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.
