Abstract
In this note we discuss features of the simplest spinning Discrete Series Unitary Irreducible Representations (UIR) of SO(1,4). These representations are known to be realised in the single particle Hilbert space of a free gauge field propagating in a four dimensional fixed de Sitter background. They showcase distinct features as compared to the more common Principal Series realised by heavy fields. Upon computing the 1 loop Sphere path integral we show that the edge modes of the theory can be understood in terms of a Discrete Series of SO(1, 2). We then canonically quantise the theory and show how group theory constrains the mode decomposition. We further clarify the role played by the second SO(4) Casimir in the single particle Hilbert space of the theory.
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References
Planck collaboration, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6 [Erratum ibid. 652 (2021) C4] [arXiv:1807.06209] [INSPIRE].
Supernova Cosmology Project collaboration, Measurements of Ω and Λ from 42 high redshift supernovae, Astrophys. J. 517 (1999) 565 [astro-ph/9812133] [INSPIRE].
SDSS collaboration, Baryon Acoustic Oscillations in the Sloan Digital Sky Survey Data Release 7 Galaxy Sample, Mon. Not. Roy. Astron. Soc. 401 (2010) 2148 [arXiv:0907.1660] [INSPIRE].
D. Anninos and D.M. Hofman, Infrared Realization of dS2 in AdS2, Class. Quant. Grav. 35 (2018) 085003 [arXiv:1703.04622] [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 D.A. Galante, Constructing AdS2 flow geometries, JHEP 02 (2021) 045 [arXiv:2011.01944] [INSPIRE].
D. Anninos, D.A. Galante and S.U. Sheorey, Renormalisation group flows of deformed SYK models, JHEP 11 (2023) 197 [arXiv:2212.04944] [INSPIRE].
L. Susskind, Entanglement and Chaos in De Sitter Space Holography: an SYK Example, JHAP 1 (2021) 1 [arXiv:2109.14104] [INSPIRE].
F. Ecker, D. Grumiller and R. McNees, dS2 as excitation of AdS2, SciPost Phys. 13 (2022) 119 [arXiv:2204.00045] [INSPIRE].
D. Anninos and E. Harris, Interpolating geometries and the stretched dS2 horizon, JHEP 11 (2022) 166 [arXiv:2209.06144] [INSPIRE].
E. Witten, Deformations of JT Gravity and Phase Transitions, arXiv:2006.03494 [INSPIRE].
V. Gorbenko, E. Silverstein and G. Torroba, dS/dS and \( T\overline{T} \), JHEP 03 (2019) 085 [arXiv:1811.07965] [INSPIRE].
A. Lewkowycz, J. Liu, E. Silverstein and G. Torroba, \( T\overline{T} \) and EE, with implications for (A)dS subregion encodings, JHEP 04 (2020) 152 [arXiv:1909.13808] [INSPIRE].
E. Coleman et al., De Sitter microstates from \( T\overline{T} \) + Λ2 and the Hawking-Page transition, JHEP 07 (2022) 140 [arXiv:2110.14670] [INSPIRE].
V. Shyam, \( T\overline{T} \) + Λ2 deformed CFT on the stretched dS3 horizon, JHEP 04 (2022) 052 [arXiv:2106.10227] [INSPIRE].
X. Dong, E. Silverstein and G. Torroba, De Sitter Holography and Entanglement Entropy, JHEP 07 (2018) 050 [arXiv:1804.08623] [INSPIRE].
E.P. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, Annals Math. 40 (1939) 149 [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 783 [hep-th/9712074] [INSPIRE].
B. de Wit and I. Herger, Anti-de Sitter supersymmetry, Lect. Notes Phys. 541 (2000) 79 [hep-th/9908005] [INSPIRE].
F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in Dh ≥ 3 Dimensions, arXiv:1601.05000 [https://doi.org/10.1007/978-3-319-43626-5] [INSPIRE].
V.K. Dobrev et al., Harmonic Analysis on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, Lect. Notes Phys. 63 (1977) [INSPIRE].
Z. Sun, A note on the representations of SO(1, d + 1), arXiv:2111.04591 [INSPIRE].
V. Schaub, Spinors in (Anti-)de Sitter Space, JHEP 09 (2023) 142 [arXiv:2302.08535] [INSPIRE].
T. Anous and J. Skulte, An invitation to the principal series, SciPost Phys. 9 (2020) 028 [arXiv:2007.04975] [INSPIRE].
T. Basile, E. Joung, S. Lal and W. Li, Character Integral Representation of Zeta function in AdSd+1: I. Derivation of the general formula, JHEP 10 (2018) 091 [arXiv:1805.05646] [INSPIRE].
