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Wake-Up Call: Amazon spends massively on ads now, and other news to know today
By Angela Doland. Published on June 25, 2018.
Dodge pays homage to history with EV logo
Credit: Julie Clopper/iStock
Welcome to Ad Age's Wake-Up Call, our daily roundup of advertising, marketing, media and digital news. You can get an audio version of this briefing on your Alexa device. Search for "Ad Age" under "Skills" in the Alexa app. What people are talking about today: The Cannes Lions International Festival of Creativity is over, to the relief of everyone who was not on the French Riviera last week. (To those who were: Thanks for sharing all your beach photos on Instagram!) Editor Brian Braiker has the final word from Ad Age on the festival: The event was shorter, agencies sent fewer people, and there was a feeling of getting back to business. As he writes, "This was the Diet Coke of Cannes: slimmer, caffeinated, slightly less bad for you."
Prizes: Who won what? The Ad Age list of Grand Prix is here. The big winner was the "Palau Pledge," from Host/Havas, which Ad Age's Ann-Christine Diaz described as "a tourism campaign that doubled as an environmental preservation effort." It was also huge publicity for the tiny Pacific Ocean nation of Palau (population 21,400.)
La fin: ICYMI, Omnicom was Cannes' holding company of the year, ending WPP's seven-year streak. Read more on that from Diaz and Megan Graham. And that's it for Cannes news. Moving on…
Amazon has quickly become one of the top ad spenders in the U.S. That's according to Ad Age's Datacenter, which crunched all the numbers for our 63rd annual Leading National Advertisers report. Bradley Johnson put together a handy list for you of the big takeaways. Here's one:
"Amazon ranked as the nation's fourth-largest advertiser in 2017, up from No. 8 in 2016 and No. 70 in 2010. Amazon last year spent an estimated $3.4 billion on U.S. advertising and promotions, up 28 percent."
Is it surprising (or deeply unsurprising?) that Amazon spent more money advertising itself in the U.S. last year than General Motors, Ford and Walmart?
In other Amazon news: The New York Times looks at the rise of Amazon's own private label brands. There are about 100 of them now, The Times writes, with names like Wag for dog food and Good Brief for men's undies. The Times quotes sources saying that some big established brands "are spending six figures each month advertising their products on Amazon's platform even as Amazon's private-label products are crowding them out."
Full-court press
What's new in sports sponsorships? A lot, to judge from Ad Age's E.J. Schultz's deep dive. Take Anheuser-Busch InBev's new incentive-based sponsorship model, where teams and leagues can get more money by hitting goals like making the playoffs. Teams also get incentives if their Budweiser-branded social videos reach a certain number of viewers. And in general, Schultz writes,
"As brands shell out millions to be everything from the official beer to the official bank, marketers are demanding more access to content and fan experiences that only teams and leagues can provide."
There are also new offerings like branded VR experiences at stadiums. What's the backstory? Marcel Marcondes, AB InBev's U.S. marketing VP, told Schultz: "TV viewership is down, venue attendance is down. For us as brand builders, it's much more challenging to grab consumers' attention."
The subscription box category is getting crowded, to the point where the monthly packages no longer bring "the unpredictability and joy they once carried," as Ad Age's Adrianne Pasquarelli reports. Everyone's doing it: New York eatery Katz's Delicatessen even has a pastrami subscription box, as Pasquarelli writes. Here's what one expert tells her:
"The problem is some brands have a relatively repetitive type of product," says Kit Yarrow, a consumer psychologist. "It's a psychological requirement that they open the box and have a moment of delight or surprise or excitement, otherwise there's no reason to continue."
Pastrami's fantastic and all, but do you really want a box of it to show up on your door every month like clockwork? Don't you have to be in the mood?
Just briefly:
Netflix: During the BET Awards, Netflix aired an ad featuring 47 of the platform's black stars and directors, including Spike Lee and Ava DuVernay. The Hollywood Reporter says "the timing of the commercial is rather ironic, given that it aired two days after the streaming giant fired its top PR chef Jonathan Friedland for using the N-word on multiple occasions at work ..."
Sorry: David Bossie, President Trump's former deputy campaign manager, said during a debate with a black guest on Fox News, "You're out of your cotton-picking mind." As Slate reports, Bossie apologized, and Fox responded that Bossie's "remarks do not reflect the sentiments of FOX News and we do not in any way condone them."
Moola: The median pay for CEOs in the media and telecoms industry last year was $28.7 million, more than double the median pay of chief execs of the S&P 500, The Wall Street Journal reports. That's despite the fact that "media and telecom companies posted a median total negative return of 3.8 percent," The Journal writes.
In the driver's seat: To mark the end of Saudi Arabia's ban on women behind the wheel, Saudi woman Aseel Al Hamad drove a Formula One car around the track ahead of the French Grand Prix. Read about it via Reuters.
Stunt of the day: ICYMI, a product called Hot Dog Water -- a bottle of water with a wiener floating in it -- went on sale at a festival in Vancouver for $37.99 a bottle. As Ad Age's Megan Mowery reports, "Its health benefits ostensibly included increased brain function and vitality, weight loss, and a younger physique." It turns out (shocker!) that this was a stunt, designed to make people think twice about marketing claims. As the project's creator says, "The message is the next time you have the urge to buy the latest quantum toilet paper or Gwyneth's magical health stickers, take a moment to reflect and ask yourself, is this Hot Dog Water?" Read more here.
Angela Doland
Angela Doland is the West Coast editor at Ad Age.
Big brands try reusable packaging. Plus, Mark Zuckerberg defends Facebook's ad model: Friday Wake-Up Call
Amazon's delivery robot, Gillette's sales and layoffs at BuzzFeed and Verizon: Thursday Wake-Up Call
IHeartMedia wins court approval for a plan to exit bankruptcy
YouTube bans risky viral challenges. Plus, many Facebook users don't grasp ad targeting: Thursday Wake-Up Call
An NFL-backed data firm is launching a sports-gambling marketing company
Nike launches an auto-lacing smart sneaker, and Netflix hikes its prices: Wednesday Wake-Up Call
President Trump plugs fast food, and a Gillette ad sparks debate: Tuesday Wake-Up Call


