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

TL;DR

This paper provides an accessible introduction to toric geometry with a focus on local Calabi-Yau varieties, emphasizing their relevance in string theory and the AdS/CFT correspondence, without requiring prior algebraic geometry knowledge.

Contribution

It offers a comprehensive, algebro-geometric overview of toric varieties and Calabi-Yau singularities, including constructions, resolutions, and deformations, tailored for physicists.

Findings

Construction of toric varieties as holomorphic quotients

Methods for resolving and deforming toric Calabi-Yau singularities

Explanation of the gauged linear sigma-model (GLSM) Kahler quotient

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.