[2107.14238] Non-semisimple TQFT and BPS $q$ series

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overview: We propose and possibly prove the exact relationship between the roots of 1 and the 3-manifold invariants associated with the quantum group of the generic q. Both types of invariants are labeled by additional data that play an important role in the proposed relationship. Bridging two aspects that until recently were developed separately using very different methods opens up many new avenues. In one direction, the invariants of the $q$ series labeled by the spin$^c$ structure can be studied (and perhaps even formulated) in non-semisimple invariants. In the opposite direction, we offer new insights and perspectives on various elements of nonsemisimple TQFTs, linking the latter to other invariants of knots and 3-manifolds recently realized in quantum field theory and superstring theory and 1 into one integration framework.