# The Moduli Space of \(N=1\) Superspheres with Tubes and the Sewing Operation

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*Katrina Barron*

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.

#### Table of Contents

# Table of Contents

## The Moduli Space of $N=1$ Superspheres with Tubes and the Sewing Operation

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. An introduction to the moduli space of N = 1 superspheres with tubes and the sewing operation 714 free
- 2.1. Grassmann algebras and superanalytic superfunctions 714
- 2.2. Superconformal ( 1, l)–superfunctions and power series 1421
- 2.3. Complex supermanifolds and super–Riemann surfaces 1825
- 2.4. Superspheres with tubes and the sewing operation 2027
- 2.5. The moduli space of superspheres with tubes 2330
- 2.6. The sewing equation 3037

- Chapter 3. A formal algebraic study of the sewing operation 3542
- 3.1. An extension of the automorphism property 3643
- 3.2. Formal supercalculus and formal superconformal power series 3744
- 3.3. The formal sewing equation and formal sewing identities 5562
- 3.4. The N = 1 Neveu–Schwarz algebra and a representation in terms of superderivations 7885
- 3.5. Modules for the N = 1 Neveu–Schwarz algebra 8087
- 3.6. Realizations of the sewing identities for general representations of the TV = 1 Neveu- Schwarz algebra 8289
- 3.7. The corresponding identities for positive–energy representations of the N = 1 Neveu–Schwarz algebra 8592

- Chapter 4. An analytic study of the sewing operation 8996
- 4.1. A reformulation of the moduli space of superspheres with tubes 9198
- 4.2. An action of the symmetric group S[sub(n)] on the moduli space SK[sub((n))] 93100
- 4.3. Supermeromorphic superfunctions on SK and supermeromorphic tangent spaces of SK 95102
- 4.4. The sewing operation and superspheres with one, two, and three tubes 96103
- 4.5. Generalized superspheres with tubes 103110
- 4.6. The sewing formulas and the convergence of associated series via the Fischer–Grauert Theorem 105112
- 4.7. An N = 1 Neveu–Schwarz algebra structure of central charge zero on the supermeromorphic tangent space of SK(1) at its identity 122129

- Bibliography 133140