2-Representations of Lie 2-groups and 2-Vector Bundles

Abstract

We develop a systematic 2-representation theory for coherent Lie 2-groups and its geometric incarnation via 2-vector bundles. For any coherent Lie 2-group $\mathcal{G}_{\bullet}$, we construct a bicategory 2Rep$(\mathcal{G}_{\bullet})$ of 2-representations valued in $k$-prelinear categories, and prove that biequivalent 2-groups induce biequivalent 2-representation bicategories. When $\mathcal{G}_{\bullet}$ arises from an action groupoid $H \ltimes G \rightrightarrows G$, we construct a functor from the category of faithful $H$-representations to 2Rep$(\mathcal{G}_{\bullet})$ that is injective on objects. Applied to the string 2-group String$_k(G)$, this yields, for each positive energy representation of the loop group, an associated 2-representation of the string 2-group. Geometrically, we construct 2-stacks of 2-vector bundles and principal 2-bundles, and establish an associated bundle construction as a strong transformation of 2-stacks. We further extend this to an equivariant framework over Lie groupoids, providing a foundation for higher gauge theories with symmetries.