#  Bootstrapping M-theory Orbifolds

###### Abstract

We analyze correlation functions of $SU(k)\times SU(2)_{F}$ flavor currents in
a family of three-dimensional ${\cal N}=4$ superconformal field theories,
combining analytic bootstrap methods with input from supersymmetric
localization. Via holographic duality, we extract gluon and graviton
scattering amplitudes of M-theory on ${\rm AdS}_{4}\times
S^{7}/\mathbb{Z}_{k}$ which contains a $\mathbb{C}^{2}/\mathbb{Z}_{k}$
orbifold singularity. From these results, we derive aspects of the effective
description of M-theory on the orbifold singularity beyond its leading low
energy limit. We also determine a threshold correction to the holographic
correlator from the combined contribution of two-loop gluon and tree-level
bulk graviton exchange.

###### Contents

  1. 1 Introduction
  2. 2 The M-theory orbifold
    1. 2.1 Effective theory and scattering amplitudes
    2. 2.2 Constraints on the Coulomb branch
    3. 2.3 AdS deformation and the holographic dual
    4. 2.4 Relation between 7d fields and flavor current multiplets in the SCFT
  3. 3 Superconformal kinematics
    1. 3.1 Correlators and multiplet structure
    2. 3.2 Central charges
    3. 3.3 Kinematics of the $SU(k)$ current 4-point function ${\mathcal{G}}^{abcd}$
    4. 3.4 Kinematics of the $SU(k)$-$SU(2)_{F}$ mixed 4-point function ${\mathcal{G}}^{abCD}$
  4. 4 Mellin amplitudes in the large $N$ expansion
    1. 4.1 Analytic bootstrap conditions and the large radius limit
    2. 4.2 Relation to 7d amplitudes
    3. 4.3 Structure of the four-gluon Mellin amplitude $M^{abcd}$
    4. 4.4 Structure of the mixed gluon-graviton Mellin amplitude $M^{abCD}$
  5. 5 BPS-protected CFT data
    1. 5.1 The 1d topological sector
    2. 5.2 Relation to the sphere partition function $Z$
    3. 5.3 Evaluating $Z$ from supersymmetric localization
    4. 5.4 The $1/N$ expansion of central charges
    5. 5.5 The $1/N$ expansion of BPS-protected OPE coefficients
  6. 6 Back to M-theory via Mellin amplitudes
    1. 6.1 The leading low energy limit in the M-theory orbifold
    2. 6.2 Constraining higher derivative corrections
    3. 6.3 The logarithmic threshold corrections
  7. 7 Conclusion
  8. A 7d ${\mathcal{N}}=1$ superamplitudes
    1. A.1 7d ${\mathcal{N}}=1$ super-spinor-helicity formalism
    2. A.2 Supervertices for vector multiplets
    3. A.3 Supervertices involving graviton multiplets
    4. A.4 Supervertices with 11d gravitons
    5. A.5 Loop amplitudes
  9. B Formulae for superblocks and Mellin amplitudes
  10. C Converting Mellin amplitudes to OPE coefficients
  11. D Matrix model calculations

##  1 Introduction

M-theory is believed to be an ultraviolet (UV)-complete theory of quantum
gravity that reduces to eleven-dimensional supergravity in its low-energy
limit [1]. Starting with the work of Hořava and Witten [2, 3], it was
recognized that M-theory also admits spacetimes with certain mild
singularities, where, typically, the singular loci carry additional light
degrees of freedom. A major, still unanswered, question is to classify
possible singularities admitted by M-theory, as well as to understand the
dynamics of their excitations. This question goes beyond the low-energy
supergravity limit, and it is also not accessible in string perturbation
theory. Presently, the only potentially systematic approach to it is through
holographic dualities.

In this paper, we will focus on codimension-four orbifold singularities,
particularly M-theory in the spacetime
$\mathbb{C}^{2}/\mathbb{Z}_{k}\times\mathbb{R}^{1,6}$, which is known to
support $SU(k)$ non-Abelian gauge fields at the seven-dimensional orbifold
singularity locus [4, 5, 6]. We will develop a framework for extracting the
dynamics of these excitations through a family of three-dimensional ${\cal
N}=4$ superconformal gauge theories, whose holographic duals involve M-theory
backgrounds that contain such orbifold singularities [7, 8, 9, 10, 11].

Our starting point will be the holographic duality [12, 13, 14] between
M-theory in the spacetime ${\rm AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$, where
the $\mathbb{Z}_{k}$ action fixes an $S^{3}\subset S^{7}$,111This is different
from the fixed-point-free $\mathbb{Z}_{k}$ quotient of $S^{7}$ that appears in
the holographic dual of Aharony-Bergman-Jafferis-Maldacena (ABJM) theory [15].
See [10, 11] for the precise supergravity background in M-theory and also its
reduction in type IIA string theory (where the 11d ${\mathbb{Z}}_{k}$ orbifold
singularity reduces to a stack of $k$ D6-branes). and the three-dimensional
${\cal N}=4$ superconformal field theory (SCFT) that is reached in the
infrared (IR) limit of a $U(N)$ gauge theory coupled to an adjoint
hypermultiplet and $k$ fundamental hypermultiplets [16, 17]. As is usually the
case in the anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence,
in the large $N$ limit, single-trace operators in the field theory are dual to
fluctuation modes of the M-theory background. Of particular interest are the
$SU(k)$ flavor symmetry currents $j_{\mu}^{a}$, with $a=1,\ldots,k^{2}-1$,
which act nontrivially on the fundamental hypermultiplets and are dual to
massless “gluons” in AdS localized at the 7-dimensional singular locus (namely
the ${\rm AdS}_{4}\times S^{3}$ that is fixed by the $\mathbb{Z}_{k}$ action),
and the $SU(2)_{F}$ flavor symmetry currents ${\widetilde{j}}_{\mu}^{A}$, with
$A=1,2,3$, which act nontrivially on the adjoint hypermultiplet and are dual
to a particular set of graviton and three-form modes in the bulk ${\rm
AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$ spacetime (see Table 1 for details).

The M-theory orbifold spacetime
$\mathbb{C}^{2}/\mathbb{Z}_{k}\times\mathbb{R}^{1,6}$ can be recovered from
the large radius limit of the ${\rm AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$
background. In particular, certain scattering amplitudes in
$\mathbb{C}^{2}/\mathbb{Z}_{k}\times\mathbb{R}^{1,6}$, involving modes
supported both at the orbifold singularity as well as in the bulk, can be
obtained from the large radius limit of the Mellin amplitudes in the dual CFT,
which are related to correlation functions via the Mellin transform [18, 19],
through the prescription of [20]. For example, from the four-point function of
the $SU(k)$ flavor currents $j^{a}_{\mu}$, one can extract the four-point
scattering amplitude of the gluons $g^{a}$ living on the $\mathbb{R}^{1,6}$
located at the orbifold singularity locus:

$\begin{split}\langle
j_{\mu}^{a}(\vec{x}_{1})j_{\nu}^{b}(\vec{x}_{2})j_{\rho}^{c}(\vec{x}_{3})j_{\sigma}^{d}(\vec{x}_{4})\rangle\xrightarrow[\text{limit}]{\text{flat
space}}{\cal A}(g^{a}g^{b}g^{c}g^{d})\,,\end{split}$ (1.1)

where the scattering amplitude depends on the momenta and polarizations of the
gluons, which originate from the kinematic configurations in the CFT
correlator.

We can also consider the correlation function of a pair of $SU(k)$ flavor
currents $j^{a}_{\mu}$ with a pair of $SU(2)_{F}$ flavor currents
$\widetilde{j}_{\mu}^{A}$. From the mixed four-point function of $SU(k)$ and
$SU(2)_{F}$ currents, one can extract, in a way to be made precise in Section
4, the scattering amplitude of two 7d gluons $g$ and two 11d gravitons
$h$:222The flat space limit for CFT operators with spin has not been worked
out explicitly, and therefore we will not use it. Instead, we will make use of
supersymmetry to relate the conserved currents to scalar operators in the same
multiplets, for which one can use the flat space limit formula of
[20].,333Note that in the limiting procedure of (1.1) and (1.2), the 11d
momenta of the gluons and gravitons are restricted to lie within the 4d
subspace of the ${\rm AdS}_{4}$ directions before taking the limit. For the
four-point amplitude of four 7d gluons, such a restriction can be made without
loss of generality, so one can then fully reconstruct the full amplitude
${\cal A}(g^{a}g^{b}g^{c}g^{d})$. For the amplitude of two gluons and two
gravitons, this is indeed a restricted kinematic configuration, so from (1.2)
one cannot reconstruct the full amplitude ${\cal A}(g^{a}g^{b}hh)$.

$\begin{split}\langle
j_{\mu}^{a}(\vec{x}_{1})j_{\nu}^{b}(\vec{x}_{2})\widetilde{j}_{\rho}^{A}(\vec{x}_{3})\widetilde{j}_{\sigma}^{B}(\vec{x}_{4})\rangle\xrightarrow[\text{limit}]{\text{flat
space}}{\cal A}(g^{a}g^{b}hh)\,.\end{split}$ (1.2)

The essential step in implementing this program is, of course, computing the
SCFT correlation functions. While the aforementioned SCFT can be realized as
the (strongly coupled) infrared limit of a Lagrangian gauge theory, thus far
the Lagrangian description has had limited use in the determination of
infrared observables, except for a class of supersymmetric observables that
can be computed exactly along the Renormalization Group (RG) flow [21, 22, 23,
24, 25]. In particular, there are two distinct integrated 4-point functions of
conserved current operators $j_{\mu}^{a}(\vec{x})$ in a 3d ${\cal N}=4$
SCFT,444For 3d ${\cal N}=4$ SCFTs, the supeconformal algebra has two distinct
representations that contain conserved currents (see, for instance, [26]). The
statement that follows assumes that all four operators belong to multiplets of
the same type, which is indeed the case for the $SU(k)$ and $SU(2)_{F}$
currents in our theory of interest (also known as the Higgs branch type).
Note, however, that our theory is also invariant under a $U(1)$ topological
symmetry that we do not consider here, and the associated $U(1)$ current
multiplet is of the opposite type (also known as the Coulomb branch type).
schematically of the form

$\begin{split}\int\left(\prod_{i=1}^{4}d^{3}\vec{x}_{i}\right)\mu^{\mu\nu\rho\sigma}_{abcd}(\vec{x}_{1},\vec{x}_{2},\vec{x}_{3},\vec{x}_{4})\,\langle
j_{\mu}^{a}(\vec{x}_{1})j_{\nu}^{b}(\vec{x}_{2})j_{\rho}^{c}(\vec{x}_{3})j_{\sigma}^{d}(\vec{x}_{4})\rangle\end{split}$
(1.3)

for suitable measure factors $\mu$, that can be determined from supersymmetric
localization. See [27, 28, 29, 30, 31, 32] for analogous examples of such
integrated correlators in supersymmetric gauge theories in various dimensions.

On the other hand, the SCFT correlators at sufficiently generic kinematic
configurations, from which we wish to extract scattering amplitudes in the
large radius limit of the holographic dual, cannot be determined from
supersymmetric localization alone. Additional ingredients, such as crossing
equations and analytic properties of the corresponding Mellin amplitudes, have
recently been combined with an extensive set of supersymmetric localization
results to constrain holographic correlators and extract M-theory or string
theory observables beyond the supergravity limit [33, 28, 27, 30, 29, 34, 35,
36, 37, 38, 39, 40, 41].555Similar localization constraints [42, 43] have also
been used to constrain higher derivative corrections to the effective
supergravity action [44]. The same strategy will be adopted in this paper to
extract the dynamics of gluons at the M-theory orbifold singularity.

One of our main results is that the $\mathop{\rm Tr}\nolimits F^{4}$ and
$(\mathop{\rm Tr}\nolimits F^{2})^{2}$ couplings, whose precise definitions
will be addressed in Section 2.1,666We will adopt a scheme-independent
definition of such effective couplings through particular terms in the
momentum expansion of the four-gluon amplitude, which are unambiguously
separated from the non-analytic terms in momenta. which are a priori allowed
in a 7d supersymmetric effective gauge theory, are absent for the gluons
supported at the $\mathbb{C}^{2}/\mathbb{Z}_{k}\times\mathbb{R}^{1,6}$
orbifold singularity of M-theory. This is reminiscent of the absence of
$D^{4}R^{4}$ term in the bulk M-theory (graviton) effective action, recently
extracted from correlators of ABJM theory [27]. Let us note that, previously,
the bulk BPS-protected higher derivative couplings had also been determined
through a different chain of logic, namely, by considering toroidal
compactification of M-theory, and constraining the moduli dependence thereof
via supersymmetry non-renormalization theorems [45, 46, 47, 48]. A similar
argument applies to the Abelian effective gauge theory on the Coulomb branch
of the M-theory orbifold, where the moduli dependence of the BPS-protected
Abelian $F^{4}$ effective coupling can be determined [49, 50, 51, 52]. The
relation between this abelian $F^{4}$ coupling, which becomes singular at the
origin of the Coulomb branch, and the non-Abelian $\mathop{\rm Tr}\nolimits
F^{4}$ and $(\mathop{\rm Tr}\nolimits F^{2})^{2}$ couplings determined in our
holographic approach, will be discussed in Section 2.

Going in the other direction, we can also combine what we know about the flat
space scattering amplitude with supersymmetric localization data in the
holographic gauge theory to learn more about the holographic correlators, or
equivalently the Mellin amplitudes, in AdS. In particular, we will determine a
certain logarithmic threshold contribution to the large $N$ expansion of the
correlator (1.1) coming from two-loop gluon exchange and tree-level bulk
graviton exchange combined.

The rest of this paper is organized as follows. In Section 2, we review the
construction of the $\mathbb{C}^{2}/\mathbb{Z}_{k}$ orbifold background of
M-theory, the general structure of the scattering amplitudes involving the
gluons supported at the orbifold singularity, and supersymmetry constraints
for the effective theory on the Coulomb branch where the orbifold singularity
is resolved. We also review the AdS counterpart, which we refer to as the AdS
orbifold, and its holographic dual 3d ${\cal N}=4$ SCFT. In Section 3, we
describe the kinematic structure and the superconformal block expansion of the
correlators for the flavor current multiplets of interest. The structure of
the large radius expansion of the corresponding Mellin amplitudes and its
connection to the flat space amplitudes are analyzed in Section 4. Sections 5
and 6 contain the key technical results of this paper, namely the
supersymmetric localization and the subsequent determination of (the absence
of) the four-derivative effective couplings in the M-theory orbifold, as well
as the determination of threshold contributions to the Mellin amplitude. We
conclude in Section 7 with a discussion of future directions. Some details
concerning 7d superamplitudes, the superconformal blocks, the algorithm of
extracting OPE data from the Mellin amplitudes, and matrix model technique
used to extract supersymmetric localization data, are presented in the
Appendices.

##  2 The M-theory orbifold

$\mathbb{R}^{1,6}$$\mathbb{C}^{2}/\mathbb{Z}_{k}$

Figure 1: A poor man’s portrait of the ${\mathbb{C}}^{2}/{\mathbb{Z}}_{k}$
orbifold singularity in M-theory.

The $\mathbb{C}^{2}/\mathbb{Z}_{k}$ or $A_{k-1}$ orbifold is the quotient
space of $\mathbb{C}^{2}$, parametrized by complex coordinates
$(z_{1},z_{2})$, with respect to the discrete group action

$\displaystyle(z_{1},z_{2})\mapsto(e^{2\pi in/k}z_{1},e^{-{2\pi
in/k}}z_{2})\,,\qquad n=0,1,\ldots,k-1\,.$ (2.1)

M-theory in the orbifold spacetime
$\mathbb{C}^{2}/\mathbb{Z}_{k}\times\mathbb{R}^{1,6}$ (portrayed schematically
in Figure 1) may be approached either as the strong coupling limit of type IIA
string theory in the orbifold background, which admits (at least in the weak
coupling regime) a well-known worldsheet CFT description [53, 54], or as a
geometrically singular limit of M-theory in the smooth ALE spacetime [4, 5],
namely $\mathbb{R}^{1,6}$ times the four-manifold equipped with the metric

$\displaystyle
ds_{4}^{2}=U(\vec{x})d\vec{x}^{2}+U(\vec{x})^{-1}(dy+\vec{\omega}\cdot
d\vec{x})\,,$ (2.2) $\displaystyle
U(\vec{x})=\sum_{i=1}^{k}{1\over|\vec{x}-\vec{x}_{i}|}\,,\qquad\nabla\times\vec{\omega}=\nabla
U\,,$

where $\vec{x}\in\mathbb{R}^{3}$, $y\sim y+4\pi$, in the limit where
$\vec{x}_{1},\ldots,\vec{x}_{k}$ collide. The agreement of these two different
descriptions and the existence of the singular orbifold as an M-theory
background are highly nontrivial and lie at the foundation of a wealth of
constructions of M-theory vacua [4, 5, 6].

###  2.1 Effective theory and scattering amplitudes

The massless degrees of freedom of M-theory supported at the orbifold
singularity are expected to be those of $SU(k)$ gluons and their superpartners
[4, 5], governed in the low-energy limit by the 7-dimensional ${\cal N}=1$
super-Yang-Mills theory coupled to the bulk 11d supergravity. The bulk
orbifold spacetime $M_{11}^{(k)}$ can be represented as a quotient
$\widetilde{M}_{11}/\mathbb{Z}_{k}$, such that the metric $G_{MN}$ lifts to a
smooth $\mathbb{Z}_{k}$-invariant metric on the covering space
$\widetilde{M}_{11}$. The 11d supergravity action takes the form

$\begin{split}S_{\text{grav}}[G_{MN},\ldots]=\int_{M_{11}^{(k)}}d^{11}x\,\sqrt{-G}\left(\frac{R}{2\kappa^{2}}+{\rm
SUSY\,completion}\right)\,,\end{split}$ (2.3)

where $\kappa$ is related to the 11d Planck length $\ell_{p}$ by the
convention $2\kappa^{2}=(2\pi)^{8}\ell_{p}^{9}$. The 7d $SU(k)$ gauge theory
action takes the form of ${\cal N}=1$ super-Yang-Mills covariantized with
respect to the restriction $g_{\mu\nu}$ of the bulk metric $G_{MN}$ to the 7d
orbifold singularity locus $M_{7}$,

$\begin{split}S_{\rm
SYM}[A_{\mu},\ldots;G_{MN},\ldots]=\int_{M_{7}}d^{7}x\,\sqrt{-g}\left(-{1\over
2g_{\rm YM}^{2}}\mathop{\rm Tr}\nolimits
F_{\mu\nu}F^{\mu\nu}+\cdots\right)\,,\end{split}$ (2.4)

where the Lie algebra trace $\mathop{\rm Tr}\nolimits$ is defined by the one
in the fundamental representation (i.e. $\mathop{\rm
Tr}\nolimits\equiv\mathop{\rm Tr}\nolimits_{\rm fund}$) throughout the paper
and the Yang-Mills coupling $g_{\rm YM}$ is related to the 11d Planck length
$\ell_{p}$ by $g_{\rm YM}^{2}=2(2\pi)^{4}\ell_{p}^{3}$.777This can be
determined by the identification of the 2+1 dimensional Yang-Mills instanton-
membranes with M2-branes located at the orbifold singularity. By matching the
instanton tension with that of the M2-brane, $T_{\rm M2}=T_{\rm inst}$, where
$T_{\rm M2}={1\over(2\pi)^{2}\ell_{p}^{3}}$, and $T_{\rm inst}={8\pi^{2}\over
g_{\rm YM}^{2}}$.

It is nontrivial to construct the supersymmetric completion $S_{\rm SYM}$ of
the covariantized 7d Yang-Mills action (see [6]). As is familiar in the
context of brane-bulk effective theories [3, 55, 56], a fully consistent
treatment of the coupling between the localized and the bulk degrees of
freedom is only expected at the quantum level. However, note that in contrast
to typical brane-bulk effective theories, the coupling between the localized
degrees of freedom at the orbifold singularity and the bulk ones does not
admit non-linearly realized translation symmetries in the transverse
directions, and there is no backreaction in the form of brane tension.

