Holographic duality, supersymmetry, and Painleve equation

TL;DR

This paper establishes a holographic duality between 2d N=2 quantum field theories and 4d N=2 supergravity, linking renormalization flows to solutions of the Painleve VI equation via anti-self-dual Einstein metrics.

Contribution

It provides a novel geometrical framework connecting holography, supersymmetry, and integrable equations through explicit solutions of the Painleve VI equation.

Findings

Holographic correspondence between 2d N=2 theories and 4d N=2 supergravity.

Renormalization flow equations interpreted as target space geometry constraints.

Exact solutions governed by Painleve VI equation.

Abstract

An example of the holographic correspondence between 2d, N=2 quantum field theories and classical 4d, N=2 supergravity theories is found. The constraints on the target space geometry of the 4d, N=2 non-linear sigma-models in N=2 supergravity background are interpreted as the renormalization flow equations in two dimensions. Our geometrical description of the renormalization flow is manifestly covariant under reparametrization of the 2d coupling constants. The proposed holography is described in terms of the (Weyl) anti-self-dual Einstein metrics, whose exact regular (Tod-Hitchin) solutions are governed by the Painleve VI equation.