Categorical and Enumerative Aspects of Mirror Symmetry

This open access book is a tribute to the profound contributions of Professor Bumsig Kim in the field of mathematics, particularly in the realm of mirror symmetry. Mirror symmetry is a fascinating duality between complex/algebraic geometry and symplectic geometry, manifesting in various ways. One of the foundational examples of this duality is the comparison of period integrals on a complex manifold with curve counts (enumerative invariants) on its mirror. This vision was ultimately realized through the Quasimap Wall-Crossing Formula, developed by Bumsig Kim and his collaborators. The book delves into the methods for counting curves, including Gromov–Witten theory and Fan–Jarvis–Ruan–Witten theory. In his final works, Kim and various groups of collaborators achieved methods for counting curves on gauged linear sigma models, effectively unifying Gromov–Witten and Fan–Jarvis–Ruan–Witten theory. This monumental effort involved constructing categorical frameworks and developing a dictionary between categorical and classical homological constructions. As a conference proceedings volume, this book is a collaborative effort by Professor Kim’s network of colleagues. It explores the intricate connections between categorical and enumerative aspects of mirror symmetry, showcasing the depth and breadth of Kim's work. Through detailed discussions and presentations, the book highlights the innovative approaches and groundbreaking results that have shaped the field. Readers will find a comprehensive overview of the latest advancements in mirror symmetry, enriched by the collaborative spirit and intellectual rigor that characterized Professor Kim's career. This volume not only honors his legacy but also serves as a valuable resource for researchers and students seeking to understand and build upon his pioneering work. Whether you are new to the field or an experienced mathematician, this book offers a wealth of knowledge and insights into the complex and beautiful world of mirror symmetry.