Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed $\mathcal{C}$-algebras.

Timothy Porter and Vladimir Turaev

Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum FieldTheories to closed $d$-manifolds endowed with extra structure in the form of homotopy classes of maps into a given `target' space, $B$. For $d = 1$,classifications of HQFTs in terms of algebraic structures are known when $B$ is a $K(G,1)$ and also when it is simply connected.Here we study general HQFTs with $d = 1$ and target a general 2-type, giving a common generalisation of the classifying algebraic structures for thetwo cases previously known. The algebraic models for 2-types that we use are crossed modules, $\mathcal{C}$, and we introduce a notion of formal$\mathcal{C}$-map, which extends the usual lattice-type constructions to this setting. This leads to a classification of `formal'2-dimensional HQFTs with target $\mathcal{C}$, in terms of crossed $\mathcal{C}$-algebras.

Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 113-159