Table of contents

The fourth annual conference of the European Research and Training Network on the Quantum Structure of Spacetime and the Geometric Nature of Fundamental Interactions was held in Copenhagen on 15-20 September 2003. As in previous conferences, each day started with an extended introductory lecture on a current topic relevant to the research of the Network. We would like to thank V Balasubramanian, G Dall'Agata, D Lüst, S Minwalla and R Russo for their effort in providing extensive lecture notes, which are included in these proceedings. In this way their lectures, which were very much appreciated by the participants of the conference, will become available to a wider audience. We thank also D Zanon for her lecture 'Non-perturbative results from perturbative field theory'. The shorter contributions are based on talks which represented research carried out within the Network. We also invited a number of outside speakers whose work was relevant to the topics covered by the conference. Their contributions are also included.

This workshop took place in the context of the European Community Research and Training Networks' activities. Financial support came from the Niels Bohr Fund, NORDITA and the RTN. Much of the practical work was done by the conference secretaries H Christensen, U Holm, H Kiilerich, A Lumholdt and A Rey, to whom we are grateful. We also acknowledge important assistance from computer managers B Nilsson and P Urkedal. Last, but not least, we would like to thank Andrew Wray from Institute of Physics Publishing for his editorial help.

The aim of this report is to give an overview of the duality between type IIB string theory on the maximally supersymmetric PP-wave and the BMN sector of the super-Yang–Mills theory. The general features of the string and the field theory descriptions are reviewed, but the main focus of this report is on the comparison between the two sides of the duality. In particular, it is first explained how free IIB strings emerge on the gauge theory side and then the generalizations of this relation to the full interacting theory are considered. A 'historical' approach is taken and various proposals presented in the literature are described.

We review the Penrose limit of the type IIB dual of softly broken N = 1 SYM in four dimensions obtained as a deformation of the Maldacena–Nùñez background. We extract the string spectrum on the resulting pp-wave background and discuss some properties of the conjectured dual gauge theory hadrons, the so-called 'annulons'.

We review recent progress made on lightcone string field theory in the plane-wave background.

We review recent progress in establishing AdS/CFT duality in non-supersymmetric sectors of states with large spins. Solitonic solutions of the classical string action on the AdS₅ × S⁵ background that carry spins in the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. In the limit of large spins J, the first subleading 1/J coefficient in the expansion of the string energy is expected to be non-renormalized to all orders in the inverse string tension expansion and thus can be directly compared to the one-loop anomalous dimensions of the corresponding composite operators in super YM theory. The latter may be found using the Bethe ansatz for special cases of the associated SO(6) spin chain.

We show that solitonic solutions of the classical string action on the AdS₅ × S⁵ background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. In the limit of large spins J, the first subleading 1/J coefficient in the expansion of the string energy is expected to be non-renormalized to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous dimensions of the corresponding composite operators in super YM theory. We identify a particular string solution whose classical energy exactly reproduces the 1-loop anomalous dimension of a certain set of SYM operators with two independent R charges J₁, J₂.

A brief review is given of the recent solution of a non-compact CFT describing a NS-supported pp-wave background. We will first explain how to compute the three- and four-point correlators using the current algebra techniques, thereby showing that some generic features of the non-compact WZW models become very clear in this simple context. We will then present the Penrose limit as a contraction of an U(1) × SU(2)_k WZW model, an approach that could prove useful in order to understand holography for pp-wave spacetimes. We will finally comment on the string amplitudes and on the existence of two flat space limits.

This paper reviews recent developments in the study of universes with a positive cosmological constant in string theory.

Correlation functions of discrete primary fields in the c = 1 boundary conformal field theory of a scalar field in a critical periodic boundary potential are computed using the underlying SU(2) symmetry of the model. Bulk amplitudes are unambigously determined and we give a prescription for amplitudes involving discrete boundary fields.

