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) × 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 exam-ples 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.