Abstract
We construct the Wightman function for symmetric traceless tensors and Dirac fermions in dSd+1 in a coordinate and index free formalism using a d + 2 dimensional ambient space. We expand the embedding space formalism to cover spinor and tensor fields in any even or odd dimension. Our goal is to furnish a self-contained toolkit for the study of fields of arbitrary spin in de Sitter, with applications to cosmological perturbation theory. The construction for spinors is shown in extensive detail. Concise expressions for the action of isometry generators on generic bulk fields, the 2-point function of bulk spinors, and a derivation of the uplift of the spinorial covariant derivative are included.
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M. Spradlin, A. Strominger and A. Volovich, Les Houches lectures on de Sitter space, in Les Houches Summer School: Session 76: Euro Summer School on Unity of Fundamental Physics: Gravity, Gauge Theory and Strings, pp. 423–453 (2001) [hep-th/0110007] [INSPIRE].
D. Anninos, de Sitter Musings, Int. J. Mod. Phys. A 27 (2012) 1230013 [arXiv:1205.3855] [INSPIRE].
D. Baumann, Inflation, in Theoretical Advanced Study Institute in Elementary Particle Physics: Physics of the Large and the Small, pp. 523–686 (2011) [DOI] [arXiv:0907.5424] [INSPIRE].
P.A.M. Dirac, The Electron Wave Equation in De-Sitter Space, Annals Math. 36 (1935) 657 [INSPIRE].
P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].
S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
J. Penedones, TASI lectures on AdS/CFT, in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pp. 75–136 (2017) [DOI] [arXiv:1608.04948] [INSPIRE].
M.S. Costa and T. Hansen, AdS Weight Shifting Operators, JHEP 09 (2018) 040 [arXiv:1805.01492] [INSPIRE].
D. Meltzer, E. Perlmutter and A. Sivaramakrishnan, Unitarity Methods in AdS/CFT, JHEP 03 (2020) 061 [arXiv:1912.09521] [INSPIRE].
M. Nishida and K. Tamaoka, Fermions in Geodesic Witten Diagrams, JHEP 07 (2018) 149 [arXiv:1805.00217] [INSPIRE].
G. Sengör and C. Skordis, Unitarity at the Late time Boundary of de Sitter, JHEP 06 (2020) 041 [arXiv:1912.09885] [INSPIRE].
G. Sengor and C. Skordis, Scalar two-point functions at the late-time boundary of de Sitter, arXiv:2110.01635 [INSPIRE].
Z. Sun, Higher spin de Sitter quasinormal modes, JHEP 11 (2021) 025 [arXiv:2010.09684] [INSPIRE].
X. Xiao, Holographic representation of local operators in de Sitter space, Phys. Rev. D 90 (2014) 024061 [arXiv:1402.7080] [INSPIRE].
C. Sleight and M. Taronna, From dS to AdS and back, JHEP 12 (2021) 074 [arXiv:2109.02725] [INSPIRE].
T. Garidi, J.P. Gazeau and M.V. Takook, ‘Massive’ spin two field in de Sitter space, J. Math. Phys. 44 (2003) 3838 [hep-th/0302022] [INSPIRE].
M.V. Takook, Quantum Field Theory in de Sitter Universe: Ambient Space Formalism, arXiv:1403.1204 [INSPIRE].
C. Fronsdal, Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space. 7, Phys. Rev. D 20 (1979) 848 [INSPIRE].
J. Fang and C. Fronsdal, Massless, Half Integer Spin Fields in de Sitter Space, Phys. Rev. D 22 (1980) 1361 [INSPIRE].
E. Huguet, J. Queva and J. Renaud, Conformally related massless fields in dS, AdS and Minkowski spaces, Phys. Rev. D 73 (2006) 084025 [gr-qc/0603031] [INSPIRE].
S. Faci, E. Huguet, J. Queva and J. Renaud, Conformally covariant quantization of Maxwell field in de Sitter space, Phys. Rev. D 80 (2009) 124005 [arXiv:0910.1279] [INSPIRE].
J. Bros and U. Moschella, Two point functions and quantum fields in de Sitter universe, Rev. Math. Phys. 8 (1996) 327 [gr-qc/9511019] [INSPIRE].
J. Bros, U. Moschella and J.P. Gazeau, Quantum field theory in the de Sitter universe, Phys. Rev. Lett. 73 (1994) 1746 [INSPIRE].
