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

TL;DR

This paper constructs a 3D topological quantum field theory from spherical fusion categories using string-net models and 2D surgery, providing an alternative to the Turaev-Viro model.

Contribution

It extends the string-net construction to a 3D TQFT by defining surgery operations, offering a new 2D graphical approach to Turaev-Viro invariants.

Findings

The TQFT is fully described by 2D string-net diagrams.

The construction matches the Turaev-Viro state sum model.

Provides a functorial 3D TQFT framework.

Abstract

If $C$ is a spherical fusion category, the string-net construction associates to each closed oriented surface $\Sigma$ the vector space $Z_\text{SN}(\Sigma)$ of linear combinations of $C$-labelled graphs on $\Sigma$ 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 Juh\'{a}sz' 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.