Beurling-Selberg Extremization and Modular Bootstrap at High Energies

Abstract

We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions $[\Delta -\delta ,\Delta +\delta ]$ at asymptotically large $\Delta$ in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval $[\Delta -\delta ,\Delta +\delta ]$ 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 $\delta \ge 0$. When $2\delta \in {\mathbb{Z}}_{\ge 0}$ 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.

arXiv comment28 pages, 3 figures; section on Virasoro primaries is added, typos fixed

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