# Tensor hierarchies, Borcherds algebras and E11

@article{Palmkvist2012TensorHB,
title={Tensor hierarchies, Borcherds algebras and E11},
author={Jakob Palmkvist},
journal={Journal of High Energy Physics},
year={2012},
volume={2012},
pages={1-20}
}
• Jakob Palmkvist
• Published 21 October 2011
• Physics
• Journal of High Energy Physics
A bstractGauge deformations of maximal supergravity in D = 11 − n dimensions generically give rise to a tensor hierarchy of p-form fields that transform in specific representations of the global symmetry group En. We derive the formulas defining the hierarchy from a Borcherds superalgebra corresponding to En. This explains why the En representations in the tensor hierarchies also appear in the level decomposition of the Borcherds superalgebra. We show that the indefinite Kac-Moody algebra E11…
Oxidizing Borcherds symmetries
• Physics, Mathematics
• 2013
A bstractThe tensor hierarchy of maximal supergravity in D dimensions is known to be closely related to a Borcherds (super)algebra that is constructed from the global symmetry group E11−D . We here
Superalgebras, constraints and partition functions
• Physics
• 2015
A bstractWe consider Borcherds superalgebras obtained from semisimple finite-dimensional Lie algebras by adding an odd null root to the simple roots. The additional Serre relations can be expressed
L∞ algebras and tensor hierarchies in Exceptional Field Theory and Gauged Supergravity
• Physics, Mathematics
Journal of High Energy Physics
• 2019
A bstractWe show how the gauge and field structure of the tensor hierarchies in Double and E7(7) Exceptional Field Theory fits into L∞ algebras. Special attention is paid to redefinitions, the role
Symmetries of M-theory and free Lie superalgebras
• Physics
Journal of High Energy Physics
• 2019
A bstractWe study systematically various extensions of the Poincaré superalgebra. The most general structure starting from a set of spinorial supercharges Qα is a free Lie superalgebra that we
The tensor hierarchy algebra
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect
$EL_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies
• Physics, Mathematics
• 2021
We define a generalized form of L8-algebras called EL8-algebras. As we show, these provide the natural algebraic framework for generalized geometry and the symmetries of double field theory as well
Forms and algebras in (half-)maximal supergravity theories
• Physics
• 2015
A bstractThe forms in D-dimensional (half-)maximal supergravity theories are discussed for 3 ≤ D ≤ 11. Superspace methods are used to derive consistent sets of Bianchi identities for all the forms
Duality Hierarchies and Differential Graded Lie Algebras
• Medicine, Physics
Communications in mathematical physics
• 2021
The algebraic structure allowing for consistent tensor hierarchies is proposed to be axiomatized by ‘infinity-enhanced Leibniz algebras’ defined on graded vector spaces generalizing Leibmars, and the structure can be reinterpreted as a differential graded Lie algebra.
Borcherds and Kac-Moody extensions of simple finite-dimensional Lie algebras
We study the Borcherds superalgebra obtained by adding an odd (fermionic) null root to the set of simple roots of a simple finite-dimensional Lie algebra. We compare it to the Kac-Moody algebra
Borcherds and Kac-Moody extensions of simple finite-dimensional Lie algebras
A bstractWe study the Borcherds superalgebra obtained by adding an odd (fermionic) null root to the set of simple roots of a simple finite-dimensional Lie algebra. We compare it to the Kac-Moody

#### References

SHOWING 1-10 OF 43 REFERENCES
E11, Borcherds algebras and maximal supergravity
• Physics
• 2010
A bstractThe dynamical p-forms of torus reductions of maximal supergravity theory have been shown some time ago to possess remarkable algebraic structures. The set (“dynamical spectrum”) of
Superconformal M2-branes and generalized Jordan triple systems
• Physics
• 2008
Three-dimensional conformal theories with six supersymmetries and SU(4) R-symmetry describing stacks of M2-branes are here proposed to be related to generalized Jordan triple systems. Writing the
Gauge Theories, Duality Relations and the Tensor Hierarchy
• Physics
• 2009
We compute the complete 3- and 4-dimensional tensor hierarchies, i.e. sets of p-form fields, with 1 ≤ p ≤ D, which realize an off-shell algebra of bosonic gauge transformations. We show how these
Gauged Supergravities, Tensor Hierarchies, and M-Theory
• Physics
• 2008
Deformations of maximal supergravity theories induced by gauging non-abelian subgroups of the duality group reveal the presence of charged M-theory degrees of freedom that are not necessarily
Kac-Moody spectrum of (half-) maximal supergravities
• Physics
• 2007
We establish the correspondence between, on one side, the possible gaugings and massive deformations of half-maximal supergravity coupled to vector multiplets and, on the other side, certain
Gauged maximal supergravities and hierarchies of nonabelian vector‐tensor systems
• Physics, Mathematics
• 2005
We describe generalizations of the manifestly E6(6) covariant formulation of five-dimensional gauged maximal supergravity with regard to the structure of the vector and tensor fields. We indicate how
The end of the p-form hierarchy
• Physics
• 2008
The introduction of a non-abelian gauge group embedded into the rigid symmetry group G of a field theory with abelian vector fields and no corresponding charges, requires in general the presence of a
Generalized conformal realizations of Kac–Moody algebras
We present a construction which associates an infinite sequence of Kac–Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n=1, this reduces to the well known
The E11 origin of all maximal supergravities. The hierarchy of field-strengths
• Physics
• 2009
Starting from E11 and the space-time translations we construct an algebra that promotes the global E11 symmetries to local ones, and consider all its possible massive deformations. The Jacobi
Maximal supergravity in D = 10: forms, Borcherds algebras and superspace cohomology
• Physics
• 2011
We give a very simple derivation of the forms of N = 2, D = 10 supergravity theories from supersymmetry and ${\text{SL}}\left( {2,\mathbb{R}} \right)$ (for IIB). Using superspace cohomology we show