Abstract: “I will report on a joint work in progress with Pablo BoixedaAlvarez, Michael McBreen and Zhiwei Yun where categories of microlocal sheaves on some affine Springer fibers are described in terms of theLanglands dual group. In particular, in the slope 1 case we recover the regular block in the category of (graded) modules over the smallquantum groups. Assuming a general formalism … Read More
Abstract: In work to appear with Ballin-Creutzig-Dimofte, we constructed vertex operator algebras associated to A and B twists of 3d N=4 abelian gauge theories. These are boundary VOAs supported on holomorphic boundary conditions of Costello-Gaiotto. For the B twist, the vertex algebra V_B is a simple current extension of an affine Lie superalgebra, and using the work of Creutzig-Kanade-McRae, we … Read More
In a landmark work, Frances Kirwan described the relation between the cohomology of a GIT quotient of a smooth projective variety X and the equivariant cohomology of X by what is known as the ‘subtraction method’: this relies on the equivariant perfection of the destabilizing stratification of X. Later work geared toward describing the cohomology ring structure relied on Poincare … Read More
Recent advances around Fukaya categories can be used to (mathematically rigorously) produce sheaves on Bun_G from smooth fibers of Hitchin fibrations. The resulting sheaves are presumably Hecke eigensheaves; I’ll explain why I don’t know how to prove this, and discuss various related questions.
In my talk I will first review the interpretation of the counts of solutions of Kapustin-Witten equations on a 3-manifold times a line as Stokes coefficients associated to the perturbative expansion of Chern-Simons theory. These counts can be naturally combined into q-series with integral coefficents, labelled by an ordered pair of flat connections. I will then present an explicit algorithm/formula … Read More
Abstract: Link homology is the space of states of BPS particles in a special 5d QFT. The BPS particles can be presented as D2 branes attached to Chern-Simons-related D4 branes and link-related NS5 branes. We assume that D2 branes form a stack split into domains by D4 and NS5 interfaces. The vibrations of the stack are described by a 3d … Read More
Abstract: Top down (i.e., stringy) constructions of quantum field theories provide a general template for constructing and studying a wide variety of different strongly coupled systems which are otherwise difficult to study using “textbook” methods based on perturbation theory of a Lagrangian field theory. In this talk we use the geometry of extra dimensions in string theory to study generalized … Read More
Abstract: Kontsevich and Soibelman suggested a correspondence between Donaldson-Thomas invariants of Calabi-Yau 3-folds and holomorphic curves in complex integrable systems. After reviewing this general expectation, I will present a concrete example related to mirror symmetry for the local projective plane (partly joint work with Descombes, Le Floch, Pioline), along with applications in enumerative geometry (partly joint work with Fan, Guo, … Read More
Abstract: Stable envelopes are correspondences useful for constructing geometric action of quantum groups and solutions to quantum Knizhnik-Zamolodchikov (qKZ) equations. I will review basic aspects of this and explain the construction in a novel class of examples consisting of certain vortex (also known as quasimap) moduli spaces. The main technical result is that K-theoretic curve counts in these varieties are controlled … Read More
Abstract: Deeper structures behind BPS counting on toric Calabi-Yau 3-folds have recently been realized mathematically in terms of the quantum loop group associated to a certain quiver drawn on a torus, which is endowed with an action on the BPS vector space via crystal melting. In this talk, we identify the annihilator of the aforementioned action, thus leading to the … Read More