# Gauge Theories: From Kaluza–Klein to noncommutative gravity theories

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}/SU(3), Sp(4)/SU(2)× U(1)

_{non-max}, SU(3)/U(1)× U(1) and the group manifold SU(2)× SU(2) [32] (see also [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]). It is worth mentioning that, contrary to the CY case, the dimensional reduction of a 10-d $\mathcal{N}=1$ supersymmetric gauge theory over a non-symmetric coset space, leads to 4-d theories which include supersymmetry breaking terms [33,34,35].

## 2. Coset Space Dimensional Reduction of a Higher-Dimensional Theory

#### 2.1. CSDR Constraints

#### 2.2. The 4-d Effective Action

## 3. Fuzzy Spaces

#### 3.1. The Fuzzy Sphere

#### 3.2. Gauge Theory on a Fuzzy Sphere

## 4. Dimensional Reduction of a Higher-Dimensional Theory with Fuzzy Extra Dimensions

#### 4.1. Ordinary Fuzzy Dimensional Reduction

#### 4.2. Fuzzy CSDR

## 5. Orbifolds and Fuzzy Extra Dimensions

#### 5.1. $\mathcal{N}=4$ SYM Field Theory and ${\mathbb{Z}}_{3}$ Orbifolds

- Maximal embedding of ${\mathbb{Z}}_{3}$ into SU(4)${}_{R}$ is excluded since it leads to vanishing supersymmetry.
- Embedding of ${\mathbb{Z}}_{3}$ in a subgroup of SU(4)${}_{R}$:
- -
- into an SU(2) subgroup leads to $\mathcal{N}=2$ supersymmetric models with SU(2)${}_{R}$R-symmetry; and
- -
- into an SU(3) subgroup leads to $\mathcal{N}=1$ supersymmetric models with U(1)${}_{R}$ R-symmetry.

#### 5.2. Dynamical Generation of Twisted Fuzzy Spheres

#### 5.3. Chiral Models after the Fuzzy Orbifold Projection—The $SU{\left(3\right)}_{c}\times SU{\left(3\right)}_{L}\times SU{\left(3\right)}_{R}$ Model

_{c}× SU(3)

_{L}× SU(3)

_{R}[174,175] (see also [159,160,176,177,178,179,180] for a string theory approach). First, the integer N is written as $N=n+3$, therefore for each SU(N), the embedding would be:

## 6. Gravity as a Gauge Theory

#### 6.1. 4-d Einstein’s Gravity as a Gauge Theory

#### 6.2. 3-d Gravity as a Gauge Theory

#### 6.3. 4-d Weyl Gravity as a Gauge Theory

## 7. 3-d Gravity as a Gauge Theory on Noncommutative Spaces

## 8. 4-d Gravity as a Gauge Theory on Noncommutative Spaces

#### 8.1. Fuzzy de Sitter Space

#### 8.2. A Noncommutative Gauge Theory of 4-d Gravity

#### 8.2.1. Determination of the Gauge Group and Representation by $4\times 4$ Matrices

- (a)
- six generators of the Lorentz transformations: ${\mathrm{M}}_{ab}=-{\textstyle \frac{i}{4}}[{\Gamma}_{a},{\Gamma}_{b}]=-{\textstyle \frac{i}{2}}{\Gamma}_{a}{\Gamma}_{b}\text{},ab$;
- (b)
- four generators for the conformal boosts: ${\mathrm{K}}_{a}={\textstyle \frac{1}{2}}{\Gamma}_{a}$;
- (c)
- four generators for the local translations: ${\mathrm{P}}_{a}=-{\textstyle \frac{i}{2}}{\Gamma}_{a}{\Gamma}_{5}$;
- (d)
- one generator for special conformal transformations: $\mathrm{D}=-{\textstyle \frac{1}{2}}{\Gamma}_{5}$; and
- (e)
- one U(1) generator: $\mathbf{1}$.

