Strings 2000

The following sections are included:

• PREFACE

• CONTENTS

We review results on several interesting phenomena in warped compactifications of M theory, as presented at Strings 2000. The behavior of gauge fields in dimensional reduction from d+1 to d dimensions in various backgrounds is explained from the point of view of the holographic duals (and a point raised in the question session at the conference is addressed). We summarize the role of additional fields (in particular scalar fields) in 5d warped geometries in making it possible for Poincare-invariant domain wall solutions to exist to a nontrivial order in a controlled approximation scheme without fine-tuning of parameters in the 5d action (and comment on the status of the singularities arising in the general relativistic description of these solutions). Finally, we discuss briefly the emergence of excitations of wrapped branes in warped geometries whose effective thickness, as measured along the Poincare slices in the geometry, grows as the energy increases.

The interpretation of closed fundamental strings as solitons in open string field theory is reviewed. Noncommutativity is introduced to facilitate an explicit construction. The tension is computed exactly and the correct spectrum is recovered at long wave length.

We derive the vertex operators that are expected to govern the emission of the massless d = 11 supermultiplet from the supermembrane in the light cone gauge. Our results immediately imply the linear coupling of matrix theory to an arbitrary supergravity background to all orders in anticommuting coordinates. Finally we address the definition of n-point tree level and one-loop scattering amplitudes. The resulting 3-point tree level amplitudes turn out to agree with d = 11 supergravity and are completely fixed by supersymmetry and the existence of a normalizable ground state.

We show how the Randall-Sundrum geometry, which has been proposed as a scenario for the universe realized as a 3-brane embedded in a 5-dimensional spacetime, arises naturally as an S⁵ dimensional reduction of a supersymmetric 3-brane of type IIB supergravity. In this construction, the S⁵ scale-setting "breathing mode" plays a crucial rôle, permitting one to evade certain "no-go" claims for the possibility of such a supersymmetric realization.

Strings in the three-dimensional anti-de Sitter space in the NS background is described by the SL(2, R) WZW model. We present a complete description of the Hilbert space of the model. We then compute the one-loop partition function for Euclidean BTZ black hole backgrounds or equivalently thermal AdS₃ backgrounds which are quotients of Euclidean AdS₃. We see long strings and discrete states in agreement with the structure of the Hilbert space.

We discuss some recent work on brane world construction and supersymmetry issues. We introduce a concept of bulk & brane supersymmetry in the framework of 5-dimensional supergravity which allows to formulate the rules for supersymmetry in singular spaces. The original fine-tuning of the Randall-Sundrum world brane which provides the flatness of the brane is shown to follow from bulk & brane supersymmetry on orbifold. Examples of RS-type domain walls are presented.

K-theory provides a framework for classifying Ramond-Ramond (RR) charges and fields.K-theory of manifolds has a natural extension to K-theory of noncommutative algebras, such as the algebras considered in noncommutative Yang-Mills theory or in open string field theory. In a number of concrete problems, the K-theory analysis proceeds most naturally if one starts out with an infinite set of D-branes, reduced by tachyon condensation to a finite set. This suggests that string field theory should be reconsidered for N = ∞.

We study the analog of the Klebanov-Strassler system. We first review the resolution of singularities by the enhançon mechanism, and the physics of fractional branes on an orbifold. We then describe the exact solution. This exhibits a duality cascade as in the case, but the singularity resolution is the characteristic enhançon. We discuss some related solutions and open issues.

We examine the Randall-Sundrum brane-world and demonstrate how it may be realized in terms of a doubled D3-brane configuration of type IIB theory. We further investigate the supersymmetry of the brane-world and show that, despite claims to the contrary, it is indeed compatible with supersymmetry. This result depends on having a higher dimensional origin for the model, which the type IIB realization provides.

The possibility of getting a zero (or small) cosmological constant within the brane world scenario is discussed in some detail. We find that the best one can do is to argue that within a certain class of compactifications (on squashed spheres) there may be a way of obtaining flat (or nearly flat) effective four dimensional theories by adjusting integration constants/parameters.

We consider asymptotically anti-de Sitter spacetimes in general dimensions. We review the origin of infrared divergences in the on-shell gravitational action, and the construction of the renormalized on-shell action by the addition of boundary counterterms. In odd dimensions, the renormalized on-shell action is not invariant under bulk diffeomorphisms that yield conformal transformations in the boundary (holographic Weyl anomaly). We obtain formulae for the gravitational stress energy tensor, defined as the metric variation of the renormalized on-shell action, in terms of coefficients in the asymptotic expansion of the metric near infinity. The stress energy tensor transforms anomalously under bulk diffeomorphisms broken by infrared divergences.

