Abstract:From the underlying non-linear realisation we compute the complete E11
invariant equations of motion in eleven dimensions, at the linearised level, up
to and including level four in the fields. Thus we include the metric, the
three and six forms, the dual graviton and three fields at level four. The
fields are linked by a set of duality equations, which are first order in
derivatives and transform into each other under the E11 symmetries. From these
duality relations we deduce second order equations of motion,… Show more
“…Surely there are an infinite number of I c (E 9 ) invariant duality relations connecting all the higher level fields to h ij and the three form A i 1 i 2 i 3 . This is consistent with the fact that the equations of motion that follow from the E 11 ⊗ S l 1 non-linear realisation, found in references [3], [4] and [18], only contain these degrees of freedom. Putting these equations of motion in light cone notation will lead to the duality relations we are discussing here.…”
Section: The Massless Casesupporting
confidence: 89%
“…where at the linearised level G c 1 ...c 9 ,a = ∂ [c 1 h c 2 ...c 9 ],a and the dot above the equals sign indicates that the above relation is modulo Lorentz transformations as explained in references [8], [9], [4] and [18]. We found in section that two gravity is described the independent variablesĥ i,j which appears in equation (2.19).…”
Section: Duality Relations In the Light Cone Formalismmentioning
confidence: 81%
“…We recall that in E theory the dual graviton obeys the irreducibility conditionĥ [a 1 ...a 8 ,b] = 0 and so the equation at the end of the last sentence is automatically true. At level four in the non-linear realisation we find the field A a 1 ...a 9 ,b 1 b 2 b 3 which obeys a duality relation with the three form field that is given by [18,11]…”
Section: Duality Relations In the Light Cone Formalismmentioning
confidence: 88%
“…The 128 degrees of freedom in the irreducible representation must be the ones that are contained in the equations of motion that follow from the E 11 ⊗ s l 1 non-linear realisation. These equations of motion were constructed in references [3] , [4] and [18] up to level four and the degrees of freedom found up to this level were just those of eleven dimensional supergravity. The fact that the irreducible representation considered in this section has the degrees of freedom of eleven dimensional supergravity and no more strongly suggests that the full equations of motion of the non-linear realisation will contain just these degrees of freedom.…”
We construct the E theory analogue of the particles that transform under the Poincare group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of E 11 with its vector representation. We show that one such irreducible representation has only the degrees of freedom of eleven dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in E theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of E 11 . We also argue that our familiar spacetimes have to be extended by additional coordinates when considering extended objects such as branes.
“…Surely there are an infinite number of I c (E 9 ) invariant duality relations connecting all the higher level fields to h ij and the three form A i 1 i 2 i 3 . This is consistent with the fact that the equations of motion that follow from the E 11 ⊗ S l 1 non-linear realisation, found in references [3], [4] and [18], only contain these degrees of freedom. Putting these equations of motion in light cone notation will lead to the duality relations we are discussing here.…”
Section: The Massless Casesupporting
confidence: 89%
“…where at the linearised level G c 1 ...c 9 ,a = ∂ [c 1 h c 2 ...c 9 ],a and the dot above the equals sign indicates that the above relation is modulo Lorentz transformations as explained in references [8], [9], [4] and [18]. We found in section that two gravity is described the independent variablesĥ i,j which appears in equation (2.19).…”
Section: Duality Relations In the Light Cone Formalismmentioning
confidence: 81%
“…We recall that in E theory the dual graviton obeys the irreducibility conditionĥ [a 1 ...a 8 ,b] = 0 and so the equation at the end of the last sentence is automatically true. At level four in the non-linear realisation we find the field A a 1 ...a 9 ,b 1 b 2 b 3 which obeys a duality relation with the three form field that is given by [18,11]…”
Section: Duality Relations In the Light Cone Formalismmentioning
confidence: 88%
“…The 128 degrees of freedom in the irreducible representation must be the ones that are contained in the equations of motion that follow from the E 11 ⊗ s l 1 non-linear realisation. These equations of motion were constructed in references [3] , [4] and [18] up to level four and the degrees of freedom found up to this level were just those of eleven dimensional supergravity. The fact that the irreducible representation considered in this section has the degrees of freedom of eleven dimensional supergravity and no more strongly suggests that the full equations of motion of the non-linear realisation will contain just these degrees of freedom.…”
We construct the E theory analogue of the particles that transform under the Poincare group, that is, the irreducible representations of the semi-direct product of the Cartan involution subalgebra of E 11 with its vector representation. We show that one such irreducible representation has only the degrees of freedom of eleven dimensional supergravity. This representation is most easily discussed in the light cone formalism and we show that the duality relations found in E theory take a particularly simple form in this formalism. We explain that the mysterious symmetries found recently in the light cone formulation of maximal supergravity theories are part of E 11 . We also argue that our familiar spacetimes have to be extended by additional coordinates when considering extended objects such as branes.
“…It has been proposed by West long ago and prior to the development of exceptional field theory that the D = 11 supergravity equations of motion should emerge from an E 11 invariant theory formulated in the framework of a non-linear realisation of E 11 in the 'vector' representation, such that the dynamics would follow from an E 11 invariant set of duality equations [35][36][37]. It has been realised recently that these first order duality equations can only hold modulo certain equivalence relations [38,39]. These ambiguities are argued to be liftable by passing to equations of motion that are eventually of arbitrarily high order in derivatives.…”
We construct an infinite system of non-linear duality equations, including fermions, that are invariant under global E 11 and gauge invariant under generalised diffeomorphisms upon the imposition of a suitable section constraint. We use finitedimensional fermionic representations of the R-symmetry K(E 11 ) to describe the fermionic contributions to the duality equations. These duality equations reduce to the known equations of E 8 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 E 11 that entails extra non-dynamical fields beyond those predicted by E 11 alone, generalising the known constrained p-forms of exceptional field theories. The construction hinges on the tensor hierarchy algebra extension of e 11 , both for the bosonic theory and its supersymmetric extension.
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.
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