Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups

Günaydın, Murat; Pavlyk, Oleksandr

doi:10.1088/1126-6708/2005/08/101

Cited by 50 publications

(151 citation statements)

References 27 publications

(96 reference statements)

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“…No other classes of non-degenerate orbits exist in this case; this is essentially related to the fact that no other real (non-compact) forms of E 6 exist in E 7(7) beside E 6 (2) and E 6(6) . The 1 8 -BPS and non-BPS non-degenerate orbits correspond to the maximal (non-compact) subgroup of E 7 (7) to be E 6(2) ⊗ U(1) and E 6(6) ⊗ SO(1, 1), respectively.…”

Section: Introductionmentioning

confidence: 99%

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Charge Orbits of Symmetric Special Geometries and Attractors

Bellucci

Ferrara

Günaydın

et al. 2006

Int. J. Mod. Phys. A

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152

409

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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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Section: Introductionmentioning

confidence: 99%

“…Under dimensional reduction to d = 4, such a correspondence gets extended to a correspondence between the field strengths of the vector fields (and their magnetic duals) and the Freudenthal triple system (FTS) [57,46] F (J ) defined over J [37,3,1,4,7], and it can be realized as 2 × 2 "matrices" : [37,38,39].…”

mentioning

confidence: 99%

Charge Orbits of Symmetric Special Geometries and Attractors

Bellucci

Ferrara

Günaydın

et al. 2006

Int. J. Mod. Phys. A

Self Cite

152

409

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“…See [189,190] for a comprehensive account. Their intimate relationship with the exceptional Lie groups is of central importance in their applications to string theory and supergravity.…”

Section: 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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“…Furthermore, the above global transformation leaves the metric g IJ invariant since it corresponds simply to a Kähler transformation of the Kähler potential ln V(z −z). In N = 2 MESGTs defined by Euclidean Jordan algebras J of degree three, one-to-one correspondence between vector fields of five dimensional theories (and hence their charges) and elements of J gets extended, in four dimensions, to a one-to-one correspondence between field strengths of vector fields plus their magnetic duals and Freudenthal triple systems defined over J [47,18,17,21,24]. An element X of Freudenthal triple system (FTS) F(J) [45,46] over J can be represented formally as a 2 × 2 "matrix":…”

Section: D N = 2 Maxwell-einstein Supergravity Theories and Freudenmentioning

confidence: 99%

“…If the FTS is defined over a Jordan algebra J of degree three we shall denote the corresponding quasiconformal groups either as QConf (F(J)) or simply as QConf (J). For N = 2 MESGTs defined by Jordan algebras of degree three, quasiconformal group actions of their three dimensional U-duality groups G 3 were given explicitly in [24], in a basis covariant with respect to U-duality groups of corresponding six dimensional supergravity theories.…”

Section: D N = 2 Maxwell-einstein Supergravity Theories and Freudenmentioning

confidence: 99%

Spectrum generating conformal and quasiconformal U-duality groups, supergravity and spherical vectors

Günaydın

Pavlyk

2010

J. High Energ. Phys.

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After reviewing the algebraic structures that underlie the geometries of N = 2 Maxwell-Einstein supergravity theories (MESGT) in five and four dimensions with symmetric scalar manifolds, we give a unified realization of their three dimensional U-duality groups as spectrum generating quasiconformal groups. They are F 4(4) , E 6(2) , E 7(−5) , E 8(−24) and SO(n + 2, 4). Our formulation is covariant with respect to U-duality symmetry groups of corresponding five dimensional supergravity theories, which are SL(3, R), SL(3, C), SU * (6), E 6(6) and SO(n − 1, 1) × SO(1, 1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character. We also give their quadratic Casimir operators and determine their values. Our work lays the algebraic groundwork for constructing the unitary representations of these groups induced by their geometric quasiconformal actions, which include the quaternionic discrete series. For rank 2 cases, SU (2, 1) and G 2(2) , corresponding to simple N = 2 supergravity in four and five dimensions, this program was carried out in arXiv:0707.1669. We also discuss the corresponding algebraic structures underlying symmetries of N ≥ 4 supergravity theories. They lead to quasiconformal realizations of split real forms of U-duality groups as a straightforward extension of the quaternionic real forms.

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Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups

Cited by 50 publications

References 27 publications

Charge Orbits of Symmetric Special Geometries and Attractors

Charge Orbits of Symmetric Special Geometries and Attractors

Black holes, qubits and octonions

Spectrum generating conformal and quasiconformal U-duality groups, supergravity and spherical vectors

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