Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

Abstract

We investigate the representation theory of the polynomial core T q S of the quantum Teichmüller space of a punctured surface S . This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S . Our main result is that irreducible finite-dimensional representations of T q S are classified, up to finitely many choices, by group homomorphisms from the fundamental group 1 .S/ to the isometry group of the hyperbolic 3-space ‫ވ‬ 3 . We exploit this connection between algebra and hyperbolic geometry to exhibit invariants of diffeomorphisms of S . 57R56; 57M50, 20G42This work finds its motivation in the emergence of various conjectural connections between topological quantum field theory and hyperbolic geometry, such as the now famous Volume Conjecture of Rinat Kashaev [22], and Hitochi Murakami and Jun Murakami [28]. For a hyperbolic link L in the 3-sphere S 3 , this conjectures relates the hyperbolic volume of the complement S 3 L to the asymptotic behavior of the N -th colored Jones polynomial J N L .e 2 i=N / of L, evaluated at the primitive N -th root of unity e 2 i=N . At this point, the heuristic evidence (Kashaev [22], Murakami, Murakami, Okamoto, Takata and Yokota [29], and Yokota [43; 44]) for the Volume Conjecture is based on the observation [22; 28] that the N -th Jones polynomial can be computed using an explicit R-matrix whose asymptotic behavior is related to Euler's dilogarithm function, which is well-known to give the hyperbolic volume of an ideal tetrahedron in ‫ވ‬ 3 in terms of the cross-ratio of its vertices. We wanted to establish a more conceptual connection between the two points of view, namely between quantum algebra and 3-dimensional hyperbolic geometry.We investigate such a relationship, provided by the quantization of the Teichmüller space of a surface, as developed by Rinat Kashaev [23], Leonid Chekhov and Vladimir Fock [12]. More precisely, we follow the exponential version of the Chekhov-Fock approach. This enables us to formulate our discussion in terms of non-commutative algebraic geometry and finite-dimensional representations of algebras, instead of Lie algebras and self-adjoint operators of Hilbert spaces. This may be physically less More precisely, let S be a surface of finite topological type, with genus g and with p > 1 punctures. An ideal triangulation of S is a proper 1-dimensional submanifold whose complementary regions are infinite triangles with vertices at infinity, namely at the punctures. For an ideal triangulation and a number q D e i" 2 ‫,ރ‬ the ChekhovFock algebra T q is the algebra over ‫ރ‬ defined by generators X˙1 1 , X˙1 2 , . . . , X˙1 n associated to the components of and by relations X i X j D q 2 ij X j X i , where the ij are integers determined by the combinatorics of the ideal triangulation . This algebra has a well-defined fraction division algebra y T q . In concrete terms, T q consists of the formal Laurent polynomials in variables X i satisfying the skew-commutativity relations This con...

Quantum hyperbolic geometry

“…For the bubble transit we just impose that the marked edge e (see Section 2.2) coincides with the above edge of H . Thanks to (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) and (2-21) we see that distinguished flat/charged I -triangulations are closed under such refined QHG transits.…”

“…Consider the connected component y ‫ރ‬ 0;0 of y ‫ރ‬ with even-valued flattenings, ie the Riemann surface of the map ' 0;0 in (2-4). We have an embedding (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) y ‫ރ‬ 0;0 ! y X .uI p; q/ 7 !…”

“…where b X i and  X i are as in (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and correspond to the space X i , and similarly for b Q 0 and  Q 0 . Note that they are invariant under any even permutation of the vertices of Q, and turn to the opposite under odd ones.…”

“…The embedding (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) and the form b X can be used to derive the combinatorial formulas obtained in Papadapoulos and Penner [25] (see also Bonahon [8] and Bonahon and Sözen [29]) for the Thurston's intersection form, which is a complex antisymmetric bilinear form on the shear-bend parameter spaces of pleated hyperbolic punctured surfaces. In this setting  X encodes the complex length differential form of [5,Section 11].…”

“…Note that a general QHG transit need not preserve even-valued flattenings, though a subfamily does (see the comment after (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)). This subfamily generates the five term relations defining the "more" extended pre-Bloch group of [24, Section 8] (see also [13]).…”