aainstitutetext: Rudolf Peierls Centre for Theoretical Physics, University of
Oxford, Parks Road, Oxford OX1 3PU, UK.bbinstitutetext: Harvard University, 17
Oxford Street, Cambridge, MA, 02139, USA.ccinstitutetext: Theory Group,
Department of Physics, University of Texas, Austin, TX 78712,
USA.ddinstitutetext: Maryland Center for Fundamental Physics, Department of
Physics, University of Maryland, College Park, MD 20742, USA.

#  Warped Compactifications in Particle Physics, Cosmology and Quantum Gravity

Prateek Agrawal b , Cari Cesarotti c , Andreas Karch b , Rashmish K. Mishra b
, Lisa Randall d , and Raman Sundrum

###### Abstract

Particle physics has evolved in the past decade through evaluating the
consequences of experimental measurements as well as exploiting theoretical
tools that permit exploration of new model building and cosmological
possibilities. Particularly due to insights from the AdS/CFT correspondence,
higher-dimensional warped compactifications, in particular, have played a big
role in recent developments by allowing a study of regimes of parameters that
would otherwise be intractable. Similarly, theoretical developments in quantum
gravity benefit from the bigger range of possibilities that can be explored
using warped geometry, allowing for constructions of string vacua with
positive cosmological constant and for the exploration of entanglement and
information transfer in arbitrary dimensions. Puzzles remain in both more
phenomenologically oriented and more theoretically oriented contexts which
form the basis for a rich research program in the future as well.

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

##  1 Introduction

The idea of extra dimensions—first introduced to physics in the early
twentieth century—had a renaissance in the late 1990s when potential
phenomenological applications were recognized and the notorious chirality
problem was solved with branes and string constructions. As an outgrowth of
that effort, new solutions to Einstein’s equations—in particular warped extra-
dimensional geometry with localized gravity–motivated in part by string
theoretical constructions with branes—were discovered whose implications have
had far-reaching consequences well beyond the initial phenomenological domain
but including those as well. An important connection and reason for this
ubiquity is the relation to AdS spacetime, where many new and exact results
have been found. Because of the diverse phenomenological and cosmological
applications, the discovery of a truly novel model building element, and the
potential to exploit dualities to better understand strong interactions and
conceptual aspects of the gravitational theory, warped extra dimensional
geometries have had a sustained and consequential impact and are likely to do
so in the future.

This white paper focuses on the relevance of warped extra-dimensional geometry
to phenomenology, cosmology, gravity wave physics, string constructions of de
Sitter space, and the black hole information paradox–all of which have been
active research topics over the past decade and which lead to intriguing
questions promising exciting developments in the future. The first few topics
relate to some initial work on the hierarchy problem, whereas the latter
topics re-emerged from work in string theory trying to find tractable
realizations of de Sitter space and the search for workable models to study
black hole entropy and evaporation (see Almheiri:2020cfm  for a review),
notably in more than two dimensions. In this white paper we summarize some of
the ways the geometry appears in a variety of contexts that can feature in the
theory initiative–particularly in its attempt to unify the cohort of
activities currently underway.

##  2 Model Building and Phenomenology

A major driving force for many model building and phenomenological activities
in high energy physics in the last few decades has been the seemingly
unnatural hierarchies in the Standard Model (SM)—the Planck Electroweak
hierarchy, and the fermion mass hierarchies (and the associated flavor
structure). The initial work in the late 90s on Randall-Sundrum (RS) models,
based on warped extra-dimensional geometry (illustrated in fig. 1), was a new
Beyond Standard Model (BSM) framework to maintain the EW hierarchy and as an
added feature could naturally account for fermion hierarchies, making it one
of the few of the electroweak theories to naturally account for both
hierarchies with a single underlying mechanism. Such theories address the
flavor structure of the SM without pushing the scale for new physics to a very
high value, which is marginally acceptable due to the RS GIM mechanism
Agashe:2004cp ; Cacciapaglia:2007fw ; Fitzpatrick:2007sa ; Perez:2008ee . The
low scale of new physics poses challenges, but also in principle makes the
framework more readily testable than most flavor models–although signals can
still be challenging. The experimental efforts to look for new physics in this
framework, both by direct production, or by precision measurements, has been
one of the drivers of our experimental programs–often in ways applicable to
other BSM scenarios.

Moreover, the framework can naturally explain neutrino masses and large mixing
angles in the lepton sector alongside small mixing angles in the quark sector.
The wide variation in the fermion masses and mixing angles in the SM is one of
the biggest challenges for any new physics scenario. Instead of models that
address parts of the puzzles separately, it is desirable to have a framework
that can give this wide spectrum in a natural way, from a same underlying
mechanism. This is particularly challenging, given anarchic neutrino masses
but hierarchical quark masses. RS models based on warped extra-dimensions
provide a natural way to generate such hierarchies using anarchic 5D Yukawa
structure, using the peculiar dependence of the fermion wavefunction on the
bulk fermion mass—exponential in the IR region, but power law in the UV
region. In particular, to explain neutrino masses and large mixing angles in
the lepton sector, ref. Perez:2008ee  considered Majorana mass for the right-
handed neutrinos and a standard type I seesaw mechanism in the warped
scenario. The intermediate seesaw scale needed in type I seesaw was naturally
realized in the 5D framework without putting in a new scale via a natural
choice of 5D mass parameters. In a recent work, ref. Agashe:2015izu
considered the same setup for generating neutrino masses, but by going to the
mass basis rather than the KK basis, they showed that the seesaw mechanism
looks inverted. The very small neutrino masses now arise from exchange of
singlet modes that are mostly Dirac, but have a highly suppressed Majorana
mass term, the latter being a natural consequence of the exponential nature of
the wavefunction.

Figure 1: Illustration of a finite extra dimension with warped geometry.

Since the flavor aspects are covered in another white paper flavorSnowmass2021
, the focus of our model building section will be recent work on naturalness.
To address the lack of experimental discoveries, new research has introduced
modifications with varying amounts of non-standard signals, so that the
present experimental analyses are not directly applicable. This aspect can be
taken as the starting point, but the implications are broader, since toy-
models in existing BSM frameworks where the resulting signal is challenging
can provide a way to develop and therefore cover a wider class of signal
sensitivity in our experimental programs.

