Abstract
The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions and holomorphic s-differentials satisfy certain consistency conditions on closed hyperbolic surfaces. These consistency conditions can be derived by using spectral decompositions to write quadruple overlap integrals in terms of triple overlap integrals in different ways. We show how to efficiently construct these consistency conditions and use them to derive upper bounds on eigenvalues, following the approach of the conformal bootstrap. As an example of such a bootstrap bound, we find a numerical upper bound on the spectral gap of closed orientable hyperbolic surfaces that is nearly saturated by the Bolza surface.
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References
P. Buser, Geometry and Spectra of Compact Riemann Surfaces, in Progress in Mathematics, Springer (1992).
P. Yang and S. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (1980) 55.
A. El Soufi and S. Ilias, Le volume conforme et ses applications d’après Li et Yau, Sémin. Théor. Spectr. Géom. 2 (1983–1984) 1.
A. Strohmaier and V. Uski, An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces, Commun. Math. Phys. 317 (2013) 827 [arXiv:1110.2150].
R. Aurich and F. Steiner, Periodic-orbit sum rules for the Hadamard-Gutzwiller model, Physica D 39 (1989) 169.
N. Afkhami-Jeddi, A. Ashmore and C. Cordova, Calabi-Yau CFTs and random matrices, JHEP 02 (2022) 021 [arXiv:2107.11461] [INSPIRE].
A. Ros, On the first eigenvalue of the laplacian on compact surfaces of genus three, J. Math. Soc. Jpn. (2021) 1 [arXiv:2010.14857].
M. Karpukhin and D. Vinokurov, An improved Yang-Yau inequality for the first Laplace eigenvalue, arXiv:2106.00627.
H. Huber, Über den ersten Eigenwert des Laplace-Operators auf kompakten Riemannschen Flächen, Comment. Math. Helv. 49 (1974) 251.
W. Hide and M. Magee, Near optimal spectral gaps for hyperbolic surfaces, arXiv:2107.05292.
P. Buser, Riemannsche Flächen mit Eigenwerten in (0, 1/4), Comment. Math. Helv. 52 (1977) 25.
M. Lipnowski and A. Wright, Towards optimal spectral gaps in large genus, arXiv:2103.07496.
Y. Wu and Y. Xue, Random hyperbolic surfaces of large genus have first eigenvalues greater than \( \frac{3}{16}-\upepsilon \), arXiv:2102.05581.
J. Bonifacio and K. Hinterbichler, Bootstrap Bounds on Closed Einstein Manifolds, JHEP 10 (2020) 069 [arXiv:2007.10337] [INSPIRE].
J. Bonifacio, Bootstrap bounds on closed hyperbolic manifolds, JHEP 02 (2022) 025 [arXiv:2107.09674] [INSPIRE].
R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
V. S. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002.
T. Hartman, D. Mazáč and L. Rastelli, Sphere Packing and Quantum Gravity, JHEP 12 (2019) 048 [arXiv:1905.01319] [INSPIRE].
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].
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].
J. Bonifacio and K. Hinterbichler, Unitarization from Geometry, JHEP 12 (2019) 165 [arXiv:1910.04767] [INSPIRE].
G. B. De Luca, N. De Ponti, A. Mondino and A. Tomasiello, Cheeger bounds on spin-two fields, JHEP 12 (2021) 217 [arXiv:2109.11560] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
A. Dymarsky, J. Penedones, E. Trevisani and A. Vichi, Charting the space of 3D CFTs with a continuous global symmetry, JHEP 05 (2019) 098 [arXiv:1705.04278] [INSPIRE].
A. Dymarsky, F. Kos, P. Kravchuk, D. Poland and D. Simmons-Duffin, The 3d Stress-Tensor Bootstrap, JHEP 02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
P. Kravchuk, D. Mazac and S. Pal, Automorphic Spectra and the Conformal Bootstrap, arXiv:2111.12716 [INSPIRE].
O. Alvarez, Theory of Strings with Boundaries: Fluctuations, Topology, and Quantum Geometry, Nucl. Phys. B 216 (1983) 125 [INSPIRE].
E. D’Hoker and D. H. Phong, On Determinants of Laplacians on Riemann Surfaces, Commun. Math. Phys. 104 (1986) 537 [INSPIRE].
E. D’Hoker and D. H. Phong, The Geometry of String Perturbation Theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
P. Sarnak, Integrals of products of eigenfunctions, Int. Math. Res. Not. 1994 (1994) 251.
Y. N. Petridis, On squares of eigenfunctions for the hyperbolic plane and a new bound on certain L-series, Int. Math. Res. Not. 1995 (1995) 111.
J. Bernstein and A. Reznikov, Analytic continuation of representations and estimates of automorphic forms, Ann. Math. 150 (1999) 329.
F. Caracciolo and V. S. Rychkov, Rigorous Limits on the Interaction Strength in Quantum Field Theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].
D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
W. Landry and D. Simmons-Duffin, Scaling the semidefinite program solver SDPB, arXiv:1909.09745 [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].
F. Jenni, Über den ersten Eigenwert des Laplace-Operators auf ausgewählten Beispielen kompakter Riemannscher Flächen, Comment. Math. Helv. 59 (1984) 193.
J. Cook, Properties of eigenvalues on Riemann surfaces with large symmetry groups, Ph.D. Thesis, Loughborough University, Loughborough U.K. (2018).
S. El-Showk and M. F. Paulos, Bootstrapping Conformal Field Theories with the Extremal Functional Method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].
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Bonifacio, J. Bootstrapping closed hyperbolic surfaces. J. High Energ. Phys. 2022, 93 (2022). https://doi.org/10.1007/JHEP03(2022)093
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DOI: https://doi.org/10.1007/JHEP03(2022)093