A Nonabelian Hodge Correspondence for Principal Bundles in Positive Characteristic

In this paper, we prove a nonabelian Hodge correspondence for principal bundles on a smooth variety X in positive characteristic, which generalizes the Ogus-Vologodsky correspondence for vector bundles. Then we extend the correspondence to logahoric torsors over a log pair $(X,D)$, where D a reduced normal crossing divisor in X. As an intermediate step, we prove a correspondence between principal bundles on root stacks X and parahoric torsors on $(X,D)$, which generalizes the correspondence on curves given by Balaji--Seshadri to higher dimensional case.