Extended Tqft’s and Quantum Gravity

This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2-functor Z : nCob2→2Vect, by analogy with the description of a TQFT as a functor Z : nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the Dijkgraaf-Witten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a higher-categorical version of Vect, denoted 2Vect, a bicategory of 2-vector spaces. Along the way, we prove several results showing how to construct 2-vector spaces of Vect-valued presheaves on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of “pullback and pushforward” of presheaves gives both the morphisms and 2-morphisms in 2Vect for the extended TQFT, and that these satisfy the axioms for a weak 2-functor. Finally, we discuss the motivation for this research in terms of Quantum Gravity. If the results can be extended from a finite group G to a Lie group, then for some choices of G this theory will recover an existing theory of Euclidean quantum gravity in 3 dimensions. We suggest extensions of these ideas which may be useful to further this connection and apply it in higher dimensions.