Interface line operators in $\mathcal{N}=4$ SYM theories and supersymmetric indices

TL;DR

The paper investigates interface line operators connecting two $ Nsize{=4}$ SYM theories with unitary gauge groups, realized via Type IIB branes, and defines line defect half-indices for NS5- and D5-type interfaces. It provides explicit matrix-integral expressions for Wilson and vortex line insertions and demonstrates strong evidence for S-duality through precise matches of these half-indices under $t o t^{-1}$, including Higgs and Coulomb limit analyses. Notably, it derives closed-form, Higgs-limit expressions using $q$-binomial coefficients and principal specializations of Schur functions, linking line defect spectra to column-strict plane partitions and Hall-Littlewood/Schur frameworks. The results span $U(N)|U(1)$, $U(N)|U(N)$, and $U(N)|U(M)$ configurations, revealing detailed dual dictionaries between Wilson and vortex lines and offering a systematic vortex-expansion approach to one-point and higher-point functions. These findings advance the understanding of BPS spectra at interfaces and have potential implications for holographic duals of line defects in $ Nsize{=4}$ SYM.

Abstract

We study configurations of two $\mathcal{N}=4$ super Yang-Mills theories of unitary gauge groups connected by the BPS interfaces involving line operators. We find strong evidence of S-duality of the configurations as precise matching of the line defect half-indices which enumerate the BPS local operators at the junctions of the interfaces and line operators. The interface line defect half-indices are expressible as intriguing closed-form formulae involving the $q$-binomial coefficients and the principal specializations of the Schur functions.

Figures (1)

• Figure 1: The NS5-type interface in 4d $U(N)\times U(M)$ bulk theory (left). The D5-type interface (right). The horizontal and vertical directions are the directions $6$ and $2$ respectively. The line operators are drawn as circles sitting at $x^6=0$, the location of the interfaces.