We develope the analogue of S-duality for linearized gravity in (3+1)-dimensions. Our basic idea is to consider the self-dual (anti-self-dual) curvature tensor for linearized gravity in the context of the Macdowell-Mansouri formalism. We find that the strong-weak coupling duality for linearized gravity is an exact symmetry and implies small-large duality for the cosmological constant.

By adding the Pontrjagin topological invariant to the gauge theory of the de Sitter group proposed by MacDowell and Mansouri we obtain an action quadratic in the fieldstrengths, of the Chern-Simons type, from which the Ashtekar formulation is derived.

We investigate the possibility of extending the Ashtekar theory to eight dimensions. Our approach relies on two notions: the octonionic structure and the MacDowell-Mansouri formalism generalized to a spacetime of signature 1+7. The key mathematical tool for our construction is the self-dual (antiselfdual) four-rank fully antisymmetric octonionic tensor. Our results may be of particular interest in connection with a possible formulation of M−theory via matroid theory.

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