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Higher-Dimensional Unified Theories with Continuous and Fuzzy Coset Spaces as Extra Dimensions

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Book cover Lie Theory and Its Applications in Physics (LT 2015)

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Abstract

We first briefly review the Coset Space Dimensional Reduction (CSDR) programme and present the results of the best model so far, based on the \(\mathcal {N} = 1\), \(d = 10\), \(E_8\) gauge theory reduced over the nearly-Kähler manifold \(SU(3)/U(1)\times U(1)\). Then, we present the adjustment of the CSDR programme in the case that the extra dimensions are considered to be fuzzy coset spaces and then, the best model constructed in this framework, too, which is the trinification GUT, \(SU(3)^3\).

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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 [45].

  6. 6.

    This embedding is achieved non-uniquely, specifically in \(p_N\) ways, where \(p_N\) is the possible ways one can partition the N into a set of non-increasing, positive integers [46].

  7. 7.

    The number of counter-terms required to eliminate the divergencies is finite.

  8. 8.

    Technically, this is possible because \(N\times N\) matrices can be decomposed on the U(N) generators.

  9. 9.

    Also modulo 3.

  10. 10.

    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 [16, 17].

  11. 11.

    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 [57].

  12. 12.

    Similar approaches have been studied in the framework of YM matrix models [59], lacking phenomenological viability.

  13. 13.

    As anomalous gaining mass by the Green-Schwarz mechanism and therefore they decouple at the low energy sector of the theory [55].

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Acknowledgements

This research is supported by the Research Funding Program ARISTEIA, Higher Order Calculations and Tools for High Energy Colliders, HOCTools and the ARISTEIA II, Investigation of certain higher derivative term field theories and gravity models (co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF)), by the European Union’s ITN programme HIGGSTOOLS, as well as by the Action MP1405 QSPACE from the European Cooperation in Science and Technology (COST). G.Z. would like to thank the organizers and ITP-Heidelberg for the warm and generous hospitality.

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  1. Physics Department, National Technical University, GR-15780, Athens, Zografou Campus, Greece

    G. Manolakos & G. Zoupanos

  2. Institut Für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120, Heidelberg, Germany

    G. Zoupanos

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    Vladimir Dobrev

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Manolakos, G., Zoupanos, G. (2016). Higher-Dimensional Unified Theories with Continuous and Fuzzy Coset Spaces as Extra Dimensions. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. LT 2015. Springer Proceedings in Mathematics & Statistics, vol 191. Springer, Singapore. https://doi.org/10.1007/978-981-10-2636-2_13

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