Instantons, black holes, and harmonic functions

We find a class of five-dimensional Einstein-Maxwell type Lagrangians which contains the bosonic Lagrangians of vector multiplets as a subclass, and preserves some features of supersymmetry, namely the existence of multi-centered black hole solutions and of attractor equations. Solutions can be expressed in terms of harmonic functions through a set of algebraic equations. The geometry underlying these Lagrangians is characterised by the existence of a Hesse potential and generalizes the very special real geometry of vector multiplets. Our construction proceeds by first obtaining instanton solutions for a class of four-dimensional Euclidean sigma models, which includes those occurring for four-dimensional Euclidean N = 2 vector multiplets as a subclass. For solutions taking values in a completely isotropic submanifold of the target space, we show that the solution can be expressed in terms of harmonic functions if an integrability condition is met. This condition can either be solved by imposing that the solution depends on a single coordinate, or by imposing that the target space is a para-Kaehler manifold which can be obtained from a real Hessian manifold by a generalized r-map. In the latter case one obtains multi-centered solutions. Moreover, if the integrability condition is met, the second order equations of motion can always be reduced to first order equations, which become gradient flow equations if the solution is further required to depend on one coordinate only. The dualization of axions into tensor fields and the lifting of four-dimensional instantons to five-dimensional solitons are used to motivate the addition of a boundary term to the action, which accounts for the instanton action. If the sigma model is coupled to gravity, and if the Hesse potential is of a suitable form which we specify, then the four-dimensional Euclidean Lagrangian can be lifted consistently to a five-dimensional Einstein-Maxwell type Lagrangian. Instanton solutions lift to extremal black hole solutions, and the instanton action equals the ADM mass.