Metrics over multi-parameter AdS vacua

We study the notion of a distance between different AdS vacua of string theory. The distance is measured by a metric that is derived by taking an off-shell quadratic variation of the effective action, and evaluating it over families of vacua. We calculate this metric for increasingly complex families of vacua. We first consider the two-parameter families of solutions of type $\mathrm{AdS}_4 \times \mathbb{C}\mathrm{P}^3$. We find that the metric is flat and positive, and so yields a well-defined distance along any path in the space of solutions. We then consider solutions of type $\mathrm{AdS}_3 \times S^3 \times \mathrm{CY}_2$ which have two (non-compact) flux parameters as well as a moduli space. We find that the space of solutions factorises between directions which vary the AdS radius, and the moduli space. The metric over AdS variations is flat and positive, and the metric over the moduli space is the usual one. Finally, we consider solutions of type $\mathrm{AdS}_3 \times S^3 \times S^3 \times S^1$ which also have a further direction in the space of solutions that is compact. We find that the metric is flat only on non-compact directions in the space of solutions. Restricting to such directions, we evaluate the metric and find it is positive definite and therefore yields a well-defined distance along any path.