11institutetext: Department of Physics, Cornell University, Ithaca, NY,
USA22institutetext: School of Natural Sciences, Institute for Advanced Study,
Princeton, NJ 08540, USA33institutetext: Walter Burke Institute for
Theoretical Physics, Caltech, Pasadena, CA 91125, USA44institutetext: CERN,
Theoretical Physics Department, CH-1211 Geneva 23, Switzerland

# Snowmass White Paper:
The Analytic Conformal Bootstrap

Thomas Hartman 2 Dalimil Mazáč 3 David Simmons-Duffin  4 Alexander Zhiboedov
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The analytic conformal bootstrap is an array of techniques to characterize,
constrain, and solve strongly interacting quantum field theories using
symmetries, causality, unitarity, and other general principles. In the last
decade, bolstered by the development of new Lorentzian methods, it has been
used to solve conformal field theories at large spin; to place bounds on
energy distributions, event shapes, operator product coefficients, and other
observables; and to understand aspects of quantum gravity in anti-de Sitter
space. We review these advances and highlight several promising areas for
future exploration. Targets include developing new methods to close the gap
between numerical and analytic bounds, extending the bootstrap beyond
conformal fixed points, applications to quantum gravity and cosmology, and
building on ties to condensed matter theory and mathematics.

##  1 Introduction

Conformal field theories (CFTs) are central to some of the deepest questions
in theoretical physics: How can we describe and solve strongly-coupled
systems? What are the building blocks of quantum field theories? What is the
space of UV-complete theories of quantum gravity? By studying CFTs, we gain a
powerful unifying perspective on all of these problems.

At the same time, these deep questions are hard to address with traditional
perturbative methods. The conformal bootstrap seeks to constrain and solve
CFTs using nonperturbative structures like symmetry, causality, and unitarity.
These nonperturbative conditions combine into a surprisingly rigid framework,
with wide-sweeping consequences. In the ’80s, bootstrap methods were applied
by Belavin, Polyakov, and Zamolodchikov to exactly solve an infinite class of
2d CFTs Belavin:1984vu . More recently, beginning with the work of Rattazzi,
Rychkov, Tonni, and Vichi in 2008 Rattazzi:2008pe , numerical investigations
of bootstrap conditions have established their power for constraining CFTs in
higher dimensions. For example, bootstrap constraints fix the critical
exponents of the 3d Ising model to high precision El-Showk:2012cjh ; El-
Showk:2014dwa ; Kos:2014bka ; Simmons-Duffin:2015qma ; Kos:2016ysd . These
promising results have renewed interest in developing a better analytical
understanding of bootstrap constraints.

### Motivating questions

CFTs are quantum field theories (QFTs) that are invariant under the conformal
group — a group that includes rotations and translations, and also rescalings
of space and time (dilatations). Conformal symmetry emerges at long distances
in a tremendous variety of physical systems, from condensed matter physics,
through statistical physics, to string theory, and CFTs describe this
universal emergent behavior.

The universality of conformal symmetry at long distances gives a framework for
organizing QFTs more generally: we can model a QFT as a renormalization group
(RG) flow from a (slightly deformed) CFT at short distances to another CFT at
long distances. In this sense, CFTs are the building blocks of QFTs. While
this framework does not encompass all known QFTs, it has some important
virtues. Firstly, it simultaneously describes both weakly-coupled and
strongly-coupled systems (even those with no known Lagrangian description).
Secondly, CFTs obey rigorous axioms and possess rich mathematical structures
that make them easier to understand than general QFTs.

Among these mathematical structures is the operator product expansion (OPE),
an algebra underlying all point-like observables in CFTs. Via the OPE, CFTs
become the ultimate algebraic objects — they are controlled by a rigid
algebra, but finding the structure constants of that algebra is equivalent to
solving the dynamics of a complex, relativistic, quantum many-body system.
Associativity of the OPE gives rise to the crossing symmetry equation,
expressing the equality of expansions in two different channels:

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\put(2.8,1.0){\line(1,-2){0.3}}
\put(1.2,1.0){\makebox(0.0,0.0)[]{${\sum}_{\mathcal{O}}^{\phantom{\mathcal{O}}}$}}
\put(4.5,1.0){\makebox(0.0,0.0)[]{${\sum}_{\mathcal{O}}^{\phantom{\mathcal{O}}}$}}
\put(3.9,1.0){\makebox(0.0,0.0)[]{$=$}}
\put(1.65,1.8){\makebox(0.0,0.0)[]{\small$1$}}
\put(1.65,0.2){\makebox(0.0,0.0)[]{\small$2$}}
\put(3.15,1.8){\makebox(0.0,0.0)[]{\small$4$}}
\put(3.15,0.2){\makebox(0.0,0.0)[]{\small$3$}}
\put(4.88,1.8){\makebox(0.0,0.0)[]{\small$1$}}
\put(4.88,0.2){\makebox(0.0,0.0)[]{\small$2$}}
\put(6.1,1.8){\makebox(0.0,0.0)[]{\small$4$}}
\put(6.1,0.2){\makebox(0.0,0.0)[]{\small$3$}}
\put(5.8,1.0){\makebox(0.0,0.0)[]{\small$\mathcal{O}$}}
\put(2.4,1.2){\makebox(0.0,0.0)[]{\small$\mathcal{O}$}}
\put(5.5,0.6){\line(0,1){0.8}} \put(5.0,0.35){\line(2,1){0.5}}
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\put(5.0,1.65){\line(2,-1){0.5}} \end{picture}\end{gathered}$ (1)

Using associativity and unitarity, analytic bootstrap methods have uncovered
important features of the OPE in simplifying limits, such as the large spin
limit. The myriad connections between CFTs and physical phenomena mean that
these results have wide applications. For example, analytic large-spin
calculations can be used to simultaneously understand gravitational bound
states in AdS, energy distributions in collider experiments, and cubic
symmetry breaking in critical magnets. Analytic bootstrap methods have also
been used to show that certain special OPE algebras saturate universal bounds
on scaling dimensions and other observables. An important goal is to further
close the gap between numerical bootstrap bounds and analytic bounds.

