Separability in Consistent Truncations

TL;DR

This paper investigates the conditions under which the Hamilton-Jacobi equation is separable in consistent supergravity truncations on spheres, linking geometric symmetries to solution integrability and providing a classification of separating isometries.

Contribution

It introduces the concept of separating isometries, classifies them for sphere truncations, and connects their presence to the separability of the Hamilton-Jacobi equation in supergravity compactifications.

Findings

Separable Hamilton-Jacobi equations occur when a consistent truncation has a separating isometry.

Gauge vectors must form an abelian subgroup of the separating isometry for separability.

The paper classifies all separating isometries for spheres $S^n$, $n=2,...,7$, and identifies corresponding Killing tensors.

Abstract

The separability of the Hamilton-Jacobi equation has a well-known connection to the existence of Killing vectors and rank-two Killing tensors. This paper combines this connection with the detailed knowledge of the compactification metrics of consistent truncations on spheres. The fact that both the inverse metric of such compactifications, as well as the rank-two Killing tensors can be written in terms of bilinears of Killing vectors on the underlying "round metric," enables us to perform a detailed analyses of the separability of the Hamilton-Jacobi equation for consistent truncations. We introduce the idea of a separating isometry and show that when a consistent truncation, without reduction gauge vectors, has such an isometry, then the Hamilton-Jacobi equation is always separable. When gauge vectors are present, the gauge group is required to be an abelian subgroup of the separating isometry to not impede separability. We classify the separating isometries for consistent truncations on spheres, $S^n$, for $n=2, \dots, 7$, and exhibit all the corresponding Killing tensors. These results may be of practical use in both identifying when supergravity solutions belong to consistent truncations and generating separable solutions amenable to scalar probe calculations. Finally, while our primary focus is the Hamilton-Jacobi equation, we also make some remarks about separability of the wave equation.