a 2-cocycle, where π1 , π2 : R t×s R → R are the projections on the first and the second coordinates, and θ is a trivialization of the line bundle. The following proposition2 shows that the analogy with a finite group can be carried through in this case. Proposition 6.1.2. To have a gerbe L over a groupoid G is the same thing as to have a central extension of groupoids 1 → U(1) → G˜ → G → 1. Lemma 6.1.3. In the case of a smooth manifold M [35, 15] we define the groupoid as in Example 4.1.2. For a line bundle L over R we get line bundles Lαβ := L|Uαβ over the θ
−1 ∼ double intersections Uαβ such that Lαβ ∼ = L−1 βα , and Lαβ Lβγ Lαγ = 1 over the triple intersections Uαβγ ; then we get a gerbe over the manifold as defined in Sect. 3.
We want to relate the discrete torsions of Y. Ruan [29] over a discrete group G and the gerbes over the corresponding groupoid. Example 6.1.4. Gerbes over a discrete group G are in 1-1 correspondence with the set of two-cocycles Z 2 (G, U(1)). We recall that G denotes the groupoid ∗×G ⇒ ∗ the trivial maps s, t and i(g) = g −1 and m(h, g) = hg (clearly we can drop the ∗ as is customary). A gerbe over G is a line β
∼ bundle L over G such that, if we call Lg the fiber at g, L−1 g = Lg −1 and Lg Lh = Lgh . So for each g, h ∈ G we have a trivialization βg,h ∈ U(1) satisfying βg.h βgh,k = βg,hk βh,k because βg,h
−1 βg,hk
βgh,k
−1 βh,k
Lg Lh Lk ∼ = Lgh Lk ∼ = Lghk ∼ = Lg Lhk ∼ = Lg Lh Lk . Then β : G×G → U(1) satisfies the cocycle condition and henceforth is a two-cocycle. It is clear how to construct the gerbe over G once we have the two-cocycle. The representations of Lαg : C(g) → U(1) defined in Sect. 2.2 for some α ∈ come from the fact that
H 2 (G, U(1))
Lg Lh
−1 αg,h αh,g
∼ =
Lh Lg ;
−1 then θ (g, h) := αg,h αh,g defines a representation θ (g, ) : C(g) → U(1) and it matches α the Lg for β = α. 2
We owe this observation to I. Moerdijk.
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6.2. The characteristic class of a gerbe. We want to classify gerbes over an orbifold. As we have pointed out before, the family of isomorphism classes of gerbes on a groupoid R ⇒ U forms a group under the operation of the tensor product of gerbes, that we will denote as Gb(R ⇒ U). Given an element [L] ∈ Gb(R ⇒ U) we can choose a representative L and such a representative will have an associated cocycle θ : R t ×s R → U (1). Two isomorphic gerbes will differ by the co-boundary of a cocycle R → U (1). ¯ ∼ Example 6.2.1. Gb(G) = H 2 (G, U(1)). ¯ we see that two Using Lemma 6.1.4 and the previous definition of the group Gb(G) isomorphic gerbes define cohomologous cycles, and vice versa. We will call the cohomology class L ∈ H 2 (R ⇒ U, C∗ ) of θ , the characteristic class of the gerbe L. As explained in Sect. 5.3 we can use the exponential sequence of sheaves to show that H 2 (R ⇒ U, C∗ ) ∼ = H 3 (R ⇒ U, Z) and then using the isomor3 3 ∼ phism 5.3.4 H (R ⇒ U, Z) = H (B(R ⇒ U), Z) we get Proposition 6.2.2. For a groupoid R ⇒ U we have the following isomorphism: Gb(R ⇒ U) ∼ = H 3 (B(R ⇒ U), Z) given by the map [L] → L that associates to a gerbe its characteristic class. In particular using 5.2.3 we have that Proposition 6.2.3. The group Gb(R ⇒ U) is independent of the Morita class of R ⇒ U. This could also have been obtained noting that a gerbe over an orbifold can be given as a sheaf of groupoids in the manner of 8.3.2. Example 6.2.4. Consider an inclusion of (compact Lie) groups K ⊂ G and consider the groupoid G given by the action of G in G/K, G/K × G ⇒ G/K. Observe that the stabilizer of [1] is K and therefore we have that the following groupoid: [1] × K ⇒ [1], which is Morita equivalent to the one above. From this we obtain Gb(G) ∼ = H 3 (K, Z). As it was explained in 2.4 in the case of a smooth manifold M we have that Gb(M) = [M, BBC∗ ], where BBC∗ = BPU (H) for a Hilbert space H. Let us write PU (H) to denote the groupoid × PU (H) → . We have the following Proposition 6.2.5. For an orbifold X given by a groupoid X we have Gb(X) = [X, PU (H)], where [X, PU (H)] represents the Morita equivalence classes of morphisms from X to PU (H)
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6.3. Differential geometry of gerbes over orbifolds and the B-field.. Just as in the case of a gerbe over a smooth manifold, we can do differential geometry on gerbes over an orbifold groupoid X = (R ⇒ U). Let us define a connection over a gerbe in this context. Definition 6.3.1. A connection (g, A, F, G) over a gerbe R ⇒ U consists of a complex valued 0-form g ∈ 0 (R t ×s R), a 1-form A ∈ 1 (R), a 2-form F ∈ 2 (U) and a 3-form G ∈ 3 (U) satisfying • G = dF , • t ∗ F − s ∗ F = dA and √ • π1∗ A + π2∗ A + m∗ i ∗ A = − −1g −1 dg. The 3-form G is called the curvature of the connection. A connection is called flat if its curvature G vanishes. The 3-curvature 2π √1 −1 G represents the integer characteristic class of the gerbe in cohomology with real coefficients; this is the Chern-Weil theory for a gerbe over an orbifold. One can reproduce now Hitchin’s arguments in [15] mutatis mutandis. In particular when a connection is flat one can speak of a holonomy class in H 2 (BX, U (1)). Hitchin’s discussion relating a gerbe to a line bundle on the loop space has an analogue that we have studied in [18]. There, for a given groupoid X we construct a groupoid LX that represents the free loops on X. The “coarse moduli space” or quotient space of this groupoid coincides with Chen’s definition of the loop space [7], but LX has more structure. In particular if we are given a gerbe L over X, using the holonomy 1 we construct a “line bundle” over LXS , the fixed subgroupoid under the action 1 of S 1 , by a groupoid homomorphism LXS −→ U(1). Let us consider the groupoid ∧X = (∧X)1 ⇒ (∧X)0 , with objects (∧X)0 = {r ∈ R|s(r) = t (r)}, and arrows λ
(∧X)1 = {λ ∈ R|r1 → r2 ⇔ m(λ, r2 ) = m(r1 , λ)}. The groupoid ∧X is certainly e´ tale, but it is not necessarily smooth. In other words the twisted sectors are an orbispace or a topological groupoid [7, 18]. Theorem 6.3.2 [18]. The orbifold 1 X defined in 2.2 is represented by the groupoid 1 ∧X. There is a natural action of S 1 on LX. The fixed subgroupoid (LX)S under this action is equal to ∧X. The holonomy line bundle over ∧X is an inner local system as defined in 2.2. From this discussion we see that in orbifolds with discrete torsion as the ones considered by Witten in [35, p. 34], what corresponds to the B-field 3-form H in [35, p. 30] is the 3-form G of this section. The analogue of K[H ] that Witten requires in [35, p. 34] will be constructed in the next section. Let us recall that the smooth Deligne cohomology groups of an orbifold X can be defined as in Sect. 5.3. To finish this section let us state one last proposition in the orbifold case. Proposition 6.3.3 [17, Prop. 3.0.6.]. The group of gerbes with connection over an orbifold X are classified by the Deligne cohomology group H 3 (X, Z(3)∞ D ).
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7. Twisted L Kgpd -Theory 7.1. Motivation. Think of a group G as the groupoid ×G ⇒ , that is to say, a category with one object and an arrow from this object to itself for every element g ∈ G:
g
The theory of representations of G consists of the study of the functor G → R(G), where R(G), is the Grothendieck ring of representations of G with direct sum and tensor product as operations. An n-dimensional representation ρ of G is a continuous assignment of a linear map Cn
ρg
for every arrow g ∈ G. Namely a representation is encoded in a map ρ : R = × G → GLn (C) or in other words is a principal GLn (C) bundle over the groupoid G = ( × G ⇒ ). For finite groups this is simply an orbibundle. For a given orbifold, the study of its Korb -theory is the exact analogue to the previous situation, in other words, the representation theory of groupoids is K-theory. The analogue of a representation is an orbibundle as in 4.1.6. Every arrow in the groupoid corresponds to an element in GLn (C), but now there are many objects so we get a copy of Cn for every object in U, namely a bundle over U with gluing information. In the case of a smooth manifold this recovers the usual K-theory. It is clear now that we can twist Kgpd (X) by a gerbe L over X in the very same manner in which R(G) can be twisted by an extension ˜ → G → 1. 1 → C∗ → G Such an extension is the same thing as a gerbe over G = ( × G ⇒ ). This twisting recovers all the twistings of K-theory mentioned before in this paper. For a moment let’s restrict our attention to the groupoid G associated to a smooth manifold X as defined in 4.1.2, and let’s see how its K-theory can be interpreted in terms of this groupoid. Let C be the (discrete) category whose objects are finite dimensional vector spaces and whose morphisms are linear injections. Then a functor of categories G −→ C assigns to every object of G a vector space and to every morphism of G a linear isomorphism in a continuous fashion. If we recall that the groupoid G is given by U= Uα R = Uα ∩ Uβ (α, β) = (β, α) α
(α,β)
then we realize that this is equivalent to giving a trivial vector over U and linear gluing instructions, that is to say, a vector bundle over M. It is also clear that the category C is equivalent to the category with one object for every non-zero integer n ∈ Z≥0 , and with morphisms generated by the isomorphisms
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GLn (C) and the arrows n → m whenever n ≤ m. The classifying space of C is Gr(C∞ ) BGL∞ (C) BU . In this case BG X [31], and we get an element in the reduced K-theory of X, [X, BU ] = K(X). This discussion is also valid in the case in which X is an orbifold and shows that our constructions do not depend on the choice of Leray e´ tale groupoid representing the orbifold that we take. This will motivate us to define the K-theory of an orbifold X given by a groupoid G by means of such functors G → C. Actually we will use groupoid homomorphisms from G to some groupoid V whose classifying space is homotopic to C, and this will allow us to generalize the definition to the twisted case. Following Segal and Quillen’s ideas in algebraic K-theory we can do better in the untwisted case. Consider the category C of virtual objects of C, namely the objects of C are pairs of vector spaces (V0 , V1 ) and a morphism from (V0 , V1 ) to (U0 , U1 ) is an equivalence of a triple [W ; f0 , f1 ] where W is a vector space and fi : Vi ⊕ W → Ui is an isomorphism. We say that (W ; f0 , f1 ) ∼ (W , f0 , f1 ) if and only if there is an isomorphism g : W → W such that fi ◦ (idVi ⊕ g) = fi . It is a theorem of Segal that B C is homotopy equivalent to the space of Fredholm operators F(H). But while for a finite group G it would be wrong to define K(G) as [BG, F(H)] it is still correct to say that K(G) is the set of isomorphism classes of func We can similarly define the K-theory of an orbifold given by a groupoid G tors G → C. by functors of the form G → C. Let us again consider the case of a smooth manifold M. With this in mind we would like to have a group model for the space F of Fredholm operators. One possible candidate is the following. Definition 7.1.1 [27]. For a given Hilbert space H by a polarization of H we mean a decomposition H = H+ ⊕ H− , where H+ is a complete infinite dimensional subspace of H and H− is its orthogonal complement. We define the group GLres (H) to be the subgroup of GL(H) consisting of operators A that when written with respect to the polarization H+ ⊕ H− look like ab A= , cd where a : H+ → H+ and d : H− → H− are Fredholm operators, and b : H− → H+ and c : H+ → H− are Hilbert-Schmidt operators. We have the following fact. Proposition 7.1.2. The map GLres (H) → F : A → a is a homotopy equivalence. Therefore K(M) = [M, GLres (H)]. Consider a gerbe L with characteristic class α as a map M → BBU(1) = BPU(H), then we get a Hilbert projective bundle Zα (M) → M. Then we form a GLres (H)-principal bundle over M as follows. We know [9] that polarized Hilbert bundles over M are classified by its characteristic class in K 1 (M), for in view of the Bott periodicity theorem such bundles are classified by maps M → BGLres (H) = BBU = U , namely by elements in K 1 (M). This produces the desired map Gb(M) = [M, BBU (1)] → [M, U ] = K 1 (M). In several applications it is easier to start detecting gerbes by means of their image under this map (in the smooth case, the relation to gerbes and quantum field theory of the GLres (H)-bundles can be found in [6]).
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7.2. The twisted theory.. In this section we are going to “twist” vector bundles via gerbes. So for the, R ⇒ U groupoid associated to the orbifold X and L a gerbe over R: Definition 7.2.1. An n-dimensional L-twisted bundle over R ⇒ U is a groupoid extension of it, R × Cn ⇒ U × Cn and a function ρ : R → GLn (C) such that i ∗ ρ = ρ −1
&
(π1∗ ρ) ◦ (π2∗ ρ) ◦ ((im)∗ ρ) = θL · I dGLn (C) ,
where θL : R × R → U(1) is the trivialization of the triple intersection (π1∗ L · π2∗ L · θL
˜ m (im)∗ L) ∼ ˜ i, ˜ are = 1, I dGLn (C) is the identity of GLn (C) and the functions s˜ , t˜, e, defined in the same way as for bundles. We have the following equivalent definition. Proposition 7.2.2. To have an n-dimensional L-twisted bundle over R ⇒ U is the same thing as to have a vector bundle E → U together with a given isomorphism L ⊗ t ∗E ∼ = s ∗ E. Notice that we then have a canonical isomorphism, m∗ L ⊗ π2∗ t ∗ E ∼ = π1∗ L ⊗ π2∗ L ⊗ π2∗ t ∗ E ∼ = π1∗ L ⊗ π2∗ (L ⊗ t ∗ E) ∼ = π1∗ L ⊗ π2∗ s ∗ E. We can define the corresponding Whitney sum of L-twisted bundles, so for and an n-dimensional L-twisted bundle with function ρ1 : R → GLn (C) and for an m-dimensional one with ρ2 : R → GLm (C), we can define a groupoid extension R × Cn+m ⇒ U × Cn+m with function ρ : R → GLn+m (C) such that: ρ1 (g) 0 ∈ GLn+m (C). ρ(g) = 0 ρ2 (g) Definition 7.2.3. The Grothendieck group generated by the isomorphism classes of L twisted bundles over the orbifold X together with the addition operation just defined is called the L twisted K-theory of X and is denoted by L Kgpd (X). Moerdijk and Pronk [24] proved that the isomorphism classes of orbifolds are in 1-1 correspondence with the classes of e´ tale, proper groupoids up to Morita equivalence. The following is a direct consequence of the definitions. Lemma 7.2.4. The construction of L Kgpd (X) is independent of the groupoid that is associated to X. Similarly as we did with bundles over groupoids in Lemma 5.1.4, we can determine when two L-twisted bundles are isomorphic. Proposition 7.2.5. An isomorphism of L-twisted bundles over R ⇒ U (with maps ρi : R → GLn (C) for i = 1, 2) is determined by a map δ : R → GLn (C) such that
R × Cn −→ R × Cn , (r, ξ ) → (r, δ(r)ξ ),
ψ
U × Cn −→ U × Cn , (u, ξ ) → (u, δ(e(u))ξ ),
satisfying δ(i(r))ρ1 (r) = ρ2 (r)δ(r) and δ(r) = δ(es(r)).
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Proof. The proof is the same as in Lemma 5.1.4. For (r, r ) ∈ R t ×s R we get: θL idGLn (C) = ρ1 (im(r, r ))ρ1 (r )ρ1 (r) = δ(m(r, r ))−1 ρ2 (im(r, r ))δ(im(r, r )) δ(i(r ))−1 ρ2 (r )δ(r ) δ(i(r))−1 ρ2 (r)δ(r) = δ(m(r, r ))−1 ρ2 (m(r, r ))ρ2 (r )ρ2 (r) δ(r) = δ(m(r, r ))−1 θL idGLn (C) δ(r) = θL idGLn (C) .
Using the group structure of Gb(R ⇒ U) we can define a product between bundles twisted by different gerbes, so for L1 and L2 gerbes over X, L1
Kgpd (X) ⊗ L2 Kgpd (X) → L1 ⊗L2 Kgpd (X),
(R × Cn , ρ1 ) ⊗ (R × Cm , ρ2 ) → (R × Cmn , ρ1 ⊗ ρ2 )), which is well defined because (im)∗ (ρ1 ⊗ ρ2 ) ◦ π2∗ (ρ1 ⊗ ρ2 ) ◦ π1∗ (ρ1 ⊗ ρ2 ) = (im)∗ (L1 ⊗ L2 ) · π2∗ (L1 ⊗ L2 ) · π1∗ (L1 ⊗ L2 · I dGLmn (C) = θL1 ⊗L2 I dGLnm (C) = θL1 I dGLn (C) ⊗ θL2 I dGLm (C) = ((im)∗ L1 · π2∗ L1 · π1∗ L1 ) · I dGLn (C) ⊗ ((im)∗ L2 · π2∗ L2 · π1∗ L2 ) · I dGLm (C) = ((im)∗ ρ1 ) ◦ (π2∗ ρ1 ) ◦ (π1∗ ρ1 ) ⊗ ((im)∗ ρ2 ) ◦ (π2∗ ρ2 ) ◦ (π1∗ ρ2 ) , and we can define the total twisted orbifold K-theory of X as L T Kgpd (X) = Kgpd (X). L∈Gb(R⇒U )
This has a ring structure due to the following proposition. Proposition 7.2.6. The twisted groups L Kgpd (G) satisfy the following properties: 1. If L = 0 then L Kgpd (G) = Kgpd (G), in particular if G represent the orbifold X then L Kgpd (G) = Korb (X). 2. L Kgpd (G) is a module over Kgpd (G). 3. If L1 and L2 are two gerbes over G then there is a homomorphism L2
Kgpd (G) ⊗ L2 Kgpd (G) −→ L1 ⊗L2 Kgpd (G).
4. If ψ : G1 −→ G2 is a groupoid homomorphism then there is an induced homomorphism ∗ L Kgpd (G2 ) −→ ψ L Kgpd (G1 ).
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Example 7.2.7. In the case when Y is the orbifold universal cover of X with orbifold fundamental group π1orb (X) = H , we can take a discrete torsion α ∈ H 2 (H, U(1)) and define the twisted K-theory of X as in Definition 2.3.3. Let’s associate to X the groupoid constructed in example 5.1.8; we want to construct a gerbe L over RY × H ⇒ UY so AR (X) of section 2.3 (we added that the twisted L Kgpd (X) is the same as the twisted α Korb AR the upperscipts to denote that is the twisted K theory defined by A. Adem and Y. Ruan [1]). The discrete torsion α defines a central extension of H −→ H −→ 1 1 −→ U(1) −→ H and doing the cartesian product with RY we get a line bundle U(1) −→ Lα = RY × H ↓ RY × H which, by Lemma 6.1.4 and the fact that the line bundle structure comes from the lifting of H , becomes a gerbe over RY × H ⇒ UY . Clearly this gerbe only depends on the one defined in Lemma 6.1.4 for the group H . For E → X an α-twisted bunAR (X), comes with an action of H in E such that it dle over Y , an element of α Korb ˜ lifts the one of H in Y ; choosing specific lifts h, g˜ ∈ H for every h, g ∈ H , and ˜ e ∈ E, we have g( ˜ h(e)) = α(g, h)gh(e). As E is a bundle over Y , it defined by a map h
ρ : RY → GLn (C), and for h ∈ H , it defines an isomorphism RY × Cn → RY × Cn , h
with ηh : RY → GLn (C) such that (r, ξ ) → (hr, ηh (r)ξ ). The Lα -twisted bundle over RY × H ⇒ UY that E determines, is given by the groupoid RY × H × Cn ⇒ UY × Cn and the map δ : RY × H → GLn (C), (r, h) → ρ(hr)ηh (r). Because h is an isomorphism of groupoids RY × Cn → RY × Cn , it commutes with the source and target maps, and this in turn implies that for h ∈ H and r ∈ RY we have ηh (r) = ηh (eY (sY (r))) and ηh (r)ρ(r) = ρ(hr)ηh (r). In order to prove that this bundle is Lα -twisted it is enough to check that the multiplication satisfies the specified conditions. We will make use of the following diagram in the calculation /y x? ?? ?? v w ??? z r
h
−→
r
/ y x > >> >> > v w >> z
j
−→
r
/ y x @ @@ @@ @ v w @@ z
where the r’s, v’s and w’s belong to RY , the x’s, y’s and z’s belong to UY and h, j ∈ H . We have that mY (r, v) = w and m((r, h), (v , j )) = (m(r, v), j h) = (w, j h).
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Also δ(v , j )δ(r, h) = ρ(j v )ηj (v )ρ(hr)ηh (r) = ηj (v )ρ(v )ρ(r )ηh (r) = ηj (v )ρ(w )ηh (r) = ηj (v )ρ(hw)ηh (w) = ηj (v )ηh (w)ρ(w) = ηj (v )ηh (r)ρ(w) = α(j, h)ηj h (w)ρ(w) = α(j, h)δ(w, j h). The fact that ηj (v )ηh (r) = α(j, h)ηj h (w) is precisely the fact that E is an α-twisted bundle. This proves that (RY × H × Cn ⇒ UY × Cn , δ) is endowed with the structure of a Lα -twisted bundle over RY × H ⇒ UY . Conversely, if we have the Lα -twisted bundle, it is clear how to obtain the maps ρ and ηh . From the previous construction we can see that when h = idH , the map δ determines ρ (i.e. δ(r, idH ) = ρ(r)); hence ηh (r) = δ(r, h)ρ(hr)−1 . Thus we can conclude, AR (X). Theorem 7.2.8. In the above example Lα Kgpd (X) ∼ = α Korb
From 3.1.1, 2.4 and 6.2.2 we have Proposition 7.2.9. If M is a smooth manifold and the characteristic class of the gerbe L is the torsion element [H ] in H 3 (M, Z) then L
Kgpd (M) ∼ = K[H ] (M).
It remains to verify that the twisting L Kgpd (M) coincides with the twisting Kα (M) defined by 2.4.2. Proposition 7.2.10. Whenever α = L is a torsion class and M is a smooth manifold then L Kgpd (M) = Kα (M). Proof. We will use the following facts. Theorem 7.2.11 Serre [11]. Let M be a CW-complex. If a class α ∈ H 3 (M, Z) is a torsion element then there exists a principal bundle Z → M with structure group PU(n) so that when seen as an element β ∈ [M, BPU(n)] → [M, BPU] = [M, BBU(1)] = [M, BK(Z, 2)] = [M, K(Z, 3)] = H 3 (M, Z), then α = β. In other words, the image of [M, BPU(n)] → H 3 (M, Z) is exactly the subgroup of torsion elements that are killed by multiplication by n. Theorem 7.2.12 Segal [32]. Let H be a G-Hilbert space in which every irreducible representation of G appears infinitely many times. Then the equivariant index map indG : [Z, F]G −→ KG (Z), is an isomorphism (where F is the space of G-Fredholm operators over H and G acts on F by conjugation).
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Lemma 7.2.13. Let X = Z/G be an orbifold where the Lie group G acts on Z. Let ˜ → G → 1. Consider the α ∈ H 2 (G, U(1)) define a group extension 1 → U(1) → G natural homomorphism ψ : KG˜ (Z) → KG˜ (G) ∼ = R(U(1)) - it can also be seen as the ˜ → R(U(1)). Let : U(1) → U(n) be the composition of KG˜ (Z) → KG˜ (∗) ∼ = R(G) diagonal embedding representation. Then α
AR Korb (X) = ψ −1 ().
Proof. The orbifold X is represented by the groupoid G = (Z × G ⇒ Z), while the gerbe Lα is represented by the central extension of groupoids (Proposition 6.1.2) → G → 1 1 → U(1) → G = (Z × G ˜ ⇒ Z). Therefore, using the fact that Kgpd (U(1)) = R(U (1)) we where G get the surjective map → R(U(1)), Kgpd (G) = K ˜ (Z) we get the result. using 7.2.1 and observing that Kgpd (G) G
Let us consider in the previous lemma the situation where X = M is smooth, Z is Serre’s principal PU(n)-bundle associated to α = L, G = PU(n) and β is the class in H 2 (PU(n), U(1)) labeling the extension 1 → U(1) → U(n) → PU(n) → 1. Then using Theorem 7.2.8 and 7.2.12 we get that L
AR Kgpd (Z/G) = β Korb (Z/G) = ψ −1 () ⊆ KU(n) (Z) = [Z, F]U(n) .
Notice that by 2.4.2 Kα (M) is defined as the homotopy classes of sections of the bundle Fα = Z ×PU(n) F. This space of sections can readily be identified with the space [Z, Fα ]PU(n) and the proposition follows from this. We should point out here that the theory so far described is essentially empty whenever the characteristic class L is a non-torsion element in H 3 (M, Z). The following is true. Proposition 7.2.14. If there is an n-dimensional L-twisted bundle over the groupoid G then Ln = 1. Proof. Consider the equations, i ∗ ρ = ρ −1
&
(π1∗ ρ) ◦ (π2∗ ρ) ◦ ((im)∗ ρ) = θL · I dGLn (C)
and take determinants in both equations, we get i ∗ det ρ = det ρ −1
&
(π1∗ det ρ) ◦ (π2∗ det ρ) ◦ ((im)∗ det ρ) = det θL · I dGLn (C) .
Defining f = det ρ we have i ∗ f = f −1
&
(π1∗ f ) ◦ (π2∗ f ) ◦ ((im)∗ f ) = θLn .
This means that the coboundary of f is θ n . This concludes the proof.
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Another way to think of this is by noticing that if we restrict the central extension 1 → U(1) → U(n) → PU(n) → 1 to the subgroup SU(n) we get the n-fold covering map 1 → Zn → SU(n) → PU(n) → 1, where the kernel Zn is the group of n-roots of unity. In any case we need to consider a more general definition when the class L is a non-torsion class. An obvious generalization of 2.4.2 would be to consider the class of the gerbe α = L ∈ H 3 (BG, Z) and consider Kα (BG) in the sense of 2.4.2. This works well for a manifold, but unfortunately for a finite group and the trivial gerbe α = 1 we have that Kα (BG) = R(G)and not R(G) as we should have (this is exactly the problem we encountered in the last section with [BG, F(H)]). Fortunately one of the several equivalent definitions of [4] can be carefully generalized to serve our purposes. To motivate this definition consider the following situation. Suppose first that the class α = L ∈ H 3 (BG, Z) is a torsion class. Take any α-twisted vector bundle ρ so that i ∗ ρ = ρ −1
&
(π1∗ ρ) ◦ (π2∗ ρ) ◦ ((im)∗ ρ) = θL · I dU(n)
and let β : G → PU(n) be the projectivization of ρ : G → U(n). Then β is a bonafide groupoid homomorphism, in other words i ∗ β = β −1
&
(π1∗ β) ◦ (π2∗ β) ◦ ((im)∗ β) = I dPU(n)
as equations in PU(n). Then α as a map BG → BPU(H) is simply obtained as the realization of the composition of β : G → PU(n) with the natural inclusion PU(n) → PU(H). Fix now and for all time a ρ0 and a β0 constructed in this way. PU(n) as the group whose elements are the Define the semidirect product U(n)× pairs (S, T ), where S ∈ U(n) and T ∈ PU(n) with multiplication (S1 , T1 ) · (S2 , T2 ) = (S1 T1 S2 T1−1 , T1 T2 ). PU(n) that make Consider the family of groupoid homomorphisms f : G → U(n)× the following diagram commutative: f
/ U(n)× PU(n) G JJ JJ JJ JJ q2 β0 JJJ % PU(n) where q1 is the projection onto U(n) and q2 the projection onto PU(n). PU(n) like above we can then write ρ = Given a homomorphism f : G → U(n)× (q1 ◦ f ) · ρ0 and verify that ρ satisfies the conditions to define a twisted vector bundle over G. Conversely given a twisted vector bundle ρ we can define a homomorphism f by means of the formula f (g) = (ρ(g)ρ0 (g)−1 , β0 (g)).
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PU(n) so Therefore in the case of a torsion class α a homomorphism f : G → U(n)× that q1 f = β0 is another way of encoding a twisted vector bundle. In the case of a non-torsion class α we need to consider infinite dimensional vector spaces. So we let K be the space of compact operators of a Hilbert space H. Let us write UK to denote the subgroup of U(H) consisting of unitary operators of the form I + K, where I is the identity operator and K is in K. If h ∈ PU(H) and g ∈ UK PU(H). We can define now the then hgh−1 ∈ UK and therefore we can define UK × K-theory for an orbifold X given by G twisted by a gerbe L with non-torsion class α : G → PU(H) (cf. 6.2.5). Definition 7.2.15. The set of isomorphism classes of groupoid homomorphisms f : G → PU(H) that make the following diagram commutative: UK × f /U × PU(H) G JJ JJJ K JJJ q2 α JJJ $ PU(H)
is L Kgpd (G) the groupoid K-theory of G twisted by L. This definition works for a gerbe whose class is non-torsion and has the obvious naturality conditions. In particular it becomes 2.4.2 if the groupoid represents a smooth manifold. The discussion immediately before the definition shows that this definition generalizes the one given before for L Kgpd G when L was torsion. Then Proposition 7.2.6 remains valid in the non-torsion context. In view of Theorem 6.3.2 and Theorem 7.2.8 we can reformulate Theorem 2.3.2 as follows. Theorem 7.2.16. Let X be a Leray groupoid representing an orbifold X/ with finite, L a gerbe over X coming from discrete torsion, and let be the holonomy inner local system defined in 6.3.2. Then L
∗ Kgpd (X) ⊗ C ∼ (X; C). = Horb,
It is natural to conjecture that the previous theorem remains true even if the gerbe L is arbitrary and X is any proper e´ tale groupoid. We will revisit this issue elsewhere. The following astonishing result of Freed, Hopkins and Teleman can be written in terms of the twisting described in this section. For more on this see [18]. Example 7.2.17 [12]. Let G be connected, simply connected and simple. Consider the groupoid G = (G × G ⇒ G), where G is acting on G by conjugation. Let h be the dual Coxeter number of G. Let L be the gerbe over G with characteristic class dim(G) + k + h ∈ HG3 (G). Then L
Kgpd (G) ∼ = Vk (G),
where Vk (G) is the Verlinde algebra at level k of G.
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7.3. Murray’s bundle gerbes. The theory described in the previous sections is interesting even in the case in which the orbifold X is actually a smooth manifold M = X. In this case M. Murray et. al. [26, 4] have recently proposed a way to interpret the twisted K-theory L K(M) in terms of bundle gerbes. Bundle gerbes are geometric objects constructed on M that give a concrete model for a gerbe over M [25]. The purpose of this section is to explain how the theory of bundle gerbes can be understood in terms of groupoids. π
Definition 7.3.1. A bundle gerbe over M is a pair (L, Y ), where Y −→ M is a surjective p submersion and L −→ Y π ×π Y = Y [2] is a line bundle satisfying • L(y,y) ∼ =C • L(y1 ,y2 ) ∼ = L∗(y2 ,y1 ) • L(y1 ,y2 ) ⊗ L(y2 ,y3 ) ∼ = L(y1 ,y3 ) . We start the translation to the groupoid language with the following definition. π
Definition 7.3.2. Given a manifold M and a surjective submersion Y −→ M we define the groupoid G(Y, M) = (R ⇒ U) by • R = Y [2] = Y π ×π Y •U =Y • s = p1 : Y [2] → Y , s(y1 , y2 ) = y1 and t = p2 : Y [2] → Y , t (y1 , y2 ) = y2 • m((y1 , y2 ), (y2 , y3 )) = (y1 , y3 ). From Definition 6.1.1 we immediately obtain the following. Proposition 7.3.3. A bundle gerbe (L,Y) over M is the same as a gerbe over the groupoid G(Y, M). We will write L(L, Y ) to denote the gerbe over G(Y, M) associated to the bundle gerbe (L, Y ). Notice that the groupoid G(Y, M) is not necessarily e´ tale, but it is Morita equivalent to an e´ tale groupoid. Let M(M, Uα ) be the e´ tale groupoid associated to a cover {Uα } of M as in 4.1.2. Proposition 7.3.4. The groupoid G(Y, M) is Morita equivalent to M(M, Uα ) for any open cover {Uα } of M. Proof. Since all groupoids M(M, Uα ) are Morita equivalent (for any two open covers have a common refinement) it is enough to consider the groupoid M(M, M) = (M ⇒ M) coming from the cover consisting of one open set. The source and target maps of M(M, M) are both identity maps. Then the proposition follows from the fact that the following diagram is a fibered square Y [2]
πs
/M
π×π
/ M ×M
s×t
(s,t)
Y ×Y
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Corollary 7.3.5. The group of bundle gerbes over M is isomorphic to the group Gb(M(M, Uα )) for the Leray groupoid M(M, Uα ) representing M. In particular there is a bundle gerbe in every Morita equivalence class of gerbes over M. Murray [26] defines a characteristic class for a bundle gerbe (L, Y ) over M as follows. Definition 7.3.6. The Dixmier-Douady class of d(P ) = d(P , Y ) ∈ H 3 (M, Z) is defined as follows. Choose a Leray open cover {Uα } of M. Choose sections sα : Uα → Y , inducing (sα , sβ ) : Uα ∩Uβ → Y [2] . Choose sections σαβ of (sα , sβ )−1 (P ) over Uα ∩Uβ . Define gαβγ : Uα ∩ Uβ ∩ Uγ → C× by σαβ σβγ = σαγ gαβγ . Then d(L) = [gαβγ ] ∈ H 2 (M, C× ) ∼ = H 3 (M, Z). Then in view of Propositions 6.2.2 and 6.2.3 we have the following. Proposition 7.3.7. The Dixmier-Douady class d(L, Y ) is equal to the characteristic class L(L, Y ) defined above 6.2.2. Moreover the assignment (L, Y ) → gαβγ realizes the isomorphism of 7.3.5. Definition 7.3.8 [26]. A bundle gerbe (L, Y ) is said to be trivial whenever d(L, Y ) = 0. Two bundle gerbes (P , Y ) and (Q, Z) are called stably isomorphic if there are trivial bundle gerbes T1 and T2 such that P ⊗ T 1 Q ⊗ T2 . The following is an easy consequence of 6.2.2, 6.2.3 and 7.3.7. Lemma 7.3.9. The Dixmier-Douady class is a homomorphism from the group of bundle gerbes over M with the operation of tensor product, and H 3 (M, Z) Corollary 7.3.10. Two bundle gerbes (P , Y ) and (Q, Z) are stably isomorphic if and only if d(P ) = d(Q) Proof. Suppose that (P , Y ) and (Q, Z) are stably isomorphic. Then P ⊗ T1 Q ⊗ T2 ; hence d(P ⊗ T1 ) = d(Q ⊗ T2 ). Therefore from the previous lemma we have d(P ) + d(T1 ) = d(Q) + d(T2 ) and by definition of trivial we get d(P ) = d(Q). Conversely if d(P ) = d(Q) then d(P ⊗Q∗ ) = 0 and then by definition T2 = P ⊗Q∗ is trivial. Define the trivial bundle gerbe T1 = Q∗ ⊗Q. Then P ⊗T1 Q⊗T2 completing the proof. Y, M) to denote the U(1) Given a bundle gerbe (L, Y ) over M we will write G(L, central groupoid extension of G(Y, M) defined by the associated gerbe, where Y, M) → G(Y, M) → 1. 1 → U(1) → G(L, As we have explained before such extensions are classified by their class in the cohomology group H 3 (BG(Y, M), Z) = H 3 (M, Z). As a consequence of this and 7.3.5 we have.
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Theorem 7.3.11. Two bundle gerbes (P , Y ) and (Q, Z) are stably isomorphic if and , Y, M) is Morita equivalent to G(Q, only if G(P Z, M). Therefore there is a one-to-one correspondence between stably isomorphism classes of bundle gerbes over M and classes in H 3 (M, Z). The category of bundle gerbes over M with stable isomorphisms is equivalent to the category of gerbes over M with Morita equivalences. Definition 7.3.12. Let (L, Y ) be a bundle gerbe over M. We call (E, L, Y, M) a bundle gerbe module if • E → Y is a hermitian vector bundle over Y . ∼ • We are given an isomorphism φ : L ⊗ π1−1 E −→ π2−1 E. • The compositions L(y1 ,y2 ) ⊗(L(y2 ,y3 ) ⊗Ey3 ) → L(y1 ,y2 ) ⊗Ey2 → Ey1 and (L(y1 ,y2 ) ⊗ L(y2 ,y3 ) ) ⊗ Ey3 → L(y1 ,y3 ) ⊗ Ey3 → Ey1 coincide. In this case we also say that the bundle gerbe (L, Y ) acts on E. The bundle gerbe Ktheory Kbg (M, L) is defined as the Grothendieck group associated to the semigroup of bundle gerbe modules (E, L, Y, M) for (L, Y, M) fixed. As a consequence of 7.3.11 and 7.2.2 we have the following fact. Theorem 7.3.13. The category of bundle gerbe modules over (L, Y ) is equivalent to the category of L(L, Y )-twisted vector bundles over G(Y, M). Moreover we have L
Kgpd (G(Y, M)) ∼ = Kbg (M, L).
Corollary 7.3.14. If the gerbe L has a torsion class [H ] then K[H ] (M) = Kbg (M, L). 8. Appendix: Stacks, Gerbes and Groupoids We mentioned at the beginning of Sect. 4 that a stack X is a space whose points can carry a “group valued” multiplicity, and that they are studied by studying the family {Hom(S, X )}S , where S runs through all possible spaces (or schemes). In fact by Yoneda’s Lemma, as is well known, even when X is an ordinary space that knowing everything for the functor Hom(, X ) is the same thing as knowing everything about X . A stack is a category fibered by groupoids, where CS = Hom(S, X ) with an additional sheaf condition. A very unfortunate confusion of terminologies occurs here. The word groupoid has two very standard meanings. One has been used along all the previous sections of this paper. But now we need the second meaning, namely a groupoid is a category where all morphisms have inverses. In this Appendix we use the word groupoid with both meanings and we hope that the context is enough to avoid confusion. Both concepts are, of course, very related. 8.1. Categories fibered by groupoids. Let C, S be a pair of categories and p : S → C a functor. For each U ∈ Ob(C) we denote SU = p−1 (U ). Definition 8.1.1. The category S is fibered by groupoids over C if • For all φ : U → V in C and y ∈ Ob(SV ) there is a morphism f : x → y in S with p(f ) = φ
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• For all ψ : V → W , φ : U → W , χ : U → V , f : x → y and g : y → z with φ = ψ ◦ χ , p(f ) = φ and p(g) = ψ there is a unique h : x → z such that f = g ◦ h and p(h) = χ , f
/y xA AA }> } g AAh } AA }} A }}} z φ /W U@ @@ χ }> ψ }} @@ } @@ } }} V From the previous conditions we see that for φ : U → V and y ∈ Ob(SV ) there exist a unique morphism f : x → y such that p(f ) = φ. Therefore x is defined uniquely from this information and then it will be called φ ∗ y. Hence φ ∗ will be a well defined functor from SV to SU . 8.2. Sheaves of categories. Definition 8.2.1. A Grothendieck Topology (G.T.) over a category C is a prescription of coverings {Uα → U }α such that: • {Uα → U }α & {Uαβ → Uα }β implies {Uαβ → U }αβ • {Uα → U }α & V → U implies {Uα ×U V → V }α ∼ =
• V −→ U isomorphism, implies {V −→ U }. A category with a Grothendieck Topology is called a Site. This definition is easier to understand through an example. Example 8.2.2. Let C be T op, and {Uα → U }α will be a cover if the Uα ’s are homeomorphic to its image and U = α im(Uα ). The prescription is deciding whether a collection of subsets form a cover or not. Definition 8.2.3. A Sheaf F over a site C is a functor p:F → C such that • For all S ∈ Ob(C), x ∈ Ob(FS ) and f : T → S ∈ Mor(C) there exists a unique φ : y → x ∈ Mor(F) such that p(φ) = f • For every cover {Sα → S}α , the following sequence is exact: FS →
FSα ⇒
FSα ×S Sβ .
Definition 8.2.4. A Stack in groupoids over C is a functor p : S → C such that • S is fibered in groupoids over C • For any U ∈ Ob(C) and x, y ∈ Ob(SU ), the functor U → Sets φ : V → U → H om(φ ∗ x, φ ∗ y) is a sheaf (Ob(U) = {(S, χ )|S ∈ Ob(C), χ ∈ H om(S, U )})
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• If φi : Vi → U is a covering family in C, any descent datum relative to the φi ’s, for objects in S, is effective. Example 8.2.5. For X a G-set (provided with a G action over it) let C = T op, the category of topological spaces, and S = [X/G] the category defined as follows: Ob([X/G])S = {f : ES → X}, the set of all G-equivariant maps from principal G-bundles ES over S ∈ Ob(T op), and Mor([X/G]) ⊆ H omBG (ES , ES ) given by ES o (proj,f )
S×X o
With the functor
/ S × S ES 1×f
/ S×X
p : [X/G] → T op, (f : ES → X) → S.
By definition [X/G] is a category fibered by groupoids, and if the group G is finite [X/G] is a stack.
8.3. Gerbes as stacks. For simplicity we will start with a smooth X and we will consider a Grothendieck topology on X induced by the ordinary topology on X as in 8.2.2. We will follow Brylinski [5] very closely. The following definition is essentially due to Giraud [13]. Definition 8.3.1. A gerbe over X is a sheaf of categories C on X so that • The category CU is a groupoid for every open U . • Any two objects Q and Q of CU are locally isomorphic, namely for every x ∈ X there is a neighborhood of x where they are isomorphic. • Every point x ∈ X has a neighborhood U so that CU is non-empty. We will require our gerbes to have as band the sheaf A = C∗ over X. This means that for every open U ⊂ X and for every object Q ∈ CU there is an isomorphism of sheaves α : Aut(Q) → AU , compatible with restrictions and commuting with morphisms of C. Here Aut(Q)|V is the group of automorphisms of P |V . The relation of this definition to the one we have used is given by the following Proposition 8.3.2. To have a gerbe in terms of data (Lαβ ) as in 3.1 is the same thing as to have a gerbe with band C∗ as a sheaf of categories. Proof. Starting with the data in 3.1 we will construct the category CU for a small open set U . Since U is small we can trivialize the gerbe L|U . The objects of CU are the set of all possible trivializations (fαβ ) with the obvious morphisms.
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Conversely suppose that you are given a gerbe as a sheaf of categories. Then we construct a cocycle cαβγ as in [5, Prop. 5.2.8]. We take an object Q ∈ CUα and an auto∗ morphism uαβ : Qα |Uαβ → Qβ |Uαβ and define hαβγ = u−1 αγ uαβ uβγ ∈ Aut(Pγ ) = C ˇ producing a Cech cocycle giving us the necessary data to construct a gerbe as in 3.1. 8.4. Orbifolds as stacks. Now we can define the stack associated to an orbifold. Let X be an orbifold with {(Vp , Gp , πp )}p∈X as orbifold structure. Let C be the category of all open subsets of X with the inclusions as morphisms and for U ⊂ X, let SU be the category of all uniformizing systems of U such that they are equivalent for every q ∈ U to the orbifold structure, in other words SU = (W, H, τ )|∀q ∈ U, (Vq , Gq , πq )&(W, H, τ ) are equivalent at q . By Lemma 2.1.3 and the definition of orbifold structure it is clear that the category S is fibered by groupoids. It is known, and this requires more work, that this system S → C is also a stack, a Deligne-Mumford stack in the smooth case. 8.5. Stacks as groupoids. The following theorem has not been used in this paper, but it is the underlying motivation for the approach that we have followed. Theorem 8.5.1 [3]. Every Deligne-Mumford stack comes from an e´ tale groupoid scheme. Moreover, there is a functor R ⇒ U → [R ⇒ U] from the category of groupoids to the category of stacks inducing this realization. When s, t are smooth we can realize in a similar manner Artin stacks [3]. Acknowledgements. We would like to use this opportunity to thank Alejandro Adem, Marius Crainic, Bill Dwyer, Paulo Lima-Filho, Bill Fulton, Mike Hopkins, Shengda Hu, Haynes Miller, Ieke Moerdijk, Mainak Poddar, Joel Robbin and Yong-Bin Ruan for very helpful discussions regarding this work that grew out of seminars at Michigan by Fulton, and at Wisconsin by Adem and Ruan. We would also thank Jouko Mickelsson, Michael Murray, Eric Sharpe, Zoran Skoda and very specially Ieke Moerdijk and Alan Weinstein for letters regarding the preliminary version of this paper.
References 1. Adem, A., Ruan, Y.: Twisted orbifold K-theory. Commun. Math. Phys. 237(3), 533–556 (2003) MR 1 993 337 2. Atiyah, M.: K-Theory Past and Present. arXiv:math.KT/0012213 3. Behrend, K., Edidin, D., Fantechi, B., Fulton, W., G¨ottsche, L., Kresch, A.: Introduction to Stacks. In preparation 4. Bouwknegt, P., Carey, A.L., Mathai, V., Murray, M.K., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228(1), 17–45 (2002) MR 2003g:58049 5. Brylinski, J.-L.: Loop spaces characteristic classes and geometric quantization. Progress in Mathematics 107. Boston, MA: Birkh¨auser Boston, Inc., 1993 6. Carey, A.L., Mickelsson, J.: The universal gerbe Dixmier-Douady class and gauge theory. Lett. Math. Phys. 59(1), 47–60 (2002) MR 2003e:58048 7. Chen, W.: A Homotopy Theory of Orbispaces. (2001) arXivmath.AT/0102020 8. Chen, W., Ruan, Y.: Orbifold Quantum Cohomology. (2000) arXiv:math.AG/0005198 9. Cohen, R.L., Jones, J.D.S., Segal, G.B.: Floer’s infinite-dimensional Morse theory and homotopytheory. In: The Floer Memorial Volume, Basel: Birkh¨auser, 1995, pp. 297–325
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10. Crainic, M., Moerdijk, I.: A homology theory for´etale groupoids. J. Reine Angew. Math. 521, 25–46 (2000) 11. Donovan, P., Karoubi, M.: Graded Brauer Groups and K-theory. IHES 38 (1970) 12. Freed, D.: The Verlinde algebra is twisted equivariant K-theory. arXiv:math.RT/0101038 13. Giraud, J.: Cohomologie non Ab´eliene. Berlin-Heidelberg-New York: Springer-Verlag, 1971 14. Grothendieck, A.: Dix expos´es sur la cohomologie des sch´emas. Amsterdam: North Holland, 1968 15. Hitchin, N.: Lectures Notes on Special Lagrangian Submanifolds. (1999) arXiv:math.DG/9907034 16. Lupercio, E., Uribe, B.: Holonomy for grebes over orbifolds. arXiv:math.AT/0307114 17. Lupercio, E., Uribe, B.: Deligne Cohomology for Orbifolds Discrete Torsion and B-fields. In: Geometric and Topological methods for Quantum Field Theory, Singapore: World Scientific, 2002 18. Lupercio, E., Uribe, B.: Loop groupoids, gerbes, and twisted sectors on orbifolds. In: Orbifolds in mathematics and physics (Madison WI 2001), Contemp. Math. Vol. 310, Providence RI: Am. Math. Soc., 2002, 163–184 MR 1 950 946 19. May, P.: Simplicial Objects in Algebraic Topology. Van Nostrand Mathematical Studies 11, Princeton N.J.: D. Van Nostrand Co. Inc., 1967 20. Moerdijk, I.: Calssifying topos and foliations. Ann. Inst. Fourier 41, 189–209 (1991) 21. Moerdijk, I.: Proof of a conjecture of A. Haefiger. Topology 37(4), 735–741 (1998) 22. Moerdijk, I.: Orbifolds as groupoids: an introduction. In: Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., Vol. 310, Providence, RI: Am. Math. Soc. (2002), pp. 205– 222 MR 1 950 948 23. Moerdijk, I., Pronk, D.A.: Orbifolds sheaves and groupoids. K-Theory 12(1), 3–21 (1997) 24. Moerdijk, I., Pronk, D.A.: Simplicial cohomology of orbifolds. Indag. Math. (N.S.) 10(2), 403–416 (1999) 25. Murray, M.K.: Bundle Gerbes. J. Lond. Math. Soc. 54(2), 403–416 (1996) 26. Murray, M.K., Stevenson, D.: Bundle Gerbes. Stable isomorphism and local theory. J. Lond. Math. Soc. 62(2), 925–937 (2000) 27. Pressley, A., Segal, G.: Loop groups. New York: The Clarendon Press Oxford University Press, Oxford Science Publications, 1986 28. Rosenberg, J.: Continuous trace C ∗ -algebras from the bundle theoretic point of view. J. Aust. Math. Soc. A47, 368 (1989) 29. Ruan, Y.: Stringy geometry and topology of orbifolds. In: Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), Contemp. Math. Vol. 312, Providence RI: Am. Math. Soc. 2002, pp. 187–233 MR 1 941 583 30. Satake, I.: The Gauss -Bonet theorem for V -manifolds. J. Math. Soc. Japan 9, 464–492 (1957) 31. Segal, G.: Classifying spaces and spectral sequences. Inst. Hautes Etudes Sci. Publ. Math. 34, 105– 112 (1968) 32. Segal, G.: Fredholm complexes. Quart. J. Math. Oxford Ser. 21(2), 385–402 (1970) 33. Segal, G.: Categories and cohomology theories. Topology 13, 292–312 (1974) 34. Thurston, W.: Three-dimensional geometry and topology. Vol. 1. Princeton Mathematical Series 35. Princeton NJ: Princeton University Press, 1997 35. Witten, E.: D-branes and K-theory. J. High Energy Phys. 12(19), pp. 41 (1998) (electronic). MR 2000e:81151 Communicated by A. Connes
Commun. Math. Phys. 245, 491–517 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1015-1
Communications in
Mathematical Physics
Tropical R and Tau Functions A. Kuniba1 , M. Okado2 , T. Takagi3 , Y. Yamada4 1
Institute of Physics, University of Tokyo, Tokyo 153-8902, Japan. E-mail: [email protected] 2 Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan. E-mail: [email protected] 3 Department of Applied Physics, National Defense Academy, Kanagawa 239-8686, Japan. E-mail: [email protected] 4 Department of Mathematics, Faculty of Science, Kobe University, Hyogo 657-8501, Japan. E-mail: [email protected] Received: 28 March 2003 / Accepted: 17 July 2003 Published online: 19 December 2003 – © Springer-Verlag 2003
Abstract: Tropical R is the birational map that intertwines products of geometric crys(1) tals and satisfies the Yang-Baxter equation. We show that the Dn tropical R introduced (2) (1) by the authors and its reduction to A2n−1 and Cn are equivalent to a system of bilinear difference equations of Hirota type. Associated tropical vertex models admit solutions in terms of tau functions of the BKP and DKP hierarchies. 1. Introduction (1)
Let B = {x = (x1 , . . . , xn )} be the set of variables. The tropical R for An−1 [HHIKTT, Ki, Y] is the birational map R : B × B → B × B specified by R(x, y) = (x , y ), Pi (x, y) =
n n k=1 j =k
xi = yi k
xi+j
Pi (x, y) , Pi−1 (x, y)
yi = xi
Pi−1 (x, y) , Pi (x, y) (1.1)
yi+j ,
j =1
where all the indices are considered to be in Z/nZ. It satisfies the inversion relation R 2 = id on B × B and the Yang-Baxter equation R1 R2 R1 = R2 R1 R2 on B × B × B, where R1 (x, y, z) = (R(x, y), z) and R2 (x, y, z) = (x, R(y, z)). Given (x, y), it is characterized as the unique solution to the equations on (x , y ): 1 1 1 1 + = + (1.2) xi yi+1 xi yi+1 with an extra constraint ni=1 (xi /yi ) = ni=1 (yi /xi ) = 1. A representation theoretical background for the tropical R is provided by the geometric crystals [BK] and their natural extrapolation into the affine setting as demonstrated xi yi = xi yi ,
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in Sect. 1 in [KOTY].To recall it consider the matrix realization, where each element x ∈ B is associated with the matrix (see also [KNY])
−1 −ζ x1−1 −1 x −1 2 M(x, ζ ) = .. −1 . −1 −1 xn (1)
involving the spectral parameter ζ . Then the structure of the An−1 geometric crystal on B is realized as simple matrix operations. For instance the geometric Kashiwara operator eic : (x1 , . . . , xn ) → (. . . , xi−1 , cxi , c−1 xi+1 , xi+2 , . . . ) is induced by a multiplication of M with certain unipotent matrices. The product of matrices M(x (1) , ζ ) · · · M(x (j ) , ζ ) corresponds to the product of geometric crystals (x (1) , . . . , x (j ) ) ∈ B × · · · × B in the sense of [BK]. Equation.(1.2) is equivalent to the matrix equation M(x, ζ )M(y, ζ ) = M(x , ζ )M(y , ζ ).
