In the SYZ program, the mirror of (Y) is the moduli space of Lagrangian branes in (Y). When (Y) is equipped with a Hamiltonian (G)-action, we prove that its mirror determines a canonical complex Lagrangian subvariety in the Coulomb branch of the 3d (mathcal{N}=4) pure (G)-gauge theory.
{"title":"SYZ Mirrors in non-Abelian 3d Mirror Symmetry","authors":"Ki Fung Chan, Naichung Conan Leung","doi":"arxiv-2408.09479","DOIUrl":"https://doi.org/arxiv-2408.09479","url":null,"abstract":"In the SYZ program, the mirror of (Y) is the moduli space of Lagrangian\u0000branes in (Y). When (Y) is equipped with a Hamiltonian (G)-action, we\u0000prove that its mirror determines a canonical complex Lagrangian subvariety in\u0000the Coulomb branch of the 3d (mathcal{N}=4) pure (G)-gauge theory.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202101","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}
Matthew Strom Borman, Mohamed El Alami, Nick Sheridan
We construct the $L_infty$ structure on symplectic cohomology of a Liouville domain, together with an enhancement of the closed--open map to an $L_infty$ homomorphism from symplectic cochains to Hochschild cochains on the wrapped Fukaya category. Features of our construction are that it respects a modified action filtration (in contrast to Pomerleano--Seidel's construction); it uses a compact telescope model (in contrast to Abouzaid--Groman--Varolgunes' construction); and it is adapted to the purposes of our follow-up work where we construct Maurer--Cartan elements in symplectic cochains which are associated to a normal-crossings compactification of the Liouville domain.
{"title":"An $L_infty$ structure on symplectic cohomology","authors":"Matthew Strom Borman, Mohamed El Alami, Nick Sheridan","doi":"arxiv-2408.09163","DOIUrl":"https://doi.org/arxiv-2408.09163","url":null,"abstract":"We construct the $L_infty$ structure on symplectic cohomology of a Liouville\u0000domain, together with an enhancement of the closed--open map to an $L_infty$\u0000homomorphism from symplectic cochains to Hochschild cochains on the wrapped\u0000Fukaya category. Features of our construction are that it respects a modified\u0000action filtration (in contrast to Pomerleano--Seidel's construction); it uses a\u0000compact telescope model (in contrast to Abouzaid--Groman--Varolgunes'\u0000construction); and it is adapted to the purposes of our follow-up work where we\u0000construct Maurer--Cartan elements in symplectic cochains which are associated\u0000to a normal-crossings compactification of the Liouville domain.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202103","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 characterize when some small Seifert fibered spaces can be the convex boundary of a symplectic rational homology ball and give strong restrictions for others to bound such manifolds. As part of this, we show that the only spherical $3$-manifolds that are the boundary of a symplectic rational homology ball are the lens spaces $L(p^2,pq-1)$ found by Lisca and give evidence for the Gompf conjecture that Brieskorn spheres do not bound Stein domains in C^2. We also find restrictions on Lagrangian disk fillings of some Legendrian knots in small Seifert fibered spaces.
{"title":"Symplectic rational homology ball fillings of Seifert fibered spaces","authors":"John B. Etnyre, Burak Ozbagci, Bülent Tosun","doi":"arxiv-2408.09292","DOIUrl":"https://doi.org/arxiv-2408.09292","url":null,"abstract":"We characterize when some small Seifert fibered spaces can be the convex\u0000boundary of a symplectic rational homology ball and give strong restrictions\u0000for others to bound such manifolds. As part of this, we show that the only\u0000spherical $3$-manifolds that are the boundary of a symplectic rational homology\u0000ball are the lens spaces $L(p^2,pq-1)$ found by Lisca and give evidence for the\u0000Gompf conjecture that Brieskorn spheres do not bound Stein domains in C^2. We\u0000also find restrictions on Lagrangian disk fillings of some Legendrian knots in\u0000small Seifert fibered spaces.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202106","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}
Matthew Strom Borman, Mohamed El Alami, Nick Sheridan
We prove that under certain conditions, a normal crossings compactification of a Liouville domain determines a Maurer--Cartan element for the $L_infty$ structure on its symplectic cohomology; and deforming by this element gives the quantum cohomology of the compactification.
