Axion Wormholes and the AdS/CFT Factorization Problem

This work investigates the relevance of Euclidean and complex axion wormholes to the AdS/CFT factorization problem. We use a framework that defines bulk gravitational path integrals by integrating over a real Lorentz-signature contour and then, as needed, perhaps further analytically continuing the resulting functions of boundary conditions. For technical reasons we focus on the case of 2+1 bulk dimensions. The AdS boundary conditions (in any dimension) require us to impose Dirichlet boundary conditions on the standard Euclidean axion $\chi_E$. Fixing its asymptotic values on two boundary spheres to $\pm \chi_{E,\infty}$, we find such wormholes to be subdominant to a UV-sensitive endpoint contribution for $\chi_{E, \infty}$ near the real axis, and that (with our conventions) they become dominant only for $\chi_{E, \infty}$ near the negative imgainary axis. Furthermore, such wormholes are irrelevant to our computation for ${\rm Im} \chi_{E, \infty} >0$ (in the sense that the associated ascent contour fails to intersect the contour of integration). The relevance of the wormhole saddle for real positive $\chi_{E, \infty}$ is in fact a matter of choice, as the saddle then lies on a Stokes' line at which the relevant intersection number changes from zero to one.