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  • Open Access

Gravitons on Nariai edges

Y. T. Albert Law1,* and Varun Lochab2,†

  • *Contact author: ytalaw@stanford.edu
  • †Contact author: vv2338@columbia.edu
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Phys. Rev. D 113, 086006 – Published 13 April, 2026

DOI: https://doi.org/10.1103/23vk-l1fl

Abstract

We show that, for any d≥3, the one-loop graviton path integral on S2×Sd−1 factorizes into bulk and edge parts. The bulk equals the thermal partition function of an ideal graviton gas in the Lorentzian Nariai geometry. The edge factor is the inverse of the path integral over two identical copies, each containing one shift-symmetric vector and three shift-symmetric scalars on Sd−1. Unlike the round Sd+1 case, all scalars are massless, indicating that graviton edge partition functions probe beyond the horizon’s intrinsic geometry—in contrast to p-form gauge theories. In the course of this work, we obtain a compact formula for the one-loop Euclidean graviton path integral on any Λ>0 Einstein manifold.

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Physics Subject Headings (PhySH)

See Also

de Sitter horizon edge partition functions

Y. T. Albert Law
Phys. Rev. D 113, 086005 (2026)

Horizon Edge Partition Functions in Λ>0 Quantum Gravity

Y. T. Albert Law and Varun Lochab
Phys. Rev. Lett. 136, 151601 (2026)

Article Text

References (115)

