Yesterday we discussed the Virasoro algebra, the central extension of the algebra of diffeomorphisms on the circle. A natural question is if this extension has a generalization to higher dimensions, e.g. to the diffeomorphism algebra on the \(d\)-dimensional torus where \(d > 1\). The answer is yes, but it is not straightforward, for two reasons:
- The diffeomorphism algebra in \(d\) dimensions has no central extension. The Virasoro extensions are not central except when \(d=1\).
- There are two independent extensions. This happens because the diffeomorphism group in \(d\) dimensions is closely related to the general linear group \(GL(d)\). There are two extensions because \(GL(d) = SL(d) \times GL(1)\) is a direct product. Only when \(d=1\) one extension vanishes because \(SL(1)\) is the trivial group consisting of a single element.
Yesterday we discussed the ordinary Virasoro algebra. The generators are \(L_m = -i\exp(imx)\) and the brackets are
\[
[L_m, L_n] = (n-m) L_{m+n} - {c\over 12} m^3 \delta_{m+n}
\]
where \(\delta_m\) is the Kronecker delta. Note that we have eliminated the linear term in the extension by a redefinition of \(L_0\).
Here comes the trick. We rewrite the ordinary Virasoro algebra as follows:
\[\begin{align}
[L_m, L_n] &= (n-m) L_{m+n} + c m^2 n S_{m+n}, \\
[L_m, S_n] &= (n+m) S_{m+n}, \\
[S_m, S_n] &= 0, \\
m S_m &\equiv 0.
\end{align}\]
To see that this is the same Lie algebra as the previous one, we notice that the last condition implies that \(S_m\) is only nonzero when \(m=0\). Hence \(S_m = {1\over 12}\delta_m\), up to a factor. We then have \([L_m, S_n] = 0\), so the extension does indeed commute with everything. Finally the extension in the first line reduces to
\[
- {c\over12} m^3 \delta_{m+n}.
\]
The advantage of the second formulation is that it can be generalized to multiple dimensions. A basis for the vector fields in \(d\) dimensions are \(L_\mu(m) = -i\exp(im\cdot x) \partial_\mu\), where \(m\) and \(x\) are now \(d\)-dimensional vectors with components \(m_\mu\) and \(x^\mu\), and \(\partial_\mu = \partial/\partial x^\mu\). We use Einstein's summation convention, so repeated indices, one up and one down, are implicitly summed over. E.g.
\[
m \cdot x = m_\mu x^\mu = \sum_{\mu = 0}^{d-1} m_\mu x^\mu.
\]
The diffeomorphism algebra on the \(d\)-dimensional torus admits the following Virasoro-like extensions:
\[\begin{align}
[L_\mu(m), L_\nu(n)] &= n_\mu L_\nu(m+n) - m_\nu L_\mu(m+n) \\
&\qquad +\ (c_1 m_\nu n_\mu + c_2 m_\mu n_\nu)\ m_\rho S^\rho(m+n), \\
[L_\mu(m), S^\nu(n)] &= n_\mu S^\nu(m+n) + \delta^\nu_\mu m_\rho S^\rho(m+n), \\
[S^\mu(m), S^\nu(n)] &= 0, \\
m_\mu S^\mu(m) &\equiv 0.
\end{align}\]
To verify that this is indeed a Lie algebra, we must check that the brackets are anti-symmetric and satisfy the Jacobi identity. We must also confirm that the last condition is preserved, and indeed it is:
\[
[L_m, n_\nu S^\nu(n)] = n_\mu (m_\nu + n_\nu) S^\nu(m+n) \equiv 0.
\]
Finally, we must show that these extensions are non-trivial, i.e. that they can not be absorbed into a redefinition of some generators. To this end, consider the subalgebra generated by the operators
\[
L_r = \alpha^\mu L_\mu(r\beta), \qquad
S_r = \beta_\nu S^\nu(r\beta),
\]
where \(\alpha^\mu\) and \(\beta_\nu\) are two constant vectors which satisfy \(\alpha^\mu\beta_\mu = 1\) and \(r\) is an integer. Then the \(L_r\) satisfies the ordinary Virasoro algebra with central charge
\[
c = 12(c_1 + c_2).
\]
Since the restrictions of the two extensions of the full multi-dimensional Virasoro algebra are non-trivial, so are the extensions themselves.
The extension proportional to \(c_1\) was discovered by Rao and Moody [Comm. Math. Phys. 159 (1994) 239-264] and the one proportional to \(c_2\) by myself [J. Phys. A. 25 (1992) 1177–1184].
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