The process known as α-induction, developed by Böckenhauer and Evans, may be thought of as a procedure that accepts the annular partition function of a boundary conformal field theory (CFT) and produces a toroidal partition function of a closed CFT. However it is stated and proved in the operator algebraic context.
It is well known that the category of representations of the vertex operator algebra associated to a CFT is a modular tensor category (MTC). The data of an annular partition function can be given as a module category over this MTC. Ostrik rephrased α-induction using this categorical language.
In my talk I will first introduce the tube category. This is a categorical analogue of the tube algebra, introduced in the operator algebraic context by Ocneanu. I will then explain how Ostrik’s categorical procedure may be reinterpreted as taking the trace of the module category and extending the result to the tube category. Finally I will describe how the required properties of a toroidal partition function (e.g. modular invariance) naturally arise from this perspective.