Via the AdS/CFT correspondence, CFTs are equivalent to UV-complete theories of
quantum gravity in Anti de Sitter (AdS) space. Outside of AdS, the question of
which UV completions exist is an elusive one. One reason is that we do not
know the axioms of quantum gravity in general spacetimes. (Some features of
effective field theory should break down — but which ones?) By contrast, the
axioms of CFT are clear and relatively well understood. Remarkably, this makes
it possible to reason with confidence about quantum gravity in AdS by studying
CFTs. For example, analytic bootstrap methods make it possible to rigorously
constrain and study gravitational $S$-matrices in AdS, despite the absence of
well-established axioms for gravitational S-matrices in flat space. Analytic
bootstrap methods have highlighted the central role of causality and unitarity
in constraining UV-completions, and these lessons may have important
implications as we seek to understand UV-completions in other contexts,
including our own universe.

##  2 Recent progress

### Lightcone bootstrap and the large-spin expansion

There are no known solvable, interacting CFTs in $d>2$ and most of our
knowledge about their properties comes from simplifying limits, such as the
$\epsilon$-expansion, weak coupling, or large $N$. The lightcone bootstrap
reveals simplicity and universality of the CFT data at large spin by analyzing
the crossing equation in the lightcone limit.

As a pair of operators becomes light-like separated, the corresponding OPE
channel is dominated by the unit operator, with small corrections coming from
low twist operators. Crossing symmetry (1) relates proximity to the light-cone
in one channel to the inverse spin of operators in the other channel. In this
way, the lightcone bootstrap leads to the conclusion that every CFT admits a
large-spin expansion, and that CFT operators organize themselves into additive
multi-twist families Alday:2007mf ; Fitzpatrick:2012yx ; Komargodski:2012ek .
It allows one to perform systematic analytic computations Fitzpatrick:2014vua
; Alday:2015eya ; Alday:2015ewa ; Alday:2016njk ; Simmons-Duffin:2016wlq , and
estimate properties of double-twist operators even in strongly coupled CFTs,
such as the 3d Ising model, when no small parameter is available Alday:2015ota
; Simmons-Duffin:2016wlq . The same idea can be generalized to many other
situations Bissi:2022mrs , such as defect CFTs Lemos:2017vnx , torus partition
functions Collier:2016cls ; Benjamin:2019stq , thermal correlators
Iliesiu:2018fao ; Iliesiu:2018zlz ; Alday:2020eua , event shapes Chen:2022jhb
; Chang:2022ryc , and higher-point correlation functions Antunes:2021kmm .

Ultimately, the lightcone bootstrap and its cousins restate the basic
structure of correlation functions dictated by locality and causality in terms
of the CFT spectral data. In the simplest case of the four-point function,
recent developments put predictions of the lightcone bootstrap on a firmer
theoretical basis by supplementing its predictions with rigorous error bars
Qiao:2017xif ; Caron-Huot:2020adz . For multi-twist operators, this is still
an open problem. In any future approach to a nonperturbative solution of CFTs,
it is desirable to make the universal large-spin structure as manifest as
possible and focus computational efforts on non-universal aspects of the
theory.

### Inversion formula and the double commutator

The OPE expresses a four-point correlation function as a discrete sum of
conformal blocks, corresponding to exchanges of the physical operators in the
theory. There is a more primitive expansion of the four-point function
(sometimes called the partial wave decomposition) in terms of a complete basis
of orthonormal single-valued functions labeled by integer spin
$J\in\mathbb{Z}_{\geq 0}$ and complex dimensions $\Delta=\frac{d}{2}+i\nu$.
These correspond to the principal series representations of the Euclidean
conformal group Dobrev:1977qv . The usual OPE is recovered from the partial
wave decomposition by deforming the $\nu$-contour and picking up poles at the
position of the physical operators Dobrev:1975ru . It turns out that the
coefficients $c(\Delta,J)$ in this decomposition can be expressed (for $J>1$)
in terms of a double commutator integrated over a Lorentzian region Caron-
Huot:2017vep ; Simmons-Duffin:2017nub . This result is known as the Lorentzian
inversion formula.

The Lorentzian inversion formula makes manifest many interesting properties of
CFT spectra. It implies that local operators and OPE coefficients are
organized into Regge trajectories Costa:2012cb  which are analytic functions
of spin that make sense even away from integer values. It is in this precise
sense that we can think of spin as an expansion parameter in the light-cone
bootstrap. Analyticity in spin ultimately stems from boundedness of
correlators in the Regge limit Maldacena:2015waa ; Caron-Huot:2017vep , a
simple but powerful result underlying the dispersive functionals discussed
below, and with other important applications, see e.g. Chowdhury:2019kaq ;
Chandorkar:2021viw .

The double commutator that enters the Lorentzian inversion formula is a much
simpler object than the full correlator, e.g. it suppresses the contribution
from double-twist operators, yet it captures most of the dynamical information
contained in the correlation function. It is also non-negative and bounded
from above, which makes it suitable for many bootstrap applications. In the
planar limit, the double commutator receives contributions from single trace
operators only and thus directly probes the spectrum of one-particle states in
the dual AdS theory. This has proven instrumental for loop computations in AdS
Aharony:2016dwx ; Alday:2017gde ; Aprile:2017bgs ; Aprile:2017xsp ;
Alday:2017vkk ; Alday:2017xua , as well as in perturbative computations
Henriksson:2017eej ; Alday:2017zzv ; Henriksson:2018myn ; Alday:2019clp ;
Henriksson:2020fqi .

