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 Mathe...

Author: M. Aizenman (Chief Editor)

26 downloads 821 Views 5MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

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, Università 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

2

F. Guerra

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)

4

F. Guerra

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.


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

6

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 .


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.

8

F. Guerra

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)


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].

10

F. Guerra

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


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).

12

F. Guerra

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., Mézard, 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. Mézard, M., Parisi, G., Virasoro, M.A.: Spin glass theory and beyond. Singapore: World Scientific, 1987 13. Mézard, 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


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

Mathematisches Institut, LMU München, 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 Schrödinger 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 Schrödinger operator. Although the fundamental properties of the one-particle and multi-particle Schrödinger 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, Fröhlich, 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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


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 .


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


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)


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


|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)


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


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)


As the next step we choose d
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


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)


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
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 Prüfer 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)


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


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)


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 Prüfer 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


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 Prüfer variables. Let now E ∈ R and uE be given by (19). The free Prüfer 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 Prüfer 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)


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 Prüfer 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 Prüfer 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 Prüfer 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 Prüfer 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


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 Prüfer 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 Prüfer 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 Prüfer 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 Prüfer variables with some initial condition, then according to (30), N (E) =

lim

N→∞

1 1 E E θω (N ) . π N

(32)


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


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 .


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 Prüfer 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 Prüfer 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 πL± k→∞ k k−1 1 1 l l = E0 S,ωl (S,ω (θ )) − S,ω (θ ) . lim πL± 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


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.


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 ) . πL±

(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 2L± 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


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 Hölder 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 )


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


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


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


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.


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 Université 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 Bièvre, S., Germinet, F.: Dynamical Localization for the Random Dimer Schrödinger Operator. J. Stat. Phys. 98, 1135–1148 (2000) Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schrödinger Operators. (Boston: Birkhäuser, 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 Schrödinger 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]


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


Communications in


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


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 ,


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


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)


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


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)


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)

58


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)


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


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)


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


−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.


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 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


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.


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)

68


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.


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)

70


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)


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 0 for n ≥ 1. Thus, Theorem 3 also holds for d > 1.


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


(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)


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


Proof. Let ϕ =

n λn P n

ϕ 0, = λn Pn and ϕ≥a = λn P n . (176) λn 0 and a constant c1 such that V (ϕ 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.

90

A. Kapustin, D. Orlov

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) .

Vertex Algebras, Mirror Symmetry, and D-Branes

91

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)

92


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,


93

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)

94


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.


95

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

96


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.


97

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 Kähler 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.

98


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)


99

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),

100


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

s0

s 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


113

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

114


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 − .


115

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].

116


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)


117

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

118


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 Kähler manifold. Let ω = GI be the corresponding Kähler 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.


119

− ¯ ¯+ ¯− 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 Kähler 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.

120


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


121

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 +

122


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.


123

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.

124


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.


125

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,

126


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)


127

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.

128


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)


129

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

130


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.


131

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

132


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,


133

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

134


∇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)).


135

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., Sénéchal, 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

136


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: Birkhäuser, 1995, pp. 120–139 24. Lüst, 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


Communications in


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) ||

138

C. Maes, E. Verbitskiy

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.

Fluctuation Symmetry for Chaotic Homeomorphisms

139

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 Gärtner-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.

140


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).


141

Example. For a hyperbolic diffeomorphism f , any Hölder 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)

142


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).


143

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 Gärtner– 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.

144


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 ),


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 Gärtner-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.

146


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


147

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)

148


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)


149

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

150


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


151

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., Netoˇcný, 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


Communications in


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.

154

J. Bourgain, E. Lindenstrauss

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,

Entropy of Quantum Limits

155

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)

156


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 .


157

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)).

158


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.


159

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 .

160


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)


161

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|
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.


163

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.

164


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 ) = ∅.


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)

166


η≤p≤ξ

w(p) 1− p

−1

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 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 θ > θ . There are two ways to build such an intertwiner from the two-soliton S-matrices:

Quantum Group Symmetry

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)

176

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


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)

178


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


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)

180


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)


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

182


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)


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.

