TASI lectures: special holonomy in string theory and M-theory

TL;DR

The notes provide an approachable tour of special holonomy in string theory and M‑theory, connecting simple geometric constructions to physical implications for compactifications with minimal supersymmetry. They explain how Calabi–Yau geometries yield massless moduli through complex and Kahler structures, how worldsheet instantons calibrate holomorphic cycles, and how heterotic standard embedding affects the low‑energy action. They then pivot to seven‑manifolds with $G_2$ holonomy, outlining the existence and properties of AC $G_2$ metrics, the Joyce construction, and the M‑theory interpretation of D6‑branes and their wrapped M2‑brane states, including gauge enhancements and potential chiral matter from brane intersections. The overarching theme is that special holonomy provides a robust framework for realizing ${\cal N}=1$ in four dimensions and for exploring non‑perturbative effects, with explicit geometric engineering via singularity resolutions and brane setups. The work underscores the deep interplay between geometry, holonomy, and the dynamics of strings and M‑theory in shaping phenomenologically relevant vacua.

Abstract

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 T^4/Z_2; 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; G_2 holonomy and its relation to N=1 supersymmetric compactifications of M-theory; certain isolated G_2 singularities and their resolution; the Joyce construction of compact manifolds of G_2 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.

Figures (7)

• Figure 1: Left: parallel transport of a vector around the tip of a cone changes its direction. Right: the same parallel transport, where the cone is thought of as a plane modded out by a discrete group.
• Figure 2: $S^3/{\bf Z}_2$ is a $U(1)$ fibration over $S^2$, and in the interior, the $U(1)$ shrinks but the $S^2$ doesn't.
• Figure 5: The Dynkin diagram for $G_2$. The weights comprising the ${\bf 7}$ are the six short roots plus one node at the origin.
• Figure 6: Parallel transport of a vector $v$ around a curve $C$. Upon returning to the point of origin $P$, $v$ has undergone some rotation, which for a seven-manifold is an element of $SO(7)$.
• Figure 7: ${\bf CP}^3$ a $S^2$ fibration over $S^4$, and in the interior, the $S^2$ shrinks but the $S^4$ doesn't.