Subsectors, Dynkin Diagrams and New Generalised Geometries

Abstract

We examine how generalised geometries can be associated with a labelled Dynkin diagram built around a gravity line. We present a series of new generalised geometries based on the groups $Spin(d,d)\times {\mathbb{R}}^{+}$ for which the generalised tangent space transforms in a spinor representation of the group. In low dimensions these all appear in subsectors of maximal supergravity theories. The case $d=8$ provides a geometry for eight-dimensional backgrounds of M theory with only seven-form flux, which have not been included in any previous geometric construction. This geometry is also one of a series of "half-exceptional" geometries, which "geometrise" a six-form gauge field. In the appendix, we consider examples of other algebras appearing in gravitational theories and give a method to derive the Dynkin labels for the "section condition" in general. We argue that generalised geometry can describe restrictions and subsectors of many gravitational theories.

42 pages, v2: minor improvements and changes, published version

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Cited by in corpus (12)

• The Geometry, Branes and Applications of Exceptional Field Theory
• Extended geometries
• Exceptional Calabi--Yau spaces: the geometry of $\mathcal{N}=2$ backgrounds with flux
• O(d+1,d+1) enhanced double field theory
• G-algebroids: a unified framework for exceptional and generalised geometry, and Poisson-Lie duality
• On symmetries and dynamics of exotic supermultiplets
• Topological G${}_2$ and Spin(7) strings at 1-loop from double complexes
• Uplifts of maximal supergravities and transitions to non-geometric vacua
• Higher curvature Bianchi identities, generalised geometry and $L_{\infty }$ algebras
• Y-algebroids and $E_{7(7)}\times {\mathbb{R}}^{+}$-generalised geometry
• Supergravity Fluxes and Generalised Geometry
• Brane Symmetries Revisited: Symmetries of Tensile and Tensionless Branes in Possibly Degenerate Metrics and their Manifestations