The Casimir Energy in Curved Space and its Supersymmetric Counterpart

TL;DR

The paper analyzes Casimir energies of $d$-dimensional CFTs on curved spaces and their deformations, highlighting that ordinary (non-supersymmetric) Casimir energy on $S^3\times\mathbb{R}$ is not intrinsic in $d=4$ due to scheme dependence, in contrast to the intrinsic $d=2$ case. It then constructs a natural, scheme-independent SUSY Casimir energy for ${\cal N}=1$ theories on $S^3\times\mathbb{R}$ by reducing to a one-dimensional supersymmetric quantum mechanics, where short multiplets yield a universal contribution and long multiplets cancel; the round-sphere result is $E_{\mathrm{susy}}=\frac{4}{27 r_3}(a+3c)$. The authors extend the analysis to deformed geometries including squashed $S^3$ (Hopf surfaces) and derive a general formula for $E_{\mathrm{susy}}(\mathfrak b)$, as well as a detailed shortening analysis for the Hopf-surface background using Barnes double zeta functions. The work connects to holographic checks for non-supersymmetric Casimir energy and discusses implications for the regularization of SUSY partition functions, providing two proofs and offering tools to match field theory results with gravity duals in diverse curved backgrounds.

Abstract

We study $d$-dimensional Conformal Field Theories (CFTs) on the cylinder, $S^{d-1}\times \mathbb{R}$, and its deformations. In $d=2$ the Casimir energy (i.e. the vacuum energy) is universal and is related to the central charge $c$. In $d=4$ the vacuum energy depends on the regularization scheme and has no intrinsic value. We show that this property extends to infinitesimally deformed cylinders and support this conclusion with a holographic check. However, for $\mathcal{N}=1$ supersymmetric CFTs, a natural analog of the Casimir energy turns out to be scheme independent and thus intrinsic. We give two proofs of this result. We compute the Casimir energy for such theories by reducing to a problem in supersymmetric quantum mechanics. For the round cylinder the vacuum energy is proportional to $a+3c$. We also compute the dependence of the Casimir energy on the squashing parameter of the cylinder. Finally, we revisit the problem of supersymmetric regularization of the path integral on Hopf surfaces.