Abstract
We formulate a model of noncommutative four-dimensional gravity on a covariant fuzzy space based on SO(1, 4), that is the fuzzy version of the dS4. The latter requires the employment of a wider symmetry group, the SO(1, 5), for reasons of covariance. Addressing along the lines of formulating four-dimensional gravity as a gauge theory of the Poincaré group, spontaneously broken to the Lorentz, we attempt to construct a four-dimensional gravitational model on the fuzzy de Sitter spacetime. In turn, first we consider the SO(1, 4) subgroup of the SO(1, 5) algebra, in which we were led to, as we want to gauge the isometry part of the full symmetry. Then, the construction of a gauge theory on such a noncommutative space directs us to use an extension of the gauge group, the SO(1, 5)×U(1), and fix its representation. Moreover, a 2-form dynamic gauge field is included in the theory for reasons of covariance of the transformation of the field strength tensor. Finally, the gauge theory is considered to be spontaneously broken to the Lorentz group with an extension of a U(1), i.e. SO(1, 3)×U(1). The latter defines the four-dimensional noncommutative gravity action which can lead to equations of motion, whereas the breaking induces the imposition of constraints that will lead to expressions relating the gauge fields. It should be noted that we use the Euclidean signature for the formulation of the above programme.
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References
H.S. Snyder, Quantized space-time, Phys. Rev. 71 (1947) 38 [INSPIRE].
C.N. Yang, On quantized space-time, Phys. Rev. 72 (1947) 874 [INSPIRE].
A. Connes, Noncommutative geometry, Academic Press inc., San Diego, CA, U.S.A. (1994).
J. Madore, An introduction to noncommutative differential geometry and its physical applications, London Math. Soc. Lect. Note Ser. 257, Cambridge University Press, Cambridge, U.K. (1999).
J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [INSPIRE].
M. Burić, T. Grammatikopoulos, J. Madore and G. Zoupanos, Gravity and the structure of noncommutative algebras, JHEP 04 (2006) 054 [hep-th/0603044] [INSPIRE].
M. Burić, J. Madore and G. Zoupanos, WKB approximation in noncommutative gravity, SIGMA 3 (2007) 125 [arXiv:0712.4024] [INSPIRE].
T. Filk, Divergencies in a field theory on quantum space, Phys. Lett. B 376 (1996) 53 [INSPIRE].
J.C. Varilly and J.M. Gracia-Bondia, On the ultraviolet behavior of quantum fields over noncommutative manifolds, Int. J. Mod. Phys. A 14 (1999) 1305 [hep-th/9804001] [INSPIRE].
M. Chaichian, A. Demichev and P. Prešnajder, Quantum field theory on noncommutative space-times and the persistence of ultraviolet divergences, Nucl. Phys. B 567 (2000) 360 [hep-th/9812180] [INSPIRE].
S. Minwalla, M. Van Raamsdonk and N. Seiberg, Noncommutative perturbative dynamics, JHEP 02 (2000) 020 [hep-th/9912072] [INSPIRE].
H. Grosse and R. Wulkenhaar, Renormalization of ϕ4 theory on noncommutative R4 to all orders, Lett. Math. Phys. 71 (2005) 13 [hep-th/0403232] [INSPIRE].
H. Grosse and H. Steinacker, Exact renormalization of a noncommutative ϕ3 model in 6 dimensions, Adv. Theor. Math. Phys. 12 (2008) 605 [hep-th/0607235] [INSPIRE].
H. Grosse and H. Steinacker, Finite gauge theory on fuzzy CP2 , Nucl. Phys. B 707 (2005) 145 [hep-th/0407089] [INSPIRE].
A. Connes and J. Lott, Particle models and noncommutative geometry, Nucl. Phys. B Proc. Suppl. 18 (1991) 29.
A.H. Chamseddine and A. Connes, The spectral action principle, Commun. Math. Phys. 186 (1997) 731 [hep-th/9606001] [INSPIRE].
