The Structure of BPS Equations for Ambi-polar Microstate Geometries

Ambi-polar metrics, defined so as to allow the signature to change from +4 to -4 across hypersurfaces, are a mainstay in the construction of BPS microstate geometries. This paper elucidates the cohomology of these spaces so as to simplify greatly the construction of infinite families of fluctuating harmonic magnetic fluxes. It is argued that such fluxes should come from scalar, harmonic pre-potentials whose source loci are holomorphic divisors. This insight is obtained by exploring the Kahler structure of ambi-polar Gibbons-Hawking spaces and it is shown that differentiating the pre-potentials with respect to Kahler moduli yields solutions to the BPS equations for the electric potentials sourced by the magnetic fluxes. This suggests that harmonic analysis on ambi-polar spaces has a novel, and an extremely rich structure, that is deeply intertwined with the BPS equations. We illustrate our results using a family of two-centered solutions.