Computation of the DeWitt-Seeley-Gilkey coefficient E4 for nonminimal operator in curved space

The demands of gauge field theories, quantum theory in external gauge and gravitational backgrounds, and quantum gravity have spurred intense study of the asymptotic heat kernel expansion for nonminimal differential operators on curved manifolds in the presence of a gauge field. There are several computer programs for calculating the coefficients in an expansion of the heat kernel. When applied to the nonminimal operator these programs appeared to be able to compute no more than the second order coefficient. We report the first complete calculation of the fourth coefficient for nonminimal operator. The calculation has been carried out on PC with the help of a program written in C. Some other new results have been also obtained by this program.