Moduli spaces of $\Lambda$-modules on abelian varieties

We study the moduli space $\mathbf{M}_X(\Lambda, n)$ of semistable $\Lambda$-modules of vanishing Chern classes over an abelian variety $X$, where $\Lambda$ belongs to a certain subclass of $D$-algebras. In particular, for $\Lambda = \mathcal{D}_X$ (resp. $\Lambda = \mathrm{Sym}^\bullet \mathcal{T}X$) we obtain a description of the moduli spaces of flat connections (resp. Higgs bundles). We give a description of $\mathbf{M}_X(\Lambda, n)$ in terms of a symmetric product of a certain fibre bundle over the dual abelian variety $\hat{X}$. We also give a moduli interpretation to the associated Hilbert scheme as the classifying space of $\Lambda$-modules with extra structure. Finally, we study the non-abelian Hodge theory associated to these new moduli spaces.