Supersymmetry first appeared in the early 1970s – in fact, one of its first appearances was within string theory itself. Once the idea of supersymmetry appeared, it began to percolate, being applied to any possible available theory. First, it was applied to particles. After particles, the next step was field theories – a process carried out from the mid-1970s. There was a machine and recipe to follow, and supersymmetric theories were constructed wherever they could be. After field theories the next step was gravity, and supersymmetry was soon being applied to general relativity, producing what are called supergravity theories. The German physicist Werner Nahm classified all possible supergravity theories in 1977. In particular, he showed that the maximal number of dimensions that supergravity theories could exist in was eleven – leading to the explicit construction of the biggest and baddest supergravity theory of them all: the writing down in 1978 of eleven-dimensional supergravity by Eugène Cremmer, Bernard Julia and (again) Joel Scherk.6 This was the apex: Nahm’s result implied that it was not possible to go beyond the eleven-dimensional theory.
Unification of the gauge forces into aspects of a single force was also one feature of the 1970s. However, the rise of the supergravity theories – which appeared increasingly special and unique as the number of dimensions reached double figure – suggested another possible type of unification: unification through geometry. The dream that geometry is responsible for the laws of physics has been passed down from Einstein. The supergravity theories offered a possible way in: the geometry of the extra dimensions would determine the physics of the lower dimensions.
The ability of extra dimensions to turn gravitational interactions into gauge interactions had been known for a long time, going back to the work of the German physicist Theodor Kaluza. Kaluza had originally showed, as far back as 1919, that Einstein’s gravitational theory in five dimensions, looked at from a four-dimensional perspective, behaved as a four-dimensional theory of gravity with an additional gauge force resembling electromagnetism. This idea had long lay dormant, but it could now find a natural home in the higher dimensional supergravity theories that were being developed. Edward Witten, recently tenured in Princeton and still on the early parts of a trajectory that would see his name become one of the most revered in the subject, caused some excitement in 1981 by showing that eleven was not only the largest number of dimensions one could obtain in supergravity, but also the smallest number of dimensions that were required to obtain the forces of the Standard Model using the approach that Kaluza had pioneered.
Over this period, research in string theory proceeded slowly. While the supersymmetric string had been developed by Pierre Ramond, André Neveu and John Schwarz, it was still not yet clear that this was a theory that fully made sense. In 1976 a modification of this was made by Ferdinando Gliozzi, Joel Scherk and David Olive, and in fact this modification produced the first fully consistent supersymmetric string theory. They also realised that the string theory contained quantum field theory, as a limit in the case where all the energies were much smaller than the characteristic energies of the strings. This was the indication that string theory contained both supersymmetry and quantum field theory within it, thereby linking the subject to the developing mainstream trends. However, the significance of this paper was something that was only appreciated later. At the time its ripples were extremely small, fading out quickly.
The paper of Gliozzi, Scherk and Olive had considered the cases of both closed strings and also open strings with endpoints. Although no one noticed then, the calculations were only correct for the case of open strings. It was not until 1981 that Michael Green and John Schwarz rectified the mistakes for the case of closed strings. Green and Schwarz also discovered that there were two possible consistent string theories in ten dimensions, which they called type IIA and type IIB.
It is actually a bit of an exaggeration to say that they knew at this time the theories were consistent. It was known that there were many subtle issues, and a full understanding of the deeper structure was still rather foggy. These ten-dimensional string theories presumably had a connection to the ten-dimensional theories of supergravity – but only one of these supergravity theories (corresponding to the type IIA theory) had actually been constructed at the time. Following Nahm’s 1977 classification, it was known that the other supergravity theory – type IIB – should exist in principle, but no one had yet written it down explicitly or determined the precise form of its equations. Indeed this was not fully accomplished until later work by Paul Howe and Peter West in 1984.
The above material has been dense, so let me step back and review where it has led to. Research in string theory in the later 1970s and early 1980s, in that it existed, was focused on understanding the theory of quantised, relativistic strings. It was not focused on applications and it was certainly not focused on matching data. It was about understanding what the theory was and what consistency conditions applied to it. This required learning the relevant calculational tools and how they could be used. There was an implicit hope that string theory could represent a theory of quantum gravity, but this hope was backed by little hard evidence. During the later part of this period, string theory began to connect back into the more mainstream parts of physics, and in particular connections appeared with supergravity theories, which were then regarded as the most promising candidates for unified quantum gravitational theories. At this stage string theory was fifteen years old, but still a very minor part of theoretical physics. While a few people were interested, it was – like popular music and the stage – not an area where a young person could be advised to make a career in.
String theory moved from the wings to centre stage in 1984, starting a period that is often called the first superstring revolution (although this name could with equal justice be applied to the period from 1968 to 1973). This year has become part of folklore for what is ultimately a sociological rather than a scientific reason – the sheer rush of people that started working on string theory then, as well as the sudden transition of the subject from a poorly studied curiosity to the most active and topical subject in theoretical particle physics. The scientific content would have been identical had the transition been slower, with a more gentle influx of numbers, or had there been greater prior interest in string theory such that its change in status was less marked. However all of these would have reduced the human impact of the story: no billionaire is quite so interesting as one who was penniless only a few months earlier.
