| Superstring theory, or string theory for short, is a sub-discipline in theoretical physics that connects to many branches in modern physics and mathematics. Its main objective and challenge is to construct a unified theory for quantum gravity and all elementary particles with the hope of making predictions for particle physics and cosmology. The mathematical structure underlying string theory is so rich and intricate, that also significant impact in present day mathematics research was made, particularly in the areas of algebraic and differential geometry. This research proposal, on geometrical aspects of string theory, is situated on the overlap of mathematics and physics, and is therefore of interdisciplinary nature. Its main goal is to invest on strategically chosen research projects with a mathematical orientation, with on the one hand the ambition to apply known mathematics to physics problems, and on the other hand, also to resolve or shed new light on problems in mathematics using string theory. Calabi-Yau (CY) spaces and the recently discovered generalizations thereof (known as generalized geometries) play a central role in this proposal. These appear in string theory as extra dimensions that are curled up to very small sizes. In this proposal, various physical aspects of string theory compactifications on CY spaces and their generalizations, called flux compactifications, will be further developed, and be given a better mathematical foundation. Special emphasis will be given on an entire new research program that was recently initiated by the applicant, on the quantum aspects of the quaternion-Kähler geometry associated with the moduli spaces of CY manifolds. This study is particularly urgent since it might lead to important new discoveries related to generalized Donaldson-Thomas invariants and wall crossing phenomena recently studied in the mathematics and physics communities. |
