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Consider a 2d CFT, e.g. a single bosonic degree of freedom.

The $TT$ OPE is

$$ T(w) T(z) = \frac{c/2}{(z-w)^4} + \frac{2 T(w)}{(z-w)^2} + \frac{\partial T(w)}{z-w} + \text{regular terms}. $$

Does it mean that under the action of the generator of special conformal transformations $l_1$ the stress-energy is conserved? And also, under $l_3, l_4, \dots$ ?

If so, does it mean that $T$ is anomalous only under the conformal transformations generated by $l_2$?

In my notation,

$$l_n = z^{n+1} \partial_z.$$

I am only asking this question to feel confident in what I think is correct. A simple "yes" would be a sufficient answer.

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$T$ is a second-level descendant of the identity operator. It is therefore annihilated by $L_n$ with $n>2$. Also, identity doesn't have any first-level descendants, so T is also annihilated by $L_1$ (which is to say that $T$ is a quasi-primary).

I am not sure whether the formula for $l_n$ you write is correct when acting on $T$, since it is not a primary field.

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