Eight-dimensional topological gravity and its correspondence with supergravity

A topological theory for Euclidean gravity in eight dimensions can be built by enforcing octonionic self-duality conditions on the spin connection. The eight-dimensional manifold must be of a special type, with G2⊂Spin(7)⊂SO(8) holonomy. The resulting theory is related to a twisted version of N=1, D=8 supergravity. The situation is comparable to that of the topological Yang–Mills theory in eight dimensions, for which the SO(8) invariance is broken down to Spin(7), but is recovered after untwisting the topological theory.