Variations of Hodge structures of a Teichmueller curve

Teichmueller curves are geodesic discs in Teichmueller space that project to an algebraic curve in the moduli space $M_g$. We show that for all $g\ge 2$ Teichmueller curves map to the locus of real multiplication in the moduli space of abelian varieties. Remark that McMullen has shown that precisely for $g=2$ the locus of real multiplication is stable under the ${\text{SL}}_2(\text{RR})$-action on the tautological bundle $\mathrm{\Omega }{\text{M}}_g$. We also show that Teichmueller curves are defined over number fields and we provide a completely algebraic description of Teichmueller curves in terms of Higgs bundles. As a consequence we show that the absolute Galois group acts on the set of Teichmueller curves.