Lectures on Special Kahler Geometry and Electric--Magnetic Duality Rotations

TL;DR

This paper reviews electric-magnetic duality rotations in even-dimensional spacetimes, focusing on four dimensions, and explores their geometric structures, dualities, and applications to supersymmetric theories and moduli space geometries.

Contribution

It provides a comprehensive review of symplectic covariance, special Kähler geometry, and dualities in four-dimensional theories, with detailed examples from Seiberg-Witten models.

Findings

Clarifies the role of symplectic covariance in dualities.

Connects moduli space geometry to space-time dualities.

Analyzes classical and non-perturbative dualities in supersymmetric theories.

Abstract

In these lectures I review the general structure of electric--magnetic duality rotations in every even space--time dimension. In four dimensions, which is my main concern, I discuss the general issue of symplectic covariance and how it relates to the typical geometric structures involved by N=2 supersymmetry, namely Special K\"ahler geometry for the vector multiplets and either HyperK\"ahler or Quaternionic geometry for the hypermultiplets. I discuss classical continuous dualities versus non--perturbative discrete dualities. How the moduli space geometry of an auxiliary dynamical Riemann surface (or Calabi--Yau threefold) relates to exact space--time dualities is exemplified in detail for the Seiberg Witten model of an $SU(2)$ gauge theory.