Stringy Bubbles Solve de Sitter Troubles

Per<EMAIL_ADDRESS>Tristan<EMAIL_ADDRESS>and
Djordje<EMAIL_ADDRESS>

1Department of Physics and Astronomy, University of New Hampshire, Durham, NH
03824, U.S.A.
2Department of Physics and Astronomy, Howard University, Washington, D.C.
20059, U.S.A.
3Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A.

###### Abstract

Finding four-dimensional de Sitter spacetime solutions in string theory has
been a vexing quest ever since the discovery of the accelerating expansion of
the universe. Building on a recent analysis of bubble-nucleation in the decay
of (false-vacuum) AdS backgrounds where the interfacing bubbles themselves
exhibit a de Sitter geometry we show that this resonates strongly with a
stringy cosmic brane construction that naturally provides for an exponential
mass-hierarchy, and localization of both gravity and matter, in addition to an
exponentially suppressed positive cosmological constant. Finally, we argue
that these scenarios can be realized in terms of a generalization of a small
resolution of a conifold singularity in the context of a (Lorentzian) Calabi-
Yau 5-fold, where the isolated (Lorentzian) two complex dimensional Fano
variety is a four-dimensional de Sitter spacetime.

###### Contents

  1. 1 Introduction
  2. 2 Nearly Singular Spacetimes
    1. 2.1 Warped Deformed Conifolds and Alike
    2. 2.2 de Sitter Bubble-Worlds
  3. 3 A Discretuum of Toy Models
  4. 4 Calabi-Yau 5-Folds
  5. 5 Summary, Outlook and Conclusions
  6. A The Effect of Defects

##  1 Introduction

For almost a quarter of a century, a specter has been haunting string theory:
the accelerated expansion of our universe implies an asymptotically and
approximately de Sitter (dS) geometry with a small but positive cosmological
constant [1, 2]. Constructing solutions in string theory with these features
has been argued to be notoriously hard if not impossible: see [3, 4, 5] for
recent comprehensive reviews. However, recent work [6, 7] and [8], as well as
[9], turn out to correspond naturally with a stringy cosmic dS-brane toy-model
[10, 11, 12, 13, 14] which we furthermore connect with a generalization of the
proposal that allows for more general spacetime varying string vacua [15].

The main ideas are the following: (1) Both the AdS-decaying [6, 7] and the so
called “axilaton” [10, 11, 12, 13, 14] scenarios are near a singularity: They
are “resolutions/smoothings” of a singular/critical model, and so are
inherently non-perturbative [8]. As such, a closer examination can (and does)
find special configurations in which the various competing contributions do
allow for a four-dimensional (sub-)spacetime of the desired dS geometry. (2)
Just as in the AdS-decaying scenario, the axilaton models can be made
dynamical by choosing their anisotropy, $\omega$, differently on the two sides
of the candidate for the observed universe, $\text{dS}^{1,3}_{z=0}$. (3)
Calabi-Yau 5-folds, $\mathfrak{X}_{5}$, provide an auxiliary Euclidean
rendition of the total Lorentzian spacetime, $\mathscr{X}^{1,9}$, and a
natural “home” for such phase-transition nucleating “bubble-worlds”: Just as
in the AdS-decaying scenario, the “exceptional” (real codimension-six) sub-
spacetimes $W^{1,3}$ naturally have positive curvature, while their complement
and local neighborhood in $\mathscr{X}^{1,9}$ is naturally hyperbolic.

The key effects of considering the theory near a singularity are discussed in
Section 2, especially focusing on the conifold in 2.1, which leads us to
review the AdS-decaying scenario in 2.2. In Section 3 we recall the salient
features of the axilaton toy models and highlight the similarities in the
overall spacetime geometry with the AdS-decaying solutions. This in turn lets
us modify the former so as to afford a dynamical scenario akin to the latter.
The global geometry of the ten-dimensional spacetime is reviewed in Section 4,
where we present evidence that the most generic of spacetime geometries in
string theory necessarily include exceptional four-dimensional sub-spacetimes
of positive curvature, the simplest of which being $\text{dS}^{1,3}$. Finally,
section 5 summarizes our key points and conclusions, while Appendix A collects
a few additional details about (spacetime) defects and their role in
understanding dS solutions in string theory.

##  2 Nearly Singular Spacetimes

The early discovery that certain singular features in spacetime are innocuous
to string dynamics [16, 17] shows that the choice of available geometries is,
in this respect, considerably more general than in conventional quantum field
theory. A large class of such singular geometries affords topology change [18,
19, 20, 21], in a milder sense also [22, 23], and so involve some form of a
phase transition. This tends to harbor radical effects both in physics and in
geometry: massless black holes [24], exoflops [22, 25], $D$-branes [26], and
orientifolds [27, 28, 29], and often involve strong coupling effects.

###  2.1 Warped Deformed Conifolds and Alike

Models that are even just near being singular then invariably involve multiple
competing contributions to the effective (target-spacetime) action, some of
which are often difficult to calculate. Nevertheless, suitable approximations
have been carefully considered and well justified, as is the case of the
warped deformations of conifold singularities [30, 31, 32]; see also [33, 34].
Here, the complex structure deformation of the conifold singularity and the
relative size of associated vanishing cycles, the $\overline{\text{D3}}$-brane
contributions, and the metric warping all affect the target spacetime physics.
A detailed examination of such a nearly singular warped direct product
compactification scenario finds a fine-tuned regime within the parameter space
in which the multiple competing contributions can be shown to induce a
metastable dS metric in the non-compact spacetime factor [8]; see however also
the review [35] and the references therein. Note that the vanishing cycles in
the deformation-singularizing complex, compact Calabi-Yau 3-folds are real
three-dimensional subspaces, matching in dimension the “probing”
$\overline{\text{D3}}$-branes. Thus, this example indicates that vacua with a
stringy dS spacetime could require a highly specialized set of circumstances
and qualities.

Mirror symmetry guarantees that the same type of phenomenon occurs near the
conifold singularity on the Kähler (or symplectic) structure side, as is the
case with very small relative sizes of so-called exceptional sets444In a
standard “blowup surgery,” these exceptional sets have long since been
familiar in physics as gravitational instantons [36, 37], while their higher
codimension “small resolution” analogues show up as the so-called worldsheet
instanton “$\mathscr{O}(-1,-1)$-curves” in Calabi-Yau 3-folds [38]. in a
Calabi-Yau 3-fold compactification. As with the near singularization by
complex structure deformation [8], one again expects a detailed balance of
various contributing factors. While exact calculations on this, Kähler, side
are much harder in general and not amenable to explicit probing and direct
computation, such mirror-symmetric configurations are qualitatively identical.
In turn, however, Calabi-Yau 3-folds are known to generically have many such
exceptional sets, real two- and four-dimensional subspaces, each one of which
“resolves” a singularity by replacing it with a complex subvariety while
preserving the complex structure outside the surgery locus [39, 40, 38]; by
mirror symmetry, the (nearly) vanishing 3-cycles are then as abundant. This
implies that the pool of such nearly singular models, on both mirror-symmetry
sides, is populous albeit sub-generic, as the individual such models involve
highly specialized arrangements.

