Abstract
We reformulate eleven-dimensional supergravity, including fermions, in terms of generalised geometry, for spacetimes that are warped products of Minkowski space with a d-dimensional manifold M with d ≤ 7. The reformulation has an E d(d) × ℝ+ structure group and it has a local $ {{\widetilde{H}}_d} $ symmetry, where $ {{\widetilde{H}}_d} $ is the double cover of the maximally compact subgroup of E d(d). The bosonic degrees for freedom unify into a generalised metric, and, defining the generalisked analogue D of the Levi-Civita connection, one finds that the corresponding equations of motion are the vanishing of the generalised Ricci tensor. To leading order, we show that the fermionic equations of motion, action and supersymmetry variations can all be written in terms of D. Although we will not give the detailed decompositions, this reformulation is equally applicable to type IIA or IIB supergravity restricted to a (d − 1)-dimensional manifold. For completeness we give explicit expressions in terms of $ {{\widetilde{H}}_4} $ = Spin(5) and $ {{\widetilde{H}}_7} $ = SU(8) representations for d = 4 and d = 7.
Highlights
In this paper we describe the reformulation of eleven-dimensional supergravity in terms of generalised geometry
This completes a programme started in [1], where we studied the formulation of type II supergravity in terms of a generalised geometry with an O(d, d) × R+ structure group first proposed by Hitchin and Gualtieri [2, 3]
In the following paper [12], we defined the analogous concepts in Ed(d) ×R+ generalised geometry, known as exceptional or extended generalised geometry [15, 16]. We showed that this construction can be used to describe eleven-dimensional supergravity restricted to d dimensions, that is, to the warped product of Minkowski space and a d-dimensional manifold
Summary
In this paper we describe the reformulation of eleven-dimensional supergravity in terms of generalised geometry. This completes a programme started in [1], where we studied the formulation of type II supergravity in terms of a generalised geometry with an O(d, d) × R+ structure group first proposed by Hitchin and Gualtieri [2, 3] Such reformulations were first given in the related “doubled” formalism in a series of papers by Hohm, Hull, Kwak and Zweibach [4,5,6], building in part on work by Siegel [7, 8]. A more recent approach is the work of Berman and Perry and collaborators [30,31,32], using the M-theory extension of double field theory [44,45,46] These authors were able to find a non-manifestly covariant form of the action in terms of derivatives of the generalised metric G. We include a number of appendices to fix our conventions and give the details of the various spinor decompositions and group actions that we use
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have