| Summary: | The type IIA and type IIB supergravity actions in d = 10 dimensions are the low-energy effective theories of type IIA and IIB string theory. In addition, the unique eleven dimensional supergravity theory is the low-energy effective action of M-theory. Higher order corrections to the low-energy effective actions of these supergravity theories contain perturbative and non-perturbative effects of the corresponding string theories and, as such, understanding the structure of the higher order terms provides an insight into the perturbative and non-perturbative formulations of string theory. The U-duality groups of type IIA/B string theory and M-theory compactified on a torus to d < 10 dimensions puts powerful constraints on the higher derivative terms in the effective actions of these theories. In particular, the higher derivative terms in d = 10 - n dimensions are required to possess coefficient functions that transform as En+1 (Z) automorphic forms. These automorphic forms are complex mathematical objects that encode all the perturbative and non-perturbative features of type II string theory and M-theory compactified on an torus to d dimensions. We investigate the structure of the higher derivative terms and their associated automorphic forms in the effective actions of type IIA/B string theory and M-theory. Constraints on automorphic forms appearing in d dimensions by dimensional reduction of arbitrary higher derivative terms in the type IIA, type IIB and M-theory effective actions to d dimensions are obtained. The behaviour of higher derivative terms in the d dimensional type II effective action in specic limits of various parameters is analysed. We derive a group theoretical interpretation for each limit. A general formula is given for a class of automorphic forms in these limits.
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