Central charge in a $d=2$ CFT I've always been confused by this very VERY basic and important fact about two-dimensional CFTs. I hope I can get a satisfactory explanation here. In a classical CFT, the generators of the conformal transformation satisfy the Witt algebra 
$$[ \ell_m, \ell_n ] = (m-n)\ell_{m+n}.$$
In the quantum theory, the same generators satisfy a different algebra 
$$[ \hat{\ell}_m,  \hat{\ell}_n ] = (m-n) \hat{\ell}_{m+n} + \frac{\hbar c}{12} (m^3-m)\delta_{n+m,0}.$$
Why is this?
How come we don't see similar things for other algebras? For example, why isn't the Poincare algebra modified when going from a classical to quantum theory?
Please, try to be as descriptive as possible in your answer. 
 A: I am not sure if this is the way you want to think about it, but I think it is worth pointing out that not having the central charge leads to a trivial quantum theory. The precise statement would be that a positive/unitary theory with c=0 has only one state, the vacuum. The details are demonstrated in 
J.F. Gomes. The triviality of representations of the Virasoro algebra with vanishing central element and L0 positive.
Phys. Lett. B 171, 75-76, 1986.
http://www.sciencedirect.com/science/article/pii/0370269386910014#
Basically, you do the usual tricks. Create some descendant whose norm you can make negative unless the primary has h=0. So you are left with the m=2, c=0, h=0 minimal model, the trivial representation.
A: The central charge term as an example of a quantum anomaly; a symmetry that is modified in the quantized version of a classical theory.  The central charge is, in fact, often referred to as the conformal anomaly.  As di-Francesco et. al. put it at the start of section 5.4.2:

The appearance of the central charge $c$, also known as the conformal anomaly, is related to the "soft" breaking of conformal symmetry by the introduction of a macroscopic scale into the system.

They then go on to show that if, for example, you consider a generic conformal field theory on $\mathbb C$, and if you map the theory onto a cylinder of circumference $L$ with coordinate $w$, then
\begin{align}
  \langle T_\mathrm{cylinder}(w)\rangle = -\frac{c\pi^2}{6L^2}
\end{align}
They also, in appendix $5A$, go on to show that when a conformal field theory is defined on a curved two-manifold, then the central charge is related to the so-called trace anomaly;
\begin{align}
  \langle T^\mu_{\phantom\mu\mu}(x)\rangle = \frac{c}{24\pi} R(x)
\end{align}
where $R$ is the Ricci scalar.  The central charge can be seen to arise naturally in radial quantization in the operator formalism of CFT:  see di-Francesco et. al chapter 6. 
Anomalies arizing from quantization aren't restricted to conformal symmetry.  See, for example, the chiral anomaly or the gauge anomaly.
