E$_9$ exceptional field theory II. The complete dynamics

TL;DR

This paper develops the first complete E9 exceptional field theory, unifying 11D and type IIB supergravity under an infinite-dimensional symmetry, and connects it to integrable structures in two-dimensional supergravity.

Contribution

It constructs a comprehensive E9 exceptional field theory with two equivalent formulations, extending the symmetry algebra and linking to integrable systems in supergravity.

Findings

Provides a pseudo-Lagrangian formulation with twisted self-duality.

Extends the symmetry algebra to include half of the Virasoro algebra.

Applicable to affine Kac-Moody groups like the Geroch group.

Abstract

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E$_9$ exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends $\mathfrak{e}_9$ by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity.