134 research outputs found
Symmetric Spaces in Supergravity
We exploit the relation among irreducible Riemannian globally symmetric
spaces (IRGS) and supergravity theories in 3, 4 and 5 space-time dimensions.
IRGS appear as scalar manifolds of the theories, as well as moduli spaces of
the various classes of solutions to the classical extremal black hole Attractor
Equations. Relations with Jordan algebras of degree three and four are also
outlined.Comment: 1+23 pages, 15 Tables. Contribution to the Proceedings of the
Conference "Symmetry in Mathematics and Physics", 18-20 Jan. 2008, IPAM
(UCLA), in celebration of V. S. Varadarajan's 70th Birthda
The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems
We review the general problem of finding a global rotation that transforms a
given set of points and/or coordinate frames (the "test" data) into the best
possible alignment with a corresponding set (the "reference" data). For 3D
point data, this "orthogonal Procrustes problem" is often phrased in terms of
minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean
distance measure relating the two sets of matched coordinates. We focus on
quaternion eigensystem methods that have been exploited to solve this problem
for at least five decades in several different bodies of scientific literature
where they were discovered independently. While numerical methods for the
eigenvalue solutions dominate much of this literature, it has long been
realized that the quaternion-based RMSD optimization problem can also be solved
using exact algebraic expressions based on the form of the quartic equation
solution published by Cardano in 1545; we focus on these exact solutions to
expose the structure of the entire eigensystem for the traditional 3D spatial
alignment problem. We then explore the structure of the less-studied
orientation data context, investigating how quaternion methods can be extended
to solve the corresponding 3D quaternion orientation frame alignment (QFA)
problem, noting the interesting equivalence of this problem to the
rotation-averaging problem, which also has been the subject of independent
literature threads. We conclude with a brief discussion of the combined 3D
translation-orientation data alignment problem. Appendices are devoted to a
tutorial on quaternion frames, a related quaternion technique for extracting
quaternions from rotation matrices, and a review of quaternion
rotation-averaging methods relevant to the orientation-frame alignment problem.
Supplementary Material covers extensions of quaternion methods to the 4D
problem.Comment: This replaces an early draft that lacked a number of important
references to previous work. There are also additional graphics elements. The
extensions to 4D data and additional details are worked out in the
Supplementary Material appended to the main tex
The r-map, c-map and black hole solutions
We consider various geometrical and physical aspects of the r-map and c-map, which are two maps induced by the dimensional reduction of 5d and 4d, N = 2 supergravity coupled to vector multiplets respectively. We treat reduction over a spacelike or timelike dimension on an equal footing, and prove, for the first time, that the target manifold in the image of the timelike c-map is para-quaternion Kahler. In order to do this we provide a new formulation of projective special Kahler geometry based on real Darboux coordinates, which is useful both mathematically and physically in its own right. As an application we investigate how the r-map and c-map can be used to generate new stationary black hole solutions. In four dimensions we construct new extremal non-BPS solutions, and in both four and five dimensions we construct new non-extremal solutions. We also take the first steps towards constructing new rotating solutions, though at this stage we only recover known solutions. The systematic and geometrical nature of these constructions allows us to gain a deeper understanding of many familiar properties of black holes in supergravity, such as the attractor mechanism and the transformation of BPS into non-BPS black holes using a field rotation matrix. We also observe an interesting and novel feature relating to non-extremal black holes: in order for solutions to correspond to non-extremal black holes with finite scalar fields we find that the number of integration constants must reduce by half. This suggests that non-extremal black holes always satisfy first order equations similar to their extremal counterparts. For STU-like models all calculations are performed explicitly
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 these
black-holes according to the H*-orbits in which the space of possible Lax
operators decomposes, H* being the isotropy group of scalar manifold
originating from time-like 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 one-field model S^3, to the algebraic classification of nilpotent
orbits and it is shown to provide a simple and practical discriminating method.
We present a detailed analysis of the S^3 model and its black hole solutions,
discussing the Liouville integrability of the corresponding dynamical system.
