Beyond E 11

Abstract

We study the non-linear realisation of E 11 originally proposed by West with particular emphasis on the issue of linearised gauge invariance. Our analysis shows even at low levels that the conjectured equations can only be invariant under local gauge transformations if a certain section condition that has appeared in a different context in the E 11 literature is satisfied. This section condition also generalises the one known from exceptional field theory. Even with the section condition, the E 11 duality equation for gravity is known to miss the trace component of the spin connection. We propose an extended scheme based on an infinite-dimensional Lie superalgebra, called the tensor hierarchy algebra, that incorporates the section condition and resolves the above issue. The tensor hierarchy algebra defines a generalised differential complex, which provides a systematic description of gauge invariance and Bianchi identities. It furthermore provides an E 11 representation for the field strengths, for which we define a twisted first order self-duality equation underlying the dynamics.

Generalized diffeomorphisms for E9

“…This extension is (up to a sign convention) the symmetry algebra G used in [30] to describe the structure of gauged supergravity in two dimensions, which we will rederive from the generalised diffeomorphisms (4.15) in section 5. Moreover, it agrees precisely with the level zero content of the tensor hierarchy algebra corresponding to e 9 , as defined in [46] for general e d . In general there is an additional highest weight module of generators, which reduces to the single element…”

“…This is suitable in the context of e.g. the tensor hierarchy algebra [32,[45][46][47]. Here, we choose to include all representations that vanish in the section, also antisymmetric ones.…”

On Brane Solutions with Intersection Rules Related to Lie Algebras

“…, k . Here, we consider exact solutions in the model (14), when vectors (U s , s ∈ S) obey the block-orthogonal decomposition (46) and (47) with scalar products defined in (45) [46]. These solutions were obtained from the corresponding solutions to the σ -model equations of motion [46].…”

“…, k ), and the relations (49) on the parameters ν s are imposed. Here, the matrix ((U s , U s )) and parameters ε s , s ∈ S , are defined in (45) and (40) …”

“…It was shown in [41] that the BPSsolutions of the oxidized theory of a simply laced group G form representations of a subgroup of the Weyl transformations of the algebra G +++ . For other aspects of very-extended Kac-Moody algebras (e.g., E 11 ), see also [42][43][44][45] and the references therein.…”