1 _International Centre for Theoretical Sciences-TIFR,
Shivakote, Hesaraghatta Hobli, Bengaluru North, 560 089, India_

2 _Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique,
91191, Gif-sur-Yvette, France
3Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA
4Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138 USA_

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

# Snowmass White Paper: Bootstrapping String Theory

Rajesh Gopakumar,1 Eric Perlmutter,2 Silviu S. Pufu,3 and Xi Yin4

###### Contents

  1. 1 Introduction
  2. 2 Bootstrapping string theory amplitudes
  3. 3 AdS vacua and string universality
  4. 4 The worldsheet theory: CFT and beyond
  5. 5 Longer-term targets

##  1 Introduction

The Veneziano amplitude, that birthed string theory, itself originated from
the analytic S-matrix bootstrap philosophy of imposing strong analyticity
constraints on scattering amplitudes to capture general principles of
locality, unitarity, and causality. A dynamical explanation of this remarkable
formula and its generalizations followed subsequently from a two-dimensional
field theory describing the worldsheet of a relativistic string. Later, in the
early eighties, it was the success of the conformal bootstrap applied to two-
dimensional CFTs which spurred the exciting progress, rooted in the
worldsheet, that expanded our understanding of the space of string theory
vacua. Thus, in some ways, string theory and bootstrap ideas, both in
spacetime and on the worldsheet, are joined at the hip.

Fast-forwarding, we once again find ourselves in the midst of a period when
bootstrap ideas have not only had a lot of fresh success with general CFTs or
QFTs, but have also begun to fruitfully and nontrivially feed into string
theory. The goal of this white paper is to articulate some important elements
of this connection, reviewing progress of the last decade with an eye toward
what we view as the most important avenues for future research.111We hasten to
state that the narrow remit of this paper does not convey the full scope of
progress in the conformal bootstrap program in recent years, nor do we claim
to summarize the status and many interesting open questions of string theory
per se.

In broad terms, this re-engagement of string theory with the bootstrap has two
strands once again. The dominant one has been a spacetime one, primarily
through the AdS/CFT correspondence. Success with bootstrapping certain CFTd
correlation functions in appropriate large $N$ regimes holographically
constrains scattering amplitudes in AdSd+1 which, by taking a suitable limit,
go over to string scattering amplitudes in flat space. Here both numerical and
analytic bootstrap techniques (often in tandem with other tools like
supersymmetric localization) play a crucial role, as we summarize in Section
2. One of the highlights here is the determination of several leading terms in
the low-energy (derivative) expansion of string theory/M-theory through this
holographic bootstrap. The other natural domain of the conformal bootstrap is
to constrain the space of CFTs. In the holographic context, this provides a
rigorous approach to charting the space of theories of quantum gravity and, in
turn, the existential questions that are difficult to settle with direct
computations. An example is the question of existence of scale separated AdS
vacua (i.e. with the non-AdS directions being small), which is important for
understanding whether low energy constructions of such vacua do indeed lift to
the full string theory. The latter are also often the crucial intermediate
stages in constructions of metastable de Sitter vacua. Questions such as these
translate, via the AdS/CFT dictionary, into whether there are CFTs with
sufficiently sparse primary spectra at low spacetime spin. We discuss these
and related topics in Section 3.

The second strand, which thus far has not made a prominent appearance, is to
bring bootstrap ideas to bear on the worldsheet description of the much more
general class of string vacua that the AdS/CFT correspondence brings to the
forefront. In Section 4, we discuss the subtleties that such a worldsheet
description will necessarily face, which would probably require us to widen
the class of theories beyond what we are conventionally familiar with. Having
such a description is important for concrete problems like a first principles
description of confining string backgrounds. More broadly, we seek techniques
to perform worldsheet calculations of the AdS ampltudes that are indirectly
obtained through the holographically dual CFT. We outline some ideas on how
one might derive mileage from “dually bootstrapping” the spacetime CFT and
worldsheet descriptions.

We conclude this paper in Section 5 with a more speculative list of stringy
contexts, accompanied by an artistic rendition of the state of the field,
where the bootstrap philosophy can potentially be usefully applied. As a
counterpoint to that, perhaps, we end this introduction with a list of some
very concrete goals, which we contextualize in the main text, that we feel can
be tackled in the short-to-medium term using bootstrap methods, as a natural
growth of the approaches summarized here:

  * •

Obtain the $D^{8}R^{4}$ terms in the type II string theory and M-theory
effective actions.

  * •

Write down the Virasoro-Shapiro amplitude for $AdS_{5}\times S^{5}$.

  * •

Definitively rule in/out scale-separated AdS vacua using large $N$ conformal
bootstrap.

  * •

Bootstrap the worldsheet description of weakly-coupled $\mathcal{N}=4$ super-
Yang-Mills theory.

