K\"ahler Uniformization from Holographic Renormalization Group Flows of M5-branes

TL;DR

This paper explores how holographic renormalization group flows in seven-dimensional supergravity lead to the uniformization of K"ahler four-manifolds, showing that arbitrary initial metrics evolve into K"ahler-Einstein metrics at fixed points.

Contribution

It derives equations governing the RG flow of K"ahler four-manifolds in M-theory and demonstrates the flow's tendency to produce K"ahler-Einstein metrics, revealing a holographic uniformization process.

Findings

RG flows wash out initial metric moduli

Infrared fixed points are K"ahler-Einstein metrics

Holographic approach links geometry and supergravity dynamics

Abstract

In this paper, we initiate the study of holographic renormalization group flows acting on the metric of four-manifolds. In particular, we derive a set of equations which govern the evolution of a generic K\"ahler four-manifold along the renormalization group flow in seven-dimensional gauged supergravity. The physical eleven-dimensional M-theory setup is given by a stack of M5-branes wrapping a calibrated K\"ahler four-cycle inside a Calabi-Yau threefold. By topologically twisting the theory in the ultraviolet, we may choose an arbitrary K\"ahler metric on the four-cycle as an asymptotic boundary condition. Along the renormalization group flow, the metric moduli are largely washed out, and at the infrared fixed point we will reach a K\"ahler-Einstein metric, which is the expected uniformization behavior.