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Computational symplectic topology and symmetric orbits in the restricted three-body problem 受限三体问题中的计算交映拓扑和对称轨道
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-27 DOI: arxiv-2407.19159
Chankyu Joung, Otto van Koert
{"title":"Computational symplectic topology and symmetric orbits in the restricted three-body problem","authors":"Chankyu Joung, Otto van Koert","doi":"arxiv-2407.19159","DOIUrl":"https://doi.org/arxiv-2407.19159","url":null,"abstract":"In this paper we propose a computational approach to proving the Birkhoff\u0000conjecture on the restricted three-body problem, which asserts the existence of\u0000a disk-like global surface of section. Birkhoff had conjectured this surface of\u0000section as a tool to prove existence of a direct periodic orbit. Using\u0000techniques from validated numerics we prove the existence of an approximately\u0000circular direct orbit for a wide range of mass parameters and Jacobi energies.\u0000We also provide methods to rigorously compute the Conley-Zehnder index of\u0000periodic Hamiltonian orbits using computational tools, thus giving some initial\u0000steps for developing computational Floer homology and providing the means to\u0000prove the Birkhoff conjecture via symplectic topology. We apply this method to\u0000various symmetric orbits in the restricted three-body problem.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Wehrheim-Woodward category of linear canonical relations between G-spaces G 空间之间线性规范关系的韦尔海姆-伍德沃德范畴
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-26 DOI: arxiv-2408.06363
Alan Weinstein
{"title":"The Wehrheim-Woodward category of linear canonical relations between G-spaces","authors":"Alan Weinstein","doi":"arxiv-2408.06363","DOIUrl":"https://doi.org/arxiv-2408.06363","url":null,"abstract":"We extend the work in a previous paper with David Li-Bland to construct the\u0000Wehrheim-Woodward category WW(GSLREL) of equivariant linear canonical relations\u0000between linear symplectic G-spaces for a compact group G. When G is the trivial\u0000group, this reduces to the previous result that the morphisms in WW(SLREL) may\u0000be identified with pairs (L,k) consisting of a linear canonical relation and a\u0000nonnegative integer.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Partially wrapped Fukaya categories of orbifold surfaces 轨道曲面的部分包裹富卡亚范畴
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-23 DOI: arxiv-2407.16358
Severin Barmeier, Sibylle Schroll, Zhengfang Wang
{"title":"Partially wrapped Fukaya categories of orbifold surfaces","authors":"Severin Barmeier, Sibylle Schroll, Zhengfang Wang","doi":"arxiv-2407.16358","DOIUrl":"https://doi.org/arxiv-2407.16358","url":null,"abstract":"We give a complete description of partially wrapped Fukaya categories of\u0000graded orbifold surfaces with stops. We show that a construction via global\u0000sections of a natural cosheaf of A$_infty$ categories on a Lagrangian core of\u0000the surface is equivalent to a global construction via the (equivariant) orbit\u0000category of a smooth cover. We therefore establish the local-to-global\u0000properties of partially wrapped Fukaya categories of orbifold surfaces closely\u0000paralleling a proposal by Kontsevich for Fukaya categories of smooth Weinstein\u0000manifolds. From the viewpoint of Weinstein sectorial descent in the sense of Ganatra,\u0000Pardon and Shende, our results show that orbifold surfaces also have Weinstein\u0000sectors of type $mathrm D$ besides the type $mathrm A$ or type\u0000$widetilde{mathrm A}$ sectors on smooth surfaces. We describe the global sections of the cosheaf explicitly for any generator\u0000given by an admissible dissection of the orbifold surface and we give a full\u0000classification of the formal generators which arise in this way. This shows in\u0000particular that the partially wrapped Fukaya category of an orbifold surface\u0000can always be described as the perfect derived category of a graded associative\u0000algebra. We conjecture that associative algebras obtained from dissections of\u0000orbifold surfaces form a new class of associative algebras closed under derived\u0000equivalence.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Shifted symplectic structure on Poisson Lie algebroid and generalized complex geometry 泊松Lie algebroid上的移动交映结构和广义复几何
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-22 DOI: arxiv-2407.15598
Yingdi Qin
{"title":"Shifted symplectic structure on Poisson Lie algebroid and generalized complex geometry","authors":"Yingdi Qin","doi":"arxiv-2407.15598","DOIUrl":"https://doi.org/arxiv-2407.15598","url":null,"abstract":"Generalized complex geometry was classically formulated by the language of\u0000differential geometry. In this paper, we reformulated a generalized complex\u0000manifold as a holomorphic symplectic differentiable formal stack in a\u0000homotopical sense. Meanwhile, by developing the machinery for shifted\u0000symplectic formal stack, we prove that the coisotropic intersection inherits\u0000shifted Poisson structure. Generalized complex branes are also studied.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a metric symplectization of a contact metric manifold 论接触计量流形的计量交映化
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-21 DOI: arxiv-2407.15057
Sannidhi Alape
