The Casimir Energy in Curved Space and its Supersymmetric Counterpart (1503.05537v2)

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.

• The paper demonstrates that the Casimir energy for supersymmetric Conformal Field Theories in curved spacetime is scheme-independent, unlike its non-supersymmetric counterpart.
• It shows that in 4D non-supersymmetric CFTs, the Casimir energy depends on regularization schemes, a contrast to the universal nature seen in 2D.
• The study leverages supersymmetric quantum mechanics and superconformal indices to prove the intrinsic nature of the supersymmetric Casimir energy, relevant for holographic dualities like AdS/CFT.

Analysis of "The Casimir Energy in Curved Space and its Supersymmetric Counterpart"

The paper investigates the Casimir energy for Conformal Field Theories (CFTs) defined on cylindrical geometries, extending the analysis to supersymmetric settings. The paper highlights contrasts between the standard Casimir energy and its supersymmetric counterpart in diverse dimensional backgrounds, presenting significant insights into the regularization scheme dependencies of these energies.

The authors begin by analyzing $d$-dimensional CFTs on the cylindrical geometry $S^{d-1} \times \mathbb{R}$, paying particular attention to the dimensional cases $d=2$ and $d=4$. It is demonstrated that in $d=2$, the Casimir energy is universal, directly connected to the central charge $c$ of the CFT. However, in $d=4$, the vacuum energy is not intrinsic as it is contingent on the choice of regularization scheme, thus lacking an absolute value. This perspective is supported through holographic methods, illustrating the non-intrinsic nature of energy reflections attributed to the Weyl anomaly coefficients $a$ and $c$ on squashed cylinders, with scheme ambiguity introduced via the counterterm $\int d^4 x \sqrt{g} R^2$.

In a more structured investigation into ${\cal N}=1$ supersymmetric CFTs, the authors establish a scheme-independent Casimir energy, highlighting its intrinsic nature in contrast to the non-supersymmetric case. This finding is supported by two distinct proofs. The first involves a supersymmetric quantum mechanics (SQM) approach whereby the Casimir energy expression for an undeformed cylinder yields a form dependent on $a + 3c$. The second proof leverages superconformal indices and consistency relations that ensure robust partition function expressions on complex manifolds with $S^3 \times S^1$ topology.

Mathematically, reducing the problem to a calculation in SQM allows for the definitive computation of the vacuum energy through $1d$ Chern-Simons terms. This reveals that each mode of the CFT corresponds to a supersymmetric quantum mechanical multiplet, organized into chiral and Fermi multiplets based on certain constraints. Interestingly, the investigation extends to squashing deformations of the cylinder, where the Casimir energy adjustments depend on additional parameters but remain intrinsic due to the preserved underlying supersymmetry.

The practical implications in theoretical physics are extensive, particularly in scenarios involving AdS/CFT correspondences. The variance in treatment between the non-supersymmetric and supersymmetric cases calls for a careful consideration regarding the implementation of SUGRA duals in holographic setups. This paper substantiates the need for accurate holographic regularization accounting for supersymmetry to predict physical quantities accurately.

Foreseeable advances in this line of research could extend to exploring further supersymmetric variations and leveraging these findings in broader quantum field theoretical and gravitational contexts, possibly inferring new insights into quantum gravity. There is also potential to unravel analogous phenomena in higher-dimensional CFTs, which may bridge current understanding gaps.

In summary, this comprehensive paper elucidates the subtle distinctions in Casimir energies encountered in CFTs and their supersymmetric counterparts. It leverages nuanced quantum field theoretical and mathematical frameworks to yield new, intrinsic insights valuable to both theoretical physics and quantum gravity domains.