{"title":"Maurer--Cartan elements in symplectic cohomology from compactifications","authors":"Matthew Strom Borman, Mohamed El Alami, Nick Sheridan","doi":"arxiv-2408.09221","DOIUrl":"https://doi.org/arxiv-2408.09221","url":null,"abstract":"We prove that under certain conditions, a normal crossings compactification\u0000of a Liouville domain determines a Maurer--Cartan element for the $L_infty$\u0000structure on its symplectic cohomology; and deforming by this element gives the\u0000quantum cohomology of the compactification.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202102","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}
Lev Buhovsky, Ben Feuerstein, Leonid Polterovich, Egor Shelukhin
We prove that autonomous Hamiltonian flows on the two-sphere exhibit the following dichotomy: the Hofer norm either grows linearly or is bounded in time by a universal constant C. Our approach involves a new technique, Hamiltonian symmetrization. Essentially, we prove that every autonomous Hamiltonian diffeomorphism is conjugate to an element C-close in the Hofer metric to one generated by a function of the height.
我们证明了二球体上的自发哈密顿流表现出以下二分法：霍弗规范要么线性增长，要么在时间上受一个普遍常数 C 的约束。从本质上讲，我们证明了每一个自发的哈密顿非同形都与霍弗公设中的一个元素 C 共轭，该元素与高度的一个函数生成的元素 C 接近。
{"title":"A dichotomy for the Hofer growth of area preserving maps on the sphere via symmetrization","authors":"Lev Buhovsky, Ben Feuerstein, Leonid Polterovich, Egor Shelukhin","doi":"arxiv-2408.08854","DOIUrl":"https://doi.org/arxiv-2408.08854","url":null,"abstract":"We prove that autonomous Hamiltonian flows on the two-sphere exhibit the\u0000following dichotomy: the Hofer norm either grows linearly or is bounded in time\u0000by a universal constant C. Our approach involves a new technique, Hamiltonian\u0000symmetrization. Essentially, we prove that every autonomous Hamiltonian\u0000diffeomorphism is conjugate to an element C-close in the Hofer metric to one\u0000generated by a function of the height.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202105","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 a monotone symplectic manifold and a smooth anticanonical divisor, there is a formal deformation of the symplectic cohomology of the divisor complement, defined by allowing Floer cylinders to intersect the divisor. We compute this deformed symplectic cohomology, in terms of the ordinary cohomology of the manifold and divisor; and also describe some additional structures that it carries.
{"title":"Symplectic cohomology relative to a smooth anticanonical divisor","authors":"Daniel Pomerleano, Paul Seidel","doi":"arxiv-2408.09039","DOIUrl":"https://doi.org/arxiv-2408.09039","url":null,"abstract":"For a monotone symplectic manifold and a smooth anticanonical divisor, there\u0000is a formal deformation of the symplectic cohomology of the divisor complement,\u0000defined by allowing Floer cylinders to intersect the divisor. We compute this\u0000deformed symplectic cohomology, in terms of the ordinary cohomology of the\u0000manifold and divisor; and also describe some additional structures that it\u0000carries.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202104","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}
The Algebra of the Infrared cite{Gaiotto:2015aoa} is a framework to construct local observables, interfaces, and categories of supersymmetric boundary conditions of massive $mathcal{N}=(2,2)$ theories in two dimensions by using information only about the BPS sector. The resulting framework is known as the ``web-based formalism.'' In this paper we initiate the generalization of the web-based formalism to include a much wider class of $mathcal{N}=(2,2)$ quantum field theories than was discussed in cite{Gaiotto:2015aoa}: theories with non-trivial twisted masses. The essential new ingredient is the presence of BPS particles within a fixed vacuum sector. In this paper we work out the web-based formalism for the simplest class of theories that allow for such BPS particles: theories with a single vacuum and a single twisted mass. We show that even in this simple setting there are interesting new phenomenon including the emergence of Fock spaces of closed solitons and a natural appearance of Koszul dual algebras. Mathematically, studying theories with twisted masses includes studying the Fukaya-Seidel category of A-type boundary conditions for Landau-Ginzburg models defined by a closed holomorphic one-form. This paper sketches a web-based construction for the category of A-type boundary conditions for one-forms with a single Morse zero and a single non-trivial period. We demonstrate our formalism explicitly in a particularly instructive example.
红外代数（Algebra of the Infrared cite{Gaiotto:2015aoa}）是一个框架，用于仅利用BPS部门的信息来构建二维中大质量$mathcal{N}=(2,2)$理论的局部观测值、界面和超对称边界条件类别。由此产生的框架被称为 "基于网络的形式主义"。在本文中，我们开始对基于网络的形式主义进行概括，以包括比（cite{Gaiotto:2015aoa}中讨论的更广泛的一类$mathcal{N}=(2,2)$量子场论：具有非三维扭曲质量的理论。新的基本要素是在一个固定的真空扇区中存在BPS粒子。在本文中，我们针对允许存在这种BPS粒子的最简单理论类别：具有单一真空和单一扭曲质量的理论，建立了基于网络的形式主义。我们的研究表明，即使在这种简单的环境中，也会出现一些有趣的新现象，包括封闭玻色子的福克空间的出现，以及科斯祖尔对偶代数的自然出现。在数学上，研究具有扭曲质量的理论包括研究由封闭全形一形式定义的朗道-金兹堡模型的 A 型边界条件的 Fukaya-Seidelcategory 。本文勾画了一个基于网络的A型边界条件类别的构造，该类别适用于具有单个莫尔兹零点和单个非三维周期的单形式。我们在一个特别有启发性的例子中明确演示了我们的形式主义。
{"title":"On the Algebra of the Infrared with Twisted Masses","authors":"Ahsan Z. Khan, Gregory W. Moore","doi":"arxiv-2408.08372","DOIUrl":"https://doi.org/arxiv-2408.08372","url":null,"abstract":"The Algebra of the Infrared cite{Gaiotto:2015aoa} is a framework to\u0000construct local observables, interfaces, and categories of supersymmetric\u0000boundary conditions of massive $mathcal{N}=(2,2)$ theories in two dimensions\u0000by using information only about the BPS sector. The resulting framework is\u0000known as the ``web-based formalism.'' In this paper we initiate the\u0000generalization of the web-based formalism to include a much wider class of\u0000$mathcal{N}=(2,2)$ quantum field theories than was discussed in\u0000cite{Gaiotto:2015aoa}: theories with non-trivial twisted masses. The essential\u0000new ingredient is the presence of BPS particles within a fixed vacuum sector.\u0000In this paper we work out the web-based formalism for the simplest class of\u0000theories that allow for such BPS particles: theories with a single vacuum and a\u0000single twisted mass. We show that even in this simple setting there are\u0000interesting new phenomenon including the emergence of Fock spaces of closed\u0000solitons and a natural appearance of Koszul dual algebras. Mathematically,\u0000studying theories with twisted masses includes studying the Fukaya-Seidel\u0000category of A-type boundary conditions for Landau-Ginzburg models defined by a\u0000closed holomorphic one-form. This paper sketches a web-based construction for\u0000the category of A-type boundary conditions for one-forms with a single Morse\u0000zero and a single non-trivial period. We demonstrate our formalism explicitly\u0000in a particularly instructive example.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202241","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 paper, we construct examples of exact Lagrangians (of "locally conformally symplectic" type) in cotangent bundles of closed manifolds with locally conformally symplectic structures and give conditions under which the projection induces a simple homotopy equivalence between an exact Lagrangian and the $0$-section of the cotangent bundle. This line of questioning follows in the footsteps of Abouzaid and Kragh, and more generally of the Arnol'd conjecture. Notably, we will see that while exact Lagrangians cannot be spheres in this setting, a naive adaptation of the Abouzaid-Kragh theorem does not hold in this generalization.
