Abstract
The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral decompositions to write quadruple overlap integrals in terms of products of triple overlap integrals in multiple ways. In this paper, we show how these consistency conditions imply bounds on the Laplacian eigenvalues and triple overlap integrals of closed hyperbolic manifolds, in analogy to the conformal bootstrap bounds on conformal field theories. We find an upper bound on the gap between two consecutive nonzero eigenvalues of the Laplace-Beltrami operator in terms of the smaller eigenvalue, an upper bound on the smallest eigenvalue of the rough Laplacian on symmetric, transverse-traceless, rank-2 tensors, and bounds on integrals of products of eigenfunctions and eigentensors. Our strongest bounds involve numerically solving semidefinite programs and are presented as exclusion plots. We also prove the analytic bound λi+1 ≤ 1/2 + 3λi + \( \sqrt{\lambda_i^2+2{\lambda}_i+1/4} \) for consecutive nonzero eigenvalues of the Laplace-Beltrami operator on closed orientable hyperbolic surfaces. We give examples of genus-2 surfaces that nearly saturate some of these bounds. To derive the consistency conditions, we make use of a transverse-traceless decomposition for symmetric tensors of arbitrary rank.
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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].
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. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [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].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002.
J. Bonifacio and K. Hinterbichler, Bootstrap Bounds on Closed Einstein Manifolds, JHEP 10 (2020) 069 [arXiv:2007.10337] [INSPIRE].
J. Bonifacio and K. Hinterbichler, Unitarization from Geometry, JHEP 12 (2019) 165 [arXiv:1910.04767] [INSPIRE].
A. Strohmaier and V. Uski, Hypermodes, http://www1.maths.leeds.ac.uk/~pmtast/hyperbolic-surfaces/hypermodes.html, (Accessed 22-01-2021).
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].
L. Pestov and V. Sharafutdinov, Integral geometry of tensor fields on a manifold of negative curvature, Sib. Math. J. 29 (1988) 427.
N.S. Dairbekov and V. Sharafutdinov, On conformal Killing symmetric tensor fields on Riemannian manifolds, Sib. Adv. Math. 21 (2011) 1.
J.W. York Jr., Conformatlly invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity, J. Math. Phys. 14 (1973) 456 [INSPIRE].
J.W. York Jr., Covariant decompositions of symmetric tensors in the theory of gravitation, OAnnales de l’I.H.P. Physique théorique 21 (1974) 319.
P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, Springer (1992).
A.L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, Heidelberg, New York, (1987).
A. Marden, Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions, Cambridge University Press, (2016). [DOI].
J.-P. Otal and E. Rosas, Pour toute surface hyperbolique de genre g, λ2g−2 > 1/4, Duke Math. J. 150 (2009) 101.
P. Buser, Riemannsche Flächen mit Eigenwerten in (0,1/4), Comment. Math. Helv. 52 (1977) 25.
H. Huber, Über den ersten Eigenwert des Laplace-Operators auf kompakten Riemannschen Flächen, Comment. Math. Helv. 49 (1974) 251.
P. Yang and S. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7 (1980) 55.
A. El Soufi and S. Ilias, Le volume conforme et ses applications d’après Li et Yau, Séminaire de théorie spectrale et géométrie 2 (1983-1984).
R. Aurich and F. Steiner, Periodic-orbit sum rules for the Hadamard-Gutzwiller model, Physica D 39 (1989) 169.
A. Ros, On the first eigenvalue of the Laplacian on compact surfaces of genus three, arXiv:2010.14857.
M. Karpukhin and D. Vinokurov, An improved Yang-Yau inequality for the first Laplace eigenvalue, arXiv:2106.00627.
M. Karpukhin, N. Nadirashvili, A.V. Penskoi and I. Polterovich, Conformally maximal metrics for Laplace eigenvalues on surfaces, arXiv:2003.02871.
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 \( \frac{3}{16}-\epsilon \), arXiv:2102.05581.
M. Mirzakhani, Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus, J. Diff. Geom. 94 (2013) 267 [arXiv:1012.2167] [INSPIRE].
A. Wright, A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces, Bull. Am. Math. Soc. 57 (2020) 359.
M. Magee, F. Naud and D. Puder, A random cover of a compact hyperbolic surface has relative spectral gap \( \frac{3}{16}-\epsilon \), arXiv:2003.10911.
W.P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc. 6 (1982) 357 [INSPIRE].
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math/0211159 [INSPIRE].
