In this paper we prove the existence of a pseudo-Kähler structure on the deformation space \({\mathcal {B}}_0(T^2)\) of properly convex \({\mathbb {R}}{\mathbb {P}}^2\) -structures over the torus. In particular, the pseudo-Riemannian metric and the symplectic form are compatible with the complex structure inherited from the identification of \({\mathcal {B}}_0(T^2)\) with the complement of the zero section of the total space of the bundle of cubic holomorphic differentials over the Teichmüller space. We show that the \(S^1\) -action on \({\mathcal {B}}_0(T^2)\) , given by rotation of the fibers, is Hamiltonian and it preserves both the metric and the symplectic form. Finally, we prove the existence of a moment map for the \({{\,\mathrm{\text {SL}}\,}}(2,{\mathbb {R}})\) -action over \({\mathcal {B}}_0(T^2)\) .