#### 8.2.2. Noncommutative Gauge Theory of Gravity

#### 8.3. The Constraints for the symmetry breaking and the action

#### 8.4. The Action and Equations of Motion

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Green, M.B.; Schwarz, J.H.; Witten, E. Superstring Theory. Vol. 1: Introduction; Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 1987; 469p. [Google Scholar]
- Gross, D.J.; Harvey, J.A.; Martinec, E.J.; Rohm, R. Heterotic String Theory. 1. The Free Heterotic String. Nucl. Phys. B
**1985**, 256, 253–284. [Google Scholar] [CrossRef] - Forgacs, P.; Manton, N.S. Space-Time Symmetries in Gauge Theories. Commun. Math. Phys.
**1980**, 72, 15–35. [Google Scholar] [CrossRef] - Kapetanakis, D.; Zoupanos, G. Coset space dimensional reduction of gauge theories. Phys. Rep.
**1992**, 219, 4–76. [Google Scholar] [CrossRef] - Kubyshin, Y.A.; Volobuev, I.P.; Mourao, J.M.; Rudolph, G. Dimensional Reduction of Gauge Theories, Spontaneous Compactification and Model Building. Lect. Notes Phys.
**1990**, 349, 1. [Google Scholar] [CrossRef] - Scherk, J.; Schwarz, J.H. How to Get Masses from Extra Dimensions. Nucl. Phys. B
**1979**, 153, 61–88. [Google Scholar] [CrossRef] - Manton, N.S. Fermions and Parity Violation in Dimensional Reduction Schemes. Nucl. Phys. B
**1981**, 193, 502–516. [Google Scholar] [CrossRef] - Chapline, G.; Slansky, R. Dimensional Reduction and Flavor Chirality. Nucl. Phys. B
**1982**, 209, 461–483. [Google Scholar] [CrossRef] - Candelas, P.; Horowitz, G.T.; Strominger, A.; Witten, E. Vacuum Configurations for Superstrings. Nucl. Phys. B
**1985**, 258, 46–74. [Google Scholar] [CrossRef] - Cardoso, G.L.; Curio, G.; Dall’Agata, G.; Lust, D.; Manousselis, P.; Zoupanos, G. NonKahler string backgrounds and their five torsion classes. Nucl. Phys. B
**2003**, 652, 5–34. [Google Scholar] [CrossRef] - Strominger, A. Superstrings with Torsion. Nucl. Phys. B
**1986**, 274, 253–284. [Google Scholar] [CrossRef] - Lust, D. Compactification of Ten-dimensional Superstring Theories Over Ricci Flat Coset Spaces. Nucl. Phys. B
**1986**, 276, 220–240. [Google Scholar] [CrossRef] - Castellani, L.; Lust, D. Superstring Compactification on Homogeneous Coset Spaces with Torsion. Nucl. Phys. B
**1988**, 296, 143–156. [Google Scholar] [CrossRef] - Becker, K.; Becker, M.; Dasgupta, K.; Green, P.S. Compactifications of heterotic theory on nonKahler complex manifolds, 1. JHEP
**2003**, 0304, 007. [Google Scholar] [CrossRef] - Becker, K.; Becker, M.; Green, P.S.; Dasgupta, K.; Sharpe, E. Compactifications of heterotic strings on nonKahler complex manifolds, 2. Nucl. Phys. B
**2004**, 678, 19–100. [Google Scholar] [CrossRef] - Gurrieri, S.; Lukas, A.; Micu, A. Heterotic on half-flat. Phys. Rev. D
**2004**, 70, 126009. [Google Scholar] [CrossRef] - Benmachiche, I.; Louis, J.; Martinez-Pedrera, D. The Effective action of the heterotic string compactified on manifolds with SU(3) structure. Class. Quantum Gravity
**2008**, 25, 135006. [Google Scholar] [CrossRef] - Micu, A. Heterotic compactifications and nearly-Kahler manifolds. Phys. Rev. D
**2004**, 70, 126002. [Google Scholar] [CrossRef] - Frey, A.R.; Lippert, M. AdS strings with torsion: Non-complex heterotic compactifications. Phys. Rev. D
**2005**, 72, 126001. [Google Scholar] [CrossRef] - Manousselis, P.; Prezas, N.; Zoupanos, G. Supersymmetric compactifications of heterotic strings with fluxes and condensates. Nucl. Phys. B
**2006**, 739, 85–105. [Google Scholar] [CrossRef] [Green Version] - Chatzistavrakidis, A.; Manousselis, P.; Zoupanos, G. Reducing the Heterotic Supergravity on nearly-Kahler coset spaces. Fortschr. Phys.
**2009**, 57, 527–534. [Google Scholar] [CrossRef] - Chatzistavrakidis, A.; Zoupanos, G. Dimensional Reduction of the Heterotic String over nearly-Kaehler manifolds. J. High Energy Phys.
**2009**, 0909, 077. [Google Scholar] [CrossRef] - Dolan, B.P.; Szabo, R.J. Dimensional Reduction and Vacuum Structure of Quiver Gauge Theory. J. High Energy Phys.
**2009**, 0908, 038. [Google Scholar] [CrossRef] - Lechtenfeld, O.; Nolle, C.; Popov, A.D. Heterotic compactifications on nearly Kahler manifolds. J. High Energy Phys.
**2010**, 1009, 074. [Google Scholar] [CrossRef] - Popov, A.D.; Szabo, R.J. Double quiver gauge theory and nearly Kahler flux compactifications. J. High Energy Phys.
**2012**, 1202, 033. [Google Scholar] [CrossRef] - Klaput, M.; Lukas, A.; Matti, C. Bundles over Nearly-Kahler Homogeneous Spaces in Heterotic String Theory. J. High Energy Phys.
**2011**, 1109, 100. [Google Scholar] [CrossRef] - Chatzistavrakidis, A.; Lechtenfeld, O.; Popov, A.D. Nearly Káhler heterotic compactifications with fermion condensates. J. High Energy Phys.
**2012**, 1204, 114. [Google Scholar] [CrossRef] - Gray, J.; Larfors, M.; Lust, D. Heterotic domain wall solutions and SU(3) structure manifolds. J. High Energy Phys.
**2012**, 1208, 099. [Google Scholar] [CrossRef] - Klaput, M.; Lukas, A.; Matti, C.; Svanes, E.E. Moduli Stabilising in Heterotic Nearly Káhler Compactifications. J. High Energy Phys.
**2013**, 1301, 015. [Google Scholar] [CrossRef] - Irges, N.; Zoupanos, G. Reduction of N=1, E
_{8}SYM over SU(3)/U(1) x U(1) x Z_{3}and its four-dimensional effective action. Phys. Lett. B**2011**, 698, 146–151. [Google Scholar] [CrossRef] - Irges, N.; Orfanidis, G.; Zoupanos, G. Dimensional Reduction of N=1, E
