Rigid Surface Operator and Symbol Invariant of Partitions

TL;DR

This paper refines the understanding of symbol invariants in the Springer correspondence, clarifies the nature of S-duality maps for rigid surface operators, and classifies problematic operators that lack duals.

Contribution

It demonstrates that S-duality maps are symbol preserving, shows the equivalence of key maps in their construction, and classifies rigid surface operators without duals.

Findings

S-duality maps are symbol preserving.

The maps $X_S$ and $Y_S$ are essentially the same.

Classified all $B_n/C_n$ rigid surface operators without duals.

Abstract

The symbol is used to describe the Springer correspondence for the classical groups by Lusztig. We refine the explanation that the $S$-duality maps of the rigid surface operators are symbol preserving maps. And we find that the maps $X_S$ and $Y_S$ used in the construction of $S$-duality maps are essentially the same. We clear up cause of the mismatch problem of the total number of the rigid surface operators between the $B_n$ and $C_n$ theories. And we construct all the $B_n/C_n$ rigid surface operators which can not have a dual. A classification of the problematic surface operators is made.