Construction of two-dimensional topological field theories with non-invertible symmetries

TL;DR

This paper develops a framework for two-dimensional topological field theories with non-invertible symmetries, providing explicit formulas, proving crossing symmetry, and constructing examples from known fusion categories.

Contribution

It introduces a systematic construction of 2D TFTs with non-invertible symmetries, including explicit formulas and methods to obtain non-regular TFTs via generalized gauging.

Findings

Derived formulas for three-point and two-point functions.

Proved crossing symmetry in the constructed TFTs.

Constructed explicit examples for Fibonacci, Ising, and Haagerup categories.

Abstract

We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup $\mathcal{H}_3$ fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.