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Siu-Cheong Lau

Publications and source records attributed to Siu-Cheong Lau.

At least 19 recordsLinked to original sources
arXiv2026

Mirror functor for deformed preprojective algebras

We study localized homological mirror symmetry associated to an immersed Lagrangian brane $\mathbb{L}$, possibly equipped with a higher rank flat bundle, of a symplectic manifold $X$. Under a certain finiteness assumption on the Floer theory of $\mathbb{L}$, we deduce a quasi-equivalence $\mathcal{D}\mathrm{Fuk}_\mathbb{L}(X) \cong \mathcal{D}_{\mathrm{fd}}(\tilde{\mathcal A}_\mathbb{L})$ using Koszul duality, where $\tilde{\mathcal{A}}_{\mathbb{L}}$ is the dual differential graded quiver algebra called the extended localized mirror. We apply this to obtain some HMS results for plumbings of cotangent bundles of spheres, and to reproduce known results of split-generation of compact objects. In the second part of the paper, we consider bulk deformation cycles of $X$ that have non-trivial intersections with $\mathbb{L}$. This gives rise to noncommutative deformations of the mirror. When applied to (framed) plumbings, we obtain mirror functors to deformed preprojective algebras (or Nakajima quiver varieties at a general complex moment-map level). For the ADHM and affine $ADE$-type immersions, our construction produces mirror functors to the noncommutative spaces studied by Kapustin-Kuznetsov-Orlov, Baranovsky-Ginzburg-Kuznetsov and Kawamata.

math.SG↗
arXiv2026

Mirror construction of Hecke correspondence between Nakajima quiver varieties

Nakajima constructed geometric representations of a deformed Kac-Moody Lie algebra using Hecke correspondences between quiver varieties. In this paper, we show that Hecke correspondences, which are holomorphic Lagrangians in products of Nakajima quiver varieties, can be obtained by applying the localized mirror construction to the morphism spaces between families of framed Lagrangian branes supported on the core of a plumbing of two-spheres. Moreover, for a non-ADE quiver, we show that the localized mirror functor is fully-faithful.

math.SG↗
arXiv2025

Equivariant Partially Wrapped Fukaya Categories on Liouville Sectors

We develop an equivariant Lagrangian Floer theory for Liouville sectors that have symmetry of a Lie group $G$. Moreover, for Liouville manifolds with $G$-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian Floer cohomology upstairs and Lagrangian Floer cohomology of its quotient. Furthermore, we study the symplectic quotient in the presence of nodal type singularities and prove that the equivariant correspondence gives an isomorphism on cohomologies which was conjectured by Lekili-Segal.

math.SG↗
arXiv2025

Mirror Symmetry for Quiver Algebroid Stacks

In this paper, we provide a new construction of quiver algebroid stacks and the associated mirror functors for symplectic manifolds. First, we formulate the concept of a quiver stack, which is a geometric structure formed by gluing multiple quiver algebras together. Next, we develop a representation theory of $A_\infty$ categories by quiver stacks. The main idea is to extend the $A_\infty$ category over a quiver stack of a collection of nc-deformed objects. The extension involves non-trivial gerbe terms. It gives an application of symplectic geometry that bridges the study of sheaves and representation theory through mirror symmetry. We provide a general framework for constructing mirror quiver stacks. In particular, we develop a novel method of gluing Lagrangians which are disjoint from each other by using quasi-isomorphisms with a `global middle agent', which is a Lagrangian immersion that produces a mirror quiver. The method relies fundamentally on the use of quiver stacks. We carry out this construction for compact immersed Lagrangians in a punctured elliptic curve, which results in a mirror nc local projective plane.

math.AG↗
arXiv2025

Equivariant Lagrangian correspondence and a conjecture of Teleman

In this paper, we study the Floer theory of equivariant Lagrangian correspondences and apply it to derive precise relations between the disc potential of an invariant Lagrangian submanifold and that of its quotient, thereby addressing a conjecture of Teleman. Furthermore, we proved that their (equivariant) Lagrangian Floer cohomologies are isomorphic. In particular, the functor by equivariant Lagrangian correspondence induces a quasi-isomorphism between the equivariant derived Fukaya category at a regular moment-map level and the derived Fukaya category of the corresponding symplectic quotient. A key step is to extend Fukaya's construction of an $A_\infty$ tri-module for Lagrangian correspondences to Borel spaces. We demonstrate that the equivariant obstruction of a Lagrangian correspondence plays an essential role, which leads to quantum corrections in the disc potentials of the quotients. We computed the disc potential of the Lagrangian correspondence in the toric setup and relate it with mirror maps for compact semi-Fano toric manifolds.

math.SG↗
arXiv2025

$T$-equivariant disc potential and SYZ mirror construction

We develop a $G$-equivariant Lagrangian Floer theory and obtain a curved $A_\infty$ algebra, and in particular a $G$-equivariant disc potential. We construct a Morse model, which counts pearly trees in the Borel construction $L_G$. When applied to a smooth moment map fiber of a semi-Fano toric manifold, our construction recovers the $T$-equivariant toric Landau-Ginzburg mirror of Givental. We also study the $\bS^1$-equivariant Floer theory of a typical singular SYZ fiber (i.e. a pinched torus) and compute its $\bS^1$-equivariant disc potential via the gluing technique developed in \cite{CHL18,HKL}.

