Letters in Mathematical Physics

, Volume 108, Issue 5, pp 1279–1306 | Cite as

ASK/PSK-correspondence and the r-map

  • Vicente Cortés
  • Peter-Simon Dieterich
  • Thomas Mohaupt


We formulate a correspondence between affine and projective special Kähler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special Kähler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special Kähler manifolds in the image of the supergravity r-map. The above one-parameter deformations are interpreted as perturbative \(\alpha '\)-corrections in heterotic and type II string compactifications with \(N=2\) supersymmetry. Also affine special Kähler manifolds with quadratic prepotential are mapped to one-parameter families of projective special Kähler manifolds with quadratic prepotential. We show that the completeness of the deformed supergravity r-map metric depends solely on the (well-understood) completeness of the undeformed metric and the sign of the deformation parameter.


Special real manifolds Special Kähler manifolds r-map 

Mathematics Subject Classification

53C26 (primary) 



This work was partly supported by the German Science Foundation (DFG) under the Research Training Group 1670 and the Collaborative Research Center (SFB) 676. The work of T.M. was partly supported by the STFC consolidated grant ST/G00062X/1. He thanks the Department of Mathematics and the Centre for Mathematical Physics of the University of Hamburg for support and hospitality during various stages of this work.


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  • Vicente Cortés
    • 1
  • Peter-Simon Dieterich
    • 1
  • Thomas Mohaupt
    • 2
  1. 1.Department of Mathematics, Center for Mathematical PhysicsUniversity of HamburgHamburgGermany
  2. 2.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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