Abstract
We consider Einstein gravity with positive cosmological constant coupled with higher spin interactions and calculate Euclidean path integral perturbatively. We confine ourselves to the static patch of the 3 dimensional de Sitter space. This geometry, when Euclideanlized is equivalent to 3-sphere. However, infinite number of topological quotients of this space by discrete subgroups of the isometry group are valid Euclidean saddles as well. The case of pure Einstein gravity is known to give a diverging answer, when all saddles are included as contribution to the thermal partition functions (also interpreted as the Hartle Hawking state in the cosmological scenario). We show how higher spins, described by metric-Fronsdal fields help making the partition function finite. We find a curious fact that this convergence is not achieved by mere inclusion of spin-3, but requires spin-4 interactions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.B. Hartle and S.W. Hawking, Wave Function of the Universe, Phys. Rev. D 28 (1983) 2960 [INSPIRE].
A. Castro, N. Lashkari and A. Maloney, A de Sitter Farey Tail, Phys. Rev. D 83 (2011) 124027 [arXiv:1103.4620] [INSPIRE].
A. Castro, N. Lashkari and A. Maloney, Quantum Topologically Massive Gravity in de Sitter Space, JHEP 08 (2011) 040 [arXiv:1105.4733] [INSPIRE].
W.P. Thurston, Three-Dimensional Geometry and Topology, Vol. 1, Ed. Silvio Levy, Princeton Univeristy Press, Princeton, New Jersey U.S.A. (1997).
R. Basu and S.K. Paul, Consistent 3D Quantum Gravity on Lens Spaces, Phys. Rev. D 85 (2012) 023520 [arXiv:1109.0793] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar and A. Saha, Quantum W -symmetry in AdS 3, JHEP 02 (2011) 004 [arXiv:1009.6087] [INSPIRE].
L.C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles and the semiclassical approximation, Commun. Math. Phys. 147 (1992) 563 [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
S.K. Hansen and T. Takata, Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras, and their asymptotic expansions, J. Knot Theory Ramifications 13 (2004) 617.
C. Fronsdal, Massless Fields with Integer Spin, Phys. Rev. D 18 (1978) 3624 [INSPIRE].
M.R. Gaberdiel, D. Grumiller and D. Vassilevich, Graviton 1-loop partition function for 3-dimensional massive gravity, JHEP 11 (2010) 094 [arXiv:1007.5189] [INSPIRE].
J.R. David, M.R. Gaberdiel and R. Gopakumar, The Heat Kernel on AdS 3 and its Applications, JHEP 04 (2010) 125 [arXiv:0911.5085] [INSPIRE].
R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
D. Anninos, T. Hartman and A. Strominger, Higher Spin Realization of the dS/CFT Correspondence, arXiv:1108.5735 [INSPIRE].
D. Anninos, F. Denef, G. Konstantinidis and E. Shaghoulian, Higher Spin de Sitter Holography from Functional Determinants, JHEP 02 (2014) 007 [arXiv:1305.6321] [INSPIRE].
S. Giombi and X. Yin, On Higher Spin Gauge Theory and the Critical O(N) Model, Phys. Rev. D 85 (2012) 086005 [arXiv:1105.4011] [INSPIRE].
G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986) 2693 [INSPIRE].
S. Giombi, I.R. Klebanov, S.S. Pufu, B.R. Safdi and G. Tarnopolsky, AdS Description of Induced Higher-Spin Gauge Theory, JHEP 10 (2013) 016 [arXiv:1306.5242] [INSPIRE].
D.V. Vassilevich, One loop quantum gravity on de Sitter space, Int. J. Mod. Phys. A 8 (1993) 1637 [INSPIRE].
D.V. Vassilevich, Gauge invariance in one loop quantum cosmology, gr-qc/9509021 [INSPIRE].
S. Bhattacharjee and P. Majumdar, Gauge-free Coleman-Weinberg Potential, Eur. Phys. J. C 73 (2013) 2348 [arXiv:1302.7272] [INSPIRE].
