Abstract
In the first part of this paper we will work out a close and so far not yet noticed correspondence between the swampland approach in quantum gravity and geometric flow equations in general relativity, most notably the Ricci flow. We conjecture that following the gradient flow towards a fixed point, which is at infinite distance in the space of background metrics, is accompanied by an infinite tower of states in quantum gravity. In case of the Ricci flow, this conjecture is in accordance with the generalized distance and AdS distance conjectures, which were recently discussed in the literature, but it should also hold for more general background spaces. We argue that the entropy functionals of gradient flows provide a useful definition of the generalized distance in the space of background fields. In particular we give evidence that for the Ricci flow the distance ∆ can be defined in terms of the mean scalar curvature of the manifold, ∆ ∼ log \( \overline{R} \). For a more general gradient flow, the distance functional also depends on the string coupling constant.
In the second part of the paper we will apply the generalized distance conjecture to gravity theories with higher curvature interactions, like higher derivative R2 and W2 terms. We will show that going to the weak coupling limit of the higher derivative terms corresponds to the infinite distance limit in metric space and hence this limit must be accompanied by an infinite tower of light states. For the case of the R2 or W2 couplings, this limit corresponds to the limit of a small cosmological constant or, respectively, to a light additional spin-two field in gravity. In general we see that the limit of small higher curvature couplings belongs to the swampland in quantum gravity, just like the limit of a small U(1) gauge coupling belongs to the swampland as well.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
H. Ooguri and C. Vafa, On the Geometry of the String Landscape and the Swampland, Nucl. Phys.B 766 (2007) 21 [hep-th/0605264] [INSPIRE].
E. Palti, The Swampland: Introduction and Review, Fortsch. Phys.67 (2019) 1900037 [arXiv:1903.06239] [INSPIRE].
N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP06 (2007) 060 [hep-th/0601001] [INSPIRE].
D. Klaewer and E. Palti, Super-Planckian Spatial Field Variations and Quantum Gravity, JHEP01 (2017) 088 [arXiv:1610.00010] [INSPIRE].
B. Heidenreich, M. Reece and T. Rudelius, The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff, Eur. Phys. J.C 78 (2018) 337 [arXiv:1712.01868] [INSPIRE].
S. Andriolo, D. Junghans, T. Noumi and G. Shiu, A Tower Weak Gravity Conjecture from Infrared Consistency, Fortsch. Phys.66 (2018) 1800020 [arXiv:1802.04287] [INSPIRE].
T.W. Grimm, E. Palti and I. Valenzuela, Infinite Distances in Field Space and Massless Towers of States, JHEP08 (2018) 143 [arXiv:1802.08264] [INSPIRE].
B. Heidenreich, M. Reece and T. Rudelius, Emergence of Weak Coupling at Large Distance in Quantum Gravity, Phys. Rev. Lett.121 (2018) 051601 [arXiv:1802.08698] [INSPIRE].
R. Blumenhagen, D. Kläwer, L. Schlechter and F. Wolf, The Refined Swampland Distance Conjecture in Calabi-Yau Moduli Spaces, JHEP06 (2018) 052 [arXiv:1803.04989] [INSPIRE].
R. Blumenhagen, Large Field Inflation/Quintessence and the Refined Swampland Distance Conjecture, PoS(CORFU2017)175 (2018) [arXiv:1804.10504] [INSPIRE].
S.-J. Lee, W. Lerche and T. Weigand, Tensionless Strings and the Weak Gravity Conjecture, JHEP10 (2018) 164 [arXiv:1808.05958] [INSPIRE].
S.-J. Lee, W. Lerche and T. Weigand, A Stringy Test of the Scalar Weak Gravity Conjecture, Nucl. Phys.B 938 (2019) 321 [arXiv:1810.05169] [INSPIRE].
T.W. Grimm, C. Li and E. Palti, Infinite Distance Networks in Field Space and Charge Orbits, JHEP03 (2019) 016 [arXiv:1811.02571] [INSPIRE].
P. Corvilain, T.W. Grimm and I. Valenzuela, The Swampland Distance Conjecture for Kähler moduli, JHEP08 (2019) 075 [arXiv:1812.07548] [INSPIRE].
S.-J. Lee, W. Lerche and T. Weigand, Modular Fluxes, Elliptic Genera and Weak Gravity Conjectures in Four Dimensions, JHEP08 (2019) 104 [arXiv:1901.08065] [INSPIRE].
A. Joshi and A. Klemm, Swampland Distance Conjecture for One-Parameter Calabi-Yau Threefolds, JHEP08 (2019) 086 [arXiv:1903.00596] [INSPIRE].