##  2 Bootstrapping string theory amplitudes

Some of the most fundamental observables in theories of quantum gravity in
asymptotically flat spacetime are the scattering amplitudes of gravitons. In
string theory, the worldsheet formalism produces, in principle, the complete
perturbative expansion of closed string amplitudes [1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11], in particular of the gravitons and their superpartners, and it also
captures certain non-perturbative corrections [12]. In practice, however,
explicit results of string amplitudes are only available up to two-loop order
in perturbation theory [13, 14, 15, 16, 17, 18, 19, 20], with the exception of
certain supersymmetry-protected terms,222Notably, these include the 3-loop
contribution to $D^{6}R^{4}$ effective coupling [21] computed in the pure
spinor formalism [7, 8, 22], and higher loop contributions to F-terms in
compactifications using topological strings [23, 24, 25]. along with pieces of
D-instanton corrections up to next-to-leading order [26, 27, 28, 29, 30, 31,
32, 33, 34, 35]. Moreover, the effective gravitational coupling is expected to
become strong at the Planck scale, obscuring the interpolation from
perturbation theory at low energies to effective descriptions of very high-
energy excitations as spacetime geometry with horizons. In M-theory, due to
the lack of a good worldvolume formulation analogous to the string worldsheet,
not much is known about the scattering amplitudes of the gravitons and their
superpartners beyond a few supersymmetry-protected terms in the low-energy
expansion [36, 37, 38, 21].

Despite the lack of a systematic framework, the non-perturbative S-matrices of
string theory and M-theory in asymptotically flat spacetimes are nevertheless
believed to exist.333Barring infrared issues in low numbers of asymptotic
dimensions [39]. A relatively novel promising direction to access them uses
conformal bootstrap techniques, which are applied as follows. First, after
replacing flat space with weakly-curved AdS space, it was shown in [40, 41,
42, 43, 44] that it is possible define AdS boundary observables that are
analogs of the flat space scattering amplitudes, and which approach the flat
space scattering amplitudes when the curvature radius of AdS is taken to
infinity. In turn, these boundary observables can be rephrased through
gauge/gravity duality [45, 46, 47] as correlation functions in the dual,
strongly coupled, large $N$ superconformal field theories, which are amenable
to the conformal bootstrap program.

To learn about string theory or M-theory using holographic CFTs, one can get
the most mileage from studying maximally supersymmetric or nearly-maximally
supersymmetric examples, such as: the 4d $SU(N)$ ${\cal N}=4$ super-Yang-Mills
(SYM) theory, which is dual to type IIB string theory on $AdS_{5}\times
S^{5}$; the 6d $(2,0)$ theory, which is dual to M-theory on $AdS_{7}\times
S^{4}$; and the 3d $U(N)_{k}\times U(N)_{-k}$ Aharony-Bergman-Jafferis-
Maldacena (ABJM) theory [48] or the more general $U(N)_{k}\times U(M)_{-k}$,
$N\neq M$, Aharony-Bergman-Jafferis (ABJ) theory [49] — with ${\cal N}=8$
superconformal symmetry when $k=1,2$ and $N=M$, and ${\cal N}=6$
superconformal symmetry otherwise — which are dual to M-theory on
$AdS_{4}\times S^{7}/\mathbb{Z}_{k}$. One reason why these holographic SCFTs
are the most promising bootstrap candidates is that the explicit Lagrangian
descriptions in the 3d and 4d examples and the large amount of supersymmetry
in all cases provide exact results that can be derived from supersymmetric
localization (see [50] for a collection of reviews and references) or from the
2d chiral algebra sectors [51, 52], and these results can be combined with the
conformal bootstrap studies. Another reason is that the stress tensor
multiplets of these maximally supersymmetric theories contain scalar
operators, whose four-point functions capture the full information contained
in the four-point functions of other operators in the same multiplet. Lastly,
the conformal bootstrap program constrains (S)CFTs based on symmetry and
consistency conditions, so one would expect that the most constrained theories
would be the most (super)symmetric ones.

To determine the flat space S-matrix in semiclassical string/M-theory, one
should extract certain coefficients in the $1/N$ expansion of correlation
functions in the SCFTs mentioned above. One approach is to first bootstrap the
full four-point function of the stress tensor multiplet superconformal primary
at finite $N$, and then consider the $1/N$ expansion of this result. Another
approach is to study the coefficients of the $1/N$ expansion order by order
using bootstrap techniques. As we now review, there has been progress in using
the first approach numerically and the second approach analytically.

### What numerical bootstrap can do for holographic theories

Figure 1: Allowed region for OPE coefficients of operators appearing in the
OPE of two stress tensor multiplet superconformal primaries for various values
of $N$ for the $U(N)_{1}\times U(N)_{-1}$ ABJM theory. Figure from [53].

The numerical bootstrap effort uses the tools being developed in the broader
conformal bootstrap program for CFTs in general spacetime dimensions (for
reviews and references, see [54, 55, 56, 57, 58]), in particular testing for
the unitarity and crossing symmetry of four-point functions using semi-
definite programming, numerically implemented, for instance, with the SDPB
semi-definite program solver [59]. Numerical studies in the holographic
theories mentioned above have been performed for 4d ${\cal N}=4$ SYM in [60,
61, 62, 63], for 3d ABJ(M) theory in [64, 65, 66, 53, 67], and for the 6d
$(2,0)$ theory in [68], and they include information obtained using
supersymmetric localization [50, 69, 70, 71, 72, 73, 74]. Of the results
obtained thus far, we would like to highlight the precise islands in OPE
coefficient space in 3d ${\cal N}=8$ ABJM theory (see Figure 1), which
represent a first step in determining the low-lying CFT data of ABJM theory
numerically.

