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Published November 2022 | public
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Scalar modular bootstrap and zeros of the Riemann zeta function

Benjamin, Nathan ORCID icon
Chang, Cyuan-Han ORCID icon
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Abstract

Using the technology of harmonic analysis, we derive a crossing equation that acts only on the scalar primary operators of any two-dimensional conformal field theory with U(1)ᶜ symmetry. From this crossing equation, we derive bounds on the scalar gap of all such theories. Rather remarkably, our crossing equation contains information about all nontrivial zeros of the Riemann zeta function. As a result, we rephrase the Riemann hypothesis purely as a statement about the asymptotic density of scalar operators in certain two-dimensional conformal field theories. We discuss generalizations to theories with only Virasoro symmetry.

Additional Information

We thank Scott Collier, Liam Fitzpatrick, Tom Hartman, Per Kraus, Yuya Kusuki, YingHsuan Lin, Aike Liu, Alex Maloney, Dalimil Mazáč, Hirosi Ooguri, Danylo Radchenko, David Simmons-Duffin, Herman Verlinde, Yifan Wang, and Xi Yin for very helpful discussions. We are especially grateful to Scott Collier and Liam Fitzpatrick for numerous discussions on related topics; to Danylo Radchenko for telling us the functionals used around (3.29); and to David Simmons-Duffin for enormous help on the numerics. We thank Liam Fitzpatrick and Tom Hartman for very helpful comments on a draft. NB is supported by the Sherman Fairchild Foundation. CHC is supported by Simons Foundation grant 488657 (Simons Collaboration on the Nonperturbative Bootstrap) and a DOE Early Career Award under grant no. DE-SC0019085. NB and CHC are grateful for the hospitality of the Bootstrap 2022 conference at the University of Porto, as well as the hospitality of United Airlines Flights 1990 and 144, during which part of this work was completed. Article funded by SCOAP3.

Additional details

Identifiers Related works Funding Dates Caltech Custom Metadata
Created:
August 22, 2023
Modified:
October 23, 2023