Why String Theory?

by Joseph Conlon

  What types of symmetries can exist? Mathematicians like to classify symmetries, and one of the most epic achievements in mathematics has been the classification of ‘all possible symmetries’. This is a theorem, technically known as the classification of finite groups, which states what all the possible allowed forms of symmetry are. This theorem was built up over decades through the work of hundreds of mathematicians, covering tens of thousands of pages across different mathematics journals.

  This theorem describes every possible form of symmetry that can arise. Some of these forms are well known, such as the rotational symmetries described here, while others are less familiar. However, almost all symmetries can be grouped into one of several large families of related symmetries. We are interested here in the small number of exceptions. These exceptional symmetries have remarkable properties. They are called sporadic symmetries, and they number twenty-six in all.

  The sporadic symmetries are not easy to grasp. One of their striking properties is that they are all large’. What is meant by large’? The symmetry of fourfold rotation is said to have order four, and the symmetry of sixfold rotations is said to have order six. The order describes how many independent ways there are of acting with the symmetry. On this count, even the smallest of the sporadic symmetries has an order of just under eight thousand. This already sounds quite impressive, until you realise that the largest of these symmetries has an order of approximately ten raised to the fifty-fourth power – a billion multiplied by itself six times. This symmetry is, for this gargantuan number of reasons, called the monster symmetry.

  How can one possibly understand a symmetry of this size? What can such a symmetry act on? What objects play the role for it that the square does for fourfold rotations? The same theorem tells us that the smallest interesting object that the monster symmetry acts on has 196 883 elements (the uninteresting object is a single element, taken to itself by every action of the symmetry). The action of the monster symmetry turns all these different elements into each another – but it is again hard to get a conceptual handle on anything with almost two hundred thousand components.

  The monster symmetry is a symmetry like no other. Its existence was first realised in the middle of the 1970s by the mathematicians Bernd Fischer and Robert Griess, and a few years later Griess was able to give an elaborate and explicit definition of the monster symmetry in terms of its actions on this object with 196 883 elements.

  As an unusual and large special case amidst the general taxonomy of symmetries, the monster symmetry might easily have remained only as an odd member of a mathematical curiosity cabinet. It did not, and the reason it did not starts with a numerological observation by the Canadian mathematician John McKay. McKay had originally worked on number theory, the branch of mathematics that deals with such questions as the nature and distribution of prime numbers. One function that appears within number theory is called Klein’s j-function. This function has been known for a long time and is named after Felix Klein, a nineteenth century German mathematician. It obeys many striking relationships, and for our purposes there is one particularly useful way of writing this function. This is as an expansion with a series of coefficients, the first of which are 1, 196 884 and 21 493 760.

  What was McKay’s flash of genius? He noted that

  196 884 = 196 883 + 1.

  Of course, this equation is trivially true. McKay’s hunch was that the closeness of these two numbers was indicative of a unifying structure connecting the monster symmetry to properties of the j-function. At first, this looked like it was as meaningful as a relationship between the value of π, the fine structure constant, and the volume of the Great Pyramid of Giza when measured in cubits. If you look at enough numbers from enough places, there will always be some odd coincidences that appear. The possibility that this actually represented a real connection – that seemed pure moonshine.

  It soon became clear that it was not. McKay told another mathematician John Thompson about his observation. Thompson was a mathematical giant who had already won the Fields Medal for previous work on the classification of symmetries. Thompson took it seriously, and discovered another coincidence. The next coefficient of the j-function was 21 493 760, while the second-smallest object that the monster symmetry can act on has a size 21 296 876. Thompson observed that

  21 493 760 = 21 296 876 + 196 883 + 1.

  It was not just the first coefficient of the j-function that had connections to the monster symmetry, but also the second one – and Thompson checked that there were similar relations that held for the next few coefficients as well. It was now clear that McKay’s observation was no fluke, and that there did exist a deep underlying relationship between these two disparate areas. These and other ‘coincidences’ were summarised in a 1979 paper by John Conway and Simon Norton entitled ‘Monstrous Moonshine’. The challenge was clear: explain why these relationships existed and where they came from.

  The person who ultimately solved this problem was Richard Borcherds. Borcherds is a British mathematician who has now moved to the United States. As is not uncommon for mathematicians, he was also a strong chess player as a child, before moving on to more productive intellectual pursuits. His work on this problem would see him awarded the Fields Medal. While the language of Borcherds’s proof was the language of mathematics, the ideas are interleaved with string theory and in part draw directly on key results from it. What follows is an unabashedly physics summary of this work.

  The string theory relevant for Borcherds’ work was the simplest string theory: the bosonic string. This was the first string theory to be constructed and has the smallest number of complications. The bosonic string theory is not viable as a description of the world – as was mentioned in chapter 5, it contains a tachyon particle that renders the theory unstable and unphysical. However, for mathematical applications this instability is of no concern.

