Abstract
We explore thermodynamic contributions to the three-dimensional de Sitter horizon originating from metric and Chern-Simons gauge field fluctuations. In Euclidean signature these are computed by the partition function of gravity coupled to matter semi-classically expanded about the round three-sphere saddle. We investigate a corresponding Lorentzian picture — drawing inspiration from the topological entanglement entropy literature — in the form of an edge-mode theory residing at the de Sitter horizon. We extend the discussion to three-dimensional gravity with positive cosmological constant, viewed (semi-classically) as a complexified Chern-Simons theory. The putative gravitational edge-mode theory is a complexified version of the chiral Wess-Zumino-Witten model associated to the edge-modes of ordinary Chern-Simons theory. We introduce and solve a family of complexified Abelian Chern-Simons theories as a way to elucidate some of the more salient features of the gravitational edge-mode theories. We comment on the relation to the AdS4/CFT3 correspondence.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Anninos, D.A. Galante and D.M. Hofman, de Sitter horizons & holographic liquids, JHEP 07 (2019) 038 [arXiv:1811.08153] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
R. Bousso, Holography in general space-times, JHEP 06 (1999) 028 [hep-th/9906022] [INSPIRE].
T. Banks, Lectures on Holographic Space Time, arXiv:1311.0755 [INSPIRE].
M. Alishahiha, A. Karch, E. Silverstein and D. Tong, The dS/dS correspondence, AIP Conf. Proc. 743 (2004) 393 [hep-th/0407125] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, The Λ-BMS4 charge algebra, JHEP 10 (2020) 205 [arXiv:2004.10769] [INSPIRE].
L. Freidel, C. Goeller and E.R. Livine, The Quantum Gravity Disk: Discrete Current Algebra, arXiv:2103.13171 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
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, arXiv:2009.12464 [INSPIRE].
Y.T.A. Law, A Compendium of Sphere Path Integrals, arXiv:2012.06345 [INSPIRE].
J.R. David and J. Mukherjee, Partition functions of p-forms from Harish-Chandra characters, arXiv:2105.03662 [INSPIRE].
S. Carlip, The Sum over topologies in three-dimensional Euclidean quantum gravity, Class. Quant. Grav. 10 (1993) 207 [hep-th/9206103] [INSPIRE].
E. Guadagnini and P. Tomassini, Sum over the geometries of three manifolds, Phys. Lett. B 336 (1994) 330 [INSPIRE].
A. Castro, N. Lashkari and A. Maloney, A de Sitter Farey Tail, Phys. Rev. D 83 (2011) 124027 [arXiv:1103.4620] [INSPIRE].
M.-I. Park, Statistical entropy of three-dimensional Kerr-de Sitter space, Phys. Lett. B 440 (1998) 275 [hep-th/9806119] [INSPIRE].
J.M. Maldacena and A. Strominger, Statistical entropy of de Sitter space, JHEP 02 (1998) 014 [gr-qc/9801096] [INSPIRE].
M. Bañados, T. Brotz and M.E. Ortiz, Quantum three-dimensional de Sitter space, Phys. Rev. D 59 (1999) 046002 [hep-th/9807216] [INSPIRE].
T.R. Govindarajan, R.K. Kaul and V. Suneeta, Quantum gravity on dS3, Class. Quant. Grav. 19 (2002) 4195 [hep-th/0203219] [INSPIRE].
X. Dong, B. Horn, E. Silverstein and G. Torroba, Micromanaging de Sitter holography, Class. Quant. Grav. 27 (2010) 245020 [arXiv:1005.5403] [INSPIRE].
J. Polchinski, The Phase of the Sum Over Spheres, Phys. Lett. B 219 (1989) 251 [INSPIRE].
D. Anninos, T. Bautista and B. Mühlmann, The two-sphere partition function in two-dimensional quantum gravity, JHEP 09 (2021) 116 [arXiv:2106.01665] [INSPIRE].
B. Mühlmann, The two-sphere partition function in two-dimensional quantum gravity at fixed area, arXiv:2106.04532 [INSPIRE].
A. Achucarro and P.K. Townsend, A Chern-Simons Action for Three-Dimensional anti-de Sitter Supergravity Theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].
E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
E. Witten, Quantization of Chern-Simons Gauge Theory With Complex Gauge Group, Commun. Math. Phys. 137 (1991) 29 [INSPIRE].
E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE].
T. Dimofte, S. Gukov, J. Lenells and D. Zagier, Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group, Commun. Num. Theor. Phys. 3 (2009) 363 [arXiv:0903.2472] [INSPIRE].
T. Dimofte, Perturbative and nonperturbative aspects of complex Chern-Simons theory, J. Phys. A 50 (2017) 443009 [arXiv:1608.02961] [INSPIRE].
C. Vafa, Fractional Quantum Hall Effect and M-theory, arXiv:1511.03372 [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
P. Fendley, M.P.A. Fisher and C. Nayak, Topological entanglement entropy from the holographic partition function, J. Statist. Phys. 126 (2007) 1111 [cond-mat/0609072] [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and the holographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE].
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].
H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
S. Ghosh, R.M. Soni and S.P. Trivedi, On The Entanglement Entropy For Gauge Theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].
J. Lin and D. Radičević, Comments on defining entanglement entropy, Nucl. Phys. B 958 (2020) 115118 [arXiv:1808.05939] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, Graduate Texts in Contemporary Physics, New York U.S.A. (1997) [DOI] [INSPIRE].
