  In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied
Yang-Mills functional over a Riemann surface from the point of view of Morse
theory. We generalize their study to all closed, compact, connected, possibly
nonorientable surfaces. We introduce the notion of "super central extension" of
the fundamental group of a surface. It is the central extension when the
surface is orientable. We establish a precise correspondence between Yang-Mills
connections and representations of super central extension. Knowing this exact
correspondence, we work mainly at the level of representation varieties which
are finite dimensional instead of the level of strata which are infinite
dimensional.
