conformal field theory - 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.

Answer

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.