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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We make a number of observations about matter-ghost string phase, which may eventually lead to a formal connection between matroid theory and string theory. In particular, in order to take advantage of the already established connection between matroid theory and Chern-Simons theory, we propose a generalization of string theory in terms of some kind of Kahler metric. We show that this generalization is closely related to the Kahler-Chern-Simons action due to Nair and Schiff. In addition, we discuss matroid/string connection via matroid bundles and a Schild type action, and we add new information about the relationship between matroid theory, D = 11 supergravity and Chern-Simons formalism.

A gauge theory of supergravity is constructed based only on the supersymmetric self-dual spin connection associated to the supergroup OSp͑1 j 4͒. We show that Jacobson's supergravity action arises naturally from our proposed action. It is formulated by taking the self-dual part of the MacDowell-Mansouri gauge theory of supergravity. In this sense, our quadratic action in the supersymmetric self-dual curvature tensor provides a relation between these two important previous extensions of supergravity. [S0031-9007(96)00197-4]

We develop a Born-Infeld type theory for gravity in any dimension. We show that in four dimensions our formalism allows a self-dual (or anti-self dual) Born-Infeld gravity description. Moreover, we show that such a self-dual action is reduced to both the Deser-Gibbons and the Jacobson-Smolin-Samuel action of Ashtekar formulation. A supersymmetric generalization of our approach is outlined.

In the context of field theory two elements seem to be necessary to search for strong-weak coupling duality. First, a gauge theory formulation and second, supersymmetry. For gravitation these two elements are present in MacDowell-Mansouri supergravity. The search for an "effective duality" in this theory presents technical and conceptual problems that we discuss. Nevertheless, by means of a field theoretical approach, which in the abelian case * Present Address: School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton NJ 08540 USA. E-mail: compean@sns.ias.edu † E-mail: nieto@uas.uasnet.mx ‡ E-mail: octavio@ifug3.ugto.mx § E-mail: cramirez@fcfm.buap.mx 1 coincides with S-duality, we exhibit a dual theory, with inverted couplings.This results in a supersymmetric non-linear sigma model of the FreedmanTownsend type.