(1.3)
Due to the presence of the spectral parameter ζ , its non-trivial solution is unique, which characterizes the tropical R as the intertwiner of the products of geometric crystals. Geometric crystals are so designed that the relevant rational functions become totally positive [BK, BFZ, L]. There is no minus sign in Pi (x, y) in (1.1) indeed. Such functions can consistently be transformed into piecewise linear ones by replacing +, × and / by max(min), + and −, respectively. Consequently, the structure of geometric crystals reduces to the one for crystal bases [K]. In the present case, the expression (1.1) leads to the piecewise linear formula [HHIKTT] for the combinatorial R [NY]. An analogous (1) result is available in [KOTY] for the Dn combinatorial R [HKOT]. The above transformation can be realized as a certain limiting procedure [TTMS] and is often called the ultradiscretization (UD). As related topics, we point out the tropical combinatorics [Ki, NoY] and the soliton cellular automata associated with crystal bases [TS, HKT1, FOY, HHIKTT, HKOTY, TNS, HKT2]. Now let us turn to the aspect of the tropical R as a classical integrable system, which is the main subject of this paper. Recall the discrete time Toda equation [HT, HTI] on the electric “current" Iit and “voltage" Vit : t Iit+1 Vit+1 = Ii+1 Vit ,
t+1 Iit+1 + Vi−1 = Iit + Vit
t , V t = V t . This difference equation is known to be intewith periodicity Iit = Ii+n i i+n t , y −1 = V t , x −1 = V t+1 and grable. Equation.(1.2) is identified with it via xi−1 = Ii+1 i i i i t+1 = I [Y]. The matrix equation (1.3) is a Lax representation in this context. In fact y −1 i i the tropical R is equivalent to a system of bilinear difference equations of Hirota type, which was effectively the base of the analyses in [HHIKTT]. To see this, introduce the functions τiJ (1 ≤ J ≤ 4, i ∈ Z/nZ) and the parameters λi , κi , and make the change of variables
xi−1 = λi δτi3 /δτi2 , −1
xi
= κi δτi3 /δτi4 ,
yi−1 = κi δτi2 /δτi1 , −1
yi
= λi δτi4 /δτi1
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J . Then the former relation in (1.2) is automatically satisfied and the with δτiJ = τiJ /τi−1 latter is translated into 2 4 3 λi τi−1 τi4 − κi τi2 τi−1 = ατi1 τi−1
(1.4)
for any nonzero parameter α independent of i. The birational map R : (x, y) → (x , y ) is induced by an automorphism τi2 ↔ τi4 , λi ↔ κi , α → −α of (1.4). Equation. (1.4) is a version of the so-called Hirota-Miwa equation [H, M], a prototype discrete soliton equation to which the well-developed machinery of free fermions and infinite dimensional Lie algebras [DJM, JM] can be applied. The solutions are provided by tau functions of the KP hierarchy with a certain reduction and discretization of time variables. (1) In this paper we elucidate the classical integrability of such type for the Dn tropical (2) (1) R introduced in [KOTY] together with its reductions to A2n−1 and Cn . We show that they are equivalent to a quartet of A type bilinear equations like (1.4), and construct (1) (2) solutions in terms of tau functions in soliton theory [JM]. The cases Dn and A2n−1 are .A related to reductions of the DKP hierarchy associated to the algebras D∞ and D∞ key is the relation between two kinds of tau functions that originate in the isomorphism . See Lemmas B.1 and B.2. The C (1) case corresponds to a further reduction D∞ D∞ n . to the BKP hierarchy associated with the algebras B∞ B∞ We shall formulate tropical vertex models, where the tropical R plays a role of local time evolution. Namely, it is the two dimensional system on a square lattice where each edge is assigned an element of a geometric crystal in such a way that those four surrounding a vertex are related by the tropical R. Our solutions to the bilinear equations are naturally extended to the tropical vertex models by duplicating the original lattice and attaching a Clifford group element to each face of it under a certain rule. The YangBaxter equation for the tropical R is naturally understood from such a point of view. See Remark 5.6. We expect that the results in this paper are glimpses of deeper relations yet to be explored between geometric crystals, crystal bases, solvable lattice models, discrete and ultradiscrete solitons and so forth. A rough picture at present looks as follows.
Uq -modules Quantum R Vertex models
q →0
Crystals Combinatorial R Soliton cellular automata
Geometric crystals
UD
Tropical R Soliton equations
In [D], Drinfeld proposed set-theoretical solutions of theYang-Baxter equation as one of the unsolved problems in quantum group theory. Since then a number of approaches have been made. See for example [ABS, ESS, JMY, MV, O, V1, V2, WX] and references therein. Our tropical R is a distinguishable example which is placed in the unique spot in the above picture and enjoys total positivity. It should be noted that the combinatorial R is also a remarkable example of the set-theoretical solution which intertwines products of finite sets. (1) The paper is organized as follows. In Sect. 2 we recall the tropical R for Dn following [KOTY]. In Sect. 3 the bilinear equations are introduced, which are divided into four groups corresponding to each vertex in Fig. 1. Uniqueness and existence of the solution are shown in Proposition 3.5. In Sect. 4 we establish the bilinearization of the tropical
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R in Theorem 4.4. In Sect. 5 we construct solutions of the bilinear equations in terms of tau functions of the DKP hierarchy formulated with two component fermions. The final (2) (1) result is given in Theorem 5.5. In Sect. 6 we introduce the tropical R for A2n−1 and Cn (1)
as reductions of the Dn case. Parallel results on the bilinearization and solutions are obtained. Appendix A provides a proof of Lemma 4.3. Appendix B contains elements of the free fermion approach [JM] as well as the key Lemmas B.1 and B.2. We leave (1) the calculation of the ultradiscrete limit of tau functions as in [HHIKTT] for An−1 as a future problem. (1)
2. Tropical R for Dn
(1)
Let us recall the tropical R for Dn from [KOTY]. Let x = (x1 , . . . , xn , x n−1 , . . . , x 1 ) (1) be a set of variables. The geometric Dn -crystal is the set B = {x} equipped with additional structures such as ϕi , εi and the geometric Kashiwara operators eic . The product B × · · · × B again becomes a geometric crystal, which is an analogue of the tensor product of crystals. The tropical R is a birational map B × B → B × B commuting with the geometric Kashiwara operators. Leaving the precise definition to [KOTY], we concentrate here on its explicit form. Set (x) = x1 x2 · · · xn x n−1 · · · x 2 x 1 ,
(2.1)
and call it the level of x. On B we introduce the involutions σ1 and σn which preserve the level. Explicitly for x = (x1 , . . . , xn , x n−1 , . . . , x 1 ) ∈ B, they are defined by σ1 : x1 ←→ x 1 , σn : xn−1 → xn−1 xn ,
x n−1 → x n−1 xn ,
xn → 1/xn ,
(2.2) (2.3)
where the variables not appearing in the above are unchanged. On B × B we introduce σ1 , σn analogously as σa (x, y) = (σa (x), σa (y))
a = 1, n.
(2.4)
Furthermore we introduce the involution σ∗ by σ∗ : xi ←→ y i ,
x i ←→ yi
(1 ≤ i ≤ n − 1),
xn ←→ yn
(2.5)
for the elements x = (x1 , . . . , x 1 ) and y = (y1 , . . . , y 1 ) of B. Note that σ1 , σn and σ∗ are all commutative, among which σ∗ is the only one that mixes x and y and interchanges the levels. Given a function F = F (x, y), we will write F σa = F (σa (x, y)) for a = 1, n, ∗. We set
n−1 m−1 n−1 yi ym y1 V0 = (x) + (x) 1+ + x n yn x i yi y1 x ym m=2 i=1 i i=1
n−1 n−1 m−1 xi x1 xm 1+ + + (y) + (y) x i yi , (2.6) x1 yi xm m=2
i=1
i=1
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and define Vi (1 ≤ i ≤ n − 1) by the recursion relations:
1 1 1 Vi−1 + ( (x) − (y)) + , (1 ≤ i ≤ n − 2), Vi = y i xi xi yi
y y yn 1 n Vn−1 = n−1 Vn−2 + ( (x) − (y)) + . yn x n−1 xn−1 y n−1
(2.7)
In terms of Vi we also define Ui (1 ≤ i ≤ n − 1) by σ∗ , U1 = V0 V0σ1 , Un−1 = Vn−1 Vn−1
−1
1 1 1 1 σ∗ σ∗ + Vi Vi−1 + Vi−1 Vi Ui = xi yi yi xi
(2.8) (2 ≤ i ≤ n − 2).
The tropical R is a birational map R(x, y) = (x , y ) on B × B given by V0σ1 V0 , x 1 = y 1 , V1 V1 V U Vi−1 i−1 i xi = yi , x i = y i , Vi Ui−1 Vi Vn−1 xn = yn σn , Vn−1 x1 = y1
y1
V0 = x1 σ∗ , V1
yi = xi yn = xn
σ∗ Vi−1
Viσ∗
y 1
,
σn Vn−1
Vn−1
(2 ≤ i ≤ n − 1),
(2.9)
V0σ1 = x 1 σ∗ , V1
y i = x i
σ∗ Vi−1 Ui
Viσ∗ Ui−1
,
(2 ≤ i ≤ n − 1),
.
In [KOTY], Ui here was denoted by Wi in Definition 4.9, and their transformation property under σa was summarized in Table 1. As shown therein, the tropical R is subtraction-free (totally positive), interchanges the levels, commutes with the geometric Kashiwara operators, and satisfies the inversion relation R(R(x, y)) = (x, y) and the Yang-Baxter relation. Remark 2.1. Let (x , y ) = R(x, y) under the Dn tropical R. It is easily seen that xn = yn = 1 is equivalent to xn = yn = 1. Similarly, (x1 , y1 ) = (x 1 , y 1 ) is equivalent to (x1 , y1 ) = (x 1 , y 1 ). (1)
3. Bilinear Equations Here we introduce a system of bilinear equations and study its properties such as existence and uniqueness of the solution. It is a preparation for Sect. 4, where the equations are related to our tropical R. In the rest of the paper we fix the elements λ = (λ1 , . . . , λ1 ) and κ = (κ1 , . . . , κ 1 ) of B, and set l = (λ),
k = (κ).
(3.1)
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We assume lk(l − k) = 0 throughout. We introduce the variables which we call the tau functions: Si , Wi , Ni , Ei
(1 ≤ i ≤ n − 2),
τi0 , τi1 , τi2 , τi3 , τi4
(0 ≤ i ≤ n).
They will be called generic if they are nonzero. Let α and β be arbitrary nonzero parameters. Our bilinear equations read as follows: κ 1 N1 τ14 − λ1 E1 τ12 = ατ00 τ01 , κ1 N1 τ04 − λ1 E1 τ02 = ατ10 τ11 , κi Ei−1 Ni − λi Ni−1 Ei = ατi0 τi1 κn−1 κn En−2 τn2 − λn−1 λn Nn−2 τn4 2 4 κn−1 En−2 τn−1 − λn−1 Nn−2 τn−1
=
(2 ≤ i ≤ n − 2),
0 ατn−1 τn1 , 1 ατn0 τn−1 .
=
The i th equation here will be referred to as 1, i (0 ≤ i ≤ n): κ1 N1 τ13 + λ1 W1 τ11 = βτ00 τ02 , κ 1 N1 τ03 + λ1 W1 τ01 = βτ10 τ12 , κ i Wi−1 Ni + λi Ni−1 Wi = βτi0 τi2 1 3 κn κ n−1 Wn−2 τn−1 + λn−1 Nn−2 τn−1 κ n−1 Wn−2 τn1 + λn−1 λn Nn−2 τn3
= =
(2 ≤ i ≤ n − 2),
0 2 βτn−1 τn−1 , 0 2 βτn τn .
The i th equation here will be referred to as 2, i (0 ≤ i ≤ n): κ1 S1 τ12 − λ1 W1 τ14 = ατ00 τ03 , κ 1 S1 τ02 − λ1 W1 τ04 = ατ10 τ13 , κ i Wi−1 Si − λi Si−1 Wi = ατi0 τi3 κ n−1 Wn−2 τn4 − λn−1 Sn−2 τn2 4 2 κn κ n−1 Wn−2 τn−1 − λn λn−1 Sn−2 τn−1
= =
(2 ≤ i ≤ n − 2),
0 ατn−1 τn3 , 3 ατn0 τn−1 .
The i th equation here will be referred to as 3, i (0 ≤ i ≤ n): κ 1 S1 τ11 + λ1 E1 τ13 = βτ00 τ04 , κ1 S1 τ01 + λ1 E1 τ03 = βτ10 τ14 , κi Ei−1 Si + λi Si−1 Ei = βτi0 τi4 1 + λn λn−1 Sn−2 τn−1 κn−1 κn En−2 τn3 + λn−1 Sn−2 τn1
3 κn−1 En−2 τn−1
= =
(2 ≤ i ≤ n − 2),
0 4 βτn−1 τn−1 , 0 4 βτn τn .
The i th equation here will be referred to as 4, i (0 ≤ i ≤ n): (1) We call { J, i | 1 ≤ J ≤ 4, 0 ≤ i ≤ n} the Dn bilinear equations. Lemma 3.1. For generic input N1 , . . . , Nn−2 , W1 , . . . , Wn−2 and τ0J , . . . , τnJ with J = 1, 2, 3, there exists a unique solution to the equations J, i (1 ≤ J ≤ 4, 1 ≤ i ≤ 0 n − 1) on the variables S1 , . . . , Sn−2 , E1 , . . . , En−2 , τ10 , . . . , τn−1 and τ04 , . . . , τn4 .
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0 . Regard 1, i and 3, i with Proof. The equations 2, 1 − 2, n − 1 fix τ10 , . . . , τn−1 1 ≤ i ≤ n − 1 as linear equations on the unknowns S1 , . . . , Sn−2 , E1 , . . . , En−2 , τ04 and τn4 . Arrange them in a matrix form Ax = αb, where the column vectors x and b are specified by
x = t (τ04 , E1 , . . . , En−2 , τn4 , Sn−2 , . . . , S1 ), 0 1 0 0 0 3 b = t (τ10 τ11 , . . . , τn−2 τn−2 , τn−1 τn1 , τn−1 τn3 , τn−2 τn−2 , . . . , τ10 τ13 ).
Then one finds that the 2n − 2 by 2n − 2 coefficient matrix A is an upper triangular Jacobi matrix except the bottom left element −λ1 W1 . Hence it is easy to see det A = (k − l)τ02 τn2 n−2 i=1 (Ni Wi ), which is nonzero for generic input. Therefore the unknowns 0 in x are uniquely determined. With τ10 , . . . , τn−1 and x at hand, one can determine the 4 4 remaining ones τ1 , . . . , τn−1 from 4, 1 − 4, n − 1 . According to Lemma 3.1, the bilinear equations J, 0 and J, n are not needed to determine the tau functions that will appear in the parameterization (4.2)-(4.3). However, they are essential in order to incorporate the involutions σa in the previous section into tau functions. The rest of this section concerns this point, leading to Propositions 3.5, 3.6, and ultimately to Proposition 4.2. Lemma 3.2. Suppose that Si , Wi , Ni , Ei (1 ≤ i ≤ n − 2) and τiJ (0 ≤ J ≤ 4, 0 ≤ i ≤ n) satisfy J, i (1 ≤ J ≤ 4, 1 ≤ i ≤ n − 1). Then we have αλ1 κ 1 τ03 τ11 + βλ1 κ 1 τ02 τ14 − αλ1 κ1 τ01 τ13 − βλ1 κ1 τ04 τ12 = 0, 3 ατn−1 τn1
2 + βλn τn4 τn−1
1 − αλn κn τn−1 τn3
4 − βκn τn2 τn−1
= 0.
(3.2) (3.3)
Proof. Write 1, 1 − 4, 1 in the matrix form:
1 κ1 τ04 −λ1 τ02 0 0 ατ1 N1 4 0 λ1 τ 3 κ1 τ 1 E1 0 βτ 0 0 = τ0 1 . 3 2 4 1 0 ατ 0 κ 1 τ0 −λ1 τ0 S1 1 W1 βτ12 κ 1 τ03 0 0 λ1 τ01 The matrix on the left-hand side is annihilated by multiplying the row vector (λ1 κ 1 τ03 , λ1 κ 1 τ02 , −λ1 κ1 τ01 , −λ1 κ1 τ04 ) from the left. So the same should happen on the right hand side, proving (3.2). The relation (3.3) is verified similarly by using 1, n − 1 − 4, n − 1 . Lemma 3.3. Suppose that Si , Wi , Ni , Ei (1 ≤ i ≤ n − 2), τiJ (1 ≤ J ≤ 4, 0 ≤ i ≤ n) and τi0 (1 ≤ i ≤ n − 1) satisfy the equations J, i (1 ≤ J ≤ 4, 1 ≤ i ≤ n − 1). Then there exists a unique solution to the equations J, 0 (1 ≤ J ≤ 4) on τ00 . Proof. It suffices to check compatibility of the four equations J, 0 (1 ≤ J ≤ 4), that is, they all lead to the same τ00 . Multiply 1, 0 by τ03 , and 3, 0 by τ01 . The difference of the resulting left-hand sides can be grouped as τ14 (κ 1 N1 τ03 + λ1 W1 τ01 ) − τ12 (λ1 E1 τ03 + κ1 S1 τ01 ).
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Due to 2, 1 and 4, 1 this vanishes, proving the compatibility of 1, 0 and 3, 0 . The compatibility of 2, 0 and 4, 0 is similarly shown by using 1, 1 and 3, 1 . Let us check the compatibility of 2, 0 and 3, 0 . We are to show ατ03 (κ1 N1 τ13 + λ1 W1 τ11 ) − βτ02 (κ1 S1 τ12 − λ1 W1 τ14 ) = 0. Upon multiplying κ 1 , the first term contains the factor κ 1 N1 τ03 and the third does κ 1 S1 τ02 . Eliminate them by using 2, 1 and 3, 1 , respectively. After a cancellation the resulting expression becomes W1 times the left-hand side of (3.2). Lemma 3.4. Suppose that Si , Wi , Ni , Ei (1 ≤ i ≤ n − 2), τiJ (1 ≤ J ≤ 4, 0 ≤ i ≤ n) and τi0 (1 ≤ i ≤ n − 1) satisfy the equations J, i (1 ≤ J ≤ 4, 1 ≤ i ≤ n − 1). Then there exists a unique solution to the equations J, n (1 ≤ J ≤ 4) on τn0 . Proof. This is verified similarly to Lemma 3.3 by using (3.3).
Since the compatibility of the two systems 1, 0 − 4, 0 and 1, n − 4, n is trivial, Lemmas 3.1, 3.3 and 3.4 lead to Proposition 3.5 (Unique existence). For generic input N1 , . . . , Nn−2 , W1 , . . . , Wn−2 and τ0J , . . . , τnJ with J = 1, 2, 3, there exists a unique solution to the bilinear equations J, i (1 ≤ J ≤ 4, 0 ≤ i ≤ n) on the remaining variables S1 , . . . , Sn−2 , E1 , . . . , En−2 and τ0J , . . . , τnJ (J = 0, 4). Let us proceed to the automorphism of the bilinear equations. For brevity the array T = (λ, κ; Si , Wi , Ni , Ei , τiJ ) consisting of all the tau functions together with (λ, κ) ∈ B × B will be called data. We extend the involutions σa (a = 1, n, ∗) on B × B defined in the previous section to the data as follows: σa (a = 1, n, ∗) acts on (λ, κ) ∈ B × B by (2.2)-(2.5) as before, σ1 : σn : σ∗ :
τ0J ←→ τ1J (0 ≤ J ≤ 4), J τn−1 ←→ τnJ (0 ≤ J ≤ 4), 0 2 4 τn−1 ←→ τn0 , τn−1 ←→ τn2 , τn−1 ←→ τn4 , 1 τi1 ←→ τi3 (0 ≤ i ≤ n − 2), τn−1 ←→ τn3 , τn1 Wi ←→ Ni ,
(3.4) (3.5) (3.6)
3 ←→ τn−1 ,
(3.7)
Si ←→ Ei (1 ≤ i ≤ n − 2).
In each involution, the tau functions not appearing in the above are unchanged. See also Proposition 4.2. It is immediate to check Proposition 3.6 (Invariance). The involutions σa (a = 1, n, ∗) defined in (3.4)-(3.7) on the data T are automorphism of the bilinear equations J, i (1 ≤ J ≤ 4, 0 ≤ i ≤ n). In particular, the composition σn σ∗ interchanges 1, i and 3, i leaving 2, i and 4, i invariant.
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4. Bilinearization of Tropical R Given µ = (µ1 , . . . , µ1 ) ∈ B and the arrays C = (C1 , . . . , Cn−2 ),
τ = (τ0 , . . . , τn ),
τ = (τ0 , . . . , τn ),
we define [µ; τ, C, τ ] = (z1 , . . . , zn , zn−1 , . . . , z1 ) ∈ B, τ0 τ1 −1 τ1 τ0 z1 = µ−1 , z = µ , 1 1 1 C1 C1 C1 τ2 C1 τ2 , z2 = µ−1 , z2 = µ−1 2 2 C 2 τ0 τ1 C2 τ0 τ1 Ci−1 τi Ci−1 τi zi = µ−1 , zi = µ−1 (3 ≤ i ≤ n − 2), i i Ci τi−1 Ci τi−1 zn−1 = µ−1 n−1 zn =
Cn−2 τn−1
τn−1 τn−2 τn−1 τn µ−1 n τn τn−1
,
zn−1 = µ−1 n−1
(4.1)
Cn−2 τn , τn−2 τn
in case n ≥ 4. For n = 3 we slightly modify it into τ0 τ1 τ1 τ0 , z1 = µ−1 , 1 C1 C1 C1 τ2 C1 τ3 , z2 = µ−1 , z2 = µ−1 2 2 τ0 τ1 τ3 τ2 τ0 τ1 τ2 τ3 . z3 = µ−1 3 τ3 τ2 z1 = µ−1 1
Note that ([µ; τ, C, τ ]) = (µ)−1 . From the data T = (λ, κ; Si , Wi , Ni , Ei , τiJ ), we construct x, y, x , y ∈ B as x = [λ; τ 3 , W, τ 2 ], y = [κ; τ 2 , N, τ 1 ], x = [κ; τ 3 , S, τ 4 ], y = [λ; τ 4 , E, τ 1 ].
(4.2) (4.3)
Note from (3.1) that (x) = (y ) = l −1 ,
(x ) = (y) = k −1 .
(4.4)
Given any pairs (x, y) and (λ, κ) such that (x) = (λ)−1 and (y) = (κ)−1 , the parameterization (4.2) is always possible due to Proposition 4.1. For any z, µ ∈ B such that (z) = (µ)−1 , the equation [µ; τ, C, τ ] = z with fixed τ , (resp. τ ) on the variables (C, τ ) (resp. (C, τ )) admits a one parameter family of solutions.
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x
τ2
N
τ1
W
τ0
E
τ3
S
τ4
y
x Fig. 1. Tau functions
It is customary to attach a vertex diagram (cross) to the relation like R(x, y) = (x , y ) for a quantum or combinatorial R, where the four edges correspond to x, y, x , y . Figure 1 is an analogue of such a diagram, which may be of help to recognize the pattern (4.2)-(4.3). Elements of B are represented there with double lines. Consequently the vertex diagram is separated into several domains where various tau functions live. It is readily seen that the transformation T → σa (T ) of the data induces the change (x, y) → σa (x, y) and (x , y ) → σa (x , y ). More generally we have Proposition 4.2. Let F = F (x, y) be any function on B × B and T = (λ, κ; Si , Wi , Ni , Ei , τiJ ) be any data obeying all the bilinear equations. Denote by F |T an expression of F resulting from the substitution of (4.2) followed by any possible application of the bilinear equations. Then one has F σa |T = σa (F |T ) (a = 1, n, ∗), where σa on the left-hand side is specified by (2.2)-(2.5), whereas on the right hand side by (3.4)-(3.7). The same fact is valid also for (4.3) and F = F (x , y ).
Proof. This is due to Propositions 3.5 and 3.6.
Lemma 4.3. Suppose that the data (λ, κ; Si , Wi , Ni , Ei , τiJ ) satisfies all the bilinear equations. Then under the substitution (4.2), the functions Ui , Vi defined in (2.6)-(2.8) are expressed as follows: V0 = V0σ∗ = V1 = V1σ1 = Vi = Viσ1 =
γ τ02 τ04 τ01 τ03
, V0σ1 =
γ S1 τ02 τ12 N1 τ03 τ13 γ Si τi2
σ1 Vn−1 = Vn−1 =
Ni τi3
γ τ12 τ14 τ11 τ13
, V1σ∗ =
, Viσ∗ =
, U1 =
γ E1 τ02 τ12 W1 τ01 τ11
γ Ei τi2 Wi τi1
γ 2 τ02 τ12 τ04 τ14 τ01 τ11 τ03 τ13
,
,
, Ui =
γ 2 τi2 τi4 τi1 τi3
(2 ≤ i ≤ n − 2),
2 τ4 2 τ 2τ 4 τ 4 γ τn−1 γ 2 τn−1 γ τn2 τn4 n n−1 n σn σ∗ n−1 , V = V = , U = , n−1 n−1 n−1 1 τ3 1 τ 1τ 3 τ 3 τn1 τn3 τn−1 τ n−1 n n−1 n n−1
where γ = (k − l)β/(lkα).
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The proof is available in Appendix A. The main result of this section is Theorem 4.4 (Bilinearization of tropical R). Suppose that the data (λ, κ; Si , Wi , Ni , Ei , τiJ ) satisfies all the bilinear equations. Then x, y, x , y specified in (4.2) and (4.3) obey the relation R(x, y) = (x , y ). Proof. Substitute (4.2) into the right-hand sides of (2.9) and apply Lemma 4.3. Then the result agrees with (4.3). Given (x, y), the tau functions τ 1 , τ 2 , τ 3 , N, W satisfying (4.2) are not unique. Theorem 4.4 guarantees that (x , y ) in (4.3) is independent of their choice under the bilinear equations. From Proposition 4.2 and the remark preceding it, we find that R(σa (x, y)) = σa (R(x, y)) in agreement with [KOTY]. We conclude that the tropical R acts on the data as the interchange λ ←→ κ, τi2 ←→ τi4 , Wi ←→ Si , Ni ←→ Ei for all i. Combined with α → −α, this is another automorphism of our bilinear equations that interchanges 2, i and 4, i leaving 1, i and 3, i invariant. This is a complementary transformation with the σn σ∗ mentioned after Proposition 3.6.
5. Solutions of Bilinear Equations There is a link between our bilinear equations and the soliton theory in terms of tau functions and infinite dimensional Lie algebras [JM]. We show in Sect. 5.1 that certain vacuum expectation values of the 2 component free fermions provide solutions to the D bilinear equations. They are parametrized with the elements of the algebras D∞ ∞ and their reduction. Then in Sect. 5.2 the solutions for the local equations are naturally extended to the tropical vertex model on the two dimensional square lattice where the tropical R plays the role of local time evolution at each vertex. As for notations and basic facts on the free fermion approach, see Appendix B.
5.1. Tau functions as vacuum expectation values. In this subsection, the letters x, y, etc. stand for the array of infinitely many time variables as in Appendix B. They should not be confused with elements in B, which have now been effectively replaced by the tau functions. Thus we refresh Fig. 1 into Fig. 2. Here we have introduced the parameters K, L, which will play a role of level in B. Each line is assigned with the array ε or ε˜ determined from these parameters according to (B.2). We assign the time variables x J on the 9 domains J = 0, 1, 2, 3, 4, N, S, W, E in Fig. 2 so that the following recursion relations are satisfied: xW − x3 = x0 − xS x2 − xW = xN − x0 xN − x2 = x0 − xW x1 − xN = xE − x0
= = = =
x E − x 4 = ε˜ (L−1 ), x 1 − x E = ε(L−1 ), x S − x 3 = ε˜ (K −1 ), x 4 − x S = ε(K −1 ).
(5.1)
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A. Kuniba, M. Okado, T. Takagi, Y. Yamada ε˜ (K −1 )
ε(K −1 )
τ2
N
τ1
W
τ0
E
τ3
S
τ4
ε(L−1 )
ε˜ (L−1 )
Fig. 2. Increment time arrays
These relations determine all the x J ’s consistently upon specifying any one of them as an initial condition. We fix x 1 to an arbitrary odd element in the sense of Appendix B.1. Then from the construction (5.1), all the time variables on the corner domains are odd, i.e., J = xJ x
J = 1, 2, 3, 4.
(5.2)
On the other hand the variables in the domains N, W, S, E satisfy N = x N + ε(K −1 ) − ε˜ (K −1 ), xS = x S + ε(K −1 ) − ε˜ (K −1 ), x W = x W + ε(L−1 ) − ε˜ (L−1 ), x E = x E + ε(L−1 ) − ε˜ (L−1 ). x
(5.3)
For i ≥ 1 define further J J J xiJ = x J + zi = (xi,1 , xi,2 , xi,3 , . . . ),
z1 = 0,
zi = −
i
ε(ak−1 ) (i ≥ 2),
(5.4)
k=2
where a2 , a3 , . . . are nonzero parameters. Note that x1J = x J and xiJ + ε(ai−1 ) = J . xi−1
Let g ∈ eD∞ and g ∈ eD∞ be the elements that are related as in (B.5). We shall use the vacuum expectation values (B.6) exclusively for odd y = (y1 , 0, y3 , 0, . . . ) with fixed y1 , y3 , . . . . Thus we simply write them as Fl1 ,l2 ;l (x; g) and fl (x; g ). Our solution to the bilinear equations is constructed in two steps. In Step 1, we construct solutions in the infinite n limit. In Step 2, we will impose a certain reduction condition on the elements g, g to make the equations close for finite n.
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Step 1. We identify the tau functions with the vacuum expectation values as Xj = F1,1 (xjX ; g) j ≥ 1 (X = S, W, N, E), fj (x1J ; g ) j = 0, 1, τjJ = (J = 1, 2, 3, 4), F1,1 (xjJ ; g) j ≥ 2, (F0,1 + (−1)j iF0,1;1 )(x10 ; g) j = 0, 1, 0 τj = F1,1 (xj0−1 ; g) j ≥ 2.
(5.5) (5.6) (5.7)
In the bilinear equations J, i in Sect. 3, consider the n infinity limit, where one just (1) forgets J, n − 1 and J, n for all J = 1, 2, 3, 4. We call the resulting system the D∞ bilinear equations. Lemma 5.1. F1,1 (x1J ) = τ0J τ1J J = 1, 2, 3, 4, 2F0,1 (x1N ; g) = τ11 τ12 ± τ01 τ02 , 2iF0,1;1 (x1N ; g) 2F0,1 (x1W ; g) = τ12 τ13 ± τ02 τ03 , 2iF0,1;1 (x1W ; g) 2F0,1 (x1S ; g) = τ13 τ14 ± τ03 τ04 , 2iF0,1;1 (x1S ; g) 2F0,1 (x1E ; g) = τ14 τ11 ± τ04 τ01 . 2iF0,1;1 (x1E ; g) Proof. The first equality follows from Lemma B.1 and (5.2). The other relations are due to Lemma B.2 and (5.3). (1)
Proposition 5.2. The parameterization (5.5)-(5.7) solves the D∞ bilinear equations with (a1 = 0) λi = L − ai , λi = L + ai (i ≥ 1), κi = K − ai , κ i = K + ai (i ≥ 1), α = K − L, β = K + L. Proof. First consider J, i with i ≥ 2. Setting l1 = l2 = 1, l = 0, x = xi0 , b3 = ai in (B.13), we have 0 + ε(b2−1 ))F1,1 (xi0 + ε(b1−1 )) (b2 − ai )F1,1 (xi−1 0 + ε(b1−1 ))F1,1 (xi0 + ε(b2−1 )) −(b1 − ai )F1,1 (xi−1 0 )F1,1 (xi0 + ε(b1−1 ) + ε(b2−1 )), = (b2 − b1 )F1,1 (xi−1
where the dependence on g ∈ eD∞ is suppressed. Due to (5.1) this yields J, i with 1 ≤ J ≤ 4 upon taking (b1 , b2 ) = (L, K), (L, −K), (−L, −K), (−L, K). Next we
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treat J, 0 and J, 1 . Setting l1 = l2 = 1, l = 0, x = x10 , b3−1 = 0 in (B.12) ∓i (B.11), we have b2 F1,1 (x10 + ε(b1−1 ))(F0,1 ∓ iF0,1;1 )(x10 + ε(b2−1 )) −b1 F1,1 (x10 + ε(b2−1 ))(F0,1 ∓ iF0,1;1 )(x10 + ε(b1−1 )) = (b2 − b1 )F1,1 (x10 + ε(b1−1 ) + ε(b2−1 ))(F0,1 ∓ iF0,1;1 )(x10 ). By setting (b1 , b2 ) = (L, K), (L, −K), (−L, −K), (−L, K), and applying Lemma 5.1, this becomes J, 1 τ0J and J, 0 τ1J for 1 ≤ J ≤ 4.
Step 2. We choose the elements g ∈ eD∞ and g ∈ eD∞ related by (B.5) as N
g = exp
bj φ(pj )φ(qj ) +
j =1
M
ˆ j ) , cj φ(pj )φ(q
(5.8)
j =1
N g = exp bj ψ (1) (pj )ψ (1)∗ (−qj ) − ψ (1) (qj )ψ (1)∗ (−pj )
(5.9)
j =1
+
M
cj ψ (1) (pj )ψ (2)∗ (−qj ) − ψ (2) (qj )ψ (1)∗ (−pj ) .
j =1
On these elements we impose the reduction condition: p j A(pj )A(−pj ) = q j 2
pj2 A(pj )A(−pj ) = qj2 A(qj )A(−qj ),
2
(5.10)
for all j , where the function A is defined by A(p) =
n−1 k=2
(1 −
p ) ak
in terms of a2 , a3 , . . . introduced in (5.4).
Lemma 5.3. Under the condition (5.10), there exists h ∈ eD∞ and h ∈ eD∞ that are related by h = ι(h )κι(h ) as in (B.5) and satisfy Fl1 ,l2 ;l (x + zn−1 , y; ω(g)) = Fl1 ,l2 ;l (x, y; h) for any x and odd y. Proof. Under the map ω specified in (B.3), ψ (α) (p) and ψ (α)∗ (q) are transformed into pψ (α) (p) and q −1 ψ (α)∗ (q). Then under the time evolution AdeH (zn−1 ,0) , they are further changed into pA(p)δα1 ψ (α) (p) and q −1 A(q)−δα1 ψ (α)∗ (q), respectively. Therefore if g in (5.9) is denoted by eX , we get AdeH (zn−1 ,0) (ω(g)) = eX with pj A(pj ) qj A(qj ) (1) bj − ψ (1) (pj )ψ (1)∗ (−qj ) + ψ (qj )ψ (1)∗ (−pj ) X = qj A(−qj ) pj A(−pj ) j
+
j
p A(p ) qj j j (1) (2)∗ (2) (1)∗ cj − ψ (p )ψ (−q ) + (q )ψ (−p ) ψ j j j j . qj pj A(−pj ) (5.11)
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Under the condition (5.10), we find X = X|bj →bj ,c→cj , bj = −
qj A(qj ) bj , pj A(−pj )
cj = −
pj A(pj ) qj
cj .
In view of the definition (B.6) and H (x + zn−1 , y) = H (x, y) + H (zn−1 , 0), the elements h and h in question are obtained by replacing bj , cj by bj , cj in g and g given in (5.8) and (5.9), respectively. Using h and h in Lemma 5.3, we now modify (5.5)–(5.7) in Step 1 into the finite n version as Xj = F1,1 (xjX ; g) 1 ≤ j ≤ n − 2 (X = S, W, N, E), J j = 0, 1, fj (x1 ; g ) J J 2 ≤ j ≤ n − 2, (J = 1, 2, 3, 4), τj = F1,1 (xj ; g) f (x J ; h ) j = n − 1, n, n−j 1 0 j j = 0, 1, (F0,1 + (−1) iF0,1;1 )(x1 ; g) 0 0 2 ≤ j ≤ n − 2, τj = F1,1 (xj −1 ; g) n−j 0 ; ω(g)) j = n − 1, n. (F0,0 + (−1) KLian−1 F−1,1;1 )(xn−2
(5.12) (5.13)
(5.14)
Lemma 5.4. J J ; ω(g)) = τn−1 τnJ F0,0 (xn−1
J = 1, 2, 3, 4,
N ; ω(g)) 2F0,0 (xn−1 2 1 ± τn2 τn−1 , = τn1 τn−1 N −1,1;1 (xn−1 ; ω(g)) W ; ω(g)) 2F0,0 (xn−1 2 3 3 2 W ; ω(g)) = τn τn−1 ± τn τn−1 , 2iL−1 F−1,1;1 (xn−1 S ; ω(g)) 2F0,0 (xn−1 4 3 3 4 S ; ω(g)) = τn τn−1 ± τn τn−1 , 2iK −1 F−1,1;1 (xn−1 E ; ω(g)) 2F0,0 (xn−1 1 4 4 1 E ; ω(g)) = τn τn−1 ± τn τn−1 . 2iL−1 F−1,1;1 (xn−1
2iK −1 F
Proof. The first relation follows from Lemma 5.3, Lemma B.1 and (5.2). From Lemma N ; ω(g)) is equal to 2F (x N ; h). 5.3, the left-hand side of the second equation 2F0,0 (xn−1 0,0 1 From (5.1), (5.3) and Lemma B.2, we get 2F0,0 (x1N ; h) = f0 (x11 ; h )f1 (x12 ; h ) + f0 (x12 ; h )f1 (x11 ; h ), which is the right-hand side. The other relations can be checked similarly. (1)
Theorem 5.5. The parameterization (5.12)-(5.14) solves the Dn with (a1 = 0)
bilinear equations
λi = L − ai , λi = L + ai (1 ≤ i ≤ n − 1), λn = 1, κi = K − ai , κ i = K + ai (1 ≤ i ≤ n − 1), κn = 1, α = K − L, β = K + L.
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Proof. Thanks to Proposition 5.2 we are left to show J, n − 1 and J, n only. Let us illustrate the proof for J = 1 since the equations for the other J follow from the same argument by taking the parameters b1 , b2 in the proof similarly to Proposition 5.2. In 0 (B.13) set l1 = l2 = l = 0, (b1 , b2 ) = (L, K), b3 = an−1 , x = xn−1 and g → ω(g). The result reads E N N E κn−1 F0,0 (xn−2 ; ω(g))F0,0 (xn−1 ; ω(g)) − λn−1 F0,0 (xn−2 ; ω(g))F0,0 (xn−1 ; ω(g)) 1 0 = (K − L)F0,0 (xn−1 ; ω(g))F0,0 (xn−2 ; ω(g)). E ; ω(g)) = F (x E ; g) = E Owing to (B.8), we know F0,0 (xn−2 1,1 n−2 n−2 and similarly N 1 ; ω(g)), F (x N ; ω(g)) F0,0 (xn−2 ; ω(g)) = Nn−2 . Rewriting F0,0 (xn−1 0,0 n−1 E ; ω(g)) by Lemma 5.4, we find that the above equation coincides with the and F0,0 (xn−1 1 + 1, n τ 1 . The other combination 1, n − 1 τ 1 − 1, n τ 1 combination 1, n − 1 τn−1 n n n−1 can be shown by using (B.10) with l1 = l2 = l = 0 similarly.
5.2. Tropical vertex model. Consider the two dimensional square lattice L equipped with the space-time coordinates (s, t) ∈ Z2 . The coordinate s (resp. t) increases rightward (resp. downward) and each vertical (resp. horizontal) line corresponds to s =constant (resp. t =constant). On each edge of L we assign an element in B so that those four surrounding a vertex obey the relation R(x, y) = (x , y ), where x, y, x , y are the ones on the east, north, south and west edges. We call the resulting two dimensional system the tropical vertex model. Unlike the usual vertex models in statistical mechanics [B], it is a deterministic system in the sense that all the edge variables are determined uniquely from their values on the northwest or southeast boundaries of L . The tropical (1) vertex model reduces to the Dn -soliton cellular automaton [HKT1, HKT2, HKOTY] in the ultradiscrete limit, where the tropical R is replaced by the combinatorial R [HKOT] (1) and B by the Dn crystal [KKM]. From (4.4) it follows that all the elements in B on the same line possess the same level. We let the levels be lt−1 (resp. ks−1 ) on the t th horizontal (resp. s th vertical) line. The bilinearization of the tropical R attained in Sect. 4 leads to another formulation of the tropical vertex model in terms of tau functions. In view of Fig. 2 we duplicate each line in L into a pair of parallel lines to form a new lattice L. Its unit structure looks like Fig. 2 where we now replace (L, K) by (L(t), K(s)). To each face of L we assign a coordinate (s, t) which now takes values in (Z/2)2 so that τ 0 : (s, t) ∈ Z2 , N : (s, t − 21 ), W : (s − 21 , t), S : (s, t + 21 ), E : (s + 21 , t), τ 1 : (s + 21 , t − 21 ), τ 2 : (s − 21 , t − 21 ), τ 3 : (s − 21 , t + 21 ), τ 4 : (s + 21 , t + 21 ). There are three types of faces depending on whether the coordinate (s, t) belongs to Z2 , (Z + 21 )2 or else. To the face of L at (s, t), we associate a tau function τi (s, t) having the components 1 ≤ i ≤ n − 2 if s + t ∈ Z + 21 and 0 ≤ i ≤ n otherwise. Then up to a boundary condition, the tropical (1) vertex model is equivalent to imposing the Dn bilinear equations on the tau functions around each face at (s, t) ∈ Z2 . Let us construct such system of tau functions {τi (s, t)} by making use of the result in Sect. 5.1, which may be regarded as a solution of our tropical vertex model. To each face of L at (s, t) ∈ (Z/2)2 we attach the time variables
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xj (s, t) = x(s, t) + zj , x(s, t) = η + ε(L(t )−1 ) + ε˜ (L(t )−1 ) + ε(K(s )−1 ) + ε˜ (K(s )−1 ), t ≥t
t >t
s <s
s ≤s
where zj is defined in (5.4) and η = η˜ is an arbitrary odd time. The sums extend over s , t ∈ Z. Let g, h ∈ eD∞ , g , h ∈ eD∞ be as in (5.12)–(5.14). Set τj (s, t) = F1,1 (xj ; g), 1 ≤ j ≤ n − 2, s + t ∈ Z + 1/2, j = 0, 1, fj (x1 ; g ) = F1,1 (xj ; g) 2 ≤ j ≤ n − 2, (s, t) ∈ (Z + 1/2)2 , f (x ; h ) j = n − 1, n, n−j 1 j j = 0, 1, (F0,1 + (−1) iF0,1;1 )(x1 ; g) F (x ; g) 2 ≤ j ≤ n − 2, = 1,1 j −1 (F + (−1)n−j ian−1 F 0,0 −1,1;1 )(xn−2 ; ω(g)) j = n − 1, n, K(s)L(t)
(s, t) ∈ Z2 ,
where xj = xj (s, t), and the time variable y suppressed here is taken as explained in the paragraph preceding (5.5). Since x(s, t) obeys the recursion relation corresponding to (1) (5.1), we conclude that {τj (s, t)} satisfies all the Dn bilinear equations if the parameters λ, κ, α, β at (s, t) ∈ Z2 are identified with those in Theorem 5.5 with (L, K) replaced by (L(t), K(s)). In particular, the levels lt−1 and ks−1 of the edge variables are determined by
lt = L(t)2
n−1
(L(t)2 − ai2 ),
ks = K(t)2
i=2
n−1
(K(s)2 − ai2 ).
i=2
Note that the formal sums over
s, t
in x(s, t) make sense under a suitable choice of η.
Remark 5.6. Consider the composition of Fig. 2 corresponding to R1 R2 R1 (x, y, z) = (x , y , z ) and R2 R1 R2 (x, y, z) = (x", y", z"). According to the bilinearization, assign x
y
y
4
4
6
z
3
x
7
6
e
a
x
b
d y
1
7
1
c
2
5
2
e
z
3
5
z
x"
d c
a b
z"
y"
the common time variables and tau functions to the faces 1, 2, . . . , 7 of the two diagrams to represent the incoming state (x, y, z) ∈ B × B × B. Then the time variables
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on the faces a, b, · · · , e are also the same in the two diagrams. Therefore one has (x , y , z ) = (x", y", z") for the associated outgoing state in B × B × B, which implies the Yang-Baxter equation R1 R2 R1 = R2 R1 R2 . (2 )
( 1)
6. Reduction to A2n−1 and Cn
(2)
(1)
Here we introduce the tropical R for A2n−1 and Cn based on Remark 2.1 and explain (1) how the results on the Dn
case can be specialized to them. We note that these tropical R have more intrinsic characterization in terms of geometric crystals as argued in [KOTY] (1) for Dn . (2)
(1)
(1)
6.1. Tropical R for A2n−1 and Cn . In this section we denote the tropical R for Dn by Rn : Bn × Bn → Bn × Bn . Let B = {x = (x1 , . . . , xn , x n , . . . , x 1 )} be the set of variables and ρ : B → Bn+1 be the embedding defined by ρ (x1 , . . . , xn , x n , . . . , x 1 ) = (x1 , . . . , xn , 1, x n , . . . , x 1 ). The tropical R for A2n−1 is the birational map Rn : B × B → B × B defined by (2)
Rn (x, y) = (x , y ) ⇔ Rn+1 (ρ(x), ρ(y)) = (ρ(x ), ρ(y )). Let B = {x = (x0 , x1 , . . . , xn , x n , . . . , x 1 )} be the set of variables and ρ : B → Bn+2 be the embedding defined by ρ (x0 , x1 , . . . , xn , x n , . . . , x 1 ) = (x0 , x1 , . . . , xn , 1, x n , . . . , x 1 , x0 ). The tropical R for Cn is the birational map Rn : B × B → B × B defined by (1)
Rn (x, y) = (x , y ) ⇔ Rn+2 (ρ (x), ρ (y)) = (ρ (x ), ρ (y )). The maps Rn and Rn are well defined due to Remark 2.1. They satisfy the inversion relation and the Yang-Baxter equation. We shall write Rn and Rn simply as R and R when the rank needs not be specified. for A2n−3 . The bilin6.2. Bilinearization of R and R . First consider the tropical Rn−1 ear equation for it is given by setting λn = κn = 1 and specializing the tau functions as (2)
J = τnJ τn−1
0≤J ≤4
(6.1)
in the Dn bilinear equations. The constraint (6.1) implies xn = yn = xn = yn = 1 in (4.2)-(4.3), which is consistent with Remark 2.1 and Proposition 3.5. It makes the bilinear equations J, n − 1 and J, n equivalent. We call the resulting subsystem (1) (2) { J, i | 1 ≤ J ≤ 4, 0 ≤ i ≤ n − 1} of the Dn case the A2n−3 bilinear equations. (1)
for Cn−2 , the bilinear equation is obtained by further setting As for the tropical Rn−2 λ1 = λ1 , κ1 = κ 1 and imposing the constraint (1)
τ0J = τ1J
0≤J ≤4
(6.2)
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on the A2n−3 bilinear equations. The constraint implies x1 = x 1 , y1 = y 1 , x1 = x 1 , y1 = y 1 in (4.2)-(4.3), which is again consistent with Remark 2.1 and Proposition 3.5. It makes the equations J, 0 and J, 1 equivalent. We call the resulting subsystem (2) (1) { J, i | 1 ≤ J ≤ 4, 1 ≤ i ≤ n − 1} of the A2n−3 case the Cn−2 bilinear equations. . Under these specializations Theorem 4.4 is still valid for Rn−1 and Rn−2 (2)
6.3. Solutions of bilinear equations for R and R . Results in Sect. 5.1 can be specialized to make (6.1) and (6.2) hold. First we deal with (6.1) for R . Let us restrict the elements (5.8) and (5.9) to the case M = 1 and take the limit q1 → 0,
p1 → am
(6.3)
for some 2 ≤ m ≤ n − 1 keeping (1 − p1 /am )/q1 2 to be an arbitrary nonzero constant. It makes the latter of (5.10) void, leaving a1 , . . . , an−1 as free parameters. All the tau functions in (5.12)-(5.14) are well defined in the limit as argued in the end of Appendix B.6. Moreover in (5.11), the coefficients p1 A(p1 )/q1 and q1 /(p1 A(−p1 )) in the second sum tend to zero. Thus the element h ∈ eD∞ in Lemma 5.3 actually belongs to eB∞ . J J Then the property (B.15) ensures that τn−1 = τn for 1 ≤ J ≤ 4 in (5.13) in agreement 0 with (6.1). The remaining relation τn−1 = τn0 in (5.14) follows from (B.16). Let us proceed to R , where the further condition (6.2) should be satisfied. The reduc(1) (1) tion from the Dn case to Cn−2 is done by setting ∀cj = 0 in (5.8)-(5.9). Namely we restrict g ∈ eD∞ and g ∈ eD∞ to g ∈ eB∞ and g ∈ eB∞ , which reduces the relevant fermion theory essentially to the single component one. The former condition in (5.10) is still present although the latter becomes void. Then obviously h ∈ eB∞ holds in Lemma 5.3, therefore (6.1) and (6.2) are valid due to (B.15), (B.9) and (B.8). Appendix A. Ui , Vi in Terms of Tau Functions Let us prove Lemma 4.3. Our task is to substitute (4.2) into the functions Ui , Vi and simplify the result by means of the bilinear equations on the tau functions. Lemma A.1. 2 βτi0 τi−1 βτ 0 βτ20 τ02 τ12 1 1 + = 1 13 (i = 1), (i = 2), (3 ≤ i ≤ n − 2), yi xi N1 W 1 Ni−1 Wi−1 τ0 τ0 0 τ 1τ 2 τ 2 βτn−1 yn 1 n n−2 n−1 + = . 1 τ2 xn−1 y n−1 κn Nn−2 Wn−2 τn−1 n
Proof. Apply 2, 1 , . . . , 2, n − 1 .
It is straightforward to check Lemma A.2. For 1 ≤ i ≤ n − 1, the ratio (y1 · · · yi )/(x1 · · · xi−1 ) is equal to τ11 τ02 (i = 1), κ1 N1
λ1 W1 τ02 τ21
κ1 κ2 N2 τ01 τ03 τ12 1 λ1 · · · λn−2 Wn−2 (τ02 )2 τn−1 (i = 2 τ2 κ1 · · · κn−1 τ01 τ03 τn−2 n−1
(i = 2), n − 1).
λ1 · · · λi−1 Wi−1 (τ02 )2 τi1 2 κ1 · · · κi Ni τ01 τ03 τi−1
(3 ≤ i ≤ n − 2),
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From Proposition 4.2, taking σ∗ in Lemma A.2 implies Lemma A.3. For 1 ≤ i ≤ n − 2, the ratio (x 1 · · · x i )/(y 1 · · · y i−1 ) is equal to τ02 τ13 λ1 W 1
(i = 1),
κ 1 N1 τ02 τ23 λ1 λ2 W2 τ01 τ03 τ12
(i = 2),
κ 1 · · · κ i−1 Ni−1 (τ02 )2 τi3 2 λ1 · · · λi Wi τ01 τ03 τi−1
(3 ≤ i ≤ n − 2).
In addition one has 1 τ3 κ 1 · · · κ n−2 κn Nn−2 (τ02 )2 τn−1 x 1 · · · x n−1 n = . 2 τ2 y 1 · · · y n−2 yn λ1 · · · λn−1 τ01 τ03 τn1 τn−2 n−1
Proof of Lemma 4.3. First we consider V0 in (2.6). In view of (2.1) and (4.4), it is expressed as V0 = l −1 v+ + k −1 v− ,
n−2 y1 · · · yi y1 · · · yn−1 1 yn 1 1 v+ = + , + + x1 · · · xi−1 xi yi x1 · · · xn−2 xn−1 y n−1 v− =
i=1 n−2 i=1
x1 · · · xi y 1 · · · y i−1
1 1 + xi yi
x 1 · · · x n−1 + y 1 · · · y n−2 yn
yn 1 + xn−1 y n−1
(A.1)
.
Using Lemmas A.1 and A.2 we get τ01 τ03 βτ02
v+ =
n−2 0 τ1 λ1 · · · λn−2 τ02 τn−1 λ1 · · · λi−1 τi0 τi1 τ10 τ11 λ1 τ02 τ20 τ21 n + + τ02 + . κ1 N 1 κ1 κ2 N 1 N 2 κ1 · · · κi Ni−1 Ni κ1 · · · κn Nn−2 τn2 i=3
0 τ 1 appearing here by means of the bilinear Rewrite τi0 τi1 (1 ≤ i ≤ n − 2) and τn−1 n equations 1, 1 − 1, n − 1 . Then all but two terms cancel out, leading to
βτ02 λ1 · · · λn τ02 τn4 4 τ0 − . v+ = κ1 · · · κn τn2 ατ01 τ03
Similarly one can derive v− =
βτ02 ατ01 τ03
−τ04
+
κ 1 · · · κ n−1 τ02 τn4
λ1 · · · λn−1 τn2
from Lemmas A.1, A.3 and the bilinear equations 3, 1 − 3, n − 1 . Substituting these expressions of v± into (A.1), we obtain V0 = β(k − l)τ02 τ04 /(lkατ01 τ03 ) as desired. Starting from this result on V0 , one can apply Lemma A.1 and the bilinear equations 3, 1 − 3, n − 1 to the recursion (2.7) successively to derive the formulas for V1 , . . . , Vn−1 . σ∗ Then the formulas for V0σ1 and V1σ∗ , . . . , Vn−1 follow from (3.4)-(3.7) and Proposition 4.2. Finally substituting these results into (2.8) and using 4, 2 − 4, n − 2 , one arrives at the desired expression for U1 , . . . , Un−1 .
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511
Appendix B. Tau Functions and Free Fermions Here we briefly recall the free fermion approach to the theory of tau functions based on [JM]. Lemma B.1 and Lemma B.2 play an important role in the main text. B.1. Single component fermions. Single component charged fermions and the Fock states are given by [ψj , ψk∗ ]+ = δj k , [ψj , ψk ]+ = [ψj∗ , ψk∗ ]+ = 0 ψ(p) = ψj p j , ψ(p)∗ = ψj∗ p −j , j
l| =
(j, k ∈ Z),
j
vac|l∗ ,
|l = l |vac , ψl−1 · · · ψ1 ψ0 , l > 0 l = 1 l=0, ψ ∗ · · · ψ ∗ ψ ∗ −2 −1 l < 0 l
∗ ∗ ∗ ψ0 ψ1 · · · ψl−1 , l > 0 ∗ l = 1 l=0, ψ ψ · · · ψ l<0 −1 −2 l
ψj |vac = 0, vac|ψj∗ = 0 (j < 0),
ψj∗ |vac = 0, vac|ψj = 0 (j ≥ 0).
