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

TL;DR

The paper develops a complete framework for constructing unitary ${\mathcal C}$-symmetric two-dimensional TFTs with non-invertible fusion-category symmetries by pairing a fusion category ${\mathcal C}$ with a module category ${\mathcal M}$ of boundary conditions. It introduces open/closed defect Hilbert spaces, boundary crossing relations, and module traces to derive all correlators from fundamental data, and proves crossing symmetry for sphere four-point functions. The authors instantiate the construction in regular TFTs for Fibonacci, Ising, and Haagerup ${\mathcal H}_3$ categories, matching known bootstrap results and detailing the bulk Frobenius algebra and defect operator data. They also explain generalized gauging as a method to obtain all non-regular TFTs from regular ones, via algebra objects and bimodule categories, clarifying dual symmetries and the Morita-like relations between theories. Altogether, the work provides a principled, comprehensive open-closed TFT formulation of non-invertible symmetry in two dimensions with concrete explicit examples and a pathway to generate non-regular theories.

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.