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Lightware releases input/output boards for TPS format
Lightware has released new products to support its TPS format in 8x8 to 80x80 MX series chassis.
TPS is the company's version of HDBaseT with extension of video, audio, Ethernet, RS232 and power over a CAT cable.
Lightware eight channel TPS input/output boards support standard video resolutions including UHD/4K.
With a full HD signal these cards can extend from 150-180m, whereas with a UHD/4K signal they extend from 100-120m over a single CAT5/6/7.
The company also offers analogue or digital audio embedding/de-embedding along with an in-built test pattern generation tool for commissioning and debugging.
Lightware control software allows users to measure multiple signal parameters and timing information and also includes a built-in tool for measuring the quality and distance of a CAT run.
Additional features include an HDCP enable/disable function, colour space conversion and equalisation settings.
The HDMI-TPS-TX/RX90 transmitter pair can be used in conjunction with the input/output boards above to provide end to end extension and routing of up to UHD/4K.
They can also be used point-to-point and provide extension of HDMI video, audio, Ethernet, RS232, IR and power over just one CAT 5/6/7.
Lightware also announced that it would soon release another version, the DVI-TPS-TX/RX90 that adds a DVI connector. This unit also carries an HDMI signal with embedded audio over the DVI connector.

redpajama_set_name: RedPajamaCommonCrawl


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\section{Introduction}
\label{s:IRS}
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~\cite{Riess:1998cb, Perlmutter:1998np}. Constructing  solutions in string theory with these features has been argued to be notoriously hard if not impossible: see~\cite{Danielsson:2018ztv,Cicoli:2018kdo,BHM:2021review} for recent comprehensive reviews.
However,  recent work~\cite{Banerjee:2018qey,Banerjee:2019fzz} and~\cite{Bento:2021nbb}, as well as~\cite{Blaback:2019zig},  turn out to correspond naturally with a stringy cosmic dS-brane toy-model~\cite{rBHM1,rBHM4,rBHM5,rBHM7,rBHM10} which we furthermore connect with a generalization of the proposal that allows for more general spacetime varying string vacua~\cite{rHitch}.

 The main ideas are the following:
 ({\small\bf1})~Both the \text{AdS}-decaying~\cite{Banerjee:2018qey,Banerjee:2019fzz}
  and the so called ``axilaton''~\cite{rBHM1,rBHM4,rBHM5,rBHM7,rBHM10} scenarios are {\em\/near\/} a singularity: They are ``resolutions/smoothings'' of a singular/critical model, and so are inherently non-perturbative
~\cite{Bento:2021nbb}.
  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.
 %
 ({\small\bf2})~Just as in the \text{AdS}-decaying scenario, the axilaton models can be made dynamical by choosing their anisotropy, $\w$, differently on the two sides of the candidate for the observed universe, $\text{dS}^{1,3}_{z=0}$.
 %
 ({\small\bf3})~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 \text{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~\ref{s:NS}, especially focusing on the conifold in~\ref{s:wdC}, which leads us to  review the \text{AdS}-decaying scenario in~\ref{s:dSB}. In Section~\ref{s:axilaton} 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~\ref{s:CY5}, 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~\ref{s:Coda} summarizes our key points and conclusions, while Appendix~\ref{s:ED} collects a few additional details about (spacetime) defects and their role in understanding dS solutions in string theory.

\section{Nearly Singular Spacetimes}
\label{s:NS}
The early discovery that certain singular features in spacetime are innocuous to string dynamics~\cite{Dixon:1985jw,Dixon:1986jc} 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~\cite{rGHC,Green:1988uw,Candelas:1989ug,Partouche:2000uq}, in a milder sense also~\cite{Aspinwall:1993nu,Aspinwall:1994zd}, and so involve some form of a phase transition. This tends to harbor radical effects both in physics and in geometry: massless black holes~\cite{Strominger:1995cz}, exoflops~\cite{Aspinwall:1993nu,Hubsch:2002st}, $D$-branes~\cite{Polchinski:1996fm}, and orientifolds~\cite{Sagnotti:1987tw,Horava:1989vt,Bianchi:1990tb}, and often involve strong coupling effects.

