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

TL;DR

This paper proposes that the 3-loop UV finiteness observed in $d=4$ ${\\cal N}=4$ supergravity arises from duality-current conservation governed by the Noether-Gaillard-Zumino identity, leveraging the universal structure of type-${\\rm E7}$ duality groups. It analyzes how duality invariance constrains possible counterterms, showing that the known duality-invariant candidates would violate NGZ unless accompanied by compensating deformations, thereby forbidding the 3-loop divergence. Through a toy model and a discussion of ${\\cal N}=4$ invariants $I^{\\cal N}$, the work argues that the non-BPS invariants cannot act as UV regulators in a way that preserves duality at the quantum level. The authors also explore Born-Infeld–type deformations as a potential route to restore duality in certain degenerate cases, highlighting the need for further data (e.g., ${\\cal N}=5$ and ${\\cal N}=6$) to test universality across 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.