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Scalar modular bootstrap and zeros of the Riemann zeta function

Benjamin, Nathan, Chang, Cyuan-Han·2022·Journal of High Energy Physics
DOI 10.1007/jhep11(2022)143
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Abstract

A bstract: 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.

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Figures

Figure 1 (a) Pole structure of the integral in (3.11) in the complex s plane. The poles are located at s = c 2 , 1+ z n 2 , 1+ z ∗ n 2 (shown here for c = 3 ), where z n are the nontrivial zeros of the Riemann zeta function with positive imaginary part. If the Riemann hypothesis is true, the tower of poles in the figure all occur at real part 3 4 , except for the pole at s = c 2 . (b) Contour deformation of the integral to Re ( s ) > c 2 .
Figure 1 (a) Pole structure of the integral in (3.11) in the complex s plane. The poles are located at s = c 2 , 1+ z n 2 , 1+ z ∗ n 2 (shown here for c = 3 ), where z n are the nontrivial zeros of the Riemann zeta function with positive imaginary part. If the Riemann hypothesis is true, the tower of poles in the figure all occur at real part 3 4 , except for the pole at s = c 2 . (b) Contour deformation of the integral to Re ( s ) > c 2 .CC-BY · From the PDF
Figure 2 Plot of a bound on the scalar gap for U(1) c CFTs with 4 derivatives, up to central charge c = 251 . The numerical data seems to be well-approximated by a quadratic function with leading coefficient 1 4 π 2 (see (3.41)).
Figure 2 Plot of a bound on the scalar gap for U(1) c CFTs with 4 derivatives, up to central charge c = 251 . The numerical data seems to be well-approximated by a quadratic function with leading coefficient 1 4 π 2 (see (3.41)).CC-BY · From the PDF
Figure 3 Plot of a bound on the scalar gap for U(1) c CFTs at odd c ≤ 27 . The colors blue, orange, green, red, brown, and purple represent the bound we get at 10 , 20 , · · · , 60 derivatives respectively. The color black represents the average Narain scalar gap, for comparison. (However, there is no a priori reason the average Narain scalar gap and the optimal U(1) c scalar gap should be similar.) See table 1 for the numerical data.
Figure 3 Plot of a bound on the scalar gap for U(1) c CFTs at odd c ≤ 27 . The colors blue, orange, green, red, brown, and purple represent the bound we get at 10 , 20 , · · · , 60 derivatives respectively. The color black represents the average Narain scalar gap, for comparison. (However, there is no a priori reason the average Narain scalar gap and the optimal U(1) c scalar gap should be similar.) See table 1 for the numerical data.CC-BY · From the PDF
Figure 4 Scalar part of the SU(3) 1 WZW model with first two leading terms subtracted, rescaled by T 1 / 4 , plotted up to T = 300 . If the Riemann hypothesis is true, then at large temperature, this function will remain bounded. However, if the Riemann hypothesis is false, at large temperatures the oscillations will grow in size and become unbounded (modulo the subtlety explained in footnote 14). In this plot, α := 2 ˆ E 1 ( e 2 πi/ 3 ) + 3 π ( γ E +log(4 π ) + 24 ζ ′ ( -1) -2) ≈ 0 . 975 (see (5.5)). By fitting this plot with oscillating functions in log( T ) , we can numerically recover the first few nontrivial zeros of the zeta function. (A similar plot can be made for any c > 1 CFT.)
Figure 4 Scalar part of the SU(3) 1 WZW model with first two leading terms subtracted, rescaled by T 1 / 4 , plotted up to T = 300 . If the Riemann hypothesis is true, then at large temperature, this function will remain bounded. However, if the Riemann hypothesis is false, at large temperatures the oscillations will grow in size and become unbounded (modulo the subtlety explained in footnote 14). In this plot, α := 2 ˆ E 1 ( e 2 πi/ 3 ) + 3 π ( γ E +log(4 π ) + 24 ζ ′ ( -1) -2) ≈ 0 . 975 (see (5.5)). By fitting this plot with oscillating functions in log( T ) , we can numerically recover the first few nontrivial zeros of the zeta function. (A similar plot can be made for any c > 1 CFT.)CC-BY · From the PDF

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Research works printed but not linked: 32

