Universality in asymptotic bounds and its saturation in $2$D CFT

TL;DR

This paper investigates the asymptotic behavior of correlators and operator spectra in 2D conformal field theories, revealing universal bounds, near-saturation by rational CFTs, and insights into thermalization and black hole physics.

Contribution

It establishes optimal bounds on operator spectra and three-point coefficients in 2D CFTs, demonstrating universality and the role of smearing in eigenstate thermalization.

Findings

Rational CFTs nearly saturate the bounds, indicating universality.

Identifies a regime separating AdS3 thermal physics from BTZ black hole physics.

Provides numerical insights into spherical conformal blocks.

Abstract

We study asymptotics of three point coefficients (light-light-heavy) and two point correlators in heavy states in unitary, compact $2$D CFTs. We prove an upper and lower bound on such quantities using numerically assisted Tauberian techniques. We obtain an optimal upper bound on the spectrum of operators appearing with fixed spin from the OPE of two identical scalars. While all the CFTs obey this bound, rational CFTs come close to saturating it. This mimics the scenario of bounds on asymptotic density of states and thereby pronounces an universal feature in asymptotics of 2D CFTs. Next, we clarify the role of smearing in interpreting the asymptotic results pertaining to considerations of eigenstate thermalization in 2D CFTs. In the context of light-light-heavy three point coefficients, we find that the order one number in the bound is sensitive to how close the light operators are from the $\frac{c}{32}$ threshold. In context of two point correlator in heavy state, we find the presence of an enigmatic regime which separates the $AdS_3$ thermal physics and the BTZ black hole physics. Furthermore, we present some new numerical results on the behavior of spherical conformal block.