Vector multiplets in N=2 supersymmetry and its associated moduli spaces

TL;DR

Van Proeyen surveys vector multiplets in both rigid and local $N=2$ supersymmetry, emphasizing special Kähler geometry defined by a holomorphic prepotential $F$ and the central role of symplectic dualities that act on the electric/magnetic sector via ${\cal N}_{AB}$ and related matrices. The Seiberg–Witten framework is used to illustrate how perturbative and nonperturbative corrections are captured by monodromies and period integrals, with the period matrix serving as a bridge to the moduli of Riemann surfaces and Calabi–Yau threefolds in string compactifications. A unified, symplectic, matrix-based formulation of rigid and local special geometry is developed, including coordinate-free descriptions, curvature relations, and holomorphic structures, and extended to Calabi–Yau moduli spaces through Picard–Fuchs equations. The c-map and the emergence of special quaternionic and very special manifolds connect four-dimensional vector-multiplet moduli to broader geometric families, highlighting the deep interplay between geometry and quantum corrections in $N=2$ theories.

Abstract

An introduction to $N=2$ rigid and local supersymmetry is given. The construction of the actions of vector multiplets is reviewed, defining special Kähler manifolds. Symplectic transformations lead to either isometries or symplectic reparametrizations. Writing down a symplectic formulation of special geometry clarifies the relation to the moduli spaces of a Riemann surface or a Calabi-Yau 3-fold. The scheme for obtaining perturbative and non-perturbative corrections to a supersymmetry model is explained. The Seiberg-Witten model is reviewed as an example of the identification of duality symmetries with monodromies and symmetries of the associated moduli space of a Riemann surface.