V.A. Letsios, The eigenmodes for spinor quantum field theory in global de Sitter space–time, J. Math. Phys. 62 (2021) 032303 [arXiv:2011.07875] [INSPIRE].
V.A. Letsios, The (partially) massless spin-3/2 and spin-5/2 fields in de Sitter spacetime as unitary and non-unitary representations of the de Sitter algebra, arXiv:2206.09851.
B. Pethybridge and V. Schaub, Tensors and spinors in de Sitter space, JHEP 06 (2022) 123 [arXiv:2111.14899] [INSPIRE].
S. Deser and R.I. Nepomechie, Gauge Invariance Versus Masslessness in De Sitter Space, Annals Phys. 154 (1984) 396 [INSPIRE].
A. Higuchi, Forbidden Mass Range for Spin-2 Field Theory in De Sitter Space-time, Nucl. Phys. B 282 (1987) 397 [INSPIRE].
S. Deser and A. Waldron, Gauge invariances and phases of massive higher spins in (A)dS, Phys. Rev. Lett. 87 (2001) 031601 [hep-th/0102166] [INSPIRE].
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].
A. Knapp, Representation theory of semisimple groups, Princeton University Press (1986).
V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals Math. 48 (1947) 568 [INSPIRE].
Harish-Chandra, Plancherel formula for the 2 × 2 real unimodular group, Proc. Natl. Acad. Sci. USA 38 (1952) 337, http://www.jstor.org/stable/88737.
L.H. Thomas, On unitary representations of the group of de sitter space, Annals Math. 42 (1941) 113, http://www.jstor.org/stable/1968990.
J. Bros, H. Epstein and U. Moschella, Scalar tachyons in the de Sitter universe, Lett. Math. Phys. 93 (2010) 203 [arXiv:1003.1396] [INSPIRE].
H. Epstein and U. Moschella, de Sitter tachyons and related topics, Commun. Math. Phys. 336 (2015) 381 [arXiv:1403.3319] [INSPIRE].
D. Anninos, T. Anous, B. Pethybridge and G. Şengör, The Discreet Charm of the Discrete Series in DS2, arXiv:2307.15832 [INSPIRE].
J. Repka, Tensor products of unitary representations of SL2(R), Am. J. MAth. 100 (1978) 747, http://www.jstor.org/stable/2373909.
O. Nachtmann, Dynamische stabilität im de-sitter-raum, (1968), https://api.semanticscholar.org/CorpusID:227809164.
D. Anninos, P. Benetti Genolini and B. Mühlmann, dS2 supergravity, JHEP 11 (2023) 145 [arXiv:2309.02480] [INSPIRE].
A. Higuchi, Linearized gravity in de Sitter space-time as a representation of SO(4,1), Class. Quant. Grav. 8 (1991) 2005 [INSPIRE].
V.A. Letsios, (Non-)unitarity of strictly and partially massless fermions on de Sitter space, JHEP 05 (2023) 015 [arXiv:2303.00420] [INSPIRE].
J.D. Bekenstein, Black holes and the second law, Lett. Nuovo Cim. 4 (1972) 737 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev. D 15 (1977) 2738 [INSPIRE].
D.A. Galante, Modave lectures on de Sitter space & holography, PoS Modave2022 (2023) 003 [arXiv:2306.10141] [INSPIRE].
G.W. Gibbons, S.W. Hawking and M.J. Perry, Path Integrals and the Indefiniteness of the Gravitational Action, Nucl. Phys. B 138 (1978) 141 [INSPIRE].
J. Polchinski, A Two-Dimensional Model for Quantum Gravity, Nucl. Phys. B 324 (1989) 123 [INSPIRE].
J. Distler and H. Kawai, Conformal Field Theory and 2D Quantum Gravity, Nucl. Phys. B 321 (1989) 509 [INSPIRE].
F. David, Conformal Field Theories Coupled to 2D Gravity in the Conformal Gauge, Mod. Phys. Lett. A 3 (1988) 1651 [INSPIRE].
T. Bautista, A. Dabholkar and H. Erbin, Quantum Gravity from Timelike Liouville theory, JHEP 10 (2019) 284 [arXiv:1905.12689] [INSPIRE].
