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Higher spin de Sitter Hilbert space

A preprint version of the article is available at arXiv.


We propose a complete microscopic definition of the Hilbert space of minimal higher spin de Sitter quantum gravity and its Hartle-Hawking vacuum state. The funda- mental degrees of freedom are 2N bosonic fields living on the future conformal boundary, where N is proportional to the de Sitter horizon entropy. The vacuum state is normalizable. The model agrees in perturbation theory with expectations from a previously proposed dS- CFT description in terms of a fermionic Sp(N) model, but it goes beyond this, both in its conceptual scope and in its computational power. In particular it resolves the apparent pathologies affecting the Sp(N) model, and it provides an exact formula for late time vac- uum correlation functions. We illustrate this by computing probabilities for arbitrarily large field excursions, and by giving fully explicit examples of vacuum 3- and 4-point functions. We discuss bulk reconstruction and show the perturbative bulk QFT canonical commuta- tions relations can be reproduced from the fundamental operator algebra, but only up to a minimal error term ∼ e−O(N ), and only if the operators are coarse grained in such a way that the number of accessible “pixels” is less than O(N ). Independent of this, we show that upon gauging the higher spin symmetry group, one is left with 2N physical degrees of freedom, and that all gauge invariant quantities can be computed by a 2N × 2N matrix model. This suggests a concrete realization of the idea of cosmological complementarity.


  1. J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [Adv. Theor. Math. Phys.2 (1998) 231] [hep-th/9711200] [INSPIRE].

  2. A.D. Linde, D.A. Linde and A. Mezhlumian, Do we live in the center of the world?, Phys. Lett.B 345 (1995) 203 [hep-th/9411111] [INSPIRE].

    ADS  Google Scholar 

  3. L. Dyson, M. Kleban and L. Susskind, Disturbing implications of a cosmological constant, JHEP10 (2002) 011 [hep-th/0208013] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  4. D.N. Page, Is our universe likely to decay within 20 billion years?, Phys. Rev.D 78 (2008) 063535 [hep-th/0610079] [INSPIRE].

  5. R. Bousso and B. Freivogel, A paradox in the global description of the multiverse, JHEP06 (2007) 018 [hep-th/0610132] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  6. A. De Simone, A.H. Guth, A.D. Linde, M. Noorbala, M.P. Salem and A. Vilenkin, Boltzmann brains and the scale-factor cutoff measure of the multiverse, Phys. Rev.D 82 (2010) 063520 [arXiv:0808.3778] [INSPIRE].

  7. R. Bousso, B. Freivogel, S. Leichenauer and V. Rosenhaus, Eternal inflation predicts that time will end, Phys. Rev.D 83 (2011) 023525 [arXiv:1009.4698] [INSPIRE].

  8. A. Borde, A.H. Guth and A. Vilenkin, Inflationary space-times are incompletein past directions, Phys. Rev. Lett.90 (2003) 151301 [gr-qc/0110012] [INSPIRE].

  9. A.H. Guth, Eternal inflation and its implications, J. Phys.A 40 (2007) 6811 [hep-th/0702178] [INSPIRE].

  10. M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys.79 (2007) 733 [hep-th/0610102] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  11. F. Denef, Les Houches Lectures on Constructing String Vacua, Les Houches87 (2008) 483 [arXiv:0803.1194] [INSPIRE].

    Google Scholar 

  12. F. Denef and M.R. Douglas, Computational complexity of the landscape. I., Annals Phys.322 (2007) 1096 [hep-th/0602072] [INSPIRE].

  13. T. Banks, TASI Lectures on Holographic Space-Time, SUSY and Gravitational Effective Field Theory, arXiv:1007.4001 [INSPIRE].

  14. S. Sethi, Supersymmetry Breaking by Fluxes, JHEP10 (2018) 022 [arXiv:1709.03554] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  15. S.W. Hawking, The Cosmological Constant Is Probably Zero, Phys. Lett.134B (1984) 403 [INSPIRE].

