Symplectic

Let X be a closed Riemann surface of genus at least 2 and let G be a semisimple Lie group. The non-abelian Hodge correspondence (in one of its forms) is a real analytic and non-symplectic diffeomorphism between the smooth moduli space of representations from pi_1(X) to G and the complex analytic moduli space of G-Higgs bundles on X. By abstract principles, the non-abelian Hodge correspondence extends to a holomorphic map between complexifications of these moduli spaces. Using the recently introduced complex harmonic maps, we'll give an explicit realization of this holomorphic extension. Time permitting, we'll discuss an application concerning the Atiyah-Bott-Goldman symplectic form. This is all joint with Christian El Emam, some in print and some in progress.