A.H. Chamseddine and A. Connes, Conceptual explanation for the algebra in the noncommutative approach to the Standard Model, Phys. Rev. Lett. 99 (2007) 191601 [arXiv:0706.3690] [INSPIRE].
C.P. Martin, J.M. Gracia-Bondia and J.C. Varilly, The Standard Model as a noncommutative geometry: the low-energy regime, Phys. Rept. 294 (1998) 363 [hep-th/9605001] [INSPIRE].
M. Dubois-Violette, J. Madore and R. Kerner, Gauge bosons in a noncommutative geometry, Phys. Lett. B 217 (1989) 485 [INSPIRE].
M. Dubois-Violette, J. Madore and R. Kerner, Classical bosons in a noncommutative geometry, Class. Quant. Grav. 6 (1989) 1709 [INSPIRE].
M. Dubois-Violette, R. Kerner and J. Madore, Noncommutative differential geometry and new models of gauge theory, J. Math. Phys. 31 (1990) 323 [INSPIRE].
J. Madore, On a quark-lepton duality, Phys. Lett. B 305 (1993) 84 [INSPIRE].
J. Madore, On a noncommutative extension of electrodynamics, Fundam. Theor. Phys. 52 (1993) 285 [hep-ph/9209226] [INSPIRE].
A. Connes, M.R. Douglas and A.S. Schwarz, Noncommutative geometry and matrix theory: compactification on tori, JHEP 02 (1998) 003 [hep-th/9711162] [INSPIRE].
N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 09 (1999) 032 [hep-th/9908142] [INSPIRE].
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A large N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/9612115] [INSPIRE].
H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Space-time structures from IIB matrix model, Prog. Theor. Phys. 99 (1998) 713 [hep-th/9802085] [INSPIRE].
M. Hanada, H. Kawai and Y. Kimura, Describing curved spaces by matrices, Prog. Theor. Phys. 114 (2006) 1295 [hep-th/0508211] [INSPIRE].
K. Furuta, M. Hanada, H. Kawai and Y. Kimura, Field equations of massless fields in the new interpretation of the matrix model, Nucl. Phys. B 767 (2007) 82 [hep-th/0611093] [INSPIRE].
B. Jurčo, S. Schraml, P. Schupp and J. Wess, Enveloping algebra valued gauge transformations for non-Abelian gauge groups on noncommutative spaces, Eur. Phys. J. C 17 (2000) 521 [hep-th/0006246] [INSPIRE].
B. Juřco, P. Schupp and J. Wess, Non-Abelian noncommutative gauge theory via noncommutative extra dimensions, Nucl. Phys. B 604 (2001) 148 [hep-th/0102129] [INSPIRE].
B. Jurčo, L. Möller, S. Schraml, P. Schupp and J. Wess, Construction of non-Abelian gauge theories on noncommutative spaces, Eur. Phys. J. C 21 (2001) 383 [hep-th/0104153] [INSPIRE].
G. Barnich, F. Brandt and M. Grigoriev, Seiberg-Witten maps and noncommutative Yang-Mills theories for arbitrary gauge groups, JHEP 08 (2002) 023 [hep-th/0206003] [INSPIRE].
M. Chaichian, P. Prešnajder, M.M. Sheikh-Jabbari and A. Tureanu, Noncommutative Standard Model: model building, Eur. Phys. J. C 29 (2003) 413 [hep-th/0107055] [INSPIRE].
X. Calmet, B. Jurčo, P. Schupp, J. Wess and M. Wohlgenannt, The Standard Model on noncommutative space-time, Eur. Phys. J. C 23 (2002) 363 [hep-ph/0111115] [INSPIRE].
P. Aschieri, B. Jurčo, P. Schupp and J. Wess, Noncommutative GUTs, Standard Model and C,P,T, Nucl. Phys. B 651 (2003) 45 [hep-th/0205214] [INSPIRE].
W. Behr, N.G. Deshpande, G. Duplancic, P. Schupp, J. Trampetic and J. Wess, The Z → γγ, gg decays in the noncommutative Standard Model, Eur. Phys. J. C 29 (2003) 441 [hep-ph/0202121] [INSPIRE].
