Calabi-Yau orbifolds over Hitchin bases

Any irreducible Dynkin diagram $\Delta$ is obtained from an irreducible Dynkin diagram $\Delta_h$ of type $\mathrm{ADE}$ by folding via graph automorphisms. For any simple complex Lie group $G$ with Dynkin diagram $\Delta$ and compact Riemann surface $\Sigma$, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair $(\Sigma, G)$ . These give rise to Calabi-Yau orbifolds over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to the Hitchin system of the same type away from singular fibers.