Conjugate $SU(r)$-connections and holonomy groups

Abstract:

In this article we show that when the structure group of the reducible principal bundle $P$ is $SU(r)$ and $Q\subset P$ is an $SO(r)$-subbundle of $P$, the rank of the holonomy group of a connection which is gauge equivalent to its conjugate connection is less than or equal to $\left [ \frac {r}{2} \right ]$, and use the estimate to show that for all odd prime $r$, if the holonomy group of the irreducible connection as above is simple and is not isomorphic to $E_8$, $F_4$, or $G_2$, then it is isomorphic to $SO(r)$.