Defects at the Intersection: the Supergroup Side

We consider two seemingly different theories in the $\mathrm{\Omega }$-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 $\mathcal{N}=1$ $\text{U}(N\mathrm{\mid }M)$ 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 $\mathcal{N}=1$ $\text{U}(1\mathrm{\mid }1)$ yields a single component defect whose partition function is reminiscent of ordinary refined Chern-Simons on a lens space.