_{8}SYM over SU(3)/U(1) x U(1) x Z_{3}and its four-dimensional effective action. PoS CORFU**2011**, 2011, 105. [Google Scholar] [CrossRef] - Butruille, J.-B. Homogeneous nearly Kähler manifolds. arXiv
**2006**, arXiv:math.DG/0612655. [Google Scholar] - Manousselis, P.; Zoupanos, G. Soft supersymmetry breaking due to dimensional reduction over nonsymmetric coset spaces. Phys. Lett. B
**2001**, 518, 171–180. [Google Scholar] [CrossRef] - Manousselis, P.; Zoupanos, G. Supersymmetry breaking by dimensional reduction over coset spaces. Phys. Lett. B
**2001**, 504, 122–130. [Google Scholar] [CrossRef] [Green Version] - Manousselis, P.; Zoupanos, G. Dimensional reduction of ten-dimensional supersymmetric gauge theories in the N=1, D=4 superfield formalism. J. High Energy Phys.
**2004**, 0411, 025. [Google Scholar] [CrossRef] - Connes, A. Noncommutative Geometry; Academic Press, Inc.: San Diego, CA, USA, 1994. [Google Scholar]
- Madore, J. An Introduction to Noncommutative Differential Geometry and Its Physical Applications; London Mathematical Society Lecture Note Series; Cambridge University Press: Cambridge, UK, 1999; Vol. 257. [Google Scholar]
- Madore, J. The Fuzzy sphere. Class. Quantum Gravity
**1992**, 9, 69–88. [Google Scholar] [CrossRef] - Buric, M.; Grammatikopoulos, T.; Madore, J.; Zoupanos, G. Gravity and the structure of noncommutative algebras. J. High Energy Phys.
**2006**, 0604, 054. [Google Scholar] [CrossRef] - Filk, T. Divergencies in a field theory on quantum space. Phys. Lett. B
**1996**, 376, 53–58. [Google Scholar] [CrossRef] - Varilly, J.C.; Gracia-Bondia, J.M. On the ultraviolet behavior of quantum fields over noncommutative manifolds. Int. J. Mod. Phys. A
**1999**, 14, 1305. [Google Scholar] [CrossRef] - Chaichian, M.; Demichev, A.; Presnajder, P. Quantum field theory on noncommutative space-times and the persistence of ultraviolet divergences. Nucl. Phys. B
**2000**, 567, 360–390. [Google Scholar] [CrossRef] - Minwalla, S.; Raamsdonk, M.V.; Seiberg, N. Noncommutative perturbative dynamics. J. High Energy Phys.
**2000**, 0002, 020. [Google Scholar] [CrossRef] - Grosse, H.; Wulkenhaar, R. Renormalization of phi**4 theory on noncommutative R**4 to all orders. Lett. Math. Phys.
**2005**, 71, 13–26. [Google Scholar] [CrossRef] - Grosse, H.; Steinacker, H. Exact renormalization of a noncommutative phi**3 model in 6 dimensions. Adv. Theor. Math. Phys.
**2008**, 12, 605–639. [Google Scholar] [CrossRef] - Grosse, H.; Steinacker, H. Finite gauge theory on fuzzy CP**2. Nucl. Phys. B
**2005**, 707, 145–198. [Google Scholar] [CrossRef] - Connes, A.; Lott, J. Particle Models and Noncommutative Geometry (Expanded Version). Nucl. Phys. Proc. Suppl.
**1991**, 18B, 29–47. [Google Scholar] [CrossRef] - Chamseddine, A.H.; Connes, A. The Spectral Action Principle. Commun. Math. Phys.
**1997**, 186, 731. [Google Scholar] [CrossRef] - Chamseddine, A.H.; Connes, A. Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model. Phys. Rev. Lett.
**2007**, 99, 191601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Martin, C.P.; Gracia-Bondia, J.M.; Varilly, J.C. The Standard model as a noncommutative geometry: The Low-energy regime. Phys. Rep.
**1998**, 294, 363–406. [Google Scholar] [CrossRef] - Dubois-Violette, M.; Madore, J.; Kerner, R. Gauge Bosons in a Noncommutative Geometry. Phys. Lett. B
**1989**, 217, 485–488. [Google Scholar] [CrossRef] - Dubois-Violette, M.; Madore, J.; Kerner, R. Classical Bosons in a Noncommutative Geometry. Class. Quantum Gravity
**1989**, 6, 1709. [Google Scholar] [CrossRef] - Dubois-Violette, M.; Kerner, R.; Madore, J. Noncommutative Differential Geometry and New Models of Gauge Theory. J. Math. Phys.
**1990**, 31, 323. [Google Scholar] [CrossRef] - Madore, J. On a quark-lepton duality. Phys. Lett. B
**1993**, 305, 84–89. [Google Scholar] [CrossRef] - Madore, J. On a noncommutative extension of electrodynamics. Fundam. Theor. Phys.
**1993**, 52, 285–298. [Google Scholar] - Connes, A.; Douglas, M.R.; Schwarz, A.S. Noncommutative geometry and matrix theory: Compactification on tori. J. High Energy Phys.
**1998**, 9802, 003. [Google Scholar] [CrossRef] - Seiberg, N.; Witten, E. String theory and noncommutative geometry. J. High Energy Phys.
