Casimir Energy and Modularity in Higher-dimensional Conformal Field Theories

Abstract

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 $T^2\times {\mathbb{R}}^{d-3}$ 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 $\tau$ of the torus. The Casimir energy is subject to constraints from modular invariance on the torus which we spell out using $PSL(2,\mathbb{Z})$ 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 $d\ge 3$.

arXiv comment48 pages, 3 figures

References in corpus (55)

• Gauge Theory Correlators from Non-Critical String Theory
• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Generalized Global Symmetries
• AdS Dual of the Critical O(N) Vector Model
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Reading between the lines of four-dimensional gauge theories
• Quantum Field Theory in the Large N Limit: a review
• The AdS/CFT Correspondence and a New Positive Energy Conjecture for General Relativity
• Field Theory Without Feynman Diagrams: One-Loop Effective Actions
• Effects of D-instantons
• Theory of Two-Dimensional Quantum Heisenberg Antiferromagnets with a Nearly Critical Ground State
• Higher Spin Gauge Theory and Holography: The Three-Point Functions
• Coupling Constant Dependence in the Thermodynamics of N=4 Supersymmetric Yang-Mills Theory
• Higher-Spin Gauge Theories in Four, Three and Two Dimensions
• Constraints on Fluid Dynamics from Equilibrium Partition Functions
• Worldline Instantons and Pair Production in Inhomogeneous Fields
• The D4-D8 Brane System and Five Dimensional Fixed Points
• Stress Tensors and Casimir Energies in the AdS/CFT Correspondence
• NS5-Branes, T-Duality and Worldsheet Instantons
• Two loops in eleven dimensions
• Cardy Formulae for SUSY Theories in d=4 and d=6
• Asymptotic density of states in 2d CFTs with non-invertible symmetries
• The Conformal Bootstrap at Finite Temperature
• Conformal field theories at non-zero temperature: operator product expansions, Monte Carlo, and holography
• Planar Yang-Mills theory: Hamiltonian, regulators and mass gap
• AdS_6 Interpretation of 5d Superconformal Field Theories
• Polylogarithm Identities in a Conformal Field Theory in Three Dimensions
• The critical O(N) CFT: Methods and conformal data
• Global Anomalies on the Hilbert Space
• Comments on the Sign and Other Aspects of Semiclassical Casimir Energies
• Modular forms and a generalized Cardy formula in higher dimensions
• Universality of Sparse $d>2$ Conformal Field Theory at Large $N$
• A Derivation of AdS/CFT for Vector Models
• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• Harmonic analysis of 2d CFT partition functions
• Critical Casimir Effect: Exact Results
• High Energy Modular Bootstrap, Global Symmetries and Defects
• Romans Supergravity from Five-Dimensional Holograms
• Casimir energy of a massive field in a genus-1 surface
• Modular-invariant large-$N$ completion of an integrated correlator in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Transition from the ${\mathbb{Z}}_2$ spin liquid to antiferromagnetic order: spectrum on the torus
• Encoding beyond cosmological horizons in de Sitter JT gravity
• A universal formula for the density of states with continuous symmetry
• An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces
• Emergent gravity from Eguchi-Kawai reduction
• Scalar Modular Bootstrap and Zeros of the Riemann Zeta Function
• Comments on a state-operator correspondence for the torus
• On Universality of Holographic Results for (2+1)-Dimensional CFTs on Curved Spacetimes
• Bootstrapping Closed Hyperbolic Surfaces
• Symmetry breaking at high temperatures in large N gauge theories
• Notes on massless scalar field partition functions, modular invariance and Eisenstein series
• Explicit holography for vector models at finite $N$, volume and temperature
• Vacuum energy is non-positive for (2+1)-dimensional holographic CFTs
• Does the Round Sphere Maximize the Free Energy of (2+1)-Dimensional QFTs?
• A Surprising Similarity Between Holographic CFTs and a Free Fermion in $(2+1)$ Dimensions

Cited by in corpus (4)

• Universal Asymptotics for High Energy CFT Data
• Generating functions and large-charge expansion of integrated correlators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Modular properties of massive scalar partition functions
• Casimir effect in critical $\mathrm{O}(N)$ models from non-equilibrium Monte Carlo simulations