Lectures on Special Kahler Geometry and Electric--Magnetic Duality Rotations
Pietro Fré
TL;DR
These lectures articulate how electric–magnetic dualities are realized geometrically in $D=4$,\ ${\cal N}=2$ theories through symplectic covariance, Special Kähler geometry for vector multiplets, and HyperKähler/Quaternionic geometry for hypermultiplets. They derive the Gaillard–Zumino period–matrix formalism ${\cal N}$ for homogeneous scalar manifolds and discuss how ${\cal N}$ governs gauge couplings and duality actions, with ST[2,n] and Calabi–Visentini coordinates illustrating both classical and quantum duality structures. A central theme is gauging: turning on couplings breaks continuous dualities to discrete ones while introducing a scalar potential via momentum maps and the (tri)holomorphic structures, tying moduli, monodromies, and non-perturbative effects to dualities. The Seiberg–Witten framework is presented as a concrete non-perturbative realization of rigid special geometry, where a dynamical Riemann surface encodes the effective action and duality data, highlighting how quantum corrections modify the symplectic embedding and the duality group.
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 Kähler geometry for the vector multiplets and either HyperKähler 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.