Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-like Formula for Near-extremal Black Holes

TL;DR

This paper develops a modular bootstrap approach to derive a Cardy-like formula for the density of near-extremal black hole microstates in 2D CFTs, revealing exponential growth of operators near a specific twist value.

Contribution

It establishes a new connection between modular invariance, Tauberian theory, and the spectrum of 2D CFTs, providing a formula for the density of states near extremality.

Findings

Operators with specific twist values grow exponentially with spin

The growth rate matches a Cardy-like formula involving the central charge

Results apply to holographic CFTs with linear twist gap growth

Abstract

We show that for a unitary modular invariant 2D CFT with central charge $c>1$ 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 $(\frac{c-1}{12}-\varepsilon,\frac{c-1}{12}+\varepsilon)$ with $\varepsilon=O(J^{-1/2}\log J)$. We establish that the number of Virasoro primary operators in such a window has a Cardy-like i.e. $\exp\left(2\pi\sqrt{\frac{(c-1)J}{6}}\right)$ growth. We make further conjectures on potential generalization to CFTs with conserved currents. 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 $J\gg c^3\gg1$. 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".