Abstract
Free differential algebras (FDAs) provide an algebraic setting for field theories with antisymmetric tensors. The “presentation” of FDAs generalizes the Cartan-Maurer equations of ordinary Lie algebras, by incorporating p-form potentials. An extended Lie derivative along antisymmetric tensor fields can be defined and used to recover a Lie algebra dual to the FDA that encodes all the symmetries of the theory including those gauged by the p-forms. The general method is applied to the FDA of D = 11 supergravity: the resulting dual Lie superalgebra contains the M-theory supersymmetry anticommutators in presence of 2-branes. MSC 2010: 53Z05, 83F50.
Highlights
Supergravity in eleven dimensions [7,8] is today considered an effective theory
More than two decades ago, it was formulated [10] as the gauging of a free differential algebra (FDA) [4,5,10,19,22], an algebraic structure that extends the Cartan-Maurer equations of an ordinary Lie algebra G by including p-form potentials, besides the usual left-invariant 1-forms corresponding to the Lie group generators of G
Some time later it was realized how to extract from the FDA the symmetries gauged by the p-forms, via a new (“extended”) Lie derivative defined along antisymmetric tensors [6]
Summary
Supergravity in eleven dimensions [7,8] is today considered an effective theory (a particular limit of M-theory, for a review see e.g. [20,21]). The resulting dual Lie superalgebra contains the supersymmetry anticommutators of M-theory coupled to a 2-brane discussed in [12], one of the extended Lie derivatives corresponding to the pseudocentral charge Zm1m2. We can apply it to an FDA containing a 3-form and a 6-form, so that both the charges ZM1M2 and ZM1−M5 enter the stage in the dual Lie algebra. This leads to the same supertranslation algebra of [10], that later was derived [17] in the context of D = 11 supergravity coupled to a 2- and a 5-brane.
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