aainstitutetext: Dipartimento di Fisica, Università di Firenze and INFN
Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy

# On protected defect correlators in 3d $\mathcal{N}\geq 4$ theories

Luigi Guerrini<EMAIL_ADDRESS>
###### Abstract

We study and compute supersymmetric observables for line defects in 3d
$\mathcal{N}\geq 4$ theories. Our setup is a novel supersymmetric
configuration involving line operators and local operators living on a linked
circle. The algebra of the local operators is described by a topological
quantum mechanics. For operators belonging to conserved current multiplets, we
propose an exact formula for their correlation functions based on a Ward
identity for integrated correlators. Our formula gives a general recipe to
compute the bremsstrahlung function for any $\frac{1}{3}$-BPS lines in
$\mathcal{N}=6$ SCFTs. We apply our relation to the $\frac{1}{2}$-BPS Wilson
loop in the ABJM model, showing the validity of previous computations.
Furthermore, our construction allows us to explore higher points correlators.
As an example, we compute the two-point function of the stress tensor
multiplet correlators in ABJM theory in the presence of the Wilson line. We
also present some perturbative checks of our formulae.

##  1 Introduction

In conformal field theories (CFTs), the fundamental observables are
correlation functions of local operators. In principle, they are completely
determined by the spectrum, i.e. the dimensions of the operators and the
operator product expansion (OPE) coefficients. Dimensions and OPE coefficients
form the CFT data. CFTs can also be enriched by extended defects preserving
the worldvolume conformal symmetry. These are conformal defects, and relevant
examples include magnetic impurities and line operators in conformal gauge
theories. Due to the reduced symmetry, more observables are available: These
include the spectrum of defect operators and the bulk-to-boundary OPE
coefficients Billo:2016cpy . Despite the remarkable progresses in recent years
based on the bootstrap approach Rattazzi:2008pe , the computation of CFT data
is still a highly non-trivial task, with or without conformal defects.

Superconformal field theories (SCFTs) are distinguished models where some CFT
data can be computed exactly. For instance, the correlation functions of
operators preserving a fraction of supersymmetry, called BPS operators, can be
protected from quantum corrections. That is the case for the two and three-
point functions of the $\frac{1}{2}$-BPS local operators in $\mathcal{N}=4$
SYM Lee:1998bxa . Building on this example, the authors of Drukker:2009sf
found a class of local operators preserving the same supersymmetries
regardless of the number of operator insertions. The crucial insight is to
endow the operators with a controlled spacetime dependence via a conformal
twist deMedeiros:2001wqm . This idea has been recently reinvestigated in
Beem:2013sza  from the superconformal bootstrap point of view and extended to
$\mathcal{N}=2$ SCFTs. Remarkably, a sector of the full operator algebra is
described by a chiral algebra provided that operators lie in the same plane
(see also Beem:2014kka  for the analogous construction in 6d). Similarly, in
3d $\mathcal{N}\geq 4$ SCFTs correlation functions of a subset of operators
placed on a line are captured by a topological field theory Chester:2014mea ;
Beem:2016cbd ; Liendo:2015cgi . We call this set the _topological sector_ ,
and it will be of the main character of this paper.

One of the advantages of the protected subsectors is the possibility of using
supersymmetric localization to access some of the associated CFT data. For
example, a localization scheme for the topological sector in UV Yang-Mills
theories flowing to interacting $\mathcal{N}\geq 4$ SCFTs was proposed in
Dedushenko:2016jxl ; Dedushenko:2017avn ; Dedushenko:2018icp  for operators on
$S^{3}$ and in Panerai:2020boq  for operators on $S^{2}\times S^{1}$. However,
a complete localization is not always accessible. Alternatively, one can
relate specific operators of the SCFT to exactly computable supersymmetric
deformations of the theory. For instance, Coulomb branch operators in 4d
$\mathcal{N}=2$ are related to marginal deformations of the SCFT
Gerchkovitz:2016gxx . Similarly, in 3d $\mathcal{N}\geq 4$ SCFTs, certain
integrated topological operators are captured by supersymmetric mass
deformations of the partition function Agmon:2017xes ; Binder:2019mpb ;
Binder:2020ckj ; Gorini:2020new ; Guerrini:2021zuk ; Bomans:2021ldw . This
strategy is particularly convenient when the deformed partition functions is
known exactly.

A natural generalization is to study superconformal field theories with
superconformal defects. They are examples of tractable conformal defects,
which can be studied with a variety of techniques, like localization,
bootstrap, integrability, and holography. Following the same logic of the
previous examples, supersymmetric setups with both non-local and local
operators would provide us with a powerful tool to investigate the physics of
the defect. Relevant examples in $\mathcal{N}=4$ SYM involve the
$\frac{1}{2}$-BPS defects, which can be the BPS Wilson line Maldacena:1998im ,
surface defects Gukov:2006jk , and boundaries/interfaces Gaiotto:2008sa ;
Gaiotto:2008ak  together with the protected local operators mentioned above
Drukker:2009sf . In these cases, localization schemes are available
Giombi:2009ds ; Wang:2020seq . Less supersymmetric examples in 4d involve
Coulomb branch operators and $\frac{1}{2}$-BPS Wilson loops in $\mathcal{N}=2$
gauge theories Billo:2018oog , and chiral algebra and surface defects in 4d
$\mathcal{N}=2$ SCFTs Bianchi:2019sxz ; Pan:2017zie .

This paper aims to explore the corresponding problem for 3d SCFTs with BPS
line operators, such as Wilson and vortex loops Gaiotto:2007qi ;
Drukker:2008jm ; Chen:2008bp ; Rey:2008bh ; Drukker:2009hy ; Drukker:2008zx ;
Kapustin:2012iw ; Drukker:2012sr . These are interesting extended operators
with holographic duals and exhibit non-trivial mapping under IR dualities
Assel:2015oxa ; Dimofte:2019zzj ; Griguolo:2021rke ; Thull:2022lif . Since in
3d the only known family of non-trivial protected local operators live in
$\mathcal{N}=4$ theories, we restrict to these models.

To fix ideas, we begin to study theories with a UV Lagrangian description.
Having in mind localization, we work directly on $S^{3}$. These theories admit
$\frac{1}{2}$-BPS Wilson and vortex loops on any great circle. Even if they
are not invariant under conformal symmetry, they are believed to flow to non-
trivial conformal defects Assel:2015oxa . From a detailed analysis of the
preserved supersymmetry algebras, we find a supersymmetric configuration with
a topological sector on the great circle linking with the loop operator. At
the CFT point, the corresponding flat space setup has the defect extended on a
straight line, and the topological operators on a circle in the orthogonal
plane, as sketched in figure 1. More precisely, for any BPS Wilson loop, we
identify a Coulomb branch topological sector, i.e. a set of operators built
out of the vector multiplet. We decribe also the mirror symmetric
configuration, with a vortex loop and a Higgs branch topological sector, whose
fields are made by scalars in the hypermultiplet. We then extend this example
to any $\frac{1}{2}$-BPS line operators in a generic $\mathcal{N}=4$ SCFT.

Then, we show how to exploit these configurations to extract CFT data. The
main tools are localization and the Ward identity relating mass deformations
and dimension one topological operators. Using the Ward identity, we argue
that the vev of the defect operator in a properly mass-deformed background is
the generating functional for integrated defect correlation functions,
generalizing the argument without defects. Using localization, one can
sometimes reduce the computation of the vev to a finite-dimensional integral,
i.e. a matrix model, which can be evaluated in different limits.

As concrete applications, we evaluate the correlation functions with dimension
one topological operators and the $\frac{1}{2}$-BPS Wilson line in the
$U(N_{1})_{k}\times U(N_{2})_{-k}$ ABJM model. The ABJM model is
$\mathcal{N}=6$ Super Chern-Simons-matter theories, dual to type IIA string
theory in AdS${}_{4}\times\mathbb{CP}_{3}$ or M-theory in AdS${}_{4}\times
S^{7}/Z_{k}$ Aharony:2008ug ; Aharony:2008gk . The relevant operator is a
specific bilinear of the matter scalar fields, which is also part of the
stress tensor multiplet. We evaluate the corresponding one and two-point
functions by taking mass derivatives of the mass-deformed vev of the Wilson
loop, which is known from localization Kapustin:2009kz ; Jafferis:2010un ;
Hama:2010av . According to Lewkowycz:2013laa , the one-point function computes
the so-called bremsstrahlung function Correa:2012at . We expand our matrix
model prediction for large values of the Chern-Simons coupling $k$. We find
perfect agreement with all the data available in the literature
Griguolo:2012iq ; Lewkowycz:2013laa ; Bianchi:2014laa ; Bianchi:2017svd ;
Bianchi:2017ozk ; Bianchi:2018bke ; Bianchi:2018scb ; Griguolo:2021rke . For
the case $N_{1}\neq N_{2}$, this provides the first complete derivation of the
bremsstrahlung, confirming the conjecture of Bianchi:2018bke . We also propose
a formula for the bremsstrahlung valid for any $\frac{1}{3}$-BPS line defect
Drukker:2022txy . As far as we know, the two-point function is a novel result,
and it is the simplest correlator exhibiting non-trivial crossing symmetry.
Finally, we test our results against a genuine Feynman diagram computation.

The paper is organized as follows. In Sec. 2 we introduce the theories and the
line operators we will study. Then, in Sec. 3, after a brief review of the
topological sector, we combine it with local operators, both for UV gauge
theories and SCFTs. In Sec. 4 we present our formula relating integrated
defect correlation functions to mass deformations, and we apply it to ABJM.
Finally, in Sec. 5 we discuss our results and compare against perturbative
computations. We end the paper with a general discussion on the results and
possible future directions. Technical details and conventions are provided in
two appendices.

Figure 1: We sketch the kinematics of the flat space configuration. The line
defect $\mathcal{D}$ extends along a straight line, while the local operators
$\mathcal{O}(\tau_{i})$ live on the circle in the orthogonal plane to the
defect.

##  2 Line operators in $\mathcal{N}\geq 4$ theories

In this section, we give an overview of BPS line operators in 3d
$\mathcal{N}\geq 4$ gauge theories, both with and without Chern-Simons terms.
Then, we discuss the observables in conformal defects, with some focus on
those related to the stress tensor.

###  2.1 Loop operators in UV gauge theories

To begin with, we discuss gauge theories without Chern-Simons terms
Dedushenko:2016jxl . Since we aim to make contact with localization, we find
it convenient to define these QFTs directly on the three-sphere $S^{3}$. We
work in toroidal coordinates $\theta\in[0,\frac{\pi}{2})$,
$\varphi\in[0,2\pi)$, $\tau\in[0,2\pi)$. The metric reads

$ds^{2}=r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}+\cos^{2}\theta\,d\tau^{2}\right)\,.$
(1)

The name toroidal is because at fixed $\theta$ the metric is a 2d torus. When
$\theta=0$, the metric shrinks to a circle $S^{1}_{\varphi}$, while at
$\theta=\pi/2$, the metric shrinks to a circle $S^{1}_{\tau}$. The subscripts
$\varphi,\,\tau$ indicate the variables parametrizing the circle in these
coordinates 111More details on the geometry of $S^{3}$ can be found in
Dedushenko:2016jxl .. These two great circles will play a central role in our
construction

The building blocks are vector multiplets and hypermultiplets. The
$\mathcal{N}=4$ vector multiplet $\mathcal{V}$ contains the gauge vector
$A_{\mu}$, the gaugino $\lambda_{\alpha,a\dot{a}}$, the dimension one scalar
$\Phi_{\dot{a}\dot{b}}$, and the auxiliary field $D_{ab}$, which has dimension
2. They transform respectively in the singlet, $(\mathbf{2},\mathbf{2})$,
$(\mathbf{3},\mathbf{1})$, and $(\mathbf{1},\mathbf{3})$ representation of the
R-symmetry group $SU(2)_{C}\times SU(2)_{H}$. All the components transform in
the adjoint representations of the gauge group $G$. The matter is organized in
hypermultiplets $\mathcal{H}$, whose field components are the $SU(2)_{H}$
doublets $q_{a}$ and $\tilde{q}^{a}$, and the fermions $\psi_{\dot{a}}$ and
$\tilde{\psi}^{\dot{a}}$, which are doublets of $SU(2)_{C}$. $q_{a}$ and
$\psi_{\dot{a}}$ transform in the unitary representation $\mathcal{R}$ of $G$,
$\tilde{q}^{a}$ and $\tilde{\psi}^{\dot{a}}$ in the complex conjugate
representation $\bar{\mathcal{R}}$.

The action for the hypers can be obtained by conformally mapping the flat
space one on $S^{3}$. We get

$\displaystyle S_{\textup{hyper}}$ $\displaystyle=\int
d^{3}x\sqrt{g}\Bigg{[}\mathcal{D}^{\mu}\tilde{q}^{a}\mathcal{D}_{\mu}q_{a}-i\tilde{\psi}^{\dot{a}}\gamma^{\mu}\mathcal{D}_{\mu}\psi_{\dot{a}}+\frac{3}{4r^{2}}\tilde{q}^{a}q_{a}+i\tilde{q}^{a}{D_{a}}^{b}q_{b}-\frac{1}{2}\tilde{q}^{a}\Phi^{\dot{a}\dot{b}}\Phi_{\dot{a}\dot{b}}q_{a}+$
$\displaystyle-i\tilde{\psi}^{\dot{a}}{\Phi_{\dot{a}}}^{\dot{b}}\psi_{\dot{b}}+i\left(\tilde{q}^{a}{\lambda_{a}}^{\dot{b}}\psi_{\dot{b}}+\tilde{\psi}^{\dot{a}}{\lambda^{b}}_{\dot{a}}q_{b}\right)\Bigg{]}\,.$
(2)

The action is invariant under (132) If we think of the gauge field as a
background field, this action is conformally invariant. Writing down a
superconformal action for the _off-shell_ vector multiplet is a hard task
Kuzenko:2015lfa , and we do not pursue it here. However, it is possible to
give up conformal invariance and write down a supersymmetric Yang-Mills term

$\displaystyle S_{\textup{YM}}=$
$\displaystyle\frac{1}{g^{2}_{\mathrm{YM}}}\int
d^{3}x\sqrt{g}\Tr[F^{\mu\nu}F_{\mu\nu}-\mathcal{D}^{\mu}\Phi^{\dot{a}\dot{b}}\mathcal{D}_{\mu}\Phi_{\dot{a}\dot{b}}+i\lambda^{a\dot{a}}\gamma^{\mu}\mathcal{D}_{\mu}\lambda_{a\dot{a}}-D^{ab}D_{ab}+$
(3)
$\displaystyle-i\lambda^{a\dot{a}}\commutator*{{\lambda_{a}}^{\dot{b}}}{\Phi_{\dot{a}\dot{b}}}-\frac{1}{4}\commutator*{{\Phi^{\dot{a}}}_{\dot{b}}}{{\Phi^{\dot{c}}}_{\dot{d}}}\commutator*{{\Phi^{\dot{b}}}_{\dot{a}}}{{\Phi^{\dot{d}}}_{\dot{c}}}-\frac{1}{2r}h^{ab}\bar{h}^{\dot{a}\dot{b}}\lambda_{a\dot{a}}\lambda_{b\dot{b}}+$
$\displaystyle+\frac{1}{r}\left({h_{a}}^{b}{D_{b}}^{a}\right)\Bigl{(}{\bar{h}^{\dot{a}}}_{\dot{b}}{\Phi^{\dot{b}}}_{\dot{a}}\Bigr{)}-\frac{1}{r^{2}}\Phi^{\dot{a}\dot{b}}\Phi_{\dot{a}\dot{b}}\bigg{]}\,.$

This action is invariant under the variations of (131), when restricted to the
Poincaré subalgebra $\mathfrak{su}(2|1)_{\ell}\oplus\mathfrak{su}(2|1)_{r}$.
Its bosonic part contains the isometry group of the sphere $SO(4)\simeq
SU(2)_{\ell}\times SU(2)_{r}$, and the Cartan of the $SO(4)$ R-symmetry group
$U(1)_{\ell}\times U(1)_{r}$. An essential ingredient in its definition is the
matrices ${h_{a}}^{b}\in\mathfrak{su}(2)_{H}$,
${\bar{h}_{\dot{a}}}^{\;\;\dot{b}}\in\mathfrak{su}(2)_{C}$ which breaks the
original R-symmetry down to the Cartan 222For explicit computation we choose
${h_{a}}^{b}=-{(\sigma_{2})_{a}}^{b}$,
${\bar{h}_{\dot{a}}}^{\;\;\dot{b}}=-{(\sigma_{3})_{\dot{a}}}^{\;\;\dot{b}}$..
The superalgebra has eight fermionic generators $Q_{\alpha}^{(\ell\pm)}$,
$Q_{\alpha}^{(r\pm)}$, closing respectively on the left and right part of the
bosonic algebra. The relevant details and conventions are summarized in the
App. B. In the end, the theory preserves eight supercharges. We postpone the
discussion on real mass and FI deformations to section 4, where they will play
a central role to perform computations.

