Phenomenology and Geometry in Heterotic String Compactifications

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The Large Hadron Collider, an experiment in Geneva, Switzerland, that collides particles at extremely high energies, is heralding a new era in particle physics. The conventional paradigm for particle physics throughout the past forty years has become known as the Standard Model . This describes an eclectic zoo of particles (like protons and electrons) together with a set of equations governing their interactions. However, the Standard Model is far from being the whole story. Above a certain energy range, its description of physics breaks down, and we are left with a fascinating puzzle of what is going on, and how to explain it - if the answer doesn't lie in the Standard Model, then where do we look?Many physicists believe the answer to this question lies within string theory. As opposed to the Standard Model, whose fundamental objects are particles (electrons and protons), the fundamental objects governing string theory are tiny strings. The strings vibrate, much like guitar strings, and their harmonics give rise to objects resembling those described in the Standard Model. In this sense, string theory has the potential to be a theory of everything . However, although promising, string theory is not without problems. For example, it predicts that we live in a ten-dimensional universe. This poses an obvious problem - we observe only four-dimensions (three space dimensions and one time direction), so where are the remaining six? The resolution lies in the notion of compactification . The idea is simple, though difficult to picture mentally: one imagines that six of the ten-dimensions are curled up, encompassing a special type of space so small that the extra dimensions are essentially invisible. This is analogous to looking at a garden hose from a great height: from this perspective, the hose appears to be a one-dimensional snake. The extra, circular dimension becomes apparent only upon closer inspection. The same phenomenon - the presence of additional dimensions perceptible only from particular perspectives - is thought to occur in string theory. The equations that govern string theory impose stringent constraints on the shape and size of the six-dimensional spaces: only certain types of spaces are allowed. On the other hand, the geometry of the space dictates precisely which experimental predictions we will observe in our four-dimensional spacetime. A loose analogy is the following: the laws of gravity imply that for a water slide to work, it must point down. On the other hand, the shape, the twists and the turns of the slide (i.e. its geometry) dictate precisely what people feel as they go down the slide. This is a wonderful example of the interplay between geometry and physics - a key theme of this project. The physics of string theory makes some bold mathematical predictions. One such prediction, known as mirror symmetry , implies that the six-dimensional spaces come in pairs, and that, from our four-dimensional point of view, they look identical. There is still much to learn about the mathematics of mirror symmetry, and it is an interesting question to ask: what can string theory teach us?In this project, we are interested in answering two complementary but related questions:1. What are the four-dimensional experimental predictions for a wide range of possible compactification spaces? Such predictions are expected to take place in the Large Hadron Collider. 2. What does string theory tell us about the mathematics and geometry of compactification spaces? Is there a general notion of mirror symmetry for every possible compactification space?By answering such questions, we will have moved further in our understanding of the fundamental structure of our physical universe, as well as in our understanding of a fundamental structure in mathematics. If the history of physics is anything to go by, these two directions go hand-in-hand.