F. Marchesano and M. Wiesner, Instantons and infinite distances, JHEP08 (2019) 088 [arXiv:1904.04848] [INSPIRE].
A. Font, A. Herráez and L.E. Ibáñez, The Swampland Distance Conjecture and Towers of Tensionless Branes, JHEP08 (2019) 044 [arXiv:1904.05379] [INSPIRE].
S.-J. Lee, W. Lerche and T. Weigand, Emergent Strings, Duality and Weak Coupling Limits for Two-Form Fields, arXiv:1904.06344 [INSPIRE].
D. Erkinger and J. Knapp, Refined swampland distance conjecture and exotic hybrid Calabi-Yaus, JHEP07 (2019) 029 [arXiv:1905.05225] [INSPIRE].
G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, de Sitter Space and the Swampland, arXiv:1806.08362 [INSPIRE].
H. Ooguri, E. Palti, G. Shiu and C. Vafa, Distance and de Sitter Conjectures on the Swampland, Phys. Lett.B 788 (2019) 180 [arXiv:1810.05506] [INSPIRE].
S.K. Garg and C. Krishnan, Bounds on Slow Roll and the de Sitter Swampland, JHEP11 (2019) 075 [arXiv:1807.05193] [INSPIRE].
G. Dvali and C. Gomez, Quantum Compositeness of Gravity: Black Holes, AdS and Inflation, JCAP01 (2014) 023 [arXiv:1312.4795] [INSPIRE].
G. Dvali and C. Gomez, Quantum Exclusion of Positive Cosmological Constant?, Annalen Phys.528 (2016) 68 [arXiv:1412.8077] [INSPIRE].
G. Dvali, C. Gomez and S. Zell, Quantum Break-Time of de Sitter, JCAP06 (2017) 028 [arXiv:1701.08776] [INSPIRE].
G. Dvali and C. Gomez, On Exclusion of Positive Cosmological Constant, Fortsch. Phys.67 (2019) 1800092 [arXiv:1806.10877] [INSPIRE].
G. Dvali, C. Gomez and S. Zell, Quantum Breaking Bound on de Sitter and Swampland, Fortsch. Phys.67 (2019) 1800094 [arXiv:1810.11002] [INSPIRE].
D. Lüst, E. Palti and C. Vafa, AdS and the Swampland, Phys. Lett.B 797 (2019) 134867 [arXiv:1906.05225] [INSPIRE].
D. Klaewer, D. Lüst and E. Palti, A Spin-2 Conjecture on the Swampland, Fortsch. Phys.67 (2019) 1800102 [arXiv:1811.07908] [INSPIRE].
R.S. Hamilton, Three-manifolds with positive ricci curvature, J. Diff. Geom.17 (1982) 255.
G. Perelman, The Entropy formula for the Ricci flow and its geometric applications, math.DG/0211159.
K. Sfetsos, Integrable interpolations: From exact CFTs to non-Abelian T-duals, Nucl. Phys.B 880 (2014) 225 [arXiv:1312.4560] [INSPIRE].
K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP12 (2014) 164 [arXiv:1410.1886] [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev.D 44 (1991) 314 [INSPIRE].
D. Lüst and E. Palti, A Note on String Excitations and the Higuchi Bound, Phys. Lett.B 799 (2019) 135067 [arXiv:1907.04161] [INSPIRE].
A. Higuchi, Forbidden Mass Range for Spin-2 Field Theory in de Sitter Space-time, Nucl. Phys.B 282 (1987) 397 [INSPIRE].
B. Chow and D Knopf, The Ricci Flow: An Introduction, AMS (2004).
B.S. DeWitt, Quantum Theory of Gravity. 1. The Canonical Theory, Phys. Rev.160 (1967) 1113 [INSPIRE].
O. Gil-Medrano and P.W. Michor, The Riemannian manifold of all Riemannian metrics, Q. J. Math.42 (1991) 183 [math.DG/9201259].
I. Bakas, Geometric flows and (some of ) their physical applications, Bulg. J. Phys.33 (2006) 091 [hep-th/0511057] [INSPIRE].
I. Bakas, Renormalization group equations and geometric flows, Ann. U. Craiova Phys.16 (2006) 20 [hep-th/0702034] [INSPIRE].
C. Gomez, Gravity as universal UV completion: Towards a unified view of Swampland conjectures, arXiv:1907.13386 [INSPIRE].
M. Larfors, A. Lukas and F. Ruehle, Calabi-Yau Manifolds and SU(3) Structure, JHEP01 (2019) 171 [arXiv:1805.08499] [INSPIRE].
