Are $S^1\times S^2$ wormholes generic with large sources?

TL;DR

This work interrogates the conjecture that large Euclidean sources generically induce wormholes in AdS/CFT with $S^1\times S^2$ boundaries by studying a cohomogeneity-1 Einstein-scalar model in which sources do not vanish on the $t=0$ surface. The authors construct a phase diagram including wormholes, thermal AdS, and Euclidean black holes, showing wormholes arise only above a critical source and exhibit Hawking-Page-like transitions, with large wormholes stable and small ones unstable. A detailed perturbative analysis across multiple sectors confirms a single negative mode for small wormholes and no negative modes for large ones, and adding an $\ell=0$ scalar component does not render wormholes generic at fixed mass in the large-sourced limit. The results parallel known Hawking-Page-like behavior for $S^3$ boundaries and highlight that a full test of Balasubramanian et al.'s conjecture requires higher-cohomogeneity constructions, as well as a careful treatment of off-shell contributions in the gravitational path integral.

Abstract

Euclidean path integrals can be used to prepare states of a Lorentzian QFT. So long as any sources are turned off on the $t=0$ surface, the resulting Lorentzian states all belong to the same Hilbert space. Constructing more states than allowed by the Lorentzian density of states means that the resulting states must be linearly dependent. For large amplitude sources and a fixed cutoff on energy, the AdS bulk dual of this effect has been conjectured to be captured by spacetime wormholes. Wormholes should then be generic in the presence of large such Euclidean sources. This hypothesis can be studied in a context with asymptotically locally AdS$_4$ boundaries of topology $S^1 \times S^2$ in which the wormhole is supported by a source for minimally-coupled massless bulk scalars. In preparation for a later more complete study, we consider here a preliminary toy version of the model in which the spacetimes are cohomogeneity-1, but with the consequence that the sources do not vanish at $t=0$. We then find that generic sources at large masses do {\it not} lead to wormholes. Along the way we map out the phase diagram for wormhole, thermal AdS, and black hole phases of our cohomogeneity-1 ansatz. We also numerically evaluate their stability by identifying negative modes. In parallel with the previously-studied case of $S^3$ boundaries, the results are analogous to those associated with the familiar Hawking-Page transition.

Figures (13)

• Figure 1: Possible contributions to bulk path integrals computing (products or moments of) inner products of dual CFT states defined by insertions of operators of the form $e^{-\beta H}A_{\rm red}$ with those defined by insertions $e^{-\beta H}A_{\rm blue}$. Insertions of $A_{\rm red}, A_{\rm blue}$ are shown respectively as red/blue dots. The disconnected solution at left generally has large Euclidean action because these fields are distinct in the bulk low-energy theory, so that the bulk inner product is small at this EFT level. The wormhole geometry is thus expected to have lower action. But the Euclidean action of the connected contribution at right need not become large in this limit and thus may be expected to dominate.
• Figure 2: Wormholes with $S^1$ boundaries with shading indicating the rotating phase of a complex scalar $\phi$. The $S^2$ factor of our setup is not shown. Blue shading indicates regions where $|{\rm Re} \phi| > |{\rm Im} \phi|$, while red indicates $|{\rm Re} \phi| < |{\rm Im} \phi|$. The left panel thus shows a case with angular momentum $n=1$ on the $S^1$ while the right panel shows $n=1/2$ (which arises if we allow antiperiodic scalars). That such boundary conditions model the insertion of $e^{-\beta H}A_{\mathrm{red/blue}}$ in a dual CFT can be seen by noting that the $n=1/2$ case may be interpreted as a less well-localized version of the right panel of figure \ref{['fig:nhalfwh2']}.
• Figure 3: The minimum radius ($r_+$ or $r_0$) of the $S^2$ for the black holes and wormholes for $\beta=1$ with either $\ell=1$ (solid lines) or $\ell=2$ (dashed lines). Blue/orange describes the large/small black hole branches, while red/green shows data for the large/small wormholes.
• Figure 4: Comparison of the actions for black holes (left) or wormholes (right) with that of thermal AdS for $\ell=1\,,\beta=1$. $S_{\rm BH}$ and $S_{\rm th}$ are the actions for a single copy of black hole or thermal AdS geometry, see \ref{['eq:bhS']} and \ref{['eq:thS']}, while $S_{\rm WH}$ is half of the action for the 2-boundary wormhole. Phase transitions occur when the colored curves cross the dashed gray line. As in figure \ref{['fig:fxibeta']}, orange/blue and green/red denote small/large black holes or wormholes.
• Figure 5: The left and right panels respectively show black hole and wormhole solutions at fixed $V$ for both $\ell=1$ and $\ell=2$. In the first case we choose $V=1$ so that black holes exist for both values of $\ell$. In the second we choose $V=7/2$ so that we find wormholes for both values of $\ell$. The vertical axis is the effective 'temperature' defined by $T=\beta^{-1}$. The color coding is the same as that in figure \ref{['fig:fxibeta']}. The solid/dashed lines (solid/open dots) are data for $\ell=1$ and $\ell=2$, respectively.