Automorphic Forms

In this work we give an introduction to automorphic forms in string theory by examining a well-known case in ten-dimensional Type IIB superstring theory. An automorphic form, constructed as a non-holomorphic Eisenstein series, is known to encode all perturbative and non-perturbative quantum corrections in the genus expansion for the R^4-term included in the asymptotic string expansion for the effective action. Furthermore, we study Type IIA superstring theory compactified on a rigid Calabi-Yau threefold. Here a discrete group, the Picard modular group, is believed to be a preserved symmetry of the quantum theory, and an automorphic form is conjectured to encode the quantum corrections to the metric of the hypermultiplet moduli space, which classically is a coset space. To read off the loop corrections arising from the string coupling, as well as the non- perturbative instanton corrections, we rewrite the Eisenstein series as a Fourier series. The Fourier series is decomposed into a constant, abelian and non-abelian part. The main complication arises when trying to identify the coefficients in the non- abelian part. We try to bring some clarity to this issue.

Autorentext

Oscar Kleväng received his B.Sc. degree in Engineering Physics from the Chalmers School of Science, Gothenburg, where he later took his M.Sc. degree in Fundamental Physics. Besides the interest in physics, he has a passion for music and plays the violoncello. He is currently working in the Gothenburg Symphony Orchestra.