arXiv subjects

Naichung Conan Leung

Publications and source records attributed to Naichung Conan Leung.

At least 19 recordsLinked to original sources

arXiv2026

A-D-E diagrams, Hodge--Tate hyperplane sections and semisimple quantum cohomology

It is known that the semisimplicity of quantum cohomology implies the vanishing of off-diagonal Hodge numbers (Hodge--Tateness). We investigate which hyperplane sections of homogeneous varieties possess either of the two properties. We provide a new efficient criterion for non-semisimplicity of the small quantum cohomology ring of Fano manifolds that depends only on the Fano index and Betti numbers. We construct a bijection between Dynkin diagrams of types A, D or E, and complex Grassmannians with Hodge-Tate smooth hyperplane sections. By applying our criteria and using monodromy action, we completely characterize the semisimplicity of the small quantum cohomology of smooth hyperplane sections in the case of complex Grassmannians, and verify a conjecture of Benedetti and Perrin in the case of (co)adjoint Grassmannians.

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

Mirror symmetry in 3d in 3d mirror symmetry

Given a compact CY3 $Y$, its A-side (resp. B-side) universal intermediate Jacobian $X$ (resp. $X^{!}$) admits a natural hyperkahler structure. Both $X$ and $X^{!}$ determines 3d Rozansky-Witten theories, in both A-model and B-model. We describe some surprising 3d mirror symmetry phenomena between $X$ and $X^{!}$. This includes (i) a 3d SYZ construction of $X^{!}$ from $X$ via moduli of certain 3d A-branes in $X$ constructed from $D^{b}\left( Y,Ω\right) $ with varying stability conditions given by $\varpi $; (ii) The 3d B-brane on $X^{!}$, constructed from varying $D^{b}\left( Y,Ω\right) $ with fixed $\varpi $, is realized as the 3d SYZ transformation of a cotangent fiber 3d A-brane in $X$. Under mirror symmetry between $Y$ and $Y^{\vee }$, the roles for $X$ and $% X^{!}$ got interchanged. We also construct 3d A- and B-branes in $X$ and $X^{!}$ via DT theory and discrete symmetries on $Y$ and $Y^{\vee }$.

hep-th↗

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

Quantization commutes with reduction for coisotropic A-branes

On a Hamiltonian $G$-manifold $X$, we define the notion of $G$-invariance of coisotropic A-branes $B$. Under neat assumptions, we give a Marsden-Weinstein-Meyer type construction of a coisotropic A-brane $B_{\operatorname{red}}$ on $X // G$ from $B$, recovering the usual construction when $B$ is Lagrangian. For a canonical coisotropic A-brane $B_{\operatorname{cc}}$ on a holomorphic Hamiltonian $G_\mathbb{C}$-manifold $X$, there is a fibration of $(B_{\operatorname{cc}})_{\operatorname{red}}$ over $X // G_\mathbb{C}$. We also show that `intersections of A-branes commute with reduction'. When $X = T^*M$ for $M$ being compact Kähler with a Hamiltonian $G$-action, Guillemin-Sternberg `quantization commutes with reduction' theorem can be interpreted as $\operatorname{Hom}_{X // G}(B_{\operatorname{red}}, (B_{\operatorname{cc}})_{\operatorname{red}}) \cong \operatorname{Hom}_X(B, B_{\operatorname{cc}})^G$ with $B = M$.

