Allowable Complex Scalars from Kaluza-Klein Compactifications and Metric Rescalings

Recently there have been discussions about which complex metrics should be allowable in quantum gravity. These discussions assumed that the matter fields were real valued. We make the observation that for compactified solutions it makes sense to demand convergence of the theory's path integral in the higher-dimensional parent theory. Upon compactification this allows for more general matter configurations in the lower-dimensional theory, in particular it allows for complex scalar fields, with a bound on their imaginary parts. Similar considerations apply to metric rescalings in the presence of higher curvature corrections. We illustrate this effect with the example of the no--boundary proposal, in which scalar fields are typically required to take complex values. We find that complex no-boundary solutions exist, and satisfy the derived bound, if the potential is sufficiently flat. For instance, for a compactification from $D$ dimensions, the bound on the imaginary part $\textrm{Im}\,(\phi)$ of the internal volume modulus reads $V_{,\phi}/V < \sqrt{\frac{D-4}{D-2}}/3\sqrt{2}.$ This leads to a mild tension with swampland conjectures.