Chern correspondence for higher principal bundles

Abstract

The classical Chern correspondence states that a choice of Hermitian metric on a holomorphic vector bundle determines uniquely a unitary 'Chern connection'. This basic principle in Hermitian geometry, later generalized to the theory of holomorphic principal bundles, provides one of the most fundamental ingredients in modern gauge theory, via its applications to the Donaldson-Uhlenbeck-Yau Theorem. In this work we study a generalization of the Chern correspondence in the context of higher gauge theory, where the structure group of the bundle is categorified. For this, we define connective structures on a multiplicative gerbe and propose a natural notion of complexification for an important class of 2-groups. Using this, we put forward a new notion of higher connection which is well-suited for describing holomorphic principal 2-bundles for these 2-groups, and establish a Chern correspondence in this way. As an upshot of our construction, we unify two previous notions of higher connections in the literature, namely those of adjusted connections and of trivializations of Chern-Simons 2-gerbes with connection.