In future numerical studies, it will be important to make use of all
information about protected correlation functions that can be obtained using
supersymmetric localization. Some of this information comes in terms of
integrated correlators, which have been used in numerical bootstrap studies in
more than two spacetime dimensions only recently [63] (see also [75] for a 2d
example). It will also be important to restrict the space of theories being
studied to those with a single stress tensor operator. This can be done by
analyzing the consistency of mixed systems of correlators involving, for
instance, the operators studied in [53, 62, 76]. With these improvements, the
goal is to obtain small allowed islands in all holographic 3d and 4d SCFT
examples mentioned above, to accurately extract the low-lying CFT data as
function of $N$, and then to expand the result in $1/N$.

### Analytic bootstrap and the low-energy (but not necessarily perturbative)
expansion

As mentioned above, another approach to large-$N$ holographic SCFTs is to
bootstrap the contribution to SCFT correlators coming from bulk interactions
with a small number of derivatives using consistency conditions instead of
performing direct computations of Witten diagrams. This approach is
perturbative in the derivative expansion, but it is not necessarily
perturbative in the string coupling constant. In 4d ${\cal N}=4$ SYM, for
instance, this approach can be used both in the ’t Hooft double scaling limit,
where $N$ is taken to infinity at fixed ’t Hooft coupling
$\lambda=g_{\text{YM}}^{2}N$, at large $\lambda$, as well as in the more
nontrivial limit where $N$ is taken to infinity while the complexified gauge
coupling $\tau$ is held fixed [77, 78, 71, 72].

Indeed, as initiated in [79, 80] and further developed in [81, 82, 83, 84, 85,
86, 87, 88, 89, 69, 71, 90, 91, 92, 93, 72, 94, 95] (see also earlier works
[96, 97]), it can be shown that crossing symmetry, supersymmetric Ward
identities, the Lorentzian inversion formula, as well as analytic properties
of Witten diagrams in Mellin space determine the four-point functions of
$1/2$-BPS external operators, up to a few undetermined constants whose number
increases with the number of derivatives in the interaction vertices. At the
first few orders in the derivative expansion, these constants can be fully
determined using exact results from supersymmetric localization or protected
sectors. By taking the large radius limit of AdS, one can connect these
results for holographic correlators to scattering amplitudes in flat space
[40, 41, 42, 43, 44], expanded at small momentum. This method has reproduced
known results [26, 36, 98, 38] for the contribution of protected higher-
derivative contact vertices to the string/M-theory S-matrix, for instance the
supersymmetrized $R^{4}$ interaction [26, 36, 98, 99] whose coefficient, as a
function of $\tau$, takes the form of a non-holomorphic Eisenstein series.

Looking forward, one should not expect this analytic bootstrap method with
input from supersymmetry-protected observables to give information about
unprotected flat space interactions. To go beyond the protected terms, one
would likely have to combine the analytic bootstrap approach with other inputs
coming, for instance, from the numerical bootstrap approach mentioned above.

### Targets for the near future

A concrete target for the near future would be to determine, or at least
bound, the coefficient with which the first unprotected interaction term,
namely the $16$-derivative interaction of the schematic form $D^{8}R^{4}$
(together with its supersymmetric completion), contributes to the graviton
S-matrix in string theory and M-theory. In string theory, this term
contributes with an unknown coefficient that is a function of $\tau$ in type
IIB string theory and a function of $g_{s}$ in type IIA. In M-theory, the
coefficient of $D^{8}R^{4}$ is just a number.444In [37], it was conjectured
that this term is absent in M-theory. A more ambitious goal would be to
determine the AdS analog of the Virasoro-Shapiro amplitude, i.e. the stress
tensor multiplet four-point function, as an expansion in $g_{\text{YM}}$ but
to all orders in $1/N$ — this would be a landmark achievement, as it would
constitute a partial but substantial solution of planar ${\mathcal{N}}=4$ SYM.
(See [100, 101] for work in this direction.)

Another goal is to determine higher-derivative contributions to the four-point
functions of the operators dual to the full Kaluza-Klein tower of modes on
$AdS_{5}\times S^{5}$, $AdS_{4}\times S^{7}$, or $AdS_{7}\times S^{4}$. In
tree-level supergravity, this was determined in [80, 79, 102, 103, 104] and
progress has been made beyond this order for instance in [105, 88, 69, 106,
107]. Since at a given order in the derivative expansion, the correlators
corresponding to the whole Kaluza-Klein tower have their origins in the same
higher derivative terms in 10 or 11 dimensions, one should search for a
compact way of encoding this information. The hidden conformal symmetry of
$AdS_{5}\times S^{5}$ supergravity [102] and other conformally flat AdS
supergravity backgrounds [108, 109] is one version of such a formulation, the
deeper meaning of which is important to understand.

