The moduli space of N=2 super-Riemann spheres with tubes

Abstract

Within the framework of complex supergeometry and motivated by two-dimensional genus-zero holomorphic N=2 superconformal field theory, we define the moduli space of N=2 super-Riemann spheres with oriented and ordered half-infinite tubes (or equivalently, oriented and ordered punctures, and local superconformal coordinates vanishing at the punctures), modulo N=2 superconformal equivalence. We develop a formal theory of infinitesimal N=2 superconformal transformations based on a representation of the N=2 Neveu-Schwarz algebra in terms of superderivations. In particular, via these infinitesimals we present the Lie supergroup of N=2 superprojective transformations of the N=2 super-Riemann sphere. We give a reformulation of the moduli space in terms of these infinitesimals. We introduce generalized N=2 super-Riemann spheres with tubes and discuss some group structures associated to certain moduli spaces of both generalized and non-generalized N=2 super-Riemann spheres. We define an action of the symmetric groups on the moduli space. Lastly we discuss the nonhomogeneous (versus homogeneous) coordinate system associated to N=2 superconformal structures and the corresponding results in this coordinate system.