One-loop aspects of de Sitter axion wormholes

Abstract

We discuss aspects of the Euclidean path integral around axion-supported de Sitter wormholes, at one-loop order. We numerically compute the phase of the path integral around these solutions, as well as for a certain "multiple wormholes" generalization, and interpret this phase in different regimes. When the geometry is well approximated by a sphere with a small handle, the wormhole admits an effective description as a sphere with two local operator insertions, whose positions fluctuate around the antipodal configuration. The antipodal configuration is an extremum of the position integral for the operators, but we show that it is an unstable one. Accordingly, the phase of the wormhole solution can be viewed as the Polchinski phase in the sphere, multiplied by an additional phase from the integral over positions of the effective local operators. Using our expressions for the one-loop determinant, we also estimate the EFT coefficients of the dual bilocal operators in odd spacetime dimensions, to one-loop order. Lastly, we also discuss "maximal flux" solutions, which have $S^{1}\times S^{D-1}$ geometry. Their Lorentzian continuations are Einstein static universes, so we call them "Einstein wormholes". In this limit, we determine the spectrum of fluctuations analytically and show that the phase of the path integral around this solution is entirely accounted for by the well-known instability of the Einstein static universe.

Figures (12)

• Figure 1: Illustration of the geometry of the Euclidean axion wormhole solution in de Sitter. The periodically identified direction is the $\tau$-direction, where in (a) the solution contains one fundamental period, and while in (b) it contains more than one fundamental period.
• Figure 2: An illustration of the sum over bi-local operator insertions that reproduces the effect of the wormhole at large distances. The sum runs over all operators $O_{i,j}$ consistent with the symmetries of the problem, and the $\mathcal{C}_{ij}$ are EFT coefficients. The wormhole drawn on the left-hand side is supposed to stand for the path integral over the small wormhole and all its moduli.
• Figure 3: (a) The norm density $\rho_1(\tau)$ of the light negative modes at $D=5, N=1$, for various $Q$. By tuning $Q$ to the near sphere limit, the wave-function becomes localized in the wormhole mouth region. (b) The eigenvalue $\lambda_1$ of the light negative modes for various $Q$ and various $D$. The dots are numerical data, and the underlying lines are a linear fit of $\lambda_1$ as a function of $Q$.(c) The norm density $\rho_2(\tau)$ of the light positive modes at $D=5$, for various $Q$. By tuning $Q$ to the near sphere limit, the wave-function becomes localized in the wormhole mouth region. (d) The eigenvalue $\lambda_2$ of the light positive modes for various $Q$ and various $D$. The dots are numerical data, and the underlying lines are a linear fit of $\lambda_2$ as a function of $Q$.
• Figure 4: The eigenvalue of negative light modes $\lambda_1$ in (a) and the positive light modes $\lambda_2$ in (b), when we change the gauge fixing parameter. To obtain this figure, we varied the gauge fixing parameters $\alpha$ and $\beta$ discussed in section (\ref{['altgf']}), along the curve $\alpha=\frac{(D-2)}{2(\beta D-1)}$. We observe that in the small $Q$ limit, the eigenvalue of negative light modes is gauge fixing invariant, while the eigenvalue of positive light modes is not. Here we take $D=6$.
• Figure 5: On the left-hand side, we have a wormhole saddle with small $\kappa Q$, which looks like a sphere with a small handle. This saddle is reproduced by the sphere with the leading effective operator insertions at antipodal positions.