# Z-graded supergeometry: Differential graded modules, higher algebroid representations, and linear structures

by Theocharis Papantonis

Date of Examination:2021-05-19

Date of issue:2021-06-21

Advisor:Prof. Dr. Madeleine Jotz Lean

Referee:Prof. Dr. Madeleine Jotz Lean

Referee:Prof. Dr. Chenchang Zhu

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## Abstract

### English

The purpose of this thesis is to present a self-standing review of $\mathbb{Z}$- (respectively $\mathbb{N}$-)graded supergeometry with emphasis in the development and study of two particular structures therein. Namely, representation theory and linear structures of $\mathcal{Q}$-manifolds and higher Lie algebroids (also known in the mathematics and physics literature as $\mathbb{Z}\mathcal{Q}$- and $\mathbb{N}\mathcal{Q}$-manifolds, respectively). Regarding the first notion, we introduce differential graded modules (or for short DG-modules) of $\mathcal{Q}$-manifolds and the equivalent notion of representations up to homotopy in the case of Lie $n$-algebroids ($n\in\mathbb{N}$). These are generalisations of the homonymous structures of the works of Vaintrob, Gracia-Saz and Mehta, and Arias Abad and Crainic, that exist already in the case of ordinary Lie algebroids, i.e. when $n=1$. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy of Lie $n$-algebroids are explained. In particular, the case of Lie $2$-algebroids is analysed in detail. The compatibility of a graded Poisson bracket with the homological vector field on a $\mathbb{Z}$-graded manifold is shown to be equivalent to an (anti-)morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of graded Poisson structures. Applying this result to symplectic Lie $2$-algebroids, gives another algebraic characterisation of Courant algebroids in terms of their adjoint and coadjoint representations. In addition, the Weil algebra of a general $\mathcal{Q}$-manifold is defined and is computed explicitly in the case of Lie $n$-algebroids over a base (smooth) manifold $M$ together with a choice of a splitting and linear $TM$-connections. Similarly to the work of Abad and Crainic, our computation involves the coadjoint representation of the Lie $n$-algebroid and the induced $2$-term representations up to homotopy of the tangent bundle $TM$ on the vector bundles of the underlying complex of the Lie $n$-algebroid given by the choice of the linear connections. The second object that we define and explore in this work is the linear structures on $\mathbb{Z}$-graded manifolds, for which we see the connection with DG-modules and representations up to homotopy. In the world of split Lie $n$-algebroids, this leads to the notion of higher VB-algebroids, which we call VB-Lie $n$-algebroids; that is, Lie $n$-algebroids that are in some sense linear over another Lie $n$-algebroid. We prove that there is an equivalence between the category of VB-Lie $n$-algebroids over a Lie $n$-algebroid $\underline{A}$ and the category of $(n+1)$-term representations up to homotopy of $\underline{A}$, generalising a well-known result from the theory of ordinary VB-algebroids over Lie algebroids, i.e., in our setting, VB-Lie $1$-algebroids over Lie $1$-algebroids.**Keywords:**graded manifold; supermanifold; Q-manifold; Lie n-algebroid; VB-algebroid; representation up to homotopy; differential graded module; adjoint representation; adjoint module; Weil algebra