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Arithmetic and Hyperbolic Structures in String Theory

Daniel Persson (Institutionen för fundamental fysik)
Göteborg : Chalmers University of Technology, 2009. ISBN: 978-91-7385-301-9.- 325 s.

This thesis consists of an introductory text followed by two separate parts which may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of spacelike singularities (the BKL-limit). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of gravity. We investigate the modification of the billiard dynamics when the original gravitational theory is formulated on a compact spatial manifold of arbitrary topology, revealing fascinating mathematical structures known as galleries. We further use the conjectured hyperbolic symmetry E10 to generate and classify certain cosmological (S-brane) solutions in eleven-dimensional supergravity. Finally, we show in detail that eleven-dimensional supergravity and massive type IIA supergravity are dynamically unified within the framework of a geodesic sigma model for a particle moving on the infinite-dimensional coset space E10/K(E10). Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are typically given by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by moduli-dependent functions which are manifestly invariant under G(Z), known as automorphic forms. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on a special class of functions known as (non-holomorphic) Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also discuss the possibility that certain generalized Eisenstein series, which are covariant under the maximal compact subgroup K(G), could play a role in determining the exact effective action for toroidally compactified higher derivative corrections. Finally, we propose that in the case of rigid Calabi-Yau compactifications in type IIA string theory, the exact universal hypermultiplet moduli space exhibits a quantum duality group given by the Picard modular group SU(2,1;Z[i]). To verify this proposal we construct an SU(2,1;Z[i])-invariant Eisenstein series, and we present preliminary results for its Fourier expansion which reveals the expected contributions from D2-brane and NS5-brane instantons.

Nyckelord: String Theory, Kac-Moody algebras, Spacelike singularities, Instantons

Avhandling avser en dubbel doktorsexamen, enligt ett officiellt avtal mellan Chalmers Tekniska Högskola och Université Libre de Bruxelles tecknat den 8:e september 2008.

Denna post skapades 2009-05-28. Senast ändrad 2013-09-25.
CPL Pubid: 94499


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Institutioner (Chalmers)

Institutionen för fundamental fysik (2005-2015)


Algebra och geometri
Matematisk fysik
Relativitetsteori, gravitation

Chalmers infrastruktur


Datum: 2009-06-12
Tid: 15:00
Lokal: Forum F, Universite Libre de Bruxelles, Bryssel, Belgien
Opponent: Boris Pioline (LPTHE Jussieu, Paris)

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