The complex geometry of holographic flows of quiver gauge theories

TL;DR

This paper explores the geometric structure of holographic flows in quiver gauge theories, deriving a master PDE for Calabi-Yau metrics that interpolate between different geometric phases, revealing new insights into supersymmetric backgrounds.

Contribution

It introduces a novel master PDE characterizing Calabi-Yau metrics in holographic flows, linking complex geometry with supersymmetric gauge theory backgrounds.

Findings

The flow solution is described by a single second order PDE.

The Pilch-Warner solution is nearly Calabi-Yau but not Kähler.

The derived PDE connects to equations governing supersymmetric flux backgrounds.

Abstract

We argue that the complete Klebanov-Witten flow solution must be described by a Calabi-Yau metric on the conifold, interpolating between the orbifold at infinity and the cone over T^(1,1) in the interior. We show that the complete flow solution is characterized completely by a single, simple, quasi-linear, second order PDE, or "master equation," in two variables. We show that the Pilch-Warner flow solution is almost Calabi-Yau: It has a complex structure, a hermitian metric, and a holomorphic (3,0)-form that is a square root of the volume form. It is, however, not Kahler. We discuss the relationship between the master equation derived here for Calabi-Yau geometries and such equations encountered elsewhere and that govern supersymmetric backgrounds with multiple, independent fluxes.