Survey on decorated marked surfaces for Calabi-Yau categories.
problem Understanding Calabi-Yau categories and related structures.
method Introducing decorations on marked surfaces to study various categories.
result Exploration of Calabi-Yau-2 and 3 categories, braid groups, quadratic differentials, and stability conditions.
New model for Calabi-Yau-X categories using decorated marked surfaces.
problem Constructing models for Calabi-Yau-X categories. method Using graded decorated marked surfaces and string models.
result Isomorphism between braid twist group and spherical twist group.
This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.
problem Mathematical definition of the algebra of BPS states.
method Construction of cohomological Hall algebras for 3-Calabi-Yau categories.
result Construction of cohomological Hall algebras and proof of Joyce's conjecture.
New map connects stability conditions to Teichmüller space.
problem Stability conditions and Teichmüller space relationship.
method Using harmonic maps and 3-Calabi-Yau categories.
result Natural map between stability conditions and Teichmüller space.
Compactifies stability space for A2 category, introducing q-deformed rational numbers.
problem Stability conditions in triangulated categories and their compactifications.
method Embedding into an infinite-dimensional projective space, using B3 braid group action. result Two orbits in the boundary correspond to q-deformed rational numbers. Kontsevich and Soibelman introduced a notion of orientation data on Calabi-Yau category. It can be viewed as a consistent choice of spin structure on moduli space of objects in the given category. The orientation data plays an important role in Donaldson-Thomas theory. Let X be a projective, simply connected and torsio…
We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi--Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also …
We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi-Yau triangulated category. The motivation/application is that the…
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
Paper calculates Donaldson-Thomas invariants for a specific category.
problem Relating quadratic differentials to stability conditions and invariants.
method Uses string and band techniques from representation theory.
result Agrees with physics predictions for degenerate ring domains.
The abstract introduces a new A∞ duality via LSFT algebra.
problem Legendrian knot duality and its A∞ extension. method Using Ng's LSFT algebra, the abstract upgrades duality to a quasi-isomorphism of A∞ bimodules over Aug+. result Explicit construction of homotopy inverse for the A∞ Sabloff map. We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair (X,W), where X is a non-compact Calabi-Yau manifold and W has compact critical set. When X is a Stein manifold (but not restricted to b…
This is a survey on two closely related subjects. First, we review the study of topological structure of `finite type' components of spaces of Bridgeland's stability conditions on triangulated categories. The key is to understand Happel-Reiten-Smalo tilting as tiling of cells. Second, we review topological realizations…
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
Study moduli space of quadratic differentials with new geometric insights.
problem Understanding the structure of moduli spaces of quadratic differentials.
method Using decorated marked surfaces, Abel-Jacobi map, and 3-Calabi-Yau categories.
result Fundamental group of moduli space equals kernel of Abel-Jacobi map.
Mutation graph of support τ-tilting modules over skew-gentle algebras is connected.
problem Understanding the structure of support τ-tilting modules over skew-gentle algebras.
method Introducing mutation of maximal rigid objects and using exchange triangles to define mutations of support τ-tilting modules.
result The mutation graph of support τ-tilting modules over a skew-gentle algebra is connected.
Study special Lagrangian sections in Calabi-Yau threefolds, showing stability conditions imply isomorphism to special Lagrangians.
problem Understanding stability conditions on Fukaya-Seidel categories of Calabi-Yau threefolds.
method Analyzing sections of special Lagrangian fibrations, constructing Bridgeland stability conditions, and relating to deformed Hermitian Yang-Mills connections.
result Semistability of L[2] implies isomorphism to special Lagrangian sections. A new method analyzes topological B-model on a torus using doubled geometry.
problem Reproduce derived category of coherent sheaves on a torus.
method Double field theory and T-duality in doubled geometry framework.
result Correctly computes BRST cohomology and derived Hom-spaces of line bundles.
New dg-algebras generalize Brauer graph algebras, with applications to stability conditions and quadratic differentials.
problem Generalizing Brauer graph algebras to new dg-algebras.
method Derived categories, mixed-angulations of surfaces, stability conditions, and quadratic differentials.
result Spaces of stability conditions on derived categories of these algebras are described in terms of spaces of quadratic differentials.
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point…
We introduce a homology theory whose Euler characteristics counts ASD bundles over four dimensional co-associative submanifolds in (almost) G_2 manifolds. As a TQFT, in relative situations, we have the Fukaya-Floer category of Lagrangians intersection in the moduli space of special Lagrangian submanifolds in CY threefo…
In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.
The book develops a new bordism-theoretic approach to understanding orientations of moduli spaces.
problem Defining and proving orientations for moduli spaces of geometric objects.
method Develops bordism categories and uses them to encode and solve orientation problems.
result Proves orientability and constructs canonical orientations for various moduli spaces.
We link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
Open 2D TFTs extend to closed theories with circle value as Hochschild homology.
problem Extending open 2D TFTs to closed theories.
method Using symmetric monoidal ∞-categories and Hochschild homology.
result Open 2D TFTs admit initial open-closed extensions.
We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle-Zhang's result for the puncture free case. As an application, we show that the intersection of the shi…
Paper corrects mistakes in moduli space orientability for Spin(7)-instantons and coherent sheaves.
problem Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds.
method Uses new theory from Joyce-Upmeier arXiv:2503.20456 to prove corrected versions of main results.
result Corrected versions of main results hold with an extra assumption on H3(X,Z). Compactifies stability conditions on triangulated categories, inspired by Teichmüller theory.
problem Moduli space of stability conditions on triangulated categories.
method Inspired by Thurston compactification of Teichmüller space, constructs maps to infinite projective space.
result Injective maps with compact closure, identifies categorical analogs of intersection functionals.
