arXiv subjects

Kwokwai Chan

Publications and source records attributed to Kwokwai Chan.

At least 19 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

Quantization of Kähler manifolds via differential operators

In this paper, we study the quantization of classical observables (i.e., functions) on a Kähler manifold $X$ as differential operators acting on holomorphic sections of tensor powers $L^{\otimes k}$ of the pre-quantum line bundle $L$. We prove two global results as follows. (1). For a general smooth function $f \in C^\infty(X)$, we construct higher order generalizations of Kostant-Souriau's pre-quantum differential operators using our Fedosov-type constructions of Bargmann-Fock sheaves in previous works. We prove that these differential operators are asymptotic to the Berezin-Toeplitz operators $T_{f,k}$ acting on the Hilbert space $H^0(X, L^{\otimes k})$ as $k \to \infty$. (2). If a smooth function $f \in C^\infty(X)$ is furthermore the symbol of a level $k$ quantizable function , then we prove that the associated Berezin-Toeplitz operator $T_{f,k}$ is a holomorphic differential operator. Conversely, Berezin-Toeplitz operators that are holomorphic differential operators all arise in this way. This gives a complete characterization of when Berezin-Toeplitz operators are holomorphic differential operators. To prove these results, we establish new orthogonality relations which generalize the classical Tuynman's Lemma, and employ various differential-geometric and analytic technqiues such as Hörmander's estimates.

math.DG↗

arXiv2026

Brane quantization and SYZ mirror symmetry

Coisotropic A-branes were introduced by Kapustin--Orlov to enlarge the Fukaya category of a symplectic manifold in a way that aligns with predictions from homological mirror symmetry. From a mathematical perspective, however, the categorical framework governing such branes remains largely undeveloped. On the other hand, Gukov--Witten's brane quantization suggests that a holomorphic deformation quantization of a holomorphic symplectic manifold $X$ arises from the endomorphism algebra $Hom_A(B_{cc},B_{cc})$ of a canonical coisotropic A-brane $B_{cc}$, which naturally acts on the morphism space $Hom_A(B,B_{cc})$ with a Lagrangian A-brane $B$ that in turn gives precisely the geometric quantization of $B$. In this paper, we consider a holomorphic symplectic manifold $X$ which admits an SYZ fibration and apply SYZ mirror symmetry to study its brane quantization. Given any semi-affine, space-filling coisotropic A-brane $B_{cc}$ on $X$, we construct the mirror B-brane $\check{B}_{cc}$ on the mirror manifold $\check{X}$ by an SYZ transform. We then present a mathematical definition of the endomorphism algebra $Hom_A(B_{cc},B_{cc})$ by constructing a distinguished non-formal holomorphic deformation quantization of $X$. Using a twisted family Toeplitz construction, we transform $Hom_A(B_{cc},B_{cc})$ to the mirror B-side and prove that this induces an isomorphism $Hom_A(B_{cc},B_{cc})\cong Hom_B(\check{B}_{cc},\check{B}_{cc})$ between the endomorphism algebras. Furthermore, taking any torus fiber of $X$ as the Lagrangian A-brane $B$, we fully realize Gukov--Witten's proposal, namely, there is a natural action of $Hom_A(B_{cc},B_{cc})$ on $Hom_A(B,B_{cc})$ which is precisely mirror to the natural action on the mirror B-side. This provides a mathematical framework which is compatible with Gukov--Witten's brane quantization proposal, SYZ mirror symmetry as well as family Floer theory.

math.DG↗

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↗

arXiv2026

A perturbative construction of primitive forms from log Landau-Ginzburg mirrors of toric manifolds

We introduce the notion of a logarithmic Landau-Ginzburg (log LG) model, which is essentially given by equipping the central degenerate fiber of the family of Landau-Ginzburg (LG) models mirror to a projective toric manifold with a natural log structure. We show that the state space of the mirror log LG model is naturally isomorphic to that of the original toric manifold. Following Li-Li-Saito, we give a perturbative construction of primitive forms by studying the deformation theory of such a log LG model, which involves both smoothing of the central degenerate fiber and unfolding of the superpotential. This yields a logarithmic Frobenius manifold structure on the base space of the universal unfolding. The primitive forms and flat coordinates we obtained are computable and closely related to the bulk-deformed Lagrangian Floer superpotential of a projective toric manifold, at least in the semi-Fano case.