  The bosonic string needs twenty-five spatial dimensions for consistency. If you twang it, there are twenty-four separate directions in which to pluck it – all dimensions except the one along the string. There are many ways to pluck the string, and the string has many harmonics. These harmonics can be counted. In the simplest case where all twenty-five spatial dimensions are identical, flat and infinite in extent, the first harmonic has a multiplicity of twenty-four, associated to all possible one-pluck notes. The second harmonic can be obtained either through simultaneously plucking the string along two directions, or by giving it an extra-hard double pluck’ in one direction. These two possibilities can be done in many more ways, and so the multiplicity of the second harmonic is greatly increased. This is a verbal description, but mathematical formulae can easily be given to count the multiplicity of harmonics up to arbitrarily high levels.

  The functions that count the harmonics also make appearances in number theory. The precise function that appears in any one case, however, depends on the geometry of the spatial dimensions. Unsurprisingly, the ways that strings can be plucked is affected by the geometry of the space that they live in. For the more conventional string theories these functions have names such as the Dedekind eta function or the Jacobi theta function – classical functions from nineteenth-century number theory.

  Where does monster symmetry come in? We are already familiar with the fact that spatial geometries can have symmetries – for example, the square has fourfold symmetry. If the geometry of string theory’s extra dimensions involved a square, that fourfold geometric symmetry would also be a symmetry of the ways that strings could be plucked. Given a single way of plucking a string, a fourfold rotational symmetry would automatically generate three more. The symmetry of the geometry would be inherited by the symmetry of the strings, and a count of the multiplicity of harmonics would have to respect that symmetry.

  The key idea of Borcherds was to identify a geometric space that had the monster symmetry as its symmetry, and then use the geometry of this space for string theory’s extra-dimensional geometry. The space Borcherds used w
as a variation on a particular twenty-four-dimensional space called the Leech lattice, which had previously been recognised as having connections to the monster symmetry. Instead of doing bosonic string theory on the simplest spaces, Borcherds was doing bosonic string theory on this bizarre Leech geometry – but a bizarre geometry that was governed by the monster symmetry.

  The fact that the monster symmetry applied to the geometry meant that the monster symmetry also applied to strings forced to vibrate within it. A string governed by a symmetry has to dance according to its tune, and when the symmetry is the monster symmetry the tune is that of the monster. Any way a string vibrated, the symmetries of the monster could turn the string into one vibrating another way. The rules of the monster constrained the allowed vibrations of the string – they had to come in groups of 1, or 196 883, or 21 296 876, or larger. The mathematics of monster symmetry allowed no other options. It was here that the real power of the symmetry entered, by enforcing these highly distinctive numerical groupings of the allowed vibrations of the string.

  Borcherds was also able to determine the function that counted the string harmonics for this geometry. In the simplest versions of the bosonic string, this function had been the Dedekind eta function. For Borcherds’ carefully chosen geometry, it was now the Klein j-function. Written as an expansion, each progressively higher coefficient of the j-function was now counting the multiplicity of the string harmonics. The coefficient 196 884 was the number of realisations of the ‘first harmonic’, the coefficient 21 493 760 the number of realisations of the ‘second harmonic’ – and so on. This gave a ‘physics’ meaning to the j-function: it was counting harmonics of the bosonic string on the Leech lattice. The fact that these coefficients could all be expressed as neat sums of the sizes of monstrous objects reflected the monster symmetry that acted on the geometry. The harmonics had to arrange themselves in this way, since the oscillating strings were forced to come in symmetry groupings. As the applicable symmetry was the monster symmetry, the sizes of these groupings had to be those of the monster.

  Through this route – although expressed in a far more mathematical language! – Richard Borcherds was able to prove the statements made in the monstrous moonshine conjectures, and for this work he was awarded the Fields Medal at the 1998 International Congress of Mathematicians in Berlin. For each award, there is a Laudation – a formal speech praising the winner and describing their work. As part of his proof, Borcherds had made crucial use of the No-Ghost theorem from string theory, which was mentioned in chapter 5 and had been proved by Peter Goddard and Charles Thorn in 1973. As a result, it was Peter Goddard who was invited to given the Laudation. The invitation left Goddard feeling ‘enormously flattered, but faced [with] the most formidable challenge’. The language of formal mathematics is often difficult even for the best physicists.

  This story of monstrous moonshine is then an example in which string theory provided some of the stones for a bridge connecting two disparate areas of mathematics. It is not yet a finished story. In recent years further examples of moonshine have been found, which connect some of the other sporadic symmetries that are tamer versions of the monster symmetry to different number-theoretic functions. These examples are not yet understood, but it is expected that it is string theory on a particular special geometry that will explain the connections. Teams of physicists and mathematicians are even now working together to uncover the string theories that can explain these links.

  9.3 MIRROR SYMMETRY

  The first mathematics we all encounter is that of counting:

  One potato, two potato, three potato, four;

  Five potato, six potato, seven potato, more!

