Lightcone Modular bootstrap and Cardy-like formula for Near-extremal black holes

The physics of near extremal black holes in AdS_3 is captured by the holographic CFTs with large central charge c, specially, by the large spin states with twist accumulating to $(c-1)/12$. We expound on this by performing a rigorous CFT analysis. First we show that for a unitary modular invariant 2D CFT with fixed 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 ( (c-1)/12 - ε , (c-1)/12 + ε ) with ε=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[2\pi \sqrt{\frac{(c-1) J}{6}] growth. A similar result is then proven for a family of holographic CFTs with the twist gap growing linearly in c and a uniform boundedness condition, in the regime J≫c^3≫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". We make further conjectures on potential generalization to CFTs with conserved currents.