Gauge theories contain various loop operators. Even if theories with extended
supersymmetry can admit loop operators on several closed paths, we shall focus
on BPS Wilson loops supported on great circles, e.g. $S^{1}_{\varphi}$. The
first example is perhaps the most well-known, namely the BPS Wilson loops
whose generic expression is obtained by adding to the standard Wilson loop a
coupling to the scalar in the vector multiplet

$W_{R}=\Tr\,P\exp\left[i\int_{\gamma}ds\left(\dot{x}^{\mu}A_{\mu}+ir\Phi_{\dot{a}\dot{b}}M_{\textup{Wislon}}^{\dot{a}\dot{b}}\right)\right]\,,$
(4)

where $R$ indicates a representation of the gauge group. Restricting the curve
$\gamma$ to be $S^{1}_{\varphi}$ and requiring SUSY, we get a
$\frac{1}{2}$–BPS Wilson loop, whose explicit form is

$W_{R}=\Tr\,P\exp\left[i\int_{S^{1}_{\varphi}}d\varphi(A_{\varphi}-ir\Phi_{\dot{1}\dot{2}})\right]\,.$
(5)

It preserves $Q^{\ell+}_{2}$, $Q^{(\ell-)}_{1}$, $Q^{(r+)}_{2}$, and
$Q^{(r-)}_{1}$, generating a $\mathfrak{su}(1|1)\oplus\mathfrak{su}(1|1)$
superalgebra.

Another distinguished class of line operators are vortex loops. They are
defect operators defined by singular classical BPS configurations for the
vector field. The field strength $F_{\mu\nu}$ is proportional to a delta
function supported on the loop worldvolume. Their BPS version was introduced
in Kapustin:2012iw ; Drukker:2012sr  and is realized by turning on a singular
imaginary profile for the auxiliary field. In the $\mathcal{N}=4$ language,
the form of an abelian BPS vortex loop of charge $q$ reads

$F=2\pi q\delta_{\ell},\quad D^{ab}=-2\pi q\star\left(\delta_{\ell}\wedge
d\ell\right)M_{\text{vortex}}^{ab}\,,$ (6)

where $d\ell$ is the one form dual to $\dot{\ell}$, and $\delta_{\ell}$ is the
Poincaré dual to the loop worldvolume. The matrix $M_{\text{vortex}}^{ab}$ is
chosen in combination with the curve $\gamma$ to set to zero the variation of
the gaugino for some specific supercharge. Again, we shall restrict to
operators supported on $S^{1}_{\varphi}$. If we also choose
$M_{\text{vortex}}^{ab}=-\mathbb{1}^{ab}$, the vortex loop preserves half of
the supersymmetry, namely $Q^{(\ell+)}_{1}$, $Q^{(\ell-)}_{2}$,
$Q^{(r+)}_{2}$, and $Q^{(r-)}_{1}$ generating another
$\mathfrak{su}(1|1)\oplus\mathfrak{su}(1|1)$ superalgebra. To get non-abelian
vortex loops, one replaces the number $q$ with an element of the Lie algebra
$\mathfrak{g}$333An alternative definition is proposed in Assel:2015oxa  (see
also Hosomichi:2021gxe  for a recent discussion). The vortex loop is
represented by adding one-dimensional local degrees of freedom on the curve
supporting the loop operators. Integrating out the 1d d.o.f. reproduces the
vortex singularity for the bulk fields..

###  2.2 BPS lines in Chern-Simons matter theory (ABJM)

Including a Chern-Simons term preserving $\mathcal{N}\geq 4$ supersymmetry
requires some more work. As far as we know, no formulation with off-shell
vector multiplets preserving the full superconformal algebra is known.
Nevertheless, Chern-Simons theories with $\mathcal{N}\geq 4$ supersymmetry
arise from specific lower supersymmetric theories after the integrations of
all the auxiliary fields. This idea was introduced in Gaiotto:2008sd  and
further generalized in Hosomichi:2008jd ; Imamura:2008dt . We will refer to
these theories as generalized Gaiotto-Witten theories. The ABJM model, which
is a $\mathcal{N}=6$ quiver theory with gauge group $U(N_{1})_{k}\times
U(N_{2})_{-k}$ (see App. A for more details), falls in this category
Aharony:2008ug ; Aharony:2008gk ; Hosomichi:2008jb . Generalized Gaiotto-
Witten theories contain many interesting BPS loop operators Cooke:2015ila .
However, since our applications will be in ABJM, we describe explicitly them
only in this case. There are no conceptual difficulties to extend our
construction to more general Chern-Simons theories.

The construction of a maximally supersymmetric Wilson loop in ABJM is somewhat
non-standard and requires to think the gauge group $U(N_{1})\times U(N_{2})$
embedded into the supergroup of $U(N_{1}|N_{2})$. Then, we consider the
holonomies of this larger group Drukker:2009hy . That is, we take line
operators of the form

$W[C]=\Tr_{\mathcal{R}}\mathcal{P}\exp(-i\int_{C}\mathcal{L})\,,$ (7)

where $\mathcal{R}$ is a representation of $U(N_{1}|N_{2})$. For a generic
smooth path $C$ in ${\mathbb{R}}^{3}$ parametrized as $x^{\mu}=x^{\mu}(\tau)$,
the general structure of $\mathcal{L}$ is given by

$\mathcal{L}=\begin{pmatrix}A_{\mu}\dot{x}^{\mu}-\frac{2\pi
i}{k}|\dot{x}|{M_{J}}^{I}C_{I}\bar{C}^{J}&i\sqrt{2\pi\frac{\ell}{k}}|\dot{x}|\eta^{\alpha}_{I}\bar{\psi}^{I}_{\alpha}\\\
-i\sqrt{2\pi\frac{\ell}{k}}|\dot{x}|\psi_{I}^{\alpha}\bar{\eta}_{\alpha}^{I}&\hat{A}_{\mu}\dot{x}^{\mu}-\frac{2\pi
i}{k}|\dot{x}|{\hat{M}_{J}}^{I}\bar{C}^{J}C_{I}\end{pmatrix}$ (8)

Sometimes we refer to the resulting operator as the “fermionic” Wilson loop
because matter fermions appear naturally in the off-diagonal entries of
$\mathcal{L}$. The possible local quantities $M_{J}^{\ \ I}(\tau)$,
$\hat{M}_{J}^{\ \ I}(\tau)$, $\eta_{I}^{\alpha}(\tau)$ and
$\bar{\eta}^{I}_{\alpha}(\tau)$ regulate the coupling of the matter fields to
the loop. Imposing local $U(1)\times SU(3)$ $R-$symmetry and local
supersymmetry invariance, we find a general expression for the couplings in
(8)

$\begin{split}&\eta_{I}^{\alpha}(\tau)=n_{I}(\tau)\eta^{\alpha}(\tau),\ \ \
\bar{\eta}^{I}_{\alpha}(\tau)=\bar{n}^{I}(\tau)\bar{\eta}_{\alpha}(\tau),\ \ \
M_{J}^{\ \ I}(\tau)=\widehat{M}_{J}^{\ \
I}(\tau)=\delta^{I}_{J}-2n_{J}(\tau)\bar{n}^{I}(\tau)\end{split}$ (9)

Following Drukker:2009hy ; Cardinali:2012ru ; Lietti:2017gtc , one can
determine the functions of $n^{I}(\tau)$ and $\eta_{\alpha}(\tau)$ on $\tau$
by imposing invariance under some superconformal transformations. Crucially,
we demand a more general BPS condition w.r.t. to Wilson loop in Yang-Mills
theories. Unlike the case of UV gauge theories, where we demand that a
supersymmetry variation annihilates the connection of the Wilson loop in (5),
in Chern-Simons matter theories we only impose the variation to be a
supergauge transformation

$\delta W=\commutator{G(\tau)}{W}\,,$ (10)

where $G(\tau)$ is a field-dependent super-gauge transformation belonging to
$U(N_{1}|N_{2})$. This weaker BPS condition makes the Wilson loop invariant
only after taking the trace, forcing us to look carefully at the form of
$G(\tau)$. Since for closed path $G(\tau)$ is not periodic, we need to modify
the definition (7) by adding a twist matrix $\mathcal{T}$ restoring the
correct periodicity of $G(\tau)$. In the end, the fermionic Wilson loop is

$\mathcal{W}=\mathrm{Str}\left(W[C]\mathcal{T}\right)$ (11)

See Cardinali:2012ru  for details444Alternatively, this issue can be cured by
introducing a classical background connection along the path, which makes the
super-gauge transformation periodic Drukker:2019bev .. When the path $C$ is
the infinite straight line parametrized by $x_{3}$, i.e. $x^{\mu}=(0,0,s)$,
$-\infty<s<+\infty$, and with the following choices for the couplings

${M_{J}}^{I}={\hat{M}_{J}}^{I}=\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\
0&0&0&1\end{pmatrix},\qquad\eta^{\alpha}_{I}=\sqrt{2}\begin{pmatrix}1\\\ 0\\\
0\\\ 0\end{pmatrix}_{I}(1,0)^{\alpha},\qquad\bar{\eta}_{\alpha}^{I}=i\sqrt{2}\
(1,0,0,0)^{I}\begin{pmatrix}1\\\ 0\end{pmatrix}_{\alpha}$ (12)

the operator preserves half of the supersymmetry charges Cardinali:2012ru ,
i.e. it defines a $1/2$ BPS linear defect. See App. B for details on the
preserved superalgebra. Even if it is not strictly required by supersymmetry,
we introduce in the definition a twist matrix $\sigma_{3}$ in the trace (see
Drukker:2009hy ; Cardinali:2012ru ). The reason is that we want to be able to
conformally map the Wilson line operator to the circular Wilson loop on
$S^{3}$ to perform computations consistently. Since the fermionic Wilson loop
on the great circle of $S^{3}$ requires the twist matrix, we insert it also
for the Wilson line. This choice amounts to taking the trace instead of the
super-trace of (7) and it is also the prescription that leads to an operator
that is dual to the 1/2 BPS string configuration in
AdS${}_{3}\times{\mathbb{CP}}^{3}$ or a 1/2 BPS M2-brane configuration in
M-theory Lietti:2017gtc .

###  2.3 Line operators as conformal defects

We review some properties of conformal defects Billo:2016cpy . By definition,
a conformal line breaks the 3d conformal symmetry down to $SO(2)\times
SL(2,\mathbb{R})$. Therefore, $\frac{1}{2}$-BPS Wilson line in ABJM is an
example of a conformal line defect. Sometimes, it is also possible to engineer
a UV avatar for a conformal line. That is the case of the lines described in
Sec. 2.1. Even if the loop operators in the UV theory do not preserve
conformal symmetry, there is strong evidence from localization and dualities
that they flow to non-trivial conformal defects in the deep IR. We will
briefly review some relevant features of the physics of conformal line
defects. The main point will be that the knowledge of the dimensions of all
bulk operators and all their three-point functions will no longer exhaust the
possible CFT data. Even if all our examples are Lagrangian, the following
discussion does not rely at all on that.

For simplicity, we limit to straight lines in flat three-dimensional space. We
split the coordinates $x^{\mu}=(x^{i},s)$, where $x^{i}$ indicates the
transverse coordinates $i=1,\dots,2$ and $s$ the coordinates along the
defects. The defect $\mathcal{D}$ is placed at the position $x_{i}=0$. We
introduce the following notation for defect correlation functions

$\langle O_{1}(x_{1}^{i_{1}})\dots
O_{n}(x_{n}^{i_{n}})\rangle_{\mathcal{D}}=\frac{1}{\langle\mathcal{D}\rangle}\langle
O_{1}(x_{1}^{i_{1}})\dots O_{n}(x_{n}^{i_{n}})\mathcal{D}\rangle\,.$ (13)

where $\langle\mathcal{D}\rangle$ is the vev of the defect. The first novelty
is that the defect one-point functions can be non-zero. For scalar operators,
it is not hard to show that

$\langle O(x)\rangle_{\mathcal{D}}=\frac{h_{O}}{|x^{i}|^{\Delta_{O}}}\,.$ (14)

where $\Delta_{O}$ is the dimension of $O$ and $|x^{i}|$ its distance from the
defect. Therefore, $h_{O}$ are new dynamical data of the CFT with the defect.

A particularly relevant example is the defect one-point function of the stress
tensor $T_{\mu\nu}$. One can show that (see e.g. Kapustin:2005py )

$\langle
T^{\mu\nu}(x)\rangle_{\mathcal{D}}=\frac{h_{T}}{|x^{i}|^{d}}H^{\mu\nu}\,.$
(15)

$H^{\mu\nu}$ is a symmetric traceless tensor

$H^{ij}=-\left(\frac{2}{3}\delta^{ij}-\frac{x^{i}x^{j}}{|x^{i}|^{2}}\right)\,,\quad
H^{ab}=\frac{1}{3}\delta^{ab}\,,\quad H^{ia}=0$ (16)

Since the stress tensor is always defined in a local CFT, $h_{T}$ is a
universal observable defined in every CFT. Higher spinning operators can have
a non-vanishing one-point function if one manages to build a tensor in the
appropriate representation. However, their discussion is beyond our purposes,
and we refer to Billo:2016cpy .

The second novelty is the possibility of defect excitations, denoted by
operators $\hat{O}(s)$ living on the defects. Their correlation functions,
denoted as

$\langle\hat{O}_{1}(s_{1})\dots\hat{O}_{k}(s_{k})\rangle_{\mathcal{D}}=\frac{1}{\langle\mathcal{D}\rangle}\langle\hat{O}_{1}(s_{1})\dots\hat{O}_{k}(s_{k})\mathcal{D}\rangle\,.$
(17)

are constrained in the usual way by the defect conformal group
$SL(2,\mathbb{R})$. Therefore, they form a peculiar conformal theory called
the defect conformal field theory (dCFT). For example, for a conformal Wilson
line $W$ in a Lagrangian theory, we can give an explicit expression for the
corresponding dCFT. The operators $\hat{O}_{i}(s)$ are combinations of
fundamental fields transforming in the same representation of the gauge group
as the Wilson loop’s connection. If we denote as $W_{s_{i},s_{j}}$ the
untraced Wilson segment from $s_{i}$ to $s_{j}$, we

$\langle\hat{O}_{1}(s_{1})\dots\hat{O}_{k}(s_{k})\rangle_{W}=\frac{1}{\langle
W\rangle}\langle\Tr W_{-\infty,s_{1}}\hat{O}_{1}(s_{1})\dots
W_{s_{k-1},s_{k}}\hat{O}_{k}(s_{k})W_{s_{k},\infty}\rangle\,.$ (18)

The dCFT satisfies the axioms of a standard CFT. Thus, correlators of the dCFT
will be determined by the defect spectrum $\Delta_{\hat{O}}$ and by the
coefficients of the defect three-point functions
$\hat{c}_{\hat{O}_{i},\hat{O}_{j},\hat{O}_{k}}$.

