# Different forms of the Ward identities in CFT

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.

On the other hand, in Blumenhagen's book "Introduction to conformal field theory", he writes that the CFT Ward identity is: (page 30)

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.

• The second equation tells you what $T(z) O_1(\sigma_1)$ is. Take that and then evaluate $- \text{Res}[ \epsilon(z)T(z) O_1(\sigma_1)]$about the point $\sigma_1$. The first equation then tells you that the result of this computation is $\delta O_1(\sigma_1)$. Check that this is the correct transformation law for a primary operator. – Prahar May 11 '18 at 12:46