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.Comment: 16+8 pages, 13 figures. v2: References added, minor edit