Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points

TL;DR

This work presents a Lorentzian quantum cosmology framework grounded in Lefschetz thimble analysis and resurgence. By resolving Stokes-line ambiguities through complex deformations, it identifies the correct thimble structure and demonstrates that, in the physically relevant parameter regime, the tunneling saddle dominates over the no-boundary saddle. A key result is the cancellation of ambiguities between the Borel resummation of perturbative series around tunneling saddles and contributions from no-boundary saddles, signaling the presence of resurgence in minisuperspace quantum cosmology. Additionally, the analysis shows that with Neumann boundary conditions the wave function reduces to Airy-function form with trivially resummable perturbative content, underscoring the role of boundary conditions in shaping the semiclassical structure of quantum cosmology.

Abstract

We revisit the path-integral approach to the wave function of the Universe by utilizing Lefschetz thimble analyses and resurgence theory. The traditional Euclidean path-integral of gravity has the notorious ambiguity of the direction of Wick rotation. In contrast, the Lorentzian method can be formulated concretely with the Picard-Lefschetz theory. Yet, a challenge remains: the physical parameter space lies on a Stokes line, meaning that the Lefschetz-thimble structure is still unclear. Through complex deformations, we resolve this issue by uniquely identifying the thimble structure. This leads to the tunneling wave function, as opposed to the no-boundary wave function, offering a more rigorous proof of the previous results. Further exploring the parameter space, we discover rich structures: the ambiguity of the Borel resummation of perturbative series around the tunneling saddle points is exactly canceled by the ambiguity of the contributions from no-boundary saddle points. This indicates that resurgence also works in quantum cosmology, particularly in the minisuperspace model.

Figures (5)

• Figure 1: Contour plot of $\textrm{Re}\left[F(x)\right]$ over the complex $x$ plane for $\alpha =1$, $\beta =-3$, $\gamma =-0.1$, and $\hbar =1$, corresponding to the parameter region (i) $12\alpha \gamma +\beta^2 >0$ with $\beta <0$. The black (red) circles represent the tunneling (no-boundary) saddle points, while the blue (red) lines denote the (dual) Lefschetz thimbles. The black horizontal line denotes the original integration contour.
• Figure 2: A similar plot to Fig. \ref{['fig:Picard-Lefschetz1']} with $\alpha =1$, $\beta =-3$, $\gamma =-1$ corresponding to the parameter region (ii), $12\alpha \gamma +\beta^2 <0$. This corresponds to the no-boundary and tunneling proposals.
• Figure 3: Complex deformation of Fig. \ref{['fig:Picard-Lefschetz2']} [in region (ii)] with $\hbar = e^{i\theta }$.
• Figure 4: A similar plot to Figs. \ref{['fig:Picard-Lefschetz1']} and \ref{['fig:Picard-Lefschetz2']} with $\alpha =1$, $\beta =3$ and $\gamma =-0.1$ corresponding to region (iii), $12\alpha \gamma +\beta^2 >0$, with $\beta >0$.
• Figure 5: Complex deformation of Fig. \ref{['fig:Picard-Lefschetz3']} [in region (iii)] with $\hbar = e^{i\theta }$.