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The exact renormalization group is applied to the world sheet theorydescribing bosonic open string backgrounds to obtain the equations of motionfor the fields of the open string. Using loop variable techniques the equationscan be constructed to be gauge invariant. Furthermore they are valid off the(free) mass shell. This requires keeping a finite cutoff. Thus we have theinteresting situation of a scale invariant world sheet theory with a finiteworld sheet cutoff. This is possible because there are an infinite number ofoperators whose coefficients can be tuned. This is in the same sense that"perfect actions" or "improved actions" have been proposed in lattice gaugetheory to reproduce the continuum results even while keeping a finite latticespacing.

Gauge Invariant Exact Renormalization Group and Perfect Actions in the Open Bosonic String Theory