arXiv subjects

Ki Fung Chan

Publications and source records attributed to Ki Fung Chan.

6 recordsLinked to original sources

arXiv2026

Quantum cohomology, shift operators, and Coulomb branches

Given a complex reductive group $G$ and a $G$-representation $\mathbf{N}$, there is an associated Coulomb branch algebra $\mathcal{A}_{G,\mathbf{N}}^\hbar$ defined by Braverman, Finkelberg and Nakajima. In this paper, we prove a new characterization of $\mathcal{A}_{G,\mathbf{N}}^\hbar$ as the largest subspace of the equivariant Borel--Moore homology of the affine Grassmannian on which shift operators (and their deformations induced by flavour symmetries) are regular, meaning that they are defined without localizations. The proofs involve showing that the defining equations of the Coulomb branch algebras precisely reflect properness of the moduli spaces used to define shift operators. As a main application, we show that if $X$ is a smooth semiprojective variety equipped with a $G$-action, and $f \colon X \to \mathbf{N}$ is a $G$-equivariant proper holomorphic map, then the equivariant big quantum cohomology $QH^\bullet_G(X)$ defines a family of closed Lagrangians in the Coulomb branch $\mathrm{Spec}\mathcal{A}_{G,\mathbf{N}}$, yielding a transformation of 3d branes in 3d mirror symmetry. Regularity of shift operators also gives way to highly efficient computations in equivariant Gromov--Witten theory; in particular, we obtain a very short proof of Peterson isomorphism.

math.AG↗

arXiv2026

2d Mirrors in nonabelian 3d Mirror Symmetry

We establish a connection between (nonabelian) equivariant 2d mirror symmetry and the geometry of Coulomb branches. In the context of 3d mirror symmetry, a Hamiltonian $G$-manifold $Y$ is expected to determine a complex Lagrangian subvariety $\mathbb{L}^G_Y$ of the Coulomb branch. Using transverse Hilbert schemes and nil-Hecke algebras, we develop an algebro-geometric framework for studying Coulomb branches and their Lagrangian subvarieties and formulate criteria for the existence of $\mathbb{L}^G_Y$ in terms of equivariant 2d mirror symmetry. We then reinterpret these criteria in terms of Lagrangian displaceability and prove the resulting statements using Lagrangian Floer theory.

math.SG↗

arXiv2026

Iwahori-Coulomb branches, stable envelopes, and quantum cohomology of cotangent bundles of flag varieties

We consider Iwahori-Coulomb branches $\mathcal{A}_{G,\mathbf{N},\mathbf{V}}^{\mathrm{Fl}}$, which are the affine flag analogs of the original Coulomb branches $\mathcal{A}_{G,\mathbf{N}}^{\mathrm{Gr}}$ defined by Braverman, Finkelberg, and Nakajima. For any conical symplectic resolution $X$, we prove that the $\mathcal{A}_{G,\mathbf{N},\mathbf{V}}^{\mathrm{Fl}}$-action on the localized equivariant quantum cohomology of $X$, induced by shift operators, satisfies a polynomiality property in terms of stable envelopes. We then study the case $X = T^*(G/P)$, the cotangent bundle of a flag variety, for which the Iwahori-Coulomb branch is isomorphic to the trigonometric double affine Hecke algebra $\mathcal{H}_{G,\hbar,k}$. The polynomiality property enables us to compute explicitly the above action in terms of the Demazure-Lusztig elements and stable envelopes. Applications include: (1) Computation of the Iwarhori-Coulomb branch action for $G/P$ by taking the confluent limit, recovering Peterson-Lam-Shimozono's theorem. (2) Construction of an explicit Namikawa-Weyl group action on the equivariant quantum cohomology of $T^*(G/P)$ that preserves the quantum product, extending a result of Li-Su-Xiong. (3) Proof of a conjecture of Braverman-Finkelberg-Nakajima stating that, up to a shift of the dilation parameter, $\mathcal{A}_{G,\mathfrak{g}^*}^{\mathrm{Gr}}$ is isomorphic to the spherical subalgebra of $\mathcal{H}_{G,\hbar,k}$.

math.AG↗

arXiv2024

3d Mirror Symmetry is Mirror Symmetry

3d mirror symmetry is a mysterious duality for certian pairs of hyperkähler manifolds, or more generally complex symplectic manifolds/stacks. In this paper, we will describe its relationships with 2d mirror symmetry. This could be regarded as a 3d analog of the paper "Mirror Symmetry is T-Duality" by Strominger, Yau and Zaslow which described 2d mirror symmetry via 1d dualities.

math-ph↗

arXiv2022

Cohomologies on almost complex manifolds and the $\partial \bar{\partial}$-lemma

We study cohomologies on an almost complex manifold $(M, J)$, defined using the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as vector-valued forms on $M$. We show how one of these, the $N$-cohomology $H^{\bullet}_N (M)$, can be used to distinguish non-isomorphic non-integrable almost complex structures on $M$. Another one, the $J$-cohomology $H^{\bullet}_J (M)$, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The $J$-cohomology encodes whether a complex manifold satisfies the $\partial \bar{\partial}$-lemma, and more generally in the non-integrable case the $J$-cohomology encodes whether $(M, J)$ satisfies the $\mathrm{d} \mathcal{L}_J$-lemma, which we introduce and motivate in this paper. We discuss several explicit examples in detail, including a non-integrable example. We also show that $H^k_J$ is finite-dimensional for compact integrable $(M, J)$, and use spectral sequences to establish partial results on the finite-dimensionality of $H^k_J$ in the compact non-integrable case.

math.DG↗

arXiv2018

The $\mathcal L_B$-cohomology on compact torsion-free $\mathrm{G}_2$ manifolds and an application to 'almost' formality

We study a cohomology theory $H^{\bullet}_φ$, called the $\mathcal L_B$-cohomology, on compact torsion-free $\mathrm{G}_2$-manifolds. We show that $H^k_φ \cong H^k_{\mathrm{dR}}$ for $k \neq 3, 4$, but that $H^k_φ$ is infinite-dimensional for $k = 3,4$. Nevertheless there is a canonical injection $H^k_{\mathrm{dR}} \to H^k_φ$. The $\mathcal L_B$-cohomology also satisfies a Poincaré duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $\mathrm{d}$ and the derivation $\mathcal L_B$, and uses both Hodge theory and the special properties of $\mathrm{G}_2$-structures in an essential way. As an application of our results, we prove that compact torsion-free $\mathrm{G}_2$-manifolds are 'almost formal' in the sense that most of the Massey triple products necessarily must vanish.

math.DG↗