arXiv math/01050182001-05math.QAmath.ATmath.CT

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

G. Rodrigues

备注：: 32 pages. Uses AMS-latex and xy-pic. Revised version: minor changes, correction of some typos, conclusions shortened. Proposition 1.3 corrected. To be published in Journal of Knot theory and its Ramifications

英文摘要：

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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