Quantum traces for representations of surface groups in SL₂(ℂ)

Abstract

We relate two different quantizations of the character variety consisting of all representations of surface groups in SL 2 . One is the Kauffman skein algebra considered by Bullock, Frohman and Kania-Bartoszyńska, Przytycki and Sikora, and Turaev. The other is the quantum Teichmüller space introduced by Chekhov and Fock and by Kashaev. We construct a homomorphism from the skein algebra to the quantum Teichmüller space which, when restricted to the classical case, corresponds to the equivalence between these two algebras through trace functions. 14D20, 57M25, 57R56Let S be an oriented surface of finite topological type. The goal of this paper is to establish a connection between two quantizations of the character variety R SL 2 .C/ .S/ D fr W 1 .S/ ! SL 2 .C/g = = SL 2 .C/;consisting of all group homomorphisms r from the fundamental group 1 .S / to the Lie group SL 2 .C/, considered up to conjugation by elements of SL 2 .C/. The double bar indicates here that the quotient is taken in the algebraic geometric sense of geometric invariant theory.The first quantization was introduced by D Bullock, C Frohman, J Kania-Bartoszyńska [12], J Przytycki and A Sikora [32] and V Turaev [35] and uses the Kauffman skein algebra S A .S/. This algebra is obtained by considering the vector space freely generated by all isotopy classes of framed links in S OE0; 1, and then taking the quotient of this space under the Kauffman skein relation; see Section 3.1. What makes S A .S / a quantization of R SL 2 .C/ .S/ is that, when A D 1, the skein algebra S 1 .S/ has a natural identification with the commutative algebra of regular functions on R SL 2 .C/ .S/ and that, as A tends to 1, the lack of commutativity of S A .S / is infinitesimally measured by the Goldman-Weil-Petersson Poisson structure [19; 20; 31; 36] on R SL 2 .C/ .S /; see [35]. There is a similar situation when A D C1, in which case S C1 .S/ has a natural identification with the algebra of functions on a twisted version of R SL 2 .C/ .S/; see Section 3.2. Guo and Liu [21]. This quantization takes advantage of the fact that, if one restricts to matrices with real coefficients, a large subset of R SL 2 .R/ .S/ with nonempty interior has a natural identification with the Teichmüller (or Fricke-Klein) space T .S/, consisting of isotopy classes of all complete hyperbolic metrics on S . Starting with an ideal triangulation of the surface, Thurston [33;34] introduced for the Teichmüller space T .S/ a set of coordinates, called shear coordinates, in which the Goldman-Weil-Petersson form is expressed in a particularly simple way. The quantum Teichmüller space is a quantization of T .S/ that is based on these shear coordinates. As this construction requires the existence of an ideal triangulation, the surface must have at least one puncture.A natural conjecture is that these two quantizations are "essentially equivalent".In the classical cases where q D 1 and A D˙1, the correspondence is relatively clear because of the identifications of S˙1.S/ and T 1 S with algebras of functions...

Quantum Teichmüller Spaces and Quantum Trace Map

“…, coincides with shear coordinate change map defined in [Hi,BW1]. Here (ψ ∆ ) −1 is defined onS bl .…”

“…A version of this statement already appeared in [BW1], and we use same proof. Every k ∈ Z∆ has a unique presentation k = u + m where m ∈ (2Z)∆ and u ∈ {0, 1}∆.…”

“…If P ⊂ ∂Σ, then (Σ, P) is called a marked surface 1 . A P-link in Σ is an immersion α : C → Σ, where C is compact 1-dimensional non-oriented manifold, such that 1 Our generalized marked surface is the same as "punctured surface with boundary" in [BW1], and our marked surfaces is the same as "marked surface" in [Mu].…”

“…If we use the face matrix Q instead of its submatrixQ, then Y (2) (∆) is the ChekhovFock algebra defined in [BW1,Liu], and Y bl (∆) is the Chekhov-Fock square root algebra of [BW1]. The skew fieldỸ (2) (∆) is considered as a quantization of a certain version of the Teichmüller space of (Σ, P), using the shear coordinates, see [CF1,BW1].…”

“…Quantum trace map through the Muller algebra in skein theory. The original construction of the quantum trace map in [BW1] involves difficult calculations, with miraculous identities. One of the goals of this paper is to offer another approach to the quantum trace map of Bonahon and Wong using Muller's extension of skein algebras.…”