Kähler uniformization from holographic renormalization group flows of M5-branes

Abstract

In this paper, we initiate the study of holographic renormalization group flows for 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 seven-dimensional 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 CalabiYau 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. We find that at the infrared fixed point, we reach a Kähler-Einstein metric, which can be interpreted as an indication of "uniformizing" behavior of the flow.

Wrapped branes and punctured horizons

“…See[19] for a generalization of this setup to M5-branes wrapped on Kähler 4-cycles 3. Some properties of the Liouville equation are summarized in appendix B.…”

A slow review of the AGT correspondence

“…For holographic duals of the 6d N = (2, 0) theory, of 4d N = 2 class S theories, and in the presence of extended operators, see [250,412,432,514,776,898,[946][947][948][949][950][951][952][953][954][955][956][957][958][959][960][961][962][963].…”

Holographic uniformization and black hole attractors

“…• Based on the physics of topologically twisted SCFTs on compact manifolds, one should expect that the holographic uniformization principle is not limited to Riemann surfaces. It would be very interesting to understand this vast generalization in particular in the context of hyperbolic three-manifolds [59], as well as four-manifolds where some initial studies were performed in [60,61]. In this context holographic uniformization and its generalizations offer another example of the close relation between the physics of RG flows and the mathematics of geometric flows, similar in spirit to Ricci flow [62,63].…”