R.M. Kashaev- Discrete Liouville Equation and Teichmuller Theory

8/3/2019 R.M. Kashaev- Discrete Liouville Equation and Teichmuller Theory

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arXiv:0810

.4352v2

[math.QA

]27Nov2008

DISCRETE LIOUVILLE EQUATION AND TEICHMULLER

THEORY

R.M. KASHAEV

Abstract. The relationship (both classical and quantum mechanical) be-

tween the discrete Liouville equation and Teichmuller theory is reviewed.

Contents

1. Introduction 12. Classical discrete Liouville equation 32.1. Discretization from the Liouville formula 32.2. Discrete Liouville equation and Teichmuller space 43. Quantum theory 73.1. The non-compact quantum dilogarithm 73.2. Quantum discrete Liouville equation 113.3. Algebra of observables and the evolution operator 113.4. Integrable structure of the quantum discrete Liouville equation 123.5. The case N = 1 134. Relation to quantum Teichmuller theory 144.1. Highlights of quantum Teichmuller theory 144.2. Quantum discrete Liouville equation and quantum Teichmuller theory 18References 24

1. Introduction

The Liouville equation [23] is a partial differential equation of the form

(1.1)2

z z=

1

2e,

which has a number of applications both in mathematics and mathematical physics.For example, it describes surfaces of constant negative curvature, thus playing in-dispensable role in uniformization theory of Riemannian surfaces of negative Euler

characteristic [24]. Indeed, let p : H be a universal covering map for a hyper-bolic surface , where H is the upper half plane with the standard Poincare metricds2, and : U H, U , a local section of p. Then, the pull-back metric ds2

in conformal form e|dz|2, z being a local complex coordinate on U, gives a solution of the Liouville equation on U.

Date: October 2008.

Work partially supported by FNS Grant No. 200020-121675.

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2 R.M. KASHAEV

In theoretical physics, the Liouville equation is often considered in analyticallycontinued form with z = x + t and z = x t as real independent coordinates.In this case, it takes the form of a classical equation of motion for a relativistic1 + 1-dimensional field theoretical system:

(1.2)2

t2

2

x2= 2e.

The invariance with respect to (holomorphic) re-parameterizations, associated withthe diffeomorphism group of the circle, make the Liouville equation relevant to two-dimensional gravity [21] and Conformal Field Theory (CFT) [7]. It is also a basicingredient in the theory of noncritical strings [25]. These are the reasons for therecent interest in the Liouville equation, especially its quantum theory [10, 11, 18,19], and its (quantum) integrability properties [2, 3, 4].

There are few, seemingly different aspects of the relationship between the Liou-ville equation and Teichmuller theory. One such aspect is purely classical, which

comes through the above mentioned uniformization theory of surfaces of negativeEuler characteristic. It exhibits even more profound features when one considersperturbative approach to quantum Liouville theory [26, 27, 28].

Another aspect is purely quantum, and it originates from a conjecture of H. Ver-linde [30], which states that there is a mapping class group equivariant isomorphismbetween the space of quantum states of the quantum Teichmuller theory on a givensurface and the space of conformal blocks of the quantum Liouville theory on thesame surface, see [29] for the up to date situation with the Verlinder conjecture.

One more aspect of the connection of the Liouville equation to Teichmuller theoryhas been considered recently in the works [15, 14, 17] through the consideration of aspecific discretized version of the Liouville equation both on classical and quantumlevels. The discrete Liouville equation has the form

(1.3) m,n1m,n+1 = (1 + m1,n)(1 + m+1,n),

where the discrete space-time is represented by integer lattice Z2 and the dynam-ical field m,n is a strictly positive real function on this lattice. To see in what sensethis is a discretized version of the Liouville equation, let us take a small positiveparameter as the lattice spacing of the discretized space-time, and consider thecombination

(x, t) = log(2m,n)

in the limit, where 0, m, n in such a way that the products x = m,and t = n are kept fixed. If a solution m,n of the discrete Liouville equation issuch that such a limit exists, then the limiting value 0(x, t) solves the dynamicalversion (1.2) of the Liouville equation.

It is worth repeating here the remark of the paper [17] that the discrete Liouvilleequation is among the simplest examples of Y-systems [33], though it is not aY-system for which is true the Zamolodchikovs periodicity conjecture.

There are also other interesting connections of (quantum) discrete integrablesystems with (hyperbolic) geometry, see for example, [8, 5, 6, 9].

The purpose of this exposition is, following the works [15, 14, 17, 22], to reviewthe relationship of the discrete Liouville equation and the Teichmuller theory bothclassically and quantum mechanically.


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DISCRETE LIOUVILLE EQUATION AND TEICHMULLER THEORY 3

Acknoledgements. It is a pleasure to thank L.D. Faddeev and A.Yu. Volkovin collaboration with whom some of the results described in this survey were ob-tained. I would also like to thank V.V. Bazhanov and V.V. Mangazeev for valuablecomments on the initial version of the paper.

2. Classical discrete Liouville equation

2.1. Discretization from the Liouville formula. The interpretation of solu-tions of the Liouville equation in terms of pull-backs of the Poincare metric leadsto the Liouville formula for a general solution of the dynamical version ( 1.2) of theLiouville equation

(2.1) e(x,t) = 4f(u)g

(u+)

(f(u) g(u+))2, u = x t

where f(x), g(x) are two arbitrary smooth functions on the real line. From thestructure of the Liouville formula it follows that it makes sense on entire real plane

provided the functions f(x), g(x) satisfy the conditions

(2.2) f(x)g(y) < 0, f(x) = g(y), (x, y) R2.

