Quasi-periodic paths and a string 2-group model from the free loop group

TL;DR

This paper investigates models for the string 2-group using free loop groups, identifies obstructions to strict models, and constructs a coherent model with explicit formulas, advancing understanding of string group representations.

Contribution

It demonstrates the impossibility of strict Lie 2-group models using free loop groups and provides a coherent alternative with explicit structure formulas.

Findings

Obstructions exist for strict models using free loop groups.

A coherent model for the string 2-group is constructed with explicit formulas.

The study highlights the importance of circle group actions in string group models.

Abstract

In this paper we address the question of the existence of a model for the string 2-group as a strict Lie-2-group using the free loop group $LSpin$ (or more generally $LG$ for compact simple simply-connected Lie groups $G$). Baez-Crans-Stevenson-Schreiber constructed a model for the string 2-group using a based loop group. This has the deficiency that it does not admit an action of the circle group $S^1$, which is of crucial importance, for instance in the construction of a (hypothetical) $S^1$-equivariant index of (higher) differential operators. The present paper shows that there are in fact obstructions for constructing a strict model for the string 2-group using $LG$. We show that a certain infinite-dimensional manifold of smooth paths admits no Lie group structure, and that there are no nontrivial Lie crossed modules analogous to the BCSS model using the universal central extension of the free loop group. Afterwards, we construct the next best thing, namely a coherent model for the string 2-group using the free loop group, with explicit formulas for all structure. This is in particular important for the expected representation theory of the string group that we discuss briefly in the end.