This award funds the research activities of Professor James Gray at the Virginia Polytechnic Institute and State University.
Two theories underlie our understanding of modern physics: quantum mechanics and Einstein's theory of gravity. Electrons experience gravity, however light they may be, and as such these quantum mechanical objects must fit within a gravitational theory. However, if one applies the usual rules of quantum mechanics to theories containing gravity, one gets nonsense. For example, the probability of certain events occurring, that is calculated in this way, turns out to be infinite. Many theories have been proposed to try and solve this problem. To date, by far the most successful is a theory called string theory. The idea is that instead of the fundamental building blocks of nature being particles, they are tiny, one dimensional, pieces of string. The troublesome probabilities mentioned above can be calculated to be proportional to one divided by the length of these strings. Thus one sees that, as one shrinks the length to zero and recovers a point like particle, one finds division by zero - an infinite answer. If the string length does not vanish, however, the theory can make sense. String theory comes with a very big catch. Mathematically, the theory only makes sense if the strings live in more than just three spatial dimensions. If some of the dimensions are wrapped up in some very small shape, we would not necessarily see them. Think of a piece of paper. This is clearly a two dimensional sheet. Now wrap the paper up into a cylinder and imagine furling it up more and more to make a smaller and smaller tube. If you could keep doing this (and if the paper was thin enough) then eventually the sheet would simply look like a line - a one dimensional object. The shape that the extra dimensions of string theory take determines the physics that someone living in the resulting three-dimensional universe would see. As an example, the volume of the shape determines the strength of gravity, as governed by Newton's constant. More subtle geometrical properties of the extra dimensions determine all of the rest of the physics of the resulting three-dimensional universe - for example, whether electrons exist and whether they carry electric charge. Given this, an obvious question arises. Is there a shape, which if used to hide the extra dimensions of string theory, gives rise to a three dimensional universe like our own? The answer to the question is still unknown. The proposed research is to use computers to study hundreds of millions of different possible shapes for the extra dimensions of string theory. In this way the PI will obtain a comprehensive survey of which three dimensional universes can be described by string theory. The PI will search this database to see if any of the shapes give rise to physics which is like that we see around us. The broader impact of this project will largely be through the technical training of graduate students, undergraduate researchers and a postdoctoral research assistant. Funding for six months of a postdoctoral researcher's tenure is included, with the other years and personnel involved being provided by resources from Virginia Tech. The PI plans to host an interdisciplinary workshop on computational algebraic geometry and string compactification at his home institution, to help develop links between these two different fields.
The project proposes to study compactifications of F-theory and heterotic string theory, which are currently the most promising candidates for reproducing known particle physics. Using modern formal methods of computational algebraic geometry, the PI will study large numbers of Calabi-Yau four- and three-fold compactifications of these theories in very fine detail. In addition, the PI will develop formalism to describe the moduli space and matter content of non-Kahler compactifications of heterotic theories. This work will try to answer two questions. First, can one find compactifications of string theory that reproduce not only the gauge group and particle content of the standard model, but also a realistic set of soft supersymmetry breaking terms? Second, if such compactifications can indeed be found, what are their common predictions for experimental observables that are yet to be measured? This research has the potential to advance mankind's knowledge of nature directly. It could help us understand whether string theory is simply a technical tool for studying some areas of mathematics and quantum field theory, or whether it is a fundamental physical theory of our universe. The work proposed will develop our understanding of string compactification on Calabi-Yau and non-Kahler manifolds in situations where the supergravity approximation is valid. The work on Calabi-Yau threefolds is designed to push the theory as close as possible to confrontation with experiment. The research on non-Kahler compactifications will lay the groundwork for more ambitious approaches to obtaining particle physics models from string theory in the future.