The neutral fermions are simple combinations thereof: ∗ ∗ ψj + (−1)j ψ−j ψj − (−1)j ψ−j ˆ = i , φ , √ √ j 2 2 [φj , φk ]+ = [φˆ j , φˆ k ]+ = (−1)k δj,−k , [φj , φˆ k ]+ = 0, ˆ φ(p) = φj p j , φ(p) φˆ j p j , =
φj =
j
j
φj |l = φˆ j |l = 0
(l = 0, 1, j < 0),
l|φj = l|φˆ j = 0
(l = 0, 1, j > 0). (B.1)
We prepare time variables and their functions: x = (x1 , x2 , x3 , . . . ), ξ(x, p) = xj p j ,
x˜ = (x1 , −x2 , x3 , −x4 , . . . ),
j ≥1
a2 a3 , , . . . ) = −˜ε (−a). 2 3 We say x is odd if x = x. ˜ With the Hamiltonians H (x) = xm ψj ψj∗+m , ε(a) = (a,
m≥1 j ∈Z
H (x, y) =
1 2
(B.2)
(−1)m+1 (xl φm φ−m−l + yl φˆ m φˆ −m−l ),
l=1,3,5,... m∈Z
the fermions exhibit the time evolutions AdeH (x) ψ(p) = eξ(x,p) ψ(p), AdeH (x) ψ ∗ (p) = e−ξ(x,p) ψ ∗ (p), ˆ ˆ = eξ(y,p) φ(p) for (x, y) = (x, ˜ y), ˜ AdeH (x,y) φ(p) = eξ(x,p) φ(p), AdeH (x,y) φ(p) where Adg(·) = g(·)g −1 .
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B.2. 2 component fermions. (β)∗
(α)
(α)
(β)
(β)∗
(α)∗
]+ = δαβ δj k , [ψj , ψk ]+ = [ψj , ψk (α) (α)∗ ψ (α) (p) = ψj p j , ψ (α)∗ (p) = ψj p −j ,
[ψj , ψk
j (α) ψj
]+ = 0 (α, β = 1, 2, j, k ∈ Z),
j
(α)∗ + (−1)j ψ−j
(α)
(α)∗
ψj − (−1)j ψ−j (α) φˆ j = i , √ 2 (α) (α) φj p j , φˆ (α) (p) = φˆ j p j , φ (α) (p) = (α)
φj
=
√ 2
,
j
H (x, y) =
l≥1
(α) ψj |vac
j (1)
(1)∗
(α)∗
= 0, vac|ψj (1)∗
(α)
(2)∗
= 0 (j < 0),
ψj
j
(2)∗
(2)
l1 , l2 | = vac|l1 l2 where l
(2)
(xl ψj ψj +l + yl ψj ψj +l ),
(α)∗
and l
(α)∗
(α)
|vac = 0, vac|ψj
= 0 (j ≥ 0),
(1)
|l2 , l1 = l2 l1 |vac ,
stand for l and l∗ with ψj , ψj∗ replaced by ψj , ψj (α)
(α)∗
.
, B , D and D . The algebras B ⊂ D are defined within B.3. Algebras A∞ , B∞ ∞ ∞ ∞ ∞ ∞ ={ the single component theory as B∞ aj k : φj φk : +d | ∃ N, aj k = 0 if |j + k| > = { N} and D∞ aj k : φj φk : + bj k : φˆ j φˆ k : + cj k : φj φˆ k : +d | ∃ N, aj k = bj k = cj k = 0 if |j + k| > N }. The algebra A∞ in the 2 component theory is defined as (α) (β)∗ : +d | ∃ N, aαi,βj = 0 if |i −j | > N }. The automorphism A∞ = { aαi,βj : ψi ψj → A are given by σ, κ, ω of A∞ and the homomorphism ι : D∞ ∞ (α)
→ (−1)j ψ−j , ψj
(α)
→ iψj , ψj
σ : ψj κ : ψj
(α)∗
(α)
(α)∗
(α)∗
(α)
→ (−1)j ψ−j , (α)∗
→ −iψj
(α) ψj
(α) (α)∗ → ψj −1 , ψj → ω: (1) ˆ (2) ι : φj → φj , φj → φj .
,
(α)∗ ψj −1 ,
(B.3)
The algebras B∞ ⊂ D∞ are defined as D∞ = {X ∈ A∞ | σ (X) = X} and B∞ = {X ∈ (α) D∞ | π(X) = X}, where π is the projection to the first component, i.e., π(ψj ) = (α)
(α)∗
(α)∗
δα1 ψj , π(ψj ) = δα1 ψj theory. The map
. Thus B∞ actually stays within the single component
−→ D∞ , ι + κι : D∞
B∞ −→ B∞
is an isomorphism and one has ι(H (x, y)) + κι(H (x, y)) = H (x, y)
if (x, y) = (x, ˜ y). ˜
(B.4)
The set of well-defined group elements corresponding to these algebras will be denoted, by abuse of notation, by eD∞ := {eX1 · · · eXk | k ≥ 0, X1 , . . . , Xk ∈ D∞ are locally nilpotent},
Tropical R and Tau Functions
513
and D . For g ∈ eD∞ , we write and similarly for A∞ , B∞ , B∞ ∞
g = eι(X )+κι(X ) = ι(g )κι(g ) ∈ eD∞ .
(B.5)
B.4. Tau functions. We are concerned with two kinds of tau functions: fl (x, y; h) = l|eH
(x,y)
Fl1 ,l2 ;l (x, y; g) = l1 , l2 |e
h|l
H (x,y)
for (x, y) = (x, ˜ y), ˜ h ∈ eD∞ ,
g|l2 − l, l1 + l
g∈e
A∞
(B.6)
.
Fl1 ,l2 :0 (x, y; g) will simply be denoted by Fl1 ,l2 (x, y; g). They enjoy the symmetry Fl1 ,l2 ;l (x, y; g) = (−1)(l1 +l2 )l F1−l1 ,1−l2 ;−l (x, ˜ y; ˜ σ (g)), Fl1 ,l2 ;l (x, y; g) = Fl1 −1,l2 −1;l (x, y; ω(g)).
(B.7) (B.8)
Clearly we have Fl1 ,l2 ;l (x, y; g) = 0
if l = 0 and g ∈ eB∞ .
(B.9)
Fl1 ,l2 ;l (x, y; g) will be denoted by Fl1 ,l2 ;l (x, y) or Fl1 ,l2 ;l (x) for simplicity. B.5. Bilinear identities. For any g ∈ eA∞ one has the hierarchy of equations
dk (−)l2 +l2 k l1 −l1 −1 eξ(x−x ,k) Fl1 −1,l2 ;l+1 (x − ε(k −1 ), y)Fl +1,l ;l −1 (x + ε(k −1 ), y ) 1 2 2π ik dk l2 −l −1 ξ(y−y ,k) −1 2 k e Fl1 ,l2 −1;l (x, y − ε(k ))Fl ,l +1;l (x , y + ε(k −1 )) = 0, + 1 2 2π ik
where l1 − l1 ≥ l − l ≥ l2 − l2 + 2 and the integration is taken along a small contour around k = ∞. This is Eq. (4.4) in [JM]. Let b1 , b2 , b3 be nonzero constants. We take y = y and suppress the dependence on it. The above equations contain the following: (b2−1 − b3−1 )Fl1 ,l2 ;l (x + ε(b2−1 ) + ε(b3−1 ))Fl1 −1,l2 +1;l+1 (x + ε(b1−1 )) + cyc = 0,
(B.10)
(b2−1 (b2−1
= 0,
(B.11)
0,
(B.12)
− b3−1 )Fl1 ,l2 ;l (x − b3−1 )Fl1 ,l2 ;l (x
(b2 − b3 )Fl1 ,l2 ;l (x
+ ε(b2−1 ) + ε(b3−1 ))Fl1 −1,l2 ;l+1 (x + ε(b1−1 )) + cyc + ε(b2−1 ) + ε(b3−1 ))Fl1 −1,l2 ;l (x + ε(b1−1 )) + cyc = + ε(b2−1 ) + ε(b3−1 ))Fl1 ,l2 ;l (x + ε(b1−1 )) + cyc = 0,
(B.13)
where +cyc means the additional two terms obtained by the cyclic permutations of b1 , b2 , b3 . They also contain F0,1 (x + ε(b2−1 ))F0,0 (x + ε(b1−1 )) − F0,1 (x + ε(b1−1 ))F0,0 (x + ε(b2−1 )) (B.14) = (b1−1 − b2−1 )F1,0;−1 (x + ε(b1−1 ) + ε(b2−1 ))F−1,1;1 (x). When both x and y are odd, Fl1 ,l2 ;l (x, y; g) with g ∈ eD∞ can be expressed in terms of fl (x, y; g ) as noted in Eq. (7.6) in [JM]. We recall this fact in
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A. Kuniba, M. Okado, T. Takagi, Y. Yamada
Lemma B.1. Let g ∈ eD∞ and g ∈ eD∞ be related as in (B.5). Suppose that x and y are both odd. Denoting Fl1 ,l2 ;l (x, y; g) and fl (x, y; g ) by Fl1 ,l2 ;l (x) and fl (x), one has F1,1 (x) = F0,0 (x) = f0 (x)f1 (x), 1 F0,1 (x) = F1,0 (x) = (f0 (x)2 + f1 (x)2 ), 2 i F0,1;1 (x) = −F1,0;−1 (x) = (f0 (x)2 − f1 (x)2 ). 2 Proof. The first equality in each line is due to (B.7). Since φ02 = φˆ 02 = 1/2, we may write ˆ contains φk , φˆ k AdeH (x,y) g = X0 + X1 φ0 + X2 φˆ 0 +X3 φ0 φˆ 0 , where Xj = Xj (φ, φ) only for k = 0 and its order is even (resp. odd) for j = 0, 3 (resp. j = 1, 2). Due to (B.1) we find fl (x; g ) = X0 + (−1)l i X3 /2 for l = 0, 1, where Xj := 0|Xj |0 = 1|Xj |1 . On the other hand from (B.4) and (B.5) it follows that AdeH (x,y) g = (X0 + (1) (2) (1) (2) (1) (2) (1) (2) X1 φ0 + X2 φ0 + X3 φ0 φ0 )(Xˆ 0 + Xˆ 1 φˆ 0 + Xˆ 2 φˆ 0 + Xˆ 3 φˆ 0 φˆ 0 ), where Xj and Xˆ are obtained from Xj by substituting the 2 component fermions into Xj as X = j
j
Xj (φ (1) , φ (2) ), Xˆ j = Xj (φˆ (1) , φˆ (2) ). Due to the property (B.1) for each component, we (1) (2) (1) (2) get F0,0 (x; g) = 0, 0|(X0 Xˆ 0 + X3 Xˆ 3 φ0 φ0 φˆ 0 φˆ 0 )|0, 0 = X0 2 + X3 2 /4 = ( X0 + i X3 /2)( X0 − i X3 /2), proving the first relation. The other relations can be shown similarly. For odd x and y, it is clear from the above proof that f0 (x, y; g ) = f1 (x, y; g )
if g ∈ eB∞ .
(B.15)
Lemma B.1 admits a generalization to
Lemma B.2. Let g ∈ eD∞ and g ∈ eD∞ be as in (B.5). Suppose y is odd and x˜ = x + ε(c−1 ) − ε˜ (c−1 ). Denoting Fl1 ,l2 ;l (x, y; g) and fl (x, y; g ) by Fl1 ,l2 ;l (x) and fl (x), one has 2F0,1 (x) −2iF0,1;1 (x) 2F0,0 (x) 2ic−1 F−1,1;1 (x)
= f0 (x + ε(c−1 ))f0 (x − ε˜ (c−1 )) ± f1 (x + ε(c−1 ))f1 (x − ε˜ (c−1 )), = f0 (x + ε(c−1 ))f1 (x − ε˜ (c−1 )) ± f1 (x + ε(c−1 ))f0 (x − ε˜ (c−1 )).
Proof. In (B.10)–(B.12), set b1 = −b2 = c and b3−1 = 0. Writing x+ = x + ε(c−1 ), x− = x − ε˜ (c−1 ), one has ˜ 0,1 (x), F1,1 (x− )F0,1 (x+ ) + F1,1 (x+ )F0,1 (x− ) = 2F1,1 (x)F ˜ 0,1;1 (x), F1,1 (x− )F0,1;1 (x+ ) + F1,1 (x+ )F0,1;1 (x− ) = 2F1,1 (x)F F1,0;−1 (x− )F0,1 (x+ ) + F1,0;−1 (x+ )F0,1 (x− ) = 2F1,0;−1 (x)F ˜ 0,1 (x). By applying (B.7) together with σ (g) = g, the right hand sides here are equal to 2F0,0 (x)F0,1 (x), 2F0,0 (x)F0,1;1 (x) and −2F0,1;1 (x)F0,1 (x), respectively. Since both
Tropical R and Tau Functions
515
x+ and x− are odd, the left hand sides are expressed in terms of f0 (x± ), f1 (x± ) by Lemma B.1. Solving the resulting three equations with respect to F0,0 (x), F0,1 (x) and F0,1;1 (x), we obtain the formulas in question up to an overall sign, which can be fixed by considering the smooth limit g → 1. The formula for F−1,1;1 (x) can be derived similarly by using (B.14). Note that in the limit c → ∞, the first three relations in Lemma B.2 reduce to Lemma B.1. B.6. Explicit formula. Let us write down the explicit formula for the vacuum expectation values (B.6). To simplify the result we introduce the parameters p1 , . . . , p2N+M and q1 , . . . , qM to modify the parameterization of g ∈ eD∞ and g ∈ eD∞ in (5.8) as N
g = exp
bi φ(pi )φ(pi¯ ) +
M
ˆ j) , cj φ(pj˜ )φ(q
j =1
i=1
N g = exp bi ψ (1) (pi )ψ (1)∗ (−pi¯ ) − ψ (1) (pi¯ )ψ (1)∗ (−pi ) i=1
+
M
cj ψ (1) (pj˜ )ψ (2)∗ (−qj ) − ψ (2) (qj )ψ (1)∗ (−pj˜ ) ,
j =1
where i¯ = 2N + 1 − i and j˜ = 2N + M + 1 − j . Given I ⊆ {1, . . . , N} and J ⊆ {1, . . . , M}, we shall write I¯ = {i¯ | i ∈ I } and J˜ = {j˜ | j ∈ J }. The cardinality of a set I is denoted by |I |. For any sets K and K we set (pµ ± pν ), ± K (p) = µ,ν∈K, µ<ν − − K (p)K (−p)
m,l K,K (p) =
µ∈K,ν∈K (pµ
+ pν )
pµm+l
(−pµ )−m−l+1 ,
µ∈K
µ∈K
and similarly for q. Let x and y be odd time variables. Then we have fl (x, y; g ) =
εl,J 2−|I |−|J | bI (x)cJ (x, y)
I,J
K = I I¯ J˜, bI (x) =
i∈I
cJ (x, y) =
εl,J =
1 i(−1)l
− − K (p)J (q)
+ + K (p)J (q)
l = 0, 1,
|J | even |J | odd,
bi exp(ξ(x, pi ) − ξ(x, −pi¯ )),
j ∈J
cj exp(ξ(x, pj˜ ) + ξ(y, qj )),
where the sum I,J extends over all the subsets I ⊆ {1, . . . , N} and J ⊆ {1, . . . , M}. For any integers l1 , l2 and l, Fl1 ,l2 ;l (x, y; g) with odd y (x is not restricted to be odd) reads
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Fl1 ,l2 ;l (x, y; g) l2 ,−l 1 ,l = (−1)|I |+(|I |+|I |)|J | bI (x)bI (x)cJ (x, y)cJ (x, y)lK,K (p)J ,J (q), I,I ,J,J ¯
K = I I J˜, K = I I¯ J˜ , bI (x) = bi exp(ξ(x, pi¯ ) − ξ(x, −pi )), i∈I
cJ (x, y)
=
j ∈J
cj exp(−ξ(x, −pj˜ ) + ξ(y, qj )),
where the sum I,I ,J,J extends over all the subsets I, I ⊆ {1, . . . , N} and J, J ⊆ {1, . . . , M} such that |J | − |J | = l. Consider the dependence of fl (x, y; g ) (l = 0, 1) and Fl1 ,l2 ;l (x, y; g) on q1 when M = 1. Apart from the factors cJ (x, y) and cJ (x, y), fl is independent of q1 and Fl1 ,l2 ;l is proportional to the single power q1l2 −l . Therefore all the functions fl and Fl1 ,l2 ;l appearing in (5.12)-(5.14) are well defined in the limit q1 → 0 with M = 1. In particular from (B.8) one has lim F−1,1;1 (x, y; ω(g)) = lim F0,2;1 (x, y; g) = 0.
q1 →0
q1 →0
(B.16)
These facts are used in Sect. 6.3. Acknowledgement. The authors thank R. Inoue for a careful reading of the manuscript. A.K. thanks M. Jimbo for explaining Eq. (7.6) in [JM]. M.O. and Y.Y. were partially supported by Grant-in-Aid for Scientific Research JSPS No.14540026 and No.11440047, respectively.
References [ABS]
Adler, V. E., Bobenko, A. I., Suris, Yu. B.: Classification of integrable equations on quadgraphs. The consistency approach. math.QA/0202024 [B] Baxter, R.J.: Exactly solved models in statistical mechanics. London:Academic Press, (1982) [BFZ] Berenstein, A., Fomin, S., Zelevinsky, A.: Parameterizations of canonical bases and totally positive matrices. Adv. in Math 122, 49–149 (1996) [BK] Berenstein, A., Kazhdan, D.: Geometric and unipotent crystals. GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal., Special Volume, I, 188–236 [DJM] Date, E., Jimbo, M., Miwa, T.: Method for generating discrete soliton equations. J. Phys. Soc. Japan. 51, 4116–4124, 4125–4131 (1982) [D] Drinfeld, V. G.: On some unsolved problems in quantum group theory. In “Quantum groups” (Leningrad 1990), Lect. Note. in Math. 1510, Berlin-Heidelberg-New York: Springer, 1992, pp. 1–8 [ESS] Etingof, P., Schedler, T., Soloviev, A.: Set-theoretical solutions to the quantum Yang-Baxter equation. Duke Math. J. 100, 169–209 (1999) [FOY] Fukuda, K., Okado, M., Yamada, Y.: Energy functions in box ball systems. Int. J. Mod. Phys. A 15, 1379–1392 (2000) (1) [HHIKTT] Hatayama, G., Hikami, K., Inoue, R., Kuniba, A., Takagi, T., Tokihiro, T.: The AM Automata related to crystals of symmetric tensors. J. Math. Phys. 42, 274–308 (2001) [HKOT] Hatayama, G., Kuniba, A., Okado, M., Takagi, T.: Combinatorial R matrices for a family of (1) (1) (2) (2) crystals: Bn , Dn , A2n , and Dn+1 cases. J. Alg. 247, 577–615 (2002) [HKOTY] Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Scattering rules in soliton cellular automata associated with crystal bases. Contemp. Math. 297, 151–182 (2002)
Tropical R and Tau Functions [HKT1] [HKT2] [H] [HT] [HTI] [JMY] [JM] [KNY] [KKM] [K] [Ki]
[KOTY] [L] [M] [MV] [NY] [NoY] [O] [TNS] [TS] [TTMS] [V1] [V2] [WX] [Y]
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Hatayama, G., Kuniba, A., Takagi, T.: Soliton cellular automata associated with crystal bases. Nucl. Phys. B577[PM], 619–645 (2000) Hatayama, G., Kuniba, A., Takagi, T.: Simple algorithm for factorized dynamics of gn automaton. J. Phys. A: Math. Gen. 34, 10697–10705 (2001) Hirota, R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan 50, 3785–3791 (1981) Hirota, R., Tsujimoto, S.: Conserved quantities of a class of nonlinear difference-difference equations. J. Phys. Soc. Japan 64, 3125–3127 (1995) Hitota, R., Tsujimoto, S., Imai, T.: Difference scheme of soliton equations. In Future directions of Nonlinear Dynamics in Physical and Biological Systems, ed. Christiansen P.L. et al, New York: Plenum, 1993, p. 7 Jiang-Hua, L., Min, Y., Yong-Chang, Z.: On the set-theoretical Yang-Baxter equation. Duke Math. J. 104, 1–18 (2000) Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS. Kyoto Univ. 19, 943–1001 (1983) (1) (1) Kajiwara, K., Noumi, M., Yamada, Y.: Discrete dynamical systems with W (Am−1 × An−1 ) symmetry. Lett. Math. Phys. 60, 211–219 (2002) Kang, S-J., Kashiwara, M., Misra, K. C.: Crystal bases of Verma modules for quantum affine Lie algebras. Compositio Math. 92, 299–325 (1994) Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73, 383–413 (1994) Kirillov, A. N.: Introduction to tropical combinatorics. In: “Physics and Combinatorics 2000” Kirillov A. N., Liskova N. (eds), Proceedings of the Nagoya 2000 International Workshop, Singapore: World Scientific, 2001, pp. 82–150 (1) Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Geometric crystal and tropical R for Dn . Int. Math. Res. Notices 48, 2565–2620 (2003) Lusztig, G.: Total positivity in reductive Groups. In “Lie theory and geometry”. Prog. in Math. 123, Boston: Birkh¨auser, 1994, pp. 531–568 Miwa, T.: On Hirota’s difference equations. Proc. Japan Acad. Ser. A Math. Sci. 58, 9–12 (1982) Moser, J., Veselov, A. P.: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun. Math. Phys. 139, 217–243 (1991) Nakayashiki, A., Yamada, Y.: Kostka polynomials and energy functions in solvable lattice models. Selecta Mathematica, New Ser. 3, 547–599 (1997) Noumi, M.,Yamada,Y.: Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions. Preprint (math-ph/0203030), to appear in Adv. Stud. in Pure Math. Odesskii, A.: Set-theoretical solutions to the Yang-Baxter relation from factorization of matrix polynomials and θ-functions. math.QA/0205051 Tokihiro, T., Nagai, A., Satsuma, J.: Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization. Inverse Problems 15, 1639–1662 (1999) Takahashi, D., Satsuma, J.: A soliton cellular automaton. J. Phys. Soc. Jpn. 59, 3514–3519 (1990) Tokihiro, T., Takahashi, D., Matsukidaira, J., Satsuma, J.: From soliton equations to integrable cellular automata through a limiting procedure. Phys. Rev. Lett. 76, 3247–3250 (1996) Veselov, A. P.: Integrable mappings. Russian Math. Surv. 46, 1–51 (1991) Veselov, A. P.: Yang-Baxter maps and integrable dynamics. math.QA/0205335 Weinstein, A., Xu, P.: Classical solutions of the quantum Yang-Baxter equation. Commun. Math. Phys. 148, 309–343 (1992) Yamada, Y.: A birational representation of Weyl group, combinatorial R-matrix and discrete Toda equation. In: “Physics and Combinatorics 2000” Kirillov A. N., Liskova N. (eds), Proceedings of the Nagoya 2000 International Workshop, Singapore: World Scientific, 2001, pp. 305–319
Communicated by L. Takhtajan
Commun. Math. Phys. 245, 519–537 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1016-0
Communications in
Mathematical Physics
Elliptic U (2) Quantum Group and Elliptic Hypergeometric Series Erik Koelink1 , Yvette van Norden1, , Hjalmar Rosengren2 1
Technische Universiteit Delft, Faculteit Elektrotechniek, Wiskunde en Informatica, Toegepaste Wiskundige Analyse, Postbus 5031, 2600 GA Delft, The Netherlands. E-mail: [email protected]; [email protected] 2 Department of Mathematics, Chalmers University of Technology and G¨ oteborg University, 412 96 G¨oteborg, Sweden. E-mail: [email protected] Received: 10 April 2003 / Accepted: 8 September 2003 Published online: 19 December 2003 – © Springer-Verlag 2003
Abstract: We investigate an elliptic quantum group introduced by Felder and Varchenko, which is constructed from the R-matrix of the Andrews–Baxter–Forrester model, containing both spectral and dynamical parameter. We explicitly compute the matrix elements of certain corepresentations and obtain orthogonality relations for these elements. Using dynamical representations these orthogonality relations give discrete biorthogonality relations for terminating very-well-poised balanced elliptic hypergeometric series, previously obtained by Frenkel and Turaev and by Spiridonov and Zhedanov in different contexts.
1. Introduction Elliptic functions appear in various solvable models in statistical mechanics and other areas of physics. A famous example is Baxter’s 8-vertex model [2], whose R-matrix, containing the Boltzmann weights, is an elliptic solution of the Yang–Baxter equation. A related face model was introduced by Andrews, Baxter and Forrester [1]. In this case the R-matrix satisfies a modified, “dynamical”, version of the Yang–Baxter equation, generalizing Wigner’s hexagon identity for the classical 6j -symbols of quantum mechanics. In the early 1980’s, the algebraic study of the Yang–Baxter equation lead to the introduction of quantum groups. The most well understood quantum groups are those constructed from the simplest, constant, solutions. Quantum groups connected to more complicated solutions, and in particular to elliptic solutions, have been more difficult to construct and study. One reason for this is that elliptic quantum groups are not Hopf algebras. Various approaches have been tried for finding a substitute; cf. [5, 8, 10, 12, 20]. In the dynamical case, a decisive step was taken by Felder and Varchenko [9], who The second author is supported by Netherlands Organisation for Scientific Research (NWO) under project number 613.006.572.
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introduced the algebra that we will study here. This example motivated Etingof and Varchenko [7] to introduce h-Hopf algebroids, a generalization of Hopf algebras adapted to studying dynamical R-matrices; cf. [15, 19] for further additions to this framework. An important mathematical application of quantum groups is their relation to basic hypergeometric series (or q-series), a class of special functions going back to work of Cauchy and Heine in the 1840’s. The input from quantum group theory has been important for the rapid development of this field during the last 20 years. To our knowledge, nobody has so far associated special functions to elliptic quantum groups in an analogous way. There is, however, a natural candidate for the special functions that should appear, namely, the elliptic or modular hypergeometric series of Frenkel and Turaev [11]. This type of sums may be used to express the elliptic 6j -symbols of Date et al. [4], which are solutions to the Yang–Baxter equation that greatly generalize the Andrews–Baxter– Forrester solution. For more information on elliptic hypergeometric series we refer to [13, 18, 17, 21, 23, 22, 24, 25, 27]. Our main aim is to give an explicit link from elliptic quantum groups to elliptic hypergeometric series. Namely, we show that 10 ω9 sums, or elliptic 6j -symbols, appear as matrix elements for an elliptic quantum group which we denote by FR (U (2)), which is the algebra of Felder and Varchenko with some extra structure. To achieve this, we first construct finite-dimensional corepresentations of FR (U (2)), analogous to the standard representations of SU (2) on spaces of homogeneous polynomials in two variables. A main result, Theorem 3.4, is an explicit expression for the matrix elements of these corepresentations. We can then calculate the action of the matrix elements in representations found by Felder and Varchenko, and show that it is given in terms of elliptic hypergeometric series. The matrix elements satisfy orthogonality relations in the non-commutative algebra FR (U (2)). Evaluating these in a representation leads to bi-orthogonality relations for 10 ω9 series. These relations were found already by Frenkel and Turaev [11]; cf. also [25]. Our new derivation of the bi-orthogonality relations shows that they can be viewed as analogues of the orthogonality relations for Krawtchouk polynomials, see [26] for the Lie group SU (2). For the quantum SU (2) group the same approach leads to quantum q-Krawtchouk polynomials, see [16]. For the dynamical quantum SU (2) group, i.e. corresponding to a trigonometric dynamical R-matrix, we get the orthogonality relations for q-Racah polynomials, see [15, §4]. So the above cases can be considered as limiting cases of the bi-orthogonality relations for elliptic 6j -symbols. The paper is organized as follows. In Sect. 2 we recall the definition of an h-Hopf algebroid and the generalized FRST-construction from [7]. Then we describe the elliptic quantum group FR (U (2)), which is obtained from the R-matrix of the Andrews–Baxter– Forrester model. In Sect. 3 we define finite-dimensional corepresentations of FR (U (2)) and compute their matrix elements explicitly. In Sect. 4 we consider representations of FR (U (2)), from which we obtain commutative versions of the orthogonality relations for matrix elements of the corepresentations. It turns out that these are in fact bi-orthogonality relations for terminating very-well-poised balanced elliptic hypergeometric 10 ω9 -series (or elliptic 6j -symbols). Notation. We denote by θ (z) the normalized Jacobi theta function θ (z) =
∞
1 − zp j
j =0
1 − p j +1 /z ,
|p| < 1,
Elliptic U (2) Quantum Group and Elliptic Hypergeometric Series
521
where p is a fixed parameter that is suppressed from the notation. It satisfies θ (pz) = θ (z−1 ) = −z−1 θ(z), and the addition formula θ (xy, x/y, zw, z/w) = θ (xw, x/w, zy, z/y) + (z/y)θ (xz, x/z, yw, y/w),
(1.1)
where we use the notation θ (a1 , . . . , an ) = θ (a1 ) · · · θ(an ). We define elliptic Pochhammer symbols by (a)n =
n−1
θ (aq 2i ),
i=0
with q another fixed parameter. We will frequently write (a1 , a2 , . . . , ak )n = (a1 )n · · · (ak )n . Elliptic binomial coefficients are defined by l θ (q 2(k−l+i) ) k = . l θ (q 2i ) i=1
Finally, the balanced very-well-poised elliptic hypergeometric series is defined by [11] r+1 ωr (a1 ; a4 , a5 , . . .
, ar+1 ) =
∞ θ (a1 q 4k ) k=0
(a1 , a4 , . . . , ar+1 )k q 2k , (1.2) θ (a1 ) (q 2 , a1 q 2 /a4 , . . . , a1 q 2 /ar+1 )k
where (a4 · · · ar+1 )2 = a1r−3 q 2(r−5) . In this paper all series terminate, i.e. one of the ai is of the form q −2n with n a nonnegative integer, so there are no convergence problems. Let us emphasize that in this paper all elliptic factorials and elliptic hypergeometric series are in base q 2 , p. 2. Elliptic U (2) Quantum Group In this section we recall the definition of h-Hopf algebroids (also known as dynamical quantum groups) and the FRST-construction. We start with the definition of the quantum dynamical Yang-Baxter equation with spectral parameter and give in (2.2) the R-matrix to which we apply the FRST-construction. The generators and relations for the resulting h-Hopf algebroid have been studied by Felder and Varchenko [9]. Let h be a finite dimensional complex vector space, viewed as a commutative Lie algebra and V = α∈h∗ Vα a diagonalizable h-module. The quantum dynamical YangBaxter equation with spectral parameter (QDYBE) is given by R 12 (λ − h(3) , z12 )R 13 (λ, z13 )R 23 (λ − h(1) , z23 ) = R 23 (λ, z23 )R 13 (λ − h(2) , z13 )R 12 (λ, z12 ).
(2.1)
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Here R : h∗ × C → End(V ⊗ V ) is a meromorphic function, h indicates the action of h, the upper indices are leg-numbering notation for the tensor product and zij = zi /zj . For instance, R 12 (λ − h(3) , z) denotes the operator R 12 (λ − h(3) , z)(u ⊗ v ⊗ w) = R(λ − µ, z)(u ⊗ v) ⊗ w for w ∈ Vµ . An R-matrix is by definition a solution of the QDYBE (2.1) which is h-invariant. In the example we study, h is one-dimensional. We identify h = h∗ = C and take V to be the two-dimensional h-module V = Ce1 ⊕ Ce−1 . In the basis e1 ⊗ e1 , e1 ⊗ e−1 , e−1 ⊗ e1 , e−1 ⊗ e−1 the R-matrix is given by
1 0 0 0 0 a(λ, z) b(λ, z) 0 R(λ, z) = R(λ, z, p, q) = , 0 c(λ, z) d(λ, z) 0 0 0 0 1
(2.2)
where a(λ, z) =
θ (z)θ (q 2(λ+2) ) , θ(q 2 z)θ (q 2(λ+1) )
θ (q 2 )θ (q 2(λ+1) z) c(λ, z) = , θ(q 2 z)θ (q 2(λ+1) )
b(λ, z)=
θ(q 2 )θ (q −2(λ+1) z) , θ(q 2 z)θ (q −2(λ+1) )
θ(z)θ(q −2λ ) d(λ, z) = . θ(q 2 z)θ (q −2(λ+1) )
(2.3)
The R-matrix defined by (2.2) satisfies the QDYBE (2.1), see [3, 1, 9, 12]. 2.1. h-Hopf algebroids. In this subsection we recall the notion of h-bialgebroids and h-Hopf algebroids (or dynamical quantum groups) originally introduced by Etingof and Varchenko [7], see also [6]. For the definition of the antipode in an h-Hopf algebroid we follow [15]. We discuss the FRST-construction that associates an h-bialgebroid to an h-invariant matrix, see [7, 6]. Let h be a finite dimensional complex vector space. Denote the field of meromorphic functions on the dual of h by Mh∗ . Definition 2.1. An h-algebra is a complex associative algebra A with 1, which is bigraded over h∗ , A = ⊕α,β∈h∗ Aαβ , and equipped with two algebra embeddings µl , µr : Mh∗ → A00 (the left and right moment map) such that µl (f )a = aµl (Tα f ),
µr (f )a = aµr (Tβ f ), for all a ∈ Aαβ , f ∈ Mh∗ ,
where Tα denotes the automorphism (Tα f )(λ) = f (λ + α). A morphism of h-algebras is an algebra homomorphism preserving the moment maps. ˜ is the h∗ -bigraded Let A and B be two h-algebras. The matrix tensor product A⊗B ˜ αβ = γ ∈h∗ (Aαγ ⊗Mh∗ Bγβ ), where ⊗Mh∗ denotes the usual vector space with (A⊗B) tensor product modulo the relations B µA r (f )a ⊗ b = a ⊗ µl (f )b, for all a ∈ A, b ∈ B, f ∈ Mh∗ .
(2.4)
The multiplication (a ⊗ b)(c ⊗ d) = ac ⊗ bd for a, c ∈ A and b, d ∈ B and the moment B ˜ maps µl (f ) = µA l (f ) ⊗ 1 and µr (f ) = 1 ⊗ µr (f ) make A⊗B into an h-algebra.
Elliptic U (2) Quantum Group and Elliptic Hypergeometric Series
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Example. Let Dh be the algebra of difference operators in Mh∗ , consisting of the operators i fi Tβi , with fi ∈ Mh∗ and βi ∈ h∗ . This is an h-algebra with the bigrading defined by f T−β ∈ (Dh)ββ and both moment maps equal to the natural embedding. ˜ ˜ h ∼ For any h-algebra A, there are canonical isomorphisms A ∼ = Dh⊗A, = A⊗D defined by x∼ = x ⊗ T−β ∼ = T−α ⊗ x, for all x ∈ Aαβ .
(2.5)
The algebra Dh plays the role of the unit object in the category of h-algebras. Definition 2.2. An h-bialgebroid is an h-algebra A equipped with two h-algebra homo˜ (the comultiplication) and ε : A → Dh (the counit) such morphisms : A → A⊗A that ( ⊗ id) ◦ = (id ⊗ ) ◦ and (ε ⊗ id) ◦ = id = (id ⊗ ε) ◦ (under the identifications (2.5)). Definition 2.3. An h-Hopf algebroid is an h-bialgebroid A equipped with a C-linear map S : A → A, the antipode, such that S(µr (f )a) = S(a)µl (f ), S(aµl (f )) = µr (f )S(a), for all a ∈ A, f ∈ Mh∗ , m ◦ (id ⊗ S) ◦ (a) = µl (ε(a)1), for all a ∈ A, m ◦ (S ⊗ id) ◦ (a) = µr (Tα (ε(a)1)), for all a ∈ Aαβ , ˜ → A denotes the multiplication and ε(a)1 is the result of applying the where m : A⊗A difference operator ε(a) to the constant function 1 ∈ Mh∗ . If there exists an antipode on an h-bialgebroid, it is unique. Furthermore, the antipode is anti-multiplicative, anti-comultiplicative, unital, counital and interchanges the moment maps µl and µr , see [15, Prop. 2.2]. The FRST-construction provides many examples of h-bialgebroids, see [7, 6, 9, 15]. We recall the construction. Let h and Mh∗ be as before, V = α∈h∗ Vα be a finite-dimensional diagonalizable h-module and R : h∗ × C → Endh(V ⊗ V ) a meromorphic function that commutes with the h-action on V ⊗ V . Let {ex }x∈X be a homogeneous basis of V , where X is an ab (λ, z) for the matrix elements of R, index set. Write Rxy ab R(λ, z)(ea ⊗ eb ) = Rxy (λ, z)ex ⊗ ey , x,y∈X
and define ω : X → h∗ by ex ∈ Vω(x) . Let AR be the unital complex associative algebra generated by the elements {Lxy (z)}x,y∈X , with z ∈ C, together with two copies of Mh∗ , embedded as subalgebras. The elements of these two copies will be denoted by f (λ) and f (µ), respectively. The defining relations of AR are f (λ)g(µ) = g(µ)f (λ), f (λ)Lxy (z) = Lxy (z)f (λ + ω(x)),
f (µ)Lxy (z) = Lxy (z)f (µ + ω(y)),
(2.6)
for all f , g ∈ Mh∗ , together with the RLL-relations xy bd Rac (λ, z1 /z2 )Lxb (z1 )Lyd (z2 ) = Rxy (µ, z1 /z2 )Lcy (z2 )Lax (z1 ), (2.7) x,y∈X
x,y∈X
for all z1 , z2 ∈ C and a, b, c, d ∈ X. The bigrading on AR is defined by Lxy (z) ∈ Aω(x),ω(y) and f (λ), f (µ) ∈ A0,0 . The moment maps defined by µl (f ) = f (λ),
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µr (f ) = f (µ) make AR into a h-algebra. The h-invariance of R ensures that the bigrading is compatible with the RLL-relations (2.7). Finally the co-unit and co-multiplication defined by ε(Lab (z)) = δab T−ω(a) , ε(f (λ)) = ε(f (µ)) = f, Lax (z) ⊗ Lxb (z), (Lab (z)) = x∈X
(f (λ)) = f (λ) ⊗ 1, (f (µ)) = 1 ⊗ f (µ),
(2.8)
(2.9)
equip AR with the structure of an h-bialgebroid, see [7]. 2.2. Elliptic U (2) quantum group. We now give the results of the generalized FRSTconstruction when applied to the R-matrix (2.2), see [9]. Let 0 < q < 1, 0 < p < 1. We assume that p, q are generic, it suffices to take p and q algebraically independent over Q. Let X = {−1, 1} and define ω : {−1, 1} → C by ω(x) = x. We denote the corresponding h-bialgebroid by FR (M(2)): it is an elliptic analogue of the algebra of polynomials on the space of complex 2 × 2-matrices. In the rest of this paper the four L-generators will be denoted by α(z) = L1,1 (z), β(z) = L1,−1 (z), γ (z) = L−1,1 (z) and δ(z) = L−1,−1 (z). Remark 2.4. Since a(λ, q 2 ) = c(λ, q 2 ) and b(λ, q 2 ) = d(λ, q 2 ) we see that the R-matrix (2.2) is singular for z = q 2 . Using (1.1) we compute θ(zq −2 ) a(λ, z) b(λ, z) det = q2 . c(λ, z) d(λ, z) θ(zq 2 ) We find that z = q 2 , up to powers of p, is the only zero of the determinant of R. For (a, b, c, d) = (1, 1, 1, −1) and (1, −1, 1, 1) in (2.7) we obtain α(z1 )β(z2 ) = a(µ, z1 /z2 )β(z2 )α(z1 ) + c(µ, z1 /z2 )α(z2 )β(z1 ), β(z1 )α(z2 ) = b(µ, z1 /z2 )β(z2 )α(z1 ) + d(µ, z1 /z2 )α(z2 )β(z1 ).
(2.10)
In the case z1 /z2 = q 2 the right-hand side of the relations in (2.10) are multiples of each other, so this also holds for the left-hand sides giving b(µ, q 2 )α(q 2 z)β(z) = a(µ, q 2 )β(q 2 z)α(z). Simplifying this identity and doing the same for (2.7) for the cases (a, b, c, d) = (±1, 1, ∓1, 1), (±1, −1, ∓1, −1) and (−1, ±1, −1, ∓1) we find θ (q −2µ ) α(q 2 z) β(z) = q 2 θ (q −2(µ+2) ) β(q 2 z) α(z), γ (z) α(q 2 z) = α(z) γ (q 2 z), δ(z) β(q 2 z) = β(z) δ(q 2 z),
(2.11a) (2.11b) (2.11c)
θ (q −2µ ) γ (q 2 z) δ(z) = q 2 θ (q −2(µ+2) ) δ(q 2 z) γ (z).
(2.11d)
From the relations (2.7) for (a, b, c, d) = (±1, ±1, ∓1, ∓1), (±1, ∓1, ∓1, ±1) we obtain three independent relations a(λ, q 2 )α(q 2 z)δ(z) + b(λ, q 2 )γ (q 2 z)β(z) = a(µ, q 2 )[γ (z)β(q 2 z) + δ(z)α(q 2 z)], a(λ, q 2 )β(q 2 z)γ (z) + b(λ, q 2 )δ(q 2 z)α(z) = b(µ, q 2 )[γ (z)β(q 2 z) + δ(z)α(q 2 z)], α(z)δ(q 2 z) + β(z)γ (q 2 z) = γ (z)β(q 2 z) + δ(z)α(q 2 z). (2.12)
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From (2.3) we see that a(λ, z), b(λ, z), c(λ, z) and d(λ, z) have a simple pole for z = q −2 . The residual relations of (2.7) are the relations obtained by multiplying by (z1 /z2 ) − q −2 and taking the limit z1 /z2 → q −2 , see [9]. By convention, we interpret (2.7) so that these are also supposed to hold. The residual relations of (2.7) for (a, b, c, d) = (1, ±1, 1, ∓1), respectively (±1, 1, ∓1, 1), (±1, −1, ∓1, −1) and (−1, ±1, −1, ∓1), are linearly dependent and simplify to (2.11). The residual relations of (2.7) for (a, b, c, d) = (±1, ±1, ∓1, ∓1), (±1, ∓1, ∓1, ±1) reduce to three independent relations of which two can be derived from (2.12). The independent relation can be written as b(µ, q 2 )γ (q 2 z)β(z) − a(µ, q 2 )δ(q 2 z)α(z) = a(λ, q 2 )[γ (z)β(q 2 z) − α(z)δ(q 2 z)]. (2.13) Note that , ε preserve the commutation relations (2.11)–(2.13). Lemma 2.5. The element F (µ) α(z)δ(q 2 z) − γ (z)β(q 2 z) det(z) = F (λ) F (µ) = δ(z)α(q 2 z) − β(z)γ (q 2 z) F (λ) q µ θ (q −2(µ+2) ) θ(q −2µ ) 2 2 = λ δ(q z)α(z) − 2 γ (q z)β(z) q θ (q −2(λ+2) ) q θ(q −2(λ+2) ) q µ θ (q −2µ ) q 2 θ (q −2(µ+2) ) 2 2 = λ α(q z)δ(z) − β(q z)γ (z) , q θ (q −2λ ) θ (q −2λ ) where F (µ) = q µ θ(q −2(µ+1) ), is a central element of FR (M(2)). Moreover, (det(z)) = det(z) ⊗ det(z) and ε(det(z)) = 1. Proof. The equality of the four expressions follows from (2.12), (2.13). The remainder of the lemma follows from [9, Theorem 13].
To FR (M(2)) we adjoin the central element det −1 (z) subject to the relation det(z)det−1 (z) = 1. The comultiplication and counit extend by (det −1 (z)) = det−1 (z) ⊗ det −1 (z),
ε(det −1 (z)) = 1.
It is easily checked that the resulting algebra, denoted by FR (GL(2, C)), is an h-bialgebroid. Note that det(z) and det −1 (z) have (0, 0)-bigrading. In the dynamical representations we consider later, det(z) does not act as id, see Remark 4.3. Therefore we do not put det(z) = 1. Lemma 2.6. The h-bialgebroid FR (GL(2, C)) is an h-Hopf algebroid with the antipode S defined by S(det−1 (z)) = det(z), F (µ) −1 −2 F (µ) −1 −2 det (q z)δ(q −2 z), S(β(z)) = − det (q z)β(q −2 z), F (λ) F (λ) F (µ) −1 −2 F (µ) −1 −2 det (q z)γ (q −2 z), S(δ(z)) = det (q z)α(q −2 z), S(γ (z)) =− F (λ) F (λ) S(f (λ)) = f (µ), S(f (µ)) = f (λ), (2.14) S(α(z)) =
on the generators and extended as an algebra antihomomorphism.
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Proof. By Proposition 2.2 of [15] we only have to check that on the generators we have
S(α(z)) S(β(z)) α(z) β(z) 10 α(z) β(z) S(α(z)) S(β(z)) = = , S(γ (z)) S(δ(z)) γ (z) δ(z) 01 γ (z) δ(z) S(γ (z)) S(δ(z)) (2.15)
and that the antipode preserves the defining relations of the algebra. The proof is straightforward, using the RLL-relations and Lemma 2.5. Note that we need the residual relation (2.13) for the second equality in (2.15).
Next we give a ∗-structure to the obtained h-Hopf algebroid. Therefore we recall the definition of a ∗-structure on h-bialgebroids, see [15]. Assuming ¯ : h → h is a conjugation, we put f (λ) = f (λ), f ∈ Mh∗ .A ∗-operator on an h-bialgebroid A is a C-antilinear, antimultiplicative involution on A satisfying µl (f ) = µl (f )∗ , µr (f ) = µr (f )∗ and (∗ ⊗ ∗) ◦ = ◦ ∗, ε ◦ ∗ = ∗Dh ◦ ε, where ∗Dh is defined by (f Tα )∗ = T−α f . We use complex conjugation on h ∼ = C. Lemma 2.7. The h-Hopf algebroid FR (GL(2, C)) has a ∗-structure defined on the generators by det−1 (z)∗ = det−1 (q −2 /z), α(z)∗ = δ(1/z),
β(z)∗ = −γ (1/z),
γ (z)∗ = −β(1/z),
δ(z)∗ = α(1/z). (2.16)
We call this h-Hopf algebroid the elliptic U (2) quantum group and denote it by FR (U (2)). Proof. We can easily check that this definition preserves the defining relations of the algebra, that it is an involution and that we have (∗ ⊗ ∗) ◦ = ◦ ∗ and ε ◦ ∗ = ∗Dh ◦ ε.
Remark 2.8. Note that S(det(z)) = det−1 (z) and det(z)∗ = det(q −2 /z). 3. Corepresentations of the Elliptic U (2) Quantum Group Before discussing a special corepresentation of the elliptic U (2) quantum group, we recall the general definition of a corepresentation of an h-bialgebroid on an h-space, see [15]. Definition 3.1. An h-space is a vector space over Mh∗ which is also a diagonalizable h-module, V = α∈h∗ Vα , with Mh∗ Vα ⊆ Vα for all α ∈ h∗ . A morphism of h-spaces is an h-invariant (i.e. grade preserving) Mh∗ -linear map. ˜ = We next define the tensor product of an h-bialgebroid A and an h-space V . Put A⊗V (A ⊗ V ), where ⊗ denotes the usual tensor product modulo the relaMh∗ α,β∈h∗ αβ Mh∗ β A ˜ )α and the extension tions µr (f )a ⊗ v = a ⊗ f v. The grading Aαβ ⊗Mh∗ Vβ ⊆ (A⊗V ˜ into an of scalars f (a ⊗ v) = µA (f )a ⊗ v, a ∈ A, v ∈ V , f ∈ M h∗ , make A⊗V l h-space.
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Definition 3.2. A (left) corepresentation of an h-bialgebroid A on an h-space V is an ˜ such that h-space morphism ρ : V → A⊗V ( ⊗ id) ◦ ρ = (id ⊗ ρ) ◦ ρ,
(ε ⊗ id) ◦ ρ = id.
(3.1)
∼ A⊗(A ˜ ⊗V ˜ = ˜ ⊗V ˜ ) The first equality is in the sense of the natural isomorphism (A⊗A) ˜ defined by f T−α ⊗ v ∼ and in the second identity we use the identification V Dh⊗V = f v, for all f ∈ Mh∗ , v ∈ Vα . Choose a homogeneous basis {vk }k of V over Mh∗ , vk ∈ Vω(k) , and introduce the corresponding matrix elements of a corepresentation ρ by ρ(vk ) = j tkj ⊗ vj . For these matrix elements we have from (3.1), (tij ) =
tik ⊗ tkj ,
ε(tij ) = δij T−ω(i) ,
k
and Definition 2.3 implies δkl =
S(tkj )tj l =
j
tkj S(tj l ).
(3.2)
j
Our next objective is to construct explicit corepresentations of FR (U (2)). Define, with the convention that the empty product is 1, vk = vk (z) = γ (z)γ (q 2 z) · · · γ (q 2(N−k−1) z) × α(q 2(N−k) z) · · · α(q 2(N−1) z),
k ∈ {0, 1, . . . , N},
(3.3)
and put V2k−N = µl (Mh∗ )vk , V = V N = N k=0 V2k−N . Then V is an h-space. Note that the grading on V is compatible with the grading of FR (M(2)). We show that ˜ N , making V N a corepresentation of FR (U (2)), see Theorem : V N → FR (U (2))⊗V 3.4. We start with the following preparatory lemma. Lemma 3.3. In the h-Hopf algebroid FR (U (2)) we have α(q 2k z)β(q 2(l−1) z) · · · β(z) θ (q 2(k−l+1) , q 2(µ+l+1) ) = β(z) · · · β(q 2(l−1) z)α(q 2k z) θ (q 2(k+1) , q 2(µ+1) ) θ (q 2 , q 2(µ+k+1) ) β(z) · · · α(q 2i z) · · · β(q 2(l−1) z)β(q 2k z), θ (q 2(k+1) , q 2(µ+1) ) l−1
+
i=0
for all k ≥ l ≥ 1. Proof. For l = 1 this is (2.10). In order to provide for the induction step we interchange the order of the β’s since β(z)β(w) = β(w)β(z) for all z, w ∈ C, use the case l = 1 and then the induction hypothesis with z → q 2 z, k → k − 1 finishes the proof using (2.6) and (2.3).
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Theorem 3.4. In the h-Hopf algebroid FR (U (2)), with vk (z) defined by (3.3), we have (vk (z)) =
N
N tkj (µ, z) ⊗ vj (z),
(3.4)
j =0 N (µ, z) are given by where the matrix-elements tkj
N (µ, z) = tkj
min(k,j )
(q 2(µ+l−j +2) )j −l k N − k (q 2(µ+N−k−2j +l+2) )l 2(µ+N−2j +2) 2(µ+N−2j −k+2l+2) ) l j −l (q )l (q j −l
l=max(0,k+j −N)
× γ (q 2(N−k−1) z) · · · γ (q 2(N−j −k+l) z)δ(q 2(N−j −k+l−1) z) · · · δ(z) × α(q 2(N−1) z) · · · α(q 2(N−l) z)β(q 2(N−l−1) z) · · · β(q 2(N−k) z). Proof. We first deal with the cases k = N and k = 0, and get the general result from the homomorphism property of the comultiplication . Claim. For all k ∈ Z≥0 , (α(z) · · · α(q 2(k−1) z)) =
k
Ckl (µ)α(q 2(k−1) z) · · · α(q 2(k−l) z)β(q 2(k−l−1) z) · · · β(z)
l=0
⊗ γ (z) · · · γ (q 2(k−l−1) z)α(q 2(k−l) z) · · · α(q 2(k−1) z), (3.5) where the coefficients Ckl (µ) ∈ Mh∗ are given by (q 2(µ−l+2) )l k Ckl (µ) = . l (q 2(µ+k−2l+2) )l Note that Ck,0 = Ck,k = 1. We prove the claim by induction on k. For k = 1 this is just (2.9) on α(z). Assume that the claim is true for k. Then we obtain from (2.9) and repeated application of (2.11b), (α(z) · · · α(q 2k z)) = (α(z) · · · α(q 2(k−1) z))(α(q 2k )z) =
k
Ck,l (µ)α(q 2(k−1) z) · · · α(q 2(k−l) z)
l=0
× β(q 2(k−l−1) z) · · · β(z)β(q 2k z) ⊗ γ (z) · · · γ (q 2(k−l) z)α(q 2(k−l+1) z) · · · α(q 2k z) +
k+1
Ck,l−1 (µ)α(q 2(k−1) z) · · · α(q 2(k−l+1) z)
l=1
× β(q 2(k−l) z) · · · β(z)α(q 2k z) ⊗ γ (z) · · · γ (q 2(k−l) z)α(q 2(k−l+1) z) · · · α(q 2k z).
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For l = 0 and l = k + 1, we have Ck+1,0 (µ) = Ck,0 (µ) = 1 and Ck+1,k+1 (µ) = Ck,k (µ) = 1 respectively. So it remains to prove that for 1 ≤ l ≤ k we have Ck+1,l (µ)α(q 2k z) · · · α(q 2(k−l+1) z)β(q 2(k−l) z) · · · β(z) = Ck,l (µ)α(q 2(k−1) z) · · · α(q 2(k−l) z)β(q 2(k−l−1) z) · · · β(z)β(q 2k z) + Ck,l−1 (µ)α(q 2(k−1) z) · · · α(q 2(k−l+1) z) × β(q 2(k−l) z) · · · β(z)α(q 2k z).
