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 ℋ3 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.
Highlights
There has been a burgeoning body of work [8,9,10,11,12] exploring the constraints on renormalization group flows by fusion category symmetries
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
Non-invertible symmetries can be directly built into the construction of anyon chains [13] and related statistical models [14, 15]; in continuum field theory, symmetry-preserving flows are triggered by relevant operators that are invariant under C, i.e. commute with the topological defect lines (TDLs) in C
Summary
A summary of the heart of this paper, section 3, is presented. In section 3.1, we explain how the open sector of a C-symmetric TFT is already captured by the structure of a module category over C, and how the closed sector can be constructed by open-to-closed maps. In the present C-symmetric case, we see that a very similar structure arises This structure can be understood intuitively via a boundary crossing relation, which we formulate, both from a physical cutting-and-sewing perspective, and mathematically from the axioms of a module category with a module trace.
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