### Energy conditions, light-ray operators, and event shapes

Local energy is not bounded from below in unitary relativistic QFTs. The same
is true for any expectation value of local operators in a bounded region. The
situation is different for non-local operators, of which light-ray operators
are simple examples. Perhaps the most famous example of a light-ray operator
is the integral of the stress-energy tensor over a null geodesic,
$\int_{-\infty}^{\infty}d\lambda T_{\alpha\beta}u^{\alpha}u^{\beta}$. The
averaged null energy condition (ANEC) states that this operator is
nonnegative. This condition is an input to many fundamental results in general
relativity Borde:1987qr  and CFT Hofman:2008ar , but in interacting field
theories it was derived only recently. Intriguingly, it was derived by two
different methods, one based on the analytic bootstrap Hartman:2015lfa ;
Hartman:2016lgu  and the other based on quantum information theory
Faulkner:2016mzt ; both methods have close ties to the emergence of causality
in the AdS/CFT correspondence Kelly:2014mra . The bootstrap method can also be
viewed as a highly boosted limit of the Lorentzian inversion formula.

In the context of CFTs, light-ray operators make a natural appearance when
thinking about a conformal collider experiment Hofman:2008ar , where they play
the role of energy and charge calorimeters placed at null infinity. More
recently, it was realized that generic points on Regge trajectories away from
integer spins have a natural interpetation in terms of light-ray operators
Kravchuk:2018htv . While being non-local, light-ray operators retain
commutativity properties of local operators in a nontrivial way
Kologlu:2019bco ; Belin:2019mnx , which leads to interesting sum rules on the
CFT data. Lightray operators built out of the stress-energy tensor form the
BMS algebra Cordova:2018ygx , as well as more general algebraic structures
Huang:2019fog ; Belin:2020lsr ; Besken:2020snx ; Korchemsky:2021htm ;
Huang:2022vcs .

Event shapes are observables used to describe hadronic events at colliders.
Traditionally, they are computed using scattering amplitudes which requires
delicate cancellations of infrared divergencies. In the context of CFTs, event
shapes become matrix elements of light-ray operators Sveshnikov:1995vi ;
Korchemsky:1997sy ; Hofman:2008ar ; Kologlu:2019mfz ; Korchemsky:2021okt , and
therefore can be efficiently studied using bootstrap techniques in both
perturbative Belitsky:2013xxa ; Belitsky:2013bja ; Belitsky:2013ofa ;
Henn:2019gkr  and nonperturbative settings. Remarkably, the machinery of the
OPE Hofman:2008ar ; Korchemsky:2019nzm ; Kologlu:2019mfz ; Chang:2020qpj  and
crossing equations Chen:2022jhb ; Chang:2022ryc  can be generalized to light-
ray operators in a nontrivial way. The light-ray OPE has interesting
applications in the study of jet substructure in QCD Dixon:2019uzg ;
Chen:2020vvp . Developing a better understanding of the space of light-ray
operators and associativity of the light-ray OPE is an important open problem
in our quest for understanding nonperturbative Lorentzian dynamics of CFTs.

### Dispersion relations and dispersive sum rules

Conformal dispersion relations express a four-point function in terms of its
double commutator Carmi:2019cub . They can be derived, for example, by
plugging the Lorentzian inversion formula into the conformal partial wave
expansion and performing the sum and integral over principal series
representations. Alternatively, conformal dispersion relations admit a
particularly simple derivation in Mellin space, where they follow from
applying Cauchy’s theorem to the nonperturbative Mellin amplitude
Penedones:2019tng . Finally, conformal dispersion relations can be thought of
in terms of an expansion in a complete set of functionals with double zeros at
double-twist locations Mazac:2019shk .

All of these closely-related constructions lead to so-called “dispersive sum
rules” Caron-Huot:2020adz , which are crossing symmetry constraints expressed
in terms of the double commutator. Perhaps the simplest example of a
dispersive sum rule is the statement that ANEC operators commute
Kologlu:2019bco . In practice, dispersive sum rules possess double zeros at
double-twist locations, except for a finite number of Regge trajectories. In
the context of holographic CFTs, double-twist operators are two-particle
states in AdS. Dispersive sum rules effectively throw out this redundant two-
particle information, focusing on the physics of light particles and the heavy
states that provide their UV completion. In the AdS bulk, they connect known
IR physics to unknown (but positive) UV data. In Caron-Huot:2021enk ,
dispersive sum rules were constructed that become flat-space dispersion
relations in an appropriate limit, allowing one to apply technology from the
$S$-matrix bootstrap to constrain holographic CFTs. This leads to detailed
bounds on the Wilson coefficients of AdS effective field theories that can be
UV-completed, discussed in more detail below. A systematic understanding of
the space of dispersive sum rules and lessons that can be learned from them is
an important open problem.

Finally, conformal dispersion relations naturally lead to an expansion of the
correlator in so-called Polyakov or Polyakov-Regge blocks. These blocks are
closely related to exchange Witten diagrams and arise by applying the
dispersion relation to a single conformal block. A similar expansion had been
used by Polyakov Polyakov:1974gs  and more recently in Sen:2015doa ;
Gopakumar:2016wkt ; Gopakumar:2016cpb ; Dey:2016mcs ; Dey:2017fab ;
Gopakumar:2018xqi  as an alternative starting point for the analytic conformal
bootstrap. Conformal dispersion relations provide a rigorous underpinning for
this idea Caron-Huot:2020adz ; Sinha:2020win ; Gopakumar:2021dvg .