**1999**, 9909, 032. [Google Scholar] [CrossRef] - Ishibashi, N.; Kawai, H.; Kitazawa, Y.; Tsuchiya, A. A Large N reduced model as superstring. Nucl. Phys. B
**1997**, 498, 467–491. [Google Scholar] [CrossRef] - Jurco, B.; Schraml, S.; Schupp, P.; Wess, J. Enveloping algebra valued gauge transformations for nonAbelian gauge groups on noncommutative spaces. Eur. Phys. J. C
**2000**, 17, 521–526. [Google Scholar] [CrossRef] - Jurco, B.; Schupp, P.; Wess, J. NonAbelian noncommutative gauge theory via noncommutative extra dimensions. Nucl. Phys. B
**2001**, 604, 148–180. [Google Scholar] [CrossRef] - Jurco, B.; Moller, L.; Schraml, S.; Schupp, P.; Wess, J. Construction of nonAbelian gauge theories on noncommutative spaces. Eur. Phys. J. C
**2001**, 21, 383–388. [Google Scholar] [CrossRef] - Barnich, G.; Brandt, F.; Grigoriev, M. Seiberg-Witten maps and noncommutative Yang-Mills theories for arbitrary gauge groups. J. High Energy Phys.
**2002**, 0208, 023. [Google Scholar] [CrossRef] - Chaichian, M.; Presnajder, P.; Sheikh-Jabbari, M.M.; Tureanu, A. Noncommutative standard model: Model building. Eur. Phys. J. C
**2003**, 29, 413–432. [Google Scholar] [CrossRef] - Calmet, X.; Jurco, B.; Schupp, P.; Wess, J.; Wohlgenannt, M. The Standard model on noncommutative space-time. Eur. Phys. J. C
**2002**, 23, 363–376. [Google Scholar] [CrossRef] - Aschieri, P.; Jurco, B.; Schupp, P.; Wess, J. Noncommutative GUTs, standard model and C,P,T. Nucl. Phys. B
**2003**, 651, 45–70. [Google Scholar] [CrossRef] - Behr, W.; Deshpande, N.G.; Duplancic, G.; Schupp, P.; Trampetic, J.; Wess, J. The Z —> gamma gamma, g g decays in the noncommutative standard model. Eur. Phys. J. C
**2003**, 29, 441–447. [Google Scholar] [CrossRef] - Aschieri, P.; Madore, J.; Manousselis, P.; Zoupanos, G. Dimensional reduction over fuzzy coset spaces. J. High Energy Phys.
**2004**, 0404, 034. [Google Scholar] [CrossRef] - Aschieri, P.; Madore, J.; Manousselis, P.; Zoupanos, G. Unified theories from fuzzy extra dimensions. Fortschr. Phys.
**2004**, 52, 718–723. [Google Scholar] [CrossRef] [Green Version] - Aschieri, P.; Madore, J.; Manousselis, P.; Zoupanos, G. Renormalizable theories from fuzzy higher dimensions. arXiv
**2004**, arXiv:hep-th/0503039. [Google Scholar] - Aschieri, P.; Grammatikopoulos, T.; Steinacker, H.; Zoupanos, G. Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking. J. High Energy Phys.
**2006**, 0609, 026. [Google Scholar] [CrossRef] - Aschieri, P.; Steinacker, H.; Madore, J.; Manousselis, P.; Zoupanos, G. Fuzzy Extra Dimensions: Dimensional Reduction, Dynamical Generation and Renormalizability. SFIN A
**2007**, arXiv:0704.28801, 25. [Google Scholar] - Steinacker, H.; Zoupanos, G. Fermions on spontaneously generated spherical extra dimensions. J. High Energy Phys.
**2007**, 0709, 017. [Google Scholar] [CrossRef] - Chatzistavrakidis, A.; Steinacker, H.; Zoupanos, G. On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes. Fortschr. Phys.
**2010**, 58, 537–552. [Google Scholar] [CrossRef] [Green Version] - Chatzistavrakidis, A.; Steinacker, H.; Zoupanos, G. Orbifolds, fuzzy spheres and chiral fermions. J. High Energy Phys.
**2010**, 1005, 100. [Google Scholar] [CrossRef] - Chatzistavrakidis, A.; Zoupanos, G. Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions. SIGMA
**2010**, 6, 063. [Google Scholar] [CrossRef] [Green Version] - Gavriil, D.; Manolakos, G.; Orfanidis, G.; Zoupanos, G. Higher-Dimensional Unification with continuous and fuzzy coset spaces as extra dimensions. Fortschr. Phys.
**2015**, 63, 442–467. [Google Scholar] [CrossRef] [Green Version] - Manolakos, G.; Zoupanos, G. The trinification model SU(3)
^{3}from orbifolds for fuzzy spheres. Phys. Part. Nucl. Lett.**2017**, 14, 322–327. [Google Scholar] [CrossRef] - Manolakos, G.; Zoupanos, G. Higher-Dimensional Unified Theories with continuous and fuzzy coset spaces as extra dimensions. Springer Proc. Math. Stat.
**2016**, 191, 203–229. [Google Scholar] [CrossRef] - Utiyama, R. Invariant theoretical interpretation of interaction. Phys. Rev.
**1956**, 101, 1597. [Google Scholar] [CrossRef] - Kibble, T.W.B. Lorentz invariance and the gravitational field. J. Math. Phys.
**1961**, 2, 212. [Google Scholar] [CrossRef] - Stelle, K.S.; West, P.C. Spontaneously Broken De Sitter Symmetry and the Gravitational Holonomy Group. Phys. Rev. D
**1980**, 21, 1466. [Google Scholar] [CrossRef] - MacDowell, S.W.; Mansouri, F. Unified Geometric Theory of Gravity and Supergravity. Phys. Rev. Lett.
**1977**, 38, 739, Erratum in**1977**, 38, 1376. [Google Scholar] [CrossRef] - Ivanov, E.A.; Niederle, J. Gauge Formulation of Gravitation Theories. 1. The Poincare, De Sitter and Conformal Cases. In Proceedings of the 9th International Colloquium on Group Theoretical Methods in Physics, Cocoyoc, Mexico, 23–27 June 1980; C80-06-23.3, pp. 545–551. Phys. Rev. D
**1982**, 25, 976. [Google Scholar] [CrossRef] - Ivanov, E.A.; Niederle, J. Gauge Formulation of Gravitation Theories. 2. The Special Conformal Case. Phys. Rev. D
**1982**, 25, 988. [Google Scholar] [CrossRef] - Kibble, T.W.B.; Stelle, K.S. Gauge theories of gravity and supergravity. In Progress In Quantum Field Theory; Ezawa, H., Kamefuchi, S., Eds.; pp. 57–81.
- Kaku, M.; Townsend, P.K.; van Nieuwenhuizen, P. Gauge Theory of the Conformal and Superconformal Group. Phys. Lett.
