Abstract We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit 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. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.
Abstract We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit 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. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.
We improve the recently discovered upper and lower bounds on the O(1) correction to the Cardy formula for the density of states integrated over an energy window (of width $2\ensuremath{\delta}$), centered at high energy in two-dimensional unitary and modular invariant conformal field theory. We prove optimality of the lower bound for $\ensuremath{\delta}\ensuremath{\rightarrow}{1}^{\ensuremath{-}}$. We prove a conjectured upper bound on the asymptotic gap between two consecutive Virasoro primaries for a central charge greater than 1, demonstrating it to be 1. Furthermore, a systematic method is provided to establish a limit on how tight the bound on the O(1) correction to the Cardy formula can be made using bandlimited functions. The techniques and the functions used here are of generic importance whenever the Tauberian theorems are used to estimate some physical quantities.
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.
Abstract We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h, h h ¯ ) → (∞, ∞) for any arbitrary spin is derived using the theorem. We further rigorously show that the error term is controlled by the twist parameter and insensitive to spin. The sensitivity of the leading piece towards spin is discussed. We identify a universal piece in microcanonical entropy when the averaging window is large. An asymptotic spectral gap on (h, h h ¯ ) plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary 2D CFT without any conserved current Ag≤24π(c−1)r2 Ag ≤ π c − 1 r 2 24 is satisfied, where g is the twist gap over vacuum and A is the minimal “areal gap”, generalizing the minimal gap in dimension to (h′, h′ h ¯ ′ ) plane and r=π43≃2.21 r = 4 3 π ≃ 2.21 . We investigate density of states in the regime where spin is parametrically larger than twist with both going to infinity. Moreover, the large central charge regime is studied. We also probe finite twist, large spin behavior of density of states.
Abstract We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with c>1 c > 1 . Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin- J operators must grow like exp(π32(c−1)J)/2J exp π 2 3 ( c - 1 ) J / 2 J in any twist interval at or above (c−1)/12 ( c - 1 ) / 12 , with a known twist-dependent prefactor. This proves that the large J spectrum becomes dense even without averaging over spins. For twists below (c−1)/12 ( c - 1 ) / 12 we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin- J operators must go to zero as J becomes large.
Abstract We show that for a unitary modular invariant 2D CFT with central charge c>1 c > 1 and having a nonzero twist gap in the spectrum of Virasoro primaries, for sufficiently large spin J , there always exist spin- J operators with twist falling in the interval (12c−1−ε,12c−1+ε) ( c - 1 12 - ε , c - 1 12 + ε ) with ε=O(J−1/2logJ) ε = O ( J - 1 / 2 log J ) . We establish that the number of Virasoro primary operators in such a window has a Cardy-like, i.e., exp(2π6(c−1)J) exp 2 π ( c - 1 ) J 6 growth. A similar result is proven for a family of holographic CFTs with the twist gap growing linearly in c and a uniform boundedness condition, in the regime J≫c3≫1 J ≫ c 3 ≫ 1 . From the perspective of near-extremal rotating BTZ black holes (without electric charge), our result is valid when the Hawking temperature is much lower than the “gap temperature.”
A bstract We derive Cardy-like formulas for the growth of operators in different sectors of unitary 2 dimensional CFT in the presence of topological defect lines by putting an upper and lower bound on the number of states with scaling dimension in the interval [∆ − δ, ∆ + δ ] for large ∆ at fixed δ . Consequently we prove that given any unitary modular invariant 2D CFT symmetric under finite global symmetry G (acting faithfully), all the irreducible representations of G appear in the spectra of the untwisted sector; the growth of states is Cardy like and proportional to the “square” of the dimension of the irrep. In the Schwarzian limit, the result matches onto that of JT gravity with a bulk gauge theory. If the symmetry is non-anomalous, the result applies to any sector twisted by a group element. For c > 1, the statements are true for Virasoro primaries. Furthermore, the results are applicable to large c CFTs. We also extend our results for the continuous U(1) group.