Quantum Teichmüller Spaces and Quantum Trace Map

Abstract

Abstract. We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov-Fock (and Kashaev) in an abstract way, can be realized as a concrete subalgebra of the skew field of the skein algebra.

Quantum Holonomies from Spectral Networks and Framed BPS States

“…Moreover, motivated by the relation with Chern-Simons theory [14,15], Turaev showed that the Poisson algebra of paths on a surface C can be quantized by the skein algebra of knots in M 3 [16]. This 3-dimensional approach was used successfully by Bonahon and Wong to define a quantum trace for SL(2, C) [11] (see also [17,18]), that is a homomorphism from the skein algebra to the quantum Teichmüller space [19]. The quantum trace maps every isotopy class of knot in M 3 to a Laurent polynomial in the quantized shear coordinates associated with an ideal triangulation of C. The isotopy invariance of the quantum trace is ensured by carefully controlling the elevation of segments of the knot, which is achieved by inserting certain transformations over the edges of the triangulation.…”

Skein algebras of surfaces

“…where I " r0, 1s, such that the boundary points of L lie in BˆI, and the framing of L at boundary points is parallel to BF . (We follow the terminology of [BW1,Mu,Le2], even though the term "tangle" may be more adequate here.) We assume additionally that each arc is isotopic mod its boundary to an arc with framing parallel to F. (Hence, adding a halftwist to it is not allowed.)…”

“…In particular, the skein algebra of a punctured surface embeds into the quantum Teichmüller space of Chekhov-Fock, [BW1], cf. [CF,Ka,Le2,Le3,Mu].…”

“…We call the quotient RSpF, Bq " SpF, Bq{S 1 pF, Bq the reduced skein algebra of pF, Bq. That algebra makes an appearance in [Le2] already and it is a quotient of the marked skein algebra of [Mu].…”

“…We say that a component γ of BF is marked if it contains at least one element of B. (The unmarked components correspond to surface punctures in [BW1,Le2,Mu]. )…”