量子特征类与Hohm度规
Quantum characteristic classes and the Hofer metric
摘要：
给定一个闭单调辛流形$M$，我们定义了自由闭路空间$L \text {Ham}(M, ω)$中取值于$QH_* (M)$的某些特征上同调类及其$S^1$等变形式.这些类推广了Seidel表示，并满足陈类的公理版本。特别是有一个惠特尼求和公式，它给出了从环$H_{*} (L\text {Ham}(M, ω), \mathbb{Q})$及其庞特里亚金积到$QH_{2n+*} (M)$及其量子积的分次环同态。作为应用，我们证明了McDuff和Slimowitz关于半自由Hamilton圈作用的Hohm度量中极小性的一个定理推广到闭路空间$L \text {Ham}(M, ω)$的高维几何中.
英文摘要：
Given a closed monotone symplectic manifold $M$, we define certain characteristic cohomology classes of the free loop space $L \text {Ham}(M, ω)$ with values in $QH_* (M)$, and their $S^1$ equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring $H_{*} (L\text {Ham}(M, ω), \mathbb{Q})$, with its Pontryagin product to $QH_{2n+*} (M)$ with its quantum product. As an application we prove an extension of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action, to higher dimensional geometry of the loop space $L \text {Ham}(M, ω)$.
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