Lorentz covariant spin two superspaces

Superalgebras including generators having spins up to two and realisable as tangent vector fields on Lorentz covariant generalised superspaces are considered. The latter have a representation content reminiscent of configuration spaces of (super)gravity theories. The most general canonical supercommutation relations for the corresponding phase space coordinates allowed by Lorentz covariance are discussed. By including generators transforming according to every Lorentz representation having spin up to two, we obtain, from the super Jacobi identities, the complete set of quadratic equations for the Lorentz covariant structure constants. These defining equations for spin two Heisenberg superalgebras are highly overdetermined. Nevertheless, non-trivial solutions can indeed be found. By making some simplifying assumptions, we explicitly construct several classes of these superalgebras.Comment: 20 pages, latex, typos correcte

Supersymmetric integrable systems from geodesic flows on superconformal groups

We discuss the possible relationship between geodesic flow, integrability and supersymmetry, using fermionic extensions of the KdV equation, as well as the recently introduced supersymmetrisation of the Camassa-Holm equation, as illustrative examples.Comment: 6 pages, late

Super self-duality for Yang-Mills fields in dimensions greater than four

Self-duality equations for Yang-Mills fields in d-dimensional Euclidean spaces consist of linear algebraic relations amongst the components of the curvature tensor which imply the Yang-Mills equations. For the extension to superspace gauge fields, the super self-duality equations are investigated, namely, systems of linear algebraic relations on the components of the supercurvature, which imply the self-duality equations on the even part of superspace. A group theory based algorithm for finding such systems is developed. Representative examples in various dimensions are provided, including the Spin(7) and G(2) invariant systems in d=8 and 7, respectively.Comment: 51 pages, late

Special Graphs

A special p-form is a p-form which, in some orthonormal basis {e_\mu}, has components \phi_{\mu_1...\mu_p} = \phi(e_{\mu_1},..., e_{\mu_p}) taking values in {-1,0,1}. We discuss graphs which characterise such forms.Comment: 8 pages, V2: a texing error correcte

Ternutator Identities

The ternary commutator or ternutator, defined as the alternating sum of the product of three operators, has recently drawn much attention as an interesting structure generalising the commutator. The ternutator satisfies cubic identities analogous to the quadratic Jacobi identity for the commutator. We present various forms of these identities and discuss the possibility of using them to define ternary algebras.Comment: 12 pages, citation adde

Democratic Supersymmetry

We present generalisations of N-extended supersymmetry algebras in four dimensions, using Lorentz covariance and invariance under permutation of the N supercharges as selection criteria.Comment: 26 pages, latex fil

Oxidation of self-duality to 12 dimensions and beyond

Using (partial) curvature flows and the transitive action of subgroups of O(d,Z) on the indices {1,...,d} of the components of the Yang-Mills curvature in an orthonormal basis, we obtain a nested system of equations in successively higher dimensions d, each implying the Yang-Mills equations on d-dimensional Riemannian manifolds possessing special geometric structures. This `matryoshka' of self-duality equations contains the familiar self-duality equations on Riemannian 4-folds as well as their generalisations on complex K\"ahler 3-folds and on 7- and 8-dimensional manifolds with G_2 and Spin(7) holonomy. The matryoshka allows enlargement (`oxidation') to a remarkable system in 12 dimensions invariant under Sp(3). There are hints that the underlying geometry is related to the sextonions, a six-dimensional algebra between the quaternions and octonions.Comment: to appear in Commun. Math. Phys., 30 pages, discussion on lower bounds to the action and references added to section 2, typos correcte

Complete integrability in classical gauge theories

We consider completely integrable classical field theory models with a view to identifying the properties which characterize their integrability. In particulars, we study the infinite sets of ‘hidden' symmetries, and the corresponding transformations carrying representations of infinite dimensional loop algebras, of the following models; the chiral-field equations in two dimensions, the self-dual sector of pure gauge theories in 4 dimensions, the functional (loop-space) formulation of 3-dimensional gauge theories, and some sectors of the extended superaymmetric gauge theories. We also construct an infinite number of conserved spinor currents for the latter theories. The (non-) integrability of the full four dimensional Yang-Mills equations is studied? and a local approximation for the non-integrable phase factor of gauge theories on an arbitrary, infinitesimally small, straight-line path is presented. Finally, we study classical gauge theories in dimensions greater than four5 and obtain, in analogy to the self-duality equations, algebraic equations for the field-strength which automatically imply the higher dimensional Yang-Mills equations as a consequence of the Bianchi identities. The most interesting sets of equations found are those in eight dimensions which have a structure related to the algebra of the octonions