sigma model. The latter corresponds to an effective field theory in
$\text{AdS}_{2}$, and correlation functions can be computed using standard
Witten diagrams [33]. The conformal bootstrap has also been applied to the
computation of the same four-point functions [35, 39], recovering and
extending the analytical results at strong coupling and studying the finite-
coupling regime numerically. The same approach yielded comparable results in a
less supersymmetric scenario [145]. More generally, line defects provide a
useful and physically interesting laboratory for the application of the
analytical techniques developed in the context of one-dimensional CFTs [146,
147, 148, 121, 55, 49]. Finite-coupling results on defect conformal data were
also obtained using integrability via the quantum spectral curve technology
[149, 79].

###  3.1 ABJM and Wilson Line Defect

In this section, we will present the study of the dCFT associated with the
1/2-BPS Wilson line in ${\cal N}=6$ super Chern-Simons theory with matter
(ABJM) [150]. These correlators of line insertions were bootstrapped in [1,
6]. The structure of Wilson loops in ABJM theory is richer than in ${\cal
N}=4$ SYM [151], admitting different realizations through the fundamental
Lagrangian fields [152, 153, 154], sometimes leading to the same quantum
expectations values through a cohomological equivalence [154]. The relevant
defect field theories should be able to fully distinguish them, possibly
describing different Wilson lines in terms of marginal deformations [155].
Further motivations are the potential existence of topological sectors that
could be associated with new supersymmetric localization procedures and the
relations with integrability that might lead to an alternative derivation of
the $h(\lambda)$ function of ABJM [156] (see also [157]) recently achieved in
[158]. The construction of supersymmetric Wilson loops in ABJ(M) theory [154,
159] is notably more intricate than in the four-dimensional relative
$\mathcal{N}=4$ SYM [140, 141, 160, 161]. For instance, when exploring the
dynamics of the 1/2-BPS heavy massive particles obtained via the Higgsing
procedure [162], one discovers that they are coupled not only to bosons (as
occurs in $D=4$) but to fermions as well. Then the low-energy theory of these
particles turns out to possess a $U(N|N)$ supergauge invariance instead of the
smaller but expected $U(N)\times\hat{U}(N)$ gauge symmetry. Therefore the
(locally) 1/2-BPS Wilson loop operator must be realized as the holonomy of a
superconnection $\mathcal{L}(t)$ living in $U(N|N)$ [154, 162, 159]:

$\mathcal{W}=\textup{Str}\left[P\exp\left(-i\oint
dt\mathcal{L}(t)\right)\mathcal{T}\right].$ (3.1)

This superconnection can be written in terms of the ABJ(M) fields

$\mathcal{L}=\begin{pmatrix}A_{\mu}\dot{x}^{\mu}-\frac{2\pi
i}{k}|\dot{x}|M_{J}{}^{I}C_{I}\bar{C}^{J}&-i\sqrt{\frac{2\pi}{k}}|\dot{x}|\eta_{I}\bar{\psi}^{I}\\\
-i\sqrt{\frac{2\pi}{k}}|\dot{x}|\psi_{I}\bar{\eta}^{I}&\hat{A}_{\mu}\dot{x}^{\mu}-\frac{2\pi
i}{k}|\dot{x}|\hat{M}^{I}{}_{J}\bar{C}^{J}C_{I}\end{pmatrix}.$ (3.2)

Here, $x^{\mu}(t)$ parametrizes the contour while the matrices
$M_{J}{}^{I}(t)$ and $\hat{M}^{I}{}_{J}(t)$ and the spinors $\eta_{I}(t)$ and
$\bar{\eta}^{I}(t)$ are local couplings, determined in terms of the circuit
$x^{\mu}(t)$ by the requirement of preserving some of the supercharges. The
invariance for this type of loop operator does not follow from imposing
$\delta_{\text{susy}}\mathcal{L}=0$ as usual, but the weaker condition

$\delta_{\text{susy}}\mathcal{L}=\mathcal{D}_{t}\mathcal{G}=\partial_{t}\mathcal{G}+i[\mathcal{L},\mathcal{G}]$
(3.3)

where $\mathcal{G}$ is a $u(N|N)$ supermatrix. Namely, the action of
supersymmetry on the connection $\mathcal{L}(t)$ can be cast as an
infinitesimal supergauge transformation of $U(N|N)$ [154, 162, 159]. This
directly implies a vanishing variation for the (super)traced Wilson loop,
provided that $\mathcal{G}$ is periodic along the contour. When $\mathcal{G}$
is not exactly periodic, one can correct the lack of periodicity either by
inserting a twist supermatrix $\mathcal{T}$ in the supertrace (see (3.1))
[159] or by adding to $\mathcal{L}(t)$ a background connection living on the
contour [151]. The explicit form of either $\mathcal{T}$ or the background
connection is irrelevant for the subsequent analysis.

We will focus on the straight-line case. The line is located at
$x^{2}=x^{3}=0$, and the couplings in (3.2) are given by

$\displaystyle\\!\\!\\!\\!\\!\\!M_{J}{}^{I}=\hat{M}^{I}{}_{J}=\left(~{}\begin{array}[]{cccc}\\!\\!\\!-1&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right)\,,\quad\mathrm{and}\quad\eta_{I}^{\alpha}=\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ 0\end{array}\right)_{\\!\\!I}\\!\\!\eta^{+},\ \
\bar{\eta}^{I}_{\alpha}=i\begin{pmatrix}1&0&0&0\end{pmatrix}^{I}\\!\\!\bar{\eta}_{+}$
(3.12)

where $\eta^{+}=\bar{\eta}_{+}^{T}=\begin{pmatrix}1&1\end{pmatrix}$. These
couplings break the original symmetry OSp$(6|4)$ to SU$(1,1|3)$: in particular
the original bosonic subsector of the supergroup containing the Euclidean
conformal group in three-dimensions Sp$(4)\simeq$ SO$(1,4)$ and the R-symmetry
group SO$(6)\simeq$ SU$(4)$ reduces to SU$(1,1)\times$ SU$(3)_{R}\times$
U$(1)_{J_{0}}$. The first factor SU$(1,1)\simeq$ SO$(2,1)$ is simply the
conformal algebra in one dimension; SU$(3)_{R}$ is the residual R-symmetry
group, and the U$(1)_{J_{0}}$ factor is a recombination of the rotations
around the line and a broken R-symmetry diagonal generator. The preserved
generators are given in Appendix B.1. The structure of the residual supergroup
implies that we have a super-dCFT living along the straight line. Its
operators are characterised by a set of four quantum numbers
$[\Delta,j_{0},j_{1},j_{2}]$ associated with the four Cartan generators of the
above bosonic subalgebra.111$\Delta$ is the conformal dimension, $j_{0}$ is
the U$(1)_{J_{0}}$ charge and $j_{1}$ and $j_{2}$ are the SU$(3)_{R}$ labels.
The structure of short and long multiplets representing this subalgebra has
been studied thoroughly in [163], and we review it in Appendix B.1. Recent
development in $1/3$-BPS lines gives additional applications to the results
below since the 1/2- and 1/3-BPS lines share some features for operator
insertions [164].

###  3.2 Line Operators and Correlators

This study will concentrate mainly on defect correlation functions (see
(1.50)) associated with the components of a short multiplet -the displacement
multiplet- which plays a fundamental role in any supersymmetric dCFT. It
contains the displacement operator $\mathbb{D}(t)$ that is supported on every
conformal defect [165] and describes infinitesimal deformations of the defect
profile, as well as other operators associated with the broken R-symmetries
and supercharges. Furthermore, since the operator is chiral, it preserves half
of the dCFT’s supercharges, making the superblock expansion accessible. For
more general operators, such as the 1/3-BPS operators studied in [6],
additional constraints from the superconformal Ward identities are needed to
constrain the superblock expansion and the full super correlator.

We will consider three different complementary realizations of this protected
multiplet that are useful to probe its properties and to calculate its
correlation functions, depending on the coupling regime and the computational
method. A first, more general, realization of the displacement multiplet is
obtained using a superfield $\Phi$ associated with a short multiplet of
SU$(1,1|3)$. A representation theory analysis shows that this superfield is
neutral under SU$(3)$ and annihilated by half of the supercharges (and thus
chiral or antichiral). In four dimensions, the analogous superfield
representation for the displacement multiplet has been derived in [34],
resulting in a superfield charged under the residual $R$-symmetry group. A
natural strong-coupling realization is provided by the AdS/CFT correspondence.
In ABJM theory, a 1/2-BPS Wilson line is dual to the fundamental open string
living in $\text{AdS}_{4}\times\mathbb{C}\textup{{P}}^{3}$, with the
appropriate boundary conditions on the straight contour. The minimal surface
spanned by the string encodes its vacuum expectation value at leading order in
the strong-coupling expansion. Fluctuations around the minimal surface
solution nicely organize in an AdS2 multiplet of transverse modes [166, 167],
whose components precisely match the quantum numbers of the displacement
multiplet. This correspondence is summarised in Table 3.1. A third realization
is obtained by inserting field operators constructed explicitly from the
elementary fields appearing in the ABJM Lagrangian. In our case, the
identification is trickier than in the ${\cal N}=4$ SYM case: the 1/2-BPS
Wilson line in ABJM is constructed as the holonomy of a superconnection living
on a $U(N|N)$ superalgebra [154, 159], and our multiplet should correspond to
the insertion of appropriate supermatrices. Perturbative weak-coupling results
for the correlation functions can be obtained in this framework with Feynman
diagrams. We construct the supermultiplet explicitly in terms of
supermatrices, taking into account the fact that the action of the relevant
supercharges is deformed by the presence of the Wilson line itself 222A basic
difference between 1/2-BPS Wilson lines in ${\cal N}=4$ SYM and ABJM is that
in the first case, the relevant connection is invariant under supersymmetry.
In contrast, in the second one, it undergoes a supergauge $U(N|N)$
transformation [154, 159]. This fact will be important in deriving the correct
field-theoretical representation of the displacement supermultiplet..

Grading | Operator | $\Delta$ | $m^{2}$
---|---|---|---
Fermion | $\mathbb{F}(t)$ | $\frac{1}{2}$ | 0
Boson | $\mathbb{O}^{a}(t)$ | $1$ | 0
Fermion | $\mathbb{\Lambda}_{a}(t)$ | $\frac{3}{2}$ | 1
Boson | $\mathbb{D}(t)$ | $2$ | 2
Table 3.1: Components of the displacement supermultiplet with their scaling
dimensions and the masses of the dual $\text{AdS}_{2}$ string excitations. The
mass is obtained through the AdS/CFT dictionary $m^{2}=\Delta(\Delta-1)$ for
the bosons and $m^{2}=(\Delta-\frac{1}{2})^{2}$ for the fermions.

####  3.2.1 Symmetry Breaking and Superconformal Algebra

The $1/2$ BPS Wilson line in ABJM breaks the three-dimensional $\mathcal{N}=6$
superconformal algebra $osp(6|4)$ to the one-dimensional $\mathcal{N}=6$
superconformal algebra $su(1,1|3)$. This algebra is generated by
$\\{D,P,K,R_{a}^{b},J;Q^{a},\bar{Q}_{a},S^{a},\bar{S}_{a}\\}$, where $a=1,2,3$
and the generators can be identified with $osp(6|4)$ generators (see Appendix
A of [1] and C of [163]). The bosonic generators $\\{D,P,K\\}$ form the one-
dimensional conformal algebra $su(1,1)\sim so(2,1)$, the traceless generators
${R_{a}}^{b}$ ($a,b=1,2,3$) form SU$(3)$, and $J_{0}$ forms the $u(1)$
R-symmetry. Their commutation relations are

$\displaystyle[P,K]$ $\displaystyle=-2D\,,$ $\displaystyle~{}~{}[D,P]$
$\displaystyle=P\,,$ $\displaystyle[D,K]$ $\displaystyle=-K$
$\displaystyle[{R_{a}}^{b},{R_{c}}^{d}]$
$\displaystyle=\delta_{a}^{d}{R_{c}}^{b}-\delta^{b}_{c}{R_{a}}^{d}\,,$ (3.13)

and $J_{0}$ commutes with all of them. The anticommutation relations for the
fermionic generators $Q^{a}$, $\overline{Q}_{a}$ and the corresponding
superconformal charges $S^{a},~{}\overline{S}_{a}$, $a=1,2,3$ are

$\displaystyle\\{Q^{a},\bar{Q}_{b}\\}$ $\displaystyle=2\delta^{a}_{b}P\,,$
$\displaystyle\\{S^{a},\bar{S}_{b}\\}$ $\displaystyle=2\delta^{a}_{b}K$ (3.14)
$\displaystyle\\{Q^{a},\bar{S}_{b}\\}$
$\displaystyle=2\delta_{b}^{a}(D+\tfrac{1}{3}J_{0})-2{R_{b}}^{a}$
$\displaystyle\\{\bar{Q}_{a},S^{b}\\}$
$\displaystyle=2\delta_{b}^{a}(D-\tfrac{1}{3}J_{0})+2{R_{a}}^{b}\,,$ (3.15)

and the non-vanishing mixed commutators are

$\displaystyle[D,Q^{a}]$ $\displaystyle=\frac{1}{2}Q^{a}$
$\displaystyle[D,\overline{Q}_{a}]$
$\displaystyle=\frac{1}{2}\overline{Q}_{a}$ $\displaystyle[K,Q^{a}]$
$\displaystyle=S^{a}$ $\displaystyle[K,\overline{Q}_{a}]$
$\displaystyle=\overline{S}_{a}$ (3.16) $\displaystyle[D,S^{a}]$
$\displaystyle=-\frac{1}{2}S^{a}$ $\displaystyle[D,\overline{S}_{a}]$
$\displaystyle=-\frac{1}{2}\overline{S}_{a}$ $\displaystyle[P,S^{a}]$
$\displaystyle=-Q^{a}$ $\displaystyle[P,\overline{S}_{a}]$
$\displaystyle=-\overline{Q}_{a}$ (3.17) $\displaystyle[{R_{a}}^{b},Q^{c}]$
$\displaystyle=\delta_{a}^{c}Q^{b}-\tfrac{1}{3}\delta_{a}^{b}Q^{c}$
$\displaystyle[{R_{a}}^{b},\overline{Q}_{c}]$
$\displaystyle=-\delta_{c}^{b}\overline{Q}_{a}+\tfrac{1}{3}\delta_{a}^{b}\overline{Q}_{c}$
$\displaystyle[J_{0},Q^{a}]$ $\displaystyle=\tfrac{1}{2}Q^{a}$
$\displaystyle[J_{0},\overline{Q}_{a}]$
$\displaystyle=-\tfrac{1}{2}\overline{Q}_{a}$ (3.18)
$\displaystyle[{R_{a}}^{b},S^{c}]$
$\displaystyle=\delta_{a}^{c}S^{b}-\tfrac{1}{3}\delta_{a}^{b}S^{c}$
$\displaystyle[{R_{a}}^{b},\overline{S}_{c}]$
$\displaystyle=-\delta_{c}^{b}\overline{S}_{a}+\tfrac{1}{3}\delta_{a}^{b}\overline{S}_{c}$
$\displaystyle[J_{0},S^{a}]$ $\displaystyle=\tfrac{1}{2}S^{a}$
$\displaystyle[J_{0},\overline{S}_{a}]$
$\displaystyle=-\tfrac{1}{2}\overline{S}_{a}.$ (3.19)

The quadratic Casimir is

$\displaystyle{\bf
C}^{(2)}=D^{2}-\frac{1}{2}\,\\{K,P\\}+\frac{1}{3}J_{0}^{2}-\frac{1}{2}{R_{a}}^{b}\,{R_{b}}^{a}+\frac{1}{4}\,[\overline{S}_{a},Q^{a}]+\frac{1}{4}\,[S^{a},\overline{Q}_{a}]~{},$
(3.20)

and, when acting on a highest weight state $[\Delta,j_{0},j_{1},j_{2}]$ of
$su(1,1|3)$, it has eigenvalue

$\displaystyle{\bf
c_{2}}=\Delta(\Delta+2)+\frac{j_{0}^{2}}{3}\,-\frac{1}{3}(j_{1}^{2}+j_{1}j_{2}+j_{2}^{2}).$
(3.21)

To compute the superconformal blocks, we must represent the above algebra as
an action on superconformal primaries. For this purpose, we use the superspace
$(t,\theta_{a},\bar{\theta}^{a})$ introduced above.333The natural superspace
to study correlation functions in the bulk was introduced in [168], and the
setup adopted here can be seen as the reduction of the one considered there.
We can then write the differential action of the $\mathcal{N}=6$ generators as

$\displaystyle P$ $\displaystyle=-\partial_{t}$ (3.22) $\displaystyle D$
$\displaystyle=\textstyle{-t\partial_{t}-\frac{1}{2}\theta_{a}\partial^{a}-\frac{1}{2}\bar{\theta}^{a}\bar{\partial}_{a}-\Delta}$
(3.23) $\displaystyle K$
$\displaystyle=-t^{2}\partial_{t}-(t+\theta\bar{\theta})\theta_{a}\partial^{a}-(t-\theta\bar{\theta})\bar{\theta}^{a}\bar{\partial}_{a}-(\theta\bar{\theta})^{2}\partial_{t}-2\,t\,\Delta+\frac{2}{3}j_{0}\,\theta\bar{\theta}$
(3.24) $\displaystyle Q^{a}$
$\displaystyle=\partial^{a}-\bar{\theta}^{a}\partial_{t}$ (3.25)
$\displaystyle\bar{Q}_{a}$
$\displaystyle=\bar{\partial}_{a}-\theta_{a}\partial_{t}$ (3.26)
$\displaystyle S^{a}$
$\displaystyle=(t+\theta\bar{\theta})\partial^{a}-(t-\theta\bar{\theta})\bar{\theta}^{a}\partial_{t}-2\bar{\theta}^{a}\bar{\theta}^{b}\bar{\partial}_{b}-2(\Delta+\frac{1}{3}j_{0})\bar{\theta}^{a}$
(3.27) $\displaystyle\bar{S}_{a}$
$\displaystyle=(t-\theta\bar{\theta})\bar{\partial}_{a}-(t+\theta\bar{\theta})\theta_{a}\partial_{t}-2\theta_{a}\theta_{b}\partial^{b}-2(\Delta-\frac{1}{3}j_{0})\theta_{a}$
(3.28) $\displaystyle J_{0}$
$\displaystyle=-\frac{1}{2}\theta_{a}\partial^{a}+\frac{1}{2}\bar{\theta}^{a}\bar{\partial}_{a}+j_{0}$
(3.29) $\displaystyle R_{a}{}^{b}$
$\displaystyle=-\theta_{a}\partial^{b}+\bar{\theta}^{b}\bar{\partial}_{a}+\textstyle{\frac{1}{3}}\,\delta_{a}^{b}\,(\theta_{c}\partial^{c}-\bar{\theta}^{c}\bar{\partial}_{c})$
(3.30)

where $\partial^{a}=\frac{\partial}{\partial\theta_{a}}$,
$\bar{\partial}_{a}=\frac{\partial}{\partial\bar{\theta}^{a}}$,
$\theta\bar{\theta}=\theta_{a}\bar{\theta}^{a}$ and we neglect the SU$(3)$
charges $j_{1}$ and $j_{2}$ because we will only be interested in neutral
superfields. The superspace is also equipped with super covariant derivatives

$\displaystyle\mathsf{D}^{a}=\partial^{a}+\bar{\theta}^{a}\partial_{t}\,,\qquad\qquad\bar{\mathsf{D}}_{a}=\bar{\partial}_{a}+\theta_{a}\partial_{t}\,.$
(3.31)

####  3.2.2 Displacement Multiplet

This section is devoted to defining and constructing the displacement super
multiplet for a line defect given by the $1/2$ BPS Wilson line in ABJM. For
completeness, we start by recapitulating some basic facts about this theory.
The gauge sector consists of two gauge fields $A_{\mu}$ and $\hat{A}_{\mu}$
belonging respectively to the adjoint of $U(N)$ and $\hat{U}(N)$. The matter
sector instead contains the complex scalar fields $C_{I}$, $\bar{C}^{I}$, and
the fermions $\psi_{I}$ and $\bar{\psi}^{I}$. The fields
$(C_{I},\bar{\psi}^{I})$ transforms in the bifundamental $(N,\bar{N})$ while
the couple $(\bar{C}^{I},\psi_{I})$ lives in the $(\bar{N},N)$. The additional
capital index $I=1,2,3,4$ labels the (anti)fundamental representation of the
R-symmetry group SU$(4)$. The kinetic term for the gauge fields consists of
two Chern-Simon actions of opposite level $(k,-k)$, while the ones for scalars
and fermions take the standard form in terms of the usual covariant
derivatives. To ensure super-conformality, the action is also endowed with a
suitable sextic scalar potential and $\psi^{2}C^{2}$ Yukawa-type interactions
explicitly spelt out in [150, 169].

