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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.

Published as SciPost Phys. 8, 088 (2020)

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.

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