Bounds on the density of states and the spectral gap in CFT$_{2}$

TL;DR

This work tightens universal bounds on the $O(1)$ corrections to the Cardy density of states in high-energy 2D CFTs via Tauberian analysis with bandlimited comparison functions. It proves the lower bound is optimal as the energy-window width $δ$ approaches $1$ from below and establishes the asymptotic gap between consecutive Virasoro primaries to be exactly $1$ for central charge $c>1$, with the Monster CFT saturating these bounds. A systematic framework is developed to quantify how tightly these bounds can be sharpened using bandlimited functions, revealing a Beurling–Selberg/sphere packing connection that bounds the attainable improvement. The results illuminate fundamental limits of Tauberian methods in CFT spectral estimates and expose deep links between modular invariance, spectral gaps, and extremal function theory.

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 $2δ$), centered at high energy in 2 dimensional conformal field theory. We prove optimality of the lower bound for $δ\to 1^{-}$. 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.

Figures (10)

• Figure 1: $\mathrm{Exp}[s_{\pm}]\ $ as a function of $\delta$, the half-width of the energy window. The blue line is the bound obtained in Baur. The orange line denotes the improved bound that we report here. The green line is the analytical lower (upper) bound on the upper (lower) bound, while the brown dots stand for the lower (upper) bound on the upper (lower) bound obtained from enforcing the positive definiteness condition on the Fourier transform of $\pm(\phi_{\pm}-\Theta)$ via MATLAB. The bound on bounds represented by the green line is thus weaker than that of represented by the brown dots. The brown shaded region is not achievable by any bandlimited function.
• Figure 2: $\mathrm{Exp}[s_{-}]\ $ as a function of $\delta$, the half-width of the energy window around $\delta=1$. The black curve makes the bounding curve continuous, i.e. the bounding curve is now given by $\text{max}\left\{\text{orange curve},\text{black curve}\right\}$.
• Figure 3: Verification of the bound on $s(\delta,\Delta)$, order one correction to entropy using $2$D Ising model (the top row) and non-chiral Monster CFT (the bottom row). We have plotted for different $\delta$, the half width of the interval under consideration. One can see for high enough $\Delta$, the bound is satisfied, indicating the asymptotic nature of the bound. The partition function for chiral Monster CFT can be reinterpreted as a S modular invariant particle function of a non chiral CFT with $c=12$. The dense cyan curve in between the red and the black line is obtained from the actual partition function.
• Figure 4: $\mathrm{Exp}[s_{\pm}]\ $: The orange line denotes the improved lower (upper) bound while the blue line is from Baur.
• Figure 5: The orange line represents the improvement on the lower bound by using the function $\phi^{\text{Sphere}}_-$ appearing in the sphere packing problem.