Two two-dimensional supergravity theories from Calabi–Yau four-folds

We consider two-dimensional supergravity theories with four supercharges constructed from compactification of Type II string theory on a generic Calabi–Yau four-fold. In type IIA and type IIB cases, respectively, new superspace formulations of N=(2,2) and N=(0,4) dilaton supergravities are found and their coupling to matter multiplets is discussed.