Abstract
We consider gravity compactifications whose internal space consists of small bridges connecting larger manifolds, possibly noncompact. We prove that, under rather general assumptions, this leads to a massive spin-two field with very small mass. The argument involves a recently-noticed relation to Bakry-Émery geometry, a version of the so-called Cheeger constant, and the theory of synthetic Ricci lower bounds. The latter technique allows generalizations to non-smooth spaces such as those with D-brane singularities. For AdSd vacua with a bridge admitting an AdSd+1 interpretation, the holographic dual is a CFTd with two CFTd−1 boundaries. The ratio of their degrees of freedom gives the graviton mass, generalizing results obtained by Bachas and Lavdas for d = 4. We also prove new bounds on the higher eigenvalues. These are in agreement with the spin-two swampland conjecture in the regime where the background is scale-separated; in the opposite regime we provide examples where they are in naive tension with it.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G.B. De Luca and A. Tomasiello, Leaps and bounds towards scale separation, JHEP 12 (2021) 086 [arXiv:2104.12773] [INSPIRE].
C. Bachas and J. Estes, Spin-2 spectrum of defect theories, JHEP 06 (2011) 005 [arXiv:1103.2800] [INSPIRE].
C. Csáki, J. Erlich, T.J. Hollowood and Y. Shirman, Universal aspects of gravity localized on thick branes, Nucl. Phys. B 581 (2000) 309 [hep-th/0001033] [INSPIRE].
C. Bachas and I. Lavdas, Quantum Gates to other Universes, Fortsch. Phys. 66 (2018) 1700096 [arXiv:1711.11372] [INSPIRE].
C. Bachas and I. Lavdas, Massive Anti-de Sitter Gravity from String Theory, JHEP 11 (2018) 003 [arXiv:1807.00591] [INSPIRE].
N. De Ponti and A. Mondino, Sharp Cheeger-Buser type inequalities in RCD(K, ∞) spaces, J. Geom. Anal. 31 (2021) 2416.
D. Klaewer, D. Lüst and E. Palti, A Spin-2 Conjecture on the Swampland, Fortsch. Phys. 67 (2019) 1800102 [arXiv:1811.07908] [INSPIRE].
C. Bachas, Massive AdS Supergravitons and Holography, JHEP 06 (2019) 073 [arXiv:1905.05039] [INSPIRE].
K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].
C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités XIX 1983/84 , Lecture Notes in Mathematics 1123, Springer (1985), pp. 177–206.
C. Villani, Synthetic theory of Ricci curvature bounds, Jpn. J. Math. 11 (2016) 219.
L. Ambrosio, Calculus, heat flow and curvature-dimension bounds in metric measure spaces, in proceedings of the International Congress of Mathematicians 2018 , Volume I. Plenary lectures, Rio de Janeiro, Brazil, 1–9 August 2018, World Scientific Publishing, Singapore (2018), pp. 301–340.
G.B. De Luca, N. De Ponti, A. Mondino and A. Tomasiello, work in progress.
L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014) 1405.
N. Gigli, On the differential structure of metric measure spaces and applications, in Memoirs of the American Mathematical Society 236, American Mathematical Society, Providence RI U.S.A. (2015).
L. Ambrosio, N. Gigli, A. Mondino and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure, Trans. Amer. Math. Soc. 367 (2015) 4661.
M. Erbar, K. Kuwada and K.-T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015) 993.
L. Ambrosio, A. Mondino and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, in Memoirs of the American Mathematical Society 262, American Mathematical Society, Providence RI U.S.A. (2019).
F. Cavalletti and E. Milman, The globalization theorem for the curvature dimension condition, Invent. Math. 226 (2021) 1.
K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006) 65.
K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006) 133.
J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. 169 (2009) 903.
J. Cheeger and T.H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Diff. Geom. 46 (1997) 406.
J. Cheeger and T.H. Colding, On the structure of spaces with Ricci curvature bounded below. II, J. Diff. Geom. 54 (2000) 13.
J. Cheeger and T.H. Colding, On the structure of spaces with Ricci curvature bounded below. III, J. Diff. Geom. 54 (2000) 37.