Despite the fact that in the dynamical version ( 1.2) of the Liouville equation thecomplex analytical aspects of the uniformization of hyperbolic surfaces are somehowlost, there is still an action of the group P SL(2,R) on functions f(x) and g(x) givenby transformations

f(x) af(x) + b

cf(x) + d, g(x)

ag(x) + b

cg(x) + d

which leave unchanged the Liouville solution (2.1). One can show that the set ofP SL(2,R)-orbits of pairs (f, g) is in bijection with the set of solutions of the Liou-ville equation. For example, in the physically interesting case of periodic solutions

(x + L, t) = (x, t)

which correspond to a space-time of the topological type of cylinder S1 R, thefunctions f(x) and g(x) are quasi-periodic:

(2.3) f(x + L) =af(x) + b

cf(x) + d, g(x + L) =

ag(x) + b

cg(x) + d

where the P SL(2,R)-matrix

T[] =

a bc d

is called the monodromy matrix associated with a periodic solution .

The basic idea of A.Yu. Volkov behind the discrete Liouville equation is to writefirst a P SL(2,R)-invariant finite difference analogue of the Liouville formula ( 2.1)by replacing its right hand side by a cross-ratio of four shifted quantities f(xt),g(x + t ). Namely, if we define

(2.4) h(x, t) = (f(u + ) g(u+ + ))(f(u ) g(u+ ))

(f(u + ) f(u ))(g(u+ + ) g(u+ )), u = x t,

where we assume conditions (2.2), then it is easily seen that the limit

(x, t) = lim0

2h(x, t)


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4 R.M. KASHAEV

exists and the formula

e(x,t) =1

(x, t)

coincides with the Liouville formula (2.1). On the other hand, one observes thatfour shifted cross-ratios h(x , t), h(x, t ) depend on six values f(x t +k), g(x + t + k), with k {0, 1}. Taking into account the P SL(2)-invarianceof the cross-ratios, this means that among those four cross-ratios only three areindependent, and one identifies the following relation

h(x, t + )h(x, t ) = (1 + h(x + , t))(1 + h(x , t))

which, after the substitutions x = m and t = n takes the form of the discreteLiouville equation (1.3) for the function

(2.5) m,n = h(m,n), (m, n) Z2.

Notice, that one and the same pair of functions (f, g) is used for constructingsolutions of both the Liouville equation and its discrete counterpart.

When the quasi-periodicity conditions (2.3) are satisfied, function (2.4) is peri-odic:

h(x + L, t) = h(x, t),

so that we have periodic solutions of the discrete Liouville equation as soon as thelattice spacing is chosen to be a rational multiple of the period L. Namely, for = LM/N with positive mutually prime integers M and N, function (2.5) satisfiesthe equation

m+N,n = m,n.

In particular, when N = 1, m,n is independent of the first argument, and thediscrete Liouville equation becomes a one dimensional discrete equation of the form

n1n+1 = (1 + n)2.

This latter equation has first appeared in this context in [ 17] where it has been in-terpreted as the evolution of the zero-modes of the continuous Liouville equation.

2.2. Discrete Liouville equation and Teichmuller space. We consider an an-nulus with N marked points on each of its boundary components (2N points intotal), labeled A1, . . . , AN for one boundary component, and B1, . . . , BN, for an-other. Additionally, choose an ideal triangulation shown in figure 1, where thevariables f1, . . . , f 2N not only serve to identify the interior edges, but also denotethe associated to the triangulation the Fock coordinates in the Teichmuller spaceof the annulus. Notice that here we consider an unusual situation where all marked

r r r

r r r

f1 f2 f3 f4 f2N f2N+1

r r

r r

p p p

p p p

A1

B1

A2

B2

A3

B3

AN

BN

AN+1

BN+1

Figure 1. An ideally triangulated annulus with N marked pointson each boundary component. The leftmost and the rightmostvertical edges are identified with equalities f2N+1 = f1, AN+1 =A1, BN+1 = B1.


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points are located on the boundary of the annulus, and for parametrization of theTeichmuller space one does not need to associate coordinates on the boundary idealarcs. This can be seen by looking at a fundamental domain in the Poincare upperplane which is an ideal polygon with 2N+ 2 ideal vertices, see figure 2 for the caseN = 2.

f1

f2

f3

f4

f5

r r r r r r

H

A1 A2 A3 B3 B2 B1

Figure 2. A fundamental domain in the Poincare upper half planefor an ideally triangulated annulus with N = 2 marked points oneach boundary component. Under the covering map p : H H/Zthe images of the biggest circle with label f1 and the middle small-est circle with label f5 coincide so that p(A1) = p(A3), p(B1) =

p(B3).

Clearly, the isometry class of such a polygon is determined by 2N 1 real

parameters corresponding to positions of 2N+ 2 vertices modulo the 3-dimensionalisometry group P SL(2,R). This is less than the number of Fock coordinates fi,1 i 2N, but one more degree of freedom comes from the gluing condition: onechooses a P SL(2,R)-matrix restricted by the condition that it should map a givenordered pair of boundary points (the extremities (A1, B1) of the half circle labeledf1) to another ordered pair (the extremities (AN+1, BN+1) of the half circle labeledf2N+1), and it is known that there is a one parameter family of such matrices.

The mapping class group of our annulus is given by all homeomorphisms preserv-ing the set of marked points, not necessarily point-wise. We are interested in theunique mapping class, denoted D1/N, which fixes the set {A1, . . . , AN} point-wiseand cyclically permutes the set {B1, . . . , BN}:

B1 B2 B3 BN B1.

As the notation suggests, the N-th power of this class is nothing else than the onlyDehn twist of the annulus which fixes the boundary point-wise.

Theorem 1 ([14]). The discrete dynamical system on the Teichmuller space of anannulus with N marked points on each boundary component, corresponding to themapping class D1/N, is described by the discrete Liouville equation (1.3) on thesublattice m + n = 1 (mod 2) with the 2N-periodic boundary condition

m+2N,n = m,n,


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6 R.M. KASHAEV

the evolution step being identified with the translation along the light-cone:

m,n m,n = m1,n+1.

