Kähler 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 Kähler four-manifold along the renormalization group flow in sevendimensional gauged supergravity. The physical eleven-dimensional M-theory setup is given by a stack of M5-branes wrapping a calibrated Kähler four-cycle inside a Calabi-Yau threefold. By topologically twisting the theory in the ultraviolet, we may choose an arbitrary Kähler 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 Kähler-Einstein metric, which is the expected uniformization behavior. ar X iv :1 71 0. 09 47 9v 1 [ he pth ] 2 5 O ct 2 01 7