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Hints: Tong is in eq. (4.26) calculating the OPE $${\cal R}[\partial X(z) :e^{ikX(w)}:]~=~\ldots\tag{4.26}$$ Note that the radial order ${\cal R}$ is implicitly written in Tong's text. He is using a nested Wick's theorem between radial order ${\cal R}$ and normal order $: :$, cf. e.g. my Phys.SE answer here. The contraction is $$ C(z,w)~=~{\cal R}[ X(z) X(...


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All the questions are basically answered in the classic paper "Supersymmetries and their representations". See also the wonderful talk: What's new with Q?. 1.- When the theory is conformal: In $D=2$ $N=(1,0)$ (heterotic and type I strings), $N=(1,1)$ (type $IIB$ string), $N=(2,0)$ (type $IIA$ string), $N=(2,2)$ ( N=2 strings), $N=(2,1)$ ($N=2$ ...


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The metrics, $g_{ab}$ and $\bar{g}_{ab}$ are conformally equivalent if we can write $$ \bar{g}_{ab} = \Omega^2g_{ab}\,, $$ where $\Omega\equiv \Omega(x)$ is a non-zero differentiable function of the space-time coordinates. In this case, their Weyl tensors are equivalent, i.e., $\bar{C}^{a}_{\,\,bcd} = {C}^{a}_{\,\,bcd}$. Setting $g_{ab}$ equal to the ...


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