Calabi–Yau Compactifications

We study a large class of exact superstring vacua obtained by compactifying a 10d SUSY string down to 4d on Riemannian spaces of SU(3) holonomy a.k.a. Calabi–Yau (CY) 3-folds. These vacua are invariant under 4d Poincaré symmetry and one-quarter of the original supersymmetries. We discuss their deep geometry and physics, aiming to be didactical but also mathematically precise and self-contained. We review all geometry we need. One may study these vacua from three viewpoints: These three approaches, despite the difference of languages, perspectives, and tools, turn out to be essentially equivalent [13].