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Supergravity: An Anthology of Solutions

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Gravity, a Geometrical Course

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Abstract

Chapter 9 presents a limited anthology of supergravity solutions aimed at emphasizing a few relevant new concepts. Relying on the special geometries described in Chap. 8 a first section contains an introduction to supergravity spherical Black Holes, to the attraction mechanism and to the interpretation of the horizon area in terms of a quartic symplectic invariant of the U duality group. The second and third sections deal instead with flux compactifications of both M-theory and type IIA supergravity. The main issue is that of the relation between supersymmetry preservation and the geometry of manifolds of restricted holonomy. The problem of supergauge completion and the role of orthosymplectic superalgebras is also emphasized. Appendices contain the development of gamma matrix algebra necessary for the inclusion of spinors, details on superalgebras and the user guide to Mathematica codes for the computer aided calculation of Einstein equations.

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William Shakespeare

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Notes

  1. 1.

    As we illustrate below the attraction mechanism corresponds to the following notable property of supergravity black holes which was discovered by Ferrara and Kallosh in 1995 [1, 2]: independently from their values at spatial infinity, the scalar fields flow to universal fixed values at the event horizon, dictated solely by the electromagnetic charges of the hole.

  2. 2.

    In the supergravity framework BPS solutions are those that preserve a certain amount of supersymmetry, namely that admit a certain number of so named Killing spinors, i.e. of supersymmetry parameters such that supersymmetry transformations along them leave the chosen solution invariant.

  3. 3.

    In [18] it was shown that every orbit of solutions contains a representative where the Taub-NUT charge is zero. Alternatively from a dynamical system point of view the Taub-NUT charge can be annihilated by setting a constraint which is consistent with the Hamiltonian and which reduces the dimension of the system by one unit. The problem of black hole physics is therefore equivalent to the sigma model based on an appropriate codimension one hypersurface in the \( \mathcal{Q} \) manifold.

  4. 4.

    See for instance the lecture notes [19].

  5. 5.

    The special overall normalization of the Poincaré metric is chosen in order to match the general definitions of special geometry applied to the present case.

  6. 6.

    By τ α we denote the gamma matrices in 7-dimensions, satisfying the Clifford algebra {τ α,τ β}=−δ αβ. With the symbol \(\tau^{\alpha_{1}\dots\alpha_{n}}\) we denote, as usual, the antisymmetrized product of n such matrices.

  7. 7.

    The theory of Sasakian manifolds, as applied to supergravity compactifications was discussed in [39]. In short an odd dimensional manifold is named Sasakian if the even dimensional cone constructed over it has vanishing first Chern class. After several manipulations this implies that the Sasakian manifold is an S 1-fibre bundle over a suitable complex base manifold.

  8. 8.

    With respect to the results obtained for the mini superspace extension of M-theory configuration everything is identical in (9.4.51)–(9.4.54) except the obvious reduction of the index range of (α,β,…) from 7 to 6-values. The only difference is in (9.4.55) where the last contribution proportional to the Kähler form is an essential novelty of this new type of compactification.

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  1. Dipartimento di Fisica Teorica, University of Torino, Torino, Italy

    Pietro Giuseppe Frè

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Frè, P.G. (2013). Supergravity: An Anthology of Solutions. In: Gravity, a Geometrical Course. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5443-0_9

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