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Quantum Curve and Bilinear Fermionic Form for the Orbifold Gromov–Witten Theory of ℙ[r] ℙ[r]的轨道格罗莫夫-维滕理论的量子曲线和双线性费米子形式
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-15 DOI: 10.1007/s10114-024-1633-4
Chong Yao Chen, Shuai Guo

We construct the quantum curve for the Baker–Akhiezer function of the orbifold Gromov–Witten theory of the weighted projective line ℙ[r]. Furthermore, we deduce the explicit bilinear Fermionic formula for the (stationary) Gromov–Witten potential via the lifting operator contructed from the Baker–Akhiezer function.

我们构建了加权投影线ℙ[r]的轨道格罗莫夫-维滕理论的贝克-阿基泽函数的量子曲线。此外,我们还通过贝克-阿基泽函数构造的提升算子,推导出了(静止)格罗莫夫-维滕势能的显式双线性费米子公式。
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引用次数: 0
On a Formation of Singularities of Solutions to Soliton Equations Represented by L, A, B-triples 论 L、A、B 三元组表示的孤子方程解奇异点的形成
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-2324-x
Iskander A. Taimanov

We discuss the mechanism of formation of singularities of solutions to the Novikov–Veselov, modified Novikov–Veselov, and Davey–Stewartson II (DSII) equations obtained by the Moutard type transformations. These equations admit the L, A, B-triple presentation, the generalization of the L, A-pairs for 2+1-soliton equations. We relate the blow-up of solutions to the non-conservation of the zero level of discrete spectrum of the L-operator. We also present a class of exact solutions, of the DSII system, which depend on two functional parameters, and show that all possible singularities of solutions to DSII equation obtained by the Moutard transformation are indeterminancies, i.e., points when approaching which in different spatial directions the solution has different limits.

我们讨论了通过 Moutard 型变换得到的 Novikov-Veselov、修正 Novikov-Veselov 和 Davey-Stewartson II(DSII)方程解的奇点形成机制。这些方程采用了 L、A、B 三重表述,即 2+1 索利顿方程的 L、A 对的广义表述。我们把解的膨胀与 L 操作符离散谱零级的不守恒联系起来。我们还提出了一类取决于两个函数参数的 DSII 系统精确解,并证明了通过 Moutard 变换得到的 DSII 方程解的所有可能奇点都是不确定的,即接近不同空间方向上的解具有不同极限的点。
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引用次数: 0
On Differential Equations of Integrable Billiard Tables 论积分台球桌的微分方程
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-2450-5
Vladimir Dragović, Andrey E. Mironov

We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire billiards, for finding surfaces in ℝ3 with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ℝ3 homeomorphic to a torus, being a table of a billiard admitting two additional first integrals.

我们介绍了一种方法,可以找到定义台球桌的函数的微分方程,从而使相关的台球系统具有局部第一积分。我们在三种情况下说明了这种方法:(局部)可积分线状台球的情况,寻找速度一阶积分的ℝ3曲面,以及寻找与环同构的ℝ3片断光滑曲面,即允许两个额外一阶积分的台球桌。
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引用次数: 0
On the Quantum Cohomology of Blow-ups of Four-dimensional Quadrics 论四维四面体胀大的量子同调
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-2236-9
Jian Xun Hu, Hua Zhong Ke, Chang Zheng Li, Lei Song

We propose a conjecture relevant to Galkin’s lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture ({cal O}) holds in these two cases.

我们提出了一个与加尔金下界猜想相关的猜想,并对四维二次元在一点或沿投影面的炸开进行了验证。我们还证明了猜想({cal O})在这两种情况下都成立。
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引用次数: 0
A Tropical View on Landau–Ginzburg Models 兰道-金兹堡模型的热带视角
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-2295-y
Michael Carl, Max Pumperla, Bernd Siebert

This paper, largely written in 2009/2010, fits Landau–Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001. This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines, previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for ℙ2. A major insight is the equivalence of properness of the Landau–Ginzburg potential with smoothness of the anticanonical divisor on the mirror side. We obtain proper superpotentials which agree on an open part with those classically known for toric varieties. Examples include mirror LG models for non-singular and singular del Pezzo surfaces, Hirzebruch surfaces and some Fano threefolds.

这篇论文主要写于2009/2010年,将兰道-金兹堡模型纳入了最后一位作者与马克-格罗斯自2001年以来共同追求的镜像对称计划。马克-格罗斯在研究ℙ2的镜像对称性时,曾在二维中引入了断裂线的概念。我们的一个重要发现是,朗道-金兹堡势的适当性与镜像侧反调和除数的平滑性是等价的。我们得到的适当超势在开放部分与环状变体的经典超势一致。例子包括非奇异和奇异 del Pezzo 表面、Hirzebruch 表面和一些法诺三褶的镜像 LG 模型。
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引用次数: 0
Gluing Affine Vortices 粘合仿射漩涡
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-2248-5
Guang Bo Xu

We provide an analytical construction of the gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition. This result is a necessary ingredient in studies of the relation between gauged sigma model and nonlinear sigma model, such as the closed or open quantum Kirwan map.

