Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 010, 71 pages      arXiv:2107.14238      https://doi.org/10.3842/SIGMA.2023.010
Contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday

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

Francesco Costantino a, Sergei Gukov b and Pavel Putrov c
a) Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse, France
b) Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA
c) The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, Trieste 34151, Italy

Received January 21, 2022, in final form February 10, 2023; Published online March 15, 2023

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 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.

Key words: 3-manifold invariants; knot invariants; TQFT.

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