A Surprising Similarity Between Holographic CFTs and a Free Fermion in $(2+1)$ Dimensions

TL;DR

This paper compares the vacuum free energy of different 2+1 dimensional CFTs on deformed spheres, revealing a surprising quantitative similarity between a holographic CFT and a free fermion, despite their different coupling regimes.

Contribution

It introduces a novel numerical approach to compute holographic CFT free energy and uncovers an unexpected close match with free fermion results across deformations.

Findings

Free energies of scalar and fermion are qualitatively similar.

Holographic CFT's free energy closely matches that of the free fermion, within 1%.

The scalar and fermion differ by up to 50% across deformations.

Abstract

We compare the behavior of the vacuum free energy (i.e. the Casimir energy) of various $(2+1)$-dimensional CFTs on an ultrastatic spacetime as a function of the spatial geometry. The CFTs we consider are a free Dirac fermion, the conformally-coupled scalar, and a holographic CFT, and we take the spatial geometry to be an axisymmetric deformation of the round sphere. The free energies of the fermion and of the scalar are computed numerically using heat kernel methods; the free energy of the holographic CFT is computed numerically from a static, asymptotically AdS dual geometry using a novel approach we introduce here. We find that the free energy of the two free theories is qualitatively similar as a function of the sphere deformation, but we also find that the holographic CFT has a remarkable and mysterious quantitative similarity to the free fermion; this agreement is especially surprising given that the holographic CFT is strongly-coupled. Over the wide ranges of deformations for which we are able to perform the computations accurately, the scalar and fermion differ by up to 50% whereas the holographic CFT differs from the fermion by less than one percent.