In 2d CFT, why the $T_{zz}$ component of energy-momentum tensor is holomorphic even at quantum level? In 2d Conformal Field Theory, the $T_{zz}$ component of energy-momentum tensor is treated as a holomorphic function $T(z)=T_{zz}$ at quantum level such as in OPE involved energy-momentum tensor.
I know, in classical theory, this conclusion comes from the traceless and conservation equation of $T_{\mu\nu}$. 
The traceless leads to $T_{z\bar{z}}=0$ and then conservation law implies
 $$\partial_{\bar z} T_{zz}=\partial_z T_{\bar z\bar z}=0$$
However, a quantum field does not need to satisfy the classical equation of motion. (Since in path integral fomalism $\int D\phi\  e^{-S[\phi ]}$, the quantum field $\phi$ can be any smooth field configuration and the energy-momentum tensor is a functional of field $\phi$.) The conservation equation is  classical。
Then, why should the $T_{zz}$ still be a holomorphic function at quantum level?
 A: I will follow Di Francesco's CFT book. By combining the Ward identities associated to conformal invariance (Eq. 5.32), you can get to
$$ \partial_{\bar{z}} \Big[ \langle T(z, \bar{z} ) X \rangle - \sum_{i=1}^n \left( \frac{1}{z-w_i} \partial_{w_i} \langle X \rangle + \frac{h_i}{(z-w_i)^2} \langle X \rangle \right) \Big] = 0  , \qquad \qquad  \text{(Eq. 5.39)}$$
where $T(z, \bar{z})$ is the "holomorphic" component of the energy-momentum tensor ( in this convention $T(z, \bar{z}) = - 2 \pi T_{zz}$ ) and $X$ is some string of quasi-primary fields $\Phi_1(w_1, \bar{w}_1) ... \Phi_n(w_n, \bar{w}_n)$ with holomorphic dimensions $h_1, ... , h_n$. This means that the correlation function $\langle T(z, \bar{z} ) X \rangle$ is actually holomorphic up to the sum
$$ \sum_{i=1}^n \left( \frac{1}{z-w_i} \partial_{w_i} \langle X \rangle + \frac{h_i}{(z-w_i)^2} \langle X \rangle \right). $$
Note that this sum is not exactly holomorphic because of the property 
$$\partial_{\bar{z}} \frac{1}{z} = \pi \delta (\mathbf{x}), $$ 
where $\delta (\mathbf{x})$ is the Dirac delta at the point $z$ but in Euclidean coordinates (its normalization may change when changing the coordinates). Therefore, if you apply this property to Eq. 5.39 you can easily see that $\partial_{\bar{z}}  \langle T(z, \bar{z} ) X \rangle$ actually vanishes as long as we don't take $z$ to coincide with some $w_i$ (these are called contact terms and are basically always present in the quantum theory). This is I think what is meant by "$T_{zz}$ is holomorphic at the quantum level".
