# Stringy Bubbles Solve de Sitter Troubles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nearly Singular Spacetimes

#### 2.1. Warped Deformed Conifolds and Alike

#### 2.2. De Sitter Bubble-Worlds

## 3. A Discretuum of Toy Models

## 4. Calabi–Yau 5-Folds

- ${\mathbb{P}}^{1}\setminus \left\{2\phantom{\rule{0.166667em}{0ex}}\mathrm{pts}.\right\}\approx {\mathbb{C}}^{*}$ (a non-compact cylinder), which is well known to be Ricci-flat.
- ${T}^{2}\setminus \left\{\mathrm{pt}.\right\}$ is a 1-handled disc, and so a hyperbolic non-compact surface.

- The Kähler class of ${\mathbb{P}}^{2}$ is positive over “all complex submanifolds,” all of which are equivalent to the ${\mathbb{P}}^{1}$ at the North Pole “infinity” (This refers to the standard cell decomposition ${\mathbb{P}}^{2}\approx {\mathbb{C}}^{2}\cup {\mathbb{P}}^{1}$);
- There is therefore a Riemannian metric that differs from the above Kähler metric only in being null over the ${\mathbb{P}}^{1}$ at the North pole;
- which is therefore a valid metric on $({S}^{4}\setminus \left\{\mathrm{Noth}\phantom{\rule{4.pt}{0ex}}\mathrm{pole}\right\})\approx {\mathbb{C}}^{2}$, and fails in those positivity requirements only at the North pole, where it vanishes—and so is nowhere negative.

## 5. Summary, Outlook and Conclusions

**1**) a careful analysis of multi-parameter and near-singular configurations, and (

**2**) focusing on exceptional sub-spacetimes within larger-dimensional spacetimes with non-factoring geometry. These two ideas in fact naturally resonate closely with the discretuum of “axilaton” models [10,12,13,14,15] that may be viewed as a non-holomorphic and non-analytic deformation of the stringy cosmic string (’brane) scenario [42,43].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Effect of Defects

**Figure A1.**A disc (

**middle**, suggesting a hyperbolic metric), may be compactified to a sphere (

**left**) by identifying the entire boundary with the “North pole.” Identifying the boundary segments in the indicated interleaved fashion produces the genus-2 Riemann surface (

**right**, a basis of 1-cycles identified with the indicated boundary segments).

## Notes

1 | |

2 | A candidate for the observable four-dimensional world with its geometry unspecified is denoted ${W}^{1,3}$, while ${M}^{1,3}$, ${\mathrm{dS}}^{1,3}$ and ${\mathrm{AdS}}^{1,3}$ specify Minkowski, de Sitter and anti de Sitter geometries, respectively. |

3 | |

4 | There is also a natural connection to the more recent and rather vast cobordism generalization [55]. |

5 | The stringy cosmic string-like [59] limit includes a total of $12+\left|\xi \right|$ supersymmetric 7-branes. |

6 | To be precise: for ${Z}_{-}\left(z\right)$ in (4a), each of the two circular boundaries of ${\mathcal{Y}}_{\perp}^{2}$ shrinks to a point as ${z}_{0}\to \infty $, thus rendering ${\mathcal{Z}}_{-}$ compact. For ${Z}_{-}\left(z\right)$ in (4a), $\mathcal{Z}\phantom{\rule{-0.166667em}{0ex}}\approx \phantom{\rule{-0.166667em}{0ex}}{\mathbb{C}}^{*}$ is non-compact and two points must be added to compactify ${\mathcal{Z}}_{+}\to {\mathbb{P}}^{1}$. |

7 | We will return to this non-trivial K3-fibration in Section 4. |

8 | The product $\mathcal{A}\phantom{\rule{-0.166667em}{0ex}}\u22ca\phantom{\rule{-0.166667em}{0ex}}\mathcal{B}$ denotes that the warp-factors in the block-diagonal metric vary over $\mathcal{B}$. |

9 | Being Fano, ${c}_{1}\left(\mathfrak{S}\right)\phantom{\rule{-0.166667em}{0ex}}>\phantom{\rule{-0.166667em}{0ex}}0$, implies the scalar curvature invariant to be positive, $R\left(\mathfrak{S}\right)>0$. |

10 | For our present purposes, a foliation $X\phantom{\rule{-0.166667em}{0ex}}\divideontimes \phantom{\rule{-0.166667em}{0ex}}Y$ means that the total space looks locally at every point as a direct product of local portions of the two factors, X and Y. |

11 | Supersymmetry is in string theory largely correlated with complex structure, and as mentioned above, $({S}^{4}\setminus \left\{\mathrm{pt}.\right\})\approx {\mathbb{C}}^{2}$ of course admits a complex structure, for which the excised point is an obstruction. |

12 | This may be pictured as a two-step process: ( 1) “open” the point into a circular boundary, then (2) identify segments of the boundary according to the template in Figure A1, middle. |

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**Figure 1.**Some possible geometries in target spacetime; these simplified illustrations hint at the variations in any of the relevant structures (complex, Kähler, symplectic, supersymmetry), and may be combined in various ways.

**Figure 2.**A depiction of the ${W}_{z}^{1,3}\u22ca{\mathcal{Y}}_{\perp}^{2}$ fibration with ${Z}_{+}\left(z\right)$ (

**left**) and ${W}_{z}^{1,3}\u22ca{\mathcal{Y}}_{\perp}^{2}$ with ${Z}_{-}\left(z\right)$ (

**right**). Far left and far right: the proper distance plotted vertically in ${\mathcal{Y}}_{\perp}^{2}$, indicating the radial dependence of the circumference—which is obscured in the two central depictions. Only the mid-radius fiber, ${W}_{z=0}^{1,3}$, of the fibration ${W}_{z}^{1,3}\u22ca{\mathcal{Y}}_{\perp}^{2}$ is depicted as a vertical cylinder, ${W}_{z=0}^{1,3}\times {S}^{1}$; it defines the “inside” and “outside’ part of the six-dimensional spacetime’; see (7), below.

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Berglund, P.; Hübsch, T.; Minic, D.
Stringy Bubbles Solve de Sitter Troubles. *Universe* **2021**, *7*, 363.
https://doi.org/10.3390/universe7100363

**AMA Style**

Berglund P, Hübsch T, Minic D.
Stringy Bubbles Solve de Sitter Troubles. *Universe*. 2021; 7(10):363.
https://doi.org/10.3390/universe7100363

**Chicago/Turabian Style**

Berglund, Per, Tristan Hübsch, and Djordje Minic.
2021. "Stringy Bubbles Solve de Sitter Troubles" *Universe* 7, no. 10: 363.
https://doi.org/10.3390/universe7100363