A relevant example of a defect operator is the _displacement operator_. The
insertion of the defect modifies the Ward identities for broken symmetries by
contact terms living localized on the worldline. The displacement operator
$\mathsf{D}^{i}$ is responsible for such contributions for the broken
transverse translations $P^{i}$. Explicitly, one can show that

$\partial_{\mu}T^{\mu
i}(x)=-\delta_{\mathcal{D}}(x^{i})\mathsf{D}^{i}(x^{a})\,,$ (19)

The definition holds inside correlation functions. The physical interpretation
is that the displacement operator describes the energy exchanges between the
bulk and the defect. We observe that the displacement is defined up to primary
defect operators. We fix them by requiring that $T^{\mu\nu}$ is a conformal
primary near the defect. In turn, this implies that the displacement is a
defect primary, whose dimension is fixed by the Ward identity to be
$\Delta_{\mathsf{D}}=2$. Being the normalization of the displacement fixed by
the Ward identity, the coefficient $B$ of its two-point function is a physical
quantity

$\langle\mathsf{D}^{i}(x^{a})\mathsf{D}^{i}(y^{b})\rangle_{\mathcal{D}}=\delta^{ij}\frac{12B}{|x-y|^{2\Delta_{\mathsf{D}}}}\,.$
(20)

The quantity $B$ plays a similar role to the coefficient of the two-point
function of the stress tensor for standard CFTs. Furthermore, for line
defects, $B$ has a physical interpretation as the bremsstrahlung function,
namely the energy emitted by a slowly moving particle Correa:2012at .

The final set of CFT data $b_{O\hat{O}}$ is needed to specify bulk-to-boundary
$OPE$. This means that the configuration with the bulk operator very close to
the defect is indistinguishable from an infinite sum of defect excitations.
For a scalar bulk operator of dimension $\Delta$, it reads

$O(x^{a},x^{i})\sim
b_{O\hat{O}}|x_{i}|^{\hat{\Delta}-\Delta}\hat{O}(x_{a})+\cdots$ (21)

where $\hat{\Delta}$ is the dimension of the operator $\hat{O}(x_{a})$. In
conclusion, we identify a set of novel CFT data specified by $h_{O}$,
$\Delta_{\hat{O}}$, $b_{O\hat{O}}$,
$\hat{c}_{\hat{O}_{i},\hat{O}_{j},\hat{O}_{k}}$. However, they are not
independent due to the constraints of crossing symmetry, which imposes non-
trivial relations among the various data. The simplest observable to impose
these constraints is the two-point functions of bulk operators. The systematic
study of these constraints leads to the defect bootstrap program Liendo:2012hy
.

All the above considerations do not depend on supersymmetry. In the rest of
the paper, we will compute some of the introduced observables in
supersymmetric theories.

##  3 Topological sector and line operators

In this section we aim to identify supersymmetric configurations involving
line operators and local operators. In $\mathcal{N}\geq 4$ theories, a family
of local protected operators were introduced in Chester:2014mea ; Beem:2016cbd
. Since their correlators restricted on a line are position-independent, we
will refer to them as the _topological sector_. These operators are the
natural candidate to play the role of local operators in our putative
construction.

To familiarize ourselves with the topological sector, we recall its
construction in $\mathcal{N}\geq 4$ superconformal field theories (SCFTs). The
3d superconformal algebra is $\mathfrak{osp}(4|4)$. Its maximal bosonic
subalgebra is the direct sum 3d conformal algebra $\mathfrak{sp}(4)$ and the
R-symmetry algebra $\mathfrak{su}(2)_{H}\oplus\mathfrak{su}(2)_{C}$, whose
generators are denoted by ${R_{a}}^{b}$ and
${{\bar{R}}_{\dot{a}}}^{\;\;\dot{b}}$, respectively. The fermionic generators
are given by the Poincarè supercharges $Q_{\alpha,a\dot{a}}$ and
superconformal charges $S_{\alpha,a\dot{a}}$ 555See again App B for more
details..

The idea for building the topological sector is to adapt the construction of
the chiral ring to 3d SCFTs. To have non-trivial correlators, we choose the
relevant nilpotent supercharge $\mathcal{Q}$ to be a linear combination of the
$Q_{\alpha,a\dot{a}}$ and $S_{\alpha,a\dot{a}}$ generators Beem:2013sza ;
Chester:2014mea . Then , we demand the existence of a $\mathcal{Q}$-exact
R-twisted translation $\hat{P}\sim P+R$, for some ordinary translation $P$
along a fixed direction. Similarly to the chiral ring case, we can move
$\mathcal{Q}$-closed operators in the $P$-direction by acting with $\hat{P}$
without changing the $\mathcal{Q}$-cohomology. Therefore, correlation
functions of $\mathcal{Q}$-closed operators placed on the 1d submanifold
generated by the translation turn out to be piecewise constant, depending at
most on the order of the insertions. Unlike the chiral ring in SCFTs, these
correlators do not have to vanish because the twisted translation generates
R-symmetry-dependent factors, which combine with the operators to make
singlets.

To be concrete, in 3d euclidean flat space we take the 1d submanifold to be
the line $x_{1}=x_{2}=0$ and focus on the construction of topological
operators in the Higgs branch first. The maximal superconformal algebra
preserved by the line is a central extension of the algebra
$\mathfrak{su}(1,1|2)$, whose bosonic subalgebra includes the
$\mathfrak{so}(2,1)$ conformal algebra and the $\mathfrak{su}(2)$ R-symmetry
algebra that we identify with $\mathfrak{su}(2)_{H}$. The bosonic generators
are $P_{3}$, $K_{3}$, $D$, ${R_{a}}^{b}$, and the fermionic ones are
$Q_{1,a\dot{2}}$, $Q_{2,a\dot{1}}$, $S_{2,a\dot{1}}$, and $S_{1,a\dot{2}}$.
The central element is $Z=iM_{12}-R_{\dot{1}\dot{2}}$, where $M_{12}$ is the
generator of rotations in the plane orthogonal to the line.

If we consider the two supercharges

$Q_{1}^{H}\equiv Q_{11\dot{2}}+\frac{1}{2r}{S^{2}}_{2\dot{2}}\,,\qquad
Q_{2}^{H}\equiv Q_{21\dot{1}}+\frac{1}{2r}{S^{1}}_{2\dot{1}}\,,$ (22)

with $r$ being an arbitrary length parameter, from the $\mathfrak{su}(1,1|2)$
algebra it is easy to see that they are both nilpotent operators. Moreover,
their anticommutator reads

$\anticommutator*{Q_{1}^{H}}{Q_{2}^{H}}=\frac{1}{r}Z\,.$ (23)

It then follows that since $Z$ must vanish on the $Q^{H}_{1,2}$ cohomology
classes, operators belonging to these classes are inserted along the fixed
point locus of $M_{12}$ and have zero $R_{\dot{1}\dot{2}}$ charge.

We now perform a topological twist by combining the $\\{P_{3},K_{3},D\\}$
generators of the $\mathfrak{so}(2,1)$ conformal algebra along the line and
the $\mathfrak{su}(2)_{H}$ R-symmetry generators. They are given by

$\displaystyle\hat{P}=P_{3}-\frac{1}{2r}{R_{2}}^{1}\,,\qquad\hat{K}=K_{3}-2r{R_{1}}^{2}\,,\qquad\hat{D}=D+{R_{1}}^{1}\,.$
(24)

One can think of them as the generators of a new one-dimensional conformal
algebra along the $x_{3}$-line. It turns out that the twisted generators are
all $Q_{1,2}^{H}$-exact. Therefore, we can first construct operators localized
at the origin from $Q_{1,2}^{H}$-closed, gauge invariant operators of the 3d
theory. Explicitly, the cohomology of the two supercharges contain local
operators $O_{a_{1},\,\dots,\,a_{n}}$ with the following properties
Chester:2014mea : they are Lorentz scalars, transform in the
$(\mathbf{n+1},\,\mathbf{1})$ of
$\mathfrak{su}(2)_{H}\oplus\mathfrak{su}(2)_{C}$, and have conformal dimension
$\Delta=-n/2$. We then move them along the line by applying the twisted
translation generator $\hat{P}$. The corresponding twisted translated operator
at position $\vec{x}=(0,\,0,\,s)$ is given by

$O(s)=e^{is\hat{P}}O_{1,\dots,1}(0)e^{-is\hat{P}}=O_{a_{1},\dots,a_{n}}(\vec{x})\big{|}_{\vec{x}=(0,0,s)}u^{a_{1}}\dots
u^{a_{n}}\,,\qquad u^{a}=\left(1,\frac{is}{2r}\right)\,.$ (25)

These operators are still $Q_{1,2}^{H}$-closed and form the Higgs topological
sector of the ${\cal N}\geq 4$ SCFT on the line. An analogous construction can
be carried on for Coulomb branch operators by exchanging dotted and undotted
indices. This amounts to exchange the role of $\mathfrak{su}(2)_{C}$ and
$\mathfrak{su}(2)_{H}$.

One of the crucial insights of Dedushenko:2016jxl  is that the topological
sector can also be defined away from the CFT point. In the end, what is needed
is a couple of supercharges $Q$, $\tilde{Q}$ such that
$\anticommutator*{Q}{\tilde{Q}}$ yields the twisted translation. We also
require $Q^{2}$ to close on a different twisted translation, which leaves
invariant the locus generated by $\anticommutator*{Q}{\tilde{Q}}$. Thus we can
define a sensible $Q$-cohomology. As in the conformal case, operators in the
$Q$-cohomology define the topological sector. For UV gauge theories on
$S^{3}$, one can define topological sectors on any great circle of $S^{3}$. If
the UV gauge theories are good in the sense of Gaiotto:2008ak , the UV
topological sectors will contain physical information on the IR SCFT.

In the following, we generalize this construction to have an additional BPS
loop operator on the great circle defined on the circle fixed $Q^{2}$, and
with a topological sector on the linked great circle. For SCFTs, we can map
the setup to flat space, where the defect extends along a straight line
direction, and the topological sector lives on the circle of radius $r$ in the
orthogonal plane. We first introduce our construction for the Wilson and
vortex loops in UV gauge theories defined in Sec. 2.1. These are the simplest
examples in which our construction is available. Then, we apply it to the
fermionic Wilson line in ABJM. From this example, we argue the existence of a
topological sector compatible with any $\frac{1}{2}$-BPS line operators in
$\mathcal{N}=4$ SCFTs.

###  3.1 Decorating the Wilson loop

The first example we investigate is the Wilson loop in UV gauge theories on
$S^{3}$, defined in Eq. (5). The explicit preserved
$\mathfrak{su}(1|1)\oplus\mathfrak{su}(1|1)$ algebra is

$\displaystyle\anticommutator*{Q_{2}^{(\ell+)}}{Q_{1}^{(\ell-)}}$
$\displaystyle=\frac{i}{r}\left(J_{12}^{(\ell)}+\frac{R_{C}-R_{H}}{2}\right)\,,$
(26) $\displaystyle\anticommutator*{Q_{2}^{(r+)}}{Q_{1}^{(r-)}}$
$\displaystyle=\frac{i}{r}\left(J_{12}^{(r)}+\frac{R_{H}+R_{C}}{2}\right)\,.$
(27)

where $J_{12}^{(\ell)}$ and $J_{12}^{(r)}$ are generators of isometries on
$S^{3}$ (see App. B) and their action on operators is given by

$P_{\tau}=i\partial_{\tau}=J_{12}^{(\ell)}+J_{12}^{(r)}\,,\qquad
P_{\varphi}=i\partial_{\varphi}=J_{12}^{(\ell)}-J_{12}^{(r)}\,,$ (28)

where $P_{\tau}$ and $P_{\varphi}$ generates translations along the $\tau$ and
$\varphi$ circle, respectively. $R_{H}$ and $R_{C}$ are R-symmetry
transformations generated by $R_{H}=\frac{1}{2}{h_{a}}^{b}{R_{b}}^{a}$ and
$R_{C}=\frac{1}{2}{\bar{h}_{\dot{a}}}^{\;\;\dot{b}}{\bar{R}_{\dot{b}}}^{\;\;\dot{a}}$
and acts on operators as

$\commutator{R_{H}}{O_{a}}=-\frac{1}{2}{h_{a}}^{b}O_{b}\,,\qquad\commutator{R_{C}}{O_{\dot{a}}}=-\frac{1}{2}{\bar{h}_{a}}^{\dot{b}}O_{\dot{b}}\,.$
(29)

We are looking for a one-parameter family of supercharges squaring on a linear
combination of $P_{\varphi}$ and an R-symmetry transformation which leaves the
Wilson loop invariant, that is $R_{H}$. We choose our supercharge to be a
linear combination of the nilpotent supercharges

$Q^{\textup{W}}_{1}=Q_{2}^{(\ell+)}+iQ_{1}^{(r-)}\,,\qquad
Q^{\textup{W}}_{2}=Q_{1}^{(\ell-)}+iQ_{2}^{(r-)}\,.$ (30)

Then, the “cohomological” supercharge is
$Q_{\beta}^{\textup{W}}=Q^{\textup{W}}_{1}+\beta\,Q^{\textup{W}}_{2}$ and
satisfies $Q_{\beta}^{\textup{W}}\propto P_{\varphi}-R_{H}$. We are looking
for a family of topological operators annihilated by $Q_{\beta}^{\textup{W}}$.
As in the case without defect, we interpret $P_{\varphi}-R_{H}$ as the central
charge $Z$. Consequently, operators in cohomology must be annihilated by
$P_{\varphi}-R_{H}$. Then, they must be Coulomb branch operators and placed on
$S^{1}_{\tau}$.

It is not hard to construct the corresponding twisted translation
$\hat{P}_{\tau}$. Because of the following anticommutators

$\displaystyle\anticommutator*{Q^{\textup{W}}_{1}}{Q^{(\ell-)}_{1}-iQ^{(r+)}_{2}}=\anticommutator*{Q^{\textup{W}}_{2}}{{Q^{(\ell+)}_{2}-iQ^{(r-)}_{1}}}=\frac{i}{r}\left(P_{\tau}+R_{C}\right)\,,$
(31)

we choose it to be $\hat{P}_{\tau}=P_{\tau}+R_{C}$. We have thus recovered the
minimal structure the topological sector. Unlike the SCFT case, we cannot rely
on representation theory to classify the cohomology of
$Q_{\beta}^{\textup{W}}$. Therefore, we need to check explicitly whether an
operator is in the cohomology or not. Limiting to operators built out of
fundamental fields, we find that 666Presumably, the cohomology contain also
interesting BPS monopole operators like in Dedushenko:2017avn ;
Dedushenko:2018icp . We do not address that problem in this paper.

$\Phi(0)=\frac{1}{2}\Phi_{\dot{1}\dot{1}}(0)-\Phi_{\dot{1}\dot{2}}(0)+\frac{1}{2}\Phi_{\dot{2}\dot{2}}(0)\,.$
(32)

is annihilated by $Q^{\textup{W}}_{i}$ at $\theta=0$, $\tau=0$. We use the
twisted translation $\hat{P}_{\tau}=P_{\tau}+R_{C}$ to move $\Phi$ along the
$\tau$-circle. The twisted translated operator is annihilated by
$\mathcal{Q}_{\beta}^{\textup{W}}$ and it reads

$\Phi(\tau)=v^{\dot{a}}(\tau)v^{\dot{b}}(\tau)\Phi_{\dot{a}\dot{b}}(\tau)\,,\qquad
v^{\dot{a}}(\tau)=\frac{1}{\sqrt{2}}\left(e^{\frac{i}{2}\tau},-e^{-\frac{i}{2}\tau}\right)\,.$
(33)

The gauge invariant polynomials of $\Phi(\tau)$ are the simplest topological
operators compatible with the BPS Wilson loop.

###  3.2 Decorating the vortex loop

In this section, we study configurations involving vortex loops defined in Eq.
(6) and topological operators. Based on mirror symmetry Intriligator:1996ex ,
we expect the existence of a supersymmetric setup in which the vortex loop is
on $S^{1}_{\varphi}$ and the Higgs branch topological sector is on
$S^{1}_{\tau}$. We briefly check that the intuition is indeed correct.

