Memoirs of the American Mathematical Society 2003; 135 pp; softcover Volume: 162 ISBN10: 0821832603 ISBN13: 9780821832608 List Price: US$61 Individual Members: US$36.60 Institutional Members: US$48.80 Order Code: MEMO/162/772
 Within the framework of complex supergeometry and motivated by twodimensional genuszero holomorphic \(N = 1\) superconformal field theory, we define the moduli space of \(N=1\) genuszero superRiemann surfaces with oriented and ordered halfinfinite 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 \(N=1\) NeveuSchwarz 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 \(N=1\) NeveuSchwarz 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 \(N=1\) NeveuSchwarz element in an \(N=1\) NeveuSchwarz 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 \(N=1\) NeveuSchwarz algebra with central charge zero. Readership Graduate students and research mathematicians interested in applications of algebraic geometry to mathematical physics. Table of Contents  Introduction
 An introduction to the moduli space of \(N=1\) superspheres with tubes and the sewing operation
 A formal algebraic study of the sewing operation
 An analytic study of the sewing operation
 Bibliography
