Why String Theory?
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We study candidate vacuum configurations in … E8 × E8 supergravity and superstring theory that have unbroken N = 1 super symmetry in four dimensions. This condition permits only a few possibilities, all of which have vanishing cosmological constant.
Today there are at least 473 800 776 such candidate possibilities known.
‘String theory’ was suddenly no longer just a remote theory of quantum gravity; it now appeared that it might also explain the structure of particle physics. It was a heady period when the frontiers of the subject were rapidly advancing, and no one knew where this advance would stop.
Throughout this euphoric period, there were some important known knowns – or so it was thought. Two of these knowns were that the correct number of dimensions was ten and that the correct objects to think about were strings. In particular, the supergravity theories that had been studied before were merely classical limits of the full string theory. It was felt that real physicists should study the full theory and not the old approximations. Older theories such as eleven-dimensional supergravity, together with their membranes with two extended spatial dimensions, were regarded as accidental and erroneous artefacts which could now be discarded.
There was one large, clear, known unknown in this period. We have already seen in chapter 3 that quantum field theory is easiest in a certain limit: the weak coupling limit. This is the limit in which particles interact only very weakly with each other. Answers are obtained by computing successive terms in a series and then summing these terms. In this limit, the later terms are merely small corrections to the first terms. While useful for precision work, they are not needed to obtain a qualitatively correct answer. This perturbative limit is the easiest limit for calculational purposes, but is not the full story.
String theory throughout the 1980s was a subject also defined in this perturbative limit. There was a single coupling, the string coupling, which was the precise analogue of the couplings of quantum field theory. The only known way to calculate was through an approximation series in this coupling, valid when the coupling was small. It was known how to calculate the first terms in the perturbative expansion, but even there the calculational difficulties rapidly grew insuperable. It was entirely unclear what lay beyond the perturbative limit, or how any kind of reliable calculational approach could be found in this regime. It was clearly desirable to move beyond perturbation theory, in the same way it was clearly desirable to end poverty and bring about world peace. The only minor issue was how. The strength of physics is in calculation, and calculations are restricted to what you can actually calculate.
Finally, there were the unknown unknowns. The radical changes in perspective experienced between 1968 and 1985 made it seem unlikely that all good ideas had been found. Edward Witten spoke about this in 1986:
One has to remember that string theory … is already eighteen years old, and looking back into the past we can see that ten or fifteen years ago there was a long road ahead, a lot of things that weren’t known that had to be known, and it’s probably still true today.
The words would be prescient – and it would turn out that it was the deprecated and discarded branes of supergravity that would be key to moving beyond the perturbative limit.
5.6 THE THEORY FORMERLY KNOWN AS STRINGS
With this attitude, string theory moved into the 1990s. After the great rush of work in the five years following 1984, the level of activity was reducing. It was a time for consolidation. It was a time for the exploitation of previous results rather than the development of new ones.
It was also a time when excitement was building elsewhere. Stung by the discovery of the W+, W– and Z0 bosons at CERN in 1983 – as the New York Times editorial put it, ‘Europe 3 United States – not even Z-zero’ – the Land of the Free and the Home of the Brave had embarked on an ambitious undertaking to take back, for good and for ever, leadership of experimental particle physics. A site in Texas had been selected for the construction of the Superconducting SuperCollider. The plan was to construct an 87 kilometre tunnel and fill it with superconducting magnets, generating at Waxahachie National Laboratory the largest and most powerful accelerator complex in the world. The project was American. The scale was Texan. It was deliberately on a size that CERN, whose own plans for the Large Hadron Collider were based on reusing an existing tunnel, could not compete with. The project was approved and construction started. Gigantic boring machines excavated kilometre after kilometre of tunnel deep underground the Texan soil. Had it been built, the SuperCollider would have been twenty times more powerful than any collider then existing, and even three times more powerful than the Large Hadron Collider following its 2015 energy upgrade. The vista of experimental particle physics would have been thrown out to the horizon, and then again.
It was not built. On November 3rd, 1992, William Jefferson Clinton was elected the forty-second president of the United States, replacing George H. W. Bush. The Superconducting SuperCollider had been conceived in the Reagan years and had also enjoyed the strong support of the one-time Texas congressman Bush. It was a project associated with the previous Republican administration and, by 1993, a project facing questions over management quality and budget escalation. With the United States in recession, and Democratic control of the Presidency, the Senate and the House of Representatives, the Superconducting Super Collider was cancelled on October 21st, 1993. This cancellation represented a blow that experimental particle physics in the United States has never recovered from. Current leadership in the subject sits, without question, at the outskirts of Geneva. In the long-term planning for the subject, possible locations for future colliders include CERN, China and Japan – but not the United States.
