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Homotopy field theory in dimension 2 and group-algebras
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abstract
We apply the idea of a topological quantum field theory (TQFT) to maps from manifolds into topological spaces. This leads to a notion of a (d+1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed d-dimensional manifolds and (d+1)-dimensional cobordisms endowed with homotopy classes of maps into a given space. For a group $\pi$, we introduce cohomological HQFT's with target $K(\pi,1)$ derived from cohomology classes of $\pi$ and its subgroups of finite index. The main body of the paper is concerned with (1+1)-dimensional HQFT's. We classify them in terms of so called crossed group-algebras. In particular, the cohomological (1+1)-dimensional HQFT's over a field of characteristic 0 are classified by simple crossed group-algebras. We introduce two state sum models for (1+1)-dimensional HQFT's and prove that the resulting HQFT's are direct sums of rescaled cohomological HQFT's. We also discuss a version of the Verlinde formula in this setting.
Forward citations
Cited by 2 Pith papers
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A Comment On Topological Degeneracy In Gauged WZW Models
The gauged WZW model is the GKO coset model coupled to a 2d topological field theory whose commutative Frobenius algebra End(B) controls an extra finite degeneracy.
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2D HQFTs and Frobenius $(\mathcal{G},\mathcal{V})$-categories
Two-dimensional homotopy quantum field theories with target pairs (X,Y) are classified by crossed loop Frobenius categories over the relative fundamental groupoid of (X,Y).
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