On Absence of 3-loop Divergence in N=4 Supergravity

Understand

We argue that N=4 supergravity is 3-loop UV finite because the relevant supersymmetric candidate counterterm is known to be SL(2, R)x SO(6) invariant, which violates the Noether-Gaillard-Zumino current conservation.

• Analogous arguments, based on the universality properties of groups of type E7, also apply to N=5,6,8 in 4,5,7 loops, respectively, since the 1/N BPS invariants break duality symmetry between Bianchi identities and quantum corrected vector field equations.