# The moduli space of superspheres with
tubes and the sewing operation

### About this Title

**Katrina Barron**

Publication: Memoirs of the American Mathematical Society

Publication Year
2003: Volume 162, Number 772

ISBNs: 978-0-8218-3260-8 (print); 978-1-4704-0370-6 (online)

DOI: http://dx.doi.org/10.1090/memo/0772

MathSciNet review: 1954484

MSC (2000): Primary 32C11; Secondary 17B68, 30F10, 32G15, 58A50

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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.

Readership

Graduate 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
= 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