### Analytic functionals and sphere packing

One of the early successes of the conformal bootstrap was the observation that
bounds on the CFT data are often close to being saturated by interesting CFTs,
such as the critical 3D Ising model El-Showk:2012cjh . The bounds are obtained
by numerically solving an optimization problem in a space of linear
functionals acting on the crossing equation (1). This leads to several
interesting questions: Do physical CFTs saturate the bounds exactly? If so, is
there an analytic proof of the exact saturation? It turns out that at least in
some cases, the answer to both questions is yes Mazac:2016qev . In these
cases, it is possible to construct an analytic functional which certifies the
exact saturation Mazac:2016qev ; Rychkov:2017tpc ; Mazac:2018mdx ;
Mazac:2018ycv ; Mazac:2018qmi ; Hartman:2019pcd ; Paulos:2019gtx ; Afkhami-
Jeddi:2020ezh ; Caron-Huot:2020adz ; Paulos:2020zxx ; Ghosh:2021ruh ;
Paulos:2021jxx . In essentially all the examples where an analytic functional
has been found, the corresponding solution to the bootstrap is a free theory.
Analytic functionals for free theory are examples of dispersive sum rules,
discussed above. It is an important open problem to understand whether
interacting CFTs exactly saturate bootstrap bounds coming from a small set of
correlators, and to construct the requisite analytic functionals.

There is a remarkably close correspondence between the analytic functionals
and the recent solution of the sphere packing problem in 8 and 24 dimensions
Hartman:2019pcd . The key observation is that a method for proving upper
bounds on the sphere packing density due to Cohn and Elkies CE  can be
formulated as a bootstrap problem. When the analytic functionals of
Mazac:2016qev  are applied to this problem, they lead to directly to
Viazovska’s solution SpherePacking8 ; SpherePacking24  of sphere packing in 8
and 24 dimension. This establishes a precise connection between the conformal
bootstrap and pure mathematics. Other examples of such connections are the
recent application of bootstrap techniques to study Laplacian spectra on
manifolds Bonifacio:2020xoc ; Bonifacio:2021msa ; Kravchuk:2021akc ;
Bonifacio:2021aqf , and explorations of 2D CFTs through the lens of quantum
codes Dymarsky:2020bps ; Dymarsky:2020qom ; Dymarsky:2020pzc ;
Dymarsky:2021xfc ; Buican:2021uyp . It is likely that many more examples will
be found in the future. In this context, a particularly important task is to
look for mathematical structures which automatically lead to solutions of the
conformal bootstrap equations Gadde:2017sjg ; Kravchuk:2021akc .

### Constraints on gravitational theories in AdS

As explained in the introduction, we can use the conformal bootstrap to study
gravitational theories in AdS space. To do so, we first need to provide a
general characterization of CFTs dual to weakly coupled gravity in AdS
Heemskerk:2009pn . The requirement of weak gravitational coupling translates
to the CFT having many degrees of freedom per site, i.e. large $N$.
Furthermore, we are typically interested in gravity theories with no light
particles of spin greater than two. In CFT language, this maps to a large
lower bound $\Delta_{\mathrm{gap}}$ on the dimension of the lightest higher-
spin single-trace operator. This can only be achieved if the CFT is strongly
coupled.

Equipped with this characterization, we can start exploring the space of CFTs
with large $N$ and large $\Delta_{\mathrm{gap}}$ using the conformal
bootstrap. Conveniently, in the $N\rightarrow\infty$ limit, any such CFT
becomes a mean field theory, and thus can be solved explicitly. One can then
expand the CFT data perturbatively in $1/N$ and $1/\Delta_{\mathrm{gap}}$. The
perturbative corrections are constrained by the CFT crossing equations. There
is a significant amount of evidence that the general solution of this
perturbative bootstrap exactly agrees with the predictions of a general
effective field theory in AdS Heemskerk:2009pn ; Penedones:2010ue ;
Fitzpatrick:2010zm ; Fitzpatrick:2011ia ; Paulos:2011ie ; Fitzpatrick:2011dm ;
Meltzer:2019nbs ; Meltzer:2020qbr . This is analogous to the statement that
the low-energy expansion of a general consistent $S$-matrix can be generated
using effective field theory in flat space Weinberg:1978kz . However, in AdS,
it is often much easier to analytically solve the bootstrap equations than it
is to evaluate Witten diagrams. This idea has led to the evaluation of a large
amount of CFT data in theories with weakly coupled bulk duals Aharony:2016dwx
; Rastelli:2016nze ; Rastelli:2017udc ; Alday:2017vkk ; Aprile:2017bgs ;
Alday:2017xua ; Aprile:2017xsp ; Caron-Huot:2018kta ; Sleight:2018epi ;
Sleight:2018ryu ; Carmi:2019ocp ; Carmi:2020ekr ; Carmi:2021dsn .

Let us reiterate that the conformal bootstrap provides a precise
nonperturbative definition of what it means to have a UV complete theory of
gravity in AdS. However, by expanding the crossing equations perturbatively in
$1/N$ and $1/\Delta_{\mathrm{gap}}$, we are throwing away valuable bootstrap
constraints relating the UV and the IR. In other words, the set of fully
consistent gravitational theories in AdS could be much smaller than the set of
effective field theories one can write down! Thus the conformal bootstrap can
provide a rigorous underpinning for the swampland program Vafa:2005ui ;
Arkani-Hamed:2006emk , which aims to identify which effective field theories
admit a UV completion; see Palti:2019pca  for a review.

