This series consists of talks in the area of Mathematical Physics.
In this talk, I'd like to explain how quantum integrability and (q-deformed) W-algebraic structure arise from the moduli space of quiver gauge theory. It'll be also shown that our construction gives rise to a new family of W-algebras.
Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk, I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution. This result builds upon work of Crawley–Boevey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.
I'll discuss an ongoing project of mine with Dinakar Muthiah and Oded Yacobi, which is based on ideas of Kreiman-Lakshmibai-Magyar-Weyman. It is aimed at developing a "nice" standard monomial theory for the affine Grassmannian in type A, where "nice" means hopefully close in spirit to Hodge's classical standard monomial theory for finite dimensional Grassmannians.
Factorization spaces (introduced by Beilinson and Drinfeld as "factorization monoids") are non-linear analogues of factorization algebras. They can be constructed using algebro-geometric methods, and can be linearised to produce examples of factorization algebras, whose properties can be studied using the geometry of the underlying spaces. In this talk, we will recall the definition of a factorization space, and introduce the notion of a module over a factorization space, which is a non-linear analogue of a module over a factorization algebra.
This is based on my joint work with Yaping Yang. In this talk, we use the equivariant elliptic cohomology theory to study the elliptic quantum groups. We define a sheafified elliptic quantum group for any symmetric Kac-Moody Lie algebra. This definition is naturally obtained using the cohomological Hall algebra associated to the equivariant elliptic cohomology. After taking suitable rational sections, the sheafified elliptic quantum group becomes a quantum algebra consisting of the elliptic Drinfeld currents.
We explain how a doubled version of the Beilinson-Bernstein localization functor can be understood using the geometry of the wonderful compactification of a group. Specifically, bimodules for the Lie algebra give rise to monodromic D-modules on the horocycle space, and to filtered D-modules on the group that respect a certain matrix coefficients filtration. These two categories of D-modules are related via an associated graded construction in a way compatible with localization, Verdier specialization, and additional structures. This is joint work with David Ben-Zvi and David Nadler.
In my talk, I will briefly review the representation theoretical construction of conformal blocks attached to an affine Kac-Moody algebra and a smooth algebraic curve with marked points. I will focus on the case when the algebraic curve is an elliptic curve. The bundle of conformal blocks carries a canonical flat connection: the Knizhnik-Zamolodchikov-Bernard (KZB) equation. There are various generalizations of the KZB equation. I will talk about one generalization that constructed by myself and Toledano Laredo recently: the elliptic Casimir connection.
By the non-abelian Hodge theory of Carlos Simpson, harmonic bundles
interpolate between bundles with connections on a curve and
Higgs bundes (precise formulations requires some additional data like parabolic structure and stability structure).
I will explain the framework for a generalization of the non-abelian Hodge theory
which unifies Simpson's story ("rational case") with those for q-difference
equations ("trigonometric case") and elliptic difference equations
("elliptic case").
Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories provided a beautiful link between the geometry of "spacetimes" (cobordisms) and algebraic structures. Combining this with the physical notion of "locality" led to the introduction of the language of higher categories into the topic.
Natural targets for extended topological field theories are higher Morita categories: generalizations of the bicategory of algebras, bimodules, and homomorphisms.