BPS equations and Non-trivial Compactifications

TL;DR

This paper constructs exact eleven-dimensional BPS supergravity solutions with non-trivial Calabi-Yau compactifications, using a local model with hyper-Kähler spaces, and explores their properties and brane distributions.

Contribution

It introduces a method to find exact BPS solutions with non-trivial Calabi-Yau manifolds by employing a local model and compensators to ensure smoothness.

Findings

Exact solutions with three electric charges in five dimensions.

Calabi-Yau manifolds fibered over space-time with compensators.

Discussion of brane distributions on less symmetric internal manifolds.

Abstract

We consider the problem of finding exact, eleven-dimensional, BPS supergravity solutions in which the compactification involves a non-trivial Calabi-Yau manifold, ${\cal Y}$, as opposed to simply a $T^6$. Since there are no explicitly-known metrics on non-trivial, compact Calabi-Yau manifolds, we use a non-compact "local model" and take the compactification manifold to be ${\cal Y} = {\cal M}_{GH} \times T^2$ where ${\cal M}_{GH}$ is a hyper-K\"ahler, Gibbons-Hawking ALE space. We focus on backgrounds with three electric charges in five dimensions and find exact families of solutions to the BPS equations that have the same four supersymmetries as the three-charge black hole. Our exact solution to the BPS system requires that the Calabi-Yau manifold be fibered over the space-time using compensators on ${\cal Y}$. The role of the compensators is to ensure smoothness of the eleven-dimensional metric when the moduli of ${\cal Y}$ depend on the space-time. The Maxwell field Ansatz also implicitly involves the compensators through the frames of the fibration. We examine the equations of motion and discuss the brane distributions on generic internal manifolds that do not have enough symmetry to allow smearing.