H. Ooguri and C. Vafa, World sheet derivation of a large N duality, Nucl. Phys. B 641 (2002) 3 [hep-th/0205297] [INSPIRE].
D. Anninos and B. Mühlmann, Notes on matrix models (matrix musings), J. Stat. Mech. 2008 (2020) 083109 [arXiv:2004.01171] [INSPIRE].
G.V. Dunne, R. Jackiw and C.A. Trugenberger, Chern-Simons Theory in the Schrödinger Representation, Annals Phys. 194 (1989) 197 [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
D. Tong, Lectures on the Quantum Hall Effect, (2016) [arXiv:1606.06687] [INSPIRE].
S. Giombi, I.R. Klebanov and G. Tarnopolsky, Conformal QEDd, F-Theorem and the ϵ Expansion, J. Phys. A 49 (2016) 135403 [arXiv:1508.06354] [INSPIRE].
R. Floreanini and R. Jackiw, Selfdual Fields as Charge Density Solitons, Phys. Rev. Lett. 59 (1987) 1873 [INSPIRE].
X.G. Wen, Chiral Luttinger Liquid and the Edge Excitations in the Fractional Quantum Hall States, Phys. Rev. B 41 (1990) 12838 [INSPIRE].
G. ‘t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
D. Das and S. Datta, Universal features of left-right entanglement entropy, Phys. Rev. Lett. 115 (2015) 131602 [arXiv:1504.02475] [INSPIRE].
M. Geiller, Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity, Nucl. Phys. B 924 (2017) 312 [arXiv:1703.04748] [INSPIRE].
G. Wong, A note on entanglement edge modes in Chern Simons theory, JHEP 08 (2018) 020 [arXiv:1706.04666] [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
A.N. Schellekens, Introduction to conformal field theory, Fortsch. Phys. 44 (1996) 605 [INSPIRE].
S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids, JHEP 05 (2008) 016 [arXiv:0802.3231] [INSPIRE].
E. Sagi and R.A. Santos, Supersymmetry in the Fractional Quantum Hall Regime, Phys. Rev. B 95 (2017) 205144 [arXiv:1610.07627] [INSPIRE].
K. Pilch, P. van Nieuwenhuizen and M.F. Sohnius, de Sitter Superalgebras and Supergravity, Commun. Math. Phys. 98 (1985) 105 [INSPIRE].
T. Anous, D.Z. Freedman and A. Maloney, de Sitter Supersymmetry Revisited, JHEP 07 (2014) 119 [arXiv:1403.5038] [INSPIRE].
Danny Birmingham, Matthias Blau, Mark Rakowski and George Thompson, Topological field theory, Phys. Rept. 209 (1991) 129.
A.S. Cattaneo, P. Cotta-Ramusino, J. Fröhlich and M. Martellini, Topological BF theories in three-dimensions and four-dimensions, J. Math. Phys. 36 (1995) 6137 [hep-th/9505027] [INSPIRE].
G.V. Dunne and R. Jackiw, ‘Peierls substitution’ and Chern-Simons quantum mechanics, Nucl. Phys. B Proc. Suppl. 33 (1993) 114 [hep-th/9204057] [INSPIRE].
V. de Alfaro, S. Fubini and G. Furlan, Conformal Invariance in Quantum Mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].
T. Anous and J. Skulte, An invitation to the principal series, SciPost Phys. 9 (2020) 028 [arXiv:2007.04975] [INSPIRE].
K. Andrzejewski and J. Gonera, On the geometry of conformal mechanics, arXiv:1108.1299 [INSPIRE].
K. Andrzejewski, Quantum conformal mechanics emerging from unitary representations of SL(2,ℝ), Annals Phys. 367 (2016) 227 [arXiv:1506.05596] [INSPIRE].
S. Carlip, The Statistical mechanics of the (2 + 1)-dimensional black hole, Phys. Rev. D 51 (1995) 632 [gr-qc/9409052] [INSPIRE].
M. Bañados and A. Gomberoff, Black hole entropy in the Chern-Simons formulation of (2+1) gravity, Phys. Rev. D 55 (1997) 6162 [gr-qc/9611044] [INSPIRE].
G. Arcioni, M. Blau and M. O’Loughlin, On the boundary dynamics of Chern-Simons gravity, JHEP 01 (2003) 067 [hep-th/0210089] [INSPIRE].
A. Castro, P. Sabella-Garnier and C. Zukowski, Gravitational Wilson Lines in 3D de Sitter, JHEP 07 (2020) 202 [arXiv:2001.09998] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
L. McGough and H. Verlinde, Bekenstein-Hawking Entropy as Topological Entanglement Entropy, JHEP 11 (2013) 208 [arXiv:1308.2342] [INSPIRE].
M.A. Rubin and C.R. Ordóñez, Symmetric Tensor Eigen Spectrum of the Laplacian on n Spheres, J. Math. Phys. 26 (1985) 65 [INSPIRE].
J.R. Fliss, X. Wen, O. Parrikar, C.-T. Hsieh, B. Han, T.L. Hughes et al., Interface Contributions to Topological Entanglement in Abelian Chern-Simons Theory, JHEP 09 (2017) 056 [arXiv:1705.09611] [INSPIRE].
L. Rozansky, Witten’s invariant of three-dimensional manifolds: Loop expansion and surgery calculus, hep-th/9401060 [INSPIRE].
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: 2106.13832
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
Anninos, D., Harris, E. Three-dimensional de Sitter horizon thermodynamics. J. High Energ. Phys. 2021, 91 (2021). https://doi.org/10.1007/JHEP10(2021)091
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2021)091