Alexandre Anahory Simoes, Leonardo Colombo, Manuel de Leon, Modesto Salgado, Silvia Souto
In this paper, we investigate the reduction process of a contact Lagrangian system whose Lagrangian is invariant under a group of symmetries. We give explicit coordinate expressions of the resulting reduced differential equations, the so-called Lagrange-Poincare-Herglotz equations. Our framework relied on the associated Herglotz vector field and its projected vector field, and the use of well-chosen quasi-velocities. Some examples are also discussed.
{"title":"Symmetry reduction and reconstruction in contact geometry and Lagrange-Poincaré-Herglotz equations","authors":"Alexandre Anahory Simoes, Leonardo Colombo, Manuel de Leon, Modesto Salgado, Silvia Souto","doi":"arxiv-2408.06892","DOIUrl":"https://doi.org/arxiv-2408.06892","url":null,"abstract":"In this paper, we investigate the reduction process of a contact Lagrangian\u0000system whose Lagrangian is invariant under a group of symmetries. We give\u0000explicit coordinate expressions of the resulting reduced differential\u0000equations, the so-called Lagrange-Poincare-Herglotz equations. Our framework\u0000relied on the associated Herglotz vector field and its projected vector field,\u0000and the use of well-chosen quasi-velocities. Some examples are also discussed.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202136","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}
For any Anosov diffeomorphims on a closed odd dimensional manifold, there exists no invariant contact structure.
对于闭合奇数维流形上的任何阿诺索夫衍射，都不存在不变的接触结构。
{"title":"There exist no cotact Anosov diffeomorphisms","authors":"Masayuki Asaoka, Yoshihiko Mitsumatsu","doi":"arxiv-2408.06965","DOIUrl":"https://doi.org/arxiv-2408.06965","url":null,"abstract":"For any Anosov diffeomorphims on a closed odd dimensional manifold, there\u0000exists no invariant contact structure.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202119","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 introduce a new version of symplectic annular Khovanov homology and establish spectral sequences from (i) the symplectic annular Khovanov homology of a knot to the link Floer homology of the lift of the annular axis in the double branched cover; (ii) the symplectic Khovanov homology of a two-periodic knot to the symplectic annular Khovanov homology of its quotient; and (iii) the symplectic Khovanov homology of a strongly invertible knot to the cone of the axis-moving map between the symplectic annular Khovanov homology of the two resolutions of its quotient.
{"title":"Symplectic annular Khovanov homology and fixed point localizations","authors":"Kristen Hendricks, Cheuk Yu Mak, Sriram Raghunath","doi":"arxiv-2408.06453","DOIUrl":"https://doi.org/arxiv-2408.06453","url":null,"abstract":"We introduce a new version of symplectic annular Khovanov homology and\u0000establish spectral sequences from (i) the symplectic annular Khovanov homology\u0000of a knot to the link Floer homology of the lift of the annular axis in the\u0000double branched cover; (ii) the symplectic Khovanov homology of a two-periodic\u0000knot to the symplectic annular Khovanov homology of its quotient; and (iii) the\u0000symplectic Khovanov homology of a strongly invertible knot to the cone of the\u0000axis-moving map between the symplectic annular Khovanov homology of the two\u0000resolutions of its quotient.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202120","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}
Our purpose here is to adapt the results of Geodesic circle foliations for Reeb flows or Hamiltonian flows on contact manifolds. Consequently, all periods are exactly the same if the contact manifold is connected and all orbits on the contact manifold are closed. We also present concrete examples of periodic flows, all of whose orbits are closed, such as Harmonic oscillators, Lotka-Volterra systems, and others. Lotka-Volterra systems, Reeb flows, and some geodesic flows have non-trivial periods, whereas the periods of Harmonic oscillators and similar systems can be easily obtained through direct calculations. As an application to quantum mechanics, we examine the spectrum of semiclassical Shr"odinger operators. Then we have one of the semiclassical analogies of the Helton-Guillemin theorem.
