Strings, Branes and Gravity

The following sections are included:

• FOREWORD

• CONTENTS

In the light of the duality between physics in the bulk of anti-de Sitter space and a conformal field theory on the boundary, we review the M2, D3 and M5 branes and how their near-horizon geometry yields the compactification of D = 11 supergravity on S⁷, Type IIB supergravity on S⁵ and D = 11 supergravity on S⁴, respectively. We discuss the "Membrane at the End of the Universe" idea and its relation to the corresponding superconformal singleton theories that live on the boundary of the AdS₄, AdS₅ and AdS₇ vacua. The massless sectors of these compactifications are described by the maximally supersymmetric D = 4, D = 5 and D = 7 gauged supergravities. We construct the non-linear Kaluza-Klein ansätze describing the embeddings of the U(1)⁴, U(1)³ and U(1)² truncations of these supergravities, which admit 4-charge AdS₄, 3-charge AdS₅ and 2-charge AdS₇ black hole solutions. These enable us to embed the black hole solutions back in ten and eleven dimensions and reinterpret them as M2, D3 and M5 branes spinning in the transverse dimensions with the black hole charges given by the angular momenta of the branes. A comprehensive Appendix lists the field equations, symmetries and transformation rules of D = 11 supergravity, Type IIB supergravity, and the M2, D3 and M5 branes.

Following is a collection of lecture notes on D–branes, which may be used by the reader as preparation for applications to modern research applications such as: the AdS/CFT and other gauge theory/geometry correspondences, Matrix Theory and stringy non–commutative geometry, etc. In attempting to be reasonably self–contained, the notes start from classical point–particles and develop the subject logically (but selectively) through classical strings, quantisation, D–branes, supergravity, superstrings, string duality, including many detailed applications. Selected focus topics feature D–branes as probes of both spacetime and gauge geometry, highlighting the role of world–volume curvature and gauge couplings, with some non–Abelian cases. Other advanced topics which are discussed are the (presently) novel tools of research such as fractional branes, the enhançon mechanism, D(ielectric)–branes and the emergence of the fuzzy/non–commutative sphere. (This is an expanded writeup of lectures given at ICTP, TASI, and BUSSTEPP.).

This is a write-up of introductory lectures on black holes in string theory given at TASI-99. Topics discussed include: black holes, thermodynamics and the Bekenstein-Hawking entropy, the information problem; supergravity actions, conserved quantum numbers, supersymmetry and BPS states, units, duality and dimensional reduction, solution-generating; extremal M-branes and D-branes, smearing, probe actions, nonextremal branes, the Gregory-Laflamme instability; breakdown of supergravity and the Correspondence Principle, limits in parameter space, singularities; making black holes with branes, intersection-ology, the harmonic function rule, explicit d = 5, 4 examples; string computations of extremal black hole entropy in d = 5, 4, rotation, fractionation; non-extremality and entropy, the link to BTZ black holes, Hawking radiation and absorption cross-sections in the string/brane and supergravity pictures.

These notes provide a brief introduction to modern cosmology, focusing primarily on theoretical issues. Some attention is paid to aspects of potential interest to students of string theory, on both sides of the two-way street of cosmological constraints on string theory and stringy contributions to cosmology. Slightly updated version of lectures at the 1999 Theoretical Advanced Study Institute at the University of Colorado, Boulder.

This is a summary of key issues in Matrix Theory and its compactifications. It is emphasized that Matrix Theory is a valid Discrete Light Cone Quantization of M Theory with at least 6 noncompact asymptotically flat dimensions and 16 or 32 Supersymmetry charges. The background dependence of the quantum mechanics of M Theory, and the necessity of working in light cone frame in asymptotically flat spacetimes are explained in terms of the asymptotic density of states of the theory, which follows from the Bekenstein-Hawking entropy formula. In four noncompact dimensions one is led to expect a Hagedorn spectrum in light cone energy. This suggests the possible relevance of "little string theories" (LSTs) to the quantum description of four dimensional compactifications, because one can argue that their exact high energy spectrum has the Hagedorn form. Some space is therefore devoted to a discussion of the properties of LSTs, which were first discovered as the proper formulation of Matrix Theory on the five torus.