Proof. Recall that under a flip the Fock coordinates transform according to theformulas:

(2.6) a = a/(1 + 1/e), d = d/(1 + 1/e), b = b(1 + e), c = c(1 + e), e = 1/e,

where the variables are shown in Fig. 3, and all other variables staying unchanged.We remark that this transformation law still applies even if some of the sides of

dd

dd

dd

dd

dd

r

rr

r

dd

dd

dd

dd

dd

r

rr

r

a b

c d

e

a b

c d

e

Figure 3. A flip transformation corresponding to equations (2.6).

the quadrilateral are a part of the boundary. The only modification is that thereis no coordinate, associated to a boundary edge, and thus there is nothing to betransformed on this edge.

Now, from figure 4 and the transformation law (2.6) it follows that the mappingclass D1/N acts in the Teichmuller space according to the following formulas

(2.7) f2j f2j = 1/f2j1, f2j+1 f

2j+1 = f2j(1 + f2j1)(1 + f2j+1).

If we identify the variables f1, . . . , f 2N with the initial data for the 2N-periodic

r r r

r r r

ppp

ppp

p p p

p p p

f2j f2j+1 f2j+2

Aj

Bj

Aj+1

Bj+1

Aj+2

Bj+2

r r r

r r r

ppp

ppp

p p p

p p p

f2j1 f2j f2j+1

Aj

Bj1

Aj+1

Bj

Aj+2

Bj+1

r r r

r r r

ppp

ppp

p p p

p p p

f2j f2j+1 f

2j+2

Aj

Bj1

Aj+1

Bj

Aj+2

Bj+1

D1/N

Figure 4. The action of the mapping class D1/N on the triangu-lated annulus: it is identical on the bottom boundary and a cyclicshift to the right by one spacing on the top boundary.

discrete Liouville equation (1.3) on the sublattice m + n = 1 (mod 2) along the


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zig-zag line n {1, 0} according to formulas

fm = m,0 if m = 1 (mod 2);1/m,1 otherwise,then, the transformation formulas (2.7) exactly correspond the the light-cone evo-lution:

m,n m,n = m1,n+1

for the time instants n {1, 0}.

3. Quantum theory

In what follows we describe quantum discrete Liouville equation and its connec-tion with quantum Teichmuller theory. Quantum theory of both the Teichmullerspace and the discrete Liouville equation is build up on the base of a particularspecial function, called non-compact quantum dilogarithm. It is a building block ofthe operators realizing elements of the mapping class group in quantum Teichmuller

theory, while in the case of quantum Liouville equation it is used for constructionof the evolution operator and the Baxters Q-operator (the operator reflecting theintegrable structure of the discrete Liouville equation). This is why we start bydescribing some of the properties of this function.

3.1. The non-compact quantum dilogarithm. Let complex b have a nonzeroreal part b = 0. Denote

cb i(b + b1)/2.

The non-compact quantum dilogarithm, b(z), z C, |z| < |cb|, is defined bythe formula

(3.1) b(z) exp

1

4

+

e2izx dx

sinh(xb) sinh(x/b)x

,

where the singularity at x = 0 is put below the contour of integration. Thisdefinition implies that b(z) is unchanged under substitutions b b1, b b.

Using this symmetry, we choose b to lay in the first quadrant of the complex plane,namely

b > 0, b 0,

which implies that cb > 0.

Remark 1. This function is closely related to double sine function of Barnes [1].In the context of quantum integrable systems and quantum groups L.D. Faddeevin [12] pointed out its remarkable properties.

3.1.1. Functional relations. In what follows, we shall use the following notation:

(3.2) inv = ei(1+2c2b)/6 = eic

2b2o , o = e

i(14c2b)/12.

Function (3.1) satisfies the inversion relation

(3.3) b(z)b(z) = eiz2inv,

and a pair of functional equations

(3.4) b(z + ib1/2) = (1 + e2zb

1

)b(z ib1/2).

The latter equations enable us to extend definition of b(z) to the entire complexplane.


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8 R.M. KASHAEV

When b is real or a pure phase, function b(z) is unitary in the sense that

(3.5) (b(z)) = 1/b(z

), (1 |b|)b = 0.

If self-adjoint operators p and q in L2(R) satisfy the Heisenberg commutation rela-tion

(3.6) pq qp = (2i)1,

then the following operator five term identity holds:

(3.7) b(q)b(p) = b(p)b(p + q)b(q).

For real b this can be proved in the C-algebraic framework [32]. See [15] for theproof in the case of complex b by the use of the integral Ramanujan identity.

3.1.2. Analytic properties. Let b2 > 0. We can perform the integration in (3.1)by the residue method. The result can be written in the form

(3.8) b(z) = (e2(zcb)b

1

; q2)/(e2(z+cb)b; q2),

where

q = eib2

, q = eib2

,

and

(x; y) j=0

(1 xyj), x, y C, |y| < 1.

Formula (3.8) defines a meromorphic function on the entire complex plane, satis-fying functional equations (3.3) and (3.4), with essential singularity at infinity. So,it is the desired extension of definition (3.1). It is easy to read off location of itspoles and zeroes:

zeroes of (b(z))1 = {(cb + mib + nib

1) : m, n Z0}.

The behavior at infinity depends on the direction along which the limit is taken:

(3.9) b(z)

|z|

1 | arg(z)| > 2 + arg(b);

eiz2

inv | arg(z)| 0.

Thus, for complex b, double quasi-periodic -functions, generators of the field ofmeromorphic functions on complex tori, describe the asymptotic behavior of thenon-compact quantum dilogarithm.


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3.1.3. Integral Ramanujan identity. Consider the following Fourier integral:

(3.10) (u, v, w) Rb(x + u)

b(x + v)e2iwx dx,

where

(3.11) (u + cb) > 0, (v + cb) > 0, (u v) < w < 0.