  I have an uncle who is a professional mathematician. When I was a very small child, in that I had any sense of his work, I thought it involved counting really, really large numbers. As I grew slightly older, I matured in my views. No longer did I think Uncle Joe’s work simply involved counting. I was now at school, and learning to multiply two-digit numbers together. What he did, I now imagined, would be the grown-up version: multiplying twenty digit numbers together, and similarly performing by hand labyrinthine long division calculations to many decimal places. If primary school mathematics was about sums, grownup mathematics would be about hard sums.

  As a child I was both wrong and right. Of course, in many ways I was wrong and immature. Real mathematics involves deep structures and arguments held together by careful chains of reasoning. While some parts of it, such as the proof that there is no largest prime number,5 are accessible to children, most is, like the drinks cabinet, out of the sight of young eyes. However, I was not entirely wrong. Some parts of grown-up mathematics – although not in fact the area my uncle worked on – do involve counting. They involve working out the total number of ways that something can happen. For example, how many ways are there of writing a whole number as a sum of smaller numbers? Two can be written as either two or one plus one. Three can be written as three, two plus one, or one plus one plus one. How does the possible number of sums grow as our original number gets very large? This is a counting problem, and it is the simplest form of many such related problems in mathematics.

  The theory of numbers is one area where counting problems appear in mathematics. Another area is in geometry. While mathematicians’ usage of the word ‘geometry’ can be broad, I will confine its meaning here to that of regular spaces. Although they may have two, three, four, five or six dimensions, these spaces are just fancy generalisations of familiar circles, spheres and ellipsoids.

  What do geometers count? One answer is curves – how many different curves are there in a geometry? Equivalently, how many ways are there to wrap an elastic band?

  What does this mean? As a more precise way of formulating this problem, we imagine an elastic band – or a string, or indeed any object with tension – that is confined to a surface and is not allowed to move off the surface. The surface could be the surface of a sphere, a cylinder or a bagel (more technically known as a torus). We imagine placing the elastic band on the surface, and asking what can happen to it. The number of possible results depends on the surface – and this is what mathematicians count. For a sphere, the band always shrinks down under tension to zero size. It has nothing for it to wrap itself around and so can never prevent itself from collapsing. For a cylinder, the band can however wrap around the cylinder – either once, or two, three or four (or more) times. Provided the band has to remain on the surface and cannot be slipped off, there is no way to untangle these wrappings. For the bagel (or torus), there are now actually two distinct ways of wrapping the band, one for each of the circles present in the bagel. Each of these circles may again be wrapped either once or many times.

  This counting problem has different answers for spheres and tori. The answer also remains the same however one kneads or moulds the surface. If a sphere is deformed into an ellipsoid, the elastic band can still shrink itself to zero size. However a band wrapped around a cylinder remains wrapped – and cannot be unwrapped, even if we squeeze the cylinder. This is useful information, as it shows that this counting problem provides a way of distinguishing surfaces from one another. As the answer does not depend on how one deforms the surface, it depends only on the topology of the surface – and therefore surfaces for which the count is different are topologically distinct from each other.

  I have introduced this geometric counting problem with one-dimensional elastic bands. More generally, this mathematics is the mathematics of counting the number of possible ways that a circle, or the surface of a sphere, or the surface of a hyper-sphere, can fit into geometries with additional dimensions. For any given geometry, one can ask how many ‘different’ ways there are to put one of these objects in the geometry. This branch of mathematics is called homology, and in a loose sense it counts holes in surfaces.

  It is interesting because it provides a way of distinguishing different geometries and quantifying their difference. As the geometries grow more complicat
ed, so do the techniques. While finding the possible ways of putting an elastic band around a circle may not be taxing, counting the number of ways of embedding a sphere in a six-dimensional Calabi-Yau geometry requires considerable thought and training.

  Calabi-Yau spaces were first mentioned in chapter 5. They are a type of geometric space that can have any even number of dimensions. They arose in the 1970s as a type of geometry of interest to mathematicians, and in particular to those who were algebraic geometers. As mentioned in chapter 5, it was realised in 1985 that six-dimensional Calabi-Yau spaces are promising starting points for particle physics models descending from string theory. Provided the additional dimensions of heterotic string theory are curled up into a Calabi-Yau space, the resulting four-dimensional theory has rough similarities to possible extensions of the Standard Model. Indeed, as seen in chapter 5 it was this result that first led to applications of string theory to particle physics.

  Before this paper, not one physicist anywhere in the world cared one whit about Calabi-Yau geometries and their properties. After it, many did. One of its key results was that the particle content in four dimensions depended on the geometric properties of the Calabi-Yau space – and in particular on the number of possible ways that spheres or hyper-spheres could fit into it. For the simplest class of models, the number of particle generations was determined by the difference between the number of ways of embedding two-dimensional surfaces and the number of ways of embedding three-dimensional surfaces. While in the end this feature does not seem to give useful insight concerning the presence of three generations in the Standard Model, it does explain why Calabi-Yau geometries came under intense investigation by a community trained as physicists – who suddenly cared about counting the number of ways to embed surfaces into such geometries.