Restricting Positive Energy Representations of Diff⁺(S ¹) to the Stabilizer of n Points

Abstract

Let G _n ⊂ 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 ≤ 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.