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

TL;DR

This paper proves a conjecture relating the topological K-theory of quasi-BPS categories for Higgs bundles to BPS cohomology, connecting geometric Langlands, mirror symmetry, and invariants of Calabi-Yau threefolds.

Contribution

It establishes an isomorphism between the topological K-theory of quasi-BPS categories and BPS cohomology, confirming a conjecture at the K-theory level.

Findings

Proved the conjecture at the level of topological K-theories.

Showed the K-theory of BPS categories is isomorphic to BPS cohomology.

Extended results to cases where rank and Euler characteristic are coprime.

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 non-commutative 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 $\chi$-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.