Generally, the examples discussed so far are static background configurations,
wherein the observable four-dimensional spacetime555A candidate for the
observable four-dimensional world with its geometry unspecified is denoted
$W^{1,3}$, while $M^{1,3}$, $\text{dS}^{1,3}$ and $\text{AdS}^{1,3}$ specify
Minkowski, de Sitter and anti de Sitter geometries, respectively., $W^{1,3}$,
and the extra dimensions ($\mathscr{Y}^{6}$) form a rigid direct product, as
sketched at Figure 1 (a), where the geometry of $\mathscr{Y}^{6}$ and of the
observable four-dimensional spacetime are independent of each other.

(a) rigid product(b) fibration (w/singularity)(c) non-analyticity(d)
exceptional subset 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

Allowing for $\mathscr{Y}^{6}$ to vary in some observable spatial directions
was shown to necessarily include singularizations of $\mathscr{Y}^{6}$ that
manifest as stringy cosmic strings (’branes) [41, 42], as sketched in Figure 1
(b). Allowing furthermore for non-analytic variations of the geometry of some
of the extra dimensions [43, 44] warps the overall geometry and can allow for
a simultaneous emergence of exponential mass-hierarchy, localization of
gravity, matter, and an exponentially suppressed cosmological constant in
$\text{dS}^{1,3}$ [10, 11, 12, 13]; see the sketch in Figure 1 (c). We will
return to these constructions in Section 3.

###  2.2 de Sitter Bubble-Worlds

Consider now, in turn, the recent dynamical proposal with the spacetime
geometry consisting of two copies of $\text{AdS}^{1,4}$, glued together non-
analytically across a 3-brane-world, $\text{dS}^{1,3}_{z=0}$ [6, 7]; see also
[45, 46, 47]; herein, we relabel coordinates so the shell is located at
$z\\!=\\!0$. The remaining five dimensions compactified on a suitable space of
positive curvature, such as $S^{5}$, complete a direct product ten-dimensional
spacetime, as expected in string theory.

The hallmark novelty in the new “AdS-decaying” proposal [6, 7] as compared
with earlier literature is that the key subspace, $\text{dS}^{1,3}_{z=0}$,
emerges as a “bubble,” a domain wall-like interface between a metastable,
false-vacuum $\text{AdS}^{1,4}_{\rm out}$ and the nucleated, expanding true-
vacuum $\text{AdS}^{1,4}_{\rm in}$. The discontinuity at $z\\!=\\!0$ in the
total space must be sourced (via standard matching conditions) by a suitably
chosen but otherwise unspecified matter distribution, which then also provides
the interfacing shell, $\text{dS}^{1,3}_{z=0}$, with a constant tension. It is
then further verified that neither bulk nor shell-localized matter invalidate
the main result, that the “shellworld,” $\text{dS}^{1,3}_{z=0}$, is indeed a
dS universe with the cosmological constant that can have phenomenologically
acceptable values. As in [44], the graviton is shown to obey a Schrödinger-
like equation, which guarantees the existence of a graviton mode duly
localized to the shell, $\text{dS}^{1,3}_{z=0}$, although here this is not the
mode trapped by the $\delta(z)$-function well [6, 46]. Note that the so-
obtained spacetime geometry is a combination of the sketches in Figure 1 (a)
and Figure 1 (d): $\text{dS}^{1,3}_{z=0}$ appears as a (non-factor) subspace
within the first five-dimensional spacetime factor of $(\text{AdS}^{1,4}_{\rm
in}\mathop{\texttt{\\#}}\limits\text{AdS}^{1,4}_{\rm out})\times S^{5}$.

Studied ever since the original proposal [48] (see [49] for a recent update),
such vacuum decay phenomena have been estimated to be generic in string theory
[50, 51]666Refs. [50, 51] in fact conjecture that all de Sitter vacua with
small cosmological constant decay faster than their horizon size; the latter
of these two notes the contradiction with the results of Ref. [52]., implying
then the same also for nucleation, a subset of which resulting in de Sitter
bubble-worlds. These phenomena are again “near” phase transitions, for which
no perturbative description can be expected to be complete. Notably, the
analysis of Refs. [6, 7] focuses on the dynamically enfolding aftermath of
such a phase transition, and so again pertains to nearly singular
configurations. Thus, both in the claimed ubiquity and in this inherently non-
perturbative nature, this is reminiscent of the string theory realization [19,
20] of “Reid’s phantasy” [40]. 777There is also a natural connection to the
more recent and rather vast cobordism generalization [53]. In fact, as we will
argue shortly, this can be made more specific in the context of spacetime
varying string vacua generalizing the early work by one of the authors [15].
To see why this is the case, we will however next turn to a brief review of
the axilaton toy model [12, 13, 14] in order to showcase the similarities with
the $\text{AdS}^{1,4}$-decay model [6, 7].

##  3 A Discretuum of Toy Models

Generally accommodating the overall geometry types sketched in Figure 1 (a–c)
and aiming for a (de Sitter) candidate four-dimensional spacetime, the metric
Ansatz has a direct sum format,
$\mathrm{d}s^{2}=A^{2}\,\mathrm{d}s^{2}_{\scriptscriptstyle
1,3}+B^{2}\,\mathrm{d}s^{2}_{\scriptscriptstyle 6}$, where
$\mathrm{d}s^{2}_{\scriptscriptstyle 1,3}$ and
$\mathrm{d}s^{2}_{\scriptscriptstyle 6}$ are appropriate line elements for the
candidate observable spacetime, $W^{1,3}$, and its co-factor,
$\mathscr{Y}^{6}$, while $A,B$ are “warp factors” that are most often allowed
to vary over at least some directions in $\mathscr{Y}^{6}$. The particular
class of toy models [10, 11, 12, 13] further specializes to

$\mathrm{d}s^{2}=A^{2}(z)\,\mathrm{d}s^{2}_{1,3}+\ell^{2}B^{2}(z)(\mathrm{d}z^{2}{+}\mathrm{d}\theta^{2})_{\mathscr{Y}^{2}_{\perp}}+\mathrm{d}s^{2}_{\mathscr{Y}^{4}},$
(1)

where $\ell$ sets the length-scale in $\mathscr{Y}^{2}_{\perp}$, so the warp
factors $A(z),B(z)$ and the coordinates $(z,\theta)\in\mathscr{Y}^{2}_{\perp}$
are dimensionless. The observable spacetime ($\mathrm{d}s^{2}_{1,3}$) is
expected to have de Sitter geometry, and the warp factors are chosen to depend
only on the log-radial coordinate, $z$; also, $\theta\simeq\theta+2\pi$ is the
standard azimuthal angle in $\mathscr{Y}^{2}_{\perp}$. The last summand,
$\mathrm{d}s^{2}_{\mathscr{Y}^{4}}$, could also be warped but is for now
assumed to be independent of the other six coordinates; see however below.

