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The very-extended trombone
High Energy Physics - Theory · Submitted 11 Jan 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim All the possible deformed algebras are in one to one correspondence with all such maximal supergravity theories.
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The abstract, sentence by sentence
Starting from the very-extended Kac-Moody algebra $E_{11}$, we consider the algebra $E_{11,D}^{local}$, obtained by adding to the non-negative level $E_{11}$ generators the $D$-dimensional momentum operator and an infinite set of additional generators that promote the global $E_{11}$ symmetries to gauge ones. We determine all the possible trombone deformations of this algebra, that is the deformations that involve the $D$-dimensional scaling operator. The Jacobi identities imply that such deformations are uniquely determined by a single tensor belonging to the same representation of the internal symmetry group as the vector generators and satisfying additional quadratic constraints. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible maximal supergravity theory in which the global scaling symmetry is gauged in any dimension. All the possible deformed algebras are in one to one correspondence with all such maximal supergravity theories. The tensor that parametrises the deformation is identified with the embedding tensor that is used to parametrise all maximal supergravity theories with gauged scaling symmetry, and the quadratic constraints that we determine exactly coincide with the field theory results.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Starting from the very-extended Kac-Moody algebra $E_{11}$, we consider the algebra $E_{11,D}^{local}$, obtained by adding to the non-negative level $E_{11}$ generators the $D$-dimensional momentum operator and an infinite set of additional generators that promote the global $E_{11}$ symmetries to gauge ones. | contribution 0.51 | 0.77 | 0.04 | asserted 0.98 |
| s1 | We determine all the possible trombone deformations of this algebra, that is the deformations that involve the $D$-dimensional scaling operator. | contribution 0.97 | 0.93 | 0.06 | asserted 0.98 |
| s2 | The Jacobi identities imply that such deformations are uniquely determined by a single tensor belonging to the same representation of the internal symmetry group as the vector generators and satisfying additional quadratic constraints. | result 0.97 | 0.85 | 0.15 | asserted 1.00 |
| s3 | The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible maximal supergravity theory in which the global scaling symmetry is gauged in any dimension. | result 0.74 | 0.76 | 0.28 | asserted 1.00 |
| s4 | All the possible deformed algebras are in one to one correspondence with all such maximal supergravity theories. | result 0.92 | 0.77 | 0.54 | asserted 1.00 |
| s5 | The tensor that parametrises the deformation is identified with the embedding tensor that is used to parametrise all maximal supergravity theories with gauged scaling symmetry, and the quadratic constraints that we determine exactly coincide with the field theory results. | result 0.79 | 0.87 | 0.34 | asserted 1.00 |
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Original abstract
Starting from the very-extended Kac-Moody algebra $E_{11}$, we consider the algebra $E_{11,D}^{local}$, obtained by adding to the non-negative level $E_{11}$ generators the $D$-dimensional momentum operator and an infinite set of additional generators that promote the global $E_{11}$ symmetries to gauge ones. We determine all the possible trombone deformations of this algebra, that is the deformations that involve the $D$-dimensional scaling operator. The Jacobi identities imply that such deformations are uniquely determined by a single tensor belonging to the same representation of the internal symmetry group as the vector generators and satisfying additional quadratic constraints. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible maximal supergravity theory in which the global scaling symmetry is gauged in any dimension. All the possible deformed algebras are in one to one correspondence with all such maximal supergravity theories. The tensor that parametrises the deformation is identified with the embedding tensor that is used to parametrise all maximal supergravity theories with gauged scaling symmetry, and the quadratic constraints that we determine exactly coincide with the field theory results.
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How the literature uses it
2 arXiv papers cite this in 3 sentences
What it cites · 82 sentences without a confident label
Builds on 2 papers
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Considering together the analysis of [ref] and the one performed in this section, we write more generally the commutator of the 2-form generator with momentum as [equation] where $W^M_2_M_1$ is related to the embedding tensor $Θ^M_1_$ as explained in [ref] .
The E(11) origin of all maximal supergravities - the hierarchy of field-strengths · The general method · 0.81
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The same applies to all supergravity theories with more than 16 supercharges [ref] .
Real forms of very extended Kac-Moody algebras and theories with eight supersymmetries · Conclusions · 0.80
Uses the method or tool of 2 papers
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Applying the same arguments of [ref] we obtain that imposing that the symmetric part in$ab$of eq.
Local E(11) · The general method · 0.94
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([ref]) we also determine the gauge transformations of these fields to be (see Section 2 and Appendix B of [ref] for a detailed derivation) [equation] where the parameters $a__1 ... _n , M_n$ are given in terms of the gauge parameters $Λ__1 ..._n-1 , M_n$ as [equation]
The E(11) origin of all maximal supergravities - the hierarchy of field-strengths · The general method · 0.88
Compares with 5 papers
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([ref]) and ([ref]) the Jacobi identity involving the 2-form and two momentum operators gives [equation] while the Jacobi identity involving the 1-form generator, momentum and the scalar generator $Θ^M_1 K + a Θ^N_1 D_, N_1 ^M_1 R^+ Θ^M_1_R^$ gives [equation] One can show that these quadratic constraints exactly reproduce the quadratic constraints in [ref] in all cases.
Supergravities without an Action: Gauging the Trombone · The general method · 0.98
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With respect to the deformations analysed in [ref] , these deformations not only involve the generators of the internal symmetry, but also the trace of the $GL(D, ℝ)$ generators.
The E(11) origin of all maximal supergravities - the hierarchy of field-strengths · Conclusions · 0.91
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This is exactly what happens also in the IIA case of [ref] , in which the trombone deformation associated to the gauged IIA theory of [ref] is not compatible with the Romans deformation.
Local E(11) and the gauging of the trombone symmetry · D=9 · 0.92
Cites as support 10 papers
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One also finds $D-1$ form generators in representations of $E_11-D$ which remarkably are the same as those fixed by the linear constraints on the embedding tensor for the maximal gauged supergravities in dimension $D$ [ref] .
The E_{11} origin of all maximal supergravities · Introduction · 0.98
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One also finds $D-1$ form generators in representations of $E_11-D$ which remarkably are the same as those fixed by the linear constraints on the embedding tensor for the maximal gauged supergravities in dimension $D$ [ref] .
E(11) and the Embedding Tensor · Introduction · 0.98
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Therefore the Romans massive IIA is naturally encoded in $E_11$ [ref] .
Massive IIA Supergravity as a Non-linear Realisation · Introduction · 0.99
Cites as background 4 papers
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This algebra was first conjectured in [ref] to be a symmetry of M-theory.
E_11 and M Theory · Introduction · 0.82
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This extension is dimension-dependent, and it was called $E_11,D^local$ in [ref] .
Local E(11) · Introduction · 0.80
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The massless maximal supergravity theory was constructed in [ref] , and possesses an $E_7(7)$ on-shell symmetry (recently, it was shown that this theory also admits a lagrangian formulation in which the $E_7(7)$ symmetry is manifest [ref] ).
E7(7) invariant Lagrangian of d=4 N=8 supergravity · D=4 · 0.85
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