Homotopy Field Theory in Dimension 2 and Group-algebras

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 π, we introduce cohomological HQFT's with target K(π, 1) derived from cohomology classes of π 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.