This class of toy models also neglects all other matter fields, but explicitly
includes the “universal” axion-dilaton (“axilaton” ) field,
$\tau\mathrel{\buildrel\rm def\over{=}}B+ig_{s}^{-1}e^{-\Phi}$, in string
theory, which exhibit an $\mathop{\textrm{SL}}(2;\mathbb{Z})$-monodromy owing
to modular invariance [54]. Assuming then that $\tau=\tau(\theta)$ separates
variables in an evidently non-holomorphic (and so non-supersymmetric) way, and
admits two 3-parameter classes of such solutions, which the
$\mathop{\textrm{SL}}(2;\mathbb{Z})$-monodromy restricts to a discretuum.
Notably, $\tau=\tau(\omega\theta)$, and the
$\mathscr{Y}^{2}_{\perp}$-anisotropy, $\omega$, appears in the Einstein-
Friedan equation [55, 56],

$\displaystyle R_{\mu\nu}$ $\displaystyle={\cal
G}_{\tau\bar{\tau}}\,(\partial_{\mu}\tau)(\partial_{\nu}\bar{\tau})\mathrel{\buildrel\rm
def\over{=}}\widetilde{T}_{\mu\nu}=\mathrm{diag}[\>\underbrace{0,0,0,0}_{\text{dS}^{1,3}},\,\underset{z}{0},\underset{\theta}{(\omega/2\ell)^{2}},\,\underbrace{0,0,0,0}_{\mathscr{Y}^{4}}\>]~{},$
(2) $\displaystyle{\cal G}_{\tau\bar{\tau}}$
$\displaystyle=\frac{-1~{}~{}}{(\tau{-}\bar{\tau})^{2}}={\cal
G}_{\bar{\tau}\tau},\quad{\cal G}_{\tau\tau}=0={\cal
G}_{\bar{\tau}\,\bar{\tau}}.$ (3)

and sources the warped geometry (1) [12, 13]:

$\displaystyle A(z)$
$\displaystyle=Z_{\pm}(z)\Big{(}1-\frac{\omega^{2}z_{0}^{2}}{160}Z_{\pm}(z)^{2}+\dots\Big{)},$
(4a) $\displaystyle B(z)$ $\displaystyle=\frac{1}{\ell
z_{0}\sqrt{\Lambda_{b}}}\Big{(}1-\frac{\omega^{2}z_{0}^{2}}{40}Z_{\pm}(z)^{2}+\dots\Big{)}~{},$
(4b) with the harmonic functions $\displaystyle Z_{\pm}(z)$
$\displaystyle\mathrel{\buildrel\rm def\over{=}}1\pm|z|/z_{0},\quad z_{0}>0,$
(4c) and the cosmological constant $\displaystyle\Lambda_{b}$
$\displaystyle=\frac{\omega^{2}-(2\xi/3z_{0})A^{2}(0)}{48\,\ell^{2}},$ (4d)

where $0\leqslant\xi\leqslant 12$ counts the source-branes888The stringy
cosmic string-like [57] limit includes a total of $12{+}|\xi|$ supersymmetric
7-branes. [58]. In [12, 13, 14], the maximal, $\xi\\!=\\!12$, was used for
simplicty. Thus, the $\mathscr{Y}^{2}_{\perp}$-anisotropy of the axilaton,
$\tau(\omega\theta)$, drives the cosmological constant (4d). Furthermore, the
inequalities $\omega^{2}\\!\geqslant\\!(2\xi/3z_{0})A^{2}(0)$ and
$\Lambda_{b}\\!\geqslant\\!0$ are saturated only in the supersymmetric limit,
where $\omega,(\xi/z_{0}),\Lambda_{b}\\!\to\\!0$ [12, 13]. Finally, the
length-scale $\ell$ emerges in (2) via dimensional transmutation (akin to
$\Lambda_{\rm QCD}$), as this equation is the condition (the vanishing of the
worldsheet QFT beta-function, to the lowest order) for the target-spacetime
metric to not renormalize.

Notably, the metric (1)–(4) is continuous at $z\to\pm z_{0}$ with both sign-
choices in (4a), where $A(z)$ with $Z_{-}(z)$ merely “bounces” as does the
flat-space line element in spherical coordinates at the origin. In stark
contrast, assuming $\mathrm{d}s^{2}_{1,3}$ to be the Minkowski line element
forces the solution with $Z_{-}(z)$ to exhibit a naked singularity at
$|z|\\!=\\!z_{0}$ [10, 58], beyond which the metric becomes complex. Thereby,
the de Sitter metric (1) explicitly desingularizes the total spacetime in this
class of toy models [12]. This is corroborated by verifying that the standard
curvature invariants for (1)–(4) are all finite, whereas the Kretschmann
(Riemann-squared) invariant diverges at $z_{0}$ in the Minkowski solution.
Finally, (4c) implies that the cosmological constant effectively specifies the
“distance” (in the parameter space) to the so-resolved singularity.

In this sense, this class of toy models also describe nearly singular
spacetimes, much as those studied in Ref. [8, 33, 34, 35] — except that the
background configurations are here restricted to a discretuum by the
$\mathop{\textrm{SL}}(2;\mathbb{Z})$-monodromy. Indeed, the axilaton,
$\tau(\omega\theta)$, varies over $\mathscr{Y}^{2}_{\perp}$ including values
that oscillate about $g_{s}e^{\Phi}=O(1)$, even a few orders of magnitude! For
the whole model, this not only implies strong coupling, but is (owing to the
$\mathop{\textrm{SL}}(2;\mathbb{Z})$-monodromy of $\tau(\omega\theta)$) also
discontinuous along a branch-cut, e.g., along $\theta\\!=\\!\pm\pi$. Across
this branch-cut, the model non-perturbatively patches a stringy weak-coupling
regime with a reciprocally strong-coupling regime — akin to “$S$-folds” [59].
Nevertheless, the effective observable string coupling, $\langle
g_{s}e^{\Phi}\rangle$, remains perturbatively small in the sub-spacetime
$\text{dS}^{1,3}_{z=0}$, preserving the impression of a weakly coupled
effective field theory within this toy-model candidate for the observable
universe [13, 14].

For future reference, we note that the effective Planck mass-scale, $M_{4}$,
in $\text{dS}^{1,3}$ is exponentially boosted [11, 14],

$M_{4}\\!^{2}=M_{6}\\!^{4}\,\ell^{2}\,z_{0}\\!^{5/8}\,\xi^{-3/8}\,e^{+\xi
z_{0}}\,2\pi\Gamma\\!_{\pm}\big{(}{\textstyle{3\over 8}};\xi z_{0}\big{)},$
(5)

as compared with the six-dimensional Planck mass-scale resulting after
compactifying on $\mathscr{Y}^{4}$. Here, $\Gamma\\!_{-}$ is the “lower”
incomplete Gamma function, and $\Gamma\\!_{+}$ its “upper” complement, and
$0\\!\leqslant\\!2\pi\Gamma\\!_{\pm}\big{(}{\textstyle{3\over 8}};\xi
z_{0}\big{)}\\!\leqslant\\!2\pi\Gamma\\!_{\pm}\big{(}{\textstyle{3\over
8}}\big{)}\\!\approx\\!14.89$. This in turn exponentially suppresses the
cosmological constant, $\Lambda_{b}\propto
z_{0}\\!^{-5/4}\,\xi^{3/4}\,e^{-2\xi z_{0}}M_{4}\\!^{2}/\ell^{2}$. As is by
now standard [44], the graviton modes are governed by an effective potential
that here depends on the source-counting parameter, $\xi$, that also appears
in (5) [11]. This potential also has a superposed $\delta(z)$-function well
induced from the $|z|$-dependence of the metric (4a), which guarantees the
existence of a $\text{dS}^{1,3}_{z=0}$-localized graviton mode. A more
detailed computation shows that the Newton potential is corrected by
$O(\ell^{3}/r^{3})$ terms [12]. This is suppressed as compared to the naïve
dimensional analysis estimate, $O(\ell^{2}/r^{2})$, based on
$\dim(\mathscr{Y}^{2}_{\perp})\\!=\\!2$, and indicates that the effect depends
on the (hyperbolic) curvature of $\mathscr{Y}^{2}_{\perp}$ near the mid-
radius, $z\\!=\\!0$. Also, this non-analyticity induces a
$\delta(z)$-contribution in the Ricci tensor, which must be matched by matter
localized to the exceptional spacetime factor $\text{dS}^{1,3}_{z=0}$ by the
same $\delta(z)$-function well in the effective potential. This populates the
candidate observable spacetime with universally $\delta(z)$-localized modes of
any matter added to this toy model.