The preserved $\mathfrak{su}(1|1)\oplus\mathfrak{su}(1|1)$ algebra is

$\displaystyle\anticommutator*{Q_{2}^{(\ell+)}}{Q_{1}^{(\ell-)}}$
$\displaystyle=\frac{i}{r}\left(J_{12}^{(\ell)}+\frac{R_{C}-R_{H}}{2}\right)\,,$
(34) $\displaystyle\anticommutator*{Q_{1}^{(r+)}}{Q_{2}^{(r-)}}$
$\displaystyle=\frac{i}{r}\left(J_{12}^{(r)}-\frac{R_{H}+R_{C}}{2}\right)\,.$
(35)

We follow the same logic as before, but we exchange the role of $R_{C}$ and
$R_{H}$. In this way, we find the nilpotent supercharges are

$\mathcal{Q}^{\textup{V}}_{1}=Q_{2}^{(\ell+)}+iQ_{1}^{(r+)}\,,\qquad\mathcal{Q}^{\textup{V}}_{2}=Q_{1}^{(\ell-)}+iQ_{2}^{(r-)}\,.$
(36)

Their anticommutator plays again the role of the central charge

$\anticommutator*{\mathcal{Q}^{\textup{V}}_{1}}{\mathcal{Q}^{\textup{V}}_{2}}=\frac{i}{r}\left(P_{\varphi}+R_{C}\right)\,.$
(37)

Then, a putative topological sector can only live on $S^{1}_{\tau}$ and
contains Higgs branch operators. Thus, the corresponding twisted translation
is easily derived from

$\displaystyle\anticommutator*{\mathcal{Q}^{\textup{V}}_{1}}{Q_{1}^{(\ell-)}-iQ_{2}^{(r-)}}=\anticommutator*{\mathcal{Q}^{\textup{V}}_{2}}{Q_{2}^{(\ell+)}-iQ_{-}^{(r+)}}$
$\displaystyle=\frac{4i}{r}\left(P_{\tau}-R_{H}\right)\,,$ (38)

and it reads $\hat{P}_{\tau}=P_{\tau}-R_{H}$. The natural candidates for a
non-trivial cohomology are the scalar fields in the hypermultiplet. By direct
inspection we find that both $q^{1}$ and $\tilde{q}^{1}$ are annihilated by
any linear combination of $\mathcal{Q}^{\textup{V}}_{1}$ and
$\mathcal{Q}^{\textup{V}}_{2}$. Then, we build the corresponding twisted
fields by acting with the twisted translation $\hat{P}_{\tau}=P_{\tau}-R_{H}$

$Q(\tau)\equiv u^{a}(\tau)q_{a}(\tau),\quad\tilde{Q}(\tau)\equiv
u^{a}(\tau)\tilde{q}_{a}(\tau)\,,\quad
u^{a}=\left(-\sin\frac{\tau}{2},-\cos\frac{\tau}{2}\right)\,.$ (39)

Therefore, the Higgs topological is made by the gauge invariant polynomials of
the twisted fields $Q$, $\tilde{Q}$.

###  3.3 The superconformal case

We now concentrate on superconformal lines in SCFTs with at least
$\mathcal{N}=4$ supersymmetry. While the construction in UV gauge theories
relies on the existence of Lagrangians, our goal is to extend it to specific
classes of line operators regardless of the definition of the SCFTs. However,
for the sake of concreteness and keeping in mind the applications of the
following sections, we begin with a detailed discussion of the fermionic
Wilson line in ABJM Drukker:2009hy . Building on this example, we will argue a
more general conclusion for $\mathcal{N}=4$ SCFTs.

To begin with, we briefly describe the
$\mathfrak{su}(1,1|3)\oplus\mathfrak{u}(1)_{b}$ superalgebra preserved by the
fermionic Wilson loops Bianchi:2017ozk . The Wilson loop preserves the
following supercharges

$\bar{Q}_{1,34}\,,\;\bar{Q}_{1,32}\,,\;\bar{Q}_{1,24}\,,\;\bar{Q}_{2,14}\,\;\bar{Q}_{2,12}\,,\;\bar{Q}_{2,13}\,,\;\bar{S}_{1,34}\,\;\bar{S}_{1,32}\,,\;\bar{S}_{1,24}\,\;\bar{S}_{2,14}\,,\;\bar{S}_{2,12}\,,\;\bar{S}_{2,13}\,,$
(40)

where $\bar{Q}_{\alpha,IJ}$ and $\bar{S}_{\alpha,IJ}$ are Poincaré and
superconformal superchages respectively, and $I,J=1,\dots,4$ are the
antisymmetric $\mathfrak{su}(4)$ indices. They close on the 1d conformal group
spanned by $P_{3}$, $K_{3}$, and $D$, and on the $\mathfrak{su}(3)$ R-symmetry
generators ${L_{a}}^{b}$, whose precise description is in the appendix. The
additional $\mathfrak{u}(1)_{b}$ factor commutes with all these generators,
and we can safely ignore it for the rest of the paper.

We want to make contact with the algebra of the Wilson loop in UV gauge
theories described starting from Eq. (30). As a first step, we break the full
$\mathcal{N}=6$ superconformal algebra to the $\mathcal{N}=4$ one. We decide
to identify the $\mathfrak{su}(2)_{H}$ factor of the residual $\mathcal{N}=4$
R-symmetry algebra with an $\mathfrak{su}(2)$ factor of the algebra preserved
by the fermionic Wilson loop. We denote these generators as ${R_{a}}^{b}$.
Then, we construct the Poincaré subalgebra
$\mathfrak{su}(2|1)_{\ell}\oplus\mathfrak{su}(2|1)_{r}$ in the usual way. The
details are rather boring and technical, and the interested reader can find
all the details in the App. B. The upshot is that we can embed the
supercharges preserved by the Wilson loop in the relevant Poincaré
superalgebra. That is, if we take the supercharges $Q^{\textup{W}}_{1}$ and
$Q^{\textup{W}}_{2}$ defined in (30), and we use the explicit embedding
derived in App. B, we find the two cohomological supercharges

$\displaystyle Q_{1}$
$\displaystyle=\frac{1}{2}\left[i\bar{Q}_{1,32}-\bar{Q}_{1,34}+\bar{Q}_{2,12}+i\bar{Q}_{2,14}+\frac{1}{2r}\left(i\bar{S}_{1,32}-\bar{S}_{1,34}-\bar{S}_{2,12}-i\bar{S}_{2,14}\right)\right]$
(41) $\displaystyle Q_{2}$
$\displaystyle=\frac{1}{2}\left[\bar{Q}_{1,32}-i\bar{Q}_{1,34}+i\bar{Q}_{2,12}+\bar{Q}_{2,14}+\frac{1}{2r}\left(i\bar{S}_{1,34}+i\bar{S}_{2,12}+\bar{S}_{2,14}-\bar{S}_{1,32}\right)\right]$
(42)

They annihilate the Wilson loop, and their anticommutator closes on
$P_{\varphi}-R_{H}$, where we are implicitly compactifying the theory on
$S^{3}$. Moreover, we can define the twisted translations from the
anticommutators $\anticommutator*{Q_{1}}{\tilde{Q}_{1}}$ and
$\anticommutator*{Q_{2}}{\tilde{Q}_{2}}$, where

$\displaystyle\tilde{Q}_{1}$
$\displaystyle=\frac{1}{2}\left[\bar{Q}_{1,32}-i\bar{Q}_{1,34}-i\bar{Q}_{2,12}-\bar{Q}_{2,12}+\frac{1}{2r}\left(i\bar{S}_{1,34}-\bar{S}_{1,32}-i\bar{S}_{2,14}-\bar{S}_{2,14}\right)\right]$
(43) $\displaystyle\tilde{Q}_{2}$
$\displaystyle=\frac{1}{2}\left[-i\bar{Q}_{1,32}+\bar{Q}_{1,34}+\bar{Q}_{2,12}+i\bar{Q}_{2,12}+\frac{1}{2r}\left(\bar{S}_{1,34}-i\bar{S}_{1,32}-\bar{S}_{2,14}-i\bar{S}_{2,14}\right)\right]$
(44)

Then, the twisted translation is again $P_{\tau}-R_{C}$, being $R_{C}$ a
combination of the original $\mathfrak{su}(4)$ R-symmetry generators.

The final step is to study the cohomology. In the conformal case, this problem
has been already addressed on the line in full generality in Chester:2014mea
for $\mathcal{N}=4$ SCFTs. Then, up to a conformal transformation that does
not affect the cohomology, we do not need to repeat their analysis. Here we
limit ourselves to writing down the operator, which will be studied in detail
in the following sections. The natural candidate to look at is gauge invariant
polynomials of the scalar fields $C_{I}$, $\bar{C}^{I}$. Indeed, studying
their supersymmetric variation (A) w.r.t. to any linear combination of
$Q^{\textup{W}}_{1}$ and $Q^{\textup{W}}_{2}$, we find that the combinations
$C_{1}+C_{3}$ and $\bar{C}^{1}-\bar{C}^{3}$ vanishes for $\tau=0$. Then, with
the familiar procedure, we identify the building blocks for gauge invariant
topological operators

$\mathcal{C}(\tau)=U^{I}(\tau)C_{I}(\tau)\qquad\bar{\mathcal{C}}(\tau)=\bar{V}_{I}(\tau)\bar{C}^{I}(\tau)\,,$
(45)

with

$U^{I}(\tau)=\frac{1}{\sqrt{2}}\left(e^{-\frac{i}{2}\tau},\,0\,,\,e^{\frac{i}{2}\tau},\,0\,\right)^{I}\,,\qquad\bar{V}_{I}(\tau)=\frac{1}{\sqrt{2}}\left(e^{\frac{i}{2}\tau},\,0\,,\,-e^{-\frac{i}{2}\tau},\,0\,\right)_{I}\,.$
(46)

For instance, the simplest operator we can study is

$\mathcal{O}(\tau)=\Tr(\bar{\mathcal{C}}(\tau)\mathcal{C}(\tau))\,,$ (47)

which is also part of the stress tensor multiplet 777The fact that the stress
tensor multiplet admits a protected topological operator is a special feature
of $\mathcal{N}=6$ SCFTs. In a generic $\mathcal{N}=4$ SCFT the dimension one
topological operator is part of a conserved current multiplet. .

As a check of the correctness of our results, we can see that it agrees with
3d IR dualities. ABJM theory with gauge group $U(N)_{1}\times U(N)_{-1}$ is
dual to a UV Yang-Mills theory coupled to one fundamental and one adjoint
hyper Kapustin:2010xq . Then, we expect the dual operator of
$\mathcal{O}(\tau)$ defined in (47) to be the Coulomb branch operator built
out of $\Phi_{\dot{a}\dot{b}}$ introduced in (33). This intuition agrees with
the results of Hayashi:2022ldo .

Now we are ready to generalize our construction to generic $\mathcal{N}=4$
SCFTs. Our argument relies on the superconformal algebra and not on its
specific representation, and it extends to all $\frac{1}{2}$-BPS line
operators in $\mathcal{N}=6$ SCFTs. Moreover, we did not only use the full
$\mathcal{N}=6$ but only its $\mathcal{N}=4$ subalgebra. Similarly, for the
Wilson line, we did not exploit the full $\mathfrak{su}(3)$ algebra, but only
an $\mathfrak{su}(2)$ factor. We conclude that our construction can be readily
extended to any BPS line operators in $\mathcal{N}=4$ SCFT preserving an
$\mathfrak{su}(2)$ factor. Looking at the classification of Agmon:2020pde , we
conclude that there is a topological sector compatible with any
$\frac{1}{2}$-BPS line operators in $\mathcal{N}=4$ SCFTs. This is the first
main result of the paper. In the following, we discuss how to calculate defect
CFT data from our setup.

##  4 Exact formula for stress tensor correlators in ABJM

In this section, we provide a formula to extract defect CFT data for BPS
lines. Our method is based on a powerful Ward identity which relates
topological operators belonging to conserved current multiplets to mass or FI
deformations of the $S^{3}$ partition function. Since the latter are amenable
of supersymmetric localization, many exact results are accessible. Elaborating
on this idea, we will argue that the vev of the defect in the properly
deformed background computes defect correlation functions of protected
operators. Finally, we will apply our formula to the maximally supersymmetric
Wilson line in ABJM.

###  4.1 The cohomological Ward identity

We review the relation between the topological sectors and supersymmetric
deformations of the $S^{3}$ partition functions. Let us begin with mass
deformations. They can be built every time a theory has a flavor symmetry by
using the correspinding current multiplet
$\Sigma^{A}=(J^{A}_{ab},\,\chi^{A}_{a\dot{a}},\,j^{A}_{\mu},\,K^{A}_{\dot{a}\dot{b}})$.888Here
$A$ is an index which runs from 1 to the rank of the flavor symmetry Lie
algebra. The dimension-one scalars $J^{A}_{ab}$ are in the
$(\mathbf{3},\mathbf{1})$ of the R-symmetry group, $\chi^{A}_{a\dot{a}}$ are
the fermion partners of dimension $3/2$ in the $(\mathbf{2},\mathbf{2})$ of
the R-symmetry group, $j_{\mu}^{A}$ are the flavor conserved currents, and
$K^{A}_{\dot{a}\dot{b}}$ are dimension-two scalars in the
$(\mathbf{1},\mathbf{3})$ of the R-symmetry group. A current multiplet can
always be coupled to a background vector field. If we take the supersymmetric
background in which $\Phi_{\dot{a}\dot{b}}=m\bar{h}_{\dot{a}\dot{b}}$,
$D_{ab}=h_{ab}m/r$, and all the other fields vanishing, the corresponding
action is called real mass deformation. This amounts to modifying the action
by the following terms

$S_{\textup{mass}}=m_{A}\int_{S^{3}}d^{3}x\sqrt{g}\left(-ih^{ab}J^{A}_{ab}+\bar{h}^{\dot{a}\dot{b}}K^{A}_{\dot{a}\dot{b}}\right)+O\left(m^{2}\right)\,.$
(48)

The order $m^{2}$ terms are needed to preserve supersymmetry, but its explicit
expression will not be relevant to us. We also observe that $J^{A}_{ab}$ has
the quantum numbers of a Higgs branch operator, and therefore it can be made
topological. The corresponding twisted operator is
$J^{A}(\tau)=J^{A}_{ab}u^{a}u^{b}$. Thus, we suspect the existence of a
relation between mass terms and dimension one topological operators. Building
on this intuition, it turns out that the mass deformation
$S_{\textup{mass}}[m]$ is almost a total supersymmetric variation w.r.t. to
the cohomological supercharge preserving the topological sector. The only non-
exact part is a boundary term, namely the “topologized” version of the
superconformal primary of the current multiplet integrated over the great
circle supporting the topological sector Guerrini:2021zuk ; Bomans:2021ldw .
In formulae we get

$\partialderivative{m}\left(S_{\textup{mass}}[m]-4\pi
r^{2}m\oint_{S^{1}_{\varphi}}J(\varphi)\right)=\anticommutator{Q_{\beta}^{H}}{\dots}\,,$
(49)

It is not hard to use the Ward identity to prove that the mass deformed
partition function is the generating functional of integrated topological
operators, that is

$\Big{\langle}\left(\int
d\varphi_{1}\,J^{A_{1}}(\varphi_{1})\right)^{k_{1}}\dots\left(\int
d\varphi_{n}\,J^{A_{n}}(\varphi_{n})\right)^{k_{n}}\Big{\rangle}=\left(-\frac{1}{4\pi
r^{2}}\right)^{\\!n}\,\frac{1}{Z}\frac{\partial^{k}Z[m_{A_{1}},\dots,m_{A_{n}}]}{\partial
m^{k_{1}}_{A_{1}}\dots\partial
m^{k_{n}}_{A_{n}}}\bigg{|}_{m_{A_{1}}\\!,m_{A_{2}}\\!,\cdots=0}$ (50)

where $k=k_{1}+\dots+k_{n}$ with $k_{1},\dots,k_{n}\in\mathbb{N}$. This
version of the Ward identity is very useful for calculating correlation
functions and extracting CFT data. For instance, the r.h.s. can be evaluated
exactly with localization, and related to the non-trivial CFT data stored in
the l.h.s.. A relevant application of this method is the evaluation of the
coefficients of the string theory and M-theory action beyond the SUGRA limit
Agmon:2017xes ; Binder:2019mpb ; Binder:2020ckj .

