# How did the two copies of the Witt algebra become two copies of the Virasoro algebra in the CFT?

The Virasoro algebra $$$$[L_m,L_n]=(m-n) L_{m+n} +\frac{c}{12} (m^3-m) \delta_{m+n,0}$$$$ of the stress energy tensor $$T$$ was said to follow from the witt algebra of the local conformal transformation $$z\rightarrow z+\epsilon_n$$ where $$\epsilon_n=-z^{n+1}$$ , and $$\ell_n=-z^{n+1} \partial _z$$ was the generator of the local conformal transformation with the Witt algebra $$$$[\ell_m ,\ell_n]=(m-n) \ell_{m+n}$$$$

However, the $$L_n=\frac{1}{2\pi i}\oint dz z^{n+1} T(z)$$ came from the mode expansion of $$T(z)$$ and the Virasoro algebra followed from the definition of the quasi primary fields ($$T(z)T(w)$$ expansion). It was not clear where did the generator of the local conformal transformation $$\ell_n$$ come to play, i.e. $$L_n$$ seemed to have nothing to do with $$\ell_n$$.

How did the two copies of the Witt algebra become two copies of the Virasoro algebra in the CFT? Especially, how does the global subgroup $$\ell_0,\ell_{\pm 1}$$ influence $$L_n$$? Was there necessary a bijection, or was it simply an algebraic coincidence?

Related: Virasoro Algebra vs Witt Algebra

$$L_n$$ seemed to have nothing to do with $$\ell_n$$.
There exists a specific relation between these two. The symmetry of the action is reflected on the correlation functions (which are the 'observables' of a theory) through Ward Identities. The general form of these identities are: $$\frac{\partial}{\partial x_{\mu}}\langle j^{\mu}(x) \Phi(x_1)\Phi(x_2)... \rangle = \sum \delta(x-x_i) \langle\Phi(x_1)...G\Phi(x_i)... \rangle$$
In the above expression, $$j^{\mu}$$ is the current associated with the symmetry and $$G\Phi(x)$$ denotes the variation of the field under the symmetry transformations.
For example, we know that the current associated with the translation (along $$x^{\nu}$$) symmetry is $$(j^{\mu}_{Trans})^{\nu} = T^{\mu\nu}$$. Similarly, the other symmetry generators are $$(j_{Lorentz}^{\mu})^{\rho\sigma} = T^{\mu\nu}x^{\rho}- T^{\mu\rho}x^{\sigma}$$ $$j^{\mu}_{Dilatation} = T^{\mu}_{\,\,\,\nu} x^{\nu}$$
This is the reason that modes of stress tensor $$L_n$$ generates conformal symmetry. Why the modes have non-zero central charge is another matter and that has to do with breaking of conformal symmetry at quantum level (see the related question).