Simple Singularities and N=2 Supersymmetric Yang-Mills Theory

TL;DR

This paper proposes a new class of hyperelliptic curves to describe the quantum moduli space of N=2 supersymmetric Yang-Mills theories for arbitrary gauge groups, extending Seiberg-Witten theory.

Contribution

It introduces a sequence of hyperelliptic Riemann surfaces for arbitrary gauge groups, generalizing previous work to include A-D-E type singularities.

Findings

Monodromy in the semiclassical regime is correctly reproduced.

The proposed curves generalize to simply laced gauge groups.

Remarks on potential connections to string theory are discussed.

Abstract

We present a first step towards generalizing the work of Seiberg and Witten on N=2 supersymmetric Yang-Mills theory to arbitrary gauge groups. Specifically, we propose a particular sequence of hyperelliptic genus $n-1$ Riemann surfaces to underly the quantum moduli space of $SU(n)$ N=2 supersymmetric gauge theory. These curves have an obvious generalization to arbitrary simply laced gauge groups, which involves the A-D-E type simple singularities. To support our proposal, we argue that the monodromy in the semiclassical regime is correctly reproduced. We also give some remarks on a possible relation to string theory.