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Instantons, black holes and harmonic functions,” arXiv:0906.3451 [hepth
"... Abstract: We find a class of fivedimensional EinsteinMaxwell type Lagrangians which contains the bosonic Lagrangians of vector multiplets as a subclass, and preserves some features of supersymmetry, namely the existence of multicentered black hole solutions and of attractor equations. Solutions c ..."
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Abstract: We find a class of fivedimensional EinsteinMaxwell type Lagrangians which contains the bosonic Lagrangians of vector multiplets as a subclass, and preserves some features of supersymmetry, namely the existence of multicentered black hole solutions and of attractor equations. Solutions can be expressed in terms of harmonic functions through a set of algebraic equations. The geometry underlying these Lagrangians is characterised by the existence of a Hesse potential and generalizes the very special real geometry of vector multiplets. Our construction proceeds by first obtaining instanton solutions for a class of fourdimensional Euclidean sigma models, which includes those occuring for fourdimensional Euclidean N = 2 vector multiplets as a subclass. For solutions taking values in a completely isotropic submanifold of the target space, we show that the solution can be expressed in terms of harmonic functions if an integrability condition is met. This condition can either be solved by imposing that the solution depends on a single coordinate, or by imposing that the target space is a paraKähler manifold which can be obtained from a real Hessian manifold by a generalized rmap. In the latter case one obtains multicentered solutions.
First order description of D = 4 static black holes and the Hamilton–Jacobi equation,” 0905.3938 [hepth
"... In this note we discuss the application of the Hamilton–Jacobi formalism to the first order description of four dimensional spherically symmetric and static black holes. In particular we show that the prepotential characterizing the flow coincides with the Hamilton principal function associated with ..."
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In this note we discuss the application of the Hamilton–Jacobi formalism to the first order description of four dimensional spherically symmetric and static black holes. In particular we show that the prepotential characterizing the flow coincides with the Hamilton principal function associated with the onedimensional effective Lagrangian. This implies that the prepotential can always be defined, at least locally in the radial variable and in the moduli space, both in the extremal and nonextremal case and allows us to conclude that it is duality invariant. We also give, in this framework, a general definition of the “Weinhold metric ” in terms of which a necessary condition for the existence of multiple attractors is given. The Hamilton–Jacobi formalism can be applied both to the restricted phase space where the electromagnetic potentials have been integrated out as well as in the case where the electromagnetic potentials are dualized to scalar fields using the socalled threedimensional Euclidean approach. We give some examples of application of the formalism, both for the BPS and the nonBPS black holes.
Most General Spherically Symmetric M2branes and Type IIB Strings
, 906
"... We obtain the most general spherically symmetric M2branes and type IIB strings, with R1,2 × SO(8) and R1,1 × SO(8) isometries, respectively. We find that there are 12 different classes of M2branes, and we study their curvature properties. In particular, we obtain new smooth M2brane wormholes that ..."
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We obtain the most general spherically symmetric M2branes and type IIB strings, with R1,2 × SO(8) and R1,1 × SO(8) isometries, respectively. We find that there are 12 different classes of M2branes, and we study their curvature properties. In particular, we obtain new smooth M2brane wormholes that connect two asymptotic regions: one is flat and the other can be either flat or AdS4 × S7. We find that these wormholes are traversable with certain timelike trajectories. We also obtain the most general Ricciflat solutions in five dimensions with R1,1 × SO(3) isometries. Contents
The Integration Algorithm of Lax equation for both Generic Lax matrices and Generic Initial Conditions
, 904
"... Several physical applications of Lax equation require its general solution for generic Lax matrices and generic not necessarily diagonalizable initial conditions. In the present paper we complete the analysis started in [arXiv:0903.3771] on the integration of Lax equations with both generic Lax oper ..."
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Several physical applications of Lax equation require its general solution for generic Lax matrices and generic not necessarily diagonalizable initial conditions. In the present paper we complete the analysis started in [arXiv:0903.3771] on the integration of Lax equations with both generic Lax operators and generic initial conditions. We present a complete general integration formula holding true for any (diagonalizable or non diagonalizable) initial Lax matrix and give an original rigorous mathematical proof of its validity relying on no previously published results.
