ar X iv : h ep - t h / 01 10 07 9 v 3 2 7 A pr 2 00 2 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 super-symmetry, 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 x µ but also by the 2-vector coordinates σ µν , 3-vector coordinates σ µνρ , 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 ∂/∂σ µν and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates σ µν 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.