Universality of the Microcanonical Entropy at Large Spin

We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with $c > 1$ . Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin- J operators must grow like $\exp \left( \pi \sqrt{\frac{2}{3}(c-1) J} \right) /\sqrt{2J}$ in any twist interval at or above $(c-1)/12$ , with a known twist-dependent prefactor. This proves that the large J spectrum becomes dense even without averaging over spins. For twists below $(c-1)/12$ we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin- J operators must go to zero as J becomes large.