\subsection{Warped Deformed Conifolds and Alike}
\label{s:wdC}
Models that are even just {\em\/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~\cite{Klebanov:2000hb,DeWolfe:2002nn,Douglas:2007tu}; see also~\cite{Douglas:2009zn,Bena:2009xk}. 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~\cite{Bento:2021nbb}; see however also the review~\cite{Bena:2017uuz} 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 K{\"a}hler (or symplectic) structure side, as is the case with very small relative sizes of so-called exceptional sets\footnote{In a standard ``blowup surgery,'' these exceptional sets have long since been familiar in physics as gravitational instantons~\cite{Eguchi:1978gw,Gibbons:1978tx}, 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~\cite{rBeast}.} in a Calabi-Yau 3-fold compactification. As with the near singularization by complex structure deformation~\cite{Bento:2021nbb}, one again expects a detailed balance of various contributing factors.
 While exact calculations on this, K{\"a}hler, 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~\cite{rH-AG,rReidK0,rBeast}; 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 {\em\/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 spacetime\footnote{A 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~\ref{f:4Cases}\,(a), where the geometry of $\mathscr{Y}^6$ and of the observable four-dimensional spacetime are independent of each other.
\begin{figure}[tbp]
 \begin{center}
  \begin{picture}(160,35
   \put(0,0){\includegraphics[width=160mm]{4Cases.pdf}}
   \put(19,-3){\makebox[0pt][c]{\footnotesize(a)\,rigid product}}
   \put(58,-3){\makebox[0pt][c]{\footnotesize(b)\,fibration (w/singularity)}}
   \put(100,-3){\makebox[0pt][c]{\footnotesize(c)\,non-analyticity}}
   \put(142,-3){\makebox[0pt][c]{\footnotesize(d)\,exceptional subset}}
  \end{picture}
 \end{center}
 \caption{Some possible geometries in target spacetime; these simplified illustrations hint at the variations in any of the relevant structures (complex, K{\"a}hler, symplectic, supersymmetry), and may be combined in various ways}
 \label{f:4Cases}
\end{figure}
Allowing for $\mathscr{Y}^6$ to vary in some observable {\em\/spatial\/} directions was shown to necessarily include singularizations of $\mathscr{Y}^6$ that manifest as stringy cosmic strings ('branes)~\cite{Greene:1989ya,rCYCY}, as sketched in Figure~\ref{f:4Cases}\,(b). Allowing furthermore for {\em\/non-analytic\/} variations of the geometry of some of the extra dimensions~\cite{Randall:1999ee,Randall:1999vf} warps the overall geometry and can allow for a {\em\/simultaneous\/} emergence of exponential mass-hierarchy, localization of gravity, matter, and an exponentially suppressed cosmological constant in $\text{dS}^{1,3}$~\cite{rBHM1,rBHM4,rBHM5,rBHM7}; see the sketch in Figure~\ref{f:4Cases}\,(c). We will return to these constructions in Section~\ref{s:axilaton}.

\subsection{de~Sitter Bubble-Worlds}
\label{s:dSB}
Consider now, in turn, the recent {\em\/dynamical\/} proposal with the spacetime geometry consisting of two copies of $\text{AdS}^{1,4}$, glued together non-analytically across a 3-brane-world, $\text{dS}^{1,3}_{z=0}$~\cite{Banerjee:2018qey,Banerjee:2019fzz}; see also~\cite{Hawking_2000, Karch_2001,Danielsson:2021tyb}; 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 ``\text{AdS}-decaying'' proposal~\cite{Banerjee:2018qey,Banerjee:2019fzz} 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~\cite{Randall:1999vf}, the graviton is shown to obey a Schr{\"o}dinger-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 {\em\/not\/} the mode trapped by the $\d(z)$-function well~\cite{Banerjee:2018qey,Karch_2001}. Note that the so-obtained spacetime geometry is a combination of the sketches in Figure~\ref{f:4Cases}\,(a) and Figure~\ref{f:4Cases}\,(d): $\text{dS}^{1,3}_{z=0}$ appears as a (non-factor) subspace {\em\/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~\cite{Coleman:1980aw} (see~\cite{Banks:2021wqu} for a recent update), such vacuum decay phenomena have been estimated to be generic in string theory~\cite{Ooguri:2016pdq,Freivogel:2016qwc}\footnote{Refs.~\cite{Ooguri:2016pdq,Freivogel:2016qwc} 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.~\cite{Danielsson:2016rmq}.}, 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.~\cite{Banerjee:2018qey,Banerjee:2019fzz} 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~\cite{Green:1988uw,Candelas:1989ug} of ``Reid's phantasy''~\cite{rReidK0}. \footnote{There is also a  natural connection to the more recent and rather vast cobordism generalization~\cite{McNamara:2019rup}.} 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~\cite{rHitch}. To see why this is the case, we will however next turn to a  brief review of the axilaton toy model~\cite{rBHM5,rBHM7,rBHM10} in order to showcase the similarities with the $\text{AdS}^{1,4}$-decay model~\cite{Banerjee:2018qey,Banerjee:2019fzz}.

