2010/07/29 by Marc Henneaux, Henneaux, Marc, Bernard L. Julia +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.1007.5241
openalex publication_date 2010/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mfVD 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 \mfV-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 E11 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 E11 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 \mfV2 and \mfV1 are new too. We also introduce and use the concept of a presentation of a Lie algebra that is covariant under a given subalgebra.