Isomorphic chain complexes of Hamiltonian dynamics on tori

Abstract

In this work we construct for a given smooth, generic Hamiltonian $H:{\mathbb{S}}^1\times {\mathbb{T}}^n⟶\mathbb{R}$ on the torus a chain isomorphism $\hat{I}{¦}_{\ast }:(C_{\ast }(H),{\partial }_{\ast }^M)⟶(C_{\ast }(H),{\partial }_{\ast }^F)$ between the Morse complex of the Hamiltonian action $A_H$ on the free loop space of the torus $\hat{I}{›}_0({\mathbb{T}}^n)$ and the Floer complex. Though both complexes are generated by the critical points of $A_H$, their boundary operators differ. Therefore the construction of $\hat{I}¦$ 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.