Introduction to String Compactification A. Font1 and S. Theisen2 1 Departamento de F´ısica, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 20513, Caracas 1020-A, Venezuela
[email protected] 2 Albert-Einstein-Institut, Am M¨uhlenberg 1, 14476 Golm, Germany
[email protected] Summary. We present an elementary introduction to compactifications with un- broken supersymmetry. After explaining how this requirement leads to internal spaces of special holonomy we describe Calabi-Yau manifolds in detail. We also discuss orbifolds as examples of solvable string compactifications. 1 Introduction The need to study string compactification is a consequence of the fact that a quantum relativistic (super)string cannot propagate in any space-time back- ground. The dynamics of a string propagating in a background geometry defined by the metric GMN is governed by the Polyakov action √ − 1 2 − αβ M N SP = d σ hh ∂αX ∂βX GMN(X) . (1.1) 4πα Σ α Here σ , α =0, 1, are local coordinates on the string world-sheet Σ, hαβ is M a metric on Σ with h = det hαβ,andX ,M=0,...,D− 1, are functions Σ→ space-time M with metric GMN(X). α is a constant of dimension 2 (length) . SP is the action of a two-dimensional non-linear σ-model with target space M, coupled to two-dimensional gravity (hαβ) where the D- dimensional metric GMN appears as a coupling function (which generalizes the notion of a coupling constant). For flat space-time with metric GMN = ηMN the two-dimensional field theory is a free theory.