Hidden symmetries and supergravity solutions

Abstract

The role of Killing and Killing-Yano tensors for studying the geodesic motion of the particle and the superparticle in a curved background is reviewed. Additionally, the Papadopoulos list [G. Papadopoulos, Class. Quantum Grav. 25, 105016 ( 2008)] for Killing-Yano tensors in G structures is reproduced by studying the torsion types these structures admit. The Papadopoulos list deals with groups G appearing in the Berger classification, and we enlarge the list by considering additional G structures which are not of the Berger type. Possible applications of these results in the study of supersymmetric particle actions and in the AdS/CFT correspondence are outlined.

Hidden symmetries of dynamics in classical and quantum physics

“…Santillan has analysed the Killing-Yano equation for a number of structures outside those of the Berger type analysed by Papadopoulos (Santillan, 2012a).…”

“…Then the irreducible U (n/2) representations are (Gillard, 2005;Santillan, 2012b) (W 1 )ᾱβγ = (Ω) [ᾱ,βγ] (W 2 )ᾱβγ = (Ω)ᾱ ,βγ − (Ω) [ᾱ,βγ] ,…”

Invariant solutions to the conformal Killing–Yano equation on Lie groups

“…The concept of conformal Killing-Yano p-forms (also known in the literature as twistor forms or conformal Killing forms) on Riemannian manifolds was introduced by Tachibana in [1] for p = 2 and later by Kashiwada in [2] for general p. Applications of these forms to theoretical physics were found related to quadratic first integrals of geodesic equations, symmetries of field equations, conserved quantities and separation of variables, among others (see, for instance, [3][4][5][6][7]). More recently, since the work of Moroianu, Semmelmann [8,9], a renewed interest in the subject arose among differential geometers (see, for instance, [10][11][12][13][14][15][16]).…”

On the characterization of infinitesimal symmetries of the relativistic phase space

“…The requirement of projectability is quite natural under the physical viewpoint, but it could be dropped in order to consider symmetries of the generalized contact structure which depend on phase space in an essentially non-projectable way, as we already mentioned in the Introduction. Noether symmetries of non-projectable type correspond to conserved quantities that, in the standard 4-velocity formalism, depend polynomially on velocities [33]. The coefficients of monomials are Killing tensors.…”

“…The most important generalization is obtained by dropping the projectability hypothesis and dealing with vector fields depending on the phase space in an essential way. Non-projectable symmetries of this type are called higher symmetries [3], generalized symmetries [29] or hidden symmetries (see, for example, [33]) because they are not related to any symmetry of spacetime. Non-projectable symmetries of our generalized contact structure would be Noether symmetries of the equation of motion, like proper affine or projective symmetries [9], and would correspond to conserved quantities which can be higher degree polynomials in velocities.…”