Modular Invariance, Tauberian Theorems, and Microcanonical Entropy

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.Comment: 39 pages, 9 figure

Factorization and complex couplings in SYK and in Matrix Models

We consider the factorization problem in toy models of holography, in SYK and in Matrix Models. In a theory with fixed couplings, we introduce a fictitious ensemble averaging by inserting a projector onto fixed couplings. We compute the squared partition function and find that at large $N$ for a typical choice of the fixed couplings it can be approximated by two terms: a "wormhole" plus a "pair of linked half-wormholes". This resolves the factorization problem. We find that the second, half-wormhole, term can be thought of as averaging over the imaginary part of the couplings. In SYK, this reproduces known results from a different perspective. In a matrix model with an arbitrary potential, we propose the form of the "pair of linked half-wormholes" contribution. In GUE, we check that errors are indeed small for a typical choice of the hamiltonian. Our computation relies on a result by Brezin and Zee for a correlator of resolvents in a "deterministic plus random" ensemble of matrices.Comment: 26 pages, 1 figur

Minimal Liouville gravity correlation numbers from Douglas string equation

We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of [1,2], where Lee-Yang series (2, 2s + 1) was studied, to (3, 3s + p 0) Minimal Liouville Gravity, where p 0 = 1, 2. We demonstrate that there exist such coordinates \u3c4 m,n on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates \u3c4 m,n are related in a non-linear fashion to the natural coupling constants \u3bb m,n of the perturbations of Minimal Lioville Gravity by the physical operators O m,n . We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature [3-5]. \ua9 2014 The Author(s)

Large p SYK from chord diagrams

Abstract The p-body SYK model at finite temperature exhibits submaximal chaos and contains stringy-like corrections to the dual JT gravity. It can be solved exactly in two different limits: “large p” SYK 1 ≪ p ≪ N and “double-scaled” SYK N, p → ∞ with λ = 2p 2/N fixed. We clarify the relation between the two. Starting from the exact results in the double-scaled limit, we derive several observables in the large p limit. We compute euclidean 2n-point correlators and out-of-time-order four-point function at long lorentzian times. To compute the correlators we find the relevant asymptototics of the U q su 1 1 $${\mathcal{U}}_q\left( su\left(1,1\right)\right)$$ 6j-symbol

Analytic Euclidean Bootstrap

We solve crossing equations analytically in the deep Euclidean regime. Large scaling dimension ∆ 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 $\frac{1}{\varDelta }$ 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 ∆ plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We check our formulas in a few examples (for CFTs and scattering amplitudes) and find perfect agreement. Moreover, in these examples we observe that the large ∆ expansion works very well already for small ∆ ∼ 1. We make predictions for the 3d Ising model. Our analysis of dispersion relations via complex tauberian theorems is very general and could be useful in many other contexts.We solve crossing equations analytically 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 $1\over \Delta$ 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 check our formulas in a few examples (for CFTs and scattering amplitudes) and find perfect agreement. Moreover, in these examples we observe that the large $\Delta$ expansion works very well already for small $\Delta \sim 1$. We make predictions for the 3d Ising model. Our analysis of dispersion relations via complex tauberian theorems is very general and could be useful in many other contexts

Matrix Models for Eigenstate Thermalization

We develop a class of matrix models that implement and formalize the eigenstate thermalization hypothesis (ETH) and point out that, in general, these models must contain non-Gaussian corrections in order to correctly capture thermal mean-field theory or to capture nontrivial out-of-time-order correlation functions (OTOCs) as well as their higher-order generalizations. We develop the framework of these ETH matrix models and put it in the context of recent studies in statistical physics incorporating higher statistical moments into the ETH ansatz. We then use the ETH matrix model in order to develop a matrix-integral description of Jackiw-Teitelboim (JT) gravity coupled to a single scalar field in the bulk. This particular example takes the form of a double-scaled ETH matrix model with non-Gaussian couplings matching disk correlators and the density of states of the gravitational theory. Having defined the model from the disk data, we present evidence that the model correctly captures the JT+matter theory with multiple boundaries and, conjecturally, at higher genus. This is a shorter companion paper to the work [D. L. Jafferis et al., companion paper, Jackiw-Teitelboim gravity with matter, generalized eigenstate thermalization hypothesis, and random matrices, Phys. Rev. D 108, 066015 (2023).PRVDAQ2470-001010.1103/PhysRevD.108.066015], serving as a guide to the much more extensive material presented there, as well as developing its underpinning in statistical physics