Proof of Geometric Langlands via Supersymmetric Gauge Theory — E8 Intelligence Research

Abstract

FINDING: The geometric Langlands correspondence is now proven (Gaitsgory et al.), with the Arinkin–Gaitsgory singular support condition shown to have a physical origin in N=4 supersymmetric gauge theory (Witten). | MATH: The correspondence is a categorical equivalence: \( \mathrm{IndCoh}_{\mathcal{N}}(\mathrm{LocSys}_G) \simeq \mathrm{D}(\mathrm{Bun}_G) \) — where the left side is ind-coherent sheaves on the stack of \(G\)-local systems with nilpotent singular support \(\mathcal{N}\), and the right is the derived category of D-modules on the moduli stack of \(G\)-bundles. The singular support condition restricts to the nilpotent cone \(\mathcal{N} \subset T^*\mathrm{LocSys}_G\). Witten's physical derivation uses the \(N=4\) super-Yang–Mills S-duality, mapping the A-brane category to the B-brane category with a twist that imposes the nilpotent condition. Key structures: root systems of \(G\), the affine Weyl group, and the Langlands dual group \({}^L G\). | CONNECTION: The nilpotent con Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com