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

TL;DR

This paper proves a conjecture linking the generating functions of Higgs and Coulomb branch operators in 3d N≥4 gauge theories to the three-sphere partition function with mass or FI deformations, based on cohomological equivalence.

Contribution

It provides a general proof of the equality between generating functions and partition functions for a broad class of 3d N≥4 theories, including those without explicit Lagrangians.

Findings

Confirmed the conjectured equality for various theories including ABJM.

Established cohomological equivalence as the basis for the equality.

Applicable to theories with and without explicit Lagrangian descriptions.

Abstract

We provide a proof for the conjectured equality of the generating function of integrated Higgs and Coulomb branch topological operators in 3d $\mathcal{N}\ge 4$ 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.