Communications in Mathematical Physics - Volume 233

Commun. Math. Phys. 233, 1–12 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0773-5

Communications in

Mathematical Physics

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model Francesco Guerra Dipartimento di Fisica, Universit`a di Roma “La Sapienza” and INFN, Sezione di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy. E-mail: [email protected] Received: 6 May 2002 / Accepted: 6 September 2002 Published online: 13 January 2003 – © Springer-Verlag 2003

Abstract: By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric Sherrington-Kirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site. 1. Introduction The main objective of this paper is to compare the free energy of the mean field spin glass model, introduced by Sherrington and Kirkpatrick in [16], with the expression given in the frame of the Parisi Ansatz [14, 12], including the complete phenomenon of spontaneous replica symmetry breaking. In previous work [6], we have limited our comparison to the replica symmetric case, by producing sum rules, where the difference between the true free energy, and its replica symmetric approximation, is expressed in terms of overlap fluctuations, with a well definite sign. As a result, the replica symmetric approximation turns out to be a rigorous lower bound for the quenched average of the free energy per site, uniformly in the size of the system. In the meantime, the old problem of proving the existence of the infinite volume limit for the thermodynamic quantities has been solved [9], by using a simple comparison argument. Here, we extend this comparison argument, by introducing an appropriate generalized partition function, as a function of a parameter t, with 0 ≤ t ≤ 1, able to interpolate

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between the true theory, at t = 1, and the broken replica Ansatz, at t = 0. Consequently, through a simple direct calculation, we can evaluate the difference between the true free energy, and its broken replica expression, still in the form of a sum rule, with the corrections, of a definite sign, expressed through overlap fluctuations, in properly chosen auxiliary states. As a result, the broken replica Ansatz turns out to be a rigorous lower bound for the quenched average of the free energy per site, uniformly in the size of the system. Moreover, the corrections, given in terms of overlap fluctuations, are in a form suitable for the exploration of the expected result of their vanishing, when the size of the system goes to infinity, along the program developed in [8]. Of course, our method does not use the replica trick in the zero replica limit, as explained for example in [12], nor the cavity method, as exploited for example in [13, 15, 5, 17]. We give only a brief sketch of the extension of our method to the Derrida p-spin model [2, 4, 3, 17]. A more detailed treatment will be presented elsewhere [10]. The organization of the paper is as follows. In Sect. 2, we will briefly recall the main features, and definitions, of the mean field spin glass model. In Sect. 3, the general structure of the Parisi spontaneously broken replica symmetry Ansatz will be described, in a form suitable for the developments of the next section. Section 4 contains the main results of the paper. Firstly, we introduce the interpolating generalized partition function. Then, we evaluate its derivative, with respect to the interpolating parameter, arriving to the sum rule. The general broken replica bound follows easily. In Sect. 5, we give a variational estimate for the infinite volume limit of the ground state energy per site. Next Sect. 6 gives the essential ingredients of the extension of this method to the p-spin model. Finally, Sect. 7 is devoted to conclusions and outlook for further developments. 2. The Basic Definitions for the Mean Field Spin Glass Model The generic configuration of the mean field spin glass model is defined through Ising spin variables σi = ±1, attached to each site i = 1, 2, . . . , N. The external quenched disorder is given by the N (N − 1)/2 independent and identically distributed random variables Jij , defined for each couple of sites. For the
sake  of simplicity, we assume each J to be a centered unit Gaussian with averages E J = 0, ij ij   E Jij2 = 1. The Hamiltonian of the model, in some external field of strength h, is given by the mean field expression  1  HN (σ, h, J ) = − √ Jij σi σj − h σi . N (i,j ) i

(1)

Here, the first sum extends to all site couples, and the second to all sites. For a given inverse temperature β, let us now introduce the disorder dependent partition function ZN (β, h, J ), the quenched average of the free energy per site fN (β, h), the Boltzmann state ωJ , and the auxiliary function αN (β, h), according to the well known definitions  exp(−βHN (σ, h, J )) , (2) ZN (β, h, J ) = σ1 ...σN

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

−βfN (β, h) = N −1 E log ZN (β, h, J ) = αN (β, h),  A exp(−βHN (σ, h, J )) , ωJ (A) = ZN (β, h, J )−1

3

(3) (4)

σ1 ...σN

where A is a generic function of the σ ’s. Replicas are introduced by considering a generic number s of independent copies of (1) (2) the system, characterized by the Boltzmann variables σi , σi , . . ., distributed accord(1) (2) (s) (α) (α) ing to the product state J = ωJ ωJ . . . ωJ . Here, all ωJ act on each one σi ’s, and are subject to the same sample J of the external noise.  (a) (b) The overlap between two replicas a, b is defined according to qab = N −1 i σi σi , with the obvious bounds −1 ≤ qab ≤ 1. For a generic smooth function F of the overlaps, we define the   averages
 F (q12 , q13 , . . .) = EJ F (q12 , q13 , . . .) , (5) where the Boltzmann averages J act on the replicated σ variables, and E is the average with respect to the external noise J . 3. The Broken Replica Symmetry Ansatz While we refer to the original paper [14], and to the extensive review given in [12], for the general motivations, and the derivation of the broken replica Ansatz, in the frame of the ingenious replica trick, here we limit ourselves to a synthetic description of its general structure, in a form suitable for the treatment of the next section, see also [5, 1]. First of all, let us introduce the convex space X of the functional order parameters x, as nondecreasing functions of the auxiliary variable q, both x and q taking values on the interval [0, 1], i.e. X  x : [0, 1]  q → x(q) ∈ [0, 1].

(6)

Notice that we call x the nondecreasing function, and x(q) its values. We introduce a metric on X through the L1 ([0, 1], dq) norm, where dq is the Lebesgue measure. Usually, we will consider the case of piecewise constant functional order parameters, characterized by an integer K, and two sequences q0 , q1 , . . . , qK , m1 , m2 , . . . , mK of numbers satisfying 0 = q0 ≤ q1 ≤ . . . ≤ qK−1 ≤ qK = 1, 0 ≤ m1 ≤ m2 ≤ . . . ≤ mK ≤ 1,

(7)

such that x(q) = m1 for 0 = q0 ≤ q < q1 , x(q) = m2 for q1 ≤ q < q2 , . . . , x(q) = mK for qK−1 ≤ q ≤ qK .

(8)

In the following, we will find it convenient to define also m0 ≡ 0, and mK+1 ≡ 1. The replica symmetric case corresponds to K = 2, q1 = q, ¯ m1 = 0, m2 = 1. The case K = 3 is the first level of replica symmetry breaking, and so on.

(9)

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Let us now introduce the function f , with values f (q, y; x, β), of the variables q ∈ [0, 1], y ∈ R, depending also on the functional order parameter x, and on the inverse temperature β, defined as the solution of the nonlinear antiparabolic equation   1
2 ∂q f (q, y) + f  (q, y) + x(q)f  (q, y) = 0, 2




with final condition

(10)

f (1, y) = log cosh(βy).

(11) f

= ∂y f Here, we have stressed only the dependence of f on q and y, and have put and f  = ∂y2 f . It is very simple to integrate Eq. (10) when x is piecewise constant. In fact, consider x(q) = ma , for qa−1 ≤ q ≤ qa , firstly with ma > 0. Then, it is immediately seen that the correct solution of Eq. (10) in this interval, with the right final boundary condition at q = qa , is given by 

 √ 1 log exp ma f qa , y + z qa − q dµ(z), (12) f (q, y) = ma where dµ(z) is the centered unit Gaussian measure on the real line. On the other hand, if ma = 0, then (10) loses the nonlinear part and the solution is given by  
√ (13) f (q, y) = f qa , y + z qa − q dµ(z), which can be seen also as deriving from (12) in the limit ma → 0. Starting from the last interval K, and using (12) iteratively on each interval, we easily get the solution of (10), (11), in the case of a piecewise constant order parameter x, as in (8). We refer to [7] for a detailed discussion about the properties of the solution f (q, y; x, β) of the antiparabolic equation (10), with final condition (11), as a functional of a generic given x, as in (8). Here we only state the following Theorem 1. The function f is monotone in x, in the sense that x(q) ≤ x(q), ¯ for all 0 ≤ q ≤ 1, implies f (q, y; x, β) ≤ f (q, y; x, ¯ β), for any 0 ≤ q ≤ 1, y ∈ R. Moreover f is pointwise continuous in the L1 ([0, 1], dq) norm. In fact, for generic x, x, ¯ we have  β2 1 |f (q, y; x, β) − f (q, y; x, ¯ β)| ≤ |x(q  ) − x(q ¯  )| dq  . 2 q This result is very important. In fact, any functional order parameter can be approximated in the L1 norm through a piecewise constant one. The pointwise continuity allows us to deal mostly with piecewise constant order parameters. Now we are ready for the following important definitions. Definition 1. The trial auxiliary function, associated to a given mean field spin glass system, as described in Sect. 2, depending on the functional order parameter x, is defined as  β2 1 q x(q) dq. (14) α(β, ¯ h; x) ≡ log 2 + f (0, h; x, β) − 2 0 Notice that in this expression the function f appears evaluated at q = 0, and y = h, where h is the value of the external magnetic field.

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

5

Definition 2. The Parisi spontaneously broken replica symmetry solution is defined by ¯ h; x), α(β, ¯ h) ≡ inf α(β,

(15)

x

where the infimum is taken with respect to all functional order parameters x. Of course, by taking the infimum only with respect to replica symmetric order parameters, as in (9), we would get the replica symmetric solution of Sherrington and Kirkpatrick, as exploited for example in the sum rules in [6], and [8]. The main motivation for the introduction of the quantities given by the definitions is the following expected tentative theorem Theorem 2 (expected). In the thermodynamic limit, for the partition function defined in (2), we have ¯ h). lim N −1 E log ZN (β, h, J ) = α(β, N→∞

Of course, the present technology is far from being able to give a complete rigorous proof. However, in the next section we will prove that α(β, ¯ h) is at least a rigorous upper bound for N −1 E log ZN (β, h, J ), uniformly in N . 4. The Main Results The main results of this paper are summarized in the following Theorem 3. For all values of the inverse temperature β, and the external magnetic field h, and for any functional order parameter x, the following bound holds: ¯ h; x), N −1 E log ZN (β, h, J ) ≤ α(β, uniformly in N , where α(β, ¯ h; x) is defined in (14). Consequently, we have also ¯ h), N −1 E log ZN (β, h, J ) ≤ α(β, uniformly in N , where α(β, ¯ h) is defined in (15). Moreover, for the thermodynamic limit, we have ¯ h), lim N −1 E log ZN (β, h, J ) ≡ α(β, h) ≤ α(β, N→∞

and

lim N −1 log ZN (β, h, J ) ≡ α(β, h) ≤ α(β, ¯ h),

N→∞

J -almost surely. The proof is long, and will be split in a series of intermediate results. Consider a generic piecewise constant functional order parameter x, as in (8), and define the auxiliary ˜ as follows partition function Z,   t  Z˜ N (β, h; t; x; J ) ≡ exp β Jij σi σj N σ1 ...σN (i,j )  K    √

a + βh σi + β 1 − t qa − qa−1 Ji σi . (16) i

a=1

i

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F. Guerra

Here, we have introduced additional independent centered unit Gaussian Jia , a = 1, . . . , K, i = 1, . . . , N. The interpolating parameter t runs in the interval [0, 1]. For a = 1, . . . , K, let us call Ea the average with respect to all random variables Jia , i = 1, . . . , N. Analogously, we call E0 the average with respect to all Jij , and denote by E averages with respect to all J random variables. Now we define recursively the random variables Z0 , Z1 , . . . , ZK , mK mK = EK ZK , . . . , Z0m1 = E1 Z1m1 , ZK = Z˜ N (β, h; t; x; J ), ZK−1

(17)

and the auxiliary function α˜ N (t) α˜ N (t) =

1 E0 log Z0 . N

(18)

Notice that, due to the partial integrations, any Za depends only on the Jij , and on the Jib with b ≤ a, while in α˜ all J noises have been completely averaged out. The basic motivation for the introduction of α˜ is given by Lemma 1. At the extreme values of the interpolating parameter t we have 1 E log ZN (β, h, J ), N α˜ N (0) = log 2 + f (0, h; x, β), α˜ N (1) =

(19) (20)

where f is as described in Sect. 3. The proof is simple. In fact, at t = 1, the Jia disappear, and Z˜ reduces to Z in (2). On the other hand, at t = 0, the two site couplings Jij disappear, while all effects of the Jia factorize with respect to the sites i. Therefore, we are essentially reduced to a one site problem, and it is immediate to see that the averages in (17) reduce to the Gaussian averages necessary for the computation of the solution of the antiparabolic problem (10), (11), as given by the repeated application of (12), with the f function evaluated at q = 0, and y = h. It is clear that now we have to proceed to the calculation of the t derivative of α˜ N (t). But we need some few additional definitions. Introduce the random variables fa , a = 1, . . . , K, Zama
, (21) fa = Ea Zama and notice that they depend only on the Jib with b ≤ a, and are normalized, E (fa ) = 1. Moreover, we consider the t-dependent state ω associated to the Boltzmannfaktor in (16), and its replicated . A very important role is played by the following states ω˜ a , ˜ a , a = 0, . . . , K, defined as and their replicated ones  ω˜ K (.) = ω(.), ω˜ a (.) = Ea+1 . . . EK (fa+1 . . . fK ω(.)) . Finally, we define the .a averages, through a generalization of (5),   ˜ a (.) . .a = E f1 . . . fa 

(22)

(23)

As it will be clear in the following, the .a averages are able, in a sense, to concentrate the overlap fluctuations around the value qa .

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

7

Now, we have all definitions in order to be able to state the following important results. Theorem 4. The t derivative of α˜ N (t) in (18) is given by d β2 α˜ N (t) = − dt 4 −

 1−

a=0

K β2 

4

K 

 (ma+1 − ma )qa2

(ma+1 − ma ) (q12 − qa )2 a .

(24)

a=0

Theorem 5. For any functional order parameter, of the type given in (8), the following sum rule holds: α(β, ¯ h; x) =

 1 K 1 β2  (q12 − qa )2 a (t) dt. E log ZN (β, h; J ) + (ma+1 − ma ) N 4 0 a=0 (25)

Clearly, Theorem 5 follows from the previous Theorem 4, by integrating with respect to t, taking into account the boundary values in Lemma 1, and the definition of α(β, ¯ h; x) given in Sect. 3. Moreover, one should use also the obvious identity    K 1  1 2 1− q x(q) dq. (ma+1 − ma ) qa = 2 0

(26)

a=0

By taking into account that all terms in the sum rule are nonnegative, since ma+1 ≥ ma , we can immediately establish the validity of Theorem 3. Now we must attack Theorem 4. The proof is straightforward, and involves integration by parts with respect to the external noises. We only sketch the main points. Let us begin with Lemma 2. The t derivative of α˜ N (t) in (18) is given by  1  d −1 ∂t ZK , α˜ N (t) = E f1 f2 . . . fK ZK dt N

(27)

where −1 −1 ˜ ZK ∂t ZK = Z˜ N ∂t Z N K  


β β  = √ Jij ω σi σj − √ qa − qa−1 Jia ω (σi ) . 2 1 − t a=1 2 tN (ij ) i

The proof is very simple. In fact, from the definitions in (17), we have, for a = 0, 1, . . . , K − 1,   −1 Za−1 ∂t Za = Ea+1 fa+1 Za+1 ∂t Za+1 . (28) The rest follows from iteration of this formula, and simple calculations.

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Clearly, now we have to evaluate K   



E . . . ∂Jij fa . . . ω σi σj E Jij f1 f2 . . . fK ω σi σj = a=1


 + E f1 . . . fK ∂Jij ω(σi σj ) , K    
  E Jia f1 f2 . . . fK ω(σi ) = E . . . ∂Jia fb . . . ω(σi ) + E f1 . . . fK ∂Jia ω(σi ) , b=1

where we have exploited standard integration by parts on the Gaussian J variables. The following lemma gives all additional information necessary for the proof of Theorem 4. Lemma 3. For the J -derivatives we have  
 t ∂Jij ω σi σj = β 1 − ω2 (σi σj ) , N   √

∂Jia ω(σi ) = β 1 − t qa − qa−1 1 − ω2 (σi ) , 
t ∂Jij fa = β ma fa ω˜ a (σi σj ) − ω˜ a−1 (σi σj ) , N ∂Jia fb = 0, if b < a, √

∂Jia fa = β 1 − t qa − qa−1 ma fa ω˜ a (σi ), √

∂Jia fb = β 1 − t qa − qa−1 mb fb (ω˜ b (σi ) − ω˜ b−1 (σi )) , if b > a.

(29) (30) (31) (32) (33) (34)

The proof of (29) and (30) is a standard calculation. On the other hand, Eq. (31) follows from the definition (21) and the easily established ∂Jij Zama = ma Zama Za−1 ∂Jij Za ,   −1 Za−1 ∂Jij Za = Ea+1 fa+1 Za+1 ∂Jij Za+1 , a = 1, . . . , K − 1,  t
−1 −1 ˜ ˜ ZK ∂Jij ZK = ZN ∂Jij ZN = β ω σi σj , N

  t t −1 Za ∂Jij Za = β Ea+1 fa+1 . . . fK ω(σi σj ) = β ω˜ a σi σj . N N In the same way, we can establish (32), (33), (34). But here we have to take into account that Zb does not depend on Jia if b < a. A careful combination of all information given by Lemma 2 and Lemma 3, finally leads to the proof of Theorem 4. On the other hand, the main Theorem 3 follows easily from Theorem 5, and the results of [9]. 5. Broken Replica Symmetry Bound for the Ground State Energy Let us consider the ground state energy density −eN (J, h) defined as −eN (J, h) ≡

ln ZN (β, h, J ) 1 inf HN (σ, h, J ) = − lim . σ β→∞ N βN

(35)

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

9

By taking the expectation values we also have eN (h) ≡ E (eN (J, h)) = lim

β→∞

x,

αN (β, h) . β

(36)

From the results of the previous section, we have, for any functional order parameter E (ln ZN (β, h, J )) ≤ β −1 α(β, ¯ h; x), βN

(37)

uniformly in N . Let us now introduce an arbitrary sequence 0≤m ¯1 ≤ m ¯2 ≤ ... ≤ m ¯ K,

(38)

and the corresponding order parameter x, ¯ defined as in (8), but with all ma replaced by m ¯ a . Notice that there is no upper bound equal to 1 for m ¯ K , and consequently for x. ¯ However, for sufficiently large β, we definitely have m ¯ K ≤ β. Therefore, we can take in (37) the order parameter x defined by x(q) = x(q)/β, ¯ with 0 ≤ x(q) ≤ 1. Then we can easily establish the following lemma: Lemma 4. In the limit β → ∞ we have lim β

−1

β→∞

1 α(β, ¯ h; x) = α(h; ˜ x) ¯ ≡ f¯(0, h; x) ¯ − 2



1

q x(q) ¯ dq,

(39)

0

where the function f¯, with values f¯(q, y; x) ¯ satisfies the antiparabolic equation
  1
2 ∂q f¯ (q, y) + f¯ (q, y) + x(q) ¯ f¯ (q, y) = 0, 2 with final condition

f¯(1, y) = |y|.

(40)

(41)

The proof is easy. In fact, the recursive solution for f , coming from (12), allows to prove immediately ¯ = f¯(q, y; x), ¯ (42) lim β −1 f (q, y; x/β) β→∞

by taking into account the elementary limβ→∞ β −1 log cosh(βy) = |y|. Therefore we have established Theorem 6. The following inequalities hold eN (h) ≤ α(h; ˜ x), ¯ eN (h) ≤ α(h) ˜ ≡ inf α(h; ˜ x), ¯ x¯

lim eN (h) ≡ e0 (h) ≤ α(h; ˜ x), ¯

N→∞

˜ e0 (h) ≤ α(h).

(43) (44) (45) (46)

A detailed study of the numerical information coming from the variational bound of Theorem 6 will be presented in a forthcoming paper [11].

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6. Broken Replica Symmetry Bounds in the p-Spin Model The methods developed in the previous sections can be easily extended to the Derrida p-spin model [2, 4, 3, 17]. We give here only a brief sketch. A more detailed treatment will be presented elsewhere [10]. Now the Hamiltonian contains a term coupling each group made of p spins

p! HN (σ, h, J ) = − 2N p−1

1  2

Ji1 ...ip σi1 · · · σip − h



σi .

(47)

i

(i1 ,...ip )

For the sake of simplicity, in the following we consider only the case of even p. Piecewise constant order parameters are introduced as in (7), (8), where now we assume qK = p/2. We still introduce the function f , defined by (10), with 0 ≤ q ≤ p/2, and final condition f (p/2, y) = log cosh(βy). (48) We also introduce the change of variables q → q, ¯ defined by 2q = p q¯ p−1 , so that, in particular, q¯K = 1. The definitions (14) and (15) must be modified as follows. Definition 3. The trial auxiliary function, associated to a given p-spin mean field spin glass system, as described before, depending on the functional order parameter x, is defined as β2 α(β, ¯ h; x) ≡ log 2 + f (0, h; x, β) − 2



p 2

q(q) ¯ x(q) dq.

(49)

0

Definition 4. The spontaneously broken replica symmetry solution for the p-spin model is defined by ¯ h; x), (50) α(β, ¯ h) ≡ inf α(β, x

where the infimum is taken with respect to all functional order parameters x. With the same procedure as described in Sect. 4, we arrive at the sum rule given by Theorem 7. In the p-spin model, for any functional order parameter, the following sum rule holds α(β, ¯ h; x) =

1 E log ZN (β, h; J ) N  1 K β2  p p−1 p + q12 − pq12 q¯a + (p − 1)q¯a a (t) dt (ma+1 − ma ) 4 0 a=0 1 +O , (51) N

where α(β, ¯ h; x) is defined in (49). Notice that the terms under the sum are still positive. The O of the p-spin models. From the sum rule we have also


1 N

correction is typical

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

11

Theorem 8. In the p-spin model, for any functional order parameter x, the following bound holds: 1 −1 N E log ZN (β, h, J ) ≤ α(β, ¯ h; x) + O , N where α(β, ¯ h; x) is defined in (49). Consequently, we have also N −1 E log ZN (β, h, J ) ≤ α(β, ¯ h) + O



1 N

,

where α(β, ¯ h) is defined in (50). Moreover, for the thermodynamic limit, we have ¯ h), lim N −1 E log ZN (β, h, J ) ≡ α(β, h) ≤ α(β,

N→∞

and ¯ h), lim N −1 log ZN (β, h, J ) ≡ α(β, h) ≤ α(β,

N→∞

J -almost surely. We refer to [10] for a more detailed treatment.

7. Conclusions and Outlook for Future Developments Without exploiting any reference to the zero replica trick, or to the cavity method, we have found a way to prove that the true free energy for the mean field spin glass model is bounded below by its spontaneously broken symmetry expression, given in the frame of the Parisi Ansatz. The method extends easily to the Derrida p-spin model. The key role is played by the auxiliary function α˜ N (t), defined in (18). Our method, in its very essence, is a generalization of the mechanical analogy introduced in [6], for the comparison with the replica symmetric approximation. As a direct application of the broken replica symmetry bound, Toninelli has shown that replica symmetry fails beyond the Almeida-Thouless line [18]. The main open problems are given by the extension of these methods to other disordered systems, as for example the mean field neural network models. Moreover, the sum rules developed here could be taken as the starting point to prove that the additional positive terms do really vanish in the infinite volume limit. This would prove rigorously the validity of the broken replica Ansatz. We plan to report on these problems in future papers. Acknowledgements. We gratefully acknowledge useful conversations with Romeo Brunetti, Enzo Marinari, and Giorgio Parisi. The strategy developed in this paper grew out of a systematic exploration of interpolation methods, developed in collaboration with Fabio Lucio Toninelli. This work was supported in part by MIUR (Italian Minister of Instruction, University and Research), and by INFN (Italian National Institute for Nuclear Physics).

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References 1. Baffioni, F., Rosati, F.: On the ultrametric overlap distribution for mean field spin glass models (I). Europ. Phys. J. B 17, 439–447 (2000) 2. Derrida, B.: Random energy model: An exactly solvable model of disordered systems. Phys. Rev. B 24, 2613–2626 (1981) 3. Gardner, E.: Spin glasses with p-spin interaction. Nucl. Phys. B 257, 747–765 (1985) 4. Gross, D.J., M´ezard, M.: The simplest spin glass. Nucl. Phys. B 240, 431–452 (1984) 5. Guerra, F.: Fluctuations and thermodynamic variables in mean field spin glass models. In: Stochastic Processes, Physics and Geometry, S. Albeverio et al. (eds), Singapore: World Scientific, 1995 6. Guerra, F.: Sum rules for the free energy in the mean field spin glass model. Fields Inst. Commun. 30, 161 (2001) 7. Guerra, F.: On the mean field spin glass model. In preparation 8. Guerra, F., Toninelli, F.L.: Quadratic replica coupling for the Sherrington-Kirkpatrick mean field spin glass model. J. Math. Phys. 43, 3704–3716 (2002) 9. Guerra, F., Toninelli, F.L.: The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230, 71–79 (2002) 10. Guerra, F., Toninelli, F.L.: Spontaneous replica symmetry breaking in the p-spin model. In preparation 11. Guerra, F., Toninelli, F.L.: About the ground state energy in the mean field spin glass model. In preparation 12. M´ezard, M., Parisi, G., Virasoro, M.A.: Spin glass theory and beyond. Singapore: World Scientific, 1987 13. M´ezard, M., Parisi, G., Virasoro, M.A.: SK Model: The Replica Solution without Replicas. Europhys. Lett. 1, 77 (1986) 14. Parisi, G.: A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A13, L-115 (1980) 15. Pastur, L., Shcherbina, M.: The absence of self-averaging of the order parameter in the SherringtonKirkpatrick model. J. Stat. Phys. 62, 1–19 (1991) 16. Sherrington, D., Kirkpatrick, S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35, 1792–1796 (1975) 17. Talagrand, M.: Spin Glasses: A Challenge for Mathematicians. Mean Field Models and Cavity Method. Berlin-Heidelberg-New York: Springer-Verlag, to appear 18. Toninelli, F.L.: About the Almeida-Thouless transition line in the Sherrington-Kirkpatrick mean field spin glass model. Europhy. Lett., 60, 764 (2002) Communicated by M. Aizenman

Commun. Math. Phys. 233, 13–26 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0787-z

Communications in

Mathematical Physics

Enhanced Binding in Non-Relativistic QED Christian Hainzl1,∗ , Vitali Vougalter2 , Semjon A. Vugalter1 1

Mathematisches Institut, LMU M¨unchen, Theresienstrasse 39, 80333 Munich, Germany. E-mail: [email protected]; [email protected] 2 Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada. E-mail: [email protected] Received: 7 January 2002 / Accepted: 8 November 2002 Published online: 13 January 2003 – © Springer-Verlag 2003

Abstract: We consider a spinless particle coupled to a photon field and prove that even if the Schr¨odinger operator p 2 + V does not have eigenvalues the system can have a ground state. We describe the coupling by means of the Pauli-Fierz Hamiltonian and our result holds in the case where the coupling constant α is small.

1. Introduction In the picture of Quantum electrodynamics (QED) atoms consist of charged particles, which are necessarily coupled to a photon field. If one neglects the radiation effects one obtains the standard Schr¨odinger operator. Although the fundamental properties of the one-particle and multi-particle Schr¨odinger operators have been successfully studied since the middle of the last century, the systematic mathematical study of the non-relativistic QED model was initiated by Bach, Fr¨ohlich, and Sigal in [BFS1, BFS2, BFS3] only a couple of years ago (a comprehensive review of results in non-relativistic QED can be found in [GLL]) and some very fundamental problems remains open still. One of these problems is the question of enhanced binding via interaction with a quantized radiation field. Consider a particle in a potential well βV (x) with V (x) ≤ 0. If the potential well is not deep enough, i.e. β is small, the corresponding Schr¨odinger operator does not have a discrete spectrum and binding does not occur. There exists a critical value β0 such that for β > β0 there is at least one bound state whereas for β ≤ β0 no particle can be bound. In a recent paper Griesemer, Lieb, and Loss ([GLL]) proved that a photon field cannot decrease the binding energy. If the Schr¨odinger operator with potential βV has an eigenvalue, the corresponding energy operator in non-relativistic QED (Pauli-Fierz Hamiltonian) has a ground state. ∗

Marie Curie Fellow

14

C. Hainzl, V. Vougalter, S. A. Vugalter

However, the physical intuition tells us that interaction with a photon field must increase binding. According to the photon cloud surrounding the particle, the effective mass of the electron increases and consequently it needs more energy to leave the potential well. The goal of this paper is to give a mathematical rigorous proof this phenomenon. Previously the enhanced binding was studied in the dipole approximation by Hiroshima and Spohn in [HS]. In this approximation it is assumed that the magnetic vector potential does not depend on the coordinates of the particle. They proved that, if the potential βV is fixed, for sufficiently large values of the coupling parameter α (which is the fine structure constant, see (1)), binding takes place. Our approach to this problem is different. On the one hand we study the Pauli-Fierz operator without additional restrictions on the magnetic vector potential, and on the other hand our results hold for small values of α (recall, that the physical value of α is about 1/137). We prove that in the case of the Pauli-Fierz operator (and for α small enough) binding starts at values of β strictly less than β0 . 2. Main Results We describe the self-energy of the particle by √ T = (p + αA(x))2 + Hf ,

(1)

acting on the Hilbert space H = L2 (R3 ) ⊗ F,

(2)

where F is the symmetric Fock space for the photon field. We use units such that  = c = 1 and the mass m = 21 . The electron charge is then √ given by e = α, with α ≈ 1/137 the fine structure constant. In the present paper α plays the role of a small, dimensionless number. Our results hold for sufficiently small values of α. The electron momentum operator is p = −i∇x , while A is the magnetic vector potential. We fix the Coulomb gauge divA = 0. The vector potential is
   χ (|k|) ελ aλ (k)eikx + aλ∗ (k)e−ikx dk, (3) A(x) = 1/2 R3 2π|k| λ=1,2

where the operators aλ , aλ∗ satisfy the usual commutation relations [aν (k), aλ∗ (q)] = δ(k − q)δλ,ν ,

[aλ (k), aν (q)] = 0.

(4)

The vectors ελ (k) ∈ R3 are two orthonormal polarization vectors perpendicular to k. Obviously, A(x) = D(x) + D ∗ (x), where D(x) =


 λ=1,2


 χ (|k|) ikx ε a (k)e dk = Gλ (k)aλ (k)eikx dk, 1/2 λ λ 3 R3 2π|k| R λ=1,2

and D ∗ is the operator adjoint to D.

(5)

(6)

Enhanced Binding in Non-Relativistic QED

15

The function χ (|k|) describes the ultraviolet cutoff for the interaction at large wavenumbers k. For convenience we choose χ to be the Heaviside function (! − |k|/ lC ), where lC = /(mc) is the Compton wavelength. In our units lC = 2. Our proof would work for any other cut-off. The photon field energy Hf is given by
 Hf = |k|aλ∗ (k)aλ (k)dk, (7) 3 λ=1,2 R

whereas Pf =


 λ=1,2

R3

kaλ∗ (k)aλ (k)dk

(8)

denotes the field momentum. To prove existence of enhanced binding in non-relativistic QED we would like to compare binding in the presence of the photon field and without it. To this end let us introduce the Schr¨odinger operator hβ = −& + βV (x)

(9)

with external potential βV (x) ∈ C(R3 ), which we assume to be radial V (x) = V (|x|), non-positive, V (x) ≤ 0, and with compact support. It is known that there is a critical value of the parameter β0 > 0 such that for β ≤ β0 there is no ground state and the operator (9) has only an essential spectrum and at the same time for all β > β0 the operator hβ has at least one eigenvalue. The corresponding operator with a quantized radiation field is Hβ = T + βV (x).

(10)

[Hi] guarantees the self-adjointness of Hβ on the domain D(p 2 + Hf ). Our goal is to show that the operator Hβ has a bound state for values of β strictly smaller than β0 . To establish the existence of a ground state of Hβ we apply the criterion of [GLL], which says that Hβ has a ground state if inf spec Hβ < inf spec T .

(11)

However, in contrast to the Schr¨odinger operator hβ , for which the infimum of the spectrum without potential is always 0, the inf spec T is a complicated function depending on α and !. To prove the inequality (11) one needs precise estimates on this function. Our first result is the following asymptotic estimate on (α = inf spec T . Theorem 1 (Localization of the spectrum of a free spinless particle). Let E0 = 0|D · D[Pf2 + Hf ]−1 D ∗ · D ∗ |0, with D ∗ = D ∗ (0) and |0 denotes the vacuum of F. Then, for small α,     (α − απ −1 !2 + α 2 E0  ≤ Cα 3 !4 , where C > 0 is an appropriate constant independent of α and !.

(12)

(13)

16

C. Hainzl, V. Vougalter, S. A. Vugalter

Remark 1. The number E0 can be computed directly through the integral  µ 2 
G (k1 ) · Gν (k2 ) E0 = 2 dk1 dk2 . |k1 + k2 |2 + |k1 | + |k2 |

(14)

µ,ν=1,2

The first to leading order is obtained by perturbation theory. One of our main goals here is to prove that perturbation theory is correct in the case when ! is fixed, which is a non-trivial problem, since there is no isolated eigenvalue and Kato’s perturbation methods cannot be applied. Recall that the operator hβ has a critical value β0 of the parameter β such that, for β ≤ β0 , hβ does not have a bound state. Using Theorem 1 we construct a variational trial function proving for small values of α the following: Theorem 2 (Enhanced binding). For all sufficiently small α there exists a number β1 (α) < β0 , such that for all β > β1 (α) the operator Hβ has a ground state. Observe that the converse statement is not proven, namely we do not obtain a β2 (α) > 0 such that for β < β2 (α) the ground state does not exist. Remark 2. Concerning the critical case β = β0 the proof of Theorem 2 in particular implies that there exists a real number ρ > 0 such that Hβ0 has a ground state for all α ∈ (0, ρ]. Of course we expect that binding holds on, or even increases, when α gets large, but we cannot prove it due to the fact that we can only control the self-energy for small α. 3. Proof of Theorem 1 Let us start with a free spinless electron. In this case the Hamiltonian is translation invariant, which means that it commutes with the total momentum p + Pf . It is therefore possible to rewrite the Hilbert space and the Hamiltonian as a direct integral  ⊕ d 3 P HP (15) H= R3

and

 T =

⊕

R3

d 3 P TP ,

(16)

with TP acting on HP . Each HP is isomorphic to F. In this representation TP is given by  2 √ TP = P − Pf + αA(0) + Hf . (17) According to [F] the minimum of inf spec TP is achieved for P = 0, which tells us that we only need to consider the operator  √ 2 T0 = Pf + αA + Hf . (18) Throughout this section we use A = A(0), D = D(0), and D ∗ = D ∗ (0). We define (α = inf spec T0 .

Enhanced Binding in Non-Relativistic QED

17

It turns out to be convenient to denote a general + ∈ H as + = {ψ0 , ψ1 , . . . , ψn , . . . },

(19)

ψn = ψn (x, k1 , . . . , kn ; λ1 , . . . , λn ).

(20)

where

In order not to overburden the paper with too many indices we will suppress the photon variables in ψn , when it does not lead to misunderstanding.

3.1. Upper bound. We take the trial state + = {|0, 0, −α[Pf2 + Hf ]−1 D ∗ · D ∗ |0, 0, 0 . . . },

(21)

where |0 ∈ C, 0|0 = 1, denotes the vacuum vector. The photon part of + can be written explicitly as −




√ α 2

1 Gλ (k1 ) · Gµ (k2 )|0. (k1 + k2 )2 + |k1 |2 + |k2 |

(22)

D · D ∗ − D ∗ · D = π −1 !2 ,

(23)

A2 = π −1 !2 + 2D ∗ · D + D · D + D ∗ · D ∗ .

(24)

λ,µ=1,2

Since

obviously

Therefore, since +2 ≥ 1, 

 +, T0 + /(+, +) ≤ απ −1 !2 − α 2 0|D · D[Pf2 + Hf ]−1 D ∗ · D ∗ |0 + 2α 3 D[Pf2 + Hf ]−1 D ∗ · D ∗ |02 .

(25)

One can easily see by scaling that the last two terms in the r.h.s. of (25) are of order !2 . 3.2. Lower bound1 . We start with some a priori estimates. Lemma 1. T0 ≥ απ −1 !2 − const.α 2 !3 + 21 (Pf2 + Hf ).

(26)

1 A different proof of the lower bound, based on partitions of unity of the photon configuration space and improved estimates for the localization errors for the relativistic energy, can be found in the preprint version [HVV].

18

C. Hainzl, V. Vougalter, S. A. Vugalter

Proof. Since [Pf , A] = 0, √ T0 = Pf2 + 2 αPf · A + αA2 + Hf .

(27)

By means of Schwarz’s inequality, √ √ 2 αPf · A = 4 α(Pf · D) ≤ 21 Pf2 + 8αD ∗ D

(28)

and α(D · D + D ∗ · D ∗ ) ≤ C −1 D ∗ · D + α 2 CD · D ∗

(29)

for any C > 0. Using (24) we obtain T0 ≥ (π −1 α!2 −Cπ α 2 !2 )+Hf



8α! −1 2 1 2 − π −C ! π

−α 2

C 1 2 ! + 2 (Pf +Hf ), π (30)

which implies the lemma with C = c!, ¯ with an appropriate c¯ > 0, and α! and α not too large.   Remark 3. We know from the upper bound that any approximate ground state +0 satisfies (+0 , T0 +0 ) ≤ απ −1 !2 + O(α 2 ). Therefore by Lemma 1 we infer the a priori estimate   (31) +0 , [Pf2 + Hf ]+0 ≤ const.α 2 !3 . Using (24) we derive (+0 , T0 +0 ) ≥ απ −1 !2 +0 2 +

∞


E[ψn , ψn+1 , ψn+2 ],

(32)

n=0

where E[ψn , ψn+1 , ψn+2 ]

   √ = (ψn+2 , Aψn+2 ) + 2 2 αPf · D ∗ ψn+1 + αD ∗ · D ∗ ψn , ψn+2 ,

(33)

A = Pf2 + Hf .

(34)

+0 = {ψ0 , ψ1 (k1 ), ...., ψn (k1 , . . . , kn ), ...}.

(35)

with

Recall, in our notation

We consider the term E[ψn , ψn+1 , ψn+2 ]. If we set √  f = A1/2 ψn+2 , g = −A−1/2 α2Pf · D ∗ ψn+1 + αD ∗ · D ∗ ψn

(36)

then by means of f 2 − 2(f, g) ≥ −g2 we derive

2

√



E[ψn , ψn+1 , ψn+2 ] ≥ − 2 αA−1/2 Pf · D ∗ ψn+1 + αA−1/2 D ∗ · D ∗ ψn .

(37)

Enhanced Binding in Non-Relativistic QED

19

Let us start the estimation of the r.h.s. of (37) with −α 2 (ψn , D · D[Pf2 + Hf ]−1 D ∗ · D ∗ ψn ).

(38)

It will be shown that it produces the first to leading order in α 2 . Recall [D ∗ · D ∗ ψn ]n+2 = √

n+2
n+2

1 Gµ (kj ) · Gλ (ki ) (n + 2)(n + 1) λ,µ=1,2 j =1 i=1 i=j

× ψn (k1 , . . . , k j , . . . , k i , . . . , kn+2 ),

(39)

where k j indicates that the j th variable is omitted. By permutational symmetry we can distinguish between three different terms,   ψn , D · D[Pf2 + Hf ]−1 D ∗ · D ∗ ψn = In + I In + I I In , (40) which come out quite naturally when we insert Eq. (39) into (40) and have in mind that the l.h.s. of (40) can be written as  ∗  D · D ∗ ψn , [Pf2 + Hf ]−1 D ∗ · D ∗ ψn . (41) First, the diagonal part In appears, when in the right-hand side of (41) as well as in the left hand side two photons Gµ (kj ) · Gλ (ki ) with the same variables ki , kj are produced, 2   λ
G (k1 ) · Gµ (k2 ) |ψn (k3 , . . . , kn+2 )|2 dk1 . . . dkn+2 . 2 (42) In =  n+2 2 n+2  ki  + |ki | λ,µ=1,2

i=1

i=1

 2  2 n+2   n+2      If we set Q =  n+2 i=3 ki + k1 + k2 + i=1 |ki | and b = 2 i=3 ki · k1 + k2 and use the expansion 1 1 1 1 1 1 1 = − b + b b , Q+b Q Q Q Q Q+b Q

(43)

then we see that the second term vanishes when integrating over k1 , k2 . Therefore, with 2  Q ≥ k1 + k2  + |k1 | + |k2 | and Q + b ≥ |k1 | + |k2 | we arrive at  λ 2 
G (k1 ) · Gµ (k2 ) 2 dk1 dk2 2 ψn  In ≤ |k1 + k2 |2 + |k1 | + |k2 | λ,µ=1,2  λ       G (k1 )2 Gµ (k2 )2 |k1 | + |k2 | 2 +4  2 |k1 + k2 |2 + |k1 | + |k2 | (|k1 | + |k2 |)  n+2 2


 ×  ki  |ψn (k3 , . . . , kn+2 )|2 dk1 . . . dkn+2 i=3

≤ 0|D · D[Pf2 + Hf ]−1 D ∗ · D ∗ |0ψn 2 + const.!Pf ψn 2 . For convenience we define the operator |D| by
 |D| = |Gλ (k)|aλ (k)dk. λ=1,2

(44)

(45)

20

C. Hainzl, V. Vougalter, S. A. Vugalter

|D|∗ denotes the operator adjoint. Obviously, [GLL, Lemma A. 4] still holds for |D|, namely |D|∗ |D| ≤

2 Hf . π

(46)

The second term I In occurs when a term Gµ (kj ) · Gλ (ki ) in the l.h.s. of (41) meets a two photon part Gµ (kj ) · Gλ (kl ) in the r.h.s. of (41). Using Pf2 + Hf ≥ Hf we evaluate     
 Gλ (k1 )Gµ (k2 )Gλ (k1 )Gµ (kn+2 ) I In ≤ (n + 1) n+2 i=1 |ki | λ,µ=1,2 × |ψn (k3 , . . . , kn+2 )||ψn (k2 , . . . , kn+1 )|dk1 . . . dkn+2      |G(k1 )|2 ≤ const. dk1 ψn , |D|∗ |D|ψn ≤ const.!2 ψn , Hf ψn . |k1 |

(47)

Finally, the third term, where the indices of produced photons in the right-hand side differ completely from the indices in the left-hand side of (41), can be bounded by     
 Gλ (k1 )Gµ (k2 )Gλ (kn+1 )Gµ (kn+2 ) I I In ≤ (n + 1) n+2 i=1 |ki | λ,µ=1,2 2

× |ψn (k3 , . . . , kn+2 )||ψ(k1 , . . . , kn )|dk1 . . . dkn+2     −1/2 −1/2 ≤ const. ψn , |D|∗ Hf |D|∗ |D|Hf |D|ψn ≤ const.! ψn , Hf ψn ,

(48)

where we used n+2
i=1

n+1 1/2 n+2 1/2 
 
     |ki | ≥  |ki |  |ki | ,     i=1

(49)

i=2

the fact that we can write  n −1/2
 −1/2 ψn (k1 , . . . , kn ) = |ki | ψn (k1 , . . . , kn ), Hf

(50)

i=1

and (46). We summarize   −α 2 ψn , D · DA−1 D ∗ · D ∗ ψn ≥ −α 2 0|D · D[Pf2 + Hf ]−1 D ∗ · D ∗ |0ψn 2  − const.! Pf ψn 2 + !(ψn , Hf ψn ) . (51)

Enhanced Binding in Non-Relativistic QED

21

The second diagonal term of (37) reads − α(Pf · D ∗ ψn+1 , A−1 Pf · D ∗ ψn+1 )  n+1  2  Gλ (kn+2 ) · |ψn+1 (k1 , . . . , kn+1 )|2
i=1 ki = dk1 . . . dkn+2 −α    n+2 2 n+2 λ=1,2  i=1 ki  + i=1 |ki |   n+2  Gλ (k1 ) · n+2 ki Gλ (kn+2 ) · i=1 i=1 ki + (n + 1)    n+2 2 n+2  i=1 ki  + i=2 |ki |

× ψn+1 (k1 , . . . , kn+1 )ψn+1 (k2 , . . . , kn+2 )dk1 . . . dkn+2  ≥ −const.α !Pf ψn+1 2 + (ψn+1 , |D|∗ |D|ψn+1 ) . (52) For the second term in the r.h.s. we used first    n+2 2  i=1 ki  ≤ 1.    n+2 2 n+2 k + |k |  i=1 i  i=2 i

(53)

By (46) and Schwarz’s inequality for the off-diagonal term in (37), as well as summing over all n and using the a priori knowledge (31) we arrive at the desired result. 4. Proof of Theorem 2 To prove the theorem we will check the binding condition of [GLL] for β = β0 . Namely, we will show that inf spec Hβ0 < (α − δα 2 + O(α 5/2 ).

(54)

The binding for all β ∈ (β1 , β0 ] with some β1 < β0 follows from (54) and the continuity of the quadratic form in β. In the proof of Theorem 1 we have seen that the trial state +n = {|0, 0, α[Pf2 + Hf ]−1 D(0)∗ · D(0)∗ |0, 0, 0, ..},

(55)

recovers the self energy up to the order α 2 . Our next goal is to modify this trial state in such a way that for the modified state + 0 ∈ H, (+ 0 , Hβ0 + 0 ) ≤ ((α − δα 2 + O(α 5/2 ))+2 ,

(56)

with some δ > 0. Throughout the previous section we worked with the operator A(0). Here our Hamiltonian depends on the electron variable x. In order to adapt our methods developed in the previous section we introduce the unitary transform U = eiPf ·x

(57)

acting n on the Hilbert space H. Applied to a n-photon function ϕn we obtain U ϕn = ei( i=1 ki )·x ϕn (x, k1 , . . . , kn ) and additionally U (D ∗ (x)ψ(x)) = G(k)ψ(x).

22

C. Hainzl, V. Vougalter, S. A. Vugalter

Since UpU ∗ = p − Pf we infer for our Hamiltonian Hβ0 , √ ¯ β0 , U Hβ0 U ∗ = (p − Pf + αA)2 + Hf + β0 V (x) ≡ H

(58)

with A = A(0). Obviously, ¯ β0 = inf spec Hβ0 . inf spec H

(59)

¯ β0 . Therefore, for convenience, we will work in the following with the operator H Next, we define our trial function √ + 0 = {f, −d αA−1 p · D ∗ f, −αA−1 D ∗ · D ∗ f, 0, 0, ...}, (60) with A = Pf2 + Hf , D = D(0), and d an appropriate constant which will be chosen later. We assume f (x) ∈ C02 (R3 ) to be a real, spherically symmetric function and to fulfill the condition √ p 2 f (x) ≤ C1 pf (x) ≤ C2 αf (x), (61) with some constants C1,2 . For short, denote the 1- and 2- photon terms in + 0 as ψ1 respectively ψ2 . Obviously, the terms (ψ1 , Pf · pψ1 ) and (ψ2 , Pf · pψ2 ) vanish. This can be seen by integrating over the field variables, keeping in mind that the reflection k → −k commutes with A. By means of (61) and Schwarz’s inequality we obtain  √  2 α(p · D ∗ ψ1 , ψ2 ) + |(ψ2 , p2 ψ2 )| ≤ + 0 2 O(α 5/2 ). (62) We now use our knowledge from the proof of Theorem 1 to obtain   απ −1 !2 + 0 2 +(ψ2 , [Pf2 +Hf ]ψ2 )+2α(D ∗ ·D ∗ f, ψ2 ) = (α +O(α 3 ) + 0 2 . (63) Taking into account that V ≤ 0 we arrive at  0  ¯ β0 + 0 ≤ (f, [p 2 + β0 V ]f ) − dα(f, p · DA−1 p · D ∗ f ) + ,H

+ αd 2 (f, p · DA−1 p · D ∗ f ) + (f, p · DA−1 p 2 A−1 p · D ∗ f ) + [(α + O(α 5/2 )]f 2 . Using the Fourier transform we are able to evaluate explicitly
 [Gλ (k) · l]2 (f, p · DA−1 p · D ∗ f ) = |fˆ(l)|2 dkdl = C3 pf 2 |k|2 + |k|

(64)

(65)

λ=1,2

and additionally get (f, p · DA−1 p 2 A−1 p · D ∗ f ) = C3 p 2 f 2 ≤ C4 pf 2 ,

(66)

where we used (61). This implies   0   ¯ β0 + 0 ≤ 1 − C3 αd + d 2 α(C3 + C4 ) (f, p 2 f ) + (f, β0 Vf ) + ,H + [(α + O(α 5/2 )]+ 0 2 .

(67)

Enhanced Binding in Non-Relativistic QED

As the next step we choose d <

23

C3 2(C3 +C4 )

which gives

  ¯ β0 + 0 ) ≤ (1 − να)pf 2 + β0 (f, Vf ) + (α + O(α 5/2 ) +0 2 , (+ 0 , H

(68)

where ν=

C32 . 4(C3 + C4 )

(69)

Due to our choice of β0 , obviously the operator −(1 − να)& + β0 V (x)

(70)

has at least one negative eigenvalue. However, the r.h.s. of (68) contains the terms which are of order O(α 5/2 ), and to prove Theorem 2 we have to provide more precise estimates on the negative eigenvalues of (70). The required estimate is given by Lemma 2 in the Appendix. Applying this lemma with να = γ completes the proof of the theorem. Appendix A. Auxiliary Lemma Lemma 2. Let β0 be the critical value (the maximal value of the constant β, for which the Schr¨odinger operator with the potential βV does not have a discrete spectrum). Then inf spec {−(1 − γ )& + β0 V (|x|)} < −δγ 2 ,

(71)

for some δ > 0 and γ small enough. Moreover, there exists a function fγ (x), real, spherically symmetric and satisfying condition (61), with constants C1,2 independent of γ , such that (1 − γ )∇fγ 2 + β0 (fγ , V (|x|)fγ ) < −δγ 2 fγ 2 .

(72)

Proof. Let us start by recalling some properties of the operator hβ = −& + βV (x),

(73)

V (x) ≤ 0, radial, and compactly supported, with critical value β = β0 . For β = β0 the operator hβ has a so-called virtual level or zero-resonance. It means that the equation −&ψ + β0 V (x)ψ = 0

(74)

has a generalized spherically symmetric solution ψ˜ with the following properties [VZ]: (i) Let B be a closure of the space C0∞ (R3 ) in the norm ψB = ∇ψ. Then ψ˜ ∈ B. From this point we assume that ψ˜ is a normalized solution in the sense that ˜ B = 1. Notice that ψ˜ ∈ L2 (R3 ), but ψ˜ ∈ ψ / L2 (R3 ). loc (ii) −&ψ˜ ∈ L2 (R3 ) and V (x)ψ˜ ∈ L2 (R3 ). (iii) Outside the support of V (x) ˜ ψ(x) = C|x|−1 holds.

(75)

24

C. Hainzl, V. Vougalter, S. A. Vugalter

The last property follows immediately from the fact that outside the support of V (x) a radial solution of (74) can be written as c1 |x|−1 + c2 , and ψ˜ ∈ B implies c2 = 0. Now we proceed directly to the proof Lemma 2. Let u ∈ C02 (R3 ), u(x) ≤ 1, u(x) = 1 for |x| ≤ 1, u(x) = 0 for |x| ≥ 2

(76)

˜ ˜ fn (x) = ψ(x)u(|x|γ n−1 )ψ(x)u(|x|γ n−1 )−1 .

(77)

and set

Obviously fn (x) = 1 and for large n, ˜ ˜ |x|≤2γ −1 n ∇fn (x) ≤ ψ(x)u(|x|γ n−1 )−1 {∇ ψ +C|x|−1 γ −1 n≤|x|≤2γ −1 n max[|∇u(|x|γ n−1 )|]} ˜ ˜ + c(γ −1 n)1/2 γ n−1 } ≤ ψ(x)u(|x|γ n−1 )−1 {∇ ψ −1 −1 ˜ ≤ 2ψ(x)u(|x|γ n ) .

(78)

˜ = C|x|−1 for |x| ≥ a0 Assume V (x) is supported in a ball of radius a0 . Then ψ(x) and  ˜ d|x| ψ(x)u(|x|γ n−1 )2 ≥ 4πC 2 = 4πC

2

a0 ≤|x|≤2γ −1 n (2γ −1 n − a0 )

≥ 4C 2 π γ −1 n

(79)

for n ≥ aγ0 . The inequalities (78) and (79) imply the second relation in (61) with the constant C2 independent of n and γ . To check the first inequality in (61) let us estimate  2  21  3 3 





∂ ψ˜ ∂u



˜ ˜ + n−1 ) ≤ &ψ dx 

& ψ(x)u(|x|γ ∂xi ∂xi  +

i=1

i=1

C &u(|x|γ n−1 )dx |x|2

1 2

.

(80)

According to (ii) the first term on the r.h.s. of (80) is bounded. The second term is also ˜ = 1) and the last term is bounded, since |∇u(|x|γ n−1 )| ≤ const., (recall that ∇ ψ 2a0 also bounded by a constant for n ≥ γ . Finally we arrive at





˜ ˜ n−1 ) ≤ C1 ∇ ψ(x),

& ψ(x)u(|x|γ

(81)

&fn (x) ≤ C1 ∇fn (x).

(82)

which implies

To prove the lemma it suffices now to show that for large n (1 − γ )∇fn 2 + β0 (fn , Vfn ) ≤ −δγ 2 fn 2 ,

(83)

Enhanced Binding in Non-Relativistic QED

25

with some δ > 0 independent of γ . This is equivalent to ˜ V ψ) ˜ ≤ −δγ 2 ψ(x)u(|x|γ ˜ ˜ n−1 )2 . (1 − γ )∇(ψ(x)u(|x|γ n−1 ))2 + β0 (ψ,

(84)

Recall that 4 ˜ ψu(|x|γ n−1 )2 ≤ c3 πa03 + 8πC 2 3



2γ −1 n a0

d|x| ≤ 2c4 γ −1 n,

(85)

˜ and for large n, where c3 = max|x|≤a0 |ψ(x)|, ˜ V ψ) ˜ = 0, ˜ 2 + β0 (ψ, ∇ ψ

(86)

which implies ˜ V ψ) ˜ ˜ (1 − γ )∇(ψ(x)u(|x|γ n−1 ))2 + β0 (ψ, 2 2 ˜ ˜ ˜ 2 − ∇(ψu) ˜ ≤ −γ [(∇ ψ)u − ψ∇u] + [∇ ψ ] 2  1 ˜ − Cγ 1/2 n−1/2 + 3∇ ψ ˜ 2 ∇ ψ ≤ −γ |x|≥γ −1 n 2 2 ˜ +2ψ|∇u| −1 −1 . γ

n≤|x|≤2γ

n

(87)

For n large we have ˜ 2 i) ∇ ψ = 4πC 2 |x|≥γ −1 n



∞ γ −1 n

|x|−2 d|x| = 4π C 2 γ n−1 ,

2 ˜ ≤ c5 γ 2 n−2 4π ii) ψ|∇u| γ −1 n≤|x|≤2γ −1 n

iii) Cγ 1/2 n−1/2 <



2γ −1 n γ −1 n

d|x| = c6 γ n−1 ,

1 1 ˜ = ∇ ψ, 4 4

(88) (89)

which implies together with (87), ˜ V ψ) ˜ ˜ − (1 − γ )∇(ψ(x)u(|x|γ n−1 ))2 + β0 (ψ, ≤ −γ /4 + 3C 2 γ n−1 + 2c6 γ n−1 ≤ −γ /8 ≤ −

γ2 ˜ ψu(|x|γ n−1 ))2 . 32C 2 π n

(90)

To complete the proof of the lemma it suffices now to choose n so large that (90) holds true (notice, that it can be done uniformly in γ for γ ≤ 1) and for this n take 1 −2 −1 −1 C π n , where C is the constant in (79), which depends on the zero-resoδ = 32 nance solution ψ˜ only.   Acknowledgement. The work was partially supported by the European Union through its Training, Research, and Mobility program FMRX-CT 96-0001. C. Hainzl has been supported by a Marie Curie Fellowship of the European Community programme “Improving Human Research Potential and the Socio-economic Knowledge Base” under contract number HPMFCT-2000-00660. C. H. and S. A. V. thank Robert Seiringer for many valuable comments.

26

C. Hainzl, V. Vougalter, S. A. Vugalter

References [BFS1] [BFS2] [BFS3] [F] [GLL] [HVV] [HS] [Hi] [VZ]

Bach, V., Fr¨ohlich, J., Sigal, I.M.: Mathematical theory of non-relativistic matter and radiation. Lett. Math. Phys. 34, 183–201 (1995) Bach, V., Fr¨ohlich, J., Sigal, I.M.: Quantum electrodynamics of confined non-relativistic particles. Adv. Math. 137, 299–395 (1998) Bach, V., Fr¨ohlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207, 249–290 (1999) Fr¨ohlich, J.: Existence of dressed one-electron states in a class of persistent models. Fort. der Phys. 22, 159–198 (1974) Griesemer, M., Lieb, E.H., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145, 557–595 (2001) Hainzl, Ch., Vougalter, V., Vugalter, S.A.: Enhanced binding in non-relativistic QED. mp arc 01–455 Hiroshima, F., Spohn, H.: Enhanced Binding through coupling to a quantum Field. Ann. Henri Poincar´e 2, 1159 (2001) Hiroshima, F., Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of coupling constants. Ann. Henri Poincar´e 3, 171 (2002) Vugalter, S.A., Zhislin, G.M.: The symmetry and Efimov’s effect in systems of three-quantum particles. Commun. Math. Phys 87(1), 89–103 (1982/1983)

Communicated by H. Spohn

Commun. Math. Phys. 233, 27–48 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0757-5

Communications in

Mathematical Physics

Delocalization in Random Polymer Models S. Jitomirskaya1 , H. Schulz-Baldes2 , G. Stolz3 1 2 3

Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA. Fachbereich Mathematik, Technische Universit¨at Berlin, 10623, Berlin, Germany. Department of Mathematics, University of Alabama at Birmingham, Birmingham, Al, 35294 USA.

Received: 2 January 2002 / Accepted: 10 September 2002 Published online: 19 December 2002 – © Springer-Verlag 2002

Abstract: A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.

1. Introduction Until quite recently, common wisdom was that one-dimensional random Schr¨odinger operators are in a strong localization phase and that there is nothing else of any interest to be discovered. In 1990, Dunlap, Wu and Phillips [DWP] studied the random dimer model. It is similar to the conventional Bernoulli-Anderson model with a Hamiltonian given by the sum of the discrete Laplacian and a random potential taking only two values, except that these potential values now always come in neighboring pairs (dimers). For suitable values of the parameters, this model has so-called critical energies at which the two transfer matrices across the dimers commute. This leads to a vanishing of the Lyapunov exponent and a divergence of the localization length at these energies. They then argued and showed numerically that the second moment of the position operator X on the lattice grows superdiffusively under the dynamics like
ψ|X 2 (t)|ψ ≈ C t 3/2 for a localized initial state ψ and any typical dimer configuration. This is likely to be responsible for the high conductivities of certain organic polymer chains [PW] and quasi-1D semiconductor superlattices [PTB].

28

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

This work considers Jacobi matrices Hω randomly composed of two building blocks modeling two different polymers. The associated polymer transfer matrices are supposed to commute at an energy lying within the spectra of the periodic operators containing only one of the polymers. It is proven that for almost every polymer configuration ω and every α > 0 there is a positive constant Cα such that
T 1 dt
0|eıH t |X|q e−ıH t |0 ≥ Cα T q− 2 −α , (1) 0 T where |0 is the state localized at the origin. For q = 2, this rigorously confirms that the heuristics of [DWP] (discussed below) provide a correct lower bound on transport. To prove the corresponding upper bound for critical energies of order 1 in the sense of Definition 2 remains an open problem, but we believe the quantitative lower bound on the Lyapunov exponent (Theorem 2) to be a central ingredient. Moreover, it is shown that for every configuration the l.h.s. is greater than or equal to C T q−1 for some C > 0. Note that (1) implies that the conductivity is infinite either at finite temperature or if the critical energy is at the Fermi level [SB]. The above results should be confronted with the fact that the spectrum of a random polymer model is almost surely pure-point with exponentially localized eigenfunctions. For the related Bernoulli-Anderson model, such spectral localization results were first proven in [CKM], later on in [SVW]. More recently, the random dimer model and the continuous Bernoulli-Anderson model were treated in [BG] and [DSS1] respectively. These works also established dynamical localization on energy intervals not containing a discrete set of special energies which includes the above critical energies. While [DSS1] considers continuum models, its approach can be carried over to prove spectral localization and dynamical localization away from the set of special energies for the polymer models studied here, see [DSS2] for the more general case of an arbitrary number of building blocks of bounded length. The fact that spectral localization can in principle coexist with quantum transport (even almost ballistic; note that ballistic is impossible with pure point spectrum [Sim]) was demonstrated by an example in [RJLS] (see also [BT]). However, those examples were rather artificial and much research since then was devoted to the program of proving dynamical localization (i.e. boundedness in time of the left-hand side of (1)) in models of physical interest with previously established spectral localization, by means of upgrading the proof of pure point spectrum (see [GK] and references therein, also [BJ]). The success of this program may have raised doubts as to the validity of the distinction between spectral and dynamical localization in physically relevant contexts. This paper demonstrates that the distinction should indeed be made as it shows that exponential localization and quantum transport coexist also in physical models. Let us sketch the heuristics of [DWP] leading to (1). It is known [Bov] and proven below that the Lyapunov exponent generically vanishes quadratically like γ (Ec + ) = c 2 + O( 3 ) in the vicinity of the critical energy Ec . The extension of the eigenstates in an -neighborhood of Ec is given by their localization length equal to the inverse of the Lyapunov exponent. Therefore the portion of the initial wave packet lying energetically in this -neighborhood spreads out ballistically up to time scales T ≈  −2 . Because it will be shown that the density of states is positive at Ec , this portion of states is proportional to . Consequently, the q th moment of the position operator should grow like T q multiplied by this factor  ≈ T −1/2 , showing that (1) should hold with high probability. The main technical tools are adequate action-angle variables, also called modified Pr¨ufer variables in the mathematical literature. Adapting techniques from [PF], one

Delocalization in Random Polymer Models

29

obtains perturbative expansions around the critical energy of both the density of states and the Lyapunov exponent, which prove positivity of the density of states and quadratic vanishing of the Lyapunov exponent near a generic critical energy. The proof of (1) requires an additional large deviations analysis for the Lyapunov exponent. The methods of proof are calculatory, quantitative and optimal. For example, they allow to show how large the moments of the position operator have to be if the commutator of the transfer matrices is small, but does not vanish. 2. Model and Main Results Let tˆ± = (tˆ± (0), . . . , tˆ± (L± − 1)) and vˆ± = (vˆ± (0), . . . , vˆ± (L± − 1)) be two pairs of finite sequences of real numbers, satisfying tˆ± (l) > 0 for all l = 0, . . . , L± − 1. These numbers are the hopping and potential terms of two different polymers. A family of random Jacobi matrices is now constructed by random juxtaposition of these polymers. More precisely, to any sequence ω = (ωl )l∈Z of signs + and − one associates sequences tω = (tω (n))n∈Z and vω = (vω (n))n∈Z by means of tω = (. . . , tˆω0 , tˆω1 , . . . ) and vω = (. . . , vˆω0 , vˆω1 , . . . ). An exact definition of the underlying probability space (, P), which also requires to randomize the position of vˆω0 (0) and tˆω0 (0), is given in Sect. 4.1. The polymer Hamiltonian Hω of the configuration ω is then defined by (Hω ψ)(n) = −tω (n + 1)ψ(n + 1) + vω (n)ψ(n) − tω (n)ψ(n − 1) ,

ψ ∈ 2 (Z) , (2)

and (Hω )ω∈ becomes a family of random operators if the signs are chosen with probabilities p+ and p− = 1 − p+ respectively. The polymer transfer matrices T±E at energy E ∈ R are introduced by   1 v −t 2 E . T± = Tvˆ± (L± −1)−E,tˆ± (L± −1) . . . Tvˆ± (0)−E,tˆ± (0) , where Tv,t = t 1 0 (3) The transfer matrices over several polymers are then TωE (k, m) = TωEk−1 · TωEk−2 · . . . · TωEm ,

k>m,

(4)

and TωE (k, m) = TωE (m, k)−1 if k < m, TωE (m, m) = 1. The Lyapunov exponent at energy E, also called inverse localization length, is then almost surely defined by (some more details are given in Sects. 3.4 and 4.1)   1   γ (E) = lim (5) log TωE (k, 0) , k→∞ k
L±  where
c±  = p+ c+ + p− c− for any complex numbers c± . Vanishing of the Lyapunov exponent is considered an indicator for possible delocalization. For a polymer chain, this happens in the following situation: Definition 1. An energy Ec ∈ R is called critical for the random family (Hω )ω∈ of polymer Hamiltonians if the polymer transfer matrices T±Ec are elliptic (i.e. |Tr(T±Ec )| < 2) or equal to ±1 and commute [T−Ec , T+Ec ] = 0 .

(6)

30

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

Remark 1. The definition does not allow the critical energy to be in a spectral gap or at the band edges of one of the periodic operators (constructed from (tˆ+ , vˆ+ ) and (tˆ− , vˆ− ) respectively) except for points of band touching (where the transfer matrix is ±1). Remark 2. The condition (6) contains 4 equations. Given a model, one can only vary the energy. Hence, in the space of polymer models existence of critical energies is a non-generic property. On the other hand, given an energy Ec , it is always possible to construct polymer models that have Ec as a critical energy. Examples. If L± = 1, the model reduces to the Bernoulli-Anderson model and there are no critical energies. If L+ = 2 and L− = 1, an example can be constructed as follows: choose t (l) = 1 for all l ∈ Z and vˆ+ = (0, 0) and vˆ− = (λ) and |λ| < 2, then Ec = 0 is the critical energy. The most prominent [DWP, Bov, BG] example is the random dimer model for which L+ = L− = 2 and vˆ+ = (λ, λ) and vˆ− = (−λ, −λ) (λ ∈ R), and t (l) = 1 for all l ∈ Z. This model has two critical energies Ec = λ and Ec = −λ as long as λ < 1. It was previously (non-rigorously) known that γ (Ec + ) = O( 2 ) [Bov] for the random dimer model. The definition of the critical energy assures that there exists a real invertible matrix M transforming T−Ec and T+Ec simultaneously into rotations by angles η− and η+ respectively:   cos(η± ) − sin(η± ) Ec −1 MT± M = . (7) sin(η± ) cos(η± ) Hence γ (Ec ) = 0. Because T±E are polynomials of degree L± in E, one can expand T±Ec + into powers of . Since MT±Ec M −1 are rotations, this implies that MT±Ec + M −1  ≤ 1 + c|| for || ≤ 0 and one deduces the following: Proposition 1. For 0 > 0 there exists a constant C < ∞ such that for all || ≤ 0 and m, k ∈ Z,    Ec +  (k, m) ≤ C eC|| |k−m| . (8) Tω In particular, |γ (Ec + )| ≤ C  || for C  > 0. Note that the bound in (8) does not depend on the configuration. To study a possible spreading of wave packets due to the divergence of the localization length, one best considers the moments of the associated probability distribution, notably the time-averaged moments of the position operator X on 2 (Z):
∞ dt − t q > 0. (9) Mω,q (T ) = e T
0|eıHω t |X|q e−ıHω t |0, T 0 The exponential time average may be replaced by a Cesaro mean without changing the asymptotics (e.g. [GSB]). Proposition 1 will lead more or less directly to the following deterministic lower bound on transport. Theorem 1. There exists a constant C such that for every configuration ω and for q ≥ 0, Mω,q (T ) ≥ C T q−1 .

(10)

Delocalization in Random Polymer Models

31

Remark 3. It is important that the initial condition in (9) is |0 and not an arbitrary state ψ ∈ 2 (Z). In fact, ψ could be an eigenstate of Hω and hence not lead to any diffusion. In order to study the behavior of the Lyapunov exponent in the vicinity of the critical energy, that is, go beyond the trivial upper bound |γ (Ec + )| ≤ C  ||, let us define the  and b by transmission and reflection coefficients a± ±   MT±Ec + M −1 v = a± v + b± v,

1 v = √ 2



1 −ı

 .

(11)

Both are polynomials in . As v is an eigenvector of all rotations, one has 0 a± = eıη± ,

0 b± = 0.

(12)



 /|a  | so that η0 = η . Furthermore let us set eıη± = a± ± ± ±

Theorem 2. Suppose that
e2ıη±  = 1 and
e4ıη±  = 1. Then the Lyapunov exponent of a random polymer chain satisfies γ (Ec + ) =

 sin(η ) − b sin(η )|2   2 p+ p− |b+ − − +  3 . + O |b |  ±
L±  |1 −
e2ıη± |2

(13)

Definition 2. A critical energy Ec ∈ R of a polymer Hamiltonian is said to be of order  | = O( r ), but not |b | = O( r+1 ) for both polymers. r if |b± ± Remark 4. If Ec is a critical energy of order r, then γ (Ec + ) = C  2r + O( 2r+1 ) for  = O(), the order of every critical energy is as some non-negative constant C. Since b± least 1, i.e. the Lyapunov exponent vanishes at least quadratically. Generically the order of a critical energy is 1 (this is the case in the dimer model). In the latter case, more explicit formulas for the coefficient in (13) invoking only the values of T±Ec and ∂E T±Ec at the critical energy can easily be written out. A comparison of (13) with a random phase approximation is made in Sect. 4.7. Finally, let us note that (13) also proves positivity of the Lyapunov exponent close to Ec whenever the numerator does not vanish. Remark 5. The conditions
e2ıη±  = 1 and
e4ıη±  = 1 in Theorems 2 and 3 below are linked to anomalies studied in√[CK]. For the dimer model, the condition
e4ıη±  = 1 is verified if and only if λ = 1/ 2. This particular value already appeared in [BG]. Theorem 2 shows how the localization length diverges at the critical energy. The next result concerns the asymptotics of the integrated density of states N (denoted IDS, its definition is recalled in Sect. 3.4 below). Let N± and N± denote the absolutely continuous IDS and their densities associated to the models with L± -periodic models composed of only one of the polymers. By definition of a critical energy, N± (Ec ) > 0. Theorem 3. Suppose that
e2ıη±  = 1. Then the IDS of a random polymer chain satisfies N (Ec + ) =


L± N± (Ec )
L± N± (Ec ) + + O( 2 ) .
L± 
L± 

(14)

32

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

The theorem states that N is linearly increasing at Ec so that there are many states in the vicinity of a critical energy. The spreading of these states is quantitatively nicely characterized by the diffusion exponents, namely the power law growth exponents of the moments Mω,q (T ) defined in (9) above: ± βω,q = lim ± T →∞

log(Mω,q (T )) . log(T q )

(15)

Here lim± denote the superior and inferior limit respectively. The main result is the following: Theorem 4. Suppose that |
e2ıη± | < 1. Then P-almost surely ± βω,q ≥ 1−

1 . 2q

(16)

It is interesting to compare Theorems 4 and 1. The latter implies for all configurations a weaker lower bound in (16) of the form 1 − q1 . One can construct configurations ω with slower transport than in (16). Therefore – seemingly paradoxically – typical random configurations do not lead the slowest possible transport for this model. Finally it is worth mentioning a large deviation result here. The IDS and Lyapunov exponent are both averaged quantities describing the behavior at the infinite volume limit. Given their asymptotics N (Ec + ) = N (Ec ) + N  (Ec ) + O( 2 ) and γ (Ec + ) = C 2 + O( 3 ) (here C = 0 if the order of Ec is p > 1), one therefore expects that typically (w.r.t. P) the following holds for the finite (but sufficiently large) size Hamiltonian Hω,N found by restricting Hω to 2 ({0, . . . , N − 1}) (with Dirichlet boundary conditions): Hω,N has cN 1/2 equally spaced eigenstates in the interval [Ec − N −1/2 , Ec + N −1/2 ] which are all spread out over the whole sample. Here we give an upper bound on the probability of the set of atypical configurations for which the average metal-like behavior of the eigenvalue spacing does not hold. This result shows on which scales there is strong level repulsion. Theorem 5. For every α > 0 there exist c > 0 and C < ∞ such that for all N ∈ N there are sets N (α) ⊂  satisfying α

P(N (α)) = O(e−cN ) , such that for every configuration ω in the complementary set N (α)c = \N (α) the following statement holds: the interval [Ec − N −1/2−α , Ec + N −1/2−α ] contains of the order of N 1/2−α eigenvalues of Hω,N which are equally spaced and have eigenfunctions spread out over the whole sample, namely adjacent eigenvalues E and E  satisfy 1 C ≤ |E − E  | ≤ , CN N

(17)

and for all normalized eigenfunctions ψ of Hω,N it holds that 1 C ≤ |ψ(k − 1)|2 + |ψ(k)|2 ≤ CN N for 0 ≤ k ≤ N − 1, where ψ(−1) = ψ(N) = 0.

(18)

Delocalization in Random Polymer Models

33

Outside of the interval [Ec − N −1/2−α , Ec + N −1/2−α ] we expect Poisson statistics. It seems unknown what the level statistics is like on the boundaries of this interval, but it is possibly not of Wigner-type. Theorems 2 and 3 are proved through a perturbation analysis of polymer phase shifts and action multipliers (essentially, appropriately modified Pr¨ufer variables). The key for the proof of Theorems 4 and 5 is Theorem 7 which states that with high probability norms of the transfer matrices TωE (k, m), 1 ≤ m ≤ k ≤ N for energies in the interval [Ec − N −1/2−α , Ec + N −1/2−α ] are uniformly bounded. This theorem is proved in Sect. 5 by establishing large-deviation estimates for random Weyl-type sums defined in terms of polymer phase shifts. Let us conclude with a brief remark about the one-dimensional Anderson model in the weak coupling limit, namely Hλ,ω = H0 + λVω , where H0 is a periodic operator, Vω the usual Anderson potential and λ a (small) coupling constant. Pastur and Figotin [PF] showed (in the case where H0 is the discrete Laplacian) that away from band-center and band edges of the periodic operator, the Lyapunov exponent grows quadratically in λ. The large deviation results and dynamical lower bounds presented here transpose in order to show that almost surely sup Mω,λ,q (T ) ≥ Cα λ−2q+α .

T >0

Hence the presented techniques allow to study in a very detailed way the metal-insulator transition driven by either the disorder strength or the sample size. This transition appears at a single energy, the critical energy, in the polymer models studied here. 3. Brief Review of Basic Formulas 3.1. Transfer matrices. Let (t (n))n∈Z be a sequence of positive numbers and (v(n))n∈Z a sequence of real numbers. As in (2) they define a Jacobi matrix H acting on 2 (Z). Given an initial angle θ 0 ∈ R and a complex energy z ∈ C, let us construct the formal solution (uz (n))n∈Z by − t (n + 1)uz (n + 1) + v(n)uz (n) − t (n)uz (n − 1) = zuz (n) , and the initial conditions



t (0) uz (0) uz (−1)



 =

cos(θ 0 ) sin(θ 0 )

(19)

 .

Using the definition (3) of the single site transfer matrices Tv,t , the transfer matrix from site k to n is introduced by T z (n, k) =

k 

Tv(l)−z,t (l) .

l = n−1

It allows to rewrite the (formal) eigenfunction equation (19) as     t (n) uz (n) t (k) uz (k) z = T (n, k) . uz (n − 1) uz (k − 1)

(20)

34

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

Note that the transfer matrices satisfy the transitivity relation T z (n, k) = T z (n, m) T z (m, k). A direct inductive argument then shows that, for ζ ∈ C, T z+ζ (n, k) = T z (n, k) − ζ

n −1

T z+ζ (n, l + 1)

l=k

1 t (l)



10 00



T z (l, k) .

(21)

Taking the norm of (21), estimating the r.h.s. and consequently taking the supremum over 0 ≤ k ≤ n ≤ m leads to the following perturbative result which in a slightly different form is given in [Sim2] (see also [DT]). Lemma 1. Suppose sup

0≤k≤n≤m

T z (n, k) ≤ C,

D =

1 . |t 0 ≤ l ≤ m −1 l | sup

Then, as long as CD|ζ |m < 1, sup

0≤k≤n≤m

T z+ζ (n, k) ≤

C . 1 − CD|ζ |m

3.2. Free Pr¨ufer variables. Let now E ∈ R and uE be given by (19). The free Pr¨ufer phases θ 0,E (n) and amplitudes R 0,E (n) > 0 are defined by     t (n)uE (n) cos(θ 0,E (n)) 0,E R (n) = , (22) sin(θ 0,E (n)) uE (n − 1) the above initial conditions as well as −

π 3π < θ 0,E (n + 1) − θ 0,E (n) < . 2 2

Note that the θ 0 -dependence of the Pr¨ufer variables is suppressed. Lemma 2. R 0,E (n)2 ∂E θ 0,E (n) =

 n−1 E 2 if n > 0 ,  l = 0 u (l) 

−

−1

E 2 l = n u (l) if n < 0 .

(23)

Proof. From the recurrence relation (19) and the definition of θ 0,E (n) one gets cot(θ 0,E (n)) = −t 2 (n − 1) tan(θ 0,E (n − 1)) + v(n − 1) − E . Differentiation leads to ∂E θ 0,E (n) =

t 2 (n − 1) sin2 (θ 0,E (n)) ∂E θ 0,E (n − 1) + sin2 (θ 0,E (n)) . cos2 (θ 0,E (n − 1))

Multiplying with R 0,E (n)2 and using the definition of R 0,E (n) and θ 0,E (n) gives R 0,E (n)2 ∂E θ 0,E (n) = R 0,E (n − 1)2 ∂E θ 0,E (n − 1) + uE (n − 1)2 .

(24)

Delocalization in Random Polymer Models

35

The above deduction of (24) has used that uE (n − 1) = 0. If uE (n − 1) = 0, then one may deduce (24) in a similar way from tan(θ 0,E (n)) =

cot(θ 0,E (n − 1)) −t 2 (n − 1) + (v(n − 1) − E) cot(θ 0,E (n − 1))

The lemma now follows by iterating (24).

.

 

Note in particular that (23) implies that ∂E θ 0,E (n) is strictly positive for n ≥ 2 and strictly negative for n ≤ −2. Furthermore, it follows from elementary considerations for transfer matrices that there are constants C1 and C2 such that



0 < C1 ≤ ∂E θ 0,E (n) ≤ C2 < ∞ , (25) where C1 and C2 can be uniformly bounded away from 0 and ∞ as long as |n| ≥ 2 and the quantities |n|, E and max|k|≤|n| {|v(k)|, t (k), 1/t (k)} remain bounded (where only the lower bound requires |n| ≥ 2). Let 2N be the projection on 2 ({0, . . . , N − 1}) and denote the associated finite-size Jacobi matrix by HN = 2N H 2N . As HN has Dirichlet boundary conditions, let us choose uE (−1) = 0 and t (0)uE (0) = 1 as initial conditions in the recurrence relation (19). This corresponds to an initial Pr¨ufer phase θ 0 = 0. The formal solution uE then gives an eigenvector (and E is an eigenvalue of HN ) if and only if t (N )uE (N ) = R 0,E (N ) cos(θ 0,E (N )) = 0, that is θ 0,E (N ) = π2 mod π (note therefore that uE (0) = 0 for any eigenvector of HN ). One checks iteratively for all n ≥ 0 that uE (n) > 0 for E sufficiently close to −∞ and limE→−∞ uE (n − 1)/uE (n) = 0. This and the definition of the Pr¨ufer phases implies that limE→−∞ θ 0,E (n) = 0 for all n ≥ 0, which one uses for n = N . As θ 0,E (N ) is monotone increasing in E, it follows that the j th eigenvalue Ej of HN (counted from below E1 < E2 < . . . < EN ) satisfies θ 0,Ej (N ) =

π + π(j − 1) , 2

θ0 = 0 .

(26)

This oscillation theorem implies immediately:

1 0,E

θ (N ) − # {negative eigenvalues of (HN − E) } ≤ 1 .

π

2

(27)



 cos(θ ) 3.3. Modified Pr¨ufer variables. Let us fix M ∈ SL(2, R). Set eθ = . Define a sin(θ ) smooth function m : R → R with m(θ + π) = m(θ ) + π and 0 < C1 ≤ m ≤ C2 < ∞, by r(θ )em(θ) = Meθ , r(θ) > 0 , m(0) ∈ [−π, π ) . Then the M-modified Pr¨ufer variables (R E (n), θ E (n)) ∈ R+ × R for initial condition θ E (0) = θ = m(θ 0 ) are given by θ E (n) = m(θ 0,E (n)) ,

(28)

36

and

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz



R E (n) cos(θ E (n)) R E (n) sin(θ E (n))



 = M

t (n) uE (n) uE (n − 1)

 ,

(29)

where the dependence on the initial phase is again suppressed. Bounds of the form (25) also hold for an M-modified Pr¨ufer phase θ E (n) because θ E (n) = m(θ 0,E (n)) leads to (min m )|∂E θ 0,E | ≤ |∂E θ E | ≤ (max m )|∂E θ 0,E |. Furthermore, as |θ E (n) − θ 0,E (n)| ≤ 2π, (27) implies that for the choice θ = m(0).



1 E

θ (N ) − # {negative eigenvalues of (HN − E) } ≤ 5 . (30)

π 2 The goal to have in mind when choosing M is to make the M-modified transfer matrices as simple as possible so that the M-modified Pr¨ufer variables are easy to calculate. Whenever E is in the spectrum, the most simple matrix to obtain is a rotation. Anything close to it can then be treated by perturbation theory. This is the strategy followed for the random polymer model below where M is chosen as in (7). Example. Let us consider an L-periodic Jacobi matrix H . If E ∈ R is in the interior of the spectrum of H , there exists a matrix M (depending on E, of course) such that MT E (L)M −1 = Rη , where Rη is the rotation by an angle η = η(E) obtained in accordance with the definition (28). The M-modified Pr¨ufer variables are then simply given by (R E (kL), θ E (kL)) = (1, kη) and the IDS is N (E) = η(E)/(Lπ ). 3.4. Covariant Jacobi matrices. Let (, T , Z, P) be a compact space , endowed with a Z-action T and a T -invariant and ergodic probability measure P. For a function  f ∈ L1 (, P), let us denote E(f (ω)) = dP(ω) f (ω). A strongly continuous family (Hω )ω∈ of two-sided tridiagonal, self-adjoint matrices on 2 (Z) is called covariant if the covariance relation U Hω U ∗ = HT ω holds where U is the translation on 2 (Z). Hω is characterized by two sequences (tω (n))n∈Z and (vω (n))n∈Z such that (2) holds. The IDS at energy E ∈ R of the family (Hω )ω∈ can P-almost surely be defined by [PF] N (E) =

lim

N→∞

1 Tr(χ(−∞,E] (2N Hω 2N )) , N

(31)

while the Lyapunov exponent γ (E) for E ∈ R is P-almost surely given by the formula   1   γ (E) = lim log TωE (N, 0) , N→∞ N where the transfer matrix TωE (N, 0) from site 0 to N is defined as in Sect. 3.1. Both the IDS and the Lyapunov exponent are self-averaging quantities, notably an average over P may be introduced before taking the limit without changing the result [PF]. For each Hω let (RωE (n), θωE (n)) denote the associated M-modified Pr¨ufer variables with some initial condition, then according to (30), N (E) =

lim

N→∞

 1 1  E E θω (N ) . π N

(32)

Delocalization in Random Polymer Models

37

1 E While it is readily seen that γ (E) ≥ lim+ N→∞ N E(log(Rω (N )), one may in general not get equality here as demonstrated by a counterexample in Sect. 4.1. This is due to the dependence of RωE (N ) on the initial phase θ . The next lemma solves this problem by (continuously) averaging over θ .

Lemma 3. For E ∈ R and any continuous (i.e. non-atomic) measure ν on RP (1) = [0, π ),
  1 γ (E) = lim dν(θ ) E log(RωE (N )) . (33) N→∞ N Proof. As TωE (N, 0)em−1 (θ)  = Mem−1 (θ) M −1 eθωE (N) RωE (N ), a change of variables and elementary estimates show that it is sufficient to show that γ (E) is equal to
1 dν(θ ) E (log(TωE (N, 0)eθ )) lim N→∞ N for any continuous probability measure ν. This is easy to see if γ (E) = 0, thus we now assume that γ (E) > 0. Suppose the contrary, that is there exists a ν such that
1 dν(θ ) E (log(TωE (N, 0)eθ )) < γ (E) . lim N→∞ N  By Fatou’s lemma this implies that dν(θ )E(limN→∞ N1 log(TωE (N, 0)eθ )) < γ (E). Because for a.e. ω the limit inside of the expectation is equal to either γ (E) or −γ (E) by Oseledec’s Theorem, there has to exist a set E ⊂ [0, π ) ×  of positive ν ⊗ P-measure such that limN→∞ N1 log(TωE (N, 0)eθ ) is equal to −γ (E) for all (θ, ω) ∈ E. Hence there exists an ω such that the set {θ ∈ [0, π ) | E eigenvalue of Hω (θ )} has positive ν-measure where Hω (θ ) is the half-line operator with θ -boundary condition. As ν is continuous, this set has to contain at least two distinct points. This is in contradiction to the fact that the difference equation Hω u = Eu has, up to constant multiples, at most one square-summable solution at +∞.   4. Asymptotics of IDS and Lyapunov Exponent Generalizing the strategy suggested by Pastur and Figotin [PF], this chapter is devoted to the calculation of the asymptotics for the IDS and the Lyapunov exponent near the critical energy of a random polymer model, that is the proof of Theorems 2 and 3. The techniques of [CS] allow to treat also the case of strongly mixing (instead of random) configurations of polymer chains giving similar formulas, containing a correction factor given by the Fourier transform of the correlation function. No further details are given here concerning this generalization. 4.1. Random polymer chains. For sake of completeness, let us briefly indicate how to construct (, T , Z, P) for the random polymer Hamiltonians defined in Chapter 2. Let 0 be the Tychonov space of two-sided sequences of signs. Set ± = {ω ∈ 0 | ω0 = ±} × {0, . . . , L± − 1} and  = + ∪ − . Now T :  →  is defined by   (ω, l + 1) if l < Lω0 − 1 , T (ω, l) =  (T ω, 0) if l = L − 1 , 0 ω0

38

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

where T0 is the left shift on 0 . Now for any set A± ⊂ 0 of codes all having ω0 = ±, one sets for all l ∈ {0 . . . L± − 1}, P({(ω, l) ∈  | ω ∈ A± }) =

P0 (A± ) ,
L± 

where P0 is the Bernoulli measure on 0 . It can then be verified that P is invariant and ergodic (the latter by mimicking the proof for (0 , T0 , P0 )). Random hopping terms and potential are then given by t(ω,l) = (. . . , tˆω0 , tˆω1 , . . . ) and v(ω,l) = (. . . , vˆω0 , vˆω1 , . . . ) with choice of origin t(ω,l) (0) = tˆω0 (l) and v(ω,l) (0) = vˆω0 (l). This leads to the covariant family (H(ω,l) )(ω,l)∈ of Jacobi matrices. It is this family which is referred to as (Hω ) in Sect. 2 and, in particular, in Theorems 2 to 5. According to Sect. 3.4 the Lyapunov exponent satisfies γ (E) =

   1 E E log T(ω,l) (N, 0) = N→∞ N lim

lim

N→∞

1 E log(T(ω,l) (N, 0)) , N (34)

 for P-a.e. (ω, l) ∈ . Here E = dP. On the other hand, there is also a Lyapunov exponent associated with random products of the unimodular matrices T±E :      1 1 E0 log TωE (k, 0) log TωE (k, 0) , = lim k→∞ k k→∞ k

γ0 (E) = lim

(35)

 ˜ 0 be the full meafor P0 -a.e. ω ∈ 0 and E0 = dP0 . To compare γ (E) and γ0 (E), let  k−1 sure set of those ω ∈ 0 such that (35) holds and also l=0 Lωl /k →
L±  as k → ∞. ˜ 0 it is easily seen that limN→∞ 1 log(T E (N, 0)) = γ0 (E)/
L± . Since For ω ∈  (ω,0) N ˜ 0 } = 1/
L±  > 0, one concludes from (34) that P{(ω, 0) | ω ∈  γ (E) =

1 γ0 (E) .
L± 

(36)

While γ0 is not defined through a covariant operator, it follows by the same argument as in Lemma 3 that for any continuous measure ν on [0, π ), 1 k→∞ k

γ0 (E) = lim




  dν(θ )E0 log MTωE (k, 0)M −1 eθ  .

(37)

Counterexample. The continuity condition on ν in Lemma 3 cannot be weakened as shown in the following example. Consider the polymer model with L± = 3, t (l) = 1 for all l ∈ Z and vˆ+ = ( 21 , 2, 0) and vˆ− = (− 21 , −2, 0), and choose M = 1. For E = 0 it is easily seen that T±0 eπ/2 = ∓ 21 eπ/2 and thus k1 log Tω0 (k, 0)eπ/2  = − 21 for all ω and k, while γ0 (0) = 21 . Hence a measure having an atom at θ = π2 will not satisfy (37). This also provides a counterexample to Lemma 3 with (, T , Z, P) as above. For this one uses that the event {ω | vω (0) = vˆω0 (0)} has probability 1/3 in .

Delocalization in Random Polymer Models

39

 (θ ) 4.2. Polymer phase shifts. For M given by (7), let the polymer action multipliers ρ± and the polymer phase shifts S,± (θ ) be the M-modified Pr¨ufer amplitude and phase for the L± -periodic polymers with initial phase θ at 0 and evaluated at L± (i.e. over a single polymer (tˆ+ , vˆ+ ) and (tˆ− , vˆ− ), respectively). By definition of the modified Pr¨ufer variables, this means  (θ )eS,± (θ) = MT±Ec + M −1 eθ ρ±

(38)

for all θ ∈ R. The iterated polymer phase shifts are then denoted by l+1 l (θ ) = S,ωl (S,ω (θ )) , S,ω

0 S,ω (θ ) = θ .

0 (θ ) = 1 and η = S From (7) it follows that (independent of θ) ρ± ± 0,± (θ ) − θ , at least up to a multiple of 2π which is hereby fixed. The former readily implies that γ (Ec ) = 0. To study the Lyapunov exponent in a vicinity of Ec , iterate (38) in order to deduce N−1        l log ρω l (S,ω (θ )) , log MTωEc + (N, 0)M −1 eθ  =

(39)

l=0

which combined with (36) and (37) gives γ (Ec + ) =

N−1
  1  1 l dν(θ ) E0 log(ρω l (S,ω lim (θ ))) .
L±  N→∞ N

(40)

l=0

To also express the IDS in terms of the polymer phase shifts, let (n(ω,l),k )k∈Z be the sequence of lower polymer nodes for a given (ω, l) ∈ , i.e. the integers determined by v(ω,l) (n(ω,l),k ) = vˆωk (0), for any choice of v. ˆ For N ∈ N, let n(ω,l),k be the polymer k (θ ) − θ is a rotation number for a matrix which arises node closest to N . Since S,ω from H(ω,l),N by a perturbation of rank bounded by C max{L− , L+ }, it follows that Ec + k (θ ) − θ )| ≤ C max{L , L } uniformly in θ . Thus it follows from (N ) − (S,ω |θ(ω,l) − + 1 k (θ ) − θ) almost surely and in expectation. (S,ω (32) that N (Ec + ) = limN→∞ πN Since k/N → 1/
L±  almost surely as N → ∞, this implies that 1 1 k (θ ) − θ) lim E0 (S,ω π
L±  k→∞ k k−1  1 1   l l = E0 S,ωl (S,ω (θ )) − S,ω (θ ) . lim π
L±  k→∞ k

N (Ec + ) =

(41)

l=0

4.3. Calculation of phase shifts and action multipliers. The aim of this paragraph is to calculate the polymer phase shifts and action multipliers needed in (41) and (40) in terms of the transmission and reflection coefficients defined in (11). Because det(MT±Ec + M −1 ) = 1, these coefficients satisfy  2  2 | − |b± | = 1. |a±

A further short calculation shows that

     2ıθ  2 (θ )2 = 1 + 2 e a± + 2|b± b± e | , ρ±

(42)

40

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

and  e−2ıθ a  + b±

. eı(S,± (θ) − θ) = ±

  e−2ıθ

a± + b±

 of a  , Now using the phase η± ± 

  2 a± = eıη± + O(|b± | ).

(43)

This leads to the following expansions:          2ıθ  2   2 4ıθ  3 log(ρ± + |b± + O |b± (θ )2 ) = 2 e a± b± e | − e (a± b± ) e | , (44) and       3ıη±  2  eıη± e2ı(S,± (θ)−θ) = e2ıη± + b± e−2ıθ − b± e e2ıθ + O |b± | .

(45)

4.4. Oscillatory sums. Proposition 2. Let c± ∈ C, j = 1, 2, and set j

j

IN (θ, ) = E0 (Iω,N (θ, )) ,

j

Iω,N (θ, ) =

N−1 

l

cωl e2ıj S,ω (θ) .

l=0

j

 |, 1). If
e2ıj η±  = 1 Let  be sufficiently small. If
e2ıj η±  = 1, then IN (θ, ) = O(N |b± for both j = 1, 2, 

IN1 (θ, ) = N
c± 

 eıη± 
b± 

1 −
e2ıη± 

 2 | , 1) . + O(N |b±

l+1 (θ ) = S l l Proof. Since S,ω ,ωl (S,ω (θ )) and S,ω (θ ) is independent of ωl , one gets 

1 IN1 (θ, ) =
e2ıη±  IN−1 (θ, ) +
c±  e2ıθ

+
c± 

N−1 

  l 2ı(η +S l (θ)) E0 e2ı S,ωl (S,ω (θ)) − e ωl ,ω .

(46)

l=1 2ı(η +θ)

 |). As I 1 (θ, ) = I 1 Equation (45) shows that e2ı S,ωl (θ) −e ωl = O(|b± N N−1 (θ, )+  2ıη 1 ± O(1) and
e  = 1 by hypothesis, one can solve for IN (θ, ) which directly implies  |, 1). Along the same lines, I 2 (θ, ) = O(N |b |, 1). Now that IN1 (θ, ) = O(N|b± ± N insert the expansion (45) in (46). Due to the above, the oscillatory terms in that formula  |2 ). Thus only the non-oscillatory term on the r.h.s. of (45) are then of order O(N |b± gives a contribution to the leading order.  

Delocalization in Random Polymer Models

41

4.5. Asymptotics of the IDS. η± + d± −  !m(c± e2ıθ ) + Proof of Theorem 3. Formula

(45) leads to S,±(θ ) − θ =  ) ıη± . Inserting this in (41) yields

O( 2 ), where d± = (∂ η± and c = (∂ b ) e ±  ± =0 =0 k−1     1 1 l (θ) 2ıS,ω 2 N (Ec +) = +O( ) . cωl e
η±  + 
d± − lim !m E0 k→∞ k π
L±  l=0

 |, 1) and thus By Proposition 2, the expectation of the oscillatory sum is of order O(k|b±

N (Ec + ) =

 1 
η±  + 
d±  + O( 2 ) . π
L± 

(47)

Setting p+ = 1 and p+ = 0 yields that in particular N± (Ec + ) = πL1 ± (η± + d± + O( 2 )), allowing to identify N± (Ec ) = η± /π L± and N± (Ec ) = d± /π L± . Using this to insert for η± and d± in (47) completes the proof.  

4.6. Asymptotics of the Lyapunov exponent. Proof of Theorem 2. Replacing (44) into (40) shows that 2
L±  γ (Ec + ) is, up to  |3 ), equal to the ν-average of corrections of order O(|b± 



N−1   1  l e lim E0 e2ı S,ω (θ) N→∞ N l=0   N−1    1 l (θ)   2 4ı S,ω − e
(a± b± )  lim . E0 e N→∞ N

 2
|b± | +2

 
a± b± 

l=0

 |2 ) By Proposition 2, the first oscillatory sum has a contribution of the order O(|b±  3 (which is given there) while the second oscillatory sum is of order O(|b± | ) and can hence be neglected. Therefore one obtains

1 γ (Ec + ) =
L± 



     eıη±  eıη±  
b± 
b±  1  2  3 . (48) |
|b± |  + e + O |b  ± 2 1 −
e2ıη± 

It can be directly verified that the given leading order term vanishes if either p+ = 0 or p+ = 1, which also follows from the fact that in this case Hω is a periodic Jacobi matrix, whose Lyapunov exponent vanishes in the interior of its spectral bands. Next rewrite  (48) as a fraction with common denominator |1 −
e2ıη± |2 . Since p− = 1 − p+ , the numerator is a polynomial of degree at most 3 in p+ vanishing at p+ = 0 and p+ = 1. Elementary but lengthy algebra shows that moreover its third derivative vanishes identically. Calculating the first order derivative allows as to conclude.  

42

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

4.7. Comments. A random phase approximation consists in supposing that the incoming l (θ ) in each summand of (40) and (41) is completely random, that is distributphases S,ω ed according to the Lebesgue measure. It can easily be checked that one actually obtains the correct answers for the derivatives of both the IDS and the Lyapunov exponent at the critical energy within this approximation. However, the lowest order non-vanishing  |2 ) and the random phase approximation gives term in the Lyapunov exponent is O(|b±  |2 /(2
L ), namely only the together with the expansion (44) that γ (Ec + ) ≈
|b± ± first term in (48). As we shall argue now, the second contribution is due to the presence of correlations (or memory) in the family of discrete time random dynamical systems (S,± , RP (1), , P)∈R . It is a result of Furstenberg [Fur] (his hypothesis can be checked here) that for each  = 0 (small enough) there exists a unique invariant measure ν on RP (1) satisfying

dν (θ )
f (S,± (θ )) , f ∈ C(RP (1)) . dν (θ ) f (θ ) = For  = 0, one invariant measure is given by the Lebesgue measure (it is unique if η+ − η− is irrational). For finite , iteration of the invariance property and Proposition 2 implies   N−1 

 eıη± 
b±  1 l 2ıθ 2ı S,ω (θ)  2 = = dν (θ ) E e | ). dν (θ ) e + O(|b±  2ıη ± N 1 −
e  

l=0

 |). These facts express the deviations of the invariant Similarly dν (θ ) e4ıθ = O(|b± measure from the Lebesgue measure and hence from the random phase approximation. Moreover, for small enough  = 0 the invariant measure ν is known to be H¨older continuous [BL, p. 161] so that one can use it in (40). Hence
  1 1  γ (Ec + ) = dν (θ ) log(ρ± (θ )2 )) . 2
L±   (θ )2 ) as in (44) then also leads to an alternative proof of (48) and the Developing log(ρ± second contribution in (48) is indeed due to the correlations as claimed above. Finally let us point out that higher order terms in  can readily be calculated, under adequate (weak) hypothesis.

5. Large Deviation Estimates Using elementary estimates on the boundary terms M and M −1 in (39), as well as (12), and the expansions (43) and (44), one obtains that for all 0 ≤ m ≤ k ≤ N ,  2    = 2δ log TωEc +δ (k, m)

sup

θ∈[0,π)

e

k−1 

l

cωl e2ı Sδ,ω (θ) + O(N δ 2 , 1) ,

(49)

l=m

δ )| where c± = eıη± (∂δ b± δ=0 . If the order of the critical energy is 1, then c± = O(1). In order to prove the delocalization results, it is necessary to show that the l.h.s. of (49) is 1 (θ, δ) of order 1 as long as O(N δ 2 ) = 1. Therefore one needs to show that sums like Iω,k √ 2 defined in Proposition 2 are with high probability of order N for O(N δ ) = 1 and |k| ≤ N. These random Weyl sums can be thought of as a discrete time (variable N )

Delocalization in Random Polymer Models

43

correlated random walk in the complex plane, the correlation being due to the presence of the dynamics Sδ,± . For the present purposes, it is sufficient to show that this sum actually behaves as a random walk on adequate time scales. Hence let us introduce, for every δ, θ,

 

1 0N (α, δ, θ ) = ω ∈ 0 ∃ k ≤ N such that |Iω,k (θ, δ)| ≥ N α+1/2 . (50) Theorem 6. If |
e2ıη± | < 1 and α > 0, there exist constants C1 and C2 such that for all θ, N and δ with N δ 2 ≤ 1: α

P0 (0N (α, δ, θ )) ≤ C1 e−C2 N .

(51)

The proof of this estimate will be given in Sect. 5.1. First, let us deduce the following consequence: Theorem 7. Let |
e2iη± | < 1 and α > 0. Then there are c, c > 0, C < ∞ such that for every N ∈ N, there exists a set N (α) ⊂  satisfying α

P(N (α)) = O(e−cN ) , and such that for every configuration (ω, l) in the complementary set N (α)c = \N (α), one has    Ec +δ+ıκ  (k, m) ≤ C , T(ω,l) for all 0 ≤ m ≤ k ≤ N and all |δ| ≤ N −α−1/2 , |κ| ≤ c /N. Proof. In order to estimate the norms of the transfer matrices using the Weyl sums, note that for any 2 × 2 matrix A, √ A = sup Aeθ  ≤ 2 maxπ Aeθ  . (52) θ∈[0,π)

θ=0, 2

Set 0N (α, δ) = 0N (α, δ, 0) ∪ 0N (α, δ, π2 ). Then, combining Theorem 6 with (49) and (52) as well as the fact that T (k, m) = T (k, 0)T (m, 0)−1 , one deduces that for all ω ∈ 0N (α, δ)c with |δ| < N −α−1/2 , norms of the transfer-matrices TωEc +δ (k, m) are uniformly bounded by a constant, not dependent on δ. Now let N (α, δ) = {(ω, l) ∈  ω ∈ 0N (α, δ)} . It follows that P(N (α, δ)) ≤

L+ + L − α P0 (0N (α, δ)) ≤ C5 e−C4 N .
L± 

Elementary estimates (based on uniform bounds on norms of transfer matrices over blocks of length no more than max L± ) imply that Ec +δ (k, m) ≤ C  , T(ω,l)

for all (ω, l) ∈ N (α, δ)c , |δ| ≤ N −α−1/2 and 0 ≤ m ≤ k ≤ N (in fact this holds for m, k up to the N th polymer node). Set  = N −α−1/2 . The theorem then follows from  Lemma 1, by taking N (α) = N  k=−N N (α, k/N ). 

44

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

5.1. Correlation bounds: Proof of Theorem 6. Lemma 4. Let κ = |
e2ıη± | < 1. Then there exists a centered complex random variable X(ω) depending on ω1 , . . . , ωr such that r

e2ı Sδ,ω (θ) = X(ω)e2ıθ + O(rδ, κ r ) . Moreover, |X(ω)| is uniformly bounded by 2. Proof. Let us set κr (ω) = exp(2ı rm=1 ηωm ). Note that |E0 (κr (ω))| = κ r . Iteration of e2ı Sδ,± (θ) = e2ı(η± +θ) + O(δ) and centering the random variable κr (ω) shows r

e2ı Sδ,ω (θ) = (κr (ω) −
e2ıη± r )e2ıθ + O(κ r , δr) , as claimed.

 

Proof of Theorem 6. Let r be the smallest integer larger than log(N −α−1/2 )/ log(κ). 1 (θ, δ) shows Applying Lemma 4 to each term (except the first r terms) of the sum Iω,k that 1 (θ, δ) = Iω,k

k−1 

l

cωl+r X(T0l ω) e2ı Sδ,ω (θ) + O(rkδ, kκ r , r) .

(53)

l=0

l+r l (θ )) was used. Recall moreover that T l ω is (Here the identity Sδ,ω (θ ) = Sδ,T l ω (Sδ,ω 0 0 the l-fold shift of ω.) Under the hypothesis of the theorem and because of the choice of r, the error term in (53) is O(N 1/2 log N ). Thus it is sufficient to prove probabilistic estimates of the appearing sum, which will be denoted by Zk (ω, δ). In order to decouple the correlations, divide {0, . . . , k−1} in 2R pieces I0 , . . . , I2R−1 of equal length [k α ], where R = [k/(2[k α ])], i.e. Is = {s[k α ], (s + 1)[k α ] − 1}, s = 0, . . . , 2R − 1. Here [x] denotes the largest integer smaller or equal to x. This excludes ck α terms which in the following can be absorbed in the error. Set for j = 0, 1: j

ZR (ω, δ) =

R−1 

Y2s+j (ω, δ) ,



Ys (ω, δ) =

s=0

l

cωl+r+1 X(T0l ω) e2ı Sδ,ω (θ) .

l∈Is

Thus Zk (ω, δ) = ZR0 (ω, δ) + ZR1 (ω, δ) + O(k α ) .

(54)

The random variable Ys (ω, δ) satisfies uniformly |Ys (ω, δ)| ≤ c1 k α . If Es denotes the averaging procedure (conditional expectation) over all random variables ωl for l ≥ s, then Lemma 4 implies Es[k α ]+1 (Ys (ω, δ)) = 0. j In the following estimates, real and imaginary parts of ZR (ω, δ) are treated sepaj rately, but in exactly the same way; hence one may suppose that ZR (ω, δ) and all the summands therein are real. For λ > 0 and β > 0, the Tchebychev and Cauchy-Schwarz inequalities imply P0 ({ω ∈ 0 | Zk (ω, δ) > λ}) ≤ e−βλ E0 (eβZk (ω,δ) ) ≤ e−βλ+Cβk

α

j

max E0 (e2βZR (ω,δ) ) .

j =0,1

Delocalization in Random Polymer Models

45

Now if −1 ≤ Y ≤ 1, by convexity 2 eβY ≤ (1 − Y )e−β + (1 + Y )eβ . Thus if Y is a real centered random variable, E(eβY (ω) ) ≤ (e−β + eβ )/2 ≤ eβ

2 /2

.

(55)

1+α

One may assume that [k α ] > r − 1 and k ≥ N 2 (otherwise it is trivially true that 1 (θ, δ)| < N α+1/2 ). Thus Z j |Iω,k R−1 (ω, δ) does not depend on the ωl with l ≥ (2(R − 1)+j )[k α ]−1 and a rescaled version of (55) can be iteratively applied to the conditional expectations, leading to   j j E0 (e2βZR (ω,δ) ) ≤ E0 E(2(R−1)+j )[k α ]−1 (e2βY2(R−1)+j (ω,δ) ) e2βZR−1 (ω,δ)   j α 2 ≤ e(2c1 k β) /2 E0 e2βZR−1 (ω,δ) ≤ ec2 β

2 k 1+α

.

Choosing β = λ/(2c2 k 1+α ) and proceeding similarly for {ω ∈ 0 | Zk (ω, δ) < −λ} thus shows (after recombining real and imaginary parts) P0 ({ω ∈ 0 | |Zk (ω, δ)| > λ}) ≤ 4 e

−λ2 /(4c2 k 1+α )+ 2cCλk 2

.

Using this estimate for λ = N α+1/2 and renormalizing the constants in order to compensate for the error terms in (53) as well as for summation over k concludes the proof.  

5.2. Eigenvalue distribution in the metalic phase. Proof of Theorem 5. Let us fix a configuration (ω, l) ∈ N (α) (see Theorem 7) and suppress its index. Using (20) and the fact that the norm of a transfer matrix is equal to the norm of its inverse yields T Ec +δ (k, 0)−1 ≤ R 0,Ec +δ (k) ≤ T Ec +δ (k, 0). Theorem 7 therefore guarantees the existence of a constant C such that for 0 ≤ k ≤ N and for −N −α−1/2 < δ < N −α−1/2 , 1 ≤ R 0,Ec +δ (k)2 ≤ C , C

1 ≤ |uEc +δ (k)|2 + |uEc +δ (k − 1)|2 ≤ C . C

This readily yields (18). Now by Lemma 2, N ≤ ∂E θ 0,Ec +δ (N ) ≤ N C . C Upon integration, one deduces N  |E − E  | ≤ |θ 0,E (N ) − θ 0,E (N )| ≤ N C |E − E  | , C for all E, E  ∈ [Ec − N −α−1/2 , Ec + N −α−1/2 ]. The oscillation theorem discussed in Sect. 3.2 now gives (17).  

46

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

6. Lower Bound on Dynamics The deterministic part of the argument presented in this section follows [DT]. Let us return to the simplified notation from Sect. 2 and write ω instead of (ω, l) since based on the results of Sect. 5 the value of l will not influence the considerations. Let us begin with some preliminaries and introduce the Green’s function

 

1 z

0 .

Gω (n) = n Hω − z Note that −tω (n + 1) Gzω (n + 1) + (vω (n) − Ec − z) Gzω (n) − tω (n) Gzω (n − 1) = δn,0 . (56) Using transfer matrices, one now has for n ≤ 0,     tω (n) Gzω (n) tω (0) Gzω (0) z = Tω (n, 0) , Gzω (n − 1) Gzω (−1) while for n ≥ 1,



tω (n) Gzω (n) Gzω (n − 1)

 =

Tωz (n, 1)



tω (1) Gzω (1) Gzω (0)

The following identity is well-known:
1 1  q Mω,q (T ) = |n| dE |Gzω (n)|2 , π T R n∈Z

(57)

 .

z = E+

(58)

ı . T

(59)

Proof of Theorem 4. For given α > 0 let c, c > 0 and C < ∞ be the constants from Theorem 7 and choose N = [c T ] and α = N −1/2−α . By Theorem 7 there exists N (α) ⊂  with P(N (α)) = O(e−cN ) and such that for ω ∈ N (α)c one has E +δ+ı/T Tω c (n, 1) ≤ C for all |δ| ≤ N −α−1/2 and n ≤ N . For such ω, because of the uniform bounds on the matrix elements tω (n) and vω (n), for n = 0 one of the three terms on the l.h.s. of (56) has to be large. Suppose first that |Gzω (0)|2 + |Gzω (1)|2 ≥ C6 > 0, then it follows from (58) and (Tωz (n, 1))−1  = Tωz (n, 1) that   C7 max |Gzω (n)|2 , |Gzω (n − 1)|2 ≥ . z Tω (n, 1)2 According to the above, as long as δ ∈ [−, ] the transfer matrices are bounded from above by C as long as n ≤ [c1 T ], in which case at least every second |Gzω (n)|2 is bigger than C7 /C 2 . Replacing this into (59),
 1 1 C7 q Mω,q (T ) ≥ n dδ 2 ≥ C8 T q  = C8 T q− 2 −α , 2πT C [−,] 0≤n≤[c1 T ]

for some constant C8 > 0. If, on the other hand |Gzω (−1)|2 ≥ C6 > 0, then one gets this estimate in the same way, but based on (57) instead of (58). This uses the fact that the analysis of Sect. 5 can also be carried out for TωEc +δ (n, 0) with negative n. A Bo− ≥ 1 − ( 1 − α)/q. Since α > 0 is arbitrary, this rel-Cantelli lemma shows that a.s. βω,q 2 finishes the proof.  

Delocalization in Random Polymer Models

47

Proof of Theorem 1. Follow the above argument by using the deterministic Proposition 1.   Acknowledgements. This paper is a heavily revised version of a preprint [JSS] which contained a first, but less direct proof of the deterministic lower bound stated in Theorem 1 below. The basic strategy (Lemma 1 and Sect. 6) of the present proof of the dynamical lower bound (Theorem 4) given the boundedness of transfer matrices (Theorem 7) was suggested by S. Tcheremchantsev. This technique, which will be published in full generality in [DT], is simpler than the Guarneri method [Gua] of proving lower bounds employed in [JSS] (and also applicable here) and allowed us to circumvent previous more intricate arguments. We greatly appreciate that S. Tcheremchantsev made his work available prior to publication. S. J. and H. S.-B. were supported by NSF grant DMS-0070755, H. S.-B. moreover by DFG grant SCHU 1358/1-1 and the SFB 288. G. S. was supported by NSF grant DMS-0070343. He would also like to acknowledge financial support of CNRS (France) and hospitality at Universit´e Paris 7, where part of this work was done.

References [BT] [BG] [BL] [Bov] [BJ] [CK] [CKM] [CS] [DSS1] [DSS2] [DT] [RJLS] [DWP] [Fur] [GK] [Gua] [GSB] [JSS] [KS] [PTB]

Barbaroux, J.-M., Tcheremchantsev, S.: Universal lower bounds for quantum diffusion. J. Funct. Anal. 168, 327–354 (1999) de Bi`evre, S., Germinet, F.: Dynamical Localization for the Random Dimer Schr¨odinger Operator. J. Stat. Phys. 98, 1135–1148 (2000) Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schr¨odinger Operators. (Boston: Birkh¨auser, 1985) Bovier, A.: Perturbation theory for the random dimer model. J. Phys. A: Math. Gen. 25, 1021– 1029 (1992) Bourgain, J., Jitomirskaya, S.: Anderson localization for the band model. In: Geometric Aspects of Functional Analysis, Lecture Notes in Math. vol. 1745, Berlin: Springer, 2000, pp. 67–79 Campanino, M., Klein, A.: Anomalies in the one-dimensional Anderson model at weak disorder. Commun. Math. Phys. 130, 441–456 (1990) Carmona, R., Klein, A., Martinelli, F.: Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108, 41–66 (1987) Chulaevsky, V., Spencer, T.: Positive Lyapunov Exponents for Deterministic Potentials. Commun. Math. Phys. 168, 455–466 (1995) Damanik, D., Sims, R., Stolz, G.: Localization of one dimensional, continuum, BernoulliAnderson models. Duke Math. J. 114, 59–100 (2002) Damanik, D., Sims, R., Stolz, G.: Localization for discrete one dimensional random word models, Preprint, mp arc 02-471 Damanik, D., Tcheremchantsev, S.: Power-Law Bounds on Transfer Matrices and Quantum Dynamics in one Dimension, Preprint, mp arc 02-270 del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum: IV. Hausdorff dimension, rank-one perturbations and localization. J. d’Analyse Math. 69, 153–200 (1996) Dunlap, D.H., Wu, H.-L., Phillips, P.W.: Absence of Localization in Random-Dimer Model. Phys. Rev. Lett. 65, 88–91 (1990) Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963) Germinet, F., Klein, A.: Bootstrap Multiscale Analysis and Localization in Random Media. Commun. Math. Phys. 222, 415–448 (2001) Guarneri, I.: Spectral properties of quantum diffusion on discrete lattices. Europhys. Lett., 10, 95–100 (1989); On an estimate concerning quantum diffusion in the presence of a fractal spectrum. Europhys. Lett. 21, 729–733 (1993) Guarneri, I., Schulz-Baldes, H.: Intermittent lower bound on quantum diffusion. Lett. Math. Phys. 49, 317–324 (1999) Jitomirskaya, S., Schulz-Baldes, H., Stolz, G.: Delocalization in polymer models. Preprint. mp-arc 02-1 Kostrykin, V., Schrader, R.: Global bounds for the Lyapunov exponent and the integrated density of states of random Schr¨odinger operators in one dimension. J. Phys. A 33, 8231–8240 (2000) Parisini, A., Tarricone, L., Bellani, V.: Parravicini, G., Diez, E., Dominguez-Adame, F., Hey, R.: Electronic structure and vertical transport in random dimer GaAs-Alx Ga1−x As superlattices. Phys. Rev. B 63, 1653218 (2001)

48 [PF]

S. Jitomirskaya, H. Schulz-Baldes, G. Stolz

Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Berlin: Springer, 1992 [PW] Phillips, P., Wu, H.-L.: Localization and Its Absence: A New Metallic State for Conducting Polymers. Science 252, 1805–1812 (1992) [SB] Schulz-Baldes, H., Bellissard, J.: Anomalous transport: a mathematical framework. Rev. Math. Phys. 10, 1–46 (1998) [Sim] Simon, B.: Absence of ballistic motion. Commun. Math. Phys. 134, 209–212 (1990) [Sim2] Simon, B.: Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schroedinger operators. Proc. Am. Math. Soc. 124, 3361–3369 (1996) [SVW] Shubin, C., Vakilian, R., Wolff, T.: Some harmonic analysis questions suggested by AndersonBernoulli models. Geom. Funct. Anal. 8, 932–964 (1998) Communicated by M. Aizenman

Commun. Math. Phys. 233, 49–78 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0721-4

Communications in

Mathematical Physics

The Existence and Stability of Noncommutative Scalar Solitons Bergfinnur Durhuus1 , Thordur Jonsson2 , Ryszard Nest1 1

Matematisk Institut, Universitetsparken 5, 2100 Copenhagen Ø, Denmark. E-mail: [email protected]; [email protected] 2 University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland. E-mail: [email protected] Received: 13 July 2001 / Accepted: 9 July 2002 Published online: 10 January 2003 – © Springer-Verlag 2003

Abstract: We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator N of the d-dimensional harmonic oscillator and sufficiently large noncommutativity parameter θ we prove the existence of a rotationally invariant soliton which depends smoothly on θ and converges to a multiple of P as θ → ∞. In the two-dimensional case we prove that these solitons are stable at large θ, if P = PN , where PN projects onto the space spanned by the N + 1 lowest eigenstates of N , and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to P = P0 for all θ in its domain of existence. Finally, for arbitrary d and small values of θ , we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on θ .

1. Introduction The mathematical structure of Riemannian geometry can be encoded in the structure of commutative algebras. Noncommutative geometry is obtained by replacing these algebras by noncommutative ones so noncommutative geometry can be regarded as a deformation of classical geometry. There are several areas of physics where noncommutative geometry is expected to be a natural tool for describing phenomena, particularly particle physics and quantum gravity. For a general background on noncommutative geometry we refer to [1–3] and for a recent overview of some of its physical applications see [4]. Quantum field theory can be defined on noncommutative spaces in a natural way and has been studied in some detail even though there is no comprehensive theory comparable to ordinary quantum field theory in Euclidean spaces, see, e.g., [5–7]. The basic idea is to replace the functions in the functional integrals that arise in ordinary quantum

50

B. Durhuus, T. Jonsson, R. Nest

field theory by elements of a noncommutative algebra which can be represented as an operator algebra. Any Riemannian geometry can be deformed to a noncommutative one and in many different ways. In this paper we are solely concerned with the standard noncommutative deformation of even dimensional Euclidean spaces and field theory on these deformed spaces. Other noncommutative spaces, in particular noncommutative spheres and tori are also of interest in physics and have been studied extensively, see [5–7] for a comprehensive list of references. Recent progress in string theory has stimulated interest in solitons in noncommutative field theories [8–10]. Several authors have found explicit solitons in gauge theories with and without matter fields [11–15]. In [16] solitons in scalar field theories were studied and it was shown that in the case of an infinite noncommutativity parameter θ , where the kinetic term in the action can be neglected, large families of solitons exist. This is in stark contrast to the commutative case where there are no solitons [17]. Various aspects of solitons in noncommutative scalar field theories are discussed in [18–26]. For background and a recent review of some of these results, see [27]. The problem we discuss can be formulated either in terms
of functions on R2d , or, by applying a quantization map, in terms of operators on L2 Rd , as explained e.g. in [16, 27]. In this paper we do not make use of the former formulation, except for some technical purposes in the final section. Thus we define solitons as critical points of the energy functional S (ϕ) = Tr

 d  k=1

 ∗

ϕ, ak [ak , ϕ] + θV (ϕ)

 ,

where ak and ak∗ are the standard annihilation and creation operators of the d-dimensional harmonic oscillator, V is a potential, θ a positive parameter
 (called the noncommutativity parameter), and ϕ is a self-adjoint operator on L2 Rd . In [28] we established the existence of spherically symmetric solitons in even dimensional scalar field theories under fairly general conditions on the potential, provided θ is sufficiently large and we proved that no spherically symmetric solutions can exist for small θ . Throughout the present paper we assume that V is twice continuously differentiable and positive, except for a second order zero at x = 0. Furthermore, we assume that V  (x) is strictly negative for x < 0 and has exactly two zeroes at positive values t and s corresponding to a local maximum and a local minimum of V , see Fig. 1. The techniques developed here can be adapted to potentials with more local maxima and minima. For the proof of Theorem 5 and for the discussion of stability in higher dimensions, we shall assume that V is analytic, although this assumption can presumably be relaxed. Our results can be divided into two classes, one concerning general solitons and another concerning solitons that are diagonal in the harmonic oscillator basis consisting of the joint eigenfunctions of ak∗ ak . In the d = 1 case the latter solitons correspond to rotationally invariant functions under the quantization map but in higher dimensions these solitons correspond to functions that are invariant under rotations in each of the d quantization planes. For d > 1 the rotationally invariant solitons are those which are functions of the number operator N . In the first category we have the following results for any nonzero critical point ϕ of S:

Existence and Stability of Noncommutative Scalar Solitons

51

• ϕ is a positive operator, whose operator norm satisfies ϕ ≤ s independently of the value of θ . • ϕ is of trace class and Tr V  (ϕ) = 0. • There exists a nonzero constant c depending only on the potential V such that the Hilbert-Schmidt norm of ϕ, denoted ϕ2 , satisfies d

ϕ2 ≥ cθ − 2 . As a corollary we find that any family ϕθ of solitons depending smoothly on the noncommutativity parameter θ (in a sense made precise in Sect. 3) has a diverging energy at some strictly positive value of θ . Hence, such families cannot exist for arbitrarily small values of θ. This result can be viewed as a noncommutative version of Derrick’s theorem [17]. Of results in the second category we mention, in particular, the following: • For any finite rank spectral projection P of the number operator N = there exists a maximal smooth family

d

∗ k=1 ak ak

(θP , ∞) θ → ϕθ of solitons such that V  (ϕθ ) > 0 and ϕθ → sP

as

θ →∞.

• If d = 1 and P equals the projection PN onto the space spanned by the N + 1 lowest eigenstates of N , the solitons ϕθ are stable for θ sufficiently large. For all other P the corresponding solitons are unstable in their full range of existence. • For P = P0 the corresponding solitons are stable for all d ≥ 1 in their full range of existence. This paper is organized as follows. In a preliminary section we describe the mathematical setting of the problem, recall results from [28] and prove some technical results on general properties of solitons. In Sect. 3 we establish the main existence theorem for solitons. We actually give two proofs, one elementary, generalizing [28], based on an analysis of the difference equation for the eigenvalues of ϕ obtained from the Euler-Lagrange equation for the variational problem for S, and another proof based on an application of a fixed point theorem. While less elementary, the latter approach has the advantage of giving smoothness of the solitons as a function of θ. A related existence proof has been obtained independently in [31]. The results on stability are proven in Sect. 4, which also contains a discussion of the extension of our approach to higher dimensions without giving full details, except in the case P = P0 . Finally, in Sect. 5 we prove non-existence of smooth families of solitons for small values of θ. It should be stressed that this result only rules out the existence of smooth families contrary to the nonexistence theorem in [28] for rotationally invariant solitons which rules out the existence of any rotationally invariant solitons for θ smaller than some positive θ0 depending only on V and d. It is an interesting unsolved question whether this stronger result also holds without the assumption of rotational invariance.

52

B. Durhuus, T. Jonsson, R. Nest

Another interesting unsolved problem concerns existence of general non-rotationally invariant solutions, in particular the so-called multi-soliton solutions described in [16]. The solitons discussed in this paper are special cases corresponding to overlapping solitons sitting at the origin. In a recent paper [32] it is shown that there does not exist a family of solutions that interpolate smoothly between two overlapping solitons at the origin and two solitons with an infinite separation. In [24 and 30] properties of moduli spaces of multi-solitons are discussed perturbatively in θ −1 . The latter paper contains a discussion of stability perturbatively to first order in θ −1 . Stability of scalar solitons under radial fluctuations is also discussed in [29]. Some of the methods of this paper have been used to establish existence and stability results for scalar solitons on the fuzzy sphere [34]. 2. General Properties of Solitons Solitons in a noncommutative 2d-dimensional scalar field theory with a potential V are finite energy solutions to the variational equation of the energy functional  d    ∗ S (ϕ) = Tr (1) ϕ, ak [ak , ϕ] + θV (ϕ) , k=1

where ak∗ and ak are the usual raising and lowering operators of the d-dimensional sim
 ple harmonic oscillator and ϕ is a self-adjoint operator on L2 Rd . We assume that the potential V is at least twice continuously differentiable with a second order zero at x = 0 and that V (x) > 0 if x = 0. Hence, finiteness of the potential energy θ TrV (ϕ) requires ϕ to belong to the space H2 of Hilbert-Schmidt operators. Consequently, S is defined and finite on the space H2,2 of self-adjoint Hilbert-Schmidt operators ϕ for which [ak , ϕ] is also Hilbert-Schmidt. We note that H2,2 is a Hilbert space with norm  · 2,2 given by  
   ϕ22,2 = Tr ϕ, ak∗ [ak , ϕ] + Tr ϕ 2 =  [ak , ϕ] 22 + ϕ22 , (2) k

k

where  · 2 denotes the Hilbert-Schmidt norm. It is easy to see that the space H0 consisting of operators that are represented by finite matrices (i.e. matrices with only finitely many non-zero entries) in the standard harmonic oscillator basis form a dense subspace of H2,2 . The variational equation of the functional (1) is 2

d   k=1

 ak∗ , [ak , ϕ] = −θV  (ϕ) .

(3)

We
regard  this equation as an equality between two Hilbert-Schmidt operators on L2 Rd . Thus, a solution ϕ to Eq. (3) belongs to H2,2 and has the property that the 1

left-hand side of Eq. (3), interpreted as a quadratic form on the domain of N 2 , where N denotes the number operator N =

d  k=1

ak∗ ak ,

Existence and Stability of Noncommutative Scalar Solitons

53

is Hilbert-Schmidt. We denote the space of such operators by D. Alternatively, D is the space of operators ϕ in H2,2 such that the linear form 
   Tr ak∗ , ψ [ak , ϕ] (4) H2,2 ψ → k

is continuous in the Hilbert Schmidt norm  · 2 . This operator theoretic formulation of the problem is the most convenient one for our discussion of the existence and stability results in Sects. 3 and 4. For the non-existence results in Sect. 5 we shall also make use of the alternative formulation in terms of ordinary functions and a quantization map (see e.g. [27]). Choosing the harmonic oscillator eigenstates |n1 , . . . , nd , ni = 0, 1, . . . , ak∗ ak |n1 , . . . , nd  = nk |n1 , . . . , nd , as the
 basis for the Hilbert space L2 Rd , rotationally symmetric functions correspond, under the standard Weyl quantization, to diagonal operators whose eigenvalues only depend on n1 + · · · + nd . If we consider a diagonal operator with eigenvalues λn , n = 0, 1, 2, . . . , Eq. (3) reduces, for d = 1, to [16, 18] θ  (5) V (λn ) , n ≥ 1, 2 θ λ1 − λ0 = V  (λ0 ) . (6) 2 Summing the second order finite difference equation for λn from n = 0 to n = m yields the first order equation (n + 1) λn+1 − (2n + 1) λn + nλn−1 =

m

λm+1 − λm =

 θ V  (λn ) , m ≥ 0. 2 (m + 1)

(7)

n=0

A necessary condition for the energy to be finite is clearly that λm → 0 as m → ∞.

(8)

Actually, this condition implies ϕ ∈ H2,2 by Lemma 1 below. In [28] we proved the existence of solutions to Eq. (7) satisfying the boundary condition (8) under fairly general conditions on the potential V . In the next section we generalize that result. In addition to the conditions on V which have been imposed above we assume that V has only one local minimum in addition to x = 0. Let the other local minimum be at s > 0. Let r ∈ (0, s) be a point where V has a local maximum and for technical convenience assume that V  does not vanish except at 0, r and s. Then V  (x) < 0 for x < 0 or x ∈ (r, s) and V  (x) > 0 for x > s or x ∈ (0, r) (see Fig. 1). The following result which will be needed in the next section was proven in [28]. We state the result for d = 1, but its generalisation to arbitrary d ≥ 1 is straightforward as explained in [28]. Lemma 1. Let {λm } be a sequence of real numbers which satisfy Eq. (7). If λn > s for some n then {λm } is increasing for m ≥ n and λm → ∞ as m → ∞. If λn ≤ 0 for some n then {λm } is decreasing for m ≥ n and λm → −∞ as m → ∞. If the sequence {λm } also satisfies the boundary condition (8) and the λm ’s are not all zero then (i) 0 < λm < s, for all m. (ii) λ m tends monotonically to 0 for m large enough. (iii) m V  (λm ) = 0 and m λm < ∞.

54

B. Durhuus, T. Jonsson, R. Nest

V’(x)

t

r

w

s x

Fig. 1. A graph of the derivative of a generic potential V which satisfies our assumptions

Dropping the assumption of rotational symmetry we have the following generalization of (i) and (iii), which, apart from being of some independent interest, we will use in Sect. 5. The remainder of the present section is not needed for the existence and stability results in the following two sections. Lemma 2. Let ϕ be a nonzero solution to Eq. (3). Then (i) the operator ϕ is positive and its norm satisfies the inequality ϕ ≤ s.  (ii) ϕ is of trace class and Tr V  (ϕ) = 0.

(9)




Before proving the above lemma we need the following result, where ϕ± denote the positive and negative parts of a bounded selfadjoint operator ϕ, defined by ϕ = ϕ+ − ϕ− , ϕ+ ϕ− = 0 , ϕ± ≥ 0 . Lemma 3. The maps

(10)

ϕ → ϕ±

are well defined and continuous from H2,2 to itself. Proof. Since ϕ± 2 ≤ ϕ2 , it suffices to show that, for all k,    ak , ϕ± 2 ≤ const  [ak , ϕ] 2 .

(11)

(12)

Existence and Stability of Noncommutative Scalar Solitons

55

√ We will prove below that this holds with the constant equal to 3. Since H0 is dense in H2,2 we can assume ϕ ∈ H0 . It is clear that the spectral projections of finite rank operators corresponding to non-zero eigenvalues belong to H0 and the same applies to the spectral projections of ϕ± . In order to estimate the norms of ϕ± it is convenient to write z 1 ϕ+ = dz , (13) 2πi γ z − ϕ where γ is a simple closed positively oriented contour in the complex plane enclosing the positive eigenvalues {λi } of ϕ but not the non-positive eigenvalues {µj }. Then   1 1 1 ak , ϕ+ = z dz . (14) [ak , ϕ] 2πi γ z − ϕ z−ϕ Denoting the spectral projection corresponding to λi by ei and the one of µj by fj , we have  1  1 1 = ei + fj . (15) z−ϕ z − λi z − µj i

j

Inserting the above identity into Eq. (14) and computing residues one obtains   ak , ϕ+ = e+ [ak , ϕ] e+ +



where e+ = Tr




ak , ϕ+

i,j



i ei

∗ 


 λi ei [ak , ϕ] fj + fj [ak , ϕ] ei , λi − µ j

(16)

is the support projection of ϕ+ . Hence,

ak , ϕ+




  = Tr e+ [ak , ϕ]∗ e+ [ak , ϕ] e+ + i,j



λi λi − µ j

2


 ×Tr ei [ak , ϕ]∗ fj [ak , ϕ] ei + fj [ak , ϕ]∗ ei [ak , ϕ] fj
 ≤ Tr e+ [ak , ϕ]∗ [ak , ϕ] e+ 
 + Tr ei [ak , ϕ]∗ fj [ak , ϕ] ei +fi [ak , ϕ]∗ ej [ak , ϕ] fi i,j


 ≤ 3 Tr [ak , ϕ]∗ [ak , ϕ] ,

(17)

where we used the fact that λi ≤ 1. (18) λi − µ j
 ∗   Clearly, the same estimate applies to Tr ak , ϕ− ak , ϕ− and the claimed result follows. 0≤

Proof of Lemma 2. (i) We first show that ϕ ≥ 0. Suppose on the contrary that ϕ− = 0. Then, since V  (−ϕ− ) < 0, we have for any integer n > 2 that 2

d  k=1


n ∗ 
n  ak , [ϕ, ak ] = θ Tr ϕ− Tr ϕ− V (ϕ) < 0 .

(19)

56

B. Durhuus, T. Jonsson, R. Nest

But, using the cyclicity of the trace,
n ∗ 
n ∗  
n ∗   Tr ϕ− ak , [ϕ, ak ] = Tr ϕ− ak , ϕ+ , ak − Tr ϕ− ak , ϕ− , ak
  
   n n = Tr ak∗ , ϕ− ϕ− , ak − Tr ak∗ , ϕ− ϕ+ , ak 
p  q  = Tr ϕ− ak∗ , ϕ− ϕ− ϕ− , ak p+q=n−1



−


p  q  Tr ϕ− ak∗ , ϕ− ϕ− ϕ+ , ak

p+q=n−1

=





Tr

p+q=n−1



+Tr

+Tr

p 2



ϕ− ak∗ , ϕ−



 q  ϕ− ϕ− , ak ϕ−2

1   n−2  ∗  1 ϕ+2 ϕ− , ak ϕ− ak , ϕ− ϕ+2

1   n−2   1 ϕ+2 ak∗ , ϕ− ϕ− ϕ− , ak ϕ+2

p



 

≥0,

(20)

which contradicts the inequality (19). To prove the inequality in (i) we note that the equation of motion (3) implies that    

ϕ−n θ Tr ϕ n V  (ϕ) = 2 Tr ϕ−n ϕ n ak∗ , [ϕ, ak ] k

= −2

 k

We also have

  ϕ−n Tr ak∗ , ϕn [ϕ, ak ] < 0.


 lim ϕ−n θ Tr ϕ n V  (ϕ) = θV  (ϕ) Tr e ,

n→∞

(21)

(22)

where e is the spectral projection of the operator ϕ corresponding to the eigenvalue ϕ. In particular, θ V  (ϕ) Tr e ≤ 0 ,

(23)

which implies the desired inequality by the assumed form of the potential V . (ii) Let Pm , m = 0, 1, 2, . . . , denote the orthogonal projection onto the eigenspace of the number operator N corresponding to eigenvalue m, and set λm = Tr (Pm ϕ) .

(24)

Then the equation of motion (3) gives 
1 θ Tr Pm V  (ϕ) = (m + 1) λm+1 − (2m + d) λm + (m + d − 1) λm−1 . 2 Summing this identity over m ≤ n we get (as in the spherically symmetric case) 
 Tr Pi V  (ϕ) , (n + 1) λn+1 − (n + d) λn = θ i≤n

(25)

(26)

Existence and Stability of Noncommutative Scalar Solitons

57

and, finally, summing over n ≤ p, λp+1 − λ0 = θ

 n≤p

  
 1 (d − 1) λn + Tr Pi V  (ϕ)  . (n + 1)

(27)

i≤n

Besides this equation we shall also make use of the fact that   V  (ϕ) = aϕ + O ϕ 2

(28)

for some positive constant a as a consequence of the assumptions made on V . Since ϕ is Hilbert-Schmidt it follows from this that V  (ϕ) is of trace class if and only if ϕ is of trace class. We first prove that this is the case if (and only if) limm→∞ λm = 0 and in this case, Tr V  (ϕ) = 0. In fact, by (28), 
  (29) Tr Pi V  (ϕ) = (aλi + ci ) , i≤n



i≤n

 where i ci is absolutely convergent while all the terms in i≤n λi are positive, since  ϕ is a positive operator by (i). It follows that the sum i≤n Tr Pi V  (ϕ) has a limit L, finite or +∞, as n → ∞. On the other hand, it follows from our assumptions that the right-hand side of Eq. (27) converges as p → ∞ and consequently, since the λm ’s are  nonnegative, L must be zero. Hence, Eq. (29) implies that i λi converges, i.e., ϕ is of trace class, and the trace L of V  (ϕ) is zero as claimed.  It remains to show that λm → 0 as m → ∞. Assume this is not the case. Then i λi = +∞ and therefore, by Eq. (29), we have  
(30) Tr Pi V  (ϕ) > 1 i≤m

for m large enough. Thus, by Eq. (27), λp ≥ θ

 n≤p−1


1  Tr Pi V  (ϕ) ≥ const ln p , n+1

(31)

i≤n

for p large enough. Repeating the argument with the inductive assumption λp ≥ const pl , for sufficiently large p, where l is a nonnegative integer, leads to λp ≥ const pl+1 for p sufficiently large. Hence, λm increases faster than any power of m, if it does not tend to zero. But this is not possible since, by the Cauchy-Schwarz inequality,     λ2m = (Tr Pm ϕ)2 ≤ Tr Pm ϕ 2 Tr Pm ≤ const md−1 Tr Pm ϕ 2 (32) and hence,    λ2  m 2 = ϕ22 < ∞ . ≤ Tr P ϕ m d−1 m m m This finishes the proof of Lemma 2.

(33)

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B. Durhuus, T. Jonsson, R. Nest

3. Existence We now proceed to discuss the existence of rotationally invariant solutions to Eq. (3). Let t be the location of the maximum of V  in the interval [0, s] and let w be the location of the minimum of V  in the same interval (see Fig. 1). As above we denote by P0 , P1 , . . . the orthogonal projections onto the eigenspaces of the number operator of the d-dimensional harmonic oscillator. The purpose of this section is to prove the following theorem.
 Theorem 1. For any projection P on L2 Rd , which is the sum of a finite number of the projections Pn , there is a unique maximal family ϕθ , θ > θP , of rotationally invariant solutions of Eq. (3), which depends smoothly on θ, i.e., is continuously differentiable with respect to the norm  · 2,2 , and fulfills V  (ϕθ ) > 0 ,

(34)

ϕθ → s P

(35)

as well as

in Hilbert-Schmidt norm as θ → ∞. Proof. We shall give two proofs of existence of solutions for sufficiently large θ. The first proof is an extension of the proof given in [28] for P = P0 . For simplicity we restrict to d = 1 and to P = P0 + · · · PN , the adaptation of the arguments to arbitrary d ≥ 1 being explained in [28]. First, assume θ is so large that θ |V  (w) | ≥ w. 2 (N + 1)

(36)

In this case we claim there is a unique λ ∈ [w , s) such that if we set λ0 = λ and define λi for i > 0 by the recursion (7) then λ0 > λ1 > · · · λN ≥ w

(37)

and λN+1 = 0. In order to prove the claim we begin by choosing λ0 close to but smaller than s so that (37) holds, which clearly is possible. Then λN > λN+1 by (7), and if λN+1 = 0 we are done. Note that all the λi ’s depend continuously on λ0 and λN+1 → s as λ0 → s. If λN+1 < 0 we increase λ0 until λN+1 = 0 and the inequalities (37) still hold because λ1 , . . . λN all increase with λ0 . If λN+1 > 0 we decrease λ0 until λN+1 = 0 and (37) still holds due to the inequality (36). This proves the existence of λ. Next take θ still larger, if necessary, so that
 V  (t) ≥ (N + 1) |V  λ |. (38)
 This is clearly possible because λ → s as θ → ∞. We now claim there exists λ¯ ∈ λ, s such that if we take λ0 = λ¯ then (37) holds and λN+1 = λN+2 , i.e. 0=

N+1  θ V  (λi ) . 2 (N + 2) i=0

(39)

Existence and Stability of Noncommutative Scalar Solitons

59

In order to verify the existence of λ¯ we note that, as a consequence of (7), for λ0 greater than but close to λ we have λN+1 is greater than but close to 0, and λN+1 increases with λ0 . Hence, in view of (38)
and  the fact that λ1 , . . . , λN are also increasing functions of λ0 , there is a λ0 ≡ λ ∈ λ, s such that 

V (λN+1 ) = −

N 

V  (λi ) ,

(40)

i=0

which establishes the claim. We note that for λ0 = λ we have λN+1 ∈ (0, t). If a sequence {λi } obeys the recursion (7) and has the property λ0 > λ1 > · · · > λp , but λp+1 ≥ λp , we say that the sequence turns at p. We note that in this case λp > 0 by Lemma 1 and if λp+1 = λp then λp+2 > λp+1 by (7). Define the set     A = λ0 ∈ λ, λ¯ : {λi } turns at some p . (41) By construction λ ∈ / A and λ¯ ∈ A. Put )0 = inf A. Since each λi depends continuously on the initial value λ0 it follows that )0 ∈ / A. Now consider the sequence defined by λ0 = )0 and Eq. (7). Since this sequence does not turn it is monotonically decreasing. In order to show that this sequence provides a solution to our problem it therefore suffices to show that λi → 0 as i → ∞. Suppose λi becomes negative for some i. Then Lemma 1 implies that λi → −∞. By the continuity of λi as a function of λ0 it follows that for λ0 sufficiently close to )0 the sequence λi tends monotonically to −∞, but this contradicts the definition of )0 . We conclude that the limit limi→∞ λi = a ≥ 0 exists and by (7) we have V  (a) =

2 lim (λi+1 − λi ) = 0. θ i→∞

(42)

Hence, a = 0 since λi < r for i > N . This completes the proof of the existence of rotationally invariant solutions ϕθ for large enough θ and it follows easily from the construction that ϕθ → sP in the operator norm as θ → ∞. It is worthwhile noting that the proof given here shows that the sequence of eigenvalues {λi } of ϕθ is strictly decreasing for θ large enough. This is special for the choice of projection P made above. The same technique can be applied to demonstrate existence of solutions converging to any projection of the type stated in the theorem, but since this result as well as the claim of differentiability are obtained in a more uniform manner by the second method of proof, we shall not discuss that approach in more detail here. Also, the above proof can easily be generalized to establish the existence of solutions which converge to finite rank operators of the form tP + sP  , P P  = 0, as θ → ∞. The second proof of existence is by use of a fixed point theorem. Let us first note that the operator *, defined by *ϕ =

d   k=1

 ak∗ , [ak , ϕ] ,

(43)

is self-adjoint and positive on H2 with domain D. Indeed,
as explained in Sect. 5, it is  unitarily equivalent to the standard Laplace operator on L2 R2d via a quantization map
 πW : L2 R2d → H2 , which justifies the notation * for this operator in the remainder

60

B. Durhuus, T. Jonsson, R. Nest


 of this proof. Given a bounded self-adjoint operator B on L2 Rd , it defines by left multiplication a bounded self-adjoint operator on H2 , which we shall also denote by B. By the Kato-Rellich theorem [33] * + B is self-adjoint with domain D. Assuming B ≥ c > 0 we have * + B ≥ c and hence * + B maps D bijectively onto H2 with bounded inverse (* + B)−1 ≤ c−1 .

(44)

The same statement holds if B is of the form B=

∞ 

bn Pn

(45)

n=0

and we restrict * + B to D = D ∩ H2 , where H2 is the Hilbert subspace of H2 consisting of diagonal operators of the form (45). This follows by using that H2 corresponds
2d 2 under the quantization map πW to rotation invariant functions in L R on which the Laplace operator is known to be self-adjoint. Alternatively, one can use the explicit form *ϕ = −

∞ 

{(n + d) λn+1 − (2n + d) λn + nλn−1 } Pn ,

(46)

n=0

where ϕ =

∞

n=0 λn Pn , ∞ 

and the domain D consists of those ϕ which fulfill

| (n + d) λn+1 − (2n + d) λn + nλn−1 |2 < ∞ .

(47)

n=0

Since * + B is a closed symmetric operator it suffices to verify that the orthogonal complement to its range is {0}. But it is easily seen that ϕ belongs to this orthogonal complement if and only if (n + d) λn+1 − (2n + d) λn + nλn−1 = bn λn ,

(48)

for n ≥ 0. The proof of Lemma 1 shows that any non-trivial solution {λn } of this recursion relation diverges to ±∞, since bn ≥ c > 0. Hence ϕ = 0 if ϕ ∈ H2 , as desired. As a consequence, we note that for ρ ≥ 0 and B and c as above, the operator ρ* + B has a bounded inverse on H2 fulfilling (ρ* + B)−1 ≤ c−1 ,

(49)

the case ρ = 0 being obvious. In view of these preparatory remarks, we rewrite Eq. (3) as ρ*ϕ + V  (ϕ) = 0 , where ρ = 2θ −1 . Then ψ0 = sP is a solution for ρ = 0. Since ψ0 ∈ H2 and   V  (ψ0 ) ≥ min V  (0) , V  (s) ≡ c0 > 0 ,

(50)

(51)

Existence and Stability of Noncommutative Scalar Solitons

61

by assumption, we can, for ρ ≥ 0, further rewrite the equation in the form
 −1   ϕ = ρ* + V  (ψ0 ) V (ψ0 ) ψ0 +V  (ψ0 ) −V  (ϕ) −V  (ψ0 ) (ψ0 −ϕ) ≡ Tρ (ϕ). (52) Since V is C 2 by assumption we have V  (ϕ) − V  (ψ0 ) − V  (ψ0 ) (ϕ − ψ0 ) 2 = o (ϕ − ψ0 2 ) ,

(53)

and also
−1  −1
V (ψ0 ) ψ0 − ψ0 2 = ρ ρ* + V  (ψ0 ) *ψ0 2 ≤ c1 ρ ,  ρ* + V  (ψ0 ) (54) where c1 = c0−1 *ψ0 2 . For ϕ in the ball

  Bε (ψ0 ) = ϕ ∈ H2 : ϕ − ψ0 2 ≤ ε ,

(55)

Tρ (ϕ) − ψ0 2 ≤ c1 ρ + o (1) ϕ − ψ0 2 ,

(56)

we then have

and hence, Tρ (ϕ) ∈ Bε (ψ0 ) if ρ and ε are sufficiently small. Similarly, one sees that Tρ (ϕ) − Tρ (ψ) 2 ≤ o (1) ϕ − ψ0 2 ,

(57)

so Tρ is a contraction on Bε (ψ0 ), if ρ and ε are sufficiently small. Fixing ε accordingly, Banach’s fixed point theorem implies the existence of a unique solution ψρ of Eq. (50) in Bε (ψ0 ) for 0 ≤ ρ ≤ δ and δ small enough. For 0 ≤ ρ, ρ0 ≤ δ, we have

 −1  ψρ − ψρ0 = ρ* + V  (ψ0 ) (ρ0 − ρ) *ψρ0 + V  ψρ0
 
−V  ψρ − V  (ψ0 ) ψρ − ψρ0 (58) from which we get
 ψρ − ψρ0 2 ≤ c2 |ρ − ρ0 | + o ψρ − ψρ0 2 ,

(59)

where the c2 depends only on ρ0 , and we have assumed ε is small enough such
constant  that V  ψρ > 0. This inequality implies that ψρ is a Lipschitz continuous function of ρ if ε is small enough. In turn, Eq. (58) implies that ψρ is differentiable in the ·2 -norm with −1
dψρ = ρ* + V  (ψ0 ) *ψρ . dρ

(60)

By standard arguments, the family ψρ , 0 ≤ ρ
< δ extends to a maximal family, differentiable in the  · 2 -norm, and such that V  ψρ > 0. It remains to establish the stronger claim of smoothness in the norm ·2,2 for ρ > 0. −1
In order to obtain this, it is sufficient to verify that the bijective operator ρ* + V  (ϕ) from H2 onto D is bounded, when D is equipped with the  · 2,2 -norm, for ρ > 0 and

62

B. Durhuus, T. Jonsson, R. Nest


−1 V  (ϕ) > 0. It is straightforward to verify that under these conditions ρ* + V  (ϕ) is bounded (and ρ* + V  (ϕ) as well, in fact), when D is equipped with the norm  1 2 , ϕ4,2 = *ϕ22 + ϕ22

(61)

which is easily seen to be stronger than  · 2,2 . In addition, simple estimates show that the derivative given by Eq. (60) is continuous in this norm. This completes the proof of the theorem with ϕθ = ψρ for ρ = 2θ −1 . We remark that the above argument can easily be generalized to prove the existence of solutions to Eq. (3) which converge to sP , where P is a projection onto the space spanned by a finite number of the joint eigenfunctions of the number operators ak∗ ak . As remarked above, these solutions are not rotationally invariant but only invariant under rotations in the d two-dimensional quantization planes. 4. Stability In this section we study the stability of solutions to Eq. (3) in the case d = 1. Extension to d > 1 is briefly discussed at the end of the section. A solution ϕ is defined to be stable if the second functional derivative of the action S at ϕ is a positive semidefinite quadratic form at ϕ, i.e.,   1 d2  4 (ω) ≡ S + 6ω) ≥0. (62) (ϕ  2 2 d6 6=0 The natural domain of definition of the quadratic form 4 depends generally both on the potential V and on ϕ. Under the previously stated assumptions on V the domain contains at least the space H0 for the rotationally symmetric solutions that we consider here. If 4 is continuous with respect to the norm  · 2,2 it is sufficient to show stability for perturbations ω in H0 . Since the kinetic term in S (ϕ) is quadratic, continuity of 4 means that the second functional derivative of V is a continuous quadratic form with respect to the Hilbert-Schmidt norm. This continuity is easy to check, using the analytic functional calculus, if V is analytic in a neighborhood of the interval [0, s] which we will assume to be the case from now on. For this reason we restrict attention below to ω ∈ H0 . Our results about stability can be summarized in the following three theorems. Theorem 2. Let ϕ be a rotationally invariant, finite energy solution to (3) and let λ0 , λ1 , . . . denote the eigenvalues of ϕ in the harmonic oscillator basis. Then ϕ is unstable unless {λn } is a decreasing sequence. This theorem implies that only the solutions corresponding to P = P0 + · · · + PN in Theorem 1 can possibly be stable and these are in fact stable in their full range of existence. By abuse of notation we denote the solution whose existence is proven in Theorem 1 by ϕN , for a fixed value on θ , in the remainder of this section. Theorem 3. The solution ϕ0 of Eq. (3) constructed in the previous section is stable for all values of θ in the maximal range. Theorem 4. For any N ≥ 0 the solution ϕN constructed in the previous section is stable for θ sufficiently large.

Existence and Stability of Noncommutative Scalar Solitons

63

We note that Theorem 3 implies Theorem 4 in the case N = 0. We choose to state and prove Theorem 3 separately because it is stronger than Theorem 4 for N = 0 and the proof is simpler. In the proof of Theorem 4 we have to rely on asymptotic expansions of the eigenvalues for large θ which are not needed in the proof of Theorem 3. We remark further that solutions with eigenvalues λn some of which lie in the region where V  < 0 are in general unstable but one can construct examples of stable solutions with eigenvalues in the region where V  < 0. Before proving the theorems we do some groundwork and establish notation. Let   K (ϕ) = Tr ϕ, a ∗ [a, ϕ] ∞ 

=

|n| [a, ϕ] |m|2

(63)

n,m=0

denote the kinetic energy functional. Let ϕ be a rotationally invariant solution of Eq. (3) with a nondegenerate spectrum. Then we can write ϕ + 6ω = U6∗ ϕ6 U6 ,

(64)

where U6 is unitary and ϕ6 is diagonal in the harmonic oscillator basis. It follows that     d2 d2   S + 6ω) = 2K + θ Tr V . (65) ) (ϕ (ω) (ϕ 6   2 2 d6 d6 6=0 6=0 Notice that the assumption ω ∈ H0 implies that only finitely many of the eigenvalues and eigenvectors of ϕ are perturbed, and we can apply standard non-degenerate perturbation theory [33]. Let λn (6) denote the eigenvalue of ϕ6 which converges to λn as 6 → 0. Then λn (6) is real analytic in 6, and  ∞    
 2  d2    Tr V = V λ V + λ . (ϕ ) (0) (λ ) (0) (λ ) 6 n n n n  d6 2 6=0

(66)

n=0

From standard perturbation theory we know that λn (0) = n|ω|n

(67)

and λn (0) = 2

 |n|ω|m|2 . λn − λ m

(68)

m=n

The condition for stability can therefore be written as 4 (ω) = K (ω) + θ

∞  |n|ω|m|2 θ V  (λn ) + |n|ω|n|2 V  (λn ) λn − λ m 2 n=0

m=n

= K (ω) + θ

 m
≥ 0.

|n|ω|m|2

V  (λ

n

) − V  (λ

λn − λ m

m)

+

∞ θ |n|ω|n|2 V  (λn ) 2 n=0

(69)

64

B. Durhuus, T. Jonsson, R. Nest

We remark that the last term in 4 is nonnegative if V  (λn ) ≥ 0 for all n. The kinetic energy term can be written ∞  √ √ | n + 1 n + 1|ω|m − m n|ω|m − 1|2 ,

(70)

n,m=0

√ where m n|ω|m − 1 is set to zero for m = 0, and we see that the kinetic energy couples the matrix elements of ω to their nearest neighbours along diagonals with n − m fixed. On the other hand, the potential part of 4 does not couple different matrix elements of ω. Note that n|ω|m = m|ω|n∗ since ω is self-adjoint but otherwise the matrix elements of ω can be chosen arbitrarily. Proof of Theorem 2. We begin by assuming that the spectrum of ϕ is nondegenerate. We will show that there exists a perturbation ω such that 4 (ω) < 0 unless the λn ’s are decreasing. We take ω such that n|ω|m = 0 for |n − m| = 1. Then we can write ∞   √ √ 4 (ω) = | n + 1 n + 1|ω|n − n n|ω|n − 1|2 n=0

 √ √ + | n + 1 n + 1|ω|n + 2 − n + 2 n|ω|n + 1|2

+θ

∞  |n|ω|n + 1|2
n=0

λn − λn+1

 V  (λn ) − V  (λn+1 ) .

(71)

The above expression is quadratic in the variables αn = n + 1|ω|n,

(72)

n = 0, 1, 2, . . . . Assuming without loss of generality that the αn ’s are real we have  qnm αn αm , (73) 4 (ω) = 2 n,m

where the symmetric matrix qnm has only nonvanishing matrix elements on the diagonal and next to the diagonal which are given by qnn = 2 (n + 1) + γn , √ √ qnn+1 = − n + 1 n + 2, √ √ qnn−1 = − n n + 1,

(74) (75) (76)

where γn =

θ V  (λn+1 ) − V  (λn ) . 2 λn+1 − λn

(77)

We need to show that qnm is a positive semidefinite matrix. This is most easily done by diagonalising qnm , using elementary row and column operations, and verifying that the diagonal entries C0 , C1 , . . . in the resulting diagonal matrix C are non-negative. In the first step we divide the first row by q00 , multiply it by −q10 and add the resulting row to the second row. Then we see that the first two diagonal entries of C are

Existence and Stability of Noncommutative Scalar Solitons

65

C0 = q00 ,

(78)

C1 = q11 −

2 q10

q00

.

(79)

Inductively we find Ck = qkk −

2 qkk−1

Ck−1

.

(80)

We can evaluate C0 and C1 directly using the equation of motion (5) and find λ2 − λ1 , λ1 − λ 0 λ3 − λ2 C1 = 3 . λ2 − λ 1

C0 = 2

(81) (82)

Now it is straightforward to prove from Eq. (80) by induction that Ck = (k + 2)

λk+2 − λk+1 , λk+1 − λk

(83)

and we conclude that Ck > 0 for all k if and only if the sequence {λn } is monotone. Obviously, the sequence cannot be increasing since λn > 0 for all n and λn → 0 as n → ∞. Suppose now that the spectrum of ϕ is degenerate. Then we can carry out the construction above and see that either one of the Ck ’s becomes negative by Eq. (83) or a pair of successive eigenvalues coincide. In the latter case assume that λn and λn+1 is the first pair of coinciding eigenvalues. Then the difference quotient in the last term in Eq. (71) must be replaced by V  (λn ). We find by the same diagonalization calculation that Cn−1 = 0 and it follows from Eqs. (74)–(76) that the upper left (n + 1) by (n + 1) corner matrix has a negative eigenvalue. Proof of Theorem 3. Let λn be the eigenvalue of ϕ0 corresponding to the eigenvector |n, n = 0, 1, 2, . . . . Since V  (λn ) ≥ 0 for all n, by hypothesis, and the kinetic energy only couples the matrix elements of ω along diagonals it is sufficient and also necessary, in view of Eq. (69), to prove that 

4k (ω) ≡ |n| [a, ω] |m|2 + |m| [a, ω] |n|2 . n−m=k

+θ|n|ω|m|2

V  (λn ) − V  (λm ) λn − λ m

 ≥0

(84)

for k ≥ 1. For each fixed k the argument is quite similar to the proof of the previous theorem. We put αn = n + k|ω|n which can be assumed to be real for the purpose of proving positivity. We see that 4k (ω) is a quadratic form 2Qk in the variables αn . As in the previous proof the matrix representing Qk has only nonvanishing matrix elements on the diagonal and next to it, and they are given by qnn = 2n + 1 + k + γn ,  qnn−1 = − n (n + k),  qnn+1 = − (n + 1) (n + 1 + k),

(85) (86) (87)

66

B. Durhuus, T. Jonsson, R. Nest

and γn =

θ V  (λn ) − V  (λn+k ) . 2 λn − λn+k

(88)

The positivity of this form is equivalent to the positivity of the numbers Cn defined inductively by C0 = 1 + k + γ 0

(89)

and Cn = 2n + 1 + k + γn −

n (n + k) , n = 1, 2, . . . Cn−1

(90)

by the same row and column argument as in the proof of Theorem 1. The case k = 1 is taken care of by the argument in Theorem 1 since the eigenvalues λn form a decreasing sequence. In order to prove the positivity of Cn for general values of k we observe, using Eq. (5), that λn+1 − λn+k+1 λn − λn+k *λn − *λn+k+1 *λn+1 − *λn+k +n + (n + k) , λn − λn+k λn − λn+k

qnn = (2n + k + 1)

(91)

where *λn = λn−1 − λn , n ≥ 1. Furthermore, *λn > *λn+1

(92)

for n ≥ 1, since V  (λn ) > 0 for n ≥ 1 in the case at hand, N = 0. We have C0 = (k + 1)

λ1 − λk+1 *λ1 − *λk +k λ0 − λ k λ0 − λ k

(93)

λ1 − λk+1 . λ0 − λ k

(94)

and therefore, since *λ1 ≥ *λk , C0 ≥ (k + 1)

Finally, using Eqs. (91) and (92), it follows by induction that Cn ≥ (n + 1 + k)

λn+1 − λn+k+1 , λn − λn+k

(95)

and the proof is complete. Proof of Theorem 4. We only need to consider N ≥ 1. As explained in the proof of Theorem 3 it suffices to show that there exists a number θc such that the Ci ’s, defined inductively by Eqs. (89) and (90), are positive for each value of k = 0, 1, 2, . . . , provided θ ≥ θc . Note that for k = 0 we simply have γi = 21 θV  (λi ). We begin by discussing the case k = 0 and choose θc such that V  (λm ) ≥ 0

(96)

for θ ≥ θc and m = 0, 1, . . . . Then C0 ≥ 1 and it follows easily by induction that Cm ≥ m + 1 for m > 0. The case k = 1 follows from the proof of Theorem 2 since {λn } is by construction monotonically decreasing.

Existence and Stability of Noncommutative Scalar Solitons

67

  In general V  (λn ) is not a positive decreasing sequence for n ≥ 1 so the argument used in the proof of Theorem 3 does not generalize and we will need to use information about the asymptotic behaviour of the eigenvalues of ϕN as θ → ∞. We begin by analysing the asymptotic behaviour of the eigenvalues of ϕN regarded as functions of θ . By Theorem 1 we can write the eigenvalues as λi (θ ) = s − ri (θ ) , i = 0, 1, . . . , N, λi (θ ) = ri (θ ) , i = N + 1, N + 2, . . . ,

(97) (98)

where ri (θ) → 0 as θ → ∞ for all i. The potential function V is assumed to be C 2 and V  (0) > 0, V  (s) > 0 so the equation of motion (7) used for m = 0 implies that r0 (θ ) − r1 (θ ) = −

 θ   V (s) r0 (θ) + o (r0 (θ)) 2

(99)

which shows that θ r0 (θ ) → 0 as θ → ∞. Repeating this argument for the next values of m we find that θ ri (θ ) → 0, i = 0, 1, . . . N − 1.

(100)

Using (100) in the equation of motion for m = 0, 1, . . . , N − 1 we find by an analogous argument that θ 2 ri (θ ) → 0, i = 0, 1, . . . N − 2.

(101)

Continuing in the same vein we obtain θ N−i ri (θ ) → 0 as i = 0, 1, . . . N − 1.

(102)

Using (102) in Eq. (7) with m = N gives θ V  (λN (θ )) → −2 (N + 1) s,

(103)

which implies   2 (N + 1) s + o θ −1 .  V (s) θ

(104)

dN dN−1 d0 , rN−1 (θ ) ∼ , . . . , r0 (θ) ∼ N+1 , θ θ2 θ

(105)

rN (θ ) = Continuing this argument we find rN (θ ) ∼ where

dN =

2 (N + 1) s . V  (s)

(106)

We do not need the explicit values of di for i = 0, . . . N − 1. Using (105) in Eq. (7) with m = N + 1 yields rN+1 (θ ) ∼

dN+1 , θ

(107)

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B. Durhuus, T. Jonsson, R. Nest

where V  (0) dN+1 = V  (s) dN = 2 (N + 1) s .

(108)

Taking now m > N + 1 in Eq. (7) we find θ ri (θ ) → 0 as θ → ∞

(109)

for i ≥ N + 2. Continuing the analysis in the same fashion as for i ≤ N we obtain the bound   (110) ri (θ ) = O θ N−i for i ≥ N + 2. This completes our discussion of the behaviour of the eigenvalues of ϕN for large θ . We now use the asymptotic behaviour of the λi ’s to find the asymptotic behaviour of the γi ’s. This is a straightforward calculation using Eq. (5) and Eqs. (105)–(110). The results can be summarized as follows: k=2 m ≤ N − 2 : γm =

θ  V (s) + O (1) , 2

θ  V (0) + O (1) , 2   (N + 2) dN+1 + dN m = N − 1 : γm = − (N + 1) + + O θ −2 , sθ   N dN − dN+1 m = N : γm = − (N + 1) + + O θ −2 . sθ m ≥ N + 1 : γm =

(111) (112) (113) (114)

k≥3

θ  V (s) + O (1) , 2 θ m ≥ N + 1 : γm = V  (0) + O (1) , 2 (N + 1) dN + (N + 2) dN+1 m + k = N + 1 : γm = − (N + 1) + sθ   −2 +O θ ,   (N + 1) dN+1 + N dN m = N : γm = − (N + 1) + + O θ −2 , sθ   N dN δk3 + (N + 2) dN+1 m + k = N + 2 : γm = − + O θ −2 , sθ   (N + 2) dN+1 δk3 + N dN m = N − 1 : γm = − + O θ −2 , sθ   −2 All other cases : γm = O θ . m + k ≤ N : γm =

(115) (116)

(117) (118) (119) (120) (121)

All the correction terms to the above asymptotic expressions are uniform in k and m for θ ≥ θc and θc sufficiently large.

Existence and Stability of Noncommutative Scalar Solitons

69

We are now ready to show that Cm > 0 for all k ≥ 2 provided θ is sufficiently large. First, we note that it is an immediate consequence of the preceding asymptotic formulae and the recursion relations (89) and (90) that Cm > 0 for n ≤ N − k and θ ≥ θc , if θc is large enough. It is convenient to separate the discussion of the remaining values of m into two cases depending on whether N − k ≥ 0 or not. Case I. N − k ≥ 0. By Eqs. (111) and (115), C0 = k + 1 + γ 0 ≥ k + 1 +

θ  V (s) + O (1) . 2

(122)

Choosing θc sufficiently large we also have γ0 , . . . , γN−k > 0

(123)

and by induction Cm ≥ m + k + 1 + γ m ≥

θ  V (s) + O (1) 2

(124)

for m = 0, 1, . . . , N − k. I.a. Assume first that k = 2. Then we find, using the asymptotic formulae above,   (N + 2) dN+1 − (N − 2) dN CN−1 = N + (125) + O θ −2 sθ and


   4dN + N 2 + 3N + 4 dN+1 CN = + O θ −2 . N sθ

(126)

Choosing θc large enough CN−1 and CN are positive and CN+1 = 2 (N + 1) + 3 + γN+1 − ≥

2θ V  (0) + O (1) . N 2 + 3N + 4

(N + 1) (N + 3) CN (127)

For θ sufficiently large, CN+1 ≥ N + 2 and it follows by induction that Cm ≥ m + 1 for m ≥ N + 2 if θc is so large that γm ≥ 0 for m ≥ N + 2. I.b. Assume next that k = 3. Then we find by a calculation similar to the one in I.a:   3dN + (N + 2) dN+1 CN−2 = N − 1 + (128) + O θ −2 , sθ   3 (N + 2) (dN + dN+1 ) N dN CN−1 = N + (129) + O θ −2 , − sθ (N − 1) sθ (N + 2) (N + 3) (dN+1 + dN ) (N + 1) dN+1 − 3dN CN = +3 sθ N (N − 1) sθ   −2 +O θ , (130) CN+1 =

θ 18 (N + 1) V  (0) + O (1) . 2 (N + 1)3 + 11N + 17

(131)

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B. Durhuus, T. Jonsson, R. Nest

Choosing θc sufficiently large the above coefficients are all positive and taking θc so large that CN+1 ≥ N + 2 and γm ≥ 0 for m ≥ N + 2, we conclude by induction that all the Cm ’s are positive. I.c. Now we consider the case k ≥ 4. The calculation is analogous to the one given −1 above for k = 2 and k = 3. We evaluate  CN +1−k , CN+2−k , . . . CN to order θ and
find that CN+1−i = N + 2 − i + O θ −1 for i = 2, . . . , k and then

   (N + k) · · · (N + 2) dN+1 CN ≥ N + 1 + k (132) + O θ −2 , N · · · (N + 2 − k) sθ  

 θ  (N + k) · · · (N + 2) −1 CN+1 ≥ V (0) 1 − (N + 1 + k) N + 1 + k + O (1) . 2 N · · · (N + 2 − k) Noting that the coefficient of θ in the last expression is positive we proceed to show by induction as before that Cm > 0 for all m provided θc is chosen large enough. Case II. k ≥ N + 1. Again it is convenient to split the argument into different subcases. II.a. If N + 1 = k = 2 then from the asymptotic formulae we find   3d2 + d1 C0 = 1 + + O θ −2 , sθ   4d1 + 8d2 C1 = + O θ −2 , sθ θ  C2 ≥ V (0) + O (1) , 4

(133) (134) (135)

and the argument can be completed by induction as before, provided θc is taken large enough. II.b. In the case N = 1 and k ≥ 3 we find

  3d2 δk3 + d1 C0 = k + 1 − + O θ −2 , sθ 

  3δk3 d2 kd1 + O θ −2 . C1 = k + 2 − + 1 + k sθ (k + 1) sθ

(136) (137)

Choosing θc sufficiently large we find that C0 > 0, C1 ≥ 2 and γm > 0 for m ≥ 2. It follows as before that Cm ≥ m + 1 for m ≥ 2. II.c. Consider N + 1 = k = 3. The crucial coefficients in this case are C2 which is of order θ −1 and C3 which diverges at large θ. We find   33d3 + 27d2 C2 = (138) + O θ −2 sθ   198 ≥  (139) + O θ −2 , V (0) θ and consequently C3 =

9V  (0) θ + O (1) . 22

Taking θc large we can now complete the argument by induction as before.

(140)

Existence and Stability of Noncommutative Scalar Solitons

71

II.d. The case N = 2 and k ≥ 4 is quite similar to II.b. We omit the details which are straightforward. II.e. Consider the case N + 1 = k ≥ 4. We calculate the Cm inductively, starting with C0 and keeping terms to order θ −1 . We find eventually

   N dN (N + 1) (N + 2) · · · (2N ) dN + dN+1 CN−1 = N + − + O θ −2 , 2 · 3 · · · (N − 1) sθ sθ (141) and after a short calculation

   (N + 1) dN+1 (N + 2) (N + 3) · · · (2N + 1) CN ≥ 1+ + O θ −2 , sθ 2 · 3···N which implies CN+1



  θ  (2N + 1)! −1 + O (1) ≥ V (0) 1 − 2 1 + 2 N ! (N + 1)!

(142)

(143)

and allows us to complete the argument by induction provided θc is large enough. II.f. The remaining cases N ≥ 3 and k ≥ N + 2 are simpler than those discussed above. One finds that none of the Cm ’s approaches zero for large θ . We omit the details. This completes the proof of Theorem 4. We end this section by commenting briefly on how to extend the stability results to dimensions d > 1. Even though the eigenvalues of the rotationally invariant operators are degenerate in this case the extension of the formula (69) for the stability functional 4 is straightforward to derive if the potential V is analytic in a neighborhood of the interval [0, s], as we are assuming.  If we have a solution ϕ = λn Pn to Eq. (3), we find by the analytic functional calculus that  ∞

 θ 4 (ω) = 2n + d + V  (λn ) Pn ωPn 22 2 n=0

  θ V  (λn ) − V  (λm ) n+m+d + Pn ωPm 22 +2 2 λ − λ n m m
d  

(nk + 1) (mk + 1)n + δk |ω|m + δk n|ω|m ,

(144)

k=1 n,m

where, as usual, the Pn are the spectral projections of the number operator, and the standard harmonic oscillator basis vectors are |n, where n = (n1 , . . . , nd ) is a multi-index of non-negative integers. Furthermore, δ1 , . . . , δd denotes the standard orthonormal basis for Rd . We see that 4 only couples the matrix elements of ω diagonally, i.e., it suffices to show that 4 (ω) ≥ 0 for n|ω|m = 0

unless n − m = ±; ,

where ; is an arbitrary integer multi-index, with |;| ≡ ;1 + · · · + ;d ≥ 0.

(145)

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B. Durhuus, T. Jonsson, R. Nest

Consider first the case |;| = 0, in which the second sum on the right hand side of Eq. (144) does not contribute. If V  (λn ) ≥ 0 for all n, we have 4 (ω) ≥ 410 (ω) + · · · + 4d0 (ω) , where 4k0 (ω) =



(146)

(nk + mk + 1) |n|ω|m|2

|n|=|m|

−2

 

(nk + 1) (mk + 1)n + δk |ω|m + δk n|ω|m .

(147)

|n|=|m|

The contribution to this expression from any fixed values of ni and mi , for i = k, is a quadratic form in the the matrix elements n|ω|m = n1 , . . . , mk + ;k , . . . , nd |ω|m1 , . . . , mk , . . . , md  ,

(148)

that may be assumed to be real. It is a simple matter to verify that this quadratic form is positive definite. Therefore, so is 4 (ω) for |;| = 0, provided the condition V  (λn ) ≥ 0 holds. For |;| = 0 the first sum on the right-hand side of Eq. (144) does not contribute. For the coefficient of Pn ωPm 22 in the second sum one obtains the value (n + m + d)

*λn − *λm+1 *λn+1 − *λm λn+1 − λm+1 +n +m λn − λ m λn − λ m λn − λ m

(149)

by using Eq. (3) in the form (n + d) *λn − n*λn−1 =

θ  V (λn ) , 2

(150)

where *λ = λn+1 − λn , n ≥ 1. This allows us to write 1 4 (ω) = 41 (ω) + · · · + 4d (ω) , 2 where 4k (ω) =

 

(nk + mk + 1)

n=m+;

(151)

λn+1 − λm+1 λn − λ m

 *λn − *λm+1 *λn+1 − *λm |n|ω|m|2 + nk + mk λn − λ m λn − λ m   −2 (nk + 1) (mk + 1)n + δk |ω|m + δk n|ω|m .

(152)

n=m+;

Considering terms with fixed values of ni , mi , i = k, in this expression one obtains a quadratic form in the matrix elements that can be handled by an analysis similar to the one that was carried out for the case d = 1. We do not elaborate further on the general case here but note that the analysis of the one-soliton case, N = 0, of Theorem 3, generalises immediately to 4k . This result is obtained by observing that the sequence {*λn } is again decreasing in this case as a consequence of Eq. (150) since V  (λn ) > 0 for n ≥ 1. Thus, Theorem 3 also holds for d > 1.

Existence and Stability of Noncommutative Scalar Solitons

73

5. Nonexistence of Smooth Families In [28] we proved that rotationally symmetric solutions to Eq. (3) do not exist for sufficiently small values of θ. The purpose of this section is to prove non-existence of smooth families of solutions for small θ without assuming rotational symmetry. By a smooth family of solutions we mean a mapping from an interval I ⊂ R to H2,2 , I θ → ϕθ ∈ H2,2 ,

(153)

which is continuously differentiable in the norm topology of H2,2 . The proof is based on three lemmas below which are most conveniently established by representing operators by functions via a quantization map. The Weyl or Weyl-Wigner quantization is perhaps the best known quantization map. It can be defined as the mapping πW which to
a function f (x, p) of 2d variables, x, p ∈ Rd , associates an  2 d operator πW (f ) on L R whose kernel KW (f ) is given by 

x+y −d KW (f ) (x, y) = (2π ) , p ei(x−y)·p dp . f (154) 2 Rd It is obvious that πW maps Schwartz functions on R2d bijectively onto operators whose kernels are Schwartz functions and also maps tempered distributions onto operators whose kernels are tempered distributions.
 More important for the following is the easily verifiable fact that πW maps L2 R2d isometrically (up to a factor (2π )d/2 ) onto the
 space of Hilbert-Schmidt operators on L2 Rd , πW (f ) 22 = |KW (f ) (x, y) |2 dxdy = (2π )−d |f (x, p) |2 dx dp. (155) R2d

R2d

We shall find it more convenient to use the so-called Kohn-Nirenberg quantization π for which the kernel K (f ) of π (f ) is given by −d f (x, p) ei(x−y)·p dp. (156) K (f ) (x, y) = (2π) Rd

The quantization map π clearly has the same properties as the ones we described for πW above. Likewise, the following properties of π are shared by πW except for the last one: (a) If π (f ) is of trace class then −d Tr π (f ) = K (f ) (x, x) dx = (2π ) Rd

R2d

f (x, p) dx dp .

(157)

(b) If g depends only on x we have π (g (x)) = g (x) , where the right-hand side is to be interpreted as a multiplication operator. (c) If h depends only on p we have

 1 π (h (p)) = h ∇x . i

(158)

(159)

74

B. Durhuus, T. Jonsson, R. Nest

(d) If g and h are as above, then



π (g (x) f (x, p) h (p)) = g (x) π (f ) h

1 ∇x i

 .

(160)

From (b) and (c) it follows that  1
1 ak = √ xk + ∂xk = √ π (xk + ipk ) 2 2

(161)

 1
1 ak∗ = √ xk − ∂xk = √ π (xk − ipk ) . 2 2

(162)

and

From the definition of π one then obtains  1
[ak , π (f )] = √ π ∂xk f + i∂pk f 2

(163)

and 

 −1
 ak∗ , π (f ) = √ π ∂xk f − i∂pk f . 2

Consequently, 2

 k

 ak∗ , [ak , π (f )] = π (*f ) ,

(164)

(165)

where * is the Laplace operator on R2d , and the (complexification of) the space D introduced in Sect. 2 is just the image under π of the domain of definition of the selfadjoint operator *. Notice, however, that contrary to πW the quantization map π does not generally map real-valued functions to self-adjoint operators. There
 is to our knowledge no known simple characterisation of the subspace of L2 R2d consisting of functions f such that π (f ) is of trace class. We shall need the following result, depending crucially on property (d) above, concerning such functions. Here  · 1 denotes the standard trace norm. Lemma 4. Suppose f is a square integrable function such that π (f ) is of trace class. Then its Fourier transform F (f ) is bounded and its uniform norm F (f ) ∞ satisfies the inequality F (f ) ∞ ≤ π (f ) 1 . (166)  
Proof. First, note that π e−iξ ·x = e−iξ ·x and π e−ip·η = e−η·∇x are unitary operators. Hence,   π e−iξ ·x f (x, p) e−ip·η = e−iξ ·x π (f ) e−η·∇x (167)


is of trace class and using properties (a) and (d) above we have F (f ) (ξ, η) = e−iξ ·x f (x, p) e−ip·η dx dp R2d      = Tr π e−iξ ·x f (x, p) e−ip·η = Tr e−iξ ·x π (f ) e−η·∇x , (168)

Existence and Stability of Noncommutative Scalar Solitons

75

and hence |F (f ) (ξ, η) | ≤ Tr (|π (f ) |) = π (f ) 1 ,

(169)

which proves the assertion. Using the above result we get the following a priori estimate relating the HilbertSchmidt and trace norms of any solution of Eq. (3). Lemma 5. There exists a constant C, depending only on V , such that any solution ϕ of Eq. (3) fulfills d

ϕ2 ≤ Cθ 2 ϕ1 .

(170)

Proof. Since both ϕ and V  (ϕ) are Hilbert-Schmidt there exist square integrable functions f and F such that ϕ = π (f ) and V  (ϕ) = π (F ). By Eq. (165) the equation of motion (3) may be written as *f + θ F = 0

(171)

or, equivalently, F (f ) (ξ, η) =

|ξ |2

−θ F (F ) (ξ, η) . + |η|2

(172)

Using Lemma 4 and the fact that for an appropriate constant c, F (F ) L2 = (2π )2d V  (ϕ) 2 ≤ cϕ2 ,

(173)

we get (2π)d ϕ22 = F (f ) 2L2 2 = |F (f ) | dξ dη + |ξ |2 +|η|2 ≤δ 2

=

|ξ |2 +|η|2 ≤δ 2

|ξ |2 +|η|2 >δ 2

|F (f ) |2 dξ dη



|F (f ) |2 dξ dη + θ 2

≤ const δ 2d F (f ) 2∞ + ≤ const δ 2d ϕ21 + c

|ξ |2 +|η|2 >δ 2

|F (F ) |2




|ξ |2 + |η|2

2 dξ dη

θ2 F (F ) 2L2 δ4

θ2 ϕ22 δ4

(174)

for some constant c. If we now let δ 4 = cθ 2 , the result follows. Our next goal is to obtain a lower bound on the Hilbert-Schmidt norm of solutions to Eq. (3). Lemma 6. There exists a constant C  , depending only on the potential V , such that any non-zero solution ϕ of Eq. (3) satisfies the inequality d

C  θ − 2 ≤ ϕ2 .

(175)

76

B. Durhuus, T. Jonsson, R. Nest

Proof. Let ϕ =



n λn P n

ϕ
be the spectral decomposition of ϕ, and set, for a > 0,   = λn Pn and ϕ≥a = λn P n . (176) λn
λn ≥a

By our assumptions about V we can fix a > 0 and a constant c1 such that V  (ϕ
(179)

where c3 = c1 c2 . From this we deduce ϕ1 = ϕ
(180)

where c4 = (1 + c3 ) /a. Finally, from (180) and the a priori estimate of Lemma 5, we get d

ϕ1 ≤ Cc4 θ 2 ϕ2 ϕ1

(181)

from which the claimed inequality follows. We are now in a position to prove the announced non-existence result. Theorem 5. Let V be analytic on a neighbourhood of the interval [0, s]. Suppose (a , b] θ → ϕθ ∈ H2,2 ,

(182)

where 0 ≤ a < b, is a smooth map such that ϕθ is a nonzero solution of the equation of motion (3) for each θ ∈ (a, b). Then a > 0. Proof. Since ϕθ is a solution to Eq. (3) the derivative of the energy S (ϕθ ) with respect to θ is given by d S (ϕθ ) = Tr V (ϕθ ) . dθ

(183)

This is easy to prove using the analytic functional calculus. Since V is positive definite, it satisfies an estimate of the form V (ϕ) ≥ const ϕ 2 ,

(184)

Existence and Stability of Noncommutative Scalar Solitons

77

and hence, by Lemma 6, d S (ϕθ ) ≥ CV θ −d , dθ

(185)

where the constant CV depends only on V (but not on the given family of solutions). Hence, for d > 1, the function θ → S (ϕθ ) +

CV −d+1 θ d −1

(186)

is increasing. Now suppose that a = 0. Then S (ϕθ ) ≤ S (ϕb ) +

 CV  −d+1 − θ −d+1 b d −1

(187)

which contradicts positivity of S (ϕθ ) for small θ. CV −d+1 θ in (186) should be replaced by −CV ln θ and For d = 1 the expression d−1 the same conclusion holds. This proves the theorem.

Acknowledgements. The work of B. D. is supported in part by MatPhySto funded by the Danish National Research Foundation. This research was partly supported by TMR grant no. HPRN-CT-1999-00161. T. J. is indebted to the Niels Bohr Institute and the CERN Theory Division for hospitality.

References 1. Connes, A.: Noncommutative Geometry. San Diego: Academic Press, 1994 2. Madore, J.: An Introduction to Noncommutative Differential Geometry and its Physical Applications. 2nd edn., Cambridge: Cambridge University Press, 1999 3. Gracia-Bondia, J.M., Varilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Boston: Birkh¨auser, 2000 4. Duplij, S., Wess, J., (eds.): Noncommutative Structures in Mathematics and Physics. NATO Science Series II, Mathematics, Physics and Chemsitry, Vol. 22, Dordrecht: Kluwer Academic Publishers, 2001 5. Konechny, A., Schwarz, A.: Introduction to (M)atrix theory and noncommutative geometry. Phys. Rep. 360, 354–465 (2002) 6. Douglas, M.R., Nekrasov, N.A.: Noncommutative field theory. Rev. Mod. Phys. 73, 977–1029 (2002) 7. Szabo, R.J.: Quantum field theory on noncommutative spaces. hep-th/0109162 8. Hoppe, J.: Membranes and integrable systems. Phys. Lett. B 250, 44 (1990) 9. Connes, A., Douglas, M., Schwarz, A.: Noncommutative geometry and matrix theory: Compactification on tori. JHEP 02, 003 (1998) [hep-th/9711162] 10. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 06, 032 (1999) [hepth/9908142] 11. Polychronakos, A.P.: Flux tube solutions in noncommutative gauge theories. Phys. Lett. B 495, 407 (2000) [hep-th/0007043] 12. Aganagic, M., Gopakumar, R., Minwalla, S., Strominger, A.: Unstable solitons in noncommutative gauge theory. JHEP 0104, 001 (2001) [hep-th/0009142] 13. Bak, D.: Exact solutions of multi-vortices and false vacuum bubbles in noncommutative AbelianHiggs theories. Phys. Lett. B 495, 251 (2000) [hep-th/0008204] 14. Gross, D.J., Nekrasov, N.A.: Solitons in noncommutative gauge theory. JHEP 0103, 044 (2001) [hep-th/0010090] 15. Harvey, J.A., Kraus, P., Larsen, F.: Exact noncommutative solitons. JHEP 0012, 024 (2000) [hepth/0010060] 16. Gopakumar, R., Minwalla, S., Strominger, A.: Noncommutative solitons. JHEP 0005, 020 (2000) [hep-th/0003160] 17. Derrick, G.: Comments on nonlinear wave equations as models for elementary partcles. J. Math. Phys. 5, 1252 (1964)

78

B. Durhuus, T. Jonsson, R. Nest

18. Zhou, C.-G.: Noncommutative scalar solitons at finite θ . hep-th/0007255 19. Gorsky, A.S., Makeenko, Y.M., Selivanov, K.G.: On noncommutative vacua and noncommutative solitons. Phys. Lett. B 492, 344 (2000) [hep-th/0007247] 20. Lindstrøm, U., Rocek, M., von Unge, R.: Non-commutative soliton scattering. JHEP 0012, 004 (2000) [hep-th/0008108] 21. Solovyov, A.: On noncommutative solitons. Mod. Phys. Lett. A 15, 2205 (2000) [hep-th/0008199] 22. Bak, D., Lee, K.: Elongation of moving noncommutative solitons. Phys. Lett. B 495, 231 (2000) [hep-th/0007107] 23. Matsuo, Y.: Toplogical charges of noncommutative soliton. Phys. Lett. B 499, 223 (2001) [hepth/0009002] 24. Gopakumar, R., Headrick, M., Spradlin, M.: On noncommutative multi-solitons. hep-th/0103256 25. Araki, T., Ito, K.: Scattering of noncommutative (n, 1) solitons. hep-th/0105012 26. Spradlin, M., Volovich, A.: Noncommutative solitons on K¨ahler manifolds. hep-th/0106180 27. Harvey, J.A.: Komaba lectures on noncommutative solitons and D-branes. hep-th/0102076 28. Durhuus, B., Jonsson, T., Nest, R.: Noncommutative scalar solitons: Existence and nonexistence. Phys. Lett. B 500, 320 (2001) [hep-th/0011139] 29. Jackson, M.G.: The stability of noncommutative scalar solitons. hep-th/0103217 30. Hadasz, L., Lindstr¨om, U., Rocek, M., von Unge, R.: Noncommutative multisolitons: Moduli spaces, quantization, finite θ effects and stability. hep-th/0104017 31. Magnus, R.: A singular perturbation problem and the existence of non-commutative, non-rotationally-symmetric scalar solitons. Preprint, Science Institute, University of Iceland, 2001 32. Durhuus, B., Jonsson, T.: A note on noncommutative scalar solitons. hep-th/0204096, to appear in Phys. Lett. B 33. Kato, T.: Perturbation Theory for Linear Operators. New York: Springer, 1966 34. Austing, P., Jonsson, T., Thorlacius, L.: Scalar solitons on the fuzzy sphere. hep-th/0206060 Communicated by A. Connes

Commun. Math. Phys. 233, 79–136 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0755-7

Communications in

Mathematical Physics

Vertex Algebras, Mirror Symmetry, and D-Branes: The Case of Complex Tori Anton Kapustin1 , Dmitri Orlov2 1

School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA. E-mail: [email protected] 2 Algebra Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin str., GSP-1, Moscow 117966, Russia. E-mail: [email protected] Received: 3 May 2001 / Accepted: 17 August 2002 Published online: 8 January 2003 – © Springer-Verlag 2003

Abstract: A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat K¨ahler metric and a closed 2-form we associate an N = 2 superconformal vertex algebra (N = 2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N = 2 SCVA’s. We show that for algebraic tori the isomorphism of N = 2 SCVA’s implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N = 2 SCVA’s related by a mirror morphism. If the 2-form is of type (1, 1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich’s conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be “twisted” by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . 1.1 Physicist’s mirror symmetry . . . . . . . . . 1.2 Homological mirror symmetry . . . . . . . . 1.3 Vertex algebras and chiral algebras . . . . . . 2. Summary of Results . . . . . . . . . . . . . . . . . 2.1 Physicist’s mirror symmetry for complex tori 2.2 Applications to homological mirror symmetry 2.3 Physical applications . . . . . . . . . . . . . 3. Superconformal Vertex Algebras . . . . . . . . . .

