Pre‐NQ Manifolds and Correspondence Spaces: the Nilmanifold Example

Courant algebroids correspond to degree‐2 symplectic differential graded manifolds or NQ‐manifolds for short. We review how a similar construction shows that locally the gauge structure of Double Field Theory corresponds to degree‐2 symplectic pre‐NQ manifolds. To illustrate first steps towards a global understanding of the pre‐NQ case, we apply the local constructions to 3‐dimensional nilmanifolds carrying an abelian gerbe. These are among prime examples where T‐duality is well‐understood and allow us to investigate classic results in the graded language. This article explains how the local gauge structures of Double Field Theory is captured by particular symplectic pre‐NQ manifolds. Furthermore, first steps towards a global understanding of Double Field Theory are presented.