Abstract
Chapter 8 is a bestiary of Supergravity Special Geometries associated with its scalar sector. The chapter clarifies the codifying role of the scalar geometry in constructing the bosonic part of a supergravity Lagrangian. The dominant role among the scalar manifolds of homogeneous symmetric spaces is emphasized illustrating the principles that allow the determination of such U/H cosets for any supergravity theory. The mechanism of symplectic embedding that allows to extend the action of U-isometries from the scalar to the vector field sector are explained in detail within the general theory of electric/magnetic duality rotations. Next the chapter provides a self-contained summary of the most important special geometries appearing in D=4 and D=5 supergravity, namely Special Kähler Geometry, Very Special Real Geometry and Quaternionic Geometry.
Incipit liber de natura quorundam animalium, et lapidum et quid significetur per eam
from a Medieval Latin Bestiary
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Notes
- 1.
Special Kähler geometry was introduced in a coordinate dependent way in the first papers on the vector multiplet coupling to supergravity in the middle eighties [2, 14]. Then it was formulated in a coordinate-free way at the beginning of the nineties from a Calabi-Yau standpoint by Strominger [5] and from a supergravity standpoint by Castellani, D’Auria and Ferrara [3, 4]. The properties of holomorphic isometries of special Kähler manifolds, namely the geometric structures of special geometry involved in the gauging were clarified by D’Auria, Frè and Ferrara in [20]. For a review of special Kähler geometry in the setup and notations of the book see [21] and Sect. 8.5 of the present chapter.
- 2.
The notion of quaternionic geometry, as it enters the formulation of hypermultiplet coupling was introduced by Bagger and Witten in [15] and formalized by Galicki in [17] who also explored the relation with the notion of HyperKähler quotient, whose use in the construction of supersymmetric \(\mathcal{N}=2\) Lagrangians had already been emphasized in [16]. The general problem of classifying quaternionic homogeneous spaces had been addressed in the mathematical literature by Alekseevski [6].
- 3.
The notion of very special geometry is essentially due to the work of Günaydin Sierra and Townsend who discovered it in their work on the coupling of D=5 supergravity to vector multiplets [9, 10]. The notion was subsequently refined and properly related to special Kähler geometry in four dimensions through the work by de Wit and Van Proeyen [11–13].
- 4.
- 5.
- 6.
The difference between the \(\mathcal{N}=7,8\) cases and the others is properly explained in the following way. As far as superalgebras are concerned the automorphism group is always \(U\left( \mathcal{N} \right)\) for all \(\mathcal{N}\) , which can extend, at this level also beyond \(\mathcal{N}=8\) . Yet for the \(\mathcal{N}=8\) graviton multiplet, which is identical to the \(\mathcal{N}=7\) multiplet, it happens that the U(1) factor in U(8) has vanishing action on all physical states since the multiplet is self-conjugate under CPT-symmetries. From here it follows that the isotropy group of the scalar manifold must be SU(8) rather than U(8). A similar situation occurs for the \(\mathcal{N}=4\) vector multiplets that are also CPT self-conjugate. From this fact follows that the isotropy group of the scalar submanifold associated with the vector multiplet scalars is SU(4)×H′ rather than U(4)×H′. In \(\mathcal{N}=4\) supergravity, however, the U(1) factor of the automorphism group appears in the scalar manifold as isotropy group of the submanifold associated with the graviton multiplet scalars. This is so because the \(\mathcal{N}=4\) graviton multiplet is not CPT self conjugate.
- 7.
Whether the ϕ I can be arranged into complex fields is not relevant at this level of the discussion.
- 8.
Actually, in order to be true, (8.3.49) requires that the normalizer of H in G be the identity group, a condition that is verified in all the relevant examples.
- 9.
From the point of view of Lie algebra theory, there are no other independent real sections of the C n Lie algebra except the non-compact \(\mathrm{Sp}(2\overline{n}, \mathbb{R})\) and the compact \(\mathrm{USp}(2\overline{n})\). So in mathematical books the Lie group \(\mathrm{USp}(\overline{n}, \overline{n})\) does not exist being simply \(\mathrm{Sp}(2\overline{n}, \mathbb {R})\). In our present discussion we find it useful to denote by this symbol the realization of \(\mathrm{Sp}(2\overline{n}, \mathbb{R})\) elements by means of complex symplectic and pseudounitary matrices as described in the main text.
- 10.
For simplicity we do not envisage the inclusion of hypermultiplets which would span additional quaternionic manifolds.
- 11.
Since 1998 a rich stream of literature has been devoted to the so called AdS/CFT correspondence. In a nut-shell such a correspondence is rooted in the double interpretation of the groups SO(2,d−1) as anti de Sitter groups in d-dimensions and as conformal groups in (d−1)-dimensions. Such double interpretation has very far reaching consequences. At the end of a long chain of arguments it enables to evaluate exactly certain Green functions of appropriate quantum gauge-theories in d−1 dimensions by means of classical gravitational calculations in d-dimensions. In more general terms this correspondence is a kind of holography where boundary and bulk calculations can be interchanged.
- 12.
We leave aside pure \( \mathcal{N}=1 \), D=4 supergravity that from the rheonomic viewpoint is a completely trivial case.
- 13.
In the ungauged theory all two-forms have been dualized to vectors.
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Frè, P.G. (2013). Supergravity: A Bestiary in Diverse Dimensions. In: Gravity, a Geometrical Course. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5443-0_8
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