math.SG↗

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

Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity

Given a smooth holomorphic symplectic variety $X$ with a Hamiltonian $G$-action, $G$-invariant Lagrangians $C's$ induce Lagrangians in the symplectic quotient $X// G$. Given clean intersections $B=C_1\cap C_2$ whose conormal sequence splits, we show that $$C_1/G\times_{X// G} C_2/G\cong T^{\vee}[-1](B/G).$$ When $det(N_{B/C_2})$ is torsion, we have $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\cong H^{\bullet}_G(B, det(N_{B/C_2})_δ)$ provided that the Hodge-to-de Rham degeneracy holds. Furthermore, we have a generalized version of Kirwan surjectivity $Ext^{\bullet}_{X// G}(\mathcal{O}_{C_1/G}, \mathcal{O}_{C_2/G})\twoheadrightarrow Ext^{\bullet}_{X^{ss}// G}(\mathcal{O}_{C_1^{ss}/G}, \mathcal{O}_{C_2^{ss}/G})$ if $B$ is proper. When $C_1=C_2$, this is the Kirwan surjectivity, which is now interpreted as the symmetry commutes with reduction problem in 3d B-model. We also obtain similar results for $K_{C_1/G}^{1/2}$ and $K_{C_2/G}^{1/2}$.

math.AG↗

arXiv2026

Grassmannian perspectives of classical Lie groups and Cartan involutions

Classical noncompact reductive Lie group $G$ admits a compactification $\overline{G}$ as a Riemannian symmetric space by He. First, we provide a unified construction of these compactifications via Grassmannian geometry and realize the group structures in terms of the geometry of configurations of linear subspaces. Second, we show that the Cartan involution $ρ$ on $G$ extends uniquely to an isometric involution $\barρ$ on $\overline{G}$ and $\overline{G}^{\barρ} = G^ρ = K$, the maximal compact subgroup of $G$. Third, we show that $η(g) = ρ(g)^{-1}$ extends uniquely to an isometric involution $\barη$ on $\overline{G}$ and $\overline{G}^{\barη} = G_c/K$, the compact symmetric space dual to $(G^η)_0 = G/K$. This provides a natural generalization of the classical Borel embeddings $G/K \hookrightarrow G_c/K$. Furthermore, $K$ and $G_c/K$ form a complementary pair of reflective submanifolds in $\overline{G}$.

math.DG↗

arXiv2025

Intersections of twisted cotangent bundles and symplectic duality

We observe that numerous symplectic resolutions can be expressed as intersections of twisted cotangent bundles. Additionally, their dual symplectic resolutions can be derived from intersections of dual twisted cotangent bundles. We determine the collection of fixed points for certain intersections that are Poisson slices, extending the computations of fixed points for parabolic Slodowy varieties.

math.RT↗

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↗

arXiv2024

DK Conjecture for Some $K$-inequivalences from Grassmannians

The DK conjecture of Bondal-Orlov and Kawamata states that there should be an embedding of bounded derived categories for any $K$-inequivalence, which is proved to be true for the toroidal case. In this paper, we construct examples of non-toroidal $K$-inequivalences from Grassmannians inspired by Kuznetsov, Kanemitsu, Ueda, and Morimura, and we show that these $K$-inequivalences satisfy the DK conjecture.

math.AG↗

arXiv2024

Symmetry in Deformation quantization and Geometric quantization

In this paper, we explore the quantization of Kähler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form $f = f_0 - \frac{\hbar}{4π}(Δf_0 + c)$ for a certain smooth (non-formal) function $f_0$. If $f_0$ is real-valued then $f_0$ corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian $G$-symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.

