Universality for Shape Dependence of Casimir Effects from Weyl Anomaly

TL;DR

This work establishes a universal link between the shape dependence of Casimir effects in BCFT and the Weyl anomaly, showing that near-boundary divergences of the renormalized stress tensor $\ angle T_{ij} \$ are fixed by bulk and boundary central charges. By deriving an integrability relation from the Weyl-anomaly variation and evaluating it in $d=3$ and $d=4$, the authors relate Casimir coefficients $\alpha_1,\beta_i$ to boundary charges $b_2,b_3,b_4$ and bulk charges $a,c$, yielding exact mappings such as $\alpha_1 = -b_2$ (3d) and $\alpha_1 = b_4/2$ (4d), with $\beta_1 = c/(2\pi^2) + b_4$, $\beta_3 = 2 b_3 + 13 b_4/6$, among others. Free BCFT data and holographic BCFT models (Takayanagi and Miao–Chu constructions) are used to verify these relations, with holography reproducing not only the near-boundary stress-tensor structure $T_{ij} = 2 \alpha_1 \bar{k}_{ij}/x^{d-1} + \dots$ but also the surface counterterm necessary for finite total energy. The findings argue for universality of the Casimir-central-charge relations in BCFT and hint at extensions to DCFT and higher-dimensional contexts, including potential applications to boundary edge phenomena and Rényi entropy.

Abstract

We reveal elegant relations between the shape dependence of the Casimir effects and Weyl anomaly in boundary conformal field theories (BCFT). We show that for any BCFT which has a description in terms of an effective action, the near boundary divergent behavior of the renormalized stress tensor is completely determined by the central charges of the theory. These relations are verified by free BCFTs. We test them with holographic models of BCFT and find exact agreement. We propose that these relations between Casimir coefficients and central charges hold for any BCFT. With the holographic models, we reproduce not only the precise form of the near boundary divergent behavior of the stress tensor, but also the surface counter term that is needed to make the total energy finite. As they are proportional to the central charges, the near boundary divergence of the stress tensor must be physical and cannot be dropped by further artificial renormalization.Our results thus provide affirmative support on the physical nature of the divergent energy density near the boundary, whose reality has been a long-standing controversy in the literature.

Figures (1)

• Figure 1: BCFT on $M$ and its dual $N$