An infinitesimal variation of the Wilson line (3.1) translates into an
operator insertion according to the identity

$\frac{\braket{(\delta\mathcal{W})\dots}}{\braket{\mathcal{W}}}=-i\int_{\mathcal{C}}dt\,\langle\delta\mathcal{L}(t)\dots\rangle_{\mathcal{W}},$
(3.32)

where on the left-hand side (l.h.s.) we are considering an arbitrary
correlator with the deformed Wilson line while on the right-hand side (r.h.s.)
we are using the definition (1.50). Notice that $\delta$ could be any
generator of a symmetry broken by the Wilson line. When $\delta$ is the action
of a broken fermionic or bosonic charge inside the parent superalgebra, the
resulting operators can be related to an element of the super-multiplet of the
displacement operator. Consider, for instance, the six broken R-symmetry
generators $J_{1}{}^{a}\equiv\mathsf{J}^{a}$ and
$J_{a}{}^{1}\equiv\bar{\mathsf{J}}_{a}$ (see Appendix B.1). Their action will
yield six bosonic defect operators by the symbolic action

$\displaystyle[\mathsf{J}^{a},\mathcal{W}]$ $\displaystyle\equiv
i\delta_{\mathsf{J}^{a}}\mathcal{W}=\int dt~{}\mathcal{W}[\mathbb{O}^{a}(t)]$
$\displaystyle[\bar{\mathsf{J}}_{a},\mathcal{W}]$ $\displaystyle\equiv
i\delta_{\bar{\mathsf{J}}_{a}}\mathcal{W}=\int
dt~{}\mathcal{W}[\bar{\mathbb{O}}_{a}(t)],$ (3.33)

where we indicate by $\mathcal{W}[\mathbb{O}^{a}(t)]$ an operator that is
inserted on the Wilson line. The equations (3.33) are Ward identities and need
to be thought of as inserted in some correlation function. The defect operator
$\mathbb{O}^{a}(t)$ has conformal dimension $\Delta=1$ since the line defect
is dimensionless and the dilatations commute with $\mathsf{J}^{a}$. The
$U(1)_{J_{0}}$ charge is $2$ and it can be read from the commutator
$[J_{0},\mathsf{J}^{a}]=-2[J_{1}{}^{1},\mathsf{J}^{a}]=2\mathsf{J}^{a}$.
Finally, it transforms in the fundamental representation of SU$(3)$. Namely,
this operator is characterised by the following set of four quantum numbers
$[1,2,1,0].$ Similarly for $\bar{\mathbb{O}}_{a}(t)$ we have $[1,-2,0,1].$

The six broken supercharges $Q^{1a}_{-}\equiv i\bar{\mathsf{Q}}^{a}$ and
$Q^{ab}_{{+}}=\epsilon^{abc}\mathsf{Q}_{c}$ define six fermionic defect
operators

$\displaystyle[\mathsf{Q}_{a},\mathcal{W}]$ $\displaystyle=\int
dt~{}\mathcal{W}[\mathbb{\Lambda}_{a}(t)]$
$\displaystyle[\bar{\mathsf{Q}}^{a},\mathcal{W}]$ $\displaystyle=\int
dt~{}\mathcal{W}[\bar{\mathbb{\Lambda}}^{a}(t)].$ (3.34)

The quantum numbers of the line defect and the commutation relations of the
broken and preserved charges fix the quantum numbers of the defect operators.
We find $[\frac{3}{2},\frac{5}{2},0,1]$ for $\mathbb{\Lambda}_{a}(t)$ and
$[\frac{3}{2},-\frac{5}{2},1,0]$ for $\bar{\mathbb{\Lambda}}^{a}(t)$. Finally,
we can consider the two broken translations $\mathsf{P}$ and
$\bar{\mathsf{P}}$ in the directions orthogonal to the defect. They define the
displacement operators

$\displaystyle[\mathsf{P},\mathcal{W}]$ $\displaystyle=\int
dt\mathcal{W}[\mathbb{D}(t)]$ $\displaystyle[\bar{\mathsf{P}},\mathcal{W}]$
$\displaystyle=\int dt\mathcal{W}[\bar{\mathbb{D}}(t)]$ (3.35)

with charges $[2,3,0,0]$ and $[2,-3,0,0]$ respectively. The above construction
does not uniquely determine the above set of operators. Since they are defined
as objects inserted in the defect, and we integrate over the position along
the loop, we can add a total derivative with respect to $t$ to the integrand
without altering the result. The set of defect operators obtained through the
action of the broken charges organizes itself as a supermultiplet. The action
of the preserved supercharges $Q^{a}=Q^{1a}_{+}$ and
$\bar{Q}_{a}=i\,\frac{1}{2}\epsilon_{abc}Q_{-}^{bc}$ on them is essentially
dictated by the commutation relations between broken and preserved charges.
For instance, to compute $[Q^{a},\mathbb{O}^{b}]$ one has simply to consider
the commutator $[Q^{a},\mathsf{J}^{b}]$ acting on $\mathcal{W}$ and we get

$\displaystyle[Q^{a},\mathsf{J}^{b}]$
$\displaystyle=Q^{ab}_{+}=\epsilon^{abc}\mathsf{Q}_{c}$
$\displaystyle\Rightarrow$ $\displaystyle[Q^{a},\mathbb{O}^{b}]$
$\displaystyle=\epsilon^{abc}\mathbb{\Lambda}_{c}.$ (3.36)

From the commutation relation of $Q_{a}$ with the broken supercharges and
translation, we immediately find

$\displaystyle\\{Q^{a},\mathbb{\Lambda}_{b}\\}$
$\displaystyle=-2\delta^{a}_{b}\mathbb{D}\qquad\text{and}\qquad[Q^{a},\mathbb{D}]=0.$
(3.37)

Fixing the action of $\bar{Q}_{a}$ on these defect operators requires more
attention. The naïve application of the above procedure would give zero since
$\bar{Q}_{a}$ commutes with all the broken charges. This is inconsistent with
$\\{Q^{a},\bar{Q}_{b}\\}=2\delta^{a}_{b}P$, where $P$ generates the
translations along the line. However, as stressed above, any result obtained
from (3.33), (3.34), and (3.35) is defined up to a derivative with respect to
$t$, which yields zero when integrated. To fix the form of these derivatives
in the commutation relations, it is more convenient to use the super-Jacobi
identities. For instance, we can determine $[\bar{Q}_{c},\mathbb{D}]$ as
follows444We choose to represent $P$ as $-\partial_{t}$ and then
$[P,\Phi]=\partial_{t}\Phi$, see the discussion in [170].

$\displaystyle
0=\\{\bar{Q}_{c},[Q^{a},\mathbb{D}]\\}-\\{Q^{a},[\mathbb{D},\bar{Q}_{c}]\\}+[\mathbb{D},\\{\bar{Q}_{c},Q^{a}\\}]=Q^{a},[\bar{Q}_{c},\mathbb{D}]+\partial_{t}\mathbb{\Lambda}_{c}$
$\displaystyle\Rightarrow\qquad[\bar{Q}_{c},\mathbb{D}]=-\partial_{t}\mathbb{\Lambda}_{c}.$
(3.38)

where we assumed that the action of the translation $P$ is realised by
$-\partial_{t}$ to be consistent with (3.22). Similarly we can show that
$\\{\bar{Q}_{a},\mathbb{\Lambda}_{b}\\}=-2\epsilon_{abc}\partial_{t}\mathbb{O}^{c}$.
Since we do not have an operator of lower dimension, it would be natural to
set $[\bar{Q}_{c},\mathbb{O}^{b}]=0$. However, this choice is inconsistent
with the super-Jacobi identity

$\displaystyle 0=$
$\displaystyle\\{\bar{Q}_{c},[Q^{a},\mathbb{O}^{b}]\\}-\\{Q^{a},[\mathbb{O}^{b},\bar{Q}_{c}]\\}+[\mathbb{O}^{b},\\{\bar{Q}_{c},Q^{a}\\}]=-2\delta_{c}^{b}\partial_{t}\mathbb{O}^{a}+\\{Q^{a},[\bar{Q}_{c},\mathbb{O}^{b}]\\}.$
(3.39)

The consistency of (3.39) suggests the existence of an additional fermionic
operator $\mathbb{F}$, that is a singlet under SU$(3)$, which obeys the
anticommutation relation:

$\displaystyle\\{Q^{a},\mathbb{F}\\}$ $\displaystyle=\mathbb{O}^{a},$ (3.40)

which in turn implies
$[\bar{Q}_{c},\mathbb{O}^{b}]=2\delta^{b}_{c}\partial_{t}\mathbb{F}$. The
Jacobi identity for this new field immediately shows that
$\\{\bar{Q}_{a},\mathbb{F}\\}$ can be consistently chosen to vanish.
Furthermore, $\mathbb{F}$ has the correct quantum numbers to be the
superprimary of the chiral multiplet
$\bar{\mathcal{B}}^{\frac{1}{2}}_{\frac{3}{2},0,0}$ (see Appendix B.1) with
the structure

$\displaystyle\bar{\mathcal{B}}^{\frac{1}{2}}_{\frac{3}{2},0,0}:$ (3.41)

where this supermultiplet is realized in terms of the fundamental fields of
ABJM theory [1]. An analogous action of the supercharges can be derived for
the conjugated operators leading to the barred version of the commutation
relations in Table 3.2.

Summary of supersymmetry transformations $\\{Q^{a},\mathbb{F}\\}=\mathbb{O}^{a}$ |  |  | $\\{\bar{Q}_{a},\mathbb{F}\\}=0$
---|---|---|---
$[Q^{a},\mathbb{O}^{b}]=\epsilon^{abc}\mathbb{\Lambda}_{c}$ |  |  | $[\bar{Q}_{a},\mathbb{O}^{b}]=2\delta^{b}_{a}\partial_{t}\mathbb{F}$
$\\{Q^{a},\mathbb{\Lambda}_{b}\\}=-2\delta^{a}_{b}\mathbb{D}$ |  |  | $\\{\bar{Q}_{a},\mathbb{\Lambda}_{b}\\}=-2\epsilon_{abc}\partial_{t}\mathbb{O}^{c}$
$[Q^{a},\mathbb{D}]=0$ |  |  | $[\bar{Q}_{c},\mathbb{D}]=-\partial_{t}\mathbb{\Lambda}_{c}$
Table 3.2: Summary of the supersymmetry transformations of the displacement
supermultiplet of a 1/2-BPS line-defect in $\mathcal{N}=6$ supersymmetric
theories in $d=3$.

The displacement supermultiplet (3.41) appearing in the dCFT1 living along the
Wilson line should match the string transverse excitations via AdS/CFT. The
expansion of the Green-Schwarz string action around the minimal surface
solution results in a multiplet of fluctuations transverse to the string [166,
167] whose components match precisely the quantum numbers of the operators in
(3.41), once the two standard relations between AdS2 masses and the
corresponding CFT1 operator dimensions are taken into account,
$m^{2}=\Delta(\Delta-1)$ for scalars and $\Delta=\frac{1}{2}+|m|$ for spinors
[171, 172]. In particular, in the scalar fluctuation sector, one finds one
massive (with $m^{2}=2$) complex scalar field $X$ in AdS4, corresponding to
the $\Delta=2$ displacement operator $\mathbb{D}$, and three massless
_complex_ scalar fluctuations $w^{a},~{}a=1,2,3$ in
$\mathbb{C}\textup{{P}}^{3}$, corresponding to the $\Delta=1$ operators
$\mathbb{O}^{a},~{}a=1,2,3$. In the fermionic sector, there are two massless
fermions which should correspond to the $\Delta=\frac{1}{2}$ fermionic
superprimary $\mathbb{F}$ of the multiplet and its conjugate, as well as six
massive fermions (of which three with mass $m_{F}=1$ and three with
$m_{F}=-1$) corresponding to the $\Delta=\frac{3}{2}$ fermionic operator
$\mathbb{\Lambda}_{a}$ and its conjugate.

####  3.2.3 Chiral Correlators

The representation theory analysis of the supergroup SU$(1,1|3)$ reviewed in
the previous section shows that the displacement operator belongs to a chiral
multiplet with R-charge $j_{0}=\frac{3}{2}$ and dimension $\Delta=\frac{1}{2}$
(the conjugate operator belongs to an antichiral multiplet with opposite
R-charge). For this section, we will consider a chiral multiplet with
arbitrary R-charge $j_{0}$ and dimension $\Delta=\frac{j_{0}}{3}$. A chiral
superfield $\Phi_{j_{0}}$ should respect the chirality condition

$\bar{\mathsf{D}}_{a}\Phi_{j_{0}}=0\,,\qquad$ (3.42)

for every value of $a$. By defining the chiral coordinate
$y=t+\theta_{a}\bar{\theta}^{a}$, such that $\bar{\mathsf{D}}_{a}y=0$, one
simply has the component expansion

$\displaystyle\Phi_{j_{0}}(y,\theta)=\phi(y)+\theta_{a}\psi^{a}(y)-\frac{1}{2}\theta_{a}\theta_{b}\,\epsilon^{abc}\,\eta_{c}(y)+\frac{1}{3}\theta_{a}\theta_{b}\theta_{c}\,\epsilon^{abc}\xi(y)\,,$
(3.43)

where the numerical coefficients of each component are fixed by the
consistency between the action of the supercharges on the superfield and the
commutation relations in Table 3.2.
Similarly, the antichiral field is expanded as

$\displaystyle\bar{\Phi}_{j_{0}}(y,\theta)=\bar{\phi}(y)+\bar{\theta}^{a}\bar{\psi}_{a}(y)+\frac{1}{2}\bar{\theta}^{a}\bar{\theta}^{b}\,\epsilon_{abc}\,\bar{\eta}^{c}(y)-\frac{1}{3}\bar{\theta}^{a}\bar{\theta}^{b}\bar{\theta}^{c}\,\epsilon_{abc}\,\bar{\xi}(y)\,,$
(3.44)

where the coefficients in the expansion are determined by the Ward identities
relating the two-point functions of different super descendants.555Or
equivalently, one can act explicitly on the superfield with the supercharges
since the commutations with the component fields were computed using the same
Ward identities. Writing the superfield as
$\Phi(y)=\mathbb{F}+c_{1}\theta^{a}\mathbb{O}_{a}+\frac{1}{2}c_{2}\theta^{a}\theta^{b}\epsilon_{abc}\Lambda^{c}+\frac{1}{3}c_{3}\theta^{3}\mathbb{D},$
(3.45) one obtains the commutation relations
$\displaystyle\\{Q_{a},\mathbb{F}\\}=c_{1}\mathbb{O}_{a}$ (3.46)
$\displaystyle[Q_{a},\mathbb{O}_{b}]=-\frac{c_{2}}{c_{1}}\epsilon_{abc}\Lambda^{c}$
(3.47)
$\displaystyle\\{Q_{a},\Lambda^{c}\\}=2\frac{c_{3}}{c_{2}}\delta_{a}^{c}\mathbb{D}$
(3.48) $\displaystyle[Q_{a},\mathbb{D}]=0,$ (3.49) which when compared to
(3.43) give the correct factors.  Using the commutation relations from above,
we can relate the constants from the two-point functions

$\langle\mathbb{F}(0)\bar{\mathbb{F}}(t)\rangle=\frac{c_{\mathbb{F}}}{t}\quad\quad\langle\mathbb{O}_{a}(0)\bar{\mathbb{O}}^{b}(t)\rangle=\frac{\delta_{a}^{b}c_{\mathbb{O}}}{t^{2}}\quad\quad\langle\Lambda(0)^{a}\bar{\Lambda}_{b}(t)\rangle=\frac{\delta^{a}_{b}c_{\Lambda}}{t^{3}}\quad\quad\langle\mathbb{D}(0)\bar{\mathbb{D}}(t)\rangle=\frac{c_{\mathbb{D}}}{t^{4}}\quad\quad$

by using the vanishing of some of the two-point correlators such as
$\langle\mathbb{F}\bar{\mathbb{O}}_{a}\rangle$. Acting on it with Q and using
the previous commutation relations gives:

$\displaystyle\quad\quad Q_{a}\langle\mathbb{F}\bar{\mathbb{O}}^{b}\rangle=0$
(3.50)
$\displaystyle\Leftrightarrow\langle\\{Q_{a},\mathbb{F}\\}\bar{\mathbb{O}}^{b}\rangle+\langle\mathbb{F}[Q_{a},\bar{\mathbb{O}}^{b}]\rangle=0$
(3.51)
$\displaystyle\Leftrightarrow\langle\mathbb{O}_{a}\bar{\mathbb{O}}^{b}\rangle+2\delta_{a}^{b}\langle\mathbb{F}\partial_{t}\bar{\mathbb{F}}\rangle=0$
(3.52)
$\displaystyle\Leftrightarrow\frac{\delta_{a}^{b}c_{\mathbb{O}}}{\tau^{2}}+2\delta_{a}^{b}\partial_{t}\frac{c_{\mathbb{F}}}{\tau}=0$
(3.53) $\displaystyle\Leftrightarrow c_{\mathbb{O}}=+2c_{\mathbb{F}}.$ (3.54)

Doing the same process on
$\langle\mathbb{O}_{a}\bar{\Lambda}_{b}\rangle\quad\langle\Lambda^{b}\bar{\mathbb{D}}\rangle$
gives the equalities:

$\displaystyle c_{\Lambda}=4c_{\mathbb{O}}$ (3.55) $\displaystyle
2c_{\mathbb{D}}=3c_{\Lambda}.$ (3.56)

The two-point function of a chiral and antichiral superfield must be expressed
in terms of the chiral distance
($\bar{\mathsf{D}}_{i}\braket{i\bar{j}}=\mathsf{D}_{j}\braket{i\bar{j}}=(Q_{i}+Q_{j})\,\braket{i\bar{j}}=(\bar{Q}_{i}+\bar{Q}_{j})\,\braket{i\bar{j}}=0$)

$\displaystyle\braket{i\bar{j}}=y_{i}-y_{j}-2{\theta_{a}}_{i}\bar{\theta^{a}}_{j}\,,$
(3.57)

and can be written as

$\displaystyle\braket{\Phi_{j_{0}}(y_{1},\theta_{1})\bar{\Phi}_{-j_{0}}(y_{2},\bar{\theta}_{2})}=\frac{c_{\Phi_{j_{0}}}}{\braket{1\bar{2}}^{\frac{2j_{0}}{3}}}\,.$
(3.58)

For the case of the displacement supermultiplet, the normalization factor
$C_{\Phi}$ has an important physical interpretation. As shown in subsection
3.2.2, most of the components in the displacement multiplet are obtained by
the action of a broken symmetry generator. Such generators have a natural
normalization in the bulk theory. Therefore the Ward identities (3.35) fix the
physical normalization of the displacement operator, making its two-point
function an important piece of dCFT data. There is concrete evidence that in
the presence of a superconformal defect, this coefficient is related to the
one-point function of the stress tensor operator [173, 174, 175]. Moreover,
for the case of the Wilson line, this coefficient is particularly important as
it computes the energy emitted by an accelerating heavy probe in a conformal
field theory [59, 176], often called Bremsstrahlung function. The relation
with the two-point function of the displacement operator in the context of
superconformal theories in three and four dimensions has allowed for the exact
computation of this quantity in a variety of examples [59, 176, 173, 166, 167,
177, 178, 179, 163, 180, 181, 174, 182, 183]. In the present context we have

$\displaystyle C_{\Phi}(\lambda)=2B_{1/2}(\lambda)\,,$ (3.59)

where $B_{1/2}(\lambda)$ is the Bremsstrahlung function associated to the
1/2-BPS Wilson line in ABJM theory (see [177, 163]). From the Graßmann
expansion of the two-point function (3.58), we extract the following two-point
functions

$\displaystyle\langle\mathbb{F}(t_{1})\bar{\mathbb{F}}(t_{2})\rangle$
$\displaystyle=\frac{C_{\Phi}}{t_{12}}$
$\displaystyle\langle\mathbb{O}^{a}(t_{1})\bar{\mathbb{O}}_{b}(t_{2})\rangle$
$\displaystyle=\frac{2\,C_{\Phi}\,\delta^{a}_{b}}{t_{12}^{2}}$ (3.60)
$\displaystyle\langle\mathbb{\Lambda}_{a}(t_{1})\bar{\mathbb{\Lambda}}^{b}(t_{2})\rangle$
$\displaystyle=\frac{8\,C_{\Phi}\,\delta_{a}^{b}}{t_{12}^{3}}$
$\displaystyle\langle\mathbb{D}(t_{1})\bar{\mathbb{D}}(t_{2})\rangle$
$\displaystyle=\frac{12\,C_{\Phi}}{t_{12}^{4}}$ (3.61)

which is consistent with the component expansion above.

