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Non-commutativity in Unified Theories and Gravity

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Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 (LT-XII/QTS-X 2017)

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Abstract

First, we briefly review the Coset Space Dimensional Reduction scheme and the results of the best model so far. Then, we present the introduction of fuzzy coset spaces used as extra dimensions and perform a dimensional reduction. In turn, we describe a construction which mimics the results of a reduction, starting from a 4-dimensional theory and we present a successful example of a dynamical generation of fuzzy spheres. Finally, we propose a construction of the 3-d gravity as a gauge theory on specific non-commutative spaces.

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Notes

  1. 1.

    A coset space is called symmetric when \(f_{ab}^c=0\).

  2. 2.

    The \(S^2\) metric can be expressed in terms of the Killing vectors as \(g^{\alpha \beta }=\dfrac{1}{R^2}\xi ^\alpha _a\xi _a^\beta \).

  3. 3.

    In general, k is a parameter related to the size of the fuzzy coset space. In the case of the fuzzy sphere, k is related to the radius of the sphere and the integer l.

  4. 4.

    Also, \(\text {Tr}\text {tr}_G\) is interpreted as the trace of the U(NP) matrices.

  5. 5.

    See also [47].

  6. 6.

    Also modulo 3.

  7. 7.

    In case of ordinary reduction of a 10-dimensional \(\mathcal {N}=1\) SYM theory, one obtains an \(\mathcal {N}=4\) SYM Yang-Mills theory in four dimensions having a global \(SU(4)_R\) symmetry which is identified with the tangent space SO(6) of the extra dimensions [14, 15, 61].

  8. 8.

    The SSB terms that will be inserted into \(V_{\mathcal {N}=1}^{proj}(\phi )\), are purely scalar. Although this is enough for our purpose, it is obvious that more SSB terms have to be included too, in order to obtain the full SSB sector [60].

  9. 9.

    Similar approaches have been studied in the framework of YM matrix models [63], lacking phenomenological viability.

  10. 10.

    As anomalous gaining mass by the Green-Schwarz mechanism and therefore they decouple at the low energy sector of the theory [58].

  11. 11.

    In the Lorentzian case there is a similar construction, where the 3-dimensional spacetime with Lorentzian signature is foliated by fuzzy hyperboloids [88].

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Acknowledgements

We acknowledge support by the COST action QSPACE MP1405. G.Z. thanks the MPI for Physics in Munich for hospitality and the A.v.Humboldt Foundation for support.

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  1. Physics Department, National Technical University, 15780, Zografou, Greece

    G. Manolakos & G. Zoupanos

  2. Max-Planck Institut für Physik, Föhringer Ring 6, 80805, Munchen, Germany

    G. Zoupanos

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    Vladimir Dobrev

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Manolakos, G., Zoupanos, G. (2018). Non-commutativity in Unified Theories and Gravity. In: Dobrev, V. (eds) Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 1 . LT-XII/QTS-X 2017. Springer Proceedings in Mathematics & Statistics, vol 263. Springer, Singapore. https://doi.org/10.1007/978-981-13-2715-5_10

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