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If you have not seen it yet, conformal bootstrap in $1+1$ is extremely powerful, and in many cases essentially determine the whole theory. Everything is done analytically. Recent works of higher-dimensional generalizations share many basic features with the $1+1$ version, so it seems not a bad idea to start from there.

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You can rescale by $x \to \lambda x$ in that case the integral becomes: $$\phi(\lambda x)=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_p}}(a_pe^{i\lambda px}+a_p^\dagger e^{-i\lambda px})$$ We want this integral now in terms of the original integral. Since we are integrating over p, we are free to redefine the this variable into anything we like. We can ...

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Tricritical Ising model belongs to the family of minimal models ($M(5,4)$). There are several different coset constructions that represent them, one of them is the following: $M(m+1,m)=SU(2)_{m-2} \times SU(2)_1/SU(2)_{m-1}$

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Let us suppress (world-sheet) time $\tau$ in what follows, i.e. consider a fixed time $\tau$. Let there be given a continuous map $\phi:\Sigma\to M$, where the world-space $\Sigma$ and the target space $M$ are both 1D manifolds. We will assume that such a 1D manifold is either a real line $\mathbb{R}$ or a circle $S^1\cong\mathbb{R}/\mathbb{Z}$. That gives ...

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Notice first that even before restricting the domain of $\phi$, we are considering the theory on the cylinder and identifying the boundary condition $\phi(x + L,t) = \phi(x,t)$. Now to explain the restriction, let's take this example. Consider a field configuration at some fixed time $\phi(x,0)$, we only have to study this in the domain $[0,L]$. Now pick ...

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So, as I now understand, it is (obviously) only in two dimensions that $h$ and $\bar{h}$ have the interpretation of the weights corresponding to left- and right- movers. And it is therefore clear that $h-\bar{h}$ is the spin. However, the spin-statistics theorem says that spin is either integer or half-integer, so it isn't quite clear how $(h-\bar{h})$ is ...

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A conformal field theory is a quantum field theory which is invariant under conformal transformations. Due to this invariance, correlation functions must obey linear equations called conformal Ward identities. Conformal blocks are not just solutions of the conformal Ward identities, but actually elements of a particular basis of solutions. Let us focus on ...

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The proof of this statement has only been established for 4-dimensions recently. See M. A. Luty, J. Polchinski, R. Rattazzi "The a-theorem and the Asymptotics of 4D Quantum Field Theory" JHEP01 (2013) 152 For two dimensions, there has been a robust proof for more than 26 years. See J. Polchinski, "Scale And Conformal Invariance in Quantum Field Theory," ...

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Your last $S$ should be an $R$. The conformal group on is the set of transformations of $R^{p+q}$ that preserve angles ona $R^{p, q}$ is a Euclidean space with $p$ normal dimensions and $q$ imaginary ones; $S^{p,q}$ is presumably the unit sphere in this space; stereographic projections from the real space to the sphere will preserve angles. Then for each ...

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Once you fix a coordinate system $X$, then for any two points A and B the distance $d(A,B)$ between the two can be defined and calculated from the metric in that particular coordinate system, allowing you to define limits of this type: For a sequence of points $P_n$ and a point $Q$, $P_n\rightarrow Q$ if and only if $d(P_n,Q)\rightarrow0$. It's true that ...

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