I will report on joint work with Yukinobu Toda (partially in progress) about the (categorical) Hall algebra of Higgs bundles on a curve. These categories have semiorthogonal decompositions in certain categories, called quasi-BPS, whose construction is inspired by the enumerative geometry of Calabi-Yau threefolds. 

I will focus on two conjectural dualitities. The first is between the Hall algebra of semistable Higgs bundles of degree zero and a ``limit” category. This equivalence aims to make precise the proposal of Donagi-Pantev of considering the classical limit of the de Rham Langlands equivalence. The second is a primitive (or BPS) version of the first, and it relates categories of sheaves on moduli of semistable Higgs bundles (for various degrees). This equivalence may be regarded as a version of the D-equivalence conjecture/ SYZ mirror symmetry. We can prove (partial) versions of these conjectures for topological K-theory of these categories.