Exact $S$-duality Map for Rigid Surface Operators

TL;DR

This paper presents a precise $S$-duality map for rigid surface operators in four-dimensional gauge theories, resolving longstanding classification mismatches by extending the map to include non-rigid operators.

Contribution

It introduces a natural $S$-duality map based on partition manipulations, addressing the mismatch problem and encompassing non-rigid operators for all gauge group ranks.

Findings

The map is realized by moving the longest row in the partition pair.

It resolves the mismatch problem between dual theories.

The approach clarifies several longstanding puzzles.

Abstract

Surface operators in four-dimensional gauge theories are two-dimensional defects, serving as natural generalizations of Wilson lines and 't Hooft line operators. They act as ideal probes for exploring the non-perturbative structure of the theory. Rigid surface operators are a specific class of surface operators characterized by the absence of continuous deformation parameters. It is expected that a closed $S$-duality map should exist among these rigid operators. While progress has been made on specific examples or subclasses by leveraging invariants and empirical conjectures, a complete picture remains elusive. A significant challenge arises when multiple rigid surface operators share identical invariants, making the determination of $S$-duality relations difficult. More critically, a mismatch exists in the number of rigid surface operators between dual theories when classified by invariants; this is referred to as the \textit{mismatch problem}. This discrepancy suggests the necessity of extending the scope of consideration beyond strictly rigid operators. In this paper, we propose a direct, natural, and precise $S$-duality map for rigid surface operators. Our map is realized by moving the longest row in the pair of partitions defining a surface operator from one factor to the other, with an additional box appended or deleted to balance the total number of boxes. This mapping naturally incorporates non-rigid surface operators, thereby resolving the mismatch problem. The proposed map is applicable to gauge groups of all ranks and clarifies several long-standing puzzles in the field.