M. Henningson and K. Sfetsos, Spinors and the AdS/CFT correspondence, Phys. Lett. B 431 (1998) 63 [hep-th/9803251] [INSPIRE].
P. Candelas and D.J. Raine, General Relativistic Quantum Field Theory-An Exactly Soluble Model, Phys. Rev. D 12 (1975) 965 [INSPIRE].
J.F. Koksma and T. Prokopec, Fermion Propagator in Cosmological Spaces with Constant Deceleration, Class. Quant. Grav. 26 (2009) 125003 [arXiv:0901.4674] [INSPIRE].
I.I. Cotaescu, Polarized Dirac fermions in de Sitter space-time, Phys. Rev. D 65 (2002) 084008 [hep-th/0109199] [INSPIRE].
I.I. Cotaescu, Integral representation of the Feynman propagators of the Dirac fermions on the de Sitter expanding universe, Eur. Phys. J. C 78 (2018) 769 [arXiv:1809.00670] [INSPIRE].
R. Camporesi, The Spinor heat kernel in maximally symmetric spaces, Commun. Math. Phys. 148 (1992) 283 [INSPIRE].
R. Camporesi and A. Higuchi, On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [INSPIRE].
T. Hertog, G. Tartaglino-Mazzucchelli and G. Venken, Spinors in Supersymmetric dS/CFT, JHEP 10 (2019) 117 [arXiv:1905.01322] [INSPIRE].
T. Kawano and K. Okuyama, Spinor exchange in AdSd+1, Nucl. Phys. B 565 (2000) 427 [hep-th/9905130] [INSPIRE].
L. Iliesiu, F. Kos, D. Poland, S.S. Pufu, D. Simmons-Duffin and R. Yacoby, Bootstrapping 3D Fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE].
S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev. D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
A. Higuchi, D. Marolf and I.A. Morrison, On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT, Phys. Rev. D 83 (2011) 084029 [arXiv:1012.3415] [INSPIRE].
V. Gorbenko and L. Senatore, λϕ4 in dS, arXiv:1911.00022 [INSPIRE].
C. Sleight and M. Taronna, Spinning Witten Diagrams, JHEP 06 (2017) 100 [arXiv:1702.08619] [INSPIRE].
C. Sleight and M. Taronna, Bootstrapping Inflationary Correlators in Mellin Space, JHEP 02 (2020) 098 [arXiv:1907.01143] [INSPIRE].
C. Sleight, A Mellin Space Approach to Cosmological Correlators, JHEP 01 (2020) 090 [arXiv:1906.12302] [INSPIRE].
C. Sleight and M. Taronna, From AdS to dS exchanges: Spectral representation, Mellin amplitudes, and crossing, Phys. Rev. D 104 (2021) L081902 [arXiv:2007.09993] [INSPIRE].
L. Di Pietro, V. Gorbenko and S. Komatsu, Analyticity and unitarity for cosmological correlators, JHEP 03 (2022) 023 [arXiv:2108.01695] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
A. Bzowski, P. McFadden and K. Skenderis, Holography for inflation using conformal perturbation theory, JHEP 04 (2013) 047 [arXiv:1211.4550] [INSPIRE].
P. McFadden and K. Skenderis, Holography for Cosmology, Phys. Rev. D 81 (2010) 021301 [arXiv:0907.5542] [INSPIRE].
G.L. Pimentel, Inflationary Consistency Conditions from a Wavefunctional Perspective, JHEP 02 (2014) 124 [arXiv:1309.1793] [INSPIRE].
D. Anninos, T. Anous, D.Z. Freedman and G. Konstantinidis, Late-time Structure of the Bunch-Davies de Sitter Wavefunction, JCAP 11 (2015) 048 [arXiv:1406.5490] [INSPIRE].
I. Mata, S. Raju and S. Trivedi, CMB from CFT, JHEP 07 (2013) 015 [arXiv:1211.5482] [INSPIRE].
D. Harlow and D. Stanford, Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
M. Hogervorst, J. Penedones and K.S. Vaziri, Towards the non-perturbative cosmological bootstrap, arXiv:2107.13871 [INSPIRE].
D. Baumann, G. Goon, H. Lee and G.L. Pimentel, Partially Massless Fields During Inflation, JHEP 04 (2018) 140 [arXiv:1712.06624] [INSPIRE].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The cosmological bootstrap: weight-shifting operators and scalar seeds, JHEP 12 (2020) 204 [arXiv:1910.14051] [INSPIRE].
