A master exceptional field theory

Abstract

We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

The E10 Wheeler-DeWitt operator at low levels

“…We expect the other kinematic constraints (diffeomorphism and Gauss-type) appearing in field theory to play a crucial role in this. However, their final significance in our present approach is not clear and might necessitate the introduction of extra fields beyond the E 10 /K(E 10 ) symmetric space, in the same way that they appear in exceptional field theory [17][18][19]. A further indication of the need for extra degrees of freedom is the apparent incompatibility of the full E 10 /K(E 10 ) model with supersymmetry [20,21].…”

“…Furthermore, it seems doubtful that the discrepancies arising at levels |ℓ| ≥ 3 can be resolved purely within the framework of E 10 alone, as already suggested by the absence of the trace of the spin connection. It has been argued from the point of view of exceptional field theory and the tensor hierarchy algebra that an appropriate extension of the E 10 coset will involve an indecomposable structure where E 10 is augmented by highest weight representations where the first one is triggered by the trace of the spin connection [24, 19,25], a fact that is also suggested by compatibility with supersymmetry [20,61].…”

“…A further indication of the need for extra degrees of freedom is the apparent incompatibility of the full E 10 /K(E 10 ) model with supersymmetry [20,21]. The parts of the Gauss-type constraints that do not contain explicit spatial derivatives have been investigated in [22,23] where it was found that they have a resemblance to Sugawara-type constructions, a fact that is also compatible with the extra fields of exceptional field theory and tensor hierarchy algebras [24,19,25].…”

A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry

“…Evidently, for any given E d(d) ExFT, one can construct or imagine multiple other 'hybrid' formulations depending on how one chooses to separate or mix the longitudinal and transverse directions. More ambitiously, one could choose to work with the recently fully constructed 'master' E 11 ExFT [55], for which no coordinate decomposition is necessary. Evidently this would eschew the technical difficulties of the latter in favour of the technicalities associated to working with an infinite-dimensional algebra.…”

“…Beyond the usual suspects, exceptional field theory also offers a way to handle the vast number of mixed symmetry tensors that appear coupling to exotic branes [60,61]. It may not be unreasonable to suggest using the E 11 'master' ExFT recently constructed in [55], as this presumably provides scope to construct an infinite number of brane scaling limits. Here there is no need to artificially split the coordinates and one can work with 11/10-dimensional quantities throughout, albeit at the obvious price of dealing with a very infinite algebra.…”