Differential graded manifolds, infinity-stacks and generalized geometries
Three lectures by Dmitry Roytenberg at the CRCG Workshop - Higher Structures in Topology and Geometry III, June 4-5 2009.
Differential graded manifolds are supermanifolds equipped with an additional grading and differential in the structure sheaf. They can be thought of as a simultaneous generalization of Lie algeboids and L-infinity algebras. DG manifolds arise naturally as first-order approximations to a very wide class of geometric constructions. Conversely, every DG manifold gives rise naturally to an infinity-stack (in the formalism of simplicial presheaves). Except for a few special cases, the question of representability of any finite truncation of this stack by a finite-dimensional simplicial manifold is still open. DG manifolds can be thought of as generalized tangent bundles, thus leading to generalized differential geometries, the ordinary geometry corresponding to the de Rham complex. The goal of these lectures is to give a brief introduction to this circle of ideas.
In the first lecture l will give an introduction to supermanifolds and dg manifolds as supermanifolds with an action of diffeomorphisms of the odd line. I will define the tangent complex of a dg manifold, generalizing the tangent bundle of a manifold and the anchor of a Lie algebroid. I will give a number of examples. Lastly, I will discuss additional structures on dg manifolds, such as differential forms and vector fields, and the formalism of derived brackets; this will be the framework for the generalized geometries mentioned above.
In the second lecture I will give a brief introduction to the language of infinity stacks in the formalism of simplicial presheaves. I will sketch Severa's theory of dg manifolds as first approximations. Conversely, I will describe a construction of an infinity-stack from a dg manifold and discuss possible approaches to the representability problem. time permitting, I will describe the computation, due to Andre Henriques, of the homotopy sheaves of a reduced dg manifold, and a generalization of the van Est homomorphism.
The third lecture will be devoted to the special case of Courant algebroids. I will show that these are equivalent to dg manifolds with a certain kind of additional sympelctic structure, and how Hitchin's generalized geometry in the presence of a gerbe can be naturally interpreted in this framework. I will also describe Bressler's obstruction-theoretic interpretation of the first Pontryagin class in terms of Courant algebroids. Lastly, I will describe an explicit algebraic construction of a differential graded algebra associated to a Courant-Dorfman algebra and give several examples.
Notes of the talks taken by Christoph Wockel.