In Tong's lecture notes on String theory, he shows the following Ward identity for CFT: (page 73)
Where $\delta$ is the variation w.r.t (infinitesimal) conformal transformations.
I don't see how/if these are related. Could someone shed some light on this? I like the clean argument that Blumenhagen makes, so it would be great to be able to get Tong's result from Blumenhagen's result.