Affine 7-brane backgrounds and five-dimensional EN theories on S1 Original Research Article

Elliptic curves for the 7-brane configurations realizing the affine Lie algebras En (1⩽n⩽8) and Ẽn (n=0,1) are systematically derived from the cubic equation for a rational elliptic surface. It is then shown that the En 7-branes describe the discriminant locus of the elliptic curves for five-dimensional (5D) N=1 En theories compactified on a circle. This is in accordance with a recent construction of 5D N=1 En theories on the IIB 5-brane web with 7-branes, and indicates the validity of the D3 probe picture for 5D En theories on R4×S1. Using the En curves we also study the compactification of 5D En theories to four dimensions.