Pub Date : 2021-04-12DOI: 10.4310/JSG.2022.v20.n6.a2
I. Biswas, J. Hurtubise, L. Jeffrey, Sean Lawton
Let G be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between G-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when G=SL(2,C). We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the 5-holed sphere, the first example for an Euler characteristic -3 surface.
{"title":"Poisson maps between character varieties: gluing and capping","authors":"I. Biswas, J. Hurtubise, L. Jeffrey, Sean Lawton","doi":"10.4310/JSG.2022.v20.n6.a2","DOIUrl":"https://doi.org/10.4310/JSG.2022.v20.n6.a2","url":null,"abstract":"Let G be a compact Lie group or a complex reductive affine algebraic group. We explore induced mappings between G-character varieties of surface groups by mappings between corresponding surfaces. It is shown that these mappings are generally Poisson. We also given an effective algorithm to compute the Poisson bi-vectors when G=SL(2,C). We demonstrate this algorithm by explicitly calculating the Poisson bi-vector for the 5-holed sphere, the first example for an Euler characteristic -3 surface.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81974045","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 : 2021-02-15DOI: 10.4310/jsg.2022.v20.n5.a6
Shira Tanny
We study the relation between spectral invariants of disjointly supported Hamiltonians and of their sum. On aspherical manifolds, such a relation was established by Humili`ere, Le Roux and Seyfaddini. We show that a weaker statement holds in a wider setting, and derive applications to Polterovich's Poisson bracket invariant and to Entov and Polterovich's notion of superheavy sets.
{"title":"A max inequality for spectral invariants of disjointly supported Hamiltonians","authors":"Shira Tanny","doi":"10.4310/jsg.2022.v20.n5.a6","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n5.a6","url":null,"abstract":"We study the relation between spectral invariants of disjointly supported Hamiltonians and of their sum. On aspherical manifolds, such a relation was established by Humili`ere, Le Roux and Seyfaddini. We show that a weaker statement holds in a wider setting, and derive applications to Polterovich's Poisson bracket invariant and to Entov and Polterovich's notion of superheavy sets.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86228840","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 : 2021-02-09DOI: 10.4310/JSG.2022.v20.n3.a1
J. Asplund, T. Ekholm
The Chekanov-Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams. It also leads to a proof of a conjectured isomorphism between partially wrapped Floer cohomology and Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space.
{"title":"Chekanov-Eliashberg $mathrm{dg}$-algebras for singular Legendrians","authors":"J. Asplund, T. Ekholm","doi":"10.4310/JSG.2022.v20.n3.a1","DOIUrl":"https://doi.org/10.4310/JSG.2022.v20.n3.a1","url":null,"abstract":"The Chekanov-Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams. It also leads to a proof of a conjectured isomorphism between partially wrapped Floer cohomology and Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78793774","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 : 2021-01-01DOI: 10.4310/JSG.2021.V19.N1.A1
M. Braverman, Yiannis Loizides, Yanli Song
We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``quantization commutes with reduction'' property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper.
{"title":"Geometric quantization of $b$-symplectic manifolds","authors":"M. Braverman, Yiannis Loizides, Yanli Song","doi":"10.4310/JSG.2021.V19.N1.A1","DOIUrl":"https://doi.org/10.4310/JSG.2021.V19.N1.A1","url":null,"abstract":"We introduce a method of geometric quantization for compact $b$-symplectic manifolds in terms of the index of an Atiyah-Patodi-Singer (APS) boundary value problem. We show further that b-symplectic manifolds have canonical Spin-c structures in the usual sense, and that the APS index above coincides with the index of the Spin-c Dirac operator. We show that if the manifold is endowed with a Hamiltonian action of a compact connected Lie group with non-zero modular weights, then this method satisfies the Guillemin-Sternberg ``quantization commutes with reduction'' property. In particular our quantization coincides with the formal quantization defined by Guillemin, Miranda and Weitsman, providing a positive answer to a question posed in their paper.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79026355","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 : 2021-01-01DOI: 10.4310/jsg.2021.v19.n3.a4
Weonmo Lee, Y. Oh, Renato Vianna
{"title":"Asymptotic behavior of exotic Lagrangian tori $T_{a,b,c}$ in $mathbb{C}P^2$ as $a+b+c to infty$","authors":"Weonmo Lee, Y. Oh, Renato Vianna","doi":"10.4310/jsg.2021.v19.n3.a4","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n3.a4","url":null,"abstract":"","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83358165","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 : 2020-12-31DOI: 10.4310/JSG.2022.v20.n6.a1
J. Andersen, A. Malusà, Gabriele Rembado
We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in K"{a}hler quantisation, with polarisations coming from the nonabelian Hodge hyper-K"{a}hler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.
