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See https://arxiv.org/pdf/hep-th/9905111v3.pdf (page 58), and references therein. I hope it helps.

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(Note that I am only starting to study these works and I may be wrong on some points.) The paper you are quoting is indeed providing a full definition of (type II and heterotic) superstring theory (type I is missing), valid at the quantum level and for both the NS and R sectors. The definition is basically following the construction of Zwiebach (arxiv:hep-...

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Within distribution theory, a mathematically rigorous formulation of OP's eq. (2) is $$\lim_{z\to 0^+} z^{\Delta-d}K_{\Delta}(z,x)~=~\delta^d(x), \tag{A}$$ where $$K_{\Delta}(z,x)~:=~ \frac{\Gamma(\Delta)}{\pi^{\frac{d}{2}} \Gamma(\Delta\!-\!\frac{d}{2})} \left( \frac{z}{z^2+x^2}\right)^{\Delta},\qquad x~\in~\mathbb{R}^d, \qquad \Delta >\frac{d}{2},\... 1 In order to show that some function is the Delta distribution, you have to show that It is zero except for where the argument of the function vanishes. It integrates to 1 when integrated over the full coordinate range. We can see these two properties explicitly. If x-x'\not=0, then$$ \lim_{z\to0}K(z,x,x')=\lim_{z\to0}\frac{z^\Delta}{(x-x')^{2\Delta}...

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The Nambu-Goto action (including normalization) is $$S_{NG} = \frac{1}{2\pi \alpha'}\int dr d\theta \frac{L^2 r}{z^2} \sqrt{1 + z'^2}=\frac{L^2}{\alpha'}\int dr \frac{r}{z^2} \sqrt{1 + z'^2}$$ The corresponding Hamiltonian is not conserved as the Lagrangian is explicitly $r$-dependent. You can however see that a solution to the equations of motion  0=\...

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