Quasiconformal Teichmuller theory as an analytical foundation for two-dimensional conformal field theory

TL;DR

This paper explores the connection between quasiconformal Teichmuller theory and conformal field theory, proposing the Weil-Petersson class parametrizations as a natural analytic setting for CFT with supporting rigorous results.

Contribution

It introduces the Weil-Petersson class parametrizations as a new analytic framework for conformal field theory, linking geometric function theory with CFT.

Findings

Weil-Petersson class parametrizations have the necessary regularity for CFT.

Rigorous analytic results support the proposed framework.

The approach bridges Teichmuller theory and conformal field theory.

Abstract

The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. We survey the recent and careful study of these objects, which has led to significant connections with quasiconformal Teichmuller theory and geometric function theory. In particular we propose that the natural analytic setting for conformal field theory is the moduli space of Riemann surfaces with so-called Weil-Petersson class parametrizations. A collection of rigorous analytic results is advanced here as evidence. This class of parametrizations has the required regularity for CFT on one hand, and on the other hand are natural and of interest in their own right in geometric function theory.