A positive metric over DGKT vacua

We study the notion of a metric over the space of AdS solution in string theory, leading to an associated distance between them. Such a distance is the idea underlying the AdS distance conjecture. We utilise the previously developed prescription for extracting such a metric: taking an off-shell quadratic variation of the string theory effective action and then evaluating it over the space of on-shell solutions. It was shown that this prescription leads to a well-defined positive metric over M-theory Freund-Rubin vacua. In this work, we use the same prescription to calculate the metric over type IIA DGKT vacua. These are much more involved, they have multiple flux parameters and exhibit scale separation. While it remains an open question whether these vacua exist as fully localised solutions of string theory, they are well-defined within the four-dimensional effective theory, which is all that is required for the calculation. We find that they also have a positive metric over them. Interestingly, this metric turns out to be independent of the many flux parameters in the solution, similarly to what happens for metrics over scalar field spaces. This non-trivial flux cancellation, as well as results from explicit vacua, lead us to propose a Swampland condition: that the metric over the space of vacua in quantum gravity, as defined by the above prescription, is always positive.