Geometrical Approaches to Particle Phenomenology: from String Compactification to Supersymmetric Gauge Theories

The eyes of the physics community are turning toward Geneva. In about a year or, the largest particle accelerator and indeed the largest machine known to man, will be turned on. Its purpose, is to smash particles at such a speed and energy that we would be taking a glimpse at the sub-atomic world in a hitherto unfathomed clarity. Data will stream in at an alarming rate. Are we prepared? In a way, we have been preparing for two decades. Since a golden era of particle physics in the 60's nad 70's, what is known as the Standard Model (SM) has been measured and tested to breathtaking accuracy in its description of the microcosm of elementary particles. However, there is a catch. The SM has a number of arbitrary parameters which hint at a more natural, unified theory. More seriously, the SM, in all its glory, has encountered uncurable problems in being compatible with the theory of gravity, the force responsible for the macrocosmic world. Is there a unified theory? Albert Einstein, with prophetic vision, had spent the last of his years trying desperately to find this theory. In comes String Theory. By the mid-80's it was realised that this theory, constituting a fundamental paradigm shift in understanding physics, was a natural unification of gravity with the SM, of the large-scale with the small-scale. It proposes that all particles are different vibration modes of tiny strings, different notes, if you will, of a cosmic symphony. In this symphony, all forces, all interactions, all particles, and indeed all space and time, harmoniously unify. Again, there is a catch. The theory only makes sense in 10 dimensions, as opposed to the three plus one (for time) with which we are familiar. Moreover, the strings are so small that we may never be able to directly detect them. My research is on where the missing 6 dimensions are (after all, 10 minus 4 is equal to 6), what are their properties, and, indeed, how they determine the world we see, assuming that string theory were to be the unified theory of everything. These extra dimensions curl up in specific geometric ways and I have been involved in applying the cutting-edge results from the mathematicians, from the higher-dimensional geometers, to constructing theories which resemble (or, hopefully, exactly reproduce) the SM. The theories which we produce, from these 6-dimensional spaces, all have a special property which is yet to be observed. This is called supersymmetry. It is a proposal that every elementary particle we have thus far seen, comes with a 'super'-partner yet to be been. I have been studying how to obtain a supersymmetric version of the SM from string theory and have had some success. Earlier this year, my collaborators and I have found a special 6-dimensional geometry which gives just the right particles! We know that such geometry is rare since of the thousands of models constructed from string theory and of the infinite number of possibilities for the 6-dimensional space, this is the only one that has exactly the right particles, no more and no less. There remains much to be done. We must work out the details of this model and especially ascertain how special our geometry is and whether there are other possibilities. A key goal of the machine at Geneva is to test signatures of supersymmetry. Checking our model against the influx of data is of vital importance. The Theoretical Physics Department at Oxford is a unique place, in that it has some of the founding members who initiated this study of trying to bring string theory to produce the SM interaction of particles, and in that it neighbours the Mathematical Institute, which is a world center for geometry. With these members of both departments I am currently collaborating. The dynamic interaction is precisely geared to my using the latest advanced in geometry to answer perhaps most pressing issue of particle physics, in light of the Geneva data: 'how does string theory produce the SM?'