Pub Date : 2024-01-15DOI: 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.
{"title":"Quantum Curve and Bilinear Fermionic Form for the Orbifold Gromov–Witten Theory of ℙ[r]","authors":"Chong Yao Chen, Shuai Guo","doi":"10.1007/s10114-024-1633-4","DOIUrl":"10.1007/s10114-024-1633-4","url":null,"abstract":"<div><p>We construct the quantum curve for the Baker–Akhiezer function of the orbifold Gromov–Witten theory of the weighted projective line ℙ[<i>r</i>]. Furthermore, we deduce the explicit bilinear Fermionic formula for the (stationary) Gromov–Witten potential via the lifting operator contructed from the Baker–Akhiezer function.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"43 - 80"},"PeriodicalIF":0.8,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140888088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 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.
{"title":"On a Formation of Singularities of Solutions to Soliton Equations Represented by L, A, B-triples","authors":"Iskander A. Taimanov","doi":"10.1007/s10114-024-2324-x","DOIUrl":"10.1007/s10114-024-2324-x","url":null,"abstract":"<div><p>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 <i>L, A, B</i>-triple presentation, the generalization of the <i>L, A</i>-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 <i>L</i>-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.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"406 - 416"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 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.
{"title":"On Differential Equations of Integrable Billiard Tables","authors":"Vladimir Dragović, Andrey E. Mironov","doi":"10.1007/s10114-024-2450-5","DOIUrl":"10.1007/s10114-024-2450-5","url":null,"abstract":"<div><p>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 ℝ<sup>3</sup> with a first integral of degree one in velocities, and for finding a piece-wise smooth surface in ℝ<sup>3</sup> homeomorphic to a torus, being a table of a billiard admitting two additional first integrals.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"417 - 424"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140888191","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 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.
{"title":"On the Quantum Cohomology of Blow-ups of Four-dimensional Quadrics","authors":"Jian Xun Hu, Hua Zhong Ke, Chang Zheng Li, Lei Song","doi":"10.1007/s10114-024-2236-9","DOIUrl":"10.1007/s10114-024-2236-9","url":null,"abstract":"<div><p>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 <span>({cal O})</span> holds in these two cases.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"313 - 328"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139393069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 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。
{"title":"A Tropical View on Landau–Ginzburg Models","authors":"Michael Carl, Max Pumperla, Bernd Siebert","doi":"10.1007/s10114-024-2295-y","DOIUrl":"10.1007/s10114-024-2295-y","url":null,"abstract":"<div><p>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 ℙ<sup>2</sup>. 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.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"329 - 382"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 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.
{"title":"Gluing Affine Vortices","authors":"Guang Bo Xu","doi":"10.1007/s10114-024-2248-5","DOIUrl":"10.1007/s10114-024-2248-5","url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"250 - 312"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 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。
{"title":"Partial Collapsing Degeneration of Floer Trajectories and Adiabatic Gluing","authors":"Yong-Geun Oh, Ke Zhu","doi":"10.1007/s10114-024-2234-y","DOIUrl":"10.1007/s10114-024-2234-y","url":null,"abstract":"<div><p>In the present paper, we study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic <i>ε</i>-family and its reversal adiabatic gluing, as the prototype of the partial collapsing degeneration of 2-dimensional (perturbed) <i>J</i>-holomorphic maps to 1-dimensional gradient segments. We consider the case when the Floer equations are <i>S</i><sup>1</sup>-invariant on parts of their domains whose adiabatic limit has positive length as <i>ε</i> → 0, which we call <i>thimble-flow-thimble</i> 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 <i>pearly complex</i> of <i>L</i> 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.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"161 - 249"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139394231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 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.
{"title":"GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back","authors":"Di Yang","doi":"10.1007/s10114-024-2258-3","DOIUrl":"10.1007/s10114-024-2258-3","url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"383 - 405"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140888199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 10.1007/s10114-024-4998-5
Huijun Fan, Xiaobo Liu, Gang Tian
{"title":"Preface of the Special Issue on Symplectic Geometry and Mathematical Physics","authors":"Huijun Fan, Xiaobo Liu, Gang Tian","doi":"10.1007/s10114-024-4998-5","DOIUrl":"10.1007/s10114-024-4998-5","url":null,"abstract":"","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"1 - 2"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139392459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-05DOI: 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.
{"title":"Normal Crossings Singularities for Symplectic Topology: Structures","authors":"Mohammad Farajzadeh-Tehrani, Mark Mclean, Aleksey Zinger","doi":"10.1007/s10114-024-2042-4","DOIUrl":"10.1007/s10114-024-2042-4","url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 1","pages":"107 - 160"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}