Beurling-Selberg extremization and modular bootstrap at high energies

Journal Article · Wed Jun 17 00:00:00 EDT 2020 · SciPost Physics

DOI:https://doi.org/10.21468/SciPostPhys.8.6.088· OSTI ID:1633669

Mukhametzhanov, Baurzhan ^[1]; Pal, Sridip ^[1]

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1. Institute for Advanced Study, Princeton University

We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions [\Delta - \delta,\Delta + \delta] $[\Delta -\delta ,\Delta +\delta ]$ at asymptotically large \Delta $\Delta$ 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] $[\Delta -\delta ,\Delta +\delta ]$ 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 $\delta \ge 0$ . When 2\delta \in \mathbb Z_{\geq 0} $2\delta \in {\mathbb{Z}}_{\ge 0}$ 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 $c>1$ theories.

• Sponsoring Organization: USDOE
• Grant/Contract Number: SC0009988
• OSTI ID: 1633669
• Alternate ID(s): OSTI ID: 1801992
• Journal Information: SciPost Physics, Journal Name: SciPost Physics Journal Issue: 6 Vol. 8; ISSN 2542-4653
• Publisher: Stichting SciPost
• Country of Publication: Netherlands
• Language: English