Elsevier

Nuclear Physics B

Volume 503, Issue 3, 20 October 1997, Pages 565-613
Nuclear Physics B

What is special Kähler geometry?

https://doi.org/10.1016/S0550-3213(97)00408-2Get rights and content

Abstract

The scalars in vector multiplets of N = 2 supersymmetric theories in four dimensions exhibit ‘special Kähler geometry’, related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the definition of special geometry. We show equivalences of some definitions and give examples which show that earlier definitions are not equivalent, and are not sufficient to restrict the Kähler metric to one that occurs in N = 2 supersymmetry. We treat the rigid as well as the local supersymmetry case. The connection is made to moduli spaces of Riemann surfaces and Calabi-Yau 3-folds. The conditions for the existence of a prepotential translate to a condition on the choice of canonical basis of cycles.

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      An alternative ‘bilagrangian’ extrinsic construction of ASK manifolds has been given in [50]. For a more detailed comparison between the approach presented in this review and alternative formulations, we use [53], where various definitions of special Kähler geometry have been collected and compared to each other, and [54], which has extended these definitions to arbitrary target space signature. The geometry of moduli spaces of Calabi–Yau three-folds provides natural realizations of special real and special Kähler geometry.

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