Spectral Curves for Super-Yang-Mills with Adjoint Hypermultiplet for General Lie Algebras

TL;DR

This paper constructs Seiberg-Witten curves for $ ext{N}=2$ supersymmetric Yang-Mills theories with an adjoint hypermultiplet for all classical and exceptional Lie algebras, using integrable systems and confirming their properties.

Contribution

It generalizes the construction of Seiberg-Witten curves to arbitrary Lie algebras using Lax pairs from elliptic Calogero-Moser systems, extending previous results.

Findings

Curves have correct group-theoretic and complex structure.

Agreement with perturbative prepotential for $D_n$.

Derived a RG-like equation relating prepotential and Hamiltonian.

Abstract

The Seiberg-Witten curves and differentials for $\N=2$ supersymmetric Yang-Mills theories with one hypermultiplet of mass $m$ in the adjoint representation of the gauge algebra $\G$, are constructed for arbitrary classical or exceptional $\G$ (except $G_2$). The curves are obtained from the recently established Lax pairs with spectral parameter for the (twisted) elliptic Calogero-Moser integrable systems associated with the algebra $\G$. Curves and differentials are shown to have the proper group theoretic and complex analytic structure, and to behave as expected when $m$ tends either to 0 or to $\infty$. By way of example, the prepotential for $\G = D_n$, evaluated with these techniques, is shown to agree with standard perturbative results. A renormalization group type equation relating the prepotential to the Calogero-Moser Hamiltonian is obtained for arbitrary $\G$, generalizing a previous result for $\G = SU(N)$. Duality properties and decoupling to theories with other representations are briefly discussed.