In a different direction, the superconformal bootstrap may also be used to
target various questions about the formal structure of string amplitudes.
Number-theoretic properties of string amplitudes are a topic of active
investigation in the string theory community: in particular, understanding the
transcendentality properties of string amplitudes at low genus [110, 111], and
constraining the types of $SL(2,\mathbb{Z})$-invariant functions that can
appear in both integrated [112, 113, 114, 115] and unintegrated [116, 117,
118, 119] worldsheet amplitudes. The conformal bootstrap may be able to lend
novel perspectives, via the properties of correlation functions in (say)
$\mathcal{N}=4$ SYM and their flat space limits.

### Defects and branes

The dynamics of branes and open strings, beyond the low-energy effective
action, can also be accessed via bootstrapping holographic theories. One
approach is through SCFTs with flavor symmetry realized through flavor branes
in the bulk. In this context, correlators of the conserved current multiplets
are related to open string scattering amplitudes of the gluons living on the
branes [120, 121, 122]. Another approach is through defect CFTs that are dual
to wrapped branes, where correlators of local operators in the presence of the
defect are related to closed string amplitudes in the presence of the brane. A
first step in tackling this problem would be to understand whether an analog
of the Mellin representation holds for correlators in the presence of defects;
see [123] for an analogous representation for holographic thermal correlators.

The characterization of membranes in M-theory beyond the low-energy limit has
been a longstanding open question. Alternatively to the holographic approach,
it is likely that the strategy for bootstrapping the S-matrix of Nambu-
Goldstone modes of the flux tube [124] can be extended to $2+1$ dimensions
with nonlinearly realized target space super-Poincaré symmetry, and used to
constrain the dynamics of M2-branes at higher orders in the momentum
expansion.

##  3 AdS vacua and string universality

Moving away from specific top-down instances of AdS/CFT, bootstrap methods can
be used more broadly to study the landscape of AdS string vacua in string
theory and M-theory. One of the main questions in pursuing such an approach is
to understand the necessary and sufficient conditions for a CFT to be
“holographic,” a colloquial shorthand to mean that the AdS dual is classical,
weakly curved, and contains Einstein gravity (or a supersymmetrization
thereof) as a low-energy limit. This notion originated with the conjecture of
Heemskerk, Penedones, Polchinski, and Sully [125] that there are, essentially,
only two such conditions: a large central charge, and a parametrically large
spectral gap to the lightest single-trace operator of spin greater than two. A
significant step was taken toward its proof in [126], where graviton three-
point vertices (stress tensor three-point functions in the dual CFT language)
were shown to be parametrically suppressed by the higher spin scale,
$\Delta_{\rm gap}$, in powers according to dimensional analysis. This was
subsequently proven and generalized to other OPE structures from conformal
bootstrap points of view [127, 128, 129, 130, 131, 132, 133, 134, 135, 136]. A
key theme in those works was to study Lorentzian scattering experiments. That
philosophy has recently been synthesized with a “bottom-up” program to
constrain effective theories of gravity from analyticity and unitarity of the
graviton S-matrix, leading to precise numerical bounds on coefficients in the
derivative expansion of four-point amplitudes around general relativistic
scattering in AdS [137, 138].

A proof of the conjecture of [125] is still lacking in complete generality,
and there are some interesting open questions. How strong of an assumption is
large $c$ factorization? Are there bounds on spectral data in the scalar
sector, so far inaccessible to Lorentzian techniques which rely on spin? What
would it take to show that arbitrary $n$-point graviton scattering matches
that of general relativity at large $\Delta_{\rm gap}$?

The conditions mentioned above for a CFT to be holographic are widely believed
to hold, partly because they yield predictions consistent with the Regge
structure of string excitations and with the empirical properties of known
top-down instances of AdS/CFT. This begs the question of string universality,
namely, whether every consistent theory of quantum gravity is a string theory.
This is a central question for modern-day research on quantum gravity. A
related question from the conformal bootstrap point of view is to what extent,
and under what conditions, CFTs exhibit structures characteristic of
string/M-theory.555This question can be further refined for the prototypical
holographic CFT, which is defined by a sequence of CFTs with increasing and
unbounded central charge labeled by some parameter, such as the rank of a
gauge group. In this case, one can sensibly ask whether the ground state of
every such CFT is dual to an AdS vacuum of string/M-theory, which can be
constructed geometrically in the parametric limit. It is hard to state
precisely how this notion would extend to the more general case where the CFT
in question is not, in any obvious way, a member of such a sequence, though
the spirit of the idea may still be meaningful. See, for instance, [139, 140,
141, 142, 143, 128, 144, 145, 146, 147] for varied progress in this direction.
For holographic CFTs, the specific role of the higher-spin scale $\Delta_{\rm
gap}$ in suppressing bulk higher-derivative interactions [126] implies the
necessity of higher-spin excitations. A more precise version of this
connection with string theory applies to all CFTs, which have been shown to
organize their spectra into families of operators that are analytic in spin,
furnishing CFT analogs of Regge trajectories [128]. These stringy properties
complement direct approaches to bootstrapping quantum gravity amplitudes [146,
147], in which the four-point graviton amplitudes of string theory are found
to lie at very special points, close to the boundaries, in the general
parameter space allowed by unitarity and crossing symmetry.