D. Anninos and B. Mühlmann, The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe), JHEP 12 (2021) 206 [arXiv:2111.05344] [INSPIRE].
B. Mühlmann, The two-sphere partition function from timelike Liouville theory at three-loop order, JHEP 05 (2022) 057 [arXiv:2202.04549] [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, 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].
D. Anninos, De Sitter Musings, Int. J. Mod. Phys. A 27 (2012) 1230013 [arXiv:1205.3855] [INSPIRE].
M. Spradlin, A. Strominger and A. Volovich, Les Houches lectures on de Sitter space, in the proceedings of the Les Houches Summer School: Session 76: Euro Summer School on Unity of Fundamental Physics: Gravity, Gauge Theory and Strings, Les Houches, France, July 30 – August 31 (2001), pp. 423–453 [hep-th/0110007] [INSPIRE].
E. Schrodinger, Expanding universes, Cambridge University Press (1956).
P.A.M. Dirac, The Electron Wave Equation in De-Sitter Space, Annals Math. 36 (1935) 657 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press (2023) [https://doi.org/10.1017/9781009253161] [INSPIRE].
U. Ottoson, A classification of the unitary irreducible representations of SO0 (N, 1), Commun. Math. Phys. 8 (1968) 228.
F. Schwarz, Unitary irreducible representations of the groups SO(n,1), J. Math. Phys. 12 (1971) 131.
M. Boers, Group theory and de Sitter QFT: the concept of mass, M.Sc. thesis, Groningen University (2013) [INSPIRE].
K. Hinterbichler and A. Joyce, Manifest Duality for Partially Massless Higher Spins, JHEP 09 (2016) 141 [arXiv:1608.04385] [INSPIRE].
S. Deser and R.I. Nepomechie, Anomalous Propagation of Gauge Fields in Conformally Flat Spaces, Phys. Lett. B 132 (1983) 321 [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].
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].
A. Higuchi, Symmetric tensor fields in de Sitter spacetime, Yale preprint YTP-85-22, Yale University Physics Deptartment, New Haven, CT, U.S.A. (1985).
N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge, U.K. (1984) [https://doi.org/10.1017/CBO9780511622632] [INSPIRE].
O. Babelon and C.M. Viallet, The Geometrical Interpretation of the Faddeev-Popov Determinant, Phys. Lett. B 85 (1979) 246 [INSPIRE].
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. B 341 (1990) 187 [INSPIRE].
Z. Bern, E. Mottola and S.K. Blau, General covariance of the path integral for quantum gravity, Phys. Rev. D 43 (1991) 1212 [INSPIRE].
D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
Harish-Chandra, On the characters of a semisimple Lie group, Bull. Am. Math. Soc. 61 (1955) 389.
Harish-Chandra, Invariant eigendistributions on semisimple Lie groups, Bull. Am. Math. Soc. 69 (1963) 117.
M. Atiyah et al., The Harish-Chandra character, in Representation Theory of Lie Groups, chapter 7, Cambridge University Press (1980) [https://doi.org/10.1017/CBO9780511662683.007].
T. Hirai, The characters of irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 41 (1965) 526.
T. Hirai, On irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 38 (1962) 258.
T. Hirai, On infinitesimal operators of irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad. 38 (1962) 83.
W. Donnelly and A.C. Wall, Unitarity of Maxwell theory on curved spacetimes in the covariant formalism, Phys. Rev. D 87 (2013) 125033 [arXiv:1303.1885] [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].
H. Casini, M. Huerta, J.M. Magán and D. Pontello, Logarithmic coefficient of the entanglement entropy of a Maxwell field, Phys. Rev. D 101 (2020) 065020 [arXiv:1911.00529] [INSPIRE].
J.S. Dowker, Entanglement entropy for even spheres, arXiv:1009.3854 [INSPIRE].
L. Lindblom, N.W. Taylor and F. Zhang, Scalar, Vector and Tensor Harmonics on the Three-Sphere, Gen. Rel. Grav. 49 (2017) 139 [arXiv:1709.08020] [INSPIRE].
G.W. Gibbons, Spectral Asymmetry and Quantum Field Theory in Curved Space-time, Annals Phys. 125 (1980) 98 [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].
