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.

arXiv comment48 pages

References in corpus (11)

• Twisted differential String and Fivebrane structures
• Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories
• Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory
• Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group
• L-infinity algebras of local observables from higher prequantum bundles
• Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations
• A Construction of String 2-Group Models using a Transgression-Regression Technique
• Towards an M5-Brane Model II: Metric String Structures
• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• Higher U(1)-gerbe connections in geometric prequantization
• Higher Gauge Theory with String 2-Groups