In this paper we propose a computational approach to proving the Birkhoff conjecture on the restricted three-body problem, which asserts the existence of a disk-like global surface of section. Birkhoff had conjectured this surface of section as a tool to prove existence of a direct periodic orbit. Using techniques from validated numerics we prove the existence of an approximately circular direct orbit for a wide range of mass parameters and Jacobi energies. We also provide methods to rigorously compute the Conley-Zehnder index of periodic Hamiltonian orbits using computational tools, thus giving some initial steps for developing computational Floer homology and providing the means to prove the Birkhoff conjecture via symplectic topology. We apply this method to various symmetric orbits in the restricted three-body problem.
{"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":"95 1","pages":""},"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}
We extend the work in a previous paper with David Li-Bland to construct the Wehrheim-Woodward category WW(GSLREL) of equivariant linear canonical relations between linear symplectic G-spaces for a compact group G. When G is the trivial group, this reduces to the previous result that the morphisms in WW(SLREL) may be identified with pairs (L,k) consisting of a linear canonical relation and a nonnegative integer.
我们扩展了与大卫-李-布兰德（David Li-Bland）合作的前一篇论文中的工作，构建了紧凑群 G 的线性交点 G 空间之间的等变线性规范关系的韦尔海姆-伍德沃德类别 WW(GSLREL)。
{"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":"23 1","pages":""},"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}
We give a complete description of partially wrapped Fukaya categories of graded orbifold surfaces with stops. We show that a construction via global sections of a natural cosheaf of A$_infty$ categories on a Lagrangian core of the surface is equivalent to a global construction via the (equivariant) orbit category of a smooth cover. We therefore establish the local-to-global properties of partially wrapped Fukaya categories of orbifold surfaces closely paralleling a proposal by Kontsevich for Fukaya categories of smooth Weinstein manifolds. From the viewpoint of Weinstein sectorial descent in the sense of Ganatra, Pardon and Shende, our results show that orbifold surfaces also have Weinstein sectors of type $mathrm D$ besides the type $mathrm A$ or type $widetilde{mathrm A}$ sectors on smooth surfaces. We describe the global sections of the cosheaf explicitly for any generator given by an admissible dissection of the orbifold surface and we give a full classification of the formal generators which arise in this way. This shows in particular that the partially wrapped Fukaya category of an orbifold surface can always be described as the perfect derived category of a graded associative algebra. We conjecture that associative algebras obtained from dissections of orbifold surfaces form a new class of associative algebras closed under derived equivalence.
{"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":"415 1","pages":""},"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}
Generalized complex geometry was classically formulated by the language of differential geometry. In this paper, we reformulated a generalized complex manifold as a holomorphic symplectic differentiable formal stack in a homotopical sense. Meanwhile, by developing the machinery for shifted symplectic formal stack, we prove that the coisotropic intersection inherits shifted Poisson structure. Generalized complex branes are also studied.
{"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":"41 1","pages":""},"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}
In this article, we investigate metric structures on the symplectization of a contact metric manifold and prove that there is a unique metric structure, which we call the metric symplectization, for which each slice of the symplectization has a natural induced contact metric structure. We then study the curvature properties of this metric structure and use it to establish equivalent formulations of the $(kappa, mu)$-nullity condition in terms of the metric symplectization. We also prove that isomorphisms of the metric symplectizations of $(kappa, mu)$-manifolds determine $(kappa, mu)$-manifolds up to D-homothetic transformations. These classification results show that the metric symplectization provides a unified framework to classify Sasakian manifolds, K-contact manifolds and $(kappa, mu)$-manifolds in terms of their symplectizations.
{"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":"42 1","pages":""},"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}
Given $f: M to N$ a homotopy equivalence of compact manifolds with boundary, we use a construction of Geoghegan and Nicas to define its Reidemeister trace $[T] in pi_1^{st}(mathcal{L} N, N)$. We realize the Goresky-Hingston coproduct as a map of spectra, and show that the failure of $f$ to entwine the spectral coproducts can be characterized by Chas-Sullivan multiplication with $[T]$. In particular, when $f$ is a simple homotopy equivalence, the spectral coproducts of $M$ and $N$ agree.
给定 $f：给定 $f: M to N$ 是有边界的紧凑流形的同调等价，我们使用 Geoghegan 和 Nicas 的构造来定义它在pi_1^{st}(mathcal{L} N, N)$ 中的 Reidemeister trace$[T] 。我们将戈尔斯基-邢斯顿共乘实现为谱的映射，并证明了$f$不能缠绕谱共乘的情况可以用查斯-沙利文与$[T]$相乘来描述。特别是，当 $f$ 是简单同调等价时，$M$ 和 $N$ 的谱共乘会一致。
{"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":"82 1","pages":""},"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}
Gwyn Bellamy, Christopher Dodd, Kevin McGerty, Thomas Nevins
We consider the category of modules over sheaves of Deformation-Quantization (DQ) algebras on bionic symplectic varieties. These spaces are equipped with both an elliptic $mathbb{G}_m$-action and a Hamiltonian $mathbb{G}_m$-action, with finitely many fixed points. On these spaces one can consider geometric category $mathcal{O}$: the category of (holonomic) modules supported on the Lagrangian attracting set of the Hamiltonian action. We show that there exists a local generator in geometric category $mathcal{O}$ whose dg endomorphism ring, cohomologically supported on the Lagrangian attracting set, is derived equivalent to the category of all DQ-modules. This is a version of Koszul duality generalizing the equivalence between D-modules on a smooth variety and dg-modules over the de Rham complex.
{"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":"8 1","pages":""},"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}
Consanguinity of entropy and complexity is pointed out through the example of coherent states of the $SL(2,C)$ group. Both are obtained from the K"ahler potential of the underlying geometry of the sphere corresponding to the Fubini-Study metric. Entropy is shown to be equal to the K"ahler potential written in terms of dual symplectic variables as the Guillemin potential for toric manifolds. The logarithm of complexity relating two states is shown to be equal to Calabi's diastasis function. Optimality of the Fubini-Study metric is indicated by considering its deformation.
{"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":"2013 1","pages":""},"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}
We provide the detailed construction of the virtual cycles needed for defining the cohomological field theory associated to a gauged linear sigma model in geometric phase.
{"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":"7 1","pages":""},"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}
We examine the holonomy-perturbed traceless SU(2) character variety of the trivial four-stranded tangle {p_1,p_2,p_3,p_4} X [0,1] in S^2 X [0,1] equipped with a strong marking, either an earring or a bypass. Viewing these marked tangles as endomorphisms in the cobordism category from the four-punctured sphere to itself, we identify the images of these endomorphisms in the Weinstein symplectic partial category under the partially defined holonomy-perturbed traceless character variety functor. We express these endomorphisms on immersed curves in the pillowcase in terms of doubling and figure eight operations and prove they have the same image.
我们研究了 S^2 X [0,1] 中的三维四链纠缠 {p_1,p_2,p_3,p_4} 的全局性扰动无痕 SU(2) 特征多样性。S^2 X [0,1] 中的 X [0,1] 带有一个强标记，要么是耳环，要么是旁路。我们把这些标记的三角形看成是从四穿刺球到其本身的共线性范畴中的内同构，并在部分定义的holonomy-perturbed traceless character variety functor 下识别出这些内同构在韦恩斯坦交点偏范畴中的映像。我们用加倍运算和图八运算来表达枕套中浸没曲线上的这些内同构，并证明它们具有相同的图像。
{"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":"37 1","pages":""},"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}
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