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Supersymmetry and Noncommutative Geometry

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 9))

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Abstract

We introduce the core concepts and formalisms that are needed in our search for a noncommutative geometric description of supersymmetric theories. We start with a concise overview of supersymmetry and the minimal supersymmetric extension of the Standard Model (MSSM). We then provide a bird’s eye view of noncommutative geometry, geared towards its applications in high-energy physics.

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Notes

  1. 1.

    The possible values for N, the number of supersymmetry generators, depend on the space-time dimension. For example, for \(d = 4\), \(N = 1, 2, 4\) or 8.

  2. 2.

    This makes it an example of \(N=1\) supersymmetry.

  3. 3.

    One should keep in mind though that Minkowski space is not an example of a Riemannian manifold. Rather it is pseudo-Riemannian since its metric is diagonal with negative entries.

  4. 4.

    The parameter \(\varLambda \) more or less serves as a cut-off, and will in the derivation of the SM (Sect. 1.2.3 ahead) be interpreted as the GUT-scale.

  5. 5.

    To be explicit, the element \((\lambda , q, m)\in \mathscr {A}_F\) acts on—say—\({\mathbf {2}} \otimes {\mathbf {3}}^o \ni v\otimes \bar{w}\) as \(qv \otimes \bar{w} m = qv \otimes \overline{m^*w}\).

  6. 6.

    An exception to this rule is when one component of the algebra acts in the same way on more than one different representations in \(\mathscr {H}_F\).

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Authors and Affiliations

  1. Faculty of Science, Radboud University Nijmegen, Nijmegen, The Netherlands

    Wim Beenakker & Walter D. van Suijlekom

  2. University of Amsterdam, Amsterdam, The Netherlands

    Wim Beenakker

  3. Nikhef, Radboud University Nijmegen, Nijmegen, The Netherlands

    Thijs van den Broek

Authors
  1. Wim Beenakker
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  2. Thijs van den Broek
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  3. Walter D. van Suijlekom
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Correspondence to Thijs van den Broek .

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Beenakker, W., van den Broek, T., van Suijlekom, W.D. (2016). Introduction. In: Supersymmetry and Noncommutative Geometry. SpringerBriefs in Mathematical Physics, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-24798-4_1

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