{"title":"On the projection of exact Lagrangians in locally conformally symplectic geometry","authors":"Adrien Currier","doi":"arxiv-2408.07760","DOIUrl":"https://doi.org/arxiv-2408.07760","url":null,"abstract":"In this paper, we construct examples of exact Lagrangians (of \"locally\u0000conformally symplectic\" type) in cotangent bundles of closed manifolds with\u0000locally conformally symplectic structures and give conditions under which the\u0000projection induces a simple homotopy equivalence between an exact Lagrangian\u0000and the $0$-section of the cotangent bundle. This line of questioning follows\u0000in the footsteps of Abouzaid and Kragh, and more generally of the Arnol'd\u0000conjecture. Notably, we will see that while exact Lagrangians cannot be spheres\u0000in this setting, a naive adaptation of the Abouzaid-Kragh theorem does not hold\u0000in this generalization.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202107","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}
The spectral diameter of a symplectic ball is shown to be equal to its capacity; this result upgrades the known bound by a factor of two and yields a simple formula for the spectral diameter of a symplectic ellipsoid. We also study the relationship between the spectral diameter and packings by two balls.
{"title":"The spectral diameter of a symplectic ellipsoid","authors":"Habib Alizadeh, Marcelo S. Atallah, Dylan Cant","doi":"arxiv-2408.07214","DOIUrl":"https://doi.org/arxiv-2408.07214","url":null,"abstract":"The spectral diameter of a symplectic ball is shown to be equal to its\u0000capacity; this result upgrades the known bound by a factor of two and yields a\u0000simple formula for the spectral diameter of a symplectic ellipsoid. We also\u0000study the relationship between the spectral diameter and packings by two balls.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202209","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 arXiv:2011.06562, the first author and Otto van Koert proved a generalized version of the classical Poincar'e-Birkhoff theorem, for Liouville domains of any dimension. In this article, we prove a relative version for Lagrangians with Legendrian boundary. This gives interior chords of arbitrary large length, provided the twist condition introduced in arXiv:2011.06562 is satisfied. The motivation comes from finding spatial consecutive collision orbits of arbitrary large length in the spatial circular restricted three-body problem, which are relevant for gravitational assist in the context of orbital mechanics. This is an application of a local version of wrapped Floer homology, which we introduce as the open string analogue of local Floer homology for closed strings.
在 arXiv:2011.06562 中，第一作者和 Otto van Koert 证明了经典 Poincar'e-Birkhoff 定理的广义版本，适用于任意维数的 Liouville 域。在本文中，我们证明了具有 Legendrian 边界的拉格朗日的相对版本。只要满足 arXiv:2011.06562 中引入的扭转条件，就能得到任意大长度的内部弦。其动机来自于在空间圆受限三体问题中寻找任意大长度的空间连续碰撞轨道，这与轨道力学背景下的引力辅助有关。这是包裹弗洛尔同源性局部版本的应用，我们将其引入为封闭弦的局部弗洛尔同源性的开弦类似物。
{"title":"A Relative Poincaré-Birkhoff theorem","authors":"Agustin Moreno, Arthur Limoge","doi":"arxiv-2408.06919","DOIUrl":"https://doi.org/arxiv-2408.06919","url":null,"abstract":"In arXiv:2011.06562, the first author and Otto van Koert proved a generalized\u0000version of the classical Poincar'e-Birkhoff theorem, for Liouville domains of\u0000any dimension. In this article, we prove a relative version for Lagrangians\u0000with Legendrian boundary. This gives interior chords of arbitrary large length,\u0000provided the twist condition introduced in arXiv:2011.06562 is satisfied. The\u0000motivation comes from finding spatial consecutive collision orbits of arbitrary\u0000large length in the spatial circular restricted three-body problem, which are\u0000relevant for gravitational assist in the context of orbital mechanics. This is\u0000an application of a local version of wrapped Floer homology, which we introduce\u0000as the open string analogue of local Floer homology for closed strings.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202108","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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