Generating functions for Higgs/Coulomb branch operators from 1d-3d cohomological equivalence

Abstract

We provide a proof for the conjectured equality of the generating function of integrated Higgs and Coulomb branch topological operators in 3d N ≥ 4 gauge theories and the three sphere partition function deformed by mass or FI parameters. The equality is the result of cohomological equivalence and applies to all theories in this class, including ABJM and other generalized Gaiotto-Witten models, and those without an explicit supersymmetric Lagrangian.

One-dimensional sectors from the squashed three-sphere

“… 47). In the presence of flavor symmetries, with global symmetry group G F acting on the hypermultiplets, one can introduce real mass parameters m valued in the Cartan of the Lie algebra of G F by coupling the hypermultiplets to a background vector multiplet with background values (2.41) with µ = m. In other words, instead of considering S hyp [H I , V], where V is a dynamical vector multiplet, one considersS hyp H I , V + V bk µ=m .…”