On the existence of heterotic-string and type-II-superstring field theory vertices

We consider the problem of the existence of heterotic-string and type-II-superstring field theory vertices in the product of spaces of bordered surfaces parameterizing the left- and right-moving sectors of these theories. It turns out that this problem can be solved by proving the existence of a solution to the BV quantum master equation in moduli spaces of bordered spin-Riemann surfaces. We first prove that for arbitrary genus

,

Neveu–Schwarz boundary components, and

Ramond boundary components such solutions exist. We also prove that these solutions are unique up to homotopy in the category of BV algebras. Furthermore, we prove that there exists a map in this category under which these solutions are mapped to fundamental classes of Deligne-Mumford stacks of associated punctured spin-Riemann surfaces. These results generalize the work of Costello on the existence of a solution to the BV quantum master equations in moduli spaces of bordered Riemann surfaces which, through the work of Sen and Zwiebach, are related to the existence of bosonic-string vertices, and their relation to fundamental classes of Deligne-Mumford stacks of associated punctured Riemann surfaces. Using the existence of solutions to the BV quantum master equation in moduli spaces of spin-Riemann surfaces, we prove that heterotic-string and type-II-superstring field theory vertices, for arbitrary genus

and an arbitrary number of any type of boundary components, exist. Furthermore, we prove the existence of a solution to the BV quantum master equation in spaces of bordered $N=1$ super-Riemann surfaces for arbitrary genus

,

Neveu–Schwarz boundary components, and

Ramond boundary components.