Special Geometry, Hessian Structures and Applications

TL;DR

This paper reviews how Hessian geometry underpins the special geometries of vector multiplets in ${ m N}=2$ theories, with applications to black holes, string theory, and dimensional reductions, highlighting the role of the Hesse potential.

Contribution

It introduces Hessian geometry as a framework for special geometry in ${ m N}=2$ theories and explores its applications and relations via the r- and c-maps.

Findings

Hessian geometry provides an elegant formulation of special geometry.

Applications include static BPS black holes and topological string theory.

Euclidean theories involve para-complex special geometry.

Abstract

The target space geometry of abelian vector multiplets in ${\cal N}= 2$ theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometries of four- and five-dimensional vector multiplets. We formulate ${\cal N}= 2$ theories in terms of Hessian structures and give various concrete applications of Hessian geometry, ranging from static BPS black holes in four and five space-time dimensions to topological string theory, emphasizing the role of the Hesse potential. We also discuss the r-map and c-map which relate the special geometries of vector multiplets to each other and to hypermultiplet geometries. By including time-like dimensional reductions, we obtain theories in Euclidean signature, where the scalar target spaces carry para-complex versions of special geometry.