The Cardy Formula from Goldstone Bosons

Abstract

Two dimensional conformal field theories, can be described by their pseudo Goldstone bosons when the conformal symmetry is spontaneously and anomalously broken. We show that the Schwarzian action of these bosons leads to the Cardy formula without using modular invariance. As a result, the Cardy formula applies to conformal field theories on a cylinder and chiral theories in one dimension. This also explains why the Cardy--Verlinde formula for theories on $S^1 \times S^{d-2}$ can be written in the form of the Cardy formula of an effective two dimensional theory.