{"title":"Genus-one complex quantum Chern–Simons theory","authors":"J. Andersen, A. Malusà, Gabriele Rembado","doi":"10.4310/JSG.2022.v20.n6.a1","DOIUrl":"https://doi.org/10.4310/JSG.2022.v20.n6.a1","url":null,"abstract":"We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in K\"{a}hler quantisation, with polarisations coming from the nonabelian Hodge hyper-K\"{a}hler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87941127","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 : 2020-12-10DOI: 10.4310/jsg.2022.v20.n6.a4
Yann Rollin
We prove that every smoothly immersed 2-torus of $mathbb{R}^4$ can be approximated, in the C0-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $mathbb{R}^4$, the surface can be approximated in the C1-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic 2-tori of $mathbb{R}^{2n}$.
{"title":"Polyhedral approximation by Lagrangian and isotropic tori","authors":"Yann Rollin","doi":"10.4310/jsg.2022.v20.n6.a4","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n6.a4","url":null,"abstract":"We prove that every smoothly immersed 2-torus of $mathbb{R}^4$ can be approximated, in the C0-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $mathbb{R}^4$, the surface can be approximated in the C1-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic 2-tori of $mathbb{R}^{2n}$.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78106769","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 : 2020-12-07DOI: 10.4310/jsg.2022.v20.n4.a4
D. Panyushev, O. Yakimova
Let $mathfrak g$ be a semisimple Lie algebra, $mathfrak hsubsetmathfrak g$ a reductive subalgebra such that $mathfrak h^perp$ is a complementary $mathfrak h$-submodule of $mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra ${mathcal S}(mathfrak g)$ by taking the subalgebra ${mathcal Z}$ generated by the bi-homogeneous components of all $Hin{mathcal S}(mathfrak g)^{mathfrak g}$. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras ${mathcal Z}$. As a by-product, we prove that ${mathcal Z}$ is Poisson commutative if $mathfrak h$ is abelian and describe ${mathcal Z}$ in the special case when $mathfrak h$ is a Cartan subalgebra. In this case, ${mathcal Z}$ appears to be polynomial and has the maximal transcendence degree $(mathrm{dim},mathfrak g+mathrm{rk},mathfrak g)/2$.
{"title":"Reductive subalgebras of semisimple Lie algebras and Poisson commutativity","authors":"D. Panyushev, O. Yakimova","doi":"10.4310/jsg.2022.v20.n4.a4","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n4.a4","url":null,"abstract":"Let $mathfrak g$ be a semisimple Lie algebra, $mathfrak hsubsetmathfrak g$ a reductive subalgebra such that $mathfrak h^perp$ is a complementary $mathfrak h$-submodule of $mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra ${mathcal S}(mathfrak g)$ by taking the subalgebra ${mathcal Z}$ generated by the bi-homogeneous components of all $Hin{mathcal S}(mathfrak g)^{mathfrak g}$. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras ${mathcal Z}$. As a by-product, we prove that ${mathcal Z}$ is Poisson commutative if $mathfrak h$ is abelian and describe ${mathcal Z}$ in the special case when $mathfrak h$ is a Cartan subalgebra. In this case, ${mathcal Z}$ appears to be polynomial and has the maximal transcendence degree $(mathrm{dim},mathfrak g+mathrm{rk},mathfrak g)/2$.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72588245","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 : 2020-12-05DOI: 10.4310/jsg.2022.v20.n6.a5
Zhengyi Zhou
We show that the minimal symplectic area of Lagrangian submanifolds are universally bounded in symplectically aspherical domains with vanishing symplectic cohomology. If an exact domain admits a $k$-semi-dilation, then the minimal symplectic area is universally bounded for $K(pi,1)$-Lagrangians. As a corollary, we show that Arnold chord conjecture holds for the following four cases: (1) $Y$ admits an exact filling with $SH^*(W)=0$ (for some ring coefficient); (2) $Y$ admits a symplectically aspherical filling with $SH^*(W)=0$ and simply connected Legendrians; (3) $Y$ admits an exact filling with a $k$-semi-dilation and the Legendrian is a $K(pi,1)$ space; (4) $Y$ is the cosphere bundle $S^*Q$ with $pi_2(Q)to H_2(Q)$ nontrivial and the Legendrian has trivial $pi_2$. In addition, we obtain the existence of homoclinic orbits in case (1). We also provide many more examples with $k$-semi-dilations in all dimensions $ge 4$.
