Organizers: Adeel Khan, Kevin Lin, Y.P. Lee
Institute of Mathematics, Academia Sinica
Let πΊ be a semisimple algebraic group and πΊβ‘/π΅ its flag variety. For πΊ =πβ’πΏπ, Givental and Lee showed that the quantum πΎ-theoretic π½-function is an eigenfunction of a π-difference Toda Hamiltonian. They also proposed a construction of commuting Hamiltonians for general πΊ. It has since been pointed out that this construction requires modification in non-simply-laced types.
The general problem is to embed the Weyl-invariant representation ring into an algebra of π-difference operators so that the π½-function is a joint eigenfunction, and to determine the resulting Hamiltonians explicitly. In this talk, I will discuss the current status of this problem and present an explicit formula for the Hamiltonian associated with a minuscule fundamental weight. This is joint work with Koushik Brahma, Yicen Huang, Takafumi Kouno, and Kohei Yamaguchi.
There are two natural ways to deform Joyce's construction of vertex algebras. One is to include tautological insertions, and the other is to work equivariantly. I introduce T-deformed vertex algebras, which unify these two refinements. For the purpose of wall-crossing, they must be treated differently because the associated Lie algebras use different expansions. I will explain this behavior explicitly by studying tautological stable pair wall-crossing and the equivariant 2-loop quiver. Using the calculus of T-deformed vertex algebras developed in an upcoming joint work with E. Bouaziz, we derive 7 new Virasoro algebras for the affine plane.
We study orbifold Calabi-Yau hypersurfaces in weighted projective stacks from the point of view of the gauged linear sigma model. We give a very brief introduction to the gauged linear sigma model which is a physics version of quasimap theory and review some general conjectures. We explain how these conjectures allow for determination of the orbifold Gromov-Witten invariants without using Gromov-Witten or quasimap theory.
Archive from Academic Year 2025-26