In this mini-series I will introduce GV invariants from a physicists perspective. We will discuss their interpretation as BPS invariants in M-theory compactifications on Calabi-Yau threefolds, their imprint as quantum corrections to the effective action of type IIA (and also IIB) string theory, and their relevance for understanding the Kähler moduli spaces of Calabi-Yau threefolds. We will introduce both the Hosono-Klemm-Theisen-Yau (HKTY) method to compute GV invariants using mirror symmetry (+ its specific implementation in CYtools), and more direct enumerative methods that apply in toric special cases.
Part I: GV-invariants are BPS-indices
Part II: GV-invariants determine birational geometry
Part III: Computing GV-invariants — explicit examples and HKTY implementation in CYtools