Restrictions (3.11) actually can be considerably relaxed by deforming the integra-tion path in the complex x plane, keeping the asymptotic directions of the two endswithin the sectors (| arg x| /2) > arg b. So, the enlarged in this way domainfor the variables u, v, w has the form:

(3.12) | arg(iz)| < arg b, z {w, u v w, v u 2cb}.

Integral (3.10) can be evaluated explicitly by the residue method, the result being

(u, v, w) =b(u v + cb)b(w cb)

b

(u v w + cb

)e2iw(vcb)1o(3.13)

=b(v + w u cb)

b(v u cb)b(w + cb)e2iw(u+cb)o,(3.14)

where the two expressions in the right hand side are related to each other throughthe inversion relation (3.3). In [15] this identity has been demonstrated to be anintegral counterpart of the Ramanujan 11 summation formula.

3.1.4. Fourier transformations. Particular values of (u, v, w) lead to the followingFourier transformation formulas:

(3.15) +(w)

R

b(x)e2iwx dx = (0, v , w)|v

= e2iwcbo/b(w + cb) = eiw21o b(w cb),

and

(3.16) (w)

R

(b(x))1e2iwx dx = (u, 0, w)|u

= e2iwcb1o b(w cb) = eiw2o/b(w + cb).

The corresponding inverse transformations read:

(3.17) (b(x))1 =

R

(y)e2ixydy,

where the pole at y = 0 is surrounded from below.

3.1.5. Other integral identities. The non-compact quantum dilogarithm satisfiesalso integral analogs of other basic hypergeometric identities, see for example [20].

For any n 1 define

(3.18) n(a1, . . . , an; b1, . . . , bn1; w)

R

dx ei2x(wcb)n

j=1

b(x + bj cb)

b(x + aj),

where bn = i0,

(bj) > 0, (cb aj) > 0,n

j=1

(bj aj cb) < (w cb) < 0.


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10 R.M. KASHAEV

The integral analog of the 11-summation formula of Ramanujan in this notationtakes the form

(3.19) 1(a; w) = o b(a + w cb)b(a)b(w).

Equivalently, we can rewrite it as follows

(3.20) 1(a; w)

R

b(x + a)

b(x + cb i0)ei2x(w+cb)dx

= ei2(a+cb)(w+cb)1(a; w) = 1o

b(a)b(w)

b(a + w + cb).

By using the integral Ramunujan formula, one can obtain an integral analog of theHeine transformation formula of the 12 basic hypergeometric series:

2(a, b; c; w) =b(c b)

b(a)2(c b, w; a + w; b).

By using the evident symmetry

2(a, b; c; w) = 2(b, a; c; w),

we come to an integral analog of the EulerHeine transformation formula

(3.21) 2(a, b; c; w)

=b(c b)b(c a)b(a + b + w c)

b(a)b(b)b(w)2(c a, c b; c; a + b + w c).

Performing the Fourier transformation on the variable w and using the equa-tion (3.19), we obtain an integral analog of the summation formula of Saalschutz:

(3.22) 3(a,b,c; d, a + b + c d cb; cb) = 3oe

id(2cbd)

b(a + b d cb)b(b + c d cb)b(c + a d cb)b(a)b(b)b(c)b(a d)b(b d)b(c d)

.

One special case of this formula is obtained by taking the limit c :

(3.23) 2(a, b; d; cb) = 3oe

id(2cbd)b(a + b d cb)

b(a)b(b)b(a d)b(b d).

From equation (3.22) one can derive an integral or non-compact analog of theBailey lemma, which is equivalent to the following: if operators p and q satisfy theHeisenberg commutation relation (3.6), then the operator valued function

(3.24) Q(u, v) = Q(u, v; p, q)

eiq2

b(u q)b(v q)b(p + u + v)

b(p)b(u + q)b(v + q)e

iq2

gives a commuting operator family in variables u and v, and it acts diagonally ona one-parameter family of vectors

(3.25) Q(u, v)|s = |sb(u + s)b(v + s)b(u s)b(v s)ei2s2,

where the vectors |s are defined by their matrix elements

(3.26) x|s =b(s x cb + i0)

b(s + x + cb i0)ei2(x+cb)s, s R0,


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with respect to the position basis x|, x R, where the operator q is diagonal,and p acts as a differentiation:

x|q = xx|, x|p = 12i

x

x|.

Some of these and other interesting properties of the non-compact quantum dilog-arithm as well as their interpretation in the context of integrable systems are alsodescribed in [31, 6].

3.2. Quantum discrete Liouville equation. The quantum version of the equa-tion (1.3) (with 2N-periodic boundary conditions) reads as

(3.27) wm,t+1wm,t1 = (1 + qwm+1,t)(1 + q1+2N,1wm1,t),

where the field variables wm,t are elements of the observable algebra (see below),satisfying the periodicity condition

wm+2N,t =

wm,t,

and q = exp(ib2), b being the couplingconstant (or square root thereof). The latteris expected to be related with the Virasoro central charge through the formula

cVir = 1 + 6(b + b1)2.

Remark 2. Notice that the case N = 1 is special, where the two terms in the righthand side of the equation (3.27) are given in terms of one and the same value of thefield variable wm+1,t = wm1,t. This is similar to the affine Cartan matrix of the

type A(1)N for N = 1. A modification shows up also in the defining commutation

relations of the observable algebra, see relations (3.29) below.

3.3. Algebra of observables and the evolution operator. The algebra of ob-servables is generated by a finite set of self-adjoint operators {r1, . . . , r2N}. We shall

think of them as an operator family parameterized by integers {rj}jZ satisfyingthe periodicity condition

(3.28) rj+2N = rj .

The defining commutation relations are as follows

(3.29) [rm, rn] =

(1)m(1 + N,1)(2i)1, if n = m 1 (mod 2N);

0, otherwise.

Taking into account the interpretation in terms of the Teichm uller space of anannulus (with N marked points on each boundary component), the operators rjcan be considered as quantized logarithmic Fock coordinates, the commutationrelations (3.29) exactly corresponding to the Poisson structure of the Teichmullerspace.