The localization formula for topological correlations function in UV gauge
theories of Dedushenko:2016jxl  gives a highly non-trivial consistency check.
According to Dedushenko:2016jxl , topological correlators are captured by a
quadratic quantum mechanics. The resulting 1d action
$S_{\sigma}[Q\,,\tilde{Q}]$ is coupled to the standard matrix model arising
from the usual localization scheme for the vector multiplet. The quantum
mechanics’ fundamental fields $Q,\,\tilde{Q}$ of the effective theory are
identified with the twisted translated operators built out of the
hypermultiplet (seef Eq. (39)). Now we turn on the mass deformation of Eq.
(48). The localized expression gets modified by adding a 1d mass term for the
1d action

$S_{\sigma}[Q\,,\tilde{Q}]\to S_{\sigma}[Q\,,\tilde{Q}]-4\pi
r^{2}m^{A}\int_{-\pi}^{\pi}d\varphi\,\tilde{Q}(\varphi)T^{A}Q(\varphi)\,.$
(51)

where $T^{A}$ are the generators of the flavor symmetry in the proper
representation. The operators
$J^{A}(\varphi)\equiv-\tilde{Q}(\varphi)T^{A}Q(\varphi)$ are the twisted
operators $J^{A}(\varphi)$ is nothing but the lowest component of the current
multiplet coupled to the background vector field. Then, derivatives w.r.t. to
$m$ reproduces (50).

The Ward identity does not require any localization but only the existence of
a flavor current. Therefore, the range of validity of (50) is extended to any
$\mathcal{N}=4$ SCFTs, regardless of their specific realization. These include
even theories where the localization argument has not been developed yet, like
Chern-Simons matter theories. This observation is crucial to apply the formula
to ABJM theory.

The novel step is to apply this identity in the presence of a defect
$\mathcal{D}$ preserving $Q$. Until the insertion is $Q$-closed, the
cohomological argument goes through. Thus we can still use Eq. (49) to relate
correlation functions of topological operators with the defect $\mathcal{D}$
to mass derivatives of the vev of the defect with a mass-deforned backgrounds.
It is straightforward, to write down the explicit formula

$\Big{\langle}\int d\varphi_{1}\,J^{A_{1}}(\varphi_{1})\dots\int
d\varphi_{n}\,J^{A_{n}}(\varphi_{n})\Big{\rangle}_{\mathcal{D}}=\left(-\frac{1}{4\pi
r^{2}}\right)^{\\!n}\,\frac{1}{\mathcal{D}}\frac{\partial^{n}\langle\mathcal{D}\rangle[m_{1},\dots,m_{n}]}{\partial
m^{A_{1}}\dots\partial
m^{A_{n}}}\bigg{|}_{m^{A_{1}}\\!,m^{A_{2}}\\!,\cdots=0}$ (52)

where $\langle\mathcal{D}\rangle$ is the vev of the defect and
$\langle\rangle_{\mathcal{D}}$ indicates the normalized correlation function
in the presence of the defect. For instance, we can use (57) to compute
correlation functions of Higgs branch operators with the vortex loop
introduced in Eq. (6). However, as in the case without the defect, our formula
is independent by the explicit realization of the theory and the defect.

Analogously, we can extend the formula for Coulomb branch operators. Here the
relevant deformation is the Fayet-Iliopoulos (FI) term. To write it down, we
think of $\Phi_{\dot{a}\dot{b}}$ as the bottom component of the current
multiplet of the topological current
$j_{\mu}\propto\epsilon_{\mu\nu\rho}F^{\nu\rho}$. The multiplet also includes
the fermions $\lambda_{a\dot{a}}$ and the auxiliary fields $D_{ab}$. For each
$U(1)$ factor of the gauge group, there is a multiplet of this type. One can
couple this multiplet to an abelian background twisted vector multiplet
$\tilde{\mathcal{V}}_{\rm back}$. The relevant coupling is simply an
$\mathcal{N}=4$ mixed Chern-Simons term 999For details see the App. B of
Guerrini:2021zuk .. In the rigid limit, we get the FI term on $S^{3}$

$S_{FI}=i\zeta\int_{S^{3}}d^{3}x\sqrt{g}\left({h_{a}}^{b}{D_{b}}^{a}-\frac{1}{r}\bar{h}\indices{{}^{\dot{a}}_{\dot{b}}}\Phi\indices{{}^{\dot{b}}_{\dot{a}}}\right)\,.$
(53)

As discussed in Guerrini:2021zuk , we can write a similar Ward identity for
such a deformation following the same logic of real masses

$S_{FI}=-4\pi
ir^{2}\int_{S^{1}_{\varphi}}d\varphi\,\Phi_{\dot{a}\dot{b}}v^{\dot{a}}v^{\dot{b}}+\delta_{\xi}\left[\int_{S^{3}}d^{3}x\sqrt{g}\,\tilde{\eta}^{a\dot{a}}\lambda_{a\dot{a}}\right]\,.$
(54)

Then, derivatives w.r.t. to the FI parameters gives 1d integrated Coulomb
branch operators. Even in this case, the formula is compatible with
localization for Coulomb branch operators. Again, any defect preserving the
cohomological supercharge can be inserted without additional complications. In
this way, we get another identity for defect correlation functions

$\langle\int_{S^{1}_{\varphi_{1}}}d\varphi_{1}\,\Phi(\varphi_{1})\cdots\int_{S^{1}_{\varphi_{n}}}d\varphi_{n}\,\Phi(\varphi_{n})\rangle_{\mathcal{D}}=\left(\frac{i}{4\pi
r^{2}}\right)^{n}\frac{\partial^{n}}{\partial\zeta^{n}}\langle\mathcal{D}\rangle[\zeta]\,,$
(55)

In UV gauge theories, the formula allows us to compute the corresponding
defect topological correlators in the presence of the Wilson loop of Eq. (5).
However, we stress again that these formulas are independent of the specific
description of the theory and the defect, and hold again for any
$\frac{1}{2}$-BPS conformal line operators in any $\mathcal{N}=4$ SCFTs.

In the rest of the section, we apply our formula to the other explicit example
we discussed, namely $\frac{1}{2}$-BPS Wilson loops ABJM.

###  4.2 Defect correlation functions in ABJM

In Sec. 3.3 we showed the existence of a supersymmetric configuration
involving the fermionic Wilson line and the topological sector in ABJM. Even
if a localization scheme in Chern-Simons matter theory is missing, we can
still apply the Ward identity. We only need to identify the correct
deformation coupled to the current multiplet with superconformal primary the
operator

${O_{I}}^{J}=\Tr(C_{I}\bar{C}^{J})-\frac{1}{4}\delta_{I}^{J}\Tr(C_{k}\bar{C}^{K})\,.$
(56)

The case of ABJM is particularly interesting, as the bilinear operator is part
of the stress tensor multiplet. Then, Ward identities can relate correlation
functions of twisted operators $\mathcal{O}(\tau)$ defined in Eq. (47) to
stress tensor correlators. For instance, as we will discuss in detail later,
the one-point function of $\mathcal{O}(\tau)$ will be related to the
bremsstrahlung function, which is a universal defect CFT data.

In our setup, we can embed $C_{1}$, $C_{3}$ into a twisted-hypermultiplet
$\tilde{\mathcal{H}}$, whose content is the same as a regular hyper, but with
the role of $\mathfrak{su}(2)_{H}$ and $\mathfrak{su}(2)_{C}$ indices
exchanged. That is, we embed $C_{1}$, $C_{3}$ in a scalar field $q_{\dot{a}}$
and $\bar{C}^{1}$, $\bar{C}^{3}$ can be organized as a $\tilde{q}^{\dot{a}}$.
The bilinear operator transforming in the adjoint of $\mathfrak{su}(2)_{C}$ is
the primary of a current for a flavor symmetry from the $\mathcal{N}=4$
perspective. Then, it can be coupled to a rigid background vector, and the
resulting mass term looks like (48), with dotted and undotted indices
exchanged. This allows us to use the Ward identity for Coulomb branch
operators and write down a formula for integrated correlators

$\Big{\langle}\int d\tau_{1}\,\mathcal{O}(\tau_{1})\dots\int
d\tau_{n}\,\mathcal{O}(\tau_{n})\Big{\rangle}_{W}=\left(\frac{i}{4\pi
r^{2}}\right)^{\\!n}\,\frac{1}{W}\frac{\partial^{n}\langle
W\rangle[m]}{\partial m^{n}}\bigg{|}_{m=0}$ (57)

where $\langle W\rangle[m]$ denotes the vev of the Wilson loop in the mass
deformed ABJM theory on $S^{3}$.

The crucial point is that the l.h.s. can be computed exactly with
localization. We recall the result from the $\mathcal{N}=2$ perspective
Benna:2008zy . We can decompose $C_{I}$ in terms of the $\mathcal{N}=2$
chirals $W_{i}$, $Z_{i}$ as follows:
$C_{I}=(W_{1},W_{2},Z^{\dagger}_{1},Z^{\dagger}_{2})$. Then, if we give mass
$m$ to $W_{1}$ and $Z^{\dagger}_{1}$ we get the mass deformation for $C_{1}$
and $C_{3}$. The localized partition function reads Kapustin:2009kz ;
Jafferis:2010un ; Hama:2010av

$\displaystyle Z_{\textup{ABJM}}=\frac{1}{N_{1}!N_{2}!}\int
d\lambda_{i}d\mu_{j}e^{ik\pi\left(\sum_{i}^{N_{1}}\lambda_{i}^{2}-\sum_{j}^{N_{2}}\mu_{j}^{2}\right)}\frac{\prod_{i\neq
j}^{N_{1}}2\sinh\pi(\lambda_{i}-\lambda_{j})\prod_{i\neq
j}^{N_{2}}2\sinh\pi(\mu_{i}-\mu_{j})}{\prod_{i,\,j}\left[2\cosh\pi(\lambda_{i}-\mu_{j})\right]\left[2\cosh(\pi(\lambda_{i}-\mu_{j}+mr))\right]}\,.$
(58)

The vev of supersymmetric operators is computed by insertions of operators in
this matrix model. For the $\frac{1}{2}$–BPS Wilson loop the insertion is
Drukker:2009hy

$\langle
W\rangle=\langle\sum_{i=1}^{N_{1}}e^{2\pi\lambda_{i}}+\sum_{j=1}^{N_{2}}e^{2\pi\mu_{j}}\rangle$
(59)

#### The weak coupling computation

We find the matrix model insertion by computing derivatives in the matrix
model. For the one-point function, we obtain

$\partialderivative{m}W[m]=\langle-\sum_{i,j}\pi
r\tanh\pi(\lambda_{i}-\mu_{j})\rangle_{W}$ (60)

where $\langle\rangle_{W}$ indicates the matrix model average with the WL
insertion.

We exploit our relation at weak coupling, namely in the limit $k\gg 1$. To
expand the matrix model, we rescale the variable by a factor $\sqrt{k}$

$x_{i}=\pi\sqrt{k}\lambda_{i}\qquad y_{j}=\pi\sqrt{k}\lambda_{j}$ (61)

We also reconstruct the $U(N_{1})\times U(N_{2})$ Haar measure.101010We recall
the Haar measure for a single $U(N)$ factor  $\int dUf(U)=\int
du_{i}\prod_{i<j}(u_{i}-u_{j})^{2}f(U)$ (62) where $u_{i}$ are the eigenvalues
of $U$. Then, the undeformed matrix model reads as

$\displaystyle Z=\frac{1}{N_{1}!N_{2}!}\int$ $\displaystyle
dXdY\,e^{i\left(\sum_{i=1}^{N_{1}}x_{i}^{2}-\sum_{j=1}^{N_{2}}y_{j}^{2}\right)}\prod_{i\neq
j}\frac{\left[\sinh g(x_{i}-x_{j})\right]\left[\sinh
g(y_{i}-y_{j})\right]}{g(x_{i}-x_{j})g(y_{i}-y_{j})}\times$
$\displaystyle\times\prod_{i,j}\left[2\cosh^{-1}g(x_{i}-y_{j})\right]\left[2\cosh^{-1}\left(g(x_{i}-y_{j})\right)\right]$
(63)

where $dX$ and $dY$ denotes the opportune Haar measure, $g=k^{-\frac{1}{2}}$
and we neglect all the overall constants. Also, the Wilson loop insertion gets
modified into

$\langle
W\rangle=\langle\sum_{i=1}^{N_{1}}e^{2gx_{i}}+\sum_{j=1}^{N_{2}}e^{2gy_{j}}\rangle$
(64)

As explained in Gorini:2020new ; Chester:2021gdw , we can expand all the
functions for $g\ll 1$. The result can be expressed as a sum of Gaussian
averages of matrix multitrace insertions

$t_{\nu}(X)=\left(\Tr X\right)^{\nu_{1}}\left(\Tr
X^{2}\right)^{\nu_{2}}\dots\left(\Tr X^{k}\right)^{\nu_{k}}\dots$ (65)

where $\nu$ is a collective notation for the set of integers
$\nu_{1},\dots,\nu_{k}$. We have similar insertions also for $Y$. The explicit
integral reads

$\langle t_{\nu}\rangle=\frac{\int
dXdY\,e^{-g(X^{2}-Y^{2})}t_{\nu}(X)t_{\lambda}(Y)}{\int
dXdY\,e^{-g(X^{2}-Y^{2})}}$ (66)

We compute the integrals using the results of Itzykson:1990zb , recently
reviewed in Chester:2021gdw . We push the expansion up to $k^{-3}$. We get

$\langle\mathcal{O}(\tau)\rangle=\frac{N_{1}N_{2}}{4kr\left(N_{1}+N_{2}\right)}-\frac{\pi^{2}N_{1}N_{2}\left(N_{1}N_{2}-3\right)}{24r\,k^{3}\left(N_{1}+N_{2}\right)}+O(k^{-4})$
(67)

With the same method, we calculate the two point function. This is captured by
the matrix model average

$\partialderivative[2]{m}W[m]=(\pi
r)^{2}\langle\left(\sum_{i,j}\tanh(\pi(\lambda_{i}-\mu_{j}))\right)^{2}-\sum_{i,j}\frac{1}{\cosh^{2}(\pi\left(\lambda_{i}-\mu_{j})\right)}\rangle_{W}$
(68)

After similar computations we find

$\langle\mathcal{O}(\tau_{1})\mathcal{O}(\tau_{2})\rangle_{W}=\frac{N_{1}N_{2}}{64\pi^{2}r^{2}}-\frac{N_{1}N_{2}\left(N_{1}^{2}+N_{2}^{2}-14\right)}{384k^{2}r^{2}}+O(k^{-4})$
(69)

One can check that the result differs from the two-point function without the
Wilson loop already in the term proportional to $k^{-2}$ Gorini:2020new . The
reason is that the topological operators have a non-trivial bulk-to-boundary
OPE with the defect.

In the following section, we perform some perturbative checks and elaborate on
the relation with the stress tensor.

##  5 Discussion and perturbative checks

As a further check of our result and to give also a somewhat different insight
into defect correlation functions, we compute the first perturbative orders
using Feynman diagrams. We will expand the action and the Wilson loop for
$k\gg 1$ and evaluate all possible contractions among the local operators. We
will limit to checking the leading order for the one-point function. For the
two-point function, we verify the vanishing of the first quantum correction.
We also discuss a general formula for the bremsstrahlung function in
$\mathcal{N}=6$ SCFTs.

###  5.1 One-point function and Bremmstrahlung

We compute $\langle\mathcal{O}(\tau)\rangle_{W}$ in perturbation theory. Since
non-zero diagrams require interaction with the Wilson loop, the first
contribution is of order $k^{-1}$. At this order in $k$, the Feynman diagrams
are those in figure 2.

Figure 2: Feynman diagrams for the leading contribution of the defect one-
point function.

It is easy to evaluate the contribution of (b). After the Wick contractions,
the diagram is proportional to $U^{I}\bar{V}_{I}\equiv 0$. Then, we conclude
that the contribution of these diagrams is vanishing, i.e. (3b)=0 111111One
can also check that the integral is vanishing for parity..