The four-point function of 1/2-BPS (chiral) operators will, in general, be a
function of four spacetime coordinates and twelve Graßmanian coordinates
subject to the constraints from three bosonic generators $(P,K,D)$ and twelve
Graßmannian generators ($Q^{a},\bar{Q}_{a},S^{a},\bar{S}_{a}$) of $su(1,1|3)$.
By simple counting, one expects the correlator to depend on a single super
cross-ratio. For the two different four-point function configurations, we have

$\displaystyle\\!\\!\braket{\Phi_{j_{0}}(y_{1},\theta_{1})\bar{\Phi}_{-j_{0}}(y_{2},\bar{\theta}_{2})\Phi_{j_{0}}(y_{3},\theta_{3})\bar{\Phi}_{-j_{0}}(y_{4},\bar{\theta}_{4})}$
$\displaystyle=\frac{C^{2}_{\Phi_{j_{0}}}}{\braket{1\bar{2}}^{\frac{2j_{0}}{3}}\braket{3\bar{4}}^{\frac{2j_{0}}{3}}}\,f(\mathcal{Z})\,,$
(3.62)
$\displaystyle\\!\\!\braket{\Phi_{j_{0}}(y_{1},\theta_{1})\bar{\Phi}_{-j_{0}}(y_{2},\bar{\theta}_{2})\bar{\Phi}_{-j_{0}}(y_{3},\bar{\theta}_{3})\Phi_{j_{0}}(y_{4},\theta_{4})}$
$\displaystyle=-\,\frac{C^{2}_{\Phi_{j_{0}}}}{\braket{1\bar{2}}^{\frac{2j_{0}}{3}}\braket{4\bar{3}}^{\frac{2j_{0}}{3}}}\,h(\mathcal{X})\,,$
(3.63)

where

$\displaystyle\mathcal{Z}$
$\displaystyle=\frac{\braket{1\bar{2}}\braket{3\bar{4}}}{\braket{1\bar{4}}\braket{3{\bar{2}}}}$
$\displaystyle\mathcal{X}$
$\displaystyle=-\frac{\braket{1\bar{2}}\braket{4\bar{3}}}{\braket{1\bar{3}}\braket{2\bar{4}}}$
(3.64)

are the two superconformal cross-ratios corresponding to the two different
correlators. These two invariants are built out of the chiral distance (3.57).
In the general case, one may have a set of additional superconformal
invariants which are nilpotent due to their Graßmann nature. For long-
multiplet four-point functions, using such nilpotent invariants guarantees a
finite truncation in the superspace expansion (see [87]). However, none of
these invariants is compatible with the chirality condition (3.42). The
absence of nilpotent invariants is expected for correlators of 1/2-BPS
operators and, in general, for four-point functions containing two chiral and
two anti-chiral operators [184]. It is important to notice that both
correlators (3.119) and (3.120) are ordered such that
$t_{1}<t_{2}<t_{3}<t_{4}$. These two correlators would be related by crossing
in higher dimensions, but this is not the case in one dimension. To make this
more concrete, let us consider the bosonic part of the cross-ratios (3.64)

$\displaystyle z$ $\displaystyle=\frac{t_{12}t_{34}}{t_{14}t_{32}}$
$\displaystyle\chi$ $\displaystyle=\frac{t_{12}t_{34}}{t_{13}t_{24}}$ (3.65)

where $z$ is the bosonic part of $\mathcal{Z}$ and $\chi$ is the bosonic part
of $\mathcal{X}$. With our ordering we have $z<0$ and $0<\chi<1$. The
transformation

$\displaystyle z=\frac{\chi}{\chi-1}\,$ (3.66)

relates the two cross-ratios. The absence of nilpotent invariants implies that
the superprimary correlators

$\displaystyle\\!\\!\braket{\phi(t_{1})\bar{\phi}(t_{2})\phi(t_{3})\bar{\phi}(t_{4})}$
$\displaystyle=\braket{\phi(t_{1})\bar{\phi}(t_{2})}\braket{\phi(t_{3})\bar{\phi}(t_{4})}\,f(z)\,,$
(3.67)
$\displaystyle\\!\\!\braket{\phi(t_{1})\bar{\phi}(t_{2})\bar{\phi}(t_{3})\phi(t_{4})}$
$\displaystyle=\braket{\phi(t_{1})\bar{\phi}(t_{2})}\braket{\bar{\phi}(t_{3})\phi(t_{4})}\,h(\chi)\,,$
(3.68)

fully determine the four-point functions of the whole superconformal
multiplet, i.e. the correlators of superconformal descendants can be obtained
by the action of differential operators on $f(z)$. Of course, one is free to
express the function $f(z)$ in terms of the cross-ratio $\chi$ by considering
$f(\frac{\chi}{\chi-1})$. One can also take the analytic continuation of
$f(z)$ for $0<z<1$, which would naively establish a relation between (3.67)
and (3.68).666$f(z)$ has branch cut singularities at coincident points, i.e.
$z=0,1,\infty$. Nevertheless, this is not the case in one dimension, and
$h(\chi)$ is not the analytic continuation of $f(z)$. Still, we will see in
subsection 3.2.4 that a relationship between these two functions exists
through their $s$-channel block expansion.

Just as in the case of the two-point function, the correlators of the super
descendants can be obtained by expanding the Graßmann coordinates of the four-
point correlator in superspace.

$\displaystyle\\!\\!\\!\\!\\!\braket{\mathbb{F}(t_{1})\bar{\mathbb{F}}(t_{2})\mathbb{F}(t_{3})\bar{\mathbb{F}}(t_{4})}=\frac{C_{\Phi}^{2}}{t_{12}t_{34}}f(z)$
(3.69)
$\displaystyle\\!\\!\\!\\!\\!\braket{\mathbb{O}^{a_{1}}(t_{1})\bar{\mathbb{O}}_{a_{2}}(t_{2})\mathbb{O}^{a_{3}}(t_{3})\bar{\mathbb{O}}_{a_{4}}(t_{4})}=\frac{4C_{\Phi}^{2}}{t_{12}^{2}t_{34}^{2}}\,\Big{[}\delta^{a_{1}}_{a_{2}}\delta^{a_{3}}_{a_{4}}\,\big{(}\,f(z)-zf^{\prime}(z)+z^{2}f^{\prime\prime}(z)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\delta^{a_{1}}_{a_{4}}\delta^{a_{3}}_{a_{2}}\,\big{(}z^{2}f^{\prime}(z)+z^{3}f^{\prime\prime}(z)\big{)}\,\Big{]}$
(3.70)
$\displaystyle\\!\\!\\!\\!\\!\braket{\mathbb{D}(t_{1})\bar{\mathbb{D}}(t_{2})\mathbb{D}(t_{3})\bar{\mathbb{D}}(t_{4})}=\frac{(12C_{\Phi})^{2}}{t_{12}^{4}t_{34}^{4}}\,\frac{1}{36}\Big{[}36f(z)-36(z^{4}+z)f^{\prime}(z)+18z^{2}(-14z^{3}+3z^{2}+1)f^{\prime\prime}(z)$
$\displaystyle\qquad\qquad\qquad\qquad-6z^{3}\left(55z^{3}-39z^{2}+3z+1\right)f^{(3)}(z)-3z^{4}\left(46z^{3}-63z^{2}+18z-1\right)f^{(4)}(z)$
$\displaystyle\qquad\qquad\qquad\qquad-3(z-1)^{2}z^{5}(7z-1)f^{(5)}(z)-(z-1)^{3}z^{6}f^{(6)}(z)\,\Big{]}\,$
(3.71)
$\displaystyle\\!\\!\\!\\!\\!\braket{\mathbb{D}(t_{1})\bar{\mathbb{D}}(t_{2})\mathbb{O}^{a_{3}}(t_{3})\bar{\mathbb{O}}_{a_{4}}(t_{4})}=\frac{24C_{\Phi}^{2}}{t_{12}^{4}t_{34}^{2}}\delta^{a_{3}}_{a_{4}}\frac{1}{6}\Big{[}\,(\\!1\\!-z)\,z^{4}f^{(4)}\\!(z)-(3z+\\!1\\!)\,z^{3}f^{(3)}\\!(z)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+3z^{2}\,f^{\prime\prime}(z)-6zf^{\prime}(z)+6f(z)\,\Big{]}\,,$
(3.72)

where for each correlator, we factorised the (squared) two-point function
contribution (3.60) arising from the double OPE. In the next section, we will
use analytic bootstrap techniques to evaluate the function $f(z)$
perturbatively to third-order at strong coupling. In section 3.3, the
superspace analysis is used to find $f(z)$ from the correlators of
superconformal descendants evaluated directly at strong coupling using Witten
diagrammatics. The consistency of these various correlators with the
superspace analysis provides a non-trivial check of the relations above.

####  3.2.4 Symmetry and Superblocks

##### Selection rules for the 1/2-BPS operators

Even though we only consider half-BPS multiplets as external operators, more
general multiplets can be exchanged when an OPE is applied to the correlator.
In our case, there are two qualitatively different OPE channels to consider,
depending on whether we take the chiral-antichiral OPE $\Phi\times\bar{\Phi}$
or the chiral-chiral OPE $\Phi\times\Phi$. Each has selection rules for
superconformal representations appearing in these two channels and
corresponding superconformal blocks. We start from the chiral-antichiral
channel. In [181], all the selection rules for the 1/6-BPS defect theory were
derived, and it was found that only the identity and long multiplets can
appear in the chiral-antichiral OPE for $su(1,1|1)$, a.k.a. $\mathcal{N}=2$
supersymmetry. For the $\mathcal{N}=6$ case of interest here, every pair
$\\{Q^{a},\bar{Q}_{a}\\}$ at fixed $a$ generates a $su(1,1|1)$ subgroup of
$su(1,1|3)$, and a chiral multiplet of $su(1,1|3)$ is also a chiral multiplet
of all the three $su(1,1|1)$ subgroups. This implies that the operators
appearing in the $\Phi\times\bar{\Phi}$ OPE must belong to long multiplets of
all the $su(1,1|1)$ subgroups, i.e. they must not be annihilated by any
supercharge. Furthermore, three-point functions with one chiral, one
antichiral and one long multiplet are non-vanishing only when the superprimary
R-charges sum to zero, namely a long multiplet enters only when its
superprimary can be exchanged. We conclude that

$\displaystyle\bar{\mathcal{B}}_{j_{0}}\times\mathcal{B}_{-j_{0}}\sim\mathcal{I}+\mathcal{A}^{\Delta}_{0,0,0}\,,$
(3.73)

where $\mathcal{I}$ is the identity, $\mathcal{B}_{-j_{0}}$ is the chiral
multiplet, and $\mathcal{A}^{\Delta}_{0,0,0}$ is an uncharged long
multiplet.777We use the notation of Appendix B.1 for the $su(1,1|3)$
supermultiplets. In this case, the unitarity bound (B.24) is $\Delta\geq 0$,
and every positive dimension is allowed for long operators.

For the chiral-chiral channel, the situation is richer. We will borrow an
argument from [185] to extract the selection rules. Consider the chiral
superprimary operator $\phi$. chirality gives

$\displaystyle[\bar{Q}_{a},\phi(t)]$ $\displaystyle=0$
$\displaystyle[\bar{S}_{a},\phi(t)]$ $\displaystyle=0$ (3.74)

for any $a$ and any $t$. The first condition is simply the definition of
chirality. In contrast, the second one comes from the requirement that $\phi$
is a superprimary at the origin together with the commutation relation
$[P,\bar{S}_{a}]=-\bar{Q}_{a}$. These two conditions imply that any operator
$\mathcal{O}$ appearing in the $\phi\times\phi$ OPE must respect

$\displaystyle[\bar{Q}_{a},\mathcal{O}(t)]$ $\displaystyle=0$
$\displaystyle[\bar{S}_{a},\mathcal{O}(t)]$ $\displaystyle=0$ (3.75)

for any $a$. It is then immediate to realize that the only superprimary
operator which is allowed to appear is a chiral operator of dimension
$\Delta_{\text{exc}}=\frac{2j_{0}}{3}$. All other multiplets will contribute
with a single super descendant (and all its conformal descendants) generated
by the repeated action of $\bar{Q}_{a}$. Concretely, a long multiplet will
contribute with the operator generated by $\bar{Q}^{3}O$ (here
$\bar{Q}^{3}=\epsilon^{abc}\bar{Q}_{a}\bar{Q}_{b}\bar{Q}_{c})$, where $O$ is
the superprimary. The complete analysis yields

$\displaystyle\bar{\mathcal{B}}_{j_{0}}\times\bar{\mathcal{B}}_{j_{0}}\sim\bar{\mathcal{B}}_{2j_{0}}+\bar{\mathcal{B}}_{2j_{0}+\frac{1}{2},1}+\bar{\mathcal{B}}_{2j_{0}+1,0,1}+\mathcal{A}^{\Delta}_{2j_{0}+\frac{3}{2},0,0}.$
(3.76)

In particular, in the OPE of the superprimary operator $\phi$, every
supermultiplet contributes with a single conformal family, whose conformal
primary has quantum numbers $[\Delta,2j_{0},0,0]$. The dimension is fixed in
terms of $j_{0}$ for short multiples. In Table 3.3, we summarize the schematic
form of the only relevant super descendant operators and make explicit their
conformal dimension, which can be easily obtained from the associated
superprimary. The dimension of the long multiplet is unfixed, but it should
respect the unitarity bound (B.24). For this specific case, we find that the
dimension of the superprimary must be $\Delta>\frac{2j_{0}}{3}+\frac{1}{2}$,
while the relevant super descendant, obtained by acting with all the
$\bar{Q}$’s, must have dimension888Here the equality is excluded because in
that case, the long multiplet decomposes as in Table B.1 and the relevant
super descendant falls back into the $\bar{\mathcal{B}}_{2j_{0}+1,0,1}$
multiplet.

$\Delta_{\text{exc}}^{\text{long}}>\frac{2j_{0}}{3}+2\,.$ (3.77)

This bound will be very important in the following, where we will focus on the
$j_{0}=\frac{3}{2}$ case. In summary, the $\phi\times\phi$ spectrum admits
three protected conformal primaries of dimensions $\frac{2j_{0}}{3}$,
$\frac{2j_{0}}{3}+1$ and $\frac{2j_{0}}{3}+2$ and infinitely many operators
with unprotected dimensions strictly higher than the protected ones.

Multiplet | Exc. | $\Delta_{\text{exc}}$
---|---|---
$\bar{\mathcal{B}}_{2j_{0}}$ | $O$ | $\frac{2j_{0}}{3}$
$\bar{\mathcal{B}}_{2j_{0}+\frac{1}{2},1}$ | $\bar{Q}O$ | $\frac{2j_{0}}{3}+1$
$\bar{\mathcal{B}}_{2j_{0}+1,0,1}$ | $\bar{Q}^{2}O$ | $\frac{2j_{0}}{3}+2$
$\mathcal{A}^{\Delta}_{2j_{0}+\frac{3}{2},0,0}$ | $\bar{Q}^{3}O$ | $\Delta+\frac{3}{2}$
Table 3.3: The multiplets contributing to the chiral-chiral OPE with a
schematic representation of the only superconformal descendant (but conformal
primary) contributing to the OPE (in this Table $O$ indicates the superprimary
of each multiplet).

##### Superblocks

We now derive the superconformal blocks associated with the two channels. In
the chiral-chiral channel, each supermultiplet only contributes with a single
conformal family. Therefore one only needs to select the $sl(2)$ conformal
blocks with the appropriate dimensions [186].

$\displaystyle g_{h}(\chi)=\chi^{h}\,_{2}F_{1}(h,h;2h;\chi).$ (3.78)

Each conformal primary listed in Table 3.3 contributes with an $sl(2)$ block
with $h=\Delta_{\text{exc}}$. Let us consider the specific correlator of
interest here999The correlator (3.67) does not admit an expansion in this
channel since one cannot take point $\phi(t_{1})$ close to $\phi(t_{3})$.

$\displaystyle\braket{\phi(t_{1})\bar{\phi}(t_{2})\bar{\phi}(t_{3})\phi(t_{4})}=\frac{C_{\Phi}^{2}}{t_{14}^{\frac{2j_{0}}{3}}t_{23}^{\frac{2j_{0}}{3}}}\hat{h}(\chi)$
(3.79)

where, comparing with (3.68) we defined
$\hat{h}(\chi)=(\frac{1-\chi}{\chi})^{\frac{2j_{0}}{3}}h(\chi)$. The
$\phi\times\phi$ channel corresponds to the $\chi\to 1$ limit and we have

$\displaystyle\\!\\!\\!\\!\hat{h}(\chi)=\mathsf{c}_{\frac{2j_{0}}{3}}g_{\frac{2j_{0}}{3}}(1-\chi)+\mathsf{c}_{\frac{2j_{0}}{3}+1}g_{\frac{2j_{0}}{3}+1}(1-\chi)+\mathsf{c}_{\frac{2j_{0}}{3}+2}g_{\frac{2j_{0}}{3}+2}(1-\chi)+\sum_{\Delta}\mathsf{c}_{\Delta}g_{\Delta}(1-\chi)$
(3.80)

where the sum runs over unprotected operators with dimension
$\Delta>\frac{2j_{0}}{3}+2$, and $\mathsf{c}_{\Delta}$’s are the squared
moduli of the OPE coefficients.101010We use a different font to distinguish
the coefficients from the chiral-chiral and the chiral-antichiral channel.

In the chiral-antichiral channel, long multiplets with quantum numbers
$[\Delta,0,0,0]$ contribute with all the allowed super descendants, and we can
decompose the four-point correlation function in terms of superconformal
blocks. The latter are always expressed as a linear combination of ordinary
$sl(2)$ blocks with shifted dimensions. Conformal blocks can be seen as
eigenfunctions of the Casimir differential operator [186]. Analogously,
superconformal blocks can be computed by considering the differential equation
generated by the action of the superconformal Casimir [184]. We start by the
$s$-channel OPE expansion of the superspace four-point function (3.119),
insert a resolution of the identity between points $t_{2}$ and $t_{3}$, and
act on each term of the sum with the quadratic Casimir ${\bf C}_{1}+{\bf
C}_{2}$, where ${\bf C}_{i}$ is the differential operator (3.20) acting on the
supercoordinates $(t_{i},\theta_{ai},\bar{\theta}^{a}_{i})$. This leads to the
block expansion

$\displaystyle f(z)=1+\sum_{\Delta}\,c_{\Delta}\,G_{\Delta}(z)$ (3.81)

where the sum runs over the dimensions $\Delta>0$ of the superprimary
operators exchanged in the chiral-antichiral channel, and each conformal block
satisfies the differential equation

$\displaystyle\big{(}-z^{2}(z-1)\partial_{z}^{2}-z(z-3)\partial_{z}\big{)}\,G_{\Delta}(z)=\Delta(\Delta+2)\,G_{\Delta}(z)\,,$
(3.82)

where $\Delta(\Delta+2)$ is the Casimir eigenvalue (3.21) with zero R-charges.
The equation above is solved by the hypergeometric function 111111The
appearance of a minus sign in (3.81) is due to our use of $z$, which takes
real negative values, rather than the more standard $\chi$, see (3.65).

$\displaystyle
G_{\Delta}(z)=(-z)^{\Delta}{}_{2}F_{1}(\Delta,\Delta,2\Delta+3;z)\,.$ (3.83)

$G_{\Delta}(z)$ can be decomposed in terms of a finite sum of $sl(2)$ blocks
(3.78).121212Here we use
$\hat{g}(z)=g(\frac{z}{z-1})=(-z)^{\Delta}\,_{2}F_{1}(\Delta,\Delta;2\Delta;z)$,
which implements the change of variable from $\chi$ to $z$.

