Defects at the Intersection: the Supergroup Side

TL;DR

This paper explores intersecting defects in 5d supersymmetric gauge theories with supergroup structures, revealing dualities, algebraic origins, and connections to refined Chern-Simons theory through localization and Higgsing techniques.

Contribution

It introduces a novel supergroup-like structure in defect theories arising from Higgsing 5d gauge theories and relates their partition functions via analytic continuation, connecting to algebraic and string dualities.

Findings

Partition functions related by analytic continuation.

Supergroup structures emerge in defect theories.

Connections to refined Chern-Simons theory and string dualities.

Abstract

We consider two seemingly different theories in the $\Omega$-background: one arises upon the most generic Higgsing of a 5d $\mathcal{N}=1$ $\text{U}(N)$ gauge theory coupled to matter, yielding a 3d-1d intersecting defect; the other one arises upon simple Higgsing of a 5d $\mathcal{N}=1$ $\text{U}(N|M)$ supergroup gauge theory coupled to super-matter, yielding another defect. The cases $N=M=1$ are discussed in detail via equivariant localization to matrix-like models. The first theory exhibits itself a supergroup-like structure, which can be motivated via non-perturbative string dualities, and in a matter decoupling limit it is argued to be dual to a supergroup version of refined Chern-Simons theory. Furthermore, it is observed that the partition functions of the two defect theories are related by analytic continuation in one of the equivariant parameters. We find a common origin in the algebraic engineering through $q$-Virasoro screening currents. Another simple Higgsing of the 5d $\mathcal{N}=1$ $\text{U}(1|1)$ yields a single component defect whose partition function is reminiscent of ordinary refined Chern-Simons on a lens space.