P. Aschieri, J. Madore, P. Manousselis and G. Zoupanos, Dimensional reduction over fuzzy coset spaces, JHEP 04 (2004) 034 [hep-th/0310072] [INSPIRE].
P. Aschieri, J. Madore, P. Manousselis and G. Zoupanos, Unified theories from fuzzy extra dimensions, hep-th/0401200 [INSPIRE].
P. Aschieri, J. Madore, P. Manousselis and G. Zoupanos, Renormalizable theories from fuzzy higher dimensions, in 3rd summer school in modern mathematical physics, (2005), pg. 135 [hep-th/0503039] [INSPIRE].
P. Aschieri, T. Grammatikopoulos, H. Steinacker and G. Zoupanos, Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking, JHEP 09 (2006) 026 [hep-th/0606021] [INSPIRE].
P. Aschieri, H. Steinacker, J. Madore, P. Manousselis and G. Zoupanos, Fuzzy extra dimensions: dimensional reduction, dynamical generation and renormalizability, SFIN A 1 (2007) 25 [arXiv:0704.2880] [INSPIRE].
H. Steinacker and G. Zoupanos, Fermions on spontaneously generated spherical extra dimensions, JHEP 09 (2007) 017 [arXiv:0706.0398] [INSPIRE].
A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes, Fortsch. Phys. 58 (2010) 537 [arXiv:0909.5559] [INSPIRE].
A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, Orbifolds, fuzzy spheres and chiral fermions, JHEP 05 (2010) 100 [arXiv:1002.2606] [INSPIRE].
A. Chatzistavrakidis and G. Zoupanos, Higher-dimensional unified theories with fuzzy extra dimensions, SIGMA 6 (2010) 063 [arXiv:1008.2049] [INSPIRE].
D. Gavriil, G. Manolakos, G. Orfanidis and G. Zoupanos, Higher-dimensional unification with continuous and fuzzy coset spaces as extra dimensions, Fortsch. Phys. 63 (2015) 442 [arXiv:1504.07276] [INSPIRE].
G. Manolakos and G. Zoupanos, The trinification model SU(3)3 from orbifolds for fuzzy spheres, Phys. Part. Nucl. Lett. 14 (2017) 322.
G. Manolakos and G. Zoupanos, Higher-dimensional unified theories with continuous and fuzzy coset spaces as extra dimensions, Springer Proc. Math. Stat. 191 (2016) 203 [arXiv:1602.03673] [INSPIRE].
R.J. Szabo, Quantum field theory on noncommutative spaces, hep-th/0109162 [INSPIRE].
M. Maceda, J. Madore, P. Manousselis and G. Zoupanos, Can noncommutativity resolve the big bang singularity?, Eur. Phys. J. C 36 (2004) 529 [hep-th/0306136] [INSPIRE].
A.H. Chamseddine and V. Mukhanov, Resolving cosmological singularities, JCAP 03 (2017) 009 [arXiv:1612.05860] [INSPIRE].
R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956) 1597 [INSPIRE].
T.W.B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961) 212 [INSPIRE].
S.W. MacDowell and F. Mansouri, Unified geometric theory of gravity and supergravity, Phys. Rev. Lett. 38 (1977) 739 [Erratum ibid. 38 (1977) 1376] [INSPIRE].
K.S. Stelle and P.C. West, Spontaneously broken de Sitter symmetry and the gravitational holonomy group, Phys. Rev. D 21 (1980) 1466 [INSPIRE].
E.A. Ivanov and J. Niederle, On gauge formulations of gravitation theories, in Proceedings, group theoretical methods in physics, Cocoyoc, Mexico (1980), pg. 545.
E.A. Ivanov and J. Niederle, Gauge formulation of gravitation theories. 1. The Poincaŕe, de Sitter and conformal cases, Phys. Rev. D 25 (1982) 976 [INSPIRE].
