BPS equations and non-trivial compactifications

Abstract

Abstract:We consider the problem of finding exact, eleven-dimensional, BPS supergravity solutions in which the compactification involves a non-trivial Calabi-Yau manifold, 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 Y = M GH × T 2 , where M GH is a hyper-Kähler, 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 Y. The role of the compensators is to ensure smoothness of the eleven-dimensional metric when the moduli of 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.

The structure of BPS equations for ambi-polar microstate geometries

“…The significant point is that q j r j V vanishes if r i = 0, for i = j, and limits to 1 as r j → 0. This means that D j (r −n k ) is not more singular that the original function, r −n k , for all values of k. One can also check [26]…”

“…The compensators for GH metrics in the coordinates (ψ, y) were computed in [25,26]. When the GH points all lie along the z-axis, one has…”