Strings '90

The following sections are included:

• INTERNATIONAL ADVISORY COMMITTEE

• PREFACE

• CONTENTS

The matrix model solution of c=1 matter coupled to two dimensional quantum gravity is reviewed, both in the case where the target space is the real line and a circle of finite radius. The role and physical significance of the nonsinglet states is analysed. The meaning of this theory as a two dimensional string theory is discussed and a fermionic field theory representation is constructed.

We reconsider the derivation of a recently found double scaling solution for 1D string field theory from matrix quantum mechanics and propose a new set of observables, which are cut-off independent, unlike the standart n-point functions. The relations between the new and old observables are found and their physical interpretation is discussed.

We review recent work giving the solution of matrix models used to define minimal models coupled to two-dimensional gravity.

I review recent results about two dimensional topological gravity, and describe some speculations relating n matrix models to topological gauge theories, W algebras, and (in the c → 1 limit) four dimensional topological gravity.

I discuss various features of the Liouville path integral, including factorization, operators and operator products, the contour of integration, and Ward identities. Many of these results have been obtained previously by Seiberg.

A non perturbative, canonical quantization program for 3+1 dimensional gravity can be carried out to completion in the 2+1 dimensional case. The resulting quantum theory in turn provides a number of conceptual and technical insights that are likely to be useful in the 3+1 theory. The main results are summarized without detailed proofs.

We summarize recent results (with Aleksandar Mikovič) on the random lattice approach to the study of the nonperturbative behavior of superstrings.

Non-critical strings with local N = 2 world-sheet supersymmetry are examined. The corresponding Liouville action, as well as the string susceptibility exponent and the scaling dimensions of all dressed chiral operators are unambiguously determined at the classical level. We also discuss the possibility of defining new “exotic” N = 2 string theories above their critical dimension D_crit = 2.

Collective effects are investigated in the covariant field theories for open and closed bosonic strings. The tachyon destabilizes the perturbative vacuum and interactions induce the formation of a condensate in a new stable vacuum. The spectrum of states is considerably different from the perturbative spectrum. In particular, there are neither massless gauge bosons nor a massless graviton. Other topics such as vacuum structure, the possibility of Lorentz symmetry breaking, and the running at tree level of the coupling constant are discussed.

We discuss the origin of quantum group features in both Wess-Zumino-Witten models and (2 + 1)-dimensional topological Chern-Simons theories. We identify certain matrices in these theories whose elements obey the commutation relations characteristic of a quantum group.

This talk describes a modification of the ten-dimensional superparticle action in which the superspace is extended to include an extra anti-commuting space-time spinor coordinate. This is the only known formulation of the superparticle which can be covariantly quantized and whose BRST cohomolgy gives a quantum spectrum which is precisely that often-dimensional super-Yang-Mills theory. The light-cone quantization is described and shown to give the correct light-cone superfield description of the physical states. By making covariant gauge choices free BRST-invariant quantum actions are obtained. In each case, the BRST cohomology class of zero ghost number gives the spectrum of N=1 super-Yang-Mills.

It is shown that membrane theories exhibit global non-compact symmetries which have their origin in duality tranformations on the three-dimensional worldvolume which rotate field equations into Bianchi identities.

Recent work on the existence of solutions to string theory with spacetime singularities is reviewed. Some extensions of these solutions are also discussed.

The standard superspace can be generalized to a loop superspace where the supercoordinates depend on a loop variable. This enables us to define a generalized covariant derivative containing a super 2-form which couples to a superstring. In this framework, the principle of light-like integrability leads, to the equations of motion of supergravity in ten dimensions. The super submanifold which plays an important role in the phenomenon of light-like integrability is closely related to the fermionic gauge symmetry, known as the κ-symmetry, of the manifestly spacetime supersymmetric string theory.

In this talk we will describe the Osp(10/4) covariant computations of the conformal anomaly of the quantized superstring. The anomaly will be shown to be a function of the graded dimension and quadratic index of a spinor representation.

We will discuss the problem of BRST cohomology of manifestly supersym-metric quantized particles and strings. We will show that the covariantly gauge-fixed action of the ten-dimensional Brink-Schwarz superparticle has a fermionic, nilpotent symmetry whose cohomology is given by the 8 + 8 supersymmetric Yang-Mills multiplet.

