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

Understand

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-covariant `Riemann' curvature, both of which can be completely covariantized after symmetrizing or contracting the ${\mathbf{SL}(N)}$ vector indices properly.