Abstract
We calculate the relevant Spencer cohomology of the minimal Poincaré superalgebra in 5 spacetime dimensions and use it to define Killing spinors via a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold. We give a definition of bosonic backgrounds in terms of this data. By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal (ungauged) 5-dimensional supergravity, but also find a set of field equations for an mathfrak{sp} (1)-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric theories on curved backgrounds. We define the Killing superalgebra of bosonic backgrounds and show that their existence is implied by the field equations. The maximally supersymmetric backgrounds are characterised and their Killing superalgebras are explicitly described as filtered deformations of the Poincaré superalgebra.
Highlights
The interplay between supersymmetry and geometry has a long and beautiful history, but it is fair to say that we are still trying to understand which geometries can support supersymmetric theories
By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal 5-dimensional supergravity, and find a set of field equations for an sp(1)-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric theories on curved backgrounds
We define the Killing superalgebra of bosonic backgrounds and show that their existence is implied by the field equations
Summary
The interplay between supersymmetry and geometry has a long and beautiful history, but it is fair to say that we are still trying to understand which geometries can support supersymmetric theories. If we assume that the definition of a Killing spinor is that it be parallel with respect to a suitable connection in the spinor bundle (possibly augmented by algebraic — i.e., non-differential — constraints), a straightforward generalisation of the result in [5] for the Killing superalgebra of eleven-dimensional supergravity backgrounds shows that the resulting superalgebra has a special algebraic structure It is naturally filtered in such a way that the associated graded superalgebra is a graded subalgebra of the Poincaré superalgebra. 6 [14] Poincaré superalgebra, the Spencer cohomology is richer: additional bosonic fields may be turned on, and the definitions of Killing spinors, supersymmetric backgrounds and Killing superalgebras may be consistently generalised to accommodate them The existence of such generalisations is intriguing, not least because they provide curved backgrounds for rigidly supersymmetric theories which do not appear to be attainable via supergravity. Appendix A is a compilation of combinatorial tensor identities used in geometric calculations
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