M.N. Chernodub, L. Faddeev and A.J. Niemi, Non-abelian Supercurrents and Electroweak Theory, JHEP 12 (2008) 014 [arXiv:0804.1544] [INSPIRE].
S. Bhattacharjee and P. Majumdar, Gauge-free electroweak theory: Radiative effects, Phys. Rev. D 83 (2011) 085019 [arXiv:1006.1712] [INSPIRE].
J.S. Dowker, Spherical universe topology and the Casimir effect, Class. Quant. Grav. 21 (2004) 4247 [hep-th/0404093] [INSPIRE].
A. Strominger, Inflation and the dS/CFT correspondence, JHEP 11 (2001) 049 [hep-th/0110087] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
A. Castro and A. Maloney, The Wave Function of Quantum de Sitter, JHEP 11 (2012) 096 [arXiv:1209.5757] [INSPIRE].
C. Krishnan, A. Raju, S. Roy and S. Thakur, Higher Spin Cosmology, Phys. Rev. D 89 (2014) 045007 [arXiv:1308.6741] [INSPIRE].
E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
S. Axelrod, S. Della Pietra and E. Witten, Geometric Quantization of Chern-Simons Gauge Theory, J. Diff. Geom. 33 (1991) 787 [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].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
S.K. Hansen, Reshetikhin-Turaev invariants of Seifert 3-manifolds and a rational surgery formula, Algebr. Geom. Topol. 1 (2001) 627.
J.M.F. Labastida and A.V. Ramallo, Operator Formalism for Chern-Simons Theories, Phys. Lett. B 227 (1989) 92 [INSPIRE].
H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Spin-3 Gravity in Three-Dimensional Flat Space, Phys. Rev. Lett. 111 (2013) 121603 [arXiv:1307.4768] [INSPIRE].
A. Bagchi, S. Lal, A. Saha and B. Sahoo, One loop partition function for Topologically Massive Higher Spin Gravity, JHEP 12 (2011) 068 [arXiv:1107.2063] [INSPIRE].
D. Anninos, F. Denef and D. Harlow, Wave function of Vasiliev’s universe: A few slices thereof, Phys. Rev. D 88 (2013) 084049 [arXiv:1207.5517] [INSPIRE].
R. Lawrence and L. Rozansky, Witten-Reshetikhin-Turaev Invariants of Seifert Manifolds, Commun. Math. Phys. 205 (1999) 287.
S. Carlip, The Sum over topologies in three-dimensional Euclidean quantum gravity, Class. Quant. Grav. 10 (1993) 207 [hep-th/9206103] [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
S. Lal and B. Sahoo, Holographic Renormalisation for the Spin-3 Theory and the (A)dS3/CFT2 correspondence, JHEP 01 (2013) 004 [arXiv:1209.4804] [INSPIRE].
C. Krishnan and S. Roy, Higher Spin Resolution of a Toy Big Bang, Phys. Rev. D 88 (2013) 044049 [arXiv:1305.1277] [INSPIRE].
A. Karch and C.F. Uhlemann, Higher-spin realization of a de Sitter static patch/cut-off CFT correspondence, Phys. Rev. D 88 (2013) 046001 [arXiv:1306.0582] [INSPIRE].
H.A. Gonzalez, J. Matulich, M. Pino and R. Troncoso, Asymptotically flat spacetimes in three-dimensional higher spin gravity, JHEP 09 (2013) 016 [arXiv:1307.5651] [INSPIRE].
H.A. Gonzalez and M. Pino, Boundary dynamics of asymptotically flat 3D gravity coupled to higher spin fields, JHEP 05 (2014) 127 [arXiv:1403.4898] [INSPIRE].
A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, Conical Defects in Higher Spin Theories, JHEP 02 (2012) 096 [arXiv:1111.3381] [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: 1507.04757
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Basu, R. Higher spin de Sitter quantum gravity. J. High Energ. Phys. 2015, 151 (2015). https://doi.org/10.1007/JHEP10(2015)151
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2015)151