Abstract
The 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 propagating p-forms has been described as a (truncation of a) real Borcherds superalgebra D that is characterized concisely by a Cartan matrix which has been constructed explicitly for each spacetime dimension 11 ≥ D ≥ 3. In the equations of motion, each differential form of degree p is the coefficient of a (super-) group generator, which is itself of degree p for a specific gradation (the -gradation). A slightly milder truncation of the Borcherds superalgebra enables one to predict also the “spectrum” of the non-dynamical (D − 1) and D-forms. The maximal supergravity p-form spectra were reanalyzed more recently by truncation of the field spectrum of E 11 to the p-forms that are relevant after reduction from 11 to D dimensions. We show in this paper how the Borcherds description can be systematically derived from the split (“maximally non compact”) real form of E 11 for D ≥ 1. This explains not only why both structures lead to the same propagating p-forms and their duals for p ≤ (D − 2), but also why one obtains the same (D−1)-forms and “top” D-forms. The Borcherds symmetries 2 and 1 are new too. We also introduce and use the concept of a presentation of a Lie algebra that is covariant under a given subalgebra.
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ArXiv ePrint: 1007.5241
Unité mixte de recherche (UMR 8549) du CNRS et de l’ENS, associée à l’Université Pierre et Marie Curie et aux Fédérations de recherche FR684 et FR2687.
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Henneaux, M., Julia, B.L. & Levie, J. E 11, Borcherds algebras and maximal supergravity. J. High Energ. Phys. 2012, 78 (2012). https://doi.org/10.1007/JHEP04(2012)078
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DOI: https://doi.org/10.1007/JHEP04(2012)078