Automorphic Forms in String Theory
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Typ
Examensarbete för masterexamen
Master Thesis
Master Thesis
Program
Publicerad
2010
Författare
Kleväng, Oscar
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
Grundläggande vetenskaper , Matematik , Elementarpartikelfysik , Basic Sciences , Mathematics , Elementary particle physics