Axial anomalies in external tensor fields

Problem of calculating the axial anomaly for Dirac fermions in external tensor fields is studied. The sequence of supersymmetric one-dimensional models is presented. Their supercharges are equal, after quantization, to Dirac operators in external tensor fields and the density of Witten's partition function gives the anomaly. It is shown that the action in the corresponding path integral differs from the classical one. Gaussian approximation gives the anomaly only in the case of the third-rank tensor with zero exterior derivative and in that case the anomaly is calculated in all dimensions. The interpretation of this field as the torsion of gravitational field and also connection with the results of Witten and Alvarez-Gaume and Attyah–Singer index theorem are discussed.