Beurling-Selberg Extremization and Modular Bootstrap at High Energies

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 \(\delta \geq 0\). When \(2\delta \in \mathbb Z_{\geq 0}\) 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.