Geometry, dualities and the string landscape

String theory is a putative theory of quantum gravity that also has the potential to give a unified description of the particles and interactions that form the fundamental building blocks of the matter and forces in our universe. In addition, string theory has a remarkable set of duality symmetries.Since according to general relativity gravity is intrinsically a theory of geometry, these dualities have had important implications for mathematics.

This project aims to bring together two important recent developments in string theory. Broadly, the first is a set of new notions of geometrical structure that appears in the string theory generalisation of Riemannian geometry and unifies the string theory degrees of freedom. The second is the study of the "string landscape", that is, the general question of which low-energy theories of particle physics can arise in string theory. Since supersymmetric string theories (and their duals, known as "M-theory") describe ten- or eleven-dimensional geometries, one must assume that part of spacetime is a small six- or seven-dimensional compact space in order to realise our own four-dimensional universe. The geometry of this "compactification space" strongly influences the nature of the low-energy theory. Thus, a central question in the string landscape programme is to understand the range of possible compact geometries, and their properties. The new developments in string geometry mean that we now have access to a much more complete set of backgrounds, and, crucially, have new tools for studying properties such as their "moduli" (geometrical deformations that cost little or no energy).

Developing our understanding of these geometries, the connections to duality and the implications for and insights from the string landscape programme are the central goals of this proposal. The medium- to long-term goals of this research are gain new understanding of the nature of quantum gravity, the fundamental symmetries of string theory, and the possibility of finding realistic unified theories of gravity and quantum mechanics. There are also potentially new and intriguing implications for mathematics, notably string-motivated extension of notions of special holonomy and algebraic geometry.