# Geometric/Visual Interpretation of Virasoro Algebra

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!

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.

• 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. Jul 8, 2011 at 9:39
• 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 $c=0$. Jul 8, 2011 at 12:41
• Did I understand correct that you are saying that I can use the same geometric interpretation of Witt Algebra and apply it to Virasoro? Jul 8, 2011 at 13:06
• 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 $c\neq 0$. Jul 8, 2011 at 14:30
• 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). Jun 10, 2013 at 7:14

There is a real Lie group $\tilde{Diff}(S^1)$ which is a $U(1)$ central extension of the real Lie group $Diff(S^1)$, and the Virasoro algebra is the Lie algebra of this Lie group.

The central extension $\tilde{Diff}(S^1)$ can be realized geometrically in two ways. The first is via a Hilbert space embedding (as in the book of Pressley-Segal), and the second is via the determinant line bundle.

This is all nicely explained in Appendix D of the book "Two-dimensional conformal geometry and vertex operator algebras" by Huang.

• Interesting! I guess the point is that the Virasoro algebra might appear as the Lie algebra of a Lie group, but presumably when it comes to the specific way it acts in the case of e.g. 1+1D CFT it cannot be interpreted as the action of such a Lie group. Dec 30, 2016 at 21:28
• No. There is no problem - the full Lie group $\tilde{Diff}(S^1)$ acts in 1+1d CFT, not just the Lie algebra. This is explained in Huang's book. It is a folk misunderstanding that there is some kind of "issue", that only "the Lie algebra" acts. Jan 9, 2017 at 9:01
• What you are referring to as a folk misunderstanding seems to be resolved differently in Schottenloher's book "A Mathematical Introduction to Conformal Field Theory", section 5.4 concerns the non-existence of the complex Virasoro group. Apr 14, 2017 at 5:32
• There is no disagreement between the way Schottenloher's book and Huang's book resolve this. See the last paragraph in Section 5.4 in Schottenloher's book. Apr 15, 2017 at 9:18
• Schottenloher's book is also a great book, but he mischaracterises the problem a bit in Section 5.4. Schottenloher conceives of the group Diff(S^1) of diffeomorphisms of the circle as being significant for conformal field theory. Physicists think like this too, and this leads to confusion. Rather it is the quotient group Diff(S^1) / Rot(S^1) of diffeomorphisms of S^1 divided by rotations which is significant for conformal field theory. Apr 15, 2017 at 9:25