N. Gigli, A. Mondino and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. 111 (2015) 1071.
A. Petrunin, Alexandrov meets Lott-Villani-Sturm, Münster J. Math. 4 (2011) 53 [arXiv:1003.5948].
J. Bertrand, C. Ketterer, I. Mondello and T. Richard, Stratified spaces and synthetic Ricci curvature bounds, Ann. Inst. Fourier 71 (2021) 123.
F. Galaz-García, M. Kell, A. Mondino and G. Sosa, On quotients of spaces with Ricci curvature bounded below, J. Funct. Anal. 275 (2018) 1368.
C. Villani, Optimal transport. Old and new, in Grundlehren der Mathematischen Wissenschaften 338, Springer (2009).
J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999) 428.
L. Ambrosio, N. Gigli and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math. 195 (2014) 289.
E.B. Davies, Spectral theory and differential operators, in Cambridge Studies in Advanced Mathematics 42, Cambridge University Press, Cambridge U.K. (1995).
B. Crampton, C.N. Pope and K.S. Stelle, Braneworld localisation in hyperbolic spacetime, JHEP 12 (2014) 035 [arXiv:1408.7072] [INSPIRE].
J.R. Lee, S. Gharan and L. Trevisan, Multiway spectral partitioning and higher-order Cheeger inequalities, J. ACM 61 (2014) 37.
K. Funano, Eigenvalues of Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds, arXiv:1307.3919.
L. Miclo, On hyperboundedness and spectrum of Markov operators, Invent. Math. 200 (2015) 311.
S. Liu, An optimal dimension-free upper bound for eigenvalue ratios, arXiv:1405.2213.
J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in analysis (Papers dedicated to Salomon Bochner, 1969), Princeton University Press, Princeton NJ U.S.A. (1970), pp. 195–199.
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, in Oxford Mathematical Monographs, Oxford University Press, Oxford U.K. (2000).
K. Bacher and K.-T. Sturm, Ricci bounds for Euclidean and spherical cones, in Singular phenomena and scaling in mathematical models, Springer (2014), pp. 3–23.
P. Buser, A note on the isoperimetric constant, Ann. Sci. Éc. Norm. Supér. 15 (1982) 213.
S. Cremonesi and A. Tomasiello, 6d holographic anomaly match as a continuum limit, JHEP 05 (2016) 031 [arXiv:1512.02225] [INSPIRE].
M. Van Raamsdonk, Cosmology from confinement?, arXiv:2102.05057 [INSPIRE].
I. Akal, Y. Kusuki, T. Takayanagi and Z. Wei, Codimension two holography for wedges, Phys. Rev. D 102 (2020) 126007 [arXiv:2007.06800] [INSPIRE].
C.F. Uhlemann, Islands and Page curves in 4d from Type IIB, JHEP 08 (2021) 104 [arXiv:2105.00008] [INSPIRE].
O. Aharony, M. Berkooz and E. Silverstein, Multiple trace operators and nonlocal string theories, JHEP 08 (2001) 006 [hep-th/0105309] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions, and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
M. Berkooz, A. Sever and A. Shomer, ‘Double trace’ deformations, boundary conditions and space-time singularities, JHEP 05 (2002) 034 [hep-th/0112264] [INSPIRE].
A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008 [hep-th/0011156] [INSPIRE].
A. Karch and L. Randall, Localized gravity in string theory, Phys. Rev. Lett. 87 (2001) 061601 [hep-th/0105108] [INSPIRE].
E. Milman, On the role of convexity in functional and isoperimetric inequalities, Proc. Lond. Math. Soc. 99 (2009) 32.
N. De Ponti, A. Mondino and D. Semola, The equality case in Cheeger’s and Buser’s inequalities on RCD spaces, J. Funct. Anal. 281 (2021) 109022.
M. Keller, S. Liu and N. Peyerimhoff, A note on eigenvalue bounds for non-compact manifolds, Math. Nachr. 294 (2021) 1134.
E. D’Hoker, J. Estes and M. Gutperle, Exact half-BPS Type IIB interface solutions. I. Local solution and supersymmetric Janus, JHEP 06 (2007) 021 [arXiv:0705.0022] [INSPIRE].