We discuss some consequences of the fact that symmetry groups appearing in compactified (super-)gravity may be non-simply connected. The possibility to add fermions to a theory results in a simple criterion to decide whether a three-dimensional coset sigma model can be interpreted as a dimensional reduction of a higher-dimensional theory. Similar criteria exist for higher-dimensional sigma models, though less decisive. A careful examination of the topology of symmetry groups rules out certain proposals for M-theory symmetries, which are not ruled out at the level of the algebras. We conclude with an observation on the relation between the 'generalized holonomy' proposal, and the actual symmetry groups resulting from E₁₀ and E₁₁ conjectures.

After briefly reviewing the methods that allow us to derive consistently new Lie (super)algebras from given ones, we consider enlarged superspaces and superalgebras, their relevance and some possible applications.

In this work we consider the full interacting effective actions for fundamental strings and D-branes in arbitrary bosonic type II supergravity backgrounds. The explicit form of these actions is given in terms of component fields, up to second order in the fermions. The results take a compact form exhibiting κ-symmetry, as well as supersymmetry in a background with Killing spinors. Also we give the explicit transformation rules for these symmetries in all cases.

In these proceedings, we review the main results concerning superspace geometries with nonanticommutative spinorial variables and field theories formulated on them. In particular, we report on the quantum properties of the WZ model formulated in the N = 1/2 nonanticommutative superspace.

In this paper we describe the general geometrical framework of brane world constructions in orientifolds of type IIA string theory with D6-branes wrapping 3-cycles in a Calabi–Yau 3-fold. These branes generically intersect at points on the internal space, and the patterns of intersections govern the chiral fermion spectra. We discuss how the open string spectra in intersecting brane models are constructed, how the standard model can be embedded, and also how supersymmetry can be realized in this class of string vacua. After the general considerations we restrict the discussion to the case of orbifold backgrounds with intersecting D6-branes and to the quintic Calabi–Yau manifold. Then, we discuss parts of the effective action of intersecting brane world models. Specifically, from the Born–Infeld action of the wrapped D-branes we compute the tree-level, D-term scalar potential, which is important for the stability of the considered backgrounds as well as for questions related to supersymmetry breaking. Second, we review the recent computation concerning gauge coupling unification and also concerning one-loop gauge threshold corrections in intersecting brane world models. Finally, we also discuss some aspects of proton decay in intersecting brane world models.

We review the supergravity derivation of some non-perturbatively generated effective superpotentials for gauge theories. Specifically, we derive the Veneziano–Yankielowicz superpotential for pure super Yang–Mills theory from the warped deformed conifold solution, and the Affleck–Dine–Seiberg superpotential for SQCD from a solution describing fractional D3-branes on a orbifold.

We discuss the role played by the divergences appearing in the interaction between a fractional D3 brane dressed with an SU(N) gauge field and a stack of N fractional D3 branes on the orbifolds C²/Z₂ and C³/(Z₂ × Z₂). In particular we show that the logarithmic divergences in the closed string channel, interpreted as due to twisted massless tadpoles, are mapped, under open/closed string duality, in the logarithmic ones in the open string channel, due to the massless states circulating in the annulus diagram and corresponding to the one-loop divergences that one finds in the gauge theory living in the worldvolume of the brane. This result provides quantitative evidence of why the chiral and scale anomalies of the supersymmetric and non-conformal gauge theories supported by the worldvolume of the branes can be inferred from supergravity calculations.

In this paper we review the Dijfgraaf–Vafa conjecture for U(N_c) super Yang–Mills theories with matter in the adjoint and fundamental representation. Secondly, we discuss a proposal to derive the effective superpotential in terms of the mesonic superfields and show how the Affleck–Dine–Seiberg non-perturbative superpotential arises directly from the matrix model. Finally, we review how the random matrix model can be used to obtain factorization data of Seiberg–Witten curves with matter.

We discuss some of the recent developments of super Yang–Mills theories in the context of the gauge–string correspondence.