(3.6)
Using Ck,l−1 (µ) = Ck,l (µ)
θ (q 2l , q 2(µ+k−2l+3) , q 2(µ+k−2l+2) ) , θ (q 2(k−l+1) , q 2(µ+k−l+2) , q 2(µ−l+2) )
and (2.6) we obtain that the right-hand side of (3.6) equals Ckl (µ)α(q 2(k−1) z) · · · α(q 2(k−l+1) z) × α(q 2(k−l) z)β(q 2(k−l−1) z) · · · β(z)β(q 2k z)
θ (q 2l , q 2(µ+k−l+2) , q 2(µ+k−l+1) ) + β(q 2(k−l) z) · · · β(z)α(q 2k z) . θ (q 2(k−l+1) , q 2(µ+k+1) , q 2(µ+1) )
By Lemma 3.3, with (k, l) replaced by (k − l, k − l) the term in square brackets equals
θ (q 2 , q 2(µ+k+1) ) β(z) · · · α(q 2n z) · · · β(q 2(k−l) z) θ (q 2(k+1) , q 2(µ+1) ) n=0 2l 2(µ+k−l+2) , q ) θ (q ×β(q 2k z) + β(z) · · · β(q 2(k−l)z )α(q 2k z) θ (q 2(k+1) , q 2(µ+1) )
θ (q 2(k+1) , q 2(µ+k−l+1) ) θ (q 2(k−l+1) , q 2(µ+k+1) )
=
k−l
θ (q 2(k+1) , q 2(µ+k−l+1) ) α(q 2k z)β(q 2(k−l) z) · · · β(z), θ (q 2(k−l+1) , q 2(µ+k+1) )
where we use Lemma 3.3 with (k, l) replaced by (k, k − l + 1) in the last step. Using α(z)α(w) = α(w)α(z) for all w, z ∈ C and (2.6) we see that the right-hand side of (3.6) equals the left-hand side using Ck+1,l (µ) = Ck,l (µ)
θ (q 2(k+1) , q 2(µ+k−2l+2) ) . θ (q 2(k−l+1) , q 2(µ+k−l+2) )
This proves the claim. Since the (α,β)- and the (γ ,δ)-commutation relations are similar we analogously have (γ (z) · · · γ (q
2(k−1)
z)) =
k
Ckl (µ)γ (q 2(k−1) z) · · · γ (q 2(k−l) z)δ(q 2(k−l−1) z) · · · δ(z)
l=0
⊗ γ (z) · · · γ (q 2(k−l−1) z)α(q 2(k−l) z) · · · α(q 2(k−1) z). (3.7)
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Using (3.5), (3.7) and that the comultiplication is a morphism we find (γ (z) · · · γ (q 2(N −k−1) z)α(q 2(N−k) z) · · · α(q 2(N−1) z)) =
k N−k
CN−k,m (µ)Ck,l (µ − 2m + N − k)γ (q 2(N−k−1) z) · · · γ (q 2(N−k−m) z)
l=0 m=0
× δ(q 2(N −k−m−1) z) · · · δ(z)α(q 2(N−1) z) · · · α(q 2(N−l) z) × β(q 2(N −l−1) z) · · · β(q 2(N−k) z) ⊗ γ (z) · · · γ (q 2(N−m−l−1) z)α(q 2(N−m−l) z) · · · α(q 2(N−1) z), where we use (2.6), (2.11b). Substituting m = j − l proves the theorem.
In the next proposition we prove that this corepresentation is unitary in a certain sense. Note that this property is an extension of unitarizability of a corepresentation introduced in [15]. N (µ, z) of the corepresentation in Theorem 3.4 Proposition 3.5. The matrix elements tkj satisfy N k (µ)S(tkj (µ, z))∗ = j (λ)tjNk (µ, q −2(N−2) /z)
N−1
det −1 (q −2i /z),
i=0
with k (µ) =
(q 2(µ−k+2) )k N k (q 2(µ+N−2k+2) )k
N−k−1 i=0
k−1 q −(µ+N−2k−i) q −(µ−k+i) . −2(µ+N−2k−i+1) ) θ (q θ(q −2(µ−k+i+1) ) i=0
N−1 −1 −2i Proof. To simplify the formulas in the proof we denote D = i=0 det (q /z), k−1 N (q 2(µ−k+2) )k GNk (µ) = 2(µ−N −2k+2) ) and Fk (µ) = i=0 F (µ + i), where F is defined in (q k k Lemma 2.5. N (µ, z) for k or j equal to 0 or From Theorem 3.4 we see that the matrix elements tkj N consist of a single term. Using Lemmas 2.6, 2.7 and the commutation relations of the elliptic quantum group proves the proposition in case j = N , N [S(tkN (µ, z))]∗ = D
FN−k (µ − k + 1) Fk (λ − k) FN−k (λ − N ) Fk (µ − k)
× α(q 2 /z) · · · α(q −2(k−2) /z)β(q −2(k−1) /z) · · · β(q −2(N−2) /z) FN−k (µ − k + 1) Fk (λ − k) N =D GNk (µ)−1 tNk (µ, q −2(N−2) /z). FN−k (λ − N ) Fk (µ − k) N (µ, z)) = N t N (µ, z) ⊗ t N (µ, z) and σ ◦ ((∗ ◦ S) ⊗ (∗ ◦ S)) ◦ = From (tkN j =0 kj jN ◦ (∗ ◦ S) we obtain N j =0
N N S(tjNN (µ, z))∗ ⊗ S(tkj (µ, z))∗ = (S(tkN (µ, z))∗ ).
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This relation gives N
N N FN−j (µ − j + 1)Fj (µ − j )GNj (µ)−1 tNj (µ, q −2(N−2) /z) ⊗ S(tkj (µ, z))∗
j =0
= [1 ⊗ D FN−k (µ − k + 1)Fk (µ − k)GNk (µ)−1 ] ×
N
N tNj (µ, q −2(N−2) /z) ⊗ tjNk (µ, q −2(N−2) /z).
j =0 N (µ, z)}N are linearly independent (this follows easily from Proposition 4.2 Since {tNj j =0 and Lemma 4.4), the identity holds termwise. So (2.4) proves the proposition.
4. Discrete Bi-Orthogonality for Elliptic Hypergeometric Series Using Proposition 3.5 we can reformulate the orthogonality relations (3.2) for the matrix elements as δkl =
N j =0
=
(tjNl (µ, z))∗
N−1 j (λ) det −1 (q −2i /z) tj k (µ, q −2(N−2) /z) k (µ)
N N−1 l (λ) N N (µ, z))∗ det −1 (q −2i /z). tlj (µ, q −2(N−2) /z)(tkj j (µ) j =0
(4.1a)
i=0
(4.1b)
i=0
To obtain commutative versions of (4.1), we need to represent the algebra FR (U (2)) explicitly. For this we need the notion of a dynamical representation of an h-algebra, see [6, 7, 9, 15]. Let V = α∈h∗ Vα be an h-space and let (Dh,V )αβ be the space of C-linear operators U on V such that U (gv) = T−β (g)U (v) and U (Vγ ) ⊆ Vγ +β−α for all g ∈ Mh∗ , v ∈ Vβ , γ ∈ h∗ . Then the space Dh,V = α,β∈h∗ (Dh,V )α,β is an h-algebra with the moment maps µl , µr : Mh∗ → (Dh,V )00 given by µl (f )(v) = T−α (f )(v) and µr (f )(v) = f v for all v ∈ Vα . Definition 4.1. A dynamical representation of an h-algebra A on an h-space V is an h-algebra homomorphism A → Dh,V . Proposition 4.2 (see [9]). Let ω ∈ C be arbitrary and Hω be the h-space with basis ∞ ω ω ω = {ek }∞ and weight decomposition H H k=0 ω−2k , Hω−2k = Mh∗ ek . Then there k=0 ω exists a dynamical representation π : FR (M(2)) → Dh,Hω , defined on the generators by π ω (α(z))(gek ) = Ak (λ, z)T−1 gek , π ω (β(z))(gek ) = Bk (λ, z)T1 gek+1 , π ω (γ (z))(gek ) = Ck (λ, z)T−1 gek−1 , π ω (δ(z))(gek ) = Dk (λ, z)T1 gek , ω ω π (µr (f ))(gek ) = f (λ)gek , π (µl (f ))(gek ) = f (λ − ω + 2k)gek ,
(4.2)
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where g ∈ Mh∗ and θ (q −2(λ+1)−2k )θ (zq ω−2k+1 ) , θ (q −2(λ+1) )θ (zq ω+1 ) θ (q 2 )θ (zq −2(λ+1)+ω−2k−1 ) Bk (λ, z) = q k , θ (q −2(λ+1) )θ (zq ω+1 ) θ (q 2k )θ (q 2(ω−k+1) )θ (zq 2(λ+1)−ω+2k−1 ) Ck (λ, z) = q −(k−1) , C0 (λ, z) = 0, θ (q 2 )θ (q 2(λ+1) )θ (zq ω+1 ) θ (q −2(λ+1−ω+k) )θ (zq −ω+2k+1 ) Dk (λ, z) = . θ (q −2(λ+1) )θ (zq ω+1 ) Ak (λ, z) = q 2k
Remark 4.3. Using the addition formula (1.1) we obtain π ω (det(z)) = q ω
θ (zq 1−ω ) id, θ (zq 1+ω )
so det(z) acts as a scalar. Note that this scalar is 1 if ω = 0. The action of a matrix element in the dynamical representation defined above can be calculated in terms of elliptic hypergeometric series. Lemma 4.4. For the dynamical representation of Proposition 4.2 we have N Nω π ω (tkj (µ, z))(gem ) = τkj m (λ, z)(TN−2j g)em+k−j , Nω (λ, z) is given by where τkj m Nω τkj m (λ, z) 3
5
= (−1)N−k θ (q 2 )k−j q 2 k(k−1)+N(N+1)+ 2 j (j +1)+2N(λ−k−2j )+m(k−j )+3j k−2kλ (q −2(λ+1) , q 2(m+k−j +1) , q 2(N−k−j +1) , q 2(ω−m−k+1) , zq 2(λ+N−2j +m+2)−ω−1 )j × (q 2 , q 2(λ+N−k−2j +2) , q 2(λ−j +2) )j ×
1 (zq −2(λ−2j +m+k)+ω−1 )k (q −2(λ+N−2j −ω+m) , zq 2(m+k)−ω+1 )N−k−j ω+1 −2(λ+N−2j ) 2(λ−j +1) (zq )N (q )k (q )N−k−j
×10 ω9 [q 2(λ+N−2j −k+1) ; q −2k , q −2j , q 2(λ−j +1) , q 2(λ+N−2j −ω+m+1) , q 2(λ+N+2+m−2j ) , zq 2(N−m−k)+ω+1 , z−1 q −2(m+k−1)+ω−1 ]. Note that if k + j ≥ N , one of the elliptic Pochhammer symbols in the denominator of (1.2), (q 2(N −k−j +1) )l , equals zero. This singularity is only apparent because of (q 2(N −k−j +1) )j in front of the 10 ω9 . Lemma 4.4 is a generalization of [16, §6] for the quantum SU (2) group and of [15, Prop. 4.5 ] for the dynamical quantum SU (2) group.
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Proof. From Proposition 4.2 and Theorem 3.4 it follows min(k,j )
N (µ, z))(gem ) = π ω (tkj
l=max(0,k+j −N)
× ×
k N − k (q 2(λ+N−k−2j +l+2) )l l j −l (q 2(λ+N−2j +2) )l
(q 2(λ+l−j +2) )j −l (q 2(λ+N−2j −k+2l+2) )j −l j −l−1
Cm+k−l−n (λ − j + l + 1 + n, q 2(n+N−k−j +l) z)
n=0
×
N−k−j +l−1
Dm+k−l (λ + N − k − 2j + 2l − 1 − n, q 2n z)
n=0
×
j −l−1
Am+k−l (λ + n + N − k − 2j + l + 1, q 2(N−l+n) z)
n=0
×
k−l−1
Bm+n (λ + N − 2j − 1 − n, q 2(n+N−k) z)
n=0
×(TN−2j g)em+k−j .
(4.3)
This gives the required form of the lemma, and it remains to show that we can idenNω (λ, z) with an elliptic hypergeometric series. From the explicit expressions of tify τkj m Proposition 4.2 we see that we can rewrite the four products in terms of elliptic factorials: j −l−1
Am+k−l (λ + n + N − k − 2j + l + 1, q 2(N−l+n) z)
n=0
= (−1)l z−l q 2l(l−N)−l(ω+l) k−l−1
(q 2(λ+N−2j +m+2) , zq 2(N−m−k)+ω+1 , q 2(λ+N−2j −k+2) )l , (q −2(N−1)−ω−1 /z)l (q 2(λ+N−2j −k+2) )2l
Bm+n (λ + N − 2j − 1 − n, q 2(n+N−k) z)
n=0 1
= (−1)l q 2l(m+k−l)+l(l+1)+ 2 (k−l)(2m+k−l−1) (q 2(λ+N −2j −k+1) , q −2(N−1)−ω−1 /z)l (zq −2(λ+k+m−2j )+ω−1 )k ×θ (q 2 )k , (q 2(λ+m−2j +1)−ω+1 )l (q −2(λ+N−2j ) , zq 2(N−k)+ω+1 )k j −l−1
Cm+k−l−n (λ − j + l + 1 + n, q 2(n+N−k−j +l) z)
n=0 1
= (−1)l θ (q 2 )−j q j (l+1−m−k)−l(m+k+2)+ 2 (j −l)(j −l−1) (q 2(λ−j +2) , zq 2(N−k−j )+ω+1 )l × −2(m+k) 2(ω−m−k+1) (q ,q , zq 2(λ+N−2j +m+2)−ω−1 )l 2(λ+N−2j +M+2)−ω−1 , q 2(m+k−j +1) ) 2(ω−m−k+1) , zq (q j × , (q 2(λ−j +2) , zq 2(N−k−j )+ω+1 )j
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N −k−j +l−1
Dm+k−l (λ + N − k − 2j + 2l − 1 − n, q 2n z)
n=0
= (−1)N−j −k+l zl ×q (N−j −k)(2λ+N−3j −k+2l+2)+l(ω+l) ×
(q −2(λ+N −2j −ω+m) , zq 2(m+k)−ω+1 )N−k−j (q 2(λ−j +1) , zq ω+1 )N−k−j
(q 2(λ+N−2j −ω+m+1) , q 2(λ−j +1) , q −2(m+k−1)+ω−1 /z)l , (q 2(λ+N−2j −k+1) )2l (zq 2(N−k−j )+ω+1 )l
(4.4)
where we use elementary transformation formulas, e.g. (aq −4l )l = (−1)l (aq −4l )l q l(l−1) (q 2 /a)2l /(q 2 /a)l for the elliptic factorials. Furthermore for the elliptic binomials and the other factor in (4.3) we have (q −2k )l k = (−1)l q 2l(k−l+1)+l(l−1) , l (q 2 )l (q −2j )l (q 2(N−k−j +1) )j N −k = (−1)l q 2l(j −l+1)+l(l−1) , j −l (q 2(N−k−j +1) )l (q 2 )j (q 2(λ+l−j +2) )j −l (q 2(λ+N−k−2j +l+2) )l = (−1)j q 2j (λ−j +2)+j (j −1) (q 2(λ+N−2j +2) )l (q 2(λ+N−2j −k+2l+2) )j −l ×
[(q 2(λ+N−k−2j +2) )2l ]2 (q −2(λ+1) )j 2(λ+N−k−2j +2) 2(λ+N−2j +2) , q 2(λ−j +2) , q 2(λ+N−k−j +2) ) (q ,q
l (q
(4.5)
2(λ+N−2j −k+2) )
Then, substituting (4.4) and (4.5) into (4.3) gives the required result.
. j
Analogously we can compute Lemma 4.5. For the dynamical representations of Proposition 4.2 we have N Nω (µ, z))∗ )(gem ) = τ˜kj π ω ((tkj m (λ, z)(T−N+2j g)em+j −k , Nω (λ, z) is given by where τ˜kj m Nω τ˜kj m (λ, z)
= q 2j
2 + 1 k(k−1)− 1 j (j −1)+k(1−m−j )+2m(N−j −k)+mj +2(N−k)(λ−k+1)−(N−k−j )(N−k−j −1) 2 2
×(−1)N−j θ (q 2 )j −k
(q 2(N−k−j +1) , q −2(λ−N+2j +1) , q −2(λ+2j −k+m)+ω+1 /z)j (q 2 )j (q 2(λ−k+2) , q −2(λ+2j −N) , q −2(N−k−1)+ω+1 /z)j
(q 2(m+j −k+1) , q 2(ω−m−j +1) , q −2(N−3−λ+k−m−j )−ω−1 /z)k (q 2(λ−k+2) , q −2(N−1)+ω+1 /z)k −2(λ+j +m+1−k) , q −2(N−k+m−1)+ω+1 /z)N−j −k (q × (q 2(λ+2−N+j ) , q −2(N−j −k)+ω+1 /z)N−j −k ×
×10 ω9 [q 2(λ−k+1) ; q −2k , q −2j , q 2(λ+j −N +1) , q 2(λ−k−ω+m+j +1) , q 2(λ+j +m−k+2) , zq 2(N−m−j )+ω−1 , q −2(m+j −1)+ω+1 /z].
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Lemmas 4.4 and 4.5 can be used to convert the relations (4.1) to bi-orthogonality relations for elliptic hypergeometric series. The resulting bi-orthogonality relations of Theorem 4.6 and 4.8 have been obtained previously by Frenkel and Turaev [11] and Spiridonov and Zhedanov [25] (see also Remark 4.9). Theorem 4.6. A bi-orthogonality relation for the elliptic hypergeometric series is given by δkl hk =
N
wj
10 ω9 [q
2(−2l−j +1)
; q −2j , q −2l , q 2(−l−N+1) , q 2(−l−ω+M+1) ,
j =0
q 2(−l+M+2) , zq 2(N−M−j −l)+ω−1 , q −2(M+j +l−2)+ω−1 /z] × 10 ω9 [q 2(−2k−j +1) ; q −2j , q −2k , q 2(−k−N +1) , q 2(−k−ω+M+1) , q 2(−k+M+2) , zq 2(N−M−j −k)+ω−3 , q −2(M+j +k−2)+ω+1 /z], where the quadratic norm hk and the weight function wj are given by hk =
(q 2 , q −2(+M+1) , q −2(−ω+M) , q −2(−N) )k (q −2(+1) )2k (q −2 )2k (q 2(M+1) , q −2N , q −2(ω−M) , q −2 )k (zq 2M−ω−1 , q 2(M−N)−ω+5 /z)k × −2(+M)+ω+1 (q /z, zq 2(N−−M)+ω−5 )k −2(−ω+2M+1) , q −2 )N (zq ω−1 , zq −ω−3 )N (q × −2(+M+1) −2(−ω+M) , (q ,q )N (zq 2M−ω−1 , zq −2M)+ω−3 )N
and wj =: w1 (j, k)w2 (j, l) with w1 (j, k) θ (q 2(−ω+2M−N+1+2j ) ) (q 2(−ω+2M−N+1) )j θ (q 2(−ω+2M−N+1) ) (q 2(−ω+2M+2) )j 2(−k+M)−ω−1 )j (zq × −2(N−2−M−k)−ω+1 (q /z)j
= q 2j −2k
×
(q 2(M+k+1) , q −2(N−k) , q −2(−k+1) , q −2(ω−M−k) )j 2 2(−N +M+2) (q , q , q 2(M+1) , q −2(−2k+1) , q −2N , q −2(ω−M) , q 2(−N−ω+M+1) )
w2 (j, l) =
, j
(q 2(M+l+1) , q −2(N−l) , q −2(−l+1) , q −2(ω−M−l) )j (q −2(−2l+1) )j −2(N−3−+l−M)−ω−1 /z)j (q × . 2(M+l)−ω−1 (zq )j
Remark 4.7. These relations are bi-orthogonality relations since there is a shift in the spectral parameter z. Omitting all other parameters the bi-orthogonality relations are in fact relations of the form δkl hk = wj Pl (j, q 2 z)Pk (j, z). j
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Proof. Applying the dynamical representation π ω of Proposition 4.2 to (4.1a) gives N
δkl em =
j =max(0,k−m)
×
N−1
q −ω
i=0
Nω τ˜j,l,m+j −k (λ, z)
j (λ − ω − N + 2m + 2j − 2k + 2l) k (λ − N + 2l)
θp (q −2i+1+ω /z) Nω τj km (λ − N + 2l, q −2(N−2) /z)em−k+l . θp (q −2i+1ω /z)
Replacing λ + 2l by , m − k by M and z by z we obtain δkl =
N j =max(0,−M)
j ( − ω + 2M + 2j − N ) −ω θp (q −2i+1+ω /z) q k ( − N ) θp (q −2i+1ω /z) N
i=0
Nω Nω ×τ˜j,l,M+j ( − 2l, z)τj,k,M+k ( − N, q −2(N−2) /z).
Using Lemmas 4.4 and 4.5 and elementary relations for the elliptic factorials proves the theorem.
Theorem 4.8. The dual bi-orthogonality relation for the elliptic hypergeometric series is given by w1 (l, j )w2 (k, j ) 2(−2j −l+1) δkl = ; q −2l , q −2j , q 2(−N−j +1) , 10 ω9 [q (hj ) j
q 2(−j −ω+M+1) , q 2(+M−j +2) , q −2(M+j +l−2)+ω+1 /z, zq −2(M+j −N+l+1)+ω−1 ] × 10 ω9 [q 2(−2j −k+1) ; q −2k , q −2j , q 2(−N−j +1) , q 2(−j −ω+M+1) , q 2(+M−j +2) , q −2(M+j +k−1)+ω+1 /z, zq −2(M+j −N+k)+ω−1 ], where w1 , w2 and hj are as in Theorem 4.6. Proof. These dual bi-orthogonality relations can be computed from (4.1b) by applying the dynamical representation. Since the bi-orthogonal system in Theorem 4.6 is known to be self-dual [25], we can also obtain the dual relations from Theorem 4.6.
Remark 4.9. In [11] an elliptic analogue of Bailey’s transformation formula is proved. Let bcdef g = a 3 q 2(n+2) and λ = a 2 q 2 /bcd. Then −2n ] 10 ω9 [a; b, c, d, e, f, g, q (aq 2 , aq 2 /ef, λq 2 /e, λq 2 /f )
−2n ]. (4.6) 10 ω9 [λ; λb/a, λc/a, λd/a, e, f, g, q (aq 2 /e, aq 2 /f, λq 2 /ef, λq 2 )n We can relate the bi-orthogonality relations of Theorem 4.6 and 4.8 to the ones given in [25]. To obtain this relation explicitly we have to apply the elliptic analogue of Bailey’s transformation formula (4.6) twice to both 10 ω9 -functions in our bi-orthogonality relations in different ways. Finally, let us emphasize that we do not need Bailey’s transformation formula to obtain the bi-orthogonality relations of Theorem 4.6 and 4.8 in the symmetric form given.
=
n
Remark 4.10. Using the dynamical representation of Proposition 4.2 we can obtain the transformation formula (4.6) from the unitarity property of the corepresentations stated in Proposition 3.5. Acknowledgement. We thank the referee for constructive remarks.
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References 1. Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SOS model and generalized RogersRamanujan-type identities. J. Statist. Phys. 35, 193–266 (1984) 2. Baxter, R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193–228 (1972) 3. Baxter, R.J.: Eight-vertex model in lattice statistics and one-dimensional anisotropic spin chain, I, II. Ann. Phys. 76, 1–24, 25–47 (1973) 4. Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Exactly solvable SOS models. II. Proof of the star-triangle relation and combinatorial identities. In: Conformal field theory and solvable lattice models, Boston, MA: Academic Press, 1988, pp. 17–122 5. Enriquez, B., Felder, G.: Elliptic quantum groups Eτ,η (sl2 ) and quasi-Hopf algebras. Commun. Math. Phys. 195, 651–689 (1998) 6. Etingof, P., Schiffmann, O.: Lectures on the dynamical Yang-Baxter equations. In: Quantum groups and Lie theory, Vol. 290 of London Math. Soc. Lecture Note Ser., Cambridge: Cambridge Univ. Press, 2001, pp. 89–129 7. Etingof, P., Varchenko, A.: Solutions of the quantum dynamicalYang-Baxter equation and dynamical quantum groups. Commun. Math. Phys. 196, 591–640 (1998) 8. Felder, G.: Elliptic quantum groups. In: XIth International Congress of Mathematical Physics, Cambridge, MA: Internat. Press, 1995, pp. 211–218 9. Felder, G., Varchenko, A.: On representations of the elliptic quantum group Eτ,η (sl2 ). Commun. Math. Phys. 181, 741–761 (1996) 10. Foda, O., Iohara, K., Jimbo, M., Kedem, R., Miwa, T., Yan, H.: An elliptic quantum algebra for sl2 . Lett. Math. Phys. 32, 259–268 (1994) 11. Frenkel, I.B., Turaev, V.G.: Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions. In: The Arnold-Gelfand mathematical seminars, Boston, MA: Birkh¨auser Boston, 1997, pp. 171–204 12. Jimbo, M., Odake, S., Konno, H., Shiraishi, J.: Quasi-Hopf twistors for elliptic quantum groups. Transform. Groups 4, 303–327 (1999) 13. Kajiwara, K., Noumi, M., Masuda, T., Ohta, Y., Yamada, Y.: 10 E9 solution to the elliptic Painlev´e equation. J. Phys. A 36, 263–272 (2003) 14. Koelink, H.T.: Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications. Acta Appl. Math. 44, 295–352 (1996) 15. Koelink, E., Rosengren, H.: Harmonic analysis on the SU(2) dynamical quantum group. Acta Appl. Math. 69, 163–220 (2001) 16. Koornwinder, T.H.: Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials. Nederl. Akad. Wetensch. Indag. Math. 51, 97–117 (1989) 17. Rosengren, H.: A proof of a multivariable elliptic summation formula conjectured by Warnaar. In: q-series with applications to combinatorics, number theory, and physics, Providence, RI: Am. Math. Soc., 2001, pp. 193–202 18. Rosengren, H.: Elliptic hypergeometric series on root systems. Adv. Math, to appear 19. Rosengren, H.: Duality and self-duality for dynamical quantum groups. Algebr. Represent. Theory, to appear 20. Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Funkt. Anal. i Prilozhen. 16, 27–34 (1982) 21. Spiridonov, V.P.: Elliptic beta integrals and special functions of hypergeometric type. In: Pakuliak, S., von Gehlen, G. (eds.), Integrable structures of exactly solvable two-dimensional models of quantum field theory, Dordrecht: Kluwer Acad. Publ., 2001, pp. 305–313 22. Spiridonov, V.P.: An elliptic incarnation of the Bailey chain. Int. Math. Res. Not. 37, 1945–1977 (2002) 23. Spiridonov, V. P.: Theta hypergeometric series. In: Malyshev, M.A., Vershik, A.M. (eds.), Asymptotic Combinatorics with Applications to Mathematical Physics, Dordrecht: Kluwer Acad. Publ., 2002, pp. 307–327 24. Spiridonov, V.P.: Theta hypergeometric integrals, math.CA/0303205 25. Spiridonov, V., Zhedanov, A.: Spectral transformation chains and some new biorthogonal rational functions. Commun. Math. Phys. 210, 49–83 (2000) 26. Vilenkin, N.Ja., Klimyk, A.U.: Representation of Lie groups and special functions. Vol. 1, Dordrecht: Kluwer Academic Publishers Group, 1991 27. Warnaar, S.O.: Summation and transformation formulas for elliptic hypergeometric series. Constr. Approx. 18, 479–502 (2002) Communicated by L. Takhtajan
Commun. Math. Phys. 245, 539–550 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1018-y
Communications in
Mathematical Physics
Remarks on the Blow-up of the Euler Equations and the Related Equations Dongho Chae Department of Mathematics, Seoul National University, Seoul 151-742, Korea. E-mail: [email protected] Received: 21 April 2003 / Accepted: 24 September 2003 Published online: 19 December 2003 – © Springer-Verlag 2003
Abstract: We consider the Euler system for inviscid incompressible fluid flows, and its perturbations in Rn , n ≥ 2. We prove global well-posedness of this perturbed Euler system in the Triebel-Lizorkin spaces for initial vorticity which is small in the critical Besov norms. Comparison type theorems about the blow-up of solutions are proved between the Euler system and its perturbations. We also study the possiblity of the self-similar type of blow-up of solutions to the equations. 1. Introduction and Main Results We are mainly concerned with the following Euler equations for the homogeneous incompressible fluid flows: ∂v + (v · ∇)v = −∇p, (x, t) ∈ Rn × (0, ∞) ∂t (E) div v = 0, (x, t) ∈ Rn × (0, ∞), v(x, 0) = v0 (x), x ∈ Rn where v = (v1 , · · · , vn ), vj = vj (x, t), j = 1, · · · , n, is the velocity of the flow, p = p(x, t) is the scalar pressure, and v0 is the given initial velocity, satisfying div v0 = 0. The local well-posedness of the solution is established by many authors in various function spaces [12–14, 5, 21, 22, 3, 4]. The question of finite (or infinite) time blow-up of such a local regular solution of (E) is an outstanding open problem in mathematical fluid mechanics. One of the most significant achievements in this direction is the celebrated Beale-Kato-Majda (BKM) criterion for the blow-up of solutions [2], which states T∗ m lim sup v(t)H = ∞ if and only if ω(s)L∞ ds = ∞, tT∗
0
540
D. Chae
where ω =curl v is the vorticity of the flows. Bahouri and Dehman also obtained similar blow-up criterion in the H¨older space [1]. Recently the BKM criterion has been refined by Kozono and Taniuchi [15], replacing the L∞ norm of vorticity by the BMO norm, and by 0 0 the author of this paper [3], replacing the L∞ norm of vorticity by the B˙ ∞,∞ (= F˙∞,∞ ) s norm and the Sobolev norm u(t)H m by the Triebel-Lizorkin norm u(t)Fp,q respecm . We also 0 tively. We note here that L∞ → BMO → B˙ ∞,∞ , and H m (Rn ) = F2,2 mention that there is a geometric type of blow-up criterion, using deep structure of the nonlinear term of the Euler equation [8]. In this paper we study the well-posednes/blow-up problems of the Euler system and its perturbations, which are supposed to be “closer” to the original Euler system than the usual Navier-Stokes perturbation. In order to optimize our results we use the Besov/Triebel-Lizorkin spaces. Our first perturbation of (E) is the following: ∂u + a(t)(u · ∇)u = −∇q, (x, t) ∈ Rn × (0, ∞) ∂t (aE) div u = 0, (x, t) ∈ Rn × (0, ∞), u(x, 0) = u0 (x), x ∈ Rn where u(x, t), q = q(x, t) are similar to the above, and u0 is a given initial vector field satisfying div u0 = 0. a(t) > 0 is a given continuous real valued function on [0, ∞). If we set a(t) ≡ 1, then the system (aE) reduces to the well-known Euler equations for homogeneous incompressible fluid flows. Below we will impose the condition that a(·) ∈ L1 (0, ∞). We observe that if we choose, say a(t) = 1 for t ∈ [0, t0 ], and 1 a(t) = 1+(t−t 2 for t ∈ (t0 , ∞), then the system (aE) coincides with (E) during the 0) time interval [0, t0 ], and distorts from (E) after that. For the system (aE) we first state the following small data global existence result. Theorem 1.1 (Small data global existence for (aE)). Let s > n/p, with (p, q) ∈ [1, ∞) × [1, ∞], or s = n with p = 1, q ∈ [1, ∞]. Suppose a(·) ∈ L1 (0, ∞). There s satisfies exists an absolute constant C0 > 0 such that if initial vorticity ω0 ∈ Fp,q ω0 B˙ 0
∞,1
< C0
−1
∞
a(t)dt
,
0
s+1 ) of (aE) exists. Moreover, the solution then a global unique solution u ∈ C([0, ∞); Fp,q satisfies the estimate
∞ C0 0 a(t)dtω0 B˙ 0 ∞,1 . s s exp
∞ sup ω(t)Fp,q (1.1) ≤ ω0 Fp,q 1 − C a(t)dtω 0≤t<∞ 0 0 0 B˙ 0 ∞,1
− 2s
s is the usual fractional order Remark 1.1. Since W s,p (Rn )(= (1−) Lp (Rn )) = Fp,2 Sobolev space, Theorem 1.1 implies immediately the global well-posedness of (aE) in W s,p (Rn ) for initial data u0 ∈ W s,p (Rn ) with ω0 B˙ 0 sufficiently small. We empha∞,1 0 size here that we need smallness only for the B˙ ∞,1 norm of vorticity. In view of the 0 ∞ ˙ embedding B∞,1 → L (see Lemma 2.1 below), it would be interesting to improve the above result to the case with the smallness assumption on the weaker norm of vorticity, ω0 L∞ .
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The following theorem states the equivalence of local existence of the Euler system with the global existence of the perturbed system with suitable modifications of the initial data. Theorem 1.2 (Comparison between (E) and (aE)). Let s > n/p+1, (p, q) ∈ [1, ∞) × [1, ∞] (s = n + 1 with p = 1, q ∈ [1, ∞] respectively). Then, the following statements (i)–(iv) are equivalent. (In (i) and (ii) below v E is the solution of the Euler system (E) with the initial data v0E ; in (iii) and (iv) below, u is the solution of (aE) with the initial T∗ data u0 (x) = ∞ a(s)ds v0E (x).) 0
s , namely (i) The solution v E blows up at t = T∗ < ∞ in Fp,q s = ∞. lim sup v E (t)Fp,q
(1.2)
tT∗
(ii) Let ωE =curl v E , then T∗ E ω (t)B˙ 0 dt = ∞ ∞,∞
0
T∗
ω (t)B˙ 0 dt = ∞ resp. . E
∞,1
0
(1.3)
(iii) The solution u blows at t = ∞, namely s lim sup u(t)Fp,q = ∞.
(1.4)
t∞
(iv) Let ω =curl u, then ∞ ω(t)B˙ 0 a(t)dt = ∞ 0
∞,∞
∞ 0
ω(t)B˙ 0 a(t)dt = ∞ resp. . ∞,1
(1.5)
s by · W s,p (Rn ) in (1.3) and (1.5). Remark 1.2. As in Remark 1.1 we can replace · Fp,q Combining Theorems 1.1 and 1.2, we obtain the following lower bound estimate of the possible blow-up time for (E), the proof of which is in the next section.
Corollary 1.1. Suppose s > n/p + 1, (p, q) ∈ [1, ∞) × [1, ∞]. Let the solution v E of s blow up in F s at T . Then, we have (E) with initial data v0E ∈ Fp,q ∗ p,q T∗ ≥
C0 , E ω0 B˙ 0 ∞,1
(1.6)
where C0 is the absolute constant introduced in Theorem 1.1 and ω0E = curl v0E . Remark 1.3. In the proof of the local existence theorem, Theorem 1.3 ([3]), we could obtain the lower bound of the possible blow-up time as T∗ ≥
C1 E s v0 Fp,q
for s > n/p + 1, (p, q) ∈ [1, ∞) × [1, ∞], where C1 = C1 (p, q). Since we have the s , we find that the lower bound (1.6) is inequalities ω0E B˙ 0 ≤ v0E B˙ 1 ≤ Cv0E Fp,q ∞,1 ∞,1 sharper in general.
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Next, we consider the following damping perturbation of the Euler equations: ∂u + (u · ∇)u = −∇q − εu, (x, t) ∈ Rn × (0, ∞) ∂t (E)ε div u = 0, (x, t) ∈ Rn × (0, ∞) u(x, 0) = u0 (x), x ∈ Rn with ε > 0, which could be considered as a “milder” perturbation of the Euler system than the usual Navier-Stokes system. Actually the system (E)ε is a special case of (aE). Indeed, if we set U (x, t) = eεt u(x, t), P (x, t) = e−εt p(x, t), then we find (U, P ) satisfies ∂U + e−εt (U · ∇)U = −∇P , (x, t) ∈ Rn × (0, ∞) ∂t div U = 0, (x, t) ∈ Rn × (0, ∞) U (x, 0) = u0 (x), x ∈ Rn , which is the system (aE) with a(t) = e−εt . Applying Theorem 1.1, we immediately obtain the following result. Corollary 1.2 (Small data global existence for (E)ε ). Let s > n/p, with (p, q) ∈ [1, ∞) × [1, ∞], or s = n with p = 1, q ∈ [1, ∞]. There exists an absolute constant s and the “damping constant” ε satisfies C0 > 0 such that if initial vorticity ω0 ∈ Fp,q ω0 B˙ 0
∞,1
<
ε , C0
s+1 ) of (E) exists. Moreover the solution then a global unique solution u ∈ C([0, ∞); Fp,q ε satisfies the estimate
C0 ω0 B˙ 0 ∞,1 s s exp sup ω(t)Fp,q . (1.7) ≤ ω0 Fp,q ε − C0 ω0 B˙ 0 0≤t<∞ ∞,1
Remark 1.4. A similar remark to Remark 1.1, concerning the changes of the function spaces into the more familiar spaces such as W s,p (Rn ), also holds for Corollary 1.2. Remark 1.5. The Euler system (E) has scaling property that if (v, p) is a solution of the system (E), then for any λ > 0 and α ∈ R, the functions v λ,α (x, t) = λα v(λx, λα+1 t),
pλ,α (x, t) = λ2α v(λx, λα+1 t)
(1.8)
are also solutions of (E) with the initial data v0λ,α (x) = λα v0 (λx). Note that ω0λ,α B˙ 0
= λα+1 ω0 B˙ 0 , while ∞,1 ∞ ∞ ωλ,α (t)L∞ dt = ω(t)L∞ dt. ∞,1
0
0
Thus, choosing α = −1, we can change the size of initial data to obtain the large data global existence results from the small data ones by application of the BKM criterion. On the other hand, (Eε ) has the scaling property that if u(x, t) is a solution with initial data u0 (x), then uλ (x, t) = λ−1 u(λx, t) is a solution with initial data uλ0 (x) = λ−1 u0 (λx). With such a scaling transform we cannot change the size, ω0 B˙ 0 of the initial data, ∞,1 and the small initial data global existence result cannot be transferred into the large data ones.
Blow-up of the Euler Equations and the Related Equations
543
Corollary 1.3 (Comparison between (E) and (E)ε ). Let s > n/p + 1, (p, q) ∈ [1, ∞) × [1, ∞] (s = n with p = 1, q ∈ [1, ∞] respectively). Then, the following statements (i)–(iv) are equivalent. (In (i) and (ii) below v E is the solution of the Euler system (E) with the initial data v0E ; in (iii) and (iv) below, λ > 0 is given, and uε is the solution of (E)ε with ε = Tλ∗ and with the initial data u0 (x) = λ1 v0E (x).) s , namely (i) The solution v E blows up at t = T∗ < ∞ in Fp,q s = ∞. lim sup v E (t)Fp,q
(1.9)
tT∗
(ii) Let ωE =curl v E , then T∗ E ω (t)B˙ 0 dt = ∞ 0
∞,∞
T∗
ω (t)B˙ 0 dt = ∞ resp. . E
∞,1
0
(1.10)
(iii) The solution uε blows at t = ∞, namely s lim sup uε (t)Fp,q = ∞.
(1.11)
t∞
(iv) Let ωε =curl uε , then ∞ ωε (t)B˙ 0 dt = ∞ 0
∞,∞
0
∞
ωε (t)B˙ 0 dt = ∞ resp. . ∞,1
(1.12)
Remark 1.6. In terms of the usual Sobolev spaces, H m (Rn ) with m > n2 + 1, Corollary 1.3 implies that if we have local solution v E ∈ C([0, T ]; H m (Rn )) to the problem (E) with initial data v0E , then necessarily we have global solution u ∈ C([0, ∞); H m (Rn )) of (E)ε with ε = T1∗ and the same initial data u0 = v0E . This resembles the comparison type of result between the Euler equations and the Navier-Stokes equations obtained by Constantin(See Theorem 1.1[7]). Regarding Remark 1.5 above it would be interesting to ask if any self-similar type of blow-up of solutions of the 3D Euler equations is possible, or not. A similar question for the 3D Navier-Stokes equations was raised by J. Leray [17], and answered negatively by ˇ ak Necas, ˇ Ruˇzicka ˇ and Sver ´ [19]. In view of the scaling properties in (1.8), we should check if there exists a nontrivial solution (v α (x, t), pα (x, t)) of (E) of the form,
1 x 1 x α α v (x, t) = , p (x, t) = q α u 1 2α 1 (T∗ − t) α+1 (T∗ − t) α+1 (T∗ − t) α+1 (T∗ − t) α+1 (1.13) for t ∈ [0, T∗ ) and α = −1. Substituting (1.13) into (E), we find that (u, q) should be a solution of the system 1 α u+ (x · ∇)u + (u · ∇)u = −∇q (SSE) α + 1 α+1 div u = 0, which could be regarded as the Euler version of Leray’s equations. We have the following result for (SSE).
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Theorem 1.3. If u ∈ H 1 (R3 ) is a classical solution of (SSE) in R3 , then either u ≡ 0, or the helicity of u is equal to zero, namely R3 u · ωdx = 0, where ω =curl u. In the final section we consider the damping perturbation of the Constantin-LaxMajda equation [9], for which we explicitly obtain the solution. 2. Preliminaries We first set our notations, and recall definitions of the Triebel-Lizorkin spaces. We follow [20]. Let S be the Schwartz class of rapidly decreasing functions. Given f ∈ S, its Fourier transform F(f ) = fˆ is defined by 1 fˆ(ξ ) = e−ix·ξ f (x)dx. (2π)n/2 Rn We consider ϕ ∈ S satisfying Supp ϕˆ ⊂ {ξ ∈ Rn | 21 ≤ |ξ | ≤ 2}, and ϕ(ξ ˆ )>0 if 23 < |ξ | < 23 . Setting ϕˆj = ϕ(2 ˆ −j ξ ) (in other words, ϕj (x) = 2j n ϕ(2j x)), we can adjust the normalization constant in front of ϕˆ so that ϕˆj (ξ ) = 1 ∀ξ ∈ Rn \ {0}. j ∈Z
Given k ∈ Z, we define the function Sk ∈ S by its Fourier transform ϕˆj (ξ ). Sˆk (ξ ) = 1 − j ≥k+1
Let s ∈ R, p, q ∈ [0, ∞]. Given f ∈ S , we denote j f = ϕj ∗ f , and then the homogeneous Triebel-Lizorkin semi-norm f F˙ s is defined by p,q j qs q 2 |j f (·)| if q ∈ [1, ∞) j ∈Z f F˙ s = . Lp p,q sup(2j s |j f (·)|) if q = ∞ j ∈Z p L
˙s Fp,q
is a quasi-normed space with the quasiThe homogeneous Triebel-Lizorkin space norm given by ·F˙ s . For s > 0, (p, q) ∈ [1, ∞)×[1, ∞] we define the inhomogeneous p,q s Triebel-Lizorkin space norm f Fp,q of f ∈ S as s f Fp,q = f Lp + f F˙ s . p,q
The inhomogeneous Triebel-Lizorkin space is a Banach space equipped with the norm, · F˙ s . Similarly, the homogeneous Besov norm f B˙ s is defined by p,q
p,q
f B˙ s
p,q
=
1 ∞ q q j qs 2 ϕ ∗ f if q ∈ [1, ∞) j Lp
−∞
sup 2j s ϕj ∗ f Lp if q = ∞ j
.
Blow-up of the Euler Equations and the Related Equations
545
s is a quasi-normed space with the quasi-norm given The homogeneous Besov space B˙ p,q s by · B˙ s . For s > 0 we define the inhomogeneous Besov space norm f Bp,q of p,q s p f ∈ S as f Bp,q = f L + f B˙ s . p,q
Lemma 2.1. Let s ∈ (0, n), (p, q) ∈ [1, ∞) × [1, ∞] and sp = n. Then the following sequence of continuous embedding hold: n s 0 0 → B˙ p,1 → B˙ ∞,1 → F˙∞,1 → L∞ . F˙1,q
(2.1)
The first embedding of Lemma 2.1 is proved in [10], while the second one is proved in [4]. The others are obvious from the definitions of the corresponding norms. 3. Proof of the Main Results We begin with the following key estimates. Lemma 3.1. Let s ∈ (0, ∞), (p, q) ∈ [1, ∞) × [1, ∞] be given. Suppose u is a solution of the system (1.1)–(1.3) with initial data u0 , and let ω, ω0 be the associated vorticities. Then, we have the following estimate: t s s s dτ ≤ ω0 Fp,q +C a(τ )∇u(τ )L∞ ω(τ )Fp,q (3.1) ω(t)Fp,q 0
for vorticities, while u(t)F˙ s ≤ u0 F˙ s + C p,q
p,q
t
a(τ )∇u(τ )L∞ u(τ )F˙ s dτ p,q
0
(3.2)
for velocities. The proof of (3.1) in the case a(t) ≡ 1 is done in [4], and the additional factor a(t) does not change the proof. The proof of (3.2) is essentially the same as that of (3.1), using the first equation of (aE) directly instead of using the vorticity formulation. In particular, the case with (p, q) ∈ [1, ∞) × [1, ∞] is done in [3], using a different argument from [4]. Proof of Theorem 1.1. We provide here a priori estimates only. The construction of solutions by iteration scheme based on the estimates in the Triebel-Lizorkin norms is similar to [3]. By the Gronwall lemma applied to (3.1), we have t s s exp C ω(t)Fp,q ≤ ω0 Fp,q a(τ )∇u(τ )L∞ dτ 0 t s exp C ≤ ω0 Fp,q a(τ )ω(τ )B˙ 0 dτ , (3.3) ∞,1
0
0 → L∞ in (2.1), and the boundedness of the where we used the embedding B˙ ∞,1 0 Calderon-Zygmund singular integral operator from B˙ ∞,1 into itself [10]. In order to estimate ωB˙ 1 we observe that from (3.2), ∞,1
u(t)B˙ 1
∞,1
≤ u0 B˙ 1
∞,1
≤ u0 B˙ 1
∞,1
t
+C
t
+C 0
a(τ )∇u(τ )L∞ u(τ )B˙ 1 dτ ∞,1
0
a(τ )u(τ )2B˙ 1 dτ. ∞,1
(3.4)
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D. Chae
By Gronwall’s lemma we derive u(t)B˙ 1
∞,1
for all t > 0 with C
t 0
u0 B˙ 1 ∞,1
t 1 − C 0 a(τ )dτ u0 B˙ 1
≤
(3.5)
∞,1
a(τ )dτ u0 B˙ 1
∞,1
sup ω(t)B˙ 0
∞,1
0≤t<∞
≤
< 1. Hence, Cω0 B˙ 0 ∞,1
∞ . 1 − C 0 a(τ )dτ ω0 B˙ 0
(3.6)
∞,1
Substituting (3.6) into (3.3), we obtain s s sup ω(t)Fp,q ≤ ω0 Fp,q
exp C
0≤t<∞
s exp ≤ ω0 Fp,q
0
∞
a(τ )dτ sup ω(t)B˙ 0
∞,1
0≤t<∞
∞
a(τ )dτ ω0 B˙ 0 ∞,1 .
∞ 1 − C 0 a(τ )dτ ω0 B˙ 0 C
0
(3.7)
∞,1
Proof of Theorem 1.2. We first recall that the equivalence between (i) and (ii) is proved in [3, 4]; for the case s > n/p with p ∈ [1, ∞), q ∈ [1, ∞], T∗ s lim sup ω(t)Fp,q = ∞ if and only if ω(t)B˙ 0 dt = ∞, (3.8) ∞,∞
0
tT∗
while, for s = n, p = 1, q ∈ [1, ∞], s lim sup ω(t)Fp,q = ∞ if and only if
tT∗
0
T∗
ω(t)B˙ 0 dt = ∞. ∞,1
(3.9)
On the other hand, by following exactly the same procedure as in [3, 4] it is easy to find that the following blow-up criterion holds for the system (aE): The solution u(t) of the s , namely system (aE) blows up at t = T∗ < ∞ in Fp,q s lim sup u(t)Fp,q = ∞,
(3.10)
t→T∗
if and only if
T∗
a(t)dt = ∞
(3.11)
for s > n/p + 1, (p, q) ∈ [1, ∞]2 , while T∗ ω(t)B˙ 0 a(t)dt = ∞
(3.12)
0
0
ω(t)B˙ 0
∞,∞
∞,1
for s = n + 1, p = 1, q ∈ [1, ∞] respectively. Thus the equivalence of (iii) and (iv) follows. We establish here the equivalence of (ii) and (iv) below, which is the main
Blow-up of the Euler Equations and the Related Equations
547
point in this proof. By direct calculations we find the following relations hold between solutions of the system (E), and the system (aE) respectively. Suppose (v E , pE ) with given initial data v0E is a classical solution of the problem (E) on Rn × [0, λaL1 (0,∞) ), then (u, q) defined by the formula
t
u(x, t) = λv (x, λ E
t
q(x, t) = λ p (x, λ 2 E
a(s)ds), 0
a(s)ds)
(3.13)
0
is a solution of the problem (aE) on Rn × [0, ∞) with the initial data u0 (x) = λv0E (x). From the formula (3.13), we have immediately
λaL1 (0,∞)
ωE (t)B˙ 0
∞,∞
0
∞
dt =
ω(t)B˙ 0
a(t)dt,
(3.14)
ω(t)B˙ 0 a(t)dt.
(3.15)
∞,∞
0
and
λaL1 (0,∞)
∞,1
0
Substituting λ =
ωE (t)B˙ 0 dt =
T∗ aL1 (0,∞) ,
T∗
0
∞,1
we find
ω (t)B˙ 0 E
∞,∞
0
∞
∞
dt =
ω(t)B˙ 0
a(t)dt,
(3.16)
ω(t)B˙ 0 a(t)dt.
(3.17)
∞,∞
0
and
T∗
ωE (t)B˙ 0 dt = ∞,1
0
∞
0
∞,1
Thus, the equivalence of (ii) and (iv) follows from (3.16) and (3.17).
Note added to the proof. In the above proof we may use · L∞ instead of · B˙ 0 in ∞,∞ (3.8), (3.11), (3.14) and (3.16). Proof of Corollary 1.1. Let us suppose that the solution v E of (E) with initial data v0E s blows up at T∗ , namely, lim suptT∗ v E (t)Fp,q = ∞. Then, we consider the system (aE) with initial data u0 , and with a(t) = χ[0,T∗ ] (t), where χA (·) is the characteristic function of the set A. Then, plugging such a function a(t) into (1.5), we find that Theorem 1.2 implies ∞ T∗ ω(t)B˙ 0 a(t)dt = ω(t)B˙ 0 dt = ∞ ∞,∞
0
∞,∞
0
for u0 = v0E . Due to Theorem 1.1, this implies that ω0E B˙ 0
∞,1
Thus (1.6) follows.
= ω0 B˙ 0
∞,1
≥ ∞ 0
C0 C0 = . T∗ a(t)dt
548
D. Chae
Proof of Corollary 1.3. The equivalence of (i) and (ii) and of (iii) and (iv) follows from [3, 4] and their obvious modifications as previously. Here we establish the equivalence of (ii) and (iv). By direct calculations we can check easily that the following relations hold between the two solutions; suppose (v E , pE ) with given initial data v0E is a classical solution of the problem (E) on Rn × [0, λε ), then (u, q) defined by the formula u(x, t) = λe−εt v E x, λε (1 − e−εt ) , (3.18) q(x, t) = λ2 e−2εt p E x, λε (1 − e−εt ) is a classical solution of the problem (E)ε on Rn × [0, ∞) with initial data u0 (x) = λv0E (x). From the formula (3.18), we have immediately
λ ε
0
ω (t)L∞ dt =
∞
E
0
ω(t)L∞ dt.
(3.19)
Substituting ε = Tλ∗ , and applying the same argument as in the proof of Theorem 1.2, we obtain the equivalence of (ii) and (iv). Proof of Theorem 1.3. If u(x) ∈ H 1 (R3 ) is a classical solution of (SSE), then both of the energy and the helicity of the flows v α (x, t) are finite and conserved. The energy of v α (x, t) during t ∈ [0, T∗ ) is 3−2α 1 1 α 2 α+1 E(t) = |v (x, t)| dx = (T∗ − t) |u(y)|2 dy, 2 R3 2 R3 while the helicity is (below ωα =curl v α ) 2−2α α α α+1 H (t) = v (x, t) · ω (x, t)dx = (T∗ − t) u(y) · ω(y)dy. R3
R3
(3.20)
(3.21)
If both the energy and the helicity of u(x) are not zero, then from (3.20) and (3.21), in order to have conservations for each of the nonzero quantities, we should have α = 23 = 1, which is absurd. 4. Remarks on the Constantin-Lax-Majda Equation As a model problem of the perturbed Euler equation we consider the Constantin-LaxMajda equation first introduced in [9]: (x, t) ∈ R × R+ ωt − H (ω)ω = 0 (CLM) ω(x, 0) = ω0 (x) x∈R with ω = ω(x, t) a scalar function, and H (f ) is the Hilbert transform of f defined by 1 f (y) H (f ) = P V dy. (4.1) π x−y For the problem (CLM), Constantin-Lax-Majda derived the following explicit solution [9](see also Sect. 5.2.1 of [18]): ω(x, t) =
4ω0 (x) (2 − tH ω0 (x))2 + t 2 ω02 (x)
.
(4.2)
Blow-up of the Euler Equations and the Related Equations
The perturbed equation we are concerned with is σt − H (σ )σ = −εσ (CLM)ε σ (x, 0) = σ0 (x)
549
(x, t) ∈ R × R+ . x∈R
Similarly to (3.18) with λ = 1, we have the following relation between the two solutions: 1 σ (x, t) = e−εt ω x, (1 − e−εt ) , (4.3) ε σ0 (x) = ω0 (x). Combining (4.3) with (4.2), we easily obtain the following explicit solution of (CLM)ε : 4ε2 σ0 (x)e−εt . 2 2ε − H σ0 (x)(1 − e−εt ) + (1 − e−εt )2 σ02 (x)
σ (x, t) =
(4.4)
The formula leads us to the following proposition: Proposition 4.1. In case H σ0 (x) ≤ 0 for all x ∈ R there is no blow up of solution. Otherwise, we consider the three cases. Let us put S = {x ∈ R : σ0 (x) = 0, H σ0 (x) > 0}. (i) If ε > 21 supx∈S H σ0 (x), then there is no blow-up. (ii) If there exists x ∈ S such that ε < 21 H σ0 (x), then th solution blows up at T∗ given by −1 1 2ε T∗ = ln 1 − . (4.5) ε supx∈S H σ0 (x) (iii) If the set S1 = {x ∈ R : σ0 (x) = 0, ε = 21 H σ0 (x)} is nonempty, and if for all x ∈ R \ S1 we have ε > 21 H σ0 (x), or σ0 (x) > 0, then the solution blows up at t = +∞. Remark 4.1. We note that (i),(iii) above are the new phenomena of (CLM)ε and did not occur in (CLM). Acknowledgement. The author would like to deeply thank the referee for kind suggestions leading to Corollary 1.1 and Remark 1.3. This research is supported partially by the grant no. 2000-2-10200-002-5 from the basic research program of the KOSEF.
