Axion Wormholes and the AdS/CFT Factorization Problem

TL;DR

The paper probes AdS/CFT factorization by analyzing Euclidean and complex axion wormholes within a Lorentzian bulk path integral, focusing on 2+1 dimensions and Dirichlet axion boundaries. It develops a regulated Lorentzian framework with constrained, two-boundary wormholes to assess connected partition functions $Z^{\tiny c}[S^d\sqcup S^d]$, showing that for real boundary values $\chi_\infty$ the leading contribution comes from UV endpoint saddles rather than wormhole saddles, while Euclidean wormholes become relevant only after analytic continuation to imaginary $\chi_\infty$ and under Stokes phenomena. Through a detailed construction of constrained wormholes, a careful treatment of the UV regulator $\lambda$, and a Picard-Lefschetz analysis of the Lorentzian path integral, the authors demonstrate that connected gravitational partition functions do not generically vanish due to wormholes, though their influence is highly contour- and boundary-condition-dependent. The findings highlight the crucial role of boundary conditions and contour choices in holographic factorization and offer a controlled, semiclassical approach to assessing wormhole contributions, with implications for higher-dimensional AdS/CFT and potential extensions to other deformations. $Z(\beta)$ and related quantities are handled with Lorentzian-contour prescriptions, and the analysis makes explicit how $L_0$, $L_\gamma$, and $T$ govern the saddle structure and the analytic continuation required to probe complex boundary data.

Abstract

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 $χ_E$. Fixing its asymptotic values on two boundary spheres to $\pm χ_{E,\infty}$, we find such wormholes to be subdominant to a UV-sensitive endpoint contribution for $χ_{E, \infty}$ near the real axis, and that (with our conventions) they become dominant only for $χ_{E, \infty}$ near the negative imgainary axis. Furthermore, such wormholes are irrelevant to our computation for ${\rm Im} χ_{E, \infty} >0$ (in the sense that the associated ascent contour fails to intersect the contour of integration). The relevance of the wormhole saddle for real positive $χ_{E, \infty}$ 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.

Figures (24)

• Figure 1: Gravitational path integrals are naturally taken to sum over possible topologies. As illustrated in the bottom line, wormhole topologies can lead to failures of factorization.
• Figure 2: A simple cartoon of a Euclidean wormhole that contributes to $Z[S^d\sqcup S^d]$. Our approach writes $Z[S^d\sqcup S^d]$ as a Lorentzian path integral where the timelike direction runs along the boundary (as indicated by the arrow marked $\tau_Z$), while that of Loges:2022nuw uses a timelike direction transverse to the boundary (indicated by the arrow labeled $\tau_C$). It is not yet clear if these choices define the same $Z[S^d\sqcup S^d]$. We hope to return to this question in future work.
• Figure 3: In two dimensions, the cylinder is the only orientable connected manifold that admit an everywhere non-vanishing vector field that is tangent to its boundaries. Here we show a disk with a vector field that is tangent to the boundary and which exhibits the expected zero.
• Figure 4: An AdS-Schwarzschild black hole with two surfaces of constant killing time shown in blue. The surfaces are related by $e^{-iHT}$ for some $T$. Cutting out the corresponding wedge of the spacetime and identifying the two surfaces yields a spacetime with a codimension-2 singularity at the would-be horizon bifurcation surface and which we may call a Lorentzian conical defect. The defect is located at the would-be bifurcation surface of the original black hole horizon (red dot).
• Figure 5: The Penrose diagram for a de Sitter spacetime with two surfaces of constant static time shown in blue. The surfaces are related by $e^{-iHT}$ for some $T$. Cutting out the corresponding wedge and identifying the two surfaces yields a spacetime with a codimension-2 singularity located at the would-be cosmological horizon bifurcation surface of the original spacetime (red dot).