Non-Semisimple TQFT's and BPS $q$-Series

TL;DR

This paper establishes a precise relationship between quantum group-based 3-manifold invariants at roots of unity and generic q, unifying different approaches and opening new research avenues in non-semisimple TQFTs and q-series invariants.

Contribution

It proposes and partially proves a relation connecting semisimple and non-semisimple 3-manifold invariants, bridging previously separate frameworks and enabling new insights into quantum invariants.

Findings

Relation between quantum group invariants at roots of unity and generic q.

Unification of non-semisimple TQFTs with other quantum invariants.

Potential to formulate q-series invariants labeled by spin^c structures.

Abstract

We propose and in some cases prove a precise relation between 3-manifold invariants associated with quantum groups at roots of unity and at generic $q$. Both types of invariants are labeled by extra data which plays an important role in the proposed relation. Bridging the two sides - which until recently were developed independently, using very different methods - opens many new avenues. In one direction, it allows to study (and perhaps even to formulate) $q$-series invariants labeled by spin$^c$ structures in terms of non-semisimple invariants. In the opposite direction, it offers new insights and perspectives on various elements of non-semisimple TQFT's, bringing the latter into one unifying framework with other invariants of knots and 3-manifolds that recently found realization in quantum field theory and in string theory.