Recent developments in high-energy physics have brought to light the need to formulate a consistent notion of distance between gravitational vacua. Specifically, within the Swampland program, the AdS Distance conjecture proposes to assign such a notion for AdS vacua in quantum gravity. Beginning with very basic AdS vacua in string theory, I will expand upon this concept, providing a more formal framework for it.
Firstly, I will introduce a precise definition of a metric for the space of conformal variations of AdS. This metric, however, turns out to be negative, resulting in an ill-defined distance, a property related to the famous conformal factor problem in quantum gravity. Nonetheless, in string theory, variations in the AdS conformal factor are accompanied by variations in the internal dimensions and background flux. Interestingly, the inclusion of these variations can change the sign of the metric over the space of AdS vacua variations. I will propose a consistent procedure for deriving the distance between AdS vacua by introducing the notion of an "action metric," which accounts for all of these variations simultaneously. I will test this procedure by focusing on AdS4 and AdS7 Freund-Rubin vacua in M-theory and demonstrate that it yields a well-defined and positive distance.