Local equivalence of representations of Diff$^+(S^1)$ corresponding to different highest weights

TL;DR

This paper proves that positive energy irreducible projective unitary representations of Diff$^+(S^1)$ with different highest weights are locally equivalent, extending the classification of superselection sectors of Virasoro nets.

Contribution

It establishes local equivalence of representations for different highest weights, completing the classification of superselection sectors of Virasoro nets.

Findings

Representations with different highest weights are locally equivalent.

The result extends previous work limited to certain regions of the parameter space.

Completes the full classification of superselection sectors of Virasoro nets.

Abstract

Let $c,h$ and $c,\tilde{h}$ be two admissible pairs of central charge and highest weight for ${\rm Diff}^+(S^1)$. It is shown here that the positive energy irreducible projective unitary representations $U_{c,h}$ and $U_{c,\tilde{h}}$ of the group ${\rm Diff}^+(S^1)$ are locally equivalent. This means that for any $I\Subset S^1$ open proper interval, there exists a unitary operator $W_I$ such that $W_I U_{c,h}(\gamma)W_I^* = U_{c,\tilde{h}}(\gamma)$ for all $\gamma \in {\rm Diff}^+(S^1)$ which act identically on $I^c\equiv S^1\setminus I$ (i.e. which can "displace" or "move" points only in $I$). This result extends and completes earlier ones that dealt with only certain regions of the "$c,h$-plane", and closes the gap in the full classification of superselection sectors of Virasoro nets.