Title. Four Dimensional Topological Quantum Field Theories from $G$-crossed Braided Categories

Abstract.  We construct a state-sum type invariant of smooth closed oriented 4-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a (3+1)-dimensional topological quantum field theory (TQFT). The invariant of 4-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant from a ribbon fusion category and Yetter's invariant from homotopy 2-types. Furthermore, a cohomology class in $H^4(G,U(1))$ can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. Very recently, Reutter and Douglas generalized the above construction to spherical fusion 2-categories, which is expected to produce the most general (3+1)-TQFTs of state-sum type.