U-gravity: SL(N)

Abstract

We construct a duality manifest gravitational theory for the special linear group, SL(N) with N ≠ 4. The spacetime is formally extended, to have the dimension 1/2 N (N - 1), yet is gauged. Consequently the theory is subject to a section condition. We introduce a semi-covariant derivative and a semi-covariant 'Riemann' curvature, both of which can be completely covariantized after symmetrizing or contracting the SL(N) vector indices properly. Fully covariant scalar and 'Ricci' curvatures then constitute the action and the 'Einstein' equation of motion. For N ≥ 5, the section condition admits duality inequivalent two solutions, one (N - 1)-dimensional and the other three-dimensional. In each case, the theory can describe not only Riemannian but also non-Riemannian backgrounds. © 2014 The Author(s).