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3.1 Quantum fields . . . . . . . . . . . . . . . . . . . . 3.2 The definition of a vertex algebra . . . . . . . . . . . 3.3 Conformal vertex algebras . . . . . . . . . . . . . . 3.4 N = 1 superconformal vertex algebras . . . . . . . . 3.5 N = 2 superconformal vertex algebras . . . . . . . . N = 2 SCVA of a Flat Complex Torus . . . . . . . . . . . 4.1 Vertex algebra structure . . . . . . . . . . . . . . . . 4.2 N = 2 superconformal structure . . . . . . . . . . . Morphisms of Toroidal Superconformal Vertex Algebras . 5.1 Isomorphisms of N = 1 SCVA’s . . . . . . . . . . . 5.2 Isomorphisms of N = 2 SCVA’s . . . . . . . . . . . 5.3 Mirror morphisms of N = 2 SCVA’s . . . . . . . . . Homological Mirror Symmetry with B-Fields . . . . . . . 6.1 Mirror symmetry and D-branes . . . . . . . . . . . . 6.2 Fukaya category with a B-field . . . . . . . . . . . . Supersymmetric σ -Model of a Flat Torus . . . . . . . . . . The Relation Between Vertex Algebras and Chiral Algebras Projectively Flat Connections and the Fundamental Group

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1. Introduction 1.1. Physicist’s mirror symmetry. A physicist’s Calabi Yau is a triple (X, G, B), where X is a compact complex manifold with a trivial canonical bundle, G is a Ricci-flat K¨ahler metric on X, and B is a class in H 2 (X, R/Z) which is in the kernel of the Bockstein homomorphism H 2 (X, R/Z) → H 3 (X, Z). The class B can be lifted to a class b ∈ H 2 (X, R), and a closed 2-form B representing it is known as the B-field. Physicists believe that there is a procedure which associates to any such triple an N = 2 superconformal vertex algebra (N = 2 SCVA). The precise definition of an N = 2 SCVA is rather complicated and will be given in Sect. 3. Roughly speaking, it is a Euclidean quantum field theory on a two-dimensional manifold R × S1 whose Hilbert space is acted upon by a unitary representation of the infinite-dimensional Lie super-al1 ¯± gebra with even generators Ln , L¯ n , Jn , J¯n , n ∈ Z, odd generators Q± r , Qr , r ∈ Z + 2 , and the following nonvanishing Lie brackets: d 3 (m − m)δm,−n , 4 d [L¯ m , L¯ n ] = (m − n)L¯ m+n + (m3 − m)δm,−n , 4 [Lm , Jn ] = −nJn+m , [Jm , Jn ] = dmδ
m m,−n, ] = − r Q± [Lm , Q± r+m , r 2± ± [Jm , Qr ] = ±Qr+m ,   + − 1 1 d r2 − Qr , Qs = Lr+s + (r − s)Jr+s + 4 8 8  + − 1 ¯r ,Q ¯ s = L¯ r+s + 1 (r − s)J¯r+s + d r 2 − Q 4 8 8 [Lm , Ln ]

= (m − n)Lm+n +

[L¯ m , J¯n ] = −nJ¯n+m , [J¯m , J¯n ] =
dmδm,−n, m ¯± ¯± [L¯ m , Q ] = −r Q r+m , r 2± ± ¯ ¯ ¯  [Jm , Qr ] = ±Qr+m , 1 δr,−s , 4 1 δr,−s . 4 (1)

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Here d = dimC X, and {·, ·} denotes the anti-commutator. This algebra is a direct sum of two copies of the celebrated N = 2 super-Virasoro algebra with central charge c = 3d. If one omits Jn , J¯n and all the odd generators from the definition of the N = 2 SCVA, one gets a structure which we call a conformal vertex algebra (CVA), and which is also known as a conformal field theory on R×S1 ∼ = C∗ . Thus an N = 2 SCVA is a conformal ∗ field theory on C with some additional structure. Heuristically, the construction of an N = 2 SCVA from a triplet (X, G, B) proceeds as follows. To any K¨ahler manifold (X, G) equipped with a closed 2-form B one can associate a two-dimensional classical field theory on R × S1 , the so-called N = 2 supersymmetric σ -model. For the reader’s convenience, the definition of the σ -model is given in Appendix A. The space of solutions of the corresponding classical equations of motion is an infinite-dimensional symplectic supermanifold with a symplectic action of two copies of the N = 2 super-Virasoro algebra with zero central charge (see [29, 8], and Appendix A). It can be argued that consistent quantization of this classical field theory is possible only for c1 (TX ) ≥ 0, e.g. when X is a Fano manifold or a Calabi-Yau manifold. In the Fano case (c1 (TX ) > 0) the quantized σ -model is an N = 2 field theory, but not a superconformal one, because only a finite-dimensional subalgebra of the classical N = 2 super-Virasoro algebra survives quantization. The same happens if c1 (TX ) = 0 but G is not Ricci-flat. If c1 (TX ) = 0 and G is Ricci flat, both N = 2 super-Virasoro algebras survive quantization (though the central charges become nonzero), and therefore the quantized σ -model is an N = 2 superconformal field theory, i.e. an N = 2 SCVA. One can also argue that this N = 2 SCVA in fact depends only on the image of B in H 2 (X, R/Z), i.e. on B. The actual quantization of the σ -model is feasible only for very special (X, G, B). In particular, if X is a complex torus, the corresponding N = 2 SCVA can be constructed for any flat G and any B ∈ H 2 (X, R/Z). The quantization of the σ -model for a flat complex torus is sketched in Appendix A. Two physicist’s Calabi-Yaus are said to be mirror if there exists an isomorphism of the corresponding conformal vertex algebras which acts on the algebra (1) as the so-called mirror involution: Ln → Ln ,

∓ Q± r → Qr ,

Jn → −Jn ,

L¯ n → L¯ n ,

¯± Q r

J¯n → J¯n .

→

¯± Q r ,

(2)

Such a morphism of N = 2 SCVA’s will be called a mirror morphism. Mirror symmetry defined in this way acts pointwise on the moduli space of physicist’s Calabi-Yaus. If one drops G and B from the definition of a physicist’s Calabi-Yau, then mirror symmetry becomes a correspondence between two families of K¨ahler manifolds with a trivial canonical bundle whose Hodge numbers are related by hp,q = h d−p,q . The latter notion of mirror symmetry is much weaker than the physicist’s mirror symmetry. Nevertheless, much of the mathematical work on mirror symmetry up to now has focused on this weaker notion, since it proved hard to make sense of the σ -model. As mentioned above, the quantum σ -model is manageable when X is a complex torus, so one could hope to understand mirror symmetry in detail in this particular case. This is what this paper aims to do. Although from the physical point of view mirror symmetry for complex tori appears to be rather trivial, we will see that its study sheds considerable light on the Homological Mirror Symmetry Conjecture (HMSC), a subject to which we now turn.

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1.2. Homological mirror symmetry. String theory makes highly nontrivial predictions about the enumerative geometry of a Calabi-Yau X in terms of its mirror X  . The success of these predictions seems quite mysterious from a purely mathematical standpoint. In an insightful paper [23], M. Kontsevich formulated a conjecture which relates the properties of a Calabi-Yau with those of its mirror and suggested that it captures the essence of mirror symmetry. Subsequently this conjecture was reinterpreted in physical terms [32]. In this subsection we recall the main features of Kontsevich’s conjecture. Let X be a complex algebraic variety (or a complex manifold). Denote by OX the sheaf of regular functions (or the sheaf of holomorphic functions). Recall that a coherent sheaf is a sheaf of OX–modules that locally can be represented as a cokernel of a morphism of holomorphic vector bundles. Coherent sheaves form an abelian category which will be denoted by Coh(X). To any abelian category we can associate a certain triangulated category called the bounded derived category. We denote by D b (X) the bounded derived category of coherent sheaves on X. Roughly speaking, the category D b (X) is a factor-category of the category of bounded complexes of coherent sheaves by the subcategory of acyclic complexes (i.e. complexes with trivial cohomology sheaves). On the other hand, it has been proposed [12, 23] that to any compact symplectic manifold Y one can associate a certain category whose objects are (roughly speaking) vector bundles on Lagrangian submanifolds equipped with unitary flat connections. The morphisms in this category have been defined when Lagrangian submanifolds intersect transversally. This conjectural category is called the Fukaya category and denoted F(Y ). The category F(Y ) is not an abelian category; rather, it is supposed to be an A∞-category equipped with a shift functor. For an introduction to A∞-categories see [21]. An A∞–category is not a category in the usual sense, because the composition of morphisms is not associative. The set of morphisms between two objects in an A∞–category is a differential graded vector space. To any A∞–category one can associate a true category which has the same objects but the space of morphisms between two objects is the 0th cohomology group of the morphisms in the A∞–category. Applying this construction to F(Y ), we obtain a true category F0 (Y ) which is also called the Fukaya category. Kontsevich [23] also constructs a certain triangulated category DF0 (Y ) out of F(Y ). We will call it the derived Fukaya category. Conjecturally, the category F0 (Y ) is a full subcategory of DF0 (Y ). A physicist’s Calabi-Yau (X, G, B) is both a complex manifold and a symplectic manifold (the symplectic form being the K¨ahler form ω = GI ). Thus we can associate to it a pair of triangulated categories D b (X) and DF0 (X). The Homological Mirror Symmetry Conjecture (HMSC) asserts that if two algebraic Calabi-Yaus (X, G, B) and (X , G , B  ) are mirror to each other, then D b (X) is equivalent to DF0 (X  ), and vice versa. The Homological Mirror Symmetry Conjecture can be reinterpreted in physical terms. To every N = 2 superconformal field theory one can associate the set of BPS D-branes, or more precisely two sets: the set of A-type D-branes and the set of B-type D-branes. This is reviewed in more detail in Sect. 6. These sets are equipped with a rather intricate algebraic structure: that of an A∞–category. This structure encodes the properties of correlators in a topological open string theory (see [16] and references therein). A mirror morphism between two N = 2 superconformal field theories identifies the A-type D-branes of the first theory with the B-type D-branes of the second theory, and vice versa. Now suppose that an N = 2 superconformal field theory originates from a physicist’s Calabi-Yau (X, G, B). In this case there is evidence that A-type D-branes are closely related to objects of the Fukaya category, while B-type D-branes are related

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to coherent sheaves on X. To prove the Homological Mirror Symmetry Conjecture it would be sufficient to show that the derived Fukaya category of (X, G, B) (resp. the derived category of X) can be recovered from the A∞-category of A-type D-branes (resp. B-type D-branes). Conversely, proving the HMSC would likely result in an improved understanding of BPS D-branes. So far the Homological Mirror Symmetry Conjecture (with some important modifi = 1, i.e. cations, see Sect. 6 for details) has been proved only for dimC X = dim XC for the elliptic curve [31]. Two features make this case particularly manageable. First, the N = 2 SCVA for the elliptic curve is known, so one knows the precise conditions under which (X, G, B) is mirror to (X , G , B  ). Second, all objects and morphisms in the Fukaya category can be explicitly described. In this paper we perform a check of the HMSC for the case when both X and X are algebraic tori of arbitrary dimension. We will see that for algebraic tori of dimension higher than one the HMSC as formulated by Kontsevich can not be true in general. The main reason is that both the derived category of coherent sheaves and the derived Fukaya category do not depend on the B-field, while in the physical mirror symmetry it plays an essential role. However, a certain modification of the HMSC which takes into account the B-field passes our check and has a good chance to be correct. This modification is suggested both by our results on the N = 2 SCVA for complex tori, and by consideration of BPS D-branes. The modified HMSC conjecture is formulated in Sect. 6. It reduces to the original HMSC when the B-field vanishes for both manifolds related by the mirror morphism. The implications of our results for BPS D-branes on Calabi-Yau manifolds are briefly described in Sect. 2 and in more detail in Sect. 6. 1.3. Vertex algebras and chiral algebras. Vertex algebras play a key role in physicist’s mirror symmetry. A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. In the mathematical literature the terms vertex algebra and chiral algebra are used interchangeably. Roughly speaking, a chiral algebra is a vector superspace V together with a map Y : V → EndV [[z, z−1 ]] satisfying a number of properties [19]. One says that Y maps states to quantum fields. The definition of a chiral algebra first appeared in the work of Borcherds [6], but its origins go back to the classic paper of Belavin, Polyakov, and Zamolodchikov [5] where an algebraic approach to two-dimensional conformal field theory was proposed. From a physical viewpoint, chiral algebras are conformal field theories such that all fields are meromorphic (do not depend on z¯ ). Only very special conformal field theories have this property. Moreover, a generic conformal field theory does not factorize as a tensor product of two chiral algebras, one depending on z and another on z¯ , despite some claims to the contrary in the physics literature. For example, the quantization of the σ -model associated to a flat torus yields a conformal field theory which factorizes in this manner only for very special values of G and B. Thus in order to give a precise meaning to physicist’s mirror symmetry, we need to find a sufficiently general definition of a vertex algebra allowing for fields which depend both on z and z¯ . To avoid confusion, we will refer to these more general objects as vertex algebras, while vertex algebras in the sense of [19] will be called chiral algebras. Once both z and z¯ are allowed, they need not enter only in integer powers, so Y will take values in a space of “fractional power series in z and z¯ with coefficients in End(V )”, rather than in End(V )[[z, z¯ , z−1 , z¯ −1 ]]. The necessity of fractional powers can be seen by inspecting the conformal field theories associated to flat tori. Because

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of this, the definition of a vertex algebra is not a trivial extension of the definition of a chiral algebra. We hope that our definition of a vertex algebra will be of some interest to physicists as well as mathematicians. Its advantage over the more standard definitions of conformal field theory is that it is purely algebraic and based on the notion of Operator Product Expansion (OPE). In contrast, other rigorous definitions take Wightman axioms as a starting point. These axioms have an analytic flavor and do not make reference to OPE. In fact, the existence of OPE does not follow from Wightman axioms (except in some very special cases), and has to be postulated separately. Another advantage of our definition is that it does not require an inner product on the state space. Thus it is capable of describing “non-unitary” conformal field theories which find applications in statistical mechanics. 2. Summary of Results 2.1. Physicist’s mirror symmetry for complex tori. Let T be a 2d-dimensional real torus U/ &, where U ∼ = R2d is a real vector space, and & ∼ = Z2d is a lattice in U. Let I be a (constant) complex structure on T , G be a flat K¨ahler metric on T , and b ∈ H 2 (T , R). We will represent b by a constant 2-form B which is uniquely determined by b. In this simple case there is a well-known explicit construction of the corresponding N = 2 SCVA which we denote V ert (&, I, G, B). We review this construction in Sect. 4. The relation of this construction to the quantized σ -model is explained in Appendix A. Our first result describes when two different quadruples (&, I, G, B) and (&  , I  , G , B  ) yield isomorphic N = 2 SCVA’s. To state it, we first introduce some notation. Let & ∗ = Hom(&, Z) be the dual lattice in U ∗ , and T ∗ be the dual torus U ∗ / & ∗ . There is natural pairing l : & ⊕ & ∗ → Z. There is also a natural Z-valued symmetric bilinear form q on & ⊕ & ∗ defined by q((w1 , m1 ), (w2 , m2 )) = l(w1 , m2 ) + l(w2 , m1 ),

w1,2 ∈ &, m1,2 ∈ & ∗ .

Given G, I, B, we can define two complex structures on T × T ∗ :   I 0 , I(I, B) = BI + I t B −I t   −I G−1 B I G−1 J (G, I, B) = . GI − BI G−1 B BI G−1

(3) (4)

The notation here is as follows. We regard I and J as endomorphisms of U ⊕ U ∗ , and write the corresponding matrices in the basis in which the first 2d vector span U, while the remaining vectors span U ∗ . In addition, G and B are regarded as elements of Hom(U, U ∗ ), and I t denotes the endomorphism of U ∗ conjugate to I. It is easy to see that J depends on G, I only in the combination ω = GI, i.e. it depends only on the symplectic structure on T and the B-field. There is also a third natural complex structure I˜ on T × T ∗ , which is simply the complex structure that T × T ∗ gets because it is a Cartesian product of two complex manifolds:   I 0 ˜ I= . 0 −I t This complex structure will play only a minor role in what follows. Note that I coincides with I˜ if and only if B (0,2) = 0.

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Theorem 2.1. V ert (&, I, G, B) is isomorphic to V ert (&  , I  , G , B  ) if and only if there  exists an isomorphism of lattices & ⊕ & ∗ and &  ⊕ & ∗ which takes q to q  , I to I  , and  J to J . Our second result describes when (T , I, G, B) is mirror to (T  , I  , G , B  ). Theorem 2.2. V ert (&, I, G, B) is mirror to V ert (&  , I  , G , B  ) if and only if there is  an isomorphism of lattices & ⊕ & ∗ and &  ⊕ & ∗ which takes q to q  , I to J  , and J to I . 2.2. Applications to homological mirror symmetry. Let us now explain the implications of these results for the HMSC. First, note that if both B and B  are of type (1, 1), the criterion for mirror symmetry is identical to the one proposed in [14]. In that work, this criterion was taken as a definition of mirror symmetry for algebraic tori. We now see that this definition agrees with the physical notion of mirror symmetry and can be generalized to non-algebraic tori and arbitrary B-fields. Second, Theorem 2.1 allows us to make a check of the HMSC. Suppose the tori (T1 , I1 , G1 , B1 ) and (T2 , I2 , G2 , B2 ) are both mirror to (T  , I  , G , B  ). Then V ert (&1 , I1 , G1 , B1 ) is isomorphic to V ert (&1 , I1 , G1 , B1 ), and by Theorem 2.1 there is an isomorphism of lattices &1 ⊕ &1∗ and &2 ⊕ &2∗ which intertwines q1 and q2 , I1 and I2 , and J1 and J2 . On the other hand, if we assume that both (T1 , I1 ) and (T2 , I2 ) are algebraic, then HMSC implies that D b ((T1 , I1 )) is equivalent to D b ((T2 , I2 )). The criterion for this equivalence is known [30, 28]: it requires the existence of an isomorphism of &1 ⊕ &1∗ and &2 ⊕ &2∗ which intertwines q1 and q2 , and I˜1 and I˜2 . Clearly, since I = I˜ in general, this condition is in conflict with the one stated in the end of the previous paragraph. Instead, we only have the following result: Corollary 2.3. If V ert (&1 , I1 , G1 , B1 ) is isomorphic to V ert (&2 , I2 , G2 , B2 ), both (T1 , I1 ) and (T2 , I2 ) are algebraic, and both B1 and B2 are of type (1, 1), then D b ((T1 , I1 )) is equivalent to D b ((T2 , I2 )). In Sect. 5 we also prove the following result. Theorem 2.4. Let (T1 , I1 , G1 , B1 ) be a complex torus equipped with a flat K¨ahler metric and a B-field of type (1, 1). Let (T2 , I2 ) be another complex torus. Let I˜ 1 and I˜ 2 be the product complex structures on T1 ×T1∗ and T2 ×T2∗ . Suppose there exists an isomorphism of lattices g : &1 ⊕ &1∗ → &2 ⊕ &2∗ mapping q1 to q2 and I˜ 1 to I˜ 2 . Then on T2 there exists a K¨ahler metric G2 and a B-field B2 of type (1, 1) such that V ert (&1 , I1 , G1 , B1 ) is isomorphic to V ert (&2 , I2 , G2 , B2 ) as an N = 2 SCVA. Combining this with Theorem 2.1 and the criterion for the equivalence of D b ((T1 , I1 )) and D b ((T2 , I2 )), we obtain a result converse to Corollary 2.3. Corollary 2.5. Let (T1 , I1 , G1 , B1 ) be an algebraic torus equipped with a flat K¨ahler metric and a B-field of type (1, 1). Let (T2 , I2 ) be another algebraic torus. Suppose D b ((T1 , I1 )) is equivalent to D b ((T2 , I2 )). Then on T2 there exists a K¨ahler metric G2 and a B-field B2 of type (1, 1) such that V ert (&1 , I1 , G1 , B1 ) is isomorphic to V ert (&2 , I2 , G2 , B2 ) as an N = 2 SCVA.

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If dimC T = 1, then the B-field is automatically of type (1, 1). Therefore the HMSC passes the check in this special case. Of course, this is what we expect, since the HMSC is known to be true for the elliptic curve [31]. On the other hand, for dimC T > 1 we seem to have a problem. Not all is lost however, and a simple modification of the HMSC passes our check. The modification involves replacing (T , I ) with a noncommutative algebraic variety, or more precisely, replacing the structure sheaf of (T , I ) with an Azumaya algebra over (T , I ). Let us recall the definition and basic facts about Azumaya algebras. Let A be an OX– algebra which is coherent as a sheaf of OX–modules. Denote by Coh(A) the abelian category of sheaves of (right) A–modules which are coherent as sheaves of OX–modules, and by D b (A) the bounded derived category of Coh(A). We will be interested in a simple case of this situation when A is an Azumaya algebra. Recall that A is called an Azumaya algebra if it is locally free as a sheaf of OX–modules, and for any point x ∈ X the restriction A(x) := A ⊗OX C(x) is isomorphic to a matrix algebra Mr (C). A trivial Azumaya algebra is an algebra of the form End(E), where E is a vector bundle. Two Azumaya algebras A and A are called similar (or Morita equivalent) if there exist vector bundles E and E  such that A ⊗OX End(E) ∼ = A ⊗OX End(E  ). It is easy to see that in this case the categories Coh(A) and Coh(A ) are equivalent, and therefore the derived categories D b (A) and D b (A ) are equivalent as well. Azumaya algebras modulo Morita equivalence generate a group with respect to the tensor product. This group is called the Brauer group of the variety and is denoted by Br(X). There is a natural map ∗ ). Br(X) −→ H 2 (X, OX This map is an embedding and its image is contained in the torsion subgroup of ∗ ). The latter group is denoted by Br  (X) and called the cohomological BraH 2 (X, OX uer group of X. The well-known Grothendieck conjecture asserts that the natural map Br(X) −→ Br  (X) is an isomorphism for smooth varieties. This conjecture was proved for abelian varieties [17]; we will assume that it is true in general. Let X be an algebraic variety over C, and let B ∈ H 2 (X, R/Z). Let β : H 2 (R/Z) → ∗ ) be the homomorphism induced by the canonical map R/Z −→ O ∗ . We 2 H (X, OX X have the following commutative diagram of sheaves: 0 −→ Z  −→ R  −−−→ R/Z  −→ 0  β exp(2πi·)

∗ −→ 0 0 −→ Z −→ OX −−−→ OX ∗ ), and consider an Azumaya algeSuppose β(B) is a torsion element of H 2 (X, OX bra AB which corresponds to this element. The derived category D b (X, AB ) does not depend on the choice of AB because all these algebras are Morita equivalent. Thus we can denote it simply D b (X, B).

Remark 2.6. It appears that a similar triangulated category can be defined even when ∗ ) gives us an O ∗ gerbe X over X. β(B) is not torsion. Any element a ∈ H 2 (X, OX a X

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b Consider the derived category DQcoh (Xβ(B) ) of quasicoherent sheaves on this gerbe. b Now our triangulated category can be defined as a full subcategory of DQcoh (Xβ(B) ) consisting of weight-1 objects with some condition of finiteness, which replaces coherence.

A sufficient condition for the equivalence of D b (X1 , B1 ) and D b (X2 , B2 ) for the case of algebraic tori is provided by the following theorem [30]. Theorem 2.7. Let (T1 , I1 ) and (T2 , I2 ) be two algebraic tori. Let B1 ∈ H 2 (T1 , R/Z) and B2 ∈ H 2 (T2 , R/Z), and suppose β maps both B1 and B2 to torsion elements. If there exists an isomorphism of lattices &1 ⊕ &1∗ and &2 ⊕ &2∗ which maps q1 to q2 , and I1 to I2 , then D b ((T1 , I1 ), B1 ) is equivalent to D b ((T2 , I2 ), B2 ). Remark 2.8. It appears plausible that this is also a necessary condition for D b ((T1 , I1 ), B1 ) to be equivalent to D b ((T2 , I2 ), B2 ). Remark 2.9. It appears plausible that the theorem remains true even when β(B1 ) and β(B2 ) have infinite order, see Remark 2.6. Combining Theorem 2.7 with our Theorem 2.1, we obtain the following result. Corollary 2.10. Suppose V ert (&1 , I1 , G1 , B1 ) is isomorphic to V ert (&2 , I2 , G2 , B2 ), both (T1 , I1 ) and (T2 , I2 ) are algebraic, and both B1 and B2 are mapped by β to torsion elements. Then D b ((T1 , I1 ), B1 ) is equivalent to D b ((T2 , I2 ), B2 ). This corollary suggests that we modify the HMSC by replacing D b (X) with D b (X, B). Once we decided to include the B-field, it seems unnatural to assume that the Fukaya category is independent of it. D-brane considerations suggest a particular way to “twist” the Fukaya category with a B-field (see Sect. 6). Let us denote this “twisted” category by F(Y, B). Here Y is a compact symplectic manifold, and B ∈ H 2 (Y, R/Z) is in the kernel of the Bockstein homomorphism H 2 (Y, R/Z) → H 3 (Y, Z). The modified HMSC asserts that if (X, G, B) is mirror to (X  , G , B  ), then D b (X, B) is equivalent to DF0 (X  , B  ). Corollary 2.10 shows that this conjecture passes the check which the original HMSC fails. If both B and B  vanish, the modified HMSC reduces to the original HMSC. Thus one could ask if it is possible to set the B-field to zero once and for all and work with the original HMSC. This is highly unnatural for the following reason. Suppose we have a mirror pair of physicist’s Calabi Yaus which both happen to have zero B-fields. Now let us start varying the complex structure of the first Calabi-Yau. It can be seen in the case of complex tori and can be argued in general that the corresponding deformation of the second Calabi-Yau generally involves both the K¨ahler form and the B-field. Thus if we have a family of Calabi-Yaus with zero B-field and varying complex structure, the mirror family of Calabi-Yaus will have nonzero B-field for almost all values of the parameter. For example, in the case of the elliptic curve, the usual Teichm¨uller parameter τ takes values in the upper half-plane. The mirror elliptic curve has vanishing B if and only if τ can be made purely imaginary by a modular transformation. In the case of the elliptic curve, the effect of the B-field on the HMSC is relatively minor. It has no effect on the derived category of coherent sheaves because h0,2 = 0. The objects of the Fukaya category are also unmodified in this case (see Sect. 6), and the only change in the definition of morphisms is to complexify the symplectic form. For higher-dimensional varieties, the modification of the Fukaya category is more serious.

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2.3. Physical applications. Transformations of the target space metric and the B-field which leave the conformal field theory unchanged are known as T-duality transformations. For a real torus T n = Rn / &, & ∼ = Zn , such transformations form a group isomorphic to O(n, n, Z) [29, 24]. The main novelty of this work is that we consider complex tori, and study transformations of G, B, and the complex structure which leave the N = 2 superconformal field theory unchanged or induce a mirror morphism. Our results have implications for the study of BPS D-branes on Calabi-Yau manifolds, a subject which received much attention recently (see [10] and references therein). They suggest that BPS D-branes of type B are best thought of as objects of the derived category of coherent sheaves when the B-field is zero. When the B-field is nonzero but ∗ ) is a torsion class, the derived category of coherent the corresponding class in H 2 (X, OX sheaves should be replaced with the derived category of a certain noncommutative alge∗) braic variety (an Azumaya algebra over X). When the class of the B-field in H 2 (X, OX has infinite order, it appears that B-type D-branes should be regarded as objects of the derived category of “coherent” sheaves on a gerbe over X. Note a similarity with the results of [20, 4] where it was shown that in the presence of a B-field D-brane charges on a smooth manifold X are classified by the K-theory of an Azumaya algebra over X, or more generally by the K-theory of a Dixmier-Douady algebra over X. The main differences are that Refs. [20, 4] work in a C ∞ -category, the D-branes are not required to be BPS, and the focus is on D-brane charges rather on D-branes themselves. In Sect. 6 we describe the effect of a closed B-field on BPS D-branes of type A (the ones associated to flat unitary bundles on special Lagrangian submanifolds in a Calabi-Yau). This subject was previously studied by Hori et al. [18] for the case of a single D-brane, i.e. when the rank of the bundle is one. Hori et al. find that the restriction of the B-field to the Lagrangian submanifold must vanish. We find that this restriction is too strong: it is sufficient to require the restriction of the B-field to have integer periods. For the higher rank case we argue that in general the unitary bundle on the Lagrangian submanfold is projectively flat rather than flat. Correspondingly, the restrictions on the B-field are even weaker. 3. Superconformal Vertex Algebras 3.1. Quantum fields. Let V be a vector superspace over C. The parity of an element a ∈ V is denoted p(a) and takes values in integers modulo 2. Definition 3.1. The space of quantum fields in one formal variable with values in End(V ) is a vector superspace whose elements have the form



−h−n −h−n¯ C(h+n,h+n) z¯ , ¯ z h∈J n,n∈ ¯ Z

where J is some subset of [0, 1) (different for different elements), C(h+n,h+n) ¯ ∈ End(V ), and the following conditions are satisfied: a) the set J is countable; b) for any element v ∈ V there is a finite subset Jv ⊂ J such that C(h+n,h+n) ¯ (v) = 0 for all h ∈ J \Jv and all n, n¯ ∈ Z;

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c) for any element v ∈ V there is an integer N such that Ch+n,h+n¯ (v) = 0 for all h ∈ J if n > N or n¯ > N. The space of quantum fields in one formal variable with values in End(V ) is denoted QF1 (V ). Given an element A(z, z¯ ) of QF1 (V ), we will denote the coefficient of z−h−n z¯ −h−n¯ in A(z, z¯ ) by A(h+n,h+n) ¯ . The intersection of QF1 (V ) with End(V )[[z, z−1 ]] (resp. End(V )[[¯z, z¯ −1 ]]) will be called the space of meromorphic (resp. anti-meromorphic) fields. We will denote by A(z) (resp. A(¯z)) meromorphic (resp. anti-meromorphic) fields. The coefficient of z−n in A(z) (resp. the coefficient of z¯ −n in A(¯z)) will be denoted A(n) . Definition 3.2. The space of quantum fields in two formal variables with values in End(V ) is a vector superspace whose elements have the form



−h−n −h−n¯ −g−m −g−m ¯ C(h+n,h+n,g+m,g+ z¯ w w¯ , ¯ m) ¯ z (h,g)∈J n,n,m, ¯ m∈ ¯ Z

where J ∈ [0, 1)2 , C(h+n,h+n,g+m,g+ ¯ m) ¯ ∈ End(V ), and the following conditions are satisfied: a ) the set J is countable; b ) for any element v ∈ V there is a finite subset Jv ⊂ J such that C(h+n,h+n,g+m,g+ ¯ m) ¯ (v) = 0 ¯ m, m ¯ ∈ Z; for all (h, g) ∈ J \Jv and all n, n, c ) for any element v ∈ V and any (h, g) ∈ J, there is an integer N such that C(h+n,h+n,g+m,g+ ¯ m) ¯ (v) = 0, C(h+m,h+m,g+n,g+ ¯ n) ¯ (v) = 0, for n > N and any n, ¯ m, m ¯ ∈ Z, and C(h+n,h+n,g+m,g+ ¯ m) ¯ (v) = 0, C(h+m,h+m,g+n,g+ ¯ n) ¯ (v) = 0, for n¯ > N and any n, m, m ¯ ∈ Z. The space of quantum fields in two formal variables with values in End(V ) is denoted QF2 (V ). ¯ Item (c ) in the definition of QF2 (V ) ensures that given an element C(z, z¯ , w, w)of QF2 (V ), one can substitute z = w, z¯ = w¯ and get a well-defined element of QF1 (V ). This element will be denoted C(w, w, ¯ w, w). ¯ Note that in general a product of two fields A(z, z¯ ) ∈ QF1 (V ) and B(w, w) ¯ ∈ QF1 (V ) does not belong to QF2 (V ), precisely because (c ) is not satisfied. In this situation one says that the product of A(z, z¯ ) and B(w, w) ¯ has a singularity for z = w, z¯ = w. ¯ If an element A(z, z¯ , w, w) ¯ ∈ QF2 (V ) does not contain nonzero powers of z¯ (resp. z) we will say that this field is meromorphic (resp. anti-meromorphic) in the first variable, and write it as A(z, w, w) ¯ (resp. A(¯z, w, w)). ¯ Fields in two variables (anti-)meromorphic in the second variable are defined in a similar way.

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3.2. The definition of a vertex algebra. We set  ∞ 

−h 1 (−1)j w j z−j −h , = j (z − w)h

iz,w

1 = (¯z − w) ¯ h

iz¯ ,w¯

1 = (z − w)h

iw,z

1 = (¯z − w) ¯ h

iw,¯ ¯ z

j =0 ∞ 

 −h (−1)j w¯ j z¯ −j −h , j

j =0

 ∞ 

−h

e−iπh (−1)j zj w −j −h ,

j

j =0

 ∞ 

−h j

j =0

eiπh (−1)j z¯ j w¯ −j −h ,

where 

 −h (−h)(−h − 1) · · · (−h − (j − 1)) = . j j!

¯ −h in These are formal power series expansions of the functions (z − w)−h and (¯z − w) the regions |z| > |w|, |z| < |w| and |¯z| > |w|, ¯ |¯z| < |w|. ¯ Definition 3.3. A vertex algebra structure on a vector superspace V consists of the following data: (i) an even vector |vac ∈ V , (ii) a pair T , T¯ of commuting even endomorphisms of V annihilating |vac, (iii) a parity-preserving linear map Y : V → QF1 (V ),

Y : a → Y (a) = a(z, z¯ ).

These data must satisfy the following requirements. 1. Y (|vac) = id ∈ End(V ). ¯ 2. [T , a(z, z¯ )] = ∂a(z, z¯ ), [T¯ , a(z, z¯ )] = ∂a(z, z¯ ). ¯ zT +¯ z T 3. a(z, z¯ )|vac = e a. 4. For any a, b ∈ V there are integers N, M, real numbers hj ∈ [0, 1), j = 1, . . . , M, and quantum fields Cj (z, z¯ , w, w) ¯ ∈ QF2 (V ), j = 1, . . . , M, such that a(z, z¯ )b(w, w) ¯ =

M

iz,w

j =1

1 1 iz¯ ,w¯ Cj (z, z¯ , w, w), ¯ h +N (z − w) j (¯z − w) ¯ hj +N (5)

(−1)p(a)p(b) b(w, w)a(z, ¯ z¯ ) =

M

j =1

iw,z

1 1 iw,¯ Cj (z, z¯ , w, w). ¯ ¯ z h +N j (z − w) (¯z − w) ¯ hj +N (6)

The map Y is called the state-operator correspondence. The coefficient of z−α z¯ −β in Y (a) is called the (α, β) component of Y (a) and denoted by a(α,β) .

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The last requirement in the definition of a vertex algebra is called the Operator Product Expansion (OPE) axiom. It contains two important ideas. The equality (5) says that the product of two fields in the image of Y has only power-like singularities for z = w, z¯ = w. ¯ The difference of (5) and (6) means, roughly speaking, that the fields in the image of Y are mutually local, in the sense that their supercommutator vanishes when z = w and z¯ = w. ¯ This is particularly clear when all hi are equal to zero. Then the supercommutator of a(z, z¯ ) and b(w, w) ¯ is proportional to 1 1 1 δ (N−1) (z − w)δ (N−1) (¯z − w) ¯ + δ (N−1) (z − w) iz¯ ,w¯ 2 ((N − 1)!) (N − 1)! (¯z − w) ¯ N 1 1 + ¯ iz,w , (7) δ (N−1) (¯z − w) (N − 1)! (z − w)N where δ (k) (z − w) is the k th derivative of the formal delta-function defined as a formal power series


z n δ(z − w) = z−1 . w n∈Z

Given any two elements of QF1 (V ), we will say that they are mutually local if for their products the OPE formulas (5,6) hold for some N, M ∈ Z, hj ∈ [0, 1), j = 1, . . . , M, and Cj ∈ QF2 (V ), j = 1, . . . , M. Vertex algebras as defined above are a generalization of chiral algebras as defined in [19] in the following sense. First, any chiral algebra is automatically a vertex algebra, with T¯ = 0 and the image of Y consisting of meromorphic fields only. Second, if we consider the subspace in V consisting of vectors which are mapped to meromorphic fields, the restriction of T and Y to this subspace specifies on it the structure of a chiral algebra. Similarly, the restriction of T¯ and Y to the anti-meromorphic sector yields another chiral algebra. Moreover, all meromorphic fields supercommute with all anti-meromorphic fields. Thus any vertex algebra contains a pair of commuting chiral subalgebras. All these facts are proved in Appendix B. The OPE formulas simplify when one of the fields is meromorphic or anti-meromorphic. For example, the OPE of a meromorphic field a(z), a ∈ V , with a general field b(w, w), ¯ b ∈ V , has the following form (see Appendix B for proof): a(z)b(w, w) ¯ =

N

iz,w

j =1

(−1)p(a)p(b) b(w, w)a(z) ¯ =

N

j =1

1 Dj (w, w)+ ¯ : a(z)b(w, w) ¯ :, (z − w)j (8)

1 iw,z Dj (w, w)+ ¯ : a(z)b(w, w) ¯ :. (z − w)j

Here N is some integer, Dj (w, w) ¯ ∈ QF1 (V ), and : a(z)b(w, w) ¯ : is an element of QF2 (V ) defined as follows: : a(z)b(w, w) ¯ := a(z)+ b(w, w) ¯ + (−1)p(a)p(b) b(w, w)a(z) ¯ −, where we set a(z)+ =

n≤0

a(n) z−n ,

a(z)− =

n>0

a(n) z−n .

(9)

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The field : a(z)b(w, w) ¯ : is called the normal ordered product of a(z) and b(w, w). ¯ Since it belongs to QF2 (V ), one can set z = w and get a well-defined field in one variable : a(w)b(w, w) ¯ : . The difference between the right-hand side of (8) and : a(z)b(w, w) ¯ : is called the singular part of the OPE. Similarly, one can define the normal ordered product of an anti-meromorphic field with a general field. The normal ordered product of two general fields is not defined. Let us consider now the OPE of two meromorphic fields a(z) and b(z). We already mentioned that meromorphic fields form a chiral algebra, thus the OPE (8) simplifies even further: a(z)b(w) =

N

iz,w

1 Dj (w)+ : a(z)b(w) :, (z − w)j

iw,z

1 Dj (w)+ : a(z)b(w) : . (z − w)j

j =1

(−1)p(a)p(b) b(w)a(z) =

N

j =1

Here Dj (w), j = 1, . . . , N, are meromorphic elements of QF1 (V ). Exchanging a(z) and b(w) we get b(w)a(z) =

N

iw,z

1 Cj (z)+ : b(w)a(z) :, (w − z)j

iz,w

1 Cj (z)+ : b(w)a(z) :, (w − z)j

j =1

(−1)p(a)p(b) a(z)b(w) =

N

j =1

where Cj (z), j = 1, . . . , N, are meromorphic elements of QF1 (V ). In general, the normal ordered product is not supercommutative, i.e. : a(z)b(w) : = (−1)p(a)p(b) : b(w)a(z) : . Neither is it associative, in the sense that in general : a(z) : b(z)c(z) :: = :: a(z)b(z) : c(z) : . We will define the normal ordered product of more than two (anti-)meromorphic fields inductively from right to left: : a1 (z)a2 (z) . . . an (z) :=: a1 (z) : a2 (z) . . . an (z) :: . An important special case where the normal ordered product of meromorphic fields is supercommutative is when the fields Dj (w) do not depend on w, i.e. are constant endomorphisms of V . This follows directly from the above OPE formulas. One can also show that if pairwise OPE’s of meromorphic fields a(z), b(z), and c(z) have this property, then their normal ordered product is associative [9]. For example, the normal ordered product of free fermion and free boson fields is supercommutative and associative [19, 9]. Another important special case is the OPE of a meromorphic field and an anti-meromorphic field. In this case one can also define two normal ordered products, : a(z)b(w) ¯ : and : b(w)a(z) ¯ : . But it follows easily from Eqs. (8) and analogous equations for the OPE of an anti-meromorphic field and a general field, that in this case the singular part of the OPE vanishes, the normal ordered product coincides with the ordinary product,

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and that consequently all meromorphic fields supercommute with all anti-meromorphic fields. Thus ¯ :. : a(z)b(w) ¯ := (−1)p(a)p(b) : b(w)a(z) This is discussed in more detail in Appendix B. The singular part of the OPE of two meromorphic fields a(z) and b(z) completely determines and is determined by the supercommutators of a(n) and b(m) for all n, m ∈ Z. Explicit formulas which enable one to pass from the OPE to the supercommutators and back can found in [19]. When writing the OPE of two meromorphic fields we will use a shortened notation in which only the singular part of the OPE is shown. To indicate this, the equality sign is replaced by ∼. In addition, we will only write the first of the OPE’s in (8), and correspondingly will omit the symbol iz,w , as is common in the physics literature. Similar notation is used for the OPE of two anti-meromorphic fields. Thus instead of a(z)b(w) =

N

iz,w

j =1

1 Dj (w)+ : a(z)b(w) : (z − w)j

we will write N

Dj (w) a(z)b(w) ∼ . (z − w)j j =1

We conclude this subsection by defining morphisms of vertex algebras. A morphism from a vertex algebra (V , |vac, T , T¯ , Y ) to a vertex algebra (V  , |vac , T  , T¯  , Y  ) is a morphism of superspaces f : V → V  such that f (|vac) = |vac ,

f T = T  f,

f T¯ = T¯  f,

and Y  (f (a))f (b) = f (Y (a)b)

∀a, b ∈ V .

3.3. Conformal vertex algebras. Definition 3.4. Let V = (V , |vac, T , T¯ , Y ) be a vertex algebra. Conformal structure on V is a pair of even vectors L, L¯ ∈ V such that (i) L(z, z¯ ) = L(z) =

n∈Z

Ln z−n−2 ,

¯ z¯ ) = L(¯ ¯ z) = L(z,



L¯ n z¯ −n−2 .

n∈Z

(ii) L−1 = T , L¯ −1 = T¯ . 2L(w) ∂L(w) c/2 + + , (iii) L(z)L(w) ∼ 4 2 (z − w) (z − w) z−w ¯ w) ¯ c/2 ¯ 2L( ¯ ∂¯ L(w) ¯ z)L( ¯ w) L(¯ ¯ ∼ + + . 4 2 (¯z − w) ¯ (¯z − w) ¯ z¯ − w¯ (iv) For any a ∈ V ¯ [L0 , a(z, z¯ )] = z∂a(z, z¯ ) + (L0 a)(z, z¯ ), [L¯ 0 , a(z, z¯ )] = z¯ ∂a(z, z¯ ) + (L¯ 0 a)(z, z¯ ). (10)

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Here c, c¯ ∈ C. A vertex algebra with a conformal structure is called a conformal vertex algebra (CVA). The numbers c and c¯ are called the holomorphic and anti-holomorphic central charges of the CVA. The reason for this name is the following. The OPE’s (10) are equivalent to the following commutation relations for all n, m ∈ Z [19]: m3 − m δm,−n , 12 m3 − m [L¯ m , L¯ n ] = (m − n)L¯ m+n + c¯ δm,−n , 12 [Ln , L¯ m ] = 0.

[Lm , Ln ] = (m − n)Lm+n + c

¯ Hence the components of L(z) and L(z) form two commuting Virasoro algebras. The Virasoro algebra is the unique central extension of the Witt algebra (the algebra of the infinitesimal diffeomorphisms of a circle). In the present case the central charges of the two Virasoro algebras are c and c. ¯ Note that Axiom 3 in the definition of a vertex algebra implies that both Ln and L¯ n annihilate |vac for all n ≥ −1. ¯ to a CVA (V  , |vac , Y  , L , L¯  ) is A morphism f from a CVA (V , |vac, Y, L, L) a morphism of the underlying vertex algebras which satisfies f (L) = L ,

¯ = L¯  . f (L)

A conformal vertex algebra is almost the same as a conformal field theory. Namely, a physically acceptable conformal field theory is a conformal vertex algebra whose state space V is equipped with a positive-definite Hermitian inner product, and the following additional constraints are satisfied: (v) The space V splits as a direct sum of the form ¯ j, ⊕j ∈J Wj ⊗ W ¯ j are unitary highest-weight modules where J is a countable set, and Wj and W over the meromorphic and anti-meromorphic Virasoro algebras, respectively. (vi) The vacuum vector is the only vector in V annihilated by both L0 and L¯ 0 . The conformal vertex algebras we will be working with satisfy these constraints and therefore are honest conformal field theories. However, we prefer not to stress the “real” aspects of conformal field theories in this paper. Furthermore, in order for a conformal field theory to admit a string-theoretic interpretation, it must be defined on a Riemann surface of arbitrary genus. (The above axioms define a conformal field theory in genus zero.) This does not require new data, but imposes additional, so-called sewing, constraints. We will work in genus zero only, and therefore will neglect the sewing constraints.

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3.4. N = 1 superconformal vertex algebras. ¯ be a conformal vertex algebra with central Definition 3.5. Let V = (V , |vac, Y, L, L) ¯ ∈V charges c, c. ¯ N = 1 superconformal structure on V is a pair of odd vectors Q, Q such that

(i) Q(z, z¯ ) = Q(z) =

r∈Z+ 21

Qr

z

, r+3/2

¯ z¯ ) = Q(¯ ¯ z) = Q(z,

(ii) The following OPE’s hold true:

r∈Z+ 21

¯r Q

z¯ r+3/2

.

3 Q(w) ∂Q(w) + , 2 (z − w)2 (z − w) 1 L(w) c/6 + , Q(z)Q(w) ∼ (z − w)3 2 (z − w) and similar OPE’s for the anti-meromorphic fields with z, w, c, ∂ replaced with ¯ z¯ , w, ¯ c, ¯ ∂. L(z)Q(w) ∼

¯ z) are called left-moving and right-moving supercurrents, The fields Q(z) and Q(¯ respectively. A CVA with an N = 1 superconformal structure is called an N = 1 superconformal vertex algebra (N = 1 SCVA). N = 1 superconformal structure is also known as (1, 1) superconformal structure. ¯ one obtains the definition of (1, 0) superconformal structure. Morphisms Omitting Q, of N = 1 SCVA’s are defined in an obvious way. ¯ z) with themselves and L(z), L(¯ ¯ z) are equivalent to the The OPE’s of Q(z), Q(¯ following commutation relations:
  ¯ r] = m − r Q ¯ r+m , [L¯ m , Q − r Qr+m , 2 2     1 c 1 1¯ c¯ 2 2 1 ¯ ¯ {Qr , Qs } = Lr+s + r − δr,−s , {Qr , Qs } = Lr+s + r − δr,−s . 2 12 4 2 12 4

[Lm , Qr ] =


m

As usual, the barred generators supercommute with the unbarred ones. Thus Ln , L¯ n , ¯ r form an infinite-dimensional Lie super-algebra which is a direct sum of two Qr , Q copies of the N = 1 super-Virasoro algebra with central charges c and c. ¯

3.5. N = 2 superconformal vertex algebras. ¯ be a conformal vertex algebra with central Definition 3.6. Let V = (V , |vac, Y, L, L) charges c, c. ¯ N = 2 superconformal structure on V is a pair of even vectors J, J¯ ∈ V ¯ +, Q ¯ − ∈ V such that and four odd vectors Q+ , Q− , Q

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Jn , zn+1 n∈Z

Q+ r + + Q (z, z¯ ) = Q (z) = , r+3/2 z 1

(i) J (z, z¯ ) = J (z) =

r∈Z+ 2

Q− (z, z¯ ) = Q− (z) =



r∈Z+ 21

Q− r , r+3/2 z

J¯n , z¯ n+1 n∈Z

Q ¯+ r + + ¯ ¯ Q (z, z¯ ) = Q (¯z) = , r+3/2 z ¯ 1 J¯(z, z¯ ) = J¯(¯z) =

r∈Z+ 2

¯ − (z, z¯ ) = Q ¯ − (¯z) = Q



r∈Z+ 21

¯− Q r ; r+3/2 z¯

(ii) the following OPE’s hold true: 3 Q± (w) ∂Q± (w) L(z)Q± (w) ∼ , + 2 2 (z − w) (z − w) ∂J (w) J (w) + , L(z)J (w) ∼ (z − w)2 (z − w) c/3 , J (z)J (w) ∼ (z − w)2 Q± (w) J (z)Q± (w) ∼ ± , (z − w) c/12 1 J (w) 1 ∂J (w) + 2L(w) , + + Q+ (z)Q− (w) ∼ 3 2 (z − w) 4 (z − w) 8 (z − w) Q± (z)Q± (w) ∼ 0, and similar OPE’s for the anti-meromorphic fields with z, w, c, ∂ replaced with ¯ z¯ , w, ¯ c, ¯ ∂. The fields J (z) and J¯(¯z) are called left-moving and right-moving R-currents, the ¯ ± (¯z) are called left-moving and right-moving supercurrents, respecfields Q± (z) and Q tively. A CVA with N = 2 superconformal structure is called an N = 2 superconformal vertex algebra (N = 2 SCVA). ¯ The above OPE’s together with the OPE’s for L(z), L(z) are equivalent to the commutation relations (1) if we set c = c¯ = 3d. N = 2 superconformal structure is also known as a (2, 2) superconformal structure. ¯ ± (¯z), one gets the definition of a If one omits the anti-meromorphic currents J¯(¯z), Q (2, 0) superconformal structure. Given an N = 2 SCVA, one can obtain an N = 1 SCVA by setting Q = Q+ + Q− , ¯ ¯+ + Q ¯ − . Thus an N = 2 SCVA can be regarded as an N = 1 SCVA with Q = Q additional structure. Morphisms of N = 2 SCVA’s are defined in an obvious way. A mirror morphism between two N = 2 SCVA’s is an isomorphism between the underlying N = 1 SCVA’s ¯ ± , J, J¯: which induces the following map on Q± , Q 







f (Q+ ) = Q− , f (Q− ) = Q+ , f (J ) = −J  , ¯ + , f (Q ¯ −) = Q ¯ − , f (J¯) = J¯ . ¯ +) = Q f (Q This map acts as an outer automorphism on the algebra (1). A composition of two mirror morphisms is an isomorphism of N = 2 SCVA’s.

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4. N = 2 SCVA of a Flat Complex Torus The purpose of this section is to describe an N = 2 SCVA canonically associated to a complex torus endowed with a flat K¨ahler metric and a constant 2-form. None of this material is new, and everything can be found, in one form or another, in standard string theory textbooks [24, 29]. We simply translate these standard constructions into the language of vertex algebras. 4.1. Vertex algebra structure. Let U be a real vector space of dimension 2d. Let & ∼ = Z2d ∗ ∗ be a lattice in U. Let & ⊂ U be the dual lattice Hom(&, Z). Let T = U/ &, T ∗ = U ∗ / & ∗ . Let G be a metric on U, i.e. a positive symmetric bilinear form on U. Let B be a real skew-symmetric bilinear form on U. Let l be the natural pairing & × & ∗ → Z. The natural pairing U × U ∗ → R will be also denoted l. Let Z∗ be the set of nonzero integers. Let the vectors e1 , . . . , e2d ∈ U be the generators of &. The components of an element w ∈ & in this basis will be denoted by w i , i = 1, . . . , 2d. The components of an element m ∈ & ∗ in the dual basis will be denoted by mi , i = 1, . . . , 2d. We also denote by Gij , Bij the components of G, B in this basis. It will be apparent that the superconformal vertex algebra which we construct does not depend on the choice of basis in &. In the physics literature & is sometimes referred to as the winding lattice, while & ∗ is called the momentum lattice. Consider a triple (T , G, B). To any such triple we will associate a superconformal vertex algebra V which may be regarded as a quantization of the supersymmetric σ -model described in Appendix A. The state space of the vertex algebra V is V = Hb ⊗C Hf ⊗C C [& ⊕ & ∗ ]. Here Hb and Hf are bosonic and fermionic Fock spaces defined below, while C [& ⊕& ∗ ] is the group algebra of & ⊕ & ∗ over C. To define Hb , consider an algebra over C with generators αsi , α¯ si , i = 1, . . . , 2d, s ∈ ∗ Z and relations ij ij

j j j [αsi , αp ] = s G−1 δs,−p , [α¯ si , α¯ p ] = s G−1 δs,−p , [αsi , α¯ p ] = 0. (11) i and α i are called left and right bosonic creators, respec¯ −s If s is a positive integer, α−s tively, otherwise they are called left and right bosonic annihilators. Either creators or annihilators are referred to as oscillators. i ,a i ,i = The space Hb is defined as the space of polynomials of even variables a−s ¯ −s 1, . . . , 2d, s = 1, 2, . . . . The bosonic oscillator algebra (11) acts on the space Hb via i → a i ·, α−s
−s ij ∂ i , αs → s G−1 j ∂a−s

i → a i ·, α¯ −s ¯ −s
ij ∂ α¯ si → s G−1 , j ∂ a¯ −s

for all positive s. This is the Fock-Bargmann representation of the bosonic oscillator algebra. The vector 1 ∈ Hb is annihilated by all bosonic annihilators and will be denoted |vacb . The space Hb will be regarded as a Z2 -graded vector space with a trivial (purely ¯ b , where Hb (resp. H ¯ b) even) grading. It is clear that Hb can be decomposed as Hb ⊗ H is the bosonic Fock space defined using only the left (right) bosonic oscillators.

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To define Hf , consider an algebra over C with generators ψsi , ψ¯ si , i = 1, . . . , 2d, s ∈ Z + 21 subject to relations ij
j {ψsi , ψp } = G−1 δs,−p ,

ij
j {ψ¯ si , ψ¯ p } = G−1 δs,−p ,

j {ψsi , ψ¯ p } = 0.

(12)

i and ψ i are called left and right fermionic creators respectively, oth¯ −s If s is positive, ψ−s erwise they are called left and right fermionic annihilators. Collectively they are referred to as fermionic oscillators. i , θ¯ i , The space Hf is defined as the space of skew-polynomials of odd variables θ−s −s i = 1, . . . , 2d, s = 1/2, 3/2, . . . . The fermionic oscillator algebra (12) acts on Hf via i → θ i ·, ψ−s
−s ij i ψs → G−1

∂ j ∂θ−s

,

i → θ¯ i ·, ψ¯ −s
−s ij i ¯ ψs → G−1

∂ , j ¯ ∂ θ−s

for all positive s ∈ Z + 21 . This is the Fock-Bargmann representation of the fermionic oscillator algebra. The vector 1 ∈ Hf is annihilated by all fermionic annihilators and will be denoted |vacf . The fermionic Fock space has a natural Z2 grading such that ¯ f , where Hf (resp. H ¯ f ) is constructed |vacf  is even. It can be decomposed as Hf ⊗ H using only the left (right) fermionic oscillators. For w ∈ &, m ∈ & ∗ we will denote the vector w ⊕ m ∈ C [& ⊕ & ∗ ] by (w, m). We will also use a shorthand |vac, w, m, for |vacb  ⊗ |vacf  ⊗ (w, m). To define V, we have to specify the vacuum vector, T , T¯ , and the state-operator correspondence Y. But first we need to define some auxiliary objects. We define the operators W : V → V ⊗ & and M : V → V ⊗ & ∗ as follows: W i : b⊗f ⊗(w, m) → w i (b⊗f ⊗(w, m)),

Mi : b⊗f ⊗(w, m) → mi (b⊗f ⊗(w, m)).

We also set ∞ j

 αs Y (z) = , szs s=−∞ j

∞ j

 α ¯s , s z¯ s s=−∞ 1
−1 j k ∂X j (z) = G Pk − ∂Y j (z), z j k
¯ j (¯z) = 1 G−1 P¯k − ∂¯ Y¯ j (¯z), ∂X z¯

ψrj ψ j (z) = , zr+1/2 1

Y¯ j (¯z) =

(13) (14) (15)

r∈Z+ 2

ψ¯ j (¯z) =

r∈Z+ 21

j ψ¯ r

z¯ r+1/2

,

(16)

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where a prime on a sum over s means that the term with s = 0 is omitted, and Pk and P¯k are defined by  1 Pk = √ (Mk + −Bkj − Gkj W j ), 2

 1 P¯k = √ (Mk + −Bkj + Gkj W j ). 2

Note that we did not define Xj (z, z¯ ) themselves, but only their derivatives. The reason is that the would-be field X j (z, z¯ ) contains terms proportional to log z and log z¯ , and therefore does not belong to QF1 (V ). The vacuum vector of V is defined by |vac = |vac, 0, 0. The operators T , T¯ ∈ End(V ) are defined by T =

j Pj α−1

+

∞

s=1

j T¯ = P¯j α¯ −1 +

∞

s=1

j Gj k α−1−s αsk

+

r= 21 , 23 ,...

j

Gj k α¯ −1−s α¯ sk +





 1 j r+ ψ−1−r ψrk , 2



r= 21 , 23 ,...

r+

 1 j ψ¯ −1−r ψ¯ rk . 2

The state-operator correspondence is defined as follows. The state space V is spanned by vectors of the form ¯

iq j1 jn j¯ j¯n¯ i1 ¯ iq¯ ¯ i¯1 α−s . . . α−s α¯ 1s1 . . . α¯ −¯ sn¯ ψ−r1 . . . ψ−rq ψ−¯r1 . . . ψ−¯rq¯ |vac, w, m, n −¯ 1

(17)

where n, n, ¯ q, q¯ are nonnegative integers, s1 , . . . , sn , s¯1 , . . . , s¯n¯ are positive integers, and r1 , . . . , rq , r¯1 , . . . , r¯q¯ are positive half-integers. This vector is mapped by Y to the following quantum field:

−1  −1 ¯ ¯  Aw,m T (w, m) pr(w ,m ) z−2G (k,k ) z¯ −2G (k,k ) (w ,m )∈&⊕& ∗


 exp kj Y j (z)+ + k¯j Y¯ j (¯z)+ :

q q¯ n n¯ ¯ ¯

∂ sl X jl (z) ∂ s¯l¯ X jl¯ (¯z) ∂ rt −1/2 ψ it (z) ∂ r¯t¯−1/2 ψ¯ it¯ (¯z)

(¯sl¯ − 1)! t=1 
exp kj Y j (z)− + k¯j Y¯ j (¯z)− . l=1

(sl − 1)!

¯ l=1

(rt − 21 )!

t¯=1

(¯rt¯ − 21 )!

: (18)

¯ k  , k¯  are elements of U ∗ defined by Here k, k,  1 kj = √ (mj + −Bj k − Gj k w k ), 2

 1 k¯j = √ (mj + −Bj k + Gj k w k ), 2

    1 1 kj = √ (mj + −Bj k − Gj k w k ), k¯j = √ (mj + −Bj k + Gj k w k ), 2 2 the operator T (w, m) is a translation on the lattice & ⊕ & ∗ : T (w, m) : (a, b) → (a + w, b + m),

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and the operators pr(w ,m ) : V → V are projections onto the subspace Hb ⊗ Hf ⊗ (w , m ). Finally, Aw,m is a sign equal to exp(iπ l(w, m )). We also recall that for any meromorphic quantum field a(z) the fields a(z)+ and a(z)− are defined by (9), and there is a similar definition for the anti-meromorphic fields. Thus Y j (z)± and Y¯ j (¯z)± are given by

αsj , Y (z)− = szs

αsj Y (z)+ = , szs

α¯ sj Y¯ j (¯z)− = , s z¯ s

α¯ sj Y¯ j (¯z)+ = . s z¯ s

j

j

s>0

s<0

s>0

s<0

One can easily check that (18) is indeed a well-defined quantum field. Furthermore, the vector (17) is unchanged when the bosonic oscillators are permuted, and is multiplied by the parity of the permutation when the fermionic oscillators are permuted. For the map Y to be well-defined, (18) must have the same property. To see that this is indeed the case, note that the OPE of the fields ψ j and ∂X j is given by  −1 j k G ∂Xj (z)∂X k (w) ∼ , (z − w)2  −1 j k G j k , (19) ψ (z)ψ (w) ∼ (z − w) ∂X j (z)ψ i (z) ∼ 0, and similarly for the anti-meromorphic fields. It follows that the singular part of the OPE ¯ j and their derivatives is proportional to the identity operator, and for ψ j , ψ¯ j , ∂Xj , ∂X therefore their normal ordered product is supercommutative. To facilitate the understanding of (18), we list a few special cases of the state-operator correspondence. j The state α−s |vac, 0, 0 is mapped by Y to 1 ∂ s X j (z). (s − 1)! j

The state α¯ −s |vac, 0, 0 is mapped to 1 ∂¯ s X j (¯z). (s − 1)! j

The state ψ−s |vac, 0, 0 is mapped to 1 (s − 21 )!

∂ s−1/2 ψ j (z).

j The state ψ¯ −s |vac, 0, 0 is mapped to

1 (s − 21 )!

∂¯ s−1/2 ψ¯ j (¯z).

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The state |vac, w, m is mapped to

−1 (k,k  )

Aw,m T (w, m) z−2G

(w ,m )∈&⊕& ∗

−1 (k, ¯ k¯  )

z¯ −2G


 exp kj Y j (z)+ + k¯j Y¯ j (¯z)+


exp kj Y j (z)− + k¯j Y¯ j (¯z)− pr(w ,m ) .

Checking that (V , |vac, T , T¯ , Y ) satisfies the vertex algebra axioms is a tedious but straightforward exercise which we leave to the reader. Implicitly, the axioms are verified in most textbooks on string theory, for example in [29, 24]. 4.2. N = 2 superconformal structure. We first define an N = 1 superconformal structure on V by setting 1 1 : G (∂X(z), ∂X(z)) : − : G (ψ(z), ∂ψ(z)) : , 2 2   ¯ z) = 1 : G ∂X(¯ ¯ z), ∂X(¯ ¯ z) : − 1 : G ψ(¯ ¯ z), ∂¯ ψ(¯ ¯ z) : , L(¯ 2 2 i Q(z) = √ : G (ψ(z), ∂X(z)) : , 2 2  i ¯ z) = √ ¯ z) : . ¯ z), ∂X(¯ Q(¯ : G ψ(¯ 2 2 L(z) =

It can be easily checked that all these fields are in the image of Y, that L−1 = T , L¯ −1 = T¯ , and that they satisfy the OPE’s specified in Definition 3.5. The central charges turn out to be c = c¯ = 3d. To define an N = 2 superconformal structure, we need to choose a complex structure I on U with respect to which G is a K¨ahler metric. Let ω = GI be the corresponding K¨ahler form. Then the left-moving supercurrents and the U (1) current are defined as follows: i 1 Q± (z) = √ : G (ψ(z), ∂X(z)) : ± √ : ω (ψ(z), ∂X(z)) : , 4 2 4 2 i J (z) = − : ω(ψ(z), ψ(z)) : . 2 ¯ ± (¯z) and J¯(¯z) are defined by the same expressions with The right-moving currents Q ¯ and ψ replaced by ψ. ¯ We omit the check that the OPE’s of these ∂X replaced by ∂X currents are as specified in Definition 3.6. In checking the OPE’s the relations (19) are useful. 5. Morphisms of Toroidal Superconformal Vertex Algebras 5.1. Isomorphisms of N = 1 SCVA’s. Let (T , G, B) and (T  , G , B  ) be a pair of 2ddimensional real tori equipped with flat metrics and constant B-fields. Given G and B, we define a flat metric on T × T ∗ by the formula   G − BG−1 B BG−1 G(G, B) = 2 . −G−1 B G−1

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The meaning of this formula is that the value of G on a pair of vectors x1 ⊕ y1 and x2 ⊕ y2 , xi ∈ U, yi ∈ U ∗ , i = 1, 2, is    G − BG−1 B BG−1  x2 . 2 x1 y1 y2 −G−1 B G−1 G(G, B) is obviously a symmetric form on U ⊕U ∗ , and its positive-definiteness follows from the positive-definiteness of G and the identity   G 0 R(G, B), (20) G = R(G, B)t 0 G where 

−1 − G−1 B R(G, B) = 1 − G−1 B

 G−1 . G−1

We will use a shorthand G(G, B) = G and G(G , B  ) = G  . Recall also that we have canonical Z-valued symmetric bilinear forms on & ⊕ & ∗ and &  ⊕ &  ∗ denoted by q and q  , respectively (see Sect. 2). In this subsection we prove Theorem 5.1. N = 1 SCVA’s corresponding to (T , G, B) and (T  , G , B  ) are isomorphic if and only if there exists an isomorphism of lattices & ⊕ & ∗ and &  ⊕ &  ∗ which takes q to q  , and G to G  . The “if” part of this theorem is proved in many string theory papers, see for example [25, 34]. Below we outline a construction of the isomorphism of N = 1 SCVA’s given an isomorphism of lattices and then prove the “only if” part of the theorem. Let g be an isomorphism of & ⊕ & ∗ with &  ⊕ &  ∗ . We will write it as follows:   ab g= , cd where a ∈ Hom(&, &  ), b ∈ Hom(& ∗ , &  ), c ∈ Hom(&, &  ∗ ), d ∈ Hom(& ∗ , &  ∗ ). The “realified” maps from U, U ∗ to U  , U  ∗ will be denoted by the same letters. Let us also set H = G + B. Both V and V  are tensor products of the group algebra of the respective lattice and bosonic and fermionic Fock spaces. The vertex algebra isomorphism f : V → V  respects this tensor product structure. C[& ⊕ & ∗ ] is mapped to C[&  ⊕ &  ∗ ] in an obvious way:      w ab w f : → . m cd m The mapping of Fock spaces is defined by the substitutions    i  j a−s i a−s , s = 1, 2, . . . , → M(g, H )j i j a¯ −s a¯ −s    i  j 1 3 θ−s i θ−s , s = , ,... , →  M(g, H ) j i j ¯θ−s ¯θ−s 2 2

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 a − bH t 0 M(g, H ) = . 0 a + bH

where

In particular, f preserves the bosonic and fermionic vacuum vectors. Let us now indicate why this mapping is an isomorphism of N = 1 SCVA’s. The statement that g takes q to q  is equivalent to a t c + ct a = bt d + d t b = 0,

a t d + ct b = id& ∗ ,

(21)

where a t denotes the conjugate of a, etc. The statement that g takes G to G  is equivalent to H  = (c + dH )(a + bH )−1 , H

(22)

G

= + B  , H = G + B. To show this, let us denote the right-hand side of where the above equation by H  , let G and B  be the symmetric and anti-symmetric parts of H  , and let G  = G(G , B  ). In view of (20) we have 



 t

G = R(G , B )



 G 0 R(G , B  ). 0 G

Let us multiply this equation by g t from the left and by g from the right and use the identity R(G , B  )g = M(g, H )R(G, B),

(23)

which can be easily proved using (21). We get    G 0 M(g, H )R(G, B). g t G  g = R(G, B)t M(g, H )t 0 G We now use another easily checked identity:   −1 −1 G = (a + bH )t G(a + bH )−1 = (a − bH t )t G(a − bH t )−1 , and obtain

(24)

g t G  g = G.

On the other hand, we know that g t G  g = G. Thus G  = G  , and hence G = G , B  = B  , H  = H  . This proves (22). As a consequence of G = G and (24), we obtain a useful formula relating G and G: −1 −1   G = (a + bH )t G(a + bH )−1 = (a − bH t )t G(a − bH t )−1 . (25) Using these relations, one can easily check that the map f intertwines Y and Y  , i.e. Y  (f (a), z, z¯ ) = f Y (a, z, z¯ )f −1 , In particular, we have

 ∂X  i (z) f = M(g, H )ij f ¯  i (z) ∂X  i  ψ (z) f −1 f = M(g, H )ij  ψ¯ i (z) 

−1

∀a ∈ V .

(26)



 ∂X j (z) , ¯ j (z) ∂X  j  ψ (z) . ψ¯ j (z)

(27)

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¯ z), Q(z), Q(¯ ¯ z) imply that the N = 1 These relations and the definition of L(z), L(¯ superconformal structure is also preserved: L (z) = f L(z)f −1 , ¯ z)f −1 , L¯  (¯z) = f L(¯

Q (z) = f Q(z)f −1 , ¯  (¯z) = f Q(¯ ¯ z)f −1 . Q

(28)

Hence f is an isomorphism of N = 1 superconformal vertex algebras. In the remainder of this subsection we prove the “only if” part of the theorem. Let (T , G, B) and (T  , G , B  ) be two real tori equipped with a flat metric and a constant B-field. Thus T = U/ & and T  = U  / &  , where U and U  are real vector spaces and & and &  are lattices of maximal rank in the respective spaces. Clearly, for the N = 1 SCVA’s to be isomorphic, the central charges of the corresponding super-Virasoro algebras must agree, hence dim U = dim U  . We pick an isomorphism of U and U  and a ¯ Q, Q) ¯ and V  = (V  , Y  , |vac , L , L¯  , Q , Q ¯ ) basis in U. Let V = (V , Y, |vac, L, L,  be the corresponding N = 1 SCVA’s. Let f : V → V be an isomorphism of N = 1 SCVA’s. This means that Eqs. (26) and (28) hold true. In particular, f preserves the form of the OPE. Consider the “Hamiltonians” L0 , L¯ 0 ∈ End(V ). A short computation yields: L0 =

1 1 G(Z, Z)− q(Z, Z)+Nb +Nf , 8 4

1 1 L¯ 0 = G(Z, Z)+ q(Z, Z)+ N¯ b + N¯ f . 8 4

Here Z = (W, M) is regarded as an element of End(H) ⊗R (U ⊕ U ∗ ), and we defined

Nb = N¯ b =

∞

s=1 ∞

s=1

G (α−s , αs ) ,

Nf =



rG (ψ−r , ψr ) ,

r=1/2,3/2,...

G (α¯ −s , α¯ s ) ,

N¯ f =



 rG ψ¯ −r , ψ¯ r .

r=1/2,3/2,...

The operators Nb , Nf , N¯ b , N¯ f commute with each other. For what follows it is important to know their spectrum in Fock space. One can show that the Fock space decomposes into a tensor sum of the joint eigenspaces of Nb , Nf , N¯ b , N¯ f , and that all the eigenvalues are nonnegative. Furthermore, the spectrum of Nb , N¯ b is integer, and the spectrum of Nf , N¯ f is half-integer. Finally, the only vector in Hb ⊗ Hf annihilated by all four operators is |vacb  ⊗ |vacf . (All of these facts are standard and can be easily proved using the commutation relations for the oscillators.) Note also that the spectrum of the operator  G(Z,Z) is nonnegative because G is a positive-definite form. The only vector in C & ⊕ & ∗ annihilated by G(Z, Z) is (0, 0). Now let us find all the eigenvectors of L0 , L¯ 0 with eigenvalues (1/2, 0). Suppose a ∈ V is such an eigenvector. Since L0 , L¯ 0 commute with Z = (W, M), we may assume that a is an eigenvector of Z with an eigenvalue z = (w, m), where w ∈ &, m ∈ & ∗ . In view of the above we have three possibilities:

Vertex Algebras, Mirror Symmetry, and D-Branes

Case 1.

Case 2.

105

Nb a = N¯ b a = Nf a = N¯ f a = 0, 1 G(z, z) − q(z, z) = 2, 2 Nb a = N¯ b a = Nf a = 0,

1 G(z, z) + q(z, z) = 0. 2   1 ¯ Nf − a = 0, 2

G(z, z) = q(z, z) = 0. Case 3.



Nb a = N¯ b a = N¯ f a = 0,



Nf −

(29)

1 a = 0, 2

G(z, z) = q(z, z) = 0. The first case is ruled out, because we must have q(z, z) = −1, in contradiction with the fact that q is an even form. In the second case, we must have z = 0. Then from the formulas for L0 , L¯ 0 we see that such a vector has eigenvalues (0, 1/2) rather than (1/2, 0). Hence this case is also ruled out. In the third case, we must have z = 0. Furthermore, it is easy to see that all vectors satisfying (29) must also satisfy αsi a = α¯ si a = 0, ψ¯ ri a = 0,

ψri a

i = 1, . . . , 2d, s = 1, 2, . . . , i = 1, . . . , 2d, r = 1/2, 3/2, . . . ,

= 0,

i = 1, . . . , 2d, r = 3/2, 5/2, . . . .

It follows that a must have the form  2d 

i a= ci ψ−1/2 |vac, 0, 0, i=1

where ci , i = 1, . . . , 2d, are arbitrary complex numbers. A similar argument shows that all eigenvectors of L0 , L¯ 0 with eigenvalues (0, 1/2) have the form  2d 

i c¯i ψ¯ −1/2 |vac, 0, 0, i=1

where c¯i , i = 1, . . . , 2d, are arbitrary complex numbers. Now recall that L0 f = f L0 and L¯ 0 f = f L¯ 0 . This implies that f identifies the (1/2, 0) eigenspace of (L0 , L¯ 0 ) with the (1/2, 0) eigenspace of (L0 , L¯ 0 ), and (0, 1/2) eigenspace of (L0 , L¯ 0 ) with the (0, 1/2) eigenspace of (L0 , L¯ 0 ). Thus there exist two invertible complex matrices Fji and F¯ji such that
 i j ψ−1/2 |vac, 0, 0 = f Fji ψ−1/2 |vac, 0, 0 ,
 i j ψ¯ −1/2 |vac, 0, 0 = f F¯ji ψ¯ −1/2 |vac, 0, 0 .

Applying Y  to both sides of this equation and using (26), we obtain: 

ψ i (z) = f Fji ψ j (z) f −1 ,

ψ¯ i (¯z) = f F¯ji ψ¯ j (¯z) f −1 .

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An immediate consequence of this is the transformation law for fermionic oscillators: 

 j ψ¯ ri = f F¯ji ψ¯ r f −1 ,

j

ψri = f Fji ψr f −1 ,

1 r ∈Z+ . 2

Compatibility with the commutation relations of the fermionic oscillators then requires: F¯ T G F¯ = G.

F T G F = G,

¯ z) with themselves (Alternatively, one may derive this by comparing the OPE of ψ(z), ψ(¯   ¯ and the OPE of ψ (z), ψ (¯z) with themselves.) Now let us turn to bosonic oscillators. Consider the OPE of Q(z) with ψ(z): i ∂Xi (w) Q(z)ψ i (w) ∼ √ . 2 2 (z − w) Since f preserves the OPE and takes Q(z) to Q (z), and ψ(w) to F −1 ψ  (w), we infer that  ∂X i (z) = f Fji ∂X j (z) f −1 . ¯ z) with ψ(w) ¯ Similarly, the OPE of Q(¯ implies that  ∂¯ X¯ i (¯z) = f F¯ji ∂¯ X¯ j (¯z) f −1 .

These formulas imply the following transformation laws for bosonic oscillators: 

j

αni = f Fji αn f −1 ,

 j α¯ ni = f F¯ji α¯ n f −1 ,

n ∈ Z.

Another consequence is the transformation law of Z: Z  = f gZ f −1 , where g ∈ HomR (U ⊕ U ∗ , U  ⊕ U  ∗ ) is defined by   F 0 g = R(G , B  )−1 R(G, B), 0 F¯ and Z and Z  are regarded as elements of End(C[& ⊕& ∗ ])⊗R (U ⊕U ∗ ) and End(C[&  ⊕ &  ∗ ])⊗R (U  ⊕U  ∗ ). Now note that Z and Z  are in fact “realifications” of some elements in End(C[& ⊕ & ∗ ]) ⊗Z (& ⊕ & ∗ ) and End(C[&  ⊕ &  ∗ ]) ⊗Z (&  ⊕ &  ∗ ). This means that g is a “realification” of an element of HomZ (& ⊕ & ∗ , &  ⊕ &  ∗ ), which we also denote g. It remains to show that g takes q to q  and G to G  . To this end notice that the transformation laws for the oscillators imply Nb = f Nb f −1 ,

Nf = f Nf f −1 ,

N¯ b = f N¯ b f −1 ,

N¯ f = f N¯ f f −1 .

Then it follows from L0 = f L0 f −1 and L¯ 0 = f L¯ 0 f −1 that for all x ∈ & ⊕ & ∗ we have q  (gx, gx) = q(x, x), G  (gx, gx) = G(x, x). This concludes the proof of the theorem.

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5.2. Isomorphisms of N = 2 SCVA’s. The goal of this subsection is to prove Theorem 2.1 which we restate below. Given a metric G on U, a compatible complex structure I on U, and B ∈ E2 U ∗ , we define a pair of commuting complex structures on U ⊕ U ∗ as follows:   I 0 , I(I, B) = BI + I t B −I t   −I G−1 B I G−1 J (G, I, B) = . GI − BI G−1 B BI G−1 The complex structure J can be expressed in terms of the K¨ahler form ω = GI and B:   ω−1 B −ω−1 . J (ω, B) = ω + Bω−1 B −Bω−1 We will use a simplified notation I(I, B) = I, I(I  , B  ) = I  , etc. The complex structures I, J and the symmetric forms G, q are related by an identity G = −2qIJ , where G and q are understood as elements of HomR (U, U ∗ ). Theorem 5.2. V ert (&, I, G, B) is isomorphic to V ert (&  , I  , G , B  ) as an N = 2  SCVA if and only if there is an isomorphism of lattices & ⊕ & ∗ and &  ⊕ & ∗ which takes    q to q , I to I , and J to J . To prove this theorem, note that f : V → V  is an isomorphism of N = 2 SCVA’s if and only if it is an isomorphism of the underlying N = 1 SCVA’s, and maps J (z) to J  (z) and J¯(¯z) to J¯ (¯z). Now suppose f is an isomorphism of N = 1 SCVA’s underlying V ert (&, I, G, B) and V ert (&  , I  , G , B  ). By Theorem 5.1 we know that there exists g ∈ Hom(& ⊕ & ∗ , &  ⊕ &  ∗ ) which takes q to q  , and G to G  . To prove the theorem, it is sufficient to show that f maps J (z), J¯(¯z) correctly if and only if g maps I to I  and J to J  . In fact, since G = −2qIJ and G  = −2q  I  J  , it is sufficient to show that f maps J (z), J¯(¯z) correctly if and only if g maps I to I  . Using the transformation law (27) for the fields and the formula (25) relating G and G , one can easily see that f maps J (z) to J  (z) if and only if I  = (a − bH t )I (a − bH t )−1 .

(30)

Similarly, f maps J¯(¯z) to J¯ (¯z) if and only if I  = (a + bH )I (a + bH )−1 . On the other hand, I(I, B) can be written as   −1 I 0 R(G, B). I(I, B) = R(G, B) 0I This and the identity (23) imply that I  = gIg −1 if and only if      I 0 I 0 = M(g, H ) M(g, H )−1 . 0I 0 I This matrix identity is equivalent to (30, 31), which proves the theorem. Let us also note the following simple corollary of this theorem.

(31)

(32)

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Corollary 5.3. Let (T , I, G, B) be a complex torus equipped with a flat K¨ahler metric and a B-field of type (1, 1). Let T  = U  / &  be another torus of the same dimension and I  be a complex structure on T  . Let I˜ and I˜ be the product complex structures on T ×T ∗   and T  × T ∗ . Suppose there exists an isomorphism of lattices g : & ⊕ & ∗ → &  ⊕ & ∗ mapping q to q  and I˜ to I˜ . Then on T  there exists a K¨ahler metric G and a B-field of type (1, 1) such that V ert (&, I, G, B) is isomorphic to V ert (&  , I  , G , B  ) as an N = 2 SCVA. To show this, we define H  using (22) and set G and B  to be the symmetric and skew-symmetric parts of H  , respectively. Then it follows from (25) that G is positivedefinite. By Theorem 5.1 the N = 1 SCVA corresponding to (T , G, B) is isomorphic to an N = 1 SCVA corresponding to (T  , G , B  ). Using the fact that g intertwines I˜ to I˜  it is easy to show that H  I  + I t H  = 0, which means that G is a K¨ahler metric and B  has type (1, 1). In particular, I˜ = I  . Then it follows from the identity G  = −2q  I  J  and the fact g intertwines G, q, I and G  , q  , I  that g also intertwines J and J  . Theorem 5.2 then implies that V ert (&, I, G, B) is isomorphic to V ert (&  , I  , G , B  ) as an N = 2 SCVA. 5.3. Mirror morphisms of N = 2 SCVA’s. In this subsection we establish a criterion for the existence of a mirror morphism between two complex tori equipped with flat K¨ahler metrics and B-fields. Theorem 5.4. V ert (&, I, G, B) is mirror to V ert (&  , I  , G , B  ) if and only if there is  an isomorphism of lattices & ⊕ & ∗ and &  ⊕ & ∗ which takes q to q  , I to J  , and J to  I. The proof is very similar to that of Theorem 5.2. Again it is sufficient to show that if f is an isomorphism of the underlying N = 1 SCVA’s, and g the corresponding isomorphism of lattices, then f J (z)f −1 = −J  (z),

f J¯(¯z)f −1 = J¯ (¯z)

is equivalent to gJ g −1 = I  .

(33)

I  = (a + bH )I (a + bH )−1 = −(a − bH t )I (a − bH t )−1 .

(34)

The first of these is equivalent to

On the other hand, J (G, I, B) can be written as   −1 −I 0 R(G, B), J (G, I, B) = R(G, B) 0 I which together with (23) and (32) implies that (33) is equivalent to      −I 0 I 0 −1 M(g, H ) M(g, H ) = . 0 I 0 I This is obviously equivalent to (34). This concludes the proof.

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6. Homological Mirror Symmetry with B-Fields 6.1. Mirror symmetry and D-branes. As explained in Sect. 2, Kontsevich’s conjecture ∗ ) is must be modified if the B-field does not vanish. When the image of B in H 2 (X, OX b torsion, our results on complex tori suggest that the bounded derived category D (X) should be replaced with D b (X, B), the bounded derived category of coherent modules over an Azumaya algebra. The similarity class of the Azumaya algebra is determined by ∗ ). (Presumably, when B does not map to a torsion class, the image of B in H 2 (X, OX the proper analogue of D b (X) is some “coherent” subcategory of the derived category of quasicoherent sheaves on a gerbe over X, see Remark 2.6.) However, this does not provide any hint as to what the modification of the Fukaya category might be. In this section we explain some string theory lore which suggests a particular definition of the Fukaya category in the presence of the B-field. A similar proposal has been made in [2]. The ordinary σ -model whose quantization yields an N = 2 superconformal vertex algebra is a classical field theory on a two-dimensional manifold F = R × S1 (“the worldsheet”). Let us replace S1 with an interval I = [0, 1] and consider the same σ -model on a worldsheet with boundaries R × I. This procedure is referred to as passing from closed to open strings. Now, in order to make the space of solutions of the Euler-Lagrange equations a symplectic supermanifold, one has to supply boundary conditions for the fields of the σ -model on both ends of the interval. In addition one requires that these boundary conditions preserve N = 2 superconformal symmetry. To be more precise, while the classical σ -model on R × S1 has two copies of the N = 2 superVirasoro algebra (with zero central charge) as its classical symmetry, the σ -model on R × I is required to be symmetric only with respect to a single N = 2 super-Virasoro algebra. There are two essentially different classes of such boundary conditions, called A and B boundary conditions. The B-type boundary conditions preserve the “diagonal” super-Virasoro subalgebra whose generators are given by Ln + L¯ n ,

Jn + J¯n ,

¯+ Q+ r + Qr ,

¯− Q− r + Qr ,

1 n ∈ Z, r ∈ Z + . 2

The A-type boundary conditions preserve a different subalgebra whose generators are Ln + L¯ n ,

−Jn + J¯n ,

¯+ Q− r + Qr ,

¯− Q+ r + Qr ,

1 n ∈ Z, r ∈ Z + . 2

Superconformally-invariant boundary conditions for a σ -model are called supersymmetric (or BPS, for Bogomolny-Prasad-Sommerfeld) D-branes. Thus we have BPS D-branes of types A and B. Note that the mirror involution (2) exchanges the two types of D-branes. D-branes are understood best when the B-field is zero. In this case one can construct examples of the B-type boundary conditions by starting from a holomorphic submanifold of the Calabi-Yau manifold X. More generally, one can start from a holomorphic submanifold M ⊂ X and a holomorphic bundle on M equipped with a compatible connection. On the other hand, examples of the A-type boundary conditions (with zero B-field) can be constructed starting from a Lagrangian submanifold L ⊂ X (with respect to the K¨ahler form), a trivial unitary bundle E on L, and a unitary flat connection on E. Note that one can choose different boundary conditions for the Euler-Lagrange equations on the two ends of the interval I. The only constraint is that both boundary conditions must be of the same type (A or B). If this condition is violated, then the symmetry of the corresponding classical field theory is only some subalgebra of the N = 2 superVirasoro algebra, namely an N = 1 super-Virasoro algebra.

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After quantization, the σ -model on R × I is supposed to yield a superconformally invariant quantum field theory on the same manifold. The axioms of such quantum field theories have not been formulated yet, and we will not attempt it here. Suffice it to say that physicists expect that any B-type D-brane can be consistently quantized, while A-type boundary conditions may lead to “anomalies,” i.e. inconsistencies in the quantization procedure. One can argue that anomalies are absent if the A-type D-brane originates from a special Lagrangian submanifold. We recall that a special Lagrangian submanifold in a Calabi-Yau manifold with a K¨ahler metric is defined by two properties: it is Lagrangian, and the restriction of a nonzero section of the canonical bundle to the submanifold is proportional to its volume form. Thus to any physicist’s Calabi-Yau with zero B-field one can associate two sets: the set of B-type D-branes, and the set of (non-anomalous) A-type D-branes. The former set has many elements in common with the set of coherent sheaves on X. The latter set resembles the set of objects of the Fukaya category of X. Moreover, there are heuristic arguments using path integrals showing that either A or B-type D-branes form an A∞–category (see [16] and references therein). Thus, conjecturally, to every physicist’s Calabi-Yau with zero B-field one can canonically associate a pair of A∞–categories, the categories of A- and B-type D-branes. Assuming there are shift functors on them, one can define the corresponding triangulated categories as in [23]. It is natural to conjecture that for B = 0 the triangulated category associated with A-type (resp. B-type) D-branes is equivalent to DF(X) (resp. D b (X)) [32, 10]. There are several pieces of evidence supporting this conjecture. First, as we have already remarked, F(X) and Coh(X) have many objects in common with the categories of A and B-type D-branes, respectively. Second, using path integrals one can argue [33] that the category of B-type D-branes is independent of the K¨ahler form, while the category of A-type D-branes is independent of the complex structure on X if ω is fixed. For further evidence see [10] and references therein. If this conjecture is true, then Kontsevich’s conjecture has a natural explanation. Suppose we have a mirror pair of physicist’s Calabi-Yaus X and X , both with zero B-field. The corresponding N = 2 SCVA’s are related by a mirror morphism. Since a mirror morphism of N = 2 SCVA’s acts on the N = 2 super-Virasoro by the mirror involution, it exchanges the A and B-type boundary conditions. Hence it induces an equivalence of D b (X) with the derived Fukaya category DF0 (X  ), and vice versa.

6.2. Fukaya category with a B-field. Now let us generalize this to nonzero B-fields. We already know the effect of a B-field on D b (X): the sheaf OX is replaced with a certain sheaf of noncommutative algebras. This agrees with the string theory lore that the B-field makes the D-brane worldvolume noncommutative [7, 11]. The effect of the B-field on the Fukaya category seems rather different. Let us start by recalling the definition of the set of objects of the Fukaya category [23]. Let (X, ω) be a symplectic manifold of dimension 2d. We fix an almost complex structure I on X compatible with ω and thereby obtain a Hermitian metric on X. (If X is a physicist’s Calabi-Yau, it automatically comes equipped with a compatible complex structure). Moreover, we assume that c1 (TXhol ) = 0. In this case the line bundle Ed (Ghol X ) is trivial and has a nowhere vanishing holomorphic section G which is called a calibration. Naively, an object of the Fukaya category should be a triple (L, E, ∇), where L is a Lagrangian submanifold, E is a trivial unitary vector bundle on L, and ∇ is a flat connection on E. From the physical point of view, such a triple allows one to define

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an A-type boundary condition for the classical σ -model, and therefore it is an A-type D-brane [33, 26]. The naive definition of an object does not allow one to define a nontrivial shift functor and A∞ structure. This difficulty can be overcome as follows [23]. For any point x ∈ L the tangent space Tx L is a Lagrangian subspace of Tx X. The Grassmannian of Lagrangian subspaces has fundamental group equal to Z. Each Lagrangian submanifold comes with a Gauss map from L to LG, where LG → X is a fibration whose fiber over →X x is the Grassmannian of Lagrangian subspaces of Tx X. Consider a fibration LG covering LG → X such that its fiber is the universal cover of the fiber of LG → X. (As mentioned in [23], there is a canonical choice of such a fibration if c1 (TXhol ) = 0.)  Not Instead of L, we will consider pairs (L, i), where i is a lift of the Gauss map to LG. every Lagrangian L admits such a lift, so not any Lagrangian submanifold can be extended to an object of the Fukaya category. Note that any Lagrangian L comes equipped with two natural d-forms: the volume form and the restriction of the calibration G. The latter is defined up to a multiplicative constant. Their quotient is a nowhere vanishing  if function f which maps L to C∗ . One can show that the Gauss map admits a lift to LG and only if the image f (L) is contractible. For example, any special Lagrangian L has a lift, because by definition of specialty the function f is constant for any such L. To summarize, we can define an object of the Fukaya category in the absence of the B-field as a quadruple (L, i, E, ∇), where L and i are as above, and (E, ∇) is a trivial complex vector bundle on L with a unitary flat connection. The natural fiberwise action  → X induces an action of Z on such quadruples. One hopes that this action of Z on LG extends to a shift functor from the Fukaya category to itself. Now let us try to guess how the definition of the Fukaya category should be modified when B = 0. Let B be a closed 2-form on X representing B ∈ H 2 (X, R/Z). (Since we assumed that B is in the kernel of the Bockstein homomorphism H 2 (X, R/Z) → H 3 (X, Z), such a 2-form exists.) Let F∇ be the curvature of a connection ∇ on a bundle E on L. If B = 0, the condition on ∇ is F∇ = 0.

(35)

On the other hand, it is a general principle of string theory that the equations of motion must be invariant with respect to a substitution B → B + dλ,

∇ → ∇ + 2π i idE λ|L ,

(36)

where λ is any real 1-form on X. This must be true because the action of the σ -model on R × I is invariant with respect to such transformations [29]. This requirement is sufficient to fix the generalization of (35) to arbitrary B: F∇ = 2πi idE B|L .

(37)

We propose that an object of the Fukaya category for B = 0 is a quadruple (L, i, E, ∇), where L and i are the same as above, E is a complex vector bundle on L, and ∇ is a connection on E satisfying (37). We can make some checks of this proposal. First, our definition of an object depends on how one lifts B ∈ H 2 (X, R/Z) to a 2-form B. However, given two different 2-forms B1 and B2 representing B, there is a one-to-one map between the corresponding sets of objects. Indeed, let f = B2 − B1 . It is easy to see that f has integral periods, and therefore there exists a line bundle N on X and a connection ∇0 on N such that the

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curvature of ∇0 is equal to 2πif. The bijection between the set of objects corresponding to B1 and the set of objects corresponding to B2 is given by L → L,

i → i,

E → E ⊗ N |L ,

∇1 → ∇1 ⊗ idN + idE ⊗ ∇0 .

(38)

Second, from Eq. (37) we see that c1 (E) = rank(E)b|L , where b is the de Rham cohomology class of B. Since c1 (E) is integral, we infer that rank(E)B|L = 0. In particular, for rank(E) = 1, we get that the restriction of B to L must vanish. This is consistent with the results of Hori et al. [18], who analyzed the A-type boundary conditions in the rank-one case. Hori et al. find that the restriction of B to L must be zero if one wants to make an A-type D-brane out of L. We found that it is sufficient to require B|L = 0. We need to address one more subtlety. The original HMSC required E to be a unitary vector bundle and ∇ to be a unitary connection [23]. This requirement naturally arises in the string theory context as well. Nevertheless, this condition is much too strong. Even in the case of the elliptic curve one has to allow for non-unitary connections on the A-side if one wants to account for all bundles on the B-side [31]. In that case, the right thing to do is to require the holonomy representation of ∇ to have eigenvalues with unit modulus. It is natural to conjecture that this is also the right requirement for dimC X > 1 or B = 0. In the absence of the B-field, any pair (L, i) can be extended (in many different ways) to an object of the Fukaya category. The situation is more complex for B = 0. Recall that to any flat connection on a manifold L one can canonically associate a finite-dimensional representation of π1 (L) (or, equivalently, a finite-dimensional representation of the group algebra of π1 (L)), and vice versa. In fact, this map is a one-to-one correspondence. Similarly, given a bundle E on L and a connection ∇ on E such that F∇ satisfies (37), one can construct a finite-dimensional representation of a twisted group algebra of π1 (L) in the following way. To (E, ∇) we can associate a projective representation R of π1 (L). To any such R one can attach an element ψR of H 2 (π1 (L), U (1)). Acting on it with the natural embedding j

H 2 (π1 (L), U (1)) → H 2 (L, U (1))

(39)

we obtain an element j (ψR ) ∈ H 2 (L, U (1)). One can show that j (ψR ) = B|L (we identify R/Z with U (1)). To any 2-cocycle ψ one can associate a twisted group algebra Cψ [π1 (L)], which is a vector space generated by the elements of π1 (L) with the following multiplication law: g · h = ψ(g, h)gh,

g, h ∈ π1 (L).

The correspondence between pairs (E, ∇) satisfying (37) and finite-dimensional representations of the twisted group algebra Cψ [π1 (L)] is one-to-one. A proof of this fact is given in Appendix C. The eigenvalues of the holonomy representation of ∇ have unit modulus if and only if the eigenvalues of g ∈ π1 (L) have unit modulus. In particular this means that a Lagrangian submanifold L can be extended to an object of the Fukaya category only if B|L is in the image of the homomorphism (39). As a by-product, we obtained an equivalent definition of an object of the Fukaya category: it is a triple (L, i, R), where L, i are the same as above, and R is a finite-dimensional

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representation of the twisted group algebra Cψ [π1 (L)] such that j (ψR ) = B|L and all the eigenvalues of R(g) have unit modulus for all g ∈ π1 (L). Morphisms in the modified Fukaya category F(X, B) are defined in analogy with [12, 23]. Let U1 = (L1 , i1 , E1 , ∇1 ) and U2 = (L2 , i2 , E2 , ∇2 ) be two objects such that L1 and L2 intersect transversally. Morphisms from U1 to U2 in F(X) form a complex of vector spaces defined by the rule  Homi (E1 |x , E2 |x ). (40) Hom· (U1 , U2 ) = x∈L1 ∩L2

It is graded in the following way. For any point x ∈ L1 ∩ L2 we have two points i1 (x) and i2 (x) on the universal cover of the Lagrangian Grassmannian of Tx X. To these two points we can associate an integer µ(i1 (x), i2 (x)) which is called the Maslov index of i1 (x), i2 (x) (see for example [3]). By definition, the space Hom(E1 |x , E2 |x ) has a grading µ(i1 (x), i2 (x)). The differential on Hom(U1 , U2 ) is defined by the rule

m1 (u; z), d(u) = z∈L1 ∩L2

where u ∈ Hom(E1 |x , E2 |x ), and m1 (u; z) ∈ Hom(E1 |z , E2 |z ) is given by     

∗ ∗ m1 (u; z) = ± exp 2πi φ (−B + iω) · P exp φ ∇ . D

φ:D→X

∂D

Here φ is an (anti)-holomorphic map from the disk D = {|w| ≤ 1, w ∈ C} to X such that φ(−1) = x, φ(1) = z and φ([x, z]) ⊂ L2 and φ([z, x]) ⊂ L1 . The path-ordered integral is defined by the following rule    x    z ∗ ∗ ∗ P exp φ ∇ := P exp φ ∇2 · u · P exp φ ∇1 . ∂D

x

z

This homomorphism from E1 |z to E2 |z can be described as follows. We take a vector e ∈ E1 |z , use the connection ∇1 transport it to E1 |x , apply the map u, and obtain an element of E2 |x . Then we transport this element to E2 |z using the connection ∇2 . The ± sign indicates the natural orientation on the space of (anti)-holomorphic maps. One expects that there are finitely many such maps if µz − µx = 1. To define the composition of morphisms, let us take u ∈ Hom(E1 |x , E2 |x ) and v ∈ Hom(E2 |y , E3 |y ), where x ∈ L1 ∩ L2 and y ∈ L2 ∩ L3 . Then the composition of u and v is defined as

v◦u= m2 (v, u; z), z∈L1 ∩L3

where m2 (v, u; z) ∈ Hom(E1 |z , E3 |z ) is given by    

m2 (v, u; z) = ± exp 2πi φ ∗ (−B + iω) · P exp D

φ:D→X

∂D

 φ∗∇ .

Here we sum over (anti)-holomorphic maps φ from a two-dimensional disk D to X, such that three fixed points p1 , p2 , p3 ∈ ∂D are mapped to x, y, z respectively, and φ([pi , pi+1 ]) ∈ Li+1 . The path-ordered integral here is calculated by the rule    p2   p1    p3 ∗ ∗ ∗ ∗ P exp φ ∇ := P exp φ ∇3 ·v·P exp φ ∇2 ·u·P exp φ ∇1 ∂D

p2

p1

p3

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In the same manner we can define higher order compositions using zero-dimensional components of spaces of maps φ from the disk D to X with φ(∂D) sitting in the union of Lagrangian submanifolds. It is easy to check that the above definition of morphisms and their compositions does not change if we replace B with another 2-form with the same image in H 2 (X, R/Z). The check makes use of (37) and (38). This confirms our claim that the Fukaya category depends only on B. The rules for computing morphisms and their compositions can be explained heuristically using the path integral for the σ -model on a worldsheet with boundaries [33]. The category F0 (X) has the same objects as F(X), but the morphisms are the degree zero cohomology groups of the complexes defined above. Note that different objects of F(X) often become isomorphic in F0 (X). For example, in the case when X is a real symplectic 2-torus, any one-dimensional submanifold is Lagrangian. Many of them admit a lift of the Gauss map. Thus the category F(X) contains many more objects than the derived category of the elliptic curve (an elliptic curve with a flat metric is a selfmirror). But in F0 (X) any object becomes isomorphic to some other object associated with a special Lagrangian submanifold (see [31]). More generally, it appears likely that working in the category F0 (X) one may restrict the set of objects of the Fukaya category and consider only special Lagrangian submanifolds with respect to a holomorphic calibration. For different L the calibrations may differ by a multiplicative constant. This restriction is also natural from the string theory point of view, since, as explained above, non-anomalous A-type D-branes are associated with special Lagrangian submanifolds in a Calabi-Yau [26]. A. Supersymmetric σ -Model of a Flat Torus In this section we define the classical field theory known in the physics literature as the N = 1 supersymmetric σ -model. The data needed to specify a σ -model consist of a C ∞ manifold M (“the target space”), a Riemannian metric G on M, and a 2-form B on M. We then discuss the problem of the quantization of the σ -model in the case when the target space is a flat torus. The superconformal vertex algebra constructed in Sect. 4 can be regarded as a solution of the quantization problem. A detailed discussion of supersymmetric σ -models can be found in [8]. Let W be a two-dimensional C ∞ manifold R × S1 (“the worldsheet”). We parametrize W by (τ, σ ) ∈ R × R/(2πZ). The coordinate τ is regarded as “time.” We endow W with a Minkowskian metric ds 2 = dτ 2 − dσ 2 and orientation dτ ∧ dσ. Thus ∗dσ = dτ, ∗dτ = dσ. The symmetric tensor corresponding to the metric will be denoted g. General coordinates on W will be denoted (y 0 , y 1 ). The invariant volume element √ dτ ∧ dσ = d 2 y − det g will be denoted dF. We denote by S + and S − = S +∗ the complexified semi-spinor representations of SO(1, 1) and by V its complexified fundamental representation. Complexified semi-spinor representations are one-dimensional complex vector spaces endowed with SO(1, 1)-invariant nondegenerate morphisms γ : S− → V ⊗ S+,

γ¯ : S + → V ⊗ S − .

(41)

These morphisms are determined up to a scalar factor, and we assume that they satisfy the Clifford algebra relation γ γ¯ + γ¯ γ = 2g −1 · idS + ⊕S − .

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Here g −1 is regarded as map C → V ∗ ⊗ V ∗ . In a suitable basis, one has     1 1 γ = , γ¯ = . −1 1 Since H 1 (W, Z2 ) = Z2 , there are two inequivalent spinor structures on W. The trivial one is called the periodic, or Ramond, spin structure in the physics literature. The nontrivial one is known as the anti-periodic, or Neveu-Schwarz, spin structure. Both spin structures play a role in string theory, but for our purposes it will be sufficient to consider the Neveu-Schwarz spin structure. The corresponding semi-spinor bundles on W will be denoted by the same letters S + , S − . The parity-reversed (i.e. odd) semi-spinor bundles will be denoted by NS + , NS − . More generally, N will denote the parity-reversal functor. The vector space morphisms γ and γ¯ give rise to a pair of bundle morphisms S − → T W ⊗ S + and S + → T W ⊗ S − which we denote by the same letters. Let M be a C ∞ manifold endowed with a Riemannian metric G and a real 2-form B. At this stage we do not require B to be closed. The indices of the tangent bundle T M will be denoted by j, k, l, . . . in the upper position. The indices of the cotangent bundle T ∗ M will be denoted by the same letters in the lower position. Summation over repeating indices is always implied. Let X be a C ∞ map from W to M. Let ψ and ψ¯ be C ∞ sections of X∗ T M ⊕ NS + and X ∗ T M ⊕ NS − , respectively. N = 1 supersymmetric σ -model with worldsheet W and target (X, G, B) is a classical field theory on W defined by the action    

1 1 j k Gj k (X) dX ∧ ∗dX + Bj k (X) dX j ∧ dX k 4π W 4π W   1 + Gj k (X)ψ j i γ¯ · ∇ψ k + Gj k (X)ψ¯ j iγ · ∇ ψ¯ k 4π W  1 j k ¯l ¯m + Rj klm (X)ψ ψ ψ ψ dF. (42) 2 Here the covariant derivatives ∇ψ and ∇ ψ¯ are sections of X∗ T M ⊗ NS ± ⊗ T ∗ W defined as follows:    3
−1 j m j j j ∇ψ = Dψ + G + (dB)klm dX k ψ l , kl 2    3
−1 j m j j j ¯ ¯ ∇ ψ = Dψ + − (dB)klm dX k ψ¯ l , G kl 2 where {j, kl} are the Christoffel symbols constructed from G, and D : S ± → S ± ⊗T ∗ W is the Levi-Civita covariant derivative constructed from g. Rj klm (X) is the curvature corresponding to the following connection 1-form on M:    3
−1 j m j + (dB)klm dx l . G kl 2 In the last term in the action we used twice the natural SO(1, 1)-invariant pairing S + ⊗ S − → C. This complicated-looking action has an elegant reformulation in terms of superfields, i.e. maps from a super-Riemann surface to M [8].

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The extrema of the action (42) are given by the solutions of the Euler-Lagrange equations. In the case when all the fields are even, it is well known that the space of solutions of the Euler-Lagrange equations is a manifold with a natural symplectic structure. This statement remains true in the supersymmetric context (see e.g. [15]). In the present case the symplectic structure is given by     1 ∂X k ∂X k j δX ∧ δ Gj k (X) + Bj k (X) 2π τ =τ0 ∂τ ∂σ  j k j k ¯ ¯ + − iGj k (X)δψ ∧ δψ + iGj k (X)δ ψ ∧ δ ψ dσ. (43) Here we used the fact the Euler-Lagrange equations are second-order in time derivatives ¯ and therefore a solution is completely of X and first-order in time derivatives of ψ, ψ, determined by the values of X, ∂X/∂τ, ψ, and ψ¯ on any circle τ = τ0 . One can check that the symplectic structure thus defined does not depend on τ0 . The space of solutions endowed with this symplectic structure is called the phase space of the σ -model. We are interested in the case when M is a torus T 2d = R2d / &, & ∼ = Z2d , with a constant metric G and a constant 2-form B. We will fix an isomorphism between & and Z2d . Without loss of generality we may assume that the action of & on R2d is x j → x j + 2πnj ,

nj ∈ Z,

j = 1, 2, . . . , 2d.

In this special case the σ -model action becomes  j    1 ∂X ∂X k ∂X j ∂X k ∂X j ∂X k Gj k − + 2Bj k 4π W ∂τ ∂τ ∂σ ∂σ ∂τ ∂σ      ∂ ∂ ∂ ∂ + iGj k ψ j + ψ k + iGj k ψ¯ j − ψ¯ k dτ dσ. ∂τ ∂σ ∂τ ∂σ The Euler-Lagrange equations have a simple form:  2    ∂ ∂2 ∂ ∂ j X − = 0, + ψ j = 0, ∂σ 2 ∂τ 2 ∂σ ∂τ In what follows we will use the notation   1 ∂ ∂ ∂− = − , 2 ∂σ ∂τ

1 ∂+ = 2





∂ ∂ − ∂σ ∂τ

∂ ∂ + ∂σ ∂τ



ψ¯ j = 0.

(45)

 .

The Poisson brackets of the fields evaluated at equal times follow from (43):   Xj (τ, σ ), Xk (τ, σ  ) = 0, P .B.   j k 
∂Xk (τ, σ  ) Xj (τ, σ ), = 2π G−1 δ σ − σ , ∂τ  P .B. ¯ ψ(τ, σ ), ψ(τ, σ  ) P .B. = 0,
j k    ψ(τ, σ ), ψ(τ, σ  ) P .B. = −2π i G−1 δ σ − σ , j k 
  ¯ ¯ ψ(τ, σ ), ψ(τ, σ  ) P .B. = −2π i G−1 δ σ − σ . The Poisson brackets between even and odd fields vanish.

(44)

(46)

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Note that neither the Euler-Lagrange equations (45) nor the symplectic structure corresponding to (46) depend on B. This happens whenever B is closed, because in this case the B-dependent terms in the action are locally total derivatives. We will see below that quantization of the σ -model introduces arbitrariness which is parametrized by a class in H 2 (M, R/Z). The usual interpretation is that while the classical σ -model does not detect a closed B-field, the quantized σ -model detects the image of B in H 2 (M, R/Z). The Euler-Lagrange equations (45) can be rewritten in the Hamiltonian form:   ∂Xj (τ, σ ) = X j (τ, σ ), H (τ ) , P .B. ∂τ  j  j   ∂X ∂ ∂X (τ, σ ) = (τ, σ ), H (τ ) , ∂τ ∂τ ∂τ P .B.   ∂ψ j , (τ, σ ) = ψ j (τ, σ ), H (τ ) P .B. ∂τ   ∂ ψ¯ j . (τ, σ ) = ψ¯ j (τ, σ ), H (τ ) P .B. ∂τ The Hamiltonian H is a function on the phase space given by  j   k ¯k ∂X ∂X k 1 ∂X j ∂X k j ∂ψ j ∂ψ ¯ H (τ0 ) = Gj k + − iψ + iψ dσ. 4π τ =τ0 ∂τ ∂τ ∂σ ∂σ ∂σ ∂σ As a consequence of the equations of motion, we have dHdτ(τ0 0 ) = 0. Hamiltonian vector fields on the phase space are those vector fields which preserve the symplectic form. They obviously form a Lie (super-)algebra with respect to the Lie bracket. We will now exhibit a subalgebra in this super-algebra which is isomorphic to the direct sum of two copies of the N = 1 super-Virasoro algebra. Recall that given a function W on the phase space, we can define a Hamiltonian vector field vW as follows: vW (·) = { · , W }P .B. One has an identity [vW , vU ]Lie = v{W,U }P .B. . We will define a set of functions on the phase space which forms a super-Virasoro algebra with respect to the Poisson bracket; then the corresponding set of Hamiltonian vector fields forms a super-Virasoro algebra with respect to the Lie bracket. The set of functions we want to define is a vector space generated over C by the following elements:    1 i −inσ j k Ln = e Gj k ∂− X ∂− X − ψ∂− ψ dσ, n ∈ Z, 2π τ =τ0 2    1 i ¯ + ψ¯ dσ, L¯ n = einσ Gj k ∂+ X j ∂+ X k + ψ∂ n ∈ Z, 2π τ =τ0 2 (47)  −i 1 Qr = e−irσ Gj k ψ j ∂− X k dσ, r ∈Z+ , 4π τ =τ0 2  i 1 ¯ Qr = eirσ Gj k ψ¯ j ∂− X k dσ, r ∈Z+ . 4π τ =τ0 2

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Two remarks are in order concerning these expressions. First, all these functions on the phase space implicitly depend on τ0 as a parameter. Second, since we picked the antiperiodic spin structure on W, the lift of ψ to the universal cover of W is an anti-periodic function of σ. This is the reason the index r runs over half-integers. The Poisson brackets of the generators can be easily computed using (46), and the nonvanishing ones turn out to be {Lm , Ln }P .B. = −i(m − n)Lm+n , 
m {Lm , Qr }P .B. = −i − r Qm+r , 2 i {Qr , Qs }P .B. = − Lr+s , 2









L¯ m , L¯ n

¯r L¯ m , Q

P .B. P .B.

  ¯ r, Q ¯s Q P .B.

= −i(m − n)L¯ m+n , 
m ¯ m+r , (48) −r Q = −i 2 i = − L¯ r+s . 2

Thus the space spanned by the generators is a Lie super-algebra isomorphic to the direct sum of two copies of the N = 1 super-Virasoro algebra (with zero central charge). Note that L0 + L¯ 0 = H. Recalling that the τ -dependence of any function F on the phase space is determined by dF = {F, H }P .B. , dτ and using (48), one can show that all the generators have a very simple dependence on τ0 : Ln (τ0 ) = e−inτ0 Ln (0), Qr (τ0 ) = e−irτ0 Qr (0),

L¯ n (τ0 ) = e−inτ0 L¯ n (0), ¯ r (0). ¯ r (τ0 ) = e−irτ0 Q Q

Thus the space spanned by the generators does not depend on τ0 . The presence of two copies of the N = 1 super-Virasoro algebra acting on the phase space is a feature of the supersymmetric σ -model with an arbitrary target (M, G, B). This fact is crucial for string theory applications of the σ -model, see [29] for details. Now let us choose a constant complex structure I on M such that G is a Hermitian metric. This makes M a K¨ahler manifold. Let ω = GI be the corresponding K¨ahler form. It turns out that we can embed each of the two N = 1 super-Virasoro algebras in a bigger N = 2 super-Virasoro algebra. The additional generators are given by   −i e−i(r+1/2)σ Gj k ∓ iωj k ψ j ∂− X k dσ, 8π τ =τ0   i ± ¯ Qr = ei(r+1/2)σ Gj k ∓ iωj k ψ¯ j ∂+ X k dσ, 8π τ =τ0 Q± r =

1 r ∈Z+ , 2 1 r ∈Z+ , 2 (49)



−i e−inσ ωj k ψ j ψ k dσ, 4π τ =τ0  −i ¯ Jn = einσ ωj k ψ¯ j ψ¯ k dσ, 4π τ =τ0 Jn =

n ∈ Z, n ∈ Z.

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− ¯ ¯+ ¯− Note that Qr = Q+ r + Qr and Qr = Qr + Qr for all r. The Poisson brackets between ± Ln , Qr , and Jn are given by



Lm , Q± r

 P .B.

=−i


m

 − r Q± r+m ,

2 {Lm , Jn }P .B. = inJn+m ,  + +   − Qr , Qs P .B. = Q− r , Qr P .B. = 0,  + − i i Qr , Qs P .B. = − Lr+s − (r − s)Jr+s , 4 8   ± Jm , Q± r P .B. = ∓ iQr+m . The Poisson brackets between the barred generators have the same form. The Poisson brackets between barred and unbarred generators are trivial, as usual. Again, the emergence of the N = 2 super-Virasoro is not limited to the particular situation we are considering: one can prove that the phase space of the supersymmetric σ -model is acted upon by the N = 2 super-Virasoro if (M, G) is an arbitrary K¨ahler manifold, and B is closed [1]. The statement can be further generalized to B-fields which are not closed [13]. Let us now look more closely at the space of solutions of the Euler-Lagrange equations. Note that any map X : W → M induces a homomorphism of the homology groups H1 (W) → H1 (M). The group H1 (W) ∼ = π1 (W) ∼ = Z has a preferred generator, namely the loop winding the S1 in the direction of increasing σ. Since H1 (T 2d ) = &, we see that to any map X : W → M we can assign an element w(X) of &. The components of w are the so-called winding numbers of the map X. Thus the phase space of the σ -model is a disconnected sum  M= Mw . w∈&

We will see in a moment that Mw is connected for all w. The Euler-Lagrange equations (45) are linear and can be solved by Fourier transform. The general solution in Mw is given by ∞ j k 
i  1
j is(σ −τ ) j αs e pk + √ + α¯ s e−is(σ +τ ) , (50) Xj = x j + σ w j + τ G−1 2 s=−∞ s

j ψr eir(σ −τ ) , (51) ψj = r∈Z+1/2

¯j

ψ =



j ψ¯ r e−ir(σ +τ ) .

(52)

r∈Z+1/2 j j j j j j j j Here αs , α¯ s are complex numbers satisfying (αs )∗ = α−s , (α¯ s )∗ = α¯ −s ; ψr , ψ¯ r are elements of NC; x j , j = 1, . . . , 2d, take values in R/(2π Z); and pj , i = 1, . . . , 2d, j j j j take values in R. The variables αs , α¯ s , ψr , ψ¯ r will be referred to as “the oscillators.” The j variables (x , pj ), j = 1, . . . , 2d, together parametrize a copy of T ∗ M ∼ = T 2d × R2d . Thus for any w ∈ & the supermanifold Mw is a product of the vector superspace spanned by αn , α¯ n , n ∈ Z, ψr , ψ¯ r , r ∈ Z + 1/2, and the cotangent bundle of M.

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The Poisson brackets of the coordinates on Mw can be computed from (46) and (50-52). The non-vanishing ones are given by 

 j k  j k j j k k αn , αm = −in G−1 δm+n , α¯ n , α¯ m = −in G−1 δm+n , P .B. P .B. 

 j k  j k j j ψr , ψsk = −i G−1 δr+s , ψ¯ r , ψ¯ sk = −i G−1 δr+s , P .B. P .B.   j x j , pk = δk . P .B.

Thus the symplectic supermanifold Mw decomposes into a product of a symplectic vector superspace spanned by the oscillators and T ∗ M with the standard symplectic structure. It is customary to continue analytically the time variable τ to the imaginary axis. If we set τ = it, then the combination v = σ + τ = σ + it becomes a complex variable. Since we identify σ ∼ σ + 2π, it is convenient to set v = i log z, where z ∈ C∗ . After analytic continuation ∂− and ∂+ become ∂v = −iz∂z and ∂¯v = i z¯ ∂¯z , respectively. The functions Xj (v(z)) are multi-valued functions of z if w = 0. But their derivatives with respect to z and z¯ are single-valued, and moreover are holomorphic and anti-holomorphic, respectively:   ∞ j ∂Xj i  α s i
−1 j k pk − w j − √ , =− G ∂z 2z 2 s=−∞ zs+1   ∞ j ∂Xj i  α¯ s i
−1 j k pk + w j − √ . =− G ∂ z¯ 2¯z 2 s=−∞ z¯ s+1

√ Note that after rescaling Xj → (i 2)X j these expressions become formally the same as (13,14), except that in (13,14) the coordinates on the phase space w k , pk , αsk , α¯ sk are replaced with the operators W k , Mk − Bkl W l , αsk , α¯ sk , respectively. This replacement is the quantization map discussed in more detail below. Similarly, after analytic continuation to imaginary τ, the sections ψ j and ψ¯ j become holomorphic and anti-holomorphic, respectively. One additional subtlety arises due to the fact that ψ and ψ¯ are sections of semi-spinor bundles. Thus the coordinate change v → z = e−iv must be accompanied by ψ j → z−1/2 ψ j , and ψ¯ j → z¯ −1/2 ψ¯ j . This accounts for the shift r → r + 21 between (51,52) and (15,16). Let us now turn to the quantization of the σ -model. This discussion provides a motivation for the constructions of Sect. 4. Since the classical phase space is a disconnected sum of identical pieces labeled by w ∈ &, the quantum-mechanical Hilbert space will be a direct sum of identical Hilbert spaces labeled by w ∈ &. Thus we only need to understand how to quantize the supermanifold Mw . In turn, Mw decomposes as a product of T ∗ M with the standard symplectic structure, and a vector superspace spanned by the oscillators. The vector superspace spanned by the oscillators can be quantized using the wellknown Fock-Bargmann prescription. The resulting Hilbert superspace is the so-called Fock space, i.e. the completion with respect to a suitable norm of the space of polyi ,a i , s = 1, 2, . . . , and odd variables θ , θ¯ , r = nomials of even variables a−s ¯ −s −r −r 1/2, 3/2, . . . . We will denote this space of polynomials HF ock . The quantization of T ∗ M is also standard and yields the Hilbert space which is the completion of the space C ∞ (M) of smooth functions on M = R2d / & with respect to

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an L2 norm. Using the Fourier transform, this Hilbert space can be identified with the completion of the group algebra of & ∗ with respect to an O2 norm. Thus the quantization procedure sketched above leads to the Hilbert space which is a suitable completion of an infinite-dimensional superspace ⊕w∈& C[& ∗ ] ⊗ HF ock . This can be written in a more symmetric form: C[& ⊕ & ∗ ] ⊗ HF ock . For our purposes, only the superspace structure, and not the Hilbert space structure, is important. Thus we need not perform the completion procedure, and can take the above superspace as the state space of the N = 2 superconformal vertex algebra corresponding to the supersymmetric σ -model. We will call this vector superspace the state space of the quantized σ -model. Finding a suitable state space is but a part of the quantization problem. Quantizing a classical field theory usually requires finding a sufficiently large subset of functions on the phase space closed under the Poisson brackets, and a map from this subset to the set of linear operators on the state space, such that the Poisson brackets are mapped to −i times the graded commutator. The choice of the subset of functions on the phase space is dictated by physical considerations. For example, for string theory applications it is imperative to have an N = 1 super-Virasoro algebra acting on the state space. Thus the distinguished subset must include the generators of the N = 1 super-Virasoro algebra (47) and their linear combinations. We will also require that the subset include the generators of the N = 2 super-Virasoro (49). Usually one also requires that the distinguished subset include the fields in terms of which the classical action is written. In our case these are Xj (σ, τ ), ψ j (σ, τ ), ψ¯ j (σ, τ ). One also wants the operator corresponding to the Hamiltonian H = L0 + L¯ 0 to have nonnegative spectrum. To quantize the fields X j , ψ j , and ψ¯ j it is sufficient to quantize the oscillators and (x j , pj ) (the coordinates on T ∗ M). The Fock-Bargmann quantization map sends oscillators with negative subscripts to multiplication operators on the space of polynomials: j

j

α¯ s → a¯ s ,

j

j

j

j j ψ¯ r → θ¯r ,

αs → as , ψr → θr ,

j

s = −1, −2, . . . , 3 1 r = − ,− ,... . 2 2

The oscillators with positive subscripts are mapped to differentiation operators on the space of polynomials:
j k j αs → s G−1
j k j ψr → G−1

∂ k ∂a−s

∂

k ∂θ−r

,

,


j k j α¯ s → s G−1
j k j ψ¯ r → G−1

∂

k ∂ a¯ −s ∂ , k ¯ ∂ θ−r

,

s = 1, 2, . . . , r=

1 3 , ,... . 2 2

It is easy to see that the (graded) commutators between these operators are equal to i times the Poisson brackets of their classical counterparts, as required. The quantization of (x j , pj ) proceeds as follows. The function x j is a multi-valued function on the phase space and cannot be quantized. But any smooth function f (x 1 , . . . , x 2d ) which is periodic, i.e. invariant with respect to shifts x j → x j +

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2π nj , nj ∈ Z, is a univalued function on the phase space. The standard quantization of T ∗ M maps such a function to a multiplication operator on C ∞ (M): f (x 1 , . . . , x 2d ) → f (x 1 , . . . , x 2d ). Actually, the vector space we are dealing with is not just C ∞ (M), but a &-graded vector space F = ⊕w∈& C ∞ (M), and therefore we should quantize a pair (f, w) rather than f. This leads to an important subtlety. If w = 0, we can assign to (f, w) a multiplication operator which acts on each of the &-homogeneous components of F in an identical manner. On the other hand, if w = 0, it does not seem right to assign to it multiplication by f, since such a quantization procedure would map different classical functions to the same quantum-mechanical operator. A natural guess for the operator corresponding to (f, w) is multiplication by f followed by an operator Tw , where Tw shifts the &-grading by w. This guess will be justified below. Under the standard quantization of T ∗ M, the function pj is mapped to a differentiation operator on F: pj → pˆ j = −i

∂ . ∂x j

(53)

If fˆw is the quantum operator corresponding to the function (f, w) ∈ F, we have the commutation relation   ∂f [fˆw , pˆ j ] = i . ∂x j w This should be compared with the classical relation {f (x), pj }P .B. =

∂f (x) . ∂x j

The Fourier transform which identifies the completion of F with the completion of C[& ⊕ & ∗ ] sends pˆ j to the following operator Mj on C[& ⊕ & ∗ ]: Mj : (w, m) → mj (w, m),

∀(w, m) ∈ & ⊕ & ∗ .

(54)

¯ j , ψ j , and ψ¯ j . Putting all this together, we obtain the quantization map for ∂Xj , ∂X It is easy to check that this √ yields the expressions (13–16) of Sect. 4 with B = 0 (after we rescale Xj by a factor i 2). Now we can also motivate the state-operator correspondence postulated in Sect. 4. The main idea that the quantization map should send local classical observables to local ¯ j , ψ j , ψ¯ j and their quantum fields belonging to the image of Y. For example, ∂Xj , ∂X derivatives are local classical observables, so the corresponding quantum fields must lie j in the image of Y. These considerations explain the mapping of the states α−s |vac, j j j α¯ −s |vac, ψ−r |vac, and ψ¯ −r |vac. Together with the axioms of vertex algebra, this uniquely fixes the mapping of other states in the subspace w = m = 0. Other natural local classical observables are suitable exponentials of X j (z, z¯ ). (The classical field Xj (z, z¯ ) itself is multi-valued and therefore should not be quantized.) Requiring that they map to local quantum fields fixes the form of Y for all (w, m) ∈ & ⊕ & ∗ . An interested reader is referred to [29] for details.

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Another important ingredient is the quantization of the N = 2 super-Virasoro algebra. Naively, one would like to define the quantum generators by the same formulas (47,49), but with the classical fields replaced by the quantum fields. This idea runs into an immediate problem since the products of quantum fields at the same point are not well-defined. The normal ordering prescription resolves this problem and leads to well-defined operators. One can easily check that this definition of the generators of the N = 2 super-Virasoro is equivalent to the one given in Sect. 4. The operators thus defined form an infinite-dimensional Lie super-algebra which is a central extension of the classical N = 2 super-Virasoro (47,49). One can also check that the spectrum of H = L0 + L¯ 0 is nonnegative. It remains to explain how to include the effect of the B-field. As remarked above, a closed B-field does not affect the classical σ -model. However, the above quantization procedure admits a modification which depends on a class in H 2 (M, R/Z). We wish to interpret this class as the cohomology class of the B-field. The modification affects the quantization of T ∗ M and consists in replacing the space of smooth functions on R2d / & with the space of smooth functions on R2d satisfying the following quasi-periodicity condition: j wk

f (x 1 + 2πn1 , . . . , x 2d + 2πn2d ) = e−2πiBj k n

f (x 1 , . . . , x 2d ),

where Bj k is a real skew-symmetric matrix which we can interpret as an element of H 2 (M, R) in a natural manner. We will denote the space of such functions Cw∞ (M, B). It is clear that Cw∞ (M, B) depends only on the image of B in H 2 (M, R/Z). Thus the modification consists of replacing F with the space F(B) = ⊕w∈& Cw∞ (M, B). Fourier transform identifies a completion of Cw∞ (M, B) with a completion of C[& ∗ ], as before, so the Hilbert space of the quantum theory is unaffected by B. But the map of the classical functions on the phase space to operators is affected. First, the product of two quasi-periodic functions f ∈ Cw∞ (M, B) and f  ∈ Cw∞ ∞ (M, B) belongs to the space Cw+w  (M, B). Hence the multiplication operators do not preserve the &-grading on F(B). Rather, multiplication by f ∈ Cw∞ (M, B) shifts the grading by w. If we want the limit B → 0 to be smooth, we have to postulate that even for B = 0 multiplication by f ∈ Cw∞ (M, B) shifts the grading by w. This provides a justification for the guess made above. Second, while the function pj is still mapped according to (53), the Fourier transform of pˆ j is different from (54). Namely, it is easy to see that the Fourier transform of the differentiation operator on Cw∞ (M, B) is given by Mj − Bj k w k . Putting these facts together, one obtains the quantization map for all classical fields in agreement with (13–16). B. The Relation Between Vertex Algebras and Chiral Algebras In this appendix we describe some properties of vertex algebras in the sense of Definition 3.3. Let (V , |vac, T , T¯ , Y ) be a vertex algebra. We prove that the subspace of V spanned by vectors which are mapped by Y to meromorphic fields has a natural structure of a chiral algebra. Furthermore, anti-meromorphic fields form another chiral algebra, and these two chiral algebras supercommute with each other. We also describe an analogue of the Borcherds (or associativity) formula for vertex algebras. Finally, we show that any chiral algebra is a vertex algebra.

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We start with the following useful lemma. Lemma B.1. Let N, M be integers and let hj , j = 1, . . . , K be distinct real numbers belonging to [0, 1). Suppose the following relation holds: K

iz,w

j =1

1 1 iz¯ ,w¯ Cj (z, z¯ , w, w) ¯ = 0, N+h j (z − w) (¯z − w) ¯ M+hj

(55)

¯ ∈ QF2 (V ). Then Cj (z, z¯ , w, w) ¯ ≡ 0 for all j. where Cj (z, z¯ , w, w) It is sufficient to prove the statement for M = N = 0. Let v ∈ V be an arbitrary vector. We are going to prove that the value of Cj on v vanishes for all j. To this end let us evaluate both sides of (55) on v and set w = zx and w¯ = z¯ x. ¯ Since Cj ∈ QF2 (V ), the expression Cj (z, z¯ , zx, z¯ x)(v) ¯ can be written as

fαβ (x, x)z ¯ −α z¯ −β , (56) α,β

where each fαβ is a finite sum of fractional powers of x, x¯ with coefficients in V . Hence the value of the left-hand side of Eq. (55) on v is a sum



z−α z¯ −β x −γ x¯ −δ Tαβγ δ , α,β

(γ ,δ)∈Jαβ

where Jαβ ⊂ R2 is a finite set for each (α, β). Each Tαβγ δ has the form K

j =1

ix

1 1 ix¯ fj (x, x), ¯ h j (1 − x) (1 − x) ¯ hj

(57)

where all fj are polynomials in x, x¯ with coefficients in V , and hj ∈ [0, 1) are distinct real numbers. The symbol ix (resp. ix¯ ) means “expand in a Taylor series around x = 0” (resp. x¯ = 0). To prove the lemma it is sufficient to show that if the expression Eq. (57) is zero, then fj ≡ 0 for all j. To prove this, we rewrite fj as a polynomial in 1 − x and 1 − x. ¯ Then Eq. (57) takes the form L

l=1

ix

1 1 ix¯ al , (1 − x)tl (1 − x) ¯ sl

where (tl , sl ) are distinct pairs of real numbers, and each al ∈ V is a coefficient of some fj . Let us denote this expression by T . We will show by induction in L that if T is equal to 0 then al = 0 for all l. This will imply that fj ≡ 0 for all j. The base of induction is evident. Suppose a1 = 0. Multiply T by ix (1 − x)t1 ix (1 − x) ¯ s1 and apply to the resulting expression an operator A(1 − x)∂x + B(1 − x)∂ ¯ x¯ , where A, B are arbitrary real numbers. We obtain a sum with L − 1 terms: L

l=2

ix

1 1 ix¯ (A(tl − t1 ) + B(sl − s1 ))al (1 − x)tl (1 − x) ¯ sl

which is equal to 0 whenever T = 0. Since A and B are arbitrary, by the induction hypothesis we get al = 0 for l = 2, . . . , L. Consequently, a1 is equal to 0 as well. This proves the lemma.

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Theorem B.2 (Uniqueness theorem). Let V be a subspace in QF1 (V ) which satisfies the following conditions: 1. any field A(z, z¯ ) ∈ V is mutually local with all fields Y (a), a ∈ V ; 2. all fields are creative, i.e. A(z, z¯ )|0 ∈ V [[z, z¯ ]]. Then the map s : V → V [[z, z¯ ]], A(z, z¯ ) → A(z, z¯ )|0 is injective. Suppose A(z, z¯ )|0 = 0. Take a vector a ∈ V and consider Y (a). From locality we know that Y (a)(z, z¯ )A(w, w) ¯ =

M

iz,w

j =1

1 1 iz¯ ,w¯ Cj (z, z¯ , w, w). ¯ h +N j (z − w) (¯z − w) ¯ hj +N

Hence we have M

j =1

iz,w

1 1 iz¯ ,w¯ Cj (z, z¯ , w, w)|0 ¯ = 0. (z − w)hj +N (¯z − w) ¯ hj +N

Using the arguments of Lemma B.1, we get Cj (z, z¯ , w, w)|0 ¯ = 0 for all j . Now from locality we obtain A(w, w)Y ¯ (a)(z, z¯ )|0 = 0. This implies that A(w, w)a ¯ = 0 for any a ∈ V . Hence A(w, w) ¯ = 0, and the theorem is proved. Corollary B.3. For any a ∈ V the following identities hold: Y (T a) = ∂Y (a),

¯ (a). Y (T¯ a) = ∂Y

Both fields Y (T a) and ∂Y (a) are mutually local with all fields Y (b). Moreover we have ¯ Y (T a)|0 = ∂Y (a)|0 = T eT z+T z¯ a. Hence by the uniqueness theorem Y (T a) = ∂Y (a). The other identity is proved similarly. We call a vector a ∈ V meromorphic (resp. anti-meromorphic) if Y (a) is meromorphic (resp. anti-meromorphic). To show that meromorphic and anti-meromorphic vectors form two supercommuting chiral algebras, it is sufficient to prove the following proposition. Proposition B.4. Let V be a vertex algebra. Then 1. the subspace of meromorphic vectors is closed with respect to Y and T , i.e. T (a) and a(n) b are meromorphic when a ∈ V and b ∈ V are meromorphic,

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2. the OPE of two meromorphic fields a(z) and b(w) can be written in the form 1 C(z, w), (z − w)N 1 C(z, w), (−1)p(a)p(b) b(w)a(z) = iw,z (z − w)N a(z)b(w) = iz,w

C(z, w) ∈ QF2 (V ),

where N is an integer, 3. If a ∈ V is meromorphic and b ∈ V is anti-meromorphic, then their OPE has the form ¯ = C(z, w), ¯ (−1)p(a)p(b) b(w)a(z)

a(z)b(w) ¯ = C(z, w), ¯

C(z, w) ¯ ∈ QF2 (V ).

Let us prove statement (1) of the proposition. From Corollary B.3. we infer that a is meromorphic if and only if T¯ a = 0. Since T and T¯ commute, this immediately implies that T a is meromorphic when a is meromorphic. Further, consider Y (a)b, where both a and b are meromorphic. We have T¯ Y (a)b = Y (a)(T¯ b) = 0. Hence T¯ (a(n) b) = 0, and all a(n) b are meromorphic as well. Statements (2) and (3) of the proposition are special cases of a more general statement which we are going to prove. Proposition B.5. Let a, b ∈ V . If a is meromorphic, then the OPE of a(z) and b(w, w) ¯ can be written in the form 1 D(z, w, w), ¯ (z − w)N 1 ¯ = iw,z D(z, w, w), ¯ (−1)p(a)p(b) b(w, w)a(z) (z − w)N a(z)b(w, w) ¯ = iz,w

(58)

where D(z, w, w) ¯ ∈ QF2 (V ), and N is an integer. This means that if a certain variable does not appear on the left-hand-side of the OPE, it does not appear on the right-hand-side either. The general form of the OPE of a(z) and b(w, w) ¯ is a(z)b(w, w) ¯ =

M

i=1

iz,w

1 1 iz¯ ,w¯ Ci (z, z¯ , w, w), ¯ (z − w)N+hi (¯z − w) ¯ N+hi

where N ∈ Z, hi , i = 1, . . . , M, are distinct real numbers which belong to [0, 1), and Ci ∈ QF2 (V ). Let us act on both sides with an operator (¯z − w) ¯ ∂∂z¯ . We get 0=

M

i=1

iz,w

1 1 iz¯ ,w¯ N+h i (z − w) (¯z − w) ¯ N+hi

 −(N + hi ) + (¯z − w) ¯

By Lemma B.1. we may conclude that for all i we have   ∂ −(N + hi ) + (¯z − w) ¯ Ci = 0. ∂ z¯

 ∂ Ci . ∂ z¯

(59)

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Now let us show that Ci ≡ 0 if hi = 0. Assume the converse. Then there is a vector v ∈ V such that

Ci (z, z¯ , w, w)(v) ¯ = c(αβγ δ) z−α z¯ −β w −γ w¯ −δ = 0. α,β,γ ,δ

Equation (59) implies (N + hi + β)c(αβγ δ) = (β − 1)c(α,β−1,γ ,δ+1) .

(60)

Since Ci ∈ QF2 (V ), we can choose α, β, γ , δ so that c(α,β,γ ,δ) = 0 and c(α,β−1,γ ,δ+1) = 0. From Eq. (60) we find that β = −(N + hi ). Furthermore, (60) implies that c(α,β+k,γ ,δ−k) =

    β +k−1 −(N + hi ) + k − 1 c(α,β,γ ,δ) c(α,β,γ ,δ) = k k

for all k ∈ N. If hi ∈ Z then the vector c(α,β+k,γ ,δ−k) ∈ V is nonzero for all k ∈ N. But this contradicts the condition Ci ∈ QF2 (V ). Since hi ∈ [0, 1) for all i, and hi = hj for i = j, we conclude that Ci = 0 for all i except maybe one, and for the latter value of i we have hi = 0. In addition, for c(α,β+k,γ ,δ−k) to be zero for k > 0, as required by the condition Ci ∈ QF2 (V ), the integer N must be nonnegative. Thus the OPE of a(z) and b(w, w) ¯ has the form a(z)b(w, w) ¯ = iz,w

1 1 iz¯ ,w¯ C(z, z¯ , w, w), ¯ (z − w)N (¯z − w) ¯ N

where C(z, z¯ , w, w) ¯ ∈ QF2 (V ) and N ≥ 0. Applying Eq. (59) to C(z, z¯ , w, w) ¯ and differentiating it with respect to z¯ , we infer that C(z, z¯ , w, w) ¯ =

1 ¯ (¯z − w) ¯ N ∂z¯N C(z, z¯ , w, w) N!

and

∂z¯N+1 C(z, z¯ , w, w) ¯ = 0.

1 N ∂z¯ C(z, z¯ , w, w) ¯ ∈ QF2 (V ) does not depend on z¯ . Let For this reason the element N! us denote it by D(z, w, w). ¯ Then the OPE of a(z) and b(w, w) ¯ takes the form

1 D(z, w, w), ¯ (z − w)N 1 (−1)p(a)p(b) b(w, w)a(z) ¯ = iw,z D(z, w, w). ¯ (z − w)N a(z)b(w, w) ¯ = iz,w

This completes the proof of Proposition B.4.. As a corollary, we have: Corollary B.6. Meromorphic and anti-meromorphic vectors form two supercommuting chiral algebras. In the theory of chiral algebras an important role is played by the so-called Borcherds formula which expresses the OPE of any two fields a(z) and b(z) in the image of Y through their normal ordered product and the Borcherds products a(n) b. We will prove an analogue of the Borcherds formula for vertex algebras.

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Note that any field D(z, w, w) ¯ ∈ QF2 (V ) meromorphic in the first variable can be expanded in a Taylor series in (z − w) to an arbitrarily high order. This means that for any integer K > 0 there exists a field DK (z, w, w) ¯ ∈ QF2 (V ) such that D(z, w, w) ¯ =

K−1

j =0

 (z − w)j ∂ j D(z, w, w) ¯  + (z − w)K DK (z, w, w). ¯  j! ∂zj z=w

To prove this, it is sufficient to show that for any D(z, w, w) ¯ ∈ QF2 (V ) we have ¯ D(z, w, w) ¯ − D(w, w, w) ¯ = (z − w)D1 (z, w, w) ¯ ∈ QF2 (V ). This fact is trivial. Note also that if D ∈ QF2 (V ) for some D1 (z, w, w) contains fractional powers of z (and therefore also depends on z¯ ), the Taylor formula need not hold. Using the Taylor formula, the OPE (58) can be rewritten in the following form: a(z)b(w, w) ¯ =

N

iz,w

j =1

1 Cj (w, w) ¯ + DN (z, w, w), ¯ (z − w)j

¯ ∈ QF1 (V ) for all j, DN (z, w, w) ¯ ∈ QF2 (V ). It is easy to see that Cj where Cj (w, w) and DN are uniquely defined by this formula. Moreover it can be easily checked that Cn (w, w) ¯ coincides with ¯ := Resz ((z − w)n−1 (a(z)b(w, w) ¯ − b(w, w)a(z))). ¯ a(w)(n) b(w, w) The analogue of the Borcherds formula provides explicit expressions for Cj and DN in terms of a and b:  ¯ = Y a(j ) b (w, w), ¯ j = 1, . . . , N, DN (z, w, w) ¯ =: a(z)b(w, w) ¯ :. Cj (w, w) (61) Here the normal ordered product : a(z)b(w, w) ¯ :∈ QF2 (V ) is defined as follows. Let



a(n) z−n , a(z)− = a(n) z−n . a(z)+ = n≤0

n>0

Then the normal ordered product of a(z) and b(w, w) ¯ is defined by : a(z)b(w, w) ¯ := a(z)+ b(w, w) ¯ + (−1)p(a)p(b) b(w, w)a(z) ¯ −. Thus the OPE of a meromorphic field and an arbitrary field takes the form a(z)b(w, w) ¯ =

N

iz,w

j =1

 1 Y a(j ) b (w, w)+ ¯ : a(z)b(w, w) ¯ :. j (z − w)

(62)

Similarly, the OPE of an anti-meromorphic field and an arbitrary field is given by a(¯z)b(w, w) ¯ =

N

j =1

iz¯ ,w¯

 1 Y a(j ) b (w, w)+ ¯ : a(¯z)b(w, w) ¯ :. j (¯z − w) ¯

(63)

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To prove the analogue of the Borcherds formula it is sufficient to show that a(w)(n) b(w, w) ¯ is mutually local with any Y (c). Indeed, it can be easily checked that ¯ = Y (a(n) b)(w, w)|0, ¯ a(w)(n) b(w, w)|0 and hence by the uniqueness theorem we obtain a(w)(n) b(w, w) ¯ = Y (a(n) b)(w, w). ¯ Lemma B.7. If a ∈ V is meromorphic, then a(z)(n) b(z, z¯ ), n ≥ 1 is mutually local with any Y (c). We have to prove that a(w)(n) b(w, w) ¯ = Resz ((z − w)n−1 (a(z)b(w, w) ¯ − b(w, w)a(z))) ¯ is mutually local with any Y (c) = c(z, z¯ ). Let us consider A = (z1 − z2 )n−1 (a(z1 )b(z2 , z¯ 2 )c(z3 , z¯ 3 ) − b(z2 , z¯ 2 )a(z1 )c(z3 , z¯ 3 )) and

B = (z1 − z2 )n−1 (c(z3 , z¯ 3 )a(z1 )b(z2 , z¯ 2 ) − c(z3 , z¯ 3 )b(z2 , z¯ 2 )a(z1 )). We know that for some sufficiently large r ∈ N the following identities hold: (z1 − z2 )r a(z1 )b(z2 , z¯ 2 ) = (z1 − z2 )r b(z2 , z¯ 2 )a(z1 ), (z1 − z3 )r a(z1 )c(z3 , z¯ 3 ) = (z1 − z3 )r c(z3 , z¯ 3 )a(z1 ).

Now let us consider (z2 − z3 )M . We have M  

M M (z2 − z1 )M−r (z1 − z3 )s . (z2 − z3 ) = s s=0

)M ,

where M ≥ 2r. We get Let us multiply A with (z2 − z3   M

M (z2 − z1 )M−r (z1 − z3 )s A. s s=0

summand in this expression is 0, because (z1 − z2 )M−s For 0 ≤ s ≤ r the  n−1 r (z1 − z2 ) = (z1 − z2 ) , where r  ≥ r. Hence the expression is equal to M  

M (z2 − z1 )M−r (z1 − z3 )s A s sth

s=r+1

=

M  

M (z2 − z1 )M−r (z1 − z3 )s (z1 − z2 )n−1 (a(z1 )b(z2 , z¯ 2 )c(z3 , z¯ 3 ) s

s=r+1

− b(z2 , z¯ 2 )a(z1 )c(z3 , z¯ 3 )) M  

M = (z2 − z1 )M−r (z1 − z3 )s (z1 − z2 )n−1 s s=r+1

× (a(z1 )b(z2 , z¯ 2 )c(z3 , z¯ 3 ) − b(z2 , z¯ 2 )c(z3 , z¯ 3 )a(z1 )) M  

M = (z2 − z1 )M−r (z1 − z3 )s (z1 − z2 )n−1 [a(z1 ), b(z2 , z¯ 2 )c(z3 , z¯ 3 )]. s s=r+1

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In the same way we find that M  

M (z2 − z3 )M B = (z2 − z1 )M−r (z1 − z3 )s [a(z1 ), c(z3 , z¯ 3 )b(z2 , z¯ 2 )]. s s=r+1

From our definition of a vertex algebra we know that

1 1 b(z2 , z¯ 2 )c(z3 , z¯ 3 ) = iz2 ,z3 iz¯ 2 ,¯z3 Ej (z2 , z¯ 2 , z3 , z¯3 ), h +N (z2 − z3 ) j (¯z2 − z¯ 3 )hj +N j c(z3 , z¯ 3 )b(z2 , z¯ 2 ) =

j

iz3 ,z2

1 1 iz¯ 3 ,¯z2 Ej (z2 , z¯ 2 , z3 , z¯ 3 ) (z2 − z3 )hj +N (¯z2 − z¯ 3 )hj +N

for some Ej from QF2 (V ). Substituting these expressions into the formulas above we find that (z2 − z3 )M (a(z2 )(n) b(z2 , z¯ 2 ))c(z3 , z¯ 3 ) 

M   M = Resz1 (z2 − z1 )M−r (z1 − z3 )s (z1 − z2 )n−1 s

s=r+1

iz2 ,z3

j

 1 1 i [a(z ), E (z , z ¯ , z , z ¯ )] , z¯ 2 ,¯z3 1 j 2 2 3 3 (z2 − z3 )hj +N (¯z2 − z¯ 3 )hj +N

and (z2 − z3 )M c(z3 , z¯ 3 )a(z2 )(n) b(z2 , z¯ 2 ) 

M   M = Resz1 (z2 − z1 )M−r (z1 − z3 )s (z1 − z2 )n−1 s

j

s=r+1

 1 1 iz3 ,z2 iz¯ 3 ,¯z2 [a(z1 ), Ej (z2 , z¯ 2 , z3 , z¯3 )] . (z2 − z3 )hj +N (¯z2 − z¯ 3 )hj +N

To prove mutual locality of a(z)(n) b(z, z¯ ) with any Y (c) one only needs to show that one can divide both sides of the above equations by (z2 − z3 )M . In fact, it is sufficient to show this for M = 1, and then use induction on M. To show that one can divide both sides by z2 − z3 , we note that the kernel of multiplication by z − w consists of expressions of the form


z n D(z, z¯ , w, w), ¯ w n∈Z

where D(z, z¯ , w, w) ¯ is a formal fractional power series with coefficients in End(V ) (but not necessarily an element of QF2 (V )). If D(z, z¯ , w, w) ¯ is not identically zero, then there exists v ∈ V such that when this expression is applied to v, one gets a fractional power series with coefficients in V containing arbitrarily large negative powers of w and z. On the other hand, applying any element of QF1 (V ) or QF2 (V ) to any v ∈ V one always obtains a fractional power series with powers bounded from below. This implies that one can divide both sides of the above equations by z2 −z3 . The Borcherds formulas are proven.

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Three remarks are in order here. First, it seems that there is no analogous way to rewrite the OPE of two fields when neither of them is meromorphic or anti-meromorphic. Consequently, the normal ordered product of two general fields is not a very useful concept. Second, given two meromorphic fields, one can define two normal ordered products: : a(z)b(w) : = a(z)+ b(w) + (−1)p(a)p(b) b(w)a(z)− , : b(w)a(z) : = b(w)+ a(z) + (−1)p(a)p(b) a(z)b(w)− . Correspondingly, there are two different OPEs that one can write down. The first one is a(z)b(w) =

N

iz,w

 1 Y a(j ) b (w)+ : a(z)b(w) :, j (z − w)

iw,z

 1 Y a(j ) b (w)+ : a(z)b(w) :, j (z − w)

iw,z

 1 Y b(j ) a (z)+ : b(w)a(z) :, j (w − z)

iz,w

 1 Y b(j ) a (z)+ : b(w)a(z) : . j (w − z)

j =1

(−1)p(a)p(b) b(w)a(z) =

N

j =1

and the second one is b(w)a(z) =

N

j =1

(−1)p(a)p(b) a(z)b(w) =

N

j =1

In general, the two normal ordered products are not related in any simple way. Third, given a meromorphic and an anti-meromorphic field, one can also define two normal ordered products. However, in this case they always coincide up to a sign: : a(z)b(w) ¯ := (−1)p(a)p(b) : b(w)a(z) ¯ :. Indeed, the OPE formulas (62,63) read in this case ¯ =: a(z)b(w) ¯ :, a(z)b(w) ¯ = (−1)p(a)p(b) b(w)a(z) b(w)a(z) ¯ = (−1)p(a)p(b) a(z)b(w) ¯ =: b(w)a(z) ¯ :. This fact also follows directly from the definition of the normal ordered product and the fact that meromorphic and anti-meromorphic fields in the image of Y supercommute. Finally, let us show that any chiral algebra is a special case of a vertex algebra with T¯ = 0 and the image of Y consisting of meromorphic fields only. The only thing which needs to be checked is the OPE axiom. For a chiral algebra, the OPE of any two fields in the image of Y has the form a(z)b(w) =

N

iz,w

 1 Y a(n) b (w)+ : a(z)b(w) :, (z − w)n

iw,z

 1 Y a(n) b (w)+ : a(z)b(w) : . (z − w)n

n=1

(−1)p(a)p(b) b(w)a(z) =

N

n=1

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Obviously, a(n) b(w) belongs to QF2 (V ). It is also easy to check that : a(z)b(w) : also belongs to QF2 (V ). Hence, the above OPE can be rewritten as a(z)b(w) = iz,w

1 C(z, w), (z − w)N

where C(z, w) ∈ QF2 (V ). Therefore the OPE axiom is satisfied. C. Projectively Flat Connections and the Fundamental Group In this appendix we establish a relation between projectively flat connections on complex vector bundles on a connected manifold and finite representations of a twisted group algebra of the fundamental group. This relation is a generalization of the well-known statement that flat connections on complex vector bundles are in one-to-one correspondence with representations of the fundamental group. Let M be a paracompact connected C ∞-manifold. Let us fix a closed real 2-form B on M. Consider a complex vector bundle E on M with a connection ∇ such that its curvature F∇ ∈ G2 ⊗ End(E) is equal to F∇ = 2πiB ⊗ idE .

(64)

Such a connection is called projectively flat, and it is flat if and only if B = 0. When B is non-zero, we can consider the condition (64) as a “twisted” variant of the flatness condition. We will prove that the set of such connections is in one-to-one correspondence with finite representations of a twisted group algebra of π1 (M) defined below. Let us fix a point x ∈ M. Since (E, ∇) is projectively flat, for any contractible closed path c starting at x the holonomy operator Hc : Ex −→ Ex is equal to tc · id, where tc is a nonzero complex number. By the Reduction Theorem (see [22]) (E, ∇) can be reduced locally to a C∗–bundle, and therefore by Stockes’ theorem    ∗ tc = exp 2πi φ B , D

where φ is a map from the two dimensional disk D to M satisfying φ(∂D) = c. Since B is a real 2-form, (E, ∇) in fact locally reduces to a U (1)-bundle. The above formula for tc is independent of the choice of φ only if    exp 2πi φ∗B = 1 (65) S2

for any map φ from the 2-dimensional sphere S 2 to M. Thus a vector bundle (E, ∇) with curvature F∇ = 2πiB ⊗ idE can exist only if the de Rham cohomology class of B belongs to the kernel of the composition homomorphism H 2 (M, R) → H 2 (M, U (1)) → Hom(π2 (M), U (1)). Let us consider the Hopf sequence π2 (M) −→ H2 (M, Z) −→ H2 (K(G, 1), Z) −→ 0,

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where G := π1 (M). This sequence induces an injective map 0 −→ H 2 (K(G, 1), U (1)) −→ H 2 (M, U (1)).

(66)

Denote by B the image of B in H 2 (M, U (1)). We showed that if B does not belong to the image of the map (66) then the set of vector bundles (E, ∇) with curvature F∇ = 2π iB ⊗ idE is empty. Assume now that B is in the image of the map (66). Let us fix a point x ∈ M and for each element g ∈ G choose a closed path cg beginning at x and representing g such that the closed path cg −1 coincides with the inverse of cg for any g. Let c(g,h) be a loop which is the union of the loops ch , cg , and c(gh)−1 This loop is contractible. Define a function ψ : G × G → U (1) by the rule    ∗ φ B , (67) ψ(g, h) = exp 2π i D

where φ is a map from the two dimensional disc D to M satisfying φ(∂D) = c(g,h) . It is easy to see that this function is a 2-cocycle on the group G. Moreover, if we choose the representatives cg differently, we obtain a cocycle which is cohomologous to ψ. The holonomy operators along the loops cg , ch , and cgh satisfy the following relation: Hcg · Hch = ψ(g, h)Hcgh . This identity has the following representation-theoretic meaning. With any 2-cocycle ψ one can associate a twisted group algebra Cψ [G], which is a vector space generated by the elements g ∈ G with the following multiplication law: g · h = ψ(g, h)gh. (Note that if two 2-cocycles are cohomological to each other, then the corresponding twisted group algebras are isomorphic.) The holonomy operators Hcg define a representation of the twisted group algebra Cψ [G] on the vector space Ex . An equivalent definition of the algebra Cψ [G] goes as follows. Let Lpx be the loop space of M with the well-known composition of loops (which is associative only up to a homotopy). Let us consider the corresponding non-associative “group” algebra C[Lpx ]. Then the algebra Cψ [G] is a factor-algebra of C[Lpx ] modulo all relations of the form    ∗ c − exp 2πi φ B · 1 = 0, D

where c is a contractible loop, and φ is a map from the disc D to M such that φ(∂D) = c. By (65) this definition does not depend on the choice of φ. For any loop c ∈ Lpx we denote by r(c) the element of the twisted group algebra which is the image of c with respect to this factorization. In this way to any vector bundle (E, ∇) satisfying the condition (64) we can associate a finite-dimensional representation of the twisted group algebra. We assert that this is a one-to-one correspondence. To show this, we describe how to construct (E, ∇) starting from a representation R of the twisted group algebra. Let CM be the sheaf of algebras of complex-valued C ∞–functions on M. Let A be a sheaf of algebras on M defined as Cψ [G] ⊗C CM . If R is a representation of the twisted group algebra, then the sheaf R = R ⊗C CM has a natural left module structure over the sheaf of algebras A. Below we construct a sheaf P of right A–modules with a connection

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∇P and set E = P ⊗A R. This sheaf is the sheaf of sections of a complex vector bundle on M, and ∇P induces a natural connection ∇ on it. τ  −→  the pull-back of the form B to Let M M be a universal covering. Denote by B   is M. It is easy to check that B belongs to the image of the map (66) if and only if B   an exact form. Let us choose a 1-form η on M such that dη = B.  = Cψ [G] ⊗C C  on M.  The tautological action of G Consider a sheaf of algebras A M   on M can be lifted to a left action on A as follows. Let cg be a loop in M based at a fixed point x ∈ M and representing the element g ∈ G, and let r(cg ) be the corresponding  Let  element of the twisted group algebra of G (see above). Let x0 be a lift of x to M. cg −1   be a path on M which covers cg , begins at g (x0 ) and ends at x0 . For any point y ∈ M −1 let us choose some path dy from y to x0 . Let  cg,y be a path from g (y) to y which is a  is cg , and dy−1 . The left action of the group G on the sheaf A composition of g −1 (dy ),  defined by the rule:    g(a ⊗ f )(y) = exp −2πi

 cg,y

η (r(cg )a ⊗ f (g −1 y)),

 where a ∈ Cψ [G] and f is a C ∞–function on M.  is G-invariThis definition does not depend on the choice of dy , because the form B ant. Nor does it depend on the choice of cg , because for any other loop cg representing g we have         η r(cg ) = exp −2π i η + 2π i φ ∗ B)r(c exp −2πi g   cg,y

 = exp −2π i

  cg,y





 cg,y

D

η)r(cg ,

 such that φ(∂D) is the composition of  where φ is a map from D to M cg and the inverse of  cg .  by the formula Furthermore, we can define a connection on A  ⊗ f ) = a ⊗ (df + 2π if η). ∇(a   and This connection is G-invariant. Indeed, let us regard cg,y η as a function on M denote it by h(y). Then we have  ⊗ f )(y) = g(a ⊗ (df + 2π if η))(y) g ∇(a = exp(−2π ih(y))r(cg )a ⊗ (df (g −1 y) + 2π if (g −1 y)η(g −1 y)). On the other hand, since dh(y) = η(y) − η(g −1 y) we obtain −1   ∇g(a ⊗ f )(y) = ∇(r(c g )a ⊗ exp(−2π ih(y))f (g y))

= exp(−2π ih(y))r(cg )a ⊗ (df (g −1 y) − 2πif (g −1 y)dh(y) + 2π if (g −1 y)η(y)) = exp(−2π ih(y))r(cg )a ⊗ (df (g −1 y) + 2π if (g −1 y)η(g −1 y)).

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 depend  and the action of the group G on A The definitions of the connection ∇   ∇)  on the choice of η. However, if we take another form η = η + df then the data (A,    and (A, ∇ ) are isomorphic under the multiplication by the function exp(−2π if ). Moreover, this isomorphism is compatible with the action of the group G.  G with a connection ∇P We define a sheaf P on M as the sheaf of invariants τ∗ (A)  induced by ∇. The sheaf P has a right module structure over A. It is locally free of rank 1 as an A-module. It follows from the preceding discussion that the datum (P, ∇P ) is unique and depends only on the form B. To any representation R of the twisted group algebra of G we attach a complex vector bundle E = P ⊗A R with the connection ∇ induced by ∇P . It is easy to see that the representation of the twisted group algebra on the space Ex corresponding to ∇ is isomorphic to R. Thus pairs (E, ∇) satisfying (64) are in one-to-one correspondence with finite-dimensional representations of Cψ [G], where the cocycle ψ is defined by (67). Acknowledgements. We are grateful to Maxim Kontsevich for valuable comments and to Markus Rosellen for pointing out a gap in the reasoning of Appendix B in the first version of the paper. We also wish to thank the Institute for Advanced Study, Princeton, NJ, for a very stimulating atmosphere. The first author was supported by DOE grant DE-FG02-90ER40542. The second author was supported in part by RFFI grant 99-01-01144 and a grant for support of leading scientific groups N 00-15-96085. The research described in this publication was made possible in part by Award No RM1-2089 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF).

References 1. Alvarez-Gaume, L., Freedman, D.: Geometrical structure and ultraviolet finiteness in the supersymmetric sigma-model. Commun. Math. Phys. 80, 443 (1981) 2. Arinkin, D., Polishchuk, A.: Fukaya category and Fourier transform. Preprint math//9811023 3. Arnold, V.I.: Sturm theorem and symplectic geometry. Funkts. Anal. i Prilozh. 19(4), 1 (1985) 4. Bouwknegt, P. and Mathai, V.: D-branes, B-fields and twisted K-theory. JHEP 0003, 007 (2000); preprint hep-th/0002023 5. Belavin, A., Polyakov, A., Zamolodchikov, A.: Infinite conformal symmetries in two-dimensional quantum field theory. Nucl. Phys. B241, 333 (1984) 6. Borcherds, R.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068 (1986) 7. Connes, A., Douglas, M., Schwarz, A.: Noncommutative geometry and Matrix theory: Compactification on tori. JHEP 9802, 003 (1998); preprint hep-th/9711162 8. Deligne, P. et al: Quantum Fields and Strings: A course for Mathematicians. Vol. 1, 2, Providence, RI: AMS, 1999 9. Di Francesco, P., Mathieu, P., S´en´echal, D.: Conformal Field Theory. New York: Springer, 1997 10. Douglas, M.: D-branes on Calabi-Yau manifolds. Preprint math//0009209 11. Douglas, M., Hull, C.: D-branes and the noncommutative torus, JHEP 9802, 008 (1998); preprint hep-th/9711165 12. Fukaya, K.: Morse Homotopy, A∞ -category and Floer Homologies. MSRI preprint No. 020-94 (1994) 13. Gates, S. J., Hull, C., Roˇcek, M.: Twisted multiplets and new supersymmetric nonlinear sigma-models. Nucl. Phys. B248, 157 (1984) 14. Golyshev, V., Lunts, V., Orlov, D.: Mirror symmetry for abelian varieties. To appear in J. Algebraic Geometry, Preprint math//9912003 15. Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton, NJ: Princeton University Press, 1992 16. Hofman, C., Ma, W.K.: Deformations of topological open strings. Preprint hep-th/0006120 17. Hoobler, R.: Brauer groups of abelian schemes. Ann. Sci. Ec. Norm. Sup. 5, 45 (1972) 18. Hori, K., Iqbal, A., Vafa, C.: D-branes and mirror symmetry. Preprint hep-th/0005247 19. Kac, V.: Vertex Algebras for Beginners. University Lecture Series, vol. 10, Providence, RI: AMS, 1997

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20. Kapustin, A.: D-branes in a topologically nontrivial B-field, Adv. Theor. Math. Phys. 4, 127 (2000); preprint hep-th/9909089 21. Keller, B.: Introduction to A∞ -algebras and modules. Preprint math//9910179 22. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vol. 1, 2, New York: John Wiley & Sons, 1996 23. Kontsevich, M.: Homological algebra of mirror symmetry, Proceedings of ICM (Zurich, 1994), Basel: Birkh¨auser, 1995, pp. 120–139 24. L¨ust, D., Theisen, S.: Lectures on String Theory. Berlin-Heidelberg: Springer-Verlag, 1989 25. Narain, K., Sarmadi, M., Witten, E.: A note on toroidal compactification of heterotic string theory. Nucl. Phys. B279, 369 (1987) 26. Ooguri, H., Oz, Y., Yin, Z.: D-branes on Calabi-Yau spaces and their mirrors. Nucl. Phys. B477, 407 (1996); preprint hep-th/9606112 27. Orlov, D.: Equivalences of derived categories and K3 surfaces. J. Math. Sci. 84, 1361 (1997); preprint math//9606006 28. Orlov, D.: On equivalences of derived categories of coherent sheaves on abelian varieties, Preprint MPI/97-49, math//9712017 29. Polchinski, J.: String Theory. Vol. 1, 2, Cambridge, UK: Cambridge University Press, 1998 30. Polishchuk, A.: Symplectic biextensions and generalization of the Fourier-Mukai transforms. Math. Res. Let. 3, 813 (1996); preprint math//9511018 31. Polishchuk, A., Zaslow, E.: Categorical mirror symmetry: The elliptic curve. Adv. Theor. Math. Phys. 2, 443 (1998); preprint math/9801119 32. Vafa, C.: Extending mirror conjecture to Calabi-Yau with bundles. Preprint hep-th/9804131 33. Witten, E.: Chern-Simons gauge theory as a string theory. Preprint hep-th/9207094 34. Kugo, T., Zwiebach, B.: Target space duality as a symmetry of string field theory. Progr. Theor. Phys. 87, 801 (1992); preprint hep-th/9201040 Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 233, 137–151 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0764-6

Communications in

Mathematical Physics

Large Deviations and a Fluctuation Symmetry for Chaotic Homeomorphisms Christian Maes1 , Evgeny Verbitskiy2 1

Instituut voor Theoretische Fysica, K.U.Leuven, Celestijnenlaan 200 D, 3001 Leuven, Belgium. E-mail: [email protected] 2 Eurandom, Technical University of Eindhoven, P.O.Box 513, 5600 MB, Eindhoven, The Netherlands. E-mail: [email protected] Received: 6 March 2002 / Accepted: 16 September 2002 Published online: 8 January 2003 – © Springer-Verlag 2003

Abstract: We consider expansive homeomorphisms with the specification property. We give a new simple proof of a large deviation principle for Gibbs measures corresponding to a regular potential and we establish a general symmetry of the rate function for the large deviations of the antisymmetric part, under time-reversal, of the potential. This generalizes the Gallavotti-Cohen fluctuation theorem to a larger class of chaotic systems. 1. Introduction Since the beginning of statistical mechanics, there has been an ongoing exchange of ideas between the theory of heat and the theory of dynamical systems. The thermodynamic formalism has become a standard chapter for studies in dynamical systems and, ever since Clausius, heat is understood as motion. More recently, there has been a fruitful revival of connecting the two theories. In particular, programs are running for understanding the effect of nonlinearities on transport coefficients and for defining nonequilibrium ensembles in terms of Sinai-Ruelle-Bowen (SRB) measures, [3, 15, 6]. It was also argued that the role of entropy production is, in strongly chaotic dynamical systems, played by the phase space contraction. Gallavotti and Cohen went on to prove a fluctuation symmetry for the distribution of the timeaverages of the phase space contraction rate and they hypothesized that this symmetry is much more general and relevant also for the construction of nonequilibrium statistical mechanics. In [10] it was emphasized that this symmetry results from the Gibbs formalism and in the more recent [11] the general relation between entropy production and phase space contraction was investigated. It was found that phase space contraction obtains its formal analogy with the physical entropy production as a source term in the potential for time-reversal breaking. For Anosov diffeomorphisms f , as were considered by Gallavotti-Cohen, this can be seen as follows: the SRB distribution is a Gibbs measure for the potential ϕ(x) = − log ||Df |E u (x) ||

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and ϕ˜1 (x) = − log ||Df |E s (x) || = −ϕ(i ◦ f (x)) is minus its time-reversal. Here E u (x) and E s (x) are the unstable and stable subspaces of the tangent space at the point x and i is the time-reversal for which f is dynamically reversible: i ◦ f = f −1 ◦ i. The unstable and stable subspaces are not orthogonal but for Anosov systems, the angle between these spaces is uniformly bounded away from 0. Therefore there exists a constant C > 0 such that n−1 n−1



σ (f k (x)) − (ϕ + ϕ˜1 )(f k (x))
≤ C
k=0

k=0

for all x and n ∈ N with σ (x) = − log ||Df (x)||, the phase space contraction rate. While, from [10, 11], the most natural analogue of entropy production rate is given by the antisymmetric part of the potential under timereversal, that is ψ0 (x) = ϕ(x) − ϕ(i(x)) for the purpose of computing the time-average and its large deviations, for Anosov systems, no distinction can be made between ψ0 , ψ1 = ϕ + ϕ˜1 and σ . Within the set-up of Gallavotti and Cohen, the ergodic averages of ψ0 and of ψ1 and of σ have exactly the same large deviation rate function. Once this is recognized, it is natural to generalize the fluctuation theorem of [6] to other dynamical systems. We believe this is interesting because it takes us away from uniform hyperbolic behavior, which is not typical for real physical systems. Moreover, physically relevant dynamical systems such as billiards or systems of hard balls have singularities, i.e., they are not everywhere differentiable. Our proof however of the fluctuation symmetry for expansive homeomorphisms with the specification property relies on the corresponding thermodynamic formalism established by Ruelle and Haydn [14, 7]. These results are valid for homeomorphisms and hence do not require differentiable systems. A natural question is how big the class is of expansive homeomorphisms with the specification property, and what are examples of dynamical systems from this class that are not covered by previous results. We refer to the last section of this paper for a detailed discussion and here we mention two expansive interval maps with specification, which can serve as a basis for a construction of multidimensional examples1 : • a) fβ (x) = βx mod 1, where β > 1; • b)fs (x) = x + x 1+s mod 1, where s > 0. There exist values of β > 1 such that fβ has a specification property, but does not allow a Markov partition, see [16]. In fact, there are uncountably many such β’s, while in order to admit a Markov partition, β must be a special algebraic number. The second example — the so-called Manneville-Pomeau map, is probably the simplest example of a non-uniformly hyperbolic dynamical system. Note that 0 is an indifferent fixed point. 1 It is necessary to go to higher dimensions, since neither a unit interval, nor a unit circle, can support an expansive homeomorphism.

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In Sect. 2 and 3 we start with some basic definitions and results. We do this in order to make the text, as much as possible, self-contained. Section 4 provides a new proof of the large deviation principle for expansive homeomorphisms with the specification property. It is based on directly checking the conditions of the G¨artner-Ellis large deviation theorem. Section 5 has the proof of the fluctuation symmetry of the time-averages of what is denoted above by ψ1 (or, equivalently, ψ0 ). We end with some further remarks. 2. Expansive Homeomorphisms with Specification The following definitions and basic results can be found in [2, 7, 14, 8]. We will always assume (X, d) to be a compact metric space. Definition 2.1. A homeomorphism f : X → X is called expansive if there exists a constant γ > 0 such that if d(f n (x), f n (y)) < γ for all n ∈ Z

then

x = y.

(2.1)

The largest γ > 0 is called the expansivity constant of f . Another important property is the following: Definition 2.2. We say that f : X → X is a homeomorphism with the specification property (abbreviated to “a homeomorphism with specification”) if for each δ > 0 there exists an integer p = p(δ) such that the following holds: if a) I1 , . . . , In are intervals of integers, Ij ⊆ [a, b] for some a, b ∈ Z and all j, b) dist(Ij , Ij ) ≥ p(δ) for j = j , then for arbitrary x1 , . . . , xn ∈ X there exists a point x ∈ X such that for every j d(f k (x), f k (xj )) < δ for all

k ∈ Ij .

Homeomorphisms that are expansive and satisfy the specification property, form a general class of “chaotic” dynamical systems. For example, the following is an immediate corollary of Theorem 18.3.9 in [8]. Theorem 2.3 ([8, Theorem 18.3.9]). If f : X → X is a transitive Anosov diffeomorphism, then f is expansive and satisfies the specification property. 3. Topological Pressure, Regular Potentials and Gibbs Distributions Definition 3.1. For every n ∈ N and x, y ∈ X define a new metric dn (x, y) =

max

j =0,... ,n−1

d(f j (x), f j (y)),

and let Bn (x, ε) = {y ∈ X : dn (x, y) < ε} for ε > 0. The set E ⊂ X is said to be (n, ε)-separated if for every x, y ∈ E such that x = y we have dn (x, y) > ε.

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For a function ϕ : X → R (to be called potential) and x ∈ X put (Sn ϕ)(x) =

n−1 

ϕ(f k (x)).

k=0

The topological pressure is defined on the space C(X) of all continuous functions on (X, d). Definition 3.2. For n ∈ N and ε > 0 define      Zn (ϕ, ε) = sup exp (Sn ϕ)(x) , E

(3.1)

x∈E

where the supremum is taken over all (n, ε)-separated sets E. The pressure is then defined as P (ϕ) = lim lim sup ε→0 n→∞

1 log Zn (ϕ, ε). n

(3.2)

The topological entropy of f , denoted by htop (f ), is by definition the topological pressure of ϕ ≡ 0. The topological entropy of an expansive homeomorphism on a compact metric space is always finite and so is the topological pressure of any continuous function. The topological pressure P : C(X) → R is a continuous and convex function. The following statement is known as the Variational Principle [18]. Theorem 3.3. Denote by M(X, f ) the set of all f -invariant Borel probability measures on X. Let ϕ ∈ C(X). Then   P (ϕ) = sup hµ (f ) + ϕdµ . µ∈M(X,f )

This result inspires the following definition. Definition 3.4. An element µ of M(X, f ) is called an equilibrium measure for the potential ϕ if  P (ϕ) = hµ (f ) + ϕdµ. The equilibrium measure for ϕ ≡ 0 (if it exists) is called a measure of maximal entropy. We impose additional conditions on the class of potentials under consideration. As we shall see later (Theorem 3.8), the corresponding equilibrium measures will then be Gibbs measures. Definition 3.5. A continuous function ϕ is called regular if for every sufficiently small ε > 0 there exists K = K(ε) > 0 such that for all n ∈ N,



d(f k (x), f k (y)) < ε for k = 0, . . . , n − 1 ⇒
(Sn ϕ)(x) − (Sn ϕ)(y)
< K. The set of all regular functions is denoted by Vf (X).

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Example. For a hyperbolic diffeomorphism f , any H¨older continuous function ϕ is in Vf (X) [8, Prop.20.2.6]. Theorem 3.6 ([2, 13, 8]). If f is an expansive homeomorphism with specification and ϕ ∈ Vf (X) then there exists a unique equilibrium measure µϕ , i.e., µϕ is the unique element of M(X, f ) such that  P (ϕ) = hµϕ (f ) + ϕdµϕ . Moreover, µϕ is ergodic, positive on open sets and mixing. This equilibrium measure µϕ can be constructed from the measures concentrated on periodic points in the following way. For every n ≥ 1 define a probability measure µϕ,n supported on the set of periodic points Fix(f n ) = {x ∈ X : f n (x) = x} as follows: µϕ,n =

1 Z(f, ϕ, n)



e(Sn ϕ)(x) δx ,

(3.3)

x∈Fix(f n )

where δx is a Dirac measure at x and Z(f, ϕ, n) =



e(Sn ϕ)(x) is a normalizing

x∈Fix(f n )

constant. Theorem 3.7 ([2, 8]). The equilibrium measure µϕ is a weak-∗ limit of the sequence {µϕ,n }, i.e., for every h ∈ C(X)   h(x)dµϕ,n → h(x)dµϕ as n → ∞. The next result gives a “local” (i.e., Gibbs) description of equilibrium measures for regular potentials, see [7] for a detailed discussion. Theorem 3.8 ([7, Prop. 2.1],[8, Th. 20.3.4]). Let f be an expansive homeomorphism with the specification property. Let ϕ ∈ Vf (X) and denote its unique equilibrium measure by µϕ . Then, for sufficiently small ε > 0, there exist constants Aε , Bε > 0 such that for all x ∈ X and n ≥ 1, 

µϕ {y ∈ X : d(f k (x), f k (y)) < ε for k = 0, . . . , n − 1} (3.4) ≤ Bε . Aε ≤ exp (−nP (ϕ) + (Sn ϕ)(x)) We have seen that for every ϕ ∈ Vf (X) there exists a unique equilibrium measure. The formula (3.3) and (3.4) make it possible to give necessary and sufficient conditions on potentials ϕ1 , ϕ2 ∈ Vf (X) to have the same equilibrium measures µ1 = µϕ1 = µϕ2 = µ2 . We add the proof for completeness, see [8, 17]. Theorem 3.9. Let f be an expansive homeomorphism with specification. The equilibrium measures µ1 and µ2 corresponding to the potentials ϕ1 , ϕ2 ∈ Vf (X) coincide if and only if there exists a constant c ∈ R such that (Sn ϕ1 )(x) = (Sn ϕ2 )(x) + nc for all x ∈ Fix(f n ) and all n.

(3.5)

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Proof. If (3.5) holds for all x ∈ Fix(f n ) and n, then, by (3.3), one has µ1,n = µ2,n for all n. Thus µ1 = µ2 . Suppose that µ1 and µ2 coincide, and let µ = µ1 = µ2 . Consider “adjusted” potentials ϕ1 = ϕ1 − P (ϕ1 ) and ϕ2 = ϕ2 − P (ϕ2 ). Let x ∈ Fix(f n ) for some n ∈ N. Applying (3.4) for sufficiently small ε > 0 we conclude that



 ϕ1 )(x) ≤ µ(Bn (x, ε)) ≤ Bε2 exp (Sn ϕ2 )(x) . A1ε exp (Sn ϕ1 )(x) ≤ (Sn ϕ2 )(x) + C for some constant C independent of x This implies that (Sn and n. Since x ∈ Fix(f kn ) for all k ∈ N we have that (Skn ϕ1 )(x) (Skn ϕ2 )(x) ϕ2 )(x). ≤ lim = (Sn k→∞ k→∞ k k

ϕ1 )(x) = lim (Sn

By symmetry we obtain the opposite inequality. Hence ϕ1 )(x) = (Sn ϕ2 )(x) (Sn for all x ∈ Fix(f n ) and n ∈ N. This implies (3.5) with c = P (ϕ1 ) − P (ϕ2 ).

 

We now recall some properties of the pressure for expansive homeomorphisms with specification. These facts will be used later in the proof of the main results. Lemma 3.10. Suppose f : X → X is an expansive homeomorphism with specification. Let ϕ, ψ ∈ Vf (X). Then the function P (ϕ + qψ), q ∈ R, is continuously differentiable with respect to q and its derivative is given by  dP (ϕ + qψ) = ψdµq , dq where µq is the equilibrium measure corresponding to the potential ϕ + qψ. Moreover, P (ϕ + qψ) is a strictly convex function of q provided the equilibrium measure for the potential ψ is not the measure of maximal entropy. When the equilibrium measure for the potential ψ is the measure of maximal entropy one has  P (ϕ + qψ) = P (ϕ) + q ψdµϕ for all q ∈ R, where µϕ is the equilibrium measure for ϕ. The proof of this lemma is almost identical to the proof of Lemma 4.1 in [17], and relies on the results of Walters [19], who showed that for expansive dynamical systems differentiability of the pressure function P (·) at ϕ is equivalent to the uniqueness of equilibrium measures for ϕ. Since the specification condition together with the regularity condition on ϕ imply uniqueness of equilibrium measures, we obtain the desired differentiability of the pressure function. In order to prove the Large Deviations result for expansive homeomorphism with specification, we will have to use some results on the convergence 1 log Zn (ϕ, ε) → P (ϕ) n in (3.2).

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Definition 3.11. We say that E is a maximal (n, ε)-separated set if it cannot be enlarged by adding new points and preserving the separation condition. The following estimates from [7] will be used later. Lemma 3.12. Let f be an expansive homeomorphisms and γ > 0 be its expansivity constant. Let ϕ ∈ Vf (X). For every finite set E put    Zn (ϕ, E) = exp (Sn ϕ)(x) . x∈E

ε, ε

E, E

< γ /2 and (1) If tively, then one has

are the maximal (n, ε)- and (n, ε )-separated sets respec-

Zn (ϕ, E) ≤ CZn (ϕ, E ), where the constant C = C(ε, ε ) is independent of n. In particular,

1 log Zn (ϕ, En ), (3.6) n where En are the arbitrary maximal (n, ε)-separated sets. (2) If f satisfies the specification property and ε < γ /2 then there exists a constant D = D(ϕ, ε) > 0 such that P (ϕ) = lim

n→∞

| log Zn (ϕ, En ) − nP (ϕ)| < D

(3.7)

for all n and all maximal (n, ε)-separated sets. (3) Suppose f is expansive and satisfies the specification property, then P (ϕ) = lim

n→∞

1 log Zn (ϕ, Fix(f n )). n

(3.8)

4. Large Deviations In this section we establish the Large Deviation Principle for expansive homeomorphisms with specification and Gibbs measures corresponding to regular potentials. In fact, one can deduce this from more general results of Young [20] or Kifer [9]. However, for our class of dynamical systems one can easily check the conditions of the G¨artner– Ellis theorem. For the sake of completeness and to stand on it in the next section, we provide the details here. Suppose, f : X → X is an expansive homeomorphism with specification, and ϕ, ψ are regular functions, i.e., ϕ, ψ ∈ Vf (X). Let µ = µϕ be an equilibrium measure for the potential ϕ. We will study the distribution of ergodic averages of ψ with respect to µϕ . Namely, we will establish that the limit n−1   1 1 lim log µϕ x ∈ X : ψ(f k (x)) ∈ A n→∞ n n k=0

exists for the appropriate intervals A ⊂ R. Our first step will be the study of the so-called free energy function. For every q ∈ R and n ∈ N define    1 cn (q) = log exp q(Sn ψ)(x) dµϕ . n We have to prove that cn (q) converges for every q, and that the limiting function c(q) is finite and sufficiently smooth in q. This is done in the next lemma.

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Lemma 4.1. For every q ∈ R the following limit exists: c(q) = lim cn (q). n→∞

The limit c(q) is also given by c(q) = P (ϕ + qψ) − P (ϕ),

(4.1)

where P (·) is the topological pressure. The free energy c(q) is finite, differentiable and convex for every q. It is strictly convex provided the equilibrium measure for ϕ is not the measure of maximal entropy. Proof. We start by giving estimates for cn (q), which will lead to (4.1). All other properties of the free energy c(q) will follow from the corresponding properties of the topological pressure. Let ε > 0 be sufficiently small. Let En = {xi } be any maximal (n, ε)-separated set. Since En is a maximal separated set, for every x ∈ X there exists xi ∈ En such that x ∈ Bn (xi , ε). Since ψ is a regular function, for x ∈ Bn (xi , ε) (i.e., dn (x, xi ) < ε) one has




(Sn ψ)(x) − (Sn ψ)(xi )
≤ K(ψ, ε) for some constant K(ψ, ε). Therefore, 

 exp(ncn (q)) = exp q(Sn ψ)(x) dµϕ  

 ≤ exp q(Sn ψ)(x) dµϕ xi ∈En Bn (xi ,ε)

≤



 exp |q|K(ψ, ε) + q(Sn ψ)(xi ) µϕ (Bn (xi , ε))

xi ∈En

≤ C exp(−nP (ϕ))



 exp q(Sn ψ)(xi ) + (Sn ϕ)(xi )

xi ∈En

= C exp(−nP (ϕ))Zn (ϕ + qψ, En ), where C = Bε exp(|q|K(ψ, ε)), and where we have used an upper estimate from (3.4) on the measures of balls Bn (xi , ε). To prove a similar lower estimate we use that for two different points xi , xj from an (n, ε)-separated set En the intersection Bn (xi , ε/2) ∩ Bn (xj , ε/2) is empty. Hence, 

 exp(ncn (q)) = exp q(Sn ψ)(x) dµϕ  

 ≥ exp q(Sn ψ)(x) dµϕ xi ∈En Bn (xi ,ε/2)

≥



 exp −|q|K(ψ, ε) + q(Sn ψ)(xi ) µϕ (Bn (xi , ε/2))

xi ∈En

≥ C exp(−nP (ϕ))

 xi ∈En

 exp q(Sn ψ)(xi ) + (Sn ϕ)(xi )

= C exp(−nP (ϕ))Zn (ϕ + qψ, En ),

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145

where C = Aε/2 exp(−|q|K(ψ, ε)), and we have used the lower estimate from (3.4). Combining together the upper and the lower estimates on cn (q) we obtain 1 C∗ 1 C∗ log Zn (ϕ + qψ, En ) − P (ϕ) + ≤ cn (q) ≤ log Zn (ϕ + qψ, En ) − P (ϕ) + n n n n for some constants C∗ , C ∗ . Since for a sufficiently small ε > 0 the limit lim

n→∞

1 log Zn (En , ϕ + qψ) n

exists (Lemma 3.12) and is equal to P (ϕ + qψ) we obtain the first part of the statement. The properties of the pressure function P (ψ + qϕ) are given by Lemma 3.10. This finishes the proof.   The rate function I (p) is obtained from the free energy c(q) by a Legendre transform: for p ∈ R put I (p) = sup(qp − c(q)). q

Since c(q) is differentiable, we can introduce the following quantities: p = sup c (q) = lim c (q), q

q→+∞

p = inf c (q) = lim c (q). q

q→−∞

(4.2)

Existence of the limits follows from the convexity of the free energy c(q). Standard arguments of convex analysis show that I (p) is finite for p ∈ (p, p) and I (p) = +∞ for p ∈ [p, p]. Moreover, since c(q) is smooth and convex, I (p) is also a smooth and convex function of p on (p, p). Now all the conditions of the G¨artner-Ellis theorem [4] are satisfied and we obtain a Large Deviations result for expansive homeomorphisms with specification. Theorem 4.2 (Large deviations). Let f : X → X be an expansive homeomorphism with specification. Let ϕ, ψ ∈ Vf (X), and let µϕ be the Gibbs measure for ϕ. Assume that µψ is not the measure of maximal entropy. Then there exists a smooth real convex function I on the open interval (p, p) such that, for every interval J with J ∩(p, p) = ∅,   1 1 log µϕ x : (Sn ψ) ∈ J = − inf I (p). n→∞ n p∈J ∩(p,p) n lim

5. Fluctuation Symmetry Choose some regular potential ϕ ∈ Vf (X) and let µϕ be the corresponding Gibbs measure. As always, f : X → X is an expansive homeomorphism with specification on a compact metric space (X, d). We make two assumptions: A) Reversibility. There exists a homeomorphism i : X → X, preserving the metric d such that i ◦ f ◦ i = f −1 and i 2 = identity.

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Fix an integer k and define ϕ˜k and ψk as follows: ϕ˜k (x) = −ϕ(i ◦ f k (x)),

ψk (x) = ϕ(x) + ϕ˜k (x).

From the point of statistical mechanics, the most natural choice is for k = 0. Yet, to connect with the phase space contraction for Anosov diffeomorphisms it is natural to take k = 1. While the proofs remain valid and unchanged for all choices of k, we choose to present the rest for k = 1 and we simply write ϕ˜ = ϕ˜1 , ψ = ψ1 . Note that ϕ˜ and ψ are also regular potentials. B) Dissipativity. Assume that the equilibrium measure for ψ ∈ Vf (X) is not the measure of maximal entropy. Assumption B can be viewed as a generalization of the corresponding dissipation condition of [6] or [15]: If the equilibrium measure for ψ = ϕ + ϕ˜ is not the measure of maximal entropy, then  ψdµϕ > 0 which expresses the breaking of time-reversal symmetry. In fact these conditions are equivalent as we will show in Theorem 5.2 below. Assumption B is quite natural because only under this assumption can one talk about a non-trivial fluctuation symmetry. To understand the role of Assumption A, it is instructive to make the following calculation. Recall the definition of the approximants µϕ,n of (3.3), see Theorem 3.7. Denote by En (g) =

1 Z(f, ϕ, n)



e(Sn ϕ)(x) g(x)

x∈Fix(f n )

the expectation of a function g with respect to µϕ,n . Let g(x) = G(x, f (x), . . . , f n−1 (x)) for some G on Xn and define g 1 (x) = G(i ◦ f n−1 (x), . . . , i ◦ f (x), i(x)) = g(i ◦ f n−1 (x)). Then, by a change of variables that leaves the set Fix(f n ) globally invariant (under Assumption A), En (g 1 ) =

1 Z(f, ϕ, n)



e(Sn ϕ)(f

1−n ◦i(x))

g(x).

x∈Fix(f n )

Again by reversibility and by the definition of ϕ, ˜ n−1  k=0

ϕ(f k+1−n ◦ i(x)) = −

n−1 

ϕ(f ˜ n−k−2 (x))

k=0

so that for x ∈Fix(f n ), Sn ϕ(f 1−n ◦ i(x)) − Sn ϕ(x) = −Sn ψ(x). We conclude that En (g 1 ) = En (ge−Sn ψ ) or, Sn ψ can be viewed as the logarithmic ratio of the probability of a trajectory and the probability of the corresponding time-reversed trajectory, see [10, 11]. These basic identities drive the following

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Theorem 5.1 (Fluctuation symmetry). Assume A)–B). There exists p ∗ > 0 such that, if |p| < p∗ , then lim lim

δ→0 n→∞

µϕ ({x : σn (x) ∈ (p − δ, p + δ)}) 1 log = p, n µϕ ({x : σn (x) ∈ (−p − δ, −p + δ)})

where n−1

σn (x) =

1 k  ψ f (x) . n k=0

Proof. The fluctuation symmetry can be expressed in terms of the rate function I of Theorem 4.2; we must show it has the symmetry I (−p) − I (p) = p

for

|p| < p ∗ .

