Ming Zhang, Yang Zhou
In this paper, we prove a K-theoretic wall-crossing formula for $ε$-stable quasimaps for all GIT targets in all genera. It recovers the genus-0 K-theoretic toric mirror theorem by Givental-Tonita and the genus-0 mirror theorem for quantum K-theory with level structure by Ruan-Zhang. The proofs are based on K-theoretic virtual localization on the master space introduced by the second author in arXiv:1911.02745.
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In this paper, we prove a wall-crossing formula for $ε$-stable quasimaps to GIT quotients conjectured by Ciocan-Fontanine and Kim, for all targets in all genera, including the orbifold case. We prove that stability conditions in adjacent chambers give equivalent invariants, provided that both chambers are stable. In the case of genus-zero quasimaps with one marked point, we compute the invariants in the left-most stable chamber in terms of the small $I$-function. Using this we prove that the quasimap $J$-functions are on the Lagrangian cone of the Gromov--Witten theory. The proofs are based on…
Higher-genus quasimap wall-crossing via localization
本文通过局部化方法和扭曲图空间，为任意亏格下不同$ε$的$ε$-稳定拟映射模空间虚拟类关联的穿墙公式提供了新证明，并预计可推广至一般规范线性sigma模型。
For each positive rational number $ε$, we define $K$-theoretic $ε$-stable quasimaps to certain GIT quotients $W\sslash G$. For $ε>1$, this recovers the $K$-theoretic Gromov-Witten theory of $W\sslash G$ introduced in more general context by Givental and Y.-P. Lee. For arbitrary $ε_1$ and $ε_2$ in different stability chambers, these $K$-theoretic quasimap invariants are expected to be related by wall-crossing formulas. We prove wall-crossing formulas for genus zero $K$-theoretic quasimap theory when the target $W\sslash G$ admits a torus action with isolated fixed points and isolated one-dimens…
Wall-crossing for quasimaps to GIT stack bundles
本文定义到 GIT 栈丛的 ε-稳定拟映射，并研究 ε 变化时 ε-拟映射理论的穿墙行为。
Motivic quasimap wall-crossing for Grassmannians
研究格拉斯曼流形的动机拟映射壁穿越，通过定义对称函数环的\(\mathbb{Q}\)-代数自同构给出公式，其\(\varepsilon \to 0\)极限有特定交换，利用相对商概型几何及对称函数理论证明壁穿越公式。
We state a wall-crossing formula for the virtual classes of epsilon-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus g descendant Gromov-Witten potential and the genus g epsilon-quasimap descendant potential is established. For the quintic threefold, our results may be interpreted as giving a rigorous and geometric interpretation of the holomorphic limit of the BCOV B-model partition function of the mirror family.
In previous work (arXiv:1304.7056) we have conjectured wall-crossing formulas for genus zero quasimap invariants of GIT quotients and proved them via localization in many cases. We extend these formulas to higher genus when the target is semi-positive, and prove them for semi-positive toric varieties, in particular for toric local Calabi-Yau targets. The proof also applies to local Calabi-Yau's associated to some non-abelian quotients.
For each positive rational number epsilon, the theory of epsilon-stable quasimaps to certain GIT quotients W//G developed in arXiv:1106.3724[math.AG] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to epsilon --> 0. For epsilon >1 one obtains the usual Gromov-Witten theory of W//G, while the other theories are new. However, they are all expected to contain the same information and in particular the numerical invariants should be related by wall-crossing formulas. In this paper we analyze the genus zero picture and find that the wall-crossing…
We study K-theoretic GLSM invariants with one-dimensional gauge group and introduce elliptic central charges that depend on an elliptic cohomology class called an elliptic brane and a choice of level structure. These central charges have an integral representation related to an interpolation problem for the elliptic brane and satisfy natural q-difference equations. Integral representaions lead to the wall crossing statement for the central charges in two different phases of the GLSM. We explain how the orbifold structure leads to differences between wall crossing of the usual (K-theoretic) and…
K-moduli wall crossing for quasimaps to a projective variety
该研究发展射影簇拟映射的K-模壁跨越理论，构造多 regime 模空间，关联稳定映射、拟映射与紧化空间，用于椭圆曲面紧化间的模插值。
The local/logarithmic correspondence and the degeneration formula for quasimaps
本文通过准映射理论研究局部几何与对数曲线计数的关系，提出新的退化公式并揭示不同相对Gromov-Witten理论表述间的联系。
We propose and prove a mirror theorem for the elliptic quasimap invariants for smooth Calabi-Yau complete intersections in projective spaces. The theorem combined with the wall-crossing formula appeared in paper (arXiv:1308.6377) implies mirror theorems of Zinger and Popa for the elliptic Gromov-Witten invariants for those varieties. This paper and the wall-crossing formula provide a unified framework for the mirror theory of rational and elliptic Gromov-Witten invariants.
