Three-Dimensional Spin TFTs from Gauging Line Defects

TL;DR

This paper constructs three-dimensional spin topological field theories from oriented TFTs with line defects using algebraic data, extending previous models and classifying abelian spin Chern-Simons theories.

Contribution

It introduces a new method to derive 3D spin TFTs from line defect data and extends earlier constructions to include a broader class of theories.

Findings

Constructed 3D spin TFTs from line defect data and Frobenius algebras.

Extended previous models to include spin structures and equivariant modules.

Reproduced classification of abelian spin Chern-Simons theories.

Abstract

From the input of an oriented three-dimensional TFT with framed line defects and a commutative $\Delta$-separable Frobenius algebra $A$ in the ribbon category of these line defects, we construct a three-dimensional spin TFT. The framed line defects of the spin TFT are labelled by certain equivariant modules over $A$, and the spin structure may or may not extend to a given line defect. Physically the spin TFT can be interpreted as the result of gauging a one-form symmetry in the original oriented TFT. This spin TFT extends earlier constructions in Blanchet-Masbaum (1996) and Blanchet (2005) [arXiv:math/0303240], and it reproduces the classification of abelian spin Chern-Simons theories in Belov-Moore (2005) [arXiv:hep-th/0505235].