Moduli of bundles over rational surfaces and elliptic curves I: simply laced cases

TL;DR

This paper extends the correspondence between del Pezzo surfaces and flat $E_n$ bundles over elliptic curves to a broader class of rational surfaces called $ADE$ surfaces, establishing a link for all simply laced, simple, compact, and simply-connected Lie groups.

Contribution

It constructs $ADE$ bundles over new rational surfaces and generalizes the correspondence to all flat $G$ bundles for simply laced Lie groups over elliptic curves.

Findings

Constructed $ADE$ bundles over $ADE$ surfaces.

Extended the correspondence to all simply laced Lie groups.

Set the stage for future work on non-simply laced groups.

Abstract

It is well-known that del Pezzo surfaces of degree $9-n$ one-to-one correspond to flat $E_n$ bundles over an elliptic curve. In this paper, we construct $ADE$ bundles over a broader class of rational surfaces which we call $ADE$ surfaces, and extend the above correspondence to all flat $G$ bundles over an elliptic curve, where $G$ is any simply laced, simple, compact and simply-connected Lie group. In the sequel, we will construct $G$ bundles for non-simply laced Lie group $G$ over these rational surfaces, and extend the above correspondence to non-simply laced cases.