TOPOLOGICAL MEANING OF GAUGE AND GRAVITATIONAL ANOMALIES AND GAUSSIAN FACTORS

We discuss a relation between the 2n-dimensional anomalies and the (2n+2)-dimensional chiral anomalies using the Gaussian factor (heat kernel) regularization. Recently, it was shown that 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 gravitational 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 gravitational 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.