ODE/IM Correspondence and Quantum Periods

Katsushi Ito, Hongfei Shu

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English

Springer Nature Switzerland AG
01 February 2025

Differential calculus & equations; Mathematical physics

Series: SpringerBriefs in Mathematical Physics

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.

By: Katsushi Ito, Hongfei Shu
Imprint: Springer Nature Switzerland AG
Country of Publication: Switzerland
Volume: 51
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9789819604982
ISBN 10: 9819604982
Series: SpringerBriefs in Mathematical Physics
Pages: 128
Publication Date: 01 February 2025
Audience: Professional and scholarly , College/higher education , Undergraduate , Further / Higher Education
Format: Paperback
Publisher's Status: Active

ODE/IM Correspondence.- Exact WKB Analysis and TBA Equations.- Massive ODE/IM Correspondence.- Integrable Models and Functional Relations.

Katsushi Ito is a full professor in Tokyo Institute of Technology. Hongfei Shu is a junior researcher in Zhengzhou University.