Abstract
We study d-dimensional Conformal Field Theories (CFTs) on the cylinder, $$ {S}^{d-1}\times \mathrm{\mathbb{R}} $$ , 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.
Highlights
Often referred to as “radial quantization.” Denoting the noncompact coordinate by τ, we can ask about the energy of the ground state E0, defined as
We study d-dimensional Conformal Field Theories (CFTs) on the cylinder, Sd−1 × R, and its deformations
The expectation value of the energy-momentum tensor is taken in the ground state of the CFT on the cylinder
Summary
Because of the separation of scales between the radius of the three-sphere and the radius of the circle, it is natural to study the reduction of the theory on the three-sphere The result of this reduction is a supersymmetric quantum mechanics with infinitely many degrees of freedom.
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