Conformal field theory and mapping class groups

TL;DR

This paper reviews how rational conformal field theories generate finite-dimensional representations of surface mapping class groups, connecting recent mathematical advances with the theory's formalism, especially in higher genus cases.

Contribution

It synthesizes recent mathematical developments within the conformal field theory framework and discusses the current understanding of higher genus cases.

Findings

Recent mathematical developments fit into the conformal field theory formalism.

The paper reviews the structure of representations in higher genus.

Connections between conformal blocks and mapping class groups are elucidated.

Abstract

Rational conformal field theories produce a tower of finite-dimensional representations of surface mapping class groups, acting on the conformal blocks of the theory. We review this formalism. We show that many recent mathematical developments can be fit into the first 2 floors of this tower. We also review what is known in higher genus.