Four dimensional topological quantum field theories from $G$-crossed braided categories

Crossed Modules and Quantum Groups in Braided Categories

Let A be a Hopf algebra in a braided category C. Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category DY (C)AA of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group (A,A,R) the correspo...

Topological Quantum Field Theories on Generalized Cobordism Categories

Topological Quantum Field Theories

We intruduce a new invariant for σ models (and foliations more generally) using the even pairing between K-homology and cyclic homology. We try to calculate it for the simplest case of foliations, namely principal bundles. We end up by discussing some possible physical applications primarily in Quantum Gravity and M-Theory. PACS classification: 11.10.-z; 11.15.-q; 11.30.-Ly

Topological Quantum Field Theories

Following my plenary lecture on ICMP2000 I review my results concerning two closely related topics: topological quantum field theories and the problem of quantization of gauge theories. I start with old results (first examples of topological quantum field theories were constructed in my papers in late seventies) and I come to some new results, that were not published yet.

Topological Quantum Field Theories

In this article, written primarily for physicists and geometers, we introduce the notion of TQFT, orbifold, and then we survey the construction of TQFTs originating from orbifolds such as Chen-Ruan cohomology and orbifold string topology.