Abstract
Superconformal geometries in spacetime dimensions D = 3, 4, 5 and 6 are discussed in terms of local supertwistor bundles over standard superspace. These natually admit superconformal connections as matrix-valued one-forms. In order to make contact with the standard superspace formalism it is shown that one can always choose gauges in which the scale parts of the connection and curvature vanish, in which case the conformal and S-supersymmetry transformations become subsumed into super-Weyl transformations. The number of component fields can be reduced to those of the minimal off-shell conformal supergravity multiplets by imposing constraints which in most cases simply consists of taking the even covariant torsion two-form to vanish. This must be supplemented by further dimension-one constraints for the maximal cases in D = 3, 4. The subject is also discussed from a minimal point of view in which only the dimension-zero torsion is introduced. Finally, we introduce a new class of supermanifolds, local super Grassmannians, which provide an alternative setting for superconformal theories.
Highlights
Dedicated to this year’s Nobel Laureate in Physics, Sir Roger Penrose, in appreciation of his many achievements, including the invention of twistor theory
In order to make contact with the standard superspace formalism it is shown that one can always choose gauges in which the scale parts of the connection and curvature vanish, in which case the conformal and S-supersymmetry transformations become subsumed into super-Weyl transformations
The superspace geometry corresponding to all D = 4 off-shell CSG multiplets was given in [8, 9] using conventional superspace with an SL(2, C)×Υ(N ) group in the tangent spaces together with real super-Weyl transformations
Summary
The objective now is to construct an element of the conformal group depending on the scale and conformal parameters, and a connection one-form with values in the conformal algebra which will transform in the required way provided that the transformation of the Schouten tensor is as given above in (2.10). Where Qb := eaQab, we observe that the primed quantities are invariant under infinitesimal X gauge transformations, for which δQa = D Xa. We can define a new curvature two-form R a := D Qa. We can interpret X as a local conformal boost parameter, Qa as the corresponding gauge field and R a as its curvature.
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