Geometric construction of elliptic integrable systems and N=1^* superpotentials

TL;DR

This paper presents a geometric construction of elliptic integrable systems, linking them to supersymmetric gauge theories and string theory, and extends known methods to a broader class of algebras.

Contribution

It generalizes the symplectic quotient construction of elliptic Calogero-Moser systems to various algebras and connects these systems to superpotentials in N=1* gauge theories.

Findings

Elliptic Calogero-Moser systems are derived from symplectic quotients for multiple algebras.

The connection between Hamiltonians and superpotentials in supersymmetric gauge theories is clarified.

The folding procedures for integrable models are systematized using orbit algebra theory.

Abstract

We show how the elliptic Calogero-Moser integrable systems arise from a symplectic quotient construction, generalising the construction for A_{N-1} by Gorsky and Nekrasov to other algebras. This clarifies the role of (twisted) affine Kac-Moody algebras in elliptic Calogero-Moser systems and allows for a natural geometric construction of Lax operators for these systems. We elaborate on the connection of the associated Hamiltonians to superpotentials for N=1* deformations of N=4 supersymmetric gauge theory, and argue how non-perturbative physics generates the elliptic superpotentials. We also discuss the relevance of these systems and the associated quotient construction to open problems in string theory. In an appendix, we use the theory of orbit algebras to show the systematics behind the folding procedures for these integrable models.