# Loop groups and diffeomorphism groups of the circle as colimits

## TL;DR

This paper demonstrates that loop groups and diffeomorphism groups of the circle can be represented as colimits of smaller groups supported on subintervals, enabling new insights into their structure and representations.

## Contribution

It introduces a novel colimit decomposition for loop and diffeomorphism groups of the circle, and connects their representations to affine Lie algebra representations.

## Key findings

- Loop groups are colimits of groups supported on subintervals.
- Established an equivalence between solitonic and locally normal representations.
- Constructed a functor linking conformal net representations to affine Lie algebra representations.

## Abstract

We show that loop groups and the universal cover of $\mathrm{Diff}_+(S^1)$ can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals of $S^1$. Analogous results hold for based loop groups and for the based diffeomorphism group of $S^1$. These results continue to hold for the corresponding centrally extended groups. We use the above results to construct a comparison functor from the representations of a loop group conformal net to the representations of the corresponding affine Lie algebra. We also establish an equivalence of categories between solitonic representations of the loop group conformal net, and locally normal representations of the based loop group.