The Cardy Formula from Goldstone Bosons

TL;DR

This paper derives the Cardy formula from the dynamics of Goldstone bosons in 2D conformal field theories, bypassing modular invariance, and extends its applicability to chiral and one-dimensional theories.

Contribution

It demonstrates that the Schwarzian action of Goldstone bosons naturally leads to the Cardy formula without relying on modular invariance, broadening its scope.

Findings

Cardy formula derived from Goldstone bosons' Schwarzian action

Applicability of Cardy formula to chiral and 1D theories shown

Explains the form of the Cardy--Verlinde formula in higher dimensions

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.