Factorization of the 3d superconformal index

TL;DR

The paper proves that the 3d ${\cal N}=2$ ${\rm U}(N)$ superconformal index, for theories with fundamentals/antifundamentals and with or without Chern-Simons terms, factorizes into a product of vortex and antivortex partition functions, Z_vortex and Z_ antivortex, by explicit residue evaluation of the index matrix integral. It shows this factorization explicitly in the Abelian case and then summarizes the nonabelian CS-enabled generalizations, including the mirror of one free chiral and the relation of Z_vortex to topological open string amplitudes. The factorized form clarifies Seiberg-like dualities (Aharony dualities) and N=4 mirror symmetry at the index level, with detailed examples for simple and general cases, and reveals connections to open/closed topological string theories and higher-dimensional defects. The results provide a concrete, analytic framework to study 3d dualities and their geometric/topological interpretations, potentially extending to broader gauge groups and defect configurations.

Abstract

We prove that 3d superconformal index for general $\mathcal N=2$ U(N) gauge group with fundamentals and anti-fundmentals with/without Chern-Simons terms is factorized into vortex and anti-vortex partition function. We show that for simple cases, 3d vortex partition function coincides with a suitable topological open string partition function. We provide much more elegant derivation at the index level for $\mathcal N=2$ Seiberg-like dualities of unitary gauge groups with fundamantal matters and $\mathcal N=4$ mirror symmetry

Figures (2)

• Figure 1: A strip geometry.
• Figure 2: A necklace $U(1)^N$ quiver theory.