U-gravity : ${\mathbf{SL}(N)}$

Abstract: We construct a duality manifest gravitational theory for the special linear group, ${\mathbf{SL}(N)}$ with $N{\neq 4}$. The spacetime is formally extended, to have the dimension $\textstyle{\frac{1}{2}} N(N-1)$, yet is gauged'. Consequently the theory is subject to a section condition. We introduce a semi-covariant derivative and a semi-covariantRiemann' curvature, both of which can be completely covariantized after symmetrizing or contracting the ${\mathbf{SL}(N)}$ vector indices properly. Fully covariant scalar and Ricci' curvatures then constitute the action and theEinstein' equation of motion. For $N\geq 5$, 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.