Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems

TL;DR

This paper provides a comprehensive overview of supersymmetric Yang-Mills theories, their dualities, and their deep connections with integrable systems like Calogero-Moser, highlighting recent solutions and mathematical structures involved.

Contribution

It offers a detailed, self-contained review linking supersymmetric gauge theories with integrable systems, including new insights into Seiberg-Witten solutions via elliptic Calogero-Moser models.

Findings

Review of N=2 super-Yang-Mills and duality structures

Construction of Lax pairs for elliptic Calogero-Moser systems

Recent solutions of Seiberg-Witten theory using integrable models

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.