Category theory for conformal boundary conditions

Details Category theory for conformal boundary conditions Category theory for conformal boundary conditions

2001-06-07

arxiv.org/abs/math/0106050v4

Fields Institute Communications 39 (2003) 25-71

Mathematics

Category Theory

47 pages, LaTeX2e + epsf + fic-l style; v2: More concise conjectures in section 6, with more comments on torus and annulus par

Scientific paper

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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