**1977**, 69B, 304–308. [Google Scholar] [CrossRef] - Fradkin, E.S.; Tseytlin, A.A. Conformal Supergravity. Phys. Rep.
**1985**, 119, 233. [Google Scholar] [CrossRef] - Freedman, D.Z.; Proeyen, A.V. Supergravity; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Chamseddine, A.H. Supersymmetry and Higher Spin Fields. Ph.D. Thesis, Department of Theoretical Physics Imperial, College of Science and Technology, London, UK, 1976. [Google Scholar]
- Chamseddine, A.H.; West, P.C. Supergravity as a Gauge Theory of Supersymmetry. Nucl. Phys. B
**1977**, 129, 39–44. [Google Scholar] [CrossRef] - Witten, E. (2+1)-Dimensional Gravity as an Exactly Soluble System. Nucl. Phys. B
**1988**, 311, 46–78. [Google Scholar] [CrossRef] - Madore, J.; Schraml, S.; Schupp, P.; Wess, J. Gauge theory on noncommutative spaces. Eur. Phys. J. C
**2000**, 16, 161–167. [Google Scholar] [CrossRef] [Green Version] - Chamseddine, A.H. Deforming Einstein’s gravity. Phys. Lett. B
**2001**, 504, 33–37. [Google Scholar] [CrossRef] - Chamseddine, A.H. SL(2,C) gravity with complex vierbein and its noncommutative extension. Phys. Rev. D
**2004**, 69, 024015. [Google Scholar] [CrossRef] - Aschieri, P.; Castellani, L. Noncommutative D=4 gravity coupled to fermions. J. High Energy Phys.
**2009**, 0906, 086. [Google Scholar] [CrossRef] - Aschieri, P.; Castellani, L. Noncommutative supergravity in D=3 and D=4. J. High Energy Phys.
**2009**, 0906, 087. [Google Scholar] [CrossRef] - Ćirić, M.D.; Nikolić, B.; Radovanović, V. Noncommutative SO(2,3)
_{★}gravity: Noncommutativity as a source of curvature and torsion. Phys. Rev. D**2017**, 96, 064029. [Google Scholar] [CrossRef] - Cacciatori, S.; Klemm, D.; Martucci, L.; Zanon, D. Noncommutative Einstein-AdS gravity in three-dimensions. Phys. Lett. B
**2002**, 536, 101. [Google Scholar] [CrossRef] - Cacciatori, S.; Chamseddine, A.H.; Klemm, D.; Martucci, L.; Sabra, W.A.; Zanon, D. Noncommutative gravity in two dimensions. Class. Quantum Gravity
**2002**, 19, 4029. [Google Scholar] [CrossRef] - Aschieri, P.; Castellani, L. Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map. J. High Energy Phys.
**2014**, 1411, 103. [Google Scholar] [CrossRef] - Banados, M.; Chandia, O.; Grandi, N.E.; Schaposnik, F.A.; Silva, G.A. Three-dimensional noncommutative gravity. Phys. Rev. D
**2001**, 64, 084012. [Google Scholar] [CrossRef] - Banks, T.; Fischler, W.; Shenker, S.H.; Susskind, L. M theory as a matrix model: A Conjecture. Phys. Rev. D
**1997**, 55, 5112. [Google Scholar] [CrossRef] - Aoki, H.; Iso, S.; Kawai, H.; Kitazawa, Y.; Tada, T. Space-time structures from IIB matrix model. Prog. Theor. Phys.
**1998**, 99, 713–746. [Google Scholar] [CrossRef] - Hanada, M.; Kawai, H.; Kimura, Y. Describing curved spaces by matrices. Prog. Theor. Phys.
**2006**, 114, 1295–1316. [Google Scholar] [CrossRef] - Furuta, K.; Hanada, M.; Kawai, H.; Kimura, Y. Field equations of massless fields in the new interpretation of the matrix model. Nucl. Phys. B
**2007**, 767, 82–99. [Google Scholar] [CrossRef] [Green Version] - Yang, H.S. Emergent Gravity from Noncommutative Spacetime. Int. J. Mod. Phys. A
**2009**, 24, 4473–4517. [Google Scholar] [CrossRef] - Steinacker, H. Emergent Geometry and Gravity from Matrix Models: An Introduction. Class. Quantum Gravity
**2010**, 27, 133001. [Google Scholar] [CrossRef] - Kim, S.W.; Nishimura, J.; Tsuchiya, A. Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions. Phys. Rev. Lett.
**2012**, 108, 011601. [Google Scholar] [CrossRef] [PubMed] - Nishimura, J. The Origin of space-time as seen from matrix model simulations. PTEP
**2012**, 2012, 01A101. [Google Scholar] [CrossRef] - Nair, V.P. Gravitational fields on a noncommutative space. Nucl. Phys. B
**2003**, 651, 313–327. [Google Scholar] [CrossRef] [Green Version] - Abe, Y.; Nair, V.P. Noncommutative gravity: Fuzzy sphere and others. Phys. Rev. D
**2003**, 68, 025002. [Google Scholar] [CrossRef] - Valtancoli, P. Gravity on a fuzzy sphere. Int. J. Mod. Phys. A
**2004**, 19, 361–370. [Google Scholar] [CrossRef] - Nair, V.P. The Chern-Simons one-form and gravity on a fuzzy space. Nucl. Phys. B
**2006**, 750, 321–333. [Google Scholar] [CrossRef] - Burić, M.; Madore, J.; Zoupanos, G. WKB Approximation in Noncommutative Gravity. SIGMA
**2007**, 3, 125. [Google Scholar] [CrossRef] [Green Version] - Burić, M.; Madore, J.; Zoupanos, G. The Energy-momentum of a Poisson structure. Eur. Phys. J. C
**2008**, 55, 489–498. [Google Scholar] [CrossRef] - Snyder, H.S. Quantized space-time. Phys. Rev.
**1947**, 71, 38. [Google Scholar] [CrossRef] - Yang, C.N. On quantized space-time. Phys. Rev.