    ADS  Google Scholar 

  16. M.J. Duff, The Cosmological Constant Is Possibly Zero, but the Proof Is Probably Wrong, Phys. Lett.B 226 (1989) 36 [INSPIRE].

  17. E. Silverstein, (A)dS backgrounds from asymmetric orientifolds, Clay Math. Proc.1 (2002) 179 [hep-th/0106209] [INSPIRE].

  18. R. Bousso and J. Polchinski, Quantization of four form fluxes and dynamical neutralization of the cosmological constant, JHEP06 (2000) 006 [hep-th/0004134] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  19. S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, de Sitter vacua in string theory, Phys. Rev.D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].

  20. X. Dong, B. Horn, E. Silverstein and G. Torroba, Micromanaging de Sitter holography, Class. Quant. Grav.27 (2010) 245020 [arXiv:1005.5403] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  21. G.W. Gibbons and S.W. Hawking, Cosmological Event Horizons, Thermodynamics and Particle Creation, Phys. Rev.D 15 (1977) 2738 [INSPIRE].

  22. T. Banks, W. Fischler and S. Paban, Recurrent nightmares? Measurement theory in de Sitter space, JHEP12 (2002) 062 [hep-th/0210160] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  23. T. Banks, Some thoughts on the quantum theory of stable de Sitter space, hep-th/0503066 [INSPIRE].

  24. N. Goheer, M. Kleban and L. Susskind, The trouble with de Sitter space, JHEP07 (2003) 056 [hep-th/0212209] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  25. M.K. Parikh and E.P. Verlinde, de Sitter holography with a finite number of states, JHEP01 (2005) 054 [hep-th/0410227] [INSPIRE].

  26. M. Alishahiha, A. Karch, E. Silverstein and D. Tong, The dS/dS correspondence, AIP Conf. Proc.743 (2004) 393 [hep-th/0407125] [INSPIRE].

    ADS  Google Scholar 

  27. D. Anninos, S.A. Hartnoll and D.M. Hofman, Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline, Class. Quant. Grav.29 (2012) 075002 [arXiv:1109.4942] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  28. D. Anninos and D.M. Hofman, Infrared Realization of dS2 in AdS2 , Class. Quant. Grav.35 (2018) 085003 [arXiv:1703.04622] [INSPIRE].

  29. E.P. Verlinde, Emergent Gravity and the Dark Universe, SciPost Phys.2 (2017) 016 [arXiv:1611.02269] [INSPIRE].

    ADS  Google Scholar 

  30. Y. Neiman, Towards causal patch physics in dS/CFT, EPJ Web Conf.168 (2018) 01007 [arXiv:1710.05682] [INSPIRE].

    Google Scholar 

  31. B. Freivogel and M. Kleban, A Conformal Field Theory for Eternal Inflation, JHEP12 (2009) 019 [arXiv:0903.2048] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  32. L. Susskind, The Census taker’s hat, arXiv:0710.1129 [INSPIRE].

  33. J. Garriga and A. Vilenkin, Holographic Multiverse, JCAP01 (2009) 021 [arXiv:0809.4257] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  34. B. Freivogel, Y. Sekino, L. Susskind and C.-P. Yeh, A holographic framework for eternal inflation, Phys. Rev.D 74 (2006) 086003 [hep-th/0606204] [INSPIRE].

  35. J. Maltz, de Sitter Harmonies: Cosmological Spacetimes as Resonances, Phys. Rev.D 95 (2017) 066006 [arXiv:1611.03491] [INSPIRE].

  36. A. Strominger, The dS/CFT correspondence, JHEP10 (2001) 034 [hep-th/0106113] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  37. E. Witten, Quantum gravity in de Sitter space, hep-th/0106109 [INSPIRE].

