# The geometry of graded cotangent bundles

## TL;DR

This paper explores the geometric structures of graded cotangent bundles associated with vector bundles, linking them to higher algebraic structures and physical models, and analyzing their symplectic and Q-structure properties.

## Contribution

It introduces a detailed study of graded cotangent bundles, connecting their geometry to higher Courant algebroids, Dirac structures, and AKSZ sigma-models, revealing new relationships.

## Key findings

- Canonical symplectic structures on graded manifolds
- Relation to higher Courant algebroids and Dirac structures
- Connections to AKSZ sigma-models

## Abstract

Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical structures, such as higher Courant algebroids on $A\oplus\bigwedge^{k-1}A^*$ and higher Dirac structures therein, semi-direct products of Lie algebroid structures on $A$ with their coadjoint representations up to homotopy, and branes on certain AKSZ $\sigma$-models.