Monday 12.16
Hiraku Nakajima (IPMU)
Title: Geometric Satake correspondences for Kac-Moody Lie algebras and
Coulomb branches of 3d N=4 gauge theories
Abstract: The usual geometric Satake correspondence gives relation
between geometry of affine Grassmannian of a complex reductive group
and representation theory of its Langlands dual group. Recently we are
trying to build a similar result for more general Kac-Moody Lie
algebras, based on Coulomb branches of 3d N=4 gauge theories defined by
Braverman, Finkelberg and myself mathematically rigorously. I will
start with the historical account for the usual one, and then survey the
current status of the generalization.
Tuesday 12.17
Eric Vasserot (IMJ)
Title: COHERENT CATEGORIFICATION OF QUANTUM LOOP ALGEBRAS
Abstract: We construct an equivalence of graded Abelian categories from a category of representations of a quiver-Hecke algebra to a category of equivariant perverse coherent sheaves on the affine Grassmannians of type A. We prove that this equivalence is monoidal. Using this, we compare the monoidal categorification of the quantum open unipotent cells of type SL(2) affine in terms of quiver-Hecke algebras given by Kashiwara-Kim-Oh-Park with the monoidal categorification in terms of equivariant perverse coherent sheaves on the affine Grassmannians given by Cautis-Williams. The proof uses a categorification of the preprojective K-theoretic Hall algebra introduced by Schiffmann-Vasserot.
Sergey Cherkis (Arizona)
Title: Moduli Spaces of Monopoles and Quantum Field Theories
Abstract: Moduli spaces of monopoles are hyperkaehler manifolds which are conjectured to be isometric to Coulomb branches of supersymmetric quantum field theories. After reviewing various cases of this correspondence, we focus on doubly periodic monopoles, called monowalls. Their spectral description suggests numerous isometries among moduli spaces of different monowalls. It also leads to a natural compactification of these moduli spaces as exploded manifolds.
We conclude by reporting our joint work with Rebekah Cross, in which we compute the asymptotic metric on the monowall moduli spaces and observe that it can be expressed in terms of a volume cut out by a plane arrangement associated to each monowall.
Wednesday 12.18
Alexei Oblomkov (U Mass, Amherst)
Title: 3D TQFT, Soergel bimodules and knot homology
Abstract: Talk is based on joint work with Lev Rozansky. I will explain a mathematical
construction of N=4 gauge 3D TQFT, known as Kapustin-Saulina-Rozansky theory.
The defects in this theory encode the braids and that allows us to categorify
Ocneanu-Jones trace and obtain a triply-graded knot homology. We show the
homology coincide with the Khovanov-Rozansky trace on the Rouquier complexes
on Soergel bimodules. Because of the geometric nature of our TQFT
construction we obtain
an interpretation of the homology as space of global sections of a
sheaf on the Hilbert
scheme of point on the plane.
Thursday 12.19
Sergei Gukov (Caltech)
Title: Logarithmic VOAs from 3d boundary chiral algebras
Abstract: I will give an overview and some of the highlights of a research program initiated in 2013 with Gadde and Putrov, the main subject of which is the study of 2d-3d combined systems that involve 3d N=2 theories with 2d N=(0,2) boundary conditions. We will see how holomorphic-topological twists of such systems lead to boundary chiral algebras, somewhat similar to Beem-Rastelli chiral algebras of 4d N=2 theories. In particular, just like the characters of Beem-Rastelli chiral algebras are computed by the so-called Schur index of 4d N=2 theories, characters of our boundary chiral algebras are computed by the "half-index" of 2d-3d combined systems, also introduced in the above-mentioned work with Gadde and Putrov and originally motivated by applications to topology.
Ryo Fujita (Kyoto)
Title: Graded quiver varieties and singularities of normalized R-matrices for fundamental modules
Abstract: The normalized R-matrix is a unique intertwining operator
between tensor products of two finite-dimensional simple modules over the quantum affine algebra.
It can be seen as a matrix-valued rational function in the spectral parameter,
whose denominator determines when the tensor product module becomes reducible.
In this talk, we present a simple unified formula expressing the denominators of the normalized
R-matrices between the fundamental modules of type $ADE$.
It has an interpretation in terms of representations of Dynkin quivers
and can be proved in a unified way using the geometry of graded quiver varieties.
As a by-product, we obtain a geometric interpretation of
Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality
functor when it arises from a family of fundamental modules.
Tatsuyuki Hikita (RIMS)
Title: Elliptic canonical bases for hypertoric varieties
Abstract: Lusztig defined a notion of canonical bases in equivariant K-theory of
Springer resolutions or Slodowy varieties by constructing certain
involutions called bar involution. In this talk, I will explain a
reformulation and conjectural generalizations of these K-theoretic
canonical bases for good conical symplectic resolutions using K-theoretic
analogue of the notion of stable envelope introduced by Maulik and
Okounkov. Then I will define an elliptic analogue of the bar involutions
using elliptic stable envelope defined by Aganagic-Okounkov. I will also
give explicit families of elements invariant under elliptic bar
involutions for hypertoric varieties which I call elliptic canonical
bases.
Friday 12.20
PeterKoroteev (UC Berkeley)
Title: q-Opers, QQ-Systems and Bethe Ansatz
Abstract: I will show the equivalence between nondegenerate
Z-twisted (G,q)-opers and solutions of generalized Bethe equations for an arbitray simply connected complex simple Lie group G. While in the simply laced case the resulting Bethe equations are the well-known Bethe equations for XXZ model, the non-simply laced cases feature different version of such equations which do not fit known XXZ models.
Our results bring together several approaches to the q-deformed version of opers, namely older works on q-difference version Drinfeld-Sokolov reduction, paper by Mukhin and Varchenko on difference Miura opers and the recent approach of the speaker together with Sage and Zeitlin.