Automorphic spectra and the conformal bootstrap

Kravchuk, Petr; Mazáč, Dalimil; Pal, Sridip

doi:10.1090/cams/26

Automorphic spectra and the conformal bootstrap
HTML articles powered by AMS MathViewer

by Petr Kravchuk, Dalimil Mazáč and Sridip Pal;

Comm. Amer. Math. Soc. 4 (2024), 1-63

DOI: https://doi.org/10.1090/cams/26

Published electronically: January 17, 2024

HTML | PDF

Abstract:

We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of $\mathrm {PSL}_2(\mathbb {R})$ and semidefinite programming, the method yields rigorous upper bounds on the Laplacian spectral gap. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is $\lambda _1\leq 3.8388976481$, while the Bolza surface has $\lambda _1\approx 3.838887258$. The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case. The ideas were closely inspired by modern conformal bootstrap.

References

V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568–640. MR 21942, DOI 10.2307/1969129
James Bonifacio and Kurt Hinterbichler, Bootstrap bounds on closed Einstein manifolds, J. High Energy Phys. 10 (2020), 069, 38. MR 4204036, DOI 10.1007/jhep10(2020)069
Svend Bundgaard and Jakob Nielsen, On normal subgroups with finite index in $F$-groups, Mat. Tidsskr. B 1951 (1951), 56–58. MR 48447
J. Barros-Neto, Spaces of vector valued real analytic functions, Trans. Amer. Math. Soc. 112 (1964), 381–391. MR 169084, DOI 10.1090/S0002-9947-1964-0169084-5
James Bonifacio, Bootstrap bounds on closed hyperbolic manifolds, J. High Energy Phys. 2 (2022), Paper No. 025, 30. MR 4407440, DOI 10.1007/jhep02(2022)025
James Bonifacio, Bootstrapping closed hyperbolic surfaces, J. High Energy Phys. 3 (2022), Paper No. 093, 18. MR 4426184, DOI 10.1007/jhep03(2022)093
Joseph Bernstein and Andre Reznikov, Subconvexity bounds for triple $L$-functions and representation theory, Ann. of Math. (2) 172 (2010), no. 3, 1679–1718. MR 2726097, DOI 10.4007/annals.2010.172.1679
Andrew R. Booker and Andreas Strömbergsson, Numerical computations with the trace formula and the Selberg eigenvalue conjecture, J. Reine Angew. Math. 607 (2007), 113–161. MR 2338122, DOI 10.1515/CRELLE.2007.047
Bogdan Bucicovschi, Seeley’s theory of pseudodifferential operators on orbifolds, 1999, arXiv:math/9912228.
A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403, DOI 10.1090/surv/067
Henry Cohn and Noam Elkies, New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. MR 1973059, DOI 10.4007/annals.2003.157.689
T. C. Chau, A note concerning Fox’s paper on Fenchel’s conjecture, Proc. Amer. Math. Soc. 88 (1983), no. 4, 584–586. MR 702279, DOI 10.1090/S0002-9939-1983-0702279-5
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), no. 3, 1017–1033. MR 3664817, DOI 10.4007/annals.2017.185.3.8
Joseph Cook, Properties of eigenvalues on Riemann surfaces with large symmetry groups, doctoral thesis, eprint: 2108.11825.
F. A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nuclear Phys. B 678 (2004), no. 1-2, 491–507. MR 2022998, DOI 10.1016/j.nuclphysb.2003.11.016
Lorenzo Di Pietro, Victor Gorbenko, and Shota Komatsu, Analyticity and unitarity for cosmological correlators, J. High Energy Phys. 3 (2022), Paper No. 023, 78. MR 4426232, DOI 10.1007/jhep03(2022)023
A. El Soufi and S. Ilias, Le volume conforme et ses applications d’après Li et Yau, Séminaire de Théorie Spectrale et Géométrie, Année 1983–1984, Univ. Grenoble I, Saint-Martin-d’Hères, 1984, pp. VII.1–VII.15 (French). MR 1046044
Sheer El-Showk, Miguel F. Paulos, David Poland, Slava Rychkov, David Simmons-Duffin, and Alessandro Vichi, Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012), 025022, arXiv:1203.6064.
S. Ferrara, A. F. Grillo, and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Ann. Physics 76 (1973), 161–188. MR 411394, DOI 10.1016/0003-4916(73)90446-6
Ralph H. Fox, On Fenchel’s conjecture about $F$-groups, Mat. Tidsskr. B 1952 (1952), 61–65. MR 53937
