# Pre-NQ Manifolds and Correspondence Spaces: the Nilmanifold Example

## TL;DR

This paper explores the geometric structures underlying Double Field Theory using pre-NQ manifolds, focusing on nilmanifolds with abelian gerbes to understand T-duality in a graded geometric framework.

## Contribution

It introduces a global perspective on pre-NQ manifolds applied to nilmanifolds, connecting local gauge structures to T-duality through graded geometry.

## Key findings

- Pre-NQ manifolds model gauge structures of Double Field Theory.
- Application to nilmanifolds illustrates T-duality in graded geometry.
- Provides a foundation for global understanding of pre-NQ structures.

## Abstract

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.