Universality of the Microcanonical Entropy at Large Spin

We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> . Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin-<i>J</i> operators must grow like <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>exp</mml:mo> <mml:mfenced><mml:mi>π</mml:mi> <mml:msqrt> <mml:mrow><mml:mfrac><mml:mn>2</mml:mn> <mml:mn>3</mml:mn></mml:mfrac> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>c</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo></mml:mrow> <mml:mi>J</mml:mi></mml:mrow> </mml:msqrt> </mml:mfenced> <mml:mo>/</mml:mo> <mml:msqrt><mml:mrow><mml:mn>2</mml:mn> <mml:mi>J</mml:mi></mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> in any twist interval at or above <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>c</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mn>12</mml:mn></mml:mrow> </mml:math> , with a known twist-dependent prefactor. This proves that the large <i>J</i> spectrum becomes dense even without averaging over spins. For twists below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>c</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mn>12</mml:mn></mml:mrow> </mml:math> we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin-<i>J</i> operators must go to zero as <i>J</i> becomes large.