Spin/Pin-structures on vector bundles have long featured prominently in differential geometry, in particular providing part of the foundation for the original proof of the renowned Atiyah–Singer Index Theory. More recently, they have underpinned the symplectic topology foundations of the so-called real sector of the mirror symmetry of string theory.

This semi-expository three-part monograph provides an accessible introduction to Spin- and Pin-structures in general, demonstrates their role in the orientability considerations in symplectic topology, and presents their applications in enumerative geometry.

Part I contains a systematic treatment of Spin/Pin-structures from different topological perspectives and may be suitable for an advanced undergraduate reading seminar. This leads to Part II, which systematically studies orientability problems for the determinants of real Cauchy–Riemann operators on vector bundles. Part III introduces enumerative geometry of curves in complex projective varieties and in symplectic manifolds, demonstrating some applications of the first two parts in the process. Two appendices review the Čech cohomology perspective on fiber bundles and Lie group covering spaces.

Contents:

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  Preface

  
  


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  About the Authors

  
  


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  Spin- and Pin-Structures:

  
  
  
  
  • Main Results and Examples of Part I
  
  
  • The Lie Groups Spin(n) and Pin±(n)
  
  
  • Proof of Theorem 1.4(1): Classical Perspective
  
  
  • Proof of Theorem 1.4(1): Trivializations Perspectives
  
  
  • Equivalence of Definitions 1.1–1.3
  
  
  • Relative Spin- and Pin-Structures
  
  
  
  


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  Orientations for Real CR-Operators:

  
  
  
  
  • Main Results and Applications of Part II
  
  
  • Base Cases
  
  
  • Intermediate Cases
  
  
  • Orientations for Twisted Determinants
  
  
  
  


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  Real Enumerative Geometry:

  
  
  
  
  • Pin-Structures and Immersions
  
  
  • Counts of Rational Curves on Surfaces
  
  
  • Counts of Stable Real Rational Maps
  
  
  • Counts of Real Rational Curves vs. Maps
  
  
  
  


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  Appendices:

  
  
  
  
  • Čech Cohomology
  
  
  • Lie Group Covers
  
  
  
  


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  Bibliography

  
  


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  Index of Terms

  
  


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  Index of Notation

  
  

Readership: Graduate students preparing for research in geometry and topology; active researchers in search of specific references on Spin/Pin-structures and orientations of determinants of real Cauchy–Riemann operators; Part I and some of Part III can be used for an advanced undergraduate reading course or seminar.

Key Features:

• This topic is very important in topology, geometry, and high energy physics


• Comprehensive treatment from different perspectives (classic, modern standard, and new), and describes explicitly the correspondences between these different perspectives


• Unique in applying properties of relative Spin- and Pin-structures to obtain properties of orientations of real Cauchy–Riemann operators on vectors


• Besides the rigorous theories, also contains many interconnected examples, and presents in detail the applications to real enumerative in the final part


• Self-contained and comes with background information; part of the book is accessible to advanced undergraduates


• One of the author Aleksey Zinger is a renowned expert in this field. The other author Xujia Chen is an emerging star