Related papers: The Open Crepant Transformation Conjecture for Tor…
We announce a result on quantum McKay correspondence for disc invariants of outer legs in toric Calabi-Yau 3-orbifolds, and illustrate our method in a special example $[\mathbb C^3 /\mathbb Z_5 (1, 1, 3)]$.
Gross-Joyce-Tanaka arXiv:2005.05637 proposed a wall-crossing conjecture for Calabi-Yau fourfolds. Assuming it, we prove the conjecture of Cao-Kool arXiv:1712.07347 for 0-dimensional sheaf-counting invariants on projective Calabi-Yau…
We prove a version of homological mirror symmetry statement for toric Calabi-Yau $3$-orbifolds, thus extending arXiv:1604.06448 to the case of orbifolds under the mirror symmetry setting considered in arXiv:1604.07123. The B-model is the…
The SYZ approach to mirror symmetry for log Calabi-Yau manifolds starts from a Lagrangian torus fibration on the complement of an anticanonical divisor. A mirror space is constructed by gluing local charts (moduli spaces of local systems on…
We construct Lagrangian sections of a Lagrangian torus fibration on a 3-dimensional conic bundle, which are SYZ dual to holomorphic line bundles over the mirror toric Calabi-Yau 3-fold. We then demonstrate a ring isomorphism between the…
Let X/G be a 3-dimensional Calabi-Yau orbifold with codimension 2 singularities. The topology of crepant resolutions of X/G is described by the McKay correspondence (Reid, Ito). We study Calabi-Yau 3-folds Y that arise by deforming the…
Let X be the toric variety (P^1)^4 associated with its four-dimensional polytope. Consider a resolution of the singular Fano variety associated with the dual polytope of X. Generically, anti-canonical sections Y of X and anticanonical…
We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the…
Let $Y$ be a smooth projective $3$-fold admitting a K3 fibration $f : Y \rightarrow \mathbb{P}^1$ with $-K_Y = f^*\mathcal{O}(1)$. We show that the pseudoautomorphism group of $Y$ acts with finitely many orbits on the codimension one faces…
In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds…
In this note, we describe a a systematic procedure to find toric crepant resolutions of orbifold vertex, and show that the generating series of certain disc invariants of the orbifold vertex can be suitably identified with the generating…
A Calabi-Yau 4-fold of Camere-Garbagnati-Mongardi is a crepant resolution of the quotient of a hyperk\"ahler 4-fold by an antisymplectic involution. In this paper, we compare two different types of holomorphic torsion invariants; one is the…
For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of…
We prove that the open Gromov-Witten invariants on K3 surfaces satisfy the Kontsevich-Soibelman wall-crossing formula. One one hand, this gives a geometric interpretation of the slab functions in Gross-Siebert program. On the other hands,…
We propose a conjecture that relates some local Gromov-Witten invariants of some crepant resolutions of Calabi-Yau 3-folds with isolated singularities with some Donaldson-Thomas type invariants of the moduli spaces of representations of…
Futaki invariants of the classical moduli space of 4d N=1 supersymmetric gauge theories determine whether they have a conformal fixed point in the IR. We systematically compute the Futaki invariants for a large family of 4d N=1…
We introduce the notion of Wall-Crossing Structure and discuss it in several examples including complex integrable systems, Donaldson-Thomas invariants and Mirror Symmetry. For a big class of non-compact Calabi-Yau 3-folds we construct…
The derived categories of toric varieties admit semi-orthogonal decompositions coming from wall-crossing in GIT. We prove that these decompositions satisfy a Jordan-Holder property: the subcategories that appear, and their multiplicities,…
In this expository paper, we discuss how Fourier-Mukai-type transformations, which we call SYZ mirror transformations, can be applied to provide a geometric understanding of the mirror symmetry phenomena for semi-flat Calabi-Yau manifolds…
In this note, we study the SYZ mirror construction for a toric Calabi-Yau manifold using instanton corrections coming from Woodward's quasimap Floer theory instead of Fukaya-Oh-Ohta-Ono's Lagrangian Floer theory. We show that the resulting…