Casimir energy and modularity in higher-dimensional conformal field theories

TL;DR

The paper studies the Casimir energy density \(E(\tau)\) for \(d>2\) CFTs on \(T^2\times \mathbb R^{d-3}\), showing it is a modular-invariant function of the torus complex-structure modulus \(\tau\). It derives a universal thin-torus expansion controlled by two perturbative coefficients related to thermal stress-tensor one-point functions, plus exponentially small worldline-instanton corrections, and illustrates the result in free scalar, large-\(N\) \(O(N)\), and holographic CFTs.

Key findings

• In the thin-torus limit \(\tau\to i\infty\), the Casimir energy density has the universal form \(E(\tau)=c_1 \tau_2^{d/2}+c_2 \tau_2^{1-d/2}+O(\tau_2^\alpha e^{-2\pi M n\tau_2}e^{2\pi i n Q\tau_1})\), with \(c_1=-b_T/d\) and \(c_2=-b_T/(d-1)\) tied to thermal one-point functions.
• Modular invariance under \(PSL(2,\mathbb Z)\) constrains \(E(\tau)\); using \(PSL(2,\mathbb Z)\) spectral theory, the paper gives a spectral decomposition and argues that higher KK-charge instanton sectors are determined by the \(Q=0\) and \(Q=\pm1\) sectors.
• For a free real scalar, \(E(\tau)=E^*_{d/2}(\tau)\), with \(c_1=\Lambda(d/2)\) and \(c_2=\Lambda((d-1)/2)\).
• For the large-\(N\) critical \(O(N)\) model in \(d=3\), \(c_1^{O(N)}=2N\zeta(3)/(5\pi)\), \(c_2^{O(N)}=0\), and \(\Delta_0=\frac1\pi\log\frac{1+\sqrt5}{2}\); for holographic CFTs with Einstein gravity duals, \(E(\tau)=\frac{(4\pi)^{d-1}\ell^{d-1}}{4d^d G_N}\frac{A^{d/2}}{\mathrm{sys}(T^2)^d}\), with \(c_1=\frac{(4\pi)^{d-1}\ell^{d-1}}{4d^d G_N}\) and \(c_2=0\).

Methods

• The paper uses effective field theory from Kaluza-Klein reduction on the small torus cycle to derive the universal thin-torus expansion, including perturbative terms and worldline-instanton corrections.
• It applies \(PSL(2,\mathbb Z)\) spectral theory to decompose the modular-invariant Casimir energy density into Eisenstein series and Maass cusp forms, extracting constraints on the spectral overlaps.
• It tests the general structure in analytic and numerical examples: free scalar CFT, the large-\(N\) critical \(O(N)\) model via the gap equation and generalized Eisenstein series, and holographic CFTs via AdS soliton solutions.

Limitations

• The paper states that non-degeneracy of Maass cusp forms is widely believed but not proven, and that a more detailed study of the spectral decomposition is left for the future.
• It assumes the thermal mass squared for KK zero modes is non-negative, noting recent counterexamples to this statement in the literature.
• Evident from the text: the positivity conjecture and the conjecture that \(E(\tau)\) has a global minimum at \(\tau=e^{2\pi i/3}\) are proposed but not proven; many explicit results rely on large-\(N\), free, or holographic limits.