The Remodeling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti relates all-genus open-closed Gromov-Witten invariants in a toric Calabi-Yau 3-manifold/3-orbifold to the Chekhov-Eynard-Orantin Topological Recursion (TR) invariants of its mirror curve. In this talk, I will describe the Remodeling Conjecture when 1) there are multiple Aganagic-Vafa Lagrangian branes, including inner...
We explain an elliptic trace formula for correlations of chiral QFT on elliptic curves. As an application, we show how this leads to a corresponding holomorphic anomaly equation and enumerative geometry on elliptic curves.
In 1980, Saito, Sekiguchi and Yano found ``flat generator system’’ on the orbit spaces of irreducible finite Coxeter Groups. Their construction can be understood in the framework of the almost duality of Frobenius manifolds proposed by Dubrovin.
In late 2010’, the story was extended to well-generated finite complex reflection groups by Arsie- Lorenzoni, Kato-Mano-Sekiguchi, and...
We prove two conjectured presentations of the quantum $K$-ring of type A partial flag varieties, one coming from quantum field theory, the other coming from quantum integrable systems. Our main tool to do this is using abelianization, which also gives a purely geometric interpretation for the Bethe Ansatz equations, which appear in both conjectures.
I will discuss a definition of refined curve counting invariants of Calabi-Yau threefolds with a C*-action in terms of stable maps on Calabi-Yau fivefolds. The corresponding disconnected generating function should conjecturally equate the Nekrasov-Okounkov K-theoretic membrane index under a refined version of the Gromov-Witten/Pandharipande-Thomas correspondence. I'll present several acid...
The genus-zero Gromov-Witten invariants of a smooth projective variety can be encoded in an infinite-dimensional Lagrangian submanifold, known as the Givental cone, within the loop space of the cohomology group. A Givental-style mirror theorem states that a certain explicit cohomology-valued hypergeometric series, called the I-function, lies in this Givental cone. In joint work with Coates,...
In a joint work with Andrei Okounkov, Yehao Zhou and Zijun Zhou. We introduce stable envelopes in critical cohomology and K-theory for symmetric quiver varieties with potentials and related geometries.
Critical stable envelopes are compatible with dimensional reductions, specializations, Hall products, and other geometric constructions. In particular, for tripled quivers with canonical...
I will describe how the quasimap approach to equivariant quantum K-theory is modified when the curve-counting parameter is sent not to unity, but to a primitive root of unity instead. In particular, this leads to the appearance of the Frobenius action on the moduli space. Upon reducing the quantum difference equation modulo primes, we arrive at the Grothendieck–Katz p-curvature and prove that...
The Tutte polynomial was introduced in the 1940s as a two-variable generalisation of the chromatic polynomial of a graph. It is the universal matroid invariant satisfying a deletion-contraction relation, and is the subject of much recent work.
I will describe a geometric realisation of the Tutte polynomial via the cohomology of a symplectic dual pair of hypertoric varieties. The same...
Topological recursion is a powerful tool in mathematical physics, applicable to various problems in enumerative geometry, such as intersections on moduli spaces and Hurwitz numbers. In my talk, I will discuss the KP integrability of topological recursion, which arises naturally in the context of the x-y swap relation. This integrability can be described through certain integral transforms,...
Br\'ezin--Gross--Witten (BGW) numbers and Witten's intersection numbers are two families of rational numbers, that both have physical origins, topological meanings and backgrounds from matrix models. Their partition functions are known to be particular tau-functions of the KdV hierarchy, satisfying Virasoro constraints. In view of integrable systems, the origin of the BGW numbers and Witten's...
In the first part, I will introduce our in progress work on the Seiberg duality conjecture in geometric level. Consider a quiver with potential. It has been proved by many people that its quasimap I function is preserved under quiver mutation in some sense. In this work, we further consider the behavior of the scheme under quiver mutation, and we have proved that the scheme is also preserved...
Given a complex reductive group G and a G-representation N, there is an associated Coulomb branch algebra defined by Braverman–Finkelberg–Nakajima. In joint work with Chan and Lam, we show that these Coulomb branch algebras can be described as the largest subcomodules of the equivariant BM-homology of the affine Grassmannian on which certain shift operators admit non-equivariant limits. I will...
This talk concerns the Langlands program for 3-manifolds, initiated by Ben-Zvi–Gunningham–Jordan–Safronov. From the physical perspective, it arises as part of S-duality in 4d N=4 supersymmetric quantum field theory. One formulation of the Langlands duality conjecture for 3-manifolds involves infinite-dimensional DT cohomology, defined as the cohomology of certain perverse sheaves on the stack...
I will report on joint work in progress with Tasuki Kinjo, Hyeonjun Park, and Pavel Safronov, regarding Joyce's conjecture on a (-1)-shifted microlocalization of the virtual fundamental class. This construction subsumes not only the usual virtual fundamental class for quasi-smooth moduli spaces (read: perfect obstruction theories), but also other virtual class type constructions appearing...
