The action with manifest E7 type symmetry

Abstract

Abstract We generalize Cremmer-Julia 1st order action of $$ \mathcal{N}=8 $$ N = 8 supergravity with manifest E 7(7) symmetry to cases of $$ \mathcal{N}=6 $$ N = 6 with manifest SO∗(12) and $$ \mathcal{N}=5 $$ 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 $$ \mathcal{N}\ge 5 $$ N ≥ 5 supergravitiesaredeformedbyacandidateUVdivergencetheghostfielddoesnot decouple. Therefore we argue that U duality and supersymmetry suggest an explanation of the mysterious cancellation of UV infinities at L = 4, $$ \mathcal{N}=5 $$ N = 5 in d=4. The same reasoning implies that, in absence of duality and supersymmetry anomalies, which still require a better understanding, $$ \mathcal{N}\ge 5 $$ N ≥ 5 perturbative supergravities may be UV finite at higher-loops.