With $Z_{+}(z)$ in (4a), the “transverse” annular region,
$\mathscr{Y}^{2}_{\perp}\approx(\mathscr{Z}\\!\times\\!S^{1})$, has
$\mathscr{Z}_{+}\\!=\\!(-\infty,\infty)$ so that
$\mathscr{Y}^{2}_{\perp}\approx\mathbb{C}^{*}$ is non-compact, as depicted in
Figure 2 (left). In turn, with $Z_{-}(z)$ in (4a),
$\mathscr{Z}_{-}\\!=\\![-z_{0},z_{0}]$ for so that $\mathscr{Y}^{2}_{\perp}$
includes its two circular boundaries; see Figure 2 (left). In both of these
solutions, the proper (log-radial) length of $\mathscr{Z}$ is infinite,
although the surface area of $\mathscr{Y}^{2}_{\perp}$ is finite. The 2-point
compactification/completion999To be precise: for $Z_{-}(z)$ in (4a), each of
the two circular boundaries of $\mathscr{Y}^{2}_{\perp}$ shrinks to a point as
$z_{0}\to\infty$, thus rendering $\mathscr{Z}_{-}$ compact. For $Z_{-}(z)$ in
(4a), $\mathscr{Z}\\!\approx\\!\mathbb{C}^{*}$ is non-compact and two points
must be added to compactify $\mathscr{Z}_{+}\to\mathbb{P}^{1}$.
$\mathscr{Y}^{2}_{\perp}\\!\hookrightarrow\\!S^{2}\\!=\\!\mathbb{P}^{1}$ then
allows us to regard $\mathscr{Y}^{2}_{\perp}$ as a 2-point puncturing of
$\mathbb{P}^{1}$. Fibering the remaining factor, $\mathscr{Y}^{4}$ (e.g., K3),
over $\mathscr{Y}^{2}_{\perp}$ shows that the unobservable space,
$\mathscr{Y}^{2}_{\perp}\\!\ltimes\\!\mathscr{Y}^{4}$ in the axilaton models
[12, 13, 14] is also obtainable as a 2-fibre puncturing of K3-fibrations over
$\mathbb{P}^{1}$ — the latter of which are well understood Calabi-Yau 3-folds,
as they are used in static, Calabi-Yau compactifications to Minkowski
spacetime [38]101010We will return to this non-trivial K3-fibration in Section
4..

$W^{1,3}_{z=0}$$\mathscr{Y}^{2}_{\perp}$(horizontally)
$\mathscr{Y}^{2}_{\perp}$ with [2pt] $Z_{+}(z)=1+|z|/z_{0}$
$\mathscr{Y}^{2}_{\perp}$ with [2pt] $Z_{-}(z)=1-|z|/z_{0}$ Figure 2: A
depiction of the $W^{1,3}_{z}\rtimes\mathscr{Y}^{2}_{\perp}$ fibration with
$Z_{+}(z)$ (left) and $W^{1,3}_{z}\rtimes\mathscr{Y}^{2}_{\perp}$ with
$Z_{-}(z)$ (right). Far left and far right: the proper distance plotted
vertically in $\mathscr{Y}^{2}_{\perp}$, indicating the radial dependence of
the circumference — which is obscured in the two central depictions. Only the
mid-radius fiber, $W^{1,3}_{z=0}$, of the fibration
$W^{1,3}_{z}\rtimes\mathscr{Y}^{2}_{\perp}$ is depicted as a vertical
cylinder, $W^{1,3}_{z=0}\times S^{1}$; it defines the “inside” and “outside’
part of the six-dimensional spacetime’; see (7), below.

The 2-fibre compactification/completion process

$\text{dS}^{1,3}_{z}\\!\rtimes\\!\left(\mathscr{Z}\\!\times\\!S^{1}\right)\ltimes\text{K3}~{}\xrightarrow[~{}z_{0}\to\infty~{}]{~{}\omega\to
0~{}}~{}M^{1,3}\\!\times\\!\left(\mathscr{Y}^{6}=(\mathbb{P}^{1}\ltimes\text{K3})\right)$
(6)

may be understood as the right-hand side depicting a static, Minkowski/CY
geometry being identified as the (double, supersymmetry-restoring) limit,
$\omega,1/z_{0}\to 0$, of the axilaton geometry on the left-hand side.
Intermediate in this process is the “decompactification” of the Minkowski/CY
geometry if $\mathscr{Y}^{2}_{\perp}\\!\approx\\!\mathbb{C}^{*}$ is a non-
compact, unbounded cylinder, or a cylinder with boundaries if
$\mathscr{Y}^{2}_{\perp}\\!\approx\\![-z_{0},z_{0}]\\!\times\\!S^{1}$. In this
sense, the geometry given on the far left-hand side in (6) is the
(supersymmetry-breaking, non-holomorphic) deformation of a punctured version
of the geometry depicted on the far right-hand side. In turn, the
$\text{dS}^{1,3}_{z=0}\\!\times\\!S^{1}\\!\times\\!\text{K3}$ configuration in
the log-radial middle, $z\\!=\\!0$, of the annulus
$\mathscr{Y}^{2}_{\perp}\\!=\\!S^{1}\\!\times\\!\mathscr{Z}$ serves as a
“cylindrical” interface between the “inside” and “outside” regions; see Figure
2.

Although somewhat different in the technical details, the overall spacetime
geometry in the axilaton toy model (1)–(5) [12, 13, 14] and in the
$\text{AdS}^{1,4}$-decay model [6, 7] are remarkably similar: The Schrödinger-
like potential governing the graviton exhibits a discontinuity at
$\text{dS}^{1,3}_{z=0}$, which then properly localizes a graviton mode. Just
as in the axilaton toy model, the $\text{AdS}^{1,4}$-decay model also shows
the metric within the so embedded $\text{dS}^{1,3}_{z=0}$ to be of the de
Sitter type, with the cosmological constant in the correct ballpark. The
respective Ricci tensors also require a balancing source by matter modes,
again localized to the same locus.

The two spacetime geometries, shown side-by-side, with the interfacing
$\text{dS}^{1,3}_{z=0}$ shown underscoring the joining
($\mathop{\texttt{\\#}}\limits$) symbol111111The product
$\mathscr{A}\\!\rtimes\\!\mathscr{B}$ denotes that the warp-factors in the
block-diagonal metric vary over $\mathscr{B}$.:

$\Big{(}\text{AdS}^{1,4}_{\rm
in}\\!\mathop{\texttt{\\#}}\limits_{\text{dS}^{1,3}_{z=0}}\\!\text{AdS}^{1,5}_{\rm
out}\Big{)}\times
S^{5}\quad\text{vs.}\quad\Big{(}\big{(}\underbrace{\text{dS}^{1,3}_{z}\\!\rtimes\\!\mathscr{Y}^{2}_{\perp,\rm
in}}_{z<0}\big{)}\\!\\!\mathop{\texttt{\\#}}\limits_{\text{dS}^{1,3}_{z=0}\times
S^{1}}\\!\\!\big{(}\underbrace{\text{dS}^{1,3}_{z}\\!\rtimes\\!\mathscr{Y}^{2}_{\perp,\rm
out}}_{z>0}\big{)}\Big{)}\times\mathscr{Y}^{4}.$ (7)

The left-hand side scenario is rendered dynamical by choosing
$\text{AdS}^{1,5}_{\rm out}$ to be the false vacuum, with higher energy than
the true-vacuum $\text{AdS}^{1,5}_{\rm in}$. This inequality provides the
interfacing $\text{dS}^{1,3}_{z=0}$ with tension and drives its spatial
expansion. The left-hand side scenario (7) thus depicts a point-nucleation of
the “true” $\text{AdS}^{1,4}_{\rm in}$ within the “false”
$\text{AdS}^{1,5}_{\rm out}$, with the four-dimensional boundary serving as
the candidate observable spacetime, $\text{dS}^{1,3}_{z=0}$ [6, 7].

