$E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory

TL;DR

This paper develops a unified geometric framework based on extended tangent spaces and $E_{d(d)} imes R^+$ symmetry to describe bosonic supergravity and M-theory in various dimensions, simplifying equations of motion and symmetries.

Contribution

It introduces a generalized geometry formalism that unifies supergravity fields, symmetries, and equations of motion across multiple dimensions, extending previous approaches.

Findings

Unified description of supergravity and M-theory sectors

Generalized Ricci scalar and tensor encode equations of motion

Connections to flux compactifications and other M-theory approaches

Abstract

We show that generalised geometry gives a unified description of bosonic eleven-dimensional supergravity restricted to a $d$-dimensional manifold for all $d\leq7$. The theory is based on an extended tangent space which admits a natural $E_{d(d)} \times \mathbb{R}^+$ action. The bosonic degrees of freedom are unified as a "generalised metric", as are the diffeomorphism and gauge symmetries, while the local $O(d)$ symmetry is promoted to $H_d$, the maximally compact subgroup of $E_{d(d)}$. We introduce the analogue of the Levi--Civita connection and the Ricci tensor and show that the bosonic action and equations of motion are simply given by the generalised Ricci scalar and the vanishing of the generalised Ricci tensor respectively. The formalism also gives a unified description of the bosonic NSNS and RR sectors of type II supergravity in $d-1$ dimensions. Locally the formulation also describes M theory variants of double field theory and we derive the corresponding section condition in general dimension. We comment on the relation to other approaches to M theory with $E_{d(d)}$ symmetry, as well as the connections to flux compactifications and the embedding tensor formalism.