Killing superalgebras for Lorentzian four-manifolds

Understand

We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of $\mathbb{Z}$-graded subalgebras with maximum odd dimension of the $N{=}1$ Poincar\'e superalgebra in four dimensions.

• Part of this calculation involves computing a Spencer cohomology group which, by analogy with a similar result in eleven dimensions, prescribes a notion of Killing spinor, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four dimensions.
• We prove that such Killing spinors always generate a Lie superalgebra, and that this Lie superalgebra is a filtered deformation of a subalgebra of the $N{=}1$ Poincar\'e superalgebra in four dimensions.
• Demanding the flatness of the connection defining the Killing spinors, we obtain equations satisfied by the maximally supersymmetric backgrounds.