The initial data for the field variables in (3.27) are exponentials of the generatingelements:

w2j+1,0 = e2br2j+1 , w2j,1 = e

2br2j .

Proposition 1. Let operatorUlc be defined by the formula

(3.30) Ulc = GNj=1

b(r2j), b(x) = 1/b(x),


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12 R.M. KASHAEV

where operatorG is defined through the system of linear equations

(3.31) Grj = (1)jrj1G, j Z.

Then, the field variables defined by the formula

(3.32) wj,t Utlce

2brj+tUtlc , j + t = 1 (mod 2),

satisfy the 2N-periodic quantum discrete Liouville equation (3.27).

Remark 3. Note that the operator Ulc is identified with the light-cone evolutionoperator in the sense that it realizes the appropriate translation in the space-timelattice through the formula

Ulcj,tU1lc = j1,t+1.

Remark 4. Because of the duality symmetry b b1 in the theory, there actuallyexist two types of exponential fields exp(2b1rm) which satisfy two dual quantumdiscrete Liouville equations.

3.4. Integrable structure of the quantum discrete Liouville equation. Thequantum discrete Liouville equation is integrable in the sense of the quantum inversescattering method [13]. That means that it admits a set of commuting operatorswith the evolution operator being one of them, and there is a system of lineardifference equations, called Baxter equations, relating these operators.

In what follows, the order in products with non-commuting entries will be indi-cated as follows:

nim

ai anan1 am+1am,

min

ai amam+1 an1an.

Consider algebra AN of operators with a generalized linear basis of the form

1i2N

e2irixi ,

where self-adjoint operators ri satisfy commutation relations (3.29), and variablesxi take real or complex values. The term generalized here means that a genericelement of the algebra AN is an integral of the form

X2Nf(x1, . . . , x2N)

1i2N

e2irixi

dx1 dx2N,

where f(x1, . . . , x2N) is a complex valued distribution (generalized function), andX2N C2N is a 2N-dimensional (over R) sub-manifold.

The ascending cyclic product is a set of linear mappings,

o+j : AN AN, j Z, o+j = o

+j+2N,

acting diagonally on the basis monomials:

o+1(

1i2N

e2irixi) e2ix2Nx1

1i2N

e2irixi o+j(

jij+2N1

e2irixi).

We define the transfer-matrices

(3.33) t() = o+1 Tr

1j2N

Lj ,


13/25


where

(3.34) L

j= e

(1)jb1(rj) e(1)jb1(+rj)[j + 1]2

e(1)j

b1

(rj) e(1)j

b1

(+rj) ,[j]2 = (1 (1)

j)/2,

and the trace is that of two-by-two matrices. We also define a Q-operator

(3.35) Q() = o+1

1j2N

w[j]2 (, rj)

G, C,

where

(3.36) e2ixb(x )wi(, x)

b(x + ) , if i = 0;

1 , if i = 1,

and G is defined by equation (3.31).

Proposition 2. The transfer-matrices (3.33) and the Q-operator (3.35) commuteamong themselves

(3.37) [t(), t()] = [Q(),Q()] = [t(),Q()] = 0, = ,

solve the following Baxter equations:

(3.38) t()Q() = Q( + ib1/2) + (1 e4b1)NQ( ib1/2),

and the evolution operatorUlc of the quantum discrete Liouville equation is givenby the formula

(3.39) Ulc = Q(0).

Formula (3.39) is verified straightforwardly, the commutativity part of the propo-sition is the standard argument by using the YangBaxter equations, while the proof

of the Baxter equations given in [22] uses a less standard argument.

Remark 5. The product of two neighboring L-operators L+2iL+2i+1 is equivalent to

the spectral parameter dependent L-operator introduced in [16] for the descriptionof the (continuous) Liouville equation in the framework of the inverse scatteringmethod.

3.5. The case N = 1. When N = 1, the algebra AN is generated by a singleHeisenberg pair of position and momentum operators p and q:

r1 = p q, r2 = p + q, [p, q] = (2i)1.

Calculation of the operators (3.33) and (3.35) at N = 1 gives the following result:

t(z) = L+(p, q) e2bp + 2 cosh(2bq),

3oQz cb

2

=R

ei2cb(x+cb)Q(x cb + i0, ; p, q)ei2xzdx

where o is defined in equation (3.2) and

Q(u, ; p, q) = eiq2

b(u + q)b(p)b(u q)eiq2,

see also equation (3.24). Calculation of the integral gives the formula

Q(0) = b(p + q)ei2(c2bq

2).


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14 R.M. KASHAEV

By acting on the vectors |s, and using equation (3.25), we obtain

(3.40) t(z)|s = L+(p, q)|s = |s2 cosh(2bs)

and

(3.41) Q

z cb

2

|s = |s

1o e

i(s2+2c2b)[z|s,

where the vector

[z|

R

dxei2xzix2

x|

is such that

[z|p = pz[z|, [z|q = qz[z|,(3.42)

pz 1

2i

z z, qz

1

2i

z.(3.43)

On the other hand, the Baxter equation (3.38) at N = 1 can formally be written

in the form(3.44) (L+(pz, qz) L

+(p, q))Q

z cb

2

= 0

which, when applied to the vector |s, is reduced to an identity by using equa-tions (3.40) (3.42).

4. Relation to quantum Teichmuller theory

4.1. Highlights of quantum Teichmuller theory.

4.1.1. Groupoid of decorated ideal triangulations. Let = g,s be an oriented sur-face of genus g with s punctures. Denote M = 2g 2 + s and assume that Ms > 0.Then surface admits ideal triangulations.

Definition 1.A decorated ideal triangulation of is an ideal triangulation , whereall triangles are provided with a marked corner, and a bijective ordering map

: {1, . . . , 2M} j j T()

is fixed. Here T() is the set of all triangles of .

