Microlocal sheaves and quiver varieties (Q341050)

Let \(G=\text{GL}_n\), let \(X\) be a (possibly nodal) compact complex oriented \(C^{\infty}\) surface, and let \(\text{LS}_{G}(X)\) be the moduli space of \(G\)-local systems on \(X\); \(\text{LS}_{G}(X)\) has the structure of a symplectic manifold. Additionally, for \(Q\) a finite oriented graph; then the associated quiver varieties can be viewed as symplectic reductions of the cotangent bundles to the moduli spaces of representations of \(Q\) with varying dimension vectors. The work described here draws a connection between these two classes of symplectic manifolds through the use of microlocal sheaves, a generalization of perverse sheaves.NEWLINENEWLINESuppose that \(X\) is nonsingular, and let \(q\in \mathbb{C}^{\times}\). Let \(\mathcal{L}\) be a line bundle on \(X\), and for \(A\) a finite set of points let \(D\mathcal{M}^{\mathcal{L},q}(X,A)\) be the category of \((\mathcal{L},q)\)-twisted microlocal complexes \(\mathcal{F}\) on \(X\) such that \(\widetilde{\mathcal{F}}^{\circ}\) has locally constant cohomology outside the preimage of \(A\), and let \(\mathcal{M}^{\mathcal{L},q}(X,A)\) be the abelian subcategory of let \(q\)-twisted microlocal sheaves. Let \(\Gamma_X\) be the intersection graph on \(X\), and let \(I\) denote its number of vertices. The author defines a multiplicative preprojective algebra \(\Lambda^{\underline{q}}(\Gamma_X)\) for each choice of \(\underline{q}\in (\mathbb{C^{\times}})^I\). The main result (in the nonsingular case) is an equivalence between the categories \(\mathcal{M}^{\mathcal{L},q}(X,\varnothing)\) and finite dimensional modules over \(\Lambda^{q^{\deg{\mathcal{L}}}}(X)\), where \(\deg\mathcal{L}=(\deg(\varpi_i^*\mathcal{L})\).NEWLINENEWLINEThe above result is then generalized to the case where \(X\) is nodal by adapting the construction of \(\Lambda^{q^{\deg{\mathcal{L}}}}(X)\).