Automorphic forms and string theory: Small automorphic representations and non-perturbative effects
Doktorsavhandling, 2017

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.

instantons

automorphic forms

automorphic representations

Eisenstein series

U-duality

non-perturbative effects

string theory

PJ-salen, Fysik Origo, Fysikgården 2
Opponent: Professor Solomon Friedberg, Department of Mathematics, Boston College, USA

Författare

Henrik Gustafsson

Chalmers, Fysik, Teoretisk fysik

Small automorphic representations and degenerate Whittaker vectors

Journal of Number Theory,; Vol. 166(2016)p. 344-399

Artikel i vetenskaplig tidskrift

Philipp Fleig, HG, Axel Kleinschmidt, Daniel Persson. Eisenstein series and automorphic representations

Olof Ahlén, HG, Axel Kleinschmidt, Baiying Liu, Daniel Persson. Fourier coefficients attached to small automorphic representations of SLn(A)

Quantum mechanics is used to understand particle physics and Einstein's theory of general relativity describes gravity, but it has long been a challenge in theoretical physics to unify them into a quantum theory of gravity. Such a theory would, for example, be able to explain the quantum properties of a black hole.

String theory is a quantum theory of gravity and its partition function carries information about the quantum states of the theory and their interactions. Because of the many symmetries of string theory, the partition function can be described by functions called automorphic forms. In this thesis, I study such automorphic forms and compute their Fourier coefficients which contain important information about instantons and black holes.

Fundament

Grundläggande vetenskaper

Ämneskategorier

Geometri

Annan fysik

Matematisk analys

ISBN

978-91-7597-609-9

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4290

Utgivare

Chalmers tekniska högskola

PJ-salen, Fysik Origo, Fysikgården 2

Opponent: Professor Solomon Friedberg, Department of Mathematics, Boston College, USA