Future work is expected to address more ambitious questions. One of these is
to definitively rule in/out scale-separated AdS vacua in classical string
theory. This is a long-standing question in the swampland program, with
differing opinions on the validity of existing string/M-theory solutions in
the mold of [148, 149, 150]. A scale-separated AdS solution exhibits a
parametric hierarchy $L_{\rm AdS}\gg L_{\mathcal{M}}$, where $\mathcal{M}$
comprises the internal dimensions of the string/M-theory compactification; the
CFT dual of a putative scale-separated solution would contain a very sparse
spectrum of single-trace operators with $\Delta\sim\mathcal{O}(1)$, dual only
to low-lying bulk “moduli” rather than Kaluza-Klein modes on $\mathcal{M}$.
(For preliminary work approaching this from the CFT perspective, see e.g.
[151, 152, 153, 154, 155].) The scale separation question may be viewed as a
refinement of the conjecture of [125] that specifies the number of large bulk
dimensions in which the theory exhibits sub-AdS locality. It is natural to
study a version of this question in the presence of supersymmetry: if a
holographic CFT possesses a continuous R-symmetry, it is not established
whether this symmetry must, in fact, be geometrized at the AdS scale,
notwithstanding standard AdS/CFT lore. This question is unanswered even for 4d
${\mathcal{N}}=4$ superconformal theories: can a “pure $AdS_{5}$”
compactification of string theory — with a 5d maximal supergravity multiplet,
but no large extra dimensions — and its exotic, non-SYM ${\mathcal{N}}=4$ SCFT
dual be ruled out? (See [156, 157] for related comments and some possible
hints.) The conformal bootstrap can rephrase [153] these questions in terms of
bounds on the parametric density of single-trace light operators at large
$\Delta_{\rm gap}$, which could plausibly be derived by clever application of
UV-IR relations, or by stronger use of the structure of Regge trajectories in
large $N$ CFTs. An extreme example of this type would be a theory of “pure
gravity” in $AdS_{3}$; the precise definition of such a theory varies
according to taste, but no satisfactory example has been found (or ruled out),
despite ongoing efforts [158, 159, 160, 161, 162, 163, 164, 165, 166, 167,
168].

Venturing one step further from the lamppost, conformal bootstrap tools will
ideally be developed to allow a scan of sporadic AdS vacua of string theory.
These sporadic vacua would be dual to isolated points in the space of CFTs,
theories with a large but finite number of degrees of freedom, that are not
members of a parametric sequence of CFTs of increasing central charge. Only
then can the conformal bootstrap properly begin to address the (AdS) landscape
question in string theory. Such a development could plausibly follow from new
approaches to motion in the space of solutions to crossing symmetry [169, 170,
171], or from continuing the parallel work directly in AdS of first principles
constraints on semiclassical gravitational scattering.

##  4 The worldsheet theory: CFT and beyond

The existence of a two-dimensional theory of gravity on the string worldsheet,
commonly referred to as the “worldsheet CFT” by a slight abuse of terminology,
has been nothing short of a miracle. Here we would like to dwell and reflect
on the power and limitations of the worldsheet formalism, which are not yet
fully understood.

The Neveu-Schwarz-Ramond formalism [1, 4, 9, 10, 11] gives a complete
formulation of superstring perturbation theory666Notably, the picture changing
operator formulation is completed through vertical integration [9, 10], which
allowed for a consistent formulation of perturbative NSR superstring field
theory [11].  in vacua of purely Neveu-Schwarz (NS) type. One generally
expects deformations of background NSNS fields to correspond to deformations
of the worldsheet “matter” CFT, although this is only precisely formulated in
time-independent backgrounds with well-defined asymptotic states. Turning on
Ramond (R) sector background field strengths necessarily couples the matter
and ghost systems on the worldsheet, and furthermore deforms the worldsheet
CFT in a non-local manner [172, 173, 174]. The pure spinor formalism [7, 175],
based on an entirely different matter and ghost structure, treats NS and R
sectors on equal footing; a complete formulation of string perturbation theory
in this framework remains an outstanding challenge.777In Minkowskian
spacetime, the full space of states and consistency of the pure spinor
worldsheet CFT remains to be clarified [8]. There appears to be no such
obstacle in $AdS_{5}\times S^{5}$, where the worldsheet CFT is expected to be
well-defined at least perturbatively in ${\alpha}^{\prime}$ (see [176] for
effort in this direction). Let us remark that the Green-Schwarz formalism [2,
3], in contrast, should be viewed as a construction of the string effective
action that nonlinearly realizes spacetime supersymmetries, and does not a
priori offer an unambiguous quantization scheme beyond leading orders in the
derivative expansion unless supplemented with additional symmetries such as
integrability [177].

