Supergroups, q-series and 3-manifolds

TL;DR

This paper introduces supergroup analogues of 3-manifold invariants, focusing on superunitary groups like SU(2|1), and explores their properties, calculations, and conjectured relations to quantum invariants.

Contribution

It develops a framework for supergroup homological blocks, provides an explicit algorithm for their computation, and investigates their mathematical properties and connections to quantum supergroups.

Findings

Defined supergroup invariants for 3-manifolds

Provided an algorithm for computing q-series for plumbed 3-manifolds

Studied quantum modularity and resurgence properties

Abstract

We introduce supergroup analogues of 3-manifold invariants $\hat{Z}$, also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups, and work out the case of SU(2|1) in details. Physically these invariants are realized as the index of BPS states of a system of intersecting fivebranes wrapping a 3-manifold in M-theory. As in the original case, the homological blocks are q-series with integer coefficients. We provide an explicit algorithm to calculate these q-series for a class of plumbed 3-manifolds and study quantum modularity and resurgence properties for some particular 3-manifolds. Finally, we conjecture a formula relating the $\hat{Z}$ invariants and the quantum invariants constructed from a non-semisimple category of representation of the unrolled version of a quantum supergroup.