a $\sigma$-model field obeying Neumann boundary conditions on the world sheet, whose zero mode is the target time.

This is the basic vertex deformation which we assume to describe the motion of a $D$-particle in a curved geometry to leading order at least, where spacetime back reaction and curvature effects are assumed weak. Such vertex deformations may be viewed as a generalization of the flat-target-space case [@dparticle].

Perhaps a formally more desirable approach towards the construction of the complete vertex operator would be to start from a T-dual (Neumann) picture, where the deformation (\[path\]) should correspond to a proper Wilson loop operator of an appropriate gauge vector field. Such loop operators are by construction independent of the background geometry. One can then pass onto the Dirichlet picture by a T-duality transformation viewed as a canonical transformation from a $\sigma$-model viewpoint [@otto]. In principle, such a procedure would yield a complete form of the vertex operator in the Dirichlet picture, describing the path of a $D$-particle in a curved geometry. Unfortunately, such a procedure is not free from ambiguities at a quantum level [@otto], which are still unresolved for general curved backgrounds. Therefore, for our purposes here, we shall consider the 