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

TL;DR

This paper argues that N=4 supergravity does not have a 3-loop divergence due to symmetry considerations, and extends similar reasoning to higher N supergravities, suggesting all are finite at their respective loop orders.

Contribution

It provides a symmetry-based argument showing the absence of 3-loop divergence in N=4 supergravity and extends this reasoning to N=5,6,8 supergravities at higher loops.

Findings

N=4 supergravity is 3-loop UV finite due to symmetry constraints.

Symmetry arguments extend to N=5,6,8 supergravities, preventing divergences at certain loops.

Duality symmetry violations relate to potential divergences in extended supergravities.

Abstract

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.