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Topological strings and their physical applications

Andrew Neitzke, Cumrun Vafa

TL;DR

This work surveys topological string theory as a bridge between mathematics and physics, detailing Calabi-Yau geometry, toric methods, and the core A- and B-models. It develops computational tools—mirror symmetry, the topological vertex, and open/closed dualities—that yield genus-by-genus topological amplitudes and illuminate their dependence on Kahler and complex moduli. The authors then connect these amplitudes to physical observables in 4D N=2 and N=1 gauge theories, as well as to BPS black-hole entropy in 4D and 5D, including a discussion of geometric transitions and holomorphic matrix models. Finally, they outline a speculative topological M-theory framework that could unify A- and B-model descriptions and offer a nonperturbative viewpoint on topological strings and their holographic duals.

Abstract

We give an introductory review of topological strings and their application to various aspects of superstrings and supersymmetric gauge theories. This review includes developing the necessary mathematical background for topological strings, such as the notions of Calabi-Yau manifold and toric geometry, as well as physical methods developed for solving them, such as mirror symmetry, large N dualities, the topological vertex and quantum foam. In addition, we discuss applications of topological strings to N=1,2 supersymmetric gauge theories in 4 dimensions as well as to BPS black hole entropy in 4 and 5 dimensions. (These are notes from lectures given by the second author at the 2004 Simons Workshop in Mathematics and Physics.)

Topological strings and their physical applications

TL;DR

This work surveys topological string theory as a bridge between mathematics and physics, detailing Calabi-Yau geometry, toric methods, and the core A- and B-models. It develops computational tools—mirror symmetry, the topological vertex, and open/closed dualities—that yield genus-by-genus topological amplitudes and illuminate their dependence on Kahler and complex moduli. The authors then connect these amplitudes to physical observables in 4D N=2 and N=1 gauge theories, as well as to BPS black-hole entropy in 4D and 5D, including a discussion of geometric transitions and holomorphic matrix models. Finally, they outline a speculative topological M-theory framework that could unify A- and B-model descriptions and offer a nonperturbative viewpoint on topological strings and their holographic duals.

Abstract

We give an introductory review of topological strings and their application to various aspects of superstrings and supersymmetric gauge theories. This review includes developing the necessary mathematical background for topological strings, such as the notions of Calabi-Yau manifold and toric geometry, as well as physical methods developed for solving them, such as mirror symmetry, large N dualities, the topological vertex and quantum foam. In addition, we discuss applications of topological strings to N=1,2 supersymmetric gauge theories in 4 dimensions as well as to BPS black hole entropy in 4 and 5 dimensions. (These are notes from lectures given by the second author at the 2004 Simons Workshop in Mathematics and Physics.)

Paper Structure

This paper contains 46 sections, 145 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Calabi-Yau spaces
  3. Definition of Calabi-Yau space
  4. Why Calabi-Yau?
  5. Examples of Calabi-Yau spaces
  6. Dimension $1$
  7. Dimension $2$
  8. Dimension $3$
  9. Conifolds
  10. Toric geometry
  11. The topological string
  12. Sigma models and ${\mathcal{N}}=(2,2)$ supersymmetry
  13. Twisting the ${\mathcal{N}}=(2,2)$ supersymmetry
  14. Constructing the string correlation functions
  15. Why Calabi-Yau threefolds?
  16. ...and 31 more sections

Figures (20)

  • Figure 1: A rectangular torus; the top and bottom sides are identified, as are the left and right sides.
  • Figure 2: A 2-torus with a more general metric; again, opposite sides of the figure are identified.
  • Figure 3: A crude representation of the local $\mathbb C\mathbb P^2$ geometry, ${\mathcal{O}}(-3) \to \mathbb C\mathbb P^2$.
  • Figure 4: A crude representation of the local $\mathbb C\mathbb P^1$ geometry, ${\mathcal{O}}(-1) \oplus {\mathcal{O}}(-1) \to \mathbb C\mathbb P^1$.
  • Figure 5: The three conifold geometries: from left to right, deformed, singular and resolved. Both geometries look like $S^2 \times S^3$ near infinity (the bottom of the figure); they are distinguished by whether the $S^2$ or the $S^3$ shrinks to zero size in the interior (the top of the figure.)
  • ...and 15 more figures

Theorems & Definitions (29)

  • Example 2.1: The complex plane
  • Example 2.2: The punctured complex plane, aka the cylinder
  • Example 2.3: The 2-torus
  • Example 2.4: Cartesian products
  • Example 2.5: K3
  • Example 2.6: Complex projective space
  • Example 2.7: ALE spaces
  • Example 2.8: The quintic threefold
  • Example 2.9: Local $\mathbb C\mathbb P^2$.
  • Example 2.10: Local $\mathbb C\mathbb P^1$.
  • ...and 19 more