We turn to (3a), and we find

$\displaystyle\langle W\mathcal{O}(0)\rangle$
$\displaystyle=\frac{2}{N_{1}+N_{2}}\Tr(U^{A}C_{A}\bar{V}_{B}\bar{C}^{B})(-i)\left(-\frac{2\pi
i}{k}\right){M_{I}}^{J}\int_{-\infty}^{\infty}ds\,\Tr(\bar{C}^{I}C_{J}(s))=$
$\displaystyle=-\frac{1}{16\pi^{2}}\frac{4\pi}{k}\frac{N_{1}N_{2}}{N_{1}+N_{2}}\frac{U^{I}{M_{I}}^{J}\bar{V}_{J}}{r}\int_{-\infty}^{\infty}ds\,\frac{1}{1+s^{2}}=\frac{1}{4kr}\frac{N_{1}N_{2}}{N_{1}+N_{2}}\,.$
(70)

where we used that
$U^{I}(\tau){M_{I}}^{J}\bar{V}_{J}(\tau)\equiv-\frac{1}{2}$. The result is in
perfect agreement with the matrix model prediction of Eq. (67).

We now give the precise physical interpretation for this one-point function.
Since $\mathcal{O}(0)$ is the twisted operator of the superconformal primary
${O_{I}}^{J}$, which is part of the stress tensor multiplet, there must be a
supersymmetric Ward identity relating $\langle\mathcal{O}(0)\rangle_{W}$ and
$\langle T_{\mu\nu}\rangle_{W}$. As explained in Sec. 2.3, all the physical
information of $\langle T_{\mu\nu}\rangle_{W}$ is contained in the constant
$h_{T}$ of Eq. (15). Then, using conformal symmetry, we conclude that

$\langle\mathcal{O}(0)\rangle_{W}\propto\frac{h_{t}}{r}\,.$ (71)

The quantity $h_{T}$ is in turn related to the bremsstrahlung function $B$
defined by Eq. (20) by the Ward identity $B=2h_{T}$ Lewkowycz:2013laa . The
explicit formula was first conjectured by separating the radiation component
from the self-energy part of the field. Then, the relation was shown to hold
in full generality by using dCFT considerations Bianchi:2018zpb ;
Bianchi:2019sxz . In the case $N_{1}=N_{2}$, the bremsstrahlung is known, and
we can compare it with our result to fix the missing constant of
proportionality. In the end, we find 121212It would be interesting to derive
the result more formally from a Ward identity like in Fiol:2015spa , perhaps
by using the superspace introduced in Liendo:2015cgi .  Griguolo:2012iq ;
Lewkowycz:2013laa ; Bianchi:2014laa ; Bianchi:2017svd ; Bianchi:2017ozk ;
Bianchi:2018bke ; Bianchi:2018scb

$\langle\mathcal{O}(0)\rangle_{W}=\frac{2h_{T}}{r}=\frac{B}{r}\,.$ (72)

However, this conclusion depends only on the structure of the $\mathcal{N}=6$
stress tensor multiplet and not on the specific theory and therefore holds in
any $\mathcal{N}=6$ SCFT. Then, we can read the bremsstrahlung function for
the fermionic Wilson line in the $U(N_{1})_{k}\times U(N_{2})_{-k}$ model

$B(k,N_{1},N_{2})=\frac{N_{1}N_{2}}{4k(N_{1}+N_{2})}-\frac{\left(\pi^{2}N_{1}N_{2}\left(N_{1}N_{2}-3\right)\right)}{24k^{3}\left(N_{1}+N_{2}\right)}+O(k^{-4})$
(73)

This formula was already conjectured by extending a prescription, valid in the
limit $N_{1}=N_{2}$, based on a worldvolume deformation of the BPS
$\frac{1}{2}$-BPS circular Wilson loop known as latitude Bianchi:2017ozk .
However, to the best of our knowledge, a rigorous argument was still missing.

Combining all these considerations, we propose a general formula for any BPS
superconformal line $\mathcal{D}$ preserving at least $\mathfrak{su}(2)$
R-symmetry factor in any $\mathcal{N}=6$ SCFT

$B=\frac{i}{4\pi
r\langle\mathcal{D}\rangle}\frac{\partial\langle\mathcal{D}\rangle[m]}{\partial
m}\bigg{|}_{m=0}$ (74)

According to the general classification of Agmon:2020pde , these are
$\frac{1}{3}$-BPS lines (see Drukker:2022txy  for a recent explicit example).
We stress again that there is no dependence on the specific realization of
theory.

###  5.2 The two-point function

We focus now on the two-point function. The computation of the tree-level is
straightforward131313We are using a shorthand notation
$\tau_{i}\to(r\cos\tau_{i},r\sin\tau_{i},0)$, $i=1,2$

$\langle\mathcal{O}(\tau_{1})\mathcal{O}(\tau_{2})\rangle_{W}^{(0)}=N_{1}N_{2}\frac{U(\tau_{1})\cdot\bar{V}(\tau_{2})U(\tau_{2})\cdot\bar{V}(\tau_{1})}{16\pi^{2}|\tau_{1}-\tau_{2}|}=\frac{N_{1}N_{2}}{r^{2}64\pi^{2}}\,.$
(75)

As a consistency check, we push the perturbative computation to 1-loop. The
diagrams contributing to this order are those in figure 3.

Figure 3: Feynman diagrams for the 1-loop correction of the defect two-point
function. Figure 4: The Feynman diagram for the 1-loop correction of vev of
the fermionic Wilson loop.

We see that in diagrams (c), (d), and (e), the operators interact with the
Wilson loop, while (f) is a vacuum correction. Therefore, (f) is canceled
against the 1-loop correction $\delta W$ to the vev of the Wilson loop of
figure 4 coming from the normalization. Indeed, since

$\mathrm{(f)}=\langle\mathcal{O}(\tau_{1})\mathcal{O}(\tau_{2})\rangle_{W}^{(0)}\delta
W\,,$ (76)

we have

$\langle\mathcal{O}(\tau_{1})\mathcal{O}(\tau_{2})\rangle_{W}=\frac{\langle\mathcal{O}(\tau_{1})\mathcal{O}(\tau_{2})\rangle^{(0)}+\langle\mathcal{O}(\tau_{1})\mathcal{O}(\tau_{2})\rangle^{(1)}}{1+\delta
W}=\langle\mathcal{O}(\tau_{1})\mathcal{O}(\tau_{2})\rangle^{(0)}+\mathrm{(c)}+\mathrm{(d)}+\mathrm{(e)}\,.$
(77)

The term $\delta W$ is both IR and UV divergent and can be found in
Griguolo:2012iq , so its cancellation is crucial to ensure supersymmetry. Let
us discuss the remaining contributions.

As explained in Gorini:2020new  on the line, diagram (c) is vanishing for the
kinematics of the Chern-Simons propagator, which always leads to an integrand
function odd in at least one of the loop integration variables.

Then we write down diagram (d), and we perform the Wick contractions

$\displaystyle(d)=$
$\displaystyle-\frac{4\pi}{k}\Tr(U^{A}C_{A}\bar{V}_{B}\bar{C}^{B})(\tau_{1})\Tr(U^{K}C_{K}\bar{V}_{L}\bar{C}^{L})(\tau_{2}){M_{I}}^{J}\int_{-\infty}^{\infty}ds\,\Tr(\bar{C}^{I}C_{J}(s))=$
$\displaystyle=$ $\displaystyle-\frac{1}{16\pi
k|\tau_{1}-\tau_{2}|}\frac{N_{1}N_{2}}{N_{1}+N_{2}}\big{(}U^{I}(\tau_{1}){M_{I}}^{J}\bar{V}_{J}(\tau_{2})U^{K}(\tau_{2})V_{k}(\tau_{1})+$
$\displaystyle
U^{I}(\tau_{1}){M_{I}}^{J}\bar{V}_{J}(\tau_{2})U^{K}(\tau_{2})V_{k}(\tau_{1})\big{)}\equiv
0\,.$ (78)

after the evaluation of the contraction of the polarization vector with
${M_{I}}^{J}$.

Finally, we consider diagram (e). We can simplify the computation using the
topologicity of the operators. Indeed, if we set $\tau_{1}=\pi/2$ and
$\tau_{2}=3\pi/2$, we get again the integral over the entire spacetime of an
odd function, which is zero. In the end, we find that there is no 1-loop
correction, as in Eq. (69). This simple result holds only for topological
operators in this specific kinematical configuration. It is not necessarily
true for any two-point function of the superconformal primary of the stress
tensor multiplet.

##  6 Conclusion and outlook

In this work, we constructed and computed supersymmetric correlation functions
involving both local operators and line operators in 3d $\mathcal{N}=4$
theories. The local operators are those forming the topological sector
Chester:2014mea , which is thus compatible with BPS line defects. We discussed
this setup for $\frac{1}{2}$-BPS line defects both in UV $\mathcal{N}=4$ super
Yang-Mills theory on $S^{3}$ and $\mathcal{N}=4$ SCFTs. For dimension one
operators, we argued that the vev of the loop operator with specific
supersymmetric flavor deformations is the generating functional for integrated
defect correlation functions. When there is a localization scheme for the vev
of the defect, the formula allows us to compute exact defect correlation
functions.

We applied this method to the $\frac{1}{2}$-BPS Wilson line in the
$U(N_{1})_{k}\times U(N_{2})_{-k}$ ABJM model, and we calculated the one and
two-point function of dimension one topological operators for Chern-Simons
level $k\gg 1$. Since this operator is part of the stress tensor multiplet, we
managed to relate the one-point function to the so-called bremsstrahlung
function, whose expression was rigorously proved only in the case of
$N_{1}=N_{2}$. Our formula extends the derivation to all $\frac{1}{3}$-BPS
lines in $\mathcal{N}=6$ SCFT, regardless of the specific details. We also
present the first result for a two-point function in ABJM with BPS line
defects. We compared our prediction with an honest Feynman diagram
computation, showing perfect agreement.

There are several directions to explore in the future. While there are several
examples of defect correlation functions in 4d SCFTs in various regimes and
with different amounts of supersymmetry, the situation in 3d is still rather
understudied. The first direction will be the improvement of the matrix model
computation. A natural candidate is the Fermi gas technique Marino:2011eh ,
which has already been developed for BPS Wilson lines Nosaka:2015iiw  and mass
deformations Klemm:2012ii , but not for mass deformations with the Wilson loop
141414See also Chester:2020jay ; Gaiotto:2020vqj ; Hatsuda:2021oxa  for
applications to the topological sector without defects. The combination with
the technology developed in this paper will allow us to access the defect
correlation function in the strongly coupled regime and compare it with the
string or M-theory dual.

It would be interesting to analyze our defect correlation functions from the
bootstrap perspective. Without defects, the topological sector represents a
consistent and much simpler truncation of the full bootstrap equation. It
turns out that the operator algebra is captured by specific algebraic
structure, namely a deformation quantization of the Higgs (or Coulomb) branch
chiral ring Beem:2016cbd . It would be interesting to understand how this
picture is modified by the presence of the defect and study the correspondent
problem along the lines of Chang:2019dzt ; Fan:2019jii .

Moreover, the protected sector can furnish useful inputs for the general
defect bootstrap problem. While the study of the defect CFT for the fermionic
Wilson line has been initiated Bianchi:2017ozk ; Bianchi:2020hsz ;
Gorini:2022jws , nothing is known for correlation functions of bulk operators
with defects. We expect the existence of superconformal Ward identities,
similar to other supersymmetric setups in 4d Liendo:2016ymz ; Liendo:2018ukf .
Perhaps it is also possible to use an inversion formula Caron-Huot:2017vep
like in Barrat:2021yvp .

Finally, it would be interesting to study these observables for other defects,
like vortex loops Drukker:2008zx ; Kapustin:2012iw ; Drukker:2012sr . A
possible difficulty is the lack of a general localization scheme for these
disorder operators in Chern-Simons matter theory. It would also be interesting
to study the same problems in $\mathcal{N}=4$ QFTs, where derivatives of the
deformed vev of the defect compute correlation functions of flavor current
multiplets (see Chang:2019dzt  for the corresponding problem without defects).

###### Acknowledgements.

It is a pleasure to thank Luca Griguolo, Domenico Seminara, Itamar Yaakov, and
Stefano Cremonesi for interesting discussions and useful insights. We are
especially grateful to Luca Griguolo and Itamar Yaakov for useful comments on
the draft. We also thank the Department of Mathematical Sciences at Durham
University and its faculty members for hospitality, discussions, and feedback
when presenting this work in the local journal club. This work has been
supported in part by Della Riccia Foundation, Italian Ministero
dell’Università e Ricerca (MUR), and by Istituto Nazionale di Fisica Nucleare
(INFN) through the “Gauge Theories, Strings, Supergravity” (GSS) and “Gauge
and String Theory” (GAST) research projects.

##  Appendix A ABJ(M) action and Feynman rules

Here we shortly summarize the basic notions about ABJ(M) theory needed in the
paper. We work in Euclidean space with coordinates
$x^{\mu}=(x^{1},x^{2},x^{3})$ and metric $\delta_{\mu\nu}$. We take the 3d
flat Clifford algebra $\gamma^{\mu}$ to be generated by the Pauli matrices
$(\gamma^{\mu})_{\alpha}^{\;\beta}\equiv(\sigma^{\mu})_{\alpha}^{\;\beta}$,
$\mu=1,2,3$. Spinorial indices are raised and lowered according to

$\psi^{\alpha}=\varepsilon^{\alpha\beta}\psi_{\beta},\qquad\psi_{\alpha}=\varepsilon_{\alpha\beta}\psi^{\beta}\qquad{\rm
with}\qquad\varepsilon^{12}=-\varepsilon_{12}=1$

The field content of the $U(N_{1})_{k}\times U(N_{2})_{-k}$ ABJ(M) theory
includes two gauge fields $(A_{\mu})_{i}^{j}$,
$(\hat{A}_{\mu})_{\hat{i}}^{\hat{j}}$ belonging to the adjoint representation
of $U(N_{1})$ and $U(N_{2})$ respectively, minimally coupled to four matter
multiplets $(C_{I},\bar{\psi}^{I})_{I=1,\dots,4}$ in the $(N_{1},\bar{N}_{2})$
representation of the gauge group and their conjugates
$(\bar{C}^{I},\psi_{I})_{I=1,\dots,4}$ in the $(\bar{N}_{1},N_{2})$.

The Euclidean gauge-fixed action is given by

$S=S_{\textup{CS}}+S_{\textup{mat}}+S_{\textup{pot}}\,,$ (79)

where $S_{\textup{CS}}$ contain the Chern-Simons kinetic terms,
$S_{\textup{mat}}$ the kinetic terms for the matter fields and the gauge-
matter interactions, and $S_{\textup{pot}}$ the Yukawa, and the potential
terms. For the explicit expressions we refer to Bianchi:2018bke ;
Gorini:2020new ; Gorini:2022jws .

