The quantum trace as a quantum non-abelianization map

Abstract

We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov-Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong's quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the SL 2 -character variety. This algebraic morphism shares many resemblance with the non-abelianization map of Gaiotto, Moore, Hollands and Neitzke. When the punctured surface is closed, we prove that this algebraic non-abelianization map induces a birational morphism between a smooth torus and the relative SL 2 character variety.

Triangular decomposition of character varieties

“…We prove that the character variety X C * (Σ) is canonically isomorphic to the relative singular cohomology group H 1 (Σ P , ∂Σ P ; C * ) and provide a simple description of its Poisson structure. The motivation to study this particular case lies in the connection, established in [KQ19b] in collaboration with A.Quesney, between this affine variety and the quantum Teichmüller space.…”

“…For δ ∈ G, we first consider the trivial 5. The case G = C * When G = C * , the character varieties have a simple description and are closely related to the quantum Teichmüller spaces (see [KQ19b]). Let c 1 , c 2 be two geometric curves in Σ P in transverse position and denote by σ 1 , σ 2 the cycles in Z 1 (Σ C , ∂Σ C ; Z) represented by c 1 and c 2 .…”

Stated skein algebras and their representations

“…The relative intersection form on H 1 ( Σ \ B, A; Z) restricts to a skew-symmetric map (still denoted by the same letter) on H σ 1 ( Σ \ B, A; Z) with integral values and Theorem 3.9. (Bonahon-Wong [BW16a] for unmarked surfaces, K.-Quesney [KQ19b] in general) The quadratic pair…”

“…A candidate for the quantum torus was proposed in [KQ19b] based on pants decomposition. An important step has been announced by Karuo [Kar] who will construct a filtration of S A (Σ g,0 ), based on pants decomposition, and prove that the associated graded algebra can be embedded into some quantum torus.…”

Mapping class group representations derived from stated skein algebras

“…that the Chebyshev-Frobenius morphism Ch A is the restriction of the Frobenius F r N through the quantum trace (this is how Ch A was first defined in [BW11]). The centers and PI-degrees of the balanced Chekhov-Fock algebras were computed in [BW17] for unmarked surfaces and in [KQ19b] for marked surfaces. In the particular case of Σ * g , it is described as follows.…”