To describe some recent progress on this front, consider the space of
gravitational effective field theories. It can be parametrized by a set of
light particles including the graviton and the infinite set of their local
interaction vertices. It is well known that causality leads to sign
constraints on some higher-derivative couplings in quantum gravity
Brigante:2007nu ; Hofman:2008ar ; Hofman:2009ug ; Camanho:2014apa . This has
been derived from the analytic bootstrap Afkhami-Jeddi:2016ntf ;
Kulaxizi:2017ixa ; Costa:2017twz ; Meltzer:2017rtf , and recently, it has been
generalized to prove new strict bounds on higher-derivative couplings in
gravitational theories in AdS Caron-Huot:2021enk . The idea relies on the
dispersive CFT sum rules reviewed above. These sum rules equate EFT couplings
with sums over the ultraviolet states. Unitarity of the UV contributions then
leads to bounds on the IR couplings. As a result, the analytic bootstrap sheds
light on the emergence of causality in quantum gravity, and helps to constrain
the landscape of consistent theories.

### More general observables

We have focused so far on vacuum four-point functions of light operators.
There are many other observables one can define in a CFT, which lead to
additional bootstrap constraints. A simple example is vacuum correlators of
$n>4$ local operators. At first sight, these may seem superfluous since
crossing symmetry of all four-point functions in principle implies consistency
of arbitrary $n$-point functions. However, in practice, higher-point functions
may package some CFT data together in a more useful way. For example, they
give us access to structure constants of multiple operators with large spin,
and multi-trace operators in large-$N$ theories Rosenhaus:2018zqn ;
Goncalves:2019znr ; Bercini:2020msp ; Antunes:2021kmm ; Barrat:2021tpn .

One can also consider correlation functions in the presence of extended
objects, such as boundaries, lines, or more general defects. It is natural to
demand that the defects preserve the maximal subgroup of the conformal group.
In that case, the set of conformal bootstrap constraints can be enlarged to
the full set of correlation functions involving local operators distributed in
spacetime, as well as on various conformal defects Cardy:1984bb ; Cardy:1989ir
; Cardy:1991tv ; Liendo:2012hy ; Billo:2016cpy . Various analytic bootstrap
methods, such as the large-spin expansion, inversion formulas and analytic
functionals can be extended to these situations Gliozzi:2015qsa ;
Rastelli:2017ecj ; Lemos:2017vnx ; Lauria:2018klo ; Bissi:2018mcq ;
Kaviraj:2018tfd ; Mazac:2018biw ; Liendo:2018ukf ; Liendo:2019jpu ; Gimenez-
Grau:2019hez ; Dey:2020jlc ; Gimenez-Grau:2021wiv ; Barrat:2021yvp .

Furthermore, correlation functions of a local CFT should be consistent not
only in flat space, but on an arbitrary manifold. A well known consequence is
modular invariance of the partition function of a 2D CFT on $S^{1}\times
S^{1}$. Famously, it leads to a universal upper bound on the dimension of the
first nontrivial Virasoro primary Hellerman:2009bu . For recent developments,
see Keller:2012mr ; Friedan:2013cba ; Collier:2016cls ; Anous:2018hjh ;
Afkhami-Jeddi:2019zci ; Hartman:2019pcd ; Benjamin:2019stq ; Alday:2019vdr ;
Benjamin:2020mfz .

In $d>2$, perhaps the simplest manifold not conformally equivalent to
$\mathbb{R}^{d}$ is $S^{1}\times\mathbb{R}^{d-1}$. Placing a CFT on this
geometry is equivalent to studying its physics at nonzero temperature, or
equivalently at fixed energy density. This is also very interesting from the
point of view of holography since finite temperature states are dual to black
holes. Demanding the consistency of the OPE with periodicity around the
thermal circle leads to constraints on finite temperature observables El-
Showk:2011yvt ; Iliesiu:2018fao ; Iliesiu:2018zlz ; Alday:2020eua . More
elaborate geometries also lead to interesting constraints and further
connections to black hole thermodynamics Shaghoulian:2015dwa ;
Shaghoulian:2015kta ; Nakayama:2016cim ; Hasegawa:2016piv ; Belin:2016yll ;
Shaghoulian:2016gol ; Horowitz:2017ifu ; Gadde:2020bov ; Giombi:2020xah .

Thermal properties of CFTs emerge from average properties of heavy operators.
To see this, note that $S^{1}\times\mathbb{R}^{d-1}$ is a limit of
$S^{1}\times S^{d-1}$ as the radius of the second factor goes to infinity. In
this limit, we are sensitive to high-energy states on $S^{d-1}$, or in other
words, to local operators with large scaling dimension. In holography,
individual heavy operators are dual to black hole microstates. It is a
fascinating question to understand the emergence of macroscopic physics from
the statistical properties of heavy operators. In the context of the
bootstrap, average properties of heavy operators can be probed rigorously
using Tauberian theorems Pappadopulo:2012jk ; Mukhametzhanov:2018zja ;
Mukhametzhanov:2019pzy ; Ganguly:2019ksp ; Pal:2019yhz ; Pal:2019zzr ;
Mukhametzhanov:2020swe . Interesting probes of heavy operators also emerge
from studying the Regge limit Caron-Huot:2020ouj , and higher-point
correlation functions Anous:2021caj . In CFTs with global symmetries, there
exists a similar macroscopic limit, where we take large charge at fixed charge
density. Recently, it was realized that this limit admits a description in
terms of an effective field theory Hellerman:2015nra . It turns out that one
can formulate the conformal bootstrap directly in this limit, which reproduces
the EFT description, while also allowing for more exotic possibilities
Jafferis:2017zna . This is yet another instance of the general principle that
effective field theory can be thought of as a device for systematically
solving the bootstrap equations in a perturbative limit.

##  3 Future directions

### Bootstrapping quantum gravity in AdS

As we reviewed, the study of consistent gravitational theories in AdS is
naturally included as a corner of the conformal bootstrap. One of the most
important directions for future research is to understand the full set of
constraints that the bootstrap imposes on this class of theories. In
particular, what are the bounds on the spectra of light particles and their
low-energy couplings in the presence of gravity?

