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 $H⋉G\rightrightarrows G$, 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.

arXiv comment52 pages.Extensively revised version. Major changes include: add Theorem 2 (biequivalence invariance of 2-representation bicategories), a new construction of 2-equivariant 2-vector bundles (Section 5), and a functorial associated bundle construction (Section 4.2)