Special Holonomy in String Theory and M-Theory

A brief, example-oriented introduction is given to special holonomy and its uses in string theory and M-theory. We discuss A_k singularities and their resolution; the construction of a K3 surface by resolving T⁴/Z₂; holomorphic cycles, calibrations, and worldsheet instantons; aspects of the low-energy effective action for string compactifications; the significance of the standard embedding of the spin connection in the gauge group for heterotic string compactifications; G₂ holonomy and its relation to N = 1 supersymmetric compactifications of M-theory; certain isolated G₂ singularities and their resolution; the Joyce construction of compact manifolds of G₂ holonomy; the relation of D6-branes to M-theory on special holonomy manifolds; gauge symmetry enhancement from light wrapped M2-branes; and chiral fermions from intersecting branes. These notes are based on lectures given at TASI '01.