\section{A Discretuum of Toy Models}
\label{s:axilaton}

Generally accommodating the overall geometry types sketched in Figure~\ref{f:4Cases}\,(a--c) and aiming for a (de~Sitter) candidate four-dimensional spacetime, the metric Ansatz has a direct sum format,
 $\mathrm{d} s^2 = A^2\,\mathrm{d} s^2_{\sss1,3} +B^2\,\mathrm{d} s^2_{\sss6}$,
where $\mathrm{d} s^2_{\sss1,3}$ and $\mathrm{d} s^2_{\sss6}$ 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~\cite{rBHM1,rBHM4,rBHM5,rBHM7} further specializes to
\begin{equation}
  \mathrm{d} s^2 = A^2(z)\,\mathrm{d} s^2_{1,3} +\ell^2 B^2(z)(\mathrm{d} z^2{+}\mathrm{d}\q^2)_{\mathscr{Y}^2_\perp}
  +\mathrm{d} s^2_{\mathscr{Y}^4},
 \label{e:metric}
\end{equation}
where $\ell$ sets the length-scale in $\mathscr{Y}^2_\perp$, so the warp factors $A(z),B(z)$ and the coordinates $(z,\q)\in\mathscr{Y}^2_\perp$ 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, $\q\simeq\q+2\p$ 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,
 $\t\mathrel{\buildrel{\rm def}\over=} B+ig_s^{-1}e^{-\F}$,
in string theory, which exhibit an $\mathop{\textrm{SL}}(2;\mathbb{Z})$-monodromy owing to modular invariance~\cite{Polchinski:1998rq}.
Assuming then that $\t=\t(\q)$ 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 {\em\/discretuum.} Notably, $\t=\t(\w\q)$, and the $\mathscr{Y}^2_\perp$-anisotropy, $\w$, appears in the Einstein-Friedan equation~\cite{rF79a,rF79b},
\begin{alignat}9
  R_{\mu\nu} &={\cal G}_{\t\bar\t}\,(\vd_{\mu}\t)(\vd_{\nu}\bar\t)
  \mathrel{\buildrel{\rm def}\over=}\widetilde{T}_{\mu\nu}
  =\mathrm{diag}[\>\underbrace{0,0,0,0}_{\text{dS}^{1,3}},\,\underset{z}{0\strut},
  \underset{\q}{(\w/2\ell)^2\strut},\,\underbrace{0,0,0,0}_{\mathscr{Y}^4}\>]~,
\label{e:EinStein}\\
  {\cal G}_{\t \bar\t} &=\frac{-1~~}{(\t{-}\bar\t)^2} ={\cal G}_{\bar\t\t},\quad
  {\cal G}_{\t\t}=0={\cal G}_{\bar\t\,\bar\t}.
\end{alignat}
and sources the warped geometry~\eqref{e:metric}~\cite{rBHM5,rBHM7}:
\begin{subequations}
 \label{e:newAB}
\begin{alignat}9
    A(z) &= Z_\pm(z) \Big(1-\frac{\w^2 z_0^2}{160} Z_\pm(z)^2 +\dots\Big),
  \label{e:newA}\\
   B(z) &= \frac{1}{\ell z_0\sqrt{\Lambda_b}}
            \Big(1-\frac{\w^2z_0^2}{40} Z_\pm(z)^2 +\dots\Big)~,
  \label{e:newB}\\
\intertext{with the harmonic functions}
   Z_\pm(z) &\mathrel{\buildrel{\rm def}\over=} 1\pm|z|/z_0,\quad z_0>0,
  \label{e:newB}
\intertext{and the cosmological constant}
   \Lambda_b&=\frac{\omega^2 - (2\xi/3z_0)A^2(0)}{48\,\ell^2},
  \label{e:Lb}
\end{alignat}
\end{subequations}
where $0\leqslant\xi\leqslant12$ counts the source-branes\footnote{The stringy cosmic string-like~\cite{Shapere:1989kp} limit includes a total of $12{+}|\xi|$ supersymmetric 7-branes.}~\cite{rBHM2}. In~\cite{rBHM5,rBHM7,rBHM10}, the maximal, $\xi\!=\!12$, was used for simplicty. Thus, the $\mathscr{Y}^2_\perp$-anisotropy of the axilaton, $\t(\w\q)$, drives the cosmological constant~\eqref{e:Lb}. Furthermore, the inequalities $\w^2\!\geqslant\!(2\xi/3z_0)A^2(0)$ and $\Lambda_b\!\geqslant\!0$ are saturated only in the supersymmetric limit, where $\w,(\xi/z_0),\Lambda_b\!\to\!0$~\cite{rBHM5,rBHM7}. Finally, the length-scale $\ell$ emerges in~\eqref{e:EinStein} via dimensional transmutation (akin to $\L_{\rm QCD}$), 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~\eqref{e:metric}--\eqref{e:newAB} is continuous at $z\to\pm z_0$ with both sign-choices in~\eqref{e:newA}, 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 $|z|\!=\!z_0$~\cite{rBHM1,rBHM2}, beyond which the metric becomes complex. Thereby, the de~Sitter metric~\eqref{e:metric} explicitly desingularizes the total spacetime in this class of toy models~\cite{rBHM5}. This is corroborated by verifying that the standard curvature invariants for~\eqref{e:metric}--\eqref{e:newAB} are all finite, whereas the Kretschmann (Riemann-squared) invariant diverges at $z_0$ in the Minkowski solution. Finally,~\eqref{e:newB} 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.~\cite{Bento:2021nbb,Douglas:2009zn,Bena:2009xk,Bena:2017uuz} --- except that the background configurations are here restricted to a discretuum by the $\mathop{\textrm{SL}}(2;\mathbb{Z})$-monodromy. Indeed, the axilaton, $\t(\w\q)$, varies over $\mathscr{Y}^2_\perp$ including values that oscillate about $g_se^{\F}=O(1)$, 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 $\t(\w\q)$) also discontinuous along a branch-cut, e.g., along $\q\!=\!\pm\p$. Across this branch-cut, the model non-perturbatively patches a stringy weak-coupling regime with a reciprocally strong-coupling regime --- akin to ``$S$-folds''~\cite{rHIS-nGeoCY}. Nevertheless, the effective observable string coupling,
 $\vev{g_se^{\F}}$, 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~\cite{rBHM7,rBHM10}.

For future reference, we note that the effective Planck mass-scale, $M_4$, in $\text{dS}^{1,3}$ is exponentially boosted~\cite{rBHM4,rBHM10},
\begin{equation}
  M_4\!^2=M_6\!^4\,\ell^2\,z_0\!^{5/8}\,\xi^{-3/8}\,e^{+\xi z_0}\,
  2\p\Gamma\!_\pm\big(\frc38;\xi z_0\big),
 \label{e:M4M6}
\end{equation}
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
 $0\!\leqslant\!2\p\Gamma\!_\pm\big(\frc38;\xi z_0\big)\!\leqslant\!2\p\Gamma\!_\pm\big(\frc38\big)\!\approx\!14.89$. 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~\cite{Randall:1999vf}, the graviton modes are governed by an effective potential that here depends on the source-counting parameter, $\xi$, that also appears in~\eqref{e:M4M6}~\cite{rBHM4}. This potential also has a superposed $\d(z)$-function well induced from the $|z|$-dependence of the metric~\eqref{e:newA}, 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~\cite{rBHM5}. This is suppressed as compared to the na{\"\i}ve 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 $\d(z)$-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 $\d(z)$-function well in the effective potential. This populates the candidate observable spacetime with universally $\d(z)$-localized modes of any matter added to this toy model.

