Pub Date : 2024-06-03DOI: 10.4310/jsg.2023.v21.n5.a4
Timothy,Perutz, Nick,Sheridan
We give a definition of Seidel’s ‘relative Fukaya category’, for a smooth complex projective variety relative to a simple normal crossings divisor, under a semipositivity assumption. We use the Cieliebak–Mohnke approach to transversality via stabilizing divisors. Two features of our construction are noteworthy: that we work relative to a normal crossings divisor which supports an effective ample divisor but need not have ample components; and that our relative Fukaya category is linear over a certain ring of multivariate power series with integer coefficients.
{"title":"Constructing the relative Fukaya category","authors":"Timothy,Perutz, Nick,Sheridan","doi":"10.4310/jsg.2023.v21.n5.a4","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n5.a4","url":null,"abstract":"We give a definition of Seidel’s ‘relative Fukaya category’, for a smooth complex projective variety relative to a simple normal crossings divisor, under a semipositivity assumption. We use the Cieliebak–Mohnke approach to transversality via stabilizing divisors. Two features of our construction are noteworthy: that we work relative to a normal crossings divisor which supports an effective ample divisor but need not have ample components; and that our relative Fukaya category is linear over a certain ring of multivariate power series with integer coefficients.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"72 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252222","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 : 2024-06-03DOI: 10.4310/jsg.2023.v21.n5.a3
Nicolò,Drago, Stefan,Waldmann
In this paper we study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and their relation with the underlying Poisson geometry in analogy to Weinstein’s seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, we focus on the case of $b$-Poisson manifolds, where we provide an almost complete characterization of the convex cone of KMS measures.
{"title":"Classical KMS functionals and phase transitions in Poisson geometry","authors":"Nicolò,Drago, Stefan,Waldmann","doi":"10.4310/jsg.2023.v21.n5.a3","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n5.a3","url":null,"abstract":"In this paper we study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and their relation with the underlying Poisson geometry in analogy to Weinstein’s seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, we focus on the case of $b$-Poisson manifolds, where we provide an almost complete characterization of the convex cone of KMS measures.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141252244","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 : 2023-12-22DOI: 10.4310/jsg.2023.v21.n4.a1
Kaoru Ikeda
We consider a new orbit method for unitary representations which determines the explicit values of the multiplicities of the irreducible components of unitary representations of the connected Lie groups. We provide the polarized symplectic affine space on which the Lie group acts. This polarization is obtained by the Hamiltonian flows of the full Kostant–Toda lattice. The Hamiltonian flows of the ordinary Toda lattice are not sufficient for constructing this polarization. In this paper we do an experiment on the case of the unitary representations of the Heisenberg groups. The irreducible representations of the Heisenberg group are obtained and classified by $mathbb{R}$ by the Stone–von Nuemann theorem. The multiplicities are obtained by using spontaneous symmetry breaking and Sato hyper‑functions.
{"title":"Applications of the full Kostant–Toda lattice and hyper-functions to unitary representations of the Heisenberg groups","authors":"Kaoru Ikeda","doi":"10.4310/jsg.2023.v21.n4.a1","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n4.a1","url":null,"abstract":"We consider a new orbit method for unitary representations which determines the explicit values of the multiplicities of the irreducible components of unitary representations of the connected Lie groups. We provide the polarized symplectic affine space on which the Lie group acts. This polarization is obtained by the Hamiltonian flows of the full Kostant–Toda lattice. The Hamiltonian flows of the ordinary Toda lattice are not sufficient for constructing this polarization. In this paper we do an experiment on the case of the unitary representations of the Heisenberg groups. The irreducible representations of the Heisenberg group are obtained and classified by $mathbb{R}$ by the Stone–von Nuemann theorem. The multiplicities are obtained by using spontaneous symmetry breaking and Sato hyper‑functions.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"33 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139022407","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 : 2023-12-22DOI: 10.4310/jsg.2023.v21.n4.a4
Vicente Cortés, Liana David
We develop a theory of $T$-duality for transitive Courant algebroids. We show that $T$-duality between transitive Courant algebroids $E to M$ and $tilde{E} to tilde{M}$ induces a map between the spaces of sections of the corresponding canonical weighted spinor bundles $mathbb{S}_E$ and $mathbb{S}_tilde{E}$ intertwining the canonical Dirac generating operators. The map is shown to induce an isomorphism between the spaces of invariant spinors, compatible with an isomorphism between the spaces of invariant sections of the Courant algebroids. The notion of invariance is defined after lifting the vertical parallelisms of the underlying torus bundles $M to B$ and $tilde{M} to B$ to the Courant algebroids and their spinor bundles. We prove a general existence result for $T$-duals under assumptions generalizing the cohomological integrality conditions for $T$-duality in the exact case. Specializing our construction, we find that the $T$-dual of an exact or a heterotic Courant algebroid is again exact or heterotic, respectively.
我们建立了一个关于反式库朗梯形的 $T$ 对偶性理论。我们证明，跨库仑梯形 $E to M$ 和 $tilde{E} to tilde{M}$ 之间的 $T$ 对偶性诱导了相应的规范加权簇的截面空间之间的映射。到 tilde{M}$ 之间诱导出一个映射，这个映射是相应的佳能加权旋量束 $mathbb{S}_E$ 和 $mathbb{S}_tilde{E}$ 交织佳能狄拉克生成算子的截面空间。结果表明，该映射诱导了不变旋量空间之间的同构，与库朗梯形不变截面空间之间的同构相容。不变性的概念是在把底层环束 $M to B$ 和 $tilde{M} to B$ 的垂直平行线提升到 Courant algebroids 之后定义的。到 B$ 到库朗特实体及其旋量束。我们证明了$T$对偶的一般存在性结果，其假设条件概括了精确情况下$T$对偶的同调积分条件。将我们的构造特殊化后，我们发现精确库仑矢或异质库仑矢的 $T$ 二重又分别是精确的或异质的。
{"title":"$T$-duality for transitive Courant algebroids","authors":"Vicente Cortés, Liana David","doi":"10.4310/jsg.2023.v21.n4.a4","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n4.a4","url":null,"abstract":"We develop a theory of $T$-duality for transitive Courant algebroids. We show that $T$-duality between transitive Courant algebroids $E to M$ and $tilde{E} to tilde{M}$ induces a map between the spaces of sections of the corresponding canonical weighted spinor bundles $mathbb{S}_E$ and $mathbb{S}_tilde{E}$ intertwining the canonical Dirac generating operators. The map is shown to induce an isomorphism between the spaces of invariant spinors, compatible with an isomorphism between the spaces of invariant sections of the Courant algebroids. The notion of invariance is defined after lifting the vertical parallelisms of the underlying torus bundles $M to B$ and $tilde{M} to B$ to the Courant algebroids and their spinor bundles. We prove a general existence result for $T$-duals under assumptions generalizing the cohomological integrality conditions for $T$-duality in the exact case. Specializing our construction, we find that the $T$-dual of an exact or a heterotic Courant algebroid is again exact or heterotic, respectively.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139022443","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 : 2023-12-22DOI: 10.4310/jsg.2023.v21.n4.a2
Bahar Acu, John B. Etnyre, Burak Ozbagci
Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions.We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer $m gt 3$, any finitely presented group can be realized as the fundamental group of some iterated planar contact manifold of dimension $m$. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps.
{"title":"Generalizations of planar contact manifolds to higher dimensions","authors":"Bahar Acu, John B. Etnyre, Burak Ozbagci","doi":"10.4310/jsg.2023.v21.n4.a2","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n4.a2","url":null,"abstract":"Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions.We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer $m gt 3$, any finitely presented group can be realized as the fundamental group of some iterated planar contact manifold of dimension $m$. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139022376","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 : 2023-12-22DOI: 10.4310/jsg.2023.v21.n4.a3
Urs Frauenfelder, Agustin Moreno
In this article, we will introduce a collection of tools aimed at studying periodic orbits of Hamiltonian systems, their (linear) stability, and their bifurcations. We will provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by anti-symplectic involutions (e.g. the circular restricted three-body problem). Such cylinders induce continuous paths which do not cross the bifurcation locus of suitable GIT quotients of the symplectic group, which are branched manifolds whose topology provide the desired obstructions. Namely, the complement of the corresponding loci consist of several connected components which we enumerate and explicitly describe; by construction these cannot be joined by a path induced by an orbit cylinder. We also provide preferred normal forms for each regular and singular component. We further introduce a notion of signature for symmetric orbits, which extends the notion from Krein theory (which only applies for elliptic orbits), to allow also for the case of symmetric orbits which are hyperbolic. This signature helps predict at which points of a symmetric orbit a bifurcation arises. This gives a general theoretical framework for the study of stability and bifurcations of symmetric orbits, with a view towards practical and numerical implementations within the context of space mission design. This is the subject of the follow-up paper $href{https://doi.org/10.1007/978-3-319-72278-8}{[7]}$, where this framework is supported by numerical work.
