## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Integrable Systems in Quantum Field Theory and Statistical Mechanics, M. Jimbo, T. Miwa and A. Tsuchiya, eds. (Tokyo: Mathematical Society of Japan, 1989), 127 - 148

### 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.

#### Article information

**Dates**

Received: 24 March 1989

First available in Project Euclid:
17 June 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1529259753

**Digital Object Identifier**

doi:10.2969/aspm/01910127

**Mathematical Reviews number (MathSciNet)**

MR1048596

**Zentralblatt MATH identifier**

0696.17011

#### Citation

Cardy, John L. Boundary Conditions in Conformal Field Theory. Integrable Systems in Quantum Field Theory and Statistical Mechanics, 127--148, Mathematical Society of Japan, Tokyo, Japan, 1989. doi:10.2969/aspm/01910127. https://projecteuclid.org/euclid.aspm/1529259753