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Applications of Polyfold Theory I: The Polyfolds of Gromov–Witten Theory


About this Title

H. Hofer, K. Wysocki and E. Zehnder

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 248, Number 1179
ISBNs: 978-1-4704-2203-5 (print); 978-1-4704-4060-2 (online)
DOI: https://doi.org/10.1090/memo/1179
Published electronically: March 20, 2017
Keywords:sc-smoothess, polyfolds, polyfold Fredholm sections, GW-invariants

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Table of Contents


Chapters

  • Chapter 1. Introduction and Main Results
  • Chapter 2. Recollections and Technical Results
  • Chapter 3. The Polyfold Structures
  • Chapter 4. The Nonlinear Cauchy-Riemann Operator
  • Chapter 5. Appendices

Abstract


In this paper we start with the construction of the symplectic field theory (SFT). As a general theory of symplectic invariants, SFT has been outlined in Introduction to symplectic field theory (2000), by Y. Eliashberg, A. Givental and H. Hofer who have predicted its formal properties. The actual construction of SFT is a hard analytical problem which will be overcome be means of the polyfold theory due to the present authors. The current paper addresses a significant amount of the arising issues and the general theory will be completed in part II of this paper. To illustrate the polyfold theory we shall use the results of the present paper to describe an alternative construction of the Gromov-Witten invariants for general compact symplectic manifolds.

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