In principle, one may formulate the effective theory of the M-theory orbifold
in a Wilsonian framework, by working with a Wilsonian effective action that
contains an infinite series of higher derivative corrections to $S_{\rm
SYM}+S_{\rm grav}$, along with a UV cutoff scheme. In practice, it can be
exceedingly complicated to implement gauge invariance and supersymmetry
constraints on the effective Lagrangian beyond the leading order in the
derivative expansion. It is often more convenient to instead formulate the
effective theory in terms of the small momentum expansion of scattering
amplitudes, where gauge redundancies are eliminated, and supersymmetry
constraints are realized linearly through Ward identities [47, 52, 48].
Loosely speaking, the analytic terms in the momentum expansion of the
amplitudes are in correspondence with the higher-derivative couplings in the
effective Lagrangian, while the non-analytic terms of the momentum expansion
are constrained by unitarity and locality. We will primarily adopt the latter
approach in this paper, and we will organize the effective couplings of the
M-theory orbifold in terms of the on-shell supervertices of the localized
gluons and the bulk gravitons.

$\displaystyle g$$\displaystyle\sim\kappa^{2}$$\displaystyle
A$$\displaystyle\sim{g_{\rm
YM}^{2}}$$\displaystyle\sim\frac{1}{\kappa^{2}}$$\displaystyle\sim\frac{1}{g_{\rm
YM}^{2}}$$\displaystyle\sim\frac{1}{g_{\rm YM}^{2}}$

Figure 2: Propagators and minimal coupling vertices for the gluons localized
at the singularity (represented as the blue-shaded subspace) and the bulk
graviton. In a Wilsonian effective theory, there are infinitely many more
higher derivative vertices, e.g. those of Figures 4 and 6.

$A$$A$$A$$A$

$A$$A$$A$$A$

$A$$A$$A$$A$

$A$$A$$A$$A$

Figure 3: Diagrams that exhibit unitarity cuts of the four-gluon amplitude
into minimal coupling vertices, of momentum scalings $s^{0},s^{3\over
2},s^{3},s^{3}$ respectively. Note that the graviton propagator in the third
diagram is not restricted to the orbifold locus, leading to an enhanced
momentum scaling.

To leading order in the low-energy limit, the scattering amplitudes can be
evaluated via tree-level Feynman diagrams (built out of the propagators and
vertices of Figure 2) of the 7d SYM coupled to 11d supergravity. In
particular, the leading low energy superamplitude of four gluon multiplets is
given, in the super-spinor-helicity convention described in Appendix A.1, by
[57]

$\displaystyle\mathfrak{A}_{F^{2}}\left(\\{p_{i},\lambda_{i},\eta_{i},a_{i}\\}_{i=1}^{4}\right)=2g_{\rm
YM}^{2}\delta^{7}(P)\delta^{8}(Q)\left[{{\mathtt{B}}^{a_{1}a_{2}a_{3}a_{4}}\over
st}+({\rm cyclic~{}permutations~{}on~{}}2,3,4)\right].$ (2.5)

Here $s,t,u=-s-t$ are 7d Mandelstam variables defined below,

$\displaystyle s=-(p_{1}+p_{2})^{2}\,,\quad t=-(p_{1}+p_{4})^{2}\,,\quad
u=-(p_{1}+p_{3})^{2}\,,$ (2.6)

$a_{i}$ are adjoint gauge indices, $P$ is the total momentum, $Q$ is the total
super-momentum, and ${\mathtt{B}}^{abcd}$ is a color factor defined by

$\displaystyle{\mathtt{B}}^{abcd}\equiv\mathop{\rm Re}\nolimits\mathop{\rm
Tr}\nolimits(T^{a}T^{b}T^{c}T^{d})\,,$ (2.7)

where $T^{a}$ are hermitian $SU(k)$ generators normalized according to
$\mathop{\rm Tr}\nolimits(T^{a}T^{b})={1\over 2}{\delta}^{ab}$.

Beyond the leading low energy limit, the momentum expansion of the full
4-gluon superamplitude is expected to take the form, where the dependence on
the super-spinor-helicity variables are implicit,

$\displaystyle\mathfrak{A}$
$\displaystyle=\mathfrak{A}_{F^{2}}+\mathfrak{A}_{F^{2}|F^{2}}+\mathfrak{A}_{F^{4}}+\left[\mathfrak{A}_{R}+\mathfrak{A}_{F^{2}|F^{2}|F^{2}}+\mathfrak{A}_{D^{2}F^{4}}\right]+\cdots\,,$
(2.8)

where the various terms are interpreted as follows. First,
$\mathfrak{A}_{F^{2}}$ is the leading low-energy amplitude (2.5) with $s^{0}$
overall momentum scaling (first diagram in Figure 3).
$\mathfrak{A}_{F^{2}|F^{2}}$, given in (A.25), represents a “1-loop” term of
$s^{3\over 2}$ overall momentum scaling, that is completely determined by its
unitarity cut into a pair of $\mathfrak{A}_{F^{2}}$ amplitudes [58, 59]
(second diagram in Figure 3). The first higher derivative “corrections” that
are not fixed by supersymmetry and perturbative unitarity considerations arise
at order $s^{2}$ (first diagram in Figure 4), and take the form

$\displaystyle\mathfrak{A}_{F^{4}}=\delta^{7}(P)\delta^{8}(Q)\Big{[}\kappa_{[F^{2}]^{2}}{\mathtt{A}}^{a_{1}a_{2}a_{3}a_{4}}+\kappa_{F^{4}}{\mathtt{B}}^{a_{1}a_{2}a_{3}a_{4}}+({\rm
cyclic~{}permutations~{}on~{}}2,3,4)\Big{]}\,,$ (2.9)

where we have introduced another color factor by

$\displaystyle{\mathtt{A}}^{abcd}\equiv\mathop{\rm
Tr}\nolimits(T^{a}T^{b})\mathop{\rm Tr}\nolimits(T^{c}T^{d})\,.$ (2.10)

This amplitude may be viewed as the contribution from the supersymmetric
completion of a term in an effective Lagrangian of the schematic form
$\kappa_{[F^{2}]^{2}}(\mathop{\rm Tr}\nolimits
F^{2})^{2}+\kappa_{F^{4}}\mathop{\rm Tr}\nolimits F^{4}$, whose Lorentzian
indices are contracted by the rank-eight $t_{8}$ tensor which is dictated by
supersymmetry [60, 61]. We will specify these effective couplings
unambiguously through their contribution (2.9) to the local (i.e. analytic in
momenta) part of the amplitude (2.8).

$A$$A$$A$$A$$F^{4}~{}$

$A$$A$$A$$A$$D^{2}F^{4}~{}$

Figure 4: Some higher-derivative supervertices (represented by the red dots)
that potentially contribute to the four-gluon amplitude. The distinct color
structures have been suppressed in the diagrams and the corresponding momentum
scalings are $s^{2}$ and $s^{3}$ respectively.

At order $s^{3}$ in momentum scaling, there are several new structure that
appear in (2.8). $\mathfrak{A}_{R}$ can be viewed as a “tree-level” gluon
amplitude that involves the exchange of a bulk graviton (third diagram of
Figure 3). The momentum power counting is unusual here: each cubic vertex
contributes $+2$, the graviton propagator contributes $-2$, and the
integration over the transverse momentum $p_{\perp}$ of the graviton
contributes $+4$. From the perspective of Wilsonian effective theory, such a
tree diagram is a priori subject to a UV divergence, schematically of the form
[55, 56]

$\displaystyle\int d^{4}p_{\perp}{p_{\parallel}^{4}\over
p_{\parallel}^{2}+p_{\perp}^{2}}\sim p_{\parallel}^{6}\log{\Lambda\over
p_{\parallel}}+p_{\parallel}^{4}\Lambda^{2},$ (2.11)

where the cutoff dependence can be absorbed into local terms. The
superamplitude $\mathfrak{A}_{R}$ is most conveniently determined, up to local
terms, by its unitarity cut into a pair of gluon-gluon-graviton vertices
(A.23), to be (cf. (A.29))

$\displaystyle\mathfrak{A}_{R}$ $\displaystyle={1\over
2}\kappa^{2}\delta^{7}(P)\delta^{8}(Q)\Big{[}{\mathtt{A}}^{a_{1}a_{2}a_{3}a_{4}}k{s\over
16\pi^{2}}\log(-\ell_{p}^{2}s)+({\rm
cyclic~{}permutations~{}on~{}}2,3,4)\Big{]},$ (2.12)

where the factor of $k$ is due to the sum over $\mathbb{Z}_{k}$ images in the
bulk graviton propagator.

The second term $\mathfrak{A}_{F^{2}|F^{2}|F^{2}}$ with the same momentum
scaling in (2.8) is, up to local terms, the two-loop gluon superamplitude
given in (A.27) (see last diagram in Figure 3). In contrast to the formal 7d
SYM 2-loop amplitude which is logarithmically divergent [57], we define
$\mathfrak{A}_{F^{2}|F^{2}|F^{2}}$ by cutting off the loop integrals in (A.27)
at Planck scale $\Lambda\sim 1/\ell_{p}$, giving a finite result with an IR
non-analyticity similar to that of (2.12). The curious fact that the gluon
2-loop amplitude arises at the same order in momentum scaling as the bulk
graviton tree-level exchange, which is tied to the cubic relation between
gravitational and Yang-Mills coupling $\kappa\sim g_{\rm YM}^{3}$, is
reminiscent of (an “M-theory upgrade” of) the relationship between 1-loop open
and tree-level closed string amplitudes in open/closed duality.

The remaining local terms of $s^{3}$ momentum scaling in (2.8) are constrained
by supersymmetry Ward identities to be of the form (see second diagram in
Figure 4)

$\displaystyle\mathfrak{A}_{D^{2}F^{4}}=\delta^{7}(P)\delta^{8}(Q)\Big{[}\kappa_{D^{2}[F^{2}]^{2}}{\mathtt{A}}^{a_{1}a_{2}a_{3}a_{4}}s+\kappa_{D^{2}F^{4}}{\mathtt{B}}^{a_{1}a_{2}a_{3}a_{4}}u+({\rm
cyclic~{}permutations~{}on~{}}2,3,4)\Big{]}.$ (2.13)

From the Wilsonian perspective, we expect an effective theory with cutoff at
the Planck scale, whose coupling coefficients cannot depend on any parameters
other than $k$ which specifies the M-theory background. Thus
$\kappa_{D^{2}[F^{2}]^{2}}$ and $\kappa_{D^{2}F^{4}}$, related to coefficients
of terms in the effective Lagrangian of the schematic form $D^{2}(\mathop{\rm
Tr}\nolimits F^{2})^{2}$ and $D^{2}\mathop{\rm Tr}\nolimits F^{4}$ (where
$D$’s stand for gauge-covariant derivatives, suitably distributed on the
$F_{\mu\nu}$’s), are proportional to $\ell_{p}^{6}$ with possibly
$k$-dependent coefficients.

$g$$A$$A$$g$

$g$$A$$A$$g$

$g$$A$$A$$g$

$g$$A$$A$$g$

Figure 5: Diagrams that exhibit unitarity cuts of two-gluon-two-graviton
amplitudes into minimal coupling vertices, of momentum scalings $s,s,s^{5\over
2},s^{9\over 2}$ respectively. Note that the last diagram is beyond the
derivative orders consider in this paper.

The mixed 4-point superamplitude of two gluon multiplets and two graviton
multiplets can be organized in a similar momentum expansion

$\displaystyle\widetilde{\mathfrak{A}}=\widetilde{\mathfrak{A}}_{R}+\widetilde{\mathfrak{A}}_{RF^{2}}+\widetilde{\mathfrak{A}}_{F^{2}|R}+\widetilde{\mathfrak{A}}_{R^{2}F^{2}}+\cdots\,,$
(2.14)

where the various terms are interpreted as follows. The leading low-energy
term $\widetilde{\mathfrak{A}}_{R}$ may be viewed as tree-level exchange of a
gluon or a graviton, and it is determined by its residues at the gluon or
graviton pole, which factorize into a pair of cubic minimal coupling vertices
(first two diagrams of Figure 5). For comparison with the holographic
correlators considered in this paper, it suffices to restrict to the case
where the gravitons do not carry transverse momenta, where the result can be
expressed in a relatively simple manner using the 7d super-spinor-helicity
notation of Appendix A.1,888This expression is derived by embedding the
${\mathcal{N}}=1$ graviton and vector multiples into an ${\mathcal{N}}=2$,
i.e. maximally supersymmetric, graviton multiplet, and projecting from the
tree-level scattering amplitude of the latter.

$\displaystyle\left.\widetilde{\mathfrak{A}}_{R}\right|_{p_{3}^{\perp}=p_{4}^{\perp}=0}$
$\displaystyle=\kappa^{2}{\rm
Tr}(T^{a_{1}}T^{a_{2}})\delta^{7}(P){\delta^{8}(Q)\over
stu}\left.\partial_{\widehat{\eta}_{1}}^{2}\partial_{\widehat{\eta}_{2}}^{2}\prod_{{\alpha}=1}^{8}\left(\lambda^{\alpha}_{1I}\widehat{\eta}_{1}^{I}+\lambda^{\alpha}_{2I}\widehat{\eta}_{2}^{I}+\widetilde{q}_{3}^{\alpha}+\widetilde{q}_{4}^{\alpha}\right)\right|_{\widetilde{\eta}_{3}^{2}|_{\bf
5}\widetilde{\eta}_{4}^{2}|_{\bf 5}}\,,$ (2.15)

where $\\{p_{i},\lambda_{i},\eta_{i},a_{i}\\}_{i=1,2}$ are the momenta, super-
spinor-helicity variables, and gauge indices of the gluons,
$\\{p_{i},\lambda_{i},\eta_{i},\widetilde{\eta}_{i}\\}_{i=1,2}$ are the 7d
momenta and super-spinor-helicity variables of the gravitons, and
$\widehat{\eta}_{1}^{I},\widehat{\eta}_{2}^{I}$ are auxiliary Grassmann
variables, with
$\partial_{\widehat{\eta}}^{2}\equiv\Omega^{IJ}{\partial\over\partial\widehat{\eta}^{I}}{\partial\over\partial\widehat{\eta}^{J}}$.

$g$$A$$A$$g$$D_{\perp}^{2}D^{-2}RF^{2}$

$g$$A$$A$$g$$D_{\perp}^{2}D^{-2}RF^{2}$

$g$$A$$A$$g$$R^{2}F^{2}$

Figure 6: Diagrams that exhibit possible unitarity cuts of the amplitude of
two gluons and two gravitons that involve higher-derivative supervertices
(represented by the red dots), with momentum scaling $s^{2},s^{2},s^{3}$
respectively.

The first possible correction to the minimal gluon-graviton coupling that is
compatible with supersymmetry arises at 4-derivative order in the effective
Lagrangian, of the schematic form $\kappa_{RF^{2}}R\mathop{\rm Tr}\nolimits
F^{2}+\cdots$, corresponding to the cubic supervertex [52]

$\displaystyle\kappa_{RF^{2}}{\mathfrak{V}}_{D_{\perp}^{2}D^{-2}RF^{2}},$
(2.16)

where ${\mathfrak{V}}_{D_{\perp}^{2}D^{-2}RF^{2}}$ is given by the $k=1$ case
of (A.24). This supervertex vanishes when the transverse momentum $p_{\perp}$
of the graviton vanishes. On the other hand, one expects at the same
derivative order a 2-gluon-2-graviton superamplitude
$\widetilde{\mathfrak{A}}_{RF^{2}}$ that is rational in momenta, whose
factorization through its gluon or graviton pole involves the cubic vertex
(2.16) (see first two diagrams in Figure 6). The vanishing of the latter at
$p_{\perp}=0$ implies that $\widetilde{\mathfrak{A}}_{RF^{2}}$, when
restricted to gravitons with vanishing transverse momenta
($p_{3}^{\perp}=p_{4}^{\perp}=0$ in the notation of (2.15)), is free of poles.
A closely related feature of the Mellin amplitude in ${\rm AdS}_{4}\times
S^{7}/\mathbb{Z}_{k}$ will be observed in Section 4.4.

The next correction in the momentum expansion (2.14) is the 1-loop term
$\widetilde{\mathfrak{A}}_{F^{2}|R}$, which can be determined by its unitarity
cut into tree amplitudes of gluons and gravitons (third diagram in Figure 5).
It can be viewed as the 1-loop superamplitude of the (2-derivative)
SYM$+$SUGRA, which is free of logarithmic divergences and has $s^{5\over 2}$
overall momentum scaling.

Finally, at 6-derivative order (momentum scaling $s^{3}$), we have the
possibility of an effective coupling of the schematic form
$\kappa_{R^{2}F^{2}}R^{2}\mathop{\rm Tr}\nolimits F^{2}+\cdots$, corresponding
to the supervertex given by the $k=0$ case of (A.20) (see last diagram in
Figure 6),

$\displaystyle\widetilde{\mathfrak{A}}_{R^{2}F^{2}}=\kappa_{R^{2}F^{2}}\delta^{7}(P)\delta^{8}(Q){\rm
Tr}(T^{a_{1}}T^{a_{2}})\widetilde{q}_{34}^{2}\,.$ (2.17)

To summarize, the effective theory of the M-theory orbifold is characterized
by a sequence of higher derivative coupling coefficients
$\kappa_{[F^{2}]^{2}}$, $\kappa_{F^{4}}$, $\kappa_{D^{2}[F^{2}]^{2}}$,
$\kappa_{D^{2}F^{4}}$, $\kappa_{RF^{2}}$, $\kappa_{R^{2}F^{2}}$ up to momentum
scaling $s^{3}$ and so forth for higher order corrections, that are thus far
unconstrained by supersymmetry, perturbative unitarity, and locality
considerations.

###  2.2 Constraints on the Coulomb branch

It is useful to compare the aforementioned non-Abelian effective theory to the
low-energy expansion of the $U(1)^{k-1}$ effective gauge theory on the Coulomb
branch of the M-theory orbifold vacua, where the $\vec{x}_{i}$’s in
(LABEL:alemetric) are separated. In particular, the BPS-protected terms in the
Abelian effective action, of the schematic form

$\displaystyle f_{abcd}^{(4)}(\vec{x}_{i})F^{a}F^{b}F^{c}F^{d}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}f_{ab,cd}^{(6)}(\vec{x}_{m})D^{2}(F^{a}F^{b})F^{c}F^{d},$
(2.18)

where $F^{a}$ stands for the field strength in the $a$th Cartan $U(1)$ factor,
are constrained by supersymmetry in such a way that $f^{(4)}$ and $f^{(6)}$
obey certain second-order differential equations on the Coulomb branch moduli
space [49, 62, 50, 51, 52],999See [63, 64] for closely parallel analyses of
similar constraints in 6d effective theories in the context of superstring
compactifications. and thereby can be determined given enough data at large
values of $|\vec{x}_{ij}|$ where the low energy expansion of the M-theory
effective action in weakly curved spacetimes is applicable.

It is also possible to determine the coefficient functions appearing in (2.18)
by considering further compactifications, where dual string theoretic
descriptions are available, and combine the supersymmetry constraints with
known results of the low energy expansion in a regime captured by string
perturbation theory [47, 52, 48]. For instance, via a circle compactification
to type IIA string theory, a set of arguments similar to those given in
Section 5 of [64] determine $f^{(4)}_{abcd}$ in (2.18) to be identical to the
coefficient of such a term in the 1-loop effective action of the 7d $SU(k)$
super-Yang-Mills theory on its Coulomb branch. In particular, $f^{(4)}_{abcd}$
vanishes in the limit $|\vec{x}_{i}|\to\infty$, and diverges at the origin
$\vec{x}_{i}=0$.

