Gauge Theory Wilson Loops and Conformal Toda Field Theory

TL;DR

This paper explores the relationship between Wilson loops in SU(N) gauge theories and monodromies in A_{N-1} Toda field theory, extending previous work on Liouville theory to higher rank cases.

Contribution

It demonstrates that Wilson loops in SU(N) gauge theories correspond to monodromies of degenerate operators in A_{N-1} Toda field theory, generalizing known results from Liouville theory.

Findings

Wilson loops are linked to monodromies of degenerate operators in Toda theory.

The orientation of the loop determines the fundamental or anti-fundamental representation.

The analysis uses properties of the hypergeometric differential equation.

Abstract

The partition function of a family of four dimensional N=2 gauge theories has been recently related to correlation functions of two dimensional conformal Toda field theories. For SU(2) gauge theories, the associated two dimensional theory is A_1 conformal Toda field theory, i.e. Liouville theory. For this case the relation has been extended showing that the expectation value of gauge theory loop operators can be reproduced in Liouville theory inserting in the correlators the monodromy of chiral degenerate fields. In this paper we study Wilson loops in SU(N) gauge theories in the fundamental and anti-fundamental representation of the gauge group and show that they are associated to monodromies of a certain chiral degenerate operator of A_{N-1} Toda field theory. The orientation of the curve along which the monodromy is evaluated selects between fundamental and anti-fundamental representation. The analysis is performed using properties of the monodromy group of the generalized hypergeometric equation, the differential equation satisfied by a class of four point functions relevant for our computation.