From Heun to Painlev\'e on Sasaki-Einstein Spaces and Their Confluent Limits

TL;DR

This paper explores the connection between Heun and Painlevé equations on Sasaki-Einstein spaces, analyzing monodromy data and limits that simplify the equations, with implications for holography and scalar field dynamics.

Contribution

It provides a detailed study of isomonodromic deformations of scalar fields on Sasaki-Einstein spaces, linking Heun and Painlevé equations and examining their confluent limits.

Findings

Derived Painlevé VI equations from Heun equations on Y^{p,q} spaces.

Identified limits leading to lower-rank Painlevé equations.

Analyzed monodromy data relevant for holographic scalar fields.

Abstract

The aim of this paper is to study the effect of isomonodromic deformations of the evolution of scalar fields in Sasaki-Einstein spaces in the context of holography. Here we analyze the monodromy data of the general Heun equation, resulting from a scalar on Y$^{p,q}$, thus obtaining the corresponding Painlev\'e VI equation. Furthermore we have considered limits leading to a coalescence of singularities, which in turn transform the original Painlev\'e VI equation, to one of lower rank. The confluent limits we have considered are Y$^{p,p}$, T$^{1,1} / \mathbb{Z}_2$ and Y$^{\infty, q}$.