It is not entirely straightforward, however, to relate the ${\rm tr}F^{4}$ and
$({\rm tr}F^{2})^{2}$ effective coupling at the origin of Coulomb branch to
the Abelian $F^{4}$ effective coupling away from the origin of the Coulomb
branch. In particular, the latter contribution by itself is singular at the
origin of the Coulomb branch, and only after summing up an infinite set of (a
priori BPS-unprotected) higher-derivative contributions to the Abelian
effective theory on the Coulomb branch can we recover the amplitude at the
origin of Coulomb branch. It is plausible, nonetheless, that a connection
between the non-Abelian $\mathop{\rm Tr}\nolimits F^{4}$, $(\mathop{\rm
Tr}\nolimits F^{2})^{2}$ couplings at the origin and the Abelian $F^{4}$
effective coupling away from the origin can be made in the framework of a non-
Abelian Wilsonian effective gauge theory. Starting with a Wilsonian effective
action of the form $S_{\rm SYM}+\int d^{4}x\left(c_{\Lambda}\mathop{\rm
Tr}\nolimits F^{4}+\cdots\right)$, where the Wilsonian coefficients
$c_{\Lambda}$, etc. depend on the cutoff scale $\Lambda$,101010Strictly
speaking, the argument that follows requires a Wilsonian cutoff scheme that
respects both gauge invariance and supersymmetry. An explicit construction of
such a scheme, which is possible through the Batalin-Vilkovisky formalism [65,
66], is beyond the scope of this paper.  we can calculate both the
$\mathop{\rm Tr}\nolimits F^{4}$ term in the 4-gluon amplitude at the origin
of the Coulomb branch and the $F^{4}$ term in the Abelian amplitude away from
the origin. The structure of the supersymmetric amplitudes computed via the
unitarity method [58, 59] and power counting suggests that at the 8-derivative
order, both the non-Abelian amplitude at the origin and the Abelian amplitude
away from the origin should only receive a 1-loop contribution from the SYM
theory with Wilsonian cutoff $\Lambda$ and a tree-level contribution that is
linear in $c_{\Lambda}$. This would then imply that the $\mathop{\rm
Tr}\nolimits F^{4}$ term in the 4-gluon amplitude vanishes, given that we
already know the Abelian $F^{4}$ term coincides with the 1-loop SYM amplitude
in the $\Lambda\to\infty$ limit.

In the rest of this paper, we will pursue a different strategy that allows for
determining the non-Abelian effective couplings directly at the origin of the
Coulomb branch, namely from the CFT data of a holographic dual.

###  2.3 AdS deformation and the holographic dual

The holographic duality of interest arises from the decoupling limit of the
system of $N$ M2-branes probing the M-theory orbifold singularity
${\mathbb{C}}^{2}/{\mathbb{Z}}_{k}\times{\mathbb{R}}^{1,6}$, or equivalently
$N$ D2-branes lying within the world-volume of a stack of $k$ D6-branes in
type IIA string theory [7, 8, 9, 10, 11]. The near horizon limit of the
gravity dual leads to ${\rm AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$, where the
$\mathbb{Z}_{k}$ action leaves fixed an $S^{3}\subset S^{7}$ and results in an
orbifold singularity locus in the quotient space. Explicitly, we can
parameterize the $S^{7}$ using four complex coordinates $z_{i}$ subject to the
constraint $\sum_{i=1}^{4}\left\lvert z_{i}\right\rvert^{2}=1$, whereas the
generator of the $\mathbb{Z}_{k}$ acts by
$(z_{1},z_{2},z_{3},z_{4})\mapsto(z_{1}e^{2\pi i/k},z_{2}e^{-2\pi
i/k},z_{3},z_{4})$ as in (2.1) and the fixed $S^{3}$ is located at
$z_{1}=z_{2}=0$. Note that this is different from the case of ABJM theory [15]
engineered from M2-branes probing a different ${\mathbb{Z}}_{k}$ orbifold
singularity ${\mathbb{C}}^{4}/{\mathbb{Z}}_{k}\times{\mathbb{R}}^{1,2}$ where
the ${\mathbb{Z}}_{k}$ acts all four complex coordinates $z_{1,2,3,4}$ in the
same way (i.e. $z_{i}\to z_{i}e^{2\pi i/k}$). After taking the decoupling
limit, the holographic dual of the ABJM theory involves a fixed-point-free
${\mathbb{Z}}_{k}$ quotient of $S^{7}$ as the internal manifold.

The low energy dynamics of the massless D2-D2 and D2-D6 open strings is
governed by a three-dimensional ${\cal N}=4$ supersymmetric $U(N)$ gauge
theory with an adjoint hypermultiplet and $k$ fundamental hypermultiplets,
which admits $SU(2)_{H}\times SU(2)_{C}$ R-symmetry as well as
$G_{F}=SU(k)\times SU(2)_{F}\times U(1)$ flavor symmetry, where the $SU(k)$
factor transforms the fundamental hypermultiplets into one another, the
$SU(2)_{F}$ factor acts on the fields of the adjoint hypermultiplet, and the
$U(1)$ factor is a topological symmetry under which the only charged operators
are monopole operators. The ${\cal N}=4$ SCFT of interest, which arises at the
infrared fixed point of this gauge theory, is the holographic dual of M-theory
on the aforementioned ${\rm AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$
orbifold.111111In the special case $k=2$, the $U(1)$ topological symmetry is
enhanced to yet another $SU(2)$ factor, and the full flavor symmetry is
$SU(2)\times SU(2)\times SU(2)$. In another case $k=1$, supersymmetry is
enhanced to ${\cal N}=8$ [17], and the resulting SCFT is equivalent [67] to
the $U(N)_{1}\times U(N)_{-1}$ ABJM theory of [15]. See for instance [16, 17,
68, 69, 70, 71, 42] for various aspects of this theory and its holographic
dual.

While the $SU(k)$ flavor symmetry is not realized geometrically, the
$SU(2)_{F}\times U(1)$ as well as the $SU(2)_{H}\times SU(2)_{C}$ R-symmetry
are. When acting on the embedding coordinates $z_{i}$, the following pairs
form doublets under the various $SU(2)$ symmetries:

$\begin{split}SU(2)_{H}:\begin{pmatrix}z_{3}\\\
-z_{4}^{*}\end{pmatrix}\,,\begin{pmatrix}z_{4}\\\
z_{3}^{*}\end{pmatrix}\,,\quad SU(2)_{C}:\begin{pmatrix}z_{1}\\\
-z_{2}^{*}\end{pmatrix}\,,\begin{pmatrix}z_{2}\\\
z_{1}^{*}\end{pmatrix}\,,\quad SU(2)_{F}:\begin{pmatrix}z_{3}\\\
z_{4}\end{pmatrix}\,,\begin{pmatrix}-z_{4}^{*}\\\
z_{3}^{*}\end{pmatrix}\,.\end{split}$ (2.19)

In addition, the $U(1)$ flavor symmetry acts with charge $+1$ on $z_{1}$ and
charge $-1$ on $z_{2}$.

The curvature radius $L$ of the ${\rm AdS}_{4}$, measured in units of the 11d
Planck length $\ell_{p}$, is related to the parameters $N$ and $k$ of the
gauge theory via [16]

$\begin{split}\frac{L^{9}}{\ell_{p}^{9}}=\frac{\pi^{3}\sqrt{2}}{4}(Nk)^{3\over
2}\,,\end{split}$ (2.20)

at leading order at large $N$. The equations of motion of 11d supergravity
imply that the radius of curvature of $S^{7}/\mathbb{Z}_{k}$ is $2L$, at least
to leading order at large $L/\ell_{p}$. In the $L\to\infty$ limit, the ${\rm
AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$ reduces to the locally flat
$\mathbb{C}^{2}/\mathbb{Z}_{k}\times\mathbb{R}^{1,6}$ orbifold.

###  2.4 Relation between 7d fields and flavor current multiplets in the SCFT

Here we briefly discuss the holographic dictionary between the 7d fields (from
the graviton and the vector multiplets) on the fixed-point locus ${\rm
AdS}_{4}\times S^{3}$ of the AdS orbifold ${\rm AdS}_{4}\times
S^{7}/\mathbb{Z}_{k}$ and operators in the flavor current multiplets (denoted
as ${\mathcal{D}}[1]$ in the following) in the dual 3d ${\mathcal{N}}=4$ SCFT.
We focus on the bosonic operators for simplicity. Each ${\mathcal{D}}[1]$
flavor multiplet contains a superconformal primary scalar $J$ of dimension
$\Delta=1$ in the representation of $(\bf 3,1)$ with respect to the
$SU(2)_{H}\times SU(2)_{C}$ R-symmetry (labeled by dimensions of the $SU(2)$
irreps). The entire multiplet transforms in the adjoint representation of a
flavor group $G_{F}$. This is also known as the Higgs branch flavor multiplet,
which contains conserved $G_{F}$ currents as superconformal descendants at
level 2 together with dimension $\Delta=2$ scalars $K$ transforming as $(\bf
1,3)$ under the $SU(2)_{H}\times SU(2)_{C}$. There is one other type of flavor
multiplet with respect to the 3d ${\mathcal{N}}=4$ superconformal algebra (see
e.g. [26]), known as the Coulomb branch flavor multiplet, where the R-symmetry
groups $SU(2)_{H}$ and $SU(2)_{C}$ switch roles (i.e. scalar operators $J$ and
$K$ switch $SU(2)_{H}$ and $SU(2)_{C}$ R-symmetry quantum numbers).

Fields | Weight $\Delta$ | Spin $\ell$ | $SU(2)_{H}$ | $SU(2)_{F}$ | $SU(2)_{C}$
---|---|---|---|---|---
$\varphi^{\bm{a}}$ | 2 | 0 | 1 | 1 | 3
| 1 | 0 | 1 | 1 | 3
$A_{\mu}$ | $1$ | 0 | 3 | 1 | 1
| $2$ | 0 | 3 | 1 | 1
$g_{\mu\nu}$ | 1 | 0 | 3 | 3 | 1
$V_{\mu}^{\bm{a}}$ | 2 | 0 | 1 | 3 | 3
Table 1: KK modes of the 7d fields reduced on $S^{3}$ that correspond to
scalar operators in the ${\mathcal{D}}[1]$ flavor multiplets. The
representations with respect to the $SU(2)_{H}\times SU(2)_{C}$ R-symmetry and
$SU(2)_{F}$ Higgs branch flavor symmetry are listed. The two rows for
$\varphi^{\bm{a}}$ and $A_{\mu}$ differ in the choice of the asymptotic
boundary condition on AdS4.

In Table 1, we list the 7d fields whose KK reductions on $S^{3}$ produce the
${\rm AdS}_{4}$ dual scalar fields for the $J$ and $K$ type operators of 3d
${\mathcal{N}}=4$ flavor multiplets. Note that for scalar fields of mass $m$
in the range $-{9\over 4}<m^{2}L^{2}<-{5\over 4}$ there are two boundary
conditions compatible with unitarity, subjected to further constraints from
supersymmetry [72, 73, 74, 70]. Here, the relevant 7d fields from the 11d
supergravity reduced on the orbifold singularity
${\mathbb{C}}^{2}/{\mathbb{Z}}_{k}$ consist of the 7d ${\mathcal{N}}=1$
graviton multiplet, $SU(k)$ vector multiplets from the massless modes
supported at the singularity, and a $U(1)$ vector multiplet for the residue
$U(1)$ isometry of the orbifold (for $k>2$, see Footnote 11 for a related
comment). The KK reduction of general vector multiplets on $S^{3}$ has been
worked out in [75]. The leading KK modes indeed give the holographic dual for
the $J$ and $K$ type operators (see Table 3 in [75]). Consistency with
supersymmetry requires that the $SU(k)$ vector multiplet and the $U(1)$ vector
multiplet have opposite boundary conditions on AdS4, as they correspond to
Higgs and Coulomb branch symmetries respectively [70].

As mentioned above, the SCFT admits in addition an $SU(2)_{F}$ flavor symmetry
of the Higgs branch type, which originates from the 7d graviton multiplet in
the bulk. In turn, this 7d graviton multiplet can be obtained by reducing the
11d graviton multiplet on the $S^{3}$ transverse to the orbifold singularity.
Thus, we can infer the relevant scalar KK modes that listed in Table 1 using
the known results of the reduction of the 11d fields on $S^{7}$ [76, 77, 78,
79], and then imposing the $\mathbb{Z}_{k}$ quotient. In the $S^{7}$
reduction, [76, 77, 78, 79] found that the lowest supermultiplet of KK modes
contains scalar operators of dimensions $\Delta=1$ and $\Delta=2$ transforming
in different $35$-dimensional representations of the $SO(8)_{R}$ R-symmetry
which arise from the reduction on $S^{7}$ of the 11d graviton and three-form
gauge potential. In the convention in which the ${\mathcal{N}}=8$ supercharges
transform in the ${\bf 8}_{v}$ of $SO(8)_{R}$, we can take the $\Delta=1$
scalars to transform in ${\bf 35}_{c}$ and the $\Delta=2$ scalars to transform
in ${\bf 35}_{s}$. These representations split under an $SU(2)_{H}\times
SU(2)_{C}\times SU(2)_{1}\times SU(2)_{2}$ subgroup of $SO(8)_{R}$ as

$\displaystyle{\bf 8}_{v}$ $\displaystyle=({\bf 2},{\bf 2},{\bf 1},{\bf
1})\oplus({\bf 1},{\bf 1},{\bf 2},{\bf 2})\,,$ (2.21) $\displaystyle{\bf
35}_{c}$ $\displaystyle=({\bf 3},{\bf 1},{\bf 3},{\bf 1})\oplus({\bf 1},{\bf
3},{\bf 1},{\bf 3})\oplus({\bf 2},{\bf 2},{\bf 2},{\bf 2})\oplus({\bf 1},{\bf
1},{\bf 1},{\bf 1})\,,$ $\displaystyle{\bf 35}_{s}$ $\displaystyle=({\bf
1},{\bf 3},{\bf 3},{\bf 1})\oplus({\bf 3},{\bf 1},{\bf 1},{\bf 3})\oplus({\bf
2},{\bf 2},{\bf 2},{\bf 2})\oplus({\bf 1},{\bf 1},{\bf 1},{\bf 1})\,.$

Note that the $SU(2)_{1}$ subgroup above is identified with the 3d
${\mathcal{N}}=4$ $SU(2)_{F}$ flavor symmetry whereas $SU(2)_{2}$ is broken by
the orbifold to $U(1)$ (for $k>2$). For determining the scalar operators in
the half-maximal supergravity in 7d, it suffices to restrict to
$SU(2)_{2}$-invariant fields in the last two lines of (2.21), leaving only the
first and last summands in these equations. The first summands on the RHS of
the second and third lines correspond, respectively, to the desired $J$ and
$K$ operators in the $SU(2)_{F}$ flavor multiplet. The last summand in the
second line corresponds to the superconformal primary scalar in the 3d
${\mathcal{N}}=4$ stress tensor multiplet.

##  3 Superconformal kinematics

The main observables of our interest in the 3D ${\cal N}=4$ SCFT are the
4-point correlation functions of flavor multiplets, which are dual to
amplitudes of gravitons and gluons in the ${\rm AdS}_{4}\times
S^{7}/\mathbb{Z}_{k}$ orbifold.

###  3.1 Correlators and multiplet structure

We will work with the $\Delta=1$ scalar superconformal primaries in the flavor
multiplets of type ${\mathcal{D}}[1]$ associated with the $SU(k)$ and
$SU(2)_{F}$ symmetries,

$\displaystyle
J^{a}(\vec{x},y)\equiv\phi^{a}_{{\alpha}{\beta}}(\vec{x})y^{\alpha}y^{\beta},$
(3.1)
$\displaystyle\widetilde{J}^{A}(x,\vec{y})\equiv\widetilde{\phi}^{A}_{{\alpha}{\beta}}(\vec{x})y^{\alpha}y^{\beta},$

where $a$ is an $SU(k)$ adjoint index, $A$ an $SU(2)_{F}$ adjoint index,
${\alpha}$ an $SU(2)_{H}$ doublet index, and $y^{\alpha}$ is an auxiliary
“spinor polarization” variable with respect to $SU(2)_{H}$. In particular, the
$SU(k)$ current $j^{a}_{\mu}$ is a superconformal descendant of
$\phi^{a}_{{\alpha}{\beta}}$, whereas the $SU(2)_{F}$ current
$\widetilde{j}^{A}_{\mu}$ is a descendant of
$\widetilde{\phi}^{A}_{{\alpha}{\beta}}$. We normalize the operators
(LABEL:yydef) such that their two-point functions are

$\begin{split}\langle
J^{a}(\vec{x}_{1},y_{1})J^{b}(\vec{x}_{2},y_{2})\rangle&=\frac{\langle
y_{1},y_{2}\rangle^{2}}{|\vec{x}_{12}|^{2}}\mathop{\rm
tr}\nolimits(T^{a}T^{b})\,,\\\
\langle\widetilde{J}^{A}(\vec{x}_{1},y_{1})\widetilde{J}^{B}(\vec{x}_{2},y_{2})\rangle&=\frac{\langle
y_{1},y_{2}\rangle^{2}}{|\vec{x}_{12}|^{2}}\delta^{AB}\,,\end{split}$ (3.2)

where $\langle
y_{1},y_{2}\rangle\equiv\epsilon_{{\alpha}{\beta}}y_{1}^{\alpha}y_{2}^{\beta}$
and $\vec{x}_{12}\equiv\vec{x}_{1}-\vec{x}_{2}$.

The correlators of interest are constrained by conformal and R-symmetries to
be121212We will not consider the 4-point function of only $\widetilde{J}$’s in
this work.

$\begin{split}\langle
J^{a}(\vec{x}_{1},y_{1})J^{b}(\vec{x}_{2},y_{2})J^{c}(\vec{x}_{3},y_{3})J^{d}(\vec{x}_{4},y_{4})\rangle&=\frac{\langle
y_{1},y_{2}\rangle^{2}\langle
y_{3},y_{4}\rangle^{2}}{|\vec{x}_{12}|^{2}|\vec{x}_{34}|^{2}}\mathcal{G}^{abcd}(U,V;\alpha)\,,\\\
\langle
J^{a}(\vec{x}_{1},y_{1})J^{b}(\vec{x}_{2},y_{2})\widetilde{J}^{C}(\vec{x}_{3},y_{3})\widetilde{J}^{D}(\vec{x}_{4},y_{4})\rangle&=\frac{\langle
y_{1},y_{2}\rangle^{2}\langle
y_{3},y_{4}\rangle^{2}}{|\vec{x}_{12}|^{2}|\vec{x}_{34}|^{2}}\mathcal{G}^{abCD}(U,V;\alpha)\,,\\\
\end{split}$ (3.3)

where the conformal cross-ratios $U,V$ and the R-symmetry cross-ratio
${\alpha}$ are defined as

$\begin{split}U\equiv\frac{\vec{x}_{12}^{2}\vec{x}_{34}^{2}}{\vec{x}_{13}^{2}\vec{x}_{24}^{2}}\equiv
z\bar{z}\,,\qquad
V\equiv\frac{\vec{x}_{14}^{2}\vec{x}_{23}^{2}}{\vec{x}_{13}^{2}\vec{x}_{24}^{2}}\equiv(1-z)(1-\bar{z})\,,\qquad\alpha=\frac{\langle
y_{1},y_{3}\rangle\langle y_{2},y_{4}\rangle}{\langle
y_{1},y_{2}\rangle\langle y_{3},y_{4}\rangle}\,.\end{split}$ (3.4)

To avoid cluttering the notation, the two types of correlators are both
denoted ${\mathcal{G}}$, distinguished only through their flavor indices
($abcd$ versus $abCD$).

With this way of representing the correlators, the superconformal Ward
identities can be expressed very compactly as [80]

$\begin{split}\left(z\partial_{z}-\frac{1}{2}\alpha\partial_{\alpha}\right)\mathcal{G}(U,V;\alpha)\big{|}_{\alpha=z^{-1}}=0\,,\end{split}$
(3.5)

where $z$ is defined as in (3.4), and ${\mathcal{G}}$ stands for either
${\mathcal{G}}^{abcd}$ or ${\mathcal{G}}^{abCD}$. The superconformal block
decomposition of the correlator takes the form

$\begin{split}{\mathcal{G}}(U,V;\alpha)=\sum_{({\mathcal{M}},R)\in{\rm
Sym}^{2}({\mathcal{D}}[1],{\bf
adj})}{\mathtt{T}}_{R}\lambda^{2}_{{\mathcal{M}},R}\mathfrak{G}_{\mathcal{M}}(U,V;\alpha)\,,\end{split}$
(3.6)

where ${\cal M}$ and $R$ label irreducible representations of the
superconformal group and the flavor group that appear in the OPE.
${\mathtt{T}}_{R}$ are invariant tensors of the flavor group that carry the
same suppressed flavor indices as ${\mathcal{G}}$ does. The corresponding OPE
coefficients are denoted by $\lambda_{{\mathcal{M}},R}$, and
$\mathfrak{G}_{\mathcal{M}}$ are superconformal blocks.