184


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)


185

so that the new conserved charges are ¯ i + î q Ti , i = Qi + Q Q

(4.11)

where, to first order in perturbation theory, λb i βˆ 2 . 2πc 1 − βˆ 2

î =

(4.12)

Note that at this stage the boundary parameters î can still take arbitrary values. However, we shall see in subsequent sections how the |î | 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 − î ). . . . , 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 + î ((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

+ î ((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 = î (q −1 − q)K i i + x K i i+1 − x −1 K i+1 i ,

0=

i

i+1 − K i , î q K i j + x −1 K i+1 j , î 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)

186


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 î = 0 for all i or that |î | = 1 for all i. If all î = 0 then K can be an arbitrary diagonal matrix. If all |î | = 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 ) = î · · · ˆj −1 (−q x)i−j +(n+1)/2 k(θ),

for j > i,

K j i (θ ) = î · · · 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 î 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)


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

188


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


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


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].

314

N.J. MacKay, B.J. Short

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 ansätze 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

Boundary Scattering, Symmetric Spaces and Principal Chiral Model

315

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].

316


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 . −∞


317

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)

318


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


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

320


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 ansätze 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 .


321

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

322


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.


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


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.


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

.

326


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

(


= −( ), = [1]σo (2θ )

= 0 and either

Sp(2n)

)t

(


= , = σp (2θ )

c τ (iπ/2−θ) τ (iπ/2+θ)

=

2ih π ,

( )t = −( ), h = [1] h2 iπc − h2 σp (2θ)

or

)t

(


SU (N ) conjugated

= −( ), = [1]σp (2θ ) )t = = σu (2θ )

(

no solution


or

(

)t = −( ), = −[1]σu (2θ )


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.


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

328


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)


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


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)


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


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


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


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)


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)

336


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.


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.

338


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 )},


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.

340


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.


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} .

342


(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.


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


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.


345

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.

346


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


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.

348


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 .


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

.

350


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.


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.

352


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 Gérard 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 Mathématiques, U de Montréal, 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., González-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 Lüscher, 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., Fröhlich, 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., Köberle, 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


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


Communications in


On the Sutherland Spin Model of BN Type and Its Associated Spin Chain F. Finkel, D. Gómez-Ullate, A. González-López ∗ , M.A. Rodr´ıguez, R. Zhdanov∗∗ Departamento de F´ısica Teórica 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

192

F. Finkel, D. Gómez-Ullate, A. González-López, 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)

194


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)


195

We shall also make use of the following commutation relations between the operators Kij , Ki and the operators Ji : i 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


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 .

198


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 .


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 .

200


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 .


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


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 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


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,


−

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 =


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)


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


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., Gómez-Ullate, D., González-López, 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., Gómez-Ullate, D., González-López, 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)


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


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 Matemática y Estad´ıstica “Prof. Ing. Rafael Laguardia” (IMERL), Facultad de Ingenier´ıa, Universidad de la República, 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 Técnico, 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)

214


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 ∈


215

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

216


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,


217

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+

218


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.


219

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

220


+ 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

Qz (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.


221

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

222


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)


223

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).

224


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


225

√ √ 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)

226


˜ 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


227

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.

228


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


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


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 República) 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


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.)

232

E. Koelink, J Kustermans

[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 quanq (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

Locally Compact Quantum Group

233

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

234


∗ 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. Schmüdgen 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


235

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,

236


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 ∗ -alq (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.


237

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 .

238


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


239

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:

240


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.


241

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◦ .

242


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].


243

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

244

E. Koelink, J Kustermans β

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).


245

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 ( apθ | 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.

246


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


247

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)

248


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.


249

˜ : 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.

250


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


251

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.

252

E. Koelink, J Kustermans ϕ

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)


253

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 )


× 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 ) |


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 )



× 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 )



× 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 ) .

254


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


=

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) .

256


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)) .


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.

258


˘ 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)∗ .


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 .

