Boundary conditions for axionic wormholes, imaginary distance bound and KSW allowability

Abstract

We study four-dimensional axion wormholes in the Lorentzian mini-superspace path integral using dual scalar and three-form flux formulations. We analyze this duality for generic metric boundary conditions and real lapse integration contour, and show that it holds on any background, including complex ones. With the Dirichlet condition on the metric, the fixed-flux path integral is evaluated exactly. In the Euclidean regime, saddle geometries organize into same-side and cross-throat segments of the Giddings-Strominger (GS) wormhole, depending on whether the two boundaries lie on the same or opposite sides of the throat. A Picard-Lefschetz analysis in the covering plane of lapse reveals that the cross-throat saddle has vanishing intersection number, and therefore does not contribute. In the asymptotically flat limit, it corresponds to the imaginary wormhole that saturates the imaginary distance bound (IDB). The same conclusion follows independently on the scalar side of the duality, and is confirmed by the exact amplitude. Replacing the Dirichlet condition with a one-parameter Neumann condition, the cross-throat saddle that leads to the imaginary wormholes remains irrelevant for the physical lift of the contour. This purely imaginary parameter characterizing the Neumann condition interpolates continuously between the half- and complete-wormhole geometries. Convergence of the sum over fixed-charge sectors imposes a corresponding interpolating bound not only on the imaginary part of the boundary axion but also on the parameter characterizing the Neumann boundary condition. An independent analysis based on the KSW criterion reproduces exactly the same bound, emphasizing its compatibility with the IDB.