Supergravity as Generalised Geometry I: Type II Theories

TL;DR

This work recasts ten-dimensional Type II supergravity as a generalised gravity theory defined by an $O(10,10)\times\mathbb{R}^+$ structure on the generalised tangent space, introducing a torsion-free, structure-preserving generalised connection $D$ and deriving a manifest $\mathit{Spin}(9,1)\times\mathit{Spin}(1,9)$ covariant formulation of the action, EOM, and SUSY variations. NSNS, RR, and fermionic sectors are unified within this geometric framework: the NSNS data inhabit an $O(d,d)$-type bundle, RR fields are packaged as Spin$(9,1)\times Spin(1,9)$ spinors (via $F$ and $F_{\#}$), and supersymmetry variations reduce to covariant derivatives with respect to $D$ combined with RR couplings. Although the generalised curvature is non-tensorial in general, tensorial measures like the generalized Ricci tensors $R_{a\bar b}$ and scalar $S$ can be defined; these reproduce the standard Type II dynamics when contracted appropriately. The reformulation clarifies the geometric meaning of duality symmetries, aligns with (and connects to) double field theory and Siegel-type approaches, and points to natural extensions to $E_{d(d)}$-type structures and non-geometric backgrounds, with potential implications for higher-derivative corrections and background construction.

Abstract

We reformulate ten-dimensional type II supergravity as a generalised geometrical analogue of Einstein gravity, defined by an $O(9,1)\times O(1,9)\subset O(10,10)\times\mathbb{R}^+$ structure on the generalised tangent space. Using the notion of generalised connection and torsion, we introduce the analogue of the Levi-Civita connection, and derive the corresponding tensorial measures of generalised curvature. We show how, to leading order in the fermion fields, these structures allow one to rewrite the action, equations of motion and supersymmetry variations in a simple, manifestly $\mathit{Spin}(9,1)\times\mathit{Spin}(1,9)$-covariant form.