Wilson loops in 5d $\mathcal{N}=1$ theories and S-duality

TL;DR

This paper investigates how S-duality transforms Wilson loop operators in 5d $ N=1$ theories, confirming string theory predictions through exact calculations of Wilson loop VEVs and revealing additional factors in the duality map.

Contribution

It provides the first exact computations of Wilson loop transformations under S-duality in 5d $ N=1$ theories, validating string theory predictions and elucidating the role of background Wilson loops.

Findings

Wilson loops map to dual Wilson loops under S-duality.

Exact VEV computations confirm string theory predictions.

Additional multiplicative factors in the duality map are identified.

Abstract

We study the action of S-duality on half-BPS Wilson loop operators in 5d $\mathcal{N}=1$ theories. The duality is the statement that different massive deformations of a single 5d SCFT are described by different gauge theories, or equivalently that the SCFT points in parameter space of two gauge theories coincide. The pairs of dual theories that we study are realized by brane webs in type IIB string theory that are S-dual to each other. We focus on $SU(2)$ SQCD theories with $N_f \le 4$ flavors, which are self-dual, and on $SU(3)$ SQCD theories, which are dual to $SU(2)^2$ quiver theories. From string theory engineering we predict that Wilson loops are mapped to dual Wilson loops under S-duality. We confirm the predictions with exact computations of Wilson loop VEVs, which we extract from the 5d half-index in the presence of auxiliary loop operators (also known as higher qq-characters) sourced by D3 branes placed in the brane webs. A special role is played by Wilson loops in tensor products of the (anti)fundamental representation, which provide a natural basis to express the S-duality action. The exact computations also reveal the presence of additional multiplicative factors in the duality map, in the form of background Wilson loops.