  We study the regularity problem of the nonlinear sigma model with gravitino
fields in higher dimensions. After setting up the geometric model, we derive
the Euler--Lagrange equations and consider the regularity of weak solutions
defined in suitable Sobolev spaces. We show that any weak solution is actually
smooth under some smallness assumption for certain Morrey norms. By assuming
some higher integrability of the vector spinor, we can show a partial
regularity result for stationary solutions, provided the gravitino is critical,
which means that the corresponding supercurrent vanishes. Moreover, in
dimension less than 6, partial regularity holds for stationary solutions with
respect to general gravitino fields.
