Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit T → ∞, and a universal spectrum in the Cardy regime, Δ → ∞. We show that a much stronger form of universality holds in theories with a large central charge c and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all $$ \Delta \ge \frac{c}{6} $$ . The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal ‘enigma’ saddlepoints in the thermodynamics of 3d gravity.
Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit T → ∞, and a universal spectrum in the Cardy regime, Δ → ∞. We show that a much stronger form of universality holds in theories with a large central charge c and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all Δ≥6c . The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal ‘enigma’ saddlepoints in the thermodynamics of 3d gravity.
We analyze the operator spectrum of Conformal Field Theories at large conformal dimensions Delta with and without global symmetries. General constraints of crossing symmetry and unitarity allow us to extract a number of universal properties of heavy operator spectrum perturbatively in 1overDelta.\nFirst, we consider four-point correlators and solve crossing equations in the deep Euclidean regime. Large scaling dimension Delta tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading 1overDelta tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex Delta plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We make predictions for the 3d Ising model.\nSecond, we apply the methods of tauberian theory to modular invariance in 2d CFTs. We derive lower and upper bounds on the number of operators within a given energy interval. At high energies we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. \nThird, we consider unitary CFTs with continuous global symmetries in d>2. We consider a state created by the lightest operator of large charge Qgg1 and analyze the correlator of two light charged operators in this state. Assuming the correlator admits a well-defined large Q expansion and, relatedly, that the macroscopic (thermodynamic) limit of the correlator exists, we find that the crossing equations admit a consistent truncation, where only a finite number N of Regge trajectories contribute at leading nontrivial order. We classify all such solutions to the crossing. For one Regge trajectory N=1, the solution is unique and given by the effective field theory of a Goldstone mode.
We study whether the relations between the Weyl anomaly, entanglement entropy (EE), and thermal entropy of a two-dimensional (2D) conformal field theory (CFT) extend to 2D boundaries of 3D CFTs, or 2D defects of D≥3 CFTs. The Weyl anomaly of a 2D boundary or defect defines two or three central charges, respectively. One of these, b, obeys a c theorem, as in 2D CFT. For a 2D defect, we show that another, d_{2}, interpreted as the defect's "conformal dimension," must be non-negative if the averaged null energy condition holds in the presence of the defect. We show that the EE of a sphere centered on a planar defect has a logarithmic contribution from the defect fixed by b and d_{2}. Using this and known holographic results, we compute b and d_{2} for 1/2-Bogomol'nyi-Prasad-Sommerfield surface operators in the maximally supersymmetric (SUSY) 4D and 6D CFTs. The results are consistent with b's c theorem. Via free field and holographic examples we show that no universal "Cardy formula" relates the central charges to thermal entropy.
The entanglement spectrum of a bipartite quantum system is given by the distribution of eigenvalues of the modular Hamiltonian. In this work, we compute the entanglement spectrum in the vacuum state for a subregion of a d-dimensional conformal field theory (CFT) admitting a holographic dual. In the case of a spherical (or planar) entangling surface, we recover known results in two dimensions, including the Cardy formula in the high energy regime. In higher dimensions d>2, we analytically determine a generalization of the Cardy formula valid at large energies and consistent with previous studies of CFT spectra in the literature. We also investigate numerically the spectrum at energy levels far above the modular ground state energy. We extend our analysis to the supersymmetric point of Einstein-Maxwell gravity, providing exact results when d=2,3, and a generalization of the Cardy formula at high energies in generic dimension d. We consider small shape deformations of a spherical entangling surface, for both the non-supersymmetric and the supersymmetric cases. In all cases we find that the high-energy scaling of the microcanonical entropy with the modular energy is unaffected by the shape deformation. This result suggests that the high-energy regime of the entanglement spectra carries universal information, independent of the shape of the entangling surface.
A bstract We prove the conjecture proposed by Hartman, Keller and Stoica (HKS) [1]: the grand-canonical free energy of a unitary 2D CFT with a sparse spectrum below the scaling dimension 12c c 12 + ϵ and below the twist 12c c 12 is universal in the large c limit for all β L β R ≠ 4 π 2 . The technique of the proof allows us to derive a one-parameter (with parameter α ∈ (0, 1]) family of universal inequalities on the unitary 2D CFT partition function with general central charge c ⩾ 0, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the inequality on the ( β L , β R ) plane. The infinite iteration of this equation gives the boundary of maximal-validity domain, which depends on the parameter α in the inequality. In the c → ∞ limit, with the additional assumption of a sparse spectrum below the scaling dimension 12c c 12 + ϵ and the twist 12αc αc 12 (with α ∈ (0, 1] fixed), our inequality shows that the grand-canonical free energy exhibits a universal large c behavior in the maximal-validity domain. This domain, however, does not cover the entire ( β L , β R ) plane, except in the case of α = 1. For α = 1, this proves the conjecture proposed by HKS [1], and for α < 1, it quantifies how sparseness in twist affects the regime of universality. Furthermore, this implies a precise lower bound on the temperature of near-extremal BTZ black holes, above which we can trust the black hole thermodynamics.
Inspired by the holographic computation of large interval entanglement entropy of two-dimensional (2D) conformal field theory at high temperature, it was proposed that the thermal entropy is related to the entanglement entropy as ${S}_{th}=\underset{l\ensuremath{\rightarrow}0}{\mathrm{lim}}({S}_{EE}(R\ensuremath{-}l)\ensuremath{-}{S}_{EE}(l))$. In this paper, we prove this relation for 2D CFT with a discrete spectrum in two different ways. Moreover, we discuss this relation for a 2D noncompact free scalar, which is a gapless CFT with continuous spectrum. We show that it could be recovered, after appropriately regularizing the theory.
We prove using invariance under the modular S- and ST-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges c,c~>1) contains a primary operator with dimension Δ1 satisfying 0<Δ1<(c+c~)/24+0.09280... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS3/CFT2 dictionary, the bound on Δ1 proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ<0 can be no heavier than 1/(8GN)+O(−Λ); the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.