The action is invariant under the $\mathcal{N}=6$ superconformal algebra
$\mathfrak{osp}(6|4)$. Its explicit realization is

$\displaystyle\delta C_{K}$ $\displaystyle=$
$\displaystyle-\bar{\xi}^{IJ,\alpha}\ \varepsilon_{IJKL}\
\bar{\psi}^{L}_{\alpha}$ $\displaystyle\delta\bar{C}^{K}$ $\displaystyle=$
$\displaystyle 2\bar{\xi}^{KL,\alpha}\ \psi_{L,\alpha}$
$\displaystyle\delta\bar{\psi}^{K,\beta}$ $\displaystyle=$ $\displaystyle
2i\bar{\xi}^{KL,\alpha}{(\gamma^{\mu})_{\alpha}}^{\beta}D_{\mu}C_{L}+\frac{4\pi
i}{k}\bar{\xi}^{KL,\beta}(C_{L}\bar{C}^{M}C_{M}-C_{M}\bar{C}^{M}C_{L})+\frac{8\pi
i}{k}\bar{\xi}^{IJ,\beta}C_{I}\bar{C}^{K}C_{J}$
$\displaystyle\quad-\,2i\bar{\eta}^{KL,\beta}C_{L}$
$\displaystyle\delta\psi^{\beta}_{K}$ $\displaystyle=$
$\displaystyle-i\bar{\xi}^{IJ,\alpha}\varepsilon_{IJKL}{(\gamma^{\mu})_{\alpha}}^{\beta}D_{\mu}\bar{C}^{L}+\frac{2\pi
i}{k}\bar{\xi}^{IJ,\beta}\varepsilon_{IJKL}(\bar{C}^{L}C_{M}\bar{C}^{M}-\bar{C}^{M}C_{M}\bar{C}^{L})$
$\displaystyle\quad+\frac{4\pi
i}{k}\bar{\xi}^{IJ,\beta}\varepsilon_{IJML}\bar{C}^{M}C_{K}\bar{C}^{L}+i\bar{\eta}^{IJ,\beta}\varepsilon_{IJKL}\bar{C}^{L}$
$\displaystyle\delta A_{\mu}$ $\displaystyle=$ $\displaystyle\frac{4\pi
i}{k}\bar{\xi}^{IJ,\alpha}{(\gamma_{\mu})_{\alpha}}^{\beta}\bigg{(}C_{I}\psi_{J\beta}-\frac{1}{2}\varepsilon_{IJKL}\bar{\psi}^{K}_{\beta}\bar{C}^{L}\bigg{)}$
$\displaystyle\delta\hat{A}_{\mu}$ $\displaystyle=$ $\displaystyle\frac{4\pi
i}{k}\bar{\xi}^{IJ,\alpha}{(\gamma_{\mu})_{\alpha}}^{\beta}\bigg{(}\psi_{J\beta}C_{I}-\frac{1}{2}\varepsilon_{IJKL}\bar{C}^{L}\bar{\psi}^{K}_{\beta}\bigg{)}$
(80)

where the parameters of the transformations are expressed in terms of
conformal Killing spinors, whose flat space expression is

$\bar{\xi}^{IJ}_{\alpha}=\bar{\theta}^{IJ}_{\alpha}-x^{\mu}(\gamma_{\mu})_{\alpha}^{\;\beta}{\bar{\eta}}^{IJ}_{\beta}\,.$
(81)

For perturbative computations, we also need the tree-level propagators. After
rescaling the gauge fields in the action read as

$A_{\mu}\to\frac{1}{\sqrt{k}}A_{\mu}\qquad,\qquad\hat{A}_{\mu}\to\frac{1}{\sqrt{k}}\hat{A}_{\mu}$
(82)

they read

  * •

Scalar propagator

$\displaystyle\langle{(C_{I})_{i}}^{\hat{j}}(x)\
{(\bar{C}^{J})_{\hat{k}}}^{l}(y)\rangle$
$\displaystyle=\delta^{J}_{I}\delta^{l}_{i}\delta^{\hat{j}}_{\hat{k}}\
\frac{\Gamma(\frac{1}{2}-\epsilon)}{{4\pi}^{\frac{3}{2}-\epsilon}}\frac{1}{{|x-y|}^{1-2\epsilon}}$
(83)

  * •

Fermion propagator

$\langle{(\psi_{\alpha I})_{\hat{i}}}^{j}(x)\
{(\bar{\psi}^{J\beta})^{\hat{l}}}_{k}(y)\rangle=\delta^{J}_{I}\delta^{\hat{l}}_{\hat{i}}\delta^{j}_{k}\
i\,\frac{\Gamma(\frac{3}{2}-\epsilon)}{{2\pi}^{\frac{3}{2}-\epsilon}}\
{(\gamma^{\mu})_{\alpha}}^{\beta}\ \frac{(x-y)_{\mu}}{{|x-y|}^{3-2\epsilon}}$
(85)

  * •

Vector propagators in Landau gauge

$\displaystyle\langle{(A_{\mu})_{i}}^{j}(x)\
{(A_{\nu})_{k}}^{l}(y)\rangle=\delta^{l}_{i}\delta^{j}_{k}\
i\,\frac{\Gamma(\frac{3}{2}-\epsilon)}{{\pi}^{\frac{1}{2}-\epsilon}}\
\varepsilon_{\mu\nu\rho}\ \frac{(x-y)^{\rho}}{{|x-y|}^{3-2\epsilon}}$
$\displaystyle\langle{(\hat{A}_{\mu})_{\hat{i}}}^{\hat{j}}(x)\
{(\hat{A}_{\nu})_{\hat{k}}}^{\hat{l}}(y)\rangle=-\,\delta^{\hat{l}}_{\hat{i}}\delta^{\hat{j}}_{\hat{k}}\
i\,\frac{\Gamma(\frac{3}{2}-\epsilon)}{{\pi}^{\frac{1}{2}-\epsilon}}\
\varepsilon_{\mu\nu\rho}\ \frac{(x-y)^{\rho}}{{|x-y|}^{3-2\epsilon}}$ (86)

##  Appendix B From $\mathcal{N}=6$ to $\mathcal{N}=4$ superalgebras

In this appendix, we spell out the details of the supersymmetry algebra
appearing in this work.

####  $\mathcal{N}=6$ superalgebra

$\mathcal{N}=6$ SCFTs in 3d are invariant under the $\mathfrak{osp}(6|4)$
superalgebra. It contains the usual 3d conformal algebra, whose commutation
relations are

$\displaystyle[M^{\mu\nu},M^{\rho\sigma}]$
$\displaystyle=\delta^{\sigma\mu}M^{\nu\rho}-\delta^{\sigma\nu}M^{\mu\rho}+\delta^{\rho\nu}M^{\mu\sigma}-\delta^{\rho\mu}M^{\nu\sigma}\
$ $\displaystyle[P^{\mu},K^{\nu}]$
$\displaystyle=2(\delta^{\mu\nu}D+M^{\mu\nu})$ (87)
$\displaystyle[P^{\mu},M^{\nu\rho}]$
$\displaystyle=\delta^{\mu\nu}P^{\rho}-\delta^{\mu\rho}P^{\nu}$
$\displaystyle[K^{\mu},M^{\nu\rho}]$
$\displaystyle=\delta^{\mu\nu}K^{\rho}-\delta^{\mu\rho}K^{\nu}$
$\displaystyle[D,P^{\mu}]$ $\displaystyle=P^{\mu}$ $\displaystyle[D,K^{\mu}]$
$\displaystyle=-K^{\mu}$

The spacetime generators act on scalar operators as

$\displaystyle\commutator{P_{\mu}}{O(x)}$ $\displaystyle=-i\partial_{\mu}O(x)$
(88a) $\displaystyle\commutator{K_{\mu}}{O(x)}$
$\displaystyle=i(-x^{2}\partial_{\mu}+2x_{\mu}x\cdot\partial+2x_{\mu}\Delta)O(x)$
(88b) $\displaystyle\commutator{M_{\mu\nu}}{O(x)}$
$\displaystyle=(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})O(x)$ (88c)
$\displaystyle\commutator{D}{O(x)}$
$\displaystyle=(-x^{\mu}\partial_{\mu}-\Delta)O(x)\,,$ (88d)

where $\Delta$ is the dimension of the operator $O(x)$. The bosonic part of
the algebra contains also an $\mathfrak{su}(4)\simeq\mathfrak{so}(6)$
R-symmetry factor, generated by the traceless matrices ${J_{I}}^{J}$, with
$I,J=1,\dots,4$. Their algebra reads

$[{J_{I}}^{J},{J_{K}}^{L}]=\delta^{L}_{I}{J_{K}}^{J}-\delta^{J}_{K}{J_{I}}^{L}$
(89)

We have defined implicitly also the action of ${J_{I}}^{J}$ on the
(anti-)fundamental representation. That is, for a generic operator $\Phi_{I}$
($\bar{\Phi}^{I}$) the action of ${J_{I}}^{J}$ reads

$[{J_{I}}^{J},\Phi_{K}]=\frac{1}{4}\delta^{J}_{I}\Phi_{K}-\delta^{J}_{K}\Phi_{I}\qquad\qquad[{J_{I}}^{J},\bar{\Phi}^{K}]=\delta^{K}_{I}\bar{\Phi}^{J}-\frac{1}{4}\delta^{J}_{I}\bar{\Phi}^{K}$
(90)

The odd generators are $Q^{IJ}_{\alpha}$, $S^{IJ}_{\alpha}$ and close the
following algebra

$\displaystyle\\{Q^{IJ}_{\alpha},Q^{KL,\beta}\\}$
$\displaystyle=\varepsilon^{IJKL}{(\gamma^{\mu})_{\alpha}}^{\beta}P_{\mu}\qquad\qquad\\{S_{\alpha}^{IJ},S^{\beta
KL}\\}=\varepsilon^{IJKL}{(\gamma^{\mu})_{\alpha}}^{\beta}K_{\mu}$ (91)
$\displaystyle\\{Q^{IJ}_{\alpha},S^{\beta KL}\\}$
$\displaystyle=\varepsilon^{IJKL}\left(\frac{1}{2}{(\gamma^{\mu\nu})_{\alpha}}^{\beta}M_{\mu\nu}+\delta^{\beta}_{\alpha}D\right)+\delta^{\beta}_{\alpha}\varepsilon^{KLMN}(\delta^{J}_{M}{J_{N}}^{I}-\delta^{I}_{M}{J_{N}}^{J})$

and similarly for $\bar{Q}_{\alpha
IJ}=\frac{1}{2}\varepsilon_{IJKL}Q^{KL}_{\alpha}$ and $\bar{S}_{\alpha
IJ}=\frac{1}{2}\varepsilon_{IJKL}S^{KL}_{\alpha}$. Finally, the mixed
commutators are

$\displaystyle={(\gamma^{\mu})_{\alpha}}^{\beta}S^{IJ}_{\beta}$
$\displaystyle[P^{\mu},S_{\alpha}^{IJ}]$
$\displaystyle={(\gamma^{\mu})_{\alpha}}^{\beta}Q^{IJ}_{\beta}$ (92)
$\displaystyle[M^{\mu\nu},Q_{\alpha}^{IJ}]$
$\displaystyle=-\frac{1}{2}{(\gamma^{\mu\nu})_{\alpha}}^{\beta}Q_{\beta}^{IJ}$
$\displaystyle[M^{\mu\nu},S_{\alpha}^{IJ}]$
$\displaystyle=-\frac{1}{2}{(\gamma^{\mu\nu})_{\alpha}}^{\beta}S_{\beta}^{IJ}$
$\displaystyle[D,Q_{\alpha}^{IJ}]$ $\displaystyle=\frac{1}{2}Q_{\alpha}^{IJ}$
$\displaystyle[D,S^{\alpha IJ}]$ $\displaystyle=-\frac{1}{2}S^{\alpha IJ}$
$\displaystyle[{J_{I}}^{J},Q^{KL}_{\alpha}]$
$\displaystyle=\delta^{K}_{I}Q^{JL}_{\alpha}+\delta^{L}_{I}Q^{KJ}_{\alpha}-\frac{1}{2}\delta^{J}_{I}Q^{KL}_{\alpha}$
$\displaystyle[{J_{I}}^{J},S^{\alpha KL}]$
$\displaystyle=\delta^{K}_{I}S^{\alpha JL}+\delta^{L}_{I}S^{\alpha
KJ}-\frac{1}{2}\delta^{J}_{I}S^{\alpha KL}$

The definition of $\delta$ in terms of $\bar{Q}_{\alpha IJ}$ and
$\bar{S}_{\alpha IJ}$ is

$\delta\Phi=\commutator*{\bar{\theta}^{IJ}\bar{Q}_{IJ}-i\bar{\eta}^{IJ}\bar{S}_{IJ}}{\Phi}\,.$
(93)

####  $\frac{1}{2}$–BPS Wilson line in ABJM

We briefly describe the superalgebra
$\mathfrak{su}(1,1|3)\oplus\mathfrak{u}(1)_{b}$ preserved by the fermionic
Wilson loop. The maximal bosonic subalgebra of $\mathfrak{su}(1,1|3)$ is
$\mathfrak{sl}(2)\oplus\mathfrak{su}(3)\oplus\mathfrak{u}(1)$, where
$\mathfrak{sl}(2)\simeq\mathfrak{su}(1,1)$ is the euclidean conformal algebra
in one dimension and $\mathfrak{su}(3)\oplus\mathfrak{u}(1)$ is the R-symmetry
algebra. The $\mathfrak{u}(1)_{b}$ factor is generated by
$B=M_{12}+2i{J_{1}}^{1}$.

The $\mathfrak{su}(1,1)$ algebra is generated by $P_{3}$, $K_{3}$, and $D$.
The $\mathfrak{su}(3)$ R-symmetry subalgebra is generated by traceless
operators ${L_{a}}^{b}$, whose explicit form reads

${L_{a}}^{b}=\begin{pmatrix}{J_{2}}^{2}+\frac{1}{3}{J_{1}}^{1}&{J_{2}}^{3}&{J_{2}}^{4}\\\
{J_{3}}^{2}&{J_{3}}^{3}+\frac{1}{3}{J_{1}}^{1}&{J_{3}}^{4}\\\
{J_{4}}^{2}&{J_{4}}^{3}&-{J_{3}}^{3}-{J_{2}}^{2}-\frac{2}{3}{J_{1}}^{1}\end{pmatrix}$
(94)

These generators satisfy the algebraic relation

$[{L_{a}}^{b},{L_{c}}^{d}]=\delta_{a}^{d}{L_{c}}^{b}-\delta_{c}^{b}{L_{a}}^{d}$
(95)

From Eq. (90) and definitions (94) it follows that the action of the $SU(3)$
R-symmetry generators on fields in the (anti-)fundamental representation is

$[{L_{a}}^{b},\Phi_{c}]=\frac{1}{3}\delta^{b}_{a}\Phi_{c}-\delta^{b}_{c}\Phi_{a}\qquad[{L_{a}}^{b},\bar{\Phi}^{c}]=\delta^{c}_{a}\bar{\Phi}^{b}-\frac{1}{3}\delta^{b}_{a}\bar{\Phi}^{c}$
(96)

The spectrum of bosonic generators of $\mathfrak{su}(1,1|3)$ is completed by a
residual $\mathfrak{u}(1)$ generator $M$, defined as

$M\equiv 3iM_{12}-2{J_{1}}^{1}$ (97)

We now move to the fermionic sector of the superalgebra. Since we have placed
the line along the $x^{3}$-direction, the fermionic generators of the one-
dimensional superconformal algebra are identified with the following
supercharges

$Q_{1}^{12},Q_{1}^{13},Q^{14}_{1},Q_{2}^{23},Q^{24}_{2},Q^{34}_{2}\qquad\text{and}\qquad
S_{1}^{12},S_{1}^{13},S^{14}_{1},S_{2}^{23},S^{24}_{2},S^{34}_{2}$ (98)

For an exhaustive description of the algebra, we refer to Gorini:2020new .

####  $\mathcal{N}=4$ superalgebra

In order to define a topological sector compatible with the Wilson line, we
need to identify the $\mathcal{N}=4$ Poincaré subalgebra introduced in
Dedushenko:2016jxl  in the ABJM superalgebra. In the following, we detail the
resulting decomposition, providing the relevant conventions.

To begin with, we choose the $\mathfrak{su}(2)_{H}\oplus\mathfrak{su}(2)_{C}$
R-symmetry algebra into $\mathfrak{su}(4)$. This can be explicitly realized by
setting the $\mathfrak{su}(2)_{H}\oplus\mathfrak{su}(2)_{C}$ ${R_{a}}^{b}$,
${\bar{R}_{\dot{a}}}^{\;\;\dot{b}}$ as follows 151515The full pattern of
symmetry breaking is $SU(4)\to SU(2)_{H}\times SU(2)_{C}\times U(1)_{F}$,
where $U(1)_{F}$ is generated by $F={J_{4}}^{4}+{J_{2}}^{2}$. It commutes with
the full $\mathfrak{osp}(4|4)$ superalgebra. Such generator will not play any
role in our construction.