Strictly Semistable Quasimaps on the Wall
针对射影空间目标，构建了严格半稳定对象出现的代数栈，并分析其Θ-分层变化，通过非交换局部化发展了ε-稳定拟映射不变量的K-理论墙交叉公式。
Wall-crossing formula and genus-one Virasoro conjecture for Fano complete intersections
该研究在射影空间光滑法诺完全交的ambient状态空间上证明了亏格一Virasoro猜想，推广了拟映射不变量的跨壁公式，为完全交与射影空间的Gromov-Witten理论建立了关键桥梁。
The elliptic quasimap potential function is explicitly calculated for Calabi-Yau complete intersections in projective spaces by Kim and Lho. We extend this result to local Calabi-Yau varieties. Using this as well as the wall crossing formula by Ciocan-Fontanine and Kim, we can calculate the elliptic Gromov-Witten potential function.
Quasimaps to moduli spaces of sheaves
本文发展了曲面层模空间的拟映射理论，证明其模空间正常且带完美障碍理论，并自然同构于三维簇上的层模空间；利用纠缠尾理论建立墙交叉公式，联系Gromov-Witten与Donaldson-Thomas理论，并在del Pezzo曲面上计算了希尔伯特概形的公式。
We extend to orbifolds the quasimap theory of arXiv:0908.4446 and arXiv:1106.3724, as well as the genus zero wall-crossing results from arXiv:1304.7056 and arXiv:1401.7417. As a consequence, we obtain generalizations of orbifold mirror theorems, in particular, of the mirror theorem for toric orbifolds recently proved independently by Coates, Corti, Iritani, and Tseng (arXiv:1310.4163).
We prove an equivalence between the Bryan--Steinberg theory of $π$-stable pairs on $Y = \mathcal{A}_{m-1} \times \mathbb{C}$ and the theory of quasimaps to $X = \mathrm{Hilb}(\mathcal{A}_{m-1})$, in the form of an equality of K-theoretic equivariant vertices. In particular, the combinatorics of both vertices are described explicitly via box counting. Then we apply the equivalence to study the implications for sheaf-counting theories on $Y$ arising from 3d mirror symmetry for quasimaps to $X$, including the Donaldson--Thomas crepant resolution conjecture.
The hybrid model is the Landau-Ginszburg-type theory that is expected, via the Landau-Ginzburg/Calabi-Yau correspondence, to match the Gromov-Witten theory of a complete intersection in weighted projective space. We prove a wall-crossing formula exhibiting the dependence of the genus-zero hybrid model on its stability parameter, generalizing the work of the second author and Ruan for quantum singularity theory and paralleling the work of Ciocan-Fontanine--Kim for quasimaps. This completes the proof of the genus-zero Landau-Ginzburg/Calabi-Yau correspondence for complete intersections of hypers…
Coulomb branch localization, quasimaps, and surface counting in Calabi--Yau fourfolds
本文提出了一种基于超对称局域化的弦论方法，用于研究局部Calabi--Yau四流形中的曲面计数，通过PT1稳定对与ADHM箭图扭曲拟映射间的谱对应，导出了K理论拟映射配分函数的留数公式，并消除了等变计算中的符号歧义。
Quantum Adams operations in quasimap K-theory
本文在准映射量子K-理论中定义量子Adams运算，作为量子Steenrod运算的K-理论类比，验证其与q-差分结构相容，并通过局部化计算识别为p-曲率算子，同时提出并验证K-理论量子Hikita猜想。
Motivated by Witten's work (arXiv:hep-th/9312104), we propose the Verlinde/Grassmannian correspondence which relates the GL Verlinde numbers to the K-theoretic quasimap invariants of the Grassmannian. We recover these two types of invariants by imposing different stability conditions on the gauged linear sigma model associated to the Grassmannian. We construct two families of stability conditions connecting the two theories and prove two wall-crossing results. We confirm the Verlinde/Grassmannian correspondence in the rank two case.
