Virtual curve counting on Calabi-Yau 3-folds: A primer on DT/PT theory

In string theory, our universe is hypothesized to be a product of Minkowski spacetime and a Calabi-Yau 3-fold. Curves in a CY 3-fold will be locations where endpoints of strings can attach. Studying the enumerative geometry of curves on CY 3-fold has thus aroused interest among both mathematicians and physicists. My essay will explain the notion of virtual counting, focussing on the Donaldson-Thomas resp. Pandharipande-Thomas invariants. We will dive into the deformation theory, derived category and the motivic Hall algebra to better understand the corresponding two compactifications of the moduli spaces of curves on CY 3-folds.