Flat Superspace Research Articles

Global properties of supermanifolds

We construct new examples of supermanifolds, and determine the vector bundle structure of the supermanifolds commonly used in physics. We show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts, and which is one of a nested sequence of foliations. We point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds. For example, a compact supermanifold with a single even dimension must have vanishing Euler characteristic. We also show that a globally defined superfield on a “nice” compact supermanifold must be constant along the leaves of the foliations. By this mechanism, the global topology of a supermanifold can be used to impose physically interesting constraints on superfields. As an example, we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral.

Graded de Sitter space-time, group and algebra

The (m+n)-dimensional flat superspace is considered and graded linear groups with their graded algebra are calculated. The (4+4)-dimensional graded de Sitter space-time is discussed geometrically by its embedding in the (5+4)-dimensional flat superspace, and its isometry subgroupOSp¼ is calculated. We have used the isomorphism between the algebra ofSp4,R andSO3,2 to express the generatorsSab ofSO3,2 in terms of the generatorshαβ of the symplectic groupSp4,R, then it is easy to obtain the graded de Sitter algebra from the orthosymplectic algebraOSp¼. Finally, by contraction we obtain the graded Poincare algebra, that is the superalgebra with which we are familiar.

SUPER KILLING EQUATION AND SUPERSYM-METRY TRANSFORMATION

In super space (x, θ), as the metric tensor field GAB(x, θ) is given, we calculated the Riemann curvature tensor RDABC of fourth rank and its generalized cyclicity.The equation that must be satisfied by isometry in the super space, i.e. super Killing equation: ξA:B+ηabξB:A=0, is deduced.In flat super space with zero curvature tensor, we have obtained the general solu-tions of the super Killing equation and the commutation relations of the corresponding generators. In the case of constant curvature, we have obtained a special solution of the super Killing equation.

Einstein equations, supersymmetry, and flat limits of curved superspace

An affine theory of unified interactions is reviewed. Einstein's free gravitational equation emerges as a special case of the proposed equations of motion on superspace. For this example the vacuum is spontaneously broken to be globally Lorentz invariant. Flat superspace is defined and the corresponding algebraic properties studied. Next the graded de Sitter group is introduced and various contractions performed. From these we obtain and discuss supersymmetry as well as some spin-3/2 algebras. Finally, form invariance under Fermi displacements is employed to develop a powerful calculational technique, an example of which is presented in the Appendix.

Flat superspace with supertorsion

We give a simple proof that a recently suggested theory of space-time as contained in a flat supermanifold with specified torsion in the Fermi variables makes good physical sense. We do this in two parts by showing firstly that supergravity is appropriately contained in such a theory and secondly that the resulting theory, on quantisation, is expected to be renormalisable. We briefly discuss the manner in which matter may be included in this theory.

Bianchi identities for supersymmetric gauge theories

The Bianchi identities for gauge theories in an extended flat superspace are evaluated. They permit better understanding of possible constraint equations, and can serve as a starting point for further constructions of gauge theories with extended supersymmetry.