One possible direction along these lines has been the construction of UV
complete models that can arise within existing BSM frameworks. In one such
approach, ref. Agashe:2016rle  constructed a 5D model with a non-standard KK
spectrum. The 5D bulk gauge and matter fields were required to propagate to
different degrees in the IR of the warped geometry, resulting in a
qualitatively different low energy effective field theory (EFT) description,
and hence drastically different signals at colliders. The resulting KK
spectrum where the matter KK modes are heavier than the gauge KK modes,
allowed the models to be safe from flavor/CP constraints. The collider
accessible modes are the KK graviton and KK gauge bosons, although with very
different decay patterns as compared to the minimal RS scenario. An important
feature was a relatively lighter radion in the spectrum. The phenomenological
consequences of these features was analyzed in more detail in refs.
Agashe:2016kfr ; Agashe:2017wss ; Agashe:2018leo , and a full exploration of
the resulting low energy EFT and their experimental consequences has been
useful from the point of covering the space of possibilities, though sometimes
requiring moderate fine-tuning in the EW sector. Dedicated experimental
analysis based on these scenarios were performed recently CMS:2021nrj ;
CMS:2021qyp ; CMS:2021qev .

In another work with the same broad motivation, ref. Csaki:2018kxb
constructed a 5D model where the KK states form a continuum beyond a certain
scale–a qualitatively different spectrum of new physics states. The explicit
construction used warped geometry in a linear dilaton background. The top and
gauge partners needed to regulate the Higgs mass sensitivity form a continuum,
though a realistic benchmark still required percent level fine-tuning. The
continuum nature of the KK states motivates broader experimental search
strategies, given that the standard resonant searches are inapplicable. Both
these works explored interesting modifications to the low energy EFT in this
framework, and their non-standard experimental signals–these directions are
well motivated independently in the search for new physics.

Unnaturalness of the cosmological constant (CC) is an even more challenging
puzzle, at least as measured by the degree of unnaturalness. Even though much
of past research has focused on them separately, it is plausible that the two
are related, and warrants constructing explicit models where such connections
can be studied. In this direction, ref. Arvanitaki:2016xds  constructed a
model based on flat as well as warped extra-dimensions, where tuning the mass
of Higgs relative to a fermion mass scale allowed the radius of an extra
dimension to become large and develop an enhanced number of vacua available to
scan the cosmological constant down to its observed value. The model has
fermions that are accessible at colliders, and an unnaturally light radion,
which evades high energy collider constraints. In another explicit model, ref.
Bloch:2019bvc  constructed a crunching mechanism that selects Hubble patches
with small CC after reheating. Patches with large CC tunnel at a fast rate to
a vacuum with large negative CC and ultimately crunch, while patches with
small CC are long-lived. A crucial ingredient is a crunching sector that
reacts to the presence of a large CC and acts to lower its value, thereby
driving the relevant patch to crunch. The crunching sector has a super-cooled
first order phase transition that was modelled by warped RS geometry. Yet
another model Csaki:2020zqz  used a similar idea of crunching away patches
with large CC, by mixing the Higgs to the dilaton of a strongly coupled CFT
(dual to radion in the RS framework), such that a meta stable minimum appears
when the Higgs VEV is non-zero and below the TeV scale. As a result, only
Hubble patches with unnaturally small values of the Higgs mass support
inflation and post-inflationary expansion, while all other patches rapidly
crunch. Interestingly, a consequence of this structure is again a very light
radion. Although the verdict is out on these models, all these directions
prompt thinking about the cosmological constant and model building more
generally in a different way, tying it to EW sector and cosmology.

In addition to explaining the various hierarchies that appear in the SM, RS
geometries can also be used to understand dynamics of hidden sectors. As an
example, RS is used in ref. Cesarotti:2020uod  to construct a phenomenological
model of a generic strongly coupled hidden sector. The motivation for this
kind of model stems from the necessity to discriminate a generic new physics
signal event from the immense SM background. Since QCD primarily results in 2
or 3-jet events, a potentially lucrative and robust approach to identifying
new physics is to search for signature of many isotropically distributed jets.
However, generating template events that look quasi-isotropic can be poorly-
motivated or intractable due to the number of free parameters in the toy
model. The authors of ref. Cesarotti:2020uod  therefore develop a 5d
simplified model using RS framework that can generate spherical radiation
patterns with a minimal number of parameters.

Figure 2: Visualizations of a characteristic final state radiation patterns
produced with the 5d simplified model framework. The length of the vectors
corresponds to the total energy of the particle. By varying the input
parameters, the simulation can produce collimated (left), uniform (center),
and intermediate (right) event shapes. Figures adapted from ref.
Cesarotti:2020uod .

This approach is motivated by the appearance of KK towers in extra dimensions
which introduce many new particles to the model without requiring a large
number of parameters. The curvature in RS is essential for cascade decays in
the model to proceed in a quasi-isotropic pattern: a flat extra dimension
would yield decays exactly at threshold, producing a high-multiplicity cloud
of particles at rest; the curvature allows slightly off-threshold, therefore
unboosted with randomly oriented non-zero momentum. The global event shape can
be tuned via the 5d scalar mass and interpolates between a wide array of
topologies, as shown in fig. 2. Ultimately this 5d simplified model approach
produces a minimal toy model to study spherical radiation patterns.
Furthermore, this work has also been applied to develop and quantify a novel
event shape observable, event isotropy Cesarotti:2020ngq , which has improved
dynamic range in the quasi-isotropic regime. It is important to note that this
is one particular study of the dynamics generated within the RS framework, and
that many other phenomenological models can stem from warped extra-dimensional
geometry to study novel cascades in hidden sectors Dienes:2018bbv ;
Brax:2019koq ; Costantino:2020msc , and motivate thinking about portals to
hidden sectors Contino:2020tix  in model-independent ways.