With $Z_+(z)$ in~\eqref{e:newA}, 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~\ref{f:WxY}\,(left).
In turn, with $Z_-(z)$ in~\eqref{e:newA},
 $\mathscr{Z}_-\!=\![-z_0,z_0]$ for
  so that $\mathscr{Y}^2_\perp$ includes its two circular boundaries; see Figure~\ref{f:WxY}\,(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/completion\footnote{To be precise: for $Z_-(z)$ in~\eqref{e:newA}, 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~\eqref{e:newA}, $\mathscr{Z}\!\approx\!\mathbb{C}^*$ 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 {\em\/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~\cite{rBHM5,rBHM7,rBHM10} 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~\cite{rBeast}\footnote{We will return to this non-trivial K3-fibration in Section~\ref{s:CY5}.}.
\begin{figure}[tbp]
 \begin{center}
  \begin{picture}(130,55)(-5,0)
   \put(-20,20){\includegraphics[height=30mm]{LanternR.pdf}}
   \put(7,0){\includegraphics[height=50mm]{CylBWorldOutIn2.pdf}}
   \put(57,0){\includegraphics[height=50mm]{CylBWorldInOut2.pdf}}
   \put(93,23){\includegraphics[height=25mm]{FunnelsR.pdf}}
   \put(50,20){\TikZ{\path[use as bounding box](0,0);
              \draw[brown!75!black,ultra thick,<->](-.3,1.5)--++(0,1);
              \draw[brown!75!black,ultra thick,<->](1.2,1.5)--++(0,1);
              \path[brown!75!black](.45,2)node{\Large$W^{1,3}_{z=0}$};
              \draw[orange,thick,->](-5,.5)to[out=-60,in=-90]++(1.5,-.4);
              \draw[blue,thick,->](5,.5)to[out=-120,in=-30]++(-1.5,-.3);
              \draw[blue,thick,->](-5,2.5)to[out=30,in=90]++(3.5,-.1);
              \draw[orange,thick,->](5,2.2)to[out=120,in=90]++(-2.5,.2);
              \path(.45,.25)node{\Large$\mathscr{Y}^2_\perp$};
              \path(.45,-.25)node{(horizontally)};
              \path(-5,-1)node{\shortstack{$\mathscr{Y}^2_\perp$ with\\[2pt]
                                           $Z_+(z)=1+|z|/z_0$}};
              \path(6,-1)node{\shortstack{$\mathscr{Y}^2_\perp$ with\\[2pt]
                                           $Z_-(z)=1-|z|/z_0$}};
              }}
  \end{picture}
 \end{center}
 \caption{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~\eqref{e:Geometries}, below.}
 \label{f:WxY}
\end{figure}
The 2-fibre compactification/completion process
\begin{equation}
  \text{dS}^{1,3}_z\!\rtimes\!\left(\mathscr{Z}\!\times\!S^1\right)\ltimes\text{K3}
  ~\xrightarrow[~z_0\to\infty~]{~\w\to 0~}~
  M^{1,3}\!\times\!\left(\mathscr{Y}^6 =
  (\mathbb{P}^1\ltimes\text{K3})\right)
 \label{e:pDefo}
\end{equation}
may be understood as the right-hand side depicting a static,
Minkowski/CY geometry being identified as the (double, supersymmetry-restoring) limit, $\w,1/z_0\to0$, 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~\eqref{e:pDefo} 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~\ref{f:WxY}.

 Although somewhat different in the technical details, the overall spacetime geometry in the axilaton toy model~\eqref{e:metric}--\eqref{e:M4M6}~\cite{rBHM5,rBHM7,rBHM10} and in the $\text{AdS}^{1,4}$-decay model~\cite{Banerjee:2018qey,Banerjee:2019fzz} are remarkably similar: The Schr{\"o}dinger-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 ($\mathop{\texttt{\#}}\limits$) symbol\footnote{The product $\mathscr{A}\!\rtimes\!\mathscr{B}$ denotes that the warp-factors in the block-diagonal metric vary over $\mathscr{B}$.}:
\begin{equation}
  \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.
 \label{e:Geometries}
\end{equation}
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~\eqref{e:Geometries} 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, $\text{dS}^{1,3}_{z=0}$~\cite{Banerjee:2018qey,Banerjee:2019fzz}.

 The right-hand side of~\eqref{e:Geometries} axilaton configuration has been presumed so far to be static and in/out-symmetric~\cite{rBHM5,rBHM7,rBHM10} --- but this need not at all be the case. Since all physical features in this model depend on the anisotropy of the axilaton, $\t(\w\q)$, it suffices to choose $\w_{\rm in}\neq\w_{\rm out}$. The two sides of the interfacing $\text{dS}^{1,3}_{z=0}\!\times\!S^1$ in~\eqref{e:Geometries} are
\begin{equation}
   \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},
\end{equation}
which are analogous to $\text{AdS}^{1,4}_{\rm in}$ and $\text{AdS}^{1,4}_{\rm out}$, as depicted in Figure~\ref{f:WxY}. 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, $\w>0$ in $\t=\t(\w\q)$, 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,''
 $\Tw{T}_{\q\q}=(\w/2\ell)^2$, in~\eqref{e:EinStein}.