**1947**, 72, 874. [Google Scholar] [CrossRef] - Heckman, J.; Verlinde, H. Covariant non-commutative space-time. Nucl. Phys. B
**2015**, 894, 58–74. [Google Scholar] [CrossRef] - Burić, M.; Madore, J. Noncommutative de Sitter and FRW spaces. Eur. Phys. J. C
**2015**, 75, 502. [Google Scholar] [CrossRef] - Sperling, M.; Steinacker, H.C. Covariant 4-dimensional fuzzy spheres, matrix models and higher spin. J. Phys. A
**2017**, 50, 375202. [Google Scholar] [CrossRef] - Burić, M.; Latas, D.; Nenadovixcx, L. Fuzzy de Sitter Space. arXiv
**2017**, arXiv:1709.05158. [Google Scholar] [CrossRef] - Steinacker, H.C. Emergent gravity on covariant quantum spaces in the IKKT model. J. High Energy Phys.
**2016**, 1612, 156. [Google Scholar] [CrossRef] - Chatzistavrakidis, A.; Jonke, L.; Jurman, D.; Manolakos, G.; Manousselis, P.; Zoupanos, G. Noncommutative Gauge Theory and Gravity in Three Dimensions. Fortschr. Phys.
**2018**, 66, 1800047. [Google Scholar] [CrossRef] - Jurman, D.; Manolakos, G.; Manousselis, P.; Zoupanos, G. Gravity as a Gauge Theory on Three-Dimensional Noncommutative spaces. PoS CORFU
**2018**, 2017, 162. [Google Scholar] [CrossRef] - Hammou, A.B.; Lagraa, M.; Sheikh-Jabbari, M.M. Coherent state induced star product on R**3(lambda) and the fuzzy sphere. Phys. Rev. D
**2002**, 66, 025025. [Google Scholar] [CrossRef] - Vitale, P. Noncommutative field theory on ${\mathbb{R}}_{\lambda}^{3}$. Fortschr. Phys.
**2014**, 62, 825. [Google Scholar] [CrossRef] - Kováčik, S.; Prešnajder, P. The velocity operator in quantum mechanics in noncommutative space. J. Math. Phys.
**2013**, 54, 102103. [Google Scholar] [CrossRef] [Green Version] - Jurman, D.; Steinacker, H. 2D fuzzy Anti-de Sitter space from matrix models. J. High Energy Phys.
**2014**, 1401, 100. [Google Scholar] [CrossRef] - Manolakos, G.; Manousselis, P.; Zoupanos, G. Four-dimensional Gravity on a Covariant Noncommutative Space. arXiv
**2019**, arXiv:1902.10922. [Google Scholar] - Kaluza, T. Zum Unitätsproblem der Physik. Sitzungsber. Preuss. Akad. Wiss. Berl. (Math. Phys.)
**1921**, 1921, 966–972, Translation, Int. J. Mod. Phys. D**2018**, 27, 1870001. [Google Scholar] [CrossRef] - Klein, O. Quantum Theory and Five-Dimensional Theory of Relativity. (In German and English). Z. Phys.
**1926**, 37, 895–906, Surv. High Energ. Phys.**1986**, 5, 241–244. [Google Scholar] [CrossRef] - Kerner, R. Generalization of the Kaluza-Klein Theory for an Arbitrary Nonabelian Gauge Group. Ann. Inst. Henr. Poincare Phys. Theor.
**1968**, 9, 143–152. [Google Scholar] - Cho, Y.M. Higher - Dimensional Unifications of Gravitation and Gauge Theories. J. Math. Phys.
**2006**, 16, 2029–2035. [Google Scholar] [CrossRef] - Cho, Y.M.; Freund, P.G.O. Nonabelian Gauge Fields in Nambu-Goldstone Fields. Phys. Rev. D
**1975**, 12, 1711. [Google Scholar] [CrossRef] - Mecklenburg, W. The Kaluza-Klein Idea: Status and Prospects. Fortschr. Phys.
**1984**, 32, 207–260. [Google Scholar] [CrossRef] - Bailin, D.; Love, A. Kaluza-klein Theories. Rep. Prog. Phys.
**1987**, 50, 1087. [Google Scholar] [CrossRef] - Salam, A.; Strathdee, J.A. On Kaluza-Klein Theory. Ann. Phys.
**1982**, 141, 316–352. [Google Scholar] [CrossRef] - Witten, E. Fermion Quantum Numbers Kaluza-Klein Theory. Conf. Proc. C
**1983**, 8306011, 227. [Google Scholar] - Horvath, Z.; Palla, L.; Cremmer, E.; Scherk, J. Grand Unified Schemes and Spontaneous Compactification. Nucl. Phys. B
**1977**, 127, 57–65. [Google Scholar] [CrossRef] - Wetterich, C. Dimensional Reduction of Weyl, Majorana and Majorana-weyl Spinors. Nucl. Phys. B
**1983**, 222, 20–44. [Google Scholar] [CrossRef] - Palla, L. On Dimensional Reduction of Gauge Theories: Symmetric Fields and Harmonic Expansion. Z. Phys. C
**1984**, 24, 195. [Google Scholar] [CrossRef] - Pilch, K.; Schellekens, A.N. Formulae for the Eigenvalues of the Laplacian on Tensor Harmonics on Symmetric Coset Spaces. J. Math. Phys.
**1984**, 25, 3455. [Google Scholar] [CrossRef] - Forgacs, P.; Horvath, Z.; Palla, L. Spontaneous Compactification To Nonsymmetric Spaces. Z. Phys. C
**1986**, 30, 261–266. [Google Scholar] [CrossRef] - Chapline, G.; Manton, N.S. The Geometrical Significance of Certain Higgs Potentials: An Approach to Grand Unification. Nucl. Phys. B
**1981**, 184, 391–405. [Google Scholar] [CrossRef] - Bais, F.A.; Barnes, K.J.; Forgacs, P.; Zoupanos, G. Dimensional Reduction of Gauge Theories Yielding Unified Models Spontaneously Broken to SU(3) X U(1). Nucl. Phys. B
**1986**, 263, 557–590. [Google Scholar] [CrossRef] - Kubyshin, Y.A.; Mourao, J.M.; Volobuev, I.P. Scalar Fields in the Dimensional Reduction Scheme for Symmetric Spaces. Int. J. Mod. Phys. A
**1989**, 4, 151–171. [Google Scholar] [CrossRef] - Harnad, J.P.; Shnider, S.; Vinet, L. The Yang-Mills System in Compactified Minkowski Space. 1. Invariance Conditions and SU(2) Invariant Solutions. J. Math. Phys.
**1979**, 20, 931. [Google Scholar] [CrossRef] - Harnad, J.P.; Vinet, L.; Shnider, S. Group Actions on Principal Bundles and Invariance Conditions for Gauge Fields. J. Math. Phys.
