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$S$ and $T$ matrices described modular transformations. Fusion coefficient and particle spins $(N^{ij}_k,s_i)$ describe the mutual and self statistics of anyons. Their relation is well known in category theory (Our papers http://arxiv.org/abs/1506.05768 and http://arxiv.org/abs/1507.04673 used those relations).

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The OPE coefficients respect the symmetries. For example, consider the OPE $$\mathcal{O}^i(x) \times \mathcal{O}^j(0)=\sum_{k} C^{ij}_{k}\left[|x|^{\Delta_k-\Delta_i-\Delta_j}\mathcal{O}^k(0)+\mathrm{descendants}\right]$$ where for the time being I am suppressing the spin index. Then the $C^{ij}_{k}$ transform as $$C^{ij}_{k}\rightarrow U^{i}_a U^{i}_b ... 2 The significance of the MÃ¶bius transformations \mathrm{PSL}(2,\mathbb{C}) in 2D conformal field theory is that they are the globally defined conformal transformations on the Riemann sphere. While the infinitesimal conformal transformations form the infinite-dimensional Witt algebra spanned by the vector fields$$ L_n = -z^{n+1}\partial_zwe must be ... 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 ... 0 OP last question essentially reads (v1): What's the meaning of the product of normal-ordered operators: \partial X(z) \partial X(z) : :\partial X(w) \partial X(w): ~?$$Strictly speaking, it is a radially ordered product of normal-ordered operators$${\cal R} \left[ : \partial X(z) \partial X(z) : :\partial X(w) \partial X(w): \right].$$However, ... 1 There is nothing to add to the first part of Jake Lebovic's answer. With regard to the second part of the question -- how to compute the OPE of two stress tensors -- one uses Wick's theorem. Normal ordering means one does not contract together the individual fields making up the normal ordered operator, in this case the two \partial X's making up ... 0 There are various ways of arguing that z should be an inverse energy scale, none of them to my knowledge very precise. In fact, it's not clear to me that it is possible to make a precise relationship between z and energy scale. That said, perhaps the simplest way of arguing for the relationship is to note that for massive particles, AdS is like a well: ... 1 You want to take the derivative with respect to both z and w. Take$${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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Let me work in the usual limit where a classical theory of gravity maps to a strongly interacting conformal field theory (CFT) with some color like parameter $N$ that is taken to be very large. In this limit, a central statement of the correspondence is that a generating function for correlation functions on the CFT side is given by the on-shell ...

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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 $m$ ...

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First notice that dimensionless and rotation invariant correlation functions $f(g,a,x)$ only depend on $g$ and $x^2/a^2=z\bar{z}/a^2$, such that $-a\partial_af/2=z\partial_zf=\bar{z}\partial_{\bar{z}}f$. Notice also that the Callan-Symanzik equation implies that any correlation function $f(g, a, x)$ in the regime $a\ll{x}$ is invariant under the ...

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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 $$\langle c(z_1) c(z_2) c(z_3) c(z_4) b(w) ... 3 Consider two vector fields a^\mu(x) and b^\mu(x) in a space-time. The local angle between the two vector fields is given by$$ \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) ...

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