The moduli space of the N = 2 supersymmetric G2 Yang-Mills theory

Abstract

We present the hyper-elliptic curve describing the moduli space of the N=2 supersymmetric Yang-Mills theory with the G 2 gauge group. The exact monodromies and the dyon spectrum of the theory are determined. It is verified that the recently proposed solitonic equation is also satisfied by our solution.

Moduli space ofN=2supersymmetricG2gauge theory

“…To be precise, hyperelliptic curves have been found for the theories with classical gauge groups A n , B n , C n , and D n and various matter fields transforming in the fundamental representation [2]. Hyperelliptic curves have also been suggested for the exceptional gauge groups G 2 , F 4 , E 6,7,8 [3,4,5]. Clearly the construction of these hyperelliptic curves was based on an ad hoc ansatz which could be justified by performing certain highly non-trivial consistency checks such as reproducing the correct weak coupling monodromies or checking the number of special points in the moduli space where the supersymmetries can be broken down to N = 1.…”

“…Also it is not known if the special Prym, which is the physically relevant subspace of the Jacobian, could be embedded in the Jacobian of a hyperelliptic curve. Thus no direct correspondence between the spectral curves of [7] and the hyperelliptic ones suggested in [3,4,5] has been set up until now. The aim of this paper is to tackle this question.…”

“…4 It can be characterized as the subgroup of SO (7) which leaves one element of the spinor representation invariant. It is of rank two but a convenient basis of its Cartan subalgebra can be given by three elements satisfying one relation.…”

“…Let us now compare this information with that contained in the Riemann surfaces of [7] and [3,4]. We start with the curve from the integrable system.…”

Picard–Fuchs ordinary differential systems in N=2 supersymmetric Yang–Mills theories

“…Seiberg and Witten's approach to N = 2 supersymmetric SU(2) Yang-Mills theory 2,3 was extended to other higher rank gauge group cases coupled with or without quark hypermultiplets, 5,6,7,8,9,10,11,12,13,14,15,16 and it was found that the quantum moduli spaces of those gauge theories could be identified with those of Riemann surfaces with certain genus. In these studies, general algorithms to get Picard-Fuchs equations were developed, 17,18,19,20,21 but these equations are in general realized as a set of simultaneous partial differential equations (PDEs) in terms of moduli derivatives.…”