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

TL;DR

The paper proves a Cardy-like counting formula for Virasoro primaries near the twist accumulation point in unitary 2D CFTs with $c>1$ and a twist gap, showing the density of such operators grows as $\exp\big(2\pi\sqrt{(c-1)J/6}\big)$ within a shrinking twist window of width $\varepsilon=O(J^{-1/2}\log J)$. It develops a modular bootstrap approach in the double lightcone limit, combining DLC$_w$ analysis with Tauberian methods to derive two-sided bounds on the weighted spectral count $\mathcal{A}_J(\beta_L,\varepsilon)$ and thus on $\mathcal{N}_J(\varepsilon)$, with precise dependence on the spin $J$, central charge $c$, and twist gap $\tau_{gap}$. The work extends to holographic CFTs at large $c$, yielding a Schwarzian-sector interpretation for near-extremal rotating BTZ black holes when the Hawking temperature is much smaller than the gap temperature, and discusses potential generalizations to CFTs with conserved currents, including cases with infinitely many currents and shifted twist accumulation points. These results reinforce the link between high-spin spectra, modular invariance, and black hole microstates, and open avenues for rigorous treatment of current algebras and $W_N$ theories within Tauberian-CFT frameworks.

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. $\exp\left(2π\sqrt{\frac{(c-1)J}{6}}\right)$ 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".

Figures (4)

• Figure 1: Illustration of the idea behind equation \ref{['AJ:prop']}. The blue lines represent the allowed positions of the spectrum, constrained by $h-\bar{h}\in\mathbb{Z}$. We aim to count the spectrum around the pink line ($h=A$). We choose two windows (shown in red) with the same width in $h$ but different widths in $\bar{h}$. Due to the integer-spin constraint, the spectrum inside the two windows is the same, as long as the windows intersect with only one of the blue lines.
• Figure 2: A typical behavior of $\rho(h,\bar{h})e^{-\bar{h}\beta_{R}}$ (with $h$ fixed) at small $\beta_R$.
• Figure 3: The three regimes of $T_{\rm H}$ compared to the "gap temperature" $c^{-1}$ (assuming $c\gg1$ and $\beta_{R}=O(c^{-1})$ or smaller). The results in our paper are valid in the pink regime. The arguments of Ghosh:2019rcj (Schwarzian sector) correspond to the green regime. The arguments of Preskill:1991tb (general regime of validity of black hole thermodynamics) correspond to the blue regime.
• Figure 4: Plot of $f(x)=\frac{1}{\left[\cos x-\frac{\sin x}{x}\right]^2}$ in the regime $\frac{\pi}{2}\leqslant x\leqslant\pi$.

Theorems & Definitions (23)

• Theorem 2.1
• Remark 2.2
• Corollary 2.3
• Remark 2.4
• Lemma 2.5
• proof
• Remark 2.6
• Remark 3.1
• Conjecture 3.2
• Remark 3.3