Mapping Class Group Representations and Generalized Verlinde Formula

TL;DR

This paper constructs unitary representations of mapping class groups using rational Hopf algebras and introduces a generalized Verlinde formula to compute traces based on fusion rules and quantum dimensions.

Contribution

It provides a new framework linking mapping class group representations with rational Hopf algebras and generalizes the Verlinde formula for trace calculations.

Findings

Unitary representations of mapping class groups are constructed via rational Hopf algebras.

A generalized Verlinde formula is proposed for trace computations.

Explicit formulae relate traces to fusion rules, quantum dimensions, and statistics phases.

Abstract

Unitary representations of centrally extended mapping class groups $\tilde M_{g,1}, g\geq 1$ are given in terms of a rational Hopf algebra $H$, and a related generalization of the Verlinde formula is presented. Formulae expressing the traces of mapping class group elements in terms of the fusion rules, quantum dimensions and statistics phases are proposed.