An important class of weakly coupled strings are flux tubes in confining large
$N$ gauge theories. While much has been learned about the long distance
dynamics of confining strings [178, 179, 180, 181, 182, 124], the nature of a
UV-complete worldsheet theory that allows for a string theoretic description
of glueball states [183, 184] remains unclear. The known superstring
backgrounds that admit confining fundamental strings, typically acquiring
spacetime geometry of the form

$ds^{2}=f(y)dx^{\mu}dx_{\mu}+g_{ij}(y)dy^{i}dy^{j}\,,$ (1)

where the warp factor $f(y)$ reaches a positive minimal value at a finite
location on the base manifold (parameterized by $y^{i}$), necessarily involve
RR flux [185, 186, 187]. It begs the question of whether a worldsheet CFT, in
the sense of a 2d quantum field theory obeying locality axioms including OPE
and modular invariance, really exists in this setting. The pure spinor
formalism strongly hints at an affirmative answer, provided that one suitably
relaxes standard assumptions on 2d unitary CFTs. To find the appropriate
bootstrap axioms for the general worldsheet “conformal theory”, bridging the
gap between recent efforts in bootstrapping unknown 2d CFTs [162, 188] and
effective string theories [124], is likely the key to making progress on this
front.

The gauge/gravity duality suggests an alternative, constructive path toward a
worldsheet theory, in which the string worldsheet emerges from large $N$ gauge
theories through Feynman diagrams and the spin chain structure of single-trace
operators. Recent work on concrete AdS/CFT dual pairs in the weak coupling
limit of the spacetime CFTd provides a tractable testing ground [189, 190,
191, 192]. In particular, the tensionless string worldsheet theories dual to
both the 2d symmetric product orbifold CFT and free ${\cal N}=4$ SYM appear to
have free field descriptions [189, 193, 191]. In the AdS3/CFT2 case, the
string worldsheet is seen to emerge [194] from the “Feynman diagrams” of the
CFT2 [195] very much as per the general program of [196, 197, 198],
particularly the construction based on Strebel lengths in [198, 199]. This
utilizes the natural appearance of the moduli space of Riemann surfaces from
the ’t Hooft large $N$ Feynman diagrams [197, 198, 199].

The tensionless string limit for ${\rm AdS}_{3}$ reveals a further striking
relation between the spacetime and the worldsheet, in which the latter plays
the role of a covering space for the former [190], concretely realizing the
structure of meromorphic branched coverings à la Lunin-Mathur [200, 195]
within the spacetime CFT itself. Recently, an interesting higher-dimensional
analog is found that involves holomorphic covering maps into (ambi-)twistor
space [201]. All of this hints at the possibility of making direct contact
between conformal data of the worldsheet and spacetime CFTs, namely the
operator spectrum and structure constants. Let us sketch a way to frame this
connection concretely and in broader generality.

Suppose a large $N$ gauge theory has a weakly coupled dual bulk string theory
that admits a worldsheet CFT description, with the spectrum of single-trace
operators mapped to single string states. The thermal partition function of
the gauge theory, at temperature $T=1/\beta$ below the Hawking-Page
transition, is expected to admit a well-defined infinite $N$ limit (see [202,
203] for a discussion, especially in the context of weakly coupled SYM)

$\displaystyle Z(\beta)=\lim_{N\to\infty}{\rm Tr}_{\rm gauge}\left[e^{-\beta
H}\right]$ (2)

that counts multi-string states in the bulk as a free Fock space of single
string states. The partition function $Z(\beta)$ should be equivalently
computed by the path integral over genus one string worldsheets that wind
around the thermal circle, and it should admit an expression in terms of
worldsheet CFT data, at least in purely NSNS backgrounds with light-like
isometries [204, 189], of the form888In particular, the single-trace
contribution to the gauge theory partition function can be extracted from the
trace over winding number 1 states in the worldsheet theory [204].

$\displaystyle\log Z(\beta)=\int_{\cal F}\frac{d^{2}\tau}{2\tau_{2}}{\rm
Tr}_{\rm WS}\left[e^{2\pi i(\tau L_{0}-\bar{\tau}\bar{L}_{0})}\right]\,.$ (3)

Here, the trace on the RHS is taken over worldsheet CFT states with only
transverse excitations, and the moduli integration is performed over the
fundamental domain of $PSL(2,\mathbb{Z})$. The compatibility of (2) and (3)
amounts to highly nontrivial constraints on the spectra of the two sides of
the duality. An expression analogous to (3) should hold whenever a worldsheet
CFT exists, and may serve as the starting point for bootstrapping this
worldsheet theory.