(5.1)

Since I (p) is the Legendre transform of c(q) = P (ϕ + qψ) − P (ϕ), the symmetry (5.1) must be reflected in a certain symmetry of the free energy c(q). We claim that c(q) = c(−1 − q) for all

q ∈ R.

(5.2)

Assume for a moment that (5.2) is true. Then from (4.2) we conclude that p = −p. Since under the assumptions of the theorem c(q) is not identically equal to a constant, we obtain p = p. Hence, the domain of the rate function I (p) is a symmetric interval containing zero. Let p ∗ = p = −p. For every p in (−p∗ , p∗ ), I (p) is finite and satisfies (5.1):     I (−p) = sup (−p)q − c(q) = sup p(−q) − c(q) q∈R



q∈R

  = sup pq − c(−q) = sup pq − c(−1 + q) q∈R



(by (5.2))

q∈R

    = sup p(−1 + q) − c(−1 + q) + p = sup pq − c(q) + p q∈R

q∈R

= I (p) + p. Now let us prove (5.2), or, what amounts to the same thing (see Lemma 4.1),



 P (q + 1)ϕ + q ϕ˜ = P −qϕ − (1 + q)ϕ˜ , where the topological pressure P (·) is obtained from (3.8).

∀q ∈ R,

(5.3)

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The “time reversing” homeomorphism i maps the set Fix(f n ) to itself. For x ∈ Fix(f n ) one has n−1 

(1 + q)ϕ(f k (x)) + q ϕ(f ˜ k (x))

k=0

= = = =

n−1 

−(1 + q)ϕ(i ˜ ◦ f k+1 (x)) − qϕ(i ◦ f k+1 (x))

k=0 n−1 

−(1 + q)ϕ(f ˜ n ◦ i ◦ f k+1 (x)) − qϕ(f n ◦ i ◦ f k+1 (x))

k=0 n−1 

−(1 + q)ϕ(f ˜ n−k−1 ◦ i(x)) − qϕ(f n−k−1 ◦ i(x))

k=0 n−1 

−(1 + q)ϕ(f ˜ m ◦ i(x)) − qϕ(f m ◦ i(x)).

(5.4)

m=0

Now, taking into account that i is a bijection on Fix(f n ), we obtain that n−1    k k exp ϕ(f (x)) + qψ(f (x)) x∈Fix(f n )

=

k=0



exp

x∈Fix(f n )

n−1 

 ϕ(f k (x)) − (1 + q)ψ(f k (x)) ,

k=0

implying c(q) = c(−1 − q) and finishing the proof.

 

Theorem 5.2 (Dissipativity conditions). Let f , i, ϕ, and ϕ˜ be as above. Then the following conditions are equivalent: 1) the equilibrium measure for ψ = ϕ + ϕ˜ is not the measure of maximal entropy; 2)  the equilibrium measure µϕ for ϕ is not the equilibrium measure for −ϕ; ˜ 3)

ψdµϕ > 0.

Proof. We first show the equivalence of conditions 1) and 2). According to Theorem 3.9, the equilibrium measure for ψ is the measure of maximal entropy if and only if there exists a constant c1 such that n−1 

ψ(f k (x)) = nc1

(5.5)

k=0

for all n ∈ N and every x ∈ Fix(f n ). Similarly, ϕ and −ϕ˜ have the same equilibrium measure if and only if for some constant c2 one has n−1  k=0

ϕ(f k (x)) = −

n−1  k=0

ϕ(f ˜ k (x)) + nc2 ,

(5.6)

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again for all n ∈ N and every x ∈ Fix(f n ). Clearly, since ψ = ϕ + ϕ, ˜ (5.5) and (5.6) are equivalent. To show that the second and the third condition are equivalent, first recall that in the proof of Theorem 5.1 we have established (5.3), and in particular, the following equality P (ϕ) = P (−ϕ). ˜

(5.7)

Now, since µϕ is an equilibrium measure for ϕ, one has  P (ϕ) = hµϕ (f ) + ϕdµϕ . On the other hand, applying the Variational Principle to −ϕ, ˜ we conclude that  P (−ϕ) ˜ ≥ hµϕ (f ) − ϕdµ ˜ ϕ, with equality if and only if µϕ is the equilibrium measure for −ϕ. ˜ Therefore  ψ dµϕ ≥ P (ϕ) − P (−ϕ) ˜ =0 ˜ This finishes the with equality if and only if µϕ is the equilibrium measure for −ϕ. proof.   6. Concluding Remarks 1) As mentioned already in the introduction, the Fluctuation Theorem of Gallavotti and Cohen says that the large deviation rate function I (p) for the time-averages of the phase space contraction in the SRB measure for dissipative and reversible Anosov diffeomorphism has the symmetry: I (−p) − I (p) = p. Our results are valid under a greater generality: not only for SRB measures, but also for the Gibbs measures for expansive homeomorphisms with the specification property. Phase space contraction then gets replaced by the antisymmetric part of the potential under time-reversal which is essentially the same as the phase space contraction rate for Anosov systems. The original proof of Gallavotti and Cohen used Markov partitions (symbolic dynamics). Clearly, this is not an option for us. On the other hand, Ruelle in [15] gave a proof, again for Anosov systems, based on shadowing and one can check that the argument from [15] goes through without many substantial modifications in the case of expansive homeomorphisms with specification. Our approach is still different and even simpler: it is different, physically, because we concentrate on the antisymmetric part of the potential under time-reversal and mathematically, we obtain a symmetry of the rate function I (p) directly from the properties of the topological pressure. It is important that in this generalization, we can also treat continuous (therefore, not necessarily differentiable) transformations. It is interesting to see that our proof comes close to the proposal in [5] where the fluctuation symmetry was first discussed. A further development of this idea, in particular the application of periodic orbit expansions, can be found in e.g. [12]. On the other hand, the fact that we are using periodic points in the proof of our main result Theorem 5.1 is non-essential. We show how the proof of Theorem 5.1 can be adapted, and at the same

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time, we deal now with the case ϕ(x) ˜ = −ϕ(i(x)), and not with ϕ(x) ˜ = −ϕ(i ◦ f (x)) as above. We mentioned earlier that the Fluctuation Symmetry is true for the physically more relevant case of ϕ(x) ˜ = −ϕ(i(x)). Lemma 3.12 states that for expansive homeomorphisms with specification, the topological pressure of any function ϕ from Vf (X) can be obtained as follows:     1 1 k P (ϕ) = lim log Zn (ϕ, En ) = lim log exp ϕ(f (x)) , n→∞ n n→∞ n x∈En

k=0

where {En } is an arbitrary sequence of maximal (n, ε)-separated sets, with ε sufficiently small. For every x ∈ X, n−1 

n−1



 



 (1 + q)ϕ f k (x) + q ϕ˜ f k (x) = −(1 + q)ϕ˜ f k (y) − qϕ f k (y) ,

k=0

k=0

where y = f −(n−1) ◦ i(x) = i ◦ f n−1 (x). To complete the proof we show that, if En is a maximal (n, ε)-separated set, then so is i ◦ f (n−1) (En ). Let u, v be arbitrary distinct points of En . Then, for any k = 0, . . . , n − 1, one has

 d f k (i ◦ f n−1 (u)), f k (i ◦ f n−1 (v)) = d(i ◦ f n−1−k (u), i ◦ f n−1−k (v)) = d(f n−1−k (u), f n−1−k (v)), where in the last equality we used that the time reversing homeomorphism i is preserving the metric d. Therefore, dn (i ◦ f n−1 (u), i ◦ f n−1 (v)) = dn (u, v), and hence, i ◦f n−1 (En ) is also an (n, ε)-separated set. Finally, since En has been chosen to be a maximal (n, ε)-separated set and since the cardinalities of En and i ◦ f n−1 (En ) are equal (i ◦ f n−1 is a homeomorphism), clearly i ◦ f n−1 (En ) is also a maximal (n, ε)-separated set. As a conclusion, Zn ((1 + q)ϕ + q ϕ, ˜ En ) = Zn (−(1 + q)ϕ˜ − qϕ, i ◦ f n−1 En ), which leads to the required equality P ((1 + q)ϕ + q ϕ) ˜ = P (−(1 + q)ϕ˜ − qϕ), see (5.3). 2) Transitive Anosov systems are expansive and do satisfy the specification property. One can find examples of smooth expansive dynamical systems with the specification property, which are not Anosov, see e.g. [1]. Unfortunately, we were not able to find any interesting reversible examples. However, this is quite a typical situation in the field: there are also not very many examples of reversible dissipative Anosov systems. Nevertheless, we think that the validity of the fluctuation symmetry for a larger class of dynamical systems is a step forward. The main reason was already mentioned: uniform hyperbolic behavior and everywhere differentiability is not typical for real physical systems. Secondly, the definition of regular potentials can be extended to include discontinuous functions which satisfy the key property n−1 n−1



d(f k (x), f k (y)) < ε for k = 0, . . . , n − 1 ⇒
ϕ(f k x) − ϕ(f k y)
< K. k=0

k=0

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It is important to understand whether hyperbolic systems with singularities, such as hard ball systems or billiards, satisfy this condition for natural potentials. If this is indeed the case, then one immediately obtains the fluctuation symmetry for such systems as well. Acknowledgement. We are grateful to David Ruelle and Floris Takens for helpful discussions. E.V. was partially supported by NWO grant 613-06-551 and EET grant K99124.

References 1. Aoki, N., Hiraide, K.: Topological Theory of Dynamical Systems. Amsterdam: North-Holland Publishing Co., 1994. Recent advances 2. Bowen, R.: Some systems with unique equilibrium states. Math. Syst. Theory 8(3), 193–202 (1974/75) 3. Dorfman, J.R.: An Introduction to Chaos in Nonequlibrium Statistical Mechanics. Cambridge: Cambridge University Press, 1999 4. Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. New York: Springer-Verlag, 1985 5. Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401–2404; “Erratum”, 71, 3616 (1993) 6. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80(5–6), 931–970 (1995) 7. Haydn, N.T.A., Ruelle, D.: Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification. Commun. Math. Phys. 148(1), 155–167 (1992) 8. Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Volume 54 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press, 1995 9. Kifer, Y.: Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc. 321(2), 505–524 (1990) 10. Maes, C.: Fluctuation theorem as a Gibbs property. J. Stat. Phys. 95, 367–392 (1995) 11. Maes, C., Netoˇcn´y, K.: Time-reversal and entropy. J. Stat. Phys. 110, 269–310 (2003) 12. Rondoni, L., Morriss, G.P.: Applications of periodic orbit theory to N-particle systems. J. Stat. Phys. 86, 991–1009 (1997) 13. Ruelle, D.: Thermodynamic formalism. In: Volume 5 of Encyclopedia of Mathematics and its Applications. Reading, Mass: Addision-Wesley, 1978 14. Ruelle, D.: Thermodynamic formalism for maps satisfying positive expansiveness and specification. Nonlinearity 5(6), 1223–1236 (1992) 15. Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95(1–2), 393–468 (1999) 16. Schmeling, J.: Symbolic dynamics for β-shifts and self-normal numbers. Ergodic Theory Dynam. Sys. 17(3), 675–694 (1997) 17. Takens, F., Verbitski, E.: Multifractal analysis of local entropies for expansive homeomorphisms with specification. Commun. Math. Phys. 203(3), 593–612 (1999) 18. Walters, P.: An introduction to ergodic theory. Volume 79 of Graduate Texts in Mathematics. New York-Berlin: Springer-Verlag, 1982 19. Walters, P.: Differentiability properties of the pressure of a continuous transformation on a compact metric space. J. Lond. Math. Soc. (2) 46(3), 471–481 (1992) 20. Young, L.-S.: Large deviations in dynamical systems. Trans. Amer. Math. Soc. 318(2), 525–543 (1990) Communicated by A. Kupiainen

Commun. Math. Phys. 233, 153–171 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0770-8

Communications in

Mathematical Physics

Entropy of Quantum Limits Jean Bourgain1 , Elon Lindenstrauss2 1

School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA. E-mail: [email protected] 2 Department of Mathematics, Stanford University, Stanford, CA 94305, USA. E-mail: [email protected] Received: 14 March 2002 / Accepted: 24 June 2002 Published online: 13 January 2003 – © Springer-Verlag 2003

Abstract: In this paper we show that any measure arising as a weak∗ limit of microlocal lifts of eigenfunctions of the Laplacian on certain arithmetic manifolds have dimension at least 11/9, and in particular all ergodic components of this measure with respect to the geodesic flow have positive entropy. 1. Introduction In this paper we report some progress towards a conjecture of Rudnick and Sarnak regarding eigenfunctions of the Laplacian  on a compact manifold M for certain special arithmetic surfaces M of constant curvature (see below for definitions): Conjecture 1.1 (QUE [5]). If M has negative curvature, then for any sequence of eigenfunctions φi of the Laplacian, normalized to have L2 -norm 1, such that the eigenvalues λi tend to −∞, the probability measures |φi (x)|2 d vol(x) converge in the weak∗ topology to the Riemannian volume vol(M)−1 d vol. (Recall that µi converge weak∗ to µ if for every continuous function with compact support,

f d µi −→ f d µ as i → ∞.) A similar conjecture can be stated also in the finite volume case [6]. Of particular number theoretic interest are manifolds of the form \H with  a congruence arithmetic lattice, in which case it is natural to assume that the eigenfunctions are Hecke-Maas forms, i.e. also eigenfunctions of all Hecke operators. We shall refer to this special case of Conjecture 1.1 as the Arithmetic Quantum Unique Ergodicity Conjecture. While most of our methods are quite general, the number theoretic argument used to prove Theorem 3.4 is specific to lattices coming from quaternion algebras over the rationals or to congruence sublattices of SL2 (Z). We plan to address the general case using a different technique in a future paper.

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It is well known (see [2, 7, 11]) that any weak∗ limit as in the above conjecture of |φi (x)|2 d vol(x) is a projection of a measure on \ SL(2, R) invariant under the geodesic flow; our main result is that if we assume that φi are all Hecke-Maas forms, then all ergodic components of this measure on \ SL(2, R) have strictly positive entropy with an explicit lower bound, namely κ  = 2/9 (where the speed of the geodesic flow is normalized so that the entropy of the Haar-Lesbegue measure is 2). This in particular implies that the support of such a measure on X has Hausdorff dimension at least 1 + κ  . The first result of this type was proved by Rudnick and Sarnak [5]. They proved that this limiting measure (or even its singular part if any) cannot be supported on a finite union of closed geodesics. Wolpert [10] gave explicit bounds (though substantially weaker than ours) on the modulus of continuity of the limiting measure for  = SL(2, Z); however he used the substantial additional assumption that the support of the singular part (if any) of the measure is compact. In [4], the second named author extended Rudnick and Sarnak’s result to more general groups and lattices, as well as strengthening it by showing that the measure of any closed geodesic is zero. While in general dimension is not preserved under projections, it can be shown that for the projection π : \ SL2 (R) → \H dimension is preserved in the following sense: if µ is invariant under the geodesic flow on \ SL2 (R) with the entropy of all ergodic components ≥ η then the dimension of π µ is at least 1 + η if η ≤ 1; if η > 1 then π µ is regular with respect to the natural measure on \H (see below for a more precise statement). This result is proved in Lindenstrauss and Ledrappier [3]. Thus our results on the dimension of the limiting measure on \ SL(2, R) immediately give bounds on the dimension of any weak∗ limit of |φi (x)|2 d vol(x). Finally, we remark that it follows from an identity of T. Watson [9] that the Grand Riemann Hypothesis (GRH) implies the Arithmetic Quantum Unique Ergodicity Conjecture, that is that any weak∗ limit as above is indeed the natural volume measure. In fact, the GRH gives a best possible rate of convergence of these measures. 2. Statement of Main Results In this paper we deal with uniform lattices that arise from quaternion algebras over Q. Thus, the following notations will be used throughout this paper: • H a quaternion division algebra over Q, split over R. • R an order in H . •  a lattice in SL2 (R) corresponding to the norm one elements of R (see below). We recall that an order R is a subring of H that spans H over Q satisfying that for every a ∈ R both the norm n(a) and the trace tr(a) are integral. Our techniques are also equally applicable to congruence sublattices of SL(2, Z), though the nonuniformity of the lattice requires some minor modifications which we present in Sect. 4. We fix once and for all an isomorphism  : H (R) ∼ = M2 (R). For α ∈ R of positive norm n(α), we let α ∈ SL(2, R) denote the matrix α = n(α)−1/2 (α). We let  be the image under  of the norm one elements in R; as is well known this  is a uniform lattice in SL(2, R). While we do not require R to be a maximal order,

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we will require that ±1 ∈ R. Set M = \ SL(2, R)/ SO(2, R) and X = \ SL(2, R) which is a 2-to-1 cover of the unit tangent bundle of M. We shall say an element α ∈ R is primitive if it cannot be written as mα  with m ∈ N\{1}. Let R(m) be the set of all primitive α ∈ R with n(α) = m, and define the Hecke operator Tm : C ∞ (X) → C ∞ (X) by  f (αx). Tm : f (x) → α∈R(1)\R(m)

Similarly, we define the Hecke points Tm (x) of a x ∈ X by Tm (x) = {αx : α ∈ R(1)\R(m)}. For all but finitely many primes, Tpk (x) (for all k ≥ 1) consists of (p + 1)pk−1 distinct points. We will assume implicitly throughout this paper that all primes considered are outside this finite set. Similarly one can define Hecke operators for SL2 (Z) (and after dropping finitely many primes also for congruence sublattices). In this case we take R  = M2 (Z) ∩ GL(2, R), and taking R  (m) to be all primitive integral matrices of determinant m, primitive being defined exactly as in the previous case. This again can be used to define Hecke operators as above with precisely the same properties. Let & < SL(2, R) be a lattice. We will denote by QL(&) the collection of all measures on &\ SL(2, R) that can be obtained as limiting measures of micro local lifts of L2 -normalized eigenfunctions of both the Laplacian and all Hecke operators on &\H. All measures in QL(&) are invariant under the geodesic flow; if & is uniform, then they are also clearly probability measures. It is a delicate and probably difficult issue to show that in the nonuniform case all measures in QL(&) are probability measures (this is however a consequence of the GRH). We will need to use the following one parameter subgroups of SL(2, R):   10 + u (x) = , x 1 1x u− (x) = , 0  t1  e 0 a(t) = . 0 e−t Set, for any ε, τ > 0, B(ε, τ ) = a((−τ, τ ))u− ((−ε, ε))u+ ((−ε, ε)) and B(ε) = B(ε, ε); all of these sets are open neighborhoods of the identity in SL(2, R). Throughout this note, we let τ0 be a small fixed number, satisfying e10τ0 + e−10τ0 < 2.5 say τ0 = 1/50.

(2.1)

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Theorem 2.1. Let & =  or a congruence sublattice of SL(2, Z). For any µ ∈ QL(&) and any compact subset of &\ SL(2, R), we have that for any x in this compact subset µ(xB(ε, τ0 ))  εκ



for κ  = 2/9. Corollary 2.2. (1) Almost every ergodic component of a measure µ ∈ QL(&) has entropy ≥ κ  . (2) The Hausdorff dimension of the support of µ is at least 1+κ  (unless & is nonuniform and µ = 0). We derive this theorem from the following estimate regarding eigenfunctions of Hecke operators on X: Theorem 2.3. Let & be as above, and - ∈ L2 (&\ SL(2, R)) be an eigenfunction of all Hecke operators with L2 -norm 1. Then for any compact subset . of &\ SL(2, R), for any x ∈ . and r > 0,
 |-(y)|2 dvol(y)  r κ . xB(r,τ0 )

Proof of Theorem 2.1 assuming Theorem 2.3. Let φi be a sequence of eigenfunctions of the Laplacian and all Hecke operators on M = &\H, and let µ be a limiting measure of the micro-local lift of the φi to the unit tangent bundle SM of M which can be identified with X = &\ SL(2, R). We recall the following important properties of the micro-local lift (see [4] for details): (1) |φi |2 dvol converge weak∗ to the projection of µ to M. (2) Let ω be the Casimir operator. Considering L2 (M) as a subset of L2 (X) one can find a sequence of Casimir eigenfunctions -i which are also eigenfunctions of all Hecke operators on L2 (X) with -i 2 = 1 such that: (a) φi and -i have the same ω-eigenvalue. (b) µ is the weak∗ limit of |-i |2 dvolX . (c) µ is invariant under the geodesic flow (under the identification SM ∼ = X this is the flow that arises from the action &g → &ga(t) ). By Theorem 2.3 for all x ∈ . and i,
 |-i (y)|2 dvolX (y)  ε κ . xB(ε,τ0 )

Since µ is the weak∗ limit of |-i |2 dvolX ,




µ (xB(ε, τ0 ))
lim

|-i (y)|2 dvolX (y),

xB(ε,τ0 )

so 

µ (xB(ε, τ0 ))  εκ .  

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Finally, we mention the following corollary of Theorem 2.1 and the results in [3]: Corollary 2.4. Let dM denote the image of the standard hyperbolic metric on H to M, and µ˜ a weak∗ limit of |φi |2 dvolM with φi a sequence of Hecke-Maas forms as above. Then for any κ  < κ  ,

d µ(x)d ˜ µ(y) ˜ < ∞.  dM (x, y)κ +1 M

We note that if one could improve the constant κ  in Theorem 2.1 to be > 1 then one would have by [3] that µ˜ is regular with respect to the Riemannian volume with an L2 Radon Nikodyn derivative. The full Quantum Unique Ergodicity Conjecture in this case is equivalent to κ  = 2. 3. On the Distribution of Hecke Points and a Proof of Theorem 2.3 for Quaternion Lattices Lemma 3.1. If α, β are two primitive commuting elements of H (Q) \ Q then Q(α) = Q(β). Proof. Since α, β commute, K = Q(α, β) is a field embedded in H (Q), and unless Q(α) = Q(β) we have that [K : Q] = 4. Let θ be a generator for K, i.e. K = Q(θ ). Then since θ ∈ H (Q) it has to satisfy the degree two polynomial with rational coefficients θ 2 − tr(θ )θ + n(θ ) = 0 – a contradiction.   Lemma 3.2. For any τ > 0 and ε ∈ (0, 0.1) we have that   B(ε, τ )B(ε, τ ) ⊂ B Oτ (ε), 2τ + Oτ (ε 2 ) , B(ε, τ )−1 ⊂ B(Oτ (ε), τ + Oτ (ε 2 )).

(3.1)

Proof. We prove only (3.1), the proof of the second equation being very similar. Let g1 = a(t1 )u− (a1 )u+ (b1 ), g2 = a(t2 )u− (a2 )u+ (b2 ), then g1 g2 = a(t1 )u− (a1 )u+ (b1 )a(t2 )u− (a2 )u+ (b2 ) = a(t1 + t2 )u− (e−2t2 a1 )u+ (e2t2 b1 )u− (a2 )u+ (b2 ). Set b˜1 = e2t2 b1 , and rewrite u+ (b˜1 )u− (a2 ) as   1 a2 u+ (b˜1 )u− (a2 ) = ˜ b1 1 + a2 b˜1      −1 

−1  = u− a2 1 + a2 b˜1 a − ln 1 + b˜1 a2 u+ b˜1 1 + a2 b˜1 ; thus g1 g2 = a(t1 + t2 + Oτ (ε 2 ))u− (Oτ (ε))u+ (Oτ (ε)).  

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Lemma 3.3. If α, β ∈ R satisfy, for some x ∈ SL(2, R), that βx, αx ∈ xB(ε, τ0 )

(3.2)

with ε < 0.1, Cε 2 ≤ [n(α)n(β)]−1 (C some constant depending only on τ0 ) then α and β commute. Proof. Take tα , tβ so that αx ∈ xa(tα )B(ε, 0) and similarly for β. Consider now   ρ = α −1 , β = αβ −1 α −1 β = ∗

1 α β¯ αβ. ¯ n(α)n(β)

A straightforward calculation using (3.2) and Lemma 3.2 shows that ρx = αβ −1 α −1 βx ∈ αβ −1 α −1 xa(tβ )B(ε, 0) ⊂ αβ −1 xa(tβ − tα )B(C1 ε, C1 ε 2 ) ⊂ ··· ⊂ xa(0)B(C2 ε, C2 ε 2 ) for some C2 that can be calculated explicitly using Lemma 3.2. However, tr(ρ) ∈ Z[1/n(α)n(β)], and for any z ∈ xB(C2 ε, C2 ε 2 )x −1 , |tr(z) − 2| ≤ C3 ε 2 . As long as C3 ε 2 ≤ [n(α)n(β)]−1 , this implies that tr(ρ) = 2, hence, since R contains no unipotent elements, ρ = 1. This shows that α and β do indeed commute.   We defer the proof of the following theorem to the next section Theorem 3.4. For any ε > 0, and any sufficiently large D and N ≥ D 1/4+ε , there exists a set W ⊂ {1, . . . , N} of size |W | ≥ N κ (κ = 4/5) of square free integers divisible by a bounded number of primes p, all with D p = −1. Remark. It is possible to improve on the value of κ (see the remark √ following Lemma 5.6); the natural limit of the argument given here seems to be e/2 − ε ≈ 0.824.

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Theorem 3.5. For any set of primes P, x ∈ \ SL(2, R) and ε > 0, there is a set W of cube free integers with the following properties: (1) (2) (3) (4)

Any n ∈ W has a bounded number of prime factors (uniformly in ε, x, δ). For any n ∈ W , p 2 |n iff p|n and p ∈ P. The sets in {yB(ε, τ0 ) : y ∈ Tn (x), n ∈ W } are pairwise disjoint.  |W |  ε−κ /4 , with κ κ = = 2/9. 2(1 + κ)

Remark. Improving κ of Theorem 3.4 to κ = 0.824 will give κ  ≈ 0.225. Proof. Let n1 ≤ n2 be a pair of integers with smallest n2 such that there are some y1 = y2 ∈ X with yb ∈ Tnb (x)

for b = 1, 2

satisfying y1 B(ε, τ0 ) ∩ y2 B(ε, τ0 ) = ∅. Choose a representative α1 ∈ R(n1 ) of the coset of R(1)\R(n1 ) sending x to y1 . By definition of τ0 (see (2.1)), there will be a unique α2 ∈ R(n2 ) such that α1 xB(ε, τ0 ) ∩ α2 xB(ε, τ0 ) = ∅. Now set α to be a primitive element of R so that α¯ 1 α2 ∈ Zα. Since y1 = y2 we have that α ∈ R(M) for some M > 1 dividing n1 n2 . By definition of α, we have that x ∈ αxB(4ε, 3τ0 ). Consider the subring Q(α) < H . Since H is a division ring, Q(α) is isomorphic to some number field L; let i : Q(α) → L be this isomorphism. Since α is primitive, α ∈ Q; since α satisfies the degree 2 polynomial over Z, t 2 − tr(α)t + n(α) = 0. L is a quadratic extension of Q, namely √

L∼ D for D = tr(α)2 − 4n(α). =Q Notice that since H splits over R, i(α) ∈ R, hence D ≥ 0. We give the following upper bound for D. By definition,

|tr(α)| −1 = |tr(α)| ∈ |tr xB(4ε, 3τ )x |  1, 0 n(α)1/2 hence |tr(α)|  n(α)1/2 and D  n(α) ≤ n1 n2 .

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We define a multiplicative function ζP by ζP (1) =  1, p if p prime ∈ /P , ζP (p) = p 2 if p ∈ P ζP (p 2 ) = 0. 

If n2 ≥ ε−2κ we can take W = {ζP (p) : p prime
n2 } , 



and we are done. Thus we may assume that D  ε−4κ , n2 ≤ ε−2κ . Take

−1/4 1/2−κ    ε 1/2−κ  D 4κ  , N ∼ ε 2 n1 n2 so in particular N  D 1/4+ (i.e. N  D 1/4+ε0 for any ε0 ). Apply Theorem 3.4 to find a set W˜ ⊂ {2, . . . , N} with   ˜ (3.3) W  N κ ≥ εκ(1/2−κ ) = εκ satisfying the conditions of that theorem. We now take W to be 

W = ζP (n) : n ∈ W˜ . By Theorem 3.4, any n ∈ W has a bounded number of prime factors, and by definition of W and ζP we have that p 2 |n iff p|n and p ∈ P. In view of (3.3) we know that the W has the prescribed number of elements. Thus it remains to be verified that the sets of the collection {yB(ε, τ0 ) : y ∈ Tl (x), l ∈ W } are all pairwise disjoint. Assume to the contrary that there are distinct zb ∈ Tlb (x)

lb ∈ W

such that z1 B(ε, τ0 ) ∩ z2 B(ε, τ0 ) = ∅. We find that there is some primitive β with 1 = n(β)|l1 l2 , x ∈ βxB(4ε, 3τ0 ),

(3.4)

so in particular, |n(β)| ≤ l1 l2 ≤ N 4 . By Lemma 3.3, since |n(α)n(β)|−1 ≥ [N 4 n1 n2 ]−1  ε2 , √ α and β commute, hence β ∈ Q(α) ∼ =i L = Q( D).

(3.5)

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Since the conjugate β¯ of β is mapped to the Galois conjugate of the image of i(β) in L, the norm n(β) is the same as the norm of the image i(β) of β in L. But by Theorem 3.4 any prime factor p of n(β) satisfies D p = −1. Thus any such prime p remains inert in the extension L : Q, and so must divide n(β) (indeed must divide the norm of any integral element of L) an even number of times. We conclude that n(β) =: A2 is a square and moreover the two ideals (in the ring of integers of L) "i(β)#L ,

"A#L

are equal. Equivalently, we have that i(β)/A is a unit of the ring of integers of L. This in turn implies that tr(β) = trL (i(β)/A) ∈ Z combining (3.4) with (2.1), and assuming, as we may, that ε is sufficiently small, we have that |tr(β)| ∈ Z ∩ [2, 5/2 + Oτ0 (ε)] = {2}, or (since H (Q) is a division domain) that β = ±1, and β is not primitive – a contradiction.   Lemma 3.6. Let T be a r + 1 regular tree (or even any r + 1 regular graph with girth ≥ 2). Let TT : CT → CT be the operator  f (y) [TT f ](x) = dT (y,x)=1

(with dT denoting the usual metric on the tree). Assume φ is an eigenfunction of TT , with eigenvalue λ. Then √    |φ(y)|2 if |λ| > 10r  |φ(x)|2  d(y,x)=1   |φ(y)|2 otherwise.  d(y,x)=2

Proof. Assume |λ| >

√

r 10 .

Then by Cauchy-Schwartz, 2  1  ≤ 1 (r + 1) |φ(x)|2 = φ(y) |φ(y)|2 2 2 |λ| |λ| dT (x,y)=1 dT (x,y)=1 2  |φ(y)| .

Now assume |λ| <

√

dT (x,y)=1

r 10 .

Then for any y with dT (x, y) = 1,  φ(z) + φ(x), λφ(y) = d(z,y)=1 d(z,x)=2

1 φ(x) = r +1

  d(y,x)=1



λφ(y) −

 1 = λ2 φ(x) − r +1



d(z,x)=2

d(z,x)=2

 φ(z) .

 φ(z)

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So

 φ(z) |φ(x)|2 = [(r + 1) − λ2 ]−2 d(z,x)=2  r(r + 1) ≤ |φ(z)|2 [(r + 1) − λ2 ]2 d(z,x)=2  2  |φ(z)| . d(z,x)=2

Note that throughout the proof, the implicit constants are absolute and do not depend on λ, r.   Corollary 3.7. Let - be an eigenfunction of all Hecke operators on X. Let n be a square free integer n = p1 · p2 · . . . · pk with k = O(1). Take where

m = p1α1 p2α2 . . . pkαk , 

1 if Tpi - = λpi - with |λpi | > αi = 2 otherwise. Then for all x ∈ X,



|-(x)|2 

√

pi 10

|-(y)|2 .

y∈Tm (x)

Proof. We prove the corollary by induction on k. The case k = 0, i.e. m = n = 1, states that |-(x)|2  |-(x)|2 , which is of course true. Now if n = p1 . . . pk−1 , αk−1 , m = p1α1 · · · pk−1 then Tm (x) = Tpαk ◦ Tm (x). Furthermore, since - restricted on the Hecke tree associk ated with pk is an eigenfunction of the tree Laplacian we may apply Lemma 3.6 to show ∀y ∈ X,  |-(y)|2  |-(z)|2 , z∈T

α (y) pk k

so |-(x)|2 



|-(y)|2

y∈Tm (x)







|-(z)|2

z∈T

α (y) y∈Tm (x) p k



k

|-(z)|2 .

z∈Tm (x)

Note that the implicit constant depends only on the bound on k.

 

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Proof of Theorem 2.3. Let λp denote the eigenvalue of - with √ respect to the Hecke operator Tp . Let P be the sets of all primes for which |λp | ≤ p/10. By Theorem 3.5,  there is a set W of cube free integers of size ≥ ε −κ such that for any n ∈ W , we have 2 that p |n iff p|n and p ∈ P, and such that yB(ε, τ0 )

y ∈ Tn (x), n ∈ W

(3.6)

are all pairwise disjoint. Since - is a Hecke eigenfunction, by Corollary 3.7, for all n ∈ W and any y ∈ X,  |-(y)|2  |-(z)|2 z∈Tn (y)

(note that the implicit constant in the above equation is universal and does not depend on any parameter) hence for any n ∈ W ,

 |-(y)|2 dvolX (y)  |-(z)|2 dvolX (y) n (y) xB(ε,τ0 ) z∈T


xB(ε,τ0 )

=



|-(y)|2 dvolX (y).

z∈Tn (x)zB(ε,τ ) 0

Summing over n ∈ W , and using the disjointness property (3.6), we get that

1   2 |-(y)| dvolX (y)  |-(y)|2 dvolX (y) |W | n∈W z∈Tn (x)zB(ε,τ ) xB(ε,τ0 ) 0
η 2 |-(y)| .
ε X

  4. The Case of Λ a Congruence Sublattice of SL(2, Z) In this section we present the modifications needed to carry out the proof of Theorem 2.3 to the nonuniform case. For simplicity we will discuss only the case of & = SL(2, Z), leaving the straightforward verification for congruence sublattices to the reader. Recall the notations R  = M2 (Z) ∩ GL(2, R), and R  (m) = all primitive integral matrices of determinant m. As before we set M = SL(2, Z)\H and X = SL(2, Z)\ SL(2, R). In order to conform more closely to the notations of the previous section, we set for α ∈ R  n(α) = det(α), and α¯ = n(α)α −1 ∈ R  . The starting point of the proof is Lemma 3.3. While the proof of this lemma essentially carries over to SL(2, Z), the final step gives only that tr(ρ) = 2, which in view of the existence of unipotents in R  does not imply ρ = 1. As an alternative we use the following: Lemma 4.1. If α, β ∈ R  satisfy, for some x ∈ SL(2, R), that βx, αx ∈ xB(ε, τ0 ), Cε2 ≤ [n(α)n(β)]−1

(4.1)

(with ε sufficiently small and C some constant depending on τ0 and on x, uniformly on x in compact subsets of X) then α and β commute.

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Proof. Let . be a compact subset of SL(2, R) with x ∈ .. Define as before tα and tβ so αx ∈ xa(tα )B(ε, 0) and similarly for β. Let B0 (@1 , @2 ) = log B(@1 , @2 ), so that B0 (@1 , @2 ) is a small neighborhood of the zero matrix in M2 (R). Then αβx ∈ xa(tα + tβ )B(Cε, Cε 2 ), βαx ∈ xa(tα + tβ )B(Cε, Cε 2 ). So [α, β]+ = βα − βα ∈ x −1 B0 (C  ε, C  ε 2 )x ⊂ B0 (C  ε, C  ε), C  some constant depending on ., τ0 . But [α, β]+ ∈ M2 (Z), so [α, β]+ ∈ B0 (C  ε, C  ε) ∩

1 M2 (Z). det(αβ)1/2

Assuming (det α det β)−1  ε2 for a sufficiently large implicit constant depending on . we have that indeed [α, β]+ = 0.   Having proved a suitable substitute to Lemma 3.3, we discuss the modifications needed to prove Theorem 3.5. As usual our result will no longer be uniform in x ∈ X but only uniform for x in an arbitrary compact subset of X. For the convenience of the reader we restate this theorem, from which Theorem 2.3 is easily derived in the same way as in the previous section. Theorem 4.2. For any compact subset . ⊂ SL(2, Z)\ SL(2, R), for any set of primes P, x ∈ . and ε > 0, there is a set W of cube free integers with the following properties: (1) (2) (3) (4)

Any n ∈ W has a bounded number of prime factors (uniformly in ε, ., x). For any n ∈ W , p2 |n iff p|n and p ∈ P. The sets in {yB(ε, τ0 ) : y ∈ Tn (x), n ∈ W } are pairwise disjoint.  |W |  ε−κ /4 , uniformly on ., with κ  = 2/9 as in Theorem 3.5.

Proof. We proceed exactly as in Theorem 3.5. Let n1 ≤ n2 be a pair of integers with smallest n2 such that there are some y1 = y2 ∈ X with yb ∈ Tnb (x)

for b = 1, 2

satisfying y1 B(ε, τ0 ) ∩ y2 B(ε, τ0 ) = ∅.

Entropy of Quantum Limits

165

Choose a representative α1 ∈ R  (n1 ) sending x to y1 . Take any α2 ∈ R  (n2 ) such that α1 xB(ε, τ0 ) ∩ α2 xB(ε, τ0 ) = ∅. Now set α to be a primitive element of R  so that α¯ 1 α2 ∈ Zα. Since y1 = y2 we have that α ∈ R  (M) for some M > 1 dividing n1 n2 . By definition of α, we have that x ∈ αxB(4ε, 3τ0 ). 



Without loss of generality, as before, we can assume n2  ε−2κ , M  ε−4κ  ε−1 (with a large implicit constant). In this case α is R-semisimple (i.e. α has two distinct real eigenvalues), since any element of B(4ε, 3τ0 ) which is not R-semisimple must lie in B(4ε, Cε) for a suitably large absolute constant C. Since x is in some fixed compact set ., we conclude that unless α is R-semisimple, α ∈ B(C. ε, C. ε) ∩ M −1/2 M2 (Z) = {1}, a contradiction. Thus again Q(α) is isomorphic to some real quadratic number field √ L = Q( D), and the rest of the proof carries out without any additional difficulties.   5. On Primes Which are Quadratic Nonresidues mod D Theorem 5.1. For any ε > 0 there is a α > 0 so that for every large enough integer 1/4+ε one has that the set P of primes D which is not a perfect square, and N ≥ D D α N ≤ p ≤ N with p = −1 satisfy  1 1 > − ε. p 2

p∈P

We cite the following standard version of Brun’s combinatorial sieve: Theorem 5.2 ([8, Theorem 3, p. 60]). Let A be a finite set of integers and let P be a set of prime numbers. Write Ad := #{a ∈ A : a ≡ 0  P (y) := p,

(mod d)},

p∈P ,p≤y

S(A, P , y) := card{a ∈ A : (a, P (y)) = 1}. Assume there exist a non-negative multiplicative function w, some real number X, and positive constants κ, A such that Ad =: Xw(d)/d + Rd

(d|P (y)),

(5.1)

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J. Bourgain, E. Lindenstrauss

  η≤p≤ξ

w(p) 1− p

−1

 <

log ξ log η

κ 

A 1+ log η

 (2 ≤ η ≤ ξ ).

Then we have, uniformly for A, X , y and u ≥ 1, S(A, P , y) = X

  w(p) {1 + O(u−u/2 )} + O  1− p



 p≤y,p∈P



(5.2)

 |Rd | .

d≤y u ,d|P (y)

(5.3) We will also use the following estimate of D. Burgess: Theorem 5.3 ([1, Theorem 2]). Let k be a cube free positive integer and let χ be a non-principal Dirichlet character belonging to the modulus k. Let SH (N ) =

N+H 

χ (n).

n=N+1

Then for any ε > 0 and r ∈ Z+ we have that |SH (N )|  H 1−1/r k {(r+1)/4r

2 }+ε

,

(5.4)

with the implicit constant depending on ε and r. Since we may have to apply Theorem 5.3 with a character modulo 8k, k odd, we note the following immediate corollary: Corollary 5.4. Suppose k = dk  with k  cube free and (d, k  ) = 1, and χ a non-principal Dirichlet character modulo k, then |SH (N )| ε,r,d H 1−1/r k 

{(r+1)/4r 2 }+ε

.

(5.5)

Proof. Write SH (N ) =

d−1 

SH,l (N )

(5.6)

l=0

with SH,l (N ) =



χ (n).

(5.7)

N
We show that for all l,   2 SH,l (N ) ε,r,d H 1−1/r k  ((r+1)/4r +ε .

We now note that χ  (m) = χ (md)

(5.8)

Entropy of Quantum Limits

167

is a non-principal Dirichlet character modulo k  , and we may apply Theorem 5.3 on 

SH,l (N ) =



χ (md) =

M+1≤m≤M+H /d

χ  (m),

M+1≤m≤M+H /d

where M is defined by dM ≡ N + l (mod k  ).

 

Proof of Theorem 5.1. We will prove thetheorem in two steps. First we show that there are many integers n ≤ N satisfying Dn = −1 which have no prime factor less than N α for a suitably chosen α using Brun’s combinatorial sieve. Then we show how this implies that  1 is large p

p∈P

for which we again use the combinatorial sieve in a somewhat degenerate case. Let P1 be the set of all primes, and     D A= n: = −1 . n We now set as in Theorem 5.2, S(B, P1 , y) = card {a ∈ A : (a, p) = 1 ∀ prime p
y} . We now set w(n) = 1

for all square free n

  and X = N/2. This choice satisfies (5.2). By quadratic reciprocity, Dn is a non-principal character modulo (at most) 8D, and we may clearly assume D is square free. Burgess’ estimate (Corollary 5.4) allows us to bound Rd of (5.1) by      1  D N ε¯ ,r (N/d)1−1/r D (r+1)/4r 2 +¯ε . (5.9) |Rd | = 1+ − dn 2d 2 1≤n≤N/d By Theorem 5.2 we know that    

 1 N  |Rd | S(A, P1 , y)  1− 1 − Cu−u/2 + O  2 p u p
y

d
y

with C independent of A, u, y, X. We fix r, ε¯ by requiring that N −1/r D



 (r+1)/4r 2 +¯ε

 N −ε/10r

and take u so that Cu−u <

ε 100

(5.10)

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and α α
ε¯ u−1 /10. We now estimate the O(·) term in (5.10) for y = N α . By (5.9)     2 |Rd |  D (r+1)/4r +¯ε d
yu d|P1 (y)

d
y u

 N 1−ε/10r



d −(1−1/r)

d
y u

N

1−ε/10r u/r

y

,

so we see that for D, N large enough

  1 (1 − ε/10)N  S(A, P1 , N ) ≥ 1− . 2 p α α

(5.11)

p≤N

The second part of the proof will use the bound on S(A, P1 , y) to show that there = −1. We remark that any n ≤ N contributing to are many primes ≤ N with D p D α S(A, P1 , N ), that is such that n = −1 and n is not divisible by a prime smaller than N α , is divisible by some prime p in     D α P = primes N ≤ p ≤ N, = −1 . p A trivial application of the combinatorial sieve for the prime set   P1p ,N α = primes
N α , p with p some prime in P shows that

     1 1 + ε/10 n
N s.t. p |n but (n, p) = 1 for all p ≤ N α
N 1− . p p α p
Summing over all

p

∈ P , we have that

S(A, P1 , N α )
(1 + ε/10) N

 

1−

p
Combining this with (5.11) gives  1 p ∈P

p



1 − ε/10  2 (1 + ε/10)

1 2

1 p

 1 .  p  p ∈P

− ε.

  Corollary 5.5 (of Theorem 5.1). Let α, D and N ≥ D 1/4+ε as in Theorem 5.1. Then there is a subset W ⊂ {1, . . . , N} of size  N κ (κ = 0.8) so that for any w ∈ W and α p|w, we have that D p = −1 and p ≥ N .

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The argument deducing Corollary 5.5 from Theorem 5.1 can be translated to the following purely combinatorial question: Lemma 5.6. For any S ⊂ R+ let
dx m(S) = , x S

FS = {s1 + s2 + · · · + sr : r ≥ 1 ∀i, si ∈ S} . Then if ε > 0 is small enough, for every S ⊂ (0, 1] with m(S) > 1/2 − ε, FS ∩ [κ, 1] = ∅ for κ = 0.8. Remark. A√ more refined analysis should probably enable improving κ from the above lemma to 2e −, which is easily seen to be optimal by taking S to be the interval [1/2, κ]. Proof. Assume that FS ∩ [0.8, 1] = ∅.

(5.12)

Since for any n ∈ N, nS ⊂ FS, we have that

  4 1 1 4 1 4  1 4  S ⊂ [0, 1] − , = , ∪ , ∪ , . 5n n 2 5 3 10 4 15 n∈N

Suppose first that

 S∩

 1 4 , = ∅, 4 15

(5.13)

and let s be any element from this set. Then by (5.12), S ∩ [0.8 − s, 1 − s] = ∅; notice that for any s ∈ [1/4, 4/15], [4/5 − s, 1 − s] ⊂ [1/2, 4/5] and m[4/5 − s, 1 − s] = ln

1−s ≥ 0.31. 4/5 − s

Thus, if (5.13) holds,         1 4 1 2 1 4 4 m(S) ≤ m , +m , +m , −m − s, 1 − s ≤ 0.41, 2 5 3 5 4 15 5 a contradiction to m(S) > 1/2−.

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Since m(S) > 1/2−, and since (5.13) does not hold, we have that      1 2 6 1 4 ,  m (S) − m ,  0.5 − ln −  0.317, m S∩ 2 5 3 5 5 hence if we define α by

 m

4 5

α,

 = 0.317,

i.e. α = 0.8e−0.317 ≤ 0.583, we would have that   1 S∩ , α = ∅. 2 ! Take s to be some element in S ∩ 21 , α . Then on the one hand   4 , 1 = ∅, (s + S) ∩ 5 and on the other hand

 s+

so S ⊂

1 4 2, 5

   4 1 2 , ⊂ ,1 , 3 5 5

!

, hence  m (S)
m

a contradiction.

1 4 , 2 5

 = ln

8 < 0.47 < 0.5−, 5

 

Proof of Corollary 5.5. Let P denote the set of primes ∈ [N α , N ] with recall that  1 ≥ 1/2 − ε. p



D p

= −1. We

p∈P

Fix δ > 0, r = 1 + δ very small depending only on ε. Let S˜ denote the integers 

S˜ = n : N α ≤ r n ≤ rN, P ∩ [r n−1 , r n ]  r n(1−δ) , and divide P into two sets: P1 = P2 =



 

P ∩ r n−1 , r n ,

n∈S˜



 

P ∩ r n−1 , r n .

n∈ / S˜

Clearly,  1
p

p∈P2

 N α
r n(1−δ) r N −δ , rn

Entropy of Quantum Limits

171

so for N large enough  1 ≥ 1/2 − 2ε. p

p∈P1

Applying the previous lemma to   (s − 1) log r s log r  , S= log N log N s∈S˜

we find that there are s1 , . . . , sk ∈ S˜ with κ≤ and we can take

log r(s1 + · · · + sk ) ≤1 log N



W = P ∩ [r s1 −1 , r s1 ] × · · · × P ∩ [r sk −1 , r sk ] .

  Acknowledgements. Both authors are very grateful to Peter Sarnak, who has introduced us independently to this question, with whom we had numerous discussions on this problem that have strongly influenced our work. The second named author would also like to thank him for his consistent helpful support and encouragement throughout his two year stay at the Institute for Advanced Study in Princeton. We are indebted to Enrico Bombieri, Henryk Iwaniec and Kannan Soundararajan for their help regarding the sieve method.

References 1. Burgess, D.A.: On character sums and L-series. II. Proc. Lond. Math. Soc. (3), 13, 524–536 (1963) 2. Colin de Verdi`ere, Y.: Ergodicit´e et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985) 3. Ledrappier, F., Lindenstrauss, E.: On the projection of measures invariant under the Geodesic flow. Preprint 4. Lindenstrauss, E.: On quantum unique ergodicity for \H × H . To appear in IMRN, 2001 5. Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161(1), 195–213 (1994) 6. Sarnak, P.: Some problems in number theory, analysis and mathematical physics. In: Mathematics: Frontiers and Perspectives, Providence RI: Am. Math. Soc., 2000, pp. 261–269 7. Schnirelman, A.I.: Ergodic properties of eigenfunctions. Usp. Math. Nauk. 29, 181–182 (1974) 8. Tenenbaum, G.: Introduction to Analytic and Probabilistic Number Theory. Cambridge: Cambridge University Press, 1995. Translated from the second French edition (1995) by C.B. Thomas 9. Watson, T.C.: PhD thesis, Princeton, 2001 10. Wolpert, S.A.: The modulus of continuity for γ0 (m)\H semi-classical limits. Commun. Math. Phys. 216(2), 313–323 (2001) 11. Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987) Communicated by P. Sarnak

Commun. Math. Phys. 233, 173–190 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0758-4

Communications in

Mathematical Physics

Quantum Group Symmetry in sine-Gordon and Affine Toda Field Theories on the Half-Line G.W. Delius, N.J. MacKay Department of Mathematics, University of York, York YO10 5DD, U.K. E-mail: gwd2, [email protected] Received: 10 December 2001 / Accepted: 7 October 2002 Published online: 19 December 2002 – © Springer-Verlag 2002

Abstract: We consider the sine-Gordon and affine Toda field theories on the half-line with classically integrable boundary conditions, and show that in the quantum theory a remnant survives of the bulk quantized affine algebra symmetry generated by non-local charges. The paper also develops a general framework for obtaining solutions of the reflection equation by solving an intertwining property for representations of certain coideal subalgebras of Uq (g). ˆ 1. Introduction Affine Toda field theories are integrable relativistic quantum field theories in 1+1 dimensions with a rich spectrum of solitons. There is one Toda theory for each affine Kac-Moody algebra gˆ and the sine-Gordon model is the simplest example, for gˆ =
= a (1) . Each of these models possesses a quantum group symmetry, generated sl(2) 1 by non-local charges. Conservation of these charges determines the S-matrices up to an overall scalar factor, as will be recalled in Sect. 2.1. The presence of a boundary breaks this quantum group symmetry, and also destroys classical integrability unless the boundary conditions are very carefully chosen. The purpose of this paper is to show that, with such classically integrable boundary conditions, a remnant of the quantum group symmetry nevertheless survives. This residual symmetry is just as powerful for determining the reflection matrices (boundary S-matrices) and boundary states as the original symmetry was for determining the particle spectrum and S-matrices in the bulk. The main results of the paper are the simple expressions (3.27) and (3.28) for the non-local symmetry charges in the sine-Gordon model on the half line and their gener(1) alisations (4.11) to an affine Toda theories. The sine-Gordon charges were first written down by Mezincescu and Nepomechie [24]. We derive the non-local symmetry charges in two very different ways. The approach in Sects. 3 and 4 treats the boundary condition as a perturbation of the Neumann

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G.W. Delius, N.J. MacKay

boundary condition and employs boundary conformal perturbation theory [18]. These sections thus generalize the approach of [2] which derives the non-local charges on the whole line. The second approach is purely algebraic and is introduced in Sect. 2.3 and applied in Sect. 4.3. While the calculations in boundary perturbation theory presuppose a knowledge of [2], the algebraic techniques described in Sect. 2 are basic and of wider significance. We confirm the correctness of our symmetry charges by using them to rederive the sine-Gordon soliton reflection matrix [18] in Sect. 3.5 and the reflection matrices for (1) the vector solitons in an affine Toda theories [17] in Sect. 4.2. We obtain the reflection matrices by solving linear equations which are a consequence of the symmetry. This is much simpler than the original approaches that relied on solving the reflection equation [4] (boundary Yang-Baxter equation), which is a non-linear equation.

2. Algebraic Techniques 2.1. Soliton S-matrices as quantum group intertwiners. This section is a review of the role of the quantum group symmetry in determining the soliton S-matrices and serves as preparation for the following section about reflection matrices. The affine Toda field theory associated to the affine Lie algebra gˆ is known [2, 14] to be symmetric under the quantum affine algebra Uq (g) ˆ with zero center. The solitons in µ affine Toda field theory are arranged in various multiplets. Let Vθ be the space spanned by the solitons in multiplet µ with rapidity θ . Each such space carries a representation µ µ πθ : Uq (g) ˆ → End(Vθ ). The asymptotic two-soliton states span the tensor product µ ν spaces Vθ ⊗ Vθ  which also carry representations of Uq (g) ˆ built using the coproduct. µ An incoming two-soliton state in Vθ ⊗ Vθν with θ > θ  will evolve during scattering µ into an outgoing state in Vθν ⊗ Vθ with scattering amplitudes given by the S-matrix µ

µ

S µν (θ − θ  ) : Vθ ⊗ Vθν → Vθν ⊗ Vθ .

(2.1)

The S-matrix has to commute with the symmetry or, in other words, the S-matrix has to be an intertwiner of the tensor product representations, µ

S µν (θ − θ  ) ◦ (πθ ⊗ πθν )((Q)) µ

ˆ = (πθν ⊗ πθ )((Q)) ◦ S µν (θ − θ  ) for all Q ∈ Uq (g).

(2.2)

Because the tensor product representations are irreducible for generic rapidities, Schur’s lemma tells us that this equation determines the S-matrix uniquely up to an overall scalar factor. This intertwiner can also be obtained from the universal R-matrix R of Uq (g) ˆ as S µν (θ − θ  ) ∝ Pˇ ◦ R µν (θ − θ  ),

µ

R µν (θ − θ  ) = (πθ ⊗ πθν )(R),

(2.3)

where Pˇ interchanges the tensor factors. The scattering of three solitons is described by an intertwiner between the tensor µ µ product representations Vθ ⊗ Vθν ⊗ Vθλ and Vθλ ⊗ Vθν ⊗ Vθ with θ > θ  > θ  . There are two ways to build such an intertwiner from the two-soliton S-matrices:

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175

µ

S µν (θ−θ  ) ⊗ id

µ

id ⊗ S µλ (θ−θ  )

µ

µ

S µλ (θ−θ  ) ⊗ id

µ

id ⊗ S µν (θ−θ  )

µ

Vθ ⊗ Vθν ⊗ Vθλ −−−−−−−−→ Vθν ⊗ Vθ ⊗ Vθλ −−−−−−−−−→ Vθν ⊗ Vθλ ⊗ Vθ     S νλ (θ  −θ  )⊗id id⊗S νλ (θ  −θ  )

Vθ ⊗ Vθλ ⊗ Vθν −−−−−−−−−→ Vθλ ⊗ Vθ ⊗ Vθν −−−−−−−−→ Vθλ ⊗ Vθν ⊗ Vθ (2.4) Because the tensor product representations are irreducible for generic rapidities, the intertwiner is unique. The above diagram is therefore commutative (up to an overall scalar factor) – that is, the S-matrices satisfy the Yang-Baxter equation. For the tensor product of two vector representations the intertwining property (2.2) was solved by Jimbo [20] for all non-exceptional quantum affine algebras (both twisted and untwisted). The intertwiners for many other tensor products have been constructed [5] using the tensor product graph method. The complete sets of soliton S-matrices for (1) (1) (2) the algebras Uq (an ) [19], Uq (cn ) [15], and Uq (a2n−1 ) [16] have been constructed. 2.2. Soliton reflection matrices as intertwiners. As reviewed in the previous section, quantum group symmetry can be used to obtain the soliton S-matrices. We now want to present a similar technique for obtaining the reflection matrices for solitons hitting a boundary. So far the only way to obtain reflection matrices has been to solve the reflection equation. Because the reflection equation is a non-linear functional matrix equation, solving it is very difficult for anything but the simplest cases. We will instead obtain a linear equation. We will now restrict the solitons to live on the left half line x ≤ 0 by imposing suitable integrable boundary conditions at x = 0. A soliton with positive rapidity θ will eventually hit the boundary and be reflected into another soliton with opposite rapidity −θ. The corresponding quantum process is described by the reflection matrix µ

µ¯

K µ (θ ) : Vθ → V−θ .

(2.5)

The multiplet µ¯ of the reflected soliton does not necessarily have to be the same as that of (1) the incoming soliton, but it has to have the same mass. It turns out [6] in the case of an th Toda theory with the usual boundary conditions that solitons in the r rank antisymmetric tensor multiplet are converted into solitons in the (n + 1 − r)th rank antisymmetric tensor multiplet. µ µ¯ ˆ intertwiner between the representations Vθ and V−θ – that is, there There is no Uq (g) µ is no K (θ ) satisfying µ

µ¯

K µ (θ )πθ (Q) = π−θ (Q)K µ (θ )

(2.6)

for all Q ∈ Uq (g). ˆ This is not surprising because the boundary should be expected to break the quantum group symmetry down to a subalgebra B of Uq (g). ˆ The intertwining property (2.6) should only hold for all Q in B. The unbroken symmetry algebra B should be “large enough” so that the intertwining condition (2.6) determines the reflection matrix uniquely up to an overall scalar factor. The subalgebra B must also be a left coideal of Uq (g) ˆ in the sense that ˆ ⊗B (Q) ∈ Uq (g)

for all Q ∈ B.

(2.7)

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G.W. Delius, N.J. MacKay

This allows it to act on multi-soliton states. Later in this paper we will construct such subalgebras B describing the residual quantum group symmetries of affine Toda field theories with integrable boundary conditions. If two solitons are incident on the boundary, they can be reflected in two different µ orders. Correspondingly there are two ways of constructing intertwiners from Vθ ⊗ Vθν µ¯ ν¯ : to V−θ ⊗ V−θ  id ⊗K ν (θ  )

µ

S µ¯ν (θ+θ  )

µ

µ

id ⊗K µ (θ)

µ¯

ν¯ ν¯ ⊗ V ν¯ ⊗ V Vθ ⊗ Vθν −−−−−−→ Vθ ⊗ V−θ −−−−−→ V−θ −−−−−→ V−θ  −   θ − −θ    µν  S ν¯ µ¯ (θ−θ  ) S (θ −θ  ) µ

id ⊗K µ (θ)

µ¯

S ν µ¯ (θ+θ  )

id ⊗K ν (θ  )

µ¯

µ¯

ν¯ Vθν ⊗ Vθ −−−−−−→ Vθν ⊗ V−θ −−−−−−→ V−θ ⊗ Vθν −−−−−−→ V−θ ⊗ V−θ  (2.8)

Provided the tensor product representations are irreducible as representations of the subalgebra B the diagram above is commutative (up to an overall scalar factor) – that is, the reflection matrix automatically satisfies the reflection equation. Due to the identification (2.3) between the S-matrix and the R-matrix the reflection equation can also be written in the form 1

2

Pˇ R ν¯ µ¯ (θ − θ  )Pˇ K µ (θ ) R µ¯ν (θ + θ  ) K ν (θ  ) 2

1

= K ν (θ  ) Pˇ R ν µ¯ (θ + θ  )Pˇ K µ (θ ) R µν (θ − θ  ), 1

(2.9)

2

where we employ the standard notation A = A ⊗ id, A = id ⊗ A. Note that there is one such reflection equation for every pair of soliton multiplets and that generally these reflection equations involve 4 different R-matrices. In affine Toda theory it is possible for solitons to bind to the boundary, thereby creating multiplets of boundary bound states. The space V [λ] spanned by the boundary bound states in multiplet [λ] will carry a representation π [λ] : B → End(V [λ] ) of the symmetry algebra B. The reflection of solitons in multiplet µ with rapidity θ off a boundary µ bound state in multiplet [λ] is described by a reflection matrix K µ[λ] (θ ) : Vθ ⊗ V [λ] → µ¯ V−θ ⊗ V [λ] which is determined by the intertwining property µ

µ¯

K µ[λ] (θ ) (πθ ⊗ π [λ] )((Q)) = (π−θ ⊗ π [λ] )((Q)) K µ[λ] (θ ),

∀ Q ∈ B. (2.10)

2.3. Construction of symmetry algebra. In this paper we will use boundary conformal perturbation theory to construct generators of the coideal subalgebras B ⊂ Uq (g) ˆ that occur as the symmetry algebras of affine Toda field theories on the half line, where  parameterizes the boundary condition. However we also have an alternative construction which we will describe in this section. The construction has the disadvantage that it requires the a priori knowledge of at least one solution of the reflection equation but it has the advantage that it does not rely on first order perturbation theory. We will use it in Sect. 4.3 to verify the expressions for the symmetry charges derived in Sect. 4.1. µ Let us assume that for one particular representation Vθ we know the reflection matrix µ µ¯ K µ (θ ) : Vθ → V−θ . We define the corresponding Uq (g)-valued ˆ L-operators [13] in

Quantum Group Symmetry

177

terms of the universal R-matrix R of Uq (g) ˆ [21],   µ µ µ Lθ = πθ ⊗ id (R) ∈ End(Vθ ) ⊗ Uq (g), ˆ   µ ¯ µ ¯ µ ¯ L¯ θ = π−θ ⊗ id (Rop ) ∈ End(V−θ ) ⊗ Uq (g). ˆ

(2.11)

Here Rop is the opposite universal R-matrix obtained by interchanging the two tensor factors. Motivated by [31] we construct the matrices µ

µ¯

µ

µ

µ¯

ˆ Bθ = L¯ θ (K µ (θ ) ⊗ 1) Lθ ∈ Hom(Vθ , V−θ ) ⊗ Uq (g).

(2.12)

It may make things clearer to introduce matrix indices: µ

µ¯

µ

ˆ (Bθ )α β = (L¯ θ )α γ (K µ (θ ))γ δ (Lθ )δ β ∈ Uq (g),

(2.13)

where we are using the usual summation convention. We will now check that for all θ µ the (Bθ )α β are elements of a coideal subalgebra B which commutes with the reflection matrices. ν¯ which satisfies the Let us first check that any reflection matrix K ν (θ  ) : Vθν → V−θ  µ appropriate reflection equation commutes with the action of the elements (Bθ )α β , i.e., that (see Eq. (2.6)) µ

µ

ν¯ α ν  K ν (θ  ) ◦ πθν ((Bθ )α β ) = π−θ  ((Bθ ) β ) ◦ K (θ ),

(2.14)

or, in index-free notation, µ

µ

ν¯ ν  (id ⊗ K ν (θ  )) ◦ (id ⊗ πθν )(Bθ ) = (id ⊗ π−θ  )(Bθ ) ◦ (id ⊗ K (θ )).

(2.15)

We observe that µ

µ

(id ⊗ πθν )(Lθ ) = (πθ ⊗ πθν )(R) = R µν (θ − θ  ),

µ¯ (id ⊗ πθν )(L¯ θ ) µ ν¯ (id ⊗ π−θ  )(Lθ ) ν¯ ¯ µ¯ (id ⊗ π−θ  )(Lθ )

= = =

µ¯ (π−θ ⊗ πθν )(Rop ) = Pˇ R ν µ¯ (θ + θ  )Pˇ , µ ν¯ µ¯ν (πθ ⊗ π−θ (θ + θ  ),  )(R) = R µ¯ ν¯ op ˇ ν¯ µ¯ (θ − θ  )Pˇ , (π−θ ⊗ π−θ  )(R ) = P R

(2.16) (2.17) (2.18) (2.19)

Substituting this into (2.15) and using Pˇ R ∝ S gives (id ⊗ K ν (θ  )) ◦ S ν µ¯ (θ + θ  ) ◦ (id ⊗ K µ (θ )) ◦ S µν (θ − θ  ) = S ν¯ µ¯ (θ − θ  ) ◦ (id ⊗ K µ (θ )) ◦ S µ¯ν (θ + θ  ) ◦ (id ⊗ K ν (θ  )),

(2.20)

which is just the reflection equation (compare with (2.8)). We thus see that every soluµ tion of the reflection equation commutes with all the generators (Bθ )α β , and vice versa: every matrix which satisfies the intertwining equation (2.15) is also a solution of the reflection equation (2.20). µ Next we need to check the coideal Under the assumption that all (Bθ )α β  µ property.  α are in B we need to show that  (Bθ ) β is in Uq (g) ˆ ⊗ B. Using that  µ α   µ α   (Lθ ) β = (πθ ) β ⊗  (R) (2.21)  µ α  = (πθ ) β ⊗ id ⊗ id (R13 R12 ) (2.22) µ

µ

= (Lθ )γ β ⊗ (Lθ )α γ ,

(2.23)

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G.W. Delius, N.J. MacKay

and similarly   µ¯ µ¯ µ¯  (L¯ θ )α β = (L¯ θ )α γ ⊗ (L¯ θ )γ β ,

(2.24)

 µ  µ¯ µ µ  (Bθ )α β = (L¯ θ )α δ (Lθ )σ β ⊗ (Bθ )δ σ ,

(2.25)

we find that

which is indeed in Uq (g) ˆ ⊗ B as required. Note that the B-matrices satisfy the quadratic relations 1

2

1

2

µ µ Pˇ R ν¯ µ¯ (θ − θ  )Pˇ Bθ R µ¯ν (θ + θ  ) Bθν = Bθν Pˇ R ν µ¯ (θ + θ  )Pˇ Bθ R µν (θ − θ  ),

(2.26)

which follow from the FRT relations satisfied by the L-matrices together with the reflection equation (2.9). Algebras with relations of this form are known as reflection equation algebras [31]. Our construction can thus be viewed as an embedding of the reflection equation algebras into the quantized enveloping algebras. 3. The sine-Gordon Model 3.1. Review of non-local charges. For the purpose of finding its quantum group symmetry charges one views the sine-Gordon model as a perturbation of a free bosonic conformal field theory by a relevant operator 'pert [32]. The (Euclidean) action on the whole line is1   1 ¯ + λ S= (3.1) d 2 z ∂φ ∂φ d 2 z 'pert (x, t) , 4π 2π with the perturbing operator ˆ

ˆ

'pert (x, t) = ei βφ(x,t) + e−i βφ(x,t) ,

(3.2)

where βˆ is the coupling constant2 . We impose the condition φ(−∞, t) = 0. The free boson field may be split into holomorphic and antiholomorphic parts, φ = ¯ = 0 = ∂ ϕ, ϕ + ϕ, ¯ where ∂ϕ ¯ and the two-point functions are ϕ(z)ϕ(w)0 = − ln(z − w),

ϕ(¯ ¯ z)ϕ( ¯ w) ¯ 0 = − ln(¯z − w), ¯

ϕ(z)ϕ( ¯ w) ¯ 0 = 0. (3.3)

The set of fields in the conformal field theory consists only of those combinations of derivatives and exponentials of the fundamental fields ϕ and ϕ¯ which do not suffer from logarithmic divergences and are local with respect to each other. See [22] for a clear account of the free bosonic theory and its perturbation into the sine-Gordon model. We use the conventions of [2] and denote Euclidean light-cone coordinates as z = (t + ix)/2 and z¯ = (t − ix)/2, where the Euclidean time t is related to Minkowski time t M by t = it M . The derivatives are then ∂ = ∂t − i∂x , ∂¯ = ∂t + i∂x . We write d 2√ z = idz d z¯ = −dx dt/2. 2 βˆ is related to the conventional β by βˆ = β/ 4π . 1

Quantum Group Symmetry

179

The Uq (slˆ2 ) symmetry of the sine-Gordon model is generated by the non-local charges [2]  ∞  ∞ 1 1 ¯ Q± = dx(J± − H± ) , Q± = dx(J¯± − H¯ ± ) , (3.4) 4π −∞ 4π −∞ where ± 2iˆ ϕ

2i

H± = λ

βˆ 2 βˆ 2 −2

∓ ϕ¯ :, J¯± =: e βˆ : ,     : exp ±i 2ˆ − βˆ ϕ ∓ i βˆ ϕ¯ : ,

H¯ ± = λ

βˆ 2 βˆ 2 −2

    ˆ : : exp ∓i 2ˆ − βˆ ϕ¯ ± i βϕ

J± =: e

β

β

β

(3.5) (3.6) (3.7)

together with the topological charge T =

 ∞ βˆ dx ∂x φ . 2π −∞

(3.8)

The time-independence of the charges follows from the current conservation equations ¯ ± = ∂H± , ∂J

∂ J¯± = ∂¯ H¯ ± ,

(3.9)

which were obtained in first-order perturbation theory in [2]. In order to derive the Uq (slˆ2 ) relations and coproduct, we set φ(−∞) = 0. The parameter q is then related to the Toda coupling constant βˆ by   q = exp 2iπ(1 − βˆ 2 )/βˆ 2 . (3.10)

3.2. Neumann boundary condition. We now want to restrict the sine-Gordon model to the half line x ≤ 0 by imposing the Neumann boundary condition ∂x φ˜ = 0 at x = 0. Note that to avoid confusion we will always decorate fields in the theory on the half-line with a tilde. Again we will consider the sine-Gordon model as a perturbation of the free boson. A simple way to describe the free bosonic field theory on the half-line with Neumann boundary condition is to identify its fields with the subset of parity-invariant fields of the theory on the whole line. Thus for every field '(x, t) on the whole line there exists ˜ a field '(x, t) on the half-line defined by ˜ ¯ '(x, t) = '(x, t) + '(−x, t)

for x ≤ 0 ,

(3.11)

¯ where '(−x, t) is the parity-transform of '(x, t). The fundamental field on the half-line, ˜ φ(x, t) = φ(x, t) + φ(−x, t), immediately satisfies the Neumann boundary condition. The two-point functions for its chiral components ϕ(x, ˜ t) = ϕ(x, t) + ϕ(−x, ¯ t)

and

˜¯ ϕ(x, t) = ϕ(x, ¯ t) + ϕ(−x, t)

(3.12)

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G.W. Delius, N.J. MacKay

follow immediately from (3.3), and are ˜¯ ϕ(w) ˜¯ ϕ(z) z − w) ¯ , ϕ(z) ˜ ϕ(w) ˜ 0 = −2 ln(z − w) , 0 = −2 ln(¯ ˜¯ w) ϕ(z) ˜ ϕ( ¯ 0 = −2 ln(z − w) ¯ .

(3.13) (3.14)

Note the non-vanishing two-point function between the holomorphic and anti-holomorphic components.  The perturbation d 2 z 'pert is invariant under parity and is thus a valid perturbation of the boson on the half-line too. Note that one does not obtain the sine-Gordon model ˜ ˜ on the half-line by perturbing with the operator exp(i βˆ φ(x, t)) + exp(−i βˆ φ(x, t)), as one might naively have thought3 . ¯ ± described in the previous section transform under parity The charges T , Q± and Q P as follows: ¯ ∓, Q± → Q

P : T → −T ,

¯ ± → Q∓ . Q

(3.15)

¯ ∓ therefore give conserved charges ˜ ± = Q± + Q The parity-invariant combinations Q ˜ ± in terms of currents on on the half-line, but T does not. We can express the charges Q the half-line, ˜± = 1 Q 4π



0

−∞

  dx J˜± − H˜ ± + J˜¯∓ − H¯˜ ∓ ,

(3.16)

where the half-line currents are defined according to (3.11), for example J˜± (x) = J± (x) + J¯∓ (−x). They satisfy the conservation equations ∂¯ J˜± = ∂ H˜ ±

and

∂ J˜¯± = ∂¯ H¯˜ ± .

(3.17)

3.3. General boundary conditions as boundary perturbations. In [30, 18] the sine-Gordon model was found to be classically integrable with the rather more general boundary condition   ˜ ˆ˜ ˆ b − ei βˆ φ(0,t)/2 ∂x φ˜ = i βλ at x = 0 . (3.18) − + e−i β φ(0,t)/2 We shall treat this as a boundary perturbation of the sine-Gordon model with Neumann boundary condition:  λb pert dt 'boundary (t) , (3.19) S = SNeumann + 2π with the boundary perturbing operator ˆ˜

ˆ˜

'boundary (t) = − ei β φ(0,t)/2 + + e−i β φ(0,t)/2 pert

= − e 3

ˆ i βφ(0,t)

+ + e

ˆ −i βφ(0,t)

.

See however [1, Appendix C] for a different treatment of the boundary perturbation

(3.20)

Quantum Group Symmetry

181

To check that S does indeed produce the boundary condition (3.18) we calculate the ˜ t) in first-order boundary conformal perturbation theory correlation functions of ∂x φ(0, [18, 29]: ˜ t) · · ·  ∂x φ(0, ˜ ˜ t) · · · N = lim ∂x φ(x, t)e−Sboundary · · · N = ∂x φ(0, x→0−  λb pert ˜ − t)'boundary (t  ) · · · N + O(λ2b ), dt  lim ∂x φ(x, 2π x→0−

(3.21)

where · · · N denotes the correlation function with Neumann boundary condition. The first term on the right-hand side vanishes of course. To evaluate the second we need the operator product expansions −2i βˆ t − t  + ix −2i βˆ pert ˜¯ ∂ ϕ(x, t)'boundary (t  ) = t − t  − ix

∂ ϕ(x, ˜ t)'boundary (t  ) = pert

 ˆ˜ ˆ˜ − ei β φ(0,t)/2 − + e−i β φ(0,t)/2 + regular terms,



 ˆ˜ ˆ˜ − ei β φ(0,t)/2 − + e−i β φ(0,t)/2 + regular terms,



and thus, using that ∂x φ = 2i (∂ ϕ˜ − ∂¯ ϕ˜¯ + O(λb ), ∂x φ(x, t)'boundary (t  )

  1 1 ˆ˜ ˆ˜ = βˆ − − ei β φ(0,t)/2 − + e−i β φ(0,t)/2 + . . . . (3.22)   t − t + ix t − t − ix pert

We can now use the identity

2π i 1 1 lim = − δ(t − t  ), ∂ n−1   n  n − (t − t + ix) (t − t − ix) (n − 1)! t x→0 to find

  ˜ ˆ˜ ˜ t) · · ·  = i βλ ˆ b − ei βˆ φ(0,t)/2 ∂x φ(0, − + e−i β φ(0,t)/2 ,

(3.23)

(3.24)

in agreement with the boundary condition (3.18).

3.4. Quantum group charges for general boundary condition. For the general boundary conditions (3.18) (that is, in the presence of the boundary perturbation (3.20)) the ˜ ± in (3.16) will no longer be conserved. We calculate the time-dependence of charges Q the charges:  0   ˜± = 1 ∂t Q dx ∂t J˜± − H˜ ± + J˜¯∓ − H˜¯ ∓ 4π −∞  0   i =− dx ∂x J˜± + H˜ ± − J˜¯∓ − H˜¯ ∓ 4π −∞  i ˜ J± (0, t) + H˜ ± (0, t) − J˜¯∓ (0, t) − H˜¯ ∓ (0, t) . =− 4π

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For the Neumann boundary condition, J± (0, t) = J¯∓ (0, t) and H± (0, t) = H¯ ∓ (0, t) ˜ ± = 0. But in general we will obtain in first order perturbation theory and thus ∂t Q ˜ + (0, t) · · ·  ∂t Q  λb  ˜ + (x, t)'pert (3.25) dt  lim ∂t Q =− boundary (t ) . . . N + . . . 2π x→0−   λb −i  ˜ pert =− J+ (x, t) − J˜¯− (x, t) 'boundary (t  ) . . . N + . . . . (3.26) dt  lim  − 2π x→0 4π Because H˜ + and H˜¯ − are themselves already of order λ they do not contribute to first order. The necessary operator product expansions are

2i 2i 1 ϕ(x,t) ϕ(−x,t) ¯ ˆ βˆ βˆ e−i βφ(0,t) : + . . . , : e + e (t − t  + ix)2

2i 2i 1 ϕ(x,t) ϕ(−x,t) ¯ ˆ pert βˆ βˆ J˜¯− (x, t)'boundary (t  ) = + e−i βφ(0,t) : + · · · . : e + e (t − t  − ix)2 pert J˜+ (x, t)'boundary (t  ) = +

Using now that at the boundary ϕ¯ = ϕ = φ/2 up to order λ terms, we obtain ˜ + (0, t) · · ·  = − λb + 1 ∂t Q 2π 2πi :e



i φ(x,t) βˆ





dt lim

x→0−

ˆ

1 1 −  2  (t − t + ix) (t − t − ix)2





e−i βφ(0,t ) : + · · · 



λb + βˆ 2 i 1 −βˆ φ(0,t) ∂t e βˆ + ··· = 2π βˆ 2 − 1 λb + βˆ 2 = ∂t q T + · · · , 2π βˆ 2 − 1 where we used that q has the value given in (3.10). It follows that the charge ˆ2 ¯ − + ˆ+ q T with ˆ+ = λb + β ˆ + = Q+ + Q Q 2π 1 − βˆ 2

(3.27)

is conserved to first order in perturbation theory. By similar calculations we obtain a second conserved charge ˆ2 ˆ − = Q− + Q ¯ + + ˆ− q −T with ˆ− = λb − β . Q 2π 1 − βˆ 2

(3.28)

These charges were first written down in [24]. They generate a coideal subalgebra of ˆ because Uq (g) ˆ ± ) = (Q± + Q ¯ ∓ ) ⊗ 1 + q ±T ⊗ Q ˆ ±. (Q

(3.29)

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183

3.5. Reflection matrices derived from quantum group symmetry. We will now use our ˆ ± to derive the soliton reflection matrix, up to an knowledge of the conserved charges Q overall factor. The sine-Gordon model only has a single two-dimensional soliton multiplet, spanned by the soliton and anti-soliton states |A± (θ ). The soliton reflection matrix describes what happens to a soliton during reflection off the boundary: a soliton of type α with rapidity θ is converted into a combination of soliton types β with opposite rapidity −θ with probability amplitudes K β α , K : |Aα (θ ) → |Aβ (−θ)K β α (θ ).

(3.30)

ˆ ± on the soliton states can be obtained from the The action of the symmetry charges Q action of the quantum group charges given in [2]. One finds (after a change of basis with respect to that used in [2]) ˆ ± : |Aα (θ ) → |Aβ (θ )πθ (Q ˆ ± )β α Q

(3.31)

with ˆ ± )+ + = ˆ± q ±1 , πθ (Q

ˆ ± )− − = ˆ± q ∓1 , πθ (Q

(3.32)

ˆ ± )+ − = c e±θ/γ , ˆ ± )− + = c e∓θ/γ , πθ (Q (3.33) πθ (Q  where γ = βˆ 2 /(2 − βˆ 2 ) and c = λγ 2 (q 2 − 1)/2π i. We know that reflection and symmetry transformations have to commute, which leads to the following set of eight homogeneous linear equations for the four entries of the reflection matrix (see (2.6)): ˆ ± )β α = π−θ (Q ˆ ± )γ β K β α , ∀ γ , α ∈ {+, −}. K γ β (θ ) πθ (Q

(3.34)

This set of equations is very easy to solve and one finds the unique solution (up to an undetermined overall factor k(θ ))   K + − (θ ) = K − + (θ ) = e2θ/γ − e−2θ/γ k(θ ), (3.35) K ±± =

 q − q −1  ˆ± eθ/γ + ˆ∓ e−θ/γ k(θ ). c

(3.36)

This agrees with the soliton reflection matrix determined in [18] by solving the reflection equation. 4. Affine Toda Theory To every affine Lie algebra gˆ of rank n there is associated an affine Toda field theory [25] with Euclidean action

  n 1 λ 1 2 2 ¯ ˆ S= d z ∂φ ∂φ + exp −i β αj · φ , (4.1) d z 4π 2π |αj |2 j =0

describing an n-component bosonic field φ in two dimensions. The exponential interaction potential is expressed in terms of the simple roots αi , i = 0, . . . , n of gˆ (projected onto the root space of g). The parameter λ sets the mass scale and βˆ is the coupling constant.

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For simplicity we shall restrict our attention to simply-laced algebras, and choose the (slightly unusual) convention that |αi |2 = 1. With g = a1 = su(2) this specializes to the sine-Gordon model of Sect. 3. 4.1. Conserved charges from conformal perturbation theory. The quantum group symmetry algebra Uq (g) ˆ is generated by the topological charges  ∞ βˆ Tj = dx αj · ∂x φ, (4.2) 2π −∞ together with the non-local conserved charges  ∞  ∞ 1 1 ¯ dx (Jj − Hj ) , Qj = dx (J¯j − H¯ j ) , j = 0, 1, . . . , n, Qj = 4π c −∞ 4π c −∞ (4.3) where Jj =: exp



2i αj βˆ

 · ϕ :,

J¯j =: exp



2i αj βˆ

 · ϕ¯ : ,

(4.4)

Hj = λ

βˆ 2 βˆ 2 −2

    ˆ j · ϕ¯ : , : exp i 2ˆ − βˆ αj · ϕ − i βα

(4.5)

H¯ j = λ

βˆ 2 2 ˆ β −2

    ˆ j · ϕ :, : exp i 2ˆ − βˆ αj · ϕ¯ − i βα

(4.6)

β

β

λγ 2 (qi2 − 1)/2π i in order to and we choose the normalization constant c = obtain the simple q-commutation relations given later in (4.26). The linear combina¯ j are parity-invariant and thus yield conserved charges on the ˜ j = Qj + Q tions Q half-line with Neumann boundary conditions. We now add to the action a boundary perturbation,  λb pert S = SNeumann + (4.7) dt 'boundary (t) , 2π where pert 'boundary (t)

=

n j =0

which leads to the boundary condition ˆ b ∂x φ˜ = −i βλ

n j =0





i βˆ ˜ t) j exp − αj · φ(0, 2 

i βˆ ˜ t) j αj exp − αj · φ(0, 2

,

(4.8)

 at x = 0.

(4.9)

By calculations entirely analogous to those of Sect. 3 we find that, due to this perturba˜ i are no longer conserved, but instead satisfy tion, the Q ˜i = ∂t Q

λb i βˆ 2 ∂t q Ti , 2πc βˆ 2 − 1

(4.10)

Quantum Group Symmetry

185

so that the new conserved charges are ¯ i + ˆi q Ti , i = Qi + Q Q

(4.11)

where, to first order in perturbation theory, λb i βˆ 2 . 2πc 1 − βˆ 2

ˆi =

(4.12)

Note that at this stage the boundary parameters ˆi can still take arbitrary values. However, we shall see in subsequent sections how the |ˆi | are fixed, leaving only a choice of signs. ˆ i , i = 0, The symmetry algebra of the boundary affine Toda theory generated by the Q T i ˆ ˆ ˆ i − ˆi ). . . . , n, is a coideal subalgebra of Uq (g) ˆ because (Qi ) = Qi ⊗ 1 + q ⊗ (Q 4.2. Reflection matrices derived from quantum group symmetry. The conserved charges (4.11) derived in the previous section can now be used to derive the soliton reflection matrices, as explained in Sect. 2.2. We will illustrate this here in the example of the (1) vector solitons in an Toda theories. The new feature that arises which was not visible in the sine-Gordon model is that solitons are converted into antisolitons upon reflection off the boundary. Thus in particular the vector solitons are reflected into solitons in the conjugate vector representation. µ Let Vθ be the space spanned by the vector solitons with rapidity θ . Choosing a suitµ able basis for Vθ and defining the elementary matrices ei j to be the matrices with a 1 in ˆ generators are the ith row and the jth column, the representation matrices of the Uq (g) µ

µ ¯ −1 i πθ (Q e i+1 , i) = x

πθ (Qi ) = x ei+1 i ,

µ

πθ (Ti ) = −ei i + ei+1 i+1 ,

(4.13)

where x = eθ/γ with γ = βˆ 2 /(2 − βˆ 2 ) and where we identify the indices n + 1 = 0, µ¯ n + 2 = 1. The representation matrices for the conjugate representation on Vθ are µ¯

µ¯

µ¯

¯ i ) = x −1 ei+1 i , πθ (Q

πθ (Qi ) = x ei i+1 ,

πθ (Ti ) = ei i − ei+1 i+1 .

(4.14)

ˆ i then immediately follow The representation matrices of the symmetry generators Q from (4.11), µ ˆ i+1 −1 i πθ (Q e i+1 + ˆi ((q −1 − 1) ei i + (q − 1) ei+1 i+1 + 1), i) = x e i +x µ¯ ˆ πθ (Q i)

=xe

i

i+1

+x

−1 i+1

e

i

i

+ ˆi ((q − 1) e i + (q

−1

− 1) e

i+1

i+1

+ 1),

(4.15) (4.16)

where 1 denotes the unit matrix. Because the representation matrices are so sparse most of the (n+1)3 components of the intertwining equation (2.6) for the K-matrix are trivial, leaving only the 2n(n + 1) equations, 0 = ˆi (q −1 − q)K i i + x K i i+1 − x −1 K i+1 i ,

0=

i

i+1 − K i , ˆi q K i j + x −1 K i+1 j , ˆi q −1 K j i + x K j i+1 ,

0=K 0=

i+1

(4.17) (4.18)

j = i, i + 1,

(4.19)

j = i, i + 1.

(4.20)

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The equations in (4.19) and (4.20) can be used to determine all upper triangular entries in terms of K 1 2 and all lower triangular entries in terms of K 2 1 . Then Eq. (4.17) determine the diagonal entries. Finally the fact that all diagonal entries must be the same according to (4.18) not only determines K 2 1 in terms of K 1 2 but also requires that either ˆi = 0 for all i or that |ˆi | = 1 for all i. If all ˆi = 0 then K can be an arbitrary diagonal matrix. If all |ˆi | = 1 then one obtains the non-diagonal solution   K i i (θ ) = q −1 (−q x)(n+1)/2 − ˆ q (−q x)−(n+1)/2

k(θ) , −q

q −1

K i j (θ ) = ˆi · · · ˆj −1 (−q x)i−j +(n+1)/2 k(θ),

for j > i,

K j i (θ ) = ˆi · · · ˆj −1 ˆ (−q x)j −i−(n+1)/2 k(θ ),

for j > i,

(4.21)

which is unique up to the overall numerical factor k(θ ). We have defined ˆ = ˆ0 ˆ1 · · · ˆn . Note that this agrees with the solution found by Gandenberger in [17]. It is interesting to note that the restriction on the possible values of the boundary parameters ˆi which we found above agrees with the restrictions which were found in [3] from the requirement of classical integrability. The reflection matrices for solitons in the multiplets corresponding to the other fundamental representations could be determined either in the same manner as above or by a fusion procedure. Furthermore there will be boundary bound states which may transform non-trivially under the quantum group symmetry and again their reflection matrices can be obtained by using the intertwining property or by boundary fusion. We refer the reader to [9] where similar calculations have been done for the rational reflection matrices which have a Yangian symmetry.

4.3. Construction of conserved charges from reflection matrices. In Sect. 4.1 we derived ˆ i by using first-order boundary conformal expressions for the symmetry generators Q perturbation theory. These symmetry generators allowed us to determine the reflection (1) matrices for the vector solitons in an Toda theory in Sect. 4.2. This success can be taken as confirmation of the correctness of the expressions (4.11) for the symmetry generators. However one might worry that higher-order perturbation theory might produce additional terms which, while not visible in the vector representation, might be required to guarantee the commutation of the generators with the reflection matrices in higher representations. In order to rule this out, we will now rederive the expressions for the ˆ i using the construction introduced in Sect. 2.3. We will show how to extract generators Q ˆ i from the B µ by expanding to first order in x = exp(θ/γ ). the Q θ The construction requires the L-matrices that are obtained from the universal Rmatrix according to (2.11). Luckily we will not need to work with the rather involved expression for the full universal R-matrix. Instead we will introduce the spectral parameter x and expand to first order in x. For this purpose let us introduce the homomorphism 9x : Uq (g) ˆ → Uq (g)[x, ˆ x −1 ], defined by ¯ i ) = x −1 Q ¯ i, 9x (Q

9x (Qi ) = x Qi ,

9x (Ti ) = Ti .

(4.22)

This will be useful later because µ

µ

πθ = π0 ◦ 9x for x = eθ/γ .

(4.23)

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187

We introduce the spectral parameter dependent universal R-matrix as R(x) = (9x ⊗ id)(R).

(4.24)

It satisfies

  R(x) ((9x ⊗ id) ◦ ) (Q) = (9x ⊗ id) ◦ op (Q)R(x)

ˆ ∀Q ∈ Uq (g).

(4.25)

We want to use this property to determine the expression for R(x) to linear order in x. For this purpose we need the relations between the Uq (g) ˆ generators: [Ti , Qj ] = αi · αj Qj ,

¯ j ] = −αi · αj Q ¯j, [Ti , Q

2Ti ¯ j − q −αi ·αj Q ¯ j Qi = δij q − 1 , Qi Q qi2 − 1

(4.26)

where qi = q αi ·αi /2 . The generators also satisfy Serre relations which we will not need however. The coproduct  is defined by (Qi ) = Qi ⊗ 1 + q Ti ⊗ Qi , (Qi ) = Qi ⊗ 1 + q Ti ⊗ Qi , (Ti ) = Ti ⊗ 1 + 1 ⊗ Ti .

(4.27)

Using this information we find that   n n 2 −T ¯ l q l q j,k=1 gj k Tj ⊗Tk + O(x 2 ), R(x) = 1 ⊗ 1 + x (1 − ql )Ql ⊗ Q

(4.28)

l=0

or, equivalently, R(x) = q



n

j,k=1 gj k Tj ⊗Tk

1⊗1+x

n l=0

 (1 − ql2 ) q −Tl

¯l Ql ⊗ Q

+ O(x 2 ), (4.29)

where gj k αk · αl = −δj l . (1) We now specialize to the vector representation of an and find the L-operators according to Eq. (2.11), µ

µ

µ

Lθ = (πθ ⊗ id)(R) = (π0 ⊗ id)(R(x))   n A −1 l+1 ¯ l + O(x 2 ), 1⊗1+x (q − q) e l ⊗ Q =q

(4.30) (4.31)

l=0 µ¯ µ¯ µ¯ L¯ θ = (π−θ ⊗ id)(Rop ) = (π0 ⊗ id)(R(x)op )   n −1 l+1 (q − q) e l ⊗ Ql q −A + O(x 2 ), = 1⊗1+x



(4.32) (4.33)

l=0

(−ej

+ ej +1

µ where A = gj k j j +1 ) ⊗ Tk . We also expand the reflection matrix K (θ ) given in (4.21) to first order in x,   n K µ (θ ) = B 1 + x ˆl (q −1 − q) el+1 l + O(x 2 ), (4.34) l=0

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G.W. Delius, N.J. MacKay

where B = −ˆ q (−q x)−(n+1)/2 /(q − q −1 ). Putting it all together according to Eq. (2.12) gives n   µ ¯ l + ˆl q Tl + O(x 2 ). Bθ = B + x (q −1 − q) el+1 l ⊗ Ql + Q (4.35) l=0

ˆ i from the non-vanishing entries of the matrix B µ Thus we can read off our generators Q θ at first order in x. This proves that their action does indeed commute with all reflection µ matrices because we had shown this for the entries of the matrix Bθ already in Sect. 2.3. (1) They thus generate a symmetry algebra of an affine Toda theory on the half-line. 5. Discussion In this paper we have derived non-local conserved charges for the sine-Gordon model (1) and an affine Toda field theories on the half-line and have shown how to use these to determine the soliton reflection matrices by solving the linear intertwining equations. The calculations in conformal boundary perturbation theory used to derive the nonlocal charges in Sects. 3.4 and 4.1 may be of interest in themselves because of the way in which the perturbation theory for the model on the half-line is embedded into that for the model on the whole line. The calculations should easily generalize to the Toda theories for arbitrary affine Kac-Moody algebras g, ˆ allowing us to derive the hitherto unknown soliton reflection (1) matrices in these theories. This has recently been carried through for the case of gˆ = dn in [10]. One can then derive also the particle reflection amplitudes, as was done for the (1) an Toda particles in [8]. The reconstruction of the symmetry algebra from the reflection matrix described in Sect. 2.3 is of wider applicability. For example we can apply it to the diagonal reflection µ µ µ matrices Kθ : Vθ → Vθ found in [11]. This gives the symmetry algebra of the su(n) spin chain on the half-line and is the subject of forthcoming work with Phil Isaac. We expect that there will also be new kinds of boundary conditions in affine Toda field theory which preserve this symmetry algebra, generalizing the boundary condition found in [7]. It will be interesting to find these, in particular as some of the corresponding soliton reflection matrices have already been calculated in the continuum limit of the su(n) spin chain [12]. We have demonstrated that one can obtain solutions of the reflection equation by solving intertwining equations for representations of suitable coideal subalgebras of Uq (g). ˆ This is of great practical importance because the linear intertwining equations are much easier to solve than the reflection equation itself. The interesting mathematical problem therefore now presents itself of classifying all relevant coideal subalgebras of Uq (g) ˆ and their representations. We expect this to lead to a classification of all trigonometric reflection matrices in analogy to the classification of trigonometric R-matrices in terms of representations of quantum affine algebras. The required properties of the coideal subalgebras are that they should be “small enough” so that the intertwiners exist, but also “large enough” so that the tensor product representations are generically irreducible. In the rational case, generators for the relevant coideal subalgebras of the Yangians Y (g) have been constructed in [9]. In that case one has to consider the involutive automorphisms σ of the Lie algebra g which lead to symmetric pairs (g, g σ ). The coideal subalgebra of Y (g) is then a quantization of the corresponding twisted polynomial

Quantum Group Symmetry

189

algebra. We shall denote these algebras Y (g, g σ ) and refer to them as twisted Yangians. Twisted Yangians for g = su(n) have already been described in [27] for g σ = so(n) and g σ = sp(n) and in [26] for g σ = su(m) ⊕ su(n − m) ⊕ u(1). One might therefore hope in the trigonometric case to arrive at a theory of twisted quantized affine algebras Uq (g, ˆ gˆ σ ). In the classical case twisting by an inner automorphism leads to isomorphic algebras, leaving only the known twisted affine algebras based on Dynkin diagram automorphisms. In the quantum case however, where a particular Cartan subalgebra is singled out by the quantization, new algebras arise. In the non-affine case the analogous construction of coideal subalgebras of Uq (g) from involutions has been studied in [23]. The motivation in this case is that they lead to quantum symmetric pairs and thus to quantum symmetric spaces and their associated q-orthogonal polynomials. These algebras include those constructed in [28] by reflection matrix techniques closer to our construction in Sect. 2.3. We will be seeking affine generalizations of these works. Preliminary results from this paper were presented at the 5th Workshop on CFT and integrable models in Bologna in September 2001 and at seminars in Tokyo, Sydney, Brisbane and York in October and November 2001. Acknowledgement. Part of this work was performed during a short stay of GWD at the University of Tokyo supported by the JSPS and the Royal Society. GWD would like to thank Alex Molev, Ruibin Zhang, Mark Gould, and Yao-Zhong Zhang for their hospitality in Sydney and Brisbane. GWD is supported by an EPSRC advanced fellowship.

References 1. Bajnok, Z., Palla, L., Takacs, G.: Boundary states and finite size effects in sine-Gordon model with Neumann boundary condition. Nucl. Phys. B 614, 405 (2001) [arXiv:hep-th/0106069] 2. Bernard, D., Leclair, A.: Quantum group symmetries and nonlocal currents in 2-D QFT. Commun. Math. Phys. 142, 99 (1991) 3. Bowcock, P., Corrigan, E.F., Dorey, P.E., Rietdijk, R.H.: Classically integrable boundary conditions for affine Toda field theories. Nucl. Phys. B 445, 469 (1995) [arXiv:hep-th/9501098] 4. Cherednik, I.V.: Factorizing Particles On A Half Line And Root Systems. Theor. Math. Phys. 61 977 (1984) [Teor. Mat. Fiz. 61, 35 (1984)] 5. Delius, G.W., Gould, M.D., Zhang, Y.Z.: On the construction of trigonometric solutions of the Yang-Baxter equation. Nucl. Phys. B 432, 377 (1994) [arXiv:hep-th/9405030] 6. Delius, G.W.: Restricting affine Toda theory to the half-line. JHEP 9809, 016 (1998) [arXiv:hepth/9807189] 7. Delius, G.W.: Soliton-preserving boundary condition in affine Toda field theories. Phys. Lett. B 444, 217 (1998) [arXiv:hep-th/9809140] 8. Delius, G.W., Gandenberger, G.M.: Particle reflection amplitudes in a(n)(1) Toda field theories. Nucl. Phys. B 554, 325 (1999) [arXiv:hep-th/9904002] 9. Delius, G.W., MacKay, N.J., Short, B.J.: Boundary remnant of Yangian symmetry and the structure of rational reflection matrices. Phys. Lett. B 522, 335–344 (2001) [arXiv:hep-th/0109115] (1) 10. Delius, G.W., George, A.: Quantum affine reflection algebras of type dn and reflection matrices. Submitted to Lett. Math. Phys. [arXiv:math.QA/0208043] 11. de Vega, H.J., Gonzalez Ruiz, A.: Boundary K matrices for the six vertex and the n(2n-1) A(n-1) vertex models. J. Phys. A 26, L519 (1993) [arXiv:hep-th/9211114] 12. Doikou, A., Nepomechie, R.I.: Soliton S matrices for the critical A(1)(N-1) chain. Phys. Lett. B 462, 121 (1999) [arXiv:hep-th/9906069] 13. Faddeev, L.D., Reshetikhin, N.Y., Takhtajan, L.A.: Quantization of lie groups and lie algebras. Lengingrad Math. J. 1, 193 (1990) [Alg. Anal. 1, 178 (1990)] 14. Felder, G., Leclair, A.: Restricted quantum affine symmetry of perturbed minimal conformal models. Int. J. Mod. Phys. A 7, 239 (1992) [arXiv:hep-th/9109019] 15. Gandenberger, G.M., MacKay, N.J.: Exact S matrices for d(N+1)(2) affine Toda solitons and their bound states. Nucl. Phys. B 457, 240 (1995) [arXiv:hep-th/9506169]

190

G.W. Delius, N.J. MacKay (1)

16. Gandenberger, G.M., MacKay, N.J., Watts, G.M.: Twisted algebra R-matrices and S-matrices for bn affine Toda solitons and their bound states. Nucl. Phys. B 465, 329 (1996) [arXiv:hep-th/9509007] 17. Gandenberger, G.M.: New non-diagonal solutions to the a(n)(1) boundary Yang-Baxter equation. arXiv:hep-th/9911178 18. Ghoshal, S., Zamolodchikov, A.B.: Boundary S matrix and boundary state in two-dimensional integrable quantum field theory. Int. J. Mod. Phys. A 9, 3841 (1994) [Erratum-ibid. A 9, 4353 (1994)] [arXiv:hep-th/9306002] 19. Hollowood, T.: Quantizing SL(N) solitons and the Hecke algebra. Int. J. Mod. Phys. A 8, 947 (1993) [arXiv:hep-th/9203076] 20. Jimbo, M.: Quantum R Matrix For The Generalized Toda System. Commun. Math. Phys. 102, 537 (1986) 21. Khoroshkin, S.M., Tolstoy, V.N.: The universal R-matrix for quantum untwisted affine Lie algebras. Funct. Anal. Appl. 26, 69–71 (1992) 22. Klassen, T.R., Melzer, E.: Sine-Gordon not equal to massive Thirring, and related heresies. Int. J. Mod. Phys. A 8, 4131 (1993) [arXiv:hep-th/9206114] 23. Letzter, G.: Coideal Subalgebras and Quantum Symmetric Pairs. [arXiv:math.QA/0103228] 24. Mezincescu, L., Nepomechie, R.I.: Fractional-spin integrals of motion for the boundary sine-Gordon model at the free fermion point. Int. J. Mod. Phys. A 13, 2747 (1998) [arXiv:hep-th/9709078] 25. Mikhailov, A.V., Olshanetsky, M.A., Perelomov, A.M.: Two-Dimensional Generalized toda Lattice. Commun. Math. Phys. 79, 473 (1981) 26. Mintchev, M., Ragoucy, E., Sorba, P., Zaugg, P.: Yangian symmetry in the nonlinear Schroedinger hierarchy. J. Phys. A 32, 5885 (1999) [arXiv:hep-th/9905105] 27. Molev, A., Nazarov, M., Olshansky, G.: Yangians and classical Lie algebras. Russ. Math. Surveys 51, 205 (1996) [arXiv:hep-th/9409025] 28. Noumi, M., Sugitani, T.: Quantum symmetric spaces and related q-orthogonal polynomials. In: Group Theoretical Methods in Physics (Toyonaka, 1994), River Edge, NJ: World Sci. Publishing, pp. 28–40, 1995 [arXiv:math.QA/9503225] 29. Penati, S., Zanon, D.: Quantum integrability in two-dimensional systems with boundary. Phys. Lett. B 358, 63 (1995) [arXiv:hep-th/9501105] 30. Sklyanin, E.K.: Boundary Conditions For Integrable Equations. Funct. Anal. Appl. 21, 164 (1987) [Funkt. Anal. Pril. 21N2, 86 (1987)] 31. Sklyanin, E.K.: Boundary Conditions For Integrable Quantum Systems. J. Phys. A 21, 2375 (1988) 32. Zamolodchikov, A.B.: Integrable Field Theory From Conformal Field Theory. Adv. Stud. Pure Math. 19, 641 (1989) Communicated by L. Takhtajan

Commun. Math. Phys. 233, 313–354 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0735-y

Communications in

Mathematical Physics

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. E-mail: [email protected]; [email protected] Received: 16 May 2001 / Accepted: 16 August 2002 Published online: 10 January 2003 – © Springer-Verlag 2003

Abstract: We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we find a connection with (type I, Riemannian, globally) symmetric spaces G/H : there is a class of integrable BCs in which the boundary field is restricted to lie in a coset of H ; these BCs are parametrized by G/H × G/H ; there are rational solutions of the BYBE in the defining representations of all classical G parametrized by G/H ; and using these we propose boundary S-matrices for the principal chiral model, parametrized by G/H × G/H , which correspond to our boundary conditions.

1. Introduction The bulk principal chiral model (PCM) – that is, the 1 + 1-dimensional, G × G-invariant nonlinear sigma model with target space a compact Lie group G – is known to have a massive spectrum of particles in multiplets which are (sometimes reducible) representations of G × G. These are irreducible representations, however, of the Yangian algebra of non-local conserved charges, and the multiplets are also distinguished by a set of local, commuting conserved charges with spins equal to the exponents of G modulo its Coxeter number. The corresponding bulk scattering (‘S-’) matrices are constructed from G-invariant (rational) solutions of the Yang-Baxter equation (YBE), whose poles determine the couplings between the multiplets. In this paper we investigate the model on the half-line – that is, with a boundary. Any proposed boundary S-matrices must satisfy the boundary Yang-Baxter equation (BYBE), for which only a limited range of solutions is known ([7–9, 11–13] is a selection). For constant solutions of the BYBE (i.e. without dependence on a spectral parameter or rapidity) there is a well-established connection with (quantum) symmetric spaces [35].

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We shall find a class of BYBE solutions corresponding to, and parametrized by, the symmetric spaces G/H . These solutions utilize for the bulk S-matrix the rational1 , G-invariant solution of the (usual, bulk) YBE in the defining (N -dimensional vector) representation of a classical G, and are themselves rational and N -dimensional, describing the scattering of the bulk vector particle off the boundary ground state. We make the ansatz that they are constant or linear in rapidity, and thus have at most two channels. The underlying algebraic structures are the twisted Yangians [36], though the relationship remains to be explored. However, we begin by investigating how our solutions might arise as boundary S-matrices, by discussing the principal chiral field on the half-line and boundary conditions which preserve its integrability (see also [31], and [10] for a more general discussion of boundary integrability). We shall find two classes of BCs which are associated with the G/H . In the first, “chiral” class, the field takes its values at the boundary in a coset G G of H , and the space of such cosets is (up to a discrete ambiguity) H ×H . Correspondingly, we use our BYBE solutions to construct boundary S-matrices, parametrized by G G H × H , which preserve the same remnant of the G × G symmetry as the integrable boundary conditions. In the second, “non-chiral” class, for which we do not generally have corresponding boundary S-matrices, the boundary field lies in a translate of the Cartan immersion of G/H in G. To summarize: a connection between boundary integrability and symmetric spaces emerges naturally in two very different ways: by seeking classically integrable boundary conditions, and by solving the BYBE. The plan of the paper is as follows. In Section Two, building naturally on the results of [3, 4] for the bulk PCM, we discuss boundary conditions which lead naturally to conservation of local charges. As mentioned, there are two classes of BC, which we call “chiral” and “non-chiral”. In Section Three we find minimal boundary S-matrices, by making ans¨atze for the BYBE solutions and applying the conditions of crossing-unitarity, hermitian analyticity and R-matrix unitarity, and explain how these are related to symmetric spaces. This section is necessarily rather long and involved, and many of the details appear in appendices. From these, in Section Four, we construct boundary Smatrices for the PCM, and find that these correspond naturally to the chiral BCs. The key statements of our results for the boundary S-matrices can be found in Sect. 3.4 (for the minimal case, without physical strip poles) and Sect. 4.2 (for the full PCM S-matrices). This paper supersedes the preliminary work of [6].

2. The Principal Chiral Model on the Half-Line 2.1. The principal chiral model on the full line. We first describe the model on the full line, without boundary. This subsection is largely drawn from [4], and full details may be found there. The principal chiral model may be defined by the lagrangian L=

 1
Tr ∂µ g −1 ∂ µ g , 2

(2.1)

where the field g(x µ ) takes values in a compact Lie group G. (We could also include an overall, coupling constant, but this may be absorbed into
, and will not be important 1

before the inclusion of scalar prefactors

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for our purposes.) It has a global GL × GR symmetry g
→ gL ggR−1 associated with conserved currents −1 j (x, t)L µ = ∂µ g g ,

−1 j (x, t)R µ = −g ∂µ g

(2.2)

which take values in the Lie algebra g of G: that is, j = j a t a (for j L or j R : henceforth we drop this superscript) where t a are the generators of g, and (with G compact) Tr(t a t b ) = −δ ab . The equations of motion are ∂ µ jµ (x, t) = 0 ,

∂µ jν − ∂ν jµ − [jµ , jν ] = 0 ,

(2.3)

which may be combined as 1 ∂− j+ = −∂+ j− = − [j+ , j− ] 2

(2.4)

in light-cone coordinates x ± = 21 (t ± x) (and thus ∂± = ∂0 ± ∂1 ). In addition to the usual spatial parity P : x
→ −x, the PCM lagrangian has further involutive discrete symmetries. The first, which we call G-parity and which exchanges L ↔ R, is π : g
→ g −1 ⇒ j L ↔ j R . (2.5) (In the usual QCD effective model, ‘parity’ is the combination P π .) Then there is g
→ α(g) where α is any involutive automorphism, though only for outer automorphisms may this have a non-trivial effect on the invariant tensors and local charges which we shall consider shortly. The canonical Poisson brackets for the model are   j0a (x), j0b (y) = f abc j0c (x) δ(x−y)   j0a (x), j1b (y) = f abc j1c (x) δ(x−y) + δ ab δ  (x−y) (2.6)   j1a (x), j1b (y) = 0 at equal time. These expressions hold for either of the currents j L or j R separately, while the algebra of j L with j R (which we shall not need here) involves only δ  (x−y) terms in the brackets of space- with time-components. This model has two distinct sets of conserved charges, and the two sets commute. The first is the extension of the GL × GR charges to the larger algebra of non-local, Yangian charges [14, 15] Y (gL ) × Y (gR ); we shall not discuss these here. There is also2 an infinite set of local, commuting charges with spins s equal to the exponents of g modulo its Coxeter number,  ∞ q±s = ka1 a2 ···an j±a1 (x)j±a2 (x) · · · j±an (x) dx (2.7) −∞

(where n = s + 1); here, unlike for the Yangian, j L and j R give the same charges (up to a change of sign), of which there is therefore only one set. The primitive invariant tensors k have to be very carefully chosen to ensure the charges commute – for the full story see [4]. Such a set appears to be precisely what is needed for quantum integrability, where it leads to the beautiful structure of masses and interactions described in [16]. 2

In this paper we restrict to the classical g, although they also exist for exceptional g [5].

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2.2. Boundary conditions for the model on the half-line. Varying the bulk action on the half-line −∞ < x ≤ 0 imposes the additional boundary equation Tr(g −1 ∂1 g.g −1 δg) = 0

at x = 0 ,

where the variation is over all δg such that g −1 δg ∈ g. Clearly the Neumann condition ∂1 g|0 = 0 solves this, as does the Dirichlet condition δg|0 = 0, or ∂0 g|0 = 0. But we can also impose mixed conditions, in any way such that (g −1 ∂1 g)a (g −1 ∂0 g)a = 0 (with the usual summation convention). We begin by considering some simple mixed boundary conditions written in terms of the currents j . A little later we shall generalize these, and write them in terms of the fields g. We take as a BC on the currents j+a (0) = R ab j−b (0) ,

(2.8)

with each j chosen independently to be either L or R; we refer to the four possibilities as LL, RR, LR and RL. The boundary equation of motion then requires that R ab be an orthogonal matrix. We would also like consistency with the Poisson brackets: if we extend the currents’ domains to x > 0 by requiring j+a (x) = R ab j−b (−x), then this further requires that R be symmetric, and give an (involutive; α 2 = 1) automorphism α of g via α : t a
→ R ab t b . Together these imply that R is diagonalizable with eigenvalues ±1, so that we may write g =h⊕k , where h and k are the +1 and −1 eigenspaces respectively. The h indices then correspond to Neumann directions j+a = j−a ⇒ j1a = 0, the k indices to Dirichlet directions j+a = −j−a ⇒ j0a = 0 (all at x = 0). Further, R’s being an automorphism implies that [h, h] ⊂ h ,

[h, k] ⊂ k ,

[k, k] ⊂ h ,

precisely the properties required of a symmetric space G/H [17], where H is the subgroup generated by h and invariant under the involution α. For integrability we require a great deal of R. As we have said, there are two infinite sets of charges. The Yangian charges appear no longer to be conserved on the half-line, even with pure Neumann BCs [19] (naively, at least, it seems that there are remnants only, as we shall see). However, we believe they are not essential for integrability, because precisely half of the local charges remain conserved, with either qs + q−s or qs − q−s surviving. Our conjecture is that these are enough to guarantee the properties of quantum integrability, such as factorizability of the S-matrix. The first charge is energy, and its conservation on the half-line is just the equation of motion, requiring R to be orthogonal. For the higher charges we take q|s| = qs ± q−s  0   = ka1 a2 ...an j+a1 ...j+an ± j−a1 ...j−an dx . −∞

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Then 

    ka1 a2 ...an (∂− + ∂1 ) j+a1 ...j+an ± (∂+ − ∂1 ) j−a1 ...j−an dx −∞   = ka1 a2 ...an j+a1 ...j+an ∓ j−a1 ...j−an |x=0
 = kb1 b2 ...bn R a1 b1 ...R an bn ∓ ka1 ...an j−a1 ...j−an ) |x=0 .

d q|s| = dt

0

That this is zero for one choice of sign follows from the result [20] that, for every R and k, kb1 b2 ...bn R a1 b1 ...R an bn = ka1 ...an ,

(2.9)

where  = ±1. (This is obvious, with  = 1, when α is an inner automorphism, but not at all obvious for outer automorphisms.) So, if we now regard α as acting on the currents, α(j±a ) = R ab j±b , we see that  q|s| =

0 −∞

  ka1 a2 ...an j+a1 ...j+an + α(j−a1 )...α(j−an ) dx

(2.10)

is the charge which remains conserved in the presence of the boundary. For LL and RR conditions its density is the combination which is invariant under the combined action P α of spatial parity P (which exchanges j+ ↔ j− ) and α, while for LR and RL it is the combination invariant under these together with G-parity, P απ . Further, these charges still commute. In the Poisson bracket of the charges constructed from tensors k (1) and k (2) , a total derivative term which vanished in the bulk now gives an additional contribution proportional to (all at x = 0)
 (2) as b1 as b1 a1 br br (1) (1) (2) a1 j k ...j j ...j −   j ...j j ...j kca + + + + − − − − 1 ...as cb1 ...br
 (1) (2) d1 a1 ds as e1 b1 er br (1) (2) d1 a1 = kcd1 ...ds kce1 ...er R ...R R ...R −  δ ...δ ds as δ e1 b1 ...δ er br × j−a1 ...j−as j−b1 ...j−br 
(1) d0 a0 d1 a1 e0 b0 e1 b1 (1) (2) d0 a0 d1 a1 e0 b0 e1 b1 = kd0 d1 ...ds ke(2) R ...R R ...−   δ δ ...δ δ ... R 0 e1 ...er × δ a0 b0 j−a1 ... j−br = 0 by property (2.9). Finally, it is precisely the q|s| of (2.10) above that still commute with

0 L/R the G× G-generating charges Q = −∞ j0 dx. 2.2.1. General chiral BCs. As we commented when first introducing the BC (2.8), in j+ = α(j− ) we may take each j as either L or R. The LL and RR conditions are then related: j+L = α(j−L ) ⇒ gj+R g −1 = α(gj−R g −1 ) (all at x = 0). In fact the most general such (we shall call it “chiral”) BC is to take g(0) ∈ kL H kR−1 .

(2.11)

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This is the Dirichlet part of the BC; when we impose the boundary equation-of-motion we supplement it with Neumann conditions within this boundary target space (which we shall henceforth refer to as the D-submanifold) so that the current conditions become
 kL−1 j+L kL = α kL−1 j−L kL , 
(at x = 0). kR−1 j+R kR = α kR−1 j−R kR The constant group elements kL and kR parametrize left- and right-cosets of H in G and may be taken to lie in the Cartan immersion of G/H in G, so that the possible BCs are parametrized by G/H × G/H . (In fact this is true only at the level of the Lie algebras: there is a further discrete ambiguity in the choice of kL , kR . For details of this, and of the Cartan immersion, we refer the reader to Appendices 6.1, 5.1.) Our earlier results about conservation and commutation of charges and consistency with the Poisson brackets still apply (generalized here by twisting the currents with an inner automorphism, which does not change the definition of the conserved charge q|s| ). Note that when kL = kR = e (where e is the identity element in G), we have g(0) ∈ H , the continuous Dirichlet boundary parameters which determine the D-submanifold are all trivial, and the residual symmetry is H × H . For any kL , kR the case H = G corresponds to the pure Neumann condition, while trivial H , the pure Dirichlet condition, is inadmissible for any non-abelian G. We should point out at this stage that we have not succeeded in finding a boundary Lagrangian for any of our mixed BCs. That is, we have no Lagrangian of which the free variation leads to our conditions. The Dirichlet conditions have to be imposed as “clamped” BCs, restricting the boundary variation of g. Let us now examine how much of the G× G symmetry survives. We can see that the BC (2.11) is invariant under kL H kL−1 × kR H kR−1 , and we can check that it is precisely the charges generating this subgroup of G× G which are conserved on the half-line. For

0 the global G-generating charges Qa = −∞ j0a (x) dx (where subscripts either all L or all R are to be understood), consider  0 d
−1  −1 ∂0 j0 dx k k Qk = k dt −∞ = k −1 j1 (0)k  1 −1 = k j+ k − k −1 j− k x=0 2  1 −1 −1 α(k j− k) − k j− k = , x=0 2

which is zero on h (only). 2.2.2. General non-chiral BCs. If we explore similarly the LR and RL conditions, we find that the condition G g(0) ∈ gL gR−1 (2.12) H (where G/H = {α(g)g −1 |g ∈ G} is the Cartan immersion of G/H in G) leads to gL−1 j0L gL = α(gR−1 j0R gR ) (with gL , gR again constant elements of G). If we then apply the boundary equation-of-motion, the condition on the currents becomes

Boundary Scattering, Symmetric Spaces and Principal Chiral Model


 gL−1 j±L gL = α gR−1 j∓R gR

319

at x = 0.

Unlike our chiral BCs (which are parametrized by G/H × G/H ), these BCs are parametrized by a single G (again quotiented by a discrete subgroup; see Appendix 6.2). In the specialization gL = gR , however, the boundary is parametrized by G/H . Note the inversion of the role of the dimension of H in determining the dimension of the D-submanifold, compared to the chiral case: there (and setting kL = kR = e) we had g(0) ∈ H , whereas here (with gL = gR = e) we have g(0) ∈ G/H . The two extreme non-chiral cases give us nothing new: with trivial H we revert to the free, pure Neumann condition, while at the other extreme of H = G we have the pure Dirichlet condition. As with the chiral case, we can check conservation of the generators of the remnant of the G× G symmetry. This time, because the Cartan immersion of G/H is invariant under Hdiag. (the diagonal subgroup g
→ hgh−1 ), the surviving global symmetry is Hdiag. conjugated (in G× G) by (gL , gR ), and we may check that   d
−1 1 −1 L gL QL gL + gR−1 QR gR = gL (j+ − j−L )gL + gR−1 (j+R − j−R )gR x=0 dt 2  1 −1 R −1 R R R gR (j+ − j− )gR − α(gR (j+ − j− )gR ) = x=0 2 which is zero precisely on h × h. At this stage it is worth comparing our results with those obtained in the WessZumino-Witten model – that is, with D-branes on group manifolds. There, initial suggestions of a connection with symmetric spaces [21] were supplanted by an understanding that the D-submanifold is actually a “twisted” or “twined” conjugacy class [22, 23], Cα (g0 ) = {α(g)g0 g −1 |g ∈ G}. This situation arises because in the WZW model there is only one pair of currents, j+L and j−R , and so only one, LR, boundary condition. In our case we have two, LR and RL, conditions, and their interplay further requires that α(g0 ) = g0−1 . But the space M of such g0 is, for the non-Grassmannian cases, precisely √ the Cartan immersion G/H (which is connected to the identity, so that we can find g0 ), and −1

−1

Cα (g0 ) = {α(g)g0 g −1 |g ∈ G} = {α(gg0 2 )(gg0 2 )−1 |g ∈ G} =

G . H

(2.13)

(For the Grasmannian cases, M is a union of disconnected components, the identity-connected component being G/H – see Appendix 5. However, each of the other components is actually a translate of the immersion of G/H  for a different H  [38], so we obtain no new BCs in this way.) The analogous BC in our case is with gL = gR = e, and the residual symmetry is Hdiag. , necessarily preserved by any BC utilizing a twisted conjugacy class, as in the WZW model. Note that the α = 1, H = G case is purely Dirichlet in our case, whereas in the WZW model g0 is unconstrained and there is still freedom at the boundary. We expect that the non-chiral BCs should remain integrable when a Wess-Zumino term (of arbitrary size) is added, and we plan to explore this in future work. Finally we note the relationship of our work to that on the Gross-Neveu model3 [25]. This model has a single global G = O(N ) invariance, broken by the BC to an H = O(M) subgroup. Their boundary S-matrix is then diagonal. 3

not the generalized chiral GN model, which remains to be investigated

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2.3. Remarks on quantization. As with the bulk model, we shall assume that our results carry through into the quantum theory: that classically conserved charges remain conserved in the quantum theory; that charges classically in involution do not develop O(
2 ) anomalies in their commutators; and that our BCs therefore lead to quantum conservation of the charges which generate the residues of the G× G symmetry. All of this leads to the expectation that boundary scattering factorizes, so that solutions of the BYBE provide boundary S-matrices. The only technique which can give evidence for the continued conservation of the local charges after quantization is Goldschmidt-Witten anomaly counting [26]. This was carried out for the bulk case in [3], where for each classical G at least one non-trivial charge was found to be necessarily conserved in the quantum theory. This was extended to boundary models in [24], and used to prove quantum conservation of the spin-3 charge for G = SO(N ) in [6] for one of our BCs. It is simple to check that for all our BCs and for all classical G, each charge which necessarily survives quantization in the bulk model also survives in the presence of the boundary. We do not give details. Finally, the form of our admissible D-submanifolds is not so surprising when we remember that bulk sigma models on symmetric spaces have particularly nice behaviour after quantization, in that the symmetric spaces preserve their shape under renormalization [39]. We would certainly expect our D-submanifolds to behave similarly nicely.

3. The Minimal Boundary S-Matrices In this section we construct boundary S-matrices which are minimal — that is, which have no poles on the physical strip. We follow the method used in the bulk case [29], where minimal S-matrices were found by solving the Yang-Baxter equation and applying unitarity, analyticity and crossing symmetry, and the desired pole structure then implemented using the CDD ambiguity. In the boundary case we seek minimal boundary S-matrices by making ans¨atze to solve the boundary Yang-Baxter equation (BYBE) and applying unitarity, analyticity and the combined crossing-unitarity relation [27]. These minimal solutions will be used to construct PCM boundary S-matrices with the appropriate pole structure in Section Four. We shall make the ansatz that the boundary S-matrix (the “K-matrix”) in the defining, N -dimensional, vector representation of a classical group G is in one of the two forms K1 (θ ) = ρ(θ )E

and

K2 (θ ) =

τ (θ) (I + cθ E) . (1 − cθ)

(3.1)

Here c and E are constants, the latter an N × N matrix; we shall explain the θ -dependent terms below. The crucial point at this stage is the equivalent physical statement that K has at most two “channels”: since its matrix structure is at most linear in rapidity θ , it will decompose into one (for K1 ) or two (for K2 ) projectors. This will prove sufficient to yield a set of solutions related to the symmetric spaces in the following Correspondence. For a given G-invariant bulk factorized S-matrix (i.e. a solution of the YBE) in the defining representation, the K-matrices of the form (3.1) fall into a set of families in 1 − 1 correspondence with the set of symmetric spaces G/H , and each family is parametrized by a space of admissible E which is isomorphic to (possibly a finite multiple of) the corresponding G/H .

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Such solutions, we believe, correspond to scattering off the boundary ground state. We would obtain solutions with many more channels by considering scattering of bulk particles in higher tensor representations or off higher boundary bound states, or both. These can be obtained by fusion from our solutions4 , and the results of this paper thus lay the foundation for future work in this direction. The calculations of this section will necessarily, because case-by-case and exhaustive, be rather involved. Our strategy is to lead the reader through the implications of the BYBE, unitarity, hermitian analyticity and crossing-unitarity, initially culminating in a precise statement of our solutions in Sect. 3.4. Details of the calculations are relegated to Appendices 7 and 8. Then in Sect. 3.5 we explain how our solutions are parametrized by the symmetric spaces; details appear in Appendix 5. 3.1. Calculating the boundary S-matrices. Throughout the rest of this paper we shall use the following (somewhat unconventional) notation for the bulk and boundary S-matrices: k l

Sijkl (θ ) :

θ

j

i

j

i K ij (φ) :

φ

t where θ is the rapidity difference between the two in-coming particles which scatter in the first diagram and φ is the rapidity of the in-coming particle reflecting in the second. We consider the BYBE for two particles in the vector representation np

Sijkl (θ − φ)(Ij m ⊗ K ln (θ ))Smo (θ + φ)(Ioq ⊗ K pr (φ)) pr

= (Iij ⊗ K kl (φ))Sjlnm (θ + φ)(Imo ⊗ K np (θ ))Soq (θ − φ).

(3.2)

(θ and φ are now the rapidities of the two particles.) We attempt to find solutions of the form (equivalent to (3.1))    h − + K1 (θ ) = ρ(θ )E , (3.3) and K2 (θ ) = τ (θ) P − P ciπ where ρ(θ ) and τ (θ ) are scalar prefactors, h is the dual Coxeter number of the group, c is a parameter and 4

with the exception of the SO(N) spinorial multiplets

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N.J. MacKay, B.J. Short

P± =

1 (I ± E) 2

[x] =

θ+ θ−

iπx h iπx h

,

where I is the identity matrix and E is a general square matrix of the same size5 . We find that constraints are imposed on the scalar prefactors and the matrix E by the BYBE and unitarity, analyticity and crossing-unitarity. These constraints are in general dependent on the choice of classical group, SU (N ), SO(N ) or Sp(N ), but in all cases involving K2 (θ ) we find that E 2 = 1, so that P ± are projectors, as the notation suggests. We note that the constraints imposed on the scalar prefactors will allow us to find them only up to the usual CDD ambiguity. This freedom is then further restricted by taking the K-matrices to be minimal – that is, taking ρ(θ ) and τ (θ) to have no poles on the physical strip. We shall also require τ (θ ) to have a zero at θ = 1c so that K2 (θ ) is finite at this point, since we shall find that 1c can lie in the physical strip. For the case of SU (N ) the vector representation is not self-conjugate. This allows us to consider the situation where a particle scattering off the boundary returns as an ¯ anti-particle (for an analogous situation see [11]). In this case the K-matrix, K i j (θ ), must satisfy a version of the BYBE which for convenience we shall refer to as the ‘conjugated’ BYBE, np ¯

Sijkl (θ − φ)(Ij m ⊗ K l n¯ (θ ))Smo¯ (θ + φ)(Io¯ q¯ ⊗ K pr¯ (φ)) ¯

¯

p¯ r¯

= (Iij ⊗ K k l (φ))Sjlnm¯ (θ + φ)(Im¯ o¯ ⊗ K np¯ (θ ))So¯ q¯ (θ − φ) .

(3.4)

We shall find that only K1 (θ ) gives a solution of the conjugated BYBE. Analyticity and unitarity become more subtle in this case. At this point we introduce the diagrammatic algebra used in the calculations we perform. We represent the matrices I , J and E, where J is the symplectic form matrix, in the following way:  I: Jt: . J: E: (Thus from the properties of J we have = −( ) and = = Matrix multiplication is achieved by concatenation of the diagrams, for example J I J t E:

=

.)

=

Using this diagrammatic algebra we can rewrite the K-matrices of (3.1) as  τ (θ ) K1 (θ ) = ρ(θ ) . K2 (θ ) = + cθ (1 − cθ)

(3.5)

This diagrammatic description is used in the calculations presented in the appendices, and makes them much clearer. We substitute either of these K-matrices into the BYBE, taking the minimal Smatrix to be that of the bulk PCM for a particular G, derived in [29] up to a few minor inconsistencies which we have corrected. This yields constraints (dependent on G and on whether we take K1 or K2 ) that E must satisfy in order for Ki to be a solution of the BYBE. In fact the constraints imposed by unitarity, analyticity and crossing-unitarity on the Ki are more restrictive than the BYBE constraints, which we do not, therefore, 5

The apparently circuitous involvement of iπ/ h ensures that x is an integer, as is usual in the literature.

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323

list here. (The reader is referred to Appendix 7.1 for details of the BYBE calculations.) Instead we go on to consider unitarity, analyticity and crossing-unitarity. 3.2. Unitarity and Analyticity 3.2.1. The non-conjugating cases. For the non-conjugating cases the K-matrices are required to satisfy the conditions of unitarity [27] and hermitian analyticity [28]  † K(θ )K(−θ ) = I, K(θ) = K(−θ ∗ ) . (3.6) Substituting K1 we obtain =

ρ(θ )ρ(−θ )

= ρ(−θ ∗ )∗ (

ρ(θ )

)† .

These matrix equations are equivalent to ρ(θ )ρ(−θ ) = =α

1 , α

ρ(θ ) = βρ(−θ ∗ )∗ ,

,

)† = β

(

,

where α and β are constants with β ∈ U (1). Recalling K1 (θ ), we see that we have the freedom in our definition of ρ(θ ) and constraints imposed on the matrix (

)† =

to set β = 1 and ensure α ∈ U (1), so that the by unitarity and analyticity become =α

and

Similarly for K2 we obtain τ (θ )τ (−θ ) (1 − c2 θ 2 )  + cθ

τ (θ ) (1 − cθ ) These are equivalent to

where α ∈ U (1).

(3.7)





=

− c2 θ 2

τ (−θ ∗ )∗ = (1 + c∗ θ)

(1 − c2 θ 2 ) , (1 − γ c2 θ 2 ) τ (θ ) τ (−θ ∗ )∗ = , (1 − cθ ) (1 + c∗ θ )

τ (θ )τ (−θ ) =



, ∗

− c θ(

=γ c

= −c∗ (

† ) .

, )† .

If we consider the expression for K2 (θ ), we see that we have the freedom to set c to be purely imaginary and choose γ ∈ U (1). Then the constraints on the matrix (

)† =

and

=γ

become

where γ ∈ U (1)

as was the case for K1 . In fact we find that the parameters α and γ cannot be freely chosen in U (1); the only choice consistent with the hermiticity of

is that they are

324

N.J. MacKay, B.J. Short

equal to 1 (see Appendix 8.1). Consequently we find that for both K1 and K2 , unitarity and analyticity impose (

)† =

and

=

(3.8)

.

The corresponding constraints imposed on ρ(θ ) and τ (θ) are ρ(θ ) = ρ(−θ ∗ )∗ , τ (θ ) = τ (−θ ∗ )∗ ,

ρ(θ )ρ(−θ) = 1, τ (θ )τ (−θ) = 1 .

(3.9) (3.10)

3.2.2. SU (N )-conjugating. In the case of the conjugated BYBE (3.4), which we call “SU (N )-conjugating”, unitarity and analyticity can no longer be applied as straightforwardly. The reason is that we are no longer dealing with a K-matrix that is an endomorphism of the vector representation space V , but rather with K(θ) : V → V¯ . In order to apply our conditions we must introduce K  (θ ) : V¯ → V and consider the space V ⊕ V¯ on which the endomorphism  0 K  (θ ) ˜ )= K(θ acts. K(θ ) 0 ˜ We can then apply analyticity and unitarity to K(θ) which yields K(θ ) = K  (−θ ∗ )† ,

K(θ )K  (−θ) = I .

(3.11)

The conjugated BYBE that K(θ ) and K  (θ ) must satisfy allows only the K1 (θ ) form for both. Thus we have K(θ ) = ρ(θ )

and

K  (θ ) = ω(θ)

,

= E and where ρ(θ ) and ω(θ ) are scalar prefactors, whilst matrices. Substituting these into our conditions yields ρ(θ ) = αω(−θ ∗ )∗ , α

=(

ρ(θ )ω(−θ) =

)† ,

=β

≡ F are constant 1 , β

(3.12)

.

We relate these via charge conjugation, whose action on the vector multiplets we represent as Cu = u¯ (where u¯ is the antiparticle of u – note that ¯ does not simply denote complex conjugation). Of course charge conjugation is self-inverse, and its action on K(θ ) : V → V¯ gives CK(θ )C, which maps V¯ → V , as does K  (θ ), and so the two are related by some change of basis. That is, K  (θ ) = ACK(θ )CB −1 , where A is the change-of-basis matrix for V and B is that for V¯ . We note that B is related to A by B = C −1 AC (since we require that charge conjugation still be represented by C after the change of basis), so that K  (θ ) = ACK(θ )A−1 C.

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

325

Substituting for K(θ ) and K  (θ ) gives ω(θ )F = ACρ(θ )EA−1 C which we split up as γ F = ACEA−1 C.

ω(θ ) = γρ(θ ),

Taking the determinant of the matrix equation we have γ N det F = det E

since det C = ±1 .

(3.13)

We now have a set of three pairs of constraints (3.12, 3.13), with each pair containing one constraint relating the matrices E and F , and another relating the scalar prefactors ρ(θ ) and ω(θ). Each pair of constraints contains an arbitrary complex parameter, α, β or γ , but we find that these parameters are related so that the total freedom is equivalent to only two complex parameters. This is exactly the freedom we have in splitting K(θ) and K  (θ ) into scalar prefactors and matrix parts, so that we can consistently set all three parameters to 1 (details in Appendix 8.2). This leaves us with E = F†

and

ρ(θ ) = ω(−θ ∗ )∗ ,

EF = I

and

ρ(θ)ω(−θ) = 1,

and

ρ(θ ) = ω(θ) .

det F = det E

(3.14)

Now F and ω(θ ) are completely fixed by E and ρ(θ ), and the boundary S-matrix for V ⊕ V¯ is  † 0 E ˜ K(θ ) = ρ(θ ) , (3.15) E 0 subject to the following constraints, derived from those above: E†E = I ∗ ∗

ρ(θ ) = ρ(−θ )

and

det E = ±1,

and

ρ(θ )ρ(−θ) = 1 .

(3.16)

3.3. Crossing-Unitarity. The boundary S-matrices must also satisfy crossing-unitarity [27], which in our notation is   iπ iπ K ij − θ = Sjil¯k¯ (2θ)K lk +θ . (3.17) 2 2 The crossed version of this, which the SU (N )-conjugating K-matrix must satisfy, is   i j¯ iπ il l k¯ iπ K − θ = Sj k (2θ)K +θ . (3.18) 2 2 Substituting the Ki into the relevant version of the crossing-unitarity equation gives constraints on

and the scalar prefactors. We tabulate these below (details in Appendix

7.2). Note that in our diagrammatic algebra T r(E) is represented by the symbol

.

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N.J. MacKay, B.J. Short

Group

K1 (θ )

SU (N ) ρ(iπ/2−θ) ρ(iπ/2+θ)

= 0, = σ¯ u (2θ )

K2 (θ ) τ (iπ/2−θ) τ (iπ/2+θ)

= 0 and either

SO(N )

)t = , = σo (2θ )

(

ρ(iπ/2−θ) ρ(iπ/2+θ)

c = 2ih , h  h π h = 2 iπc − 2 σ¯ u (2θ) c

τ (iπ/2−θ) τ (iπ/2+θ)

=

2ih π ,

( )t = ,    h = h2 iπc − h2 σo (2θ)

or

)t

(

ρ(iπ/2−θ) ρ(iπ/2+θ)

= −( ), = [1]σo (2θ )

= 0 and either

Sp(2n)

)t

(

ρ(iπ/2−θ) ρ(iπ/2+θ)

= , = σp (2θ )

c τ (iπ/2−θ) τ (iπ/2+θ)

=

2ih π ,

( )t = −( ),    h = [1] h2 iπc − h2 σp (2θ)

or

)t

(

ρ(iπ/2−θ) ρ(iπ/2+θ)

SU (N ) conjugated

= −( ), = [1]σp (2θ ) )t = = σu (2θ )

(

no solution

ρ(iπ/2−θ) ρ(iπ/2+θ)

or

(

)t = −( ), = −[1]σu (2θ )

ρ(iπ/2−θ) ρ(iπ/2+θ)

The functions σ and σ¯ are scalar prefactors for the bulk S-matrices (see Appendix 7).

3.4. The boundary S-matrices. We have obtained a series of constraints on and on ρ(θ ) and τ (θ ) which must be satisfied if the proposed K1 (θ ) and K2 (θ ) are to be boundary S-matrices. The constraints on the scalar prefactors enable us to determine them exactly for each G, providing we assume some extra minimality conditions, namely that they should be meromorphic functions of θ with no poles on the physical strip Im θ ∈ [0, π2 ]. Having calculated the scalar prefactors (we do not include details of the calculations, as the reader can simply check the results if required) we obtain the boundary S-matrices below. We note that in the case of K2 (θ ) there is also the possibility that the pole at θ = 1c lies on the physical strip and so the scalar prefactor τ (θ ) may be required to have a zero at this point. In fact we shall find that one of ± 1c always lies on the physical strip, and the expressions given below for K2 (θ ), for each case, are valid when 1c lies on the physical strip. When − 1c lies on the physical strip instead, the correct expressions for the minimal K2 -matrices can be obtained by the interchange c ↔ −c,

↔−

.

(Note that this leaves c unchanged.) In Sect. 4 we shall add CDD factors making the PCM boundary S-matrices invariant under this interchange.

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

327

3.4.1. SU (N ). The minimal boundary S-matrices for SU (N ) are

K1 (θ ) =



θ  2iπ −θ  2iπ



+

1 2 1 2

+

+ +





1 −θ 2h  2iπ  1 θ 2h  2iπ

+ +



1 2 1 2

= 0,

with

(3.19)

where  is the gamma function, and  θ + −1  2iπ K2 (θ ) =  −θ (1 − cθ )  2iπ + ×( with c

=

+ cθ

1 2 1 2

+ +









1 −θ θ 2h  2iπ  2iπ   1 θ −θ 2h  2iπ  2iπ

+ +





1 −θ 2iπc  2iπ  1 θ 2iπc  2iπ

+

1 2 1 2

+

+ +



1 2iπc 1 2iπc

(3.20)

)

2ih π .

(Note that in the limit c → ∞, with c

fixed, K2 (θ ) → K1 (θ ), as we would expect.)

3.4.2. SO(N ). There are two types of minimal boundary S-matrix of the form K1 (θ ) for SO(N ), one symmetric and the other antisymmetric, as well as the minimal solution of the form K2 (θ ). (This last was investigated in [8] and, for M = 1, in [34].) They are K1 (θ ) =

with (

)t =

K1 (θ ) =

with (



θ  2iπ −θ  2iπ



)t

+







3 −θ 4  2iπ  3 θ 4  2iπ

+ +





1 θ 2  2iπ  1 −θ 2  2iπ

+ +

1 2 1 2

+

1 2 1 2

+

+



+ +





1 −θ 4  2iπ  1 θ 4  2iπ

 ), ⇒

= −(

+ +





1 θ 2  2iπ  1 −θ 2  2iπ

+

)t =

+ +

+

+

,c

=





1 −θ 2h  2iπ  1 θ 2h  2iπ

1 4 1 4

+

3 4 3 4

+

+



1 2h 1 2h

(3.21)

+ +

+



1 2h 1 2h

(3.22)

= 0 , and

 θ  −θ  θ  + 43  2iπ + −1  2iπ K2 (θ ) =  θ  2iπ  −θ 3 −θ (1 − cθ )  2iπ + 4  2iπ  2iπ +  θ  −θ 1 1  2iπ  2iπ + 21 + 2iπc + 2iπc ×  −θ  θ ( 1 1  2iπ + 21 + 2iπc  2iπ + 2iπc with (



1 −θ 2h  2iπ  1 θ 2h  2iπ

= 0,

,

θ  2iπ −θ  2iπ



+

1 2 1 2

+ +





1 −θ 2h  2iπ  1 θ 2h  2iπ

+ cθ

)

+ +

1 4 1 4

+ +



1 2h 1 2h

(3.23)

2ih π .

(Note that K2 (θ ) is symmetric, so that in the limit c → ∞, with c to the symmetric K1 (θ ), again as expected.)

fixed, K2 (θ ) tends

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N.J. MacKay, B.J. Short

3.4.3. Sp(N ). For Sp(N ) there are again two minimal solutions of the form K1 (θ ), as well as the minimal K2 (θ ) S-matrix,  −θ  θ  −θ  θ 1 1 + 41  2iπ + 21  2iπ + 21 + 2h  2iπ + 41 + 2h  2iπ K1 (θ ) =  −θ (3.24)  θ  −θ  θ 1 1  2iπ + 41  2iπ + 21  2iπ + 21 + 2h  2iπ + 41 + 2h  t with ( ) = , ⇒ =0 , K1 (θ ) = with



+

θ  2iπ −θ  2iπ



−1 K2 (θ ) = (1 − cθ )  θ  2iπ ×  −θ  2iπ with (The with c





+



1 θ 2  2iπ  1 −θ 2  2iπ

+

+ +

= 0, and 

),

1 2 1 2

+ +



=

), c

matrix in K2 (θ ) satisfies



1 −θ 2h  2iπ  1 θ 2h  2iπ



θ −θ θ  2iπ  2iπ + 43  2iπ +    −θ 3 θ −θ  2iπ + 4  2iπ  2iπ +  −θ 1 1  2iπ + 21 + 2iπc + 2iπc  θ ( 1 1  2iπ + 21 + 2iπc + 2iπc

= −(

)t

(



= −(

)t

(

+



3 −θ 4  2iπ  3 θ 4  2iπ

1 2 1 2

+ +

+ +

3 4 3 4

+ +



1 −θ 2h  2iπ  1 θ 2h  2iπ

+ cθ



1 2h 1 2h

+ +

(3.25)

3 4 3 4

+ +



1 2h 1 2h

(3.26)

)

2ih π .

)t

(

= −(

) and we find that as c → ∞,

fixed, K2 (θ ) tends to the K1 (θ ) with this property, as expected.)

3.4.4. SU (N )-conjugating. Here it is only K1 (θ ) that provides valid boundary S-matrices. There are two minimal possibilities, with symmetric and antisymmetric , K1 (θ ) = K1 (θ ) =



θ  2iπ −θ  2iπ





θ  2iπ −θ  2iπ



+ + + +





1 −θ 4  2iπ  1 θ 4  2iπ





1 −θ 4  2iπ  1 θ 4  2iπ

+ + + +

1 4 1 4 3 4 3 4



+

1 2h 1 2h

+ + +

with (

)t =

with (

)t = −(

, and

(3.27)



1 2h 1 2h

).

(3.28)

We have been unable to make contact between our solutions and a rational limit of the trigonometric solutions in [11]. 3.5. Constraints on E: The symmetric-space correspondence. We now turn our attention to the constraints imposed on the matrices E. We recall that E† = E

and

E2 = I

(3.29)

were to be imposed in all cases, except that of SU (N )-conjugating, due to unitarity and analyticity. For the case of SU (N )-conjugating (3.29) is replaced by E†E = I

and

det E = ±1 .

The further constraints particular to the different groups were

(3.30)

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

Group SU (N )

K1 (θ ) T r(E) = 0

SO(N )

T r(E) = 0 and E t = ±E T r(E) = 0 and J E t J = ±E E t = ±E

Sp(2n) SU (N )

K2 (θ ) c T r(E) =

329

2ih π 2ih π ,

c T r(E) = Et = E c T r(E) = 2ih π , J E t J = −E no solution

conjugated

3.5.1. SU (N ). We begin by considering SU (N ), where in addition to the constraints imposed by unitarity and analyticity we have a single extra constraint on the trace of E. From the first two constraints we can express E as the conjugate of a diagonal matrix X by an SU (N ) matrix, E = Q† XQ where X is of the form

 X=

with Q ∈ SU (N ),

IM 0 0 −IN−M

,

IM is the M × M identity matrix and 0 ≤ M ≤ N (see Appendix 5.2.1). By the cyclicity of trace, if we impose the trace condition for K1 then N must be even and equal to 2M. If we impose the condition for K2 we obtain a condition on c, and so find  K1 (θ ) † IN/2 0 Q1 E = Q1 0 −IN/2

† E = Q2



K 2 (θ ) IM 0 Q2 0 −IN−M

2ih where c = π(2M−N)

with Qi ∈ SU (N ). We can see that the case K1 corresponds to the limit of K2 in which we take M = N2 , and hence to c → ∞, as we would expect. Thus we have parametrized the possibilities for E with a matrix Q ∈ SU (N ) and an integer M. Once M is fixed, the suitable E form a space isomorphic to the symmetric space SU (N ) , S(U (M) × U (N − M)) where the correspondence is between an element E = Q† XQ and the left coset of H = S(U (M) × U (N − M)) by Q. In the same way the possible E for K1 (θ ) form a space isomorphic to SU (N ) S(U (N/2) × U (N/2))

(only for N even).

3.5.2. SO(N ). We now consider SO(N ), where in addition to the constraints associated with SU (N ) we also have E t = ±E. We consider first the case E t = E: E † = E , E t = E ⇒ E ∗ = E.

330

N.J. MacKay, B.J. Short

So E is a symmetric real matrix, and we can diagonalize it by conjugating with a matrix R ∈ SO(N ). Since E squares to the identity the diagonal matrix must be of the form  IM 0 X= . 0 −IN−M Then imposing the constraints on the trace of E is the same as for SU (N ) and we have K 2 (θ )  K1 (θ )  I I 0 0 N/2 M 2ih R1 E = R2t R2 where c = π(2M−N) E = R1t 0 −IN/2 0 −IN−M with Ri ∈ SO(N ). In a similar way to the SU (N ) case, once M is fixed, the space of matrices E is isomorphic to SO(N ) , S(O(M) × O(N − M)) with the E for K1 (θ ) isomorphic to SO(N ) . S(O(N/2) × O(N/2)) Thus, in the same way as for SU (N ), we have an isomorphism between the space of allowed E and the symmetric spaces (see Appendix 5.2.2 for more details). The remaining case to consider is that of the antisymmetric K1 (θ ). We find in this case (see Appendix 5.2.6) that the matrices E form a space isomorphic to two copies of the symmetric space SO(N ) . U (N/2) 3.5.3. Sp(N ). In addition to the SU (N ) constraints, for Sp(N ) we also have J E t J = ±E. We consider first the case J E t J = −E, with E2 = I

and

J E t J = −E ⇒ J E t J E = −I ⇒ E t J E = J

so that, since we also know E ∈ U (N ), we must have E ∈ Sp(N ). After appealing to an argument involving quarternionic matrices (Appendix 5.2.3) we find that the space of allowed E for K2 is isomorphic to Sp(N ) . Sp(M) × Sp(N − M) In the case of K1 (θ ) we again require M =

N 2

and the E-space is isomorphic to

Sp(N ) . Sp(N/2) × Sp(N/2) For K1 (θ ) with J E t J = +E, E is conjugate over C to  IN/2 0 0 −IN/2 as T r(E) = 0. We find (see Appendix 5.2.7) that the allowed E form a space isomorphic to Sp(N ) . U (N/2)

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

331

3.5.4. SU (N )-conjugating. The last case to consider is that of SU (N )-conjugating. We recall that in this case the constraints due to unitarity and analyticity were slightly modified, to E†E = I

and

det E = ±1 .

Crossing-unitarity imposed E t = ±E. Taking the symmetric case first, the allowed E form a set {E|E † E = I, E t = E, det E = ±1} which turns out to be isomorphic to {1, ω} ×

SU (N ) SO(N )

where any ω s.t. ωN = −1 is chosen.

That is, the sets {E|E † E = I, E t = E, det E = 1} and {E|E † E = I, E t = E, det E = −1} are both separately isomorphic to SU (N ) SO(N ) and so their union produces two copies of the symmetric space (see Appendix 5.2.4). Lastly, we turn to the antisymmetric case {E|E † E = I, E t = −E, det E = ±1} . This we find (Appendix 5.2.5) is isomorphic to {1, ω, ω2 , ω3 } ×

SU (N ) . Sp(N )

4. The PCM Boundary S-Matrices In this section we construct the boundary S-matrices for the principal chiral model. We recall [29] that the bulk model S-matrix has G × G symmetry and is constructed as SP CM (θ ) = X11 (θ ) (SL (θ ) ⊗ SR (θ )) ,

(4.1)

where X11 (θ ) is the CDD factor for the PCM and SL,R (θ ) are left and right copies of the minimal S-matrix possessing G-symmetry. Following this prescription, we shall use the minimal K-matrices from the previous section to construct boundary Smatrices for the PCM on a half-line. We then go on to explore their symmetries and make connection with the classical results of Sect. 2.

332

N.J. MacKay, B.J. Short

4.1. The CDD factors. Introducing the CDD factor, X11 (θ ), into the bulk S-matrix for the PCM requires that we introduce an extra factor, Y11 (θ ) (or Y11¯ (θ ) in the case of SU (N ) conjugating), into the boundary S-matrix in order to satisfy crossing-unitarity. We construct the boundary S-matrix for the PCM as KP CM (θ ) = Y11 (θ ) (KL (θ ) ⊗ KR (θ )) ,

(4.2)

where KL,R (θ ) are left and right copies of the same type of minimal K-matrix, chosen from among the possibilities classified in Section Three. In order that KP CM (θ ) satisfy the crossing-unitarity equation with SP CM (θ ) we require   iπ Y11 ( iπ Y − θ ) ( − θ ) ¯ 2 2 or 11 iπ = X11¯ (2θ ) = X11 (2θ) for SU (N ) conjugating . Y11 ( iπ + θ ) Y ( ¯ 11 2 + θ ) 2 (4.3) Note that in the cases of SO(N ) and Sp(N ), for which the particles are self-conjugate, X11¯ (θ ) = X11 (θ ). The CDD factors for the bulk PCM S-matrices are X11 (θ ) = (2)θ = X11¯ (iπ − θ ) X11 (θ ) = −(2)θ (h − 2)θ X11 (θ ) = −(2)θ (h − 2)θ where

(x)θ =

sinh ( θ2 +

SU (N ), SO(N ), Sp(N ), iπx 2h ) iπx 2h )

sinh ( θ2 − We find [30] the following candidates for the Y functions: Y11 (θ ) = (1 − h)θ   h h Y11 (θ ) = +2 + 1 (1 − h)θ 2 θ 2 θ   h h +2 + 1 (1 − h)θ Y11 (θ ) = 2 2 θ  θ  h h +2 +1 Y11¯ (θ ) = 2 θ 2 θ

(4.4)

.

SU (N ), SO(N ), (4.5) Sp(N ), SU (N ) conjugated,

and note that none of these factors have poles on the physical strip. We still have freedom to multiply by an arbitrary boundary CDD factor. That is, we can replace any of the above Y (θ ) factors by g(θ)Y (θ), where g(θ) is a CDD factor. This allows us to introduce simple poles into the PCM boundary S-matrices. In the case where we construct a PCM S-matrix using left and right copies of K2 (θ ), we wish to introduce the CDD factor6   h h h− (4.6) g(θ ) = − ciπ θ ciπ θ which gives a simple pole at θ = 1c , corresponding to the formation of a boundary bound state. We are assuming, as in Sect. 3.4, that M ≤ N2 so that 1c is on the physical strip. As stated in Sect. 3.4, the resulting PCM boundary S-matrix will possess a c ↔ −c symmetry and so will be correct for M ≥ N2 also. 6

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

333

4.2. The boundary S-matrices. We now list the full PCM boundary S-matrices for the various G. We make use of the relation π (z)(1 − z) = . sin (π z) We will also require the scalar factors  θ  −θ 1  2iπ + 21 + 2h  2iπ + 21 η(θ) =  −θ  θ , 1  2iπ + 21 + 2h  2iπ + 21

ν(θ ) =



θ  2iπ −θ  2iπ



+ +

1 2 1 2

+ +







1 −θ 2h  2iπ  1 θ 2h  2iπ

,

µ(θ ) =

 −θ  θ  θ  −θ 1 1 1 1  2iπ + 21 − 2iπc  2iπ + 2iπc  2iπ − 2iπc  2iπ + 21 + 2iπc −1  θ  −θ  −θ ,  θ 1 1 1 1 4π 2 c2  2iπ  2iπ + 21 − 2iπc  2iπ + 1 + 2iπc  2iπ + 1 − 2iπc + 21 + 2iπc  θ  −θ 1  2iπ + n4  2iπ + m4 + 2h 2ih and n,m (θ ) = where c =  −θ  θ . 1 π(2M − N )  2iπ + n4  2iπ + m4 + 2h The PCM S-matrices can 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 η(θ )EL ⊗ η(θ )ER , where7 EL/R ∈ and

SU (N ) , S(U (N/2) × U (N/2))


 KP CM (θ ) = (1 − h)θ µ(θ) ν(θ )(I + cθ EL ) ⊗ ν(θ )(I + cθ ER ) ,

where EL/R ∈

(4.7)

(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 KP CM (θ ) = +2 + 1 (1−h)θ η(θ )1,3 (θ )EL ⊗η(θ )1,3 (θ )ER , (4.9) 2 θ 2 θ where8

SO(N ) × {+1, −1} , U (N/2)  
 h h KP CM (θ ) = +2 + 1 (1 − h)θ η(θ )3,1 (θ )EL ⊗ η(θ )3,1 (θ )ER , 2 θ 2 θ (4.10) EL/R ∈

7 We are using the symmetric space notation G/H here to denote the relevant translated Cartan immersion. Details are given in Appendix 5. 8 The factor {+1, −1} indicates that the space containing E L/R is a twofold copy of the symmetric space – no group structure is implied. See Appendix 5.2.6 for details.

334

N.J. MacKay, B.J. Short

where EL/R ∈ and

SO(N ) , S(O(N/2) × O(N/2))

 h h +2 +1 KP CM (θ ) = 2 θ 2
θ  × (1 − h)θ µ(θ) ν(θ )3,1 (θ )(I + cθ EL ) ⊗ ν(θ)3,1 (θ )(I + cθ ER ) , (4.11) where SO(N ) EL/R ∈ . S(O(N − M) × O(M)) 

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 η(θ )1,1 (θ )EL ⊗ η(θ )1,1 (θ )ER , KP CM (θ ) = 2 θ 2 θ (4.12) where Sp(N ) EL/R ∈ , U (N/2)  
 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 (4.13) where Sp(N ) EL/R ∈ , Sp(N/2) × Sp(N/2)) and   h h +2 +1 KP CM (θ ) = 2 θ 2
θ  × (1 − h)θ µ(θ) ν(θ )3,3 (θ )(I + cθ EL ) ⊗ ν(θ )3,3 (θ )(I + cθ ER ) , (4.14) where Sp(N ) EL/R ∈ . Sp(N − M) × Sp(M)) 4.2.4. SU (N )-conjugating. Lastly, we have found two types of representation-conjugating boundary S-matrix for SU (N ) 
  h h KP CM (θ ) = 1,1 (θ )EL ⊗ 1,1 (θ )ER , (4.15) +2 +1 2 θ 2 θ where EL/R ∈ {1, ω} × 

and KP CM (θ ) =

h +2 2

 θ

SU (N ) SO(N )

(ωN = −1) ,


 h 1,3 (θ )EL ⊗ 1,3 (θ )ER , +1 2 θ

(4.16)

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

335

where EL/R ∈ {1, ω, ω2 , ω3 } ×

SU (N ) . Sp(N )

4.3. Symmetries of the PCM boundary S-matrices. We now consider the symmetries possessed by the PCM boundary S-matrices. Before looking at the surviving group symmetries at the boundary, we first point out a symmetry possessed by those S-matrices constructed from the K2 (θ )-type minimal solution. 4.3.1. M ↔ N − M symmetry. This was first noted, for SU (N ) diagonal boundary scattering, in [33]. The Grassmannian symmetric spaces SU (N ) S(U (M) × U (N − M))

SO(N ) S(O(M) × O(N − M))

Sp(N ) Sp(M) × Sp(N − M)

are all invariant under M ↔ N − M. Consequently, we might expect that the PCM boundary S-matrices constructed using matrices EL/R lying in translated Cartan constructions of these symmetric spaces would also respect this symmetry. This is exactly what we find for the KP CM (θ ) matrices (4.8), (4.11) and (4.14). To see this invariance, we consider how the exchange M ↔ N − M affects the degrees of freedom in the K-matrices. The matrices EL/R are constructed as −1 EL/R = UL/R XUL/R ,

 X=

IM 0 0 −IN−M

where UL/R ∈ SU (N ), SO(N ) or Sp(N ) and

,

X→
Xˇ =

so under M ↔ N − M



(4.17)

IN−M 0 0 −IM

.

ˇ −1 . Taking traces we see that Thus under M ↔ N − M, EL/R
→ Eˇ L/R = UL/R XU L/R c ↔ −c under the exchange. Now note that the scalar factor µ(θ) is invariant under c ↔ −c, and that no other scalar factor depends on c. So KP CM (θ )
→ Kˇ P CM (θ ), where (I + cθ EL/R )
→ (I − cθ Eˇ L/R ) . Now

ˇ −1 = UL/R OXO −1 U −1 −Eˇ L/R = UL/R (−X)U L/R L/R

where

 O=

0 IN−M ±IM 0

.

We choose the sign ± to ensure that det O = 1 and (noting that when N and M are even O ∈ Sp(N )) we see that UL/R ∈ G = SU (N ), SO(N ) or Sp(N ) ⇒ (UL/R O) ∈ G. Thus, if we denote by KP CM (θ ; UL , UR ) the matrix constructed using −1 EL/R = UL/R XUL/R

(4.18)

and by Kˇ P CM (θ ; UL , UR ) the image of this under the exchange M ↔ N − M, we have Kˇ P CM (θ ; UL , UR ) = KP CM (θ ; UL O, UR O) .

(4.19)

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N.J. MacKay, B.J. Short

So we see that the KP CM (θ ) matrices do respect this invariance of the symmetric spaces, in the sense that the action of the exchange M ↔ N − M on the K-matrices is simply a translation in the parameter space. A further consequence of this emerges if we consider the pole structure of these K-matrices. There is exactly one simple pole on the physical strip at either θ = 1c or θ = − 1c (since M = N2 ). If we interpret the simple pole as the formation of a boundary bound state at this rapidity, then the bound state is in a representation projected onto by either PL+ ⊗ PR+ or PL− ⊗ PR− , respectively, where ± = PL/R

1 (I ± EL/R ) = UL/R 2



1 −1 . (I ± X) UL/R 2

(4.20)

We note 1 (I + X) = 2



IM 0 0 0

and

1 (I − X) = 2



0 0 0 IN−M

.

We find that ± 1c lies on the physical strip as M ≶ N2 , and so the boundary bound state representation is always the smaller of the two projection spaces. We plan to investigate further the spectrum of boundary bound states in future work, but for the moment we return to consider the surviving remnant of group symmetry at the boundary and make connections with the classical boundary conditions of Section Two. 4.3.2. Boundary group symmetry in the non-conjugating cases. We recall [29] that the principal chiral model in the bulk possesses a global G × G symmetry, respected by the bulk S-matrices. In Section Two we saw that the introduction of a boundary in the classical PCM generally breaks the G×G symmetry, so that only a remnant survives, the nature of which is dictated by the boundary condition. In particular we saw in Sect. 2.2.1 that the boundary condition (2.11), g(0) ∈ kL H kR−1

where kL/R parametrize left/right cosets of H ∈ G,

preserves kL H kL−1 × kR H kR−1 . Turning our attention to the PCM boundary S-matrices, we find that KP CM (θ; kL , kR ) is invariant under exactly this symmetry. That is,

 KP CM (θ ; kL , kR ), kL H kL−1 × kR H kR−1 = 0 .

(4.21)

We begin with the Grassmannian cases, where it is enough to show that  (I + cθ EL ) ⊗ (I + cθ ER ), kL hL kL−1 × kR hR kR−1 = 0 ,

where (for subscripts L and R) h are arbitrary elements of H , E = kXk −1 and k ∈ G = SU (N ), SO(N ) or Sp(N ). But this is immediate: Xh = hX, since H is constructed to be precisely those elements in G which commute with X.

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337

For the case G/H = SO(2n)/U (n) (respectively G/H = Sp(2n)/U (n)) we note (Appendix 5.2.6, resp. 5.2.7) that E = ikJ k −1 , where k ∈ G, and that H = U (n) is constructed as those elements in G satisfying J h = hJ , giving the required result. 4.3.3. Boundary group symmetry in the SU (N )-conjugating case. The cases of SU (N )/SO(N ) and SU (N )/Sp(N ) are a little more subtle. Performing similar calculations to the above (and again leaving the L/R suffix implicit) we find, on constructing E as in Appendices 5.2.4, 5.2.5, that E khk −1 = (khk −1 )∗ E, which implies KP CM (θ ; kL , kR ) (kL HL kL−1 × kR HR kR−1 ) = (kL HL kL−1 × kR HR kR−1 )∗ KP CM (θ; kL , kR ) .

(4.22)

Such a result is not surprising, since in this case KP CM (θ ) : VL ⊗ VR → V¯L ⊗ V¯R . It is straightforward to obtain a symmetry relation in which, as earlier in section 3.2.2, we consider a boundary S-matrix which is an endomorphism of (VL ⊕ V¯L ) ⊗ (VR ⊕ V¯R ). We do not give details. 4.4. Concluding summary. When G/H is a symmetric space, the classical boundary condition g(0) ∈ kL H kR−1 preserves the local PCM conserved charges necessary for integrability. Thus, as stated in Sect. 2.2.1, the possible BCs are parametrized by a moduli space G/H × G/H 9 . We have also found boundary S-matrices which are parametrized by G/H × G/H . Further, we find that the global symmetry which survives in the presence of this BC is precisely that which commutes with the boundary S-matrix KP CM (θ; kL , kR )10 . So we finish with this Claim. The principal chiral model on G is classically integrable with boundary condition g(0) ∈ kL H kR−1 , where kL/R ∈ G and G/H is a symmetric space; and it remains integrable at the quantum level, where its boundary S-matrix is KP CM (θ ; kL , kR ). 5. Appendix: Symmetric Spaces and the Cartan Immersion Under the action of an involutive automorphism α (which may or may not be inner), a Lie algebra splits into eigenspaces g = h ⊕ k of eigenvalue +1 (h) and −1 (k), with [h, h] ⊂ h , [h, k] ⊂ k , [k, k] ⊂ h . The subgroup H generated by h is compact, and we have taken G to be compact (type I) rather than maximally non-compact (type III). For the classical groups these are the groups G = SU (N ), SO(N ) and Sp(N ) (where the argument of Sp is understood always to be even) themselves, along with those described in the table below. The dimension is dimG−dimH , and the automorphism is given by its action on U in the defining representation, where X is the diagonal matrix with M +1s and N − M  −1s and J is the symplectic form matrix, which is block-diagonal with N/2 blocks 01 −10 and satisfies J 2 = −IN . 9 10

up to a discrete ambiguity, as further explained in Appendix 6.1. with the subtlety noted above in the case of SU (N)-conjugated.

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N.J. MacKay, B.J. Short

symmetric space

dimension automorphism

SU (N )/S(U (N − M) × U (M))

2M(N − M) U
→ XU X

SO(N )/SO(N − M) × SO(M)

M(N − M)

U
→ XU X

Sp(N )/Sp(N − M) × Sp(M)

M(N − M)

U
→ XU X U
→ U ∗

SU (N )/SO(N )

N(N+1) 2

−1

SU (N )/Sp(N )

N(N−1) 2

− 1 U
→ −J U ∗ J

SO(2n)/U (n)

n(n − 1)

U
→ −J U J

Sp(2n)/U (n)

n(n + 1)

U
→ −J U J

(We refer to the first three as the “Grassmannian” cases.) 5.1. The Cartan immersion. The Cartan immersion constructs G/H as a subspace of G (due to Cartan, and described briefly in [17] or more fully in [18] (Vol.II, Sect.10, Prop.4). Lifting α in the natural way from the algebra to the group (so that α(h) = h for all h ∈ H ), under we have

gH →
α(g)g −1 G/H = {α(g)g −1 |g ∈ G} .

(This statement, of course, depends crucially on the fact that we have chosen H so that it consists of all elements of G invariant under α; for the more general case see [37, 38].) G We then have α(k) = k −1 for all k ∈ H → G. Defining M = {k ∈ G|α(k) = k −1 } ,

(5.1)

it turns out [37, 38] that G/H is in 1-1 correspondence with M0 , the identity-connected component of M. In the non-Grassmannian cases M = M0 , but in the Grassmannian cases M is a union of disconnected components, each of which is the Cartan immersion of a different G/H  [38]. In order to make connections with Subsect. 3.5 we consider translations of the Cartanimmersed G/H . In the Grassmannian cases we translate by left-multiplying by the diagonal matrix X. (In the unitary and orthogonal cases if det X = −1 then the resulting construction is no longer a subset of SU (N ) or SO(N ), but lies in the determinant −1 part of U (N) or O(N ), respectively.) In the case of SU (N )/SO(N ) we do not need to translate the Cartan construction. For SU (N )/Sp(N ) we translate by J (which has determinant 1). In the remaining cases of SO(2n)/U (n) and Sp(2n)/U (n) we translate by iJ , which has determinant (−1)n and so will be a translation into the determinant −1 part of O(2n) if and only if n is odd. The full set of translated Cartan immersions is then SU (N ) S(U (M) × U (N − M)) SO(N ) S(O(M) × O(N − M))

∼ = {U XU † |U ∈ SU (N )}, ∼ = {U XU t |U ∈ SO(N )},

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

Sp(N ) Sp(M) × Sp(N − M) SU (N ) SO(N ) SU (N ) Sp(N ) SO(2n) U (n) Sp(2n) U (n)

339

∼ = {U XU −1 |U ∈ Sp(N )}, ∼ = {U ∗ U † |U ∈ SU (N )}, ∼ = {U ∗ J U † |U ∈ SU (N )}, ∼ = {iU J U t |U ∈ SO(2n)}, ∼ = {iU J U −1 |U ∈ Sp(2n)} ,

and we treat each of these in turn in Appendices 5.2.1–5.2.7. 5.2. The boundary S-matrix constraints. The aim of this appendix is to show in every case that the above constructions of the symmetric spaces can be described in terms of constraints (those from Subsect. 3.5) on a single complex N × N matrix E ∈ Gl(N, C). 5.2.1. {U XU † |U ∈ SU (N )} = {E|E † = E, E 2 = I, T r(E) = 2M − N }. {U XU † |U ∈ SU (N)} ⊆ {E|E † = E, E 2 = I, T r(E) = 2M − N } is obvious. Now if E † = E then ∃ U ∈ SU (N ) s.t. E = U DU † , where D is diagonal. E 2 = I ⇒ D 2 = I so D has diagonal entries ± 1. Thus, after possible reordering of the diagonal entries (which we absorb into U )  IM˜ 0 D= . 0 −IN−M˜ The constraint on T r(E) implies D = X and so we have E = U XU † , as required. 5.2.2. {U XU t |U ∈ SO(N )} = {E|E † = E, E 2 = I, E t = E, T r(E) = 2M − N }. {U XU t |U ∈ SO(N )} ⊆ {E|E † = E, E 2 = I, E t = E, T r(E) = 2M − N } is obvious. Now E † = E and E t = E imply that E is a real symmetric matrix, therefore ∃ U ∈ SO(N ) s.t. E = U DU t , where D is diagonal. As in the case above the conditions E 2 = I and T r(E) = 2M − N require that D = X, so we have E = U XU t , as required. 5.2.3. {U XU −1 |U ∈ Sp(N )} = {E|E † = E, E 2 = I, J E t J = −E, T r(E) = 2M − N}. {U XU −1 |U ∈ Sp(N )} ⊆ {E|E † = E, E 2 = I, J E t J = −E, T r(E) = 2M − N} is obvious. In order to show the converse we consider the following correspondence11 . Consider the representation of H by the 2 × 2 complex matrices a = a0 e + a1 i + a2 j + a3 k
→ A = a0 E + a1 I + a2 J + a3 K, 11

Thanks to Ian McIntosh for providing this suggestion.

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N.J. MacKay, B.J. Short

where

 E=

10 01



 I=

i 0 0 −i



 J=

0 −1 1 0



 K=

0 −i −i 0

.

If we define complex conjugation on H in the standard way (note that this corresponds to hermitian conjugation of the matrices), and consider quaternions of unit length aa∗ = (a02 + a12 + a22 + a32 )e = e these correspond to elements of SU (2) = Sp(2) under the map. (Recall our definition of Sp(2n), as being the subset of U (2n) satisfying the condition AJ At = J . In the n = 1 case this is simply the constraint det A = 1, and so SU (2) = Sp(2).) This correspondence can be generalized to Sp(2n) in the following way. Consider     A11 A12 . . . A1n a11 a12 . . . a1n  A21 A22 . . . A2n   a21 a22 . . . a2n     →
A=  .. . . . .. . . ..  = A,  .. .. . . . ..   . . .  . an1 an2 . . . ann An1 An2 . . . Ann
Aij , the 2 × 2 block, in the way described above. We define where the quaternion, aij →   J 0 ... 0  0 J 0 0 −1   J =. . where J = ; ..  1 0  ..  00 J then the conditions AA† = I (where I is the quaternionic identity matrix) and A ∈ Sp(2n) are in exact correspondence. We now appeal to the fact that the (quaternionic) unitary matrix A can be diagonalized Q† AQ = D,

where Q, D ∈ U (n, H) and D is diagonal.

Under the isomorphism we have established between U (n, H) and Sp(2n) this statement corresponds to Q† AQ = D, where Q, D ∈ Sp(2n) and D is of the form   D1 0     0 D2 . . .   with Di ∈ SU (2) . D=  . . .. .. 0   0 Dn We can diagonalize each SU (2) block Di by conjugating by some Pi ∈ SU (2). If we form the matrix P by placing these SU (2) blocks, in order, down the diagonal then we have P † DP = X, where X is diagonal . Since SU (2) = Sp(2) we find that P ∈ Sp(2n). Thus, for A ∈ Sp(2n), we have obtained the conjugation Q† P † AP Q = X,

where Q, P , X ∈ Sp(2n) and X is diagonal .

Taking U = P Q, we have shown the following to be true: For all A ∈ Sp(2n) ∃ U ∈ Sp(2n) s.t. U −1 AU = X, where X is diagonal.

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

341

Now, recalling the set {E|E † = E, E 2 = I, J E t J = −E, T r(E) = 2M − N }, we have E 2 = I and J E t J = −E ⇒ EJ E t = J . Thus E ∈ Sp(N ) and we apply our result above to give E = U XU −1 for some U ∈ Sp(N ). We find that X2 = I and T r(X) = 2M − N , so we have  IM 0 X= . 0 −IN−M (Note: any X satisfying the above is conjugate to this X via some Sp(2n) matrix, which we absorb into U .) Thus we have E = U XU −1 with the required X. 5.2.4. {U ∗ U † |U ∈ SU (N )} = {E|E † E = I, E t = E, det E = 1}. {U ∗ U † |U ∈ SU (N )} ⊆ {E|E † E = I, E t = E, det E = 1} is obvious. Now if E † E = I then ∃ Q ∈ SU (N ) s.t. E = QDQ† , where D is diagonal. If we impose the condition E t = E, we have Q∗ DQt = QDQ† ⇒ DQt Q = Qt QD . Since D is diagonal, we can find a diagonal matrix C s.t. C 2 = D and with the property CQt Q = Qt QC. Thus, we have E = QC 2 Q† = Q∗ Qt QC 2 Q† = Q∗ CQt QCQ† = U ∗U †

setting

Q∗ CQt = U ∗ .

We see that det U = det C ∗ = ±1, and so we have U ∈ {V ∈ U (N )| det V = ±1}. Next, we show that the two sets {U ∗ U † |U U † = I, det U = 1} and {U ∗ U † |U U † = I, det U = −1} are in fact the same. This is trivially true for N = 1, so we assume N ≥ 2 and suppose we have an element E belonging to the first set, that is E = U ∗ U † for U ∈ SU (N ). We consider U  = U T where   01 0  T = 1 0 (note: T ∈ O(N ), det T = −1), 0 IN−2 and we see that ∗ † U  U  = U ∗T T t U † = U ∗U † = E.

We note that U  ∗ U  † is a member of the second set, right multiplying by T in this way is a self-inverse operation, and thus we have shown the two sets to be equal. Thus, we have established the equality {U ∗ U † |U ∈ SU (N )} = {E|E † E = I, E t = E, det E = 1} .

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N.J. MacKay, B.J. Short

(Note that {E|E † E = I, E t = E, det E = −1} = ω{E|E † E = I, E t = E, det E = 1} where ωN = −1, since the map E
→ ωE is an isomorphism between the sets. Thus we have {E|E † E = I, E t = E, det E = ±1} = {1, ω} × {U ∗ U † |U ∈ SU (N )} as required for Sect. 3.5.4). 5.2.5. {U ∗ J U † |U ∈ U (N ), det U = ±1} = {E|E † E = I, E t = −E, det E = 1}. {U ∗ J U † |U ∈ U (N ), det U = ±1} ⊆ {E|E † E = I, E t = −E, det E = 1} is obvious. Now if E † E = I then ∃ Q ∈ SU (N ) s.t. E = QDQ† , where D is diagonal. If we impose the condition E t = −E, we have Q∗ DQt = −QDQ† ⇒ DQt Q = −Qt QD . We denote the diagonal entries of D by Di , so that D has entries Di δij . If we denote the entries of Qt Q by Pij then the condition above becomes Di δij Pj k = −Pij Dj δj k ⇒ Di Pik = −Dk Pik . We see that Pik = 0 ⇒ Di = −Dk . By a suitable rearrangement of the diagonal entries of D (which we absorb by redefining Q) we take D to be of the form 

0

d1 In1

  0   D =  ...   .  .. 0

−d1 Im1 0 ...

... ... 0 ..

.

0 .. . .. .

d2 In2 .. .. . . 0 . . . 0 −dk Imk

        

where ni ≥ mi ≥ 0, ni ≥ 1, dj = ±di if j = i.

Then Qt Q (which is symmetric) must have the form 

0  P1t    0   t QQ= 0   .  ..   0 0

 P1 0 0 . . . 0 0 0 0 0 ... 0 0   .. ..  . .  0 0 P2  .. ..  t . .  0 P2 0   .. . . .. . . 0  .  0 . . . . . . . . . 0 Pk  0 . . . . . . 0 Pkt 0

where Pi is of size ni rows by mi columns.

We recall that Qt Q ∈ SU (N ), so that the ni rows of Pi are orthonormal with respect to the inner product on Cmi . Consequently, mi ≥ ni and so we must have ni = mi , with Pi ∈ SU (ni ), for all i.

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

343

˜ where We now decompose D into two diagonal matrices D = DE     iIn1 0 −id1 I2n1 0 ... 0    0 −iIn1  ..     . 0 −id2 I2n2 .. , E =  . D˜ =  .     .. ..     . . 0 iInk 0 0 ... 0 −idk I2nk 0 −iInk These matrices satisfy ˜ t Q = Qt QD˜ DQ

EQt Q = −Qt QE .

Recall det E = 1 ⇒ det D = 1, so since det E = 1 we also have det D˜ = 1. We now choose a diagonal matrix C such that C 2 = D˜ and CQt Q = Qt QC whose determinant will be ±1. (Note that all possible C will have the same determinant.) Now we consider E = QDQ† = QC 2 EQ† = Q∗ Qt QCECQ† CE = EC as C and E are diagonal = Q∗ CQt QECQ† . We set R = QEQ† , then QE = RQ so that E = Q∗ CQt RQCQ† . Now we consider the properties of R, R † = QE † Q† = −QEQ† = −R, R t = Q∗ EQt = Q∗ EQt QQ† = −Q∗ Qt QEQ† = −R R 2 = QEQ† QEQ† = −I as E 2 = −I .

⇒ R ∗ = R,

Further, det R = 1 so we have R ∈ SO(N ) and R 2 = −I . Thus, appealing to the argument contained in the next Sect. 5.2.6, we can find a matrix O ∈ O(N ) such that R = OJ O t . Substituting this into our expression for E we have E = Q∗ CQt OJ O t QCQ† . Setting U ∗ = Q∗ CQt O → U † = O t QCQ† we have E = U ∗J U †

where U ∈ U (N ), det U = ±1 .

We have shown {U ∗ J U † |U ∈ U (N ), det U = ±1} = {E|E † E = I, E t = −E, det E = 1}, as required. However, unlike the last Subsect. 5.2.4, here the det U = ±1 subsets are different, for consider: suppose U ∗ J U † = V ∗ J V † where U, V ∈ U (N ), det U = 1, det V = −1 ⇒ det (V t U ∗ ) = −1 : then U ∗ J U † = V ∗ J V † ⇒ V t U ∗ J U † V = J ⇒ V t U ∗

∈ Sp(N ) ⇒ det (V t U ∗ ) = 1  

The subsets of U (N ) such that det = ±1 are isomorphic via multiplication by ω where ωN = −1. Thus {U ∗ J U † |U ∈ U (N ), det U = −1} = ω2 {U ∗ J U † |U ∈ SU (N )}.

344

N.J. MacKay, B.J. Short

So we have {E|E † E = I, E t = −E, det E = 1} = {1, ω2 } × {U ∗ J U † |U ∈ SU (N )} and, as required by Subsect. 3.5.4, {E|E † E = I, E t = −E, det E = ±1} = {1, ω, ω2 , ω3 } × {U ∗ J U † |U ∈ SU (N )}. 5.2.6. {iU J U t |U ∈ O(2n)} = {E|E † = E, E 2 = I, E t = −E, } {iU J U t |U ∈ O(2n)} ⊆ {E|E † = E, E 2 = I, E t = −E, }. is obvious. Now, E † = E, E t = −E ⇒ E ∗ = −E, so we consider F = −iE. Then F is a real matrix satisfying FtF = I and F 2 = −I . Since F is orthogonal there exists a matrix R ∈ SO(2n) such that R t F R has the form   O1 0    0 O2 . . .    where Oi ∈ O(2) .   . . .. .. 0   0 On Since F 2 = −I each Oi2 = −I2 , the only solutions for which Oi ∈ O(2) are  0 −1 Oi = ± = ± . 1 0 We note that it is possible to conjugate − by the matrix  01 ∈ O(2) 10 to obtain . Thus we can find a matrix R  ∈ O(2)⊗n ⊂ O(2n) such that R t R t F RR  = J . If we set U = RR  then we have E = iU J U t

for U ∈ O(2n) as required.

We note that {iU J U t |U ∈ O(2n)} = {iU J U t |U ∈ SO(2n)}, since iU J U t = iV J V t ⇒ V tUJUtV = J ⇒ V t U ∈ Sp(2n) ⇒ det U = det V .

for U, V ∈ O(N )

Both sets {iU J U t |U ∈ SO(2n)} and {iU J U t |U ∈ O(2n), det U = −1} are constructions of the symmetric space SO(2n) U (n) † 2 t and so {E|E = E, E = I, E = −E} is isomorphic to two copies of the symmetric space, as stated in Sect. 3.5.2.

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

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5.2.7. {iU J U −1 |U ∈ Sp(2n)} = {E|E † = E, E 2 = I, J E t J = E}. {iU J U −1 |U ∈ Sp(2n)} ⊆ {E|E † = E, E 2 = I, J E t J = E} is obvious. To establish the converse, we consider the conditions on E, E 2 = I and J E t J = E ⇒ EJ E t = −J, E † = E and E 2 = I ⇒ E † E = I . If we let F = −iE then the conditions on F are F † F = I,

F 2 = −I

and

FJFt = J .

Thus F ∈ Sp(2n), so from 5.2.3 we can find a V ∈ Sp(2n) such that F = V DV −1 , where D is a diagonal matrix. Since F 2 = −I we must also have D 2 = −I , so the entries of D must be ±i. As D ∈ Sp(2n) we cannot choose the signs of these diagonal entries completely arbitrarily and we find we are restricted to   ±I 0     0 ±I . . .  i 0   D= recalling I = , 0 −i .. ..  . . 0  0 ±I and with each ± freely chosen. However, we have K IK† = −I and so we can further conjugate in Sp(2n) (which we absorb into V ) to ensure that D has all n ± signs set to +. Now we notice that J ∈ Sp(2n) and so, as above, there exists some W ∈ Sp(2n) such that D = W J W −1 . If we set U = V W then we have obtained F = U J U −1 , where U ∈ Sp(2n). Recalling that E = iF , we see that we have obtained E = iU J U −1

as required.

6. Appendix: Discrete Ambiguities in Boundary Parameters First, recall the G× G invariance of the Lagrangian under g
→ gL ggR−1 . Of course this should really be G×G Z(G) (where Z(G) is the centre of G, which is finite), since for z ∈ Z(G) we have g = zgz−1 . The physics literature for the bulk principal chiral model generally is not concerned with this. In the same way we do not explore in the text the ambiguities in our boundary parameters, but we wish here at least to state them precisely, and to prove that they are finite in number.

6.1. Chiral BCs. The ambiguity here arises because there may be non-trivial g1 , g2 such that g1 H g2−1 = H . This requires

and thence

α(g1 )hα(g2−1 ) = g1 hg2−1

∀h ∈ H ,

g1−1 α(g1 )h = hg2−1 α(g2 )

∀h ∈ H.

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N.J. MacKay, B.J. Short

So (by setting h = e) we see that g1−1 α(g1 ) = g2−1 α(g2 ) = k,

where k ∈

G ∩ C(H ) , H

where C(H ) < G is the centralizer of H . But, for a symmetric space, H is a maximal Lie subgroup (there is no Lie subgroup of greater dimension which contains H ), so G H ∩ C(H ) is finite, and so the solutions g1 ∈ H x, g2 ∈ Hy have x = y (since the Cartan immersion is 1-1) and are also finite in number.

6.2. Non-chiral BCs. Here the potential ambiguity is that there may be non-trivial g1 , g2 G G G −1 such that g1 H g2 = H . In contrast to the chiral case, we can push g1 through H : g1 {α(g)g −1 |g ∈ G}g2−1 = {α(α(g1 )g)g −1 |g ∈ G}g2−1 = {α(α(g1 )g)(α(g1 )g)−1 |g ∈ G}α(g1 )g2−1 G = α(g1 )g2−1 . H G G So the boundary is parametrized by G, up to g0 such that H g0 = H . This requires −1 −1 −1 −1 −1 α(kg0 ) = g0 k for all k ∈ G/H → G, so k α(g0 ) = g0 k . This must hold for k = e, so g0 ∈ M of (5.1), and commutes with every element of G/H → G. Such g0 form a group, which must be finite: for suppose not, that its algebra is generated by k0 ⊂ k. Then [k0 , k] = 0. Also [[h, k0 ], k] ⊂ [[k, k0 ], h] + [[h, k], k0 ], both of which are empty, so [h, k0 ] ⊂ k0 . Thus [k0 , g] ⊂ k0 , and k0 is an ideal, and is therefore trivial. Note the specialization (mentioned in the text) when g1 = g2 : the boundary is then parametrized by α(g1 )g1−1 and thus by G/H (again quotiented, here by Z(G/H ), those elements of G/H which commute with all of G/H ). It is straightforward to propose a compatible PCM boundary S-matrix, though we do not do so here.

7. Appendix: Boundary Yang Baxter and Crossing-Unitarity Calculations In this section we include a representative selection of the BYBE and crossing-unitarity calculations required to obtain the various constraints on the boundary S-matrices that we have considered in this paper. We hope they will be sufficiently illustrative that the interested reader can perform any calculations not presented here for themselves. First we list the minimal bulk S-matrices derived in [29]  σu (θ ) hθ SU (N ) : , −  hθ 2iπ 1 − 2iπ   hθ σo (θ ) hθ 2iπ SO(N ) :  , − + h hθ hθ 2iπ 1 − 2iπ 2 − 2iπ   hθ σp (θ ) hθ 2iπ Sp(N ) :  , − + h hθ hθ 2iπ 1 − 2iπ 2 − 2iπ

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347

where h is the dual Coxeter number and the scalar prefactors σ are  −θ  θ  2iπ + h1  2iπ σu (θ ) =  −θ  θ ,  2iπ + h1  2iπ  θ  −θ  −θ  θ + h1  2iπ + 21  2iπ  2iπ + 21 + h1  2iπ σo (θ ) = −  −θ  −θ  θ  θ ,  2iπ + h1  2iπ + 21  2iπ  2iπ + 21 + h1  θ  −θ  −θ  θ  2iπ + 21 + h1  2iπ + h1  2iπ + 21  2iπ σp (θ ) =  −θ  −θ  θ  θ ,  2iπ + 21 + h1  2iπ + h1  2iπ + 21  2iπ where  is the gamma function, and  SU (N ) N h = N − 2 SO(N )  N + 2 Sp(N ) . Note that

σ¯ u (θ ) = σu (iπ − θ) .

7.1. BYBE calculations. Recall the boundary Yang Baxter equation
  np Sijkl (θ − φ) Ij m ⊗ K ln (θ ) Smo (θ + φ) Ioq ⊗ K pr (φ) 
 pr = Iij ⊗ K kl (φ) Sjlnm (θ + φ) Imo ⊗ K np (θ ) Soq (θ − φ) . For clarity of the calculations we introduce the notation u=

hθ , 2iπ

v=

hφ , 2iπ

u0 =

h . 4

In any BYBE calculation the scalar prefactors cancel, and we consider here only the matrix part of the equation. 7.1.1. The SU (N ) case with K1 boundary S-matrix Substituting into the BYBE with the bulk S-matrix for the SU (N ) PCM and K1 gives   − (u − v) − (u + v)   = − (u + v) − (u − v) Expanding out and cancelling where possible, we are left with = Thus, for the equation to be satisfied we require the condition :=α

for some constant α.

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N.J. MacKay, B.J. Short

7.1.2. The Sp(N ) case with K2 boundary S-matrix. Substituting into the BYBE with the bulk S-matrix for the Sp(N ) PCM and K2 gives 

 −(u − v)  × + cv ˜





+t (u − v) + cu ˜ − (u + v) + t (u + v)   = + cv ˜ − (u + v) + t (u + v)   × , + cu ˜ − (u − v) + t (u − v)

where t (u) = 2u0u−u and c˜ is related to the original S-matrix constant c by the relation c c˜ = 2iπ h . Expanding out, cancelling where possible (noting that the terms involving and cancel after some simple algebra) and rearranging (some less trivial algebra!) we are left with 



−cuvt ˜ (u − v)t (u + v) 2 + c˜  + 2cuvt ˜ (u − v)t (u + v)  2 c˜ uv(u − v) −  −c˜2 uv(u + v)t (u − v)  2 −c˜ uv(u − v)t (u + v)

− −





+ c˜ uvt (u − v) − = 0. −

−

2

In order for this to hold we are forced to have =α

for some constant α

=β

i.e.

)t = β

(

.

(Note: then β 2 = 1.) We find that the equation is then satisfied provided 2β − 2 − c˜

= 0.

Since β ± 1 we must have :=α (

)t =

⇐⇒ c˜

=0

or

and

(

)t = − (

) ⇐⇒ c˜

= −4 .

