Tauberian-Cardy formula with spin
Sridip Pal, Zhengdi Sun
Abstract
A bstract We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight ( h, $$ \overline{h} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>h</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ) → ( ∞, ∞ ) for any arbitrary spin is derived using the theorem. We further rigorously show that the error term is controlled by the twist parameter and insensitive to spin. The sensitivity of the leading piece towards spin is discussed. We identify a universal piece in microcanonical entropy when the averaging window is large. An asymptotic spectral gap on ( h, $$ \overline{h} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>h</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ) plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary 2D CFT without any conserved current $$ Ag\le \frac{\pi \left(c-1\right){r}^2}{24} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ag</mml:mi> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>π</mml:mi> <mml:mfenced> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> <mml:msup> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mn>24</mml:mn> </mml:mfrac> </mml:math> is satisfied, where g is the twist gap over vacuum and A is the minimal “areal gap”, generalizing the minimal gap in dimension to ( h ′ , $$ \overline{h}^{\prime } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>h</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>′</mml:mo> </mml:math> ) plane and $$ r=\frac{4\sqrt{3}}{\pi}\simeq 2.21 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>4</mml:mn> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:mrow> <mml:mi>π</mml:mi> </mml:mfrac> <mml:mo>≃</mml:mo> <mml:mn>2.21</mml:mn> </mml:math> . We investigate density of states in the regime where spin is parametrically larger than twist with both going to infinity. Moreover, the large central charge regime is studied. We also probe finite twist, large spin behavior of density of states.