The right-hand side of (7) axilaton configuration has been presumed so far to
be static and in/out-symmetric [12, 13, 14] — but this need not at all be the
case. Since all physical features in this model depend on the anisotropy of
the axilaton, $\tau(\omega\theta)$, it suffices to choose $\omega_{\rm
in}\neq\omega_{\rm out}$. The two sides of the interfacing
$\text{dS}^{1,3}_{z=0}\\!\times\\!S^{1}$ in (7) are

$\big{(}\text{dS}^{1,3}_{z}\\!\rtimes\\!\mathscr{Y}^{2}_{\perp,\rm
in}\big{)}_{z<0}\qquad\text{and}\qquad\big{(}\text{dS}^{1,3}_{z}\\!\rtimes\\!\mathscr{Y}^{2}_{\perp,\rm
out}\big{)}_{z>0},$ (8)

which are analogous to $\text{AdS}^{1,4}_{\rm in}$ and $\text{AdS}^{1,4}_{\rm
out}$, as depicted in Figure 2. Indeed, the log-radial $z$-dependence renders
the proper distance between $z\\!=\\!0$ and $z_{0}$ infinite while keeping the
surface area of $\mathscr{Y}^{2}_{\perp}$ finite, implying a hyperbolic
geometry. This geometry is driven by the axilaton anisotropy, $\omega>0$ in
$\tau=\tau(\omega\theta)$, and differs from the well-known Ricci-flat geometry
of the annular $\mathscr{Y}^{2}_{\perp}$ that one might have expected: the
negative curvature of $\mathscr{Y}^{2}_{\perp}$ is forced by balancing the
anisotropy-driven “pressure,”
$\widetilde{T}_{\theta\theta}=(\omega/2\ell)^{2}$, in (2).

With different values of the axilaton anisotropy, the regions
$\text{dS}^{1,3}_{z<0}\rtimes\mathscr{Y}^{2}_{\perp,\rm in}$ and
$\text{dS}^{1,3}_{z>0}\rtimes\mathscr{Y}^{2}_{\perp,\rm out}$ harbor different
“pressures,” which drives the expansion of one region into the other. By
analogy then, the right-hand side configuration (7) depicts a cosmic string-
sourced nucleation of an analogous phase transition with a
$\text{dS}^{1,3}_{z=0}\times S^{1}$-shaped “cylindrical”
$(3{+}1{+}1)$-dimensional interface. However, unlike in the AdS-decay model
[6, 7], the $\mathop{\textrm{SL}}(2;\mathbb{Z})$-monodromy of the axilaton
models [12, 13, 14] restricts the anisotropy $\omega$ to discrete values. This
allows only discrete variations, i.e tunneling from one configuration to
another.

##  4 Calabi-Yau 5-Folds

Let us reconsider then the overall, total spacetime in string theory, relying
only on the most general of restrictions while keeping the models of the
foregoing discussion in the back of our mind. To this end, first generalize
the geometry in the axilaton model by allowing for the K3 as in (6) to be
fibered over the $\mathscr{Z}\\!\times\\!S^{1}$, and hence leading to a
K3-fibered Calabi-Yau 3-fold in the $z_{0}\to\infty,~{}\omega\to 0$ limit. In
addition, we consider an auxiliary, “Euclideanized” rendition, which affords
relating algebro-geometric features of this depiction to the Lorentzian
features of the original. This affords an old idea [15] a fresh look, and will
turn out to improve this proposal.

To start, string dynamics is generally understood to require the overall
target spacetime, $\mathscr{X}^{1,9}$, to be Ricci-flat — i.e., to admit a
metric the Ricci tensor of which is a total derivative. There are in general
stringy corrections, but the metric is generally expected to be perturbatively
“close” to Ricci-flat. This understanding follows from several related, but
rather distinct vantage points:

  1. 1.

the inherent semi-infinite cohomology in string theory [60],

  2. 2.

the free loop space as the configuration space of (closed) strings [61, 62,
63, 64, 65, 66, 67, 68, 69],

  3. 3.

the worldsheet quantum field theory approach [70].

The orientability of closed strings [54] (including zero-modes [71, 72, 73]!)
implies that the second-listed analysis should be extended to the orientation-
doubled loop-space and (Super)Diff$(S^{1})/S^{1}$. In lieu of this extension,
we note that it will perforce include the results of the original loop-space
approach. (Owing to the still very much developing nature of string theory, a
technically precise meaning of this notion is still unclear, and the
conceptual understanding itself may require some adjustments.) The above
studies show that the free loop-space of $\mathscr{X}^{1,9}$ must be Ricci-
flat, which then determines the geometry of $\mathscr{X}^{1,9}$ in turn [38].
Consider then a suitable Euclidean Wick-rotation of $\mathscr{X}^{1,9}$ that
is, at least in Type-II theories (owing to their $N\\!=\\!2$ supersymmetry)
guaranteed to admit both a complex structure and a Kähler metric. This assigns
to the ten-dimensional spacetime, $\mathscr{X}^{1,9}$, a corresponding
auxiliary Calabi-Yau 5-fold, $\mathfrak{X}_{5}$; various dualities extend this
argument throughout other types of string vacua.

Just as Calabi-Yau 3-folds generically contain numerous exceptional, isolated
so-called “$\mathscr{O}(-1,-1)$ curves” [40], Calabi-Yau 5-folds generically
contain numerous exceptional, isolated Fano ($c_{1},R_{\mu\bar{\nu}}>0$)
compact complex surfaces, $\mathfrak{S}_{2}$, i.e., real four-dimensional
subspaces [15]. It the Lorentzian original, (the preimage of) at least some of
these isolated subspaces are soliton-like (rather than instanton-like), and so
admit a metric of the $(1,3)$-signature, perhaps singular at some (sub)locus.
Owing to their positive scalar curvature121212Being Fano,
$c_{1}(\mathfrak{S})\\!>\\!0$, implies the scalar curvature invariant to be
positive, $R(\mathfrak{S})>0$., the simplest of these will then admit a de
Sitter metric and may serve as candidates for the observable spacetime. Recall
that for all Einstein spacetimes, $R_{\mu\nu}=\Lambda g_{\mu\nu}$ defines
$1/\sqrt{\Lambda}$ as the characteristic length scale, relating it to the
curvature, and via $\int\\!\sqrt{\det(g_{\mu\nu})}$ also to the volume. Wick
rotation changes the overall coefficient in this tensorial equation, and so
preserves the relation. More precisely, it suffices for $\mathfrak{S}_{2}$ to
admit a metric that equals the de Sitter metric within the past light-cone of
a typical/current observer; see Appendix A for more detail. For such
exceptional sets at least, $\Lambda_{b}>0$ parametrizes the relative size of
such an exceptional, isolated real 4-manifold,
$(\text{dS}^{1,3}\mapsto\mathfrak{S}_{2})\subset\mathscr{X}^{1,9}$, — very
much akin to the de Sitter desingularizing deformation (1)–(5).

