Two dimensional sigma models describe (harmonic) maps of Riemann surfaces to Riemannian manifolds. I will present the motivations to study the complexification of this problem. I will present the novel approach, developed in my work with Igor Krichever, allowing to construct essentially all twisted complexified harmonic maps of two-torus to spheres and projective spaces. -- Part of the London Integrability Journal Club. If you are a new participant please register at integrability-london.weebly.com. The link will be emailed.