Table of contents

Chairmen's Preface In 1984 we obtained a grant from the Nobel Foundation to organize a Nobel Symposium on "Unification of the Fundamental Interactions". In our proposal which we submitted in the fall of 1983 we stated that we wanted to cover the various attempts to unification such as GUT'S, supergravity, Kaluza–Klein theories and superstrings. What has happened in particle physics since then is already history. With the realization that certain superstring theories could be anomaly free, it became clear that these models could encompass earlier attempts to unification as well as solving the fundamental problem of quantum gravity. The excitement that some of us had felt for some time now spread through most of the particle physics community and this excitement certainly was evident during the Symposium. With the international advisory committee we originally chose a list of around 30 invitees which could best represent the various subjects listed above. When it came to the final planning of the programme essentially all talks dealt with superstrings!

We were very fortunate that almost all of the invitees managed to come to the Symposium. From the western world only three were unable to participate, André Neveu, Steven Weinberg and Bruno Zumino. We certainly missed them during the meeting. We were particularly happy that Stephen Hawking managed to take part actively. Our real problem was to get participants from the Soviet Union. Out of eight invitations only one came through. We were very happy to have Renata Kallosh, who really did her utmost to enlighten us about not only her own work but also about recent progress in the USSR, However, we were very sorry that in spite of all our letters, telegrammes and endless attempts to get telephone calls through and despite the good relations between the Swedish and Soviet Academies of Sciences we had to miss Ludwig Faddeev, Valodja Gribov, Andrej Linde, Victor Ogievetsky, Sasha Polyakov, Misha Shifman and Arkadij Vainshtein. We had hoped that the Nobel Symposium should be a signal for improved relations between East and West, but in this respect we failed.

The Symposium was held in the small town of Marstrand, a summer resort on an island outside Göteborg. The idea was to have it relatively close to home and to show the participants perhaps the best part of the local nature. Another motive was to keep the participants semi-isolated to provide ample time for discussions. With the somewhat heavy programme we found that even so, the days were not long enough.

This Symposium was the second Nobel Symposium on elementary particle physics. The first one, also organized by our group, was held in 1968. We hope and believe that the next one need not be awaited another 18 years!

The Symposium was made possible by a generous grant from the Nobel Foundation. An additional grant from the Royal Swedish Academy of Sciences was also essential. A grant from Nordita made it possible to invite some ten Nordic observers. We also gratefully acknowledge help with the organization of the Symposium from the city of Goteborg, the city of Kungälv, Volvo, Skandinaviska Enskilda banken, Ericsson AB and IBM Sweden. We thank all the participants for all their efforts, to come to our remote part of the world, to give excellent talks, to write up their talks astonishingly quickly and to share with us so much of their knowledge and expertise.

Last but not least we are indebted to the members of the organizing committee, members of the group and the secretarial staff.

Goteborg, August 1986

Editors' Note In these Proceedings the lectures are given in the order of their oral presentation at the Symposium. Regrettably, a few contributors could not participate (V. Gribov, A. Linde, V. Ogievetsky, M. Shifman and A. M. Polyakov). Their contributions have been placed close to the end of the Proceedings. As these authors did not have the opportunity of correcting the proofs, responsibility for printing errors must rest with the Editors.

It is a pleasure to thank all the authors for their enthusiastic and rapid response in our efforts to bring the contributions to the stage of printing.

Without the efficient and fruitful collaboration with Physica Scripta the publication of these Proceedings would have been substantially delayed.

The last Nobel Symposium on elementary particle physics ended on May 25 1968. The main topics discussed at that meeting were: high energy scattering of hadrons, infinite component field theory and electroweak unification. The last of these topics is the one which is best remembered since it is associated with the revolution in our understanding of the field theoretic description of the weak and electromagnetic forces. The first two topics, however, are closely associated with the origins of string theory. This talk will survey the development of string theory from that time up to the summer of 1984, since when there has been an explosion of activity in superstring theory which forms the main topic of this meeting.

The formulation of the string theory of unified elementary particle interactions in terms of operators in a Fock space is now seen to relate to the representation theory of certain infinite dimensional algebras. This insight has enhanced the understanding of the physical and mathematical theories involved and furthermore has led to applications in other branches of theoretical physics. A brief account of the new results is given here.

This lecture is divided into two parts. In the first part we review the application of index theory and differential topology to the understanding of local and global anomalies. In the second part we review some recent applications of algebraic geometry to unravel the structure of multiloop string diagrams.

The structure of the heterotic string which is widely regarded as the most promising candidate for a unified theory is reviewed together with the attempts to reconstruct the standard model from it. A progress report on recent developments is also given.

We review the old covariant quantization by writing the equations, that characterize the physical states and their vertex operators.

