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1

Recall that the (global) conformal group is given by $${\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \},\tag{1} $$ cf. e.g. this Phys.SE post. Using the embedding $\imath: \mathbb{R}^{p,q}\hookrightarrow \overline{\mathbb{R}^{p,q}}$ into the conformal compactifification $\overline{\mathbb{R}^{p,q}}$, one may show after a short calculation that the ...


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Comments to the post (v2): Ref. 1 is considering the $d$-dimensional real Euclidean space $(\mathbb{R}^d,|\cdot|^2)$ with the standard norm $$|x|^2~:=~\sum_{\mu=1}^d (x^{\mu})^2~=~\sum_{\mu,\nu=1}^d x^{\mu}\eta_{\mu\nu}x^{\nu}, \qquad \eta_{\mu\nu} ~=~{\rm diag}(1,\ldots, 1),\tag{A}$$ and inner product $$\langle x ,y\rangle~:=~\sum_{\mu,\nu=1}^d x^{\mu}\...


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Conceptually, the idea is that planes and spheres are equivalent from the point of conformal geometry. Conformal transformations map {planes, spheres} to {planes, spheres}, and in fact do this transitively -- any object in the set {planes, spheres} can be obtained from any other by a conformal transformation. Inversion is a reflection against a sphere, and ...


0

Thinking of the sphere $S^n$ as the one-point compactification of $\mathbb{R}^n$, we can consider the stereographic projection from the plane defined by $x^0 = 0$ to the unit sphere $\{x\in \mathbb{R}^n\,:\,|x| = 1\}$. This map is actually defined on $\mathbb{R}^n\cup\{\infty\}$, it takes the point $\infty$ to the north pole of the unit sphere. Moreover, ...


2

Well while it has similarities with the OPE, it is more than that. In fact, it satisfies the OPE limit when $z\to w_j$ for any $j$, since the OPE you are talking about tells you only the singular terms, while there are also infinitely many non-singular terms, i.e., schematically $$ T(z)\phi(w,\bar w)=\frac{h_\phi}{(z-w)^2}\phi(w,\bar w)+\frac{1}{z-w}\...


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We can have CFTs with $c \neq {\tilde c}$ as long as $$ c - {\tilde c} \in 24 {\mathbb Z} $$ This condition arises from modular invariance of the CFT when it is put on the torus. PS - In radially quantized CFTs, the adjoint condition is $L_m^\dagger = L_{-m}$ and ${\bar L}_m^\dagger = {\bar L}_{-m}$.


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There is a certain difference between the Euclidean and Lorenzian cases. In Euclidean QFT the correlation functions, the Schwinger functions, are all in a sense time-ordered (see Osterwalder–Schrader axioms). Intuitively one can say that this is because all separations are space-like. Say, if you consider the correlation function of one scalar, they are ...


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A more "pedestrian" application is that the low-energy physics of a Heisenberg antiferromagnetic spin chain is described by a WZW theory. This is a very simple and concrete model which has been shown to accurately describe many real materials. See for example http://arxiv.org/abs/hep-th/9802014v1, http://arxiv.org/abs/1211.5421v1, or section 7.10 of ...



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