The central charge counts the number of degrees of freedom only for matter fields living on a flat manifold (or supermanifold in the case of superstrings).
An example where this counting argument fails for matter fields is the case of strings moving on a group manifold $G$ whose central charge is given by the Gepner-Witten formula:

$c = \frac{k\mathrm{dim}(G)}{k+\kappa(G)}$

Where $k$ is the level and $\kappa$ is the dual coxeter number. Please see the following
article by Juoko Mickelsson.

One of the best ways to understand this fact (and in addition the ghost sector central extension) is to follow the Bowick-Rajeev approach described in a series of papers, please see for example the following scanned preprint. I'll try to explain their apprach in a few words.

Bowick and Rajeev use the geometric quantization approach. They show that the Virasoro central charges are curvatures of line bundles over $Diff(S^1)/S^1$ called the vacuum bundles.

Bowick and Rajeev quantize the space of loops living on the matter field manifold.
This is an infinite dimensional Kaehler manifold. One way to think about it
is as a collection of the Fourier modes of the string, the Fourier modes corresponding to positive frequencies are the holomorphic coordinates and vice versa. In addition, in order to define an energy operator (Laplacian) on this manifold one needs a metric (this causes the distinction between the flat and curved metric cases where the dimension counting is valid or not. The reason that the counting argument works in the flat case is that the Laplacian in this case has constant coefficients).

The quantization of a given loop results Fock space in which all the negative frequency modes are under the Dirac sea. However this Fock space is not invariant under a reparametrization of the loop. One can imagine that over each point of $Diff(S^1)/S^1$, there is a Fock
space labeled by this point. This is the Fock bundle whose collection of vacuum vectors is a line bundle called the vacuum bundle. Bowick and Rajeev proved that the central charge is exactly the curvature of this line bundle.

The situation for the ghosts is different. Please see the Bowick-Rajeev
refence above. Their contribution to the central charge is equal to the
curvature of the canonical bundle. This bundle appears in geometric quantization due to the
noninvariance of the path integral measure on $Diff(S^1)/S^1$.

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user David Bar Moshe