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String Theory on Calabi-Yau Manifolds

Brian Greene

TL;DR

This work surveys how string theory reshapes geometry through quantum effects, dualities, and nonperturbative degrees of freedom, focusing on Calabi–Yau compactifications and N = 2 superconformal theories. It develops a unified framework linking perturbative CY sigma-models, Landau–Ginzburg models, and minimal models, with a central emphasis on moduli spaces and mirror symmetry. The author then demonstrates that topology change in string theory—via flops and conifold transitions—can be physically smooth and is best understood through the enlarged Kähler and complex structure moduli spaces, especially when analyzed with toric geometry. The text culminates by showing how toric methods provide concrete, calculable bridges between complex-structure data on a manifold and Kähler data on its mirror, enabling precise verifications of mirror symmetry and insights into quantum geometry.

Abstract

These lectures are devoted to introducing some of the basic features of quantum geometry that have been emerging from compactified string theory over the last couple of years. The developments discussed include new geometric features of string theory which occur even at the classical level as well as those which require non-perturbative effects. These lecture notes are based on an evolving set of lectures presented at a number of schools but most closely follow a series of seven lectures given at the TASI-96 summer school on Strings, Fields and Duality.

String Theory on Calabi-Yau Manifolds

TL;DR

This work surveys how string theory reshapes geometry through quantum effects, dualities, and nonperturbative degrees of freedom, focusing on Calabi–Yau compactifications and N = 2 superconformal theories. It develops a unified framework linking perturbative CY sigma-models, Landau–Ginzburg models, and minimal models, with a central emphasis on moduli spaces and mirror symmetry. The author then demonstrates that topology change in string theory—via flops and conifold transitions—can be physically smooth and is best understood through the enlarged Kähler and complex structure moduli spaces, especially when analyzed with toric geometry. The text culminates by showing how toric methods provide concrete, calculable bridges between complex-structure data on a manifold and Kähler data on its mirror, enabling precise verifications of mirror symmetry and insights into quantum geometry.

Abstract

These lectures are devoted to introducing some of the basic features of quantum geometry that have been emerging from compactified string theory over the last couple of years. The developments discussed include new geometric features of string theory which occur even at the classical level as well as those which require non-perturbative effects. These lecture notes are based on an evolving set of lectures presented at a number of schools but most closely follow a series of seven lectures given at the TASI-96 summer school on Strings, Fields and Duality.

Paper Structure

This paper contains 75 sections, 3 theorems, 336 equations, 44 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The State of String Theory
  3. What is Quantum Geometry?
  4. The Ingredients
  5. Some Classical Geometry
  6. Manifolds
  7. Equivalences
  8. Tangent Spaces
  9. Differential Forms
  10. Cohomology and Harmonic Analysis --- Part I
  11. Metrics: Hermitian and Kähler Manifolds
  12. Kähler Differential Geometry
  13. Holonomy
  14. Cohomology, Harmonic Analysis --- Part II
  15. Examples of Kähler Manifolds
  16. ...and 60 more sections

Key Result

Theorem 1

Any submanifold of ${\Bbb C} P^n$ can be realized as the zero locus of a finite number of homogeneous polynomial equations.

Figures (44)

  • Figure 1: The deformation from classical dynamics to quantum field theory.
  • Figure 2: The deformation from classical Riemannian geometry to quantum stringy geometry.
  • Figure 3: The charts of a manifold $X$.
  • Figure 4: The charts for a complex manifold. Notice that in this case the cooordinates are complex numbers.
  • Figure 5: The stereographic projection of the sphere from the north pole.
  • ...and 39 more figures

Theorems & Definitions (6)

  • Theorem 1: Chow
  • Definition 1
  • Theorem 2: Yau
  • Theorem 3
  • Definition 2
  • Definition 3