 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~\eqref{e:Geometries} 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 \text{AdS}-decay model~\cite{Banerjee:2018qey,Banerjee:2019fzz}, the
  $\mathop{\textrm{SL}}(2;\mathbb{Z})$-monodromy of the axilaton models~\cite{rBHM5,rBHM7,rBHM10} restricts the anisotropy $\w$ to discrete values. This allows only discrete variations, i.e tunneling from one configuration to another.

\section{Calabi-Yau 5-Folds}
\label{s:CY5}
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~(\ref{e:pDefo}) to be fibered over the $\mathscr{Z}\!\times\!S^1$, and hence leading to a K3-fibered Calabi-Yau 3-fold in the $z_0\to\infty,~\w\to 0$ 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~\cite{rHitch} 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 {\em\/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:
\begin{enumerate}
 \item the inherent semi-infinite cohomology in string theory~\cite{rFrGaZu86},
 \item the free loop space as the configuration space of (closed) strings~\cite{Bowick:1990wt,Bowick:1988nj,rBowRaj87b,rBowRaj88,rBowRaj87a,rBowRaj87,rAlGoRe87,Oh:1987sq,rHHRR-sDiffS1},
 \item the worldsheet quantum field theory approach~\cite{Pilch:1987eb}.
\end{enumerate}
The orientability of closed strings~\cite{Polchinski:1998rq} (including zero-modes~\cite{Freidel:2015pka,Freidel:2017wst,Freidel:2017nhg}!) 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~\cite{rBeast}.
 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 K{\"a}hler 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''~\cite{rReidK0}, Calabi-Yau 5-folds generically contain numerous exceptional, isolated Fano ($c_1,R_{\mu\bar\nu}>0$) compact complex {\em\/surfaces,} $\mathfrak{S}_2$, i.e., real four-dimensional subspaces~\cite{rHitch}. 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 curvature\footnote{Being 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, $R_{\mu\nu} = \L g_{\mu\nu}$ defines $1/\sqrt{\L}$ 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~\ref{s:ED} 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~\eqref{e:metric}--\eqref{e:M4M6}.

Reid's argument for 3-folds~\cite{rReidK0} 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~\eqref{e:Geometries}. In the coarse classification discussed in Section~\ref{s:NS} and referring to Figure~\ref{f:4Cases}, the scenario $\text{dS}^{1,3}_{z=0}\subset\mathscr{X}^{1,9}$ that is the preimage of the ``Euclideanizing'' Wick rotation assignment
\begin{equation}
  \text{dS}^{1,3}_{z=0}\subset\mathscr{X}^{1,9}
  ~\mapsto~
 \mathfrak{S}_2\subset\mathfrak{X}_5
\end{equation}
 is depicted in Figure~\ref{f:4Cases}\,(d): the total spacetime, $\mathscr{X}^{1,9}$, is in no way assumed to factorize, nor even foliate\footnote{For 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~\cite{rTY1,rTY2}:
\begin{alignat}9
  \underbrace{\text{compact Fano$_{n{+}1}$}}_{c_1>0} &~\smallsetminus~
  \underbrace{\text{compact CY$_n$-fold}}_{c_1=0}    &~=~
  \underbrace{\text{non-compact CY$_{n{+}1}$-fold}}_{c_1=0}. \label{e:F-CY=CY}
\intertext{Here, the non-compact space on the right-hand side is known to admit a suitable K{\"a}hler 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:}
  \underbrace{\text{compact CY$_{n{+}1}$}}_{c_1=0}   &~\smallsetminus~
  \underbrace{\text{compact Fano$_n$-fold}}_{c_1>0}  &~\overset?=~
  \underbrace{\text{non-compact Hyp$_{n{+}1}$-fold}}_{c_1<0},
 \label{e:CY-F=AdS}
\end{alignat}
where Hyp$_{n+1}$ 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 $n\!=\!0$:
\begin{enumerate}
 \item $\mathbb{P}^1\smallsetminus\{2\,\text{pts.}\}\approx\mathbb{C}^*$ (a non-compact cylinder),
  which is well known to be Ricci-flat.
 \item $T^2\smallsetminus\{\text{pt.}\}$ is a 1-handled disc, and so a hyperbolic non-compact surface.
\end{enumerate}
Furthermore, we will need the following specializing generalization
\begin{equation}
   \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}.
 \label{e:CY-kF=AdS}
\end{equation}
 Locally, this result follows from the so-called adjunction theorem~\cite{rBeast}: 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 {\em\/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~\eqref{e:CY-kF=AdS} 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~\eqref{e:CY-kF=AdS} then implies that the non-compact complex 5-fold $\mathfrak{X}_5\smallsetminus\mathfrak{S}_2$ admits a metric of negative Ricci curvature. (Again, {\em\/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 \text{AdS}-decaying scenario~\eqref{e:Geometries}. However, these exceptional 4-manifolds, $W^{1,3}\subset\mathscr{X}^{1,9}$, have {\em\/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 \text{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~\cite{Witten:1981gj} and also~\cite{Dibitetto:2020csn}.

In the Euclidean (and holomorphic) rendition, the standard K{\"a}hler 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 K{\"a}hler cone.
 In the simplest case, take the bubble to be $\mathbb{P}^2$. Then:
\begin{enumerate}
 \item The K{\"a}hler 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 $\mathbb{P}^2\approx\mathbb{C}^2\cup\mathbb{P}^1$.)
 \item There is therefore a Riemannian metric that differs from the above
   K{\"a}hler metric only in being null over the $\mathbb{P}^1$ at the North pole,
 \item 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.
\end{enumerate}
Together, these imply that $W^{1,3}$ has positive curvature, the simplest (most symmetric) of which is $\text{dS}^{1,3}$; see also Appendix~\ref{s:ED}.

