This series consists of talks in the area of Mathematical Physics.
This is a joint work with A.Kuznetsov and L.Rybnikov.
We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety.
We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.
We will discuss a (conjectural) explicit presentation for the equivariant cohomology of Nakajima quiver varieties of type ADE. This presentation arises as a shadow of the expected symplectic duality between slices to Schubert varieties in the affine Grassmannian and Nakajima quiver varieties (a.k.a. the expected Coulomb and Higgs branches for a quiver gauge theory).
Categorical symplectic geometry studies an invariant of symplectic manifolds called the "Fukaya (A-infinity) category", which consists of the Lagrangian submanifolds and a symplectically-robust intersection theory of these Lagrangians. Over the last two decades the Fukaya category has emerged as a powerful tool: for instance, it has produced inroads to Arnol'd's Nearby Lagrangians Conjecture, and it allowed Kontsevich to formulate the the Homological Mirror Symmetry conjecture.
Let $S$ be a surface, $G$ a semi-simple group of type B, C or D. I will explain why the moduli space of framed local systems $A_{G,S}$ defined by Fock and Goncharov has the structure of a cluster variety, and fits inside a larger structure called a cluster ensemble. This was previously known only in type A. This gives a more direct proof of results of Fock and Goncharov for the symplectic and spin groups, and also allows one to quantize higher Teichmuller space in these cases.
I will review the possible role in Geometric Langlands
of N=4 boundary conditions in four-dimensional supersymmetric Yang Mills theory.
The action of S-duality on such boundary conditions can be understood
in terms of symplectic duality.
K-theoretical/Cohomological Hall algebras, associated with the stack of zero-dimensional sheaves on $\mathbb{C}^2$, play a prominent role in the proof, given by Schiffmann and Vasserot, of the AGT conjecture for
We study the representation theory of truncated shifted Yangians. These algebras arise as quantizations of slices to Schubert varieties in the affine Grassmannian. We will describe the combinatorics of their highest weights, which is encoded in Nakajima's monomial crystal. We also prove Hikita's conjecture in this context.
I'll discuss the two topological twists of 3d N=4 theories, and explain how to understand them in the AKSZ/BV formalism, and how they relate to twists of 4d N =2 theories. Symplectic duality then takes the form of an equivalence between 3d N=4 theories which interchanges the two topological twists. I'll also introduce monopole operators and explain the role they play in symplectic duality.