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arXiv:1910.07727 · Journal of High Energy Physics

Tauberian-Cardy formula with spin

Sridip Pal, Zhengdi Sun

Published Jan 21, 2020• 51 citations
arXivHTMLPDF

Abstract

Abstract We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h, $$ \overline{h} $$ h ¯ ) → (∞, ∞) 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, $$ \overline{h} $$ h ¯ ) 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 $$ Ag\le \frac{\pi \left(c-1\right){r}^2}{24} $$ Ag ≤ π c − 1 r 2 24 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′, $$ \overline{h}^{\prime } $$ h ¯ ′ ) plane and $$ r=\frac{4\sqrt{3}}{\pi}\simeq 2.21 $$ r = 4 3 π ≃ 2.21 . 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.

DOI
10.1007/jhep01(2020)135

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Abstract We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h, h‾\overline{h}h h ¯ ) → (∞, ∞) 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, h‾\overline{h}h h ¯ ) 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≤π(c−1)r224Ag\le \frac{\pi \left(c-1\right){r}^2}{24}Ag≤24π(c−1)r2​ Ag ≤ π c − 1 r 2 24 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′, h‾′\overline{h}^{\prime }h′ h ¯ ′ ) plane and r=43π≃2.21r=\frac{4\sqrt{3}}{\pi}\simeq 2.21r=π43​​≃2.21 r = 4 3 π ≃ 2.21 . 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.
We improve the recently discovered upper and lower bounds on the O(1)O(1)O(1) correction to the Cardy formula for the density of states integrated over an energy window (of width $2\ensuremath{\delta}$), centered at high energy in two-dimensional unitary and modular invariant conformal field theory. We prove optimality of the lower bound for $\ensuremath{\delta}\ensuremath{\rightarrow}{1}^{\ensuremath{-}}$. We prove a conjectured upper bound on the asymptotic gap between two consecutive Virasoro primaries for a central charge greater than 1, demonstrating it to be 1. Furthermore, a systematic method is provided to establish a limit on how tight the bound on the O(1)O(1)O(1) correction to the Cardy formula can be made using bandlimited functions. The techniques and the functions used here are of generic importance whenever the Tauberian theorems are used to estimate some physical quantities.
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