Even richer constraints remain to be disentangled from the data of the three-
point functions. In Minkowskian spacetime, the unitarity and locality property
of perturbative string amplitudes are closely tied to the OPE in the
worldsheet CFT. For string theories in AdSd+1, one might expect a similar
relationship between the OPE of single-trace operators in the dual spacetime
CFTd and some suitable notion of OPE in the worldsheet theory [205, 206]. It
remains a major challenge to identify the appropriate bootstrap conditions on
the putative worldsheet OPE. The task is to recast these “dual bootstrap”
constraints into an effective form like in the conventional bootstrap. A
concrete aim would be to obtain the worldsheet description of weakly coupled
${\cal N}=4$ SYM using this approach.

##  5 Longer-term targets

The next generation of bootstrap techniques will require new ideas. Let us
mention a few possible avenues that may be relevant for string theory. In the
context of the superconformal bootstrap, one would like to acquire a better
understanding of the implications of (generalized) S-duality, for
${\mathcal{N}}=4$ SYM and other SCFTs [207, 208, 209, 210, 211]. Many SCFTs
have exactly marginal couplings on which S-duality symmetries act; in
holographic duality with string theory, these are identified with U-duality
symmetries preserved by an AdS compactification. The bootstrap program has
thrived in part because of the efficient progress in identifying and
processing symmetries: for example, computations of conformal blocks and Ward
identities, and the discovery of unobvious symmetries [51, 52, 212, 65, 213,
102]. To extract the physical consequences of S-duality for CFT data and the
AdS dual string quantities, the bootstrap should incorporate these symmetries
as efficiently as possible (see [214] for a new approach). Similar comments
apply to integrability and emergent Yangian symmetry at large $N$: for
${\mathcal{N}}=4$ SYM [215] and other maximally supersymmetric SCFTs [216,
217], the techniques and results of planar integrability should be combined
with, or fed into, the conformal bootstrap at large $N$.

We close this paper by describing some research directions that are further
afield; speculations on the future; and a visual depiction of the past,
present, and possible future of bootstrapping string theory.

### Beyond the low-energy expansion

Current efforts in bootstrapping supersymmetric holographic CFTs largely focus
on the four-point correlator of BPS multiplets. The latter, being merely an
analytic function of the cross ratios $z,\bar{z}$ and dependent on the central
charges and couplings, in principle contains a huge amount of information
including graviton $2\to 2$ S-matrix element at all energies and impact
parameters, capturing Planckian and black hole physics. While analytic
bootstrap efforts have concentrated on the lightcone limit [218, 219, 220,
221, 222, 223, 128] and the Regge limit [140, 129, 224],999The lightcone limit
amounts to taking $z\to 0$, fixing $\bar{z}$. The Regge limit is defined as
$z\to 0$, $z/\bar{z}$ fixed, after performing a monodromy of $\bar{z}$ around
1 on the complex plane. The bulk point limit is defined by taking
$z\to\bar{z}$ after performing the same monodromy of $\bar{z}$ around 1.
Euclidean correlators, on the other hand, correspond to complex conjugate
$z,\bar{z}$. numerical bootstrap results that determine internal operator
dimensions and OPE coefficients within small islands are a priori applicable
in the Euclidean regime where the conformal block expansion converges
absolutely. To access high-energy scattering in the flat space limit requires
taking the bulk point limit, and at the same time $N\to\infty$ with finite
(gauge) coupling. Whether the bootstrap can pin down correlators in the bulk
point limit, possibly through high precision numerics, remains a major
challenge.

Another trans-Planckian phenomenon of interest in string theory is the
annihilation/decay of branes [225, 226, 227]. A possibly tractable example
through holography is the annihilation of a pair of giant gravitons in
$AdS_{5}\times S^{5}$ [228] into closed string radiation. Even though such
amplitudes are in principle related to the scattering of closed strings off a
D-brane via analytic continuation and crossing, the physical process occurs at
an energy scale far beyond the realm of string perturbation theory. The answer
is in principle contained in correlation functions involving determinant
operators (see [229] for recent progress on their OPE coefficients) and may be
amenable to the bootstrap. Vis-à-vis string universality, a more abstract goal
is to bootstrap the necessity of branes themselves.

### Thermal bootstrap, matrix bootstrap, and black holes

So far, the conformal bootstrap has had comparatively little to say about
finite temperature physics [230, 231], and even less to say about thermal
phases of string theory specifically. A partial exception to this is in CFT2,
thanks to the power of modular invariance on the torus; however, we note that
the modular bootstrap has made little contact with string theory or string
universality.

Perhaps the main goal is to describe the quantum states or resonances of a
black hole that account for its macroscopic entropy [232, 233]. The successful
microscopic explanation of the entropy for certain supersymmetric black holes
[234, 235, 236, 237, 238, 239] demonstrates convincingly that string theory
and holographic dualities capture the fundamental degrees of freedom of
quantum gravity. Far less is understood concerning far-from-supersymmetric
black holes, with the notable exception of results from Monte Carlo simulation
of strongly coupled matrix quantum mechanics [240]. Recently, a new bootstrap
based on loop equations and positivity constraints [241, 242, 243, 244, 245]
has been successfully applied to large $N$ matrix models. While still in its
infancy, this approach opens a new possible pathway toward probing black hole
states from strongly coupled matrix models.

