Conformal symmetry, unitarity, and the equations of motion for strings in background fields

We impose the constraint of the unitarity of the S -matrix on the theory of a closed bosonic string propagating in a background of some of its modes. The equations of motion for not only the massless but also for some of the massive modes of the string are thus consistently obtained in a weak field expansion. To a given order in the weak field expansion, they contain contributions to all orders in α′. We also show that conformal invariance guarantees that the S -matrix be unitary.