Mathematics - Symplectic Geometry Publications (50)


Mathematics - Symplectic Geometry Publications

Given an open book decomposition of a contact three man-ifold (M, $\xi$) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k n , we construct a Legendrian knot $\Lambda$ close to the stable foliation of a page, together with a small Legendrian pushoff $\Lambda$. When k $\ge$ 5, we apply the techniques of [CH2] to show that the strip Legen-drian contact homology of $\Lambda$ $\rightarrow$ $\Lambda$ is well-defined and has an exponential growth property. The work [Al2] then implies that all Reeb vector fields for $\xi$ have positive topological entropy. Read More

Inspired by Katok's examples of Finsler metrics with a small number of closed geodesics, we present two results on Reeb flows with finitely many periodic orbits. The first result is concerned with a contact-geometric description of magnetic flows on the 2-sphere found recently by Benedetti. We give a simple interpretation of that work in terms of a quaternionic symmetry. Read More

This paper shows that the method of uncertainty quantification via the introduction of Stratonovich cylindrical noise in the Hamiltonian formulation introduces stochastic Lie transport into the dynamics in the same form for both electromagnetic fields and fluid vorticity dynamics. Namely, the resulting stochastic partial differential equations (SPDE) retain their unperturbed form, except for an additional term representing induced Lie transport by a set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases, via pairing data correlation vector fields for cylindrical noise with the momentum map for the deterministic motion, which is responsible for the well-known analogy between hydrodynamics and electromagnetism. Read More

Given an algebraic hypersurface $H=f^{-1}(0)$ in $(\mathbb{C}^*)^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau-Ginzburg model. We propose an enriched version of this picture which also features the wrapped Fukaya category of the complement $(\mathbb{C}^*)^n\setminus H$ and the Fukaya-Seidel category of the Landau-Ginzburg model $((\mathbb{C}^*)^n,f)$. We illustrate our speculations on simple examples, and sketch a proof of homological mirror symmetry for higher-dimensional pairs of pants. Read More

In the setting of symplectic manifolds which are convex at infinity, we use a version of the Aleksandrov maximum principle to derive uniform estimates for Floer solutions that are valid for a wider class of Hamiltonians and almost complex structures than is usually considered. This allows us to extend the class of Hamiltonians which one can use in the direct limit when constructing symplectic homology. As an application, we detect elements of infinite order in the symplectic mapping class group of a Liouville domain, and prove existence results for translated points. Read More

In the first part of the article we study Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$ which are generated by Lipschitz functions and prove a rigidity result for the image of coisotropic cylinders. The tools are Viterbo's symplectic capacities and a series of inequalities coming from their relation with symplectic reduction. In the second part we consider the Sine-Gordon equation and treat it as an infinite-dimensional Hamiltonian system. Read More

It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409. Read More

We extend Noether's theorem to the setting of multisymplectic geometry by exhibiting a correspondence between conserved quantities and continuous symmetries on a multi-Hamiltonian system. We show that a homotopy co-momentum map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We apply our results to generalize the theory of the classical momentum and position functions from the phase space of a given physical system to the multisymplectic phase space. Read More

Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. Read More

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. Read More

In this paper, we develop a differential-graded symplectic (Batalin-Vilkovisky) version of the framework of Crawley-Boevey, Etingof and Ginzburg on noncommutative differential geometry based on double derivations to construct non-commutative analogues of the Courant algebroids introduced by Liu, Weinstein and Xu. Adapting geometric constructions of \v{S}evera and Roytenberg for (commutative) graded symplectic supermanifolds, we express the BRST charge, given in our framework by a `homological double derivation', in terms of Van den Bergh's double Poisson algebras for graded bi-symplectic non-commutative 2-forms of weight 1, and in terms of our non-commutative Courant algebroids for graded bi-symplectic non-commutative 2-forms of weight 2 (here, the grading, or ghost degree, is called weight). We then apply our formalism to obtain examples of exact non-commutative Courant algebroids, using appropriate graded quivers equipped with bi-symplectic forms of weight 2, with a possible twist by a closed Karoubi-de Rham non-commutative differential 3-form. Read More

In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to $b$-symplectic manifolds: $b$-contact manifolds. Read More

A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold -- which is viewed as the obstruction to the existence of a Q-invariant Berezin volume -- is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds. Read More

We apply the geometric quantization procedure via symplectic groupoids proposed by E. Hawkins to the setting of epistemically restricted toy theories formalized by Spekkens. In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. Read More

