Developments in special geometry

TL;DR

This paper reviews recent advances in special geometry related to N=2 supersymmetric theories, including new formulations, generalizations, and applications such as solution construction and geometric properties preservation.

Contribution

It introduces a new formulation of the local c-map using the Hesse potential, and discusses generalizations to non-supersymmetric theories and Euclidean versions.

Findings

The local r-map and c-map preserve geodesic completeness.

Construction of static solutions via dimensional reduction.

Unified view of real, complex, and quaternionic special geometry.

Abstract

We review the special geometry of N = 2 supersymmetric vector and hypermultiplets with emphasis on recent developments and applications. A new formulation of the local c-map based on the Hesse potential and special real coordinates is presented. Other recent developments include the Euclidean version of special geometry, and generalizations of special geometry to non-supersymmetric theories. As applications we disucss the proof that the local r-map and c-map preserve geodesic completeness, and the construction of four- and five-dimensional static solutions through dimensional reduction over time. The shared features of the real, complex and quaternionic version of special geometry are stressed throughout.