D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization, SciPost Phys. 11 (2021) 071 [arXiv:2005.04234] [INSPIRE].
D. Baumann, C. Duaso Pueyo and A. Joyce, Bootstrapping Cosmological Correlations, AAPPS Bull. 30 (2020) 2.
N. Arkani-Hamed, D. Baumann, H. Lee and G.L. Pimentel, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities, JHEP 04 (2020) 105 [arXiv:1811.00024] [INSPIRE].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological Polytopes and the Wavefunction of the Universe, arXiv:1709.02813 [INSPIRE].
H. Goodhew, S. Jazayeri, M.H. Gordon Lee and E. Pajer, Cutting cosmological correlators, JCAP 08 (2021) 003 [arXiv:2104.06587] [INSPIRE].
S. Jazayeri, E. Pajer and D. Stefanyszyn, From locality and unitarity to cosmological correlators, JHEP 10 (2021) 065 [arXiv:2103.08649] [INSPIRE].
J. Bonifacio, E. Pajer and D.-G. Wang, From amplitudes to contact cosmological correlators, JHEP 10 (2021) 001 [arXiv:2106.15468] [INSPIRE].
H. Isono, On conformal correlators and blocks with spinors in general dimensions, Phys. Rev. D 96 (2017) 065011 [arXiv:1706.02835] [INSPIRE].
Y. Choquet-Bruhat, C. DeWitt-Morette and M. Dillard-Bleick, Analysis, manifolds, and physics, North-Holland Pub. Co., Amsterdam, The Netherlands (1982).
M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
S. Curry and A.R. Gover, An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity, arXiv:1412.7559 [INSPIRE].
I.G. Macdonald, Hypergeometric functions I, arXiv:1309.4568.
R. Bousso, A. Maloney and A. Strominger, Conformal vacua and entropy in de Sitter space, Phys. Rev. D 65 (2002) 104039 [hep-th/0112218] [INSPIRE].
B. Allen, Vacuum States in de Sitter Space, Phys. Rev. D 32 (1985) 3136 [INSPIRE].
M. Sasaki, T. Tanaka and K. Yamamoto, Euclidean vacuum mode functions for a scalar field on open de Sitter space, Phys. Rev. D 51 (1995) 2979 [gr-qc/9412025] [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
P. Adshead, R. Easther and E.A. Lim, The ‘in-in’ Formalism and Cosmological Perturbations, Phys. Rev. D 80 (2009) 083521 [arXiv:0904.4207] [INSPIRE].
A. Trautman, Spinors and the dirac operator on hypersurfaces. i. general theory, J. Math. Phys. 33 (1992) 4011.
A. Trautman, The Dirac operator on hypersurfaces, Acta Phys. Polon. B 26 (1995) 1283 [hep-th/9810018] [INSPIRE].
D. Anninos, F. Denef, R. Monten and Z. Sun, Higher Spin de Sitter Hilbert Space, JHEP 10 (2019) 071 [arXiv:1711.10037] [INSPIRE].
S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001) 577 [hep-th/0103198] [INSPIRE].
S. Deser and A. Waldron, Arbitrary spin representations in de Sitter from dS/CFT with applications to dS supergravity, Nucl. Phys. B 662 (2003) 379 [hep-th/0301068] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight Shifting Operators and Conformal Blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].
R.L. Arnowitt, S. Deser and C.W. Misner, The Dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].
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].
D. Anninos, G.S. Ng and A. Strominger, Future Boundary Conditions in de Sitter Space, JHEP 02 (2012) 032 [arXiv:1106.1175] [INSPIRE].
M. Visser, How to Wick rotate generic curved spacetime, arXiv:1702.05572 [INSPIRE].
D. Schlingemann, From Euclidean field theory to quantum field theory, Rev. Math. Phys. 11 (1999) 1151 [hep-th/9802035] [INSPIRE].
D. Schlingemann, Euclidean field theory on a sphere, hep-th/9912235 [INSPIRE].
J.R. David and J. Mukherjee, Partition functions of p-forms from Harish-Chandra characters, JHEP 09 (2021) 094 [arXiv:2105.03662] [INSPIRE].
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].
T. Banks, B. Fiol and A. Morisse, Towards a quantum theory of de Sitter space, JHEP 12 (2006) 004 [hep-th/0609062] [INSPIRE].
Y.T.A. Law, A compendium of sphere path integrals, JHEP 12 (2021) 213 [arXiv:2012.06345] [INSPIRE].
