Exact renormalization flow and domain walls from holography

TL;DR

This paper explores the holographic relationship between 2d N=2 quantum field theories and 4d N=2 supergravity, providing an exact geometric description of renormalization flows and domain walls via solutions to the Painlevé VI equation.

Contribution

It establishes a covariant geometric framework for 2d renormalization group flows in N=2 theories using holography, and presents explicit exact solutions involving Painlevé VI equations.

Findings

Exact solutions to 2d RG flow equations are obtained.

Domain walls are described by solutions to Painlevé VI.

The approach links geometric structures to renormalization group dynamics.

Abstract

The holographic correspondence between 2d, N=2 quantum field theories and classical 4d, N=2 supergravity coupled to hypermultiplet matter is proposed. The geometrical constraints on the target space of the 4d, N=2 non-linear sigma-models in N=2 supergravity background are interpreted as the exact renormalization group flow equations in two dimensions. Our geometrical description of the renormalization flow is manifestly covariant under general reparametrization of the 2d coupling constants. An explicit exact solution to the 2d renormalization flow, based on its dual holographic description in terms of the Zamolodchikov metric, is considered in the particular case of the four-dimensional NLSM target space described by the SU(2)-invariant (Weyl) anti-self-dual Einstein metrics. The exact regular (Tod-Hitchin) solutions to these metrics are governed by the Painlev'e VI equation, and describe domain walls.