Gromov--Witten Invariants of Non-Convex Complete Intersections in Weighted Projective Stacks
本文利用 quasimap wall-crossing 与可逆镜像映射，突破凸性限制，计算加权射影叠中 Calabi–Yau 三重完全交的亏格 0 orbifold Gromov–Witten 不变量。
Gromov-Witten/Hurwitz wall-crossing
本文提出 $\varepsilon$-容许稳定性并证明 GW/Hurwitz 穿墙公式，进而将相对 PT/GW 对应与 Ruan 扩张 crepant 消解猜想等价化，并验证环面 del Pezzo 曲面情形。
Euler transformation for multiple $q$-hypergeometric series from wall-crossing formula of $K$-theoretic vortex partition function
本文通过展示多重q-超几何级数的变换公式与K-理论旋涡分区函数的墙穿越公式一致，揭示了欧拉变换在三维N=2和N=4超对称 gauge 理论中的几何意义。
K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces
该研究利用对数拟映射的K-模空间构造模插值，确定其壁跨越，通过Borcherds乘积构造新自守形式，证明特定区间的K-模空间为半环面紧化。
For a Fermat quasi-homogeneous polynomial $W$, we study a family of K-theoretic quantum invariants parametrized by a positive rational number $ε$. We prove a wall-crossing formula by showing the generating functions lie on the Lagrangian cone of the permutation-equivariant K-theoretic FJRW theory of $W$.
Equivariant K-theoretic enumerative invariants and wall-crossing formulae in abelian categories
本文将Joyce的同调泛穿墙公式提升至K理论并引入等变性，为模栈的等变K理论枚举不变量建立了通用穿墙框架，并指出拓扑K-同调构成等变乘性顶点代数。
Formal splitting and stack-theoretic normal crossings desingularization
本文通过形式分裂定理和Abramovich等人算法，展示如何用加权吹胀获得保持正常交叉的堆叠理论除 singularities。
$K$-theoretic wall-crossing formulas and multiple basic hypergeometric series
该研究通过壁穿现象分析箭图模空间的K-理论积分，聚焦A₁型手锯箭图簇，推导函数方程与配分函数显式公式，为含Kajihara变换等多重基本超几何级数变换公式提供几何解释。
Generalized K-theoretic invariants and wall-crossing via non-abelian localization
本文定义了广义K-理论不变量并证明其满足墙穿越公式，引入K-Hall代数并利用非交换局部化定理进行推导，扩展了Joyce-Liu的墙穿越到非标准心的D^b(X)中。
We define a set theoretic "analytic continuation" of a polytope defined by inequalities. For the regular values of the parameter, our construction coincides with the parallel transport of polytopes in a mirage introduced by Varchenko. We determine the set-theoretic variation when crossing a wall in the parameter space, and we relate this variation to Paradan's wall-crossing formulas for integrals and discrete sums. As another application, we refine the theorem of Brion on generating functions of polytopes and their cones at vertices. We describe the relation of this work with the equivariant i…
We conjecture that the Joyce-Song wall-crossing formula for Donaldson-Thomas invariants arises naturally from an asymptotic expansion in the field theoretic work of Gaiotto, Moore and Neitzke. This would also give a new perspective on how the formulae of Joyce-Song and Kontsevich-Soibelman are related. We check the conjecture in many examples.
Wall-crossing formula for framed quiver moduli
本文研究框定箭袋表示模空间的穿墙现象，针对 A 型旗流形、仿射 Laumon 空间、Nakajima 箭袋簇及射影平面相关层模空间，建立 Euler 类积分的穿墙公式。
Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather complicated transformation rules and non-trivial combinatorial problems. In this paper we propose a framework, reminiscent of collections and plethysms in the theory of operads, that conceptualizes those transformation rules. As an application we obtain new streamlined proofs of some existing wall-crossing formulas as well as some new formulas related to attractor…
Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and $K$-theoretic invariants for many moduli stacks of interest, including $K$-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the $K$-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory…
Application of Lagrange inversion to wall-crossing for Quot schemes on surfaces
本文利用拉格朗日反演技巧证明了分别由几何虚拟局部化与顶点代数得到的两个幂级数相等，从而使 Quot 概形计数不变量工作独立于先前文献，并给出新的组合恒等式。
Partial desingularization up to normal-crossings in characteristic 0 and 2
研究特征0和2下正常交叉保持的奇点消解，特征0中允许引入堆叠理论加权吹脱后可实现，而特征2中分类了pinch points并证明加权吹脱无法实现正常交叉保持消解。
We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behavior away from a Donaldson divisor. We define geometric operations called mutations of Lagrangian tori in del Pezzo surfaces and in toric Fano varieties of higher dimension, and study the corresponding wall-crossing formulas that compute the disk potential of a mutated torus from that of the original one. In the case of del Pezzo surfaces, this justifies the connection between Vianna's tori and the theory of mutations of Landau-Ginzburg seeds…
We prove the following results for toric Deligne-Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch-Riemann-Roch theorem; the Fourier--Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.
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