Introduction and Summary of Results

Abstract

AbstractIn the last decade quantum field theory and string theory have strongly impacted many areas of mathematics, especially the geometry and topology of low dimensional manifolds. In particular, a wealth of intriguing mathematical structures were discovered to be inherent to so called topological quantum field theories (TQFT’s) and conformal field theories (CFT’s). Originally, these notions refer to a class of concrete physical quantum field theories, among which three dimensional Chern- Simons theory and two dimensional rational conformal field theory are some of the most prominent ones. It was soon realized that the abstract setting of category theory makes it possible to efficiently organize the zoo of data and structures of these field theories. Eventually, TQFT’s evolved into purely mathematical notions, defined axiomatically in the language of categories and functors. Axiomatic TQFT’s and similar theories are, therefore, in nature rather similar to other functors in algebraic topology, such as homology. Atiyah was the first mathematician to cast the notion of TQFT’s into an axiomatic framework in his seminal work [Ati88]. Independently and at about the same time G. Segal [Seg88] formulates a mathematical definition of CFT’s, which very similarly based on categories and functors. The notion of extended TQFT’s that we will introduce here and on which our constructions will be based involves higher category theory, namely double categories and double functors. It thus contains both Atiyah’s notion of a TQFT in dimension three and Segal’s notion of CFT as special cases, though they appear on different categorical levels. The definition will not only be a natural and conceptual unification of previous theories, but further abstractions will allow us to construct new classes of TQFT’s, namely non-semisimple TQFT’s, that are manifestly different from other combinatorially defined ones and in some cases describe TQFT’s based on classical gauge theories.KeywordsHopf AlgebraWilson LineConformal Field TheoryMapping Class GroupAbelian CategoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.