Matrix string models for exact (2,2) string theories in RR backgrounds

We formulate matrix string models on a class of exact string backgrounds with non-constant RR-flux parameterized by a holomorphic prepotential function and with manifest (2,2) supersymmetry. This lifts these string theories to M-theory exposing the non-perturbative string interaction which is studied by generalizing the instanton asymptotic expansion, well established in the flat background case, to this more general case. We obtain also a companion matrix model with four manifest supersymmetries in eleven dimensions.