Abstract
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. We solve these equations, arriving at the classification of maximally supersymmetric backgrounds whose associated Killing superalgebras are precisely the filtered deformations we classify in this paper.
Highlights
A number of impressive exact results [1,2,3,4,5,6,7,8,9] obtained in recent years via supersymmetric localisation have motivated a more systematic exploration of quantum field theories with rigid supersymmetry in curved space
We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of Z-graded subalgebras with maximum odd dimension of the N = 1 Poincare 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 have considered the supersymmetries of rigid supersymmetric field theories on Lorentzian four-manifolds from the viewpoint of their Killing superalgebras
Summary
A number of impressive exact results [1,2,3,4,5,6,7,8,9] obtained in recent years via supersymmetric localisation have motivated a more systematic exploration of quantum field theories with rigid supersymmetry in curved space. The most systematic strategy for identifying curved backgrounds which support some amount of rigid supersymmetry has hereto been that pioneered by Festuccia and Seiberg in [12] In four dimensions, they described how a large class of rigid supersymmetric non-linear sigma-models in curved space can be obtained by taking a decoupling limit (in which the Planck mass goes to infinity) of the corresponding locally supersymmetric theory coupled to minimal off-shell supergravity. These results are contained in Theorem 7 in section 3.2 and Proposition 8, respectively. It is worth pointing out that ([18] section 2.1) contains several useful identities (e.g., integrability conditions and covariant derivatives of Killing spinor bilinears) that we encounter in our construction of the Killing superalgebra for minimal off-shell supergravity backgrounds
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