##  3 Cosmology

The cosmological history of the universe provides a way to probe very high
energy dynamics, and has become increasingly relevant as we enter the era of
gravitational wave (GW) detection and precision cosmology. GWs can probe phase
transitions in the early universe before the CMB epoch, and have the potential
to transmit vital information about the degrees of freedom and the dynamics of
the phase transitions. The RS model provides a rich playground for novel
cosmological signals. This is an exciting arena with several open questions in
phenomenology and model building, and deep questions in quantum field theory
such as the confinement-deconfinement transition. This has led to a very
active ongoing exploration of cosmology in RS models with many open questions
that we describe in this section. The high temperature phase of the RS model
is described by a 5-dimensional black hole which can have significant impact
on the mechanisms to populate cosmological relics that rely on a global
symmetry. The phase transition to the low-temperature phase is generically
first-order and occurs around the TeV scale, producing GWs peaked at mHz
frequency, which is exactly in the sensitivity range of upcoming space-based
GW measurements, such as those expected from LISA Randall:2006py .

The holographic dual of the 5D geometry in the RS model is expected to be a
nearly-conformal gauge theory confining at around the TeV scale. Thus we
expect a confinement transition in RS at high temperatures, analogous to the
QCD confinement crossover at $T\sim 100$ MeV. The seminal work of ref.
Witten:1998zw  laid the groundwork for studying confining phase transitions in
holographic theories. This framework was applied more explicitly to the RS
phase transition in ref. Creminelli:2001th . As the universe gets reheated
after the end of inflation, the thermodynamically preferred phase of the RS
model is described by a background geometry in which the IR brane is replaced
by a black brane–a horizon with translational symmetry along the transverse
directions. The geometry is of a Schwarzschild blackhole in global-AdS,
referred to as the AdSS geometry. In equilibrium, the location of the black
brane is determined by the temperature of the universe. As the temperature
drops, the black brane moves towards the AdS horizon (in the deep IR). Below a
critical temperature $T_{c}$, the black brane moves past the putative location
of the IR brane (whose location is decided by the Planck-Electroweak
hierarchy), and the standard RS geometry becomes the thermodynamically
preferred phase. The phase transition between these two phases is of first
order, since the two geometries are both solutions of Einstein’s equations and
hence are local saddles of the path integral, thereby having a barrier
separating them in the configuration space. By AdS/CFT duality, these two
phases are described by the confined and deconfined phase of the dual CFT.

The details of the phase transition are hard to calculate in general, but can
be estimated in models where the radion remains light Creminelli:2001th . The
transition occurs through the nucleation of a true vacuum bubble in the AdSS
phase, where the RS geometry inside connects with the AdSS geometry
surrounding it with a gravitational instanton. The tunneling exponent can be
then computed under the assumption that the action of the bubble can be well-
approximated by the radion contribution. Even when the RS geometry becomes the
stable state of the theory, the rate of phase transition $\Gamma$ (probability
per unit four-volume) of transition to this state is heavily suppressed, and
schematically scales as

$\displaystyle\Gamma\sim\text{exp}\left(-\frac{M_{5}^{3}/k^{3}}{\delta}\right)\sim
e^{-N_{c}^{2}/\delta}\>,$ (1)

where $M_{5}$ is the 5D Planck mass, $k$ is the AdS curvature and $\delta$ is
a dimensionless parameter related to the stabilization mechanism of the RS
geometry. By AdS/CFT duality, it can be written as the second term in eq. (1),
where $N_{c}$ is the number of colors of the dual CFT, and $\delta$ is now a
parameter related to an explicit breaking of the CFT. For perturbative control
of the 5D description, we need $(M_{5}/k)^{3}\sim N_{c}^{2}\gg 1$, while the
requirement of stabilizing the geometry with a large hierarchy requires
$\delta\lesssim 1$, both of which reduce the rate of transition exponentially.
The requirement for a phase transition at $T\lesssim T_{c}$ translates to a
very stringent bound on $M_{5}/k$:

$\displaystyle(M_{5}/k)^{3}\lesssim\delta\log(M_{\text{Pl}}/T_{c})\>,$ (2)

where $M_{\text{Pl}}$ is the 4D Planck scale. To explain the electroweak
hierarchy, $T_{c}\sim\mathcal{O}(\text{TeV})$ which translates to
$M_{5}/k\lesssim 10$, even for $\delta\sim\mathcal{O}(1)$ \- the low energy
gravitational description being barely under theoretical control. The way out
is to require the universe to undergo some amount of supercooling so that the
phase transition completes at $T_{n}\ll T_{c}$, and the rate for transition is
enhanced by some function of the ratio $T_{c}/T_{n}$. However, this barely
helps with making $M_{5}/k$ parametrically large, and further, $T_{n}$ can not
be arbitrarily small if this description is to fit the standard BBN picture.

Assuming these issues are addressed in more complete models (such as those
referenced below), the features of being a first order phase transition and
requiring supercooling can have dramatic observational consequences. The first
order nature of the phase transition can lead to gravitational wave (GW)
emission Randall:2006py ; vonHarling:2017yew ; Baratella:2018pxi ;
Agashe:2020lfz ; Baldes:2021aph  which can be observed in present and upcoming
GW experiments such as LISA Caprini:2015zlo ; Caprini:2019egz , DECIGO
Seto:2001qf  and BBO Corbin:2005ny . This is an exciting opportunity on many
frontiers – as the theoretical and experimental advances in the GW detection
program develop over coming decade, the opportunity to probe early universe
dynamics in this BSM framework will become increasingly relevant. The
supercooling feature affects the relic density of the relics of the cosmic
past since there is a period for which the universe inflates, driven by the
vacuum energy in the deconfined phase, and dilutes all the relic densities.
This opens up parts of parameter space hitherto ruled out in the standard
scenario. This is a further opportunity for novel model building and
exploration of theories and mechanisms that become relevant in this context.

The theoretical activities of the past decade have focused broadly on how
robust this feature of a suppressed rate of phase transition and the
associated supercooling is. There are two broad categories under which the
recent approaches can be classified.

