The Kahler Structure of Supersymmetric Holographic RG Flows

TL;DR

This work shows that the moduli-space metric experienced by a brane probe in holographic RG flows is naturally Kahler across a variety of ten- and eleven-dimensional backgrounds, and it provides explicit Kahler potentials $K$ along the flows. A remarkably simple differential equation governs $K$ (the “niceK” relation), and exact solutions are obtained in multiple examples, including ${ m N}=1$ and ${ m N}=2$ flows in $D=4$ and corresponding flows in $D=3$ with $S^5$ and $T^{1,1}$ moduli spaces. The authors connect the Kahler structure to scaling dimensions and $U(1)$ R-symmetries, and show that the same underlying equation applies to purely Coulomb-branch flows in both ten and eleven dimensions, suggesting a universal feature of holographic RG flows. These results provide a concrete bridge between gauge-theory data and Kahler geometry in highly nontrivial holographic settings, with potential implications for extracting IR physics from strongly coupled flows. The work also highlights the need for careful variable choices to reveal simple underlying structures in complex supergravity backgrounds.

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.