E.A. Ivanov and J. Niederle, Gauge formulation of gravitation theories. 2. The special conformal case, Phys. Rev. D 25 (1982) 988 [INSPIRE].
T.W.B. Kibble and K.S. Stelle, Gauge theories of gravity and supergravity, in Progress in quantum field theory, H. Ezawa and S. Kamefuchi eds., (1985), pg. 57 [INSPIRE].
F. Brandt, C.P. Martin and F. Ruiz, Anomaly freedom in Seiberg-Witten noncommutative gauge theories, JHEP 07 (2003) 068 [hep-th/0307292] [INSPIRE].
C.E. Carlson, C.D. Carone and R.F. Lebed, Bounding noncommutative QCD, Phys. Lett. B 518 (2001) 201 [hep-ph/0107291] [INSPIRE].
I. Hinchliffe, N. Kersting and Y.L. Ma, Review of the phenomenology of noncommutative geometry, Int. J. Mod. Phys. A 19 (2004) 179 [hep-ph/0205040] [INSPIRE].
B.P. Dolan, D. O’Connor and P. Prešnajder, Matrix ϕ4 models on the fuzzy sphere and their continuum limits, JHEP 03 (2002) 013 [hep-th/0109084] [INSPIRE].
D. O’Connor and B. Ydri, Monte Carlo simulation of a NC gauge theory on the fuzzy sphere, JHEP 11 (2006) 016 [hep-lat/0606013] [INSPIRE].
Y. Kimura, Noncommutative gauge theory on fuzzy four sphere and matrix model, Nucl. Phys. B 637 (2002) 177 [hep-th/0204256] [INSPIRE].
J. Medina, I. Huet, D. O’Connor and B.P. Dolan, Scalar and spinor field actions on fuzzy S4 : fuzzy CP3 as a \( {S}_F^2 \) bundle over \( {S}_F^4 \), JHEP 08 (2012) 070 [arXiv:1208.0348] [INSPIRE].
J. Medina and D. O’Connor, Scalar field theory on fuzzy S4 , JHEP 11 (2003) 051 [hep-th/0212170] [INSPIRE].
A. Chatzistavrakidis, L. Jonke, D. Jurman, G. Manolakos, P. Manousselis and G. Zoupanos, Noncommutative gauge theory and gravity in three dimensions, Fortsch. Phys. 66 (2018) 1800047 [arXiv:1802.07550] [INSPIRE].
G. Manolakos and G. Zoupanos, Non-commutativity in unified theories and gravity, Springer Proc. Math. Stat. 263 (2017) 177 [arXiv:1809.02954] [INSPIRE].
D. Jurman, G. Manolakos, P. Manousselis and G. Zoupanos, Gravity as a gauge theory on three-dimensional noncommutative spaces, PoS(CORFU2017)162 (2018) [arXiv:1809.03879] [INSPIRE].
H. Grosse and P. Prešnajder, The construction on noncommutative manifolds using coherent states, Lett. Math. Phys. 28 (1993) 239 [INSPIRE].
A. Géré, P. Vitale and J.-C. Wallet, Quantum gauge theories on noncommutative three-dimensional space, Phys. Rev. D 90 (2014) 045019 [arXiv:1312.6145] [INSPIRE].
D. Jurman and H. Steinacker, 2D fuzzy anti-de Sitter space from matrix models, JHEP 01 (2014) 100 [arXiv:1309.1598] [INSPIRE].
J. Heckman and H. Verlinde, Covariant non-commutative space-time, Nucl. Phys. B 894 (2015) 58 [arXiv:1401.1810] [INSPIRE].
M. Burić and J. Madore, Noncommutative de Sitter and FRW spaces, Eur. Phys. J. C 75 (2015) 502 [arXiv:1508.06058] [INSPIRE].
M. Sperling and H.C. Steinacker, Covariant 4-dimensional fuzzy spheres, matrix models and higher spin, J. Phys. A 50 (2017) 375202 [arXiv:1704.02863] [INSPIRE].
