Heterotic horizons, Monge-Ampère equation and del Pezzo surfaces

Abstract

Heterotic horizons preserving 4 supersymmetries have sections which are T ² fibrations over 6-dimensional conformally balanced Hermitian manifolds. We give new examples of horizons with sections S ³ × S ³ × T ² and SU(3). We then examine the heterotic horizons which are T ⁴ fibrations over a Kähler 4-dimensional manifold. We prove that the solutions depend on 6 functions which are determined by a non-linear differential system of 6 equations that include the Monge-Ampére equation. We show that this system has an explicit solution for the Kähler manifold S ² × S ². We also demonstrate that there is an associated cohomological system which has solutions on del Pezzo surfaces. We raise the question of whether for every solution of the cohomological problem there is a solution of the differential system, and so a new heterotic horizon. The horizon sections have topologies which include ((k − 1)S ² × S ⁴# k(S ³ × S ³) × T ² indicating the existence of exotic black holes. We also find an example of a horizon section which gives rise to two different near horizon geometries.