Note on KSW-allowability of Wine-Glass Geometry

Abstract

In this note we consider no-boundary instantons and wine-glass geometries which are of interest in the context of quantum cosmology. While the former usually appears as a dominant saddle in the path-integral, the wineglass geometry can become dominant saddle in some situations. The later has been argued to have a longer inflationary phase of the Universe. Kontsevich-Segal-Witten (KSW)-allowability criterion which classifies geometries on the basis of the requirement of having a meaningful QFT on it, pushes one to analyse the allowability of the various geometries. In this note we do a simple study to seek answer to the allowabilty of no-boundary instantons and wine-glass geometries, where the later is obtained via analytical continuation of Lorentzian deSitter in pure gravity. Our simple analysis which make use of a milder version of KSW allowability criterion shows that no-boundary instanton is KSW allowed while wine-glass geometries obtained via such analytic continuation in pure-gravity are KSW disallowed. This study however doesn't covers wineglass saddles arising in gravity coupled with matter theories.

Figures (5)

• Figure 1: A visualization of Hartle-Hawking instanton (fig (a)) and the corresponding time contour (fig (b)). In fig(a) the cyan-region denotes Euclidean sphere, while the white region represent Lorentzian deSitter. In fig(b) the horizontal line correspond to euclidean deSitter which is the hemisphere, while the vertical line correspond to Lorentzian deSitter.
• Figure 2: KSW-allowability of Hartle-Hawking no-boundary instanton via Ridge criterion (fig a) and via a more rigorous procedure of extremal-curves (fig b). In fig a, the green region is KSW-allowed while the light orange region is KSW disallowed. In fig b, the red and blue curve denotes the extremal curves starting from $\tau=0$ and asymptote to $\mathcal{R}e(\tau) = \pi/2$. The green curve illustrates one such allowable path, along which the KSW function remains bounded by $\pi/\epsilon < \pi$ where $\epsilon \approx 1.036934$ throughout the length connecting $\nu_{0}=0$ and $\nu_f=\pi/2 \pm i 0.5$. From both fig (a) and (b) one can notice that the HH instanton is KSW allowed.
• Figure 3: A visualization of wineglass geometry and the corresponding time contour. In fig (a): the light-orange correspond to Euclidean AdS, the gray-region corresponds to complex geometry, is the wormhole joining EAdS to EdS, the blue-region is the section of $4$-sphere which is the Euclidean deSitter, the white-region is the Lorentzian deSitter. Fig (b) correspond to the time-contour: the vertical red line correspond to EAdS, the circular region depicts the wormhole with throat, the horizontal black line correspond to section of spherical geometry, while the vertical blue line correspond to Lorenztian dS.
• Figure 4: KSW-allowability of Wine-glass instanton via $q=0$ condition (fig a) and via Ridge criterion analysis (fig b). In fig a), the green region is $\mathcal{R}e(\sqrt{g}) >0$ while light orange region denotes $\mathcal{R}e(\sqrt{g}) < 0$ region. In fig b), green region denotes Ridge criterion allowed region, while light orange represent Ridge criterion disallowed thereby KSW disallowed region.
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