Reid’s argument for 3-folds [40] generalizes straightforwardly to higher
dimensions, so Calabi-Yau 5-folds are expected to abound in such exceptional
real four-dimensional subspaces, making the pool of candidates abundant. Being
the exceptional sets of small resolutions of conifold singularities the
dynamical evolution of the Lorentzian of these four-dimensional subspaces
$\text{dS}^{1,3}\mapsto\mathfrak{S}_{2}$ may well enfold akin to the
$\text{dS}^{1,3}_{z=0}$ bubble-worlds discussed above (7). In the coarse
classification discussed in Section 2 and referring to Figure 1, the scenario
$\text{dS}^{1,3}_{z=0}\subset\mathscr{X}^{1,9}$ that is the preimage of the
“Euclideanizing” Wick rotation assignment

$\text{dS}^{1,3}_{z=0}\subset\mathscr{X}^{1,9}~{}\mapsto~{}\mathfrak{S}_{2}\subset\mathfrak{X}_{5}$
(9)

is depicted in Figure 1 (d): the total spacetime, $\mathscr{X}^{1,9}$, is in
no way assumed to factorize, nor even foliate131313For our present purposes, a
foliation $X\\!\divideontimes\\!Y$ means that the total space looks locally at
every point as a direct product of local portions of the two factors, $X$ and
$Y$., and $\text{dS}^{1,3}\mapsto\mathfrak{S}_{2}$ is an isolated, exceptional
sub-spacetime. On general grounds, such non-factorizing spacetimes ought to be
the generic case in string theory.

Now consider the auxiliary Calabi-Yau 5-fold, $\mathfrak{X}_{5}$, a particular
exceptional Fano surface in it, $\mathfrak{S}_{2}\subset\mathfrak{X}_{5}$, and
its non-compact complement, $\mathfrak{X}_{5}\smallsetminus\mathfrak{S}_{2}$,
as well as their respective Lorentzian counterparts: $\mathscr{X}^{1,9}$,
$\text{dS}^{1,3}$ and $\mathscr{X}^{1,9}\smallsetminus\text{dS}^{1,3}$. Note
that the Ricci curvature is not additive for such triples (nor is then the
scalar curvature), but we have the Tian-Yau theorem [74, 75]:

$\displaystyle\underbrace{\text{compact Fano${}_{n{+}1}$}}_{c_{1}>0}$
$\displaystyle~{}\smallsetminus~{}\underbrace{\text{compact
CY${}_{n}$-fold}}_{c_{1}=0}$ $\displaystyle~{}=~{}\underbrace{\text{non-
compact CY${}_{n{+}1}$-fold}}_{c_{1}=0}.$ (10) Here, the non-compact space on
the right-hand side is known to admit a suitable Kähler metric/form, $J$, and
a holomorphic volume-form, $\Omega$ (which in turn implies the existence of a
covariantly constant spinor, i.e., global supersymmetry), that satisfy the
standard relationship $J^{n}=\bar{\Omega}\\!\wedge\\!\Omega$ asymptotically,
near the locus where the compact Calabi-Yau $n$-fold was excised. It would
then seem reasonable to expect that, analogously:
$\displaystyle\underbrace{\text{compact CY${}_{n{+}1}$}}_{c_{1}=0}$
$\displaystyle~{}\smallsetminus~{}\underbrace{\text{compact
Fano${}_{n}$-fold}}_{c_{1}>0}$
$\displaystyle~{}\overset{?}{=}~{}\underbrace{\text{non-compact
Hyp${}_{n{+}1}$-fold}}_{c_{1}<0},$ (11)

where Hypn+1 denotes a “hyperbolic” complex $n{+}1$-fold with negative Ricci
curvature, again with analogous asymptotics near the locus where the compact
Fano $n$-fold was excised. Both of these relations are well-nigh trivial and
easily seen for $n\\!=\\!0$:

  1. 1.

$\mathbb{P}^{1}\smallsetminus\\{2\,\text{pts.}\\}\approx\mathbb{C}^{*}$ (a
non-compact cylinder), which is well known to be Ricci-flat.

  2. 2.

$T^{2}\smallsetminus\\{\text{pt.}\\}$ is a 1-handled disc, and so a hyperbolic
non-compact surface.

Furthermore, we will need the following specializing generalization

$\underbrace{\text{compact
CY${}_{n{+}1}$}}_{c_{1}=0}~{}\smallsetminus~{}\underbrace{\text{isolated
comp.\
Fano${}_{n-k}$-fold}}_{c_{1}>0}~{}\overset{?}{=}~{}\underbrace{\text{non-
compact Hyp${}_{n{+}1}$-fold}}_{c_{1}<0}.$ (12)

Locally, this result follows from the so-called adjunction theorem [38]: For
example, $c_{1}(\mathbb{P}^{1})\\!=\\!2\\!>\\!0$ and
$c_{1}(\text{CY}_{3})\\!=\\!0$ imply that the local neighborhood of
$\mathbb{P}^{1}\subset\text{CY}_{3}$ is any one of the rank-2 bundles
$\mathscr{O}_{\mathbb{P}^{1}}(\ell{-}2,{-}\ell)$, with arbitrary
$\ell\\!\in\\!\mathbb{Z}$. Depending on the choice of $\ell$, the curvature of
this normal bundle will be negative along some fibers but positive along
others. However, we seek an isolated $\mathbb{P}^{1}$, which has no local
holomorphic deformations, and which happens precisely when $\ell\\!=\\!1$. In
that exceptional case, the local neighborhood of that
$\mathbb{P}^{1}\subset\text{CY}_{3}$ has a uniformly negative Ricci curvature.
The claim (12) then aims to generalize this, albeit perhaps in an appropriate
average sense, throughout the complement
$\text{CY}_{3}\\!\smallsetminus\\!\mathbb{P}^{1}$ — and more generally,
throughout $\mathfrak{X}_{5}\smallsetminus\mathfrak{S}_{2}$; we are not aware
of a rigorous global result either way.

The so-generalized claim (12) then implies that the non-compact complex 5-fold
$\mathfrak{X}_{5}\smallsetminus\mathfrak{S}_{2}$ admits a metric of negative
Ricci curvature. (Again, locally, near an isolated
$\mathfrak{S}_{2}\subset\mathfrak{X}_{5}$, this is a consequence of the
adjunction theorem.) With suitable boundary conditions to match the excised
subspace $W^{1,3}\mapsto\mathfrak{S}_{2}$, the Lorentzian counterpart,
$\mathscr{X}^{1,9}\smallsetminus W^{1,3}$, should then also admit a metric
with negative curvature — analogous to $\text{AdS}^{1,4}_{\rm out}$ in the
AdS-decaying scenario (7). However, these exceptional 4-manifolds,
$W^{1,3}\subset\mathscr{X}^{1,9}$, have nothing “inside,” much as a circle
inside $\mathbb{R}^{3}$ does not carve it into two separate parts: The
complement, $\mathscr{X}^{1,9}\\!\smallsetminus\\!W^{1,3}$ is a single-
component connected space since $W^{1,3}\subset\mathscr{X}^{1,9}$ has real
codimension 6. (This is unlike the real codimension-1 subspace in the AdS-
decay scenario that does carve the total spacetime,
$\text{AdS}^{1,5}\\!\times\\!S^{5}$, into two disconnected parts, an “outside”
and an “inside.”) Having only the “outside,” $\mathscr{X}^{1,9}\smallsetminus
W^{1,3}$, these exceptional sub-spacetimes $W^{1,3}\subset\mathscr{X}^{1,9}$
are rather literally akin to “bubbles of nothing”; see [76] and also [77].