Stephen S. Gelbart, Automorphic forms on adèle groups, Annals of Mathematics Studies, No. 83, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1975. MR 379375, DOI 10.1515/9781400881611
I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Generalized Functions, vol. 6, Academic Press, Inc., Boston, MA, 1990. Translated from the Russian by K. A. Hirsch; Reprint of the 1969 edition. MR 1071179
J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésiques, J. Math. Pures Appl. 4 (1898), 27–73.
F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640, DOI 10.1515/jnum-2012-0013
Joachim A. Hempel, On the uniformization of the $n$-punctured sphere, Bull. London Math. Soc. 20 (1988), no. 2, 97–115. MR 924235, DOI 10.1112/blms/20.2.97
Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
Will Hide and Michael Magee, Near optimal spectral gaps for hyperbolic surfaces, Ann. of Math. (2) 198 (2023), no. 2, 791–824. MR 4635304, DOI 10.4007/annals.2023.198.2.6
Thomas Hartman, Dalimil Mazáč, and Leonardo Rastelli, Sphere packing and quantum gravity, J. High Energy Phys. 12 (2019), 048, 66. MR 4075697, DOI 10.1007/jhep12(2019)048
Daniel Harlow, Jonathan Maltz, and Edward Witten, Analytic continuation of Liouville theory, J. High Energy Phys. 12 (2011), 071, i, 104. MR 2935613, DOI 10.1007/JHEP12(2011)071
Matthijs Hogervorst, Joao Penedones, and Kamran Salehi Vaziri, Towards the non-perturbative cosmological bootstrap, J. High Energy Phys. 2 (2023), Paper No. 162, 75. MR 4552025, DOI 10.1007/jhep02(2023)162
Matthijs Hogervorst and Slava Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013), no. 10, 106004, DOI:10.1103.
Heinz Huber, On the spectrum of the Laplace operator on compact Riemann surfaces, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, RI, 1980, pp. 181–184. MR 573433
Mikhail Karpukhin, On the Yang-Yau inequality for the first Laplace eigenvalue, Geom. Funct. Anal. 29 (2019), no. 6, 1864–1885. MR 4034923, DOI 10.1007/s00039-019-00518-z
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001. An overview based on examples; Reprint of the 1986 original. MR 1880691
Filip Kos, David Poland, and David Simmons-Duffin, Bootstrapping mixed correlators in the 3D Ising model, J. High Energy Phys. 11 (2014), 109, DOI:10.1007.
Filip Kos, David Poland, and David Simmons-Duffin, Bootstrapping the $O(N)$ vector models, JHEP 06 (2014), 091, DOI:10.1007, J. High Energy Phys. 06(2014)091.
Filip Kos, David Poland, David Simmons-Duffin, and Alessandro Vichi, Precision islands in the Ising and $O(N)$ models, J. High Energy Phys. 8 (2016), 036, front matter+15. MR 3555601, DOI 10.1007/JHEP08(2016)036
Mikhail Karpukhin and Denis Vinokurov, The first eigenvalue of the Laplacian on orientable surfaces, Math. Z. 301 (2022), no. 3, 2733–2746. MR 4437337, DOI 10.1007/s00209-022-03009-4
R. P. Langlands, Problems in the theory of automorphic forms, Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin-New York, 1970, pp. 18–61. MR 302614
Zhijin Li and David Poland, Searching for gauge theories with the conformal bootstrap, J. High Energy Phys. 3 (2021), Paper No. 172, 48. MR 4262643, DOI 10.1007/jhep03(2021)172
Walter Landry and David Simmons-Duffin, Scaling the semidefinite program solver SDPB, (Sept 2019), arXiv:1909.09745.
Dalimil Mazáč, Analytic bounds and emergence of $\textrm {AdS}_2$ physics from the conformal bootstrap, J. High Energy Phys. 4 (2017), 146, front matter+43. MR 3650122, DOI 10.1007/JHEP04(2017)146
J. S. Milne, Modular functions and modular forms, https://www.jmilne.org/math/CourseNotes/MF110.pdf.
Dalimil Mazáč and Miguel F. Paulos, The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices, J. High Energy Phys. 2 (2019), 162, front matter + 50. MR 3925258, DOI 10.1007/jhep02(2019)162
Philippe Michel and Akshay Venkatesh, The subconvexity problem for $\textrm {GL}_2$, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171–271. MR 2653249, DOI 10.1007/s10240-010-0025-8
Paul D. Nelson, Bounds for standard $L$-functions, (Sept. 2021), arXiv:2109.15230.
