On the Geometric Properties of AdS Instantons

TL;DR

This paper investigates the geometric properties of AdS instantons, focusing on harmonic functions and Killing vectors, to understand their stability and boundary conditions in the context of AdS/CFT correspondence.

Contribution

It provides a detailed analysis of the harmonic functions and Killing vectors of AdS instantons, revealing their geometric structure and implications for vacuum stability.

Findings

Existence of d-2 harmonic functions on d-dimensional instantons.

Instantons with restrictive boundary conditions have d-1 commuting Killing vectors.

Most Killing vectors are duals of harmonic one-forms without fixed points.

Abstract

According to the positive energy conjecture of Horowitz and Myers, there is a specific supergravity solution, AdS soliton, which has minimum energy among all asymptotically locally AdS solutions with the same boundary conditions. Related to the issue of semiclassical stability of AdS soliton in the context of pure gravity with a negative cosmological constant, physical boundary conditions are determined for an instanton solution which would be responsible for vacuum decay by barrier penetration. Certain geometric properties of instantons are studied, using Hermitian differential operators. On a $d$-dimensional instanton, it is shown that there are $d-2$ harmonic functions. A class of instanton solutions, obeying more restrictive boundary conditions, is proved to have $d-1$ Killing vectors which also commute. All but one of the Killing vectors are duals of harmonic one-forms, which are gradients of harmonic functions, and do not have any fixed points.