New developments in special geometry

TL;DR

This paper reviews recent advances in special geometry, focusing on real coordinates, para-complex geometry in supersymmetry, and the variational principles related to BPS black hole entropy in supergravity.

Contribution

It introduces new insights into the role of real coordinates in special geometry and connects black hole entropy with the Legendre transform of the Hesse potential.

Findings

Para-complex geometry of vector and hypermultiplets analyzed

Black hole entropy expressed as Legendre transform of Hesse potential

Enhanced understanding of scalar field geometry in supergravity

Abstract

We review recent developments in special geometry, emphasizing the role of real coordinates. In the first part we discuss the para-complex geometry of vector and hypermultiplets in rigid Euclidean N=2 supersymmetry. In the second part we study the variational principle governing the near horizon limit of BPS black holes in matter-coupled N=2 supergravity and observe that the black hole entropy is the Legendre transform of the Hesse potential encoding the geometry of the scalar fields.