Exact Lagrangians in conical symplectic resolutions

Filip Zivanovic
Thu, Jun 25, 2020
Qolloquium: A One-Day Conference on Quivers, Representations, Resolutions

Conical symplectic resolutions are a vast family of holomorphic symplectic manifolds that appear in representation theory, algebraic and differential geometry, and also in theoretical physics. Their typical examples arise from the hyperkähler quotient construction (quiver and hypertoric varieties) but also from the representation theory of Lie algebras (resolutions of Slodowy varieties, slices in affine Grassmannians). In this talk, I will focus on their symplectic topology. In particular, we find families of non-isotopic exact Lagrangian submanifolds in them arising from different C*-actions. These Lagrangians have a very nice symplectic topology; in particular, we conjecture (work in progress) that all of their Floer-theoretic invariants are completely determined by their topology. At the end of the talk, I will discuss the special cases of Nakajima quiver varieties and resolutions of Slodowy varieties, where their count becomes feasible and interesting in its own.