65 research outputs found
Conformal Triality of the Kepler problem
We show that the Kepler problem is projectively equivalent to null geodesic
motion on the conformal compactification of Minkowski-4 space. This space
realises the conformal triality of Minkwoski, dS and AdS spaces.Comment: 4 pages, no figures. Some modification
Hidden Symmetries of Dynamics in Classical and Quantum Physics
This article reviews the role of hidden symmetries of dynamics in the study
of physical systems, from the basic concepts of symmetries in phase space to
the forefront of current research. Such symmetries emerge naturally in the
description of physical systems as varied as non-relativistic, relativistic,
with or without gravity, classical or quantum, and are related to the existence
of conserved quantities of the dynamics and integrability. In recent years
their study has grown intensively, due to the discovery of non-trivial examples
that apply to different types of theories and different numbers of dimensions.
Applications encompass the study of integrable systems such as spinning tops,
the Calogero model, systems described by the Lax equation, the physics of
higher dimensional black holes, the Dirac equation, supergravity with and
without fluxes, providing a tool to probe the dynamics of non-linear systems.Comment: 54 pages, review article. To be published in Rev. Mod. Phy
Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux
The Eisenhart-Duval lift allows embedding non-relativistic theories into a
Lorentzian geometrical setting. In this paper we study the lift from the point
of view of the Dirac equation and its hidden symmetries. We show that
dimensional reduction of the Dirac equation for the Eisenhart-Duval metric in
general gives rise to the non-relativistic Levy-Leblond equation in lower
dimension. We study in detail in which specific cases the lower dimensional
limit is given by the Dirac equation, with scalar and vector flux, and the
relation between lift, reduction and the hidden symmetries of the Dirac
equation. While there is a precise correspondence in the case of the lower
dimensional massive Dirac equation with no flux, we find that for generic
fluxes it is not possible to lift or reduce all solutions and hidden
symmetries. As a by-product of this analysis we construct new Lorentzian
metrics with special tensors by lifting Killing-Yano and Closed Conformal
Killing-Yano tensors and describe the general Conformal Killing-Yano tensor of
the Eisenhart-Duval lift metrics in terms of lower dimensional forms. Lastly,
we show how dimensionally reducing the higher dimensional operators of the
massless Dirac equation that are associated to shared hidden symmetries it is
possible to recover hidden symmetry operators for the Dirac equation with flux.Comment: 18 pages, no figures. Version 3: some typos corrected, some
discussions clarified, part of the abstract change
Ricci-flat spacetimes admitting higher rank Killing tensors
Ricci-flat spacetimes of signature (2,q) with q=2,3,4 are constructed which
admit irreducible Killing tensors of rank-3 or rank-4. The construction relies
upon the Eisenhart lift applied to Drach's two-dimensional integrable systems
which is followed by the oxidation with respect to free parameters. In four
dimensions, some of our solutions are anti-self-dual.Comment: 12 page
Classification of supersymmetric spacetimes in eleven dimensions
We derive, for spacetimes admitting a Spin(7) structure, the general local
bosonic solution of the Killing spinor equation of eleven dimensional
supergravity. The metric, four form and Killing spinors are determined
explicitly, up to an arbitrary eight-manifold of Spin(7) holonomy. It is
sufficient to impose the Bianchi identity and one particular component of the
four form field equation to ensure that the solution of the Killing spinor
equation also satisfies all the field equations, and we give these conditions
explicitly.Comment: 9 pages, latex. v2: change of title (formerly known as "Spin(7)
structures in eleven dimensions"); short section on integrability conditions
added, various minor changes. To appear in Phys.Rev.Let
On Integrability of spinning particle motion in higher-dimensional black hole spacetimes
We study the motion of a classical spinning particle (with spin degrees of
freedom described by a vector of Grassmann variables) in higher-dimensional
general rotating black hole spacetimes with a cosmological constant. In all
dimensions n we exhibit n bosonic functionally independent integrals of
spinning particle motion, corresponding to explicit and hidden symmetries
generated from the principal conformal Killing--Yano tensor. Moreover, we
demonstrate that in 4-, 5-, 6-, and 7-dimensional black hole spacetimes such
integrals are in involution, proving the bosonic part of the motion integrable.
We conjecture that the same conclusion remains valid in all higher dimensions.
Our result generalizes the result of Page et. al. [hep-th/0611083] on complete
integrability of geodesic motion in these spacetimes.Comment: Version 2: revised version, added references. 5 pages, no figure
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