Which version of the universe's wave function develops a phase problem?

The Hartle-Hawking wave function—a proposal for the quantum state of the universe—has two versions in Anti-de Sitter space: one where the boundary geometry is summed over (fully dynamical) and one where it is held fixed, as in AdS/CFT. Computing one-loop quantum corrections, the authors find that only the fully dynamical version acquires a nontrivial imaginary phase, analogous to a known issue in de Sitter quantum gravity. Fixing the boundary eliminates the phase. Applying the same logic in reverse to de Sitter space, fixing its equatorial geometry cancels the phase there too. The upshot is that troublesome phase factors in gravitational path integrals are a symptom of leaving boundary degrees of freedom dynamical, not an intrinsic feature of the geometry.