THE MODULI SPACE OF N=1 SUPERSPHERES WITH TUBES AND THE SEWING OPERATION by Katrina Barron ( 9780821832608 )

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Title:

THE MODULI SPACE OF N=1 SUPERSPHERES WITH TUBES AND THE SEWING OPERATION

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Format:

Paperback

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ISBN 10:

0821832603

ISBN 13:

9780821832608

Publisher:

AMERICAN MATHEMATICAL SOCIETY

Pub. date:

15 February, 2003

Series:

Memoirs of the American Mathematical Society v. 162

Pages:

135

Description:

Within the framework of complex supergeometry and motivated by two-dimensional genus-zero holomorphic $N = 1$ superconformal field theory, this book defines the moduli space of $N=1$ genus-zero super-Riemann surfaces with oriented and ordered half-infinite tubes, modulo superconformal equivalence.

Synopsis:

Within the framework of complex supergeometry and motivated by two-dimensional 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.

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Imprint:

American Mathematical Society

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Returnable

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