Abstract
We continue earlier discussions on loop variables and the exact renormalization group on the string world sheet for closed and open string backgrounds. The world sheet action with a UV regulator is written in a generally background covariant way by introducing a background metric. It is shown that the renormalization group gives background covariant equations of motion – this is the gauge invariance of the graviton. Interaction is written in terms of gauge invariant and generally covariant field strength tensors. The basic idea is to work in Riemann normal coordinates and covariantize the final equation. It turns out that the equations for massive modes are gauge invariant only if the space–time curvature of the (arbitrary) background is zero. The exact RG equations give quadratic equations of motion for all the modes including the physical graviton. The level (2,2¯) massive field equations are used to illustrate the techniques. At this level there are mixed symmetry tensors. Gauge invariant interacting equations can be written down. In flat space an action can also be written for the free theory.
Highlights
The Renormalization Group (RG) approach to obtaining equations of motion for the fields of string theory has a long history [[1]-[14]]
In III a world sheet action was written down that gave gauge invariant equations of motion in the form of the exact renormalization group (ERG)
The ERG is reproduced here for convenience: 3As mentioned in Section 3, in quantum field theory there are proofs that the on-shell S-matrix is independent of the choice of background fields [60, 61, 62]
Summary
The Renormalization Group (RG) approach to obtaining equations of motion for the fields of string theory has a long history [[1]-[14]]. For open strings the gauge transformations are those of the free theory and are not modified by interactions, unlike Witten’s BRST string field theory In this sense the theory looks Abelian, until Chan-Paton factors are introduced. The gauge transformations need to be modified to include a ”non-Abelian rotation” This is the usual transformation induced on tensor fields by general coordinate transformations. The coordinate invariance of the theory can be maintained, if one modifies the transformations laws for the massive fields to include additional non tensorial terms. We will show in some detail that in the case of flat background metric the massive mode equations will be both gauge and general covariant. The details of the dimensional reduction, field content and gauge transformation are worked out Since this level involves fields of mixed symmetry, this is interesting quite independent of string theory.
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