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 withcentral charge c >1 and having a nonzero twist gap in the spectrum ofVirasoro primaries, for sufficiently large spinJ, there always exist spin-Joperators with twist falling in the interval (c-1\12 -epsilon,c-1\12+epsilon) with epsilon=O(J(-1\2)logJ). We establish that the number of Virasoro primary op-erators in such a window has a Cardy-like, i.e., exp(2 pi root(c-1)J\6)growth. A similar result is proven for a family of holographic CFTs with the twistgap growing linearly incand a uniform boundedness condition, in theregimeJ >> c(3)>> 1. From the perspective of near-extremal rotating BTZblack holes (without electric charge), our result is valid when the Hawkingtemperature is much lower than the "gap temperature."