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.

我们计算了 5 个时空维度中最小庞加莱超代数的相关 Spencer 上同调，并使用它通过 5 维洛伦兹自旋流形的旋量丛上的连接来定义杀死旋量。我们根据这些数据给出了玻色子背景的定义。通过对旋量连接的曲率施加约束，我们恢复了最小（未测量的）5 维超重力的场方程，但也找到了一组\( \mathfrak{sp} \)(1)-valued one-form 我们将其解释为弯曲背景上一类刚性超对称理论的玻色子数据。我们定义了玻色子背景的 Killing 超代数，并表明它们的存在是由场方程暗示的。表征了最大超对称背景，并且它们的 Killing 超代数被明确描述为 Poincaré 超代数的过滤变形。