Seifert fibering operators in 3d $\mathcal{N}=2$ theories

TL;DR

This paper derives an exact formula for supersymmetric partition functions of 3d $ =2$ theories on Seifert manifolds, introducing fibering operators as fundamental building blocks and confirming duality invariance.

Contribution

It introduces fibering operators as a novel method to compute partition functions on Seifert manifolds, generalizing previous results and connecting to holomorphic blocks.

Findings

Exact formula for Seifert manifold partition functions

Fibering operators as key computational tools

Partition functions match across dualities

Abstract

We study 3d $\mathcal{N}=2$ supersymmetric gauge theories on closed oriented Seifert manifold---circle bundles over an orbifold Riemann surface---, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the "fibering operators." These operators are half-BPS line defects, whose insertion along the $S^1$ fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space $L(p,q)_b$ with rational squashing parameter $b^2 \in \mathbb{Q}$, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.