In this thesis we investigate the role of odd Poisson brackets in related areas of supergeometry. In particular we study three different cases of their appearance: Courant algebroids and their homotopy analogues, weak Poisson structures and their relation to foliated manifolds, and the structure of odd Poisson manifolds and their modular class. In chapter 2 we introduce the notion of a homotopy Courant algebroid, a subclass of which is suggested to stand as the double objects to $\L$bialgebroids. We provide explicit formula for the higher homotopy Dorfman brackets introduced in this case, and the higher relations between these and the anchor maps. The homotopy Loday structure is investigated, and we begin a discussion of what other constructions in the theory of Courant algebroids can be carried out in this homotopy setting. Chapter 3 is devoted to lifting a weak Poisson structure corresponding to a local foliation of a submanifold to a weak Koszul bracket, and interpreting the results in terms of the cohomology of an associated differential. This bracket is shown to produce a bracket on coexact differential forms. In chapter 5 studies classes of second order differential operators acting on semidensities on an arbitrary supermanifold. In particular, when the supermanifold is odd Poisson, we given an explicit description of the modular class of the odd Poisson manifold, and provide the first nontrivial examples of such a class. We also introduce the potential field of a general odd Laplacian, and discuss its relation to the geometry of the odd Poisson manifold and its status as a connectionlike object.
Date of Award  1 Aug 2017 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Theodore Voronov (Supervisor) & Hovhannes Khudaverdyan (Supervisor) 

 Modular class
 Weak Poisson
 Odd Laplacian
 Homotopy Courant
 Homotopy Poisson
 Odd Poisson
 Courant algebroid
Odd Poisson supermanifolds, Courant algebroids, homotopy structures, and differential operators
Peddie, M. (Author). 1 Aug 2017
Student thesis: Phd