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Category theory for conformal boundary conditions

J. Fuchs·C. Schweigert
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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.

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J. Fuchs, C. Schweigert. 2006-08-04. Category theory for conformal boundary conditions. https://arxiv.org/abs/math/0106050

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