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

349

7.1.3. The SU (N )-conjugating case Recall the conjugated BYBE np ¯

Sijkl (θ − φ)(Ij m ⊗ K l n¯ (θ ))Smo¯ (θ + φ)(Io¯ q¯ ⊗ K pr¯ (φ)) ¯

¯

p¯ r¯

= (Iij ⊗ K k l (φ))Sjlnm¯ (θ + φ)(Im¯ o¯ ⊗ K np¯ (θ ))So¯ q¯ (θ − φ) . The K2 boundary S-matrix does not satisfy the above equation, but K1 does, under some constraints. Substituting in and using our simplifying notation we get   − (u − v) − (2u0 − u − v)   . = − (2u0 − u − v) − (u − v) Expanding out and cancelling all possible terms leaves =

.

So to satisfy the conjugated BYBE we must have ( )t = ± . There are two other BYBEs to consider in addition to the V ⊗ V → V¯ ⊗ V¯ case considered so far. The V¯ ⊗ V¯ → V ⊗ V BYBE will be similar to the above, so that if we denote by the matrix part of the V¯ → V boundary S-matrix then we must have (

. The last case to consider is V ⊗ V¯ → V¯ ⊗ V , where the BYBE is

 ¯ np k l¯ ln pr¯ (θ − φ) I ⊗ K (θ ) S (θ + φ) I ⊗ K (φ) Sij j m oq mo ¯

 ¯ ¯ pr¯ ¯ = Ii¯j¯ ⊗ K k l (φ) Sjl¯n¯m¯ (θ + φ) Im¯ o¯ ⊗ K np (θ ) Soq ¯ (θ − φ) .

)t = ±

Substituting into this, again with simplified notation, we get     − (2u0 − u + v) − (u + v)     = − (u + v) − (2u0 − u + v)

.

Expanding out and cancelling all the terms we can we have − (2u0 − u + v) =

− (u + v)

− (2u0 − u + v)

− (u + v)

.

This equation is satisfied provided =

=α

=

and

=β .

Since ( )t = ± and ( )t = ± , we have β = ±α and so have only one independent parameter. The conditions imposed for the conjugated SU (N ) case are thus (

)t = ±

and

(

)t = ±

and

=

=α

.

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N.J. MacKay, B.J. Short

7.2. Crossing-unitarity calculations. Recall the crossing-unitarity equation  K ij

iπ −θ 2



= Sjil¯k¯ (2θ)K lk

iπ +θ 2

.

We note that it is Sjil¯k¯ (2θ ) that is required here, which is a crossed and transposing version of the standard S-matrix. We obtain it by taking the standard S-matrix substituting iπ − 2θ for 2θ , turning the matrix diagrams through 90o and then postmultiplying by (the transposition operator). (We note that the process of crossing doesn’t alter the S-matrix for the SO(N ) and Sp(N ) cases, but we go through the process anyway to illustrate the SU (N ) cases.) We again make use of some simplifying notation. 7.2.1. The SO(N ) case with K2 boundary S-matrix. Substituting into the crossingunitarity equation we have 

τ ( iπ 2 − θ) (1 − c( iπ 2 − θ )) =

+ c(u ˜ 0 − u) 

σo (iπ − 2θ )τ ( iπ 2 + θ)

− 2(u0 − u)

(1 − 2uo + 2u)(1 − c( iπ + θ ))  2 × + c(u ˜ 0 + u)

⇒

τ ( iπ 2 − θ) τ ( iπ 2 + θ)

+

u0 − u u







 + c(u ˜ 0 − u) 

N + c˜

(u0 + u)   σo (iπ − 2θ )  −2(u − u) + u0 − u iπ 0 ×(1 − c( 2 − θ ))  u  = (1 − 2uo + 2u)   −2c(u ˜ 0 − u)(u0 + u)  ×(1 − c( iπ c(u ˜ 0 − u)(u0 + u) 2 + θ )) + u In order for this to be satisfied it is necessary to impose ( coefficients of the τ ( iπ 2 − θ) τ ( iπ 2 + θ)

)t = ±

   .   

. Considering

terms we find =

 (1 − c( iπ σo (iπ − 2θ )(u0 + u) 1 2 − θ)) . ± −2 iπ (1 − 2u0 + 2u) u (1 − c( 2 + θ))

For the coefficients of the and a constraint on



terms to be consistent with this we require the + sign choice,

, so altogether we have the matrix constraints (

)t =

and

c˜

= −4.

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

351

From the crossing symmetry of the S-matrix we can simplify the constraint on the scalar prefactor, obtaining τ ( iπ 2 − θ) τ ( iπ 2 + θ)

=

(u0 + u)(1 − c( iπ 2 − θ)) (u0 − u)(1 − c( iπ 2 + θ))

which can be written as τ ( iπ 2 − θ)

=

τ ( iπ 2 + θ)

σo (2θ)

   h h h − σo (2θ) . 2 ciπ 2

7.2.2. The SU (N )-conjugating case The K-matrix for SU (N )-conjugating must satisfy a conjugated version of the crossing-unitarity equation,   ¯ iπ ¯ iπ K ij − θ = Sjilk (2θ)K l k +θ . 2 2 For this case it is not the crossed S-matrix we require, but the standard S-matrix. Substituting in, we have    iπ iπ σu (2θ ) ρ ρ = − 2u . −θ +θ 2 (1 − 2u) 2 )t = ±

On expanding this, we see that ( scalar prefactor becomes

ρ( iπ 2 − θ) ρ( iπ 2

+ θ)

which can be written as ρ( iπ 2 − θ) ρ( iπ 2 + θ)

=

  

=

is required. Then the condition on the

(1 ∓ 2u) σu (2θ) , (1 − 2u)

σu (2θ )

(

−[1]σu (2θ) (

)t = )t = −

.

8. Appendix: Unitarity and Hermitian Analyticity Calculations In this section we prove the two results, concerning complex parameters that could consistently be set to 1, stated in Sect. 3.2. 8.1. The non-conjugating case. We start from (3.7), E† = E

and

E 2 = αI

where α ∈ U (1).

Since E is hermitian we can diagonalize it as D = QEQ† where Q ∈ SU (N ). Then we have Further, Now

D∗

D † = QE † Q† = QEQ† = D ⇒ D ∗ = D.

D 2 = QEQ† QEQ† = QE 2 Q† = αI. = D ⇒ α ∈ R+ so α = 1, as stated in 3.2.1.

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N.J. MacKay, B.J. Short

8.2. The conjugating case. We start from (3.12) and (3.13), αE = F †

and

EF = βI

and

γ N det F = det E

and

ρ(θ ) = αω(−θ ∗ )∗ , 1 ρ(θ )ω(−θ) = , β γρ(θ ) = ω(θ) .

We note that α ∗ EE † = βI β ⇒ EE † = ∗ I α β + ⇒ ∗ ∈R α and ⇒ ⇒ ⇒

ω(θ ) = αγ ω(−θ ∗ )∗ ω(−θ ∗ )∗ = α ∗ γ ∗ ω(θ ) ω(−θ ∗ )∗ = α ∗ γ ∗ αγ ω(−θ ∗ )∗ αγ ∈ U (1) .

Thus, if we use the rescaling freedom in K(θ ) = ρ(θ )E to set α = 1, we have β ∈ R+ and γ ∈ U (1). Therefore we have enough freedom in rescaling K  (θ ) = ω(θ )F to set both β and γ to 1 also, as stated in 3.2.2. Acknowledgments. We should like to thank Tony Sudbery and Ian McIntosh for discussions of symmetric spaces. NJM would like to thank Patrick Dorey, Ed Corrigan and G´erard Watts for helpful discussions, and Bernard Piette, Paul Fendley and Jonathan Evans for email exchanges. BJS would like to thank Gustav Delius, Brett Gibson and Mark Kambites for discussions. Finally NJM thanks the Centre de Recherches Math´ematiques, U de Montr´eal, where this work was begun during the ‘Quantum Integrability 2000’ program, for hospitality and financial support, and BJS thanks the UK EPSRC for a D.Phil. studentship.

References 1. Belavin, A., Drinfeld, V.: Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funct. Anal. 17, 220 (1983) 2. Drinfeld, V.: Hopf algebras and the quantum Yang-Baxter equation. Sov. Math. Dokl. 32, 254 (1985) 3. Evans, J.M., Hassan, M., MacKay, N.J., Mountain, A.: Conserved charges and supersymmetry in principal chiral models. hep-th/9711140 4. Evans, J.M., Hassan, M., MacKay, N.J., Mountain, A.: Local conserved charges in principal chiral models. Nucl. Phys. B561, 385 (1999), hep-th/9902008 5. Evans, J.M.: Integrable sigma-models and Drinfeld-Sokolov hierarchies. hep-th/0101231, Nucl. Phys. B608, 591 (2001); From PCM to KdV and back. hep-th/0103250 6. MacKay, N.J.: The SO(N) principal chiral field on a half-line. J. Phys. A32, L189 (1999), hep-th/9903137 7. Cherednik, I.V.: Factorizing particles on a half-line and root systems. Theor. Math. Phys. 61, 977 (1983) 8. MacKay, N.J.: Fusion of SO(N) reflection matrices. Phys. Lett. B357, 89 (1995), hep-th/9505124 9. Abad, J., Rios, M.: Non-diagonal solutions to reflection equations in su(n) spin chains. Phys. Lett. B352, 92 (1995), hep-th/9502129 10. Corrigan, E.: Reflections. hep-th/9601055

Boundary Scattering, Symmetric Spaces and Principal Chiral Model (1)

11. Gandenberger, G.M.: On a2 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

32.

33. 34. 35.

36.

353

reflection matrices and affine Toda theories. Nucl. Phys. B542, 659 (1)

(1999), hep-th/9806003, and New non-diagonal solutions to the an boundaryYang-Baxter equation. hep-th/9911178 de Vega, H.J., Gonz´alez-Ruiz, A.: Boundary K-matrices for the six-vertex and the n(2n − 1) An−1 vertex models. J. Phys. A26, L519 (1993), hep-th/9211114, and Boundary K-matrices for the XY Z, XXZ and XXX spin chains, J. Phys. A27, 6129 (1994), hep-th/9306089 Lima-Santos, A.: Reflection K-matrices for 19-vertex models. Nucl. Phys. B558, 637 (2000), solvint/9906003 L¨uscher, M.: Quantum non-local charges and the absence of particle production in the 2D non-linear σ -model. Nucl. Phys. B135, 1 (1978) Bernard, D.: Hidden Yangians in 2D massive current algebras. Commun. Math. Phys. 137, 191 (1991) Dorey, P.E.: Root systems and purely elastic S-matrices. Nucl. Phys. B358, 654 (1991) Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. 2nd ed. New York: Academic Press, 1978 Fomenko, A.T.: The Plateau Problem, Vols. I and II, London: Gordon and Breach, 1990 Mourad, M.F., Sasaki, R.: Non-linear sigma models on a half plane. Int. J. Mod. Phys. A11, 3127 (1996), hep-th/9509153 Evans, J.M., Mountain, A.J.: Commuting charges and symmetric spaces. Phys. Lett. B483, 290 (2000), hep-th/0003264 Kato, M., Okada, T.: D-branes on group manifolds. Nucl. Phys. B499, 583 (1997), hep-th/9612148 Felder, G., Fr¨ohlich, J., Fuchs, J., Schweigert, C.: The geometry of WZW branes. J. Geom. Phys. 34, 132 (2000), hep-th/9909030 Stanciu, S.: D-branes on group manifolds. hep-th/9909163 Moriconi, M., de Martino, A.: Quantum integrability of certain boundary conditions. Phys. Lett. B447, 292 (1999), hep-th/9809178 Moriconi, M., de Martino, A.: Boundary S-matrix for the Gross-Neveu model. Phys. Lett. B451, 354 (1999), hep-th/9812009 Goldschmidt, Y.Y., Witten, E.: Conservation laws in some two-dimensional models. Phys. Lett. B91, 392 (1980) Ghoshal, S., Zamolodchikov, A.: Boundary S-matrix and boundary state in two-dimensional integrable quantum field theory. Int. J. Mod. Phys. A9, 3841 (1994), hep-th/9306002 Luis Miramontes, J.: Hermitian analyticity versus real analyticity in two-dimensional factorized S-matrix theories. hep-th/9901145 Ogievetsky, E., Reshetikhin, N., Wiegmann, P.: The principal chiral field in two dimensions on classical lie algebras: the Bethe-ansatz solution and factorized theory of scattering. Nucl. Phys. B280, 45–96 (1987) Fring, A., K¨oberle, R.: Affine Toda field theory in the presence of reflecting boundaries. Nucl. Phys. B419, 647 (1994), hep-th/9309142 Corrigan, E., Sheng, Z.-M.: Classical integrability of the O(N) nonlinear sigma model on a half-line. Int. J. Mod. Phys. A12, 2825 (1997), hep-th/9612150 Prata, J.: Reflection factors for the principal chiral model. Phys. Lett. B438, 115 (1998), hepth/9806222 Klimcik, C., Severa, P.: Open strings and D-branes in WZNW models. Nucl. Phys. B488, 653 (1997), hep-th/9609112 Recknagel, A., Schomerus, V.: D-branes in Gepner models. Nucl. Phys. B531, 185 (1998), hepth/9712186 Alekseev, A., Schomerus, V.: D-branes in the WZW model. Phys. Rev. D60, 61901 (1999), hepth/9812193 Doikou, A., Nepomechie, R.: Bulk and boundary S-matrices for the SU (N) chain. Nucl. Phys. B521, 547 (1998), hep-th/9803118 Ghoshal, S.: Boundary S-matrix of the SO(N)-symmetric non-linear sigma model. Phys. Lett. B334, 363 (1994), hep-th/9401008 Donin, J., Shnider, S.: Quantum symmetric spaces. J. Pure. Appl. Algebra 100, 103 (1995), hepth/9412031 Dijkhuizen, M.: Some remarks on the construction of quantum symmetric spaces. math.QA/9512225 Noumi, M., Sugitani, T.: Quantum symmetric spaces and related q-orthogonal polynomials. math.QA/9503225 Molev, A., Nazarov, M., Ol’shanskii, G.: Yangians and classical Lie algebras. Russ. Math. Surveys 51, 205 (1996) Molev, A., Olshanski, G.: Centralizer construction for twisted Yangians. q-alg/9712050

354

N.J. MacKay, B.J. Short

37. Eichenherr, H.: More about nonlinear sigma models on symmetric spaces. Nucl. Phys. B164, 528 (1980), err.ibid. B282, 745 (1987) 38. Antoine, J.-P., Piette, B.: Classical nonlinear sigma models on Grassmann manifolds of compact or noncompact type. J. Math. Phys. 28, 2753 (1987) 39. Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Oxford: OUP, 3rd edition, – see ch.14 and references therein Fendley, P.: Integrable sigma models and perturbed coset models. hep-th/0101034 40. Moriconi, M.: Integrable boundary conditions and reflection matrices for the O(N) nonlinear sigma model. hep-th/0108039. Nucl. Phys. B619, 396 (2001) Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 233, 191–209 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0742-z

Communications in

Mathematical Physics

On the Sutherland Spin Model of BN Type and Its Associated Spin Chain F. Finkel, D. G´omez-Ullate, A. Gonz´alez-L´opez ∗ , M.A. Rodr´ıguez, R. Zhdanov∗∗ Departamento de F´ısica Te´orica II, Universidad Complutense, 28040 Madrid, Spain Received: 14 February 2002 / Accepted: 19 June 2002 Published online: 10 December 2002 – © Springer-Verlag 2002

Abstract: The BN hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited to derive a complete family of commuting integrals of motion of the model, thus establishing its integrability. The Dunkl operators are shown to possess a common flag of invariant finite-dimensional linear spaces of smooth scalar functions. This implies that the Hamiltonian of the model preserves a corresponding flag of smooth spin functions. The discrete spectrum of the restriction of the Hamiltonian to this spin flag is explicitly computed by triangularization. The integrability of the hyperbolic Sutherland spin chain of BN type associated with the dynamical model is proved using Polychronakos’s “freezing trick”.

1. Introduction Since the publication of the pioneering papers of Calogero [5] and Sutherland [32, 33], the study of solvable and integrable quantum many-body problems has become a fruitful field of research with multiple connections in many branches of contemporary mathematics and physics. From a mathematical standpoint, one of the key developments in the field was the discovery by Olshanetsky and Perelomov of an underlying AN root system structure for both the Calogero and Sutherland models [25]. The integrability of these models follows by expressing the Hamiltonian as one of the radial parts of the Laplace–Beltrami operator in a symmetric space associated with the given root system. It was also shown in this paper that the original inverse square (Calogero) and trigonometric/hyperbolic (Sutherland) potentials arise as appropriate limits of the most general potential in this class, given by the Weierstrass ℘-function, and that integrable models associated to other root systems also exist. The rational and trigonometric Calogero–Sutherland (CS) models are also exactly solvable, in the sense that their ∗ ∗∗

Corresponding author. E-mail: [email protected] On leave of absence from Institute of Mathematics, 3 Tereshchenkivska St., 01601 Kyiv – 4 Ukraine

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F. Finkel, D. G´omez-Ullate, A. Gonz´alez-L´opez, M.A. Rodr´ıguez, R. Zhdanov

eigenfunctions and eigenvalues can be computed algebraically. In fact, the study of the eigenfunctions of these models has led to significant advances in the theory of multivariate orthogonal polynomials [1, 11, 23]. Apart from their mathematical interest, CS models have found numerous applications in diverse areas of physics such as soliton theory [22, 29], fractional statistics and anyons [4, 6], random matrix theory [35], and Yang–Mills theories [9, 16], to name only a few. During the last decade, CS models with internal degrees of freedom have been actively explored by a variety of methods, including the exchange operator formalism [24], the Dunkl operators approach [2,8,11], reduction by discrete symmetries [30], and construction of Lax pairs [19–21]. Historically, the first CS models with spin discussed in the literature were related to the original models of AN type introduced by Calogero and Sutherland [2, 17, 19, 20, 24, 34]. The integrability of these CS spin models was established in some cases by relating the Hamiltonian to a quadratic combination of Dunkl operators of AN type [7, 10, 26]. The BN counterpart of the AN CS spin models mentioned above were first considered by Yamomoto [38]. In this paper, the spectrum of the rational BN spin model was explicitly determined, and its integrability was shown by means of the Lax pair approach. In Ref. [39], Yamamoto and Tsuchiya presented an alternative proof of the integrability of this model using Dunkl operators of BN type. The same operators were later employed by Dunkl to construct a complete basis of eigenfunctions [11]. In contrast, the trigonometric/hyperbolic BN spin model has received remarkably little attention. In our recent paper [13] we proved that this model is exactly solvable in the sense of Turbiner [36, 37], meaning that its Hamiltonian leaves invariant a known infinite increasing sequence (or flag) of finite-dimensional linear spaces of smooth spin functions. In fact, in [13] we developed a systematic method for constructing exactly (or in some cases partially) solvable BN -type CS models with spin by combining several families of Dunkl operators. The key elements of this method – first introduced in the AN case in [12] – are: i) the definition of a new family of Dunkl operators, and ii) the construction of a very wide class of quadratic combinations of these operators and those in the other two families considered by Dunkl in [11]. The interest on CS spin models has been further enhanced by their close connections with integrable spin chains of Haldane–Shastry type [18, 31]. Spin chains describe a fixed arrangement of particles that interact through their spins. A well-known example is the Heisenberg spin chain, whose spins are equally spaced and interact only with their nearest neighbors. The Haldane–Shastry model was actually the first one-dimensional spin chain with long range interactions whose spectrum could be computed exactly. In this model, the spin sites are equally spaced in a circle and interact with each other with strength decreasing as the inverse square of the chord distance between the sites. The integrability of the Haldane–Shastry spin chain was proved by Fowler and Minahan in [14]. Polychronakos later realized that the commuting conserved quantities of the Haldane–Shastry spin chain can be elegantly deduced from those of the (dynamical) Sutherland spin model of AN type by applying what he called the “freezing trick” [27] (see also [34]). This corresponds to taking the strong coupling limit in the Sutherland spin model and restricting to states with no momentum excitations, so that the internal degrees of freedom remain the only relevant variables in the problem and the particles are “frozen” at their classical equilibrium positions. This observation is, in principle, valid for any integrable spin Calogero–Sutherland model. For instance, in Ref. [27] the freezing trick is applied to the spin Calogero model with rational interaction to construct a new integrable spin chain of rational type in which the sites are no longer equally spaced. The spectrum of this chain was later calculated by Frahm [15] and Polychro-

On the Sutherland Spin Model of BN Type

193

nakos [28]. Bernard, Pasquier and Serban [3] studied the spin chain associated with the trigonometric Sutherland model, establishing its integrability for certain values of the parameters in the Hamiltonian. The aim of this paper is twofold. In the first place, we prove the integrability of the hyperbolic Sutherland spin model of BN type, from which we are also able to deduce the integrability of the spin chain associated with this model. Secondly, we give an explicit formula for the eigenvalues of the dynamical model whose corresponding (square-integrable) eigenfunctions lie in the invariant flag mentioned above. The paper is organized as follows. In Sect. 2 we introduce a family of commuting Dunkl operators of BN type. We show that the sums of even powers of these Dunkl operators generate a complete set of commuting integrals of motion of the Hamiltonian of the model. The commutation relations satisfied by the Dunkl operators and the usual permutation and sign reversing operators, which possess a richer structure than in the rational case, play a key role in the proof of this result. In Sect. 3 we analyze the spectrum of the Hamiltonian for any value of the spin. Our analysis is based on the fact that the Dunkl operators leave invariant a flag of finite-dimensional linear spaces of smooth scalar functions. This flag yields a corresponding invariant flag of smooth spin functions for the hyperbolic BN Sutherland Hamiltonian. We construct a partially ordered basis of this “spin” flag in which the Hamiltonian is represented by a triangular matrix. In this way we can explicitly compute the eigenvalues of the restriction of the Hamiltonian to the (finite-dimensional) intersection of the spin flag with the Hilbert space of the system. We shall use the term algebraic in what follows to refer to these eigenvalues and its corresponding eigenfunctions. It remains an open problem to determine whether the algebraic sector of the spectrum actually coincides with the discrete spectrum. We also study in detail the (algebraic) ground state, determining its degeneracy for all values of the spin. In Sect. 4 we define the spin chain associated with the hyperbolic Sutherland spin model of BN type, and apply the freezing trick to derive a complete family of commuting integrals of motion of this chain. 2. Integrability of the Sutherland Spin Model of BN Type The Hamiltonian of the hyperbolic BN Sutherland spin model is defined by 

 H∗ = − ∂x2i + 2a sinh−2 xij− (a + Sij ) + sinh−2 xij+ (a + S˜ij ) i

+b


i

i<j

sinh

−2

xi (b + Si ) − b




  cosh−2 xi b + Si ,

(1)

i

where xij± = xi ± xj and a, b, b are real parameters. Here and in what follows, any summation or product index without an explicit range will be understood to run from 1 to N , unless otherwise constrained. The operators Sij and Si in Eq. (1) act on the finite-dimensional Hilbert space    1  S = |s1 , . . . , sN   si = −M, −M + 1, . . . , M; M ∈ N , (2) 2 associated to the particles’ internal degrees of freedom, as follows: Sij |s1 , . . . , si , . . . , sj , . . . , sN  = |s1 , . . . , sj , . . . , si , . . . , sN  , Si |s1 , . . . , si , . . . , sN  = |s1 , . . . , −si , . . . , sN  . We have also used the customary notation S˜ij = Si Sj Sij .

(3)

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The operators Sij and Si are represented in S by (2M + 1)N -dimensional Hermitian matrices, and obey the following algebraic relations: Sij2 = 1, Si2

= 1,

Sij Sj k = Sik Sij = Sj k Sik , Si Sj = Sj Si ,

Sij Skl = Skl Sij ,

Sij Sk = Sk Sij ,

Sij Sj = Si Sij ,

(4)

where the indices i, j, k, l take distinct values in the range 1, . . . , N. The algebra S generated by the operators Sij , Si is thus isomorphic to the group algebra of the Weyl group WN of type BN , also known as the hyperoctahedral group. We shall also make use of the permutation operators Kij = Kj i and the sign reversing operators Ki (1 ≤ i = j ≤ N ), whose action on a function f (x), with x = (x1 , . . . , xN ) ∈ RN , is defined as follows: (Kij f )(x1 , . . . , xi , . . . , xj , . . . , xN ) = f (x1 , . . . , xj , . . . , xi , . . . , xN ) , (Ki f )(x1 , . . . , xi , . . . , xN ) = f (x1 , . . . , −xi , . . . , xN ) .

(5)

The operators Kij and Ki obey algebraic identities analogous to (4). We shall denote by K  S the algebra generated by the coordinate permutation and sign changing operators Kij and Ki . Note also that the operators ij = Kij Sij and i = Ki Si generate an algebra W isomorphic to K and S. From now on we shall identify the abstract group WN with its realizations generated by the operators Kij , Ki on C ∞ (RN ), Sij , Si on S, or ij , i on C ∞ (RN ) ⊗ S, depending on the context. The Hamiltonian (1) describes a system of N identical particles, whose physical states are therefore either totally symmetric or totally antisymmetric under particle exchange. Moreover, since H ∗ clearly commutes with the family of commuting operators i (i = 1, . . . , N), we can choose a basis of common eigenfunctions of H ∗ and all the operators i . Given an element ψk of this basis, it follows from the commutation relations of the sign reversing operators i with the permutation operators ij that i ψk = k ψk , independently of i. In principle, the parity k could depend on k. However, we shall see in the following section that all the algebraic eigenfunctions have the same parity, and that this parity is determined by the sign of the parameter b. From now on we shall assume, for definiteness, that we are dealing with a system of fermions whose algebraic states are also antisymmetric under sign reversal of each particle’s spatial and internal coordinates. This covers what is perhaps the physically most interesting case, namely that of a system of spin 1/2 particles, for which the internal degrees of freedom are naturally interpreted as the particles’ spin. The results of this paper can be easily modified to treat any other choice of the particles’ statistics and parity. In the rest of this section we shall prove the integrability of the model (1) by expressing the Hamiltonian in terms of the following family of BN -type Dunkl operators: 
 (1 + coth xij− ) Kij + (1 + coth xij+ ) K˜ ij Ji = ∂xi − a j =i




+ b (1 + coth xi ) + b (1 + tanh xi ) Ki + 2a Kij ,

(6)

j
where K˜ ij = Ki Kj Kij . The operators Ji are related to the operators introduced by Yamamoto [38] in connection with the trigonometric BCN spin Sutherland model. A key property of the Dunkl operators (6) is their commutativity: [Ji , Jj ] = 0 ,

i, j = 1, . . . , N .

(7)

On the Sutherland Spin Model of BN Type

195

We shall also make use of the following commutation relations between the operators Kij , Ki and the operators Ji :  i





Kij Jj − Ji Kij = 2a 1 +



Kij Kj l  ,

(10)

i
[Ki , Jj ] = 2aKij (Kj − Ki ) , [Kj , Ji ] = 0 ,
{Ki , Ji } = −2(b + b ) − 2a Kil (Ki + Kl ) ,

(11) (12)

l>i

where i < j and k are three distinct indices ranging from 1 to N . Note that the Dunkl operators of rational type considered in Ref. [39] satisfy the latter equations with the right-hand side replaced by zero. The nonvanishing of the right-hand side of Eqs. (8)–(12) gives rise to some nontrivial technical points in the proof of the integrability of the Hamiltonian (1), as we shall see below. Another important property of the Dunkl operators Ji is that they preserve the linear space     
 Rm = µ(x) exp 2 ni xi  ni = −m, −m + 1, . . . , m , i = 1, . . . , N , i

(13) where µ(x) =

a      |sinh xi |b |cosh xi |b , sinh xij− sinh xij+  · i<j

(14)

i

for any nonnegative integer m. This fact will prove crucial for the calculation of the spectrum of the hyperbolic BN Sutherland spin model. We define a mapping ∗ : D ⊗ K → D ⊗ S, where D denotes the algebra of linear differential operators on C ∞ (RN ), as follows. If D ∈ D, we set ∗  D Kα1 · · · Kαr = (−1)r D Sαr · · · Sα1 , (15) where αk stands for ij or i. The mapping ∗ is then linearly extended to D ⊗ K. For instance, the Hamiltonian of the hyperbolic BN Sutherland spin model (1) is obtained by applying the “star” mapping to the operator 

 H =− sinh−2 xij− (a − Kij ) + sinh−2 xij+ (a − K˜ ij ) ∂x2i + 2a i

+b


i

i<j

sinh

−2

xi (b − Ki ) − b


i

  cosh−2 xi b − Ki .

(16)

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Let 0 be the antisymmetrisation operator, defined by the relations 20 = 0 and ij 0 = −0 , j > i = 1, . . . , N. More explicitly, 1
l P l , N! N!

0 =

l=1

where Pl denotes an element of the realization of the symmetric group generated by the operators ij , and l is the signature of Pl . For instance, if N = 2, 3 the antisymmetriser 0 is given by N =2: N =3:

 1 1 − 12 , 2 1 0 = (1 − 12 − 13 − 23 + 12 13 + 12 23 ) . 6

0 =

The total antisymmetriser  with respect to the action of the operators ij and i is determined by the relations 2 =  and ij  = −,

i  = −,

It may be easily shown that =

1 2N

j > i = 1, . . . , N.

(17)

  (1 − i ) 0 . i

Since Kij2 = Ki2 = 1, the relations (17) are equivalent to Kij  = −Sij ,

Ki  = −Si ,

j > i = 1, . . . , N.

(18)

From these relations and the definition of the star mapping it follows immediately that A  = A∗ 

(19)

for every operator A ∈ D ⊗ K. The proof of the integrability of the hyperbolic BN Sutherland spin model is based on the following lemmas. Lemma 1. If B ∈ D ⊗ S satisfies B = 0, then B = 0. Proof. The operator B ∈ D ⊗ S is of the form
B= fi (x) Bi ∂ i ,

(20)

i∈I

where Bi ∈ S, fi ∈ C ∞ (RN ), i = (i1 , . . . , iN ) is a multiindex belonging to a finite subset I ⊂ N0N (with N0 = {0, 1, 2, . . . }), and ∂ i = ∂1i1 · · · ∂NiN . Let us denote by Wl , with l ∈ L = {1, . . . , 2N N!}, the elements of the realization of the Weyl group WN generated by the operators ij and i . The action of the total antisymmetrisation operator  over a factored state ψ = ϕ(x) |s, with ϕ ∈ C ∞ (RN ) and |s ∈ S, is given by
ψ = l (Wl ϕ) Wl |s , (21) l∈L

On the Sutherland Spin Model of BN Type

197

where l = ±1 is the parity of the total number of generators ij , i in any decomposition of Wl . By hypothesis, B(ψ) =




  l fi (x)∂ i Wl ϕ · Bi (Wl |s) = 0 .

(22)

i∈I,l∈L

  Applying the latter equation to a family of functions ϕj (x) j ∈I ×L satisfying the condition   i∈I,l∈L det ∂ i Wl ϕj = 0 , j ∈I ×L

(23)

we obtain Bi (Wl |s) = 0 ,

for all i ∈ I, l ∈ L .

In particular (taking l ∈ L so that Wl is the identity) Bi |s = 0 for all i ∈ I . Since |s ∈ S is arbitrary, it follows that Bi = 0 for all i ∈ I , and hence B vanishes identically.

Lemma 2. If B ∈ D ⊗ K commutes with  then (AB)∗ = A∗ B ∗ for all A ∈ D ⊗ K. Proof. Using Eq. (19) and the hypothesis repeatedly we obtain: (AB)∗  = AB = AB = A∗ B = A∗ B = A∗ B ∗  . The statement follows from the previous lemma.



We shall often make use of the following immediate consequence of Lemma 2: Lemma 3. If A, B ∈ D ⊗ K commute with  then [A, B]∗ = [A∗ , B ∗ ]. We shall now construct a complete family of commuting integrals of motion for the hyperbolic BN Sutherland spin Hamiltonian (1). The construction is based on the observation that the operator H in Eq. (16) can be expressed as H =−




Ji2 .

(24)

i

By (7), the operators Ip =




2p

Ji ,

p ∈ N,

(25)

i

commute with one another. In view of the previous lemma, it suffices to prove that Ip commutes with the total antisymmetriser  for all p ∈ N to conclude that the star operators Ip∗ (p ∈ N) form a commuting family of integrals of motion of H ∗ = −I1∗ . We shall in fact prove the following stronger result: Lemma 4. The operators Ip (p ∈ N) commute with the permutation and sign changing operators Kij and Ki .

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F. Finkel, D. G´omez-Ullate, A. Gonz´alez-L´opez, M.A. Rodr´ıguez, R. Zhdanov

Proof. First of all, the elementary permutation Ki,i+1 commutes with Ip for i = 1, . . . , N − 1. Indeed, Ki,i+1 commutes with Jj for j = i, i + 1 by (8), while for j = i, i + 1 we have 2p

Ki,i+1 Ji

2p

2p

= Ji+1 Ki,i+1 − 2a 2p

Ki,i+1 Ji+1 = Ji Ki,i+1 + 2a

2p−1


r=0 2p−1


2p−r−1 r Ji+1 ,

Ji

2p−r−1 r Ji+1 .

Ji

r=0

Since an arbitrary permutation can be expressed as the product of elementary permutations, this shows that Kij commutes with Ip for all i = j . Secondly, the sign reversing operator KN commutes with Ip for all p, since KN Ji = Ji KN , while

if

i < N,

KN JN2 = −JN KN JN − 2(b + b )JN = JN2 KN .

This implies that Ki commutes with Ip for an arbitrary i = 1, . . . , N, since





0 = KiN KN KiN , Ip = KiN KiN Ki , Ip = Ki , Ip . The operators Ip∗ (p ∈ N) thus form an infinite commuting family. Moreover, by examining the terms of highest order in the partial derivatives one can easily conclude that the set {Ip∗ }N p=1 is algebraically independent. We have thus proved the main result of this section: Theorem 1. The operators {Ip∗ }N p=1 form a complete family of commuting integrals of motion for the hyperbolic BN Sutherland spin Hamiltonian H ∗ = −I1∗ . We also note that the constants of motion Ip∗ (p ∈ N) commute with the total permutation and sign changing operators ij and i . This is a consequence of Lemma 4 and the following general fact: Lemma 5. If A ∈ D ⊗ K commutes with Kij (resp. Ki ) then A∗ commutes with ij (resp. i ). Proof. We can write A=




Dγ K γ ,

(26)

γ ∈

where Kγ is a monomial in Kkl and Kl , Dγ ∈ D, and is a finite set such that {Kγ | γ ∈ } is linearly independent. By hypothesis
A = Kij AKij = Kij (Dγ ) Kij Kγ Kij , (27) γ ∈

where Kij (Dγ ) is the image of Dγ under the natural action of Kij in D. Comparing (27) with (26) we conclude that for each γ ∈ there exists γ  ∈ such that Kij Kγ Kij = Kγ  , and Kij (Dγ ) = Dγ  .

On the Sutherland Spin Model of BN Type

199

On the other hand we have


ij A∗ ij = Kij (Dγ ) Sij Kγ∗ Sij = Kij (Dγ ) Kγ∗ = Dγ  Kγ∗ . γ ∈

γ ∈

γ ∈

Since (γ  ) = γ , we have  = , and therefore the right-hand side of the previous

formula equals A∗ . The equality i A∗ i = A∗ is established in a similar way. 3. Spectrum of the Sutherland Spin Model of BN Type We shall now study the algebraic sector of the spectrum of the BN -type Sutherland spin model (1). The starting point in our discussion is the invariance under H of the space Rm for all m = 0, 1, . . . , which is an immediate consequence of Eq. (24) and the definition of the operators Ji . We shall construct a basis of the H -invariant space Rm with respect to which the matrix of H |Rm is upper triangular, thereby obtaining an exact formula for the spectrum of this operator. To derive the spectrum of H ∗ from that of H , we shall make use of the identity H ∗ [ (ϕ|s)] =  [(H ϕ)|s] ,

(28)

where ϕ ∈ C ∞ (RN ) and |s ∈ S. The latter identity, which is an immediate consequence of Eq. (19) and Lemma 4, implies that the spaces Mm = (Rm ⊗ S) ,

m = 0, 1, . . . ,

(29)

are invariant under H ∗ . From the basis of Rm triangularizing H |Rm we shall construct a basis of Mm with respect to which H ∗ |Mm is also represented by an upper triangular matrix. In this way we shall determine the spectrum of the Sutherland spin model of BN type (1). Let us start by computing the spectrum of the operator H . Following closely the approach of Ref. [2] for spin models of AN type, we shall define a suitable partial ordering in the set of (scaled) exponential monomials  
fn (x) = µ(x) exp 2 n = (n1 , . . . , nN ) , −m ≤ ni ≤ m , (30) ni xi , i

spanning the subspaces Rm . We shall then show that the operator H is represented by a triangular matrix in any partially ordered basis of Rm . The partial ordering in the basis (30) is defined as follows. Given a multiindex n = (n1 , . . . , nN ) ∈ ZN , we define the nonnegative and nonincreasing multiindex [n] by         [n] = ni1  , . . . , niN  , where ni1  ≥ · · · ≥ niN  . (31)

If n, n ∈ ZN are nonnegative and nonincreasing multiindices, we shall say that n ≺ n if n1 − n1 = · · · = ni−1 − ni−1 = 0 and ni < ni . For two arbitrary multiindices n, n ∈ ZN , by definition n ≺ n if and only if [n] ≺ [n ]. The partial ordering ≺ in ZN induces a partial ordering in the exponential monomial basis (30), namely fn ≺ fn if and only if n ≺ n . The action of the Weyl group on the basis (30) preserves this partial ordering, i.e., if fn ≺ fn then Wfn ≺ Wfn for all W ∈ WN .

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If n = (n1 , . . . , nN ) ∈ ZN and s ∈ Z, we shall use the following notation: #(s) = card{i : ni = s} , (s) = min{i : ni = s} , with (s) = +∞ if ni = s for all i = 1, . . . , N. For instance, if n = (5, 2, 2, 1, 1, 1, 0) then #(1) = 3 and (1) = 4. The computation of the spectrum of H is based on the following result:

Proposition 1. If n ∈ ZN is a nonnegative and nonincreasing multiindex, the following identity holds:
 n Ji fn = λn,i fn + cn,i fn , (32) n ∈ZN n ≺ n 

n ∈ R and where cn,i    2ni + b + b + 2a N + i + 1 − #(ni ) − 2(ni ) , λn,i = −b − b + 2a(i − N ) ,

ni > 0, ni = 0.

(33)

Proof. After some algebra one readily obtains the following expression:    n −n 1−n −n
αijj i − 1 βij i j − 1   + Ji fn = fn 2ni + b + b + 2a(N − 1) − 2a αij − 1 βij − 1 j i  1−2ni zi1−2ni − 1 z + 1 , − 2b − 2b i (34) zi + 1 zi − 1 where

zi = e2xi ,

αij = zi zj−1 ,

βij = zi zj .

Consider, for instance, the first term in αij in Eq. (34). Since j < i, nj ≥ ni . If nj = ni this term vanishes. If nj > ni we have   n −n nj −ni −1
αijj i − 1 fn (35) zj−r zir  , = fn 1 + αij − 1 r=1

where the last sum only appears if nj − ni > 1. In this case we have 0 < max{nj − r, ni + r} < nj for all r = 1, . . . , nj − ni − 1, so the multiindices of the monomials in the summation symbol in Eq. (35) satisfy (n1 , . . . , nj − r, . . . , ni + r, . . . , nN ) ≺ n . It may be likewise verified that the multiindices n of the monomials arising from the remaining terms in Eq. (34) either coincide with n or satisfy n ≺ n. The value of λn,i given in Eq. (33) can be computed by evaluating the constant part of the expression in square brackets in the right-hand sideof Eq. (34).  For instance, the first term in αij in Eq. (34) contributes the quantity −2a (ni ) − 1 to λn,i .

On the Sutherland Spin Model of BN Type

201



Note that Eq. (32) does not hold if n does not belong to ZN , so that Proposition 1 in general does not determine the spectrum of the restriction of Ji to Rm . On the other hand, for an arbitrary multiindex n ∈ ZN we shall only need the following weaker result: Corollary 1. If n ∈ ZN then Ji fn =






n γn,i fn ,

(36)

n ∈ZN [n ][n] 

n . for some real constants γn,i

Proof. We have Ji fn = Ji Wf[n] , where W is any element of WN such that fn = Wf[n] . Equation (36) then follows from the previous proposition, the commutation relations (8)–(12) and the invariance of the partial ordering ≺ under the action of the Weyl group.

The algebraic spectrum of H can be computed in closed form using the previous results, which imply the following proposition: Proposition 2. For all n ∈ ZN the following identity holds:

  Hfn = − λ2[n],i fn + cnn fn , with cnn ∈ R . i

(37)

n ∈ZN n ≺ n

Proof. Let W be any element of WN such that fn = Wf[n] . Since H = −I1 commutes with W by Lemma 4, from (32) we obtain


n n Hfn = W Hf[n] = − λ2[n],i fn − λ[n],i c[n],i Wfn − c[n],i W Ji fn . i

n ∈ZN, i n ≺ [n]

n ∈ZN, i n ≺ [n]

Equation (37) follows immediately from the latter equation, Corollary 1 and the invariance of the partial ordering ≺ under the action of the Weyl group.

  Let Bm = fn(j ) | j = 1, . . . , (2m + 1)N be any exponential monomial basis of the linear space Rm partially ordered according to ≺, i.e., such that if n(j ) ≺ n(k) then j < k. The previous proposition implies that the matrix of the restriction of H to Rm with respect to Bm is upper triangular. The eigenvalues of this matrix are its diagonal elements
En = − λ2[n],i ; −m ≤ nj ≤ m , j = 1, . . . , N . (38) i

It should be noted, however, that the algebraic eigenfunctions of H must satisfy appropriate boundary conditions that we shall now discuss. In the first place, since the potential of the BN -type spin Sutherland Hamiltonian (1) diverges on the hyperplanes xi ± xj = 0, 1 ≤ i ≤ j ≤ N , as (xi ± xj )−2 , we must require that the eigenfunctions

202

F. Finkel, D. G´omez-Ullate, A. Gonz´alez-L´opez, M.A. Rodr´ıguez, R. Zhdanov

of H vanish faster than (xi ± xj )1/2 near these hyperplanes. This yields the conditions (cf. Eq. (14)) a, b >

1 . 2

(39)

Secondly, the eigenfunctions must be square-integrable on their domain, which (without loss of generality) shall be taken as the open set X ⊂ RN given by 0 < xN < · · · < x1 .

(40)

The algebraic eigenfunctions lying in Rm will satisfy this condition if and only if 1 (b + b ) + a(N − 1) + m < 0 . 2

(41)

The latter inequality implies that the number of algebraic levels of H is finite, since m cannot exceed the integer m1 defined by    1  (42) m1 = max m ∈ N0  (b + b ) + a(N − 1) + m < 0 . 2 Note, in particular, that there are no algebraic eigenfunctions unless the parameters in the potential verify the inequality 1 (b + b ) + a(N − 1) < 0 . 2

(43)

From now on, we shall work on the maximal H -invariant subspace Rm1 . Remark 1. We could also have considered algebraic eigenfunctions of the Hamiltonian (1) antisymmetric under permutations but even under sign reversals. On these eigenfunctions, the action of the operators Sij and Si coincides with that of the operators −Kij and Ki , respectively. Therefore Eq. (15) in the definition of the star mapping should be replaced by ∗   D Kα1 · · · Kαr = (−1)r D Sαr · · · Sα1 , where r  is the number of permutation operators in the monomial Kα1 . . . Kαr . As a consequence, the Hamiltonian (1) is the image under the new star mapping of the operator H (−b, −b ), with H (b, b ) given by Eq. (16). It follows from Eqs. (39) and (43) that H ∗ possesses algebraic eigenfunctions of even parity if and only if a>

1 , 2

1 b<− , 2

1 − (b + b ) + a(N − 1) < 0 . 2

(44)

In particular, these inequalities and Eqs. (39) and (43) imply that H ∗ cannot have both odd and even parity algebraic eigenfunctions for any values of the parameters. Let us turn now to the algebraic spectrum of the BN -type Sutherland spin model (1). First of all, if Eqs. (39) and (43) are satisfied the wavefunctions in Mm1 are normalizable and well behaved near the singular hyperplanes. Secondly, if ϕ ∈ Rm1 is an eigenfunction of H with E and |s ∈ S is an arbitrary spin state, it follows  eigenvalue  from Eq. (28) that  ϕ|s ∈ Mm1 is either zero or an eigenfunction of H ∗ with the same energy E. The algebraic spectrum of H ∗ is thus a subset of   the algebraic spectrum of H . Instead of studying the conditions under which  ϕ|s does not vanish, in the

On the Sutherland Spin Model of BN Type

203

next proposition we shall directly construct from Bm1 a basis of Mm1 with respect to which the matrix of H ∗ |Mm1 is upper triangular. We shall use the following notation: m0 =

N −M , 2M + 1

(45)

where x denotes the smallest integer greater than or equal to x ∈ R. Proposition 3. The H ∗ -invariant space Mm1 is nonzero if and only if m0 ≤ m1 . If this condition holds, a basis of Mm1 consists of states of the form    fn |s1 , . . . , sN  , (46)

where n ∈ ZN and |s1 , . . . , sN  satisfy:  2M + 1 , if 0 < ni ≤ m1 i) #(ni ) ≤ (47) M, if ni = 0 ; ii) si > sj , iii) si > 0 ,

if ni = nj

and i < j ;

if ni = 0 .

(48) (49)

Proof. In the first place, since W = (W ) for any element W of the realization of   WN generated by ij , i , the space Mm1 is spanned by states of the form  fn |s , N

where n ∈ Z and |s ∈ S. Moreover, from the definition of the total antisymmetriser  it follows that a state of the form (46) with n ∈ [ZN ] vanishes if and only if either si = sj when ni = nj > 0 and i = j , or si = ±sj when ni = nj = 0 and i = j , or si = 0 when ni = 0. In particular, the condition (47) is necessary to ensure that the state (46) does not vanish. Since this condition cannot hold if n1 < m0 , and n1 ≤ m1 for all states in Mm1 , it follows that Mm1 is trivial if m1 < m0 . On the other hand, if m0 ≤ m1 all the states (46)–(49) are nonzero, and it is immediate to show that they are also linearly independent. Moreover, any nonzero state of the form (46) with n ∈ [ZN ] can be written as  

   W fn |s1 , . . . , sN  = (W )  fn |s1 , . . . , sN  , where n and |s1 , . . . , sN  satisfy (47)–(49), and W ∈ WN is an element of the stabilizer of fn .

Corollary 2. If m0 ≤ m1 , the dimension of the H ∗ -invariant space Mm1 is given by     m1 (2M + 1) + M dim Mm1 = . (50) N Proof. Indeed, from Eqs. (48)–(49) it easily follows that     
  M 2M + 1 2M + 1 ... dim Mm1 = N0 N1 Nm1 N0 +···+Nm1 =N

 m1 (2M + 1) + M . = N 



204

F. Finkel, D. G´omez-Ullate, A. Gonz´alez-L´opez, M.A. Rodr´ıguez, R. Zhdanov

The algebraic spectrum of the hyperbolic BN Sutherland spin model (1) follows directly from Proposition 3: Theorem 2. If m0 ≤ m1 , the algebraic energies of H ∗ are given by En∗ = −




λ2n,i ,

(51)

i



where n ∈ ZN satisfies the condition (47) and λn,i is given by Eq. (33).   Proof. Let ψn,s =  fn |s1 , . . . , sN  be an element of the basis (46)–(49) of Mm1 . Using Eqs. (37) and (28) we easily obtain H ∗ ψn,s = −


i

λ2n,i ψn,s +




   cnn  fn |s1 , . . . , sN  .

(52)

n ∈ZN n ≺ n

  The state  fn |s1 , . . . , sN  is proportional to a basis element of the form ψn ,s  , with fn = Wfn for some element W of WN and n ≺ n. Therefore the matrix of H ∗ |Mm1 in the basis (46)–(49) is also upper triangular, with diagonal elements given by (51).

The algebraic ground state of the Hamiltonian H ∗ and its degeneracy d can be determined using Proposition 3 and the previous theorem: Proposition 4. The multiindex n yielding the algebraic ground state and the degeneracy of this state are given by   M n = 0, d = i) N ≤ M: ; N ii) N > M: r 2M+1 2M+1 M ! "  ! "  ! "  ! "  (53) n = m0 , . . . , m0 , m0 − 1, . . . , m0 − 1, . . . , 1, . . . , 1, 0, . . . , 0 ,   2M + 1 d= , r where r = N − (m0 − 1)(2M + 1) − M. Proof. Note, first of all, that from the definition (45) of m0 it follows that 1 ≤ r ≤ 2M+1. Let n ∈ ZN be a nonnegative and nonincreasing multiindex satisfying n1 ≤ m1 and Eq. (47). The contribution to the algebraic energy (51) of all the terms λ2n,i such that ni is equal to a fixed value k ∈ {nj }N j =1 can be easily evaluated in closed form. Indeed, denoting for brevity i0 = (k) , i1 = (k) + #(k) − 1 ,   k 1 αk = α − , b + b + 2a(N − 1) , α=− 2a a from (33) we have, for k > 0,

On the Sutherland Spin Model of BN Type

−

i1


λ2n,i = −4a 2

i=i0

i1


205

(i − i0 − i1 − αk + 1)2

i=i0

   = −4a 2 #(k) αk2 + i0 + i1 − 2 αk +

  1  1 2 i0 + i0 i1 + i12 − 7i0 + 5i1 + 1 . 3 6

(54)

Similarly, the contribution to the algebraic energy of all the terms in Eq. (51) with k = 0, −

i1


λ2n,i = −4a 2

i=i0

i1


(i + α − 1)2 ,

(55)

i=i0

is easily seen to equal the right-hand side of Eq. (54), since α0 = α. The derivative of the right-hand side of Eq. (54) with respect to k, with i0 and i1 fixed, is given by 4a#(k)(2αk + i0 + i1 − 2) .

(56) m1 a

> 0 on account of This is clearly positive, since i0 + i1 − 2 ≥ 0 and αk ≥ α − (42). Hence the energy decreases if k decreases, i0 and i1 being fixed. It follows that any multiindex n corresponding to the minimum value of the algebraic energy must be of the form   n = m, . . . , m, m − 1, . . . , m − 1, . . . , , . . . ,  , (57) where  = 0 or  = 1 and m0 ≤ m ≤ m1 . Let k be an integer in the range 1 to m. We shall consider next the change in the algebraic energy associated to the multiindex (57) when #(k) decreases by 1, while #(k − 1) increases by 1 (including the case in which k =  = 1 and therefore #(k −1) = #(0) = 0). Suppose, for instance, that k ≥ 2. Denoting i2 = (k − 1) + #(k − 1) − 1, the change in the algebraic energy is given by  i
i2 1 −1
2 2 (i − i0 − i1 − αk + 2) − (i − i1 − i2 − αk−1 + 1)2 4a − i=i0

+

i1


i=i1

(i − i0 − i1 − αk + 1)2 +

i2




(i − i1 − i2 − αk−1 )2 

i=i1 +1

i=i0

= −4 (1 + 2a(αk + i1 − 1)) ≤ −4(1 + 2aαk ) < 0 .

(58)

It may be similarly verified that when k = 1 and either  = 0 or  = 1 the change in the algebraic energy is negative. This implies that the multiindex n yielding the algebraic ground state is of the form (53) if N > M, and zero otherwise. The degeneracy of the algebraic ground state then follows immediately from Proposition 3.

The algebraic ground energy can be easily obtained from the previous proposition and Eqs. (33) and (38). Indeed, denoting ν = 2M + 1 ,

c = b + b + 2m0 − a ,

the algebraic ground energy for even ν (that is, half-integer M) is given by

206 ∗ E0,e =

F. Finkel, D. G´omez-Ullate, A. Gonz´alez-L´opez, M.A. Rodr´ıguez, R. Zhdanov

  1 1 4 νm0 4m20 (aν − 1) − 6cm0 − aν − 2 + (a 2 − 3c2 )N + 2acN 2 − a 2 N 3 , 3 3 3

while for odd ν (integer M), the algebraic ground energy reads   ∗ ∗ = E0,e + m0 a + 2c + 2m0 (1 − aν) . E0,o 4. The BN -Type Sutherland Spin Chain In this section we shall introduce a quantum spin chain closely related to the hyperbolic BN Sutherland spin model (1). We shall establish the integrability of this chain by explicitly constructing a complete family of commuting integrals of motion associated with the integrals Ip∗ of the Hamiltonian (1). The starting point in this construction is the following expansion of the hyperbolic BN Sutherland spin Hamiltonian (1) in terms of the parameter a:
∂x2i + a H∗ + a 2 U (x) , (59) H∗ = − i

where H∗ =


 i=j

 $
# sinh−2 xij− Sij + sinh−2 xij+ S˜ij + β sinh−2 xi − β  cosh−2 xi Si , i

(60) U (x) =


# i=j

$
# $ sinh−2 xij− + sinh−2 xij+ + β 2 sinh−2 xi − β 2 cosh−2 xi

(61)

i

and

b b , β = . a a The Hamiltonian of the hyperbolic Sutherland spin chain of BN type is by definition the operator H∗0 , where the superscript 0 means that the coordinates xi are replaced by the equilibrium points xi0 of the potential U , which satisfy the system β=

∂U 0 0 (x , . . . , xN ) = 0, ∂xi 1

i = 1, . . . , N .

(62)

A necessary condition for the system (62) to have a solution in the region xi > 0, i = 1, . . . , N, is that β 2 > β 2 + 2(N − 1). In fact, there is strong numerical evidence that a solution exists if and only if |β  | > |β| + 2(N − 1). Note that this inequality corresponds to the condition (43) (when b > 0) or (44) (when b < 0) necessary for the existence of square-integrable algebraic eigenfunctions of the dynamical model, a fact certainly deserving further study. Let us define the operator Ji ∈ C ∞ (RN ) ⊗ K by Ji = ∂xi − a Ji , We shall also denote Ip =


i

2p

Ji ,

i = 1, . . . , N . p ∈ N.

(63)

(64)

On the Sutherland Spin Model of BN Type

207

Note that I1 is the coefficient of a 2 in I1 = −H , and thus equals −U (x) by Eq. (59). We shall prove below that the operators {Ip∗0 }N p=1 form a complete family of commuting integrals of motion for the Sutherland BN spin chain Hamiltonian H∗0 . Let us begin by establishing the commutativity of the operators Ip∗0 for all p ∈ N. In fact, the following stronger result holds: Proposition 5. The operators Ip∗ (p ∈ N) form a commuting family. Proof. The proposition follows directly from the commutativity of the operators Ip∗ , taking into account that Ip∗ is the coefficient of a 2p in the expansion of Ip∗ in powers of a.

We show next that Ip∗0 commutes with H∗0 for all p ∈ N. Note that this is not a consequence of the previous proposition, since H∗0 is not proportional to I1∗0 = −U (x0 ). Proposition 6. The operators Ip∗0 (p ∈ N) commute with the BN Sutherland spin chain Hamiltonian H∗0 . Proof. From (24) it follows that [H, Ji ] = 0 ,

i = 1, . . . , N .

Using Eqs. (6) and (59) in the previous identity and equating to zero the coefficient of a 2 in the resulting expression we obtain [H , Ji ] = −

∂U , ∂xi

i = 1, . . . , N .

It follows that [H , Ip ] = −

N 2p−1
2p − 1


Jir

r

i=1 r=0

∂U 2p−r−1 J ≡ Cp . ∂xi i

The operator C1 = −[H, U (x)] vanishes identically, since H does not contain partial derivatives and U (x) is a symmetric even function of x. Note, however, that Cp need not vanish for p > 1. Expanding in powers of a the identities [H, ] = [Ip , ] = 0 we obtain [H, ] = [Ip , ] = 0 , p ∈ N. By Lemma 3 we have ∗ ∗

p ∈ N. (65) H , Ip = Cp∗ , From the symmetry of the function U (x) with respect to permutations and sign changes it follows that ∂U/∂xi commutes with Kj k and Kj for j, k = i, while Kij

∂U ∂U = Kij , ∂xi ∂xj

Ki

∂U ∂U =− Ki . ∂xi ∂xi

From these identities one can easily show that Cp∗ is of the form Cp∗ =


∂U i

∂xi

∗ Cp,i ,

p ∈ N.

The commutativity of H∗0 with Ip∗0 follows from the latter equation, Eq. (65) and the definition of the equilibrium points (62).

208

F. Finkel, D. G´omez-Ullate, A. Gonz´alez-L´opez, M.A. Rodr´ıguez, R. Zhdanov

Finally, we prove the algebraic independence of the set {Ip∗0 }N p=1 , thus establishing the integrability of the hyperbolic Sutherland spin chain of BN type: Theorem 3. The operators {Ip∗0 }N p=1 form a complete family of commuting integrals of motion for the BN Sutherland spin chain Hamiltonian H∗0 . 0 Proof. The set {Ip0 }N p=1 is algebraically independent, since the operators Ji (i = 1, . . . , N) are linearly independent and commute with each other. The counterpart of Lemma 2 for the inverse of the star operator implies that the family {Ip0∗ }N p=1 is also 0∗ algebraically independent. The lemma now follows from the identity Ip = Ip∗0 .

Note also that the constants of motion Ip∗0 (p ∈ N) commute with the total permutation and sign reversing operators ij and i . This follows from the identities [Ip∗ , ij ] = [Ip∗ , i ] = 0 by taking the coefficient of a 2p . Acknowledgements. This work was partially supported by the DGES under grant PB98-0821. R. Zhdanov would like to acknowledge the financial support of the Spanish Ministry of Education and Culture during his stay at the Universidad Complutense de Madrid. The authors would also like to thank the referee for several helpful remarks.

References 1. Baker, T.H., Forrester, P.J.: The Calogero–Sutherland model and generalized classical polynomials. Commun. Math. Phys. 188, 175–216 (1997) 2. Bernard, D., Gaudin, M., Haldane, F.D.M., Pasquier, V.: Yang–Baxter equation in long-range interacting systems. J. Phys. A: Math. Gen. 26, 5219–5236 (1993) 3. Bernard, D., Pasquier, V., Serban, D.: Exact solution of long-range interacting spin chains with boundaries. Europhys. Lett. 30, 301–306 (1995) 4. Basu-Mallick, B.: Spin-dependent extension of Calogero–Sutherland model through anyon-like representations of permutation operators. Nucl. Phys. B 482, 713–730 (1996) 5. 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) 6. Carey, A.L., Langmann, E.: Loop groups, anyons and the Calogero–Sutherland model. Commun. Math. Phys. 201, 1–34 (1999) 7. Cherednik, I.: A unification of Knizhnik–Zamolodchikov and Dunkl operators via affine Hecke algebras. Invent. Math. 106, 411–431 (1991) 8. Cherednik, I.: Integration of quantum many-body problems by affine Knizhnik–Zamolodchikov equations. Adv. Math. 106, 65–95 (1994) 9. D’Hoker, E., Phong, D.H.: Calogero–Moser systems in SU(N) Seiberg–Witten theory. Nucl. Phys. B 513, 405–444 (1998) 10. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989) 11. Dunkl, C.F.: Orthogonal polynomials of types A and B and related Calogero models. Commun. Math. Phys. 197, 451–487 (1998) 12. Finkel, F., G´omez-Ullate, D., Gonz´alez-L´opez, A., Rodr´ıguez, M.A., Zhdanov, R.: AN -type Dunkl operators and new spin Calogero–Sutherland models. Commun. Math. Phys. 221, 477–497 (2001) 13. Finkel, F., G´omez-Ullate, D., Gonz´alez-L´opez, A., Rodr´ıguez, M.A., Zhdanov, R.: New spin Calogero–Sutherland models related to BN -type Dunkl operators. Nucl. Phys. B 613, 472–496 (2001) 14. Fowler, M., Minahan, A.: Invariants of the Haldane–Shastry SU (N) chain. Phys. Rev. Lett. 70, 2325–2328 (1993) 15. Frahm, H.: Spectrum of a spin chain with inverse square exchange. J. Phys. A: Math. Gen. 26, L473–L479 (1993) 16. Gorsky, A., Nekrasov, N.: Hamiltonian systems of Calogero type, and 2-dimensional Yang–Mills theory. Nucl. Phys. B 414, 213–238 (1994) 17. Ha, Z.N.C., Haldane, F.D.M.: Models with inverse-square exchange. Phys. Rev. B 46, 9359–9368 (1992)

On the Sutherland Spin Model of BN Type

209

18. Haldane, F.D.M.: Exact Jastrow–Gutzwiller resonating-valence-bond ground state of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r 2 exchange. Phys. Rev. Lett. 60, 635–638 (1988) 19. Hikami, K., Wadati, M.: Integrability of Calogero–Moser spin system. J. Phys. Soc. Japan 62, 469– 472 (1993) 20. Hikami, K., Wadati, M.: Integrable spin- 21 particle systems with long-range interactions. Phys. Lett. A 173, 263–266 (1993) 21. Inozemtsev, V.I., Sasaki, R.: Universal Lax pairs for spin Calogero–Moser models and spin exchange models. J. Phys. A: Math. Gen. 34, 7621–7632 (2001) 22. Kasman, A.: Bispectral KP solutions and linearization of Calogero–Moser particle systems. Commun. Math. Phys. 172, 427–448 (1995) 23. Lapointe, L., Vinet, L.: Exact operator solution of the Calogero–Sutherland model. Commun. Math. Phys. 178, 425–452 (1996) 24. Minahan, J.A., Polychronakos, A.P.: Integrable systems for particles with internal degrees of freedom. Phys. Lett. B 302, 265–270 (1993) 25. Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep. 94, 313–414 (1983) 26. Polychronakos, A.P.: Exchange operator formalism for integrable systems of particles. Phys. Rev. Lett. 69, 703–705 (1992) 27. Polychronakos, A.P.: Lattice integrable systems of Haldane–Shastry type. Phys. Rev. Lett. 70, 2329– 2331 (1993) 28. Polychronakos, A.P.: Exact spectrum of SU(n) spin chain with inverse-square exchange. Nucl. Phys. B 419, 553–566 (1994) 29. Polychronakos, A.P.: Waves and solitons in the continuum limit of the Calogero–Sutherland model. Phys. Rev. Lett. 74, 5153–5157 (1995) 30. Polychronakos, A.P.: Generalized Calogero models through reductions by discrete symmetries. Nucl. Phys. B 543, 485–498 (1999) 31. Shastry, B.S.: Exact solution of an S = 1/2 Heisenberg antiferromagnetic chain with long-ranged interactions. Phys. Rev. Lett. 60, 639–642 (1988) 32. Sutherland, B.: Exact results for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019–2021 (1971) 33. Sutherland, B.: Exact results for a quantum many-body problem in one dimension. II. Phys. Rev. A 5, 1372–1376 (1972) 34. Sutherland, B., Shastry, B.S.: Solution of some integrable one-dimensional quantum systems. Phys. Rev. Lett. 71, 5–8 (1993) 35. Taniguchi, N., Shastry, B.S., Altshuler, B.L.: Random matrix model and the Calogero–Sutherland model: A novel current-density mapping. Phys. Rev. Lett. 75, 3724–3727 (1995) 36. Turbiner, A.V.: Quasi-exactly solvable problems and sl(2) algebra. Commun. Math. Phys. 118, 467–474 (1988) 37. Turbiner, A.: Lie algebras and polynomials in one variable. J. Phys. A: Math. Gen. 25, L1087–L1093 (1992) 38. Yamamoto, T.: Multicomponent Calogero model of BN -type confined in a harmonic potential. Phys. Lett. A 208, 293–302 (1995) 39. Yamamoto, T., Tsuchiya, O.: Integrable 1/r 2 spin chain with reflecting end. J. Phys. A: Math. Gen. 29, 3977–3984 (1996) Communicated by L. Takhtajan

Commun. Math. Phys. 233, 211–230 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0752-x

Communications in

Mathematical Physics

Bernoulli Elliptical Stadia Gianluigi Del Magno1,∗ , Roberto Markarian2 1 2

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. E-mail: [email protected] Instituto de Matem´atica y Estad´ıstica “Prof. Ing. Rafael Laguardia” (IMERL), Facultad de Ingenier´ıa, Universidad de la Rep´ublica, Montevideo, Uruguay. E-mail: [email protected]

Received: 10 May 2002 / Accepted: 24 June 2002 Published online: 10 January 2003 – © Springer-Verlag 2003

Abstract: Let Qa,h be a convex region of the plane whose boundary consists of two semiellipses joined by two (straight) lines parallel to the major axis of the semiellipses (elliptical stadium). The axes of the have length 2 and 2a, a > 1, and the
semiellipses √ √ lines have length 2h. For 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, we give a complete proof of the following result: the billiard map in the elliptical stadium Qa,h is ergodic, K-mixing and Bernoulli with respect to the natural billiard measure. 1. Introduction Let Q = Qa,h be a convex region of the plane bounded by two semiellipses with axes of length 2 and 2a, a > 1 and joined by two straight lines of length 2h which are parallel to the major axis of the semiellipses. The region Q is called elliptical stadium. Let T be the dynamical system describing the free motion of a point mass in Q with elastic reflections at the boundary according to the law: the angle of incidence equals the angle of reflection. In [M-O-P], T has non-zero Lyapunov exponents (L.E.)
it was√shown that the map √ 2 2 for 1 < a < 4 − 2 2 and h > 2a a − 1 by constructing a suitable T -invariant cone field. In this paper, for the same values of the parameters a, h, we give a complete proof of the following result: the billiard map T of the elliptical stadium Qa,h is ergodic, K-mixing and Bernoulli with respect to the natural billiard measure. The scheme of the proof of the ergodicity of T is the same as the one used in [De] for truncated ellipses and consists of two main steps. First we introduce a restricted phase space and the first return map
on it induced by T . Then we show that
is locally ergodic (i.e., each ergodic component of
is open (mod 0)) by using the version of the Fundamental Theorem proved in [L-W]. In this part of the proof, the main difference ∗ Present address: Department of Mathematics, Instituto Superior T´ecnico, 1049-001 Lisbon, Portugal. E-mail: [email protected]

212

G. Del Magno, R. Markarian

between this paper and [De] is the proof of the noncontraction property for blocks of type 2 (Lemma 9). The second step in the proof of the ergodicity consists in proving that
has only one ergodic component. We prove that
has only a finite number of ergodic components (Lemma 11), and construct a finite collection of trajectories (possibly disjoint) with the property that starting from any ergodic component we can reach any other ergodic component by travelling along these trajectories (Theorem 3). The Kolmogorov and Bernoulli properties are proved at the end of the paper using the general results contained in [Pe77] and [C-H, O-W]. 2. Background Material on Billiards Let Q be an open bounded and connected subset of the plane whose boundary consists of a finite number of closed C k+1 -curves i , k ≥ 2. The billiard in Q is the dynamical system describing the free motion of a point mass inside Q with elastic reflections at the boundary  = ∪i i . The study of billiards turns out to be simpler if we consider the so-called billiard map associated to the billiard in Q. Let n(q) be the unit normal of the curve  at the point q ∈  pointing toward the interior of Q. Consider the set M  = {(q, v) : q ∈ , |v| = 1, v, n(q) ≥ 0}. Let π denote the projection of M  onto Q, i.e., π(q, v) = q. We introduce the set of coordinates (s, θ ) on M  where s is the arc length parameter along  and θ is the angle between v and the tangent to the boundary at q. Clearly 0 ≤ θ ≤ π and n(q), v = sin θ. A natural probability measure on M  is dν = c sin θ dsdθ , where c = (2||)−1 is the normalizing factor and || stands for the total length of . The billiard map T : M  → M  is defined by T (q0 , v0 ) = (q1 , v1 ), where q1 is the point of  hit first by the oriented line through (q0 , v0 ) and v1 is the velocity vector after the reflection at q1 . Formally, v1 = v0 − 2n(q1 ), v0 n(q1 ). We denote by zi = (qi , vi ) ∈ M  , i ∈ N, the i th iterate of z0 = (q0 , v0 ) under the map T . These points represent the successive collisions with the  of a trajectory beginning at z0 = (q0 , v0 ) so that T (qi , vi ) = (qi+1 , vi+1 ). The angle between vi and the boundary at qi is denoted by θi , and the Euclidean distance between the bouncing points qi and qi+1 is denoted by ti . Since the speed of the point mass is one, ti is also the time between qi and qi+1 . The negative iterates zi = (qi , vi ), i < 0 of z0 are defined analogously. The main relations are T zi = zi+1 and qi+1 = qi + ti vi with i ∈ Z. The map T is piecewise C k . It is not well defined at z0 if n(q1 ) is not defined or if the oriented line through z0 is tangent to some k (θ1 = 0, π ). Finally T is continuous but not differentiable at z0 if  is C 1 but not C 2 at q1 . The measure ν is preserved by T . The sets of points x = (q, v) ∈ M  whose forward or backward trajectory is tangent to  or ends in i ∩ j have µ-measure zero. If T is well defined and differentiable at z˜ 0 = (q˜0 , v˜0 ), then for all z0 = (q0 , v0 ) in a small neighborhood of z˜ 0 the derivative matrix of T is given by   t K − sin θ  Dz0 T = 

0

0

0

sin θ1 t0 K0 − sin θ0 K1 − K0 sin θ1

t0 sin θ1

K1 t0 sin θ1

−1

 ,

(1)

where Ki = K(zi ), i = 0, 1, is the curvature of  at qi . If both q0 , q1 do not belong to straight lines, then (1) can be rewritten as

Bernoulli Elliptical Stadia

213



t0 − d0  r0 sin θ1  t0 − d0 − d1 r0 d1

t0 sin θ1 t0 −d1 d1

  ,

(2)

where ri = 1/Ki , i = 0, 1, is the radius of curvature of  at qi and di = ri sin θi , i = 0, 1. Note that if Ki > 0 (focusing component), then di is the length of the subsegment of q0 q1 contained in the disk D(qi ) tangent to  at qi with radius ri /2 (half-osculating disk). We remark the main differences with other usual conventions related with these formulas (see, for example [C-M]): the curvature of the ellipse is positive, the angle θ , measured from the boundary to v, is always positive and increases counterclockwise. 2.1. Elliptical billiards. In this section, we recall some elementary geometrical properties of elliptical billiards which we will use extensively in the sequel. Consider an ellipse with semiaxis of length 1 and a > 1 given by x(u) = a cos u, y(u) = sin u,

0 ≤ u ≤ 2π.

The curvature of the ellipse at the point (x(u), y(u)) is given by a K(u) = 2 2 . (a sin u + cos2 u)3/2 An ellipse may be also parameterized by the coordinate ϕ which is the angle made by the line tangent to the ellipse at the point (x, y) with the horizontal line y = 0. The equations of our ellipse and its curvature parameterized by ϕ are given by y(ϕ) = ±


1 1 + a 2 tan2 ϕ

,

x(ϕ) = −a 2 y(ϕ) tan ϕ,

(1 + a 2 tan2 ϕ)1/2 (1 + tan2 ϕ + a 2 tan2 ϕ + a 2 tan4 ϕ) . a 2 (1 + tan2 ϕ)5/2 The relation between the coordinates s and ϕ reads K(s)ds = dϕ from which we immediately obtain that dθ/dϕ = K(s)−1 dθ/ds. Now consider the billiard in our ellipse. Its phase space may be parameterized by the two sets of coordinates (s, θ ) and (ϕ, θ ). For the moment we use the second set of coordinates. The function √ cos2 θ −  2 cos2 ϕ a2 − 1 G(ϕ, θ ) = [ = is the eccentricity of the ellipse], 1 −  2 cos2 ϕ a (3) K(ϕ) =

is a first integral for the billiard map of the ellipse, meaning that G is constant (≤ 1) along the orbits of the billiard map. The phase space of the elliptical billiard is foliated by the level curves of G. It is important to compute the slope of the tangent lines to these level curves. We will give the answer in coordinates (s, θ ). The slope of the level curve G = G(z) at z = (s, θ ) is given by p(z) :=

dθ K(s) 2 sin 2ϕ = (1 − G). ds sin 2θ

(4)

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Level curves in the phase space are associated to curves in the plane called caustics. A caustic  is characterized by the property: if a segment (or its continuation) of a trajectory is tangent to , then any other segment (or its continuation) of the trajectory will be tangent to . It is well known that an ellipse E has two continuous families of caustics consisting of ellipses and hyperbolae confocal to E. Hence we can divide the level curves of G in two classes: elliptic and hyperbolic level curves. Definition 1. Denote by E the subset of M  consisting of elliptic level curves and by H the subset of M  consisting of hyperbolic level curves. Clearly, points in E have trajectories with elliptical caustic, and points in H have trajectories with hyperbolic caustic. We have 0 < G < 1 on E, and 1 − a 2 ≤ G < 0 on H. Along trajectories that pass through the foci, whose union forms a saddle connection in the phase space, we have G = 0. In the next lemma, we restate the results of Lemma 2 and Corollary 1 of [M-O-P]. This lemma illustrates an important property of the trajectories in ellipses on which the results of this paper rely.
√ Lemma 1. If 1 < a < 4 − 2 2 and q−1 , q0 , q1 , q2 are successive bouncing points of a billiard trajectory in the ellipse with hyperbolic caustic such that q0 and q1 are on one semiellipse, then i) only q0 and q1 belong to the same semiellipse (there are at most two successive reflections at the same semiellipse), ii) the tangency points of the segments q−1 q0 , q0 q1 , q1 q2 with the hyperbolic caustic occur inside the ellipse, iii) the distance between the tangency point of q0 q1 with the hyperbolic caustic and the boundary of the semiellipse is bounded below by a positive constant independent of q0 , q1 . Proof. We only need to prove part (iii), since (i) is Lemma 2 and (ii) is Corollary 1 of [M-O-P]. Suppose that we can find a sequence of trajectories like those in the statement of the lemma such that the distance between the tangency point of q0 q1 with the caustic (hyperbola) and the boundary of the elliptical billiard is arbitrarily close to 0. When a tangency point is very close to the boundary of the elliptical billiard, let us say to the point q1 , then the Reflection Formula (6) (see Sec. 3.1) implies that the tangency point between q1 q2 with the caustic occurs outside the ellipse, contradicting part (ii).

2.2. Elliptical stadia. We study now the billiard in the elliptical stadium. Let us denote by 1 , 2 the two semiellipses of the stadium with length m and semiaxis of length 1 and a > 1. In this representation 1 is the semiellipse contained in the half-plane {x ≥ 0} and the major semiaxis of the ellipses lies on the x-axis of the plane. The lines joining the two semiellipses have length 2h. We denote the focusing component of the boundary of the stadium by + = 1 ∪ 2 and the neutral component consisting of the two lines by 0 . Let a1 , a2 , a3 , a4 be the intersections of the semiellipses with the lines starting from the bottom right and moving counterclockwise, their s-coordinates are, respectively, 0, m, m + 2h, 2m + 2h (the total length of the  is 2m + 4h). We call such points corners of . We partition M  in the two rectangles M+ = π −1 (+ ) and M0 = π −1 (0 ) which are the subset of the phase space corresponding to the focusing and neutral components of . The set M+ is partitioned in two subsets M1 = π −1 (1 ) and M2 = π −1 (2 ). For i = 1, 2, we define / Mi }, Vi = Mi \ Ui = {z ∈ Mi : T −1 z ∈ Mi or T z ∈ Mi } Ui = {z ∈ Mi : T −1 z, T z ∈

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and Yi = {z ∈ Mi : T z ∈ / Mi }. In words, Ui is the set of points z ∈ Mi which reflect only once at i , while Vi is the set of points z ∈ Mi which have at least two consecutive reflections at i , and Yi is the set of points z ∈ Mi which may have several consecutive reflections backward in time at i but their next bounce is not at i . Note that Yi includes Ui and the points of Vi which leave Mi . We might say that Vi \ Yi is the set of “sliding” trajectories along a semiellipse. We define U = U1 ∪ U2 , V = V1 ∪ V2 and Y = Y1 ∪ Y2 . Let BN , N ≥ 0, be the subset of M0 consisting of vectors not perpendicular to 0 which have at least N consecutive reflections at 0 both in the past and in the future before reaching + . Clearly B0 = M0 . The need for defining all these sets is explained in Remark 6. For a fixed N ≥ 0, we define M = M+ ∪ BN . The set M is the reduced phase space on which we will work. The actual value of N will be chosen in the next section (see Remark 6). Let T denote the billiard map on M  . The map
: M → M is the first return map on M induced by T ,
(y) = T A(y) y, where A(y) = inf{i ≥ 1 : T i y ∈ M}. The measure µ = (ν(M))−1 ν|M is an invariant probability measure for
. Remark 1. Oseledec’s Theorem can be applied to T (see [K-S]) and gives the existence ν-a.e. of L.E. for T . It follows immediately that
has L.E. µ-a.e. and these are proportional to the L.E. of T with constant of proportionality equal to the average (with respect to µ) of the return time A(y) over M (see [Wo85], Lemma 2.2). Also note that j  ∪∞ j =0 T M = M which implies that T is ergodic if and only if
is ergodic (see [C-F-S], Chapter 1, §5). The maps T and
are piecewise analytic diffeomorphisms. We denote by S + the subset of M where
fails to be C 1 and call it the singular set of
. It is easy to see that S + consists of points whose next reflection is at corners of the stadium or at the boundary of BN . The singular set S − for the map
−1 is defined similarly but this time we consider reflections in the past. The set Sn+ = S + ∪
−1 S + ∪ · · · ∪
−n+1 S + , n ≥ 1 is the singularity set for
n , and Sn− = S − ∪
S − ∪ · · · ∪
n−1 S − is the singularity + = ∪S + and S − = ∪S − . Then S = S + ∩ S − is the set of points set of
−n . Let S∞ ∞ n ∞ n ∞ ∞ whose trajectories hit a corner or the boundary of BN in the future and in the past. We denote by S + and S − the singular sets of T and T −1 which are subsets of M  consisting of points whose next reflection forward and backward in time is at a corner of the stadium. Similarly Sn+ and Sn− are the singular sets of T n and T −n for every n ≥ 0. ± Remark 2. The singularity sets Sn± of
are directly related to the singularity sets Sm + + of T . In fact, if z ∈ S ∩ M+ , then we have z ∈ S (if there are no bounces on M0 ) + + or z ∈ S2N+1 (if T i z ∈ M0 for i = 1, . . . , 2N ). If z ∈ S + ∩ BN , then z ∈ SN+1 . + + Finally, if z ∈ Sn ∩ M+ , then z ∈ S(2N+1)n . Similar relations are obtained for Sn− . ± . Hence Sn± ⊂ S(2N+1)n

3. Hyperbolicity 3.1. Geometric optics. In order to understand the dynamics of a billiard, we study the dynamics of infinitesimal families of billiard trajectories (variations). For a planar billiard, it turns out that variations are parameterized by a projective quantity called focusing

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time. A complete description of the dynamics of a variation is given by the law of reflection which explains how the focusing time of a variation changes after reflecting at the boundary of the billiard table. A variation η(α, t) is an one-parameter smooth family of lines in R2 . We say that η(α, t) focuses if ∂η/∂α|α=0 = 0 for some t ∈ R which we call the focusing time of η(α, t). Let u ∈ Tz M, z ∈ M and ξ : (−ε, ε) → M be a curve such that ξ(0) = z and ξ  (0) = u. We associate with u the variations η+ (α, t) = q(α) + tv(α), t ∈ R,

η− (α, t) = q(α) + t v(α), ˜ t ∈ R,

where ξ(α) = (q(α), v(α)) and v(a) ˜ is obtained by reflecting v(α) at q(α) ∈ . Although for each vector u ∈ Tz M, we can construct infinitely many distinct variations η+ (α, t) (η− (α, t)), all of them focus and their focusing time is the same. We will call forward (backward) focusing time of u the focusing time of the variation η+ (α, t) (η− (α, t)). Let u = us ∂/∂s + uθ ∂/∂θ ∈ Tz M with us , uθ ∈ R, its forward and backward focusing times f+ (u), f− (u) are given by (see [Wo86])  sin θ if us = 0 uθ f± (u) = K± us , (5) 0 if us = 0 where K ≥ 0 is the curvature of  at π(z). Reflection Law. Let z0 = (q0 , v0 ) ∈ M \ S + and u ∈ Tz0 M. If f0 is the forward focusing time of u and f1 is the forward focusing time of Dz0 T u, then the relation between f0 and f1 is given by 1 1 2K1 + = , f1 t0 − f0 sin θ1

(6)

where K1 is the curvature of  at q1 (the other symbols are explained at the beginning of Sec. 2). Now we compare the evolution of two vectors after a reflection. The following lemma, which follows easily from (6), shows that the focusing time is monotone (under some conditions). This property will be very useful in the proof of the hyperbolicity of
. (1)

(2)

Lemma 2. Same notations as above. Let u1 , u2 ∈ Tz0 M. Denote by f0 , f0 the for(1) (2) ward focusing times of u1 , u2 and by f1 , f1 the forward focusing times of Dz0 T u1 , (1) (2) (2) Dz0 T u2 . Furthermore, assume that 0 < f0 < f0 < t (z) and 0 < f1 . Then (1) (2) 0 < f1 < f1 .
√ 3.2. Invariant cone field. We restrict ourselves to the case 1 < a < 4 − 2 2 and √ h > 2a 2 a 2 − 1. At this point N is any positive number. Later on in this section and in Sect. 4 (see Lemma 7), we will impose some conditions on N . Our final goal is to prove the ergodicity of the map
which, in turn, will give the ergodicity of T . The first step we take is the construction of a cone field on M which is eventually strictly
-invariant and has some additional properties required by the Fundamental Theorem (see Remark 6). As a consequence of the existence of this cone field, we have that
and therefore T have a.e. non-zero L.E. This property implies,

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by Pesin’s theory ([K-S]), that T has a.e. local stable and unstable manifolds which are absolute continuous. We define the cone field {C} on M as follows. For z ∈ V the outgoing and incoming segments of z are both tangent to a caustic (ellipse or hyperbola). Let P+ (z) and P− (z) be the corresponding points of tangency. We define C(z) as the set of all u ∈ Tz M which focus between π(z) and P+ (z), C(z) = {u ∈ Tz M : 0 ≤ f+ (u) ≤ d(π(z), P+ (z))}. Given z ∈ M+ , call Q+ (z) and Q− (z) the points where the outgoing and the incoming segments of z intersect the boundary of the half-osculating disk of  at π(z). For z ∈ U , we define C(z) = {u ∈ Tz M : 0 ≤ f+ (u) ≤ d(π(z), Q+ (z))}. Finally, for z ∈ BN , we define C(z) as the collection of all “divergent” vectors u ∈ Tz M, C(z) = {u ∈ Tz M : f+ (u) ≤ 0}. This definition is equivalent to the following one. C(z) is the set of vectors whose slopes satisfy: dθ dθ (z) ≥ 0, if z ∈ U ; (z) ≥ p(z), if z ∈ V ; ds ds

and

dθ (z) ≤ 0, if z ∈ BN . ds

In [M-O-P], a similar cone field was defined (it differs on M0 ) and proved to be eventually strictly invariant. One edge of C(z) is the line {u ∈ Tz M : us = 0}, while the second edge depends on z ∈ M. The next lemma shows that the slope of the second edge is uniformly bounded from −1/r(z) for z ∈ M+ . Note that −1/r(z) is the slope of a variation consisting of outgoing parallel rays. Lemma 3. There exists a 0 < δ < 1/rmax such that uθ /us ≥ −1/r(z) + δ for all z ∈ M+ and all u ∈ C(z). Proof. We need an estimate of the slope m(z) of the non-vertical edge of the cones C(z), z ∈ M+ . For z ∈ U , we have m(z) = 0, and the lemma is true for any fixed 0 < δ < 1/rmax on U . Now let z = (s, θ ) ∈ V . In this case m(z) = p(z). Using formula (4), we see that |m(z)| ≤ B() tan θ/r(z), where B() =  2 /2(1 −  2 ). Then it is clear that we can find θ¯ and 0 < δ1 < 1/rmax for which the lemma is true on the set {(s, θ ) ∈ V : θ ∈ (0, θ¯ ) ∪ (π − θ¯ , π)}. It remains to prove the lemma on the complementary set {(s, θ ) ∈ V : θ ∈ [θ¯ , π − θ¯ ]}. For a z belonging to this set, formula (5) and ¯ +>0 the fact that f+ (u) > 0 imply that m(z) ≥ −1/r(z) + δ2 , where δ2 = sin θ/m (m+ is finite). The proof is finished by taking δ = min{δ1 , δ2 }.

Let C  (z) be the closure of the complementary cone of C(z) in Tz M. Define F+ (z) = sup f+ (u)

and

F− (z) = sup f− (u).

m+ = sup F+ (z)

and

m− = sup F− (z).

u∈C(z)

u∈C  (z)

Now let z∈M+

z∈M+

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By Lemma 3 or directly by construction of {C(z)}, it follows that m+ and m− are finite and that m+ = m− because of the symmetry of the billiard table. The value of N . We choose the value of N in such a way that the minimum length l(N ) of all trajectories starting at M+ and ending in BN is greater than m+ . In fact, this condition needs to be satisfied only by trajectories which start at points of U , because for the trajectories starting at V the vectors in C (C  ) focuses forward (backward) inside the table (it is obvious for points in E, and use Lemma 1 for points in H). For trajectories starting at U an upper bound for m+ is given by rmax = a 2 < 2. Since the distance between the segment is 2, it is clear that we can take N = 1. This value is enough to guarantee that {C} is eventually strictly invariant. However, we may need a larger N for the noncontraction property to hold (as explained in the proof of Lemma 7).
√ √ Proposition 1. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, the cone field {C(z)} is eventually strictly
-invariant. Proof. First we note that {C(z)} is measurable because it is piecewise continuous by construction. We make now an observation which will simplify the proof of the strict invariance of C. Let X1 (z) and X2 (z) be two vectors belonging to the edges of C(z) such that 0 = f+ (X1 (z)) < f+ (X2 (z)). By construction of C(z), it is easy to see that f+ (X2 (z)) < t (z). Now if we apply Lemma 2 to the vectors of C(z), we obtain that Dz
X2 (z) ∈ C(
z) implies Dz
u ∈ int(C(
z)) for any u ∈ C(z) and u = X2 (z). In other words, to check that C is (strictly) invariant, we only need to check that the boundary vectors X2 are mapped (strictly) inside C. We have to check the invariance of C in the following cases: 1) z,
z ∈ Vi , 2) z ∈ Vi and
z ∈ Vj with i = j , 3) z,
z ∈ U , 4) z ∈ V and
z ∈ U , 5) z ∈ U and
z ∈ V , 6) z ∈ M+ ,
z ∈ BN or z ∈ BN ,
z ∈ M+ , 7) z,
z ∈ BN . With the exception of cases 6 and 7, these verifications were done in the Appendix of [M-O-P]. So here we only need to check that we have invariance (actually strict invariance) in cases 6 and 7. The proof is easier if we recall that C is (strict)
-invariant if and only if C  is (strict)
−1 -invariant. By our choice of N, it follows that a N > 0 such that for every z ∈ M+ ∩
−1 BN (z ∈ M+ ∩
BN ), the vector u ∈ C(z) (u ∈ C  (z)) focuses forward (backward) before x reflects at π(
x) (π(
−1 x)). In other words Dz
C(z) (Dz
−1 C  (z)) consists of divergent (convergent) vectors so that Dz
C(z) (Dz
−1 C  (z)) is strictly contained in C(
z) (C  (
−1 (z))). Let A be the set of points z ∈ M for which {C(z)} is not eventually strictly invariant. + and A ∩ S + = ∅. Clearly µ(A ) = 0. We can write A = A1 ∪ A2 , where A1 ⊂ S∞ 2 1 ∞ From Propositions 2-6 in the Appendix of [M-O-P], we see that the positive semitrajectory of a point in A2 does not cross the table from one semiellipse to the other and reflects only at one arc. Hence, it must be the periodic trajectory along the minor axis of the semiellipses. So µ(A2 ) = 0, and therefore µ(A) = 0.

By Theorem 1 in [Wo86], we conclude that
√ √ Theorem 1. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then
has non-zero Lyapunov exponents µ-a.e. on M. We show now that for the values of a and h considered in this paper, trajectories in the elliptical stadium have uniform defocusing time when they cross the table from one arc to the other.

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Consider a trajectory which crosses the stadium only once and whose endpoints are on distinct arcs. More precisely, consider a finite orbit {z, T z, . . . , T n z}, n > 0 such that T z, . . . , T n−1 z ∈ M0 if n > 1 and assume, without loss of generality that, z ∈ M1 and T n z ∈ M2 . We denote by L(z) the length of the trajectory, i.e., L(z) = n−1 i=0 ti .
√ √ Lemma 4. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then there exists a positive ˜ number d˜ such that L(z) − F+ (z) − F− (T n z) ≥ d. Proof. Let z = T n z. If z ∈ Ui , z ∈ Uj , then F+ (z) = d1 (z), F− (z ) = d2 (z) and the lemma follows from Proposition 6 in [M-O-P]. If z ∈ Vi , z ∈ Vj , then F+ (z) = d(π(z), P+ (z)), F− (z ) = d(π(z ), P− (z )). It is easy to see that Propositions 2 and 3 in [M-O-P] imply that P+ (z) precedes P− (z ) along the piece of trajectory [z, z ] and that their distance d(P+ (z), P− (z )) is bounded below by a positive constant not depending on z. In this case and in the following one, recall Lemma 1. Finally if z ∈ Vi and z ∈ Ui , the Lemma follows from Propositions 4 and 5 in [M-O-P]. The symmetric case can be proved in the same way.

4. Local Ergodicity 4.1. General setting. We are going to use the version of the Fundamental Theorem proved in [L-W] so the first thing to do is to “redefine” the map
and its phase space in agreement with the formalism introduced in [L-W]. Phase Space and Symplectic Boxes. It is easy to see that each set Ui , i = 1, 2, is path-connected, and that each set Vi , i = 1, 2, has two path-connected components Vi,j , j = 1, 2, where Vi,1 = {(s, θ ) : s ∈ Vi and θ < π/2} and Vi,2 = {(s, θ ) : s ∈ Vi and θ > π/2}. It also easy to see that BN has four disjoint path-connected components BN,j , j = 1, . . . , 4, each consisting of vectors with a base point at the same segment of 0 and forming with it an angle which is either greater than π/2 or less than π/2. The closure of the sets Vi,j , i, j = 1, 2, Ui , i = 1, 2, and BN,j , j = 1, . . . , 4 are the symplectic boxes (see [L-W] for the definition of a symplectic box) forming the phase space of
. We will denote them by Ai , i = 1, . . . , 10 and simply refer to them as boxes of M. The map
is an analytic diffeomorphism from the interior of each box to its image, but it might not be well defined at points belonging to the boundary of several boxes. However we can extend
from the interior of each box up to its boundary. By doing so,
becomes a multivalued map at points belonging to the boundary of several boxes. From now on, when we refer to
we will have in mind this multivalued map. The singular set S + divides each box Ai into a finite collection of sets. We obtain − a new partition of M, and we denote the closure of its elements by A+ n . Similarly S decomposes M into subsets whose closure is denoted by A− (note that the cardinality m of these two partitions is the same). − We have that
maps the interior of each set A+ n into the interior of a set Am . As we + have done before, we can extend
from the interior of each set An up to its closure, and by doing so we obtain a multivalued map also on S + . Similarly, by extending
−1 − from the interior of each set A− n up to its closure, we obtain a multivalued map on S . To summarize, we have decomposed the phase space M into boxes Ai . Each box − is decomposed in a finite number of sub-boxes A+ n (Am ) whose boundary consists of + − subsets of S (S ) and ∂Ai ,
is a diffeomorphism from the interior of any A+ n to

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+ the interior of a A− m and
is a multivalued map on S . Finally the cone field {C} is continuous in the interior of each Ai . Let X1 (z), X2 (z) be the unitary boundary vectors of C(z) = {u ∈ Tz M : u = aX1 (z) + bX2 (z), ab ≥ 0}. To each cone C(z), there is an associated quadratic form given by Qz (u) = ab. The quantity

Q
z (Dz
u) inf σ (Dz
) = u∈int(C(z)) Qz (u)

measures the amount of expansion generated by Dz
. Similarly we can define σ for
−1 by replacing C(z) with its complementary cone. Definition 2. A point z ∈ M is called sufficient if there exists a n > 0 such that z ∈ / Sn+ ∩ Sn− and σ (Dz
n ) > 3 or σ (Dz
−n ) > 3. Remark 3. Since {C} is eventually strictly
-invariant, the set of sufficient points of M has full µ-measure [L-W, §7F]. In the remaining part of this section, we prove the following theorem. Theorem 2. Every sufficient point in a box Ai of M has a neighborhood in Ai that belongs (mod 0) to an ergodic component of
. To prove this result we use the Fundamental Theorem [L-W]. Remark 4. The proof of the Fundamental Theorem for
, which is based on Hopf’s argument, relies on the existence µ-a.e. of local stable and unstable manifolds (LM in brief) of
and their absolute continuity. If a piecewise smooth map satisfying some general conditions (see Part I, [K-S]) has non-zero L.E. with respect to some invariant Borel probability measure λ, then Pesin’s theory ([K-S]) implies the existence λ-a.e. of LM of the map and their absolute continuity. Note that, although the maps T and T −1 satisfy the conditions of [K-S] (this is proved in Part V of [K-S] for a very general class of billiard maps to which T belongs), the map
(or any power of T ) does not necessarily. However, we do not need to check that
satisfies these conditions to prove the existence of LM for
and their absolute continuity. In fact, the LM of T are LM of
(and therefore they are absolutely continuous). We give only the proof that the local unstable manifolds (LUM) of T are LUM of
; the argument for local stable manifolds is the same. Consider a LUM W u of T at z ∈ M  . Suppose that W u is not a LUM of − must cut W u , and therefore, by Remark 2, S − cuts W u as well. But this is
. Then S∞ ∞ impossible, since W u is a LUM of T . The proof that LM are also LM for T n , n ∈ Z is identical. To apply the Fundamental Theorem, we need to verify the following conditions: 1. (Monotonicity) The cone field {C(z)} is eventually strictly
-invariant, and the restriction of C to the interior of any box of M is continuous. 2. (Proper Alignment) The tangent space of S − at any point z ∈ S − is strictly contained in C(z), and the tangent space at any point z ∈ S + is strictly contained in the complementary cone C  (z). 3. (Regularity) For any n ≥ 1, the sets Sn+ and Sn− are regular, i.e., they are finite unions of smooth arcs (closed) which only intersect at their endpoints.

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4. (Noncontraction property) Let || · || be the standard Riemannian metric on M in the coordinates (s, θ ). There exists a constant 0 < ρ such that for every n ≥ 1 and every z ∈ M \ Sn+ , Dz
n u ≥ ρu for every u ∈ C(z). 5. (Sinai-Chernov Ansatz) Let µS be the 1-dimensional Riemannian volume on S − ∪S + . For µS -a.e. z ∈ S − (S + ), lim

n→+∞(−∞)

σ (Dz
n ) = +∞.

Remark 5. Note that the standard Riemannian metric in coordinates (s, θ ) does not generate the invariant area element as it is required by the Fundamental Theorem [L-W] (see §7, p. 36). However the symplectic area sin θ ds dθ is smaller than the Riemannian area ds dθ , and this fact makes the Fundamental Theorem work also in this situation (see also the remark in §14.A, p. 73 of [L-W]). The proof of the Sinai-Chernov Ansatz is exactly the same as in Lemmas 14 and 15 of [De].
√ √ Lemma 5. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then Conditions 1 and 2 are satisfied. Proof. Condition 1 follows from the definition of {C(z)} and Proposition 1. We now prove Condition 2. As a consequence of Remark 2 and the invariance of the cone field, it is sufficient to prove Condition 2 for the singular sets S ± of T . Let z ∈ S + , and assume that z belongs only to one arc of S + . Pick a vector v ∈ Tz S + . The positive semitrajectory of every point contained in a small neighborhood of z in S + hits a corner ai . It follows that v focuses at ai . Therefore v must be strictly contained in the complement of C(z). Note that if z belongs to several arcs of S + , then the same argument works for the tangent space of each arc. Similarly we can prove that the tangent space of S − at any point z ∈ S − is strictly contained in C(z).


√ √ Lemma 6. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then Condition 3 is satisfied. Proof. Exactly as explained in the proof of the previous lemma, it is enough to prove the lemma for the singularity sets of T (recall Remark 2). We only need to prove that Sn+ is regular. In fact, the regularity of Sn− follows from the regularity of Sn+ because Sn− = RSn+ , where R is the map given by R(s, θ) = (s, π −θ). − and S + intersect transversally for every m, n ≥ 0. In fact by Let us note that Sm n Proper Alignment and invariance of {C}, it follows that Tz (
m S − ) is strictly contained in C(z) for every z ∈
m S − , and Tz (
−n S + ) is strictly contained in C  (z) for every − and S + intersect transversally for all z ∈
−n S + . Hence we easily conclude that Sm n m, n ≥ 0. To prove the regularity of Sn+ we use induction. The sets S + and S − are regular. Now let n > 1, and assume that Sn+ is regular. Since S − and Sn+ are transversal, their intersection consists of finitely many points. Thus
−1 is continuous everywhere on Sn+ except for a finite number of points, and so
−1 Sn+ is a finite union of smooth arcs that intersect only at their endpoints, i.e.,
−1 Sn+ is regular. The same conclusion clearly + = S + ∪
−1 Sn+ .

holds for Sn+1

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4.2. Noncontraction property Definition 3. Given a finite orbit γ = {z,
z, . . . ,
n z} with n > 0, we say that the noncontraction property holds along γ if there is a constant λ > 0 such that Dz
n u ≥ λu for every u ∈ C(z). Consider the following finite orbits where z ∈ / Sn+ , n > 0: 1. 2. 3. 4. 5.

{z,
z, . . . ,
n z} ∈ BN , {z,
z, . . . ,
n z} ∈ M+ , {z,
z} with z ∈ BN and
z ∈ M+ , {z,
z} with z ∈ M+ and
z ∈ BN , {z,
z, . . . ,
n z} with z,
n z ∈ Y .

We call these orbits blocks. It is not difficult to see that every finite orbit consists at most of 8 blocks. Namely given a finite orbit γ = {z,
z,
2 z, . . . ,
n z}, n > 0, there exists a 1 ≤ k¯ ≤ 8 such that γ =

k¯ 

{
ni−1 z, . . . ,
ni z},

i=1

where n0 = 0 < n1 < · · · < nk¯ = n, and {
ni−1 z, . . . ,
ni z} is one of the five blocks listed above. Assume for the moment that the noncontraction property holds along each block with the constant 0 < αj ≤ 1, j = 1, . . . , 5. If λ = (min1≤i≤5 αi )8 , then we have Dz
n u ≥ λu for every u ∈ C(z). The constant λ does not depend on the orbit considered and we conclude that the noncontraction property is satisfied. It remains to prove the noncontraction along the five blocks.
√ √ Lemma 7. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then the noncontraction property holds along the blocks of type 1,3, 4 and 5. Proof. The proof of the noncontraction property for blocks of type 5 is the same as the proof of Lemma 13 in [De]. Let {z,
z} be a block of type 4. As z ∈ M+ and
z ∈ BN , then, by (1), we have   L − d0 L Dz
=  r0 sin θ1 sin θ1  , − r10 −1 where L is the length of the segment [z,
z], r0 = r(z), d0 = r(z) sin θ(z) and θ1 = θ (
z). By Lemma 3, there exists 0 < δ < 1/rmax such that −1/r(z) + δ ≤ uθ /us for all u = (us , uθ ) ∈ C(z) and all z ∈ M+ . If uθ /us ≥ 0, then

 1 1 Dz
u ≥ u . us + uθ ≥ min 1, (7) r(z) r(z)

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If −1/r(z) + δ ≤ uθ /us < 0, then we have 1 1 uθ Dz
u ≥ us + uθ ≥ |us | · + ≥ |us |δ. r(z) r(z) us

(8)

Now from −1/r(z) + δ ≤ uθ /us < 0, we easily obtain


r(z) 1 + r 2 (z)

≤

|us | ≤1 u

so that (8) becomes Dz
u ≥


δr(z) 1 + r 2 (z)

u .

(9)

If rmax = a 2 and rmin = 1/a are the maximum and the minimum of the radius of curvature of + , then, combining (7) and (9), we conclude that    1 rmin  u . Dz
u ≥ min ,δ  rmax 2  1 + rmin Consider now a block {z,
z} of type 3. This time we have   sin θ 0 L −  sin θ1 sin θ1  Dz
=  sin θ0 L−d1  , − d1 d1 where L is the length of [z,
z], θ0 = θ (z), θ1 = θ (
z) and d1 = r(
z) sin θ1 . The cone C(z) consists of divergent vectors (us uθ ≤ 0 for every u = (us , uθ ) ∈ C(z)). Hence we have

 2 sin θ0 L Dz
u ≥ − us + uθ ≥ min{sin θ0 , L} u (10) sin θ1 sin θ1 for every u ∈ C(z). For N ≥ 1 there exist two positive numbers t and L¯ for which t ≤ sin θ (z) and L¯ ≤ L(z) on {z ∈ BN :
z ∈ M+ }. We conclude that ¯ u . Dz
u ≥ min{t, L}

(11)

Finally consider a block {z,
z, . . . ,
n z} of type 1. For this case, we have   −1 sinLθ1 , D z
n = 0 −1 where L is the length of the segment [z,
n z] and θ1 = θ (
z). All cones C(
k z), k = 1, . . . , n, consist of divergent vectors so that

 2   L Dz
n u ≥ −us + uθ ≥ min{1, L} u ≥ u (12) sin θ1 for every u ∈ C(z) (in this case L(z) > 2).



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Remark 6. We introduced the set BN (and the map
) for two reasons. The first one is that in order to apply the Fundamental Theorem we need a continuous cone field on every symplectic box of M. Since m+ , m− are finite, the cone field C on BN can be chosen to be the constant cone field consisting of divergent vectors. The second reason is that the noncontraction property does not hold along trajectories that start at M0 , end at M+ and are almost horizontal to the segments 0 . By introducing BN with N > 0, we only need to consider trajectories which make an angle uniformly large with the horizontal direction and for these trajectories the noncontraction property holds as proved in Lemma 7. We introduce a new set of coordinates (J, J  ) in the tangent planes of M which are connected to the coordinates (us , uθ ) by the formulas J = sin θ us , J  = − 1r us − uθ . J is the restriction to M of a transversal Jacobi field along a billiard trajectory and J  is its derivative (see [L-W]). We recall briefly some properties of the coordinates (J, J  ) which we will use later. For the first property see [Do91, Wo94]. – The evolution of (J, J  ) along a segment of trajectory of length τ between two consecutive collisions is given by   1 τ . (13) 0 1 At a reflection, (J, J  ) is transformed by the map   −1 0 , 2 −1 d

(14)

where d = r sin(θ ). – In coordinates (J, J  ), the cone C(z), z ∈ M+ is given by C(z) = {(J, J  ) : J  ≤ −J /F+ (z)}. – The function u = (J, J  ) → |J  | defines a seminorm on M. We will denote it by | · | to distinguish it from the standard norm  · . In the next lemma we prove that these seminorms are equivalent on C(z), z ∈ M+ . Lemma 8. The seminorms | · | and  ·  are equivalent on C(z) for z ∈ M+ . Proof. We need to show that there are two positive numbers c1 ≤ c2 such that c1 u2 ≤  J 2 ≤ c2 u2 for all u ∈ ∪z∈M+ C(z). One inequality can be proved easily. We have  2 , 1}(u2 + u2 ), and therefore we can take J 2 = u2s /r 2 + u2θ + 2us uθ /r ≤ 2 max{1/rmin s θ 2 c2 = 2 max{1/rmin , 1}. To prove the other inequality we consider two cases. If us uθ ≥ 0,  2 , 1}(u2 + u2 ). Lemma 3 tells us that the then we obtain J 2 ≥ u2s /r 2 + u2θ ≥ min{1/rmax s θ other case we need to consider is us > 0 and 0 > uθ /us ≥ −1/r + δ. This time, we get  2 J 2 = (us /r +uθ )2 = (1/r +uθ /us )2 u2s ≥ δ 2 u2s . Since u2θ ≤ (−1/r +δ)2 u2s ≤ u2s /rmin 2 2 2 2 2 2 2 2 2 (and writing us δ = 2us δ /2), we obtain J ≥ (us + uθ )δ min{1, rmin }/2. Therefore 2 , δ 2 /2, δ 2 r 2 /2}.

we can take c1 = min{1, 1/rmax min

Bernoulli Elliptical Stadia

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√ √ Lemma 9. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then the noncontraction property holds along the blocks of type 2. Proof. We may think of blocks of type 2 as finite trajectories of the elliptical billiard because each subset Mi , i = 1, 2, of the elliptical stadium is naturally identified to the phase space of a semiellipse. In the sequel, when we refer to an element originally defined on M+ (T ,
, C, etc.) we will think of it as defined on the phase space of the elliptical billiard as well. By virtue of the equivalence of  ·  and | · | on M+ , it is enough to show that the noncontraction property is verified with respect to the semi-norm |J  |. The proof consists of two parts. In the first part, we prove the statement for θ close to 0 or π (small θ ), and in the second part, we prove it for θ bounded away from 0 and π (large θ ). Small θ. For z0 = (s0 , θ0 ) in the phase space of the elliptical billiard, define n(z0 ) ≥ 0 as the number of consecutive reflections of z0 along the semiellipse to which s0 belongs. Let zn = (sn , θn ) = T n z, 0 ≤ n ≤ n(z0 ). Finally denote by θmax and θmin the maximum and the minimum of θ along the invariant curve G = G(s, θ ) containing z0 . The next lemma is formulated only for θ ≤ π/2 (close to 0) but a similar result holds for π/2 ≤ θ (close to π). Sublemma 1. There exist θ¯ > 0 and β > 1 such that if θk < θ¯ for some 0 ≤ k ≤ n(z0 ), then θn ≤ βθk ,

0 ≤ n ≤ n(z0 ).

Proof. If the block is in H, the result is obvious (in fact, n = 1 and θ is close to π/2). If the block is in E, consider the function G = G(s, θ ), s = s(ϕ) defined in (3). For 0 < θ < π/2 and G > 0, the invariant curve G = G(s, θ ) is the graph of the function
θ = cos−1 G +  2 cos2 ϕ(1 − G). Thus θmax (G) = cos−1 and θmin (G) = cos−1




√

G = sin−1

√

G +  2 (1 − G) = sin−1

1−G




(1 − G)(1 −  2 ).

As G goes to 1, we obtain θmax (G) 1 . →√ θmin (G) 1 − 2 Now since G → 1 as θmin → 0, there exist θ¯ > 0 and β > 1 such that if θmin < θ¯ , then θmax /θmin ≤ β. This easily implies θn ≤ θmax ≤ βθmin ≤ βθk .

Let z = (s, θ ) and z˜ = (˜s , θ˜ ) = T z. Given (J, J  ) ∈ C(z), let (J˜, J˜  ) = Dz T (J, J  ). According to the construction of C(z), we have |J˜/J  | ≥ F− (˜z) and |J˜/J˜  | ≤ F+ (˜z) so that J˜  F (˜z) − .  ≥ J F+ (˜z)

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˜ with 0 < θ˜ < θ¯  and Sublemma 2. There exists a θ¯  > 0 such that for every z˜ = (˜s , θ)  every (J, J ) ∈ C(z), we have J˜  A = A() = 2 2 /(1 −  2 ). (15)  ≥ 1 − A tan θ˜ , J Proof. By the previous remarks we have (see formulas (5) and (4)) 1+ F− (˜z) = F+ (˜z) 1−

 2 sin 2ϕ˜ (1 − G) sin 2θ˜ .  2 sin 2ϕ˜ (1 − G) sin 2θ˜

Let f (˜s , θ˜ ) =

 2 sin 2ϕ˜ ˜ (1 − G(˜s , θ)) sin 2θ˜

then ˜ F− (˜z) f (˜s , θ) −1=2 . ˜ F+ (˜z) 1 − f (˜s , θ) For every s˜ , we have 1 − G(˜s , θ˜ ) =

1 − cos2 θ˜ 1 − cos2 θ˜ ≤ 1 −  2 cos2 ϕ˜ 1 − 2

so that |f (˜s , θ˜ )| ≤

 2 1 − cos2 θ˜ 2 ˜ tan θ. = 1 −  2 sin 2θ˜ 2(1 −  2 )

Note that for θ˜ small, we have

f (˜s , θ˜ ) ˜ 2 ≤ 4|f (˜s , θ)|. 1 − f (˜s , θ˜ )

Therefore, provided that θ˜ is small enough, we have f F− (˜z) ≥ 1 − 4|f | ≥ 1 − A tan θ. ˜ ≥ 1 − 2 F+ (˜z) 1−f

Let z0 = (s0 , θ0 ), and zn = (sn , θn ) = T n z0 for 0 < n ≤ n(z0 ). Pick a (J0 , J0 ) ∈ C(z), and let (Jn , Jn ) = Dz T n (J0 , J0 ) for 0 < n ≤ n(z0 ). ¯ ¯ β and θ¯  are the It follows easily from Sublemma 1, that if θ0 ≤ min{θ/β, θ¯  } (θ, same as in Sublemmas 1 and 2), then θn ≤ θ¯  for all 0 ≤ n ≤ n(z0 ). Therefore we can apply Sublemma 2 to each factor of the expression    Jn Jn J1 = J  J  . . . J  , 0 n−1 0

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obtaining  n Jn  ≥ (1 − A tan θi ) J 0

for all 0 < n ≤ n(z0 ).

i=1

By Sublemma 1, then we have  Jn ≥ (1 − A tan βθ0 )n(z0 ) . J 0 In [Do91, p. 242], Donnay gave an estimate of the number of consecutive reflections along a focusing arc for small angles. He proved that there exists a C > 0 such that for all (s0 , θ0 ) with θ0 sufficiently small we have n(s0 , θ0 ) ≤ C/ sin θ20 . Thus we get    θ0 −1 Jn ≥ (1 − A tan βθ0 )C sin 2 . J 0 It is easy to see that lim (1 − A() tan βθ )C

θ→0+



sin

θ 2

−1

= exp(−2CAβ).

(16)

¯ such that for any Hence given a small 0 < δ < exp(−2CAβ), there exists a θ˜ ≤ θ/β ˜ (s0 , θ0 ) with θ0 ≤ θ we have  Jn ≥ exp(−2CAβ) − δ > 0 for all 0 < n ≤ n(s0 , θ0 ). J 0 We conclude the first part of the proof by noticing that the lower bound exp(−2CAβ)− δ is independent of n. Large θ. Now let θ˜ < θ0 < π − θ˜ . By using contradiction and Sublemma 1, we obtain that θ˜ /β < θn for all 0 ≤ n ≤ n(z0 ). We claim that F− is uniformly bounded away from ˜ This is quite obvious for blocks of type 0 along blocks of type 2 with θ˜ < θ0 < π − θ. 2 in E, while it follows from part (iii) of Lemma 1 for blocks of type 2 in H. Also it is easy to see that the number of reflections n(z0 ) is bounded, i.e., there exists a n¯ such that n(z0 ) ≤ n. ¯ Since m+ is finite, we conclude that there is a < 1 such that F− /F+ ≥ a. Hence we have    Jn Jn J1 n n¯ = for all 0 < n ≤ n(z0 ).

J  J  . . . J  ≥ a ≥ a 0 n−1 0 Conditions 1–5 are verified so that Theorem 2 is proved. 

Remark 7. Consider the Riemannian metric ρ given by J 2 + J 2 on the tangent bundle of M  . It can be proved that the symplectic form ω = sin θ ds ∧ dθ becomes J ∧ J  in coordinates (J, J  ), thus the volume form induced by the metric ρ coincides with the symplectic form ω. Also it can be checked that the noncontraction property holds also for the metric ρ. Therefore, by introducing the metric ρ from the very beginning, we could have applied directly the Fundamental Theorem to
without using Remark 5.

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5. Ergodicity, Kolmogorov and Bernoulli Properties In this section we conclude the proof of the ergodicity for the map
. In the previous section, we have shown that each sufficient point has a neighborhood contained in a box of M which belongs (mod 0) to an ergodic component of
. To finish our proof, we need first to show that each box belongs (mod 0) to an ergodic component, and then that the orbits of any pair of boxes are not disjoint (mod 0). The first part is accomplished by proving that the set of sufficient points contained in a box is topologically rich (pathconnected in our case), while for the second part we need to construct special trajectories which have the property that starting from one box and traveling along these trajectories we can reach every other box.
√ √ Lemma 10. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then the subset of sufficient points contained in a box of M is path-connected. Proof. Let Ai be a box of M. The proof of the Sinai-Chernov Ansatz (Lemma 14 in [De]) includes the following result: the subset of non-sufficient points of Ai is con+ ∩ S − . From the regularity of the sets S ± , it follows immediately that the tained in S∞ ∞ n + − is at most countable. If we remove a countable set from a path-connected set S∞ ∩ S∞ set, we still obtain a path-connected set.

By a standard argument (see for instance Corollary 4.3 part (a) of [M93]) involving Theorem 2, we obtain that each box of M belongs to an ergodic component of
.
√ √ Lemma 11. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then every box of M belongs (mod 0) to an ergodic component of
. We can finish now the proof of the ergodicity of
.
√ √ Theorem 3. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then the map
is ergodic. Proof. We have to show that all boxes of M belong (mod 0) to the same ergodic component of
. To do this it is enough to check that the three boxes U1 , U2 , V2,2 belong (mod 0) to the same ergodic component. If this is true, then, by the symmetry of Q, any other triple U1 , U2 , Vi,j , 1 ≤ i, j ≤ 2, will belong (mod 0) to the same ergodic component. We also need to show that BN belongs to the same ergodic component of U and V . This easily follows from µ((
BN ) ∩ M+ ) > 0. Two boxes Ai , Aj , i = j belong (mod 0) to the same ergodic component if there is an orbit γ which intersects Ai and Aj . If such an orbit exists, then there will be an open set in Ai which is mapped into an open set in Aj , and open sets contain sets of sufficient points of positive measure. It is easy to see that, practically, what we need to do is to show that there are two orbits such that one intersects U1 and U2 , and the other one intersects V2,2 and U1 . The first orbit is given by the periodic orbit along the x-axis. To construct the second orbit, we proceed as follows. We claim that if z0 ∈ M0 , z1 = T z0 ∈ M+ , qi = π(zi ), 0 ≤ i ≤ 1 and q0 q1 intersects the x-axis, then z1 ∈ Y . In fact, assume, without loss of generality, / Y . If z2 = T 2 z0 , q2 = π(z2 ), then q1 and q2 belong that z1 ∈ M2 and suppose that z1 ∈ to the same semiellipse. Let t be the line containing q0 q1 , and t  be the line obtained by reflecting t about the x-axis. The segments q0 q1 and q1 q2 are tangent to the same confocal ellipse E  and q1 q2 is contained in the set bounded by t  and the semiellipse. E  is tangent to t, so E  must be tangent to t  as well by symmetry. However, the segment

Bernoulli Elliptical Stadia

229

q1 q2 lies to the left side of t  , and therefore we conclude that E  intersects t  transversally, obtaining a contradiction. Consider the orbit γ = {z0 , z1 , . . . , zn }, zi = T i z0 , n > 2 such that π(z0 ) = a4 ; z1 ∈ M2 ; π(z2 ) = a3 ; zi ∈ M0 , 2 < i < n; zn ∈ M1 . Note that the alternate segments of γ are parallel. By the claim it is easy to see that for any neighborhood W of z0 , we have T W ∩ Y2 = ∅ and T W ∩ Y2 c = ∅. We analyze several cases: a) if zn ∈ Y1 , then, by continuity of T k , 1 ≤ k ≤ n, we can find a z0 ∈ M2 close to z0 such that i) z0 , z1 ∈ M2 , ii) z1 ∈ Y2 , iii) zi ∈ M0 , 1 ≤ i < n and i = n + 1, and iv) zn ∈ Y1 . Hence we have z0 ∈ V2 and zn ∈ U1 . b) If zn ∈ / Y , then we can find a z0 ∈ M0 close to z0 such that i) z1 ∈ Y , ii) zn , zn ∈ M1 and iii) zn ∈ / Y (again by the continuity of T k , 1 ≤ k ≤ n). Thus z1 ∈ U2 and zn ∈ V1 . c) If π(zn ) is a corner, then zn+1 ∈ M1 and we can find a z0 ∈ M0 close to z0 such that i) z1 ∈ Y2 and ii) zn ∈ / Y . The last fact follows from the continuity of T k , 1 ≤ k ≤ n. We have again z1 ∈ U2 and zn ∈ V1 .

As an easy corollary, we obtain the ergodicity of the billiard map T .
√ √ Corollary 1. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then T is ergodic. The next theorem shows that T has indeed stronger ergodic properties.
√ √ Theorem 4. If 1 < a < 4 − 2 2 and h > 2a 2 a 2 − 1, then the billiard map T is a K-system and Bernoulli. Proof. The Bernoulli property follows from the K-property by the general results in [C-H] or [O-W]. Thus we only need to show that T is a K-system. Since the billiard map T has non-zero Lyapunov exponents ν-a.e. and is ergodic, Theorem 7.9 in [Pe77] (see also Theorem 13.1 in [K-S]) implies the existence of a finite partition {C1 , . . . , Cm } of M  with the following properties: 1) ν(Ci ) > 0 for i = 1, . . . , m, 2) T Ci = Ci+1 , i = 1, . . . , m − 1, and T Cm = C1 , 3) T m |Ci is K-system for i = 1, . . . , m. It is clear that T is a K-system if T n is ergodic for every integer n > 0. Now the ergodicity of T n can be proved using the same idea used for T . We consider the same ˜ : M → M,
(z) ˜ reduced phase space M = M+ ∪BN and the first return time map
= n B(z) n n j (T ) z induced on M by T , where B(z) = inf{j > 0 : (T ) z ∈ M}. First we ˜ is local ergodic, and then, by constructing special trajectories of
, ˜ that
˜ show that
has only one ergodic component. Finally we need to show ν( k T kn M) = 1. Clearly ˜ preserves the measure µ.
˜ has LSM and LUM µ-a.e. which are absolutely Recalling Remark 4, we see that
˜ continuous. Furthermore note that the cone field C is eventually strictly invariant for
. ˜ Now the local ergodicity for
can be proved exactly as we did for
. ˜ To show We obtain that each box Ai belongs (mod 0) to an ergodic component of
. ˜ has only one ergodic component, we show, as in Lemma 18, that for any pair of that
˜ intersecting both. In fact, we only need to check boxes of M there is a trajectory of
the existence of the following trajectories: 1) a trajectory intersecting V1,i and V2,j for each i, j = 1, 2; 2) a trajectory intersecting Ui and Vj,k for each i, j, k = 1, 2; 3) a trajectory intersecting each connected component of BN and Vi,j for each i = 1, 2. We only prove the existence of trajectories intersecting V1,1 and V2,1 because the existence of the other trajectories can be proved in the same way. The ergodicity of T implies that ν-a.e. trajectory of T covers densely M. Hence we can find a z0 ∈ V1,1 with

230

G. Del Magno, R. Markarian

the property that for some m > 0 we have zm = (sm , θm ) with sm ∈ 2 and θm so small that the next n reflections take place at 2 , i.e., sm+1 , . . . , sm+n ∈ 2 . By the definition ˜ we conclude that there exists an integer k such that
˜ k z0 ∈ V2,1 . of
, kn Finally we need to show that ν( k T M) = 1. This follows, again, from the ergodicity of T . Suppose, in fact, that there is a set A ⊂ M, ν(A) > 0 such that ν(A ∩ k T kn M) = 0. The ergodicity of T implies that ν-a.e. z ∈ A has a dense trajectory in M. Then, by using the same argument used in the previous paragraph, we see that there is an integer k such that T kn z ∈ M for ν-a.e. z ∈ A which contradicts

ν(A ∩ k T kn M) = 0. Acknowledgements. R.M. acknowledges the support of CSIC (Universidad de la Rep´ublica) and Proyecto “Clemente Estable” Nro. 4086, Uruguay. G. Del Magno would like to thank L. Bunimovich, C. Liverani and M. Lenci for useful discussions.

References [Bu79]

Bunimovich, L.A.: On ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65, 295–312 (1979) [C-H] Chernov, N.I., Haskell, C.: Nonuniformly hyperbolic K-systems are Bernoulli. Ergod. Th. & Dynam. Sys. 16, 19–44 (1996) [C-M] Chernov, N.I., Markarian, R.: Introduction to the ergodic theory of chaotic billiards. IMCA, Lima, 2001 [C-F-S] Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Berlin: Springer, 1982 (Original Ed. 1980) [De] Del Magno, G.: Ergodicity of a class of truncated elliptical billiard. Nonlinearity 14, 1761– 1786 [Do91] Donnay, V.: Using integrability to produce chaos: Billiards with positive entropy. Commun. Math. Phys. 141, 225–257 (1991) [K-S] Katok, A., Strelcyn, J.-M.: Invariant manifolds, entropy and billiards; smooth maps with singularities. Lect. Notes Math. vol. 1222, New York: Springer, 1986 [L-W] Liverani, C., Wojtkowski, M.: Ergodicity in Hamiltonian systems. Suny Stony Brook Preprint # 1992/16. Dynamics Reported, Dynam. Report. Expositions Dynam. Systems (N.S.) vol. 4, Berlin: Springer, 1995, pp. 130–202 [M93] Markarian, R.: New ergodic billiards: Exact results. Nonlinearity 6, 819–841 (1993) [M-O-P] Markarian, R., Oliffson Khamporst, S., Pinto de Carvalho, S.: Chaotic properties of the elliptical stadium. Commun. Math. Phys. 174, 661–679 (1996) [O-W] Ornstein, D.S., Weiss, B.: On the Bernoulli nature of systems with hyperbolic structure. Ergod. Th. & Dynam. Sys. 18, 441–456 (1998) [Pe77] Pesin,Ya.B.: Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv. 32, 55–114 (1977) [Wo85] Wojtkowski, M.: Invariant families of cones and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 5, 145–161 (1985) [Wo86] Wojtkowski, M.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys. 105, 391–414 (1986) [Wo94] Wojtkowski, M.: Two applications of Jacobi fields to the billiard ball problem. J. Differ. Geom. 40, 155–164 (1994) Communicated by G. Gallavotti

Commun. Math. Phys. 233, 231–296 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0736-x

Communications in

Mathematical Physics

A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2,C) Erik Koelink1 , Johan Kustermans2,∗ 1

Technische Universiteit Delft, Faculteit ITS, Afdeling Toegepaste Wiskundige Analyse, Mekelweg 4, 2628CD Delft, The Netherlands. E-mail: [email protected] 2 Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B, 3001 Leuven, Belgium. E-mail: [email protected] Received: 28 June 2001 / Accepted: 25 July 2002 Published online: 10 December 2002 – © Springer-Verlag 2002

Abstract: S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently
(1, 1) of SU (1, 1) in by Woronowicz gave strong indications that the normalizer SU SL(2, C) is a much better quantization candidate than SU (1, 1) itself. In this paper we
q (1, 1), a new example of a unishow that this is indeed the case by constructing SU modular locally compact quantum group (depending on a parameter 0 < q < 1) that
(1, 1). After defining the underlying von Neumann algebra of is a deformation of SU
q (1, 1) we use a certain class of q-hypergeometric functions and their orthogonality SU relations to construct the comultiplication. The coassociativity of this comultiplication is the hardest result to establish. We define the Haar weight and obtain simple formulas for the antipode and its polar decomposition. As a final result we produce the underlying
q (1, 1). The proofs of all these results depend on various properties of C∗ -algebra of SU q-hypergeometric 1 ϕ1 functions. Introduction Arguably one of the most important and simplest non-compact Lie groups is the SU (1, 1) group, which is isomorphic to SL(2, R). In 1990, one of the first attempts to construct a quantum version of SU (1, 1) was made in [22] and [23] by T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, Y. Saburi and K. Ueno and independently, in [10] by L. Vaksman and L. Korogodsky. We follow [22] and [23] because these expositions are more elaborate. Their starting point is a real form Uq (su(1, 1)) of the quantum universal enveloping algebra Uq (sl(2, C)) (defined in [22, Eq. (1.9)]). Intuitively, one should view Uq (su(1, 1)) as a quantum universal enveloping algebra of the “quantum Lie algebra” of the still to be constructed locally compact quantum group SUq (1, 1). The dual A of Uq (su(1, 1)) is turned into some sort of topological Hopf ∗ -algebra ([22, Sect. 2] and ∗

Post-doctoral researcher of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.)

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[23, Eq. (2.6)]) which is naively referred to as a topological quantum group. In a next step the coordinate Hopf ∗ -algebra Aq (SU (1, 1)) is given as a ∗ -subalgebra of A that also inherits the comultiplication, counit and antipode from A ([22, Eq. (2.7)] and [23, Eq. (0.9)]). In this philosophy, they first introduce infinite-dimensional infinitesimal representations of quantum SUq (1, 1) (see [23, Eq. (1.1)]) which they then exponentiate to infinitedimensional unitary corepresentations of A (see [23, Eq. (1.2)]) providing hereby the quantum analogues of the discrete, continuous and complementary series of SU (1, 1) but also a new strange series of corepresentations. In an attempt to get hold of the C∗ -algebra B that should be viewed as the quantum analogue of the space C0 (SU (1, 1)), they follow the strategy of first transforming the generators and relations of Aq (SU (1, 1)) formally into other elements that produce a ∗ -algebra B (not inside Aq (SU (1, 1)) !) definable by generators and relations that, unlike in the case of Aq (SU (1, 1)), can be faithfully represented by bounded operators. The C∗ -algebra B is then defined to be the universal enveloping C∗ -algebra of B. Other important contributions (other than the ones that will be mentioned later on) to the study of different aspects of quantum SU (1, 1) have been made by T. Kakehi, T. Masuda, K. Ueno in [8], by D. Shklyarov, S. Sinel’shchikov & L. Vaksman in [26]. The authors of [23] seem to get a little bit overenthusiastic by claiming the existence of a comultiplication on B turning B into a Hopf ∗ -algebra (see [23, Prop. 7]). This is in fact in stark contrast with the result, proven by S.L. Woronowicz in 1991, that showed that quantum SU (1, 1) does not exist as a locally compact quantum group (see [29, Thm. 4.1 and Sect. 4.C]). Not surprisingly, this was considered to be quite a setback for the theory of locally compact quantum groups in the operator algebra setting. ∗ Recall  thatSU (1, 1) is the linear Lie group { X ∈ SL(2, C) | X U X = U }, where 1 0 U = . In 1994, a breakthrough in this stalemate was forced by L.I. Korogod0 −1
(1, 1) instead of that of sky (see [9]) who studied deformations of the linear Lie group SU
(1, 1) denotes the Lie group { X ∈ SL(2, C) | X∗ U X = U or X ∗ U X SU (1, 1). Here, SU = −U } which is in fact the normalizer of SU (1, 1) in SL(2, C). In this paper, we will follow up on Korogodsky’s and Woronowicz’ ideas to introduce a locally compact quan
q (1, 1), depending on a parameter 0 < q < 1, that is the quantum version tum group SU
(1, 1). In doing so, we believe we also make a strong case for the use of this group SU of q-hypergeometric functions in the operator algebra approach to quantum groups. But what is actually meant by a locally compact quantum group? This question has kept a lot of people busy for the last 20 years and the most satisfying answer has been given by S. Vaes and the second author in [20] (see the introduction of this same paper for an extensive account of the history and the importance of different people in the development of the theory of locally compact quantum groups). The paper [20] is written in the C∗ -algebra setting but it will be easier to use the von Neumann algebraic approach as introduced in [21]. Both approaches are completely equivalent and the last result of this paper produces the generic C∗ -algebraic quantum group out of the von Neumann algebraic one. In order to see what we are aiming for, we formulate the definition of a von Neumann algebraic quantum group as defined in [21].

Definition 1. Consider a von Neumann algebra M together with a unital normal ∗ -homomorphism  : M → M ⊗ M such that ( ⊗ ι) = (ι ⊗ ). Assume moreover the existence of

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1. a normal semifinite faithful weight ϕ on M that is left invariant: ϕ((ω ⊗ ι)(x)) = ϕ(x)ω(1) for all ω ∈ M∗+ and x ∈ M+ ϕ. 2. a normal semifinite faithful weight ψ on M that is right invariant: ψ((ι⊗ω)(x)) = ψ(x)ω(1) for all ω ∈ M∗+ and x ∈ M+ ψ. Then we call the pair (M, ) a von Neumann algebraic quantum group. For a discussion about the consequences of this definition and related notations we refer to [21, Sect. 1]. For quite a while, the major drawback of the theory of non-compact quantum groups was the lack of a whole array of different examples of non-compact locally compact quantum groups. So the importance of this paper lies mainly in the fact that we add an example to the rather short list of existing atomic non-compact quantum groups like quantum E(2), quantum ax + b and quantum az + b. Also note that this new locally compact quantum group is the first analogue of a non-compact semi-simple Lie group. By atomic we mean that these examples (up till now) can not be constructed out of simpler quantum groups through different existing theoretical construction procedures of which the most important one is arguably the double crossed product construction. In
q (1, 1) can now serve as one of the ingredients in these conturn, this new example SU struction procedures. It is moreover very conceivable that the importance of SU (1, 1) in
q (1, 1) in the quantum group the classical group theory will be parallelled by that of SU setting.
q (1, 1) and the corepresentations of SU
q (1, 1) The dual von Neumann algebra of SU appearing in the direct integral decomposition of the left regular corepresentations have been calculated (see [13]). In the near future, we should also be able to fit the rest of the unitary corepresentations of [23] rigorously in the framework of locally compact quantum groups. It is to be expected that new results for q-hypergeometric functions can emerge this way. This paper is rather technical and an overview of the most important results of it (and of [13]) that is light on q-special functions (but without proofs) can be found in [14]. Since the ideas of Korogodsky are a chief motivation for this paper, we will briefly discuss the most important results of [9]. Let (Aq , ) be the Hopf ∗ -algebra associated
q (1, 1) (in [9], Aq is denoted by S). We give a precise definition of (Aq , ) in to SU Eqs. (1.1) and (1.2). The ∗ -algebra Aq is generated by elements α, γ and a central self-adjoint involution e. Define two central orthogonal projections p± in Aq as p± = 21 (1 ± e) ± + − and define the ∗ -subalgebras A± q of Aq as Aq = p± Aq . Thus, Aq = Aq ⊕ Aq (in [9], ± ± Aq are denoted by R ). Let us also set γ± = p± γ . + + Define a ∗ -homomorphism + : A+ q → Aq ⊗ Aq such that + (a) = (p+ ⊗ + ∗ p+ )(a) for all a ∈ A+ q . Then (Aq , + ) is a Hopf -algebra which should be thought of ∗ as the q-deformation of the coordinate Hopf -algebra associated to the group SU (1, 1). Most of the relevant representations of Aq are infinite-dimensional so that a lot of care has to be taken to make the notion of representations of Aq more precise (see [25]). Consider a Hilbert space H , a dense subspace D of H and a unital algebra representation π of Aq on D such that π(a)v, w = v, π(a ∗ )w for all a ∈ Aq and v, w ∈ D. Then we call π a ∗ -representation of Aq in H . We set D(π ) = D. We call π well-behaved if π(γ ∗ γ )   is an essentially self-adjoint operator in H and if the spectrum σ π(γ ∗ γ ) ⊆ q 2Z , where π(γ ∗ γ ) denotes the closure of π(γ ∗ γ ) as an operator in H . Similar definitions are used for ∗ -representations of A± q in H (where γ is replaced by γ± ). The spectral condition is not present in [9], a weaker spectral condition was introduced in [32]. It is

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∗ clear that we can identify ∗ -representations of A± q with -representations π of Aq such that π(p∓ ) = 0, or equivalently π(e) = ±1. Since e belongs to the center of Aq , each irreducible ∗ -representation of A± q is nec− . Korogodsky classified all essarily an irreducible ∗ -representation of either A+ or A q q well-behaved ∗ -representations of Aq in [9, Prop. 2.4], most of them are infinite-dimensional. K. Schm¨udgen pointed us to the necessity of imposing some sort of spectral condition for this classification result to be true (and which is completed in [32]). Given two well-behaved infinite-dimensional irreducible ∗ -representations π1 , π2 of Aq in Hilbert spaces H1 , H2 respectively, Korogodsky showed in [9, Thm. 6.1] that there does not exist a well-behaved ∗ -representation π of Aq in H1 ⊗ H2 such that D(π1 ) D(π2 ) ⊆ D(π ) and π(a) v = (π1 π2 )((a)) v for all a ∈ Aq and v ∈ D(π1 ) D(π2 ), where denotes the algebraic tensor product. Thus, loosely speaking, the tensor product of two well-behaved infinite-dimensional irreducible ∗ representations π1 and π2 of Aq does not exist. Applied to two well-behaved infinitedimensional ∗ -representations of A+ q , this result corresponds to the non-existence of quantum SU (1, 1) as a locally compact quantum group as proven in [29]. However, Korogodsky also showed that the situation is not as bad as the above result on first sight suggests. Therefore consider two well-behaved ∗ -representations π1 , π2 of Aq in Hilbert spaces H1 , H2 such that for i = 1, 2, πi is the finite direct sum of well-behaved infinite-dimensional irreducible ∗ -representations of Aq for which the number of direct summands of πi that are ∗ -representations of A+ q equals the number of direct summands of πi that are ∗ -representations of A− . Then [9, Thm. 6.2 q and 6.3] imply the existence of a well-behaved ∗ -representation π of Aq in H1 ⊗ H2 such that D(π1 ) D(π2 ) ⊆ D(π ) and π(a) v = (π1 π2 )((a)) v for all a ∈ Aq and v ∈ D(π1 ) D(π2 ). It should however be pointed out that π is not unique! Around 1996, Woronowicz picked up on this observation to start his study of quantum
q (1, 1) on the Hilbert space level (see [32]). Instead of focusing on all well-behaved SU ∗ -representations of A , he redirected his attention to ∗ -representations that are direct q sums (or more precisely, direct integrals) of the sort described above. For this purpose,
q (1, 1)-quadruple on a Woronowicz introduces in [32, Def. 1.1] the notion of an SU Hilbert space. Let us use some slightly different terminology in order to stay closer to Korogodsky’s viewpoint. Let π be a well-behaved ∗ -representation of Aq in some Hilbert space H and Y a closed linear operator in H such √  that Phase π(γ ) commutes with Y , the domain of Y and
q (1, 1)-representation in H . Y ∗ agree and π q e γ ∗ −α ⊆ Y . Then we call (π, Y ) a SU
It follows that for such a SU q (1, 1)-representation, the quadruple (π(α), π(γ ), π(e), Y )
q (1, 1) of unbounded type in the sense of [32]. forms a SU
q (1, 1)-quadruple of unbounded type, it follows from [32, If (α, ˜ γ˜ , e, ˜ Y˜ ) is a SU
q (1, 1)-representation. Moreover, [32, Thm. 1.3] Thm. 3.7] that it arises from such a SU gives a precise meaning to the statement that π contains as much irreducible well-be− haved ∗ -representations of A+ q as of Aq . In [32, Thm. 1.4] it is then shown that, in
q (1, 1)accordance with the above formal discussion, the tensor product of two SU
quadruples can be defined as a new SU q (1, 1)-quadruple. Recast in the terminology
q (1, 1)-representations (H1 , π1 , Y1 ), of this introduction this means that given two SU
(H2 , π2 , Y2 ) it is possible to construct a new SU q (1, 1)-representation (H1 ⊗ H2 , π, Y ) such that D(π1 ) D(π2 ) ⊆ D(π ) and π(a) v = (π1 π2 )((a)) v for all a ∈ Aq and

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v ∈ D(π1 ) D(π2 ). The extra information contained in Y moreover allows to get rid of the uniqueness problems mentioned before. There is however one drawback to [32]. At the moment of writing this paper, there was
q (1, 1)still a gap in the proof of the associativity of the tensor product construction SU quadruples. Although it is difficult to estimate the seriousness of this gap, it corresponds to the most difficult part of the proof of the coassociativity of the comultiplication in this paper. In 1999, J. Stokman together with the first author studied the properties of a left invariant weight h on the C∗ -algebra C of quantum SU (1, 1) (see [16]). However, remember that it is impossible to define a comultiplication on C turning it into a locally compact quantum group. Instead, one looks at a suitable dense subspace B of C ∗ so that for ω1 , ω2 ∈ B it is possible to define the product ω1 ω2 so that on a formal level, ω1 ω2 = (ω1 ⊗ ω2 ). It turns out that the invariant weight h can be written as a sum of positive functionals in B and as a consequence (ω h)(x) can be defined as a pointwise convergent sum for suitable ω ∈ B and suitable x ∈ C. The left invariance statement then becomes (ω h)(x) = ω(1) h(x). The benefit of [16] to this paper lies in the fact that the definition of ω1 ω2 depends on a class of special functions that arise formally as Clebsch-Gordan coefficients. These special functions can be extended in such a way that they serve as the principal set of data to construct the locally compact
q (1, 1). Moreover, the formula for the left Haar weight of SU
q (1, 1) quantum group SU is a straightforward generalization of the formula defining h. A naive response to this all would be to presume that given all these existing results it
q (1, 1) as a full blown locally compact quantum should be not so difficult to construct SU group. But this seems to be far from the truth. The family of special functions produced
(1, 1), in [16] can be easily extended from the context of SU (1, 1) to the context of SU but the orthogonality and completeness of the relevant family of functions needs some non-trivial special function theory that can be found in [2] and [3] (see Proposition 3.2). This is in fact the only place where we heavily rely on the theory of q-hypergeometric not covered in the appendix of this paper. Given these special functions it is not hard to
q (1, 1), the hard part of the construction of SU
q (1, 1) define the comultiplication of SU lies in the proof of the coassociativity of the comultiplication. Similarly, the left Haar weight is pretty easy to define, but checking that it is left invariant in the sense of [21] requires some non-trivial quantum group techniques. Whereas [9] is mainly a motivational paper for this one, [32] is not only motivational (especially the use of the reflection operator u introduced in notation 2.2) but also gives
q (1, 1). The advantage of the approach of [32] lies mainly an alternative approach to SU in the fact that the role of the generators and relations is more explicit and constructional. In order to do so, Woronowicz also presents a beautiful operator theoretic theory of balanced extensions, similar to the theory of self-adjoint extension of symmetric operators. As a consequence of all this, it should not be so difficult to prove that the C∗ -algebra introduced in Proposition 4.15 is generated by the generators and relations, introduced in [32, Def. 1.1], and in the sense of [30]. We on the other hand take a more pragmatic stance and approach the subject by using special functions. As a disadvantage, it follows that the role of generators and relations is less clear (but still lurking in the background), but as a clear advantage we can prove the coassociativity of the comultiplication, prove the left invariance of the left Haar weight and also give an explicit formula for the multiplicative unitary. It also allows us to circumvent the theory of balanced extensions and stick to more standard operator theoretic techniques. As a consequence, the proofs of this paper are different from the ones in [32]. A notable exception is the use of an operator,

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introduced in [32], in the proof of the last (albeit simplest) step in the proof of Theorem 3.11. The paper is organized as follows. In the first section we introduce the Hopf ∗ -al
q (1, 1) and realize it as a ∗ -algebra of unbounded operators on gebra associated to SU some Hilbert space. In the subsequent section the underlying von Neumann algebra is introduced. We define the comultiplication in Sect. 3 and claim its coassociativity at the end of the section but the proof of this fact is given in Sect. 5 in order to enhance the readability of the paper. In Sect. 4 we construct the Haar weights and prove their
q (1, 1) is indeed a locally compact quantum group. invariance, thereby proving that SU We end Sect. 4 by calculating the underlying C∗ -algebraic quantum group. The appendix contains most of the basic results that we will need from the theory of special functions. Notations and Conventions The set of all natural numbers, not including 0, is denoted by N. Also, N0 = N ∪ {0}. Fix a number q > 0. Let a ∈ C. If n ∈ N0 , the q-shifted factorial (a; q)n ∈ C  i (so (a; q)0 = 1). From now on, we asis defined as (a; q)n = n−1 i=0 (1 − q a) sume that q < 1. Then (a; q)∞ ∈ C is defined as (a; q)∞ = limn→∞ (a; q)n . Using some basic infinite product theory, one checks that this limit exists and that the function a ∈ C → (a; q)∞ is analytic. Also, (a; q)∞ = 0 if and only if a ∈ q −N0 . We also use the notation (a1 , . . . , am ; q)k = (a1 ; q)k . . . (am ; q)k if a1 , . . . , am ∈ C and k ∈ N0 ∪ {∞}. If a, b, z ∈ C, we define  

∞  1 (a; q)n (b q n ; q)∞ a ; q, z = (a; b; q, z) := (−1)n q 2 n(n−1) zn . b (q ; q)n

(1)

n=0

We collected some further basic information about these  function in the appendix. See [6] for an extensive treatment on q-hypergeometric functions. Let B1 , . . . , Bn be sets such that Bi = T or Bi = −q Z ∪ q Z . Set I = { i ∈ {1, . . . , n} | Bi = −q Z ∪ q Z }. Consider a set K ⊆ (−q Z ∪ q Z )I and a function f : { (x1 , . . . , xn ) ∈ B1 × . . . × Bn | (xi )i∈I ∈ K } → C. Then we set f (x) := 0 for (x1 , . . . , xn ) ∈ B1 × . . . × Bn such that (xi )i∈I ∈ K. (2) If f is a function, the domain of f will be denoted by D(f ). If X is a set, the identity mapping on X will be denoted by ιX and most of the time even by ι. If H is a Hilbert space, 1H denotes ιH . The set of all complex valued functions on X is denoted by F(X), the set of all elements in F(X) having finite support, is denoted by K(X). Let V be a vector space and S a subset of V . Then S denotes the linear span of S in V . Consider a locally compact space with a regular Borel measure µ on it. If g ∈ L2 ( , µ), we denote the class of g in L2 ( , µ) by [g]. If f is a measurable function on , we define the linear operator Mf in L2 ( , µ) such that D(Mf ) = { [g] | g ∈ L2 ( , µ) such that f g ∈ L2 ( , µ) } and Mf ([g]) = [f g] for all such classes [g] ∈ D(Mf ). Let S, T be two linear operators acting in a Hilbert space H . We say that S ⊆ T if D(S) ⊆ D(T ) and S(v) = T (v) for all v ∈ D(S). Following [32] we call S balanced if D(S) = D(S ∗ ). The symbol will be used to denote the algebraic tensor product of vector spaces and linear mappings. The symbol ⊗ on the other hand will denote the tensor product of Hilbert spaces, von Neumann algebras and sufficiently continuous linear mappings.

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If n ∈ N and X is locally compact space, X n will denote the n-fold product set = X × . . . × X. Consider von Neumann algebras M, N acting on Hilbert spaces H, K respectively and π : M → N a normal ∗ -homomorphism. Let T be a densely defined, closed linear operator in H affiliated with M (in the von Neumann algebraic sense) and T = U |T | the polar decomposition of T . Then there exists a unique positive operator P in K such that f (P ) = π(f (|T |)) for all f ∈ L∞ (R+ ). Now we set π(T ) = π(U ) P . Xn


(1, 1) 1. The Hopf∗ -Algebra Underlying Quantum SU In order to resolve the problems surrounding quantum SU (1, 1), Korogodsky proposed
(1, 1) instead of constructing the quantum in [9] to construct the quantum version of SU version of SU (1, 1) itself. He suggested that the Hopf ∗ -algebra of this quantum group should be the one that we describe now. Throughout this paper, we fix a number 0 < q < 1. Define Aq to be the unital ∗ -algebra generated by elements α , γ and e and relations 0 0 0 α0† α0 − γ0† γ0 = e0 , γ0† γ0 = γ0 γ0† , α0 γ0 = q γ0 α0 , α0 γ0† = q γ0† α0 , α0 e0 = e0 α0 , γ0 e0 = e0 γ0 ,

α0 α0† − q 2 γ0† γ0 = e0 , e0† = e0 , e02 = 1,

(1.1)

where † denotes the ∗ -operation on Aq (in order to distinguish this kind of adjoint with the adjoints of possibly unbounded operators in Hilbert spaces). There exists a unique unital ∗ -homomorphism 0 : Aq → Aq Aq such that 0 (α0 ) = α0 ⊗ α0 + q (e0 γ0† ) ⊗ γ0 , 0 (γ0 ) = γ0 ⊗ α0 + (e0 α0† ) ⊗ γ0 , 0 (e0 ) = e0 ⊗ e0 .

(1.2)

The pair (Aq , 0 ) turns out to be a Hopf ∗ -algebra with counit ε0 and antipode S0 determined by ε0 (α0 ) = 1, S0 (α0 ) = e0 α0† , S0 (α0† ) = e0 α0 , ε0 (γ0 ) = 0, S0 (γ0 ) = −q γ0 , ε0 (e0 ) = 1, S0 (γ0† ) = − q1 γ0† , S0 (e0 ) = e0 . As always we want to represent this Hopf ∗ algebra Aq by possibly unbounded operators in some Hilbert space in order to produce a locally compact quantum group in the sense of Definition 1 of the Introduction. In Proposition 2.4 of [9], Korogodsky produces a family of irreducible ∗ -representations of the ∗ -algebra Aq . We glue part of this family of irreducible ∗ -representations together to a ∗ -representation of Aq on one Hilbert space that will be the Hilbert space that our locally compact quantum group will act upon. For this purpose we define Iq = { −q k | k ∈ N } ∪ { q k | k ∈ Z } . Set Iq− = { x ∈ Iq | x < 0 } and Iq+ = { x ∈ Iq | x > 0 }. We will use the discrete topology on Iq .

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Let T denote the group of complex numbers of modulus 1. We will consider the uniform measure on Iq and the normalized Haar measure on T. Our ∗ -representation of Aq will act in the Hilbert space H defined by H = L2 (T) ⊗ L2 (Iq ). We let ξ denote the identity function on Iq , whereas ζ denotes the identity function on T. If p ∈ −q Z ∪ q Z , we define δp ∈ F(Iq ) such that δp (x) = δx,p for all x ∈ Iq (note that δp = 0 if p ∈ Iq ). The family ( δp | p ∈ Iq ) is the natural orthonormal basis of L2 (Iq ). Also recall the natural orthonormal basis ( ζ m | m ∈ Z ) for L2 (T). Instead of looking at the algebra Aq as the abstract algebra generated by generators and relations we will use an explicit realization of this algebra as linear operators on the dense subspace E of H defined by E =  ζ m ⊗ δx | m ∈ Z, x ∈ Iq ,  ⊆ H . Of course, E inherits the inner product from H . Let L+ (E) denote the ∗ -algebra of adjointable operators on E (see [25, Prop. 2.1.8]), i.e. L+ (E) = { T ∈ End(E) | ∃S ∈ End(E), ∀v, w ∈ E : T v, w = v, Sw } . The ∗ -operation in L+ (E) (and L+ (E E) for that matter) will be denoted by † . So if T ∈ L+ (E), the operator T † ∈ L+ (E) is defined to be the operator S in the above definition. It follows that T † ⊆ T ∗ where T ∗ is the usual adjoint of T as an operator in the Hilbert space H . It also follows that T is a closable operator in H . Define linear operators α0 , γ0 , e0 on E such that

α0 (ζ m ⊗ δp ) = sgn(p) + p −2 ζ m ⊗ δqp , γ0 (ζ m ⊗ δp ) = p−1 ζ m+1 ⊗ δp , e0 (ζ m ⊗ δp ) = sgn(p) ζ m ⊗ δp ,

(1.3)

for all p ∈ Iq , m ∈ Z. Then α0 , γ0 and e0 belong to L+ (E), e0† = e0 and    α0† ζ m ⊗ δp = sgn(p) + q 2 p −2 ζ m ⊗ δq −1 p , (1.4)   γ0† ζ m ⊗ δp = p−1 ζ m−1 ⊗ δp   for all p ∈ Iq , m ∈ Z. Note that α0† ζ m ⊗ δ−q = 0. Then Aq is the ∗ -subalgebra of L+ (E) generated by α0 , γ0 and e0 . Since L+ (E)

+ L (E) is canonically embedded in L+ (E E), we obtain Aq Aq as a ∗ -subalgebra of L+ (E E). As such, the comultiplication  is, according to Eqs. (1.2), given by 0 (α0 ) = α0 α0 + q (e0 γ0† ) γ0 , 0 (γ0 ) = γ0 α0 + (e0 α0† ) γ0 , 0 (e0 ) = e0 e0 ,

(1.5)

where denotes the algebraic tensor product of linear mappings.
(1, 1) 2. The von Neumann Algebra Underlying Quantum SU In this section we introduce the von Neumann algebra acting on H that underlies the von
q (1, 1). The reason for looking Neumann algebraic version of the quantum group SU

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first at the von Neumann algebraic picture is twofold. The most important one is the lack of density conditions in the definition of von Neumann algebraic quantum groups (Definition 1 of the introduction), these are automatically satisfied. Moreover, in this case the von Neumann algebra turns out to be very simple. Towards the end of this paper
q (1, 1). we will use [21, Prop. 1.6] to produce the C∗ -algebraic version of SU In order to get into the framework of operator algebras, we need to introduce the topological versions of the algebraic objects α0 , γ0 and e0 as possibly unbounded operators in the Hilbert space H . So let α denote the closure of α0 , γ the closure of γ0 and e the closure of e0 , all as linear operators in H . So e is a bounded linear operator on H , whereas α and γ are unbounded, closed, densely defined linear operators in H . Note also that α ∗ is the closure of α0† and that γ ∗ is the closure of γ0† . Let us comment on these closures and their adjoints a little bit further by looking at α (the operator γ is treated in a similar way). From the discussion about the adjointable operators on E in the previous section we already know that α0† ⊆ α0∗ = α ∗ . Hence, the closure of α0† is a restriction of α ∗ . One easily checks that the domain of α consists of all elements v ∈ L2 (T) ⊗ L2 (Iq ) for which the sum m∈Z,p∈Iq (sgn(p) + p−2 ) |v, ζ m ⊗ δp |2 is convergent and for such v,

sgn(p) + p −2 v, ζ m ⊗ δp  ζ m ⊗ δqp . α(v) = m∈Z,p∈Iq

Of course, a similar characterization exists for the closure of α0† . Using this characterization for the closure of α0† and using the vectors ζ m ⊗ δp inside the domain of α, one shows that α ∗ is a restriction of the closure of α0† . Thus, α ∗ is the closure of α0† . Lemma 2.1. The domains of α, α ∗ , γ and γ ∗ coincide. This is also true for the domains of the tensor products 1H ⊗ α, 1H ⊗ α ∗ , 1H ⊗ γ and 1H ⊗ γ ∗ , where H can be any Hilbert space. Proof. Since 1H ⊗ γ is normal, D(1H ⊗ γ ∗ ) = D(1H ⊗ γ ). We know that α0† α0 = γ0† γ0 +e0 , thus (1H α0 )v2 = (1H γ0 )v2 +(1H ⊗e)v2 for all v ∈ H E. Since 1H ⊗e is bounded, 1H ⊗α is the closure of 1H α0 and 1H ⊗γ is the closure of 1H γ0 , it follows that D(1H ⊗ α) = D(1H ⊗ γ ). Using the equality α0 α0† = q 2 γ0† γ0 + e0 , one proves in a similar way that D(1H ⊗ α ∗ ) = D(1H ⊗ γ ).   All these operators can easily be realized as a combination of shift and multiplication operators. Consider p ∈ −q Z ∪ q Z . We define a translation operator Tp on F(T × Iq ) such that for f ∈ F(T × Iq ), λ ∈ T and x ∈ Iq , we have that (Tp f )(λ, x) = f (λ, px). By discussion (2) in Notations and conventions, we get that (Tp f )(λ, x) = 0 if px ∈ Iq . If p, t ∈ Iq and g ∈ F(T), then Tp (g ⊗δt ) = g ⊗δp−1 t , thus, Tp (g ⊗δt ) = 0 if p−1 t ∈ Iq . For instance, note that if f ∈ F(T × Iq ), g ∈ F(T) and λ ∈ T, • (Tq −1 f )(λ, −q) = 0 and Tq (g ⊗ δ−q ) = 0, • (T−1 f )(λ, x) = 0 for all x ∈ Iq such that x ≥ 1 ; T−1 (g ⊗ δp ) = 0 for all p ∈ Iq such that p ≥ 1. Notation 2.2. Define the mapping ρ : −q Z ∪ q Z → B(H ) such that ρp equals the partial isometry on H induced by Tp for all p ∈ −q Z ∪ q Z . Let us also single out the following special cases:

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1) Define w = ρq −1 , which is an isometry on B(H ). 2) Define u = ρ−1 , which is a self-adjoint partial isometry on B(H ). In terms of multiplication and shift operators, our closed linear operators in H are easily recognized as α = w (1 ⊗ M√sgn(ξ )+ξ −2 ),

γ = Mζ ⊗ Mξ −1 ,

e = 1 ⊗ Msgnξ . (2.1)

These tensor products are obtained by closing the algebraic tensor product mappings with respect to the norm topology on H . Let us recall the following natural terminology. If T1 , . . . , Tn are closed, densely defined linear operators in H , the von Neumann algebra N on H generated by T1 , . . . , Tn is the one such that N  = { x ∈ B(H ) | xTi ⊆ Ti x and xTi∗ ⊆ Ti∗ x for i = 1, . . . , n } . Almost by definition, N is the smallest von Neumann algebra acting on H so that T1 , . . . , Tn are affiliated with M in the von Neumann algebraic sense. If w1 , . . . , wn are the partial isometries obtained from the polar decompositions of T1 , . . . , Tn respectively, then N is also the von Neumann algebra on H generated by n

{wi } ∪ { f (Ti∗ Ti ) | f ∈ L∞ (σ (Ti∗ Ti )) } .

(2.2)

i=1

It is now very tempting to define the von Neumann algebra underlying quantum
q (1, 1) as the von Neumann algebra generated by α, γ and e. However, for reasons SU that will become clear later (see the discussion in the beginning of the next section and the remark after Proposition 3.8), the underlying von Neumann algebra will be the one generated by α, γ , e and u. Definition 2.3. We define Mq to be the von Neumann algebra on H generated by α, γ , e and u.
q (1, 1). For convenience, we So Mq will be the von Neumann algebra underlying SU will also introduce Nq as the von Neumann algebra on H generated by α, γ and e. Lemma 2.4. 1) Nq is generated by {w} ∪ { Mf | f ∈ L∞ (T × Iq ) }, 2) Mq is generated by {w, u} ∪ { Mf | f ∈ L∞ (T × Iq ) }, 3) Mq = L∞ (T) ⊗ B(L2 (Iq )). Proof. By Eq. (2.2) we know that Nq is generated by the set {w, Mζ ⊗1 , M1⊗sgn(ξ ) } ∪ { f (α ∗ α)|f ∈ L∞ (σ (α ∗ α)) } ∪ { f (γ ∗ γ )|f ∈ L∞ (σ (γ ∗ γ )) }. Since γ ∗ γ = 1⊗Mξ −2 , it follows that the von Neumann algebra generated by {M1⊗sgn(ξ ) } ∪ { f (γ ∗ γ )|f ∈ L∞ (σ (γ ∗ γ )) } equals the von Neumann algebra generated by {M1⊗sgn(ξ ) } ∪ { M1⊗f (|ξ |) |f ∈ L∞ (Iq+ ) }. Thus, the von Neumann algebra generated by {M1⊗sgn(ξ ) } ∪ { f (γ ∗ γ )|f ∈ L∞ (σ (γ ∗ γ )) } equals { M1⊗f | f ∈ L∞ (Iq ) }. Because α ∗ α = 1 ⊗ Msgn(ξ )+ξ −2 , it now follows that Nq is generated by {w, Mζ ⊗1 } ∪ { M1⊗f | f ∈ L∞ (Iq ) }. Therefore the first statement holds, and as a consequence also the second one holds.

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Define the partial isometries w0 , u0 on L2 (Iq ) such that (w0 f )(x) = f (q −1 x) and (u0 f )(x) = f (−x) for all f ∈ L2 (Iq ) and x ∈ Iq . Letting M0 denote the von Neumann algebra generated by {w0 , u0 } ∪ L∞ (Iq ), the second statement implies that Mq = L∞ (T) ⊗ M0 . It is not so difficult to see that every rank one projector of the form L2 (Iq ) → L2 (Iq ) : v → v, δp  δt , where p, t ∈ Iq , belongs to M0 , thus M0 = B(L2 (Iq )).   So we get in particular that ρp ∈ Mq for all p ∈ −q Z ∪ q Z . Let us collect some further elementary results about the map ρ. If p ∈ −q Z ∪ q Z , we define ip = ρp∗ ρp , the initial projection of ρp . Consider f ∈ L2 (T × Iq ) and define g ∈ L2 (T × Iq ) such that for λ ∈ T, x ∈ Iq , we have that g(λ, x) = f (λ, x) if p −1 x ∈ Iq and g(λ, x) = 0 if p−1 x ∈ Iq . Then ip ([f ]) = [g]. Lemma 2.5. Consider p, t ∈ −q Z ∪ q Z and f ∈ L∞ (T × Iq ), then ρp ρt = ρp t it

and

ρp∗ = ρp−1

and ρp Mf = MTp f ρp

and

ρp MTp−1 f = Mf ρp .

Proof. We only prove the first equality, the other ones are even more straightforward to check. Take f ∈ L2 (T × Iq ) such that f (λ, x) = 0 for all λ ∈ T and x ∈ Iq such that t −1 x ∈ Iq . Now (ρp ρt )([f ]) = [Tp (Tt (f ))], whereas ρp t ([f ]) = [Tp t (f )]. Let y ∈ Iq and ν ∈ T. We consider 4 different cases: (1) If py ∈ Iq and p ty ∈ Iq , then (Tp (Tt (f ))(ν, y) = Tt (f )(ν, py) = f (ν, p ty). Thus Tp t (f )(ν, y) = f (ν, p ty) = (Tp (Tt (f ))(ν, y). (2) Now suppose that py ∈ Iq and p ty ∈ Iq . Then (Tp (Tt (f )))(ν, y) = 0. On the other hand, Tp t (f )(ν, y) = f (ν, p ty). Since t −1 (p ty) = py ∈ Iq , we have that f (ν, p ty) = 0. Thus Tp t (f )(ν, y) = 0 = (T  p (Tt (f ))(ν,  y). (3) Suppose that py ∈ Iq and p ty ∈ Iq . Then T p (Tt (f )) (ν,  y) = Tt (f )(ν, py) = 0 since t (py) ∈ Iq . Thus Tp t (f )(ν, y) = 0 = Tp (Tt (f )) (ν,y).  (4) If py ∈ Iq and p ty ∈ Iq , then clearly Tp t (f )(ν, y) = 0 = Tp (Tt (f )) (ν, y). So we see that Tp (Tt (f )) = Tp t (f ) for this kind of function f . Thus ρp ρt it = ρp t it .   So we see that ρ is almost a group representation, but not quite. It is possible to use the results in the previous lemma to perform a partial cross product construction (which should not be confused with the crossed product used in [9]). Since we do not need this approach in the rest of the paper, we will not pursue this matter any further. Instead we focus on a more useful and simpler picture of Mq : For every p, t ∈ Iq and m ∈ Z we define (m, p, t) = ρp−1 t Mζ m ⊗δt ∈ Mq . So if x ∈ Iq and r ∈ Z, then (m, p, t) (ζ r ⊗ δx ) = δx,t ζ m+r ⊗ δp .

(2.3)

Define Mq◦ =  (m, p, t) | m ∈ Z, p, t ∈ Iq . Using Eq. (2.3), it is easy to see that the family ( (m, p, t) | m ∈ Z, p, t ∈ Iq ) is a linear basis of Mq◦ .

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The multiplication and ∗ -operation are easily expressed in terms of these basis elements: (m1 , p1 , t1 ) (m2 , p2 , t2 ) = δp2 ,t1 (m1 + m2 , p1 , t2 ), (m, p, t)∗ = (−m, t, p)

(2.4)

for all m, m1 , m2 ∈ Z, p, p1 , p2 , t, t1 , t2 ∈ Iq . So we see that Mq◦ is a σ -weakly dense sub∗ -algebra of Mq .
(1, 1) 3. The Comultiplication of Quantum SU
q (1, 1). In the first part we start In this section we introduce the comultiplication of SU with a motivation for the formulas appearing in Definition 3.1. Although the discussion
q (1, 1), it is important and clarifying to know is not really needed in the build up of SU how we arrived at the formulas in Definition 3.1. But first we introduce two auxiliary functions (1) χ : −q Z ∪ q Z → Z such that χ (x) = logq (|x|) for all x ∈ −q Z ∪ q Z , (2) κ : R → R such that κ(x) = sgn(x) x 2 for all x ∈ R. Our purpose is to define a comultiplication  : Mq → Mq ⊗ Mq . Assume for the moment that this has already been done. It is natural to require  to be closely related to the comultiplication 0 on Aq as defined in Eq. (1.5). The least that we expect is 0 (T0 ) ⊆ (T ) and 0 (T0† ) ⊆ (T )∗ for T = α, γ . In the rest of this discussion we will focus on the inclusion 0 (γ0† γ0 ) ⊆ (γ ∗ γ ), where 0 (γ0† γ0 ) ∈ L+ (E E). Because γ ∗ γ is self-adjoint, the element (γ ∗ γ ) would also be self-adjoint. So the hunt is on for self-adjoint extensions of the explicit operator 0 (γ0† γ0 ). Unlike in the case of quantum E(2) (see [29]), the operator 0 (γ0† γ0 ) is not essentially self-adjoint. But it was already known in [9] that 0 (γ0† γ0 ) has self-adjoint extensions (this follows easily because the operator in (3.2) commutes with complex conjugation, implying that the deficiency spaces are isomorphic). Let us make a small detour to quantum SU (1, 1). In this case, the operators α0 and γ0 are replaced by their restrictions to L2 (T ⊗ Iq+ ). Then 0 (γ0† γ0 ) still has self-adjoint extensions for each of which the domain is obtained by imposing a boundary condition on functions in its domain (this boundary condition is a simple relation between the function and its Jackson derivative in the limit towards 0, see [9, Eq. (6.2)]). However, in this case the closure of the operator 0 (α0 ) does not leave the domain of such a self-adjoint extension invariant because it distorts the boundary condition. From a practical point of view, it should also be said that an explicit manageable spectral decomposition in terms of special functions for these self-adjoint extensions is missing, cf. [15, Rem. 2.7].
(1, 1). Although 0 (γ † γ0 ) has a selfNow we return to the case of quantum SU 0 adjoint extension, it is not unique. We have to make a choice for this self-adjoint extension, but we cannot extract the information necessary to make this choice from α and γ alone. This is why we do not work with Nq but with Mq which has the above extra extension information contained in the element u. These kind of considerations were
q (1, 1). In this already present in [33] and were also introduced in [32] for quantum SU paper, this principle is only lurking in the background but it is treated in a fundamental and rigorous way in [32].

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Now we get into slightly more detail in our discussion about the extension of 0 (γ0† γ0 ). Define a linear map L : F(T × Iq × T × Iq ) → F(T × Iq × T × Iq ) such that (Lf )(λ, x, µ, y) = [ x −2 (sgn(y) + y −2 ) + (sgn(x) + q 2 x −2 ) y −2 ] f (λ, x, µ, y)

¯ x −1 y −1 (sgn(x) + x −2 )(sgn(y) + y −2 ) f (λ, qx, µ, qy) + sgn(x) q −1 λµ

+ sgn(x) q λµ¯ x −1 y −1 (sgn(x) + q 2 x −2 )(sgn(y) + q 2 y −2 ) f (λ, q −1 x, µ, q −1 y) for all λ, µ ∈ T and x, y ∈ Iq . A straightforward calculation reveals that if f ∈ E E, then 0 (γ0† γ0 ) [f ] = [L(f )]. From this, it is a standard exercise to check that [f ] ∈ D(0 (γ0† γ0 )∗ ) and 0 (γ0† γ0 )∗ [f ] = [L(f )] if f ∈ L2 (T × Iq × T × Iq ) and L(f ) ∈ L2 (T × Iq × T × Iq ) (without any difficulty, one can even show that D(0 (γ0† γ0 )∗ ) consists precisely of such elements [f ]). If θ ∈ −q Z ∪ q Z , we define θ = { (λ, x, µ, y) ∈ T × Iq × T × Iq | y = θx }. We consider L2 (θ ) naturally embedded in L2 (T × Iq × T × Iq ). It follows easily from the above discussion that 0 (γ0† γ0 )∗ leaves L2 (θ ) invariant. Thus, if T is a self-adjoint extension of 0 (γ0† γ0 ), the obvious inclusion T ⊆ 0 (γ0† γ0 )∗ implies that T also leaves L2 (θ ) invariant. Therefore every self-adjoint extension T of 0 (γ0† γ0 ) is obtained by choosing a selfadjoint extension Tθ of the restriction of 0 (γ0† γ0 ) to L2 (θ ) for every θ ∈ −q Z ∪ q Z and setting T = ⊕θ∈−q Z ∪q Z Tθ . Therefore fix θ ∈ −q Z ∪ q Z . Define Jθ = { z ∈ Iq 2 | κ(θ) z ∈ Iq 2 } which is a q 2 -interval around 0. On Jθ we define a measure νθ such that νθ ({x}) = |x| for all x ∈ Jθ . Now define the unitary transformation Uθ : L2 (T × T × Jθ ) → L2 (θ ) such that Uθ ([f ]) = [g], where f ∈ L2 (T × T × Jθ ) and g ∈ L2 (θ ) are such that g(λ, z, µ, θz) = (λµ) ¯ χ(z) (−sgn(θ z))χ(z) |z| f (λ, µ, κ(z))

(3.1)

for all λ, µ ∈ T and z ∈ Iq such that θ z ∈ Iq . Define the linear operator Lθ : F(Jθ ) → F(Jθ ) such that 1   (Lθ f )(x) = 2 2 − (1 + x)(1 + κ(θ) x) f (q 2 x) θ x

−q 2 (1 + q −2 x)(1 + q −2 κ(θ) x) f (q −2 x)  +[(1 + κ(θ ) x) + q 2 (1 + q −2 x)] f (x)

(3.2)

for all f ∈ F(Jθ ) and x ∈ Jθ . Then an easy calculation shows that Uθ∗ 0 (γ0† γ0 )
E E Uθ = 1 Lθ
K(Jθ ) . So our problem is reduced to finding self-adjoint extensions of Lθ
K(Jθ ) . This operator Lθ
K(Jθ ) is a second order q-difference operator for which eigenfunctions in terms of q-hypergeometric functions are known. We can use a reasoning similar to the one in [15, Sect. 2] to get hold of the self-adjoint β β extensions of Lθ
K(Jθ ) : Let β ∈ T. Then we define a linear operator Lθ : D(Lθ ) ⊆ 2 2 L (Jθ , νθ ) → L (Jθ , νθ ) such that

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D(Lθ ) = { f ∈ L2 (Jθ , νθ ) | Lθ (f ) ∈ L2 (Jθ , νθ ), f (0+) = β f (0−) and (Dq f )(0+) = β (Dq f )(0−) } β

β

and Lθ is the restriction of Lθ to D(Lθ ). Here, Dq denotes the Jackson derivative, that is, (Dq f )(x) = (f (qx) − f (x))/(q − 1)x for x ∈ Jθ . Also, f (0+) = β f (0−) is an abbreviated form of saying that the limits limx↑0 f (x) and limx↓0 f (x) exist and β limx↓0 f (x) = β limx↑0 f (x). Then Lθ is a self-adjoint extension of Lθ
K(Jθ ) . If β

β, β  ∈ T and β = β  , then Lθ = Lθ . It is tempting to use the extension L1θ to construct our final self-adjoint extension for 0 (γ0† γ0 ) (although there is no apparent reason for this choice). However, in order to obtain a coassociative comultiplication, it turns out that we have to use the extension sgn(θ) Lθ to construct our final self-adjoint extension. This is reflected in the fact that the expression s(x, y) appears in the formula for ap in Definition 3.1. This all would be only a minor achievement if we could not go any further. But the results and techniques used in the theory of q-hypergeometric functions will even allow sgn(θ) us to find an explicit orthonormal basis consisting of eigenvectors of Lθ . These eigenvectors are, up to a unitary transformation, obtained by restricting the functions ap in Definition 3.1 to θ , which is introduced after this definition. The special case θ = 1 was already known to Korogodsky (see [9, Prop. A.1]), although the proof in [9] seems to contain quite a few gaps. For instance, the presentation in [3] shows that a lot of more care has to be taken to solve these kind of problems. In order to compress the formulas even further, we introduce two other auxiliary functions 1 (1) ν : −q Z ∪ q Z → R+ such that ν(t) = q 2 (χ(t)−1)(χ(t)−2) for all t ∈ −q Z ∪ q Z . (2) Another auxiliary function s : R0 × R0 → {−1, 1} is defined such that  −1 if x > 0 and y < 0 s(x, y) = for all x, y ∈ R0 . 1 if x < 0 or y > 0 β

Let us also collect some basic manipulation rules. Therefore take x, y, z ∈ R0 . (1) If x > 0 then s(x, y) = sgn(yz) s(x, z) and s(y, x) = s(z, x). If x < 0, then s(x, y) = s(x, z) and s(y, x) = sgn(yz) s(z, x). So if sgn(xyz) = 1, then s(x, y) = s(x, z). (2) It is clear that s(x, y) = −sgn(x) s(x, −y) and s(x, y) = sgn(y) s(−x, y). (3) One easily checks that s(x, y) = sgn(xy) s(y, x). As a consequence we get also that s(−x, y) = s(−y, x) and s(x, −y) = s(y, −x).     a a If a, b, z ∈ C, we define  ; z and (a; b; z) to be equal to  ; q 2 ; z . b b √ We will also use the normalization constant cq = ( 2 q (q 2 , −q 2 ; q 2 )∞ )−1 . Definition 3.1. If p ∈ Iq , we define a function ap : Iq × Iq → R such that ap is supported on the set { (x, y) ∈ Iq × Iq | sgn(xy) = sgn(p) } and ap (x, y) is given by  (−κ(p), −κ(y); q 2 )∞ χ(p) χ(x) cq s(x, y) (−1) (−sgn(y)) |y| ν(py/x) (−κ(x); q 2 )∞   2 −q /κ(y) 2 × 2 ; q κ(x/p) q κ(x/y) for all (x, y) ∈ Iq × Iq satisfying sgn(xy) = sgn(p).

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The presence of the expression (−sgn(y))χ(x) in this same formula can be traced back to the defining formula (3.1) for Uθ . The extra vital information that we need is contained in the following proposition. For θ ∈ −q Z ∪ q Z we define θ = { (x, y) ∈ Iq × Iq | y = θx }. Proposition 3.2. Consider θ ∈ −q Z ∪q Z . Then the family ( ap
θ | p ∈ Iq such that sgn (p) = sgn(θ ) ) is an orthonormal basis for l 2 (θ ). This result implies also a dual result, stemming from the following simple duality principle (in special function theory, these are referred to as dual orthogonality relations). Consider a set I and suppose that l 2 (I ) has an orthonormal basis (ej )j ∈J . For every i ∈ I , we define a function fi on J by fi (j ) = ej (i). Then (fi )i∈I is an orthonormal basis for l 2 (J ). If we apply this principle to the line θ , the previous proposition implies the next one. Proposition 3.3. Consider θ ∈ −q Z ∪ q Z and define J = Iq+ if θ > 0 and J = Iq− if θ < 0. For every (x, y) ∈ θ we define the function e(x,y) : J → R such that e(x,y) (p) = ap (x, y) for all p ∈ J . Then the family ( e(x,y) | (x, y) ∈ θ ) forms an orthonormal basis for l 2 (J ). The first of these two propositions will be shortly used to define the comultiplication on Mq , both of them pop up in the proof of the left invariance of the Haar weight. For the proofs of Propositions 3.2 and 3.3 we refer to the literature. For 1 ≥ θ > 0 Proposition 3.3 is [3, Thm. 4.1] and Proposition 3.2 follows from Corollary 4.2 and the following remark of [3] in base q 2 and with (c, q 2α ) replaced by (1, κ(θ )). For θ ≥ 1 we derive Propositions 3.2 and 3.3 similarly from [3, Thm. 4.1 and Cor. 4.2] in base q 2 and with (c, q 2α ) replaced by (1, κ(θ )−1 ). The case θ ≥ 1 can also be reduced to the case 0 < θ ≤ 1 using the second symmetry property of Proposition 3.5. The functions ap (x, y)
θ are eigenfunctions of the operator Lθ as considered in (3.2). The general theory used in the first half of [3] is explained in [12]. For θ < 0 the statements in Propositions 3.2 and 3.3 reduce to statements on specific classes of orthogonal polynomials. For θ < 0 the orthogonality relations in Proposition 3.2 are directly obtainable from the (a) orthogonality relations for the Al-Salam and Carlitz polynomials Un (x; q) in base q 2 with a replaced by κ(θ ), see Al-Salam’s and Carlitz’s original paper [2] or references in [11]. This time we have that ap (x, y)
θ are eigenfunctions of the operator Lθ , due to the second order q-difference equation for the Al-Salam and Carlitz polynomials, see [11]. The dual result in Proposition 3.3, which follows immediately from Proposition 3.2 for θ < 0 since the corresponding moment problem is determinate, can also be matched to orthogonal polynomials after applying elementary transformation formulas for basic hypergeometric series. The dual orthogonality relations split up in three summation results. Two of these summations can be matched to the orthogonality relations for the q-Charlier polynomials given in [11, Eq. (3.23.3)], but for different parameters. The remaining summation is an easy consequence of [6, Eq. (1.3.16)]. The functions ap (x, y) can be thought of as Clebsch-Gordan coefficients for the tensor product decomposition of the representation described in (2.1) with itself. Note that the corresponding Clebsch-Gordan coefficients for the quantum SU (2) group are given in terms of Wall polynomials, see Koornwinder [17, Rem. 4.2], and we can write the Wall polynomials in terms of the functions ap (x, y), precisely for y ∈ −q −N0 , using [6, Eq. (III.3)]. There is a nice symmetry in ap (x, y) with respect to interchanging x, y and p. One of these symmetries is easy to see while the proof of the other one requires an extra lemma.

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Lemma 3.4. Consider x, y, p ∈ Iq 2 and m ∈ Z such that y/x = sgn(p) q 2m . Then   (−y; q 2 )∞  −q 2 /y 2  x/p ; q 2 q x/y (−x; q 2 )∞   (−x; q 2 )∞  −q 2 /x 2 logq 2 (|p|)+1 2 m = sgn(p) (−q /p)  y/p . ; q q 2 y/x (−y; q 2 )∞ Proof. If p > 0, this follows easily from Proposition 6.6(1). Now assume that p < 0. By Result 6.4, the above equation is equivalent to  2  −q /p 2  2 ; q x/y q x/p  (−x; q 2 )∞  −q 2 /p 2 log (|p|)+1 = sgn(p) q 2 (−q 2 /p)m  2 ; q y/x . 2 q y/p (−y; q )∞ This equality follows easily from Proposition 6.6(2).

 

With this lemma in hand, one verifies the second of the next symmetries. Use Result 6.4 to prove the first symmetry, the third one is then a consequence of the previous two symmetries. For the first two symmetries you will also need two of the manipulation rules for s(x, y) discussed before Definition 3.1, namely the last one of (1) and the first one of (3). Proposition 3.5. If x, y, p ∈ Iq , then ap (x, y) = (−1)χ(yp) sgn(x)χ(x) |y/p| ay (x, p), ap (x, y) = sgn(p)χ(p) sgn(x)χ(x) sgn(y)χ(y) ap (y, x), ap (x, y) = (−1)χ(xp) sgn(y)χ(y) |x/p| ax (p, y) . Now we produce the eigenvectors of our self-adjoint extension of 0 (γ0† γ0 ) (see the remarks after the proof of Proposition 3.9 ). We will use these eigenvectors to define a unitary operator that will induce the comultiplication. The presence of λχ(y) and µχ(x) in the formulas for
r,s,m,p can be traced back to Eq. (3.1). The dependence of
r,s,m,p on r, s and p is chosen in such a way that Proposition 3.9 is true. Definition 3.6. Consider r, s ∈ Z, m ∈ Z and p ∈ Iq . We define the element
r,s,m,p ∈ H ⊗ H such that  ap (x, y) λr+χ(y/p) µs−χ(x/p) if y = sgn(p) q m x
r,s,m,p (λ, x, µ, y) = 0 otherwise for all x, y ∈ Iq and λ, µ ∈ T. We know from Proposition 3.2 that the family (
r,s,m,p | r, s ∈ Z, m ∈ Z, p ∈ Iq ) forms an orthonormal basis for H ⊗ H . As a consequence, we get the following result. Proposition 3.7. If x, y ∈ Iq and r, s ∈ Z, then, using L2 -convergence, ζ r ⊗ δx ⊗ ζ s ⊗ δy = ap (x, y)
r−χ(y/p),s+χ(x/p),χ(y/x),p . p∈Iq


q (1, 1). Now we are ready to introduce the comultiplication of quantum SU

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Proposition 3.8. Define the unitary transformation V : H ⊗ H → L2 (T) ⊗ L2 (T) ⊗ H such that V (
r,s,m,p ) = ζ r ⊗ ζ s ⊗ ζ m ⊗ δp for all r, s ∈ Z, m ∈ Z and p ∈ Iq . Then there exists a unique injective normal ∗ -homomorphism  : Mq → Mq ⊗ Mq such that (a) = V ∗ (1L2 (T) ⊗ 1L2 (T) ⊗ a)V for all a ∈ Mq . Proof. Define the ∗ -homomorphism  : Mq → B(H ⊗H ) such that (a) = V ∗ (1L2 (T) ⊗ 1L2 (T) ⊗ a)V for all a ∈ Mq . Fix a ∈ Mq . If r, s, m ∈ Z, p ∈ Iq , then Definition 3.6 implies that (1H ⊗ Mζ ⊗ 1L2 (Iq ) )
r,s,m,p =
r,s+1,m,p . It follows that V (1H ⊗Mζ ⊗1L2 (Iq ) ) = (1L2 (T) ⊗Mζ ⊗1H )V .As a consequence, V (1H ⊗b⊗1L2 (Iq ) ) = (1L2 (T) ⊗ b ⊗ 1H )V for all b ∈ L∞ (T). It follows that (a)(1H ⊗ b ⊗ 1L2 (Iq ) ) = (1H ⊗ b ⊗ 1L2 (Iq ) )(a) for all b ∈ L∞ (T). Thus, (a) ∈ (1H ⊗ L∞ (T) ⊗ 1L2 (Iq ) ) = B(H ) ⊗ L∞ (T)⊗B(L2 (Iq )) = B(H )⊗Mq by Lemma 2.4. In a similar way, (a) ∈ Mq ⊗B(H ). Hence (a) ∈ Mq ⊗ Mq by the commutator theorem for tensor products.   The requirement that (Mq ) ⊆ Mq ⊗ Mq is the primary reason for introducing the extra generator u. We cannot work with the von Neumann algebra Nq that is generated only by α, γ and e, because (Nq ) ⊆ Nq ⊗ Nq . This definition of  and Eq. (2.1) imply easily that 
r,s,m,p | r, s ∈ Z, m ∈ Z, p ∈ Iq  is a core for (α), (γ ) and  (α)
r,s,m,p = sgn(p) + p −2
r,s,m,pq , (3.3) (γ )
r,s,m,p = p −1
r,s,m+1,p for r, s ∈ Z, m ∈ Z and p ∈ Iq . Since
r,s,m,p (λ, x, µ, y) = 0 if sgn(x) sgn(y) = sgn(p) it follows that (e ⊗ e)Fr,s,m,p = sgn(p) Fr,s,m,p . Hence V (e ⊗ e) = (1L2 (T2 ) ⊗ e)V . As a consequence, (e) = e ⊗ e. Recall the linear operators 0 (α0 ), 0 (γ0 ) acting on E E (Eq. (1.5)). Also recall the distinction between ∗ and †. Since (α) = V ∗ (1L2 (T) ⊗ 1L2 (T) ⊗ α)V and (γ ) = V ∗ (1L2 (T) ⊗ 1L2 (T) ⊗ γ )V , Lemma 2.1 implies that (α) and (γ ) are balanced. For the proof of the next proposition q-contiguous relations for q-hypergeometric functions are essential (see Lemma 6.5). Proposition 3.9. The following inclusions hold: 0 (α0 ) ⊆ (α), 0 (α0 )† ⊆ (α)∗ , 0 (γ0 ) ⊆ (γ ) and 0 (γ0 )† ⊆ (γ )∗ . Proof. (1) The first step is to prove for i, j, m ∈ Z, p ∈ Iq and r, s ∈ Z, x, y ∈ Ip , (α)
i,j,m,p , ζ r ⊗ δx ⊗ ζ s ⊗ δy  = 
i,j,m,p , 0 (α0† )(ζ r ⊗ δx ⊗ ζ s ⊗ δy ) , (3.4) and this boils down to the q-contiguous relations in Lemma 6.5. First we can rewrite Eq. (6.2) in terms of the functions ap of Definition 3.1 by a straightforward verification as

sgn(p) + p −2 aqp (x, y)

= (sgn(x) + q 2 /x 2 )(sgn(y) + q 2 /y 2 ) ap (q −1 x, q −1 y) + sgn(x) (q/xy) ap (x, y) . (3.5)

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Here we apply Eq. (6.2) in base q 2 with a,b,z replaced by −q 2 /κ(y), q 2 κ(x/y), κ(x/p) respectively (and take some care if x = −q or y = −q). Now consider the left-hand side of Eq. (3.4). By Eq. (3.3) and Definition 3.6, the inner product on the left-hand side of Eq. (3.4) is zero unless |y/x| = q m , sgn(p) = sgn(xy), i+ χ (y/qp) = r and j − χ (x/qp) = s in which case, this inner product equals sgn(p) + p −2 aqp (x, y). (3.6) † + For the right-hand side of Eq. (3.4) we recall that 0 (α0 ) ∈ L (E E) is given by 0 (α0† ) = q e0 γ0 γ0† + α0† α0† . From Eqs. (1.3) and (1.4), it follows that the right hand side of Eq. (3.4) equals sgn(x) (q/xy) 
i,j,m,p , ζ r+1 ⊗ δx ⊗ ζ s−1 ⊗ δy 

+ (sgn(x) + q 2 /x 2 )(sgn(y) + q 2 /y 2 ) 
i,j,m,p , ζ r ⊗ δq −1 x ⊗ ζ s ⊗ δq −1 y  . (3.7) By Definition 3.6, this implies that the right-hand side of Eq. (3.4) is zero unless |y/x| = q m , sgn(p) = sgn(xy), i + χ (y/qp) = r and j − χ (x/qp) = s, which proves Eq. (3.4) if these conditions are violated. If, on the other hand, these conditions are satisfied, Definition 3.6 guarantees that (3.7) equals

sgn(x) (q/xy) ap (x, y) + (sgn(x) + q 2 /x 2 )(sgn(y) + q 2 /y 2 ) ap (q −1 x, q −1 y) . Therefore (3.6) and Eq. (3.5) imply Eq. (3.4) also in this case. So we have established Eq. (3.4) in all possible cases. Since the elements
i,j,m,p form a core for (α), we conclude that ζ r ⊗ δx ⊗ ζ s ⊗ δy ∈ D((α)∗ ) and (α)∗ (ζ r ⊗ δx ⊗ζ s ⊗δy ) = 0 (α0† )(ζ r ⊗δx ⊗ζ s ⊗δy ), thus proving that 0 (α0† ) ⊆ (α)∗ . Taking the adjoint of this equation, we see that (α) ⊆ 0 (α0† )∗ . In general, 0 (α0 ) ⊆ 0 (α0† )∗ . Since D((α)) = D((α)∗ ) ⊇ E E, we conclude that the inclusion 0 (α0 ) ⊆ (α) also holds. (2) The inclusions regarding (γ ) and (γ )∗ are treated in the same way but this time one applies Eq. (6.3) in base q 2 with a,b,z replaced by −q 2 /κ(y), q 2 κ(x/y), q 2 κ(x/p) respectively, to obtain

p−1 ap (x, y) = sgn(x) y −1 sgn(x) + 1/x 2 ap (qx, y)

+x −1 sgn(y) + q 2 /y 2 ap (x, q −1 y) for all x, y, p ∈ Iq .

 

This proposition implies also that (γ ∗ γ ) is an extension of 0 (γ0† γ0 ). We also know that 
r,s,m,p | r, s ∈ Z, m ∈ Z, p ∈ Iq  is a core for (γ ∗ γ ) and (γ ∗ γ )
r,s,m,p = p−2
r,s,m,p for r, s, m ∈ Z, p ∈ Iq . Using this information it is not so difficult to sgn(θ) for all θ ∈ −q Z ∪ q Z , but we will not make check that Uθ∗ (γ ∗ γ )
θ Uθ = 1 ⊗ Lθ any use of this fact in this paper. In the next step we investigate the behavior of V with respect to the flip map  : H ⊗ H → H ⊗ H . Later on, this will guarantee that the unitary antipode (see the discussion after the proof of [21, Prop. 1.4]) commutes with the comultiplication up to the flip map. It will also reduce some of the calculations in the proof of the coassociativity. This behavior is directly related to the second symmetry property of Proposition 3.5.

