Quasi-BPS categorical duality for Higgs bundles

Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$ of degree $l$, and let $g^{\rm{sp}}$ be the genus of the spectral curve. For integers $(r,\chi,w)$, the BPS condition means that $(r,\chi,w+1-g^{\rm{sp}})$ is primitive. The quasi-BPS category is denoted by $\mathbb{T}^L(r,\chi)_w$. Quasi-BPS categorical duality. If $(r,\chi,w)$ satisfies the BPS condition, then there is an equivalence

$$\mathbb{T}^L(r,w+1-g^{\rm{sp}})_{-\chi+1-g^{\rm{sp}}}\stackrel{\sim}{\to}\mathbb{T}^L(r,\chi)_w.$$

This is motivated by relative Fourier–Mukai duality over the locus of smooth spectral curves and proposes an extension of that equivalence to categorical Calabi–Yau compactifications over the Hitchin base. The general equivalence remains conjectural.