The tensor product of a pair of ${\mathcal{D}}[1]$ flavor multiplets can be
decomposed into irreducible superconformal and flavor group representations,
in the notation of [81], as

$\begin{split}{\rm Sym}^{2}(\mathcal{D}[1],{\bf adj})&=1+(\mathcal{D}[1],{\bf
adj})+\sum_{R\in\text{sym}}(\mathcal{D}[2],R)+(\mathcal{B}[0],1)+\sum_{R\in\text{asym}}\sum_{\text{odd
}j}(\mathcal{B}[1]_{j},R)\\\ &~{}~{}~{}+\sum_{R\in\text{sym}}\sum_{\text{even
}j}\sum_{\Delta>j+1}(\mathcal{L}[0]_{\Delta,j},R)\,.\end{split}$ (3.7)

Here, 1 stands for the trivial (identity) representation. $\mathcal{B}[0]$
(denoted as ${\mathcal{B}}[0]_{0}$ in [81]) is the stress tensor multiplet,
whose superprimary is a $\Delta=1$ scalar.131313For free theories we would
also have the ${\mathcal{B}}[0]_{\ell}$ multiplets in the notation of [81]),
corresponding to conserved currents of higher spins $\ell+2$.
$({\mathcal{D}}[2],R)$ are half-BPS multiplets, whose superprimaries are
$\Delta=2$ scalars in flavor irreps $R$ in the symmetric product of two
adjoint representations. $({\mathcal{B}}[1]_{j},R)$ are $\Delta=j+2$ semi-
short multiplets of odd integer spin $j$, and flavor irrep $R$ in the anti-
symmetric product of two adjoints. Finally, $(\mathcal{L}[0]_{\Delta,j},R)$
are long multiplets of even integer spin $j$ and flavor irrep $R$ in the
symmetric product of two adjoints, subject to the unitarity bound
$\Delta>j+1$.

The explicit formulae for the superblocks $\mathfrak{G}_{{\mathcal{M}}}$ of
the BPS multiplets
${\mathcal{M}}={\mathcal{D}}[1],{\mathcal{D}}[2],{\mathcal{B}}[0]$, relevant
for the observables extracted in this paper, are given in Appendix B.

###  3.2 Central charges

The central charges $c_{J},c_{\widetilde{J}}$ and $c_{T}$ are defined through
the 2-point functions of the canonically normalized141414Note that the
normalization of $j^{a}_{\mu}$ here differs from that of [82] by a factor
$2^{-{1\over 2}}$ due to our convention of the $SU(k)$ generators $T^{a}$.
flavor currrents $j^{a}_{\mu}$, $\widetilde{j}^{A}_{\mu}$, and the stress
tensor $T_{\mu\nu}$151515We use the conventions where $c_{T}=1$ for a free
scalar or Majorana fermion. This differs from the $c_{T}$ normalization of
[81] by the factor $3\over 2$.

$\displaystyle\langle
j_{\mu}^{a}(x)j_{\nu}^{b}(0)\rangle=\frac{c_{J}}{(4\pi)^{2}}\frac{{\rm
Tr}(T^{a}T^{b})I_{\mu\nu}(x)}{x^{4}}\,,\qquad\langle\widetilde{j}_{\mu}^{A}(x)\widetilde{j}_{\nu}^{B}(0)\rangle=\frac{c_{\widetilde{J}}}{(4\pi)^{2}}\frac{\delta^{AB}I_{\mu\nu}(x)}{x^{4}}\,,$
(3.8) $\displaystyle\langle
T_{\mu\nu}(x)T_{\sigma\rho}(0)\rangle=\frac{c_{T}}{(4\pi)^{2}}\frac{\left[\left(\frac{1}{2}I_{\mu\sigma}(x)I_{\nu\rho}(x)+I_{\mu\rho}(x)I_{\nu\sigma}(x)\right)-\frac{\delta_{\mu\nu}\delta_{\sigma\rho}}{3}\right]}{x^{6}}\,,$

where $I_{\mu\nu}\equiv\delta_{\mu\nu}-2\frac{x_{\mu}x_{\nu}}{x^{2}}$.

###  3.3 Kinematics of the $SU(k)$ current 4-point function
${\mathcal{G}}^{abcd}$

The decomposition (3.6) for the $SU(k)$ current 4-point function
${\mathcal{G}}^{abcd}$ involves the representations (3.7) with $R$ labeling
one of the following $SU(k)$ irreps161616Here, ${\bf S}$ is the irrep with
Dynkin label $[20\cdots 02]$ for $k>2$ and $[4]$ for $k=2$, and ${\bf
S}^{\prime}$ is the irrep with Dynkin label $[010\cdots 010]$ for $k>3$ and
$[020]$ for $k=3$, and ${\bf A}$ is the irrep with Dynkin label $[20\cdots
010]$ for $k>3$ and $[30]$ for $k=3$.

$\begin{split}{\bf adj}\otimes{\bf adj}={\bf 1}_{s}\oplus{\bf S}_{s}\oplus{\bf
S}^{\prime}_{s}\oplus{\bf A}_{a}\oplus{\bf\bar{A}}_{a}\oplus{\bf
adj}_{a}\oplus{\bf adj}_{s}\,,\end{split}$ (3.9)

for $k\geq 4$, where the subscript $s$ or $a$ refers to irreps appearing in
the symmetric or anti-symmetric tensor product, respectively. In the special
case $k=3$ we should remove ${\bf S}^{\prime}_{s}$, which does not exist,
whereas in the case $k=2$ we keep only ${\bf 1}_{s}$, ${\bf adj}_{a}$, and
${\bf S}$.

It will be convenient to adopt a notation in which all values of $k$ are
treated on equal footing by rewriting (3.6) as

$\begin{split}{\mathcal{G}}^{abcd}(U,V;\alpha)&=\sum_{{\mathcal{M}}\in{\mathcal{D}}[1]\times{\mathcal{D}}[1]}\sum_{R={\bf
1},{\bf 2},{\bf 3},{\bf 4},{\bf adj}_{a},{\bf
asym}_{a}}{\mathtt{T}}^{abcd}_{R}\lambda^{2}_{{\mathcal{M}},R}\mathfrak{G}_{\mathcal{M}}(U,V;\alpha)\,,\\\
\end{split}$ (3.10)

where, by a slight abuse of notation, we label by $R$ the following invariant
tensors ${\mathtt{T}}_{R}$:

$\begin{split}&{\mathtt{T}}_{\bf
1}^{abcd}={\mathtt{A}}^{abcd}\,,\qquad{\mathtt{T}}_{\bf
2}^{abcd}={\mathtt{A}}^{acdb}+{\mathtt{A}}^{adbc}\,,\quad{\mathtt{T}}_{\bf
3}^{abcd}={\mathtt{B}}^{abcd}+{\mathtt{B}}^{acdb}\,,\quad{\mathtt{T}}_{\bf
4}^{abcd}={\mathtt{B}}^{adbc}\,,\\\ &{\mathtt{T}}_{{\bf
adj}_{a}}^{abcd}={\mathtt{B}}^{abcd}-{\mathtt{B}}^{acdb}\,,\qquad{\mathtt{T}}_{{\bf
asym}_{a}}^{abcd}={\mathtt{A}}^{abcd}-{\mathtt{A}}^{acdb}\,,\end{split}$
(3.11)

where ${\mathtt{A}}^{abcd}$ and ${\mathtt{B}}^{abcd}$ are the color factors
defined in (2.10) and (2.7).

In this convention, the flavor current and stress tensor OPE coefficients are
related, by conformal Ward identities, to the central charge $c_{J}$ and
$c_{T}$ as [83]

$\begin{split}\lambda^{2}_{\mathcal{D}[1],{\bf
adj}_{a}}=\frac{16}{c_{J}}\,,\qquad\lambda^{2}_{\mathcal{B}[0],{\bf
1}}=\frac{64}{c_{T}}\,.\\\ \end{split}$ (3.12)

###  3.4 Kinematics of the $SU(k)$-$SU(2)_{F}$ mixed 4-point function
${\mathcal{G}}^{abCD}$

The decomposition (3.6) for the mixed current multiplet 4-point function
${\mathcal{G}}^{abCD}$, in the $ab\to CD$ channel, involves only singlets of
the flavor group (“$R=1$”). This immediately excludes the superconformal
representations ${\mathcal{B}}[1]_{j}$ and ${\mathcal{D}}[1]$ from the OPE.
The only flavor invariant tensor that appear in this OPE channel is

$\displaystyle{\mathtt{T}}_{\bf 1}^{abCD}=\mathop{\rm
Tr}\nolimits(T^{a}T^{b})\delta^{CD}\,,$ (3.13)

with the $\mathcal{B}[0]$ channel coefficient

$\displaystyle\lambda^{2}_{{\mathcal{B}}[0]}={64\over c_{T}}\,.$ (3.14)

In the $aC\to bD$ or $aD\to bC$ channel, on the other hand, $R$ is necessarily
the adjoint representation with respect to $SU(k)\times SU(2)_{F}$, and the
superconformal representations ${\mathcal{B}}[1]_{j}$, rather than
${\mathcal{B}}[0]$, appear in the OPE.

##  4 Mellin amplitudes in the large $N$ expansion

The holographic interpretation of the CFT correlators of interest is most
easily understood through its Mellin transform [18, 19, 20], whose $1/N$
expansion is naturally organized through the derivative expansion of a bulk
effective field theory [84].

To begin with, at strictly $N=\infty$ the correlators ${\cal G}^{abcd}$ and
${\cal G}^{abCD}$ reduce to those of the generalized free field theory (GFFT),

$\begin{split}{\cal
G}^{abcd}_{\text{GFFT}}(U,V;\alpha)={\mathtt{A}}^{abcd}+\alpha^{2}U{\mathtt{A}}^{dbca}+(1-\alpha)^{2}\frac{U}{V}{\mathtt{A}}^{cbad}\,,\quad{\cal
G}^{abCD}_{\text{GFFT}}(U,V;\alpha)={\mathtt{T}}_{\bf 1}^{abCD}\,.\end{split}$
(4.1)

The (reduced) Mellin amplitudes $M^{abcd}(s,t;{\alpha})$ and
$M^{abCD}(s,t;{\alpha})$ are defined by first subtracting from
${\mathcal{G}}(U,V;{\alpha})$ the GFFT correlator, and rewriting the
“connected correlator”
$\mathcal{G}_{\text{conn}}\equiv\mathcal{G}-\mathcal{G}_{\text{GFFT}}$ as the
integral transform

$\begin{split}\mathcal{G}_{\text{conn}}(U,V;\alpha)=\int\frac{ds\,dt}{(4\pi
i)^{2}}U^{\frac{s}{2}}V^{\frac{t}{2}-1}M(s,t;\alpha)\Gamma^{2}\left[1-\frac{s}{2}\right]\Gamma^{2}\left[1-\frac{t}{2}\right]\Gamma^{2}\left[1-\frac{u}{2}\right]\,,\end{split}$
(4.2)

where $u\equiv 4-s-t$, and the $s$, $t$ integration contours run parallel to
the imaginary axis with ${\rm Re}(s),{\rm Re}(t),{\rm Re}(u)<2$, so that all
poles of the Gamma functions lie on one side or the other of the contours.
$M(s,t;{\alpha})$ is expected to be well-defined, order by order in the $1/N$
expansion, with suitable analyticity assumptions that are motivated from the
bulk effective field theory. These assumptions, combined with the
superconformal Ward identities and crossing invariance, impose nontrivial
constraints on the Mellin amplitude at each order in $1/N$, dubbed the
“analytic bootstrap” [85].

###  4.1 Analytic bootstrap conditions and the large radius limit

The superconformal Ward identities (3.5) can be translated into the Mellin
representation via the replacement rule

$\begin{split}{\cal G}(U,V;\alpha)\to
M(s,t)\,,~{}~{}~{}U\partial_{U}\to\widehat{U\partial_{U}}\,,~{}~{}~{}V\partial_{V}\to\widehat{V\partial_{V}}\,,~{}~{}~{}U^{m}V^{n}\to\widehat{U^{m}V^{n}}\,.\end{split}$
(4.3)

Here the hatted operators are defined to act on the Mellin amplitude $M(s,t)$
as [86]

$\begin{split}\widehat{U\partial_{U}}{M}(s,t)&=\frac{s}{2}{M}(s,t)\,,\\\
\widehat{V\partial_{V}}{M}(s,t)&=\left[\frac{t}{2}-1\right]{M}(s,t)\,,\\\
\widehat{U^{m}V^{n}}{M}(s,t)&={M}(s-2m,t-2n)\left(1-\frac{s}{2}\right)_{m}^{2}\left(1-\frac{t}{2}\right)_{n}^{2}\left(1-\frac{u}{2}\right)_{-m-n}^{2}\,,\end{split}$
(4.4)

where $(x)_{n}\equiv\Gamma(x+n)/\Gamma(x)$ is the Pochhammer symbol.

Crossing invariance of the CFT correlators amounts to

$\displaystyle
M^{abcd}(s,t;\alpha)=M^{bacd}(s,u;1-\alpha)=(\alpha-1)^{2}M^{cbad}(t,s;\alpha/(\alpha-1))\,,$
(4.5) $\displaystyle M^{abCD}(s,t;\alpha)=M^{baCD}(s,u;1-\alpha)\,.$

At each order in the $1/N$ expansion, the leading large $s,t$ asymptotic
behavior of the Mellin amplitude is controlled by the derivative expansion of
the bulk effective field theory. Their precise relation is captured by the
large radius limit [20], where the Mellin amplitude turns into a simple
integral transform of a suitable “flat space” scattering amplitude

$\begin{split}32\pi^{9\over
2}\lim_{N\to\infty}L^{3}M^{abcd}(L^{2}s,L^{2}t;\alpha)&=(u+s\alpha)^{2}\int_{0}^{\infty}d\beta\,\beta^{3\over
2}e^{-\beta}{\cal A}^{abcd}\left({2\beta}s,{2\beta}t\right)\,,\\\
\frac{128\pi^{13\over
2}}{3k}\lim_{N\to\infty}L^{7}M^{abCD}(L^{2}s,L^{2}t;\alpha)&=\delta^{CD}(u+s\alpha)^{2}\int_{0}^{\infty}d\beta\,\beta^{3\over
2}e^{-\beta}\widetilde{\cal
A}^{ab}\left({2\beta}s,{2\beta}t\right)\,.\end{split}$ (4.6)

On the LHS, $L$ is the AdS radius, related to $N$ by (2.20). Note importantly
that $M(s,t;{\alpha})$ itself has, of course, highly nontrivial $N$-dependence
which is not shown explicitly in (4.6). The power of $L$ is dictated by the
number of transverse dimensions of the relevant scattering amplitudes, whereas
the overall factor is fixed by comparison with the reduced SYM+SUGRA tree-
level amplitudes (4.11) and (LABEL:gravampol) presented in Section 4.2.

On the RHS, ${\mathcal{A}}^{abcd}(s,t)$ comes from the 7d scattering amplitude
of gluons supported at the M-theory orbifold singularity, with momenta
restricted to a 4d Minkowskian subspace and polarization vectors restricted to
the transverse 3d. A special choice of polarizations [33, 87], resulting from
the flat space limit of the wave functions of the KK modes in ${\rm
AdS}_{4}\times S^{3}$ produces the factor $(u+s\alpha)^{2}$.171717An analogous
factor for the scattering of graviton KK modes in M-theory on ${\rm
AdS}_{4}\times S^{7}$ was derived in Appendix C of [33].

Similarly, $\widetilde{\mathcal{A}}^{ab}(s,t)$ comes from the scattering
amplitude of a pair of 7d gluons with a pair of 11d gravitons, whose momenta
are restricted to 4d and polarizations in the transverse 3d. Their precise
relation, which involves a choice of polarizations that corresponds to the
integration over KK wave functions and produces the factor
$\delta^{CD}(u+s\alpha)^{2}$, will be explained in the next subsection.

###  4.2 Relation to 7d amplitudes

We now explain the precise relation between the 7d gluon and graviton
amplitudes in the locally flat M-theory orbifold, considered in Section 2.1,
and ${\mathcal{A}}^{abcd}(s,t)$, $\widetilde{\mathcal{A}}^{ab}(s,t)$ that
arise in the flat space limit (4.6) of Mellin amplitudes in ${\rm
AdS}_{4}\times S^{3}$.

We begin with the four-gluon superamplitude (2.8), and restrict to the bosonic
gluon states with momenta $p_{i}$ in a 4d subspace, and polarization vectors
$\epsilon_{i}$ restricted to the 3 transverse directions. The integration over
the relevant KK wave functions in ${\rm AdS}_{4}\times S^{3}$ amounts to
further specializing the polarization vector $\epsilon_{i}$ in a way that is
correlated with the $SU(2)_{H}$ spinor polarization $y^{\alpha}$ in
(LABEL:yydef), via [87]

$\displaystyle(\epsilon_{i})^{{\alpha}{\beta}}=y_{i}^{\alpha}y_{i}^{\beta},$
(4.7)

where we have used the 3d bispinor notation for $\epsilon_{i}$. In terms of
the super helicity variables introduced in Appendix A.1, this amounts to
specializing to the super spinor helicities

$\displaystyle\lambda_{i({\mathfrak{a}},{\alpha})(+,\gamma)}=\lambda_{i\mathfrak{a}}\epsilon_{{\alpha}\gamma}\,,~{}~{}~{}\lambda_{i({\mathfrak{a}},{\alpha})(-,\gamma)}=\lambda_{i(\dot{\mathfrak{a}},{\alpha})(+,\gamma)}=0\,,~{}~{}~{}\lambda_{i(\dot{\mathfrak{a}},{\alpha})(-,\gamma)}=\widetilde{\lambda}_{i\dot{\mathfrak{a}}}\epsilon_{{\alpha}\gamma}\,,$
(4.8)
$\displaystyle\eta_{i(\varkappa,\gamma)}=\eta_{i\varkappa}y_{i\gamma}\,,$

where we have traded the $SO(1,6)$ Lorentz spinor index ${\alpha}$ with
$SO(1,3)\times SO(3)$ spinor indices $({\mathfrak{a}},{\alpha})$ and
$(\dot{\mathfrak{a}},{\alpha})$, and $SO(5)$ little group spinor index $I$
with $SO(2)\times SO(3)$ indices $(\varkappa=\pm,\gamma)$.
$(\lambda_{i{\mathfrak{a}}},\widetilde{\lambda}_{i\dot{\mathfrak{a}}},\eta_{i\varkappa})$
are the 4d super-spinor-helicity variables associated with the $i$-th
particle. The super momentum of the $i$-th particle is now written as

$\displaystyle
q_{i({\mathfrak{a}},{\alpha})}=\lambda_{i\mathfrak{a}}\eta_{i+}y_{i{\alpha}},~{}~{}~{}q_{i(\dot{\mathfrak{a}},{\alpha})}=\widetilde{\lambda}_{i\dot{\mathfrak{a}}}\eta_{i-}y_{i{\alpha}}\,,$
(4.9)

resulting in the polarization factor below which can be fixed easily by going
to a specific kinematic frame (see also [87])

$\displaystyle\delta^{8}(Q)\propto\left(\prod_{i=1}^{4}\eta_{i+}\eta_{i-}\right)\det_{i,{\mathfrak{a}}{\alpha}}(\lambda_{i{\mathfrak{a}}}y_{i{\alpha}})\det_{i,{\dot{\mathfrak{a}}}{\beta}}(\widetilde{\lambda}_{i\dot{\mathfrak{a}}}y_{i{\beta}})=\left(\prod_{i=1}^{4}\eta_{i+}\eta_{i-}\right)\langle
y_{1},y_{2}\rangle^{2}\langle y_{3},y_{4}\rangle^{2}(u+s{\alpha})^{2}.$ (4.10)

Therefore, up to an overall normalization convention, the reduced 4d amplitude
${\mathcal{A}}^{abcd}$ appearing in (4.6) is obtained from the superamplitude
$\mathfrak{A}$ (2.8) by simply stripping off the $\delta^{8}(Q)$ factor. We
adopt the normalization convention in which the SYM tree-level contribution is

