Gwyn Bellamy (University of Glasgow) - online talk

Minimal degenerations for quiver varieties.

For any symplectic singularity, one can consider the minimal degenerations between symplectic leaves - these are the relative singularities of a pair of adjacent leaves in the closure relation. I will describe a complete classification of these minimal degenerations for Nakajima quiver varieties. It provides an effective algorithm for computing the associated Hesse diagrams. In examples, these are dual to the Hesse diagram produced by physicists for the Coulomb branch of the quiver gauge theory.  The talk is based on joint work in progress with Alastair Craw and Travis Schedler. 

Amihay Hanany (Imperial College London)

Coulomb Branch: HyperKähler Quotients

TBA

Young-Hoon Kiem (KIAS)

Counting sheaves in algebraic geometry

By the Kobayashi-Hitchin correspondence, gauge theoretic invariants such as Donaldson invariant, Seiberg-Witten invariant and Vafa-Witten invariant have counterparts in sheaf counting in algebraic geometry. I will discuss algebraic theories of sheaf counting and singularities of relevant moduli spaces.

Hee-Cheol Kim (POSTECH)

Blowup Equations for 5d/6d theories 

I will talk about the blowup equations for 5d/6d supersymmetric QFTs and little string theories which generalize Nakajima-Yoshioka's blowup equations for the instanton partition functions of the 4d/5d gauge theories on Omega background.

Dmytro Matvieievskyi (University of Tokyo / Kavli IPMU)

Towards the symplectic dual of nilpotent orbit covers

Let G be a semisimple complex Lie group and Gv be the Langlands dual group. In joint work with Ivan Losev and Lucas Mason-Brown (arxiv:2108.03453) we proposed that the nilpotent Slodowy slice to a nilpotent orbit in Lie algebra gv and an affinization of a certain cover of a special nilpotent orbit  in g should be symplectically dual. Namely, we define an injective map d from the set of nilpotent orbits in gv to the set of covers of special nilpotent orbits in g, and the nilpotent slice to Ov should be dual to the affinization of d(Ov). It raises a natural question. What should be dual to a more general cover, in particular to a cover of non-special nilpotent orbit? In joint work with Lucas Mason-Brown and Shilin Yu we generalize the earlier construction of the map d to get some covers of non-special orbits. I will explain the main ideas of both constructions, and speculate on what partial answer it gives to the question posted above.

Hiraku Nakajima (University of Tokyo / Kavli IPMU)

Special Colloquium:
Gauge Theories and Geometric Representation Theory

 Gauge theories study connections and sections on bundles over manifolds. As usual in quantum field theories, physicists consider Feynman integrals over infinite dimensional spaces of all connections and sections. It is a difficult problem to justify these integrals in a mathematically rigorous way. Nevertheless, for supersymmetric gauge theories, we have succeeded in reducing infinite dimensional integrals to finite dimensional ones, and hence obtained mathematically well-defined integrals. A famous example is Donaldson invariants for 4-manifolds. These successes led Atiyah and Segal to formulate topological quantum field theories. In lower dimensions 2 or 3, topological quantum field theories have some connections with the so-called affine Grassmannian manifolds, which have been studied in geometric representation theory. I would like to give an example of a connection, my recent study on Coulomb branches of 3d SUSY gauge theories.

Workshop Talk:
Coulomb branches and S-dual varieties 

 For a given pair of a reductive group $G$ and a symplectic manifold $M$ with hamiltonian $G$-action, we have considered Higgs and Coulomb branches. We usually consider cases when $M$ is a symplectic representation, but the definition make sense in general. In some cases, we have observed that it is better to assign another symplectic manifold $M^\vee$ with hamiltonian action of the Langlands dual group $G^\vee$, after breaking $M$ into basic building blocks. They are called S-dual varieties. Examples arose in quiver gauge theories of type A, and their orthosymplectic analog. Moreover, they conjecturally fit with a recent preprint by Ben-Zvi, Sakellaridis and Venkatesh on relative Langlands duality. 

Yoshinori Namikawa (Kyoto University, RIMS)

A remark on isolated symplectic singularities with trivial local fundamental group

Recently, Bellamy et al constructed an infinite series of 4-dimensional isolated symplectic sngularities with trivial local fundamental group, inspired by a question of Beauville. In this talk, we introduce an easy construction  of  isolated symplectic singularities (of any dimension) with trivial local fundamental group.  We will use a toric hyperkaehler construction.

Ryo Yamagishi (University of Bath)

Symplectic singularities on compact sympletic varieties

Considering the compact symplectic (or hyperkähler) manifolds constructed by resolving symplectic singularities such as the ones due to O'Grady, the study of symplectic singularities turns out to be useful even if we are only interested in compact symplectic manifolds. In this talk, after explaining some known results about singularities appearing in moduli spaces of sheaves on a K3 surface, I will give an interesting example of a singular moduli space with a symplectic involution and discuss the (non)existence of a symplectic resolution of the involution quotient. This talk is based on joint work in progress with Hsueh-Yung Lin.