Exceptional Jordan matrix models, octonionic strings/branes and star product deformations

Abstract

A brief review of the essentials behind the construction of a Chern-Simons-like brane action from the Large N limit of Exceptional Jordan Matrix Models paves the way to the construction of actions for membranes and p-branes moving in octonionic-spacetime backgrounds endowed with octonionic-valued metrics. It is found that the action of a membrane moving in spacetime backgrounds endowed with an octonionic-valued metric is not invariant under the usual diffeomorphisms of its world volume coordinates σa→σ′a(σb), but instead, it is invariant under the rigid E6(−26) transformations which preserve the volume (cubic) form. An extensive study of the bosonic octonionic string moving in flat backgrounds, and its quantization, follows. One of the most significant findings (without invoking supersymmetry) is that the number of degrees of freedom of three (massless) fermion generations is the same as the number of transverse dimensions of an octonionic string moving in D=26 octonionic dimensions. The star-product deformations of octonionic p-branes follow. In particular, we focus on the octonionic membrane along with the phase space quantization methods developed by [30] within the context of Nonassociative Quantum Mechanics. We finalize with an analysis of two octonionic gravitational actions and with some concluding remarks on Double and Exceptional Field theories, Nonassociative Gravity and A∞,L∞ algebras.