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˜ : Let us define anti-unitary permutation operators   : L2 (T2 ) → L2 (T2 ) and  2 4  ˜ 1 ⊗ v2 ⊗ v3 ⊗ v4 ) = → L (T ) such that  (v1 ⊗ v2 ) = v¯2 ⊗ v¯1 and (v v¯4 ⊗ v¯3 ⊗ v¯2 ⊗ v¯1 for all v1 , v2 , v3 , v4 ∈ L2 (T). We also introduce the anti-unitary operator J˘ on H such that J˘ [f ] = [g], where f, g ∈ L2 (T × Iq ) are such that g(λ, x) = sgn(x)χ(x) f (λ, x) for all x ∈ Iq and λ ∈ T.

L2 (T4 )

Proposition 3.10. We have that V  = (  ⊗ J˘)V (J˘ ⊗ J˘) and ˜ ⊗ J˘)(1L2 (T2 ) ⊗ V )(V ⊗ 1H )(J˘ ⊗ J˘ ⊗ J˘) . V13 (1H ⊗ V )13 = ( Proof. Take r, s, m ∈ Z and p ∈ Iq . Proposition 3.5 implies that ap (x, y) = sgn(p)χ(p) sgn(x)χ(x) sgn(y)χ(y) ap (y, x) for all x, y ∈ Iq . If x, y ∈ Iq and λ, µ ∈ T, then Definition 3.6 implies that ((J˘ ⊗ J˘)
r,s,m,p )(λ, x, µ, y) = ((J˘ ⊗ J˘)
r,s,m,p )(µ, y, λ, x) = sgn(x)χ(x) sgn(y)χ(y) Fr,s,m,p (µ, y, λ, x) = δx,sgn(p)yq m sgn(x)χ(x) sgn(y)χ(y) µr+χ(x/p) λs−χ(y/p) ap (y, x) = δy,sgn(p)xq −m sgn(p)χ(p) λ−s+χ(y/p) µ−r−χ(x/p) ap (x, y) = sgn(p)χ(p)
−s,−r,−m,p (λ, x, µ, y) . Thus, V (J˘ ⊗ J˘)
r,s,m,p = sgn(p)χ(p) V
−s,−r,−m,p = sgn(p)χ(p) ζ −s ⊗ ζ −r ⊗ ζ −m ⊗ δp = (  ⊗ J˘)(ζ r ⊗ ζ s ⊗ ζ m ⊗ δp ) = (  ⊗ J˘)V
r,s,m,p . From this all we conclude that V  = (  ⊗ J˘)V (J˘ ⊗ J˘). Notice that 13 = 23 12 23 . Thus V13 (1H ⊗ V )13 = V13 (1H ⊗ V )23 12 23 = V13 (1H ⊗ [(  ⊗ J˘)V (J˘ ⊗ J˘)])12 23 . ˆ : L2 (T2 ) ⊗ L2 (T2 ) → L2 (T2 ) ⊗ L2 (T2 ) denote the flip map. If we let  : Let  2 L (T2 ) ⊗ H ⊗ H → H ⊗ L2 (T2 ) ⊗ H denote the permutation map defined by (u ⊗ v ⊗ w) = w ⊗ u ⊗ v for all u ∈ L2 (T2 ) and v, w ∈ H , we get that V13 (1H ⊗ V )13 = V13  ([(  ⊗ J˘)V (J˘ ⊗ J˘)] ⊗ 1H ) ˆ ⊗ 1H )(1L2 (T2 ) ⊗ V ) ([(  ⊗ J˘)V (J˘ ⊗ J˘)] ⊗ 1H ) = ( ˆ ⊗ 1H )(1L2 (T2 ) ⊗ [(  ⊗ J˘)V (J˘ ⊗ J˘)]) ([(  ⊗ J˘)V (J˘ ⊗ J˘)] ⊗ 1H ) = ( ˆ  ⊗   ) ⊗ J˘)(1L2 (T2 ) ⊗ V ) (V ⊗ 1H )(J˘ ⊗ J˘ ⊗ J˘) = (( ˜ ⊗ J˘)(1L2 (T2 ) ⊗ V )(V ⊗ 1H )(J˘ ⊗ J˘ ⊗ J˘). = (

 

We end this section with the statement that the comultiplication is coassociative, the result of this paper that is the most technical to prove.

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Theorem 3.11. The ∗ -homomorphism  : Mq → Mq ⊗Mq is coassociative, i.e. (⊗ι)  = (ι ⊗ ). In order to enhance the readability of this paper, the proof will be given in Sect. 5 but we stress that the results of Sect. 4 do not play any role therein. 4. The Haar Weight on (Mq , ∆) In this section we construct the left Haar weight on (Mq , ) and prove its left invariance through the construction of a partial isometry that afterwards turns out to be the multi
(1, 1). We also establish the unimodularity of (Mq , ). plicative unitary of quantum SU Since Mq = L∞ (T) ⊗ B(L2 (Iq )) we can consider the trace Tr on Mq given by Tr = Tr L∞ (T) ⊗ Tr B(L2 (Iq )) , where Tr L∞ (T) and Tr B(L2 (Iq )) are the canonical traces on L∞ (T) and B(L2 (Iq )) which we choose to be normalized in such a way that Tr L∞ (T) (1) = 1 and Tr B(L2 (Iq )) (P ) = 1 for every rank one projection P in B(L2 (Iq )). Given a weight η on Mq , we use the following standard concepts from weight theory: + M+ η = { x ∈ Mq | η(x) < ∞ },

Mη = linear span of M+ η,

Nη = { x ∈ Mq | η(x ∗ x) < ∞ } . Next we introduce a GNS-construction for the trace Tr. Define K = H ⊗ L2 (Iq ) = 2 m q ) ⊗ L (Iq ). If m ∈ Z and p, t ∈ Iq , we set fm,p,t = ζ ⊗ δp ⊗ δt ∈ K. Thus ( fm,p,t | m ∈ Z, p, t ∈ Iq ) is an orthonormal basis for K. Now define (1) a linear map Tr : NTr → K such that Tr (a) = p∈Iq (a ⊗ 1L2 (Iq ) )f0,p,p for a ∈ NTr . (2) a unital ∗ -homomorphism π : Mq → B(K) such that π(a) = a ⊗ 1L2 (Iq ) for all a ∈ Mq . L2 (T) ⊗ L2 (I

Then (K, π, Tr ) is a GNS-construction for Tr. Now we are ready to define the weight that will turn out to be left- and right invariant with respect to . Use the remarks before [20, Prop. 1.15] to define a linear map  = (Tr )γ ∗ γ : D() ⊆ Mq → K. Let us be a little bit more precise and recall that 1 |γ | = (γ ∗ γ ) 2 . The set of elements a ∈ Mq for which the composition a |γ | is bounded and the unique extension a · |γ | ∈ B(H ) of a |γ | belongs to NTr is a core for . For such an element a, the vector (a) is defined as Tr (a · |γ |). Definition 4.1. We define the faithful normal semi-finite weight ϕ on Mq as ϕ = Trγ ∗ γ . By definition, (K, π, ) is a GNS-construction for ϕ. See [28] for more details about this definition. This definition of ϕ is of course compatible with the usual construction of absolutely continuous weights (see [24]). So ϕ we already know that the modular automorphism group σ ϕ of ϕ is such that σs (x) = 2is −2is |γ | x |γ | for all x ∈ Mq and s ∈ R. The modular conjugation of ϕ with respect to (K, π, ) will be denoted by J , the modular operator of ϕ will be denoted by ∇. Since ϕ = Tr γ ∗ γ and  = (Tr )γ ∗ γ , J equals the modular conjugation of Tr with respect to (K, π, Tr ). Let us recall how these modular objects are related to the modular group. So fix ϕ ϕ ϕ a ∈ Nϕ . If t ∈ R, then σt (a) ∈ Nϕ and ∇ it (a) = (σt (a)). Provided a ∈ D(σ i ), 2

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 ϕ  ϕ ϕ the element σ i (a)∗ ∈ Nϕ and J (a) =  σ i (a)∗ . For x ∈ D(σ i ), we have that 2

ax ∈ Nϕ and (ax) = J π(σ i (x)∗ )J (a). ϕ

2

2

2

Let us first establish some easy formulas that make working with the weight ϕ pretty easy. Lemma 4.2. Consider m ∈ Z and p, t ∈ Iq . Then (1) (m, p, t) ∈ Nϕ and ((m, p, t)) = |t|−1 fm,p,t , (2) (m, p, t) ∈ Mϕ and ϕ((m, p, t)) = δm,0 δp,t t −2 , ϕ (3) (m, p, t) is analytic with respect to σ ϕ and σz ((m, p, t)) = |p −1 t|2iz (m, p, t) for all z ∈ C. Proof. (1) By Eq. (2.4), (m, p, t)∗ (m, p, t) = (0, t, t) = M1⊗δt which clearly belongs to M+ Tr . Thus (m, p, t) ∈ NTr and ((m, p, t) ⊗ 1)f0,p ,p = fm,p,t . (4.1) Tr ((m, p, t)) = p ∈Iq

It is clear that (m, p, t) |γ | is bounded and that the extension of this element to an element of B(H ) is given by |t|−1 (m, p, t) ∈ NTr . Therefore the remarks before [20, Prop. 1.15] imply that (m, p, t) ∈ Nϕ and ((p, m, t)) = |t|−1 Tr ((m, p, t)) = |t|−1 fm,p,t . (2) By Eq. (2.4), (m, p, t) = (0, p, p)∗ (m, p, t), thus ϕ((m, p, t)) = ( (m, p, t)), ((0, p, p)). Hence (2) follows from (1). (3) Choose s ∈ R. By definition, (m, p, t) = ρp−1 t Mζ m ⊗δt . Thus Lemma 2.5 implies that |γ |2is (m, p, t) = M1⊗|ξ |−2is ρp−1 t Mζ m ⊗δt = ρp−1 t MT

−2is ) pt −1 (1⊗|ξ |

Mζ m ⊗δt

= |p −1 t|2is ρp−1 t M1⊗|ξ |−2is Mζ m ⊗δt = |p −1 t|2is ρp−1 t Mζ m ⊗δt M1⊗|ξ |−2is = |p −1 t|2is (m, p, t) |γ |2is , from which it follows that σs ((m, p, t)) = |p −1 t|2is (m, p, t) and the claim follows.   ϕ

These results imply easily the next lemma which is needed to extend results (involving ) that have been proven for the elements (m, p, t) to the whole of Nϕ . Corollary 4.3. If a ∈ Nϕ , there exists a bounded net (ai )i∈I in  (m, p, t) | m ∈ Z, p, t ∈ Iq  such that (ai )i∈I converges to a in the strong∗ topology and ((ai ))i∈I converges to (a) in the norm topology. Proof. Define C =  (m, p, t) | m ∈ Z, p, t ∈ Iq . Set B = { b ∈ Mq | b ≤ a } and equip B with the strong∗ topology. In B × K, we consider the set G that is the closure of { (b, (b)) | b ∈ B ∩ C }. Let F (Iq ) denote the set of all finite subsets of Iq and turn F (Iq ) into a directed set by inclusion. For L ∈ F (Iq ), we define the projection PL = v∈L (0, v, v). Thus, (PL )L∈F (Iq ) converges strongly to 1. Fix L ∈ F (Iq ) for the moment. By Lemma 4.2(1) we see that PL ∈ Nϕ . Since (bPL ) = π(b) (PL ) for all b ∈ Mq , Kaplansky’s density theorem (see the proof of [7, Thm. 5.3.5]), applied to the ∗ -algebra C, implies that (aPL , (aPL )) ∈ G.

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ϕ

It is clear from Lemma 4.2(3) that PL ∈ D(σ i ) and σ i (PL ) = PL . Thus, (aPL ) = 2 2  J π(PL )J (a). It follows that (aPL , (aPL )) L∈F (I ) → (a, (a)), implying that q (a, (a)) ∈ G and the lemma follows.   Corollary 4.4. Consider m ∈ Z and p, t ∈ Iq . Then 1) Jfm,p,t = f−m,t,p , 2) fm,p,t ∈ D(∇) and ∇fm,p,t = |p −1 t|2 fm,p,t . Proof. Recall that J is also the modular conjugation of the trace Tr. So Eq. (4.1) in the previous proof implies that Jfm,p,t = J Tr ((m, p, t)) = Tr ((m, p, t)∗ ) = Tr ((−m, t, p)) = f−m,t,p . Let s ∈ R. By Lemma 4.2, (m, p, t) ∈ Nϕ and ϕ 2is σs ((m, p, t))  = |p −1 t| p, t). So we get that ∇ is (  (m, ϕ −1 (m, p, t)) =  σs ((m, p, t)) = |p t|2is ((m, p, t)). Thus, since fm,p,t = |t| ((m, p, t)), ∇ is fm,p,t = |p −1 t|2is fm,p,t and also the second claim follows.   Now we know enough about the weight ϕ to proceed to the proof of the left invariance of ϕ with respect to . For this purpose we introduce a partial isometry that will later turn out to be the multiplicative unitary of (Mq , ). The  formula defining ∗ ((m , p , t )) ⊗ W ∗ in the next proposition is obtained by formally calculating W 1 1 1  ((m2 , p2 , t2 )) = ( ⊗ )(((m2 , p2 , t2 ))((m1 , p1 , t1 ) ⊗ 1)), but at this stage we do not know whether ϕ is left invariant so we cannot use this formula, as is done in the general theory, to define W ∗ . Proposition 4.5. There exists a unique surjective partial isometry W on K ⊗ K such that W ∗ (fm1 ,p1 ,t1 ⊗ fm2 ,p2 ,t2 ) =

|t2 /y| at2 (p1 , y) ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 )

y, z ∈ Iq sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq

× fm1 +m2 −χ(p1 p2 /t2 z),z,t1 ⊗ fχ(p1 p2 /t2 z),sgn(p2 t2 )(yz/p1 )q m2 ,y for all m1 , m2 ∈ Z and p1 , p2 , t1 , t2 ∈ Iq . Proof. The proof of this proposition is based on the properties of the functions ap , in particular on Propositions 3.2, 3.3 and 3.5. We now give details. Fix m1 , m2 ∈ Z and p1 , p2 , t1 , t2 ∈ Iq . If y, z, y  , z ∈ Iq such that sgn(p2 t2 )(yz/p1 ) q m2 , sgn(p2 t2 )(y  z /p1 )q m2 ∈ Iq and (y, z) = (y  , z ), then it is clear that the element fm1 +m2 −χ(p1 p2 /t2 z),z,t1 ⊗fχ(p1 p2 /t2 z),sgn(p2 t2 )(yz/p1 )q m2 ,y is orthogonal to the element fm1 +m2 −χ(p1 p2 /t2 z ),z ,t1 ⊗ fχ(p1 p2 /t2 z ),sgn(p2 t2 )(y  z /p1 )q m2 ,y  . By Propositions 3.2, 3.3 and 3.5 we get moreover |t2 /y|2 |at2 (p1 , y)|2 |ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 )|2 y, z ∈ Iq sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq

=



y∈Iq

≤

y∈Iq



|t2 /y|2 |at2 (p1 , y)|2

|ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 )|2

z ∈ Iq sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq

|t2 /y|2 |at2 (p1 , y)|2 =



y∈Iq

|ay (p2 , t1 )|2 = 1 .

(4.2)

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From this, it follows that we can define the element v(m1 , p1 , t1 ; m2 , p2 , t2 ) ∈ K ⊗ K as

|t2 /y| at2 (p1 , y) ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 )

y, z ∈ Iq sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq

× fm1 +m2 −χ(p1 p2 /t2 z),z,t1 ⊗ fχ(p1 p2 /t2 z),sgn(p2 t2 )(yz/p1 )q m2 ,y . Note that v(m1 , p1 , t1 ; m2 , p2 , t2 ) is just the right-hand side of the formula by which we want to define W ∗ . Now choose also m1 , m2 ∈ Z and p1 , p2 , t1 , t2 ∈ Iq . Notice that inequality (4.2) together with the Cauchy-Schwarz equality implies that

| (t2 /y) (t2 /y) at2 (p1 , y) at2 (p1 , y) ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 )  ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 ) |

y, z ∈ Iq sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq 

sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq

is finite, which allows us to compute the next sum in any order we want: v(m1 , p1 , t1 ; m2 , p2 , t2 ), v(m1 , p1 , t1 ; m2 , p2 , t2 )

=

|t2 /y| |t2 /y| at2 (p1 , y) at2 (p1 , y) ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 )

y, z ∈ Iq sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq 

sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq 

× ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 ) fm1 +m2 −χ(p1 p2 /t2 z),z,t1 , fm1 +m2 −χ(p1 p2 /t2 z),z,t1  × fχ(p1 p2 /t2 z),sgn(p2 t2 )(yz/p1 )q m2 ,y , f



χ(p1 p2 /t2 z),sgn(p2 t2 )(yz/p1 )q m2 ,y



=



|t2 /y| |t2 /y| at2 (p1 , y) at2 (p1 , y) ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 )

y, z ∈ Iq sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq 

sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq 

× ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 ) δm1 +m2 ,m1 +m2 δt1 ,t1 δ|p1 p2 /t2 |,|p1 p2 /t2 |

× δ



sgn(p2 t2 )q m2 /p1 ,sgn(p2 t2 )q m2 /p1

 = δm1 +m2 ,m1 +m2 δt1 ,t1 δp1 p2 /t2 ,p1 p2 /t2 δ sgn(p2 t2 )q m2 /p1 ,sgn(p2 t2 )q m2 /p1 |t2 /y| |t2 /y| at2 (p1 , y)at2 (p1 , y)

y∈Iq



z ∈ Iq sgn(p2 t2 )(yz/p1 )q m2 ∈ Iq

ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 )ap2 (z, sgn(p2 t2 )(yz/p1 )q m2 ) .

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Thus, using the orthogonality relation in Proposition 3.2 together with Proposition 3.5, we get that v(m1 , p1 , t1 ; m2 , p2 , t2 ), v(m1 , p1 , t1 ; m2 , p2 , t2 )  = δm1 +m2 ,m1 +m2 δt1 ,t1 δp1 p2 /t2 ,p1 p2 /t2 δ sgn(p2 t2 )q m2 /p1 ,sgn(p2 t2 )q m2 /p1 |t2 /y| |t2 /y| at2 (p1 , y) at2 (p1 , y)δp2 ,p2 δsgn(y),sgn(p1 t2 ) y∈Iq   = δm1 +m2 ,m1 +m2 δt1 ,t1 δp1 /t2 ,p1 /t2 δ δ sgn(t2 )q m2 /p1 ,sgn(t2 )q m2 /p1 p2 ,p2   (−1)χ(yt2 ) sgn(p1 )χ(p1 ) (−1)χ(yt2 ) sgn(p1 )χ(p1 ) ay (p1 , t2 ) ay (p1 , t2 )

y∈Iq

= δm1 +m2 ,m1 +m2 δt1 ,t1 δp1 /t2 ,p1 /t2 δ 

  δ sgn(t2 )q m2 /p1 ,sgn(t2 )q m2 /p1 p2 ,p2 

(−1)χ (t2 t2 ) sgn(p1 )χ(p1 ) sgn(p1 )χ(p1 )



ay (p1 , t2 ) ay (p1 , t2 ) .

y∈Iq

Because of the factor δp1 /t2 ,p1 /t2 in the above expression, Proposition 3.3 implies that v(m1 , p1 , t1 ; m2 , p2 , t2 ), v(m1 , p1 , t1 ; m2 , p2 , t2 ) = δm1 +m2 ,m1 +m2 δt1 ,t1 δp1 /t2 ,p1 /t2 δ m2 sgn(t2 )q





/p1 ,sgn(t2 )q m2 /p1 

δp2 ,p2 (−1)χ(t2 t2 ) sgn(p1 )χ(p1 ) sgn(p1 )χ(p1 ) δp1 ,p1 δt2 ,t2

= δm1 ,m1 δm2 ,m2 δp1 ,p1 δp2 ,p2 δt1 ,t1 δt2 ,t2

= fm1 ,p1 ,t1 ⊗ fm2 ,p2 ,t2 , fm1 ,p1 ,t1 ⊗ fm2 ,p2 ,t2  . Now the lemma follows easily.

 

The next proposition deals with the essential step towards the left invariance of the weight ϕ and the realization that W is the multiplicative unitary associated to quantum
(1, 1). SU Proposition 4.6. Consider ω ∈ B(K)∗ and a ∈ Nϕ . Then (ωπ ⊗ ι)(a) ∈ Nϕ and ((ωπ ⊗ ι)(a)) = (ω ⊗ ι)(W ∗ ) (a). Proof. Choose m, m1 , m2 ∈ Z, p, t, p1 , t1 , p2 , t2 ∈ Iq . Take also v ∈ Iq and let us calculate the element ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t))((m1 , p1 , t1 ) ⊗ (0, v, v)). Choose r, s ∈ Z and x, y ∈ Iq . Then Proposition 3.7, Definition 3.6 and Eqs. (2.3), (2.4) tell us that ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t)) (ζ r ⊗ δx ⊗ ζ s ⊗ δy ) = az (x, y) ((−m2 , t2 , p2 ) ⊗ 1)V ∗ (1 ⊗ (m, p, t)) z∈Iq

× V
r−χ(y/z),s+χ(x/z),χ(y/x),z

Locally Compact Quantum Group

=



255

az (x, y) ((−m2 , t2 , p2 ) ⊗ 1)V ∗ (1 ⊗ (m, p, t))(ζ r−χ(y/z) ⊗ ζ s+χ(x/z)

z∈Iq

⊗ζ χ(y/x) ⊗ δz ) = at (x, y) ((−m2 , t2 , p2 ) ⊗ 1)V ∗ (ζ r−χ(y/t) ⊗ ζ s+χ(x/t) ⊗ ζ χ(y/x)+m ⊗ δp ) = at (x, y) ((−m2 , t2 , p2 ) ⊗ 1)
r−χ(y/t),s+χ(x/t),χ(y/x)+m,p = at (x, y) ap (z, sgn(p)|y/x|q m z) ((−m2 , t2 , p2 ) ⊗ 1) z ∈ Iq sgn(p)|y/x|q m z ∈ Iq

=

× ζ r−χ(y/t)+χ(q

m zy/px)

⊗ δz ⊗ ζ s+χ(x/t)−χ(z/p) ⊗ δsgn(p)|y/x|q m z

at (x, y) ap (z, sgn(pt)(y/x)q m z) ((−m2 , t2 , p2 ) ⊗ 1)

z ∈ Iq sgn(pt)(y/x)q m z ∈ Iq

× ζ r+m−χ(px/zt) ⊗ δz ⊗ ζ s+χ(px/zt) ⊗ δsgn(pt)(y/x)q m z by Eq. (2.3). So we see that ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t)) (ζ r ⊗ δx ⊗ ζ s ⊗ δy ) = 0 if sgn(pt)(y/x)q m p2 ∈ Iq . If, on the other hand, sgn(pt)(y/x)q m p2 ∈ Iq , then ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t)) (ζ r ⊗ δx ⊗ ζ s ⊗ δy ) = at (x, y) ap (p2 , sgn(pt)(y/x)q m p2 ) (ζ r+m−m2 −χ(px/p2 t) ⊗ δt2 ⊗ ζ s+χ(px/p2 t) ⊗δsgn(pt)(y/x)q m p2 ) . Now we have for r, s ∈ Z and x, y ∈ Iq that ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t))((m1 , p1 , t1 ) ⊗ (0, v, v)) (ζ r ⊗ δx ⊗ ζ s ⊗ δy ) = δt1 ,x δv,y ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t))(ζ m1 +r ⊗ δp1 ⊗ ζ s ⊗ δv ) . So we see that ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t))((m1 , p1 , t1 ) ⊗ (0, v, v)) = 0 if sgn(pt)(v/p1 )q m p2 ∈ Iq . If, on the other hand, sgn(pt)(v/p1 )q m p2 ∈ Iq , then ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t))((m1 , p1 , t1 ) ⊗ (0, v, v)) (ζ r ⊗ δx ⊗ ζ s ⊗ δy ) = δt1 ,x δv,y at (p1 , v) ap (p2 , sgn(pt)(v/p1 )q m p2 ) ζ r+m1 +m−m2 −χ(pp1 /p2 t) ⊗ δt2 ⊗ ζ s+χ(pp1 /p2 t) ⊗ δsgn(pt)(v/p1 )q m p2 , and thus ((m2 , p2 , t2 )∗ ⊗ 1)((m, p, t))((m1 , p1 , t1 ) ⊗ (0, v, v)) = at (p1 , v) ap (p2 , sgn(pt)(v/p1 )q m p2 ) × (m1 + m − m2 − χ (pp1 /p2 t), t2 , t1 ) ⊗(χ (pp1 /p2 t), sgn(pt)(v/p1 )q m p2 , v) . Applying ϕ ⊗ ι to this equation and using Lemma 4.2, we see that [ (ω((m1 ,p1 ,t1 )),((m2 ,p2 ,t2 )) π ⊗ ι)(((m, p, t))) ] (0, v, v) = δm1 +m−m2 ,χ(pp1 /p2 t) δt1 ,t2 (t1 )−2 at (p1 , v) ap (p2 , sgn(pt)(v/p1 )q m p2 ) ×(m + m1 − m2 , sgn(pt)(v/p1 )q m p2 , v) .

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Or, again by Lemma 4.2, (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι)(((m, p, t))) (0, v, v) = δm1 +m−m2 ,χ(pp1 /p2 t) δt1 ,t2 |t2 /t1 | at (p1 , v) ap (p2 , sgn(pt)(v/p1 )q m p2 ) ×(m + m1 − m2 , sgn(pt)(v/p1 )q m p2 , v) , so we conclude that if sgn(pt)(v/p1 )q m p2 ∈ Iq , the element (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι)(((m, p, t))) (0, v, v) belongs to Nϕ and    (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι)(((m, p, t))) (0, v, v) = δm1 +m−m2 ,χ(pp1 /p2 t) δt1 ,t2 |v|−1 at (p1 , v) ap (p2 , sgn(pt)(v/p1 )q m p2 ) × fm+m1 −m2 ,sgn(pt)(v/p1 )q m p2 ,v . (4.3) Also recall that (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι)(((m, p, t))) (0, v, v) = 0 if sgn(pt) (v/p1 )q m p2 ∈ Iq . Notice that Corollary 4.4 implies that J π((0, v, v))Jfn,x,y = J π((0, v, v)) f−n,y,x = δv,y Jf−n,y,x = δv,y fn,x,y . So we get by Proposition 4.5 that J π((0, v, v))J (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 ⊗ ι)(W ∗ ) fm,p,t = δt1 ,t2 δm1 +m−χ(p1 p/tp2 ),m2 |t/y| at (p1 , y) ap (p2 , sgn(pt)(yp2 /p1 )q m ) y ∈ Iq sgn(pt)(yp2 /p1 )q m ∈ Iq

× J π((0, v, v))J fχ(p1 p/tp2 ),sgn(pt)(yp2 /p1 )q m ,y = δt1 ,t2 δm1 +m−χ(p1 p/tp2 ),m2 |t/y| at (p1 , y) ap (p2 , sgn(pt)(yp2 /p1 )q m ) y ∈ Iq sgn(pt)(yp2 /p1 )q m ∈ Iq

× δv,y fm+m1 −m2 ,sgn(pt)(yp2 /p1 )q m ,y . So we see that J π((0, v, v))J (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 ⊗ ι)(W ∗ ) fm,p,t = 0 if sgn(pt) (vp2 /p1 )q m ∈ Iq . If, on the other hand, sgn(pt)(vp2 /p1 )q m ∈ Iq , J π((0, v, v))J (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 ⊗ ι)(W ∗ ) fm,p,t = δt1 ,t2 δm1 +m−χ(p1 p/tp2 ),m2 |t/v| at (p1 , v) ap (p2 , sgn(pt)(vp2 /p1 )q m ) ×fm+m1 −m2 ,sgn(pt)(vp2 /p1 )q m ,v . From Eq. (4.3) and Lemma 4.2, we conclude that    (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι)(((m, p, t))) (0, v, v)

= J π((0, v, v))J (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 ⊗ ι)(W ∗ ) ((m, p, t)) .

We use the net of projections (PL )L∈F (Iq ) introduced in the proof of Corollary 4.3. If L ∈ F (Iq ), we know by the previous equation that (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι)(((m, p, t))) PL ∈ Nϕ and    (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι)(((m, p, t))) PL = J π(PL )J (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 ⊗ ι)(W ∗ ) ((m, p, t)) .

Locally Compact Quantum Group

257

Therefore the σ -strong∗ closedness of  implies that (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι) (((m, p, t))) ∈ Nϕ and    (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π ⊗ ι)(((m, p, t))) = (ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 ⊗ ι)(W ∗ ) ((m, p, t)) .

Since the linear span of such linear functionals ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 π is norm dense in (Mq )∗ , the proposition now follows easily from Corollary 4.3 and the σ -strong∗ closedness .   Now, the proof of the left invariance of ϕ can be dealt with as in the proof of [20, Prop. 8.15]. Proposition 4.7. The weight ϕ is left invariant with respect to . Proof. Take a ∈ Nϕ and ω ∈ (Mq )+ ∗ . Since π(Mq ) is in standard form with respect an orthonormal to K, there exists a vector v ∈ K such that ω = ωv,v π . Take also  basis (ei )i∈I for K. It follows from the proof of [20, Lem. A.5] that i∈L (ωv,ei π ⊗ ι)((a))∗ (ωv,ei π ⊗ ι)((a)) L∈F (I ) is an increasing net in Mq+ that converges strongly to (ω ⊗ ι)(a ∗ a) (as before, F (I ) denotes the set of finite subsets of I , directed by inclusion). Therefore the σ -weak lower semi-continuity of ϕ implies that     ϕ (ωv,ei π ⊗ ι)((a))∗ (ωv,ei π ⊗ ι)((a)) . ϕ (ω ⊗ ι)(a ∗ a) = i∈I

By the previous proposition, this implies that        (ωv,ei π ⊗ ι)((a)) ,  (ωv,ei π ⊗ ι)((a))  ϕ (ω ⊗ ι)(a ∗ a) = i∈I

=



(ωv,ei ⊗ ι)(W ∗ )(a), (ωv,ei ⊗ ι)(W ∗ )(a)

i∈I

=



(ωv,ei ⊗ ι)(W ∗ )∗ (ωv,ei ⊗ ι)(W ∗ )(a), (a) .

i∈I

So if we use first [20, Lem. A.5] and afterwards the fact that W ∗ is an isometry (Proposition 4.5), we see that   ϕ (ω ⊗ ι)(a ∗ a) = (ωv,v ⊗ ι)(W W ∗ )(a), (a) = ωv,v (1) (a), (a) = ω(1) ϕ(a ∗ a), and we have proven the proposition.

 

We have introduced the anti-unitary operator J˘ on H before Proposition 3.10. Thus, ˘ J (ζ n ⊗ δx ) = sgn(x)χ(x) ζ −n ⊗ δx for all n ∈ Z and x ∈ Iq . This implies easily that J˘(m, p, t)J˘ = sgn(p)χ(p) sgn(t)χ(t) (−m, p, t) for m ∈ Z and p, t ∈ Iq . It shows first of all J˘Mq J˘ = Mq . This allows us to define an anti-∗ -isomorphism R˘ on Mq such ˘ ˘ ˘ that R(a) = J˘a ∗ J˘ for all a ∈ Mq . Proposition 3.10 guarantees that χ (R˘ ⊗ R) = R, where χ denotes the flip map χ : Mq ⊗ Mq → Mq ⊗ Mq . Later on, R˘ turns out to be intimately connected with the unitary antipode.

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˘ Thus ϕ is also right invariant. Proposition 4.8. We have that ϕ = ϕ R. ˘ Proof. If m ∈ Z and p, t ∈ Iq , then R((m, p, t))∗ = J˘(m, p, t)J˘ = sgn(p)χ(p) ˘ sgn(t)χ(t) (−m, p, t). Hence, Lemma 4.2 implies that R((m, p, t))∗ ∈ Nϕ . This same lemma implies moreover for m, m ∈ Z and p, t, p , t  ∈ Iq :  ˘ ˘ (R((m, p, t))∗ ), (R((m , p , t  ))∗ ) 







= sgn(p)χ(p) sgn(t)χ(t) sgn(t  )χ(t ) sgn(p  )χ(p ) ×((−m, p, t)), ((−m , p , t  )) = sgn(p)χ(p) sgn(t)χ(t) sgn(t  )χ(t ) sgn(p  )χ(p ) |tt  |−1 δm,m δp,p δt,t  = |tt  |−1 δm,m δp,p δt,t  = ((m , p , t  )), ((m, p, t)) . ˘ ∗ ∈ Nϕ If a ∈  (m, p, t) | m ∈ Z, p, t ∈ Iq , the above result implies that R(a) ∗ ∗ ˘ and (R(a) ) = (a). By Corollary 4.3 and the σ -strong closedness of  this ˘ ∗ ∈ Nϕ and (R(a) ˘ ∗ ) = (a), thus implies that for all a ∈ Nϕ , the element R(a) ∗ ∗ ∗ ˘ a)) = ϕ(R(a) ˘ R(a) ˘ ˘ ϕ(R(a ) = (R(a) ) = (a) = ϕ(a ∗ a). Since R˘ 2 = ι, it ˘ Because χ (R˘ ⊗ R) ˘ follows that ϕ = ϕ R. = R˘ and ϕ is left invariant, we get that ϕ is right invariant.   Combined with Theorem 3.11, we have proven the main result of this paper (see Definition 1 of the Introduction for the terminology used): Theorem 4.9. The pair (Mq , ) is a von Neumann algebraic quantum group which we
q (1, 1). denote by SU By Proposition 4.6 and the remark after [21, Thm.1.2], we also conclude that Proposition 4.10. The element W is the multiplicative unitary of (Mq , ) with respect to the GNS-construction (K, π, ). The unitarity can also be proved directly from Propositions 4.5, 3.2, 3.3 and 3.5. We denote the antipode, unitary antipode and scaling group of (Mq , ) by S, R and τ respectively. All these objects are defined after the proof of [21, Prop. 1.4]. The modular element and scaling constant of a quantum group are introduced at the same place. Because (Mq , ) is unimodular, that is, possesses a weight that is left and right invariant, we get the following result. Proposition 4.11. We have that ϕR = ϕ. Furthermore, the modular element of (Mq , ) is the identity operator and the scaling constant of (Mq , ) equals 1. Proof. We denote the modular element and scaling constant of (Mq , ) by δ and ν reϕ spectively, see the discussion after the proof of [21, Prop. 1.4]. We know that σs (δ) = ν s δ for all s ∈ R (*) and ϕR = ϕδ . Because ϕ is right invariant, the uniqueness of right Haar weights implies the existence of λ > 0 such that ϕR = λ ϕ. Thus, the uniqueness of the Radon-Nykodim derivative guarantees that δ = λ 1. But, since (δ) = δ ⊗ δ, it follows that δ = 1. Therefore, (*) implies that ν = 1.   In the next part of this section we calculate explicit expressions for S, R and τ acting on elements (m, p, t). For this purpose we calculate some explicit slices (ι ⊗ ω)(W ∗ ), where ω ∈ B(K)∗ .

Locally Compact Quantum Group

259

Consider m, m1 , m2 ∈ Z, p, p1 , p2 , t, t1 , t2 ∈ Iq . It follows easily from Proposition 4.5 that (ι ⊗ ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 )(W ∗ ) fm,p,t

= |t1 /t2 | at1 (p, t2 ) ap1 (sgn(p1 t1 )(pp2 /t2 )q −m1 , p2 ) fm1 −m2 +m,sgn(p1 t1 )(pp2 /t2 )q −m1 ,t (4.4)

if sgn(p1 t1 )(pp2 /t2 )q −m1 ∈ Iq and χ (p1 t2 /p2 t1 ) = m2 − m1 . If, on the other hand, sgn(p1 t1 )(pp2 /t2 )q −m1 ∈ Iq or χ (p1 t2 /p2 t1 ) = m2 − m1 , then (ι ⊗ ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 ) (W ∗ ) fm,p,t = 0. Proposition 4.12. Consider m1 , m2 , m3 , m4 ∈ Z and p, t ∈ Iq . Then we define ω ∈ B(K)∗ as the absolutely convergent sum ω=



sgn(x)χ(x) sgn(y)χ(y) (−1)χ(xy) |xy|ap (sgn(p)q m3 x, x)

x, y ∈ Iq sgn(p)q m3 x ∈ Iq sgn(t)q m4 y ∈ Iq × at (sgn(t)q m4 y, y)ωfm

m m ,f 1 +χ (x/y),sgn(p)q 3 x,sgn(t)q 4 y m2 +χ (x/y),x,y

(4.5)

.

Then (ι⊗ω)(W ∗ ) = t 2 q −m1 −m3 (−1)m2 δχ(t/p),m1 δm3 −m4 ,m2 −m1 π((m1 −m2 , p, t)). Proof. Let s ∈ Iq and m ∈ Z. By Definition 3.1, we have for z ∈ Iq satisfying sgn(s) q m z ∈ Iq ,  |as (sgn(s)q m z, z)| ≤ cq ν(sq −m )

(−κ(s); q 2 )∞ |z| (−κ(z); q 2 )∞ (q 2 ; q 2 )∞

|(−q 2 /κ(s); (q 2(1+m) /s 2 )κ(z); sgn(s)q 2(m+1) )| , which by Lemma 6.8 implies that z∈Iq ,sgn(s)q m z∈Iq |as (sgn(s)q m z, z)| < ∞. It follows that the sum in the statement of this proposition is absolutely convergent. So, as a norm convergent limit of finite sums of vector functionals, we obtain ω ∈ B(K)∗ . Take n ∈ Z and c, d ∈ Iq . Now (ι ⊗ ω)(W ∗ ) fn,c,d = sgn(x)χ(x) sgn(y)χ(y) (−1)χ(xy) |xy|ap (sgn(p)q m3 x, x) x, y ∈ Iq sgn(p)q m3 x ∈ Iq sgn(t)q m4 y ∈ Iq

× at (sgn(t)q m4 y, y) (ι ⊗ ωfm

m m ,f 1 +χ (x/y),sgn(p)q 3 x,sgn(t)q 4 y m2 +χ (x/y),x,y

)(W ∗ ) fn,c,d .

By Eq. (4.4), we get immediately that (ι ⊗ ω)(W ∗ ) fn,c,d = 0 if sgn(pt)cq −m1 ∈ Iq or m3 − m4 = m2 − m1 . Now suppose that sgn(pt)cq −m1 ∈ Iq and m3 − m4 = m2 − m1 .

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Then Eq. (4.4) and Proposition 3.5 imply that (ι ⊗ ω)(W ∗ ) fn,c,d =



sgn(x)χ(x) sgn(y)χ(y) (−1)χ(xy) |xy|ap (sgn(p)q m3 x, x)

x, y ∈ Iq sgn(p)q m3 x ∈ Iq sgn(t)q m4 y ∈ Iq

× at (sgn(t)q m4 y, y)q m4 asgn(t)q m4 y (c, y) asgn(p)q m3 x (sgn(pt)cq −m1 , x) × fm1 −m2 +n,sgn(pt)cq −m1 ,d =



sgn(x)χ(x) sgn(y)χ(y) (−1)χ(xy) |xy|ap (sgn(p)q m3 x, x)

x, y ∈ Iq sgn(p)q m3 x ∈ Iq sgn(t)q m4 y ∈ Iq

× at (sgn(t)q m4 y, y)q m4 (−1)χ(cy)+m4 sgn(y)χ(y) |c/q m4 y| × ac (sgn(t)q m4 y, y) (−1)χ(cx)+m3 +m1 sgn(x)χ(x) |cq −m1 /q m3 x| × asgn(pt)cq −m1 (sgn(p)q m3 x, x) fm1 −m2 +n,sgn(pt)cq −m1 ,d =



c2 q −m1 −m3 (−1)m1 +m3 +m4 ap (sgn(p)q m3 x, x)

x, y ∈ Iq sgn(p)q m3 x ∈ Iq sgn(t)q m4 y ∈ Iq

× asgn(pt)cq −m1 (sgn(p)q m3 x, x)at (sgn(t)q m4 y, y) ac (sgn(t)q m4 y, y) × fm1 −m2 +n,sgn(pt)cq −m1 ,d . This sum is absolutely convergent so we can compute it in any order we deem useful. Therefore the orthogonality relations in Proposition 3.2 imply that (ι ⊗ ω)(W ∗ ) fn,c,d = c2 q −m1 −m3 (−1)m1 +m3 +m4 δp,sgn(pt)cq −m1 δt,c fn+m1 −m2 ,sgn(pt)cq −m1 ,d = t 2 q −m1 −m3 (−1)m1 +m3 +m4 δ|p|,|t|q −m1 δt,c fn+m1 −m2 ,p,d = t 2 q −m1 −m3 (−1)m1 +m3 +m4 δ|p|,|t|q −m1 π((m1 − m2 , p, t)) fn,c,d ,

(4.6)

where π is the GNS-representation of ϕ. If sgn(pt)cq −m1 ∈ Iq , then δ|p|,|t|q −m1 π((m1 − m2 , p, t)) fn,c,d = δ|p|,|t|q −m1 δt,c fm1 −m2 +n,p,d = 0 , implying that Eq. (4.6) also holds if sgn(pt)cq −m1 ∈ Iq and m3 − m4 = m2 − m1 . Therefore the lemma follows.   Now we can easily calculate the action of S on elements of the form (m, p, t). Proposition 4.13. Consider m ∈ Z and p, t ∈ Iq . Then (m, p, t) ∈ D(S) and S((m, p, t)) = sgn(p)χ(p) sgn(t)χ(t) (−1)m q m (m, t, p).

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Proof. Define ω ∈ B(K)∗ as in Eq. (4.5) for m1 = χ (t/p), m2 = χ (t/p) − m, m3 = −χ (t/p) and m4 = −χ (t/p) + m. Then Proposition 4.12 implies that π((m, p, t)) = t −2 (−1)m+χ(pt) (ι ⊗ ω)(W ∗ ). Thus [20, Prop. 8.3] implies that (m, p, t) ∈ D(S −1 ) and   π S −1 ((m, p, t)) = t −2 (−1)m+χ(pt) (ι ⊗ ω)(W ) ∗ ∗ = t −2 (−1)m+χ(pt) (ι ⊗ ω)(W ¯ ) , (4.7) where, as is customary, ω¯ ∈ B(K)∗ is given by ω(a) ¯ = ω(a ∗ ) for all a ∈ B(K). By definition, ω= sgn(x)χ(x) sgn(y)χ(y) (−1)χ(xy) |xy|ap ((p/|t|)x, x) x, y ∈ Iq (p/|t|)x ∈ Iq (|p|/t)q m y ∈ Iq

× at ((|p|/t)q m y, y)ωfχ (t/p)+χ (x/y),(p/|t|)x,(|p|/t)q m y ,f−m+χ (t/p)+χ (x/y),x,y . Thus, ω¯ =



sgn(x)χ(x) sgn(y)χ(y) (−1)χ(xy) |xy| ap ((p/|t|)x, x)

x, y ∈ Iq (p/|t|)x ∈ Iq (|p|/t)q m y ∈ Iq

× at ((|p|/t)q m y, y)ωf−m+χ (t/p)+χ (x/y),x,y ,fχ (t/p)+χ (x/y),(p/|t|)x,(|p|/t)q m y =



(−1)χ(xy) |xy| sgn(p)χ(p) sgn((p/|t|)x)χ((p/|t|)x) ap (x, (p/|t|)x)

x, y ∈ Iq (p/|t|)x ∈ Iq (|p|/t)q m y ∈ Iq m

× sgn(t)χ(t) sgn((|p|/t)q m y)χ((|p|/t)q y) at (y, (|p|/t)q m y) × ωf−m+χ (t/p)+χ (x/y),x,y ,fχ (t/p)+χ (x/y),(p/|t|)x,(|p|/t)q m y , where we used Proposition 3.5 in the last equation. A simple change in summation variables x  = (p/|t|)x and y  = (|p|/t)yq m then reveals that  ω¯ = sgn(p)χ(p) sgn(t)χ(t) (−1)m q −m (t 2 /p 2 ) sgn(x  )χ(x ) x  , y  ∈ Iq (|t|/p)x  ∈ Iq (t/|p|)q −m y  ∈ Iq 

 

× sgn(y  )χ(y ) (−1)χ(x y ) |x  y  |ap ((|t|/p)x  , x  ) at ((t/|p|)q −m y  , y  ) × ωfχ (t/p)+χ (x  /y  ),(|t|/p)x  ,(t/|p|)q −m y  ,fm+χ (t/p)+χ (x  /y  ),x  ,y  , which upon close inspection shows that ω¯ is sgn(p)χ(p) sgn(t)χ(t) (−1)m q −m (t 2 /p 2 ) times the functional in Eq. (4.5) where m1 = χ (t/p), m2 = χ (t/p) + m, m3 = χ (t/p) and m4 = χ (t/p) − m. By Proposition 4.12 this implies that ∗ (ι ⊗ ω)(W ¯ ) = sgn(p)χ(p) sgn(t)χ(t) (−1)m q −m (t 2 /p 2 ) t 2 (p 2 /t 2 )

(−1)m+χ(pt) π((−m, p, t)) = sgn(p)χ(p) sgn(t)χ(t) (−1)m q −m t 2 (−1)m+χ(pt) π((−m, p, t)) .

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Therefore Eq. (4.7) implies that  S −1 ((m, p, t)) = sgn(p)χ(p) sgn(t)χ(t) (−1)m q −m (m, t, p).  It is now possible to recognize the polar decomposition of the antipode. For this purpose, define the anti-unitary operator I on H and the strictly positive operator Q in H such that  ζ n ⊗δx | n ∈ Z, x ∈ Iq  is a core for Q and I (ζ n ⊗δx ) = (−1)n sgn(x)χ(x) ζ −n ⊗ δx and Q(ζ n ⊗ δx ) = q 2n ζ n ⊗ δx for all n ∈ Z and x ∈ Iq . Recall that the unitary antipode R and scaling group τ are introduced after the proof of [21, Prop. 1.4]. Proposition 4.14. The unitary antipode R and the scaling group τ for (Mq , ) are such that R(a) = I a ∗ I and τs (a) = Qis aQ−is for all a ∈ Mq and s ∈ R. Let m ∈ Z and p, t ∈ Iq . Then R((m, p, t)) = sgn(p)χ(p) sgn(t)χ(t) (−1)m (m, t, p). If z ∈ C, then the element (m, p, t) belongs to D(τz ) and τz ((m, p, t)) = q 2miz (m, p, t). Proof. By Proposition 4.11 and the discussion after the proof of [21, Prop. 1.4] we know that there exists a strictly positive operator P on K such that P is (a) = (τs (a)) for all a ∈ Nϕ and s ∈ R. Note that π(τs (a)) = P is π(a)P −is for all s ∈ R. Choose m ∈ Z and p, t ∈ R. Since S = Rτ− i and R and τ commute, we see that τ−i = S 2 . Thus, Prop2

osition 4.13 implies that (m, p, t) ∈ D(τ−i ) and τ−i ((m, p, t)) = q 2m (m, p, t). Thus, Lemma 4.2 and the von Neumann algebraic version of [18, Prop. 4.4] guarantee that ((m, p, t)) ∈ D(P ) and P ((m, p, t)) = q 2m ((m, p, t)). Therefore, Lemma 4.2 tells us that fm,p,t ∈ D(P ) and Pfm,p,t = q 2m fm,p,t = (Q ⊗ 1)fm,p,t . Because such elements fm,p,t form a core for Q ⊗ 1, we conclude that Q ⊗ 1 ⊆ P . Taking the adjoint of this inclusion, we see that P ⊆ Q⊗1. As a consequence, P = Q⊗1. So if a ∈ Mq and s ∈ R, we see that τs (a)⊗1 = π(τs (a)) = P is π(a)P −is = Qis aQ−is ⊗1, implying that τs (a) = Qis aQ−is . Note that Qis (ζ n ⊗ δx ) = q 2nis ζ n ⊗ δx for all n ∈ Z, x ∈ Iq and s ∈ R. Now a straightforward calculation reveals that τs ((m, p, t)) = Qis (m, p, t)Q−is = q 2mis (m, p, t) for all m ∈ Z, p, t ∈ Iq and s ∈ R. Fix m ∈ Z and p, t ∈ Iq for the moment. By definition S = Rτ− i . The previous 2 paragraph implies that τ− i ((m, p, t)) = q m (m, p, t). Therefore Proposition 4.13 2

guarantees that R((m, p, t)) = q −m S((m, p, t)) = sgn(p)χ(p) sgn(t)χ(t) (−1)m (m, t, p). Using the definition of I , one checks easily that also I (m, p, t)∗ I = I (−m, t, p)I = sgn(p)χ(p) sgn(t)χ(t) (−1)m (m, t, p) and thus by the previous calculation, R((m, p, t)) = I (m, p, t)∗ I . Since the von Neumann algebra Mq is generated by such elements (m, p, t), we conclude that R(a) = I a ∗ I for all a ∈ Mq .  

˘ One easily checks that R and R˘ are connected through the formula R((m, p, t)) = (−1)m R((m, p, t)) if m ∈ Z and p, t ∈ Iq . Consider z ∈ C. Since the ∗ -algebra  (m, p, t) | m ∈ Z, p, t ∈ Iq  is dense in Mq for the σ -strong∗ topology and this same ∗ -algebra is clearly invariant under each τs , where s ∈ R, it follows from the von Neumann algebraic version of [19, Cor. 1.22] that  (m, p, t) | m ∈ Z, p, t ∈ Iq  is a σ -strong∗ core for τz . If n ∈ Z, the fact that S n = R τ−n i implies that  (m, p, t) | 2 m ∈ Z, p, t ∈ Iq  is a σ -strong∗ core for S n .

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Following [21, Prop. 1.6] we associate to the von Neumann algebraic quantum group (Mq , ) a reduced C∗ -algebraic quantum group in the sense of [20]. Therefore we define the C∗ -subalgebra Aq of Mq such that π(Aq ) is the norm closure, in B(K), of the set { (ι ⊗ ω)(W ∗ ) | ω ∈ B(H )∗ }. Then [21, Prop. 1.6] guarantees that (Aq , 
Aq ) is a reduced C∗ -algebraic quantum group in the sense of [20]. In the next proposition we give an explicit description for Aq . We will use the following notation. For f ∈ C(T × Iq ) and x ∈ Iq we define fx ∈ C(T) such that fx (λ) = f (λ, x) for all λ ∈ T. Proposition 4.15. Denote by C the C ∗ -algebra of all functions f ∈ C(T × Iq ) such that (1) fx converges uniformly to 0 as x → 0 and (2) fx converges uniformly to a constant function as x → ∞. Then Aq is the norm closed linear span, in B(H ), of the set { ρp Mf | f ∈ C, p ∈ −q Z ∪ q Z }. Proof. Let p, t ∈ Iq and define Fp,t ∈ F(Iq ) such that Fp,t (x) = ap (x, t) for all x ∈ Iq . The reason for introducing the function Fp,t stems from the fact that (ι ⊗ ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 )(W ∗ ) = δχ(p1 t2 /p2 t1 ),m2 −m1 |t1 /t2 |  × π ρsgn(p1 t1 )(t2 /p2 )q m1 Mζ m1 −m2 ⊗T



sgn(p1 t1 )(p2 /t2 )q −m1

(Fp1 ,p2 )Ft1 ,t2

(4.8)

for p1 , p2 , t1 , t2 ∈ Iq and m1 , m2 ∈ Z (and which follows from Eq. (4.4) ). Since B(K)∗ is the closed linear span of vector functionals, Aq is the closed linear span of elements of the form (4.8). Let p, t ∈ Iq . Take m ∈ Z such that |p/t| = q m . We consider two different cases: sgn(p) = sgn(t) : Note first of all that in this case, Fp,t (x) = 0 for all x ∈ Iq− . Definition 3.1 implies the existence of a bounded function h : Iq+ → R such that Fp,t (x) = h(x) 

ν(pt/x) (−κ(x); q 2 )∞

(−q 2 /κ(t); q 2 κ(x/t); q 2 κ(x/p))

for all x ∈ Iq+ . If x → 0, we have that ν(pt/x) → 0, (−κ(x); q 2 )∞ → 1 and (−q 2 /κ(t); q 2 κ(x/t); q 2 κ(x/p)) → (−q 2 /κ(t); 0; 0). It follows that Fp,t (x) → 0 as x → 0. Proposition 3.5 and Definition 3.1 imply for x ∈ Iq+ , Fp,t (x) = ap (x, t) = sgn(p)χ(pt) ap (t, x) 1

−1

2 (−κ(t); q 2 )∞2 x ν(px/t) = (−1)χ(pt) sgn(p)χ(pt) cq (−κ(p); q 2 )∞ 1

2 (−q 2 /κ(x); q 2 κ(t/x); q 2 κ(t/p)). (−κ(x); q 2 )∞

(4.9)

If x → ∞, then (−q 2 /κ(x); q 2 κ(t/x); q 2 κ(t/p)) → (0; 0; q 2 κ(t/p)) = (q 2 κ(t/p); q 2 )∞ , where we used [6, Eq. (1.3.16)]. (4.10)

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Let x ∈ Iq+ and choose s ∈ Z such that x = q s . Then Result 6.1 implies that 1

2 x ν(px/t) (−κ(x); q 2 )∞ 1

= q s q 2 (s+m−1)(s+m−2) 1

1



(−q 2s ; q 2 )∞ 1

1

−1

2 = q s q 2 s(s−1) q 2 (m−1)(m−2) q (m−1)s q − 2 s(s−1) (−1, −q 2 ; q 2 )∞ (−q 2(1−s) ; q 2 )∞2 1 √ 1 − = q 2 (m−1)(m−2) 2 (−q 2 ; q 2 )∞ x m (−q 2 /x 2 ; q 2 )∞2 . 1

2 → So if p = t (and thus m = 0), x ν(px/t) (−κ(x); q 2 )∞

√ 2 q (−q 2 ; q 2 )∞ as x → 1

2 ∞. If, on the other hand, p > t (and thus m < 0), we see that x ν(px/t) (−κ(x); q 2 )∞ → 0 as x → ∞. If we √ combine this with Eq. (4.9) and the convergence in (4.10), we conclude, since cq−1 = 2 q (q 2 , −q 2 ; q 2 )∞ , that

(1) Fp,p (x) → 1 as x → ∞, (2) if p > t, then Fp,t (x) → 0 as x → ∞ . But Proposition 3.5 implies that |Fp,t | = |t/p| |Ft,p |. By the convergence in (2), this implies that if p < t, also Fp,t (x) → 0 as x → ∞. sgn(p)
= sgn(t): Note that in this case, Fp,t (x) = 0 for all x ∈ Iq+ . Completely similar as in the beginning of the first part of this proof, one shows that Fp,t (x) → 0 as x → 0. From the previous discussion we only have to remember that Fp,t (x) → 0 as x → 0. Provided p = t, we have that Fp,t (x) → 0 as x → ∞. And Fp,p (x) → 1 as x → ∞. (4.11) Let B denote the norm closed linear span of { ρp Mf | f ∈ C, p ∈ −q Z ∪ q Z } in B(H ). Use the notation of Eq. (4.8). If Tsgn(p1 t1 )(p2 /t2 )q −m1 (Fp1 ,p2 )Ft1 ,t2 ∈ C0 (Iq ), then Eqs. (4.8) and (4.11) immediately guarantee that (ι ⊗ ωfm1 ,p1 ,t1 ,fm2 ,p2 ,t2 )(W ∗ ) ∈ π(B). If, on the other hand, Tsgn(p1 t1 )(p2 /t2 )q −m1 (Fp1 ,p2 )Ft1 ,t2 ∈ C0 (Iq ), we must by (4.11) necessarily have that t1 = t2 and p1 = p2 . Thus, Eq. (4.8) implies in this case that (ι ⊗ ωfm1 ,p1 ,t1 ,fm2 ,p1 ,t1 )(W ∗ )   = δ0,m2 −m1 π ρq m1 |t1 /p1 | Mζ m1 −m2 ⊗T −m1 (F )F q |p1 /t1 | p1 ,p1 t1 ,t1   = δ0,m2 −m1 π ρq m1 |t1 /p1 | M1⊗Tq −m1 |p /t | (Fp1 ,p1 )Ft1 ,t1 ∈ π(B) , 1 1

where we used (4.11) in the last relation. Hence π(Aq ) ⊆ π(B), thus Aq ⊆ B. Proposition 4.12 implies that ρt −1 p Mζ m ⊗δp = (m, t, p) ∈ Aq for all p, t ∈ Iq and m ∈ Z. It follows that ρx Mζ m ⊗δp ∈ Aq for all m ∈ Z, x ∈ −q Z ∪ q Z and p ∈ Iq . Thus, ρx Mf ∈ Aq for all x ∈ −q Z ∪ q Z and f ∈ C0 (T × Iq ). If x ∈ −q Z and f ∈ C, it is easy to see that there exists g ∈ C0 (T × Iq ) such that ρx Mf = ρx Mg . Hence, ρx Mf ∈ Aq in this case. Now fix x ∈ q Z and f ∈ C. Take m ∈ Z such that x = q m . Because f ∈ C, there exists c ∈ C such that fy → c 1T uniformly as y → ∞. Consequently, f − c (1T ⊗ Tx −1 (F1,1 )F1,1 ) belongs to C0 (T × Iq ) by (4.11). As such, ρx Mf −c (1T ⊗Tx −1 (F1,1 )F1,1 ) ∈

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Aq . Moreover, Eq. (4.8) guarantees that π(ρx M1⊗Tx −1 (F1,1 )F1,1 ) = (ι ⊗ ωfm,1,1 ,fm,1,1 ) (W ∗ ) ∈ π(Aq ). Hence ρx M1⊗Tx −1 (F1,1 )F1,1 ∈ Aq . It follows that ρx Mf ∈ Aq . Hence B ⊆ Aq , proving that B = Aq .   5. Proof of the Coassociativity of the Comultiplication This section is devoted to the proof of Theorem 3.11. First one proves that (⊗ι)(γ ) = (ι ⊗ )(γ ). It is easy to prove that both operators agree on the intersection of their domain, but it takes a lot of work to establish the equality of their domains. In order to enhance the continuity of the exposition, we give the proof in the second half of this section. Once we have proven that ( ⊗ ι)(γ ) = (ι ⊗ )(γ ), it is straightforward to show that ( ⊗ ι)(x) = (ι ⊗ )(x) for all x ∈ Nq . In the final step of the proof of the coassociativity it is shown that ( ⊗ ι)(u) = (ι ⊗ )(u). Since (x) = V ∗ (1L2 (T2 ) ⊗ x)V for all x ∈ Mq , we get for any closed densely defined operator T in H , affiliated with Mq , ( ⊗ ι)(T ) = (V ∗ ⊗ 1H )(1L2 (T2 ) ⊗ V ∗ )(1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ T )(1L2 (T2 ) ⊗ V )(V ⊗ 1H ) (5.1) and ∗ (1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ T )V13 (1H ⊗ V ), (ι ⊗ )(T ) = (1H ⊗ V ∗ )V13

(5.2)

where V13 : H ⊗ L2 (T2 ) ⊗ H → L2 (T2 ) ⊗ L2 (T2 ) ⊗ H is defined as usual. (2) (2) Using the map 0 : Aq → Aq Aq Aq defined by 0 = (0 ι)0 = (2) (2) (2) (2) (ι 0 )0 , we get adjointable operators 0 (α0 ), 0 (α0† ), 0 (γ0 ), 0 (γ0† ) in + L (E E E). Proposition 3.9 will imply the next result. Proposition 5.1. If T = α or T = γ , then (2)

0 (T0 ), ⊆ ( ⊗ ι)(T ), (2) 0 (T0† ) ⊆ ( ⊗ ι)(T ∗ ),

(2)

0 (T0 ) ⊆ (ι ⊗ )(T ), (2) 0 (T0† ) ⊆ (ι ⊗ )(T ∗ ) .

Proof. We only prove the inclusion for T = α. The other ones are dealt with in the same way. Proposition 3.9 tells us that α0 α0 + q e0 γ0† γ0 ⊆ V ∗ (1L2 (T2 ) ⊗ α)V and thus, 1L2 (T2 ) α0 α0 + q 1L2 (T2 ) e0 γ0† γ0

⊆ (1L2 (T2 ) ⊗ V ∗ )(1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ α)(1L2 (T2 ) ⊗ V ) .

(5.3)

By Lemma 2.1, D(1L2 (T2 ) ⊗ α) = D(1L2 (T2 ) ⊗ eγ ∗ ). Let v ∈ D(1L2 (T2 ) ⊗ α) and ∞ 2 2 w ∈ E. There exists a sequence (vn )∞ n=1 in L (T ) E such that (vn )n=1 → v and ∞  † (1L2 (T2 ) α0 )(vn ) n=1 → (1L2 (T2 ) ⊗ α)(v). Since (1L2 (T2 ) α0 )(1L2 (T2 ) α0 ) = (1L2 (T2 ) e0 γ0 )(1L2 (T2 ) e0 γ0† )+1L2 (T2 ) e0 , 1L2 (T2 ) e0 is bounded and 1L2 (T2 ) ⊗eγ ∗  ∞ is closed, it follows that (1L2 (T2 ) e0 γ0† )(vn ) n=1 → (1L2 (T2 ) ⊗ eγ ∗ )(v). Therefore the net ∞  (1L2 (T2 ) α0 α0 + q 1L2 (T2 ) e0 γ0† γ0 )(vn ⊗ w) n=1

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converges to ( (1L2 (T2 ) ⊗ α) α0 + q (1L2 (T2 ) ⊗ eγ ∗ ) γ0 )(v ⊗ w). Hence, inclusion (5.3) and the fact that the operator on the right hand side of this inclusion is closed imply that v ⊗ w ∈ D (1L2 (T2 ) ⊗ V ∗ )(1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ α)(1L2 (T2 ) ⊗ V ) and ( (1L2 (T2 ) ⊗ α) α0 + q (1L2 (T2 ) ⊗ eγ ∗ ) γ0 )(v ⊗ w)

= (1L2 (T2 ) ⊗ V ∗ )(1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ α)(1L2 (T2 ) ⊗ V )(v ⊗ w) .

In other words, (1L2 (T2 ) ⊗ α) α0 + q (1L2 (T2 ) ⊗ eγ ∗ ) γ0

⊆ (1L2 (T2 ) ⊗ V ∗ )(1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ α)(1L2 (T2 ) ⊗ V ) .

As a consequence, Proposition 3.9 implies that 0 (α0 ) = 0 (α0 ) α0 + q 0 (e0 γ0† ) γ0 ⊆ V ∗ (1L2 (T2 ) ⊗ α)V α0 (2)

+ q V ∗ (1L2 (T2 ) ⊗ eγ ∗ )V γ0 ⊆ (V ∗ ⊗ 1H )(1L2 (T2 ) ⊗ V ∗ )(1L2 (T2 ) ⊗1L2 (T2 ) ⊗ α)(1L2 (T2 ) ⊗ V )(V ⊗ 1H ) = ( ⊗ ι)(α) .

  For later purposes we introduce some extra notation: if z, w ∈ C, we set φ(w; z) =

∞ (wq 2r ; q 2 )∞ r=0

(q 2 ; q 2 )r

q 2r(r−1) zr .

(5.4)

Note that this φ-function is closely related to the 0 ϕ2 -function (see [6, Def. (1.2.22)]). The next step of the proof consists of showing that Proposition 5.2. ( ⊗ ι)(γ ) = (ι ⊗ )(γ ). The proof of this result is a tedious but not so difficult task and is given in the second half of this section, after Proposition 5.4. It is however important to remember that this result is needed for the last two parts (Corollary 5.3 and Proposition 5.4) of the proof of the coassociativity of . Recall that we introduced Nq after Definition 2.3. Corollary 5.3. We have that ( ⊗ ι)(x) = (ι ⊗ )(x) for all x ∈ Nq . Proof. The previous proposition implies that     D ( ⊗ ι)(γ ) = D (ι ⊗ )(γ )

(5.5)

By Eq. (5.1), we know that   D ( ⊗ ι)(T ) = (V ∗ ⊗ 1H )(1L2 (T2 ) ⊗ V ∗ ) D(1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ T ) for T = γ and T = α. Lemma 2.1 guarantees that D(1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ γ ) =     D(1L2 (T2 ) ⊗ 1L2 (T2 ) ⊗ α). As a consequence, D ( ⊗ ι)(γ ) = D ( ⊗ ι)(α) . In     a similar way oneshows that D (ι ⊗ )(γ ) = D (ι ⊗ )(α) . By Eq. (5.5), this  guarantees that D ( ⊗ ι)(α) = D (ι ⊗ )(α) .

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Proposition 5.1 tells us that 0 (α0† ) ⊆ (⊗ι)(α ∗ ) and 0 (α0† ) ⊆ (ι⊗)(α ∗ ). Taking the adjoints of this inclusion, we see that ( ⊗ ι)(α) and (ι ⊗ )(α) are both (2) restrictions of 0 (α0† )∗ . The previous discussion taught us that also the domains of ( ⊗ ι)(α) and (ι ⊗ )(α) agree. Thus, ( ⊗ ι)(α) = (ι ⊗ )(α). Since Mζ ⊗ 1 is the unitary in the polar decomposition of γ and ( ⊗ ι)(γ ) = (ι⊗)(γ ), the uniqueness of the polar decomposition implies that (⊗ι)(Mζ ⊗1) = (ι ⊗ )(Mζ ⊗ 1) (*) and ( ⊗ ι)(|γ |) = (ι ⊗ )(|γ |). Since (e) = e ⊗ e, ( ⊗ ι)(e) = (ι⊗)(e), so we get that (⊗ι)(e |γ |) = (ι⊗)(e |γ |). Applying functional calculus to this equation, we see that ( ⊗ ι)(M1⊗g ) = (ι ⊗ )(M1⊗g ) for all g ∈ L∞ (Iq ). Combining this with (*), it follows that (⊗ι)(π(f )) = (ι⊗)(π(f )) for all f ∈ L∞ (T × Iq ) (cf. the proof of Lemma 2.4). Because w is the isometry in the polar decomposition of α and ( ⊗ ι)(α) = (ι ⊗ )(α), the uniqueness of the polar decomposition implies that ( ⊗ ι)(w) = (ι ⊗ )(w). The corollary follows by Lemma 2.4.   (2)

(2)

In order to finalize the proof of the coassociativity we rely on formulas discovered by S.L. Woronowicz. In particular, we borrow the use of the extended Hilbert space, the definition of ˜ as in (5.10) and Eq. (5.11) in the proof below. These formulas can be found in [32, Sect. 5] (Eq. (5.11) is a slight variation of the one in [32], where e is replaced by 1). S.L. Woronowicz told us that he has a proof for these identities, not included in the version of [32] yet. Our proof of the strong convergence in (5.10) and Eq. (5.11) are probably different from his ones, due to the difference of both approaches. The proof of the equality ( ⊗ ι)(u) = (ι ⊗ )(u) is not present in [32], but it is possible that S.L. Woronowicz also knows a proof for this. However, the
q (1, 1)-quadruples that corresponds to the equality property of tensor products of SU D(( ⊗ ι)(γ )) = D((ι ⊗ )(γ )) is still not proven within the approach of [32]. Proposition 5.4. The ∗ -homomorphism  : Mq → Mq ⊗ Mq is coassociative. Proof. Since Mq is generated by Nq ∪ {u}, it only remains to show that ( ⊗ ι)(u) = (ι ⊗ )(u) by the previous corollary. Define first of all the isometry v ∈ Nq as v = e w (Mζ∗ ⊗ 1). Set Kq = −q Z ∪ q Z and use the uniform measure on Kq . Define H˜ = L2 (T) ⊗ L2 (Kq ). Then H˜ is obviously the orthogonal sum of H and L2 (T) ⊗ L2 (Kq \ Iq ). Let L be a Hilbert space. Then we define a normal ∗ − homomorphism B(H ⊗ L) → B(H˜ ⊗ L) : a → aˆ such that a
ˆ H ⊗L = a and a
ˆ (H ⊗L)⊥ = 0. If L1 , L2 are two Hilbert spaces, (b ⊗ c)ˆ = bˆ ⊗ c for all b ∈ B(H ⊗ L1 ), c ∈ B(L2 ). (5.6) On H˜ we define unitary operators e, ˜ w˜ and u˜ such that for f ∈ L2 (T × Kq ), (ef ˜ )(λ, x) = sgn(x) f (λ, x),

(wf ˜ )(λ, x) = f (λ, q −1 x),

(uf ˜ )(λ, x) = f (λ, −x),

for almost all (λ, x) ∈ T × Kq . Now define the unitary element v˜ in B(H˜ ) as v˜ = e˜ w˜ (Mζ∗ ⊗ 1). Since (L2 (T) ⊗ L2 (Kq \ Iq )) ⊗ H is separable, we can extend the orthonormal basis for H ⊗ H given by the functions of Definition 3.6 to an orthonormal basis (
r,s,m,p | r, s, m ∈ Z, p ∈ Kq ) for H˜ ⊗ H . If p ∈ Kq , we define δ˜p ∈ L2 (Kq ) such that δ˜p (x) = δx,p for all x ∈ Kq . Now define the unitary operator

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E. Koelink, J Kustermans

V˜ : H˜ ⊗ H → L2 (T2 ) ⊗ H˜ such that V˜
r,s,m,p = ζ r ⊗ ζ s ⊗ ζ m ⊗ δ˜p for all r, s, m ∈ Z and p ∈ Kq . So the restriction of V˜ to H ⊗H equals V and V˜ (H ⊥ ⊗H ) = L2 (T2 )⊗H ⊥ . ˜ : B(H˜ ) → B(H˜ ⊗ H ) such that Define the normal injective ∗ -homomorphism  ∗ ˜ ˜ a) (a) = V˜ (1L2 (T2 ) ⊗ a)V˜ for all a ∈ B(H˜ ). Thus, if a ∈ Mq , then ( ˆ = (a)ˆ. By ˆ = ( ⊗ ι)(b)ˆ for all b ∈ Mq ⊗ B(H ). ˜ ⊗ ι)(b) Eq. (5.6), this implies that ( ˜ v)
Take r, s, m ∈ Z and p ∈ Kq . Choose n ∈ N. One easily checks that ( ˜ r,s,m,p = ˜ v) sgn(p)
r,s,m−1,pq . Thus, ( ˜ n
r,s,m,p = sgn(p)n
r,s,m−n,pq n . Notice that p q n ∈ Iq if n ≥ 1 − χ (p). Assume that n ≥ 1 − χ (p). Choose x ∈ Kq , y ∈ Iq and λ, µ ∈ T. If q n x ∈ Iq and y/x = sgn(p) q m , then y/(q n x) = sgn(pq n ) q m−n which by Definition 3.6 and Result 6.4 guarantees that   −n ˜ v) ˜ n
r,s,m,p (λ, x, µ, y) = sgn(p)n [(v˜ −n ⊗ 1)
r,s,m−n,pq n ](λ, x, µ, y) (v˜ ⊗ 1)( = sgn(p)n sgn(x)n λn
r,s,m−n,pq n (λ, q n x, µ, y) n

= sgn(y)n λn aq n p (q n x, y) λr+χ(y/pq ) µs−χ(xq

n /pq n )

= sgn(y)n aq n p (q n x, y) λr+χ(y/p) µs−χ(x/p) = λr+χ(y/p) µs−χ(x/p) sgn(y)n cq s(q n x, y) (−1)χ(q

n p)

n

× (−sgn(y))χ(q x) |y| ν(q n py/q n x)   (−κ(q n p), −κ(y); q 2 )∞  −q 2 /κ(q n p) 2 n  κ(q x/y) ; q × q 2 κ(q n x/q n p) (−κ(q n x); q 2 )∞ = λr+χ(y/p) µs−χ(x/p) cq s(x, y) (−1)χ(p) (−sgn(y))χ(x) |y| ν(py/x)   (−κ(q n p), −κ(y); q 2 )∞  −q 2 /κ(q n p) 2 n  κ(q x/y) × ; q 2 q κ(x/p) (−κ(q n x); q 2 )∞ = λr+χ(y/p) µs−χ(x/p) cq s(x, y) (−1)χ(p) (−sgn(y))χ(x) |x| q m ν(q m p)   (−κ(q n p), −sgn(p) q 2m κ(x); q 2 )∞  −q 2 /κ(q n p) 2(1−m+n) .  ; sgn(p) q × q 2 κ(x/p) (−κ(q n x); q 2 )∞ (5.7) If, on the other hand, 

q nx 

∈ Iq or y/x =

sgn(p) q m ,

then

˜ v) ˜ n
r,s,m,p (λ, x, µ, y) = sgn(p)n [(v˜ −n ⊗ 1)
r,s,m−n,pq n ](λ, x, µ, y) (v˜ −n ⊗ 1)( = sgn(p)n sgn(x)n λn
r,s,m−n,pq n (λ, q n x, µ, y) =0 .

Using notation (5.4), we define the function fr,s,m,p : T × Kq × T × Iq → C such that for x ∈ Kq , y ∈ Iq , λ, µ ∈ T, fr,s,m,p (λ, x, µ, y) equals s−χ(x/p) c s(x, y) (−1)χ(p) (−sgn(y))χ(x) |y| ν(q m p) λr+χ(y/p) q  µ × (−κ(y); q 2 )∞ φ(q 2 κ(x/p); −q 2(2−m) /p 2 )

(5.8)

if y/x = sgn(p) q m and fr,s,m,p (λ, x, µ, y) = 0 if y/x = sgn(p) q m . Then it is  ˜ clear from the above computations (and the proof of Lemma 6.2) that (v˜ −n ⊗ 1)  ∞ n (5.9) (v) ˜
r,s,m,p n=1 converges pointwise to fr,s,m,p .

Locally Compact Quantum Group

269

Define J = { x ∈ Kq | |x| ≤ q 1−m or sgn(x) = sgn(p) }. By Lemma 6.8 there exists a function H ∈ l 2 (J )+ such that

| |x| (−sgn(p) q 2m κ(x); q 2 )∞ (−q 2 /κ(q n p); q 2 κ(x/p); sgn(p)q 2(1−m+n) ) | ≤ H (x) for all x ∈ J and n ∈ N. 1 2 There exists clearly a number C > 0 such that q m ν(q m p) (−κ(q n p); q 2 )∞ (q 2 ; −1

q 2 )∞2 ≤ C for all n ∈ N. Let (x, y) ∈ Kq × Iq such that y/x = sgn(p)q m . If |x| > q 1−m then |q m x| ≥ 1 and since sgn(p)q m x = y ∈ Iq , this implies that sgn(x) = sgn(p). Hence, x belongs to J . So we can define an element G ∈ L2 (T × Kq × T × Iq ) such that for λ, µ ∈ T and x ∈ Kq , y ∈ Iq , we have that G(λ, x, µ, y) = 0 if y/x = sgn(p)q m and G(λ, x, µ, y) = C H (x) if y/x = sgn(p)q m . Then Eq. (5.7) and the remarks thereafter ˜ v) ˜ n
r,s,m,p | ≤ G for all n ∈ N such that n ≥ 1 − χ (p). Thereimply that |(v˜ −n ⊗ 1)( fore the convergence in (5.9) and the dominated convergence theorem imply that fr,s,m,p  ∞ ˜ v) belongs to H˜ ⊗ H and that the sequence (v˜ −n ⊗ 1)( ˜ n
r,s,m,p n=1 converges to fr,s,m,p in H˜ ⊗ H .  ∞ ˜ v) Since the sequence (v˜ −n ⊗ 1)( ˜ n n=1 is a sequence of isometries and the linear span of such elements
r,s,m,p is dense in H˜ ⊗ H , we conclude that there exists an is ∞ ˜ v) ometry ˜ : H˜ ⊗ H → H˜ ⊗ H such that the sequence (v˜ −n ⊗ 1)( ˜ n n=1 converges strongly to ˜ . (5.10) Thus, ˜
r,s,m,p = fr,s,m,p for all r, s, m ∈ Z and p ∈ Kq . Using Eq. (5.8) and ˜ and u, the definition of  ˜ one easily checks that (u˜ ⊗ e)fr,s,m,p = fr,s,m,−p and ˜ (u)
˜ r,s,m,p =
r,s,m,−p for all r, s, m ∈ Z and p ∈ Kq (note that s(−x, y) = sgn(y) s(x, y) for all x, y ∈ R \ {0}). ˜ u) As a consequence, ˜ ( ˜ = (u˜ ⊗ e) ˜ . (5.11) ˜ Define ˆ ∈ B(H ⊗ H ) such that ˆ
H ⊗H = ˜
H ⊗H and ˆ
H ⊥ ⊗H = 0. One sees ˜ v)
that (v)
r,s,m,p = sgn(p)
r,s,m−1,qp = ( ˜ r,s,m,p for all r, s, m ∈ Z, p ∈ Iq , ˜ implying that (v) is the restriction of (v) ˜ to H ⊗ H . So we get for every n ∈ N that  n ˜ v) ˜ v) ˜ v) ( ˆ n
H ⊗H = (v)ˆ
H ⊗H = ( ˜ n
H ⊗H , whereas ( ˆ n
H ⊥ ⊗H = 0. It follows  −n ∞ ˜ v) that ˆ is the strong limit of the sequence of isometries (v˜ ⊗ 1)( ˆ n n=1 . As such, ˆ belongs to B(H˜ ) ⊗ Mq . We set P = u∗ u, which belongs to Nq because it is a multiplication operator. Mul˜ Pˆ ) and using the fact that u˜ Pˆ = u, tiplying Eq. (5.11) from the right by ( ˆ we see that ˆ ˆ ˆ ˜ ( ˜ u) ˜ ˜ ˆ ˆ ˆ = (u˜ ⊗ e) (P ). Thus, (u) = (u˜ ⊗ e) (P ) . (5.12) Restricting the previous equation to H ⊗ H , we get ˆ (u) = (u˜ ⊗ e) ˆ (P ). Applying ι ⊗  to this equality, it follows that (ι ⊗ )( ˆ ) (ι ⊗ )((u)) = (u˜ ⊗ e ⊗ e)(ι ⊗ )( ˆ )(ι ⊗ )(P ). (5.13) ˜ ⊗ ι to Eq. (5.12), we get ( ˜ ⊗ ι)( ˆ )( ⊗ ι)((u))ˆ = (( ˜ u) If we apply  ˜ ⊗ ˆ ˜ ˆ e)( ⊗ ι)( )( ⊗ ι)((P )) . So the restriction of this equation to H ⊗ H ⊗ H gives ˜ ˆ )(⊗ι)(u) = (( ˜ u)⊗e)( ˜ ˆ )(⊗ι)(P ). Multiplying this equation (⊗ι)( ˜ ⊗ι)( from the left by ˜ ⊗ 1, we see that

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E. Koelink, J Kustermans

˜ ⊗ ι)( ˆ )( ⊗ ι)(u) = ( ˜ ( ˜ u) ˜ ⊗ ι)( ˆ )( ⊗ ι)(P ) ( ˜ ⊗ 1) ( ˜ ⊗ e)( ˜ ⊗ ι)( ˆ )(ι ⊗ )(P ), = (u˜ ⊗ e ⊗ e)( ˜ ⊗ 1)( (5.14) where in the last equality we used Eq. (5.11), the fact that P ∈ Nq and Corollary 5.3. ˜ ⊗ ι)( ˆ ). Let n ∈ N. Since v belongs to Nq , Corollary Now we calculate ( ˜ ⊗ 1) ( ˜ ⊗ ι)( ˜ v) 5.3 guarantees that ( ⊗ ι)(v) = (ι ⊗ )(v) which implies that ( ˆ = ˜ (ι ⊗ )(v). ˆ Hence,   ˜ v˜ n )] ⊗ 1)( ˜ ⊗ ι) (v˜ −n ⊗ 1)( ˜ vˆ n ) = (v˜ −n ⊗ 1 ⊗ 1) ( ˜ ⊗ ι)( ˜ vˆ n ) ( [(v˜ −n ⊗ 1)( ˜ vˆ n ) . = (v˜ −n ⊗ 1 ⊗ 1) (ι ⊗ )( If we let n tend to infinity in this equality, the expression we started with converges ˜ ⊗ ι)( ˆ ) whereas the expression we end up converges strongly strongly to ( ˜ ⊗ 1)( ˆ ˜ ⊗ ι)( ˆ ) = (ι ⊗ )( ˆ ). Inserting this in Eq. (5.14), to (ι ⊗ )( ). Thus, ( ˜ ⊗ 1)( ˆ we conclude that (ι ⊗ )( ) ( ⊗ ι)(u) = (u˜ ⊗ e ⊗ e)(ι ⊗ )( ˆ ) (ι ⊗ )(P ). Comparing this with Eq. (5.13), we conclude that (ι ⊗ )( ˆ ) (ι ⊗ )(u) = (ι ⊗ )( ˆ ) ( ⊗ ι)(u) .

(5.15)

The initial projection of ˆ is given by X ⊗ 1, where X ∈ B(H˜ ) is the orthogonal projection onto H . In other words, ˆ ∗ ˆ = X ⊗ 1. Thus, (ι ⊗ )( ˆ )∗ (ι ⊗ )( ˆ ) = X ⊗ 1 ⊗ 1, implying that (ι ⊗ )( ˆ ) is a partial isometry with initial space H ⊗ H ⊗ H . Therefore Eq. (5.15) guarantees that (ι ⊗ )(u) = ( ⊗ ι)(u).   It is clear from the end of the proof above that we need ˆ to be an isometry on H ⊗ H ⊗ H . This is precisely the reason for using the extended Hilbert space H˜ , a ∞  definition of ˆ as the strong limit of ((v ∗ )n ⊗ 1)(v)n n=1 would not result in an isometry. The need for using ˜ should be clear from the fact that the range of the operator ˜ ⊗ ι)( ˆ )( ⊗ ι)(u) in Eq. (5.14) is not contained in H ⊗ H ⊗ H . ( In the last part of this section we provide the proof of Proposition 5.2. The most intricate part of this proof is contained in Proposition 5.8. The results before this proposition reduce the problem to a simpler form. We stress that this exposition does not depend on Corollary 5.3 or Proposition 5.4. In order to filter out some distracting features we will change our Hilbert space H ⊗3 . For this purpose, define a measure on Iq32 such that ({(x, z, y)}) = |x| |y|

(−x, −y; q 2 )∞ (−z; q 2 )∞

for all x, y, z ∈ Iq 2 . In this section, all L2 -spaces that involve Iq32 are defined by this measure. Define the unitary transformation U : H ⊗3 → L2 (T3 ) ⊗ L2 (Iq32 , ) such that U [f ] = [g], where the functions f ∈ L2 ((T × Iq )3 ) and g ∈ L2 (T3 × Iq32 ) are such that

Locally Compact Quantum Group

271 − logq 2 |x|

g(λ, µ, η, x, z, y) = s(xz, y) s(z, x) λ × sgn(z)

logq 2 |x|

sgn(y)

µ

logq 2 |x/y| logq 2 |y|

logq 2 |xz|

η

(−1) 

|x|−1/2 |y|−1/2

logq 2 |z|

(−z; q 2 )∞ (−x, −y; q 2 )∞

× f (λ, κ −1 (x), µ, κ −1 (z), η, κ −1 (y)) for all λ, µ, η ∈ T and x, z, y ∈ Iq 2 . If θ ∈ −q 2Z ∪q 2Z , we define the sets (see Fig. 1 and Fig. 2 in the proof of Proposition 5.8) K(θ ) = { (x, y) | x, y ∈ Iq 2 such that θxy ∈ Iq 2 } and L(θ ) = { (x, θxy, y) | (x, y) ∈ K(θ) } . k ∈ F(K(sgn(p)q 2k )) such Consider m, k ∈ Z and p ∈ Iq 2 . Define the function Fm,p 2k that for (x, y) ∈ K(sgn(p)q ),

1 k Fm,p (x, y) = cq2 q 2 (m+k+1)(m+k+2) ν(κ −1 (p)q m ) (−p; q 2 )∞   −q 2 /p  2(1−m) ; sgn(p) q 2(1−m) q y/|p|   2 −q /x  (5.16) ; q 2(1+m+k) x 2(1+k) sgn(p) q y k (x, y) = 0 if sgn(p)q −2m y ∈ I . We also define if sgn(p)q −2m y ∈ Iq 2 and Fm,p q2 k k 2k k the function Gm,p ∈ F(K(sgn(p)q )) by Gm,p (x, y) = F−m,p (y, x) for all (x, y) ∈ K(sgn(p)q 2k ). k ,G ˜ km,p ∈ F(I 32 ) both of which have support in Next we define functions F˜m,p q

L(sgn(p)q 2k ) and satisfy k k (x, sgn(p)q 2k xy, y) = Fm,p (x, y) and F˜m,p

˜ km,p (x, sgn(p)q 2k xy, y) = Gkm,p (x, y) G

for all (x, y) ∈ K(sgn(p)q 2k ). Define the unitary element ς ∈ C(T3 ) such that ς (λ, η, µ) = λµη¯ for all λ, η, µ ∈ T. k ˜ km,p | p ∈ Iq 2 , m, k ∈ Z } Lemma 5.5. The families { F˜m,p | p ∈ Iq 2 , m, k ∈ Z } and { G are both orthonormal bases of L2 (Iq32 , ). Moreover, k | v ∈ L2 (T3 ), p ∈ Iq 2 , m, k ∈ Z  is a core for U ( ⊗ 1) the space  v ⊗ F˜m,p ∗ ι)((γ )) U , ˜ km,p | v ∈ L2 (T3 ), p ∈ Iq 2 , m, k ∈ Z  is a core for U (ι ⊗ 2) the space  v ⊗ G ∗ )((γ )) U , 3) if v ∈ L2 (T3 ), p ∈ Iq 2 and m, k ∈ Z, then k k ) = |p|− 2 Mς v ⊗ F˜m+1,p , (U ( ⊗ ι)((γ ))U ∗ ) (v ⊗ F˜m,p 1

˜ km,p ) = |p|− 2 Mς v ⊗ G ˜ km+1,p . (U (ι ⊗ )((γ ))U ∗ ) (v ⊗ G 1

272

E. Koelink, J Kustermans

Proof. The proof of this fact is pretty straightforward. For r, s, t, n, m ∈ Z and p ∈ Iq , r,s,t,n ⊗3 as , Gr,s,t,n we define Fm,p m,p ∈ H r,s,t,n = (V ∗ ⊗ 1H )(1L2 (T2 ) ⊗ V ∗ )(ζ r ⊗ ζ s ⊗ ζ t ⊗ ζ n ⊗ ζ m ⊗ δp ), Fm,p

∗ ∗ r s t n m Gr,s,t,n m,p = (1H ⊗ V )V13 (ζ ⊗ ζ ⊗ ζ ⊗ ζ ⊗ ζ ⊗ δp ) .

r,s,t,n From Proposition 3.8, we get immediately that the families ( Fm,p | r, s, t, n, m ∈ r,s,t,n Z, p ∈ Iq ) and ( Gm,p | r, s, t, n, m ∈ Z, p ∈ Iq ) are both orthonormal bases of H ⊗3 . Moreover, Eqs. (2.1), (5.1) and (5.2) imply that r,s,t,n 1) the space  Fm,p | r, s, t, n, m ∈ Z, p ∈ Iq  is a core for the operator (⊗ι)((γ )), 2) the space  Gr,s,t,n m,p | r, s, t, n, m ∈ Z, p ∈ Iq  is a core for the operator (ι⊗)((γ )), 3) if r, s, t, m, n ∈ Z and p ∈ Iq , then r,s,t,n r,s,t,n ( ⊗ ι)((γ )) Fm,p = p −1 Fm+1,p

and

−1 Gr,s,t,n . (ι ⊗ )((γ )) Gr,s,t,n m,p = p m+1,p

(5.17)

By Definition 3.6, we get the following equalities (in which the infinite sums are L2 convergent), r,s,t,n Fm,p

= (V ∗ ⊗ 1H )(1L2 (T2 ) ⊗ V ∗ )(ζ r ⊗ ζ s ⊗ ζ t ⊗ ζ n ⊗ ζ m ⊗ δp ) = (V ∗ ⊗ 1H )(ζ r ⊗ ζ s ⊗
t,n,m,p ) = ap (sgn(p)q −m y, y) (V ∗ ⊗ 1H ) y ∈ Iq sgn(p)q −m y ∈ Iq −m

× (ζ r ⊗ ζ s ⊗ ζ t+χ(y/p) ⊗ δsgn(p)q −m y ⊗ ζ n−χ(q y/p) ⊗ δy ) ap (sgn(p)q −m y, y)
r,s,t+χ(y/p),sgn(p)q −m y ⊗ ζ n+m+χ(p/y) ⊗ δy = y ∈ Iq sgn(p)q −m y ∈ Iq

=





asgn(p)q −m y (x, sgn(p)sgn(y)q t+χ(y/p) x)

y ∈ Iq x ∈ Iq sgn(p)q −m y ∈ Iq sgn(p)sgn(y)q t+χ (y/p) x ∈ Iq

× ap (sgn(p)q −m y, y) ζ r+t+χ(y/p)+χ(x)+m−χ(y) ⊗ δx ⊗ ζ s−χ(x)−m+χ(y) ⊗ × δsgn(p)sgn(y)q t+χ (y/p) x ⊗ ζ n+m+χ(p/y) ⊗ δy =





ap (sgn(p)q −m y, y) asgn(p)q −m y (x, (q t /p) xy)

y ∈ Iq x ∈ Iq sgn(p)q −m y ∈ Iq (q t /p) xy ∈ Iq

× ζ r+t+m+χ(x/p) ⊗ δx ⊗ ζ s−m+χ(y/x) ⊗ δ(q t /p) xy ⊗ ζ n+m+χ(p/y) ⊗ δy .

(5.18)

Choose λ, µ, η ∈ T and x, z, y ∈ Iq 2 . Look first at the case where sgn(p) q −2m y ∈ Iq 2 and z = sgn(p) q 2(t−χ(p)) xy. Set x  = κ −1 (x), y  = κ −1 (y) and z = κ −1 (z). Thus,

Locally Compact Quantum Group

273 





sgn(p)q −m y  ∈ Iq and (q t /p)x  y  = z ∈ Iq . Set d = λr+t+m+χ(x /p) µs−m+χ(y /x )  ηn+m+χ(p/y ) . The above equation, Proposition 3.5 and Definition 3.1 imply that r,s,t,n (λ, κ −1 (x), µ, κ −1 (z), η, κ −1 (y)) Fm,p

= d ap (sgn(p)q −m y  , y  ) asgn(p)q −m y  (x  , z ) 





= d (sgn(p)sgn(y  ))m+χ(y ) sgn(z )χ(z ) sgn(x  )χ(x ) ap (sgn(p)q −m y  , y  ) asgn(p)q −m y  (z , x  ) 

= d (sgn(p) sgn(y  ))m+χ(y ) 





× sgn(z )χ(x ) (sgn(p) sgn(x  ) sgn(y  ))χ(p)+χ(y )+t sgn(x  )χ(x ) 



× cq s(sgn(p)q −m y  , y  ) (−1)χ(p)+m+χ(y ) sgn(y  )m+χ(y ) |y  |    (−κ(p), −y; q 2 )∞ −q 2 /κ(p) m 2(1−m)  ν(pq ) ; sgn(p) q q 2(1−m) y/|κ(p)| (−sgn(p)q −2m y; q 2 )∞ 





× cq s(z , x  ) (−1)m+χ(y )+χ(z ) sgn(x  )χ(z ) |x  |   (−sgn(p)q −2m y, −x; q 2 )∞  −q 2 /x 2(1+m)  z/y × ν(q −m y  x  /z ) ; sgn(p) q q 2 z/x (−z; q 2 )∞ 

= d cq2 ν(pq m ) ν(q −m−t+χ(p) ) s(xz, y) s(z, x) (−1)χ(p)+χ(z ) 

 

1

1

× sgn(p)m+t+χ(p) sgn(z )χ(x ) sgn(y  )χ(x z ) |x| 2 |y| 2   (−κ(p), −x, −y; q 2 )∞  −q 2 /κ(p) 2(1−m) ×  ; sgn(p) q 2(1−m) q y/|κ(p)| (−z; q 2 )∞   −q 2 /x 2(1+m+t−χ(p)) × x ; q sgn(p) q 2(1+t−χ(p)) y logq 2 |x|

=λ

µ

logq 2 |y/x| − logq 2 |y|

logq 2 |z|

η

s(xz, y) s(z, x)

logq 2 |x|

log |xz|

1

1

× (−1) sgn(z) sgn(y) q 2 |x| 2 |y| 2  (−x, −y; q 2 )∞ × (−1)χ(p) sgn(p)m+t+χ(p) λr+t+m−χ(p) µs−m (−z; q 2 )∞ t−χ(p) × ηn+m+χ(p) F˜m,κ(p) (x, z, y) .

If sgn(p) q −2m y ∈ Iq 2 or z = sgn(p) q 2(t−χ(p)) xy, then Eq. (5.18) implies immedir,s,t,n ately that Fm,p (λ, κ −1 (x), µ, κ −1 (z), η, κ −1 (y)) = 0. So we see that t−χ(p) r,s,t,n U Fm,p = (−1)χ(p) sgn(p)m+t+χ(p) ζ r+t+m−χ(p) ⊗ ζ s−m ⊗ ζ n+m+χ(p) ⊗ F˜m,κ(p) .

Proposition 3.10 implies that ∗ ∗ ˜ ˘ ˘ ˘ ˘ U Gr,s,t,n m,p = U 13 (J ⊗ J ⊗ J )(V ⊗ 1H )(1L2 (T2 ) ⊗ V )( ⊗ J )

× (ζ r ⊗ ζ s ⊗ ζ t ⊗ ζ n ⊗ ζ m ⊗ δp )

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E. Koelink, J Kustermans

= sgn(p)χ(p) U 13 (J˘ ⊗ J˘ ⊗ J˘)(V ∗ ⊗ 1H )(1L2 (T2 ) ⊗ V ∗ ) × (ζ −n ⊗ ζ −t ⊗ ζ −s ⊗ ζ −r ⊗ ζ −m ⊗ δp ) −n,−t,−s,−r = sgn(p)χ(p) U 13 (J˘ ⊗ J˘ ⊗ J˘)F−m,p .

Define the anti-unitary operator I˘ on L2 (T3 ) ⊗ L2 (Iq32 , ) such that I˘ [f ] = [g], where f, g ∈ L2 (T3 × Iq32 ) are such that g(λ, µ, η, x, z, y) = sgn(xyz)

logq 2 |xyz|

f (η, µ, λ, y, z, x) for all x, y, z ∈ Iq 2 and λ, µ, η ∈ T. Let x, y, z ∈ Iq 2 . If z > 0, it follows from the considerations before Definition 3.6 that s(xz, y) = sgn(xy) s(yz, x) and s(z, x) = sgn(xy) s(z, y). If on the other hand z < 0, we have s(xz, y) = s(yz, x) and s(z, x) = s(z, y). Thus, in both cases s(xz, y) s(z, x) = s(yz, x) s(z, y). Now it is straightforward to check that U 13 (J˘ ⊗ J˘ ⊗ J˘) = I˘U . Thus, −n,−t,−s,−r χ(p) ˘ U Gr,s,t,n I U F−m,p m,p = sgn(p)

= (−1)χ(p) sgn(p)m+s I˘ (ζ −n−s−m−χ(p) ⊗ ζ −t+m −s−χ(p) ⊗ζ −r−m+χ(p) ⊗ F˜ ). −m,κ(p)

−s−χ(p)

Since F˜−m,κ(p) is supported on the set L(q −2s /κ(p)) and sgn(xyz) sgn(p)s+χ(p) for all (x, y, z) ∈ L(q −2s /κ(p)), we get that

logq 2 |xyz|

=

−s−χ(p)

χ(p) ˜ U Gr,s,t,n sgn(p)m+χ(p) ζ r+m−χ(p) ⊗ ζ t−m ⊗ ζ n+s+m+χ(p) ⊗ G m,p = (−1) m,κ(p)

.

This implies for r  , s  , t  ∈ Z, k, m ∈ Z and p ∈ Iq 2 , 





k ζ r ⊗ ζ s ⊗ ζ t ⊗ F˜m,p 

= (−1) 

logq 2 |p |



r  −m−k,s  +m,k+logq 2 |p |,t  −m−logq 2 |p |

sgn(p  )m+k U Fm,κ −1 (p )



˜k  ζr ⊗ ζs ⊗ ζt ⊗ G m,p = (−1)

logq 2 |p |

sgn(p  )

m+logq 2 |p |

r  −m+log 2 |p |,−k−logq 2 |p |,s  +m,t  −m+k

U Fm,κ −1 (p ) q

.

From these two equations and the properties listed in (5.17) in the beginning of the proof, all the claims in the statement of this lemma easily follow.   Set D =  ζ r | r ∈ Z  ⊆ L2 (T). Lemma 5.6. The operator U 0 (γ0 )U ∗ is an adjointable operator on D 3 K(Iq 2 ) 3 (2)

such that (U 0 (γ0 )U ∗ )† = U 0 (γ0† )U ∗ . Moreover (2)

(2)

(U 0 (γ0 )U ∗ g)(λ, µ, η, x, z, y) 1 λµη ¯  = q |xy/z| 2 (1 + q −2 y) g(λ, µ, η, q 2 x, z, q −2 y) xy (2)

− (1 + q −2 y) g(λ, µ, η, x, q −2 z, η, q −2 y)  − (1 + z) g(λ, µ, η, q 2 x, q 2 z, y) + g(λ, µ, η, x, z, y) for g ∈ D 3 K(Iq 2 ) 3 , λ, µ, η ∈ T and x, z, y ∈ Iq 2 .

Locally Compact Quantum Group

275

Proof. Since the domain of 0 (γ0 ) and 0 (γ0† ) is E 3 and clearly U (E 3 ) = (2) D 3 K(Iq 2 ) 3 , the statements about the adjointability of 0 (γ0 ) follow easily. Using Eqs. (1.5), one checks that (2)

(2)

(2)

0 (γ0 ) = γ0 α0 α0 + e0 α0† γ0 α0 + e0 α0† e0 α0† γ0 + q γ0 e0 γ0† γ0 . So we get for f ∈ E 3 , x, y, z ∈ Iq and λ, µ, η ∈ T that (2)

(0 (γ0 ) f )(λ, x, µ, z, η, y)



λ = sgn(z) + q 2 /z2 sgn(y) + q 2 /y 2 f (λ, x, µ, q −1 z, η, q −1 y) x



µ sgn(x) + 1/x 2 sgn(y) + q 2 /y 2 f (λ, qx, µ, z, η, q −1 y) + sgn(x) z



η 2 sgn(x) + 1/x sgn(z) + 1/z2 f (λ, qx, µ, qz, η, y) + sgn(xz) y qλµη ¯ + sgn(z) f (λ, x, µ, z, η, y) xyz



q 2λ = sgn(yz) 1 + q −2 κ(z) 1 + q −2 κ(y) f (λ, x, µ, q −1 z, η, q −1 y) xyz

qµ  1 + κ(x) 1 + q −2 κ(y) f (λ, qx, µ, z, η, q −1 y) + sgn(y) xyz  η  + 1 + κ(x) 1 + κ(z) f (λ, qx, µ, qz, η, y) xyz qλµη ¯ + sgn(z) f (λ, x, µ, z, η, y) . xyz The inverse U ∗ of U is such that (U ∗ f )(λ, x, µ, z, η, y) = s(xz, y) s(z, x) λlogq |x| µlogq |y/x| η− logq |y| (−1)logq |z| sgn(z)logq |x| sgn(y)logq |xz|  (−κ(x), −κ(y); q 2 )∞ |x| |y| f (λ, µ, η, κ(x), κ(z), κ(y)) (−κ(z); q 2 )∞ for f ∈ D 3 K(Iq ) 3 , λ, µ, η ∈ T and x, z, y ∈ Iq . Then the following rules are easily established for g ∈ D 3 K(Iq 2 ) 3 , x, y, z ∈ Iq 2 and ω = (λ, η, µ) ∈ T3 . (1) If f ∈ D 3 K(Iq ) 3 , then (U Mf U ∗ g)(ω, x, z, y) = f (λ, κ −1 (x), µ, κ −1 (z), η, κ −1 (y)) g(ω, x, z, y). (2) (U (1 Tq −1 Tq −1 )U ∗ g)(ω, x, z, y)  ¯ (1 + q −2 y)(1 + q −2 z)−1 g(ω, x, q −2 z, q −2 y). = −q −1 sgn(y) µη (3) (U (Tq 1 Tq −1 )U ∗ g)(ω, x, z, y)  = sgn(yz) λµ¯ 2 η (1 + q −2 y)(1 + x)−1 g(ω, q 2 x, z, q −2 y). (4) (U (Tq Tq 1)U ∗ g)(ω, x, z, y) = −q sgn(z) λµ¯ (1 + z)(1 + x)−1 g(ω, q 2 x, q 2 z, y).

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E. Koelink, J Kustermans

This implies that for g ∈ D 3 K(Iq 2 ) 3 , x, y, z ∈ Iq 2 and ω = (λ, η, µ) ∈ T3 , (U 0 (γ0 )U ∗ g)(ω, x, z, y) qλµη ¯ = −sgn(z) −1 (1 + q −2 y) g(ω, x, q −2 z, q −2 y) κ (xyz) qλµη ¯ +sgn(z) −1 (1 + q −2 y) g(ω, q 2 x, z, q −2 y) κ (xyz) qλµη ¯ −sgn(z) −1 (1 + z) g(ω, q 2 x, q 2 z, y) κ (xyz) qλµη ¯ +sgn(z) −1 g(ω, x, z, y) . κ (xyz) 1 λµη ¯  = q |xy/z| 2 (1 + q −2 y) g(ω, q 2 x, z, q −2 y) xy (2)

−(1 + q −2 y) g(ω, x, q −2 z, q −2 y)  −(1 + z) g(ω, q 2 x, q 2 z, y) + g(ω, x, z, y) .

 

Let us also get rid of part of the operators acting on L2 (T3 ). For this purpose we introduce normal operators γl on γr in L2 (Iq3 , ) such that k | p ∈ Iq 2 , m, k ∈ Z is a core for γl , (1) the space  F˜m,p k ˜ (2) the space  Gm,p | p ∈ Iq 2 , m, k ∈ Z is a core for γr , 1 k k ˜ km,p = |p|− 21 G ˜k = |p|− 2 F˜m+1,p and γr G (3) if p ∈ Iq 2 and m, k ∈ Z, then γl F˜m,p m+1,p .

By Lemma 5.5 we know that U ((⊗ι)(γ ))U ∗ = Mς ⊗γl and U ((ι⊗)(γ ))U ∗ = Mς ⊗ γr . So in order to prove that ( ⊗ ι)(γ ) = (ι ⊗ )(γ ), it is enough to show that γl = γr . The description of γl and γr in terms of eigenvectors will not be sufficient to solve this problem. We will also need to know the explicit action of γl and γr on functions in its domain. For this purpose we introduce the linear operators γ˜ , γ˜ † belonging to End(F(Iq32 )) such that 1

(γ˜ f )(x, z, y) = q |xy/z| 2

1  (1 + q −2 y) f (q 2 x, z, q −2 y) xy

− (1 + q −2 y)f (x, q −2 z, q −2 y)−(1 + z)f (q 2 x, q 2 z, y)+f (x, z, y)



and 1  (1 + q −2 x)f (q −2 x, z, q 2 y)−(1 + z)f (x, q 2 z, q 2 y) xy  − (1 + q −2 x) f (q −2 x, q −2 z, y) + f (x, z, y) 1

(γ˜ † f )(x, z, y) = q |xy/z| 2

for all f ∈ F(Iq32 ) and x, y, z ∈ Iq 2 . Note that γ˜ † is obtained from γ˜ by interchanging x and y.

Locally Compact Quantum Group

277

Lemma 5.7. If f ∈ D(γl ) and g ∈ D(γr∗ ), then γl (f ) = γ˜ (f ) and γr∗ (g) = γ˜ † (g). Proof. Define the sesquilinear form . , . : K(Iq32 ) × F(Iq32 ) → C such that

f, g =

f (x, z, y) g(x, ¯ z, y) |x| |y|

(x,z,y)∈I 32

(−x, −y; q 2 )∞ (−z; q 2 )∞

q

for all f ∈ K(Iq32 ) and g ∈ F(Iq32 ). Under this pairing, F(Iq32 ) is anti-linearly isomorphic to the algebraic dual of K(Iq32 ). So there exist an anti-linear anti-homomorphism

◦

: End(K(Iq32 )) → End(F(Iq32 )) such that T (f ), g = f, T ◦ (g) for all f ∈ K(Iq32 )

and g ∈ F(Iq32 ). If f ∈ K(Iq32 ) and g ∈ L2 (Iq32 , ) then f, g equals the inner product

in L2 (Iq32 , ). So if T ∈ End(K(Iq32 )), then T ∗ (f ) = T ◦ (f ) for all f ∈ D(T ∗ ) (and the

domain of T ∗ consists of all f ∈ L2 (Iq32 , ) such that T ◦ (f ) ∈ L2 (Iq32 , ) ). Moreover, T ∈ End(K(Iq32 )) is adjointable if and only if T ◦ (K(Iq32 )) ⊆ K(Iq32 ) in which case T †

is the restriction of T ◦ to K(Iq32 ).

If p = (p1 , p2 , p3 ) ∈ (−q 2Z ∪ q 2Z )3 and g ∈ F(Iq32 ), we define T˜p , M˜ g ∈

End(F(Iq32 )) such that T˜p (f )(x, z, y) = f (p1 x, p2 z, p3 y) and M˜ g (f )(x, z, y) = g(x, z, y) f (x, z, y) for all f ∈ F(Iq32 ) and x, z, y ∈ Iq 2 . We denote the restrictions of M˜ g , T˜p to K(Iq32 ) by Mˆ g , Tˆp ∈ End(K(Iq32 )) respectively. Let ξ1 , ξ2 , ξ3 denote the 3 coordinate functions on Iq32 . Define also an auxiliary func-

tion ω : Iq32 → R such that for x, y, z ∈ Iq 2 , ω(x, z, y) = (1 + q −2 z)−1 if z = −q 2 and ω(x, z, y) = 0 if z = −q 2 . It is not so difficult to check that Tˆq◦2 ,1,1 = q −2 M˜ 1+q −2 ξ1 T˜q −2 ,1,1 , ◦ ˜ ˜ Tˆ1,q 2 ,1 = Mω T1,q −2 ,1 , Mˆ g◦ = M˜ g¯ if g ∈ F(I 32 ) .

◦ 2 ˜ ˜ Tˆ1,1,q −2 = q M(1+ξ3 )−1 T1,1,q 2 , ◦ ˜ ˜ Tˆ1,q −2 ,1 = M1+ξ2 T1,q 2 ,1 ,

q

Thus Tˆq◦2 ,q 2 ,1 = q −2 M˜ ω(1+q −2 ξ1 ) T˜q −2 ,q −2 ,1 ,

◦ 2 ˜ ˜ Tˆ1,q −2 ,q −2 = q M(1+ξ2 )(1+ξ3 )−1 T1,q 2 ,q 2 ,

Tˆq◦2 ,1,q −2 = M˜ (1+q −2 ξ1 )(1+ξ3 )−1 T˜q −2 ,1,q 2 . Let γˆ denote the restriction of γ˜ to K(Iq32 ). Clearly, γˆ = (1, q −2 , q −2 ),

(q 2 , 1, q −2 ),

p

Tˆp Mˆ gp , where p

(1, 1, 1) and gp belongs to F(Iq3 ). ranges over Therefore γˆ is adjointable. Moreover, γˆ ◦ = p Mˆ gp Tˆp◦ . Now an easy calculation reveals that γˆ ◦ = γ˜ † . Similarly, (γˆ † )◦ = γ˜ . (q 2 , q 2 , 1),

278

E. Koelink, J Kustermans

Lemma 5.6 implies that U 0 (γ0 )U ∗ = Mς
D 3 γˆ and U 0 (γ † )U ∗ = Mς∗
D 3

γˆ † . By Proposition 5.1 and the remarks before this lemma we get that (2)

(2)

Mς∗
D 3 γˆ † = U 0 (γ0† )U ∗ ⊆ U (( ⊗ ι)(γ ∗ ))U ∗ = Mς∗ ⊗ γl∗ . (2)

Multiplying this inclusion from the right with Mς ⊗ 1L2 (I 3

q2

1L2 (T3 ) ⊗ γl∗ .

, ) , we get that 1D 3 γˆ

†

⊆

Fix a unit vector w ∈ D 3 . Define the bounded linear operator T : L2 (T3 ) → C such that T (x) = x, w for all x ∈ L2 (T3 ). Then (T ⊗ 1L2 (I 3 , ) ) (1L2 (T3 ) ⊗ γl∗ ) ⊆ γl∗ (T ⊗ 1L2 (I 3

q2

q2

3 ∗ , ) ). So if v ∈ K(Iq 2 ), then w ⊗ v ∈ D(1L2 (T3 ) ⊗ γl ) and (1L2 (T3 ) ⊗

γl∗ )(w ⊗ v) = w ⊗ γˆ † v. Thus, v = (T ⊗ 1L2 (I 3

q2

γl∗ v = γl∗ (T ⊗ 1L2 (I 3

q2

= (T ⊗ 1L2 (I 3

q2

, ) )(w

, ) )(w

, ) )(w

⊗ v) = (T ⊗ 1L2 (I 3

q2

⊗ v) ∈ D(γl∗ ) and

, ) )(1L2 (T3 )

⊗ γl∗ )(w ⊗ v)

⊗ γˆ † v) = γˆ † v .

So we have shown that γˆ † ⊆ γl∗ . Taking the adjoint of this inclusion we see that γl ⊆ (γˆ † )∗ . By the remarks in the beginning of the proof, this implies for f ∈ D(γl ), (2) γl (f ) = (γˆ † )∗ (f ) = (γˆ † )◦ (f ) = γ˜ (f ). Starting from the inclusion 0 (γ0 ) ⊆ (ι ⊗ ∗ )(γ ), the statement concerning γr is proven in the same way.   Let us further reduce the 3-dimensional problem to a 2-dimensional one. If θ ∈ q 2Z ∪ −q −2Z , we define L2θ (Iq32 , ) to be the closed subspace of L2θ (Iq32 , ) consisting of all functions that have support in L(θ ). Then L2 (Iq32 , ) is the orthogonal k ,G ˜ km,p ∈ L2 (I 3 , ) for all sum ⊕θ∈q 2Z ∪−q −2Z L2θ (Iq32 , ). We know that F˜m,p sgn(p) q 2k q 2 p ∈ Iq 2 , m, k ∈ Z. log |θ |

2 Thus, Lemma 5.5 implies that both families ( F˜m,pq

log |θ| ˜ m,pq 2 sgn(θ ), m ∈ Z ) and ( G normal bases of L2θ (Iq32 , ). It is invariant under γl and γr .

| p ∈ Iq 2 such that sgn(p) =

| p ∈ Iq 2 such that sgn(p) = sgn(θ ), m ∈ Z ) are orthofollows from the definition of γl and γr that L2θ (Iq32 , )

Before starting the proof of the next result, let us first introduce some notation. Therefore suppose that C ⊆ (−q 2Z ∪ q 2Z ) × (−q 2Z ∪ q 2Z ) and consider a function f : C → C. Then we define new functions ∂1 f, ∂2 f : C → C such that (∂1 f )(x, y) = 2 f (q 2 x,y)−f (x,y) (x,y) and (∂2 f )(x, y) = f (x,q y)−f for all (x, y) ∈ C (recall discussion x y (2) in Notation and conventions). Proposition 5.8. We have that γl = γr . Proof. Fix θ ∈ −q 2Z ∪ q 2Z . Set k = logq 2 |θ|. On K(θ) we define the measure θ such that (−x, −y; q 2 )∞ θ ({(x, y)}) = |x| |y| (−θxy; q 2 )∞

Locally Compact Quantum Group

279

for all (x, y) ∈ K(θ ). Define a unitary transformation  : Lθ (Iq32 , ) → L2 (K(θ ), θ ) such that (f )(x, y) = f (x, θ xy, y) for all f ∈ Lθ (Iq32 , ) and (x, y) ∈ K(θ). Nok k ˜ km,p = Gkm,p for all m ∈ Z and p ∈ Iq 2 such that = Fm,p and G tice that F˜m,p θ sgn(p) = sgn(θ ). Set γr =  γr
Lθ (I 3 , ) ∗ and γlθ =  γl
Lθ (I 3 , ) ∗ . Then, q2

q2

k | p ∈ Iq 2 such that sgn(p) = sgn(θ ), m ∈ Z  is a core of γlθ , (1) the space  Fm,p (2) the space  Gkm,p | p ∈ Iq 2 such that sgn(p) = sgn(θ ), m ∈ Z  is a core of (γrθ )∗ . (5.19)

Choose f ∈ D(γlθ ) and g ∈ D((γrθ )∗ ). By Lemma 5.7, we know that for (x, y) ∈ K(θ ), 1

(γlθ f )(x, y) = q |θ |− 2

1  (1 + q −2 y) f (q 2 x, q −2 y) xy

 − (1 + q −2 y) f (x, q −2 y) − (1 + θxy) f (q 2 x, y) + f (x, y) (5.20) and 1  (1 + q −2 x) g(q −2 x, q 2 y) − (1 + θxy) g(x, q 2 y) xy  − (1 + q −2 x) g(q −2 x, y) + g(x, y) . (5.21) 1

((γrθ )∗ g)(x, y) = q |θ |− 2

Note the symmetry between the above formulas for γlθ and (γrθ )∗ with respect to interchanging x and y. Define the auxiliary function h : K(θ) → C such that h(x, y) = (−x,−y;q 2 )∞ for all (x, y) ∈ K(θ ). (−θxy;q 2 ) ∞

We want to show that γlθ f, g = f, (γrθ )∗ g. The difference of these inner products are sums over the whole area K(θ ) but we will start by approximating these sums by sums over a finite subset of K(θ ) obtained by cutting off K(θ) by 4 rectangles (see Fig. 1 and Fig. 2 later in the proof). Then we let these rectangles get bigger and bigger. In this way, these finite sums converge certainly to the sum over the whole of K(θ). In the second part of the proof, we will then prove that these finite sums converge to 0, implying that γlθ f, g must be equal to f, (γrθ )∗ g. So let us first calculate the contribution of one rectangle, lying in one of the quadrants, to the difference of the above inner product. Therefore let a, b, c, d ∈ Iq 2 such that sgn(a) = sgn(b), sgn(c) = sgn(d), |a| < |b| and |c| < |d| and set C(a, b; c, d) =





(γlθ f )(x, y) g(x, ¯ y)

(x, y) ∈ K(θ) x between a and b y between c and d

 −f (x, y) ((γrθ )∗ g)(x, y) |x| |y| h(x, y) ,

(5.22)

where the statement “x between a and b” allows for x to be equal to a or b as well (similarly for y).

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E. Koelink, J Kustermans

Using Eqs. (5.20) and (5.21) and recalling discussion (2) in Notations and conventions, we get that 1

q −1 |θ| 2 sgn(ac) C(a, b; c, d) =

b d

(1 + q −2 y) f (q 2 x, q −2 y) g(x, ¯ y) h(x, y)

x=a y=c

−

b d

(1 + q −2 x) f (x, y) g(q ¯ −2 x, q 2 y) h(x, y)

x=a y=c

−

b d

(1 + q −2 y) f (x, q −2 y) g(x, ¯ y) h(x, y)

x=a y=c

+

b d

(1 + θ xy) f (x, y)g(x, ¯ q 2 y) h(x, y)

x=a y=c

−

d b

(1 + θ xy) f (q 2 x, y) g(x, ¯ y) h(x, y)

x=a y=c

+

b d

(1 + q −2 x) f (x, y) g(q ¯ −2 x, y) h(x, y)

x=a y=c −2

=

q d b

(1 + y) f (q 2 x, y) g(x, ¯ q 2 y) h(x, q 2 y)

x=a y=q −2 c

−

−2 b q

d

(1 + x) f (q 2 x, y) g(x, ¯ q 2 y) h(q 2 x, y)

x=q −2 a y=c −2

−

q d b

(1 + y) f (x, y) g(x, ¯ q 2 y) h(x, q 2 y)

x=a y=q −2 c

+

b d

(1 + θ xy) f (x, y)g(x, ¯ q 2 y) h(x, y)

x=a y=c

−

b d

(1 + θ xy) f (q 2 x, y) g(x, ¯ y) h(x, y)

x=a y=c

+

−2 b q

d

(1 + x) f (q 2 x, y) g(x, ¯ y) h(q 2 x, y) .

(5.23)

x=q −2 a y=c

Define the set S = { (x, y) ∈ (−q 2Z ∪ q 2Z ) × (−q 2Z ∪ q 2Z ) | q 2 θxy ∈ Iq 2 }. Define a (−x,−y;q)∞ for all (x, y) ∈ S. new auxiliary function h : S → C such that h (x, y) = (−q 2 θxy;q 2 ) ∞

Locally Compact Quantum Group

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Now observe the following basic facts: (1 + y)h(x, q 2 y) = h (x, y) if(x, q 2 y) ∈ K(θ), (1 + x)h(q 2 x, y) = h (x, y) if (q 2 x, y) ∈ K(θ), (1 + θ xy)h(x, y) = h (x, y) if (x, y) ∈ K(θ) . Remembering that f and g are defined on K(θ), these facts combined with Eq. (5.23) imply that 1

q −1 |θ | 2 sgn(ac) C(a, b; c, d) −2

=

q d b

2

x=a y=q −2 c q d b



f (x, y) g(x, ¯ q y) h (x, y) + 2

x=a y=q −2 c

−

b d

d

f (q 2 x, y) g(x, ¯ q 2 y) h (x, y)

x=q −2 a y=c

−2

−

−2 b q



f (q x, y) g(x, ¯ q y) h (x, y) − 2

b d

f (x, y)g(x, ¯ q 2 y) h (x, y)

x=a y=c 

f (q x, y) g(x, ¯ y) h (x, y) + 2

−2 b q

d

f (q 2 x, y) g(x, ¯ y) h (x, y) .

x=q −2 a y=c

x=a y=c

In the above sum, most of the terms of the sums in the left column are cancelled by the terms of the sums in the right column. What remains is 1

q −1 |θ | 2 sgn(ac) C(a, b; c, d) =

b

2

f (q x, q

−2



d) g(x, ¯ d) h (x, q

−2

d) +

b

f (q 2 x, c) g(x, ¯ q 2 c) h (x, c) −

x=q −2 a

−

b

d y=c

=

b

d

f (b, y) g(q ¯ −2 b, q 2 y) h (q −2 b, y)

y=c

f (x, q −2 d) g(x, ¯ d) h (x, q −2 d) +

x=a

−

f (q 2 a, y) g(a, ¯ q 2 y) h (a, y)

y=q −2 c

x=a

−

d

b

f (x, c)g(x, ¯ q 2 c) h (x, c)

x=a

f (q 2 a, y) g(a, ¯ y) h (a, y) +

d

f (b, y) g(q ¯ −2 b, y) h (q −2 b, y)

y=c

(f (q 2 x, q −2 d) − f (x, q −2 d) ) g(x, ¯ d) h (x, q −2 d)

x=a

−

d y=c

f (b, y) (g(q ¯ −2 b, q 2 y) − g(q ¯ −2 b, y) ) h (q −2 b, y)

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+f (a, c) g(a, ¯ q 2 c)h (a, c) − ¯ c)h (a, c) + −f (q 2 a, c)g(a,

b

(f (q 2 x, c) − f (x, c) ) g(x, ¯ q 2 c) h (x, c)

x=q −2 a d

f (q 2 a, y)(g(a, ¯ q 2 y)− g(a, ¯ y))h (a, y).

y=q −2 c

(5.24) Set I = { v ∈ Iq+2 | v < q 2 , v < |θ |−1 and v < |θ |− 2 q }. If v ∈ I we define v  = 1

q 2 /v|θ |. Let us calculate the contribution of the 4 rectangles depicted in Fig. 1 and Fig. 2 to γlθ f, g − f, (γrθ )∗ g. Therefore set C(v) = C(v, v  ; v, v  ) + C(−v, −q 2 ; v, v  ) + C(−v, −q 2 ; −v, −q 2 )+C(v, v  ; −v, −q 2 ). It is clear from Eq. (5.22) and the dominated convergence theorem that C(v) converges to γlθ f, g − f, (γrθ )∗ g as v → 0. (5.25) If θ < 0, we set −q 2

C∞,1 (v) =

(∂1 f )(x, q −2 v  ) g(x, ¯ v  ) h (x, q −2 v  ) |x|

x=−v −q 2

−

f (v  , y) (∂2 g)(q ¯ −2 v  , y) h (q −2 v  , y) |y|

y=−v

and if θ > 0, we set 

C∞,1 (v) =

v

(∂1 f )(x, q −2 v  ) g(x, ¯ v  ) h (x, q −2 v  ) |x|

x=v 

−

v

f (v  , y) (∂2 g)(q ¯ −2 v  , y) h (q −2 v  , y) |y| .

y=v

Fig. 1. θ > 0

Fig. 2. θ < 0

Locally Compact Quantum Group

283

Regardless of the sign of θ, we set ¯ −2 v  , −sgn(θ ) q 2 v) h (q −2 v  , −sgn(θ ) v) C∞,0 (v) = sgn(θ ) f (v  , −sgn(θ ) v) g(q −sgn(θ ) f (−sgn(θ ) q 2 v, q −2 v  )g(−sgn(θ) ¯ v, v  ) h (−sgn(θ) v, q −2 v  ) and C0,0 (v) = f (v, v)g(v, ¯ q 2 v) h (v, v) − f (q 2 v, v) g(v, ¯ v) h (v, v) −f (−v, v) g(−v, ¯ q 2 v) h (−v, v) + f (−q 2 v, v) g(−v, ¯ v) h (−v, v) 2  2 +f (−v, −v) g(−v, ¯ −q v) h (−v, −v) − f (−q v, −v) g(−v, ¯ −v) h (−v, −v) −f (v, −v) g(v, ¯ −q 2 v) h (v, −v) + f (q 2 v, −v) g(v, ¯ −v) h (v, −v) , 

C0,1 (v) =

v

f (q 2 v, y) (∂2 g)(v, ¯ y) h (v, y) |y|

y=q −2 v 

−

v

f (−q 2 v, y) (∂2 g)(−v, ¯ y) h (−v, y) |y| ,

y=q −2 v −q 2

C0,2 (v) =

f (q 2 v, y) (∂2 g)(v, ¯ y) h (v, y) |y|

y=−q −2 v −q 2

−

f (−q 2 v, y) (∂2 g)(−v, ¯ y) h (−v, y) |y| ,

(5.26)

y=−q −2 v 

C0,3 (v) = −

v

(∂1 f )(x, v) g(x, ¯ q 2 v) h (x, v) |x|

x=q −2 v 

+

v

(∂1 f )(x, −v) g(x, ¯ −q 2 v) h (x, −v) |x| ,

x=q −2 v −q 2

C0,4 (v) = −

(∂1 f )(x, v) g(x, ¯ q 2 v) h (x, v) |x|

x=−q −2 v −q 2

+

(∂1 f )(x, −v) g(x, ¯ −q 2 v) h (x, −v) |x|.

x=−q −2 v

A bookkeeping exercise based on Eq. (5.24) reveals that 1

q −1 |θ| 2 C(v) = C∞,0 (v) + C∞,1 (v) +

4 i=0

C0,i (v) .

(5.27)

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E. Koelink, J Kustermans

In order to show all the nitty gritty work involved, let us calculate C(−v, −q 2 ; v, v  ) in the case of θ < 0 (keep Fig. 1 in mind). So in Eq. (5.24) we have to set a = −v, b = −q 2 , c = v and d = v  . / K(θ). If (1) If x ∈ Iq 2 ∩ [−q 2 , −v], then f (x, q −2 v  ) = 0 since (x, q −2 v  ) ∈ / K(θ). x ∈ Iq 2 ∩ [−q 2 , −v) then also f (q 2 x, q −2 v  ) = 0 since (q 2 x, q −2 v  ) ∈ 2 −2  But note that (−q v, q v ) ∈ K(θ ). So the first sum of expression (5.24) equals f (−q 2 v, q −2 v  ) g(−v, ¯ v  ) h (−v, q −2 v  ). (2) For all y ∈ Iq 2 ∩ [v, v  ], clearly g(−1, ¯ q 2 y) = g(−1, ¯ y) = 0. Thus the second sum in expression (5.24) equals 0. (3) For all x ∈ Iq 2 ∩[−q 2 , −q −2 v], we have that f (q 2 x, v)−f (x, v) = −|x| (∂1 f )(x, v) ¯ q 2 y)−g(−v, ¯ y) = |y| (∂2 g)(−v, ¯ y). and for all y ∈ Iq 2 ∩[v, v  ], we have that g(−v, Putting all these results into Eq. (5.24), we see that 1

−q −1 |θ| 2 C(−v, −q 2 ; v, v  ) = f (−q 2 v, q −2 v  ) g(−v, ¯ v  ) h (−v, q −2 v  ) +f (−v, v) g(−v, ¯ q 2 v) h (−v, v) − f (−q 2 v, v) g(−v, ¯ v) h (−v, v) −q 2

+

(∂1 f )(x, v) g(x, ¯ q 2 v) h (x, v) |x|

x=−q −2 v 

+

v

f (−q 2 v, y) (∂2 g)(−v, ¯ y) h (−v, y) |y| .

y=q −2 v

Similarly, one calculates C(v, v  ; v, v  ), C(−v, −q 2 ; −v, −q 2 ) and C(v, v  ; −v, −q 2 ). Adding these four results together, one finds Eq. (5.27). The case θ < 0 is treated similarly. Now we are going to make specific choices for our functions f and g. Therefore take m, n ∈ Z, p, t ∈ Iq 2 such that sgn(p) = sgn(t) = sgn(θ ) and define f and g such that for (x, y) ∈ K(θ ), 

f (x, y) = 

 

−q 2 /p ; sgn(θ ) q 2(1−m) 2(1−m) q y/|p|



−sgn(θ ) q 2(1+m) /y ; sgn(θ ) q 2(1+k) y q 2(1+m+k) x

 (5.28)

if sgn(θ ) q −2m y ∈ Iq 2 and f (x, y) = 0 if sgn(θ ) q −2m y ∈ Iq 2 , and on the other hand, g(x, y) = 

 



−q 2 /t

; q 2(1−n) x/|t|

sgn(θ ) q 2(1−n)



−sgn(θ ) q 2(1+n) /x ; sgn(θ ) q 2(1+k) x q 2(1+k+n) y

 (5.29)

if sgn(θ ) q −2n x ∈ Iq 2 and f (x, y) = 0 if sgn(θ ) q −2n x ∈ Iq 2 . By Eq. (5.16) and Result k ,Gk 6.4, Fm,p −n,t are proportional to f ,g respectively. In the next part we will show that for this choice of f and g, C(v) → 0 as v → 0.

Locally Compact Quantum Group

285

Notice that g is obtained from f by interchanging x and y and replacing the parameters m,p by n,t. There is also some symmetry with respect to interchanging x and y in the formulas defining C∞,0 (v) and C∞,1 (v). Similarly, note the symmetry with respect to this variable interchange between C0,1 (v), C0,2 (v) and C0,3 (v), C0,4 (v) respectively. This is of course in correspondence with the symmetry observed in Eqs. (5.20) and (5.21). We will make use of this symmetry to reduce our work. Let a ∈ C. Then ( (aq 2 z; q 2 )∞ − (az; q 2 )∞ )/z = a (aq 2 z; q 2 )∞ for all z ∈ C. Using this simple fact, a careful inspection reveals that for (x, y) ∈ K(θ), (∂1 f )(x, y) = q 2(1+m+k)   



−q 2 /p

; q 2(1−m) y/|p|

sgn(θ ) q 2(1−m)

−sgn(θ ) q 2(1+m) /y ; sgn(θ ) q 2(2+k) y q 2(2+m+k) x





if sgn(θ ) q −2m y ∈ Iq 2 and (∂1 f )(x, y) = 0 if sgn(θ ) q −2m y ∈ Iq 2 . On the other hand, (∂2 g)(x, y) = q 2(1+n+k)   



−q 2 /t

; q 2(1−n) x/|t|

sgn(θ ) q 2(1−n)

−sgn(θ ) q 2(1+n) /x ; sgn(θ ) q 2(2+k) x q 2(2+n+k) y



 (5.30)

if sgn(θ ) q −2n x ∈ Iq 2 and (∂2 g)(x, y) = 0 if sgn(θ ) q −2n x ∈ Iq 2 . Now we will show that the different summands of C(v) as described in (5.27) converge to 0 as v → 0. (1) First we quickly check that C0,0 (v) → 0 as v → 0. Let ( (xr , yr ) )∞ r=1 be a sequence in K(θ ) that converges to (0, 0). Then sgn(θ ) q −2m yr ∈ Iq 2 and sgn(θ ) q −2n xr ∈ Iq 2 for r big enough. Therefore Eqs. (5.28) and (5.29) imply, as in the proof of Lemma 6.2, that f (xr , yr ) → (−q 2 /p; 0; sgn(θ )q 2(1−m) ) φ(0; −q 2(2+m+k) ) g(xr , yr ) → (−q /t; 0; sgn(θ )q 2

2(1−n)

) φ(0; −q

2(2+n+k)

)

as r → ∞ , as r → ∞ ,

where we used notation (5.4). We have also that h (xr , yr ) → 1 as r → ∞. From all this, we easily conclude that C0,0 (v) → 0 as v → 0. (2) Let us now deal with C∞,1 (v). First consider the case where θ < 0. Assume for the moment that v ≤ q −2m /|θ| and v ≤ q −2n /|θ|. Then q −2m (q −2 v  ) ≥ 1 and q −2n (q −2 v  ) ≥ 1. Hence, sgn(θ ) q −2m (q −2 v  ) and sgn(θ ) q −2n (q −2 v  ) do not belong to Iq 2 . As a consequence, (∂1 f )(x, q −2 v  ) = (∂2 g)(q −2 v  , x) = 0 for all x ∈ Iq 2 ∩ [−q 2 , −v]. We conclude that C∞,1 (v) = 0 if v ≤ q −2m /|θ| and v ≤ q −2n /|θ |. Thus C∞,1 (v) → 0 as v → 0. Now we look at the more challenging case where θ > 0. Referring to Definition 3.1, Eq. (5.30) and Result 6.4, we get the existence of a constant D > 0, only depending on m, p and θ such that | (∂1 f )(x, q −2 y)|2 h (x, q −2 y) x y = D |aκ −1 (p) (κ −1 (q −2m )κ −1 (q −2 y), κ −1 (q −2 y))|2 |aκ −1 (q −2(m+1) y) (κ −1 (θy) κ −1 (x), κ −1 (x))|2

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E. Koelink, J Kustermans

for all x, y ∈ Iq+2 . Therefore Proposition 3.2 implies that

| (∂1 f )(x, q −2 y)|2 h (x, q −2 y) x y

y∈I +2 x∈I +2 q

q

=D



|aκ −1 (p) (κ −1 (q −2m )κ −1 (q −2 y), κ −1 (q −2 y))|2

y∈I +2 q



×

|aκ −1 (q −2(m+1) y) (κ −1 (θy) κ −1 (x), κ −1 (x))|2

x∈I +2 q

≤D



|aκ −1 (p) (κ −1 (q −2m )κ −1 (q −2 y), κ −1 (q −2 y))|2 ≤ D .

y∈I +2 q

Thus,



y

y∈I +2



q

implies that

x∈I +2 q

x∈I +2 q

On the other hand,

|(∂1 f )(x, q −2 y)|2 h (x, q −2 y) x < ∞, which clearly

| (∂1 f )(x, q −2 y)|2 h (x, q −2 y) x → 0 as y → ∞.

|g(x, y)|2 h (x, q −2 y) x

y ∈ I +2 x∈I +2 q q y≥1

=





|g(x, y)|2

y ∈ I +2 x∈I +2 q q y≥1

≤ (1 + q −2 )





(−x, −y; q 2 )∞ (1 + q −2 y) x (−θ xy; q 2 )∞

|g(x, y)|2

y ∈ I +2 x∈I +2 q q y≥1

(−x, −y; q 2 )∞ xy <∞, (−θxy; q 2 )∞

since g belongs to L2 (K(θ ), θ ). So we also get that



x∈I +2 q

h (x, q −2 y) x → 0 as y → ∞. By the Cauchy-Schwarz inequality, we have

|g(x, y)|2



v  2  (∂1 f )(x, q −2 v  ) g(x, ¯ v  ) h (x, q −2 v  ) x  x=v

≤



| (∂1 f )(x, q −2 v  )|2 h (x, q −2 v  ) x

x∈I +2 q



 |g(x, v  )|2 h (x, q −2 v  ) x .

x∈I +2 q

Therefore the considerations above imply that 

v x=v

(∂1 f )(x, q −2 v  ) g(x, ¯ v  ) h (x, q −2 v  ) x → 0

as v → 0 .

Locally Compact Quantum Group

287

Hence, the obvious symmetry between f and g guarantees that also 

v

f (v  , y) (∂2 g)(q ¯ −2 v  , y) h (q −2 v  , y) y → 0

as v → 0 .

y=v

Thus, we conclude that C∞,1 (v) → 0 as v → 0. (3) In the next step we prove that C0,1 (v) + C0,2 (v) → 0 as v → 0. (5.31) Let v ∈ I such that v ≤ q 2(n+1) . Then |q −2n v| ≤ q 2 , thus ±sgn(θ ) q −2n v ∈ Iq 2 . Therefore Eqs. (5.28), (5.30) and Result 6.4 imply for y ∈ Iq 2 ∩ ([−q 2 , −q −2 v] ∪ [q −2 v, v  ]), f (±q 2 v, y) (∂2 g)(±v, ¯ y) h (±v, y) |y| = q 2(1+n+k) (∓v, −y; q 2 )∞ (∓q 2 θvy; q 2 )−1 ∞ |y| (−q 2 /p; q 2(1−m) y/|p|; sgn(θ )q 2(1−m) ) (∓1/v; sgn(θ )q 2(1+k) y; ±q 2(2+m+k) v) (−q 2 /t; ±q 2(1−n) v/|t|; sgn(θ ) q 2(1−n) ) (∓sgn(θ )q 2(1+n) /v; q 2(2+n+k) y; ±sgn(θ )q 2(2+k) v)

(5.32)

if sgn(θ )q −2m y ∈ Iq 2 and f (±q 2 v, y) (∂2 g)(±v, ¯ y) h (±v, y) |y| = 0 if sgn(θ )q −2m y ∈ Iq 2 (since f (±q 2 v, y) = 0 in this case). If v ∈ I and y ∈ Iq 2 ∩ ([−q 2 , −q −2 v] ∪ [q −2 v, v  ]), then (±v, y) ∈ K(θ), thus ±q 2 θ vy ≥ −q 2 , so (∓q 2 θ vy; q 2 )∞ ≥ (q 2 ; q 2 )∞ . So we get easily the existence of a constant C > 0 such that 2 2(1−n) |q 2(1+n+k) (∓v; q 2 )∞ (∓q 2 θ vy; q 2 )−1 v/|t|; sgn(θ )q 2(1−n) )| ≤ C ∞ (−q /t; ±q

for all v ∈ I such that v ≤ q 2(n+1) and y ∈ Iq 2 ∩ ([−q 2 , −q −2 v] ∪ [q −2 v, v  ]). Moreover, Lemma 6.8 implies the existence of (a) a number D > 0 such that |(∓1/v; sgn(θ )q 2(1+k) y; ±q 2(2+m+k) v)| ≤ D for all y ∈ Iq 2 such that sgn(θ ) q −2m y ∈ Iq 2 and v ∈ I such that v ≤ q 2(n+1) , 1

1

(b) a function H1 ∈ l 2 (Iq 2 )+ such that |y| 2 |(−y; q 2 )∞ | 2 |(−q 2 /p; q 2(1−m) y/ |p|; sgn(θ )q 2(1−m) )| ≤ H1 (y) for all y ∈ Iq 2 , 1

1

(c) a function H2 ∈ l 2 (Iq 2 )+ such that |y| 2 |(−y; q 2 )∞ | 2 |(∓sgn(θ )q 2(1+n) /v; q 2(2+n+k) y; ±sgn(θ )q 2(2+k) v)| ≤ H2 (y) for all y ∈ Iq 2 and v ∈ I such that v ≤ q 2(n+1) . Now define H ∈ l 1 (Iq 2 )+ as H = C D H1 H2 . Then Eq. (5.32) implies that ¯ y) h (±v, y) |y| | ≤ H (y) | f (±q 2 v, y) (∂2 g)(±v, for v ∈ I such that v ≤ q 2(n+1) and y ∈ Iq 2 ∩ ([−q 2 , −q −2 v] ∪ [q −2 v, v  ]). Define the function G : Iq 2 → C such that for y ∈ Iq 2 , G(y) = q 2(1+n+k) (−y; q 2 )∞ |y| (−q 2 /t; 0; sgn(θ ) q 2(1−n) ) (−q 2 /p; q 2(1−m) y/|p|; sgn(θ ) q 2(1−m) ) φ(sgn(θ ) q 2(1+k) y; −q 2(2+m+k) ) φ(q 2(2+n+k) y; −q 2(3+n+k) )

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if sgn(θ ) q −2m y ∈ Iq 2 and G(y) = 0 if sgn(θ ) q −2m y ∈ Iq 2 (here we use the notation introduced in (5.4) ). If we fix y ∈ Iq 2 , we see that y ∈ [−q 2 , −q −2 v]∪[q −2 v, v  ] if v is small enough. By ¯ y) h (±v, y) |y| → G(y) Eq. (5.32), we have moreover that f (±q 2 v, y) (∂2 g)(±v, as v → 0. Therefore the dominated convergence theorem implies that G ∈ l 1 (Iq 2 ) and 

v

f (±q 2 v, y) (∂2 g)(±v, ¯ y) h (±v, y) |y|

y=q −2 v

+

−2 −q v

f (±q 2 v, y) (∂2 g)(±v, ¯ y) h (±v, y) |y| →

y=−q 2



G(y)

y∈Iq 2

as v → 0. Thus, it follows that C0,1 (v) + C0,2 (v) → 0 as v → 0. The aforementioned symmetry between f and g then guarantees that also C0,3 (v) + C0,4 (v) → 0 as v → 0. (4) In the last step we look at C∞,0 (v). First assume that θ < 0. If v ∈ I and v ≤ q −2m /|θ | then sgn(θ ) q −2m (q −2 v  ) = −q −2m /v|θ | ≤ −1. Thus Eq. (5.28) implies that f (−sgn(θ ) q 2 v, q −2 v  ) g(−sgn(θ) ¯ v, v  ) h (−sgn(θ) v, q −2 v  ) = 0. So 2 −2  in this case, f (−sgn(θ ) q v, q v ) g(−sgn(θ ¯ ) v, v  )  −2  h (−sgn(θ) v, q v ) → 0 as v → 0. Now we look at the more challenging case where θ > 0. If v ∈ I and v ≤ q 2n+2 , then sgn(θ )q −2m (q −2 v  ) and −q −2n v clearly belong to Iq 2 ; thus Eqs. (5.28) and (5.29) and Result 6.4 imply that |f (−sgn(θ ) q 2 v, q −2 v  ) g(−sgn(θ) ¯ v, v  ) h (−sgn(θ ) v, q −2 v  )| 2 2(1−m) = (−v, −1/vθ ; q 2 )∞ (q 2 ; q 2 )−1 /vθp; q 2(1−m) ) | ∞ | (−q /p; q

| (1/v; q 2(1+k) /vθ ; −q 2(2+m+k) v) || (−q 2 /t; −q 2(1−n) v/t; q 2(1−n) ) | | (q 2(1+n) /v; q 2(2+n+k) /vθ ; −q 2(1+k) v) | . By Lemma 6.7, there exists a number C > 0 such that | (−q 2 /p; q 2(1−m) /vθp; q 2(1−m) ) | ≤ C for all v ∈ I . Therefore the above equality and Lemma 6.9 allow us to conclude that ¯ v, v  ) h (−sgn(θ) v, q −2 v  ) → 0 f (−sgn(θ ) q 2 v, q −2 v  ) g(−sgn(θ)

as v → 0

also in this case. The symmetry between f and g thus guarantees that also the first term of C∞,0 (v) → 0 as v → 0. Therefore C∞,0 (v) → 0 as v → 0. Together with Eq. (5.27) the convergence results proven in parts (1),(2),(3) and (4) imply that C(v) → 0 as v → 0. By (5.25), this implies that γlθ f, g − f, (γrθ )∗ g = 0. k , Gk k θ ∗ k So, by our choice of f and g, we find that γlθ Fm,p −n,t  = Fm,p , (γr ) G−n,t . θ    θ ∗ Therefore the conditions stated in (5.19) imply that γl x , y  = x , (γr ) y   for all x  ∈ D(γlθ ) and y  ∈ D((γlθ )∗ ). Consequently, (γrθ )∗ ⊆ (γlθ )∗ (*). Since γlθ and γrθ are normal, this implies that D(γrθ ) = D((γrθ )∗ ) ⊆ D((γlθ )∗ ) = D(γlθ ). But taking the  adjoint of the inclusion (*), we arrive at the inclusion γlθ ⊆ γrθ . Hence, γlθ = γrθ . 

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Since U (( ⊗ ι)(γ )) U ∗ = Mζ ⊗ γl and U ((ι ⊗ )(γ )) U ∗ = Mζ ⊗ γr (see the remarks before Lemma 5.7) the previous proposition entails that ( ⊗ ι)(γ ) = (ι ⊗ )(γ ) and we have proven Proposition 5.2. We can finally explain why the comultiplication  is defined in such a way that sgn(θ) ∗ Uθ (γ ∗ γ )
θ Uθ = 1 ⊗ Lθ (see the discussions surrounding Eq. (3.2) and after Proposition 3.9). Recall that this choice is reflected in the presence of the factor s(x, y) in Definition 3.1. In the proof above we showed that C0,1 (v) → 0 as v → 0, cf. (5.31). Looking at the defining formula (5.26) for C0,1 (v), it is not too hard to imagine that for this to be true, we need at least that the functions f and g defined in Eqs. (5.28) and (5.29) have the same limit behavior when crossing the Y -axis, going from the first to the the second quadrant (for these particular functions, these are just continuous transitions). Note that the relevance of f and g to the coassociativity of  is obvious from Lemma 5.5 (and that all potential discontinuous transitions are filtered out by U ). If one changes the comultiplication by leaving out the factor s(x, y) in Definition 3.1, and thus opting for the equality Uθ∗ (γ ∗ γ )
θ Uθ = 1 ⊗ L1θ , it turns out that f and g have different limit transitions implying that C0,1 (v) does not converge to 0 as v → 0. From this it would follow that ( ⊗ ι)(γ ) and (ι ⊗ )(γ ) have different domains in this case. From Lemma 5.5 we have two orthonormal bases for the same space L2 (Iq32 , );  k   k  ˜ m,p and G . Hence, there exists a unitary operator R on F˜m,p p∈Iq 2 ,m,k∈Z

p∈Iq 2 ,m,k∈Z

L2 (Iq32 , ) mapping the first basis on the second. Its matrix elements are given by ˜ km2 ,p  2 3 R(m1 , k1 , p1 ; m2 , k2 , p2 ) = F˜mk11 ,p1 , G 2 2 L (I

q2

, )

and these matrix elements of R can be thought of as Racah coefficients. It would be desirable to find an explicit expression for the Racah coefficients and to show that R(m1 , k1 , p1 ; m2 , k2 , p2 ) = 0 for p1 = p2 , since this would immediately imply γl = γr and hence yield an alternative proof for Proposition 5.8 and hence of Proposition 5.2. However, we have not been able to carry out this programme. 6. Appendix In this appendix we collect the basic properties of the -functions defined in Eq. (1) of the Introduction. Most of them stem from special function theory, so the proofs in this section are mainly intended for people that are not too familiar with this theory. As a general reference for q-hypergeometric functions we use [6]. Let us fix a number 0 < q < 1. 2 Note first of all that the presence of the factor q k in the series (1) implies that if we keep two of the parameters a,b,z fixed, the series converges uniformly on compact subsets in the remaining parameter. As a consequence, (a; b; q, z) is analytic in the remaining parameter. Also, the function C3 → C : (a, b, z) → (a; b; q, z) is analytic. The -functions are closely related to the 1 ϕ1 -functions (see [6, Def. (1.2.22)]) in the sense that (a; b; q, z) = (b; q)∞ 1 ϕ1 (a; b; q, z) if b ∈ q −N0 (if b ∈ q −N0 , 1 ϕ1 (a; b; q, z) is not defined in general). As a consequence, a lot of the identities that are known for 1 ϕ1 - functions extend by analytic continuation in the parameters to similar identities involving -functions. A very basic but useful formula is known as the

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Result 6.1 (θ -product identity). Consider a ∈ C \ {0}, k ∈ Z. Then 1

(aq k ; q)∞ (q 1−k /a; q)∞ = (−a)−k q − 2 k(k−1) (a; q)∞ (q/a; q)∞ .

(6.1)

Proof. If a ∈ q Z , both sides of the above equation are easily seen to be equal to 0. Now we look at the case where a ∈ q Z . Suppose that k ≥ 0. Then we have for n ∈ N, k−1 −i (aq k ; q)n ( q 1−k /a; q)n+k i=0 (1 − q /a) =  k−1 i (a; q)n+k ( q/a; q)n i=0 (1 − aq )   k−1 i (−a)−k i=0 q −i k−1 i=0 (1 − aq ) = k−1 i i=0 (1 − aq ) 1

= (−a)−k q − 2 k(k−1) so if we let n tend to infinity, Eq. (6.1) follows. If k ≤ 0, we apply Eq. (6.1), where we replace k by −k and a by q/a.   The most important low order q-hypergeometric functions are the 2 ϕ1 functions. If a, b, c, z ∈ C satisfy c ∈ q −N0 and |z| < 1, these are defined as (see [6, Def. (1.2.22)] and the discussion thereafter)  2 ϕ1 (a, b; c; q, z) = 2 ϕ1

 ∞ (a; q)n (b; q)n n a b ; q, z = z . c (c; q)n (q; q)n n=0

If we keep a,b and c fixed, the function z, |z| < 1 → 2 ϕ1 (a, b; c; q, z) has an analytic extension to the set C \ [1, ∞) (see the beginning of [6, Sect. 4.3]). Of course, the value of this extension at z ∈ C \ [1, ∞) is also denoted by 2 ϕ1 (a, b; c; q, z). We will not use this fact directly in this paper but point to this extension because it is used in [3]. A connection between 2 ϕ1 -functions and 1 ϕ1 -functions is obtained by the following basic limit transition. Lemma 6.2. Consider a, b, c, z ∈ C such that c ∈ q −N0 and (xi )∞ i=1  a sequence in C \ {0} that converges to 0 and satisfies |xi z| < 1 for all i ∈ N. Then 2 ϕ1 (a, b/xi ; ∞ c; q, xi z) i=1 converges to 1 ϕ1 (a; c; q, bz). ∞ (a;q)n n Proof. By definition, we have that 2 ϕ1 (a, b/xi ; c; q, xi z) = n=0 (c;q)n (q;q)n z  k th (b/xi ; q)n xin and (b/xi ; q)n xin = n−1 k=0 (xi − bq ). It follows that the n term of 1

(a;q)n this series converges to the number (c;q) (−1)n q 2 n(n−1) (bz)n as i → ∞. Take n (q;q)n ε > 0 such that |εz| < 1. So there exists i0 ∈ N such that |xi | < ε/2 for all i ∈ Z≥i0 . It is not so difficult to see that there exists M > 0 such that |(b/xi ; q)n xin ε −n | ≤ M for  (a;q)n n all n ∈ N0 and i ∈ Z≥i0 . Thus, since ∞ n=0  (c;q)n (q;q)n (εz)  < ∞, the dominated ∞  convergence theorem implies that the sequence 2 ϕ1 (a, b/xi ; c; q, xi z) i=1 converges (a;q)n n 21 n(n−1) (bz)n = ϕ (a; c; q, bz). to ∞   1 1 n=0 (c;q)n (q;q)n (−1) q

This limit transition is used to get a more direct relationship between 2 ϕ1 -functions and -functions.

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Lemma 6.3. Let a, b, z ∈ C such that |z| < 1. Then (z; q)∞ 2 ϕ1 (a, b; 0; q, z) = (a; az; q, bz). Proof. First suppose that a = 0 and az ∈ q −N0 . Take a sequence (xi )∞ i=1 in C \ ({0} ∪ q −N0 ) such that (xi )∞ → 0 and |x /a| < 1 for all i ∈ N. Then Heine’s transi i=1 formation formula [6, Eq. (1.4.5)] implies that 2 ϕ1 (a, b; xi ; q, z) = (xi /a, az; q)∞ (xi , z; q)−1 ∞ 2 ϕ1 (abz/xi , a; az; q, xi /a). If we let i tend to ∞ in this equality, we obtain by the previous lemma the equality 2 ϕ1 (a, b; 0; q, z) = (az; q)∞ (z; q)−1 ∞ 1 ϕ1 (a; az; q, bz) = (z; q)−1 ∞ (a; az; q, bz). The general result follows by analytic continuation.   The following formula will be used throughout the paper. Result 6.4. Consider a, b, z ∈ C such that b = 0. Then (a; b; q, z) = (az/b; z; q, b). Proof. First assume that b and z do not belong to q −N0 . Take a sequence (xi )∞ i=1 in C \ ({0} ∪ q −N0 ) such that (xi )∞ → 0, |zx | < 1 and |bx | < 1 for all i ∈ N. Let i i i=1 i ∈ N and apply Heine’s transformation formula [6, Eq. (1.4.5)] for a  a, b  1/xi , c  b and z  xi z. Thus, 2 ϕ1 (a, 1/xi ; b; q, xi z) = (bxi , z; q)∞ (b, xi z; q)−1 ∞ 2 ϕ1 (az/b, 1/xi ; z; q, bxi ). If we let i tend to ∞ in this equality, Lemma 6.2 implies that 1 ϕ1 (a; b; q, z) = (z; q)∞ (b; q)∞−1 ϕ1 (az/b; z; q, b). Thus, (a; b; q, z) = 1

(az/b; z; q, b), still under the assumption that b, z ∈ q −N0 . The general formula now follows from analytic continuation.  

Roughly speaking, q-contiguous relations are relations between expressions of the form (ra; sb; q, tz), where r, s, t ∈ {1, q, q −1 }. We need the following two. Lemma 6.5. Consider a, b, z ∈ C, then (a; b; q, z) = (1 − a) (qa; b; q, z) + a (a; b; q, qz) , (a; b; q, z) = (a − b) (a; qb; q, qz) + (1 − a) (qa; qb; q, z) .

(6.2) (6.3)

Proof. By definition, we have (1 − a) (qa; b; q, z) =

∞

(1 − a)

n=0

=

∞ (a; q)n (1 − aq n ) (q n b; q)∞

(q; q)n

n=0

=

1 (qa; q)n (q n b; q)∞ (−1)n q 2 n(n−1) zn (q; q)n

∞ (a; q)n (q n b; q)∞

(q; q)n

n=0

−a

(q; q)n

1

(−1)n q 2 n(n−1) zn

∞ (a; q)n (q n b; q)∞ n=0

1

(−1)n q 2 n(n−1) zn

1

(−1)n q 2 n(n−1) (qz)n

= (a; b; q, z) − a (a; b; q, qz) .

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On the other hand, (a − b) (a; qb; q, qz) + (1 − a) (qa; qb; q, z) ∞ 1 (a; q)n (q n+1 b; q)∞ = (a − b) (−1)n q 2 n(n−1) (qz)n (q; q)n n=0 ∞

+ =

(1 − a)

n=0 ∞

(a − b)q n

n=0 ∞

+ = =

n=0 ∞ n=0

1 (a; q)n (q n+1 b; q)∞ (−1)n q 2 n(n−1) zn (q; q)n

(1 − q n a)

n=0 ∞

1 (qa; q)n (q n+1 b; q)∞ (−1)n q 2 n(n−1) zn (q; q)n

(1 − q n b)

1 (a; q)n (q n+1 b; q)∞ (−1)n q 2 n(n−1) zn (q; q)n

1 (a; q)n (q n+1 b; q)∞ (−1)n q 2 n(n−1) zn (q; q)n

1 (a; q)n (q n b; q)∞ (−1)n q 2 n(n−1) zn = (a; b; q, z) . (q; q)n

 

We need the following transformation formulas. Proposition 6.6. Consider a, b, z ∈ C and k ∈ Z. Then (1) (q k+1 /a; q)∞ (aq −k ; q −k+1 ; q, z) = (q/a; q)∞ (az/q)k (a; q k+1 ; q, zq k ) if a, z = 0, (2) (q k+1 a/z; q)∞ (q −k ; a; q, z) = (z/q)k (q k a; q)∞ (q −k ; qa/z; q, q 2 /z) if z = 0 and k ≥ 0. Proof. (1) Suppose first that k ≥ 0. Note that (q n+1−k ; q)∞ = 0 for all n ∈ Z≤k−1 . Hence, (q k+1 /a; q)∞ (aq −k ; q −k+1 ; q, z) ∞ 1 (aq −k ; q)n (q n+1−k ; q)∞ = (q 1+k /a; q)∞ (−1)n q 2 n(n−1) zn (q; q)n n=k

= (q k+1 /a; q)∞

∞ (aq −k ; q)n+k (q n+1 ; q)∞ n=0

(q; q)n+k

1

(−1)n+k q 2 (n+k)(n+k−1) zn+k

1 2 k(k−1)

= (−1)k q zk (q k+1 /a; q)∞ (a/q; q −1 )k ∞ 1 (a; q)n (q n+k+1 ; q)∞ (−1)n q nk q 2 n(n−1) zn (q; q)n n=0

1

1

= (−1)k q 2 k(k−1) zk (q k+1 /a; q)∞ (q/a; q)k (−a)k q −k q − 2 k(k−1) (a; q k+1 ; q, zq k ) = (q/a; q)∞ (az/q)k (a; q k+1 ; q, zq k ) . If k < 0, we apply (1) where we replace k by −k, a by aq −k and z by zq k to obtain (1) in this case.

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293

(2) First we assume that a = 0 and |q k+1 a/z| < 1. Since (q −k ; q)n = 0 if n ≥ k + 1, the series terminates and (q k+1 a/z; q)∞ (q −k ; a; q, z) = (q k+1 a/z; q)∞

k 1 (q −k ; q)n (q n a; q)∞ (−1)n q 2 n(n−1) zn (q; q)n n=0

= (q k+1 a/z; q)∞

k 1 (q −k ; q)k−n (q k−n a; q)∞ (−1)k−n q 2 (k−n)(k−n−1) zk−n (q; q)k−n n=0

= (q

k+1

1

a/z; q)∞ (−1)k q 2 k(k−1) zk

k 1 (q −k ; q)k−n (q k−n a; q)∞ (−1)n q −kn q 2 n(n+1) z−n . (q; q)k−n

(6.4)

n=0

Now, (q k−n a; q)∞ = (q k a; q)∞ (q k−1 a; q −1 )n 1

(q −k ; q)k−n (q; q)k−n

= (q k a; q)∞ q nk q − 2 n(n+1) (−a)n (q 1−k /a; q)n −1 i k i i=−k (1 − q ) i=k−n+1 (1 − q ) = −1  k i i i=1 (1 − q ) i=−n (1 − q ) 1

= (−1) q k

− 21 k(k+1)

(−1)n q kn q − 2 n(n−1) (q −k ; q)n 1

(−1)n q − 2 n(n+1) (q; q)n 1 (q −k ; q)n = (−1)k q − 2 k(k+1) q (k+1)n , (q; q)n which upon substitution in Eq. (6.4) implies that (q k+1 a/z; q)∞ (q −k ; a; q, z) = (q k+1 a/z; q)∞ (q k a; q)∞ (z/q)k ×

k (q −k ; q)n (q 1−k /a; q)n k+1 (q a/z)n (q; q)n n=0

= (q k+1 a/z; q)∞ (q k a; q)∞ (z/q)k 2 ϕ1 (q −k , q 1−k /a; 0; q, q k+1 a/z) = (q k a; q)∞ (z/q)k (q −k ; qa/z; q, q 2 /z) , where we used Lemma 6.3 in the last step. The general result now follows by analytic continuation.   In most cases it is not necessary to know the precise expression for the -functions, sometimes only the asymptotic behavior matters. We need the following estimate (see [16, Lem. 2.8]). Lemma 6.7. Consider a, z ∈ C, k ∈ Z≥0 . Then |(a; q 1−k ; q, z)| ≤ (−|a|; q)∞ 1 (−|z|; q)∞ |z|k q 2 k(k−1) .

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Proof. For n ∈ N0 , we have |(a; q)n | ≤ (−|a|; q)n ≤ (−|a|; q)∞ . Also notice that (q n−k+1 ; q)∞ = 0 for all n ∈ N0 satisfying n < k. Hence, by the definition of , we see that ∞   (q n−k+1 ; q)∞ 1 n(n−1) n   q2 |z|  (a; q 1−k ; q, z)  ≤ (−|a|; q)∞ (q; q)n n=0

= (−|a|; q)∞

∞ (q n−k+1 ; q)∞

(q; q)n

n=k ∞ (q n+1 ; q)∞ n=0

(q; q)n+k

= (−|a|; q)∞ |z|k

1

q 2 (n+k)(n+k−1) |z|n+k ∞ (q n+k+1 ; q)∞

(q; q)n

n=0

≤ (−|a|; q)∞ |z| q k

1

q 2 n(n−1) |z|n = (−|a|; q)∞

1 2 k(k−1)

∞ n=0

1

1

q 2 n(n−1) q nk q 2 k(k−1) |z|n

1 1 q 2 n(n−1) (q k |z|)n (q; q)n 1

= (−|a|; q)∞ (−|z|q ; q)∞ |z|k q 2 k(k−1) , k

where we used [6, Eq. 1.3.16] in the last equality. Now the lemma follows.

 

The last two estimates play a vital role in the proof of the coassociativity of the comultiplication. Lemma 6.8. Consider a, b, c ∈ C, p ∈ −q Z ∪ q Z and β > 0. Let m ∈ [1, ∞], M > 0, ε > 0 and define the q-interval J as J = { x ∈ −q Z ∪ q Z | |x| ≤ M or sgn(x) = sgn(p) }. Then there exist N > 0 and f ∈ l m (J )+ such that |(a/λ; px; q, bλ)| ≤ N

and

1

|x|β |(cx; q)∞ | 2 |(a/λ; px; q, bλ)| ≤ f (x)

for all x ∈ J and λ ∈ C \ {0} such that |λ| ≤ ε. Proof. Choose k ∈ Z such that |p| = q k . Take r, s > 0 such that r > |b|(ε + |a|) and s > |c|. Let λ ∈ C \ {0} such that |λ| ≤ ε and x ∈ J . Since (pxq r ; q)∞ ≥ 0 for all r ∈ Z, we get that | (a/λ; px; q, bλ) | ≤

∞ |(a/λ; q)n | (pxq n ; q)∞ n=0

(q; q)n

1

q 2 n(n−1) |bλ|n

∞ n−1 1 (pxq n ; q)∞  = ( |λ − q i a| )q 2 n(n−1) |b|n (q; q)n n=0

≤

i=0

∞ (pxq n ; q)∞ n=0

(q; q)n

1

q 2 n(n−1) (|b| (|λ| + |a|))n

≤ (0; px; q, −r) .

(6.5) 1

This implies the existence of C > 0 such that | (a/λ; px; q, bλ) | ≤ C and |(cx; q)∞ | 2 ≤ C for all x ∈ J such that |x| ≤ max{M, q 2 /|p|} and λ ∈ C \ {0} such that |λ| ≤ ε. (6.6)

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Define the function f : J → R+ such that for x ∈ J , we have that f (x) = C 2 |x|β if |x| ≤ max{M, q 2 /|p|} and 1

1

1

1

2 2 (−q/s; q)∞ q 2 k(k−1) q (k+β)t r 1−k−t s − 2 q 4 t (t−1) f (x) = (−r; q)∞ (−s; q)∞ t

if |x| > max{M, q 2 /|p|} and t ∈ Z is such that |x| = q t . Then it is clear that f ∈ l m (J )+ . Take x ∈ J such that |x| > max{M, q 2 /|p|}. Choose t ∈ Z such that |x| = q t . Since x ∈ J and |x| > M, we have that sgn(x) = sgn(p), hence px = q k+t . It is also clear that t ≤ 1 − k, thus 1 − k − t ≥ 0. By estimate (6.5) and Lemma 6.7, we get for all λ ∈ C \ {0} such that |λ| ≤ ε, | (a/λ; px; q, bλ) | ≤ (0; px; q, −r) = (0; q 1−(1−k−t) ; −r) 1

≤ (−r; q)∞ r 1−k−t q 2 (k+t−1)(k+t) 1

1

= (−r; q)∞ r 1−k−t q 2 k(k−1) q kt q 2 t (t−1) .

(6.7)

Also, Lemma 6.1 guarantees that 1 1 2

1 2

1 2

| (cx; q)∞ | ≤ (−|cx|; q) ≤ (−sq ; q)∞ = 1 2

t

1 2

≤ (−s; q)∞ (−q/s; q)∞ s

− 2t

q

1

2 2 (−q/s; q)∞ (−s; q)∞ 1 2

1

s − 2 q − 4 t (t−1) t

(−q 1−t /s; q)∞

− 41 t (t−1)

. 1

This estimate, together with the estimates in (6.6) and (6.7) imply that |x|β |(cx; q)∞ | 2 |(a/λ; px; q, bλ)| ≤ f (x) for all x ∈ J and λ ∈ C \ {0} such that |λ| ≤ ε. Estimate (6.7) also implies the existence of D > 0 such that |(a/λ; px; q, bλ)| ≤ D for all x ∈ J such that |x| > max{M, q 2 /|p|} and λ ∈ C \ {0} such that |λ| ≤ ε. So if we set N = max{C, D}, we find by the estimate in (6.6) that |(a/λ; px; q, bλ)| ≤ N for all x ∈ J and λ ∈ C \ {0} such that |λ| ≤ ε.   The next result is an easy consequence of the previous lemma. Lemma 6.9. Consider a, b, c ∈ C, p ∈ −q Z ∪ q Z . Let (xi )i∈I be a net in −q Z ∪ q Z  1 for all i ∈ I and (x ) → ∞. Then the net |(cxi ; q)∞ | 2 such that sgn(xi ) = sgn(p) i i∈I  |(axi ; pxi ; q, b/xi )| i∈I converges to 0. Acknowledgement. We would like to thank S. L. Woronowicz for insightful discussions on quantum
(1, 1), providing the preliminary version of [32] and giving a series of lectures on [32] in Trondheim SU (April 2000), which the second author attended.

References 1. Abe, E.: Hopf algebras. Cambridge Tracts in Mathematics, vol 74, Cambridge: Cambridge University Press, 1980 2. Al-Salam, W.A., Carlitz, L.: Some orthogonal q-polynomials. Math. Nach. 30, 47–61 (1965) 3. Ciccoli, N., Koelink, E., Koornwinder, T.: q-Laguerre polynomials and big q-Bessel functions and their orthogonality relations. Meth. Appl. Anal. 6(1), 109–127 (1999) 4. Dixmier, J.: Les alg`ebres d’op´erateurs dans l’espace hilbertien. Deuxi`eme e´ dition. Paris: Gauthier-Villars, 1969 5. Enock, M., Schwartz, J.-M.: Kac Algebras and Duality of Locally Compact Groups. Berlin: Springer-Verlag, 1992

296

E. Koelink, J Kustermans

6. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge: Cambridge University Press, 1990 7. Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras I. Graduate Studies in Mathematics, 15. Am. Math. Soc. (1997) 8. Kakehi, T., Masuda, T., Ueno, K.: Spectral analysis of a q-difference operator which arises from the quantum SU (1, 1) group. J. Operator Theory 33, 159–196 (1995) 9. Korogodsky, L.I.: Quantum Group SU (1, 1)
Z2 and super tensor products. Commun. Math. Phys. 163, 433–460 (1994) 10. Korogodsky, L.I., Vaksman, L.L.: Spherical functions on the quantum group SU (1, 1) and the q-analogue of the Mehler-Fock formula. (Russian) Funkt. Anal. i Prilozhen. 25(1), 60–62 (1991). Translation in Funct. Anal. Appl. 25(1), 48–49 (1991) 11. Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98-17, Delft Univ. Technology (1998). http://aw.twi.tudelft.nl/ koekoek/ askey.html 12. Koelink, E.: Spectral theory and special functions. Lecture notes for the 2000 Laredo Summer school of the SIAM Activity Group (2000). math.CA/0107036
(1, 1). In preparation (2002) 13. Koelink, E., Kustermans, J.: The Pontryagin dual of quantum SU
(1, 1) and its Pontryagin dual. To appear in the Proceed14. Koelink, E., Kustermans, J.: Quantum SU ings of ‘La 69`eme rencontre entre physiciens et th´eoriciens et math´ematiciens. IRMA, Strasbourg’ (2002) 15. Koelink, E., Stokman, J.V.: The big q-Jacobi function transform. To appear in Constructive Approximation (2002). #math.CA/9904111 16. Koelink, E., Stokman, J.V.: With an appendix by M. Rahman,: Fourier transforms on the quantum SU (1, 1) group. Publ. RIMS Kyoto Univ. 37(4), 621–715 (2001) #math.QA/9911163 17. Koornwinder, T.H.: The addition formula for little q-Legendre polynomials and the SU (2) quantum group. SIAM J. Math. Anal. 22, 295–301 (1991) 18. Kustermans, J.: KMS-weights on C ∗ -algebras. Preprint Odense Universitet (1997) #functan/9704008 19. Kustermans, J.: One-parameter representations on C*-algebras. Preprint Odense Universitet (1997) #funct-an/9707009 ´ Norm. Sup. 4`e s´erie, 20. Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. scient. Ec. t.33, 837–934 (2000) 21. Kustermans, J., Vaes, S.: Locally compact quantum groups in the von Neumann algebraic setting. To appear in Math. Scand. (2000) #math.QA/0005219 22. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Saburi, Y., Ueno, K.: Unitary representations of the quantum group SUq (1, 1): Structure of the dual space of Uq (sl, (2)). Lett. in Math. Phys. 19, 187–194 (1990) 23. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Saburi, Y., Ueno, K.: Unitary representations of the quantum group SUq (1, 1): II-Matrix elements of unitary representations and the basic hypergeometric functions. Lett. in Math. Phys. 19, 195–204 (1990) 24. Pedersen, G.K., Takesaki, M.: The Radon-Nikodym theorem for von Neumann algebras. Acta Math. 130, 53–87 (1973) 25. Schm¨udgen, K.: Unbounded operator algebras and representation theory. In: Operator Theory 37, Basel-Boston: Birkh¨auser, 1990 26. Shklyarov, D., Sinel’shchikov, S., Vaksman, L.L.: On function theory in quantum discs: Invariant kernels. Preprint Institute for Low Temperature Physics & Engineering (1998). #math.QA/9808047 27. Stratila, S.: Modular Theory in Operator Algebras. Turnbridge Wells, England: Abacus Press, 1981 28. Vaes, S.: A Radon-Nikodym theorem for von Neumann algebras. J. Operator Theory 46, 477–489 (2001) #math.OA/9811122 29. Woronowicz, S.L.: Unbounded elements affiliated with C ∗ -algebras and non-compact quantum groups. Commun. Math. Phys. 136, 399–432 (1991) 30. Woronowicz, S.L.: C ∗ -algebras generated by unbounded elements. Rev. Math. Phys. 7(3), 481–521 (1995) 31. Woronowicz, S.L.: From multiplicative unitaries to quantum groups. Int. J. Math. 7(1), 127–149 (1996) 32. Woronowicz, S.L.: Extended SU (1, 1) quantum group. Hilbert space level. Preprint KMMF. In preparation (2000) 33. Woronowicz, S.L., Zakrzewski, S.: Quantum ax + b group. Preprint KMMF (1999)

Communicated by L. Takhtajan

Commun. Math. Phys. 233, 297–311 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0750-z

Communications in

Mathematical Physics

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations Dongho Chae, Jihoon Lee School of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea. E-mail: [email protected]; [email protected] Received: 24 April 2002 / Accepted: 29 July 2002 Published online: 10 December 2002 – © Springer-Verlag 2002

Abstract: We consider the quasi-geostrophic equation with the dissipation term, κ(−)α θ , 0 ≤ α ≤ 21 . In the case α = 21 , Constantin-Cordoba-Wu [6] proved the global existence of strong solution in H 1 and H 2 under the assumption of small L∞ -norm of initial data. In this paper, we prove the global existence in the scale invariant Besov space, 2−2α 2−2α B˙ 2,1 , 0 ≤ α ≤ 21 for initial data small in the B˙ 2,1 norm. We also prove a global 1 . stability result in B˙ 2,1 1. Introduction We are concerned with the two dimensional dissipative quasi-geostrophic equation:   ∂t θ + (u · ∇)θ + κ(−)α θ = 0, R2 × R+ , α ≥ 0, 1 (DQG)α u = ∇ ⊥ (−)− 2 θ,  θ (0, x) = θ0 , where the scalar θ represents the potential temperature, and u is the fluid velocity. It is well-known that (DQG)α is very similar to the three dimensional Navier-Stokes equations (see e.g. [7]). Besides its similarity to the three dimensional fluid equations, (DQG)α represents the potential temperature dynamics of atmospheric and ocean flow [14]. In the case κ = 0, Resnick [15] obtained global existence of weak solutions of (DQG)α with L2 initial data in the periodic domain or whole R2 . The case α > 21 is called sub-critical, and the case α = 21 is critical, while the case 0 ≤ α < 21 is super-critical, respectively. In the sub-critical cases, a unique existence of a global in time regular solution was proved. This result and many other studies for sub-critical case can be found in [8, 15, 17–19]. In the super-critical or critical case, it is not known whether regular local solutions develop finite time singularities or not. For the critical case(α = 21 ), Constantin, Cordoba, and Wu [6] proved the existence and uniqueness

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of global classical solutions of the (DQG) 1 on the spatial periodic domain when the 2 initial data has small L∞ norm. Recently, there are many studies on the small data global existence and the large data local existence problems using the scaling invariant Besov and the Triebel-Lizorkin spaces (see e.g. [2–5, 9, 10, 20] and references therein). Considering scaling analysis, we have if θ (x, t) and u(x, t) are solutions of (DQG)α , then θλ (x, t) := λ2α−1 θ (λx, λ2α t) and uλ (x, t) = λ2α−1 u(λx, λ2α t) are also solutions 2−2α of (DQG)α . Thus B˙ 2,1 is the scaling invariant function space. Our first main result of this paper is the global existence and uniqueness result for the initial-value prob2−2α lem (DQG)α with the initial data small in B˙ 2,1 norm. The precise statement is as follows. Theorem 1 [Global existence]. There exists a constant  > 0 such that for any θ0 ∈ 2−2α B2,1 with θ0 B˙ 2−2α < , the IVP (DQG)α with 0 ≤ α ≤ 21 has a unique global 2,1     β  2−2α  2 ∩ L1 0, ∞; B˙ 2,1 ∩ C [0, ∞); B2,1 , solution θ, which belongs to L∞ 0, ∞; B2,1 where β = max{2 − 2α − δ1, 1} for any δ1 > 0. Moreover, for any σ > 0, θ  γ  2+2α  2 ∞ 1 ˙ also belongs to L σ, ∞; B2,1 ∩ L σ, ∞; B2,1 ∩ C (0, ∞); B2,1 , where γ =  2 − δ2 , if 0 ≤ α < 21 , for any δ2 > 0. Furthermore the solution θ satisfies the 2, if α = 21 , following estimates:  ∞ θ(t)B˙ 2 dt sup θ (t)B 2−2α + Cκ 2,1

0≤t<∞

  ≤ θ0 B 2−2α exp C 2,1

and

0 ∞

0

2,1



θ(t)B˙ 2 dt , 2,1

 ∞ θ(t)B˙ 2+2α dt sup θ (t)B 2 + Cκ 2,1 2,1 σ ≤t<∞ σ   ∞ ≤ θ (σ )B 2 exp C θ(t)B˙ 2 dt . 2,1

σ

2,1

Remark 1. We note that the smallness assumption is made only on the homogeneous norm of the initial data. Remark 2. If we try to follow the Fujita–Kato method ([12] and [13]) to prove the above 2

+1−2α

p type of theorem for θ0 ∈ Bp,q , we need to prove the continuity of the bilinear operator  t

 1 α B(θ, θ) = e−κ(t−s)(−) div θ(s)∇ ⊥ (−)− 2 θ(s) ds

0 2

+1−2α

p in C([0, ∞); Bp,q ), and we found there are technical difficulties for such a continuity estimate for all 1 ≤ p, q ≤ ∞ when 0 ≤ α ≤ 21 . Thus we could not adapt Fujita-Kato’s argument to prove the above theorem.

Our second main theorem is concerned with the global stability result for the critical case, which could be viewed as a generalization of Theorem 1 in this case.

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations

299

Theorem 2 [Global stability]. Assume that θ 1 is a solution of the IVP (DQG) 1 2 1 ) ∩ L1 (0, ∞; B ˙ 2 ). Then there exists a positive consatisfying θ 1 ∈ C([0, ∞); B2,1 2,1 stant 0 = 0 (κ, θ01 B˙ 1 , θ 1 L1 (0,∞;B˙ 2 ) ) such that if θ01 − θ02 B˙ 1 < 0 , there exists 2,1 2,1 2,1 a unique global solution θ 2 ∈ C([0, ∞); B 1 ) ∩ L1 (0, ∞; B˙ 2 ) ∩ C((0, ∞); B 2 ) of 1 . (DQG) 1 with initial data θ02 ∈ B2,1

2,1

2,1

2,1

2

2. Littlewood-Paley Decomposition We first set our notations, and recall definitions of the Besov spaces. We follow [16]. Let S be the Schwartz class of rapidly decreasing functions. Given f ∈ S, its Fourier transform F(f ) = fˆ is defined by  1 ˆ f (ξ ) = e−ix·ξ f (x) dx. (2π)n/2 ˆ )≥0 We consider ϕ ∈ S satisfying Supp ϕˆ ⊂ {ξ ∈ Rn | 21 ≤ |ξ | ≤ 2}, and ϕ(ξ 1 −j j n j if 2 < |ξ | < 2. Setting ϕˆj = ϕ(2 ˆ ξ ) (in other words, ϕj (x) = 2 ϕ(2 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 Sˆk (ξ ) = 1 − ϕˆj (ξ ). j ≥k+1

We observe Supp ϕˆj ∩ Supp ϕˆj  = ∅

if |j − j  | ≥ 2.

Let s ∈ R, p, q ∈ [0, ∞]. Given f ∈ S  , we denote j f = ϕj ∗ f . Then the homogeneous Besov semi-norm f B˙ s is defined by p,q

f B˙ s

p,q

  1 ∞ q q   2j qs ϕj ∗ f Lp if q ∈ [1, ∞) −∞   = j s  if q = ∞.  sup 2 ϕj ∗ f Lp j

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 f ∈ S as f Bp,q = f Lp + f B˙ s . Let us now state some basic properties for the p,q Besov spaces.

Proposition 1. (i) Bernstein’s Lemma : Assume that f ∈ Lp , 1 ≤ p ≤ ∞, and supp fˆ ⊂ {2j −2 ≤ |ξ | < 2j }, then there exists a constant Ck such that the following inequality holds: Ck−1 2j k f Lp ≤ D k f Lp ≤ Ck 2j k f Lp .

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Dongho Chae, Jihoon Lee

(ii) We have the equivalence of norms D k f B˙ s

p,q

∼ f B˙ p,q s+k .

(iii) Let s > 0, q ∈ [1, ∞], then there exists a constant C such that the following inequality holds : 

, f gB˙ s ≤ C f Lp1 gB˙ s + gLr1 f B˙ s p,q

p2 ,q

r2 ,q

for homogeneous Besov spaces, where p1 , r1 ∈ [1, ∞] such that 1 r1

+

1 r2 .

Let s1 , s2 ≤ and

N p



−

N p

s +s2 − N p

1 B˙ p,1

+

1 p2

=

≤ Cf B˙ s1 gB˙ s2 . p,1

p,1

 − 1, Np , then we have

[u, q ]wLp ≤ cq 2−q(s+1) u q∈Z cq

1 p1

s +s2 − N p

(iv) If s satisfies s ∈



=

1 s1 s2 such that s1 + s2 > 0, f ∈ B˙ p,1 and g ∈ B˙ p,1 . Then f g ∈ B˙ p,1

f g

with

1 p

N +1

p B˙ p,1

wB˙ s

p,1

≤ 1. In the above, we denote [u, q ]w = uq w − q (uw). N

p (v) (Embedding). B˙ p,1 (RN ) is an algebra included in the space C0 of continuous functions tending to 0 at infinity. Let s ∈ R,  > 0, and suppose p, q ∈ [1.∞]. Then it holds

s s B˙ p,1 → H˙ ps → B˙ p,∞ ,

and s+ s → Bp,q . Bp,∞

(vi) (Interpolation). We have the following interpolation inequalities for s1 , s2 ∈ R, θ ∈ [0, 1]: , u ˙ θ s1 +(1−θ )s2 ≤ Cuθ˙ s1 u1−θ ˙ s2 Bp,1

Bp,1

Bp,1

and u

θ s +(1−θ )s2

Bp,11

≤ Cuθ s1 u1−θ s2 . Bp,1

Bp,1

Proof. The proof of (i)–(vi) is rather standard and we can find the proof in many references. The proof of (i) can be found in [5], Lemma 2.1.1. (ii) is straightforward from (i). The proof of the first part of (iii) can be found in [3] and the second part of (iii) can be N

p found in [4]. We can find the proof of (iv) in [10]. The fact that B˙ p,1 is embedded in C0 can be found in [1]. We can find the proof of the last part of (v) and (vi) in [16].  

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations

301

3. Proof of Theorems 1

For notational simplicity, we denote (−) 2 by  in the following.: Proof of Theorem 1. Step 1. A priori estimates. Applying the operator q to the first equation of (DQG)α , we easily infer that   ∂t q θ + (u · ∇)q θ + κ2α q θ = − q , u · ∇θ. (1) Multiplying q θ on both sides of (1) and integrating over R2 , we have 1 d q θ 2L2 + Cκ22αq q θ 2L2 ≤ [q , u] · ∇θL2 q θ L2 2 dt ≤ Ccq 2−(2−2α)q uB˙ 2 ∇θ B˙ 1−2α q θL2 . 2,1

2,1

(2)

For the above last inequality, we used the commutator estimate (iv) of Proposition 1. Dividing both sides of (2) by q θ L2 , multiplying 2(2−2α)q on both sides of (2), and taking the summation over q ∈ Z, we obtain that d θ (t)B˙ 2−2α + C1 κθ (t)B˙ 2 ≤ Cu(t)B˙ 2 θ(t)B˙ 2−2α 2,1 2,1 2,1 2,1 dt ≤ C2 θ(t)B˙ 2 θ(t)B˙ 2−2α . 2,1

(3)

2,1

For the above second inequality, we used the Calderon-Zygmund type inequality[11] u(t)B˙ 2 ≤ Cθ(t)B˙ 2 . 2,1

2,1

By Gronwall’s inequality, we have    ∞ sup θ (t)B˙ 2−2α + C1 κ θ (t)B˙ 2 dt ≤ θ0 B˙ 2−2α exp C2

0≤t<∞

2,1

2,1

0

2,1

∞ 0

θ(t)B˙ 2 dt . 2,1

(4) And we note that θ (t)L2 ≤ θ0 L2 . Thus we have the following apriori estimate:  sup θ (t)B 2−2α + C1 κ 2,1

0≤t<∞



≤ θ0 B 2−2α exp C2 2,1

 0

0 ∞

∞

θ(t)B˙ 2 dt 2,1



θ(t)B˙ 2 dt . 2,1

Step 2. Iteration and Uniform estimates. We define the following sequences:  n+1 + (un · ∇)θ n+1 + κ2α θ n+1 = 0,  R2 × R+ , 0 ≤ 2α ≤ 1,  ∂t θ 1 (I) un = ∇ ⊥ (−)− 2 θ n ,   θ n+1 (0, x) = θ n+1 (x) =  q≤n+1 q θ0 . 0

(5)

302

Dongho Chae, Jihoon Lee

Setting (θ 0 , u0 ) = (0, 0), and solving the linear system, we can find (θ n , un ) for all n. Similarly to a priori estimates, we have  ∞   n+1   θ (t)B˙ 2 dt sup θ n+1 (t)B˙ 2−2α + C1 κ 2,1

0≤t<∞

   n+1    exp C2 ≤ θ0 B˙ 2−2α 2,1

2,1

0 ∞

 θ n (t) ˙ 2 dt . B

(6)

2,1

0

If θ0 B˙ 2−2α ≤ , then for all n, we will show that 2,1

  sup θ n+1 (t)B˙ 2−2α + C1 κ 2,1

0≤t<∞



∞

0

 θ n+1 (t) ˙ 2 dt ≤ M, B

(7)

2,1

for some M > 0. We choose  so small that  C2 M ≤ M. exp C1 κ (This is possible if we take M > 1.) Assume that θ n L∞ 0,T ;B˙ 2−2α  + C1 κθ n L1 0,T ;B˙ 2  ≤ M. From the estimate of 2,1

2,1

(6), we have   sup θ n+1 (t)B˙ 2−2α + C1 κ

0≤t<∞

2,1







 θ n+1 (t) ˙ 2 dt B

∞

0 ∞

   θ n (t) ˙ 2 dt ≤ θ0n+1 B˙ 2−2α exp C2 B2,1 2,1 0    C2 M ≤ θ0n+1 B˙ 2−2α exp 2,1 C1 κ ≤ M.

2,1



This implies (7). Step 3. Equations of difference and existence. Define δθ n+1 = θ n+1 − θ n , and δun+1 = un+1 − un , then we have the following system of the differences:   ∂t δθ n+1 + (un · ∇)δθ n+1 + (δun · ∇)θ n + κ2α δθ n+1 = 0, R2 × R+ , 0 ≤ 2α ≤ 1, 1  (I ) δun = ∇ ⊥ (−)− 2 δθ n ,  n+1 δθ

(0, x) = n+1 θ0 .

Similarly to a priori estimates, we have (It is possible to estimate the solutions similarly 2−2α−η 1 below. η can be chosen to be 1 − 2α ≤ η < 2 − 2α. Here in B2,1 replacing B2,1 we choose η = 1 − 2α for concreteness.)    1 d q δθ n+1 2 2 + C3 κ22αq q δθ n+1 2 2 L L 2 dt  n+1    −q n n+1     ≤ C2 u B˙ 2 δθ q δθ 1 L2 B˙ 2,1 2,1   +Cq (δun · ∇θ n )L2 q δθ n+1 L2 .

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations

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In the above inequality, we used the commutator estimates (iv) of Proposition 1, i.e.     n [u , q ]∇δθ n+1  2 ≤ Ccq 2−q un  ˙ 2 ∇δθ n+1  ˙ 0 . B B L 2,1

2,1

Dividing both sides of the above inequality by q δθ n+1 L2 , multiplying 2q , and taking summation over q, we infer that d δθ n+1 B˙ 1 + C3 κδθ n+1 B˙ 1+2α 2,1 2,1 dt ≤ Cun B˙ 2 δθ n+1 B˙ 1 + C(δun · ∇)θ n B˙ 1 2,1

2,1

2,1

≤ C4 θ n B˙ 2 δθ n+1 B˙ 1 + C5 δθ n B˙ 1 ∇θ n B˙ 1 . 2,1

2,1

2,1

2,1

In the above inequality, we used the second part of (iii) of Proposition 1 and the Calderon-Zygmund type inequality, i.e. (δun · ∇)θ n B˙ 1 ≤ Cδun B˙ 1 ∇θ n B˙ 1 , 2,1

2,1

2,1

and δun B˙ 1 ≤ Cδθ n B˙ 1 . 2,1

2,1

Gronwall’s inequality gives us that  ∞ n+1 sup δθ (t)B˙ 1 + C3 κ δθ n+1 B˙ 1+2α dt 2,1

0≤t<∞



≤ δθ0n+1 B˙ 1 exp C5 2,1

0 2,1



θ n (t)B˙ 2 dt 2,1



+C4 sup δθ n (t)B˙ 1 0≤t<∞

2,1

0 ∞



∞

  θ n (t)B˙ 2 dt exp C5 2,1

0

∞ 0

θ n (τ )B˙ 2 dτ . 2,1

From the uniform estimates (7), we can choose  so small that   ∞ exp C5 θ n (t)B˙ 2 dt ≤ 2 2,1

0

and



∞

C4 0

θ n (t)B˙ 2 dt < 2,1

1 8

∞ for all n. (This is possible because we obtain from (7) that 0 θ n+1 (t)B˙ 2 dt ≤ 2,1 Then we have  ∞ sup δθ n+1 (t)B˙ 1 + C3 κ δθ n+1 (t)B˙ 1+2α dt 0≤t<∞

2,1

≤ 2δθ0n+1 B˙ 1 + 2,1

0

1 sup δθ n (t)B˙ 1 . 2,1 4 0≤t<∞

2,1

M C1 κ .)