Other Fano complex surface candidates for the bubble $\mathfrak{S}_2$ will contain more than one exceptional (complex) curve, each isomorphic to $\mathbb{P}^1\!\approx\!S^2$~\cite{rBeast}. 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 $3,4,{\cdots},12$~\cite{rGHSAR,rBeast}.
 (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 {\em\/signature\/} of these (real) four-dimensional spaces is $\s = 4{-}\chi_E$.
 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~\ref{s:ED}.

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 {\em\/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, $\mathscr{C}_{t,\e}\subset(\mathscr{X}^{1,9}\!\smallsetminus\!W^{1,3})$, such that $\lim_{\e\to0}\mathscr{C}_{t,\e}=\mathscr{C}_t\subset W^{1,3}$. (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 {\em\/away from\/} $\text{dS}^{1,3}_{\rm us}$ but their restriction to $\text{dS}^{1,3}_{\rm us}$ will fail to be covariantly constant {\em\/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'' K{\"a}hler metric; it needs an $\mathfrak{S}_2$-localized ``correction'' to be positive everywhere on $\mathfrak{X}_5$; see~\cite{rGHSAR} 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 structure\footnote{Supersymmetry 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.

\section{Summary, Outlook and Conclusions}
\label{s:Coda}
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,~\cite{Banerjee:2018qey,Banerjee:2019fzz,Bento:2021nbb}. Notably, these indicate two separate aspects that render such work more fruitful:
 ({\small\bf1})~a careful analysis of multi-parameter and near-singular configurations,
and
 ({\small\bf2})~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~\cite{rBHM1,rBHM4,rBHM5,rBHM7,rBHM10} that may be viewed as a non-holomorphic and non-analytic deformation of the stringy cosmic string ('brane) scenario~\cite{Greene:1989ya,rCYCY}.

 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~\cite{Banerjee:2018qey,Banerjee:2019fzz}. 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~\cite{rHitch} and~\cite{rBHM7}. 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~\eqref{e:Geometries} in the \text{AdS}-decaying scenario of Section~\ref{s:dSB}.

 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.

\noindent
{\bf 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.

redpajama_set_name: RedPajamaArXiv


Trade Deadline Day is a tangle of rumors, uncertainty and frayed nerves—so how does it feel to be told you're heading to a new team twice in the same day? The players who have been swapped in quick succession describe it being more surreal than anything else.
It was wedding week at his summer cottage and Sam Gagner was busy getting ready. Relatives were arriving. Friends needed directions. His future wife Rachel had already handled most of the logistics for the ceremony and reception—including an extravagant Great Gatsby-themed boat cruise around Lake Muskoka—but a few final tasks remained. "Just making sure that everything was all settled," Gagner says.
Then his cellphone buzzed. Again and again and again.
The latest indoctrinate arrived Sunday when Nashville acquired Mark Letestu from Edmonton and then swiftly flipped him to Columbus, less than 24 hours before Monday's trade deadline. Defenseman Mark Streit spent roughly one hour as a member of the Tampa Bay Lightning before heading to Pittsburgh in a separate swap last March. Blueliner Victor Bartley experienced a double-deal day in Jan. 2016 (Predators to Coyotes, Coyotes to Canadiens), as did winger Marty McInnis in Oct. 1998 (Flames to Blackhawks, Blackhawks to Ducks). Of course, no mention of multiple moves would be complete without Mike "Suitcase" Sillinger, the most traded player in league history, who went from Columbus to Dallas to Phoenix on July 22, 2003.
Luckily for those involved, NHL news travels fast. No one has been airborne, bound for one particular city, when the second trade goes down and an entirely new destination beckons upon landing. In fact, most of these bang-bang moves happen behind the scenes, absent of the particular player's knowledge. On Feb. 1, 1971, Maple Leafs forward Mike "Shakey" Walton was relaxing at a ski chalet near Toronto, away from the team over a dispute about playing time. The landline rang. It was someone claiming to work for the Bruins.
Almost a quarter-century later, Claude Lemieux was getting dental work done on four dislodged front teeth, the nasty result of a high stick taken during the '94–95 playoffs with the Devils. A contract dispute had ensured Lemieux's inevitable departure from New Jersey, but the winger wasn't entirely sure where he was headed. "It happened quickly," he recalls. "It's not like I was traded to the Islanders and became a member of the organization for a day and then traded later to Colorado. It was quick. It just felt like I got traded straight-up.
So Gagner waited, thrust into trade limbo, stressing over his ultimate destination. Hockey is a business, everyone says, but mantras don't make moving any easier. Minutes felt like months. At last, word washed onto the lake shores: Gagner to Arizona, via Tampa Bay, summarized by Roberto Luongo's tweet of an igloo emoji, followed by a palm tree and finally a cactus.
Like Streit last spring, Gagner only joined the Lightning for several hours before moving elsewhere. But the experience left a lasting impression. When the Coyotes later visited Tampa Bay during the 2014–15 season, teammates coaxed Gagner into putting money on the board—a bet of sorts traditionally foisted upon players facing their former teams.
"I was expecting a tribute video too," Gagner says. "When it didn't come, it upset me a little bit. But I'll move on."


