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

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.