In this talk I will describe an attempt to bridge string theory and large N QCD, as obtained in a recent paper [1]. This is based on a relation between area preserving diffeomorphisms and SU(∞). The reduced model of QCD takes the form of a version of string theory that is related to ordinary string theory in the gauge det(γ) = −1, where γ_iy is the world sheet metric. This relationship may open the way to cross fertilization between these two theories. In order to discuss these points we briefly review some highlights of reduced models and area preserving diffeomorphisms and then apply the formalism to gauge theories and matrix models.

We consider applications of recent results describing Kosterlitz-Thouless type phase transitions of vortices and monopoles defined on a two dimensional space of spherical topology. It is argued that bosonic Liouville theory, and its N = 1 supersymmetric generalization, are in the strong coupling phase unless D < 1. For the N = 2 theory, however, it is unlikely that such a restriction is necessary; the theory being in the weak phase for any value of D.

The following sections are included:

• Compactification on Abstract (2,2)-Super-Conformal Field Theories

• (2,2)_9,9-Theories on Groupfolds

• The Free Fermion Construction of 4D Superstring Vacua

• Classification of the (2,2)-Vacua Corresponding to Class I Groupfolds

• References

In the early universe finite temperature effects in string theories could have played a decisive role in determining the suitability of a given string theory and of a particular dimensionality of space-time. We review some recent results on the thermal mass shifts of open string theories as a function of space-time dimension.

The use of a superspace approach to realise BRST and anti-BRST symmetry is re-examined. It is shown that Sp(2)-superfield prepotentials always arise upon solving the constraints imposed upon the generalized Sp(2)-superfield field strengths. It is proposed that such ‘toy’ models contain features which should be incorporated into “geometrical” formulations of covariant string field theories.

A model space for the Virasoro group is constructed and some remarks on its properties are given.

The logical framework of closed string field theory is discussed. Such field theory requires the existence of vertices, namely, sets of surfaces, which appear to be generators of an algebra of Riemann surfaces. The consistency conditions on these vertices take the form of geometrical recursion relations. Conformal field theories yield representations of the vertices and allow the construction of string actions. The recursion relations turn into the Batalin-Vilkovisky master equation.

Brief comments are made on a variety of topics in string theory. Some recent thoughts on supersymmetric string actions are described.

We review recent developments in the theory of W_∞ algebras, which are N → ∞ limits of W_N extended conformal algebras, and indicate some possible directions for further investigation.

Using the Feigin-Fuchs representation of the W-algebra extended conformal field theories allows us to obtain their braiding matrices and fusion relations. The braiding matrices are shown to coincide with a certain limit of the Boltzmann weights of the IRF models.

We review the construction of super-W algebras in the operator product expansion method and the Lie super algebraic approach to generalized (N = 1 and N = 2) super-KdV equations and the associated (N = 1 and N = 2) super-W algebras.

The W₃ algebra is gauged by using the Noether method. This algebra contains spin 2 Virasoro generators L_m and spin 3 generators W_m, and squares of the Virasoro generators appear on the right-hand side of some commutators. The theory contains n scalar matter fields φⁱ ≡ φⁱ(x⁺, x⁻) coupled to spin 2 gauge fields h⁺⁺, h⁻⁻ and spin 3 gauge fields B⁺⁺⁺, B⁻⁻⁻.

The Noether results through third order in the expansion parameter suggest the notion of a “nested covariant derivative”, and by interpreting it as a field equation, we integrate the latter and find the complete, infinitely nonlinear, action in one stroke. The result is very simple, see eq. (20). The transformation laws are then completed by requiring this action to be invariant. The results, again infinitely nonlinear, are given in closed form in eqs. (16) and (23). Recent results concerning W_N and w_∞ gauge theories, as well as some open problems, are mentioned at the end.

In analogy with the construction of the Z₂-twisted bosonic conformal field theory, whose discrete automorphism group is the Monster group, the largest finite sporadic simple group, an explicit construction is given for the Z₂-twisted fermionic conformal field theory comprised of a Neveu-Schwarz and a Ramond sector. A triality element, a discrete symmetry of the bosonic conformal field theory which is responsible for promoting Conway's group to the Monster, is a generic feature of twisted conformal field theory.

Properties of the heterotic string which possesses at M_pℓ E₆ symmetry in the physical sector and matter parity symmetry are discussed. If E₆ → [SU(3)]³ breaking occurs at M_pℓ followed by intermediate scale breaking to SU(3) × SU(2) × U(1) triggered by SUSY soft breaking masses, then there exists at least two (and generally more) new light SU(3) × SU(2) × U(1) singlets. These new particles interact with leptons and Higgs bosons giving rise to neutrino masses, neutrino oscillations and lepton number violating decays μ → eγ, τ → μγ accessible to experimental detection. Constraints for the Gepner three generation model to be phenomenologically viable are given.

