Isomorphic chain complexes of Hamiltonian dynamics on tori

Abstract

In this work we construct for a given smooth, generic Hamiltonian $H : \mathbb{S}^1\times\mathbb{T}^n \longrightarrow \mathbb{R}$ on the torus a chain isomorphism $ \Phi_* : \big(C_*(H),\partial^M_*\big) \longrightarrow \big(C_*(H),\partial^F_*\big)$ between the Morse complex of the Hamiltonian action $A_H$ on the free loop space of the torus $\Lambda_0(\mathbb{T}^n)$ and the Floer complex. Though both complexes are generated by the critical points of $A_H$, their boundary operators differ. Therefore the construction of $\Phi$ is based on counting the moduli spaces of hybrid type solutions which involves stating a new non-Lagrangian boundary value problem for Cauchy-Riemann type operators not yet studied in Floer theory.