This lecture covers the calculation of the instanton pre-factor, the twisted partition function, and the integration of zero modes in the context of the double well potential. The instructor explains the dilute instanton gas approximation and the dominance of single instantons for specific conditions. Various representations of the twisted partition function are discussed, along with the concept of saddle points and the motion in the invented potential. The lecture also delves into the path integral representation and the change in zero mode integration variables.