Special Year on Symplectic Duality and 3d Mirror Symmetry

The wild character varieties M_B play the role of "Lie groups" in global Lie theory. In some cases such spaces have a "Lie algebra" M^* which is isomorphic to a Nakajma quiver variety, and one can then view the quiver as a "Dynkin diagram" for such spaces. In this talk I'll explain how to generalise this story by defining a diagram for any algebraic connection on the affine line, even if there is no corresponding Nakajima quiver variety. (It is known that any moduli space of meromorphic connections on the Riemann sphere is isomorphic to one on the affine line, so no generality is lost.) This is joint work with D. Yamakawa.

As is well known the matrix exponential map can be viewed as taking the monodromy of a connection Adz/z on a punctured disk. Considering more general, irregular, connections leads to many generalisations of the exponential map, known as Riemann-Hilbert-Birkhoff maps in general. In this talk I'll describe some of the symplectic geometry of the moduli spaces M^* of connections that play the role of Lie algebras in this story. This leads to the quiver modularity theorem, that a large class of Nakajma quiver varieties arise as such additive moduli spaces/"Lie algebras" M^*. It was conjectured and proved in the simply-laced case by the speaker (arxiv 0806, Pub. Math. IHES 2012) and proved in general by Hiroe-Yamakawa, Adv. Math. 2014.

We discuss methods to construct mirror pairs from string theory. For example, Hanany-Witten move. Outline

We discuss different brane/string theory constructions of 3d N=4 SCFTs, and how to construct mirror pairs from string theory. Outline

We continue on the overview, and emphasis on different constructions of 3d N=4 SCFTs. Outline

Outline

Representation theory is the study of vector spaces with prescribed symmetries. Replacing vector spaces by more complicated structures (of categorical, topological or geometrical origin) leads to new directions in algebra. This sheds a new light on geometric representation theory. More general moduli spaces constructions and enumerative invariants should fit in that framework. An additional motivation is the construction of invariants of low-dimensional manifolds from quantum field theories.

TBA

I will discuss the existence of very stable Higgs bundles: how it implies a precise formula for the multiplicity of the components of the nilpotent cone and its relationship to mirror symmetry. Joint project with Nigel Hitchin.

TBA