$\begin{split}{\mathcal{A}}^{abcd}_{F^{2}}(s,t)&=2g_{\rm
YM}^{2}\left(\frac{{\mathtt{B}}^{abcd}}{st}+\frac{{\mathtt{B}}^{acdb}}{su}+\frac{{\mathtt{B}}^{adbc}}{tu}\right)\,.\end{split}$
(4.11)

Similarly, for the two-gluon-two-graviton amplitude (2.14), we will restrict
to bosonic gluon and graviton states with momenta $p_{i}$ in a 4d subspace,
and polarization tensors restrict to 3 transverse directions along the
orbifold locus. The effect of integration over KK wave functions on ${\rm
AdS}_{4}\times S^{3}$ leads to, in the flat space limit, the restrictions
(LABEL:lambdarestr), together with

$\displaystyle\widetilde{\eta}_{i(\varkappa,\gamma)}=\widetilde{\eta}_{i\varkappa}\widetilde{y}_{i\gamma}\,,$
(4.12)

where $\widetilde{y}_{3},\widetilde{y}_{4}$ are $SU(2)_{F}$ polarization
spinors. This results in the polarization factor below (for the restricted
super-spinor-helicity variables),

$\displaystyle\delta^{8}(Q)\left.\partial_{\widehat{\eta}_{1}}^{2}\partial_{\widehat{\eta}_{2}}^{2}\prod_{{\alpha}=1}^{8}\left(\lambda^{\alpha}_{1I}\widehat{\eta}_{1}^{I}+\lambda^{\alpha}_{2I}\widehat{\eta}_{2}^{I}+\widetilde{q}_{3}^{\alpha}+\widetilde{q}_{4}^{\alpha}\right)\right|_{\widetilde{\eta}_{3}^{2}|_{\bf
5}\widetilde{\eta}_{4}^{2}|_{\bf 5}}$ (4.13)
$\displaystyle\propto\left(\prod_{i=1}^{4}\eta_{i+}\eta_{i-}\right)\det_{i,{\mathfrak{a}}{\alpha}}(\lambda_{i{\mathfrak{a}}}y_{i{\alpha}})\det_{i,{\dot{\mathfrak{a}}}{\beta}}(\widetilde{\lambda}_{i\dot{\mathfrak{a}}}y_{i{\beta}})\cdot\left(\prod_{i=3,4}\widetilde{\eta}_{i+}\widetilde{\eta}_{i-}\right)tu\langle\widetilde{y}_{3},\widetilde{y}_{4}\rangle^{2}$
$\displaystyle=\,\left(\prod_{i=1}^{4}\eta_{i+}\eta_{i-}\right)\left(\prod_{i=3,4}\widetilde{\eta}_{i+}\widetilde{\eta}_{i-}\right)\langle
y_{1},y_{2}\rangle^{2}\langle
y_{3},y_{4}\rangle^{2}\langle\widetilde{y}_{3},\widetilde{y}_{4}\rangle^{2}tu(u+s{\alpha})^{2}\,,$

where the first line implements the projection of the four-point
supersymmetric invariant ${\delta}^{16}(Q)$ for the 7d maximal supergravity
(see [48] for details) to the 2-gluon-2-graviton component (see Footnote 8).
The second line of (LABEL:gravampol) follows easily after using the
$SO(5)_{R}$ invariance of the four-point invariant ${\delta}^{16}(Q)$ with
momentum conservation [48] which allows us to replace
$\partial^{2}_{\hat{\eta}_{2}}$ by $\partial^{2}_{\hat{\eta}_{1}}$ in the
first line. To obtain the final equality in (LABEL:gravampol) we have used
(4.10).

Applying (LABEL:gravampol) to $\widetilde{\mathfrak{A}}_{R}$ (2.15), we obtain
the leading low energy contribution to the reduced 4d amplitude
$\widetilde{\mathcal{A}}^{ab}$ appearing in (4.6),

$\begin{split}\widetilde{\mathcal{A}}^{ab}_{R}(s,t)&=-\kappa^{2}{\rm
Tr}(T^{a}T^{b})\,{1\over s}\,.\end{split}$ (4.14)

Note that this reduced amplitude no longer has the gluon pole in the $t$ or
$u$ channel.

###  4.3 Structure of the four-gluon Mellin amplitude $M^{abcd}$

The analytic bootstrap conditions constrain the $1/N$ or large radius
expansion of the gluon Mellin amplitude $M^{abcd}(s,t;\alpha)$ to be of the
form

$\begin{split}M^{abcd}&=\frac{M^{abcd}_{F^{2}}}{c_{J}}+\frac{M^{abcd}_{F^{2}|F^{2}}}{c_{J}^{2}}+\frac{\kappa_{[F^{2}]^{2}}M^{abcd}_{[F^{2}]^{2}}+\kappa_{F^{4}}M^{abcd}_{F^{4}}}{N^{7\over
6}}\\\
&{}+\Bigg{[}\frac{M^{abcd}_{R}}{c_{T}}+\frac{M^{abcd}_{F^{2}|F^{2}|F^{2}}}{c_{J}^{3}}+\frac{\widetilde{\kappa}_{1}M^{abcd}_{[F^{2}]^{2}}+\widetilde{\kappa}_{2}M^{abcd}_{F^{4}}+\widetilde{\kappa}_{3}M^{abcd}_{D^{2}[F^{2}]^{2}}+\widetilde{\kappa}_{4}M^{abcd}_{D^{2}F^{4}}}{N^{3\over
2}}\Bigg{]}\\\ &{}+\frac{\log
c_{J}}{c_{J}^{3}}\Big{[}\kappa^{\prime}_{\log,[F^{2}]^{2}}M^{abcd}_{[F^{2}]^{2}}+\kappa^{\prime}_{\log,F^{4}}M^{abcd}_{F^{4}}+\kappa_{\log,D^{2}[F^{2}]^{2}}M^{abcd}_{D^{2}[F^{2}]^{2}}+\kappa_{\log,D^{2}F^{4}}M^{abcd}_{D^{2}F^{4}}\Big{]}\\\
&{}+O(N^{-{11\over 6}})\,,\end{split}$ (4.15)

where $c_{J}$ and $c_{T}$, defined by (LABEL:cJcT) with the explicit large $N$
expressions given in (5.20) and (5.22), scale with $N$ as $\sim N^{1\over 2}$
and $\sim N^{3\over 2}$ respectively. $M_{F^{2}}^{abcd}$,
$M_{F^{2}|F^{2}}^{abcd}$, and $M_{F^{2}|F^{2}|F^{2}}^{abcd}$ are the tree-
level, one-loop, and (the finite part of) two-loop Yang-Mills Mellin
amplitudes, respectively, with the Yang-Mills coupling constant stripped off.
$M^{abcd}_{[F^{2}]^{2}}$ and $M^{abcd}_{F^{4}}$ are contributions to the gluon
Mellin amplitude from supersymmetric effective couplings of the schematic form
$(\mathop{\rm Tr}\nolimits F^{2})^{2}$ and $\mathop{\rm Tr}\nolimits F^{4}$,
respectively. Likewise, $M^{abcd}_{D^{2}[F^{2}]^{2}}$ and
$M^{abcd}_{D^{2}F^{4}}$ capture the contribution from the analogous effective
couplings with two extra derivatives. The coefficients $\kappa_{[F^{2}]^{2}}$,
$\kappa_{F^{4}}$, etc., to be determined later, are by definition independent
of $s,t$ and do not scale with $N$.

The leading term on the RHS of (4.15), namely the tree-level Mellin amplitude
${1\over c_{J}}M^{abcd}_{F^{2}}$, comes from the Witten diagram with a gluon
exchange, shown in the first diagram of Figure 3. In terms of the
superconformal block expansion of the CFT correlator, it involves the flavor
multiplet superblock $\mathfrak{G}_{{\mathcal{D}}[1]}$ appearing on the RHS of
(3.10), whose coefficient is $\lambda^{2}_{{\mathcal{D}}[1]}\propto 1/c_{J}$
(3.12). The explicit expression of $M^{abcd}_{F^{2}}$, derived in [87], is

$\displaystyle M^{abcd}_{F^{2}}(s,t;{\alpha})$
$\displaystyle=\left[{16}\left({\mathtt{B}}^{abcd}-{\mathtt{B}}^{acdb}\right)\left(P_{1}(2\alpha-1)G^{M}_{1,0}(s,t)-G^{M}_{2,1}(s,t)\right)+4{\mathtt{B}}^{abcd}\frac{2-\alpha^{2}-2\alpha}{\pi^{2}}\right]$
(4.16)
$\displaystyle~{}~{}~{}+\left.\Big{[}\cdots\Big{]}\right|_{(a\leftrightarrow
b,~{}t\leftrightarrow u,~{}{\alpha}\to
1-{\alpha})}+({\alpha}-1)^{2}\left.\Big{[}\cdots\Big{]}\right|_{(a\leftrightarrow
c,~{}s\leftrightarrow t,~{}{\alpha}\to{{\alpha}\over{\alpha}-1})}\,,$

where $P_{n}(x)$ is the $n$-th Legendre polynomial, and
$G^{M}_{\Delta,j}(s,t)$ is the singular part of the Mellin space conformal
blocks of weight $\Delta$ and spin $j$ [86], given explicitly in (B.3). The
second line of (4.16) contains two terms related by crossing, where the
$\left[\cdots\right]$ represents the same expression as in the first line.

In particular, (4.16) contains poles in $s$ that correspond to the exchange of
single trace operators in the flavor multiplet with all possible odd positive
integer twists. Its large $s,t$ asymptotic expression is

$\begin{split}M^{abcd}_{F^{2}}\to~{}&\frac{8}{\pi^{2}}(u+s\alpha)^{2}\left[\frac{{\mathtt{B}}^{abcd}}{st}+\frac{{\mathtt{B}}^{acdb}}{su}+\frac{{\mathtt{B}}^{adbc}}{tu}\right],~{}~{}~{}s,t\to\infty.\\\
\end{split}$ (4.17)

The overall $s^{0}$ momentum scaling is expected from the first diagram of
Figure 3, where each Yang-Mills vertex contributes 1 to the momentum power
counting, whereas the propagator subtracts 2. Indeed, one can apply the 7d
flat space limit (4.6) to ${1\over c_{J}}M^{abcd}_{F^{2}}$, and recover the
expected flat space tree-level gluon scattering amplitude (with specialized
polarizations) (4.11).

The second term on the RHS of (4.15), ${1\over
c_{J}^{2}}M^{abcd}_{F^{2}|F^{2}}$, is a 1-loop Mellin amplitude that admits a
unitarity cut into a pair of tree-level Yang-Mills “$F^{2}$” amplitudes, as
depicted in the third diagram of Figure 3. In principle, this 1-loop Mellin
amplitude can be determined from the “tree-level”, i.e. leading large $N$, CFT
data, using the AdS unitarity cut method of [88]. Its explicit computation
requires unmixing a rather involved set of degenerate tree-level data, which
we will not pursue here.181818We briefly discuss its expected flavor structure
in Appendix B. Its flat space limit is governed by (A.25), with $s^{3\over 2}$
overall momentum scaling.

The third term on the RHS of (4.15), of order $N^{-{7\over 6}}$, corresponds
to Witten diagrams that involve contact interaction vertices, or effective
couplings, of the form $\kappa_{[F^{2}]^{2}}(\mathop{\rm Tr}\nolimits
F^{2})^{2}$ and $\kappa_{F^{4}}\mathop{\rm Tr}\nolimits F^{4}$, where the
precise contractions of tensor indices are fixed by supersymmetry. The large
$N$ scaling follows from the AdS/CFT dictionary (2.20) and derivative
counting. In practice, the Mellin amplitudes $M^{abcd}_{[F^{2}]^{2}}$ and
$M^{abcd}_{F^{4}}$ can be determined from the expectation that they are degree
2 polynomials in $s$ and $t$, and that they obey the superconformal Ward
identities and crossing invariance. Their explicit expressions are

$\begin{split}&M^{abcd}_{[F^{2}]^{2}}(s,t;\alpha)=\Theta(s,t,\alpha)\left({\mathtt{A}}^{abcd}+{\mathtt{A}}^{acdb}+{\mathtt{A}}^{adbc}\right)\,,\\\
&M^{abcd}_{F^{4}}(s,t;\alpha)=\Theta(s,t,\alpha)\left({\mathtt{B}}^{abcd}+{\mathtt{B}}^{acdb}+{\mathtt{B}}^{adbc}\right)\,,\end{split}$
(4.18)

where

$\displaystyle\Theta(s,t,\alpha)=\frac{1}{3}\left(16\alpha^{2}-8\alpha+3(\alpha-1)^{2}s^{2}-2(\alpha-1)s(7\alpha+3t-5)+3t^{2}-2(2\alpha+5)t+8\right)\,.$
(4.19)

Note that in the large $s,t$ limit, $\Theta(s,t,\alpha)\to(u+s\alpha)^{2}$.
The coefficients $\kappa_{[F^{2}]^{2}}$ and $\kappa_{F^{4}}$ are the same as
those appearing in the flat space amplitude (2.9), and are thus far
unconstrained by superconformal symmetry and the general structure of the
large radius expansion. We will determine them in the next section with the
additional input of supersymmetric localization data.

In the second line of (4.15), we find three different structures that arise at
order $N^{-{3\over 2}}$: the tree-level Mellin amplitude with a graviton
exchange ${1\over c_{T}}M^{abcd}_{R}$, the gluon two-loop Mellin amplitude
${1\over c_{J}^{3}}M^{abcd}_{F^{2}|F^{2}|F^{2}}$, and additional contributions
that can be viewed as due to contact interactions. The graviton exchange term
$M^{abcd}_{R}$ contains the stress tensor multiplet superblock
$\mathfrak{G}_{{\mathcal{B}}[0]}$ in the CFT correlator (3.10), whose overall
coefficient is $\lambda^{2}_{{\mathcal{B}}[0]}\propto 1/c_{T}$ (3.12),
together with an infinite tower of long multiplet superblocks whose
coefficients also scale like $1/c_{T}$.191919The $\mathcal{N}=8$ half-BPS KK
modes on ${\rm AdS}_{4}\times S^{7}$ appear in rank $p$ traceless symmetric
irreps of $SO(8)_{R}$ with $\Delta=p$. Upon reduction to $\mathcal{N}=4$
multiplets of ${\rm AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$, only even $p$ modes
include a long multiplet that will contribute to $M^{abcd}_{R}$. The gluon
2-loop term $M^{abcd}_{F^{2}|F^{2}|F^{2}}$ can in principle be fixed by AdS
unitarity cuts and lower order amplitudes, up to polynomial ambiguities which
can be absorbed into the coefficients $\widetilde{\kappa}_{i}$ of the contact
Mellin amplitudes.

Up to local terms, the flat space limit of $M^{abcd}_{R}$ is governed by
(2.12), whereas the flat space limit of $M^{abcd}_{F^{2}|F^{2}|F^{2}}$ is
governed by (A.27), both having $s^{3}\log s$ scaling at large momenta.
However, unlike the flat space graviton exchange and gluon 2-loop amplitudes
whose log divergences are cutoff at Planck scale $1/\ell_{p}$, $M^{abcd}_{R}$
and $M^{abcd}_{F^{2}|F^{2}|F^{2}}$ involve only the AdS radius scale $1/L$.
The comparison with the flat space limit indicates that there must be
additional contact interaction terms in the Mellin amplitude of order
$N^{-{3\over 2}}\log(L/\ell_{p})$ (or equivalently $\log c_{J}\over
c_{J}^{3}$), that are proportional to a pair of degree 3 polynomials
$M^{abcd}_{D^{2}[F^{2}]^{2}},M^{abcd}_{D^{2}F^{4}}$ corresponding to the
$D^{2}(\mathop{\rm Tr}\nolimits F^{2})^{2}$ and $D^{2}\mathop{\rm Tr}\nolimits
F^{4}$ supervertices in the flat space limit, up to degree 2 ambiguities
proportional to $M^{abcd}_{[F^{2}]^{2}},M^{abcd}_{D^{2}F^{4}}$ (4.18). We can
determined the explicit expression of $M^{abcd}_{D^{2}[F^{2}]^{2}}$ and
$M^{abcd}_{D^{2}F^{4}}$ by solving the analytic bootstrap conditions,

$\displaystyle M^{abcd}_{D^{2}[F^{2}]^{2}}(s,t;\alpha)$
$\displaystyle={\mathtt{A}}^{abcd}\left[(\alpha-1)^{2}s^{3}+\frac{s}{5}\left(4\left(7\alpha^{2}+4\alpha-4\right)+5t^{2}-24\alpha
t\right)\right.$ (4.20)
$\displaystyle\left.\qquad+\frac{4}{5}(t-2)(-2\alpha+2t-3)-\frac{2}{5}(\alpha-1)s^{2}(12\alpha+5t-4)\right]+({\rm
cross~{}terms}),$ $\displaystyle M^{abcd}_{D^{2}F^{4}}(s,t;\alpha)$
$\displaystyle=-\frac{s+t-2}{5}{\mathtt{B}}^{abcd}\left[28\alpha^{2}+5(\alpha-1)^{2}s^{2}-2(\alpha-1)s(12\alpha+5t-4)+5t^{2}-8(2\alpha+1)t\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}+({\rm cross~{}terms}),$

where the cross terms are related by the same manner as in (4.16). The large
$s,t$ asymptotic expressions of (4.20) are

$\displaystyle{}M^{abcd}_{D^{2}[F^{2}]^{2}}$
$\displaystyle\to(u+s\alpha)^{2}\left(s{\mathtt{A}}^{abcd}+u{\mathtt{A}}^{acdb}+t{\mathtt{A}}^{adbc}\right),$
(4.21) $\displaystyle M^{abcd}_{D^{2}F^{4}}$
$\displaystyle\to(u+s\alpha)^{2}\left(u{\mathtt{B}}^{abcd}+t{\mathtt{B}}^{acdb}+s{\mathtt{B}}^{adbc}\right).$

Without the explicit expressions of $M^{abcd}_{R}$ and
$M^{abcd}_{F^{2}|F^{2}|F^{2}}$, we cannot meaningfully determine the
$\widetilde{\kappa}$ coefficients in the second line of (4.15). We can,
however, unambiguously determine the order $\log c_{J}\over c_{J}^{3}$ term
coefficients $\kappa^{\prime}_{\log,[F^{2}]^{2}}$,
$\kappa^{\prime}_{\log,F^{4}}$, $\kappa_{\log,D^{2}[F^{2}]^{2}}$,
$\kappa_{\log,D^{2}F^{4}}$ appearing in the third line of (4.15), as will be
explained in Section 6.3.

###  4.4 Structure of the mixed gluon-graviton Mellin amplitude $M^{abCD}$

We next consider the mixed gluon-graviton Mellin amplitude
$M^{abCD}(s,t;\alpha)$, whose large radius expansion is constrained by the
analytic bootstrap to be of the form

$\begin{split}M^{abCD}&=\frac{M^{abCD}_{R}}{c_{T}}+\frac{\kappa_{RF^{2}}M_{RF^{2}}^{abCD}}{N^{11\over
6}}+\Bigg{[}\frac{M^{abCD}_{F^{2}|R}}{c_{J}c_{T}}+\frac{\widetilde{\kappa}M^{abCD}_{RF^{2}}}{N^{2}}\Bigg{]}\\\
&+\frac{\kappa^{\prime}_{RF^{2}}M^{abCD}_{RF^{2}}+\kappa_{R^{2}F^{2}}M^{abCD}_{R^{2}F^{2}}}{N^{13\over
6}}+O(N^{-{15\over 6}})\,.\end{split}$ (4.22)

Here $M^{abCD}_{R}$ and $M^{abCD}_{F^{2}|R}$ are tree-level and one-loop
gluon-graviton Mellin amplitudes respectively, whereas $M^{abCD}_{RF^{2}}$ and
$M^{abCD}_{R^{2}F^{2}}$ (given in (4.25)) are degree 2 and 3 polynomials in
$s,t$. The coefficients $\kappa_{RF^{2}}$, $\kappa_{R^{2}F^{2}}$ etc., to be
determined, are independent of $s,t$ and do not scale with $N$.

