On duality symmetries of supergravity invariants

TL;DR

This work investigates how duality symmetries constrain counterterms in maximal supergravity, using dimensional reduction, superspace measure considerations, and the ectoplasm/closed-superform framework. It shows that $E_{7(7)}$ invariance cannot be maintained for the known $R^{4}$- and related BPS-type invariants in $D=4$ with $N=8$, forcing the first duality-invariant candidate to appear only at seven loops as a non-BPS $ abla^{8}R^{4}$ term (the superspace volume). Extending the analysis to $N=5$ and $N=6$, the authors demonstrate the non-invariance of the corresponding $R^{4}$ and $ abla^{2}R^{4}$ invariants under the duality groups, yielding finiteness at three loops for $N=5,6$ and at four loops for $N=6$, with duality arguments pushing possible divergences to higher orders. The combination of dimensional-reduction Laplace equations, harmonic-measure obstructions, and closed-superform cocycles provides a cohesive, field-theoretic explanation for the observed finiteness patterns and the structure of potential higher-loop invariants in these theories.

Abstract

The role of duality symmetries in the construction of counterterms for maximal supergravity theories is discussed in a field-theoretic context from different points of view. These are: dimensional reduction, the question of whether appropriate superspace measures exist and information about non-linear invariants that can be gleaned from linearised ones. The former allows us to prove that F-term counterterms cannot be E7(7)-invariant in D=4, N=8 supergravity or E6(6)-invariant in D=5 maximal supergravity. This is confirmed by the two other methods which can also be applied to D=4 theories with fewer supersymmetries and allow us to prove that N=6 supergravity is finite at three and four loops and that N=5 supergravity is three-loop finite.