Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems

TL;DR

This collection of four lectures develops a cohesive framework linking 4D ${\cal N}=1,2,4$ SUSY Yang-Mills theory to classical integrable systems via Seiberg–Witten theory. By interpreting low-energy effective actions through holomorphic prepotentials and spectral curves, the notes connect nonperturbative gauge dynamics to Lax pairs, spectral curves, and monodromies of elliptic Calogero–Moser and Toda systems. The treatment extends from SU($N$) to general gauge algebras, incorporating adjoint matter and twisted/non-twisted Calogero–Moser constructions, with explicit results for perturbative and instanton sectors. The work highlights deep structural correspondences among quantum field theory, algebraic geometry, and integrable systems, with broad implications for dualities, monodromy, and nonperturbative dynamics in gauge theories.

Abstract

We present a series of four self-contained lectures on the following topics: (I) An introduction to 4-dimensional 1\leq N \leq 4 supersymmetric Yang-Mills theory, including particle and field contents, N=1 and N=2 superfield methods and the construction of general invariant Lagrangians; (II) A review of holomorphicity and duality in N=2 super-Yang-Mills, of Seiberg-Witten theory and its formulation in terms of Riemann surfaces; (III) An introduction to mechanical Hamiltonian integrable systems, such as the Toda and Calogero-Moser systems associated with general Lie algebras; a review of the recently constructed Lax pairs with spectral parameter for twisted and untwisted elliptic Calogero-Moser systems; (IV) A review of recent solutions of the Seiberg-Witten theory for general gauge algebra and adjoint hypermultiplet content in terms of the elliptic Calogero-Moser integrable systems.