Abstract
We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E9 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.
Highlights
This paper completes the work started in [1], constructing the exceptional field theory for the Kac-Moody group E9
We have constructed the complete dynamics of E9 exceptional field theory in two different formulations
This formulation of E9 exceptional field theory allows us to naturally reproduce D = 2 supergravity and its linear system by setting to zero the internal derivative ∂| as well as all constrained fields B(k) and χγ1|. In this formulation, a proper gauge connection for generalised diffeomorphisms can be defined, allowing for instance the construction of covariant external currents for the scalar fields, and the vector fields and their field strengths transform in a manner familiar from lower-rank Exceptional field theories (ExFT)
Summary
This paper completes the work started in [1], constructing the exceptional field theory for the Kac-Moody group E9. The spectral parameter of the linear system introduced in [31, 32] depends on the D = 2 space-time coordinates, a property that is a priori incompatible with the definition of fields in E9 lowest weight modules like AM μ in ExFT This was not yet an issue in the construction of the ExFT scalar potential [1] which corresponds to a truncation of the theory in which all duality equations are consistently projected out. The compensating transformation belongs to one K(E9) = K(E8) subgroup of E8 and leaves the monodromy matrix invariant It is defined in terms of a field-dependent anti-involution which acts by inversion of γ(w) and differs from the Hermitian conjugation we introduced in (2.10) which, in the spectral parameter representation, acts by inversion of w. Notice that the generators L0, L−1 do not commute with the loop algebra but normalise it
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