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Liouville central charge in quantum Teichmuller theory

In the quantum Teichmuller theory, based on Penner coordinates, the mapping class groups of punctured surfaces are represented projectively. The case of a genus three surface with one puncture is worked out explicitly. The projective factor is calculated. It is given by the exponential of the Liouville central charge.


Introduction
In [5] L.D. Faddeev suggested to use as a building block for the evolution operator in the quantum Liouville model on discrete space-time [6] the following non-compact version of the quantum dilogarithm function: where the singularity at z = 0 is put below of the contour of integration, and λ is a real parameter. Among other things this function satisfies an inversion relation Faddeev conjectured that the inversion factor in this equation (the constant part of the r.h.s) is related to the Liouville central charge [7]. The purpose of this paper is to prove Faddeev's conjecture. Our approach is based on the quantum theory of Teichmüller spaces of punctured surfaces suggested recently in [13] and independently by Fock, Chekhov and Frolov [8,9]. It is known [17] that the Teichmüller spaces appear as the classical phase spaces of Chern-Simons theory with gauge group SL(2, R). This theory is related to 2 + 1 dimensional gravity with cosmological constant [1,18,17], which in turn induces the Liouville theory at spatial infinity [2,3].
The quantum theory of the Teichmúller spaces, described in [13], is based on Penner's coordinates [16]. The mapping class (or modular) group of a punctured surface is projectively represented in the corresponding Hilbert space of physical states in terms of the non-compact quantum dilogarithm. We work out explicitly the case of a genus three surface with one puncture, and calculate the corresponding projective factor which, being essentially given by the inversion factor of the non-compact quantum dilogarithm, appears to be the exponential of the quantum Liouville central charge. This is in agreement with the interpretation of the Hilbert space of quantum states as Virasoro conformal blocks [17], since the projective factors in the representations of the mapping class groups, coming from two-dimensional conformal field theories, are known to be given by the exponential of the corresponding Virasoro central charges [10,15].
2 The non-compact quantum dilogarithm

Notation
For any natural m define embeddings where a stands in the i-th position, which mean that the operator a i acts on the i-th argument as a, and identically on the others: The permutation group S m is represented faithfully in End L 2 (R m ):

Some algebraic relations
Let p, q ∈ End L 2 (R) be defined by They satisfy the Heisenberg commutation relation The non-compact quantum dilogarithm, defined by eqn (1), solves a pair of functional equations and, as was argued in [5], satisfies the following five-term operator equation: This fact can be proved rigorously [19].
We consider a unitary operator T ∈ End L 2 (R 2 ), defined by Eqn (4) is equivalent to while the inversion relation (2) leads to the equation where unitary operator R ∈ End L 2 (R) is defined by and Besides these we have a symmetry property with the same operator R defined by eqn (8). Note that R satisfies the following equations: It can also be written in an operator form

Projective representations of mapping class groups
We call a two-cell in CW complex triangle if exactly three boundary points of the corresponding two-disk are mapped to the zero-skeleton. We shall also call zero-cells and one-cells vertices and edges, respectively. Let Σ = Σ g,r be a closed oriented Riemann surface of genus g with a set V of r > 0 marked points such that M ≡ 2g − 2 + r > 0. By ideal triangulation (i.t.) τ of Σ we mean the isotopy class of a representation of Σ as a CW complex with V as the set of vertices, and all two-cells being triangles. It is easy to count that there are 2M triangles and 3M edges in any i.t.. We shall denote by E(τ ) and T (τ ) the sets of edges and triangles respectively in i.t. τ .
if each triangle is provided by a marked corner and a bijective ordering mappinḡ is fixed. Denote the set of all d.i.t. as ∆(Σ). The permutation group S 2M naturally acts in ∆(Σ) from the right: where τ σ differs from τ only in ordering: We also define two elementary transformations ρ t , ω e : ∆(Σ) → ∆(Σ) as follows.
Let t be a triangle in d.i.t. τ . We define d.i.t. τ t ≡ ρ t (τ ) by changing the marked corner of triangle t as is shown in Figure 1 and leaving the ordering mapping unchanged:τ =τ t . Let now e be an edge in d.i.t. τ shared by two distinct triangles s, t with marked corners as is shown in the l.h.s. of Figure 2. Then d.i.t. τ e ≡ ω e (τ ) is obtained from τ by replacing the cells e, s, t by new cells e ′ , s ′ , t ′ as is shown in the r.h.s. of Figure 2. The ordering mappingτ e is defined byτ For each d.i.t. τ and for each 1 ≤ i ≤ 2M define Let i = j be such that triangles s ≡τ (i) and t ≡τ (j) share an edge e as in the l.h.s. of Figure 2. Then we define Finally, for any permutation σ ∈ S 2M define where P σ is defined by eqn (3). We claim that due to relations (6), (10) and (7), F can be extended to a unitary operator valued function F(τ, τ ′ ) on ∆(Σ)×∆(Σ) such that for any d.
The mapping class or modular group M Σ of Σ naturally acts in ∆(Σ). By construction we have the following invariance property of the function F: This enables us to construct a unitary projective representation of M Σ : Indeed, where we denote by ≃ an equality up to a numerical factor. In the next section we calculate the projective factor in the case of the surface Σ 3,1 of genus three and with one puncture.
4 The case of Σ 3,1 To find the central charge we shall need a suitable presentation of the mapping class group. First, following [14] introduce some notation. Let S = S(Σ) be the set of isotopy classes of simple closed curves on the surface Σ. We denote by D α , α ∈ S, the Dehn twist along α. Define I(α, β) = min{|a ∩ b|: a ∈ α, b ∈ β}, ∀α, β ∈ S.
We shall use α ∩ β = ∅ to denote I(α, β) = 0; α ⊥ β to denote I(α, β) = 1; and α ⊥ 0 β to denote I(α, β) = 2 so that their algebraic intersection number is zero. a ∪ b at p from a to b is defined as follows. Fix any orientation on a and use the orientation on the surface to determine an orientation on b. Then resolve the intersection according to the orientations, see Figure 3. The resolution is independent on the choice of the orientations on a. If α ⊥ β or α ⊥ 0 β, take a ∈ α, b ∈ β so that |a ∩ b| = I(α, β). Then the curve obtained by resolving all intersection points in a ∩ b from a to b is again simple closed curve. Denote by αβ its' isotopy class. It follows that when α ⊥ β then αβ ⊥ α, β and when α ⊥ 0 β then αβ ⊥ 0 α, β. Also the Dehn twist of β along α is given by D α (β) = αβ if α ⊥ β, and D α (β) = α(αβ) if α ⊥ 0 β. Let N (a) and N (b) be small regular neighborhoods of a and b. Then N (a ∪ b) = N (a) ∪ N (b) is homeomorphic to a torus with one boundary component when α ⊥ β , and to a sphere with four boundary components when α ⊥ 0 β. Denote by ∂(α, β) the isotopy class of the curve system ∂N (a ∪ b).

