Conformal theory with zero central charge

Question

Suppose there is a conformal field theory which has the global conformal symmetry namely $SL(2,R).$ and after central extension it is enhanced to Virasoro algebra with central charge, $c=0$ (also known as Witt algebra). What does this physically signify?

In 2D CFT $c$ has different physical interpretations : it is a measure of degrees of freedom, Weyl anomaly etc. What is the implication of having a CFT with $c=0$?

Edit : I am particularly interested about $D=1$. In this case there are papers (e.g, hep/th 9901139) which says the global conformal symmetry enhances to Virasoro algebra whose central charge, $c=0$.

Answer 1

Here an easy way to think about your problem in QFTs (EDIT: I thank ACuriousMind for having pointed out that my argument does not hold in string theory)

Is c=0 meaningful? Consider the case $d=2$.

Unitary CFT's have positive central charge. Let's assume the central charge may be zero for some strange CFT.

Then, the $c-$theorem tells you the central charge has to decrease along the RG flow and if you depart from $c=0$ this would give you a theory with negative central charge. For sure, this theory cannot be unitary but at the same thime the RG flow preserves unitarity.

Note that you can always do a relevant deformation of a CFT.

My argument would tell you that a CFT with $c=0$ (assuming this statement is true) cannot be perturbed.

Probabily there is a better answer by discussing Verma modules in $d=2$.

Answer 2

For standard QFT, the other answer is correct - a theory with vanishing central charge is not unitary, hence not a viable quantum theory in itself.

However, CFTs with vanishing total central charge play a central role in string theory, since the vanishing of the Weyl anomaly is precisely the condition for the quantization of the string to be consistent. Here, "total" means you have some CFTs with positive central charges - the theories of the physically meaningful fields - and others with negative central charges - the theories of the ghosts incurred by gauge symmetry, which decouple from all physical processes and whose (non-)unitarity is therefore of no concern - and the sum of all these central charges yields an overall CFT with vanishing central charge.

The constraint that the Weyl anomaly must vanishing for the quantum theory of the string to be consistent directly leads to (super)string theory's prediction for the number of extra dimensions, see also this answer of mine.