Geometric structures on moduli spaces in physical theories

Abstract

The moduli space of solutions to the equations of motion of a physical theory often carries geometric structures that encode interesting features of the theory such as its symmetries. If the physical theory is defined on some manifold, then the symmetries of the theory place nontrivial constraints on the geometry of the manifold. In particular, there is a fascinating interplay between the geometric structures on the moduli space of solutions and those on the manifold it is defined on. So far, the main setting in which this interplay has been studied has been Kaehler Ricci-flat metrics on complex manifolds with trivial canonical bundle (i.e. Calabi-Yau manifolds), but many aspects of this interplay (such as mirror symmetry relating enumerative invariants of symplectic manifolds with Hodge-theoretic periods of their mirrors) are expected to generalise to other settings. The goal of this project is to investigate potential such generalisations in the context of the Hull-Strominger system on non-Kaehler Calabi-Yau manifolds using a wide variety of tools, such as generalised geometry in the sense of Hitchin and special geometry in the sense of Freed.