The reason for this mild digression is to explain some of the background reasons that caused so many young theorists to work on string theory in the period I am about to describe. In part, this was driven by a set of important breakthroughs that occurred around 1995. However it would be wrong to ignore the other external circumstances. The short-to-medium term future for experimental high-energy physics had just turned far bleaker, and it was natural that work tending towards predicting and understanding data became less attractive to any young would-be hotshot.9
The prevailing view of string theory changed in 1995 as problems that had appeared far-off and intractable were suddenly revealed to be nearby and soluble. This occurred not through any one single discovery, but through a new perspective on a variety of existing known results. The traditional dating of this change is to a talk by Edward Witten at the Strings ’95 conference at the University of Southern California on March 14th, 1995: both the talk and Witten’s subsequent paper summarising it brought a complete change of outlook. However, it was not that everything Witten said was new; and it was not that everything Witten talked about was specifically his own work. The rapid and dramatic change in outlook was due both to Witten’s synthesis of disparate results into a clear single picture and to the commanding personal influence that Witten had in the field.
Let me try to summarise briefly the changes of 1995. First, it was realised that the five different string theories then known were all related. They were indeed more than related – they were connected. It was possible to interpolate continuously from one theory to another. The five theories were merely individual parts on a continuously connected web in theory space. As mentioned earlier, in each theory the strength of the interactions is parametrised by something called the string coupling. The string coupling is the analogue in string theory of the coupling constants in quantum field theory: it determines how likely it is that two strings will interact, and how likely it is that a single string can split into two distinct strings. The calculations of the 1980s were performed at weak coupling, a limit where these interactions occur rarely. The question was: what happened at strong coupling, where the techniques of the 1980s all broke down?
The surprising answer of 1995 was that the theory at strong coupling simply turned into either the same theo
ry at weak coupling, viewed from a different light, or into a different theory – but also at weak coupling and in the regime where it was easy to calculate. These are examples of the dualities described in chapter 3.
What is meant by the same theory at weak coupling? It means exactly the same theory, with exactly the same equations – but with some of the labels changed. For example, suppose the original theory, defined at weak coupling, had equations involving two terms ‘B2’ and ‘C2’. The equations for this theory at strong coupling are exactly the same, but wherever we wrote ‘B2’, we now write ‘C2’. If we relabel ‘B’ as ‘C’ and ‘C’ as ‘B’, we are back where we started.
We can imagine moving along a line starting from weak string coupling (infinitesimally close to zero) all the way to strong string coupling (where the coupling becomes formally infinite). The point we end at is in no way exotic. In some cases, the end of the line is the same as its start: it resembles the Circle Line rather than the District Line. This is indeed actually the case for what is called type IIB string theory – the limit of this theory at strong string coupling is exactly the same as the theory at weak coupling, except with some labels interchanged.
In other cases, the strong coupling limit of one theory turns out to be a different theory at weak coupling. This applied for the type I SO(32) theory and the heterotic SO(32) theory: one theory taken to strong coupling turned out to be the other theory at weak coupling. Although these two theories had appeared quite distinct – involving very different particles and interactions – they emerged as the same theory, simply in different regimes. The part of each theory that appeared impenetrable was simply the other theory at its most accessible point.
These results did not just materialise out of the mist. These dualities extended previous examples of the phenomenon. A related question was the behaviour of theories at small and large values of the radii of the extra dimensions. The physical conditions here seem very different: a cylinder may remain a cylinder irrespective of its size, but a large cylinder is surely a different cylinder than a small one. However, there are again dualities that relate small and large radius: for example, the type IIA theory on a cylinder R times larger than the length of a string is exactly the same theory as the type IIB theory on a cylinder R times smaller than the length of a string. This duality is called T-duality, and it was already a well-known result that had been discovered back in the 1980s.
The idea of S-duality – the statement that a theory at strong coupling could be identical to a theory at weak coupling – went back to 1977 work on quantum field theory by Claus Montonen and David Olive. Olive had been one of the pioneers of string theory, but like many others had moved on to other topics during the late 1970s. In 1990 the idea that superstrings exhibited strong-weak coupling duality had been put forward by Anamaria Font, Luis Ibáñez, Dieter Lüst and Fernando Quevedo – although within the ultimately incorrect context of the heterotic string.
However, how could these ideas ever be checked, given the difficulties of computing at strong coupling? A key step was to find properties of the theory that were, in a technical sense, protected’: they could be calculated at weak coupling and the result extrapolated to strong coupling. In 1994 Ashoke Sen, an Indian physicist as brilliant as he is modest, found an example of such an object. S-duality required this object, which had both electric and magnetic charge, to exist – and Sen was able to show it did exist, as required by duality. The claim of an existence of a larger duality called U-duality, unifying previous examples, had also been put forward six months before Witten’s talk by Chris Hull of Queen Mary and Westfield College in London and Paul Townsend of Cambridge.