E. D’Hoker, J. Estes and M. Gutperle, Exact half-BPS Type IIB interface solutions. II. Flux solutions and multi-Janus, JHEP 06 (2007) 022 [arXiv:0705.0024] [INSPIRE].
B. Assel, C. Bachas, J. Estes and J. Gomis, Holographic Duals of D = 3 N = 4 Superconformal Field Theories, JHEP 08 (2011) 087 [arXiv:1106.4253] [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Stat. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
N. Bobev and P.M. Crichigno, Universal RG Flows Across Dimensions and Holography, JHEP 12 (2017) 065 [arXiv:1708.05052] [INSPIRE].
I. Bah, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].
F. Apruzzi, M. Fazzi, A. Passias and A. Tomasiello, Supersymmetric AdS5 solutions of massive IIA supergravity, JHEP 06 (2015) 195 [arXiv:1502.06620] [INSPIRE].
I. Bah, A. Passias and P. Weck, Holographic duals of five-dimensional SCFTs on a Riemann surface, JHEP 01 (2019) 058 [arXiv:1807.06031] [INSPIRE].
F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].
C. Couzens, N.T. Macpherson and A. Passias, \( \mathcal{N} \) = (2, 2) AdS3 from D3-branes wrapped on Riemann surfaces, arXiv:2107.13562 [INSPIRE].
A. Legramandi and C. Núñez, Holographic description of SCFT5 compactifications, arXiv:2109.11554 [INSPIRE].
M. Pernici and E. Sezgin, Spontaneous Compactification of Seven-dimensional Supergravity Theories, Class. Quant. Grav. 2 (1985) 673 [INSPIRE].
A. Rota and A. Tomasiello, AdS4 compactifications of AdS7 solutions in type-II supergravity, JHEP 07 (2015) 076 [arXiv:1502.06622] [INSPIRE].
F. Apruzzi, M. Fazzi, A. Passias, A. Rota and A. Tomasiello, Six-Dimensional Superconformal Theories and their Compactifications from Type IIA Supergravity, Phys. Rev. Lett. 115 (2015) 061601 [arXiv:1502.06616] [INSPIRE].
D. Gaiotto and J.M. Maldacena, The Gravity duals of N = 2 superconformal field theories, JHEP 10 (2012) 189 [arXiv:0904.4466] [INSPIRE].
P. Buser, Geometry and spectra of compact Riemann surfaces, Springer (2010).
R. Schoen, A lower bound for the first eigenvalue of a negatively curved manifold, J. Diff. Geom. 17 (1982) 233.
K. Chen, M. Gutperle and C.F. Uhlemann, Spin 2 operators in holographic 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 06 (2019) 139 [arXiv:1903.07109] [INSPIRE].
J.-P. Otal and E. Rosas, Pour toute surface hyperbolique de genre g, λ2g−2 > 1/4, Duke Math. J. 150 (2009) 101.
J. Bonifacio, Bootstrap Bounds on Closed Hyperbolic Manifolds, arXiv:2107.09674 [INSPIRE].
J. Bonifacio and K. Hinterbichler, Bootstrap Bounds on Closed Einstein Manifolds, JHEP 10 (2020) 069 [arXiv:2007.10337] [INSPIRE].
C. Córdova, G.B. De Luca and A. Tomasiello, New de Sitter Solutions in Ten Dimensions and Orientifold Singularities, JHEP 08 (2020) 093 [arXiv:1911.04498] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
D. Orlando and S.C. Park, Compact hyperbolic extra dimensions: a M-theory solution and its implications for the LHC, JHEP 08 (2010) 006 [arXiv:1006.1901] [INSPIRE].
D. Borthwick, Spectral theory of infinite-area hyperbolic surfaces, Springer (2007).
W. Ballmann, H. Matthiesen and S. Mondal, Small eigenvalues of surfaces of finite type, Compos. Math. 153 (2017) 1747 [arXiv:1506.06541].
O. Foster, Lectures on Riemann Surfaces, Springer (1981).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2109.11560
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
De Luca, G.B., De Ponti, N., Mondino, A. et al. Cheeger bounds on spin-two fields. J. High Energ. Phys. 2021, 217 (2021). https://doi.org/10.1007/JHEP12(2021)217
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)217