Quasi-BPS categories for Higgs bundles

We introduce quasi-BPS categories for twisted Higgs bundles, which are building blocks of the derived category of coherent sheaves on the moduli stack of semistable twisted Higgs bundles over a smooth projective curve. Under some condition (called BPS condition), the quasi-BPS categories are non-commutative analogues of Hitchin integrable systems. We begin the study of these quasi-BPS categories by focusing on a conjectural symmetry which swaps the Euler characteristic and the weight. Our conjecture is inspired by the Dolbeault Geometric Langlands equivalence of Donagi--Pantev, by the Hausel--Thaddeus mirror symmetry, and by the $\chi$-independence phenomenon for BPS invariants of curves on Calabi-Yau threefolds. We prove our conjecture in the case of rank two and genus zero. In higher genus, we prove a derived equivalence of rank two stable twisted Higgs moduli spaces as a special case of our conjecture. In a separate paper, we prove a version of our conjecture for the topological K-theory of quasi-BPS categories and we discuss the relation between quasi-BPS categories and BPS invariants of the corresponding local Calabi-Yau threefold.