Integrable Many-Body Systems and Gauge Theories
A. Gorsky, A. Mironov
TL;DR
The paper surveys how integrable many-body systems encode the nonperturbative structure of gauge theories, revealing a dictionary between spectral curves, generating differentials, and prepotentials on moduli spaces. It develops the Seiberg-Witten framework and extends it through brane realizations, Whitham hierarchies, and topological data, unifying 4d–6d theories with a network of integrable models (Toda, Calogero, Ruijsenaars, and Dell chains). Perturbative prepotentials, nonperturbative configurations, and dualities are treated in tandem, showing how dimensional uplift and adjoint/fundamental matter map to distinct integrable systems (e.g., XXX/XXZ/XYZ spin chains, elliptic Calogero, and double-elliptic models). The review highlights the role of Whitham dynamics and WDVV equations in organizing the RG flow and topological sectors, and it discusses the rich web of dualities, including S-, mirror-like dualities and separation of variables, that connect gauge theories to brane constructions and to different integrable pictures. Overall, the work clarifies how finite-dimensional integrable systems govern low-energy actions and reveal hidden symmetries across dimensions, with practical implications for computing prepotentials and spectra in supersymmetric gauge theories.
Abstract
The review studies connections between integrable many-body systems and gauge theories. It is shown how the degrees of freedom in integrable systems are related with topological degrees of freedom in gauge theories. The relations between families of integrable systems and N=2 supersymmetric gauge theories are described. It is explained that the degrees of freedom in the many-body systems can be identified with collective coordinates of D-branes, solitons in string theory.