An important goal is to test the hypothesis of string universality: Do all
weakly-coupled, consistent theories of gravity come from string theory, or are
there counterexamples? The analytic conformal bootstrap toolkit is mature
enough to begin addressing this question.

One promising approach is to apply dispersive sum rules Penedones:2019tng ;
Caron-Huot:2020adz ; Gopakumar:2021dvg . Even for a single four-point
function, their consequences are far from fully understood. To make further
progress, it will also be essential to generalize dispersive techniques to
systems of correlators and higher-point functions.

It is important to stress that consistent gravitational theories satisfy a
much larger set of constraints than ordinary quantum field theories. Indeed,
the presence of dynamical gravity in AdS implies locality of the dual CFT. Not
only do such CFTs contain a stress tensor, but their correlation functions
must be consistent on general manifolds, in contrast with theories dual to
non-gravitational QFTs in AdS.

The additional constraining power of more general geometries is apparent
already in AdS3/CFT2, where modular invariance, i.e. consistency on the torus,
implies a universal upper bound on $\Delta^{\text{2d}}_{\text{gap}}$, the
dimension of the lightest nontrivial primary Hellerman:2009bu . Conceivably,
studying local higher-d CFTs on various manifolds will lead to similarly far-
reaching consequences. Staying in the realm of 2d CFTs, what is the optimal
upper bound on $\Delta^{\text{2d}}_{\text{gap}}$ at large central charge? In
particular, how can we go beyond the current analytical record
$\Delta^{\text{2d}}_{\text{gap}}\leq c/8.503$ Hartman:2019pcd  and even beyond
the numerical estimate $\Delta^{\text{2d}}_{\text{gap}}\lesssim c/9.1$
Afkhami-Jeddi:2019zci  coming from the spinless modular bootstrap?

The task of charting the space of UV-completable gravitational effective field
theories is also the goal of the swampland program Vafa:2005ui ; Palti:2019pca
. This program has generated a large set of far-reaching conjectures
concerning properties of quantum gravity. These conjectures are supported, to
a varying degree of rigor, by physical arguments as well as exhaustive
searches of examples in string theory. In contrast, bounds coming from the
bootstrap have typically been less dramatic, but benefit from a high degree of
rigor since they only rely on unitarity, causality and symmetry. Through a
combination of ideas from the two sides, the gap between bootstrap bounds and
swampland conjectures will continue to narrow, and should ultimately
disappear.

### Bootstrapping holographic duality beyond AdS/CFT

The prototypical examples of holographic duality involve a $d$-dimensional CFT
on the conformal boundary of anti-de Sitter spacetime, and in these cases
there is a direct map from the analytic bootstrap to properties of quantum
gravity. However, holographic duality is believed to extend beyond AdS, and
there are many potential insights to be gained by applying bootstrap methods
to quantum gravity in other settings.

One route is to take the flat-space limit of AdS/CFT. This has already been
used to develop $S$-matrix bootstrap methods and to derive bounds on
gravitational scattering Caron-Huot:2021rmr ; Caron-Huot:2021enk , but the
bounds are not sharp in $D=4$ due to infrared divergences that appear when the
AdS size becomes infinite. Can CFT methods be used to construct good IR-safe
boundary observables for four-dimensional gravity in asymptotically flat
space?

Another perspective on holography in flat spacetime starts from the
observation that flat-space scattering amplitudes have CFT-like kinematics on
the celestial sphere. This has led to the notion of celestial holography
Pasterski:2016qvg . Can the methods of the analytic bootstrap such as
functionals, dispersion relations and OPE inversion be applied to celestial
CFTs? Additionally, how are the light-ray operators of the analytic bootstrap
related to the conservation laws, soft theorems, and memory effects of
celestial CFT?

The bootstrap can be applied to observables in de Sitter spacetime, which is
relevant to our universe during the early stages of expansion as well as the
present-day era of dark energy. There are various proposals for holographic
descriptions of de Sitter in which the CFT lives, for example, on the
spacelike future boundary, at the edge of a flat-space bubble, or on the de
Sitter horizon. How are unitarity and causality manifested in the CFT
description? There has been recent progress on this question from various
points of view, involving both CFT methods and scattering amplitudes. The most
interesting question is perhaps how best to extract constraints on
inflationary observables; see e.g. Baumann:2015nta ; Cordova:2017zej ;
Melville:2019wyy ; Baumann:2019ghk ; Ye:2019oxx ; Pajer:2020wnj ;
deRham:2020zyh ; Pajer:2020wxk ; Grall:2021xxm ; Goodhew:2021oqg ;
Meltzer:2021zin ; Hogervorst:2021uvp ; DiPietro:2021sjt ; Melville:2022ykg
for recent steps in this direction.

### Conformal manifolds and deformations

The QFT landscape has isolated points with conformal symmetry; plateaus
parameterized by exactly marginal couplings; and renormalization group flows
activated by relevant or irrelevant couplings, or by moving into the moduli
space of vacua. On the plateaus — including the ${\cal N}=4$ SCFT in four
dimensions which features in the AdS/CFT correspondence — the existing methods
of the analytic bootstrap are applicable, but most of them fail to take
advantage of the conformal manifold. Instead, they treat a fixed CFT
irrespective of its deformations. New methods are needed to take advantage of
the vast additional data encoded elsewhere in the conformal manifold, and to
address questions about the manifold itself, such as whether there are
universal properties of CFTs at infinite distance Perlmutter:2020buo . See
Behan:2017mwi ; Chester:2021aun ; Collier:2022emf  for recent works exploring
conformal manifolds from the bootstrap perspective.