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6. Chemin, J.Y.: Perfect incompressible fluids. Oxford: Clarendon Press, 1998 7. Constantin, P.: Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations. Commun. Math. Phys. 104, 311–326 (1986) 8. Constantin, P., Fefferman, C., Majda, A.: Geometric constraints on potential singularity formation in the 3-D Euler equations. Commun. P. D. E. 21(3-4), 559–571 (1996) 9. Constantin, P., Lax, P.D., Majda, A.: A simple one-dimensional model for the three-dimensional vorticity equations. Commun. Pure Appl. Math. 38, 715-724 (1985) 10. Frazier, M., Torres, R., Weiss, G.: The boundedness of Caleron-Zygmund operators on the spaces α,q F˙p . Revista Mathem´atica Ibero-Americana 4(1), 41–72 (1988) 11. Jawerth, B.: Some observations on Besov and Lizorkin-Triebel Spaces. Math. Sacand. 40, 94–104 (1977) 12. Kato, T.: Nonstationary flows of viscous and ideal fluids in R3 . J. Funct. Anal. 9, 296–305 (1972) 13. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988) 14. Kato, T., Ponce, G.: Well posedness of the Euler and Navier-Stokes equations in Lebesgue spaces Lsp (R2 ). Revista Mathem´atica Ibero-Americana, 2, 73–88 (1986) 15. Kozono, H., Taniuchi, Y.: Limiting case of the Sobolev inequality in BMO, with applications to the Euler equations. Commun. Math. Phys. 214, 191–200 (2000) 16. Kozono, H., Ogawa, T., Taniuchi, T.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semilinear evolution equations. Math Z. 242(2), 251–278 (2002) 17. Leray, J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) 18. Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge: Cambridge Univ. Press, 2002 ˇ ak, V.: On Leray’s self-similar solutions of the Navier-Stokes equations. 19. Neˇcas, J., Ruˇziˇcka, M., Sver´ Acta Math. 176, 283–294 (1996) 20. Triebel, H.: Theory of Function Spaces. Basel-Boston: Birkh¨auser, 1983 21. Vishik, M.: Hydrodynamics in Besov spaces. Arch. Rational Mech. Anal. 145, 197–214 (1998) 22. Vishik, M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. ´ Norm. Sup., 4e s´erie. 32, 769–812 (1999) Ann. Scient. Ec. Communicated by P. Constantin
Commun. Math. Phys. 245, 551–576 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1024-0
Communications in
Mathematical Physics
2 ) on ∞-Cycles Form Factors and Action of U√−1 (sl M. Jimbo1 , T. Miwa2 , E. Mukhin3 , Y. Takeyama2 1
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan. E-mail: [email protected] 2 Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan. E-mail: [email protected]; [email protected] 3 Department of Mathematics, Indiana University-Purdue University-Indianapolis, 402 N.Blackford St., LD 270, Indianapolis, IN 46202, USA. E-mail: [email protected] Received: 26 May 2003 / Accepted: 5 August 2003 Published online: 6 February 2004 – © Springer-Verlag 2004
Abstract: Let p = {Pn,l } n,l∈Z≥0 be a sequence of skew-symmetric polynomials in n−2l=m
X1 , · · · , Xl satisfying degXj Pn,l ≤ n − 1, whose coefficients are symmetric Laurent polynomials ∞-cycle if in z1 , · · · , zn . We call p an Pn+2,l+1 X =z−1 ,z =z,z =−z = z−n−1 la=1 (1 − Xa2 z2 ) · Pn,l holds for all n, l. n n−1 l+1 These objects arise in integral representations for form factors of massive integrable field theory, i.e., the SU (2)-invariant Thirring model and the sine-Gordon model. The variables αa = − log Xa are the integration variables and βj = log zj are the rapidity variables. To each ∞-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the ∞-cycles. In this paper, we define an action of U√−1 ( sl2 ) on the space of ∞-cycles. There are two sectors of ∞-cycles depending on whether n is even or odd. Using this action, we show that the character of the space of even (resp. odd) ∞-cycles which are polynomials in z1 , · · · , zn is equal to the level (−1) irreducible character of sl2 with lowest weight −0 (resp. −1 ). We also suggest a possible tensor product structure of the full space of ∞-cycles. 1. Introduction First let us recall the form factor bootstrap approach to massive integrable models [10] in field theory. A form factor is a tower of meromorphic functions f = (fn (β1 , . . . , βn ))n≥0 in the variables β1 , . . . , βn ∈ C, satisfyng a certain set of axioms, to be referred to as Axiom 1–3. There are two sectors in the space of form factors: the even sector where fn = 0 for all odd n, and the odd sector where fn = 0 for all even n. With each form factor is associated a local field in the theory. Axiom 1 describes the exchange relation of βj and βj +1 in fn , and Axiom 2 relates the analytic continuation fn (β1 , . . . , βn−1 , βn + 2π i)
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to the cyclic shift fn (βn , β1 , . . . , βn−1 ). These two axioms imply that, for each n, fn (β1 , . . . , βn ) is a solution to the quantum Knizhnik-Zamolodchikov (qKZ) equation. Axiom 3 stipulates that fn−2 is determined by the residue of fn at the simple pole βn = βn−1 + π i. In this paper we consider the SU (2) invariant Thirring model (ITM). For the details of the axioms in this case, see Sect. 4. One of the basic issues in the theory of form factors is to describe all form factors satisfying the three axioms. In recent papers [6, 7], Nakayashiki solved this problem for the case of ITM under some assumptions. Subsequently, in [4] the case of the restricted sine-Gordon model was studied by a different method based on representation theory of quantum affine algebras. For the purpose of introducing the subject, and for making a comparison between the two methods, let us briefly review the results of [6, 7, 4]. Nakayashiki’s approach consists of three steps. The first step [8] is to construct solutions of the qKZ equation by exploiting hypergeometric integrals (cf. Sect. 5 for the explicit formulas). The solutions are sl2 -singular vectors. They are parameterized by polynomials Pn,l (X1 , . . . , Xl |z1 , . . . , zn ) in two sets of variables, X1 , . . . , Xl and z1 , . . . , zn . The variables Xa are related to the integration variables αa by Xa = e−αa , and the variables zj to βj by zj = eβj . The polynomial Pn,l is skew-symmetric in X1 , . . . , Xl of degree less than or equal to n − 1 in each Xa , and is symmetric in z1 , . . . , zn . We call Pn,l a deformed cycle. Among them there are null cycles, i.e., those which give rise to vanishing integrals. The problem of finding all null cycles has been solved by Smirnov [11] and Tarasov [13]. The second step [6] is to characterize deformed cycles which give rise to minimal form factors in the following sense. A form factor f = (fn )n≥0 is called N -minimal if fn = 0 for all n < N . If this is the case, then the function fN satisfies the zero residue condition, resβN =βN −1 +πi fN = 0.
(1.1)
We say that a deformed cycle PN,l is N -minimal if the corresponding function fN given by the hypergeometric integral satisfies Axiom 1, Axiom 2 and (1.1). Nakayashiki observed that a sufficient condition for PN,l to be minimal is given by PN,l (X1 , . . . , Xl−1 , z−1 |z1 , . . . , zN−2 , z, −z) = 0.
(1.2)
He assumed that (modulo null cycles) the condition (1.2) is also necessary, and constructed an explicit basis of the space of deformed cycles satisfying it. The third step [7] is to construct a tower f, starting from fN corresponding to each of the basis elements of minimal cycles. If fn and fn+2 are given by deformed cycles Pn,l and Pn+2,l+1 , respectively, then Axiom 3 (with n replaced by n + 2) is valid if Pn+2,l+1 (X1 , . . . , Xl , z−1 |z1 , . . . , zn , z, −z) = z−n−1
l
(1 − Xa2 z2 ) · Pn,l (X1 , . . . , Xl |z1 , . . . , zn ).
(1.3)
a=1
In [7], Nakayashiki constructed directly a tower of deformed cycles {Pn,l } satisfying the linking condition (1.3), starting from each minimal cycle he has constructed in [6]. In [4], a different approach was taken in the construction of minimal cycles. In that paper, the restricted sine-Gordon model (RSG) was considered. The hypergeometric integrals for ITM and RSG have different form, but the deformed cycles are exactly the same. Let us denote the space of minimal deformed cycles with fixed N by
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553
WN = ⊕0≤l≤N WN,l . In [4], the space WN was constructed as a representation space of a subalgebra of U√−1 ( sl2 ) over the ring RN of symmetric polynomials in z1 , . . . , zN , and it was shown that the total space WN is created from the constant polynomial 1N ∈ WN,0 by the RN -linear action of this subalgebra. In the present paper we continue the study of form factors along the same direction. We generalize the result in several respects. First, we consider deformed cycles Pn,l which are symmetric Laurent polynomials in the variables z1 ,. . . ,zn . Namely, we consider both chiralities simultaneously. We denote the space of deformed cycles in this n,l , and that of minimal deformed cycles by W n,l . Second, we conextended sense by C √ n,l . By doing sider the action of the full quantum algebra U −1 (sl2 ) on Wn = ⊕0≤l≤n W so, we no longer need to introduce multiplication by symmetric Laurent polynomials ‘by n,0 . Finally, and hand’, since it is incorporated as a part of this action on the subspace W √ most importantly, we consider also the action of U −1 (sl2 ) on towers of polynomials. Let us elucidate the last point. We are interested in a tower of deformed cycles p = (Pn,l )n−2l=m satisfying the linking condition (1.3), where m is a fixed integer. As we have seen, such a sequence gives rise to a form factor satisfying Axiom 3 as well. We will refer to p as an ∞-cycle of weight m. (For the precise definition, see Sect. 3.1.) The following is an example of ∞-cycles of weight m, 1m = (1m , Xm+1 , Xm+1 ∧ X m+3 , Xm+1 ∧ X m+3 ∧ X m+5 , . . . ). Here we have used the wedge notation for skew-symmetric polynomials (see (2.11) n,l ⊗ C(z1 , · · · , zn ) below). The action of the quantum algebra extends to the space C of polynomials whose coefficients are rational functions in z1 , . . . , zn . Therefore it acts naturally on sequences of polynomials componentwise. Consider the orbit of the particular ∞-cycles given above with m = 0, 1, √ := U√ ( Z −1 sl2 ).10 + U −1 (sl2 ).11 .
(1.4)
We show that any element in this space is an ∞-cycle, that is, a sequence of deformed cycles (in particular, they are symmetric Laurent polynomials in z1 , · · · , zn ), satisfying N consisting of the linking condition (1.3). The space (1.4) is filtered by submodules Z N-minimal ∞-cycles p = (Pn,l ) with Pn,l = 0 for n < N . We show that the natural map N /Z N+1 −→ ⊕N≥0 W N ⊕N≥0 Z is an isomorphism of U√−1 ( sl2 )-modules. In particular, any minimal deformed cycle can be lifted to an ∞-cycle, and hence gives rise to a form factor. This gives an alternative proof of Nakayashiki’s result [7] and extends it in the presence of both chiralities. These are the main results of the present paper. As was shown in [6], the character of the space of even, polynomial ∞-cycles, i.e., ⊕N:even WN , coincides with the character of the level 1 basic representation for sl2 . In our convention, it is more natural to think of this character as the level (−1)-character. When we introduce negative powers in zj , the character becomes ill defined. Instead of dealing we fix a non-negative integer L ∈ Z≥0 and consider the subspace with the full space Z, consisting of N-minimal deformed cycles PN,l such that (z1 · · · zN )L PN,l are polynomials in z1 , . . . , zN . Then the sum of the corresponding characters (over N ≡ i mod 2 with i = 0, 1 fixed) has the product form χi (q −1 , z) · χ0 (q, z; L), where χi (q −1 , z) is the level (−1)-character and χ0 (q, z; L) denotes the character of a Demazure subspace of the level 1 irreducible module with highest weight 0 . The latter tends to χ0 (q, z) as
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is isomorphic as a U√ ( L → ∞. This leads us to conjecture that Z −1 sl2 )-module to the tensor product of level (−1)- and level 1-modules. We plan to address this point in our next paper. Let us give some remarks. First, the space of ∞-cycles is not the same as that of form factors. On one hand we should take into account the quotients by null cycles, and on the other hand we should incorporate form factors other than the singular vectors with respect to the sl2 -action. At the level of characters, these two effects cancel each other. The problem of determining the symmetries of the space of form factors themselves is beyond the scope of the present paper. Second, lifting of a minimal form factor fN into a tower is not unique. Starting from a given lifting, we can add infinitely many ∞-cycles that are minimal and of increasing degrees of minimality, without changing the degree N part fN . Note, however, that such It is not clear an infinite sum of ∞-cycles is not an ∞-cycle in our definition of Z. (though it is clear that it belongs if Nakayashiki’s extension belongs to our space Z Since form factors are determined only up to null cycles, the to the completion of Z). identification of form factors is not a simple problem. We illustrate this point by some examples. Form factors corresponding to the su(2) currents given in [10] belong to the This is true at the level of ∞-cycles. On the other hand, the known formula space Z. but [10] for form factors corresponding to the energy-momentum tensor arise from Z, only modulo null cycles. The plan of the paper is as follows. In Sect. 2, we introduce our notation on the quantum loop algebra at roots of unity, and give an action of U√−1 ( sl2 ) on the space of deformed cycles. In Sect. 3 we introduce the space of ∞-cycles. The linking condition is preserved by the action of U√−1 ( sl2 ). The main result is stated in Theorem 3.4. We also calculate the character of the truncated space in terms of the level one irreducible characters and the Demazure characters. In Sect. 4, we apply the results on the ∞-cycles to the form factors. Some technical matters are given in Appendices. Appendix A is devoted to the derivation of the U√−1 ( sl2 ) action on the tensor product of evaluation modules. In Appendix B we give a proof of a lemma regarding certain null cycles.
2. The Space of Deformed Cycles sl2 ) 2.1. Quantum loop algebra. In this subsection, we recall some basic facts about Uq ( at roots of unity. Our basic reference is [2]. Let C(q) be the field of rational functions in indeterminate q. The quantum loop sl2 ) is a C(q)-algebra generated by xk± (k ∈ Z), an (n ∈ Z\{0}) and t1±1 , algebra Uq ( with the defining relations [t1 , an ] = 0, [am , an ] = 0, t1 xk± t1−1 = q ±2 xk± , [2n] ± x , [an , xk± ] = ± n k+n ± ± ± ± xk+1 xl± − q ±2 xl± xk+1 = q ±2 xk± xl+1 − xl+1 xk± , [xk+ , xl− ] =
+ ϕk+l
q
− − ϕk+l . − q −1
(2.1) (2.2) (2.3) (2.4) (2.5)
Form Factors of Massive Integrable Field Theory
Here k∈Z
555
± k ϕ±k z
=
t1±1 exp
±(q − q
−1
)
∞
a±n z
n
,
n=1
and [j ] = (q j − q −j )/(q − q −1 ). We use the notation x (n) =
xn , [n]!
[n]! =
n
[j ].
j =1
sl2 ) generated by (xn± )(r) (n ∈ Z, r ∈ Let Uqres ± be the C[q, q −1 ]-subalgebra of Uq (
±1 t1 ; n 0 res −1 Z≥0 ). Let Uq be the C[q, q ]-subalgebra generated by t1 , (n ∈ Z, r ∈ Z≥0 ) r and a˜ n (n ∈ Z\{0}), where
r t1 q n−s+1 − t1−1 q −n+s−1 n t1 ; n an , . = a˜ n = r [n] q s − q −s s=1
Let further Uqres be the C[q, q −1 ]-subalgebra generated by Uqres ± and Uqres 0 . We have the triangular decomposition ([2], Prop. 6.1) Uqres = Uqres − · Uqres 0 · Uqres + . We will use also C[q, q −1 ]-subalgebras Bq± generated by the following elements: Bq+ : (xn+ )(r) , (xn− )(r) , a˜ m , t1±1 , + (r) − (r) Bq− : (x−n ) , (x−n ) , a˜ −m , t1±1 ,
where n, r run over Z≥0 and m over Z>0 , respectively. Introduce the generating series + + X≥0 (t) := xn+ (q −1 t)n , X<0 (t) := xn+ (q −1 t)n , n≥0
− (t) X>0
:=
n<0
xn− (q −1 t)n ,
− X≤0 (t)
n>0
:=
xn− (q −1 t)n .
n≤0
By Corollary 4.6 in [2] and the Remark below it, Uqres + (resp. Uqres − ) is also generated + + − over C[q, q −1 ] by the coefficients of (X≥0 (t))(r) and (X<0 (t))(r) (resp. (X>0 (t))(r) and − (r) (X≤0 (t)) ), where r runs over Z≥0 . For any non-zero complex number , the specialization U is defined to be Uqres ⊗C[q,q −1 ] C by the ring homomorphism C[q, q −1 ] → C sending q to . The subalgebreas U± , U0 and B± are defined similarly. In this paper we restrict to the case U√−1 . Define in Uqres , a± (t) :=
±n>0
a˜ n t n ,
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± 0 and denote their images in U√−1 by the same symbols. The subalgebras U√ , U√ −1 −1 ± and B√ are generated over C by (the coefficients of) the following elements: −1 − − − − − U√ : X>0 (t), X>0 (t)(2) , X≤0 (t), X≤0 (t)(2) , −1
t1 ; n ±1 0 U√ : t , (n ∈ Z), a± (t), 1 −1 2
(2.6) (2.7)
+ + U√ : X≥0 (t), −1
+ X≥0 (t)(2) ,
+ X<0 (t),
+ X<0 (t)(2) ,
(2.8)
+ + B√ : X≥0 (t), −1
+ X≥0 (t)(2) ,
− X>0 (t),
− X>0 (t)(2) ,
(2.9)
− X≤0 (t)(2) ,
(2.10)
x0− , − + B√ : X<0 (t), −1
x0+ ,
(x0− )(2) ,
a+ (t),
+ X<0 (t)(2) ,
(x0+ )(2) ,
t1±1 ,
− X≤0 (t),
a− (t),
t1±1 .
We assign the degree and weight to U√−1 as follows. deg(xn± )(r) = nr, deg a˜ m = m, deg t1±1 = 0, wt (xn± )(r) = ±2r, wt a˜ m = 0,
wt t1±1 = 0.
2.2. Representation of U√−1 on the space of polynomials. Until the end of this section, we fix a non-negative integer n. Let Kn = C(z1 , · · · , zn ) be the field of rational functions n be its subring consisting of symmetric Laurent polynomials. in z1 , . . . , zn , and let R Consider the vector space over Kn , j An := ⊕nl=0 An,l , An,l := ∧l ⊕n−1 j =0 Kn X . By definition we set An,l = 0 if l < 0. Note that An,0 = Kn and An,l = 0 if l > n. For 0 ≤ l ≤ n, we identify an element P ∈ An,l with a skew-symmetric polynomial in the variables X1 , . . . , Xl with coefficients in Kn , of degree at most n − 1 in each Xj . We use the wedge product notation for P1 (X1 , · · · , Xl1 ) ∈ An,l1 and P2 (X1 , · · · , Xl2 ) ∈ An,l2 , P1 ∧ P2 (X1 , · · · , Xl1 +l2 ) 1 := Skew P1 (X1 , · · · , Xl1 )P2 (Xl1 +1 , · · · , Xl1 +l2 ) , l1 !l2 !
(2.11)
where Skew stands for the skew-symmetrization Skew f (X1 , · · · , Xm ) := (sgn σ )f (Xσ (1) , · · · , Xσ (m) ). σ ∈Sm
We shall introduce a Kn -linear action of U√−1 on An . For that purpose, let us prepare some notation. Set n (X) :=
n
(1 − zj X),
j =1
n (X1 , X2 ) := n (X1 ) n (X2 ) − n (−X1 ) n (−X2 ).
Form Factors of Massive Integrable Field Theory
557
Define also n (t, −X) t , 2(X − t) n (t) t t X1 − X 2 Fn(2) (t|X1 , X2 ) := n (X1 , X2 ) X 1 + t X 2 + t X1 + X 2 t t n (−X2 ) + n (t, −X1 ) X1 − t X2 + t n (t) t t n (−X1 ) − n (t, −X2 ). X2 − t X1 + t n (t) Fn (t|X) :=
Fn (t|X) is a polynomial in X of degree n − 1 because n (t, −X) is divisible by X − t. At t ±1 = 0, it has a power series expansion in t ±1 whose coefficients are symmetric (2) polynomials in z1±1 , · · · , zn±1 . Fn (t|X1 , X2 ) has similar properties. Let us define the action of the generators g ∈ U√−1 on P ∈ An,l . For l = 0 or 1 and + g ∈ U√ , we set −1 + (t).P = 0, X≥0
+ X<0 (t).P = 0
+ (t)(2) .P = 0, X≥0
(l = 0),
+ X<0 (t)(2) .P = 0
(l = 0, 1).
In the other cases, we define the action as follows. t1±1 .P := i ±(n−2l) .P ,
(2.12)
:= Fn (t) ∧ P ,
(2.13)
− X>0 (t).P − −4iX>0 (t)(2) .P
:=
a+ (t).P :=
Fn(2) (t) ∧ P , n zj t j =1
1 − zj t
(2.14)
P (X1 , · · · , Xl )
l
t n (Xp ) P (X1 , · · · , t , · · · , Xl ) Xp − t n (t) p=1 −P (X1 , . . . , Xl ) + (t → −t) ,
+
a− (t).P := −
n j =1
p
(2.15)
1 P (X1 , · · · , Xl ) 1 − zj t
l
t n (Xp ) P (X1 , · · · , t , · · · , Xl ) Xp − t n (t) p=1 Xp − (2.16) P (X1 , . . . , Xl ) + (t → −t) , t 1 + i 1−n X≥0 P (X1 , · · · , Xl−1 , t), (t).P := (2.17) n (t) n P (X1 , · · · , Xl−2 , −u, u) du + i(−1)n+1 X≥0 (t)(2) .P := resu=za−1 . (2.18) n (−u) n (u) u−t
−
a=1
p
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M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
In (2.13),(2.14),(2.15),(2.17) and (2.18), the right-hand sides stand for the power series expansion in t, and in (2.16) the expansion in t −1 . Note that the factor Xp ± t in − the denominator of (2.15) or (2.16) divides the numerator. Define also −X≤0 (t).P , − + + (2) 1−n n+1 (2) −4iX≤0 (t) .P , −i X<0 (t).P and i(−1) X<0 (t) .P by the expansion at t = ∞ of the right hand side of (2.13), (2.14), (2.17), (2.18), respectively. Note, in particular, that a˜ m .P :=
n
zjm P
(m ∈ Z : odd).
(2.19)
j =1
Proposition 2.1. With the above rule, An is a U√−1 -module. This action of U√−1 is essentially the one on the n-fold tensor product of two-dimensional evaluation modules in disguise. We give a proof of Proposition 2.1 in Appendix A. The following lemma will be used later. Lemma 2.2. On the subspace An,0 = Kn , we have a˜ m .P =
n
zjm .P
(m ∈ Z\{0}, P ∈ An,0 ).
j =1 0 n denotes the unit. n = U√ .1 , where 1n ∈ R We have R −1 n
Proof. This is an immediate consequence of (2.15)–(2.16) and (2.19).
2.3. Deformed cycles. For the application to form factors, we introduce several submodules of An , n ⊃ D n ⊃ W n An ⊃ C to be defined as follows. Set n := ⊕nl=0 C n,l , C
n,l := ∧l ⊕n−1 Rˆ n X j . C j =0
n will be referred to as a deformed cycle. We say that P ∈ C n,l is weakly An element of C minimal if −1 P = 0 if Xl−1 = −Xl−1 = zn−1 = −zn ,
(2.20)
P = 0 if Xl−1 = zn−1 = −zn .
(2.21)
and minimal if
These conditions arise in the study of form factors (see Lemma 3.2 below). Denote by n,l (resp. W n,l ) the subspace of weakly minimal (resp. minimal) elements of C n,l . We D set n := ⊕nl=0 D n := ⊕nl=0 W n,l , W n,l . D n consisting of symmetric polynomials. ReplacLet further Rn be the subring of R n by Rn in the above, we define the spaces Cn = ⊕n Cn,l , Dn = ⊕n Dn,l , ing R l=0 l=0 Wn = ⊕nl=0 Wn,l .
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559
n , W n are U√ -submodules. Lemma 2.3. (i) The spaces D −1 + (ii) The spaces Dn , Wn are B√ -submodules. −1 n are stable under the n , W Proof. From the formulas (2.12)–(2.18), it is clear that D + + action of all generators listed in (2.6), (2.7) and (2.8), except for X≥0 (t)(2) , X<0 (t)(2) . n , they give The only subtle point about the latter elements is that, when acted on P ∈ C rise to symmetric rational functions in z1 , · · · , zn which have a pole on za + zb = 0 n these poles do not appear, and that the in general. We show that for elements P ∈ D (weak-) minimality condition is preserved. + n,l with l ≥ 2. As an example, consider the case Q = i(−1)n X≥0 (t)(2) .P , P ∈ D Explicitly we have 1 1 1 2 2 2 b(=a) (1 − zb /za ) 1 − za t n
Q(X1 , · · · , Xl−2 |z1 , · · · , zn ) =
a=1
×P (X1 , · · · , Xl−2 , za−1 , −za−1 |z1 , · · · , zn ). Since the right-hand side is symmetric in z1 , · · · , zn , the only possible poles are those at za + zb = 0 (a = b). However, those poles are absent because of (2.20). Let us verify n,l−2 [[t]]. If l < 4, there is nothing to show. Suppose l ≥ 4. Setting zn−1 = z that Q ∈ D and zn = −z we find Q(X1 , · · · , Xl−2 |z1 , · · · , zn−2 , z, −z) 1 P (X1 , · · · , Xl−2 , za−1 , −za−1 |z1 , · · · , zn−2 , z, −z) 1 1 + n−2 2 2 2 2 2 2 4 b=1 (1 − zb2 /z2 ) (1 − tza )(1 − z /za ) 1≤b≤n−2 (1 − zb /za ) n−2
=
a=1
b(=a)
z ∂P × (X1 , · · · , Xl−2 , z−1 , −z−1 |z1 , · · · , zn−2 , z, −z) 1 + tz ∂zn−1 z ∂P (X1 , · · · , Xl−2 , z−1 , −z−1 |z1 , · · · , zn−2 , z, −z) . + 1 − tz ∂zn Under further specialization Xl−3 = −Xl−2 = z−1 , the first term vanishes by (2.20), and the rest is 0 by skew symmetry. + In the same way, (2.21) is also preserved. The case X<0 (t)(2) and the remaining assertions can be shown similarly. n be the unit element. Then we have Proposition 2.4. Let 1n ∈ C n = U√ .1n , W −1
+ Wn = B√ .1 . −1 n
n . It is known [4] that Wn is generated from Proof. By Lemma 2.3, U√−1 acts on W − − (2) − − 1n by the action of x0 , (x0 ) , X>0 (t), X>0 (t)(2) and the multiplication by elements + + + 0 of Rn . By Lemma 2.2, Rn = (B√−1 ∩ U√−1 ).1n . Since X≥0 (t), X≥0 (t)(2) kill 1n , and + since the action of U√−1 is Rn -linear, we obtain Wn = B√ .1 . The other assertion −1 n follows in a similar manner.
Consider √ the tensor product of evaluation representation ρz1 ⊗ · · · ⊗ ρzn specialized n is to q = −1 (see (A.1)). As explained in Appendix A, the action of U√−1 on W ⊗n induced from ρz1 ⊗ · · · ⊗ ρzn under the identification of 1n with v+ . Proposition 2.4 implies that
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n is isomorphic to the subrepresentation of ρz1 ⊗ Corollary 2.5. As U√−1 -module, W ⊗n · · · ⊗ ρzn generated by v+ . n,l by Define the degree on C deg0 Xa = −1,
deg0 zj = 1.
(2.22)
n and W n are n,l . With this definition, D We also assign a weight m = n − 2l to P ∈ C √ bi-graded U −1 -modules. 2 ) on the Space of ∞-Cycles 3. Action of U√−1 (sl In this section we introduce certain sequences of cycles which we call ∞-cycles, and define an action of U√−1 on them. n,l , Pn+2,l+1 ∈ C n+2,l+1 . We say that the pair 3.1. Links of cycles. Let Pn,l ∈ C (Pn,l , Pn+2,l+1 ) is a link if Pn+2,l+1 (X1 , · · · , Xl , z−1 |z1 , · · · , zn , z, −z) = z−n−1
l
(1 − Xa2 z2 ) · Pn,l (X1 , · · · , Xl |z1 , · · · , zn ).
(3.1)
a=1
The condition (3.1) appeared in [9] in the following equivalent form. Lemma 3.1. A pair (Pn,l , Pn+2,l+1 ) is a link if and only if there exists a ∗ (X1 , · · · , Xl |Xl+1 |z1 , · · · , zn |z) Pn,l
with the following properties: ∗ is a skew-symmetric polynomial in X , · · · , X with deg P ∗ ≤ n − 1 (1 ≤ (i) Pn,l 1 l Xi n,l ∗ ≤ n + 1. i ≤ l). It is a polynomial in Xl+1 with degXl+1 Pn,l ∗ is a symmetric Laurent polynomial in z , · · · , z , and an even Laurent polyno(ii) Pn,l 1 n mial in z. (iii) We have
(3.2) Pn+2,l+1 (X1 , · · · , Xl+1 |z1 , · · · , zn , z, −z) 1 ∗ = Skew (1 − Xa2 z2 ) Pn,l (X1 , · · · , Xl |Xl+1 |z1 , · · · , zn |z) , l! 1≤a≤l
∗ Pn,l (X1 , · · ·
, Xl |z−1 |z1 , · · · , zn |z) = z−n−1 Pn,l (X1 , · · · , Xl |z1 , · · · , zn ). (3.3)
Here Skew in (3.2) stands for the skew-symmetrization with respect to X1 , . . . , Xl+1 .
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561
Proof. The ‘if’ part is evident. To see the ‘only if’ part, suppose (3.1) holds. Since Pn+2,l+1 (X1 , · · · , Xl+1 |z1 , · · · , zn , z, −z) n 1 n+1 − Skew Xl+1 (1 − Xa2 z2 ) · Pn,l (X1 , · · · , Xl |z1 , · · · , zn ) l! a=1
has a zero at Xl+1 = ±z−1 , it can be written as · · · , zn |z). Setting
l+1
2 2 a=1 (1 − Xa z ) · Q(X1 , · · ·
, Xl+1 |z1 ,
∗ (X1 , · · · , Xl |Xl+1 |z1 , · · · , zn |z) Pn,l n+1 = Xl+1 Pn,l (X1 , · · · , Xl |z1 , · · · , zn ) 1 2 z2 )Q(X1 , · · · , Xl+1 |z1 , · · · , zn |z), (1 − Xl+1 + l+1
one easily verifies the conditions mentioned above.
The following lemma is obvious from the definition. Lemma 3.2. (i) If (Pn,l , Pn+2,l+1 ) is a link, then Pn,l and Pn+2,l+1 are weakly minimal. n , (0, Pn,l ) is a link if and only if Pn,l is minimal. (ii) For Pn,l ∈ C Let us study the action of U√−1 on links. n,l , Pn+2,l+1 ∈ C n+2,l+1 and g ∈ U√ . If the pair Proposition 3.3. Let Pn,l ∈ C −1 (Pn,l , Pn+2,l+1 ) is a link, then (g.Pn,l , g.Pn+2,l+1 ) is also a link. Using the formulas for the action (2.13)–(2.18), it is straightforward to verify the linking condition (3.1). We omit the proof. 3.2. ∞-cycles. Let m be an integer. We call a sequence of cycles p = (Pn,l )n,l≥0,n−2l=m ,
n,l Pn,l ∈ C
an ∞-cycle if (Pn,l , Pn+2,l+1 ) is a link for any n ≥ 0. The integer m is called the weight of p. We denote by Z[m] the space of ∞-cycles of weight m, and set Z = ⊕m∈Z Z[m]. Often we write the component Pn,l as (p)n,l . For an ∞-cycle p ∈ Z[m], define the action of g ∈ U√−1 by g.p = g.Pn,l )n,l≥0,n−2l=m . Example. For each m ∈ Z≥0 , there is a distinguished ∞-cycle 1m ∈ Z[m] given by (1m )m,0 = 1m ,
1m+2l,l = Xm+1 ∧ · · · ∧ X m+2l−1
(l ≥ 1).
From the formula (2.13) we find + (t).1m )m+2l,l−1 = t m+1 X m+3 ∧ · · · ∧ X m+2l−1 + O(t m+2 ), (i −1−m X≥0
and hence + 1m = 1m+2 , (−1)m+1 xm+1
xn+ 1m = 0
(n ≤ m).
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M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
In particular, for all m ≥ 0 we have + (−1)k x2k−1 · · · x3+ x1+ 10 1m = + + + x2k−2 · · · x2 x0 11
(m = 2k), (m = 2k − 1).
(3.4)
− · · · x3− x1− 10 ∈ Z[2k] as a nontrivial example. For Next we give a formula for x2k−1 k = 1, 2 the ∞-cycles are given by e2a+1 X 2a ∧ X ∧ X 3 ∧ · · · ∧ X 2l+1 , (x1− 10 )2+2l,2+l = i 1/2 0≤a
(x3− x1− 10 )4+2l,4+l
e2a +1 e2a +2 e2a +3 2 2 2 2a e2a1 +1 e2a1 +2 e2a1 +3 X 1 ∧ X 2a2 ∧ X ∧ X 3 ∧ · · · ∧ X 2l+3 , = 0 1 e1 0≤a1
where ea is the a th elementary symmetric polynomial in zj ’s. To write down the formula for general k, we set (0)
w2n,r := (−1) ×
r(r−1) 2
S(2(r−1),... ,2,0|2ar ,... ,2a1 ) (z1 , . . . , z2n )X 2a1 ∧ · · · ∧ X 2ar .
0≤a1 <···<ar
Here S(2(r−1),... ,2,0|2ar ,... ,2a1 ) (z1 , . . . , z2n ) is the Schur polynomial associated with the Young diagram given in the Frobenius notation by (2(r − 1), . . . , 2, 0|2ar , . . . , 2a1 ). Then the formula is given by − (x2k−1 · · · x3− x1− 10 )2k+2l,2k+l = i k
2 /2
(0)
w2k+2l,k ∧ X ∧ X 3 ∧ · · · ∧ X 2(k+l)−1 .
In particular we have − (x2k−1 · · · x3− x1− 10 )2k,2k = i k
2 /2
(zj + zj ) X0 ∧ X 1 ∧ · · · ∧ X 2k−1 .
1≤j <j ≤2k
It is clear that this cycle is minimal.
The central object of our study are the following modules. := Z (0) ⊕ Z (1) , Z
(i) := U√ .1i Z −1
(i = 0, 1),
Z := Z (0) ⊕ Z (1) ,
+ Z (i) := B√ .1 −1 i
(i = 0, 1).
∩ Z[m] and likewise for Z. We also define the = ⊕m∈Z Z[m], Z[m] := Z We have Z degree of an ∞-cycle p = (Pn,l ) by deg p =
n2 + deg0 Pn,l , 4
(3.5)
where deg0 is defined in (2.22). Since any pair (Pn,l , Pn+2,l+1 ) in p is a link, the righthand side above is independent of n.
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563
is We say an ∞-cycle p = (Pn,l ) is N -minimal if Pn,l = 0 for n < N. The space Z N consisting of N-minimal ∞-cycles, filtered by U√−1 -submodules Z = Z 0 ⊃ Z 1 ⊃ · · · . Z + N of Z. If p = Similarly we have a filtration by B√ -submodules ZN = Z ∩ Z −1 N , then PN,l is N -minimal, where l = N−m . We have therefore (Pn,l )n,l≥0,n−2l=m ∈ Z 2 natural injective homomorphisms
:= ⊕N≥0 Z N /Z N+1 → ⊕N≥0 W N , ϕˆ : gr Z ϕ : gr Z := ⊕N≥0 ZN /ZN+1 → ⊕N≥0 WN ,
(3.6) (3.7)
which respect the bi-grading (d, m) given by −deg and weight. (See Sect. 3.3 for the reason of the minus sign.) We are now in a position to state the main result of this paper. Theorem 3.4. The map (3.6) (resp. (3.7)) is an isomorphism of bi-graded U√−1 -mod+ ules (resp. B√ -modules). −1 N . By Proposition 2.4, there Proof. It suffices to show the surjectivity. Take any P ∈ W exists g ∈ U√−1 such that P = g.1N . From (3.4), we can choose g ∈ U√−1 and i ∈ {0, 1} such that 1N = g .1i . Then p = gg .1i is an ∞-cycle which is sent to P under the map (3.6). The case (3.7) is completely parallel. Theorem 3.4 gives an alternative proof of Nakayashiki’s result [7] and extends it to the case including negative powers of z1 , · · · , zn . Remark. Fix m. For any sequence pj ∈ Z[m] such that (pj )n,(n−m)/2 = 0 for n < j , the infinite sum j ≥0 pj is well defined and belongs to Z[m]. Theorem 3.4 implies that conversely any p ∈ Z[m] can be written as such an infinite sum with pj ∈ Z[m]. By a character of 3.3. Characters. In this subsection we study the characters of Z, Z. a bi-graded vector space V = ⊕d,m Vd,m , we mean the generating series q d zm dim Vd,m . (3.8) chq,z V = d,m
Below we use the bi-grading (d, m) by −deg and weight. In order to match our characters to the irreducible characters of sl2 we put the minus sign for the degree. Theorem 3.5. For i = 0 or 1, we have chq,z Z (i) = χi (q −1 , z),
(3.9)
where χi (q, z) :=
1 (q)∞
2 /4
qm
zm
m∈2Z+i
is the character of the level 1 integrable module with the highest weight i of sl2 .
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M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
Proof. This is a consequence of the isomorphism (3.7) and Nakayashiki’s result [6] on the character of Wn . Note that (3.9) implies that the character of Z (i) is equal to that of the level −1 integrable module with the lowest weight −i . (i) is ill-defined since each weight subspace is On the other hand, the character of Z infinite dimensional. To get around this inconvenience, we follow the idea of [5] and introduce the truncated character. For each L ∈ 21 Z≥0 , consider the space N,L := W
N
zj
−L
· WN .
j =1
N,0 ⊂ W N,1 ⊂ · · · and W N = ∪L∈Z W N,L . We will use below the standard We have W ≥0 q-binomial symbol (q)m (m ≥ n ≥ 0), m = (q)n (q)m−n n 0 (otherwise), j where (z)n = n−1 j =0 (1 − q z). Recall that each integrable sl2 -module has a family of Demazure subspaces parametrized by the elements of the affine Weyl group. Their characters 2L 2 m /4 m χi (q, z; L) = z (i ≡ 2L mod 2) m q m∈2Z+i
L+ 2
give polynomial finitizations of the full characters in the sense that limL→∞ χi (q, z; L) = χi (q, z). Proposition 3.6. Let L ∈ 21 Z≥0 , i, j = 0, 1 with j ≡ 2L mod 2. Then we have N,L = χi (q −1 , z)χj (q, z; L). chq,z ⊕ N ≥0 W (3.10) N ≡i mod2
Proof. Without loss of generality, we prove (3.10) in the summed form over i = 0, 1. Theorem 3.5 is equivalent to the statement chq −1 ,z WN = q N
2 /4
N l=0
1 N N−2l z . (q)N l
Hence, taking the sum over i = 0, 1, the left hand side of (3.10) with q replaced by q −1 becomes q N 2 /4−NL N
q lm+l(l−2L) 2 zN−2l = zm q m /4−Lm . (3.11) l (q)N (q)l (q)l+m l,m∈Z N≥l≥0
l≥0,−m
The sum over l can be simplified by using the formula ([4], Lemma 6.2) (q l+1 z)∞ 2L
l(l−2L) l q z = zs , s q −1 (q)l l≥0
s≥0
(3.12)
Form Factors of Massive Integrable Field Theory
565
2L signifies the q-binomial symbol with q replaced by q −1 . Specializing s q −1 z = q m in (3.12), we find that the right hand side of (3.11) becomes
2L m m2 /4−Lm 1 z q q ms . (q)∞ s q −1
wherein
m,s∈Z
Changing s to s + L − j/2, and changing m to m − 2s + j afterwards, we obtain
1 m m2 /4 2 2L z q z−2s+j q −(s−j/2) . j s+L− (q)∞ 2 q −1 m∈Z
Changing q to
q −1
s∈Z
we obtain the desired formula.
Letting L → ∞ with L ∈ Z, we obtain from (3.10) a formal but suggestive expression χi (q −1 , z)χ0 (q, z) (i) . for the would-be “character” of Z We refer the reader to [1] for similar results on product formulas for shifted characters. 4. Form Factors and ∞-Cycles In this section we discuss the relation between the space of ∞-cycles and form factors of the SU (2) invariant Thirring model (ITM). 4.1. Form factor axioms. First we recall the setting. Let V = Cv+ ⊕ Cv− be the vector representation of sl2 = CE ⊕ CF ⊕ CH . The action is given by Ev+ = 0, Ev− = v+ , F v+ = v− , F v− = 0, H v± = ±v± . Denote by P ∈ End(V ⊗2 ) the permutation operator P (u ⊗ v) = v ⊗ u. The S-matrix of ITM is the linear operator acting on V ⊗2 defined by S(β) =
ζ (−β) β − π iP . ζ (β) β − π i
Here ζ (β) is a certain meromorphic scalar function that accounts for the overall normalization. The precise formula can be found e.g. in [9], Eq.(16). With each local field O in the theory is associated a tower of functions f O = (fn (β1 , . . . , βn ))n≥0 called the form factor of O. The function fn (β1 , . . . , βn ) takes values in the tensor product V ⊗n . Form factor (fn )n≥0 should satisfy the following axioms: Axiom 1 :fn (. . . , βj +1 , βj , . . . ) = Pj,j +1 Sj,j +1 (βj − βj +1 )fn (. . . , βj , βj +1 , . . . ), Axiom 2 :fn (β1 , . . . , βn−1 , βn + 2πi) nπ i
= e 2 Pn,n−1 · · · P2,1 fn (βn , β1 , . . . , βn−1 ), Axiom 3 : resβn =βn−1 +πi fn (β1 , . . . , βn ) = (I + e− 2 Sn−1,n−2 (βn−1 − βn−2 ) · · · Sn−1,1 (βn−1 − β1 )) ×fn−2 (β1 , . . . , βn−2 ) ⊗ (v+ ⊗ v− − v− ⊗ v+ ). nπ i
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M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
Here Pj,j +1 is the permutation operator acting on the j th and (j + 1)st components, and Sj,j (β) is the operator acting on the tensor product of the j th and j th components of V ⊗n as S(β). The operators Si,j (β) and Pi,j commute with the action of sl2 . If (fn )n≥0 is a form factor, then (Ffn )n≥0 is also a form factor. For that reason, we restrict our considerations to only form factors satisfying the highest weight condition, Hfn (β1 , . . . , βn ) = mfn (β1 , . . . , βn ), Efn (β1 , . . . , βn ) = 0
(4.1) (4.2)
for some m ∈ Z≥0 for all n. Other form factors can be obtained from these ones by the action of F ∈ sl2 . 4.2. The integral formula and ∞-cycles. A large class of form factors of the ITM is given in terms of the hypergeometric integral [8]. It has the following structure: ψP (β1 , . . . , βn ) :=
#M=l
vM
l C l p=1
dαp
l
φ(αp ; β1 , . . . , βn )
p=1
×wM (α1 , . . . , αl |β1 , . . . , βn )P (X1 , . . . , Xl |z1 , . . . , zn ), (4.3) where 0 ≤ l ≤ n. Let us explain the notation. First, vM ∈ V ⊗n denotes the vector vM := v1 ⊗ · · · ⊗ vn ∈ V ⊗n , where the j ’s are related to the index set M = {m1 , . . . , ml } (1 ≤ m1 < · · · < ml ≤ n) by M = {j |j = −}. Second, φ and wM are fixed functions given by α−βj +πi n ( −2πi ) 1 φ(α; β1 , . . . , βn ) := , α−β −(α−β ) j j 1−e ( −2πi ) j =1 wM := Skewα1 ,... ,αl gM , mp −1 l αp − β j + π i 1 gM (α1 , . . . , αl |β1 , . . . , βn ) := αp − βmp αp − β j p=1 j =1 × (αp − αp + π i). 1≤p
In the second line, Skewα1 ,... ,αl stands for the skew symmetrization with respect to n,l is a deformed cycle, wherein the variables Xa , zj are α1 , . . . , αl . Third, P ∈ C related to the variables αa , βj by Xa = e−αa ,
zj = eβj .
Lastly, the integration contour C goes along the real axis, except that the simple poles of the integrand at αp = βj − 2πiZ≥0
Form Factors of Massive Integrable Field Theory
567
are located below C, and those at αp = βj − πi + 2πiZ≥0 above C. These are the only poles of the integrand. The integral converges absolutely if 2l ≤ n. n,l we set For a cycle Pn,l ∈ C n n ζ (βj − βj ) · ψPn,l (β1 , . . . , βn ), fPn,l (β1 , . . . , βn ) := cn,l e 4 j =1 βj 1≤j <j ≤n
where cn,l is a constant defined by l d2k+2a,a , (m = 2k), cn,l := a=0 l d a=1 2k+2a−1,a , (m = 2k + 1), n 2π (−2πi)−l− 2 l!. dn,l := ζ (−πi) The precise connection between ∞-cycles and form factors is given by the following theorem proved in [9]. Theorem 4.1. For an ∞-cycle p = (Pn,l ) of weight m ∈ Z≥0 , the tower fp = (fPn,l ) satisfies the form factor axioms as well as the highest weight conditions (4.1), (4.2). A local field O is said to have Lorentz spin s if the homogeneous property fnO (β1 + , . . . , βn + ) = es fnO (β1 , . . . , βn ) holds for its form factor. It is easy to see that the Lorentz spin of fp is equal to deg p introduced in (3.5). Suppose f O = (fn ) is a form factor of a local field. Then there exists an N ≥ 0 such that fn = 0 if n < N. From Axiom 3 we have Axiom 3 :
resβN =βN −1 +πi fN (β1 , . . . , βN ) = 0.
It can be shown [6] that the function fPN,l associated with a minimal cycle PN,l satisfies Axiom 3 . As in [6, 4], we expect the converse to be true, namely that for any N -minimal N,l form factor f = (fn )n≥0 satisfying (4.1), (4.2), there exists a minimal cycle PN,l ∈ W such that fPN,l = fN . Under this assumption, Theorem 3.4 states that any f (with fixed weight m) can be written as fp for some ∞-cycle p ∈ Z. 4.3. Realization of the space of local operators. The space of ∞-cycles is not the same as the space of local fields, since there are null cycles. In [13] it is proved that n,l = ker Pn,l → fPn,l . (4.4) (x0− .An,l−1 + (x0− )(2) .An,l−2 ) ∩ C Now we set
n,l := C n,l / (x − .An,l−1 + (x − )(2) .An,l−2 ) ∩ C n,l M 0 0
and consider the projection n,l −→ M n,l , C
Pn,l → P¯n,l .
Set L[m] := {p¯ = (P¯n,l )|p = (Pn,l ) ∈ Z[m]}.
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M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
Lemma 4.2. If m < 0, then we have L[m] = {0}. Proof. Let p = (Pn,l ) ∈ Z[m] be N -minimal and set l = N,l . In [4] it is proved that W N,l / x − .W N,l −1 + (x − )(2) .W N,l −2 = 0 W 0 0
N−m 2 . Then
if
we have PN,l ∈
N − 2l < 0.
Hence there exists an ∞-cycle + (x − )(2) .Z q ∈ x0− .Z 0 such that p − q is (N + 2)-minimal. Repeating this argument, we find that for each n, l we have P¯n,l = 0. Therefore p¯ = 0. Now we set L = ⊕m≥0 L[m]. Denote by F the space of form factors satisfying (4.1) and (4.2). Then the following map is well defined: : L −→ F,
p¯ = (P¯n,l ) → fp = (fPn,l ).
We conjecture that the map is an isomorphism. In the end, let us give some examples discussed in [9]. Identity operator. The form factor of the identity operator I is of weight zero and zero minimal. It is given by f I = (1, 0, 0, . . . ). Note that f2 = 0 and f0 = 0 does not violate Axiom 3, since the latter becomes trivial for n = 2, resβ2 =β1 +π i f2 (β1 , β2 ) = (I + (−1)I )(f0 ⊗ (v+ ⊗ v− − v− ⊗ v+ )) = 0. The form factor f I is obtained from the ∞-cycle 10 = (10 , X, X ∧ X 3 , X ∧ X 3 ∧ X 5 , . . . ). This can be seen from (4.4) and the following lemma. Lemma 4.3. Set A¯ 2l,l = {P (X1 , . . . , Xl ) ∈ A2l,l |P (X1 , . . . , Xl−1 , 0) = 0}. Then A¯ 2l,l ⊂ x0− .A2l,l−1 + (x0− )(2) .A2l,l−2 . For the proof, see Appendix Appendix B.
Form Factors of Massive Integrable Field Theory
569
su(2) currents. The ∞-cycles associated with su(2) currents jσ+ (σ = ±) are of weight two and two minimal. They are given by (j++ )2l+2,l = (−1)l ( (j−+ )2l+2,l
2l+2
zj−1 )X ∧ X 3 ∧ · · · ∧ X 2l−1 ,
j =1
= X ∧ X 5 ∧ · · · ∧ X 2l+1 3
(l ≥ 0).
It is easy to see that + j++ = −x−1 I,
j−+ = x1+ I.
Energy-momentum tensor. Denote by Tz and Tz¯ the holomorphic and anti-holomorphic part of the energy momentum tensor, respectively. They are of weight zero and two minimal. The form factors are obtained from the following sequences of deformed cycles: (Tz )2l,l = (
2l
zj )X 0 ∧ X 3 ∧ X 5 ∧ . . . ∧ X 2l−1 ,
j =1
(Tz¯ )2l,l = (−1)l−1 (
2l
zj−1 )(
j =1
2l
zj−1 )X 0 ∧ X 1 ∧ X 3 ∧ · · · ∧ X 2l−3
(l ≥ 1).
j =1
Note that these sequences are not ∞-cycles. However the following holds modulo null cycles (4.4): Tz = −ix1− x1+ I,
+ − Tz¯ = −ix−1 x−1 I.
These equalities can be checked by using Lemma 4.3. Appendix A. Polynomial Realization of Evaluation Modules In this appendix, we derive an action of the quantum algebra U√−1 on the space An = ⊕nl=0 An,l . The definition of the action is given in Subsect. 2.2. Here we give the origin of the definition and some details of its derivation. Recall that the space An,l is the space of skew-symmetric polynomials in the variables X1 , . . . , Xl of degree less than or equal to n−1 with coefficients in Kn = C(z1 , . . . , zn ). We use the wedge notation defined in (2.11) for skew-symmetric polynomials. For 1 ≤ a ≤ n we set Ga (X) =
a−1
n
j =1
j =a+1
(1 + zj X)
(1 − zj X).
First note the following simple fact (see Remark below Eq. (3.12) in [4]). Lemma A.1. The skew-symmetric polynomials Gp1 ∧ · · · ∧ Gpl (1 ≤ p1 < · · · < pl ≤ n) constitute a Kn -basis of An,l .
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M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama
Set V = Cv+ ⊕ Cv− . Define operators σ z , σ ± acting on V : 1 0 01 00 z + − σ = ,σ = ,σ = . 0 −1 00 10 When we consider the tensor product of V , we denote by σa∗ the operator σ ∗ acting on n = Kn ⊗ C[q, q −1 ], the a th tensor component. Extending the coefficient ring we set K n = An ⊗ C[q, q −1 ]. We define the action of g ∈ Uqres (see Sect. 3.1) on V ⊗n ⊗ K n A (n−1) res by (ρz1 ⊗ · · · ⊗ ρzn ) ◦ (g), where ρz is the evaluation representation of Uq on V ⊗ C[q, q −1 ][z, z−1 ] such that e1 → σ + , f1 → σ − , t1 → q σ , e0 → zσ − , f0 → z−1 σ + , t0 → q −σ z
z
(A.1)
and is the coproduct defined by (ei ) = ei ⊗ 1 + ti ⊗ ei ,
(fi ) = fi ⊗ ti−1 + 1 ⊗ fi ,
(ti ) = ti ⊗ ti .