  1. D. Anninos, F. Denef, Y. T. A. Law, and Z. Sun, Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions, J. High Energy Phys. 01 (2022) 088.
  2. T. Hirai, On irreducible representations of the Lorentz group of n-th order, Proc. Jpn. Acad. 38, 258 (1962).
  3. T. Hirai, On infinitesimal operators of irreducible representations of the Lorentz group of n-th order, Proc. Jpn. Acad. 38, 83 (1962).
  4. Harish-Chandra, The characters of semisimple Lie groups, Trans. Am. Math. Soc. 83, 98 (1956).
  5. Harish-Chandra, Invariant eigendistributions on semisimple Lie groups, Bull. Am. Math. Soc. 69, 117 (1963).
  6. T. Hirai, The characters of irreducible representations of the Lorentz group of n-th order, Proc. Jpn. Acad. 41, 526 (1965).
  7. M. Grewal and Y. T. A. Law, Real-time observables in de Sitter thermodynamics, J. High Energy Phys. 10 (2025) 052.
  8. M. Grewal, Y. T. A. Law, and V. Lochab, Spinning thermal static patch correlators (to be published).
  9. Y. T. A. Law and K. Parmentier, Black hole scattering and partition functions, J. High Energy Phys. 10 (2022) 039.
  10. Y. T. A. Law, Characters, quasinormal modes, and quantum de Sitter thermodynamics, Proc. Sci. CORFU2022 (2023) 130 [arXiv:2304.01471].
  11. Y. T. A. Law, companion paper, de Sitter horizon edge partition functions, Phys. Rev. D 113, 086005 (2026).
  12. Y. T. Albert Law and V. Lochab, companion Letter, Horizon edge partition functions in Λ>0 quantum gravity, Phys. Rev. Lett. 136, 151601 (2026).
  13. W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, J. High Energy Phys. 09 (2016) 102.
  14. M. Geiller, Edge modes and corner ambiguities in 3d Chern–Simons theory and gravity, Nucl. Phys. B924, 312 (2017).
  15. A. J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, J. High Energy Phys. 02 (2018) 021.
  16. M. Geiller, Lorentz-diffeomorphism edge modes in 3D gravity, J. High Energy Phys. 02 (2018) 029.
  17. L. Freidel, E. R. Livine, and D. Pranzetti, Gravitational edge modes: From Kac–Moody charges to Poincaré networks, Classical Quantum Gravity 36, 195014 (2019).
  18. T. Takayanagi and K. Tamaoka, Gravity edges modes and Hayward term, J. High Energy Phys. 02 (2020) 167.
  19. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, J. High Energy Phys. 11 (2020) 026.
  20. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part II. Corner metric and Lorentz charges, J. High Energy Phys. 11 (2020) 027.
  21. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part III. Corner simplicity constraints, J. High Energy Phys. 01 (2021) 100.
  22. W. Donnelly, L. Freidel, S. F. Moosavian, and A. J. Speranza, Gravitational edge modes, coadjoint orbits, and hydrodynamics, J. High Energy Phys. 09 (2021) 008.
  23. L. Ciambelli and R. G. Leigh, Isolated surfaces and symmetries of gravity, Phys. Rev. D 104, 046005 (2021).
  24. S. Carrozza and P. A. Hoehn, Edge modes as reference frames and boundary actions from post-selection, J. High Energy Phys. 02 (2022) 172.
  25. L. Ciambelli, R. G. Leigh, and P.-C. Pai, Embeddings and integrable charges for extended corner symmetry, Phys. Rev. Lett. 128 (2022).
  26. S. Carrozza, S. Eccles, and P. A. Hoehn, Edge modes as dynamical frames: Charges from post-selection in generally covariant theories, SciPost Phys. 17, 048 (2024).
  27. L. Ciambelli and R. G. Leigh, Universal corner symmetry and the orbit method for gravity, Nucl. Phys. B986, 116053 (2023).
  28. T. G. Mertens, J. Simón, and G. Wong, A proposal for 3D quantum gravity and its bulk factorization, J. High Energy Phys. 06 (2023) 134.
  29. G. Wong, A note on the bulk interpretation of the quantum extremal surface formula, J. High Energy Phys. 04 (2024) 024.
  30. W. Donnelly, L. Freidel, S. F. Moosavian, and A. J. Speranza, Matrix quantization of gravitational edge modes, J. High Energy Phys. 05 (2023) 163.
  31. K.-S. Lee, A. Sivakumar, and J. Yoon, Gravitational edge mode in N=1 Jackiw-Teitelboim supergravity, J. High Energy Phys. 08 (2024) 011.
  32. A. Blommaert and S. Colin-Ellerin, Gravitons on the edge, J. High Energy Phys. 03 (2025) 116.
  33. J. R. Fliss, A. Frenkel, S. A. Hartnoll, and R. M. Soni, Minimal areas from entangled matrices, SciPost Phys. 18, 171 (2025).
  34. D. N. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B453, 281 (1995).
  35. W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85, 085004 (2012).
  36. W. Donnelly and A. C. Wall, Do gauge fields really contribute negatively to black hole entropy?, Phys. Rev. D 86, 064042 (2012).
  37. C. Eling, Y. Oz, and S. Theisen, Entanglement and thermal entropy of gauge fields, J. High Energy Phys. 11 (2013) 019.
  38. D. Radicevic, Notes on entanglement in Abelian gauge theories, arXiv:1404.1391.
  39. W. Donnelly, Entanglement entropy and non-Abelian gauge symmetry, Classical Quantum Gravity 31, 214003 (2014).