{"title":"On a metric symplectization of a contact metric manifold","authors":"Sannidhi Alape","doi":"arxiv-2407.15057","DOIUrl":"https://doi.org/arxiv-2407.15057","url":null,"abstract":"In this article, we investigate metric structures on the symplectization of a\u0000contact metric manifold and prove that there is a unique metric structure,\u0000which we call the metric symplectization, for which each slice of the\u0000symplectization has a natural induced contact metric structure. We then study\u0000the curvature properties of this metric structure and use it to establish\u0000equivalent formulations of the $(kappa, mu)$-nullity condition in terms of\u0000the metric symplectization. We also prove that isomorphisms of the metric\u0000symplectizations of $(kappa, mu)$-manifolds determine $(kappa,\u0000mu)$-manifolds up to D-homothetic transformations. These classification\u0000results show that the metric symplectization provides a unified framework to\u0000classify Sasakian manifolds, K-contact manifolds and $(kappa, mu)$-manifolds\u0000in terms of their symplectizations.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory 通过参数化定点理论实现环共积同调不变性的障碍
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-18 DOI: arxiv-2407.13662
Lea Kenigsberg, Noah Porcelli
{"title":"Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory","authors":"Lea Kenigsberg, Noah Porcelli","doi":"arxiv-2407.13662","DOIUrl":"https://doi.org/arxiv-2407.13662","url":null,"abstract":"Given $f: M to N$ a homotopy equivalence of compact manifolds with boundary,\u0000we use a construction of Geoghegan and Nicas to define its Reidemeister trace\u0000$[T] in pi_1^{st}(mathcal{L} N, N)$. We realize the Goresky-Hingston\u0000coproduct as a map of spectra, and show that the failure of $f$ to entwine the\u0000spectral coproducts can be characterized by Chas-Sullivan multiplication with\u0000$[T]$. In particular, when $f$ is a simple homotopy equivalence, the spectral\u0000coproducts of $M$ and $N$ agree.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141739024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties 仿生交点变体上的拉格朗日斯克莱塔和科斯祖尔对偶性
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-18 DOI: arxiv-2407.13286
Gwyn Bellamy, Christopher Dodd, Kevin McGerty, Thomas Nevins
{"title":"Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties","authors":"Gwyn Bellamy, Christopher Dodd, Kevin McGerty, Thomas Nevins","doi":"arxiv-2407.13286","DOIUrl":"https://doi.org/arxiv-2407.13286","url":null,"abstract":"We consider the category of modules over sheaves of Deformation-Quantization\u0000(DQ) algebras on bionic symplectic varieties. These spaces are equipped with\u0000both an elliptic $mathbb{G}_m$-action and a Hamiltonian $mathbb{G}_m$-action,\u0000with finitely many fixed points. On these spaces one can consider geometric\u0000category $mathcal{O}$: the category of (holonomic) modules supported on the\u0000Lagrangian attracting set of the Hamiltonian action. We show that there exists\u0000a local generator in geometric category $mathcal{O}$ whose dg endomorphism\u0000ring, cohomologically supported on the Lagrangian attracting set, is derived\u0000equivalent to the category of all DQ-modules. This is a version of Koszul\u0000duality generalizing the equivalence between D-modules on a smooth variety and\u0000dg-modules over the de Rham complex.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141739025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On entropy and complexity of coherent states 关于相干态的熵和复杂性
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-18 DOI: arxiv-2407.13327
Koushik Ray
{"title":"On entropy and complexity of coherent states","authors":"Koushik Ray","doi":"arxiv-2407.13327","DOIUrl":"https://doi.org/arxiv-2407.13327","url":null,"abstract":"Consanguinity of entropy and complexity is pointed out through the example of\u0000coherent states of the $SL(2,C)$ group. Both are obtained from the K\"ahler\u0000potential of the underlying geometry of the sphere corresponding to the\u0000Fubini-Study metric. Entropy is shown to be equal to the K\"ahler potential\u0000written in terms of dual symplectic variables as the Guillemin potential for\u0000toric manifolds. The logarithm of complexity relating two states is shown to be\u0000equal to Calabi's diastasis function. Optimality of the Fubini-Study metric is\u0000indicated by considering its deformation.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141739023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Gauged linear sigma model in geometric phase. II. the virtual cycle 几何相位中的测量线性西格玛模型。二、虚拟循环
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-16 DOI: arxiv-2407.14545
Gang Tian, Guangbo Xu
{"title":"Gauged linear sigma model in geometric phase. II. the virtual cycle","authors":"Gang Tian, Guangbo Xu","doi":"arxiv-2407.14545","DOIUrl":"https://doi.org/arxiv-2407.14545","url":null,"abstract":"We provide the detailed construction of the virtual cycles needed for\u0000defining the cohomological field theory associated to a gauged linear sigma\u0000model in geometric phase.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An endomorphism on immersed curves in the pillowcase 枕头套中沉浸曲线的内态性
arXiv - MATH - Symplectic Geometry Pub Date : 2024-07-15 DOI: arxiv-2407.11247
Christopher M. Herald, Paul Kirk
{"title":"An endomorphism on immersed curves in the pillowcase","authors":"Christopher M. Herald, Paul Kirk","doi":"arxiv-2407.11247","DOIUrl":"https://doi.org/arxiv-2407.11247","url":null,"abstract":"We examine the holonomy-perturbed traceless SU(2) character variety of the\u0000trivial four-stranded tangle {p_1,p_2,p_3,p_4} X [0,1] in S^2 X [0,1] equipped\u0000with a strong marking, either an earring or a bypass. Viewing these marked\u0000tangles as endomorphisms in the cobordism category from the four-punctured\u0000sphere to itself, we identify the images of these endomorphisms in the\u0000Weinstein symplectic partial category under the partially defined\u0000holonomy-perturbed traceless character variety functor. We express these\u0000endomorphisms on immersed curves in the pillowcase in terms of doubling and\u0000figure eight operations and prove they have the same image.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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