304

Dongho Chae, Jihoon Lee

Thus we have for any positive integer N > 0, N

N   sup δθ n+1 (t)B˙ 1 + C3 κ 2,1

n=1 0≤t<∞

≤2

N n=1

B2,1

n=2

N     n+1 θ0  ˙ 1 + 1 sup δθ n (t)B˙ 1 B2,1 2,1 4 0≤t<∞ n=2

≤ 2Cθ0 B˙ 1 + 2,1

1 4

N

2,1

sup δθ n (t)B˙ 1

2,1

n=2 0≤t<∞

≤ 2Cθ0 B˙ 1 + · · · + ≤

n=1 0

 δθ n+1 (t) ˙ 1+2α dt

∞

N N   n+1   δθ  ˙1 + 1 sup δθ n (t)B˙ 1 0 B2,1 2,1 4 0≤t<∞ n=1

≤2



2C θ0 B˙ 1 2,1 4N−1

8C θ0 B˙ 1 . 2,1 3

Multiplying both sides of the first equation of (I ) by δθ n+1 and integrating over R2 , we have        1 d δθ n+1 2 2 + κ δθ n+1 2˙ α ≤ C (δun · ∇)θ n  2 δθ n+1  2 L H L L 2 dt  n   n   n+1       2.  ≤ C δu L2 ∇θ L∞ δθ L Using (v) of Proposition 1 and Calderon-Zygmund inequality, we obtain for some constant C3 > 0,      d δθ n+1  2 ≤ C δun  2 θ n  ˙ 2 L L B dt  n   n  2,1     ≤ C6 δθ L2 θ B˙ 2 . 2,1

Choose  so small that C6 obtain that

∞ 0

θ n (t)B˙ 2 dt < 2,1

1 4.

Using Gronwall’s inequality, we

      sup δθ n+1 (t)L2 ≤ δθ0n+1 L2 + C6 sup δθ n (τ )L2

0≤t<∞

0≤τ <∞

    1 sup δθ n (τ )L2 . ≤ δθ0n+1 L2 + 4 0≤τ <∞ Thus we have for any positive integer N > 0,



∞ 0

 θ n (t) ˙ 2 dt B 2,1

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations N n=1

305

N N   n+1     δθ  2+1 sup δθ n+1 (t)L2 ≤ sup δθ n (t)L2 0 L 4 0≤t<∞ 0≤t<∞ n=1

≤

N

n=2

  n+1 θ0 

L2

+

n=1

N   1 sup δθ n (t)L2 4 0≤t<∞ n=2

  1 1   ≤ θ0 B 1 + θ0 B 1 + · · · + N−1 θ0 B 1 2,1 2,1 2,1 4 4  4 ≤ θ0 B 1 . 2,1 3    1+2α  1 By iteration, θ n converges to θ in L∞ 0, ∞; B2,1 ∩ L1 0, ∞; B˙ 2,1 . From the     2−2α 2 . Uniqueness can uniform estimates we have θ ∈ L∞ 0, ∞; B2,1 ∩ L1 0, ∞; B˙ 2,1 β

be proved similarly. We now prove θ (t) is continuous with values in B2,1 , where β = max{2 − 2α − δ1 , 1} for any δ1 > 0. θ n+1 satisfies ∂t θ n+1 = −(un · ∇)θ n+1 − 2α θ n+1 . 1 ). We readily have The right-hand side of the above equation belongs to L1 (0, ∞; B2,1   1 n+1 belongs to C [0, ∞); B2,1 . Since we have θ

θ (t) − θ (s)B 1 ≤ θ (t) − θ n+1 (t)B 1 + θ n+1 (t) − θ n+1 (s)B 1 2,1

2,1

+θ 

n+1

2,1

(s) − θ (s)B 1 , 2,1



1 by the three  argument. By interpolation, we have θ ∈ θ belongs to C [0, ∞); B2,1   β C [0, ∞); B2,1 , where β = max{1, 2 − 2α − δ1 } for any δ1 > 0. We still have to prove     γ  2+2α  2 θ belongs to L∞ σ, ∞; B2,1 ∩ L1 σ, ∞; B˙ 2,1 ∩ C (0, ∞); B2,1 , where σ > 0 and  2 − δ2 , if 0 ≤ α ≤ 21 , for any δ2 > 0. We provide only a priori estimates. Since γ = 2, if α = 21 ,   2 , we can choose σ > 0 such that we already know that θ belongs to L1 0, ∞; B˙ 2,1 2 . Similarly to Step 1, we have θ (σ ) ∈ B2,1

     d q θ  2 + Cκ22αq q θ  2 ≤ Ccq 2−2q u ˙ 2 θ  ˙ 2 . B2,1 L L B2,1 dt Multiplying 22q , taking the summation over q, and using Calderon-Zygmund type estimate, we have d θ B˙ 2 + CκθB˙ 2+2α ≤ Cθ 2B˙ 2 . 2,1 2,1 2,1 dt Using Gronwall’s inequality, we have 

∞

θ(t)B˙ 2+2α dt sup θ (t)B 2 + Cκ 2,1 2,1 σ ≤t<∞ σ   ∞ ≤ θ (σ )B 2 exp C θ(t)B˙ 2 dt . 2,1

σ

2,1

306

Dongho Chae, Jihoon Lee

   2+2α  2 Thus θ belongs to L∞ σ, ∞; B2,1 ∩ L1 σ, ∞; B˙ 2,1 . By the interpolation theorem,  2−δ2  for any δ2 > 0 if 0 ≤ α < 21 . If α = 21 , θ satwe have θ ∈ C [σ, ∞); B2,1   2 , θ belongs to isfies ∂t θ = −(u · ∇)θ − θ . Since (u · ∇)θ , θ ∈ L1 σ, ∞; B˙ 2,1   2 . C [σ, ∞); B2,1  

This completes the proof of Theorem 1.

  Proof of Theorem 2. We derive a differential inequality for θ 1 (t) − θ 2 (t)B˙ 1 . The 2,1

difference  = θ 1 − θ 2 satisfies the following system:  1 1 2  ∂t  + (U · ∇)θ − (U · ∇) + (u · ∇) + κδ = 0, R × R+ , 1 (A) U = ∇ ⊥ (−)− 2 ,  (0, x) = θ01 − θ02 , where U = u1 − u2 .

1. Iteration  and uniform estimates. We define the following iterating sequences Step n+1 , U n+1 :  n+1 + (U n+1 · ∇)θ 1 − (U n · ∇)n+1 + (u1 · ∇)n+1 + κδn+1  ∂t  1 (II) = 0, R2 × R+ , U n+1 =∇ ⊥ (−)− 2 n+1  ,   n+1  (0, x) = q≤n+1 q θ01 − q θ02 . Since (θ 1 , u1 ) is given, we can obtain n and U n for all n by setting (0 , U 0 )=(0, 0). Taking q operator on both sides of the first equation of (II), we have ∂t q n+1 + κq n+1 − (U n · ∇)q n+1 + (u1 · ∇)q n+1     = −[U n , q ] · ∇n+1 + u1 , q · ∇n+1 − q (U n+1 · ∇)θ 1 .

(8)

Multiplying q n+1 on both sides of (8) and integrating over R2 , then we have        1 d q n+1 2 2 + Cκ2q q n+1 2 2 ≤ [U n , q ] · ∇n+1  2 q n+1  2 L L L L 2 dt  1    n+1   n+1   + [u , q ] · ∇ 2 q  2  L  L n+1 1 n+1 +q ((U · ∇)θ ) 2 q   2 . L

L

(9) Dividing both sides of (9) by q taking summation over q, we obtain that

n+1 

L2 ,

2q

multiplying

on both sides of (9), and

d n+1 B˙ 1 + C7 κn+1 B˙ 2 2,1 2,1 dt n n+1 ≤ CU B˙ 2  B˙ 1 + Cu1 B˙ 2 n+1 B˙ 1 + C(U n+1 · ∇)θ 1 B˙ 1 2,1

≤ C8  B˙ 2  n

2,1

2,1

n+1

2,1

B˙ 2 + C9 θ B˙ 2  1

2,1

2,1

2,1

n+1

B˙ 1 . 2,1

2,1

(10)

In the above inequality, we used (iii) and (iv) of Proposition 1 and the Calderon-Zygmund type estimate. If we choose 0 so small that   ∞ C7 κ 1 0 exp C9 θ B˙ 2 dt < , 2,1 2C8 0

Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations

then we show that (Pn )

307

 κC7 ∞ sup  (t) + n (t)B˙ 2 dt 2,1 2 0 0≤t<∞   ∞ θ 1 B˙ 2 dt . ≤ 0 exp C9 n

2,1

0

Suppose that (Pn ) is satisfied and let us prove that (Pn+1 ) is also true. From (10), we obtain d n+1 B˙ 1 + (C7 κ − C8 sup n (t)B˙ 1 )n+1 B˙ 2 2,1 2,1 2,1 dt 0≤t<∞ ≤ C9 θ 1 B˙ 2 n+1 B˙ 1 . 2,1

2,1

Thus we have C7 κ d n+1 B˙ 1 + n+1 B˙ 2 ≤ C9 θ 1 B˙ 2 n+1 B˙ 1 . 2,1 2,1 2,1 2,1 dt 2 Using Gronwall’s inequality, we have sup n+1 (t)B˙ 1 + 2,1

0≤t<∞

  ∞ C7 κ 1  exp C θ  dt . n+1 B˙ 2 ≤ n+1 1 2 9 B˙ 2,1 B˙ 2,1 0 2,1 2 0

Multiplying n+1 on both sides of the first equation of (II) and integrating over R2 , we infer that d n+1 2L2 ≤ CU n+1 L2 ∇θ 1 B˙ 1 n+1 L2 2,1 dt 1 n+1 2 ≤ C∇θ B˙ 1  L2 . 2,1

Using Gronwall’s inequality, we have    exp C sup n+1 L2 ≤ Cn+1 2 L 0

0≤t<∞

∞

0

θ 1 B˙ 2 dt . 2,1

(11)

Thus we obtain the uniform estimates. Step 2. Difference and existence. Setting δn+1 = n+1 − n and δU n+1 = U n+1 − U n , we have the following equations of the difference:  ∂t δn+1 − (δU n · ∇)n + (δU n+1 · ∇)θ 1    −(U n · ∇)δn+1 + (u1 · ∇)δn+1 + κδn+1 = 0, R2 × R , + (II ) 1  δU n+1 = ∇ ⊥ (−)− 2 δn+1 ,   n+1 δ (0, x) = (n+1 θ01 − n+1 θ02 ). Taking q operator on both sides of the first equations of (II ), we infer that ∂t q δn+1 + κq δn+1 − (U n · ∇)q δn+1 + (u1 · ∇)q δn+1 = −[U n , q ] · ∇δn+1 + [u1 , q ] · ∇δn+1 +q ((δU n · ∇)n ) − q ((δU n+1 · ∇)θ 1 ).

(12)

308

Dongho Chae, Jihoon Lee

Multiplying q δn+1 on both sides of (12) and integrating over R2 , we have 1 d q δn+1 2L2 + Cκ2q q δn+1 2L2 2 dt ≤ [U n , q ] · ∇δn+1 L2 q δn+1 L2 +[u1 , q ] · ∇δn+1 L2 q δn+1 L2 +q ((δU n · ∇)n )L2 q δn+1 L2 +q ((δU n+1 · ∇)θ 1 )L2 q δn+1 L2 .

(13)

Dividing both sides of (13) by q δn+1 L2 , multiplying 2q on both sides, taking summation over q, and using (iii) and (iv) of Proposition 1, we obtain d δn+1 B˙ 1 + C7 κδn+1 B˙ 2 2,1 2,1 dt n n+1 ≤ CU B˙ 2 δ B˙ 1 + Cu1 B˙ 2 δn+1 B˙ 1 2,1

2,1

2,1

+CδU B˙ 1 ∇ B˙ 1 + CδU n

n

2,1

2,1

≤ C B˙ 2 δ n

n+1

2,1

2,1

B˙ 1 + Cθ B˙ 2 δ 2,1

2,1

n+1

2,1

≤ C10  B˙ 2 δ

n+1

2,1

2,1

n+1

2,1

n

n

B˙ 1 ∇θ B˙ 1 1

1

+Cδ B˙ 1  B˙ 2 + Cδ n

2,1

n+1

B˙ 1

2,1

B˙ 1 θ B˙ 2 1

2,1

2,1

B˙ 1 + C11 θ B˙ 2 δ 1

2,1

n+1

2,1

B˙ 1

2,1

+C12  B˙ 2 δ B˙ 1 . n

n

2,1

2,1

Using Gronwall’s inequality, we have sup δ

n+1

(t)B˙ 1 + C7 κ 2,1

0≤t<∞



≤ δn+1 0 B˙ 1 exp +C12 × exp

0

∞

δn+1 (t)B˙ 2 dt 2,1

0



∞

C10 n (t)B˙ 2 + C11 θ 1 (t)B˙ 2 dt 2,1 2,1 2,1 0  ∞ sup δn (t)B˙ 1 n (t)B˙ 1 dt

0≤t<∞  ∞ 0

From (Pn ), we have  ∞



2,1

2,1

0

C10 n (t)B˙ 2 + C11 θ 1 (t)B˙ 2 dt . 2,1

 (t)B˙ 2 n

2,1

(14)

2,1

  ∞ 20 1 ≤ exp C9 θ (t)B˙ 2 dt . 2,1 κC7 0

(15)

Substituting (15) into (14), we infer that  ∞ sup δn+1 (t)B˙ 1 + C7 κ δn+1 (t)B˙ 2 dt 2,1

0≤t<∞



≤

δn+1 1 0 B˙ 2,1 

2,1

0



+ C12 sup δ (t)B˙ 1

∞

n

0≤t<∞

2,1

0

  (t)B˙ 1 dt n

2,1

20 C10 1 1 × exp exp C9 θ L1 (0,∞;B˙ 2 ) + C11 θ L1 (0,∞;B˙ 2 ) . (16) 2,1 2,1 κC7



Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations

309

Setting  M1 = exp



20 C10 exp C9 θ 1 L1 (0,∞;B˙ 2 ) + C11 θ 1 L1 (0,∞;B˙ 2 ) , 2,1 2,1 κC7

choose 0 so small that

  ∞ 20 exp C9 0 θ 1 B˙ 2 dt 2,1

κC7

≤

1 4C12 M1

.

Then we have from (16),  sup δ 0≤t<∞

n+1

(t)B˙ 1 + C7 κ 2,1

≤ M1 δn+1 0 B˙ 1 + 2,1

∞

0

δn+1 (t)B˙ 2 dt 2,1

1 sup δn (t)B˙ 1 . 2,1 4 0≤t<∞

We have for any positive integer N > 0, N

sup δn+1 (t)B˙ 1 + C7 κ 2,1

n=1 0≤t<∞

≤ CM1

N

N  n=1 0

2n+1 n+1 0 L2 +

n=1

≤ CM1 0 B˙ 1 + · · · + 2,1

≤

∞

δn+1 (t)B˙ 2 dt 2,1

1 sup δn (t)B˙ 1 2,1 4 0≤t<∞

CM1 0 B˙ 1 2,1 4N−1

4 CM1 0 B˙ 1 . 2,1 3

Multiplying δn+1 on both sides of (II ) and integrating over R2 , we have 1 1 d δn+1 (t)2L2 + κ 2 δn+1 (t)2L2 2 dt ≤ ∇n (t)L∞ δU n (t)L2 δn+1 (t)L2

+∇θ 1 (t)L∞ δU n+1 (t)L2 δn+1 (t)L2 ≤ C13 n (t)B˙ 2 δn (t)L2 δn+1 (t)L2 2,1

+C14 θ (t)B˙ 2 δn+1 (t)2L2 . 1

(17)

2,1

Dividing both sides of (17) by δn+1 L2 and using Gronwall’s inequality, we obtain   ∞ 1 sup δn+1 (t)L2 ≤ δn+1  exp C θ (t) dt 2 2 14 L B˙ 0 0≤t<∞

0

+C13 sup δn (t)L2 0≤t<∞





∞

× exp C14 0

2,1



∞ 0

n (t)B˙ 2 dt

θ 1 (t)B˙ 2 dt . 2,1

2,1

(18)

310

Dongho Chae, Jihoon Lee

 ∞ Setting M2 = exp C14 0 θ 1 B˙ 2 dt , choose 0 so small that

20 exp C9

2,1

∞ 0

θ 1 (t)B˙ 2 dt



2,1

κC7

≤

1 4C13 M2

.

Thus we have from (18), sup δn+1 (t)L2 ≤ M2 δn+1 0 L2 +

0≤t<∞

1 sup δn (t)L2 . 4 0≤t<∞

1 )∩L1 (0, ∞; B ˙ 2 ). From the uniform By iteration, n converges to  in L∞ (0, ∞; B2,1 2,1 1 2 ∞ 1 ˙ estimates, we have  ∈ L (0, ∞; B2,1 ) ∩ L (0, ∞; B2,1 ). Uniqueness can be proved 1 (so θ 2 is continuous). n+1 similarly. We still have to prove  is continuous in B2,1 satisfies

∂t n+1 = −(U n+1 · ∇)θ 1 + (U n · ∇)n+1 − (u1 · ∇)n+1 − κn+1 . 1 ), we readily have Since the right-hand side of the above belongs to L1 (0, ∞; B2,1 1 . By using the three  argument as before n+1 (t) is continuous with values in B2,1 1 we find  ∈ C([0, ∞); B2,1 ). Similarly to the proof of Theorem 1 and the apriori esti2 ). This completes the proof of Theorem mates of solution, we have θ 2 ∈ C((0, ∞); B2,1 2.  

Acknowledgements. This research is supported partially by the grant no.2002-2-10200-002-5 from the basic research program of the KOSEF and J. Lee is supported by BK 21 project.

References 1. Bourdaud, G.: Re´alisations des espaces de Besov homog`enes. Arkiv f¨or matematik 26, 41–54 (1988) 2. Chae, D.: On the Well-Posedness of the Euler Equations in the Triebel-Lizorkin Spaces. Comm. Pure Appl. Math. 55, 654–678 (2002) 3. Chae, D.: Local existence and blow-up criterion for the Euler equations in the Besov spaces. RIMGARC Preprint no. 01–7 4. Chemin, J.-Y.: About Navier-Stokes system. Pr´epublication du Laboratorie d’analyse num´erique de Paris 6, R96023 (1996) 5. Chemin, J.Y.: Perfect Incompressible Fluids. Oxford: Clarendon Press, 1998 6. Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative Quasi-geostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001) 7. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1459–1533 (1994) 8. Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 148, 937–948 (1999) 9. Danchin, R.: Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141, 579–614 (2000) 10. Danchin, R.: Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm. Partial Differ. Eqs. 26, 1183–1233 (2001) 11. Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley theory and the study of function spaces. AMSCBMS Regional Conference Series in Mathematics 79, (1991) 12. Fujita, H., Kato, T.: On the Navier-Stokes initial value problem I .Arch. Rat. Mech.Anal. 16, 269–315 (1964) 13. Kato, T.: Strong Lp solutions of the Navier-Stokes equations in Rm with applications to weak solutions. Math. Z. 187, 471–480 (1984) 14. Pedlosky, J.: Geophysical Fluid Dynamics. New-York: Springer-Verlag, 1987

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15. Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph. D. thesis University of Chicago, Chicago (1995) 16. Triebel, H.: Theory of Function Spaces. Basel-Boston: Birkh¨auser, 1983 17. Wu, J.: Quasi-geostrophic-type equations with initial data in Morrey spaces. Nonlinearity 10, 1409– 1420 (1997) 18. Wu, J.: Dissipative quasi-geostrophic equations with Lp data. Electronic J. Differ. Eqs. 2001, 1–13 (2001) 19. Wu, J.: On solutions of three quasi-geostrophic models. Preprint 20. Vishik, M.: Hydrodynamics in Besov spaces. Arch. Rat. Mech. Anal. 145, 197–214 (1998) Communicated by P. Constantin

Commun. Math. Phys. 233, 355–381 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0734-z

Communications in

Mathematical Physics

On Noncommutative Multi-Solitons Rajesh Gopakumar, Matthew Headrick, Marcus Spradlin Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA. E-mail: [email protected] Received: 29 May 2001 / Accepted: 16 August 2002 Published online: 7 November 2002 – © Springer-Verlag 2002

Abstract: We find the moduli space of multi-solitons in noncommutative scalar field theories at large θ, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/θ is a consequence of a Bogomolnyi bound obeyed by the kinetic energy of the θ = ∞ solitons. In two spatial dimensions, the parameter space for k solitons is a K¨ahler de-singularization of the symmetric product (R2 )k /Sk . We exploit the existence of this moduli space to construct solitons on quotient spaces of the plane: R2 /Zk , cylinder, and T 2 . However, we show that tori of area less than or equal to 2π θ do not admit stable solitons. In four dimensions the moduli space provides an explicit K¨ahler resolution of (R4 )k /Sk . In general spatial dimension 2d, we show it is isomorphic to the Hilbert scheme of k points in Cd , which for d > 2 (and k > 3) is not smooth and can have multiple branches. Contents 1. Introduction . . . . . . . . . . . . . . . . . 2. The Moduli Space at Infinite θ . . . . . . . 2.1. Solitons and projection operators . . . 2.2. Geometry of the Grassmannian . . . . 3. The Moduli Space at Finite θ . . . . . . . . 3.1. The Bogomolnyi bound . . . . . . . . 3.2. Topology . . . . . . . . . . . . . . . 3.3. Geometry . . . . . . . . . . . . . . . 4. The Effective Potential on the Moduli Space 5. Solitons on Quotient Spaces . . . . . . . . 5.1. R2 /Zk . . . . . . . . . . . . . . . . . 5.2. Cylinder . . . . . . . . . . . . . . . . 5.3. T 2 . . . . . . . . . . . . . . . . . . . 6. Solitons in Higher Dimensions . . . . . . .

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356 357 358 359 360 360 361 362 365 367 367 368 369 372

356

R. Gopakumar, M. Headrick, M. Spradlin

6.1. 4 + 1 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Hilbert schemes of points . . . . . . . . . . . . . . . . . . . . . 6.3. An exotic example . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Completeness and Smoothness of the Finite θ Moduli Space A.1. Some elementary machinery . . . . . . . . . . . . . . . . . . . A.2. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Construction of the Soliton on the Integral Torus . . . . . . .

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372 374 374 375 375 377 378 378

1. Introduction Among the many reasons to study noncommutative field theories is that they might be useful for studying stringy behavior in a controlled manner. In the context of string theory, the noncommutativity is a remnant of the noncommutative geometry of open string field theory [1–4]. An interesting class of excitations in the field theory are noncommutative solitons [5]. The existence and form of these classical solutions are fairly independent of the details of the theory, making them useful probes of stringy behavior. Coincident solitons exhibit a non-abelian enhancement of the zero mode spectrum, and in fact these solitons are the D-branes of string theory manifested in a field theory limit while still capturing many of their stringy features. Note that, in contrast, D-branes do not appear as finite energy excitations in conventional (commutative) field theory limits of string theory. The fact that one can see D-branes in a simpler field theoretic context has had important consequences, primarily in the study of tachyon condensation in open string field theory [6, 7]. Many of Sen’s conjectures [8, 9] on this subject have been beautifully confirmed using properties of noncommutative solitons. Similarly, unstable solitons in noncommutative gauge theory [10, 11] can be interpreted as codimension two (or higher) D-branes localized on other D-branes [12, 13]. The gauge theory successfully reproduces the dynamics of tachyon condensation in this system. Other studies of scalar solitons include [14–23]. In this paper we will begin to investigate some aspects of how D-branes see space time by exploring the moduli space of solitons in noncommutative scalar field theory in 2d +1 dimension. The study of multi-solitons itself turns out to have many interesting features. At θ = ∞, because of the presence of the U(∞) symmetry in the dominant potential energy term, there is an infinite dimensional moduli space of solutions corresponding to arbitrary hermitian projection operators on a Hilbert space H. Since this symmetry is not preserved by the kinetic term, one might expect the degeneracy to be completely lifted when one includes this leading 1/θ correction. Surprisingly, we find that there remains a smaller yet non-trivial moduli space at this order, which roughly corresponds to individual gaussian solitons free to roam the plane. The lack of a force between static solitons at this order in 1/θ is due to a Bogomolnyi bound for the kinetic energy of projection operators. This moduli space, however, is not protected by any symmetry, and is lifted by a classical effective potential which is generated at the next order in 1/θ , and presumably by quantum corrections as well. The classical effective potential (which is bounded both above and below) leads to an attractive force among the solitons. Thus in the true (exact) moduli space all k solitons are coincident. Nevertheless, the approximate moduli space is a useful description of the dynamics of multiple solitons within a certain range of energies [25].

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Why is this moduli space interesting if it is only an approximate one? The answer is twofold. Firstly, this moduli space is very closely related to the symmetric product spaces (R2d )k /Sk , which arise in the study of supersymmetric vacua of Dp-Dp  systems in string theory. In fact, we will see that our moduli space (for a scalar field theory with noncommutativity in 2d spatial directions) is precisely the same as the so-called Hilbert scheme of k points in R2d . This is, for d ≤ 2, a smooth resolution of the symmetric product space which is potentially singular when points coincide. For the case of d = 2 we see that the solitons resolve the singularity in an interesting way: when any two of them are brought together, the final configuration depends on the complex direction in the relative R4 by which they approach each other, so that there is a hidden P1 at the putative singular point. This is also exactly what D-branes see as a stringy resolution of the singularity. For d > 2, interestingly, the Hilbert scheme is not a manifold in any sense and has exotic branches which also show up in brane systems. In our context, these branches reflect the fact that coincident solitons in higher dimension have a large number of moduli associated with the shape of the lump solution. The second reason why these approximate moduli spaces are interesting is that they enable us to construct infinite arrays of solitons in R2d that respect some discrete symmetry. Such an array can be viewed as a single soliton on the quotient space. The lowest energy stable solutions on the quotient space—the analogues of the gaussian soliton on R2 —are constructed this way. For the torus we encounter the somewhat unexpected fact that solitons do not exist when the area is less than or equal to 2π θ. The moduli spaces also inherit an interesting geometric structure from the noncommutativity. For instance, in the simplest case of two spatial dimensions, the k soliton moduli space is—as a complex manifold—simply Symk (R2 ) ≡ (R2 )k /Sk , the symmetric product of the single soliton moduli space R2 . Geometrically, however, the solitons smooth out the conical singularities that occur on Symk (R2 ) where two or more points come together, as the explicit K¨ahler metric shows. (The K¨ahler metric on the moduli space of k = 2 solitons has appeared in the work of Lindstr¨om et al. [24].) Similarly, the smooth Hilbert scheme resolution of Symk (R4 ) also inherits a K¨ahler metric, distinct from the hyperk¨ahler one which arises in the case of instanton physics. That noncommutative solitons have such a rich structure in their moduli space is very encouraging and makes them worthy of further exploration. The way apparent singularities are resolved is very stringy, and it is interesting that the noncommutative algebra of projection operators sees the resolved geometry in a simple way, which the commutative algebra of functions cannot. This is perhaps a clue that noncommutative algebras might play a fundamental role in understanding geometry in string theory. Some of the results in this paper were announced at Strings 2001 [39]. The paper [28] by E. Martinec and G. Moore, which has appeared in the meanwhile, has overlap with the discussion in Sect. 5. K. Lee and collaborators have studied the resolution of moduli spaces of instantons and vortices on noncommutative spaces in [29–31]. 2. The Moduli Space at Infinite θ In this section we briefly review the construction of solitons in 2 + 1 dimensional noncommutative scalar field theory at large θ . The moduli space at θ = ∞ (defined precisely below) is isomorphic to the space of projection operators1 on an infinite-dimensional Hilbert space. We review the relevant mathematical details about the geometry of this space (known as the Grassmannian). 1

Throughout this paper, by “projection operator” we mean “hermitian projection operator.”

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V (φ )

φ λ

0

Fig. 1. The potential for φ is assumed to have a global minimum at φ = 0 and a local minimum at φ = λ

2.1. Solitons and projection operators. Until the last section of this paper, we work in a 2 + 1 dimensional scalar field theory with spacelike noncommutativity:  
1 2 2 2 ˙ S = dt d w (2.1) φ − ∂w φ∂w¯ φ − m V (φ)∗ , 2 √ where w = (x 1 + ix 2 )/ 2. The subscript on the potential indicates that it is evaluated with the Moyal star product, with noncommutativity parameter θ ww¯ = −iθ (for details, see [5]). It will be convenient to let the factor m2 multiply the entire potential, so we assume V  (0) = 1. The existence of stable solitons in this theory depends on the potential having a second, local minimum, which we put at φ = λ, with  V (λ) > V (0) = 0 (Fig. 1). The energy functional for static configurations, E[φ] = d 2 w(∂w φ∂w¯ φ + m2 V (φ) √  ), may be conveniently rewritten using the Weyl-Moyal correspondence, w → θa, ∂w → − √1 [a † , · ], etc., which maps the star product into operator multiplication in the θ Hilbert space H of a 1-dimensional particle:   ˆ θ m2 ] = 2π TrH [a, φ][ ˆ φ, ˆ a † ] + θm2 V (φ) ˆ . E[φ, (2.2) Fixing the function V (φ), it is the dimensionless parameter θm2 that controls the relative importance of the kinetic and potential terms in (2.2). Exact solutions to the equation of motion are known in the limit θ m2 → ∞, when the kinetic term may be neglected compared to the potential term [5]. In this section we describe the moduli space of such solitons. In the next section we include the kinetic term as a perturbation, and see how it lifts the moduli space down to a much smaller but still non-trivial one. To define the theory in the limit θ m2 → ∞ we must rescale the energy:   1 ˆ ˆ ≡ lim ˆ θ m2 = 2π Tr V (φ). (2.3) E φ, E0 [φ] θm2 →∞ θ m2 ˆ = 0 take the form Stable solutions to the resulting equation of motion V  (φ) φˆ = λP ,

(2.4)

where P is any projection operator. The energy of such a solution is E0 = 2π kV (λ), where k is the rank of P , so we will assume for now that this rank is finite. (In Sect. 5

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we will also discuss projection operators of infinite rank.) If one interprets the rank one solutions as single solitons, then the rank k solutions may be thought of as corresponding to k non-interacting solitons, each of energy 2π V (λ). This interpretation will become more meaningful in the next section. Note that (2.3) has an invariance under arbitrary unitary transformations U of the ˆ † . Since any two projection operators of rank k Hilbert space, under which φˆ → U φU can be continuously connected by U(∞) transformations, there is an infinite dimensional moduli space of solutions with energy 2π kV (λ). In fact, the rank k projection operators on H (or equivalently, the k-dimensional hyperplanes in H) form a manifold known as the Grassmannian Gr(k, H), which can also be described as the coset space U(∞) , U(k) × U(∞ − k)

(2.5)

where U(∞) acts on the entire space, while U(∞ − k) acts only on the orthogonal complement of a k-dimensional hyperplane. The U(∞) symmetry protects the moduli space against corrections. In the next subsection, we describe the geometry of the Grassmannian, partly because of its interest as the moduli space of solitons in the limit θm2 → ∞, but more importantly because this will give us the tools to study the geometry of the moduli space of solitons at finite θ m2 , which is a submanifold of the Grassmannian. 2.2. Geometry of the Grassmannian. The Grassmannian has a natural complex structure, which it inherits from H. Points (vectors) in H may be parametrized by a (infinite) set of holomorphic coordinates za ; these could be the coefficients of the vector in some particular basis. If we now have a set of k linearly independent vectors |ψi  ∈ H which depend holomorphically on the za , then the hyperplane they span is also considered to depend holomorphically on the za . The Grassmannian also has a natural K¨ahler structure, which can be computed explicitly as follows.2 Defining the matrix hij ≡ ψi |ψj 

(2.6)

and its inverse hij , the hermitian operator that projects onto the hyperplane spanned by the |ψi  is P = |ψi hij ψj |.

(2.7)

The matrix hij is the metric on the image of P . It is straightforward to show that the metric on the Grassmannian is K¨ahler, ga b¯ ≡ Tr(∂a P ∂b¯ P ) = ∂a ∂b¯ K,

(2.8)

and the K¨ahler potential is given simply by K = ln det(hij ).

(2.9)

2 For the mathematical cognoscenti: the natural K¨ahler form may be obtained as the curvature of a certain line bundle. Let E be the tautological bundle whose fiber over a point in Gr(k, H) is simply the k-dimensional hyperplane that it is. The inner product ·|· on E induces a natural metric on the determinant bundle det(E): the norm of a section ψ = |ψ1  ∧ · · · ∧ |ψk  of det(E) is simply ||ψ||2 = det( ψi |ψj ). The curvature form i∂a ∂b¯ ln det( ψi |ψj ) of this bundle is the natural K¨ahler form on Gr(k, H).

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There is an ambiguity in choosing the |ψi , with any two choices being related by a GL(k) matrix that depends holomorphically on the coordinates. The respective K¨ahler potentials will differ by a holomorphic plus an anti-holomorphic function, leaving the metric unchanged. Although the derivation of the moduli space metric has been presented as a mathematical triviality, we emphasize that the physical moduli space metric, which arises as the kinetic term for time-dependent moduli za (t), differs from (2.8) only by an overall factor. After rescaling the energy as in (2.3), we find the physical metric
2πλ2 2π λ2 1 2 d w ∂ φ∂ φ = Tr(∂ P ∂ P ) = ∂a ∂b¯ K, (2.10) ¯ ¯ a a b b θ m2 m2 m2 with K given by the formula (2.9). 3. The Moduli Space at Finite θ The U(∞) symmetry that protects the moduli space in the limit θm2 → ∞ is broken in the scalar theory at finite θ m2 by the kinetic term. One might expect that the infinite dimensional moduli space Gr(k, H) discussed in the previous section is completely lifted at finite θ . Remarkably, we find that despite the lack of symmetry, the leading 1/(θ m2 ) correction to the energy satisfies a Bogomolnyi-like bound which preserves a finite dimensional subspace of the space of projection operators of rank k. We show that the remaining moduli space Mk corresponds to k individual solitons which are free to roam the plane, and that Mk has a K¨ahler metric which is smooth even when the solitons come together. 3.1. The Bogomolnyi bound. In a perturbation expansion in 1/(θ m2 ), 1 φˆ 1 + · · · , θ m2 1 E = θ m2 E0 + E1 + E2 + · · · , θm2 the first correction to the energy is just the kinetic term: φˆ = φˆ 0 +

E1 [φˆ 0 ] = 2π Tr[a, φˆ 0 ][φˆ 0 , a † ].

(3.1)

(3.2)

Due to the fact that V  (φˆ 0 ) = 0, E1 is independent of the correction φˆ 1 . E1 will act like a potential on the space of minima of E0 described in the previous section. The minima of this potential will form the moduli space at finite θm2 , which may then be further corrected at higher orders in 1/(θ m2 ). A reasonable guess would be that a minimum of the kinetic energy is achieved only by rotationally symmetric solutions. Rotations (about the origin, for simplicity) are generated by the harmonic oscillator Hamiltonian a † a + 1/2, so rotational symmetry translates, for operators, into being diagonal in a basis of harmonic oscillator eigenstates |n. Indeed, it was shown in [5] that any sum of operators of the form λ|n n| extremizes E1 (within the infinite θ m2 moduli space), while the operator λ

k−1 n=0

|n n|

(3.3)

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was conjectured to minimize it.3 In particular, for k = 1 this would be achieved by the gaussian soliton λ|0 0|. Using the (exact) translational symmetry of the theory to center such a solution at any point on the plane, the moduli space (for any given k) appears to be simply the plane itself, and the moduli space dynamics completely trivial. However, the story is not so simple. It turns out that there are non-rotationally symmetric minima of E1 . Indeed, the full moduli space Mk has an interesting structure, large enough to allow non-trivial dynamics. This unexpected fact is a consequence, not of any symmetry possessed by E1 , but rather of a Bogomolnyi-like bound that it satisfies:

 E1 [φˆ 0 ] = 2πλ2 Tr[a, P ][P , a † ] = 2πλ2 Tr P + 2F (P )† F (P ) ≥ 2π λ2 k, (3.4) where F (P ) ≡ (1 − P )aP .

(3.5)

The bound is saturated when F (P ) = 0, in other words when the image of P is an invariant subspace of the operator a.4 The projection operators satisfying this condition define a subspace Mk of the Grassmannian Gr(k, H). In the next subsection we will see that it is a finite dimensional subspace, and that the field configurations corresponding to these projection operators have a natural interpretation in terms of separated solitons. In the following subsection we will then investigate the geometry of Mk , showing in particular that it is smooth. 3.2. Topology. Starting with the simplest case of k = 1, it is clear that any 1-dimensional invariant subspace of a must be spanned by an eigenstate of that operator, i.e. by † a coherent state. We use the non-normalized coherent states |z ≡ eza |0, their virtue for our purposes being that they depend holomorphically upon the eigenvalue z, so that they form a complex submanifold of H. This in turn implies that the moduli space M1 is a (1-dimensional) complex submanifold of Gr(1, H). It is therefore K¨ahler, and in fact the metric (2.8) is simply ds 2 = dz d z¯ . The solution φˆ z = λ

|z z|

z|z

(3.6)

maps back under the Weyl-Moyal correspondence to the function √ θ−z|2

¯ = 2λe−2|w/ φz (w, w)

(3.7) √ This is just a translated gaussian soliton localized around w = θz. Its moduli space is naturally isomorphic to the physical plane. At higher k, the most obvious way to construct an invariant subspace of a is as a direct sum of such 1-dimensional invariant subspaces, each spanned by a different coherent state |zi . We can think of this as k solitons, each with independent collective coordinate zi . (Indeed, if the zi are far from each other, then the respective coherent states are nearly orthogonal and the corresponding field configuration is approximately .

3 The authors of [5] showed that λ|0 0| indeed minimizes E , and that infinitesimal unitary trans1 formations which mix |0 and |1 are zero modes of E1 around that extremum, thus establishing the conjecture for k = 1, 2. 4 A similar equation has arisen in a different context in [40].

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the sum of distant Gaussian solitons (3.7).) The moduli space is, at least naively, the k-fold symmetric product of the single-soliton moduli space, Symk (C) ≡ Ck /Sk (symmetric because permuting the zi leaves the configuration unchanged; the solitons are like identical particles). However, such symmetric products can be singular at the locus of coincidence, and need to be looked at with care. Let us consider next the case of k = 2 and see what happens when two solitons are brought together. The description of the corresponding projection operator in terms of the subspace spanned by coherent states {|z1 , |z2 } becomes bad since it seems as if we have only one independent vector at the coincident point. Actually, as z2 approaches z1 , it is not the subspace that becomes singular but simply our description of it. Instead we should describe it as spanned, for example, by the vectors |z1  and (|z2 −|z1 )/(z2 −z1 ). This basis has the virtue that it is non-singular in the limit z2 → z1 ; in fact the second vector goes to ∂z1 |z1  = a † |z1 . Now it is clear that we are still in the moduli space in this limit, since the limiting plane, spanned by |z1  and a † |z1 , is obviously an invariant subspace of a. It is straightforward to generalize this proof for any number n of solitons coming together. In Appendix A we show that

(3.8) lim span {|z1 , . . . , |zn } = span |z, a † |z, . . . , (a † )n−1 |z . zi →z

Again, the right-hand side is obviously an invariant subspace of a. Conversely, one can see that any finite dimensional invariant subspace of a is a direct sum of spaces of the form (3.8), simply by writing the restriction of a to the subspace in Jordan form. That the moduli space Mk does not degenerate in any way at the coincidence locus is a reflection of the mathematical statement that Symk (C) is smooth everywhere. The coordinates zi are bad coordinates near the coincidence locus but in fact Symk (C) is isomorphic to Ck . Moreover, Mk is isomorphic to Ck as a complex manifold since the spanning vectors |zi  depend holomorphically on the coordinates zi . The isomorphism of complex structures extends to the coincidence locus since the change of basis matrix used in Appendix A to go to the non-singular description also depends holomorphically on the zi . This also implies that Mk is a complex submanifold of Gr(k, H). It is straightforward, using (2.7) and the inverse Weyl-Moyal transformation, to find the physical field configuration φ(w, w) ¯ corresponding to any given point in Mk . For example, with the zi all distinct, we have φ(w, w) ¯ = 2λ

k

√ √ θ−¯zi )(w/ θ−zj )

¯ hij hj i e−2(w/

.

(3.9)

i,j =1

Figure 2 illustrates how the process of two solitons coming together appears in the real space, while Fig. 3 shows an example of a quadruple soliton. 3.3. Geometry. The flat metric ds 2 = dzi d z¯ i on Symk (C) has conical singularities on the coincidence locus. The solitons are not pointlike, however, and the smoothness of the merging process depicted in Fig. 2 suggests that they somehow round out these singularities. Indeed, the moduli space Mk has a smooth K¨ahler metric on it. In this subsection we make some general observations about the K¨ahler metric and investigate in detail the simplest case of k = 2 (which has appeared in [24]), but the general proof of smoothness is relegated to Appendix A.

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The fact that Mk is a complex submanifold of the K¨ahler manifold Gr(k, H) implies that Mk itself is K¨ahler. The K¨ahler potential is given by (2.9):

 K = ln det zi |zj  = ln det(ez¯ i zj ).

(3.10)

2 1 -4

0 2 1

-2 0 -1

0 2

4-2

2 1

2 1

0 -2

0

-1 0

-1

1 2

-2

2 1

2 1

0 -2

0

-1 0

-1

1 2

-2

Fig. 2. The double soliton at various separations. Top: At large separation (here z = ±2), the solitons have the same shape as single solitons (3.7). Middle: As they come together they begin to coalesce (here z = ±1). Bottom: When they coincide they form √ a rotationally symmetric “level 2” soliton. The vertical axis is φ(w, w)/λ ¯ and the horizontal plane is w/ θ

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2 1 2

0 -2 2

1 0

0

-1

2 4 -2

Fig. 3. A quadruple soliton consisting of a triple soliton at the origin and a single soliton at z = 2. The axes are as in figure 2

(There is an overall factor of 2πθ λ2 in the physical moduli space metric, which we will ignore here.) Two important features of the geometry are immediate consequences of this form for K¨ahler potential. Firstly, as the translational symmetry of the field theory implies, the  center of mass coordinate c ≡ k1 i zi factors out, and the geometry for this modulus is the flat plane:   K = k|c|2 + ln det ey¯i yj , (3.11) where yi = zi − c are the relative coordinates. Secondly, when the separations |zi − zj | are large, the determinant is dominated by the diagonal, so the metric reduces to the flat one on Symk (C): K≈

k

|zi |2

(|zi − zj |  1).

(3.12)

i=1

A third property of the K¨ahler potential is also apparent from Eq. (3.10): it diverges on the coincidence locus. This is merely a coordinate singularity. The problem is the same as the one we faced in the last subsection: the basis of coherent states |zi  is degenerate when two or more of the zi coincide. The solution is also the same: make a holomorphic change of basis to a non-degenerate basis. We show in  Appendix A that the Jacobian of this change of basis is the Vandermonde determinant i>j (zi − zj ), and when the K¨ahler potential (3.10) is calculated in this basis one obtainss K = K − ln |zi − zj |2 . (3.13) i>j

The K¨ahler potentials K  and K yield the same metric away from the coincidence locus, but only the metric obtained from the correct potential K  extends smoothly to the locus.

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Let us see in detail how this works in the case of k = 2. Since the center of mass coordinate plays no role, we will put the two solitons at z1 = z and z2 = −z respectively. But z is not a good coordinate in the neighborhood of z = 0, due to the identification z ∼ −z. The good coordinate is instead σ = z2 , in terms of which the metric is perfectly well behaved near σ = 0:   2 2 sinh 2|z|2 K  = ln = |σ |2 + O(|σ |4 ), (3.14) 4|z|2 3 so

 ds 2 =

 2 + O(|σ |2 ) dσ d σ¯ . 3

(3.15)

The space retains the symmetry of the cone, as well as its geometry far from the vertex, but the vertex itself is rounded out. The fact that the moduli space is curved implies that there are velocity-dependent forces among the solitons.5 One can use the metric derived from (3.13) to study the motion of these solitons on the plane, by calculating geodesics on the moduli space [25]. For instance, two-soliton scattering at zero impact parameter has been shown to lead to ninety degree deflection [24]. This can be understood as an immediate consequence of the smoothness and rotational symmetry of the moduli space: A process where two solitons start at z = ±z0 and move toward each other is represented in the moduli space by starting at the point σ = z02 and moving toward the origin. The system will pass smoothly through the origin, and by symmetry then pass through the point σ = −z02 , representing the configuration where the solitons are at z = ±iz0 . For k > 2 the metric is complicated and therefore difficult to study, but one interesting phenomenon which appears is that the velocity-dependent forces tend to spread coincident solitons apart when they move in the presence of other solitons. For example, if a double soliton moves in the presence of a stationary third soliton, the pair will spread apart along the direction of their motion. 4. The Effective Potential on the Moduli Space The moduli space described in the previous section is not protected by any symmetry and is therefore lifted, by classical and presumably quantum effects. A classical effective potential on the moduli space arises from the fact that the solitons φˆ 0 we have been discussing so far are not exact solutions, but receive O(1/(θ m2 )) corrections (the term φˆ 1 in (3.1)). We will calculate these corrections explicitly in this section and show that the effective potential leads to an attractive force among the solitons. As we will see, this potential, in addition to being parametrically suppressed, is also well behaved. It is bounded both from below and from above and hence is a small perturbation (for large θ m2 ) to the moduli space that we studied in the previous section.

5 In the paper [41] the question was raised as to whether there are bound-state wave-functions due to the curvature of the moduli space. At least for the case k = 2, we may answer this question in the negative. By using a Weyl transformation to bring the relative moduli space to the plane, it is straightforward to show that the eigenvalue equation for the laplacian admits no normalizable solutions.

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In the perturbation series (3.1) for the energy, the first sub-leading correction is   1 2  † ˆ ˆ ˆ ˆ ˆ ˆ (4.1) E2 [φ0 , φ1 ] = 2π Tr 2[a, φ0 ][φ1 , a ] + φ1 V (φ0 ) . 2 (We have eliminated the term involving φˆ 2 due to the vanishing of V  (φˆ 0 ).) Fixing φˆ 0 , the equation of motion for φˆ 1 obtained from (4.1) is easily solved:6 φˆ 1c = −2[a † , [a, φˆ 0 ]]V  (φˆ 0 )−1 .

(4.2)

Plugging this solution back into (4.1), we find, after some algebra, that the correction to the energy is Veff (P ) ≡

1 E2 [φˆ 0 , φˆ 1c (φˆ 0 )] = −ETr G(P )2 , θ m2

where E≡

4πλ2 θ m2

 1+

1 V  (λ)

(4.3)

 (4.4)

and G(P ) ≡ P a(1 − P )a † P = P + [P a † , aP ].

(4.5)

The first way of writing G(P ) makes it obvious that it is non-negative definite, while the second way makes it obvious that Tr G(P ) = k. Together these two facts imply that Tr G(P )2 is bounded below by k and above by k 2 , i.e. the effective potential is bounded above by −kE and below by −k 2 E. The upper bound is achieved when G(P ) = P , i.e. [P a † , aP ] = 0; this occurs only in the limit that the solitons are infinitely far from each other, since only in this limit is aP unitarily diagonalizable. The lower bound, on the other hand, is achieved when G(P ) has only a single nonzero eigenvalue, equal to k. Let the corresponding eigenvector be |ψ, so that G(P ) = k|ψ ψ|. Then for every |ψ   orthogonal to |ψ, (1 − P )a † P |ψ   = 0,

(4.6)

which in turn implies that the image of P is of the form of the right-hand side of (3.8). (Otherwise, if the image of P were the direct sum of two or more spaces of this form, then every linear combination of the states (a † )n1 −1 |z1  and (a † )n2 −1 |z2  would violate (4.6). Then G(P ) would not be of the desired form because it would have two nonzero eigenvalues.) So the effective potential is minimized by configurations in which all k solitons are coincident. The moduli space of such configurations is obviously just the 6

We have ignored the issue of operator ordering, which is justified a posteriori by the fact that (as one can readily verify) the solution φˆ 1c commutes with φˆ 0 . There is in fact a continuous family of solutions   to the correct equation of motion 2[a † , [a, φˆ 0 ]] + φˆ 1 V  (φˆ 0 ) = 0 (where the subscript s indicates s

symmetrization over orderings of φˆ 1 and φˆ 0 ), of which φˆ 1c is the unique one that commutes with φˆ 0 . The other solutions differ from φˆ 1c by an operator of the form P B(1 − P ) + (1 − P )B † P for some B. One way to understand this is that operators of this form perturb the solution φˆ 0 in an “angular” direction (parallel to the infinite θ moduli space), and the second derivatives of Tr V vanish in those directions.

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-2 -2.5 -3 -3.5

1

2

3

4

Fig. 4. The two-soliton effective potential (4.7): Veff /E versus |σ | = |z1 − z2 |2 /4

plane itself. In other words, all the moduli yi for relative motion are lifted. The remaining center-of-mass modulus c is exact by the translational symmetry of the theory, and cannot be lifted at any order in 1/(θ m2 ), or by quantum corrections. The two-soliton effective potential, in terms of the coordinate σ = (z1 − z2 )2 /4, is  4|σ |2 , Veff (σ ) = −2E 1 + sinh2 (2|σ |) 

(4.7)

which is plotted in Fig. 4. The force is attractive, but falls off exponentially when the √ solitons are more than a few multiples of θ apart. At higher k, the functional form of Veff is more complicated, but it retains these features.

5. Solitons on Quotient Spaces We can exploit the results of Sect. 3 for multi-solitons to construct solitons on quotients of R2 by various discrete symmetry groups. The basic principle is simple: construct a multi-soliton on the covering space that respects the quotienting symmetry.

5.1. R2 /Zk . As a simple example, consider the orbifold R2 /Zk . We may put k solitons at the vertices of a regular polygon: zi = ωi z0 , where ω ≡ e2πi/k . Such configurations form a submanifold of Mk , parametrized by the single modulus z0 ; by the rotational symmetry of the theory this submanifold is totally geodesic. This 2-dimensional moduli space is the same orbifold R2 /Zk , since z0 is identified with ωz0 . Geometrically, however, its K¨ahler structure is deformed, with the conical singularity at the orbifold fixed point smoothed out, as described in Subsect. 3.3. This is an example of the stringy behavior of noncommutative solitons: as non-local probes, they see a singular geometry in a non-singular way. The attractive potential of order λ2 /(θ m2 ) between the soliton and its images, described in Sect. 4, will draw it toward the fixed point.

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√ 5.2. Cylinder. A soliton on a cylinder of circumference θl√can be represented as an infinite array of solitons on the plane, located at zj = z0 +ij l/ 2, j ∈ Z, where z0 is√the modulus. Since the moduli space is quotiented under the identification z0 ∼ z0 + il/ 2, it has the same geometry as the underlying cylinder. Furthermore, this moduli space is exact by the translational symmetry of the theory— the attraction between the soliton and its images on one side balance the attraction of those on the other side. In finding the explicit field configuration φ(w, w) ¯ of this soliton, Eq. (3.9) is not of much use, since it requires inverting an infinite dimensional matrix. We will therefore present an alternative construction. For simplicity we will assume in √ what follows a a that z0 = 0. It will √ be convenient to use real coordinates y ≡ x / θ (recall that w = (x 1 + ix 2 )/ 2), and the corresponding hermitian operators yˆ a . The discrete trans1 lations by which the plane is quotiented are generated by the unitary operator U ≡ eil yˆ . We thus wish to construct the projection operator whose image is spanned by the set {U j |0 : j ∈ Z} of (normalized) coherent states. To do this, we will find an orthonormal basis for this hyperplane of the form {U j |ψ : j ∈ Z}, where

ψ|U j |ψ = δj 0 .

(5.1)

This will allow us to decompose P as a sum of orthogonal projection operators: P =



U j |ψ ψ|U −j .

(5.2)

j

The inverse Weyl-Moyal transformation then yields     2πλ −2πijy 2 / l ∗ 1 πj πj 1 φ(y , y ) = ψ y + , e ψ y − l l l 1

2

(5.3)

j

where ψ(y 1 ) ≡ y 1 |ψ is the wave function of |ψ in the basis of yˆ 1 eigenstates. We will find the vector |ψ by finding its coefficients in the U j |0 basis, or more precisely, the Fourier transform of those coefficients: |ψ =



cj U j |0 = c(l ˜ yˆ 1 )|0.

(5.4)

j

Working in terms of the wave functions, with ψ0 (y 1 ) ≡ y 1 |0 = π −1/4 e−(y

1 )2 /2

(5.5)

,

we impose the orthonormality condition (5.1),
δ

j 0



=

1

dy 1 eij ly |c(ly ˜ 1 )|2 ψ0 (y 1 )2


2π/ l

= 0



1

dy 1 eij ly |c(ly ˜ 1 )|2

j

 ψ0 y 1 +

2πj l

2 .

(5.6)

On Noncommutative Multi-Solitons

We can now solve for c˜ (choosing it to be real) and thus for ψ:  l 1  ψ0 (y 1 ) ψ(y ) = 2π j |ψ0 (y 1 + 2πj/ l)|2  l , = 2πϑ00 (2iy 1 / l, τ ) where τ ≡ 4πi/ l 2 .7 Plugging this into (5.3), we find   ϑ10 (2y 2 / l, τ ) ϑ00 (2y 2 / l, τ ) + . φ(y 1 , y 2 ) = λ ϑ00 (2iy 1 / l, τ ) ϑ10 (2iy 1 / l, τ )

369

(5.7)

(5.8)

This is plotted in Fig. 5 for various values of l. One can show by a modular transformation that in the decompactification limit l → ∞, (5.8) goes over to the single soliton on the plane. 5.3. T 2 . Just as in the case of the cylinder, a soliton on the torus can be represented √ as a lattice of solitons on the plane. With τ the modular parameter of the torus and θl its circumference in the x 1 direction, place the solitons in the z-plane at zj1 j2 = (j1 + √ j2 τ )l/ 2, j1 , j2 ∈ Z. Again, the forces between the soliton and its images will balance out, and the moduli space (which is the torus itself) is exact. In principle there is no problem defining the projection operator whose image is the infinite dimensional hyperplane spanned by {|zj1 j2  : j1 , j2 ∈ Z}, for any size and shape of torus. In practice, however, we have only been able to find an explicit expression for the field configuration corresponding to such a projection operator in the special case where the parameter A ≡ τ2 l 2 /2π (which is the area of the torus in units of 2π θ) is an 2 integer. A major simplification occurs in this case because then the operators U1 ≡ e−il yˆ 1 2 and U2 ≡ eil(τ2 yˆ −τ1 yˆ ) generating the two lattice translations commute, and the same method as that used for the cylinder above can be employed. The details of the derivation are presented in Appendix B; the final result is:      

  ∗ n+1/2 n+1/2  ∗

n n ϑ ν − ϑ ν + 00 00 A A λ  ϑ00 ν − A ϑ00 ν + A φ(w, w) ¯ =  + 

  2    n 2 2   n+1/2 ϑ00 ν + A ϑ00 ν + A      

  ∗ n+1/2 n+1/2  ∗

n n ϑ10 ν + A ϑ10 ν − A ϑ10 ν − A ϑ10 ν + A  + + , 

  2   2  n    n+1/2 ϑ10 ν + A ϑ10 ν + A  (5.9) √ √ where ν ≡ 2w/( θ l), all theta functions take τ/A as their second argument, and all sums run from n = 0 to A − 1. It is not obvious how to generalize such a formula to the 7 In fact, this construction is a very general one, applicable to constructing an arbitrary soliton on the cylinder at θ = ∞. In other words we can construct arbitrary projection operators of the form (5.2) which respect the discrete translation symmetry. All one needs to do is replace |0 in (5.4) and (5.5) by ˜ The solution for the coefficients c(l an arbitrary ket |ψ. ˜ yˆ 1 ) is then as in the first line of (5.7). Examples of such general solitons on the cylinder have also been constructed by V. Balasubramaniam and A. Naqvi (unpublished).

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case where A is not an integer. The shape of this soliton for the case A = 2 is plotted in Fig. 6. Consider now the alternative case where 1/A is an integer. Here the Moyal star product is in fact equivalent to ordinary pointwise multiplication of functions, and so

2 1 0

2 -2

1 0

0 2

-1

2

1 -2

0 1 0.5

0 2

0 -0.5

-2 0 2 2

1

0 0.5 0.25 0 -0.25 Fig. 5. The cylinder soliton (5.8) at various circumferences: l = 3 (top), l = 2 (middle), and l = 1 (bottom). One and a half copies of the soliton are displayed. The axes are as in Fig. 2

On Noncommutative Multi-Solitons

371

2 1 2

0 0 0 2 6 4 2 0

-2 22 1 0 0 1 2 Fig. 6. The soliton (5.9) on the torus of area 4πθ (A = 2) for two values of the modular parameter: the square torus τ = i (top), and τ = 3i + 1/2 (bottom). Four copies of the torus are displayed. The axes are as in Fig. 2

a small puzzle arises: No non-trivial continuous solutions to the equation of motion φ  φ = λφ are possible, yet one can certainly define the projection operator P whose image is the hyperplane spanned by {|zj1 j2  : j1 , j2 ∈ Z}. It is not hard to guess the resolution: this lattice of coherent states actually spans the entire Hilbert space, so P is just the identity, and the “soliton” is the constant solution φ = λ. Indeed, the formula (5.9) shows explicitly that this is the case when A = 1. But we can say even more: it is known [32–34] that any lattice of coherent states with A ≤ 1 is a complete (actually, overcomplete) basis for the Hilbert space.8 We thus conclude that there are no stable scalar solitons on any torus—rational or irrational—of area less than or equal to 2π θ. 8 The case A = 1 is known as a von Neumann lattice, since it was first discussed by him in [42], where the claim was made (without proof) that such a lattice forms a complete basis. An amusing fact about von Neumann lattices is that they are overcomplete by exactly one state, that is, there is exactly one linear relation among the infinitely many states. Lattices with A < 1 are overcomplete by infinitely many states.

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6. Solitons in Higher Dimensions The analysis of this paper easily generalizes to 2d + 1 dimensional scalar field theory with spacelike noncommutativity, although we will see that the topological structure of the moduli space Mdk in higher dimensions is much richer than in d = 1. We choose complex coordinates wr , r = 1, . . . , d, in which the noncommutativity parameter takes the form θ r s¯ = −iδrs θs . The Weyl-Moyal correspondence maps the field φ to an operator φˆ on the Hilbert space Hd ≡ H ⊗ · · · ⊗ H of a particle in d √ dimensions, with wr → θr ar and ∂r → − √1θ [ar† , · ]. The moduli space at infinite9 r θ again consists of operators of the form φˆ = λP for any projection operator P of a given rank k. The Bogomolnyi bound (3.4) on the leading contribution from the kinetic energy takes the form E1 [φˆ 0 ] = (2π)d λ2 θ = (2π)d λ2 θ

d 1 Tr[ar , P ][P , ar† ] θr r=1 d

 1  Tr P + 2Fr (P )† Fr (P ) θr

r=1 d

≥ (2π)d λ2 θ k

r=1

1 , θr

(6.1)

where θ ≡ θ1 · · · θd and Fr (P ) ≡ (1 − P )ar P . The moduli space Mdk at finite θ is therefore the space of projection operators on Hd whose image is an invariant subspace of all of the ar . A large class of such operators may be constructed by letting P project onto the †  Such a P corresponds space spanned by k independent coherent states |zi  ≡ ezi · a |0. to a multi-centered gaussian with peaks at zi . Naively, the moduli space of such solitons seems to be Symk (Cd ), as we saw is indeed the case in d = 1. However, in dimension d > 1 coincident solitons have more moduli than are present in the symmetric product, and we will see that the full moduli space Mdk is the so-called Hilbert scheme Hilbk (Cd ) of k points in Cd . Before introducing the general machinery of Hilbert schemes, we turn in the next subsection to the relatively simple case of d = 2 in order to gain some insight into the geometry of the moduli space when higher dimensional solitons come together. 6.1. 4 + 1 dimensions. Consider first the case k = 2, with two separated solitons described by the projection operator onto the space spanned by |z1  and |z2 . When the two solitons come together we have   z1 − z2 lim span{|z1 , |z2 } = span |z, γ · a † |z , where γ ≡ lim . (6.2) z1 − z2 | zi →z zi →z | Thus the “origin” of the relative moduli space is not a single point, but rather a P1 parametrized by the complex direction γ along which the two solitons came together. This 9 By this we mean the limit in which all of the θ are taken to infinity with their ratios held fixed, while r the energy is rescaled by a factor of m2 θ1 · · · θd , generalizing (2.3).

On Noncommutative Multi-Solitons

373

is in contrast to the d = 1 case studied in Sect. 3, where we found that two solitons brought together along any direction end up at the unique point span {|z, a † |z}. The physical significance of the sphere hiding at the coincidence locus is made clear by applying the Weyl-Moyal correspondence to the solution φˆ γ = λP (γ ), where P (γ )  γ · a † |0}.  This yields the function is the projection operator onto span {|0, √ √ |w1 γ1 / θ1 + w2 γ2 / θ2 |2 2 2 φγ (w)  = 16λe−2|w1 | /θ1 −2|w2 | /θ2 , (6.3) |γ |2 from which we see that the modulus γ encodes information about the shape of the level two lump. In other words, whereas two coincident solitons in d = 1 have (other than the overall translational mode) only a modulus corresponding to separating the two solitons, in d = 2 there is in addition a modulus corresponding to deforming the lump. Factoring out the C2 center of mass, we can parametrize the relative moduli space  by letting with coordinates z ∈ C and γ ∈ C2 \ {0} im P (z, γ ) = span{|zγ , |−zγ }.

(6.4)

These coordinates are subject to the identification (z, γ ) ∼ (z/λ, λγ ) for λ ∈ C \ {0}, which we recognize as defining the complex line bundle O(−1) over P1 . The base P1 sits at z = 0, which can be seen from limz→0 im P (z, γ ) = P (γ ) as defined above. Plugging (6.4) into (2.9) gives the K¨ahler potential on the moduli space  

2 sinh 2|z|2 |γ |2  K = ln , (6.5) |z|2 where the factor in the denominator arises as in (3.13) from the Jacobian of the transformation to a basis which remains nondegenerate as z → 0. Note that the K¨ahler potential (6.5) induces the familiar Fubini-Study metric on the P1 at z = 0. We conclude that two solitons in 4 + 1 dimensions resolve the singular configuration space Sym2 (C2 ) into a smooth complex manifold with topology C2 × OP1 (−1) and a smooth K¨ahler metric given by (6.5). In particular, the P1 at the origin has finite area (2π )3 λ2 θ1 θ2 . For k > 2 solitons in 4 + 1 dimensions the topology of the moduli space is more complicated. For example, for k = 3 the relative moduli space has, in addition to the  × OP1 (−1) when any two solitons come together, a P1 nontrivally fibered (C2 \ {0}) 1 over P at the “origin” of the relative moduli space where all three solitons merge. These moduli again correspond to deformations of the shape of the soliton, in contrast to the situation in d = 1 where the only moduli (other than overall translations) of a collection of coincident solitons correspond to separating some of the solitons from each other. As we will see in the next subsection, M2k has the same topology as the moduli space of k U(1) instantons on noncommutative R4 [37], which is a smooth complex manifold for any k. However, two important differences with that case warrant mention. Firstly, the metric on the scalar soliton moduli space M2k is only K¨ahler and not hyperk¨ahler. Secondly, nothing in our analysis requires that the noncommutativity parameter θ in 4 dimensions be self-dual. Indeed, the precise values of θ1 and θ2 scale out of the problem—their only role is to set the scale of the physical coordinates wr with respect to the dimensionless moduli space coordinates zi . The moduli space thus possesses an accidental SU(2) symmetry which is not possessed by the full theory when θ1 = θ2 .

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R. Gopakumar, M. Headrick, M. Spradlin

6.2. Hilbert schemes of points. In this subsection we unify the discussion of the moduli space of k solitons in 2d + 1 dimensions by showing that Mdk is isomorphic to a mathematical object known as the Hilbert scheme Hilbk (Cd ) of k points in Cd . Hilbk (Cd ) is defined as the set of ideals I of codimension k in the polynomial ring C[x1 , . . . , xd ]. The correspondence between projection operators and ideals is intuitively clear: if f is a polynomial in some ideal I and g is any polynomial, then the polynomial f g is still in I. Therefore the polynomials in an ideal I may be thought of roughly as projection operators from all of C[x1 , . . . , xd ] into I. The precise correspondence we demonstrate is motivated by a nearly identical correspondence between projection operators on a Fock space and ideals in polynomial rings (see for example [37, 36]) which appears in the construction of noncommutative instantons on R4 . For any polynomial f ∈ C[x1 , . . . , xd ], we may construct the ket  ∈ Hd . |f  ≡ f (a1† , . . . , ad† )|0

(6.6)

The one-to-one correspondence between Hilbk (Cd ) and the moduli space Mdk goes as follows. For an ideal I ⊂ C[x1 , . . . , xd ], we let 1 − P be the projection operator onto (the closure of) the linear subspace spanned by {|f  : f ∈ I}. Conversely, given P ∈ Mdk , we may recover I = {f ∈ C[x1 , . . . , xd ] : P |f  = 0}. This I is indeed an ideal by virtue of the fact that P ar† = P ar† P . For example, if P projects onto the subspace spanned by k independent coherent states |zi , then the corresponding ideal is simply the set of all polynomials which vanish at the k points zi . If n of the points zi come together at z, then the ideal becomes the set of polynomials with an order n zero at z. For d = 1 we have simply Hilbk (C) ∼ = Ck , in agreement with the result of Sect. 3. For k = 2 but arbitrary d, a trivial generalization of the argument in the previous subsection shows that the origin of the relative moduli space is not just a point but a whole Pd−1 . The total space Md2 in this case is the center of mass Cd times the complex line bundle O(−1) over Pd−1 . For d = 2 but arbitrary k, the Hilbert scheme Hilbk (C2 ) is a smooth manifold of complicated topology which arises as the moduli space of k U(1) instantons on noncommutative R4 [35–38]. For k > 3 and d > 2, however, the Hilbert scheme is not smooth, and in fact it is not even a manifold, having in general several different branches of varying dimension. We present an example in the next subsection.

6.3. An exotic example. It is not difficult to construct exotic branches of the moduli space Mdk = Hilbk (Cd ) for sufficiently large k and d. Let us consider k = 12 solitons in d = 8.10 If the moduli space M812 only contained information about the location of 12 solitons free to move about in C8 , then we would expect it to be 96-dimensional. However, we will now construct a 99-dimensional submanifold of M812 which opens up when all of the solitons coincide. We work in a harmonic oscillator basis H8 = span{|n1 , . . . , n 8  : nr ≥ 0}. Let us denote by |α, α = 1, . . . , 36, the 36 basis vectors which have nr = 2. Then for any three linearly independent 36-component vectors w1α , w2α and w3α , we can define the projection operator P by 10 A nearly identical construction works in many other cases—in d = 3, for example, with as few as k = 97 solitons.

On Noncommutative Multi-Solitons

375

 |1, . . . , 0, · · · , |0, . . . , 1, w1α |α, w2α |α, w3α |α}, im P (w1 , w2 , w3 ) = span{|0, (6.7) with summation over the α indices implied. This operator projects onto a 12-dimensional subspace of H8 which is invariant under all of the ar . Therefore P ∈ M812 for any three vectors w. These vectors represent the choice of a three dimensional subspace of the 36-dimensional space spanned by |α, and hence they parametrize the Grassmannian Gr(3, C36 ). Now Gr(3, C36 ) is 99-dimensional, so (6.7) gives a 99 parameter family of projection operators inside of M812 . The branch parametrized by projection operators of the form (6.7) cannot be smoothly connected to the 96-dimensional branch where all of the solitons are separated. The analysis of the global structure of the Hilbk (Cd ) for general k > 3 and d > 2, including details about how the various branches connect to each other, is a problem that mathematicians have only begun to tackle. Acknowledgements. It is a pleasure to thank R. Glauber, T. Graber, D. Gross, K. Hori, S. Katz, D. Khosla, A. Mikhailov, S. Minwalla, A. Naqvi, N. Nekrasov, K. Oh, B. Pioline, S. Sethi, A. Strominger, L. Thorlacius, C. Vafa, A. Volovich, M. Wijnholt, and S. Zhukov for helpful conversations. We are especially grateful to S. Minwalla for comments on the manuscript. This work was supported in part by DOE grant DE-FG02-91ER40654 and the David and Lucile Packard Foundation. One of us (R.G.) would like to thank the I.T.P. for hospitality and acknowledge the support of NSF grant PHY99-07949.

Appendix A. Completeness and Smoothness of the Finite θ Moduli Space In this appendix we prove the various statements made in Sect. 3 about the topology and geometry of the moduli space Mk when n points za come together at a point z, with the other k − n points fixed. Note that throughout this appendix we will use indices a, b = 1, . . . , n while the indices i, j will continue to run from 1 to k, as in the body of the paper. The completeness of the moduli space follows trivially from the Hilbert scheme analysis of Sect. 6, but nevertheless it is useful to see explicitly how to construct a basis of states which is nondegenerate as two or more solitons come together. In particular, this is necessary to prove the assertion made above (3.13) that the Jacobian of the required change of basis is the Vandermonde determinant, and to verify that the metric obtained from (3.13) extends smoothly to the coincidence locus.

A.1. Some elementary machinery. The proofs are completely straightforward but require some rather heavy notation, which we now introduce. Define ua = za − z. We will make use of the n × n Vandermonde matrix V with entries Vab = ub−1 a ,

(A.1)

which has determinant  ≡ det V =

 a>b

(ua − ub ).

(A.2)

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R. Gopakumar, M. Headrick, M. Spradlin

We will also make use of the polynomials defined by  c1 c1  u  1 · · · un   .. , F{ca } (ua ) =  ... .   cn u · · · ucnn  1

(A.3)

where {ca } is a set of n ordered nonnegative integers. It is clear that F{ci } is a homogeneous polynomial in the ui which vanishes unless the ci are distinct. The lowest degree non-zero one is therefore just the Vandermonde determinant: F0,... ,n−1 (ua ) = .

(A.4)

Since the polynomial F{ca } vanishes if any two of the ua are equal, it must be divisible by ua − ub for all a and b, and hence by . Therefore Q{ca } (ua ) ≡ F{ca } (ua )/

(A.5)

is again a homogeneous polynomial. Furthermore, since both  and F{ca } are antisymmetric under the interchange of any two of the ua , Q{ca } must be a symmetric polynomial in the ua . Now introduce an auxiliary variable u and consider the determinant   1 ··· 1   1   n n  u1 · · · un   −u  (u + ua ) =  .. ub F0,... , (A.6)  .. ..  = b,... ,n (ua ), . . .   a=1 b=0  (−u)n un · · · un  n 1 where the notation  b means that the index b is to be omitted. Upon dividing both sides of (A.6) by  we learn that the Q{ca } of degree ≤ n, (u ), σb (ua ) ≡ Q0,... ,
n−b,... ,n a

0 ≤ b ≤ n,

(A.7)

are precisely the elementary symmetric polynomials in the ua : σ0 = 1, σ1 = ua , a

σ2 =



ua ub ,

a
.. . σn =

(A.8) n 

ua .

a=1

The σa are good coordinates on Symk (C) in a neighborhood of ua = 0 and are therefore the coordinates of interest as we take n points za to z keeping the other k − n points fixed. Note that each Q{ca } of degree greater than n can be expressed as a polynomial in the σa .

On Noncommutative Multi-Solitons

377

The inverse of the Vandermonde matrix is (V −1 )ab =

(−1)a+b a , . . . , un ). F0,... , b,... ,n (u1 , . . . , u 

(A.9)

Finally, we will use the identity Q0,... ,b,... ˆ ,n−1,p (ua ) = =

n (−1)n p (−1)a ua F0,... , a , . . . , un ) b,... ,n−1 (u1 , . . . , u 

a=1 n p (−1)n+b ua (V −1 )ab a=1

(A.10)

which is obtained by expanding out the determinant defined by the left-hand side in the p last row (the row containing ua ).

A.2. Completeness. We start with the basis {|zi }. Expanding out the exponential in † |za  = eua a |z gives |za  =

n

|z; bVab +

∞

p−1

|z; pua

,

(A.11)

p=n+1

b=1

where |z; m ≡

(a † )m−1 |z. (m − 1)!

(A.12)

Multiplying (A.11) by V −1 gives |za (V −1 )ab = |z; b +

∞

|z; p

p=n+1

= |z; b + (−1)n+b

n

p−1

ua

(V −1 )ab

a=1 ∞

|z; pQ0,... , b,... ,n−1,p−1 (ua ),

(A.13)

p=n+1

using (A.10). The polynomial multiplying |p in (A.13) has degree greater than zero for any p > n, so that when we take all the ua to zero these terms vanish and we have simply lim |za (V −1 )ab = |z; b.

za →z

(A.14)

This establishes both (3.8) and the fact that the Jacobian of the necessary change of basis is the Vandermonde determinant, as advertised above (3.13).

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A.3. Smoothness. Let yi = zi − c be the relative coordinates on Mk as in Sect. 3.3. Repeated application of the exponential expansion yields the formula det(ey¯i yj ) =

1 k!

∞ n1 ,... ,nk =0

  1 Fn ,... ,n (y1 , . . . , yk )2 . 1 k n1 ! · · · nk !

Since each F is divisible by the Vandermonde determinant  = (A.5), the K¨ahler potential (3.13) is simply given by 



i>j (yi

1 k!

− yj ) as in

∞

  1 Qn ,... ,n (y1 , . . . , yk )2 1 k n ! · · · nk ! n1 ,... ,nk =0 1   k 1 ck (0) + = ck (i) |σi (y)|2 + O(σi3 ) , k!

eK =

(A.15)

(A.16)

i=1

where the ck (i) are some positive numbers. It is manifest from the second line of (A.16) that K  is well behaved as the solitons are brought together, i.e. as σi (y) → 0. We see directly from this derivation how dividing eK by the Vandermonde determinant renders the K¨ahler potential K  nonsingular on the coincident locus. Appendix B. Construction of the Soliton on the Integral Torus √ We consider a torus with periodicities l and τ l (in units of θ ), where A ≡ τ2 l 2 /2π is assumed to be an integer. The generators of the fundamental group are represented by the unitary operators U1 ≡ e−il yˆ , 2

U2 ≡ eil(τ2 yˆ

1 −τ yˆ 2 ) 1

,

(B.1)

which commute for such an integral torus. We wish to construct the projection operator whose image is spanned by the lattice of coherent states, j

j

U1 1 U2 2 |0,

(j1 , j2 ) ∈ Z2 .

(B.2)

Following the same strategy as on the cylinder, we will find a particular linear combination, j j |ψ = cj1 j2 U1 1 U2 2 |0, (B.3) j1 ,j2

that satisfies ! j j ψ|U1 1 U2 2 |ψ = δj1 0 δj2 0 . The projection operator we seek is then j j −j −j U1 1 U2 2 |ψ ψ|U2 2 U1 1 . P = j1 ,j2

(B.4)

(B.5)

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We employ a generalization of the so-called kq representation [34, 43], which provides a basis of simultaneous eigenstates of U1 and U2 : " |kq ≡

l −iτ1 (yˆ 2 )2 /2τ2 ij lk e |q + j l, e 2π

(B.6)

j

where the ket on the right is a yˆ 1 eigenstate. We thus have U1 |kq = e−ilk |kq,

U2 |kq = eilτ2 q |kq.

(B.7)

The set {|kq : 0 ≤ k < 2π/ l, 0 ≤ q < l} forms an orthonormal and complete basis for the Hilbert space. In terms of wave functions in the kq representation, (B.3) becomes Cψ (k, q) ≡ kq|ψ =



cj1 j2 e−ij1 lk+ij2 lτ2 q kq|0 = c(k, ˜ q)C0 (k, q),

(B.8)

j1 ,j2

where C0 (k, q) ≡ kq|0 =

    q + kτ/τ2 τ τ 2 , . (B.9) k + ikq ϑ exp − √ 00 2iτ2 l A π 1/4 l 1

Note that c˜ is doubly periodic: c(k ˜ + 2π/ l, q) = c(k, ˜ q + l/A) = c(k, ˜ q). The orthonormality condition (B.4) becomes
δj1 0 δj2 0 =




2π/ l

l

dk


0

0




2π/ l

=

l/A

dk 0

 2 dq e−ij1 lk+ij2 lτ2 q Cψ (k, q) dq e

−ij1 lk+ij2 lτ2 q

0

|c(k, ˜ q)|

2

 A−1 n=0

 2   C0 k, q + n l  .  A  (B.10)

Hence (choosing c˜ real), Cψ (k, q) = #

C0 (k, q) . A−1 2π/A n=0 |C0 (k, q + nl/A)|2

(B.11)

With |ψ now in hand, we would like to find the field configuration corresponding to the projection operator (B.5). The inverse Weyl-Moyal transformation yields a Fourier expansion on the torus: %  $ λ 2πi j 2 − τ 1 j1 2 1 |ψ j1 yˆ +

ψ| exp yˆ φ(y1 , y2 ) = A l τ2 j1 ,j2 $ %  2πi j 2 − τ 1 j1 2 1 × exp − . j1 y + y l τ2

(B.12)

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To find the Fourier coefficients in terms of the kq wave function of |ψ, we need the following result, which may be derived from (B.6): %  $ 2πi j 2 − τ 1 j1 2

k  q  | exp |kq j1 yˆ 1 + yˆ l τ2   πij1 j2 2πij1 q  = + + ij lk exp l A j     j2 l 2πj    ×δ q − q − + jl . (B.13) δ k−k − A l  j

One can then show that φ(y 1 , y 2 )

     2A−1 π λ −2πij2 y 2 /(τ2 l) ∗ 2 1 τ1 2 j2 l τ1 j2 l Cψ y 2 , y 1 − y 2 + Cψ y , y − y − e A τ2 2A τ2 2A j2 =0     π τ1 j2 l π τ1 j2 l +Cψ∗ y 2 + , y 1 − y 2 − Cψ y 2 + , y 1 − y 2 + . (B.14) l τ2 2A l τ2 2A

=

Combining this formula with (B.11) and (B.9), one finally arrives at the expression (5.9). References 1. Witten, E.: Noncommutative Geometry And String Field Theory. Nucl. Phys. B268, 253 (1986) 2. Seiberg, N., Witten, E.: String Theory and Noncommutative Geometry. JHEP 09, 032 (1999), [hep-th/9908142] 3. Witten, E.: Noncommutative Tachyons and String Field Theory. [hep-th/0006071] 4. Schnabl, M.: String field theory at large B-field and noncommutative geometry. JHEP 11, 031 (2000), [hep-th/0010034] 5. Gopakumar, R., Minwalla, S., Strominger, A.: Noncommutative solitons. JHEP 05, 020 (2000), [hep-th/0003160] 6. Dasgupta, K., Mukhi, S., Rajesh, G.: Noncommutative tachyons. JHEP 06, 022 (2000), [hep-th/0005006] 7. Harvey, J.A., Kraus, P., Larsen, F., Martinec, E.J.: D-branes and Strings as Non-commutative Solitons. JHEP 07, 042 (2000), [hep-th/0005031] 8. Sen, A.: Descent Relations Among Bosonic D-branes. Int. J. Mod. Phys. A14, 4061 (1999), [hep-th/9902105] 9. Sen, A.: Universality of the Tachyon Potential. JHEP 12, 027 (1999), [hep-th/9911116] 10. Polychronakos, A.P.: Flux tube solutions in noncommutative gauge theories. Phys. Lett. B495, 407 (2000), [hep-th/0007043] 11. Bak, D.: Exact Solutions of Multi-Vortices and False Vacuum Bubbles in Noncommutative AbelianHiggs Theories. Phys. Lett. B495, 251 (2000), [hep-th/0008204] 12. Aganagic, M., Gopakumar, R., Minwalla, S., Strominger, A.: Unstable solitons in noncommutative gauge theory. JHEP 04, 001 (2001), [hep-th/0009142] 13. Gross, D.J., Nekrasov, N.A.: Solitons in noncommutative gauge theory. JHEP 03, 044 (2001), [hep-th/0010090] 14. Jatkar, D.P., Mandal, G., Wadia, S.R.: Nielsen-Olesen Vortices in Noncommutative Abelian Higgs Model. JHEP 09, 018 (2000), [hep-th/0007078] 15. Bak, D., Lee, K.: Elongation of Moving Noncommutative Solitons. Phys. Lett. B495, 231 (2000), [hep-th/0007107] 16. Gorsky, A.S., Makeenko, Y.M., Selivanov, K.G.: On noncommutative vacua and noncommutative solitons. Phys. Lett. B492, 344 (2000), [hep-th/0007247] 17. Zhou, C.: Noncommutative scalar solitons at finite θ . [hep-th/0007255] 18. Solovyov, A.: On Noncommutative Solitons. Mod. Phys. Lett. A15, 2205 (2000), [hep-th/0008199]

On Noncommutative Multi-Solitons

381

19. Harvey, J.A., Kraus, P., Larsen, F.: Exact noncommutative solitons. JHEP 12, 024 (2000), [hep-th/0010060] 20. Durhuus, B., Jonsson, T., Nest, R.: Noncommutative scalar solitons: Existence and nonexistence. Phys. Lett. B500, 320 (2001), [hep-th/0011139] 21. Li, M.: Quantum Corrections to Noncommutative Solitons. [hep-th/0011170] 22. Kiem, Y., Kim, C., Kim, Y.: Noncommutative Q-balls. [hep-th/0102160] 23. Jackson, M.G.: The Stability of Noncommutative Scalar Solitons. [hep-th/0103217] 24. Lindstr¨om, U., Roˇcek, M., von Unge, R.: Non-commutative Soliton Scattering. JHEP 12, 004 (2000), [hep-th/0008108] 25. Manton, N.S.: A remark on the scattering of BPS monopoles. Phys. Lett. B110, 54 (1982) 26. Bars, I., Kajiura, H., Matsuo, Y., Takayanagi, T.: Tachyon Condensation on Noncommutative Torus. Phys. Rev. D63, 086001 (2001), [hep-th/0010101] 27. Sahraoui, E.M., Saidi, E.H.: Solitons on compact and noncompact spaces in large noncommutativity. [hep-th/0012259] 28. Martinec, E.J., Moore, G.: Noncommutative Solitons on Orbifolds. [hep-th/0101199] 29. Bak, D., Lee, K., Park, J.-H.: Noncommutative vortex solitons. [hep-th/0011099] 30. Lee, K., Tong, D., Yi, S.: The moduli space of two U(1) instantons on noncommutative R 4 and R 3 × S 1 . [hep-th/0008092] 31. Lee, K., Yi, P.: Quantum Spectrum of Instanton Solitons in five dimensional noncommutative U(N) theories. [hep-th/9911186] 32. Perelomov, A.M.: On the completeness of a system of coherent states. Teor. Mat. Fiz. 6, 213 (1971) 33. Bargmann, V., Butera, P., Girardello, L., Klauder, J.R.: On the completeness of the coherent states. Rep. on Math. Phys. 2, 221 (1971) 34. Bacry, H., Grossman, A., Zak, J.: Proof of completeness of lattice states in the kq representation. Phys. Rev. B12, 1118 (1975) 35. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. American Mathematical Society University Lecture Series, Providence, RI: AMS 1999 36. Furuuchi, K.: Instantons on Noncommutative R 4 and Projection Operators. Prog. Theor. Phys. 103, 1043 (2000), [hep-th/9912047] 37. Nekrasov, N.A.: Noncommutative instantons revisited. [hep-th/0010017] 38. Braden, H.W., Nekrasov, N.A.: Instantons, Hilbert Schemes and Integrability. [hep-th/0103204] 39. Talk by R. Gopakumar at Strings 2001, Mumbai, http://theory.tifr.res.in/strings/ Proceedings/gkumar/. 40. Lechtenfeld, O., Popov, A.D., Spendig, B.: Noncommutative solitons in open N = 2 string theory. [hep-th/0103196] 41. Hadasz, L., Lindstr¨om, U., Roˇcek, M., von Unge, R.: Noncommutative Multisolitons: Moduli Spaces, Quantization, Finite θ Effects, and Stability. [hep-th/0104017] 42. von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton, 1955 43. Zak, J.: In: Solid state Physics, Vol. 27 edited by Ehrenreich, H., Seitz, F., and Turnbull D. New York: Academic, 1972 Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 233, 383–402 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0731-2

Communications in

Mathematical Physics

Jack Superpolynomials, Superpartition Ordering and Determinantal Formulas Patrick Desrosiers1 , Luc Lapointe2 , Pierre Mathieu1 1

D´epartement de Physique, Universit´e Laval, Qu´ebec, Canada, G1K 7P4. E-mail: [email protected]; [email protected] 2 Department of Mathematics and Statistics, McGill University, Montr´eal, Qu´ebec Canada, H3A 2K6. E-mail: [email protected] Received: 11 May 2001 / Accepted: 16 August 2002 Published online: 7 November 2002 – © Springer-Verlag 2002

Abstract: We call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model. We obtain an ordering on superpartitions from the explicit action of the model’s Hamiltonian on monomial superfunctions. This allows to define Jack superpolynomials as the unique eigenfunctions of the model that decompose triangularly, with respect to this ordering, on the basis of monomial superfunctions. This further leads to a simple and explicit determinantal expression for the Jack superpolynomials. 1. Introduction Orthogonal polynomials that are eigenfunctions of Hamiltonians of physical interest certainly deserve consideration. Among this class of orthogonal polynomials are the eigenfunctions of integrable quantum many-body systems of the Calogero-Moser-Sutherland (CMS) type [1–4], such as the Jack polynomials [5, 6], the eigenfunctions of the trigonometric CMS model. In [7], we launched the study of a generalized version of Jack polynomials that are eigenfunctions of the supersymmetric extension of the trigonometric CMS model [8] (stCMS model). In addition to the bosonic variables xi (i = 1, · · · , N , where N is the number of particles), this model contains fermionic degrees of freedom described by the variables θi . Therefore, the generalized version of Jack polynomials incorporates Grassmannian (i.e., fermionic) variables. The usual Jack polynomials are not uniquely defined as being eigenfunctions of the trigonometric CMS model. An extra ingredient is required to characterize them: they need to decompose triangularly on the basis of monomial functions. Such a triangular decomposition entails an ordering. In this case, it is the standard dominance ordering [6, 9]. A similar situation holds for the superanalogues: in order to characterize precisely the Jack superpolynomials, it is necessary to require a triangular decomposition in terms of

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P. Desrosiers, L. Lapointe, P. Mathieu

some sort of monomial superfunctions. The first step in this case thus amounts to finding a fermionic extension of the monomial functions. Fortunately, there is a rather natural way of defining the symmetric monomial superfunctions once the general symmetry properties of the stCMS eigenfunctions are understood. These monomial superfunctions are indexed by superpartitions (see [7] and Sect. 2), which are, roughly, ordered pairs of partitions (with one of the partitions required to have distinct parts). The second step then reduces to the formulation of a proper ordering among superpartitions allowing to characterize the triangular decomposition of the Jack superpolynomials. Relying on the usual dominance ordering on standard partitions, it proved possible to find such an ordering [7]. Although this dominance ordering is technically sufficient to calculate the Jack superpolynomials, it does not provide the most precise characterization of the Jack superpolynomials. Indeed, it does not rule out monomials that do not actually appear in the decomposition of a given Jack superpolynomial. The aim of the present paper is to improve this situation by introducing a weaker ordering (i.e., in the sense that fewer elements are comparable) on superpartitions. The basic clue to the discovery of the ordering lies in the following observation. If a triangular decomposition exists for the Jack superpolynomials in terms of the symmetric monomial superfunctions, the action of the Hamiltonian itself on the basis of symmetric monomial superfunctions is likely to be triangular (and the converse is also true). The key point then is that there are usually fewer (and certainly never more) terms in the expansion of the action of the Hamiltonian on a given symmetric supermonomial (in terms of symmetric superpolynomials) than there are in the expansion of this superpolynomial (again in terms of symmetric supermonomials).1 Therefore, it is simpler to unravel the ordering at work by considering the action of the Hamiltonian on supermonomials. This is how the ordering presented here was found. The analysis of the action of the Hamiltonian on the supermonomial basis had an important offshoot: all the expansion coefficients could be calculated explicitly. This provided a rather nontrivial generalization of the results of [10], which are recovered here as a special case. As already stated, the triangular nature of the action of the Hamiltonian on the supermonomial basis implies a triangular decomposition of the Jack superpolynomials. This, combined with the fact that the superpartitions entering in the decomposition of a given Jack superpolynomial on the monomial superfunction basis are associated to distinct eigenvalues, readily implies that the Jack superpolynomials can be written as determinants. The entries of the determinants being the known eigenvalues and the known expansion coefficients of the action of the Hamiltonian on the supermonomial basis, one thus ends up with a remarkably simple and rather explicit expression for the Jack superpolynomials that generalizes the one obtained in [11] in the non-supersymmetric case. The article is organized as follows. In Sect. 2, we briefly review the stCMS model as well as some results of [7] that are required for the present article, namely superpartitions, the symmetric supermonomials and the Jack superpolynomials. In Sect. 3 we present our main result: the explicit action of the Hamiltonian on the supermonomial basis. The resulting monomials are characterized by partitions that can be obtained from () the original one via the action of a two-particle lowering operator Rij (the labels i and 1 In the terminology of this paper, the terms that can appear in the action of the Hamiltonian on the supermonomial basis are obtained from a single rearrangement. In the Jack superpolynomial decomposition, there are in addition terms associated to superpartitions generated from multiple rearrangements.

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385

j being those of the particles on which the operator acts - with “intensity” specified by ). The new ordering is also formulated directly in terms of this lowering operator. Section 4 is devoted to computing explicitly the action of the stCMS Hamiltonian on supermonomials and thereby proving the result stated in Sect. 3. In the last section, we reformulate the definition of the Jack superpolynomials in terms of this new ordering and present the determinantal formula. We would like to stress that all our results provide genuine extensions of non-supersymmetric ones, the latter being associated to the zero-fermion sector. In this sector, a superpartition reduces to a partition and a monomial superfunction reduces to an ordinary monomial symmetric function. In this regard, the detailed proof of the action of the stCMS Hamiltonian on the supermonomial basis includes an explicit proof of the main result of [10]. 2. The Supersymmetric CMS Model and Jack Superpolynomials 2.1. The Hamiltonian. The Hamiltonian of the stCMS model reads [8]: H=−

  † N 1
2  π 2
β(β − 1 + θij θij ) πβ 2 N (N 2 − 1) − ∂xi + , 2 L L 6 sin2 (π xij /L) i=1

(1)

i<j

where xij = xi − xj , θij = θi − θj and θij† = ∂θi − ∂θj . The θi ’s, with i = 1, · · · , N , are the anticommuting variables that, added to the standard variables xi , make the model supersymmetric. That is to say, for i, j ∈ {1, . . . , N}, we have [xi , xj ] = 0 ,

{θi , θj } = θi θj + θj θi = 0 ,

[xi , θj ] = 0 .

(2)

The supersymmetric invariance is enforced in the very construction of this Hamiltonian, i.e., in defining it as the anticommutator of two supersymmetric charges Q and Q† : H=

1 {Q, Q† } 2

with

Q=




∂θj (∂xj − ij (x)),

Q† =

j




θj (∂xj + ij (x)),

j

(3) where j (x) =

 x  πβ
jk . cot π L L

(4)

k=j

The supersymmetric charges are fixed by the requirement that they be nilpotent and that, in the absence of fermionic variables (that is, by setting θi = ∂θi = 0), the Hamiltonian reduce to the tCMS Hamiltonian. An observation that proves to be crucial in [7] is that the term encapsulating the dependency upon the fermionic variables is a fermionic-exchange operator [8]: κij ≡ 1 − θij θij† = 1 − (θi − θj )(∂θi − ∂θj ).

(5)

This means that its action on any monomial function f (θi , θj ) is such that κij f (θi , θj ) = f (θj , θi ) κij .

(6)

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This allowed the powerful exchange-operator formalism [12] to be used to study the properties of the model in [7]. Note that when we refer to exchange-operators (κij , Kij , Kij in this article) we understand operators that satisfy relations of the type: κij = κj i

,

κij κj k = κik κij = κj k κki

,

κij2 = 1.

(7)

It proves convenient to remove the contribution of the ground-state wave function, πx   jk sinβ ψ0 (x) = β (x) ≡ , (8) L j
from the Hamiltonian. The transformed Hamiltonian (which is still supersymmetric) becomes then   1 L 2 −β ¯ H≡  Hβ . (9) 2 π When written in terms of the new bosonic variables zj = e2πixj /L , it finally reads H¯ =



zij
zi zj (zi ∂i )2 + β (zi ∂i − zj ∂j ) − 2β (1 − κij ), 2 zij zij i

i<j

(10)

i<j

where ∂i = ∂zi and zij = zi + zj . The Jack superpolynomials will be eigenfunctions of this Hamiltonian. 2.2. Superpartitions and the monomial symmetric superfunctions. As explained in [7], since H¯ leaves invariant the space of polynomials of a given degree in z and a given degree in θ , we can look for eigenfunctions of the form:
A(m) (z, θ; β) = θi1 · · · θim A(i1 ...im ) (z; β) , m = 0, 1, 2, 3, . . . , 1≤i1
(11) where A(i1 ...im ) is a homogeneous polynomial in z. Due to the presence of m fermionic variables in its expansion, A(m) is said to belong to the m-fermion sector. The solutions A(m) , being symmetric superpolynomials, must be invariant under the combined action of κij and Kij , the exchange operator acting on the zi variables: Kij f (zi , zj ) = f (zj , zi , )Kij .

(12)

In other words, A(m) must be invariant under the action of Kij , where Kij ≡ κij Kij .

(13)

Given that the θ products are antisymmetric, the functions A(i1 ...im ) must satisfy Kij A(i1 ...im ) (z; β) = −A(i1 ...im ) (z; β) ∀ i and j ∈ {i1 . . . im } , Kij A(i1 ...im ) (z; β) = A(i1 ...im ) (z; β) ∀ i and j ∈ {i1 . . . im } ,

(14)

i.e., A(i1 ...im ) must be completely antisymmetric in the variables {zi1 , . . . , zim }, and totally symmetric in the remaining variables z/{zi1 , . . . , zim }.

Jack Superpolynomials, Superpartition Ordering, Determinantal Formulas

387

Superpartitions (to be denoted by capital Greek letters) provide the proper labelling of symmetric superpolynomials [7]. A superpartition in the m-fermion sector is a sequence of integers that generates two standard partitions separated by a semicolon:  = (1 , . . . , m ; m+1 , . . . , N ) = (λa ; λs ),

(15)

the first one being associated to an antisymmetric function, meaning that its parts are all distinct: λa = (1 , . . . , m ),

i > i+1 ≥ 0 ,

i = 1, . . . m − 1,

(16)

and the second one, to a symmetric function: λs = (m+1 , . . . , N ),

i ≥ i+1 ≥ 0 ,

i = m + 1, . . . , N − 1 ,

(17)

with equal parts then being allowed. In the zero-fermion sector, the semicolon is omitted and  reduces to λs . We often write the degree of a superpartition as n = || = N i=1 i . Note finally that to any superpartition , we associate a unique standard partition ∗ obtained by rearranging the parts of the superpartition in decreasing order. For instance, the rearrangement of  = (4, 2, 1; 5, 3, 3, 1) is the partition ∗ = (5, 4, 3, 3, 2, 1, 1). The monomial symmetric superpolynomials have been introduced in [7] as a natural basis of the ring of symmetric superfunctions (we will also refer to this basis as the supermonomial basis): m (z, θ ) = m(1 ,... ,m ;m+1 ,... ,N ) (z, θ ) =




θ σ (1,... ,m) zσ () ,

(18)

σ ∈SN

where the prime indicates that the summation is restricted to distinct terms, and where 







σ (m+1) zσ () = z1 σ (1) · · · zmσ (m) zm+1 · · · zN σ (N )

and

θ σ (1,... ,m) = θσ (1) · · · θσ (m) . (19)

Given a reduced decomposition σ = σi1 · · · σin of an element σ of the symmetric group SN , let Kσ stand for Ki1 ,i1 +1 · · · Kin ,in +1 . In this notation, we can rewrite the monomial superfunction m as m =

 1
Kσ θ1 · · · θm z , f

(20)

σ ∈SN

where the normalization constant f is f = fλs = nλs (0)! nλs (1)! nλs (2)! · · · , with nλs (i) the number of i’s in λs , the symmetric part of  = (λa ; λs ).

(21)

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2.3. Jack superpolynomials. The Jack superpolynomials can now be defined. Definition 1. The Jack superpolynomials are the unique functions satisfying the following two properties [7]: (i) : H¯ J (z, θ ; β) = ε J (z, θ ; β) ,
(ii) : J (z, θ ; β) = m (z, θ ) +

c,' (β)m' (z, θ) .

(22)

'; '∗ <∗

In the definition, the ordering on partitions is the usual dominance ordering, λ≥ω

iff λ1 + λ2 + · · · + λi ≥ ω1 + ω2 + · · · + ωi ,

∀i ,

(23)

and the eigenvalue ε , which turns out to be independent of the fermionic sector, is given by: 
2 ∗j + β(N + 1 − 2j )∗j , (24) ε = ε∗ = j

where ∗i ≡ (∗ )i . The uniqueness property, rather obviousfrom a computational point of view, is established in Sect. 5. Many explicit examples of Jack superpolynomials can be found in [7]. As stressed in [7], the mere existence and uniqueness of functions satisfying the two conditions of Definition 1 is remarkable. Later in this article, we will show that this definition does in fact provide a genuine characterization of the Jack superpolynomials. However, it was also observed that the dominance ordering of the partitions ∗ is not precise enough in the sense that some monomials m' with '∗ < ∗ do not appear in the sum. A partial ordering among superpartitions that fixes this problem will be presented in the following section. 3. Action on the Monomial Basis and Ordering of Superpartitions In order to motivate the dominance ordering presented at the end of this section, we first display the explicit action of H¯ on the monomial superpolynomials. This, in turn, will lead to a simple expression for the Jack superpolynomials in the form of determinants (cf. Sect. 5). As in the previous section, let  = (λa ; λs ) = (1 , . . . , m ; m+1 , . . . , N ) be a superpartition in the m-fermion sector. In order to characterize the monomials that are generated by the action of H¯ on m , () we introduce the operator2 Rij whose action, for i < j and  ≥ 0, is given by: ()

Rij (1 , . . . , i , . . . , j , . . . , N ) (1 , . . . , i − , . . . , j + , . . . , N ) if i > j , = (1 , . . . , i + , . . . , j − , . . . , N ) if j > i .

(25)

2 The non-supersymmetric version of this operator was introduced in [9] in the case  = 1 (more precisely, it is the raising version of the operator that is used), and in [11] in the general case.

Jack Superpolynomials, Superpartition Ordering, Determinantal Formulas

389

()

This action of Rij is non-zero only in the following cases: I: II : III :

 − −1

i, j ∈ {1, . . . , m} and  i 2 j  ≥  , i ∈ {1, . . . , m} , j ∈ {m + 1, . . . , N} and |i − j | − 1 ≥  ,  − i, j ∈ {m + 1, . . . , N} and  i 2 j  ≥  ,

(26)

where x stands for the largest integer smaller or equal to x. Otherwise, it is understood () () that Rij annihilates the superpartition , i.e., Rij  = ∅. In the following, we will say that a pair (i, j ) is of type I if i, j ∈ {1, . . . , m}, of type II if i ∈ {1, . . . , m} and j ∈ {m + 1, . . . , N}, and of type III if i, j ∈ {m + 1, . . . , N}. In the m-fermion sector, given a sequence γ = (γ1 , . . . , γm ; γm+1 , . . . , γN ), we will denote by γ the superpartition whose antisymmetric part is the rearrangement of (γ1 , . . . , γm ) and whose symmetric part is the rearrangement of (γm+1 , . . . , γN ). For example, we have (1, 3, 2; 2, 3, 1, 2) = (3, 2, 1; 3, 2, 2, 1).

(27)

Also, σγ will stand for the element of SN that sends γ to γ , that is σγ γ = γ . Note that we can always choose σγ such that σγ = σγa σγs , with σγa and σγs permutations of {1, . . . , m} and {m + 1, . . . , N} respectively. We now state the explicit action of the stCMS Hamiltonian on monomials. The proof will be presented in the following section. Theorem 2. H¯ acts on m (z, θ ) as follows:
H¯ m (z, θ ) = ε m (z, θ ) + v' (β) m' (z, θ) , (28) '= ()

where ε is given in (24) and v' (β) is non-zero only if ' = (ωa ; ωs ) = Rij , for a ()

given Rij with  > 0. In this case, the coefficient v' (β) reads:    2β sgn(σ a () ) i − j − δ n(i − , j + ) if i > j Rij   v' (β) = , (29)  2β sgn(σ a () ) j − i − δ n(i + , j − ) if j > i Rij 

where

sgn(σ a () ) Rij 

stands for the sign of the permutation σ a () . The parameter δ is Rij 

equal to 2, 1 or 0 if the pair (i, j ) is of type I, II or III in (26) respectively. Finally n(a, b) is a symmetry factor given by:  1 i, j of type I   n s (b) i, j of type II ω n(a, b) = n s (a)n s (b) (30) i, j of type III and a = b . ω   1ω s s 2 nω (a) (nω (a) − 1) i, j of type III and a = b Since in (29), 1 ≤  ≤ max{i , j }, the symmetry factor n(a, b) of the nondiagonal coefficients is never invoked with arguments a or b equal to zero. This has the important consequence that the coefficients v' (β) in (28) do not depend on the number N of variables, as long as N is large enough so that m' (z, θ ) = 0. Indeed, with ' = (ωa ; ωs ), the number of variables only influences the number of parts equal to zero in the partition ωs , which as we said does not influence the value of the symmetry factor n(a, b).

390

P. Desrosiers, L. Lapointe, P. Mathieu ( )

()

Remark 3. If ' = Rij  = Ri  j  , we must have  =  , i = i  and j = j  . The choice of i, j or i  , j  is thus irrelevant in the computation of v' (β). We now look at an example. For N ≥ 5, we have H¯ m(2,1;4,2) = ε(2,1;4,2) m(2,1;4,2) + 2β m(3,1;3,2) − 4β m(3,2;2,2) + 4β m(2,1;3,3) +8β m(2,1;3,2,1) + 24β m(2,1;2,2,2) + 4β m(2,1;4,1,1) .

(31)

The coefficient −4β in front of m(3,2;2,2) is explained first by identifying the proper rearrangement that links (3, 2; 2, 2) to (2, 1; 4, 2), namely (2)

(3, 2; 2, 2) = R2,3 (2, 1; 4, 2) = (2, 3; 2, 2)

(32)

(which is a type-II case) and then by evaluating the corresponding coefficient v,' from Theorem 2:  a 2β sgn(σ(2,3;2,2) ) 4 − 1 − 2 n(3, 2) = −4β , (33) since n(3, 2) = n(2,2) (2) = 2 (there are two 2’s in (2, 2) ) and the sign of the permutation that sends (2, 3) to (3, 2) is −1. Knowing the action of H¯ on the monomial basis allows us to define a partial ordering on superpartitions. ( )

( )

Definition 4. We say that  ≥s ' iff ' = Rik ,jk k . . . Ri1 ,j1 1 , for a given sequence of ( )

( )

operators Ri1 ,j1 1 , . . . , Rik ,jk k (the equality occurs in the case of the null sequence). 3 Proposition 5. The relation ≥s provides a partial ordering on superpartitions.

Proof. To show that ≥s is a genuine ordering, we have to verify the following three properties: (1) reflexivity:  ≥s ; (2) transitivity: if . ≥s ' and ' ≥s , then . ≥s ; (3) antisymmetry: if  ≥s ' and ' ≥s , then  = '. Reflexivity follows from the consideration of a null sequence of lowering operators. The transitivity property is also immediate since two sequences can be glued together to form a longer sequence. In order to prove antisymmetry, we first make the following two simple observations from (25): (i) :  ≥s ' ⇒ ∗ ≥ '∗ ,

(ii) :  ≥s '

and

∗ = '∗ ⇒  = ' , (34)

where as usual ∗ and '∗ are the partitions associated to the superpartitions  and ' respectively. Now, let  ≥s ' and ' ≥s . Then, from the first observation, ∗ ≥ '∗ and '∗ ≥ ∗ , i.e., ∗ = '∗ . Therefore, we have  ≥s ' and ∗ = '∗ , which lead to  = ' from the second observation.   3 The dominance ordering on partitions is a special case of this ordering on superpartitions. We draw () attention to the technical importance of formulating an ordering using the lowering operator Rij , especially in the case of the usual dominance ordering (as opposed to its formulation in terms of inequalities).

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391

Using this new partial ordering on superpartitions, we now have that the action (28) of H¯ on m (z, θ ) is triangular. Corollary 6. H¯ m (z, θ ) = ε m (z, θ ) +




v' (β) m' (z, θ ) .

(35)

'<s 

Notice that not all '’s such that ' <s  appear in this expansion with a non-zero coefficient. Only those that can be obtained from  by the application of a single lowering operator have non-zero coefficients. ¯ on the Monomial Basis: A Detailed Proof 4. The Action of H We will first focus on the non-diagonal coefficients in the action of H¯ on m (z, θ ), that is, on the coefficients v,' for  = ', and wait until the end of the section before computing the diagonal coefficient ε . We will show that the coefficients v,' are indeed given by the expression displayed in Theorem 2 for the special case N = 2, for the pairs (i, j ) of types I, II and III. And, in a second step, we will prove the theorem in general. () The reason why the two-particle case plays such a central role is that the operator Rij acts only on two parts of a partition, that is, on N = 2 particles. 4.1. Non-diagonal coefficients: The case N = 2. We will use the following notation: Bij =

zij (zi ∂i − zj ∂j ), zij

Fij = −2

zi zj (1 − κij ), 2 zij

(36)

(B for boson and F for fermion). The part of the Hamiltonian acting non-diagonally on the monomial basis is β(B + F ), where

Bij , F = Fij . (37) B= 1≤i<j ≤N

1≤i<j ≤N

In order to simplify the notation, we will set 1 = r, 2 = s. Case III. In this case, we must have m = 0. This gives  = (; r, s) = (r, s), and m(r,s) =

1 f(r,s)

(1 + K12 ) z1r z2s =

1 f(r,s)



z1r z2s + z1s z2r .

(38)

Clearly, the action of F12 vanishes since there are no θ terms. The action of B12 is simply  1 z12 (r − s)(z1 z2 )s z1r−s − z2r−s f(r,s) z12   z12 (r − s)(z1 z2 )s z12 z1r−s−1 + z1r−s−2 z2 + · · · + z1 z2r−s−2 + z2r−s−1 = z12   

= (r − s)(z1 z2 )s z1r−s + 2 z1r−s−1 z2 + · · · + z1 z2r−s−1 + z2r−s , (39)

B12 m(r,s) =

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P. Desrosiers, L. Lapointe, P. Mathieu

where we have set f(r,s) = 1, since the action vanishes in the case r = s. Therefore, by recombining to construct monomial symmetric functions, we get βB12 m(r,s) = βBm(r,s) = β(r − s)m(r,s) +

(r−s)/2


2β(r − s)m(r−,s+) .

(40)

=1 ()

()

If we let ' = (r − , s + ), we have ' = R12  = R12 . We thus recover the right value 2β(1 − 2 ) = 2β(r − s) for the coefficient v,' , for 1 ≤  ≤ (1 − 2 )/2 (see (29) with δ = 0). Case II. We now have m = 1. Consider first the action of B + F on m(r;s) , supposing that r > s. Given 1 (1 + K12 ) θ1 z1r z2s = θ1 z1r z2s + θ2 z1s z2r (41) m(r;s) = f(r;s) (since f(r;s) = 1), we get B12 m(r;s) =

 z12 (r − s)(z1 z2 )s θ1 z1r−s − θ2 z2r−s z12

(42)

and z 1 z2 (θ1 − θ2 )(z1r z2s − z1s z2r ) 2 z12  2 =− (θ1 − θ2 ) (z1 z2 )s z1r−s z2 + · · · + z1 z2r−s . z12

F12 m(r;s) = −2

(43)

In the sum B12 + F12 , we now concentrate on the term θ1 :     θ1 (z1 z2 )s (r − s)z12 z1r−s − 2 z1r−s z2 + · · · + z1 z2r−s . (B12 + F12 )m(r;s) θ = 1 z12 (44) To proceed further, we use the following identity proved in the appendix. Identity 7.   r−s r−s 12 r−s (r − s)z z1 − 2(z1 z2 + · · · + z1 z2 ) = z12 (r − s)z1r−s +

r−s−1


 2(r − s − )z1r−s− z2 . (45)

=1

Taking into account the θ2 terms (which are recovered by symmetry from the θ1 terms under z1 ↔ z2 ), we have: β(B12 + F12 )m(r;s) = β(B + F )m(r;s) = β(r − s)m(r;s) +

r−s−1


2β(r − s − )m(r−;s+) .

(46)

=1

We thus get the right coefficient, namely 2β(1 − 2 − δ), with δ = 1. Moreover, the upper limit on  shows that 1 − 2 > , as it should. The derivation in the case r < s is identical.

Jack Superpolynomials, Superpartition Ordering, Determinantal Formulas

393

Case I. Since we now have m = 2, we consider the action of B + F on m(r,s;0) . Given that, for r > s, m(r,s;0) =

1 f(r,s;0)

(1 + K12 ) θ1 θ2 z1r z2s = θ1 θ2 (z1r z2s − z1s z2r ) ,

(47)

z12 (r − s)(z1 z2 )s (z1r−s + z2r−s ) , z12

(48)

a direct computation yields B12 m(r,s;0) = θ1 θ2 and since (1 − κ12 )θ1 θ2 = 2θ1 θ2 , z1 z2 r s (z1 z2 − z1s z2r ) 2 z12 z1 z2 = −4θ1 θ2 2 (z1 z2 )s z12 (z1r−s−1 + z1r−s−2 z2 + · · · + z2r−s−1 ) z12 1 = −4θ1 θ2 (z1 z2 )s (z1r−s z2 + z1r−s−1 z22 + · · · + z1 z2r−s ) . (49) z12

F12 m(r,s;0) = −4θ1 θ2

Therefore, we obtain (B12 +F12 )m(r,s;0) =

θ 1 θ2 (z1 z2 )s {(r −s)z12 (z1r−s +z2r−s )−4(z1r−s z2 + · · · + z1 z2r−s )} . z12 (50)

To proceed, we need the following identity also proved in the appendix. Identity 8. 

 (r − s)z12 (z1r−s + z2r−s ) − 4(z1r−s z2 + · · · + z1 z2r−s )   (r−s−1)/2  
= z12 (r − s)(z1r−s − z2r−s ) + 2(r − s − 2)(z1r−s− z2 − z1 z2r−s− ) .   =1

(51) Using the identity, we get β(B12 + F12 )m(r,s;0) = β(B + F )m(r,s;0) = β(r − s)m(r,s;0) +

(r−s−1)/2
=1

2β(r − s − 2)m(r−,s+;0) . (52)

From this expression, the value of the coefficient v,' is given immediately. It is indeed 2β(1 − 2 − δ), with δ = 2. The upper limit in the summation implies that (1 − 2 − 1)/2 ≥ , as it should.

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4.2. Non-diagonal coefficients: The case N > 2. Let ij denote the restriction of a superpartition to its i th and j th entries, that is,   (i , j ; 0) if i, j is of type I (53) ij = (i ; j ) if i, j is of type II .  ( ,  ) if i, j is of type III i j Also, in accordance with (26), we define Sij , to be   − −1 i j   if i, j of type I  2  Sij = |i − j | − 1 if i, j of type II .   i −j if i, j of type III  2 

(54)

For an arbitrary number of particles, we have thus β(B + F ) m
 1
=β (Bij + Fij ) Kσ θ1 · · · θm z f 1≤i<j ≤N σ ∈SN  

β = Kσ  (Bij + Fij ) θ1 · · · θm z  f σ ∈SN

β = f




1 1 2 f

1 = f

=

1 f

1 = f 1 = f

 Kσ (Bij + Fij ) θ1 · · · θm z

1≤i<j ≤N σ ∈SN

1 β = 2 f =

1≤i<j ≤N










 Kσ (Bij + Fij )(1 + Kij ) θ1 · · · θm z

1≤i<j ≤N σ ∈SN




1≤i<j ≤N =0




Sij




v

Sij







Sij




1≤i<j ≤N =0


ij





() ij Rij  ij



()



ij



()

ij Rij 

f

()



Rij 

ij

ij

 Kσ 

1 f

sgn(σ a () ) Rij 

 Kσ θ1 · · · θm z

() Rij 

 

ij ()

Rij 



σ ∈SN


σ ∈SN

  () sgn(σ a () ) Kσ Kσ a () Kσ s () θ1 · · · θm zRij  Rij 

  () ij Rij 

f

()



Rij 

ij





v sgn(σ a () ) Rij 

()

(1 + Kij ) θ1 · · · θm z



Rij 

σ ∈SN






v

S

ij


1≤i<j ≤N =0




  () ij Rij 

Rij 

v

1≤i<j ≤N =0

v

f

1≤i<j ≤N =0






S

ij


()

ij Rij 

f



() Rij  ij

Rij 




ij

Rij 

 Kσ  θ1 · · · θm z

()

Rij 



σ  ∈SN ij

f

()

Rij 

m

()

Rij 

. (55)

Jack Superpolynomials, Superpartition Ordering, Determinantal Formulas

395

Let us be more specific about some of these steps. In the fourth equality, we have introduced a factor (1 + Kij )/2 in order to generate two terms out of θ1 · · · θm z because the action of Bij + Fij is defined on two terms (cf. Eqs.(38), (41) and (47)). We have thus generated an intermediate N = 2 problem in order to use the N = 2 results derived previously. The factor β is now reabsorbed in the coefficient v  ()  : ij Rij 

  () ij Rij 



v



= 2β |i − j | − δ ,

ij

 = 0 .

(56)

ij

Observe that the non-diagonal coefficients correspond to the cases where  > 0, but that the action of B + F also includes a diagonal contribution, to be analysed later. In the sixth equality, we reexpress the outcome in terms of a single term. However, the ordering of the variables may not be adequate for this to be a genuine supermonomial leading term. Some reordering may be required, forcing the appearance of extra factors Kσ a Kσ s in the next equality. Moreover, reordering the anticommuting variables may generate a sign given by the factor sgn(σ a ). In the following step, we redefine the summation variable as follows: σ  = σ σ a σ s . In the final equality, we simply use the definition of supermonomials. We are now in a position to compute the coefficient v' . Let #(', ) denote the ()

number of distinct ways we can choose (i, j ), i < j , such that ' = Rij . Note that such “symmetries” can only occur on the symmetric side, that is, if there is more than one possible choice of i (or j ), then i > m (or j > m). Therefore, sgn(σ a () ) is inRij 

dependent of the particular choice of the pair (i, j ) leading to ', since the underlying degeneracy is independent of the antisymmetric side. The coefficient of m' is thus v'

v  ()  ij Rij  1 ij = sgn(σ a () )  f' #(', ) .  R  f f () ij Rij 

(57)

ij

The factor #(', ) comes from the different values of the pair (i, j ) leading to the same superpartition '. From Remark 3, these distinct values all lead to the same coefficient v  ()  /f ()  . Since it has also just been mentioned that these different choices ij Rij 

ij

Rij 

ij

do not affect the value of sgn(σ a () ), v' is simply given by the coefficient of m Rij 

()

Rij 

in (55) (disregarding the sum) multiplied by #(', ). Let us now evaluate explicitly the various factors entering in the expression of v' , for the three cases treated separately. We first consider Case III, that is, the case where i, j ∈ {m + 1, . . . , N}. If i −  = j + , when going from  to ', the number of i and j decreases by 1, whereas the number of i −  and j +  increases by 1 (while the other k ’s remain unaffected). Therefore, the ratio f' /f depends only upon the multiplicity factors involving these parts, that is, nωs (i − )! nωs (j + )! nωs (i )! nωs (j )! f' = f nλs (i − )! nλs (j + )! nλs (i )! nλs (j )! nωs (i − )! nωs (j + )! nωs (i )! nωs (j )!    =  nωs (i − ) − 1 ! nωs (j + ) − 1 ! nωs (i ) + 1 ! nωs (j ) + 1 ! nωs (i − )nωs (j + )  . =  (58) nωs (i ) + 1 nωs (j ) + 1

396

P. Desrosiers, L. Lapointe, P. Mathieu

  Finally, since in that case #(', ) is equal to nωs (i ) + 1 nωs (j ) + 1 (the product of the number of i ’s and j ’s in the symmetric part of ), and since f ()  = 1, Rij 

we get v' = sgn(σ a () ) v Rij 

  () ij Rij 

nωs (i − )nωs (j + ) ,

ij

(59)

ij

which agrees with the expression given in Theorem 2. In the case where i − = j +, when going from  to ', the number of i and j decreases by 1, whereas the number of i −  increases by 2, that is,  nωs (i − ) nωs (i − ) − 1 f'  . =  (60) f nωs (i ) + 1 nωs (j ) + 1   The number #(', ) is still nωs (i )+1 nωs (j )+1 , but now f ()  = 2, giving Rij 

v' =

ij

 1 sgn(σ a () ) v  ()  nωs (i − ) nωs (i − ) − 1 ,  R  R  ij 2 ij ij ij

(61)

as announced. Let us now consider Case II with i > j . This case is similar to Case III with i −  = j + . The only difference is that since i now belongs to the antisymmetric sector, it does not influence the value of f' /f . Therefore, in this case, nωs (j + ) f' , = f nωs (j ) + 1

(62)

and v' = sgn(σ a () ) v Rij 

  () ij Rij 

nωs (j + ).

(63)

ij

The situation when i < j is similar. Finally, in Case I, i and j both belong to the antisymmetric sector, and thus f' = f . This gives a v' = sgn(σi,j, )v

  () ij Rij 

.

(64)

ij

4.3. The diagonal coefficients. To complete the study of the action of the Hamiltonian on the supermonomial basis, we now evaluate the diagonal coefficient explicitly. The full stCMS Hamiltonian reads H¯ = A + β(B + F ) ,

with

A=

N


(zi ∂i )2 .

(65)

i=1

The action of A on the supermonomial basis is diagonal:  N
A m = 2i m . i=1

(66)

Jack Superpolynomials, Superpartition Ordering, Determinantal Formulas

397

There is also a contribution coming from the part β(B + F ) of the Hamiltonian. It can be obtained from (55) for the case  = 0 with v

  (0) ij Rij 

= vij ij = β|i − j | .

ij

(67)

This value is independent of the sector – cf. Eqs. (40), (46) and (52). The diagonal contribution of β(B + F )m is thus v =

1 f

=β


1≤i<j ≤N


1≤i<j ≤N




vij ij f = β

|i − j |

1≤i<j ≤N

(∗i − ∗j ) = β

N
k=1

(N + 1 − 2k)∗k .

(68)

The step in which we eliminate the absolute value provides the technical reason why the eigenvalue is ultimately expressed in terms of the parts of ∗ . The last equality is obtained as follows [3]. Let L be such that ∗L = 0 but ∗j >L = 0. We can write


1≤i<j ≤N

(∗i − ∗j ) = =




1≤i≤L<j ≤N




(N

∗i +

− L)∗i




1≤i<j ≤L

+

1≤i≤L

=




(∗i − ∗j )




[(L − k) − (k − 1)]∗k

1≤k≤L

(N + 1 − 2k)∗k .

(69)

1≤k≤L

To evaluate the double sum, we simply count the number of occurrences of ∗k with positive and negative signs: it arises N − k times with a plus sign and k − 1 times with a minus sign. Note finally that the last sum can be extended from L to N without changing its value. Combining the diagonal contribution of β(B + F ) to that of A yields ε =

N
k=1

[2k + β(N + 1 − 2k)∗k ] =

N
k=1

[∗k 2 + β(N + 1 − 2k)∗k ] .

(70)

This is also the eigenvalue of the Jack superpolynomial J , which is thus computed here without the help of Dunkl operators. 5. Determinantal Expression for the Jack Superpolynomials 5.1. A new definition of the Jack superpolynomials. Given that we have obtained a partial ordering on superpartitions, it is natural to ask whether we can replace Definition 1 of the Jack superpolynomials by a stronger definition involving the weaker ordering on superpartitions. That is, we would like to know whether the Jack superpolynomials can be characterized by the two conditions:  (i) : J = m + ';'<s  c' m' , (71) (ii) : H¯ J = ε J .

398

P. Desrosiers, L. Lapointe, P. Mathieu

The interest of such a definition is that, given the triangular action of H¯ (see Corollary 6) and the fact that ε = ε' if ' <s , we can obtain, as we will see later, a determinantal expression for J . Therefore, there exist functions satisfying conditions (71). That we can indeed characterize the Jack superpolynomials using these conditions relies on the uniqueness of the Jack superpolynomials (cf. Definition 1), which we now establish. Lemma 9. There exists a unique function J satisfying the two conditions: (i) : (ii) :

 J = m + '; '∗ <∗ c' m' , H¯ J = ε J .

(72)

Proof. The existence of such a function will follow from the determinantal expression that we will present later on. The uniqueness is proved as follows. Let J and J¯ be two functions satisfying the conditions of the theorem. Then, if we suppose that J = J¯ , we have J − J¯ =




d' m' ,

(73)

'; '∗ <∗

for some coefficients d' . Now, let '(1)
n
i=1

d'(i) m'(i) =

n
i=1

 d'(i) ε'(i) m'(i) +




 v'(i) . m. = ε

.<s '(i)

n
i=1

d'(i) m'(i) . (74)

In the middle expression, the coefficient of m'(n) is simply equal to ε'(n) d'(n) , since '(n) dominates '(i) , i = 1, . . . , n − 1, in the order on superpartitions. But in the last expression of (74), the coefficient of m'(n) is equal to ε d'(n) . We must thus have ε d'(n) = ε'(n) d'(n) , with d'(n) = 0, that is we must have ε = ε'(n) . But this is  ∗ impossible because, from (73), ∗ > '(n) , which implies that ε = ε'(n) . Therefore, a contradiction arises and J cannot be different from J¯ .    It is easy to see, from (34)(i), that if a functionJ is such that J = m + '<s  c' m' , then it is also such that J = m + '; '∗ <∗ e' m' , for some coefficients e' . Therefore, the previous lemma (combined with the existence of functions satisfying conditions (71)) leads directly to the following theorem. Theorem 10. The Jack superpolynomials can be defined by the two conditions: (i) : (ii) :

 J = m + ';'<s  c' m' , H¯ J = ε J .

(75)

Jack Superpolynomials, Superpartition Ordering, Determinantal Formulas

399

5.2. Determinantal formulas. As mentioned earlier, an important offshoot of this new definition is that we can obtain determinantal expressions for the Jack superpolynomials. Indeed, because we are looking for Jack superpolynomials with a triangular expansion
J = m + c' m' , (76) '<s 

the fact that ε' = ε if ' <s  combined with the triangular action (35)
v' m' , H¯ m = ε m +

(77)

'<s 

leads to a determinantal expression. Theorem 11. If (1) , (2) , . . . , (n) =  is a total ordering (compatible with the ordering on superpartitions) of all superpartitions ≤s , then the Jack superpolynomial J is given by the following determinant:   m(1) m(2) ··· ··· m(n−1) m(n)     v(n) (1)   ε(1) − ε(n) v(2) (1) · · · · · · v(n−1) (1)  0 ε(2) − ε(n) · · · · · · v(n−1) (2) v(n) (2)     .. .. .. .. (78) J = c  , . . 0 . .     .. .. .. .. ..   . . . . .    0 0 · · · 0 ε (n−1) − ε (n) v (n) (n−1)  







where the constant of proportionality is c = (−1)

n−1

n−1  i=1

1 . ε(i) − ε(n)

(79)

For a proof that this determinant is in fact a non-zero eigenvector of H¯ with eigenvalue ε , the reader is referred to [11]. The proof found in this article can be applied to our case as well. Notice that in the determinantal expression, the relative ordering of the superpartitions not related by the partial ordering ≥s is irrelevant. It might be important to remark that, since the coefficients v' do not depend on the number of variables N , and since the differences 
 |'| = || , ('∗k 2 − ∗k 2 ) − 2 k β ('∗k − ∗k ) , ε' − ε = ε'∗ − ε∗ = k

(80) also do not depend on the number of variables, the determinantal expression for J does not depend on N . For example, with H¯ m(3;1) = (10 + 4 β N − 6 β) m(3;1) + 2 β m(2;2) + 8 β m(2;1,1) + 2 β m(1;2,1) , H¯ m(2;2) = (8 + 4 β N − 8 β) m(2;2) + 4 β m(2;1,1) + 2 β m(1;2,1) , H¯ m(2;1,1) = (6 + 4 β N − 10 β) m(2;1,1) + 6 β m(1;1,1,1) ,

H¯ m(1;2,1) = (6 + 4 β N − 10 β) m(1;2,1) + 12 β m(1;1,1,1) , H¯ m(1;1,1,1) = (4 + 4 β N − 16 β) m(1;1,1,1) ,

(81)

400

P. Desrosiers, L. Lapointe, P. Mathieu

we have

J(3;1)

   m(1;1,1,1) m(1;2,1) m(2;1,1) m(2;2) m(3;1)    6β 0 0   −6 − 10β 12β 1   0 −4 − 4β 0 2β 2β  =  3  64(3 + 5β)(1 + β) 0 0 −4 − 4β 4β 8β    0 0 0 −2 − 2β 2β  β β(2 + 3β) β(1 + 2β) m(2;1,1) + m(1;2,1) = m(3;1) + m(2;2) + 2 1+β (1 + β) 2(1 + β)2 3β 2 + m(1;1,1,1) . (82) (1 + β)2

6. Conclusion Although unravelling the natural generalization of the dominance ordering at the level of superpartitions was our original motivation for this work, we have obtained much more than that: we ended up with a simple determinantal expression for the Jack superpolynomials. In a sense, the mere discovery of a determinantal expression for Jack superpolynomials ensures that the pivotal concepts underlying their construction, namely monomial symmetric superpolynomials and superpartitions (as well as their associated ordering) are the right ones. There is a natural supersymmetric extension of the physical scalar product with respect to which the Jack polynomials are orthogonal. However, the Jack superpolynomials are not orthogonal with respect to this supersymmetric scalar product. The next step is thus to seek a general pattern for constructing linear combinations of Jack superpolynomials sharing the same Hamiltonian eigenvalue that would be orthogonal. Also, we believe the present formalism to be extendable to the case of the Hi-Jack polynomials (or generalized Hermite polynomials) and their supersymmetric counterparts. We hope to report elsewhere on this subject. Note Added. After this article was accepted for publication, we completed the program of constructing orthogonal Jack superpolynomials. These results will be presented in the article Jack polynomials in superspace. Acknowledgements. This work was supported by FCAR and NSERC. L.L. wishes to thank Luc Vinet for his financial support. P.D. would also like to thank the Fondation J.A Vincent for a student fellowship.

A. Proofs of Identity 7 and 8 Proof of Identity 7. Expanding the factors z12 and z12 , Identity 7 is equivalent to (r − s)



z1r−s+1

− z1r−s z2



+

r−s−1


2(r − s − )z1r−s−+1 z2

=1

−

r−s−1
=1

r−s  
2(r − s − )z1r−s− z2+1 = (r − s) z1r−s+1 + z1r−s z2 − 2 z1r−s−+1 z2 . =1

(83)

Jack Superpolynomials, Superpartition Ordering, Determinantal Formulas

401

Now, doing simple transformations on the indices of summation, we have   r−s−1
(r − s) z1r−s+1 − z1r−s z2 + 2(r − s − )z1r−s−+1 z2 =1

−

r−s−1


2(r − s − )z1r−s− z2+1

=1

  r−s−1
= (r − s) z1r−s+1 − z1r−s z2 + 2(r − s − )z1r−s−+1 z2 =1

−

r−s


2(r − s −  + 1)z1r−s−+1 z2

=2 r−s−1  
= (r −s) z1r−s+1 − z1r−s z2 +2(r − s − 1)z1r−s z2 − 2z1 z2r−s − 2 z1r−s−+1 z2 =2





= (r − s) z1r−s+1 + z1r−s z2 − 2

r−s


z1r−s−+1 z2 ,

(84)

=1

 

which proves (83) and, consequently, Identity 7.

Proof of Identity 8. Using Identity 7 twice (once with z1 ↔ z2 ), we have r−s
 (r − s)z12 z1r−s + z2r−s − 4 z1r−s−+1 z2 =1


 r−s−1 = z12 (r − s) z1r−s − z2r−s + 2(r − s − )z1r−s− z2 −

r−s−1


=1

2(r − s

!

− )z1 z2r−s−

.

(85)

=1

Finally, after a sequence of simple transformations, r−s−1


2(r − s − )z1r−s− z2 −

=1

= =

r−s−1
=1 r−s−1



=1

2(r − s − )z1 z2r−s−

=1

2(r − s − )z1r−s− z2 −

r−s−1


2  z1r−s− z2

=1

2(r − s − 2)z1r−s− z2

=1  r−s−1 2 

=

r−s−1


2(r − s − 2)z1r−s− z2 +

r−s−1
= r−s−1 2 +1

2(r − s − 2)z1r−s− z2

402

P. Desrosiers, L. Lapointe, P. Mathieu  r−s−1 2 

=




=1  r−s−1 2 

=




2(r − s

− 2)z1r−s− z2

r−s−1− r−s−1 2 

−




2(r − s − 2)z1 z2r−s−

=1

  2(r − s − 2) z1r−s− z2 − z1 z2r−s− ,

(86)

=1

we obtain Identity 8. Note that in the final step, when r − s is even, we have r − s − 1 − r−s−1 r−s−1  r−s−1 2  =  2  + 1. The extra case corresponding to  =  2  + 1 = (r − s)/2 can be ignored since it has a factor (r − s − 2) = 0.   References 1. Calogero, F.: Ground state of one-dimensional N body system. J. Math. Phys. 10, 2197 (1969); Solution of a three body problem in one-dimension. J. Math. Phys. 10, 2191 (1969); Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419 (1971) 2. Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Surv. in Appl. Math. 235–258 (1976) 3. Sutherland, B.: Quantum many body problem in one-dimension: Ground state. J. Math. Phys. 12, 246 (1971); Exact results for a quantum many body problem in one-dimension. Phys. Rev. A4, 2019 (1971); Exact results for a quantum many body problem in one-dimension. II. Phys. Rev. A5, 1372 (1972) 4. Olshanetsky, M.A., Perelomov, A.M.: Classical integrable finite dimensional systems related to Lie algebras. Phys. Rept. 71, 313 (1981); Quantum integrable systems related to Lie algebras. Phys. Rept. 94, 313 (1983) 5. Jack, H.: A class of symmetric polynomials with a parameter. Proc. Roy. Soc. Edinburgh Sect. A, 69, 1–18 (1970/1971); A surface integral and symmetric functions. Proc. Roy. Soc. Edinburgh Sect. A, 69, part 4, 347–364 (1972) 6. Stanley, R.P.: Some combinatorial properties of Jack symmetric functions. Adv. Math.,77, 76–115 (1988) 7. Desrosiers, P., Lapointe, L., Mathieu, P.: Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials. Nucl. Phys. B 606, 547 (2001) 8. Sriram Shastry, B., Sutherland, B.: Superlax pairs and infinite symmetries in the 1/r 2 system. Phys. Rev. Lett. 70, 4029 (1993), cond-mat/9212029 9. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. 2ieme edition, Oxford: The Clarendon Press, Oxford University Press, 1995; Symmetric Functions and Orthogonal Polynomials. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, American Mathematical Society, 1998 10. Sogo, K.: Eigenstates of Calogero-Sutherland-Moser model and generalized Schur functions. J. Math. Phys. 35, 2282 (1994) 11. Lapointe, L., Lascoux, A., Morse, J.: Determinantal formula and recursion for Jack polynomials, Electro. J. Comb. 7, 467 (2000) 12. Polychronakos, A.P.: Exchange operator formalism for integrable systems of particles. Phys. Rev. Lett. 69, 703 (1992), hep-th/9202057; Minahan, J.A., Polychronakos, A.P.: Integrable systems for particles with internal degrees of freedom, Phys. Lett. B302, 299 (1993), hep-th/9206046 Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 233, 403–421 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0756-6

Communications in

Mathematical Physics

N = 1 Supersymmetric Sigma Model with Boundaries, I Cecilia Albertsson1 , Ulf Lindstr¨om1 , Maxim Zabzine2 1

Institute of Theoretical Physics, Stockholm Centre for Physics, Astronomy and Biotechnology, University of Stockholm, 106 91 Stockholm, Sweden. E-mail: [email protected], [email protected] 2 INFN Sezione di Firenze, Dipartimento di Fisica, Via Sansone 1, 50019 Sesto F.no (FI), Italy. E-mail: [email protected] Received: 27 November 2001 / Accepted: 16 August 2002 Published online: 19 December 2002 – © Springer-Verlag 2002

Abstract: We study an N = 1 two-dimensional non-linear sigma model with boundaries representing, e.g., a gauge fixed open string. We describe the full set of boundary conditions compatible with N = 1 superconformal symmetry. The problem is analyzed in two different ways: by studying requirements for invariance of the action, and by studying the conserved supercurrent. We present the target space interpretation of these results, and identify the appearance of partially integrable almost product structures.

1. Introduction The two-dimensional non-linear sigma model with boundaries and N = 1 worldsheet supersymmetry plays a prominent role in the description of open Neveu-SchwarzRamond (NSR) strings. Studying this model may thus help to understand those aspects of D-brane physics where the sigma model description is applicable. The model is also interesting in its own right; for instance, it is a well known fact that supersymmetric models have intriguing relations with geometry. Here we will show that the supersymmetric sigma model with boundaries naturally leads to the appearance of partially integrable almost product geometry. The idea is to look at the non-linear sigma model with the minimal possible amount of worldsheet supersymmetry, i.e. with only one spinor parameter in the supersymmetry transformations. It turns out that even this minimal amount of supersymmetry leads to interesting restrictions on the allowed boundary conditions. These worldsheet restrictions can be reinterpreted in terms of the target space manifold, with the result that the open strings are allowed to end only on (pseudo) Riemannian submanifolds of the target space. Special cases of the kind of conditions we present here are often adopted in the literature (e.g. [1, 3]), without derivation. The aim of this paper is to show, in a pedagogical manner, how one arrives at these conditions via systematic analysis of the action and

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the conserved currents. However, the conditions we find are more general than the ones usually assumed. The paper is organised as follows. In Sect. 2 we consider the non-linear sigma model action defined on a (pseudo) Riemannian target space manifold. We derive the general boundary conditions that simultaneously set to zero the field variation and the supersymmetry variation of the action. We check that these conditions are compatible with the supersymmetry algebra, and then consider a special case. In Sect. 3 we rederive the boundary conditions using a different approach, namely requiring N = 1 superconformal invariance at the level of the conserved currents. In Sect. 4 we interpret the worldsheet conditions in terms of the target space manifold, which leads to some interesting properties of D-branes. In Sect. 5 we briefly touch on the interesting question of additional supersymmetries, and finally, in Sect. 6, we give a summary of the present work and an outline of our plans for future investigations. 2. The N = 1 σ -Model Action The first approach we take in finding the boundary conditions is to analyze the action. Field variations of the action yield boundary field equations, which must be compatible with the vanishing of the supersymmetry variation of the action. Making a general (linear) ansatz for the relation between worldsheet fermions on the boundary, this compatibility requirement imposes restrictions on our ansatz. 2.1. Boundary equations of motion. We start from the non-linear sigma model action
S = d 2 ξ d 2 θ D+ µ D− ν gµν (), (1) where gµν is a Riemannian1 metric and we use the standard 2D superfield notation (see Appendix A). Varying S with respect to the fields µ we obtain a boundary term
 µ ν µ ν δS = i dτ (δψ+ ψ+ − δψ− ψ− )gµν +  ν ρ ν ρ +δXµ (igµν (∂++ X ν − ∂= X ν ) + (ψ− ψ− − ψ + ψ+ )νµρ ) σ = 0,π , (2) where νµρ is the Levi-Civita connection, νρµ = gνσ  σρµ =

1 (gνµ,ρ + gρν,µ − gρµ,ν ). 2

(3)

We now make a general linear ansatz for the fermionic boundary conditions2 (see [4] for a discussion of general fermionic ans¨atze), µ

ν ψ− = ηR µν (X) ψ+ |σ = 0,π ,

(4)

µ

where R ν is a (1, 1) “tensor” which squares to one, R µν R νρ = δ µρ , 1

(5)

Hereafter, whenever we use the word “Riemannian,” we mean “Riemannian or pseudo Riemannian.” From now on we shall drop the notation |σ =0,π , since all conditions in this paper are understood to hold on the boundary. 2

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and we have found it convenient to introduce the parameter η which takes on the values ±1. The property (5) can be justified by worldsheet parity, i.e., the theory should be invariant under interchange of ψ+ and ψ− . Relaxation of this property leads to the appearance of generalised torsion (i.e., the combination B + dA of a B-field and gauge field); however, we focus here on a torsion-free background.3 µ We view R ν as a formal object, for the moment avoiding to specify whether it is defined globally or only locally; such issues are discussed in Sect. 4. Substituting the µ ansatz (4), as well as its field variation, into (2) yields conditions on R ν if the variation µ δS is to vanish. The first such condition comes from cancelling the δψ+ -terms and says µ that R ν must preserve the metric, gµν = R σµ R ρν gσρ .

(6) ρ

Note that this condition, together with (5), implies that Rµν ≡ gµρ R ν is symmetric. After imposing (6), we are left with the δXµ -part of (2),
  σ γ ψ+ , (7) δS = i dτ δXµ igµν (∂++ X ν − ∂= X ν ) + R νσ gνρ ∇µ R ργ ψ+ where ∇µ is the spacetime Levi-Civita covariant derivative. Before exploring what conditions the vanishing of (7) gives us, we first remark on µ the interpretation of R ν . The open string may either move about freely, in which case its ends obey Neumann boundary conditions; alternatively, the ends of the string may be confined to a subspace, corresponding to Dirichlet conditions. Consider a d-dimensional target space with Dp-branes, i.e., there are d − (p + 1) Dirichlet directions along which the field X µ is frozen (∂0 X i = 0, i = p + 1, ..., d − 1). At any given point on a Dp-brane we can choose local coordinates such that X i are the directions normal to the brane and Xm (m = 0, ..., p) are coordinates on the brane. We call such a coordinate system adapted to the brane. In this basis the fermionic boundary conditions are [7] m m = η ψ+ , ψ−

whence R µν =



i i ψ− = −η ψ+ ,

δ nm 0 0 −δ i j

 .

(8) µ

(This tensor is also used in boundary state formalism, see e.g. [8, 9].) So R ν has a clear physical interpretation: its (+1) eigenvalues correspond to Neumann conditions µ and its (−1) eigenvalues correspond to Dirichlet conditions. Thus the general tensor R ν represents the boundary conditions covariantly at the given point. Given this setup of Neumann and Dirichlet directions, it is convenient to introduce µ µ the objects P ν and Q ν as [9] P µν =

1 µ (δ + R µν ), 2 ν µ

Qµν =

1 µ (δ − R µν ). 2 ν

(9)

µ

Because R squares to one, P ν and Q ν are orthogonal projectors, defining the Neumann and Dirichlet directions, respectively, in a covariant way. They satisfy P µρ P ρν = P µν , 3

Qµρ Qρν = Qµν ,

The general case will be treated in a forthcoming publication [5] (see also [6]).

(10)

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and Qµν P νρ = P µν Qνρ = 0,

P µν + Qµν = δ µν .

(11)

Continuing the analysis of the field variation (7), we note that since, on the boundary, the X-field is frozen along the Dirichlet directions, its corresponding variation vanishes. µ µ That is, Q ν δX ν = 0, and we may write δX µ = P ν δX ν . Thus the set of general parity invariant boundary conditions implied by the boundary equations of motion are  µ µ ν =0  ψ− − ηR ν ψ+ µ µ ρ σ ψγ = 0 , (12) gδµ P ν (∂++ X ν − ∂= X ν ) − iP δ R νσ gνρ ∇µ R γ ψ+ +  µ Q ν (∂= X ν + ∂++ X ν ) = 0 µ

where R ν satisfies



µ

µ

R ν R νρ = δ ρ . ρ gµν = R σµ R ν gσρ

(13)

2.2. Supersymmetry variations. In general one does not expect the boundary conditions derived as field equations to be supersymmetric, since the action is invariant under the supersymmetry algebra only up to a boundary term. To ensure worldsheet supersymmetry the boundary conditions should make the supersymmetry variation of the action µ vanish. The constraints on R ν implied by this condition must then be made compatible with (12) and (6). Assuming the standard (1,1) supersymmetry variation (63) with supersymmetry parameter $ + = η$ − , the variation of (1) yields the boundary term
 µ µ ν ν ν − η∂= X ν ψ+ ) + igµν F+− (ψ+ + ηψ− ) δs S = $ − dτ gµν (∂++ X µ ψ−  µ ν ρ µ ν ρ − iψ+ ψ− (14) ψ+ µνρ − iηψ+ ψ− ψ− νµρ . Inserting the ansatz (4), δs S simplifies to
µ δs S = η$ − dτ ψ+ gµν (R νσ ∂++ X σ − ∂= X ν )

 σ σ γ ρ ν − 2iψ+ ψ+ R σ P µ νργ . + 2iηgµν P νσ F+−

(15)

We now have two choices; either to stay completely off-shell, or to make use of the (algebraic) bulk field equations for the auxiliary field F .4 If we stay off-shell and plug the conditions (12) into (15) we get a condition involving the F -field. However, this condition is not uniquely determined; for example, one can always add a term of the ν ψ ρ , where K form Kµνρ ψ+ µνρ is symmetric either in the first two indices or in the first + and the last (for instance, one could choose Kµνρ = gµσ ∇ν R σρ ). Therefore, staying off-shell, we do not obtain a unique form of the boundary condition for F . (Even if this F -condition were unique, we still need it to be compatible with the F -field equation on the boundary.) Hence the way to proceed is to go partly on-shell and use the F -field equation of motion, ρ

ν ν gµν F+− + ψ+ ψ− µνρ = 0, 4

Note that there are no boundary contributions in deriving the F -field equations.

(16)

N = 1 Supersymmetric Sigma Model with Boundaries

407

restricted to the boundary (i.e., with the ansatz (4) inserted). Substituting this in (15) yields the supersymmetry variation
 µ δs S = η$ − dτ ψ+ gµν (R νσ ∂++ X σ − ∂= X ν ) , which, using the bosonic conditions from (12), reduces to
µ σ γ δ ν ψ+ P µ R σ gνρ ∇δ R ργ . δs S = iη$ − dτ ψ+ ψ+

(17)

µ

δs S vanishes only if R ν satisfies P δµ R νσ gνρ ∇δ R ργ + P δσ R νγ gνρ ∇δ R ρµ + P δγ R νµ gνρ ∇δ R ρσ = 0,

(18)

which, by contraction with Q, leads to ρ

P σγ P µν ∇[σ Qδµ] = P µγ P νσ Q [µ,ν] = 0.

(19)

This is the integrability condition for P (cf. Appendix D). In conclusion, the boundary conditions (12) are consistent with worldsheet supersymmetry, in the sense discussed above, when P is integrable. The geometrical meaning of this integrability condition is discussed in Sect. 4. 2.3. Compatibility with the algebra. We may rephrase the question about boundary supersymmetry and study compatibility with the supersymmetry algebra. This means that the boundary conditions should be consistent with the supersymmetry transformation of our fermionic ansatz (4). Applying the transformations (63) with $ + = η$ − to the ansatz one finds a bosonic boundary condition5 ρ

ν ν ∂= X µ = R µν ∂++ X ν + 2iηP µν F+− − 2iR µν,σ P σρ ψ+ ψ+ .

(20)

Compatibility with the supersymmetry algebra is attained when the boundary field equations (12) satisfy (20). Note that (20) is stronger than the requirement of zero supersymmetry variation of the action; this may be seen by substituting (20) into (15) to get
µ σ γ − dτ ψ+ ψ+ δs S = −2iη$ ψ+ gσ ν P ρµ ∇ρ R νγ , (21) ρ

which vanishes identically because gσ ν P µ ∇ρ R νγ is symmetric in σ and γ (by (6)). To show that the boundary equations of motion are consistent with (20), we first go on-shell by using the F -field equation (16) restricted to the boundary, obtaining γ

ν = 0. ∂= X µ − R µν ∂++ X ν + 2iP σγ ∇σ R µν ψ+ ψ+

(22) µ

Contraction with Q then gives the integrability condition for P , after using that Q ν ∂0 X ν = 0. On the other hand, contracting (22) with P and using that P is integrable, we arrive at the second (bosonic) condition in (12). Thus we conclude that (12) are the general parity invariant boundary conditions compatible with the supersymmetry algebra, provided that R squares to one and preserves the metric, and that P is integrable. 5 The set of supersymmetric boundary conditions can be written in terms of 1D superfields, see Appendix B.

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2.4. Preservation of the metric. The properties (13) allow us to draw some conclusions µ about the form of R ν . For a general metric gµν there is only one solution for R that µ µ squares to one and preserves the metric, namely R ν = δ ν . Thus there can be no Dirichlet directions, i.e., this general background cannot support D-branes, or worldsheet supersymmetry is broken. If, on the other hand, the metric is not of a completely general form, there may be µ other solutions for R ν , depending on the form of gµν . In the presence of Dp-branes, the metric must not mix Neumann and Dirichlet directions. To see why, go to adapted coordinates (Xm , Xi ) at some point in the target space; then R = diag(1, −1) at this point, and preservation of the metric, gµν = R σµ R ρν gσρ , tells us that the only backgrounds that allow Dp-branes are those satisfying [1] gim = 0.

(23)

Thus we see that, at the given point, the metric must be such that the Neumann and Dirichlet directions decouple, if the strings attached to the brane are to remain supersymmetric. There remains the important question of whether the adapted system of coordinates at a point can be extended to a neighbourhood along the Dp-brane. This is where integrability comes in; we address this issue in Sect. 4. 2.5. A special case. One may ask what is required for the two-fermion term in the bosonic boundary conditions in (12) to vanish, so that the boundary conditions take the simple form  µ µ ν ψ− = ηR ν ψ+ (24) µ ∂= X µ = R ν ∂++ X ν often assumed in the literature; see, e.g., [1]. From (12) it is easily seen that in addition to (13) one needs to impose P ρµ ∇ρ R νγ = 0,

(25)

a condition that also implies P -integrability (cf. Eq. (19)). µ µ Note that for a general metric, where we have R ν = δ ν , the conditions (24) reduce to  µ µ ψ− − ηψ+ = 0 (26) µ ∂= X − ∂++ X µ = 0 which corresponds to the freely moving open string, its ends satisfying Neumann conditions in all directions (corresponding to space-filling D9-branes). 3. N = 1 Superconformal Symmetry The second route to finding the boundary conditions allowed by the supersymmetric sigma model involves studying the conserved currents. Here we derive these currents and the conditions they must satisfy on the boundary, showing how this yields constraints µ on R ν .

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3.1. Conserved currents. We want to retain classical superconformal invariance in the presence of boundaries. To do this, the appropriate objects to study are the currents corresponding to supertranslations in (1, 1) superspace. These supercurrents may be derived using superspace notation (we sketch the main steps in Appendix C), and we obtain µ ν T+− + = D+  ∂++  gµν ,

(27)

T=+ = D− µ ∂= ν gµν ,

(28)

obeying the conservation laws D− T+− + = 0.

D+ T=+ = 0,

Note that these conserved currents are defined only up to the equations of motion. The components of the supercurrents (27) and (28) correspond to the supersymmetry current and stress tensor as follows, µ

ν G+ = T+− + | = ψ+ ∂++ X gµν ,

(29)

µ

G− = T=+ | = ψ− ∂= X ν gµν ,

(30) µ

µ ν ν T++ = −iD+ T+− + | = ∂++ X ∂++ X gµν + iψ+ ∇+ ψ+ gµν , µ

ν gµν . T−− = −iD− T=+ | = ∂= X µ ∂= X ν gµν + iψ− ∇− ψ−

(31) (32)

The covariant derivative acting on the worldsheet fermions is given by ν ν σ ∇± ψ+ = ∂++ ψ+ +  νρσ ∂++ X ρ ψ+ , =

=

ν ν σ ∇± ψ − = ∂++ ψ− +  νρσ ∂++ X ρ ψ− , =

=

(33)

where  νρσ is the Levi-Civita symbol. Moreover, the conservation laws in components acquire the following form, ∂++ G− = 0, ∂= G+ = 0,

∂++ T−− = 0, ∂= T++ = 0.

To ensure superconformal symmetry on the boundary we need to impose boundary conditions on the currents (29)–(32). To see what these conditions look like, consider the conserved charge,
0 = ∂0 Q = dσ ∂0 J 0 , (34) where the current J is any of the currents G and T , and J 0 is the τ -component of J . Then current conservation, ∂α J α = 0, implies that
0 = − dσ ∂1 J 1 , resulting in the boundary condition J + − J − = 0.

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Applying this to our currents G and T , we arrive at the boundary conditions6 G+ − ηG− = 0,

T++ − T−− = 0.

(35)

At the classical level, these conditions are just saying that the left-moving super-Virasoro algebra coincides with the right-moving one. We emphasize that classically the conditions (35) can make sense only on-shell. This means that we may (and should) make use of the field equations in our analysis of the current constraints. 3.2. Boundary conditions. To examine how the current conditions affect the choice of µ R ν in the fermionic boundary conditions, we begin by rewriting the stress tensor in a suitable form. The problem is that the stress tensor has both normal and tangential fermionic derivatives. This is remedied by defining 2∇0 ≡ ∇+ + ∇− and using the fermionic equations of motion, µ µ ν ν gµν (ψ+ ∇− ψ+ − ψ − ∇+ ψ − ) = 0, to rewrite T++ − T−− as follows, T++ − T−− = ∂++ X µ ∂++ X ν gµν − ∂= X µ ∂= X ν gµν µ

µ

µ

µ

ν ν + i(ψ+ − ηψ− )∇0 (ψ+ + ηψ− )gµν

ν ν + i(ψ+ + ηψ− )∇0 (ψ+ − ηψ− )gµν .

(36)

The general form of the boundary conditions which satisfy (35) are found by again making the ansatz (4), µ

ν , ψ− = ηR µν ψ+

R µν R νσ = δ µσ ,

(37)

recalling that the last property is needed if we want worldsheet parity as a symmetry of the boundary conditions. Using (36) and (37), the conditions (35) imply the following two properties, gµν = R σµ R ρν gσρ ,

P σγ P µν ∇[σ Qδµ] = 0,

(38)

i.e., R preserves the metric and P is integrable, just as we saw in Sect. 2. The first of these properties is straightforwardly obtained from the T -condition; the second is obtained by using the first property to rewrite the G- and T -conditions as σ (g ∂ X ν − R µ ∂ X ν g ) = 0 G+ − ηG− = ψ+ σ ν ++ µν σ =  µ σ ψγ = 0 T++ − T−− = 2∂0 X ρ gρν (∂++ X ν − ∂= X ν ) − iP δρ R σ gµν ∇δ R νγ ψ+ + (39) µ

µ

µ

and inserting δ ν = P ν + Q ν between ∂0 X ρ and the parenthesis in the T -condition (we assume that there are Dirichlet directions, i.e., that there is a non-vanishing Q such that µ Q ν ∂0 X ν = 0). We then use the result in the G-condition, thus obtaining the second of Eqs. (38). A remark may be in order on the fact that we obtain the individual conditions (38), rather than some more general condition involving all the worldsheet fields. This may be 6

The η appears here due to supersymmetry.

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411

understood by recognising that, after using the fermionic boundary condition   and the curµ rent conditions, we have reduced the original set of four independent fields ψ± , ∂++ X µ =

µ

to a set of only two independent fields, say ψ+ and ∂++ X µ . But since these remaining fields are truly independent, terms of different structure that appear in the current conditions must vanish separately. Thus, for example in the analysis of the G-condition, a term µ ψ+ ∂++ X ν (gµν − R σµ R ρν gσρ ) will appear which must vanish independently, giving the first of Eqs. (38), and the second condition is obtained analogously. We conclude that the complete set of boundary conditions that solve the current boundary conditions (35) are  µ µ ν ψ− − ηR ν ψ+ =0    µ µ ρ  ν σ ψγ = 0  gδµ P ν (∂++ X − ∂= X ν ) − iP δ R νσ gνρ ∇µ R γ ψ+  +    P σ P µ ∇ Qδ = 0 γ ν [σ µ] . (40) µ  Q ν (∂= X ν + ∂++ X ν ) = 0    ρ   g = R σµ R ν gσρ   µν  µ µ R ν R νρ = δ ρ

These conditions are identical to the boundary conditions (12), (13) derived from the action in Sect. 2. In particular, it is clear that again P needs to be integrable. As an alternative to the above procedure, one may derive the conditions (40) by imposing compatibility with the supersymmetry algebra on the currents in a more direct manner. That is, we can use the (on-shell) supersymmetry transformation (22) of the fermionic boundary conditions. The result is again that the G- and T -conditions require, in addition to the fermionic boundary condition and its supersymmetry transformation, µ that R ν satisfy (38), i.e., we arrive at the boundary conditions (40). Note that now P -integrability may also be derived in the same way as discussed in Sect. 2.3, contracting (22) with Q. 4. Geometric Interpretation In this section we discuss the target space interpretation of the boundary conditions we µ have found. All information about the boundary conditions is encoded in R ν , so we µ focus on the restrictions on R ν alone. There are two ways of viewing the boundary µ conditions, locally and globally; either R ν is defined only locally in the target space, or it is defined globally. µ

4.1. Locally defined conditions. We first take the local point of view, assuming that R ν is defined only in some region of the target space manifold. The first boundary condition µ µ we consider is the integrability of P . Take an arbitrary point x0 in the region where R ν µ µ is defined. We may then write any contravariant vector dX at x0 as dXµ = P µν dX ν + Qµν dX ν , µ

µ

(41)

µ

where we have used the fact that δ ν = P ν + Q ν . It follows that the distribution P is defined by Qµν (X)dX ν = 0,

(42)

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since this leaves dX µ = P µν dX ν .

(43)

The definition (43) leads to P -integrability, by acting on it with the exterior derivative d, 0 = d 2 X µ = P µν,ρ dX ρ ∧ dX ν = whence

ρ

1 µ ρ P P P ν dX λ ∧ dX σ , 2 [ν,ρ] λ σ

(44)

µ

P λ P νσ Q [ν,ρ] = 0,

which is the P -integrability condition (cf. Eq. (78)). It turns out that P -integrability is the very condition necessary and sufficient for the differential equations (42) to be completely solvable in a neighbourhood. To see this, µ note that for Q ν (X)dX ν = 0 to be integrable, it is necessary and sufficient that in a neighbourhood, 1 µ 1 ρ µ Q [ν,ρ] dX ρ ∧ dX ν = P λ P νσ Q [ν,ρ] dX λ ∧ dX σ . (45) 2 2 If the equations (42) are completely integrable, then they admit rank(Q) independent solutions. Going to a coordinate basis where the equations take the form d X˜ i = 0, we may write the solutions as 0 = d(Qµν (X)dX ν ) =

X˜ i (X) = q i ,

i = 0, ..., rank(Q) − 1,

(46)

where q i are constants. Interpreting the above result in terms of the target space manifold, we see that the coordinates (46) define a rank(P )-dimensional submanifold. If P has rank p + 1, then the submanifold is (p + 1)-dimensional, i.e., it is a Dp-brane, and {X˜ i } is the system of coordinates adapted to the brane. Now P -integrability allows us to extend this system of coordinates to a neighbourhood (i.e., it is defined not only at one point) along the Neumann directions. Note that this result arises in a purely algebraic fashion from the requirement of minimal worldsheet supersymmetry (see Sect. 2). Next we interpret the requirement that R preserve the metric, gµν = R σµ R ρν gσρ .

(47)

Unlike P -integrability, this property is not purely algebraic; it requires additional information, for instance conserved currents (see Sect. 3). As we saw in Sect. 2.4, preservation of the metric implies that in coordinates adapted to the brane the metric must take a block diagonal form, gµν = diag(gmn , gij ), on the worldsheet boundary. Given that P is integrable, we can extend the adapted coordinate system to a neighbourhood along the brane, so that R = diag(1, −1) in this neighbourhood. Thus the metric is block diagonal in this neighbourhood, and the metric along the Neumann directions, gnm , could in principle serve as a metric on the Dp-brane. In conclusion, we see that requiring N = 1 superconformal invariance for open strings results in that they are allowed to end only on Riemannian submanifolds of the target space manifold. Of course, this implies no restrictions on the “bulk” target space; the conditions here only apply to the worldsheet boundary, telling us where the strings may end, not what the rest of the background looks like. As an aside, note that if applied to the case where both P and Q are integrable, the above discussion leads to the existence of both Dp- and D(d − (p + 1) − 1)-branes, in a d-dimensional target space.

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4.1.1. Confined geodesics. As we discussed in Sect. 2.5, the fermionic boundary condition and its supersymmetry transformation simplify to (24) when the extra requirement (25) is imposed. This requirement has a nice geometrical interpretation when defined in a neighbourhood of a brane. Since ∇ρ R νγ = −2∇ρ Qνγ , one can rewrite (25) as P ρµ ∇ρ Qνγ = 0.

(48)

When this condition is satisfied, P is called parallel [10] with respect to the LeviCivita connection. Indeed, if P is integrable one can always construct a symmetric affine connection such that (48) holds. µ µ To understand the physical meaning of (48), take a point x0 and a vector v µ at x0 , µ ν which is contained in P (i.e., Q ν v = 0). Then the autoparallel curve with respect to µ the Levi-Civita connection, v µ ∇µ v ν = 0, is uniquely determined by the initial point x0 µ and the initial direction v . The condition (48) ensures that the path thus determined is µ always contained in P . This is easily seen by inserting v µ = P ν v ν into the curve (where now v µ is any vector along the curve),   0 = v µ ∇µ v ν = v µ ∇µ P νσ v σ = v µ v σ ∇µ P νσ + P νσ v µ ∇µ v σ µ

= −v λ v σ P λ ∇µ Qνσ .

Thus the autoparallel curve is compatible with the requirement that v µ be contained in P , if (48) holds. A distribution P such that autoparallel curves starting out in P always remain in P is called geodesic7 [10]. Physically it means that if a geodesic starts on a Dp-brane, then it will always remain on the brane. Hence particles cannot escape from the brane. 4.2. Globally defined conditions. Turning now to global issues, we ask when it makes sense to interpret the boundary conditions (12), (13) globally and what kind of restricµ tions it implies for the target space. If the tensor R ν is defined globally on the target µ ν µ space, and if the property R ν R ρ = δ ρ holds globally, then the target space manifold µ µ is said to admit an almost product structure R ν (see Appendix D). If in addition R ν preserves the metric, then the manifold is called an almost product Riemannian manifold. Thus our boundary conditions tell us that when the target space is an almost product Riemannian manifold with the property that P is integrable, then it admits Dp-brane solutions at any point, and that these solutions respect N = 1 superconformal symmetry. If the target space is a locally product manifold, i.e., both P and Q are integrable, then the sigma model admits both Dp- and D(d − (p + 1) − 1)-brane solutions at any point of the manifold, and they are all compatible with N = 1 superconformal invariance. 4.2.1. Warped product spacetimes. Consider again the case where P is a geodesic distribution satisfying (48). There is an interesting example of a geometry for which this condition is fulfilled globally, namely the warped product spacetimes [10]. Given two Riemannian manifolds (Mi , gi ), i = 1, 2, and a smooth function f : M = M1 × M2 → R, construct the metric g = g1 ⊕ ef g2 on M. Then the manifold (M, g)

7

This property is a weaker condition than (48).

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C. Albertsson, U. Lindstr¨om, M. Zabzine

is a warped product manifold.8 Thus, if (48) is defined globally, we know that the target space is a warped product spacetime, and we also know from Sect. 2.5 that the only boundary conditions preserving N = 1 superconformal symmetry in this case are µ

ν , ψ− = ηR µν ψ+

∂= X µ = R µν ∂++ X ν ,

although now they need to be globally defined. 5. Additional Supersymmetry In this section we discuss the non-linear sigma model with N = 2 supersymmetry. Although our discussion is somewhat sketchy we show that the solution of the problem is more general than usually assumed in the literature. We have shown that for the N = 1 supersymmetric sigma model (1) the following set of parity invariant boundary conditions,  µ µ ν ψ− − ηR ν ψ+ = 0, (49) µ µ γ ν ∂= X µ − R ν ∂++ X ν + 2iP σγ ∇σ R ν ψ+ ψ+ = 0, respect N = 1 superconformal invariance. The fermionic and bosonic boundary conditions are related to each other through supersymmetry transformations. Due to the integrability of P this solution admits a nice geometrical interpretation in terms of a submanifold. Now if the model (1) has a second supersymmetry the target space must be a K¨ahler manifold. In real coordinates the second on-shell supersymmetry can be written as follows,  + ν ν )J µ + $2− ψ−  δ2 X µ = −($2 ψ+ ν µ µ µ ρ ν µ + ν ψ ρ + $−J µ ψ ν ψ ρ δ2 ψ+ = −i$2 ∂++ X ν J ν + $2− J σ  σνρ ψ− ψ+ + $2+ J ν,ρ ψ+ + 2 ν,ρ + − ,  µ µ µ ρ µ − µ ν + $+J ν ρ ν ρ δ2 ψ− = −i$2− ∂= X ν J ν − $2+ J σ  σνρ ψ− ψ+ 2 ν,ρ ψ− ψ+ + $2 J ν,ρ ψ− ψ− (50) µ

µ

µ

where J ν is a complex structure (i.e., J ν J νρ = −δ ρ and J is integrable) with the propµ erty ∇ρ J ν = 0. The new symmetry gives rise to several new questions. For instance, what are the boundary conditions that ensure survival of this symmetry on the boundary? And what restrictions, if any, does it imply for the conditions (49)? Here we address the latter of these questions. To this end, we start from the fermionic boundary condition in (49) and use the second supersymmetry variation to derive the corresponding bosonic boundary conditions,   γ ν ρ ∂= X µ +(ηη2 )J µσ R σν J νγ ∂++ X γ −i (ηη2 )J µσ ∇ρ R σν J ργ +J µσ ∇ρ R σν J λ R λγ ψ+ ψ+ = 0, (51) where we assume that $2+ = η2 $2− . The equations (49) and (51) should be equivalent. Comparing the X-part we get the following condition (which also is part of the requirement for the second boundary supersymmetry), J µγ R γν J νσ = −(ηη2 )R µσ ,

(52)

8 In the present context it is interesting that many exact solutions of Einstein equations, e.g., Schwarzschild, Robertson-Walker, Reissner-Nordstr¨om, etc., and also p-brane solutions, are examples of warped product spacetimes.

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415

or equivalently, J µγ R γν = (ηη2 )R µγ J γν .

(53)

The case ηη2 = 1 corresponds to the so-called B-type conditions and ηη2 = −1 to A-type conditions as defined in [1]. One can show that the B-type conditions correspond to K¨ahler submanifolds and the A-type conditions to coisotropic submanifolds. We emphasize that integrability of P plays a crucial role in this argument. Using the property (52) we rewrite Eq. (51) as follows,   γ ν µ ∂= X µ − R µν ∂++ X ν − i(ηη2 )J λ J ργ ∇ρ R λν + ∇σ R λν R σρ ψ+ ψ+ = 0.

(54)

Comparing the two-fermion terms in (49) and (54) we get the following condition, 

γ ν µ = 0. (55) P ργ ∇ρ R µν + (ηη2 )J λ J ργ P σρ ∇σ R λν ψ+ ψ+ This condition does not imply that the two-fermion term in the bosonic boundary condition vanishes altogether. However, it does put some restrictions on this term. As an example we consider the B-type conditions. In this case it is convenient to go ¯ ¯ ¯ to complex coordinates such that J ij = iδ ij , J ij¯ = −iδ ij¯ and J ij¯ = J ij = 0. In these coordinates the property (52) implies ¯

R ij¯ = R ij = 0,

¯

∇ρ R ij¯ = ∇ρ R ij = 0.

(56)

The conditions (55) take the form ¯

j

l P s¯l¯∇s¯ R ij ψ+ ψ+ = 0,

j¯

¯

l P sl ∇s R ij¯ ψ+ ψ+ = 0,

(57)

whence the bosonic boundary condition may be written as j

l ψ+ = 0, ∂= X i − R ij ∂++ X j + 2iP sl ∇s R ij ψ+

(58)

in holomorphic coordinates. The structure in front of the two-fermion term is related to the second fundamental form of a submanifold [5]. We would like to stress that the solutions we have discussed are more general than those usually considered in the literature (see e.g. [1] and [2]) and that they admit a nice geometrical interpretation. As one could see from our previous discussion, the requirements of zero variation (2) and of conformal invariance on the boundary (i.e., T++ − T−− = 0) are completely equivalent. However, in the literature one encounters restrictions on the boundary that are stronger than the vanishing of (2). For instance, in [2] the following conditions are imposed on the boundary, δX µ gµν ∂1 X ν = 0, µ ν µ ν gµν (δψ+ ψ+ − δψ− ψ− ) = 0.

(59)

(Note that in normal coordinates the two-fermion term in (2) is absent.) This is certainly a solution of the problem, but not the most general one. The topic deserves further study and we plan to return to it in future work.

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6. Conclusions In this paper we have investigated what boundary conditions on an N = 1 sigma model are required for supersymmetry. First, vanishing of the boundary supersymmetry variations together with invariance of the action under general variations give us a set of boundary conditions. These turn out to be compatible with the supersymmetry algebra on the boundary in that the fermionic and bosonic boundary conditions form a supersymmetry multiplet on the (auxiliary) F -field shell. We further studied the boundary conditions required for left and right currents to agree on the boundary. The conditions derived in this way are identical to those we find using our first method. Our boundary conditions are more general than those usually adopted in the literature, and they are derived here in a systematic manner. We believe that this makes them useful. One interesting feature of our results is the occurrence of a mixed second rank tensor µ R ν that squares to the identity and preserves the metric. We have shown that the proµ jector in the Neumann directions, formed from R ν , satisfies an integrability condition. This condition has the natural interpretation that it is possible to choose coordinates µ along a D-brane such that R ν is constant and diagonal. Our boundary conditions give information about how D-branes may be embedded in spacetime and what the corresponding local geometry looks like. In particular, we showed that minimal boundary supersymmetry requires open strings to end only on (pseudo) Riemannian submanifolds of the target space. This fact is usually assumed in the literature, but we have derived it in a rigorous way. µ Mathematically it may also be of interest to consider the case when R ν is globally defined. Then it has the geometric interpretation of an almost product structure, and we briefly discussed this. In this paper we have only considered sigma models in a non-trivial background metric. The boundary conditions will get modified if a background antisymmetric B-field is also included [6]. We turn to this case in a future publication [5]. Other questions of interest to us are related to treating the full theory, i.e., to include the 2D supergravity fields. And, even in the gauge-fixed case, to include contributions from the ghost fields, e.g., to the currents. A. (1,1) Supersymmetry Throughout the paper we use µ, ν, ... as spacetime indices, (+ +, =) as worldsheet indices, and (+, −) as two-dimensional spinor indices. We also use superspace conventions, where the pair of spinor coordinates of two-dimensional superspace are labelled θ ± , and the covariant derivatives D± and supersymmetry generators Q± satisfy 2 2 D+ = i∂++ , D− = i∂= , Q± = −D± + 2iθ ± ∂++ , =

{D+ , D− } = 0, (60)

where ∂++ = ∂0 ± ∂1 . In terms of the covariant derivatives, a supersymmetry transfor= mation of a superfield  is then given by δ ≡ (ε + Q+ + ε − Q− )

N = 1 Supersymmetric Sigma Model with Boundaries

417

= −(ε + D+ + ε − D− ) + 2i(ε + θ + ∂++ + ε − θ − ∂= ).

(61)

The components of a superfield  are defined via projections as follows, | ≡ X,

D± | ≡ ψ± ,

D+ D− | ≡ F+− ,

(62)

where a vertical bar denotes “the θ = 0 part of ”. Thus, in components, the (1, 1) supersymmetry transformations are given by  µ µ δXµ = −$ + ψ+ − $ − ψ−    µ µ δψ+ = −i$ + ∂++ X µ − $ − F−+ . (63) µ µ  δψ− = −i$ − ∂= X µ − $ + F+−   µ µ µ δF+− = −i$ + ∂++ ψ− + i$ − ∂− ψ+ B. 1D Superfield Formalism One may view the 2D supersymmetry algebra as a combination of two 1D algebras (for similar considerations see [11]). To see this, we rewrite the (1, 1) supersymmetry algebra in terms of 1D supermultiplets. Assuming that $ + = η$ and $ − ≡ $, (63) becomes  µ µ δXµ = −$(ηψ+ + ψ− )    µ µ δ(ηψ+ + ψ− ) = −2i$∂0 X µ . (64) µ µ µ δ(ψ− − ηψ+ ) = −2$(ηF+− − i∂1 X µ )    µ µ µ µ δ(ηF+− − i∂1 X ) = −i$∂0 (ψ− − ηψ+ ) Introducing a new notation for the following combinations of fields, 1 µ µ 4 µ ≡ √ (ηψ+ + ψ− ), 2

1 µ µ ˜ µ ≡ √ (ψ− 4 − ηψ+ ), 2

µ

f µ ≡ ηF+− − i∂1 X µ , (65)

and redefining $ =

√ 2$, the algebra (64) takes on the simple form  µ δX = −$4 µ    δ4 µ = −i$∂ X µ 0 . ˜ µ = −$f µ δ4    µ ˜µ δf = −i$∂0 4

(66)

Clearly, (66) is a decomposition of the 2D algebra into two 1D supermultiplets. We introduce a 1D superfield notation for these multiplets, K µ = Xµ + θ 4 µ ,

˜ µ + θf µ , Sµ = 4

(67)

where θ is the single Grassmann coordinate of the respective 1D superspace, and the corresponding 1D superderivative is now D, satisfying D 2 = i∂0 . The fermionic boundary condition (4) and its supersymmetry transformation may be rewritten in terms of the 1D supermultiplets,  µ µ 4 = P ν 4ν ,    ∂ X µ = iP µ ∂ X ν + 1 Qµ N ν 4 σ 4 ρ , 0 ν 0 ν σρ 4 (68)  µ = Qµν 4 ν , 4    µ µ µ ρ , f = Q ν f ν + 41 Q ν N νσρ 4 σ 4

418

C. Albertsson, U. Lindstr¨om, M. Zabzine µ

where N νσρ is the Nijenhuis tensor for R ν , (79). In terms of the 1D superfields, these may be concisely written as DK µ = P µν (K)DK ν , S µ = Qµν (K)S ν .

(69)

It is clear from conditions (69) that the multiplet (X µ , 4 µ ) may be thought of as living ˜ µ , f µ ) lives in the Dirichlet along the Neumann directions, whereas the multiplet (4 directions. C. Supercurrents Here we briefly sketch how to derive the supercurrents that define the supersymmetry currents and stress tensor discussed in Sect. 3. Our derivation here includes a nonvanishing background B-field; to obtain the currents relevant to the B = 0 case, one just puts Bµν,ρ = Bµν = 0 in the result below. We start from the superspace formulation of the non-linear sigma model, where we have promoted the worldsheet to a superspace by supplementing the ordinary worldsheet coordinates ξ ± with a pair of Grassmann coordinates θ ± . The model has the same form as (1), except now we make it locally supersymmetric by introducing the supervielbein EMA as well as replacing the flat superderivatives D± by covariant ones, ∇± . We have
S = d 2 ξ d 2 θ E∇+ µ ∇− ν eµν (), (70) where E is the determinant of the supervielbein. The indices run over the lightcone coordinates and their Grassmann counterparts, M and A taking values (++, =, +, −). The superfield µ is defined in Appendix A, and eµν is the superfield whose lowest component is the spacetime metric plus B-field, eµν = gµν + Bµν . Varying (70) with respect to the independent components of the supervielbein, we obtain an expression of the form
 1 δS = d 2ξ d 2θ E (−1)A TAB HBA , (71) 2 A

where TAB are the supercurrents. We take the independent variations to be (H++ , H−− , H±++ , H±= ), where HBA ≡ δEAM EMB [12]. Using the equations of motion,   1 ∇+ ∇− ν gµν − ∇+ ρ ∇− ν gµρ,ν + gρν,µ − gµν,ρ +Bµν,ρ + Bρµ,ν + Bνρ,µ = 0, 2 + as well as their ∇± derivatives, all components TAB vanish except T+− + and T= . To revert to the case with global supersymmetry, we reduce the covariant derivatives to flat ones again, and we get

i µ ν µ ν ρ T+− (72) + = D+  ∂++  gµν − D+  D+  D+  Bµν,ρ , 2 i T=+ = D− µ ∂= ν gµν + D− µ D− ν D− ρ Bµν,ρ . (73) 2 These are the supercurrents that define the supersymmetry current and stress tensor via Eqs. (29)–(32).

N = 1 Supersymmetric Sigma Model with Boundaries

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D. Almost product manifolds Here we review the relevant mathematical definitions pertaining to almost product manifolds. In our use of terminology we closely follow Yano’s books [10, 13]; however, the reader should be aware that often a different terminology is used in the literature. µ Let M be a d-dimensional manifold with a (1, 1) tensor R ν such that, globally, R µν R νρ = δ µρ .

(74) µ

Then M is an almost product manifold with almost product structure R ν . If M admits a Riemannian metric gµν such that gµν = R ρµ R σν gρσ ,

(75)

then M is an almost product Riemannian manifold9 . One may define the following two tensors on M, P µν =

1 µ (δ + R µν ), 2 ν

Qµν = µ

µ

1 µ (δ − R µν ). 2 ν

(76)

µ

P and Q are globally defined and satisfy R ν = P ν − Q ν . They are orthogonal projectors, i.e., Qµν P νρ = P µν Qνρ = 0 and

P µν P νρ = P µρ ,

Qµν Qνρ = Qµρ .

The eigenvalues of R with respect to P and Q are +1 and −1, respectively. Thus P and Q define two complementary distributions such that tangent vectors v µ on M with R-eigenvalue +1 belong to P , and vectors with R-eigenvalue −1 belong to Q, P :

{v µ : R µν v ν = v µ },

Q:

{v µ : R µν v ν = −v µ }.

Clearly, rank(P ) equals the number of +1 eigenvalues of R, and rank(Q) is the number of −1 eigenvalues of R. In fact, this implies that the holonomy group of an almost product Riemannian manifold is10 O(rank(P )) × O(rank(Q)). µ At any one point x0 in M, it is always possible to find a local coordinate basis such that R, P and Q take their canonical form    m  m   δn 0 00 δn 0 µ µ R µν = , P . (77) = = , Q ν ν 0 −δ ij 0 δ ij 0 0 The almost product structure R as well as the distributions P and Q may or may not be integrable. The integrability condition for P and Q are stated as follows.11 P is completely integrable if ρ

P µγ P νσ Q [µ,ν] =

1 ρ 1 (N γ σ − R ρµ N µγ σ ) = Qρµ N µγ σ = 0, 8 4

9 This is not much of a restriction, since if the manifold ρ σ R µ R ν g˜ ρσ will satisfy (75). 10 For a pseudo Riemannian manifold the holonomy group 11

(78)

allows a metric g˜ µν , then gµν = g˜ µν +

is O(1, rank(P ) − 1) × O(rank(Q)). Integrability may also be expressed in terms of the Frobenius theorem.

420

C. Albertsson, U. Lindstr¨om, M. Zabzine µ

where N γ σ is the Nijenhuis tensor for R, ρ

ρ

N ρµν = R γµ R [ν,γ ] − R γν R [µ,γ ] .

(79)

Q is completely integrable if ρ

Qµγ Qνσ P [µ,ν] =

1 ρ 1 (N γ σ + R ρµ N µγ σ ) = P ρµ N µγ σ = 0. 8 4

(80)

ρ

R is integrable if both Q and P are integrable, i.e., if N µν = 0. In this case M is an integrable almost product manifold, also called a locally product manifold. Integrability determines the extent to which R, P and Q may keep their canonical µ form in a local neighbourhood of the point x0 . On an integrable almost product manifold, R, P and Q can always be brought to the form (77) in a whole neighbourhood µ of x0 . However, if only one of P and Q is integrable, then the canonical form can be extended only in the corresponding directions. Thus, if P is integrable, one can adopt (77) along the P -directions, and similarly for Q-integrability. In terms of transition functions on M, R-integrability means that there is a system of coordinate neighbourhoods with coordinates X µ splitting into (X n , Xi ) such that the transition functions f µ are of the form X˜ n = f n (X m ) and X˜ i = f i (X j ) so that f n,i = 0 and f i,n = 0. This is analogous to the case of almost complex manifolds (where R 2 = −1); there R-integrability implies that one can find local (anti)holomorphic coordinates with (anti)holomorphic transition functions. µ If, for a locally product manifold M, R ν is a covariantly constant tensor (i.e., µ ∇ρ R ν = 0 with respect to some connection), then the manifold is called a locally decomposable Riemannian manifold. The warped product manifold (see discussion at the end of Sect. 4) is an example of a locally product manifold which is not locally decomposable. Acknowledgements. We are grateful to Ingemar Bengtsson and Andrea Cappelli for discussions and comments. MZ would like to thank the ITP, Stockholm University, where part of this work was carried out. UL acknowledges support in part by EU contract HPNR-CT-2000-0122 and by NFR grant 650-1998368.

References 1. Ooguri, H., Oz, Y., Yin, Z.: D-branes on Calabi-Yau spaces and their mirrors. Nucl. Phys. B 477, 407 (1996) [hep-th/9606112] 2. Hori, K., Iqbal, A., Vafa, C.: D-branes and mirror symmetry. arXiv:hep-th/0005247 3. Alvarez, E., Barb´on, J.L.F., Borlaf, J.: T-duality for open strings. Nucl. Phys. B 479, 218–242 (1996) [hep-th/9603089] 4. Borlaf, J., Lozano, Y.: Aspects of T-duality in open strings. Nucl. Phys. B 480, 239–264 (1996) [hep-th/9607051] 5. Albertsson, C., Lindstr¨om, U., Zabzine, M.: N=1 supersymmetric sigma model with boundaries, II. [hep-th/0202069] 6. Haggi-Mani, P., Lindstr¨om, U., Zabzine, M.: Boundary conditions, supersymmetry and A-field coupling for an open string in a B-field background. Phys. Lett. B 483, 443 (2000) [hep-th/0004061] 7. Polchinski, J.: TASI Lectures on D-Branes. [hep-th/9611050] 8. Hashimoto, A., Klebanov, I.: Scattering of strings from D-branes. Nucl. Phys. Proc. Suppl. 55B, 118 (1997) [hep-th/9611214] 9. Fotopoulos, A.: On (α  )2 corrections to the D-brane action for non-geodesic world-volume embeddings. JHEP 0109, 005 (2001) [hep-th/0104146] 10. Yano, K.: Differential Geometry on Complex and Almost Complex Spaces. Oxford: Pergamon, 1965 11. Hori, K.: Linear Models of Supersymmetric D-Branes. [hep-th/0012179]

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12. Bastianelli, F., Lindstr¨om, U.: Induced chiral Supergravities in 2D. Nucl. Phys. B 416, 227 (1994) [hep-th/9303109] 13. Yano, K., Kon, M.: Structures of manifolds. Series in Pure Mathematics, Vol. 3, Singapore: World Scientific, 1984 Communicated by R.H. Dijkgraaf

Commun. Math. Phys. 233, 423–438 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0760-x

Communications in

Mathematical Physics

The Kernel of the Modular Representation and the Galois Action in RCFT P. Bantay ∗ Institute for Theoretical Physics, Rolland Eotvos University, Budapest, Hungary. E-mail: [email protected] Received: 28 January 2002 / Accepted: 16 August 2002 Published online: 8 January 2003 – © Springer-Verlag 2003

Abstract: It is shown that for the modular representations associated to Rational Conformal Field Theories, the kernel is a congruence subgroup whose level equals the order of the Dehn-twist. An explicit algebraic characterization of the kernel is given. It is also shown that the order of the Dehn-twist is bounded by a function of the number of primary fields, allowing for a systematic enumeration of the modular representations coming from RCFTs. Restrictions on the spectrum of the Dehn-twist and arithmetic properties of modular matrix elements are presented. 1. Introduction It has been known since the work of Cardy [1] that a most important feature of Conformal Field Theory is modular invariance: the genus one characters of the chiral algebra afford a unitary representation of the modular group
(1) = SL(2, Z), and the torus partition function is an invariant sesquilinear combination of them. In fact, this is not simply a unitary representation, because it comes with a distinguished basis formed by the genus one characters of the primary fields. The corresponding representation matrices enjoy many intriguing algebraic and arithmetic properties, the most important ones being summarized in Verlinde’s theorem [2, 3], but one may also cite the trace identities of [4] and other related results. These special features are related to the fact that the modular representation provides part of the defining data of a 3D Topological Field Theory [5, 6]. A most important characteristic of the modular representation is its kernel, the normal subgroup of
(1) whose elements are represented by the identity matrix. It has been conjectured by several authors [7–10] that the kernel is a congruence subgroup, i.e. it contains some principal congruence subgroup
(N ). This would have important consequences for a better understanding of the analytic and arithmetic properties of the genus one characters, e.g. it would make available the apparatus of Hecke operators. No ∗

Work supported by grant OTKA T32453.

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general proof of the above congruence subgroup property may be found in the literature, the best known result being that the congruence property holds if the Dehn-twist T is represented by a matrix of odd order [11], but for generic RCFTs the order of T tends to be even, as can be seen on the example of (Virasoro) minimal models. The aim of the present paper is to present a proof of the congruence subgroup property for an arbitrary RCFT. We shall do this by studying the Galois action [12, 13] in the RCFT and in suitable permutation orbifolds of it. By exploiting the knowledge of the modular representation of these permutation orbifolds, we’ll be able to give a simple description of the Galois action on arbitrary modular matrices. This in turn will allow us to write down a simple expression Eq. (27) for the matrix elements of an arbitrary modular transformation, leading to a straightforward characterization of the kernel and an immediate proof of the congruence subgroup property. It should be emphasized that the results we present are much more powerful than the congruence property itself, for they allow a more refined study of the interplay between the arithmetic properties of modular matrix elements and the group theoretic properties of the modular representation, leading for example to an arithmetic characterization of the kernel. 2. The Galois Action In this section we’ll review the basic aspects of the Galois action in RCFT, beginning with some standard facts about the modular representation, mostly to set the notation. Let’s consider a Rational Conformal Field Theory C with a finite set I of primary fields. The genus one characters χp (τ ) of the primaries p ∈ I transform
 according to a ab unitary representation of the modular group
(1), i.e. for m = ∈
(1) we have cd
  aτ + b q χp (1) = Mp χq (τ ) . cτ + d q In the sequel, we shall always denote by M the matrix representing m ∈
(1) in the basis of genus one characters.
 11 Of special interest are the matrices T and S representing t = and s = 01
 0 −1 . As s and t generate the modular group
(1), any representation matrix M 1 0 may be written in terms of S and T , although the resulting expression may be quite cumbersome. It follows from the defining relations of
(1) that ST S = T −1 ST −1 , S 4 = 1.

(2) (3)

Moreover, it is known that S 2 is the charge conjugation operator, i.e. q

(S 2 )p = δp,q ,

(4)

where q denotes the charge conjugate of the primary q. The most important properties of S and T are summarized in the celebrated theorem of Verlinde [2, 3]: 1. T is diagonal of finite order.