Here we use the Chevalley generators ei , fi , ti , i.e., + = t1−1 f0 , x0− = f1 , x1− = e0 t1 , t0 = t1−1 . x0+ = e1 , x−1
n from the action on V ⊗n ⊗ K n by using an isomorphism We induce an action of Uqres on A ⊗n n and A n . The isomorphism is given as follows. Let ψ1 , . . . , ψn be a set between V ⊗ K of Grassmann variables. We denote by n the Grassmann algebra generated by them. It is an irreducible module over the fermion algebra n generated by ψa , ψa∗ (1 ≤ a ≤ n) such that [ψa , ψb∗ ]+ = δa,b , [ψa , ψb ]+ = [ψa∗ , ψb∗ ]+ = 0. n : There are isomorphisms of vector spaces over K n n ⊗ K n A n . V ⊗n ⊗ K
(A.2)
The first isomorphism is given by the Jordan-Wigner transformation ψa∗ = σa+
n
(−iσjz ),
ψa = σa−
j =a+1
n
(iσjz )
j =a+1
⊗n and the identification of v+ ∈ V ⊗n with 1 ∈ n . The second isomorphism is given by n with the wedge product the identification of the left multiplication of ψa on n ⊗ K Ga ∧ on An (see Lemma A.1). − − + + (t), X≤0 (t), X≥0 (t), X<0 (t), Our goal is to compute the actions of the operators X>0 − − + + (2) (2) (2) (2) X>0 (t) , X≤0 (t) , X≥0 (t) , X<0 (t) , a± (t) (see Sect. 2.2) in the limit ε → 0 by setting q = ieε . The computation is elementary but long. We do not give all the details of computation. By a straightforward calculation we obtain − Lemma A.2. We have the expansion of the end term of the half current X>0 (t) up to the order ε, z −1 − q x1 = za ψ a + za ψa σb ε + O(ε 2 ). (A.3) a
a
Form Factors of Massive Integrable Field Theory
571
Similarly, we have the expansion of the end term of the half current a+ (t) up to the order ε2 , iq −1 a1 = α0 + α1 ε + α2 ε 2 + O(ε 3 ), where α0 =
α1 = −
za ,
a
α2 − α0 /2 = 4
za σaz + 4
a
(A.4)
za ψa ψb∗ ,
a
za ψa σbz ψc∗ .
a0 (t) are determined recursively by the equation − (t) − q −1 tx1− = X>0
it − (t)]. [iq −1 a1 , X>0 [2]
(A.5)
Using (A.3) and (A.4), one can check Proposition A.3. We have the expansion − (t) = γ0 (t) + γ1 (t)ε + O(ε 2 ), X>0
where γ0 (t) =
Aa (t)ψa , γ1 (t) =
a
Ba,b (t)ψa σbz +
a
Ca,b,c (t)ψa ψb ψc∗ ,
(A.6)
(A.7)
a
and Aa (t) =
za t 1 + z j t , 1 − za t 1 − zj t j >a
Ca,b,c (t) = 8Ca,b (t)
Ac (t) , zc t
1 − zb t Aa (t), 1 + zb t za t 1 + zj t zb t Ca,b (t) = . 1 − zj t 1 − zb t 1 − za t Ba,b (t) =
a<j
The following equality is useful in this calculation. 1 + zj t 1 Ab (t) = − 1 . 2 1 − zj t b>a
(A.8)
j >a
− + + Instead of repeating similar calculations for X≤0 (t), X≥0 (t), X<0 (t), we can use n given by symmetries. Let α be an anti-algebra map of the algebra n ⊗ K
α(za ) = zn+1−a ,
−1 ∗ α(ψa ) = zn+1−a ψn+1−a ,
α(ψa∗ ) = zn+1−a ψn+1−a .
(A.9)
n [[t, t −1 ]] of formal series in t by setting α(t) = t. We extend it to the space n ⊗ K Similarly, we define an algebra map β by −1 , β(za ) = zn+1−a
∗ β(ψa ) = (−1)a+1 ψn+1−a ,
β(ψa∗ ) = (−1)a+1 ψn+1−a , (A.10)
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and extend it to formal series in t by setting β(t) = t −1 . Then we have α 2 = id, β 2 = (−1)(n+1) id, αβ = (−1)(n+1) βα, where is the parity in the fermion algebra n . Namely, = 0 on the even part of n , and = 1 on the odd part. We also have α(T ) = T , β(T ) = T −1 , where T =
n
(iσaz ).
a=1
Proposition A.4. We have + − X≥0 (t) = −iT t −1 α(X>0 (t)) mod ε2 , + X<0 (t) − X≤0 (t)
= =
− iT β(X>0 (t)) mod ε2 , − tβα(X>0 (t)) mod ε2 . −1
(A.11) (A.12) (A.13)
Proof. We use the equalities α(q −1 a1 ) = q −1 a1 mod ε3 , β(q −1 a1 ) = −qa−1 mod ε3 .
(A.14) (A.15)
Let us prove (A.11). It is easy to check − iT t −1 α(x1− q −1 t) = x0+ mod ε2 . − By using (A.14), Eq. (A.5), which determines X>0 (t), is transformed into − − iT t −1 α(X>0 (t)) − x0+ = −
it − (t))] mod ε 2 . [iq −1 a1 , − iT t −1 α(X>0 [2]
From the defining relations of the Drinfeld currents we have + (t) − x0+ = − X≥0
it + [iq −1 a1 , X≥0 (t)]. [2]
Therefore, we obtain (A.11). Other cases are similar.
From Propositions A.3 and A.4, a short calculation leads to n , the actions of the half currents in the limit Proposition A.5. On the space n ⊗ K √ q → −1 are given as follows. We write them on the subspace isomorphic to An,l by (A.2), and show the equalities as rational functions in t. However, they should be − + properly understood as equalities of power series in t for X>0 (t), X≥0 (t), and in t −1 − + for X≤0 (t), X<0 (t), − − (t) = −X≤0 (t) = X>0
− − X>0 (t)(2) = X≤0 (t)(2)
za t 1 + z j t (A.16) ψa , 1 − za t 1 − zj t a j >a 1 + zj t za t zb t ψa ψb , (A.17) =i 1 − za t 1 − zj t 1 − zb t a
+ + X≥0 (t) = −X<0 (t) = −i n−2l−1
a<j
a
1 + zj t 1 ψ ∗, 1 − za t 1 − zj t a j
(A.18)
Form Factors of Massive Integrable Field Theory
+ + X≥0 (t)(2) = X<0 (t)(2) = (−1)n i
a
573
1 + zj t 1 1 ψ ∗ψ ∗. 1 − za t 1 − zj t 1 − zb t a b a<j
(A.19) We set
b± (t) :=
an (q −1 t)n ,
±n>0 (2) b± (t)
:=
±n>0
a2n (q −1 t)2n . q + q −1
(2)
Proposition A.6. The actions of b± (t), b± (t) in the limit ε → 0 are given as follows: m a za m b+ (t) = −i (A.20) t , m m≥1,odd za t (2) 2b+ (t) = the even part of σaz 1 + z t a a 1 − zj t za t 1 ψa ψb∗ , −4 (A.21) 1 + za t 1 + zj t 1 + zb t a
b− (t) = −β(b+ (t)),
a<j
(2) b− (t)
(2)
= −β(b+ (t)).
(A.22)
Proof. We use the equality b+ (t) =
1 − log(1 + (q − q −1 )t1−1 [x0+ , X>0 (t)]). q − q −1
A simple calculation shows (A.20) and (A.21). For b− (t), we use the properties of the algebra map β, (A.17) and β(x0+ ) = −iT −1 x0− mod ε2 ,
β(t1−1 ) = t1 ,
and obtain β(b+ (t)) =
1 + log(1 + (q − q −1 )t1 [x0− , X<0 (t)]) mod ε2 . q − q −1
Since the expression in the right-hand side is exactly −b− (t), we have (A.22) in the limit ε → 0. Proof of Proposition 2.1. The final step is to rewrite the actions by the second part of the isomorphisms (A.2). It is often useful in the work to note that these actions are symmetric with respect to z1 , . . . , zn on An . This fact follows from the construction in [14] as explained in [4]. The actions (2.13) and (2.14) follow from the equalities Aa (t)Ga (x) = Fn (t, X), (A.23) a
4
a
Cab (t)Ga (X) ∧ Gb (X) = Fn(2) (t|X1 , X2 ).
(A.24)
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M. Jimbo, T. Miwa, E. Mukhin, Y. Takeyama (2)
Note that there are no poles at t = X in Fn (t, X), or t = X1 , X2 in Fn (t|X1 , X2 ). By induction, we can check the equalities at the pole t = zn−1 (and therefore at t = za−1 for all a), and at t = ∞. The action (2.19) follows from (A.20). Let us prove (2.15). We rewrite as σaz = 1 − 2ψa ψa∗ in (A.21). Then, the proof reduces to the case l = 1, which is equivalent to the equality 1 − zj t Gb (X) zb t za t Ga (X) − 2 − 1 + za t 1 + zb t 1 + zj t 1 + za t b
a
1 + za t 1 ψ ∗. 1 − za t 1 − za t a j
This operator sends Ga (X) to 1 + za t Ga (t) 1 = . 1 − za t 1 − za t n (t) j
Computing the effect of T , we obtain (2.17). Similarly, the proof of (2.18) reduces to the equality for b < a,
Ga (−u)Gb (u) − Ga (u)Gb (−u) du n (−u) n (u) u−t c 1 + zj t 1 1 = . 1 − zb t 1 − zj t 1 − za t resu=zc−1
b<j
The first term with Ga (−u)Gb (u) vanishes because there are no poles at u = zc−1 for any c. The second term with −Ga (u)Gb (−u) can be calculated by the residue at u = t because there are no poles at u = −zc−1 . Appendix B. Proof of Lemma 4.3 Following the line of [13], we give a proof of Lemma 4.3 by using the fermionic realization of the space An,l given in Appendix A. Let ψ1 , . . . , ψn be a set of Grassmann variables. Denote by Kn [ψ1 , . . . , ψn ] the exterior algebra generated by them over Kn = C(z1 , . . . , zn ). Introduce the degree defined by deg ψa = 1 and denote by Kn [ψ1 , . . . , ψn ]l the homogeneous component of degree l. Then we have the isomorphism ∼
Cn,l : Kn [ψ1 , . . . , ψn ]l −→ An,l .
Form Factors of Massive Integrable Field Theory
575
See Lemma A.2 for the definition of this isomorphism. The action of x0− and (x0− )(2) on An = ⊕nl=0 An,l is intertwined with multiplication on Kn [ψ1 , . . . , ψn ] with the following elements: x0− ↔ 1 =
n
(−1)n−a ψa ,
i(x0− )(2) ↔ 2 =
(−1)a+b ψa ψb .
1≤a
a=1
Now let us prove the lemma. Set n = 2l in the construction above. Introduce a set of new generators {ϕa }2l a=1 of Kn [ψ1 , . . . , ψ2l ] given by ϕa = ψa − ψ2l (a = 1, . . . , 2l − 1),
ϕ2l = ψ2l .
The elements 1 and 2 are represented in terms of these generators as follows: 1 =
2l−1
(−1)a ϕa ,
(B.1)
˜ 2 − 1 (ϕ2l−1 − ϕ2l ), 2 =
(B.2)
a=1
where ˜2 =
(−1)a+b ϕa ϕb .
1≤a
From the definition of the isomorphism C2l,l we can see that C2l,l (Kn [ϕ1 , . . . , ϕ2l−1 ]l ) = A¯ 2l,l . From (B.1) we have that Kn [ϕ1 , . . . , ϕ2l−1 ]l = 1 · Kn [ϕ1 , . . . , ϕ2l−2 ]l−1 + Kn [ϕ1 , . . . , ϕ2l−2 ]l . (B.3) In [13] it is proved that the map Kn [ϕ1 , . . . , ϕ2l−2 ]l−2 −→ Kn [ϕ1 , . . . , ϕ2l−2 ]l ,
˜2 ·x x →
is an isomorphism. From this fact, (B.2) and (B.3), we find that Kn [ϕ1 , . . . , ϕ2l−1 ]l ⊂ 1 · Kn [ϕ1 , . . . , ϕ2l ]l−1 + 2 · Kn [ϕ1 , . . . , ϕ2l−2 ]l−2 . This completes the proof.
Acknowledgements. We thank Masaki Kashiwara for helpful discussions. JM is partially supported by the Grant-in-Aid for Scientific Research (B2) no.12440039, and TM is partially supported by (A1) no.13304010, Japan Society for the Promotion of Science. EM is partially supported by the National Science Foundation (NSF) grant DMS-0140460. YT is supported by the Japan Society for the Promotion of Science.
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References 1. Berkovich, A., Paule, P.: Variants of the Andrews-Gordon Identities. The Ramanujan J. 5, 391–404 (2001); Lattice Paths, q-Multinomials and Two Variants of the Andrews-Gordon Identities. ibid., 409–424 2. Chari, V., Pressley, A.: Quantum affine algebras at roots of unity. Representation Theory (electronic) 1, 280–382 (1997) 3. Jimbo, M., Miwa, T.: Quantum KZ equation with |q| = 1 and correlation functions of the XXZ model in the gapless regime. J. Phys. A: Math. Gen. 29, 2923–2958 (1996) 4. Jimbo, M., Miwa, T., Takeyama, Y.: Counting minimal form factors of the restricted sine-Gordon model. math-ph/0303059, to appear in Moscow Math. J. 5. Koubek,A.: The space of local operators in perturbed conformal field theories. Nucl. Phys. B435[FS], 703–734 (1995) 6. Nakayashiki, A.: Residues of q-hypergeometric integrals and characters of affine Lie algebras. math.QA/0210168 7. Nakayashiki, A.: The chiral space of local operators in SU (2)-invariant Thirring model. math.QA/0303192 8. Nakayashiki, A., Pakuliak, S., Tarasov, V.: On solutions of the KZ and qKZ equations at level 0. Ann. Inst. Henri Poincar´e 71, 459–496 (1999) 9. Nakayashiki, A., Takeyama, Y.: On form factors of the SU (2) invariant Thirring model. In: MathPhys Odyssey 2001, Integrable Models and Beyond-in honor of Barry M. McCoy, M. Kashiwara and T. Miwa, (eds.), Progr. in Math. Phys., Basel-Boston: Birkh¨auser, 2002, pp. 357–390 10. Smirnov, F.: Form factors in completely integrable models in quantum field theory. Singapore: World Scientific, 1992 11. Smirnov, F.: On the deformation of Abelian integrals. Lett. Math. Phys. 36, 267–275 (1996) 12. Smirnov, F.: Counting the local fields in SG theory. Nucl. Phys. B453[FS], 807–824 (1995) 13. Tarasov, V.: Completeness of the hypergeometric solutions of the qKZ equations at level zero. Am. Math. Soc. Translations Ser. (2) 201, 309–321 (2000) 14. Tarasov, V., Varchenko, A.: Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras. Inventiones Math. 128, 501–588 (1997) Communicated by L. Takhtajan
Commun. Math. Phys. 245, 577–609 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1011-5
Communications in
Mathematical Physics
The Stability of the Non-Equilibrium Steady States Yoshiko Ogata Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan. E-mail: [email protected] Received: 27 May 2003 / Accepted: 4 September 2003 Published online: 23 January 2004 – © Springer-Verlag 2004
Abstract: We show that the non-equilibrium steady state (NESS) of the free lattice Fermion model far from equilibrium is macroscopically unstable. The problem is translated to that of the spectral analysis of the Liouville Operator. We use the method of positive commutators to investigate it. We construct a positive commutator on the lattice Fermion system, whose dispersion relation is ω(k) = cos k − γ . 1. Introduction The investigation of equilibrium states in statistical physics has a long history. In a mathematical framework, an equilibrium state is defined as a state which satisfies the Kubo-Martin-Schwinger condition. This is a generalization of the Gibbs equilibrium state. Many researches to justify this definition have been made [HKT, PW, R1]. One of them is the investigation of “return to equilibrium”; an arbitrary initial state that is normal with respect to the equilibrium state will converge to the equilibrium state. Recall that the notion of normality represents the macroscopic equivalence in a quasi-local system. Although it is a physically fundamental phenomenon, to prove it rigorously is not easy. A complete proof of return to equilibrium in the two-sided XY-model was given by H. Araki [A4]. The other example is an open quantum system, which consists of a finite subsystem and an infinitely extended reservoir in equilibrium [JP1, JP2, M]. The open quantum system converges to the asymptotic state, which is the equilibrium state of the coupled systems. The notable fact here is that this state is normal to the initial one, i.e., macroscopically equivalent to the initial state. This shows that the equilibrium state of the reservoir is macroscopically stable when a finite subsystem is connected to. Recently non-equilibrium steady state (NESS) far from equilibrium has attracted considerable interests. The NESS is introduced as a state asymptotically realized from an inhomogeneous initial state [JP3, JP4, R2]. A question rises naturally here; is the NESS macroscopically stable? As an analogy of return to equilibrium in an open system, we connect a finite small system to the NESS through a bounded interaction. Will the NESS
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Y. Ogata
converge to a state that is normal to the initial state or not? We consider this problem about a free Fermion model on one-dimensional lattice. The explicit form of the NESS is known on this model [HA, AP]. We show if the NESS is far from equilibrium, it is macroscopically unstable. This result is due to the following fact: for the NESS far from equilibrium, the number of the particles with momentum k is different from the number of the particles with momentum −k, although they have the same energy. Technically, the investigation corresponds to the study of the spectral property about the Liouville operator, which represents the dynamics on the Gelfand Naimark Segal (GNS) Hilbert space of the initial state. We use the positive commutator method to analyze the spectrum. It is well-known that a radiation field system with energy dispersion relation ω(k) = |k| has a nice covariance property and the positive commutator is constructed with the aid of it. However, in our system, the dispersion relation is ω(k) = cos k, a new method for constructing a positive commutator is required. We shall construct such a positive commutator and investigate the macroscopic stability of the NESS in this paper. The paper is organized as follows. In the next section, we introduce basic definitions and notations, then state the main theorems. In Sect. 3, we will explain the strategy of the proof. In Sect. 4, we review the role of the standard theory in the research of the NESS, and introduce the modular structure of our model. In Sect. 5, we introduce the rescaling group which is the key to construct the positive commutator. In order for the positive commutator method to work, we have to introduce the cut off of the interaction. This is done in Sect. 6. Section 7 is devoted to the construction of the positive commutator. Then in Sect. 8, we derive the spectral property of the Liouville operator using the method of M.Merkli [M] on the Virial Theorem. We complete the proof in Sect. 9. Below, for a self-adjoint operator A, we denote by P (A ⊂ I ) the spectral projection of A onto the subset I . 2. Main Results In this section, we introduce the basic definitions, and state the main results. 2.1. The C ∗ -algebraic framework. A C ∗ -dynamical system is a pair (O, τ ) where O is a C ∗ -algebra, and τ is a strongly continuous one-parameter group of automorphisms of O. The elements of O describe observables in the physical system and τ specifies their time evolution. Below we assume that O has an identity. A physical state is described as a positive linear functional with norm 1. Let ω be a state on O with GNS triple (H, π, ). The notion of ω-normal is defined as follows: Definition 2.1. A state η is said to be ω-normal if there exists a density matrix ρ on H such that η(·) = Trρπ(·). If a state is not ω-normal, it is called ω-singular. For a quasi-local algebra, ω-normality means that η is approximated in norm topology by local perturbation of ω. So in a quasi-local algebra, we can make the following interpretation: if η is ω-normal and ω is η-normal, ω and η are macroscopically equivalent. The NESS of dynamics τt associated with the state ω are the weak−∗ accumulation points of the set of states 1 T ω ◦ τt dt T 0
Stability of the Non-Equilibrium Steady States
579
as T → ∞. We denote the set of the NESS by τ (ω). As the set of states on O is weak ∗ -compact by Alaoglu’s Theorem, τ (ω) is a non-empty set whose elements are τ -invariant. 2.2. Macroscopic instability. In this subsection, we introduce the notion of macroscopic instability. Let (Om , τm ) be a C ∗ -dynamical system. Let ω0 be an initial state over Om and let ωm be the NESS corresponding to the pair (ω0 , τm ), i.e., ωm ∈ τm (ω0 ). Now we are interested in the stability of ωm . To investigate it, we add an external finite C ∗ -dynamical system (OS , τS ). The combined system (OS ⊗ Om , τS ⊗ τm ) is also a C ∗ -dynamical system. Let us introduce a bounded interaction V between OS and Om , and denote by τV the perturbed dynamics. We shall define the macroscopic instability of ωm as follows: Definition 2.2. The NESS ωm ∈ τm (ω0 ) is macroscopically unstable under a perturbation V , if for any state ωS over OS , no element in τV (ωS ⊗ ωm ) is ωS ⊗ ωm -normal. The NESS ωm ∈ τm (ω0 ) is macroscopically unstable if ωm is macroscopically unstable under some bounded perturbation V . Let us explain it in detail for our system. First we divide the one-dimensional Fermion lattice to the left and the right, and consider a state that each side is in equilibrium at different temperature. This is the initial state ω0 . The corresponding NESS τm (ω0 ) under the time evolution of the free lattice Fermion τm = α f consists of only one point: τm (ω0 ) = {ωρ }. To investigate the stability of ωm = ωρ , we prepare an external finite system described in a finite dimensional Hilbert space Cd . We connect it to ωm through a bounded interaction V . If there is no ωS ⊗ ωm -normal state in τV (ωS ⊗ ωm ), for any state ωS over OS , ωm is macroscopically unstable. 2.3. The model. In this paper we consider the free lattice Fermion system in one dimension. The explicit form of the NESS is known for this system [HA, AP]. Let h ≡ l 2 (Z) be a Hilbert space of a single Fermion. By Fourier transformation, it is unitary equivalent to L2 ([−π, π)). The Hamiltonian h of a single Fermion is given by (hf ) (n) =
1 (f (n − 1) + f (n + 1)) − γf (n) , 2
(1)
on h which is described in Fourier representation as (k) = ω (k) fˆ (k) , hf ω (k) = cos (k) − γ . The γ -term represents the interaction with the external field, and γ is a parameter in (−1, 1). The free lattice Fermi gas R is described as the CAR-algebra Of over h. And its dynamics is given by f αt (a (f )) = a eith f . (2) In the initial state ω0 , the lattice is separated into the left and the right. And they are kept at different inverse temperature β− , β+ , respectively. The NESS ωρ associated with ω0 is realized as the asymptotic state under the dynamics α f (2). The explicit form of
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Y. Ogata
ωρ was obtained in [HA] and [AP], independently: ωρ is a state whose n-point functions have a structure ωρ a(fn )∗ · · · a(f1 )∗ a(g1 ) · · · a(gm ) = δnm det( fi , ρgj ),
(3)
where ρ is represented as a multiplication operator, −1 1 + eβ+ ω(k) k ∈ [0, π ) , ρ(k) = −1 1 + eβ− ω(k) k ∈ [−π, 0)
(4)
ω(k) = cos (k) − γ , in the Fourier representation. If β+ = β− , we will say that the NESS ωρ is far from equilibrium. In this paper, the stability of this state is considered. The observables of the small system are described as a C ∗ -algebra OS ≡ B(HS ) on a finite d-dimensional Hilbert space HS = Cd . We denote by HS the free Hamiltonian of the system on HS . The free dynamics αtS is given by αtS (A) = eitHS Ae−itHS . The combined system S + R is described as the C ∗ -algebra O ≡ OS ⊗ O f . f
The free dynamics of the combined system is given by αt0 = αtS ⊗ αt . We denote by δ0 the derivation of αt0 . Let us consider the dynamics including the interaction between S and R. In this paper, we define the interaction term V by V = λ · Y ⊗ a (f ) + a ∗ (f ) ,
(5)
where f ∈ h is called a form factor. Here λ is a coupling constant and Y is a self-adjoint operator on HS . Note that V is an element of O. The perturbed dynamics αt is generated by δ = δ0 + i[V , ·] with D(δ) = D(δ0 ). αt is expanded as follows; αt (A) ≡ αt0 (A) + × 0
n≥1
tn−1
t
in
t1
dt1 0
dt2 · · ·
0
dtn αt0n (V ) , · · · , αt01 (V ) , αt0 (A) .
(6)
The right-hand side converges in norm topology in O. αt is a strongly continuous one parameter group of automorphisms.
Stability of the Non-Equilibrium Steady States
581
2.4. Main Theorem. In the analysis, we carry out the variable transformation from k ∈ [−π, π ) to t ∈ R, by t (k) = tan 2k . Under this variable translation, we identify h = L2 ([−π, π ), dk) with L2 (R, t 2 2+1 dt). We need several assumptions on the small system and the interaction V . Assumption 2.1. Under the identification h = L2 ([−π, π ), dk) = L2 (R, t 2 2+1 dt), let u(θ ) be a strongly continuous one parameter unitary group on h defined by θ t2 + 1 θ 2 2 (−∞, ∞), t , θ ∈ R, g ∈ L g e dt , (u(θ )g) (t) ≡ e 2 e2θ t 2 + 1 t2 + 1 and let p be the generator of u(θ ) = eiθp . For a constant 0 < v < 1, let v be the interval of R defined by 4t 2 ≥ v . v = t ∈ R ; s1 (t) = (1 + t 2 )2 We assume that the form factor f ∈ h in (5) is in the domain D(p3 ). Furthermore, we assume that there exists 0 < v < 1 such that the support of f satisfies suppf ⊂ v . Assumption 2.2. Let ei , ej be an arbitrary pair of eigenvalues of LS = HS ⊗1−1⊗HS on HS ⊗ HS such that ei = ej .There exists kij such that ω(±kij ) = cos kij − γ = ei − ej , 0 ≤ kij , t (kij ) ∈ v . Assumption 2.3. 1. The Hamiltonian HS has no degenerated eigenvalue. 2. There exists a > 0 such that min |f (±kij )| ≥ a.
ei =ej
Assumption 2.4. Let ϕn be the nth eigenvector of HS with eigenvalue En , and let pn be the spectral projection onto ϕn on HS . We denote the eigenvalue of LS = HS ⊗1−1⊗HS as En,m ≡ En − Em , and introduce the following subsets for each eigenvalue e of LS : (j )
Nl
(i)
≡ {i; Ei,j = e}, Nr ≡ {j ; Ei,j = e}, (j ) Nl ≡ Nl , Nr ≡ Nr(i) . j
For a set N , we define a projection pN ≡
n∈N
i
pn .
1. For e = 0, we assume
2δ0 ≡ min inf σ pN (n) Y¯ pNrc Y¯ pN (n) n∈Nl
r
r
(n) HS Nr
p
+ min inf σ pN (m) YpNlc YpN (m) m∈Nr
l
l
(m) HS Nl
p
> 0,
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where σ (A) represents the spectrum of A, and Y¯ is the complex conjugation of Y with respect to the orthonormal basis consisting of eigenvectors of HS . 2. Let Ymn ≡ ϕm |Y |ϕn . We assume |Ymn | > 0 for all m = n. An example which satisfies all of the Assumptions is represented in Appendix A. Here is the main theorem of this paper. Theorem 2.3. Let ωρ be a NESS far from equilibrium, i.e., the state given by (3) with inverse temperatures β+ = β− . Suppose that Assumption 2.1 to 2.4 are satisfied. Then there exists a λ1 > 0 s.t. if 0 < |λ| < λ1 , then ωρ is macroscopically unstable under the perturbation V (5). Especially, ωρ is macroscopically unstable. Furthermore, assume β0 < β+ , β− < β1 , pf , f ≤ b for any fixed 0 < β0 < β1 < ∞ and 0 < b < ∞. Here p is the generator of u(θ ) = eiθp . Then if we fix β+ and 50 β− , we have λ1 ∼ O(v 9 ) as v goes to 0. On the other hand, if we fix v then we have 200 λ1 ∼ O(|β+ − β− | 11 ) as β+ − β− → 0. Theorem 2.4. Let ωρ be an equilibrium state, i.e. the state given in (3) with β ≡ β+ = β− . Then there exists a β−KMS state ωV w.r.t. the perturbed dynamics αt (6), which is normal to ωS ⊗ ωρ for an arbitrary faithful state ωS of OS . Suppose that Assumption 2.1 to 2.4 are satisfied. Then there exists a λ1 > 0 such that if 0 < |λ| < λ1 , any ωV -normal state η exhibits return to equilibrium in an ergodic mean sense, i.e. 1 T η (αt (A)) dt = ωV (A) lim T →∞ T 0 for all A ∈ O. Furthermore, assume β0 < β < β1 , pf , f ≤ b for any fixed 0 < β0 < β1 < ∞ 50 and 0 < b < ∞. Then we have λ1 ∼ O(v 9 ) as v goes to 0. 3. The Strategy of the Proof In this section, we explain the strategy and the organization of the proof. 3.1. The kernel of Liouville operator. By the standard theory, the problem of macroscopic instability is translated into the spectral problem of the so-called Liouville operator L (see Sect. 4, Proposition 4.3) : if KerL = {0}, ω is macroscopically unstable. In our model, the Liouville operator L is an operator on H = (HS ⊗ HS ) ⊗ F(h ⊕ h), given by L = (HS ⊗ 1) ⊗ 1 − (1 ⊗ HS ) ⊗ 1 + 1 ⊗ d (h ⊕ −h) + λI0 .
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Here F(h⊕h) is a Fermi Fock space over h⊕h and d (h ⊕ −h)is a second quantization of the multiplication operator h ⊕ −h. λI0 is the interaction term. The main part of this paper is to prove the following theorem on the eigenvector of L: Theorem 3.1. For each eigenvalue e˜ of LS = HS ⊗ 1 − 1 ⊗ HS , define an operator (e) ˜ on the space P (LS = e) ˜ · HS ⊗ HS by π (e) ˜ ≡ dk m(k, 1)∗ P (LS = e) ˜ δ (ω (k) + LS − e) ˜ m(k, 1) −π π + dk m(k, 2)∗ P (LS = e) ˜ δ (−ω (k) + LS − e) ˜ m(k, 2) −π
m(k, i) ≡ Y ⊗ 1 · g1i (k) − 1 ⊗ Y¯ · g2i (k) i = 1, 2.
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j
Here gi are defined by 1
g11 = (1 − ρ) 2 f,
g12 = ρ 2 f¯, 1
1
g21 = ρ 2 f,
g22 = (1 − ρ) 2 f¯, 1
and f¯ is the complex conjugation of f in the Fourier representation. Let γe˜ be a strictly positive constant such that (e) ˜ ≥ γe˜ · (P ((e) ˜ = 0))⊥ . Let P˜e˜ be ˜ · P ((e) ˜ = 0) ⊗ Pf , P˜e˜ = P (LS = e) where Pf is the projection onto the vacuum f of F(h ⊕ h). For e ∈ R, let e(e) ˜ ≡ e if e is an eigenvalue of LS , and let e(e) ˜ be an eigenvalue of LS which is next to e, if e is not an eigenvalue of LS . Suppose that Assumptions 2.1 and 2.2 are satisfied. Then there exists λ1 > 0 s.t.,if 0 < |λ| < λ1 , then there is no eigenvector ˜ ˜ = 0, e is not an with eigenvalue e which is orthogonal to P˜e(e) ˜ . In particular, if Pe(e) eigenvalue of L. Furthermore, if we assume β0 < β+ , β− < β1 , pf , f ≤ b for any fixed 0 < β0 < β1 < ∞ and 0 < b < ∞, we can choose λ1 as 100 182 100 100 100 v . (9) , (γe(e) λ1 = C min v 26 , ˜ ) 11 , (vγe(e) ˜ ) 18 γe(e) ˜ Here C is a constant which depends on β0 , β1 and b, but is independent of v and β+ −β− . By Theorem 3.1, the existence of the eigenvector of L is determined by the kernel of (e). We have the following theorem on it (Sect. 9): Theorem 3.2. Suppose that Assumption 2.1 to 2.4 are satisfied. Then if β+ = β− , Ker(e) = {0} for any eigenvalue e of LS . If β+ = β− , Ker(e) = {0} for any non-zero eigenvalue e = 0 of LS , but Ker(0) is non-trivial one-dimensional space. The following fact will develop in the proof: the non-existence of the non-trivial kernel of (0) for the far from equilibrium case β+ = β− is caused by the fact that the number of the particles with the momentum k, −k are different although they have the same energy ω(k) = ω(−k). Note that the same fact induces the existence of current for NESS. Combining Theorem 3.1 and 3.2, we obtain the instability of NESS far from equilibrium. 3.2. The proof of Theorem 3.1. To prove the theorem, we use the positive commutator method. The idea of the method is as follows: suppose that there exists an anti-selfadjoint operator A such that [L, A] ≥ c > 0. If L has an eigenvector ψ with eigenvalue e, we have 0 = 2Re (L − e)ψ, Aψ = ψ, [L − e, A]ψ ≥ c > 0, which is a contradiction. Hence L has no eigenvector. This is called the Virial Theorem. However this argument works only formally and indeed we have to take care of the domain question.
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The proof of Theorem 3.1 is divided into two steps: the first one is to construct the positive commutator (Sect. 5 to Sect. 7), and the second one is to justify the above arguments rigorously (Sect. 8). To construct the positive commutator is a non-trivial problem, and we have to work it out for each model. Now, our field has the dispersion relation ω(k) = cos k − γ . Under the variable translation k → t, we show that there exists a strongly continuous one parameter group of unitaries U (θ ) on H, which satisfies U (−θ ) (1 ⊗ d (h ⊕ −h)) U (θ ) = 1 ⊗ d h−θ ⊕ −hθ , where hθ is a multiplication operator on L2 (R, t 2 2+1 dt) defined by hθ (t) ≡ h(eθ t). This means U (θ ) induces the rescaling of the multiplication operator in h. Let A0 be the generator of U (θ ). This A0 satisfies [L0 , A0 ] = S1 = 1 ⊗ d (s1 ⊕ s1 ) ≥ 0, where S1 is a second quantization of multiplication operator (s1 f ) (t) = s1 (t)f (t),
s1 (t) =
4t 2 . (1 + t 2 )2
Hence by considering rescaling of multiplication operators with respect to t, we obtained the positive commutator [L0 , A0 ] = S1 ≥ 0. However, what is really needed is the strictly positive commutator. If S1 has a spectral gap, following the well-known procedure, we can construct a strictly positive commutator. But S1 does not have a gap now, and we need to overcome this problem. Let cv be the complement of v , and let Ncv be the number operator of particles whose momentum is included in cv . Furthermore, let P be the spectral projection onto the subspace Ncv = 0, and set P¯ = 1 − P . If we restrict ourselves to P H, P S1 P has a spectral gap v > 0, so if L strongly commutes with Ncv , we can construct the positive commutator for L in P H. Furthermore, if the subspace P¯ H includes no eigenvector of L, we just need to analyze L on P H. To make the spectral localization possible, we introduce a cut off of the form factor f ∈ h with respect to v , i.e., suppf ⊂ v . By this cut off, L strongly commutes with Ncv , and furthermore, there is no eigenvector of L, in the subspace P¯ H. Hence we obtain a positive commutator which is sufficient to analyze the eigenvector of L. To carry out the second step, the rigorous justification of the Virial Theorem, we used the new method introduced by M.Merkli. By approximating the eigenvector of L by vectors in the domain of the total number operator N and A0 , we can carry out the arguments rigorously. 4. The Standard Theory and NESS In this section, we explain the role of the standard theory (see [BR1]) in the investigation of the NESS and introduce the modular structure in our model.
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4.1. The standard theory. Let M be a von Neumann algebra on a Hilbert space H with a cyclic and separating vector . A positive linear functional ω on M is said to be normal if there exists a positive traceclass operator ρ such that ω (·) = Tr (ρ·) . We denote by M∗+ the set of all normal positive linear functionals. We can define an operator S0 on a dense set M by S0 x = x ∗ . S0 is closable and we represent the polar decomposition of the closure S¯0 as S¯0 = J 2 . 1
J is called the modular conjugation and is called the modular operator. By the Tomita-Takesaki Theory we have J MJ = M , where M is the commutant of M. Let j : M → M be the antilinear ∗-isomorphism defined by j (x) = J xJ . We define the natural positive cone P by the closure of the set {xj (x) ; x ∈ M}. For ξ ∈ P, define the normal positive functional ωξ ∈ M∗+ by ωξ (x) = ξ, xξ
∀x ∈ M.
We have the following theorems: Theorem 4.1. For any ω ∈ M∗+ , there exists a unique ξ(ω) ∈ P such that ω = ωξ (ω) . Theorem 4.2. For any ∗-automorphism α of M, there exists a unique unitary operator U (α) on H satisfying the following properties: 1. U (α) xU (α)∗ = α (x) , x ∈ M, 2. U (α) P ⊂ P and U (α) ξ (ω) = ξ α −1∗ (ω) , ω ∈ M∗+ , where (α ∗ ω)(A) ≡ ω(α(A)), 3. [U (α) , J ] = 0. Let αt be a one parameter group of automorphisms and let U (t) be a one parameter group of unitaries associated with αt . If U (t) is strongly continuous, it is written as U (t) = eitL with self adjoint operator L. We call L the Liouville operator of αt . 4.2. The NESS and the Liouville operator. Now we move from W ∗ -dynamical systems to C ∗ -dynamical systems, and explain the role of the standard theory in the investigation of the NESS. Let αt be a one parameter group of automorphisms which describes the dynamics of a unital C ∗ -algebra O, and let ω be a state of O. Let (H, π, ) be the GNS triple of ω. Suppose that is a cyclic and separating vector for the von Neumann algebra π(O) , and that there exists an extension α˜ t of αt to π(O) . Then by Theorem 4.2 there is a one parameter group of unitary operators Ut such that α˜ t (x) = Ut xUt∗ ,
x ∈ π(O) ,
Ut∗ P ⊂ P.
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In addition, suppose that Ut is strongly continuous and let L be the Liouville operator. If ω∞ ∈ α (ω) is ω-normal, there exists a vector ξ∞ in P s.t. ω∞ (A) = ξ∞ , π (A) ξ∞ ,
A ∈ O,
by Theorem 4.1. As ω∞ ∈ α (ω), ω∞ is invariant under αt : ω∞ ◦ αt = ω∞ . Hence we have ∗ Ut ξ∞ , π (A) Ut∗ ξ∞ . = ω∞ ◦ αt (A) = ω∞ (A) = ξ∞ , π (A) ξ∞ , A ∈ O. As ξ∞ , Ut∗ ξ∞ ∈ P, we get Ut∗ ξ∞ = ξ∞ , by Theorem 4.1. This means ξ∞ ∈ KerL. In other words, we have the following proposition. Proposition 4.3. If KerL = {0}, any state in α (ω) is ω-singular i.e., ω is macroscopically unstable. 4.3. The modular structure of the model. Now we introduce the modular structure of our model. As the structure is the same as that of [JP3], we just state the results. Let ωS be a state over OS , defined by ωS = Tr(ρS ·) with a density matrix pi |ψi ψi | , ρS = i
on HS . Here {ψi } are orthogonal unit vectors on HS . And let ωρ be a quasi-free state over Of defined in (3). Let ψ¯ ∈ HS (resp. A¯ ∈ B(HS )) be a complex conjugation of ψ ∈ HS (resp. A ∈ B(HS )) with respect to the orthogonal basis of HS , that is given by eigenvectors of HS . We denote the Fermi Fock space over h ⊕ h by F(h ⊕ h). Then we have the following proposition: Proposition 4.4. Suppose that ωS is faithful, and 0 < ρ < 1. The GNS triple (H, π, ) associated with the state ω = ωS ⊗ ωρ is given by H = HS ⊗ Hf ,
= S ⊗ f ,
π = πS ⊗ πf ,
where HS = HS ⊗ HS ,
S =
1
pi2 ψi ⊗ ψ¯ i ,
πS (A) = A ⊗ 1,
A ∈ OS ,
Hf = F (h ⊕ h) , f : vacuum vector of F (h ⊕ h) , 1 1 πf (a (f )) = a (1 − ρ) 2 f ⊕ 0 + a ∗ 0 ⊕ ρ 2 f¯ , 1 1 πf (a ∗ (f )) = a ∗ (1 − ρ) 2 f ⊕ 0 + a 0 ⊕ ρ 2 f¯ , f ∈ h, and is a cyclic and separating vector for the von Neumann algebra π(O) . The dynamics αt (6) defined on the C ∗ -algebra O extends to a weakly continuous one parameter
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group of automorphisms α˜ t over π(O) . α˜ t is implemented by a strongly continuous one parameter group of unitaries and the corresponding Liouville operator L on H is L = (HS ⊗ 1) ⊗ 1 − (1 ⊗ HS ) ⊗ 1 + 1 ⊗ d h˜ + λ (Y ⊗ 1) ⊗ a (g1 ) + a ∗ (g1 ) − λ 1 ⊗ Y¯ ⊗ (−1)N a (g2 ) − a ∗ (g2 ) , (11) where h˜ = h ⊕ −h, g1 = g11 ⊕ g12 ,
g2 = g21 ⊕ g22 , 1
g11 = (1 − ρ) 2 f,
g12 = ρ 2 f¯, 1
g22 = (1 − ρ) 2 f¯.
1
1
g21 = ρ 2 f,
(12) (13) (14)
˜ Here, d h˜ is the second quantization of h.
We denote the Fermion number operator on H by N . We define the system Liouville operator LS , the field Liouville operator Lf , the free Liouville operator L0 , and the interaction I0 operator by LS = (HS ⊗ 1) ⊗ 1 − (1 ⊗ HS ) ⊗ 1, Lf = 1 ⊗ d h˜ , L0 = LS + Lf ,
I0 = (Y ⊗ 1) ⊗ a (g1 ) + a ∗ (g1 ) − 1 ⊗ Y¯ ⊗ (−1)N a (g2 ) − a ∗ (g2 ) . (15)
on H. 5. Rescaling Group In order to construct the positive commutator for the lattice system, we introduce a strongly continuous one parameter group of unitaries. By the variable transformation k → t, we have h = L2 ([−π, π ) , dk) = L2 ((−∞, ∞) , dµ) , where µ is the positive measure on R given by 2 dt. +1 The multiplication operator m(k) on h is transformed as m(k) → m(k(t)), where k(t) ≡ 2 tan−1 t. In particular, the single particle energy h is transformed to dµ (t) =
t2
2 − 1 − γ. +1 Now we introduce a unitary operator u(θ ) on h defined by θ t2 + 1 f eθ t , θ ∈ R, f ∈ L2 ((−∞, ∞), dµ). (u(θ )f ) (t) ≡ e 2 2θ 2 e t +1 h (k) = cos k − γ
→
h (k (t)) =
t2
By an elementary calculation, we have the following lemma;
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Lemma 5.1. u(θ ) is a strongly continuous one parameter group of unitaries. From this lemma, u(θ ) is written as u(θ ) = eiθp with a selfadjoint operator p. The action of p on C0∞ (R) is (pf ) (t) = −i
t2 1 f (t) − 2 f (t) + tf (t) , 2 t +1
f ∈ C0∞ (R) .
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Let u(θ ˜ ) be a unitary operator on h ⊕ h defined by u(θ ˜ ) ≡ u(θ ) ⊕ u(−θ ). The generator of u(θ ˜ ) is p˜ ≡ p ⊕ −p. We define a unitary operator U (θ) on H by the second quantization (u(θ ˜ )) of u(θ ˜ ): U (θ) ≡ 1 ⊗ (u˜ (θ )) . Since U (θ ) is a strongly continuous one parameter unitary group, it is written as U (θ) = ˜ is an anti-selfadjoint operator. eθA0 , where A0 = 1 ⊗ d(i p) Now we introduce the commutator of Lf and A0 as a quadratic form on the dense set (D(N ) ∩ D(A0 )) × (D(N ) ∩ D(A0 )). By straightforward calculation, the action of u(θ ˜ ) on a multiplication operator m1 ⊕ m2 on h ⊕ h is (u˜ (−θ ) (m1 ⊕ m2 ) u˜ (θ )) (t) = m1 e−θ t ⊕ m2 eθ t . Then, the action of U (θ ) on the second quantization d(m1 ⊕ m2 ) of m1 ⊕ m2 is θ U (−θ ) (1 ⊗ d (m1 ⊕ m2 )) U (θ ) = 1 ⊗ d m−θ , (17) ⊕ m 2 1 where (mθi f ) (t) = mi eθ t f (t). We have the following proposition: Proposition 5.2. Suppose that m1 , m2 are bounded multiplication operators on L2 (R, dµ) which are differentiable and satisfy supt∈R |qmi (t)| < ∞, where qmi (t) ≡ −tmi (t) . Then we have 1 ⊗ d (m1 ⊕ m2 ) ϕ, A0 ψ + A0 ϕ, 1 ⊗ d (m1 ⊕ m2 ) ψ
= ϕ, 1 ⊗ d (qm1 ⊕ −qm2 ) ψ , for all ϕ, ψ ∈ D(N ) ∩ D(A0 ).
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Proof. As U (θ ) preserves number, we have U (θ )D(N ) ⊂ D(N ) and have the following relation for all ϕ, ψ ∈ D(N) ∩ D(A0 ): θ − d d m−θ ⊕ m (m1 ⊕ m2 ) 2 1 ψ lim ϕ, 1 ⊗ θ→0 θ U (θ ) − 1 = lim (1 ⊗ d (m1 ⊕ m2 )) U (θ) ϕ, ψ θ→0 θ U (θ ) − 1 + (19) ϕ, 1 ⊗ d (m1 ⊕ m2 ) ψ . θ Using the dominated convergence theorem, we have the following statements: 1. For each ψ ∈ D(N) and bounded operator m on h ⊕ h, lim (1 ⊗ d (m)) U (θ ) ψ = (1 ⊗ d (m)) ψ.
θ→0
2. For each ψ ∈ D(A0 ), U (θ ) − 1 ψ = A0 ψ. θ→0 θ lim
3. For each ψ ∈ D(N), −θ θ − d d m ⊕ m (m1 ⊕ m2 ) 1 2 − d (qm1 ⊕ −qm2 ) ψ lim 1 ⊗ = 0. θ→0 θ Substituting these to (19), we obtain the statement of the proposition.
Substituting m1 = h, m2 = −h to (18), we obtain Lf ϕ, A0 ψ + A0 ϕ, Lf ψ = ϕ, S1 ψ , for all ϕ, ψ ∈ D(N ) ∩ D(A0 ). Here S1 is the second quantization 1 ⊗ d (s1 ⊕ s1 ) of s1 ⊕ s1 on h ⊕ h, with s1 defined by 4t 2 . (1 + t 2 )2 Note that S1 is positive. This is the main point of this commutator. Similarly, we have (s1 f ) (t) = s1 (t)f (t),
s1 (t) =
S1 ϕ, A0 ψ + A0 ϕ, S1 ψ = ϕ, S2 ψ , S2 ϕ, A0 ψ + A0 ϕ, S2 ψ = ϕ, S3 ψ , where S2 = 1 ⊗ d (s2 ⊕ −s2 ) , (s2 f ) (t) = s2 (t)f (t), S3 = 1 ⊗ d (s3 ⊕ s3 ) , (s3 f ) (t) = s3 (t)f (t), Note that S1 , S2 , S3 are all N -bounded.
8 −1 + t 2 t 2 s2 (t) = 3 , 1 + t2
16 t 2 − 4t 4 + t 6 s3 (t) = . 4 1 + t2
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The commutator of A0 with the interaction term can also be considered. As the form factor f ∈ h satisfies f ∈ D(p3 ), suppf ⊂ v and ρ is differentiable in v , we have g1 , g2 ∈ D(p˜ 3 ). We get ϕ, I0 A0 ψ + A0 ϕ, I0 ψ = ϕ, I1 ψ , on D(A0 ) × D(A0 ). Here I1 is defined by ˜ 1 ) + a ∗ (−i pg I1 ≡ (Y ⊗ 1) ⊗ a (−i pg ˜ 1) − 1 ⊗ Y¯ ⊗ (−1)N a (−i pg ˜ 2 ) − a ∗ (−i pg ˜ 2) . Similarly, we have ϕ, I1 A0 ψ + A0 ϕ, I1 ψ = ϕ, I2 ψ , with
ϕ, I2 A0 ψ + A0 ϕ, I2 ψ = ϕ, I3 ψ ,
I2 ≡ (Y ⊗ 1) ⊗ a −p˜ 2 g1 + a ∗ −p˜ 2 g1 − 1 ⊗ Y¯ ⊗ (−1)N a −p˜ 2 g2 − a ∗ −p˜ 2 g2 , I3 ≡ (Y ⊗ 1) ⊗ a i p˜ 3 g1 + a ∗ i p˜ 3 g1 − 1 ⊗ Y¯ ⊗ (−1)N a i p˜ 3 g2 − a ∗ i p˜ 3 g2 .
6. Cut Off of the Interaction and Eigenvector In case the dispersion relation of the field Lf is ω(k) = |k|, A0 is taken as the generator of the shift operator. And the commutator is given by the number operator N : [Lf , A0 ] = N . Note that the dispersion relation of N has a strictly positive spectral gap 1 > 0. However in our case, the dispersion relation is ω(k) = cos(k) − γ , and the commutator is [Lf , A0 ] = S1 . Note that the dispersion relation of S1 is positive, but it does not have a spectral gap: it attains zero at t = 0 and t = ±∞. The existence of the spectral gap is essential in the application of the positive commutator method as seen in the arguments in [M]. So this situation causes a problem. To overcome this difficulty, we assume Assumption 2.1 on the interaction. In this section, we see that as the result of the cut off suppf ⊂ v , any eigenvector of L is in the range of P = P (Ncv = 0). Now let h˜ = h ⊕ h. We decompose h˜ with respect to R = v ⊕ cv : h˜ v
h˜ = h˜ v ⊕ h˜ cv , = L2 (v , dµ) ⊕ L2 (v , dµ), h˜ c = L2 (c , dµ) ⊕ L2 (c , dµ). v
v
v
Note that h˜ (12) is decomposed into h˜ = h˜ v ⊕ h˜ cv , with respect to this decomposition of the Hilbert space, because it is a multiplication operator. Let Nv , Ncv be Nv ≡ 1 ⊗ d(1v ⊕ 0),
Ncv ≡ 1 ⊗ d(0 ⊕ 1cv ),
on H with respect to this decomposition. Below for a Hilbert space K, we denote by U(K) a set of unitary operators on K, and by F(K) the Fock space over K. Furthermore, (u) is the second quantization of u ∈ U(K), and for a self-adjoint operator κ on K, dK(κ) is the second quantization of κ. We have the following proposition:
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Proposition 6.1. There is a unitary operator U : F(h˜ v ⊕ h˜ cv ) → F(h˜ v ) ⊗ F(h˜ cv ) which satisfies the following conditions: 1. For any uv ∈ U(h˜ v ) and ucv ∈ U(h˜ cv ), U (uv ⊕ ucv )U ∗ = (uv ) ⊗ (ucv ). 2. For any f ∈ h˜ v and g ∈ h˜ cv , U a(f ⊕ g)U ∗ = a(f ) ⊗ 1 + (−1)Nv ⊗ a(g). By this proposition, we have ˜
˜
˜
˜
U (eit hv ⊕ eit hcv )U ∗ = (eit hv ) ⊗ (eit hcv ). Recall that L is given by (11). As Assumption 2.1 suppf ⊂ v implies g1 , g2 ∈ h˜ v , we obtain the following unitary equivalence; U LU ∗ = LS + 1 ⊗ dv (h˜ v ) ⊗ 1 + 1 ⊗ 1 ⊗ dcv (h˜ cv ) +λ (Y ⊗ 1) ⊗ a (g1 ) + a ∗ (g1 ) ⊗ 1 −λ 1 ⊗ Y¯ ⊗ (−1)Nv a (g2 ) − a ∗ (g2 ) ⊗ (−1)Ncv . By this equivalence, we have the following lemma, Lemma 6.2. L and Ncv strongly commute. Similarly, Ncv strongly commutes with LS , Lf , L0 , λI0 N , S1 . Using this fact, we obtain the following proposition: Proposition 6.3. Suppose ψ is an eigenvector of L. Then ψ satisfies P Ncv = 0 ψ = ψ, where P Ncv = 0 is the spectral projection of Ncv corresponding to Ncv = 0. Proof. Let Pe (resp.Po ) be the spectral projection of Ncv onto Ncv = even (resp.Ncv = odd). Let us decompose the Hilbert space H as H = Pe H ⊕ Po H. Then by Lemma 6.2, L is decomposed into L = Pe L ⊕ Po L,
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with respect to the subspaces Pe H and Po H. In particular, if ψ is an eigenvector of L, Pe ψ is an eigenvector of Pe L and Po ψ is an eigenvector of Po L. With respect to the decomposition H = HS ⊗ F(h˜ v ) ⊗ F(h˜ cv ), Pe L on Pe H is Pe L|Pe H = LS + 1 ⊗ dv h˜ v + V1 + V2 ⊗ 1 + (1 ⊗ 1) ⊗ dcv h˜ cv |Pe H
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with V1 = λ (Y ⊗ 1) ⊗ a (g1 ) + a ∗ (g1 ) , V2 = −λ 1 ⊗ Y¯ ⊗ (−1)Nv a (g2 ) − a ∗ (g2 ) , because Pe = 1 ⊗ P (Ncv = even). On the other hand, Po L on Po H is Po L|Ho = LS + 1 ⊗ dv h˜ v + V1 − V2 ⊗ 1 + (1 ⊗ 1) ⊗ dcv (h˜ cv )|Po H because Po = 1 ⊗ P (Ncv = odd). Note that dcv (h˜ cv ) and Ncv strongly commute on F(h˜ cv ). So dcv (h˜ cv ) is decomposed into Pe dcv (h˜ cv ) ⊕ Po dcv (h˜ cv ). On the other hand, dcv (h˜ cv ) in F(h˜ cv ) has a unique eigenvector cv which is the vacuum vector of F(h˜ cv ). So dcv P F (h˜ c ) e v has the unique eigenvector c , while dc has no eigenvector. Hence by ˜ v
v
Po F (hcv )
Theorem B.1, if there exists an eigenvector ψe of Pe L, it is of the form ψe = ϕ ⊗ cv , and Po L has no eigenvector. Hence if ψ is the eigenvector of L, we have Pe ψ = ϕ ⊗ cv ,
Po ψ = 0.
That is, ψ = ϕ ⊗ cv . This means P (Ncv = 0)ψ = ϕ ⊗ P (Ncv = 0)cv = ϕ ⊗ cv = ψ.
7. Positive Commutator In this section, we construct a strictly positive commutator. First we define the operator [L, A0 ] on D([L, A0 ]) ≡ D(N) by [L, A0 ] ≡ S1 + λI1 . Note that this is defined as an operator of the right-hand side, and not as a commutator LA0 − A0 L. However the arguments in Sect. 5 guarantees Lϕ, A0 ψ + A0 ϕ, Lψ = ϕ, (S1 + λI1 )ψ = ϕ, [L, A0 ] ψ , for all ψ, ϕ ∈ D(A0 ) ∩ D(N).