The tree-level term ${1\over c_{T}}M^{abCD}_{R}$, with the explicit expression

$\begin{split}M^{abCD}_{R}(s,t;\alpha)&={\mathtt{T}}_{\bf
1}^{abCD}\Big{[}{128}\left(G^{M}_{1,0}(s,t)-2P_{1}(2\alpha-1)G^{M}_{2,1}(s,t)+G^{M}_{3,2}(s,t)\right)\\\
&+\frac{8}{\pi^{2}}\left(\left(-8\alpha^{2}+16\alpha-7\right)s+4\left(4\alpha^{2}-12\alpha+(4\alpha-2)t+3\right)\right)\Big{]}\,,\\\
\end{split}$ (4.23)

contains graviton exchange poles (in the $s$-channel), corresponding to the
second diagram of Figure 5. From the point of view of superconformal block
decomposition of the mixed correlator ${\mathcal{G}}^{abCD}$(3.3) in the
$ab\to CD$ channel, the graviton exchange involves the stress tensor ${\cal
B}[0]$ superblock, whose coefficient is $\lambda^{2}_{{\mathcal{B}}[0]}\propto
1/c_{T}$ (3.12). In the large radius limit, using (4.6) one can recover from
$\frac{1}{c_{T}}M^{abCD}_{R}$ the flat space tree-level mixed gluon-graviton
amplitude (with specialized polarizations) (4.14).

Note that $M^{abCD}_{R}$ does not contain gluon exchange poles (in the $t$\-
or $u$-channel), which one might have naively expected from the first diagram
of Figure 5. However, as seen in Section 4.2, the kinematic restriction of the
corresponding flat space amplitude is such that the gluon poles drop out. From
the perspective of the superconformal block decomposition of
${\mathcal{G}}^{abCD}$ in the cross channels $aC\to bD$ or $aD\to bC$, the
tree-level Mellin amplitude involves the twist-2 ${\mathcal{B}}[1]_{j}$
superblocks, which do not have poles in $t$ or $u$ as explained in [86].

The possibility of a flat space $R\mathop{\rm Tr}\nolimits F^{2}$ 4-derivative
effective coupling [52] suggests a potential contribution to $M^{abCD}$ at
order $N^{-{11\over 6}}$. Naively, one might expect that such a Mellin
amplitude contains gluon or graviton poles. However, as explained following
(2.16), the $R\mathop{\rm Tr}\nolimits F^{2}$ 3-point vertex in fact vanishes
when the graviton momentum is restricted to 7d, and thus we expect the
$R\mathop{\rm Tr}\nolimits F^{2}$ contribution to the Mellin amplitude,
denoted by $M^{abCD}_{RF^{2}}$, to be free of poles and takes the form of a
degree 2 polynomial in $s$ and $t$. The only candidate expression that solves
the superconformal Ward identities is

$\displaystyle M^{abCD}_{RF^{2}}(s,t;\alpha)={\mathtt{T}}_{\bf
1}^{abCD}\Theta(s,t,\alpha)\,,$ (4.24)

where $\Theta$ is defined in (4.19).

At order $N^{-2}$, we expect that the only contribution to the gluon-graviton
Mellin amplitude arises from the 1-loop Witten diagram of SYM+SUGRA (third
diagram of Figure 5), as the latter is free of logarithmic divergences, and
there should be no independent contact interactions that contribute at this
order. In practice, one may determine the non-polynomial part of the 1-loop
contribution, ${1\over c_{J}c_{T}}M^{abCD}_{F^{2}|R}$, via AdS unitarity cuts,
in terms of tree-level data. Note that it has $s^{5\over 2}$ overall momentum
scaling in the large radius limit. Such a unitarity cut computation would
leave a polynomial ambiguity, in this case of degree 2 in $s$ and $t$, which
is necessarily proportional to $M^{abCD}_{RF^{2}}$.202020An analogous
ambiguity exists for the 1-loop amplitude in 3d $\mathcal{N}=8$ ABJM theory,
which in that case was fixed using localization to a value such that the
corresponding CFT data is analytic for all spins [40, 38].

At order $N^{-{13\over 6}}$, one encounters potential contributions from
contact interactions of the form $R^{2}\mathop{\rm Tr}\nolimits F^{2}$, which
take the form of a degree 3 polynomial in $s$ and $t$, and are further
determined by superconformal Ward identities to be a linear combination of
$M^{abCD}_{RF^{2}}$ (4.24) and

$\begin{split}&M^{abCD}_{R^{2}F^{2}}={\mathtt{T}}_{\bf
1}^{abCD}\Big{[}(\alpha-1)^{2}s^{3}+\frac{s}{5}\left(4\left(7\alpha^{2}+4\alpha-4\right)+5t^{2}-24\alpha
t\right)\\\
&\qquad\qquad\qquad+\frac{4}{5}(t-2)(-2\alpha+2t-3)-\frac{2}{5}(\alpha-1)s^{2}(12\alpha+5t-4)\Big{]}\,.\end{split}$
(4.25)

The coefficient of the latter, $\kappa_{R^{2}F^{2}}$, is the same as the
coefficient appearing in the flat space amplitude (2.17).

##  5 BPS-protected CFT data

###  5.1 The 1d topological sector

Every 3d ${\mathcal{N}}=4$ SCFT contains two “1d topological sectors,” each
generated by a special family of half-BPS operators [89, 90, 23, 24, 25]. The
1d topological operators of interest to this work belong to one of the two
sectors and are constructed as

$\begin{split}\widehat{J^{a}}(\tau)&={\sqrt{c_{J}}\over
16\sqrt{2}\pi}\left.J^{a}(\vec{x};y)\right|_{\vec{x}=(0,\tau,0),y=(1,\tau)},\\\
\widehat{\widetilde{J}^{A}}(\tau)&={\sqrt{c_{\widetilde{J}}}\over
16\sqrt{2}\pi}\left.\widetilde{J}^{A}(\vec{x};y)\right|_{\vec{x}=(0,\tau,0),y=(1,\tau)}\,,\end{split}$
(5.1)

where the normalization was chosen such that these operators couple in a
simple way to the mass parameters that we will introduce later. As shown in
[89], the variable $\tau$ may be viewed as valued in
$\mathbb{R}\cup\\{\infty\\}\simeq S^{1}$, such that the correlators of
$\widehat{J^{a}}$ and $\widehat{\widetilde{J}^{A}}$ depend only on the cyclic
ordering of their $\tau$-coordinates.

Indeed, the 1d topological 4-point function of $\widehat{J^{a}}$ and
$\widehat{\widetilde{J}^{A}}$ can be obtained from ${\mathcal{G}}(U,V;\alpha)$
(defined as in (3.3)) by simply setting

$\displaystyle\eta=1,~{}~{}~{}~{}\alpha=1/\sqrt{U},$ (5.2)

where the $\eta$ variable is defined in (B.2). This is such that only the
half-BPS superblocks survive, and that the dependence on the remaining cross-
ratio drops out.

The 1d topological correlator associated with ${\mathcal{G}}^{abcd}$ is given,
for $k\geq 4$, by212121The $c_{J}^{2}$ factor outside the parantheses comes
from the noncanonical two-point function (LABEL:cJcT), and the extra factors
of $\frac{1}{2}$ and $\frac{3}{8}$ come from the
$P_{r}\left(2{\alpha}-1\right)$ in the definition of the superblocks (B.1),
where only the conformal block associated with the superconformal primary
survives.

$\begin{split}\left\langle\widehat{J^{a}}(\tau_{1})\widehat{J^{b}}(\tau_{2})\widehat{J^{c}}(\tau_{3})\widehat{J^{d}}(\tau_{4})\right\rangle=&\frac{c_{J}^{2}}{(512\pi^{2})^{2}}\Bigg{[}{\mathtt{T}}_{\bf
1}^{abcd}+\frac{1}{2}\mathop{\rm
sgn}\nolimits(\tau_{12}\tau_{13}\tau_{24}\tau_{34})\lambda^{2}_{\mathcal{D}[1],{\bf
adj}_{a}}{\mathtt{T}}_{{\bf adj}_{a}}^{abcd}\\\
&+\frac{3}{8}\Big{(}\lambda^{2}_{\mathcal{D}[2],{\bf 1}}{\mathtt{T}}_{\bf
1}^{abcd}+\lambda^{2}_{\mathcal{D}[2],{\bf 2}}{\mathtt{T}}_{\bf
2}^{abcd}+\lambda^{2}_{\mathcal{D}[2],{\bf 3}}{\mathtt{T}}_{\bf
3}^{abcd}+\lambda^{2}_{\mathcal{D}[2],{\bf 4}}{\mathtt{T}}_{\bf
4}^{abcd}\Big{)}\Bigg{]}\,,\end{split}$ (5.3)

where $\tau_{ij}\equiv\tau_{i}-\tau_{j}$, and the various OPE coefficients and
flavor tensors are defined as in (3.10) and (3.11). The superconformal
multiplets that appear in the OPE include the identity, the flavor multiplet
${\mathcal{D}}[1]$, as well as four ${\mathcal{D}}[2]$ multiplets that come in
the four $SU(k)$ irreps that appear in the symmetric part of the tensor
product (3.9). In the special case $k=3$, (5.3) still holds with the term
associated with the representation $({\mathcal{D}}[2],{\bf 4})$ dropped,
whereas in the case $k=2$ both $({\mathcal{D}}[2],{\bf 3})$ and
$({\mathcal{D}}[2],{\bf 4})$ terms are dropped.

Swapping $(a,\tau_{1})\leftrightarrow(c,\tau_{3})$ on the LHS of (5.3) leads
to the nontrivial crossing equations222222Here we have substituted
$\lambda^{2}_{{\mathcal{D}}[1],{\bf adj}_{a}}$ with its expression in terms of
$c_{J}$ (3.12).

$\begin{split}&k\geq 2:\quad 0=\frac{8}{3}+\lambda^{2}_{\mathcal{D}[2],{\bf
1}}-\lambda^{2}_{\mathcal{D}[2],{\bf 2}},\\\ &k\geq 4:\quad
0=\lambda^{2}_{\mathcal{D}[2],{\bf 3}}-\lambda^{2}_{\mathcal{D}[2],{\bf
4}}-\frac{64}{3c_{J}}\,,\\\ &k=3:\quad 0=\lambda^{2}_{\mathcal{D}[2],{\bf
3}}-\frac{1}{2}\lambda^{2}_{\mathcal{D}[2],{\bf
1}}-\frac{4}{3}+\frac{64}{9c_{J}}\,,\end{split}$ (5.4)

so there is just one crossing equation for $k=2$, but two otherwise.

The 1d topological correlator associated with ${\mathcal{G}}^{abCD}$ can be
derived similarly, and gives

$\begin{split}&\left\langle\widehat{J^{a}}(\tau_{1})\widehat{J^{b}}(\tau_{2})\widehat{\widetilde{J}^{C}}(\tau_{3})\widehat{\widetilde{J}^{D}}(\tau_{4})\right\rangle=\frac{c_{J}c_{\widetilde{J}}}{(512\pi^{2})^{2}}{\mathtt{T}}_{\bf
1}^{abCD}\big{(}1+\frac{3}{8}\lambda^{2}_{\mathcal{D}[2]}\big{)}\,,\end{split}$
(5.5)

where $\lambda_{{\mathcal{D}}[2]}$ is the OPE coefficient of the flavor
singlet ${\mathcal{D}}[2]$ multiplet that appears in (3.6) and (3.7), and the
flavor tensor ${\mathtt{T}}_{\bf 1}^{abCD}$ is defined as in (3.13). Note that
if we considered the $aC\to bD$ or $aD\to bC$ channels, where the only flavor
channel is the adjoint, then we would not have an identity block contribution.

After taking into account (3.12) and (5.4), there remain two independent OPE
coefficients that appear in the 1d topological correlator (5.3) in the case
$k\geq 4$, or one independent OPE coefficient in (5.3) in the case $k=2,3$,
and one OPE coefficient that appear in the mixed 1d topological correlator
(5.5). These will be determined using supersymmetric localization.

###  5.2 Relation to the sphere partition function $Z$

A 3d ${\mathcal{N}}=4$ gauge theory with hypermuiltiplet matter can be defined
on $S^{3}$ via an action of the schematic form $S=S_{\rm hyper}[{\cal H},{\cal
V}]+S_{\rm YM}[{\cal V}]$ [23], where the hypermultiplet coupling $S_{\rm
hyper}[{\cal H},{\cal V}]$ preserves $osp(4|4)$ superconformal symmetry, while
the Yang-Mills action $S_{\rm YM}[{\cal V}]$, which preserves half of the
supersymmetries, is irrelevant in the IR. One can further deform the theory on
$S^{3}$ by hypermultiplet mass terms in way that preserves 8 out of the 16
supercharges. The partition function $Z$ of this theory, on one hand, can be
evaluated exactly via supersymmetric localization, and on the other hand,
captures the 1d topological correlators through its dependence on the mass
parameters.

Specializing to the 3d ${\cal N}=4$ $U(N)$ gauge theory of interest, namely
one with 1 adjoint and $k$ fundamental hypermultiplets, let $m_{a}$ be the
mass parameters for the fundamental hypermultiplets, where $a$ is an adjoint
index of the $SU(k)$ flavor symmetry, and let $M_{A}$ be the mass parameters
for the adjoint hypermultiplet, where $A$ is an adjoint index of $SU(2)_{F}$.
The $S^{3}$ partition function of this mass deformed theory is denoted
$Z(m,M)$. It relates to the central charges $c_{J}$ and $c_{\widetilde{J}}$ by
[91]

$\begin{split}\frac{\partial_{m_{a}}\partial_{m_{b}}Z}{Z}\Big{|}_{m,M=0}=-{\pi^{2}\over
8}c_{J}{\rm
Tr}(T^{a}T^{b})\,,\qquad\frac{\partial_{M_{A}}\partial_{M_{B}}Z}{Z}\Big{|}_{m,M=0}=-{\pi^{2}\over
8}c_{\widetilde{J}}\delta^{AB}\,.\end{split}$ (5.6)

The partition function $Z(m,M)$ can also be related to correlation functions
of the topological operators (5.1), as follows. A Weyl transformation from
$\mathbb{R}^{3}$ to a round $S^{3}$ shows that the 1d topological sector,
originally defined in (5.1) on a straight line in $\mathbb{R}^{3}$ that passes
through the origin, can also be defined on $S^{3}$. Since a straight line
passing through the origin gets mapped to a great circle on $S^{3}$, the
topological sector on $S^{3}$ will be defined on this great circle. As shown
in [23] in a more general setup that includes the theory of interest here, one
can use supersymmetric localization for the theory on $S^{3}$ to show that the
1d sector is described by a gauged quantum mechanics theory with the same
partition function $Z(m,M)$ as the 3d theory. Furthermore, if one deforms the
$S^{3}$ theory by real mass parameters $m_{a}$ and $M_{A}$, the 1d gauged
quantum mechanics gets modified by the addition of the operator

$\begin{split}4\pi i\int
d\varphi\,\left(m_{a}\widehat{J^{a}}(\varphi)+M_{A}\widehat{J^{A}}(\varphi)\right)\end{split}$
(5.7)

to the 1d action.232323For a general proof that does not rely on
supersymmetric localization, see [92, 93]. The normalizations of the operators
$\widehat{J^{a}}$ and $\widehat{J^{A}}$ in (5.1) were chosen precisely such
that (5.7) holds without additional prefactors. One can check that this is the
case by reproducing (5.6) from the point of view of the 1d theory. Indeed, we
have

$\begin{split}\frac{\partial_{m_{a}}\partial_{m_{b}}Z}{Z}\Big{|}_{m,M=0}&=(4\pi
i)^{2}\int
d\varphi_{1}\,d\varphi_{2}\,\left\langle\widehat{J^{a}}(\varphi_{1})\widehat{J^{b}}(\varphi_{2})\right\rangle\,,\\\
\frac{\partial_{M_{A}}\partial_{M_{B}}Z}{Z}\Big{|}_{m,M=0}&=(4\pi i)^{2}\int
d\varphi_{1}\,d\varphi_{2}\,\left\langle\widehat{\widetilde{J}^{A}}(\varphi_{1})\widehat{\widetilde{J}^{B}}(\varphi_{2})\right\rangle\,.\end{split}$
(5.8)

The definition (5.1) and the normalization condition (3.2) imply that the two-
point functions of the 1d operators are
$\left\langle\widehat{J^{a}}(\varphi_{1})\widehat{J^{b}}(\varphi_{2})\right\rangle=\frac{c_{J}}{512\pi^{2}}\mathop{\rm
tr}\nolimits(T^{a}T^{b})$ and
$\left\langle\widehat{\widetilde{J}^{A}}(\varphi_{1})\widehat{\widetilde{J}^{B}}(\varphi_{2})\right\rangle=\frac{c_{\widetilde{J}}}{512\pi^{2}}\delta^{AB}$.
Plugging these expressions into (5.8) and performing the integrals, one can
recover (5.6).

Using a similar expression to (5.8) for the four-point function gives

$\displaystyle\frac{\partial_{m_{a}}\partial_{m_{b}}\partial_{m_{c}}\partial_{m_{d}}Z}{Z}\Big{|}_{m,M=0}$
$\displaystyle=(4\pi)^{4}\int_{0}^{2\pi}d\varphi_{1}d\varphi_{2}d\varphi_{3}d\varphi_{4}\left\langle\widehat{J^{a}}(\varphi_{1})\widehat{J^{b}}(\varphi_{2})\widehat{J^{c}}(\varphi_{3})\widehat{J^{d}}(\varphi_{4})\right\rangle$
(5.9) $\displaystyle={\pi^{4}c_{J}^{2}\over 64}\left[{\mathtt{T}}_{\bf
1}^{abcd}+{1\over 6}\lambda^{2}_{\mathcal{D}[1],{\bf
adj}_{a}}{\mathtt{T}}^{abcd}_{{\bf adj}_{a}}+{3\over 8}\sum_{{\bf i}={\bf
1},{\bf 2},{\bf 3},{\bf 4}}\lambda^{2}_{\mathcal{D}[2],{\bf
i}}{\mathtt{T}}^{abcd}_{\bf i}\right]\,.$

Recall that $\lambda_{{\mathcal{D}}[1],{\bf adj}_{a}}$ has already been
determined in (3.12), and that $\lambda_{{\mathcal{D}}[2],{\bf i}}$ obey the
crossing constraints (5.4). For $k\geq 4$, (5.9) gives two new independent
constraints on the OPE coefficients appearing on the RHS, associated with the
two quartic Casimirs of $su(k)$. It suffices to consider mass deformation
parameters $m_{1}$ and $m_{3}$ associated with the following two Cartan
generators of $su(k)$, expressed in the fundamental representation as diagonal
$k\times k$ matrices

$\begin{split}(T^{1})_{I}{}^{J}&=\frac{1}{2}\text{diag}\\{1,-1,0,\cdots,0\\}\,,\qquad(T^{3})_{I}{}^{J}=\frac{1}{2\sqrt{6}}\text{diag}\\{1,1,1,-3,0\cdots,0\\}\,,\end{split}$
(5.10)

which lead to the independent constraints

$\begin{split}&{256\over\pi^{4}c_{J}^{2}}\frac{\partial_{m_{1}}^{4}Z}{Z}\Big{|}_{m,M=0}=1+\frac{3}{16}\left(2\lambda^{2}_{\mathcal{D}[2],{\bf
1}}+4\lambda^{2}_{\mathcal{D}[2],{\bf 2}}+2\lambda^{2}_{\mathcal{D}[2],{\bf
3}}+\lambda^{2}_{\mathcal{D}[2],{\bf 4}}\right)\,,\\\
&{256\over\pi^{4}c_{J}^{2}}\frac{\partial_{m_{3}}^{4}Z}{Z}\Big{|}_{m,M=0}=1+\frac{3}{16}\left(2\lambda^{2}_{\mathcal{D}[2],{\bf
1}}+4\lambda^{2}_{\mathcal{D}[2],{\bf 2}}+{7\over
3}\lambda^{2}_{\mathcal{D}[2],{\bf 3}}+{7\over
6}\lambda^{2}_{\mathcal{D}[2],{\bf 4}}\right)\,.\end{split}$ (5.11)

For $k=2,3$, (5.9) produces a single new independent constraint on the BPS-
protected OPE coefficients, in addition to the two crossing constraints in the
$k=3$ case, or the one crossing constraint in the $k=2$ case.

