Abstract: Integrable models are known to keep reemerging over time in various mathematical incarnations. Recently, such models based on quantum groups naturally appeared in the framework of enumerative geometry. In this context the so-called Bethe ansatz equations, instrumental for finding the spectrum of the XXZ model Hamiltonian, naturally show up as constraints for the quantum K-theory ring of quiver varieties. … Read More

Many problems in mathematical physics, from the WKB method to knot theory, involve quantum versions of algebraic curves. In this talk I will review an approach to the quantization of local mirror curves which makes it possible to reconstruct topological string theory on toric Calabi-Yau manifolds. In this approach, the quantization of the mirror curve leads to a trace class … Read More

Abstract: Braverman, Finkelberg, and Nakajima define the K-theoretic Coulomb branch of a 3d \mathcal N=4 SUSY gauge theory as the affine variety \mathcal M_{G,N} arising as the equivariant K-theory of certain moduli space \mathscr R_{G,N}, labelled by the complex reductive group G and its complex representation N. It was conjectured by Gaiotto, that (quantized) K-theoretic Coulomb branches bear the structure … Read More

Abstract: I will explain a physically motivated construction of Ricci-flat K3 metrics via a hyper-Kähler quotient, which yields the first examples of explicit Ricci-flat metrics on compact non-toroidal Calabi-Yau manifolds. I will also relate it — both physically and mathematically — to a second such construction, which is as yet not completely explicit: the missing data is the BPS index … Read More

The monodromy of quantum difference equations is closely related to elliptic stable envelopes invented by M.Aganagic and A. Okounkov. In the talk I will explain how to extract these equations from the monodromy using the geometry of the variety X and of its symplectic dual Y. In particular, I will discuss how to extend the action of representation-theoretic objects on … Read More