Why String Theory?

by Joseph Conlon

  Mathematicians tend to be abstract. Physicists like examples. One of the first activities of these physicists was to attempt to construct as many Calabi-Yau geometries as possible. In collaboration with mathematicians, new techniques were sought to increase the number of known Calabi-Yau spaces. With these techniques, the number of such known spaces grew first from a few to tens, then from tens to hundreds and then from hundreds to thousands. As the numbers grew, it became possible to examine the properties of this large collection of geometries.

  As this was done, an interesting property was encountered. In technical notation, the numbers used to count the ways of embedding two-dimensional and three-dimensional surfaces are called h1,1 and h2,1. These numbers can reach from unity up to values of almost a thousand. However, a curious pairing was noticed. For example, there was a space for which h1,1 was one and h2,1 was one hundred and one. However there was another space for which h1,1 was one hundred and one and h2,1 was one. As more and more Calabi-Yaus were enumerated, it was noticed that this pairing structure was ubiquitous. If all known values of h1,1 and h2,1 were plotted on a plane, the result looked as if it had been reflected about a central mirror. For every Calabi-Yau space, there appeared to be another mirror partner space for which the values of h1,1 and h2,1 were interchanged.

  These mirror pairs exhibited a further striking feature. It was realised that the equations obtained from starting with type IIA string theory, and compactifying it on one Calabi-Yau space, were exactly identical to the equations obtained from starting with type IIB string theory and compactifying it on the mirror partner of the original space. This was another example of a duality – one theory on one geometry was identical to a second theory on a different geometry.

  It is this last fact that led to the reason why mirror symmetry became of such interest to mathematicians. As with other dualities, it is also true here that quantities that are hard to compute on one side of the duality become easy to compute on the other. Quantities that were hard to compute for IIA string theory on one Calabi-Yau were easy’ to compute for IIB string theory on the mirror space. In particular, it was realised that the easy’ classical computations on one side were equivalent to full quantum calculations on the other side, including both perturbative and non-perturbative effects.

  These full quantum computations included quantities of clear mathematical interest. In particular, they contained a count of what can be roughly described as the number of ways of writing curves within the Calabi-Yau space. This is a similar but distinct quantity to the number of ways, alluded to above, of fitting surfaces and hyper-surfaces within such a space. For the simplest such curves, said to be of degree one, this count had been performed in the nineteenth century. For more complicated curves of degree two and degree three, the count had only been performed more recently – but the techniques were complicated.

  What was exciting about mirror symmetry was that the string theory calculations provided an expression for the number of curves – not just of degree one and two, but of degree four, five, six and higher. In one go, it produced an expression that purported to count the number of curves in a Calabi-Yau space up to arbitrarily high degree. Philip Candelas, then of Texas and now of Oxford, presented these results to a conference of mathematicians in Berkeley in 1991. For the simplest example the numbers went as follows:

  Degree

  Number of curves

  1

  2 875

  2

  609 250

  3

  317 206 375

  4

  242 467 530 000

  5

  229 305 888 887 625

  While the results agreed with the existing mathematical results for curves of degree one and two, they disagreed for curves of degree three. For a few weeks there was an impasse – but then the mathematicians who had obtained this result rechecked their codes and found a bug. The new result agreed precisely with the physical result presented by Candelas.

  This made it clear that string theory was both able to say deep things about Calabi-Yau spaces and also able to obtain results about them that were inaccessible to conventional mathematical techniques. This was exciting both to mathematicians and physicists – and by now has led to an enormous amount of work in this area, going far beyond these original results. Mathematicians started learning string theory and trying to understand its tools and methods – because it was able to tell them new results about geometric spaces they were interested in.

  9.4 CULTS IN PHYSICS

  When considering the influence of string theory and related ideas on mathematics, there is one name that comes up more than any other. That name is Edward Witten of the Institute for Advanced Study in Princeton. Witten has become an icon of the subject, despite only settling on a career in physics after a meandering start, including an undergraduate major in history, a foray into journalism and dropping out of an economics doctoral program after one term. He went to Princeton and started as a graduate student of David Gross. He was in fact the second graduate student of Gross, the first being Frank Wilczek, whose Nobel Prize-winning doctoral work featured in the introductory chapter.6 Witten was thus one of the first members of that generation whose entire careers have been subsequent to the construction of the Standard Model.

  Witten has made numerous contributions to quantum field theory, general relativity, string theory and mathematics. Witten’s contributions to modern mathematics have been sufficiently numerous that he was awarded the Fields Medal in 1990 – with the Laudation given by Michael Atiyah – thereby becoming the only physicist so far to win mathematics’ highest award. Various results led to this prize. He had shown how the theory of knots – what are the different knots you can make with a piece of string? – is related to quantum field theory, and how some of the interesting properties of knots can be computed using quantum field theory. He had provided a remarkably simple proof of the fact that any gravitational system has positive energy. He had pioneered an entirely novel approach to Morse theory, a branch of topology – and all these were only the achievements in his second subject.