Kernel of Modular Representation and Galois Action in RCFT

2. S is symmetric. 3. The quantities Npqr =

425

 Sps Sqs Srs s∈I

S0s

are non-negative integers, being the dimension of suitable spaces of holomorphic blocks. Here and in the sequel, the label 0 refers to the vacuum of the theory. The basic idea in the theory of the Galois action [12, 13] is to look at the field F obtained by adjoining to the rationals Q the matrix elements of all modular transformations. One may show that, as a consequence of Verlinde’s theorem, F is a finite Abelian extension of Q. By the celebrated theorem of Kronecker and Weber this meansthat F is a subfield of some cyclotomic field Q [ζn ] for some integer n, where ζn = exp 2πi is a primitive n nth root of unity. We’ll call the conductor of C the smallest n for which F ⊆ Q [ζn ] and which is divisible by the order of the Dehn-twist. Among other things, the above results imply that the Galois group Gal (F /Q) is a homomorphic image of the Galois group Gn = Gal (Q [ζn ] /Q). But it is well known that Gn is isomorphic to the group (Z/nZ)∗ of prime residues modulo n, its elements being the Frobenius maps σl : Q [ζn ] → Q [ζn ] that leave Q fixed, and send ζn to ζnl for l coprime to n. Consequently, the maps σl are automorphisms of F over Q. According to [13], we have (for l coprime to the conductor)  q πq σl Sp = εl (q)Sp l (5) for some permutation πl ∈ Sym (I) of the primaries and some function εl : I → {−1, +1}. In other words, upon introducing the orthogonal monomial matrices q

(Gl )p = εl (q)δp,πl q

(6)

and denoting by σl (M) the matrix that one obtains by applying σl to M elementwise, we have σl (S) = SGl = G−1 (7) l S. Note that for l and m both coprime to the conductor πlm = πl πm , Glm = Gl Gm . The Galois action on T is even simpler, for T is diagonal, and its eigenvalues are roots of unity, consequently σl (T ) = T l (8) 3. Galois Action on Λ Matrices In this section we’ll study the Galois action in some appropriate permutation orbifold [14]. According to the Orbifold Covariance Principle [15], all the properties of the Galois action reviewed in the previous section should hold in the permutation orbifold, in particular the Galois action on the S-matrix elements may be described via suitable permutations πl of the primaries of the orbifold and signs εl . This will in turn allow us to determine the Galois action on #-matrices.1 1

For the definition and basic properties of #-matrices, see the Appendix.

426

P. Bantay

We fix a positive integer N , and consider the group $ generated by the cyclic permutation (1, . . . , N). One can then form the permutation orbifold C $ according to [16, 17], which is a new RCFT with explicitly known genus one characters and modular transformation matrices. Among the primary fields of the permutation orbifold C $ there is a subset J of special relevance to us. The primaries in J are labeled by triples [p, n, k], where p ∈ I is a primary of C, while n and k are integers mod N . The subset of those [p, n, k], where n is coprime to N will be denoted by J0 . It follows from the general theory of permutation orbifolds [16, 17] that the primary fields [p, n, k] ∈ J0 have vanishing S-matrix elements with the primaries not in J , while for [q, m, l] ∈ J we have
 1 −(km+ln) q mnˆ [q,m,l] S[p,n,k] = ζN #p , (9) N N   where nˆ denotes the mod N inverse of n and ζN = exp 2πi N . Let’s now consider the action of a Galois transformation σl with l coprime to N . According to the results of the previous section, we have 2   π˜ [p,n,k] [p,n,k] = ε˜ l (p, n, k) Sq l σl Sq (10) for some permutation π˜ l of the primaries of C $ and some signs ε˜ l . Lemma 1. The set J is invariant under the permutations π˜ l , i.e. π˜ l (J ) = J . For a primary [p, n, k] ∈ J0 one has  (11) π˜ l [p, n, k] = πl p, ln, k˜ for some function k˜ of l, p, n and k, and ε˜ l (p, n, k) = εl (p),

(12)

Proof. First, let’s fix [p, n, k] ∈ J . According to Eq. (9), we have   1 p n [p,n,k] S[q,1,0] = ζN−k #q N N and this expression differs from 0 for at least one q ∈ I, by the unitarity of #-matrices. Select such a q ∈ I, and apply σl to both sides of the equation. One gets that   π˜ l [p,n,k] [p,n,k] ε˜ l (p, n, k) S[q,1,0] = σl S[q,1,0] differs from 0, but this can only happen if π˜ l [p, n, k] ∈ J because [q, 1, 0] ∈ J0 . Next, for [p, n, k] ∈ J0 consider 1 −nm p Sq . ζ N N Applying σl to both sides of the above equation we get from Eq. (5), [p,n,k]

S[q,0,m] =

π˜ [p,n,k]

l ε˜ l (p, n, k)S[q,0,m] =

(13)

1 −lnm πp εl (p)Sq l . ζ N N

2 We’ll always assume that l is coprime to the conductor, but this is no loss of generality, as it can always be achieved according to Dirichlet’s theorem on primes in arithmetic progressions.

Kernel of Modular Representation and Galois Action in RCFT

427

But the lhs equals ε˜ l (p, n, k)

1 −nm p˜ ζ ˜ Sq N N

˜ Equating both sides we arrive at according to Eq. (13) if π˜ l [p, n, k] = [p, ˜ n, ˜ k]. p˜

πp −m(n−ln) ˜ Sq l .

Sq = εl (p)˜εl (p, n, k)ζN As the lhs is independent of m, we must have

n˜ ≡ ln (mod N ) and

p˜ = πl p

as well as

ε˜ l (p, n, k) = εl (p).

  Proposition 1. For l coprime to the denominator of r, ˆ σl (# (r)) = # (lr) Gl Zl (r ∗ ) = Zl (r) G−1 l #(lr),

(14)

where Zl (r) is a diagonal matrix whose order divides the denominator N of r, and lˆ is the mod N inverse of l. Proof. Write r = Nn , with n coprime to N , and recall that r ∗ = Nnˆ , where nˆ is the mod N inverse of n. Consider 1 −(k+nm) p  n  [p,n,k] S[q,1,m] = ζN #q . (15) N N Applying σl to both sides of the above equation and taking into account Eqs. (11,12), we get 

πl p,ln,k˜

εl (p)S[q,1,m]



=

1 −l(k+nm)  p  n  σl #q ζ . N N N

But from Eq. (15), the lhs equals εl (p)

1 −(k+lnm) ˜ πp #q l ζ N N




ln N



so after rearranging, we get
    ˜ p n π p ln (lk−k) σl #q = εl (p)ζN . #q l N N

(16)

As the lhs is independent of k k˜ − lk ≡ k0 (l, p, r) (mod N ) .

(17)

Of course, the point is that k0 is independent of k. If we introduce the diagonal matrix

428

P. Bantay

−k0 (l,p,r ∗ )

q

Zl (r)p = δp,q ζN

,

(18)

then it is obvious that the order of Zl (r) divides the denominator of r and the first equality in Eq. (14) holds. The second follows simply from # (r)T = # (r ∗ ).   Note that it follows from the definition of Zl (r) that Zl (0) = I and Zl (r + 1) = Zl (r). Lemma 2.

−m ˆ G−1 l Zm (lr)Gl = Zlm (r)Zl (r),

(19)

whenever both l and m are coprime to the denominator of r. Proof. This follows at once from σlm = σl σm applied to #(r).

 

Lemma 3. If n is coprime to the denominator of r, then Zln (r) = Zl (nr) Proof. Let r =

m N,

(20)

with m coprime to N . According to Eq. (9), [q,nm,0]

q

N S[p,n,0] = #p (r) . Applying σl to both sides we get  q  −nk0 (l,q, nm πq [πl q,lnm,k0 ] N ) = εl (q)ζN # (lr)pl , σl #p (r) = N εl (q)S[p,n,0] i.e.

   ∗ # (lr) Gl Zl r ∗ = # (lr) Gl Zln nr ˆ

by Eq. (14), i.e.

  Zl (r) = Zln nr ˆ .

But this is equivalent to the assertion.

 

Theorem 1. For all l coprime to the conductor l G−1 l T Gl = T . 2

(21)

Proof. Take N equal to the order of T . Clearly, N divides the conductor. By Eq. (31) of the Appendix,
 1 1 2 1 # = T − N S −1 T −N ST − N = T − N . N But







 1 l 1 l 1 1 l σl2 # # Z = σl # Gl Zl = Zl G−1 G l l l N N N N N N


 l = T N because both Zl and T are diagonal, hence they commute. If N is even, then taking the N th power of the last equation gives the result. If N is odd, then there exists k such that 2 2k ≡ 1 (mod N ), and taking the kN th power of the equation gives again Eq. (21).   or in other words

2

− 2lN

− N2 G−1 l T

Gl Zl2

Kernel of Modular Representation and Galois Action in RCFT

429

Remark 1. This result has been conjectured in [11], where some of its consequences (to be reviewed in the next section) had been derived. Corollary 1. 2 G−1 l MGl = σl (M)

(22)

for any modular matrix M. Proof. We have just seen that it holds for T , and it also holds for S by Eq. (7). But s and t generate
(1), consequently the claim should hold for any m ∈
(1).   Proposition 2. If l is coprime to both the conductor and the denominator of r, then r l r l G−1 Zl (r) . l T Gl = T 2

Proof. Let’s write r =

n N . We

(23)

know from [16, 17] that ω[p,n,k] = ζNnk ωp , 1/N

   c where ωp = exp 2πi (p − 24 is the exponentiated conformal weight of the primary p, i.e. the matrix element Tpp . By Eq. (21) l 2 /N

ωp i.e. Zll




= ωπ˜ l [p,1,0] = ω[πl p,l,k0 ] = ζNlk0 ωπl p ,

1 N

1/N



−lk0 (l,p,1/N)

pp

or in other words Zll




= ζN

1 N



1/N

−l 2 /N

= ωπl p ωp

l2

−N N = G−1 . l T Gl T 1

Taking the nth power of the last equation and using Eq. (20) gives the result, because Zl and T commute.   Lemma 4. Suppose that l is coprime to the denominator of both r1 and r2 . Then Zl (r1 )Zl (r2 ) = Zl (r1 + r2 ) .

(24)

Proof. Tl

2 (r +r ) 1 2

r1 +r2 Zll (r1 )Zll (r2 ) = G−1 Gl = T l l T

according to Proposition 2.

 

2 (r +r ) 1 2

Zll (r1 + r2 )

430

P. Bantay

4. The Galois Action Revisited In this section we translate the results of the previous section about the Galois action on #-matrices to statements about the Galois action on modular matrix elements. Proposition 3. For l coprime to the conductor, ˆ

Gl = S −1 T l ST l ST l ,

(25)

where lˆ denotes the inverse of l modulo the conductor. Proof. Let’s apply σl to both sides of the modular relation Eq. (2). One gets −l −l SGl T l G−1 l S = T SGl T .

After rearranging and using Eq. (21) we get the assertion.

 

Proposition 4. The conductor equals the order N of T , and F = Q [ζN ]. Proof. Clearly, N divides the conductor, for the eigenvalues of T are roots of unity. But for l ≡ 1 (mod N) we have σl (T ) = T l = T and

ˆ

σl (S) = T l ST l ST l = S

according to Eqs. (8,7,25), consequently all such l fixes F , i.e. F ⊆ Q [ζN ]. On the other hand, if σl fixes F then l ≡ 1 (mod N ), i.e. Gal (F /Q) = Gal (Q [ζN ] /Q).   Remark 2. The above two propositions had been derived in [11], upon postulating Eq. (21). Earlier, they have been conjectured in [18]. Lemma 5. If Gl is diagonal, then Gl = ±I. Proof. Gl diagonal means that πl is the identity permutation. If we apply σl to S0p , we get   σl S0p = εl (0) S0p = εl (p) S0p because πl p = p. But S0p > 0, consequently εl (p) = εl (0) = ±1, proving the lemma.   Proposition 5. Let N0 denote the order of the matrix ω0−1 T , i.e. the least common multiple of the denominators of the conformal weights. Then N = eN0 , where the integer e divides 12. Moreover, the greatest common divisor of e and N0 is either 1 or 2. Proof. That N0 divides N is obvious. If we consider the matrix T N0 , then it will equal ζ = ω0N0 times the identity matrix. But N is the exact order of T , consequently ζ is a primitive eth root of unity. On the other hand, Eq. (21) implies N0 G−1 Gl = T l l T

for all l coprime to N , i.e.

ζ l = ζ. 2

2N 0

Kernel of Modular Representation and Galois Action in RCFT

431

In other words, the exponent of (Z/eZ)∗ should divide 2, and this can only happen if e divides 24. Next, let’s consider l = 1 + 6N0 . By the above result, l is coprime to both N0 and e, consequently it is also coprime to N . If lˆ denotes its inverse mod N , then we have lˆ =



1 + 6N0 if N0 is odd, 1 − 6N0 if N0 is even.

According to Eq. (25), ˆ

Gl = ST l S −1 T l ST l = ζ 6(2±1) I. But if Gl is diagonal, then it equals ±I, so that ζ 12(2±1) = 1. Together with ζ 24 = 1, this yields ζ 12 = 1. Finally, let D denote the gcd of e and N0 , and consider l = 1+ N D . It is straightforward N ˆ that l is coprime to N, and its inverse modulo N is l = 1 − D . But e

Gl = ζ D I, and once again this should equal ±I, consequently D divides 2.

 

Remark 3. As it turns out, there are no further restrictions on the value of N/N0 , i.e. each divisor of 12 occurs for some RCFT. Techniques similar to the above yield that N0 is odd if 4 divides e. Corollary 2. N0 times the central charge is an even integer.   c . But ω0N = 1, so by the Proof. If c denotes the central charge, then ω0 = exp −π i 12 12N0 above ω0 = 1, proving the claim.   Proposition 6. There exists a function N (r) such that the conductor of a theory with r primary fields divides N (r). Proof. Let’s denote by s(r) the exponent of the symmetric group Sr of degree r. As s(r) is the identity permutation, i.e. the permutations πl belong to Sr , it follows that πl s(r) s(r) Gl is diagonal. By Lemma 5 this means that Gl = ±I, which in turn implies that l 2s(r) ≡ 1 (mod N) for all l, i.e. the exponent of the group (Z/N Z)∗ divides 2s(r). Taking N(r) to be the greatest integer satisfying this last condition proves the proposition.   According to this proposition, for a given number of primary fields one has only a finite number of consistent choices for the conductor N and the matrix T . The upper bound for N(r) given in the proof may be greatly improved by exploiting the known properties of the matrices Gl (e.g. that they commute), which leads to the following table for small values of r:

432

P. Bantay r 2 3 4 5

N(r) 240 5040 10080 1441440

5. The Kernel of the Modular Representation Recall that the kernel K consists of those modular transformations which are represented by the identity matrix, i.e.  q K = m ∈
(1) | Mp = δp,q .
 ab Proposition 7. Let m = ∈ SL(2, Z), l coprime to the conductor, lk = 1 + αc cd and
 la b + αad m ˆ = . c kd Then

ˆ l T −αld . σl (M) = MG

(26)

Proof. According to Eq. (14) 

σl (M) = σl T because

 a ∗ c

a/c

#

a 

= dc . But

c


#

al c

T 

d/c



=T

al/c




al # c




 d Gl Zl T dl/c c

ˆ −kd/c = T −al/c MT

so ˆ −kd/c Gl Zl σl (M) = MT


 d T ld/c . c

From Eq. (23) T −kd/c Gl = Gl T −l

2 kd/c


Zl

−lkd c

 .

Putting all this together, we get ˆ l T −ld(1+αc)/c Zl σl (M) = MG




−d c




 d Zl T ld/c c

which proves the claim according to Eq. (24).  
 ab Corollary 3. If m = ∈
(1) with d coprime to the conductor, then cd σd (M) = T b S −1 T −c σd (S) . Proof. If we take l = d and k = a in Eq. (26), then α = b and m ˆ = t b s −1 t −c st b .

(27)  

Kernel of Modular Representation and Galois Action in RCFT

433


Theorem 2. Let d be coprime to the conductor N . Then kernel K if and only if

ab cd

 ∈
(1) belongs to the

σd (S) T b = T c S.

Proof. If m ∈ K, then σd (M) is the identity matrix, and Eq. (27) implies the result.

(28)  

Remark 4. Apparently, the above criterion does only apply in case d is coprime to the conductor. But if m ∈ K, then 
a − kc b − k 2 c + k(a − d) −k k t mt = c d + kc also belongs to K. Dirichlet’s theorem on primes in arithmetic progressions ensures that d + kc is coprime to the conductor for infinitely many k, reducing the general case to the coprime one. For the next result, recall that




ab
1 (N ) = ∈
(1) | a, d ≡ 1 (mod N ) , c ≡ 0 (mod N ) cd and



(N ) =

ab cd



∈
1 (N ) | b ≡ 0 (mod N ) .

Theorem 3. K ∩
1 (N ) =
(N ) . In particular, K is a congruence subgroup of level N .
 ab Proof. If ∈
1 (N), then Eq. (28) reduces to ST b = S, whose only solution is cd b ≡ 0 (mod N).   The above result means that the modular representation factors through SL2 (N ) ∼ =
(1)/
(N ), i.e. there exists a representation D of SL2 (N ) such that the modular representation is the composite map D ◦ µN , where we denote by µN the natural homomorphism
(1) → SL2 (N ). The representation D gives us an elegant description of the Galois action. Definition 1. For l coprime to N , define the automorphism τl : SL2 (N ) → SL2 (N ) by

 a lb ab τl = ˆ , (29) cd lc d where lˆ is the mod N inverse of l. Theorem 4. σl ◦ D = D ◦ τl .

(30)

434

P. Bantay

Proof. As a homomorphic image of
(1), SL2 (N ) is generated by µN (s) and µN (t), so it is enough to verify Eq. (30) for these matrices. For t, the lhs reads σl ◦ D ◦ µN (t) = T l while the rhs is


D

1l 01



= Tl

and these are clearly equal. For the case of s, the lhs reads ˆ

σl (S) = SGl = T l ST l ST l according to Eq. (25), while the rhs is
D It is easy to check that

proving the claim.




0 −l lˆ 0



0 −l lˆ 0

 .

ˆ

≡ t l st l st l (mod N ) ,

 

Note that in terms of the representation D we have
 lˆ 0 Gl = D , 0l i.e. the matrices Gl represent the diagonal subgroup of SL2 (N ). 6. Examples Let’s look at a couple of simple examples to illustrate the above results. First, let’s consider the Lee-Yang model, i.e. the minimal model M (5, 2). It has central charge 1 c = − 22 5 and two primary fields, whose conformal weights are 0 and − 5 . The S matrix reads

  4π 2π sin − sin  2 5 5 

 . S=√   2π  4π 5 sin sin 5 5 The conductor N equals 60, while N0 = 5, providing an example where the ratio N/N0 attains the upper bound of the Lemma. As it turns out, the image K¯ = µN (K) of the kernel is a non-Abelian group of order 192, whose center is Z22 , and whose derived subgroup is Z32 . A small generating set for K¯ is provided by the matrices




 19 5 31 35 56 5 , , . 5 14 5 56 35 31

Kernel of Modular Representation and Galois Action in RCFT

435

As a second example, let’s consider the Ising model, i.e. the minimal model M (4, 3). The central charge is c = 21 , there are 3 primary fields of conformal weights 0, 21 and 1 16 , and √   1 1 √2 1 S= 1 1 − 2. √ 2 √ 2− 2 0 The conductor equals 48, while N0 = 16. In this case K has order 64, its derived subgroup is Z2 , while its center is Z22 × Z4 . A small generating set for K is given by the SL2 (48) matrices





 43 40 29 40 21 8 35 40 , , , . 40 35 40 37 40 45 40 43

Finally, the following table contains the relevant data for some minimal models, the pq th entry giving respectively the conductor N , the ratio N/N0 , and the index [K :
(N)] = |µN (K)| for the minimal model M(q, p) (the first entry in the first column being the Lee-Yang model discussed above).

p\q 2 3 4 5 6 7 8 9 10

5 60;12;192 40;2;16 240;3;64

6

120;1;4

7 42;6;48 168;6;128 336;3;128 840;6;256 168;1;8

8

9 36;4;24

32;1;4

120;3;32

480;3;32 672;3;64

144;1;8 360;2;32 504;2;32 288;1;4

840;3;64 360;1;4

As an aside, we note that for all minimal models M (q, p)   q p=2 4q p = 3 N0 =  4pq p > 3 and N/N0 = for p > 3, while

10

 6   3 N/N0 = 2   1

6 Gcd(6, pq) q q q q

≡1 ≡4 ≡5 ≡2

(mod 6) (mod 6) (mod 6) (mod 6)

11 33;3;16 88;2;16 528;3;128 1320;6;256 264;1;8 1848;6;256 1056;3;64 792;2;32 1320;3;64

436

for p = 3 and

P. Bantay

 12    6    4 N/N0 = 3     2   1

q ≡ 1, 5, 13, 17 (mod 24) q ≡ 7, 23 (mod 24) q ≡ 9, 21 (mod 24) q ≡ 11, 19 (mod 24) q ≡ 15 (mod 24) q ≡ 3 (mod 24)

for p = 2. 7. Summary As we have seen in the previous sections, the theory of the Galois action supplemented by the Orbifold Covariance Principle leads to a host of interesting results about the arithmetic and group theoretic properties of the modular representation. In particular, we have been able to show that the kernel of the modular representation is always a congruence subgroup, whose level equals the conductor, i.e. the order of the Dehn-twist T . Equation (28) gives a simple characterization of the kernel, which allows for efficient algorithms to determine the kernel explicitly. We have also determined the Galois action on arbitrary modular matrices, and gave an effective formula to compute the representation matrix of any modular transformation, Eq. (27). Moreover, the results of Sect. 4 put severe restrictions on the spectrum of the Dehn-twist T , and imply that, for a given number of primary fields, there is only a finite number of allowed modular representations. It seems unlikely that the above results could be derived without using the theory of the Galois action, and provide a nice example of the use of arithmetic in mathematical physics. Of course, a host of questions remain open. An obvious one is to find a direct characterization of the kernel. While Eq. (28) gives a simple criterion to decide whether a modular transformation belongs to the kernel in case its diagonal elements are coprime to the conductor, and in principle the general case can be reduced to this one, a simple criterion valid for any modular transformation would be welcome. One might also wonder to find a small set of numerical data that would characterize the kernel completely. As we have seen, these would include the parameters N and N0 , which already determine the kernel to a great extent, but are not enough to characterize it completely. Another line of study is to investigate the algebro-geometric properties of the modular curve X (K) associated to the kernel. This is a compact Riemann-surface, whose complex structure can be described by standard techniques, e.g. one can determine the period matrix, etc. It is also known that the field of meromorphic functions on X (K) is a Galois extension of C (j ) with Galois group
(1)/K, where j is the classical modular invariant. As the genus one characters of the primaries are meromorphic functions on this modular curve, it is tempting to expect that some deeper properties of the RCFT should be encoded in structure of the curve X (K). Still another interesting question concerns the representation D of SL2 (N ) introduced in Sect. 5. We know already some of its properties, e.g. its kernel and its behavior under Galois transformations, Eq. (30), but there is clearly much more to learn about it. As the representation theory of SL2 (N ) is known, one would be interested to know which irreducible representations appear in D and with which multiplicities, etc. Of course, one should bear in mind that D is not simply a representation, but a representation with a choice of a distinguished basis.

Kernel of Modular Representation and Galois Action in RCFT

437

Finally, let’s make some comments about the relevance of the above on the classification program. According to Proposition 6, for a given number of primary fields it is in principle possible to enumerate all matrices S and T that are compatible with the results of this work, i.e. those which can occur as the generators of the modular representation of a consistent Rational Conformal Field Theory. Of course, it may happen that such a list would contain spurious entries which do not correspond to any RCFT, but one may hope that using all known properties of the modular representation, these spurious solutions can be discarded, leading to a complete list. Clearly, even if the above program succeeds, it would be just a first step toward the classification, for the modular data only provide a partial set of invariants of RCFTs. 8. Appendix In this appendix we review the definition and the basic properties of the #-matrices used in Sect. 3. See [19] for more details. Let r = nk be a rational number in reduced form, i.e. with n > 0 and k and n coprime.
 ky Choose integers x and y such that kx − ny = 1, and define r ∗ = xn . Then m = nx belongs to
(1), and we define the matrix # (r) via q

q

# (r)p = ωp−r Mp ωq−r or symbolically

∗

∗

#(r) = T −r MT −r . Of course, one should fix some definite branch of the logarithm to make the above definition meaningful, but different choices lead to equivalent results. It is a simple matter to show that #(r) is well defined, i.e. does not depend on the actual choice of x and y. In the same vein, #(r) is periodic in r with period 1, i.e. #(r + 1) = #(r). For r = 0 we just get back the S matrix #(0) = S, and for a positive integer n we have
 1 1 1 = T − n S −1 T −n ST − n . # n Finally, one can show that and

(31)

 q p # r ∗ p = #(r)q q

q

# (−r)p = # (r)p . Acknowledgement. Many thanks to Terry Gannon for explaining to me his work on the Galois action and the congruence subgroup problem.

438

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Cardy, J.: Nucl. Phys. B270, 186 (1986) Verlinde, E.: Nucl. Phys. B300, 360 (1988) Moore, G., Seiberg, N.: Commun. Math. Phys. 123, 177 (1989) Bantay, P.: hep-th/0007164 Witten, E.: Commun. Math. Phys. 121, 351 (1989) Fr¨ohlich, J., King, C.: Int. J. Mod. Phys. A20, 5321 (1989) Moore, G.: Nucl. Phys. B293, 139 (1987) Eholzer, W.: Commun. Math. Phys. 172, 623 (1995) Eholzer, W., Skoruppa, N.-P.: Commun. Math. Phys. 174, 117 (1995) Bauer, M., Coste, A., Itzykson, C., Ruelle, P.: J. Geom. Phys. 22, 134 (1997) Coste, A., Gannon, T.: math-QA/9909080 de Boere, J., Goeree, J.: Commun. Math. Phys. 139, 267 (1991) Coste, A., Gannon, T.: Phys. Lett. B323, 316 (1994) Klemm, A., Schmidt, M.G.: Phys. Lett. B245, 53 (1990) Bantay, P.: Phys. Lett. B488, 207 (2000) Borisov, L., Halpern, M.B., Schweigert, C.: Int. J. Mod. Phys. A13, 125 (1998) Bantay, P.: Phys. Lett. B419, 175 (1998) Bauer, M.: Advanced Series in Mathematical Physics. 24, 152 (1997) Bantay, P.: Nucl. Phys. B633, 365 (2002)

Communicated by R. H. Dijkgraaf

Commun. Math. Phys. 233, 439–461 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0775-3

Communications in

Mathematical Physics

Linear Stability in an Ideal Incompressible Fluid Y. Latushkin1,∗ , M. Vishik2,∗∗ 1 2

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: [email protected] Department of Mathematics, University of Texas, Austin, TX 78712-1082, USA. E-mail: [email protected]

Received: 20 May 2002 / Accepted: 5 September 2002 Published online: 10 January 2003 – © Springer-Verlag 2003

Abstract: We give an explicit construction of approximate eigenfunctions for a linearized Euler operator in dimensions two and three with periodic boundary conditions, and an estimate from below for its spectral bound in terms of an appropriate Lyapunov exponent. As a consequence, we prove that in dimension 2 the spectral and growth bounds for the corresponding group are equal. Therefore, the linear hydrodynamic stability of a steady state for the Euler equations in dimension 2 is equivalent to the fact that the spectrum of the linearized operator is pure imaginary. In dimension 3 we prove the estimate from below for the spectral bound that implies the same equality for every example where the relevant Lyapunov exponents could be effectively computed. For the kinematic dynamo operator describing the evolution of a magnetic field in an ideally conducting incompressible fluid we prove that the growth bound equals the spectral bound in dimensions 2 and 3. 1. Introduction In this paper we study spectral properties of a linearized Euler equation in dimensions d = 2 or d = 3. The main tool in our analysis is a system of 3d ordinary differential equations called the bicharacteristic amplitude system. Its dynamics is related to the essential spectrum of the linearized Euler operator. The bicharacteristic amplitude system was studied in [FV, V, VF], see also related work in [E, ES, FSV, LH1, LH2, Lu], as well as a general discussion of the linearized Euler equations in [C, DR, L, Y]. We give an explicit geometric construction for approximate eigenfunctions of the linearized Euler operator. This construction is of independent interest, and also leads to ∗ The first author was partially supported by the Twinning Program of the National Academy of Sciences and National Science Foundation, and by the Research Council and Research Board of the University of Missouri. ∗∗ The second author was partially supported by the National Science Foundation grant DMS 9876947 and CRDF grant RM1-2084.

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an estimate from below for the spectral bound of this operator in terms of the maximal Lyapunov exponent of the bicharacteristic amplitude system. This estimate implies that the spectral bound for the linearized Euler operator in dimension 2 coincides with the growth bound for the corresponding group. Thus, as in the classical Lyapunov theory, the linear stability of a steady state for the Euler equation is characterized in terms of the spectrum of the corresponding linearized operator. In dimension 3 we get an estimate of the spectral bound from below through a modified Lyapunov exponent of the bicharacteristic amplitude system restricted to an invariant subbundle. Also, this estimate gives a direct proof of a general sufficient condition for linear hydrodynamic instability obtained in [FV, V, VF] using the technique of Fourier integral operators. For a strongly continuous semigroup {etA }t≥0 on a Hilbert space we let s(A) = sup{Rez : z ∈ σ (A)}

and

ω(A) = lim t −1 log etA

t→∞

respectively, denote its spectral and growth bound. Note that for each t0 > 0 we have ω(A) = t0−1 log r(et0 A ). Here and below r(·) is the spectral radius and σ (·) is the spectrum. Under certain (and rather restrictive) assumptions on A the exponential stability of the semigroup {etA } is controlled by the position of σ (A), that is, the equality ω(A) = s(A) holds. For instance, ω(A) = s(A) provided A is a generator of an analytic semigroup. However, generally only the inequality ω(A) ≥ s(A) holds. There exist numerous examples where the inequality ω(A) > s(A) is strict, see [CL, EN, vN] for a detailed discussion of this phenomenon. The strict inequality ω(A) > s(A) occurs even for the generator A that appears as a result of first-order perturbation for the two-dimensional wave equation, see [R]. For d = 2 or d = 3 let u : Td → Rd denote a smooth divergence-free steady state solution to the nonlinear Euler equation on the torus Td . On the space Ls2 = Ls2 (Td ; Rd ) of divergence-free vector-fields w : Td → Rd consider the linearized Euler operator, L, defined as follows: Lw = − u, ∇w − w, ∇u − ∇p. Here divw = 0, and p : Td → R is the pressure. The divergence-free condition is understood in the sense of distributions. We say that the steady-state u is linearly hydrodynamically stable if ω(L) = 0. A natural question arises if the linear hydrodynamic stability is determined by the position of the spectrum σ (L), that is, if the equality ω(L) = s(L) holds. In this paper we give an affirmative answer to this question. Note that the spectral mapping theorem σp (etL ) = etσp (L) , t = 0, always holds for the point spectrum σp (·). Thus, if L has an eigenvalue with a positive real part, then, trivially, u is not linearly hydrodynamically stable. Also, since u is a steady-state for the nonlinear Euler equation, u satisfies the equation Lu = 0. Thus, s(L) ≥ 0. The equality s(L) = ω(L) shows that the condition s(L) = 0 is not only necessary, but also sufficient for the linear hydrodynamic stability of u. To prove the equality ω(L) = s(L), we give an explicit construction for a sequence of approximate eigenfunctions for L. That is, we find a point z ∈ σap (L), the approximate point spectrum, and a sequence of wn ∈ Ls2 with wn = 1 such that (L − z)wn = O(n−1 ) as n → ∞. This construction allows us to give a direct estimate for s(L) from below in terms of a Lyapunov exponent for the following bicharacteristic amplitude system:

Linear Stability in an Ideal Incompressible Fluid

x˙ = u(x), x(0; x0 ) = x0 , x0 ∈ Td ;
T ∂u ˙ξ = − (x(t; x0 )) ξ, ξ(0; x0 , ξ0 ) = ξ0 ; ∂x ∂u b˙ = − (x(t; x0 ))b ∂x  ∂u ξ(t; x0 , ξ0 ) ξ(t; x0 , ξ0 ) +2 (x(t; x0 ))b, , ∂x |ξ(t; x0 , ξ0 )| |ξ(t; x0 , ξ0 )|

441

(1.1) (1.2)

b(0; ξ0 , b0 ) = b0 . (1.3)

Here and throughout the paper ·, · and | · | denote the scalar product and norm in Rd , “T ”-transposition, ξ0 ∈ (Rd )∗ is a covector, b0 ∈ Rd , and ξ0 ⊥b0 . Let  µ = limt→∞ t −1 log sup |b(t; x0 , ξ0 , b0 )| : (x0 , ξ0 , b0 ) ∈ Td × (Rd )∗ × Rd ,  (1.4) |ξ0 | = |b0 | = 1 and ξ0 ⊥b0 denote the maximal Lyapunov exponent for (1.1)–(1.3). Note that the limit in (1.4) exists by a standard subadditivity argument. It was proved in [FV, VF] that ω(L) ≥ µ and in [V] that µ = t0−1 log ress (et0 L ), t0 > 0, (1.5) where ress (·) is the essential spectral radius. In this paper we use an elementary dynamical construction to relate the spectral bound s(L) and a Lyapunov exponent for the bicharacteristic amplitude system. Let  µ⊥ = limt→∞ t −1 log sup |b(t; x0 , ξ0 , b0 )| : (x0 , ξ0 , b0 ) ∈ Td × (Rd )∗ × Rd ,  u(x0 )⊥ξ0 , |ξ0 | = |b0 | = 1 and ξ0 ⊥b0 . (1.6) The difference with (1.4) is the requirement u(x0 )⊥ξ0 . We stress that if d = 2 then µ⊥ = µ, see Lemma 2.4 (iv) below. Remark 1.1. In all examples we know for d = 3 where both quantities µ and µ⊥ could be effectively computed, the condition µ = µ⊥ holds. We do not know if this holds for all smooth solutions to the stationary Euler equation on T3 .  Theorem 1.2. If d = 2 or d = 3 then for the spectral bound s(L) of the linearized Euler operator and the Lyapunov exponent µ⊥ of the bicharacteristic amplitude system one has: s(L) ≥ µ⊥ . (1.7) Remark 1.3. We may and will assume throughout the paper that µ⊥ > 0. Indeed, since u is a steady state solution for the Euler equation, Lu = 0. Thus, if µ⊥ = 0 then s ≥ 0 = µ⊥ , and Theorem 1.2 trivially holds.  Due to the general inequality ω(L) ≥ s(L), if d = 2 or if d = 3 and µ = µ⊥ , then Theorem 1.2 gives an alternative proof of the inequality ω(L) ≥ µ from [FV, VF]. The plan of the proof of Theorem 1.2 is outlined in Sect. 2; technical details are given in Sects. 3 and 4. The proof of Theorem 1.2 also yields the following fact.

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Proposition 1.4. Let µ⊥ > 0. Assume d = 2 or d = 3 and the set of the periodic points in T3 for the flow induced by the x-equation (1.1) has an empty interior. Then µ⊥ + iR ⊂ σap (L). Theorem 1.2 and (1.5) imply the following corollary, saying that the linear hydrodynamic stability is, in fact, controlled by the position of the spectrum of L. Corollary 1.5. If d = 2 or if d = 3 and µ = µ⊥ [see Remark 1.1], then the spectral and growth bounds for the linearized Euler operator coincide: s(L) = ω(L). Proof. Recall that the spectral mapping theorem σp (et0 L ) = et0 σp (L) , t0 = 0, holds for the point spectrum (see, e.g., [EN]). Therefore, aiming to prove the inequality s(L) ≥ ω(L), we may assume that r(et0 L ) = ress (et0 L ). Now, using Theorem 1.2, equality µ = µ⊥ , and (1.5), we conclude: s(L) ≥ µ⊥ = µ = t0 −1 log ress (et0 L ) = t0 −1 log r(et0 L ) = ω(L).   Finally, we consider on Ls2 the “push-forward” group {etM }t∈R generated by M, the kinematic dynamo operator for an ideally conducted fluid, defined as follows: Mw = − u, ∇w + w, ∇u,

div w = 0, div u = 0.

(1.8)

Equation w˙ = Mw is the equation of magneto-hydrodynamics; it describes the evolution of a magnetic field in an ideally conducting fluid flow, see, e.g., [AK, CG] for further references. Theorem 1.6. On Ls2 one has ω(M) = s(M). More results of this sort on the space of continuous divergence-free vector-fields can be found in [CLM1, CLM2]. 2. Main Construction Let χ t : x0 → x(t; x0 ), t ∈ R, denote the flow on Td generated by the x-equation (1.1). Note, that the ξ -equation (1.2) defines a linear skew-product flow over χ t on  := Td × (Rd )∗ . Let t (x0 ) = ([Dχ t (x0 )]T )−1 ,

x0 ∈ Td ,

denote the corresponding cocycle; here Dχ t is the differential. In other words, ξ(t; x0 , ξ0 ) = t (x0 )ξ0 , and t+τ (x0 ) = t (χ τ x0 )τ (x0 ), x0 ∈ Td , ξ0 ∈ (Rd )∗ , t, τ ∈ R. The system of x- and ξ -equations (1.1)–(1.2) defines a flow {ϕ t }t∈R on Td × (Rd )∗ by the rule ϕ t : (x0 , ξ0 ) → (χ t x0 , t (x0 )ξ0 ). Since ξ(t; x0 , ξ0 ), u(x(t; x0 )) = const is a first integral for (1.1)–(1.2), the flow {ϕ t } leaves invariant the set ⊥ ⊂  defined as follows: ⊥ := {(x0 , ξ0 ) ∈ Td × ((Rd )∗ \ 0) : ξ0 ⊥u(x0 )}. Due to the homogeneous form of the b-equation (1.3), that is, to the identity b(t; x0 , ξ0 , b0 ) = b(t; x0 , ±

ξ0 , b0 ), |ξ0 |

(2.1)

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443

only the direction [ξ0 ] ∈ P((Rd )∗ ), the projective space, of ξ0 matters in (1.3). Thus, one can think that ⊥ ⊂ Td × P((Rd )∗ ), ϕ t (x0 , ξ0 ) = (χ t x0 , [t (x0 )ξ0 ]), etc. Note that b(t; x0 , ξ0 , b0 ), ξ(t; x0 , ξ0 ) = const is also a first integral of (1.1)–(1.3). Therefore, the b-equation (1.3) defines a linear skew-product flow over the flow {ϕ t } on the bundle B with the base ⊥ and fibers {b0 : b0 ⊥ξ0 }. Let B t (x0 , ξ0 ) denote the corresponding cocycle. In other words, b(t; x0 , ξ0 , b0 ) = B t (x0 , ξ0 )b0 , and B t+τ (x0 , ξ0 ) = B t (ϕ τ (x0 , ξ0 ))B τ (x0 , ξ0 ), t, τ ∈ R. Thus, µ⊥ is the maximal Lyapunov exponent for the cocycle {B t }. We will make use of the following first integrals for system (1.1)–(1.3). If d = 2 then |b(t; x0 , ξ0 , b0 )||ξ(t; x0 , ξ0 )| = const,

t ∈ R,

(2.2)

for each (x0 , ξ0 ) ∈  and b0 ⊥ξ0 , see (4.7) in [FSV, Prop. 4.3] for the proof. If d = 3 then   (2.3) Area b(t; x0 , ξ0 , b0 ), b(t; x0 , ξ0 , b1 ) |ξ(t; x0 , ξ0 )| = const, t ∈ R, for each (x0 , ξ0 ) ∈  and any two linearly independent b0 and b1 that are perpendicular to ξ0 , see [FV, Prop.2.1] for the proof. To prove Theorem 1.2, we will show that for some z ∈ C with Re z = µ⊥ one has z ∈ σap (L). That is, we need to prove that for each n ∈ N there exists a nonzero w = w(n) ∈ Ls2 such that

Lw − zw Ls2 ≤ cn−1 w Ls2 .

(2.4)

Here and throughout the paper c stands for a generic constant. Denote L0 w = − u, ∇w − w, ∇u. We will be looking for the approximate eigenfunction w = w(n) as in (2.4), supported in a thin flow-box for the flow χ t , and of the form

eis/$ a . (2.5) w = $∇ × ∇s × |∇s|2 Here the phase s : Td → R, the amplitude a : Td → Rd , and $ > 0 will be defined below. Suppose that we have selected C ∞ –smooth s and a in (2.5) (independent of $) in a way that ∇s⊥a. An elementary calculation, using the formula ∇ × (A × B) =

B, ∇A − A, ∇B + AdivB − BdivA, shows that for any $ > 0 one has:

a is/$ is/$ (2.6) w = −ie a + $e w0 , where w0 = ∇ × ∇s × |∇s|2 does not depend on $. Note that div w = 0. Set   ∇s ∂u and a, q=2 ∂x |∇s|2

p = $qeis/$ .

(2.7)

Then, with this choice of pressure, and using (2.6), after a short calculation, for each z ∈ C one has: Lw − zw = L0 w − ∇p − zw = u, ∇seis/$ [−a/$ − iw0 ] −ieis/$ [(L0 − z)a + q∇s] + $eis/$ [L0 w0 − zw0 − ∇q].

(2.8)

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Choose $ so small that $ · L0 w0 − zw0 − ∇q L2 ≤ cn−1 a||L2 .

(2.9)

Suppose that z, s and a were selected such that u, ∇s = 0 and, for q as in (2.7), one has:

(L0 − z)a + q∇s L2 ≤ cn−1 a L2 . (2.10) Note that w L2 ≥ 21 a L2 for $ > 0 sufficiently small according to (2.6). Applying (2.9), (2.10), and (2.8), we have (2.4). The choice of z, s and a, satisfying (2.10), will be based on an appropriate choice of a point x∗ ∈ Td , u(x∗ ) = 0, a covector ξ∗ ∈ (Rd \ 0)∗ with ξ∗ ⊥u(x∗ ), and a vector b∗ with b∗ ⊥ξ∗ . We will choose a, supported in a thin flow-box ' = {χ t σ : σ ∈ '0 } through a cross-section '0  x∗ such that a(x) = γ (t, σ )b(t; σ, ∇s(σ ), b(σ )),

x = χ t σ ∈ ',

σ ∈ '0 ,

(2.11)

where γ is an appropriate smooth scalar “cut-off” function, and ∇s(σ ) and b(σ ), b(σ )⊥∇s(σ ), are appropriate initial data for ξ – and b–equations (1.2) and (1.3). Note that u, ∇s = 0 implies T ∂u

u, ∇∇s = − ∇s. (2.12) ∂x d ∇s(χ t σ ), we conclude that ξ(t; σ ; ∇s(σ )) = ∇s(χ t σ ) dt solves the ξ -equation (1.2). Remark, that ∇s(σ )⊥b(σ ) implies ∇s(x)⊥a(x) for all x ∈ ', since a(x) for x = χ t σ is parallel to b(t; σ, ∇s(σ ), b(σ )) and b, ξ  = const is a first integral for (1.1)–(1.3). With the choice of q as in (2.7) and a as in (2.11) one has:   d ∂u ∇s ∂u (L0 − z)a + q∇s = − a(χ t σ ) − ∇s a − za + 2 a, dt ∂x ∂x |∇s|2
   ∂u ∇s ∂u = −γ˙ b − γ b˙ + ∇s − zγ b, b−2 b, ∂x ∂x |∇s|2 Since [ u, ∇∇s](χ t σ ) =

where, for brevity, b = b(t; σ, ∇s(σ ), b(σ )) and γ˙ = ∂γ /∂t. Since b solves the b-equation (1.3), the expression in the square brackets is identically zero. Thus, (2.10) holds as soon as we have the following estimate:  

 γ˙ + zγ b(t; σ, ∇s(σ ), b(σ )) 2 ≤ cn−1 a L2 . (2.13) L Therefore, Theorem 1.2 is implied by the following proposition. Proposition 2.1. There exists a sequence zj ∈ C, j = 1, 2, . . ., with Re zj → µ⊥ such that for every z = zj and for each n ∈ N there exist a point x∗ ∈ Td , u(x∗ ) = 0, a flow-box ' through a cross-section '0  x∗ , a choice of vectors b(σ ), σ ∈ '0 , a phase s : ' → R, a function γ = γ (t, σ ) for χ t σ ∈ ' such that for the amplitude a as in (2.11) the following holds: (i) ∇s(σ )⊥b(σ ), σ ∈ '0 ; (ii) u(x), ∇s(x) = 0, x = χ t σ ∈ '; (iii) estimate (2.13) holds.

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445

The proof of Proposition 2.1 will be given in the next two sections. The point (x∗ , ξ∗ ) ∈ ⊥ with ξ∗ = ∇s(x∗ ) and the vector b∗ = b(x∗ ) with b∗ ⊥ξ∗ , needed for the proof, will be chosen close to the so-called Ma˜ne point and Ma˜ne vector. We say that a point (x0 , ξ0 ) ∈ ⊥ is a Ma˜ne point and a vector b0 , b0 ⊥ξ0 , is a Ma˜ne vector for the cocycle {B t } provided sup{|e−µ⊥ t B t (x0 , ξ0 )b0 | : t ∈ R} < ∞,

|b0 | = 1.

(2.14)

Remark 2.2. The existence of a Ma˜ne point and a Ma˜ne vector for any continuous cocycle is proved in [CL, Lemma 6.29], see a related comment in the proof of Lemma 2.4 below, and [CL] for a general discussion concerning Ma˜ne points in the context of evolution operators.  For any x0 ∈ Td let p(x0 ) = inf{t > 0 : χ t (x0 ) = x0 } denote the prime period for x0 and define p(x0 ) = ∞ in case x0 is not periodic. Note that p(·) is lower semicontinuous. Define the continuous period function pc (·) as pc (x0 ) = inf{supy∈U p(y) : U is an open set containing x0 }. In particular, p(x0 ) = ∞ implies pc (x0 ) = ∞. Remark that pc (x0 ) = np(x0 ) for some n ∈ {0, 1, . . .} for each periodic point x0 ∈ Td . Formally, there are two possibilities for the point x0 ∈ Td as in (2.14). • Periodic case: There exists a constant p¯ ∈ R+ and a neighborhood U  x0 such that pc (y) ≤ p¯ for all y ∈ U ; • Aperiodic Case: For every p¯ ∈ R+ and every neighborhood U  x0 there exists x ∈ U such that p(x) > p. ¯ Remark 2.3. Observe that the periodic case implies that the cocycle t (x0 ) can not have exponential growth or decay. This follows from the fact that the differential Dχ t (x0 ) cannot grow or decay exponentially; otherwise the periodic orbit through x0 or the stagnation point x0 would have an stable or unstable manifold, which contradicts the assumption that x0 has a neighborhood consisting entirely of periodic orbits with uniformly bounded periods. In other words, for each δ > 0 there exists a constant Cδ such that

t (x0 ) ≤ Cδ eδ|t| for all t ∈ R. (2.15)  We claim that if d = 2, then the periodic case for the Ma˜ne point satisfying (2.14) is not possible. Indeed, fix δ ∈ (0, µ⊥ ). Using 2.15 and 2.2 we have |b0 ||ξ0 | = |b(t; x0 , ξ0 , b0 )||ξ(t, x0 , ξ0 )| ≤ ceµ⊥ t Cδ eδ|t| → 0 as t → −∞, and the claim is proved. The proof of Proposition 2.1 for the periodic case (when d = 3) and aperiodic case (when d = 2 or d = 3), respectively, will be given in Sect. 3 and 4. Similar to (2.5) construction of approximate eigenfunctions for the operator M as in (1.8) is given in Sect. 5. To conclude this section we show that the largest exponential growth µ⊥ for the bicharacteristic amplitude system, actually, attains at some point (x0 , ξ0 ) and vector b0 . Lemma 2.4. (i) If d = 2 or d = 3, then there exist a point (x0 , ξ0 ) ∈  and a vector b0 , b0 ⊥ξ0 , |b0 | = 1, such that µ = lim t −1 log |b(t; x0 , ξ0 , b0 )| = lim t −1 log |b(t; x0 , ξ0 , b0 )|. t→∞

t→−∞

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(ii) If d = 2 or d = 3 then there exist a point (x0 , ξ0 ) ∈ ⊥ and a vector b0 , b0 ⊥ξ0 , |b0 | = 1, such that µ⊥ = lim t −1 log |b(t; x0 , ξ0 , b0 )| = lim t −1 log |b(t; x0 , ξ0 , b0 )|. t→∞

t→−∞

(iii) If d = 2 or if d = 3 and µ = µ⊥ , then the point (x0 , ξ0 ) in (i) can be selected such that (x0 , ξ0 ) ∈ ⊥ ; (iv) If d = 2 then µ = µ⊥ . Proof. We will prove (ii), the proof of (i) is similar, and (iii) follows from (ii). Let ⊥ := {(x0 , ξ0 ) ∈ ⊥ : |ξ0 | = 1}. If ξ solves (1.2), then η(t) = ξ(t)/|ξ(t)| solves the  equation  ∂u T  ∂u T  η˙ = − η+ η, η η, η(0; x0 , η0 ) = η0 . ∂x ∂x  ⊥ . Due Let  ϕ t : (x0 , η0 ) → (x(t; x0 ), η(t; x0 , η0 )) denote the corresponding flow on  to 2.1, the cocycle over the flow { ϕ t } induced by the b-equation (1.3) is the same as the cocycle over the flow {ϕ t }.  of continuous sections F of the bundle B  with the base   ⊥ and On the space C(B) the fibers B(x0 ,ξ0 ) = {b0 : b0 ⊥ξ0 } consider the operator T defined by the rule (T F )(x0 , η0 ) = B 1 ( ϕ −1 (x0 , η0 ))F ( ϕ −1 (x0 , η0 )),

⊥ . (x0 , η0 ) ∈ 

The operator T is called a Mather, or evolution operator, see [CL] for more information concerning evolution operators. It is easy to see that µ⊥ = log r(T ). Thus, r(e−µ⊥ T ) = 1. By Lemma 6.29 in [CL] this implies the existence of a Ma˜ne point and a Ma˜ne vector as in (2.14). ϕ )}, see [CL, Thm.8.15]. Here Erg( ϕ) It is known that log r(T ) = sup{λ1ν : ν ∈ Erg(  ⊥ , and λ1ν is the maxdenotes the set of Borel  ϕ -ergodic probability measures on  imal Lyapunov-Oseledets exponent for the cocycle {B t }t∈R over { ϕ t }t∈R , given by the Oseledets Multiplicative Ergodic Theorem [O1]. According to this theorem, for ⊥,ν ⊂   ⊥ of full ν-measure such that for each each ν ∈ Erg( ϕ ) there is a subset  ⊥,ν the fiber B(x0 ,ξ0 ) can be decomposed as B(x0 ,ξ0 ) = ⊕j W j (x0 , ξ0 ) ∈  (x0 ,ξ0 ) such that j

j

λν = limt→±∞ t −1 log |B t (x0 , ξ0 )b0 | if and only if b0 ∈ Wν . By [CL, Rem.8.18] there ϕ )}. Thus, µ⊥ = λ1ν , is, in fact, a measure ν ∈ Erg( ϕ ) such that λ1ν = sup{λ1ν : ν ∈ Erg( 1 ⊥,ν and b0 ∈ W and for any (x0 , ξ0 ) ∈  (x0 ,ξ0 ) , |b0 | = 1, assertion (ii) holds. To see the inequality µ ≤ µ⊥ in (iv), take (x0 , ξ0 ) and b0 as indicated in (i). Since d = 2, we can use 2.2: 0 = lim t −1 log |b(t; x0 , ξ0 , b0 )| + lim t −1 log |ξ(t; x0 , ξ0 )| t→∞

t→∞

= µ + lim t t→∞

−1

log |ξ(t; x0 , ξ0 )|.

Since µ ≥ µ⊥ > 0 by the assumption, we have |ξ(t; x0 , ξ0 )| ≤ ce−µt . Since ξ(t; x0 , ξ0 ), u(x(t; x0 )) = const is a first integral for (1.1)–(1.2), and u(·) is bounded, we have: | ξ0 , u(x0 )| = | ξ(t; x0 , ξ0 ), u(x(t; x0 ))| ≤ ce−µt → 0 and u(x0 )⊥ξ0 .

 

as

t → ∞,

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3. Periodic Case In this section we assume that d = 3 and the periodic case holds for the Ma˜ne point (x0 , ξ0 ) ∈ ⊥ . Recall that u(x0 )⊥ξ0 . The plan of this section is as follows. At first, we assume that u(x0 ) = 0. Under this assumption, we derive in Subsect. 3.1 certain spectral properties of the monodromy operator B pc (x0 ) (x0 , ξ0 ). In Subsect. 3.2 we construct the phase s with the properties (i)–(ii) as in Proposition 2.1. In Subsect. 3.3 we construct the amplitude a that satisfies (iii) in Proposition 2.1. The main construction described in Sect. 2 then gives the approximate eigenfunction for L and proves the inequality s(L) ≥ µ⊥ , required in Theorem 1.2, provided the assumption u(x0 ) = 0 holds. Finally, in Subsect. 3.4 we show how to reduce the proof of Theorem 1.2 for the case when x0 is a stagnation point, that is, u(x0 ) = 0, to the case u(x0 ) = 0 considered before. 3.1. Properties of B. Assume that u(x0 ) = 0. Since we are in the periodic case, and p(·) is lower semicontinuous, we can select a small open set U  x0 in Td such that ¯ p ≤ p(y) ≤ p,

y ∈ U,

for some

p, ¯ p > 0,

(3.1)

and U belongs to a single chart on Td , that is, U ⊂ R3 . Let f = fx0 be a “parallelization” of the flow, that is, a diffeomorphism of U on f (U ) ⊂ R3 such that fx0 (x0 ) = 0

and Dfx0 (x)u(x) = e1 ,

x ∈ U,

(3.2)

where e1 = (1, 0, 0)T . For small |t| we let χ¯ t = f ◦ χ t ◦ f −1 and observe that ¯ = x¯ + te1 , χ¯ t (x)

x¯ = f (x),

x ∈ U.

(3.3)

(0) and x0 → Dfx0 (x0 ) are continuous. Also, note that the functions x0 → Dfx−1 0 Set ' 0 = fx0 (U ) ∩ {x = (0, x 2 , x 3 )T ∈ R3 }, and consider the small cross-section '0 = {x ∈ U : fx0 (x) ∈ ' 0 } through x0 . For σ ∈ ' 0 denote σ = fx−1 (σ ) ∈ '0 . 0 Lemma 3.1. Assume that x0 ∈ Td is a χ t -periodic point such that (3.1) holds for an open set U  x0 . Further, assume that ξ0 ∈ (R3 )∗ is such that u(x0 )⊥ξ0 . Then pc (x0 ) (x0 )ξ0 = ξ0 . Proof. We will need a coordinate representation for t (x0 ) = ([Dχ t (x0 )]T )−1 . To this end, for σ = (0, x 2 , x 3 )T ∈ ' 0 define the function ρ(x 2 , x 3 ) = pc (x0 ) − pc (fx−1 (0, x 2 , x 3 )T ). 0 ∂ρ Note that ρ(0, 0) = 0 and let ρi = ∂x (0, 0), i = 2, 3. i p (x c 0 ) x. Note that α(x ) = x . Hence, there exConsider the map α : x → χ 0 0 $ ists a smaller neighborhood U ⊂ U such that α(U $ ) ⊂ U . For a small |τ |, take x = σ + τ e1 ∈ f (U $ ), σ = (0, x 2 , x 3 )T ∈ ' 0 . Denote α = f ◦ α ◦ f −1 . Then, using (3.3), one has:

α(x) = f ◦ α ◦ f −1 (χ τ (σ )) = f ◦ χ pc (x0 ) ◦ χ τ (σ ) = f ◦ χ pc (x0 ) ◦ χ τ ◦ χ −pc (σ ) σ = f ◦ χ [pc (x0 )−pc (σ )+τ ] ◦ f −1 σ = χ [pc (x0 )−pc (σ )+τ ] (σ ).

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However, by the continuity of pc (·), we may assume that U $ is so small that for t = pc (x0 ) − pc (σ ) + τ formula (3.3) holds. This gives α(x) = σ + [pc (x0 ) − pc (σ ) + τ ]e1 = x + ρ(x 2 , x 3 )e1 . For the differentials,   1 ρ2 ρ3 Dα(0) =  0 1 0  , 0 0 1

and

(0) · Dα(0) · Dfx0 (x0 ). Dα(x0 ) = Dfx−1 0

Since [Df (x0 )]T [Df −1 (0)]T = I , we have: pc (x0 ) (x0 ) = ([Dα(x0 )]T )−1 = [Dfx0 (x0 )]T ([Dα(0)]T )−1 [Dfx−1 (0)]T 0   1 00 = [Df (x0 )]T A[Dfx−1 (0)]T , where A =  −ρ2 1 0  . 0 −ρ3 0 1 2

3

Set ξ 0 = [Dfx−1 (0)]T ξ0 . Since ξ0 ⊥u(x0 ), we have ξ 0 ⊥e1 . If ξ 0 = (0, ξ 0 , ξ 0 )T , then 0 Aξ 0 = ξ 0 . Therefore, pc (x0 ) (x0 )ξ0 = [Dfx0 (x0 )]T Aξ 0 = [Dfx0 (x0 )]T ξ 0 = ξ0 .   Lemma 3.1 implies that ϕ pc (x0 ) (x0 , ξ0 ) = (x0 , ξ0 ) for the Ma˜ne point (x0 , ξ0 ). Denote Bµ = e−µ⊥ pc (x0 ) B pc (x0 ) (x0 , ξ0 ). Lemma 3.2. Exactly one of the following two statements holds: (i) Bµ has a simple eigenvalue λ(x0 ) = 1; (ii) Bµ has a simple eigenvalue λ(x0 ) = −1. The other eigenvalue belongs to (−1, 1). Proof. For the Ma˜ne vector b0 condition (2.14) implies sup{|Bµk b0 | : k ∈ Z} < ∞, since B t is a cocycle and (x0 , ξ0 ) is ϕ-periodic. Therefore, σ (Bµ ) ∩ {z ∈ C : |z| = 1} = ∅. Recall that Bµ acts on vectors b, orthogonal to ξ . Therefore, Bµ is, in fact, a (2 × 2) matrix with real entries, and the eigenvalues λ1 , λ2 of Bµ are either complex conjugate or both real. To fix notations, assume |λ1 | = 1. Take two linearly independent eigenvectors bi for Bµ such that Bµ bi = λi bi and bi ⊥ξ0 , i = 1, 2. Using 2.3 for vectors b1 and b2 , and t = pc (x0 ), and Lemma 3.1 one has: |ξ0 | = |ξ(pc (x0 ); x0 , ξ0 )| = e−2µ⊥ pc (x0 ) |λ2 |−1 |ξ0 |. (3.4)  Therefore, |λ2 | < 1 and λ2 ∈ R.  Since Bµ = Bµ (x0 , ξ0 ) has a simple eigenvalue, by the standard perturbation theory, there exists a small neighborhood V  (x0 , ξ0 ) in ⊥ such that for each (x∗ , ξ∗ ) ∈ V the matrix Bµ (x∗ , ξ∗ ) := e−µ⊥ pc (x∗ ) B pc (x∗ ) (x∗ , ξ∗ ) (3.5) has a simple eigenvalue close to that of Bµ (x0 , ξ0 ). We will say that the Ma˜ne point (x0 , ξ0 ) is of type 1 if (i) holds, and is of type −1 if (ii) holds in Lemma 3.2.

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Proposition 3.3. For each n ∈ N there exist a point x∗ ∈ Td , a covector ξ∗ ∈ (R3 )∗ \ 0 with ξ∗ ⊥u(x∗ ), a neighborhood V0  (x∗ , ξ∗ ) in ⊥ such that for each (σ, ξσ ) ∈ V0 there exist λ(σ ) ∈ C and vectors bσ , bσ ⊥ξσ with the following properties: (a) pc (σ ) = p(σ ), (b) B p(σ ) (σ, ξσ )bσ = eµ⊥ p(σ ) λ(σ )bσ , (c) | log |λ(σ )|| ≤ cn−1 , (d) λ(σ ) < 0 provided (x0 , ξ0 ) is the Ma˜ne point of type −1, and λ(σ ) > 0 provided (x0 , ξ0 ) is the Ma˜ne point of type 1. In addition, if σ → ξσ is a continuous family, then σ → λ(σ ), σ → bσ can be chosen to be continuous as well. Proof. Recall that by (3.1) the range of the function pc (·)/p(·) consists of finitely many values in {1, 2, . . .}. We check that the set V := {x ∈ U : pc (x)/p(x) = 1} is open and dense in U . To see that V is open, assume there is a sequence yn → x ∈ V such that p(yn ) ≤ pc (yn )/2. Since pc (yn ) → pc (x) = p(x), by (3.1) and the definition of p(·) choose tn ∈ [p, 23 p(x)] such that χ tn (yn ) = yn . For a limiting point t0 = lim tn ∈ [p, 23 p(x)] we have χ t0 (x) = x, in contradiction with the definition of p(x). To see that V is dense, pick x ∈ U and use the definition of pc (x) to find yn → x such that p(yn ) → pc (x). Since pc (yn ) → pc (x), we have pc (yn )/p(yn ) → 1, and so yn ∈ V for large enough n. By Lemma 3.2 one of the eigenvalues, say, λ(x0 ), of the matrix Bµ (x0 , ξ0 ) is simple, and λ(x0 ) = 1, resp., λ(x0 ) = −1, for the Ma˜ne point of type 1, resp. −1. Note that for Bµ (x∗ , ξ∗ ) in (3.5) one has Bµ (x∗ , ξ∗ ) → Bµ (x0 , ξ0 ) as (x∗ , ξ∗ ) → (x0 , ξ0 ). Using the argument at the beginning of the proof, choose a point (x∗ , ξ∗ ) ∈ ⊥ with pc (x∗ ) = p(x∗ ) so close to (x0 , ξ0 ) that inequalities (c)–(d) hold for σ = x∗ and eigenvalues λ(x∗ ) of Bµ (x∗ , ξ∗ ), and, in addition, such that the corresponding eigenvector bx∗ of Bµ (x∗ , ξ∗ ) is close to that of Bµ (x0 , ξ0 ). This choice is possible by a standard perturbation theory. Selecting a small enough neighborhood V0 gives (a)–(d). We note that the eigenvalues of Bµ (x∗ , ξ∗ ) are real and simple.   3.2. Phase. In this subsection we construct a flow-box ' through a cross-section '0 and define the phase s : ' → R to satisfy Proposition 2.1. In fact, we look for s that solves the Cauchy problem

u, ∇s = 0,

s|'0 = s0 ,

with the initial data s0 on '0  x∗ chosen such that ∇s0 (x∗ ) = ξ∗ . Take a point (x∗ , ξ∗ ) ∈ ⊥ , u(x∗ )⊥ξ∗ , and its neighborhood V0 as indicated in Proposition 3.3. Choose the “parallelization” fx∗ as in the beginning of Subsect. 3.1, that is, fx∗ (x∗ ) = 0, Dfx∗ (x)u(x) = e1 , and small cross-sections ' 0 = {(0, x 2 , x 3 )}

and '0 = {x : fx∗ (x) ∈ ' 0 } ⊂ V0 .

By the choice of V0 we have pc (σ ) = p(σ ) for all σ ∈ '0 . For x from a narrow tubular neighborhood U0 = {χ t σ : σ ∈ ' 0 , |t| << 1} set s 0 (x) = ξ ∗ , x,

where ξ ∗ = ([Dfx∗ (x∗ )]T )−1 ξ∗ ,

(3.6)

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and denote s0 (x) = s 0 (fx∗ (x)) for x ∈ fx−1 (U 0 ). Note that ∗ ∇s0 (x) = [Dfx∗ (x)]T ∇s 0 (fx∗ (x)). Since ∇s 0 (x) = ξ ∗ , one has: ∇s0 (x) = [Dfx∗ (x)]T ξ ∗ = [Dfx∗ (x)]T ([Dfx∗ (x∗ )]T )−1 ξ∗ , and, in particular, ∇s0 (x∗ ) = ξ∗ . Define the phase s : ' → R in the flow-box ' := {x = χ t σ : t ∈ [0, p(σ )], σ ∈ '0 } as follows: s(x) = s0 (σ ), Clearly, for x ∈

fx−1 (U 0 ) ∗

x = χ t σ,

t ∈ [0, p(σ )],

σ ∈ '0 .

(3.7)

we have:

∇s(x), u(x) = ∇s0 (x), u(x) = [Dfx ∗ (x)]T ([Dfx∗ (x∗ )]T )−1 ξ∗ , u(x) = ξ∗ , [Dfx∗ (x∗ )]−1 ([Dfx∗ (x)]u(x))

= ξ∗ , [Dfx∗ (x∗ )]−1 e1  = ξ∗ , u(x∗ ) = 0. From (3.7) ∇s(x)⊥u(x) for all x ∈ '. Thus, (ii) in Proposition 2.1 holds. Denote ξσ := (∇s)(σ ), σ ∈ '0 .

(3.8)

Note that ξσ → ξ∗ as σ → x∗ . Using Bµ (σ, ξσ ) → Bµ (x∗ , ξ∗ ) as σ → x∗ , make '0 small enough and, for Bµ (σ, ξσ ), select the corresponding eigenvalues λ(σ ) with properties (c)–(d) and the corresponding eigenvectors b(σ ) := bσ that satisfy (b) in Proposition 3.3. In particular, b(σ )⊥ξσ . Since ξσ = (∇s)(σ ) for σ ∈ '0 , condition (i) in Proposition 2.1 holds. Thus, (3.7) gives the required phase. In addition, using (3.1), we may select '0 so small that |p(σ )−1 − p(x∗ )−1 | ≤ cn−1 ,

σ ∈ '0 .

(3.9)

3.3. Amplitude. In this subsection we define the approximate eigenvalue z for L, an appropriate amplitude a as in (2.11), and prove (iii) in Proposition 2.1. Take '0 as indicated in the previous subsection and a “bump”-function α ∈ C0∞ ('0 ), α : '0 → [0, 1], with α(x∗ ) = 1. For ξσ as in (3.8) and the corresponding eigenvectors b(σ ) and eigenvalues λ(σ ) for the matrix e−µ⊥ p(σ ) B p(σ ) (σ, ξσ ), we set: γ (t, σ ) = α(σ )|λ(σ )|−t/p(σ ) e−it arg λ(σ )/p(σ ) e−µ⊥ t ,

σ ∈ '0 . (3.10) If (x0 , ξ0 ) is the Ma˜ne point of type 1, we denote z = µ⊥ and recall that arg λ(σ ) = 0 in (3.10). If (x0 , ξ0 ) is the Ma˜ne point of type −1, we denote z = µ⊥ + iπ/p(x∗ ), and recall that arg λ(σ ) = π, see Proposition 3.3. Finally, define a as in (2.11), that is, set a(x) = γ (t, σ )B t (σ, ξσ )b(σ ),

where ξσ = ∇s(σ ),

t ∈ [0, p(σ )],

t ∈ [0, p(σ )],

We claim that estimate (2.13) holds. If (x0 , ξ0 ) is of type 1, then     |γ˙ + zγ | = −p(σ )−1 log |λ(σ )| · γ  ≤ cn−1 |γ |

σ ∈ '0 .

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by the choice of λ(σ ) as in (c)–(d) of Proposition 3.3, and the boundedness of p(·) as in 3.1. If (x0 , ξ0 ) is of type −1, then          γ˙ + zγ  =  − p(σ )−1 [log |λ(σ )| + iπ] + iπ p(x∗ )−1 γ        =  − p(σ )−1 log |λ(σ )| − iπ p(σ )−1 − p(x∗ )−1 γ  ≤ cn−1 |γ | by the choice of λ(σ ) as in (c)–(d) of Proposition 3.3, the boundedness of p(·) as in 3.1, and the choice of '0 as in 3.9. Since a = γ b as in 2.11, the estimate (2.13) holds. This concludes the proof of Theorem 1.2 for d = 3 and the periodic case of Ma˜ne point (x0 , ξ0 ) under the assumption u(x0 ) = 0. 3.4. Stagnation point. In this subsection we assume that d = 3, the periodic case holds for the Ma˜ne point (x0 , ξ0 ), and that x0 is a stagnation point, that is, u(x0 ) = 0. We show how to reduce the proof of the inequality s(L) ≥ µ⊥ in Theorem 1.2 to the case u(x0 ) = 0. Proposition 3.4. There exists a sequence {(xj , ξj )}∞ j =1 ⊂ ⊥ , such that ξj ⊥u(xj ),

lim (xj , ξj ) = (x0 , ξ0 ),

j →∞

u(xj ) = 0,

and

lim µj = µ⊥ ,

j →∞

where µj :=

1 log max{|z| : z ∈ σ (B pc (xj ) (xj , ξj ))}. pc (xj )

(3.11)

Assume that Proposition 3.4 holds. Then σ (e−µj pc (xj ) B pc (xj ) (xj , ξj )) ∩ {z ∈ C : |z| = 1} = ∅ and there is a vector bj , |bj | = 1, such that sup{|e−µj t B t (xj , ξj )bj | : t ∈ R} < ∞,

j = 1, 2, . . . .

Therefore, (xj , ξj ) is a Ma˜ne point, see 2.14, such that u(xj ) = 0. Using results in Subsects. 3.1–3.3, we conclude that Proposition 2.1 holds for zj ∈ C with Re zj = µj . By the main construction of the approximate eigenfunction in Sect. 2 we conclude that s(L) ≥ µj → µ⊥ , as required in Theorem 1.2. It remains to prove Proposition 3.4. For this, we will need two lemmas. ∂u Denote, for brevity, A = ∂x (x0 ). Since div u = 0, the Jacobi matrix A is a 3 × 3 matrix with real entries and zero trace. Since Dχ t (x0 ) = etA , and the periodic case holds for the Ma˜ne point (x0 , ξ0 ), by Remark 2.3 we have σ (etA ) ⊂ {z ∈ C : |z| = 1}. Therefore, σ (A) = {−iθ, 0, iθ } for some real θ ≥ 0. In addition to this, A2 is symmet ric. Indeed, u solves the stationary Euler equation 3j =1 uj ∂j ui = −∂i p, i = 1, 2, 3. Differentiating and using u(x0 ) = 0 we conclude that A2 = [−∂k ∂j p(x0 )]3k,j =1 . Lemma 3.5. If the periodic case for the Ma˜ne point (x0 , ξ0 ) holds, then θ > 0. Proof. We will make use of the first integral 2.3 for the Ma˜ne point (x0 , ξ0 ), Ma˜ne vector b0 , and any vector b1 , nonparallel to b0 . By Remark 2.3, for each δ > 0 there exists Cδ

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such that |ξ(t; x0 , ξ0 )| ≤ Cδ eδ|t| , t ∈ R. Since |B t (x0 , ξ0 )b0 | ≤ ceµ⊥ t , t ∈ R, due to 2.14, formula 2.3 gives |B t (x0 , ξ0 )b1 | ≥ c(δ)e−µ⊥ t−δ|t| ,

t ∈ R,

(3.12)

for each b1 nonparallel to b0 . Let us suppose that σ (A) = {0}. If A = 0 then b(t; x0 , ξ0 , b1 ) = b1 , in contradiction with 3.12 for δ < µ⊥ and t → −∞. Assume A = 0. Since A2 is symmetric, it follows T A2 = 0. From ξ(t; x0 , ξ0 ) = e−tA ξ0 it follows that  ξ0 if AT ξ0 = 0; ξ(t; x0 , ξ0 ) |ξ0 | , (3.13) = lim T A ξ0 t→±∞ |ξ(t; x0 , ξ0 )| ∓ |A if AT ξ0 = 0. T ξ |, 0

Denote

ξ(t; x0 , ξ0 ) ξ(t; x0 , ξ0 ) ⊗ − I d, |ξ(t; x0 , ξ0 )| |ξ(t; x0 , ξ0 )| and re-write the b-equation 1.3 as P (t) = 2

˙ = P (t)Ab(t). b(t)

(3.14)

We have limt→±∞ P (t) = P∞ . Notice that P∞ A = −A. Indeed, in the first case of 3.13 we have 2 ξ|ξ0 ⊗ξ|20 Av − Av = −Av, v ∈ R3 ; in the second case of 3.13 A2 = 0 implies T

T

0

ξ0 Av − Av = −Av, v ∈ R3 . Let b(t) be the solution to the Cauchy problem 2 A |Aξ0T⊗A ξ0 |2 for 3.14 with b(0) = b1 . Using variation of parameters,

b(t) = e

P∞ At

t b1 +

= e−At b1 +



eP∞ A(t−τ ) (P (τ ) − P∞ )Ab(τ ) dτ

0 t

e−A(t−τ ) (P (τ ) − P∞ )Ab(τ ) dτ  t = b1 − tAb1 + {(P (τ ) − P∞ )Ab(τ ) − (t − τ )A(P (τ ) − P∞ )Ab(τ )} dτ. 0

0

Therefore, ˙ |b(t)| ≤ c + g(t)|b(t)| + g(t)



t

|b(τ )| dτ,

0

where g(t) > 0 for all t and g(t) → 0 as |t| → ∞. Let δ > 0 be small. Let, e.g., t > 0 and k(t) = |b(t)|e−δt . Then,  t ˙k(t) + δk(t) ≤ ce−δt + g(t)k(t) + g(t) eδ(τ −t) k(τ ) dτ. 0

Therefore, for large t, ˙ ≤ ce−δt + g(t) k(t)



t 0

eδ(τ −t) k(τ ) dτ ≤ δ + δ



t

k(τ ) dτ.

0

This implies k(t) ≤ δeδt for large t. Therefore, |b(t)| ≤ δe2δt . Similarly, |b(t)| ≤ δe2δt as t → −∞. This contradicts 3.12 for δ < µ⊥ /3 as t → −∞.  

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Lemma 3.6. If the periodic case for the Ma˜ne point (x0 , ξ0 ) holds, then the solutions ξ(t; x0 , ξ0 ) of the ξ -equation 1.2 are 2π/θ-periodic. Proof. We claim that there is an a ≥ 1 and a basis in R3 in which A has the following form:   0 a −1 θ 0 A = −aθ (3.15) 0 0 . 0 0 0 2 2 2 To prove claim,  we recall that A is symmetric with σ (A ) = {−θ , 0}. Thus,  the 2 −θ I 0 A2 = in some orthogonal basis. Since A has zero trace, in this basis we 0 0 have:   α β 0   α β 2 = −θ 2 I, α 2 + βγ = −θ 2 . A = γ −α 0 , where γ −α 0 0 0

Rotating the basis, we may assume that α = 0, and 3.15 holds. Then the solution ξ(t; x0 , ξ0 ) of 1.2 is as follows: ξ1 = ξ10 cos θt + ξ20 a sin θt, ξ2 = −ξ10 a −1 sin θt + ξ20 cos θ t, ξ3 = ξ30 .   Proof of Proposition 3.4. Since the periodic case for the Ma˜ne point (x0 , ξ0 ) holds, there is a small neighborhood U  x0 that consists of {χ t }-periodic orbits with uniformly bounded periods: p(y) ≤ p for all y ∈ U . In every neighborhood of x0 there exists a point y that is not a stagnation point: u(y) = 0. Otherwise u ≡ 0 in some neighborhood of x0 and 2.14 with µ⊥ > 0 fails. First, we claim that each orbit in U contains a point x such that ξ0 ⊥u(x). Indeed, pick an orbit through y ∈ U and parameterize it as {γ (t) : t ∈ [0, p(y)], γ (0) = γ (p(y))} so that u(γ (t)) = γ˙ (t). Then,  p(y)  p(y)

ξ0 , u(γ (t))dt = ξ0 , γ˙ (t)dt = ξ0 , [γ (p(y)) − γ (0)] = 0. 0

0

Therefore, the function ξ0 , u(γ (·)) cannot be strictly positive (or negative) on [0, p(y)], and for some x on the orbit we have ξ0 ⊥u(x), as claimed. Let ξj = ξ0 and choose xj → x0 , j = 1, 2, . . ., such that u(xj )⊥ξ0 and u(xj ) = 0, using the claim just proved. It remains to prove that µj → µ⊥ for µj defined in 3.11. ∂u Since ∂x (·) is continuous, p(xj ) ≤ p, and xj → x0 , we have that for each T > 0, lim max

j →∞ 0≤t≤T

∂u t ∂u (χ xj ) − (x0 ) = 0. ∂x ∂x

(3.16)

For the sequence xj with limj →∞ xj = x0 we denote p0 = pc (x0 ) = lim pc (xj ) j →∞

and consider two cases. Case p0 = 0. Since u(xj ) = 0 and ξ0 ⊥u(xj ), from Lemma 3.1 applied to xj we have that (xj , ξ0 ) is {ϕ t }–periodic with period pc (xj ), i. e., pc (xj ) (xj )ξ0 = ξ0 . Passing to the limit xj → x0 and using continuity of the cocycle {t } we have p0 (x0 )ξ0 = ξ0 , i. e., (x0 , ξ0 ) is {ϕ t }–periodic with period p0 . Using 2.14 with t = kp0 , k ∈ Z, we obtain a z ∈ C, |z| = 1, such that eµ⊥ p0 z ∈ σ (B p0 (x0 , ξ0 )). By continuity of the

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cocycle {B t } we have σ (B pc (xj ) (xj , ξ0 )) → σ (B p0 (x0 , ξ0 )). Therefore, there is a sequence zj ∈ σ (B pc (xj ) (xj , ξ0 )), j = 1, 2, . . ., such that zj → eµ⊥ p0 z. This yields log |zj | → µ⊥ p0 and 1 µj ≥ log |zj | → µ⊥ . (3.17) pc (xj ) Case p0 = 0. Since pc (xj ) → 0 as j → ∞, there is a sequence n = n(j ) → ∞, n ∈ N, such that npc (xj ) → 2π/θ as j → ∞. By Lemma 3.6, (x0 , ξ0 ) is a 2π/θ -periodic point for the flow {ϕ t }. Since {B t } is a cocycle over this flow, and 2.14 holds, for the monodromy matrix B 2π/θ (x0 , ξ0 ) of the b-equation 1.3 we have: σ (e−µ⊥ 2π/θ B 2π/θ (x0 , ξ0 )) ∩ {z ∈ C : |z| = 1} = ∅.

(3.18)

Using Lemma 3.6 and Lemma 3.1, we have lim

max

j →∞ t∈[0,4π/θ ]

dist(ϕ t (x0 , ξ0 ), ϕ t (xj , ξj )) = 0.

This implies that limj →∞ e−µ⊥ npc (xj ) B npc (xj ) (xj , ξj ) = e−µ⊥ 2π/θ B 2π/θ (x0 , ξ0 ). Using 3.18, find zj ∈ σ (B pc (xj ) (xj , ξj )) such that 1 = limj →∞ |e−µ⊥ npc (xj ) zjn |. By definition 3.11 of µj ,

(3.19) 0 = lim npc (xj ) log |zj |/pc (xj ) − µ⊥ ≤ 2π ( lim µj − µ⊥ )/θ. j →∞

j →∞

Since µ⊥ ≥ µj by the definition of the maximal Lyapunov exponent µ⊥ , 3.17 and 3.19 give µ⊥ = limj →∞ µj , as required.   4. Aperiodic Case In this section we assume that d = 2 or d = 3 and the aperiodic case holds for the Ma˜ne point (x0 , ξ0 ) ∈ ⊥ as in 2.14. Recall that u(x0 )⊥ξ0 . We will prove Proposition 2.1 by constructing the phase s and the amplitude a supported in a long and thin flow-box ' that depends on a given n ∈ N. We present the arguments for d = 3, the case d = 2 is treated similarly. Fix n ∈ N. 4.1. Phase. In this subsection we define the flow-box ' through a cross-section '0 and a phase s : ' → R. The main idea is again to consider the Cauchy problem

u, ∇s = 0,

s|'0 = s0 ,

(4.1)

where the initial data s0 is chosen such that ∇s(x∗ ) = ξ∗ . Fix N ≥ n and, for the Ma˜ne point (x0 , ξ0 ), take any sufficiently close to (x0 , ξ0 ) point (x∗ , ξ∗ ) ∈ ⊥ such that p(x∗ ) = pc (x∗ ) ≥ 5N . The specific choice of (x∗ , ξ∗ ), ξ∗ ⊥u(x∗ ), will be indicated later in Subsect. 4.2. As in the beginning of Subsect. 3.1, let fx∗ : U → Rd be the “parallelization” of u in a small neighborhood U of x∗ , that is, fx∗ (x∗ ) = 0, and Dfx∗ (x)u(x) = e1 , x ∈ U . Let ' 0 = fx∗ (U ) ∩ {x = (0, x 2 , x 3 )}, and '0 = fx−1 (' 0 ) be a small cross-section through ∗ x∗ . We may and will assume that '0 so small that, for U0 := {χ t σ : σ ∈ '0 , 0 ≤ t < 1}, one has: χ k (U0 ) ∩ χ j (U0 ) = ∅, Consider the flow-box ' =

{χ t σ

k = j,

k, j ∈ Z,

: σ ∈ '0 , |t| ≤ 3N }.

|k|, |j | ≤ 4N.

(4.2)

Linear Stability in an Ideal Incompressible Fluid

455

Lemma 4.1. There exists s : ' → R such that ∇s(x∗ ) = ξ∗ and

u(x), ∇s(x) = 0,

x ∈ '.

(4.3)

Proof. The construction of s is almost identical to that in Subsect. 3.2. For x in a narrow tubular neighborhood of ' 0 for the flow {χ t }, defined in 3.3, we set s 0 (x) = ξ ∗ , x (cf. 3.6). For x in a narrow tubular neighborhood of '0 for the flow {χ t } let s0 (x) = s 0 (fx∗ (x)). For x ∈ ' we then define (cf. 3.7) s(x) = s0 (σ ), where x = χ t (σ ), σ ∈ '0 , |t| ≤ 3N. Calculations in Subsect. 3.2 show that s is as needed in the lemma.   Take s as in Lemma 4.1. Denote ξσ = ∇s(σ ), σ ∈ '0 . Taking the gradient in 4.3 we observe that T ∂u 0 = u(x), ∇ ∇s(x) + ∇s(x). ∂x Thus, the function t → ξ(t; σ, ξσ ) = ∇s(χ t σ ) solves the ξ -equation 1.2. We remark that s just constructed satisfies (ii) in Proposition 2.1. 4.2. Amplitude. In this subsection we construct the amplitude a as in 2.11. Note that a similar construction has been used in [LS] for the evolution operators. We finish the proof of Proposition 2.1 for the aperiodic case in Subsect. 4.3. Let (x0 , ξ0 ) be the Ma˜ne point as in 2.14. For the fixed n ∈ N choose δ > 0 so small that max{|1 − eδ |, |1 − e−δ |} < n−1 , (4.4) and denote:

bk = e−µ⊥ k e−δ|k| B k (x0 , ξ0 )b0 ,

k ∈ Z,

(4.5)

where b0 is the Ma˜ne vector as in 2.14. If C denotes the supremum in 2.14 then |bk | ≤ Ce−δ|k| , and (bk )k∈Z ∈ l 2 (Z; R3 ). Select N ≥ n so large that N  k=−N

|bk |2 ≥ 2−1

∞ 

|bk |2 .

(4.6)

k=−∞

Choose a small neighborhood W  (x0 , ξ0 ) in ⊥ such that for all (x∗ , ξ∗ ) ∈ W , u(x∗ )⊥ξ∗ , one has: 2N  k=−2N N  k=−N

|e−µ⊥ k B k (x∗ , ξ∗ )b0 |2 e−2δ|k| ≤ 2

2N 

|bk |2 ,

k=−2N

|e−µ⊥ k B k (x∗ , ξ∗ )b0 |2 e−2δ|k| ≥ 2−1

N 

|bk |2 .

(4.7)

k=−N

Since the aperiodic case for the Ma˜ne point holds, we can fix (x∗ , ξ∗ ) ∈ W such that p(x∗ ) = pc (x∗ ) ≥ 5N and, in addition, ξ∗ ⊥b0 . Recall that u(x∗ )⊥ξ∗ . Use this choice of (x∗ , ξ∗ ) to construct the flow-box ' and s : ' → R as indicated in the previous subsection. Recall that the function σ → ξσ := ∇s(σ ) is continuous, and ξx∗ = ξ∗ by Lemma 4.1. Take any continuous in σ family b(σ ) ∈ R3 , σ ∈ '0 , such that b(x∗ ) = b0 and b(σ )⊥ξσ , σ ∈ '0 . Using 4.7 and the continuity of the function

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Y. Latushkin, M. Vishik

σ → ξσ and σ → b(σ ), and passing, if necessary, to a smaller '0 we may and will assume that for each σ ∈ '0 the following inequalities hold: 2N 

|e

−µ⊥ k

k

2 −2δ|k|

B (σ, ξσ )b(σ )| e

k=−2N N 

2N 

≤3

|bk |2 ,

k=−2N

|e−µ⊥ k B k (σ, ξσ )b(σ ))|2 e−2δ|k| ≥ 3−1

k=−N

N 

|bk |2 .

(4.8)

k=−N

Our next step in finding a as in 2.11 is to define the “cut-off” function γ (t, σ ), χ t σ ∈ '. First, select an open subset '0$ ⊂ '0 such that for U0$ = {x = χ t σ : σ ∈ '0$ , 2/3 ≤ t < 1} one has: m(U0 ) ≤ cm(U0$ ), (4.9) where m is the {χ t }-invariant Lebesgue measure on Td and U0 is as in 4.2. Choose α ∈ C0∞ ('0 ), α : '0 → [0, 1], such that α|'0$ = 1. Second, choose a smooth function β1 : R → [0, 1] such that supp β1 ⊂ [−2N, 2N ], and β1 (t) = 1 for |t| ≤ N + 1 with β1$ ∞ ≤ cN −1 . Third, fix a smooth β2 : [0, 1] → [0, 1] such that β2 (τ ) = 0 for τ ∈ [0, 1/3] and β2 (τ ) = 1 for τ ∈ [2/3, 1]. Finally, for t = τ + k, k ∈ Z, τ ∈ [0, 1], set: (4.10) β(t) = β1 (t)[(1 − β2 (τ ))e−δ|k−1| + β2 (τ )e−δ|k| ]. Since δ is as small as in 4.4 and N −1 ≤ n−1 , our choice of β1 implies that supp β ⊂ [−2N, 2N] and |β $ (τ + k)| ≤ cn−1 e−δ|k| ,

τ ∈ [0, 1],

k ∈ Z.

(4.11)

On the other hand, by the choice of β2 one has: β(τ + k) = β1 (τ + k)e−δ|k|

for

τ ∈ [2/3, 1],

k ∈ Z.

(4.12)

We are ready to define the amplitude a(x) for x = χ t σ , |t| ≤ 3N , σ ∈ '0 . Set γ (t, σ ) = e−µ⊥ t α(σ )β(t). Take s as in Lemma 4.1. Recall that '0 , ξσ = ∇s(σ ), and b(σ )⊥ξσ , σ ∈ '0 , are selected such that 4.8 holds. Also, assertions (i) and (ii) in Proposition 2.1 are satisfied. Finally, with this choice of ξσ and b(σ ), define a(x) = γ (t, σ )B t (σ, ξσ )b(σ ) for x = χ t σ ∈ ' and a(x) = 0 otherwise. 4.3. Estimates for the norm. In this subsection we give estimates to prove (iii) in Proposition 2.1. Let z = µ⊥ . We claim that (2.13) holds. Step 1. (Estimate from above). First, we estimate the LHS of (2.13) from above. Recall, that U0 = {χ τ (σ ) : σ ∈ '0 , τ ∈ [0, 1)}. Since χ k (U0 ) are disjoint due to 4.2, one has, with b = B t (σ, ξσ )b(σ ):  

  γ˙ + zγ b2 2 = |α(σ )β $ (t)e−µ⊥ t b|2 dm(x) L k ∪2N k=−2N χ (U0 )

=

2N  

k=−2N

χ k (U0 )

|α(σ )β $ (t)e−µ⊥ t b|2 dm(x).

Linear Stability in an Ideal Incompressible Fluid

457

For x ∈ ' we define t = t (x), t (x) ∈ [−3N, 3N ], such that x = χ t (x) σ , σ ∈ '0 . Since m is χ t -invariant, the last expression is equal to 

2N  U0 k=−2N

|α(σ )β $ (t (χ k x))e−µ⊥ t (χ

k x)

B t (χ

k x)

(σ, ξσ )b(σ )|2 dm(x).

(4.13)

But, for the cocycle Bµt (x, ξ ) = e−µ⊥ t B t (x, ξ ) over {ϕ t }, one has for x ∈ U0 : e−µ⊥ t (χ

k x)

B t (χ

k x)

(σ, ξσ )b(σ ) = Bµτ (x) (ϕ k (σ, ξσ ))Bµk (σ, ξσ )b(σ ).

Here we write t (χ k x) = t (χ k+τ σ ) = k +τ (x), x ∈ U0 with τ (x) ∈ [0, 1). This implies: k x)

|e−µ⊥ t (χ |e−µ⊥

t (χ k x)

k x)

(σ, ξσ )b(σ )| ≤ c|e−µ⊥ k B k (σ, ξσ )b(σ )|,

(4.14)

t (χ k x)

(σ, ξσ )b(σ )| ≥ c|e−µ⊥ k B k (σ, ξσ )b(σ )|.

(4.15)

B t (χ B

Using the estimate 4.14 and the estimate 4.11 for |β $ |, we conclude that the integral in 4.13 is bounded from above by the expression  c

2N 

U0 k=−2N

|n−1 e−δ|k| e−µ⊥ k B k (σ, ξσ )b(σ )|2 dm(x)



≤ cn−2

3

U0

2N  k=−2N

|bk |2 dm(x) ≤ cn−2 (bk )k∈Z 2C2 · m(U0 ),

where 4.8 was also used. Thus, for the LHS of (2.13) we have:  

 γ˙ + zγ b(t; σ, ∇s(σ ), b(σ )) 2 ≤ cn−1 (bk )k∈Z C2 · m(U0 )1/2 . L

(4.16)

Step 2. (Estimate from below). We estimate a L2 from below as follows. Similarly to 4.13, one has:

a 2L2 = e−µ⊥ t α(σ )β(t)B t (σ, ξσ )b(σ ) 2L2  2N  k k = |α(σ )β(k + τ (x))e−µ⊥ t (χ (x)) B t (χ (x)) (σ, ξσ )b(σ )|2 dm(x). U0 k=−2N

(4.17) Recall that 4.12 holds for x = χ t σ , t = k + τ , τ ∈ [2/3, 1]. Also, |k| ≤ N implies β1 (k + τ (x)) = 1 due to the choice of β1 . Therefore, with U0$ = {x = χ t σ : σ ∈ '0$ , 2/3 ≤ t < 1}, the RHS of 4.17 is greater than or equal to the quantity  U0$

N 

k x)

B t (χ

k x)

(σ, ξσ )b(σ )|2 dm(x)

k=−N



≥c

|e−δ|k| e−µ⊥ t (χ N 

U0$ k=−N

|e−δ|k| e−µ⊥ k B k (σ, ξσ )b(σ )|2 dm(x),

458

Y. Latushkin, M. Vishik

where the estimate 4.15 was used. Due to 4.6, 4.8 and 4.9, we have:

a 2L2 ≥ c m(U0$ )3−1

N  k=−N

|bk |2 ≥ c m(U0 ) (bk )k∈Z 2C2 .

Combining this with 4.16, we have (2.13), and the proof of Proposition 2.1 and Theorem 1.2 is completed for the aperiodic case as well. Proof of Proposition 1.4. Let z = µ⊥ + iρ for arbitrary ρ ∈ R. Define γ (t, σ ) = e−zt α(σ )β(t), see the end of Subsect. 4.2, and a as in 2.11. As in Subsect. 4.3 estimate (2.13) holds. The main construction in Sect. 2 shows that z ∈ σap (L). 5. Kinematic Dynamo Operator In this section we prove Theorem 1.6 for the kinematic dynamo operator M defined in 1.8. To this end for a z ∈ C with Re z = ω(M) we construct an approximate eigenfunction w for M. That is, for each n ∈ N we find a vector-field w ∈ Ls2 , supported in a thin flow-box, such that

(M − z)w Ls2 ≤ cn−1 w Ls2 . (5.1) Recall that the group {etM } acts on Ls2 by the rule (etM w)(x) = Dχ t (χ −t x)w(χ −t x), where χ t : Td → Td , as above, is the flow generated by the equation x˙ = u(x) with div u = 0 on Td , and Dχ t is the differential. It is known that ν := ω(M) is the maximal Lyapunov exponent for the cocycle {Dχ t } over {χ t }, acting on the tangent bundle T Td , see [O2, CL or CLM1]. We may and will assume that ν > 0. Otherwise, since Mu = 0, one has 0 = ω(M) ≤ s(M). As in Remark 2.2, there exist a Ma˜ne point x0 ∈ Td and Ma˜ne vector v0 ∈ Tx0 Td , |v0 | = 1, such that sup{|e−νt Dχ t (x0 )v0 | : t ∈ R} < ∞.

(5.2)

Formally, again, there are two cases: • Periodic case: there exists a constant p ∈ R+ and U  x0 such that pc (y) ≤ p for all y ∈ U ; • Aperiodic case: for every p ∈ R+ and every U  x0 there exists x ∈ U such that p(x) > p. By Remark 2.3 in the periodic case the differential at x0 cannot grow exponentially. Thus, assumption ν > 0 leads to a contradiction, and the periodic case is not possible. In what follows we will assume that the aperiodic case holds for the Ma˜ne point as in 5.2. We will be looking for the vector-field w (cf. 2.5) of the form

eis/$ w(x) = $∇ × ∇s × a , x = χ t σ ∈ ', σ ∈ '0 . |∇s|2 This vector-field is supported in a thin and long flow-box ' through the cross-section '0  x∗ , where x∗ is a point close to x0 . Here the phase s, the amplitude a and the value of the small parameter $ > 0 will be defined below.

Linear Stability in an Ideal Incompressible Fluid

459

Suppose s and a are selected such that ∇s⊥a. Then 2.6 holds. If, in addition, ∇s, u = 0 then

a , Mw = −ieis/$ Ma + $eis/$ Mw0 , with w0 = ∇ × ∇s × |∇s|2 and

Mw − zw = −ieis/$ [Ma − za] + $eis/$ [Mw0 − zw0 ].

Choosing $ small enough, we have 5.1 provided z and a are chosen such that

Ma − za L2 ≤ cn−1 a L2 .

(5.3)

In contrast to w in 5.1, we do not require div a = 0. To satisfy 5.3, we will be looking for a (cf. 2.11) of the form a(x) = γ (t, σ )Dχ t (σ )v(σ ),

x = χ t σ ∈ ',

σ ∈ '0 ,

(5.4)

where γ is an appropriate smooth scalar “cut-off” function, and v(σ ) are appropriate initial data. For this choice of a we have: Ma − za =



d ∂u a(χ t σ ) − a(χ t σ ) − za(χ t σ ) = − γ˙ + zγ Dχ t (σ )v(σ ). ∂x dt

Note that u, ∇s = 0 implies that the function ξ(t, σ, ∇s(σ )) = ∇s(χ t σ ) solves the ξ -equation 1.2, see 2.12. Then Dχ t (σ )v(σ )⊥∇s(χ t σ ) if v(σ )⊥∇s(σ ), σ ∈ '0 . But a in 5.4 is parallel to Dχ t (σ )v(σ ). Thus, 5.3 and Theorem 1.6 will be proved as soon as the following fact is proved (cf. Proposition 2.1 and recall that ν = ω(M), the growth bound). Proposition 5.1. There exists z ∈ C with Re z = ν such that for each n ∈ N there exist a point x∗ ∈ Td , a flow-box ' through a cross-section '0  x∗ , a choice of vectors v(σ ) ∈ Tσ Td , σ ∈ '0 , a phase s : ' → R, a function γ = γ (t, σ ) for χ t σ ∈ ' such that for the amplitude a as in 5.4 one has: (i) ∇s(σ )⊥v(σ ), σ ∈ '0 , (ii) u(x), ∇s(x) = 0, x = χ t σ ∈ '; (ii) the following estimate holds:  

 γ˙ + zγ Dχ t (σ )v(σ ) 2 ≤ cn−1 a L2 . L

(5.5)

Proof. To construct s : ' → R on ' = {x = χ t σ : |t| ≤ 3N, σ ∈ '0 }, let s(x) = s0 (σ ), where s0 : '0 → R is chosen as in Lemma 4.1 starting with an arbitrary ξ∗ ⊥u(x∗ ). Note that ∇s⊥u on ', since s(χ t σ ) does not depend on t. As in Subsect. 4.2, fix n ∈ N, choose δ to satisfy 4.4, denote (cf. 4.5) vk = e−νk e−δ|k| Dχ k (x0 )v0 ,

k ∈ Z,

for v0 as in 5.2, choose N ≥ n (cf. 4.6) such that N  k=−N

|vk |2 ≥ 2−1

∞  k=−∞

|vk |2 ,

460

Y. Latushkin, M. Vishik

and a point x∗ , close to x0 (cf. 4.7) such that p(x∗ ) = pc (x∗ ) ≥ 5N . On the small cross-section '0  x∗ select a continuous family of vectors v(σ )⊥∇s(σ ), σ ∈ '0 , such that (cf. 4.8) for each σ ∈ '0 the following inequalities hold: 2N  k=−2N N  k=−N

|e−νk Dχ k (σ )v(σ )|2 e−2δ|k| ≤ 3

2N 

|vk |2 ,

k=−2N

|e−νk Dχ k (σ )v(σ )|2 e−2δ|k| ≥ 3−1

N 

|vk |2 .

k=−N

To choose a, select α and β as in Subsect. 4.2, set γ (t, σ ) = e−νt α(σ )β(t). Since ξ(t; σ, ∇s(σ )) = ∇s(χ t σ ) solves the ξ -equation 1.2, and ξ(t; σ, ∇s(σ )), Dχ t (σ ) v(σ ) = const, we conclude that ∇s(x)⊥a(x) for a as in 5.4. The same line of arguments as in Subsect. 4.3 with µ⊥ replaced by ν and B t (σ, ξσ )b(σ ) replaced by Dχ t (σ )v(σ ) gives the desired estimate 5.5 for z = ν.   Acknowledgements. The authors thank Susan Friedlander for useful discussions.

References [AK]

Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. New York: Springer-Verlag, 1998 [C] Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press, 1961 [CL] Chicone, C., Latushkin, Y.: Evolution semigroups in dynamical systems and differential equations. Math. Surv. Monogr. vol. 70. Providence, RI: AMS, 1999 [CLM1] Chicone, C., Latushkin, Y., Montgomery-Smith, S.: The spectrum of the kinematic dynamo operator for an ideally conducting fluid. Commun. Math. Phys. 173, 379–400 (1995) [CLM2] Chicone, C., Latushkin, Y., Montgomery-Smith, S.: The annular hull theorems for the kinematic dynamo operator for an ideally conducting fluid. Indiana Univ. Math. J. 45, 361–379 (1996) [CG] Childress, S., Gilbert, A.: Stretch, Twist, Fold : The Fast Dynamo. Berlin/New York: SpringerVerlag, 1995 [DR] Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge: Cambridge Univ. Press, 1981 [E] Eckhoff, K.S.: On stability for symmetric hyperbolic systems. I: J. Diff. Eqns. 40, 94–115 (1981); II: J. Diff. Eqns. 43, 281–304 (1982) [ES] Eckhoff, K.S., Storesletten, L.: On the stability of rotating compressible and inviscid fluids. J. Fluid Mech. 99, 433–448 (1980) [FSV] Friedlander, S., Strauss, W., Vishik, M.: Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincar´e, Anal. Non lin´eaire 14, 187–209 (1997) [FV] Friedlander, S., Vishik, M.: Instability criteria for steady flows of a perfect fluid. Chaos 2, 455–460 (1992) [LS] Latushkin, Y., Schnaubelt, R.: The spectral mapping theorem for evolution semigroups on Lp associated with strongly continuous cocycles. Semigroup Forum 59, 404–414 (1999) [LH1] Lifschitz, A., Hameiri, E.: Local stability conditions in fluid dynamics. Phys. Fluids A 3, 2644–2651 (1991) [LH2] Lifschitz, A., Hameiri, E.: Localized instabilities of vortex rings with swirl. Commun. Pure Appl. Math. 46, 1379–1408 (1993) [L] Lin, C.C.: The Theory of Hydrodynamic Stability. Cambridge: Cambridge Univ. Press, 1955 [Lu] Ludwig, D.: Exact and asymptotic solutions of the Cauchy problem. Commun. Pure Appl. Math. 13, 473–508 (1960) [M] Mather, J.: Characterization of Anosov diffeomorphisms. Indag. Math. 30, 479–483 (1968) [EN] Engel, K., Nagel, R.: One-parameter Semigroups for Linear Evolution Equations. New York: Springer-Verlag, 2000 [O1] Oseledets, V.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968)

Linear Stability in an Ideal Incompressible Fluid [O2] [R] [vN] [V] [VF] [Y]

461

Oseledets, V.: D-entropy and the antidynamo theorem. In: Proc. 6th Intern. Symp. Inform. Theory, Part III. Tashkent, 1984, pp. 162–163 Renardy, M.: On the linear stability of hyperbolic PDEs and viscoelastic flows. Z. Angew. Math. Phys. 45, 854–865 (1994) van Neerven, J.M.A.M.: The asymptotic behavior of semigroups of linear operators. Operator Theory Adv. Appl. vol. 88. Basel: Birkh¨auser, 1996 Vishik, M.M.: Spectrum of small oscillations of an ideal fluid and Lyapunov exponents. J. Math. Pures et Appl. 75, 531–558 (1996) Vishik, M., Friedlander, S.: Dynamo theory methods for hydrodynamic stability. J. Math. Pures Appl. 72, 145–180 (1993) Yudovich, V.: The linearization method in hydrodynamical stability theory. Transl. Math. Monogr. vol. 74. Providence, RI: AMS, 1989

Communicated by P. Constantin

Commun. Math. Phys. 233, 463–491 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0761-9

Communications in

Mathematical Physics

Equilibrium Fluctuations for Interacting Ornstein-Uhlenbeck Particles∗ Stefano Olla1 , Christel Tremoulet2 1

Ceremade, UMR 7534 CNRS, Universit´e Paris IX-Dauphine, Place de Lattre de Tassigny, 75775 Paris cedex 16, France. E-mail: [email protected] 2 Universit´e de Cergy Pontoise, D´epartement de Math´ematiques, 2 Av. Adolphe Chauvin, B.P.222, Pontoise, Cergy-Pontoise-Cedex, France. E-mail: [email protected] Received: 10 July 2001 / Accepted: 9 September 2002 Published online: 8 January 2003 – © Springer-Verlag 2003

Abstract: We study the hydrodynamic density fluctuations of an infinite system of interacting particles on Rd . The particles interact between them through a two body superstable potential, and with a surrounding fluid in equilibrium through a random viscous force of Ornstein-Uhlenbeck type. The stationary initial distribution is the Gibbs measure associated with the potential and with a given temperature and fugacity. We prove that the time-dependent density fluctuation field converges in law, under diffusive scaling of space and time, to the solution of a linear stochastic partial differential equation driven by white noise. 1. Introduction Consider an aqueous suspension of particles in equilibrium at temperature T = 1/β. Let the interaction between the particle be given by a two body potential V . We assume that V is superstable, positive and smooth with compact support. The interaction with the fluid is modeled by an Ornstein-Uhlenbeck type force (linear viscosity plus white noise) such that the particle velocities are maintained in equilibrium, i.e. distributed by a Maxwellian distribution of temperature T (cf. [14, 20]). The dynamics of this model is given by the solution of the following infinite system of stochastic differential equations: dxj (t) = vj (t)dt, dvj (t) = −


i=j

∇V (xj (t) − xi (t))dt − γ vj (t)dt +

 2γ dwj (t). β

(1.1)

∗ We thank J. Fritz for fruitful discussions, in particular about the existence of the infinite dynamics. A special thanks to L. Bertini for help in the proof of the spectral gap estimate (cf. Appendix B).

464

S. Olla, C. Tremoulet

Here {wj (t)} are independent standard Wiener processes, and γ > 0 is the friction coefficient. We set here the mass of each particle equal to 1. We refer to this system as the interacting Ornstein-Uhlenbeck particles. Consider the grand canonical Gibbs measures µz,β corresponding to the formal Hamiltonian H=


vj2 j

2

+

1
V (xj − xi ) 2

(1.2)

i=j

with fugacity z and inverse temperature β. As in the case of the corresponding deterministic Hamiltonian system (γ = 0), the proof of the existence of the dynamics given by (1.1) for a wide set of initial configurations is a challenging problem. In [8], J. Fritz proves the existence and uniqueness of the solution of (1.1) for a class of initial configurations that has probability 1 for any grand canonical Gibbs measure. The results contained in [8] are limited to dimension d ≤ 2. For our purposes it is enough to consider the existence of the equilibrium dynamics: for a given µz,β there exists a set of initial configurations and a set of realizations of the Wiener processes {wj } such that they have full measure and for which (1.1) has a non-exploding solution (see Sect. 2 for a precise definition). We prove this existence theorem in Appendix A (Sect. 7), by approximation with finite local dynamics. The proof, that has the advantage that it works in any dimension, follows the classical approach of Lanford ([11, 13]) adapted to our stochastic case. A certain care should be done in this proof for existence of the dynamics when dealing with random evolutions. In fact an approximation by local dynamics defined using reflection on hard walls would bring difficulties concerning the existence of these local dynamics (cf. the paper of R. Lang with the addition of T. Shiga [12], where a similar problem arises in the context of the interacting Brownian motions). In order to avoid such problems, we apply an idea of J. Fritz ([9, 10]: define special smooth local dynamics such that the stationary Gibbs measures stay unmodified (cf. Eqs. (7.1)). The purpose of this paper is to study the macroscopic behavior of this system in equilibrium. Let us consider the positions of the particles under diffusive rescaling: xjε (t) = εxj (ε −2 t),

(1.3)

where ε is a scaling parameter. We are interested in the macroscopic limit as ε → 0. The empirical distribution of the particles is a random positive measure on Rd defined by nεt (G) = εd


j

G(xjε (t)),

where G is a smooth function with compact support. If the particles are distributed (in the original microscopic coordinates xj ) by the equilibrium Gibbs measure µz,β , by the law of large numbers, we have  µz,β − a.s., nεt (G) → ρ G(q)dq where ρ = ρ(z, β) is the average density of particles per unit volume for µz,β .

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Then we consider the density fluctuation field    ε −d/2 ε nt (G) − ρ G(q)dq . ξt (G) = ε In equilibrium, the positions of the particles are distributed by µz,β , and because of the mixing properties of this Gibbs measure one can show (cf. [1]) that ξtε (J ), for a fixed t, converges in law to a Gaussian field ξ on Rd with mean zero and covariance  < ξ(f )ξ(g) >= χ f (q)g(q) dq, (1.4) where χ = χ (z, β) is the compressibility of µz,β . We prove in this paper that ξtε , as a distribution valued process, converges in law to the solution ξt of the stochastic linear partial differential equation:  2ρ ∇ · j, (1.5) ∂t ξ = D(ρ)#ξ + γβ where {ji }i=1,...,d are δ-correlated space–time white noises, i.e. d-vector Gaussian fields on Rd+1 with covariance < ji (q, t)jh (q , s) >= δ(q − q )δ(t − s)δi,h . The diffusion coefficient D(ρ) is identified as the derivative of the thermodynamic pressure as a function of the density, as in [17]. Equation (1.5) should be intended in the weak sense, i.e. for any smooth test function G,  t ξs (D#G) ds + Mt (∇G), ξt (G) − ξ0 (G) = 0

where Mt (∇G) is a continuous martingale with constant quadratic variation given by 2ρt  2 dq. Since (1.5) is obtained in equilibrium, the initial condition ξ is |∇G(q)| 0 γβ distributed by the Gaussian field with covariance given by (1.4), which is in fact the invariant law for the evolution given by (1.5). This implies the following identity for the bulk diffusion coefficient: ρ . (1.6) D= γβχ If we consider the solution of (1.1) in a long time scale, the velocities will relax to equilibrium. If we suppress them we obtain a closed evolution on the positions given by  2 1
∇V (xj (t) − xi (t)) dt + (1.7) dwj (t). dxj (t) = − γ γβ i=j

We refer to this system as the interacting Brownian particles. In [20] Spohn proved the convergence of the density fluctuation field ξtε for (1.7). The limit equation is still given by (1.5). The equivalence of the bulk diffusion in the two systems could also be seen from the hydrodynamic limit out of equilibrium (cf. [24, 17]).

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As noticed first by Rost [19], the main point in the proof of a theorem of equilibrium fluctuations for a system with conserved quantities is the so-called BoltzmannGibbs principle. This principle is an estimate of the space-time variance of the difference between a non-conserved quantity (typically the flux of the density fluctuation field) and its linearization along the conserved quantities (here the density fluctuation field itself). This principle has been proven in various reversible models (in [2] for zero-range models, in [6] for speed-change exclusion models and in [20] for the interacting Brownian particle system (1.7)). In all these papers the approach to the proof of the Boltzmann-Gibbs principle is the following: one defines an Hilbert space of functions of the configurations, with a scalar product related to the static variance of these functions with respect to the stationary measure of the process. Then one needs to identify the subspace that is invariant under the action of the semigroup of the generator of the process with the one dimensional subspace generated by the density fluctuation field. This identification follows by a strong version of the equivalence of the canonical and grand canonical Gibbs measures. One of the difficulties in this approach is the selfadjoint extension of the generator of the process in this new Hilbert space (observe that all these results concern reversible models). In [4], C. C. Chang introduced a different, and simpler, approach to the proof of the Boltzmann-Gibbs principle. In Chang’s approach, by using the same strong form of the equivalence of ensembles, the problem is reduced to the estimate of the space-time variance in the canonical measure in a fixed finite volume. Then this estimate can be done by standard finite dimensional analysis (cf. [7] Chapter 11, for a clear exposition of the method). Besides its simplicity, this approach has two other advantages: it avoids problems in defining the dynamics in a different Hilbert space, and it can be extended more easily to non-reversible models (cf. [5]). Our proof follows the direction of this second approach. The difficulty here is due to the fact that the generator of our Markov process is degenerate in the directions of the positions of the particles (noise acts only on the velocities). This poses a problem even in the finite dimensional analysis needed to estimate the variance in the canonical measure on a finite box. We solve this problem by relating the (degenerate) generator of our process to the strictly elliptic and symmetric generator of the interacting Brownian particles process (cf. Proof of Proposition 4.2). Then we need an estimate of the spectral gap of this symmetric operator, that we prove in Appendix B. For the strong form of the equivalence of ensembles, which is in fact a local central limit theorem for the Gibbs measures, we use the results by Spohn contained in [20] (recently improved in [3], cf. Sect. 2). The paper is organized as follows: in Sect. 2 we give a precise definition of the model and of the dynamics in equilibrium. In Sect. 3 we study the evolution equations of the fluctuation field. In Sect. 4 we prove the Boltzmann-Gibbs principle for our system. In Sect. 5 we prove the tightness of the distribution of the fluctuation field as a stochastic process with values in a certain negative Sobolev space. In Sect. 6, by using the Boltzmann-Gibbs principle and the compactness result, we prove the convergence to the solution of (1.5). In Appendix A we prove the existence of the dynamics in equilibrium. In Appendix B we prove a lower bound on the spectral gap of the generator of the interacting Brownian motions (cf. (1.7)) in a finite box. We use this bound in order to prove some estimates on the derivatives of the solutions of the Poisson equation related to the generator of the interacting Brownian motions. We need these estimates in the proof of the Boltzmann-Gibbs principle.

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2. The Dynamics in Equilibrium We assume that V : Rd → R is a pair potential satisfying the following assumptions: (i) (ii) (iii) (iv)

V V V V

is central (i.e. depends only on |q|) and symmetric: V (q) = (−q). is positive: V ≥ 0, and V (0) > 0. has finite range: V (q) = 0 if |q| ≥ R. is two times continuously differentiable.

Notice that the positivity implies immediately the lower regularity condition (LR) of [16]. Furthermore observe that (ii) is equivalent to superstability (cf. condition (SS) in [16]). The configuration space ( is defined as the set of locally finite labeled configurations of particles ω = {(xi , vi ), i ∈ N}, where xi = xi (ω) and vi = vi (ω) have values in Rd , and the sequence xi = xi (ω) has no accumulation points. We will use the notation ωx = {xi (ω)}. Let ( be equipped with the weak topology: limn ωn = ω means that limn xi (ωn ) = xi (ω) and limn vi (ωn ) = vi (ω). We denote by B the corresponding Borel σ -algebra of subsets of (. A grand canonical Gibbs state for V at temperature β −1 and activity (or fugacity) z is a probability measure µ on ((, B) that distribute ωx according to a grand canonical Gibbsian point field with pair interacting potential βV and activity z, while velocities are independent of positions and are identically distributed as independent Gaussian variables of zero mean and variance β −1 . This means that the µ-conditional probability to find n particles in a finite volume region + ⊂ Rd with configuration {(x1 , v1 ), . . . , (xn , vn )}, conditioned to the configuration outside +, i.e. ω+c = {(xi (ω), vi (ω)), xi (ω) ∈ +c , i ∈ N}, has density with respect to the Lebesgue measure on (+ × Rd )n given by the DLR equations: µ+ ((x1 , v1 ), . . . , (xn , vn )|ω+c )   zn 1 x exp −βHn ((x1 , v1 ), . . . , (xn , vn ); ω+ = c) , x Z+ (ω+c ) n! where x Hn ((x1 , v1 ), . . . , (xn , vn ); ω+ c) =

(2.1)

 n  v2
j j =1

 n 
1
+ V (xj − xi ) + V (xj − yi ) 2  2 x i=1

yi ∈ω+c

x ) is the corresponding normalization. and Z+ (ω+ c In order to simplify notations we fix the values γ = β = 1, and, since the activity z ∈ (0, z0 ) is fixed, we will omit. Writing the explicit dependence on z and we write µ = µz,β , and ρ = ρ(z, β) for the corresponding density of particles. In the following we will assume that   −1 0 < z < z0 = 0.28 e (1 − e−V (q) ) dq . (2.2)

We need this condition on the activity in order to apply Spohn’s results on the equivalence of ensembles (cf. [20]). As a consequence we are in the low fugacity regime where the grand canonical Gibbs measure is unique. A recent paper by Cancrini and Tremoulet (cf. [3]) permits to improve (2.2) substituting the constant 0.28 with 1/3, that

468

S. Olla, C. Tremoulet

is the sufficient condition for the exponential L2 -mixing of the Gibbs measure needed by Spohn in Lemma 4 in [20]. We give now a precise definition of the dynamics in equilibrium associated to (1.1) and to a grand canonical Gibbs measure µ. Let W be the probability measure induced on  = C([0, ∞), Rd )N by the infinite independent Wiener processes {wi (·), i ∈ N}. On ( ×  we define the product measure Pµ = µ ⊗ W, and we denote Eµ the corresponding expectation. On ( ×  we also define the increasing filtration {Ft }t>0 , where Ft is the σ -algebra generated by {wi (s), s ≤ t, i ∈ N} and B. Given an initial configuration ω(0), a solution of (1.1) is a Ft -adapted continuous stochastic process {ω(t)} with values in (, which satisfies  t vi (ω(s)) ds, xi (ω(t)) = xi (ω(0)) + 0  t
vi (ω(t)) = vi (ω(0)) − ∇V (xi (ω(s)) − xj (ω(s)) ds (2.3) 0 j =i



t

− 0

vi (ω(s))ds +

√

2 wi (t).

In Appendix A it is proven that there exists a set M ⊂ ( ×  such that Pµ (M) = 1, and such that if (ω(0), {wi (·)}i∈N ) ∈ M, then (2.3) has a solution. This way it is possible to define a strongly continuous semigroup of contraction operators P t on L2 ((, B, µ). A straightforward calculation shows that the generator of this process (1.1) can be written as a sum of a symmetric operator and an antisymmetric one (with respect to µ) : L = A + S, A = {H, ·} = S=







 vj · ∇xj −

j

 #vj − vj · ∇vj .


i=j

 ∇V (xj − xi ) · ∇vj  ,

(2.4)

j

Observe that the antisymmetric operator A is given by the Poisson brackets with the Hamiltonian H, i.e. is the generator of the corresponding deterministic Hamiltonian dynamics. 3. Time Evolution of the Fluctuation Field For every configuration ω ∈ ( and ε > 0, the density fluctuation field is the distribution on S (Rd ) (the dual space of the Schwartz space S(Rd ) of smooth rapidly decreasing functions) defined by   
G(εxj (ω)) − ρ G(q) dq  , ξ ε (G)(ω) = ε −d/2 ε d j

where G ∈ S(Rd ). Defining Gε (q) = εd/2 G(εq), observe that ξ ε (G)(ω) = ξ 1 (Gε )(ω). It is easy to see that ξ ε (G) ∈ L2 ((, µ). In fact we have the following bound:

Equilibrium Fluctuations

469

Lemma 3.1. There exists a constant B = B(ρ) < ∞ such that for any G:   sup ξ ε (G)2 ≤ BG2L2 . ε>0

(3.1)

Proof. Using the 2-point correlation function ρ2 of the grand canonical Gibbs measure µ, and the translation invariance of µ:    Gε (q1 )Gε (q2 )(ρ2 (q1 , q2 ) − ρ 2 ) dq1 dq2 ξ ε (G)2 =   1 ≤ Gε (q1 )2 + Gε (q2 )2 |ρ2 (q1 , q2 ) − ρ 2 | dq1 dq2 2    = Gε (q)2 dq |ρ2 (0, q )−ρ 2 | dq = G2L2 |ρ2 (0, q ) − ρ 2 | dq . This last quantity is finite by Lemma 4.4.8 of [15].

 

In order to simplify notations, we will denote ξtε = ξ ε (ω(ε −2 t)). For a fixed arbitrary T > 0, we denote by P ε the probability distribution, under Pµ , of {ξtε , 0 ≤ t ≤ T } in C [0, T ], S (Rd ) .

Consider now a smooth test function G with compact support. By a simple calculation we have  ε−2 t
ε ε ξt (G) − ξ0 (G) = ε d/2+1 ∇G(εxj (s)) · vj (s) ds 0

√ d/2+1  = 2ε 0

− εd/2+1

j

ε−2 t

 ∇G(εxj (s)) · dwj (s) + ε

j

2 0

ε−2 t

γε (ω(s)) ds

 ∇G(εxj (ε −2 t)) · vj (ε −2 t) − ∇G(xjε (0)) · vj (0) ,


 j




(3.2) where γε (ω) = εd/2




 

d


α,σ =1

j

∂α ∂σ G(εxj )vjα vjσ − ε −1




 ∇G(εxj ) · ∇V (xj − xi )

i=j

(3.3) . with q α the α th component of q and ∂α = ∂q α . It is easy to see that variance of the last two terms in the rhs of (3.2) converges to 0 as ε → 0. In fact, by stationarity we have     2
  ∇G(εxj (ε −2 t)) · vj (ε −2 t) − ∇G(xjε (0)) · vj (0) Eµ  ε d/2+1 j

≤ 2ε < (εd/2 2


j

∇G(εxj ) · vj )2 >µ

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S. Olla, C. Tremoulet

and by the independence of the velocities: 

d/2 2 d 2 < (ε ∇G(εxj ) · vj ) >µ = ε < |∇G(εxj )| >µ = ρ |∇G(q)|2 dq. j

j

(3.4) The main step we need to do, in order to obtain (1.5) as a macroscopic equation, is to prove that:  2  t −2 ε lim Eµ = 0. (3.5) (γε (ω(ε s)) − Dξs (#G)) ds ε→0

0

For any continuous function f on Rd with compact support, let us define the local function  1
σ σ ϒα,σ (f )(ω) = (xi − xj )Vα (xi − xj ) f (λxi + (1 − λ)xj )dλ, (3.6) 0

i,j

where Vα (q) = ∂q α V (q).

 Lemma 3.2. Let f be a continuous function with compact support such that f (q) dq = 0. Then there exists a constant C = C(ρ) such that < (ϒα,σ (f ))2 >µ ≤ C f 2L2 .  Proof. This is basically (4.13) in [20]. Since f (q) dq = 0, we have  1  < ϒα,σ (f ) > = dλ (q1σ − q2σ )Vα (q1 − q2 ) 0

× f (λq1 + (1 − λ)q2 )ρ2 (q1 , q2 ) dq1 dq2  1   = dλ dz zσ Vα (z)ρ2 (0, z) dq2 f (λz + q2 ) = 0. 0

Let us denote by ρ˜4 the truncated 4-point correlation function of µ, i.e. ρ˜4 (q1 , q2 , q3 , q4 ) = ρ4 (q1 , q2 , q3 , q4 ) − ρ2 (q1 , q2 )ρ2 (q3 , q4 ). In order to simplify notations, let us define g(q1 , q2 , λ) = (q1σ − q2σ )Vα (q1 − q2 )f (λq1 + (1 − λ)q2 ) . Then, by using the Jensen inequality, we have < (ϒα,σ (f ))2 >µ = < (ϒα,σ (f )− < ϒα,σ (f ) >)2 >µ 2   1 
≤ dλ  g(xi , xj , λ) − g(q1 , q2 , λ)ρ2 (q1 , q2 ) dq1 dq2  0



1

=

(i,j )

 dλ

µ

dq1 dq2 dq3 dq4 g(q1 , q2 , λ) g(q3 , q4 , λ) ρ˜4 (q1 , q2 , q3 , q4 ).

0

(3.7)

Equilibrium Fluctuations

471

By a linear change of variables in the quadruple integration and translation invariance, the right-hand side of (3.7) is equal to  1  dλ z1σ Vα (z1 ) z3σ Vα (z3 ) f (λz1 + z2 ) f (λz3 + z2 + z4 ) 0

× ρ4 (z1 , 0, z3 + z4 , z4 ) dz1 dz2 dz3 dz4 , which is bounded by    2 f (z) dz dz1 dz3 dz4 |ρ˜4 (z1 , 0, z3 + z4 , z4 )|. CV  

This last integral is finite by Thm. 4.4.8 of [15].

In the following Gα,σ (q) = ∂q α ∂q σ G(q). Defining Gεα,σ (q) = ε d/2 Gα,σ (εq), then using the symmetry of V we can rewrite

εd/2−1

εd/2−1 ∇G(εxj ) · ∇V (xj − xi )) = (∇G(εxj ) − ∇G(εxi )) · 2 j

i=j

∇V (xj − xi ) =

1
2

α,σ

j

i=j

ϒα,σ (Gεα,σ ).

Let h(q) be a positive continuous function with support in B(0, 1/2), the ball centered at the origin and of radius 1/2, and with total integral equal to 1. Since Gεα,σ − Gεα,σ ∗ hL2 converges to 0 as ε → 0, by Lemma 3.2 we have lim < (ϒα,σ (Gεα,σ − Gεα,σ ∗ h))2 >µ −→ 0.

ε→0

ε→0

(3.8)

Let {τq , q ∈ Rd } the shift operator on (, defined by τq ω = {(xj (ω)+q, vj (ω)), j ∈ N}. By (3.8), Schwarz inequality and stationarity, we can substitute in (3.5) γε with 
d/2 Gα,σ (εq)φα,σ (τq ω) dq, γ˜ε (ω) = ε α,σ

where φα,σ is the local function given by
1 φα,σ (ω) = h(xj )vjα vjσ − ϒα,σ (h). 2

(3.9)

j

Taking the expectation with respect to µ and using the translation invariance we have < φα,σ >µ

  1 1 dq1 dq2 ρ2 (q1 , q2 )(q1σ −q2σ )Vα (q1 −q2 ) h(λq1 +(1−λ)q2 ) dλ 2 0     1 1 dqρ2 (q, 0)q σ Vα (q) = δα,σ ρ − dq ρ2 (q, 0)q · ∇V (q) − 2 2d

= ρδα,σ − = ρδα,σ

and observe that, by the virial theorem (cf. [18, 23]), this last quantity is equal to δα,σ P (ρ), where P is the thermodynamic pressure expressed as a function of the density ρ. Consequently D = P (ρ). The limit (3.5) is a direct consequence of the Boltzmann-Gibbs principle that we enunciate and prove in the next section.

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S. Olla, C. Tremoulet

4. Boltzmann-Gibbs Principle In this section we will consider the following set of smooth local functions on (. We say that a measurable function φ : ( → R is in E if it can be written as  

φ(ω) = H f1 (xi (ω), vi (ω)), . . . , fm (xi (ω), vi (ω)) , i

i

where {fk (q, v), k = 1, . . . , m} are smooth functions on Rd × Rd with compact support in q and growth at most linearly in v (i.e. |f (q, v)| ≤ C(1 + |v|)), and H (y1 , . . .! , ym ) ∈ m C ∞ (Rm ) such that there exists a constant c < ∞ such that |H (y1 , . . . , ym )| ≤ ec k=1 |yk | . If + is a finite region containing all the q-supports of fk , then there exists a finite constant C such that for any configuration ω,  
|φ(ω)| ≤ exp C (1 + |vj (ω)|) . (4.1) xj (ω)∈+

By (4.1) and by superstability estimate (cf. [16]), it follows that φ ∈ Lp (µ) for any p < ∞. We denote its expectation by &(ρ) =< φ >µρ φ &(ρ) is a smooth function of ρ. In fact, as a function of the density ρ. Observe that φ &(ρ) is a smooth function of the activity z, which always by the superstability estimates, φ is a smooth function of the density ρ in the ranges defined by (2.2) (cf. [15], Theorem 4.2.3, p.76). Let us define  
& dφ &(ρ) − ?(ω) = φ(ω) − φ (ρ)  h(xj ) − ρ  , dρ j

where h : Rd → R+ is a positive smooth function with support  in B(0, 1/2), the ball centered at the origin and of radius 1/2, and with total integral h(q)dq = 1. The main result of this section is contained in the following proposition: Proposition 4.1 (Boltzmann-Gibbs principle). Let φ ∈ E and ? be defined as above. Then for any smooth function G with compact support on Rd , ' 2 (  t  d/2 −2 lim Eµ ε = 0. ds dq G(εq)?(τq ω(ε s)) ε→0

0

We will make wide use of the following estimate: Lemma 4.1. There exists a finite constant C, depending only on V and ρ, such that for any G in L2 (Rd ) and for any local function ψ(ω) in L2 (µ) such that < ψ >= 0 and whose support is contained in a finite set + ⊂ Rd , we have:  2  G(q)ψ(τq ω) dq ≤ C|+| < ψ 2 > G(q)2 dq. (4.2)

Equilibrium Fluctuations

473

Proof. This is a consequence of the exponential decay of the correlations, cf. Lemma 4 in [20], and the proof is very close to Lemma 9 in [20].  2 G(q)ψ(τq ω) dq 

G(q)G(q ) < ψ(τq ω)ψ(τq ω) > dq dq    1 ≤ G(q)2 + G(q )2 | < ψ(τq ω)ψ(τq ω) > | dq dq 2   = dq G(q)2 dq | < ψ(τq−q ω)ψ(ω) > |. =

By Lemma 4 in [20] there exists c and α positive constants, depending only on V and ρ, such that ¯ ¯ +) ¯ |+|e ¯ −αd(τq +, | < ψ(τq ω)ψ(ω) > | ≤ < ψ 2 >µ min{1, c|+|e e

¯ +) ¯ −αd(τq +,

},

¯ = {q ∈ Rd : d(q, +) ≤ R}, (R being the radius where d is the distance on Rd , and + of the support of the interaction V ). Then performing the integration in q one obtains (4.2).   Proposition 4.1 will be proven in several steps. In the first one, we will condition ? on the positions configuration that we denote by ωx . We will use then the following lemma: Lemma 4.2. Let A be a local function on L2 (µ) such that its µ-expectation conditioned on the positions configuration ωx is < A|ωx >= 0

µ − a.s.

Then for any smooth function G with compact support on Rd , ' 2 (  t  lim Eµ ε d/2 = 0. ds dq G(εq)A(τq ω(ε −2 s)) ε→0

0

Proof. Observe that the operator S, the symmetric part of the generator of the process defined by (2.4), has a spectral gap equal to 1. Since < A|ωx >= 0, S −1 A(ω) is a function in L2 (µ). Recall we have defined ε d/2 G(εq) = Gε (q). It follows that S −1 dq Gε (q)A(τq ω) ∈ L2 (µ). Then (cf. Theorem 2.2 in [22]): '  (  Eµ

t

ds

0

≤ 8tε2

)

dq Gε (q)A(τq ω(ε −2 s))

2

  * dq Gε (q)A(τq ω) (−S)−1 dq Gε (q)A(τq ω) .

By the spectral gap of S this last quantity is bounded by: + +2 + + + dq G (q)A(τ ω) ≤ 8tε2 + ε q + + 2

L (µ)

.

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S. Olla, C. Tremoulet

By Lemma 4.1 this is bounded by 8tε 2 G2L2 C(A).   As a consequence of Lemma 4.2, we only need to prove Proposition 4.1 for the function ˜ x ) =< ?|ωx > . ?(ω ˜ is still a local smooth function satisfying (4.1). Observe that ? ˜ For l large, let + be a centered box of size l. The µ canonical expectation of ? conditioned on the configuration of the positions of the particles outside + (denoted by x ), and on the number of particles in + is defined by: ω+ c x ˜ + (ω), ω+ ˜ x ) = < ?|N C+ ?(ω c > . x and it does not depend on ρ. Always By DLR equations, it can be defined for every ω+ c ˜ is smooth on the set of configurations where by DLR equations, one can see that C+ ? there are no particles on the border of +. Since this set has µ full measure, we have that C+ is smooth µ-a.s. Again by the DLR equations one has  
N+ (ω) C+  h(xj ) = , (4.3) |+| j

consequently ˜ x ) = C+ φ(ω ˜ x) − φ &(ρ) − C+ ?(ω

  & dφ N+ (ω) (ρ) −ρ , dρ |+| &

d<φ>

dρ Recall the thermodynamic relation d(logz) = χ. Then ddρφ (ρ) = χz dz µz . It is proven in [20], Eq. (8.5), that, if z(ρ) satisfies (2.2), one has the following strong equivalence of ensembles:

˜ 2 > = 0. lim |+| < (C+ ?)

+↑Rd

(4.4)

Indeed in [3] an even stronger statement is proved under a weaker condition than (2.2). Then, using Lemma 4.1, (4.4) implies  2  ˜ q ωx ) >= 0. lim lim < ε d/2 dq G(εq)C+ ?(τ

+↑Rd ε→0

˜ around its canonical expectation (this is basically Lemma 9 in [20]). So we can recenter ? ˜ and Proposition 4.1 will follow from the following one: C+ ?,

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475

Proposition 4.2. For any smooth function G with compact support on Rd and for any finite box +, ' 2 (  t  d/2 x −2 ˜ q ω (ε s)) ˜ − C+ ?)(τ lim Eµ ε = 0. ds dq G(εq)(? ε→0

0

Proof. We denote by x x ˜ ˜ − C+ ?)(ω f (x1 , . . . , xN+ ; ω+ ) c ) = (?

and we will consider f as a function of the positions of the particles inside +n , for x and any fixed number of particles N = n inside any fixed exterior configuration ω+ c + +. Since the set of configurations with particles on the boundary of + have null meax without particle on the sure with respect to µ, we can consider only configurations ω+ c x boundary of +, and the dependence of f on ω+c is smooth on this set of configurations. x fixed, f is a smooth function of the positions (x , . . . , x ) Furthermore, for any such ω+ c 1 n n on the interior of + , x and N = n fixed, we consider the elliptic operator on +n , Then, for any ω+ c +   n

W #xj − Ln,ωx c = (4.5) (∇V )(xj − xi ) · ∇xj  +

j =1

i=j

with Neumann boundary conditions. Then, by the properties of f , a smooth function x ) solution of u(x1 , . . . , xn ; ω+ c x −LW n,ωx c u = f (x1 , . . . , xn ; ω+c ) +

(4.6)

exists. We consider now u as a local function of ωx (i.e. as a function of {x1 , . . . , xN+ , x }). ω+ c x By Lemma 9.1 in Appendix C, there exist c1 , c2 finite constant independent of n, ω+ c and u such that  n  ,  
x x 2 x C+ |∇xk u|2 (n, ω+ (n, ω+ c ) ≤ c1 n exp c2 (n + N∂ + + (ω+c )) C+ f c ), R

k=1

where ∂R+ + = +(l + R) \ +. Integrating with respect to µ we obtain   
|∇xk u|2 ≤ c1 N+ ec2 N+(l+R) f 2 ,

(4.7)

xk ∈+

and the left-hand side is bounded by (4.1) and superstability estimates. The trick now is to relate LW x to the generator of our process L (cf. (2.4)). Define n,ω+ c the local function
(ω) = vj · ∇xj u. xj ∈+

476

S. Olla, C. Tremoulet

By (4.7),  ∈ L2 (µ), and by the results contained in Appendix C it is in the domain of the generator L. Then we apply L to  and we obtain



L(ω) =

(vk · ∇xk )(vj · ∇xj )u −

xj ∈+ k







∇V (xj − xi ) · ∇xj u − .





xj ∈+

i=j

(4.8) The first term on the rhs of (4.8) can be rewritten as d


xj ∈+ k

α,σ

vkα vjσ ∂xkα ∂xjσ u =




#xj u + A1 (ω)

xj ∈+

where, by Lemmas 9.2 and 9.3, A1 (ω) is a local function in L2 (µ) such that < A1 |ωx > = 0

µ − a.s.

Observe that also  has the same property. In conclusion we can write x ˜ − C+ ?)(ω ˜ (? ) = LW N+ ,ωx c u = L(ω) + A(ω), +

where A(ω) is a local function in L2 (µ) such that < A|ωx > = 0

µ − a.s.

So we can apply Lemma 4.2, and we are left to prove '

ε d/2

lim Eµ

ε→0



t

 ds

0

2 (

dq G(εq)(L)(τq ω(ε −2 s))

= 0.

(4.9)

Integrating in time we can rewrite εd/2



t

 ds

0

=ε

d/2+2



dq G(εq)(L)(τq ω(ε −2 s)) (4.10) dq G(εq)((τq ω(ε

−2

t)) − (τq ω(0))) + Mεt ,

where Mεt is a martingale such that 2  
 ε 2 d+2 E (Mt ) = tε dq G(εq)∂vk (τq ω) > < = tεd+2



k

dq dq G(εq) G(εq ) <


k

∂vk (τq ω)∂vk (τq ω) >,

Equilibrium Fluctuations

477

and by the Schwarz inequality this is bounded by 1/2  1/2  

d+2 2 2 tε dqdq |G(εq)||G(εq )| < (∂vk (τq ω)) (∂vk (τq ω)) >  = tε2 < εd/2



k

k

1/2 2
 >. dq|G(εq)| (∂vk (τq ω))2 

k

Since




(∂vk (ω))2 =




(∇xk u)2 ,

xk ∈+

k

  then, by (4.7), this is a local function in L1 (µ), and by Lemma 4.1, E (Mεt )2 converges to 0 as ε → 0. It follows by Lemma 4.2 that the variance of the first term on the rhs of (4.10) goes to 0 as ε → 0.   5. Tightness For any k ≥ 0 and f, g ∈ C ∞ (Rd ) consider the scalar product   k (g, f )k = g(q) |q|2 − # f (q) dq Rd

(5.1)

and denote by Hk the corresponding closure. For any positive k we denote by H−k its dual space with respect to the L2 (Rd ) ≡ H0 scalar product. It is convenient to represent the scalar product (·, ·)k in the ON base of the Hermite polynomials, which are the eigenfunctions of |q|2 − #. Let n be a multi-index ! of (Z+ )d and | n| = di=1 n(i). We denote by λn(i) = 2n(i) + 1 for n(i) ∈ Z+ and !d .d th λn = i=1 λn(i) . Define hn (q) = i=1 hn(i) (qi ), where hm is the m normalized Hermite polynomial of order m in R. We have then for every k ≥ 0 and f ∈ L2 ,  2 
f 2k = f (q)(|q|2 − #)k f (q) dq = λkn f (q)hn (q) dq . Rd

Rd

n∈(Z+ )d

This is valid also for negative k. So the H−k -norm of a distribution ξ on Rd can be written as
2 λ−k (5.2) ξ 2−k = n ξ(hn ) . n∈(Z+ )d

Observe that for k > k, the injection J of H−k in !H−k is compact. In fact it can be approximated by the finite range operators Jm ξ = |n|≤m ξ(hn )hn , and it is easy to see that the operator norm of the difference is bounded by

J − Jm  ≤ (2m + d)−(k −k) . ε Recall that for a fixed arbitrary T > 0, wehave denoted by  P the distribution, under ε ε −2 d Pµ , of {ξt = ξ (ω(ε t)), 0 ≤ t ≤ T } in C [0, T ], S (R ) .

478

S. Olla, C. Tremoulet

Proposition 5.1. For any k > d + 1 and every T > 0, the sequence of probability measures {P ε , ε ∈ (0, 1]} has support in C([0, T ], H−k ) and is relatively compact in this space. By the compactness of the injections H−k H→ H−k for k < k , and standard compactness arguments, Proposition 5.1 is a consequence of the following proposition. Proposition 5.2. For any k > d + 1 and every T > 0, we have that (i)  sup Eµ

ε∈(0,1)

 sup

t∈[0,T ]

||ξtε ||2−k

< +∞.

(ii) For any R > 0, 



lim lim sup Pµ  sup ||ξtε − ξsε ||−k > R  = 0.

δ→0

ε→0

t,s∈[0,T ] |t−s|≤δ

To prove Proposition 5.2, we need the following key estimate: Lemma 5.3. Let G ∈ S(Rd ). Then there exists a constant B = B(ρ, T ) < ∞ such that      ε 2 sup Eµ sup ξt (G) ≤ B G(q)2 + |∇G(q)|2 dq. t∈[0,T ]

ε∈(0,1)

Proof. Let us define Fε (ω) =

!

j

∇G(εxj ) · vj . By (1.1), we have

ξtε (G) = ξ0ε (G) + ε d/2+1 Then  Eµ

sup ξtε (G)2

t∈[0,T ]









ε−2 t

0



Fε (ω(s)) ds.

'

≤ 2 ξ ε (G)2 + 2Eµ  sup

t∈[0,T ]

ε d/2+1



ε−2 t

0

(5.3)

(2  Fε (ω(s)) ds  .

/ 0 By Lemma 3.1, there exists a constant B such that ξ ε (G)2 ≤ B G2L2 . Since Fε = −SFε , by Theorem 2.2 of [22]  ' (2   ε−2 t Eµ  sup ε d/2+1 Fε (ω(s)) ds  ≤ 8T ε d < Fε (−S)−1 Fε > t∈[0,T ]

0

= 16T εd <




|∇G(εxj )|2 >

j

= 16T ρ∇G2L2 .

 

Equilibrium Fluctuations

479

Proof of Proposition 5.2. We start by proving (i). By (5.2) and Lemma (5.3) we have ' ( ' (
Eµ sup ||ξtε ||2−k ≤ sup ξtε (hn )2 λ−k n Eµ t∈[0,T ]

n

≤B

t∈[0,T ]


n

2 λ−k n (1 + ∇hn L2 )

≤B




2 λ−k n (1 + hn 1 )

n

≤B

(5.4)


n

λ−k n (1 + λn ).

This series converges provided k > d + 1 and therefore (i) is proven. We are left with the proof of (ii). Always by (5.2) we have    
   sup ξtε (hn ) − ξsε (hn ) 2  . Eµ  sup ||ξtε − ξsε ||2−k  ≤ λ−k n Eµ t,s∈[0,T ] |t−s|≤δ

t,s∈[0,T ] |t−s|≤δ

n∈Nd

(5.5)

For every R ≥ 1, 


n∈Nd :| n|≥R

 sup λ−k n Eµ

2

'

n∈Nd :| n|≥R

λ−k n Eµ



ξtε (hn ) − ξsε (hn ) 

t,s∈[0,T ] |t−s|≤δ




≤4



( sup (ξtε (hn ))2 t∈[0,T ]

.

By (5.4), since k > d + 1, lim lim sup

R→∞

ε→0

'


n∈Nd :| n|≥R

λ−k n Eµ

( sup (ξtε (hn ))2 t∈[0,T ]

= 0.

(5.6)

Consequently, in order to prove (ii), we only need to show that for every n,   lim lim sup Eµ  sup (ξtε (hn ) − ξsε (hn ))2  = 0.

δ→0

ε→0

t,s∈[0,T ] |t−s|≤δ

By (2.3) ξtε (hn ) − ξsε (hn ) = I1ε (t, s) + I2ε (t, s) + I3ε (t, s), where I1ε (t, s) =

√ d/2+1  2ε

ε−2 t

ε−2 s


j

∇hn (εxj (τ )) · dwj (τ ),

(5.7)

480

S. Olla, C. Tremoulet

I2ε (t, s)

 =

ε−2 t

ε −2 s

γε (ω(u)) du,

where γε is given by  


 ∂α ∂σ hn (εxj ) vjα vjσ − ε −1 ∇hn (εxj ) · ∇V (xi − xj ) γε (ω) = εd/2 j

and

α,σ

i=j

  I3ε (t, s) = − ηε (ω(ε −2 t)) − ηε (ω(ε −2 s)) ,

where we have defined d

ηε (ω) = ε 2 +1




∇hn (εxj (ω)) · vj (ω).

j

Observe that, by Lemma 3.2, γε is uniformly bounded in L2 (µ), then by the Cauchy-Schwarz inequality and stationarity    T   Eµ γε (ω(ε −2 τ ))2 dτ ≤ δT < γε2 >≤ δT c( n) Eµ  sup I2ε (t, s)2  ≤ δ t,s∈[0,T ] |t−s|≤δ

0

(5.8) with c( n) a constant independent of ε and δ. It follows that this term vanishes to 0 as δ goes to 0. For what concerns I1ε (t, s), we will prove that for any R > 0,   lim lim sup Pµ  sup |I1ε (t, s)| > R  = 0.

δ→0

ε→0

t,s∈[0,T ] |t−s|≤δ

This follows if we prove 1 lim lim sup Pµ δ→0 ε→0 δ

 sup s≤t≤s+δ

|I1ε (t, s)|

 > R = 0.

By stationarity this is equivalent to prove   1 ε lim lim sup Pµ sup |I1 (t, 0)| > R = 0. δ→0 δ ε→0 0≤t≤δ Let us write I1ε (t, s) as the difference Mtε −Msε , where Mtε is a martingale with quadratic variation given by  t
Aε (t) = 2εd (∇hn )2 (εxj (ε −2 u)) du. 0

j

Equilibrium Fluctuations

481

Observe that supε < Aε (t) >≤ Ctρn. For any K > 0 define the stopping time τε,K = inf{t : Aε (t) > K} ε &tε,K is a martingale with quadratic &tε,K = Mt∧τ . Then for any fixed K, M and let M ε,K variation bounded by K, and by a standard estimate   R2 &tε,K | ≥ R ≤ 2e− 2δK . (5.9) Pµ sup |M 0≤t≤δ

Then for any fixed K and any R > 0,     R2 Pµ sup |Mtε | > R ≤ 2e− 2δK + Pµ τε,K < δ 0≤t≤δ

and   Cρnδ . Pµ τε,K < δ ≤ Pµ (Aε (δ) > K) ≤ K It follows that for any R > 0,   1 lim lim sup Pµ sup |Mtε | ≥ R = 0. δ→0 ε→0 δ 0≤t≤δ In order to study I3ε (t, s), observe that  



Eµ  sup I3ε (t, s)2  ≤ 2Eµ

sup

t∈[0,T ]

t,s∈[0,T ] |t−s|≤δ

Then the evolution equations for ηε say ηε (ω(ε

−2

t)) = e

−ε−2 t



t

+



ηε (ω(0)) + e−ε

−2 (t−τ )

0

t

2



e−ε

ηε (ω(ε −2 t))

−2 (t−τ )

0

 .

γε (ω(ε −2 τ )) dτ

dMτε .

For the first term of the rhs of the above expression, observe that < ηε2 >→ 0 as ε → 0 (cf. (3.4)). About the second term, by Schwarz inequality we have   t 2  −2 e−ε (t−τ ) γε (ω(ε −2 τ )) dτ Eµ sup t∈[0,T ]

0



≤ T < γε2 > sup

t

t∈[0,T ] 0

e−2ε

About the martingale term, by Doob’s inequality:      Eµ

t

sup

t∈[0,T ]

0

e

−ε−2 (t−τ )

dMτε

that converges to 0 as ε → 0.

 

T

2

≤8 0

e

−2 (t−τ )

dτ −→ 0.

−2ε−2 (T −τ )

ε→0

  dτ ρ (∇hn )2 (q) dq

482

S. Olla, C. Tremoulet

6. The Macroscopic Equation As a consequence of the results contained in Sects. 3 and 4 we have that for any test function G ∈ C 2 (Rd ) with compact support  1 12   t 1 ε 1 ε ε ε lim Eµ sup 11ξt (G) − ξ0 (G) − ξs (D#G) ds − Mt (G)11 = 0, ε→0 0

0≤t≤T

where the martingale Mtε is given by the stochastic integral,  ε−2 t √
2ε d/2+1 (∇G)(εxj (s)) · dwj (s). Mtε (G) = 0

The quadratic variation of

j

Mtε

is given by  ε−2 t
εd |(∇G)(εxj (s))|2 ds. Aεt = 2ε2 0

j

 It is easy to see that Aεt converges uniformly for t ∈ [0, T ] to 2ρt |(∇G)(q)|2 dq in L1 (Pµ ) and a.s., i.e.  1 1  1 ε 1 2 Eµ sup 11At − 2ρt |(∇G)(q)| dq 11 = 0. 0≤t≤T

Consequently the law of Mtε(G) in C([0, T ], R) converges to a brownian motion Mt (G) with variance given by 2ρt |(∇G)(q)|2 dq. By the results in Sect. 5, the laws P ε of ξtε are tight in C([0, T ], H−k ) for any k > d + 1. It follows that any limit point P of P ε is concentrated on the solutions of the equation  t ξt (G) − ξ0 (G) = ξs (D#G) ds + Mt (G) ∀G ∈ Cc2 (Rd ). (6.1) 0

Then the conditions of Theorem 11.0.2 in [7] are satisfied and Holley-Stroock theory can be applied to identify ξt as the corresponding generalized Ornstein-Uhlenbeck process. Since these limits are obtained in the stationary state µ, for each fixed time t ∈ [0, T ] the marginal distribution of ξt is the law of the centered Gaussian field on Rd with covariance (cf. [1])  EP (ξt (G)ξt (F )) = χ G(q)F (q) dq. This permits to identify P as the distribution of the stationary solution of (6.1), and D with ρ/χ. We summarize the final result in the following theorem. Theorem 6.1. Let k > d + 1 and T > 0. The law ξtε on C([0, T ], H−k ) converges to the law of the Gaussian process with covariance given by  EP (ξt (G)ξs (F )) = χ G(q)(e|t−s|D# F )(q) dq with D =

ρ χ.

Equilibrium Fluctuations

483

7. Appendix A: Existence of the Dynamics We prove here the existence of the equilibrium dynamics for the system of stochastic differential equations defined by (2.3). The idea is to approximate the infinite dynamics by some local dynamics that have µ as invariant measure. For that, let us introduce a once continuously differentiable function a : Rd → [0, 1] with compact support + ⊂ Rd such that |∇a(q)| ≤ 1 for all q ∈ Rd . For every cutoff a, we consider the following system of stochastic differential equations: dxi (t) = a(xi (t))vi (t)dt,
∇V (xj (t) − xi (t))dt + ∇a(xi (t))dt dvi (t) = − a(xi (t)) j =i

− a(xi (t))vi (t)dt +

2

(7.1)

2a(xi (t))dwi (t)

with initial conditions xi (0) = xi (ω) and vi (0) = vi (ω). Observe that particles outside + are frozen, while in the region where a = 1 our particles follow the original equations of motion. The generator of the Markov process associated to (7.1) can be written as L+ =




  eH+ (ω) ∂vi a(xi )e−H+ (ω) ∂vi + ∂vi a(xi )e−H+ (ω) ∂xi − ∂xi a(xi )e−H+ (ω) ∂vi ,

i

where we have defined 
 vj (ω)2
1
H+ (ω) = V (xj (ω)−xi (ω)) + V (xj (ω) − xk (ω)) . + 2 2 c xj (ω)∈+

xi (ω)∈+

xk (ω)∈+

x , the canonical Gibbs measure µ( · |N = For any m ∈ N and any fixed configuration ω+ c + x m, ω+c ) is invariant for (7.1). Consequently µ is also invariant. We shall write ωa (t) = (x a (t), v a (t)) for any solution of (7.1). Since for almost every configuration with respect to µ there are only finitely many particles in +, the global existence of these dynamics for µ-a.e. initial configuration does not pose any problem.

Step 1. We prove first some bounds on the velocities and the densities that are valid with Pµ -probability 1. In the following, we shall write BL for the ball centered at the origin and of radius 2L . Moreover, we shall denote B(x, r) the ball centered at x and of radius r.

Proposition 7.1. Let T ∈ R+ . For any constant C > 0 we have 



Pµ ∃L0 > 0 : ∀L ≥ L0 sup sup

sup

a i:xi ∈BL 0≤t≤T

|via (t)|

≤ CL = 1.

(7.2)

484

S. Olla, C. Tremoulet

Proof. Let us fix L > 2T /C and consider  Pµ sup

sup

sup

a i:xi (ω(0))∈BL 0≤t≤T



|via (t)|

 > CL





≤ Eµ NBL (ω(0))Pµ sup sup

a 0≤t≤T

|v1a (t)|

1 1 > CL1NBL (ω(0))

.

(7.3)

Since sup sup

a 0≤t≤T

|v1a (t)|

≤

|v1a (0)| +

√

 2 sup |w1 (t)| + 0≤t≤T



T

+ T + 0

0

T

1
1 1 ∇V (x a (s) − x a (s))1 ds 1

j =1

j

|v1a (s)| ds,

(7.3) is bounded above by  < NBL 1[|v1 |>CL/8] >µ +C1 ρ2 





+ Eµ NBL (ω(0))Pµ  

T 0



+ Eµ NBL (ω(0))Pµ

T 0

Ld

E

CL sup |w1 (t)| > 8 0≤t≤T



 1 1
1 1 CL 1 1 1 NB (ω(0)) ∇V (x1a (s) − xja (s))1 ds > 1 L 8 1 j =1

|v1a (s)|

1  CL 11 NBL (ω(0)) ds ≥ , 8 1 (7.4)

where C1 is a constant depending only on the dimension d. Since, under µ, v1 is Gaussian distributed and is independent from the positions, < NBL 1[|v1 |>CL/8] >µ ≤ C1 ρ2Ld e−C2 L . 2

The second term involving w1 (t), by a standard estimate on Brownian motion, satisfies a similar bound. Using the Chebichef inequality, the third term of (7.4), is bounded, for any α > 0, by    2     T 1
  1 1  1   − αC162 L2 a a  Eµ NBL exp α ∇V (x1 (s) − xj (s))1 ds . 1 e   0   j =1 By the Schwarz and Jensen inequalities this is bounded by    2      T  
1 2   exp αT 2  ∇V (x1a (s) − xja (s)) ds  e−αC3 L , Eµ NBL   T 0   j =1

Equilibrium Fluctuations

485

where C3 depends only on C and on T . Then by stationarity this is equal to   2      
2 2  e−αC3 L . ∇V (x1 − xj ) NBL exp αT     j =1 Since the range of the interaction is R, posing η = α∇V ∞ T 2 , this last quantity is bounded by     1/2   1/2 2 2 −αC3 L2 2 2 NBL exp ηNB(x ≤ N e−αC3 L e exp 2ηN B B(x ,R) ,R) L 1 1   1/2 2 2 ≤ C5 ρ22Ld exp 2ηNB(0,1∨2R) e−αC3 L . (7.5)    2 is finite for η small enough. Then we By superstability estimates exp 2ηNB(0,1∨2R)

can correspondingly choose α small and we obtain that (7.5) is bounded by C6 ρ2Ld 2 e−αC3 L . We treat in a similar way the 4th term in (7.4) and, for α small enough, we can bound it by   2 2 2 2 NBL eαT |v1 | e−αC3 L ≤ C7 ρ2Ld e−αC3 L . Then (7.2) follows by Borel-Cantelli lemma.

 

Proposition 7.2. 



µ ∃l > 0, ∀L : sup NB(x,L) ≤ lL x∈BL

d

= 1.

Proof. This is a consequence of Ruelle’s superstability estimates (cf. [16, 13]).

(7.6)  

Step 2. To prove the existence of limiting solutions when the cutoff is removed, we have to compare the solution corresponding to different cutoffs a. Let AL denote the set of all once continuously differentiable a : Rd → [0, 1] with compact support such that |∇a(q)| ≤ 1 ∀q ∈ Rd and such that a(q) = 1 for all |q| ≤ 2L . For a and a¯ in AL , let us consider ωa (t) = (x a (t), v a (t)) and ωa¯ (t) = (x a¯ (t), v a¯ (t)) the corresponding solutions to (7.1) with the same initial value ω(0) = (xi , vi )i∈N . For some r > 0 and t > 0 let us define .
C1 (a, a; ¯ r, t) = sup |xia (s) − xia¯ (s)| i:|xi |≤r 0≤s≤t

and

.
¯ r, t) = C2 (a, a;

sup |via (s) − via¯ (s)|.

i:|xi |≤r 0≤s≤t

We prove that

486

S. Olla, C. Tremoulet

Lemma 7.3. For any T > 0 and any r > 0, lim

sup C1 (a, a; ¯ r, T ) + C2 (a, a; ¯ r, T ) = 0

L→+∞ a,a∈A ¯ L

Pµ − a.e.

Proof. We recall that by Proposition 7.1, for a given C > 0, with Pµ probability one there exists a real L0 such that for any L ≥ L0 , any t ≤ T and any i such that xi ∈ BL we have |xia (t) − xi | ≤ CLt and |xia¯ (t) − xi | ≤ CLt. So, take L ≥ L0 in the following. Fix r0 > 0, and take L > L0 sufficiently large such that r0 + R + 2CLT ≤ 2L . Then sup |xia (t)| ≤ 2L

sup

sup |xia¯ (t)| ≤ 2L .

sup

i:|xi |≤r0 0≤t≤T

i:|xi |≤r0 0≤t≤T

By a simple computation, if |xi (0)| ≤ r0 , |xia (t) − xia¯ (t)|



t

≤ 0

|via (s) − via¯ (s)| ds,

which implies immediately 

t

¯ r0 , t) ≤ C1 (a, a;

C2 (a, a; ¯ r0 , s) ds.

0

Then, always for |xi (0)| ≤ r0 , |via (t) − via¯ (t)| ≤



t 0








CV

xj ∈B(xi ,R+2CLT )



t

+ 0

|xja (s) − xja¯ (s)| ds

|via (s) − via¯ (s)| ds,

where CV = #V ∞ . Therefore,   t

¯ r0 , t) ≤ C2 (a, a; 0



CV sup NB(x,R+2CLT ) C1 (a, a; ¯ r1 , s) + C2 (a, a; ¯ r0 , s) x∈BL

ds,

where r1 = r0 + R + 2CLT ≤ 2L . By Proposition 7.2 with Pµ probability one, there exists a finite l such that supx∈BL NB(x,R+2CLT ) ≤ C1 lLd T d . So we have proven that, with Pµ probability one, there exist two finite constants l, L0 > 0 such that for any L ≥ L0 , any t ≤ T and any a, a¯ ∈ AL , ¯ r0 , t) + C2 (a, a; ¯ r0 , t) ≤ C lT d Ld C1 (a, a;



t

¯ r1 , s) + C2 (a, a; ¯ r1 , s)) ds. (C1 (a, a;

0

We finish the proof by iterating the above relation for a number of times given by 

2 L − r0 hL = R + 2CT L



Equilibrium Fluctuations

487

and we obtain C1 (a, a; ¯ r0 , T ) + C2 (a, a; ¯ r0 , T )  th  hL  T   L ≤ C lLd T d dt1 · · · dthL C1 (a, a; ¯ 2L , thL ) + C2 (a, a; ¯ 2L , thL ) . 0

0

Since C1 (a, a; ¯ 2L , thL ) ≤ 2L and C2 (a, a; ¯ 2L , thL )) ≤ 2CL we have  h C1 2L C lT d Ld L hL +1 ¯ r0 , T ) + C2 (a, a; ¯ r0 , T ) ≤ T sup C1 (a, a; hL ! a,a¯ that converges to 0 as L → ∞.

 

Step 3. Conclusion. Now, we are able to prove the existence of limiting solutions to (1.1). Let us consider a sequence of a partial solution ωn (t) for n ∈ N of (7.1) with a common initial value ωn (0) = (xi , vi ). The corresponding cutoff function an is taken in An , i.e. an (q) = 1 if |q| ≤ 2n . By Lemma 7.3, with Pµ probability 1, ωn converges to some limit ω(t) for each t < T . Elementary considerations following the argument of the previous lemma show that this limit is a solution of (1.1) and it does not depend on the particular cutoff chosen. 8. Appendix B: Spectral Gap for Interacting Brownian Particles We prove here the spectral gap bound we used in Sect. 4. In this Appendix + is a fixed centered cube of sidelength 2l ( |+| = (2l)d ). We fix the number of particles in the box x outside +. Let m = m(ωx ) be + to be equal to n, and an arbitrary configuration ω+ c +c x that are at distance less than or the number of particles of the outside configuration ω+ c equal to R, the radius of the range of the interaction V . x ) is the corresponding canonical Gibbs expectation. Recall that C+ (·)(n, ω+ c x ) = 0. Theorem 8.1. Let f (x1 , . . . , xn ) be a C 1 (+n ) function such that C+ (f )(n, ω+ c Then n 1
12  x 2 4(2n+m)V ∞ x C+ (f 2 )(n, ω+ C+ 1∇xi f 1 (n, ω+ (8.1) c ) ≤ 8dl ne c ). i=1

x ) the canonical measure corProof. We denote by µ+ (dx) = µ+ (dx1 , . . . , dxn |n, ω+ c x ). Observe that we can rewrite the canonical variance of f responding to C+ (·)(n, ω+ c as x C+ (f 2 )(n, ω+ c)   2 1 f (x) − f (x ) µ+ (dx)µ+ (dx ) = n n 2 + ×+  2 x 1 x f (x) − f (x ) e−Hn (x,ω+c )−Hn (x ,ω+c ) dx dx , = 2 2Z +n ×+n

where we denoted x Hn (x, ω+ c)

=

n
(i,j )

V (xi − xj ) +

n



V (xi − yj )

x i=1 yj ∈ω+ c

x ) is the corresponding canonical partition function. and Z = Z+ (n, ω+ c

(8.2)

488

S. Olla, C. Tremoulet

For each couple of configurations x, x we will choose a particular piecewise differentiable path that will connect these two configurations. The choice of this transformation follows a simple rule: we move each particle one by one along the coordinates axis. So, in order to simplify notation, it is convenient to consider x, x as points in [−l, l]nd and }. Then defining write x = {y1 , . . . , ynd } and x = {y1 , . . . , ynd ξα (t) = yα + t (yα − yα )

t ∈ [0, 1)

α = 1, . . . , nd

we can rewrite the difference f (x) − f (x ) =

nd 


1

α=1 0

d f (y1 , . . . , yα−1 , ξα (t), yα+1 , . . . , ynd ) dt dt

and by the Schwarz inequality 

2 f (x) − f (x ) nd  1
1 1 √ 1fα (y , . . . , y , ξα (t), yα+1 , . . . , ynd )12 |ξ˙α (t)|2 1 − t dt, ≤ 2nd 1 α−1

(8.3)

α=1 0

where we have denoted by fα the partial derivative of f with respect to yα and ξ˙α (t) = yα − yα . Since |ξ˙α (t)| ≤ 2l we have that the right-hand side of (8.3) is bounded by 8l 2 nd

nd 
α=1 0

11

1fα (y , . . . , y 1

12 √ 1 1 − t dt.

α−1 , ξα (t), yα+1 , . . . , ynd )

(8.4)

Since the interaction V is bounded, for any ξα ∈ [−l, l] we have the uniform bound x x Hn (x, ω+ c ) + Hn (x , ω+c ) x ≥ −4(2n + m)V ∞ + Hn (y1 , . . . , yα−1 , ξα , yα+1 , . . . , ynd , ω+ c) x +Hn (y1 , . . . , yα−1 , yα , yα+1 , . . . , ynd , ω+ c ).

To prove this, observe that for any θ, η ∈ + we can rewrite x x Hn (x, ω+ c ) + Hn (x , ω+c )

x = Hn (x1 , . . . , xi−1 , θ, x1+1 , . . . , xn , ω+ c)

x + Hn (x1 , . . . , xi−1 , η, x1+1 , . . . , xn , ω+ c)

+ +

i−1
6 k=1 n


 7 (V (xk − xi ) − V (xk − θ )) + V (xk − xi ) − V (xk − η) 6

k=i+1

+

 7 (V (xk − xi ) − V (xk − η)) + V (xk − xi ) − V (xk − θ)


6   7 V (xi − yj ) − V (θ − yj ) + V (xi − yj ) − V (η − yj ) .

x yj ∈ω+ c

Then, by an appropriate choice of θ, η, (8.5) follows.

(8.5)

Equilibrium Fluctuations

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Now, by the simple change of variable yα → ξα (t), we have  12 1 1 dy1 . . . dynd dy1 . . . dynd , ξα (t), yα+1 , . . . , ynd )1 fα (y1 , . . . , yα−1 +n ×+n x )−H (y ,...,y x −Hn (y1 ,...,yα−1 ,ξα (t),yα+1 ,...,ynd ,ω+ n 1 α−1 ,yα ,yα+1 ,...,ynd ,ω+c ) c

×e

 =

l −l



×

 dy1 . . . l −l

l −l

dy1 · · ·



 dyα−1 l −l

l −l

dyα−1

dyα



l −l



l −l

 l ,ωx ) ,...,ynd −Hn (y1 ,...,yα−1 ,yα ,yα+1 +c dyα+1 . . . dynd e 

dyα+1 · · ·

−l

l −l



dynd

(1−t)l+tyα −(1−t)l+tyα

dξα 1−t

1 12 x × 1fα (y1 , . . . , yα−1 , ξα , yα+1 , . . . , ynd )1 e−Hn (y1 ,...,yα−1 ,ξα ,yα+1 ,...,ynd ,ω+c ) ≤

Z 1−t



x

+n

dx |fα (x)|2 e−Hn (x,ω+c ) .

(8.6)

Putting together (8.2), (8.4), (8.5) and (8.6), we obtain (8.1).

 

9. Appendix C: Regularity of the Solution of the Poisson Equation We prove here the regularity of the solution of Eq. (4.6), as a consequence of the spectral gap bound of the previous section. Let + be again the centered box of sidelength 2l, and let +R be the centered box of sidelength 2(l + R). In +R we consider configurations with n + m particles such that there are n particles in + (whose positions will be denoted by {x1 , . . . , xn }) and m particles in +R \ + (whose positions will be denoted by x = {y , . . . , y }). We do not consider configurations with particles on the boundary ω+ c 1 m ∂+. Let g(x1 , . . . , xn ; y1 , . . . , ym ) be a smooth function of these configurations such x ) = 0. Recall we have defined the elliptic that the canonical expectation C+ (g)(n, ω+ c operator LW n,ωx c = +

n


 #xj − (∇xj H+ ) · ∇xj ,

j =1

where ∇xj H+ =

n
i=j

∇V (xj − xi ) +

m


∇V (xj − yi ).

i

Let u(x1 , . . . , xn ; y1 , . . . , ym ) be the solution of the equation LW n,ωx c u(x1 , . . . , xn ; y1 , . . . , ym ) = g(x1 , . . . , xn ; y1 , . . . , ym ) , +

(x1 , . . . , xn ) ∈ +n (9.1)

x = with Neumann boundary conditions on ∂+n . The position of the exterior particles ω+ c m (y1 , . . . , ym ) ∈ (+R \ +) should be considered as exterior parameters in Eq. (9.1).

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Lemma 9.1.

 C+ 

n
j =1

 x c2 (n+m) x |∇xj u|2  (n, ω+ C+ (g 2 )(n, ω+ c ) ≤ c1 ne c ),

(9.2)

where c1 = 8dl 2 and c2 = 8V ∞ . Proof. Multiplying (9.1) by u and integrating respect to the canonical Gibbs measure we obtain   n
x C+  |∇xj u|2  (n, ω+ c) j =1

  x = C+ u(−LW u) (n, ω+ x c) n,ω+c    1/2   2  1/2 x 2 x x = C+ (ug) (n, ω+ C+ g (n, ω+ . (n, ω+ c ) ≤ C+ u c) c) By the spectral gap bound (8.1),

  n  
x c2 (n+m) x C+ u2 (n, ω+ C+  |∇xj u|2  (n, ω+ c ) ≤ c1 ne c ). j =1

Inserting this in the previous inequality, we obtain the bound (9.2). Lemma 9.2. There exists a constant c3 such that n




C+ |∇xj ∇xk u|

j,k=1

2



x (n, ω+ c)



2 2c2 (n+m)

≤ c3 n e

C+ n g + 2 2

n
k=1

   x |∇xk g| (n, ω+ c ).

(9.3) Proof. This is just a standard elliptic regularity argument. Fix a k = 1, . . . , n. Let {e1 , · · · , ed } be the canonical base in Rd . The function uk,α = eα · ∇xk u satisfies the equation LW n,ωx c uk,α = eα · ∇xk g − +

n


V (xj − xk )uj,α .

j =1

Defining gk,α , the right-hand side of this equation, we observe that d
α=1

2 C+ (gk,α ) ≤ 2C+ (|∇xk g|2 ) + 2V 2∞ nC+ (

≤ 2C+ (|∇xk g|

2

n


|∇xj u|2 )

j =1

) + 2c1 V 2∞ n2 ec2 (n+m) C+ (g 2 ).

By reapplying the same argument as in the proof of Lemma 9.1, we have   n
2 C+  |∇xj uk,α |2  ≤ c1 nec2 (n+m) C+ (gk,α ). j =1

and (9.3) follows from these last two inequalities.

 

Equilibrium Fluctuations

491

Lemma 9.3. n m

k=1 j =1







x 2 2c2 (n+m) C+ |∇xj ∇yk u|2 (n, ω+ C+ nmg 2 + c ) ≤ c3 n e

m
k=1

 x |∇yk g|2 (n, ω+ c ).

(9.4) Proof. The proof follows the same argument as in the one of Lemma 9.1, starting with the equations for eα · ∇yk u.   References 1. Brox, T.: Theses, Universit¨at Heidelberg, 1980 (in German, unpublished) 2. Brox, T., Rost, H.: Equilibrium fluctuations of stochastic particle systems: The role of conserved particles. Ann. Prob. 12, 742–759 (1984) 3. Cancrini, N., Tremoulet, C.: Comparison of finite volume canonical and grand canonical Gibbs measures under a mixing condition: The continuous case. Preprint (2002) 4. Chang, C-C.: Equilibrium fluctuations of gradient reversible particle systems. Probab. Theor. Relat. Fields 100, 269–283 (1994) 5. Chang, C-C., Landim, C., Olla, S.: Equilibrium Fluctuation of asymmetric simple exclusion processes in dimension d ≥ 3. Probab. Theor. Relat. Fields 119, 381–409 (2001) 6. De Masi, A., Presutti, E., Spohn, H., Wick, D.: Asymptotic equivalence for fluctuation fields for reversible exclusion processes with speed change. Ann. Prob. 14(2), 409–423 (1986) 7. Kipnis, C., Landim, C.: Scaling Limit of Interacting Particle Systems. Berlin: Springer Verlag, 1999 8. Fritz, J.: Stochastic dynamics of two-dimensional infinite-particle systems. J. Stat. Phys. 20(4), 351–369 (1979) 9. Fritz, J.: Gradient dynamics of infinite point systems. Ann. Prob. 15, 478–514 (1987) 10. Fritz, J, Liverani, C., Olla, S.: Reversibility in infinite Hamiltonian systems with conservative noise. Commun. Math. Phys. 189, 481–496 (1997) 11. Lanford, O.E.: Time evolution of large classical systems. Lecture Notes in Physics, vol. 38. BerlinHeidelberg-New York: Springer, 1975 12. Lang, R.: Wahrscheinlichkeitstheor Verw. Geb. 38, 55 (1977); with addition: Shiga, T., Z: Wahrscheinlichkeitstheor Verw. Geb. 47, 299 (1979) 13. Marchioro, C., Pellegrinotti, A., Presutti, E.: Existence of time evolution in ν-dimensional Statistical Mechanics. Commun. Math. Phys. 40, 175–185 (1975) 14. Pusey, P.N., Tough, R.J.A.: Langevin approach to the dynamic of interacting Brownian particles. J. Phys. A15, 1291–1308 (1982) 15. Ruelle, D.: Statistical Mechanics: Rigorous Results. New York-Amsterdam: W.A. Benjamin, Inc., (1969) 16. Ruelle, D.: Superstable interactions in classical statistical mechanics, Commun. Math. Phys. 18, 127–159 (1970) 17. Olla, S., Varadhan, S.R.S.: Scaling limits for interacting Ornstein-Uhlenbeck diffusions. Commun. Math. Phys. 135, 355–378 (1991) 18. Presutti, E.: A mechanical definition of thermodynamic pressure. J. Stat. Phys. 13, 301–314 (1975) 19. Rost, H.: Lecture Notes in Mathematics, vol. 1031, Berlin-Heidelberg-Newyork: Springer, 1983 20. Spohn, H.: Equilibrium fluctuations for interacting Brownian particles. Commun. Math. Phys. 103, 1–33 (1986) 21. Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin-Heidelberg-New York: Springer, 1991 22. Sethuraman S., Varadhan, S.R.S.,Yau, H.T.: Diffusive limit of a tagged particle in asymmetric simple exclusion. Comm. Pure Appl. Math. 53, 972–1006 (2000) 23. Uchiyama, K.: Pressure in classical statistical mechanics and interacting brownian particles in multi-dimensions. Ann. H. Poincare-Phys. Th. 6, 1159–1202 (2000) 24. Varadhan, S.R.S.: Scaling limits for interacting diffusions. Commun. Math. Phys. 135, 313–353 (1991) Communicated by H. Spohn

Commun. Math. Phys. 233, 493–512 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0759-3

Communications in

Mathematical Physics

The Weyl Quantization and the Quantum Group Quantization of the Moduli Space of Flat SU (2)-Connections on the Torus are the Same R˘azvan Gelca1,2 , Alejandro Uribe3,∗ 1

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA. E-mail: [email protected] 2 Institute of Mathematics of the Romanian Academy, Bucharest, Romania, 3 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA. E-mail: [email protected] Received: 27 January 2002 / Accepted: 9 September 2002 Published online: 19 December 2002 – © Springer-Verlag 2002

Abstract: We prove that, for the moduli space of flat SU (2)-connections on the 2-dimensional torus, the Weyl quantization and the quantization performed using the quantum group of SL(2, C) are the same. This is done by comparing the matrices of the operators associated through the two quantizations to cosine functions. We also discuss the ∗-product of the Weyl quantization and show that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus. 1. Introduction Quantization is a procedure for replacing functions on the phase space of a physical system (classical observables) by linear operators. While understood in many general situations, this procedure is far from being algorithmic. Some more exotic spaces whose quantizations are of interest to mathematicians are the moduli spaces of flat connections on a surface. Among them the case of the moduli space of flat SU (2)-connections on a torus is a particularly simple example of an algebraic variety that fails to be a manifold, yet is very close to being one. In this paper we compare two methods of quantizing the moduli space of flat SU (2)connections on the torus. The first uses the quantum group of SL(2, C). This quantization scheme arose when Reshetikhin and Turaev constructed a topological quantum field theory associated to the Jones polynomial of a knot. It describes both the quantum observables and the Hilbert spaces in terms of knots and links colored by representations of the quantum group of SL(2, C). Heuristically, the operators of the quantization were defined by Witten using path integrals for the Chern-Simons action. On the other hand, the moduli space of flat SU (2)-connections on the torus is the same as the character variety of SU (2)-representations of its fundamental group, so it admits a covering by the plane. Therefore one can apply a classical quantization procedure of the plane, in an equivariant manner, to obtain a quantization of the moduli space. ∗

Research supported in part by the NSF, award No. DMS 0070690

494

R. Gelca, A. Uribe

The first such procedure was introduced by Hermann Weyl. It consists of a simple rule for assigning differential operators to exponential functions, then this is extended to all smooth functions via the inverse Fourier transform. Our main result is the following Theorem. The Weyl quantization and the quantum group quantization of the moduli space of flat SU (2)-connections on the torus are unitary equivalent. The paper is structured as follows. In Sect. 2 we describe the geometric realization and the K¨ahler structure of the moduli space of flat SU (2)-connections on the torus. In Sect. 3 we review Witten’s description of the quantization for the particular case of the torus with a path integral of the Chern-Simons action, then explain the rigorous construction of Reshetikhin and Turaev using quantum groups. We also describe the matrices of the operators associated to cosine functions in the basis of the Hilbert space consisting of the colorings of the core of the solid torus by irreducible representations. We then explain the Weyl quantization of the moduli space (Sect. 4). This is done in the holomorphic setting, which can be related to the classical, real setting through the Bargmann transform. A distinguished basis of the Hilbert space is introduced in terms of odd theta functions. Section 5 contains the main result of the paper (Theorem 5.3). It shows that the two quantizations are unitary equivalent. The unitary equivalence maps the distinguished basis consisting of odd theta functions to the basis consisting of the colored cores of the solid torus. In Sect. 6 we discuss the ∗-product that arises from this quantization. We conclude with some final remarks about the quantization scheme based on the Kauffman bracket skein module, for which the result does not hold due to a sign obstruction. 2. The Phase Space We are Quantizing Throughout the paper T2 will denote the 2-dimensional torus. The moduli space of flat SU (2)-connections on a surface is the same as the character variety of SU (2)-representations of the fundamental group of the surface [2], i.e. the set of the morphisms of the fundamental group of the surface into SU (2) modulo conjugation. This is a complex algebraic variety. In the case of the torus, morphisms from π1 (T2 ) = Z ⊕ Z to SU (2) are parameterized by the images of the two generators of Z ⊕ Z, i.e. by two commuting matrices. The two matrices can be simultaneously diagonalized. Moreover conjugation can permute simultaneously the entries in the two diagonal matrices. So the moduli space of flat SU (2)-connections on the torus can be described geometrically as M = {(s, t)

| |s| = |t| = 1}/(s, t) ∼ (¯s , t¯).

This set is called the “pillow case”. It has a 2-1 covering by the torus, with branching points (1, 1), (1, −1), (−1, 1), (−1, −1). We will think of the moduli space as the quotient of the complex plane by the group generated by the translations z → z + 1 and z → z + i, and by the symmetry with respect to the origin σ (z) = −z. Off the four singularities M is a K¨ahler manifold, with K¨ahler form ω induced by −π dz ∧ d z¯ = 2πidx ∧ dy on C. Note that πdz ∧ d z¯ = ∂ ∂¯ ln h0 (z, z¯ ), where the K¨ahler potential h0 (z, z¯ ) = e−π |z| is nothing but the weight of the Bargmann measure on the plane. Also, note that the K¨ahler form ω is the genus one case of Goldman’s symplectic form defined in [11]. 2

Quantization of the Moduli Space of Flat SU (2)-Connections the Torus

495

The classical observables are the C ∞ functions on this variety. Using the covering map we identify the algebra of observables on the character variety with the algebra of functions on C = R ⊕ R generated by cos 2π(px + qy), p, q ∈ Z. Another family of important functions on M are sin 2π n(px +qy)/ sin 2π(px +qy) where p and q are coprime. As functions of connections, these associate to a connection the trace in the n-dimensional irreducible representation of SU (2) of the holonomy of the connection around the curve of slope p/q on the torus. To quantize M means to replace classical observables f by linear operators op(f ) on some Hilbert space, satisfying Dirac’s conditions: 1. op(1) = I d, 2. op({f, g}) =

1 i¯h [op(f ), op(g)] + O(¯h).

Here {f, g} is the Poisson bracket induced on M by the form ω, which is
 1 ∂f ∂g ∂f ∂g {f, g} = − . 2πi ∂x ∂y ∂y ∂x Also [A, B] is the commutator of operators, and h ¯ is Planck’s constant. Since in our case the phase space is an orbifold covered by the plane, and this orbifold is K¨ahler off singularities, it is natural to perform equivariant quantization of the plane. We do this using Weyl’s method and then compare the result with the quantization from [17] which was done using the quantum group of SL(2, C). 3. Review of the Quantum Group Approach The quantization of the moduli space of flat SU (2)-connections on a surface performed using the quantum group of SL(2, C) at roots of unity is an offspring of Reshetikhin and Turaev’s construction of quantum invariants for 3-manifolds [17]. Their work was inspired by Witten’s heuristic explanation of the Jones polynomial using Chern-Simons topological quantum field theory. We present below Witten’s idea for the particular case of the torus, and then show how the Reshetikhin-Turaev construction yields a quantization of M. 3.1. Path integrals. In [22], Witten outlined a way of quantizing the moduli space of flat connections on a trivial principal bundle with gauge group a simply connected compact Lie group. This space is identified with the symplectic quotient of the total space of connections under the action of the group of gauge transformations [2]. Let us recall how this is done when the group is SU (2) and the principal bundle lies over the cylinder over the torus M = T2 × [0, 1]. For A an SU (2)-connection on M define the Chern-Simons Lagrangian to be  2 1 tr(A ∧ dA + A ∧ A ∧ A), L= 4π M 3 where tr is the trace on the 2-dimensional irreducible representation of su(2). The Lagrangian is invariant under gauge transformations up to the addition of an integer. Now, using Witten’s idea we associate operators to the observables of the form 2 cos 2π(px + qy) and sin 2π n(px + qy)/ sin 2π(px + qy) on the torus p, q, n ∈ Z.

496

R. Gelca, A. Uribe

Let N be some fixed integer called the level of the quantization. As such, Planck’s constant is h ¯ = N1 . Assume that p and q are arbitrary integers, and let n be their greatest common divisor. Denote p  = p/n, q  = q/n. Consider the cylinder over the torus T2 × [0, 1] and let C be the curve of slope p  /q  in T2 × { 21 }. The operator associated by the quantization to the function sin 2π n(p x + q  y)/ sin 2π(p x + q  y) and denoted shortly by S(p, q) (S from sine) is the integral with kernel defined by the following path integral:  < A1 |S(p, q)|A2 >= eiN L(A) trV n (holC (A))DA, MA1 ,A2

where A1 , A2 are conjugacy classes of connections on T2 , A is a conjugacy class of connections under the action of the gauge group on T2 × [0, 1] such that A|T2 ×{0} = A1 and A|T2 ×{1} = A2 , and trV n (holC (A)), known as the Wilson line, is the trace of the n-dimensional irreducible representation of SU (2) evaluated on the holonomy of A around C of slope p /q  . Here the “average” is taken over all conjugacy classes of connections modulo the gauge group. With the same notations one defines the operator C(p, q) (C from cosine) representing the quantization of the function 2 cos 2π(px + qy) by  < A1 |C(p, q)|A2 >= eiN L(A) (trV n+1 − trV n−1 )(holC (A))DA. MA1 ,A2

This is so because of the formula 2 cos nx =

sin(n + 1)x sin(n − 1)x − . sin x sin x

We only discuss briefly the Hilbert space of the quantization, assuming the reader is familiar with [1] and [22]. The next section will make these ideas precise. The Hilbert space is spanned by the quantum invariants (i.e. partition functions) of all 3-manifolds with boundary equal to the torus. Since any 3-manifold can be obtained by performing surgery on a link that lies in the solid torus, it follows that the Hilbert space of the quantization is spanned by the partition functions of pairs of the form (S 1 × D2 , L), where L is a (colored) link in the solid torus S 1 × D2 . 3.2. Quantization using the quantum group of SL(2, C). The quantization of the character variety of the torus using the quantum group of SL(2, C) is a particular consequence of the topological quantum field theory constructed in [17]. Let us discuss its essential features. πi Fix a level r ≥ 3 of the quantization, and let t = e 2r . Comparing with the previous section, N = 2r. Quantized integers are defined by the formula [n] = (t 2n −t −2n )/(t 2 − t −2 ). The quantum algebra of sl(2, C), denoted by Ut is a deformation of its universal enveloping algebra and has generators X, Y, K subject to the relations KX = t 2 XK,

KY = t −2 Y K,

Xr = Y r = 0,

K 4r = 1.

XY − Y X =

K 2 − K −2 , t 2 − t −2

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This algebra is Hopf, so its representations form a ring under the operations of direct sum and tensor product. Reducing modulo summands of quantum trace zero, this ring contains a subring generated by finitely many irreducible representations V 1 , V 2 , . . . , V r−1 . Here V k has dimension k, basis e−(k−1)/2 , e−(k−3)/2 . . . , e(k−1)/2 and the action of Ut is defined by Xej = [m + j + 1]ej +1 , Y ej = [m − j + 1]ej −1 , Kej = t 2j ej . The idea originating in [15] and further developed in [17] is to color any link in a 3-dimensional manifold by such irreducible representations. For a knot K we denote by V n (K) its coloring by V n . Motivated by the representation theory of Ut we extend formally the definition of the V n (K) to all integers n by the rules V r (K) = 0, V n+2r (K) = V n (K) and V r+n (K) = −V r−n (K). Here the negative sign means that we color the knot by V r−n , then consider the vector with opposite sign in V (T2 ). There is a rule, for which we refer the reader to [15] and [17], for associating numerical invariants to colored links in the 3-sphere. Briefly, the idea is to use a link diagram such as the one in Fig. 1, with the local maxima, minima and crossings separated by horizontal lines, and then define an automorphism of C by associating to minima maps of the form C → V n ⊗ V n , to maxima maps of the form V n ⊗ V n → C, to crossings the quasitriangular R matrix of the quantum algebra in the n-dimensional irreducible representation, and to parallel strands tensor products of representations. The automorphism is then the multiplication by a constant and the link invariant is equal to that constant. For K a knot in the 3-sphere, V 2 (K) is its Jones polynomial [14] evaluated at the specific root of unity, and for n ≥ 2, V n (K) is called the colored (or generalized) Jones polynomial. We are now able to describe the quantization of the torus. First consider the vector space freely spanned by all colored links in the solid torus. On this vector space define the pairing [·, ·] induced by the operation of gluing two solid tori such that the meridian of the first is identified with the longitude of the second and vice versa, as to obtain a 3-sphere (Fig. 2). The pairing of two links [L1 , L2 ] is equal to the quantum invariant of the resulting link in the 3-sphere. The Hilbert space of the quantization is obtained by factoring the vector space by all linear combinations of colored links λ such that [λ, λ ] = 0 for all λ in the vector space. This quotient, denoted by V (T2 ) by quantum topologists, is finite dimensional. It is the Hilbert space of the quantization. Let α be the core S 1 × {0} of the solid torus S 1 × D, D = {z, |z| ≤ 1}. A basis of V (T2 ) is given by V k (α), k = 1, 2, . . . , r − 1. The inner product is determined by requiring that this basis is orthonormal. The pairing [·, ·] is not the inner product. These basis elements play an important role in the construction of the ReshetikhinTuraev invariants of closed 3-manifolds. As suggested in [22] and rigorously done in [17], the quantum invariant of a 3-manifold obtained by performing surgery on a link is calculated by gluing to the complement of the link solid tori with cori colored by irreducible representations, computing the colored link invariants and then summing over all possible colorings. The choice of the inner product is not accidental. The orthonormal basis V k (α), k = 1, 2, . . . , r − 1 arises, by applying Walker’s axioms for a topological quantum field theory with corners [20], from the basis βkk of the vector space of the annulus with  boundary labeled by k, with the inner product < βkk , βkk >= m [m]/[k]. This inner product on the annulus is defined using the trace pairing for representations of Ut [8].

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Fig. 1. Knot diagram

Fig. 2. Heegaard decomposition of the sphere

Consider now two integers p and q, let n be their greatest common divisor, and let also p = p/n, q  = q/n. The operator S(p, q) associated to the classical observable sin 2π n(p x + q  y)/ sin 2π(p x + q  y) is obtained by coloring the curve of slope p  /q  in the cylinder over the torus by the representation V n . As explained in the previous section, the operator that quantizes 2 cos 2π(px + qy) is C(p, q) = C(np , nq  ) = S((n + 1)p  , (n + 1)nq  ) − S((n − 1)p  , (n − 1)q  ). Their action on the Hilbert space of the solid torus is defined by gluing the cylinder over the torus to the solid torus. The operator associated to an arbitrary function in C ∞ (M) is defined by approximating the function with trigonometric polynomials in cosines, quantizing those, then passing to the limit. 3.3. The matrices of observables for the quantum group quantization. In this section we compute the matrices of the quantum group quantization of the observables cos 2π(px + qy), p, q ∈ Z. Theorem 1. In any level r and for any integers p, q and k the following formula holds   C(p, q)V k (α) = t −pq t 2qk V k−p (α) + t −2qk V k+p (α) .

