Oxidizing Borcherds symmetries

Abstract

The 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 E 11−D . We here explain how the Borcherds algebras in different dimensions are embedded into each other and can be constructed from a unifying Borcherds algebra. The construction also has a natural physical explanation in terms of oxidation. We then go on to show that the Hodge duality that is present in the tensor hierarchy has an algebraic counterpart. For D > 8 the Borcherds algebras we find differ from the ones existing in the literature although they generate the same tensor hierarchy.

Kac-Moody and Borcherds symmetries of six-dimensional chiral supergravity

“…This is the Cartan matrix of a Borcherds algebra with generatorsh i , e i and f i , which is isomorphic to our algebra. It turns out that this Cartan matrix is very similar to the Cartan matrix for type IIB obtained by following the same procedure [49] in the sense that both contain one timelike simple root and one spacelike simple root. A difference lies in the grading of the generators.…”

“…In section 6, we determine the Borcherds structure of the p-form V -duality algebra. In that analysis, we follow the method of [49] to eliminate some ambiguities, which requires the determination of the V-duality algebras for the dimensionally reduced models in spacetime dimensions lower than 6. Section 7 is devoted to the conclusions where further comments on manifest duality symmetry are provided.…”

“…To that end, we follow the method of [49], which consists in starting from the Borcherds superalgebra in lower dimensions where there is no ambiguity and oxidizing according to a well-defined procedure.…”

Exceptional geometry and Borcherds superalgebras

“…Alternatively, we could extend e n`1 to B n`1 by adding an odd null root γ´1 such that pγ´1, α 0 q "´1 and γ´1 is orthogonal to the simple roots of e n . The embedding of B n in B n`1 would then be given by identifying β 0 in B n with pγ´1`α 0 q in B n`1 , as described in [46]. The two different Dynkin diagrams of B n`1 are related by a so-called odd reflection mapping γ´1 and´β´1 to each other [29,46].…”

“…The embedding of B n in B n`1 would then be given by identifying β 0 in B n with pγ´1`α 0 q in B n`1 , as described in [46]. The two different Dynkin diagrams of B n`1 are related by a so-called odd reflection mapping γ´1 and´β´1 to each other [29,46]. We choose the former approach here, corresponding to the left Dynkin diagram of B n`1 , so that the subscripts 0 and´1 of any Chevalley generators always refer to β 0 and β´1.…”