math.AG↗

arXiv2024

Smoothing, scattering, and a conjecture of Fukaya

In 2002, Fukaya proposed a remarkable explanation of mirror symmetry detailing the SYZ conjecture by introducing two correspondences: one between the theory of pseudo-holomorphic curves on a Calabi-Yau manifold $\check{X}$ and the multi-valued Morse theory on the base $\check{B}$ of an SYZ fibration $\check{p}: \check{X}\to \check{B}$, and the other between deformation theory of the mirror $X$ and the same multi-valued Morse theory on $\check{B}$. In this paper, we prove a reformulation of the main conjecture in Fukaya's second correspondence, where multi-valued Morse theory on the base $\check{B}$ is replaced by tropical geometry on the Legendre dual $B$. In the proof, we apply techniques of asymptotic analysis developed in our previous works to tropicalize the pre-dgBV algebra which governs smoothing of a maximally degenerate Calabi-Yau log variety introduced in another of our recent work. Then a comparison between this tropicalized algebra with the dgBV algebra associated to the deformation theory of the semi-flat part $X_{\text{sf}} \subseteq X$ allows us to extract consistent scattering diagrams from appropriate Maurer-Cartan solutions.

math.AG↗

arXiv2023

Quantizable functions on Kähler manifolds and non-formal quantization

Applying the Fedosov connections constructed in our previous work, we find a (dense) subsheaf of smooth functions on a Kähler manifold $X$ which admits a non-formal deformation quantization. When $X$ is prequantizable and the Fedosov connection satisfies an integrality condition, we prove that this subsheaf of functions can be quantized to a sheaf of twisted differential operators (TDO), which is isomorphic to that associated to the prequantum line bundle. We also show that examples of such quantizable functions are given by images of quantum moment maps.

math.QA↗

arXiv2022

A geometric construction of representations of the Berezin-Toeplitz quantization

For a Kähler manifold $X$ equipped with a prequantum line bundle $L$, we give a geometric construction of a family of representations of the Berezin-Toeplitz deformation quantization algebra $(C^\infty(X)[[\hbar]],\star_{BT})$ parametrized by points $z_0 \in X$. The key idea is to use peak sections to suitably localize the Hilbert spaces $H^{0}\left(X,L^{\otimes m}\right)$ around $z_{0}$ in the large volume limit.

math.QA↗

arXiv2022

From deformation theory to tropical geometry

This is a write-up of the author's invited talk at the Eighth International Congress of Chinese Mathematicians (ICCM) held at Beijing in June 2019. We give a survey on joint works with Naichung Conan Leung and Ziming Nikolas Ma where we study how tropical objects arise from asymptotic analysis of the Maurer-Cartan equation for deformation of complex structures on a semi-flat Calabi-Yau manifold.

math.AG↗

arXiv2022

Kapranov's $L_\infty$ structures, Fedosov's star products, and one-loop exact BV quantizations on Kähler manifolds

We study quantization schemes on a Kähler manifold and relate several interesting structures. We first construct Fedosov's star products on a Kähler manifold $X$ as quantizations of Kapranov's $L_\infty$-algebra structure. Then we investigate the Batalin-Vilkovisky (BV) quantizations associated to these star products. A remarkable feature is that they are all one-loop exact, meaning that the Feynman weights associated to graphs with two or more loops all vanish. This leads to a succinct cochain level formula in de Rham cohomology for the algebraic index.

math.QA↗

arXiv2022

Tropical Lagrangian multi-sections and smoothing of locally free sheaves over degenerate Calabi-Yau surfaces

We introduce the notion of tropical Lagrangian multi-sections over a $2$-dimensional integral affine manifold $B$ with singularities, and use them to study the reconstruction problem for higher rank locally free sheaves over Calabi-Yau surfaces. To certain tropical Lagrangian multi-sections $\mathbb{L}$ over $B$, which are explicitly constructed by prescribing local models around the ramification points, we construct locally free sheaves $\mathcal{E}_0(\mathbb{L},{\bf{k}}_s)$ over the singular projective scheme $X_0(B,\mathscr{P},s)$ associated to $B$ equipped with a polyhedral decomposition $\mathscr{P}$ and a gluing data $s$. We then find combinatorial conditions on such an $\mathbb{L}$ under which the sheaf $\mathcal{E}_0(\mathbb{L},{\bf{k}}_s)$ is simple. This produces explicit examples of smoothable pairs $(X_0(B,\mathscr{P},s),\mathcal{E}_0(\mathbb{L},{\bf{k}}_s))$ in dimension 2.