### IR modification

The suppression from the small parameter $\delta$ in eq. (1) is tied to the
shape of the radion potential. Close to the temperatures at which the phase
transition becomes feasible, a change in the radion potential can modify the
estimate for the rate of the phase transition. The idea of ref.
vonHarling:2017yew  was to use the corrections to the radion potential coming
from the QCD condensate, which reduced the barrier depth, and hence enhanced
the rate for the phase transition. The authors in Baratella:2018pxi  further
required the 4D CFT degrees of freedom to flow to another fixed point after
QCD confinement, thereby gaining more theoretical control, and gave a
geometrical picture for this situation where the mass of the GW scalar is
tachyonic Pomarol:2019aae . The more recent work in Agashe:2019lhy ;
Agashe:2020lfz  used similar ingredients where they modelled the IR flow to
another fixed point by a special form of the potential for the Goldberger-Wise
(GW) field, without the need for QCD corrections to the low energy radion EFT.

### High Temperature modification

An alternative approach to the phase transition is to modify the description
of the theory at high temperatures. The authors of ref. Agrawal:2021alq
considered additional scalars in the low energy EFT apart from the radion, and
calculated finite temperature corrections to the combined potential. The
resulting potential has a meta-stable minimum at high temperatures, whose
location moves with temperature, eventually merging to the temperature
independent minimum in a standard radion potential. Additionally requiring
that the state of the universe after preheating is not AdSS but rather the
modified RS, the need of a phase transition and the issue of a small rate are
avoided altogether, at the cost of the initial condition. The model is stable
against a reverse phase transition to the AdSS phase, simply because of the
suppressed rate, turning this feature into an advantage.

The observational consequences of this modification are striking. Early
universe cosmological phenomena such as WIMP freeze-out, axion abundance,
baryogenesis, phase transitions, and gravitational wave signatures are
qualitatively modified. In particular, the temperature dependent KK scale, to
be contrasted with the IR modifications, has novel consequences, which were
pointed out in ref. Agrawal:2021alq .

### Future directions

Several interesting questions remain if we are to better understand the range
of cosmological possibilities. The role of the radion and the related low
energy EFT, on which the calculations for the rate are based, has not been
fully explored. In particular, the presence of other light fields in the EFT
can change the picture significantly. Other warped geometries such as
Klebanov-Tseytlin (KT) and Klebanov-Strassler (KS) geometries can also change
the description significantly, both by a different low energy EFT description,
as well as a richer high temperature phase (corresponding to the AdSS geometry
in simplified description) and symmetry breaking patterns Buchel:2021yay .
Perhaps more drastically, a better understanding of low energy dynamics
derived from warped compactifications might be required.

The observational consequences of alternative scenarios is another fruitful
direction. We already saw that the cosmological history and the relic
abundances are very different in the UV and IR modifications that have been
considered in the literature. A more detailed modeling of the GW signal can
tell us more about these scenario. Further exploration of these ideas can
significantly broaden our understanding of possible cosmological histories
consistent with present observations, and may give us a peek into the UV.

##  4 Warped Geometry in String Theory

Warped geometries with branes have found important applications in addressing
fundamental questions in quantum gravity and string theory. Crucial for this
connection is the fact that these braneworlds naturally result in geometries
that are locally AdS space and so can be interpreted with the AdS/CFT
correspondence. The latter asserts that gravity on AdS is equivalent to a
conformal field theory living on its boundary, providing a setting in which
quantum gravity can be reliably studied. This connection to AdS/CFT allows an
interpretation of warped compactifications with branes in a dual field theory
language with interesting consequences (non-perturbative RG, compositeness).
One application that was recognized early on is that this connection provides
a generalization of AdS/CFT to spaces with boundaries Karch:2000ct ;
Karch:2000gx . As a direct offshoot of this investigation, the first example
of a consistent theory of interacting massive gravity was uncovered which
paved the way for many of the advances that followed in this arena. This
holographic duality between conformal field theories with boundaries and
warped geometries with branes has since then been significantly deepened in
Takayanagi:2011zk ; Fujita:2011fp , where it was advanced as a general duality
between AdS and boundary conformal field theories, AdS/BCFT for short. The
basic geometry of a braneworld dual to a BCFT is summarized in figure 3.

(a) Vertical Cross Section

(b) Horizontal Cross Section

Figure 3: Horizontal and vertical cross section of global Anti-de Sitter space
with embedded branes realizing a BCFT. The red brane itself also has an Anti-
de Sitter worldvolume the (red) shaded areas are excised by the brane.

Given the interesting aspects of warped geometries that have been uncovered in
terms of effective theories of general relativity coupled to brane models,
efforts soon followed to understand these “bottom-up” models from the “top-
down”. The main purpose of these studies was to embed warped compactifications
in the framework of string theory. Ref. Verlinde:1999fy  laid out the earliest
ideas of how to realize warped compactifications with Minkowski slicing in
string theory, where it was argued that stacks of branes that are required to
cancel tadpoles arising from flux sources in string theory can pull out an
AdS-like throat from an otherwise compact manifold, giving essentially a
realization of a UV brane. A subsequent important development was recognizing
other types of warped geometries that have natural embedding in string theory,
such as that proposed by Klebanov and Strassler Klebanov:2000hb ; these warped
throats nicely allowed internal geometries to truncate, giving a top-down
realization of an IR brane. Ref Giddings:2001yu  formalized the framework of
warped compactifications with fluxes, realizing naturally large hierarchies,
which served as the starting point for many interesting string constructions
over the years. From the string theory point of view, a very different route
led to the realization of top-down AdS/BCFT models within string theory. Here
early suggestions for how to realize a BCFT in string theory from a system of
D3 branes ending on 5-branes was made in Karch:2001cw , with a concrete
realization of these ideas being worked out in a series of papers
DHoker:2007hhe ; DHoker:2007zhm  several years later. The techniques of this
latter work have since then laid the foundation for constructing string theory
embeddings of supersymmetric boundary conformal field theories in a variety of
dimensions and with varying amount of supersymmetry, as laid out in dozens of
papers.