As in 1995 Witten brought both his own insights and all these other results together, the picture he outlined included a further shock to the system, in that the answer to ‘What is string theory?’ was suddenly no longer ‘First, a theory of strings.’ The answer was now ‘a limit of M-theory’. What was M-theory? It was the unifying theory of which all string theories were simply small parts. The different string theories became seen as different limits of one object. That one object was M-theory – and whatever M-theory was, the understood string theories were just one limit of it.10
One of the most surprising aspects of this was that there was another limit of M-theory involving a classical gravity theory in eleven – not ten – spacetime dimensions. This was the eleven-dimensional supergravity theory that had first been constructed in 1978. This limit contained no strings. Instead, the basic extended excitations were not strings, but rather objects with two spatial dimensions – membranes. The theory of these membranes had been worked out by Eric Bergshoeff, Ergin Sezgin and Paul Townsend in 1987. The eleven-dimensional limit was accessed by starting with the ten-dimensional type IIA theory and making its coupling strong. In this limit, the coupling itself morphs into an extra spatial dimension – another realisation of the dynamical and mutable nature of stringy geometry. 11
From this new perspective, string theory became at once a smaller and a larger subject. It was smaller because the previously distinct string theories were now part of a single whole: there was only one thing’ to be understood, and everything else was just a different limit of this single thing. It was a larger subject because the underlying equations of M-theory were, and are, unknown. M-theory is known from its boundaries. The Picts and the Numidians could infer the existence of Rome from their battles with its legions, but could never know that it has seven hills. M-theory is in essence defined as the entity whose different limits give either the various known string theories or eleven-dimensional supergravity: but its underlying equations are not known.
The connection of string theory to eleven-dimensional supergravity was not unprecedented either. Indeed, it was as early as 1987 that Michael Duff, Paul Howe, Takeo Inami and Kelly Stelle, all then at CERN, had pointed out that you could make a ten-dimensional type IIA superstring by taking a membrane in eleven dimensions and viewing it in the limit where the eleventh dimension was very small. 12 For those who had continued working on supergravity during the 1980s, and even more so for those who worked on eleven-dimensional supergravity in this period, this was a time of intellectual vindication. For a decade they had had to face a certain amount of condescension for not making the shift to strings – Michael Duff reports having to deal with comments such as:
I want to cover up my ears every time I hear the word membrane’.
In 1995 then, ideas that had been mostly ignored suddenly came to prominence: these different pre-existing elements were drawn together by Witten into a new picture of string theory and how the subject worked.
The membranes were also not confined to supergravity and were not confined to eleven dimensions. Membrane solutions had also existed in the ten-dimensional supergravity theories that were the classical limits of string theory. However, whether these solutions meant anything in string theory was highly unclear. Soon after Witten’s talk, Joe Polchinski of the University of Texas showed that membranes also played a crucial role in string theory (under the name of D-branes). Brane solutions had already been known within the supergravity theories that existed as classical limits of string theory. Polchinski and others had also studied the role of branes within string theory, where they appeared to be exotic objects with no simple classical description. Polchinski was able to join these descriptions up. He showed that the supergravity branes admitted an extremely simple description in string theory – and this description also allowed plentiful calculations to be done.
It was soon realised that as the coupling strength was dialled from weak to strong, the string states present at weak coupling turned into the brane states of strong coupling – and vice-versa. This implied a conceptual rethink, as fundamental’ string states morphed smoothly into brane states. What this showed was that, even in string theory, strings were ultimately no more fundamental than branes. The foundational nature of one-dimensional extended objects – strings – was only an illusion coming from
the most calculationally accessible regions.
Let us then return to the question of What is string theory?’ and ask how this would be viewed in 1996. The answer would have been that the five string theories that had been identified in the 1980s were all really different manifestations of a single underlying object. The underlying theory was called M-theory, but like a great desert it was known only from its boundaries. There were no known fundamental defining equations of M-theory, and strings entered M-theory only in certain limits. The differences between the string theories of the 1980s – the type I and type II strings, and the heterotic string – were like the differences between steam, water, ice and snow. While very different, they all arise from molecules of water. The different string theories were not fundamentally different theories – they were instead different limits of the same underlying object. Although converting an iceberg to superheated steam would require an enormous input of energy, it is in principle possible in a way that converting the iceberg to laughing gas is not. Likewise, although the energies required to move the universe from one phase’ of M-theory to another phase would be unimaginably large, it could in principle be done. Such a change would represent a dramatic alteration in the affairs of the universe, but it would not modify the fundamental laws under which it operated.