{"title":"Circle Foliations Revisited: Periods of Flows whose Orbits are all Closed","authors":"Yoshihisa Miyanishi","doi":"arxiv-2408.06056","DOIUrl":"https://doi.org/arxiv-2408.06056","url":null,"abstract":"Our purpose here is to adapt the results of Geodesic circle foliations for\u0000Reeb flows or Hamiltonian flows on contact manifolds. Consequently, all periods\u0000are exactly the same if the contact manifold is connected and all orbits on the\u0000contact manifold are closed. We also present concrete examples of periodic\u0000flows, all of whose orbits are closed, such as Harmonic oscillators,\u0000Lotka-Volterra systems, and others. Lotka-Volterra systems, Reeb flows, and\u0000some geodesic flows have non-trivial periods, whereas the periods of Harmonic\u0000oscillators and similar systems can be easily obtained through direct\u0000calculations. As an application to quantum mechanics, we examine the spectrum\u0000of semiclassical Shr\"odinger operators. Then we have one of the semiclassical\u0000analogies of the Helton-Guillemin theorem.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202129","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 prove the generic existence of spectral networks for a large class of spectral data.
我们为一大类光谱数据证明了光谱网络的一般存在性。
{"title":"On the generic existence of WKB spectral networks/Stokes graphs","authors":"Tatsuki Kuwagaki","doi":"arxiv-2408.05399","DOIUrl":"https://doi.org/arxiv-2408.05399","url":null,"abstract":"We prove the generic existence of spectral networks for a large class of\u0000spectral data.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202141","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}
Nickolas Castro, Gabriel Islambouli, Jie Min, Sümeyra Sakallı, Laura Starkston, Angela Wu
We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings. We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, the Weinstein $mathcal{L}$-invariant, that is a symplectic analogue of the Kirby-Thompson's $mathcal{L}$-invariant of smooth $4$-manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein $mathcal{L}$-invariant. We present topological and geometric constraints of Weinstein domains with $mathcal{L}=0$. We also give two families of examples of multisections with divides that have arbitrarily large $mathcal{L}$-invariant.
{"title":"The contact cut graph and a Weinstein $mathcal{L}$-invariant","authors":"Nickolas Castro, Gabriel Islambouli, Jie Min, Sümeyra Sakallı, Laura Starkston, Angela Wu","doi":"arxiv-2408.05340","DOIUrl":"https://doi.org/arxiv-2408.05340","url":null,"abstract":"We define and study the contact cut graph which is an analogue of Hatcher and\u0000Thurston's cut graph for contact geometry, inspired by contact Heegaard\u0000splittings. We show how oriented paths in the contact cut graph correspond to\u0000Lefschetz fibrations and multisection with divides diagrams. We also give a\u0000correspondence for achiral Lefschetz fibrations. We use these correspondences\u0000to define a new invariant of Weinstein domains, the Weinstein\u0000$mathcal{L}$-invariant, that is a symplectic analogue of the Kirby-Thompson's\u0000$mathcal{L}$-invariant of smooth $4$-manifolds. We discuss the relation of\u0000Lefschetz stabilization with the Weinstein $mathcal{L}$-invariant. We present\u0000topological and geometric constraints of Weinstein domains with\u0000$mathcal{L}=0$. We also give two families of examples of multisections with\u0000divides that have arbitrarily large $mathcal{L}$-invariant.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202130","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}
Williason's theorem states that if $A$ is a $2n times 2n$ real symmetric positive definite matrix then there exists a $2n times 2n$ real symplectic matrix $M$ such that $M^T A M=D oplus D$, where $D$ is an $n times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of $A$. The theorem is known to be generalized to $2n times 2n$ real symmetric positive semidefinite matrices whose kernels are symplectic subspaces of $mathbb{R}^{2n}$, in which case, some of the diagonal entries of $D$ are allowed to be zero. In this paper, we further generalize Williamson's theorem to $2n times 2n$ real symmetric matrices by allowing the diagonal elements of $D$ to be any real numbers, and thus extending the notion of symplectic eigenvalues to real symmetric matrices. Also, we provide an explicit description of symplectic eigenvalues, construct symplectic matrices achieving Williamson's theorem type decomposition, and establish perturbation bounds on symplectic eigenvalues for a class of $2n times 2n$ real symmetric matrices denoted by $operatorname{EigSpSm}(2n)$. The set $operatorname{EigSpSm}(2n)$ contains $2n times 2n$ real symmetric positive semidefinite whose kernels are symplectic subspaces of $mathbb{R}^{2n}$. Our perturbation bounds on symplectic eigenvalues for $operatorname{EigSpSm}(2n)$ generalize known perturbation bounds on symplectic eigenvalues of positive definite matrices given by Bhatia and Jain textit{[J. Math. Phys. 56, 112201 (2015)]}.
