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Beurling-Selberg Extremization and Modular Bootstrap at High Energies

Baur Mukhametzhanov, Sridip Pal

TL;DR

The paper links Beurling–Selberg extremization to modular bootstrap bounds in 2d CFTs by replacing the interval indicator with bandlimited majorants/minorants and using finite Fourier support to localize the high-energy density. It derives tight asymptotic bounds on the number of operators in a scaling window, showing optimality when the window width is an integer multiple (2δ ∈ Z) and saturation by partition functions with integer-spaced spectra; it also extends the framework to fixed spin and Virasoro primaries, and develops non-integer δ results via Littmann’s generalized Poisson summation. The key contributions are explicit extremal constructions for φ±, Beurling–Selberg bounds for operator counts, and their saturation by concrete modular-invariant theories, providing a rigorous analytic bridge between number-theoretic extremization and fine-grained CFT spectra. This approach sharpens understanding of fine-grained spectral features and offers precise, scalable bounds relevant for universal aspects of 2d CFTs and their holographic duals.

Abstract

We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions $[Δ- δ,Δ+ δ]$ at asymptotically large $Δ$ in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval $[Δ- δ,Δ+ δ]$ and have Fourier transforms of finite support. The optimization of the bounds over this choice turns out to be exactly the Beurling-Selberg extremization problem, widely known in analytic number theory. We review solutions of this problem and present the corresponding bounds on the number of operators for any $δ\geq 0$. When $2δ\in \mathbb Z_{\geq 0}$ the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in $c>1$ theories.

Beurling-Selberg Extremization and Modular Bootstrap at High Energies

TL;DR

The paper links Beurling–Selberg extremization to modular bootstrap bounds in 2d CFTs by replacing the interval indicator with bandlimited majorants/minorants and using finite Fourier support to localize the high-energy density. It derives tight asymptotic bounds on the number of operators in a scaling window, showing optimality when the window width is an integer multiple (2δ ∈ Z) and saturation by partition functions with integer-spaced spectra; it also extends the framework to fixed spin and Virasoro primaries, and develops non-integer δ results via Littmann’s generalized Poisson summation. The key contributions are explicit extremal constructions for φ±, Beurling–Selberg bounds for operator counts, and their saturation by concrete modular-invariant theories, providing a rigorous analytic bridge between number-theoretic extremization and fine-grained CFT spectra. This approach sharpens understanding of fine-grained spectral features and offers precise, scalable bounds relevant for universal aspects of 2d CFTs and their holographic duals.

Abstract

We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions at asymptotically large in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval and have Fourier transforms of finite support. The optimization of the bounds over this choice turns out to be exactly the Beurling-Selberg extremization problem, widely known in analytic number theory. We review solutions of this problem and present the corresponding bounds on the number of operators for any . When the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in theories.

Paper Structure

This paper contains 16 sections, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Review
  3. Beurling-Selberg problem and Paley-Wiener theorem
  4. Extremal functions for $2\delta \in {\mathbb Z}$
  5. A bound from Poisson summation
  6. Extremal functions
  7. Saturation at $c=4,8,12,\ldots$
  8. A simple non-optimal bound for $2\delta \notin {\mathbb Z}$
  9. Extremal functions for $2\delta \notin \mathbb Z$
  10. Generalized Poisson summation
  11. Extremal functions
  12. Fixed spin
  13. Virasoro primaries
  14. Majorization and Minorization by $\phi_{\pm}$
  15. Degeneracy of states for integer spaced spectra
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Functions $\phi_{\pm}(x)$ majorising and minorising the indicator function of the interval $[-2,2]$.
  • Figure 2: Spectral formfactor in 2d Ising ($\beta =1$). It represents a typical behavior: it is large at early times and the initial decay is controlled by the vacuum in the S-dual channel. After a certain time a recurrence, generically only partial, happens. In chaotic theories the recurrence time is typically very long.
  • Figure 3: The solid blue lines are the optimal values (\ref{['Opt+']},\ref{['Opt-']}) of $2\pi\widehat{\phi}_\pm(0)$ as functions of $\delta$. The number of operators in a window of size $2\delta$ is bounded by $2\pi \widehat{\phi}_-(0) \rho_0(\Delta) \leq \int_{\Delta - \delta}^{\Delta + \delta} d\Delta' \rho(\Delta') \leq 2\pi \widehat{\phi}_+(0) \rho_0(\Delta)$. The dashed red lines are from the bounds (\ref{['intwindowNonopt']}), which are optimal only for $2\delta \in \mathbb Z$. The blue and red lines touch when $2\delta \in \mathbb Z$.