Graphically, the marked corner of a triangle is indicated by an asterisk and thecorresponding number is put inside the triangle. The set of all decorated idealtriangulations of is denoted .

Recall that if a group G freely acts in a set X then there is an associated groupoiddefined as follows. The objects are the G-orbits in X, while morphisms are G-orbits in X X with respect to the diagonal action. Denote by [x] the objectrepresented by element x X and [x, y] the morphism represented by pair ofelements (x, y) X X. Two morphisms [x, y] and [u, v], are composable if and

only if [y] = [u] and their composition is [x, y][u, v] = [x,gv], where g G is theunique element sending u to y. The inverse and the identity morphisms are givenrespectively by [x, y]1 = [y, x] and id[x] = [x, x].

Remarking that the mapping class group M of freely acts in , denote byG the corresponding groupoid, called the groupoid of decorated ideal triangulations.It admits a presentation with three types of generators and four types of relations.

The generators are of the form [, ], [, i], and [, i,j], where is obtainedfrom by replacing the ordering map by the map , where S2M is a


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permutation of the set {1, . . . , 2M}, i is obtained from by changing the markedcorner of triangle i as in figure 5, and i,j is obtained from by applying the fliptransformation in the quadrilateral composed of triangles

iand

jas in figure 6.

d

dr

r

ri

ddr

r

r

ii

Figure 5. Transformation i.

dd

dd

r

r

r

ri j

dd

dd

r

r

r

r

i

j

i,j

Figure 6. Transformation i,j.

These generators satisfy the following relations:

[, () ] = [, ], , S2M,(4.1)

[, iii] = id[],(4.2)

[, j,ki,ki,j] = [, i,jj,k],(4.3)

[, j,iii,j] = [, ij(ij)].(4.4)

The first two relations are evident, while the other two are shown graphically infigures 7, 8.

ff

fff

555

5

r r

r r

r

i

j

k

ff

fff

555

5

r r

r r

r

i

j

k

ff

fff

555

5

r r

r r

r

i

j

k

ff

fff

55

55

&&

&&

&&

r r

r r

r

i j

kf

ff

ff

55

55

&&

&&

&&

r r

r r

r

i

j

k

j,k j,k

i,j

i,j

i,k

Figure 7. Pentagon relation (4.3).

4.1.2. Hilbert spaces of square integrable functions. In what follows, we work withHilbert spaces

H L2(R), Hn L2(Rn).


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16 R.M. KASHAEV

d

d

dd

r

r

r

ri j

d

d

dd

r

r

r

r

i

j

i,j

(i, j) i j i

dd

dd

r

r

r

rj i

dd

dd

r

r

r

r

i

j

j,i

Figure 8. Inversion relation (4.4).

Any two self-adjoint operators p and q, acting in H and satisfying the Heisenbergcommutation relation (3.6), can be realized as differentiation and multiplicationoperators. Such coordinate realization in Diracs bra-ket notation has the form

(4.5) x|p = 12i

x

x|, x|q = xx|.

Formally, the set of vectors {|x}xR forms a generalized basis of H with thefollowing orthogonality and completeness properties:

x|y = (x y),

R

|xdxx| = 1.

For any 1 i m we shall use the following notation

i : End H a ai = 1 1 i1 times

a 1 1 End Hm.

Besides that, ifu End Hk for some 1 k m and {i1, i2, . . . , ik} {1, 2, . . . , m},then we shall write

ui1i2...i2 i1 i2 ik (u).The permutation group Sm naturally acts in H

m:

(4.6) P(x1 xi ) = x1(1) x1(i) . . . , Sm.

4.1.3. Semi-symmetric T-matrix. Fix self-conjugate operators p, q satisfying theHeisenberg commutation relation (3.6). Choose a parameter b satisfying the con-dition

(1 |b|)b = 0,

and define two unitary operators

A ei/3ei3q2

ei(p+q)2

End H,(4.7)

T ei2p1q2b(q1 + p2 q2) End H2.(4.8)

They satisfy the following relations characterizing a semi-symmetric T-matrix:A3 = 1,(4.9)

T12T13T23 = T23T12,(4.10)

T12A1T21 = A1A2P(12),(4.11)

where

(4.12) eic2b/3, cb =

i

2(b + b1),


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and operator P(12) is defined by equation (4.6) in the case when

S2 = (12): 1 2 1.

Operator A is characterized (up to a normalization factor) by the equations

AqA1 = p q, ApA1 = q.

Note that equations (4.9)(4.11) correspond to relations (4.2)(4.4). This fact isthe base of using the former to realize the latter.

4.1.4. Useful notation. For any operator a End H we shall denote

(4.13) ak AkakA1k , ak A

1k akAk.

It is evident thatak

= ak = ak, ak =ak, ak = ak,

where the last two equations follow from equation (4.9). In particular, we have

pk = qk, qk = pk qk,(4.14)

pk = qk pk, qk = pk.(4.15)

Besides that, it will be also useful to use the notation

(4.16) P(kl...mk) AkP(kl...m), P(kl...mk) A1k P(kl...m),

where (k l . . . m) is the cyclic permutation

(k l . . . m) : k l . . . m k.

Equation (4.11) in this notation takes a rather compact form

T12T21 = P(121).(4.17)

Remark 6. One can derive the following symmetry property of the T-matrix:

T12 = T12T2

1

T1

21

= P(12

1)

T1

21

= T1

12

P(12

1)

= T112

P(121) = T112T12T21 = T21.

4.1.5. Quantum functor. Quantum Teichmuller theory, being a three-dimensionalTQFT, is defined by a quantum functor,

F : G End H2M,

which means that we have a operator valued function

F : End H2M,

satisfying the equations

(4.18) F(, ) = 1, F(, )F(, )F(, ) C \ {0}, , , ,

(4.19) F(f(), f()) = F(, ), f M,

(4.20) F(, i) Ai,

(4.21) F(, i,j) Tij ,

(4.22) F(, ) P, S2M,

where operator P is defined by equation (4.6). Consistency of these equations isensured by the consistency of equations (4.9)(4.11) with relations (4.2)(4.4).