  40. W. Donnelly and A. C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114, 111603 (2015).
  41. K.-W. Huang, Central charge and entangled gauge fields, Phys. Rev. D 92, 025010 (2015).
  42. S. Ghosh, R. M. Soni, and S. P. Trivedi, On the entanglement entropy for gauge theories, J. High Energy Phys. 09 (2015) 069.
  43. L.-Y. Hung and Y. Wan, Revisiting entanglement entropy of lattice gauge theories, J. High Energy Phys. 04 (2015) 122.
  44. S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba, and H. Tasaki, On the definition of entanglement entropy in lattice gauge theories, J. High Energy Phys. 06 (2015) 187.
  45. W. Donnelly and A. C. Wall, Geometric entropy and edge modes of the electromagnetic field, Phys. Rev. D 94, 104053 (2016).
  46. D. Radičević, Entanglement in weakly coupled lattice gauge theories, J. High Energy Phys. 04 (2016) 163.
  47. M. Pretko and T. Senthil, Entanglement entropy of U(1) quantum spin liquids, Phys. Rev. B 94, 125112 (2016).
  48. R. M. Soni and S. P. Trivedi, Aspects of entanglement entropy for gauge theories, J. High Energy Phys. 01 (2016) 136.
  49. F. Zuo, A note on electromagnetic edge modes, arXiv:1601.06910.
  50. R. M. Soni and S. P. Trivedi, Entanglement entropy in (3+1)-d free U(1) gauge theory, J. High Energy Phys. 02 (2017) 101.
  51. C. Delcamp, B. Dittrich, and A. Riello, On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity, J. High Energy Phys. 11 (2016) 102.
  52. A. Agarwal, D. Karabali, and V. P. Nair, Gauge-invariant variables and entanglement entropy, Phys. Rev. D 96, 125008 (2017).
  53. A. Blommaert, T. G. Mertens, H. Verschelde, and V. I. Zakharov, Edge state quantization: Vector fields in Rindler, J. High Energy Phys. 08 (2018) 196.
  54. A. Blommaert, T. G. Mertens, and H. Verschelde, Edge dynamics from the path integral—Maxwell and Yang-Mills, J. High Energy Phys. 11 (2018) 080.
  55. L. Freidel and D. Pranzetti, Electromagnetic duality and central charge, Phys. Rev. D 98, 116008 (2018).
  56. A. Ball, Y. T. A. Law, and G. Wong, Dynamical edge modes and entanglement in Maxwell theory, J. High Energy Phys. 09 (2024) 032.
  57. G. Araujo-Regado, P. A. Höhn, F. Sartini, and B. Tomova, Soft edges: The many links between soft and edge modes, J. High Energy Phys. 07 (2025) 180.
  58. J. S. Dowker, Renyi entropy and CT for p-forms on even spheres, arXiv:1706.04574.
  59. U. Moitra, R. M. Soni, and S. P. Trivedi, Entanglement entropy, relative entropy and duality, J. High Energy Phys. 08 (2019) 059.
  60. J. R. David and J. Mukherjee, Partition functions of p-forms from Harish-Chandra characters, J. High Energy Phys. 09 (2021) 094.
  61. J. Mukherjee, Entanglement entropy and the boundary action of edge modes, J. High Energy Phys. 06 (2024) 113.
  62. J. S. Dowker, Note on entanglement and edge modes, arXiv:2406.15434.
  63. A. Ball and Y. T. A. Law, Dynamical edge modes in p-form gauge theories, J. High Energy Phys. 02 (2025) 182.
  64. D. Kapec, Y. Law, and C. Toldo, Quasinormal corrections to near-extremal black hole thermodynamics, J. High Energy Phys. 06 (2025) 069.
  65. M. Grewal, Y. T. A. Law, and K. Parmentier, Black hole horizon edge partition functions, J. High Energy Phys. 06 (2023) 025.
  66. Y. T. A. Law, A compendium of sphere path integrals, J. High Energy Phys. 12 (2021) 213.
  67. X. Shi and G. J. Turiaci, The phase of the gravitational path integral, J. High Energy Phys. 07 (2025) 047.
  68. V. Ivo, J. Maldacena, and Z. Sun, Physical instabilities and the phase of the Euclidean path integral, arXiv:2504.00920.
  69. H. Nariai, On some static solutions of Einstein’s gravitational field equations in a spherically symmetric case, Sci. Rep. Tohoku Univ. Eighth Ser. 34, 160 (1950).
  70. P. H. Ginsparg and M. J. Perry, Semiclassical perdurance of de Sitter space, Nucl. Phys. B222, 245 (1983).
  71. R. Bousso and S. W. Hawking, (Anti)evaporation of Schwarzschild-de Sitter black holes, Phys. Rev. D 57, 2436 (1998).
  72. M. A. Rubin and C. R. Ordónez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, J. Math. Phys. (N.Y.) 25, 2888 (1984).
  73. D. Anninos, S. A. Hartnoll, and D. M. Hofman, Static patch solipsism: Conformal symmetry of the de Sitter worldline, Classical Quantum Gravity 29, 075002 (2012).
  74. D. Anninos, D. A. Galante, and C. Maneerat, Gravitational observatories, J. High Energy Phys. 12 (2023) 024.
  75. L. Vanzo and S. Zerbini, Asymptotics of quasinormal modes for multihorizon black holes, Phys. Rev. D 70, 044030 (2004).
  76. A. Lopez-Ortega, Electromagnetic quasinormal modes of D-dimensional black holes. II, Gen. Relativ. Gravit. 40, 1379 (2008).
  77. A. Lopez-Ortega, The Dirac equation in D-dimensional spherically symmetric spacetimes, Lat. Am. J. Phys. Educ. 3, 578 (2009).
  78. J. Venâncio and C. Batista, Spin-2 quasinormal modes in generalized Nariai spacetimes, Phys. Rev. D 101, 084037 (2020).
