Index iteration theory for symplectic paths with applications to nonlinear Hamiltonian systems

Torus fibrations and localization of index I

We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Witten's deformation applied to a Hilbert bundle.

Discrete dynamical systems from mutation-periodic quivers: examples and reduction

Several new mutation-periodic quivers of period higher than 1 are introduced as well as the associated discrete dynamical systems. The reduction of these systems is developed using either a presymplectic or a Poisson approach. The presymplectic approach leads to a reduced system whose iteration map is symplectic with respect to a log symplectic form. In the Poisson approach we build a Poisson structure invariant under the iteration map, leading to a reduced system whose variables are the Casimirs of such structure.

Symplectic Surgery and the Spin-C Dirac operator

Let $G$ be a compact connected Lie group, and $M$ a compact Hamiltonian $G$-space, with moment map $J$. For each $G$-equivariant Hermitian vector bundle $E$ over $M$, one has an associated twisted Spin-C Dirac operator, whose equivariant index is a symplectic invariant of $E$. In the present paper, we study gluing properties of the equivariant index under "symplectic cutting" operations. Our main application is a proof of the Guillemin-Sternberg conjecture, which says that if $E=L$ is a quantizing line bundle and $0$ a regular value of $J$, the multiplicity of the trivial representation in the…

The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds

In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold $(M,F)$ with a bumpy, irreversible Finsler metric $F$ and $H^*(M;{\bf Q})\cong T_{d,n+1}(x)$ for some even integer $d\ge 2$ and integer $n\ge 1$, there exist at least $\frac{dn(n+1)}{2}$ distinct non-hyperbolic closed geodesics with odd Morse indices, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic…

重论辛几何中的马斯洛夫指标

Maslov Indicies In Symplectic Geometry Revisited

本文重论辛几何中的马斯洛夫指标，在已有四种定义方法的基础上提出两种新方法，并与Her和Zhong基于早期工作定义的辛矩阵路径不变量进行了比较。

Singular Reduction and Quantization

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bun…

On the Maslov index of symplectic paths that are not transversal to the Maslov cycle. Semi-Riemannian index theorems in the degenerate case

We use the notion of generalized signatures at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a real-analytic path having possibly non transversal intersections. We discuss some applications of the theory, with special emphasis on the study of the Jacobi equation along a semi-Riemannian geodesic. The research work exposed in this paper originated from a suggestion given by the third author relating the spectral flow with the partial signatures. He pointed out several references and formulated the stateme…

On the geometric quantization of contact manifolds

Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie group $G$ acts on $M$ transverse to the contact distribution $E$. In an earlier paper, we defined a $G$-transversally elliptic Dirac operator $\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on the symplectic vector bundle $E\rightarrow M$, whose equivariant index is well-defined as a generalized function on $G$, and gave a formula for its index. By analogy with the geometric quantization of symplectic manifolds, the $\mathbb{Z}_2$-graded Hilbert space $Q(M)=\ker \dirac \oplus \ker \dirac^{*}$ …

Bergman kernels and symplectic reduction

We generalize several recent results concerning the asymptotic expansions of Bergman kernels to the framework of geometric quantization and establish an asymptotic symplectic identification property. More precisely, we study the asymptotic expansion of the $G$-invariant Bergman kernel of the spin^c Dirac operator associated with high tensor powers of a positive line bundle on a symplectic manifold. We also develop a way to compute the coefficients of the expansion, and compute the first few of them, especially, we obtain the scalar curvature of the reduction space from the $G$-invariant Bergma…

轨道折叠上的拓扑量子力学与轨道折叠指标

Topological Quantum Mechanics on Orbifolds and Orbifold Index

本文通过研究辛轨道折叠上的拓扑量子力学模型，构造了量子HKR映射的轨道折叠版本，并利用精确半经典近似给出了轨道折叠代数指标的几何与量子场论解释。

Instability of semi-Riemannian closed geodesics

A celebrated result due to Poincaré affirms that a closed non-degenerate minimizing geodesic $γ$ on an oriented Riemannian surface is hyperbolic. Starting from this classical theorem, our first main result is a general instability criterion for timelike and spacelike closed semi-Riemannian geodesics on a (non)oriented manifold. A key role is played by the spectral index, a new topological invariant that we define through the spectral flow (being the Morse index truly infinite) of a path of selfadjoint Fredholm operators. A major step in the proof of this result is a em new spectral flow formul…

Positive paths in the linear symplectic group

A positive path in the linear symplectic group $\Sp(2n)$ is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-independent Hamiltonians, and which all lie in the set $\Uu$ of diagonalizable matrices with eigenvalues on the unit circle. However, as was shown by Krein, the eigenvalues of a general positive path can move off the unit circle. In this paper, we extend Krein's theory: we investigate …