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 [∆ − δ, ∆ + δ] 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 [∆ − 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 ≥ 0. When 2δ ∈ Z≥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.