Poincaré Duality and G +++ Algebras

Abstract

Theories with General Relativity as a sub-sector exhibit enhanced symmetries upon dimensional reduction, which is suggestive of “exotic dualities”. Upon inclusion of time-like directions in the reductions one can dualize to theories in different space-time signatures. We clarify the nature of these dualities and show that they are well captured by the properties of infinite-dimensional symmetry algebras (G +++- algebras), but only after taking into account that the realization of Poincaré duality leads to restrictions on the denominator subalgebra appearing in the non-linear realization. The correct realization of Poincaré duality can be encoded in a simple algebraic constraint, that is invariant under the Weyl-group of the G +++-algebra, and therefore independent of the detailed realization of the theory under consideration. We also construct other Weyl-invariant quantities that can be used to extract information from the G +++-algebra without fixing a level decomposition.