{"title":"On GIT quotients of the symplectic group, stability and bifurcations of periodic orbits (with a view towards practical applications)","authors":"Urs Frauenfelder, Agustin Moreno","doi":"10.4310/jsg.2023.v21.n4.a3","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n4.a3","url":null,"abstract":"In this article, we will introduce a collection of tools aimed at studying periodic orbits of Hamiltonian systems, their (linear) stability, and their bifurcations. We will provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by anti-symplectic involutions (e.g. the circular restricted three-body problem). Such cylinders induce continuous paths which do not cross the <i> bifurcation locus</i> of suitable GIT quotients of the symplectic group, which are branched manifolds whose topology provide the desired obstructions. Namely, the complement of the corresponding loci consist of several connected components which we enumerate and explicitly describe; by construction these cannot be joined by a path induced by an orbit cylinder. We also provide preferred normal forms for each regular and singular component. We further introduce a notion of signature for symmetric orbits, which extends the notion from Krein theory (which only applies for elliptic orbits), to allow also for the case of symmetric orbits which are hyperbolic. This signature helps predict at which points of a symmetric orbit a bifurcation arises. This gives a general theoretical framework for the study of stability and bifurcations of symmetric orbits, with a view towards practical and numerical implementations within the context of space mission design. This is the subject of the follow-up paper $href{https://doi.org/10.1007/978-3-319-72278-8}{[7]}$, where this framework is supported by numerical work.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139022404","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 : 2023-09-28DOI: 10.4310/jsg.2023.v21.n2.a2
Russell Avdek
We study moduli spaces $mathcal{M}$ of holomorphic maps $U$ to $mathbb{R}^4$ with boundaries on the Lagrangian cylinder over a Legendrian link $Lambda subset (mathbb{R}^3, xi_{std})$. We allow our domains, $dot{Sigma}$ , to have non-trivial topology in which case $mathcal{M}$ is the zero locus of an obstruction function $mathcal{O}$, sending a moduli space of holomorphic maps in $mathbb{C}$ to $H^1 (dot{Sigma})$. In general, $mathcal{O}^{-1} (0)$ is not combinatorially computable. However after a Legendrian isotopy $Lambda$ can be made left-right-simple, implying that any $U$ 1) of index $1$ is a disk with one or two positive punctures for which $pi_mathbb{C} circ U$ is an embedding. 2) of index $2$ is either a disk or an annulus with $pi_mathbb{C} circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $Lambda$ is combinatorially computable using only disks with $leq 2$ positive punctures.
{"title":"Simplified SFT moduli spaces for Legendrian links","authors":"Russell Avdek","doi":"10.4310/jsg.2023.v21.n2.a2","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n2.a2","url":null,"abstract":"We study moduli spaces $mathcal{M}$ of holomorphic maps $U$ to $mathbb{R}^4$ with boundaries on the Lagrangian cylinder over a Legendrian link $Lambda subset (mathbb{R}^3, xi_{std})$. We allow our domains, $dot{Sigma}$ , to have non-trivial topology in which case $mathcal{M}$ is the zero locus of an obstruction function $mathcal{O}$, sending a moduli space of holomorphic maps in $mathbb{C}$ to $H^1 (dot{Sigma})$. In general, $mathcal{O}^{-1} (0)$ is not combinatorially computable. However after a Legendrian isotopy $Lambda$ can be made <i>left-right-simple</i>, implying that any $U$ 1) of index $1$ is a disk with one or two positive punctures for which $pi_mathbb{C} circ U$ is an embedding. 2) of index $2$ is either a disk or an annulus with $pi_mathbb{C} circ U$ simply covered and without interior critical points. Therefore any SFT invariant of $Lambda$ is combinatorially computable using only disks with $leq 2$ positive punctures.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"20 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536588","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 : 2023-04-24DOI: 10.4310/jsg.2022.v20.n5.a2
Fan Ding, Youlin Li, Zhongtao Wu
In this paper, sufficient conditions for contact $(+1)$-surgeries along Legendrian knots in contact rational homology $3$-spheres to have vanishing contact invariants or to be overtwisted are given. They can be applied to study contact $(pm 1)$-surgeries along Legendrian links in the standard contact $3$-sphere. We also obtain a sufficient condition for contact $(+1)$-surgeries along Legendrian twocomponent links in the standard contact $3$-sphere to be overtwisted via their front projections.
{"title":"Contact $(+1)$-surgeries on rational homology $3$-spheres","authors":"Fan Ding, Youlin Li, Zhongtao Wu","doi":"10.4310/jsg.2022.v20.n5.a2","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n5.a2","url":null,"abstract":"In this paper, sufficient conditions for contact $(+1)$-surgeries along Legendrian knots in contact rational homology $3$-spheres to have vanishing contact invariants or to be overtwisted are given. They can be applied to study contact $(pm 1)$-surgeries along Legendrian links in the standard contact $3$-sphere. We also obtain a sufficient condition for contact $(+1)$-surgeries along Legendrian twocomponent links in the standard contact $3$-sphere to be overtwisted via their front projections.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536589","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 : 2023-01-01DOI: 10.4310/jsg.2023.v21.n2.a3
Daniel Álvarez-Gavela, Yakov Eliashberg, David Nadler
{"title":"Arboreal models and their stability","authors":"Daniel Álvarez-Gavela, Yakov Eliashberg, David Nadler","doi":"10.4310/jsg.2023.v21.n2.a3","DOIUrl":"https://doi.org/10.4310/jsg.2023.v21.n2.a3","url":null,"abstract":"","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135801412","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：