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.