Restricting Positive Energy Representations of Diff+(S1) to the Stabilizer of n Points

Let Gn ⊂ Diff+(S1) be the stabilizer of n given points of S1. 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 Gn? 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 ≤ 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 ≤ 3. Moreover, an example is given for c > 2 and certain values of $$h \neq \tilde{h}$$ such that $$U^c_{h}|_{G_1}\simeq U^c_{\tilde{h}}|_{G_1}$$ . It is also concluded that for these values $$U^c_{h}|_{G_n}$$ cannot be irreducible for n ≥ 2. For further values of c, h and n, the question is left open. Nevertheless, the example already shows that, on the circle, there are conformal QFT models in which local and global intertwiners are not equivalent.