Refining Geometric Langlands via Limit Categories on Cotangent Stacks — E8 Intelligence Research

Abstract

FINDING: The geometric Langlands conjecture (GLC) is being actively refined and proven via derived categories of D-modules on the moduli stack Bun_G, with a dual equivalence to coherent sheaves on the Langlands dual stack Loc_G; recent work (arXiv:2508.19624) introduces "limit categories" for cotangent stacks to bridge classical limits and D-module categories. MATH: - Core conjecture (Beilinson-Drinfeld, refined by Arinkin–Gaitsgory): \[ \text{IndCoh}(\text{Loc}_G) \simeq \text{D-mod}(\text{Bun}_G) \] where \(\text{Loc}_G = \text{Map}(X, {}^L G)/G\) (flat \({}^L G\)-bundles), \({}^L G\) is the Langlands dual group (root system \(\Phi\) ↔ coroot system \(\Phi^\vee\)). - Key categorical structures: D-modules (holonomic, constructible) ↔ coherent sheaves (ind-coherent, quasi-coherent); the equivalence is a *categorical* Fourier–Mukai transform. - Limit categories (arXiv:2508.19624): For a smooth stack \(\mathcal{X}\), define \(\lim\)-categories \(\mathcal{C}_{\lim}(T^*\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com