In the Euclidean (and holomorphic) rendition, the standard Kähler class, $J$,
(every metric in that cohomology class) of any “bubble” (desingularizing
exceptional set) $\mathfrak{S}_{2}$ is both positive over every complex
submanifold of $\mathfrak{S}_{2}$ and has a positive square (“volume”):
$\int_{C\subset\mathfrak{S}}J>0$ and $\int_{\mathfrak{S}}J^{2}>0$ are the
standard conditions on the Kähler cone. In the simplest case, take the bubble
to be $\mathbb{P}^{2}$. Then:

  1. 1.

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
$\mathbb{P}^{2}\approx\mathbb{C}^{2}\cup\mathbb{P}^{1}$.)

  2. 2.

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,

  3. 3.

which is therefore a valid metric on $(S^{4}\smallsetminus\\{\text{Noth
pole}\\})\approx\mathbb{C}^{2}$, and fails in those positivity requirements
only at the North pole, where it vanishes — and so is nowhere negative.

Together, these imply that $W^{1,3}$ has positive curvature, the simplest
(most symmetric) of which is $\text{dS}^{1,3}$; see also Appendix A.

Other Fano complex surface candidates for the bubble $\mathfrak{S}_{2}$ will
contain more than one exceptional (complex) curve, each isomorphic to
$\mathbb{P}^{1}\\!\approx\\!S^{2}$ [38]. Their Lorentzian preimages,
$W^{1,3}$, will contain more than one corresponding exceptional real two-
dimensional sub-spacetime, with a mutual intersection pattern characteristic
to the Euclidean (and complex) $\mathfrak{S}_{2}$. With a Lorentzian metric on
$W^{1,3}$, these sub-spacetimes can serve as specific loci of interest, such
as space-like horizons at infinite past/future. A further exploration of such
correlations between the Lorentzian metric structure of topologically
nontrivial spacetimes and the (almost) complex structure of their auxiliary
Euclidean renditions, but beyond our present scope.

As it is, (almost) Fano complex surfaces are
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, or a blow-up of $\mathbb{P}^{2}$ at
$0,{\cdots},9$ points. Their Euler numbers ($\int\\!c_{2}$) are, in the given
order: 4, or $3,4,{\cdots},12$ [78, 38]. (The projective space,
$\mathbb{P}^{2}$, itself may be regarded as a blowup of $S^{4}$, by replacing
a point in $S^{4}$ by the exceptional hyperplane
$\mathbb{P}^{1}\subset\mathbb{P}^{2}$.) By the Todd-Chern-Hirzebuch theorem
for Fano surfaces, $\int(c_{1}\\!^{2}{+}c_{2})=12$, so that
$\int\\!c_{1}^{2}=12{-}\\!\int\\!c_{2}$, and all characteristic classes (and
their integrals) are controlled by $\int\\!c_{2}$, i.e., the Euler number. For
completeness, we note that $p_{1}=c_{1}\\!^{2}{-}2c_{2}=3(4{-}c_{2})$, so the
signature of these (real) four-dimensional spaces is $\sigma=4{-}\chi_{E}$.
For all (almost) Fano surfaces except $\mathbb{P}^{1}\times\mathbb{P}^{1}$,
the above argument shows that they admit a metric that is nowhere negative,
and merely becomes null at select $1,{\cdots},10$ exceptional locations, which
in such a metric become “points”; the total space of these algebraic surfaces
is then diffeomorphic to $S^{4}\smallsetminus\\{\text{points}\\}$. In the
“ruled (complex) surface,” $\mathbb{P}^{1}\times\mathbb{P}^{1}$, the meridians
of either $\mathbb{P}^{1}$ may be identified as “time,” which identifies the
“Southern” and “Northern” instance of the other $\mathbb{P}^{1}$ as the
infinite past and future horizon, respectively. It seems plausible to expect a
Lorentzian metric to exist wherein those exceptional loci are at infinite
past/future, and whereby the positive curvature of of these surfaces (just as
of $S^{4}$) should translate into a positive cosmological constant, for the
most part of $W^{1,3}\mapsto\mathfrak{S}_{2}$; see Appendix A.

Even without computational details about the embedding
$W^{1,3}\subset\mathscr{X}^{1,9}$, the fact that this is (a Lorentzian
preimage of) a small resolution smoothing has strong consequences: Considering
$W^{1,3}$ from “outside,” as a (metrically) isolated as
$W^{1,3}\subset\mathscr{X}^{1,9}$ may be, at least locally near its locus
within $\mathscr{X}^{1,9}$, the whole spacetime $\mathscr{X}^{1,9}$ must
itself admit a Lorentzian metric with a co-laminar class of time coordinates.
That is, any particular time-like geodesic, $\mathscr{C}_{t}\subset
W^{1,3}\subset\mathscr{X}^{1,9}$, will have infinitesimally near time-like
geodesics,
$\mathscr{C}_{t,\epsilon}\subset(\mathscr{X}^{1,9}\\!\smallsetminus\\!W^{1,3})$,
such that $\lim_{\epsilon\to 0}\mathscr{C}_{t,\epsilon}=\mathscr{C}_{t}\subset
W^{1,3}$. (This may well become ill-defined at certain special locations of
$W^{1,3}\subset\mathscr{X}^{1,9}$, such as past/future horizons.) That is, the
Lorentzian metric of $W^{1,3}$ must extend almost everywhere smoothly into the
local neighborhood of $W^{1,3}\subset\mathscr{X}^{1,9}$ and then $W^{1,3}$
itself. Thereby, the above-implied class of “time-sliced, static” snapshots
and associated phase transition nucleation interpretation also extend to the
ten-dimensional spacetime $\mathscr{X}^{1,9}$, at least in the local
neighborhood of $W^{1,3}\subset\mathscr{X}^{1,9}$.

Finally, let us note that in this (putatively de Sitter) “exceptional sub-
spacetime, $\text{dS}^{1,3}_{\rm us}\subset\mathscr{X}^{1,9}$,” scenario
$\mathscr{X}^{1,9}$ is Ricci-flat, and so admits a standard, supersymmetry-
preserving, spacetime metric. The putatively negative-curvature complement,
$\mathscr{X}^{1,9}\\!\smallsetminus\\!\text{dS}^{1,3}_{\rm us}$, may well
admit even multiple covariantly-constant spinors away from
$\text{dS}^{1,3}_{\rm us}$ but their restriction to $\text{dS}^{1,3}_{\rm us}$
will fail to be covariantly constant within $\text{dS}^{1,3}_{\rm us}$. Now,
in the Euclideanized complex rendition, an isolated
$\mathfrak{S}_{2}\subset\mathfrak{X}_{5}$ is metrically null in the “bulk”
Kähler metric; it needs an $\mathfrak{S}_{2}$-localized “correction” to be
positive everywhere on $\mathfrak{X}_{5}$; see [78] for a lower-dimensional
explicit construction to this end. This isolation then implies that
$\mathscr{X}^{1,9}$ admits a metric in which $\text{dS}^{1,3}_{\rm us}$ is
null, i.e., a point, so that for a typical “bulk”-$\mathscr{X}^{1,9}$
observer, supersymmetry is broken only at a point — a singularity of the
supersymmetry structure141414Supersymmetry is in string theory largely
correlated with complex structure, and as mentioned above,
$(S^{4}\smallsetminus\\{\text{pt.}\\})\approx\mathbb{C}^{2}$ of course admits
a complex structure, for which the excised point is an obstruction..

