In Tong's lecture notes on String theory, he shows the following Ward identity for CFT: (page 73)

enter image description here enter image description here

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) enter image description here

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.

  • $\begingroup$ 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. $\endgroup$ – Prahar May 11 '18 at 12:46

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