  79. H. Kodama and A. Ishibashi, A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions, Prog. Theor. Phys. 110, 701 (2003).
  80. D. Borthwick, Spectral Theory of Infinite-Area Hyperbolic Surfaces, Progress in Mathematics Vol. 318 (Springer International Publishing, New York, 2016).
  81. Z. Sun, Higher spin de Sitter quasinormal modes, J. High Energy Phys. 11 (2021) 025.
  82. G. S. Ng and A. Strominger, State/operator correspondence in higher-spin dS/CFT, Classical Quantum Gravity 30, 104002 (2013).
  83. D. L. Jafferis, A. Lupsasca, V. Lysov, G. S. Ng, and A. Strominger, Quasinormal quantization in de Sitter spacetime, J. High Energy Phys. 01 (2015) 004.
  84. M. R. Tanhayi, Quasinormal modes in de Sitter space: Plane wave method, Phys. Rev. D 90, 064010 (2014).
  85. G. W. Gibbons and M. J. Perry, Quantizing gravitational instantons, Nucl. Phys. B146, 90 (1978).
  86. S. M. Christensen and M. J. Duff, Quantizing gravity with a cosmological constant, Nucl. Phys. B170, 480 (1980).
  87. E. S. Fradkin and A. A. Tseytlin, One loop effective potential in gauged O(4) supergravity, Nucl. Phys. B234, 472 (1984).
  88. B. Allen, Phase transitions in de Sitter space, Nucl. Phys. B226, 228 (1983).
  89. T. R. Taylor and G. Veneziano, Quantum gravity at large distances and the cosmological constant, Nucl. Phys. B345, 210 (1990).
  90. P. A. Griffin and D. A. Kosower, Curved spacetime one-loop gravity in a physical gauge, Phys. Lett. B 233, 295 (1989).
  91. P. O. Mazur and E. Mottola, Absence of phase in the sum over spheres, Technical Report No. LA-UR-89-2118, 1989.
  92. D. V. Vassilevich, One loop quantum gravity on de Sitter space, Int. J. Mod. Phys. A 08, 1637 (1993).
  93. M. S. Volkov and A. Wipf, Black hole pair creation in de Sitter space: A complete one loop analysis, Nucl. Phys. B582, 313 (2000).
  94. J. Polchinski, The phase of the sum over spheres, Phys. Lett. B 219, 251 (1989).
  95. D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rep. 388, 279 (2003).
  96. O. Babelon and C. M. Viallet, The geometrical Interpretation of the Faddeev-Popov determinant, Phys. Lett. 85B, 246 (1979).
  97. P. O. Mazur and E. Mottola, The gravitational measure, solution of the conformal factor problem and stability of the ground state of quantum gravity, Nucl. Phys. B341, 187 (1990).
  98. Z. Bern, E. Mottola, and S. K. Blau, General covariance of the path integral for quantum gravity, Phys. Rev. D 43, 1212 (1991).
  99. M. R. Gaberdiel, D. Grumiller, and D. Vassilevich, Graviton 1-loop partition function for 3-dimensional massive gravity, J. High Energy Phys. 11 (2010) 094.
  100. M. R. Gaberdiel, R. Gopakumar, and A. Saha, Quantum W-symmetry in AdS3, J. High Energy Phys. 02 (2011) 004.
  101. M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Jpn. 14, 333 (1962).
  102. G. W. Gibbons, S. W. Hawking, and M. J. Perry, Path integrals and the indefiniteness of the gravitational action, Nucl. Phys. B138, 141 (1978).
  103. J. Maldacena, Real observers solving imaginary problems, arXiv:2412.14014.
  104. D. Anninos, C. Baracco, S. Brian, and F. Denef, Features of the partition function of a Λ>0 universe, J. High Energy Phys. 01 (2026) 141.
  105. Y. Kluth and D. F. Litim, Heat kernel coefficients on the sphere in any dimension, Eur. Phys. J. C 80, 269 (2020).
  106. Z. Sun, A note on the representations of so (1, d+1), Rev. Math. Phys. 37, 2430007 (2025).
  107. G. Goon, K. Hinterbichler, and M. Trodden, Symmetries for Galileons and DBI scalars on curved space, J. Cosmol. Astropart. Phys. 07 (2011) 017.
  108. G. Goon, K. Hinterbichler, and M. Trodden, A new class of effective field theories from embedded branes, Phys. Rev. Lett. 106, 231102 (2011).
  109. C. Burrage, C. de Rham, and L. Heisenberg, de Sitter Galileon, J. Cosmol. Astropart. Phys. 05 (2011) 025.
  110. T. E. Clark, S. T. Love, M. Nitta, and T. ter Veldhuis, AdSd+1→AdSd, J. Math. Phys. (N.Y.) 46, 102304 (2005).
  111. T. E. Clark, S. T. Love, M. Nitta, T. ter Veldhuis, and C. Xiong, Gravitating p-branes, Phys. Rev. D 75, 065028 (2007).
  112. J. Bonifacio, K. Hinterbichler, A. Joyce, and R. A. Rosen, Shift symmetries in (anti) de Sitter space, J. High Energy Phys. 02 (2019) 178.
  113. J. Bonifacio, K. Hinterbichler, L. A. Johnson, and A. Joyce, Shift-symmetric spin-1 theories, J. High Energy Phys. 09 (2019) 029.
  114. A. Ball and L. Ciambelli, Dynamical edge modes in Yang-Mills theory, SciPost Phys. 20, 013 (2026).
  115. A. Higuchi, Symmetric tensor spherical harmonics on the N sphere and their application to the de Sitter group SO(N, 1), J. Math. Phys. (N.Y.) 28, 1553 (1987); 43, 6385(E) (2002).

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Phys. Rev. D 113, 086006– Published 13 April, 2026
  • Received 5 October 2025
  • Accepted 30 January 2026
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