Abstract:

We will discuss the general notion of symplectic duality (also known as 3D mirror symmetry) between symplectic resolutions of singularities and give examples. We will then formulate the Hikita-Nakajima conjecture describing (equivariant) cohomology ring of a symplectic resolution in terms of the dual variety. We will consider the example of the Hilbert scheme of points on the affine plane and discuss the proof of the Hikita-Nakajima conjecture in this particular case. Time permitting, we will discuss the general approach towards the proof of Hikita-Nakajima conjecture for other symplectically dual pairs (such as Higgs and Coulomb branches of certain quiver gauge theories). Various interesting objects (for example, integrable systems on Coulomb branches) appear naturally in the proof. Based on a joint work with Pavel Shlykov.