An analog of a modular functor from quantized Teichm"uller theory

On the relation between quantum Liouville theory and the quantized Teichm"uller spaces

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmüller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture.

An analog of a modular functor from quantized Teichmüller theory, I

This paper has been withdrawn by the author(s). The material contained in the paper will be published in a subtantially reorganized form, part of it is now included in math.QA/0510174

$A_n$型量子泰希米勒理论中的代数模函子猜想

The algebraic modular functor conjecture in type $A_n$ quantum Teichmüller theory

该研究针对Fock与Goncharov提出的高阶泰希米勒理论量子化的代数模函子猜想，通过拓展Fock-Goncharov框架，证明了$G = \mathrm{PGL}_{n+1}$（即$A_n$型）情形下猜想成立，构造了等变典范切割同构以重构量子代数。

Teichmüller 空间中的量化测地长度：代数方面

Quantized Geodesic Lengths for Teichmüller Spaces: Algebraic Aspects

本文基于Chekhov-Fock理论，研究量化测地长度算子的代数结构，通过量子迹映射构造量化迹-单胞映射，并验证Teschner的递推关系及不相交循环的量子迹-单胞的强交换性。

Homotopy Coherent Mapping Class Group Actions and Excision for Hochschild Complexes of Modular Categories

Given any modular category $\mathcal{C}$ over an algebraically closed field $k$, we extract a sequence $(M_g)_{g\geq 0}$ of $\mathcal{C}$-bimodules. We show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ of $\mathcal{C}$ with coefficients in $M_g$ carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus $g+1$. The ordinary Hochschild complex of $\mathcal{C}$ corresponds to $CH(\mathcal{C};M_0)$. This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor $\mathfra…