Moufang plane and octonionic Quantum Mechanics

We study the critical points of the black hole scalar potential V BH in N = 2, d = 4 supergravity coupled to n V vector multiplets, in an asymptotically flat extremal black hole background described by a 2 (n V + 1)-dimensional dyonic charge vector and (complex) scalar fields which are coordinates of a special Kähler manifold.For the case of homogeneous symmetric spaces, we find three general classes of regular attractor solutions with non-vanishing Bekenstein-Hawking entropy. They correspond to three (inequivalent) classes of orbits of the charge vector, which is in a 2 (n V + 1)-dimensional representation R V of the U-duality group. Such orbits are non-degenerate, namely they have non-vanishing quartic invariant (for rank-3 spaces). Other than the 1 2 -BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge.The three species of solutions to the N = 2 extremal black hole attractor equations give rise to different mass spectra of the scalar fluctuations, whose pattern can be inferred by using invariance properties of the critical points of V BH and some group theoretical considerations on homogeneous symmetric special Kähler geometry.

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“…respectively over the octonions O, quaternions H, complex numbers C and real numbers R [37,38,39,48,49,50,51].…”

Section: (28)mentioning

confidence: 99%

Charge Orbits of Symmetric Special Geometries and Attractors

Bellucci

Ferrara

Günaydın

et al. 2006

Int. J. Mod. Phys. A

152

409

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“…From the analysis performed in [10], only six N = 2, d = 4 maximal magic supergravities 7 exist which can be obtained by consistently truncating N = 8, d = 4 supergravity; they are given by table 5. After [57], we also include the case of ST U model [58][59][60] [46][47][48][49][61][62][63][64].…”

Section: Jhep06(2012)074mentioning

confidence: 99%

Degeneration of groups of type E 7 and minimal coupling in supergravity

Ferrara

Каллош

Marrani

2012

J. High Energ. Phys.

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Abstract:We study properties of D = 4 N 2 extended supergravities (and related compactifications of superstring theory) and their consistent truncation to the phenomenologically interesting models of N = 1 supergravity. This involves a detailed classification of the "degenerations" of the duality groups of type E 7 , when the corresponding quartic invariant polynomial built from the symplectic irreducible representation of G 4 "degenerates" into a perfect square. With regard to cosmological applications, minimal coupling of vectors in consistent truncation to N = 1 from higher-dimensional or higher-N theory is non-generic. On the other hand, non-minimal coupling involving vectors coupled to scalars and axions is generic. These features of supergravity, following from the electric-magnetic duality, may be useful in other applications, like stabilization of moduli, and in studies of non-perturbative black-hole solutions of supergravity/string theory.

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“…In the exceptional octonionic case this corresponds to a F 4 transformation, as was shown explicitly in [194]. When A = C this operation is related to the Schmidt decomposition of a two-qutrit system.…”

Section: U-duality Orbits and Jordan Algebrasmentioning

confidence: 76%

“…For large black holes, those with non-vanishing entropy corresponding to rank 4 elements of J O s 3 , E 6(6) acts transitively on elements of a given entropy (cubic norm) I 3 [193,182]. Any element J 3 (P ) ∈ J O s 3 may be diagonalised using a F 4(4) transformation [194] and the representative elements of each of the orbits [30] may be chosen as in Table 18 (where k = I 3 = 0). These will appear as the charge orbits of Table 33 in the conventional black hole/qutrit Table 33 for details.…”

Section: U-duality Orbits and Jordan Algebrasmentioning

confidence: 99%

Black holes, qubits and octonions

et al. 2009

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We review the recently established relationships between black hole entropy in string theory and the quantum entanglement of qubits and qutrits in quantum information theory. The first example is provided by the measure of the tripartite entanglement of three qubits (Alice, Bob and Charlie), known as the 3-tangle, and the entropy of the 8-charge ST U black hole of N = 2 supergravity, both of which are given by the [SL(2)] 3 invariant hyperdeterminant, a quantity first introduced by Cayley in 1845. Moreover the classification of three-qubit entanglements is related to the classification of N = 2 supersymmetric ST U black holes. There are further relationships between the attractor mechanism and local distillation protocols and between supersymmetry and the suppression of bit flip errors. At the microscopic level, the black holes are described by intersecting D3-branes whose wrapping around the six compact dimensions T 6 provides the string-theoretic interpretation of the charges and we associate the three-qubit basis vectors, |ABC (A, B, C = 0 or 1), with the corresponding 8 wrapping cycles. The black hole/qubit correspondence extends to the 56 charge N = 8 black holes and the tripartite entanglement of seven qubits where the measure is provided by Cartan's E 7 ⊃ [SL(2)] 7 invariant. The qubits are naturally described by the seven vertices ABCDEF G of the Fano plane, which provides the multiplication table of the seven imaginary octonions, reflecting the fact that E 7 has a natural structure of an O-graded algebra. This in turn provides a novel imaginary octonionic interpretation of the 56 = 7 × 8 charges of N = 8: the 24 = 3 × 8 NS-NS charges correspond to the three imaginary quaternions and the 32 = 4 × 8 R-R to the four complementary imaginary octonions. We contrast this approach with that based on Jordan algebras and the Freudenthal triple system. N = 8 black holes (or black strings) in five dimensions are also related to the bipartite entanglement of three qutrits (3-state systems), where the analogous measure is Cartan's E 6 ⊃ [SL(3)] 3 invariant. Similar analogies exist for magic N = 2 supergravity black holes in both four and five dimensions. Despite the ubiquity of octonions, our analogy between black holes and quantum information theory is based on conventional quantum mechanics but for completeness we also provide a more exotic one based on octonionic quantum mechanics. Finally, we note some intriguing, but still mysterious, assignments of entanglements to cosets, such as the 4-way entanglement of eight qubits to E 8 /[SL (2)] 8 .

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Moufang plane and octonionic Quantum Mechanics

Cited by 97 publications

References 15 publications

Charge Orbits of Symmetric Special Geometries and Attractors

Charge Orbits of Symmetric Special Geometries and Attractors

Degeneration of groups of type E 7 and minimal coupling in supergravity

Black holes, qubits and octonions

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