Exceptional SW geometry from ALE fibrations

Abstract

We show that the genus 34 Seiberg-Witten curve underlying N = 2 Yang-Mills theory with gauge group E 6 yields physically equivalent results to the manifold obtained by fibration of the E 6 ALE singularity. This reconciles a puzzle raised by N = 2 string duality.

Whitham Deformations of Seiberg–witten Curves for Classical Gauge Groups

“…This means that the curve C is no longer hyperelliptic, but can be written in a special quartic polynomial in z with rational coefficients. Well known examples of curves of this form [3,44] are the Seiberg-Witten curves of SU(5) (A 4 ) in the 10-dimensional anti-symmetric representation, E 6 in the 27-dimensional minimal representation, and G 2 in the 7-dimensional minimal representation.…”

Prepotentials in N = 2 supersymmetric SU(3) YM theory with massless hypermultiplets

“…Subsequently, generalizations to gauge groups SU(N c ) without [6,7,8] and with matter (in the fundamental representation) [9,10,11,12] have been worked out. Extensions to other groups, SO(N c ) and Sp(N c ) [13,14] as well as to exceptional groups [15,16] are also known.…”

N = 2 superconformal field theory with ADE global symmetry on a D3-brane probe

“…This reflects the compactification of Type II string theory on a K3 fibered Calabi-Yau threefold. From this point of view, our calculation for the fundamental of E 6 in section 5.3 is indeed equivalent to that in [37] to obtain the SW curve for the N = 2 E 6 Yang-Mills theory from the fibration of the E 6 ALE space. Hence our computations in section 5 can be viewed as the determination of the SW curves in the fundamental and adjoint representations for N = 2 Yang-Mills theory with ADE gauge symmetries.…”