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Fujikawa's form of the non-Abelian anomaly is related to a topological object in a direct manner. Their argument seems general and independent of concrete forms of index, anomaly and other details of models. In this paper we extend the analysis to the V, A coupling gauge theories, the gravita­ tional theory and also to the open superstring theory. In application to the gravitational anomaly, we obtain a family of consistent anomalies from the higher dimensional index, which are all equivalent to Fujikawa's form. Our construction of gravita­ tional anomalies leads naturally to counterterms which relate different forms of anomalies to one another. In the case of Type-I superstring theory in 10 dimensions, we show that the gauge anomaly in this theory is related to a index of a kind of Dirac operator in 12 dimensions. Our consideration might provide hints to geometrical understanding of string anomaly.

Topological Meaning of Gauge and Gravitational Anomalies and Gaussian Factors