Category theory for conformal boundary conditions

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.

47 pages, LaTeX2e + epsf + fic-l style; v2: More concise conjectures in section 6, with more comments on torus and annulus partition functions, and on NIM-reps in section 7; v3: Dropped assumption of semisimplicity in former lemma 5.24; lemma moved to section 4, is now lemma 4.15; v4: corrected part (ii) of proof of proposition 5.1