We discuss the recent progress made in deriving the low-energy structure of the string-derived flipped SU(5) model. The results obtained so far include: hierarchical quark and lepton masses, two light Higgs doublets, hidden sector bound state particles called cryptons, controlled d = 5 baryon decay operators, a preliminary renormalization group analysis of the gauge and Yukawa couplings of the model which predicts new exotic particles at the scale of , a new mechanism to explain the origin of the KM matrix, and testable predictions for the rare flavor violating decay τ → μγ. These precise statements follow from a detailed analysis of all relevant cubic and higher order terms in the superpotential.

I describe recent results on the intrinsic formulation of the local geometry of continuous deformations of (2,2) superconformal vacua for superstring theories compactified to four space-time dimensions.

We present an accessible account of the local geometry of the parameter space of Calabi–Yau manifolds. It is shown that the parameter space decomposes, at least locally, into a product with one factor the space of parameters of the complex structure and the other a complex extension of the parameter space of the Kähler class. It is also shown that each of these spaces is itself a Kähler manifold and is moreover a Kähler manifold of restricted type. There is a remarkable symmetry in the intrinsic structures of the two parameter spaces and the relevance of this to the conjectured existence of mirror manifolds is discussed. The two parameter spaces behave differently with respect to modular transformations and it is argued that the role of quantum corrections is to restore the symmetry between the two types of parameters so as to enforce modular invariance.

We address the study of geometrical and topological properties of string vacua. In particular, the distance in the space of couplings of two-dimensional quantum field theories is studied, as specified by the Zamolodchikov metric. We show that for world-sheet supersymmetric theories, the Witten index (target space Euler number) cannot be changed while moving a finite distance, and illustrate this for N = 1 and N = 2 minimal series as well as for Calabi-Yau manifolds. We also comment on the properties of such theories when they are coupled to two-dimensional gravity.

We consider string compactifications obtained by modding tensor products of N = 2 superconformal models by discrete symmetries. The presence of discrete torsion modifies the usual generalized GSO projection and gives rise to (2,2), (0,2) or non-supersymmetric compactifications depending on its (quantized) value. We construct left-right asymmetric compactifications by twisting differently left- and right-movers of the N = 2 blocks. Some of these constructions provide a generalization of the concept of asymmetric orbifolds to non-toroidal (Calabi-Yau) varieties. We prove that all these models can be interpreted as left-right symmetric compactifications in the presence of discrete torsion. This suggests that all presently known four-dimensional strings may be equivalently understood as compactifications of the 10-dimensional Heterotic string in the presence of background fields.

We discuss the calculation of nonrenormalizable terms in the free fermionic formulation of the heterotic string in four dimensions. In this formulation such terms can be calculated exactly in σ-model and remain uncorrected to all orders in string loops, therefore greatly enhancing the predictive power of the theory. We obtain general results on the type of N-point functions which may be nonvanishing and consider their contributions to the superpotential.

This paper reviews our legacy for the next millennium as far as the experimental high energy physics is concerned; Supercolliders, can we get out of the SU(3) × SU(2) × U(1)-trap, new detecting techniques are the main aspects of this legacy discussed in this paper.

Experimentally observable consequences of supersymmetry may involve CP-violating processes, such as the neutron electric dipole moment d_n, that are generically enhanced in supersymmetric models. We investigate the contributions to d_n both for very low and for moderate scales of supersymmetry breaking. In particular, we consider the contribution of the dimension-six CP-violating operator proposed recently by Weinberg. We show that this operator cannot be supersymmetrised and hence its contribution is suppressed in models with only very weakly broken super-symmetry. For more realistic scales of supersymmetry breaking, this operator does contribute appreciably to d_n, but the dominant contributions turn out to be those of the dimension-five quark electric dipole moment and the quark color electric dipole moment operators.

Cosmology can be used as a probe of particle physics at energy scales close to the Planck scale. Some of the most relevant new data is reviewed and compared with the predictions of three different theoretical models for the formation of structure: the model based on adiabatic perturbations, the cosmic string theory, and a new model using global textures.

Watch out! You slob! How dare you eat your lunch while reading our paper! (“Abstract” does not mean abstract art.) No wonder the copier doesn't work anymore.

In this talk we will discuss everything. We will begin at an introductory level that anyone can understand, and finish at an advanced level that even we don't understand. Familiarity with our book is not necessary, but buy it anyway.