These equations are explicitly solved by constructing the most general physical state together with the corresponding vertex operator, that is then used to construct the interaction term used in the light-cone string field theory.

Finally we construct an extension of the Sciuto Vertex, that includes the ghost coordinates and that is invariant under BRST transformations.

Supersymmetric theories postulate the existence of superpartners for all particles: photinos and gluinos, winos and zinos, spin-0 leptons and quarks, etc. They also provide relations between massive gauge bosons and Higgs bosons. We summarize the present status of experimental searches for these new particles. We also discuss extended supersymmetric theories, and their formulation in a higher dimensional spacetime. Both the grand-unification and supersymmetry breakings may then by induced, in four dimensions, by the momenta carried by GUT or SUSY particles along extra space dimensions.

Attempts to link the superstring with the reality of physics in four dimensions at energies ⩽ 1 TeV are discussed. Phenomenological considerations bearing on the choice of compactification scheme are presented, and possible experimental signatures are mentioned.

Some recent developments in string field theory are discussed and a perspective on the subject is given.

Quantum string theory is written as integrable analytic geometry on the universal moduli space of Riemann surfaces.

A covariant field theory for the interacting string recently proposed by a Kyoto group is discussed.

After a canonical transformation on the string field Φ = Σ_n=0^∞sⁿ[₀φ_n(x) + ψ_n(x)], an action for the free open spinning string is constructed containing only two string fields ₀(x) and ₁(x). This result extends previous work by Siegel, Zwiebach and Witten for the free bosonic string, and agrees with work by Yamron.

The critical dimension of the spinning string in superspace and of non-linear sigma models in x-space is determined by Fujikawa's method, using non-Abelian bosonization.

We review the construction of the free equations of motion for open and closed strings in 26 dimensions, using the methods of the Florida Group. Differing from previous treatments, we argue that the constraint L₀ - ₀ = 0 should not be imposed on all the fields of the closed string in the gauge invariant formalism; we show that it can be incorporated in the gauge invariant formalism at the price of being unable to extract the equations of motion from a Langrangian. We then describe our purely algebraic method to introduce interactions, which works equally well for open and closed strings. Quartic interactions are absent except in the Physical Gauge. Finally, we speculate on the role of the measure of the open string path functional.

An outline is given of the derivation of the formula for the n-loop string amplitude. We use the light-cone frame, and we obtain an explicit, manifestly covariant formula. The method of derivation ensures that the amplitude is unitary. We compare our approach with that based on the Polyakov ansatz.

We review the progress which has recently been made in determining the spacetime backgrounds in which strings may consistently propagate. At the no-string-loop level, satisfactory backgrounds correspond to conformally invariant non-linear signal models and the condition of conformal invariance can be expressed as a field equation for the background spacetime. The problem of finding the effect of string loops on this picture is discussed.

The scale-invariant superspace formulation of d = 10 Einstein-Yang-Mills system is considered which gives a most economical and clear presentation of the full on-shell content of the theory. Discussion of supersymmetric higher order corrections to the effective action of string theory is based on the corresponding supergeometry.

A geometrical formulation of the heterotic string theory in superspace with 10 + 496 bosonic and 16 fermionic coordinates is presented. Minimal constraints on this superspace are found which provide the local fermionic and two types of bosonic world-sheet symmetries of the heterotic string. The relation between this superspace and the anomaly-free d = 10 Einstein-Yang-Mills supergravity is discussed.

The existence of a self-dual world-sheet torsion, which is required by the world-sheet symmetries, explains the geometrical origin of the absence of right movers in the gauge degrees of freedom of the heterotic string. The action of the heterotic string in the full curved background has a particularly simple form:

.

The orbifold approach to string compactification is reviewed. The construction of non-supersymmetric, modular invariant string theories in ten dimensions is also discussed.

Recent attempts to construct higher curvature supergravity theories are reported and some implications concerning the effective low-energy action for superstring theories discussed.

We obtain the Einstein action plus quadratic curvature corrections generated by closed bosonic, heterotic and supersymmetric strings by matching the four-graviton amplitude (to first order in the slope parameter and fourth power of momenta) with an effective local gravitational action. The resulting corrections are first shown to be of the Gauss-Bonnet form. It is then noted that, by the very nature of the slope expansion, the field-redefinition theorem applies. Consequently, only the curvature-squared term is determined, while squares of its contractions are explicitly seen not to contribute. This latter property has a generalization to all orders which implies that the effective gravitational action is unavoidably ghost-free.

