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I figured it out. $\mathcal{V}^{0}$ is the vertex operator integrated over $\theta$. Integrating $\phi$ over $\theta$ gives us $\Psi$, and we can use equations (12.3.15) to rewrite $\Psi$ as $G_{-1/2}\cdot \mathcal{O}$. In order to get the $\tilde G_{-1/2}$ factor for the $\mathcal{V}^{0,0}$ operator we must expand $\phi$ further, $$ \phi = \mathcal{O} ...



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