eBook ISBN:  9781470403706 
Product Code:  MEMO/162/772.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 
eBook ISBN:  9781470403706 
Product Code:  MEMO/162/772.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $39.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 162; 2003; 135 ppMSC: Primary 17; 32; 58; 81; Secondary 30
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.
ReadershipGraduate students and research mathematicians interested in applications of algebraic geometry to mathematical physics.

Table of Contents

Chapters

1. Introduction

2. An introduction to the moduli space of $N$ = 1 superspheres with tubes and the sewing operation

3. A formal algebraic study of the sewing operation

4. An analytic study of the sewing operation


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
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.
Graduate students and research mathematicians interested in applications of algebraic geometry to mathematical physics.

Chapters

1. Introduction

2. An introduction to the moduli space of $N$ = 1 superspheres with tubes and the sewing operation

3. A formal algebraic study of the sewing operation

4. An analytic study of the sewing operation