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Noncommutative \(SO(2,3)_{\star }\) gauge theory of gravity

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Abstract

Topological gravity (in the sense that it is metric-independent) in a 2n-dimensional spacetime can be formulated as a gauge field theory for the AdS gauge group \(SO(2,2n-1)\) by adding a multiplet of scalar fields. These scalars can break the gauge invariance of the topological gravity action, thus making a connection with Einstein’s gravity. This review is about a noncommutative (NC) star-product deformation of the four-dimensional AdS gauge theory of gravity, including Dirac spinors and the Yang–Mills field. In general, NC actions can be expanded in powers of the canonical noncommutativity parameter \(\theta\) using the Seiberg–Witten map. The leading-order term of the expansion is the classical action, while the higher-order \(\theta\)-dependent terms are interpreted as new types of coupling between classical fields due to spacetime noncommutativity. We study how these perturbative NC corrections affect the field equations of motion and derive some phenomenological consequences, such as NC-deformed Landau levels of an electron. Finally, we discuss how topological gravity in four dimensions (both classical and noncommutative) appears as a low-energy sector of five-dimensional Chern–Simons gauge theory in the sense of Kaluza–Klein reduction.

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Notes

  1. However, this fact does not imply that there are no propagating local degrees of freedom, and thorough consideration shows that such degrees of freedom do exist [20].

  2. It is important to note that NC deformation does not commute with the gauge fixing. Therefore, one must first expand the NC action in powers of \(\theta\) using the SW map and then apply the gauge fixing condition.

  3. As we saw in the previous section, Minkowski space can be considered as a classical solution if we exclude the cosmological constant by a suitable choice of coefficients. Also, Minkowski space receives NC corrections at the second-order in \(\theta\), and we can therefore use only the classical Minkowski metric when working at the first NC order.

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Acknowledgements

MDĆ thanks the editors Konstantinos Anagnostopoulos, Peter Schupp and George Zoupanos for the invitation to contribute to this special issue of ”Noncommutativity and Physics”. The authors acknowledge funding provided by the Faculty of Physics, University of Belgrade, through the grant by the Ministry of Education, Science, and Technological Development of the Republic of Serbia (number 451-03-68/2022-14/200162).

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  1. Faculty of Physics, University of Belgrade, Studentski Trg 12-16, 11000, Belgrade, Serbia

    Marija Dimitrijević Ćirić, Dušan Ɖorđević, Dragoljub Gočanin, Biljana Nikolić & Voja Radovanović

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  1. Marija Dimitrijević Ćirić
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Noncommutativity and Physics. Guest editors: George Zoupanos, Konstantinos Anagnostopoulos, Peter Schupp.

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Ćirić, M.D., Ɖorđević, D., Gočanin, D. et al. Noncommutative \(SO(2,3)_{\star }\) gauge theory of gravity. Eur. Phys. J. Spec. Top. 232, 3747–3760 (2023). https://doi.org/10.1140/epjs/s11734-023-00833-5

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