Explicit Action of E7(7) on N=8 Supergravity Fields

TL;DR

This work provides an explicit, all-orders nonlinear realization of the hidden symmetry $E_{7(7)}$ in on-shell ${\mathcal N}=8$ supergravity within the unitary gauge, together with a conserved Noether-Gaillard-Zumino current for duality. By formulating the transformation laws for scalars, vectors, and spinors in terms of the coset coordinates $y_{ij,kl}$ and establishing a field-dependent $SU(8)$ compensator, the authors produce closed expressions valid to all orders in the gravitational coupling. A 133-component Noether current is derived, split into a linear $SU(8)$ part and a non-linear sector, and its conservation is demonstrated on-shell; a linearized $SU(8)$ charge is also extracted for asymptotic states. The results illuminate the structure of classical symmetries in ${\cal N}=8$ supergravity, including a conformal realization with a dilatation generator, and provide a framework for probing potential anomalies and UV properties relevant to finiteness scenarios. Overall, the paper strengthens the bridge between duality symmetries, S-matrix constraints, and UV behavior in maximal supergravity.

Abstract

We present an explicit, exact to all orders in gravitational coupling E7(7) symmetry transformations of on-shell N=8 supergravity fields in the gauge with 70 scalars in E7(7)/SU(8) coset space, the local SU(8) symmetry being fixed. The non-linear realization of E7(7) includes a field-dependent SU(8) transformation preserving the unitary gauge. We find the conserved Noether-Gaillard-Zumino current of E7(7) symmetry, the linear part of it being a chiral SU(8) symmetry. We comment on the conformal realization of the E7(7) algebra which includes a dilatation operator. We hope that these results can be useful for studies of anomalies/absence of anomalies and the UV behavior of N=8 supergravity.