Colloquium

Fusion categories

by Victor Ostrik, C. L. E. Moore Instructor, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts

Abstract

I will talk about my joint work with P. Etingof and D. Nikshych.A categorification is a procedure in which one replaces integer numbersby vector spaces, vector spaces by categories, maps betweenvector spaces by functors etc. Surprisingly enough such anabstract procedure is related to physics (here is a typicalslogan: a categorification of d-dimensional topologicalfield theory is (d+1)-dimensional topological field theory).In this talk I will explain the simplest way to categorifyring theory. In this theory rings are replaced by fusioncategories (= semisimple rigid monoidal categories with finitely many simple objects) and modules over rings are replaced bymodule categories. Our main result is the following Theorem:fusion categories and module categories over them admit nodeformations. Thus it is reasonable to try to classify theseobjects. The problem of classification of fusion categories and module categories over them appears to be closely relatedto Operator Algebras and to Conformal Field Theory. I will reviewsome results in this direction.