Katrina Barron Moduli Space of $N=1$ Superspheres with Tubes and the Sewing Operation

Annotation

Within the framework of complex supergeometry and motivated by two-dimen-sional genus-zero holomorphic $N = 1$ superconformal field theory, we define the moduli space of $N=1$ genus-zero super-Riemann surfaces with oriented and ordered half-infinite tubes, modulo superconformal equivalence. We define a sewing operation on this moduli space which gives rise to the sewing equation and normalization and boundary conditions. To solve this equation, we develop a formal theory of infinitesimal $N = 1$ superconformal transformations based on a representation of the $N=1$ Neveu-Schwarz algebra in terms of superderivations. We solve a formal version of the sewing equation by proving an identity for certain exponentials of superderivations involving infinitely many formal variables. We use these formal results to give a reformulation of the moduli space, a more detailed description of the sewing operation, and an explicit formula for obtaining a canonical supersphere with tubes from the sewing together of two canonical superspheres with tubes. We give some specific examples of sewings, two of which give geometric analogues of associativity for an $N=1$ Neveu-Schwarz vertex operator superalgebra. We study a certain linear functional in the supermeromorphic tangent space at the identity of the moduli space of superspheres with $1 + 1$ tubes (one outgoing tube and one incoming tube) which is associated to the $N=1$ Neveu-Schwarz element in an $N=1$ Neveu-Schwarz vertex operator superalgebra. We prove the analyticity and convergence of the infinite series arising from the sewing operation. Finally, we define a bracket on the supermeromorphic tangent space at the identity of the moduli space of superspheres with $1+1$ tubes and show that this gives a representation of the $N=1$ Neveu-Schwarz algebra with central charge zero.