Supergravity and the Knitting of the Kalb--Ramond Two-Form in Eight-Dimensional Topological Gravity

Topological euclidean gravity is built in eight dimensions for manifolds with $Spin(7) \subset SO(8)$ holonomy. In a previous work, we considered the construction of an eight-dimensional topological theory describing the graviton and one graviphoton. Here we solve the question of determining a topological model for the combined system of a metric and a Kalb--Ramond two-form gauge field. We then recover the complete $N=1, D=8$ supergravity theory in a twisted form. We observe that the generalized self-duality conditions of our model correspond to the octonionic string equations.