Pub Date : 2024-01-15DOI: 10.1007/s10114-024-1492-z
Jian Zhou
We explain how to construct a quantum deformation of a spectral curve associated to a tau-function of the KP hierarchy. This construction is applied to Witten–Kontsevich tau-function to give a natural explanation of some earlier work. We also apply it to higher Weil–Petersson volumes and Witten’s r-spin intersection numbers.
我们解释了如何构建与 KP 层次的 tau 函数相关的谱曲线的量子变形。我们将这一构造应用于威滕-康采维奇 tau 函数,从而自然地解释了一些早期工作。我们还将其应用于更高的魏尔-彼得森卷和威滕的 r-旋交数。
{"title":"Emergent Geometry of KP Hierarchy","authors":"Jian Zhou","doi":"10.1007/s10114-024-1492-z","DOIUrl":"https://doi.org/10.1007/s10114-024-1492-z","url":null,"abstract":"<p>We explain how to construct a quantum deformation of a spectral curve associated to a tau-function of the KP hierarchy. This construction is applied to Witten–Kontsevich tau-function to give a natural explanation of some earlier work. We also apply it to higher Weil–Petersson volumes and Witten’s r-spin intersection numbers.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140888197","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 : 2024-01-15DOI: 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":"https://doi.org/10.1007/s10114-024-1633-4","url":null,"abstract":"<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>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"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":"https://doi.org/10.1007/s10114-024-2324-x","url":null,"abstract":"<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>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"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":"https://doi.org/10.1007/s10114-024-2450-5","url":null,"abstract":"<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>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"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/s10114-024-2248-5
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":"https://doi.org/10.1007/s10114-024-2248-5","url":null,"abstract":"<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>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"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/s10114-024-2295-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":"https://doi.org/10.1007/s10114-024-2295-y","url":null,"abstract":"<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>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"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/s10114-024-2258-3
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":"https://doi.org/10.1007/s10114-024-2258-3","url":null,"abstract":"<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>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"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/s10114-024-2042-4
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":"https://doi.org/10.1007/s10114-024-2042-4","url":null,"abstract":"<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>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"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}
Pub Date : 2023-12-29DOI: 10.1007/s10114-023-1681-1
Juan Pan, Xian Kun Ren, Yun Hua Zhou
A diffeomorphism is non-uniformly partially hyperbolic if it preserves an ergodic measure with at least one zero Lyapunov exponent. We prove that a C1-smooth ℤd-action has the quasi-shadowing property if one of the generators is C1+α (α > 0) non-uniformly partially hyperbolic.
{"title":"Quasi-shadowing for ℤd-actions","authors":"Juan Pan, Xian Kun Ren, Yun Hua Zhou","doi":"10.1007/s10114-023-1681-1","DOIUrl":"https://doi.org/10.1007/s10114-023-1681-1","url":null,"abstract":"<p>A diffeomorphism is non-uniformly partially hyperbolic if it preserves an ergodic measure with at least one zero Lyapunov exponent. We prove that a <i>C</i><sup>1</sup>-smooth ℤ<sup><i>d</i></sup>-action has the quasi-shadowing property if one of the generators is <i>C</i><sup>1+<i>α</i></sup> (<i>α</i> > 0) non-uniformly partially hyperbolic.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139067344","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 : 2023-12-29DOI: 10.1007/s10114-023-2608-6
Xiang Qi Qiang, Cheng Jun Hou
We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms, and characterize them in terms of the corresponding transformation groupoids and their operator algebras. In particular, we show that two essentially free semigroup actions on totally disconnected compact spaces are continuously orbit equivalent if and only if there is a canonical abelian subalgebra preserving C*-isomorphism between the associated transformation groupoid C*-algebras. We also give some examples of orbit equivalence, consider the special case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of the associated group actions.
{"title":"Continuous Orbit Equivalence of Semigroup Actions","authors":"Xiang Qi Qiang, Cheng Jun Hou","doi":"10.1007/s10114-023-2608-6","DOIUrl":"https://doi.org/10.1007/s10114-023-2608-6","url":null,"abstract":"<p>We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms, and characterize them in terms of the corresponding transformation groupoids and their operator algebras. In particular, we show that two essentially free semigroup actions on totally disconnected compact spaces are continuously orbit equivalent if and only if there is a canonical abelian subalgebra preserving <i>C</i>*-isomorphism between the associated transformation groupoid <i>C</i>*-algebras. We also give some examples of orbit equivalence, consider the special case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of the associated group actions.</p>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139067346","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}