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One way of defining conformal transformations are by a (positive) local scaling of the metric, of the form $e^{2 \phi}$. Such a transformation always preserves the sign of spacetime distances. In particular, the light-cone remains unchanged since null distances map to null distances. What was inside the lightcone stays inside and things outside stay outside. ...

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I am talking about Euclidian signature. It is true that you cannot obtain $$x_i \mapsto \frac{x_i}{x^2}$$ from special conformal transformations, dilatations, translations and rotations because, as Olof has noted, this map changes the orientations, whereas the listed transformations preserve it. However, what you can obtain is e.g. $$x_i \mapsto ... 2 The special conformal transformations as well as the translations, dilations and rotations are all continuously connected to the identity. This means that they contain parameters such that at some particular value the trasformation becomes trivial. For example, for b=0 the special conformal trasformation you write is simply x_i\mapsto x_i. The inversion ... 0 Edit. I just realized that I didn't answer your question; I'll leave this up in case it helps answer your real question... Let use denote the operations of inversion and translation by$$ I(x) = \frac{x}{x^2}, \qquad T_b(x) = x+b Then notice what happens when we perform the following composite transformation: \begin{align} I\circ T_{-b}\circ I (x) ... 2 The most physical and understandable definition of Nekrasov's partition function to me uses five-dimensional gauge theories. Namely, any 4d N=2 susy gauge theory has a 5d version with the same matter content, so that compactifying it on a small S^1 brings it back to the original 4d theory. Then we put the theory on the so-called Omega background: it is ... 1 Complex-valued eigenvalues can be used to introduce the concept of electromagnetic mass and charge. Below I will provide an example of the Lorentz invariant model that uses complex-valued eigenvalues as a key component, but nevertheless allows for well defined momentum density with real valued mass density square. The “complexity” of the eigenvalues in this ... 1 First, a reference article, by Witten, http://arxiv.org/pdf/hep-th/9802150v2.pdf I'll try to expose the basic idea, with a flat space-time. Suppose you have a relativistic scalar field theory, on a flat space-time domain, with boundary. The equation of the field is :\square \Phi(x) = 0$$(fields on-shell) Now, define the partition function$$Z = ...

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Fist, let me just say that this book will answer (almost) all of your questions with beautiful precision and wonderful detail; I highly highly recommend that you read the chapter on the conformal group from which I am stealing the following information. Let me repeat one of the definitions that directly answers your question in the case of the ...

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The statement can be understood in terms of the GKPW-formula (named after Gubser, Klebanov, Polyakov and Witten), which does exactly that: it relates correlation functions on the CFT side (boundary) to string amplitudes on the AdS side (bulk). Assume that $\phi(\vec{x},z)$ is some field in the bulk, where $z$ is the so-called "holographic coordinate", which ...

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If you add the hypothesis of unitarity you have that the conformal weights of your field $\Delta_i$ must obey the following inequality: $$\Delta_i \geq \frac{d-2}{2} + l$$ where $l$ is the spin of the operator. The conformal bootstrap equation give you also constraints on the coefficients $C_{ijk}$ appearing in the 3-point correlation function. As far as I ...

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