Homotopy quantum field theories (HQFTs) generalize topological quantum field theories (TQFTs) by replacing manifolds by maps from manifolds to a fixed target space $X$. For example, any cohomology class in $H^3(X)$ defines a 3-dimensional HQFT with target $X$. If $X$ is aspherical, that is $X = K(G, 1)$ for some group $G$, then this cohomological HQFT is related to the Dijkgraaf-Witten invariant and is computed as a Turaev-Viro state sum via the category of $G$-graded vector spaces. More generally, the state sum Turaev-Viro TQFT and the surgery Reshetikhin-Turaev TQFT extend to HQFTs (using graded fusion categories) which are related via the graded categorical center.

This is joint work with V. Turaev.