The Poisson bracket invariant of a cover of a closed symplectic manifold measures how much a collection of smooth functions forming a partition of unity subordinate to the cover, can become close to being Poisson commuting. We introduce a new approach to this invariant in dimension 2, which enables us to significantly improve previously known lower bounds. Read More

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an orientable closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Ma\~n\'e critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, i.e. Read More

In the restricted three-body problem, consecutive collision orbits are those orbits which start and end at collisions with one of the primaries. Interests for such orbits arise not only from mathematics but also from various engineering problems. In this article, we show that there are infinitely many consecutive collision orbits in the planar circular restricited three-body problem using Floer homology. Read More

We prove that certain non-exact magnetic Hamiltonian systems on products of closed hyperbolic surfaces and with a potential function of large oscillation admit non-constant contractible periodic solutions of energy below the Ma\~n\'e critical value. For that we develop a theory of holomorphic curves in symplectizations of non-compact contact manifolds that arise as the covering space of a virtually contact structure whose contact form is bounded with all derivatives up to order three. Read More

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Read More

Given a compact oriented surface, we classify log Poisson bi-vectors whose degeneracy loci are locally modeled by a finite set of lines in the plane intersecting at a point. Further, we compute the Poisson cohomology of such structures and discuss the relationship between our classification and the second Poisson cohomology. Read More

In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic, group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex sheets. Read More

We define mixed states associated with submanifolds with probability densities in quantizable closed K{\"a}hler manifolds. Then, we address the problem of comparing two such states via their fidelity. Firstly, we estimate the sub-fidelity and super-fidelity of two such states, giving lower and upper bounds for their fidelity, when the underlying submanifolds are two Lagrangian submanifolds intersecting transversally at a finite number of points, in the semiclassical limit. Read More

Kostant gave a model for the real geometric quantization associated to polarizations via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed counting integral points inside the associated Delzant polytope. Read More

In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In addition, in the appendix by K. Read More

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. Read More

We define a quantitative invariant of Liouville cobordisms with monotone filling through an action-completed symplectic cohomology theory. We illustrate the non-trivial nature of this invariant by computing it for annulus subbundles of the tautological bundle over $\mathbb{C} P^1$ and give further conjectural computations based on mirror symmetry. We prove a non-vanishing result in the presence of Lagrangian submanifolds with non-vanishing Floer homology. Read More

In this article, we will discuss a localization formulas of equivariant cohomology about two Killing vector fields on the set of zero points ${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$ As application, we use it to get formulas about characteristic numbers and to get a Duistermaat-Heckman type formula on symplectic manifold. Read More

A central problem in symplectic topology is to classify symplectic forms up to strong isotopy. This problem has been studied on compact manifolds, using a method invented by J\"urgen Moser in the 1960s, now known as Moser's trick. Our goal is to give a version of Moser's trick in the noncompact setting, and some applications to Euclidean space, manifolds exhausted by a proper function, and manifolds with cylindrical ends. Read More

We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce invariants of periodic orbits in planar Stark-Zeeman systems and study their behaviour. Read More

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underline{\omega}$ of symplectic forms which at every point span a maximal positive-definite subspace of $\Lambda^2$ for the wedge product. This article is motivated by a conjecture of Donaldson: when $X$ is compact $\underline{\omega}$ can be deformed through cohomologous hypersymplectic structures to a hyperk\"ahler triple. We approach this via a link with $G_2$-geometry. Read More

The aim of the article is to construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity. Hence, overtwisted contact structures in each odd dimension can have a rigid behavior as far as the problem of deformations of contactomorphisms is concerned. Read More

The Gelfand-Cetlin system $\Phi_\lambda : \mathcal{O}_\lambda \rightarrow \mathbb{R}^n$ is a completely integrable system on a partial flag manifold $(\mathcal{O}_\lambda,\omega_\lambda)$ whose image is a convex polytope $\triangle_\lambda \subset \mathbb{R}^n$. In the first part of this paper, we are concerned with the topology of Gelfand-Cetlin fibers. We first show that every Gelfand-Cetlin fiber is an isotropic submanifold of $(\mathcal{O}_\lambda, \omega_\lambda)$ and it is an iterated bundle where the fiber at each stage is either a point or a product of odd dimensional spheres. Read More

These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart will also be presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can you recover the classical system? Read More

Givental's non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena such as orderability, unboundedness of the discriminant and oscillation metrics, and a contact geometric version of the Arnold conjecture. In this article we give an analogue for lens spaces of Givental's construction and its applications. Read More