We briefly review some modern developments in higher spin field theory and their links with superstring theory. The analysis is based on various BRST constructions allowing us to derive the Lagrangians for massive and massless higher spin fields on flat or constant curvature backgrounds of arbitrary dimensions.

We review in a pedagogical way the computation of gravitational corrections to the effective superpotential of w supersymmetric gauge theories coupled to supergravity, by using the method of the generalized Konishi anomaly. We focus on the first non-trivial gravitational corrections to U(N) gauge theories with a single adjoint chiral multiplet, and briefly comment on some generalizations.

We begin with a brief introduction on gauge theories, focusing on the importance of the effective superpotential in light of the new techniques to compute it systematically. We then proceed to consider theories for which the Konishi anomaly proves to be enough to solve exactly for the effective superpotential. As an example, we study a chiral SO(10) gauge theory, where we also discuss the occurrence of dynamical supersymmetry breaking.

We review the main features of string compactifications in the presence of non-trivial fluxes for various form fields in ten dimensions. After an elementary introduction to the topic, we focus the discussion on the geometrical description of the vacua and on the potential for the effective four-dimensional theory. We will especially review and emphasize the use of group structures to describe the ten-dimensional vacua and that of gauged supergravities for the effective theories.

We construct maximal D = 8 gauged supergravities by the reduction of D = 11 supergravity over three-dimensional group manifolds. Such manifolds are classified into two classes, A and B, and eleven types. This Bianchi classification carries over to the gauged supergravities. The class A theories have 1/2 BPS domain wall solutions that uplift to purely gravitational solutions consisting of 7D Minkowski and a 4D Euclidean geometry. These geometries are generically singular. The two regular exceptions correspond to the near-horizon limit of the single- or double-centre Kaluza–Klein monopole. In contrast, the class B supergravities are defined by a set of equations of motion that cannot be integrated to an action and allow for no 1/2 BPS domain walls.

We review the latest progress in understanding the phase structure of static and neutral Kaluza–Klein black holes, i.e. static and neutral solutions of pure gravity with an event horizon and with asymptotics being d-dimensional Minkowski space.

Current–current deformations for WZW models of semi-simple compact groups are discussed in a sigma model approach. We start with the Abelian rank one group U(1). Afterwards, we keep the rank one but allow for non-Abelian structures by considering SU(2). Finally, we present the general case of rank larger than one.

We report on the construction of four-dimensional gauged supergravity models that can be interpreted as type IIB orientifold compactification in the presence of 3-form fluxes and D3-branes. We mainly address our attention to the symplectic embedding of the U-duality group of the theory and the consequent choice of the gauge group, whose four-dimensional Killing vectors are the remnants of the ten-dimensional fluxes. We briefly discuss the structure of the scalar potential arising from the gauging and the properties of the Killing vectors in order to preserve some amount of supersymmetry.

In flux compactifications of M-theory a superpotential is generated whose explicit form depends on the structure group of the seven-dimensional internal manifold. In this paper, we discuss superpotentials for the structure groups: G₂, SU(3) or SU(2). For the G₂ case all internal fluxes have to vanish. For SU(3) structures, the non-zero flux components entering the superpotential describe an effective one-dimensional model and a Chern–Simons model if there are SU(2) structures.

These notes are an expanded version of a review paper on closed string tachyon condensation at the RTN workshop in Copenhagen in September 2003. We begin with a lightning review of open string tachyon condensation, and then proceed to review recent results on localized closed string tachyon condensation, focusing on two simple systems, C/Z_n orbifolds and twisted circle compactifications.

We describe holographic open/closed string dualities in two dimensions. In particular, we summarize recent descriptions of two-dimensional type 0 strings in terms of one-dimensional matrix models. The relevant matrix models arise by considering large-N systems of unstable D0-branes in the theories. They provide a concrete framework to analyse the non-perturbative structure of two-dimensional type 0 theory.