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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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