Massive deformations trigger renormalization group flows. Can the methods of
Lorentzian CFT be extended to encompass these deformations? Is there a
Lorentzian inversion formula for the data of effective field theories? What is
the relation, if any, between the sign constraints obtained through the
lightcone bootstrap, such as the ANEC, and the universal inequalities known as
$C$-theorems that govern renormalization group flows?

The existence of moduli spaces of exact supersymmetric vacua gives rise to
powerful tools for understanding the rich physics of strongly-coupled QFTs
Seiberg:1994rs ; Seiberg:1994aj . What can the conformal bootstrap teach us
about the moduli spaces of vacua? For example, how can we tell, purely from
the CFT data, that a CFT possesses a nontrivial moduli space? See Beem:2014zpa
for a discussion in the context of $\mathcal{N}=2$ SCFTs.

### Averaging and randomness

Several independent directions, including condensed matter applications, the
eigenstate thermalization hypothesis (ETH) PhysRevA.43.2046 ;
Srednicki:1994mfb , exciting recent developments in holographic duality, and
hints from mathematics all suggest an important role for randomness in the
conformal bootstrap.

In 2019, it was discovered that two-dimensional gravity is holographically
dual to random matrix theory Saad:2019lba . This is fundamentally different
from other known examples of holography because the boundary theory is not a
single CFT, but an ensemble of quantum theories. Randomness, at some level,
also appears to be an important feature of higher-dimensional black holes
Cotler:2016fpe . How does this manifest in the conformal bootstrap? Are there
universal constraints on the statistics of CFT data that can be derived
analytically? The answer is almost certainly yes, and this may provide a
handle on complex, chaotic phenomena in CFT that are inaccessible to standard
numerical and analytic methods. Tauberian theorems that underlie the Cardy
formula Cardy:1986ie  and the asymptotics of OPE coefficients
Pappadopulo:2012jk  always involve some averaging Mukhametzhanov:2018zja ;
Mukhametzhanov:2019pzy , so these two subjects are closely linked. Other
bootstrap tools that are likely to play a significant role are the fusion
kernel and, in 2d CFT, large-$c$ methods that reveal aspects of 3d gravity
from a CFT perspective Hartman:2013mia ; Fitzpatrick:2014vua ; Asplund:2014coa
; Collier:2018exn ; Collier:2019weq . For initial steps in these directions,
see e.g. Belin:2020hea ; Belin:2020jxr ; Belin:2021ibv ; Anous:2021caj ;
Pollack:2020gfa ; Cotler:2020ugk ; Cotler:2020hgz ; Heckman:2021vzx .

Similar questions have been explored in the context of eigenstate
thermalization, which is believed to govern the statistics of certain OPE
coefficients responsible for linear response at finite temperature or density.
If there are universal statistics in CFT accessible to the analytic bootstrap,
what are their consequences for non-holographic theories like the 3D Ising
model? Can the analytic bootstrap be applied to disordered systems relevant to
condensed matter?

From the mathematics side, the hints of randomness come from spectral analysis
and analytic number theory. See e.g. mazur2008finding  for a sampling of the
relevant mathematics, and Benjamin:2021ygh ; Collier:2022emf  for recent
discussions in the context of the bootstrap. Can a wider array of mathematical
tools be applied to conformal field theory, and vice-versa?

It was recently shown that aspects of holographic duality can be explained by
averaging over the conformal manifold of CFTs in a two-dimensional toy model
Maloney:2020nni ; Afkhami-Jeddi:2020ezh  and then in ${\cal N}=4$ super-Yang
Mills Collier:2022emf . It has also been suggested that the randomness of
black hole microstates is related to averaging over $N$ Schlenker:2022dyo .
Another possibility is to input some CFT data, such as the scaling dimensions
and OPE coefficients of light operators, then choose a subset of bootstrap
constraints and average over a ‘swampland’ of CFT data that satisfy the
constraints but do not necessarily have a UV completion. These ideas are
developing rapidly in various directions; what are the best approaches to
averaging in realistic theories of quantum gravity and higher-dimensional
CFTs?

### Closing the gap between analytics and numerics

The analytic bootstrap, in its simplest form, determines asymptotic data at
large spin and/or scaling dimension. It can be improved iteratively to extend
its reach toward lighter operators. Numerical methods, meanwhile, effectively
work upwards, starting with low-lying operators and producing more data for
higher dimension operators as more constraints are included. This offers the
tantalizing possibility that if these two approaches can be combined
effectively, harnessing analytic results as input to numerics, it could lead
to highly efficient methods to solve for CFT data throughout the spectrum.

For example, in CFTs like the 3d Ising model, among the hundreds of scaling
dimensions and OPE coefficients that can be found by semidefinite programming,
most can be fit to analytics with the input of a few pieces of numerical data
Simmons-Duffin:2016wlq ; Liu:2020tpf ; Caron-Huot:2020ouj ; Atanasov:2022bpi .
This suggests that much of the computational power utilised by current methods
may be spent on operators that are amenable to analytics. One of the issues
with combining analytics and numerics is that error bars in analytic bootstrap
results are currently not under precise control. At the same time, introducing
errors into the setup of a numerical bootstrap computation can destroy the
rigor of the approach. Improvements in the analysis of errors in large spin-
expansions, and creative methods for taking those errors into account in
numerics are needed, see Su:2022xnj  for recent work in this direction.

In large-$N$ theories, standard numerical methods are less effective, but
analytic techniques that take advantage of the $1/N$ expansion are more
powerful. This is therefore another area where closer integration of analytics
and numerics would be helpful.

It seems unlikely that there will ever be a fully analytic solution of chaotic
CFTs. Nonetheless, it may be possible to find fast numerical algorithms
powered by analytic bootstrap results that converge quickly for many
observables. A near-term goal in this direction is to better understand the
role of multi-twists in the analytic bootstrap, and to incorporate these into
iterative methods to determine spectra through a combination of analytics and
numerics.