B. Mühlmann, The two-sphere partition function in two-dimensional quantum gravity at fixed area, JHEP 09 (2021) 189 [arXiv:2106.04532] [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].
D. Anninos, D.A. Galante and D.M. Hofman, de Sitter horizons & holographic liquids, JHEP 07 (2019) 038 [arXiv:1811.08153] [INSPIRE].
D. Anninos and E. Harris, Three-dimensional de Sitter horizon thermodynamics, JHEP 10 (2021) 091 [arXiv:2106.13832] [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, JHEP 01 (2022) 088 [arXiv:2009.12464] [INSPIRE].
C.P. Herzog and K.-W. Huang, Boundary Conformal Field Theory and a Boundary Central Charge, JHEP 10 (2017) 189 [arXiv:1707.06224] [INSPIRE].
C.P. Herzog and V. Schaub, A sum rule for boundary contributions to the trace anomaly, JHEP 01 (2022) 121 [arXiv:2107.11604] [INSPIRE].
A. David, N. Fischer and Y. Neiman, Spinor-helicity variables for cosmological horizons in de Sitter space, Phys. Rev. D 100 (2019) 045005 [arXiv:1906.01058] [INSPIRE].
S. Caron-Huot and Y.-Z. Li, Helicity basis for three-dimensional conformal field theory, JHEP 06 (2021) 041 [arXiv:2102.08160] [INSPIRE].
S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations, Cambridge University Press, Cambridge, U.K. (2005) [DOI] [INSPIRE].
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].
E. Witten, Quantum gravity in de Sitter space, in Strings 2001: International Conference, (2001) [hep-th/0106109] [INSPIRE].
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].
Z. Sun, A note on the representations of SO(1, d + 1), arXiv:2111.04591 [INSPIRE].
E. Thieleker, On the quasi-simple irreducible representations of the lorentz groups, Trans. Am. Math. Soc. 179 (1973) 465.
E.A. Thieleker, The unitary representations of the generalized lorentz groups, Trans. Am. Math. Soc. 199 (1974) 327.
E. Joung, J. Mourad and R. Parentani, Group theoretical approach to quantum fields in de Sitter space. I. The Principle series, JHEP 08 (2006) 082 [hep-th/0606119] [INSPIRE].
E. Joung, J. Mourad and R. Parentani, Group theoretical approach to quantum fields in de Sitter space. II. The complementary and discrete series, JHEP 09 (2007) 030 [arXiv:0707.2907] [INSPIRE].
T. Anous and J. Skulte, An invitation to the principal series, SciPost Phys. 9 (2020) 028 [arXiv:2007.04975] [INSPIRE].
T.D. Newton, A note on the representations of the de sitter group, Annals Math. 51 (1950) 730.
T. Basile, X. Bekaert and N. Boulanger, Mixed-symmetry fields in de Sitter space: a group theoretical glance, JHEP 05 (2017) 081 [arXiv:1612.08166] [INSPIRE].
D. Simmons-Duffin, Projectors, Shadows, and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
E. Joung and K. Mkrtchyan, Partially-massless higher-spin algebras and their finite-dimensional truncations, JHEP 01 (2016) 003 [arXiv:1508.07332] [INSPIRE].
A. Van Proeyen, Tools for supersymmetry, Ann. U. Craiova Phys. 9 (1999) 1 [hep-th/9910030] [INSPIRE].
C. Stahl, E. Strobel and S.-S. Xue, Fermionic current and Schwinger effect in de Sitter spacetime, Phys. Rev. D 93 (2016) 025004 [arXiv:1507.01686] [INSPIRE].
A. Higuchi, Symmetric tensor fields in de sitter space-time, YTP-85-22 (1985).
A. Higuchi, Forbidden Mass Range for Spin-2 Field Theory in de Sitter Space-time, Nucl. Phys. B 282 (1987) 397 [INSPIRE].
A. Higuchi, Symmetric Tensor Spherical Harmonics on the N Sphere and Their Application to the de Sitter Group SO(N, 1), J. Math. Phys. 28 (1987) 1553 [Erratum ibid. 43 (2002) 6385] [INSPIRE].
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Pethybridge, B., Schaub, V. Tensors and spinors in de Sitter space. J. High Energ. Phys. 2022, 123 (2022). https://doi.org/10.1007/JHEP06(2022)123
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DOI: https://doi.org/10.1007/JHEP06(2022)123