These lectures discuss some of the general issues in developing a phenomenology for Superstring Theory/M Theory. The focus is on the question: how might one obtain robust, generic predictions. For example, does the theory predict low energy supersymmetry breaking? In the course of these explorations, basics of supersymmetry and supersymmetry breaking, string moduli, cosmological issues, and other questions are addressed. The notion of approximate moduli and their possible role plays a central role in the discussion.

This is an introductory review of the AdS/CFT correspondence and of the ideas that led to its formulation. We show how comparison of stacks of D3-branes with corresponding supergravity solutions leads to dualities between conformal large N gauge theories in 4 dimensions and string backgrounds of the form AdS₅ × X₅ where X₅ is an Einstein manifold. The gauge invariant chiral operators of the field theory are in one-to-one correspondence with the supergravity modes, and their correlation functions at strong 't Hooft coupling are determined by the dependence of the supergravity action on AdS boundary conditions. The simplest case is when X₅ is a 5-sphere and the dual gauge theory is the supersymmetric SU(N) Yang-Mills theory. We also discuss D3-branes on the conifold corresponding to X₅ being a coset space T^1,1 = (SU(2) × SU(2))/U(1). This background is dual to a certain superconformal field theory with gauge group SU(N) × SU(N).

We describe the moduli spaces of theories with 32 or 16 supercharges, from several points of view. Included is a review of backgrounds with D-branes (including type I' vacua and F-theory), a discussion of holonomy of Riemannian metrics, and an introduction to the relevant portions of algebraic geometry. The case of K3 surfaces is treated in some detail.

We review the geometry of the moduli space of N = 2 theories in four dimensions from the point of view of superstring compactification. The cases of a type IIA or type IIB string compactified on a Calabi–Yau threefold and the heterotic string compactified on K3 × T² are each considered in detail. We pay specific attention to the differences between N = 2 theories and N > 2 theories. The moduli spaces of vector multiplets and the moduli spaces of hypermultiplets are reviewed. In the case of hypermultiplets this review is limited by the poor state of our current understanding. Some peculiarities such as "mixed instantons" and the non-existence of a universal hypermultiplet are discussed.

In this set of lectures various properties of D-branes are discussed. After reviewing the basics, we discuss unstable D-brane/anti-D-brane systems, a subject pioneered by Sen. Following him, we discuss the construction of the non-BPS D0-brane in type I theory. This state is stable since it carries a conserved Z₂ charge. The general classification of D-brane charges using K-theory is discussed. The results for the type I theory, and the T-dual type I' theory, are emphasized. Compactification of type I on a circle or torus gives a theory with 16 supersymmetries in 9d or 8d. In each case the moduli space has three branches. The spectrum of non-BPS D-branes are different for each of these branches. We conclude by pointing out some problems with the type I D7-brane and D8-brane predicted by K-theory.

In these lectures, two different aspects of brane world scenarios in 5d gravity or string theory are discussed. In the first two lectures, work on how warped compactifications of 5d gravity theories can change the guise of the gauge hierarchy problem and the cosmological constant problem is reviewed, and a discussion of several issues which remain unclear in this context is provided. In the next two lectures, microscopic constructions in string theory which involve D-branes wrapped on cycles of Calabi-Yau manifolds are described. The focus is on computing the superpotential in the brane worldvolume field theory. Such calculations may be a necessary step towards understanding e.g. supersymmetry breaking and moduli stabilization in stringy realizations of such scenarios, and are of intrinsic interest as probes of the quantum geometry of the Calabi-Yau space.

These TASI lectures review the Holographic principle. The first lecture describes the puzzle of black hole information loss that led to the idea of Black Hole Complementarity and subsequently to the Holographic Principle itself. The second lecture discusses the holographic entropy bound in general space-times. The final two lectures are devoted to the ADS/CFT duality as a special case of the principle. The presentation is self contained and emphasizes the physical principles. Very little technical knowledge of string theory or supergravity is assumed.