Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A *Higgs bundle* on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-modules $\phi: E\to E\otimes \Omega^1_X$ (the Higgs field) such that $$\phi\wedge \phi=0:  E\to E\otimes \Omega^2_X.$$

If we replace the holomorphic vector bundle $E$ by a coherent sheaf $\mathcal{E}$, then we get a *Higgs sheaf* on $X$. See for example [On Gieseker stability for Higgs sheaves][1].

Since the bounded derived category of coherent sheaves has been intensively studied in algebraic geometry, I wonder whether the bounded derived category of Higgs sheaves (or related concept) has ever been defined or studied in literatures.


  [1]: https://arxiv.org/abs/1603.03100