  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.
