  Let X be a compact connected Riemann surface of genus g > 0 equipped with a
nonzero holomorphic 1-form. Let M denote the moduli space of semistable Higgs
bundles on X of rank r and degree r(g-1)+1; it is a complex symplectic
manifold. Using the translation structure on the open subset of X where the
1-form does not vanish, we construct a natural deformation quantization of a
certain nonempty Zariski open subset of M.
