Quasi-Conformal Actions, Quaternionic Discrete Series and Twistors: SU(2, 1) and G 2(2)

Günaydın, Murat; Neitzke, Andrew; Pavlyk, Oleksandr; Pioline, Boris

doi:10.1007/s00220-008-0563-9

Cited by 50 publications

(141 citation statements)

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“…Using the results in [21], one may check that this is precisely the condition (dz +P )/dρ = 0, where P is the projectivized SU(2) connection on the quaternionic-Kähler manifold M 3 . Therefore, the motion can be lifted to a holomorphic geodesic on the twistor space Z = G 2(2) / SU(2) × U(1) [25,24].…”

Section: The Bmpv and Taub-nut Black Holesmentioning

confidence: 99%

“…The same sigma model (up to analytic continuation) has appeared in the study of 4D black holes [20], and is in fact related to the present one by a flip of the t and ψ directions, corresponding to a Weyl reflection in a Sl(2) subgroup of G 3 . The quaternionic geometry of G 2(2) / SO(4) was studied in great detail in [21], whose notations we follow.…”

Section: Jhep05(2008)045mentioning

confidence: 99%

“…The viel-bein V Aα corresponds to the projection of the right-invariant form θ = dg · g −1 on the non-compact part p of the Lie algebra of G 2(2) , parameterized in the Iwasawa gauge as in [21],…”

Section: Geometry Of G 2(2) / So(4)mentioning

confidence: 99%

“…As indicated below (2.20), we are taking τ 1 , ζ 0 ,ζ 1 , σ to be purely imaginary, so that we can use the same expressions as in [21] which was taylored for the Riemannian space G 2(2) / SO(4). Our notations for the components of the right-invariant form θ, as well as for the Killing vectors of the right-action to be discussed presently, are summarized in figure 2.…”

Section: Geometry Of G 2(2) / So(4)mentioning

confidence: 99%

“…The metric (3.2) on the symmetric space G 2(2) / SO(4) is by construction invariant under the right-action of G 2(2) on the coset representative e, compensated by a left-action of its maximal compact subgroup such as to preserve the Iwasawa gauge (3.4). The corresponding Killing vectors were computed in [21]. Replacing the vector field ∂ φ a by the momentum p φ a conjugate to φ a , we find that the conserved charges associated to the right-action of G 2(2) are given by 7 E k =p σ (3.7a)…”

Section: Geometry Of G 2(2) / So(4)mentioning

confidence: 99%

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5D black holes and non-linear sigma models

Berkooz

Pioline

2008

J. High Energy Phys.

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Stationary solutions of 5D supergravity with U(1) isometry can be efficiently studied by dimensional reduction to three dimensions, where they reduce to solutions to a locally supersymmetric non-linear sigma model. We generalize this procedure to 5D gauged supergravity, and identify the corresponding gauging in 3D. We pay particular attention to the case where the Killing spinor is non constant along the fibration, which results, even for ungauged supergravity in 5D, in an additional gauging in 3D, without introducing any extra potential. We further study SU(2) × U(1) symmetric solutions, which correspond to geodesic motion on the sigma model (with potential in the gauged case). We identify and study the algebra of BPS constraints relevant for the Breckenridge-Myers-Peet-Vafa black hole, the Gutowski-Reall black hole and several other BPS solutions, and obtain the corresponding radial wave functions in the semi-classical approximation.

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Section: The Bmpv and Taub-nut Black Holesmentioning

confidence: 99%

Section: Jhep05(2008)045mentioning

confidence: 99%

Section: Geometry Of G 2(2) / So(4)mentioning

confidence: 99%

Section: Geometry Of G 2(2) / So(4)mentioning

confidence: 99%

Section: Geometry Of G 2(2) / So(4)mentioning

confidence: 99%

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5D black holes and non-linear sigma models

Berkooz

Pioline

2008

J. High Energy Phys.

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Octonionic Magical Supergravity, Niemeier Lattices, and Exceptional & Hilbert Modular Forms

Günaydin,

Kidambi

2024

Fortschritte der Physik

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The quantum degeneracies of Bogomolny‐Prasad‐Sommerfield (BPS) black holes of octonionic magical supergravity in five dimensions are studied. Quantum degeneracy is defined purely number theoretically as the number of distinct states in charge space with a given set of invariant labels. Quantum degeneracies of spherically symmetric stationary BPS black holes are given by the Fourier coefficients of modular forms of exceptional group . Their charges take values in the lattice defined by the exceptional Jordan algebra over integral octonions. The quantum degeneracies of rank 1 and rank 2 BPS black holes are given by the Fourier coefficients of singular modular forms and . The rank 3 (large) BPS black holes will be studied elsewhere. Following the work of N. Elkies and B. Gross on embeddings of cubic rings A into the exceptional Jordan algebra we show that the quantum degeneracies of rank 1 black holes described by such embeddings are given by the Fourier coefficients of the Hilbert modular forms (HMFs) of . If the discriminant of the cubic ring A is with p a prime number then the isotropic lines in the 24 dimensional orthogonal complement of A define a pair of Niemeier lattices which can be taken as charge lattices of some BPS black holes. The current status of the searches for the M/superstring theoretic origins of the octonionic magical supergravity is also reviewed.

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Lectures on Spectrum Generating Symmetries and U-Duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

Günaydın

2010

Springer Proceedings in Physics

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We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N = 2 Maxwell-Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras ,J, the spectrum generating symmetry groups are the conformal groups Conf (J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N = 2 MESGTs are the quasiconformal groups QConf (J) associated with J that are isomorphic to their U-duality groups in three dimensions. We then review the work on spectrum generating symmetries of spherically symmetric stationary 4D BPS black holes, based on the equivalence of their attractor equations and the equations for geodesic motion of a fiducial particle on the target spaces of corresponding 3D supergravity theories obtained by timelike reduction. We also discuss the connection between harmonic superspace formulation of 4D , N = 2 sigma models coupled to supergravity and the minimal unitary representations of their isometry groups obtained by quantizing their quasiconformal realizations. We discuss the relevance of this connection to spectrum generating symmetries and conclude with a brief summary of more recent results. Table 2: Above we list the orbits of spherically symmetric stationary BPS black hole solutions in 5D MESGTs defined by Euclidean Jordan algebras J of degree three. U-duality and stability groups are given by the Lorentz ( reduced structure ) and rotation ( automorphism) groups of J.SO(9) R ⊕ Γ (1,n−1) SO(n − 1, 1) × SO(1, 1)/SO(n − 2, 1) SO(n − 2)M 5 = M 4 = M 3 = J Str 0 (J) / Aut (J) Conf (J) / Str (J) QConf(F(J))/ Conf (J) × SU (2)

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Quasi-Conformal Actions, Quaternionic Discrete Series and Twistors: SU(2, 1) and G 2(2)

Cited by 50 publications

References 44 publications

5D black holes and non-linear sigma models

5D black holes and non-linear sigma models

Octonionic Magical Supergravity, Niemeier Lattices, and Exceptional & Hilbert Modular Forms

Lectures on Spectrum Generating Symmetries and U-Duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

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