Calabi-Yau Varieties: Arithmetic, Geometry and Physics: Lecture Notes on Concentrated Graduate Courses

Description

This volume presents a lively introduction to the rapidly developing and vast research areas surrounding Calabi–Yau varieties and string theory. With its coverage of the various perspectives of a wide area of topics such as Hodge theory, Gross–Siebert program, moduli problems, toric approach, and arithmetic aspects, the book gives a comprehensive overview of the current streams of mathematical research in the area. The contributions in this book are based on lectures that took place during workshops with the following thematic titles: “Modular Forms Around String Theory,” “Enumerative Geometry and Calabi–Yau Varieties,” “Physics Around Mirror Symmetry,” “Hodge Theory in String Theory.”  The book is ideal for graduate students and researchers learning about Calabi–Yau varieties as well as physics students and string theorists who wish to learn the mathematics behind these varieties. Front Matter....Pages i-x Front Matter....Pages 1-1 The Geometry and Moduli of K3 Surfaces....Pages 3-43 Picard Ranks of K3 Surfaces of BHK Type....Pages 45-63 Reflexive Polytopes and Lattice-Polarized K3 Surfaces....Pages 65-79 Front Matter....Pages 81-81 An Introduction to Hodge Structures....Pages 83-130 Introduction to Nonabelian Hodge Theory....Pages 131-171 Algebraic and Arithmetic Properties of Period Maps....Pages 173-208 Front Matter....Pages 209-209 Mirror Symmetry in Physics: The Basics....Pages 211-278 Front Matter....Pages 279-279 Introduction to Gromov–Witten Theory....Pages 281-301 Introduction to Donaldson–Thomas and Stable Pair Invariants....Pages 303-313 Donaldson–Thomas Invariants and Wall-Crossing Formulas....Pages 315-333 Front Matter....Pages 335-335 Enumerative Aspects of the Gross-Siebert Program....Pages 337-420 Front Matter....Pages 421-421 Introduction to Modular Forms....Pages 423-444 Lectures on BCOV Holomorphic Anomaly Equations....Pages 445-473 Polynomial Structure of Topological String Partition Functions....Pages 475-500 Front Matter....Pages 501-501 Introduction to Arithmetic Mirror Symmetry....Pages 503-539 Back Matter....Pages 541-547