Quasi-BPS categories for Higgs bundles

TL;DR

This paper introduces quasi-BPS categories for twisted Higgs bundles, exploring their properties, symmetries, and derived equivalences, with proofs in specific cases and conjectures supported by geometric Langlands and mirror symmetry insights.

Contribution

It defines quasi-BPS categories for twisted Higgs bundles, studies their symmetries, and proves conjectures in rank two, genus zero cases, advancing understanding of Higgs moduli and BPS invariants.

Findings

Proved conjecture for rank two, genus zero cases.

Established derived equivalence for higher genus rank two moduli.

Connected quasi-BPS categories to BPS invariants and mirror symmetry.

Abstract

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.