The Virasoro algebra is a Lie algebra that appears in several places in theoretical physics, e.g. in string theory and 2D statistical physics. It was discovered independently by physicists and mathematicians in 1968.
The infinite conformal group in 2D is the group of analytic functions in the complex plane. A basis for the corresponding Lie algebra is given by
\[
L_m = z^{m+1} {d\over dz},
\]
where \(z\) is a point in the complex plane and \(m\) is an integer.
The algebra of diffeomorphisms on the circle has a Fourier basis
\[
L_m = -i \exp(imx) {d\over dx}.
\]
These two algebras are in fact the same (they are isomorphic), because the generators \(L_m\) have the same bracket.
\[
[L_m, L_n] = (n-m) L_{m+n}.
\]
Recall that \([A,B] = AB - BA\). My sign convention is appropriate for lowest-weight representations. In the physics literature the opposite convention is more common.
This algebra of vector fields is not so interesting in itself. What is interesting is its central extension, the Virasoro algebra. So what is a central extension? An extension is a modification of the bracket that is still a Lie algebra. The extension is non-trivial if it cannot be eliminated with a change of basis, and it is central if its bracket with everything in sight vanishes. The Virasoro algebra is defined by
\[
[L_m, L_n] = (n-m) L_{m+n} - {c\over 12}(m^3 - m)\delta_{m+n},
\]
where \(\delta_m\) is the Kronecker delta, i.e. \(\delta_0 = 1\) and \(\delta_m = 0\) for all \(m \neq 0\). It is straightforward to verify that this bracket satisfies the conditions of a Lie algebra, i.e. anti-symmetry and the Jacobi identify still hold. The linear term in the extension is trivial, because it is eliminated if we redefine
\[
L_0 \Rightarrow L_0 + {c\over 24}.
\]
The cubic term can not be eliminated by a change of basis, so it is a non-trivial extension.
Tomorrow I will explain how to generalize the Virasoro algebra to higher dimensions. The infinite conformal group can not be generalized because it is specific to 2D, but the diffeomorphism algebra on the circle can. But the generalization is not straightforward; it took me several years and I still only got a partial answer at the time.
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Sidor
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