Calabi-Yau Manifolds and Particle Physics

Superstring theory is only consistent when it is critical which requires 10 space-time dimensions (the non supersymmetric bosonic version requires 26 dimensions to be consistent).  This is regarded not as a curse but as an opportunity.  String theory can give rise to theories in four space-time dimensions by taking string theories in a 10 dimensional space in which four of the dimensions are those of ordinary space-time and 6 are those of a 6 dimensional manifold that is understood as being extremely small, of the order of the Planck length, 10^{-35}m, 20 orders of magnitude smaller than a proton.   This is known as compactification and this tiny 6 dimensional manifold provides the internal degrees of freedom required to explain particle physics.  

The idea of compactification goes back to 1926, in the work of Kaluza and Klein. Suppose that there is a fifth dimension of space, which is periodic - that is, if you move some distance in this fifth direction, you arrive back where you started. Topologically, this extra dimension is then a circle. One can apply Fourier analysis, as studied for example in undergraduate mathematics courses, to the dependence of all fields on this direction. One finds that the zero modes - those modes that do not depend on the fifth direction at all - behave as fields in four spacetime dimensions, while the infinite set of higher Fourier modes all appear as massive particles in four dimensions: the smaller the circle, the larger the mass. Very massive particles tend to decay very quickly into smaller mass particles, and the upshot is that the only way to "see" that there is a very small extra dimension would be to produce its massive Fourier excitations in a particle accelerator. There are some (more complicated) models of extra dimensions which predict that such particles really will be observed in the Large Hadron Collider at CERN!

In string theory there are six extra spatial dimensions, not one. We then suppose that these form a very small, compact six-dimensional space. By similar reasoning to that above, these dimensions will be very difficult to observe directly. The equations of string theory imply that this six-dimensional space has to be curved in a very particular way, and spaces satisfying these conditions are called Calabi-Yau manifolds. (In fact this is an oversimplification: there is a higher-dimensional analogue of the electromagnetic field in string theory, and Calabi-Yau manifolds solve the string equations only when this field is zero.) Calabi-Yau manifolds have also been studied intensively by mathematicians, in particular Shing-Tung Yau who proved that they exist! (Yau won his Fields Medal in part for this work.)

The theory of strings on Calabi-Yau manifolds was first initiated by Philip Candelas, in collaboration with Horowitz, Strominger and Witten. This has grown into a rich subject, with an intricate interplay between the geometric and topological properties of Calabi-Yau manifolds and particle physics in four dimensions. Indeed, one of the remarkable features of string theory is that it naturally includes the correct ingredients for particle physics, as well as gravity. One finds that different Calabi-Yau manifolds, with different topological shapes, lead to different models of particle physics in four spacetime dimensions. For example, in the simplest models, the number of generations of elementary particles (three in the Standard Model) is related to the Euler number of the Calabi-Yau manifold.

Oxford is one of the world's leading centres for research in Calabi-Yau compactification. The figure on the right hand side is a plot of the known Calabi-Yau manifolds. These are convenArrayiently classified by their Hodge numbers, the sum and difference of which are plotted vertically and horizontally, which measure certain topological properties of the manifolds. Notice how sparsely populated the "tip" of the distribution is. The Oxford group of Philip Candelas, Xenia de la Ossa, Yang-Hui He (now at City University), and Balazs Szendroi recently published a paper explaining the special relationship between this corner of the distribution of Calabi-Yau manifolds and three-generation models of particle physics. Candelas and Rhys Davies recently published papers with constuctions of new Calabi-Yau manifolds with small Hodge numbers, and corresponding new three-generation models. The group has also been working on how the space of all Calabi-Yau manifolds is connected. One of the many remarkable features of string theory is that spacetime itself can change topology in a smooth manner - something which by definition is impossible in classical General Relativity. It has been conjectured by the Warwick mathematician Miles Reid that the space of all Calabi-Yaus is connected by such topology-changing transitions. In a different direction, Andre Lukas has been working with Oxford postdocs and D.Phil students on new constructions of Calabi-Yau particle physics models using very sophisticated methods in algebraic geometry.

The distribution plot of Calabi-Yau manifolds is symmetric about the vertical axis. This is a manifestation of mirror symmetry. Again, members of the Oxford group have made key contributions to this subject. String theory predicts that for every Calabi-Yau manifold, there should be an associated different Calabi-Yau manifold which is its mirror. Although this sounds simple, in fact it is an extremely deep property of Calabi-Yau manifolds, which has led to entirely new areas of mathematics. The Homological Mirror Symmetry Group, which is part of the Geometry Group at Oxford, is entirely devoted to one aspect of this mirror symmetry. Candelas and de la Ossa, in collaboration with Green and Parkes, gave the first predictions for numbers of rational curves in the so-called quintic Calabi-Yau three-fold via a computation in its mirror. The image at the top of this page is a three-dimensional projection of a section of a quintic Calabi-Yau manifold.