It is worth unpacking the reasons for this change in attitude. At the time, supergravity theories had been regarded as the best approach towards both quantum gravity and possible unifications of gravity with the other forces. The volte-face came about through the realisation that these supergravity theories had severe problems – and string theory had just the right structure to solve them. These problems were of two sorts. The first was a problem of finiteness, and the second was a problem of consistency. The problem of finiteness was that when supergravity was used to compute the probabilities to scatter particles off one another, the resulting answers were infinite. In itself, this was not new: the same feature is encountered throughout the Standard Model. However, no one regarded the Standard Model as fundamental. In the Standard Model there is, as described in chapter 3, a conceptual understanding for why these infinities occur. The Standard Model is not the correct theory of nature on the smallest distance scales, and so it is natural that it produces infinities when the calculations are extended to these scales. In the correct theory of nature at the smallest scales, these infinities should go away – but they were not going away in supergravity.
In addressing any major open problem, calculation is normally not enough. Hard calculations are never enough: it is too easy to go wrong and create the illusion of a result. What convinces is calculation combined with physical reasoning. What caused excitement was that in string theory the same calculations gave finite answers – and there was a good physics argument why they should give finite a
nswers. This physical argument came from the fact that strings are strings: they are not points. The finite size of the string smears out all interactions, and the infinities which are caused by point-like constituents go away when the constituents are string-like. This explained why the string calculations were getting finite answers – and it also gave a motivational reason for why the calculations should always give finite answers.
Of course, this is not a logical proof. The Nobel Laureate Steven Weinberg expressed this in 1988 as follows:
There are hand-waving arguments that are not at all rigorous, perhaps not entirely convincing, that the theory ought to be finite to all orders. Then when one works out how it really looks in the lowest order of perturbation theory, one finds that those hand-waving arguments really work … With superstring theory I think finiteness is a reasonable guess. I’d be more surprised than not if it weren’t finite.
Despite this, it was not these finiteness results that would trigger the rush towards string theory.
The second problem with supergravity was the problem of consistency. A subtle aspect of quantum mechanics is the existence of anomalies. Anomalies are effects that can render a theory entirely inconsistent as a quantum mechanical theory – even though the same theory makes perfect sense in classical physics. While anomalies are subtle, their effects are drastic: a theory that fails the anomaly check is like the applicant for intelligence work who fails the security check. It is the end: no other extraordinary features or properties can ever be enough to compensate for this failure, and such a theory is inconsistent and irredeemable.7
A key feature of anomalies is also that they are a low-energy effect rather than a high-energy effect. To determine whether a theory is anomalous, it is sufficient to know only the spectrum of light particles, with no need to know how the theory behaves at the highest energies. A thorough study of the effects of anomalies, with respect to both gravitational and other forces, was performed in 1983 by Luis Alvarez-Gaumé and Edward Witten. They found that the most interesting supergravity theories, the ones with the potential to incorporate both gravity and forces resembling those of the Standard Model, were anomalous – and therefore incompatible with quantum mechanics.
Influential physicists were then politely sceptical that string theory could improve on supergravity. String theory appeared to modify supergravity theories, but only at high energies. It was the low-energy properties of supergravity theories, apparently shared by string theory, that seemed to render them inconsistent.
It would turn out to be the success of string theory in addressing this problem that caused the change in attitude. The reason for this impact was not purely the result by itself – it was also that the result showed a significant incompleteness in a quantitative calculation by some of the most respected theoretical physicists in the world. It was the computation of anomaly cancellation in superstring theory by Michael Green and John Schwarz in the autumn of 1984 that propelled string theory into the active brain of theoretical physics – a position that it has never left. The calculation they performed did not find any error in the calculations of Alvarez-Gaumé and Witten. What they did find was that in string theory this calculation always had an extra term – a term that had the precise effect of cancelling the supergravity anomalies that had been found by Alvarez-Gaumé and Witten. The supergravity that was the limit of string theory was not quite the same. String-flavoured supergravity had one crucial extra term in it – and this precisely cancelled the anomalies. The same extra term then also served to remove the fatal anomalies in all other calculations as well.
It could in principle have been possible to add this term into supergravity in an ad hoc way, although this possibility had not been anticipated in advance. However, it was the way that this term just dropped out of string theory that gave the structure both plausibility and consistency. At the point where string theory had to fail and be marked as inconsistent, it succeeded – and in a way that had not been foreseen in advance as possible.
At the time there was another aspect to their result that appeared striking. Previous calculations had shown that superstring theory worked in ten, and precisely ten, spacetime dimensions. The anomaly cancellation equations of Green and Schwarz also revealed something similar. The cancellations that they found worked only for a single choice of the forces present in the ten-dimensional string theory. 8 It seemed that the more that was learned about string theory, the more unique its structure appeared.
