Anomalies and elliptic operators

The coefficients of the asymptotic expansion of $\operatorname{Sp}\exp(-tA)$ in the limit $t\to0$ for the operators of quantum field theory are calculated and it is shown how the obtained results can be applied to the calculation of the axial and conformal anomalies, charge renormalization in gauge theories, and to the investigation of two-dimensional electrodynamics.