We define an iterative procedure to obtain a non-abelian generalization of the Born-Infeld action. This construction is made possible by the use of the severe restrictions imposed by kappa-symmetry. We have calculated all bosonic terms in the action up to terms quartic in the Yang-Mills field strength and all fermion bilinear terms up to terms cubic in the field strength. Already at this order the fermionic terms do not satisfy the symmetric trace-prescription.

The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. An immediate consequence of this is that all fields get complexified. By applying this idea to gravity one discovers that the metric becomes complex. Complex gravity is constructed by gauging the symmetry U(1, D - 1). The resulting action gives one specific form of nonsymmetric gravity. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. It is argued that for this theory to be consistent one must prove the existence of generalized diffeomorphism invariance. The results are easily generalized to noncommutative spaces.

Much has been learned about string theory over the last few years by studying properties of cycles and branes in a given background geometry. Here we discuss three situations (quantum volume, attractor flows/D-brane stability, and dynamical topology change) in which cycles in a Calabi-Yau background evolve and/or degenerate in some manner, yielding various insights into aspects of quantum geometry.

We study the fluctuation spectrum of linearized gravity around non-fine-tuned branes. We focus on the case of an AdS₄ brane in AdS₅. In this case, for small cosmological constant, the warp factor near the brane is essentially that of a Minkowski brane. However, far from the brane, the metric differs substantially. The space includes the AdS₅ boundary, so it has infinite volume. Nonetheless, for sufficiently small AdS₄ cosmological constant, there is a bound state graviton in the theory, and four-dimensional gravity is reproduced. However, it is a massive bound state that plays the role of the four-dimensional graviton.

After reviewing the Green-Schwarz superstring using the approach of Siegel, the superstring is covariantly quantized by constructing a BRST operator from the fermionic constraints and a bosonic pure spinor ghost variable. Physical massless vertex operators are constructed and, for the first time, N-point tree amplitudes are computed in a manifestly ten-dimensional super-Poincaré covariant manner. Quantization can be generalized to curved supergravity backgrounds and the vertex operator for fluctuations around AdS₅ × S⁵ is explicitly constructed.

The computation of exact vertex operators for the Type IIB superstring in an AdS₃ × S³ background with Ramond-Ramond flux is described. The components of these vertex operators are shown to satisfy the supergravity linearized equations of motion for the six-dimensional (2,0) theory of a supergravity and tensor multiplet expanded around AdS₃ × S³ spacetime.

In the first part of this paper we find supergravity solutions corresponding to branes on worldvolumes of the form R^d × Σ where Σ is a Riemann surface. These theories arise when we wrap branes on holomorphic Riemann surfaces inside K3 or CY manifolds. In some cases the theory at low energies is a conformal field theory with two less dimensions. We find some non-singular supersymmetric compactifications of M-theory down to AdS₅. We also propose a criterion for permissible singularities in supergravity solutions.

In the second part of this paper, which can be read independently of the first, we show that there are no non-singular Randall-Sundrum or de-Sitter compactifications for large class of gravity theories.

We develop an approximation scheme for the quantum mechanics of N D0-branes at finite temperature in the 't Hooft large-N limit. The entropy of the quantum mechanics calculated using this approximation agrees well with the Bekenstein-Hawking entropy of a ten-dimensional non-extremal black hole with 0-brane charge. This result is in accord with the duality conjectured by Itzhaki, Maldacena, Sonnenschein and Yankielowicz. Our approximation scheme provides a model for the density matrix which describes a black hole in the strongly-coupled quantum mechanics.

The following sections are included:

• Introduction

• Early Motivation for Large Extra Dimensions

• Antimicrobial coatings
  • Type I/I' string theory and D-branes
  • Type II theories
  • Relation between Type I/I' and Type II – heterotic

• Theoretical Implications
  • U.V./I.R. correspondence
  • Unification
  • Supersymmetry breaking and scales hierarchy

• References

We study the dynamics of D-branes in the near-horizon geometry of NS fivebranes. This leads to a holographically dual description of the physics of D-branes ending on and/or intersecting NS5-branes. We use it to verify some properties of such D-branes which were deduced indirectly in the past, and discuss some instabilities of non-supersymmetric brane configurations. Our construction also describes vacua of Little String Theory which are dual to open plus closed string theory in asymptotically linear dilaton spacetimes.

Gravitational properties of domain walls in fundamental theory and their implications for the trapping of gravity are reviewed. In particular, the difficulties to embed gravity trapping configurations within gauged supergravity is reviewed and the status of the domain walls obtained via the breathing mode of sphere reduced Type IIB supergravity is presented.