  When I was a student, there was almost a cult about Witten. The name was mentioned with slightly hushed tones. He was someone on a different plane, perhaps even from a different planet. Students who had attended summer schools at Princeton would return and report on exactly what he had said when asked questions in his talk, and the precise words he used in answering. It was joked, only partly in jest, that he was a member of a superior alien species masquerading as a human. The unusual delivery for such a tall man, with a high-pitched, slightly squeaky voice – an oracle speaking as a countertenor – only reinforced the sense of difference. He had the ability to make, or break, an area. What he worked on was automatically fashionable – if he thought something was important, it was. There were many other smart people in the subject, but there was only one Big Ed.

  The collective memory contained stories of scientists who had made their careers during the 1980s by chasing after anything Witten had written. In physics, research articles are circulated in preliminary form prior to publication, originally through paper copies and later via electronic mail. Once such a preprint’ appeared with Witten’s name on it, these scientists would read it, digest it, and then quickly write a follow-up paper on the same topic – take what Witten had done, make a small perturbation, and re-solve for a different but closely related example. The final step was to send the hurried composition off to a journal; and then await the next preprint to repeat the same trick.

  How did this arise? Why was one man held in such awe? Part of the answer is certainly Witten’s prodigious talents and ability. However these talents also had a rich field to harvest. As mentioned previously, the particle physics from 1945 to the middle of the 1970s was not opposed to mathematics, but it was also not really interested in mathematics. The development of the Standard Model had stimulated great advances in the tools and theories available to particle physics. The great names of the age
cared more, though, about pursuing with vigour experimental data and its possible explanations. A full understanding of the deep structure of the equations they came up with was not what truly excited them.

  What this meant is that by the late 1970s the area at the interface of mathematics and physics lay entirely ready to be exploited by someone with the right set of talents. Edward Witten was exactly that person. He had the ideal combination of abilities. Few physicists have a similar command of mathematics, and few mathematicians can match his deep physical insight. Witten has been extraordinarily productive at the area at the boundary between mathematics and physics. He was someone who was enormously gifted – and also someone whose talents peaked in an area that had lain fallow for decades and was now fertile soil for those with the ability to cultivate it. In all areas of science, the first to the orchard gathers abundant fruit from the ground and from the low-lying branches, while those who arrive later need elaborate ladders for much smaller pickings.

  Witten’s achievements and influence have been such that for a long time he represented the model of how to do string theory, and even to a certain extent the model of how to do theoretical particle physics. Sheldon Glashow, one of the architects of the Standard Model, a 1979 Nobel Prize winner, and a trenchant opponent of string theory throughout the 1980s and 1990s, wrote a poem that ended,

  Please heed our advice that you too are not smitten –

  The book is not finished, the last word is not Witten.

  Glashow had attempted to keep string theorists out of Harvard, where he was a professor. He did not succeed; he left Harvard and has now moved to nearby Boston University.

  Cults have their dangers. The dangers of a scientific cult is that the model of the ‘ideal scientist’ can remain the same even while the science shifts, and the skills required for the next big breakthrough change. A previous cult had existed around Richard Feynman – the fun-loving, safe-cracking, girl-seducing Dick Feynman described in ‘Surely You’re Joking, Mr Feynman?’. The Feynman cult still exists in attenuated form today, and I see it every year in the personal statements dutifully compiled by British teenagers seeking entrance to university. To many, Feynman is still the model – if not of how to behave, at least of how to do physics. Feynman was indeed a great physical scientist – but he was also one whose highly intuitive approach and style rendered it impossible for him to make any of the more mathematically oriented breakthroughs of the 1970s and 1980s. It is not just that Feynman did not happen to be the physicist who got these results. He could not. His scientific style made him constitutionally incapable of it.

  Feynman may or may not have been a genius, but if everyone tried to be like him, none of these results would have been found. The same is true of Witten – not every interesting problem belongs at the intersection of mathematics and physics. Different skills are needed for different questions, and no one ever became a principal dancer at the Royal Ballet through years of practising the oboe.

  What is the summary of this chapter? The summary is that there are many scientists – some would argue too many – for whom time spent on string theory is time spent for the greater glory of mathematics. These scholars fall into several groups. There are those whose training is in physics, and who in a long-distant past did indeed torture the apparatus in laboratory courses. They use physical reasoning and argumentation to provide, for what are essentially problems in mathematics, novel lines of attack and fresh calculational methods. The benefits of looking at a problem from a new angle are clear, as it reshuffles what is ‘obvious’ and what is ‘hard’.

  There are also those who are trained and work as mathematicians. They may take the intuitive style of physical argumentation and toughen it, by alloying to it the rigour of professional mathematics. They seek to understand, in their own way and according to their own discipline, the underlying mathematical structures that enables physical reasoning to find the right answer to mathematics problems.

