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Beurling-Selberg Extremization and Modular Bootstrap at High Energies

by Baur Mukhametzhanov, Sridip Pal

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Submission summary

Authors (as registered SciPost users): Baurzhan Mukhametzhanov · Sridip Pal
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-05-29
Date submitted: 2020-05-25 02:00
Submitted by: Mukhametzhanov, Baurzhan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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.

List of changes

1) Formulas (9), (10) and discussion around them is added to clarify the precise meaning of the formula (8) and its analogs.
2)"#" is added in the RHS of (7) to emphasize that the prefactor is to be made precise later in the paper.
3) Formula (12) is added and footnote 6 is modified to reduce the dependence of this work on [25].
4) Paragraph after (17) is slightly modified to make the discussion more explicit and less dependent on [25].
5) Formula (16) is added to give explicit definition of Z_H and Z_L and reduce dependence on [25].
6) Paragraph before (18) is added about HKS bound. It was said in [25] that one can use HKS bound. Here we emphasize that it is not necessary and only high-temperature asymptotic of the partition function is needed.
7) Paragraph after (47) is modified to emphasize that we consider Klein's j-function as a non-holomorphic partition function that is S-invariant, but not necessarily $SL(2,Z)$ invariant.
8) Reference [42] is added after (50).
9) Section 7 about Virasoro primaries is added to make claims about this generalization more explicit and reduce dependence on [25].
10) Typos pointed out in both reports are fixed.

Published as SciPost Phys. 8, 088 (2020)

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