Manchester Geometry Seminar 2014/2015

Thursday 2 October 2014. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm

The "Nonlinear Pullback" of Functions and a Formal Category Extending the Category of Supermanifolds

Theodore Voronov (University of Manchester)

We introduce a mapping of function spaces on smooth (super)manifolds, which is generally nonlinear and which generalizes the usual pullback with respect to a smooth map. The construction is based on symplectic geometry of cotangent bundles. It uses canonical relations and generating functions. Here arises a certain "formal category", which extends the usual category of supermanifolds and their smooth maps. (It is very close to the category of symplectic micromanifolds and their micromorphisms considered recently by A. Weinstein and A. Cattaneo - B. Dherin - A. Weinstein.) There is a parallel construction based on odd symplectic geometry.

As an application, we show that this mapping gives an $L_{\infty}$-morphism of the algebras of functions on homotopy Schouten manifolds if the corresponding "master Hamiltonians" are related by a canonical relation. (There is a parallel statement for homotopy Poisson manifolds.) There may be applications to $L_{\infty}$-bialgebroids and higher Koszul brackets.

The talk is based on the recent work arXiv:1409.6475 [math.DG].