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Cited by 27Black Holes and Theoretical PhysicsQuantum Electrodynamics and Casimir EffectAlgebraic structures and combinatorial models

The DOI matches its Crossref record; 97 of the 98 references it cites were found in a registry or an index; none of the 35 sources its claims cite is checked for support yet.

Casimir energy and modularity in higher-dimensional conformal field theories

Open access
Luo, Conghuan, Wang, Yifan·2023·Journal of High Energy Physics
DOI 10.1007/jhep07(2023)028
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Abstract

A bstract: An important problem in Quantum Field Theory (QFT) is to understand the structures of observables on spacetime manifolds of nontrivial topology. Such observables arise naturally when studying physical systems at finite temperature and/or finite volume and encode subtle properties of the underlying microscopic theory that are often obscure on the flat spacetime. Locality of the QFT implies that these observables can be constructed from more basic building blocks by cutting-and-gluing along a spatial slice, where a crucial ingredient is the Hilbert space on the spatial manifold. In Conformal Field Theory (CFT), thanks to the operator-state correspondence, we have a non-perturbative understanding of the Hilbert space on a spatial sphere. However it remains a challenge to consider more general spatial manifolds. Here we study CFTs in spacetime dimensions d > 2 on the spatial manifold T 2 × ℝ d− 3 which is one of the simplest manifolds beyond the spherical topology. We focus on the ground state in this Hilbert space and analyze universal properties of the ground state energy, also commonly known as the Casimir energy, which is a nontrivial function of the complex structure moduli τ of the torus. The Casimir energy is subject to constraints from modular invariance on the torus which we spell out using PSL(2 , ℤ) spectral theory. Moreover we derive a simple universal formula for the Casimir energy in the thin torus limit using the effective field theory (EFT) from Kaluza-Klein reduction of the CFT, with exponentially small corrections from worldline instantons. We illustrate our formula with explicit examples from well-known CFTs including the critical O ( N ) model in d = 3 and holographic CFTs in d ≥ 3.

Overview

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TL;DR

The paper studies the Casimir energy density \(E(\tau)\) for \(d>2\) CFTs on \(T^2\times \mathbb R^{d-3}\), showing it is a modular-invariant function of the torus complex-structure modulus \(\tau\). It derives a universal thin-torus expansion controlled by two perturbative coefficients related to thermal stress-tensor one-point functions, plus exponentially small worldline-instanton corrections, and illustrates the result in free scalar, large-\(N\) \(O(N)\), and holographic CFTs.

Key findings

  • In the thin-torus limit \(\tau\to i\infty\), the Casimir energy density has the universal form \(E(\tau)=c_1 \tau_2^{d/2}+c_2 \tau_2^{1-d/2}+O(\tau_2^\alpha e^{-2\pi M n\tau_2}e^{2\pi i n Q\tau_1})\), with \(c_1=-b_T/d\) and \(c_2=-b_T/(d-1)\) tied to thermal one-point functions.
  • Modular invariance under \(PSL(2,\mathbb Z)\) constrains \(E(\tau)\); using \(PSL(2,\mathbb Z)\) spectral theory, the paper gives a spectral decomposition and argues that higher KK-charge instanton sectors are determined by the \(Q=0\) and \(Q=\pm1\) sectors.
  • For a free real scalar, \(E(\tau)=E^*_{d/2}(\tau)\), with \(c_1=\Lambda(d/2)\) and \(c_2=\Lambda((d-1)/2)\).
  • For the large-\(N\) critical \(O(N)\) model in \(d=3\), \(c_1^{O(N)}=2N\zeta(3)/(5\pi)\), \(c_2^{O(N)}=0\), and \(\Delta_0=\frac1\pi\log\frac{1+\sqrt5}{2}\); for holographic CFTs with Einstein gravity duals, \(E(\tau)=\frac{(4\pi)^{d-1}\ell^{d-1}}{4d^d G_N}\frac{A^{d/2}}{\mathrm{sys}(T^2)^d}\), with \(c_1=\frac{(4\pi)^{d-1}\ell^{d-1}}{4d^d G_N}\) and \(c_2=0\).

Methods

  • The paper uses effective field theory from Kaluza-Klein reduction on the small torus cycle to derive the universal thin-torus expansion, including perturbative terms and worldline-instanton corrections.
  • It applies \(PSL(2,\mathbb Z)\) spectral theory to decompose the modular-invariant Casimir energy density into Eisenstein series and Maass cusp forms, extracting constraints on the spectral overlaps.
  • It tests the general structure in analytic and numerical examples: free scalar CFT, the large-\(N\) critical \(O(N)\) model via the gap equation and generalized Eisenstein series, and holographic CFTs via AdS soliton solutions.