Stability of the Non-Equilibrium Steady States
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As seen in the previous section, we consider the subspace P H. If we restrict ourselves to P (1 ⊗ Pf )⊥ , we have P (1 ⊗ Pf )⊥ [L, A0 ](1 ⊗ Pf )⊥ P ≥ (v − λI1 )P (1 ⊗ Pf )⊥ , and [L, A0 ] is strictly positive on P (1 ⊗ Pf )⊥ H for λ small enough. But for P (1 ⊗ Pf )H, we have P (1 ⊗ Pf )[L, A0 ](1 ⊗ Pf )P = 0. So we need to modify the commutator to make it strictly positive on P (1 ⊗ Pf )H, too. This is done by introducing a bounded operator b: ¯ ¯ 2 I0 Q − QI0 R2 Q), b = b(e) = θ λ(QR − 1 2 , R = R (e) = (L0 − e)2 + 2 ¯ ≡ 1 − Q and I0 is defined at (15). The parameters where Q ≡ P (LS = e) ⊗ Pf , Q ¯ by R¯2 . We define [L, b(e)] on D(N ) θ and will be determined later. We denote R2 Q by [L, b(e)] ≡ Lb(e) − b(e)L. Note that LS b(e) − b(e)LS and λI0 b(e) − b(e)λI0 are bounded. For all ψ ∈ D(N), # $ # $ ¯ 2 Lf , I0 Q − Q Lf , I0 R2 Q ¯ ψ [Lf , b(e)]ψ = θ λ QR ¯ ψ, ¯ 2 I˜1 Q − QI˜1 R2 Q = θ λ QR where
˜ 1 + a ∗ hg ˜ 1 + 1 ⊗ Y¯ ⊗ (−1)N a hg ˜ 2 + a ∗ hg ˜ 2 . I˜1 ≡ (Y ⊗ 1) ⊗ −a hg As I˜1 is bounded, it can be extended to the whole H. We denote the extension by the same symbol [L, b(e)]. We define [L, A] on D(N) by [L, A] = [L, A0 ] + [L, b] = S1 + λI1 + [L, b] . By the above choice of b, we have now P (1 ⊗ Pf )[L, A](1 ⊗ Pf )P = 2θ λ2 P (P (LS = e) ⊗ Pf )I0 R¯ 2 I0 (P (LS = e) ⊗ Pf )P = 0. In this section, we prove the following theorem: Theorem 7.1. Let be the interval of R whose interior includes an eigenvalue e of LS , ¯ i.e, σ (LS ) ∩ ¯ = {e}. Let ς ∈ C ∞ (R) be a and no other eigenvalue is included in , 0 smooth function s.t.ς = 1 on and supp ς ∩ σ (LS ) = {e}. Let θ and be 44
= λ 100 ,
26
θ = λ 100 .
Let P be the spectral projection of Ncv onto {0}. Suppose that Assumption 2.1 and 2.2 are satisfied. Then there exists λ1 > 0 s.t. 182 1 P ς (L) [L, A] ς (L)P ≥ · λ 100 · γe P ς (L) 1 − 14P˜e ς (L)P , (21) 2
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for any 0 < λ < λ1 . Furthermore, if we assume β0 < β+ , β− < β1 , pf , f ≤ b for any fixed 0 < β0 < β1 < ∞ and 0 < b < ∞, we can choose λ1 as 100 100 100 100 v 182 λ1 = C min v 26 , , (γe ) 11 , (vγe ) 18 . γe Here C is a constant which depends on β0 , β1 and b, but is independent of v and β+ −β− . We carry out the proof in two steps. First, we show the strict positivity of [L, A] with respect to the spectral localization in L0 . Second we derive the strict positivity of [L, A] with respect to the spectral localization in L.
7.1. Positive commutator with respect to the spectral localization in L0 . In this subsection, we take the first step. Here is the main theorem of this subsection; Theorem 7.2. Let be an interval of R, containing exactly one eigenvalue e of LS and 0 be the spectral projection of L onto . Suppose that Assumption 2.1 and 2.2 let E 0 are satisfied. Then there exists 0 < ti , i = 1, · · · 5 such that, if λ, , θ satisfy λ + θ λ2 −2 < t1 ,
θ < t2 ,
θ λ2 −1 < t3 ,
< t4 ,
θ −1 + θλ2 −3 < t5 ,
then 0 0 P E [L, A]E P ≥
θ λ2 0 P. γe 1 − 7P˜e E
The proof goes parallel to [M]. The difference emerges from the difference of the commutator [L0 , A0 ]: in [M], it was [L0 , A0 ] = N , while we have [L0 , A0 ] = S1 . Whereas N has a spectral gap, S1 has no spectral gap. Since the positive commutator method makes use of the finite spectral gap, the lack of the gap causes a difficulty. To overcome this, we introduced the cut off in Sect. 6. To prove the theorem, we use the Feshbach map theorem [BFS1]; Theorem 7.3. Let H be a closed operator densely defined on a Hilbert space H, and let P be a projection operator such that RanP ⊂ D(H ). We set P¯ = 1 − P . Then HP¯ ≡ P¯ H P¯ is a densely defined operator on P¯ H. Let z be an element in the resolvent set ρ(HP¯ ) of HP¯ on P¯ H. Assume −1 −1 P¯ < ∞, P¯ HP¯ − z P¯ H P < ∞. P H P¯ HP¯ − z Then the Feshbach map fP (H − z) ≡ P (H − z)P − P H P¯ (HP¯ − z)−1 P¯ H P is well defined on P H. fP (H − z) and H have the isospectral property in the sense that z ∈ σ (H ) ⇐⇒ 0 ∈ σ (fP (H − z)),
z ∈ σP P (H ) ⇐⇒ 0 ∈ σP P (fP (H − z)).
Here σ and σP P represent spectrum and pure point spectrum respectively.
Stability of the Non-Equilibrium Steady States
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To apply the theorem, we introduce χν ≡ P (N ≤ ν). 0 By this χν , we can consider Bij below as bounded operators. Projections P˜e , χν , P , Q, E are in relation that
P˜e ≤ χν ,
P˜e ≤ P ,
P˜e ≤ Q,
0 Q ≤ E ,
Q ≤ P.
Hence we have the following: ¯ P˜e = 0, Q
¯ P˜e⊥ = Q, ¯ Q
0 E P χν P˜e = P˜e .
0 , Q, P , χ commute each As L0 , Ncv , N, LS strongly commute, the projections E ν other. So we can define the following projections 0 Q1 ≡ QE P χν ,
0 ¯ Q2 ≡ QE P χν ,
(22)
which satisfy Q1 P˜e = P˜e ,
Q2 P˜e = 0,
Q2 P˜e⊥ = Q2 .
(23)
The relation NP = Nv P implies 0 0 ¯ ¯ Nv Q2 = Nv P QE χν = N P QE χν = N Q2 .
As P (N = 0) = 1 ⊗ Pf , and ∩ σ (LS ) = {e}, we have ¯ = (1 − P (LS = e)) ⊗ Pf = P (N = 0)Q P (LS = f ) ⊗ Pf ≤ P (L0 ⊂ c ), f =e
which implies 0 ¯ P (N = 0)Q2 = P (N = 0)QE P χν = 0.
It follows that N Q2 = Nv Q2 ≥ Q2 .
(24)
Proof of Theorem 7.2. We apply Theorem 7.3 to the operator B defined by B ≡ [L, A] − δe · P˜e⊥ . Here, δe is a parameter which will be determined later. We define bounded operators Bij by Bij ≡ Qi B Qj
i, j = 1, 2.
), then If ϑ ∈ ρ(B22 ε (ϑ) ≡ B11 − B12 (B22 − ϑ)−1 B21 ,
is well-defined and
ε (ϑ) = fQ1 B − ϑ + ϑQ1 .
in order to study the resolvent set ρ(B ). First we investigate the lower bound of B22 22
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Proposition 7.4. We have
≥ v − δe − C1 λ + θ λ2 −2 · Q2 , B22
where C1 = I1 + 2 · I0 2 . Proof. We estimate each term of = Q2 (S1 + λI1 + [L, b] − δe · P˜e⊥ )Q2 . B22
From the inequality S1 ≥ v · Nv and the inequality (24), the first term is bounded below as Q2 S1 Q2 ≥ vQ2 . The lower bound of the second term is given by λI1 ≥ −λ·I1 , as I1 is bounded. Let us estimate the norm of the third term. Substituting L = L0 +λI0 , L0 Q = ¯ QR ¯ 2 I0 QI0 )Q2 . QL0 and Q2 Q = 0, we obtain Q2 [L, b] Q2 = −θ λ2 Q2 (I0 QI0 R2 Q+ 2 2 −2 Then it is estimated as Q2 [L, b] Q2 ≤ 2 · I0 θ λ , since we have R2 ≤ −2 . The fourth term is Q2 P˜e⊥ Q2 = Q2 by (23). Hence we obtain ≥ v − C1 λ + θ λ2 −2 − δe Q2 , B22 where C1 = I1 + 2 · I0 2 .
Suppose λ, θ, satisfies λ + θ λ2 −2 < t1 , ) of B where t1 = 5Cv 1 . Then if ϑ satisfies ϑ ≤ 21 v − δe , it is in the resolvent set ρ(B22 22 by this proposition. Next we estimate the lower bound of ε (ϑ).
Proposition 7.5. Suppose λ, θ, satisfies λ + θ λ2 −2 < t1 , ) of B where t1 = 5Cv 1 . Then if ϑ satisfies ϑ ≤ 21 v − δe , it is in the resolvent set ρ(B22 22 and the following inequality holds: % & 1 2 2 ⊥ ¯ ˜ δe · Pe Q 1 ε (ϑ) ≥ 2θ λ Q1 I0 R I0 − 2θ λ2 −C2 θ 2 λ2 Q1 I0 R¯ 2 I0 Q1 + (λ2 + θ 2 λ4 −4 )Q1 ,
where C2 = 10v −1 I1 2 + 20v −1 + 80v −1 I0 4 . is given by Proof. First B11
= Q1 S1 + λI1 + [L, b] − δe · P˜e⊥ Q1 . B11
¯ 1 = 0, As S1 Q = 0 and aQ = Qa ∗ = 0, the first and the second term vanish. Using QQ L0 Q = QL0 , we obtain % & 1 ⊥ ˜ Q1 . = 2θ λ2 Q1 I0 R¯ 2 I0 − δ · P (25) B11 e e 2θ λ2 −1 Second, Q2 B22 −ϑ Q2 is evaluated as follows;
Stability of the Non-Equilibrium Steady States
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Lemma 7.6. Suppose λ + θ λ2 −2 < t1 . Then for all ϑ ≤ 21 v − δe , ϑ is in the resolvent ) of B and set ρ(B22 22 ' ( −1 −1 0 ≤ ψ, Q2 B22 −ϑ Q2 ψ ≤ 5S1 2 Q2 ψ2 ,
∀ψ ∈ H.
Proof. As S1 Q2 ≥ vNv Q2 ≥ vQ2 , S1 is invertible on Q2 H. The proof is similar to that of Eq. (41) in [M]. In fact, it is easier because we are considering the Fermion system, whose interaction terms are bounded. We omit the details. − 21 Third, S1 B21 ψ is estimated as follows; Lemma 7.7. − 21 2 S B ψ ≤ C θ 2 λ2 R¯ I0 Q1 ψ2 + θ 2 λ4 −4 ψ2 + λ2 ψ2 , 21 2 1 where C2 = 2v −1 I1 2 + 4v −1 + 16v −1 I0 4 . Proof. The proof is the same as that of [M] (p. 342).
Now let us complete the proof of Proposition 7.5. Combining Lemma 7.6 and 7.7, we obtain the following evaluation: if λ, θ, satisfies λ + θλ2 −2 < t1 , then for all ) and ϑ ≤ 21 v − δe , we have ϑ ∈ ρ(B22 ' 2 −1 ( B21 ψ ≤ C2 θ 2 λ2 R¯ I0 Q1 ψ + (λ2 + θ 2 λ4 −4 )ψ2 , B22 − ϑ 0 ≤ ψ, B12 where C2 = 10v −1 I1 2 + 20v −1 + 80v −1 I0 4 . Combining this with (25), we obtain the required estimation. Now let us complete the proof of Theorem 7.2. Suppose that λ + θλ2 −2 < t1 . Then ) and by Proposition 7.5, for all ϑ ≤ 21 v − δe , we have ϑ ∈ ρ(B22 ε (ϑ) ≥ D, where % & 1 ⊥ ˜ Q1 δ · P D = 2θ λ2 Q1 I0 R¯ 2 I0 − e e 2θ λ2 − C2 θ 2 λ2 Q1 I0 R¯ 2 I0 Q1 + (λ2 + θ 2 λ4 −4 )Q1 . Note that D is ϑ-independent. Now let ϑ0 ≡ inf σ (B |(Q1 ⊕Q2 )H ). Here we have two cases, 1. ϑ0 ≥ 21 v − δe .
598
2.
Y. Ogata ) and ε (ϑ ) is well-defined. Then f (B −ϑ ) > ϑ0 . In this case, ϑ0 ∈ ρ(B22 0 Q1 0 satisfies 1 2 v−δe
fQ1 (B − ϑ0 ) = ε (ϑ0 ) − ϑ0 · Q1 ≥ (inf σ (D) − ϑ0 ) · Q1 . On the other hand, note that ϑ0 ∈ σ (B |(Q1 ⊕Q2 )H ). By the Feshbach Theorem 7.3, this implies 0 ∈ σ (fQ1 (B − ϑ0 )). Hence fQ1 (B − ϑ0 ) is not invertible, and we get ϑ0 ≥ inf σ (D). So we have
ϑ0 ≥ min
1 v − δe , inf σ (D) , 2
i.e., B |(Q1 ⊕Q2 )H ≥ min
1 v − δe , inf σ (D) . 2
(26)
The next problem is to investigate the lower bound of D. For the purpose, we estimate Q1 I0 R¯ 2 I0 Q1 . The annihilation operator with respect to g 1 ⊕ g 2 ∈ h ⊕ h is represented by the operator valued distribution ak1 , ak2 as a g 1 ⊕ g 2 = dk g¯ 1 (k)ak1 + g¯ 2 (k)ak2 . j∗
Here, aki , aki∗ satisfies {aki , ap } = δij δ(k − p). By the pull-through formula, ak1 Lf = (Lf + ω(k))ak1 ,
ak2 Lf = (Lf − ω(k))ak2 ,
we get a(g 1 ⊕ g 2 )R2 (e) =
dk g¯ 1 (k)R2 (e − ω(k)) ak1 + g¯ 2 (k)R2 (e + ω(k)) ak2 .
Using this relation, we obtain the following bound: Proposition 7.8. Under Assumption 2.2 we have
1 π Q1 I0 R¯ 2 I0 Q1 ≥ Q1 (e) ⊗ 1 − C3 4 Q1 . Here C3 is a positive constant which is independent of β+ − β− and v. Assume that θ, λ, satisfies the following: Assumption 7.1. λ + θ λ2 −2 < t1 ,
θ < t2 ,
θ λ2 −1 < t3 ,
< t4 ,
θ −1 + θλ2 −3 < t5 ,
where t2 ≡
1 , C2
t3 ≡
v , 8πγe
t4 ≡ (
γe 4 ) , 4C3
t5 ≡
π γe . 4C2
Stability of the Non-Equilibrium Steady States
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The second assumption implies 2θ λ2 − C2 θ 2 λ2 > θ λ2 . Using this and the lower bound of I0 R¯ 2 I0 of Proposition 7.8, we obtain
1 π D ≥ θ λ2 Q1 (e) ⊗ 1 − C3 ( 4 ) Q1 − δe · Q1 P˜e⊥ Q1 − C2 λ2 + θ 2 λ4 −4 Q1 . (27) Recall that (e) is bounded below as (e) ≥ γe · (P ((e) = 0))⊥ . Substituting this to (27), we get % & 1 C2 −1 ˜⊥ 2π 2 −3 4 · Pe − C3 ( ) − Q1 . (28) θ + θλ D ≥ θ λ Q1 γ e − δ e πθ λ2 π Now we determine δe as πθ λ2 (γe + a)
δe = Then we have
with
0 < a < γe .
& C2 −1 2 −3 Q1 . −a − C3 ( ) − θ + θλ D ≥ θλ π 2π
%
1 4
Hence inf σ (D) is bounded below by % & 1 π C2 −1 −a − C3 ( 4 ) − θ + θλ2 −3 Q1 . inf σ (D) ≥ θ λ2 π Recall the lower bound (26) of B . As the third condition of Assumption 7.1 implies 1 1 v − δe ≥ v > 0, 2 4 and
& C2 −1 2 −3 , −a − C3 ( ) − θ + θλ 0 ≥ θλ π 2π
%
1 4
the lower bound is
& C2 −1 2 −3 . −a − C3 ( ) − θ + θλ B |(Q1 ⊕Q2 )H ≥ θ λ π
2π
%
1 4
Substituting [L, A] = B + δe P˜e⊥ , and the last two conditions of Assumption 7.1, we get 0 0 P E [L, A]E P % & 1 C2 −1 2π 0 2 −3 ⊥ 0 ˜ 4 + (γe + a) · Pe E ≥ θ λ P E −a − C3 ( ) − θ + θλ P π % & 1 π C2 −1 0 0 = θ λ2 P E γe − C3 ( 4 ) − θ + θ λ2 −3 − (γe + a) · P˜e E P π
0 0 ≥ θ λ2 −1 P E γe 1 − 7P˜e E P.
Hence we obtain Theorem 7.2.
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7.2. Positive Commutator with respect to the spectral localization in L. In this subsection we complete the proof of Theorem 7.1. Let be an interval s.t. ⊂ , ∩σ (LS ) = {e} and suppς ⊂ . Let F ∈ C0∞ (R) be a smooth function 0 ≤ F ≤ 1 which satisfies F = 1 on supp ς and supp F ⊂ . We set F0 ≡ F (L0 ), and F¯0 ≡ 1 − F0 . We evaluate each term of the following equation: P ς [L, A] ςP = P ςF0 [L, A]F0 ς P + P ς F¯0 [L, A]F0 ς P + h.c. + P ς F¯ 0 [L, A]F¯ 0 ς P .
(29) (30) (31)
For the first part (29), we have the following lemma. Lemma 7.9. Suppose that λ, , θ satisfy Assumption 7.1. Then we have θ λ2 γe P ς (1 − λc1 − 7P˜e )ς P , where c1 is a constant which depends only on F and I0 . (29) = P ς F0 [L, A]F0 ςP ≥
Proof. The proof is the same as that of inequality (63) in [M].
Next we evaluate the second part (30). Lemma 7.10. 2
θλ (30) = P ς F¯0 [L, A]F0 ςP + h.c. ≥ −c2 (θ −1 + + λ −1 )P ς 2 P , where c2 is a constant which depends only on F and I0 . Proof. We divide (30) into three parts; (30) = P ς F¯0 S1 F0 ςP + h.c. + λP ς F¯ 0 I1 F 0 ςP + h.c.
+ P ς F¯0 [L, b]F0 ςP . + h.c.
(32) (33) (34)
Using the fact that 0 ≤ F0 ≤ 1, we get (32) ≥ 0 as in the case (57) of [M]. To evaluate (33), we note ς F¯0 = ς (L)(F¯ (L0 ) − F¯ (L)). By operator calculus, we have the estimation ς F¯0 ≤ c1 |λ|. As I1 is bounded, we obtain (33) ≥ −c1 λ2 . Here, c1 , c1 depends only on I0 and F¯0 . The last part (34) can be estimated in the manner of Proposition 5.2 in [M]: (34) = P ς F¯0 [L, b]F0 ςP + h.c. ) ) ) ¯ ¯ = Pς F (L0 ) − F (L) F¯0 [L, b]F0 ς P + h.c. 2 θ λ ≥ −c2 θ λ2 + θ λ3 −2 ς 2 P = −c2 + λ −1 ς 2 P . Hence we have (30) = (32) + (33) + (34) ≥ −c2 where c2 depends only on I0 and F .
θ λ2 −1 θ + + λ −1 ς 2 P ,
Stability of the Non-Equilibrium Steady States
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Finally, the third part (31) can be estimated as follows. Lemma 7.11. P ς F¯0 [L, A]F¯0 ςP ≥ −c3
θ λ2 −1 (θ + λ −1 )ς 2 P ,
where c3 depends only on I0 and F .
Proof. The same as that of Proposition 5.2 in [M].
Now let us complete the proof of Theorem 7.1. By Lemma 7.9 to 7.11, if λ, and θ satisfy Assumption 7.1, we obtain P ς [L, A] ς P = (29) + (30) + (31) θ λ2 C5 −1 −1 ˜ ≥ (θ + + λ ) ςP, γe P ς 1 − 7Pe − λC4 + γe where C4 , C5 depends only on F and I0 . Let , θ be 44
= λ 100 ,
26
θ = λ 100 .
Then 18
56
θ −1 = λ 100 , λ −1 = λ 100 ,
182
θ λ2 −1 = λ 100 ,
138
θ λ2 −2 = λ 100 ,
94
θ λ2 −3 = λ 100 ,
and if λ is sufficiently small, Assumption 7.1 is satisfied. Furthermore, for λ small enough, 1 C5 −1 λC4 + θ + + λ −1 < (35) γe 2 is satisfied. Hence we obtain P ς [L, A] ς P ≥
182 γe θ λ2 γe P ς 1 − 14P˜e ςP = λ 100 P ς 1 − 14P˜e ς P . 2 2
Now let us estimate the range of λ, which satisfies Assumption 7.1 and (35). We assume β0 < β+ , β− < β1 for any fixed 0 < β0 < β1 < ∞, and the bound of the form factor f ∈ h pf , f ≤ b for fixed 0 < b < ∞. Under these bounds, one can easily check that there exists a constant C which is independent of v and β+ − β− , such that if λ satisfies 100 100 100 100 v 182 λ < C min v 26 , , (γe ) 11 , (vγe ) 18 , γe then Assumption 7.1 and (35) are satisfied. Let 100 100 100 100 v 182 λ1 = C min v 26 , , (γe ) 11 , (vγe ) 18 , γe and we obtain Theorem 7.1.
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8. Virial Theorem In this section, we complete the proof of Theorem 3.1. We apply the new method introduced by M.Merkli [M] to treat the domain question. He solved the problem by approximating the eigenvector of L by vectors in the domain of N and A0 . Assume that ψ is a normalized eigenvector of L with eigenvalue e. Let f be a bounded C ∞ -function such that f ≥ 0, f (0) = 1 and let g be a bounded C0∞ -function with sup* port in the interval [−1, 1]. Define the operators fα ≡ f (iαA0 ), hα ≡ f (iαA0 ) and gν ≡ (νN ). When α, ν goes to zero, hα , gν strongly converges to 1. By approximating the eigenvector ψ by hα gν ψ, we can carry out the arguments rigorously. As the proof goes parallel to [M], we just comment on the differences. We define e(e) ˜ as in Theorem 3.1. For simplicity, we use the notations K ≡ [L, A0 ] = S1 + I1λ , A ψ ≡ ψ, Aψ , ψα,ν ≡ hα gν ψ, and Ijλ ≡ λIj . The proof is done by evaluating the upper and the lower bound of K ψα,ν = ψα,ν , Kψα,ν . The estimation of the upper bound is done by the expansion of commutators, using operator calculus: ' ( K ψαν = −α −1 νIm ψ|gν fα I˜0 ψ + α 2 ψ|gν Rgν ψ + Re ψ|gν R gν ψ . (36) Here, I˜0 , R and R are given by ˜ ˜ − νN)−1 adN1 (I0λ )(z − νN )−1 , I0 = 2 d g(z)(z 1 R=− d f˜ (z)(z − iαA0 )−1 adA3 0 (L0 + I0λ )(z − iαA0 )−1 4
1 ˜ − d h(z)(z − iαA0 )−2 hα , adA3 0 (L0 + I0λ ) (z − iαA0 )−1 2 1 ˜ d h(z)(z − iαA0 )−1 hα adA3 0 (L0 + I0λ )(z − iαA0 )−1 , + 2 R = d f˜(z)(z − iαA0 )−3 adA3 0 (L0 + I0λ )(z − iαA0 )−1 , k is a k−fold commutator [· · · [·, M], M, · · · , M] with M. In the case of [M], where adM the commutators are given by
adA1 0 (L0 ) = N,
adA2 0 (L0 ) = adA3 0 (L0 ) = 0,
1
and adAk 0 (I0λ ) and adN1 (I0λ ) are N 2 -bounded. Hence K ψαν is estimated as 1
1
K ψαν ∼ O(α 2 ν − 2 + ν 2 α −1 ).
(37)
In our case, the k-fold commutators don’t vanish: adA1 0 (L0 ) = S1 ,
adA2 0 (L0 ) = S2 ,
adA3 0 (L0 ) = S3 .
On the other hand, the interaction terms adAk 0 (I0 ), and adN1 (I0λ ) are bounded. Hence K ψαν is estimated as K ψαν ∼ O(α 2 ν −1 + α −1 ν).
(38)
Stability of the Non-Equilibrium Steady States
603
The difference emerges from two factors: the non-commutativity of Si with A0 , and the boundedness of the interaction. The first one makes things worse, while the second one makes it easier. The estimation of the lower bound also goes parallel to [M]. As the positive commutator is localized with respect to the spectrum of L, we need to decompose the Hilbert space. Let be an interval which contains e. Suppose that contains exactly one eigenvalue e(e) ˜ of LS i.e., ∩ σ (LS ) = {e(e)}, ˜ and e(e) ˜ belongs to the interior of . In [M], M.Merkli introduced a partition of unity 2 2 + χ¯ = 1. χ
Here χ ∈ C0∞ (R) is a smooth function s.t.χ = 1 on and supp χ ∩σ (LS ) = {e(e)}. ˜ By Theorem 7.1, we have ˜˜ P χ (L) [L, A] χ (L)P ≥ be(e) (39) χ (L)P , ˜ P χ (L) 1 − 14Pe(e) 182
for 0 < |λ| < λ1 , where be = 21 · λ 100 · γe . Another partition of unity is also needed: χ 2 + χ¯ 2 = 1, where χ ∈ C ∞ satisfies χ (t) = 1 for |t| ≤ 21 and χ (t) = 0 for 1 ≤ |t|. And the set χn = χ (N/n), χ¯ n2 = 1 − χn2 for 0 < n < 1/ν. With respect to the partition of unity, he obtained the lower bound 2 K ψα,ν ≥ be(e) P˜0 χ χn ψα,ν 2 ˜ ψα,ν − Cbe(e) ˜ δe(e),0 ˜ 3
1
−1 −O(η−1 + η + αn 2 + n− 2 ) − Cbe(e) + n−3 + α 2 n2 ), ˜ (n
where C is a positive constant and η, > 0 are arbitrary positive parameter which satisfy n −1 −2 ≥ b e(e) ˜ . Here we represented the result in our notations. 2 − Cη The argument can be carried out parallel in our case. We just need to take care of the projection P = P (Ncv = 0), because the positive commutator is localized to the range of it. This is easily done by the strong commutativity of Ncv with and K and N : K ψα,ν ≥ K P ψα,ν − I1λ · P¯ ψα,ν 2 . Here we used P¯ S1 P¯ ≥ 0. Then we obtain O(α 2 ν −1 + α −1 ν) ∼ K ψα,ν 2 ≥ be(e) χ P χn ψαν 2 − 14P˜e(e) − I1λ P χ¯ χn ψα,ν 2 ˜ ˜ χ P χn ψαν − C3 (n−1 + ν + α · n + α) − I1λ · P¯ ψα,ν 2 − I1λ · χ¯ n ψα,ν 2 − n−2 · C2 . (40) 3
1
Substituting ν = α 2 and n = α − 2 and taking the α → 0 limit, we obtain 2 0 = lim K ψα,ν ≥ be(e) ψ2 − 14P˜e(e) . ˜ ˜ ψ α→0
(41)
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If P˜e(e) ˜ ψ = 0, the inequality (41) is a contradiction. Hence for 0 < |λ| < λ1 , there is no eigenvector of L with eigenvalue e which is orthogonal to P˜e(e) ˜ . Recalling that λ1 can be taken as (9), we obtain Theorem 3.1. One might wonder if χn is necessary in our case where the interaction term is bounded. Note that the right-hand side of (40) has a term of order αn, while the left-hand side has a term of order α −1 ν. Without χn , αn is replaced by αν −1 . We can’t make αν −1 and α −1 ν converge to zero simultaneously, in any choice of ν. So χn is still required. 9. The Stability of the NESS In this section we investigate our physical interest: the stability of the NESS. By proving Theorem 3.2, we complete the proof of Theorem 2.3 and 2.4. Recall that the NESS of the free Fermion model is given by n-point functions (3) with a distribution function (4). We show if the NESS is far from equilibrium, i.e., the inverse temperature β− and β+ are different, it is macroscopically unstable (Theorem 2.3). This result is due to the following fact: for the NESS far from equilibrium, the number of the particles with momentum k is different from the number of the particles with momentum −k, although they have the same energy. On the other hand, for a class of interaction, we show return to equilibrium (Theorem 2.4).
9.1. Instability of the NESS. In this subsection, we investigate the instability of NESS under the interaction with small system. For our purpose, we study the kernel of (e). (j ) (i) Recall the definition of ϕn , En , Enm , Nl , Nr , Nl , Nr , given in Assumption 2.4. As c HS is finite dimensional, Nrc = ∅ and Nl = ∅ for e = 0. By a straightforward calculation, we obtain π 1∗ 1 (e) = dk Fn,m (k)Fn,m (k)δ ω (k) + En,m − e −π
En,m =e
2∗ 2 + Fn,m (k)Fn,m (k)δ
−ω (k) + En,m − e ,
(42)
where i Fn,m (k) = pn YpN (m) ⊗ pm g1i (k) − pn ⊗ pm Y¯ pN (n) g2i (k). r
l
First let us consider the e = 0 case. Note that each term of (42) is positive. So we take the sum over the following subset: ˙ Nlc × Nr , (n, m) ∈ Nl × Nrc ∪ and ignore the other contributions to estimate the lower bound. Note that Nl , Nr , Nlc and Nrc are all non-empty in case e = 0. We have the following relations: if n ∈ Nl , then (n) (m) (n) Nr = ∅, if m ∈ Nr , then Nl = ∅, if n ∈ Nlc , then Nr = ∅, and if m ∈ Nrc , (m) ˙ Nlc × Nr . Hence for all then Nl = ∅. Especially, En,m = e for (n, m) ∈ Nl × Nrc ∪ (n, m) ∈ Nl × Nrc , we have i Fn,m (k) = −pn ⊗ pm Y¯ pN (n) g2i (k), r
Stability of the Non-Equilibrium Steady States
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and for (n, m) ∈ Nlc × Nr , we have i Fn,m (k) = pn YpN (m) ⊗ pm g1i (k). l
Then under Assumption 2.3 and 2.4, we have pn ⊗ pN (n) Y¯ pNrc Y¯ pN (n) + pN (m) YpNlc YpN (m) ⊗ pm (e) ≥ b0 r
r
n∈Nl
m∈Nr
l
l
≥ b0 · P (LS = e)δ0 = γe · P (LS = e), with some b0 > 0 and γe = b0 δ0 > 0. Here, b0 is a constant which is independent of β+ , β− in the interval (β0 , β1 ) for fixed 0 < β0 < β1 < ∞. Hence for e = 0, we have Ker(e) = {0}, and P˜e = 0. Next let us consider e = 0 case. In this case, we can not apply the above arguments because Nlc = Nrc = ∅. By the first condition of Assumption 2.3, a vector in P (LS = 0) is of the form ϕ= ci ϕi ⊗ ϕ i . i (n)
Note that Assumption 2.3 imply Nl ϕ |(0)| ϕ = |Ymn |2
(n)
= Nr
= {n}. We have
n =m
% 2 1 1 2 2 2 × cm · (1 − ρ) (qmn ) − cn · ρ (qmn ) · |f (qmn )| ·
π
dkδ ω (k) + En,m
0
0 2 1 1 dkδ ω (k) + En,m + cm · (1−ρ) 2 (−qmn )−cn · ρ 2 (−qmn ) · |f (−qmn )|2 · −π π 2 1 1 + cm · ρ 2 (qnm ) − cn · (1 − ρ) 2 (qnm ) · |f (qnm )|2 · dkδ −ω (k)+En,m 0
2 1 1 + cm · ρ 2 (−qnm )−cn · (1−ρ) 2 (−qnm ) · |f (−qnm )|2 ·
0
& dkδ −ω (k) + En,m ,
−π
(43) where ω(qmn ) = Emn . By Assumption 2.2 and Assumption 2.3, f (qmn ), f (−qmn ), f (qnm ), f (−qnm ) are non-zero, and the integrals including the δ-function give strictly positive contributions. So there exist strictly positive a0 , such that (43) ≥
2 1 1 a0 cm · (1 − ρ) 2 (qmn ) − cn · ρ 2 (qmn )
n =m
2 1 1 + cm · (1 − ρ) 2 (−qmn ) − cn · ρ 2 (−qmn ) 2 1 1 + cm · ρ 2 (qnm ) − cn · (1 − ρ) 2 (qnm ) 2 1 1 + cm · ρ 2 (−qnm ) − cn · (1 − ρ) 2 (−qnm ) .
(44)
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Hence the necessary condition for Ker(0) to be non-trivial is the existence of {cn } which satisfies 2 1 1 cm · (1 − ρ) 2 (qnm ) − cn · ρ 2 (qmn ) 2 1 1 = cm · (1 − ρ) 2 (−qmn ) − cn · ρ 2 (−qmn ) 2 1 1 = cm · ρ 2 (qnm ) − cn · (1 − ρ) 2 (qnm ) 2 1 1 = cm · ρ 2 (−qnm ) − cn · (1 − ρ) 2 (−qnm ) = 0. This implies the following: 1
1
1
1
ρ 2 (−qnm ) ρ 2 (qnm ) cn (1 − ρ) 2 (qmn ) (1 − ρ) 2 (−qmn ) = = = = 1 1 1 1 cm ρ 2 (qmn ) ρ 2 (−qmn ) (1 − ρ) 2 (qnm ) (1 − ρ) 2 (−qnm ) (45) for n = m. Recall that the Fermion distribution ρ in the NESS is given by −1 1 + eβ+ (cos(k)−γ ) , k ∈ [0, π ), ρ(k) ≡ 1 + eβ− (cos(k)−γ ) −1 , k ∈ [−π, 0). The condition (45) requires eβ− Enm = eβ+ Enm , i.e., β− = β+ . That is, (0) has non-trivial kernel only if the NESS is an equilibrium state indeed. Otherwise, we have (0) ≥ γ0 · 1 > 0. So, if β− = β+ , we have P˜e = 0 for all eigenvalue e of LS and obtain the first statement of Theorem 3.2. Now let us complete the proof of Theorem 2.3. Combining Theorem 3.1 and 3.2, the Liouville operator L corresponding to the NESS does not have any eigenvector for 0 < |λ| < λ1 . Hence the NESS is macroscopically unstable by Proposition 4.3. We fix the bound β0 < β+ , β− < β1 , pf , f ≤ b for any fixed 0 < β0 < β1 < ∞ and 0 < b < ∞. Let us estimate the dependence of λ1 (9) on β+ − β− for fixed v. For e = 0, we have γe = b0 δ0 as seen above, which is independent of β+ − β− . On the other hand, γ0 converges to 0 as β+ − β− goes to 0. Substituting e− cn = *
β− 2 En
Zβ−
,
Zβ− =
e−β− En
n
to (44), we can estimate (43) ≥ C(β+ − β− )2 ≥ γ0 > 0, i.e. γ0 ∼ O(β+ − β− )2 . 200 Hence we have λ1 ∼ O|β+ − β− | 11 . On the other hand, if we fix β+ and β− , we have 50 λ1 (v) ∼ v 9 → 0 as v → 0. Then, we obtain Theorem 2.3.
Stability of the Non-Equilibrium Steady States
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9.2. Return to equilibrium. In this section we investigate the equilibrium case, i.e., β+ = β− . By the result of the previous subsection, we can show return to equilibrium for the class of interaction we introduced. The following theorem was shown by H.Araki [A1, A3]: Theorem 9.1. Let (A, τ ) be a C ∗ -dynamical system and let ω be a (β, τ )-KMS state with GNS-representation (H, π, ). Let L be the Liouville operator corresponding to τ . If P = P ∗ ∈ A then ∈ D(eβ(L+π(P ))/2 ). Let τ P be the perturbed automorphism group by P , and let P ≡ eβ(L+π(P ))/2 . Then the state ωP defined by ωP (A) =
(P , AP ) (P , P )
is a (β, τ P )-KMS state. On the other hand, the following theorem concerning return to equilibrium is known [BFS2]: Proposition 9.2. Let (A, τ ) be a C ∗ -dynamical system and let ω be a (β, τ )-KMS state with the GNS-representation (H, π, ). Assume that the Liouville operator L of τ has a simple eigenvalue 0 corresponding to the eigenvector , and that the rest of the spectrum of L is continuous. Then for any ω-normal state η, we have the return to equilibrium in an ergodic mean sense: 1 T lim η (αt (A)) dt = ω(A), A ∈ A. T →∞ T 0 Let us return to our system. Now we have β ≡ β− = β+ . First let us investigate the kernel of (e). As in the previous subsection, Ker(e) = {0} for e = 0. On the other hand, Eq. (43) implies that Ker(0) is one-dimensional and is spanned by a vector of the form ϕ= ci ϕi ⊗ ϕ i , i
with β
e− 2 En cn = * , Zβ
Zβ =
e−βEn .
n
Hence we obtain Theorem 3.2. f Second, note that ωρ is a (β, αt )-KMS state of Of . We denote by ωβS the (β, αS )KMS state over OS . The state ωβS ⊗ ωρ is a (β, α0 )-KMS state over OS ⊗ Of . As our perturbation V is an element of O, the nontriviality of KerL is guaranteed by Theorem 9.1. The fact that 0 is the simple eigenvalue of L can be derived as follows: let ψ1 , ψ2 be eigenvectors of L with eigenvalue 0. By Theorem 3.1, we have P˜0 ψi = ci ϕ ⊗ , with ci = 0. Then ψc11 − ψc22 is an eigenvector of L with eigenvalue 0, which is orthogonal to P˜0 . By Theorem 3.1, this entails ψ1 = cc21 ψ2 . Hence 0 is the simple eigenvalue of L. We denote the corresponding eigenvector by V and the state corresponding to V by ωV . By Theorem 9.1, ωV is a (β, α)-KMS state. On the other hand, as the kernel of (e) is trivial for e = 0, L has no other eigenvalue. So the rest of the spectrum of L is continuous. Accordingly, from Proposition 9.2, we obtain the return to equilibrium Theorem 2.4.
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Acknowledgement. The author thanks A. Arai, F. Hiroshima, and T. Matsui for useful advice. Many thanks also go to M.Merkli for helpful arguments and comments.
A. An Example which Satisfies Assumptions 2.1 to 2.4 We give an example in dimension d = 2. We consider the γ = 0 case. We fix some 0 < v < 1, and define 0 ≤ kv ≤ π/2 by sin2 kv = v. Let HS = bσz , with 0 < b < (cos kv )/4, and define 0 ≤ k4b < k2b < π2 by 4b = cos k4b , 2b = cos k2b . We have then 0 < kv < k4b < k2b ≤ π2 , and S = {k4b , −k4b , π − k4b , −π + k4b , k2b , −k2b , π − k2b , −π + k2b } are all in v . We choose f as a smooth function with support in v , which takes non-zero values on S. As the interaction, we take Y as 01 Y = . 10 Because p acts on C ∞ as (16), f is in D(p 3 ) and Assumption 2.1 is satisfied. The eigenvalue of LS is 2b, 0, −2b. We have cos(k4b ) = 4b, cos(k2b ) = 2b, cos(π − k2b ) = −2b, cos(π − k4b ) = −4b and as S is included in v , Assumption 2.2 is satisfied. Assumption 2.3 is trivial. Let us check Assumption 2.4. Let ϕ0 , ϕ1 be eigenvectors (0) of HS , corresponding to the eigenvalue −b, b respectively. For e = 2b, Nl = {1}, (1) (0) (1) Nl = ∅, Nr = ∅ and Nr = {0}. Hence we have pN (1) Y¯ pNrc Y¯ pN (1) = |ϕ0 ϕ0 | , r
r
pN (0) YpNlc YpN (0) = |ϕ1 ϕ1 | . l
l
Then we obtain δ0 = 1 > 0, and Assumption 2.4 is satisfied. We can check for the e = −2b case in the same way. B. Tensor Product of Linear Operators About the tensor product of linear operators, we have the following theorem. It can be proved using the spectral theorem given in [RS]. Theorem B.1. Let Hi (i = 1, 2) be separable Hilbert spaces, and let Li be self-adjoint operators on Hi . Let L be the self-adjoint operator on H1 ⊗ H2 , defined by L ≡ L1 ⊗ 1 + 1 ⊗ L2 . Suppose that L2 has a unique eigenvector . Then every eigenvector of L is of the form ϕ ⊗ ,
ϕ ∈ H1 .
References [A1] [A2]
Araki, H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Publ. R.I.M.S., Kyoto Univ. 9, 165–209 (1973) Araki, H.: On the equivalence of the KMS condition and the variational principal for quantum lattice systems. Commun. Math. Phys. 38, 1–10 (1974)
Stability of the Non-Equilibrium Steady States [A3]
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Araki, H.: Positive cone, Radon-Nikodym theorems, relative Hamiltonians and the Gibbs condition in statistical mechanics. In: C*-Algebras and their Applications to Statistical Mechanics and Quantum Field Theory, Kastler, D. (ed) Bologna: Editrice Compositori, 1975 [A4] Araki, H.: On the XY -model on two-sided infinite chain. Publ. RIMS, Kyoto Univ. 20, 277–296 (1984) [AP] Aschbacher, W.H., Pillet, C.A.: Non-Equilibrium steady states of the XY chain. mp-arc /02-459 (2002) [BFS1] Bach, V., Fr¨ohlich, J., Sigal, I.M.: Quantum electrodynamics of confined nonrelativistic particles. Adv. Math. 137(2), 299–395 (1998) [BFS2] Bach, V., Fr¨ohlich, J., Sigal, I.M.: Return to equilibrium. J. Math. Phys. 41(6), 3985–4060 (2000) [BR1] Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics 1. Berlin-Heidelberg-New york: Springer-Verlag, 1986 [BR2] Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics 2. Berlin-Heidelberg-New york: Springer-Verlag, 1996 [HA] Ho, T.G., Araki, H.: Asymptotic time evolution of a partitioned infinite two-sided isotropic XY -chain. Proc. Steklov. Inst. Math. 228, 191–204 (2000) [HKT] Haag, R., Kastler, D., Trych-Pohlmeyer, B.: Stability and equilibrium states. Commun. Math. Phys. 38, 173–193 (1974) [JP1] Jakˇsi´c, V., Pillet, C.A.: On a model for quantum friction. II: Fermi’s golden rule and dynamics at positive temperature. Commun. Math. Phys. 176, 619–644 (1996) [JP2] Jakˇsi´c, V., Pillet, C.A.: On a model for quantum friction. III: Ergodic properties of the spin-boson system. Commun. Math. Phys. 178, 627–651 (1996) [JP3] Jakˇsi´c, V., Pillet, C.A.: Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs. Commun. Math. Phys. 226(1), 131–162 (2002) [JP4] Jakˇsi´c, V., Pillet, C.A.: Mathematical theory of non-equilibrium quantum statistical mechanics. J. Statist. Phys. 108(5-6), 787–829 (2002) [LR] Lanford, O., Robinson, D.W.: Statistical mechanics of quantum spin systems. III. Commun. Math. Phys. 9,327–338 (1968) [M] Merkli, M.: Positive commutators in non-equilibrium quantum statistical mechanics. Commun. Math. Phys. 223, 327–362 (2001) [PW] Pusz, W., Woronowicz, S.L.: Passive states and KMS states for general quantum systems. Commun. Math. Phys. 58, 273–290 (1978) [R1] Ruelle, D.: A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule. Commun. Math. Phys. 5, 324–329 (1967) [R2] Ruelle, D.: Natural nonequilibrium states in quantum statistical mechanics. J. Stat. Phys. 98, 57–75 (2000) [RS] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis. Second edition. New York: Academic Press, Inc., 1980 Communicated by A. Kupiainen
Commun. Math. Phys. 245, 611–626 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1027-x
Communications in
Mathematical Physics
Macdonald’s Identities and the Large N Limit of Y M2 on the Cylinder Steve Zelditch Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA. E-mail: [email protected] Received: 28 May 2003 / Accepted: 19 September 2003 Published online: 6 February 2004 – © Springer-Verlag 2004
Abstract: The purpose of this paper is to determine the large N asymptotics of the free energy FN (a, U |A) of Y M2 (two-dimensional Yang Mills theory) with gauge group GN = SU (N ) on a cylinder where a is a so-called principal element of type ρ. Mathematically, 1 FN (U1 , U2 |A) = 2 log HGN (A/2N, U1 , U2 ), N where HGN is the central heat kernel of GN . We find that FN (aN , UN |A) ∼
N (dθ, dσ ), A
where is an explicit quadratic functional in the limit distribution dσ of eigenvalues of UN , depending only on the integral geometry of SU (2). The factor of N contradicts some predictions in the physics literature on the large N limit of Y M2 on the cylinder (due to Gross-Matytsin, Kazakov-Wynter, and others). 1. Introduction The purpose of this note is to determine the large N asymptotics of many cases of the free energy FN (U1 , U2 |A) of Y M2 (two-dimensional Yang Mills theory) with gauge group SU (N ) on a cylinder. Interest in this large N limit problem was raised around ten years ago in a series of physics papers by Douglas, Kazakov, Wynter, Gross, Matytsin and others [DK, KW, GM1, GM2]. They predicted, on the basis of physical and formal mathematical arguments, that the large N limit of FN (UC1 , UC2 |A) should be well-defined and related to the action along a path of the complex Burgers equation with boundary densities determined by the limiting eigenvalue densities of UC1 , UC2 .
Research partially supported by NSF grant #DMS-0071358 and by the Clay Mathematics Institute.
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In certain cases they predict a phase transition between a weak coupling and a strong coupling regime. Our results give the asymptotics of FN (aN , UN |A), where aN is a principal element of type ρ of SU (N ) in the sense of [Ko] (see Sect. 2), and where the second sequence {UN } can be any sequence of elements of SU (N ) which possesses a limiting eigenvalue density dσ . The limiting eigenvalue density of aN is uniform measure dθ on S 1 . Our asymptotics reveal some surprising results: • FN (aN , UN |A) ∼ N A (dθ, dσ ) for a certain (explicit) functional , unlike the predictions that it tends to a limit functional as N → ∞. The extra factor of N signals an error in some of the heuristic physics arguments, and (as emphasized to the author by M. Douglas and C. Vafa) calls into question in what sense the large N limit of Y M2 on the cylinder is actually a string theory as discussed in [GM1, GM2, KW] and elsewhere. At least, it indicates that extra hypotheses on the matrix pairs U1 , U2 are necessary for the conjectured asymptotics to be correct. • The principal asymptotic term is visibly analytic in A, i.e. it never exhibits a phase transition to leading order, even though dσ could be any probability measure on S 1 . This aspect of the results is consistent with the physics predictions, in that the boundary condition dθ puts the system in its strong coupling regime and it should therefore not exhibit a phase transition between weak and strong coupling [D]. The calculation presented here appears to be the first rigorous calculation of the large N limit of the partition function for Y M2 on the cylinder. Hence, it is unclear at this time whether the anomalous factor of N only occurs in the special case where UC1 = aN is a principal element of type ρ or whether it holds more generally in this context. We should emphasize that the other rigorous mathematical results (of which we are aware) have largely confirmed the large N limit picture developed in [DK, GM1, GM2, KW] and elsewhere. The Douglas-Kazakov phase transition of the genus 0 partition function at A = π 2 has been proved by Boutet de Monvel - Shcherbina in [BS]. A large deviations analysis of spherical integrals by Guionnet-Zeitouni in [GZ] justifies some of the predictions by Matytsin [M] and Gross-Matytsin [GM1] on the asymptotics of characters χR (U ). However, the results of [GZ] pertain to the analytic continuation of χR to positive elements of GL(N, C), and Gross-Matytsin’s predictions fail when U = aN . (cf. [TZ]). To state our results, we introduce some notation. The partition function of Y M2 on a cylinder, with gauge group equal to G, is given by Migdal’s formula (see [W, W2]): A ZG (U1 , U2 |A) = χR (U1 )χR (U2∗ )e− 2N 2πC2 (R) . (1) R∈G
Here, A ≥ 0 is the area of the cylinder, and the sum runs over the irreps (irreducible representations) of G, with χR the character of R and with C2 (R) equal to the eigenvalue A of the central of the Casimir in the irrep R. Thus, ZG is the value at time t = 2N heat kernel of G: χR (U1 )χR (U2∗ )e−2πtC2 (R) , (2) HG (t, U1 , U2 ) = R∈G
i.e. the kernel of the heat operator acting on the space of central functions on SU (N ). It is obtained from the usual heat kernel by averaging both variables over conjugacy classes. Since ZG (U1 , U2 |A) is conjugacy invariant, one may assume that U1 , U2 are diagonal
Macdonald’s Identities and the Large N Limit of Y M2 on the Cylinder
613
j
j
and we write Uj = D(eiθ1 , . . . , eiθ ), where is the rank of G (i.e. the dimension of its maximal torus). The main quantity of interest is the free energy, defined by FG (U1 , U2 |A) =
1 ln ZG (UC1 , UC2 |A). N2
(3)
We now consider the large N limit of the partition function and free energy. The large N limit refers to an increasing sequence GN of groups, e.g. the classical groups GN = U (N ), SU (N ), SO(N ), Spin(N ). For the sake of simplicity we restrict attention to SU (N ) and we abbreviate FSU (N) by FN (etc.). The limit we are interested in is a pointwise limit of the central heat kernel, which obviously requires some discussion since the space on which the heat kernel is defined changes with N . The large N limit of FN is defined as follows: take a pair of sequences {UNj }, UNj ∈ SU (N ) (j = 1, 2) of elements whose eigenvalue distributions dσNj :=
N Nj 1 δ eiθk ∈ M(S 1 ) N k=1
tend to a limit measures σj . Here, N = N −1 denotes the rank of the group, and M(S 1 ) denotes the probability measures on the unit circle, and convergence is in the weak sense of measures. We will denote this situation by UNj → σj . That is, UNj → σj ⇐⇒
N Nj 1 δ eiθk → σj ∈ M(S 1 ), (j = 1, 2). N
(4)
k=1
Conjecture 1.1 [GM1] (pp. 8-9). Assume that U1 → σ1 , U2 → σ2 . Then FN (U1 , U2 |A) → F (σ1 (θ ), σ2 (θ )|A) = S(σ1 (θ ), σ2 (θ )|A) − 21
S1 S1
σ1 (θ )σ1 (φ) log | sin
θ−φ 2 |dθ dφ
−
1 2 S1 S1
σ2 (θ )σ2 (φ) log | sin
θ−φ 2 |dθdφ,
where the functional S is a solution of the Hamilton Jacobi equation
2 ∂S 1 2π ∂ δS π2 2 = σ1 (θ ) − σ (θ ) . ∂A 2 0 ∂θ δσ1 (θ ) 3 1 It is often assumed in the physics articles that the limit measures have densities. The function S can be identified as the action along the solution of the boundary value problem for the complex Hopf-Burgers equation ∂f ∂f ∂t + f ∂x = 0,
f (0, x) = σ1 , f (A, x) = σ2 .
Under certain assumptions on the limit densities, it is further conjectured that a third order phase transition between the weak and strong coupling regimes should occur. In the case of the disc (the cylinder with U1 = I ), Kazakov-Wynter and Gross-Matytsin have predicted a phase transition point at dσ (θ ) π . , (θ ) := A= 2 (π) θ − θ
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Here, dσ is the limit distribution for U2 and the integral is presumably to be understood as the Hilbert transform on S 1 (although it is written as the Hilbert transform on R). The weak coupling regime is characterized by a gap in the support of the limit density (i.e. an interval of S 1 on which dσ1 vanishes). The mathematical evidence for these conjectures is largely based on approximating ˆ by a continuous (and only partially defined) integral over the discrete sum (1) over R ∈ G a space of “densities of Young tableaux”, to which the saddle point method is applied (cf. §3). These arguments are not rigorous and also make some implicit assumptions (see the discussion after Theorem 1.3). The main observation of the present article is that there exists a simple, rigorous alternative for calculating the large N limit when one of the arguments is a so-called principal element of type ρ. This alternative method is based on the use Macdonald’s identities (in Kostant’s formulation) to factor the partition function as a product over positive roots. Background on Macdonald’s identities will be provided in §2 (see also [MAC, Ko, F, Fr, PS]). The first result one obtains this way is a limit formula in which the time variable in ¯ (x) for the measure the partition function is not rescaled. We use the notation dσ ∗ dσ defined by ¯ (x) := f (eix )dσ ∗ dσ f (ei(x−y) )dσ (x)dσ (y). (5) S1
S1
S1
Thus, ∗ denotes convolution of measures and d σ¯ is short for dσ (x). ¯ For notational simplicity we sometimes denote eix ∈ S 1 more simply by x. Theorem 1.2. Let kN ∈ su(N ) be diagonal matrices with entries θjN , and assume N dσN := 1N jN=1 δ(eiθj ) → σ. Then, as N → ∞, 1 N2
log HSU (N) (t, aN , ekN ) → − 21 log η(it) +
iπ 1 ix 2 2 R log HSU (2) (t, e , e
¯ (x). )dσ ∗ dσ
Here, HSU (2) is the central heat kernel of SU (2) and eix is short for the diagonal iπ iπ matrix D(eix , e−ix ). The element a = D(e 2 , e− 2 ) is the principal element of type ρ of SU (2). As discussed in [F] (Prop. 1.3), the central heat kernel of SU (2) at these special values is given by iπ
HSU (2) (t, eix , e 2 ) =
θ1 (eiπx , it) , 2e−πt/4 sin π x
(6)
where θ1 (z, t) = 2
∞
(−1)n sin{(2n + 1)z}e−π(n+1/2)
2t
(7)
n=0
is Jacobi’s theta function. Macdonald’s identities are more often stated in terms of Jacobi theta functions, but we find it easier to work directly with heat kernels. We note that the factor of π in the exponent π(n + 1/2)2 t is responsible for the appearance of π t in many expressions to follow in the heat kernel. At first sight, this result seems to explain the normalization N12 log ZN . However, the large N limit conjectures concern the scaling limit of the central heat kernel under
Macdonald’s Identities and the Large N Limit of Y M2 on the Cylinder
615
A ¯ N → dσ ∗ dσ ¯ t → 2N . This puts into play a simultaneous limit process in dσN ∗ dσ and in the asymptotics of theta functions. We find that for our cases of the problem, this rescaling changes the growth rate of the free energy.
