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.

中文翻译：

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特殊几何、Hessian 结构和应用

${\cal N}= 2$ 理论中四维和五维时空的阿贝尔向量多重态的目标空间几何称为特殊几何。它可以用 Hessian 几何来优雅地表述。在这篇综述中，我们介绍了 Hessian 几何，重点关注与四维和五维向量多重态的特殊几何相关的方面。我们根据 Hessian 结构制定了 ${\cal N}= 2$ 理论，并给出了 Hessian 几何的各种具体应用，从四维和五维时空的静态 BPS 黑洞到拓扑弦理论，强调了黑森潜力。我们还讨论了 r-map 和 c-map，它们将向量多重态的特殊几何相互关联并与超多重态几何相关联。通过包括类似时间的降维，