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Equation (2.4.6): $T(z)X^\mu(0)\sim \frac{1}{z}\partial X^\mu(0)$ means that the RHS is the most singular term of the LHS. $T(z) = -\frac{1}{\alpha'} :\partial X^{\mu} \partial X_{\mu}:\tag{2.4.4}$ So \begin{align*} T(z)X^{\mu}(0) & =-\frac{1}{\alpha'}:\partial X^{\nu}(z)\partial X_{\nu}(z):X^{\mu}(0)\\ & =-\frac{2:\partial ...


I feel like this is an over-simplification. I agree. IMHO it's a popscience analogy that's presented as a genuine explanation, but it isn't. What is lost in the actual AdS/CFT correspondence in this analogy? A dimension. You can't just discard this. If you do, you lay yourself wide open to dropping another dimension, and saying the flat 2D image is ...

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