We can similarly extract constraints on the 1d mixed topological correlator
(5.5) from the $S^{3}$ partition function by taking the derivative of the
latter with respect to both $m_{a}$ and $M_{A}$,

$\displaystyle\frac{\partial_{m_{a}}\partial_{m_{b}}\partial_{M_{C}}\partial_{M_{D}}Z}{Z}\Big{|}_{m,M=0}$
$\displaystyle=(4\pi)^{4}\int_{0}^{2\pi}d\varphi_{1}d\varphi_{2}d\varphi_{3}d\varphi_{4}\left\langle\widehat{J^{a}}(\varphi_{1})\widehat{J^{b}}(\varphi_{2})\widehat{\widetilde{J}^{C}}(\varphi_{3})\widehat{\widetilde{J}^{D}}(\varphi_{4})\right\rangle$
(5.12) $\displaystyle={\pi^{4}c_{J}c_{\widetilde{J}}\over 64}{\mathtt{T}}_{\bf
1}^{abCD}\big{(}1+\frac{3}{8}\lambda^{2}_{\mathcal{D}[2]}\big{)}\,,$

where as before we assumed the $ab\to CD$ channel, and if we considered the
$aC\to bD$ or $aD\to bC$ channels, then we would not have an identity block
contribution.

###  5.3 Evaluating $Z$ from supersymmetric localization

An essential property of the $S^{3}$ partition function $Z(m,M)$ is that its
path integral representation can be reduced, via the supersymmetric
localization method [21, 22] (for reviews, see [94, 95]), to a finite
dimensional matrix integral. It suffices to restrict to mass parameters
$m_{\alpha}$ associated with a set of $k-1$ Cartan generators $T^{\alpha}$,
${\alpha}=1,\cdots,k-1$, which we take to be diagonal $k\times k$ matrices,
and write $m_{\alpha}(T^{\alpha})_{I}{}^{J}\equiv
m_{{\alpha},I}\delta_{I}^{J}$. Similarly, we can restrict the mass parameters
for the adjoint hypermultiplet to only one non-zero parameter, $M_{3}=M$. The
localized matrix integral expression for $Z(m,M)$ for this choice of $m$ can
be further reduced to an $N$-dimensional integral over eigenvalues,

$\begin{split}Z(m,M)=\frac{1}{N!}\int\prod_{i=1}^{N}\frac{dx_{i}}{2^{k+1}}\prod_{I=1}^{k}\frac{1}{\cosh[\pi(x_{i}+\sum_{\alpha=1}^{k-1}m_{\alpha,I})]}\frac{\prod_{i<j}\sinh^{2}\left(\pi(x_{ij})\right)}{\prod_{i,j}\cosh\left(\pi(x_{ij}-M)\right)}\,,\end{split}$
(5.13)

where $x_{ij}\equiv x_{i}-x_{j}$. The large $N$ expansion of this expression
can be extracted from its interpretation as the partition function of a non-
interacting Fermi gas of $N$ particles [96, 69, 97], leading to

$\begin{split}Z(m,M)=e^{A}C^{-\frac{1}{3}}\text{Ai}\left[C^{-\frac{1}{3}}(N-B)\right]+O(e^{-N})\,,\end{split}$
(5.14)

where $A,B,C$ are functions of $m_{\alpha},M$, and $k$. The Fermi gas method
[96, 97], summarized in Appendix D, produces $A,B,C$ a priori as power series
in $\hbar={1\over 2\pi}$. Here we simply state the results for $B$ and $C$,

$\begin{split}C=\frac{2}{\pi^{2}k(1+4M^{2})}\,,\qquad
B=\frac{\pi^{2}C}{3}+\frac{k}{24}-\frac{1}{6k}-\frac{1}{1+4M^{2}}\Big{(}\frac{k}{6}+\sum_{\alpha=1}^{k-1}{m_{\alpha}^{2}}\Big{)}\,,\end{split}$
(5.15)

and give $A$ as an expansion in $m_{\alpha}$ to order $m^{2}$,242424The exact
formula for $A$ at $m=0$ was guessed in [97].

$\begin{split}A&=\frac{k^{2}}{4}\big{[}{\mathcal{A}}(1-2iM)+{\mathcal{A}}(1+2iM)\big{]}+\frac{{\mathcal{A}}(k)}{2}+k\sum_{\alpha=1}^{k-1}{m_{\alpha}^{2}}\bigg{[}\frac{1}{4M^{2}+1}-\frac{\pi^{2}}{72}\\\
&~{}~{}~{}-\frac{\pi^{4}\left(12M^{2}-1\right)}{21600}-\frac{\pi^{6}(1-40M^{2}+80M^{4})}{1270080}+\dots\bigg{]}+O(m^{4})\,,\end{split}$
(5.16)

where ${\mathcal{A}}(x)$ on the RHS is the “constant map”, defined in [98] as

$\begin{split}{\cal
A}(x)=&\frac{2\zeta(3)}{\pi^{2}x}\left(1-\frac{x^{3}}{16}\right)+\frac{x^{2}}{\pi^{2}}\int_{0}^{\infty}dy\frac{y}{e^{xy}-1}\log\left(1-e^{-2y}\right)\\\
=&\frac{2\zeta(3)}{\pi^{2}x}-\frac{x}{12}-\frac{x^{3}\pi^{2}}{4320}+\frac{x^{5}\pi^{4}}{907200}-\frac{x^{7}\pi^{6}}{50803200}+\frac{x^{9}\pi^{8}}{1437004800}+O(x^{13})\,.\\\
\end{split}$ (5.17)

We have also computed the order $m^{4}$ terms of $A$ at $M=0$, with the result

$\begin{split}\partial_{m_{1}}^{4}A|_{m,M=0}=-2\pi^{2}(6+k)\mathfrak{b}\,,\qquad\partial_{m_{3}}^{4}A|_{m,M=0}=-2\pi^{2}(6+{7\over
6}k)\mathfrak{b}\,,\end{split}$ (5.18)

where $\mathfrak{b}$ is the constant252525This closed form expression is
guessed from the series
$\mathfrak{b}=\frac{\pi^{2}}{12}+\frac{\pi^{4}}{720}-\frac{\pi^{6}}{60480}+\frac{\pi^{8}}{1814400}-\frac{\pi^{10}}{31933440}+\cdots$.

$\begin{split}\mathfrak{b}=-\frac{\pi^{2}}{2}(x{\mathcal{A}}(x))^{\prime\prime}\big{|}_{x=1}=-\frac{1}{2}\pi^{2}\left(\frac{\pi^{2}}{32}-\frac{1}{2}\right)\,.\end{split}$
(5.19)

###  5.4 The $1/N$ expansion of central charges

We can now evaluate the $1/N$ expansion of the central charges $c_{J}$ and
$c_{\widetilde{J}}$ using (5.6) and (5.14). The lowest order terms are

$\begin{split}c_{J}&=\frac{8\sqrt{2Nk}}{\pi}-\frac{16k}{\pi^{2}}\mathfrak{a}+O(N^{-{1\over
2}})\,,\\\ c_{\widetilde{J}}&=\frac{32\sqrt{2}N^{3\over
2}\sqrt{k}}{3\pi}+\frac{10\sqrt{2}\sqrt{N}(4-k^{2})}{3\pi\sqrt{k}}+O(N^{0})\,,\\\
\end{split}$ (5.20)

where the constant $\mathfrak{a}$ is evaluated using (5.16) to be

$\begin{split}\mathfrak{a}={1\over
2k}\partial_{m}^{2}A|_{m,M=0}=1-\frac{\pi^{2}}{72}+\frac{\pi^{4}}{21600}-\frac{\pi^{6}}{1270080}+\cdots\,.\end{split}$
(5.21)

For the purpose of numerical evaluations, the expansion on the RHS of (5.21)
converges quickly. In the case $k=2$, we have tested the numerical integral of
(5.13) in $N=2$ case against the large $N$ asymptotic formulae (5.20) together
with (5.21) (truncated to the order shown), and the results (shown in Table 2)
are very close already.

| asymptotic | exact | error ($\%$)
---|---|---|---
$c_{J}$ | $4.5824$ | $4.5894$ | $0.15$
$c_{\widetilde{J}}$ | $20.634$ | $20.589$ | $0.21$
$\lambda^{2}_{{\mathcal{D}}[2]_{1}}$ | $0.4866$ | $0.48318$ | $1.12$
$\lambda^{2}_{{\mathcal{D}}[2]}$ | $2.7626$ | $2.7819$ | $0.70$
Table 2: Comparison of the large $N$ asymptotic formulae (5.20) against the
exact values at $N=2$, in the $k=2$ case, for various CFT data computable from
localization, with surprisingly good agreement.

The large $N$ expansion of $c_{T}$ has been previously obtained from the
squashed 3-sphere partition function [42], with the result

$\begin{split}c_{T}=\frac{64\sqrt{2}N^{3\over
2}\sqrt{{k}}}{3\pi}+\frac{4\sqrt{2}\sqrt{N}\left(7{k}^{2}+8\right)}{3\pi\sqrt{{k}}}+O(N^{0})\,.\end{split}$
(5.22)

Note that the ratio $c_{T}/c_{\widetilde{J}}=2+{\cal O}(N^{-1})$ has
nontrivial dependence on $1/N$ at the subleading order.262626In the case
$k=1$, the supersymmetry enhances to $\mathcal{N}=8$ and the flavor multiplet
combines with the stress tensor multiplet, leading to an exact relation
$c_{T}=2c_{\widetilde{J}}$.

###  5.5 The $1/N$ expansion of BPS-protected OPE coefficients

We can now determine, using the crossing relation (5.4), the relation between
the sphere partition function $Z$ and the 1d topological correlator (5.11),
and the large $N$ expansion of $Z$ (5.14), all of the protected OPE
coefficients that appear on the RHS of (5.3). The first few order terms in the
$1/c_{J}\sim 1/\sqrt{N}$ expansion are272727For $k=3$, the same formula holds
with $\lambda^{2}_{\mathcal{D}[2],{\bf 4}}$ omitted, whereas for $k=2$ both
$\lambda^{2}_{\mathcal{D}[2],{\bf 3}}$ and $\lambda^{2}_{\mathcal{D}[2],{\bf
4}}$ are to be omitted.

$\begin{split}k\geq 2:\quad\lambda^{2}_{\mathcal{D}[2],{\bf
1}}&={8192\over\pi^{4}}{\mathfrak{b}\over
c_{J}^{2}}-\frac{8192}{3\pi^{4}}{k\over
c_{J}^{3}}+\frac{131072}{3\pi^{6}}{\mathfrak{a}k^{2}\over
c_{J}^{4}}+O(c_{J}^{-5})\,,\\\ \lambda^{2}_{\mathcal{D}[2],{\bf
2}}&=\frac{8}{3}+\frac{8192}{\pi^{4}}{\mathfrak{b}\over
c_{J}^{2}}-\frac{8192}{3\pi^{4}}{k\over
c_{J}^{3}}+\frac{131072}{3\pi^{6}}{\mathfrak{a}k^{2}\over
c_{J}^{4}}+O(c_{J}^{-5})\,,\\\ k=3:\quad\lambda^{2}_{\mathcal{D}[2],{\bf
3}}&=\frac{4}{3}+\frac{64}{9}{1\over
c_{J}}+{4096\over\pi^{4}}{\mathfrak{b}\over
c_{J}^{2}}-\frac{4096}{3\pi^{4}}{k\over
c_{J}^{3}}+\frac{65536}{3\pi^{6}}{\mathfrak{a}k^{2}\over
c_{J}^{4}}+O(c_{J}^{-5})\,,\\\ k\geq 4:\quad\lambda^{2}_{\mathcal{D}[2],{\bf
3}}&=\frac{64}{9}{1\over c_{J}}+\frac{8192}{3\pi^{4}}{\mathfrak{b}k\over
c_{J}^{2}}+O(c_{J}^{-5})\,,\\\ \lambda^{2}_{\mathcal{D}[2],{\bf
4}}&=-\frac{128}{9}{1\over c_{J}}+\frac{8192}{3\pi^{4}}{\mathfrak{b}k\over
c_{J}^{2}}+O(c_{J}^{-5})\,,\\\ \end{split}$ (5.23)

where $\mathfrak{a},\mathfrak{b}$ are given in (5.21), (5.19) respectively.
Recall that for $k=3$, the second crossing equation in (5.4) trivially relates
$\lambda^{2}_{\mathcal{D}[2],{\bf 3}}$ to $\lambda^{2}_{\mathcal{D}[2],{\bf
1}}$ and $\lambda^{2}_{\mathcal{D}[2],{\bf 2}}$, while for $k\geq 4$ they are
independent.

We can also determine from (5.12) the only protected OPE coefficient
$\lambda_{{\mathcal{D}}[2]}$ (in the $ab\to CD$ channel) that appears in the
mixed correlator (5.5). The first two terms in its $1/N$ expansion are

$\begin{split}\lambda^{2}_{{\mathcal{D}}[2]}&=\frac{4\sqrt{2}}{\pi\sqrt{k}N^{3\over
2}}+\frac{3}{4\pi^{2}kN^{2}}(\mathfrak{a}+\mathfrak{d})+O(N^{-{5\over
2}})\,,\\\ \end{split}$ (5.24)

where the constant $\mathfrak{d}$ is obtained from (5.16) to be

$\begin{split}\mathfrak{d}={1\over
2}\partial_{m}^{2}\partial_{M}^{2}A|_{m,M=0}=-8-\frac{\pi^{4}}{900}+\frac{\pi^{6}}{15876}+\dots\,.\end{split}$
(5.25)

As discussed above, in the $aC\to bD$ or $aD\to bC$ channels the unique
protected OPE coefficient would differ by the identity contribution $8\over
3$.

##  6 Back to M-theory via Mellin amplitudes

We now return to the question of determining the effective theory of massless
degrees of freedom of M-theory in the orbifold background
$\mathbb{R}^{1,6}\times\mathbb{C}^{2}/\mathbb{Z}_{k}$ by taking the large
radius limit of Mellin amplitudes in ${\rm AdS}_{4}\times
S^{7}/\mathbb{Z}_{k}$, as constrained by the known conformal data of the dual
3d ${\cal N}=4$ SCFT.

###  6.1 The leading low energy limit in the M-theory orbifold

To proceed, we will compare the $1/N$ expansion of the BPS-protected OPE
coefficients (3.12), (3.14), (5.23), and (5.24) against that of the gluon
Mellin amplitude $M^{abcd}$ (4.15) and the mixed gluon-graviton Mellin
amplitude $M^{abCD}$ (4.22). This is achieved by first expressing the Mellin
transform of (the relevant term of) the Mellin amplitude as an expansion in
conformal blocks $G_{\Delta,j}(U,V)$ and their derivatives
$\partial_{\Delta}G_{\Delta,j}(U,V)$, taking the lightcone limit $U\to 0$ with
fixed $V$, and then projecting onto a given lightcone block using
orthogonality relations, thereby expressing the contribution to the OPE
coefficient as a contour integral of the said Mellin amplitude that can be
evaluated using residues. The algorithm, following [99], is outlined in
Appendix C. Below we simply state the results.

For the 4-gluon Mellin amplitude $M^{abcd}(s,t;{\alpha})$ (4.15), the leading
SYM tree-level term ${1\over c_{J}}M_{F^{2}}^{abcd}$ (given explicitly in
(4.16)) contributes to $\lambda^{2}_{{\mathcal{D}}[1],{\bf adj}_{a}}$,
$\lambda^{2}_{{\mathcal{D}}[2],{\bf 3}}$, and
$\lambda^{2}_{{\mathcal{D}}[2],{\bf 4}}$, producing precisely the order
$1/c_{J}$ terms of these OPE coefficients. While we have not determined the
explicit $s,t$-dependence of the 1-loop term ${1\over
c_{J}^{2}}M_{F^{2}|F^{2}}^{abcd}$, the $k$-dependence of its known general
structure [100] (similar to its flat space analog (A.25)) matches that of
(5.23) at order $1/c_{J}^{2}$.

For the mixed gluon-graviton Mellin amplitude $M^{abCD}(s,t;{\alpha})$ (4.22),
the tree-level graviton exchange term ${1\over c_{T}}M_{R}^{abCD}$ contributes
to $\lambda^{2}_{{\mathcal{B}}[0]}$ (3.14) and
$\lambda^{2}_{{\mathcal{D}}[2]}$ (5.24), and the results precisely match at
order $N^{-{3\over 2}}$.

These results confirm, in a highly nontrivial manner, that the leading order
low energy effective description of the M-theory orbifold, namely the 7d
$SU(k)$ super-Yang-Mills theory coupled to 11d supergravity, is consistent
with the holographic duality.

###  6.2 Constraining higher derivative corrections

Going beyond the leading low energy limit, we turn to the thus far
undetermined coefficients on the RHS of (4.15) and (4.22).

Comparing the 4-gluon Mellin amplitude (4.15) to the $1/c_{J}$ expansion of
the OPE coefficients $\lambda^{2}_{{\mathcal{D}}[2],i}$ (5.23), we observe
that the latter do not admit terms of order $N^{-{7\over 6}}$ that would a
priori receive contribution from the Mellin amplitudes proportional to
$M_{[F^{2}]^{2}}^{abcd}$ and $M_{F^{4}}^{abcd}$. This allows us to determine

$\displaystyle\kappa_{[F^{2}]^{2}}=\kappa_{F^{4}}=0\,.$ (6.1)

Through the large radius limit (4.6), it follows that the momentum expansion
of the gluon amplitude ${\mathcal{A}}^{abcd}(s,t)$ (with specialized
polarization and the factor $(u+s{\alpha})^{2}$ stripped off) contains no
terms of $s^{0}$ scaling, and consequently the gluon superamplitude
$\mathfrak{A}^{abcd}$ in the M-theory orbifold background does not contain the
4-derivative supervertices (2.9) corresponding to $(\mathop{\rm Tr}\nolimits
F^{2})^{2}$ or $\mathop{\rm Tr}\nolimits F^{4}$ effective couplings.282828This
is in contrast to, for instance, the non-Abelian Born-Infeld action of
D-branes which contains such effective couplings.

We further observe that the $1/c_{J}$ expansion of OPE coefficients (5.23)
does not contain any terms with logarithmic dependence on $c_{J}$, and in
particular there are no terms of order ${\log c_{J}\over c_{J}^{3}}$, in
contrast to the Mellin amplitude (4.15) that does contain order ${\log
c_{J}\over c_{J}^{3}}$ terms. It follows that the coefficients
$\kappa^{\prime}_{\log,[F^{2}]^{2}},\kappa^{\prime}_{\log,F^{4}},\kappa_{\log,D^{2}[F^{2}]^{2}},\kappa_{\log,D^{2}F^{4}}$
must be such that the contribution from the third line of (4.15) to the BPS-
protect OPE coefficients (5.23) cancels, from which we deduce the linear
relations

$\displaystyle\kappa^{\prime}_{\log,[F^{2}]^{2}}=-\frac{12}{5}\kappa_{\log,D^{2}[F^{2}]^{2}},\qquad\kappa^{\prime}_{\log,F^{4}}=-\frac{6}{5}\kappa_{\log,D^{2}F^{4}}\,.$
(6.2)

We will determine the coefficients
$\kappa_{\log,D^{2}[F^{2}]^{2}},\kappa_{\log,D^{2}F^{4}}$ by comparison with
the flat space amplitudes in the next subsection.

Next, let us compare the mixed gluon-graviton Mellin amplitude (4.22) to the
$1/N$ expansion of the OPE coefficient $\lambda_{{\mathcal{D}}[2]}^{2}$ (5.24)
appearing in the mixed correlator (5.5). We observe that (5.24) does not
contain terms of order $N^{-{11\over 6}}$ nor $N^{-{13\over 6}}$. It follows
that

$\displaystyle\kappa_{RF^{2}}=0\,,\qquad\kappa^{\prime}_{RF^{2}}=-\frac{12}{5}\kappa_{R^{2}F^{2}}\,.$
(6.3)

In particular, we conclude that the 4-derivative $R\mathop{\rm Tr}\nolimits
F^{2}$ coupling corresponding to the supervertex (2.16) is absent in the
M-theory orbifold background. On the other hand, the second relation in (6.3)
means that the known holographic CFT data do not pin down the
$R^{2}\mathop{\rm Tr}\nolimits F^{2}$ coupling coefficient
$\kappa_{R^{2}F^{2}}$, corresponding to the supervertex (2.17).292929It is
likely that a non-renormalization property of the Coulomb branch effective
theory similar to the one outlined in Section 2.2 can be used to argue that
$\kappa_{R^{2}F^{2}}$ itself vanishes.

