Special K\"ahler geometry of the Hitchin system and topological recursion

We investigate the special K"ahler geometry of the base of the Hitchin integrable system in terms of spectral curves and topological recursion. The Taylor expansion of the special K"ahler metric about any point in the base may be computed by integrating the $g=0$ Eynard-Orantin invariants of the corresponding spectral curve over cycles. In particular, we show that the Donagi-Markman cubic is computed by the invariant $W_3^{(0)}$. We use topological recursion to go one step beyond this and compute the symmetric quartic of second derivatives of the period matrix.