math.DG↗

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↗

arXiv2024

Limit of geometric quantizations on Kähler manifolds with T-symmetry

A compact Kähler manifold $\left( M,ω,J\right) $ with $T$-symmetry admits a natural mixed polarization $\mathcal{P}_{\mathrm{mix}}$ whose real directions come from the $T$-action. In \cite{LW1}, we constructed a one-parameter family of Kähler structures $\left( ω,J_{t}\right) $'s with the same underlying Kä hler form $ω$ and $J_{0}=J$, such that (i) there is a $T$-equivariant biholomorphism between $\left( M,J_{0}\right) $ and $\left( M,J_{t}\right) $ and (ii) Kähler polarizations $\mathcal{P} _{t}$'s corresponding to $J_{t}$'s converge to $\mathcal{P}_{\mathrm{mix}}$ as $t$ goes to infinity. In this paper, we study the quantum analog of above results. Assume $L$ is a pre-quantum line bundle on $\left( M,ω\right) $. Let $\mathcal{H}_{t}$ and $ \mathcal{H}_{\mathrm{mix}}$ be quantum spaces defined using polarizations $\mathcal{P}_{t}$ and $\mathcal{P}_{\mathrm{mix}}$ respectively. In particular, $\mathcal{H}_{t}=H_{\bar{\partial}_{t}}^{0}\left( M,L\right) $. They are both representations of $T$. We show that (i) there is a $T$-equivariant isomorphism between $\mathcal{H}_{0}$ and $\mathcal{H}_{\mathrm{mix}}$ and (ii) for regular $T$-weight $λ$, corresponding $λ$-weight spaces $ \mathcal{H}_{t,λ}$'s converge to $\mathcal{H}_{\mathrm{mix},λ}$ as $t$ goes to infinity.

math.SG↗

arXiv2024

Geodesic rays in the space of Kähler metrics with $T$-symmetry

Let $(M, ω, J)$ be a Kähler manifold, equipped with an effective Hamiltonian torus action $ρ: T \rightarrow \mathrm{Diff}(M, ω, J)$ by isometries with moment map $μ: M \rightarrow \mathfrak{t}^{*}$. We first construct a singular mixed polarization $\mathcal{P}_{\mathrm{mix}}$ on $M$. Second, we construct a one-parameter family of complex structures $J_{t}$ on $M$ which are compatible with $ω$. Furthermore, the path of corresponding Kähler metrics $g_{t}$ is a complete geodesic ray in the space of Kähler metrics of $M$, when $M$ is compact. Finally, we show that the corresponding family of Kähler polarizations $\mathcal{P}_{t}$ associated to $J_{t}$ converges to $\mathcal{P}_{\mathrm{mix}}$ as $t \rightarrow \infty$.

math.SG↗

arXiv2023

On Derived Categories of Generalized Grassmannian Flips

In this paper, we construct and classify a new family of flips, called generalized Grassmannian flips, by generalizing the construction of standard flips for $\mathbb{P}^m\times \mathbb{P}^n$ to any generalized Grassmannian $G/P$, where $P$ is a maximal parabolic subgroup of a complex semi-simple algebraic group. In addition, we show that a 9-fold generalized Grassmannian flip for $Sp(6, \mathbb{C})$ satisfies the DK flip conjecture by Bondal-Orlov and Kawamata via mutation techniques by Kuznetsov and Thomas' chess game method.

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↗

arXiv2023

Geometric quantizations of mixed polarizations on Kähler manifolds with T-symmetry

Let $M$ be a compact Kähler manifold equipped with a pre-quantum line bundle $L$. In [9], using $T$-symmetry, we constructed a polarization $\mathcal{P}_{\mathrm{mix}}$ on $M$, which generalizes real polarizations on toric manifolds. In this paper, we obtain the following results for the quantum space $\mathcal{H}_{\mathrm{mix}}$ associated to $\mathcal{P}_{\mathrm{mix}}$. First, $\mathcal{H}_{\mathrm{mix}}$ consists of distributional sections of $L$ with supports inside $μ^{-1}(\mathfrak{t}^{*}_{\mathbb{Z}})$. This gives $\mathcal{H}_{\mathrm{mix}}=\bigoplus_{λ\in \mathfrak{t}^{*}_{\mathbb{Z}} } \mathcal{H}_{\mathrm{mix}, λ}$. Second, the above decomposition of $\mathcal{H}_{\mathrm{mix}}$ coincides with the weight decomposition for the $T$-symmetry. Third, an isomorphism $\mathcal{H}_{\mathrm{mix}, λ} \cong H^{0}( M//_λT, L//_λT)$, for regular $λ$. Namely, geometric quantization commutes with symplectic reduction.

math.SG↗