  38. J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP05 (2003) 013 [astro-ph/0210603] [INSPIRE].

  39. M. Spradlin, A. Strominger and A. Volovich, Les Houches lectures on de Sitter space, hep-th/0110007 [INSPIRE].

  40. D. Anninos, de Sitter Musings, Int. J. Mod. Phys.A 27 (2012) 1230013 [arXiv:1205.3855] [INSPIRE].

  41. R. Bousso, TASI Lectures on the Cosmological Constant, Gen. Rel. Grav.40 (2008) 607 [arXiv:0708.4231] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  42. D. Anninos, T. Hartman and A. Strominger, Higher Spin Realization of the dS/CFT Correspondence, Class. Quant. Grav.34 (2017) 015009 [arXiv:1108.5735] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  43. M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions, Phys. Lett.B 243 (1990) 378 [INSPIRE].

  44. M.A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett.B 567 (2003) 139 [hep-th/0304049] [INSPIRE].

  45. S.R. Das and A. Jevicki, Large N collective fields and holography, Phys. Rev.D 68 (2003) 044011 [hep-th/0304093] [INSPIRE].

  46. M.R. Douglas, L. Mazzucato and S.S. Razamat, Holographic dual of free field theory, Phys. Rev.D 83 (2011) 071701 [arXiv:1011.4926] [INSPIRE].

  47. I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett.B 550 (2002) 213 [hep-th/0210114] [INSPIRE].

  48. S. Giombi and X. Yin, Higher Spin Gauge Theory and Holography: The Three-Point Functions, JHEP09 (2010) 115 [arXiv:0912.3462] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  49. 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].

  50. S. Banerjee et al., Topology of Future Infinity in dS/CFT, JHEP11 (2013) 026 [arXiv:1306.6629] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  51. S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for conformal algebra. Vacuum expectation values and operator products, Lett. Nuovo Cim.4S2 (1972) 115 [INSPIRE].

  52. D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP04 (2014) 146 [arXiv:1204.3894] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  53. A.E. Lipstein and L. Mason, Amplitudes of 3d Yang-Mills Theory, JHEP01 (2013) 009 [arXiv:1207.6176] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  54. B.S. DeWitt, Quantum Theory of Gravity. 1. The Canonical Theory, Phys. Rev.160 (1967) 1113 [INSPIRE].

  55. A. Higuchi, Quantum linearization instabilities of de Sitter space-time. 1, Class. Quant. Grav.8 (1991) 1961 [INSPIRE].

  56. A. Higuchi, Quantum linearization instabilities of de Sitter space-time. 2, Class. Quant. Grav.8 (1991) 1983 [INSPIRE].

  57. M.G. Eastwood, Higher symmetries of the Laplacian, Annals Math.161 (2005) 1645 [hep-th/0206233] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  58. A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].

  59. E. Joung and K. Mkrtchyan, Notes on higher-spin algebras: minimal representations and structure constants, JHEP05 (2014) 103 [arXiv:1401.7977] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  60. A.Y. Segal, Conformal higher spin theory, Nucl. Phys.B 664 (2003) 59 [hep-th/0207212] [INSPIRE].

  61. S.B. Giddings and D. Marolf, A global picture of quantum de Sitter space, Phys. Rev.D 76 (2007) 064023 [arXiv:0705.1178] [INSPIRE].

  62. D. Marolf and I.A. Morrison, Group Averaging for de Sitter free fields, Class. Quant. Grav.26 (2009) 235003 [arXiv:0810.5163] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  63. C. Fronsdal, Massless Fields with Integer Spin, Phys. Rev.D 18 (1978) 3624 [INSPIRE].

  64. D. Anninos and F. Denef, Cosmic Clustering, JHEP06 (2016) 181 [arXiv:1111.6061] [INSPIRE].

    ADS  Google Scholar 

  65. F. Denef, TASI lectures on complex structures, arXiv:1104.0254 [INSPIRE].

  66. E. Shaghoulian, FRW cosmologies and hyperscaling-violating geometries: higher curvature corrections, ultrametricity, Q-space/QFT duality and a little string theory, JHEP03 (2014) 011 [arXiv:1308.1095] [INSPIRE].

