A-Genus and the Sigma Model

This set of lecture notes presents a pedantic derivation of the connection between the ${\hat A}$-genus of spacetime's loop space and the genus one partition function of the $N=1/2$ sigma model. It concludes with some remarks on possible generalizations of the ${\hat A}$-genus which follow naturally from the `stringy' point-of-view but have yet to be explored mathematically. This set of lecture notes is geared towards a mathematical audience unfamiliar with the $N=1/2$ sigma model.