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 of 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 [r/2], 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).