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On the Classification of Two Center Orbits for Magical Black Holes

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Supersymmetric Gravity and Black Holes

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 142))

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Abstract

We report on recent work [4] concerning the determination of the two-centered generic charge orbits of magical \(\mathcal{N} = 2\) and maximal \(\mathcal{N} = 8\) supergravity theories in four dimensions.

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Notes

  1. 1.

    Here U-duality is referred to as the “continuous” symmetries of [11]. Their discrete versions are the U-duality non-perturbative string theory symmetries introduced by Hull and Townsend [32].

  2. 2.

    As it holds for the magical \({J}_{3}^{\mathbb{R}}\) model, see Table I.

  3. 3.

    A necessary but not sufficient condition for Eq. (5.2) to hold is p < f, such that the p dyonic charge vectors can all be taken to be linearly independent.

  4. 4.

    Multi-center configurations with “small” constituents [5, 10, 29] can be treated as well, and they will be considered elsewhere.

  5. 5.

    In this respect, we observe that the N = 3 theory coupled to three vector multiplets, whose scalar manifold, SU(3, 3) ∕ S(U(3) ×U(3)), coincides with the one of the magic model based on \({J}_{3}^{\mathbb{C}}\), cannot be included in the discussion, since the bosonic field content of the two theories is different.

  6. 6.

    As mentioned above, the irreducible rank-1 cubic case (the so-called \(\mathcal{N} = 2\), d = 4 t 3 model, associated to the trivial degree-1 Jordan algebra ℝ) has been treated in [26].

  7. 7.

    With respect to the treatment given in [36], we fix the overall normalization constant of the K-tensor to the value \(\xi = - \frac{1} {3\tau } = -\frac{f\left (f+1\right )} {6d}\), as needed for consistency reasons.

  8. 8.

    We remark that relation (5.87) characterizes \({\mathcal{Q}}_{abc}\) as the two-center generalisation of the so-called Freudenthal dual of the dyonic charge vector \({\mathcal{Q}}^{M}\), introduced (with a different normalisation) in [8]. Thus, \({\mathcal{Q}}_{abc}\) can be regarded as the (polynomial) two-center Freudenthal dual of the dyonic charge vector \({\mathcal{Q}}_{d}\). Furthermore, Eqs. (5.82), (5.86) and (5.97) imply that, under the formal interchange \({\mathcal{Q}}_{a}^{M} \leftrightarrow {\mathbb{C}}^{MN}{\mathcal{Q}}_{N\mid abc}\), I abcd is invariant and \(\mathcal{W}\leftrightarrow {\mathbf{I}}_{6}\).

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Acknowledgements

The present contribution is based on [4] made in collaboration with Alessio Marrani and Mario Trigiante. The work of S. F. is supported by the ERC Advanced Grant no. 226455, “Supersymmetry, Quantum Gravity and Gauge Fields” (SUPERFIELDS). The work of L.A. and R.D’A. is supported in part by the MIUR-PRIN contract 2009-KHZKRX.

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Authors and Affiliations

  1. Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129, Torino, Italy

    Laura Andrianopoli & Riccardo D’Auria

  2. INFN, Sezione di Torino, Torino, Italy

    Laura Andrianopoli & Riccardo D’Auria

  3. Physics Department, Theory Unit, CERN, CH-1211, Geneva 23, Switzerland

    Sergio Ferrara

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  1. Laura Andrianopoli
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  2. Riccardo D’Auria
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Correspondence to Laura Andrianopoli .

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  1. Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, via E. Fermi 40, Frascati RM, 00044, Italy

    Stefano Bellucci

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Andrianopoli, L., D’Auria, R., Ferrara, S. (2013). On the Classification of Two Center Orbits for Magical Black Holes. In: Bellucci, S. (eds) Supersymmetric Gravity and Black Holes. Springer Proceedings in Physics, vol 142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31380-6_5

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