Wall-crossing and quantum invariants of Calabi-Yau threefolds

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Background: Calabi-Yau threefolds are central objects of research in both the mathematics of algebraic geometry and the physics of string theory. In the last few decades, subtle invariants of Calabi-Yau threefolds have arisen, often having parallel descriptions in math and physics. In math, the invariants are born from the geometry of various moduli spaces associated to the Calabi-Yau threefold; in physics, the arise out of various quantum field theories associated to the Calabi-Yau threefold. These invariants have turned out to have amazingly rich structure and surprisingly many connections to other branches of mathematics. They have consequently become objects of intense study in the last 15 years. The primal instances of the invariants are those from Gromov-Witten theory and from Donaldson-Thomas theory. These theories arise from fundamentally different aspects of the geometry of the Calabi-Yau: Gromov-Witten theory comes from the moduli space of maps to the threefold, Donaldson-Thomas theory comes from the moduli of sheaves on the threefold. However, these two theories are famously conjectured to be equivalent in a remarkable way. This correspondence is a mathematical version of a deep correspondence between string theory and gauge theory in physics. The goal of this project is to refine the basic Gromov-Witten/Donaldson-Thomas correspondence to include many of the structures recently discovered in Donaldson-Thomas theory. Specifically, the motivic refinements of Donaldson-Thomas invariants and the associated wall-crossing phenomena should have manifestations in Gromov-Witten theory. This wall-crossing is a mathematical manifestation of the idea of marginal stability in physics: as the parameters of a theory change, particles can become unstable or stable and consequently will disappear or appear from the particle spectrum.