math.AG↗

arXiv2021

Bargmann-Fock sheaves on Kähler manifolds

Fedosov used flat sections of the Weyl bundle on a symplectic manifold to construct a star product $\star$ which gives rise to a deformation quantization. By extending Fedosov's method, we give an explicit, analytic construction of a sheaf of Bargmann-Fock modules over the Weyl bundle of a Kähler manifold $X$ equipped with a compatible Fedosov abelian connection, and show that the sheaf of flat sections forms a module sheaf over the sheaf of deformation quantization algebras defined $(C^\infty_X[[\hbar]], \star)$. This sheaf can be viewed as the $\hbar$-expansion of $L^{\otimes k}$ as $k \to \infty$, where $L$ is a prequantum line bundle on $X$ and $\hbar = 1/k$.

math.DG↗

arXiv2021

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

Given a degenerate Calabi-Yau variety $X$ equipped with local deformation data, we construct an almost differential graded Batalin-Vilkovisky (dgBV) algebra $PV^{*,*}(X)$, producing a singular version of the extended Kodaira-Spencer differential graded Lie algebra (dgLa) in the Calabi-Yau setting. Assuming Hodge-to-de Rham degeneracy and a local condition that guarantees freeness of the Hodge bundle, we prove a Bogomolov-Tian-Todorov--type unobstructedness theorem for smoothing of singular Calabi-Yau varieties. In particular, this provides a unified proof for the existence of smoothing of both $d$-semistable log smooth Calabi-Yau varieties (as studied by Friedman and Kawamata-Namikawa and maximally degenerate Calabi-Yau varieties (as studied by Kontsevich-Soibelman and Gross-Siebert). We also demonstrate how our construction yields a logarithmic Frobenius manifold structure on a formal neighborhood of $X$ in the extended moduli space by applying the technique of Barannikov-Kontsevich.

math.AG↗

arXiv2021

Quantization of Kähler manifolds

This is a survey on our recent works which reveal new relationships among deformation quantization, geometric quantization, Berezin-Toeplitz quantization and BV quantization on Kähler manifolds.

math.DG↗

arXiv2020

Smoothing pairs over degenerate Calabi-Yau varieties

We apply the techniques developed in our previous work with Leung to study smoothings of a pair $(X,\mathfrak{C}^*)$, where $\mathfrak{C}^*$ is a bounded perfect complex of locally free sheaves over a degenerate Calabi-Yau variety $X$. In particular, if $X$ is a projective Calabi-Yau variety admitting the structure of a toroidal crossing space and with the higher tangent sheaf $\mathcal{T}^1_X$ globally generated, and $\mathfrak{F}$ is a locally free sheaf over $X$, then we prove, using the recent results of Felten-Filip-Ruddat, that the pair $(X,\mathfrak{F})$ is formally smoothable when $\text{Ext}^2(\mathfrak{F},\mathfrak{F})_0 = 0$ and $H^2(X,\mathcal{O}_X) = 0$.

math.AG↗

arXiv2020

A Note on Disk Counting in Toric Orbifolds

We compute orbi-disk invariants of compact Gorenstein semi-Fano toric orbifolds by extending the method used for toric Calabi-Yau orbifolds. As a consequence the orbi-disc potential is analytic over complex numbers.

math.SG↗

arXiv2020

Tropical counting from asymptotic analysis on Maurer-Cartan equations

Let $X = X_Σ$ be a toric surface and $(\check{X}, W)$ be its Landau-Ginzburg (LG) mirror where $W$ is the Hori-Vafa potential. We apply asymptotic analysis to study the extended deformation theory of the LG model $(\check{X}, W)$, and prove that semi-classical limits of Fourier modes of a specific class of Maurer-Cartan solutions naturally give rise to tropical disks in $X$ with Maslov index 0 or 2, the latter of which produces a universal unfolding of $W$. For $X = \mathbb{P}^2$, our construction reproduces Gross' perturbed potential $W_n$ which was proven to be the universal unfolding of $W$ written in canonical coordinates. We also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of $W_n$ across walls of the scattering diagram formed by the Maslov index 0 tropical disks originally observed by Gross (in the case of $X = \mathbb{P}^2$).

math.AG↗

arXiv2020

Geometric quantization via SYZ transforms

The so-called quantization problem in geometric quantization is asking whether the space of wave functions is independent of the choice of polarization. In this paper, we apply SYZ transforms to solve the quantization problem in two cases: (1) semi-flat Lagrangian torus fibrations over complete compact integral affine manifolds, and (2) projective toric manifolds. More precisely, we prove that the space of wave functions associated to the real polarization is canonically isomorphic to that associated to a complex polarization via SYZ transforms in both cases.

math.SG↗