Open membranes, ribbons and deformed Schild strings

We analyze open membranes immersed in a magnetic three-form field-strength $C$. While cylindrical membranes in the absence of $C$ behave like tensionless strings, when the $C$ flux is present the strings polarize into thin membrane ribbons, locally orthogonal to the momentum density, thus providing the strings with an effective tension. The effective dynamics of the ribbons can be described by a simple deformation of the Schild action for null strings. Interactions become non-local due to the polarization, and lead to a deformation of the string field theory, whereby string vertices receive a phase factor proportional to the volume swept out by the ribbons. In a particular limit, this reduces to the non-commutative loop space found previously.