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Toric geometry and local Calabi-Yau varieties: An introduction to toric geometry (for physicists)

Cyril Closset

TL;DR

These notes introduce toric geometry with a focus on local Calabi–Yau varieties pertinent to string theory and AdS/CFT. They develop toric varieties as holomorphic quotients and via gauged linear sigma-models, and explain how to resolve and deform toric Calabi–Yau singularities, including the Altmann Minkowski-deformation framework. The text blends concrete combinatorial data (cones, fans, toric diagrams) with algebraic constructs (coordinate rings, divisors, canonical bundle) and physical viewpoints (D-branes, quiver gauge theories). It highlights how CY conditions manifest in toric data, how singularities are treated through resolutions or deformations, and how GLSM parameters map to Kähler moduli, providing a practical toolkit for building toric local CY backgrounds in physics. Overall, the notes offer a concrete, self-contained bridge between toric geometry and its uses in modern string-theoretic constructions, especially in AdS/CFT contexts.

Abstract

These lecture notes are an introduction to toric geometry. Particular focus is put on the description of toric local Calabi-Yau varieties, such as needed in applications to the AdS/CFT correspondence in string theory. The point of view taken in these lectures is mostly algebro-geometric but no prior knowledge of algebraic geometry is assumed. After introducing the necessary mathematical definitions, we discuss the construction of toric varieties as holomorphic quotients. We discuss the resolution and deformation of toric Calabi-Yau singularities. We also explain the gauged linear sigma-model (GLSM) Kahler quotient construction.

Toric geometry and local Calabi-Yau varieties: An introduction to toric geometry (for physicists)

TL;DR

These notes introduce toric geometry with a focus on local Calabi–Yau varieties pertinent to string theory and AdS/CFT. They develop toric varieties as holomorphic quotients and via gauged linear sigma-models, and explain how to resolve and deform toric Calabi–Yau singularities, including the Altmann Minkowski-deformation framework. The text blends concrete combinatorial data (cones, fans, toric diagrams) with algebraic constructs (coordinate rings, divisors, canonical bundle) and physical viewpoints (D-branes, quiver gauge theories). It highlights how CY conditions manifest in toric data, how singularities are treated through resolutions or deformations, and how GLSM parameters map to Kähler moduli, providing a practical toolkit for building toric local CY backgrounds in physics. Overall, the notes offer a concrete, self-contained bridge between toric geometry and its uses in modern string-theoretic constructions, especially in AdS/CFT contexts.

Abstract

These lecture notes are an introduction to toric geometry. Particular focus is put on the description of toric local Calabi-Yau varieties, such as needed in applications to the AdS/CFT correspondence in string theory. The point of view taken in these lectures is mostly algebro-geometric but no prior knowledge of algebraic geometry is assumed. After introducing the necessary mathematical definitions, we discuss the construction of toric varieties as holomorphic quotients. We discuss the resolution and deformation of toric Calabi-Yau singularities. We also explain the gauged linear sigma-model (GLSM) Kahler quotient construction.

Paper Structure

This paper contains 53 sections, 93 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A digression: supersymmetric gauge theories and D-branes at toric singularities
  3. Outline of the lectures
  4. Algebraic geometry: the gist of it
  5. Affine varieties
  6. Examples:
  7. Example: the conifold.
  8. Projective varieties
  9. Spectrum and scheme, in two words
  10. The Calabi-Yau condition
  11. Holomorphic vector bundles and line bundles
  12. Calabi-Yau manifolds. Kähler and complex moduli
  13. Kähler structure.
  14. CY Theorem :
  15. Kähler moduli space.
  16. ...and 38 more sections

Figures (7)

  • Figure 1: The toric fan of $\mathbb{C}\mathrm{P}^2$. Notice that it contains 7 cones: three top-dimensional ones (of dimension 2), three 1-dimensional ones (generated by the vectors), and a single 0-dimensional one (the point in the center).
  • Figure 2: Some examples of toric fans in dimension two.
  • Figure 3: Some examples of toric diagrams for local CY threefolds.
  • Figure 4: pq-webs
  • Figure 5: The two possible resolutions of the conifold.
  • ...and 2 more figures