$\displaystyle G_{\Delta}(z)$
$\displaystyle=\hat{g}_{\Delta}(z)+\frac{3\Delta}{2(2\Delta+3)}\hat{g}_{\Delta+1}(z)+\frac{3\Delta^{2}(\Delta+1)}{4(2\Delta+3)(2\Delta+1)(\Delta+2)}\hat{g}_{\Delta+2}(z)$
$\displaystyle+\frac{\Delta^{2}(\Delta+1)}{8(2\Delta+3)^{2}(2\Delta+5)}\hat{g}_{\Delta+3}(z)$
(3.84)

To find a connection between the two correlators (3.67) and (3.68), it is
important to consider the $s$-channel expansion of (3.68)

$\displaystyle h(\chi)$
$\displaystyle=1+\sum_{\Delta}\,\tilde{c}_{\Delta}\,\tilde{G}_{\Delta}(\chi)$
(3.85)

where

$\displaystyle\tilde{G}_{\Delta}(\chi)=\chi^{\Delta}{}_{2}F_{1}(\Delta,\Delta,2\Delta+3;\chi)\,.$
(3.86)

The similarity between this block expansion and (3.81) is apparent, especially
when considering the relation between $\tilde{c}_{\Delta}$ and $c_{\Delta}$.
Denoting by $\mathcal{O}_{\Delta}$ the exchanged operator of dimension
$\Delta$, the expressions of $c_{\Delta}$ and $\tilde{c}_{\Delta}$ are given
by

$\displaystyle c_{\Delta}$
$\displaystyle=f_{\phi\bar{\phi}\mathcal{O}_{\Delta}}f_{\phi\bar{\phi}\mathcal{O}_{\Delta}}$
$\displaystyle\tilde{c}_{\Delta}=f_{\phi\bar{\phi}\mathcal{O}_{\Delta}}f_{\bar{\phi}\phi\mathcal{O}_{\Delta}}$
(3.87)

where $f_{\phi\bar{\phi}\mathcal{O}_{\Delta}}$ is the three-point coefficient
determining $\braket{\phi\bar{\phi}\mathcal{O}_{\Delta}}$. In one-dimensional
CFT, the OPE coefficients depend on the signature of the permutation. In
particular, despite there being no continuous group of rotation, there is a
$\mathbb{Z}_{2}$ parity transformation $t\to-t$.131313This symmetry was called
S-parity in [92]. operators are charged under this symmetry, and one has

$\displaystyle\braket{\mathcal{O}_{1}(t_{1})\mathcal{O}_{2}(t_{2})\mathcal{O}_{3}(t_{3})}=(-1)^{T_{1}+T_{2}+T_{3}}\braket{\mathcal{O}_{3}(-t_{3})\mathcal{O}_{2}(-t_{2})\mathcal{O}_{1}(-t_{1})}$
(3.88)

where $T_{1}$, $T_{2}$ and $T_{3}$ are the charges of the operators under
parity. For our example, if $\phi$ is a bosonic operator, we have

$\displaystyle
f_{\phi\bar{\phi}\mathcal{O}_{\Delta}}=(-1)^{T_{\mathcal{O}}}f_{\bar{\phi}\phi\mathcal{O}_{\Delta}},$
(3.89)

whereas for a fermionic $\phi$

$\displaystyle
f_{\phi\bar{\phi}\mathcal{O}_{\Delta}}=(-1)^{1+T_{\mathcal{O}}}f_{\bar{\phi}\phi\mathcal{O}_{\Delta}}.$
(3.90)

Therefore the two coefficients $c_{\Delta}$ and $\tilde{c}_{\Delta}$, for a
fermionic $\phi$ are related by

$\displaystyle c_{\Delta}=(-1)^{T_{\mathcal{O}}+1}\tilde{c}_{\Delta}.$ (3.91)

As an example, which will be useful in the following, operators of the
schematic form $\mathcal{O}_{n}=\phi\partial_{t}^{n}\bar{\phi}$ have charge
$T_{\mathcal{O}}=n$. In section 3.5, these relations will allow us to
establish a precise connection between the correlators (3.67) and (3.68) in a
perturbative expansion around the free theory result.

###  3.3 Diagrammatic Results

In this section, we use the superconformal symmetry to relate the four-point
correlation function of the massless bosonic excitations
$\\{w^{a}{\bar{w}}_{a}\\}$ to that of the fundamental operator
$\\{\mathbb{F},\bar{\mathbb{F}}\\}$. The Type IIA background
$\text{AdS}_{4}\times\mathbb{C}\textup{{P}}^{3}$ is defined by

$\displaystyle
ds^{2}=R^{2}\left(ds^{2}_{\textrm{AdS}_{4}}+4ds^{2}_{\mathbb{C}\textup{{P}}^{3}}\right)\,,\qquad
e^{\phi}=\frac{2R}{k}\,,\qquad R^{2}\equiv{\widetilde{R}^{3}\over
4k}=\frac{k^{2}}{4}e^{2\phi}\,,$ (3.92) $\displaystyle
F_{2}=2\,k\,J_{\mathbb{C}\textup{{P}}^{3}}\,,\qquad
F_{4}=\frac{3}{8}R^{3}\,\operatorname{vol}(\text{AdS}_{4})\,,$ (3.93)

where $R$ is the $\text{AdS}_{4}$ radius, $\phi$ the dilaton, $k$ results from
the compactification of the original M-theory on an
$\text{AdS}_{4}\times\textup{{S}}^{7}/Z_{k}$ background.141414$k$ coincides in
the dual theory with the Chern-Simons level number [150]. Above, $F_{2}$,
$F_{4}$ are the two-form and four-form field strengths with
$J_{\mathbb{C}\textup{{P}}^{3}}$ the Kähler form on
$\mathbb{C}\textup{{P}}^{3}$. As they only play a role in the fermionic part
of the Lagrangian we only report them here for completeness.151515At first
order in perturbation, it is sufficient to consider the bosonic part of the
action to compute bosonic correlators. Using the Poincaré patch for the AdS4
metric

$\displaystyle ds^{2}_{\text{AdS}_{4}}=\frac{dz^{2}+dx^{r}dx^{r}}{z^{2}}\,,$
(3.94)

where $x^{r}=(x^{0},x^{1},x^{2})$ parametrize the Euclidean three-dimensional
boundary of $\text{AdS}_{4}$ and $z$ is the radial coordinate, the bosonic
part of the superstring action in AdS${}_{4}\times\,\mathbb{CP}^{3}$ reads

$S_{B}=\frac{1}{2}T\int
d^{2}\sigma\sqrt{h}\,h^{\mu\nu}\Big{[}\frac{1}{z^{2}}\left(\partial_{\mu}x^{r}\partial_{\nu}x^{r}+\partial_{\mu}z\partial_{\nu}z\right)+4\,G_{MN}^{\mathbb{C}\textup{{P}}^{3}}\,{\partial_{\mu}Y^{M}\partial_{\nu}Y^{N}}\Big{]}\
.$ (3.95)

Here, $\sigma^{\mu}=(t,s)$ are Euclidean world-sheet coordinates, and $T$ is
the effective string tension. In its original “dictionary” proposal [150] it
is related to the effective ’t Hooft coupling $\lambda$ of the dual
$\mathcal{N}=6$ superconformal Chern-Simons theory (realized in the limit of
$k$ and $N$ large with their ratio fixed) by equation 1.34 which we remind
here:

$\displaystyle
T=\frac{R^{2}}{2\pi\alpha^{\prime}}=\sqrt{\frac{\lambda}{2}}\,,\qquad\qquad\lambda=\frac{N}{k}\,.$
(3.96)

As we are interested in the leading and tree-level orders in perturbation
theory, we may disregard the corrections to the effective string tension $T$
due to the geometry of the background [150, 187], which start at order
$\frac{1}{\sqrt{\lambda}}$ [157]. The classical solution to (3.95), which is
relevant here, is the minimal surface corresponding to the straight Wilson
line at the boundary

$\displaystyle z$ $\displaystyle=s$ $\displaystyle x^{0}$ $\displaystyle=t$
$\displaystyle x^{i=1,2}$ $\displaystyle=0$ $\displaystyle Y^{M}$
$\displaystyle=0.$ (3.97)

This is the embedding of AdS2 in the AdS4 background of the solution of [188,
33]. The induced AdS2 metric is

$\displaystyle g_{\mu\nu}d\sigma^{\mu}d\sigma^{\nu}={1\over
s^{2}}(dt^{2}+ds^{2})\,.$ (3.98)

We will consider correlators of small fluctuations of “transverse” string
coordinates near this minimal surface.161616The transverse coordinates are
$x^{i},\,i=1,2$ and the $\mathbb{C}\textup{{P}}^{3}$ coordinates. The bosonic
symmetry of the defect conformal field theory associated with the 1/2-BPS
Wilson line is SU$(1,1)\times$ SU$(3)\times$U$(1)_{J_{0}}$, and it turns out
to be the manifest symmetry of the bosonic string action (3.95). The
SU$(1,1)\simeq$ SO$(2,1)$ symmetry can be made manifest by fixing a static
gauge where $z$ and $x^{0}$ do not fluctuate. AdS2 is embedded [33] into AdS4
using the following parametrization

$\displaystyle
ds^{2}_{4}=\frac{(1+\frac{1}{2}|X|^{2})^{2}}{(1-\frac{1}{2}|X|^{2})^{2}}ds^{2}_{2}+\frac{dXd\bar{X}}{(1-\frac{1}{2}|X|^{2})^{2}}\
,\ \ \ \ \ \ \ \ \ \ \ \ ds^{2}_{2}=\frac{1}{z^{2}}(dx_{0}^{2}+dz^{2})\ .$
(3.99)

The complex combination $X=\frac{1}{\sqrt{2}}(x^{1}+ix^{2})$ was introduced in
terms of the transverse AdS coordinates $x^{i}$ . $X$ and $\bar{X}$ have
opposite charge under $U(1)_{J_{0}}$. Finally, adopting the following
parametrization of the $\mathbb{C}\textup{{P}}^{3}$ metric

$\displaystyle
ds^{2}_{\mathbb{C}\textup{{P}}^{3}}=\frac{d\bar{w}_{a}\,dw^{a}}{1+|w|^{2}}-\frac{d\bar{w}_{a}\,w^{a}\,d{w}^{b}\,\bar{w}_{b}}{(1+|w|^{2})^{2}}\,,\qquad|w|^{2}=\bar{w}_{a}\,w^{a}\,,\qquad
a,b=1,2,3\,,$ (3.100)

the preserved SU$(3)$ subgroup of the SU$(4)$ global symmetry of
$\mathbb{C}\textup{{P}}^{3}$ is manifest.

The Nambu-Goto action with fixed static gauge reads then reads

$S_{B}=T\int
d^{2}\sigma\sqrt{\det\Big{[}\frac{(1+1/2|X|^{2})^{2}}{(1-1/2|X|^{2})^{2}}g_{\mu\nu}+2\frac{\partial_{\mu}X\partial_{\nu}\bar{X}}{(1-1/2|X|^{2})^{2}}+4\Big{(}\,\frac{\partial_{\mu}\bar{w}_{a}\partial_{\nu}w^{a}}{1+\lvert
w\rvert^{2}}-\frac{\partial_{\mu}\bar{w}_{a}w^{a}\bar{w}_{b}\partial_{\nu}w^{b}}{(1+\lvert
w\rvert^{2})^{2}}\Big{)}\,\Big{]}}.$ (3.101)

Above, $g_{\mu\nu}={1\over s^{2}}\delta_{\mu\nu}$ is the background AdS2
metric. Along the lines of [33], (3.101) can be interpreted as the action of a
straight fundamental string in AdS${}_{4}\times\mathbb{C}\textup{{P}}^{3}$
stretched from the boundary towards the AdS4 center (so, stretched along $z$),
as well as the action for a $2d$ “bulk” field theory of 1+3 complex scalars in
AdS2 geometry with SO$(2,1)\times[$U$(1)\times$SU$(3)]$ as manifest symmetry.
From the AdS/CFT point of view, this second interpretation leads to a CFT1
dual living at the $z=s=0$ boundary, namely the defect CFT defined by operator
insertions on the straight Wilson line.

Expanding the action above in powers of $X$ and $w^{a}$ one gets

$\displaystyle S_{B}\equiv$ $\displaystyle T\int
d^{2}\sigma\sqrt{g}\,L_{B}\,,\qquad
L_{B}=\,L_{2}+L_{4X}+L_{2X,2w}+L_{4w}+...\,$ (3.102) $\displaystyle L_{2}=$
$\displaystyle\textstyle
g^{\mu\nu}\partial_{\mu}X\partial_{\nu}\bar{X}+2|X|^{2}+g^{\mu\nu}\partial_{\mu}w^{a}\partial_{\nu}{\bar{w}}_{a}\
,$ (3.103) $\displaystyle L_{4X}=$ $\displaystyle\ \textstyle
2|X|^{4}+|X|^{2}\,(g^{\mu\nu}\partial_{\mu}X\partial_{\nu}\bar{X})-\frac{1}{2}\,(g^{\mu\nu}\partial_{\mu}X\partial_{\nu}X)\,(g^{\rho\kappa}\partial_{\rho}\bar{X}\partial_{\kappa}\bar{X})\,$
(3.104) $\displaystyle L_{2X,2w}=$ $\displaystyle\
\textstyle(g^{\mu\nu}\partial_{\mu}X\partial_{\nu}\bar{X})\,(g^{\rho\kappa}\partial_{\rho}w^{a}\partial_{\kappa}{\bar{w}}_{a})-(g^{\mu\nu}\partial_{\mu}X\partial_{\nu}w^{a})\;(g^{\rho\kappa}\partial_{\rho}\bar{X}\partial_{\kappa}{\bar{w}}_{a})$
$\displaystyle-(g^{\mu\nu}\partial_{\mu}\bar{X}\partial_{\nu}w^{a})\;(g^{\rho\kappa}\partial_{\rho}X\partial_{\kappa}{\bar{w}}_{a})\,$
(3.105) $\displaystyle L_{4w}=$ $\displaystyle\
-\textstyle\frac{1}{2}\textstyle(w^{a}{\bar{w}}_{a})(g^{\mu\nu}\partial_{\mu}w^{b}\partial_{\nu}{\bar{w}}_{b})-\frac{1}{2}\textstyle(w^{a}{\bar{w}}_{b})(g^{\mu\nu}\partial_{\mu}w^{b}\partial_{\nu}{\bar{w}}_{a})+\frac{1}{2}\,(g^{\mu\nu}\partial_{\mu}w^{a}\partial_{\nu}{\bar{w}}_{a})^{2}$
$\displaystyle\textstyle-\frac{1}{2}(g^{\mu\nu}\partial_{\mu}w^{a}\partial_{\nu}{\bar{w}}_{b})\;(g^{\rho\kappa}\partial_{\rho}{\bar{w}}_{a}\partial_{\kappa}w^{b})-\frac{1}{2}(g^{\mu\nu}\partial_{\mu}w^{a}\partial_{\nu}w^{b})\;(g^{\rho\kappa}\partial_{\rho}{\bar{w}}_{a}\partial_{\kappa}{\bar{w}}_{b})\
.$ (3.106)

There is one massive ($X$ with $m^{2}=2$) and three massless
($w^{a},~{}a=1,2,3$) complex scalar fields propagating in AdS2 that correspond
to the bosonic elementary CFT1 insertions represented in the displacement
supermultiplet.171717Respectively, these are the dual fields to the $\Delta=2$
displacement operator $\mathbb{D}$ and to the $\Delta=1$ operators
$\mathbb{O}^{a},~{}a=1,2,3$. As written above in subsection 3.2.2, one must
also consider the fermionic fluctuations to obtain the AdS/CFT dual of the
full displacement supermultiplet. The fermionic spectrum has been worked out
at quadratic level in [166, 167]. It consists of two massless and six massive
fermions (of which three with mass $m_{F}=1$ and three with $m_{F}=-1$), which
should correspond, respectively, to the $\Delta=\frac{1}{2}$ fermionic
superprimary $\mathbb{F}$ of the multiplet and its conjugate and to the
$\Delta=\frac{3}{2}$ fermionic operators $\mathbb{\Lambda}_{a}$ and their
conjugates. Expanding the full Type IIA Green-Schwarz action in
$\text{AdS}_{4}\times\mathbb{C}\textup{{P}}^{3}$ background [189, 190] around
the solution (3.97) up to quartic order in fermions would yield the
interaction vertices from which to evaluate directly, via Witten diagrams, the
four-point functions of fermionic fluctuations. Below we will limit our
analysis to the direct calculation of bosonic four-point functions from the
vertices in (3.102) above and compare with the bootstrap results through the
superspace analysis of section 3.2.3.181818The computation for the massless
fluctutations is below and the mixed and massive correlators can be found in
Appendix B.2. However, since the the function $f(z)$ is the four-point
correlator (3.69) of the fermionic superprimary, _all_ four-point functions
can be found from the Graßmann-expansion of the correlator.

Below, we will use these vertices of the AdS2 bulk theory to compute the
corresponding tree-level Witten diagrams in AdS2, with bulk-to-boundary
propagators ending at points ${t_{n}}$ on the boundary. As in the
AdS${}_{5}\times\text{S}^{5}$ case, no cubic terms appear in the bosonic
Lagrangian above, so at this level of perturbation theory, the correlation
functions are only a sum of four-point “contact” diagrams with four bulk-to-
boundary propagators.

#### Four-point Function of Massless Fluctuations in
$\mathbb{C}\textup{{P}}^{3}$

Here we compute the tree-level four-point Witten diagram of the
$\mathbb{C}\textup{{P}}^{3}$ fluctuations $w,{\bar{w}}$ appearing in the AdS2
action in (3.102)-(3.106). As discussed above, these are AdS/CFT dual to the
scalar operator insertions $\mathbb{O}^{a},\bar{\mathbb{O}}_{a},~{}a=1,2,3$
with protected dimension $\Delta=1$. Due to the $SO(2,1)$ conformal
invariance, the four-point function is expected to take the form

$\langle
w^{a_{1}}(t_{1})\,{\bar{w}}_{a_{2}}(t_{2})\,w^{a_{3}}(t_{3})\,{\bar{w}}_{a_{4}}(t_{4})\rangle=\frac{\big{[}C_{w}(\lambda)\big{]}^{2}}{t_{12}^{2}t_{34}^{2}}G^{a_{1}\,a_{3}}_{a_{2}\,a_{4}}(\chi).$
(3.107)

Here, $\chi$ is the conformally invariant cross-ratio defined in (3.65), and
we used for the two-point function

$\langle
w^{a_{1}}(t_{1}){\bar{w}}_{a_{2}}(t_{2})\rangle=\delta^{a_{1}}_{a_{2}}\frac{C_{w}(\lambda)}{t_{12}^{2}}\,.$
(3.108)

The function $G^{a_{1}\,a_{3}}_{a_{2}\,a_{4}}(\chi)$ in (3.107) does not
depend on the normalization of the $w^{a}$ fields, and thus on
$C_{w}(\lambda)$. One can of course choose $C_{w}(\lambda)\equiv
4B_{1/2}(\lambda)$, so as to realise a direct identification of $w^{a}$ with
the $\mathbb{O}^{a}$ in view of (3.60).191919This is the formal choice of
[33], where the analogue relation is to the $\mathcal{N}=4$ SYM Bremsstrahlung
function [59, 191, 192]. see also a related discussion in [193]. This would
correspond to an overall rescaling of the fields.202020At tree level, this
amounts to the rescaling $w^{a}\rightarrow\sqrt{2T}w^{a}$. This can be found
from the leading strong coupling value of the Bremsstrahlung function
$B_{1/2}(\lambda)=\frac{\sqrt{2\lambda}}{4\pi}\equiv\frac{T}{2\pi}$, and the
canonical choice (1.42) of the propagator. By perturbatively evaluating the
two-point function (3.108) (calculating loop corrections to the boundary-to-
boundary propagator) one should then be able to verify that the elementary
excitations $w^{a}$ are protected, as well as reproduce the strong coupling
expansion of the corresponding 1/2-BPS Bremsstrahlung function (3.59).The
disconnected part of the four-point function (3.107) originates from Wick
contractions illustrated in Figure 3.1, and is

$\displaystyle\langle
w^{a_{1}}(t_{1})\,{\bar{w}}_{a_{2}}(t_{2})\,w^{a_{3}}(t_{3})\,{\bar{w}}_{a_{4}}(t_{4})\rangle_{\text{disconn.}}$
$\displaystyle=\big{[}C_{w}(\lambda)\big{]}^{2}\,\Big{[}\frac{\delta^{a_{1}}_{a_{2}}\delta^{a_{3}}_{a_{4}}}{t_{12}^{2}t_{34}^{2}}+\frac{\delta^{a_{1}}_{a_{4}}\delta^{a_{3}}_{a_{2}}}{t_{14}^{2}t_{23}^{2}}\Big{]}$
$\displaystyle=\frac{\big{[}C_{w}(\lambda)\big{]}^{2}}{t_{12}^{2}t_{34}^{2}}\,\Big{[}\delta^{a_{1}}_{a_{2}}\delta^{a_{3}}_{a_{4}}+\frac{\chi^{2}}{(1-\chi)^{2}}\,\delta^{a_{1}}_{a_{4}}\delta^{a_{3}}_{a_{2}}\,\Big{]}\,.$
(3.109)

X$w(t_{1})$$\bar{w}(t_{2})$$w(t_{3})$$\bar{w}(t_{4})$

X$w(t_{1})$$\bar{w}(t_{2})$$w(t_{3})$$\bar{w}(t_{4})$

Figure 3.1: Witten diagram for the disconnected contribution to the four-point
function (3.107).

