Super J-holomorphic curves

The compactification of the moduli space of super Riemann surface has been studied by Deligne (1987) and more recently by Donagi and Witten (2015). We aim to extend those results by constructing a moduli space of stable maps.

Let M be a super Riemann surface with holomorphic distribution D and N a symplectic manifolds with compatible almost complex structure J. We call a map Φ: M-> N a super J-holomorphic curve if its differential maps the almost complex structure on D to J. Such a super J-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super J-holomorphic curves as a smooth subsupermanifold of the space of maps M->N. The reduced space of the moduli space coincides with the moduli space of J-holomorphic curves from the reduction of M to N.