### Lorentzian signature and analytic structure of correlators

Most of the analytic bootstrap methods discussed above require studying CFTs
in Lorentzian signature. In principle, Lorentzian constraints should follow
from CFT axioms, which are usually formulated in Euclidean signature. However,
Lorentzian signature packages important constraints in an accessible and
useful way — in particular by allowing one to formulate and explore the
condition of causality Adams:2006sv ; Camanho:2014apa ; Hartman:2015lfa .
Relatedly, Lorentzian signature admits a much richer array of singularities
than Euclidean signature, for example the lightcone, Regge, and “bulk point”
limits Gary:2009ae ; Maldacena:2015iua ; Iliesiu:2018fao ; Dodelson:2020lal .
See Bercini:2021jti  for a recent bootstrap analysis of gauge theory
correlators near a Lorentzian singularity, and Engelhardt:2016wgb ;
Engelhardt:2016crc ; Hernandez-Cuenca:2020ppu  for a discussion of Lorentzian
singularities in the context of bulk reconstruction. It was conjectured in
Maldacena:2015iua  that Lorentzian singularities should be in one-to-one
correspondence with position-space Landau diagrams. (Furthermore, emergent
singularities in holographic CFTs should correspond to Landau diagrams in the
bulk.) However, we are far from being able to verify that conjecture in $d>2$
dimensions using bootstrap methods.

Many of the Lorentzian results discussed above (including the bootstrap proof
of the ANEC Hartman:2016lgu , the proof of commutativity of ANEC operators
Kologlu:2019bco , and arguments for the convergence of dispersive sum rules
Caron-Huot:2020adz ) require technical assumptions about the behavior of CFT
correlators under analytic continuation to the so-called “second sheet,” see
Kravchuk:2021kwe  for a discussion. To place these results on a completely
rigorous footing, it will be crucial to prove bounds on analytically-continued
correlators, ideally with controlled expansions around Lorentzian
singularities. Recent progress in this direction was made in Kravchuk:2020scc
; Qiao:2020bcs ; Kravchuk:2021kwe , which proved (Lorentzian) Wightman axioms
for CFT four-point functions starting from Euclidean CFT axioms (bypassing
technical assumptions in a famous theorem of Osterwalder and Schrader
Osterwalder:1973dx ; Osterwalder:1974tc ). Can old results in multi-
dimensional complex analysis be applied fruitfully to this problem, or will
new methods be needed?

Finally, it is an interesting problem to understand the analytic structure of
CFT correlators in momentum space. See Gillioz:2018mto ; Gillioz:2019lgs ;
Gillioz:2020mdd ; Meltzer:2021bmb ; Gillioz:2021sce  for a sampling of
momentum-space methods and results in CFT.

### Quantum information, algebraic methods, and higher symmetries

Many results of the analytic bootstrap are fundamentally consequences of
locality. For example, the existence of double-twist operators reflects the
fact that two operators inserted far apart will have little influence on each
other, and so there must be composite primaries with this property. On the
other hand, locality is not manifest in the bootstrap, and the operator
language of CFT is quite different from other approaches to the physics of
interacting quantum field theory where locality is built into the axioms. How
does the bootstrap connect to these other approaches, such as algebraic QFT,
tensor network-inspired methods, or topological quantum field theory?

Approaches to QFT that make locality manifest also more naturally incorporate
extended objects, higher symmetry groups, and anomalies. It is an open
question how to better use these structures in bootstrap calculations. In two
dimensions, anomalies and line defects lead to refined modular bootstrap
results, see e.g. Lin:2019kpn ; Lin:2021udi ; Grigoletto:2021zyv ;
Grigoletto:2021oho . However, in higher-dimensions, bootstrap calculations of
local operator spectra have so far only accessed anomalies that appear in
local correlation functions. How do we improve bootstrap methods to explore
the full structure of extended QFT?

Both algebraic QFT and tensor methods are intimately connected to quantum
information theory. They rely on cutting manifolds into spatial subregions,
i.e., across codimension-2 separating surfaces, and therefore the spatial
organization of entanglement plays a central role. Connections to quantum
gravity give another hint that quantum information has a larger part to play
in the conformal bootstrap. One concrete goal in this area is to understand
the relationship between the bootstrap derivation of the averaged null energy
condition (ANEC) Hartman:2016lgu  and the information-theoretic derivation
Faulkner:2016mzt . Another target is to unify disparate approaches to
$C$-theorems in various dimensions: How does the bootstrap derivation of the
four-dimensional $a$-theorem Komargodski:2011vj  relate to the methods based
on quantum entropy Casini:2017vbe  or holographic duality Myers:2010xs ?

##  4 Outlook

Developments over the past decade have demonstrated the surprising power of
the basic principles of unitarity, causality and symmetry in constraining
conformal field theories. At the same time, it seems likely that we have only
been able to harness a small fraction of the full potential offered by these
principles. Here, we have reviewed the main existing results and suggested
several promising directions for future research. Whether or not our universe
is the unique solution of the ultimate bootstrap constraints, it is clear that
the conformal bootstrap will continue to illuminate the path towards a deeper
understanding of quantum field theory and quantum gravity.

## Acknowledgements

We would like to thank Simon Caron-Huot for helpful discussions. TH is
supported by the Simons Foundation and NSF grant PHY-2014071. DM acknowledges
funding provided by Edward and Kiyomi Baird as well as the grant DE-SC0009988
from the U.S. Department of Energy. DSD is supported by Simons Foundation
grant 488657 (Simons Collaboration on the Nonperturbative Bootstrap) and a DOE
Early Career Award under grant no. DE-SC0019085. AZ received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement number 949077).

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