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We consider two problems associated to the space of Riemann surfaces that areclosely related to string theory. We first consider the problem of existence ofheterotic-string and type-II-superstring field theory vertices in the productof spaces of bordered surfaces parameterizing left- and right-moving sectors ofthese theories. It turns out that this problem can be solved by proving theexistence of a solution to the BV quantum master equation in moduli spaces ofbordered spin-Riemann surfaces. We first prove that for arbitrary genus $g$,$n_{\text{NS}}$ Neveu-Schwarz boundary components, and $n_{\text{R}}$ Ramondboundary components such solutions exist. We also prove that these solutionsare unique up to homotopies in the category of BV algebras and are mapped tofundamental classes of Deligne-Mumford stacks of associated punctured spincurves. These results generalize the work of Costello on the existence of asolution to the BV quantum master equations in moduli spaces of borderedordinary Riemann surfaces. Using this, we prove that heterotic-string andtype-II-superstring field theory vertices exist. Furthermore, we prove theexistence of a solution to the BV quantum master equation in spaces of bordered$\mathscr{N}=1$ super-Riemann surfaces for arbitrary genus $g$, $n_{\text{NS}}$Neveu-Schwarz boundary components, and $n_{\text{R}}$ Ramond boundarycomponents. Secondly, we turn to the moduli problem of stable $\mathscr{N}=1$SUSY curves. Using Vaintrob's deformation theory of complex superspaces andDeligne's model for the compactification of the space of $\mathscr{N}=1$ SUSYcurves, we prove the representability of this moduli problem by a proper smoothDeligne-Mumford superstack. This is a generalization of the work of Deligne andMumford on the representability of the moduli problem of stable ordinary curvesby a Deligne-Mumford stack.

On the Existence of Heterotic-String and Type-II-Superstring Field Theory Vertices