A. Strominger, Superstrings with Torsion, Nucl. Phys.B 274 (1986) 253 [INSPIRE].
C.M. Hull, Compactifications of the Heterotic Superstring, Phys. Lett.B 178 (1986) 357 [INSPIRE].
G. Lopes Cardoso, G. Curio, G. Dall’Agata, D. Lüst, P. Manousselis and G. Zoupanos, Non-Kähler string backgrounds and their five torsion classes, Nucl. Phys.B 652 (2003) 5 [hep-th/0211118] [INSPIRE].
D. Lüst and D. Tsimpis, Supersymmetric AdS4compactifications of IIA supergravity, JHEP02 (2005) 027 [hep-th/0412250] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Cosmological Event Horizons, Thermodynamics and Particle Creation, Phys. Rev.D 15 (1977) 2738 [INSPIRE].
G. Dvali, C. Gomez and D. Lüst, Classical Limit of Black Hole Quantum N-Portrait and BMS Symmetry, Phys. Lett.B 753 (2016) 173 [arXiv:1509.02114] [INSPIRE].
T. Banks, Cosmological breaking of supersymmetry?, Int. J. Mod. Phys.A 16 (2001) 910 [hep-th/0007146] [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, in proceedings of the Strings 2001: International Conference, Tata Institute of Fundamental Research, Mumbai, India, 5–10 January 2001, hep-th/0106109 [INSPIRE].
L. Álvarez-Gaumé, A. Kehagias, C. Kounnas, D. Lüst and A. Riotto, Aspects of Quadratic Gravity, Fortsch. Phys.64 (2016) 176 [arXiv:1505.07657] [INSPIRE].
A.A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett.30 (1979) 682 [Pisma Zh. Eksp. Teor. Fiz.30 (1979) 719] [INSPIRE].
E.A. Bergshoeff, O. Hohm and P.K. Townsend, Massive Gravity in Three Dimensions, Phys. Rev. Lett.102 (2009) 201301 [arXiv:0901.1766] [INSPIRE].
B. Gording and A. Schmidt-May, Ghost-free infinite derivative gravity, JHEP09 (2018) 044 [Erratum JHEP10 (2018) 115] [arXiv:1807.05011] [INSPIRE].
S. Ferrara, A. Kehagias and D. Lüst, Aspects of Weyl Supergravity, JHEP08 (2018) 197 [arXiv:1806.10016] [INSPIRE].
S. Ferrara, A. Kehagias and D. Lüst, Bimetric, Conformal Supergravity and its Superstring Embedding, JHEP05 (2019) 100 [arXiv:1810.08147] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys.B 405 (1993) 279 [AMS/IP Stud. Adv. Math.1 (1996) 655] [hep-th/9302103] [INSPIRE].
I. Antoniadis, E. Gava, K.S. Narain and T.R. Taylor, Topological amplitudes in string theory, Nucl. Phys.B 413 (1994) 162 [hep-th/9307158] [INSPIRE].
G. Lopes Cardoso, G. Curio, D. Lüst, T. Mohaupt and S.-J. Rey, BPS spectra and nonperturbative gravitational couplings in N = 2, N = 4 supersymmetric string theories, Nucl. Phys.B 464 (1996) 18 [hep-th/9512129] [INSPIRE].
S.W. Hawking, The Information Paradox for Black Holes, arXiv:1509.01147 [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett.116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
A. Averin, G. Dvali, C. Gomez and D. Lüst, Gravitational Black Hole Hair from Event Horizon Supertranslations, JHEP06 (2016) 088 [arXiv:1601.03725] [INSPIRE].
A. Averin, G. Dvali, C. Gomez and D. Lüst, Goldstone origin of black hole hair from supertranslations and criticality, Mod. Phys. Lett.A 31 (2016) 1630045 [arXiv:1606.06260] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Superrotation Charge and Supertranslation Hair on Black Holes, JHEP05 (2017) 161 [arXiv:1611.09175] [INSPIRE].
S. Haco, S.W. Hawking, M.J. Perry and A. Strominger, Black Hole Entropy and Soft Hair, JHEP12 (2018) 098 [arXiv:1810.01847] [INSPIRE].
Q. Bonnefoy, L. Ciambelli, D. Lüst and S. Lüst, Infinite Black Hole Entropies at Infinite Distances and Tower of States, arXiv:1912.07453 [INSPIRE].
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1910.00453
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Kehagias, A., Lüst, D. & Lüst, S. Swampland, gradient flow and infinite distance. J. High Energ. Phys. 2020, 170 (2020). https://doi.org/10.1007/JHEP04(2020)170
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2020)170