Theorem 1.3. Suppose that ekN is a sequence such that dσN → dσ . Then, FN (aN , ekN |A) =
1 2 S1
iπ
log HSU (2) (A/2N, eix , e 2 )dσN ∗ dσN (eix )
iA log η( 2N ) + O(1/N ) N 1 ∼ − A { S 1 π min{d(eix , eiπ/2 ), d(eix , e−iπ/2 )}2 dσ ∗ dσ (eix )−
−
1 2
π 12 },
where d(eix , eiy ) is the distance along S 1 . The minimal distance above arises as the distance d(C(a), eix ) in SU (2) of the diagonal element D(eix , e−ix ) to the conjugacy class C(a) of a of the principal element of type ρ, which is conjugate to D(eiπ/2 , e−iπ/2 ). We note that there are two terms of opposite sign in the leading order term. They dθ cancel when U1 = a = U2 , since then dσ ∗ dσ = 2π and the first term reduces to N 1 4 1 π 3 π − A { π 2π 3 ( 2 ) − 12 } = 0. This appears to be the kind of case studied in [GM1]. But the two terms cannot cancel in all cases, and indeed, the leading term does not not cancel in the simplest case, where kN = 0 for all N , i.e. where U2 = I d. We then have (see [F1] or [F], Prop. 1.2) HSU (N) (t, aN , I ) = e2π dim SU (N) t/24 η(it)dim SU (N) ,
(8)
where η(t) is Dedekind’s η-function. As is easy to see (and will be verified below), the dθ eigenvalue distribution of aN tends to 2π , while that of I is obviously δ1 . In this case, the asymptotic mass equals 1 and the continuous term equals zero. We separate out this special case since the result is most easily checked on this example: Proposition 1.4. When U1 = aN and U2 = I , so that σ1 = dθ, σ2 = δ1 , then 1 N2
π log ZSU (N) (aN , 1|A) = − 2N A { 12 } −
1 2
A log( 2N ) − A/48N + O(e−cN ).
Note that Theorem 1.3 reduces to Proposition 1.4 when dσ = δ0 . Some final comments on the anomalous extra factor of N . The prediction that 1 log ZN (A) should have a limit determined by a variational problem is one of many N2 predictions of this kind in field theory and statistical mechanics and it seems strange that an extra factor of N should appear in our calculations. From discussions with M. Douglas and V. Kazakov, it seems plausible that the anomaly is due to the unusual behavior of the character values χR (a) at the principal element of type ρ. It was proved by Kostant [Ko] that the only character values at this element are χR (a) = −1, 0, 1. It appears that the physics predictions implicitly assumed a non-oscillatory behaviour in character values, and in particular no oscillation in the signs of character values. The heuristic calculations in [GM1, GM2, KW] of the large N limit of the free energy on the cylinder were based on special cases (such as U2 = U1∗ ) which do not have such oscillations in sign. The sign oscillation causes much more cancellation than expected, and this could explain why
616
S. Zelditch
our asymptotics have the form e−N rather than e−N . These special values χR (a) are also inconsistent with Matytsin’s character asymptotics (cf. [TZ]). This paper is organized as follows. In Sect. 2, we review Macdonald’s identity and associated objects which we will use in the proof of the main result. The main results are proved in Sect. 3. In Sect. 4, we discuss some aspects of the proof and mention some other results which could be proved by the methods of this paper. 3
2
2. Background on Macdonald’s Identities In this section, we review the η function and Macdonald’s identity. Our main references are the articles [Fr] of I. Frenkel and those [F, F1] of H. Fegan. Since the articles employ very different, possibly confusing, notational conventions, we highlight the main ones: • In [Fr], the η-function is defined in a non-standard way in the lower half-plane (see §4.4) and consequently the heat kernel is evaluated at time t = 4π ib, where b < 0. We will quote results of [Fr] in terms of t and we will use the usual definition of η as a modular form on the upper half plane. • In [Fr], the special element of type ρ is denoted e4πiρ , but the exponent refers to the dual element of the Cartan subalgebra rather than the linear functional ρ. To avoid confusion, we denote it simply by a as in [F]. • Fegan works with the Schr¨odinger equation rather [F1] (1.1) rather than the heat equation, so our formulae differ from his in that his it is our t.
2.1. η function. The Dedekind eta-function [I, A] is defined by 2πinz η(z) = e2πiz/24 ∞ ), n=1 (1 − e
z > 0.
(9)
It is a modular cusp form of weight 1/2, i.e. it satisfies η(γ z) = θ (γ )jγ (z)1/2 η(z),
if γ ∈ SL(2, Z),
where θ (γ ) is a certain multiplier which we will not need to know in detail (see [I], §2.8). When γ z = −1/z, we have 1 η(− ) = (iz)1/2 η(z). z In studying the free energy, we are particularly interested in log η(iy) as y → 0+. The transformation law for log η(iy) goes as follows [A]: For real numbers y > 0 we have: π log η(iy) = − 12y −
1 2
log y +
∞
1 1 m=1 m 1−e2π m/y .
(10)
Macdonald’s Identities and the Large N Limit of Y M2 on the Cylinder
617
2.1.1. Jacobi’s theta function. Although we mainly use the heat kernel parametrix on SU (2) to obtain asymptotic formulae, the alternative in terms of Jacobi’s theta function (6) can be used to check the details. The small time expansion of the heat kernel can be obtained from Jacobi’s imaginary transformation law: θ1 (−x/τ, −1/τ ) = i(−iτ )1/2 e−ix Thus, we have:
iA x, 2N
θ1
=i
2N A
1/2 e
−2x 2 N/Aπ
2 /τ
θ1
θ(x, τ ). 2iN 2iN x, A A
.
(11)
Here, we use that A 2N iA 2N ⇒ −1/τ = i ⇒ e−π 2N → e−π A . 2N A The large N or small time asymptotics now follow from (7).
τ=
2.2. Central heat kernels. Let G be any compact, connected Lie group. We denote by R the root system of (gC , hC ), where h is its Cartan subalgebra. We denote by R+ the positive roots, by P the lattice of weights and by P++ ⊂ P the dominant weights. We recall that the eigenvalue of the Casimir operator (bi-invariant Laplacian) of G in the representation with highest weight λ is equal to |Vλ = (||λ + ρ||2 − ||ρ||2 )I d|Vλ , ρ = 1/2 α. α∈R+
The fundamental solution of the heat equation is given by k(t, x, y) = ν(xy −1 , t), where ν(g, t) =
(dim Vλ ) χλ (g)e−2πt (||λ+ρ||
2 −||ρ||2 )
.
(12)
λ∈P++
As in [Fr], §4.3 it then follows that the central heat kernel is given by 2 2 HG (t, h, k) = χλ (eh )χλ (e−k )e−2πt (||λ+ρ|| −||ρ|| ) .
(13)
λ∈P++
This is simply a different notation for (2). 2.3. SU (2). As was recognized clearly by H. Fegan and others, the central heat kernel of SU (2) plays an important role in Macdonald’s identities. There is one positive root α, which can be identified with 1 if we choose it as the basis of the Cartan dual subalgebra. Then ρ = 21 and the Killing form is B(x, y) = 21 xy. In this case, the principal element a of type ρ has eigenvalues e± 2 . The weight lattice is 21 Z. The character of the irreducible representation of highest weight λ is χλ (x) = sin(2λ+1)x sin πx . We have: −1, λ is an odd integer χλ (a) = 0, is not an integer 1, is an even integer. iπ
The eigenvalues of the Casimir are c(λ) = 21 λ(λ + 1).
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S. Zelditch
The central heat kernel H (t, a, y) at the special point a may be expressed in terms of Jacobi’s theta function as in (6). As a special case of Macdonald’s identities, we further have (cf. [F1], (3.12)): HSU (2) (t, a, 1) = (e−πt/12 η(it))3 .
(14)
2.4. Macdonald identities. We briefly review Macdonald’s identities for a compact, semi-simple, simply connected Lie group G and its relation (proved by Kostant [Ko]) to the central heat kernel. They are thus valid for SU (N ). We follow [F, F1, Fr]. Macdonald’s identity involves a conjugacy class Ca of special elements of G, which we will denote by a, which are conjugate to e2xρ , where xρ is dual under the Killing form to ρ. Such elements are called “principal elements of type ρ”. A principal element of type ρ is characterized in [Ko] (see 1.3, p. 181) as a regular element such that the order of Ad(a) equals h (the order of the Coxeter element). One has h = N for SU (N ), so the eigenvalues of Ad(a) are the distinct N th roots of unity. Hence aN →
dθ , 2π
(15)
i.e. the eigenvalues of aN become uniformly distributed in the large N limit. A detailed description of such elements can be found in [Ko, F1, AF]. In Kostant’s formulation, as described by Frenkel, the Macdonald’s identities take the form ([Fr], Prop. (4.4.5)) 2 θ (−k, it) = σ (k) χλ (a)χλ (e−k )e−2πt||λ+ρ|| , (16) λ∈P++
where (see [Fr], (1.1.8) σ (k) = α∈R+ (e α,k/2 − e− α,k/2 ), where θ (k, it) is the theta-function defined in Definition (4.4.1) of [Fr]: For t > 0, θ (k, it) = e−2πt||ρ|| α∈R+ (e 2
α,k 2
− e−
α,k 2
)
−2πnt · ∞ ) α∈R (1 − e−2πnt+ α,k ). n=1 (1 − e
(17)
For our purposes the key formula is the following ([Fr], Prop. (4.4.4)): θ (k, it) = η(it)−|R+ |+ α∈R+ θ1 ( α, k), it).
(18)
θ1 (x,it) We put θ˜ (x, it) = 2e−π t/4 sin πx . The following is the version of Macdonald’s identities proved in Theorem 1.5 of [F1] (together with its Proposition 1.3). As above, an element eiθ is identified with a diagonal element D(eiθ , e−iθ ) of SU (2).
Lemma 2.1. Let G be compact, connected, semi-simple and simply connected, and let a be an element of type ρ. Then HG (t, a, ek ) = (eπt/12 η(it))−|R+ |+ α∈R+ HSU (2) (t, ei α,k , eiπ/2 ).
Macdonald’s Identities and the Large N Limit of Y M2 on the Cylinder
619
Proof. Putting together (16), (18) and (13), we get HG (t, a, ek ) = σ (k)−1 e2πt||ρ|| θ (k, it) 2
˜ k, it). = e2πt||ρ|| η(it)−|R+ |+ α∈R+ θ( α, 2
The stated result now follows from (6).
(19)
As mentioned above (8), the simplest case (20) comes from combining Lemma 2.1 and (14): HG (t, a, I ) = e2π dim G t/24 η(it)dim G .
(20)
3. Large N Limit of ZSU (N ) (a, U |A): Proof of Theorem 1.3 3.1. Simplest case. We first consider the simplest case (20). Since k = 0 (i.e. U2 = I ), we are essentially dealing with Y M2 on the disc. This case is discussed in detail in [GM2], §4. Proposition 3.1. When U1 = a and U2 = 1, so that σ1 = dθ, σ2 = δ1 , then A π 1 1 −cN ). log ZSU (N) (a, 1|A) = − 2N A 24 − 2 log 2N − A/48N + O(e N2 Thus, no phase transition occurs. Proof. Macdonald’s identity (8)–(20) gives: ZSU (N) (a, 1|A) = eπ dim SU (N)A/24N η(iA/2N )dim SU (N) . The free energy is then 1 dim SU (N ) log HSU (N) (A/2N, a, I ) = N2 N2 We note that (10) to get:
dim SU (N) N2
=
1 2
2Aπ + log η(iA/2N ) . 48N
(21)
(22)
+ O( N1 ). We substitute y = A/2N in the right side of
∞ 1 1 A 1 2N π . (23) − log + 4Nπm/A A 24 2 2N m1−e m=1 1 1 −4N(π/A)m ) for N The summand of the sum ∞ m=1 m 1−e4N π m/A is smaller than 1/2e sufficiently large, so we may bound the sum by the geometric series and obtain a bound of 1 − 1 ≤ Ce−4N(π/A) , for N sufficiently large. 1 − e−4N(π/A) Thus, the sum is an exponentially small correction, and we have log η(iA/2N ) = −
1 log ZSU (N) (a, 1|A) N2 πN dim SU (N ) 1 A Aπ − = − log + + O(e−cN/A ) N2 6A 2 2N 24N π 2N 1 A Aπ ∼− − log + . 24 A 2 2N 48N
(24)
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S. Zelditch
We see clearly the anomalous factor of N. Also, we see that there is no phase transition, despite the fact that the two eigenvalues densities, δ(1), dθ have very different support properties. In almost exactly this case, the lack of phase transition is predicted by Gross-Matytsin in [GM2], §4. They explain that when the support of σ1 is the whole circle and the support of σ2 is a single point, then the system is always in the strong coupling phase. They further explain that when σ1 = δ(θ = 0), the solution of the Hopf equation can be obtained as the solution of the integral equation −θ σ1 (θ )dθ
t f (t, θ ) = + . 1− A A θ − θ − tf (t, θ ) In our case, σ1 = 1 so the equation is simply t −θ 1− f (t, θ ) = − log(θ − tf (t, θ )). A A It would be interesting to solve this equation and check that the solution is analytic in A. 3.2. Proof of Theorem (1.2). We may take ek to be a diagonal matrix with entries e2πiλj (N) . First, we put t = A/2N . By Lemma 2.1 we have: 1 1 πt k (−|R log H (t, a, e ) = | + N )[− + log η(−it/4π ) + SU (N) N2 N2 12 1 + 2 log HSU (2) (t, ei α,k , eiπ/2 ). (25) N α>0
We note that both sides of this equation are real, so that we take take the real part without changing the equation. We now specialize to the case of SU (N ). Its roots are ei − ej and its positive roots satisfy i < j . Hence, α, k = λi − λj . Hence log HSU (2) (t, ei α,k , eiπ/2 ) = log HSU (2) (t, ei(λi −λj ) , eiπ/2 ) α∈R+
i<j
=
1 log HSU (2) (t, ei(λi −λj ) , eiπ/2 ). 2
(26)
i=j
In the last equality we use that HSU (2) (t, ei(λi −λj ) , eiπ/2 ) = HSU (2) (t, ei(λj −λi ) , eiπ/2 ). We further have log HSU (2) (t, ei(λi −λj ) , eiπ/2 ) = log HSU (2) (t, ei(λi −λj ) , eiπ/2 ) i=j
i,j
− N log HSU (2) (t, 1, eiπ/2 ). Since N ∼ N for SU (N ), we have 1 log HSU (2) (t, ei α,k , eiπ/2 ) N2 α∈R+ 1
log HSU (2) (t, eix , eiπ/2 )dσN (ei(x−x ) )dσN (eix )− log HSU (2) (t, 1, eiπ/2 ) = N S1 S1
Macdonald’s Identities and the Large N Limit of Y M2 on the Cylinder
=
S1
log HSU (2) (t, eix , eiπ/2 )dσN ∗ dσN (eix ) −
where
1 log HSU (2) (t, 1, eiπ/2 ), N
dσN ∗ dσN (eix ) =
S1
621
dσN (ei(x−x ) ) dσN (eix ).
If σN :=
N 1 δ(e2πiλj (N) ) → σ ∈ M(S 1 ), N j =1
then dσN ∗ dσN → dσ ∗ dσ , since the Fourier coefficients of the left side tend to those of the right side. Thus, we obtain the stated limit. A . 3.3. Proof of Theorem 1.3. We now re-do the calculation but make the scaling t = 2N We thus have A ix iπ/2 1 k dσN ∗ dσN (eix ) ,e ,e log HSU (2) FSU (N) (a, e ; A) = 2 S1 2N 1 A iπ/2 − log HSU (2) . (27) , 1, e N 2N
We now obtain the limit by using the uniform off-diagonal asymptotics of the central heat kernel. The central heat kernel is related to the actual heat kernel by kSU (2) (t, eix , g −1 ag)dg, log HSU (2) (t, eix , a) = log SU (2)
where kSU (2) (t, x, y) is the heat kernel. Therefore, we are interested in the uniform asymptotics of A ix A ix −1 , e , a = log , e , g ag dg kSU (2) (28) log HSU (2) 2N 2N SU (2) in x for each A. There exists a uniform heat kernel parametrix for kSU (2) given by: d(u,v)2
(29) kSU (2) (t, u, v) ∼ t −3/2 e− π t V (t, u, v), ∞ where V (t, u, v) ∼ j =0 Vj (u, v)t j . The amplitude V is to leading order the volume density in normal coordinates. We will never be evaluating it at a pair of conjugate points, so it is uniformly bounded below by a positive constant and introduces only lower order terms into the logarithm. See for instance [Ka] for general results of this kind. Instead of a parametrix, the reader might prefer to use (6) in terms of Jacobi’s theta function. This formula and also (14) can be used to check the various constants in the formula.
622
S. Zelditch
Thus, we have log HSU (2)
A ix ,e ,a 2N
∼ log
N A
3/2
2N
e− π A d(e
ix ,g −1 ag)2
SU (2)
× V (A/2N, eix , g −1 ag)dg ,
(30)
where V (A/2N, eix , g −1 ag) is a semiclassical amplitude in N . The asymptotics are determined by the minimum point of the phase d(eix , g −1 ag)2 , namely by the distance d(eix , Ca ) from eix to the conjugacy class of a. We note that the conjugacy class C(a) = {g −1 ag : g ∈ SU (2)} is a great (equatorial) 2-sphere of radius π/2 from (the north pole) I . There is a somewhat different expansion accordingly as eix = ±1, eix ∈ C(a) or for eix not of this form, which we will call a general element. If eix is a general element, then there is a unique closest point g −1 ag and the phase is non-degenerate, so we obtain 2N ix −1 2 e− π A d(e ,g ag) V (A/2N, x, g −1 ag)dg SU (2) 2N
∼ A−3/2 e− π A d(e
ix ,C )2 a
·
1 det H ess(A−1
d(eix , g −1 ag)2 )
.
(31)
We note that the powers of A cancel. If eix ∈ C(a), then since eix also represents a point in the maximal torus, eix = a or eix = a −1 , hence the unique point of minimal distance to eix in C(a) is of course eix itself. There is no essential change in the calculation except 1 that the exponent vanishes. We now consider the behavior of √ as ix −1 2 det H ess(d(e ,g
ag) )
eix → ±1. In the case eix → 1), for instance, det H ess(A−1 d(eix , g −1 ag)2 ) ∼ |x|3 and hence log det H ess(d(eix , g −1 ag)2 ) ∼ log |x| as x → 0. At eix = ±1, the entire C(a) becomes a critical manifold for the phase. Exactly at the poles, we have 3/2 A N 2 HSU (2) (t, ±1, a) = kSU (2) e−2Nπ /4πA . (32) , ±1, a ∼ 2N A Thus, as eix → 1, HSU (2)
2N ix 2 |x|−3/2 e− π A d(e ,Ca ) , x = 0 A ix ,e ,a ∼ 2N N 3/2 e−2Nπ 2 /4πA , x = 0. A
(33)
Similarly at x = π. We note that the exponent is continuous and only the power of N changes at the special points x = 0, π. 2N ix 2 Since we are interested in log asymptotics, the factor e− π A d(e ,Ca ) is dominant as long as the remainder can be integrated against dσN ∗ d σ¯ N . If this measure has a point mass at either 0 or π, there are singularities at x = 0, π in the coefficients of the asymptotics which are not integrable dσ ∗ d σ¯ . We now discuss this point in detail.
Macdonald’s Identities and the Large N Limit of Y M2 on the Cylinder
623
We fix a positive constant C > 0 and break up the integral (35) as A ix iπ/2 ,e ,e log HSU (2) ( )dσN ∗ dσN (eix ) 2N S 1 \[−C/N,C/N]∪[π−C/N,π+C/N] A ix iπ/2 + ,e ,e log HSU (2) ( )dσN ∗ dσN (eix ). (34) 2N [−C/N,C/N]∪[π −C/N,π+C/N] It follows from the pointwise asymptotics in (33) that A ix iπ/2 log HSU (2) ( )dσN ∗ dσN (eix ) ,e ,e 2N S 1 \[−C/N,C/N]∪[π −C/N,π+C/N] 2N ∼− d(C(a), eix )2 dσN ∗ dσ N (eix ). π A S 1 \[−C/N,C/N]∪[π−C/N,π+C/N]
(35)
C C For the second integral of (34) over [− N , N ] and [π − C/N, π + C/N ], we use the scaling asymptotics of the heat kernel rather than its pointwise asymptotics. Since the C C details are similar for both intervals we only carry them out around the interval [− N , N ]. u Namely, we write x = N with |u| ≤ C. We have: π 2 u u + Q(u, N, A, g −1 ag) d(ei N , g −1 ag)2 = 2 N for a bounded smooth function Q, hence A u log HSU (2) , ei N , a 2N u N 3/2 − 2N π 2 −1 2 ∼ e πA 4 e−Q(u,A,N)g ag) V (A/2N, ei N , g −1 ag)dg A SU (2) N 3/2 2N π 2 1 ∼ e− A 4 [q(u, A) + q1 (u, A) + · · · ), (36) A N
where q = e−Q(u,A,0 ) is strictly positive. Substituting (36) into the second term of (34), we get C u C u 3 N 2N π 2 − log − . dσN ∗ d σ¯ N Q(u, A)dσN ∗ d σ¯ N − πA 4 2 A N N −C −C (37) Since Q is bounded and dσN ( Nu ) ∗ d σ¯ N ( Nu ) has mass at most 1, the last term is O(1) as N → ∞. Thus, the first term of (37) dominates. Thus, we have A ix iπ/2 dσN ∗ dσN (eix ) log HSU (2) ,e ,e 2N [−C/N,C/N]∪[π−C/N,π+C/N] /N 3 N 2N π 2 dσN ∗ d σ¯ N . (38) − log ∼ − πA 4 2 A −C/N We note that d(Ca , 1) = d(Ca , −1) = π2 . Since d(Ca , eix ) is continuous, we may rewrite (37)-(38) to leading order as C/N 2N d(eix , Ca )2 dσN ∗ d σ¯ N . (39) − πA −C/N
624
We add this back to (35) to obtain the leading order term 2N d(C(a), eix )2 dσN ∗ dσ N (eix ) − πA S 1
S. Zelditch
(40)
iA ) which are indepenplus the canonical terms N1 log HSU (2) (A/2N, 1, a) and log η( 2N ix ix ±iπ/2 )}, completing the dent of dσ . We then recognize that d(e , Ca ) = min{d(e , e proof.
Remark. • It would be interesting to know when the leading order term cancels, but even this cancellation would not cure the anomoly, since there is no control over the rate of the weak convergence dσN → dσ . Hence, there is no well-defined growth rate of the lower order terms. • As noted above, the exponent in (33) is continuous. Since we are taking the logarithm, only the exponent is important to leading order, and that is why we obtain a unified formula, even when dσ has a point mass at eix = ±1. 4. Final Remarks Some final comments and remarks. • It would of course be desirable to remove the assumption that one of the matrices U1 should be a prinicpal element of type ρ. In [F2], H. Fegan states a product formula for the full fundamental solution, but unfortunately this formula is erroneous. It would be interesting to see if there exists some form of Macdonald’s identities for the full central heat kernel. • It would also be desirable to extend the results of this paper to more general groups such as U (N ), SO(N ), Sp(N ). We specialize to GN = SU (N ) because it amply illustrates the main point of this paper. Further, Macdonald’s identities pertain to simply connected compact semi-simple groups, and presumably require some modifications for U (N ), SO(N ), etc. • If we ignore the unexpected growth in N, the phase transition conjecture suggests that the limit free energy should equal θ−φ 1 S(dθ, dσ |A) − 21 S 1 S 1 log sin θ−φ 2 dσ (θ )dσ (φ) − 2 S 1 S 1 log sin 2 dθdφ. The last two terms can be obtained from our limit formula by unravelling the denominator in the logarithm. Thus, aside from orders of magnitude, the phase transition conjecture appears to state in these cases that 1 d(C(a), eix )2 dσ ∗ dσ (eix ). S(dθ, dσ |A) = 2A S 1 We have not been able to compare our results directly with the physics predictions by solving the complex Burghers equation. • In addition to considering “pointwise” asymptotics of the partition function, one could consider weak asymptotics where one averages in various ways over the spectra of U1 , U2 , considers the variance of the integrals and so on.
Macdonald’s Identities and the Large N Limit of Y M2 on the Cylinder
625
2
A • It is plausible that HN ( 2N , eA , eB ) ∼ eN F (σ1 ,σ2 ) when A, B are Hermitian rather than skew-Hermitian, by analogy with the results of Guionnet-Zeitouni [GZ]. • As mentioned in the introduction, an anomaly also occurs in Matytsin’s conjectured asymptotics [M] at the special character values χR (a), see [TZ].
Acknowledgements. The author would like to thank M. Douglas, P. Etingof, V. Kazakov, B. Kostant, N. Reshetikhin, T. Tate, and C. Vafa for going over some of the details of the calculations and for comments (quoted above) on the relations between our results and the predictions in the physics papers.
References [AF]
Adin, R.A., Frumkin, R.: Rim hook tableaux and Kostant’s η-function coefficients. arXiv: Math: CO/0201003 [A] Apostol, T.: Modular Functions and Dirichlet Series in Number Theory. Berlin-HeidelbergNew York: Springer Verlag, 1979 [BS] Boutet de Monvel, A., Shcherbina, M.V.: On free energy in two-dimensional U(n)-gauge field theory on the sphere. (Russian) Teoret. Mat. Fiz. 115(3), 389–401 (1998); translation in Theoret. Math. Phys. 115(3), 670–679 (1998) [BtD] Brocker, T., tom Dieck, T.: Representations of Compact Lie Groups. Grad. Texts in Math., Berlin: Springer-Verlag, 1985 [CdAMP] Caselle, M., D’Adda, A., Magnea, L., Panzeri, S.: Two-dimensional QCD on the sphere and on the cylinder. High energy physics and cosmology (Trieste, 1993), ICTP Ser. Theoret. Phys. 10, River Edge, NJ: World Sci. Publishing, 1994, pp. 245–255 [CMR] Cordes, S., Moore, G., Ramgoolam, S.: Large N 2DYang-Mills theory and topological string theory. Commun. Math. Phys. 185(3), 543–619 (1997) [D] Douglas, M.R.: Conformal field theory techniques in large N Yang-Mills theory. In: Quantum Field Theory and String Theory (Cargèse, 1993), NATO Adv. Sci. Inst. Ser. B Phys. 328, New York: Plenum, 1995, pp. 119–135 [D1] Douglas, M.R.: Private communication [D2] Douglas, M.R.: Large N gauge theory—expansions and transitions. In: String Theory, Gauge Theory and Quantum Gravity (Trieste, 1994). Nuclear Phys. B Proc. Suppl. 41, 66–91 (1995) [D3] Douglas, M.R.: Large N quantum field theory and matrix models. In: Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun. 12, Providence, RI: Am. Math. Soc., 1997, pp. 21–40 [D4] Douglas, M.R.: Large N in D > 2. In Strings ’95 (Los Angeles, CA, 1995), River Edge, NJ: World Sci. Publishing, 1996, pp. 187–197 [DK] Douglas, M.R., Kazakov, V.A.: Large N phase transition in continuum QCD2 . Phys. Lett. B319, 219–230 (1993) [F] Fegan, H.D.: The heat equation on a compact Lie group. Trans. Am. Math. Soc. 246, 339–357 (1978) [F1] Fegan, H.D.: The heat equation and modular forms. J. Differ. Geom. 13(4), 589–602 (1978) [F2] Fegan, H.D.: The fundamental solution of the heat equation on a compact Lie group. J. Differ. Geom. 18(4), 659–668 (1983) [Fr] Frenkel, I.G.: Orbital theory for affine Lie algebras. Invent. Math. 77(2), 301–352 (1984) [G] Gross, D.J.: Two dimensional QCD as a string theory. Nucl. Phys. B400, 161–180 (1993) [GM1] Gross, D.J., Matytsin, A.: Instanton induced large N phase transitions in two and four dimensional QCD. Nucl. Phys. B429, 50–74 (1994) [GM2] Gross, D.J., Matytsin, A.: Some properties of large N two dimensional Yang–Mills theory. Nucl. Phys. B437, 541–584 (1995) [GZ] Guionnet, A., Zeitouni, O.: Large deviations asymptotics for spherical integrals. J. Funct. Anal. 188(2), 461–515 (2002) [I] Iwaniec, H.: Topics in Classical Automorphic Forms. Grad. Studies in Math 17, Providence, RI: AMS, 1977 [Kac] Kac, V.G.: Infinite-dimensional algebras, Dedekind’s η function, classical Mobius function and the very strange formula. Adv. in Math. 30, 85–136 (1978) [Ka] Kannai, Y.: Off diagonal short time asymptotics for fundamental solutions of diffusion equations. Commun. Partial Differ. Equations 2(8), 781–830 (1977) [KW] Kazakov, V. A., Wynter, T. : Large N phase transition in the heat kernel on the U(N) group. Nucl. Phys. B 440(3), 407–420 (1995)
626 [Ko] [MAC] [M] [PS] [TZ] [U] [W] [W2]
S. Zelditch Kostant, B.: On Macdonald’s η-function formula, the Laplacian and generalized exponents. Adv. in Math. 20(2), 179–212 (1976) Macdonald, I.G.: Affine root systems and Dedekind’s η-function. Invent. Math. 15, 91–143 (1972) Matytsin, A: On the large-N limit of the Itzykson-Zuber integral. Nucl. Phys. B 411(2-3), 805–820 (1994) Pressley, A., Segal, G.: Loop Groups. Oxford Mathematical Monographs, Oxford Science Publications, Oxford: Clarendon Press, 1988 Tate, T., Zelditch, S.: Counter-example to conjectured SU(N) character asymptotics, arXiv Preprint hep-th/0310149 Urakawa, H.: The heat equation on a compact Lie group. Osaka J. Math. 12, 285–297 (1975) Witten, E.: On quantum gauge theories in two dimensions, Commun. Math. Phys. 121, 153– 209 (1991) Witten, E.: Two-dimensional gauge theories revisited. J. Geom. Phys. 9, 303–368 (1992)
Communicated by M. R. Douglas
Commun. Math. Phys. 245, 627–637 (2004) Digital Object Identifier (DOI) 10.1007/s00220-003-1029-8
Communications in
Mathematical Physics
Critical Percolation in Annuli and SLE6 Julien Dub´edat Laboratoire de Math´ematiques, Bˆat. 425, Universit´e Paris-Sud, 91405 Orsay cedex, France. E-mail: [email protected] Received: 4 June 2003 / Accepted: 29 August 2003 Published online: 20 January 2004 – © Springer-Verlag 2004
Abstract: Building on the identification of the scaling limit of the critical percolation exploration process as a Schramm-Loewner Evolution, we derive a PDE characterization for the crossing probability of an annulus. 1. Introduction Percolation is arguably the simplest example of a planar “critical” model, i.e. a random planar graph (generally speaking a subgraph of a regular lattice, eventually with spins on sites) with a small mesh such that the probabilities of geometrically meaningful macroscopic events have non-degenerate limits when the mesh goes to zero. Recall that percolation consists in removing each edge (or each vertex) in a lattice with a given probability p; bond percolation on the square lattice and site percolation on the triangular lattice are critical for p = 1/2. The probabilities of macroscopic events are generally believed to be conformally invariant: the limiting probability when the mesh of the lattice goes to zero should not change if one applies a conformal equivalence to the corresponding geometric configuration. An important related example is the conformal invariance of hitting probabilities (harmonic measure) for planar Brownian motion, which is the scaling limit of simple random walks. Using techniques from Conformal Field Theory (CFT) and Coulomb gas models, physicists have proposed several intriguing formulas for the limiting probabilities of critical percolation. Unfortunately, it does not seem easy to make their arguments rigorous. We now review some of these predictions. • Simply connected domains Mark four points on the boundary of a simply connected domain to get a conformal rectangle; up to conformal equivalence, there is only one degree of freedom, namely the aspect ratio of the rectangle. The probability that two opposite sides of the rectangle are connected by a percolation cluster is given by Cardy’s formula ([Ca92]).
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Watts’ formula ([Wa96]) describes the probability of a double crossing, i.e. the two pairs of opposite sides are all connected by one cluster. Cardy has also studied the occurrence of n disjoint clusters connecting two opposite sides ([Ca01]). • Multiply connected domains A n-connected domain is a plane connected domain the complement in C of which has n connected components (its fundamental group is the free group on (n − 1) generators). A natural question concerns the probability that two given connected components of the boundary are connected by a cluster inside the domain. The simplest case is the existence of a crossing from the outer to the inner boundary of an annulus. More generally, a law for the number of crossings of alternate colors has been proposed by Cardy ([Ca02]). • Compact Riemann surfaces Here there is no boundary; tori are the simplest example. Since the conformal structure is of crucial importance, it may be better to think of (complex) elliptic curves. Pinson ([Pin94]) has studied the image of the (first) homology group of clusters of a given color in the homology group of the torus (i.e. a random subgroup of Z × Z). Since a continuous random graph does not really make sense, one has first to clarify what these limiting probabilities do correspond to. A natural way to understand the “continuous scaling limit of critical percolation” is to focus on the interfaces between clusters, which are random curves. The limiting probabilistic objects (random curves) are the Schramm-Loewner Evolution (SLE), introduced by Schramm in [Sch00] (see [RohSch01, Wer02] for some background on SLE). It describes the only possible conformally invariant scaling limits of these interfaces. For critical site percolation on the triangular lattice, Smirnov ([Smi01]) has proved that the cluster interface indeed converges to the so-called chordal SLE6 process. His proof in fact relies on establishing directly Cardy’s formula for single crossings in conformal rectangles. Hence it may be interesting to derive some percolation probabilities in the SLE6 framework. Cardy’s formula itself is easily derived for the SLE6 , as pointed out by Schramm (see for instance [Wer02]). The general idea is that if one wants to compute probabilities of macroscopic events in terms of the SLE6 process, one uses the fact that the conditional probabilities (using the filtration associated to the SLE6 ) of this event are a martingale. This leads (usually) to a partial differential equation (in terms of the “conformal coordinates” of the problem), that characterize fully these probabilities. This method has been used in [Sch01] to derive a new formula for critical percolation, and also in [LSW01a, LSW01b, LSW02a, LSW02e] to derive the value of the corresponding critical exponents (that describe the asymptotic decay of the probabilities of some events). In the present paper we derive analytic characterizations of some percolation probabilities in annuli using Smirnov’s result on convergence of the percolation exploration process to SLE6 . The computations are somewhat tedious, but the underlying probabilistic ideas are extremely simple, and may be of use for more general problems (in particular for n-connected domains, n ≥ 3). Let U be an n-connected domain and ∂ be one connected component of its boundary (a Jordan closed curve, say). Pick two points x and y on ∂ and set the following boundary conditions on ∂: the arc (x, y) is colored in blue and the arc (y, x) in yellow (in clockwise order). A chordal event is a percolation event depending only on (U, x, y). For instance, consider the event that there exists a blue crossing between (x, y) and ∂1 and a yellow crossing between (y, x) and ∂2 , where ∂i are connected components of
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∂U . Then “grow” a small percolation (resp. SLE6 ) hull at x. Let Kt be this hull (t is a measure of its size) and xt be its “tip” (see e.g. [Smi01]); then (U \Kt , xt , y) is the perturbed domain. For a well chosen chordal event, the event holds in (U, x, y) if and only if it holds in (U \Kt , xt , y). Now consider the set of the (g + 1)-connected domain with two points marked on a connected component of the boundary modulo conformal equivalence (a Teichm¨uller space); it is a manifold of dimension (3g − 1) if g ≥ 2. Then t → (U \Kt , xt , y) (modulo conformal equivalence) should be, up to time change, a diffusion in the Teichm¨uller space; as percolation is local, it does not “feel” the boundary before actually touching it. The probability of the chordal event defines a function on this space and this function is harmonic for the diffusion. If one is able to compute a SDE for the diffusion and, crucially, to work out boundary conditions for the harmonic function, this yields an analytic characterization of the chordal event probability (as a function on the Teichm¨uller space). In this paper, we essentially carry out this program for annuli. In the first section, we recall some facts of complex analysis on the annulus (mainly the solution to the Dirichlet problem). Next, we derive the SDE for the diffusion in the Teichm¨uller space (which is isomorphic to R− × (0, 2π)). In the third section, we characterize the law of the number of nested clusters of alternate colors wrapped around the inner disk of an annulus, and make the connection with Cardy’s results (see [Ca02]). Defining SLE on Riemann surfaces has also been recently (and independently) the subject of [Z03, FK03], in different settings. 2. Annuli Let q < 1; define the annulus Aq = {z : q < |z| < 1}. It is classical that for q = q , the annuli Aq and Aq are not conformally equivalent, and that the conformal automorphisms of Aq are the maps z → uz and z → quz−1 , u ∈ U. Moreover, any doubly connected domain (i.e. a connected domain the complement in C of which has two connected components) is conformally equivalent to an annulus. Thus, one may identify the Teichm¨uller space of doubly connected domains with the set {Aq }0
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where the sum is on the vertices of the lattice 2ω1 Z+2ω2 Z. Then, for any z, ζ (z+2ω1 ) = ζ (z) + 2η1 , ζ (z + 2ω2 ) = ζ (z) + 2ω2 , with η1 = ζ (ω1 ), η2 = ζ (ω2 ). Let ω3 = ω1 + ω2 and η3 = η1 + η2 . Define also: ζ3 (z) = η3 − ζ (ω3 − u). Note that z → log |z| defines a real harmonic function on an annulus which is not the real part of an holomorphic function on this annulus. So assume that: 2π 2π φ(θ )dθ = ψ(θ)dθ. 0
0
Then one may define ω iω1 2π ω1 1 (z) = 2 φ(θ )ζ log(z) − 2 θ dθ π iπ π 0 2π ω1 ω1 iω1 ψ(θ )ζ3 log(z) − θ dθ. − 2 π iπ π 0 2π Along a clockwise loop in Aq , the first integral is increased by iω1 /π 2 0 φ(θ )η1 dθ and the second integral is increased by the same amount; so there is no problem with the logarithm’s determination. This function is holomorphic on Aq ; moreover, for all θ , (exp(iθ )) = φ(θ ) and (q exp(iθ )) = ψ(θ). We apply this result with the conditions φ(θ )dθ = 2π δθ0 , and ψ(θ ) = 1 for all θ , thus getting a holomorphic function which is well defined up to the addition of an imaginary constant. Let x = exp(iθ0 ) and y ∈ U, y = x. The holomorphic vector field: ω 2iω1 ω1 1 z ζ log(z/x) − ζ log(y/x) Vx,y (z) = π iπ iπ 2π ω dθ ω1 2iω1 1 z log(ze−iθ ) − ζ3 log(ye−iθ ) ζ3 − π iπ iπ 2π 0 is equal to z (z), so it is well defined by the following properties: • • • • • •
Vx,y is holomorphic on Aq , Vx,y may be extended continuously to the boundary, except at x, (Vx,y (z)/z) is constant on qU, (Vx,y (z)/z) = 0 on U\{x}, Vx,y (y) = 0, and Vx,y has a simple pole at x with residue −2x 2 .
3. SLE6 in an Annulus A chordal SLEκ going from x ∈ U to 1 in the unit disk D may be defined (up to linear time change) by the equations: dgt (w) = LWt (gt (w))dt,
√ where W is κ times a standard real Brownian motion starting from i(x + 1)/(1 − x) and the holomorphic vector field Ly is defined on D by:
Critical Percolation in Annuli and SLE6
631
Ly (w) = −
i(1 − w)2 . i(1 + w)/(1 − w) − y
The map gt is a conformal equivalence between D\Kt and D that fixes 1, where Kt is the SLE hull at time t. The time parameter corresponds to half of the capacity of the image of Kt under the homography w → i(w + 1)/(1 − w) seen from infinity in the half-plane. Let q < 1. For small enough t, D\gt (qD) is a doubly connected set; hence there exists a unique qt ≥ q and a unique conformal equivalence ht between D\gt (qD) and Aqt such that ht (1) = 1. Let ξt = (Wt − i)/(Wt + i), λt = ht (ξt ), and ft = ht ◦ gt . Now w → (dft )(ft−1 (w)) defines a holomorphic vector field on Aqt . It is easily seen that this vector field is proportional to Vλt ,1 (we omit the qt parameter). Thus, there exists a function at such that: d(ft (w)) = Vλt ,1 (ft (w))dat . Moreover, d log(qt ) = dat , so one may pick a(t) = log(q(t)). Using the chain rule, one gets: d(ft (w)) = (dht )(gt (w)) + ht (gt (w))dgt (w). It follows that:
dht (w) = Vλt ,1 (ht (w))dat − LWt (w)ht (w)dt.
As the poles in this expression should cancel out, necessarily: dat = ht (ξt )2
(1 − ξt )4 dt, 4λ2t
and this is indeed real. Now let gt (w) = ξt + u, with small u. Then, LWt (ξt + u) = −
(1 − ξt )4 3 + (1 − ξt )3 + O(u), 2u 2
ht (ξt + u)LWt (ξt + u) = −ht (ξt )
(1 − ξt )4 (1 − ξt )4 3 − ht (ξt ) + ht (ξt )(1 − ξt )3 + O(u) 2u 2 2
and iπ λt ht (ξt + u) 1 λt ht (ξt ) ht (ξt )2 ω1 = + O(u). log − ζ − iπ λt ω1 ht (ξt ) u 2ht (ξt ) λt λ2t Finally Vλt ,1 (ht (ξt + u)) 2λt 2iω1 ω1 1 λt ht (ξt ) ht (ξt )2 = λt − + − ) − ζ log(λ t ht (ξt ) u 2ht (ξt ) λt π iπ λ2t 2π 2iω1 ω1 ω1 dθ + ζ3 log(e−iθ ) − ζ3 log(λt e−iθ ) − 2λt + O(u). π iπ iπ 2π 0
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Notice that: 2π ζ3 0
dθ ω log(λt ) 1 log(e−iθ ) − ζ3 log(λt e−iθ ) =− 2η1 . iπ iπ 2π 2iπ
ω
1
Hence w → dht (w) is smooth at ξt and: 2ω log(λ )η ht (ξt )2 (1 − ξt )4 2iω1 ω1 1 t 1 (dht )(ξt ) = dt ζ log(λt ) − 4λt π iπ π2 3 3ht (ξt )2 (1 − ξt )4 3 4 3 + − ht (ξt )(1 − ξt ) dt. h (ξt )(1 − ξt ) − 4 t 4λt 2 √ Recall that ξt = (Wt −i)/(Wt +i), dWt = κdBt , where B is a standard real Brownian motion. Hence: (1 − ξt )2 √ (1 − ξt )3 dξt = κdBt + κdt. 2i 4 Since λt = ht (ξt ), an appropriate version of Itˆo’s formula yields: 1 dλt = (dht )(ξt ) + ht (ξt )dξt + ht (ξt )d ξt 2 h (ξt )(1 − ξt )2 √ = t κdBt 2i 2ω log(λ )η ht (ξt )2 (1 − ξt )4 2iω1 ω1 1 t 1 dt ζ log(λt ) − + 4λt π iπ π2 3 κ (1 − ξt )4 3h (ξt )2 (1 − ξt )4 + ht (ξt ) dt. − ht (ξt )(1 − ξt )3 − dt − t 2 2 4 4λt Let κ = 6 and exp(iν. ) = λ. , ν. ∈ [0, 2π]. We perform a time change using the increasing function t → a(t): √ νa 2ω1 νa dνa = − κd B˜ a + ζ (ω1 ) − ζ (ω1 ) da, π π π where (B˜ a )a≥0 is a standard real Brownian motion. Note that the zeta-functions, as well as the half-period ω1 depend implicitly on a. Using Jacobi’s theta-function, one may rewrite this SDE as (see e.g. [Cha84], Theorem V.2): √ 1 θ νa ia ˜ dνa = − κd Ba + ,− da, π θ 2π π where θ designates the derivative of the bivariate function θ(v, z) with respect to the first variable. Note that this SDE is invariant under ν ↔ 2π − ν, which is obvious from the definition of ν. The principle of this computation is closely related to the approach of locality/restriction in [LSW02d]. The fact that one gets a (time-inhomogeneous) diffusion for ν is a feature of locality for κ = 6. Note also that we could have begun with any configuration conformally equivalent to (Aq , λ, 1), getting the same dynamics for ν, hence the same percolation probabilities.
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4. Crossing of an Annulus The previous computations may be used to study various critical percolation probabilities in the annulus. For instance, consider the following crossing probability: F (ν, a) is the probability that there exists a blue crossing between the arc (1, exp(iν)) ⊂ U and the inner circle qU in the annulus Aq and a yellow crossing between (exp(iν), 1) and qU, with a = log(q). It follows from the previous section that: 1 θ νa ia (4.1) ,− F ν + Fa = 0 3Fν,ν + π θ 2π π and F satisfies the boundary conditions: F (0, a) = F (2π, a) = 0, F (ν, 0) = 1 for all a < 0, ν ∈ (0, 2π ). This last condition is a consequence of the Russo-Seymour-Welsh theory. In the rest of this section, we discuss events related to nested circuits of alternate colors around the inner disk of an annulus for critical percolation. Consider (νa , a)a as a two dimensional diffusion in the half-strip S = {z : z < 0, 0 < z < 2π }. This half-strip may be identified with the Teichm¨uller space of doubly connected plane domains with two marked boundary points (on the same connected component of the boundary) modulo conformal equivalence. The diffusion is stopped when it hits R− ∪ [0, 2iπ] and is instantaneously reflected on 2iπ + R− . For any z ∈ S, this defines a harmonic measure seen from z on R− ∪ [0, 2iπ ]. The restriction of this probability measure to R− is a finite measure (not a probability measure), which we ˜ ˜ + 2iπ, .). Thus we have defined shall note K(z, .). For y ∈ R− , note K(y, .) = K(y a defective Markov kernel on R− (identified with doubly connected domains modulo conformal equivalence). From a probabilistic point of view, this is equivalent to a (discrete time) Markov chain on R− ∪ {∂}, where ∂ is a cemetery state. The number of steps taken by this Markov chain (starting from y < 0) before reaching ∂ corresponds to the number of downcrossings and upcrossings of ν (starting from 2π at time y). For a ˜ + iν, y) is a solution of the PDE (4.1) with boundary conditions: given y, (ν, a) → K(a ˜ ˜ y) = δy , the Dirac mass at y. On K(iν, y) = 0 for all ν ∈ (0, 2π), and, on R− , K(., 2iπ + R− , there is a Neumann boundary condition: K˜ ν (2iπ + x, y) for all x < 0, just as in [LSW02e], Lemma 2.3. We now interpret downcrossings and upcrossings of ν in terms of the exploration process. See also [LSW02e] for a discussion of percolation events in annuli in relation with the exploration process (and nice figures !). For the sake of simplicity, consider critical site percolation on the triangular lattice: each vertex of the triangular lattice (or each hexagon of a honeycomb lattice) is colored in blue or yellow with probability 1/2. Consider a portion of this lattice (with small mesh) that approximates the annulus Aq . Two points are marked on the outer boundary, say x and 1; the arc (1, x) is colored in blue and the arc (x, 1) is colored in yellow. The exploration process from x to 1 (which is well defined as long as it does not hit qU) is the path with only blue hexagons on its left-hand and yellow ones on its right-hand. The exploration process completes a clockwise loop if there is a blue circuit of hexagons around the inner disk qD and a counterclockwise loop if there is a yellow circuit. It hits the inner disk before completing a circuit if there is a blue crossing from (1, x) to qU and a yellow one from (x, 1) to qU. In the following, a circuit will always be a circuit around qD, and a crossing will always be a crossing between U and qU in Aq . In the continuous setting, starting from exp(iνa ) = x, if ν. reaches 0, then the boundary “seen from the inner disk” is completely yellow; this means that the exploration
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process has completed a counterclockwise loop (including the yellow arc (x, 1)). Hence, starting from νa = 2iπ, the process ν reaches 0 before time 0 if and only if there is a yellow circuit inside the annulus (the outer circle is blue). Then ν reflects instantaneously on R− , which means that the exploration process proceeds towards qD in a conformal annulus the outer boundary of which (i.e. the circuit) is yellow. Thus a downcrossing for nu means that there exists a yellow circuit (here U is blue); a downcrossing followed by an upcrossing means that there exists a yellow circuit and a blue circuit nested in the yellow one; and so on. Now set free boundary conditions. Let N denote the event that there is no circuit (blue or yellow); equivalently, there is a blue crossing and a yellow one. For n ≥ 1, let Bn be the event that there is exactly n disjoint clusters of alternate colors wrapped around qD and the outermost is blue; Yn is the corresponding events with colors changed (see figure).
Fig. 1. The events N, B2 , Y3
Obviously P(Bi ) = P(Yi ) and: P(N ) + 2
∞
P(Bi ) = 1.
n=1
It follows from the previous discussion that the probability cn that ν starting from y +2iπ makes at least n alternate downcrossings and upcrossings is: cn = (K n )(y, 1), where designates the convolution of Markov kernels and 1 is the constant function. Note that we have analytically characterized K. Here one should be cautious because of the boundary conditions. The number cn is the probability to see at least n circuits of alternate colors, not counting the outermost one if it is blue. This outermost circuit is either blue or yellow, so: cn =
∞
P(Yk ) +
k=n
= P(Bn ) + 2
∞
k=n+1 ∞
P(Bk )
P(Bk ).
k=n+1
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Determining P(N ), P(Bi ) from the ci is then a trivial problem of (infinite dimensional) linear algebra. In the Fr´echet space of rapidly decaying series, consider the vectors v = (P(N ), P(B1 ), . . . , P(Bn ) . . . ) and w = (1, c1 , . . . , cn , . . . ), and the bounded operator: M = Id +2J + 2J 2 + · · · + 2J n + . . . , where J is the left shift: J (u0 . . . un . . . ) = (u1 . . . un+1 . . . ). Then Mv = w. As (a little formally) M = 2(Id −J )−1 − Id, one gets M −1 = Id −2(Id +J )−1 . Hence: P(Bi ) = ci + 2
∞
(−1)n cn+i
n=1
with the conventions B0 = N, c0 = 1. Note that (cn ) decays exponentially: once a circuit has been completed, the “new annulus” has a greater modulus than the initial one, and the greater the modulus, the lesser the probability that there exist circuits. Hence cn (q) < c1 (q)n , and c1 (q) < 1 from the RSW theory (one may also argue using the BK inequality rather than the previous “renewal” argument). Similarly, (P(Bn )) decays exponentially, which justifies the previous formal argument. Let M be the number of downcrossings and upcrossings completed by the diffusion ν starting from 2iπ + log q, and be the random sign: =1+2
∞
1{M≥i} .
i=1
The sum is finite a.s. From the preceding discussion, P(N ) = E() as functions of a = log q. Note that = 1 if the last visited boundary half-line is the top one, and = −1 in the other case. Since ν and 2iπ − ν have the same law, it follows that: Eiπ+log q () = 0 for all q ∈ (0, 1). Moreover, if ν does not reach the half-line iπ + R− , then = 1 (since paths are continuous). Hence, by the Markov property: P(N )(q) = E2iπ+log q () = P2iπ+log q (ν does not reach iπ + R− ). So we can characterize P(N ) as the solution of a first exit problem: consider the diffusion ν starting from z, z < 0, 0 < z < π (using the symmetry ν ↔ 2iπ − ν), stopped at time τ when it exits the half-strip {z : z < 0, 0 ≤ z < π }, the bottom part of the boundary being instantaneously reflecting; either ντ = π or ντ = 0 a.s. Then: P(N )(q) = Plog q (ντ = 0). Let H (z) = Pz (ντ = 0), z = a + iν, defined on the half-strip {z : z < 0, 0 < z < π}. It is easily seen that in the interior of the domain, ia 1 θ νa 3Hν,ν + ,− Hν + Ha = 0, π θ 2π π and H satisfies the boundary conditions: H equals 1 on i(0, π ), 0 on iπ + R− , and the normal derivative of H vanishes on R− . This mixed Dirichlet-Neumann problem characterizes completely the function q → P(N )(q). This should be compared with the following formula, derived by Cardy ([Ca02]) using Coulomb gas techniques, in link with Conformal Field Theory:
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Conjecture 1 (Cardy). Let τ = −ia/2π. With the previous notations: P(N ) =
√ η(τ )η(6τ )2 3 , η(3τ )η(2τ )2
where η designates Dedekind’s eta function. So far, we have not (yet?) been able to derive this from our characterization of P(N ). In a subsequent paper ([Dub03]), we intend to study percolation problems in multiplyconnected domains, as well as SLE8/3 in such domains. Acknowledgements. I wish to thank Wendelin Werner for his help and advice, and Vincent Beffara for very enlightening discussions.
References [Ahl79] [Ca92] [Ca01] [Ca02]
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Communicated by A. Kupiainen
