Automorphic forms and string theory: Small automorphic representations and non-perturbative effects

Abstract: This compilation thesis stems from a project with the purpose of determining non-perturbative contributions to scattering amplitudes in string theory carrying important information about instantons, black hole quantum states and M-theory. The scattering amplitudes are functions on the moduli space invariant under the discrete U-duality group and this invariance is one of the defining properties of an automorphic form. In particular, the leading terms of the low-energy expansion of four-graviton scattering amplitudes in toroidal compactifications of type IIB string theory are captured by automorphic forms attached to small automorphic representations and their Fourier coefficients describe both perturbative and non-perturbative contributions.In this thesis, Fourier coefficients of automorphic forms attached to small automorphic representations of higher-rank groups are computed with respect to different unipotent subgroups allowing for the study of different types of non-perturbative effects. The analysis makes extensive use of the vanishing properties obtained from supersymmetry described by the global wave-front set of the automorphic representation. Specifically, expressions for Fourier coefficients of automorphic forms attached to a minimal or next-to-minimal automorphic representation of SLn, with respect to the unipotent radicals of maximal parabolic subgroups, are presented in terms of degenerate Whittaker coefficients. Additionally, it is shown how such an automorphic form is completely determined by these Whittaker coefficients.The thesis also includes some partial results for automorphic forms attached to small automorphic representations of E6, E7 and E8.