Overview
- Presents a new review of the ODE/IM correspondence, including recent developments
- Provides a new application of the exact WKB method in quantum mechanics
- Covers a quick guide of integrability
Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 51)
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About this book
This book is intended to review some recent developments in quantum field theories and integrable models. The ODE/IM correspondence, which is a nontrivial relation between the spectral analysis of ordinary differential equations and the functional relation approach to two-dimensional quantum integrable models, is the main subject. This correspondence was first discovered by Dorey and Tateo (and Bazhanov, Lukyanov, and Zamolodchikov) in 1998, where the relation between the Schrodinger equation with a monomial potential and the functional equation called the Y-system was found. This correspondence is an example of the mysterious link between classical and quantum integrable systems, which produces many interesting applications in mathematical physics, including exact WKB analysis, the quantum Seiberg–Witten curve, and the AdS–CFT correspondence. In this book, the authors explain some basic notions of the ODE/IM correspondence, where the ODE can be formulated as a linear problem associated with affine Toda field equations. The authors then apply the approach of the ODE/IM correspondence to the exact WKB periods in quantum mechanics with a polynomial potential. Deformation of the potential leads to wall-crossing phenomena in the TBA equations. The exact WKB periods can also be regarded as the quantum periods of the four-dimensional N=2 supersymmetric gauge theories in the Nekrasov–Shatashvili limit of the Omega background. The authors also explain the massive version of the ODE/IM correspondence based on the affine Toda field equations, which also has an application to the minimal surface, and the gluon scattering amplitudes in the AdS/CFT correspondence.
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Table of contents (3 chapters)
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Front Matter
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Back Matter
Authors and Affiliations
About the authors
Katsushi Ito is a full professor in Tokyo Institute of Technology.
Hongfei Shu is a junior researcher in Zhengzhou University.
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Bibliographic Information
Book Title: ODE/IM Correspondence and Quantum Periods
Authors: Katsushi Ito, Hongfei Shu
Series Title: SpringerBriefs in Mathematical Physics
DOI: https://doi.org/10.1007/978-981-96-0499-9
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025
Softcover ISBN: 978-981-96-0498-2Published: 01 February 2025
eBook ISBN: 978-981-96-0499-9Published: 31 January 2025
Series ISSN: 2197-1757
Series E-ISSN: 2197-1765
Edition Number: 1
Number of Pages: X, 128
Number of Illustrations: 11 b/w illustrations, 18 illustrations in colour
Topics: Mathematical Physics, Analysis