Tauberian-Cardy formula with spin

TL;DR

This paper establishes a Tauberian theorem in 2D conformal field theory to analyze the asymptotic density of states across all spins, deriving universal bounds and spectral gaps, with implications for large central charge and finite twist regimes.

Contribution

It introduces a rigorous 2D Tauberian theorem for conformal field theories, providing universal bounds, spectral gaps, and insights into density of states for arbitrary spin and large central charge.

Findings

Derived asymptotic density of states for all spins using the theorem.

Established a universal inequality relating twist gap and minimal areal gap.

Identified a universal microcanonical entropy piece in large averaging windows.

Abstract

We prove a $2$ dimensional Tauberian theorem in context of $2$ dimensional conformal field theory. The asymptotic density of states with conformal weight $(h,\bar{h})\to (\infty,\infty)$ 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 $(h,\bar{h})$ plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary $2$D CFT without any conserved current $Ag\leq \frac{\pi(c-1)r^2}{24}$ is satisfied, where $g$ is the twist gap over vacuum and $A$ is the minimal "areal gap", generalizing the minimal gap in dimension to $(h',\bar{h}')$ plane and $r=\frac{4\sqrt{3}}{\pi}\simeq 2.21$. 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.