H.A. Kastrup, Position operators, gauge transformations, and the conformal group, Phys. Rev. 143 (1966) 1021 [INSPIRE].
A.H. Chamseddine, Invariant actions for noncommutative gravity, J. Math. Phys. 44 (2003) 2534 [hep-th/0202137] [INSPIRE].
A.H. Chamseddine, Deforming Einstein’s gravity, Phys. Lett. B 504 (2001) 33 [hep-th/0009153] [INSPIRE].
A.H. Chamseddine, A. Connes and W.D. van Suijlekom, Grand unification in the spectral Pati-Salam model, JHEP 11 (2015) 011 [arXiv:1507.08161] [INSPIRE].
F. Fathizadeh and M. Khalkhali, Curvature in noncommutative geometry, arXiv:1901.07438 [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, in Strings 2001: international conference, (2001) [hep-th/0106109] [INSPIRE].
B. Ydri, Review of M(atrix)-theory, type IIB matrix model and matrix string theory, arXiv:1708.00734 [INSPIRE].
H.C. Steinacker, Scalar modes and the linearized Schwarzschild solution on a quantized FLRW space-time in Yang-Mills matrix models, Class. Quant. Grav. 36 (2019) 205005 [arXiv:1905.07255] [INSPIRE].
M. Sperling and H.C. Steinacker, Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model, JHEP 07 (2019) 010 [arXiv:1901.03522] [INSPIRE].
F.W. Hehl, P. Von Der Heyde, G.D. Kerlick and J.M. Nester, General relativity with spin and torsion: foundations and prospects, Rev. Mod. Phys. 48 (1976) 393 [INSPIRE].
P.G.O. Freund, Introduction to supersymmetry, Cambridge University Press, Cambridge, U.K. (2012) [INSPIRE].
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge theory of the conformal and superconformal group, Phys. Lett. B 69 (1977) 304 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge, U.K. (2012).
A.H. Chamseddine, Supersymmetry and higher spin fields, Ph.D. thesis, Department of Theoretical Physics, Imperial College of Science and Technology, London, U.K. (1976).
A.H. Chamseddine and P.C. West, Supergravity as a gauge theory of supersymmetry, Nucl. Phys. B 129 (1977) 39 [INSPIRE].
L.-F. Li, Group theory of the spontaneously broken gauge symmetries, Phys. Rev. D 9 (1974) 1723 [INSPIRE].
A. Singh and S.M. Carroll, Modeling position and momentum in finite-dimensional Hilbert spaces via generalized Pauli operators, arXiv:1806.10134 [INSPIRE].
A. Barut, From Heisenberg algebra to conformal dynamical group, in Conformal groups and related symmetries.physical results and mathematical background, A. Barut and H.D. Doener eds., Springer, Berlin, Heidelberg, Germany (1985), pg. 3.
L. Álvarez-Gaumé, F. Meyer and M.A. Vazquez-Mozo, Comments on noncommutative gravity, Nucl. Phys. B 753 (2006) 92 [hep-th/0605113] [INSPIRE].
J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C 16 (2000) 161 [hep-th/0001203] [INSPIRE].
P. Aschieri and L. Castellani, Noncommutative D = 4 gravity coupled to fermions, JHEP 06 (2009) 086 [arXiv:0902.3817] [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory. Volume 2: loop amplitudes, anomalies and phenomenology, Cambridge University Press, Cambridge, U.K. (1987).
E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
L. Smolin and A. Starodubtsev, General relativity with a topological phase: an action principle, hep-th/0311163 [INSPIRE].
H.S. Yang, Emergent gravity from noncommutative spacetime, Int. J. Mod. Phys. A 24 (2009) 4473 [hep-th/0611174] [INSPIRE].
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Manolakos, G., Manousselis, P. & Zoupanos, G. Four-dimensional gravity on a covariant noncommutative space. J. High Energ. Phys. 2020, 1 (2020). https://doi.org/10.1007/JHEP08(2020)001
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DOI: https://doi.org/10.1007/JHEP08(2020)001