260


Then Eq. (4.4) and Proposition 3.5 imply that (ι ⊗ ω)(W ∗ ) fn,c,d =


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 =



× 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)


× 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).


261

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)) .

262


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 .


263

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)

264


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 ) ∈


265

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

266


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) .


267

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

268


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 .


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

270


˜ ⊗ 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


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


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,


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 )

274


= 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 .


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).

276


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.


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


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 )∞


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).

280


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 ) ∞


281

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)

282


+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


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)

284


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.


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

286


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 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) )

288


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 .


289

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

290


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.


291

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) .

292


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 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.


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) .

294


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)


295

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 algèbres d’opérateurs dans l’espace hilbertien. Deuxième 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


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 69ème rencontre entre physiciens et théoriciens et mathématiciens. 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. 4è série, 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. Schmüdgen, K.: Unbounded operator algebras and representation theory. In: Operator Theory 37, Basel-Boston: Birkhäuser, 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


Communications in


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

298

Dongho Chae, Jihoon Lee

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 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

Communications in Mathematical Physics - Volume 233

Communications in Mathematical Physics - Volume 221

Communications in Mathematical Physics - Volume 220

Communications in Mathematical Physics - Volume 235

Communications in Mathematical Physics - Volume 223

Communications In Mathematical Physics - Volume 283

Communications In Mathematical Physics - Volume 270

Communications in Mathematical Physics - Volume 208

Communications in Mathematical Physics - Volume 186

Communications In Mathematical Physics - Volume 294

Communications in Mathematical Physics - Volume 217

Communications In Mathematical Physics - Volume 274

Communications in Mathematical Physics - Volume 239

Communications in Mathematical Physics - Volume 306

Communications in Mathematical Physics - Volume 264

Communications in Mathematical Physics - Volume 227

Communications in Mathematical Physics - Volume 184

Communications in Mathematical Physics - Volume 261

Communications in Mathematical Physics - Volume 225

Communications In Mathematical Physics - Volume 263

Communications in Mathematical Physics - Volume 211

Communications In Mathematical Physics - Volume 293

Communications in Mathematical Physics - Volume 246

Communications In Mathematical Physics - Volume 298

Communications in Mathematical Physics - Volume 234

Communications In Mathematical Physics - Volume 288

Communications in Mathematical Physics - Volume 304

Communications In Mathematical Physics - Volume 292

Communications in Mathematical Physics - Volume 253

Communications in Mathematical Physics - Volume 222

Communications in Mathematical Physics - Volume 260

Communications in Mathematical Physics - Volume 233

Communications in Mathematical Physics - Volume 221

Communications in Mathematical Physics - Volume 220

Communications in Mathematical Physics - Volume 235

Communications in Mathematical Physics - Volume 223

Communications In Mathematical Physics - Volume 283

Communications In Mathematical Physics - Volume 270

Communications in Mathematical Physics - Volume 208

Communications in Mathematical Physics - Volume 186

Communications In Mathematical Physics - Volume 294

Communications in Mathematical Physics - Volume 217

Communications In Mathematical Physics - Volume 274

Communications in Mathematical Physics - Volume 239

Communications in Mathematical Physics - Volume 306

Communications in Mathematical Physics - Volume 264

Communications in Mathematical Physics - Volume 227

Communications in Mathematical Physics - Volume 184

Communications in Mathematical Physics - Volume 261

Communications in Mathematical Physics - Volume 225

Communications In Mathematical Physics - Volume 263

Communications in Mathematical Physics - Volume 211

Communications In Mathematical Physics - Volume 293

Communications in Mathematical Physics - Volume 246

Communications In Mathematical Physics - Volume 298

Communications in Mathematical Physics - Volume 234

Communications In Mathematical Physics - Volume 288

Communications in Mathematical Physics - Volume 304

Communications In Mathematical Physics - Volume 292

Communications in Mathematical Physics - Volume 253

Communications in Mathematical Physics - Volume 222

Communications in Mathematical Physics - Volume 260

Recommend Documents