Some obvious but worthwhile targets for the conformal bootstrap are to derive
the famous factor of $3/4$ relating the thermal free energies of planar
$\mathcal{N}=4$ SYM at weak and strong ’t Hooft coupling [246], and likewise
for the viscosity-to-entropy-density ratio $\eta/s=1/4\pi$ [247]. Deriving
either of these universal properties of gravity at long wavelengths would
require extending bootstrap methods to new types of observables.

It is also worth stressing that the role of string theory in resolving the
black hole information paradox and related puzzles is not yet known. Recent
computations (see [248, 249, 250] and references therein) of the Page curve
for certain evaporating black holes using the path integral of semiclassical
gravity use surprisingly little of the microscopic details of the bulk theory;
the cost is the absence of an explicit accounting of the microstates under
time evolution. It would be very satisfying if a full solution of this paradox
could be shown to require stringy effects, properly understood. Perhaps the
conformal bootstrap at large $N$ can have something to say about this
interesting problem or its AdS avatars.

### The Polyakov-Mellin bootstrap and string theory

An alternative to the conventional conformal block expansion of correlators
and imposing crossing symmetry constraints was proposed long ago by Polyakov
[251]. Here one expands in a manifestly crossing-symmetric basis of functions
whereupon the constraints now consist of requiring the contributions of
certain spurious double-trace operators to vanish. This approach has been
revisited [252, 253, 254] and it was realised that a) Polyakov’s crossing
symmetric basis for CFTd is nothing other than the all-channel sum of exchange
Witten diagrams in AdSd+1 (together with additional contact Witten diagrams
that have recently been determined by consistency [255]); and b) the
constraints are simplest to impose in Mellin space where they become the
vanishing of residues of certain spurious single and double poles. These
conditions are equivalent to those obtained from conventional dispersion
relations for Mellin amplitudes, which now have a rigorous definition that
allows for a careful specification of the Polyakov conditions (see [256, 257,
258]).

Both of the above features are very suggestive of a reorganisation of
arbitrary CFTd amplitudes in terms of a string (string field?) theory
description in AdSd+1. It has been seen that this expansion is analytically
very efficient at obtaining the first few orders results, in the
$\epsilon$-expansion, for the dimensions and OPE coefficients of the $O(N)$
Wilson-Fisher fixed points [259, 252, 253, 260]. In SCFTs, such an expansion
provides a clearer interpretation of various quantities that can be computed
using supersymmetric localization [261]. So one might hope that this AdS
reorganization of CFTs is not just a technical convenience, but pointing to a
stringy picture, much as in the long sought-after string theory dual to the 3d
Ising model [262].

More generally, the previous paragraphs describe one specific recasting of
what it means to apply the conformal bootstrap method to correlation
functions, where crossing symmetry is traded for analyticity; can we find
others, that can be algorithmically solved?

### What is (admissible in) string theory?

A large class of constructions of string vacua is based on F-theory
compactification [263, 264, 265], which takes into account S-duality monodromy
branes of type IIB string theory and goes beyond both effective field theory
and string perturbation theory. While geometric singularities play an
essential role in the F-theory paradigm, it would be desirable to have a first
principles justification or classification of the admissible singularities, as
well as a framework for dynamics beyond what is governed by the moduli space
of vacua and BPS objects.

A basic example is D7-brane in type IIB string theory. In its presence, string
perturbation theory based on a worldsheet theory with Dirichlet boundary
condition breaks down, reflecting the fact that the D7-brane cannot exist on
its own but must be modified into the stringy cosmic string solution [266].
Questions concerning the dynamics of the stringy cosmic string may be good
targets for the S-matrix bootstrap.

### Bootstrapping de Sitter quantum gravity

Quantum gravity in de Sitter space, even the question of in what sense it
exists if at all, remains a mystery [267, 268, 269, 270, 271]. A natural set
of observables in de Sitter space are the late-time correlation functions
(“cosmological correlators”) [267, 272, 273, 274, 275, 276, 277, 278, 279,
280, 281, 282] that may be amenable to a bootstrap approach. While these
correlators are subject to (Euclidean) conformal symmetry, the analog of the
locality/OPE property as well as the meaning of (bulk) unitarity are not
understood. Recently steps in this direction have been taken in the context of
quantum field theory in de Sitter space [283, 284], where the notion of OPE
and bootstrap axioms are formulated. The outstanding challenge is to extend
such axioms to quantum gravity.

## Acknowledgements

XY would like to thank Ying-Hsuan Lin for discussions. RG is supported in part
by the J.C. Bose National Fellowship of the DST-SERB, by the DAE project no.
RTI4001 and the broad framework of support for basic sciences from the people
of India. EP is supported by ERC Starting Grant 853507. SSP is supported in
part by the Simons Foundation Grant No. 488653, and in part by the US NSF
under Grant No. 2111977. XY is supported by a Simons Investigator Award from
the Simons Foundation, by the Simons Collaboration Grant on the Non-
Perturbative Bootstrap, and by DOE grant DE-SC0007870.

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