Skein (3+1)-TQFTs from Non-Semisimple Ribbon Categories

Abstract

We define a (3+1)-TQFT associated with possibly non-semisimple finite unimodular ribbon tensor categories using skein theory. This gives an explicit realization of a TQFT predicted by the cobordism hypothesis, based on recent results on dualizability. State spaces are given by admissible skein modules, and we prescribe the TQFT on handle attachments. We give some explicit algebraic conditions on the input category to define this TQFT, namely to be ''chromatic non-degenerate''. As a by-product, we obtain an invariant of 4-manifolds equipped with a ribbon graph in their boundary, and in the ''twist non-degenerate'' case, an invariant of 3-manifolds. Our construction generalizes the Crane-Yetter-Kauffman TQFTs in the semi-simple case, and the Lyubashenko (hence also Hennings and WRT) invariants of 3-manifolds. The whole construction is very elementary, and we can easily characterize the invertibility of the TQFTs, study their behavior under connected sums and provide some examples.