Universality of the microcanonical entropy at large spin

TL;DR

This work uses the modular invariance of 2D unitary non-rational CFTs with $c>1$ to derive rigorous, non-perturbative bounds on the large-spin spectrum of Virasoro primaries. By decomposing the torus partition function into vacuum and non-vacuum contributions and analyzing the Laplace transforms in the off-diagonal (large-$J$) regime, the authors prove a universal large-$J$ density growth $\rho_J(J+2h) \sim \frac{1}{\sqrt{2J}} e^{\pi\sqrt{\frac{2}{3}(c-1)J}} \rho_c(h)$ for twists $2h \ge (c-1)/12$, with slower growth below the threshold. A key result is that the large-$J$ spectrum becomes dense without averaging over spins, and that the maximal spin-$J$ gap scales as $o(J^{-1/4+\varepsilon})$, implying increasingly fine spectral spacing. The analysis yields a theorem about the modular bootstrap that the vacuum term dominates the large-$J$ density and that, upon smearing with shrinking test functions, the non-vacuum contributions are subleading to all orders, leading to a Cardy-like universal entropy $S_J(h)$ for $h\ge (c-1)/24$. These insights advance the understanding of universality in the high-energy regime of 2D CFTs and have potential implications for chaos, ETH, and holography in rotating black hole contexts.

Abstract

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 $\exp\left( π\sqrt{\frac{2}{3}(c-1) J} \right)/\sqrt{2J}$ 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.

Figures (2)

• Figure 1: In red we show the integration contour in the complex $w$ plane. It follows the steepest descent contour, in blue, in the vicinity of the saddle point, marked with blue dot. Along the rest of the contour the integrand is exponentially suppressed because it lies entirely in the gray shaded region, which is given by equation \ref{['exponentiallysmall']}.
• Figure 2: An example of compactly supported smooth functions $\varphi^+$ (red) and $\varphi^-$ (blue). They are constructed from bump functions of the form $\varphi^+(x)=\exp\left(-\frac{b_+}{a_+^2 - x^2} + \frac{b_+}{a_+^2 - 1}\right)$ (for $\left\lvert x\right\rvert\leqslant a_+$) and $\varphi^-(x)=\exp\left(-\frac{b_-}{a_-^2 - x^2} + \frac{b_-}{a_-^2}\right)$ (for $\left\lvert x\right\rvert\leqslant a_-$), with appropriately chosen parameters.

Theorems & Definitions (22)

• Proposition 3.1
• Lemma 3.2
• proof : Proof of lemma \ref{['lem:Fvaclimit']}(a)
• proof : Proof of lemma \ref{['lem:Fvaclimit']}(b)
• Proposition 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Proposition 3.6