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Gauge Theory Wilson Loops and Conformal Toda Field Theory

Filippo Passerini

TL;DR

This work extends the AGT correspondence to Wilson loops in SU(N) gauge theories by identifying their two-dimensional Toda realization with monodromies of a degenerate chiral operator in ${\cal W}_N$ Toda theory. The authors derive the loop operators from the monodromy of the generalized hypergeometric equation governing four-point functions with two degenerate insertions, showing that the fundamental and anti-fundamental representations correspond to monodromies with opposite orientation. In the limit $b\to 1$, the results reproduce Pestun's matrix-model expectations, and for $N=2$ they reduce to the Liouville theory; orientation dependence remains a distinctive feature for $N>2$. Overall, the paper provides a concrete CFT realization of loop operators within the higher-rank Toda framework and strengthens the link between gauge theory dualities and monodromy data in conformal field theory.

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.

Gauge Theory Wilson Loops and Conformal Toda Field Theory

TL;DR

This work extends the AGT correspondence to Wilson loops in SU(N) gauge theories by identifying their two-dimensional Toda realization with monodromies of a degenerate chiral operator in Toda theory. The authors derive the loop operators from the monodromy of the generalized hypergeometric equation governing four-point functions with two degenerate insertions, showing that the fundamental and anti-fundamental representations correspond to monodromies with opposite orientation. In the limit , the results reproduce Pestun's matrix-model expectations, and for they reduce to the Liouville theory; orientation dependence remains a distinctive feature for . Overall, the paper provides a concrete CFT realization of loop operators within the higher-rank Toda framework and strengthens the link between gauge theory dualities and monodromy data in conformal field theory.

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.

Paper Structure

This paper contains 7 sections, 42 equations, 3 figures.

Table of Contents

  1. Introduction and Discussion
  2. $A_{N-1}$ Conformal Toda Field Theory
  3. Four Point Correlation Function
  4. Wilson Loops in Conformal Toda Field Theory
  5. Some Lie Algebra Notions
  6. $A_{N-1}$ Algebra
  7. The Hypergeometric Monodromy Group

Figures (3)

  • Figure 1: A sphere with four punctures in a given pair of pants decomposition. The tube in between the two couples of punctures is associated to a gauge group and the closed curve is associated to a Wilson loop operator.
  • Figure 2: The prescription to compute the Wilson loop include a change of basis from the $t$-channel to the $s$-channel, a monodromy around $z=0$ and finally a change of basis from $s$-channel to the $t$-channel. Dashed lines represent identity states.
  • Figure 3: A Wilson loop in the fundamental representation ${\cal L}$ and a Wilson loop in the anti-fundamental representations $\bar{\cal L}$ are associated to curves with opposite orientation. In this example we consider the gauge theory associated to the sphere with two simple punctures and two full punctures.