Thus, these metrically isolated, codimension-six sub-spacetimes,
$\text{dS}^{1,3}_{\rm us}\subset\mathscr{X}^{1,9}$, are exceptional sets of
local small resolution smoothings of the ostensibly most generic rendition of
ten-dimensional string vacua; in these “measure-zero,” but generically
abundant sub-spacetimes, supersymmetry is broken.

##  5 Summary, Outlook and Conclusions

In this paper, we have briefly reviewed some recent work on constructing
concrete superstring models that do exhibit a sub-spacetime with a
phenomenologically acceptable de Sitter geometry and cosmological constant,
[6, 7, 8]. Notably, these indicate two separate aspects that render such work
more fruitful: (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, 11,
12, 13, 14] that may be viewed as a non-holomorphic and non-analytic
deformation of the stringy cosmic string (’brane) scenario [41, 42].

In particular, we have shown herein that this class of models exhibits both of
these hallmark characteristics, and can also be adapted so as to model the
dynamical scenario of [6, 7]. In this variant, the model represents a stringy
cosmic string-sourced nucleation in a phase transition, where the candidate
observable world, $\text{dS}^{1,3}_{z=0}$, is in the interfacing boundary.

Inspired by these resonances, we have reexamined the global geometry of the
ten-dimensional spacetime in string theory, $\mathscr{X}^{1,9}$, following
[15] and [13]. This finds plenty of exceptional sub-spacetimes that map to
exceptional sets, $\mathfrak{S}_{2}\subset\mathfrak{X}_{5}$, of small
resolution desingularizations of the auxiliary Euclidean, Calabi-Yau 5-fold
re-rendering of the total spacetime,
$\mathscr{X}^{1,9}\mapsto\mathfrak{X}_{5}$. Given their positive Ricci
curvature, $c_{1}(\mathfrak{S}_{2})\\!>\\!0$, the simplest of these have a de
Sitter geometry and so can serve as candidates for the observed four-
dimensional spacetime. Furthermore, the four-dimensional bubble-worlds,
$\text{dS}^{1,3}_{z=0}\mapsto\mathfrak{S}_{2}$ are metrically isolated,
codimension-six sub-spacetimes, that have no “inside,” and their local
neighborhood in $\mathscr{X}^{1,9}$ has a negative Ricci curvature — on par
with $\text{AdS}^{1,4}_{\rm in},\text{AdS}^{1,4}_{\rm out}$ in (7) in the AdS-
decaying scenario of Section 2.2.

It is striking to note that the above features pertain to the generic Ricci-
flat ten-dimensional stringy spacetimes, each of which in turn contains a very
large number of such sub-spacetimes. That is, no special requirement or
restriction on the total ten-dimensional (super)stringy spacetime were needed
for the above conclusions to hold, and only the general characteristics of
(super)string spacetimes and their Euclideanized Calabi-Yau counterpart have
been presumed.

Acknowledgments: We thank Giuseppe Dibitetto and Ivonne Zavala for their kind
invitation to contribute to this special issue. PB would like to thank the
CERN Theory Group for their hospitality over the past several years. TH is
grateful to the Department of Mathematics, University of Maryland, College
Park MD, and the Physics Department of the Faculty of Natural Sciences of the
University of Novi Sad, Serbia, for the recurring hospitality and resources.
DM is grateful to Perimeter Institute for hospitality and support. The work of
PB is supported in part by the Department of Energy grant DE-SC0020220. The
work of DM is supported in part by Department of Energy (under DOE grant
number DE-SC0020262) and the Julian Schwinger Foundation.

##  Appendix A The Effect of Defects

As a toy example, a sphere is of course positively curved and evidently admits
no global metric of non-positive curvature. Excising a point (e.g., the North
pole) leaves a disc, which then admits all kinds of metrics, pretty much
solely depending on (controlled by) the boundary conditions. For example,
identifying all the points of the boundary with a single point maps back to
the sphere, identifying the disc boundary with, e.g., the North Pole; see the
left-hand side of Figure 3. In turn, dividing the boundary of the disk into
eight segments, which are then identified in the interleaved fashion depicted
in the middle of Figure 3 produces a genus-2 Riemann surface depicted on the
right-hand side of Figure 3.

$a$$a$$b$$b$$c$$c$$d$$d$ Figure 3: 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).

Within the differentiable category of smooth metrics, the metric on the three
depicted objects need only differ in the infinitesimal neighborhood of the
North pole of the sphere: Passing from the sphere (Figure 3, left) to the
“double torus” (Figure 3, right) shows that the North Pole of the sphere
($H_{1}(S^{2},\mathbb{Z})=0$) has effectively been “blown up”151515This 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
3, middle. into the interleaving network of four cycles of the double torus,
which span
$H_{1}(T^{2}{\mathop{\texttt{\\#}}\limits}T^{2},\mathbb{Z})\approx\mathbb{Z}^{4}$.
That is to say, the entire topological (and curvature) difference between
$S^{2}$ and $T^{2}{\mathop{\texttt{\\#}}\limits}T^{2}$ can be concentrated
into the “$\epsilon$-neighborhood” of the North pole, replacing it by a
2-handled $\epsilon$-disc:

(13)

In the Section 4 scenario where the candidate for our own Universe is a
bubble-world, $W^{1,3}$, within the Lorentzian preimage of the auxiliary
Calabi-Yau 5-fold, $\mathscr{X}^{1,9}\mapsto\mathfrak{X}_{5}$, we are
experiencing our bubble-world with time passing along a “meridian.” For
example, in Figure 3 (left), our bubble-world cosmic time may well be aligned
with the meridians, starting at the South pole (identified as the Big Bang)
and advancing towards the North Pole (identified as The End). In that case,
the topological “handles” confined within an $\epsilon$-neighborhood of the
North pole could not have been observed by any observer, throughout almost all
of the history of such a bubble-world. The same then applies to any
topological obstruction to a de Sitter metric. Therefore, it suffices that a
candidate for our spacetime, $W^{1,3}$, admit a Lorentzian metric that is
indistinguishable from the de Sitter metric within our past light cone.

Furthermore, since in such a scenario the observed matter spectrum correlates
with certain specific (co)homology groups valued both in intrinsic
($T_{W},T^{*}_{W}$) and in extrinsic ($N_{W\subset\mathscr{X}}$) structures
pertaining to our bubble-world, $W^{1,3}\subset\mathscr{X}^{1,9}$, at least
some of those may well pertain to topological (and (co)homological, etc.)
features that are still outside of our past light cone, i.e., they could not
have been observed yet. This reasoning implies it possible that, say, we have
not observed a 4th generation of Standard Model particles yet, but may observe
them sometime in the future, once the requisite topological “handle” in
$W^{1,3}$ enters our past light cone. Since the requisite (co)homology
elements are localized to those “handles,” they will then be the sources of
such “new” matter. The “time-sliced, static” snapshots then appear to show a
nucleation of a phase transition: with a 3-generation Standard Model vacuum
all around, and a 4-generation vacuum nucleating and emanating from that
particular “handle.”

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