Three-dimensional TQFTs via string-nets and two-dimensional surgery

If $C$ is a spherical fusion category, the string-net construction associates to each closed oriented surface $\mathrm{\Sigma }$ the vector space $Z_{\text{SN}}(\mathrm{\Sigma })$ of linear combinations of $C$-labelled graphs on $\mathrm{\Sigma }$ modulo local relations, in a way which is functorial with respect to orientation-preserving diffeomorphisms of surfaces. We show how to extend this assignment to a 3-dimensional topological quantum field theory (TQFT), by defining how the surgery generators in Juhász’ presentation of the oriented 3-dimensional bordism category act on the string-net vector spaces. We show that the resulting TQFT, which is formulated completely in the two-dimensional graphical language of string-nets, is an alternative description of the Turaev-Viro state sum model.