{"title":"On generalization of Williamson's theorem to real symmetric matrices","authors":"Hemant K. Mishra","doi":"arxiv-2408.04894","DOIUrl":"https://doi.org/arxiv-2408.04894","url":null,"abstract":"Williason's theorem states that if $A$ is a $2n times 2n$ real symmetric\u0000positive definite matrix then there exists a $2n times 2n$ real symplectic\u0000matrix $M$ such that $M^T A M=D oplus D$, where $D$ is an $n times n$\u0000diagonal matrix with positive diagonal entries known as the symplectic\u0000eigenvalues of $A$. The theorem is known to be generalized to $2n times 2n$\u0000real symmetric positive semidefinite matrices whose kernels are symplectic\u0000subspaces of $mathbb{R}^{2n}$, in which case, some of the diagonal entries of\u0000$D$ are allowed to be zero. In this paper, we further generalize Williamson's\u0000theorem to $2n times 2n$ real symmetric matrices by allowing the diagonal\u0000elements of $D$ to be any real numbers, and thus extending the notion of\u0000symplectic eigenvalues to real symmetric matrices. Also, we provide an explicit\u0000description of symplectic eigenvalues, construct symplectic matrices achieving\u0000Williamson's theorem type decomposition, and establish perturbation bounds on\u0000symplectic eigenvalues for a class of $2n times 2n$ real symmetric matrices\u0000denoted by $operatorname{EigSpSm}(2n)$. The set $operatorname{EigSpSm}(2n)$\u0000contains $2n times 2n$ real symmetric positive semidefinite whose kernels are\u0000symplectic subspaces of $mathbb{R}^{2n}$. Our perturbation bounds on\u0000symplectic eigenvalues for $operatorname{EigSpSm}(2n)$ generalize known\u0000perturbation bounds on symplectic eigenvalues of positive definite matrices\u0000given by Bhatia and Jain textit{[J. Math. Phys. 56, 112201 (2015)]}.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937964","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}
For an oriented manifold $M$ and a compact subanalytic Legendrian $Lambda subseteq S^*M$, we construct a canonical strong smooth relative Calabi--Yau structure on the microlocalization at infinity and its left adjoint $m_Lambda^l: operatorname{mu sh}_Lambda(Lambda) rightleftharpoons operatorname{Sh}_Lambda(M)_0 : m_Lambda$ between compactly supported sheaves on $M$ with singular support on $Lambda$ and microsheaves on $Lambda$. We also construct a canonical strong Calabi-Yau structure on microsheaves $operatorname{mu sh}_Lambda(Lambda)$. Our approach does not require local properness and hence does not depend on arborealization. We thus obtain a canonical smooth relative Calabi-Yau structure on the Orlov functor for wrapped Fukaya categories of cotangent bundles with Weinstein stops, such that the wrap-once functor is the inverse dualizing bimodule.