Defines special Joyce structures for ASK manifolds encoding real HK structures.
problem Defining Joyce structures for ASK manifolds.
method Introducing special Joyce structures and showing they encode real HK structures.
result Special Joyce structures encode real hyperkähler structures on ASK manifolds.
We study the 3-Calabi-Yau categories D arising from quivers with potential associated to a decorated marked surface S△ introduced by the first author. We prove two conjectures in the prequel, that under a bijection between certain objects in D and certain arcs in $\mathb…
Study spherical twists on K3 surfaces, compute their centers.
problem Understanding autoequivalence groups of K3 surfaces.
method Introduced spherical twists, studied their intersection numbers, and classified subgroups.
result Computed the center of autoequivalence groups of K3 surfaces.
Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy Kähler moduli space. A few basic examples are studied. In particular, we identify …
Survey on generalized holomorphic Cartan geometries.
problem Classifying holomorphic Cartan geometries on compact Calabi-Yau manifolds.
method Introducing branched holomorphic Cartan geometries and classifying them.
result Classification of branched holomorphic Cartan geometries on compact Calabi-Yau manifolds.
A formula connects two algebraic structures derived from a category.
problem Connecting two algebraic structures derived from a category.
method Using differential graded modular functors and Calabi-Yau structures.
result The action of a specific mapping class group element transforms one algebraic structure into another.
A new method studies symplectic configurations in rational 4-manifolds using computer-aided techniques.
problem Understanding symplectic configurations in rational 4-manifolds.
method Computer-aided approach combining Cremona transformations and pseudoholomorphic curves.
result Nonexistence of Fano planes in the symplectic category.
Constructs cohomology decompositions for symmetric stacks.
problem Cohomology of symmetric stacks.
method Constructs decompositions of cohomology, Borel--Moore homology, and vanishing cycle cohomology.
result Defines BPS cohomology and proves its equivalence to intersection cohomology for smooth stacks.
Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.
problem Characterizing Vaisman manifolds with vanishing first Chern class.
method Categorization into three types based on Bott-Chern class sign, showing canonical metrics, quasi-regularity, stability, and automorphism group behavior.
result Vaisman manifolds with non-positive Bott-Chern class admit canonical metrics and are stable under deformations.
Proves stability condition for Lagrangian sections in toric weak Fano manifolds.
problem Stability of Lagrangian sections in Calabi-Yau fibrations.
method SYZ transform, toric gamma theorem, Nakai-Moishezon criterion.
result Hamiltonian isotopy of Lagrangian sections under stability condition.
Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.
problem Creating complete Calabi-Yau metrics on non-compact manifolds.
method Extending Székelyhidi's work, constructing metrics with varying complex structures and possible singularities.
result Produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Let X be a Calabi-Yau 3-fold, T=D^b(coh(X)) the derived category of coherent sheaves on X, and Stab(T) the complex manifold of Bridgeland stability conditions Z on T. It is conjectured that one can define rational numbers J^a(Z) for Z in Stab(T) and a in the numerical Grothendieck group K(T) generalizing Donaldson-Thom…
We discuss the deformation theory of special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. This category allows for the simultaneous presence of conical singularities and of non-compact, asymptotic…
We are interested in the 3-Calabi-Yau categories D arising from quivers with potential associated to a triangulated marked surface S (without punctures). We prove that the spherical twist group ST of D is isomorphic to a subgroup (generated by braid twists) of the mapping class group …
New examples of Calabi-Yau 3-folds with unique properties.
problem Finding new Calabi-Yau 3-folds with specific properties.
method Constructing complete Calabi-Yau metrics on smoothings of 3-dimensional Calabi-Yau cones with orbifold singularities.
result Examples of Calabi-Yau 3-folds with maximal volume growth and orbifold singularities.
Uniform estimates for Calabi-Yau degenerations proved.
problem Calabi-Yau degenerations of polarised algebraic manifolds.
method Uniform Skoda and L∞-estimates for Kähler potentials. result Uniform Skoda type estimate and L∞-estimate for Calabi-Yau Kähler potentials proved. Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.
problem Understanding non-Gorenstein involutions on Calabi-Yau threefolds.
method Classification of Calabi-Yau threefolds with specific properties.
result Classification of Calabi-Yau threefolds with Picard rank one and non-Gorenstein involutions.
Researchers use gluing method to describe collapsing Calabi-Yau metrics on K3 fibred 3-folds.
problem Describing collapsing Calabi-Yau metrics on K3 fibred 3-folds.
method Gluing method
result Refined description of collapsing Calabi-Yau metrics.
Polyhomogeneous expansions for Calabi-Yau metrics near singularities.
problem Analyzing metrics near conical singularities of Calabi-Yau conifolds.
method Weighted Melrose-type blow-ups, gluing, and solving complex Monge-Ampère equations.
result Polyhomogeneous expansions of smooth Calabi-Yau metrics on resolutions and smoothings.
New Calabi-Yau metrics constructed with detailed geometry at infinity.
problem Constructing complete Calabi-Yau metrics with specific properties.
method Weighted blow-up and Hölder spaces for Laplacian analysis.
result Examples of Calabi-Yau metrics with conical singularities and non-uniqueness of tangent cones.