Abstract
We improve the recently discovered upper and lower bounds on the $O(1)$ correction to the Cardy formula for the density of states integrated over an energy window (of width $2\delta$), centered at high energy in 2 dimensional conformal field theory. We prove optimality of the lower bound for $\delta\to 1^{-}$. 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 $O(1)$ 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.
Highlights
Modular invariance is a powerful constraint on the data of two-dimensional (2D) conformal field theory (CFT)
Using the fact that the low temperature behavior of the 2D CFT partition function is universal and controlled by a single parameter c, the central charge of the CFT, we can deduce the universal behavior of the partition function ZðβÞ at high temperature (β → 0)
We can derive the asymptotic behavior of the density of states, which controls the high temperature behavior of a 2D CFT [1]
Summary
Modular invariance is a powerful constraint on the data of two-dimensional (2D) conformal field theory (CFT). One cannot theoretically obtain a universal correction, which is further suppressed compared to the Oð1Þ number without assuming anything beyond unitarity and modularity This approach can be contrasted to the one taken in [17] where a convergent Rademacher sum is written down for the Fourier coefficient of Klein invariant function; such convergent sums can be derived for any CFT with holomorphic modular invariant partition function. One needs a CFT spectra saturating (1.3) It turns out, as we will explain shortly, that the numbers sÆ can be derived using functions with bounded Fourier support a.k.a bandlimited functions. We will show that for a specific value of the energy width, it is possible to achieve the optimal lower bound using bandlimited functions only
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