Abstract
We construct an infinite system of non-linear duality equations, including fermions, that are invariant under global E11 and gauge invariant under generalised diffeomorphisms upon the imposition of a suitable section constraint. We use finite-dimensional fermionic representations of the R-symmetry K(E11) to describe the fermionic contributions to the duality equations. These duality equations reduce to the known equations of E8 exceptional field theory or eleven-dimensional supergravity for appropriate (partial) solutions of the section constraint. Of key importance in the construction is an indecomposable representation of E11 that entails extra non-dynamical fields beyond those predicted by E11 alone, generalising the known constrained p-forms of exceptional field theories. The construction hinges on the tensor hierarchy algebra extension of {mathfrak{e}}_{11} , both for the bosonic theory and its supersymmetric extension.
Highlights
Exceptional field theories [1,2,3,4] are based on generalised exceptional geometries in which diffeomorphisms are unified with tensor gauge transformations in such a way that the closure of the local transformations require constraints on the fields, known as section constraints [1, 5, 6]
We use finite-dimensional fermionic representations of the R-symmetry K(E11) to describe the fermionic contributions to the duality equations. These duality equations reduce to the known equations of E8 exceptional field theory or eleven-dimensional supergravity for appropriate solutions of the section constraint
The possibility of making this fermionic modification of the first-order duality equation rests on a non-trivial relation between the unfaithful spinors and the tensor hierarchy algebra that we demonstrate at low levels
Summary
Exceptional field theories [1,2,3,4] are based on generalised exceptional geometries in which diffeomorphisms are unified with tensor gauge transformations in such a way that the closure of the local transformations require constraints on the fields, known as section constraints [1, 5, 6]. In [47], it was explained that constructing linearised gauge invariant first order field equations with E11 symmetry requires the fields to satisfy the section constraint as well as the introduction of additional fields that do not appear in the E11 coset space This construction is based on an infinite-dimensional super-algebra T (e11), that includes e11 as a subalgebra and that generalises the tensor hierarchy algebra T (en) introduced in [48] for n ≤ 8 to the Kac-Moody case. Given the unfaithful spinors Ψ and of K(E11), we show how their bilinears relate to the tensor hierarchy algebra and how this can be used to define supersymmetry transformation rules and a consistent supersymmetry algebra We show that this consistency connects to the reducible gauge structure of the E11 generalised Lie derivative and introduces yet more additional bosonic fields into the theory in order to make all symmetries manifest. Dynkin diagram of E11 with labelling of nodes used in the text
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