What is Special K\"ahler Geometry ?

TL;DR

This paper clarifies the definition of special Kähler geometry in N=2 supersymmetric theories, resolving existing ambiguities and connecting it to moduli spaces of Riemann surfaces and Calabi-Yau manifolds.

Contribution

It establishes equivalences among various definitions of special Kähler geometry and demonstrates their limitations, providing a clearer framework for its application in supersymmetry.

Findings

Clarified the definition of special Kähler geometry.

Showed that earlier definitions are not equivalent or sufficient.

Connected special Kähler geometry to moduli spaces of Riemann surfaces and Calabi-Yau 3-folds.

Abstract

The scalars in vector multiplets of N=2 supersymmetric theories in 4 dimensions exhibit `special Kaehler geometry', related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the definition of special geometry. We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kaehler metric to one that occurs in N=2 supersymmetry. We treat the rigid as well as the local supersymmetry case. The connection is made to moduli spaces of Riemann surfaces and Calabi-Yau 3-folds. The conditions for the existence of a prepotential translate to a condition on the choice of canonical basis of cycles.