Significant progress has also been made in the other direction: using branes
within string theory to aid in the construction of novel string theory
models–most notably in helping generate a tunable vacuum energy that can be
used for de Sitter models. The observation of a positive cosmological vacuum
energy is a big challenge in string theory in that the only controlled models
that resemble our world rely on spacetime supersymmetry, which can never have
a positive energy vacuum. Breaking supersymmetry in a controlled fashion when
the associated energy is far less than that of electroweak physics introduces
a big challenge. One way to address this is to generate a small parameter
through warping—one that competes with a supersymmetry breaking energy that is
suppressed by mass scales. Most prominent among these attempts is the
development of controlled models of de Sitter space, such as the one famously
pioneered by Kachru, Kallosh, Linde, and Trivedi Kachru:2003sx . Many details
of these construction have since been fleshed out with progress towards an
understanding of the 10d solutions, appearing in Kachru:2019dvo ;
Demirtas:2021nlu . This theory has introduced a good deal of interest and
controversy and has in part catalyzed the swampland studies which are quite
prevalent now and will continue to dominate some studies of string vacua.
While the full solutions of this type appear complicated at first sight, many
of their essential properties can be understood and systematically worked out
in a low energy effective description in terms of 5d warped geometries with
branes Randall:2019ent .

A different route to de Sitter space built on the idea of warped
compactifications with branes is provided by the dS/dS correspondence
Alishahiha:2004md . Very similar to the AdS realization of BCFTs, which are
based on AdS spaces sliced in a warped compactification via lower dimensional
AdS space, the dS/dS correspondence is built on a slicing of dS space by dS
slices. It was shown in Karch:2003em  that, when viewed as a warped
compactification, de Sitter space is “autolocalizing” – it has effectively a
built in UV brane even in the absence of any brane sources. Based on this
observation, the dS/dS correspondence provides a dual description of de Sitter
gravity in terms of lower de Sitter gravity by interpreting it as a brane
world with localized gravity. Based on this idea, string theory construction
of de Sitter spaces based on intersecting brane models can be given
Dong:2010pm . More recently, this duality has been put on a firm footing using
the tool of the so called TTbar deformation Gorbenko:2018oov . These latter
ideas have since then been applied to work towards a holographic understanding
of de Sitter space also directly in the static patch Coleman:2021nor .

##  5 Applications to the black hole information paradox

The particular case of warped AdS branes has also found interesting
applications in addressing questions about black hole evaporation in a fully
realized setting. Recent developments using quantum information tools to
understand the recovery of information from an evaporating black hole have led
to remarkable progress, most prominent among them the calculation of the so
called Page curve, describing the initial loss of information and its
subsequent recovery. The basic story of the Page curve calculation was first
developed in the context of low dimensional gravity. Using the concept of a
quantum entangling surface, a Page curve for an evaporating black hole
consistent with unitarity was obtained using semi-classic methods. Much of
this progress is summarized in a separate white paper dedicated to this topic
Bousso:2022ntt . While claims have been put forward that this mechanism may be
more widely applicable, most concrete calculations have been performed in the
context of gravity coupled to an external non-gravitating bath. How important
this coupling to the bath is is a topic of active debate. In fact, recent work
suggest that the non-trivial Page curve disappears if the bath is removed and
is replaced by an entirely trivial flat Page curve Laddha:2020kvp ;
Raju:2021lwh ; Geng:2021hlu .

Braneworlds, which in this context are often referred to as end-of-the-world
branes, naturally provide a setting in which AdS gravity can be coupled to an
external bath, allowing the study of black hole evaporation in the well-
controlled setting of AdS/CFT. In fact, all examples of Page curve
calculations in more than 1+1 dimensions (starting with the seminal paper
Almheiri:2019psy ), as well as some of the lower dimensional constructions
Almheiri:2019hni , are based on the realization of gravity localized on a
brane in Anti-de Sitter space. This includes the first embedding into full
fledged string theory of a unitary Page curve in more than 1+1 dimensions
Uhlemann:2021nhu . As already described in the first papers on warped AdS
branes Karch:2000ct ; Karch:2000gx , one has three dual descriptions of the
system in this case: the gravitational description in terms of classical
Einstein equations with a brane source, which can be solved classically, the
BCFT dual, which in principle gives a UV complete quantum mechanical
description of the system, as well as the one of interest for black hole
evaporation: the “intermediate” picture of gravity on the brane coupled to the
bath represented by the BCFT ambient space. This double-holographic dictionary
allows many new perspectives on the evaporation problem. For example, the
emergence of so called islands in the resolution of the black hole information
paradox can be understood from a purely BCFT point of view as a change of
dominance of two different exchange channels in the OPE of the BCFT
Rozali:2019day .

One very interesting feature of these particular realizations of evaporating
black holes on branes is that the setups automatically yield a theory of
massive gravity, as recently emphasized in Geng:2020qvw . The mass of the
graviton was first found from the point of view of the classical higher
dimensional gravity from solving the small fluctuation problem in brane
sourced Einstein equations in Karch:2000ct ; Karch:2000gx  and was understood
in terms of a loop induced mass for the graviton from the boundary conditions
connecting to the bath by Porrati in Porrati:2001db . From the BCFT point of
view, the graviton mass is given by the anomalous dimension of the boundary
stress tensor, which is not conserved due to the interactions with the BCFT
ambient space Aharony:2003qf . In fact, this field theory explanation makes it
clear that a graviton mass in an unavoidable consequence of coupling gravity
to any bath Aharony:2006hz ; Kiritsis:2006hy . Whether this mass is a crucial
part of the Page curve story, as argued in Geng:2020qvw ; Geng:2020fxl ;
Geng:2021hlu , or just a technicality that has no bearing on the information
recovery from the black hole, is a topic under active investigation. No matter
the ultimate answer, warped brane-worlds will be one of the crucial tools we
have in order to address this question.

While the main role of the end of the world branes in these recent black hole
studies has been to realize AdS gravity coupled to a bath, they have also
played a separate role in at least one of the most studied toy models of black
hole evaporation: the west coast model Penington:2019kki . In this work, end
of the world branes were not merely the canvas on which the gravitational
universe lives, they were models for the matter that lives inside the black
hole, realizing some of the micro-states of the black hole.

Besides giving simple existence proofs of unitarily evaporating black holes in
higher dimensions, warped brane worlds have also been used to address other
aspects of this problem. In many cases, an interesting phase structure exists
on when and how entanglement islands form in order to unitarize the theory.
These phase structures can easily be mapped out by performing calculations
using the classical bulk gravity Chen:2020hmv ; Chen:2020uac ; Geng:2021mic .