The first connected contribution to the four-point function comes from the
tree-level connected Witten diagrams illustrated in Figure 3.2. These are
obtained from the four-point interaction vertices $L_{4w}$ in (3.106) with
four bulk-to-boundary propagators attached and therefore are subleading in
$1/T$.

X$w(t_{1})$$\bar{w}(t_{2})$$w(t_{3})$$\bar{w}(t_{4})$$\mathcal{L}_{4w}$ Figure
3.2: Witten diagram for the connected contribution to the four-point function
(3.107).

Then the connected correlator reads

$\langle
w^{a_{1}}(t_{1})\,{\bar{w}}_{a_{2}}(t_{2})\,w^{a_{3}}(t_{3})\,{\bar{w}}_{a_{4}}(t_{4})\rangle_{\text{conn}}=\frac{1}{T}\,\big{(}\mathcal{C}_{\Delta=1}\big{)}^{4}\,\Big{[}\mathcal{Q}_{1}\,\delta^{a_{1}}_{a_{2}}\delta^{a_{3}}_{a_{4}}+\mathcal{Q}_{2}\,\delta^{a_{1}}_{a_{4}}\delta^{a_{3}}_{a_{2}}\Big{]}$
(3.110)

where ${\cal C}_{\Delta=1}={1\over\pi}$ and
$\mathcal{Q}^{a_{1}\,a_{2}}_{a_{3}\,a_{4}}$ is built out of the
$D$-functions212121Appendix D.1 contains a list of such $D$-functions. [194,
56, 97]:

$\displaystyle\mathcal{Q}_{1}$ $\displaystyle=$
$\displaystyle\,3D_{1111}+t_{12}^{2}D_{2211}+t_{34}^{2}D_{1122}-2t_{24}^{2}D_{1212}-2t_{13}^{2}D_{2121}$
(3.111)
$\displaystyle-3t_{23}^{2}\,D_{1221}-3t_{14}^{2}D_{2112}+4(t_{14}^{2}t_{23}^{2}+t_{13}^{2}t_{24}^{2}-t_{12}^{2}t_{34}^{2})D_{2222}\,$
$\displaystyle\mathcal{Q}_{2}$ $\displaystyle=$
$\displaystyle\,3D_{1111}+t_{14}^{2}D_{2112}+t_{23}^{2}D_{1221}-2t_{13}^{2}D_{2121}-2t_{24}^{2}D_{1212}$
(3.112)
$\displaystyle-3t_{12}^{2}D_{2211}-3t_{34}^{2}D_{1122}+4(t_{12}^{2}t_{34}^{2}+t_{13}^{2}t_{24}^{2}-t_{14}^{2}t_{23}^{2})D_{2222}\,.$

This leads to the expression

$\displaystyle\\!\\!\\!\\!\\!\\!\langle
w^{a_{1}}(t_{1})\,{\bar{w}}_{a_{2}}(t_{2})\,w^{a_{3}}(t_{3})\,{\bar{w}}_{a_{4}}(t_{4})\rangle_{\text{conn.}}$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi
T}\,\frac{\big{[}\mathcal{C}_{\Delta=1}\big{]}^{2}}{t_{12}^{2}t_{34}^{2}}\,\Big{[}\delta^{a_{1}}_{a_{2}}\delta^{a_{3}}_{a_{4}}\,G_{1}(\chi)+\delta^{a_{1}}_{a_{4}}\delta^{a_{3}}_{a_{2}}\,G_{2}(\chi)\Big{]}\,$
(3.113)

where, according to (3.107), we factored out
$[\mathcal{C}_{\Delta=1}\big{]}^{2}$, and

$\displaystyle G_{1}(\chi)$ $\displaystyle=$
$\displaystyle-3+\frac{1}{(\chi-1)}-\frac{\chi^{2}}{(1-\chi)^{2}}\log\chi+\left(1-\frac{4}{\chi}\right)\log(1-\chi)$
(3.114) $\displaystyle G_{2}(\chi)$ $\displaystyle=$
$\displaystyle-\frac{\chi(3\chi+1)}{(1-\chi)^{2}}+\frac{\chi^{2}(\chi+3)}{(\chi-1)^{3}}\,\log\chi-\log(1-\chi).$
(3.115)

Identifying these to the differential equation in (3.70) leads to the
following system of second-order differential equations222222 The functions
above can be expressed in terms of the invariant $z$ using (3.66).

$\displaystyle f(z)-zf^{\prime}(z)+z^{2}f^{\prime\prime}(z)$
$\displaystyle=1+\,\frac{1}{4\pi
T}\big{[}-4+z-z^{2}\,\log(-z)+\big{(}z^{2}-\frac{4}{z}+3\big{)}\,\log(1-z)\,\big{]}$
(3.116) $\displaystyle-z^{2}f^{\prime}(z)-z^{3}f^{\prime\prime}(z)$
$\displaystyle=z^{2}\,+\frac{1}{4\pi
T}\,\big{[}\,z-4\,z^{2}-z^{2}(3-4z)\,\log(-z)+(1+3z^{2}-4z^{3})\,\log(1-z)\,\big{]}\,$
(3.117)

whose unique solution agrees with the bootstrap solution (3.166)

$\displaystyle f(z)=1-z+\frac{1}{4\pi
T}\,\big{[}z-1+z(3-z)\log(-z)-\frac{(1-z)^{3}}{z}\log(1-z)\big{]}\,.$ (3.118)

One can now repeat the analysis for the correlator $\langle
w^{a_{1}}(t_{1})\,{\bar{w}}_{a_{2}}(t_{2})\,{\bar{w}}^{a_{3}}(t_{3})\,w_{a_{4}}(t_{4})\rangle$.
The result coincides with the one obtained using on the $f(z)$ above with the
replacement $z\to\chi$, and neglecting the imaginary part of the logarithm.
This perfectly agrees with what was observed in the bootstrap analysis of
section 3.5. In Appendix B.2, the $f(z)$ evaluated above also solves the
corresponding differential equations for the correlators of massive and mixed
worldsheet excitations once the normalization factors defining the
corresponding two-point functions are identified with the ones of their field
theory dual.

###  3.4 Setting up the Bootstrap

In the following, the four-point function of chiral 1/2-BPS operators is
bootstrapped the quantity of interest is

$\displaystyle\\!\\!\braket{\Phi_{j_{0}}(y_{1},\theta_{1})\bar{\Phi}_{-j_{0}}(y_{2},\bar{\theta}_{2})\Phi_{j_{0}}(y_{3},\theta_{3})\bar{\Phi}_{-j_{0}}(y_{4},\bar{\theta}_{4})}$
$\displaystyle=\frac{C^{2}_{\Phi_{j_{0}}}}{\braket{1\bar{2}}^{\frac{2j_{0}}{3}}\braket{3\bar{4}}^{\frac{2j_{0}}{3}}}\,f(\mathcal{Z})\,,$
(3.119)
$\displaystyle\\!\\!\braket{\Phi_{j_{0}}(y_{1},\theta_{1})\bar{\Phi}_{-j_{0}}(y_{2},\bar{\theta}_{2})\bar{\Phi}_{-j_{0}}(y_{3},\bar{\theta}_{3})\Phi_{j_{0}}(y_{4},\theta_{4})}$
$\displaystyle=-\,\frac{C^{2}_{\Phi_{j_{0}}}}{\braket{1\bar{2}}^{\frac{2j_{0}}{3}}\braket{4\bar{3}}^{\frac{2j_{0}}{3}}}\,h(\mathcal{X})\,,$
(3.120)

where the supercross-ratios reduce to $\chi$ and $z=\frac{\chi}{\chi-1}$ when
the fermionic coordinates are set to 0. From this point on, all the analysis
can be done in terms of bosonic quantities of a single cross-ratio $\chi$
where the constraints come from the full superspace analysis above. As such,
we consider the bosonic part of this super correlator obtained by setting

$\displaystyle\theta^{a}=\bar{\theta}_{a}=0$ (3.121)

which corresponds to the four-point function of the lowest weight field in the
supermultiplet

$\displaystyle\langle\mathbb{F}(t_{1})\bar{\mathbb{F}}(t_{2})\mathbb{F}(t_{3})\bar{\mathbb{F}}(t_{4})\rangle$
$\displaystyle=\frac{C_{\mathbb{F}}^{2}}{t_{12}t_{34}}f(\chi)$ (3.122)
$\displaystyle\langle\mathbb{F}(t_{1})\bar{\mathbb{F}}(t_{2})\bar{\mathbb{F}}(t_{3})\mathbb{F}(t_{4})\rangle$
$\displaystyle=\frac{C_{\mathbb{F}}^{2}}{t_{12}t_{34}}h(\chi).$ (3.123)

These functions have an OPE corresponding to the bosonic part of those
described in subsection 3.2.4, which are

$\displaystyle f(\chi)$
$\displaystyle=1+\sum_{h}c_{h}\chi^{h}{}_{2}F_{1}(h,h+3,2h+3;\chi)$ (3.124)
$\displaystyle=\left(\frac{\chi}{1-\chi}\right)\left(1+\sum_{h}c_{h}(1-\chi)^{h}{}_{2}F_{1}(h,h+3,2h+3;1-\chi)\right),$
(3.125)

and

$\displaystyle h(\chi)$
$\displaystyle=\sum_{h}\tilde{c}_{h}\chi^{\Delta}{}_{2}F_{1}(h,h,2h+3;\chi)$
(3.126)
$\displaystyle=\frac{\chi}{1-\chi}\left(\sum_{\Delta_{S}=\\{1,2,3\\}}\mathbb{a}_{\Delta}g_{\Delta}(1-\chi)+\sum_{\Delta_{L}>3}\mathbb{a}_{\Delta}g_{\Delta}(1-\chi)\right),$
(3.127) $\displaystyle g_{\Delta}$
$\displaystyle=\chi^{\Delta}{}_{2}F_{1}(\Delta,\Delta,2\Delta;\chi).$ (3.128)

Above, the bounds for the long operators and the presence of the short
operators is explained above and given in equation (3.77).232323In our case,
$j_{0}=\frac{3}{2}$ and the short operators of dimension $\Delta=1$ and
$\Delta=3$ cannot appear due to the fermionic nature of the excitations at
strong coupling. Additionally, from the S-parity of the correlators, we can
relate $a_{\Delta}$ to $\tilde{a}_{\Delta}$ through the charges of the
operators under S-parity. In practice, this means that

$\displaystyle a_{\Delta}=(-1)^{\Delta^{(0)}}\tilde{a}_{\Delta}.$ (3.129)

This allows us to relate the perturbative expansions of $f(\chi)$ and
$h(\chi)$ in the same way as the usual 1d braiding in [35, 49, 39]. Combined
with the crossing relations, this gives the following braiding/crossing
symmetries:

$\displaystyle f(\chi)$ $\displaystyle=\frac{\chi}{1-\chi}f(1-\chi),$ (3.130)
$\displaystyle h^{(i)}(\chi)|_{\log(\chi)^{(l)}}$
$\displaystyle=f^{(i)}(\frac{-\chi}{1-\chi})|_{\log(-\chi)^{(l)}},$ (3.131)
$\displaystyle h^{(i)}(\chi)|_{\log(1-\chi)^{(l)}}$ $\displaystyle=-\chi
h(\frac{1}{\chi})|_{\log(\chi-1)^{(l)}}.$ (3.132)

For convenience, we will write the braiding relations using “$\simeq$”, for
example

$\displaystyle h(\chi)$ $\displaystyle\simeq-\chi h(\frac{1}{\chi}).$ (3.133)

####  3.4.1 Ansatz

At strong coupling, the perturbative correlators can be obtained by computing
contributions from Witten diagrams in AdS2 see [33, 4]. Just as in flat space,
there is a natural organisation in terms of functions of increasing
transcendentality where for the four-point function, the disconnected diagram
is rational, the contact diagrams are logarithmic, and the exchange diagrams
have polylogarithms up to $Li_{3}$. Therefore, we begin with the 1d HPL Ansatz
as in [49, 39, 35]

$\displaystyle f^{(i)}(\chi)=\sum_{p_{i}}r(\chi)T_{p_{i}}(\chi)$ (3.134)

where $r_{p}$ are rational functions and $T_{p}$ are Harmonic polylogarithms
of transcendentality $p_{i}<2i-1$. For the third-order bootstrap, the Ansatz
should be defined more rigorously, as braiding for polylogarithms is a little
more subtle. In this case and in $\mathcal{N}=4$ SYM [39, 49], the
polylogarithmic part of the solution can be constrained further by requiring
the correct bosonic symmetry considerations, analogous to the function found
by considering a special subset of functions.242424One can repeat the
procedure as above with simple polylogarithms. However, braiding relations
become more intricate and this only leads to more terms to fix with the
integrated correlators.252525The author thanks Pietro Ferrero for an
introduction to these concepts. These obey special crossing properties such
that the result is always real and single valued, no matter what bosonic
crossing is considered. These are the functions:

$LL_{n}(\chi)=\sum_{s=0}^{n-1}\frac{(-\log(|\chi|))^{s}}{s!}Li_{n-s}(\chi)+\frac{\left(-\log(|\chi|)\right)^{n-1}}{n!}\log(|1-\chi|)$
(3.135)

and the crossing and braiding relations are

$\displaystyle LL_{3}(\chi)=LL_{3}(\tfrac{1}{\chi})$ (3.136) $\displaystyle
LL_{3}(\tfrac{\chi}{\chi-1})+LL_{3}(1-\chi)+LL_{3}(\chi)=\zeta(3).$ (3.137)

#### Growth of the Anomalous Dimension

The growth of the anomalous dimension of heavy exchanged operators is bounded
and related to the relevance of the bulk interactions on the string
worldsheet[130, 131]. The condition that $\gamma^{(l)}_{\Delta}$ be bounded in
the large-$\Delta$ limit is relaxed to

$\displaystyle\gamma_{\Delta}^{(l)}\sim_{\Delta\rightarrow\infty}\Delta^{l+1}$
(3.138)

where $l$ is the order in perturbation theory and $\Delta$ is the weight of
the exchanged operator. In this Regge limit, the leading order should be the
same for all operators. The reason behind this [7] is that the large-$\Delta$
behaviour or the anomalous dimension is dominated by the flat space
contribution where only isotropic terms contribute; therefore, excitations in
different directions are equivalent.

####  3.4.2 Integrated Correlator Conditions

In the case of $\mathcal{N}=4$ SYM, the topological sector known non-
perturbatively through localisation allows this integration constant to be
fixed. The analysis in [1] shows us that such a sector does not exist for the
displacement supermultiplet we are considering. Instead, one can use the
integral relations which relate the integrated four-point correlators of the
displacement multiplet to the Bremsstrahlung function of the corresponding
Wilson line. A similar relation was used in [79] in addition to the quantum
spectral curve to bootstrap the weak coupling spectrum. The relevant relation
derived in [195] is

$\displaystyle 2\left(\int
f(\chi)-\frac{h(\chi)}{\chi(1-\chi)}\right)=\frac{1}{4B_{1/2}(\lambda)}.$
(3.139)

This relation provides the physical input to match the expansion parameter
with the coupling constant. It also constrains the type of functions in the
correlator through the convergence of the integral. For example, terms that
violate the large-$\Delta$ bound in (3.138) lead to power divergences in the
integral (3.139), as do terms that are not in the minimal set described above
in 3.4.1.

The exploration of additional integrated correlator constraints such as in
[196] would be essential to computing higher-order correlators.

#### Mixing

At first order, since the relevant vertices at strong coupling have four legs,
the solving of the mixing only needs to include two-particle and four-particle
operators.262626See B.4 for a full detailed analysis of the mixing. In
particular, the protected operator
$\mathcal{O}_{2}=\mathbb{F}\partial\mathbb{F}$ exchanges both two-particle and
four-particle operators in free theory.272727Note that this is a conformal
primary, but a not a superprimary; it is a super descendent of
$\bar{\mathcal{B}}_{\frac{7}{2},1}\sim\mathbb{F}w^{a}$.  The strong coupling
free theory result is obtained by Wick contracting the elementary
fields282828In this case, the elementary field is $\mathbb{F}$. This differs
from the generalised free field result since $\mathcal{O}_{2}$ is a composite
operator, not an elementary excitation. In the free theory, length-two and
length-four operators will be exchanged.

$\displaystyle\langle\mathcal{O}_{2}\bar{\mathcal{O}}_{2}\bar{\mathcal{O}}_{2}\mathcal{O}_{2}\rangle$
$\displaystyle=\lim_{\\{x_{2},x_{4},x_{6},x_{8}\\}\rightarrow\\{x_{1},x_{3},x_{5},x_{7}\\}}\partial_{x_{2}}\partial_{x_{4}}\partial_{x_{6}}\partial_{x_{8}}\langle\mathbb{F}\mathbb{F}\bar{\mathbb{F}}\bar{\mathbb{F}}\bar{\mathbb{F}}\bar{\mathbb{F}}\mathbb{F}\mathbb{F}\rangle,$
(3.140)

which gives

$\displaystyle f^{(0)}_{\mathcal{O}_{2}}(\chi)$
$\displaystyle=\frac{1}{(1-\chi)^{4}}$ $\displaystyle
h^{(0)}_{\mathcal{O}_{2}}(\chi)$ $\displaystyle=(1-\chi)^{4}.$ (3.141)

In particular, even though the unitarity bound is $h_{L}>4$ in the charged
channel, the lightest operator exchanged in the free theory has dimension
$h_{L}=8$. This weight is the same as the operator
$\mathbb{F}\partial\mathbb{F}\partial^{2}\mathbb{F}\partial^{3}\mathbb{F}$.

Using the same constraints as for $\mathbb{F}$, the specific ones for the
chiral ring detailed in seeSection: chiral ring, and the first-order CFT data
from the four-point correlator of the displacement multiplet, the first-order
perturbation can be fixed entirely, giving292929In particular, the anomalous
dimension of $\mathbb{F}\bar{\mathbb{F}}$ can be found from both these
correlators as can the OPE coefficient
$c_{\mathcal{O}_{2},\mathcal{O}_{2}\mathbb{F}\mathbb{F}}$of $f(\chi)$.

