Category theory for conformal boundary conditions

TL;DR

The paper develops a universal categorical framework for conformal boundary conditions by using haploid special Frobenius algebras $A$ in a (possibly non-semisimple) tensor category $ olinebreak[4]C$. It proves that the module category $ olinebreak[4]C_{ olinebreak[4]}A$ inherits essential structures from $ olinebreak[4]C$, including abelianness, tensor product, and dualities, and shows sovereignty under appropriate FS-indicator conditions. The authors connect these structures to modular invariants and NIM-reps, provide concrete group and exceptional-invariant examples, and discuss interpretations in relation to vertex operator algebras and extended conformal field theories. The framework offers a model-independent route to boundary data, fusion rules, and correlation functions, potentially extending beyond rational theories and linking to VOAs and orbifold/CFT constructions. Overall, it unifies simple-current and exceptional modular invariants within a categorical setting and offers a path toward deriving correlation functions and boundary operator algebras from purely categorical data.

Abstract

We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product we obtain results about the Frobenius-Schur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in two-dimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIM-rep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.