Generalised Geometry for M-Theory

TL;DR

This work develops a unified extended-geometrical framework for M-theory flux backgrounds by enlarging the tangent structure to an E_n action and introducing extended tangent and spin bundles. It systematically constructs M-, IIA-, and IIB-like geometries as specialisations or reductions with cosets $E_n/H_n$ parameterised by a metric $G$, a 3-form $C$, and, for higher n, a 6-form $ ilde C$, capturing both geometric and non-geometric fluxes. The paper provides explicit constructions for dimensions $n=4$ through $7$, including detailed group-theoretic decompositions, transformation rules (3-form and 6-form shifts), and the role of gerbes in patching, thereby enabling analysis of supersymmetry via generalized holonomy. It further connects these extended geometries to four-dimensional flux vacua and non-geometric backgrounds, highlighting the interplay between duality symmetries and supersymmetry in a coherent geometric language.

Abstract

Generalised geometry studies structures on a d-dimensional manifold with a metric and 2-form gauge field on which there is a natural action of the group SO(d,d). This is generalised to d-dimensional manifolds with a metric and 3-form gauge field on which there is a natural action of the group $E_{d}$. This provides a framework for the discussion of M-theory solutions with flux. A different generalisation is to d-dimensional manifolds with a metric, 2-form gauge field and a set of p-forms for $p$ either odd or even on which there is a natural action of the group $E_{d+1}$. This is useful for type IIA or IIB string solutions with flux. Further generalisations give extended tangent bundles and extended spin bundles relevant for non-geometric backgrounds. Special structures that arise for supersymmetric backgrounds are discussed.