$\displaystyle
f^{(1)}_{\mathcal{O}_{2}}(\chi)=4\epsilon\frac{(\chi-4)(\chi-1)^{2}\log(1-\chi)-\chi(-4\chi+(\chi+3)\chi\log(\chi)+4)}{(\chi-1)^{5}\chi}$
(3.142)

where

$\displaystyle\epsilon=-\frac{1}{2\pi\sqrt{2\lambda}}$ (3.143)

is the strong-coupling expansion parameter and $\lambda$ the ’t Hooft coupling
for ABJM. This gives the following CFT data:

$\displaystyle\tilde{c}^{(0)}_{h}$
$\displaystyle=\frac{\sqrt{\pi}2^{-2h-3}(h+1)^{2}(h+2)(h(h+2)+9)\Gamma(h+3)}{9\Gamma\left(h+\frac{3}{2}\right)}$
(3.144) $\displaystyle\mathbb{c}^{(0)}_{h}$
$\displaystyle=\frac{\sqrt{\pi}4^{-h}\left((-1)^{h}+1\right)(h-6)(h-4)(h-2)(h+1)(h+3)(h+5)\Gamma(h)}{9\Gamma\left(h-\frac{1}{2}\right)}$
(3.145) $\displaystyle\gamma_{s}^{(1)}$ $\displaystyle=\epsilon h(h+2)$
(3.146) $\displaystyle\gamma_{t}^{(1)}$ $\displaystyle=\epsilon(h(h-1)-20).$
(3.147)

In particular, this means that both the two-particle and the four-particle
operators with vanishing $U(1)$ charge have anomalous dimensions proportional
to the quadratic Casimir and we have

$\displaystyle\langle(\gamma^{(1)})^{n}\rangle=\langle\gamma^{(1)}\rangle^{n}.$
(3.148)

###  3.5 Bootstrap Results

In this section, by imposing symmetries and consistency with the OPE
expansion, we extract the strong-coupling corrections to the four-point
function of the displacement supermultiplet. We then extract the anomalous
dimensions and the OPE coefficients from this expression. Let us start from
the expressions of the four-point functions of the superprimary (3.67) and
(3.68), which for this specific case, are

$\displaystyle\braket{\mathbb{F}(t_{1})\bar{\mathbb{F}}(t_{2})\mathbb{F}(t_{3})\bar{\mathbb{F}}(t_{4})}$
$\displaystyle=\frac{C_{\Phi}^{2}}{t_{12}t_{34}}f(\chi)$ (3.149)
$\displaystyle\braket{\mathbb{F}(t_{1})\bar{\mathbb{F}}(t_{2})\bar{\mathbb{F}}(t_{3})\mathbb{F}(t_{4})}$
$\displaystyle=\frac{C_{\Phi}^{2}}{t_{12}t_{34}}h(\chi)$ (3.150)

where for the bosonic part, we have decided to use the same cross-ratio; when
considering the full superspace correlator, one must revert to the formalism
in (3.119).

####  3.5.1 Leading Order Result

The leading order result at strong coupling can be easily obtained from Wick
contractions and is

$\displaystyle f^{(0)}(\chi)$ $\displaystyle=1-\frac{\chi}{\chi-1}$
$\displaystyle h^{(0)}(\chi)$ $\displaystyle=1-\chi.$ (3.151)

Looking at the $s$-channel expansion of the two correlators (3.81) and (3.85)
and using the well-known fact that in free theory, the only exchanged
operators are of the schematic form
$[\mathbb{F}\bar{\mathbb{F}}]_{n}\sim\mathbb{F}\partial_{t}^{n}\bar{\mathbb{F}}$
with dimension $h_{n}=1+n$. The mismatch factor $(-1)^{1+n}$ between (3.85)
and (3.81) is compensated by an identical factor from the relation (3.91) for
the OPE coefficients, which in the case of the operators
$[\mathbb{F}\bar{\mathbb{F}}]_{n}$ is

$\displaystyle c^{(0)}_{n}=(-1)^{1+n}\tilde{c}^{(0)}_{n}.$ (3.152)

As a result, in free theory, the two expansions (3.81) and (3.85) are
identical up to the braiding $\chi\leftrightarrow\frac{\chi}{\chi-1}$. The
explicit form of the OPE coefficient at leading order is extracted

$\displaystyle
c^{(0)}_{n}=\frac{\sqrt{\pi}2^{-2n-3}\Gamma(n+4)}{(n+1)\Gamma\left(n+\frac{5}{2}\right)}.$
(3.153)

The same can be done in the chiral-chiral channel

$\displaystyle\mathsf{c}^{(0)}_{n}$
$\displaystyle=\frac{\sqrt{\pi}2^{1-2n}\Gamma(n+1)}{\Gamma\left(n+\frac{1}{2}\right)}$
$\displaystyle n\text{ odd}$ (3.154) $\displaystyle\mathsf{c}^{(0)}_{n}$
$\displaystyle=0$ $\displaystyle n\text{ even}.$ (3.155)

As expected, in the OPE of two identical fermions, only operators of the
schematic form $[\mathbb{F}\partial^{n}\mathbb{F}]_{n}$ with _odd_ values of
$n$ appear. In the following, a first-order perturbation of this result is
considered.

####  3.5.2 Next-to-Leading Order

We are interested in finding the first-order strong coupling correction to the
correlator. We expand the function $\tilde{f}(\chi)$ as

$\displaystyle\hat{f}(\chi)$
$\displaystyle=\hat{f}^{(0)}(\chi)+\epsilon\,\hat{f}^{(1)}(\chi)$
$\displaystyle h(\chi)$ $\displaystyle=h^{(0)}(\chi)+\epsilon\,h^{(1)}(\chi)$
(3.156)

where $\epsilon$ is a small parameter whose precise relation with the string
tension cannot be predicted by symmetry considerations. Following [35, 49], we
start with the following Ansatz for the first order correction to $\hat{f}$
and $\hat{h}$

$\displaystyle f^{(1)}(\chi)$
$\displaystyle=\chi\left(r(\chi)\log(1-\chi)+r(1-\chi)\log\chi+q(\chi)\right)$
(3.157) $\displaystyle\hat{h}^{(1)}(\chi)$
$\displaystyle=-r\left(\tfrac{1}{1-\chi}\right)\log\chi+\left[r\left(\tfrac{\chi}{\chi-1}\right)+r\left(\tfrac{1}{1-\chi}\right)\right]\log(1-\chi)-q\left(\tfrac{\chi}{1-\chi}\right),$
(3.158)

where $r(\chi)$ and $q(\chi)$ are rational functions and

$\displaystyle q(\chi)=q(1-\chi).$ (3.159)

The OPE expansions constrain the behaviour of both functions as
$\chi\rightarrow 0$ and $\chi\rightarrow 1$

$\displaystyle f(\chi)=1+O(\chi)$ (3.160) $\displaystyle h(\chi)=O(\chi)$
(3.161) $\displaystyle\left(\frac{1-\chi}{\chi}\right)h(\chi)=O((1-\chi))$
(3.162)
$\displaystyle\left(\frac{1-\chi}{\chi}\right)h(\chi)|_{\log(1-\chi)}=O((1-\chi)^{4}).$
(3.163)

Since the identity operator receives no contributions, the rational function
$r(1-\chi)$ is regular at $\chi=0$. Since it should only have poles for
colliding operators ($\chi\rightarrow\\{0,1,\infty\\}$) one can expand it as a
Laurent series

$\displaystyle r^{(1)}(1-\chi)=\sum_{k}r_{k}(1-\chi)^{k}.$ (3.164)

The large-$\Delta$ behaviour of the anomalous dimension

$\displaystyle\gamma_{\Delta}^{(1)}\sim_{\Delta\rightarrow\infty}\Delta^{2}$
(3.165)

in all OPE channels is extremely constraining and sets almost all coefficients
$r_{k}$ to 0 except for $k=-2$ and $k=-1$. The boundary behaviour from the OPE
shown above is then sufficient to fix the function up to an overall constant
which can be reabsorbed in the expansion parameter $\epsilon$

$\displaystyle\epsilon f^{(1)}(\chi)$
$\displaystyle=\epsilon\left(\left(\frac{1}{\chi}+2\right)\log(1-\chi)+\frac{-\chi+(3-2\chi)\chi\log(\chi)+1}{(\chi-1)^{2}}\right)$
(3.166) $\displaystyle\epsilon h^{(1)}(\chi)$
$\displaystyle=-\frac{\epsilon\left((\chi-1)^{3}\log(1-\chi)+\chi(\chi-((\chi-3)\chi\log(\chi))-1)\right)}{\chi}.$
(3.167)

This parameter can be fixed using the integrated correlator constraint in
3.4.2

$\displaystyle 2\epsilon\left(\int
f^{(1)}(\chi)-\frac{h^{(1)}(\chi)}{\chi(1-\chi)}\right)=\frac{\pi}{\sqrt{2\lambda}}$
$\displaystyle\Rightarrow\qquad\epsilon$
$\displaystyle=-\frac{1}{2\pi\sqrt{2\lambda}}.$ (3.168)

The remaining configurations of the fields can be found through crossing and
braiding

$\displaystyle<\mathbb{F}\bar{\mathbb{F}}\mathbb{F}\bar{\mathbb{F}}>$
$\displaystyle=<\bar{\mathbb{F}}\mathbb{F}\bar{\mathbb{F}}\mathbb{F}>\simeq\frac{1}{(t_{12}t_{34})}h(\frac{\chi}{\chi-1})$
(3.169) $\displaystyle<\mathbb{F}\bar{\mathbb{F}}\bar{\mathbb{F}}\mathbb{F}>$
$\displaystyle=<\bar{\mathbb{F}}\mathbb{F}\mathbb{F}\bar{\mathbb{F}}>=\frac{1}{(t_{12}t_{34})}h(\chi)$
(3.170) $\displaystyle<\bar{\mathbb{F}}\bar{\mathbb{F}}\mathbb{F}\mathbb{F}>$
$\displaystyle=<\mathbb{F}\mathbb{F}\bar{\mathbb{F}}\bar{\mathbb{F}}>=\frac{1}{(t_{12}t_{34})}\chi
h(1-\chi).$ (3.171)

####  3.5.3 Leading Logarithms and Double Scaling Limit

Given the absence of mixing for $\gamma^{(1)}$, the leading logarithm powers
are entirely fixed by tree-level data presented in section 3.6 and are:

$\displaystyle h^{(0)}|_{\log(\chi)^{0}}=1-\chi,$ (3.172) $\displaystyle
h^{(1)}|_{\log(\chi)^{1}}=(\chi-3)\chi$ (3.173) $\displaystyle
h^{(2)}|_{\log(\chi)^{2}}=\frac{1}{2}\chi((11-4\chi)\chi-9)$ (3.174)
$\displaystyle
h^{(3)}|_{\log(\chi)^{3}}=\frac{1}{6}\chi(2\chi-1)(\chi(18\chi-43)+27)$
(3.175) $\displaystyle
h^{(4)}|_{\log(\chi)^{4}}=\frac{1}{24}\chi(\chi(803-4\chi(18\chi(8\chi-25)+487))-81),$
(3.176) and (3.177) $\displaystyle h^{(0)}|_{\log(1-\chi)^{0}}=1-\chi$ (3.178)
$\displaystyle h^{(1)}|_{\log(1-\chi)^{1}}=-\frac{(\chi-1)^{3}}{\chi}$ (3.179)
$\displaystyle
h^{(2)}|_{\log(1-\chi)^{2}}=-\frac{(\chi-1)^{3}\left(2\chi^{2}+\chi+2\right)}{\chi^{2}}$
(3.180) $\displaystyle
h^{(3)}|_{\log(1-\chi)^{3}}=-\frac{2(\chi-1)^{3}(9\chi^{4}+2\chi^{3}+3\chi^{2}+2\chi+9)}{3\chi^{3}}$
(3.181) $\displaystyle
h^{(4)}|_{\log(1-\chi)^{4}}=-\frac{(\chi-1)^{3}(72\chi^{6}-18\chi^{5}+6\chi^{4}+5\chi^{3}+6\chi^{2}-18\chi+72)}{3\chi^{4}}.$
(3.182)

Since these are generated by acting with the superconformal Casimir in the
different channels, they are straightforward to generate at high orders. So
one can look at the double-scaling limit [40] of these correlators in the out-
of-time-ordered configuration. To match their conventions, we will look at the
$\log(1-\chi)$ term of $h(\chi)$ and $h(1-\chi)$ related to the correlators
given above. This should incorporate all the different configurations, as only
two OPE limits are available. In this limit, the leading term will be given by
the leading power of $\log(1-\chi)$ of the transformed function

$\displaystyle\tilde{h}(\chi)$ $\displaystyle=\chi
h\left(\frac{1}{\chi}\right).$ (3.183)

Note that, given the braiding properties of $h(\chi)$, this simply corresponds
to a continuation of $h(\chi)$ with the replacement
$\log(1-\chi)\rightarrow\log(\chi-1)$. We therefore drop the tilde notation in
the following for convenience. Taking the limit $t\rightarrow\infty$ of the
function evaluated in the cross-ratio

$\displaystyle\chi=\frac{2}{1-i\sinh(t)},$ (3.184)

while keeping $\kappa=\frac{\text{e}^{t}}{4\sqrt{2\lambda}}$ fixed gives this
double-scaling limit. Organising the correlator as

$\displaystyle h(\chi)$ $\displaystyle=\sum_{n}\epsilon^{n}h^{n}(\chi)$
(3.185) $\displaystyle h^{n}|_{\log^{l}(\chi-1)}$
$\displaystyle=\frac{1}{\chi^{l}}\sum_{k}a_{k}^{n,l}\chi^{k}$ (3.186)

Given, the OPE is naturally written in terms of

$\displaystyle\hat{h}(\chi)=\frac{1-\chi}{\chi}h(\chi)$ (3.187)

we will use it in the following. The fact that the leading anomalous dimension
is proportional to the eigenvalue of the Quadratic Casimir allows us to write
the leading power of logarithms as

$\displaystyle\gamma^{(1)}_{\Delta}$ $\displaystyle=\epsilon\lambda_{\Delta}$
(3.188) $\displaystyle\hat{h}^{n}|_{\log^{n}(\chi-1)}$
$\displaystyle=\frac{\left(\mathcal{C}^{(2)}\right)^{n}}{n!}\hat{h}^{(0)}(\chi)$
(3.189)

where $\mathcal{C}^{(2)}$ is the differential expression of the quadratic
Casimir in the $\chi\rightarrow 1$ channel of $h(\chi)$, and the subleading
power of logarithms as

$\displaystyle\hat{h}^{n}|_{\log^{n-1}(\chi-1)}$
$\displaystyle=\frac{\left(\mathcal{C}^{(2)}\right)^{n-2}}{(n-2)!}\left(\hat{h}^{(2)}|_{\log(1-\chi)}(\chi)\right)-\frac{\chi}{1-\chi}\frac{(n-2)\left(\mathcal{C}^{(2)}\right)^{n-1}}{(n-1)!}\left(\hat{h}^{(1)}|_{\log(1-\chi)^{0}}(\chi)\right).$
(3.190)

These expression give recursion relations between the the $a_{k}^{n,l}$
coefficients for $l=n$ and $l=n-1$ in terms of the lower perturbative orders.
This gives the series

$\displaystyle
h_{\text{DS}}(\kappa)=\sum_{n}(-\kappa)^{n}\Gamma(n+1)+O(\text{e}^{-t}).$
(3.191)

where the subleading terms give a correction to the Lyapunov exponent:

$\displaystyle\kappa=\frac{\text{e}^{t(1-\frac{1}{4\pi\sqrt{2\lambda}})}}{4\sqrt{2\lambda}}$
(3.192)

This series is Borel resummable where the Borel transform and Borel
resummation are303030see [125] or [126] for an introduction to resurgence and
non-perturbative methods in QFT.

$\displaystyle\hat{h}_{\text{DS}}(\zeta)$ $\displaystyle=\frac{1}{1+\zeta}$
$\displaystyle h_{\text{DS}}(\mu)$
$\displaystyle=\kappa^{-1}U\left(1,1,\kappa^{-1}\right),$ (3.193)

Here $U$ is a confluent hypergeometric function of the second kind and one can
expand it around $\kappa^{-1}=0$ which gives a series in $\sqrt{\lambda}$ as
opposed to $\frac{1}{\sqrt{\lambda}}$. This matches equation (3.20) in [40] if
we identify $\Delta_{V}=\Delta_{W}=\frac{1}{2}$, corresponding to the
conformal weight of the superconformal primary, and
$\kappa=\frac{\text{e}^{(1-\frac{1}{4\pi\sqrt{2\lambda}})t}}{4\sqrt{2\lambda}}$.
Note that we also observe a correction to the Lyapunov exponent at this order.
This formalism in terms of conformal blocs is a reorganising of the
perturbative series, where the (partially unmixed) conformal data is in
reality contained in all perturbative orders as the leading powers of
logarithms. In this way, one can access a limit of the non-perturbative
correlator simply with first-order CFT data and exponentially suppressed terms
with second-order data.

####  3.5.4 Second Order Bootstrap

The next order has an Ansatz with functions up to transcendentality three:

$\displaystyle h^{(2)}(\chi)$
$\displaystyle=q^{(2)}(\chi)+r_{1}^{(2)}(\chi)\log(\chi)+r_{2}^{(2)}(\chi)\log(1-\chi)+$
(3.194) $\displaystyle
s_{1}^{(2)}(\chi)\log(1-\chi)\log(\chi)+s_{2}^{(2)}(\chi)\log^{2}(\chi)+s_{3}^{(2)}(\chi)\log^{2}(1-\chi)+s_{4}^{(2)}(\chi)(\text{Li}_{2}(\chi)-\text{Li}_{2}(1-\chi))$

Polylogarithm identities were used to eliminate redundant terms, and the
polynomiality of $\gamma^{(1)}$ was used to discount non-rational functions
multiplying $\log^{2}$ in both OPE limits. Applying crossing/braiding to the
known leading logarithms eliminates all transcendentality-three terms and
fixes $s_{1}^{(2)}(\chi)$. As with the first order bootstrap, the growth of
the anomalous dimension imposes conditions on the rational functions
$\\{q,r_{i},s_{4}\\}$ such that we have a finite sum of terms which are then
fixed by the boundary conditions set by the OPE.313131It was noted here that
relaxing this condition to $\gamma^{(2)}\sim\Delta^{4}$ only leads to a single
additional term which leads to a power divergence in the integrated
correlator.  This fixes the function up to a constant $a$

$\displaystyle h^{(2)}(\chi)$
$\displaystyle=\epsilon^{2}\left(-\frac{(\chi-1)^{3}\left(2\chi^{2}+\chi+2\right)\log^{2}(1-\chi)}{\chi^{2}}-\frac{1}{2}\chi(4\chi^{2}-11\chi+9)\log^{2}(\chi)+\right.$
$\displaystyle\left.+\frac{(8\chi^{4}-21\chi^{3}+19\chi^{2}-\chi-1)\log(1-\chi)\log(\chi)}{2\chi}+\frac{1}{2}(\chi(2\chi-3)-1)\log(\chi)+\right.$
$\displaystyle\left.-\frac{(\chi-1)\left(\chi^{2}+1\right)\log(1-\chi)}{\chi}+\chi-1\right)+\epsilon
ah^{(1)}(\chi),$ (3.195)

and equivalently

$\displaystyle f^{(2)}$
$\displaystyle=\epsilon^{2}\chi\left(\frac{\left(-2\chi^{5}+7\chi^{4}-9\chi^{3}\right)\log^{2}(\chi)}{2(\chi-1)^{3}\chi^{3}}-\frac{(1-2\chi)^{2}\log(1-\chi)\log(\chi)}{4(\chi-1)^{2}\chi^{2}}+\right.$
(3.196)
$\displaystyle\left.+\frac{\left(-2\chi^{5}+7\chi^{4}-6\chi^{3}+\chi^{2}\right)\log(\chi)}{2(\chi-1)^{3}\chi^{3}}+\frac{1}{(\chi-1)\chi}+\chi\leftrightarrow
1-\chi\right)+\epsilon af^{(1)}(\chi).$

The integrated correlator condition fixes this final free parameter to
$a=0$.323232Note that the resulting function only has terms up to
transcendentality two and only products of logarithms.

####  3.5.5 Third Order Bootstrap

For the next order correlator, the $\log^{2}$ terms can also be computed
without the preoccupation of mixing.333333There is no mixing from
$\gamma^{(1)}$ and the corresponding perturbative OPE expansion term is linear
in $\gamma^{(2)}$.

$\displaystyle
h^{(3)}|_{\log^{2}}=\gamma^{(1)}h^{(2)}|_{\log}-\frac{1}{2}\left(\gamma^{(1)}\right)^{2}h^{(2)}|_{\log^{0}}$
(3.197)

where this is to be understood in the sense of the terms in the OPE expansion;
the anomalous dimension must be summed over the weights of the primaries. The
precise evaluation of these terms is a little more involved than for the
leading power of logarithms but is well-defined in terms of known CFT data.

