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

TL;DR

This work analyzes how S-duality acts on half-BPS Wilson loops in 5d $ ext{N}=1$ theories, showing that Wilson loops transform covariantly into dual Wilson loops with background dressing factors, rather than mapping to 't Hooft-like operators as in some other dimensions. By combining Type IIB brane-web constructions with exact half-indices (via SQM/qq-characters) and residue extraction, the authors compute Wilson-loop VEVs for self-dual $SU(2)$ theories with $N_f\le4$ and for $SU(3)$ theories dual to $SU(2)\times SU(2)$ quivers, confirming the predicted S-duality maps. The results reveal a natural basis of tensor-product fundamental representations in which the duality action is simplest, and they organize the loop data into characters of enhanced global symmetries $E_{N_f+1}$ or related groups, supporting symmetry enhancement at the SCFT point. A generalization to larger quivers and to $SU(M)$ theories is proposed, suggesting a universal covariant S-duality map for Wilson loops with background dressing factors controlled by background Wilson loops.

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.

Figures (16)

• Figure 1: a) Brane realization of the pure $SU(2)$ theory. b) Half-BPS strings excitations for W-bosons and instanton particles.
• Figure 2: Hanany-Witten string creation/annihilation effects as we move a D3 brane (green dot) in the $x^{56}$ plane.
• Figure 3: 5d-1d quiver theory corresponding to the addition of a D3 brane in the center. The circle indicates the 1d flavor symmetry gauged with 5d fields. The dashed line indicates a bifundamental Fermi multiplet (two fermions).
• Figure 4: The (0,0) and (1,1) string configurations (on the left) are related to the two configurations for the fundamental Wilson loop insertion (on the right) by Hanany-Witten moves.
• Figure 5: Adding $n$ D3 branes realizes a quiver theory with a Fermi multiplet in the bifundamental of the $SU(2)\times U(n)_f$ flavor symmetry.