New geometry from string dualities

The aim of our proposed research is to develop new ideas in geometry using structures that naturally arise in string theory, which in turn feeds back to advance our understanding of the nature of gravity and particle physics. String theory is a putative quantum theory of gravity, that defines particular, natural extensions of General Relativity, Einstein's description of gravity in terms of curved geometry. These extensions include analogues of the electromagnetic field, collectively known as fluxes. Certain special spacetimes have additional symmetries (known as supersymmetries) that give additional structure to the geometry. Such structures, such as Kahler, Calabi-Yau, Sasaki-Einstein and Joyce manifolds have long been studied by mathematicians. In some cases powerful theorems exist, where the existence of solutions to the differential equations the structures have to satisfy can be translated into a more algebraic condition known as stability. In addition, there can be remarkable duality symmetries between spaces with such structures (notably mirror symmetry of Calabi-Yau manifolds, first discovered in the context of string theory). The theme of this proposal is that both ideas of stability and mirror symmetry have physical interpretations using string theory and furthermore have extensions to a larger class of natural string structures. A key ingredient is the remarkable duality between gravitational theories on certain spacetimes and certain conventional (non-gravitational) quantum field theories, known as the AdS-cft correspondence. We hope to develop these ideas in multiple ways: to ask if one can propose new existence conjectures which ultimately will be important for building string models of particle physics; to understand the relation between stability and the notion of quantum corrections in the dual quantum field theory and the connection to the algebraic structure of field theory defining so-called Calabi-Yau algebras; and to understand extensions of the topological string theories that underlie mirror symmetry.