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

TL;DR

This paper develops a category of 2-representations for coherent Lie 2-groups, constructs models of equivariant 2-vector bundles, and explores their relation to traditional representations, advancing the understanding of string 2-groups.

Contribution

It introduces a new category of 2-representations for coherent Lie 2-groups and constructs models of equivariant 2-vector bundles, providing explicit formulas and relations to existing representation categories.

Findings

Constructed a coherent model of the string 2-group using the free loop group.

Developed a category of 2-representations for coherent Lie 2-groups.

Built models of equivariant 2-vector bundles and analyzed the adjoint action.

Abstract

Murray, Roberts and Wockel showed that there is no strict model of the string 2-group using the free loop group. Instead, they construct the next best thing, a coherent model for the string 2-group using the free loop group, with explicit formulas for all structure. Based on their expectations, we build a category of 2-representations for coherent Lie 2-groups and some concrete examples. We also discuss the relation between this category of 2-representations and the category of representations. In addition, we construct a model of equivariant 2-vector bundles. At the end, we discuss the adjoint action on the string 2-representations.