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I've been trying to gain some intuition about Virasoro Algebras, but have failed so far.

The Mathematical Definition seems to be clear (as found in http://en.wikipedia.org/wiki/Virasoro_algebra). I just can't seem to gain some intuition about it. As a central extension to Witt Algebras, I was hoping that there has to be some geometric interpretation, as I can imagine Witt Algebras rather well.

If anyone has some nice Geometric or Visual Interpretation of Virasoro Algebra, I'd greatly appreciate it!

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The simplest visual representation of the Lie group associated with the Virasoro (Lie) algebra is the group of reparametrizations of a circle.

Imagine that $\sigma$ is a periodic variable with the periodicity $2\pi$. An infinitesimal diffeomorphism is specified by a periodic function $\Delta \sigma(\sigma)$ with the periodicity $2\pi$. So the generators of the reparameterizations may be written as $f(\sigma)\partial / \partial \sigma$.

The possible functions $f(\sigma)$ may be expanded to the Fourier series, so a natural basis of the generators of the reparametrizations of the circle are $$ L_m = i \exp(im\sigma) \frac{\partial}{\partial \sigma} $$ As an exercise, calculate that the commutator $[L_m,L_n]$ is what it should be according to the Virasoro algebra, namely $(m-n)L_{m+n}$.

The Virasoro algebra for a closed string has two copies of the algebra above - and for the open string, it's only one copy but it's different than the "holomorphic" derivatives I used above. There are various related ways to represent the algebra but the reparameterizations of the circle are the simplest example.

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4  
Aren't you talking about the Witt algebra? I think Knoten had a problem of visualizing the central extension of this. I understand that $\text{Diff}(S^1)$, the group of diffeomorphisms on the unit circle, is the Group associated to the Witt algebra (as you say). But do you know if such a group exist for the Virasoro algebra? Many books seem to suggest that there doesn't, but I don't think I have seen a proof of this. –  Heidar Jul 8 '11 at 9:39
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Oh, I see. There is obviously no out-of-Hilbert-space visual representation of the central extension that would differ from the $c=0$ algebra. The reason is that the central extension has $c$-numbers in the commutators. ;-) Any $c$-numbers may only be represented as the transformation of phase in the Hilbert space, and a transformation of phase of a vector in the Hilbert space doesn't change the character of this state "physically" or "geometrically" - it's just about the normalization. So central extensions are just central extensions and they share the original visualizations with the $c=0$. –  Luboš Motl Jul 8 '11 at 12:41
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Did I understand correct that you are saying that I can use the same geometric interpretation of Witt Algebra and apply it to Virasoro? –  Michael Jul 8 '11 at 13:06
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Sure, the central extension of an algebra is just a very subtle modification of the original algebra that doesn't change its physical meaning. For every central extension, one may obtain the original algebra simply by setting all the $c$-number generators to zero. This preserves the Jacobi identity etc. because the $c$-number generators commuted with anything else, anyway - well, that what it means that it was "central". ;-) In string theory, the Virasoro algebra is still the algebra of reparameterizations of the world sheet, even for $c\neq 0$. –  Luboš Motl Jul 8 '11 at 14:30
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Maybe I should have said, two years ago, that the right hand side of the Virasoro algebra - and others - contains proper operators, and those correspond to the Poisson brackets; and them it may contain the $c$-numbers. They're similar to $i\hbar$ in $[x,p]$. More generally, they are multiplied by a higher power of $\hbar$. At any rate, these $c$-number terms disappear - even relatively to the operator-valued terms - in the classical $\hbar\to 0$ limit which means that the classical interpretation is independent of these $c$-number terms (it's the same for a central extension). –  Luboš Motl Jun 10 '13 at 7:14

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