The properties of solutions to these corrected theories are then examined. First neglecting dilatons, we find the explicit "Schwarzschild" metrics. Both asymptotically flat and de Sitter solutions are present. The latter are however shown to be unstable. The former have horizons and singularities which are respectively smaller and less violent than in Einstein gravity; the correct sign of the slope parameter also ensures absence of naked singularities. When dilatons are included, the cosmological vacua are gratifyingly excluded.

The quantum properties of black holes are compared with those of elementary and composite particles. As argued before by this author, it is desirable to search for a theory of black holes in which quantum mechanical "information" is not drained away by the horizon, but such a theory requires a drastically new approach in formulating general coordinate transformations with horizons in a quantum theory. It is subsequently shown that a closed string with string tension T = 1/8πG reproduces in a remarkable way the horizon fluctuations so that a new geometric interpretation of strings is suggested.

The quantum state of the universe may be described by a wave function which is a functional of the metric and matter fields of a hypersurface which divides the spacetime into two parts. If the spacetime is not simply connected, this hypersurface will in general have more than one connected component. Most physical observables depend only on a single connected component and can be calculated from a density matrix obtained by integrating over the unobserved surfaces. On the "no boundary" proposal for the quantum state of the universe, this density matrix is given by a path integral over all metrics, connected and disconnected that are bounded by two surfaces. The extension of these ideas to string theory is discussed.

For nonsemisimple groups it is possible to find new topological terms which can be viewed as generalizations of Chern-Simons terms, depending on noninvariant constant tensors that belong to nontrivial Lie-algebra cohomology classes. We describe how these terms arise in N = 2 supergravity and present the mathematical ingredients relevant for their construction.

The goal of random dynamics is the derivation of the laws of Nature as we know them (standard model) from inessential assumptions. The inessential assumptions made here are expressed as sets of general models at extremely high energies: gauge glass and spacetime foam. Both sets of models lead tentatively to the standard model.

A new hypothesis on the nature of quark confinement and the structure of infrared behaviour in gluodynamics is proposed.

It is shown that the large-scale quantum fluctuations of the scalar field φ generated in the chaotic inflation scenario lead to an infinite process of self-reproduction of inflationary mini-universes. A model of an eternally existing chaotic inflationary universe is suggested. It is pointed out that whereas the universe locally is very homogeneous as a result of inflation, which occurs at the classical level, the global structure of the universe is determined by quantum effects and is highly non-trivial. The universe consists of exponentially large number of different mini-universes, inside which all possible (metastable) vacuum states and all possible types of compactification are realized. The picture differs crucially from the standard picture of a one-domain universe in a "true" vacuum state. Our results may serve as a justification of the anthropic principle in the inflationary cosmology. These results may have important implications for the elementary particle theory as well. Namely, since all possible types of mini-universes, in which inflation may occur, should exist in our universe, there is no need to insist (as it is usually done) that in realistic theories the vacuum state of our type should be the only possible one or the best one.

General actions for self-interacting N = 2 tensor multiplets are considered in the harmonic superspace approach. All of them are shown to be equivalent, by superfield duality transformations, to some restricted class of the hypermultiplets actions. In particular, the improved tensor multiplet theory is dual to a free hypermultiplet one. We present also superspace couplings of these matter multiplets to conformal supergravity.

We consider two problems in supersymmetric gauge theories - the anomaly supermultiplet and the calculation of the exact β functions in perturbation theory - which turn out to be interrelated. The main findings are:

(i) generalization of the non-renormalization theorems for gauge F terms ∫ d²θ W²;

(ii) solution of the anomaly puzzle and superextension of the Adler-Bardeen theorem;

(iii) a constructive algorithm for computation of all-order terms in β functions;

(iv) a new bosonic anomaly counting the number of the bosonic zero modes. We also show that Z factors of the matter superfields affect the magnitude of the observable gauge coupling. The basic technical element which ensured the breakthrough is the use of the Wilson operator expansion and distinction between the Wilson effective action S_w and the generator of 1PI vertices Γ.

Strings are important objects for physics, although it has not been proved yet. It is a common hope, that they are fundamental elements of grand unification. Much less common is the hope that they describe color-electric fluxes and help to solve gluodynamics and also that they are relevant for the solution of the three-dimensional Ising model.

In my talk I shall describe various topics, important for all three problems. I believe that in order to reach understanding it is useful to keep them in mind simultaneously.

Unfortunately, my presentation will be fragmentary, because my understanding of the subject is far from complete.

Future progress in string theory will derive from various different approaches to the subject, as well as deeper understandings of the relationships among them. Once a fundamental principle and its implications for "string geometry" are understood, there will be progress on many exciting questions including space-time singularities, early universe cosmology, and compactification. The era of string theory is likely to last for many decades and have far-reaching consequences for the understanding of fundamental physics and mathematics.

Some general remarks about the present state of superstring theory are presented.