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This paper considers the higher derivative terms in the effective action oftype II string theory and in particular the behaviour of the automorphic formsthey contain in all the different possible limits of the string parameters. Theautomorphic forms are thought to obey Poisson equations which contain theLaplacian defined on the coset space to which the scalars fields belong and wecompute this Laplacian in all the possible string theory limits. We alsoconsider these Poisson equations in the decompactification limit of a singledimension and by making two assumptions, one on the generic form of thisequation and the other on the behaviour of the automorphic forms in this limit,we find strong constraints on the allowed form of this differential equation.We show that these constraints allow one to recover much of what was previouslyknown about the automorphic forms corresponding to terms in the effectiveaction that have fourteen or fewer space-time derivatives in a simple way.

Poisson equations, higher derivative automorphic forms and string parameter limits