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On page 14 of the survey article Kac-Moody and Virasoro algebras in relation to quantum physics by Goddard and Olive, the authors show that smooth selfmaps of the circle form a Lie group corresponding to the Virasoro algebra.

I didn't fully follow their argument, and then I became more confused after looking at Schottenloher's book A Mathematical Introduction to Conformal Field Theory. In this book there are numerous comments to the effect that there does not exist an "abstract Virasoro group", i.e. one whose Lie algebra is the Virasoro algebra. In particular, section 5.4 is entitled "Does There Exist a Complex Virasoro Group?" and the answer given by the author is "no", in the form of the following two theorems:

Theorem 5.3. $\text{Diff}_+(\mathbb S)$ has no complexification. In particular, there does not exist a real Lie group whose Lie algebra is $\text{Vect}(\mathbb S)^{\mathbb C}$.

Theorem 5.4. There does not exist a complex Lie group whose Lie algebra is the (locally convex) completion of the Virasoro algebra.


At first glance, these two sources seem to be at odds with one another. My suspicion is that neither is incorrect, and that there is something going on with the complexification of the Lie group that causes the issue raised by Schottenloher. My question is, how does the group described by Goddard and Olive get around the non-existence proof of Schottenloher?

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  • $\begingroup$ Your references are subpar, that's what. Start with due diligence, e.g. Ginsparg 1988. It is linked to the 2D conformal group. $\endgroup$ – Cosmas Zachos Apr 14 '17 at 14:13
  • $\begingroup$ I read through the relevant sections (primarily section 1.2) of Ginsparg's lecture notes which you have linked to, thank you for that. However these notes do not resolve the confusion in my mind. For one thing, they define the "global conformal group" in the obvious way, then make reference (top of page 9) to a "local conformal group" in 2 dimensions, which is left undefined. (In fact, they seem to suggest that such a thing does not exist.) In any case, this leaves my question entirely unanswered, since there is no attempt to show that there cannot exist some group with that Lie algebra. $\endgroup$ – pre-kidney Apr 14 '17 at 22:43
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    $\begingroup$ Yes, nobody has constructed such a thing as that local group, so physics concentrates on the algebra and its irreps. If there were such an unlikely object, it would have been examined by mathematicians. But there is enduring lack of excitement for nonexistence proof in these waters. $\endgroup$ – Cosmas Zachos Apr 15 '17 at 0:28

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