Microlocal sheaves and quiver varieties

Abstract: We relate Nakajima Quiver Varieties (or, rather, their multiplicative version) with moduli spaces of perverse sheaves. More precisely, we consider a generalization of the concept of perverse sheaves: microlocal sheaves on a nodal curve X. They are defined as perverse sheaves on normalization of X with a Fourier transform condition near each node and form an abelian category M(X). One has a similar triangulated category DM(X) of microlocal complexes. For a compact X we show that DM(X) is Calabi-Yau of dimension 2. In the case when all components of X are rational, M(X) is equivalent to the category of representations of the multiplicative pre-projective algebra associated to the intersection graph of X. Quiver varieties in the proper sense are obtained as moduli spaces of microlocal sheaves with a framing of vanishing cycles at singular points. The case when components of X have higher genus, leads to interesting generalizations of preprojective algebras and quiver varieties. We analyze them from the point of view of pseudo-Hamiltonian reduction and group-valued moment maps.