1999/10/04 by Vladimir Turaev, Turaev, Vladimir
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Quantum Algebra (math.QA) #math.AT #math.QA
paper · pdf · doi:10.48550/arxiv.math/9910010
arxiv created 1999/10/04 · arxiv updated 2009/11/30
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.