TASI Lectures on Geometric Tools for String Compactifications

TL;DR

This paper offers a comprehensive overview of geometric methods, challenges, and recent advances in string compactifications, focusing on heterotic strings, Calabi-Yau manifolds, and related moduli problems.

Contribution

It provides a modern, self-contained introduction to tools and open questions in string compactifications, including progress on classifying backgrounds and understanding moduli spaces.

Findings

Progress on bounding and enumerating string backgrounds

Overview of Calabi-Yau constructions and classifications

Insights into moduli problems for holomorphic bundles

Abstract

In this work we provide a self-contained and modern introduction to some of the tools, obstacles and open questions arising in string compactifications. Techniques and current progress are illustrated in the context of smooth heterotic string compactifications to 4-dimensions. Progress is described on bounding and enumerating possible string backgrounds and their properties. We provide an overview of constructions, partial classifications, and moduli problems associated to Calabi-Yau manifolds and holomorphic bundles over them.