Restricting positive energy representations of ${\rm Diff}^+(S^1)$ to the stabilizer of $n$ points

Abstract

Let $G_n \subset {\rm Diff}^+(S^1)$ be the stabilizer of $n$ given points of $S^1$. How much information do we lose if we restrict a positive energy representation $U^c_h$ associated to an admissible pair $(c,h)$ of the central charge and lowest energy, to the subgroup $G_n$? The question, and a part of the answer originate in chiral conformal QFT. The value of $c$ can be easily ``recovered'' from such a restriction; the hard question concerns the value of $h$. If $c\leq 1$, then there is no loss of information, and accordingly, all of these restrictions are irreducible. In this work it is shown that $U^c_{h}|_{G_n}$ is always irreducible for $n=1$, and if $h=0$, it is irreducible at least up to $n\leq 3$. Moreover, an example is given for certain values $c>1$ and $h,\tilde{h}>0$ such that $U^c_{h}|_{G_1}\simeq U^c_{\tilde{h}}|_{G_1}$. It follows that for these values $U^c_{h}|_{G_n}$ cannot be irreducible for $n\geq 2$. For further values of $c,h$ and $n$, the question is left open. Nevertheless, the example already shows, that in general, local and global intertwiners in a QFT model may not be equivalent.