    ADS  Google Scholar 

  67. D.A. Roberts and D. Stanford, On memory in exponentially expanding spaces, JHEP06 (2013) 042 [arXiv:1210.5238] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  68. E.S. Fradkin and M.A. Vasiliev, Cubic Interaction in Extended Theories of Massless Higher Spin Fields, Nucl. Phys.B 291 (1987) 141 [INSPIRE].

  69. E.S. Fradkin and M.A. Vasiliev, On the Gravitational Interaction of Massless Higher Spin Fields, Phys. Lett.B 189 (1987) 89 [INSPIRE].

  70. M.A. Vasiliev, Cubic Vertices for Symmetric Higher-Spin Gauge Fields in (A)dSd , Nucl. Phys.B 862 (2012) 341 [arXiv:1108.5921] [INSPIRE].

  71. M.A. Vasiliev, Higher spin gauge theories: Star product and AdS space, hep-th/9910096 [INSPIRE].

  72. C. Iazeolla, E. Sezgin and P. Sundell, Real forms of complex higher spin field equations and new exact solutions, Nucl. Phys.B 791 (2008) 231 [arXiv:0706.2983] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  73. J.B. Hartle and S.W. Hawking, Wave Function of the Universe, Phys. Rev.D 28 (1983) 2960 [INSPIRE].

  74. J.B. Hartle, S.W. Hawking and T. Hertog, No-Boundary Measure of the Universe, Phys. Rev. Lett.100 (2008) 201301 [arXiv:0711.4630] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  75. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett.B 428 (1998) 105 [hep-th/9802109] [INSPIRE].

  76. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  77. J. Dixmier, Representations integrables du groupe de de Sitter, Bull. Soc. Math. Fr.89 (1961) 9.

    MathSciNet  MATH  Google Scholar 

  78. V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, Lect. Notes Phys.63 (1977) 1 [INSPIRE].

    MATH  Google Scholar 

  79. E. Joung, J. Mourad and R. Parentani, Group theoretical approach to quantum fields in de Sitter space. I. The principle series, JHEP08 (2006) 082 [hep-th/0606119] [INSPIRE].

    MathSciNet  Google Scholar 

  80. E. Joung, J. Mourad and R. Parentani, Group theoretical approach to quantum fields in de Sitter space. II. The complementary and discrete series, JHEP09 (2007) 030 [arXiv:0707.2907] [INSPIRE].

    MathSciNet  Google Scholar 

  81. A. Guijosa and D.A. Lowe, A new twist on dS/CFT, Phys. Rev.D 69 (2004) 106008 [hep-th/0312282] [INSPIRE].

  82. A. Chatterjee and D.A. Lowe, dS/CFT and the operator product expansion, Phys. Rev.D 96 (2017) 066031 [arXiv:1612.07785] [INSPIRE].

  83. E. Sezgin and P. Sundell, Holography in 4D (super) higher spin theories and a test via cubic scalar couplings, JHEP07 (2005) 044 [hep-th/0305040] [INSPIRE].

  84. O. Aharony, G. Gur-Ari and R. Yacoby, d = 3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories, JHEP03 (2012) 037 [arXiv:1110.4382] [INSPIRE].

  85. S. Giombi, S. Minwalla, S. Prakash, S.P. Trivedi, S.R. Wadia and X. Yin, Chern-Simons Theory with Vector Fermion Matter, Eur. Phys. J.C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].

  86. C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ Triality: from Higher Spin Fields to Strings, J. Phys.A 46 (2013) 214009 [arXiv:1207.4485] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  87. T. Hertog and J. Hartle, Holographic No-Boundary Measure, JHEP05 (2012) 095 [arXiv:1111.6090] [INSPIRE].

    ADS  Google Scholar 

  88. D. Anninos, R. Mahajan, D. Radǐcevíc and E. Shaghoulian, Chern-Simons-Ghost Theories and de Sitter Space, JHEP01 (2015) 074 [arXiv:1405.1424] [INSPIRE].

