Loop Variables and Gauge Invariant Exact Renormalization Group Equations for (Open) String Theory

The sigma model renormalization group formalism is manifestly background independent and is a possible way of obtaining a background independent string field theory. An exact renormalization group equation is written down for the world sheet theory describing the bosonic open string in general backgrounds and loop variable techniques are used to make the equation gauge invariant. The equations are quadratic in fields as in open string field theory. Some explicit examples are given and results are also given for curved space time. In contrast to BRST string field theory, the gauge transformations are not modified by the interactions. As in the Dirac-Born-Infeld action for massless fields, the interactions for massive fields can also be written in terms of gauge invariant field strengths.