Compactification, Geometry and Duality: N=2

TL;DR

This paper reviews the geometry of moduli spaces in N=2 supersymmetric theories from string compactifications, focusing on differences with higher N theories and discussing specific cases like Calabi-Yau and K3xT2 compactifications.

Contribution

It provides a detailed review of the moduli space geometry in N=2 theories, highlighting differences from higher N theories and discussing peculiar features like mixed instantons.

Findings

Analysis of vector and hypermultiplet moduli spaces

Discussion of peculiarities such as mixed instantons

Identification of gaps in understanding of hypermultiplet moduli

Abstract

These are notes based on lectures given at TASI99. We review the geometry of the moduli space of N=2 theories in four dimensions from the point of view of superstring compactification. The cases of a type IIA or type IIB string compactified on a Calabi-Yau threefold and the heterotic string compactified on K3xT2 are each considered in detail. We pay specific attention to the differences between N=2 theories and N>2 theories. The moduli spaces of vector multiplets and the moduli spaces of hypermultiplets are reviewed. In the case of hypermultiplets this review is limited by the poor state of our current understanding. Some peculiarities such as ``mixed instantons'' and the non-existence of a universal hypermultiplet are discussed.