math.SG↗
arXiv2025

A logifold structure on measure space

In this paper,we develop a local-to-global and measure-theoretical approach to understand datasets. The idea is to take network models with restricted domains as local charts of datasets. We develop the mathematical foundations for these structures, and show in experiments how it can be used to find fuzzy domains and to improve accuracy in data classification problems.

math.DG↗
arXiv2024

Mirror Construction for Nakajima Quiver Varieties

In this paper, we construct the ADHM quiver representations and the corresponding sheaves as the mirror objects of formal deformations of the framed immersed Lagrangian sphere decorated with flat bundles. More generally, we construct Nakajima quiver varieties as localized mirrors of framed nodal unions of Lagrangian spheres in dimension two. This produces a mirror functor from the Fukaya category of a framed plumbing of surfaces to the dg category of complexes of bundles over the corresponding Nakajima quiver varieties. For affine ADE quivers in specific multiplicities, the corresponding (unframed) Lagrangian immersions are homological tori, whose moduli of stable deformations are asymptotically locally Euclidean (ALE) spaces. We show that framed stable Lagrangian branes are transformed into monadic complexes of framed torsion-free sheaves over the ALE spaces. A main ingredient is the notion of framed Lagrangian immersions and their Maurer-Cartan deformations. Moreover, using the formalism of quiver algebroid stacks, we find isomorphisms between the moduli of stable Lagrangian immersions and that of special Lagrangian fibers of an SYZ fibration in the affine $A_n$ cases.

math.AG↗
arXiv2024

Logifold: A Geometrical Foundation of Ensemble Machine Learning

We present a local-to-global and measure-theoretical approach to understanding datasets. The core idea is to formulate a logifold structure and to interpret network models with restricted domains as local charts of datasets. In particular, this provides a mathematical foundation for ensemble machine learning. Our experiments demonstrate that logifolds can be implemented to identify fuzzy domains and improve accuracy compared to taking average of model outputs. Additionally, we provide a theoretical example of a logifold, highlighting the importance of restricting to domains of classifiers in an ensemble.

cs.LG↗
arXiv2024

Big Quantum cohomology of orbifold spheres

We construct a Kodaira-Spencer map from the big quantum cohomology of a sphere with three orbifold points to the Jacobian ring of the mirror Landau-Ginzburg potential function. This is constructed via the Lagrangian Floer theory of the Seidel Lagrangian and we show that Kodaira-Spencer map is a ring isomorphism.

math.SG↗
arXiv2024

SYZ mirror symmetry for del Pezzo surfaces and affine structures

We prove that the Landau--Ginzburg superpotential of del Pezzo surfaces can be realized as a limit of their hyperKähler rotation toward the large complex structure limit point. As a corollary, we compute the limit of the complex affine structure of the special Lagrangian fibrations constructed by Collins--Jacob--Lin in $\mathbf{P}^1\times \mathbf{P}^1$ arXiv:1904.08363 and compare it with the integral affine structures used in the work of Carl--Pumperla--Siebert arXiv:2205.07753. We also construct the Floer-theoretical Landau--Ginzburg mirrors of smoothing of $A_n$-singularities and monotone del Pezzo surfaces, by using the gluing method of Cho--Hong--Lau arXiv:1810.02045 and Hong--Kim--Lau arXiv:1805.11738. They agree with the result of hyperKähler rotation.

math.AG↗
arXiv2024

Disk potential functions for polygon spaces

We derive a Floer theoretical SYZ mirror for an equilateral and generic polygon space. The disk potential function of the monotone torus fiber of the caterpillar bending system is calculated by computing non-trivial open Gromov--Witten invariants from the structural result of the monotone Fukaya category, the topology of fibers of completely integrable systems, and toric degenerations. Then, combining the result with the work of Nohara--Ueda [NU20] and Marsh--Rietsch [MR20], we obtain the disk potential functions of bending systems and produce a mirror cluster variety of type A without frozen variables via Lagrangian Floer theory.

math.SG↗
arXiv2022

Quantum Finite Automata and Quiver Algebras

We find an application in quantum finite automata for the ideas and results of [JL21] and [JL22]. We reformulate quantum finite automata with multiple-time measurements using the algebraic notion of near-ring. This gives a unified understanding towards quantum computing and deep learning. When the near-ring comes from a quiver, we have a nice moduli space of computing machines with metric that can be optimized by gradient descent.

cs.FL↗
arXiv2021

Noncommutative homological mirror functor

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

math.AG↗
arXiv2021

Kähler Geometry of Quiver Varieties and Machine Learning

We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli which are compatible with the GIT quotient construction by the general linear group, and show that their Ricci curvatures give a Kähler metric on the moduli. Moreover, we use toric moment maps to construct activation functions, and prove the universal approximation theorem for the multi-variable activation function constructed from the complex projective space.

math.AG↗
arXiv2020

On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces

Given any smooth cubic curve $E\subseteq \mathbb{P}^2$, we show that the complex affine structure of the special Lagrangian fibration of $\mathbb{P}^2\setminus E$ constructed by Collins--Jacob--Lin arXiv:1904.08363 coincides with the affine structure used in Carl--Pomperla--Siebert for constructing mirror. Moreover, we use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which is shown to agree with the mirror constructed by Carl--Pomperla--Siebert.

math.DG↗
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↗