Given the Ansatz described above in (3.134), the OPE constraints and the Regge
bound fix all terms up to 2. One of them can be fixed by noticing that the
leading large-$\Delta$ behaviour of the anomalous dimension
$\gamma^{(3)}_{\Delta}$ should be the same for the
$\mathbb{F}\times\mathbb{F}$ channel and for the
$\mathbb{F}\times\bar{\mathbb{F}}$ channel from flat-space limit
considerations (see section 2.5.2)

$\displaystyle\gamma_{\Delta}^{(3)}$
$\displaystyle\rightarrow_{\Delta\rightarrow\infty}\left(\frac{1}{6}\left(\pi^{2}-30\right)\epsilon^{3}+\frac{1}{4}x_{2,1}\right)\Delta^{4}$
(3.198) $\displaystyle\upgamma_{\Delta}^{(3)}$
$\displaystyle\rightarrow_{\Delta\rightarrow\infty}\left(+\frac{1}{6}\left(\pi^{2}-30\right)\epsilon^{3}-\frac{1}{4}x_{2,1}\right)\Delta^{4}.$
(3.199)

The final term can be fixed with the integrated correlator giving the result

$\displaystyle
f^{(3)}(\chi)=\epsilon^{3}\chi\left(\frac{\chi\left(2\chi\left(-2\chi^{2}+\chi+7\right)-5\right)+1}{2(\chi-1)^{3}\chi^{2}}LL_{3}(\chi)-\frac{(\chi+1)(\chi(2\chi-11)+27)}{6(\chi-1)^{4}}\log^{3}(\chi)\right.$
$\displaystyle\left.+\frac{(\chi((17-6\chi)\chi-12)+9)^{3}}{(\chi-1)^{3}}\log^{2}(\chi)+\frac{\left(14\chi^{4}-37\chi^{3}+23\chi^{2}-5\chi+1\right)}{6(\chi-1)^{3}\chi^{2}}\log^{2}(\chi)\log(1-\chi)\right.$
$\displaystyle\left.+\frac{\left(12\chi^{4}-24\chi^{3}+5\chi^{2}+7\chi-3\right)}{2(\chi-1)^{2}\chi^{2}}\log(\chi)\log(1-\chi)+\frac{\left(-12(\chi-3)\chi+\pi^{2}(2\chi-3)\chi-21\right)}{6(\chi-1)^{2}\chi}\log(\chi)\right.$
$\displaystyle\left.+\frac{\left(12\chi^{4}-24\chi^{3}-9\chi^{2}+21\chi-8\right)\zeta(3)}{4(\chi-1)^{3}\chi^{3}}+\frac{\left(\pi^{2}-12\right)}{12(\chi-1)\chi}+\chi\rightarrow
1-\chi\right).$ (3.200)

Where the only presence of polylogarithms is through the single-valued
function $LL_{3}(\chi)$ presented in 3.4.1.

###  3.6 CFT Data

Before computing the values of anomalous dimensions and OPE coefficients, we
need to comment on the class of operators one expects to appear in this
context. When classifying operators that can be exchanged in a given channel,
one is interested in the eigenstates of the dilatation operator. When
perturbing the leading order result, one can think of building operators out
of the fundamental fields in the worldsheet Lagrangian (see section 3.3),
which are in one-to-one correspondence with the components of the super
displacement multiplet. Two-particle operators of the schematic form
$O_{n}\sim\mathbb{F}\partial^{n}\bar{\mathbb{F}}$ will have a three-point
function $\braket{\mathbb{F}\bar{\mathbb{F}}O_{n}}$ that is leading compared
to higher particle operators. Nevertheless, these two-particle operators are
not well-defined eigenstates of the dilatation operator. A simple example is
the mixing between two- and four-particle operators
$\mathbb{F}\partial^{2}\bar{\mathbb{F}}$ and
$\mathbb{F}\bar{\mathbb{F}}\mathbb{F}\bar{\mathbb{F}}$. Only a linear
combination of these two operators will be an eigenstate of the one-loop
dilatation operator, but both of them are allowed to appear in the
$\mathbb{F}\times\bar{\mathbb{F}}$ OPE.343434We would like to thank Pietro
Ferrero, Shota Komatsu and Carlo Meneghelli for very useful discussions on
this point. Of course, the situation becomes increasingly more intricate for
heavier operators. Therefore, we conclude that any operator with a two-
particle contribution will appear in the leading order OPE. Therefore, the
anomalous dimension we extract is a linear combination of the anomalous
dimensions of these operators, weighted by their OPE coefficients. Given that
part of this degeneracy is lifted by decomposing in terms of superconformal
blocks instead of ordinary conformal blocks, there are a few cases where we
can be certain that a single long multiplet with a given dimension can appear.
These are the $n=0$ and $n=1$ case for the $\mathbb{F}\times\bar{\mathbb{F}}$
channel associated to the operators $\mathbb{F}\bar{\mathbb{F}}$ and
$\mathbb{F}\partial\bar{\mathbb{F}}$ and the $n=1$ case in the
$\mathbb{F}\times\mathbb{F}$ channel associated to
$\mathbb{F}\partial\mathbb{F}$, which is protected. The mixing was solved at
the first order in the previous section. Still, to solve the mixing beyond the
first order for heavier operators, one must study a larger class of
correlators, particularly the 1/3 BPS operators studied in [6], a task still
underway.

In the chiral-anti-chiral OPE channel, the superconformal bloc basis, Casimir
differential equation, eigenvalue and orthogonality relation needed for the
superblock expansion are

$\displaystyle G_{n}=\chi^{n}{}_{2}F_{1}(n,n,2n+3;\chi)$ (3.201)
$\displaystyle\mathcal{L}=-\chi^{2}(\chi-1)\partial_{\chi}^{2}-\chi(\chi-3)\partial_{\chi}$
(3.202) $\displaystyle\lambda_{h}=h(h+3)$ (3.203)
$\displaystyle\oint\frac{d\chi}{2\pi
i}\frac{\chi}{(1-\chi)^{3}}G_{1+n}(\chi)G_{-3-n^{\prime}}(\chi)=\delta_{n,n^{\prime}}\,,$
(3.204)

where the contour circles $0$ counterclockwise. Expanding the blocks as in
(2.69) and inverting the expansion, we find the following CFT data. The
anomalous dimension is

$\displaystyle\gamma^{(1)}_{\Delta}$ $\displaystyle=-\epsilon\Delta(\Delta+2)$
(3.205) $\displaystyle\gamma^{(2)}_{\Delta}$
$\displaystyle=\epsilon^{2}\left(\frac{1}{2}\left(4\Delta^{3}+4\Delta^{2}-9\Delta+\frac{5}{\Delta+1}-\frac{4}{\Delta+2}+\frac{4}{\Delta}+3\right)+\left(\Delta(\Delta+2)-4\right)H_{\Delta+1}\right.$
(3.206)
$\displaystyle\left.-(-1)^{\Delta}\frac{3+(1+\Delta)^{4}}{\Delta(\Delta+1)(\Delta+2)}\right)$
$\displaystyle\gamma^{(3)}_{\Delta,even}$
$\displaystyle=\epsilon^{3}\left(\frac{1}{6}\left(-3\Delta^{2}-6\Delta+4\right)\left(-3\psi^{(1)}_{\frac{\Delta}{2}}+3\psi^{(1)}_{\frac{\Delta+1}{2}}-\pi^{2}\right)+\left(-\Delta^{2}-2\Delta+4\right)\left(H_{\Delta-1}\right){}^{2}\right.$
(3.207)
$\displaystyle\left.-\frac{5\Delta^{7}+19\Delta^{6}+13\Delta^{5}-15\Delta^{4}-26\Delta^{3}-3\Delta^{2}+48\Delta+16}{\Delta(\Delta+1)(\Delta+2)}-\frac{1}{6}\left(\pi^{2}-3\right)\Delta(\Delta+2)\right.$
$\displaystyle\left.-\frac{\left(4\Delta^{6}+16\Delta^{5}+3\Delta^{4}-30\Delta^{3}-12\Delta^{2}-8\Delta+8\right)H_{\Delta-1}}{\Delta(\Delta+1)(\Delta+2)}+\Delta(\Delta+2)\left(\Delta^{2}+2\Delta-4\right)H_{\Delta-1}^{(2)}\right)$
$\displaystyle\gamma^{(3)}_{\Delta,odd}=\epsilon^{3}$
$\displaystyle\left(\frac{1}{6}\left(3\Delta^{2}+6\Delta-4\right)\left(-3\psi^{(1)}_{\frac{\Delta}{2}}+3\psi^{(1)}_{\frac{\Delta+1}{2}}+\pi^{2}\right)+\Delta(\Delta+2)\left(\Delta^{2}+2\Delta-4\right)H_{\Delta-1}^{(2)}\right.$
(3.208)
$\displaystyle\left.-\frac{\Delta\left(5\Delta^{5}+19\Delta^{4}+19\Delta^{3}+7\Delta^{2}+2\Delta-7\right)}{(\Delta+1)(\Delta+2)}-\frac{1}{6}\left(\pi^{2}-3\right)\Delta(\Delta+2)\right.$
$\displaystyle\left.-\frac{\left(4\Delta^{6}+16\Delta^{5}+11\Delta^{4}+2\Delta^{3}+36\Delta^{2}+24\Delta-24\right)H_{\Delta-1}}{\Delta(\Delta+1)(\Delta+2)}+\left(-\Delta^{2}-2\Delta+4\right)\left(H_{\Delta-1}\right){}^{2}\right).$

The OPE coefficients are

$\displaystyle c_{\Delta}^{(0)}$
$\displaystyle=\frac{\sqrt{\pi}2^{-2\Delta-1}(\Delta+1)(\Delta+2)\Gamma(\Delta)}{\Gamma\left(\Delta+\frac{3}{2}\right)}$
(3.209) $\displaystyle c_{\Delta}^{(1)}$
$\displaystyle=\partial_{\Delta}\left(c_{\Delta}^{(0)}\gamma^{(1)}_{\Delta}\right)$
(3.210) $\displaystyle c_{\Delta}^{(2)}$
$\displaystyle=\partial_{\Delta}\left(c_{\Delta}^{(0)}\upgamma^{(2)}_{\Delta}+c_{\Delta}^{(1)}\upgamma^{(1)}_{\Delta}\right)-\partial^{2}_{\Delta}\left(c_{\Delta}^{(0)}(\upgamma^{(1)}_{\Delta})^{2}\right)$
(3.211)
$\displaystyle+\tfrac{1}{2}\epsilon^{2}c_{\Delta}^{(0)}\left(\Delta^{4}+4\Delta^{3}+3\Delta^{2}-2\Delta+4\right)\left(\psi^{(1)}_{\frac{\Delta+1}{2}}-\psi^{(1)}_{\frac{\Delta}{2}+2}\right)$
(3.212)
$\displaystyle+\tfrac{1}{2}\epsilon^{2}c_{\Delta}^{(0)}\left(\frac{1+(-1)^{\Delta}}{2}\left(-6\Delta(\Delta+1)-\frac{2}{(\Delta+1)^{2}}+6\right)+\right.$
$\displaystyle\qquad\left.+\frac{1+(-1)^{\Delta}}{2}\left(\frac{3(\Delta+1)\Delta^{3}-4\Delta+8}{\Delta^{2}}+\frac{11}{(\Delta+1)^{2}}+\frac{4}{\Delta+2}+\frac{8}{(\Delta+2)^{2}}-3\right)\right).$
(3.213)

A similar analysis can be carried out in the chiral-chiral channel, with
conformal scalar blocks. This gives

$\displaystyle\upgamma^{(1)}_{\Delta}$
$\displaystyle=-\epsilon(\Delta(\Delta-1)-2)$ (3.214)
$\displaystyle\upgamma^{(2)}_{\Delta}$
$\displaystyle=\epsilon^{2}\left(2\Delta^{3}-7\Delta^{2}-\Delta-\frac{4}{\Delta+1}-\frac{2}{\Delta}+9+((\Delta-1)\Delta+2)H_{\Delta+1}\right)$
(3.215) $\displaystyle\upgamma^{(3)}_{\Delta}$
$\displaystyle=\epsilon^{3}\left(21-5\Delta^{4}+26\Delta^{3}-\frac{63\Delta^{2}}{2}-\frac{27\Delta}{2}-\frac{1}{6}\left(\pi^{2}-3\right)(\Delta-2)(\Delta+1)\right.$
$\displaystyle\left.+\frac{1}{12}\left(-5\Delta^{2}+5\Delta+6\right)\left(3\psi^{(1)}_{\frac{\Delta}{2}}-3\psi^{(1)}_{\frac{\Delta+1}{2}}+\pi^{2}\right)+\left(-\Delta^{2}+\Delta-2\right)\left(H_{\Delta-1}\right){}^{2}\right.$
$\displaystyle\left.-2\left(2\Delta^{3}-7\Delta^{2}+5\Delta-4\right)H_{\Delta-1}+(\Delta-2)(\Delta+1)\left(\Delta^{2}-\Delta+2\right)H_{\Delta-1}^{(2)}\right),$
(3.216)

where the anomalous dimension vanishes for $\Delta=2$, as expected. The OPE
coefficients are

$\displaystyle\mathsf{c}_{\Delta}^{(0)}$
$\displaystyle=\frac{\sqrt{\pi}2^{3-2\Delta}\Gamma(\Delta)}{\Gamma\left(\Delta-\frac{1}{2}\right)}$
(3.217) $\displaystyle\mathsf{c}_{\Delta}^{(1)}$
$\displaystyle=\partial_{\Delta}\left(\mathsf{c}_{\Delta}^{(0)}\upgamma^{(1)}_{\Delta}\right)$
(3.218) $\displaystyle\mathsf{c}_{\Delta}^{(2)}$
$\displaystyle=\partial_{\Delta}\left(\mathsf{c}_{\Delta}^{(0)}\upgamma^{(2)}_{\Delta}+\mathsf{c}_{\Delta}^{(1)}\upgamma^{(1)}_{\Delta}\right)-\partial^{2}_{\Delta}\left(\mathsf{c}_{\Delta}^{(0)}(\upgamma^{(1)}_{\Delta})^{2}\right)$
(3.219)
$\displaystyle+\epsilon^{2}\mathsf{c}_{\Delta}^{(0)}\left((\Delta-3)(\Delta+2)-\frac{1}{2}\left(\Delta^{4}-2\Delta^{3}-4\Delta^{2}+5\Delta+2\right)\left(\psi^{(1)}_{\frac{\Delta}{2}}-\psi^{(1)}_{\frac{\Delta+1}{2}}\right)\right)$

The functional form of the 3rd-order OPE coefficients is neither compact nor
illuminating, so it will not be listed here. However, the weight of the
lightest uncharged operator and the OPE coefficient from the protected
operator $\mathsf{c}_{\Delta=2}$ is not affected by mixing and is the
following

$\displaystyle\Delta_{\mathbb{F}\bar{\mathbb{F}}}$
$\displaystyle=2+\frac{2\sqrt{2}}{\pi\sqrt{\lambda}}+\frac{65}{24\pi^{2}\lambda}+\frac{179+8\pi^{2}}{96\sqrt{2}\pi^{3}\lambda^{3/2}}+O\left(\lambda^{-2}\right),$
(3.220) $\displaystyle
c^{2}_{\mathbb{F}\mathbb{F}(\bar{\mathbb{F}}\partial\bar{\mathbb{F}})}$
$\displaystyle=1+\frac{3}{2\pi\sqrt{2\lambda}}+\frac{12\zeta(3)+\pi^{2}-28}{2(2\pi\sqrt{2\lambda})^{3}}+O\left(\lambda^{-2}\right).$
(3.221)

###  3.7 Comments

This series of analyses and results for the ABJM 1/2-BPS Wilson line dCFT is
extensive and leads to several conclusions. These will be listed here. The
first concerns the structure of the supermultiplet and of the supercorrelator,
the second is the obtaining of results through the analytic bootstrap. The
third result is the addition of the integrated correlator which frees the
bootstrap result from needing an input from the string-theory description. The
fourth is the explicit computation of worldsheet fluctuation correlators on
the string side of the AdS/CFT duality, these match the bootstrap results and
are consistent with the superspace analyis. The fifth is the analysis of the
mixing at first-order which not only enables the higher-order perturbative
bootstrap, but also a part of the all-order solution. This constitutes the
sixth and final conclusion, which is the double-scaling limit of the OTOC,
which is an all-loop result.

The first important result of this analysis concerns the general structure of
the four-point functions of the displacement supermultiplet. The superfield
formalism, together with the underlying superconformal symmetry, enables us to
determine all the non-vanishing correlators for a given ordering of the
external superfields in terms of a single function $f(\chi)$ of the relevant
conformal cross-ratio $\chi=\frac{t_{12}t_{34}}{t_{13}t_{24}}$, as well as
fixing the superconformal block expansion, essential to bootstrap process.
This function appears directly in the correlator of the superconformal primary
$\mathbb{F}(t)$ and its conjugate $\bar{\mathbb{F}}(t)$

$\displaystyle\braket{\mathbb{F}(t_{1})\bar{\mathbb{F}}(t_{2})\mathbb{F}(t_{3})\bar{\mathbb{F}}(t_{4})}$
$\displaystyle=\frac{C_{\Phi}^{2}}{t_{12}t_{34}}f(z)\,.$ (3.222)

Above, $C_{\Phi}$ is the normalization of the superfield two-point function
and has a physical interpretation in terms of the Bremsstrahlung function,
discussed in subsection 3.2.2. In one-dimensional CFTs, correlators come with
a specific ordering, and one can take OPEs only for neighbouring operators.
Therefore, crossing symmetry implies that the exchange $1\leftrightarrow 3$ is
a symmetry of the correlator (3.222), whereas $1\leftrightarrow 2$ is not.
This means that different orderings correspond to different functions of the
cross-ratio. However, the OPE expansion relates these functions, often through
braiding symmetry. Having encoded all the information into this function, the
analytic bootstrap process is sufficient to compute $f(\chi)$ up to the third
order in perturbation around the generalized free field theory result obtained
by Wick contractions. The two OPEs of this correlator correspond to the OPE
channels, charged and uncharged under $U(1)$. In the chiral-chiral OPE, each
superconformal primary only contributes one operator in the tower of super
descendants. Therefore, the scalar blocks are sufficient. In this channel,
short (protected) multiplets appear in the block expansion, constraining the
long operators’ dimension in the same channel. In the chiral-antichiral OPE
(uncharged), every descendent contributes. Hence, the superconformal block is
a sum of a finite number of conformal blocks corresponding to the contribution
from the superconformal descendants as seen in (3.84). The explicit form of
the superblocks is explicitly obtained by diagonalizing the superconformal
Casimir operator.

This constitutes the second important series of results. The symmetry
considerations and a minimal input are sufficient to perturbatively find the
strong coupling correlator up to three orders, which is beyond the current
capabilities in perturbative diagrammatics and is the first prediction at this
order. Using the constraints from the conformal bootstrap detailed in the
previous section, the symmetries constrain the four-point function up to the
third order.

One of the new elements in this analysis compared to [1], beyond the
additional two orders in perturbation, was the ability to fix the bootstrap
parameter through the integrated correlators in [195]. The parameter
$\epsilon$ controls the expansion around generalized free field theory, and it
will be interpreted as the inverse string tension $T$ appearing in the
effective AdS2 sigma model. The precise relation between the $\epsilon$
parameter and the string tension $T$ can be established by comparing the
explicit Witten diagram computation to the bootstrap result. However,
integrated correlator results [195] allow for an explicit fixing of this
coefficient by comparison to the ABJM Bremsstrahlung function[181, 163, 180,
151] which is also sufficient to fix remaining integration constant up to
third-order and to justify some of the bootstrap requirements.

The third element of this analysis concerns the mixing of operators. At
leading order (generalized free field theory), the operators exchanged in the
OPE channels are “two-particle” operators of the schematic form
$\mathbb{F}\partial^{n}_{t}\bar{\mathbb{F}}$ in the chiral-antichiral channel,
and $\mathbb{F}\partial^{n}_{t}{\mathbb{F}}$ (with odd $n$) in the chiral-
chiral channel.353535At strong coupling these operators should represent
worldsheet bound states, made of two of the corresponding fluctuations, as
discussed in [33]. The CFT data can then be extracted from these analytic
expressions of the correlator. At first order, the correction to the CFT data
corresponds to the leading corrections to the classical dimension of the two-
particle operators defined above (or their supersymmetric generalization).
However, at higher-order, two-particle operators generally mix with multi-
particle operators, and the results should correspond to linear combinations
of the actual anomalous dimensions weighted by OPE coefficients, except for a
handful of operators for which there is no mixing.