**1980**, 21, 2719. [Google Scholar] [CrossRef] - Harnad, J.P.; Shnider, S.; Tafel, J. Group Actions on Principal Bundles and Dimensional Reduction. Lett. Math. Phys.
**1980**, 4, 107–113. [Google Scholar] [CrossRef] - Farakos, K.; Koutsoumbas, G.; Surridge, M.; Zoupanos, G. Dimensional Reduction and the Higgs Potential. Nucl. Phys. B
**1987**, 291, 128–140. [Google Scholar] [CrossRef] - Farakos, K.; Koutsoumbas, G.; Surridge, M.; Zoupanos, G. Geometrical Hierarchy and Spontaneous Symmetry Breaking. Phys. Lett. B
**1987**, 191, 135–140. [Google Scholar] [CrossRef] - Barnes, K.J.; Forgacs, P.; Surridge, M.; Zoupanos, G. On Fermion Masses in a Dimensional Reduction Scheme. Z. Phys. C
**1987**, 33, 427–431. [Google Scholar] [CrossRef] - Harland, D.; Kurkcuoglu, S. Equivariant reduction of Yang-Mills theory over the fuzzy sphere and the emergent vortices. Nucl. Phys. B
**2009**, 821, 380–398. [Google Scholar] [CrossRef] - Arkani-Hamed, N.; Cohen, A.G.; Georgi, H. Electroweak symmetry breaking from dimensional deconstruction. Phys. Lett. B
**2001**, 513, 232. [Google Scholar] [CrossRef] - Arkani-Hamed, N.; Cohen, A.G.; Georgi, H. (De)constructing dimensions. Phys. Rev. Lett.
**2001**, 86, 4757. [Google Scholar] [CrossRef] - Andrews, R.P.; Dorey, N. Deconstruction of the Maldacena-Nunez compactification. Nucl. Phys. B
**2006**, 751, 304–341. [Google Scholar] [CrossRef] - Kachru, S.; Silverstein, E. 4-D conformal theories and strings on orbifolds. Phys. Rev. Lett.
**1998**, 80, 4855. [Google Scholar] [CrossRef] - Steinacker, H. Quantized gauge theory on the fuzzy sphere as random matrix model. Nucl. Phys. B
**2004**, 679, 66–98. [Google Scholar] [CrossRef] [Green Version] - Kim, J.E. Z(3) orbifold construction of SU(3)**3 GUT with sin**2 (theta0(W)) = 3/8. Phys. Lett. B
**2003**, 564, 35–41. [Google Scholar] [CrossRef] - Choi, K.S.; Kim, J.E. Three family Z(3) orbifold trinification, MSSM and doublet triplet splitting problem. Phys. Lett. B
**2003**, 567, 87–92. [Google Scholar] [CrossRef] - Maalampi, J.; Roos, M. Physics of Mirror Fermions. Phys. Rep.
**1990**, 186, 53. [Google Scholar] [CrossRef] - Bailin, D.; Love, A. Orbifold compactifications of string theory. Phys. Rep.
**1999**, 315, 285–408. [Google Scholar] [CrossRef] - Douglas, M.R.; Greene, B.R.; Morrison, D.R. Orbifold resolution by D-branes. Nucl. Phys. B
**1997**, 506, 84–106. [Google Scholar] [CrossRef] [Green Version] - Aldazabal, G.; Ibanez, L.E.; Quevedo, F.; Uranga, A.M. D-branes at singularities: A Bottom up approach to the string embedding of the standard model. J. High Energy Phys.
**2000**, 0008, 002. [Google Scholar] [CrossRef] - Lawrence, A.E.; Nekrasov, N.; Vafa, C. On conformal field theories in four-dimensions. Nucl. Phys. B
**1998**, 533, 199–209. [Google Scholar] [CrossRef] - Kiritsis, E. D-branes in standard model building, gravity and cosmology. Phys. Rep.
**2005**, 421, 105–190, Erratum in**2006**, 429, 121–122. [Google Scholar] [CrossRef] [Green Version] - Sohnius, M.F. Introducing Supersymmetry. Phys. Rep.
**1985**, 128, 39–204. [Google Scholar] [CrossRef] - Brink, L.; Schwarz, J.H.; Scherk, J. Supersymmetric Yang-Mills Theories. Nucl. Phys. B
**1977**, 121, 77–92. [Google Scholar] [CrossRef] - Gliozzi, F.; Scherk, J.; Olive, D.I. Supersymmetry, Supergravity Theories and the Dual Spinor Model. Nucl. Phys. B
**1977**, 122, 253–290. [Google Scholar] [CrossRef] - Djouadi, A. The Anatomy of electro-weak symmetry breaking. II. The Higgs bosons in the minimal supersymmetric model. Phys. Rep.
**2008**, 459, 1–241. [Google Scholar] [CrossRef] - Steinacker, H. Gauge theory on the fuzzy sphere and random matrices. Springer Proc. Phys.
**2005**, 98, 307–311. [Google Scholar] [CrossRef] - Grosse, H.; Lizzi, F.; Steinacker, H. Noncommutative gauge theory and symmetry breaking in matrix models. Phys. Rev. D
**2010**, 81, 085034. [Google Scholar] [CrossRef] - Steinacker, H. Emergent Gravity and Noncommutative Branes from Yang-Mills Matrix Models. Nucl. Phys. B
**2009**, 810, 1–39. [Google Scholar] [CrossRef] - Glashow, S.L. Trinification of All Elementary Particle Forces. In Proceedings of the Fifth Workshop on Grand Unification, Providence, Rhode Island, 12–14 April 1984. Print-84-0577 (BOSTON). [Google Scholar]
- Rizov, V.A. A Gauge Model of the Electroweak and Strong Interactions Based on the Group SU(3)
_{L}X SU(3)_{R}X SU(3)_{c}. Bulg. J. Phys.**1981**, 8, 461–477. [Google Scholar] - Heinemeyer, S.; Ma, E.; Mondragon, M.; Zoupanos, G. Finite SU(3)**3 model. AIP Conf. Proc.