By means of the Kostant-representation of a generic Lie algebra element, we
were able to develop an algorithm which produces the necessary number of
hamiltonians in involution required by Liouville integrability of generic
orbits. The degenerate orbits correspond to extremal black-holes and are
nilpotent. We analyze these orbits in some detail working out different
representatives thereof and showing that the relation between H* orbits and
critical points of the geodesic potential is not one-to-one. Finally we present
the conjecture that our newly identified tensor classifiers are universal and
able to label all regular and nilpotent orbits in all homogeneous symmetric
Special Geometries.Comment: Analysis of nilpotent orbits in terms of tensor classifiers in
section 8.1 corrected. Table 1 corrected. Discussion in section 11 extende
Domain Walls, Black Holes, and Supersymmetric Quantum Mechanics
Supersymmetric solutions, such as BPS domain walls or black holes, in four-
and five-dimensional supergravity theories with eight supercharges can be
described by effective quantum mechanics with a potential term. We show how
properties of the latter theory can help us to learn about the physics of
supersymmetric vacua and BPS solutions in these supergravity theories. The
general approach is illustrated in a number of specific examples where scalar
fields of matter multiplets take values in symmetric coset spaces.Comment: 35 pages, added references and corrections, version to appear in NP
N=4 BPS black holes and octonionic twistors
Stationary, spherically symmetric solutions of N=2 supergravity in 3+1
dimensions have been shown to correspond to holomorphic curves on the twistor
space of the quaternionic-K\"ahler space which arises in the dimensional
reduction along the time direction. In this note, we generalize this result to
the case of 1/4-BPS black holes in N=4 supergravity, and show that they too can
be lifted to holomorphic curves on a "twistor space" Z, obtained by fibering
the Grassmannian F=SO(8)/U(4) over the moduli space in three-dimensions
SO(8,n_v+2)/SO(8)xSO(n_v+2). This provides a kind of octonionic generalization
of the standard constructions in quaternionic geometry, and may be useful for
generalizing the known BPS black hole solutions, and finding new non-BPS
extremal solutions.Comment: 30 pages, one figure, uses JHEP3.cl
The importance of the Selberg integral
It has been remarked that a fair measure of the impact of Atle Selberg's work
is the number of mathematical terms which bear his name. One of these is the
Selberg integral, an n-dimensional generalization of the Euler beta integral.
We trace its sudden rise to prominence, initiated by a question to Selberg from
Enrico Bombieri, more than thirty years after publication. In quick succession
the Selberg integral was used to prove an outstanding conjecture in random
matrix theory, and cases of the Macdonald conjectures. It further initiated the
study of q-analogues, which in turn enriched the Macdonald conjectures. We
review these developments and proceed to exhibit the sustained prominence of
the Selberg integral, evidenced by its central role in random matrix theory,
Calogero-Sutherland quantum many body systems, Knizhnik-Zamolodchikov
equations, and multivariable orthogonal polynomial theory.Comment: 43 page
Quantum Attractor Flows
Motivated by the interpretation of the Ooguri-Strominger-Vafa conjecture as a
holographic correspondence in the mini-superspace approximation, we study the
radial quantization of stationary, spherically symmetric black holes in four
dimensions. A key ingredient is the classical equivalence between the radial
evolution equation and geodesic motion of a fiducial particle on the moduli
space M^*_3 of the three-dimensional theory after reduction along the time
direction. In the case of N=2 supergravity, M^*_3 is a para-quaternionic-Kahler
manifold; in this case, we show that BPS black holes correspond to a particular
class of geodesics which lift holomorphically to the twistor space Z of M^*_3,
and identify Z as the BPS phase space. We give a natural quantization of the
BPS phase space in terms of the sheaf cohomology of Z, and compute the exact
wave function of a BPS black hole with fixed electric and magnetic charges in
this framework. We comment on the relation to the topological string amplitude,
extensions to N>2 supergravity theories, and applications to automorphic black
hole partition functions.Comment: 43 pages, 6 figures; v2: typos and references added; v3: published
version, minor change
Gauged Supergravities in Three Dimensions: A Panoramic Overview
Maximal and non-maximal supergravities in three spacetime dimensions allow
for a large variety of semisimple and non-semisimple gauge groups, as well as
complex gauge groups that have no analog in higher dimensions. In this
contribution we review the recent progress in constructing these theories and
discuss some of their possible applications.Comment: 32 pages, 1 figure, Proceedings of the 27th Johns Hopkins workshop:
Goteborg, August 2003; references adde
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