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

Shouvik Ganguly, Sridip Pal|arXiv (Cornell University)|May 29, 2019

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper improves bounds on the O(1) correction to the Cardy formula for the density of states in 2D unitary modular invariant CFTs using complex Tauberian theorems and bandlimited functions. It proves the optimality of the lower bound as δ → 1⁻ and establishes the conjectured upper bound of 1 on the asymptotic gap between consecutive Virasoro primaries, which is saturated by the Monster CFT, ruling out Hadamard-type gaps in CFT spectra.

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 $δ o 1^{-}$. 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.

Motivation & Objective

• To improve the upper and lower bounds on the O(1) correction to the Cardy formula for the density of states in 2D CFTs.
• To prove the optimality of the lower bound in the limit δ → 1⁻.
• To establish the conjectured upper bound of 1 on the asymptotic gap between consecutive Virasoro primaries for c > 1.
• To develop a systematic method for estimating the tightness of bounds derived from bandlimited functions.
• To explore the connection between the sphere packing problem and the problem of finding optimal lower bounds using bandlimited functions.

Proposed method

• Uses complex Tauberian theorems, particularly Ingham’s theorem, to relate high-energy density of states to modular properties of the partition function.
• Applies bandlimited functions (with bounded Fourier support) to derive bounds on the O(1) correction term s(δ, Δ), treating them as extremal functionals.
• Derives a 'bound on bounds' to determine the theoretical limit of tightness achievable with bandlimited functions.
• Leverages the positive definiteness of autocorrelation functions and power spectral densities to constrain the form of extremal functions.
• Compares the achievable bound with the bound on bounds to assess potential for further improvement.
• Uses the Monster CFT as a saturation example to confirm the optimality of the spectral gap bound.

Experimental results

Research questions

• RQ1 Can the O(1) correction to the Cardy formula be bounded more tightly using bandlimited functions in 2D CFTs?
• RQ2 Is the conjectured upper bound of 1 on the asymptotic gap between Virasoro primaries optimal and universally achievable?
• RQ3 What is the theoretical limit of tightness for bounds on the O(1) correction when restricted to bandlimited functions?
• RQ4 Does the sphere packing problem provide a meaningful analogy or constraint for deriving optimal lower bounds on the O(1) correction?
• RQ5 Can the bound on bounds be saturated, and if so, under what conditions?

Key findings

• The lower bound on the O(1) correction is proven optimal in the limit δ → 1⁻, with no improvement possible beyond this value.
• The upper bound of 1 on the asymptotic gap between consecutive Virasoro primaries is proven optimal and saturated by the Monster CFT.
• The bound on bounds derived from bandlimited functions sets a theoretical limit on how tight the O(1) correction bounds can be.
• For δ = 1, the achievable bound matches the bound on bounds, indicating saturation and optimality.
• A novel connection is uncovered between the sphere packing problem and the problem of minimizing the upper bound on the O(1) correction using bandlimited functions.
• The method demonstrates that relaxing the bandlimited constraint does not yield better bounds, confirming the optimality of the δ → 1⁻ lower bound without such restrictions.