2016/06/28 by Dani Ben‐Zvi, David Ben-Zvi, David Nadler +3 · 3 citations
Mathematics · Medicine · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Automorphic form #Betti number #Combinatorics #Computer science #Conjecture #FOS: Mathematics #Geometry #Homogeneous space #Langlands dual group #Langlands program #Mathematics #Neonatal Health and Biochemistry #Pure mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Stack (abstract data type) #Topology (electrical circuits) #math.AG #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.1606.08523
published in arXiv (Cornell University) (Cornell University)
arxiv created 2016/06/28 · openalex publication_date 2016/06/28 · arxiv updated 2016/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of Kapustin-Witten in supersymmetric gauge theory. The conjecture proposes an automorphic category associated to a compact Riemann surface X and complex reductive group G is equivalent to a spectral category associated to the underlying topological surface S and Langlands dual group G^. The automorphic category consists of suitable C-sheaves on the moduli stack BunG(X) of G-bundles on X, while the spectral category consists of suitable O-modules on the character stack LocG^(S) of G-local systems on S. The conjecture is compatible with and constrained by the natural symmetries of both sides coming from modifications of bundles and local systems. On the one hand, cuspidal Hecke eigensheaves in the de Rham and Betti sense are expected to coincide, so that one can view the Betti conjecture as offering a different "integration measure" on the same fundamental objects. On the other hand, the Betti spectral categories are more explicit than their de Rham counterparts and one might hope the conjecture is less challenging. The Betti program also enjoys symmetries coming from topological field theory: it is expected to extend to an equivalence of four-dimensional topological field theories, and in particular, the conjecture for closed surfaces is expected to reduce to the case of the thrice-punctured sphere. Finally, we also present ramified, quantum and integral variants of the conjecture, and highlight connections to other topics, including representation theory of real reductive groups and quantum groups.