I'm reading some notes on CFT, and there's a strange topic that I find quite confusing. We define the Witt algebra to be the generators of conformal transformations on the complex plane.

$l_n = -z^{n+1}\partial_z$

$[l_n,l_m] = (m-n)l_{m+n}$

But then we perform something called the central extension which changes our algebra to the Virasoro algebra.

I'm okay with that. But then we go on to define the generators of conformal symmetry as contour integrals of the stress-energy tensor, and these satisfy not the Witt algebra but the Virasoro algebra. Is there a good reason for this?

  • $\begingroup$ which notes are you reading?, I've never came across Witt algebra in my CFT readings $\endgroup$ – CGH May 20 '16 at 19:16
  • $\begingroup$ Actually, it's Blumenhagen's text, Intro to CFT with applications to string theory. They're also mentioned in Ginsparg's notes, though I don't know if he calls it the Witt algebra. $\endgroup$ – Aurey May 20 '16 at 19:18

Basically the reason is that the classical conformal symmetry no longer holds at the quantum level due to the presence of the trace anomaly. More precisely, the tracelessness of the quantum stress-energy tensor is incompatible with the normal ordering needed to define it. By cohomological reasons, the trace of the stress-energy tensor, although non-vanishing, must be a central element of the "quantized Witt algebra", i.e. it must commute with all its generators. This implies that the "quantized Witt algebra" must be a non-trivial central extension of the Witt algebra, which in this particular case must be unique up to isomorphism (thanks tp Eh-whaaa for recalling this point below) - that is, a Virasoro algebra.

An exposition I particularly like on such matters is the little book of Martin Schottenloher, "A Mathematical Introduction to Conformal Field Theory" (2nd. edition), Lecture Notes in Physics 759 (Springer--Verlag, 2008).

  • $\begingroup$ Just to add to the answer you wrote, the central extension for the Witt algebra is unique, although it doesn't have to be necessarily unique for other Lie algebras. $\endgroup$ – user106422 May 20 '16 at 20:29
  • $\begingroup$ I am a little confused by your answer. Is there a direct relationship between the trace anomaly (which is non-zero only in curved backgrounds and I am not sure that it is related to normal-ordering as directly as you seem to imply) and the central extension of the algebra? In 4D CFTs the conformal anomaly is still present, while the conformal algebra is simple and free of central extensions. On the other hand the central extensions arise due to physical possibility of projective representations. (continued) $\endgroup$ – Peter Kravchuk May 21 '16 at 6:13
  • $\begingroup$ (continued) The shortest relation between the $c$ in Virasoro algebra and $c$ in trace anomaly I know is that Virasoro determines correlation functions of $T$ and these in turn (not quite trivially) determine the trace anomaly in curved background. Is there a more direct relationship? $\endgroup$ – Peter Kravchuk May 21 '16 at 6:15
  • $\begingroup$ Hmm... I'm not recalling right now the details of the relation and I'm currently away from my office. I'll answer your point when I get there. $\endgroup$ – Pedro Lauridsen Ribeiro May 21 '16 at 13:02

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