August 21, 2000
After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs $(A,F)$ of conformal field theories, where $F$ has a finite group $G$ of global symmetries and $A$ is the fixpoint theory. The comparison of the representation categories of $A$ and $F$ is strongly intertwined with various issues related to braided tensor categories. We explain that, given the representation category of $A$, the representation category of $F$ can be computed (up to equivalence) by a purely categorical construction. The latter is of considerable independent interest since it amounts to a Galois theory for braided tensor categories. We emphasize the characterization of modular categories as braided tensor categories with trivial center and we state a double commutant theorem for subcategories of modular categories. The latter implies that a modular category $M$ which has a replete full modular subcategory $M_1$ is equivalent to $M_1 \times M_2$ where $M_2=M\cap M_1$ is another modular subcategory. On the other hand, the representation category of $A$ is not determined completely by that of $F$ and we identify the needed additional data in terms of soliton representations. We comment on `holomorphic orbifold' theories, i.e. the case where $F$ has trivial representation theory, and close with some open problems. We point out that our approach permits the proof of many conjectures and heuristic results on `simple current extensions' and `holomorphic orbifold models' in the physics literature on conformal field theory.
Keywords:rational chiral conformal field theories, representation categories, modular categories