Equivariant differential characters and symplectic reduction

Equivariant Integration formulæ in HyperKähler Geometry

Lisa Jeffrey and Frances Kirwan developed an integration theory for symplectic reductions. That is, given a symplectic manifold with symplectic group action, they developed a way of pulling the integration of forms on the reduction back to an integration of group-equivariant forms on the original space. We seek an analogue of the symplectic integration formula as developed by for the hyperKahler case. This is almost straightforward, but we have to overcome such obstacles as the lack of a hyper-Darboux theorem and the lack of compactness in the case of hyperKahler reduction.

Katok 示例的定量辛拓扑与等变辛嵌入

Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings

本文计算了 Katok 示例中 Randers 度量族对应的单位余余切丛辛嵌入圆余余切丛的参数集合，发现其呈无限阶梯结构，并建立了与等变椭球嵌入的等价关系。

凸域的正（$S^1$-等变）辛同调、更高容量与Clarke对偶

Positive ($S^1$-equivariant) symplectic homology of convex domains, higher capacities, and Clarke's duality

本文证明凸域的滤过正（$S^1$-等变）辛同调与Clarke对偶泛函诱导的滤过奇异同调自然同构，据此确定Gutt-Hutchings容量与Ekeland-Hofer谱不变量一致，并刻画Besse凸域、推导条形码熵的下界。

Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory

A local normal form theorem for smooth equivariant maps between Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence of this equivariant normal form theorem, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi spac…

On semifree symplectic circle actions with isolated fixed points

Let $M$ be a symplectic manifold, equipped with a semifree symplectic circle action with a finite, nonempty fixed point set. We show that the circle action must be Hamiltonian, and $M$ must have the equivariant cohomology and Chern classes of $(P^1)^n$.

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…

Differential invariants on symplectic spinors in contact projective geometry

We present a complete classification and the construction of $\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the principal series representations, associated to the contact projective geometry on $\mathbb{RP}^{2n+1}$ and induced from the irreducible $\mathrm{Mp}(2n,\mathbb{R})$-submodules of the Segal-Shale-Weil representation twisted by a one-parameter family of characters. The proof is based on the classification of homomorphisms of generalized Verma modules for the Segal-Shale-Weil representation twisted by a one-parameter family of characters, together with a ge…

Virtual neighborhoods and pseudo-holomorphic curves

We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for nondegenerate Hamiltonian symplectomorphisms.

Surjectivity for Hamiltonian G-spaces in K-theory

Let G be a compact connected Lie group, and (M,ω) a Hamiltonian G-space with proper moment map μ. We give a surjectivity result which expresses the K-theory of the symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M, under certain technical conditions on μ. This result is a natural K-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the K-theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian G-spaces. We discuss this lemma in detail and highlight the diffe…

Induction of Hamiltonian Poisson actions

We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifolds with symmetry, we show how one can induce a Poisson action admitting an equivariant momentum map. We prove that, under certain conditions, the dressing orbits of a Poisson-Lie group can be obtained by Poisson induction from the dressing orbits of a Poisson-Lie subgroup.

Examples of equivariant Lagrangian mean curvature flow

In this expository note we describe important examples of Lagrangian mean curvature flow in $\mathbb{C}^2$ which are invariant under a circle action. Through these examples, we see compact and non-compact situations, long-time existence, singularities forming via explicit models, and significant objects in Riemannian and symplectic geometry, including the Clifford torus, Chekanov torus, Whitney sphere and Lawlor necks.

Deformation Quantization of Symplectic Fibrations

A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove that the classical moment map could be quantized if there exists an equivariant connection. Along the way we touch upon the general question of quantization with values in a bundle of algebras. We consider Fedosov's construction of deformation quantization in general. In the App…