将${\rm Diff}^+(S^1)$的正能量表示限制为$n$点的稳定子
Restricting positive energy representations of ${\rm Diff}^+(S^1)$ to the stabilizer of $n$ points
摘要：
设$G_n \subset {\rm Diff}^+(S^1)$是$n$的稳定子，$S^1$的给定点是$n$的稳定子.如果我们把一个正能量表示$U^c_h$与一个中心电荷和最低能量的容许对$(c,h)$相关联，限制到子群$G_n$，我们会损失多少信息？这个问题和部分答案起源于手征共形QFT。$c$的价值可以很容易地从这样的限制中“恢复”;困难的问题涉及$h$的价值。如果是$c\leq 1$，则没有信息损失，因此，所有这些限制都是不可约的。在这项工作中，它表明，$U^c_{h}|_{G_n}$总是不可约的$n=1$，如果$h=0$，它是不可约的至少高达$n\leq 3$。此外，给出了对于某些值$c>1$和$h,\tilde{h}>0$的示例，使得$U^c_{h}|_{G_1}\simeq U^c_{\tilde{h}}|_{G_1}$。因此，对于这些值，$U^c_{h}|_{G_n}$对于$n\geq 2$不能是不可约的。对于$c,h$和$n$的进一步值，问题仍然是开放的。尽管如此，这个例子已经表明，一般来说，在QFT模型中，局部和全局交织器可能是不等价的。
英文摘要：
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.
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