**2010**, 1200, 568. [Google Scholar] [CrossRef] - Ma, E.; Mondragon, M.; Zoupanos, G. Finite SU(N)**k unification. J. High Energy Phys.
**2004**, 0412, 026. [Google Scholar] [CrossRef] - Lazarides, G.; Panagiotakopoulos, C. MSSM from SUSY trinification. Phys. Lett. B
**1994**, 336, 190–193. [Google Scholar] [CrossRef] [Green Version] - Babu, K.S.; He, X.G.; Pakvasa, S. Neutrino Masses and Proton Decay Modes in SU(3) X SU(3) X SU(3) Trinification. Phys. Rev. D
**1986**, 33, 763. [Google Scholar] [CrossRef] - Leontaris, G.K.; Rizos, J. A D-brane inspired U(3)(C) x U(3)(L) x U(3)(R) model. Phys. Lett. B
**2006**, 632, 710–716. [Google Scholar] [CrossRef] - Heinemeyer, S.; Mondragon, M.; Zoupanos, G. The LHC Higgs boson discovery: Implications for Finite Unified Theories. Int. J. Mod. Phys. A
**2014**, 29, 1430032. [Google Scholar] [CrossRef] - Heinemeyer, S.; Mondragon, M.; Tracas, N.; Zoupanos, G. Reduction of Couplings in Quantum Field Theories with applications in Finite Theories and the MSSM. Springer Proc. Math. Stat.
**2014**, 111, 177. [Google Scholar] [CrossRef] - Heinemeyer, S.; Mondragon, M.; Zoupanos, G. Finite Unification: Theory and Predictions. SIGMA
**2010**, 6, 049. [Google Scholar] [CrossRef] [Green Version] - Mondragon, M.; Zoupanos, G. Finite unified theories: Theoretical basis and phenomenological implications. Phys. Part. Nucl. Lett.
**2011**, 8, 173. [Google Scholar] [CrossRef] - Heinemeyer, S.; Mondragón, M.; Tracas, N.; Zoupanos, G. Reduction of Couplings and its application in Particle Physics. arXiv
**2019**, arXiv:1904.00410. [Google Scholar] [CrossRef] - Chatzistavrakidis, A.; Steinacker, H.; Zoupanos, G. Orbifold matrix models and fuzzy extra dimensions. In Proceedings of the School and Workshops on Elementary Particle Physics and Gravity, Corfu, Greece, 4–18 September 2011. [Google Scholar] [CrossRef]
- Chatzistavrakidis, A.; Steinacker, H.; Zoupanos, G. Intersecting branes and a standard model realization in matrix models. J. High Energy Phys.
**2011**, 1109, 115. [Google Scholar] [CrossRef] - Chepelev, I.; Makeenko, Y.; Zarembo, K. Properties of D-branes in matrix model of IIB superstring. Phys. Lett. B
**1997**, 400, 43. [Google Scholar] [CrossRef] - Fayyazuddin, A.; Smith, D.J. P-brane solutions in IKKT IIB matrix theory. Mod. Phys. Lett. A
**1997**, 12, 1447–1454. [Google Scholar] [CrossRef] - Aoki, H.; Ishibashi, N.; Iso, S.; Kawai, H.; Kitazawa, Y.; Tada, T. Noncommutative Yang-Mills in IIB matrix model. Nucl. Phys. B
**2000**, 565, 176–192. [Google Scholar] [CrossRef] - Iso, S.; Kimura, Y.; Tanaka, K.; Wakatsuki, K. Noncommutative gauge theory on fuzzy sphere from matrix model. Nucl. Phys. B
**2001**, 604, 121–147. [Google Scholar] [CrossRef] [Green Version] - Kimura, Y. Noncommutative gauge theories on fuzzy sphere and fuzzy torus from matrix model. Prog. Theor. Phys.
**2001**, 106, 445. [Google Scholar] [CrossRef] - Kitazawa, Y. Matrix models in homogeneous spaces. Nucl. Phys. B
**2002**, 642, 210–226. [Google Scholar] [CrossRef] [Green Version] - Aoki, H.; Iso, S.; Suyama, T. Orbifold matrix model. Nucl. Phys. B
**2002**, 634, 71–89. [Google Scholar] [CrossRef] [Green Version] - Li, L.F. Group Theory of the Spontaneously Broken Gauge Symmetries. Phys. Rev. D
**1974**, 9, 1723. [Google Scholar] [CrossRef] - Manolakos, G.; Zoupanos, G. Non-commutativity in Unified Theories and Gravity. Springer Proc. Math. Stat.
**2017**, 263, 177–198. [Google Scholar] [CrossRef] - DeBellis, J.; Saemann, C.; Szabo, R.J. Quantized Nambu-Poisson Manifolds in a 3-Lie Algebra Reduced Model. J. High Energy Phys.
**2011**, 1104, 075. [Google Scholar] [CrossRef] - Kimura, Y. Noncommutative gauge theory on fuzzy four sphere and matrix model. Nucl. Phys. B
**2002**, 637, 177–198. [Google Scholar] [CrossRef] - Singh, A.; Carroll, S.M. Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Clifford Algebra. arXiv
**2018**, arXiv:1806.10134. [Google Scholar] - Barut, A. From Heisenberg algebra to Conformal Dynamical Group. In Conformal Groups and related Symmetries. Physical Results and Mathematical Backgrounf; Lecture Notes in Physics; Barut, A., Doener, H.D., Eds.; Springer: Berlin/Heidelberg, Germany, 1985. [Google Scholar]
- Alvarez-Gaume, L.; Meyer, F.; Vazquez-Mozo, M.A. Comments on noncommutative gravity. Nucl. Phys. B
**2006**, 753, 92–127. [Google Scholar] [CrossRef] - Green, M.B.; Schwarz, J.H.; Witten, E. Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies and Phenomenology; Cambridge Monographs On Mathematical Physics; Cambridge University Press: Cambridge, UK, 1987; 596p. [Google Scholar]

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Manolakos, G.; Manousselis, P.; Zoupanos, G.
Gauge Theories: From Kaluza–Klein to noncommutative gravity theories. *Symmetry* **2019**, *11*, 856.
https://doi.org/10.3390/sym11070856

**AMA Style**

Manolakos G, Manousselis P, Zoupanos G.
Gauge Theories: From Kaluza–Klein to noncommutative gravity theories. *Symmetry*. 2019; 11(7):856.
https://doi.org/10.3390/sym11070856

**Chicago/Turabian Style**

Manolakos, George, Pantelis Manousselis, and George Zoupanos.
2019. "Gauge Theories: From Kaluza–Klein to noncommutative gravity theories" *Symmetry* 11, no. 7: 856.
https://doi.org/10.3390/sym11070856