  There is another, slightly looser, class of mathematician. These are those who might use string theory or its structures as a kind of muse, but who are not directly engaged with it. For all scientists there exists an intellectual penumbra of ideas and topics that one knows loosely, and watches, but does not work on actively. Such mathematicians have no professional interest in quantum gravity, and even less in the problems of the Standard Model, which they would be hard pushed to define. However they do care about the mathematics of Kac-Moody algebras, and techniques to give information about the geometry of Calabi-Yau spaces. For such people stringy mathematics is, even if they do not work on it directly, a part of their wider scientific culture.

  The style of work discussed in this chapter has been called ‘physical mathematics’ by Greg Moore, a physicist-cum-mathematician at Rutgers University in New Jersey. The problems to be solved are ultimately mathematical in nature – the answers are not contingent on any observation or experiment. However the tools used to attack these problems were all forged within theories devised for the purpose of studying nature.

  It should be noted that in ‘physical mathematics’, ‘physical’ qualifies ‘mathematics’ – and not the other way round. Such researchers will be happy if the results they discover are eventually relevant to nature. The satisfaction they feel, though, is complete in itself. Their pleasure in their research is not tainted by any unfulfilled yearning for messy data or experimental discovery. Understanding our world is not the province of mathematics, and it is not the metric by which mathematicians rank each other. Mathematics is done for the sake of mathematics and not for utilitarian purposes. The mathematician’s mathematician G. H. Hardy famously wrote in a A Mathematician’s Apology,7

  I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.8

  For Hardy and many others, the value of their subject is not to be found in its ability to address anyone else’s concerns.

  Scientists with a career work on a combination of what others find interesting, what they find interesting and what they can make progress on. Happiness occurs when all three coincide. For certain areas of mathematics and certain flavours of mathematician, string theory has been the cause of this coincidence.

  1As the waggish statement goes, physicists used to be both smarter and more arrogant than biologists – now they are just smarter.

  2This does not stop people from trying to disprove these statements. I was a graduate student in the maths department in Cambridge, which maintained a pigeonhole for the many such crank proofs that got sent in. Anyone was welcome to read, and indeed reply, to the letters in the pigeonhole. One was particularly touching. The envelope was rich to the touch, and inside there was fine craft paper, beautifully and neatly covered with precise writing in an East Asian script. I could read not a word of it – but I knew what the carefully drawn diagram involving a circle and a square meant, and this showed that the neat writing was beyond redemption.

  3Cauchy was born one month after the French Revolution and over the next seventy years lived through all the subsequent vicissitudes of France. One of the greats, his eight hundred papers covered all of mathematics and mathematical physics. Catholic and royalist, he had little sympathy towards the revolutionaries and their successors, who had little sympathy for him, leading to him being rejected for the mathematics professorship at the Collège de France in favour of the serial kleptomaniac and mathematical non-entity Guglielmo Libri.

  4Technically, these are called instantons or sphalerons.

  5To see this, suppose that there was actually a largest prime number and thus a finite set of prime numbers. One can consider the number made by multiplying all of the prime numbers together and then adding one. A little thought convinces you that this is also a prime number, contradicting the original assumptions – and therefore the supposition that there is a largest prime number is mistaken.

  6With two such predece
ssors, one can only feel sympathy for the third graduate student of Gross!

  7Hardy was for eleven years a fellow of the same Oxford college – New College – of which I am a member, living only a few metres from where I type this. He was a more than usually reserved member of that generation of British men for whom the use of first names represented almost carnal knowledge. His contemporary, the economist Lionel Robbins, described him then as ‘No depiction known to me of a saint receiving the stigmata shows greater intensity than did Hardy’s features when plunged into meditation. Nor is it easier to conceive a more vivid exhibition of the meaning of the word ‘illumination’ than was afforded by the same features lit up by the play of wit or intent in kindly badinage. And nothing in the man belied the appearance.’

  8Despite all his good intentions, Hardy’s work on number theory now has numerous applications within cryptography.

  CHAPTER 10

  Why Strings? Cosmology and Particle Physics

  We have seen in the last two chapters various reasons to care about string theory. However, none of these reasons have involved the idea that string theory makes statements that are really true about this world at the smallest possible distances. What makes string theory interesting to those who care about the deepest laws of this world?

  This is the purpose of this chapter. I aim to explain how string theory can connect to known physics while suggesting novel ideas for going beyond it. There are several ways to approach this, but as this is what I work on, in this chapter I shall take the indulgence of writing about my own research.

  10.1 EXTRA DIMENSIONS AND MODULI

  String theory is famous as a theory of gravity, and in particular as a theory of quantum gravity. However we have also seen that to string theory, it is gravity in ten dimensions – and not four – that is the favoured first-born child. In addition to the time dimension, the familiar three spatial dimensions are extended by a further six dimensions.1 These dimensional appendages are intrinsic features of the theory – the form of gravity string theory gives us is, of necessity, one involving extra spatial dimensions.