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Betti Geometric Langlands

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 Bun_G(X) of G-bundles on X, while the spectral category consists of suitable O-modules on the character stack Loc_G^(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 c