In addition to using brane worlds as a tool to study the black hole
information paradox, recent work has also used it to study parallel
information paradoxes in cosmology Geng:2021wcq , and also drawn attention to
the fact that several fundamental aspects of the brane worlds themselves, in
particular their holographic dictionary, are still poorly understood. Only
very recently has an explicit dictionary defining the intermediate picture in
terms of bulk quantities been proposed Neuenfeld:2021wbl . Furthermore, a very
intriguing apparent non-locality in the intermediate picture has been pointed
out in Omiya:2021olc . Clearly these questions deserve further investigation.

In an orthogonal but somewhat related recent development, RS2 modeling has
been invoked as a key feature in the construction of smooth traversable
wormhole solutions Maldacena:2020sxe . The subject of asymptotic symmetries
has also enjoyed renewed interest in the last several years. In this context,
novel infinite-dimensional asymptotic symmetries of gauge and gravitational
theories in 4D asymptotically AdS spacetime were discovered by exploiting an
RS2-like construction Mishra:2017zan . In a weak gauging limit, the AdS
correlators can be related to those of a 2D CFT, defined at the intersection
of the asymptotic boundary and a probe brane. This makes the underlying
Virasoro and Kac-Moody Ward identities manifest. The connection of this
construction to questions about information paradox, and the relevance to the
existence of a Page curve, is an interesting direction to pursue.

##  6 Summary

In this white paper, we have presented a brief sampling of some of the
important implications of warped geometry and its underlying resurgence and
persistence in recent years. Both theoretically and phenomenologically this
framework yields tractable testable examples that can be rigorously studied in
detail. Warped geometries has had and will continue to have an enormous impact
on theoretical particle physics.

## Acknowledgements

We thank Hao Geng and Shamit Kachru for useful discussions and comments on the
draft.

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# A monolithic interferometer for high-sensitive strictly-local detection of
orbital angular momentum states of light

Mirko Siano  Dipartimento di Fisica, Università degli Studi di Milano and INFN
Sezione di Milano, via G. Celoria 16, 20133 Milan, Italy  Corresponding
author<EMAIL_ADDRESS>Bruno Paroli  Dipartimento di Fisica, Università
degli Studi di Milano and INFN Sezione di Milano, via G. Celoria 16, 20133
Milan, Italy  Simone Cialdi  Dipartimento di Fisica, Università degli Studi di
Milano and INFN Sezione di Milano, via G. Celoria 16, 20133 Milan, Italy
Stefano Olivares  Dipartimento di Fisica, Università degli Studi di Milano and
INFN Sezione di Milano, via G. Celoria 16, 20133 Milan, Italy  Matteo G. A.
Paris  Dipartimento di Fisica, Università degli Studi di Milano and INFN
Sezione di Milano, via G. Celoria 16, 20133 Milan, Italy  Edoardo Suerra
Dipartimento di Fisica, Università degli Studi di Milano and INFN Sezione di
Milano, via G. Celoria 16, 20133 Milan, Italy  Marco A. C. Potenza
Dipartimento di Fisica, Università degli Studi di Milano and INFN Sezione di
Milano, via G. Celoria 16, 20133 Milan, Italy

###### Abstract

We propose an innovative monolithic interferometer to distinguish the
topological charge of radiation carrying orbital angular momentum. Remarkably,
our method requires to access only a small portion of the entire wavefront.
The proposed scheme relies on a monolithic birefringent crystal, and as such
it is intrinsically stable and does not require any feedback or thermal drift
compensation. An experimental setup has been realized to prove the
effectiveness of the proposed method down to the photon counting regime.

††journal: ol

Radiation with Orbital Angular Momentum (OAM) formalized at the end of the
twentieth century [1] has in recent decades aroused considerable interest in
various scientific disciplines [2, 3, 4, 5, 6, 7, 8], effectively leading to
the formation of what is now known as singular optics [9].

Common to the various disciplines and applications, detection of OAM radiation
at the level of single photon in high-sensitive detectors is of particular
interest in both classical [10, 11] and quantum realms [12, 13]. Another
fundamental aspect is the ability of measuring the topological charge using
only a small portion of the propagating beam [8, 14, 15, 16]. Recognizing OAM
states in a completely local manner would open important perspectives, not yet
explored, for a novel class of detectors and receivers. However, achieving
high-sensitive detection of OAM states in a strictly local manner is still a
challenge, as generally the entire wavefront of radiation is necessary both
with refractive [17] and diffractive [18, 19] state-of-the-art techniques.

In this Letter, we report for the first time a novel high-sensitive local
detection method that exploits only a small portion of the beam (also far from
the singularity) to distinguish different OAM states. We also prove that the
proposed scheme performs extremely well and reliably even with a very small
collection of photons.

The method exploits the peculiar phase properties of OAM radiation in a two-
point detection scheme. Let $x-y$ be a plane transverse to the propagation
direction of an OAM beam. The phase $\phi$ of the associated electric field
varies as a function of the azimuthal angle $\theta$ on the $x-y$ plane
according to $\phi=l\theta$, where $l$ is the topological charge. Therefore, a
measurement of the phase difference $\Delta\phi=\phi_{2}-\phi_{1}$ at two
different angular positions $\theta_{1}$ and $\theta_{2}$ separated by
$\Delta\theta=\theta_{2}-\theta_{1}$ provides the OAM state
$l=\Delta\phi/\Delta\theta$. Here we show how to measure the phase difference
$\Delta\phi$ with an extremely robust and strictly local monolithic
interferometer based on a birefringent calcite crystal. The proposed scheme
exhibits high resilience to external perturbations and enables working in
photon counting regime for long times without the need of any drift
compensation, an essential requirement for high-sensitive detection.

