Rabinowitz Floer Homology

Rabinowitz作用泛函是拉格朗日乘子泛函，其临界点是固定能量的周期轨道。本文作者首先构造了该作用泛函的花同源性。与此同时，Rabinowitz Floer同源性已经找到了许多应用，例如，参见该提议的作者之一在ICM 2022上的谈话。临界点处的拉格朗日乘子对应于周期，其中负值意味着周期轨道在时间上向后穿越。这一特征将Rabinowitz Floer同调与辛同调和辛场论区别开来，在辛场论中，周期轨道只能向前穿越。这导致了Rabinowitz Floer同调与Tate同调和庞加莱对偶的深刻关系，这也是Rabinowitz Floer同调具有梯度拓扑量子场论结构的原因。本提案的目标是更深入地理解这种结构，研究它如何扩展到哈密顿延迟方程，并探索这种结构在半经典极限下的量子力学应用。

The Rabinowitz action functional is a Lagrange multiplier functional whose critical points are periodic orbits of fixed energy. A Floer homology for this action functional was first constructed by the au-thors of this proposal. Meanwhile Rabinowitz Floer homology has found numerous applications, see e.g. the ICM 2022 talk of one of the authors of this proposal. The Lagrange multiplier at a critical point corresponds to the period, where a negative value means that the periodic orbit is traversed back-wards in time. This feature distinguishes Rabinowitz Floer homology from symplectic homology and symplectic field theory where periodic orbits can only be traversed in forward time. It leads to deep relations of Rabinowitz Floer homology with Tate homology and Poincare duality, and it is the reason that Rabinowitz Floer homology has the structure of a graded Topological Quantum Field Theory. The goal of this proposal is to understand this structure in more depth, to study how it extends to Hamil-tonian delay equations, and to explore applications of this structure to quantum mechanics in the semiclassical limit.