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\(E_7\), \(F_4\) and Supergravity Scalar Potentials

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Advances in Geometry and Lie Algebras from Supergravity

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Abstract

As we mentioned in previous chapters, theIRexceptional Lie algebras, for long time regarded as mathematical curiosities, came to the forefront of research with the advent of supergravity.

To him who looks upon the world rationally, the world in its turn presents a rational aspect. The relation is mutual.

Georg Wilhelm Friedrich Hegel

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Notes

  1. 1.

    For instance, we have \(\gamma _5 \, \varepsilon _A= \varepsilon _A\) and \(\gamma _5 \, \varepsilon ^A = - \varepsilon ^A\).

  2. 2.

    We denote by \( \lambda ^i\) the right handed chiral projection while \(\lambda _i\) are the left handed ones.

  3. 3.

    As already stressed, the v.e.v.s of all the fermions are zero and Eq. (7.1.8) guarantees that they remain zero under supersymmetry transformations of parameters (7.1.9).

  4. 4.

    Here we have used the notation, \(\phi ^{ABCD} \equiv (\phi _{ABCD})^*\).

  5. 5.

    In the rheonomy approach closure of the Bianchi identities.

  6. 6.

    For example, in the de Wit–Nicolai theory, where one gauges \(G_{gauge} =\mathrm {SO(8)}\) we have:

    $$ t^{(1)}_{\varOmega \varSigma }{}^{\varPi \varGamma }{}_{\varDelta \varLambda }= \delta _{[\varDelta }^{[\varPi }\delta _{\varLambda ][\varOmega } \delta _{\varSigma ]}^{\varGamma ]}\,,\qquad t^{(2)}_{\varOmega \varSigma }{}^{\varPi \varGamma \varDelta \varLambda }=0 \,. $$
  7. 7.

    Note that some of the 28 generators of \({\mathscr {G}}_{gauge}(h,\ell ) \, \subset \, \mathrm {SL}(8,\mathbb {R})\) may be represented trivially in the adjoint representation, but in this case also the corresponding group transformations leave the embedding matrix invariant.

  8. 8.

    Just as in [56] we mention scalar fields that typically have non canonical kinetic terms.

  9. 9.

    Note that here, for simplicity we have dropped the suffix \(\mathscr {SK}\). This is done for simplicity since there is no risk of confusion.

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  1. Pietra Marazzi, Alessandria, Italy

    Pietro Giuseppe Fré

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Fré, P.G. (2018). \(E_7\), \(F_4\) and Supergravity Scalar Potentials. In: Advances in Geometry and Lie Algebras from Supergravity. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-74491-9_7

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