Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1+1

Details Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1+1 Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1+1

2001-05-02

arxiv.org/abs/math/0105018v4

Mathematics

Quantum Algebra

32 pages. Uses AMS-latex and xy-pic. Revised version: minor changes, correction of some typos, conclusions shortened. Proposit

Scientific paper

We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n,X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord}(n,X), including the fact that it only depends on the (n+1)-homotopy type of X. We also prove that an HQFT with target space X and in dimension n+1 implies the existence of geometrical structures in X; in particular, flat gerbes make their appearance. We give a complete characterization of HCobord(n,X) for n=1 (or the 1+1 case) and X the Eilenberg-Maclane space K(G,2). In the final section we derive state sum models for these HQFT's.

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