We discuss light-cone gauge description of type IIB Green-Schwarz superstring in AdS₅ × S⁵ with a hope to make progress towards understanding spectrum of this theory. As in flat space, fixing light-cone gauge consists of two steps: (i) fixing kappa symmetry in such a way that the fermionic part of the action does not depend on x^-; (ii) fixing 2-d reparametrizations by x⁺ = τ and a condition on 2-d metric. In curved AdS space the latter cannot be the standard conformal gauge and breaks manifest 2-d Lorentz invariance. It is natural, therefore, to work in phase-space framework, imposing the GGRT light-cone gauge conditions x⁺ = τ, P⁺ =const. We obtain the resulting light-cone superstring Hamiltonian.

This is the text of my talk at the Strings 2000 Conference at the University of Michigan. It consists of three interlocking conjectures about the nature of SUSY breaking and the Cosmological Constant, as well as evidence from a toy model that demonstrates the validity of the claim that systems with a UV/IR connection can exhibit much larger SUSY breaking than local field theory.

We study some aspects of open string theories on D-branes with critical electric fields. We show that the massless open string modes that move in the direction of the electric field decouple. In the 1+1 dimensional case the dual theory is U(N) SYM with electric flux, and the decoupling of massless open strings is dual to the decoupling of the U(1) degrees of freedom. We also show that, if the direction along the electric field is compact, then there are finite energy winding closed string modes. They are dual to Higgs branch excitations of the SYM theory, and their energetics works accordingly. These properties provide new non-trivial evidence for the duality.

Summary of a talk explaining three ways in which string theory and M-theory are related to the mathematics of K-theory.

We analyze the nature of space-time nonlocality in string theory. After giving a brief overview on the conjecture of the space-time uncertainty principle, a (semi-classical) reformulation of string quantum mechanics, in which the dynamics is represented by the noncommutativity between temporal and spatial coordinates, is outlined. The formalism is then compared to the space-time noncommutative field theories associated with nonzero electric B-fields.

We discuss the nonabelian world-volume action which governs the dynamics of N coincident Dp-branes. In this theory, the branes' transverse displacements are described by matrix-valued scalar fields, and so this is a natural physical framework for the appearance of noncommutative geometry. One example is the dielectric effect by which Dp-branes may be polarized into a noncommutative geometry by external fields. Another example is the appearance of noncommutative geometries in the description of intersecting D-branes of differing dimensions, such as D-strings ending on a D3- or D5-brane.

Stable non-BPS states can be constructed and studied in a variety of contexts in string theory. Here we review some interesting constructions that arise from suspended and wrapped branes. We also exhibit some stable non-BPS states that have holographic duals.

We perform a general analysis of representations of the superconformal algebras OSp(8/4,ℝ) and OSp(8* /2N) in harmonic superspace. We present a construction of their highest-weight UIR's by multiplication of the different types of massless conformal superfields ("supersingletons"). In particular, all "short multiplets" are classified. Representations undergoing shortening have "protected dimension" and may correspond to BPS states in the dual supergravity theory in anti-de Sitter space. These results are relevant for the classification of multitrace operators in boundary conformally invariant theories as well as for the classification of AdS black holes preserving different fractions of supersymmetry.

The presentation at Strings 2000 was intended to be in two main parts, but there was only time for part one. However both parts appeared on the online proceedings, and are also included in this document. The first part concerns an exploration of the connection between the physics of the "enhançon" geometry arising from wrapping N D6–branes on the K3 manifold in Type IIA string theory and that of a charge N BPS multi–monopole. This also relates to the physics of 2+1 dimensional SU(N) gauge theory with eight supercharges. The main results uncovered by this exploration are: a) better insight into the non–perturbative geometry of the enhançon; b) the structure of the moduli space geometry, and its characterisation in terms of a family of Atiyah–Hitchin–like manifolds; c) the use of Nahm data to describe aspects of the geometry, showing that the enhançon locus itself has a description as a fuzzy sphere. Part two discusses the addition of extra D2–branes into the geometry. Two probe computations show the difference between the geometry as seen by D2–branes and that seen by wrapped D6–branes, and the accompanying gauge theory interpretations are discussed.

We discuss toroidal compactifications of maximal supergravity coupled to an extended configuration of BPS states which transform consistently under the U-duality group. Under certain conditions this leads to theories that live in more than eleven space-time dimensions, with maximal supersymmetry but only partial Lorentz invariance. We demonstrate certain features of this construction for the case of nine-dimensional N = 2 supergravity.

In 1900 the world-renowned mathematician David Hilbert presented twenty-three problems at the International Congress of Mathematicians in Paris. These problems have inspired mathematicians throughout the last century. As a piece of millennial madness, all participants of the Strings 2000 Conference were invited to help formulate the ten most important unsolved problems in fundamental physics. Each participant was allowed to submit one candidate problem for consideration. To qualify, the problem must not only have been important but also well-defined and stated in a clear way. The best ten problems were selected at the end of the conference by a selection panel consisting of David Gross, Edward Witten and myself.