The Kahler Structure of Supersymmetric Holographic RG Flows

Abstract

We study the metrics on the families of moduli spaces arising from probing with a brane the ten and eleven dimensional supergravity solutions corresponding to renormalisation group flows of supersymmetric large n gauge theory. In comparing the geometry to the physics of the dual gauge theory, it is important to identify appropriate coordinates, and starting with the case of SU(n) gauge theories flowing from N=4 to N=1 via a mass term, we demonstrate that the metric is Kahler, and solve for the Kahler potential everywhere along the flow. We show that the asymptotic form of the Kahler potential, and hence the peculiar conical form of the metric, follows from special properties of the gauge theory. Furthermore, we find the analogous Kahler structure for the N=4 preserving Coulomb branch flows, and for an N=2 flow. In addition, we establish similar properties for two eleven dimensional flow geometries recently presented in the literature, one of which has a deformation of the conifold as its moduli space. In all of these cases, we notice that the Kahler potential appears to satisfy a simple universal differential equation. We prove that this equation arises for all purely Coulomb branch flows dual to both ten and eleven dimensional geometries, and conjecture that the equation holds much more generally.