我们提供了上半平面上稳定仿射旋涡的胶合图的分析构造,并具有拉格朗日边界条件。这一结果是研究测量西格玛模型和非线性西格玛模型(如封闭或开放量子柯尔旺图)之间关系的必要条件。
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引用次数: 0
Partial Collapsing Degeneration of Floer Trajectories and Adiabatic Gluing 浮子轨迹的部分塌缩退化与绝热胶合
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-2234-y
Yong-Geun Oh, Ke Zhu

In the present paper, we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic ε-family and its reversal adiabatic gluing, as the prototype of the partial collapsing degeneration of 2-dimensional (perturbed) J-holomorphic maps to 1-dimensional gradient segments. We consider the case when the Floer equations are S1-invariant on parts of their domains whose adiabatic limit has positive length as ε → 0, which we call thimble-flow-thimble configurations. The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configuration. In particular, our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse–Bott version of Lagrangian intersection Floer complex of L by Fukaya–Oh–Ohta–Ono and the pearly complex of L Lalonde and Biran–Cornea. It also provides another proof of the present authors’ earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing.

在本文中,我们研究了绝热ε族的哈密顿扰动弗洛尔轨迹的部分塌缩退化及其逆转绝热胶合,这是二维(扰动)J-荷尔摩态映射到一维梯度线段的部分塌缩退化的原型。我们考虑的情况是弗洛尔方程在其域的部分上是 S1 不变的,其绝热极限在 ε → 0 时长度为正,我们称之为顶针-流-顶针构型。我们证明的主要胶合定理也适用于有拉格朗日边界的情况,例如从珠光构型中恢复全形盘的问题。特别是,我们的胶合定理直接证明了深谷-奥塔-奥诺(Fukaya-Oh-Ohta-Ono)的莫尔斯-波特版拉格朗日交点弗洛尔复数(Lagrangian intersection Floer complex of L)与拉隆德(Lalonde)和比兰-科尔内亚(Biran-Cornea)的珠光复数(pearly complex of L)之间的链同构性质。它还为本文作者早先对 PSS 映射同构性质的证明提供了另一个证明,而无需涉及目标重定标和尺度相关胶合。
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引用次数: 0
GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back 通过弗罗贝尼乌斯流形的 GUE。I. 从矩阵引力到拓扑引力再到拓扑引力
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-2258-3
Di Yang

Dubrovin establishes a certain relationship between the GUE partition function and the partition function of Gromov–Witten invariants of the complex projective line. In this paper, we give a direct proof of Dubrovin’s result. We also present in a diagram the recent progress on topological gravity and matrix gravity.

杜布罗文在复射线的 GUE 分区函数和格罗莫夫-维滕不变式的分区函数之间建立了某种关系。在本文中,我们给出了杜布罗文结果的直接证明。我们还以图表的形式介绍了拓扑引力和矩阵引力的最新进展。
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引用次数: 0
Preface of the Special Issue on Symplectic Geometry and Mathematical Physics 交映几何与数学物理》特刊序言
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-4998-5
Huijun Fan, Xiaobo Liu, Gang Tian
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引用次数: 0
Normal Crossings Singularities for Symplectic Topology: Structures 交点拓扑学的法线交叉奇异性:结构
IF 0.8 3区 数学 Q2 MATHEMATICS Pub Date : 2024-01-05 DOI: 10.1007/s10114-024-2042-4
Mohammad Farajzadeh-Tehrani, Mark Mclean, Aleksey Zinger

Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures have applications in constructions and analysis of various moduli spaces. As a corollary of the Chern class formula for the logarithmic tangent bundle, we refine Aluffi’s formula for the Chern class of the tangent bundle of the blowup at a complete intersection to account for the torsion and extend it to the blowup at the deepest stratum of an arbitrary normal crossings divisor.

我们之前的论文介绍了正交折射分项和变项的拓扑概念,证明它们在适当意义上等价于相应的几何概念,并建立了正交折射变项的拓扑平滑性准则。本文构造了与正交交映骰子自然相关的吹积、复线束和对数切线束,并确定了最后一个束的切恩类。这些结构在各种模空间的构造和分析中都有应用。作为对数切线束的切恩类公式的推论,我们完善了阿鲁菲关于完全交点处炸开的切线束的切恩类公式,以考虑扭转,并将其扩展到任意法向交点除数的最深层炸开。
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引用次数: 0
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Acta Mathematica Sinica-English Series
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