The Action with Manifest E7 Type Symmetry

We generalize Cremmer-Julia 1st order action of $N=8$ supergravity with manifest $E_{7(7)}$ symmetry to cases of $N=6$ with manifest $SO^*(12)$ and $N=5$ with manifest $SU(1,5)$ duality symmetries. These U dualities belong to groups of type E7 which do not admit a symmetric bilinear invariant for vector fields. Therefore the 2d order classical action derived from the one with manifest E7 type duality has a ghost vector field which, under appropriate boundary conditions, decouples. We show that when classical $N\geq 5$ supergravities are deformed by a candidate UV divergence the ghost field does not decouple. Therefore we argue that U duality and supersymmetry suggest an explanation of the mysterious cancellation of UV infinities at $L=4$, $N=5$ in d=4. The same reasoning implies that, in absence of duality and supersymmetry anomalies, which still require a better understanding, $N\geq 5$ perturbative supergravities may be UV finite at higher-loops.