Special Geometry, Hessian Structures and Applications
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The target space geometry of abelian vector multiplets in four and five space-time dimensions is called special geometry and can be elegantly formulated in terms of Hessian 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.read more
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References
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Journal ArticleDOI
Black hole entropy is the Noether charge
TL;DR: The results show that the validity of the "second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory.
Journal ArticleDOI
Some properties of Noether charge and a proposal for dynamical black hole entropy
Vivek Iyer,Robert M. Wald +1 more
TL;DR: It is proved that the first law of black hole mechanics holds for arbitrary perturbations of a stationary black hole, and a local, geometrical prescription is proposed for the entropy, $S_{dyn}$, of a dynamical black hole.