Vacuum States of the Heterotic String

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String theory is believed to be a theory capable of describing all the known forces of nature, and provides a solution to the venerable problem of finding a theory of gravity consistent with quantum mechanics. To a first approximation, the world we observe corresponds to a vacuum of this theory. String theory admits many of these vacuum states and the class that is most likely to describe the observed world are the so-called `heterotic vacua'. Analysing these vacua requires the application of sophisticated tools drawn from mathematics, particularly from algebraic geometry. If history is any guide, the synthesis of these mathematical tools with observations drawn from physics will lead not only to significant progress in physics, but also important advances in mathematics. An example of such a major insight in mathematics, that arose from string theory, is mirror symmetry. This is the observation that within in a restricted class of string vacua, these arise in `mirror pairs'. This has the consequence that certain mathematical quantities, which are both important and otherwise mysterious, can be calculated in a straightforward manner. The class of heterotic vacua, of interest here, are a wider class of vacua, and an important question is to what extent mirror symmetry generalises and how it acts on this wider class.In a more precise description, the space of heterotic vacua is the parameter space of pairs (X,V) where X is a Calabi-Yau manifold and V is a stable holomorphic vector bundle on X. This space is a major object of study in algebra and geometry. String theory tells us that it is subject to quantum corrections. To understand the nature of these corrections is the key research problem in this proposal and any advance in our understanding will have a important impact in both mathematics and physics. By now it is widely understood that string theory and geometry are intimately related with much to be learned from each other, yet this relationship is relatively unexplored in the heterotic string. This fact, together with recent developments that indicate that longstanding problems have recently become tractable, means that the time is right to revisit the geometry of heterotic vacua.