Shin Nayatani and Toshihiro Shoda, Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian, C. R. Math. Acad. Sci. Paris 357 (2019), no. 1, 84–98 (English, with English and French summaries). MR 3907601, DOI 10.1016/j.crma.2018.11.008
A. M. Polyakov, Non-Hamiltonian approach to conformal quantum field theory, Ž. Èksper. Teoret. Fiz. 66 (1974), 23–42 (Russian, with English summary); English transl., Soviet Physics JETP 39 (1974), 10–18. MR 395598
David Poland, Slava Rychkov, and Alessandro Vichi, The conformal bootstrap: theory, numerical techniques, and applications, Rev. Modern Phys. 91 (2019), no. 1, 015002, 74. MR 3942977, DOI 10.1103/RevModPhys.91.015002
David Poland, David Simmons-Duffin, and Alessandro Vichi, Carving out the space of 4D CFTs, J. High Energy Phys. 05 (2012), 110, DOI:10.1007.
Antonio Ros, On the first eigenvalue of the Laplacian on compact surfaces of genus three, J. Math. Soc. Japan 74 (2022), no. 3, 813–828. MR 4484231, DOI 10.2969/jmsj/85898589
Riccardo Rattazzi, Vyacheslav S. Rychkov, Erik Tonni, and Alessandro Vichi, Bounding scalar operator dimensions in 4D CFT, J. High Energy Phys. 12 (2008), 031, 49. MR 2469909, DOI 10.1088/1126-6708/2008/12/031
Riccardo Rattazzi, Slava Rychkov, and Alessandro Vichi, Bounds in 4D conformal field theories with global symmetry, J. Phys. A 44 (2011), no. 3, 035402, 24. MR 2749080, DOI 10.1088/1751-8113/44/3/035402
Slava Rychkov and Ning Su, New developments in the numerical conformal bootstrap, doctoral thesis, 2023, arXiv:2311.15844.
Peter Sarnak, Integrals of products of eigenfunctions, Internat. Math. Res. Notices 6 (1994), 251 ff., approx. 10 pp.}, issn=1073-7928, review= MR 1277052, doi=10.1155/S1073792894000280, DOI 10.1155/S1073792894000280
Peter Sarnak, Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 441–478. MR 1997348, DOI 10.1090/S0273-0979-03-00991-1
Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288. MR 1776084, DOI 10.1007/BF02803524
David Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, J. High Energy Phys. 06 (2015), 174, DOI:10.1007.
David Simmons-Duffin, The conformal bootstrap, Proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015) (Boulder, CO, USA, June 1–26, 2015), 2017, pp. 1–74, DOI:10.1142, http://inspirehep.net/record/1424282/files/arXiv:1602.07982.pdf.
Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Inst. Fund. Res., Bombay, 1960, pp. 147–164. MR 130324
Atle Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, RI, 1965, pp. 1–15. MR 182610
Carl Ludwig Siegel, Some remarks on discontinuous groups, Ann. of Math. (2) 46 (1945), 708–718. MR 14088, DOI 10.2307/1969206
Stanislav Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244 (English, with English and French summaries). MR 1851632, DOI 10.1016/S0764-4442(01)01991-7
Alexander Strohmaier and Ville Uski, An algorithm for the computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces, Comm. Math. Phys. 317 (2013), no. 3, 827–869. MR 3009726, DOI 10.1007/s00220-012-1557-1
R. Schoen, S. Wolpert, and S. T. Yau, Geometric bounds on the low eigenvalues of a compact surface, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, RI, 1980, pp. 279–285. MR 573440
Maryna S. Viazovska, The sphere packing problem in dimension 8, Ann. of Math. (2) 185 (2017), no. 3, 991–1015. MR 3664816, DOI 10.4007/annals.2017.185.3.7
Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer-Verlag, New York-Heidelberg, 1972. MR 498999
Thomas C. Watson, Rankin triple products and quantum chaos, 2008, arXiv:0810.0425.
Paul C. Yang and Shing Tung Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 55–63. MR 577325
Don Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 1, 57–75. K. G. Ramanathan memorial issue. MR 1280058, DOI 10.1007/BF02830874
A. Zamolodchikov and Al. Zamolodchikov, Conformal bootstrap in Liouville field theory, Nuclear Phys. B 477 (1996), no. 2, 577–605. MR 1413469, DOI 10.1016/0550-3213(96)00351-3