18/25

18 R.M. KASHAEV

A particular case of equation (4.18) corresponds to = :

(4.23) F(, )F(, ) C \ {0}.

As an example, we can calculate the operator F(, 1i,j ()). Denoting 1i,j ()and using equation (4.23), as well as definition (4.21), we obtain

(4.24) F(, 1i,j ()) = F(i,j(), ) (F(, i,j(

)))1 = T1ij ,

where means equality up to a numerical multiplicative factor.Projective unitary representation of the mapping class group M is realized as

follows:M f F(, f()) End H

2M.

Indeed,

F(, f())F(, h()) = F(, f())F(f(), f(h())) F( , f h()).

4.2. Quantum discrete Liouville equation and quantum Teichmuller the-ory. Again, as in the classical case, we consider an annulus with N marked pointson each of its boundary components and choose a decorated ideal triangulation Nshown in figure 9. Equivalently, N can be thought of as an infinite triangulated

r r r

r r r

1 3

2 4

r r

r r

2N 1

2N

p p p

p p p

Figure 9. A decorated ideal triangulation of an annulus with Nmarked points on each of the boundary components. The leftmostand the rightmost vertical edges are identified.

strip, where the triangles are numerated according to figure 9 with the periodicity

conditionN(n + 2N) = N(n), n Z.

Recall that Dn/N, n Z, is the mapping class twisting the top boundary componentwith respect to the bottom one by the angle 2n/N so that the marked points onthe top component are cyclically translated by n spacings. When n = N we get apure Dehn twist DN/N = D. Clearly,

Dm/N Dn/N = D(m+n)/N.

From figure 10 it follows that the quantum realization of the transformation D1/N

has the form

(4.25) F

N, D1/N(N)

D1/N N6/NP(...j,j+1...)

2N

k=1Ak

N

l=1T2l+1,2l,

where the normalization factor is chosen in accordance with the standard normal-ization of Dehn twists in quantum Teichmuller theory. We define

Dn/N (D1/N)n, n Z.

Consider the following faithful reducible realization of the observable algebra ANin L2(R2N):

(rj) =

pj + pj1, if j = 0 (mod 2);qj + qj1, otherwise,


19/25


r r r

r r r

2j 1 2j + 1

2j 2j + 2

ppp

ppp

p p p

p p p

r r r

r r r

2j 2 2j

2j 1 2j + 1

ppp

ppp

p p p

p p p

r r r

r r r

2j 2 2j

2j 1 2j + 1

ppp

ppp

p p p

p p p

r r r

r r r

2j 2 2j

2j 1 2j + 1

ppp

ppp

p p p

p p p

D1/N

Q

j 1

j

Q

j 1

2j+1,2j (. . . , j , j 1, . . .)

Figure 10. Transformation D1/N as a morphism in the groupoidof ideal triangulations.

(G) = N+6/Nei2PN

j=1 p2jp2j+1

2Nk=1

A1k P(...,l,l1,...).

Theorem 2 ([14]). One has the following equality

(4.26) (Ulc) = D1/N.

4.2.1. Working in another triangulation. Here we consider a special decorated idealtriangulation, where the Dehn twist D is represented by a single T-operator. Ithappens so that with respect to this triangulation all operators Dn/N, 0 < n < N,are represented in terms of product of only N + 1 T-operators.

Consider a decorated ideal triangulations of the form

n:N n+1 n+2 N(N), 1 n < N,

where transformations n are defined in figure 11. Transformation n acts non-

r r r

r r r

2n 2 2n

2n 1 1

ppp

ppp

p p p

p p p

r r r

r r r

2n 2

2n

2n 1

1

ppp

ppp

p p p

p p p

r r r

r r r

2n 2

2n

2n 1

1

ppp

ppp

p p p

p p p

2n,2n1

12n1, 2n2

12n,1 n

%

Figure 11. Transformation n reduces n to n1 and two trian-gles attached to the boundary.

trivially only in the minimal annular part of n:N, which by itself is nothing elsebut the triangulated annulus n.

Quantum realizations of any element of the (extended) mapping class groupwith respect to decorated ideal triangulations and are conjugated to each other


20/25

20 R.M. KASHAEV

by operator F (, ):

F (

, (

)) F (

, ()) F ((), (

)) F (, ) F (, ())F (, ) = Ad (F (, ))F (, ())

In particular, from figure 11 it follows that

(4.27) F(N, n:N) W(n)

Nj>n

(T2j,2j1T12j1, 2j2

T12j,1

).

We would like to find the realization of the transformations Dn/N with respect todecorated ideal triangulation 1:N:

F

1:N, Dn/N(1:N)

Dn/N W(1)1Dn/NW(1)

Proposition 3. Formula

(4.28) W(n)1D1/NW(n) = n1+6/Nn1j=1

T12j+1,2j

N>k>n

T2N1,2k1

T2N1,2nT11,2N1T 2n1,2n

2n2l=1

A1l PQ

Nm=n(2m1,2m)

P(...,s,s1,...)

holds true for 1 n < N. In particular, when n = 1,

(4.29) D1/N = 6/N

N>j>1

T2N1,2j1T2N1,2T11,2N1T1,2P(...,2k+1,2k1,...)