G.D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Publications Mathématiques de l’IHÉS 34 (1968) 53.
I. Agol, The Virtual Haken Conjecture, Doc. Math. 18 (2013) 1045.
P.J. Callahan, Spectral geometry of hyperbolic 3-manifolds, Ph.D. Thesis, University of Illinois at Urbana-Champaign, U.S.A. (1994). http://hdl.handle.net/2142/22992.
R. Schoen, A lower bound for the first eigenvalue of a negatively curved manifold, J. Diff. Geom. 17 (1982) 233.
K.T. Inoue, Computation of eigenmodes on a compact hyperbolic space, Class. Quant. Grav. 16 (1999) 3071 [astro-ph/9810034] [INSPIRE].
N. Cornish and D. Spergel, On the eigenmodes of compact hyperbolic 3-manifolds, math/9906017.
K.T. Inoue, Numerical study of length spectra and low-lying eigenvalue spectra of compact hyperbolic 3-manifolds, Class. Quant. Grav. 18 (2001) 629.
D. Gabai, R. Meyerhoff and P. Milley, Minimum volume cusped hyperbolic three-manifolds, J. Am. Math. Soc. 22 (2007) 1157.
K. Hinterbichler, J. Levin and C. Zukowski, Kaluza-Klein Towers on General Manifolds, Phys. Rev. D 89 (2014) 086007 [arXiv:1310.6353] [INSPIRE].
S. Dyatlov, F. Faure and C. Guillarmou, Power spectrum of the geodesic flow on hyperbolic manifolds, Analysis & PDE 8 (2015) 923.
J. Simons, Minimal Varieties in Riemannian Manifolds, Annals Math. 88 (1968) 62.
A. Higuchi, Symmetric Tensor Spherical Harmonics on the N Sphere and Their Application to the de Sitter Group SO(N, 1), J. Math. Phys. 28 (1987) 1553 [Erratum ibid. 43 (2002) 6385] [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].
I.Y. Arefeva and I.V. Volovich, Hyperbolic Manifolds as Vacuum Solutions in Kaluza-Klein Theories, Nucl. Phys. B 274 (1986) 619 [INSPIRE].
N. Kaloper, J. March-Russell, G.D. Starkman and M. Trodden, Compact hyperbolic extra dimensions: Branes, Kaluza-Klein modes and cosmology, Phys. Rev. Lett. 85 (2000) 928 [hep-ph/0002001] [INSPIRE].
G.B. De Luca, E. Silverstein and G. Torroba, Hyperbolic compactification of M-theory and de Sitter quantum gravity, arXiv:2104.13380 [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, Annals Math. 150 (1999) 329.
J. Bernstein and A. Reznikov, Subconvexity bounds for triple L-functions and representation theory, Annals Math. 172 (2006) 1679.
C. Csáki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Gauge theories on an interval: Unitarity without a Higgs, Phys. Rev. D 69 (2004) 055006 [hep-ph/0305237] [INSPIRE].
A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Symp. Pure Math. 8 (1965).
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].
J. Cook, Properties of eigenvalues on Riemann surfaces with large symmetry groups, Ph.D. Thesis, Loughborough University, U.K. (2018). https://hdl.handle.net/2134/36294.
A. Strohmaier, Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces, arXiv:1604.02722.
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Bootstrapping 3D Fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [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].
J. Rong and N. Su, Scalar CFTs and Their Large N Limits, JHEP 09 (2018) 103 [arXiv:1712.00985] [INSPIRE].
M. Baggio, N. Bobev, S.M. Chester, E. Lauria and S.S. Pufu, Decoding a Three-Dimensional Conformal Manifold, JHEP 02 (2018) 062 [arXiv:1712.02698] [INSPIRE].
C.A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compactifications, Commun. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE].
Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang and X. Yin, \( \mathcal{N} \) = 4 superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126 [arXiv:1511.04065] [INSPIRE].
C. Guillarmou, A. Kupiainen, R. Rhodes and V. Vargas, Conformal bootstrap in Liouville Theory, arXiv:2005.11530 [INSPIRE].
J.M. Martín-García, xAct: Efficient tensor computer algebra for the Wolfram Language, http://www.xact.es.
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Bonifacio, J. Bootstrap bounds on closed hyperbolic manifolds. J. High Energ. Phys. 2022, 25 (2022). https://doi.org/10.1007/JHEP02(2022)025
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DOI: https://doi.org/10.1007/JHEP02(2022)025