Modular forms and a generalized Cardy formula in higher dimensions

TL;DR

This work extends the Cardy formula to higher dimensions by leveraging higher-dimensional modular invariance of a CFT on $\mathbb{T}^d\times\mathbb{R}$. By swapping cycles on the torus, the authors show that high-temperature thermodynamics and the asymptotic density of states are fixed by the vacuum energy on $S^1\times\mathbb{R}^{d-1}$, with the central result that $\log Z$ transforms as the absolute value of a modular form of weight $d-1$ under a restricted $SL(d+1,\mathbb{Z})$ action. They develop generalizations to hyperscaling-violation and anisotropic scaling, and validate the formula in several examples, including a free massless scalar, free Maxwell theory (recovering Casimir forces), and free $\mathcal{N}=4$ SYM. The findings connect vacuum (Casimir) energy to thermodynamic entropy and state-counting, offering potential applications to entanglement entropy and holography. The framework provides a unified, nonperturbative handle on high-energy density of states in a broad class of higher-dimensional CFTs.

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.