  89. C.-M. Chang, A. Pathak and A. Strominger, Non-Minimal Higher-Spin DS4/CFT3, arXiv:1309.7413 [INSPIRE].

  90. X. Bekaert, E. Joung and J. Mourad, Effective action in a higher-spin background, JHEP02 (2011) 048 [arXiv:1012.2103] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  91. A.C. Petkou, Evaluating the AdS dual of the critical O(N) vector model, JHEP03 (2003) 049 [hep-th/0302063] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  92. D. Anninos, F. Denef, G. Konstantinidis and E. Shaghoulian, Higher Spin de Sitter Holography from Functional Determinants, JHEP02 (2014) 007 [arXiv:1305.6321] [INSPIRE].

    ADS  Google Scholar 

  93. C. Sleight and M. Taronna, Higher Spin Interactions from Conformal Field Theory: The Complete Cubic Couplings, Phys. Rev. Lett.116 (2016) 181602 [arXiv:1603.00022] [INSPIRE].

    ADS  Google Scholar 

  94. M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP11 (2011) 071 [arXiv:1107.3554] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  95. S. Giombi, S. Prakash and X. Yin, A Note on CFT Correlators in Three Dimensions, JHEP07 (2013) 105 [arXiv:1104.4317] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  96. A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, JHEP03 (2014) 111 [arXiv:1304.7760] [INSPIRE].

    ADS  MATH  Google Scholar 

  97. A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev.D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].

  98. X. Xiao, Holographic representation of local operators in de Sitter space, Phys. Rev.D 90 (2014) 024061 [arXiv:1402.7080] [INSPIRE].

  99. D. Sarkar and X. Xiao, Holographic Representation of Higher Spin Gauge Fields, Phys. Rev.D 91 (2015) 086004 [arXiv:1411.4657] [INSPIRE].

  100. A. Albrecht, N. Kaloper and Y.-S. Song, Holographic limitations of the effective field theory of inflation, hep-th/0211221 [INSPIRE].

  101. T. Banks and W. Fischler, An upper bound on the number of e-foldings, astro-ph/0307459 [INSPIRE].

  102. N. Arkani-Hamed, S. Dubovsky, A. Nicolis, E. Trincherini and G. Villadoro, A measure of de Sitter entropy and eternal inflation, JHEP05 (2007) 055 [arXiv:0704.1814] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  103. T. Falk, R. Rangarajan and M. Srednicki, The angular dependence of the three point correlation function of the cosmic microwave background radiation as predicted by inflationary cosmologies, Astrophys. J.403 (1993) L1 [astro-ph/9208001] [INSPIRE].

    ADS  Google Scholar 

  104. J.M. Maldacena and G.L. Pimentel, On graviton non-Gaussianities during inflation, JHEP09 (2011) 045 [arXiv:1104.2846] [INSPIRE].

    ADS  MATH  Google Scholar 

  105. P. McFadden and K. Skenderis, Holographic Non-Gaussianity, JCAP05 (2011) 013 [arXiv:1011.0452] [INSPIRE].

    ADS  Google Scholar 

  106. P. McFadden and K. Skenderis, Cosmological 3-point correlators from holography, JCAP06 (2011) 030 [arXiv:1104.3894] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  107. I. Mata, S. Raju and S. Trivedi, CMB from CFT, JHEP07 (2013) 015 [arXiv:1211.5482] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  108. D. Anninos, T. Anous, D.Z. Freedman and G. Konstantinidis, Late-time Structure of the Bunch-Davies de Sitter Wavefunction, JCAP11 (2015) 048 [arXiv:1406.5490] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  109. N. Arkani-Hamed and J. Maldacena, Cosmological Collider Physics, arXiv:1503.08043 [INSPIRE].

  110. N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological Polytopes and the Wavefunction of the Universe, arXiv:1709.02813 [INSPIRE].

  111. D. Seery, M.S. Sloth and F. Vernizzi, Inflationary trispectrum from graviton exchange, JCAP03 (2009) 018 [arXiv:0811.3934] [INSPIRE].

