Abstract:We construct the non-linear realisation of the semi-direct product of E11 and
its vector representation in eleven dimensions and find the dynamical equations
it predicts at low levels. These equations are completely determined by the
non-linear realisation and when restricted to contain only the usual fields of
supergravity and the usual space-time we find precisely the equations of motion
of eleven dimensional supergravity. This paper extends the results announced in
arXiv:1512.01644 and in particular it cont… Show more
“…The approach of reference [31] focused on finding duality equations which were first order in derivatives and it found the correct equations for the forms which were uniquely determined but there were unresolved issues with the graviton sector. In references [33,34] the invariant second order equations were found, they were unique and when one retained only the low levels fields and the level zero coordinates they were precisely those of eleven dimensional supergravity. This essentially proved the E 11 conjecture.…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
confidence: 99%
“…As a result, the non-linear realisation carried out with these limitations did not lead uniquely to eleven dimensional supergravity and one had to fix several constants whose values were not determined by the calculation. A more systematic approach was taken in references [31,48,33] and [34] where the non-linear realisation of E 11 ⊗ s l 1 at low levels was constructed for the fields up to an including the dual graviton as well as the low level coordinates of the l 1 representation. These references enforced not only the Lorentz group symmetries of I c (E 11 ) but also the much more powerful symmetries at the next levels.…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
confidence: 99%
“…The local I c (E 11 ) variations of the Cartan forms are straightforward to compute, using the E 11 algebra and they are given by [31,33,34]…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
confidence: 99%
“…As explained above if this index is made into a tangent index, that is, G A,α = (E −1 ) A Π G Π,α it transforms only under the local I c (E 11 ) transformations, the transformation just being that for the inverse vielbein of equation (6.8). One finds that the Cartan forms, when referred to the tangent space, transforms on their l 1 index as [31,33,34] δG a,…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
confidence: 99%
“…Of course to get the full transformation one must combine the transformations of equation (6.18) The detailed construction of the equations of motion which follow from the E 11 ⊗ s l 1 non-linear realisation was given in reference [34] following earlier results in references [33] and [31]. We refer the reader to this reference and confine ourselves here to stating the result.…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of non-linear realisations and Kac-Moody algebras, I explain how to construct the non-linear realisation based on the Kac-Moody algebra E 11 and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a spacetime with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of spacetime, lead to precisely the equations of motion of eleven dimensional supergravity theory. By taking different group decompositions of E 11 we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the non-linear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the E 11 conjecture given many years ago.
“…The approach of reference [31] focused on finding duality equations which were first order in derivatives and it found the correct equations for the forms which were uniquely determined but there were unresolved issues with the graviton sector. In references [33,34] the invariant second order equations were found, they were unique and when one retained only the low levels fields and the level zero coordinates they were precisely those of eleven dimensional supergravity. This essentially proved the E 11 conjecture.…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
confidence: 99%
“…As a result, the non-linear realisation carried out with these limitations did not lead uniquely to eleven dimensional supergravity and one had to fix several constants whose values were not determined by the calculation. A more systematic approach was taken in references [31,48,33] and [34] where the non-linear realisation of E 11 ⊗ s l 1 at low levels was constructed for the fields up to an including the dual graviton as well as the low level coordinates of the l 1 representation. These references enforced not only the Lorentz group symmetries of I c (E 11 ) but also the much more powerful symmetries at the next levels.…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
confidence: 99%
“…The local I c (E 11 ) variations of the Cartan forms are straightforward to compute, using the E 11 algebra and they are given by [31,33,34]…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
confidence: 99%
“…As explained above if this index is made into a tangent index, that is, G A,α = (E −1 ) A Π G Π,α it transforms only under the local I c (E 11 ) transformations, the transformation just being that for the inverse vielbein of equation (6.8). One finds that the Cartan forms, when referred to the tangent space, transforms on their l 1 index as [31,33,34] δG a,…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
confidence: 99%
“…Of course to get the full transformation one must combine the transformations of equation (6.18) The detailed construction of the equations of motion which follow from the E 11 ⊗ s l 1 non-linear realisation was given in reference [34] following earlier results in references [33] and [31]. We refer the reader to this reference and confine ourselves here to stating the result.…”
Section: ) As a Results The Rigid And Local Transformations Of The mentioning
I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of non-linear realisations and Kac-Moody algebras, I explain how to construct the non-linear realisation based on the Kac-Moody algebra E 11 and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a spacetime with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of spacetime, lead to precisely the equations of motion of eleven dimensional supergravity theory. By taking different group decompositions of E 11 we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the non-linear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the E 11 conjecture given many years ago.
We develop in detail the general framework of consistent Kaluza-Klein truncations from D=11 and type II supergravities to gauged maximal supergravities in two dimensions. In particular, we unveil the complete bosonic dynamics of all gauged maximal supergravities that admit a geometric uplift. Our construction relies on generalised Scherk-Schwarz reductions of E9 exceptional field theory. The application to the reduction of D=11 supergravity on S8 × S1 to SO(9) gauged supergravity is presented in a companion paper.
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