Modular forms and a generalized Cardy formula in higher dimensions

TL;DR

This paper derives a universal asymptotic density of states formula for conformal field theories on a torus, based on modular invariance and vacuum energy, extending to non-Lorentz invariant theories and confirming with free field examples.

Contribution

It provides a new generalized Cardy-like formula for higher-dimensional CFTs derived from modular invariance and vacuum energy considerations, including non-Lorentz invariant cases.

Findings

Derived a formula for density of states in higher-dimensional CFTs.

Confirmed the formula with free scalar, Maxwell, and super Yang-Mills theories.

Connected Maxwell case to Casimir's original electromagnetic force calculation.

Abstract

We derive a formula which applies to conformal field theories on a spatial torus and gives the asymptotic density of states solely in terms of the vacuum energy on a parallel plate geometry. The formula follows immediately from global scale and Lorentz invariance, but to our knowledge has not previously been made explicit. It can also be understood from the fact that $\log Z$ on $\mathbb{T}^2\times \mathbb{R}^{d-1}$ transforms as the absolute value of a non-holomorphic modular form of weight $d-1$, which we show. The results are extended to theories which violate Lorentz invariance and hyperscaling but maintain a scaling symmetry. The formula is checked for the cases of a free scalar, free Maxwell gauge field, and free $\mathcal{N}=4$ super Yang-Mills. The case of a Maxwell gauge field gives Casimir's original calculation of the electromagnetic force between parallel plates in terms of the entropy of a photon gas.