Duality symmetry in four-dimensional string actions

We reduce the dual version of D = 10, N = 1 supergravity coupled to n vector fields to four dimensions, and derive the SL (2, R) × O (6,6 + n) transformations which leave the equations of motion invariant. For n = 0 SL (2, R) is also a symmetry of the action, but for n > 0 only those SL(2,R) transformations which act linearly on all fields leave the action invariant. The resulting four-dimensional theory is related to the bosonic part of the usual formulation of N = 4 supergravity coupled to matter by a duality transformation.