S-duality of boundary lines in $\mathcal{N}=4$ SYM theories and supersymmetric indices

TL;DR

This paper explores the S-duality of boundary lines in $ 4$ SYM theories by analyzing supersymmetric defect indices, demonstrating index matching under dual configurations, and deriving exact formulas using Macdonald polynomials.

Contribution

It provides a detailed analysis of boundary line operators and their indices in $ 4$ SYM, establishing precise duality relations and exact index computations.

Findings

Matching of indices for S-dual boundary configurations

Exact closed-form expressions for defect indices

Application of Macdonald polynomials in index calculations

Abstract

We analyze the supersymmetric defect indices of $\mathcal{N}=4$ super Yang Mills theories which are simultaneously decorated by the BPS line operators and the boundary conditions. We demonstrate that the two-point functions of the boundary 't Hooft lines of magnetic charges associated with the minuscule representations in the presence of the regular Nahm pole boundary conditions can be obtained by applying the Higgsing prescription to the half-indices of the Dirichlet boundary conditions. Accordingly, we find precise matching of the indices for pairs of the S-dual configurations with the Wilson lines and Neumann boundary conditions and those with the 't Hooft lines and the regular Nahm pole boundary conditions. Alternatively, we analytically compute the indices by means of the inner product of the Macdonald polynomials to find the exact closed-form expressions.