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

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.