The complete $D=11$ embedding of SO(8) supergravity

TL;DR

This paper provides a complete non-linear embedding of $D=11$ supergravity on $S^7$ into $D=4$ ${ m SO}(8)$-gauged supergravity, including explicit formulas for the three-form potential in terms of four-dimensional fields.

Contribution

It introduces a fully explicit, consistent truncation formula for the $D=11$ supergravity three-form potential in terms of $D=4$ fields, extending previous partial results.

Findings

Complete non-linear embedding formulae derived.

Explicit expressions for the three-form potential provided.

Consistency of the truncation demonstrated.

Abstract

The truncation formulae of $D=11$ supergravity on $S^7$ to $D=4$ ${\cal N} =8$ SO(8)-gauged supergravity are completed to include the full non-linear dependence of the $D=11$ three-form potential $\hat A_{(3)}$ on the $D=4$ fields, and their consistency is shown. The full embedding into $\hat A_{(3)}$ is naturally expressed in terms of a restricted version, still ${\cal N} =8$ but only SL(8)--covariant, of the $D=4$ tensor hierarchy. The redundancies introduced by this approach are removed at the level of the field strength $\hat F_{(4)}$ by exploiting $D=4$ duality relations. Finally, new expressions for the full consistent truncation formulae are given that are, for the first time, explicit in all $D=4$ fields.