K3 Surfaces and String Duality

TL;DR

This paper explores the moduli space of various string theories compactified on K3 surfaces, emphasizing string duality and geometric properties, providing a comprehensive review of classical geometry and conformal field theory approaches.

Contribution

It offers a detailed analysis of string compactifications on K3 surfaces, integrating classical geometry with string duality concepts, and reviews conformal field theory descriptions for multiple string types.

Findings

K3 surfaces serve as a key setting for string dualities.

The paper clarifies geometric aspects of K3 in string theory.

It connects classical geometry with modern string duality insights.

Abstract

The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review "old string theory" on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 x E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface itself. These are an extended form of the notes from lectures given at TASI 96.