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S 2

499

T

2 1

1

2 1

Fig. 3. S and T maps

Proof. The proof uses topological quantum field theory with corners. This is a concept introduced by K. Walker which refines Atiyah’s axioms of a TQFT [1] to allow gluings with corners. Details about this can be found in [20, 10]. The idea of the proof is to reduce the problem to the computation of the quantum invariant of a link complement, then construct the link complement from simple manifolds and use Walker’s axioms to compute its invariant. In Walker’s theory a key role is played by decompositions of the boundaries of manifolds into disks, annuli and pairs of pants together with some curves (seams) that keep track of twistings. These structures on the boundary are called DAP-decompositions. At the level of the vector space they correspond to choices of a basis. Gluings along subsurfaces (i.e. along a subset of the disks, annuli and pairs of pants of the decomposition) yield contractions of the Hilbert space of the quantization.  r−1 2 Here are some facts needed below. Let X = j =1 [j ] (not to be confused with j

the generator of Ut ). The vector space of an annulus has basis βj , j = 1, 2, . . . , r − 1, with pairing given by j

< βj , βkk >= δj,k X/[j ]. Gluing the boundaries of an annulus we obtain a torus, with the same basis for the vector j space. The pairing on the torus will make βj an orthonormal basis, but we don’t need this now. The moves S and T on the torus are described in Fig. 3. The (m, n)-entry of 2 the matrix of S is [mn]. The move T is diagonal and its j th entry is t j −1 . The quantum invariant of the cylinder over a surface is the identity matrix. So the  [n] n n n n quantum invariant of the cylinder over an annulus is r−1 n=1 X βn ⊗ βn ⊗ βn ⊗ βn (when taking the cylinder over the annulus the boundary of the solid torus will be canonically decomposed into four annuli). If the DAP-decomposition of a 3-manifold involves two disjoint annuli, its invariant can be written in the form r−1 k,j =1

j

βkk ⊗ βj ⊗ vk,j .

Gluing the two annuli produces a 3-manifold that in the newly obtained DAP-decom X position has the invariant equal to r−1 k=1 [k] vk,k . In the proof of the theorem we use the following formula, which can be checked using the fact that the sum of the roots of unity is zero. Lemma 1. Let a, b, c, d and e be integers. Then

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Fig. 4. Link complement r−1

[ax]t bx [cy]([x(y + d)]t 2ey + [x(y − d)]t −2ey ) 2

x,y=1

= X2 t bc

2 +be2 −2de

([a(c + e)]t 2(be−d)c + [a(c − e)]t −2(be−d)c ).

It suffices to check that the two sides of the equality yield the same results when paired with all V m (α). Recalling the pairing [·, ·] on the solid torus, we must show that [C(p, q)V k (α), V m (α)]  t −pq  2(qk−pm+km) 2(qk+pm−km) 2(−qk+pm+km) 2(−qk−pm−km) = 2 t . − t + t − t t − t −2 Let d to be the greatest common divisor of p and q, p = p/d and q  = q/d. We concentrate first on the computation of [S((d + 1)p  , (d + 1)q  )V k (α), V m (α)]

(1)

[S((d − 1)p  , (d − 1)q  )V k (α), V m (α)].

(2)

and

Here is the place where we use the topological quantum field theory with corners from [10]. The expression in (1) is the invariant of the link that has one component equal to the (p , q  )-curve on a torus colored by V d+1 , and the other two components the cores of the two solid tori that lie on one side and the other of the torus knot (see Fig. 4), colored by V k (the one inside) and V m (the one outside). The expression in (2) is the invariant of the same link but with the (p  , q  )-curve colored by V d−1 . It was shown in [10] that d+1 m of the vector that d ⊗ βkk ⊗ βm this number is equal to X−1 times the coordinate of βd+1 is the quantum invariant of the link complement. Let us produce the complement of this link by gluing together two simple 3-manifolds, whose quantum invariants are easy to compute. Consider first the cylinder over

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an annulus A and glue its ends to obtain the manifold A × S 1 . In the basis of the vector 2 ) determined by the DAP-decomposition ∂A × {1} the invariant space of V (T2 × T of this manifold is k βkk ⊗ βkk . Take another copy of the same manifold. Change the decomposition curves of the exterior torus of the first manifold to the p  /q  -curve and of the exterior torus of the second manifold to the longitude. Of course, to do this on the second manifold we apply the S-move and so the invariant of the second manifold changes to 1 jn+1 [djn+1 ]βδδ ⊗ βjn+1 . X δ,jn+1

With the first manifold the story is more complicated. Consider the continued fraction expansion q 1 =−  p −a1 −

1 −a2 −··· −a1 n

.

The required move on the boundary is then ST −an ST −an−1 S · · · ST −a1 S. So the invariant of the first manifold in the new DAP-decomposition is 2 2 jn+1 X−n−1 [jn+1 jn ]t −an (jn −1) [jn jn−1 ] · · · [j2 j1 ]t −a1 (j1 −1) [j1 k]βkk ⊗ βjn+1 . j1 ,···,jn+1

Now expand one annulus in the exterior tori of each of the two manifolds. Then glue just one annulus from the the first manifold to one annulus from the second. This way we obtain the complement of the link in the discussion. One of its boundary tori is decomposed into two annuli. Contract one of them. Since the gluing introduced a factor of X/[jn+1 ], the invariant of the manifold is X−n−1

j1 ,···,jn+1 ,δ

[δjn+1 ] 2 2 [jn+1 jn ]t −an (jn −1) [jn jn−1 ] · · · [j2 j1 ]t −a1 (j1 −1) [jn+1 ] j

n+1 × [j1 k]βjn+1 ⊗ βδδ ⊗ βkk .

At this moment we have the right 3-manifold but with the wrong DAP-decomposition. We need to fix the DAP-decomposition of the torus that corresponds to the basis elejn+1 ment βjn+1 (the boundary of the regular neighborhood of the (p  , q  )-curve) such as to transform the decomposition curve into the meridian of the link component. For this we apply the move (ST −an ST −an1 S · · · ST −a1 S)−1 . We obtain the following expression for the invariant of the extended manifold 2 [δjn+1 ] X−2n−2 [mj2n+2 ][j2n+2 j2n+1 ]t a1 j2n+1 · · · [jn+2 jn+1 ] [jn+1 ] j1 ,...,j2n+2 ,δ,k,m

m × [jn+1 jn ]t −an jn [jn jn−1 ] · · · [j2 j1 ]t −a1 j1 [j1 k]βδδ ⊗ βkk ⊗ βm 2

2

(in this formula we already reduced t ak and t −ak , 1 ≤ k ≤ n). It is important to observe that after performing the described operations the seams came right, so no further twistings are necessary.

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d±1 m. Now fix k and m, let δ = d ± 1 and focus on the coefficients of βd±1 ⊗ βkk ⊗ βm   Multiplied by X these are the colored Jones polynomial of the link [10] with the (p , q )curve colored by the d + 1-, respectively d − 1-dimensional irreducible representation of Ut . Since C(p, q) = S((d + 1)p  , (d + 1)dq  ) − S((d − 1)p  , (d − 1)q  ) and also

[(d + 1)jn+1 ] [(d − 1)jn+1 ] − = t 2jn+1 + t −2jn+1 , [jn+1 ] [jn+1 ] we deduce that the value of [C(p, q)V k (α), V m (α)] is equal to 2 X−2n−1 [mj2n+2 ][j2n+2 j2n+1 ]t a1 j2n+1 · · · [jn+2 jn+1 ](t 2djn+1 + t −2djn+1 ) j1 ,...,j2n+2

×[jn+1 jn ]t −an jn [jn jn−1 ] · · · [j2 j1 ]t −a1 j1 [j1 k]. 2

2

We want to compute these iterated Gauss sums. We apply successively Lemma 1 starting with x = jn , y = jn+1 , then x = jn−1 , y = jn+2 and so on to obtain 2 2 t −an d [mj2n+2 ][j2n+2 j2n+1 ]t a1 j2n+1 · · · [jn+3 jn+2 ] X−2n+1 j1 ,...,j2n+2

×([jn−1 (jn+2 + d)]t −2an djn+2 + [jn−1 (jn+2 − d)]t 2an djn +2 ) ×t −an−1 jn−1 [jn−1 jn−2 ] · · · [j2 j1 ]t −a1 j1 [j1 k] 2 2 = X−2n+3 t −an (an an−1 −1)d [mj2n+2 ][j2n+2 j2n+1 ]t a1 j2n+1 · · · [jn+4 jn+3 ] 2

2

j1 ,...,j2n+2

×([jn−2 (jn+3 + an d)]t −2(an an−1 −1)djn +3 + [jn−2 (jn+3 −an d)]t 2(an an−1 −1)djn +3 ) ×t −an−2 jn−2 [jn−2 jn−3 ] · · · [j2 j1 ]t −a1 j1 [j1 k] = · · ·   2 t −p q d [mj2n+2 ][j2n+2 j2n+1 ] = X−1 2

2

j2n+1 ,j2n+2 



×([kj2n+1 + kdq  ]t −2dp j2n+1 + [kj2n+1 − kdq  ]t 2dp j2n+1 ) = X−1 t −pq

r−1

[mj2n+2 ][j2n+2 j2n+1 ]

j2n+1 ,j2n+2 =1

×([kj2n+1 + kq]t −2pj2n+1 + [kj2n+1 − kq]t 2pj2n+1 ). This sum is equal to t −pq ([k(m + q)]t −2mp + [k(m − q)]t 2mp ) and the theorem is proved. 4. The Weyl Quantization The first general quantization scheme was introduced by Weyl in 1931. This scheme applies to functions on R2n and postulates that the function e2πi(pxj +qyj ) corresponds to the 1 ∂ operator e2π(pXj +qDj ) , where Xj is multiplication by the variable xj and Dj = 2πi ∂xj . In general, the operator associated to a function is a pseudo-differential operator with

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symbol equal to the function. Since we are quantizing a K¨ahler manifold, we will convert to the complex picture using the Bargmann transform. Here and throughout the paper we choose for Planck’s constant h ¯ = N1 , where N = 2r is an even integer. This is done so that Weil’s integrality condition is satisfied, and so that the Reshetikhin-Turaev topological quantum field theory is well defined. We would like the Hilbert space of the quantization to be the space of square integrable holomorphic sections of a line bundle with first Chern class equal to −N ω = −2π iN dx ∧ dy. It is finite dimensional since the phase space is compact and the Heisenberg uncertainty principle shows that particles occupy boxes of positive volume. It suffices to find the line bundle L for N = 1, and then let the bundle for an arbitrary N be L⊗N . 4.1. The line bundle. Recall that the moduli space M is obtained by factoring the complex plane by the group generated by z → z + 1, z → z + i and z → σ (z) = −z. The line bundle on L then lifts to a trivial line bundle on the complex plane. This shows that the line bundle L is defined by some cocycle χ : C × → C\{0} as the quotient C × C/ ∼ under the equivalence (z, a) ∼ (w, b) if there is λ ∈ such that (w, b) = (λz, χ (z, λ)a). We use the multiplicative notation since the group is not commutative. The cocycle condition is χ (z, λ)χ (λz, µ) = χ (z, µλ). Now we want the cocycle to be holomorphic off the singular points of the character variety, to get a holomorphic line bundle. Also we want it to be compatible with the hermitian structure. Brian Hall suggested us to work with a different hermitian structure than the one induced by the Bargmann measure. This simplifies computations, and makes the relationship with theta functions more transparent. Thus we consider the K¨ahler potential 2 h(z) = e−2π|Im z| which still has the property that −iπ dz ∧ d z¯ = ∂ ∂¯ ln h(z). Requiring the line bundle to have curvature iπ dz ∧ d z¯ yields h(z) = |χ (z, λ)|2 h(λ−1 z). To find the cocycle χ we first determine χ (z, m + in). Since h(z + m + in) = exp(−4π(−inz + in¯z + n2 ))h(z), it follows that |χ (z, m + in)| = exp 2π(−inz + in¯z + n2 ). From the fact that χ is holomorphic, it follows that   χ (z, m + in) = exp π −2inz + n2 · exp(iα(m, n)). The cocycle condition yields exp(iα(m, n) + iα(p, q) − iπ(mq − np)) = exp(iα(m + p, n + q)). This shows that exp(α(m, n) − iπ mn) is a morphism from Z × Z to S 1 . We obtain χ (z, m + in) = (−1)mn exp π(−2inz + n2 ) exp(−2π i(µm + νn)) for some real numbers µ and ν.

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Recall that σ denotes the symmetry of the complex plane with respect to the origin. We have |χ (z, σ )| = h(z)/ h(−z) = 1. Since χ is holomorphic in z it follows that χ(z, σ ) = exp(iπβ) for some β. We want to determine β. Use the model of the torus obtained by identifying opposite sides of a square. The action of σ maps 21 to − 21 , and the two correspond to the same point on the character variety. So 

1 1 ,σ = χ , −1 , χ 2 2 and hence e2πiµ = eiπβ . This shows that µ = β/2. The same argument with χ ( 21 i, σ ) and χ ( 21 i, −i) shows that ν = β/2. Also 

1 1 1 1 + i, σ = χ + i, −1 − i = eπi[2(µ+ν)−1] , χ 2 2 2 2 which implies that modulo 2, 2(µ + ν) − 1 = β. Therefore β = 1. We conclude that χ (z, (m + in)) = (−1)mn exp π(−2inz + n2 ) χ (z, σ ) = −1. 4.2. The Hilbert space of the quantization. Recall that the line bundle L corresponds to the case where the Planck’s constant is equal to 1. To get the general case with h ¯ = 1/N , we consider the line bundle L⊗N . This bundle has the hermitian metric defined by hN (z) = (h(z))N = exp(N π|Im z|2 ), and is given by the cocycle χN = (χ )N . The Hilbert space of the quantization consists of the sections of the line bundle over C that are holomorphic and whose pull-backs to the plane are square integrable with respect to the measure exp(2N πy 2 )dx dy and satisfy f (λz) = χN (z, λ)f (z), λ ∈ . The Hilbert space can thus be identified with that of holomorphic functions on the plane subject to the conditions f (z + m + in) = (−1)mnN eNπ(n

2 −2inz)

f (z)

and f (−z) = −f (z). This is nothing but the space of odd theta functions. If we drop the second condition, then we get the space of classical theta functions ?N = {f

|

f (z + m + in) = eNπ(n

2 −2inz)

f (z)}

It can be seen in [5] that the torus has other possible quantization spaces, which arise by twisting the line bundle with flat bundles. This is not the case with M. The Hilbert space of the quantization is HN = {f ∈ ?N

| f (z) = −f (−z)}.

Both ?N and HN are endowed with the inner product  2 f (z)g(z)e−2Nπ|Im z| dxdy. < f, g >= T2

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To be more accurate, the integral that defines the inner product on HN should be performed over a fundamental domain of the group , however it is more convenient to compute on the torus. Recall that an orthogonal basis of ?N is given by θj , j = 0, 1, . . . , 2r − 1, where θj (z) =

∞

e−π(Nn

2 +2j n)+2πiz(j +Nn)

.

n=−∞

The formula makes sense for all j . We have θj +N (z) = eπ(N+2j ) θj (z) and θ−j (−z) = θj (z), where the second equality follows by replacing n by −n. An orthonormal basis of HN is given by

4 N −πj 2 /N ζj = e (θj − θ−j ), j = 1, 2, . . . , r − 1. 2 To see why these vectors are indeed orthonormal note that < θk − θ−k , θj − θ−j > = < θk , θj > + < θ−k , θ−j > − < θ−k , θj > − < θk , θ−j > = δj k  θj 2 . Here we used the fact that the θj ’s can be shifted to have indices equal to one of the numbers 0, 1, . . . , 2r − 1 and the latter form an orthonormal basis in the Hilbert space associated to the torus. Using the same formula we extend the definition of ζk for all k ∈ Z. Clearly ζr has to be equal to zero, since θr (z) − θ−r (z) = θr (z) − e−π(2r−2r) θr (z) = 0. Also, for 1 ≤ k ≤ r − 1 we have θr+k − θ−r−k = eπ(2r−2(−r+k)) θ−r+k − e−π(2r+2(r−k)) θr−k = −e2rkπ (θr−k − θ−r+k ). Thus normalizing we get that ζr+k = −ζr−k . Finally, since θj +2r is a multiple of θj it follows that ζj +2r = ζj , for all integers j . 4.3. The operators of the quantization. Each observable f : M → R yields a sequence of operators indexed by the level N = 2r. Whenever there is no danger of confusion we omit the index N. We relate Weyl quantization to Toeplitz quantization and then work with Toeplitz operators, for which computations are easier. Let us first consider the case of the complex plane. Modulo some adjustments to suit the notations of this paper, Propositions 2.96 and 2.97 in [7] (see also [13]) show that the operator associated by the Toeplitz quantization to a function f on C is equal to the operator associated by the Weyl quantization to the function  1 2 σ (z, z¯ ) = e−2π|z−u| N f (u, u)dud ¯ u. ¯ N C Note that B

σ (z, z¯ ) = e 4N f (z, z¯ ),

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where 1 Bf = 2π




∂ 2f ∂ 2f + ∂x 2 ∂y 2

 .

Here we used the notation z = x + iy, z¯ = x − iy. Hence doing Weyl quantization B with symbol f is the same as doing Toeplitz quantization with symbol e− 4N f . Thinking equivariantly, we now define the operators of the quantization of the character variety. Let CN : L2 (M, L⊗N ) → HN be the orthogonal projection from the space of square integrable sections with respect 2 to the measure e−2Nπy dxdy onto the space HN . To a sufficiently regular function f on the character variety we associate the operator opN (f ) (in level N ) given by  B   opN (f ) : HN → HN , g → CN e− 4N f g . The operator g → CN (f g) is the Toeplitz operator of symbol f , denoted by Tf . An important family of operators are the ones associated to the functions 2 cos 2π(px+ qy), which we denote by C(p, q). These are the same as the Toeplitz operators with symbols B

2e− 4N cos 2π(px + qy) = 2e

p 2 +q 2 2N π

cos 2π(px + qy).

5. Weyl Quantization Versus Quantum Group Quantization To simplify the computation, we pull back everything to the line bundle on the torus. Hence we do the computations in ?N . We start with two lemmas that hold on the torus. They were inspired by [5]. Let j, k, p be integers such that −r + 1 ≤ j, k ≤ r − 1, p = p0 + γ N with γ an integer. There are two possibilities −r +1 ≤ j +p0 < N = 2r or N = 2r ≤ j +p0 < N +1. Let also u(y) be a bounded continuous function. Lemma 2. Assume that j + p0 < N . Then < e2πipx u(y)θj , θk > is different from zero if and only if k = j + p0 and in this case it is equal to π

e− 2N p

∞

2 +(j +p /2)2 /N 0

e−πm

2 /2N

e−2πi(j +p/2)m/N u(m), ˆ

m=−∞

where u(m) ˆ is the mth Fourier coefficient of u. Proof. Separating the variables we obtain  2 < exp(2πpx)u(y)θj , θk > = exp(2πpx)u(y)θj θk e−2Nπy dxdy T2

=



e

−π(Nm2 +2j m+Nn2 +2kn)

 0

1

e2πix(N(m−n)+p+j −k) dx

0

m,n

×



1

e−2πy(j +Nn+k+Nm)−2πNy u(y)dy. 2

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The first integral is equal to zero unless n = m + γ and k = p0 + j . If k = p0 + j the expression becomes e



−πNγ 2 −2πj γ −2πp0 γ

1

e−2π(Ny

2 +(2j +p +Nγ )y) 0

0

×



e



 p −2π Nm2 +Nmγ +2m j + 20 +Ny

u(y)dy.

m

After completing the square in the exponent of the third exponential we obtain that this is equal to e

π 2 − 2N p +(j +p0 /2)2 /N



1 0

e

2  −2πN m+y+ j +p/2 N

u(y)dy.

m

  2 Using the Poisson formula ( m f (m) = m fˆ(m)) for the function e−x we transform the sum of the exponentials into

e

  j +p/2 m −πm2 /2N 2πi y+ N

e

.

m

It follows that the inner product we are computing is equal to π

e− 2N p

2 +π(j +p /2)2 /N 0



2 /2N

e−πm

e2πi(j +p/2)m/N



1

e2πimy u(y)dy

0

m

which proves the lemma. Lemma 3. Assume that j + p0 ≥ N and denote p1 = p − (γ + 1)N . Then < e2π ipx u(y)θj , θk > is different from zero if and only if k = j + p1 and in this case it is equal to π

e− 2N p

2 +π(j +p /2)2 /N 1

∞

e−πm

2 /2N

e−2πi(j +p/2)m/N u(m), ˆ

n=−∞

where u(m) ˆ is the mth Fourier coefficient of u. Proof. We start with a computation like the one in the proof of Lemma 2 to conclude that k = p0 + j − N = p1 + j and m = n − γ − 1. From here the same considerations apply mutatis mutandis to yield the conclusion. Theorem 2. The quantum group and the Weyl quantizations of the moduli space of flat SU (2)-connections on the torus are unitary equivalent. The unitary isomorphism establishing this equivalence maps the orthonormal basis V j (α), j = 1, 2, . . . , r − 1, of V (T2 ) to the orthonormal basis ζj , j = 1, 2, . . . , r − 1, of HN .

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Proof. One has to show that the matrices of the operators associated by the two quantizations to a given function are the same. Since linear combinations of cosines are dense in the algebra of functions on the moduli space, it suffices to check this property for cosines. The quantum group quantization associates to cos 2π(px + qy) the operator C(p, q), while Weyl quantization associates to it the operator C(p, q). We verify that the matrix of the operator C(p, q) in the basis ζj is the same as the matrix of the operator C(p, q) in the basis V j (α). We have C(p, q) = e

p 2 +q 2 2N

T2 cos 2π(px+qy) ,

where T2 cos 2π(px+qy) is the Toeplitz operator of symbol 2 cos 2π(px + qy). Let us pull back everything to the torus using the covering map T2 → M so that we can work with exponentials. We do first the case j + p0 < N . If in Lemma 2 we let u(y) = e2πiqy we obtain π

Te2π ipx+2π iqy θj = e− 2N p

2 +(j +p /2)2 /N 1

Using this formula and the fact that

ζj =

4

e−πq

2 /2N

e−2πi(j +p/2)q/N θj +p0 .

N −πj 2 /N (θj − θ−j ) e 2

after doing the algebraic computations we arrive at   p 2 +q 2 e 2N π T2 cos 2π(px+qy) ζj = t −pq t 2j q ζj −p0 + t −2j q ζj +p0 . If j + p0 ≥ N , we let p1 = p − (γ + 1)N , and an application of Lemma 3 shows that in this case   p 2 +q 2 e 2N π T2 cos 2π(px+qy) ζj = t −pq t 2j q ζj −p1 + t −2j q ζj +p1 . But we have seen that ζj +N = ζj for all j , so in the above formulas p0 and p1 can be replaced by p. It follows that for all j , 1 ≤ j ≤ r − 1, and all integers p and q we have   p 2 +q 2 e 2N π T2 cos 2π(px+qy) ζj = t −pq t 2j q ζj −p + t −2j q ζj +p . Theorem 1 shows that

  C(p, q)V j (α) = t −pq t 2j q V j −p (α) + t −2j q V j +p (α) .

Hence the unitary isomorphism defined by V j (α) → ζj transforms the operator C(p, q) into the operator C(p, q), and the theorem is proved. From now on we identify the two quantizations and use the notation C(p, q) for the operators. Recall the notation t = eiπ/N . As a byproduct of the proof of the theorem we obtain the following product-to-sum formula for C(p, q)’s (see also [9]). Proposition 1. For any integers m, n, p, q one has mn

mn

C(m, n) ∗ C(p, q) = t |pq | C(m + p, n + q) + t −|pq | C(m − p, n − q), where |mn pq | is the determinant.

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We conclude this section by noting that in Witten’s picture the operator associated by Weyl quantization to the function sin 2π(n + 1)(p  x + q  y)/ sin 2π(p x + q  y) (p  , q  relatively prime) is the same as the quantum group quantization of the Wilson line around the curve of slope p  /q  on the torus in the n-dimensional irreducible representation of SU (2). 6. The Star Product 6.1. Definition of the star product. Let (M, ω) be a symplectic manifold. A ∗-product on M is a binary operation on C ∞ (M)[[N −1 ]] which is associative, and for all f, g ∈ C ∞ (M) satisfies N −k f ∗ g = f ∗ N −k g = N −k (f ∗ g) and also f ∗g =

∞

N −k Bk (f, g).

k=0

The operators Bk (f, g) are bi-differential operators from C ∞ (M)×C ∞ (M) to C ∞ (M), such that B0 (f, g) = f g, and such that Dirac’s correspondence principle B1 (f, g) − B1 (g, f ) = {f, g} is satisfied. Here {f, g} stands for the Poisson bracket induced by the symplectic form. One says that C ∞ (M)[[N −1 ]] is a deformation of C ∞ (M) in the direction of the given Poisson bracket. We use N for the variable of the formal series to be consistent with the rest of the paper. The character variety M is a symplectic manifold off the four singularities. Proposition 2. The formula 2 cos 2π(mx + ny) ∗ 2 cos 2π(px + qy) mn

= t |pq | 2 cos 2π((m + p)x + (n + q)y) mn

+t −|pq | 2 cos 2π((m − p)x + (n − q)y) defines a ∗-product on C ∞ (M)[[N −1 ]], which is a deformation quantization in the direction of the K¨ahler form iπ dz ∧ d z¯ . In these formulas the exponentials should be expanded formally into power series in N −1 . Proof. We have 2 cos 2π(mx + ny) ∗ 2 cos 2π(px + qy) − 2 cos 2π(mx + ny) ∗ 2 cos 2π(px + qy) = π(imq − inp)N −1 2 cos 2π((m + p)x + (n + q)y) + π(inp − imq)N −1 2 cos 2π((m − p)x + (n − q)y) − π(ipn − iqm)N −1 2 cos 2π((p + m)x + (q + n)y) − π(imq − inp)N −1 2 cos 2π((p − m)x + (q − n)y) + O(N −2 ) = 2π iN −1 (mq − np)2 cos 2π((m + p)x + (n + q)y) − 2π iN −1 (mq − np)2 cos 2π((m − p)x + (n − q)y) + O(N −2 ) = N −1 {2 cos 2π(mx + ny), 2 cos 2π(px + qy)} + O(N −2 )

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so the correspondence principle is satisfied. The coefficients Bk (f, g) are bidifferential operators since    ∂ ∂  1  ∂x1 ∂y1  det  ∂ ∂  f (x1 , y1 )g(x2 , y2 )| x1 =x2 =x B1 (f, g) = y1 =y2 =y  ∂x2 ∂y2  4πi and Bk = B1k /k!. We would like to point out that this ∗-product is different from the one that would arise if we applied the quantization methods outlined in [3] since Berezin’s ideas correspond to the anti-normal (respectively normal) ordering of the operators. Thinking now of the deformation parameter as a fixed natural number we see that the ∗-algebra defined in Proposition 6.1 is a subalgebra of Rieffel’s noncommutative torus [18] with rational Planck’s constant. That is, our ∗-product is the restriction of Rieffel’s ∗-product to trigonometric series in cosines. 6.2. The generalized Hardy space. In [12] Guillemin has shown that for each compact symplectic prequantizable manifold M there exists a ∗-product and a circle bundle S 1 E→ P → M with a canonical volume µ such that L2 (P , µ) contains finite dimensional vector spaces HN of N-equivariant functions (mod the action of S 1 ) satisfying for all f, g ∈ C ∞ (M): CN Mf CN Mg CN = CN M(f ∗g)N CN + O(N −∞ ), where CN is the orthogonal projection onto HN , Mf is the multiplication by f and (f ∗ g)N is to be understood as the ∗-product for a certain fixed integer value of N . Our operators are not Toeplitz so they won’t fit exactly Guillemin’s construction. However, along the same lines we will construct a representation of the ∗-algebra from the previous section onto an infinite dimensional Hilbert space that contains all HN ’s as direct summands. The reader can find a detailed account on how these things are done in general in [6]. Let L be the line bundle constructed in 4.1, and let Z ⊂ L∗ be the unit circle bundle in the dual of L. Z is an S 1 -principal bundle. A point in Z is a pair (x, φ), where x ∈ C and φ is a complex valued functional with |φ(x)| = x (the length of x being given by the hermitian structure). More precisely Z = {(z, ξ );

|ξ | = e−π|z|

2 /2

}.

The map (z, ξ ) → (z, ξ/|ξ |) identifies Z with C × S 1 . Let θ be the argument of ξ and consider the volume form dθ dz on Z. Using this volume form we can define the space L2 (Z). Now let N be an even integer. Note that L⊗N  Z ×N C, where Z ×N C is the quotient of Z × C by the equivalence (p · eiNθ , z) ∼ (p, eiNθ z). A complex valued smooth function f on Z is called N -equivariant if for all (x, φ) ∈ Z, f (x, φ · eiθ ) = eiNθ f (x, φ).

Quantization of the Moduli Space of Flat SU (2)-Connections the Torus

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The set of all N -equivariant functions is denoted by C ∞ (Z)N . There exists an isomorphism C ∞ (M, L⊗N )  C ∞ (Z)N , which transforms a section s ∈ C ∞ (M, L⊗N ) to a function f , with f (z, ξ ) = ξ N s(z). Here of course s has to be viewed as a C-valued function subject to the equivariance conditions from Sect. 4. It is easy to see that this map gives rise to a unitary isomorphism between the space of L2 -sections of L⊗N and the L2 -completion of C ∞ (Z)N . A little  imθ Fourier analysis (involving integrals of the form e e−inθ dθ ) shows that for different N’s, the images of the corresponding L2 spaces are orthogonal. As a result, the spaces HN are embedded as mutually orthogonal subspaces of L2 (Z). Define  H= HN . N even This space is the generalized version of the classical Hardy space. We denote by C the orthogonal projection of L2 (Z) onto H. Let f be a smooth function on M, which can be viewed as the limit (in the C ∞ topology) of a sequence of trigonometric polynomials in cos(px + qy), p, q ∈ Z. Define the operator  B   op(f ) : H → H, g → C e− 4N f g . B

If e− 4N f is a C ∞ function on the character variety, this operator is bounded. The restriction of op(f ) to HN coincides with opN (f ), for all N ≥ 1. Moreover, the product-to-sum formula from Proposition 5.4 shows that the ∗-product on C ∞ (M)[[N −1 ]] defined by the multiplication of these operators is the same as the ∗-product introduced in Sect. 6.1. 7. Final Remarks An alternative approach to the Reshetikhin-Turaev theory was constructed in [4] using Kauffman bracket skein modules. This approach also leads to a quantization of the moduli space of flat SU (2)-connections on the torus. Do we obtain the Weyl quantization in that situation as well? The answer is no. Indeed, the analogues of the basis vectors V n (α) are the colorings of the core of the solid torus by Jones-Wenzl idempotents. More precisely, to the vector V n (α) corresponds the vector Sn−1 (α), where Sn−1 (α) is the coloring of α by the n−1st Jones-Wenzl idempotent. It follows from Theorem 5.6 and the discussion preceding it in [9] that the action of the operator associated to 2 cos 2π(px + qy), denoted by (p, q)T , on these basis elements is (p, q)T Sn−1 (α) = (−1)q t −pq (t 2qk Sk−p−1 (α) + t −2qk Sk+p−1 (α)). The factor (−1)q does not appear in the formula from Theorem 1, proving that this quantization is different. However, both quantizations yield the same ∗-algebra, as shown in [9]. Finally, although quantum field theory is intimately related to Wick quantization, the present paper shows that this is not the case with the topological quantum field theory of Witten, Reshetikhin and Turaev.

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References 1. Atiyah, M.F.: The Geometry and Physics of Knots. Lezioni Lincee, Cambridge: Cambridge Univ. Press, 1990 2. Atiyah, M.F., Bott, R.: The Yang-Mills equations over a Riemann surface. Phil. Tras. Royal Soc. A 308, 523–615 (1982) 3. Berezin, F.A.: Quantization. Math. USSR Izvestija 8, 1109–1165 (1974) 4. Blanchet, C., Habegger, N., Masbaum, G., Vogel, P.: Topological quantum field theories derived from the Kauffman bracket. Topology 34, 883–927 (1995) 5. Bloch, A., Golse, F., Paul, T., Uribe, A.: Dispersionless Toda and Toeplitz operators. Preprint 6. Borthwick, D.: Introduction to K¨ahler quantization. In: First Summer School in Analysis and Mathematical Physics, Cuernavaca Morelos, Mexico. Contemporary Math. 260, 91–132 (2000) 7. Folland, G.: Harmonic Analysis in Phase Space. Princeton, NJ: Princeton, University Press, 1989 8. Frohman, Ch., Kania-Bartoszy´nska, J.: SO(3) topological quantum field theory. Commun. Anal. Geom. 4, 589–676 (1996) 9. Frohman, Ch., Gelca, R.: Skein modules and the noncommutative torus. Trans Am. Math. Soc. 352, 4877–4888 (2000) 10. Gelca, R.: SL(2, C)-topogical quantum field theory with corners. J. Knot Theor. Ramif. 7, 893–906 (1998) 11. Goldman, W.: Invariant functions on Lie groups and Hamiltonian flow of surface group representations. Invent. Math. 85, 263–302 (1986) 12. Guillemin, V.: Star Products on Compact Pre-quantizable Symplectic Manifolds. Lett. Math. Phys. 35, 85–89 (1995) 13. Hall, B.C.: Holomorphic methods in analysis and mathematical physics. In: First Summer School in Analysis and Mathematical Physics, Cuernavaca Morelos, Mexico. Contemporary Math. 260, 1–59 (2000) 14. Jones, V.F.R.: Polynomial invariants of knots via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985) 15. Kirillov, A.N., Reshetikhin, N.Y.: Representation of the algebra Uq (sl2 ), q-orthogonal polynomials and invariants of links. In: Infinite Dimensional Lie Algebras and Groups V.G. Kac (ed.), Adv. Ser. in Math. Phys. vol. 7, Singapore: World Scientific, 1988, pp. 285–339 16. Reshetikhin, N.Yu., Takhtajan, L.A.: Deformation quantization of K¨ahler manifolds. Preprint 17. Reshetikhin, N.Yu., Turaev, V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–597 (1991) 18. Rieffel, M.: Deformation quantization of Heisenberg Manifolds. Commun. Math. Phys. 122, 531– 562 (1989) 19. Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. Paris: de Gruyter, 1994 20. Walker, K.: On Witten’s 3-manifold invariants. Preprint 1991 21. Weitsman, J.: Quantization via real polarization of the moduli space of flat connections and ChernSimons gauge theory in genus one. Commun. Math. Phys. 137, 175–190 (1991) 22. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989) Communicated by A. Connes

Commun. Math. Phys. 233, 513–543 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0762-8

Communications in

Mathematical Physics

Classification of Integrable Equations on Quad-Graphs. The Consistency Approach V.E. Adler1 , A.I. Bobenko2 , Yu.B. Suris2 1

Landau Institute of Theoretical Physics, 12 Institutsky pr., 142432 Chernogolovka, Russia. E-mail: [email protected] 2 Institut f¨ ur Mathematik, Technische Universit¨at Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany.E-mail: [email protected]; [email protected] Received: 14 February 2002 / Accepted: 22 September 2002 Published online: 8 January 2003 – © Springer-Verlag 2003

Abstract: A classification of discrete integrable systems on quad–graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three–dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature representation with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three–leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three–dimensional integrable systems and to the quantum context are also discussed. 1. Introduction The idea of consistency (or compatibility) is in the core of integrable systems theory. One is faced with it already at the very definition of the complete integrability of a Hamiltonian flow in the Liouville–Arnold sense, which means exactly that the flow may be included into a complete family of commuting (compatible) Hamiltonian flows [1]. Similarly, it is a characteristic feature of soliton (integrable) partial differential equations that they appear not separately but are always organized in hierarchies of commuting (compatible) systems. It is impossible to list all applications or reincarnations of this idea. We mention only a couple of them relevant for our present account. A condition of existence of a number of commuting systems may be taken as the basis of the symmetry approach which is used to single out integrable systems in some general classes and to classify them [24]. With the help of the Miwa transformation one can encode a hierarchy of integrable equations, like the KP one, into a single discrete (difference) equation [25]. Another way of relating continuous and discrete systems, connected with the idea of compatibility, is based on the notion of B¨acklund transformations and the Bianchi permutability theorem for them [4]. This notion, born in the classical differential geometry, found its place in the modern theory of discrete integrable systems [30]. A

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V.E. Adler, A.I. Bobenko, Yu.B. Suris

sort of synthesis of the analytic and the geometric approach was achieved in [5] and is being actively developed since then, see a review in [6]. These studies have revealed the fundamental importance of discrete integrable systems; it is nowadays a commonly accepted idea that they may be regarded as the cornerstone of the whole theory of integrable systems. For instance, one believes that both the differential geometry of “integrable” classes of surfaces and their transformation theory may be systematically derived from the multidimensional lattices of consistent B¨acklund transformations [6]. So, the consistency of discrete equations steps to the front stage of the integrability theater. We say that a d–dimensional discrete equation possesses the consistency property, if it may be imposed in a consistent way on all d–dimensional sublattices of a (d + 1)–dimensional lattice (a more precise definition will be formulated below). As it is seen from the above remarks, the idea that this notion is closely related to integrability, is not new. In the case d = 1 it was used as a possible definition of integrability of maps in [34]. A clear formulation in the case d = 2 was given recently in [28]. A decisive step was made in [8]: it was shown there that the integrability in the usual sense of the soliton theory (as existence of the zero curvature representation) follows for two–dimensional systems from the three– dimensional consistency. So, the latter property may be considered as a definition of integrability, or its criterion which may be checked in a completely algorithmic manner starting with no more information than the equation itself. Moreover, in case when this criterion gives a positive result, it delivers also the transition matrices participating in the discrete zero curvature representation. (Independently, this was found in [26].) In the present paper we give a further application of the consistency property: we show that it provides an effective tool for finding and classifying all integrable systems in certain classes of equations. We study here integrable one–field equations on quad–graphs, i.e. on cellular decompositions of surfaces with all faces (2–cells) being quadrilateral [8]. In Sect. 2 we formulate our main result (Theorem 1), consisting of a complete classification of integrable systems on quad–graphs. Of course, we provide also a detailed discussion of the assumptions of Theorem 1. Sections 3, 4 are devoted to the proof of the main theorem. In Sect. 5 we discuss the so called three–leg forms [8] of all integrable equations from Theorem 1. This device, artificial from the first glance, proves to be extremely useful in several respects. First, it allows to establish a link with the discrete Toda type equations introduced in full generality in [3]. Second, it provides a mean to the most natural derivation of invariant symplectic structures for evolution problems generated by equations on quad–graphs. This is discussed in Sect. 6. Further, Sect. 7 contains a brief discussion of the relation of the equations listed in Theorem 1 with B¨acklund transformations for integrable partial differential equations. Finally, in Sect. 8 we discuss some perspectives for further work, in particular the consistency approach for three–dimensional systems, and discrete quantum systems.

2. Formulation of the Problem; Results Basic building blocks of systems on quad–graphs are equations on quadrilaterals of the type Q(x, u, v, y; α, β) = 0,

(1)

Classification of Integrable Equations on Quad-Graphs v

α

β x

515 y β

α

u

Fig. 1. An elementary quadrilateral; fields are assigned to vertices

where the fields x, u, v, y ∈ C are assigned to the four vertices of the quadrilateral, and the parameters α, β ∈ C are assigned to its edges, as shown on Fig. 1. A typical example is the so called “cross-ratio equation” (x − u)(y − v) α = , (u − y)(v − x) β

(2)

where on the left-hand side one recognizes the cross-ratio of the four complex points x, u, y, v. We shall use the cross-ratio equation to illustrate various notions and claims in this introduction. Roughly speaking, the goal of the present paper is to classify equations (1) building integrable systems on quad-graphs. We now list more precisely the assumptions under which we solve this problem. First of all, we assume that Eq. (1) can be uniquely solved for any one of its arguments x, u, v, y ∈
C. Therefore, the solutions have to be fractional-linear in each of their arguments. This naturally leads to the following condition. 1) Linearity. The function Q(x, u, v, y; α, β) is linear in each argument (affine linear): Q(x, u, v, y; α, β) = a1 xuvy + · · · + a16 ,

(3)

where coefficients ai depend on α, β. Notice that for the cross-ratio equation (2) one can take the function on the left–hand side of (1) as Q(x, u, v, y; α, β) = β(x − u)(y − v) − α(u − y)(v − x). Second, we are interested in equations on quad-graphs of arbitrary combinatorics, hence it will be natural to assume that all variables involved in Eqs. (1) are on equal footing. Therefore, our next assumption reads as follows. 2) Symmetry. Equation (1) is invariant under the group D4 of the square symmetries, that is function Q satisfies the symmetry properties Q(x, u, v, y; α, β) = εQ(x, v, u, y; β, α) = σ Q(u, x, y, v; α, β)

(4)

with ε, σ = ±1. Of course, due to the symmetries (4) not all coefficients ai in (3) are independent, cf. formulae (25), (26) below. We are interested in integrable equations of the type (1), i.e. those admitting a discrete zero curvature representation. We refer the reader to [3, 7, 8], where this notion was defined for systems on arbitrary graphs. As pointed out above, in the third of these papers it was shown that the integrability can be detected in an algorithmic manner starting with no more information than the equation itself: the criterion of integrability of an equation is its three-dimensional consistency. This property means that Eq. (1)

516

V.E. Adler, A.I. Bobenko, Yu.B. Suris α

v

y

β

β x

u

α

Fig. 2. D4 symmetry

may be consistently embedded in a three-dimensional lattice, so that similar equations hold for all six faces of any elementary cube, as on Fig. 3 (it is supposed that the values of the parameters αj assigned to the opposite edges of any face are equal to one another, so that, for instance, all edges (x2 , x12 ), (x3 , x31 ), and (x23 , x123 ) carry the label α1 ): To describe more precisely what is meant under the three-dimensional consistency, consider the Cauchy problem with the initial data x, x1 , x2 , x3 . The equations Q(x, xi , xj , xij ; αi , αj ) = 0

(5)

allow one to determine uniquely the values x12 , x23 , x31 . After that one has three different equations for x123 , coming from the faces (x1 , x12 , x31 , x123 ), (x2 , x23 , x12 , x123 ), and (x3 , x31 , x23 , x123 ). Consistency means that all three values thus obtained for x123 coincide. For instance, consider the cross-ratio equation (2). It is not difficult to check that it possesses the property of the three-dimensional consistency, and x123 =

(α1 − α2 )x1 x2 + (α3 − α1 )x3 x1 + (α2 − α3 )x2 x3 . (α3 − α2 )x1 + (α1 − α3 )x2 + (α2 − α1 )x3

(6)

Looking ahead, we mention a very amazing and unexpected feature of the expression (6): the value x123 actually depends on x1 , x2 , x3 only, and does not depend on x. In other words, four black points on Fig. 3 (the vertices of a tetrahedron) are related by a well-defined equation. One can rewrite Eq. (6) as (x1 − x3 )(x2 − x123 ) α1 − α 3 , = (x3 − x2 )(x123 − x1 ) α2 − α 3 x23

x123

x3

x31

α3 α2 x

x12

x2 α1

x1

Fig. 3. Three-dimensional consistency

(7)

Classification of Integrable Equations on Quad-Graphs

517

which also has an appearance of the cross-ratio equation for the four points (x1 , x3 , x2 , x123 ) with the parameters α1 − α3 assigned to the edges (x1 , x3 ), (x2 , x123 ) and α2 − α3 assigned to the edges (x2 , x3 ), (x1 , x123 ). This property, being very strange from first glance, holds actually not only in this but in all known nontrivial examples. We take it as an additional assumption in our solution of the classification problem. 3) Tetrahedron property. The function x123 = z(x, x1 , x2 , x3 ; α1 , α2 , α3 ), existing due to the three-dimensional consistency, actually does not depend on the variable x, that is, zx = 0. Under the tetrahedron condition we can paint the vertices of the cube into black and white ones, as on Fig. 3, and the vertices of each of two tetrahedrons satisfy an equation of the form
1 , x2 , x3 , x123 ; α1 , α2 , α3 ) = 0; Q(x

(8)


may be also taken it is easy to see that under the assumption 2) (linearity) the function Q linear in each argument. (Clearly, formulas (6), (7) may be also written in such a form.) Actually, the tetrahedron condition is closely related to another property of Eq. (1), namely to the existence of a three-leg form of this equation [8]: ψ(x, u; α) − ψ(x, v; β) = φ(x, y; α, β).

(9)

The three terms in this equation correspond to three “legs”: two short ones, (x, u) and (x, v), and a long one, (x, y). The short legs are the edges of the original quad-graph, while the long one is not. (We say that the three-leg form (9) is centered at the vertex x; of course, due to the symmetries of the function Q there have to exist also similar formulas centered at the other three vertices involved.) For instance, the cross-ratio equation (2) is equivalent to the following one: α β α−β − = . u−x v−x y−x

(10)

The three-leg form gives an explanation for the equation for the “black” tetrahedron from Fig. 3. Consider three faces adjacent to the vertex x123 on this figure, namely the quadrilaterals (x1 , x12 , x31 , x123 ), (x2 , x23 , x12 , x123 ), and (x3 , x31 , x23 , x123 ). A summation of three-leg forms (centered at x123 ) of equations corresponding to these three faces leads to the equation φ(x123 , x1 ; α2 , α3 ) + φ(x123 , x2 ; α3 , α1 ) + φ(x123 , x3 ; α1 , α2 ) = 0. This equation, in any event, relates the fields in the “black” vertices of the cube only, i.e. has the form of (8). For example, for the cross-ratio equation this formula reads: α2 − α3 α3 − α1 α1 − α2 + + = 0, x1 − x123 x2 − x123 x3 − x123 and a simple calculation convinces that this is equivalent to (6). So, the tetrahedron property is a necessary condition for the existence of a three-leg form. On the other hand, a verification of the tetrahedron property is much more straightforward than finding the three-leg form, since the latter contains two a´ priori unknown functions ψ, φ. It remains to mention that, as demonstrated in [8], the existence of the three-leg form allows one to immediately establish a relation to discrete systems of the Toda type [3].

518

V.E. Adler, A.I. Bobenko, Yu.B. Suris x23

x2

x3 α3 x34

x23

α4

α2 x1

x34

x12

α3

α2 α1

α4 α 5

α5

x51

x51

x5

x4

α1 α2

α1

x

x12

x45

x45

Fig. 4. Faces adjacent to the vertex x

Fig. 5. The star of the vertex x in the black subgraph

Indeed, if x is a common vertex of n adjacent quadrilateral faces (x, xk , xk,k+1 , xk+1 ), k = 1, 2, . . . , n, with the parameters αk assigned to the edges (x, xk ) (cf. Fig. 4), then the fields in the point x and in the “black” vertices of the adjacent faces satisfy the following equation: n 

φ(x, xk,k+1 , αk , αk+1 ) = 0.

(11)

k=1

This is a discrete Toda type equation (equation on stars) on the graph whose vertices are the “black” vertices of the original quad-graph, and whose edges are the diagonals of the faces of the original quad-graph connecting the “black” vertices. The parameters α are naturally assigned to the corners of the faces of the “black” subgraph. See Fig. 5. (Of course, a similar Toda type equation holds also for the “white” subgraph.) For instance, in the case of the cross-ratio equation, the discrete Toda type system (11) reads as n  αk − αk+1 = 0. xk,k+1 − x

(12)

k=1

In the next section we will show that the tetrahedron condition naturally separates one of two subcases of the general problem of classification of integrable equations on quad-graphs. The second subcase will not be considered in this paper; the corresponding subclass of equations is certainly not empty, but we are aware only of trivial (linearizable) examples. By solving the classification problem we identify equations related by certain natural transformations. First, acting simultaneously on all variables x by one and the same M¨obius transformation does not violate our three assumptions. Second, the same holds for the simultaneous point change of all parameters α → ϕ(α). Our results on the classification of integrable equations on quad-graphs are given by the following statement.

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Theorem 1. Up to common M¨obius transformations of the variables x and point transformations of the parameters α, the three-dimensionally consistent quad-graph equations (5) with the properties 1), 2), 3) (linearity, symmetry, tetrahedron property) are exhausted by the following three lists Q, H, A (u = x1 , v = x2 , y = x12 , α = α1 , β = α2 ). List Q: (Q1)

α(x − v)(u − y) − β(x − u)(v − y) + δ 2 αβ(α − β) = 0,

(Q2)

α(x − v)(u − y) − β(x − u)(v − y) + αβ(α − β)(x + u + v + y) −αβ(α − β)(α 2 − αβ + β 2 ) = 0,

(Q3)

(β 2 − α 2 )(xy + uv) + β(α 2 − 1)(xu + vy) − α(β 2 − 1)(xv + uy) −δ 2 (α 2 − β 2 )(α 2 − 1)(β 2 − 1)/(4αβ) = 0,

(Q4)

a0 xuvy + a1 (xuv + uvy + vyx + yxu) + a2 (xy + uv) + a¯ 2 (xu + vy) +a˜ 2 (xv + uy) + a3 (x + u + v + y) + a4 = 0,

where the coefficients ai are expressed through (α, a) and (β, b) with a 2 = r(α), b2 = r(β), r(x) = 4x 3 − g2 x − g3 , by the following formulae: a0 = a + b, a1 = −βa − αb, a2 = β 2 a + α 2 b, ab(a + b) g2 + β 2 a − (2α 2 − )b, a¯ 2 = 2(α − β) 4 ab(a + b) g2 a˜ 2 = + α 2 b − (2β 2 − )a, 2(β − α) 4 2 g g3 g2 a3 = a0 − a1 , a4 = 2 a0 − g3 a1 . 2 4 16 List H : (H1) (H2) (H3)

(x − y)(u − v) + β − α = 0, (x − y)(u − v) + (β − α)(x + u + v + y) + β 2 − α 2 = 0, α(xu + vy) − β(xv + uy) + δ(α 2 − β 2 ) = 0.

List A: (A1) (A2)

α(x + v)(u + y) − β(x + u)(v + y) − δ 2 αβ(α − β) = 0, (β 2 − α 2 )(xuvy + 1) + β(α 2 − 1)(xv + uy) − α(β 2 − 1)(xu + vy) = 0.

Remark. 1) The list A can be dropped down by allowing an extended group of M¨obius transformations, which act on the variables x, y differently than on u, v (white and black sublattices on Figs. 1,3). In this manner Eq. (A1) is related to (Q1) (by the change u → −u, v → −v), and Eq. (A2) is related to (Q3) with δ = 0 (by the change u → 1/u, v → 1/v). So, really independent equations are given by the lists Q and H. 2) In both lists Q, H the last equations are the most general ones. This means that Eqs. (Q1)–(Q3) and (H1), (H2) may be obtained from (Q4) and (H3), respectively, by certain degenerations and/or limit procedures. So, one could be tempted to shorten down these lists to one item each. However, on the one hand, these limit procedures are outside of our group of admissible (M¨obius) transformations, and on the other hand, in many situations the “degenerate” equations (Q1)–(Q3) and (H1), (H2) are of interest for

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themselves. This resembles the situation with the list of six Painlev´e equations and the coalescences connecting them, cf. [15]. 3) Parameter δ in Eqs. (Q1), (Q3), (H3) can be scaled away, so that one can assume without loss of generality that δ = 0 or δ = 1. 4) It is natural to set in Eq. (Q4) (α, a) = (℘ (A), ℘ (A)) and, similarly, (β, b) = (℘ (B), ℘ (B)). So, this equation is actually parametrized by two points of the elliptic curve µ2 = r(λ). The appearance of an elliptic curve in our classification problem is by no means obvious from the beginning, its origin will become clear later, in the course of the proof. For the cases of r with multiple roots, when the elliptic curve degenerates into a rational one, Eq. (Q4) degenerates to one of the previous equations of the list Q; for example, if g2 = g3 = 0 then the inversion x → 1/x turns (Q4) into (Q2). Bibliographical remarks. It is difficult to track down the origin of the equations listed in Theorem 1. Probably, the oldest ones are (H1) and (H3)δ=0 , which can be found in the work of Hirota [14] (of course not on general quad–graphs but only on the standard square lattice with the labels α constant in each of the two lattice directions; similar remarks apply also to other references in this paragraph). Equations (Q1) and (Q3)δ=0 go back to [30], see also a review in [27]. Equation (Q4) was found in [2]. A Lax representation for (Q4) was found in [26] with the help of the method based on the three–dimensional consistency, identical with the method introduced in [8]. Equations (Q2) and (Q3)δ=1 are particular cases of (Q4), but seem to have not appeared explicitly in the literature. The same holds for (H2) and (H3)δ=1 . 3. Classification: Analysis In principle, the three-dimensional consistency turns, under the assumptions 1), 2), into some system of functional equations for the coefficients ai of the functions Q (see (3)). However, this system is difficult to analyze and we will take a different route. For the first step, we consider the problem of the three-dimensional consistency in the following general setting: find triples of functions f1 , f2 , f3 of three arguments such that if x23 = f1 (x, x2 , x3 ),

x31 = f2 (x, x3 , x1 ),

x12 = f3 (x, x1 , x2 )

(13)

then x123 := f1 (x1 , x12 , x31 ) ≡ f2 (x2 , x23 , x12 ) ≡ f3 (x3 , x31 , x23 )

(14)

identically in x, x1 , x2 , x3 . In other words, we ignore for a moment the conditions 1) and 2) and look to what consequences the tetrahedron condition leads. The proof of the following statement demonstrates that this condition separates just one of two possible subcases in the general problem. Proposition 2. For the functions f1 , f2 , f3 compatible in the sense (13), (14) and satisfying the tetrahedron condition, the following relation holds: f3,x2 f2,x1 f1,x3 = −f3,x1 f2,x3 f1,x2 identically in x, x1 , x2 , x3 .

(15)

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Proof. Denote x123 = z(x, x1 , x2 , x3 ), and also denote for short the functions with shifted arguments (14) by capitals: F1 = f1 (x1 , x12 , x31 ), etc. Then differentiating (14) with respect to x2 , x3 and x yields the following system which is linear with respect to the derivatives of F1 :

f3,x F1,x12

f3,x2 F1,x12 = zx2 , f2,x3 F1,x31 = zx3 , + f2,x F1,x31 = zx ,

and two analogous systems for F2 , F3 obtained by the cyclic shift of indices. The above system is overdetermined, and, excluding the derivatives of F1 , we come to the first equation of the following system (the other two come from the similar considerations with F2 , F3 ): f2,x3 f3,x zx2 +f3,x2 f2,x zx3 = f3,x2 f2,x3 zx , f1,x3 f3,x zx1 +f3,x1 f1,x zx3 = f1,x3 f3,x1 zx , f1,x2 f2,x zx1 +f2,x1 f1,x zx2 = f2,x1 f1,x2 zx . Now, the tetrahedron condition zx = 0 implies that the r.h.s. vanishes, and therefore the determinant has to vanish as well; this is equivalent to (15).   The second possibility, which we do not consider here, would be that zx = 0 and Eq. (15) does not hold. In this case the above system can be solved to give zxi = &i zx , i = 1, 2, 3, where &i are expressed through fj,x , fj,xk . The compatibility of the latter three equations can be expressed as a system of differential equations for &i , and therefore for fj , which are much more complicated than (15). It certainly deserves a further investigation, see discussion in Sect. 8. The necessary condition (15) will provide us with a finite list of candidates for the three-dimensional consistency. First of all, we rewrite the relation (15) in terms of the polynomial Q using both the linearity and the symmetry assumptions. Proposition 3. Relation (15) is equivalent to g(x, x1 ; α1 , α2 )g(x, x2 ; α2 , α3 )g(x, x3 ; α3 , α1 ) = −g(x, x1 ; α1 , α3 )g(x, x2 ; α2 , α1 )g(x, x3 ; α3 , α2 ),

(16)

where g(x, u; α, β) is a biquadratic polynomial in x, u defined by either of the formulas g(x, u; α, β) = QQyv − Qy Qv , g(x, v; β, α) = QQyu − Qy Qu ,

(17) (18)

where Q = Q(x, u, v, y; α, β). The polynomial g is symmetric: g(x, u; α, β) = g(u, x; α, β).

(19)

Proof. The equivalence of the definitions (17), (18) follows from the first symmetry property in (4) (α ↔ β, u ↔ v), while the second one (x ↔ u, y ↔ v) implies (19). To prove (16), let Q = p(x, u, v)y + q(x, u, v), so that y = f (x, u, v) = −q/p. Then fv /fu = (qv p − qpv )/(qu p − qpu ), and substituting p = Qy , q = Q − yQy , we obtain QQyv − Qy Qv fv = , fu QQyu − Qy Qu

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which yields: fk,xj fk,xi

=

g(αi , αj ; x, xi ) , g(αj , αi ; x, xj )

Now (16) follows from (15).

(i, j, k) = c.p.(1, 2, 3).

 

The biquadratic polynomials (17) and (18) are associated to the edges of the basic square. One can consider also the polynomial G(x, y; α, β) = QQuv − Qu Qv ,

(20)

associated to the diagonal. They have the following important property. Lemma 4. The discriminants of the polynomials g = g(x, u; α, β), g¯ = g(x, v; β, α), and G = G(x, y; α, β), considered as quadratic polynomials in u, v, and y, respectively, coincide: gu2 − 2gguu = g¯ v2 − 2g¯ g¯ vv = G2y − 2GGyy .

(21)

Proof. This follows solely from the fact that the function Q is linear in each argument. Indeed, calculate gu2 − 2gguu = ((QQyv − Qy Qv )u )2 − 2(QQyv − Qy Qv )(QQyv − Qy Qv )uu , taking into account that Quu = 0. The result reads: gu2 − 2gguu = Q2 Q2uvy + Q2u Q2vy + Q2v Q2uy + Q2y Q2uv + 4QQuv Quy Qvy −2QQuvy (Qu Qvy + Qv Quy + Qy Quv ) − 4Qu Qv Qy Quvy −2Qu Qv Quy Qvy − 2Qu Qy Quv Qvy − 2Qv Qy Quv Quy . It remains to notice that this expression is symmetric with respect to all indices.

 

In the formula (16) the variables are highly separated, and it can be effectively analyzed further on. In the next statement we demonstrate that this functional equation relating values of g with different arguments implies some properties for a single polynomial g. Proposition 5. The polynomial g(x, u; α, β) can be represented as g(x, u; α, β) = k(α, β)h(x, u; α),

(22)

where the factor k is antisymmetric, k(β, α) = −k(α, β),

(23)

and the coefficients of the polynomial h(x, u; α) depend on parameter α in such a way that its discriminant r(x) = h2u − 2hhuu does not depend on α.

(24)

Classification of Integrable Equations on Quad-Graphs

523

Proof. Relation (16) implies that the fraction g(x, x1 ; α1 , α2 )/g(x, x1 ; α1 , α3 ) does not depend on x1 , and due to the symmetry (19) it does not depend on x as well. Therefore g(x, x1 ; α1 , α2 ) κ(α1 , α2 ) = , g(x, x1 ; α1 , α3 ) κ(α1 , α3 ) where, because of (16), the function κ satisfies the equation κ(α1 , α2 )κ(α2 , α3 )κ(α3 , α1 ) = −κ(α2 , α1 )κ(α3 , α2 )κ(α1 , α3 ). This equation is equivalent to κ(β, α) = −

φ(α) κ(α, β), φ(β)

that is, the function k(α, β) = φ(α)κ(α, β) is antisymmetric. We have: g(x, u; α, β) g(x, u; α, γ ) = κ(α, β) κ(α, γ )

⇒

g(x, u; α, β) g(x, u; α, γ ) = , k(α, β) k(α, γ )

which implies (22). To prove the last statement of the proposition, we notice that, according to (22), (23), k(α, β)h(x, u; α) = g(x, u; α, β),

−k(α, β)h(x, v; β) = g(x, v; β, α).

Due to the identity (21), we find: h2u − 2hhuu = h¯ 2v − 2h¯ h¯ vv , and therefore r does not depend on α.

h = h(x, u; α),

h¯ = h(x, v; β),

 

Thus, the three-dimensional consistency with the tetrahedron property implies the following remarkable property of the function Q which will be called property (R): (R) the determinant g = QQvy − Qv Qy is factorizable as in (22), (23), and the discriminant r = h2u − 2hhuu of the corresponding quadratic polynomial h does not depend on parameters at all. It remains to classify all functions Q with this property. This will be done in the next section. A finite list of functions Q with the property (R) consists, therefore, of candidates for the three-dimensional consistency. The final check is straightforward, and shows that the property (R) is not only necessary but also almost sufficient for the threedimensional consistency with the tetrahedron property (the list of functions with the property (R) consists of a dozen items; only in two of them one finds functions violating the consistency). As a preliminary step, we consider more closely the coefficients of the polynomial Q. The symmetry property (4) easily implies that two cases are possible, with σ = 1 and σ = −1, respectively: Q = a0 xuvy + a1 (xuv + uvy + vyx + yxu) + a2 (xy + uv) + a¯ 2 (xu + vy) + a˜ 2 (xv + uy) + a3 (x + u + v + y) + a4 , Q = a1 (xuv + uvy − vyx − yxu) + a2 (xy − uv) + a3 (x − u − v + y),

(25) (26)

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where ai (β, α) = εai (α, β),

a˜ 2 (α, β) = ε a¯ 2 (β, α),

ε = ±1.

(27)

It is easy to prove that the case (26) is actually empty. Indeed, a light calculation shows that in this case g(x, u; α, β) = g(x, u; β, α) = −P (x; α, β)P (u; α, β), where P (α, β; x) = a1 x 2 − a2 x − a3 , so that the relation (16) becomes P (x1 ; α1 , α2 )P (x2 ; α2 , α3 )P (x3 ; α3 , α1 ) = −P (x1 ; α3 , α1 )P (x2 ; α1 , α2 )P (x3 ; α2 , α3 ). Equating to zero the coefficients at (x1 x2 x3 )2 , x1 x2 x3 , and the free term yields a1 = a2 = a3 = 0 – a contradiction. Turning to the case (25), we denote the coefficients of the polynomials h, r as follows: h(x, u; α) = b0 x 2 u2 + b1 xu(x + u) + b2 (x 2 + u2 ) + bˆ2 xu + b3 (x + u) + b4 , (28) r(x) = c0 x 4 + c1 x 3 + c2 x 2 + c3 x + c4 ,

(29)

where bi = bi (α). So, we consequently descended from the polynomial Q (4 variables, 7 coefficients depending on 2 parameters) to the polynomial h (2 variables, 6 coefficients depending on 1 parameter; also take into account the factor k), and then to the polynomial r (1 variable, 5 coefficients, no parameters). It remains to go the way back, i.e. to reconstruct k, h and Q from a given polynomial r (which does not depend on the parameter α). This will be done in the next section. 4. Classification: Synthesis First of all, we factor out the action of the simultaneous M¨obius transformations of the variables x. The action x → (ax + b)/(cx + d) transforms the polynomials h, r as follows:   ax + b au + b h(x, u; α) → (cx + d)2 (cu + d)2 h , ;α , cx + d cu + d   ax +b r(x) → (cx + d)4 r . cx + d Using such transformations one can bring the polynomial r into one of the following canonical forms, depending on the distribution of its roots: • • • • • •

r(x) = 0; r(x) = 1 (r has one quadruple zero); r(x) = x (r has one simple zero and one triple zero); r(x) = x 2 (r has two double zeroes); r(x) = x 2 − 1 (r has two simple zeroes and one double zero); r(x) = 4x 3 − g2 x − g3 , / = g23 − 27g32 = 0 (r has four simple zeroes).

Next, we find for these canonical polynomials r all possible polynomials h.

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Proposition 6. For a given polynomial r(x) of the fourth degree, in one of the canonical forms above, the symmetric biquadratic polynomials h(x, u) having r(x) as their discriminants, r(x) = h2u − 2hhuu , are exhausted by the following list: 1 r=0: h = (x − u)2 ; (q0) α h = (γ0 xu + γ1 (x + u) + γ2 )2 ; (h1) α 1 (x − u)2 − ; (q1) r=1: h= 2α 2 h = γ0 (x + u)2 + γ1 (x + u) + γ2 , γ12 − 4γ0 γ2 = 1; r=x: r = x2 : r = x 2 − δ2 : r = 4x 3 − g2 x − g3 :

(h2) α3

1 α (x − u)2 − (x + u) + ; 4α 2 4 h = γ0 x 2 u2 + γ1 xu + γ2 , γ12 − 4γ0 γ2 = 1; h=

α (x 2 + u2 ) − xu + δ 2 1 − α2 1 − α2 2 1  h= √ xu + α(x + u) + g2 /4 r(α)  − (x + u + α)(4αxu − g3 ) .

h=

Proof. We have to solve the system of the form b12 − 4b0 b2 2b1 (bˆ2 − 2b2 ) − 4b0 b3 bˆ22 − 4b22 − 2b1 b3 − 4b0 b4 2b3 (bˆ2 − 2b2 ) − 4b1 b4 b32 − 4b2 b4

1 + α2

1 − α2 4α

(q2) (h3) ;

(q3)

(q4)

= c0 , = c1 , = c2 , = c3 , = c4 ,

where bk are the coefficients of h(x, u) and ck are the coefficients of r(x) (see (28), (29)). This is done by a straightforward analysis. For example, consider in detail the case r = 4x 3 − g2 x − g3 . We have: c0 = 0, c1 = 4, hence b0 = 0. Set b0 = ρ −1 , b1 = −2αρ −1 , then b2 = α 2 ρ −1 . Next, use the second and the third equations to eliminate b3 and b4 , then the last two equations give an expression for bˆ2 and the constraint ρ 2 = r(α).   Remarks . 1) Notice that for all six canonical forms of r we have a one-parameter family of h’s, denoted in the list of Proposition 6 by (q0)–(q4) (the one-parameter family for r = x 2 is not explicitly written down since it coincides with (q3) at δ = 0). It will turn out that the polynomials Q reconstructed from these h’s belong to the list Q (so that h in (qk) corresponds to Q in (Qk); h from (q0) corresponds to (Q1)δ=0 ). For the polynomials r = 0, 1, x 2 we have additional two- or three-parameter families of h’s denoted in the list of Proposition 6 by (h1)–(h3). They will lead to the correspondent Q’s in (H1)–(H3), as √ well as to (A1), (A2). 2 2) The expression r(α) in Eq. (q4) clearly shows √ that the elliptic curve µ = r(λ) comes into play at this point. One can uniformize r(α) = ℘ (A), α = ℘ (A), where A is a point of the elliptic curve, and ℘ is the Weierstrass elliptic function.Actually, the polynomial h(x, u; α) is well–known in the theory of elliptic functions and represents the addition theorem for the ℘–function. Namely, the equality h(℘ (X), ℘ (U ); ℘ (A)) = 0 is

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equivalent to A = ±X±U modulo the lattice of periods of the function ℘. To summarize: symmetric biquadratic polynomials h(x, u) with the discriminant h2u −2hhuu = r(x) are parametrized (in the non-degenerate case) by a point of the elliptic curve µ2 = r(λ). This is the origin of the elliptic curve in the parametrization of Eq. (q4). As a consequence, the spectral parameter in the discrete zero curvature representation of the latter equation also lives on the elliptic curve. This may be considered also as the ultimate reason for the spectral parameter of the Krichever–Novikov equation to live on an elliptic curve ([20], see also Sect. 7). It remains to reconstruct polynomials Q for all h’s from Proposition 6. Proof of Theorem 1. For the polynomial (25) we have: g(x, u; α, β) = (a¯ 2 a0 − a12 )x 2 u2 + (a1 (a¯ 2 − a˜ 2 ) + a0 a3 − a1 a2 )xu(x + u) + (a1 a3 − a2 a˜ 2 )(x 2 + u2 ) + (a¯ 22 − a˜ 22 + a0 a4 − a22 )xu + (a3 (a¯ 2 − a˜ 2 ) + a1 a4 − a2 a3 )(x + u) + a¯ 2 a4 − a32 , and an analogous expression for g(x, u; β, α) is obtained by the replacement a¯ 2 ↔ a˜ 2 . Using (22) and denoting bi = bi (α), bi = bi (β), k = k(α, β), we come to the following system for the unknown quantities ak : a¯ 2 a0 − a12 a1 (a¯ 2 − a˜ 2 ) + a0 a3 − a1 a2 a1 a3 − a2 a˜ 2 a¯ 22 − a˜ 22 + a0 a4 − a22 a3 (a¯ 2 − a˜ 2 ) + a1 a4 − a2 a3 a¯ 2 a4 − a32

= kb0 , = kb1 , = kb2 = k bˆ2 , = kb3 , = kb4 ,

a˜ 2 a0 − a12 a1 (a˜ 2 − a¯ 2 ) + a0 a3 − a1 a2 a1 a3 − a2 a¯ 2 a˜ 22 − a¯ 22 + a0 a4 − a22 a3 (a˜ 2 − a¯ 2 ) + a1 a4 − a2 a3 a˜ 2 a4 − a32

= −kb0 , = −kb1 , = −kb2 , = −k bˆ2 , = −kb3 , = −kb4 .

Of course, the quantity k is also still unknown here. Since we are looking for the function Q up to arbitrary factor, it is convenient to denote ai
2 = a¯ 2 − 1 = a˜ 2 + 1 , K = k (30) , A a a 2 a 2 a2 (it is easy to see that a ≡  0 since otherwise h ≡ 0). These functions are skew-symmetric: a = a¯ 2 − a˜ 2 ,

Ai =

Ai (β, α) = −Ai (α, β),


2 (β, α) = −A
2 (α, β), A

K(β, α) = −K(α, β),

and the above system can be rewritten, after some elementary transformations, as follows: K[(bˆ2 + bˆ2 )(b0 + b0 ) − (b1 + b1 )2 ] = 2(b0 − b0 ), K[(b0 + b0 )(b3 + b3 ) − (b1 + b1 )(b2 + b2 )] = b1 − b1 , K[(b1 + b1 )(b3 + b3 ) − (b2 + b2 )(bˆ2 + bˆ2 )] = 2(b2 − b2 ), 1 K[(b0 + b0 )(b4 + b4 ) − (b2 + b2 )2 ] = (bˆ2 − bˆ2 ), 2 K[(b1 + b1 )(b4 + b4 ) − (b2 + b2 )(b3 + b3 )] = b3 − b3 , K[(bˆ2 + bˆ )(b4 + b ) − (b3 + b )2 ] = 2(b4 − b ), 2

4

3

4

A0 = K(b0 + b0 ), 2A1 = K(b1 + b1 ), A2 = K(b2 + b2 ),
2 = K(bˆ2 + bˆ2 ), 2A3 = K(b3 + b3 ), A4 = K(b4 + b4 ). 4A

Classification of Integrable Equations on Quad-Graphs

527

The first six lines here form a system of functional equations for bi and K (call it the (K, b)–system), while the last two lines split away, and should be considered just as definitions of Ai . So, to any solution of the (K, b)–system there corresponds a function Q whose coefficients are given by the last two lines of the system above, and the formulas
2 + 1/2, A¯ 2 = A

2 = A
2 − 1/2, A

which follow from (30). By construction, this function has the property (R) and is, therefore, a candidate for the three-dimensional consistency with the tetrahedron property. Consider first the cases (q0), (q1), (q2), (q3), (q4), i.e. when we have a one-parameter family of polynomials h. A straightforward, although tedious, check proves that in these cases all six equations of the (K, b)–system lead to one and the same function K, provided the functions bi = bi (α) are defined as in (q0)–(q4). Calculating the corresponding coefficients Ai , we come to the functions Q given in Theorem 1 by the formulas (Q1)δ=0 , (Q1)δ=1 , (Q2), (Q3), and (Q4), respectively. A further straightforward check convinces us that all these functions indeed pass the three-dimensional consistency test with the tetrahedron propery. It remains to consider the cases (h1), (h2), (h3). A thorough analysis of the (K, b)– system shows that in these cases we have the following solutions. In the case (h1): • either h does not depend on parameters at all, and then it has to be of the form  2 h = (ε0 x + ε1 )(ε0 u + ε1 ) , and K is arbitrary; performing a suitable M¨obius transformation, we can achieve that h ≡ 1; the outcome of this subcase is the following function Q which is a candidate for the three-dimensional consistency: Q = (x − y)(u − v) + k(α, β),

(H1)

where k is an arbitrary skew-symmetric function, k(α, β) = −k(β, α); • or h = (1/α)(ε0 xu + ε1 (x + u) + ε2 )2 with arbitrary constants εi . In this case we can use M¨obius transformations to achieve h = (1/α)(x + u)2 , and the correspondent function Q coincides with (A1)δ=0 from Theorem 1. This function passes the three-dimensional consistency test with the tetrahedron property. In the case (h2): • either h = x + u + α, and the correspondent function Q coincides with (H2) from Theorem 1; • or h = (1/2α)(x + u)2 − (α/2), and the correspondent function (Q) is given in (A1)δ=1 of Theorem 1. Both functions are three-dimensionally consistent with the tetrahedron property. Finally, in the case (h3): • either h does not depend on parameters at all, then it has to be equal to h = xu, and K is arbitrary; we have in this subcase the following function Q which is a candidate for the three-dimensional consistency: Q=

1+k 1−k (xu + vy) − (xv + uy), 2 2

0) (H3

where k is an arbitrary skew-symmetric function, k(α, β) = −k(β, α); • or h = xu + α (possibly, upon application of the inversion x → 1/x, u → 1/u); the correspondent function Q is (H3)δ=1 of Theorem 1;

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α 1 + α2 (x 2 u2 +1)− xu; the correspondent function Q is given 2 1−α 1 − α2 in (A2) of Theorem 1.

• or, finally, h =

In the two last subcases the three-dimensional consistency condition is fulfilled with the tetrahedron property. To finish the proof of Theorem 1, we have to consider the functions Q given by the

and (H3

0 ). They depend on an arbitrary skew–symmetric function k, and formulas (H1) have the property (R) for any choice of the latter. As it turns out, these are the only situations when the property (R) does not automatically imply the three-dimensional consistency.

gives a map which is three-dimenA direct calculation shows that the function (H1) sionally consistent, with x123 =

k(α1 , α2 )x1 x2 + k(α2 , α3 )x2 x3 + k(α3 , α1 )x3 x1 , k(α3 , α2 )x1 + k(α1 , α3 )x2 + k(α2 , α1 )x3

(31)

if and only if k(α1 , α2 ) + k(α2 , α3 ) + k(α3 , α1 ) = 0.

(32)

To solve this functional equation, differentiate it with respect to α1 and α2 : kα1 α2 (α1 , α2 ) = 0, which together with the skew–symmetry of k yields k(α1 , α2 ) = f (α2 )−f (α1 ). A point transformation of the parameter f (α) → α allows us to take simply k(α, β) = β − α. Thus, we arrive at the case (H1) of Theorem 1.

0 ). Denote, for brevity, Finally, consider the formula (H3 4(α, β) =

1 + k(α, β) , 1 − k(α, β)

so that

4(β, α) = 1/4(α, β).

We will write also 4ij for 4(αi , αj ). A straightforward inspection shows that the function

0 ) gives a three-dimensionally consistent map with (H3 x123 =

(421 − 412 )x1 x2 + (432 − 423 )x2 x3 + (413 − 431 )x3 x1 , (423 − 432 )x1 + (431 − 413 )x2 + (412 − 421 )x3

(33)

if and only if 4(α1 , α2 )4(α2 , α3 )4(α3 , α1 ) = 1.

(34)

Just as above, up to a point transformation of the parameter, the solution of this functional equation is given by 4(α, β) = α/β, which leads to the equation (H3)δ=0 of Theorem 1. The proof of Theorem 1 is now complete.  

Classification of Integrable Equations on Quad-Graphs

529

5. Three-Leg Forms Theorem 1 classifies the three-dimensionally consistent equations under the tetrahedron condition. In Sect. 1 we have seen that the latter condition is necessary for the existence of a three-leg form ψ(x, u; α) − ψ(x, v; β) = φ(x, y; α, β)

(35)

of Eq. (1). Recall that the formula (35) implies the validity of equations on stars (11), or equations of the discrete Toda type, for the fields x at the vertices of the “black” sublattice; the same holds for the “white” sublattice. We prove now that a three–leg form indeed exists for all equations listed in Theorem 1. After that, in Sect. 6 we demonstrate some further applications of the three–leg forms, establishing the variational and symplectic structures for Eqs. (1) and (11). The theorem below provides three-leg forms for all equations of the lists Q and H (the results for the list A follow from these ones). As it turns out, in almost all cases it is more convenient to write the three-leg equation (35) in the multiplicative form 5(x, u; α)/5(x, v; β) = &(x, y; α, β).

(36)

For the list Q the functions 5 and & corresponding to the “short” and to the “long” legs, respectively, essentially coincide. One has in these cases: 5(x, u; α) = F (X, U ; A),

&(x, y; α, β) = F (X, Y ; A − B),

(37)

where some suitable point transformations of the field variables and of the parameters are introduced: x = f (X), u = f (U ), y = f (Y ), and α = ρ(A), β = ρ(B). Theorem 7. The three–leg forms exist for all equations from Theorem 1. For the lists Q, H they are listed below. (Q1)δ=0: An additive three–leg form with φ(x, y; α, β) = ψ(x, y; α − β), ψ(x, u; α) =

α . x−u

(38)

(Q1)δ=1: A multiplicative three-leg form with &(x, y; α, β) = 5(x, y; α − β), 5(x, u; α) =

x−u+α . x−u−α

(39)

(Q2): A multiplicative three-leg form with &(x, y; α, β) = 5(x, y; α − β), 5(x, u; α) =

(X + U + α)(X − U + α) , (X + U − α)(X − U − α)

(40)

where x = X 2 , u = U 2 . (Q3)δ=0: A multiplicative three-leg form with &(x, y; α, β) = 5(x, y; α/β), 5(x, u; α) =

αx − u . x − αu

(41)

Under the point transformations x = exp(2X), α = exp(2A), etc., there holds (37) with 5(x, u; α) = F (X, U ; A) =

sinh(X − U + A) . sinh(X − U − A)

(42)

530

V.E. Adler, A.I. Bobenko, Yu.B. Suris

(Q3)δ=1: A multiplicative three-leg form with (37), 5(x, u; α) = F (X, U ; A) =

sinh(X + U + A) sinh(X − U + A) , sinh(X + U − A) sinh(X − U − A)

(43)

where x = cosh(2X), α = exp(2A), etc. (Q4): A multiplicative three-leg form with (37), 5(x, u; α) = F (X, U ; A) =

σ (X + U + A)σ (X − U + A) , σ (X + U − A)σ (X − U − A)

(44)

where x = ℘ (X), α = ℘ (A), etc. (H1): An additive three-leg form ψ(x, u; α) = x + u,

φ(x, y; α, β) =

α−β . x−y

(45)

(H2): A multiplicative three–leg form 5(x, u; α) = x + u + α,

&(x, y; α, β) =

x−y+α−β . x−y−α+β

(46)

(H3): A multiplicative three–leg form 5(x, u; α) = xu + δα = exp(2X + 2U ) + δα, βx − αy sinh(X − U − A + B) &(x, y; α, β) = = , αx − βy sinh(X − U + A − B)

(47) (48)

where x = exp(2X), α = exp(2A), etc. Proof. We start with the list H, for which the situation is somewhat simpler. Finding the three-leg forms of Eqs. (H1) and (H3)δ=0 is almost immediate: these equations are equivalent to u−v =

α−β x−y

and

βx − αy u = , v αx − βy

(49)

respectively. For other equations of the list H one uses the following simple formula which will also be quoted on several occasions later on. Lemma 8. The relation Q(x, u, v, y; α, β) = 0 yields h(x, u; α) Qv . =− h(x, v; β) Qu

(50)

Proof. −

 QQyv − Qy Qv h(x, u; α) Qv  g(x, u; α, β) = . = = g(x, v; β, α) QQyu − Qy Qu Qu Q=0 h(x, v; β)

 

Classification of Integrable Equations on Quad-Graphs

531

This lemma immediately yields the three-leg forms for the cases when Quv = 0: then the right–hand side of (50) does not depend on u, v and can therefore be taken as &(x, y; α, β), while 5(x, u; α) := h(x, u; α). This covers the cases (H2) and (H3). Observe also that under the condition Quv = 0 we have G(x, y; α, β) = QQuv −Qu Qv = −Qu Qv . Thus, in these cases the function 5 associated to the “short” legs is nothing but the polynomial h, while the function & associated to the “long” legs, is just a ratio of two linear factors of the quadratic polynomial G. The situation with the list Q is a bit more tricky. The inspection of formulas (q1), (q2), (q3), (q4) shows that in these cases h(x, u; α) is a quadratic polynomial in u, and, similarly, the polynomial G(x, y; α, β) is a quadratic polynomial in y. We try a linear–fractional ansatz for 5, &: 5(x, u; α) =

p+ (x, u; α) , p− (x, u; α)

&(x, y; α, β) =

s+ (x, y; α, β) , s− (x, y; α, β)

(51)

where the functions p± (x, u, α) are linear in u, and the functions s± (x, y; α, β) are linear in y, and p+ (x, u; α)p− (x, u; α) = h(x, u; α), s+ (x, y; α, β)s− (x, y; α, β) = G(x, y; α, β). According to (21), the √ coefficients of polynomials p± (·, u; ·) and s± (·, y; ·,·) are rational functions of x and r(x). This justifies “uniformizing” changes of variables, namely x = X2 in the case (Q2), when r(x) = x, x = cosh(2X) in the case (Q3) with δ = 1, when r(x) = x 2 − 1, x = ℘ (X) in the case (Q4), when r(x) = 4x 3 − g2 x − g3 . The ansatz (51) turns out to work. For the cases (Q1)–(Q3) identify the functions p± from (51) with the numerators and denominators of the expressions listed in (39)–(43) (for the case (Q3)δ=0 take the fraction (41), i.e. the expression through x, u). In the case (Q4) the functions p± linear in u are obtained by dividing the numerators and denominators of the fractions in (44) by σ 2 (X)σ 2 (U ). Make similar identifications for the functions s± from (51). So, under the point transformations used in the formulation of Theorem 7 we have: p ± (x, u; α) = P (X, U ; ±A),

s± (x, y; α, β) = P (X, U ; ±A ∓ B),

where (Q1)δ=1 : (Q2) : (Q3)δ=0 : (Q3)δ=1 :

P (x, u; α) = x − u + α, P (X, U ; α) = (X + U + α)(X − U + α), P (X, U ; A) = sinh(X − U + A), P (X, U ; A) = sinh(X + U + A) sinh(X − U + A), σ (X + U + A)σ (X − U + A) (Q4) : P (X, U ; A) = . σ 2 (X)σ 2 (U )

A straightforward computation shows that p+ (x, u; α)p− (x, v; β)s− (x, y; α, β) − p− (x, u; α)p+ (x, v; β)s+ (x, y; α, β) = ρ(x; α, β)Q(x, u, v, y; α, β), (52)

532

V.E. Adler, A.I. Bobenko, Yu.B. Suris

with some factor ρ depending only on x. (Obviously, the left–hand side of the latter equation is linear in u, v, y.) Concretely, we find: ρ = 2 in the case (Q1)δ=1 ; ρ = 4X in the case (Q2); ρ = x in the case (Q3)δ=0 ; ρ = 2αβ sinh(2X) in the case (Q3)δ=1 ; finally, ρ = σ 4 (A)σ 4 (B)

σ (B − A) σ (2X) · σ (B + A) σ 4 (X)

in the case (Q4). So, in all these cases Q = 0 is equivalent to vanishing of the left–hand side of (52), which, in turn, is equivalent to the multiplicative three–leg formula (36) with the ansatz (51). Finally, we comment on the origin of the additive three–leg form of (Q1)δ=0 . Rescaling in (39) the parameters as α → δα, we come to the three–leg form of Eq. (Q1) with a general δ = 0. Sending δ → 0, we find: 5 = 1 + 2δψ + O(δ 2 ), & = 1 + 2δφ + O(δ 2 ), with the functions ψ, φ from (38), and thus we arrive at the additive formula for the case δ = 0.   Remark. The notion of the three–leg equation was formulated in [8]. The three–leg forms for Eqs. (Q1), (Q3)δ=0 , (H1), (H3)δ=0 were also found there. The results for (Q2), (Q3)δ=1 , (Q4), (H2), (H3)δ=1 are given here for the first time. 6. Lagrangian Structures Recall that the three–leg form of Eq. (1) imply that the discrete Toda type equation (11) holds on the black and on the white sublattices. (Of course, for multiplicative equations (36) we set ψ(x, u; α) = log 5(x, u; α), φ(x, y; α, β) = log &(x, y; α, β).) We show now that all these Toda systems may be given a variational (Lagrangian) interpretation. Notice that only the functions φ(x, y; α, β) related to the “long” legs enter the Toda equations, and that the Toda systems for the cases (H1)–(H3) are the same as in the cases (Q1), (Q3)δ=0 . Consider the point changes of variables x = f (X) listed in Theorem 7; in the cases (Q1), (H1), (H2), when no such substitution is listed, set just x = X. For the sake of notational simplicity, we write in the present section ψ(x, u; α) for ψ(f (X), f (U ); α), etc. Lemma 9. For all equations from Theorem 1 there exist symmetric functions L(X, U ; α) = L(U, X; α) and :(X, Y ; α, β) = :(Y, X; α, β) such that ∂ L(X, U ; α), ∂X ∂ φ(x, y; α, β) = φ(f (X), f (Y ); α, β) = :(X, Y ; α, β). ∂X ψ(x, u; α) = ψ(f (X), f (U ); α) =

(53) (54)

Proof. It is sufficient to notice that the functions (∂/∂U )ψ(x, u; α) are symmetric with respect to the permutation X ↔ U , and similarly for φ.   This observation has the following immediate corollary. Proposition 10. Let
be the “black” subgraph, and let E(
) be the set of its (non-oriented) edges. Let the pairs of labels (α, β) be assigned to the edges from E(
) according to Fig. 5, so that, e.g., the pair (α1 , α2 ) corresponds to the edge (x, x12 ). Then for all

Classification of Integrable Equations on Quad-Graphs u1 β1

v2 α1

y1 α1

533

β1

β2 x

v1

α2 y2

α2

β2 u2

Fig. 6. Two elementary quadrilaterals of the square lattice

equations from Theorem 1 the discrete Toda type equations (11) are the Euler–Lagrange equations for the action functional  :(X, Y ; α, β). (55) S= (X,Y )∈E(
)

This result could be anticipated, since it is quite natural to expect from a system consisting of equations on stars to have a variational origin. Our next result is, on the contrary, somewhat unexpected, since it gives a sort of a variational interpretation for the original system consisting of the equations on quadrilaterals (1). This is possible not on arbitrary quad–graphs but only on special regular lattices. We restrict ourselves here to the case of the standard square lattice. For the cases (H1), (H3)δ=0 (the discrete KdV equation and the Hirota equation) our results coincide with those found in [10], for all other equations from Theorem 1 they seem to be new. For the sake of convenience orient the square lattice as in Fig. 6. Denote by E1 (E2 ) the subset of edges running from south–east to north–west (resp. from south–west to north–east). Add to the edges of the original quad–graph the set E3 of the horizontal diagonals of all elementary quadrilaterals. Let the labels α be assigned to the edges from E1 , and let the labels β be assigned to the edges from E2 . It is natural to assign to the diagonals from E3 the pairs (α, β) from the sides of the correspondent quadrilateral. Proposition 11. Solutions of all equations from Theorem 1 are critical for the following action functional:    L(X, U ; α) − L(X, V ; β) − :(X, Y ; α, β). (56) S= (X,U )∈E1

(X,V )∈E2

(X,Y )∈E3

Proof. For any vertex x, the correspondent Euler–Lagrange equation relates the vertices of two elementary quadrilaterals, as in Fig. 6. Due to Lemma 9, this equation reads: ψ(x, u1 ; α1 ) − ψ(x, v1 ; β1 ) − φ(x, y1 ; α1 , β1 )+ ψ(x, u2 ; α2 ) − ψ(x, v2 ; β2 ) − φ(x, y2 ; α2 , β2 ) = 0. This holds, since (35) is fulfilled on both elementary quadrilaterals.

 

The variational interpretation allows one to find invariant symplectic structures for reasonably posed Cauchy problems for Eqs. (1). As is well–known, one way to set the Cauchy problem is to prescribe the values x on the zigzag line like in Fig. 7 and to impose periodicity in the horizontal direction (so that one is dealing with the square lattice on a cylinder) [10, 13]. Then Eq. (1) defines the evolution in the vertical direction, i.e. the map {xi }i∈Z/2N Z → { xi }i∈Z/2N Z .

534

V.E. Adler, A.I. Bobenko, Yu.B. Suris x0

x2

x2N

x1 x0

x2N ≡ x0

x2 x1

x2N−1

Fig. 7. The Cauchy problem on a zigzag

Proposition 12. Let the edges (xi , xi+1 ) carry the labels αi , and let the edges ( xi , xi+1 ) carry the labels αi , so that α2k = α2k−2 , and α2k−1 = α2k+1 . Denote s(X, U ; α) =

∂ ∂2 ψ(x, u; α) = L(X, U ; α), ∂U ∂X∂U

(57)

so that s(X, U ; α) = s(U, X; α). Then the following relation holds for the map {xi } → { xi }:   i , X i+1 ; i ∧ d X i+1 . s(Xi , Xi+1 ; αi )dXi ∧ dXi+1 = s(X αi )d X i∈Z/2N Z

i∈Z/2N Z

(58) Proof. We use the argument methodologically close to [23]. Consider the action functional S defined by the same formula as (56) but with the summations restricted to the edges depicted in Fig. 7, and restricted to such fields X which satisfy Eq. (1) on each elementary quadrilateral. Consider the differential dS =

 all vertices x

∂S dX. ∂X

Due to Proposition 11, the expression ∂S/∂X vanishes for all vertices where six edges x2k . So, we find: meet, i.e. for all vertices except x2k+1 and dS =

 k∈Z/N Z

−

∂L(X2k , X2k+1 ; α2k ) ∂L(X2k+1 , X2k+2 ; α2k+1 ) dX2k+1 − dX2k+1 ∂X2k+1 ∂X2k+1

2k−1 , X 2k ; 2k+1 ; 2k , X ∂L(X α2k−1 ) α2k ) ∂L(X d X2k + d X2k . ∂ X2k ∂ X2k

Differentiating this 1–form and taking into account d 2 S = 0 yields (58).

 

Remarks . 1) Notice that only the functions ψ related to the “short” legs are present in formula (58). i }. 2) One is tempted to interpret formula (58) as symplecticity of the map {Xi } → {X However, the 2–form (58) is degenerate. One finds a genuine symplectic form, if one considers a quasi-periodic initial value problem instead of a periodic one, and extends the phase space by the correspondent monodromies (cf. [13, 21] for the case (H3)δ=0 (the Hirota equation)). Actually, the formula (58) means the invariance of a degenerate 2–form only when αi = αi . This is easy to achieve by setting all αi = α, or, more generally, all α2k = α and all α2k+1 = β.

Classification of Integrable Equations on Quad-Graphs v = xk+1

α

β x = xk

535 y = uk+1 β

α

u = uk

Fig. 8. To the construction of B¨acklund transformations

We close this section with a local form of Proposition 12, also based on the three–leg form of Eqs. (1). The particular case of the Hirota equation was given in [21]. Proposition 13. Associate the two–form ω(e) = s(X, U ; α)dX ∧ dU to every (oriented) edge e = (x, u) of the quad–graph. Then Eq. (1) with the three–leg form (35) implies that the sum of ω’s along the boundary of any elementary quadrilateral, and therefore along any cycle homotopic to zero, vanishes. Proof. Differentiating (35) and wedging the result with dX, we have: ∂ ∂ ∂ ψ(x, u; α)dX ∧ dU + ψ(x, v; β)dV ∧ dX = φ(x, y; α)dX ∧ dY. ∂U ∂V ∂Y On the other hand, starting with the three–leg equation centered at y (or, in other words, flipping x ↔ y, u ↔ v), we arrive at ∂ ∂ ∂ ψ(y, v; α)dY ∧ dV + ψ(y, u; β)dU ∧ dY = φ(y, x; α)dY ∧ dX. ∂V ∂U ∂X Adding these two equations, and taking into account Lemma 9 and the notation (57), we come to the statement of the proposition.   The statement of Proposition 12 is a particular case of Proposition 13, since the boundary of the domain in Fig. 7 (a cylindrical strip) is homotopic to zero. 7. Relation to B¨acklund Transformations In this section we interpret the integrable quad-graph equations of Theorem 1 as nonlinear superposition principles (NSP) of B¨acklund transformations for the KdV–type equations. First of all, we show how B¨acklund transformations themselves can be derived from our equations. Towards this aim, consider Eq. (1) on one vertical strip of the standard square lattice: S = {0, 1} × Z. The fields on S0 = {0} × Z will be denoted by xk , while the fields on S1 = {1} × Z will be denoted by uk . So, a single square of the strip S looks as in Fig. 8. Suppose now that xk = x(kB), k ∈ Z, where x(ξ ) is a smooth function, and similarly for uk . In particular, this means that one has to set in Eq. (1) v = x + Bxξ + O(B 2 ) and y = u + Buξ + O(B 2 ). If, in addition, the parameter β is chosen properly, then Eq. (1) approximates in the limit B → 0 some differential equation which relates the functions x(ξ ) and u(ξ ). For the equations of the list Q the result is the most straightforward.

536

V.E. Adler, A.I. Bobenko, Yu.B. Suris

Proposition 14. Set in Eq. (Q1) β = B 2 /2, in Eq. (Q2) β = B 2 /4, in Eq. (Q3) β = 1 − B 2 /2, and in Eq. (Q4) β = ℘ (B 2 ). Then in the limit B → 0 just described these equations tend to xξ uξ = h(x, u; α),

(59)

with the correspondent polynomials h listed in the formulas (q1), (q2), (q3), (q4) of Proposition 6. Proof. The statement is almost obvious in the cases (Q1), (Q2). To see that it holds for Eq. (Q3), the latter can be first rewritten as β(α 2 − 1)(x − v)(u − y) = α(β 2 − 1)(xv + uy) + (1 − β)(α 2 + β)(xy + uv) +(δ 2 /4αβ)(α 2 − β 2 )(α 2 − 1)(β 2 − 1). If β = 1 − B 2 /2, then the above equation approximates (α 2 − 1)xξ uξ = −α(x 2 + u2 ) + (α 2 + 1)xu − (δ 2 /4α)(α 2 − 1)2 . Finally, in the most intricate case (Q4) one starts by rewriting the equation as a¯ 2 − a2 a1 (x − v)(u − y) = xuvy + (xuv + uvy + vyx + yxu) 2a0 a0 a¯ 2 + a2 + (x + v)(u + y) 2a0 a˜ 2 a3 a4 + (xv + uy) + (x + u + v + y) + . a0 a0 a0 From expressions for the coefficients ai given in Theorem 1, it is easy to see that if β = ℘ (B 2 ) ∼ B −4 , so that b = ℘ (B 2 ) ∼ −2B −6 , then the left–hand side of the above equation tends to axξ uξ , while the right–hand side tends to g2 x 2 u2 − 2αxu(x + u) − 2α 2 − xu + α 2 (x 2 + u2 ) 2

g2 g2 + g3 + α (x + u) + 2 + g3 α = ah(x, u; α). 2 16 This proves the proposition.

 

Equation (59), read as a Riccati equation for u with the coefficients dependent on x, describes a transformation x → u, which turns out [2] to be a B¨acklund transformation for the Krichever–Novikov equation [20]: xt = xξ ξ ξ −

3 (x 2 − r(x)), 2xξ ξ ξ

(60)

with the polynomial r(x) being the discriminant of h(x, u; α). In other words, if x is a solution of (60), and u is related to x by (59), then u is also a solution of (60). It should be noticed that, in turn, the partial differential equation (60) may be derived from (59), either through a sort of continuous limit, or as a higher symmetry. In any way, it would be fair to say that the whole theory of the Eq. (60) and its B¨acklund transformations (59) is contained in the correspondent quad-graph equation (1) from the list Q. To complete

Classification of Integrable Equations on Quad-Graphs

537

the picture, we demonstrate that all Eqs. (1) listed in Theorem 1, in turn, can be interpreted as nonlinear superposition principles for the B¨acklund transformations. To this end consider the system of four differential equations of the type (59) corresponding to four sides of the quadrilateral on Fig. 1: xξ uξ = h(x, u; α), xξ vξ = h(x, v; β),

uξ yξ = h(u, y; β), vξ yξ = h(v, y; α).

(61) (62)

We will consider it for functions h corresponding to all cases listed in Theorem 1, except for those two when h actually does not depend on parameters, namely (H1) with h(x, u) = 1, and (H3)δ=0 with h(x, u) = xu. In all other cases the system (61), (62) makes perfect sense and its consistency is quite nontrivial. Proposition 15. The equation Q(x, u, v, y; α, β) = 0 is a sufficient condition for the consistency of the system of differential equations (61), (62). Moreover, it is compatible with this system, i.e. Qξ |Q=0 = 0.

(63)

Proof. The consistency condition of the differential equations (61), (62) reads: Q := h(x, u; α)h(v, y; α) − h(x, v; β)h(u, y; β) = 0.

(64)

The left–hand side Q of this equation is a polynomial of degree 1 in each variable for the equations of the list H, and of degree 2 in each variable for the equations of the list Q. In the former case it is directly seen that Eq. (64) exactly coincides with Q(x, u, v, y; α, β) = 0. In the latter case it can be shown that the polynomial Q divides the left–hand side of (64). More precisely, it is verified that, up to a constant factor, Q = QP , where in the cases (Q1), (Q2) one has P = Q|β→−β , in the case (Q3) one has P = Q|β→1/β , and in the case (Q4) one has P = Q|(β,b)→(β,−b) . This proves the first statement of the proposition. The second one reads (Qx xξ + Qu uξ + Qv vξ + Qy yξ )|Q=0 = 0. We will prove that actually both terms Qx xξ +Qy yξ and Qu uξ +Qv vξ vanish separately. For example, (Qu uξ + Qv vξ )|Q=0 =

 1  Qu h(x, u; α) + Qv h(x, v; β) Q=0 = 0. xξ

The last step follows from Lemma 8.

 

In the two cases when the system (61), (62) becomes trivial, there still exist B¨acklund transformations of a different kind, such that the equation Q = 0 serves as their NSP. Namely, in the cases (H1), (H3)δ=0 the following equations come to replace (59): xξ + uξ = (x − u)2 + α,

(65)

xξ uξ α x u = + + , (66) u x 2 u x respectively. The second of these equations is probably better known in the coordinates x = exp(X), u = exp(U ): Xξ + Uξ = α cosh(X − U ).

(67)

538

V.E. Adler, A.I. Bobenko, Yu.B. Suris a23

b13 c1

α a2 β b1

b3

a3

c

c12 b β

c2 b1

a α

a2

b

a

Fig. 9. An elementary quadrilateral; both Fig. 10. Three–dimensional consistency; fields assigned to fields and labels are assigned to edges edges

Equation (65) defines a B¨acklund transformation for the potential KdV equation xt = xξ ξ ξ − 6xξ2 .

(68)

Similarly, Eq. (67) defines a B¨acklund transformation for the potential MKdV equation Xt = Xξ ξ ξ − 2Xξ3 ,

(69)

or, alternatively, for the sinh–Gordon equation which belongs to the same hierarchy. 8. Conclusions and Perspectives Three–dimensional consistency as integrability criterium; Yang–Baxter maps. One of the traditional but rather ad hoc definitions of the integrability of two–dimensional systems is based on the notion of the zero–curvature representation. For a system on a quad–graph consisting of Eqs. (1) with fields associated to the vertices of the elementary quadrilateral on Fig. 1, the zero curvature representation is usually encoded in the formula like L(y, u, β; λ)L(u, x, α; λ) = L(y, v, α; λ)L(v, x, β; λ), where λ is the spectral parameter, so that the matrices L take values in some loop group. In [8] and independently in [26] it was demonstrated how to derive the zero curvature representation from the three–dimensional consistency. It should be mentioned, however, that to assign fields to the vertices is not the only possibility. Another large class of two–dimensional systems on quad–graphs build those with the fields assigned to the edges, see Fig. 9. In this situation it is natural to assume that each elementary quadrilateral carries a map R : X 2 → X 2 , where X is the space where the fields a, b take values, so that (a2 , b1 ) = R(a, b; α, β). The question on the three–dimensional consistency of such maps is also legitimate and, moreover, began to be studied recently. The corresponding property can be encoded in the formula R23 ◦ R13 ◦ R12 = R12 ◦ R13 ◦ R23 ,

(70)

where each Rij : X 3 → X 3 acts as the map R on the factors i, j of the cartesian product X 3 and acts identically on the third factor. This equation should be understood as follows. The fields a, b are supposed to be attached to the edges parallel to the 1st and

Classification of Integrable Equations on Quad-Graphs

539

the 2nd coordinate axes, respectively. Additionally, consider the fields c attached to the edges parallel to the 3rd coordinate axis. Then the left–hand side of (70) corresponds to the chain of maps along the three rear faces of the cube on Fig. 10: (a, b) → (a2 , b1 ),

(a2 , c) → (a23 , c1 ),

(b1 , c1 ) → (b13 , c12 ),

while its right–hand side corresponds to the chain of the maps along the three front faces of the cube: (b, c) → (b3 , c2 ),

(a, c2 ) → (a3 , c12 ),

(a3 , b3 ) → (a23 , b13 ).

So, Eq. (70) assures that two ways of obtaining (a23 , b13 , c12 ) from the initial data (a, b, c) lead to the same results. The maps with this property were introduced by Drinfeld [12] under the name of “set–theoretical solutions of the Yang–Baxter equation”, an alternative name is “Yang–Baxter maps” used by Veselov in the recent study [35], see also references therein. Under some circumstances, systems with the fields on vertices can be regarded as systems with the fields on edges or vice versa (this is the case, e.g., for the systems (Q1), (Q3)δ=0 , (H1), (H3)δ=0 of our list, for which the variables X enter only in combinations like a = X − U for edges (x, u)), but in general the two classes of systems should be considered as different. The notion of the zero curvature representation makes perfect sense for Yang–Baxter maps: such a map can be called integrable, if it is equivalent to L(b, β; λ)L(a, α; λ) = L(a2 , α; λ)L(b1 , β; λ). The problem of integrability of Yang–Baxter maps in the sense of existence of a zero–curvature representation is under current investigation [33]. Also the problem of classification of Yang–Baxter maps, like the one achieved in the present paper, is of great importance and interest. A different direction for the development of the ideas of the present paper constitute quantum systems, or, more generally, systems with non–commutative variables. To remain in the frame of the present paper, these are systems (1), where the fields (x, u, v, y) take values in an arbitrary associative (not necessary commutative) algebra with a unit. It turns out that the notion of the three–dimensional consistency can be formulated also for such non–commutative systems. Also the derivation of the zero curvature representation can be extended to the non–commutative framework [9]. It should be mentioned that in the area of the three–dimensional consistency of classical systems there also remains a number of interesting open problems. For instance, one of the assumptions under which the classification was carried out in the present paper, was less natural, namely the tetrahedron condition. As we pointed out in Sect. 3, there exist three–dimensionally consistent equations without the tetrahedron property, however all examples we are aware of are trivial (linear or linearizable): Q(x, u, v, y) = x + y − u − v = 0

or

Q(x, u, v, y) = xy − uv = 0,

or those obtained from these two by the action of a M¨obius transformation on all variables. These examples do not contain parameters, and thus the three–dimensional consistency does not give a zero curvature representation with a spectral parameter for them (their integrability is anyway obvious). It would be interesting to find out whether there exist nontrivial examples violating the tetrahedron property. There is also a vast field of multi–field integrable equations on quad–graphs. Existing examples indicate that their study is very promising.

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Four–dimensional consistency of three–dimensional systems. Very promising is also the application of the consistency approach to the three–dimensional integrability. The role of an ad hoc definition of integrability, played in two dimensions by the zero curvature representation, now goes to the so called local Yang–Baxter equation, introduced by Maillet and Nijhoff [22]. The role of the transition matrices from the zero curvature representation is played in this novel structure by certain tensors attached to the elementary two–dimensional plaquettes of the three–dimensional lattice. There exist a number of results on finding this sort of structure for some three–dimensional integrable systems [19, 16, 17]. It would be desirable to relate this ad hoc notion of integrability to some constructive one. In the spirit of the present paper, this constructive notion should be the four–dimensional consistency. In the three–dimensional context there are a priori many kinds of systems, according to where the fields are defined: on the vertices, on the edges, or on the elementary squares of the cubic lattice. Consider first the situation when the fields are sitting on the elementary squares. Attach the fields a, b, c to the two–dimensional faces parallel to the coordinate planes 12, 23, 13, respectively, so that a, b, c are sitting on the bottom, the left and the front faces of a cube Fig. 3, and a3 , b1 , c2 on the top, the right and the back faces. The system under consideration is a map S : X 3 → X 3 attached to the cube, so that S(a, b, c) = (a3 , b1 , c2 ). The condition of the four–dimensional consistency of such a map can be encoded in the formula S134 ◦ S234 ◦ S124 ◦ S123 = S123 ◦ S124 ◦ S234 ◦ S134 .

(71)

This equation should be understood as follows. Additionally to the fields a, b, c, consider the fields d, e, f , attached to the two–dimensional faces parallel to the coordinate planes 24, 14, 34 of the four–dimensional hypercubic lattice. Each map of the type Sij k in (71) is a map on X 6 (a, b, c, d, e, f ) acting as S on the factors of the cartesian product X 6 corresponding to the variables sitting on the faces parallel to the planes ij , j k, ik, and acting trivially on the other three factors. Thus the left–hand side of (71) corresponds to the chain of maps (a, b, c) → (a3 , b1 , c2 ), (a3 , d, e) → (a34 , d1 , e2 ), (b1 , d1 , f ) → (b14 , d13 , f2 ), (c2 , e2 , f2 ) → (c24 , e23 , f21 ), while the right–hand side of (71) corresponds to the chain of maps (c, e, f ) → (c4 , e3 , f1 ), (b, d, f1 ) → (b4 , d3 , f12 ), (a, d3 , e3 ) → (a4 , d13 , e23 ), (a4 , b4 , c4 ) → (a34 , b14 , c24 ). Equation (71) expresses then the fact that two different ways of obtaining the data (a34 , b14 , c24 , d13 , e23 , f12 ) from the initial data (a, b, c, d, e, f ) lead to identical results. This equation is known in the literature as the functional tetrahedron equation [19, 17]. (Note that the standard notation used in the literature on the tetrahedron equation is different: the indices 1 ≤ α, β, γ ≤ 6 of Sαβγ numerate the two–dimensional coordinate planes.) The paper [17] contains also a list of solutions of this equation with X = C, possessing local Yang–Baxter representations with a certain ansatz for the participating tensors. One of the most remarkable examples is the star–triangle map: a3 = −

a , ab − bc − ca

b1 = −

b , ab − bc − ca

c2 = −

c . ab − bc − ca

(72)

Classification of Integrable Equations on Quad-Graphs

541

x234

x34

x23 x3 x

x1234

x134 x123

x13 x2 x1

x12

x24 x4

x124

x14 Fig. 11. Hypercube

(Usually this equation is written in a more symmetric form which is obtained by changing c → −c, with all plus signs in all denominators; however in that form the map does not satisfy Eq. (71).) See also [16] for an alternative localYang–Baxter representation for this system. In [31, 32] an important solution of the functional tetrahedron equation with X = C2 was introduced and studied in detail. There seems to be no method available for deriving the local Yang–Baxter representation for a given map satisfying the functional tetrahedron equation. Further, consider three–dimensional systems with the fields sitting on the vertices. In this case each elementary cube carries just one equation Q(x, x1 , x2 , x3 , x12 , x23 , x13 , x123 ) = 0,

(73)

relating the fields in all its vertices. Such an equation should be solvable for any of its arguments in terms of the other seven ones. The four–dimensional consistency of such equations is defined as follows: – Starting with the initial data x, xi (1 ≤ i ≤ 4), xij (1 ≤ i < j ≤ 4), Eq. (73) allows us to uniquely determine all fields xij k (1 ≤ i < j < k ≤ 4). Then we have four different possibilities to find x1234 , corresponding to four three–dimensional cubic faces adjacent to the vertex x1234 of the four–dimensional hypercube, see Fig. 8. All four values actually coincide. So, one can consistently impose Eqs. (73) on all three–dimensional cubes of the lattice Z4 . It is tempting to accept the four–dimensional consistency of equations of type (73) as the constructive definition of their integrability. It will be very important to solve the correspondent classification problem. We expect that this definition will allow one to derive the local Yang–Baxter representation, as a replacement for the zero curvature representation characteristic for the two–dimensional integrability. We give here some examples. Consider the equation (x1 − x3 )(x2 − x123 ) (x − x13 )(x12 − x23 ) = . (x3 − x2 )(x123 − x1 ) (x13 − x12 )(x23 − x)

(74)

This equation appeared for the first time in [29, 18], along with a geometric interpretation. It is not difficult to see that Eq. (74) admits a symmetry group D8 of the cube. This equation can be uniquely solved for a field at an arbitrary vertex of a three–dimensional cube, provided the fields at the other seven vertices are known. The fundamental fact not mentioned in [18] is: • Eq. (74) is four–dimensionally consistent in the above sense.

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This is closely related (in fact, almost synonymous) to the functional tetrahedron equation for the star–triangle map (72), since the plaquette variables (a, b, c) of the latter can be “factorized” into combinations of vertex variables x of Eq. (74). More precisely, given a solution of (74) and setting x12 − x x23 − x x13 − x a= , b= , c= x1 − x2 x2 − x 3 x1 − x 3 we arrive at a solution of (72). A different “factorization” of the plaquette variables into the vertex ones leads to another remarkable three–dimensional system known as the discrete BKP equation [25, 18]. For any solution x : Z4 → C of (74), define a function τ : Z4 → C by the equations τi τj xij − x = , i < j. (75) τ τij xi − x j Equation (74) assures that this can be done in an essentially unique way (up to initial data on coordinate axes whose influence is a trivial scaling of the solution). The function τ satisfies on any three–dimensional cube the discrete BKP equation: τ τij k − τi τj k + τj τik − τk τij = 0,

i < j < k.

(76)

The following holds: • Equation (76) is four–dimensionally consistent. Moreover, for the value τ1234 one finds a remarkable equation: τ τ1234 − τ12 τ34 + τ13 τ24 − τ23 τ34 = 0,

(77)

which essentially reproduces the discrete BKP equation. So, τ1234 does not actually depend on the values τi , 1 ≤ i ≤ 4. This can be considered as an analog of the tetrahedron property of Sect. 2. Notice that usually the discrete BKP equation (76) is written in a slightly different and more symmetric form, with all plus signs on the left–hand side. On every three–dimensional subspace these two forms are easily transformed into one another. However, this cannot be done on the whole of Z4 . Equation (76) with all plus signs on the left–hand side does not possess the property of the four–dimensional consistency. Further, we mention systems of the geometrical origin (discrete analogs of conjugate and orthogonal coordinate systems) [11, 6], which also have the property of four–dimensional consistency. We plan to address various aspects of four–dimensional consistency in our future publications. Acknowledgements. This research was partly supported by DFG (Deutsche Forschungsgemeinschaft) in the frame of SFB 288 “Differential Geometry and Quantum Physics”. V.A. was also supported by the RFBR grant 02-01-00144. He thanks TU Berlin for warm hospitality during the visit when part of this work has been fulfilled.

References 1. Arnold, V.I.: Mathematical Methods of Classical Mechanics. New York: Springer, 1978 2. Adler, V.E.: B¨acklund transformation for the Krichever-Novikov equation. Int. Math. Res. Notices 1, 1–4 (1998) 3. Adler, V.E.: Discrete equations on planar graphs. J. Phys. A: Math. Gen. 34, 10453–10460 (2001) 4. Bianchi, L.: Vorlesungen u¨ ber Differenzialgeometrie. Leipzig: Teubner, 1899 5. Bobenko, A.I., Pinkall, U.: Discrete surfaces with constant negative Gaussian curvature and the Hirota equation. J. Diff. Geom. 43, 527–611 (1996)

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6. Bobenko, A.I., Pinkall, U.: Discretization of surfaces and integrable systems. In: Discrete Integrable Geometry and Physics, A.I. Bobenko, R. Seiler (eds) Oxford: Clarendon Press, 1999, pp. 3–58 7. Bobenko, A.I., Hoffmann, T., Suris, Yu.B.: Hexagonal circle patterns and integrable systems: Patterns with the multi-ratio property and Lax equations on the regular triangular lattice. Int. Math. Res. Notices 3, 111–164 (2002) 8. Bobenko, A.I., Suris,Yu.B.: Integrable systems on quad-graphs. Int. Math. Res. Notices 11, 573–611 (2002) 9. Bobenko, A.I., Suris,Yu.B.: Integrable non–commutative equations on quad-graphs. Consistency approach. Preprint nlin.SI/0206010, at: http://www.arXiv.org. To appear in Letters Math. Phys. 10. Capel, H.W., Nijhoff, F.W., Papageorgiou, V.G.: Complete integrability of Lagrangian mappings and lattices of KdV type. Phys. Lett. A 155, 377–387 (1991) ¯ 11. Doliwa, A., Manakov, S.V., Santini, P.M.: ∂-reductions of the multidimensional quadrilateral lattice. The multidimensional circular lattice. Commun. Math. Phys. 196, 1–18 (1998) 12. Drinfeld, V.G.: On some unsolved problems in quantum group theory. Lecture Notes Math. 1510, 1–8 (1992) 13. Faddeev, L.D., Volkov, A.Yu.: Hirota equation as an example of an integrable symplectic map. Lett. Math. Phys. 32, 125–135 (1994) 14. Hirota, R.: Nonlinear partial difference equations. I. A difference analog of the Korteweg–de Vries equation. III. Discrete sine-Gordon equation. J. Phys. Soc. Japan 43, 1423–1433, 2079–2086 (1977) 15. Iwasaki, K., Kimura, H., Shimomura, Sh., Yoshida, M.: From Gauß to Painlev´e. A modern theory of special functions. Braunschweig: Vieweg, 1991 16. Kashaev, R.M.: On discrete three-dimensional equations associated with the local Yang-Baxter relation. Lett. Math. Phys. 38, 389–397 (1996) 17. Kashaev, R.M., Korepanov, I.G., Sergeev, S.M.: The functional tetrahedron equation. Theor. Math. Phys. 117, 1402–1413 (1998) 18. Konopelchenko, B.G., Schief, W.K.: Reciprocal figures, graphical statics and inversive geometry of the Schwarzian BKP hierarchy. Stud. Appl. Math. 109, 89–124 (2002) 19. Korepanov, I.G.: Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics. Preprint solv-int/9506003 at http://www.arXiv.org 20. Krichever, I.M., Novikov, S.P.: Holomorphic bundles over algebraic curves and nonlinear equations. Russ. Math. Surv. 35(6), 53–79 (1980) 21. Kutz, N.: Lagrangian description of doubly discrete sine–Gordon type models. In: Discrete Integrable Geometry and Physics, A.I. Bobenko, R. Seiler (eds), Oxford: Clarendon Press, 1999, pp. 235–243 22. Maillet, J.M., Nijhoff, F.: Integrability for multidimensional lattice models. Phys. Lett. B 224, 389–396 (1989) 23. Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199, 351–395 (1998) 24. Mikhailov, A.V., Shabat, A.B., Yamilov, R.I.: The symmetry approach to the classification of nonlinear equations. Complete lists of integrable systems. Russ. Math. Surv. 42(4), 1–63 (1987) 25. Miwa, T.: On Hirota’s difference equations. Proc. Japan Acad., Ser. A: Math. Sci. 58(1), 9–12 (1982) 26. Nijhoff, F.W.: Lax pair for the Adler (lattice Krichever-Novikov) system. Phys. Lett. A 297, 49–58 (2002) 27. Nijhoff, F.W., Capel, H.W.: The discrete Korteweg-de Vries equation. Acta Appl. Math. 39, 133–158 (1995) 28. Nijhoff, F.W., Walker, A.J.: The discrete and continuous Painlev´e hierarchy and the Garnier system. Glasgow Math. J. 43A, 109–123 (2001) 29. Nimmo, J.J.C., Schief, W.K.: An integrable discretization of a 2+1-dimensional sine–Gordon equation. Stud. Appl. Math. 100, 295–309 (1998) 30. Quispel, G.R.W., Nijhoff, F.W., Capel, H.W., van der Linden, J.: Linear integral equations and nonlinear difference-difference equations. Physica A 125, 344–380 (1984) 31. Sergeev, S.M.: 3D symplectic map. Phys. Lett. A 253, 145–150 (1999) 32. Sergeev, S.M.: On exact solution of a classical 3D integrable model. J. Nonlin. Math. Phys. 7, 57–72 (2000) 33. Suris, Yu.B., Veselov, A.P.: Lax pairs for Yang–Baxter maps. In preparation 34. Veselov, A.P.: Integrable mappings. Russ. Math. Surv. 46(5), 1–51 (1991) 35. Veselov, A.P.: Yang–Baxter maps and integrable dynamics. Preprint math.QA/0205335, at: http://www.arXiv.org Communicated by L. Takhtajan

Commun. Math. Phys. 233, 545–569 (2003) Digital Object Identifier (DOI) 10.1007/s00220-002-0765-5

Communications in

Mathematical Physics

On the Motion of a Charged Particle Interacting with an Infinitely Extended System∗ Paolo Butt`a , Emanuele Caglioti, Carlo Marchioro Dipartimento di Matematica, Universit`a di Roma ‘La Sapienza’, P.le Aldo Moro 2, 00185 Roma, Italy. E-mail: [email protected]; [email protected]; [email protected] Received: 7 March 2002 / Accepted: 23 September 2002 Published online: 8 January 2003 – © Springer-Verlag 2003

Abstract: We study the time evolution of a charged particle moving in a medium under the action of a constant electric field E. In the framework of fully Hamiltonian models, we discuss conditions on the particle/medium interaction which are necessary for the particle to reach a finite limit velocity. We first consider the case when the charged particle is confined in an unbounded tube of R3 . The electric field E is directed along the symmetry axis of the tube and the particle also interacts with an infinitely many particle system. The background system initial conditions are chosen in a set which is typical for any reasonable thermodynamic (equilibrium or non-equilibrium) state. We prove that, for large E and bounded interactions between the charged particle and the background, the velocity v(t) of the charged particle does not reach a finite limit velocity, but it increases to infinite as: |v(t)−Et| ≤ C0 (1+t), where C0 is a constant independent of E. As a corollary we obtain that, if the initial conditions of the background system are distributed according to any Gibbs state, then the average velocity of the charged particle diverges as time goes to infinite. This result is obtained for E large enough in comparison with the mean energy of the Gibbs state. We next study the one-dimensional case, in which the estimates can be improved. We finally discuss, at an heuristic level, the existence of a finite limit velocity for unbounded interactions, and give some suggestions about the case of small electric fields. 1. Introduction In the present paper we study the time evolution of a charged particle moving in a medium under the action of a constant electric field E. If the particle moves in the vacuum the motion is trivial and the velocity of the particle grows linearly in time. Our goal is to discuss some conditions on the interaction between the charged particle and the medium under which the former may reach a bounded asymptotic velocity. In kinetic theory it is ∗

Work partially supported by the GNFM-INDAM and the Italian Ministry of the University.

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common knowledge that the total cross section is not allowed to decrease too fast with increasing energy if a steady state has to be maintained. In the present paper we obtain similar results for an Hamiltonian system. We consider a charged particle interacting with the electric field E and with a background of infinitely many particles. The system charged particle + background evolves via Newton’s law. We introduce and discuss the following conjecture: a necessary condition for the charged particle to reach a bounded asymptotic velocity is that its interaction with the background is singular. It would be nice to prove such a result for a generic system of infinitely many particles in R3 , but it is too difficult at the present stage of knowledge (a few words on this point are spent in Sect. 5). Conversely we rigorously prove the conjecture in two specific models. In the first one we consider the particles posed in an infinitely extended tube of R3 and the electric field E is parallel to the symmetry axis of the tube. The background system initial conditions are chosen in a set large enough to be the support of any reasonable thermodynamic (equilibrium or non-equilibrium) state. Then, under assumptions which will be given in the next section, we rigorously prove that a bounded interaction cannot give rise to a finite limit velocity if the external field is sufficiently large. This is the main statement of the present paper (Theorem 2.2 below). Moreover the dependence on the initial data is under control in such a way that, if the initial conditions of the background are distributed according to any Gibbs state and the external field is chosen large enough, then also the average velocity of the charged particle diverges. We point out that this model has been chosen in order to have a quasi-one-dimensional case, where it is possible to obtain some sharp estimates on the long time behavior, but avoiding the typical pathologies of the true one-dimensional systems, in which a particle cannot overcome another one without interacting with it. When we give up this more realistic geometry and consider, as in the second model we study, the one-dimensional case, the estimates can be improved. The plan of the paper is the following: the results are stated in Sect. 2, proved in Sect. 3 for the case of the tube, and in Sect. 4 for the one-dimensional case. In Sect. 5 we discuss, at an heuristic level, the case of unbounded interactions. Moreover we shortly consider the case of small electric fields, by investigating a simple model which suggests that also in this case the drift of the charged particle is infinite for bounded interactions with background. We finally remark that in the present paper the main results are proved by assuming that the strength of the electric field is large enough, but we believe that this assumption is only a technical one. Research for obtaining rigorous results for small electric fields is in progress.

2. Notation and Statement of the Results Let ν be a fixed unit vector in R3 . For any ξ ∈ R3 , we denote by ξ ⊥ = ξ − (ξ · ν)ν its . orthogonal projection. Given L > 0, let  = {ξ ∈ R3 : |ξ ⊥ | < L} be the infinite tube of radius L and symmetry axis ν. We consider the following infinitely extended mechanical system in three dimensions. A charged particle of mass M and charge q is subject to a constant electric field E = Eν, E > 0, and it is coupled with an infinite system of neutral particles of unit mass by means of a non-negative, symmetric, twice differentiable, short-range, two-body potential . Without loss of generality we assume that q/M = 1 (it is just a redefinition of the electric field).

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The particles interact among themselves by means of a non-negative, symmetric, short-range, two-body potential  of the form (ξ ) = 1 (ξ ) + a|ξ |−b ,

(2.1)

where a, b are non-negative constants and 1 is twice differentiable and symmetric. If  is finite at the origin we assume (0) > 0, which guarantees  to be superstable [15]. Without loss of generality we assume that both and  have range not greater than one, i.e. (ξ ) = 0,

(ξ ) = 0

if |ξ | > 1.

(2.2)

We force the system to stay confined inside the tube , by requiring that all the particles are subject to a one-body potential of the form

(ξ ) =

θh (|ξ ⊥ |)

(L − |ξ ⊥ |)γ

,

ξ ∈ ,

(2.3)

where γ > 0, h ∈ (0, L), and θh (s), s ∈ R+ , is a non-negative, twice differentiable function, identically zero for s ≤ h and strictly positive at s = L. The state of the system is determined by the position and velocity x = (r, v) of the charged particle and those x i = (r i , v i ), i ∈ N, of the other particles. We denote by X = {x i }i∈N the state of the infinite extended system, which is assumed to have a locally finite density and energy. In particular it is well defined, for any µ ∈ R and R > 0,  2  vi 1
.
Q(X; µ, R) = + (r i ) + χi (µ, R) (r i − r j ) + 1 , 2 2 i

(2.4)

j :j =i

where χi (µ, R) = χ (|r i · ν − µ| ≤ R) and χ (A) denotes the characteristic function of the set A. In order to consider configurations which are typical for the thermodynamic states, we allow initial data with logarithmic divergences in the velocities and local densities. More precisely, by defining . Q(X) = sup µ

Q(X; µ, R) , 2R R:R>log(e+|µ|) sup

(2.5)

. the set X = {X : Q(X) < +∞} has a full measure w.r.t. any Gibbs state [6]. The time evolution t → (x(t), X(t)) is defined by the solutions of the Newton equations:   r¨ (t) = F (x(t), X(t)) r¨ i (t) = F i (x(t), X(t)), i ∈ N  x(0) = (0, 0), X(0) = X, where (recall we set q/M = 1)

(2.6)

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P. Butt`a, E. Caglioti, C. Marchioro


 . F (x, X) = −M −1 ∇ (r − r j ) + ∇(r) + E,

(2.7)

j


. F i (x, X) = − ∇(r i − r j ) − ∇ (r i − r) − ∇(r i ),

i ∈ N,

(2.8)

j :j =i

and, without loss of generality, we assumed that the charged particle is initially located at r = 0 with velocity v = 0. The Cauchy problem for this system of infinite equations is well posed when the initial condition X is chosen in the set X , and the solution is constructed by means of the following limiting procedure. . . Given X ∈ X and n ∈ N, let In = {i ∈ N : r i ∈ (0, n)}, where (µ, R) = {ξ ∈  : |ξ · ν − µ| ≤ R}. We define the n-partial dynamics t → (x (n) (t), X(n) (t)), (n) X(n) (t) = {x i (t)}i∈In , as the solution of the differential system:  (n)  r¨ (t) = F (x (n) (t), X(n) (t)), (n) r¨ (t) = F i (x (n) (t), X(n) (t)), i ∈ In ,  i(n) x (0) = (0, 0), X(n) (0) = {x i }i∈In .

(2.9)

Then, by using the same techniques of [8] and [7], the following theorem can be proved (for smooth interactions there are also the pioneering papers by Lanford [11, 12]). Theorem 2.1. For each X ∈ X there exists a unique flow t → (x(t), X(t)), X(t) = {x i (t)}i∈N ∈ X satisfying (2.6). Moreover, for any t ≥ 0, lim x (n) (t) = x(t),

n→+∞

(n)

lim x i (t) = x i (t) ∀ i ∈ N.

n→+∞

(2.10)

We can now state our first result. Theorem 2.2. There exist C0 , C1 , C2 > 0 such that for any X ∈ X the following holds. Let t → (x(t), X(t)) be the unique solution of Eqs. (2.6). i) If [log(e + E)]−1 E > C0 Q(X) then, for any t ≥ 0,

log(e + E) |v(t) − Et| ≤ C0 Q(X) +t √ E

(2.11)

and, for any i ∈ N, |v i (t)| ≤ C1



Q(X) log(e + |r i · ν| + E) + Q(X)t .

(2.12)

ii) If [log(e + E)]−1 E ≤ C0 Q(X) then, for any t ≥ 0 and i ∈ N, 

|v(t)| + |v i (t)| ≤ C2 log(e + Q(X)) Q(X) log(e + |r i · ν|) + Q(X)t . (2.13) We shall prove Theorem 2.2 by showing analogous estimates for the n-partial dynamics which are uniform in n. In fact the same existence and uniqueness of the infinite dynamics will be a simple corollary of these bounds.

Motion of a Charged Particle Interacting with an Infinitely Extended System

549

Remark 2.1. The non-negativity of is needed only in the proof of inequality (2.13). We expect the result to be true also without this assumption, but this extension probably requires some more technical effort. Remark 2.2. As already observed the assumption that the charged particle is initially at rest in the origin is not essential: in fact, by a Galilean transformation we can cover the general case. On the other hand, the initial distribution of the background and then the threshold in the strength of the electric field have to be changed accordingly. The explicit dependence on Q(X) in the bounds of Theorem 2.2 allows to control the long time behavior of the average velocity of the charged particle, relative to statistical solutions of the Newton equations obtained for suitable random initial data X. In particular: Corollary 2.1. For any Gibbs state · of the background system there exists a threshold ¯ E¯ > 0 such that, for any E > E, lim v(t) · ν = +∞.

t→+∞

(2.14)

The above corollary is an immediate consequence of Theorem 2.2; recall in fact that for any Gibbs state · there exist A, B > 0 such that χ (Q(X) > κ) ≤ A exp[−Bκ] for any κ > 0, see e.g. [6]. We next consider the one-dimensional model. We denote by x = (r, v) [resp. xi = (ri , vi )] the position and velocity of the charged particle [resp. neutral particles], with now r, ri ∈ R, and let X = {(ri , vi )}i∈N . The Newton equations now read:
  r¨ (t) = −M −1 ∇ (r − rj ) + E,     j
(2.15) r¨i (t) = − ∇(ri − rj ) − ∇ (ri − r), i ∈ N,    j :j =

i   x(0) = (0, 0), X(0) = X, where E > 0 and , are non-negative, symmetric, twice differentiable, and shortrange (in particular (x) = (x) = 0 for |x| > 1). We further assume that  is strictly positive at the origin. Analogously to the previous case the Cauchy problem (2.15) is well posed for X such that Q(X) < +∞, where Q(X; µ, R) , 2R µ R:R>log(e+|µ|)  2  v 1
.
Q(X; µ, R) = χ (|ri − µ| ≤ R) i + (ri − rj ) + 1 . 2 2 . Q(X) = sup

i

sup

j :j =i

Then: Theorem 2.3. There exist C¯ 0 , C¯ 1 , C¯ 2 > 0 such that for any X with Q(X) < ∞ the following holds. Let t → (x(t), X(t)) be the unique solution of Eqs. (2.15). i) If [log(e + E)]−1 E > C¯ 0 Q(X) then, for any t ≥ 0 and E ≥ 1,   log(e + E) log(1 + Et 2 )

¯ |v(t) − Et| ≤ C0 Q(X) + Q(X) . (2.16) + √ E E

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Moreover, for any t ≥ 0 and i ∈ N, 

|vi (t)| ≤ C¯ 1 Q(X) log(e + |ri | + E) + Q(X)t . ii) If [log(e + E)]−1 E ≤ C¯ 0 Q(X) then, for any t ≥ 0 and i ∈ N, 

|v(t)| + |vi (t)| ≤ C¯ 2 log(e + Q(X)) Q(X) log(e + |ri |) + Q(X)t .

(2.17)

(2.18)

We conclude the section with a notation warning: in Sects. 3 and 4, if not further specified, we shall denote by Ci , i = 3, . . . , positive constants whose numerical value may possibly depend only on the interactions , , and . 3. Proof of Theorem 2.2 Before starting with the rigorous proof let us first briefly discuss the main ideas of our approach. Disregarding for the moment the interaction √of the charged particle with the background, we observe that, after a time of order 1/ E, the charged particle reaches √ a velocity of order E. On the other hand, during this time, its displacement is bounded by a constant so that the charged particle could interact only with a finite number (not depending on E) of particles. Since the interaction is assumed to be bounded, we may expect that, even taking into account this interaction, if E is very large √ then the √ velocity of the charged particle is still of order E at a time of order 1/ E. After this time another mechanism takes place: since it can be proved that the velocities of the particles of the background may increase at most linearly, the charged particle is now much faster than all the particles it meets. Hence it interacts with each of them for a very small time and, since the interaction is bounded, this implies that also the momentum transferred during the scattering process is very small. Clearly, as time goes by, the number of particles which may interact with the charged one increases. But for large enough electric fields we may expect the charged particle to accelerate so rapidly that, in unit time, the momentum transferred by the other particles (which is of order [number of collisions] · [time of collision]) remains bounded by a constant which is smaller than E: if this is the case the charged particle will accelerate indefinitely. We shall prove that the above picture is true by using the following strategy. We consider the maximal time for which the horizontal velocity of the charged particle remains close enough to Et, and the absolute velocity of the particles which may interact with the former is much smaller than Et. By choosing E large enough, this maximal time is positive. We then study the dynamics up to this time and obtain sharper estimates implying, by a continuity argument, that this time is actually infinite. After that, the inequalities (2.11) and (2.12) follow easily from the above bounds. If the external field E is not large, the above mechanism does not work, and the charged particle can exchange a large part of its energy with the background. In any case the velocity of both the charged and the other particles may increase at most linearly in time, and this is proved by the inequality (2.13). We cannot directly work with the infinite dynamics since we need to control the explicit dependence on E and Q(X) in the limiting procedure. Then we shall rather work with the n-partial dynamics, by obtaining bounds which are uniform in n. As already noticed in the previous section, the same existence and uniqueness of the infinite dynamics will be a byproduct of our estimates.

Motion of a Charged Particle Interacting with an Infinitely Extended System

3.1. Proof of (2.11) and (2.12). Let  √ 6Et 2  . (n) (n) Un (t) = max G inf |r i (s) · ν| − Et − sup |v i (s)|, s∈[0,t] i 5 s∈[0,t]

551

(3.1)

where G ∈ C(R) is not increasing and satisfying: G(x) = 1 for x ≤ 1, G(x) = 0 for x ≥ 2. We next define   √ Es . ∀ s ∈ [0, t] , Tn = sup t ≥ 0 : max{Un (s); |v (n) (s) · ν − Es|} ≤ E + 5 (3.2) setting Tn = 0 if the above set is empty. Observe that by (2.2), for any t ∈ [0, Tn ), the i th particle can interact with the charged one during the time [0, t] only if i ∈ An (t), where   √ 6Et 2 . (n) An (t) = i ∈ In : inf |r i (s) · ν| ≤ 1 + Et + . (3.3) s∈[0,t] 5 Observe also that Un (·) is a continuous and non-decreasing function such that max

(n)

sup |v i (s)| ≤ Un (t) ≤ max

where

(n)

sup |v i (s)|,

i∈A¯ n (t) s∈[0,t]

i∈An (t) s∈[0,t]

  √ 6Et 2 . (n) ¯ . An (t) = i ∈ In : inf |r i (s) · ν| ≤ 2 + Et + s∈[0,t] 5

(3.4)

(3.5)

Since, by definition (2.5), |v i | ≤ 2 Q(X) log(e + |r i · ν|) ∀ i ∈ N, (3.6) . it follows that, setting C ∗ = 16 log(e + 2), √ E Un (0) ≤ ∀ n ∈ N ∀ E ≥ C ∗ Q(X). 2 Recalling that v(0) = 0, by continuity we conclude that if E ≥ C ∗ Q(X) then Tn > 0 for all n ∈ N. We now study the n-partial dynamics for t ∈ [0, Tn ). A basic tool will be an estimate on the growth in time of the local density and energy, which is the content of the following lemma, whose proof is reported in the Appendix. Lemma 3.1. There exists C3 > 0 such that, for any X ∈ X , E ≥ C ∗ Q(X), and n ∈ N, sup Q(X (n) (t); µ, Rn (t)) ≤ C3 Q(X)Rn (t) µ

where . Rn (t) = log(e + n) +



t 0

∀ t ∈ [0, Tn ),

ds Vn (s)

(3.7)

(3.8)

and . (n) Vn (t) = max sup |v i (s) · ν|. i∈In s∈[0,t]

(3.9)

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P. Butt`a, E. Caglioti, C. Marchioro

Proposition 3.1. There exist C4 > 0 such that, for any X ∈ X , E ≥ C ∗ Q(X), n ∈ N, and i ∈ In ,

 (n) Q(X) log(e + n) + Q(X)t ∀ t ∈ [0, Tn ). (3.10) |v i (t)| ≤ C4 Proof. For any t ∈ [0, Tn ) let Nn (µ, t) be the number of the particles i ∈ In such that (n) (n) r i (t) ∈ (µ, Rn (t)) and |v i (t) · ν| > bn (t), with Vn (t) , bn (t) = C5 Q(X) log(e + n) + 2

(3.11)

where C5 > 0 will be fixed later and Vn is defined in (3.9). Clearly: Q(X(n) (t); µ, Rn (t)) >

bn (t)2 Nn (µ, t) 2

so that, by inequality (3.7) and using the definitions (3.8) and (3.9), Nn (µ, t) < 8C3 Q(X) 



log(e + n) + tVn (t)

2 ,

2C5 Q(X) log(e + n) + Vn (t)

from which, after neglecting some positive terms, Nn (µ, t) <

2C3 8C3 Q(X)t + . √ 2 2C5 Q(X) + Vn (t) C5

(3.12)

√ We now choose C5 = 2 2C3 ; by (3.12), if Vn (t) ≥ 32C3 Q(X)t then Nn (µ, t) < 1/2, i.e. Nn (µ, t) = 0. The above argument is independent of µ, so that Vn (t) ≥ 32C3 Q(X)t (n) actually implies |v i (t) · ν| ≤ bn (t) for all i ∈ In . Since bn (t) is not decreasing, we have in fact Vn (t) ≤ bn (t) when Vn (t) ≥ 32C3 Q(X)t. Recalling the definition (3.11), we conclude that Vn (t) ≤ 2C5 Q(X) log(e + n) + 64C3 Q(X)t ∀ t ∈ [0, Tn ). (3.13) On the other hand, by definition (2.4), the inequality (3.7), and the definition of Rn (t), for any i ∈ In , (n)

v i (t)2 ≤ 2C3 Q(X)[log(e + n) + tVn (t)]

∀ t ∈ [0, Tn ).

By (3.13), the inequality (3.10) follows for a suitable positive constant C4 .

 

Proposition 3.2. There exist C1 > 0 such that, for any X ∈ X , E ≥ C ∗ Q(X), i ∈ N, and n ≥ |r i · ν|,

 (n) |v i (t)| ≤ C1 Q(X) log(e + |r i · ν| + E) + Q(X)t ∀ t ∈ [0, T¯n ), (3.14) where . T¯n = min Tn .  n ≤n

(3.15)

Motion of a Charged Particle Interacting with an Infinitely Extended System

553

Proof. The bound (3.14) follows from (3.10) if n is not too large. In order to obtain the result for all n we then need to compare different partial dynamics. Let . (n) (n−1) (n) (n−1) δi (n, t) = |r i (t) − r i (t)| + |v i (t) − v i (t)| + |r (n) (t) − r (n−1) (t)| + |v (n) (t) − v (n−1) (t)|.

From the equations of motion in integral form we have:   t  (n)   v i (t) = v i + ds F i (x (n) (s), X(n) (s)) 0  t  (n)  ds (t − s)F i (x (n) (s), X(n) (s))  r i (t) = r i + v i t +

(3.16)

(3.17)

0

and the analogous expressions for the pair (r (n) (t), v (n) (t)) with (r i , v i ) and F i replaced by (r, v) = (0, 0) and F respectively. From (3.16) and (3.17) it follows that, for any i ∈ In−1 ,  t    |δi (n, t)| ≤ (1 + t) ds F i (x (n) (s), X(n) (s)) − F i (x (n−1) (s), X(n−1) (s)) 0   (n)  + F (x (s), X(n) (s)) − F (x (n−1) (s), X(n−1) (s)) . (3.18) By (3.2) and (3.10), if t ∈ [0, T¯n ), each particle i ∈ In [resp. the charged particle] may interact, during the time [0, t], only with the particles initially contained in (r i · ν, pn (t)) [resp. (0, pn (t))], where 

√ 6Et 2 . pn (t) = 1 + 2C4 t Q(X) log(e + n) + Q(X)t + Et + . (3.19) 5 We now fix k ∈ N and define . (3.20) n(k) = min{m ∈ N : n > 1 + k + pn (t) ∀ n ≥ m}. For n ≥ n(k) each particle i ∈ Ik and the charged one do not interact (during the time [0, t] for t ∈ [0, T¯n )) with the particles j ∈ In \ In−1 . Moreover, since  and  are of the form (2.1) and (2.3) respectively, and is twice differentiable, there exists C6 > 0 such that:

 b+2 b+2 |∇(ξ ) − ∇(ζ )| ≤ C6 (ξ ) b + (ζ ) b + χ1 (ξ ) + χ1 (ζ ) |ξ − ζ |,   γ +2 γ +2 γ γ |∇(ξ ) − ∇(ζ )| ≤ C6 (ξ ) + (ζ ) + χ1 (ξ ) + χ1 (ζ ) |ξ − ζ |, (3.21) |∇ (ξ ) − ∇ (ζ )| ≤ C6 |ξ − ζ |,

(3.22)

where χ1 (·) denotes the characteristic function of the unit ball of with center at the origin. Then, recalling definitions (2.7) and (2.8), there exists C7 > 0 such that, for any n ≥ n(k), s ∈ [0, T¯n ), and i ∈ Ik ,   F i (x (n) (s), X(n) (s)) − F i (x (n−1) (s), X(n−1) (s)) 
∗  (n) (n−1) (n) (n) ≤ C7 (r i (s))η + (r i (r i (s) − r j (s))η (s))η + 1 δi (n, s) + R3

j :j =i

(n−1) (n−1) + (r i (s) − r j (s))η

  + 1 δi (n, s) + δj (n, s) ,

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P. Butt`a, E. Caglioti, C. Marchioro

where η = 1 + 2 max{b−1 ; γ −1 }, and   F (x (n) (s), X(n) (s)) − F (x (n−1) (s), X(n−1) (s))

 
∗ ≤ C7 (r (n) (s))η + (r (n−1) (s))η + 1 δi (n, s) + δj (n, s) . In the above inequalities

∗

j :j =i [resp. (n)

∗

j ] denotes (n−1)

j

the sums restricted to all the parti-

(s) [resp. r (n) (s) or r (n−1) (s)]. Using the cles j ∈ In−1 closer than 1 to r i (s) or r i definition (2.4) and introducing . uk (n, t) = sup δi (n, t), (3.23) i∈Ik

by (3.18) and the above bounds there exists C8 > 0 such that, for any t ∈ [0, T¯n ),  t 

η  uk (n, t) ≤ C8 (1 + t) ds sup Q(X (n) (s); µ, 1) + Q(X (n−1) (s); µ, 1) 0

µ

 η  + Nn (s) + En (s) + En−1 (s) uk1 (n, s),

(3.24)

where k1 = Int [k + pn (t)] + 1 (Int [·] denotes the integer part of ·), Nn (s) is the number of particles j ∈ In−1 closer than 1 to r (n) (s) or r (n−1) (s), and . M (n) 2 En (s) = v (s) + (r (n) (s)). 2 Let us evaluate the terms appearing on the right-hand side of (3.24). From (3.7) and (3.10), for any s ∈ [0, t],  

 Q(X(n) (s); µ, 1) ≤ C3 Q(X) log(e + n) + C4 t Q(X) log(e + n) + Q(X)t . (3.25) As already observed, the charged particle may interact, during the time [0, t], only with the particles initially contained in (0, pn (t)); hence, by definitions (2.4) and (2.5), for any s ∈ [0, t],   Nn (s) ≤ 2Q(X) log(e + n) + pn (t) . (3.26) Finally, since




(n) E˙n (s) = v (n) (s) · − ∇ (r (n) (s) − r j (s)) + ME j

and  is non-negative, it follows that   2En (s)  |E˙n (s)| ≤ ∇ ∞ Nn (s) + ME . M Then, recalling that En (0) = 0 and using (3.26), we obtain there exists C9 > 0 such that, for any s ∈ [0, t],      s |En (s)| ≤ C9 Q(X) log(e + n) + pn (t) + E dτ En (τ ), 0

Motion of a Charged Particle Interacting with an Infinitely Extended System

555

which can be solved by getting, for any s ∈ [0, t],    2 t 2 |En (s)| ≤ C92 Q(X) log(e + n) + pn (t) + E . (3.27) 4 By (3.24), (3.25), (3.26), and (3.27) we obtain the following integral inequality (valid for any t ∈ [0, T¯n )):  t (3.28) uk (n, t) ≤ gn (t) ds uk1 (n, s), 0

where

 2η   . gn (t) = C10 (1 + t)2η+1 Q(X) log(e + n) + pn (t) + E

(3.29)

with C10 > 0 large enough. Setting kq = Int [kq−1 + pn (t)] + 1, q ∈ N, and k0 = k, we can iterate the inequality (3.28) 6 times, with   n−k−1 . (3.30) 6 = Int 1 + pn (t) (which ensures n > n(k6−1 )). Since uk (n, t) ≤ an (t) with   

6Et √ . Q(X) log(e + n) + Q(X)t + + E + 2L an (t) = 2(1 + t) C4 5

(3.31)

(recall L is the radius of the tube ), we finally get the following bound, valid for all t ∈ [0, T¯n ): uk (n, t) ≤ an (t)

[gn (t)t]6 . 6!

(3.32)

We can now prove (3.14). We choose k = Int [|r i · ν|] + 1 and, for t ∈ [0, T¯n ), let 

. (3.33) n∗ = Int α(k 2 + E 4η+2 )et , (n)

with α > 1 to be fixed later. From (3.10) and (3.33), if n ≤ n∗ then v i (t) satisfies a (n∗ ) bound like (3.14). Conversely, let us suppose that n > n∗ ; then T¯n ≤ T¯n∗ so that v i (t) satisfies a bound like (3.14) for t ∈ [0, T¯n ). On the other hand, by (3.23), (n)

(n∗ )

|v i (t) − v i

(t)| ≤

n


uk (n , t)

∀ t ∈ [0, T¯n ).

(3.34)

n =n∗

From definitions (3.20) and (3.33) it is easy to check that there exists α0 such that if α ≥ α0 then n∗ ≥ n(k) for all k ≥ 1, E ≥ C ∗ Q(X) and t ∈ [0, T¯n ). Since T¯n ≤ T¯n for all n ≤ n, we can use (3.32) to bound each term in the sum on the right-hand side of (3.34). Moreover, from (3.33) and recalling definitions (3.19), (3.29), (3.31), and (3.30), there exists C11 > 1 such that, for all E ≥ C ∗ Q(X) and n ≥ n∗ , t ≤ log(e + n ), gn (t) ≤ C11 (1 + E)4η log6η+1 (e + n ), n − k − 1 . 6≥ 2C11 (1 + E) log2 (e + n )

pn (t) ≤ C11 (1 + E) log2 (e + n ), an (t) ≤ C11 (1 + E) log2 (e + n ),

(3.35)

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P. Butt`a, E. Caglioti, C. Marchioro

Inserting the bounds above in (3.32) and using the Stirling formula we get:   n − k − 1  uk (n , t) ≤ C11 (1 + E) exp −6 log . (3.36) 3 (1 + E)4η+2 log6(η+1) (e + n ) 2eC11 Since n∗ ≥ α(k 2 + E 4η+2 ), there is α1 ≥ α0 such that the log in the square brackets on the right-hand side of (3.36) is not smaller than 1 for all E ≥ C ∗ Q(X), k ∈ N, α ≥ α1 , and n ≥ n∗ . Hence, from (3.34), the last bound in (3.35), and (3.36) we obtain, for all α ≥ α1 ,  
n − k − 1 (n) (n∗ ) . (3.37) exp − |v i (t) − v i (t)| ≤ C11 (1 + E) 2 (e + n ) 2C (1 + E) log 11  ∗ n ≥n Since n∗ ≥ α(k 2 + E 4η+2 ), by choosing α ≥ α1 large enough, the right-hand side is bounded uniformly in E ≥ C ∗ Q(X) and k ∈ N. The proposition is proved.   Proposition 3.3. There exists C + ≥ C ∗ such that, for each given X ∈ X , if [log(e + E)]−1 E ≥ C + Q(X) then Tn = +∞ for any n ∈ N. Proof. We shall prove that there exists C + ≥ C ∗ such that, for any X ∈ X , [log(e + E)]−1 E ≥ C + Q(X), n ∈ N, and k ≤ n,

1 √ Et Uk (t) ≤ E+ (3.38) ∀ t ∈ [0, T¯n ), 2 5

1 √ Et |v (k) (t) · ν − Et| ≤ E+ (3.39) ∀ t ∈ [0, T¯n ). 2 5 By continuity this implies that Tk > T¯n for all k ≤ n (which contradicts the definition of T¯n ) unless T¯n = +∞, and the proposition is proved. In the sequel we shall assume E ≥ C12 Q(X) with C12 = max{C ∗ ; 4C1 }. Let n ∈ N and k ≤ n. By Proposition 3.2 and recalling the definitions of Tk and A¯ k (t), see (3.2) and (3.5), for t ∈ [0, T¯n ) the initial position r i of each particle i ∈ A¯ k (t) has to verify the inequality: √ 6E 2 |r i · ν| − C1 t Q(X) log(e + |r i · ν| + E) − C1 Q(X)t 2 ≤ 2 + Et + t . 5 Since E ≥ max{C ∗ ; 4C1 }Q(X) and C ∗ > 1, the above inequality implies √ √ |r i · ν| − C1 Et log(e + |r i · ν| + E) ≤ 2 + Et + 2Et 2 ,

(3.40)

from which it follows that there exists ρ > 1 such that, for any k ≤ n and t ∈ [0, T¯n ),

 . i ∈ A¯ k (t) ⇒ r i ∈ (0, RE (t)) with RE (t) = ρ log(e + E) + Et 2 , (3.41) and hence, by (2.4) and (2.5), recalling also that Ak (t) ⊆ A¯ k (t), |Ak (t)| ≤ |A¯ k (t)| ≤ 2Q(X)RE (t), where for any finite set B we denote by |B| its cardinality.

(3.42)

Motion of a Charged Particle Interacting with an Infinitely Extended System

By (3.4), (3.41), and Proposition 3.2, there exists C13 > 0 such that:   2 Uk (t) ≤ C13 Q(X) log(e + E + Et ) + Q(X)t ∀ t ∈ [0, T¯n ).

557

(3.43)

Consider now the difference |v (k) (t) · ν − Et|. By the equations of motion, since ∇ · ν = 0, we have:  ∇ ∞
t (k) |v (k) (t) · ν − Et| ≤ ds χ (|r (k) (s) − r j (s)| ≤ 1). (3.44) M 0 j

Letting . 20 tE = √ , E

(3.45)

we notice that by the definition of Tk , if Tk > tE then: 

 Es (k) v (k) (s) − v i (s) · ν ≥ 2

∀ i ∈ Ak (s)

∀ s ∈ (tE , Tk ).

(3.46)

We next estimate, for any t ∈ [0, T¯n ), |v (k) (t) · ν − Et| ≤

 ∇ ∞  |Ak (min{t; tE })| tE + Fk (t) , M

where Fk (t) = χ (T¯n > t > tE )




t tE

j

(k)

ds χ (|r (k) (s) − r j (s)| ≤ 1).

(3.47)

(3.48)

By (3.42) and the definition (3.45), there exists C14 > 0 such that |Ak (min{t; tE })| tE ≤ C14 Q(X)

log(e + E) . √ E

(3.49)

To bound the term Fk (t) we use (3.46). Given K > 0 to be fixed later, let q0 ∈ N be such that 2K+q0 < RE (t) ≤ 2K+q0 +1 (RE (t) as in (3.41)) and define: N¯ = {j ∈ N : |r j · ν| ≤ 2K }, Nq = {j ∈ N : 2K+q < |r j · ν| ≤ 2K+q+1 },

q = 0, . . . , q0 .

By (3.46), for any t ∈ [0, T¯n ), q0

Fk (t) ≤

2|N¯ |
2|Nq | ¯ + χ (Tn > tE ), EtE Etk,q q=0

where, for T¯n > tE , . (k) tk,q = min inf{s ∈ [tE , T¯n ) : |r (k) (s) − r j (s)| ≤ 1}, j ∈Nq

(3.50)

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P. Butt`a, E. Caglioti, C. Marchioro

setting tk,q = +∞ if the above set is empty for all j ∈ Nq . We choose K such that 2K = c¯ log(e+E) with c¯ to be fixed below and so large that 2K+x−1 > log(e+2K+x+1 ) for all x ≥ 0 and E ≥ 0. Then, since 2K+q+1 − 2K+q = 2K+q , by (2.4) and (2.5) we have |Nq | ≤ 2Q(X)2K+q . On the other hand, by inequality (3.40) which is valid for all i ∈ Ak (t), the time tk,q has to satisfy the condition: 2

K+q

√



− C1 Etk,q log(e + 2K+q+1 + E) ≤ 2 +

√

2 Etk,q + 2Etk,q .

It follows that if c¯ is chosen sufficiently large, then there exists C15 > 0 such that tk,q ≥ C15 2K+q /E. Finally, again by (2.4) and (2.5), we have |N¯ | ≤ 2Q(X)2K . Inserting all the previous bounds in (3.50) we finally obtain, for some C16 , C17 > 0,   q0  C16 Q(X)  K
(K+q)/2  C16 Q(X)  K Fk (t) ≤ ≤ 2 2 + 2 + 4 RE (t) √ √ E E q=0

log(e + E) ≤ C17 Q(X) +t , (3.51) √ E √ where we used 2(K+q0 )/2 ≤ RE (t). By (3.47), (3.49), and (3.51), we conclude that there exists C18 > 0 such that, for any E ≥ C12 Q(X) and k ≤ n,

log(e + E) |v (k) (t) · ν − Et| ≤ C18 Q(X) (3.52) +t ∀ t ∈ [0, T¯n ). √ E By (3.43), (3.52), and by choosing C + ≥ C12 large enough, the inequalities (3.38) and (3.39) are verified for all E such that [log(e + E)]−1 E ≥ C + Q(X). The proposition is proved.   We can now conclude the proof of the estimates (2.11) and (2.12). The latter holds with C1 as in Proposition 3.2, and C0 ≥ C + large enough. In fact, for [log(e+E)]−1 E ≥ C + Q(X), since T¯n = +∞, the bounds (3.14) and (3.52) hold for all t ≥ 0. The former implies the inequality (2.12) for the corresponding infinite dynamics. The inequality (2.11) follows from (3.52) and an analogous upper bound for |v (n) (t)⊥ |, which we next prove. Let . M (n) ⊥ 2 En⊥ (t) = |v (t) | + (r (n) (t)). 2 From the equations of motion, since ∇ · ν = 0, E˙n⊥ (t) = −


j

(n)

∇ (r (n) (t) − r j (t)) · v (n) (t)⊥ ,

and hence, since  is non-negative, $ ⊥
 ⊥  (n) E˙ (t) ≤ ∇ ∞ 2En (t) χ (|r (n) (t) − r j (t)| ≤ 1). n M j

Motion of a Charged Particle Interacting with an Infinitely Extended System

559

Then, setting E¯n⊥ (t) = sups∈[0,t] En⊥ (s) and using En⊥ (0) = 0, we obtain: $  2∇ ∞ t
2E¯n⊥ (t) (n) ≤ ds χ (|r (n) (s) − r j (s)| ≤ 1). |v (n) (t)⊥ | ≤ M M 0 j

We have already found an upper bound for the right-hand side of the above inequality, see the analysis done starting from (3.44) to prove (3.52) (but now with T¯n = +∞). We conclude that, for [log(e + E)]−1 E ≥ C + Q(X),

log(e + E) |v (n) (t)⊥ | ≤ 2C18 Q(X) +t ∀ t ≥ 0. (3.53) √ E By choosing C0 = max{C + ; 3C18 }, the inequality (2.11) follows from (3.52) and (3.53) for any E such that [log(e + E)]−1 E ≥ C0 Q(X).   3.2. Proof of (2.13). When [log(e + E)]−1 E ≤ C0 Q(X) the long time behavior of the charged particle does not differ from the one of the others. We then prove the estimate (2.13) by applying the strategy used to get (3.14) to the whole system [charged particle] + [background]. With this in mind we modify the previous notation, denoting by ˆ = (x 0 , X), x 0 = (r 0 , v 0 ) the position and velocity of the charged particle and setting X (n) ˆ (t) = (x (n) (t), X(n) (t)). X The first step is a control on the growth of the local energy and density of the system which is now defined by   (M − 1)δi,0 + 1 2 1
.
ˆ X; ˆ µ, R) = ˆ i,j + 1 , χi (µ, R)  v i + (r i ) + Q( 2 2 i

j :j =i

(3.54) ˆ i,j = (r i − r j ) if i, j ≥ 1 and  ˆ i,j = (r i − r j ) if i = 0 or j = 0. Note that where  ˆ i,j is superstable and non-negative. In the Appendix we prove that there exists Cˆ 3 > 0 such that, for any X ∈ X , [log(e + E)]−1 E ≤ C0 Q(X), and n ∈ N, ˆ sup Q(X µ

(n)

(t); µ, Rˆ n (t)) ≤ Cˆ 3 log(e + Q(X))Q(X)Rˆ n (t)

where . Rˆ n (t) = log(e + n) +



t 0

ds Vˆn (s)

∀ t ≥ 0,

(3.55)

(3.56)

and . (n) Vˆn (t) = max sup |v i (s) · ν|, i∈Iˆn s∈[0,t]

(3.57)

with Iˆn = In ∪{i = 0}. By (3.55) and arguing as in the proof of Proposition 3.1, it can be proved that there exists Cˆ 4 > 0 such that, for any X ∈ X , [log(e + E)]−1 E ≤ C0 Q(X), n ∈ N, i ∈ Iˆn , and t ≥ 0, 

(n) log(e + Q(X))Q(X) log(e + n) + log(e + Q(X))Q(X)t . (3.58) |v i (t)| ≤ Cˆ 4

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From the bound (3.58) it follows there exists Cˆ 2 > 0 such that, for any X ∈ X , [log(e + E)]−1 E ≤ C0 Q(X), i ∈ N ∪ {0}, n ≥ |r i · ν|, and t ≥ 0, (n) |v i (t)| ≤ Cˆ 2 log(e + Q(X))Q(X) log(e + |r i · ν| + Q(X)) +Cˆ 2 log(e + Q(X))Q(X)t,

(3.59)

which implies (2.13) for the infinite dynamics. The proof of (3.59) is achieved by following the same strategy used to prove Proposition 3.2.We first observe that, for k =  . Int [|r i · ν|] + 1, setting nˆ ∗ = Int α(k 2 + Q(X)8η+4 )et , any n-partial dynamics with n ≤ nˆ ∗ satisfies a bound like (3.59). We next compare different n-partial dynamics, by showing that the difference is negligible for n > nˆ ∗ . Let us remark that in this case each particle i ∈ Iˆn can interact, during the time [0, t], only with particles initially contained in (r i · ν, pˆ n (t)), where 

. log(e + Q(X))Q(X) log(e + n) + log(e + Q(X))Q(X)t . pˆ n (t) = 1 + 2Cˆ 4 t ˆ X; ˆ µ, R), the latter is sufficient Moreover, since (r 0 ) is now included in the energy Q( by itself to control the Lipschitz constant of the force. We omit the details.   Remark 3.1. As announced in the previous section, the existence of the infinite dynamics is essentially contained in our proofs. If [log(e + E)]−1 E > C0 Q(X), since Tn = +∞, this follows from definitions (3.16), (3.23), and the absolute convergence of the series  −1 n uk (n, t). The same reasoning applies in the case [log(e + E)] E ≤ C0 Q(X) by exploiting the proof of (2.13) (which we have only sketched above). We omit the details.

4. The One-Dimensional Case In this section we prove Theorem 2.3. We recall that in this case we further assume that the interaction  among the background particles is regular. This assumption is necessary for our argument to hold, but we expect the result to be true for singular interactions too. The solution to Eqs. (2.15) is obtained as the limit (in the local topology, see Theorem 2.1) of the n-partial dynamics t → (x (n) (t), X (n) (t)) obtained by neglecting all the particles initially outside the interval [−n, n]. Clearly all the results of the previous section can be proved also in this case (actually with some simplifications due to the one dimension and to the non singularity of the potential ). In particular, there exist C0 , C1 , C2 positive such that, for any X with Q(X) < +∞, the following holds. If [log(e + E)]−1 E > C0 Q(X), for any n ∈ N,

log(e + E) |v (n) (t) − Et| ≤ C0 Q(X) +t ∀ t ≥ 0, √ E 

(n) Q(X) log(e + |ri | + E) + Q(X)t ∀ t ≥ 0 ∀ i ∈ In , |vi (t)| ≤ C1 (n)

|v (n) (t) − vi (t)| ≥

Et 2

∀ t ≥ tE

(4.1) (4.2)

∀ i ∈ An (t), (4.3)

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561

where tE is defined in (3.45), In = {i ∈ N : |ri | ≤ n}, and An (t) is the obvious analogue of (3.3) in one dimension. If [log(e + E)]−1 E ≤ C0 Q(X) then, for any i ∈ N and n ≥ |ri |, 

(n) |v (n) (t)| + |vi (t)| ≤ C2 log(e + Q(X)) Q(X) log(e + |ri |) + Q(X)t . (4.4) We have only to prove a bound like (2.16) for large E. From now on we assume E ≥ 1 and [log(e + E)]−1 E > C0 Q(X). Let n0 = Int[αE 6 et ] with α large enough. By reasoning as in the proof of Proposition 3.2 (with η = 1 since  is now bounded and  is absent), we can show that there exists C19 > 0 such that |v (n) (t) − v (n0 ) (t)| ≤ e−C19 E

∀ n ≥ n0 .

(4.5)

Note in fact that, by definition (3.16), the right-hand side of (3.37) is also an upper bound ∗ for |v (n) (t) − v (n ) (t)|, and it is exponentially small for E → +∞. Let us consider the case n ≤ n0 . From here to the end of the section we use the shortened notation (x(t), X(t)) for (x (n) (t), X (n) (t)). For t ≥ tE we define
(r(t) − ri (t)) . . p(t) = v(t) − Et + M(v(t) − vi (t)) i

By the equations of motion,
(r(t) − ri (t))∇ (r(t) − rj (t)) p(t) ˙ =+ M 2 [v(t) − vi (t)]2 i,j

−


(r(t) − ri (t))∇(ri (t) − rj (t)) M[v(t) − vi (t)]2

i =j

−


(r(t) − ri (t))[ME − ∇ (ri (t) − r(t))] M 2 [v(t) − vi (t)]2

i

and therefore, by (4.3), there exists C20 > 0 such that, for t ≥ tE ,    t N (s) 1 N (s) ds √ |p(t) − p(tE )| ≤ C20 + 3/2 , √ E s Es Es tE

,

(4.6)

where N(s) = {i ∈ In : |ri (s) − r(s)| ≤ 2}. Since  is superstable, by the definition of Q(X; µ, R), there exists C21 > 0 such that N (s)2 ≤ C21 Q(X(s); r(s), 2); then, by (the one-dimensional version of) Lemma 3.1, for suitable C22 , C23 > 0, 

N (s)2 ≤ C22 Q(X) log(e + n) + s max sup |vi (τ )| i∈In τ ∈[0,s]

 ≤ C23 Q(X) log(e + E) + [1 + Q(X)]s 2 , 

(4.7)

where in the last inequality we used (4.2) and n ≤ n0 . We next use (4.7) to bound the term N (s)/(E 3/2 s) in (4.6), by getting (recall E ≥ max{1; C0 Q(X)}), for a suitable C24 > 0,  t
 t N (s) 1 |p(t) − p(tE )| ≤ C24 ds √ ds √ χ (|r(s) − rj (s)| ≤ 2). = C24 Es Es tE tE j

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P. Butt`a, E. Caglioti, C. Marchioro

An upper bound for the right-hand side of the above inequality can be obtained by the same reasoning used for the estimate of the term Fk (t) defined in (3.48). Due to the extra √ term ( Es)−1 we obtain in this case, for some C25 > 0, |p(t) − p(tE )| ≤ C25 Q(X)

log(e + E) + log R¯ E (t) , E

(4.8)

. where R¯ E (t) = ρ[log(e ¯ + E) + Et 2 ] for a suitable ρ¯ > 1. We omit the details. We now conclude the proof of (2.16); for t ∈ [0, tE ] the inequality follows by (4.1). For t > tE we estimate: |v(t) − Et| ≤ |v(t) − Et − p(t)| + |p(t) − p(tE )| + |p(tE )|. By (4.3), (4.7), √of p(t), the first term on the right-hand side is bound√ and the definition ed by C26 Q(X) log(e + E)/ E for any t > tE and some C26 > 0. By (4.8), the second term is bounded √ by C27 Q(X) log(e +√E + Et 2 )/E, with C27 > 0. Finally |p(tE )| ≤ C28 [Q(X) + Q(X)] log(e + E)/ E, with C28 > 0. We thus obtain the inequality (2.16) for C¯ 0 ≥ C0 large enough and [log(e + E)]−1 E > C¯ 0 Q(X). Clearly, for [log(e + E)]−1 E ≤ C¯ 0 Q(X), the estimate (2.18) (for a suitable C¯ 2 > 0) is a consequence of (4.1), (4.2), and (4.4). Theorem 2.3 is proved.   5. Comments In the previous sections we proved that the force due to a large electric field cannot be balanced by a bounded interaction between the charged particle and the surrounding medium. What happens when this interaction becomes singular? No rigorous results are available on these topics, but it is possible to formulate some conjectures. The first problem we discuss is the case when an hard core interaction between the charged particle and the other ones is present. In this case the “folklore” suggests that the charged particle reaches a limit velocity (whose intensity depends, of course, on the background state and on the electric field intensity). It is not possible (as we know) to give a general proof of this statement, but only to report the investigation of some particular models that suggest this result. There are some one-dimensional models [1, 2, 13, 16, 17], but, due to the topology of the space, they have the nonphysical property that a charged particle cannot overcome another one. To avoid this difficulty some higher dimensional models have been studied [4, 5, 9]. Let us recall the model studied by Boldrighini and Soloveitchik [4, 5]: in the plane a rigid rod of length L, mass M, and charge q, stays parallel to a fixed axis (say y) and moves along the x-axis under the action of an external electric field E. It hits with a gas of free particles of mass m via the usual law of the elastic collision. The authors assume the free gas to be distributed according to a slight modification of the Poisson law: they neglect the particles with a small y-component of the velocity. This assumption is technical, being needed to control the number of possible re-collisions. They prove that the charged particle has a finite limit velocity, and the Ohm’s law and the Einstein relation between drift and diffusion hold for small electric fields. The main objections to this model are the constraint on the rod and the fact that it takes into account only in part the difficult phenomena of the re-collisions. We however believe that the latter do not play an essential role in the existence of a limit velocity. Of course the extension of these results to an actual hard sphere gas is quite hard, and only further rigorous results may clarify the problem.

Motion of a Charged Particle Interacting with an Infinitely Extended System

563

It is not worthless to remark that the limit m → ∞ cannot be exchanged with t → ∞. In fact for the so-called Lorentz gas, in which the background is composed by scatterers of infinite mass, the (average) velocity of the charged particle vanishes as t → ∞ although its absolute value diverges [14]. In conclusion, the state of the art is the following: for bounded interactions |v(t)| → ∞ as t → ∞, while for hard core it seems that |v(t)|average < ∞ as t → ∞. What happens for interactions diverging as r −α , α > 0? We formulate the conjecture that the threshold effect corresponds to α = 2: for α < 2 the long time behavior is similar to the one of the bounded case, and for α > 2 is similar to the one of the hard core case. The arguments to justify this conjecture are very rough and perhaps too naive. We first discuss the scattering of the charged particle with a particle of the background alone. The asymptotic behavior depends on the “impact parameter” λ: if it is small, say λ < Const. a, the scattering behavior is similar to the hard core case, while if it is large, say λ > Const. a, it is similar to the bounded case; the above parameter a is the minimal distance reached during the scattering between the two particles. When the two particles reach their minimal reciprocal distance, part of the initial kinetic energy EK = Mv 2 /2 is transformed in potential energy Const. a −α . Inverting the relation Mv 2 /2 ≈ Const. a −α , we have a ≈ Const. |v|−2/α , i.e. the scattering process behaves as in the hard core case if λ < λ0 = Const. |v|−2/α . For the moment we neglect the scattering process with large λ, which will be discussed later on. We are thus reduced to consider a model “like” the one in [4], where in three dimensions the rod becomes a disk of radius L and in which the length L depends on the velocity v of the disk as L ≈ Const. |v|−2/α .

(5.1)

On the other hand it is possible to discuss roughly in which case a model like this one may have a finite limit velocity v∞ . In fact, if the disk reaches a stationary state with a velocity v∞ , then the work LW done by the electric field E during time T is LW = v∞ T qE.

(5.2)

We now compute the energy lost by the disk during time T . The disk hits the particles of the background contained in a tube with base of area π L2 and height v∞ T , whose number is proportional to πL2 v∞ T . We assume that M > m, so that in each collision with a slower particle the disk loses a fraction of its energy Mv 2 /2. Hence (neglecting re-collisions and the collision with fast particles) the disk loses an energy proportional to 3 L2 v∞ T.

(5.3)

We observe that in a stationary state (5.3) must be equal to (5.2): 3 v∞ T qE ≈ Const. L2 v∞ T,

(5.4)

which implies Lv∞ ≈ Const.

√

E.

(5.5)

Putting (5.1) in (5.5), we obtain a threshold for α = 2. We want now to take into account also the scattering process with large impact parameter. We modify the previous model in the following way. A disk of radius equal to the range of the interaction is divided into a central disk of radius λ0 and circular

564

P. Butt`a, E. Caglioti, C. Marchioro

concentric rings of internal radius λ0 2k , k ∈ N. If a particle of the background hits on the central circle then it has an elastic collision, if it hits on a ring of internal radius λ0 2k then it feels a force of intensity Fk = (λ0 2k )−α−1 during a collision time which −1 . The slowing down due to the elastic collisions has been previis of order λ0 2k v∞ ously calculated. It is easy to evaluate the effect due to the collisions on the k-ring: v∞ · Fk · [time of collision] · [number of collisions]. The last number is proportional to the volume of a tube with base of area proportional to (λ0 2k )2 and height v∞ T . In conclusion the slowing down effect is proportional to ∗

3 Const. λ20 v∞ T + Const. v∞

k


(λ0 2k )−α−1 (λ0 2k )2 T ,

(5.6)

k=1

where −2/α

λ0 ≈ Const. v∞

(5.7)

and k ∗ is chosen such that ∗

λ0 2k ≈ [range of the interaction].

(5.8)

∗ Remembering that kk=1 (λ0 2k ) ≈ [range of the interaction], a comparison of (5.6) with (5.4) gives the threshold α = 2 as in the previous estimate. Of course the whole justification of the conjecture is extremely rough and the task to prove it rigorously seems too hard. It is interesting to remark that, following this conjecture, the Coulomb-like interaction behaves, with respect to the limit velocity, as a bounded one. Let us now briefly discuss the case without any confinement, i.e. when the particles move in the whole space R2 or R3 . In this case the problem discussed in the present paper is clearly interesting, but the rigorous results are very scanty. Even the existence of the dynamics for infinite particles (i.e. the fact that the evolution remains quasi-local) is not obvious. In fact, while for interactions like those assumed in Sect. 2 the existence of the dynamics has been proved in R2 [10], the extension in R3 is very hard and it has only recently been obtained for bounded interactions [8]. The central point in the proofs is the control of the local density and energy of the particles. In two and three dimensions this control is very bad for large times and an analogue of our proof is not possible. We are only able to study a simpler model: the motion of a charged particle interacting by a bounded force with an infinite free gas. The main problem in this case is the choice of good initial data, in order to avoid that the charged particle might have interacted with large concentrations of the background. Initial data like Q(X) < ∞ are too large and a good choice needs a deeper investigation. It seems in any case reasonable to believe that, with such a choice of initial data, the charged particle does not have a finite limit velocity. We finally discuss the case in which the electric field E is small. Common experience suggests that in real systems the charged particle still reaches a finite limit velocity. Moreover, by the linear response theory, the drift should be proportional to the electric field (Ohm’s law). It is natural to wonder whether the singularity of the interaction is essential for these phenomena to occur. The models investigated in the present paper are too complicated to give a hint in this direction. However the analysis of a simpler model suggests that singular interactions are necessary for Ohm’s law to hold.

Motion of a Charged Particle Interacting with an Infinitely Extended System

565

We consider a charged particle in one dimension interacting via a bounded short-range potential with a background of free particles. We suppose that initially the charged particle is at rest in the origin and the background has a free particle distribution with inverse temperature β. The system evolves via Newton’s law, where only the interactions of the charged particle with an external field E and with each particle of the background are present. The drift d(E) is defined by x(t) , (5.9) t where · is the average with respect to the initial Gibbs distribution and x(t) is the position of the charged particle. In the framework of the classical diffusion theory accepted in Physics, the motion of the charged particle, in a diffusive space-time scaling, is described as the sum of a drift term plus a diffusion term. Moreover both the drift d(E) and the diffusivity σ 2 (E) are continuous functions of E and the Einstein relation holds, i.e. d(E) = lim

t→∞

d(E) β = σ 2 (0). E→0 E 2 lim

(5.10)

On the contrary, we assert the drift to be infinite in the above model. A rigorous mathematical proof of this is out of the aim of the present paper. We instead give some arguments to convince the reader of the validity of our assertion. Let Q0 > 0 be so large that the set A = {X : Q(X) ≤ Q0 } has a non-vanishing measure. For any L > 0 the subset BL ⊂ A defined by all the configurations for which there are no particles in the interval [−L, L] has a non-vanishing measure. We claim that, if L is large enough and the initial distribution is chosen in BL , then x(t) diverges as Et 2 /2. In fact, if L is large enough, the charged particle reaches the position L/2 with large velocity V without interacting with any other particle. On the other hand, if V is large enough, by means of the same mechanism discussed in Sect. 4, the charged particle cannot be slowed down by the background, and its asymptotic dynamics is a uniformly accelerated motion. Since x(t) = x(t)χ (BL )+x(t)χ (BLc ), it remains to convince ourselves that the second term on the right-hand side cannot diverge as −Const t 2 . This is very reasonable. Note in fact that, by symmetry, it vanishes for E = 0, and it is very unlikely for a positive electric field to produce a negative average acceleration. Appendix In this appendix we prove Lemma 3.1 and the bound (3.55). A notation warning: in the sequel we shall denote by C a generic positive constant whose numerical value may change from line to line and it may possibly depend only on the interactions , , and . Proof of Lemma 3.1. We introduce a mollified version of Q(X; µ, R) by defining  2  1
.
µ,R v i W (X; µ, R) = + (r i ) + fi (r i − r j ) + 1 , (A.1) 2 2 i

j :j =i

where µ,R fi

=f

|r i · ν − µ| R

(A.2)

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P. Butt`a, E. Caglioti, C. Marchioro

and f ∈ C ∞ (R+ ) is not increasing and satisfies: f (x) = 1 for x ∈ [0, 1], f (x) = 0 for x ≥ 2, and |f  (x)| ≤ 2. Clearly: Q(X; µ, R) ≤ W (X; µ, R) ≤ Q(X; µ, 2R). For 0 ≤ s ≤ t < Tn , we define . Rn (t, s) = log(e + n) +



t 0

 dτ Vn (τ ) +

t s

dτ Vn (τ )

(note that Rn (t, t) = Rn (t) and Rn (t, 0) ≤ 2Rn (t)) and compute


 µ,R (t,s) ∂s W (X (n) (s); µ, Rn (t, s)) = κi (t, s)εi (s) + fi n ε˙ i (s) ,

(A.3)

(A.4)

(A.5)

i

µ

where, denoting by ri (s) the sign of r i (s) · ν − µ, 

 µ ri (s)v i (s) · ν ∂s Rn (t, s)  |r i (s) · ν − µ| κi (t, s) = f |r i (s) · ν − µ| , − Rn (t, s) Rn (t, s) Rn (t, s)2 εi (s) =

 v i (s)2 1
  r i (s) − r j (s) + 1, + (r i ) + 2 2 j :j =i

and, to simplify notation, we have omitted the explicit dependence on n of r i , v i , κi , and εi . Since f  (|y|) ≤ 0, f  (|y|) = 0 if |y| ≤ 1, ∂s Rn (t, s) = −Vn (s), and |v i (s) · ν| ≤ Vn (s), then κi (t, s) ≤ 0. On the other hand, from the equations of motion,
v i (s) + v j (s) ε˙ i (s) = −∇ (r i (s) − r(s)) · v i (s) − ∇(r i (s) − r j (s)) · . 2 j :j =i

Then, by (A.5) and using ∇ is odd, ∂s W (X (n) (s); µ, Rn (t, s))
µ,R (t,s) fi n ∇ (r i (s) − r(s)) · v i (s) ≤− i

1
 µ,Rn (t,s) µ,R (t,s)  − fj n − fi ∇(r i (s) − r j (s)) · v i (s). 2

(A.6)

i =j

From (2.1) we have |ξ | · |∇(ξ )| ≤ C[1 + (ξ )] for any ξ = 0. Then, by the inequality  µ,R  |r i − r j |  µ,R  f χi (µ, 2R) + χj (µ, 2R) , − fj  ≤ 2 i R and since Rn (t, s) > 1, the modulus of the double sum in the right-hand side of (A.6) can be bounded from above by ∂s Rn (t, s)
−C [1 + (r i (s) − r j (s))]χi (µ, 4Rn (t, s))χj (µ, 4Rn (t, s))χi,j (s), Rn (t, s) i =j

(A.7)

Motion of a Charged Particle Interacting with an Infinitely Extended System

567

where we shortened χi,j (s) = χ (|r i (s) − r j (s)| ≤ 1). Since  is superstable, by arguing as in the proof of Eq. (2.15) in [8], the double sum in the right-hand side of (A.7) can be bounded by CW (X(n) (s); µ, 4Rn (t, s)); moreover, setting . W (X; R) = sup W (X; µ, R),

(A.8)

W (X; µ, 2R) ≤ CW (X; R)

(A.9)

µ

it can be proved that

(see e.g. [8, 7]), and hence, by (A.6),


∂s W (X (n) (s); µ, Rn (t, s)) ≤

i

−C

 ∇ (r i (s) − r(s)) · v i (s)

µ,Rn (t,s) 

fi

∂s Rn (t, s) W (X (n) (s); Rn (t, s)), Rn (t, s)

from which, by integrating and taking the supremum on µ, W (X(n) (s); Rn (t, s)) ≤ W (X(n) (0); Rn (t, 0))  s
µ,R (t,τ )   ∇ (r i (τ ) − r(τ )) · v i (τ ) + sup dτ fi n µ

0



s

−C 0

i

∂τ Rn (t, τ ) dτ W (X (n) (τ ); Rn (t, τ )). Rn (t, τ ) (A.10)

In the sum on the right-hand side of (A.10), only the particles which are initially in (µ, 4Rn (t, 0)) can contribute; the number of these particles is bounded by W (X (n) (0); 4Rn (t, 0)). By (3.46) for k = n we have, if Tn > tE with tE as in (3.45),  Eτ v(τ ) − v i (τ ) · ν ≥ 2

 Then:

 0

s

dτ


i

∀ i ∈ An (τ ) ∀ τ ∈ (tE , Tn ).

(A.11)

 ∇ (r i (τ ) − r(τ )) · v i (τ )

µ,Rn (t,τ ) 

fi

≤ W (X(n) (0); µ, 4Rn (t, 0))∇ ∞  + χ (Tn > tE ) max i

Tn

tE



√ EtE E+ 5

tE 

dτ χ(|r i (τ ) − r(τ )| ≤ 1)|v i (τ )| .

(A.12)

We observe that if Tn > tE then the bound (A.11) implies that for each i ∈ N there exists an interval [si1 , si2 ] ⊆ [tE , Tn ) such that:   χ τ ∈ [tE , Tn ) χ (|r i (τ ) − r(τ )| ≤ 1) ≤ χ ([si1 , si2 ]),

|si2 − si1 | ≤

2 . Esi1

568

P. Butt`a, E. Caglioti, C. Marchioro

Since |v i (τ )| ≤  χ (Tn > tE )

√ E + Eτ/5, it follows that Tn

tE

dτ χ (|r i (τ ) − r(τ )| ≤ 1)|v i (τ )| ≤

2 Esi1



√

E+

2 Esi1 + 5 5si1

,

and the right-hand side of the above inequality is bounded by a constant since Esi1 ≥ EtE . Hence, by (A.12), we conclude that, for any µ ∈ R,  s
µ,R (t,τ )   ∇ (r i (τ ) − r(τ )) · v i (τ ) ≤ CW (X (n) (0); µ, 4Rn (t, 0)). dτ fi n 0

i

(A.13) Inserting (A.13) in (A.10) we obtain a differential inequality which can be solved getting (for some positive constants c1 and c2 )

Rn (t, 0) c2 (n) (n) . W (X (s); Rn (t, s)) ≤ c1 W (X (0); Rn (t, 0)) Rn (t, s) Setting s = t and using that Rn (t, 0) ≤ 2Rn (t, t) = 2Rn (t), W (X (n) (t); Rn (t)) ≤ CW (X (n) (0); Rn (t)). Then, from (A.3), (A.8), and definition (2.5), we conclude that, for any t ∈ [0, Tn ), Q(X(n) (t); µ, Rn (t)) ≤ CW (X (n) (0); Rn (t)) ≤ C sup Q(X (n) (0); µ, 2Rn (t)) µ

≤ 4CQ(X)Rn (t), which proves (3.7).

 

ˆ X; ˆ µ, R) of Q( ˆ µ, R) defined Proof of (3.55). We introduce the mollified version Wˆ (X; analogously to (A.1). By arguing as before and using the equations of motion we have in this case: ˆ (n) (s); µ, Rˆ n (t, s)) ≤ −f µ,Rn (t,s) E · v 0 (s) ∂s W (X 0 1
 µ,Rˆ n (t,s) µ,Rˆ (t,s)  ˆ − fi ∇ i,j · v i (s), − fj n 2

(A.14)

i =j

where Rˆ n (t, s) is defined as Rn (t, s) in (A.4) with Vn (·) replaced by Vˆn (·). Noticing that  s µ,R (t,s) dτ f0 n |E · v 0 (τ )| ≤ E Rˆ n (t), 0

by integrating the differential inequality (A.14) we get in this case:   ˆ (n) (t); µ, Rˆ n (t)) ≤ C E Rˆ n (t) + sup Wˆ (X ˆ (n) (0); µ, Rˆ n (t)) . sup Wˆ (X µ

µ

(A.15)

Motion of a Charged Particle Interacting with an Infinitely Extended System

569

ˆ X; ˆ X; ˆ µ, R) ≤ Wˆ (X; ˆ µ, R) ≤ Q( ˆ µ, 2R) and, if R > log(e + |µ|), Since Q(
ˆ X; ˆ µ, R) ≤ Q(X; µ, R) +

(r j ) ≤ 2(1 +  ∞ )Q(X)R, Q( j =0

by (A.15) we finally obtain that, for any [log(e + E)]−1 E ≤ C0 Q(X),   ˆ X ˆ X ˆ (n) (t); µ, Rˆ n (t)) ≤ C E Rˆ n (t) + sup Q( ˆ (n) (0); µ, 2Rˆ n (t)) sup Q( µ

µ

≤ C log(e + Q(X))Q(X)Rˆ n (t), which proves (3.55).

 

References 1. Boldrighini, C.: Bernoulli property for a one-dimensional system with localized interaction. Commun. Math. Phys. 103, 499–514 (1986) 2. Boldrighini, C., Cosimi, G.C., Frigio, S., Nogueira, A.: Convergence to a stationary state and diffusion for a charged particle in a standing medium. Probab. Theory Relat. Fields 80, 481–500 (1989) 3. Boldrighini, C., Pellegrinotti, A., Presutti, E., Sinai, Ya.G., Soloveychik, M.R.: Ergodic properties of a semi-infinite one-dimensional system of statistical mechanics. Commun. Math. Phys. 101, 363–382 (1985) 4. Boldrighini, C., Soloveitchik, M.: Drift and diffusion for a mechanical system. Probab. Theory Relat. Fields 103, 349–379 (1995) 5. Boldrighini, C., Soloveitchik, M.: On the Einstein relation for a mechanical system. Probab. Theory Relat. Fields 107, 493–515 (1997) 6. Dobrushin, R.L., Fritz, J.: Non-equilibrium dynamics of one-dimensional infinite particle systems with a hard-core interaction. Commun. Math. Phys. 55, 275–292 (1977) 7. Caglioti, E., Marchioro, C.: On the long time behavior of a particle in an infinitely extended system in one dimension. J. Stat. Phys. 106, 663–680 (2002) 8. Caglioti, E., Marchioro, C., Pulvirenti, M.: Non-equilibrium dynamics of three-dimensional infinite particle systems. Commun. Math. Phys. 215, 25–43 (2000) 9. Calderoni, P., Durr, D.: The Smoluchowski limit for a simple mechanical model. J. Stat. Phys. 55, 695–738 (1989) 10. Fritz, J., Dobrushin, R.L.: Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction. Commun. Math. Phys. 57, 67–81 (1977) 11. Lanford, O.E.: Classical Mechanics of one-dimensional systems with infinitely many particles. I An existence theorem. Commun. Math. Phys. 9, 176–191 (1968) 12. Lanford, O.E.: Classical Mechanics of one-dimensional systems with infinitely many particles. II Kinetic Theory. Commun. Math. Phys. 11, 257–292 (1969) 13. Pellegrinotti, A., Sidoravicius, V., Vares, M.E.: Stationary state and diffusion for a charged particle in a one-dimensional medium with lifetimes. Teor. Veroyatnost. i Primenen. 44, 796–825 (1999); translation in Theory Probab. Appl. 44, 697–721 (2000) 14. Piasecki, J., Wajnryb, E.: Long-time behavior of the Lorentz electron gas in a constant, uniform electric field. J. Stat. Phys. 21, 549–559 (1979) 15. Ruelle, D.: Statistical Mechanics. Rigorous Results. New York-Amsterdam: W.A. Bejamin, Inc, 1969 16. Sidoravicius, V., Triolo, L., Vares, M.E.: Mixing properties for mechanical motion of a charged particle in a random medium. Commun. Math. Phys. 219, 323–355 (2001) 17. Sinai, Ya.G., Soloveychik, M.R.: One-dimensional classical massive particle in the ideal gas. Commun. Math. Phys. 104, 423–443 (1986) Communicated by J.L. Lebowitz