Beurling-Selberg extremization and modular bootstrap at high energies
Baur Mukhametzhanov, Sridip Pal
SciPost Phys. 8, 088 (2020) · published 17 June 2020
- doi: 10.21468/SciPostPhys.8.6.088
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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≥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.
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