May the forces be with us?

Hugh Aldersley-Williams reports on a conference offering fresh clues to a grand unified theory

How often do you find an eminent name such as Stephen Hawking going on a crash course in maths? Not often - but that's what has been happening for the past four weeks, and continues for the next two, at the Isaac Newton Institute for Mathematical Sciences in Cambridge.

Professor Hawking is just one of a group of physicists and mathematicians attending a conference with the daunting title of "Four-dimensional geometry and quantum field theory". But they aren't doing it out of casual interest. It could be a key step on the way to an idea which eluded Albert Einstein: a grand unified theory, joining the four forces - the strong and weak nuclear forces, gravity and magnetism.

If a "GUT" (as it is known) does emerge from all this, then an important part will have been played by an Oxford mathematician who is almost unknown outside his field: Simon Donaldson.

For years mathematics, and especially topology - the study of the relations between forms - has tended to drift apart from physics and other sciences. But now they are coming together again, due in part to Professor Donaldson's work at Oxford's Mathematical Institute. In 1983, before he had even obtained his doctorate, Donaldson discovered certain "exotic" phenomena unique to four-dimensional space. Unusually for a pure mathematician, Donaldson made his discovery with the aid of ideas from theoretical physics.

He took equations from physics describing the behaviour of fundamental particles and applied them to mathematics, specif-ically in four-dimensional topology. "What he did was to use these ideas to establish a result in pure mathematics which was totally unexpected," says Sir Roger Penrose, the mathematician and author. "Unique among the dimensions, four is the only one with this property." The results are described below; but what has been almost more interesting has been their effect.

Donaldson's findings are now filtering through to cosmology and quantum theory - where four dimensions are significant, being the three of space, plus time. One way of describing gravity, for example, is to say that it produces local distortions in the geometry of space-time. If four-dimensional geometry is "badly behaved", as Donaldson's work indicates, it makes life more difficult for physicists - but also more interesting.

Recently, Nat Seiberg and Edward Witten of the Institute for Advanced Studies at Princeton have kneaded Donaldson's equations into quantum physics, to produce a workable topological quantum field theory for four dimensions. In so doing, they showed that the idea of physical duality, which relates electricity and magnetism, could be extended to the weak and strong forces between fundamental particles.

Links like that are important in building towards a grand unified theory, which has evaded physicists and mathematicians for decades. But the Princeton result gave the impetus for the conference now in progress.

So what does its title, "Four-dimensional geometry and quantum field theory", mean? Quantum field theory seeks to explain the interrelations of fundamental particles. On occasion, experiment has borne out theories for which the mathematics was lacking. But Seiberg and Witten have turned this process on its head with a mathematical "experiment" that seems to produce the right results. "We don't really know rigorously that it's OK to make these substitutions," says Donaldson. "In a sense, mathematics is the laboratory for these things."

The significance of this work to mathematics, and now to physics, has been duly recognised. Mathematics has no Nobel prize, but Donaldson has more than made up the difference - winning both its substitute, the Fields Medal, in 1986, and the Crafoord Prize of the Royal Swedish Academy of Sciences in 1994. Witten won the Fields Medal the next time it was awarded, in 1990. Mathematicians typically do their best work while young, and Donaldson, aged 39, calls himself (with characteristic modesty) "now rather a has-been".

"His work is as pure mathematical as you can get," says Sir Michael Atiyah, past president of the Royal Society, and Donaldson's research supervisor at the time of his breakthrough. "But it draws on techniques in theoretical physics and it impinges back on theoretical physics. [Sir Isaac] Newton's work impinged in the same way."

Donaldson describes how it happened with a sense of the history of his subject rarely found among experimental scientists. A century ago, Henri Poincare launched the study of manifolds - curves and surfaces and their analogues in higher dimensions. Not long before, James Clerk Maxwell had revealed the symmetries describing the respective actions of magnetic and electric forces - in other words, their duality. Both areas have since seen much progress, but along largely separate tracks.

"My own work was really to bring together these two strands," says Donaldson. The conjecture that duality might be able to be extended to cover other forces dates back 20 years. "But two years ago, this idea was resurrected and souped up, and it is this that has led to dramatic advances in quantum field theory," he says. "One aspect has impinged on my work [in topology]. Other aspects have physical merit."

Surfaces and other manifolds may be grouped according to the properties that they share. In three dimensions, a sphere and a cube are topologically equivalent. A doughnut is different, because it has a hole in it.

In five and higher dimensions, forms that are otherwise equivalent are no longer interchangeable. Here, their surface quality - smooth like a sphere, or sharp like a cube - must be taken into account. Mathematicians have nevertheless been able to reduce topological puzzles in higher dimensions to problems of (comparatively) simple algebra.

But what about the intervening realm of four dimensions? Donaldson showed that the quality of "smoothness" in fact matters more than in any other dimension. Not only that, but the whole idea of topological equivalence, based on counting holes, was suddenly invalid. Even in the simplest cases, it turns out that there are infinitely many four-manifolds which - although equivalent according to classical topology - Donaldson revealed as being very different in terms of their smooth structure. What's more, there is no way of telling whether a smooth manifold will remain smooth or become "rough", or vice versa, under topological transformation. Four-manifolds have a specialness, rather like knots in three dimensions. You can't have a two- or a four-dimensional knot; and four-manifolds have properties not found in other dimensions.

If it is any consolation, Donaldson, too, finds it counterintuitive that the four-dimensional picture should be more complicated than not only the lower dimensions, but also the higher ones. Is it coincidental that this anomalous world seems to be our own? This question may not be on the agenda at the Newton Institute, but it is likely to be at the back of many of the minds meeting there.

An introduction to topological ideas, with animated movies, can be found on the World Wide Web at http://geom.umn.edu/