We exploit the birefringent nature of the calcite crystal to realize a
monolithic wavefront-division interferometer endowed with extreme stability
and resistance to external perturbations, according to the following scheme.
We consider two points $P_{1}$ and $P_{2}$ on the front face of the crystal,
where $P_{1}$ and $P_{2}$ represent the input ports of the interferometer.
They are at the same radial distance $r$ from the OAM singularity and are
separated by $\Delta x$, thus they subtend the angle $\Delta\theta\sim\Delta
x/r$ with respect to the center of the OAM beam. We then cover either $P_{1}$
or $P_{2}$ with a $\lambda/2$ wave plate oriented at 45 degrees with respect
to the polarization axis of the incoming OAM beam, which is made coincident
with the polarization axis of the ordinary ray of the crystal. Thanks to this
configuration, the electric fields at $P_{1}$ and $P_{2}$ have orthogonal
polarizations and propagate through the crystal along the ordinary and
extraordinary paths, respectively, and superimpose on an observation plane
downstream the crystal. By projecting both fields along a common polarization
direction with a linear polarizer, interference fringes form:

$I(x)=4I_{0}\frac{\cos{(k_{f}x+l\Delta\theta+\phi_{0})}+1}{2},$ (1)

where $I_{0}$ is the intensity of the incident beam, $k_{f}=2\pi/\lambda_{f}$,
$\lambda_{f}$ being the spatial period of the interference fringes, and
$\phi_{0}$ is the phase shift due to the difference between the ordinary and
extraordinary paths. In Eq. (1) we assume, for simplicity, that fringes are
oriented along the vertical direction $y$, with $x$ the horizontal axis.

Figure 1: (a) Experimental setup. SF-BE: spatial filter and beam expander; M1,
M2, M3: mirrors; DMD: Digital Micromirror Device; L1, L2: positive lenses;
$\lambda/2$: half-wavelength wave plate; CRYS: birefringent calcite crystal;
P: 45 degrees polarizer; S: narrow slit; F: single-mode optical fiber coupled
to L2 and to a fast acquisition system PicoScope. (b) Detail of the monolithic
interferometer based on the birefringent calcite crystal. The case of a
perfectly collimated incident beam is depicted. (c) and (d) Intensity profiles
of the OAM beams with $l=1$ and $l=2$, respectively, at the entrance of the
monolithic interferometer.

Notice that the dependence on the topological charge in Eq. (1) is given only
by the linear term $l\Delta\theta$, since $k_{f}$ and $\phi_{0}$ are
independent on $l$. Therefore, the OAM state can be directly detected either
from the absolute position of the fringes, or from their relative intensity
values at a particular observation point $x$.

Figure 2: Top: Portion of the interference pattern of the monolithic
interferometer for the topological charges $l=-1$ and $l=-4$. Interference
fringes generated by the superposition of the beams propagating along the
ordinary and extraordinary paths are clearly visible to the right. Bottom:
Intensity profiles of the interference patterns integrated along the fringe
direction.

A proof-of-principle setup was realized to prove the effectiveness of the
novel local detection method as sketched in Fig. 1 (a). A He-Ne laser beam
with $\lambda$=632.8 nm is spatially filtered and collimated. The resulting
Gaussian beam has a diameter of 3.7 mm (1/$e^{2}$ intensity value) and
illuminates a Digital Micromirror Device (DMD). A computer-generated fork
hologram is projected onto the DMD to convert the Gaussian beam into an OAM
beam. The first diffraction order is selected, corresponding to a OAM beam
with the desired topological charge. The OAM beam is then properly aligned
along the horizontal and vertical planes and sent to the interferometer by
means of two mirrors. The beam is expanded through a positive lens before
impinging onto a birefringent calcite crystal (Thorlabs BD40). The beam
diameter at the entrance of the crystal is roughly 33 mm for an $l=0$ OAM
state corresponding to a standard Gaussian beam (1/$e^{2}$ intensity value).
Half of the birefringent crystal is covered by a $\lambda/2$ wave plate,
oriented at 45 degrees with respect to the vertical polarization of the
incoming OAM beam. This ensures that the polarization of the transmitted
portion of the OAM wavefront is rotated along the horizontal plane. The
vertically and horizontally polarized portions of the incoming OAM beam
propagate inside the calcite crystal along the directions parallel to the
ordinary and extraordinary rays, respectively, and superimpose downstream the
crystal. A 45 degrees polarizer ensures interference between these two
wavefronts. Interference fringes are acquired with a standard CMOS camera
(Thorlabs DCC1240M, 1280 x 1024 pixels, 5.3 x 5.3 µm2 pixel size).

A detailed view of the monolithic interferometer is sketched in Fig. 1 (b).
For the case of a perfectly collimated incident beam, the ordinary and
extraordinary rays which, at the entrance of the crystal, are separated by a
suitable $\Delta x$, exit the crystal from the same point and propagate
collinearly and perfectly overlapped along the same direction, giving rise to
a uniform intensity distribution upon interference
($\lambda_{f}\rightarrow\infty$ in Eq. (1)). The particular value of $\Delta
x$ for which this occurs depends on the crystal and the radiation wavelength.
It is roughly 4.3 mm in our case.

Contrarily, if the incident wavefront is endowed with a finite curvature, the
two rays emerge from the crystal from two different points and superimpose on
the observation plane with a finite angle $\Theta$, thus forming interference
fringes with finite periodicity $\lambda_{f}\sim\lambda/\Theta$. In such case,
the two rays originate from portions of the incoming wavefront whose
separation $\Delta x$ is smaller compared to the case of a perfectly
collimated beam, the exact value depending on the actual wavefront curvature.
In our case, for a radius of curvature of 2.6 m, the actual separation between
the two points $P_{1}$ and $P_{2}$ is 3.3 mm.

The interference patterns acquired with the CMOS camera at the exit of the
monolithic interferometer are shown in Fig. 2 (top) for $l=-1$ and $l=-4$.
They are numerically integrated along the direction of the interference
fringes to extract the interference profiles shown in Fig. 2 (bottom), while
at the same time reducing noise. Notice that the rightmost portion of the
pattern is characterized by the expected interference fringes due to the
superposition of the ordinary and extraordinary rays from the crystal
(projected along a common polarization direction), while the leftmost portion
shows distortions introduced by diffraction at the edge of the $\lambda/2$
wave plate.

