Partition Functions and Automorphic Forms

This book provides an introduction to the research in several recently discovered and actively developing mathematical and mathematical physics areas, including Feynman integrals, hyperbolic and Lorentzian Kac-Moody algebras, superconformal indices, moonshine, Jacobi forms, elliptic genus, and string theory. It is intended for graduate students and young postdocs interested in the interaction between quantum field theory and mathematics.

Format: Paperback / softback
Length: 415 pages
Publication date: 10 July 2021
Publisher: Springer Nature Switzerland AG

This comprehensive book provides an in-depth exploration into various emerging and rapidly advancing fields of mathematics and mathematical physics. It delves into intricate topics such as:
1. Feynman Integrals and Modular Functions: This chapter offers a foundational introduction to Feynman integrals and modular functions, which play a crucial role in quantum field theory and mathematical physics. It explores their properties, applications, and connections to other areas of mathematics.
2. Hyperbolic and Lorentzian Kac-Moody Algebras: This chapter delves into the study of hyperbolic and Lorentzian Kac-Moody algebras, which are mathematical structures associated with quantum gravity and have profound implications for understanding the universe's fundamental properties. It discusses their algebraic structures, automorphic forms, and applications to quantum gravity.
3. Superconformal Indices and Elliptic Hypergeometric Integrals: This chapter explores the realm of superconformal indices and elliptic hypergeometric integrals, which are mathematical tools used to study partition functions in supersymmetric and ordinary quantum field theories in curved space-times of various dimensions. It discusses their mathematical properties, connections to other areas of mathematics, and their applications to quantum field theories.
4. Moonshine: This chapter delves into the mathematical aspects of moonshine, a fascinating theory that unifies several branches of mathematics, including representation theory, number theory, and geometry. It discusses moonshine's arithmetic properties, Jacobi forms, elliptic genera, and its connections to string theory.
5. Theory and Applications of the Elliptic Painleve Equation: This chapter explores the theory and applications of the elliptic Painleve equation, a mathematical equation that arises in various physical contexts, including quantum field theories and statistical mechanics. It discusses the equation's mathematical properties, solutions, and their connections to other areas of mathematics and physics.
6. Aspects of Painleve Equations in Quantum Field Theories: This chapter examines the aspects of Painleve equations in quantum field theories, particularly their connections to supersymmetric and ordinary quantum field theories in curved space-times of different dimensions. It discusses the mathematical techniques used to study Painleve equations and their applications to understanding the behavior of quantum systems.

The book is designed to cater to graduate students and young postdocs who are interested in the intersection of quantum field theory and mathematics related to automorphic forms, representation theory, number theory, and geometry. It employs multidisciplinary methods, including localization, Borcherds products, theory of special functions, Cremona maps, and more, to treat a wide range of partition functions. By presenting these topics in a clear and accessible manner, the book aims to foster understanding and collaboration between these fields.

Overall, this book serves as a valuable resource for researchers and scholars seeking to delve into the cutting-edge research of mathematics and mathematical physics. Its comprehensive coverage of diverse topics and multidisciplinary approach make it an essential tool for advancing our knowledge of the universe's fundamental laws.

Weight: 658g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030424022
Edition number: 1st ed. 2020