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I have some questions concerning the first two chapters of Ginsparg's text on CFT, which can be found here

1. In equation (2.1) primary fields of conformal weight [itex](h,\overline{h})[/itex] are introduced as fields that transform the following way:

[itex]\Phi(z,\overline{z}) \to \left(\frac{\partial f}{\partial z}\right)^h \left(\frac{\partial \overline{f}}{\partial \overline{z}}\right)^\overline{h}\Phi(f(z), \overline{f}(\overline{z}))[/itex]

Then, [itex]\Phi(z,\overline{z})dz^h d\overline{z}^\overline{h}[/itex] is supposed to be invariant. I don't understand that statement. [itex]ds^2[/itex] transforms the following way: [itex]ds^2 \to \left(\frac{\partial f}{\partial z}\right) \left(\frac{\partial \overline{f}}{\partial \overline{z}}\right)ds^2[/itex]

Therefore, I suppose [itex]dz \to \left(\frac{\partial f}{\partial z}\right)dz, d\overline{z} \to \left(\frac{\partial \overline{f}}{\partial \overline{z}}\right)d\overline{z}[/itex]

That leads to [itex]\Phi(z,\overline{z})dz^h d\overline{z}^\overline{h} \to \left(\frac{\partial f}{\partial z}\right)^{2h} \left(\frac{\partial \overline{f}}{\partial \overline{z}}\right)^{2\overline{h}}\Phi(f(z), \overline{f}(\overline{z}))dz^h d\overline{z}^\overline{h}[/itex]

I don't see why this transformation behaviour should be called invariant.

2. In the section on two dimensional CFT Ginsparg compactifies the spatial coordinate in oder to eliminate infrared divergences. Is there an easy way to understand why this compactification should lead to the elimination of infrared divergences? Why does this not further constrain the generalitiy of the theory?

3. After the compactification of the spatial coordinate Ginsparg maps the resulting cylinder to the complex plane. There we can use the known tools of complex analysis to proceed, in particular complex line integration. He introduces the radial ordering operator

[itex]\begin{equation}R(A(z)B(w))=

\left\{

\begin{aligned}

A(z)B(w) & \quad |z|>|w|\\

B(w)A(z) & \quad |z|<|w|

\end{aligned}

\right.

\end{equation} [/itex]

Then, Ginsparg claims, the equal-time commutator of a local operator [itex]A[/itex] with the spatial integral of an operator [itex]B[/itex] becomes the contour integral of the radially ordered product:

[itex]\left[\int dx B, A\right]_{e.t.} \to \oint dz \: R(B(z)A(w))[/itex]

I don't see why this is the case.

I am happy about answers to any of the three questions.

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# Ginsparg Applied Conformal Field Theory

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