A Torelli Theorem for the Moduli Space of Higgs Bundles on a Curve

Let X be a smooth projective curve over C, and let MXⁿ,^ξ be the moduli space of stable Higgs bundles on X (with genus g > 1), of rank n and fixed determinant ξ, with n and deg(ξ) coprime. Let X′ and ξ′ be another such curve and line bundle. We prove that if MXⁿ,^ξ and MX′ⁿ,^ξ′ are isomorphic as algebraic varieties, then X and X′ are isomorphic.