A fourth element concerns the strong coupling perturbative results. As
mentioned above, the worldsheet fluctuations around the (AdS2) minimal surface
corresponding to the 1/2-BPS Wilson line are in direct correspondence with the
components of the displacement supermultiplet. Through AdS/CFT, correlators of
these AdS2 fields evaluated at the boundary correspond to correlation
functions of the dual defect operators [33]. Their boundary-to-boundary
propagator is free for large string tension $T$, leading to a generalized free
field theory result for their four-point function. The $1/T$ expansion for the
Nambu-Goto string action involves non-trivial bulk interactions, and the
associated boundary correlators are evaluated via AdS2 Witten diagrams
363636We remark that, compared to the usual $1/N$ expansion in Witten diagrams
for higher-dimensional AdS/CFT, we are expanding the large-$N$ string sigma-
model in inverse powers of the string tension. see related discussion in
[33].. Having derived the effective Lagrangian governing the interactions of
the AdS2, the computation of all bosonic correlators confirms the result of
$f^{(0)}(\chi)$ and $f^{(1)}$, obtained by bootstrap. The analogous
computation for higher order quantities quickly becomes unpractical as the
number of terms already reaches $\sim 10^{4}$ for $f^{(2)}$ hence the power of
the conformal bootstrap. Another issue in explicit computations is
regularization subtleties in AdS2 models with derivative interactions (see
discussion in [197, 198, 199, 200, 201, 202]). However, independent
verifications could be done through integrability along the lines of [149].

A non-perturbative result stemming from this analysis is the double scaling
limit of the correlator, which can be found at all orders from unmixed first-
order CFT data and gives

$\displaystyle h_{\text{DS}}(\mu)$
$\displaystyle=\kappa^{-1}U\left(1,1,\kappa^{-1}\right)$ (3.223)

where $\kappa=\frac{\text{e}^{t}}{4\sqrt{2\lambda}}$ and $U$ is a confluent
hypergeometric function of the second kind, which matches equation (3.20) in
[40] for the fermionic displacement superfield with $\Delta=\frac{1}{2}$ and a
correction to the Lyupanov exponent.

A natural development of this work would be to solve the mixing at this order
to push the bootstrap process to higher orders and find the unmixed spectrum.
This could be done along the lines of [39] by looking at mixed four-point
functions of protected operators and long (unprotected) operators. Another
method is to look at higher-point functions. The next section will present
this in the case of $\mathcal{N}=4$ SYM.

An interesting direction concerns topological sectors, hopefully amenable to
localization, These could appear considering other supermultiplets [6]. A
fascinating case is that of the 1/3 BPS operators, of which
$\mathbb{F}\mathbb{O}^{a}$ is an example; they are charged under R-symmetry
and seem to have similar properties to the 1/2-BPS operators on the Wilson
line defect in $\mathcal{N}=4$ SYM studied in[33, 35, 39]. The R-symmetry
offers the potential of constructing topological operators and computing exact
correlation fucntions through localisation [203, 37, 36].

For less symmetric systems where all-order data may not be available to fix
the bootstrap expansion parameter, finite-coupling data for the correlators of
interest here may also be obtained with lattice field theory methods applied
to the string worldsheet, discretizing the Lagrangian of [204] expanded around
the minimal surface corresponding to the 1/2-BPS line, and using Monte Carlo
techniques on the lines of [205, 206, 207, 208, 209] for the correlators of
the worldsheet excitations.

##  Chapter 4 Bootstrapping Higher-Point Functions in $\mathcal{N}=4$ SYM dCFT

Une valse à mille temps
Offre seule aux amants
Trois cent trente-trois fois le temps
De bâtir un roman

– Jacques Brel, La valse a mille temps T.3

The 1/2-BPS Wilson line defect in $\mathcal{N}=$4 SYM has extensively been
studied at weak [210] and strong coupling [33] using Feyman diagrammatics
[119], Witten diagrams [33], the analytic bootstrap [34, 35, 39] and
integrability through the quantum spectral curve [38]. The extensive work on
this defect motivates the search for more complex quantities to compute, such
as higher $n$-point functions.111At the time of writing and submitting the
Thesis, the important advances made in [211] had not yet been published In
[119], the correlators at weak coupling of operator insertions were computed
through a recursion relation relating 1-loop lower point functions to 1-loop
$n-$point functions. At strong-coupling, such a relation would be trivial at
the same loop order since the corresponding Witten diagrams are disconnected.
As such, $n$-point functions factorise into products of lower point functions
until the perturbative order of the first connected diagram.222The first
connected diagram appears at the $\tfrac{n}{2}-1^{\text{st}}$ order for
massless excitations. However, the analysis of higher-point correlators[8],
even at GFF or at the next order, can be illuminating in terms of resolving
the mixing and finding the free OPE data.

In this Chapter, the non-vanishing five-point correlator333The correlator we
study corresponds to a pinching limit of the six-point function of the
displacement multiplet. of 1/2-BPS insertions on the 1/2-BPS Wilson line is
studied. The superblocks are obtained by requiring the multipoint Ward
Identities in [119] to be satisfied. The CFT data for the free theory is
obtained, and the first-order solution is bootstrapped, which confirms the
statement in [39] that the anomalous dimension does not mix at first order.

###  4.1 Symmetry and Operators

The 1/2-BPS Wilson in $\mathcal{N}=4$ SYM breaks the overall symmetry to
$OSP(4^{*}|4)$. The displacement multiplet preserves half the supersymmetries
of the line and is the lightest 1/2-BPS operators in the family
$\mathcal{D}_{k}$ [34] where $k$ labels the R-symmetry ($SP(4)$) irrep of the
superconformal primary. The structure of the multiplets $\mathcal{D}_{1}$ is
[35].

$\displaystyle\mathcal{D}_{1}:\qquad[0,1]^{s=0}_{\Delta=1}\rightarrow[1,0]^{s=\tfrac{1}{2}}_{\Delta=\tfrac{3}{2}}\rightarrow[0,0]^{s=1}_{\Delta=2}$
(4.1)

where the displacement operator is a superconformal descendent of the lowest
weight operator $[0,1]$. The family of operators can be obtained by
considering the chiral ring generated by $\mathcal{D}_{1}$ and can be obtained
explicitly by pinching several of these operators together, a process used in
[119] to relate higher-point functions to higher-weight four-point functions
at weak coupling. In the following, we will consider the four-point correlator

$\displaystyle\langle\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{2}\rangle=\tilde{\mathcal{A}}(y_{i},x_{i},\theta_{i})$
(4.2)

Where the R-symmetry dependence is encapsulated in the cross-ratios
$\\{r_{1},r_{2},s_{1},s_{2},t\\}$ obtained by contracting the free indices of
the operators with a null vector $y_{a}$. This correlator is a function of the
superspace coordinates [34]

$\displaystyle X=\begin{pmatrix}x\epsilon^{ab}&\theta^{a\beta}\\\
\theta^{b\alpha}&y^{(\alpha\beta)}\end{pmatrix}$ (4.3)

which are useful to derive the Ward Identities satisfied by the correlator and
to find the correlation functions of all the superconformal descendants
through differential equations such as above in (3.69). However, if we instead
consider the bosonic part of this correlation function, it will be a function
of two spacetime cross-ratios and 5 R-symmetry cross-ratios

$\langle 11112\rangle=\frac{(u_{1}\cdot u_{4})(u_{2}\cdot u_{5})(u_{3}\cdot
u_{5})}{\tau_{14}^{2}\tau_{25}^{2}\tau_{35}^{2}}\mathcal{A}(\chi_{1},\chi_{2};r_{1},r_{2},s_{1},s_{2},t)\,,$
(4.4)

where

$\displaystyle\chi_{1}$
$\displaystyle=\frac{\tau_{12}\tau_{45}}{\tau_{14}\tau_{25}}\,$
$\displaystyle\chi_{2}$
$\displaystyle=\frac{\tau_{13}\tau_{45}}{\tau_{14}\tau_{35}}$ (4.5)
$\displaystyle r_{1}=$
$\displaystyle\,\zeta_{1}\zeta_{2}=\frac{y_{12}^{2}y_{45}^{2}}{y_{14}^{2}y_{25}^{2}}\,,$
$\displaystyle s_{1}$
$\displaystyle=(1-\zeta_{1})(1-\zeta_{2})=\frac{y_{15}^{2}y_{24}^{2}}{y_{14}^{2}y_{25}^{2}}\,,$
(4.6) $\displaystyle r_{2}=$
$\displaystyle\,\eta_{1}\eta_{2}=\frac{y_{13}^{2}y_{45}^{2}}{y_{14}^{2}y_{35}^{2}}\,,$
$\displaystyle s_{2}$
$\displaystyle=(1-\eta_{1})(1-\eta_{2})=\frac{y_{15}^{2}y_{34}^{2}}{y_{14}^{2}y_{35}^{2}}\,,$
(4.7) $\displaystyle t=$
$\displaystyle\,\frac{y_{15}^{2}y_{23}^{2}y_{45}^{2}}{y_{14}^{2}y_{25}^{2}y_{35}^{2}}\,,$
(4.8)

These should be related to the bosonic pieces of the eigenvalues of the
superMatrix cross-ratios. The correlation function above can be written
explicitly as

$\mathcal{A}=F_{0}+\frac{r_{1}}{\chi_{1}^{2}}F_{1}+\frac{s_{1}}{(1-\chi_{1})^{2}}F_{2}+\frac{r_{2}}{\chi_{2}^{2}}F_{3}+\frac{s_{2}}{(1-\chi_{2})^{2}}F_{4}+\frac{t}{\chi_{12}^{2}}F_{5}\,,$
(4.9)

where each of the $F_{i}$ are functions of the two spacetime cross-ratios
$\\{\chi_{1},\chi_{2}\\}$.

###  4.2 Ward Identities

An extension to the Ward identities in [34] for higher-point correlation
functions was conjectured in [119] as :

$\displaystyle\sum_{k=1}^{n-3}\left(\frac{1}{2}\partial_{\chi_{k}}+\alpha_{k}\partial_{r_{k}}-(1-\alpha_{k})\partial_{s_{k}}\right)\mathcal{A}_{\Delta_{1}...\Delta_{n}}\raisebox{-5.69046pt}{$\Biggr{|}$}_{\raisebox{5.69046pt}{$\begin{subarray}{l}r_{i}\rightarrow\alpha_{i}\chi_{i}\\\
s_{i}\rightarrow(1-\alpha_{i})(1-\chi_{i})\\\
t_{ij}\rightarrow(\alpha_{i}-\alpha_{j})(\chi_{i}-\chi_{j})\end{subarray}$}}=0$
(4.10)

where

$\displaystyle\chi_{i-1}$
$\displaystyle=\frac{x_{1i}x_{n-1,n}}{x_{in}x_{1,n-1}}$ (4.11) $\displaystyle
r_{i-1}$ $\displaystyle=\frac{(u_{1}\cdot u_{i})(u_{n-1}\cdot
u_{n})}{(u_{i}\cdot u_{n})(u_{1}\cdot u_{n-1})}$ (4.12) $\displaystyle
s_{i-1}$ $\displaystyle=\frac{(u_{1}\cdot u_{n})(u_{i}\cdot
u_{n-1})}{(u_{i}\cdot u_{n})(u_{1}\cdot u_{n-1})}$ (4.13) $\displaystyle
t_{i-1,j-1}$ $\displaystyle=\frac{(u_{i}\cdot u_{j})(u_{1}\cdot
u_{n})(u_{n-1}\cdot u_{n})}{(u_{i}\cdot u_{n})(u_{j}\cdot u_{n})(u_{1}\cdot
u_{n-1})},$ (4.14)

This was found under the minimal assumptions the conformal prefactors should
vanish under its action, that it should be linear, have single derivatives and
be related to the topological sector. This identity was verified for various
correlators at weak and strong coupling [4].

This Ward identity applied to the five-point correlator reduces the problem of
6 independent functions of two variables to one of three functions and a
constant which corresponds to the topological sector

$\displaystyle\mathcal{A}(\chi_{1},\chi_{2};\zeta_{1},\zeta_{2},\eta_{1},\eta_{2},t)=\mathds{F}+\sum_{i=1,2,3}\mathds{D}_{i}f_{i}(\chi_{1},\chi_{2})\,,$
(4.15)

where we have defined the differential operators

$\displaystyle\mathds{D}_{1}:=$
$\displaystyle\,(v_{1}+v_{2})+v_{1}v_{2}(\partial_{\chi_{1}}+\partial_{\chi_{2}})\,,$
(4.16) $\displaystyle\mathds{D}_{2}:=$
$\displaystyle\,(w_{1}+w_{2})+w_{1}w_{2}(\partial_{\chi_{1}}+\partial_{\chi_{2}})\,,$
(4.17) $\displaystyle\mathds{D}_{3}:=$
$\displaystyle\,z+\chi_{12}(v_{1}v_{2}-w_{1}w_{2})(\partial_{\chi_{1}}+\partial_{\chi_{2}})\,,$
(4.18) $\displaystyle v_{i}=$ $\displaystyle\chi_{1}-\zeta_{i}\qquad
w_{i}=\chi_{2}-\eta_{i}\qquad z=\chi_{12}^{2}-t.$ (4.19)

with the use of more convenient variables in the topological limit. The
functions are related by

$\displaystyle f^{0}$ $\displaystyle=\sum_{i=0}^{5}F_{i}(\chi_{1},\chi_{2})$
(4.20) $\displaystyle f_{1}$
$\displaystyle=-\frac{F_{1}(\chi_{1},\chi_{2})}{\chi_{1}}+\frac{F_{2}(\chi_{1},\chi_{2})}{1-\chi_{1}}$
(4.21) $\displaystyle f_{2}$
$\displaystyle=-\frac{F_{3}(\chi_{1},\chi_{2})}{\chi_{2}}+\frac{F_{4}(\chi_{1},\chi_{2})}{1-\chi_{2}}$
(4.22) $\displaystyle f_{3}$
$\displaystyle=-\frac{F_{5}(\chi_{1},\chi_{2})}{(\chi_{1}-\chi_{2})^{2}}$
(4.23)

The different solutions to ward identities correspond to the different blocks
exchanged in the relevant OPE limits. Therefore, the Ward Identities should be
enough to fully constrain the form of the superconformal blocks. This analysis
is shown below.

###  4.3 Superconformal Blocks

$\mathcal{D}_{2}(X_{5})$$\mathcal{D}_{1}(X_{3})$$\mathcal{D}_{1}(X_{2})$$\mathcal{D}_{1}(X_{4})$$\mathcal{D}_{1}(X_{1})$$\mathcal{I}+\mathcal{D}_{2}+\mathcal{L}_{h_{1}}$$\mathcal{I}+\mathcal{D}_{2}+\mathcal{L}_{h_{2}}$
Figure 4.1: OPE diagram for the five-point function
$\langle\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{1}\mathcal{D}_{2}\rangle$
for the symmetric OPE channel, where the same operators are exchanged from
(12) and (34).

The R-symmetry selection rules [34] and the Ward Identities [119] fix the
conformal blocks entirely. In the symmetric channel, which corresponds to the
OPE between points $(12)$ and $(34)$, the operators exchanged are

$\displaystyle\mathcal{D}_{1}\times\mathcal{D}_{1}\sim\mathcal{I}+\mathcal{D}_{2}+\sum_{h}\mathcal{L}^{h>1}_{0,[0,0]}$
(4.24)

which can be summarized in the OPE diagram in Figure 4.1

In this expansion, the non-vanishing blocks are given by

$\displaystyle\mathcal{A}=$ $\displaystyle
c_{011}c_{022}c_{112}\left(\mathcal{G}_{\mathcal{I},\mathcal{D}_{2}}+\mathcal{G}_{\mathcal{D}_{2},\mathcal{I}}\right)+c_{112}^{2}c_{222}\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{2}}$
$\displaystyle+\sum_{h_{2}}c_{112}c_{11h_{2}}c_{22h_{2}}\mathcal{G}_{\mathcal{D}_{2},\mathcal{L}_{h_{2}}}+\sum_{h_{1}}c_{112}c_{11h_{1}}c_{22h_{1}}\mathcal{G}_{\mathcal{L}_{h_{1}},\mathcal{D}_{2}}$
$\displaystyle+\sum_{h_{1},h_{2}}c_{11h_{1}}c_{11h_{2}}c_{h_{1}h_{2}2}\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{L}^{h_{2}}_{0,[0,0]}}$
(4.25)

where the OPE coefficient $c_{ijk}$ are labelled by the conformal weights of
the Short/Long operators and the superblocks $\mathcal{G}_{A,B}$ are functions
of the spacetime cross-ratios as well as the R-symmetry cross-ratios. Their
explicit form will be presented below. They represent the sum of the conformal
blocks from the descendants of the exchanged fields. The multiplets for the
exchanged operators can be found in from [39]. Each conformal block will be
weighted by an R-symmetry block corresponding to the representations of the
given descendent of operators being exchanged. In this case, the R-symmetry
selection rules between these and the remaining operator ($\mathcal{D}_{2}$)
will restrict the possible terms. Additionally, only the uncharged terms under
the $SO(3)$ transverse spin will contribute. The relevant scalar blocks are
(see A.1)

$\displaystyle
g_{h_{1},h_{2}}=\chi_{1}^{h_{1}-2}(1-\chi_{2})^{h_{2}-2}F_{2}(h_{1}+h_{2}-2,h_{1},h_{2},2h_{1},2h_{2};\chi_{1},1-\chi_{2})$
(4.26)

where $F_{2}$ is the Appell Hypergeometric function. Since the correlator can
be expressed in terms of a constant topological term and three functions, it
is useful to write the expression of the blocks in terms of this
decomposition. This leads to the results in the following sections, where the
superblocks have the same complexity as the scalar blocks leading to efficient
expansions and relatively elegant expressions. Additionally, the vanishing of
the topological term for superblocks of long operators is a non-trivial check
of the validity of these expressions. In this channel, the topological limit
is given by

$\displaystyle
f^{0}=c_{\mathcal{D}_{2},\mathcal{I}}+c_{\mathcal{I},\mathcal{D}_{2}}+c_{\mathcal{D}_{2},\mathcal{D}_{2}}$
(4.27)

The process of fixing these correlators is somewhat intricate and long, in
addition to being needed for over ten different superblocks (counting those in
the asymmetric channel), so the derivation will be illustrated with the
simplest non-trivial superblock
$\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{2}}$. The process was repeated with
all the blocks listed in the Appendices C.2 and C.3 where the superconformal
blocks are all fully constrained by the Ward Identity above, including
$\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{L}^{h_{2}}_{0,[0,0]}}$
and
$\mathcal{G}_{\mathcal{L}^{h_{1}}_{0,[0,0]},\mathcal{L}^{h_{2}}_{0,[0,1]}}$
which are each a sum of over 50 scalar conformal blocks. This reveals the true
power of these multipoint Ward identities, which fix all the unknown
constants.

#### $\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{2}}$

The superblock $\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{2}}$ includes the
contributions from the superprimary $[0,2]^{\Delta=2}_{0}$ as well as all its
super descendants with vanishing transverse spin (charge under the $SO(3)$).
These are

$\displaystyle[0,2]^{2}_{0}$ $\displaystyle[2,0]_{0}^{3}$
$\displaystyle[0,0]^{4}_{0}$ (4.28)

where, in the following, we will drop the transverse spin label as it will
always vanish. There is an additional selection rule from the R-symmetry being
applied at the remaining OPE vertex $\mathcal{D}_{2}(5)$ in 4.1 where the
superconformal primary $[0,2]^{2}$ imposes the constraint that the operator
exchanged on the right must be in the product of the two on the left, i.e

$\displaystyle\begin{matrix}[0,2]\\\ [2,0]\\\
[0,0]\end{matrix}\subset[0,2]\otimes\begin{matrix}[0,2]\\\ [2,0]\\\
[0,0]\end{matrix}$ (4.29)

Applying these selection rules, we obtain the following decomposition

$\displaystyle\mathcal{G}_{\mathcal{D}_{2},\mathcal{D}_{2}}$
$\displaystyle=\beta_{1}h_{[0,0][0,2]}g_{4,2}+\beta_{2}h_{[0,2][0,0]}g_{2,4}+\beta_{3}h_{[0,2][2,0]}g_{2,3}$
$\displaystyle+\beta_{4}h_{[2,0][0,2]}g_{3,2}+\beta_{5}h_{[0,2][0,2]}g_{2,2}+\beta_{6}h_{[2,0][2,0]}g_{3,3}$
(4.30)

where $g_{h_{1},h_{2}}$ are the scalar conformal blocks in (4.26),