Boundary Conditions in Conformal Field Theory

Abstract

We consider conformal field theories on manifolds with a boundary, and the constraints placed by modular invariance on their partition functions. In particular, the partition functions on an annulus with particular boundary conditions are given by the fusion rules. This leads to a simple derivation of the Verlinde formula. We note the remarkable fact that, for some integrable models, these partition functions have the same form away from criticality, with the modular parameter $q$ of the annulus replaced by a temperature-like variable, and give a partial explanation of this in the case of the Ising model.