Topological K‐theory of quasi‐BPS categories for Higgs bundles

Abstract In a previous paper, we introduced quasi‐BPS categories for moduli stacks of semistable Higgs bundles. Under a certain condition on the rank, Euler characteristic, and weight, the quasi‐BPS categories (called BPS in this case) are noncommutative analogues of Hitchin integrable systems. We proposed a conjectural equivalence between BPS categories which swaps Euler characteristics and weights. The conjecture is inspired by the Dolbeault Geometric Langlands equivalence of Donagi–Pantev, by the Hausel–Thaddeus mirror symmetry, and by the ‐independence phenomenon for BPS invariants of curves on Calabi–Yau threefolds. In this paper, we show that the above conjecture holds at the level of topological K‐theories. When the rank and the Euler characteristic are coprime, such an isomorphism was proved by Groechenig–Shen. Along the way, we show that the topological K‐theory of BPS categories is isomorphic to the BPS cohomology of the moduli of semistable Higgs bundles.