Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-Like Formula for Near-Extremal Black Holes

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