Higher derivative gravity and torsion from the geometry of C-spaces

We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, but its straightforward extension—the so-called Clifford space (C-space), a manifold of points, lines, areas, etc.; physical quantities are Clifford algebra valued objects, called polyvectors. This provides a natural framework for description of supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. The geometry of curved C-space is investigated. It is shown that the curvature in C-space contains higher orders of the curvature in the underlying ordinary space. A C-space is parametrized not only by 1-vector coordinates xμ but also by the 2-vector coordinates σμν, 3-vector coordinates σμνρ, etc., called also holographic coordinates, since they describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto the coordinate planes. A remarkable relation between the “area” derivative ∂/∂σμν and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates σμν this indicates the presence of torsion, and if a vector valued quantity depends so, this implies non vanishing curvature. We argue that such a deeper understanding of the C-space geometry is a prerequisite for a further development of this new theory which in our opinion will lead us towards a natural and elegant formulation of M-theory.