Journal Article10.48550/arxiv.2501.09886
Holographic Bound of Casimir Effect in General Dimensions
Rong-Xin Miao
- 16 Jan 2025
TL;DR: This paper generalizes the holographic bound on the Casimir effect from 3D to higher dimensions, finding universal lower bounds for Einstein, DGP, and Gauss-Bonnet gravity, and verifying these bounds for various theories and boundary shapes.
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Abstract: Recently, it has been proposed that holography imposes a universal lower bound on the Casimir effect for 3d BCFTs. This paper generalizes the discussions to higher dimensions. We find Einstein gravity, DGP gravity, and Gauss-Bonnet gravity sets a universal lower bound of the strip Casimir effect in general dimensions. We verify the holographic bound by free theories and $O(N)$ models in the $\epsilon$ expansions. We also derive the holographic bound of the Casimir effect for a wedge and confirm free theories obey it. It implies holography sets a lower bound of the Casimir effect for general boundary shapes, not limited to the strip. Finally, we briefly comment on the impact of mass and various generalizations and applications of our results.
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Figures
Figure 5: (−κ1/CD) for GB-DGP gravity with α = −1/24 (left) and α = 1/8 (right). We focus on normal phase (63) and d = 4. The blue and green curves correspond to upper and lower bounds of (63). All curves except the blue ones approach the universal holographic limit (purple curve) for ρ → −∞. The blue curve corresponds to the critical point of normal and singular phases. For α = −1/24 and λ = −1/6, (−κ1/CD) (green curve) cannot approach zero. 
Figure 7: (−f(Ω)/CD) for 3d BCFTs. The blue, green, red and yellow curves correspond to AdS/BCFT with ρ = −∞,−0.3, 0, and free scalar. It shows holography with T → −2 (ρ → −∞) (blue curve) sets the lower bound of wedge Casimir effect. 
Figure 1: Geometry of holographic strip: a portion of AdS soliton. The region between red/blue curves (branes) and the black line is the bulk dual of strip I with negative brane tension T ≤ 0; its complement in bulk is the gravity dual for strip II (green line) with T > 0. The gravity dual of strip II contains the ‘horizon’ z = zh. The angle θ should be periodic to remove the conical singularity on it. Thus, the green lines for strip II are connected. Without loss of generality, we focus on strip I with T ≤ 0. We have (ρ < 0) and (ρ > 0) for the blue and red curves, respectively. zmax and zc denote the turning points with θ ′(zmax) = ∞ and θ′(zc) = 0. 
Figure 8: (−f(Ω)/CD) for 4d BCFTs. The blue, yellow, red, green, and purple curves correspond to AdS/BCFT with ρ = −∞,−1, 0, scalar and Maxwell field. It shows holography with T → −3 (ρ → −∞) (blue curve) sets the lower bound of wedge Casimir effect. 
Table 1: (−κ1/CD) for various 3d and 4d BCFTs 
Figure 4: (−κ1/CD) for GB gravity with d = 4. The blue, orange, green, red curves denote GB gravity with typical couplings α = (− 124 , 0, 1 8), and holographic limit (2), respectively. It shows GB gravity with various couplings α approach the same value in the limit ρ → −∞.