To check the effectiveness of the interferometer, we precisely measure the
phase shift of the fringes upon changing the topological charge from $l=-1$ to
$l=-4$. We report in Fig. 3 (left) the same integrated intensity profiles as
in Fig. 2 (bottom) in the range $5.6-6.6$ mm, properly normalized according to

$I_{n}=\frac{I-\text{min}\\{I\\}}{\text{max}\\{I-\text{min}\\{I\\}\\}},$ (2)

where $I$ is the intensity profiles in Fig. 2 (bottom) and $I_{n}$ is the
corresponding normalized version. Normalization according to Eq. (2) allows to
distinguish the OAM states from the position of the fringes despite the
absolute intensity values of the $l=-1$ and $l=-4$ OAM beams substantially
changes. The measured phase shift is $0.513\pm 0.007$ rad, in excellent
agreement with the calculated value $0.47\pm 0.05$ rad. The main source of
uncertainty for the calculated value is the position of the local
interferometer with respect to the singularity ($r=21\pm 2$ mm). The
experimental phase shift was evaluated using the phase parameters extracted by
a least squares method to Eq. (1) and the experimental error stems from the
fit uncertainty.

To test the performances of the interferometer in detecting OAM states with
high-sensitivity, we push the system into the photon counting regime. To this
aim, a small portion of the interference pattern is selected through a narrow
slit with aperture 50 µm, much smaller than the fringe period. The light
transmitted by the slit is collected by a single-mode fiber coupled to a
positive lens. The extremely small aperture of the fiber (roughly 8 µm) and
the tight alignment requirements for proper lens-fiber coupling ensure the
photon counting regime. The resulting photon counts are read by a
photomultiplier and converted to electric pulses measured by a fast
acquisition card (Picoscope 4224A).

We adjust the lateral position of the slit across the interference plane to
properly set the working point of the detector close to the region of the
interference pattern with the highest intensity derivative, to maximize
sensibility [16]. Notice that this is made possible by the small, albeit
finite wavefront curvature imposed by the positive lens before the
interferometer, which results in interference fringes with a finite
periodicity. Contrarily, for a perfectly collimated OAM beam with vanishing
curvature, a uniform intensity pattern would form downstream the crystal, as
previously discussed. Therefore, based on Eq. (1), we optimize the states
separation by experimentally setting the working point of the interferometer
to $k_{f}x+\phi_{0}\approx\pi(m+1/2)$, where $m$ is an integer [16]. We do so
by sending an $l=0$ state (corresponding to a standard Gaussian beam) to the
interferometer, and by adjusting the slit position halfway between the maximum
and the minimum intensity of an interference fringe [16].

(a)  (b)

Figure 3: Left: Normalized intensity $I_{n}$ of the integrated interference
pattern generated with the monolithic interferometer for the OAM states
$l=-1$, $l=-4$. The red and green curves represent fits to Eq. (1) to extract
the phase parameters. Right: Normalized counts obtained with the monolithic
interferometer working in photon counting mode for different OAM states.
Dashed lines show the statistical errors. Absolute photon counts are shown in
brackets.

After this initial calibration, counts relative to the different OAM states
$l=-1$, $l=1$, $l=-2$, $l=2$ have been measured, each in an interval time of
$100$ ms. For each OAM state, counts are then normalized to their maximum
value, which is obtained when the slit is positioned in correspondence of an
intensity maximum of the fringe pattern. This enables direct comparison among
the different OAM states, regardless of the overall intensity level $I_{0}$ in
Eq. (1) for the different OAM beams, similarly to the procedure adopted in Eq.
(2) and below it. Results are shown in Fig. 3 (right). The four OAM states are
well distinguishable by using $\leq$ 160 photon counts only. Notice that the
splitting of the states $l=\pm 2$ is twice as much that of the states $l\pm
1$, as predicted by theory.

Finally, to quantify the stability of the interferometer, we monitored any
eventual drift of the interference fringes generated by either the $l=-1$ or
the $l=-4$ OAM beam over long acquisitions. In the worst case, we measured a
maximum phase change of less than 0.25 rad over more than one hour,
corresponding to a phase drift of 70 µrad/s, likely in presence of mechanical
or thermal relaxations. The impact on our measurements is definitely
negligible, since this maximum drift would introduce a phase shift 30 times
smaller than the measured phase change of 0.513 rad between the two images
acquired with a temporal delay of 240 s in Fig. 3 (left), and orders of
magnitude smaller than the phase changes between either the $l=\pm 1$ states
or the $l=\pm 2$ states during the photon counting experiments, with
acquisitions being performed over a total time of 200 ms. This proves that the
monolithic interferometer fulfills the requirements of stability during normal
operations, and does not require any thermal drift compensation to distinguish
different OAM states over minutes-long acquisitions.

In conclusions, we have shown a novel, intrinsically stable monolithic
interferometer based on a birefringent calcite crystal, capable of detecting
the topological charge of radiation carrying orbital angular momentum by
exploiting a very small portion $f=5\%$ of the entire wavefront only
($f=2\Delta xL/\pi r^{2}$, $L=11$ mm being the vertical size of the calcite
crystal). An experimental setup has been realized to prove the effectiveness
of the proposed method through the absolute measurement of the phase shift
introduced by changing the OAM states. Experimental results are in good
agreement with theory. Furthermore, the high stability of the interferometer
allows photon counting measurements of OAM states. Results prove that the
states $l=-1$, $l=1$, $l=-2$ and $l=2$ can be detected with only a few
hundreds of photon counts, opening new perspectives for high-sensitive local
measurements of OAM radiation without the need of thermal drifts compensation
and with high resilience to mechanical vibrations.

In the current setup, the photon counting experiment has been conducted by
operating the monolithic interferometer in combination with a single slit.
Therefore, the sensibility can be further increased by employing e.g. a
periodic array of slits, or a grating, having the same fringe periodicity.
Acting on several fringes simultaneously, the overall accuracy and sensibility
can then be greatly enhanced. This is not necessary when the wavefront
curvature approaches zero, since the light intensity concentrates in a single
fringe downstream the crystal exit port.

The proposed method can be advantageously applied to detect OAM photon states
at large distances from the source, where it is not possible to access the
entire wavefront of the radiation beam, or in most applications where the
position of the singularity cannot be precisely aligned, especially in long-
distance free-space communications. Our findings can also foster the
development of new detection schemes in the quantum regime.

Acknowledgments This work was supported by the National Institute for Nuclear
Physics (INFN) (project: ADAMANT).

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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