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Geometric Construction of the r-Map: From Affine Special Real to Special Kähler Manifolds

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Abstract

We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N = TM carries a canonical structure of (affine) special Kähler manifold. This gives an intrinsic description of the r-map as the map \({M \mapsto N=TM}\) . On the physics side, this map corresponds to the dimensional reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We generalise this construction to the case when M is any Hessian manifold.

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Authors and Affiliations

  1. The University of Edinburgh and Maxwell Institute for Mathematical Sciences, JCMB, The King’s buildings, Edinburgh, EH9 3JZ, UK

    D. V. Alekseevsky

  2. Department Mathematik und Zentrum für Mathematische Physik, Universität Hamburg, Bundesstraße 55, D-20146, Hamburg, Germany

    V. Cortés

Authors
  1. D. V. Alekseevsky
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  2. V. Cortés
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Correspondence to V. Cortés.

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Communicated by G.W. Gibbons

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Alekseevsky, D.V., Cortés, V. Geometric Construction of the r-Map: From Affine Special Real to Special Kähler Manifolds. Commun. Math. Phys. 291, 579–590 (2009). https://doi.org/10.1007/s00220-009-0803-7

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  • DOI: https://doi.org/10.1007/s00220-009-0803-7

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