The Structure of BPS Equations for Ambi-polar Microstate Geometries

TL;DR

This paper clarifies the cohomology of ambi-polar spaces used in BPS microstate geometries, revealing that harmonic fluxes derive from scalar pre-potentials linked to the Kahler structure, simplifying their construction.

Contribution

It introduces a new understanding of the harmonic fluxes in ambi-polar geometries through Kahler structure analysis, enabling simplified construction of BPS solutions.

Findings

Harmonic fluxes originate from scalar pre-potentials on holomorphic divisors.

Differentiating pre-potentials w.r.t. Kahler moduli solves BPS equations.

Illustrated with two-centered ambi-polar solutions.

Abstract

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.