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

Abstract

The Seiberg-Witten curves and differentials for $\mathbb{N}=2$ supersymmetric Yang-Mills theories with one hypermultiplet of mass $m$ in the adjoint representation of the gauge algebra $\text{G}$, are constructed for arbitrary classical or exceptional $\text{G}$ (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 $\text{G}$. 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 $\text{G}=D_n$, 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 $\text{G}$, generalizing a previous result for $\text{G}=SU(N)$. Duality properties and decoupling to theories with other representations are briefly discussed.

Comments: 27 pages, Plain TeX; minor typos corrected, 5 refs added