Generalized Superconformal Index for Three Dimensional Field Theories

TL;DR

This work formulates a generalized $S^2 \times S^1$ superconformal index by coupling background gauge fields to global symmetries and introducing a discrete monopole flux, enabling gauging of global symmetries directly at the index level. The authors define $I(t_a,n_a;x)$ by promoting selected flavor $U(1)$ symmetries to dynamical gauges (with holonomies $z_a$ and fluxes $s_a$) and modifying chiral multiplet contributions to depend on $|\rho(s)+\sum f_a(\Phi)n_a|$, while also incorporating the topological $U(1)_J$ sector; this framework allows systematic derivations of dualities via index-level gauging. They explicitly verify abelian mirror symmetry for $N_f=1$ and general $N_f$ by relating the generalized index and the $S^3$ partition function, starting from the base dual pair (SQED with one flavor and the XYZ theory) and gauging global currents to generate the full family of duals, with precise mappings of mass and FI parameters and of global symmetries. The results establish a concrete, gauge-invariant method to test and derive three-dimensional dualities at the level of indices and partition functions, with potential to uncover additional dual pairs by generalizing the gauging procedure. Overall, the work connects generalized index constructions, BF couplings, and matrix-model partition functions to provide a robust toolkit for probing 3D N=2 dualities.

Abstract

We introduce a generalization of the S^2 x S^1 superconformal index where background gauge fields with magnetic flux are coupled to the global symmetries of the theory. This allows one to gauge a global symmetry at the level of the index, which we use to show the matching of the superconformal index for N=2 SQED with N_f flavors and its mirror dual.