An example of an asymptotically $AdS_2\times S^2$ metric satisfying NEC but which is not exactly $AdS_2\times S^2$

TL;DR

This paper constructs an asymptotically $AdS_2 imes S^2$ metric satisfying the Null Energy Condition that is not exactly $AdS_2 imes S^2$, providing a counterexample to a conjecture by Maldacena.

Contribution

It presents the first explicit example of such a metric that satisfies NEC but differs from the exact $AdS_2 imes S^2$, challenging previous assumptions.

Findings

Counterexample to Maldacena's conjecture

Metric satisfies NEC but is not exactly $AdS_2 imes S^2$

Includes an example with supercovariantly constant spinors

Abstract

We give an example of an asymptotically $AdS_2\times S^2$ metric, in the sense of \cite{GG}, which satisfies the Null Energy Condition but is not exactly $AdS_2\times S^2$. It is therefore a counterexample to a conjecture of Maldacena mentioned in \cite{GG}, but it does not satisfy field equations. In an appendix we give an example admitting supercovariantly constant spinors as in \cite{t1}, which is asymptotically $AdS_2\times S^2$ on one side only.