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We consider super Yang-Mills Theory in $N=1$ conformal supergravity. Usingthe background field method and the Feddeev-Popov procedure, the quantizedaction of the theory is presented. Its one-loop effective action is studiedusing the heat kernel method. We shall develop a non-iterative scheme,generalizing the non-supersymmetric case, to obtain the super heat kernelcoefficients. In particular, the first three coefficients, which govern theone-loop divergence, will be calculated. We shall also demonstrate how toschematically derive the higher order coefficients. The method presented herecan be readily applied to various quantum theories. We shall, as anapplication, derive the full one-loop divergence of SYM in conformalsupergravity.

Super Heat Kernel and One-Loop Divergence of Super Yang-Mills Theory in Conformal Supergravity