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Self-Dual N=(1,0) Supergravity in Eight Dimensions with Reduced Holonomy Spin(7)

We construct chiral N=(1,0) self-dual supergravity in Euclidean eight-dimensions with reduced holonomy Spin(7), including all the higher-order interactions in a closed form. We first establish the non-chiral N=(1,1) superspace supergravity in eight-dimensions with SO(8) holonomy without self-duality, as the foundation of the formulation. In order to make the whole computation simple, and the generalized self-duality compatible with supersymmetry, we adopt a particular set of superspace constraints similar to the one originally developed in ten-dimensional superspace. The intrinsic properties of octonionic structure constants make local supersymmetry, generalized self-duality condition, and reduced holonomy Spin(7) all consistent with each other.