    ADS  Google Scholar 

  112. A. Ghosh, N. Kundu, S. Raju and S.P. Trivedi, Conformal Invariance and the Four Point Scalar Correlator in Slow-Roll Inflation, JHEP07 (2014) 011 [arXiv:1401.1426] [INSPIRE].

    ADS  Google Scholar 

  113. A. Bzowski, P. McFadden and K. Skenderis, Holographic predictions for cosmological 3-point functions, JHEP03 (2012) 091 [arXiv:1112.1967] [INSPIRE].

    ADS  MATH  Google Scholar 

  114. C.A. Tracy and H. Widom, On orthogonal and symplectic matrix ensembles, Commun. Math. Phys.177 (1996) 727 [solv-int/9509007] [INSPIRE].

  115. M. Chiani, Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution, J. Multiv. Anal.129 (2014) 69 [arXiv:1209.3394].

    MathSciNet  MATH  Google Scholar 

  116. J.A. Dominguez-Molina, The Tracy-Widom distribution is not infinitely divisible, Statist. Probab. Lett.123 (2017) 56 [arXiv:1601.02898].

    MathSciNet  MATH  Google Scholar 

  117. P. Forrester and S. Ole, The importance of the Selberg integral, Warnaar Bulletin (New Series) Of The American Mathematical Society S 0273-0979(08)01221-4.

  118. M. Marin˜o, Lectures on non-perturbative effects in large N gauge theories, matrix models and strings, Fortsch. Phys.62 (2014) 455 [arXiv:1206.6272] [INSPIRE].

  119. R.R. Metsaev, Shadows, currents and AdS, Phys. Rev.D 78 (2008) 106010 [arXiv:0805.3472] [INSPIRE].

  120. R.R. Metsaev, Gauge invariant two-point vertices of shadow fields, AdS/CFT and conformal fields, Phys. Rev.D 81 (2010) 106002 [arXiv:0907.4678] [INSPIRE].

  121. S. Weinberg and E. Witten, Limits on Massless Particles, Phys. Lett.96B (1980) 59 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  122. J. Maldacena and G.L. Pimentel, Entanglement entropy in de Sitter space, JHEP02 (2013) 038 [arXiv:1210.7244] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  123. M. Srednicki, Entropy and area, Phys. Rev. Lett.71 (1993) 666 [hep-th/9303048] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  124. S. Monnier, Finite higher spin transformations from exponentiation, Commun. Math. Phys.336 (2015) 1 [arXiv:1402.4486] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  125. A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].

  126. D. Anninos, G.S. Ng and A. Strominger, Asymptotic Symmetries and Charges in de Sitter Space, Class. Quant. Grav.28 (2011) 175019 [arXiv:1009.4730] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  127. D. Anninos, G.S. Ng and A. Strominger, Future Boundary Conditions in de Sitter Space, JHEP02 (2012) 032 [arXiv:1106.1175] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  128. A. Ashtekar, Implications of a positive cosmological constant for general relativity, Rept. Prog. Phys.80 (2017) 102901 [arXiv:1706.07482] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  129. G.S. Ng and A. Strominger, State/Operator Correspondence in Higher-Spin dS/CFT, Class. Quant. Grav.30 (2013) 104002 [arXiv:1204.1057] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  130. D. Anninos and T. Anous, A de Sitter Hoedown, JHEP08 (2010) 131 [arXiv:1002.1717] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  131. D.L. Jafferis, A. Lupsasca, V. Lysov, G.S. Ng and A. Strominger, Quasinormal quantization in de Sitter spacetime, JHEP01 (2015) 004 [arXiv:1305.5523] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

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Anninos, D., Denef, F., Monten, R. et al. Higher spin de Sitter Hilbert space. J. High Energ. Phys. 2019, 71 (2019).

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  • 1/N Expansion
  • Gauge-gravity correspondence
  • Higher Spin Gravity
  • Models of Quantum Gravity