Automorphic Forms in String Theory

Abstract

In this thesis we give an introduction to automorphic forms in string theory by examining a well-known case in ten-dimensional Type IIB superstring the- ory. An automorphic form, constructed as a non-holomorphic Eisenstein series ESL(2,Z), is known to encode all perturbative and non-perturbative quantum 3/2 4 corrections in the genus expansion for the R -term included in the asymptotic string expansion for the effective action. Explicit calculations are shown, and group- and algebra theoretical aspects are thoroughly explained. Furthermore, we study Type IIA superstring theory compactified on a rigid Calabi-Yau threefold, which is the topic of the recent paper [9]. Here, our focus is more on the explicit calculations and less on physical interpretations. A discrete group SU(2, 1; Z[i]), called the Picard modular group, is believed to be a preserved symmetry of the quantum theory, and an automorphic form, constructed as a non-holomorphic Eisenstein series ESU(2,1;Z[i]), is conjectured 3/2 to encode the quantum corrections to the metric of the hypermultiplet moduli space, which classically is a coset space SU(2, 1)/(SU(2) ␣ U(1)). To read off the loop corrections arising from the string coupling gs = eφ, as well as the non- perturbative instanton corrections, we want to rewrite the Eisenstein series as a Fourier series. The general Fourier series is decomposed into a constant, abelian and non-abelian part, referring to the action of the maximal nilpotent subgroup H3 (Z) ␣ SU(2, 1; Z[i]). The main complication arises when trying to identify the coefficients in the non-abelian part of the Fourier expansion. We try to bring some clarity to this issue.