Abstract

We studied the moduli space of flat connections over a nonorientable surface via a Morse theory approach adapted from Atiyah and Bott’s work. We defined a Yang-Mills functional on the space of all connections over a nonorientable surface and obtained a Morse stratification of the space. We defined “super central extension” of the fundamental group of the surface. The representation varieties defined using super central extension correspond to gauge equivariant Yang-Mills connections and enable us to obtain reduction formulas for equivariant strata. To describe the phenomenon of the stratification here, we defined “anti-perfect Morse stratification”, which is the case when the discrepancy (i.e. the difference between Morse series and Poincare series) reaches its maximal possible value while the perfect Morse stratification is when the discrepancy reaches its minimal possible value zero.

This is a joint work with Chiu-Chu Melissa Liu.