Supersymmetric Wilson Loops, Instantons, and Deformed ${\cal W}$-Algebras

TL;DR

This paper explores the interplay between supersymmetric Wilson loops, instantons, and deformed ${ m W}$-algebras in 5d gauge theories, revealing connections to quantum affine algebras, vortex dualities, and deformed Toda theories.

Contribution

It introduces a novel framework linking 5d supersymmetric gauge theories with quantum affine algebras and deformed Toda theories via string theory and dualities.

Findings

Partition function as a $qq$-character of quantum affine algebra

Equivalence of gauge theory partition function and deformed Toda correlator

Dual descriptions via vortex perspective and string theory constructions

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra. We study 1/2-BPS Wilson loops of supersymmetric 5d $\mathfrak{g}$-type quiver gauge theories on a circle, in a non-trivial instanton background. The Wilson loops are codimension 4 defects of the gauge theory, and their interaction with self-dual instantons is captured by a modified 1d ADHM quantum mechanics. We compute the partition function as its Witten index. This index is a "$qq$-character" of a finite-dimensional irreducible representation of the quantum affine algebra $U_q(\hat{\mathfrak{g}})$. Using gauge/vortex duality, we can understand the 5d physics in 3d gauge theory terms. Namely, we reinterpret the 5d theory with vortex flux from the point of view of the vortices themselves. This vortex perspective has an advantage: it has yet another dual description in terms of deformed $\mathfrak{g}$-type Toda Theory on a cylinder, in free field formalism. We show that the gauge theory partition function is equal to a chiral correlator of the deformed Toda Theory, with stress tensor and higher spin operator insertions. We derive all the above results from type IIB string theory, compactified on a resolved $ADE$ singularity times a cylinder with punctures, with various branes wrapping the blown-up 2-cycles.