Geometry and string theory

THERE is something almost mystical about the connection between geometry and the laws of nature. An encounter at the seashore with the perfection of a logarithmic spiral in a seashell can be a source of wonder. A wonder which is multiplied when the self-same spirals appear in telescopic images of distant galaxies. That Nature is, at various levels, amenable to a geometrical understanding has been a rich theme in science. Though little more than a belief, it has consistently yielded beautiful insights. But it is perhaps fair to say that at no time in the past has geometry played as profound a role as it currently does in string theory’s endeavours to understand quantum gravity. This should not be too surprising. Since the time when Einstein taught us that gravity is geometry, we have learnt to think of space–time not as a passive arena for events, but rather as a dynamical entity. Hence any theory of quantum gravity such as string theory must describe the quantum fluctuations of geometry. Much of the elusiveness of a consistent quantum theory of gravity has to do with this notion of describing quantum fluctuations of geometry. Recall that in the quantum mechanics of particles, instead of classical trajectories from point A to point B, one explores all possible paths from A to B, each weighted by some amplitude. Amplitudes for paths far from the classical one are suppressed if their action is large compared to Planck’s constant (h) (note 1). In this regime, a nearly classical (‘semi-classical’) description can be applied. However, when the action is comparable to Planck’s constant, then trajectories far from the classical contribute equally. Such quantum trajectories are by and large far from being smooth – they are more like the paths in Brownian motion (see Figure 1). The situation is very analogous with gravity. Einstein’s equations of gravity determine the classical geometry of space–time, just as Newton’s laws fix particle trajectories. This is a good approximation for macroscopic systems of the kind we are accustomed to. However, again quantum fluctuations are important when the action is of order h. For gravity, this is true at distance scales set by Newton’s constant GN, namely lp = (hGN/c ) ≈ 10 cm (called the Planck length scale) or equivalently at characteristic energies which are enormously large Ep = (hc/lp) ≈ 10 eV (the Planck energy scale). At these scales, which would only have been encountered in the very hot, very early universe, one must come to grips with what fluctuations in geometry mean. As for trajectories, one expects contributions from geometries which are wild and Brownian, so to say. In other words, we no longer have the reassurance of a fixed (classical) background space–time and moreover the majority of the many space–times that enter into a quantum process are far from the smooth (or near-smooth) manifolds conventionally studied by mathematicians. How are we to describe these weird geometries? Are there geometrical structures that can replace our conventional notions of space–time? Or perhaps geometry emerges only as an approximate notion at distances large compared to the Planck length. Though we are still far from having all the answers, string theory, in its attempt to address these issues, has given us some inklings as to what to expect, while also expanding our notion of geometry. Geometry has also played a central role in string theory’s attempts to understand other questions of particle physics like the number of generations, the origin of gauge symmetries, the hierarchy between the scale of the standard model and that of gravity. At its most ambitious, it could perhaps even explain the number of space–time dimensions we live in. And, almost as an aside, it has had fruitful interaction with mathematics leading to a renewal of the traditional links between the two subjects. In this article I will try to convey something about this varied interplay between geometry and string theory. We will walk through several layers of this interplay, proceeding from the relatively well-understood towards the more novel contours of our picture of quantum gravity that is still emerging. More operatively, we