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It could be a story that is simply of wide interest to viewers and has little impact otherwise. Many times, breaking news is used after the news organization has already reported on the story.
Government proclamations, concerning royal ceremonies, laws, taxes, public health, have been dubbed news since ancient times.
Humans exhibit a nearly universal desire to learn and share news, which they satisfy by talking to each other and sharing information.
Technological and social developments, often driven by government communication and espionage networks, have increased the speed with which news can spread, as well as influenced its content.
The genre of news as we know it today is closely associated with the newspaper.Another corollary of the newness of news is that, as new technology enables new media to disseminate news more quickly, 'slower' forms of communication may move away from 'news' towards 'analysis'.
News is simply one input, along with paper or an electronic server necessary to prepare a final product for distribution.
According to some theories, "news" is whatever the news industry sells Journalism, broadly understood along the same lines, is the act or occupation of collecting and providing news.
From a commercial perspective, news is simply one input, along with paper (or an electronic server) necessary to prepare a final product for distribution. A news agency supplies this resource "wholesale" and publishers enhance it for retail.
Quantitatively, the internet has massively expanded the sheer volume of news items available to one person. The speed of news flow to individuals has also reached a new plateau. This insurmountable flow of news can daunt people and cause information overload.
We can call this period the "technetronic era", in which "global reality increasingly absorbs the individual, involves him, and even occasionally overwhelms him."
Online journalism is news that is reported on the Internet
In cases of government crackdowns or revolutions, the Internet has often become a major communication channel for news propagation; while it's a (relatively) simple act to shut down a newspaper, radio or television station, mobile devices such as smartphones and netbooks are much harder to detect and confiscate.
The propagation of internet-capable mobile devices has also given rise to the CITIZEN JOURNALIST, who provide an additional perspective on unfolding events.
A professor at the University Graduate School of Journalism, has said that " everything we thought we once knew about journalism needs to be rethought in the DIGITAL AGE." Today the work of journalism can be done from anywhere and done well. It requires no more than a reporter and a laptop.
In that way, journalistic authority seems to have become more individual- and less institution-based.
Online news has also changed the geographic reach of individual news stories, diffusing readership from city-by-city markets to a potentially global audience. Steve Jobs – Apple Worldwide Developers' Conference, 1997
A media organisation should adapt to meet the challenges and opportunities presented by changing audience behaviour in our module entitled "Newsroom evolution from digital denial to digital first".
According to some theories, "news" is whatever the news industry sells Journalism, broadly understood along the same lines.
News values are the professional norms of journalism.
Because each individual has a particular point of view, it is recognized that there can be no absolute objectivity in news reporting.
Journalists can collectively shift their opinion over what is a controversy up for debate and what is an established fact. Some commentators on news values have argued that journalists' training in news values itself represents a systemic bias of the news.
The importance of social media in news production and news dissemination.
The challenges and opportunities presented by social media
Social media is an increasingly disruptive force on the media landscape. It challenges traditional, mainstream media to reconsider how they operate.
Social media often releases information about which mainstream media might not have been aware, and information that mainstream media might have tried to ignore.
It can offer a wider, more diverse perspective on life than that covered by traditional media.
An empowered audience
Now we look at what a social media strategy could mean for a media organisation. But first, let's look at how we got to this stage in media's development.
The media is in a constant state of change, or at least it should be. Technological advances, leading to changing audience behaviour, resulting in altered attitudes to consuming and sharing news, which means that a media organisation can't afford to stand still.
Innovation is needed, but only if it makes business sense. There have been many stages of media evolution over the years, below we look at three.
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The deployment will create an active-duty force comparable in size to the U.S. military contingent in Iraq, as Trump's administration draws attention to a caravan of migrants that is trekking through Mexico toward the United States.
General Terrence O'Shaughnessy, the head of U.S. Northern Command, said 800 U.S. troops were already en route to the Texas border and more were headed to the borders in California and Arizona.
"The president has made it clear that border security is national security," O'Shaughnessy said, as he detailed a much larger deployment that the 800 to 1,000 troops predicted by U.S. officials last week.
O'Shaughnessy said some soldiers would be armed although it was unclear who, beyond U.S. military police, might need those weapons. U.S. officials have stressed that the troops would not police the border and instead carry out support roles like building tents and barricades, and flying U.S. customs personnel to locations along the border.
Trump railed against illegal immigration to win the 2016 U.S. presidential election and has seized on the caravan of Central American migrants at campaign rallies in the run-up to next week's vote, firing up support for his Republican Party.
Although Trump's supporters in Congress praised the deployment of troops, the American Civil Liberties Union derided it as a political stunt.
"President Trump has chosen just before midterm elections to force the military into furthering his anti-immigrant agenda of fear and division," said Shaw Drake, policy counsel for the American Civil Liberties Union Border Rights Center in El Paso, Texas.
Trump said on Twitter on Monday that the military would be waiting for the procession -- suggesting a far more direct role in confronting the migrants than the Pentagon described.
"Many Gang Members and some very bad people are mixed into the Caravan heading to our Southern Border," Trump tweeted.
"Please go back, you will not be admitted into the United States unless you go through the legal process. This is an invasion of our Country and our Military is waiting for you!" he added.
Kevin McAleenan, the U.S. commissioner of Customs and Border Protection, said a group of approximately 3,500 immigrants were travelling through southern Mexico with the intent of reaching the U.S. border. A second caravan of about 3,000 people were at the Guatemala-Mexico border, McAleenan said.
At the same time, over the last three weeks, border agents have encountered nearly 1,900 people per day either crossing the border illegally or presenting themselves at ports of entry, with over half of them being children alone or parents and children travelling together, McAleenan said.
"We are already facing a border security and humanitarian crisis at our southwest border," he said.
Trump's decision to call in the military appears to be a departure from past practice, at least in recent years, in which such operations were carried out by National Guard forces -- largely part-time military members who are often called upon to serve in response to domestic emergencies.
The decision to send active duty forces this time gives the Pentagon the ability to more rapidly mobilise greater capability than would be immediately available with the Guard, officials told Reuters.


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