{"title":"Relative Calabi-Yau structure on microlocalization","authors":"Christopher Kuo, Wenyuan Li","doi":"arxiv-2408.04085","DOIUrl":"https://doi.org/arxiv-2408.04085","url":null,"abstract":"For an oriented manifold $M$ and a compact subanalytic Legendrian $Lambda\u0000subseteq S^*M$, we construct a canonical strong smooth relative Calabi--Yau\u0000structure on the microlocalization at infinity and its left adjoint\u0000$m_Lambda^l: operatorname{mu sh}_Lambda(Lambda) rightleftharpoons\u0000operatorname{Sh}_Lambda(M)_0 : m_Lambda$ between compactly supported sheaves\u0000on $M$ with singular support on $Lambda$ and microsheaves on $Lambda$. We\u0000also construct a canonical strong Calabi-Yau structure on microsheaves\u0000$operatorname{mu sh}_Lambda(Lambda)$. Our approach does not require local\u0000properness and hence does not depend on arborealization. We thus obtain a\u0000canonical smooth relative Calabi-Yau structure on the Orlov functor for wrapped\u0000Fukaya categories of cotangent bundles with Weinstein stops, such that the\u0000wrap-once functor is the inverse dualizing bimodule.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969023","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 study homological mirror symmetry for $(mathbb{P}^2, Omega)$ viewed as an object of birational geometry, with $Omega$ the standard meromorphic volume form. First, we construct universal objects on the two sides of mirror symmetry, focusing on the exact symplectic setting: a smooth complex scheme $U_mathrm{univ}$ and a Weinstein manifold $M_mathrm{univ}$, both of infinite type; and we prove homological mirror symmetry for them. Second, we consider autoequivalences. We prove that automorphisms of $U_mathrm{univ}$ are given by a natural discrete subgroup of $operatorname{Bir} (mathbb{P}^2, pm Omega)$; and that all of these automorphisms are mirror to symplectomorphisms of $M_mathrm{univ}$. We conclude with some applications.
{"title":"A universal mirror to $(mathbb{P}^2, Ω)$ as a birational object","authors":"Ailsa Keating, Abigail Ward","doi":"arxiv-2408.03764","DOIUrl":"https://doi.org/arxiv-2408.03764","url":null,"abstract":"We study homological mirror symmetry for $(mathbb{P}^2, Omega)$ viewed as\u0000an object of birational geometry, with $Omega$ the standard meromorphic volume\u0000form. First, we construct universal objects on the two sides of mirror\u0000symmetry, focusing on the exact symplectic setting: a smooth complex scheme\u0000$U_mathrm{univ}$ and a Weinstein manifold $M_mathrm{univ}$, both of infinite\u0000type; and we prove homological mirror symmetry for them. Second, we consider\u0000autoequivalences. We prove that automorphisms of $U_mathrm{univ}$ are given by\u0000a natural discrete subgroup of $operatorname{Bir} (mathbb{P}^2, pm Omega)$;\u0000and that all of these automorphisms are mirror to symplectomorphisms of\u0000$M_mathrm{univ}$. We conclude with some applications.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938003","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 consider chiral, generally nonlinear density waves in one dimension, modelling the bosonized edge modes of a two-dimensional fermionic topological insulator. Using the coincidence between bosonization and Lie-Poisson dynamics on an affine U(1) group, we show that wave profiles which are periodic in time produce Berry phases accumulated by the underlying fermionic field. These phases can be evaluated in closed form for any Hamiltonian, and they serve as a diagnostic of nonlinearity. As an explicit example, we discuss the Korteweg-de Vries equation, viewed as a model of nonlinear quantum Hall edge modes.
{"title":"Berry Phases in the Bosonization of Nonlinear Edge Modes","authors":"Mathieu Beauvillain, Blagoje Oblak, Marios Petropoulos","doi":"arxiv-2408.03991","DOIUrl":"https://doi.org/arxiv-2408.03991","url":null,"abstract":"We consider chiral, generally nonlinear density waves in one dimension,\u0000modelling the bosonized edge modes of a two-dimensional fermionic topological\u0000insulator. Using the coincidence between bosonization and Lie-Poisson dynamics\u0000on an affine U(1) group, we show that wave profiles which are periodic in time\u0000produce Berry phases accumulated by the underlying fermionic field. These\u0000phases can be evaluated in closed form for any Hamiltonian, and they serve as a\u0000diagnostic of nonlinearity. As an explicit example, we discuss the Korteweg-de\u0000Vries equation, viewed as a model of nonlinear quantum Hall edge modes.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938005","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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