Virtual Morse theory on $\Omega Ham(M,\omega)$

TL;DR

This paper explores the Morse theoretic properties of the Hofer length functional on the loop space of Hamiltonian diffeomorphisms, linking quantum characteristic classes to topology and geometry of symplectic manifolds.

Contribution

It introduces a virtual perfect Morse-Bott framework for the Hofer length functional and applies it to study the topology and geometry of Hamiltonian groups.

Findings

The Hofer length functional behaves virtually as a perfect Morse-Bott functional.

Predictions for the index of certain geodesics were made and partially verified.

Connections between quantum characteristic classes and Morse theory are established.

Abstract

We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on $\ls$. As an application we prove a theorem which can be interpreted as stating that this functional behaves "virtually" as a perfect Morse-Bott functional with a flow. This can be applied to study topology and Hofer geometry of $ \text {Ham}(M, \omega)$. We also use this to give a prediction for the index of some geodesics for this functional, which was recently partially verified by Yael Karshon and Jennifer Slimowitz.