Table of contents

M theory on a manifold of G₂-holonomy is a natural framework for obtaining vacua with four large spacetime dimensions and = 1 supersymmetry. In order to obtain, within this framework, the standard features of particle physics, namely non-Abelian gauge groups and chiral fermions, we consider G₂-manifolds with certain kinds of singularities at which these features reside. The aim of this paper is to describe in detail how the above picture emerges. Along the way we will see how interesting aspects of strongly coupled gauge theories, such as confinement, receive relatively simple explanations within the context of M theory. We conclude with a brief discussion of the particle physics features of these vacua of M theory.

We explore aspects of the physics of de Sitter (dS) space that are relevant to holography with a positive cosmological constant. First, we display a non-local map that commutes with the de Sitter isometries, transforms the bulk–boundary propagator and solutions of free wave equations in de Sitter onto the same quantities in Euclidean anti-de Sitter (EAdS) space, and takes the two boundaries of dS to the single EAdS boundary via an antipodal identification. Second, we compute the action of scalar fields on dS as a functional of boundary data. Third, we display a family of solutions to three-dimensional gravity with a positive cosmological constant in which the equal time sections are arbitrary genus Riemann surfaces, and compute the action of these spaces as a functional of boundary data. These studies suggest that if de Sitter space is dual to a Euclidean conformal field theory (CFT), this theory should involve two disjoint, but possibly entangled factors. We argue that these CFTs would be of a novel form, with unusual hermiticity conditions relating left movers and right movers. After exploring these conditions in a toy model, we combine our observations to propose that a holographic dual description of de Sitter space would involve a pure entangled state in a product of two of our unconventional CFTs associated with the de Sitter boundaries. This state can be constructed to preserve the de Sitter symmetries and its decomposition in a basis appropriate to antipodal inertial observers would lead to the thermal properties of a static patch. To conclude, we discuss the one-parameter family of de Sitter-invariant vacua for a massive free scalar field, and their thermodynamic properties. At the free field level, we find no obvious thermodynamic reason to favour one vacuum over the other.

These lectures give a brief overview of recent developments in supersymmeric gauge theories. Topics covered include electric–magnetic duality, the AdS/ CFT correspondence and Seiberg–Witten theory.

An overview of some cosmological aspects of string theory is presented. Recent developments are emphasized, especially the attempts to derive inflation or alternatives to inflation from the dynamics of branes in string theory. Time-dependent backgrounds with potential cosmological implications, such as those provided by negative tension branes and S-branes and the rolling string tachyon are also discussed.

These lectures provide an introduction to the microscopic description of branes in curved backgrounds. After a brief reminder of the flat space theory, the basic principles and techniques of (rational) boundary conformal field theory are presented in the second lecture. The general formalism is then illustrated through a detailed discussion of branes on compact group manifolds. In the final lecture, many more recent developments are reviewed, including some results for non-compact target spaces.

We review the formalism of holographic renormalization. We start by discussing mathematical results on asymptotically anti-de Sitter (AdS) spacetimes. We then outline the general method of holographic renormalization. The method is illustrated by working all details in a simple example: a massive scalar field on anti-de Sitter spacetime. The discussion includes the derivation of the on-shell renormalized action, holographic Ward identities, anomalies and renormalization group (RG) equations, and the computation of renormalized one-, two- and four-point functions. We then discuss the application of the method to holographic RG flows. We also show that the results of the near-boundary analysis of asymptotically AdS spacetimes can be analytically continued to apply to asymptotically de Sitter spacetimes. In particular, it is shown that the Brown–York stress energy tensor of de Sitter spacetime is equal, up to a dimension-dependent sign, to the Brown–York stress energy tensor of an associated AdS spacetime.

This issue of Classical and Quantum Gravity contains the proceedings of the RTN European Winter School, which was held at Utrecht University during 17-22 January 2002. The school was part of the RTN programme `The quantum structure of spacetime and the geometric nature of fundamental interactions' of the European Union. The school was attended by 119 participants.

The lectures reflect the latest developments in string theory and aim at providing a pedagogical introduction as well as at presenting an in-depth treatment of some more specific issues. In addition to the lectures, a number of discussion sessions were organized which offered ample opportunity for the students to ask questions and for the lecturers to elaborate on the material discussed during the lectures of that day. The topics covered are M-theory, G_2-manifolds and four-dimensional physic by Bobby Acharya, de Sitter spaces by Jan de Boer, Nonperturbative gauge theories by Nick Dorey, D-branes in curved spaces by Volker Schomerus, Brane cosmology in string theory by Fernando Quevedo and Aspects of AdS-CFT by Kostas Skenderis. We should thank the lecturers for their beautiful and stimulating lectures and for their willingness to write these proceedings. Their enthusiasm and that of the students pervaded the school. We are confident that these proceedings will be appreciated by a wider readership.

The organization was made possible by the financial support of the RTN network, the Institute for Theoretical Physics, the Spinoza Institute and the Faculty of Physics and Astronomy of Utrecht University, and of the FOM programme `String Theory and Quantum Gravity'. Our gratitude goes out to the local organizers, Boris Körs, Francisco Morales, Niels Obers, Henning Samtleben, Geertje Speelman, Marika Taylor and Mario Trigiante, to Leonie Silkens of the Symposion office, and to the managing director of the ITP/Spinoza Institute, Biene Meijerman.

B de Wit and S Vandoren

Utrecht University