Let $\Lambda$ be a Legendrian submanifold of the 1-jet space of a smooth manifold. Associated to $\Lambda$ there is a Legendrian invariant called Legendrian contact homology, which is defined by counting rigid pseudo-holomorphic disks of $\Lambda$. Moreover, there exists a bijective correspondence between rigid pseudo-holomorphic disks and rigid Morse flow trees of $\Lambda$, which allows us to compute the Legendrian contact homology of $\Lambda$ via Morse theory. Read More

In this paper, I introduce weak representations of a Lie groupoid $G$. I also show that there is an equivalence of categories between the categories of 2-term representations up to homotopy and weak representations of $G$. Furthermore, I show that any VB-groupoid is isomorphic to an action groupoid associated to a weak representation on its kernel groupoid; this relationship defines an equivalence of categories between the categories of weak representations of $G$ and the category of VB-groupoids over $G$. Read More

We give the 3-dimensional Sklyanin algebras $S$ that are module-finite over their center $Z$ the structure of a Poisson $Z$-order (in the sense of Brown-Gordon). We show that the induced Poisson bracket on $Z$ is non-vanishing and is induced by an explicit potential. The ${\mathbb Z}_3 \times \Bbbk^\times$-orbits of symplectic cores of the Poisson structure are determined (where the group acts on $S$ by algebra automorphisms). Read More

This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent non-trivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Read More

The purpose of this short note is to explain how classical 5-dimensional cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered simply-connected 4-manifold X_K becoming diffeomorphic to X after a single stabilization by connected summing with CP^2 # -CPb^2 or alternatively with S^2 x S^2, and almost complete decomposability of X_K for many almost completely decomposable X, such as the elliptic surfaces. Read More

Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^* M$, asymptotic to a Legendrian submanifold $\Lambda \subset T^{\infty} M$. We study a locally constant sheaf of $\infty$-categories on $L$, called the sheaf of brane structures or $\mathrm{Brane}_L$. Its fiber is the $\infty$-category of spectra, and we construct a Hamiltonian invariant, fully faithful functor from $\Gamma(L,\mathrm{Brane}_L)$ to the $\infty$-category of sheaves of spectra on $M$ with singular support in $\Lambda$. Read More

We introduce a new action $S_{standard}^{(\rho,h; \Phi,g,B,C)}$ for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This `standard action' is abstractly a non-Abelian gauged sigma model --- based on maps $\varphi: (X^{\!A\!z},E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ with a fundamental module $E$ with a connection $\nabla$ to $Y$ --- enhanced by the dilaton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples $(\varphi,\nabla)$ to Ramond-Ramond field. In a special situation, this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. Read More

We extend results about $n$-shifted coisotropic structures from part I of this work to the setting of derived Artin stacks. We show that an intersection of coisotropic morphisms carries a Poisson structure of shift one less. We also compare non-degenerate shifted coisotropic structures and shifted Lagrangian structures and show that there is a natural equivalence between the two spaces in agreement with the classical result. Read More

The purpose of this paper is to carry out a classical construction of a non-constant holomorphic disk with boundary on (the suspension of) a Lagrangian submanifold in $\mathbb{R}^{2 n}$ in the case the Lagrangian is the lift of a coisotropic (a.k.a. Read More

A new relation between homoclinic points and Lagrangian Floer homology is presented: In dimension two, we construct a Floer homology generated by primary homoclinic points. We compute two examples and prove an invariance theorem. Moreover, we establish a link to the (absolute) flux and growth of symplectomorphisms. Read More

Given a principal bundle on an orientable closed surface with compact connected structure group, we endow the space of based gauge equivalence classes of smooth connections relative to smooth based gauge transformations with the structure of a Fr\'echet manifold. Using Wilson loop holonomies and a certain characteristic class determined by the topology of the bundle, we then impose suitable constraints on that Fr\'echet manifold that single out the based gauge equivalence classes of central Yang-Mills connections but do not directly involve the Yang-Mills equation. We also explain how our theory yields the based and unbased gauge equivalence classes of all Yang-Mills connections and deduce the stratified symplectic structure on the space of unbased gauge equivalence classes of central Yang-Mills connections. Read More

This article is the second part of the article we promised to write at the end of Section 1 of [FOOO15] (arXiv:1209.4410). (Part I appeared in [Part I] (arXiv:1503. Read More

In this note, we study the SYZ mirror construction for a toric Calabi-Yau manifold using instanton corrections coming from Woodward's quasimap Floer theory instead of Fukaya-Oh-Ohta-Ono's Lagrangian Floer theory. We show that the resulting SYZ mirror coincides with the one written down via physical means (as expected). Read More

We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups. Read More

In this note we study the expected value of certain symplectic capacities of randomly rotated centrally symmetric convex bodies in the classical phase space. Read More