D. Schlingemann, Euclidean field theory on a sphere, hep-th/9912235 [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, Principal and Complementary Series Representations at the Late-Time Boundary of de Sitter, Springer Proc. Math. Stat. 396 (2022) 269 [arXiv:2205.11550] [INSPIRE].
G. Şengör, Particles of a de Sitter Universe, Universe 9 (2023) 59 [arXiv:2212.10626] [INSPIRE].
E. Witten, On S duality in Abelian gauge theory, Selecta Math. 1 (1995) 383 [hep-th/9505186] [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
D. Anninos and E. Harris, Three-dimensional de Sitter horizon thermodynamics, JHEP 10 (2021) 091 [arXiv:2106.13832] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
G. Tian and S.-T. Yau, Kahler-einstein Metrics on Complex Surfaces With C(1) > 0, Commun. Math. Phys. 112 (1987) 175 [INSPIRE].
G. Tian, On calabi’s conjecture for complex surfaces with positive first chern class, Invent. Math. 101 (1990) 101.
G. Kallen, On the definition of the Renormalization Constants in Quantum Electrodynamics, Helv. Phys. Acta 25 (1952) 417 [INSPIRE].
H. Lehmann, On the Properties of propagation functions and renormalization contants of quantized fields, Nuovo Cim. 11 (1954) 342 [INSPIRE].
M. Loparco, J. Penedones, K. Salehi Vaziri and Z. Sun, The Källén-Lehmann representation in de Sitter spacetime, arXiv:2306.00090 [INSPIRE].
S. Hollands, Massless interacting quantum fields in deSitter spacetime, Annales Henri Poincare 13 (2012) 1039 [arXiv:1105.1996] [INSPIRE].
L. Di Pietro, V. Gorbenko and S. Komatsu, Analyticity and unitarity for cosmological correlators, JHEP 03 (2022) 023 [arXiv:2108.01695] [INSPIRE].
J. Bros, Complexified de Sitter space: analytic causal kernels and Kallen-Lehmann type representation, Nucl. Phys. B Proc. Suppl. 18 (1991) 22 [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 et al., Triangular invariants, three-point functions and particle stability on the de Sitter universe, Commun. Math. Phys. 295 (2010) 261 [arXiv:0901.4223] [INSPIRE].
P.V. Buividovich and M.I. Polikarpov, Numerical study of entanglement entropy in SU(2) lattice gauge theory, Nucl. Phys. B 802 (2008) 458 [arXiv:0802.4247] [INSPIRE].
D. Anninos, V. Letsios, A. Rios Fukelman, M. Sempe and G. Silva, in preparation.
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, F. Denef, R. Monten and Z. Sun, Higher Spin de Sitter Hilbert Space, JHEP 10 (2019) 071 [arXiv:1711.10037] [INSPIRE].
P. Goldbart and M. Stone, Mathematics for physics: a guided tour for graduate students, Cambridge University Press (2009), https://doi.org/10.1017/CBO9780511627040.
G.W. Gibbons, Part III: applications of Differential Geometry to Physics, DAMTP Lecture notes, unpublished.
K. Kumar and O. Lechtenfeld, On rational electromagnetic fields, Phys. Lett. A 384 (2020) 126445 [arXiv:2002.01005] [INSPIRE].
U.H. Gerlach and U.K. Sengupta, Homogeneous Collapsing Star: tensor and Vector Harmonics for Matter and Field Asymmetries, Phys. Rev. D 18 (1978) 1773 [INSPIRE].
I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors, JHEP 05 (2012) 036 [arXiv:1112.5342] [INSPIRE].
Acknowledgments
We thank D. Anninos, D. Galante V. Letsios and S. Vitouladitis for interesting discussions on this and related topics and feedback on the manuscript of this paper. We thank also A. Higuchi for correspondence and sharing with us his PhD Thesis. GAS would like to thank King’s College, London and the Royal Society for hospitality and financial support.
This work was funded by CONICET grants PIP-UE084 and UNLP grant X791 and PICT 2020-03826. MS is supported by a CONICET fellowship. The work of ARF was supported by the Royal society grant RF/ERE/210168 which is part of the Royal Society URF grant “The Atoms of a de Sitter Universe”.
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Fukelman, A.R., Sempé, M. & Silva, G.A. Notes on gauge fields and discrete series representations in de Sitter spacetimes. J. High Energ. Phys. 2024, 11 (2024). https://doi.org/10.1007/JHEP01(2024)011
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DOI: https://doi.org/10.1007/JHEP01(2024)011