Identity in CFT I heard and read couple of times reference to a certain identity in conformal field theory (maybe specific to two dimensions). The identity relates the trace of stress-energy tensor to the beta functions of all the operators in the simplest way:
$$T^{\mu}_{\mu} = \sum_{i}\beta({\cal{O_i}}){\cal{O_i}}$$
It's an operator identity and the sum runs over all operators (primary and descendants).
It makes senses since vanishing beta functions and "tracelessness" are two marks of conformal symmetry.
If it exists, I would find it really elegant (who cares?) and I would appreciate to read the demonstration, it strangely isn't part of the textbook material on the subject.
 A: This identity is not special to $2d$. Actually it is this identity that makes CFTs interesting, relating the two notions of conformal invariance, as a group-theoretic property, and fixed points, as a property of quantum theories along the renormalization-group flow. In order to prove it let me first state a couple of things.
1) Noether theorem: Consider an action $S=\int d^d x \mathcal{L} \big( \Phi, \partial_\nu \Phi \big)$ which depends on some fields and their derivatives. The basic statement of Noether's theorem is that if this action is invariant under an (active) rigid transformation $\delta x^\nu = \omega^\nu$ (meaning the $\omega^\nu$ don't depend on the point), then under a non-rigid transformation $\delta x^\nu = \omega^\nu( x) $ the variation of the action can be written as
$$\delta S = \int d^d x \, \partial_\nu j^\nu_a \, \omega^a$$
where $j^\nu_a$ is some current associated with the transformation. Now, for classical theories, the fields are on-shell and the action must be invariant under any trasformation, also under non-rigids ones. So we have the usual formula $\partial_\nu j^\nu_a =0$. For quantum theories it is a bit more subtle, and you get Ward identities, which is what we want. But before going on let me say something about RG flow.
2) Wilsonian RG flow: (see e.g. these lecture notes) The Wilsonian approach to renormalization consists in taking seriously the idea of a cutoff. This means that whatever (classical) action we have, we only consider it valid until some high energy cutoff $\Lambda$. In the usual renormalization language, this means either that we treat all theories as effective theories, assuming some underlying high energy theory like string theory, or that we assume the existence of some small lattice, as is the case in statistical field theory. In any case, when we want to see how the theory looks at some energy scale $\Lambda_R < \Lambda$, we integrate the fast degrees of freedom and we find a low energy theory, where the couplings have acquired a dependence on $\mu = \Lambda_R/\Lambda$.
The proof: Consider a classical CFT, which satisfies $\delta S= 0$ under conformal trasformations. Now perform a (Wilsonian) renormalization procedure, such that the couplings $\{ g_i \}$ of the Lagrangian acquire a dependence on the energy scale $\mu$. Now, if we do a scale transformation, we have 
$$ x^\nu \rightarrow (1 + \epsilon ) x^\nu \\
g_i(\mu) \rightarrow g_i \Big( \frac{\mu}{1+\epsilon} \Big) = g_i(\mu - \epsilon \mu) = g_i(\mu) - \epsilon \beta_i$$
where we have introduced the beta function $\beta_i = \beta (g_i) = \frac{d g_i}{d \log \mu}$, which only depends on $\mu$ via $g_i$ because of the renormalization group. This transformation of the couplings give a new (quantum) contribution to the variation of the action:
$$\delta S = \delta S_{\text{classic}} + \delta S_{\text{quantum}} \\ = \int d^d x \, \partial_\nu j^\nu_a \, \omega^a + \int d^d x \frac{\partial \mathcal{L}}{\partial g_i} \delta g_i = \int d^d x \, \partial_\nu j^\nu_a \, \omega^a - \int d^d x \frac{\partial \mathcal{L}}{\partial g_i}  \,\epsilon \, \beta_i$$
where we have used Noether's theorem (the formula above) for $\delta S_{\text{classical}}$. Setting $\delta S = 0$ we find
$$ \partial_\nu j^\nu = \frac{\partial \mathcal{L}}{\partial g_i}  \,\beta_i. $$
Using that the current for a scale transformation is $j^\nu = T^\nu_\lambda x^\lambda$ gives your Ward identity.
