Basic Concepts of String Theory pp 427-520 | Cite as

# String Compactifications

## Abstract

An alternative to describing compactifications via a solvable conformal field theory is the perturbative approach around a geometric supergravity background at large radius. For this purpose one analyzes the string equations of motion at leading order in a typical length scale \(L/\sqrt{\alpha \prime }\). We describe this approach in detail for a class of backgrounds which preserve some amount of space-time supersymmetry in four-dimensions: compactification on Calabi-Yau manifolds. But we start with a brief discussion of the string equations of motion as the requirement of vanishing beta-functions of the non-linear sigma model for a string moving in a curved background. We then derive a generalization of T-duality to manifolds with isometries. This leads to the so-called Buscher rules. We then introduce some of the mathematical tools which are required for an adequate treatment of Calabi-Yau compactifications. With them at hand we consider compactifications of the type II and heterotic superstring on Calabi-Yau manifolds and discuss the structure of their moduli spaces. In an appendix we fix our notation and review some concepts of Riemannian geometry. The derivations of some results which are used in the main text are also relegated to the appendix.