arXiv subjects

Hai-Long Her

Publications and source records attributed to Hai-Long Her.

7 recordsLinked to original sources

arXiv2025

On the Maslov-type index for general paths of symplectic matrices

In this article, we define an index of Maslov type for general symplectic paths which have two arbitrary end points. This Maslov-type index is a partial generalization of the Conley-Zehnder-Long index in the sense that the degenerate set of symplectic matrices is larger. The method of constructing the index is direct without taking advantage of Maslov index of Lagrangian paths and consistent no matter whether the starting point of the path is identity or not, which is different from the ones for Long's Maslov-type index and Liu's $L_0$-index. Some natural properties for the index are verified. We review other versions of Maslov indices and compare them with our definition. In particular, this Maslov-type index can be regarded as a realization of Cappell-Lee-Miller index for a pair of Lagrangian paths from the point of view of index for symplectic paths.

math.SG↗

arXiv2023

Sum of Hamiltonian manifolds

For any compact connected Lie group $G$, we study the Hamiltonian sum of two compact Hamiltonian group $G$-manifolds $(X^+,ω^+,μ^+)$ and $(X^-,ω^-,μ^-)$ with a common codimension 2 Hamiltonian submanifold $Z$ of the opposite equivariant Euler classes of the normal bundles. We establish that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds. We also compare the equivariant first Chern class of the Hamiltonian sum with the equivariant first Chern classes of $X^\pm$.

math.SG↗

arXiv2015

A Double Poisson Algebra Structure on Fukaya Categories

Let $M$ be an exact symplectic manifold with $c_1(M)=0$. Denote by $\mathrm{Fuk}(M)$ the Fukaya category of $M$. We show that the dual space of the bar construction of $\mathrm{Fuk}(M)$ has a differential graded noncommutative Poisson structure. As a corollary we get a Lie algebra structure on the cyclic cohomology $\mathrm{HC}^\bullet(\mathrm{Fuk}(M))$, which is analogous to the ones discovered by Kontsevich in noncommutative symplectic geometry and by Chas and Sullivan in string topology.

math.SG↗

arXiv2014

Relatively Open Gromov-Witten Invariants for Symplectic Manifolds of Lower Dimensions

Let $(X,ω)$ be a compact symplectic manifold, $L$ be a Lagrangian submanifold and $V$ be a codimension 2 symplectic submanifold of $X$, we consider the pseudoholomorphic maps from a Riemann surface with boundary $(Σ,\partialΣ)$ to the pair $(X,L)$ satisfying Lagrangian boundary conditions and intersecting $V$. In some special cases, for instance, under the semi-positivity condition, we study the stable moduli space of such open pseudoholomorphic maps involving the intersection data. If $L\cap V=\emptyset$, we study the problem of orientability of the moduli space. Moreover, assume that there exists an anti-symplectic involution $ϕ$ on $X$ such that $L$ is the fixed point set of $ϕ$ and $V$ is $ϕ$-anti-invariant, then we define the so-called "relatively open" invariants for the tuple $(X,ω,V,ϕ)$ if $L$ is orientable and dim$X\le 6$. If $L$ is nonorientable, we define such invariants under the condition that dim$X\le4$ and some additional restrictions on the number of marked points on each boundary component of the domain.

math.SG↗

arXiv2012

Cyclic Homology of Fukaya Categories and the Linearized Contact Homology

Let $M$ be an exact symplectic manifold with contact type boundary such that $c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya category of $M$ has the structure of an involutive Lie bialgebra. Inspired by a work of Cieliebak-Latschev we show that there is a Lie bialgebra homomorphism from the linearized contact homology of $M$ to the cyclic cohomology of the Fukaya category. Our study is also motivated by string topology and 2-dimensional topological conformal field theory.

math.SG↗

arXiv2007

Floer Homology for Symplectomorphism

Let (M,ω) be a compact symplectic manifold, and ϕbe a symplectic diffeomorphism on M, we define a Floer-type homology FH_*(ϕ) which is a gen- eralization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕup to a Hamiltonian isotopy.

math.SG↗

arXiv2007

Symplectic Energy and Lagrangian Intersection Under Legendrian Deformations

Let M be a compact symplectic manifold, and L be a closed Lagrangian submanifold which can be lifted to a Legendrian submanifold in the contactization of M. For any Legendrian deformation of L satisfying some given conditions, we get a new Lagrangian submanifold L'. We prove that the number of intersection of L and L' can be estimated from below by the sum of $Z_2$-Betti numbers of L, provided they intersect transversally.

math.SG↗