M-theory, exceptional generalised geometry and superpotentials

TL;DR

The paper develops exceptional generalised geometry (EGG) for seven-dimensional manifolds to geometrise eleven-dimensional supergravity in a framework governed by the U-duality group E_{7(7)}7. It shows that N=1 four-dimensional supersymmetry singles out an SU(7) structure on the exceptional generalized tangent space, encoded by a special element φ in the 912 representation of E_{7(7)}7, and that the four-dimensional superpotential W can be written in a manifestly E_{7(7)}7- invariant form as a holomorphic function of φ. The work provides two equivalent formulations of W: (i) a concrete SU(7)-covariant expression in terms of φ, fluxes and derivatives, and (ii) a covariant E_{7(7)}7 description using a derivative operator D acting on φ to produce an adjoint action (D·φ)φ ∈ 912, proportional to φ. Together with a detailed construction of the exceptional generalized tangent space, twisted by A and Ã, and the exceptional Courant bracket, this framework unifies M-theory flux backgrounds and clarifies the geometric meaning of the four-dimensional effective theory, with potential links to topological M-theory functionals and extended background geometries.

Abstract

We discuss the structure of "exceptional generalised geometry" (EGG), an extension of Hitchin's generalised geometry that provides a unified geometrical description of backgrounds in eleven-dimensional supergravity. On a d-dimensional background, as first described by Hull, the action of the generalised geometrical O(d,d) symmetry group is replaced in EGG by the exceptional U-duality group E_d(d). The metric and form-field degrees of freedom combine into a single geometrical object, so that EGG naturally describes generic backgrounds with flux, and there is an EGG analogue of the Courant bracket which encodes the differential geometry. Our focus is on the case of seven-dimensional backgrounds with N=1 four-dimensional supersymmetry. The corresponding EGG is the generalisation of a G_2-structure manifold. We show it is characterised by an element φin a particular orbit of the 912 representation of E_7(7), which defines an SU(7) (subset of E_7(7)) structure. As an application, we derive the generic form of the four-dimensional effective superpotential, and show that it can be written in a universal form, as a homogeneous E_7(7)-invariant functional of φ.