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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!

Qmechanic
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Michael
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2 Answers2

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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.

Luboš Motl
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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.

Bruce Bartlett
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