Instanton Counting, Quantum Geometry and Algebra

This book provides a pedagogical introduction to gauge theory, focusing on four-dimensional N=2 supersymmetric gauge theory and its applications to mathematics, such as algebraic geometry, geometric representation theory, and vertex operator algebras. It discusses the concept of instantons, solutions to the anti-self-dual Yang-Mills equation, and their generalization to various situations. The book also explores the algebraic geometric description of supersymmetric gauge theory, known as the Seiberg-Witten theory, and its connection to integrable systems. It further discusses the quantum algebraic structure of supersymmetric gauge theory, including the free field realization and the construction of quiver W-algebras.

Format: Paperback / softback
Length: 285 pages
Publication date: 07 July 2022
Publisher: Springer Nature Switzerland AG

This comprehensive book delves into the intricate realm of gauge theory, focusing on four-dimensional N = 2 supersymmetric gauge theory, and its connections to various branches of mathematics, including algebraic geometry, geometric representation theory, and vertex operator algebras. The central concept at play is the instanton, a remarkable solution to the anti-self-dual Yang-Mills equation in four dimensions.

In the first part of the book, the authors begin by providing a systematic description of instantons, elucidating how to integrate out the instanton moduli space and introducing the equivariant localization formula. They then demonstrate the generalization of this formalism to diverse scenarios, such as quiver and fractional quiver gauge theory, supergroup gauge theory, and more.

The second part of the book delves into the algebraic geometric perspective of supersymmetric gauge theory, commonly referred to as the Seiberg-Witten theory. The authors explore this theory from the perspective of string/M-theory, discussing how to quantize the geometric structure through the Ω-deformation of gauge theory. This part also highlights the relationship between supersymmetric gauge theory and integrable systems, shedding light on the quantization process.

The third and final part of the book delves into the quantum algebraic structure of supersymmetric gauge theory. After introducing the free field realization of gauge theory, the authors delve into the underlying infinite-dimensional algebraic structure, emphasizing its connection with representation theory of quivers. This connection leads to the notion of quiver W-algebra, a fundamental concept in the study of supersymmetric gauge theory. Furthermore, the authors clarify that this gauge theory construction of the algebra naturally gives rise to further affinization and elliptic deformation of W-algebra, expanding the understanding of this complex mathematical framework.

Overall, this book offers a comprehensive and in-depth exploration of recent developments in gauge theory, particularly four-dimensional N = 2 supersymmetric gauge theory, and its profound connections to mathematics. It serves as a valuable resource for researchers and students alike, providing a deep understanding of the key concepts and techniques involved in this field.

Weight: 480g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030761929
Edition number: 1st ed. 2021