Abstract
Abstract This work is to point out the relationship between the geometric approach to the string theories of Bowick and Rajeev, and Pilch and Warner and the recent research on the index of the Dirac operator in loop space. We consider character-valued Dirac index bundles instead of the vacuum bundle over some parameter space such as Diff S1/S1. Thus, some necessary conditions for the absence of the Virasoro anomaly can be derived from some formulas presented here.
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