Skein (3+1)-TQFTs from non-semisimple ribbon categories

Abstract: Using skein theory very much in the spirit of the Reshetikhin--Turaev constructions, we define a $(3+1)$-TQFT associated with possibly non-semisimple finite unimodular ribbon tensor categories. State spaces are given by admissible skein modules, and we prescribe the TQFT on handle attachments. We give some explicit algebraic conditions on the 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 invertibility of the TQFTs, study their behavior under connected sum and provide some examples.