# Virasoro Algebra vs Witt Algebra

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?

• which notes are you reading?, I've never came across Witt algebra in my CFT readings – CGH May 20 '16 at 19:16
• 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. – Aurey May 20 '16 at 19:18

• (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? – Peter Kravchuk May 21 '16 at 6:15