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.
Highlights
While simple, this setting is of physical interest
We find that the free energy of the two free theories is qualitatively similar as a function of the sphere deformation, but we 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
Because the bulk geometries become more difficult to obtain as the boundary spheres approach becoming singular, we only present results for which we are confident in the holographic CFT calculation to about 0.01% or better
Summary
An appropriate transformation of θ allows us to write (2.4) in a form conformal to the round sphere; the space of axisymmetric geometries in which we’re interested is parametrized by a single function of θ This space is impossible to comprehensively explore numerically; we will be satisfied with considering various one-parameter families of geometries which smoothly deform from the round sphere at = 0 to a singular geometry at some = 0. Our primary purpose is to compute the vacuum energy of a holographic CFT for these classes of geometries, and to compare to the results obtained for the conformally coupled scalar and the massless free Dirac fermion. These latter two CFTs have Euclidean actions given by. Our focus turns to the evaluation of the vacuum energy of the holographic CFT
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