{"title":"On the minimal symplectic area of Lagrangians","authors":"Zhengyi Zhou","doi":"10.4310/jsg.2022.v20.n6.a5","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n6.a5","url":null,"abstract":"We show that the minimal symplectic area of Lagrangian submanifolds are universally bounded in symplectically aspherical domains with vanishing symplectic cohomology. If an exact domain admits a $k$-semi-dilation, then the minimal symplectic area is universally bounded for $K(pi,1)$-Lagrangians. As a corollary, we show that Arnold chord conjecture holds for the following four cases: (1) $Y$ admits an exact filling with $SH^*(W)=0$ (for some ring coefficient); (2) $Y$ admits a symplectically aspherical filling with $SH^*(W)=0$ and simply connected Legendrians; (3) $Y$ admits an exact filling with a $k$-semi-dilation and the Legendrian is a $K(pi,1)$ space; (4) $Y$ is the cosphere bundle $S^*Q$ with $pi_2(Q)to H_2(Q)$ nontrivial and the Legendrian has trivial $pi_2$. In addition, we obtain the existence of homoclinic orbits in case (1). We also provide many more examples with $k$-semi-dilations in all dimensions $ge 4$.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78600464","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 : 2020-09-24DOI: 10.4310/jsg.2021.v19.n3.a3
Yves Martinez-Maure
Classical (real) hedgehogs can be regarded as the geometrical realiza-tions of formal di¤erences of convex bodies in R n+1. Like convex bodies, hedgehogs can be identi…ed with their support functions. Adopting a pro-jective viewpoint, we prove that any holomorphic function h : C n ! C can be regarded as the 'support function' of a complex hedgehog H h in C n+1. In the same vein, we introduce the notion of evolute of such a hedgehog H h in C 2 , and a natural (but apparently hitherto unknown) notion of complex curvature, which allows us to interpret this evolute as the locus of the centers of complex curvature. It is of course permissible to think that the development of a 'Brunn-Minkowski theory for complex hedgehogs' (replacing Euclidean volumes by symplectic ones) might be a promising way of research. We give …rst two results in this direction. We next return to real hedgehogs in R 2n endowed with a linear complex structure. We introduce and study the notion of evolute of a hedgehog. We particularly focus our attention on R 4 endowed with a linear Kahler structure determined by the datum of a pure unit quaternion. In parallel, we study the symplectic area of the images of the oriented Hopf circles under hedgehog parametrizations and introduce a quaternionic curvature function for such an image. Finally, we consider brie ‡y the convolution of hedgehogs, and the particular case of hedgehogs in R 4n regarded as a hyperkahler vector space.
{"title":"Real and complex hedgehogs, their symplectic area, curvature and evolutes","authors":"Yves Martinez-Maure","doi":"10.4310/jsg.2021.v19.n3.a3","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n3.a3","url":null,"abstract":"Classical (real) hedgehogs can be regarded as the geometrical realiza-tions of formal di¤erences of convex bodies in R n+1. Like convex bodies, hedgehogs can be identi…ed with their support functions. Adopting a pro-jective viewpoint, we prove that any holomorphic function h : C n ! C can be regarded as the 'support function' of a complex hedgehog H h in C n+1. In the same vein, we introduce the notion of evolute of such a hedgehog H h in C 2 , and a natural (but apparently hitherto unknown) notion of complex curvature, which allows us to interpret this evolute as the locus of the centers of complex curvature. It is of course permissible to think that the development of a 'Brunn-Minkowski theory for complex hedgehogs' (replacing Euclidean volumes by symplectic ones) might be a promising way of research. We give …rst two results in this direction. We next return to real hedgehogs in R 2n endowed with a linear complex structure. We introduce and study the notion of evolute of a hedgehog. We particularly focus our attention on R 4 endowed with a linear Kahler structure determined by the datum of a pure unit quaternion. In parallel, we study the symplectic area of the images of the oriented Hopf circles under hedgehog parametrizations and introduce a quaternionic curvature function for such an image. Finally, we consider brie ‡y the convolution of hedgehogs, and the particular case of hedgehogs in R 4n regarded as a hyperkahler vector space.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87191255","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}
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