Supergravity Black Holes and Billiards and Liouville integrable structure of dual Borel algebras †
, 903
"... In this paper we show that the supergravity equations describing both cosmic billiards and a large class of blackholes are, generically, both Liouville integrable as a consequence of the same universal mechanism. This latter is provided by the Liouville integrable Poissonian structure existing on t ..."
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In this paper we show that the supergravity equations describing both cosmic billiards and a large class of blackholes are, generically, both Liouville integrable as a consequence of the same universal mechanism. This latter is provided by the Liouville integrable Poissonian structure existing on the dual Borel algebra BN of the simple Lie algebra AN−1. As a by product we derive the explicit integration algorithm associated with all symmetric spaces U/H ⋆ relevant to the description of timelike and spacelike pbranes. The most important consequence of our approach is the explicit construction of a complete set of conserved involutive hamiltonians {hα} that are responsible for integrability and provide a new tool to classify flows and orbits. We believe that these will prove a very important new tool in the analysis of supergravity black holes and billiards.
Integrability of Supergravity Black Holes and new Tensor Classifiers of Regular and Nilpotent Orbits
"... In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of N = 2 supergravities with symmetric Special Geometry. Our main goal is the classification of th ..."
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In this paper we apply in a systematic way a previously developed integration algorithm of the relevant Lax equation to the construction of spherical symmetric, asymptotically flat black hole solutions of N = 2 supergravities with symmetric Special Geometry. Our main goal is the classification of these blackholes according to the H? orbits in which the space of possible Lax operators decomposes. By H? one denotes the isotropy group of the coset UD=3/H? which appears in the timelike dimensional reduction of supergravity from D = 4 to D = 3 dimensions. The main result of our investigation is the construction of three universal tensors, extracted from quadratic and quartic powers of the Lax operator, that are capable of classifying both regular and nilpotent H? orbits of Lax operators. Our tensor based classification is compared, in the case of the simple onefield model S3, to the algebraic classification of nilpotent orbits and it is shown to provide a simple discriminating method. In particular we present a detailed analysis of the S3 model, constructing explicitly its solutions and discussing the Liouville integrability of the corresponding dynamical system. By means of the Kostantrepresentation of a generic Lie algebra element, we were able to develop an
Black Hole Nilpotent Orbits and Tits Satake Universality Classes
"... In this paper we consider the problem of classification of nilpotent orbits for the pseudoquaternionic coset manifolds U/H obtained in the timelike dimensional reduction of N = 2 supergravity models based on homogeneous symmetric special geometries. Within the D = 3 approach this classification am ..."
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In this paper we consider the problem of classification of nilpotent orbits for the pseudoquaternionic coset manifolds U/H obtained in the timelike dimensional reduction of N = 2 supergravity models based on homogeneous symmetric special geometries. Within the D = 3 approach this classification amounts to a classification of regular and singular extremal black hole solutions of supergravity. We show that the pattern of such orbits is a universal property depending only on the TitsSatake universality class of the considered model, the number of such classes being five. We present a new algorithm for the classification and construction of the nilpotent orbits for each universality class which is based on an essential use of the Weyl group W of the Tits Satake subalgebra UTS ⊂ U and on a certain subgroup thereof WH ⊂ W. The splitting of orbits of the full group U into suborbits with respect to the stability subgroup H? ⊂ U is shown to be governed by the structure of the discrete coset W/WH. For the case of the universality class SO(4, 5)/SO(2, 3) × SO(2, 2) we derive the complete list of nilpotent orbits which happens to contain 37 elements. We also show how the universal orbits are regularly embedded in all the members of the class that are infinite in number. As a matter of check we apply our new algorithm also to the Tits Satake class G(2,2)/SL(2) × SL(2) confirming the previously obtained result encompassing 7 nilpotent orbits. Perspectives for future developments based on the obtained results are outlined. 1Prof. Fre ́ is presently fulfilling the duties of Scientific Counsellor of the Italian Embassy in the Russian Federation, Denezhnij pereulok, 5, 121002 Moscow, Russia. ar X iv