Spin/pin-structures And Real Enumerative Geometry

This three-part monograph provides an accessible introduction to Spin- and Pin-structures in differential geometry,demonstrates their role in symplectic topology,and presents their applications in enumerative geometry. It is suitable for an advanced undergraduate reading seminar and may be of interest to researchers in mathematics.

Format: Hardback
Length: 468 pages
Publication date: 07 January 2024
Publisher: World Scientific Publishing Co Pte Ltd

Spin/Pin-structures on vector bundles have played a significant role in differential geometry, particularly in providing a foundation for the original proof of the renowned Atiyah-Singer Index Theory. More recently, they have played a crucial role in the symplectic topology foundations of the so-called real sector of string theory's mirror symmetry. This comprehensive three-part monograph offers an accessible introduction to Spin- and Pin-structures, showcasing their significance in symplectic topology and enumerative geometry. Part I provides a systematic treatment of Spin/Pin-structures from various topological perspectives, making it suitable for an advanced undergraduate reading seminar. This leads to Part II, which delves into the study of 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 symplectic manifolds, demonstrating how the first two parts apply in this context. Two appendices review the Čech cohomology perspective on fiber bundles and Lie group covering spaces.

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.

ISBN-13: 9789811278532