  1. S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE]. · DOI
  2. C.A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE]. · DOI
  3. D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE]. · DOI
  4. N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, Universal Bounds on Charged States in 2d CFT and 3d Gravity, JHEP 08 (2016) 041 [arXiv:1603.09745] [INSPIRE]. · DOI
  5. S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE]. · DOI
  6. J.-B. Bae, S. Lee and J. Song, Modular Constraints on Conformal Field Theories with Currents, JHEP 12 (2017) 045 [arXiv:1708.08815] [INSPIRE]. · DOI
  7. N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast Conformal Bootstrap and Constraints on 3d Gravity, JHEP 05 (2019) 087 [arXiv:1903.06272] [INSPIRE]. · DOI
  8. B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP 10 (2019) 261 [arXiv:1904.06359] [INSPIRE]. · DOI
  9. T. Hartman, D. Mazáč and L. Rastelli, Sphere Packing and Quantum Gravity, JHEP 12 (2019) 048 [arXiv:1905.01319] [INSPIRE]. · DOI
  10. N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, Light-cone modular bootstrap and pure gravity, Phys. Rev. D 100 (2019) 066029 [arXiv:1906.04184] [INSPIRE]. · DOI
  11. S. Pal and Z. Sun, Tauberian-Cardy formula with spin, JHEP 01 (2020) 135 [arXiv:1910.07727] [INSPIRE]. · DOI
  12. L.F. Alday and J.-B. Bae, Rademacher Expansions and the Spectrum of 2d CFT, JHEP 11 (2020) 134 [arXiv:2001.00022] [INSPIRE]. · DOI
  13. N. Afkhami-Jeddi, H. Cohn, T. Hartman, D. de Laat and A. Tajdini, High-dimensional sphere packing and the modular bootstrap, JHEP 12 (2020) 066 [arXiv:2006.02560] [INSPIRE]. · DOI
  14. S. Collier, D. Mazac and Y. Wang, Bootstrapping Boundaries and Branes, arXiv:2112.00750 [INSPIRE]. · arXiv
  15. A. Terras, Harmonic Analysis on Symmetric Spaces — Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane, Springer, New York, U.S.A. (2013). · DOI
  16. N. Benjamin, S. Collier, A.L. Fitzpatrick, A. Maloney and E. Perlmutter, Harmonic analysis of 2d CFT partition functions, JHEP 09 (2021) 174 [arXiv:2107.10744] [INSPIRE]. · DOI
  17. N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE]. · DOI
  18. N.A. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000) 275 [hep-th/9903113] [INSPIRE]. · DOI
  19. C. Angelantonj, I. Florakis and B. Pioline, A new look at one-loop integrals in string theory, Commun. Num. Theor. Phys. 6 (2012) 159 [arXiv:1110.5318] [INSPIRE]. · DOI
  20. A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE]. · DOI
  21. Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE]. · DOI
  22. A. Bondarenko, D. Radchenko and K. Seip, Fourier interpolation with zeros of zeta and L-functions, arXiv:2005.02996. · arXiv
  23. D. Mazac, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE]. · DOI
  24. D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE]. · DOI
  25. W. Landry and D. Simmons-Duffin, Scaling the semidefinite program solver SDPB, arXiv:1909.09745 [INSPIRE]. · arXiv
  26. A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE]. · DOI
  27. C.A. Keller and A. Maloney, Poincaré Series, 3D Gravity and CFT Spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE]. · DOI
  28. N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, Twist gap and global symmetry in two dimensions, Phys. Rev. D 101 (2020) 106026 [arXiv:2003.02844] [INSPIRE]. · DOI
  29. D. Platt and T. Trudgian, The Riemann hypothesis is true up to 3∙1012, Bull. Lond. Math. Soc. 53 (2021) 792 [arXiv:2004.09765]. · DOI
  30. S. Collier and E. Perlmutter, Harnessing S-duality in $$ \mathcal{N} $$ = 4 SYM & supergravity as SL(2, ℤ)-averaged strings, JHEP 08 (2022) 195 [arXiv:2201.05093] [INSPIRE]. · DOI
  31. M.B. Green, S.D. Miller and P. Vanhove, SL(2, ℤ)-invariance and D-instanton contributions to the D6R4 interaction, Commun. Num. Theor. Phys. 09 (2015) 307 [arXiv:1404.2192] [INSPIRE]. · DOI
  32. R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, On Moduli Spaces of Conformal Field Theories with c ≥ 1, in Perspectives in String Theory: Proceedings of the Niels Bohr Institute/Nordita Meeting, Copenhagen, Denmark (1987), World Scientific (1988). · DOI

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Metadata

Authors
Benjamin, N., & Chang, C.-H.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2022-11-24
Volume
2022
Issue
11
ISSN
1029-8479
DOI
10.1007/jhep11(2022)143
Cited by
20
References
2
Language
en
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Benjamin, N., & Chang, C.-H. (2022). Scalar modular bootstrap and zeros of the Riemann zeta function. Journal of High Energy Physics. https://doi.org/10.1007/jhep11(2022)143

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