Super $J$-holomorphic Curves: Construction of the Moduli Space

TL;DR

This paper constructs the moduli space of super J-holomorphic curves from super Riemann surfaces to symplectic manifolds, using super differential equations and transversality to establish a smooth supermanifold structure.

Contribution

It introduces a novel construction of the moduli space of super J-holomorphic curves as a smooth subsupermanifold, expanding the theory of super Riemann surfaces and holomorphic curves.

Findings

Successfully constructed the moduli space as a smooth subsupermanifold

Applied super differential equations and transversality arguments

Provided a framework for future studies of super holomorphic curves

Abstract

Let $M$ be a super Riemann surface with holomorphic distribution $\mathcal{D}$ and $N$ a symplectic manifold with compatible almost complex structure $J$. We call a map $\Phi\colon M\to N$ a super $J$-holomorphic curve if its differential maps the almost complex structure on $\mathcal{D}$ 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 $M\to N$.