###  6.3 The logarithmic threshold corrections

Even though we have not explicitly computed the gluon 2-loop and graviton
exchange contributions to the Mellin amplitude $M^{abcd}$, namely ${1\over
c_{J}^{3}}M_{F^{2}|F^{2}|F^{2}}$ and ${1\over c_{T}}M_{R}$ in (4.15), they are
expected to behave as $N^{-{3\over 2}}s^{3}\log(L^{2}s)$ in the flat space
limit. By contrast, the gluon 2-loop and graviton exchange terms in the flat
space amplitude, namely $\mathfrak{A}_{F^{2}|F^{2}|F^{2}}$ and
$\mathfrak{A}_{R}$ in (2.8), behave as $s^{3}\log(-\ell_{p}^{2}s)$, where the
logarithmic dependence on $\ell_{p}$ follows from the expectation of a
Wilsonian effective description with cutoff scale $\Lambda\sim 1/\ell_{p}$
(see the comment after (2.13)). The difference between them is accounted for
by the order ${\log c_{J}\over c_{J}^{3}}$ terms in the Mellin amplitude
$M^{abcd}$ (third line of (4.15)), more precisely,

$\displaystyle{\log c_{J}\over
c_{J}^{3}}\left[\kappa_{\log,D^{2}[F^{2}]^{2}}M^{abcd}_{D^{2}[F^{2}]^{2}}+\kappa_{\log,D^{2}F^{4}}M^{abcd}_{D^{2}F^{4}}\right],$
(6.4)

whose flat space limit is governed by the logarithmic threshold terms in
$\mathfrak{A}_{F^{2}|F^{2}|F^{2}}+\mathfrak{A}_{R}$.

The contribution of $\mathfrak{A}_{F^{2}|F^{2}|F^{2}}$ to the reduced
amplitude ${\mathcal{A}}^{abcd}(s,t)$ appearing in (4.6), as follows from
(A.27) with the substitution (4.10), is

$\displaystyle\mathcal{A}^{abcd}_{F^{2}|F^{2}|F^{2}}(s,t)$
$\displaystyle=-2g_{\rm
YM}^{6}\big{(}s\mathtt{e}_{s_{1}}I_{1234}^{P}+s\mathtt{e}_{s_{2}}I_{1243}^{P}+t\mathtt{e}_{t_{1}}I_{1432}^{P}+t\mathtt{e}_{t_{2}}I_{1432}^{P}$
(6.5)
$\displaystyle~{}~{}~{}+u\mathtt{e}_{u_{1}}I_{1324}^{P}+u\mathtt{e}_{u_{2}}I_{1342}^{P}+2s\mathtt{f}_{s}I_{1234}^{NP}+2t\mathtt{f}_{t}I_{1423}^{NP}+2u\mathtt{f}_{u}I_{1324}^{NP}\big{)}\,,$

where $\mathtt{e}$ and $\mathtt{f}$ are the flavor tensor structures defined
in (A.28). $I^{P}(s,t)$ and $I^{NP}(s,t)$ are the planar $I^{p}(s,t)$ and non-
planar $I^{NP}(s,t)$ 2-loop integrals defined in [57], with UV cutoff
$\Lambda\sim 1/\ell_{p}$. For our purpose of determining (6.4), it suffices to
evaluate the logarithmic “divergence” of the 2-loop integrals, giving

$\begin{split}I_{1234}^{P}|_{\log}=-\log(\ell_{p}^{2})\frac{\pi}{10(4\pi)^{7}}\,,\qquad
I_{1234}^{NP}|_{\log}=-\log(\ell_{p}^{2})\frac{\pi}{15(4\pi)^{7}}\,.\end{split}$
(6.6)

The corresponding logarithmic dependence on $\ell_{p}$ of (6.5) is given by
(after substituting the explicit expressions of $\mathtt{e},\mathtt{f}$ from
(A.28))

$\displaystyle\mathcal{A}^{abcd}_{F^{2}|F^{2}|F^{2}}|_{\log}$
$\displaystyle={g_{\rm YM}^{6}\over
2^{13}\pi^{6}}\log(\ell_{p}^{2})\Big{[}sk{\mathtt{A}}^{abcd}+uk{\mathtt{A}}^{acdb}+tk{\mathtt{A}}^{adbc}$
(6.7) $\displaystyle\quad-(2+{k^{2}\over
10})u{\mathtt{B}}^{abcd}-(2+{k^{2}\over
10})t{\mathtt{B}}^{acdb}-(2+{k^{2}\over 10})s{\mathtt{B}}^{adbc}\Big{]}\,.$

Similarly, from (2.12) we deduce the logarithmic threshold contribution from
the graviton exchange,

$\displaystyle{\mathcal{A}}^{abcd}_{R}|_{\log}={\kappa^{2}\over
32\pi^{2}}{\log(\ell_{p}^{2})}\left[sk{\mathtt{A}}^{abcd}+uk{\mathtt{A}}^{acdb}+tk{\mathtt{A}}^{adbc}\right].$
(6.8)

Using the relations $g_{\rm YM}^{2}=2(2\pi)^{4}\ell_{p}^{3}$ and
$\kappa^{2}={1\over 2}(2\pi)^{8}\ell_{p}^{9}$, we find that the terms
proportional to $\mathtt{A}^{abcd}$ in (6.7) and (6.8) are equal and combine
in their contribution to the logarithmic threshold correction. Comparing the
sum of (6.7) and (6.8) with the flat space limit of (6.4), we determine

$\begin{split}&\kappa_{\log,D^{2}[F^{2}]^{2}}=\frac{1920}{\pi^{4}}k\,,\qquad\kappa_{\log,D^{2}F^{4}}=-\frac{96}{\pi^{4}}(20+k^{2})\,.\end{split}$
(6.9)

##  7 Conclusion

In this work we have introduced a framework to study the dynamic excitations
of M-theory singularities beyond the leading low-energy limit. Our strategy is
based on the holographic duality that maps M-theory in an AdS background, with
singularity in its internal space, to a superconformal field theory whose OPE
data are constrained by the conformal bootstrap with input from supersymmetric
localization. In particular, the $1/N$ corrections to the CFT correlators
translate to the higher derivative corrections to the scattering amplitudes in
the AdS, including those of the gravitons modes in the bulk, as well as of the
quanta localized at the singularity. By taking the large radius limit, where
the M-theory spacetime is locally flat away from the singularity, we can
extract corrections to the effective theory of the M-theory singularity.

We demonstrated this strategy in the example of M-theory on an ${\rm
AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$ orbifold, whose holographic dual is a
certain 3d $\mathcal{N}=4$ CFT with $SU(k)\times SU(2)_{F}\times U(1)$ flavor
symmetry. In the large radius limit, the M-theory background becomes an
$\mathbb{C}^{2}/\mathbb{Z}_{k}\times\mathbb{R}^{1,6}$ orbifold. We analyzed
the holographic correlators of $SU(k)\times SU(2)_{F}$ flavor currents to
subleading orders in the $1/N$ expansion, combining analytic bootstrap methods
and exact results of BPS-protected OPE coefficients from localization, and
determined, in the large radius limit, the absence of 4-derivative effective
couplings of the type $(\mathop{\rm Tr}\nolimits F^{2})^{2}$, $\mathop{\rm
Tr}\nolimits F^{4}$, $R\mathop{\rm Tr}\nolimits F^{2}$ in the $SU(k)$ gauge
theory of the M-theory orbifold.

A curious feature of the scattering amplitude of $SU(k)$ gluons at the
M-theory orbifold singularity is that there are logarithmic threshold
corrections from the 2-loop gluon exchange and tree-level bulk graviton
exchange at the same order in the momentum expansion. We have computed these
threshold corrections applying unitarity cut methods of flat space amplitudes,
and we used the results to determine the order $N^{-{3\over 2}}\log N$ terms
in the AdS gluon Mellin amplitude, which amounts to a new prediction for the
$1/N$ expansion of the dual CFT correlator.

The next order higher derivative effective couplings of the M-theory orbifold,
for instance the $D^{2}(\mathop{\rm Tr}\nolimits F^{2})^{2}$ and
$D^{2}\mathop{\rm Tr}\nolimits F^{4}$ terms of the $SU(k)$ effective gauge
theory, cannot be unambiguously separated from the gluon 2-loop and bulk
graviton exchange amplitudes. To determine, say, the 6-derivative order gluon
effective couplings from the dual CFT, requires knowing up to 2-loop
corrections to the Mellin amplitudes. While the latter can in principle be
fixed in terms of tree-level data via the AdS unitarity method [88], the
necessary computations have not been performed and will be left for future
work. To begin with, the 1-loop correction to the gluon Mellin amplitude in
${\rm AdS}_{4}\times S^{3}$ which should be computable by the method of [100,
38, 40], has no contact term ambiguity and should provide a highly nontrivial
check against the localization constraints derived in this work.

It would also be interesting to determine the tree-level bulk graviton
exchange Mellin amplitude ($M_{R}^{abcd}$ in Section 4.3), which receives
contributions from an infinite tower of graviton KK modes. The lowest mode is
in the BPS-protected stress tensor multiplet, while all higher modes are in
unprotected long multiplets. To fix the relative coefficients between the
single trace exchange diagrams for each mode, one would need to compute the
3-point functions of two gluon and one graviton KK mode from the ${\rm
AdS}_{4}\times S^{7}/\mathbb{Z}_{k}$ geometry, which should be proportional to
the inverse central charge $1/c_{T}$ for all KK modes. A consistency check is
that the flat space limit must match the 7d amplitude determined in this
paper. A similar graviton exchange diagram appeared in gluon scattering on
branes in other examples of AdS/CFT for CFTs in four [100, 37], five, and six
dimensions [101], which should be re-evaluated along similar lines.303030The
computation in [100, 101] only considered the contribution from the leading KK
mode, which by itself does not produce the expected flat space limit of the
graviton exchange amplitude.

Looking ahead, we would like to determine the unprotected higher derivative
corrections, none of which are known in M-theory, with or without orbifold
singularities. To this end, we can employ numerical bootstrap, rather than the
current form of analytic bootstrap, combined with localization input. If we
can pin down the CFT correlator/OPE data to sufficiently high numerical
precision at finite values of $N$, it would be possible to numerically extract
$1/N$ corrections in the large $N$ expansion. This has been partially achieved
in [102, 89, 103] for the stress tensor multiplet correlator in the 3d
$\mathcal{N}=8$ ABJM CFT [15] where the CFT data, determined to high numerical
precision at finite $N$, have been successfully matched with the $1/N$
expansion of Mellin amplitudes at both tree [99] and loop level
[40].313131Without the input of localization data of ABJM theory, the
analogous numerical bootstrap bounds are saturated by pure supergravity in
${\rm AdS}_{4}$ [104]. For the 3d $\mathcal{N}=4$ SCFT considered in this
work, the numerical bootstrap analysis for the flavor multiplet correlators
has been initiated in [82], and should be improved using the BPS-protected OPE
data that we have determined to all orders in $1/N$.

Finally, the orbifold singularities ${\mathbb{C}}^{2}/{\mathbb{Z}}_{k}$
analyzed in this work are special cases of general half-BPS singularities
${\mathbb{C}}^{2}/\Gamma$ in M-theory defined by finite subgroups
$\Gamma\subset SU(2)$ which follow an $ADE$ classification (here we have
focused on the $A_{k-1}$ type). The singular locus supports 7d non-abelian
gauge fields for the corresponding $ADE$ Lie algebra [4]. It would be
interesting to extend our analysis to general $D$\- and $E$-type
singularities, which can be achieved by studying 3d ${\mathcal{N}}=4$ SCFTs
whose holographic duals contain such singularities. The 3d ${\mathcal{N}}=4$
gauge theory studied here is also known as the rank $N$ 3d ADHM gauge theory
[105] for $SU(k)$ which has a natural generalization for classical Lie
algebras including the $D$-type [106, 107], whose holographic dual contains
the desired singularity [68]. Alternatively, there is a class of 3d
${\mathcal{N}}=4$ affine $ADE$ quiver gauge theories [106, 107, 108] that
realize the $ADE$ singularities in their holographic duals and are related to
the ADHM gauge theories of $A$\- and $D$\- types by mirror duality [109]. In
all examples mentioned above, the dual SCFTs can be engineered by M2 branes
probing transversely the singularity of interest. Note that the singularity
has to be non-isolated in the eight transverse directions to the M2 branes in
order to survive the decoupling limit.

To probe singularities of more generic types in M-theory, which typically
preserve fewer supersymmetries, we would need to study correlation functions
in dual SCFTs with few supersymmetries. For example, complex three-fold
Calabi-Yau singularities in M-theory can be analyzed in this way using 3d
${\mathcal{N}}=2$ SCFTs. The simplest example is perhaps the conifold
singularity ${\mathcal{C}}\times{\mathbb{R}}^{4,1}$ where the relevant 3d
gauge theory can be either a modification of the 3d ${\mathcal{N}}=8$ maximal
super-Yang-Mills or the ABJM theory by adding certain 3d ${\mathcal{N}}=2$
chiral multiplets with corresponding superpotentials [16, 110]. Similarly, in
order to extract the effective theory of real codimension-seven singularities
preserving $G_{2}$ holonomy, we would need to study correlators in the dual 3d
${\mathcal{N}}=1$ SCFTs. A class of $G_{2}$ singularities can be found in [5]
and the dual 3d ${\mathcal{N}}=1$ CFTs can be potentially identified by
repeating the analysis of [16, 110] for ${\mathcal{N}}=1$ gauge theories (see
also [111]). In the cases of ${\mathcal{N}}<4$, the constraints from
supersymmetry are much weaker and numerical bootstrap will be important to
analyze such SCFTs in order to extract the dynamics associated with the
corresponding M-theory singularities. Nonetheless, we still expect
localization constraints (if available) will play a crucial role in this
analysis, especially when the singularities are constrained by isometry to a
few parameters. This is possible in the ${\mathcal{N}}=2$ cases where certain
integrated correlation functions can be determined from the supersymmetric
$S^{3}$ partition function of the SCFT with squashing and mass deformations
which can be employed to fix the parameters of the M-theory singularity.

## Acknowledgments

We thank Liam Fitzpatrick, Joao Penedones, and Juan Maldacena for useful
conversations. This work was performed in part at Aspen Center for Physics,
which is supported by National Science Foundation grant PHY-2210452. SMC is
supported by the Royal Society under the grant URF\R1\221310. SSP is supported
in part by the US NSF under Grant No. 2111977. YW is supported in part by the
NSF grant PHY-2210420 and by the Simons Junior Faculty Fellows program. XY is
supported by DOE grant DE-SC0007870.

##  Appendix A 7d ${\mathcal{N}}=1$ superamplitudes

###  A.1 7d ${\mathcal{N}}=1$ super-spinor-helicity formalism

Here we introduce the 7d super-spinor-helicity formalism to describe
superamplitudes of massless particles in the 7d half-maximal supergravity (see
[48] for the super-spinor-helicity formalism for 7d maximal supergravity and
[64, 52] for such a formalism in half-maximal gauged supergravities in other
dimensions).

We denote spinor-helicity variables by 7d symplectic Majorana fermions
$\lambda_{{\alpha}I}$ where ${\alpha}=1,2,\dots,8$ is the $SO(1,6)$ spinor
index and $I=1,2,3,4$ the spinor index for the $SO(5)$ little group. They
satisfy the following quadratic relations with the 7d momentum $p_{\mu}$:

$\displaystyle\lambda_{{\alpha}I}\gamma_{\mu}^{{\alpha}{\beta}}\lambda_{{\beta}J}=p_{\mu}\Omega_{IJ}\,,\quad\lambda_{{\alpha}I}\lambda_{{\beta}J}{\mathcal{C}}^{{\alpha}{\beta}}=0\,,\quad\lambda_{{\alpha}I}\lambda_{{\beta}J}\Omega^{IJ}={1\over
2}p_{\mu}\gamma^{\mu}_{{\alpha}{\beta}}$ (A.1)

where ${\mathcal{C}}^{{\alpha}{\beta}}$ (similarly
${\mathcal{C}}_{{\alpha}{\beta}}$) is the symmetric charge conjugation matrix
for $SO(1,6)$ spinors and $\Omega_{IJ}$ (similarly $\Omega^{IJ}$) is the anti-
symmetric invariant tensor of the little group $SO(5)$ and they are used to
raise and lower the spinor indices as below

$\displaystyle\lambda_{{\alpha}I}=\Omega_{IJ}\lambda_{\alpha}^{J}={\mathcal{C}}_{{\alpha}{\beta}}\lambda^{\beta}_{I}\,,\quad\Omega^{IJ}\Omega_{JK}={\delta}^{I}_{K}\,,\quad{\mathcal{C}}^{{\alpha}{\beta}}{\mathcal{C}}_{{\beta}\gamma}={\delta}^{\alpha}_{\gamma}.$
(A.2)

To take into account the supermultiplet structure, we introduce 4 Grassmann
variables $\eta^{I}$ (Grassmann odd coordinates for on-shell superspace). The
massless one-particle states are labeled by representations of $SO(5)\times
SU(2)_{R}$ as $(\bf m,n)$ according to the dimensions of the irreps.

In this notation, the $8$ bosonic and 8 fermionic states in the vector
multiplet transform in the following representations of $SO(5)\times
SU(2)_{R}$,

$\displaystyle\begin{array}[]{* {5} {c}}(\bf 5,1)&\oplus&(\bf 1,3)&\oplus&(\bf
4,2)\\\
\eta_{I}\eta_{J}|_{[0,1]}&&\eta^{2{\bm{a}}-2}&&\eta_{I}\eta^{2{\bm{i}}-2}\\\
A_{\mu}&&\varphi^{\bm{a}}&&\chi_{{\alpha}}^{\bm{i}}\end{array}$ (A.3)

where ${\bm{a}}=1,2,3$ and ${\bm{i}}=1,2$ are adjoint and spinor indices for
$SU(2)_{R}$ respectively. We have listed the monomials in the Grassmann
variables representing these states in the second line above and the
corresponding 7d fields in the third line. We have also defined the following
little group invariant $\eta^{2}\equiv\eta_{I}\eta_{J}\Omega^{IJ}$ and
covariant $\eta_{I}\eta_{J}|_{\bf 5}\equiv\eta_{[I}\eta_{J]}+{1\over
4}\Omega_{IJ}\eta^{2}$.

Similarly the super-graviton multiplet consists of $40+40$ states in the
following representations,

$\displaystyle\begin{array}[]{* {11} {c}}(\bf 14,1)&\oplus&(\bf
5,3)&\oplus&~{}(\bf 1,1){}&\oplus&(\bf 10,1)&\oplus&(\bf 16,2)&\oplus&(\bf
4,2)\\\ \eta_{I}\eta_{J}\widetilde{\eta}_{K}\widetilde{\eta}_{L}|_{\bf
14}&&\widetilde{\eta}_{K}\widetilde{\eta}_{L}|_{\bf
5}\eta^{2{\bm{a}}-2}&&(\eta\widetilde{\eta})^{2}&&\eta_{I}\eta_{J}\widetilde{\eta}_{K}\widetilde{\eta}_{L}|_{\bf
10}&&\eta_{I}\widetilde{\eta}_{K}\widetilde{\eta}_{L}|_{\bf
16}\eta^{2{\bm{i}}-2}&&\eta_{I}\widetilde{\eta}^{2}\eta^{2{\bm{i}}-2}\\\
g_{\mu\nu}&&V_{\mu}^{\bm{a}}&&\phi&&B_{[\mu\nu]}&&\psi_{{\alpha}\mu}^{\bm{i}}&&\xi_{\alpha}^{\bm{i}}\end{array}$
(A.4)

where we have introduced an auxiliary Grassmann variable
$\widetilde{\eta}_{I}$, which, together with $\eta_{I}$ can be used to write