Proof. The proof is by induction in n. First, let us check that equation (4.28) holdstrue at n = N 1. We write

N6/NW(N 1)1D1/NW(N 1) = T2N,1T2N1, 2N2T1

2N1,2NT1

2N,1

N2j=1

T12j+1,2j

T12N2, 2N3T12N, 2N1

2Nk=1

A1k P(...,l,l1,...)

where we canceled one pair of T-operators. Applying now the pentagon equationto the underlined fragment and slightly reshuffling the commuting terms, we createanother pair ofT-operators:

=

N2j=1

T12j+1,2j

T2N1, 2N2T1

2N1,1T 2N3,2N2

T12N3,2N2T1

2N2, 2N3T12N1,2N

T12N, 2N1

2Nk=1

A1k P(...,l,l1,...)

finally, applying twice the inversion relation to eliminate four Ts in the second line,we arrive at the desired result.


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Now, assuming that formula (4.28) holds true for 1 < n + 1 < N, we prove thatit holds also for n:

n6/NAd W(n)1D1/N= n6/NAd(T2n+2,1T2n+1,2nT

12n+2,2n+1

) AdW(n + 1)1

D1/N

= T2n+2,1T2n+1,2nT1

2n+2,2n+1T1

2n+1,2n

n1j=1

T12j+1,2j

N>k>n+1

T2N1,2k1T2N1, 2n+2T11,2N1T 2n+1,2n+2

T 2n+2,2nT12n, 2n1

T12n+2, 2N1

2nl=1

A1l PQ

Nm=n+1(2m1,2m)

P(...,s,s1,...)

applying the pentagon relation to the underlined fragment,

=n1j=1

T12j+1,2j

N>k>n+1

T2N1,2k1

T2n+2,1T1

2n+2,2nT12n+1,2n+2T2n+2, 2N1T

11,2N1T 2n+1,2n+2

T 2n+2,2nT12n, 2n1

T12n+2, 2N1

2nl=1

A1l PQ

Nm=n+1(2m1,2m)

P(...,s,s1,...)

again applying the pentagon equation and hiding one T into the product over k,

=n1

j=1T1

2j+1,2j N>k>nT2N1,2k1T2n+2,1T

12n,2n+2T2n+2, 2N1T

11,2N1

T2n,2n+2T12n, 2n1

T12N1, 2n+2

2nl=1

A1l PQ

Nm=n+1(2m1,2m)

P(...,s,s1,...)

two more pentagon relations with subsequent cancelation of two pairs ofTs

=n1j=1

T12j+1,2j

N>k>n

T2N1,2k1T2N1,2nT11,2N1T

12n, 2n1

2nl=1

A1l PQN

m=n+1(2m1,2m)P(...,s,s1,...)

finally, application of the inversion relation to the last T gives equation (4.28).

Formula (4.29) is a particular case of (4.28) corresponding to n = 1.


(4.30) D(nN)/N = 6(Nn)/N

nj>1

T2n+1,2j1T2n+1,2T11,2n+1T1,2

n


22/25

22 R.M. KASHAEV

holds true for 1 n < N, while

(4.31) D = 6T11,

2

.

Proof. Again we use induction in n. Equation (4.30) at n = N 1 coincides withequation (4.29). Assuming the statement is true for 1 < n + 1 < N, we derive itfor n:

6(nN)/ND(nN)/N = 6(nN)/ND1/ND(n+1N)/N =

N>j>1

T2N1,2j1

T2N1,2T11,2N1T1,2

nl>1

T2n+1,2l1T2n+1,1T2n+1,2T12N1,2n+1

T 2N1,2

n+1n+1

T2N1,2j1T2N1,2n+1

nl>1

(T2N1,2l1T2n+1,2l1)

T2N1,2T2n+1,2T11,2N1T

12N1,2n+1T1,2T 2N1,2

n+1n+1

T2N1,2j1 nl>1

T2n+1,2l1T2n+1,2T11,2n+1

T11,2N1T2,1T2,2N1

n+1l>n+1

T2N1,2l1T11,2N1

n+1


23/25


4.2.2. Description in terms of variables of the discrete Liouville equation. Defineoperators

gj,k k

l=j+1

rj .

Notice that the operator, corresponding to transformation (4.27), can be writtenin the form

W(n) = (Wf(n))Wo(n),

where

Wf(n) =

Nj>n

(b(r2j)b(r2j1)b(g2j,2N+1)) , b(x) (b(x))1,

Wo(n) = Nj>n

ei2q2jq2j1ei2(p2j1p2j2+p2jp1) .Similarly, formula (4.28) can be written as

Ad(W(n)1)D1/N = Ad(Wo(n)1)

Ad(Wf(n)

1)Ulc

.

The following propositions can be proved in similar manner as propositions 3 and4.


Ad(Wf(n)1)Ulc =

nN1n

j=2 b(r2j1) N>k>n b(g2k1,2N1) b(g2n,2N1)b(g2N1,2N+1)b(r2n)e

iPN

l=n r22lG, e i/6,

holds true for 1 n < N. In particular,

Ulc Ad(Wf(1)1)Ulc =

N

N>k>1

b(g2k1,2N1)

b(g2,2N1)b(g2N1,2N+1)b(r2)eiPN

l=1 r22lG.


UNnlc = (nN)N

nj>1 b(g2j1,2n+1)b(g2,2n+1)b(g2n+1,2N+1)b(r2)

n


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24 R.M. KASHAEV

References

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(1901), 265-388[2] Bazhanov, V.V.; Lukyanov, S.L.; Zamolodchikov, A.B.: Integrable structure of conformal

field theory, quantum KdV theory and thermodynamic Bethe ansatz. Comm. Math. Phys.177 (1996), no. 2, 381398

[3] Bazhanov, V.V.; Lukyanov, S.L.; Zamolodchikov, A.B.: Integrable structure of conformalfield theory. II. Q-operator and DDV equation. Comm. Math. Phys. 190 (1997), no. 2, 247278

[4] Bazhanov, V.V.; Lukyanov, S.L.; Zamolodchikov, A.B.: Integrable structure of conformalfield theory. III. The Yang-Baxter relation. Comm. Math. Phys. 200 (1999), no. 2, 297324

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R.M. Kashaev- Discrete Liouville Equation and Teichmuller Theory

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