Theodore Voronov (University of Manchester)
theodore.voronov@manchester.ac.uk
We have introduced a "thickening" of the category of smooth manifolds to a certain (formal) category with the same class of objects but with more morphisms. The characteristic feature of these new thick morphisms of (super)manifolds (which include ordinary smooth maps as a particular case) is that they still induce pullbacks on smooth functions, but these pullbacks are no longer linear, in general. Rather, they are formal nonlinear mappings between the linear spaces of smooth functions such that their derivatives at each point are algebra homomorphisms.
Thick morphisms are defined by methods of symplectic geometry as particular formal canonical relations between cotangent bundles. Motivation for this construction comes from homotopy algebras, for which nonlinear pullbacks induced by thick morphisms can produce $L_{\infty}$-morphisms.
In the talk I will speak about this theory and its quantum version. "Quantum thick morphisms" between (super)manifolds are particular oscillatory integral operators. They give "thick morphisms" described above in the limit $\hbar\to 0$.
(See preprints arXiv:1506.02417 and arXiv:1512.04163.)