I will talk about a construction of higher dimensional loop Grassmannians, in
the framework of local spaces over Hilbert scheme of points. The notion of local
spaces is a refined version of the factorization space of Beilinson-Drinfeld. It
has many applications, examples including loop Grassmannians associated to
Kac-Moody groups, geometric Langlands duality in higher dimensions, and
non-abelian higher class field theory.
In dimension 1, this construction yields the recent reconstruction by Ivan
Mirkovic of the usual loop Grassmannians from semi-infinite orbits. During the
talk, I will mainly focus on the construction applied to dimension 2. I will
briefly mention the difficulties in dimension 3.
This talk is based on my joint work with Ivan Mirkovic and Gufang Zhao.