A wildly ramified Betti Geometric Langlands correspondence

Xin Jin (Boston College)

I will present a Betti Geometric Langlands correspondence with wild ramifications on $\mathbb{CP}^1$ , which can be viewed as a case of homological mirror symmetry. The automorphic side is the Fukaya category (equivalently microlocal sheaf category) on a (smooth) moduli spaces of wild Higgs bundles $M_G$ (associated to every complex reductive group $G$ ), and the spectral side is the dg-category of coherent sheaves on a wild character variety (for the Langlands dual group). The proof has two main ingredients. One is a gluing formula for the Fukaya category of $M_G$ . The other is a characterization of the dg-category of coherent sheaves for certain Weil restrictions, which might be of independent interest. This is joint work with Zhiwei Yun.