Enumerative Geometry and String Theory
About this Title
Sheldon Katz, University of Illinois at Urbana-Champaign, Urbana, IL
Publication: The Student Mathematical Library
Publication Year 2006: Volume 32
ISBNs: 978-0-8218-3687-3 (print); 978-1-4704-2143-4 (online)
MathSciNet review: MR2218550
MSC: Primary 14N10; Secondary 14N35, 81T30
Perhaps the most famous example of how ideas from modern physics have revolutionized mathematics is the way string theory has led to an overhaul of enumerative geometry, an area of mathematics that started in the eighteen hundreds. Century-old problems of enumerating geometric configurations have now been solved using new and deep mathematical techniques inspired by physics!
The book begins with an insightful introduction to enumerative geometry. From there, the goal becomes explaining the more advanced elements of enumerative algebraic geometry. Along the way, there are some crash courses on intermediate topics which are essential tools for the student of modern mathematics, such as cohomology and other topics in geometry.
The physics content assumes nothing beyond a first undergraduate course. The focus is on explaining the action principle in physics, the idea of string theory, and how these directly lead to questions in geometry. Once these topics are in place, the connection between physics and enumerative geometry is made with the introduction of topological quantum field theory and quantum cohomology.
Undergraduate and graduate students interested in algebraic geometry or in mathematical physics.
Table of Contents
- Chapter 1. Warming up to enumerative geometry
- Chapter 2. Enumerative geometry in the projective plane
- Chapter 3. Stable maps and enumerative geometry
- Chapter 4. Crash course in topology and manifolds
- Chapter 5. Crash course in $C^\infty $ manifolds and cohomology
- Chapter 6. Cellular decompositions and line bundles
- Chapter 7. Enumerative geometry of lines
- Chapter 8. Excess intersection
- Chapter 9. Rational curves on the quintic threefold
- Chapter 10. Mechanics
- Chapter 11. Introduction to supersymmetry
- Chapter 12. Introduction to string theory
- Chapter 13. Topological quantum field theory
- Chapter 14. Quantum cohomology and enumerative geometry