While thunderstorms raged amidst the Colorado mountains outside, Green and Schwarz developed their result at the Aspen Center for Physics, announcing it in the autumn of 1984. This result rapidly reached Edward Witten in Princeton, then in the early stages of his period as the single person of maximal influence in theoretical particle physics, a period that would endure for the next twenty years or so. He embraced string theory. Everyone else followed, and the backwater became a torrent.
In 1984 Ronald Reagan swept to a landslide victory against Walter Mondale, famine raged in Ethiopia while in Britain the miners’ strike began its long, slow, tragic collapse. What was string theory? It had just gone from a failed theory of the strong interactions to the number one hottest topic in theoretical particle physics.
5.5 THE LIMELIGHT
The five years from 1984 onwards would see a huge increase in the number of physicists working on string theory. However this is not a history chapter for the sake of history. It is rather to address the question of what ‘string theory’ is, and part of the answer involves knowing how the understanding of the subject has evolved through time – and what string theory was in this period was very definitely still the theory of quantum mechanical relativistic strings. String theory was a theory of strings, with aspirations towards being the correct quantum mechanical account of gravity and the other forces. This wish had been vocalised in 1974; while the rate of work increased, research in the subject in the mid-to-late 1980s was still guided by this earlier goal.
Much effort went into developing the most efficient and economical tools to analyse strings, as well as learning the mathematics that went along with them. The powerful physical and mathematical technology that was thrown at the subject in this period was however intellectually the continuation of work originating in the early 1970s. The number of people working on the subject increased dramatically: the general theme of research remained the same.
What was the big intellectual picture of the subject in this period? There were five consistent string theories known. There were the two type II theories – type IIA and type IIB – that had been formulated by Michael Green and John Schwarz in 1981. There was also the type I theory: this was the one in which Green and Schwarz had performed their by now famous anomaly cancellation calculations. In this paper however, Green and Schwarz had also observed that there was in principle a second solution to the anomaly cancellation equations. One solution had corresponded to the existing type I string theory. What did the other correspond to? Green and Schwarz suspected that it represented another, as yet unknown, string theory, and they speculated as much in their paper. They were right. This was the heterotic string, found in January 1985 by the so-called Princeton string quartet of David Gross, Jeffrey Harvey, Emil Martinec and Ryan Rohm; the name arose as it was a (consistent) hybrid of a bosonic string and a superstring.
The heterotic string theory was also found to come in two variants. One variant was similar to the existing type I theory – in particular, it contained the same kinds of forces. The other represented the ‘other’ solution to the anomaly cancellation equations: the E8 × E8 heterotic string.
Five string theories, each apparently consistent, were known. No fundamental principle was known that could select between the five. What to do next? There was at least a practical way of choosing. Some of the theories the two type II theories – appeared to exist as purely gravitational theories. They were theories of gravity, and that was all. There did not seem any prospect of connecting these type II theorie
s into observed particle physics. The other theories – the type I and heterotic theories – contained a particle content that, viewed from a distance in the fog and without glasses, resembled that of the Standard Model. One theory in particular, the so-called heterotic E8 × E8 theory, seemed particularly promising in this respect. The forces it contained were more elaborate versions of those present in the Grand Unified Theories that were thought to lie behind the forces of the Standard Model.
The heterotic E8 × E8 theory was still a theory defined in ten dimensions. To make a model of four-dimensional physics, it was necessary to ‘compactify’ the theory. This involved curling up six of the extra dimensions to be small and unobservable, so that their existence would have escaped our notice until now. Once this is done, this produces a theory that looks like a four-dimensional one – with particles, forces and interactions that depend on exactly how the curling up is done. It was then realised that if the extra dimensions had the geometry of what is called a Calabi-Yau space, the four-dimensional theory and its particle content would make a passable imitation of the supersymmetric grand unified theories that were then in vogue.
Having taken a decade’s leave of absence from particle physics, this suddenly offered a re-entry route for string theory back into the traditional enterprise of physics: understanding, explaining and predicting experimental data.
The structure of ten-dimensional string theories had been uniquely constrained through a variety of consistency conditions. The number of dimensions had been fixed by consistency, and the allowed types of forces present had also been fixed by consistency.
It was hoped that similar consistency conditions would apply on the reduction from ten to four dimensions. Amidst a wave of exuberance, it was thought that string theory may be on the verge of explaining the forces, masses and particle content of the Standard Model. There was no actual calculation that suggested this, but with a rush of new results coming in, it seemed that it might be only a matter of months before another new result would make it possible to compute the mass of the electron from first principles. This sense that a major understanding of the Standard Model might be just around the next corner can be felt in the abstract of one of the first papers on superstring models of particle physics from the spring of 1985, entitled Vacuum Configurations of Superstrings, and written by Philip Candelas, Gary Horowitz, Andrew Strominger and Edward Witten.
Why String Theory? Page 13