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Consider two vector fields $a^\mu(x)$ and $b^\mu(x)$ in a space-time. The local angle between the two vector fields is given by \cos\theta(x) = \frac{ a(x) \cdot b(x) }{ \|a(x)\| \|b(x)\| } = \frac{ g_{\mu\nu}(x) a^\mu(x) b^\nu(x) }{ \left| g_{\alpha\beta} a^\alpha(x) a^\beta(x) \right|^{\frac{1}{2}} \left| g_{\rho\sigma} (x) b^\rho(x) b^\sigma(x) ... 2 The definition of D requires sum over all the anyons, not just the "basic" ones -- after all, you can not really say which ones are "more basic" than others. In certain realizations, like in toric code, it is easy to write down the operators that create e and m, but there is no sense that they are more basic than \psi, since you can as well take m ... 2 Suppose \left|\psi\right\rangle =\psi\left(0\right)\left|0\right\rangle  is another primary state, created by the primary operator \psi\left(z\right). Now you want to calculate \begin{align*} \phi\left(z\right)\left|\psi\right\rangle & =\phi\left(z\right)\psi\left(0\right)\left|0\right\rangle \\ & =\sum_{p}C_{\phi\psi ... 2 The significance of the Möbius transformations \mathrm{PSL}(2,\mathbb{C}) in 2D conformal field theory is that they are the globally defined conformal transformations on the Riemann sphere. While the infinitesimal conformal transformations form the infinite-dimensional Witt algebra spanned by the vector fields L_n = -z^{n+1}\partial_z$$we must be ... 1 You want to take the derivative with respect to both z and w. Take$${X^\mu }\left( z \right){X^\nu }\left( w \right) \sim - {1 \over 4}{\eta ^{\mu \nu }}\ln \left( {z - w} \right)$$and use the following derivative$${{{\partial ^2}} \over {\partial w\partial z}}\left[ {{X^\mu }\left( z \right){X^\nu }\left( w \right)} \right] = {\partial \over ...

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There is nothing to add to the first part of Jake Lebovic's answer. With regard to the second part of the question -- how to compute the OPE of two stress tensors -- one uses Wick's theorem. Normal ordering means one does not contract together the individual fields making up the normal ordered operator, in this case the two $\partial X$'s making up ...

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The answer seems to be that, technically at least, the two point function $\langle b(z) c(w) \rangle$ does vanish on the sphere. In the context of the standard $bc$ ghost system that shows up in bosonic string theory, the simplest nonzero correlation function on the sphere that involves both $b$ and $c$ is  \langle c(z_1) c(z_2) c(z_3) c(z_4) b(w) ...

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