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The heat kernel in curved space-time is computed to fourth order in a strictexpansion in the number of covariant derivatives. The computation is made forarbitrary non abelian gauge and scalar fields and for the Riemann connection inthe coordinate sector. The expressions obtained hold for arbitrary tensorrepresentations of the matter field. Complete results are presented for thediagonal matrix elements and for the trace of the heat kernel operator. Inaddition, Chan's formula is extended to curved space-time. As a byproduct, thebosonic effective action is also obtained to fourth order.

Derivative expansion of the heat kernel in curved space