Geometric construction of the r-map: from affine special real to special K\"ahler manifolds

TL;DR

This paper provides an intrinsic geometric construction of the r-map, showing how affine special real manifolds can be embedded as Hessian domains and how their tangent bundles naturally acquire special K"ahler structures, linking geometry with physics.

Contribution

It introduces an intrinsic definition of affine special real manifolds and demonstrates that their tangent bundles are naturally special K"ahler, generalizing the r-map construction.

Findings

Any affine special real manifold can be realized as a Hessian domain in affine space.

The tangent bundle of such a manifold carries a canonical special K"ahler structure.

The construction extends to general Hessian manifolds, broadening the scope of the r-map.

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 K\"ahler 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.