Limitations

  • The paper states that non-degeneracy of Maass cusp forms is widely believed but not proven, and that a more detailed study of the spectral decomposition is left for the future.
  • It assumes the thermal mass squared for KK zero modes is non-negative, noting recent counterexamples to this statement in the literature.
  • Evident from the text: the positivity conjecture and the conjecture that \(E(\tau)\) has a global minimum at \(\tau=e^{2\pi i/3}\) are proposed but not proven; many explicit results rely on large-\(N\), free, or holographic limits.

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Integrity signals

Registry

Crossref · 10.1007/jhep07(2023)028

Cites

97 of 97 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Figures

Figure 1 Toroidal Casimir energy of the O ( N ) scalar CFT as a function of τ , computed in the standard PSL(2 , Z ) fundamental domain. In the left figure, we zoom into the region around τ 2 = 1 to show the τ 1 dependence of the Casimir energy, which is more prominent at smaller τ 2 . In the right figure, we project the 3D plot along the -τ 1 direction to highlight the suppression of τ 1 independence in this fundamental domain. E ( τ ) grows at large τ 2 as τ 3 / 2 2 , which becomes dominant as early as τ 2 ∼ 1 in the O ( N ) scalar CFT. The τ 1 dependence is negligible in this domain, except for minor dependence near the boundary at | τ | = 1 , which is mostly contributed by the lowest nonzero KK mode.
Figure 1 Toroidal Casimir energy of the O ( N ) scalar CFT as a function of τ , computed in the standard PSL(2 , Z ) fundamental domain. In the left figure, we zoom into the region around τ 2 = 1 to show the τ 1 dependence of the Casimir energy, which is more prominent at smaller τ 2 . In the right figure, we project the 3D plot along the -τ 1 direction to highlight the suppression of τ 1 independence in this fundamental domain. E ( τ ) grows at large τ 2 as τ 3 / 2 2 , which becomes dominant as early as τ 2 ∼ 1 in the O ( N ) scalar CFT. The τ 1 dependence is negligible in this domain, except for minor dependence near the boundary at | τ | = 1 , which is mostly contributed by the lowest nonzero KK mode.CC-BY · From the PDF
Figure 2 Rescaled induced mass ∆ of the O ( N ) scalar CFT as a function of τ , computed in the standard PSL(2 , Z ) fundamental domain. The induced mass hardly depends on τ 1 in this PSL(2 , Z ) frame, and approaches a constant at large τ 2 .
Figure 2 Rescaled induced mass ∆ of the O ( N ) scalar CFT as a function of τ , computed in the standard PSL(2 , Z ) fundamental domain. The induced mass hardly depends on τ 1 in this PSL(2 , Z ) frame, and approaches a constant at large τ 2 .CC-BY · From the PDF
Figure 3 Casimir energy of O ( N ) scalar CFT as a function of τ 1 ∈ [ -1 / 2 , 1 / 2] at fixed τ 2 =1 .
Figure 3 Casimir energy of O ( N ) scalar CFT as a function of τ 1 ∈ [ -1 / 2 , 1 / 2] at fixed τ 2 =1 .CC-BY · From the PDF

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Citations

Cites

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Research works printed but not linked: 6

  1. G. Segal, The definition of conformal field theory, in the proceedings of the Symposium on Topology, Geometry and Quantum Field Theory (Segalfest), (2002), p. 421–575 [INSPIRE].
  2. S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge Books Online, Cambridge University Press (1988).
  3. R.A. Rankin, A minimum problem for the epstein zeta-function, Glasgow Mathematical Journal 1 (1953) 149.
  4. J.W. Cassels, On a problem of rankin about the epstein zeta-function, Glasgow Mathematical Journal 4 (1959) 73.
  5. V. Ennola, A lemma about the epstein zeta-function, Glasgow Mathematical Journal 6 (1964) 198.
  6. P. Diananda, Notes on two lemmas concerning the epstein zeta-function, Glasgow Mathematical Journal 6 (1964) 202.

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Metadata

Authors
Luo, C., & Wang, Y.
Affiliations
New York University (US)
Corresponding author
Luo, Conghuan
ORCID
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2023-07-04
Volume
2023
Issue
7
ISSN
1029-8479 (ISSN-L)
DOI
10.1007/jhep07(2023)028
OpenAlex
W4383369103
Semantic Scholar
0bd2d0eb13b8afb393d2eb1144ee64047f89ed14
Cited by
27
References
98
Citations by year
2026: 7 · 2025: 5 · 2024: 8 · 2023: 7
FWCI
1.67
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of High Energy Physics (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion, cc-by)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University) (cc-by)
Funders
National Science Foundation (2210420)
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Electrodynamics and Casimir Effect
  • Physical Sciences › Mathematics › Geometry and Topology › Algebraic structures and combinatorial models
Keywords
Casimir energy, modular invariance, higher-dimensional conformal field theory, complex structure moduli, Kaluza-Klein reduction, worldline instantons, effective field theory, holographic CFT, ground state energy

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