Realization of the Dehn twists
To begin with 1 , we describe the simplest case of the Dehn twist of an annulus along the only simple non-contractible curve denoted in Figure 5 as α. From the same Figure 5 it follows that ω e • D α = id. So, using definition (11), we obtain where the normalization remains to be fixed. Now we consider the curves on Σ 3,1 , shown in Figure 6. We choose the Figure 6: The Dehn twists along these curves generate the mapping class group of the surface Σ 3,1 d.i.t. τ , obtained by cutting the surface into three handles and a triangle as shown in Figures 7, 8. The corresponding Dehn twists can be calculated by choosing appropriate d.i.t. where the annular neighborhoods of the curves look like as in Figure 5, and using formula (15). The result of such calculations reads: ak ≡ Ad(R k )a k , aǩ ≡ Ad(R −1 k )a k , see eqn (8) for the definition of the operator R. Note that the mutual normalization of these operators is fixed by the relations of the type (14): while the overall normalization is fixed by the lantern relation: Thus, we do not have any other freedom in normalization. Checking the chain relation we recover the projective factor:

Relation to the Liouville central charge
In the projective representations of the mapping class groups, associated with quantum CFT, the projective factors have the form exp(2πinc/24), where n is an integer and c is the Virasoro central charge [10,15]. According to the result of [13] on quantization of the Teichmüller spaces of punctured surfaces, the representation given by the operators F α is not quite the representation, corresponding to the quantum Teichmüller space of Σ 3,1 . There are additional degrees of freedom, given by the constraints associated with homologies of the surface. Explicit form of six basis constraints, corresponding to oriented contours α i , β i , is as follows: All the constraints, corresponding to other contours, are linear combinations of the basis ones: To get rid of these degrees of freedom, we note the following action of the Dehn twists on the constraints: for oriented contours α, β, α • β being their algebraic intersection index. The constraints themselves satisfy the following commutation relations: From eqns (17), (18) it follows that the combinations commute with the constraints and satisfy the defining relations for the Dehn twists with the projective factor ξ D = −ξ F in eqn (16). Taking into account eqn (9) we obtain where is the Virasoro central charge in quantum Liouville theory. This is in agreement with interpretation of physical states of quantum Teichmüller theory as Virasoro conformal blocks.

Summary
In the quantum Teichmüller theory of punctured surfaces, based on Penner coordinates, the mapping class groups are represented projectively in terms of non-compact quantum dilogarithm. Algebraically the representations are based on three operator equations (6), (7), and (10). We have calculated the projective factor for the case of genus three surface with one puncture. The reason for considering the genus three surface is due to the fact that for lower genus the projective factor can be absorbed into a redefinition of generators of the mapping class group. Only starting from genus three one has simultaneously the lantern (12) and chain (13) relations for the Dehn twists along non-separating curves 2 , which are not homogeneous in generators.
The result, given by eqns (19), (20), is the exponentiated Liouville central charge. This is in agreement with the connection of the SL(2, R) Chern-Simons and Liouville theories on the classical level, as well as with the expected interpretation of the Hilbert space of states in quantum Teichmüller theory as the space of Virasoro conformal blocks. Note that the right answer has been obtained only after elimination of the non-physical (Gaussian) degrees of freedom associated with the homologies of the surface.