Towards an Ashtekar formalism in 12 dimensions

Abstract

We discuss the Ashtekar formalism from the point of view of 12 dimensions. We first focus on the 2 + 10 spacetime signature and then we consider the transition 2 + 10 → (2 + 2) + (0 + 8). We argue that both sectors 2 + 2 and 0 + 8, which are exceptional signatures, can be analyzed from the point of view of a self-dual action associated with the Ashtekar formalism.Keywords Ashtekar theory · 12 dimensions · Two time physics Over the years it has become clear that considering gravitational and gauge theories with more than one time coordinate is a very interesting and useful idea for understanding different aspects of traditional gravitational and gauge field theories with only one time (see [1] and references therein). In particular, the 2 + 10-dimensional spacetime signature has emerged as an interesting possibility for the understanding of both supergravity and super Yang-Mills theory in D = 11 (see [2]). Thus, by seriously taking a 2 + 10-dimensional gravity one may be interested in various possibilities offered by this theory. For instance, one may be interested in a realistic theory in four-dimensions via the symmetry braking 2 + 10 → (1 + 3) + (1 + 7). However, this may not be the only attractive possibility. In fact, one may think on the alternative transition 2 + 10 → (2 + 2) + (0 + 8).(1) Dedicated to Octavio Obregón on the occasion of his 60th birthday.

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