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Beurling-Selberg Extremization and Modular Bootstrap at High Energies
by Baur Mukhametzhanov, Sridip Pal
Submission summary
| Authors (as registered SciPost users): | Baurzhan Mukhametzhanov · Sridip Pal |
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| Preprint Link: | https://arxiv.org/abs/2003.14316v3 (pdf) |
| Date accepted: | May 29, 2020 |
| Date submitted: | May 25, 2020, 2 a.m. |
| Submitted by: | Baurzhan Mukhametzhanov |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
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)