Quantum hyperbolic geometry

Baseilhac, Stéphane; Benedetti, Riccardo

doi:10.2140/agt.2007.7.845

Cited by 26 publications

(75 citation statements)

References 25 publications

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“…In this paper we evaluate 1-dimensional state-integrals at rational points in terms of the Rogers dilogarithm, the cyclic (quantum) dilogarithm and truncated state-sums at roots of unity. Our formulas are syntactically similar with (a) the constant terms of the power series that appear in the Quantum Modularity Conjecture of Zagier [Zag10,GZa], (b) the 1-loop terms of the perturbation expansion of complex Chern-Simons theory [Dim14b], (c) the state-sums of quantum Teichmüller theory [Kas94, Kas95,Kas97] and also [BB07,Sec.6].…”

Section: Introductionmentioning

confidence: 86%

Evaluation of state integrals at rational points

Garoufalidis

Kashaev

2015

Communications in Number Theory and Physics

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Abstract. Multi-dimensional state-integrals of products of Faddeev's quantum dilogarithms arise frequently in Quantum Topology, quantum Teichmüller theory and complex Chern-Simons theory. Using the quasi-periodicity property of the quantum dilogarithm, we evaluate 1-dimensional state-integrals at rational points and express the answer in terms of the Rogers dilogarithm, the cyclic (quantum) dilogarithm and finite state-sums at roots of unity. We illustrate our results with the evaluation of the state-integrals of the 4 1 , 5 2 and (−2, 3, 7) pretzel knots at rational points.

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Section: Introductionmentioning

confidence: 86%

Evaluation of state integrals at rational points

Garoufalidis

Kashaev

2015

Communications in Number Theory and Physics

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“…As this work was being developed, the type of functions occurring in explicit computations hinted at a connection between the invariant of Theorem 4, the Kashaev 6j -symbols developed in [21], and the link invariants introduced by Kashaev [21;22], Baseilhac and Benedetti [4;5;6;7]; see also . This connection has now been elucidated by the authors and Hua Bai Kashaev, Baseilhac and Benedetti [6] associate to the hyperbolic metric of the mapping torus M ' .…”

Section: Francis Bonahon and Xiaobo Liumentioning

confidence: 99%

Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

2007

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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...

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“…More recently, Baseilhac and Benedetti [3] constructed a (2 + 1)-dimensional so-called 'quantum hyperbolic field theory' (QHFT), which includes invariants H d (W, L, ρ) (with d 1 being any odd integer), where W is an arbitrary oriented closed 3-manifold, and either L is a non-empty link in W and ρ is a principal flat PSL(2, C)-bundle on W (up to gauge transformation) [1,2] or L is a framed link and ρ is defined on W \ L. The state sums giving H d (W, L, ρ) for d > 1 are in fact non-commutative generalisations (see [2]) of known simplicial formulae (see [18]) for CS(ρ) + iVol(ρ), the Chern-Simons invariant and the volume of ρ, which coincide with the usual geometric ones when ρ corresponds to the holonomy of a finite-volume hyperbolic 3-manifold. Each H d (W, L, ρ) is well defined up to the dth roots of unity multiplicative factors.…”

Section: Introductionmentioning

confidence: 97%

“…It is a non-trivial fact (see [9]) that when W = S 3 and ρ is the trivial flat PSL(2, C)-bundle on S 3 , the Kashaev invariants L d coincide with H d (S 3 , L, ρ 0 ) up to the above phase ambiguity. We recall that in [1][2][3] different instances of general 'Volume Conjectures' have been formulated in the QHFT framework. Roughly speaking, these predict that, in suitable geometric situations (for example, related to hyperbolic Dehn filling and the convergence of closed hyperbolic 3-manifolds to manifolds with cusps), the quantum state sums asymptotically recover a classical simplicial formula when d → ∞.…”

Section: Introductionmentioning

confidence: 99%