# Clarification on “central charge equals number of degrees of freedom”

It's often stated that the central charge c of a CFT counts the degrees of freedom: it adds up when stacking different fields, decreases as you integrate out UV dof from one fixed point to another, etc... But now I am puzzled by the fact that certain fields have negative central charge, for example a b/c system has $c=-26$. How can they be seen has negatives degrees of freedom? Is it because they are fictional dof, remnant of a gauge symmetry? On their own, they would describe a non unitary theory, incoherent at the quantum level.

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Yes, unitary CFTs have a positive central charge. The $bc$ system has $c=-26$ but the theory is only unitary after switching to BRST cohomologies, and then only the light-cone degrees of freedom with $c=24$ survive. So the bosonic string has $24$ degrees of freedom. –  Luboš Motl Sep 26 '12 at 18:43
–  Qmechanic Sep 26 '12 at 19:20
Thanks for both comments, I read the two question page you are referring to Qmechanic but I don't feel like it answers my question, maybe it wasn't clearly formulated. My point was only asking why those fields (b/c system) give a algebraically negative contribution to the "number of degrees of freedom of your theory"-number. Is it because in that case, the intuitive association central charge = number of dof is a bit abusive? Is it because the b/c are fictitious dof (accounting for a gauge symmetry)? Maybe something else? –  Just_a_wannabe Sep 26 '12 at 19:39
I already went through the light cone quantization of bosonic string theory, and the path integral formalism but the answer did not appear (clear) to me... –  Just_a_wannabe Sep 26 '12 at 19:41
Right, your question is different. I was merely linking to the (so far) only other Phys.SE post that discusses the central charge $c=-26$ for the $bc$ ghost system. –  Qmechanic Sep 26 '12 at 20:24

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 $Diff(S^1)/S^1$.

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