Non-Semisimple TQFT's and BPS 𝑞-Series

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 us to study (and perhaps even to formulate) 𝑞-series invariants labeled by spinᶜ 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.