On Crossing Symmmetry and Modular Invariance in Conformal Field Theory and S Duality in Gauge Theory

TL;DR

This paper investigates the connections between crossing symmetry, modular invariance in conformal field theory, and S-duality in gauge theories, revealing how partition functions relate across these frameworks using Liouville theory.

Contribution

It demonstrates that partition functions of S-dual N=2 gauge theories can be derived from crossing symmetry in Liouville theory, linking conformal and gauge theories.

Findings

Partition functions of S-dual theories are connected via crossing symmetry.

Liouville four-point functions encode gauge theory dualities.

Liouville partition functions on torus relate to N=4 gauge theory.

Abstract

In this note, we explore the relation between crossing symmetry and modular invariance in conformal field theory and S-duality in gauge theory. It is shown that partition functions of different S dual theories of N=2 SU(2) gauge theory with four fundamentals can be derived from the crossing symmetry of the Liouville four point function. We also show that the partition function of N=4 SU(2) gauge theory can be derived from the Liouville partition function on torus.