Fixed points of the mapping class group in the $SU(n)$ moduli spaces

Abstract:

Let $\Sigma$ be a compact oriented surface with or without boundary components. In this note we prove that if $\chi (\Sigma ) < 0$ then there exist infinitely many integers $n$ such that there is a point in the moduli space of irreducible flat $SU(n)$ connections on $\Sigma$ which is fixed by any orientation preserving diffeomorphism of $\Sigma$. Secondly we prove that for each orientation preserving diffeomorphism $f$ of $\Sigma$ and each $n\ge 2$ there is some $m$ such that $f$ has a fixed point in the moduli space of irreducible flat $SU(n^m)$ connections on $\Sigma$. Thirdly we prove that for all $n\geq 2$ there exists an integer $m$ such that the $m$’th power of any diffeomorphism fixes a certain point in the moduli space of irreducible flat $SU(n)$ connections on $\Sigma$.