# Question about correlation functions of 2d CFTs

I have a question regarding equation (2.22) in Ginsparg's lecture notes on CFTs. Equation (2.22) is $$\langle T(z) \phi_1(w_1, {\bar w}_1) \cdots \rangle = \sum_{i=1}^n \left( \frac{h_i}{(z-w_i)^2} + \frac{1}{z-w_i} \frac{\partial}{ \partial w_i} \right) \langle \phi_1(w_1, {\bar w}_1) \cdots \rangle$$ Here, $T(z)$ is the stress tensor of the CFT and $\phi_i$ is a primary operator of weight $(h_i,0)$ which transforms under conformal transformations as $$\delta_\epsilon \phi_i = \left( h_i \partial \epsilon + \epsilon \partial \right) \phi_i$$ He derives (2.22) from (2.21) which reads $$\langle \oint \frac{dz}{2\pi i} \epsilon(z) T(z)\phi_1(w_1, {\bar w}_1) \cdots \rangle = \sum_{i=1}^n \langle \phi_1(w_1, {\bar w}_1) \cdots \delta_\epsilon\phi_i(w_i, {\bar w}_i) \cdots \rangle$$ by setting $\epsilon(x) = \frac{1}{x-z}$.

My question is - Is (2.22) correct?

Here are my reasons to believe that it is not -

1. I believe he derives (2.22) from (2.21) by setting $\epsilon(x) = \frac{1}{x-z}$ in (2.21). (2.22) is then derived if the following holds $$\langle \oint \frac{dx}{2\pi i} \frac{T(x)}{x-z} \phi_1(w_1, {\bar w}_1) \cdots \rangle = \langle T(z)\phi_1(w_1, {\bar w}_1) \cdots \rangle$$ This would be true if the integrand on the LHS had only a pole at $x-z$. However, it has also has poles at each $x = w_i$, but those contributions aren't considered.

2. I can try and derive (2.22) in a different way - namely via contractions. I start with the LHS of (2.22) and contract $T(z)$ with each $\phi_i$. Each contraction is replaced with the operator product $$T(z) \phi_i(w_i {\bar w}_i) = \frac{h_i \phi_i(w_i {\bar w}_i) }{ ( z - w_i )^2 } + \frac{ \partial \phi_i(w_i {\bar w}_i) }{ z - w_i } + : T(z) \phi_i(w_i {\bar w}_i) :$$ Again, if I only consider the singular terms, I reproduce the RHS of (2.22). But what about $: T(z) \phi_i(w_i {\bar w}_i) :$?? In a general CFT, conformal normal ordering $:~:$ is not equivalent to creation-annihilation normal ordering ${}^\circ_\circ~{}^\circ_\circ$. The latter would vanish in a correlation function, but not the former. So, I believe in general there would be extra terms on the right of (2.22).

What am I misunderstanding?

• I don't think you should be setting $\epsilon=\frac{1}{x-z}$. Have a look at Tong's string theory notes page 76 and see if it helps... Commented Jan 15, 2015 at 10:56
• @Heterotic - I don't think that helps. For instance, I'm still concerned about the "+ non-singular" terms in the first equation on page 76 of the notes. Naively, I would think they give non-trivial contributions to the correlation function. Commented Jan 15, 2015 at 16:50

The integral will give you the residue, which is the coefficient of the $\frac1{z-w}$ term. Nothing else will contribute (this is known form complex analysis). So we need to tag carefully the $\frac1{z-w}$ term of $\epsilon(z) T(z)\phi_1(w_1, {\bar w}_1)$.

Using $\epsilon(z)=\epsilon(w)+\epsilon'(w)(z-w)+\cdots$

and $$T(z) \phi_i(w_i {\bar w}_i) = \frac{h_i \phi_i(w_i {\bar w}_i) }{ ( z - w_i )^2 } + \frac{ \partial \phi_i(w_i {\bar w}_i) }{ z - w_i } + : T(z) \phi_i(w_i {\bar w}_i) :$$

we see that

$$\epsilon(z) T(z)\phi_1(w_1, {\bar w}_1)=\cdots+\frac{\epsilon(w) \partial \phi_1(w_1, {\bar w}_1) }{ z - w_1 }+ \frac{\epsilon'(w) h_1 \phi_1(w_1, {\bar w}_1) }{ z - w_1 } +\cdots$$ where dots denote other terms that behave differently than $\frac1{z-w}$.

Then, doing the integral we obtain $$\oint \frac{dz}{2\pi i} \epsilon(z) T(z)\phi_1(w_1, {\bar w}_1)=\epsilon(w) \partial \phi_1(w_1, {\bar w}_1)+\epsilon'(w) h_1 \phi_1(w_1, {\bar w}_1) =\delta_\epsilon \phi_1.$$

"Putting the integral in brackets" and by re-instating the other (spectator) $\phi_i$'s we get (2.22).

Well in this last step we also have to use that the integral actually breaks as $$\oint_{around\ everything}=\oint_{around\ w_1}+\cdots+\oint_{around\ w_n}$$ and when doing the i-th integral the only poles in that region will be $\frac{1}{z-w_i}$ and nothing more, (ie the other w's do not contribute. We can always choose the contours this way.)

I hope that 8 years later, you're still interested in hearing a different version of an answer. (@Heterotic, unfortunately I couldn't fully understand your explanation.)

Let me start by warning that $$\epsilon$$ has to be strictly holomorphic, not meromorphic (such as $$x\mapsto\frac{1}{x-z}$$).

Now, the argument to pass from the Noether charge identity (2.21) to the Noether current identity (2.22) -the unintegrated form- is in complete analogy to the respective step on page 19, namely: $$\oint\frac{dz}{2\pi i}\epsilon(z)RT(z)\Phi(w)=h\partial\epsilon(w)\Phi(w)+\epsilon(w)\partial\Phi(w) \iff RT(z)\Phi(w)=\frac{h}{(z-w)^2}\Phi(w)+\frac{\partial\Phi(w)}{z-w}+\operatorname{reg}_z$$ ($$\operatorname{reg}_z$$ and $$\epsilon(z)$$ mean arbitrary holomorphic functions in $$z$$, $$\epsilon$$ small.) Here is how I think this implication is argued for:

1. The reverse direction "$$\Longleftarrow$$" follows directly from Cauchy's differentiation formula, $$f\,\text{holomorphic}\implies f'(w)=\oint\frac{dz}{2\pi i}\frac{f(z)}{(z-w)^2}\quad\&\quad f(w)=\oint\frac{dz}{2\pi i}\frac{f(z)}{z-w},$$ used for $$f=\epsilon$$. (Here, a meromorphic $$\epsilon$$ could totally screw up the argument.)
2. "$$\Longrightarrow$$" I cannot make as rigorous right now. When I read Ginsparg's notes, I was satisfied by the fact that $$\epsilon$$ is arbitrary among small holomorphic functions (an infinite dimensional algebra), maybe this suffices for you as well.

If you believe this "$$\Longleftrightarrow$$" then "(2.21)$$\iff$$(2.22)" is immediate as already noted ($$f=\epsilon$$ again, only a $$\sum_{j=1}^n$$ has to added).

Let me add that the important step in this section 2.4 on conformal Ward identities you're talking about is not this passing from charge to current equation, I think. Instead, it's that we learn that products of primaries $$\phi_1\phi_2\dots\phi_n$$ transform in a Leibniz rule fashion. The cool thing is that we are able to infer this by means of results from contour integration of meromorphic functions - the first step in (2.21).