3d-3d correspondence for mapping tori

TL;DR

This paper advances the understanding of 3d-3d correspondence by proposing a systematic approach to describe 3d N=2 SCFTs for various 3-manifolds, enabling the recovery of known invariants and the computation of new q-series invariants.

Contribution

It introduces a gauge theory framework with non-linear matter fields to describe 3d-3d correspondence without relying on geometric structures.

Findings

Recovered Turaev torsion and WRT invariants from twisted indices.

Proposed new methods to compute q-series invariants (Z) for manifolds with b_1 > 0.

Applied techniques to genus-1 mapping tori and extended to general 3-manifolds.

Abstract

One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d $N=2$ SCFT $T[M_3]$ --- or, rather, a "collection of SCFTs" as we refer to it in the paper --- for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on $M_3$ and, secondly, is not limited to a particular supersymmetric partition function of $T[M_3]$. In particular, we propose to describe such "collection of SCFTs" in terms of 3d $N=2$ gauge theories with "non-linear matter'' fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of $T[M_3]$, and propose new tools to compute more recent $q$-series invariants $\hat Z (M_3)$ in the case of manifolds with $b_1 > 0$. Although we use genus-1 mapping tori as our "case study," many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.