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1

I recently stumbled upon a good comment about this in Jared Kaplan's AdS/CFT notes Any quantum field theory which has hope of having an UV-completion can be viewed as as effective theory at point in the RG flow from an UV complete theory. Field theories at the UV fixed point are conformal. Hence all 'well-defined' field theories are either CFTs or points ...


1

For clarity let's work with a Lorentzian signature. Our $g$ is a metric for a 2 dimensional Lorentzian manifold $M$. It is well-known that any two dimensional pseudo-Riemannian manifold is conformally flat, that is $$g = e^{2\omega}\eta$$ Where $\eta$ is the flat 2D Minkovski metric. Your Lightcone gauge example You didn't define your $x^\pm$s but I ...


2

Comments to the question (v2): To be specific, let us assume that the underlying 2D manifold is the Riemann sphere $S^2\cong \mathbb{C}\cup\{\infty\}$. The group of globally defined conformal transformations is the 6-dimensional group $PSL(2,\mathbb{C})$ of Moebius transformations. Mathematically speaking, one should consider the groupoid of locally ...


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This extra factor arises from the analogy of the conformal factor $\alpha'\omega$ term in (6.2.16). The required $\omega$ is $$\omega = \ln \left ( \frac{2\pi}{\partial_\nu\vartheta_1}\right) $$ and substituting it to the exponential we get $$ \exp\left( -\frac{\alpha'}{2}\sum_ik_i^2 \cdot \ln \frac{2\pi}{\partial_\nu\vartheta_1} \right) = ...


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This is an old question, but in case anyone is still interested, I'll just note a small edit to the discussion above. The general variation for the metric should actually be given as $\delta g_{\mu \nu} = \nabla_{\mu} \epsilon_{\nu} + \nabla_{\nu} \epsilon_{\mu}$, where $\nabla_{\alpha}$ denotes a covariant derivative. This is because the variation must ...


2

I think the original source of this claim is the famous unpublished paper of Luescher and Mack. Everyone's citing it. It is more rigorous mathematically and general (they don't assume parity) than Di Francesco. It starts on pages 1-2 of the manuscript. The proof below is basically the same proof, just with added details and a little bit different notation. ...


0

I) Recall first the $\phi\phi$-Operator Product Expansion (OPE): $$\tag{A} {\cal R}\left\{\phi(z,\bar{z})\phi(w,\bar{w})\right\} ~-~: \phi(z,\bar{z})\phi(w,\bar{w}): ~=~C(z,\bar{z};w,\bar{w}) ~{\bf 1}, $$ where the contraction is assumed to be a $c$-number: $$\tag{B} C(z,\bar{z};w,\bar{w})~=~ \langle 0 | {\cal ...


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Here $\begin{equation} \langle ik\phi(x)ik\phi(x)\rangle = \frac{\alpha'k^2}{\pi} \text{ln}(a/2R), \end{equation}$ where $a$ is an UV cutoff. Now we can write (as all the $\phi$'s are located at $x$ i.e. Radial ordered $\{\phi^n(x)\}=\phi^{n}(x)~$) $\begin{align} \{ik\phi\}^{n}(x) ~&=~ :\{ik\phi\}^n(x): +\sum_{\text{all contractions}} \\ ...


0

$\partial_a T^{ab}=0$ implies $\partial_1 T^{11}+\partial_2 T^{21}=0$. You don't sum over the uncontracted index $b$.


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Yes, eq. (2.193) is a classical formula, and the symmetry of the (Hilbert) stress-energy-momentum tensor $T^{\mu\nu}$ is only valid classically. Quantum mechanically, the symmetry of $T^{\mu\nu}(x)$ is broken by the presence of other fields in positions $x_1,x_2,\ldots$ in the (time-ordered) correlator $$\langle T\left\{ (\hat{T}^{\mu \nu}(x) - \hat{T}^{\nu ...


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This is a very broad question and therefore impossible to answer completely, but I will try to answer some questions and refer to some literature. Your statement of the AdS/CFT correspondence was not quite complete: Type IIb string theory on asymptotically $AdS_5\times S^5$ is equivalent to $\mathcal{N}=4$ super Yang-Mills theory In the limit of large ...



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