Kac–Moody $${\widetilde{E}_k}$$ -bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Abstract

Given an elliptic curve Σ, flat Ek-bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an Ad-singularity containing Σ as an anti-canonical curve and Kac–Moody \({\widetilde{E}_{k}}\)-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding \({\widetilde{E}_k}\)-bundles over Σ can be reduced to Ek-bundles.