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.
- 2.
As it holds for the magical \({J}_{3}^{\mathbb{R}}\) model, see Table I.
- 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.
- 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.
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.
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.
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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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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