$\displaystyle{R_{a}}^{b}=\begin{pmatrix}-\frac{{J_{4}}^{4}-{J_{2}}^{2}}{2}&{J_{2}}^{4}\\\
{J_{4}}^{2}&\frac{{J_{4}}^{4}-{J_{2}}^{2}}{2}\end{pmatrix}\,,\qquad{\bar{R}_{\dot{a}}}^{\;\;\dot{b}}=\begin{pmatrix}-\frac{{J_{3}}^{3}-{J_{1}}^{1}}{2}&{J_{1}}^{3}\\\
{J_{3}}^{1}&\frac{{J_{3}}^{3}-{J_{1}}^{1}}{2}\end{pmatrix}$ (99)

The embedding makes $\mathfrak{su}(2)_{H}$ coincide with an $\mathfrak{su}(2)$
factor preserved by the Wilson line, namely the one spanned by ${L_{1}}^{3}$,
${L_{3}}^{1}$, ${L_{1}}^{1}-{L_{3}}^{3}$. Specifically, we choose the one used
in the topological line of Gorini:2020new . The resulting algebra is

$\commutator*{{R_{a}}^{b}}{{R_{c}}^{d}}=\delta_{a}^{d}{R_{c}}^{b}-\delta_{c}^{b}{R_{a}}^{d}\,,\qquad\commutator*{\bar{R}_{\dot{a}}^{\;\;\dot{b}}}{\bar{R}_{\dot{a}}^{\;\;\dot{b}}}=\delta_{\dot{a}}^{\dot{d}}\bar{R}_{\dot{c}}^{\;\;\dot{b}}-\delta_{\dot{c}}^{\dot{b}}\bar{R}_{\dot{a}}^{\;\;\dot{d}}\,.$
(100)

So far, we have broken the $\mathcal{N}=6$ R-symmetry group down to the
$\mathcal{N}=4$. To identify the full $\mathfrak{osp}(4|4)$ algebra, we take
the supercharges charged under our $\mathcal{N}=4$ R-symmetry subalgebra. They
can be organized in the following way

$Q_{\alpha,a\dot{a}}=\begin{pmatrix}\bar{Q}_{\alpha,12}&\bar{Q}_{\alpha,32}\\\
Q_{\alpha,14}&\bar{Q}_{\alpha,34}\end{pmatrix}_{a\dot{a}}\,,\qquad
S_{\alpha,a\dot{a}}=\begin{pmatrix}\bar{S}_{\alpha,12}&\bar{S}_{\alpha,32}\\\
S_{\alpha,14}&\bar{S}_{\alpha,34}\end{pmatrix}_{a\dot{a}}\,.$ (101)

We find

$\commutator*{{R_{a}}^{b}}{Q_{\alpha,c\dot{a}}}=\frac{1}{2}\delta_{a}^{b}Q_{\alpha,c\dot{a}}-\delta_{c}^{b}Q_{\alpha,a\dot{a}}\,,\qquad\commutator*{\bar{R}_{\dot{a}}^{\;\;\dot{b}}}{Q_{\alpha,a\dot{c}}}=\frac{1}{2}\delta_{\dot{a}}^{\dot{b}}Q_{\alpha,a\dot{c}}-\delta_{\dot{c}}^{\dot{b}}Q_{\alpha,a\dot{a}}\,.$
(102)

For a more direct interpretation on $S^{3}$, we find it convenient to express
the Poincaré generators using spinorial indices, and we shall use

$\displaystyle{P_{\alpha}}^{\beta}=$ $\displaystyle
P^{\mu}{(\gamma_{\mu})_{\alpha}}^{\beta}=\begin{pmatrix}P_{3}&P_{1}-iP_{2}\\\
P_{1}+iP_{2}&-P_{3}\end{pmatrix}\,,$ (103)
$\displaystyle{K_{\alpha}}^{\beta}=$ $\displaystyle
K^{\mu}{(\gamma_{\mu})_{\alpha}}^{\beta}=\begin{pmatrix}K_{3}&K_{1}-iK_{2}\\\
K_{1}+iK_{2}&-K_{3}\end{pmatrix}\,,$ (104)
$\displaystyle{M_{\alpha}}^{\beta}=$
$\displaystyle\frac{i}{2}\epsilon_{\mu\nu\lambda}M^{\mu\nu}{(\gamma_{\lambda})_{\alpha}}^{\beta}=\frac{1}{2}M^{\mu\nu}{(\gamma_{\mu\nu})_{\alpha}}^{\beta}=\begin{pmatrix}iM_{12}&-M_{13}+iM_{23}\\\
M_{13}+iM_{23}&-iM_{12}\end{pmatrix}$ (105)

Their algebra reads as

$\displaystyle\commutator*{{P_{\alpha}}^{\beta}}{{K_{\gamma}}^{\delta}}$
$\displaystyle=2\left((2\delta_{\alpha}^{\delta}\delta_{\gamma}^{\beta}-\delta_{\alpha}^{\beta}\delta_{\gamma}^{\delta})D+\delta^{\beta}_{\gamma}{M_{\alpha}}^{\delta}-\delta^{\delta}_{\alpha}{M_{\gamma}}^{\beta}\right)\,,$
(106) $\displaystyle\commutator*{{M_{\alpha}}^{\beta}}{{M_{\gamma}}^{\delta}}$
$\displaystyle=\delta_{\gamma}^{\beta}{M_{\alpha}}^{\delta}-\delta_{\alpha}^{\delta}{M_{\gamma}}^{\beta}\,,$
(107) $\displaystyle\commutator*{{M_{\gamma}}^{\delta}}{{P_{\alpha}}^{\beta}}$
$\displaystyle=\delta_{\alpha}^{\beta}{P_{\gamma}}^{\delta}+\delta_{\gamma}^{\delta}{P_{\alpha}}^{\beta}-2\delta_{\gamma}^{\beta}{P_{\alpha}}^{\delta}\,,$
(108) $\displaystyle\commutator*{{M_{\gamma}}^{\delta}}{{K_{\alpha}}^{\beta}}$
$\displaystyle=\delta_{\alpha}^{\beta}{K_{\gamma}}^{\delta}+\delta_{\gamma}^{\delta}{K_{\alpha}}^{\beta}-2\delta_{\gamma}^{\beta}{K_{\alpha}}^{\delta}\,,$
(109)

Then, we can finally write down the $\mathcal{N}=4$ algebra. The even-odd part
reads

$\displaystyle\commutator*{{P_{\alpha}}^{\beta}}{S_{\gamma,a\dot{a}}}$
$\displaystyle=2\delta_{\gamma}^{\beta}Q_{\alpha,a\dot{a}}-\delta_{\alpha}^{\beta}Q_{\gamma,a\dot{a}}\,,$
$\displaystyle\commutator*{{K_{\alpha}}^{\beta}}{Q_{\gamma,a\dot{a}}}$
$\displaystyle=2\delta_{\gamma}^{\beta}S_{\alpha,a\dot{a}}-\delta_{\alpha}^{\beta}S_{\gamma,a\dot{a}}\,,$
(110) $\displaystyle\commutator*{{M_{\alpha}}^{\beta}}{Q_{\gamma,a\dot{a}}}$
$\displaystyle=\delta_{\gamma}^{\beta}Q_{\alpha,a\dot{a}}-\frac{1}{2}\delta_{\alpha}^{\beta}Q_{\gamma,a\dot{a}}\,,$
$\displaystyle\commutator*{{M_{\alpha}}^{\beta}}{S_{\gamma,a\dot{a}}}$
$\displaystyle=\delta_{\gamma}^{\beta}S_{\alpha,a\dot{a}}-\frac{1}{2}\delta_{\alpha}^{\beta}S_{\gamma,a\dot{a}}\,,$
(111) $\displaystyle\commutator*{{D}^{\beta}}{Q_{\alpha,a\dot{a}}}$
$\displaystyle=\frac{1}{2}^{\beta}Q_{\alpha,a\dot{a}}\,,$
$\displaystyle\commutator*{{D}^{\beta}}{S_{\alpha,a\dot{a}}}$
$\displaystyle=-\frac{1}{2}S_{\alpha,a\dot{a}}\,.$ (112)

For the odd-odd part we get

$\displaystyle\anticommutator*{Q_{\alpha,a\dot{a}}}{{Q^{\beta}}_{b\dot{b}}}$
$\displaystyle=\epsilon_{a\dot{a}}\epsilon_{b\dot{b}}{P_{\alpha}}^{\beta}\,,\qquad\anticommutator*{S_{\alpha,a\dot{a}}}{{S^{\beta}}_{b\dot{b}}}=\epsilon_{a\dot{a}}\epsilon_{b\dot{b}}{K_{\alpha}}^{\beta}\,,$
(113)
$\displaystyle\anticommutator*{Q_{\alpha,a\dot{a}}}{{S^{\beta}}_{b\dot{b}}}$
$\displaystyle=\epsilon_{a\dot{a}}\epsilon_{b\dot{b}}\left({M_{\alpha}}^{\beta}+\delta_{\alpha}^{\beta}D\right)-\delta_{\alpha}^{\beta}\left(\epsilon_{ab}\bar{R}_{\dot{a}\dot{b}}+\epsilon_{\dot{a}\dot{b}}R_{ab}\right)\,.$
(114)

where $R_{ab}=\epsilon_{bc}{R_{a}}^{c}$ and
$\bar{R}_{\dot{a}\dot{b}}=\epsilon_{\dot{b}\dot{c}}{R_{\dot{a}}}^{\dot{c}}$.

We are now in business to give up conformal invariance. We define the
spacetime generators

${J^{(\ell)}_{\alpha}}^{\beta}=\frac{r}{2}\left(\frac{1}{r}{M_{\alpha}}^{\beta}-{P_{\alpha}}^{\beta}-\frac{1}{4r^{2}}{K_{\alpha}}^{\beta}\right)\,,\quad{J^{(r)}_{\alpha}}^{\beta}=\frac{r}{2}\left(\frac{1}{r}{M_{\alpha}}^{\beta}+{P_{\alpha}}^{\beta}+\frac{1}{4r^{2}}{K_{\alpha}}^{\beta}\right)\,,$
(115)

which satisfy the algebra

$\commutator*{{J^{(\ell/r)}_{\alpha}}^{\beta}}{{J^{(\ell/r)}_{\gamma}}^{\delta}}=\delta_{\gamma}^{\beta}{J^{(\ell/r)}_{\alpha}}^{\delta}-\delta_{\alpha}^{\delta}{J^{(\ell/r)}_{\beta}}^{\gamma}\,,\qquad\commutator*{{J^{(\ell)}_{\alpha}}^{\beta}}{{J^{(r)}_{\gamma}}^{\delta}}=0\,.$
(116)

If we map our theory on $S^{3}$, ${J^{(\ell)}_{\alpha}}^{\beta}$ and
${J^{(r)}_{\alpha}}^{\beta}$ are the generators of the isometry algebra on
$S^{3}$ $\mathfrak{su}(2)_{\ell}$, $\mathfrak{su}(2)_{r}$, respectively. We
can interpret $\mathfrak{su}(2)_{\ell}\oplus\mathfrak{su}(2)_{r}$ as the
bosonic part of the $\mathfrak{su}(2|1)_{\ell}\oplus\mathfrak{su}(2|1)_{r}$,
which is the $\mathcal{N}=4$ Poincaré superalgebra on $S^{3}$. More precisely,
if we define

$\displaystyle{J^{(\ell/r)}_{\alpha}}^{\beta}=\begin{pmatrix}J_{3}^{(\ell/r)}&J_{1}^{(\ell/r)}+iJ_{2}^{(\ell/r)}\\\
J_{1}^{(\ell/r)}-iJ_{2}^{(\ell/r)}&-J_{3}^{(\ell/r)}\end{pmatrix}\,,$ (117)

where $J_{i}^{(\ell/r)}$ closes the algebra

$\commutator*{J_{i}^{(\ell/r)}}{J_{j}^{(\ell/r)}}=i\epsilon_{ijk}J_{k}^{(\ell/r)}\,,$
(118)

and their action on a generic operator $O$ is

$\commutator*{J_{i}^{(\ell)}}{O}=\mathcal{L}_{v_{i}^{\ell}}O\,,\qquad\commutator*{J_{i}^{(r)}}{O}=\mathcal{L}_{v_{i}^{r}}O\,,$
(119)

where $\mathcal{L}_{v_{i}^{\ell}}$ ($\mathcal{L}_{v_{i}^{r}}$) is the Lie
derivative w.r.t. by the left (right) invariant vectors fields of $S^{3}$

$\displaystyle v_{1}^{\ell}$
$\displaystyle=\frac{i}{2}\left(-\cos(\tau+\varphi)\partial_{\theta}-\tan(\theta)\sin(\tau+\varphi)\partial_{\tau}+\cot(\theta)\sin(\tau+\varphi)\partial_{\varphi}\right)\,,$
(120a) $\displaystyle v_{2}^{\ell}$
$\displaystyle=\frac{i}{2}\left(\sin(\tau+\varphi)\partial_{\theta}-\tan(\theta)\cos(\tau+\varphi)\partial_{\tau}+\cot(\theta)\cos(\tau+\varphi)\partial_{\varphi}\right)\,,$
(120b) $\displaystyle v_{3}^{\ell}$
$\displaystyle=\frac{i}{2}\left(\partial_{\tau}+\partial_{\varphi}\right)\,,$
(120c) $\displaystyle v_{1}^{r}$
$\displaystyle=\frac{i}{2}\left(\cos(\tau-\varphi)\partial_{\theta}+\tan(\theta)\sin(\tau-\varphi)\partial_{\tau}+\cot(\theta)\sin(\tau-\varphi)\partial_{\varphi}\right)\,,$
(120d) $\displaystyle v_{2}^{r}$
$\displaystyle=\frac{i}{2}\left(-\sin(\tau-\varphi)\partial_{\theta}-\tan(\theta)\cos(\tau-\varphi)\partial_{\tau}+\cot(\theta)\cos(\tau-\varphi)\partial_{\varphi}\right)\,,$
(120e) $\displaystyle v_{3}^{r}$
$\displaystyle=\frac{i}{2}\left(\partial_{\tau}-\partial_{\varphi}\right)\,.$
(120f)

The last ingredient to describe the algebra are two matrices
${h_{a}}^{b}\in\mathfrak{su}(2)_{H}$ and
${\bar{h}_{\dot{a}}}^{\;\;\dot{b}}\in\mathfrak{su}(2)_{C}$, which select a
Cartan of the $\mathcal{N}=4$ R-symmetry group as $U(1)_{H}\times
U(1)_{C}\subset SU(2)_{H}\times SU(2)_{C}$

$R_{H}=\frac{1}{2}{h_{a}}^{b}{R_{b}}^{a}\,,\qquad
R_{C}=\frac{1}{2}{\bar{h}_{\dot{a}}}^{\;\;\dot{b}}{\bar{R}_{\dot{b}}}^{\;\;\dot{a}}\,.$
(121)

To perform the construction in full generality, we introduce two vectors
$u_{\pm}^{a}$, $\bar{u}_{\pm}^{\dot{a}}$ in the R-symmetry space, defined by
the following conditions

$\displaystyle
u_{+}^{a}u_{-a}=\bar{u}_{+}^{\dot{a}}\bar{u}_{-\dot{a}}=1\,,\quad{h_{a}}^{b}=u_{+a}u_{-}^{b}+u_{-a}u_{+}^{b}\,,\quad{\bar{h}_{\dot{a}}}^{\;\;\dot{b}}=\bar{u}_{+\dot{a}}\bar{u}_{-}^{\dot{b}}+\bar{u}_{-\dot{a}}\bar{u}_{+}^{\dot{b}}\,.$
(122)

Together, they imply that

$u^{a}_{\pm}{h_{a}}^{b}=\pm
u^{b}_{\pm}\,,\qquad\bar{u}^{\dot{a}}_{\pm}{\bar{h}_{\dot{a}}}^{\;\;\dot{b}}=\pm\bar{u}^{\dot{b}}_{\pm}$
(123)

Then, our supercharges are given by

$Q^{(\ell\pm)}_{\alpha}=\frac{1+i}{2}u_{\pm}^{a}\bar{u}^{\dot{a}}_{\mp}\left(Q_{\alpha,a\dot{a}}-\frac{1}{2r}S_{\alpha,a\dot{a}}\right)\,,\quad
Q^{(r\pm)}_{\alpha}=\frac{1+i}{2}u_{\mp}^{a}\bar{u}^{\dot{a}}_{\mp}\left(Q_{\alpha,a\dot{a}}+\frac{1}{2r}S_{\alpha,a\dot{a}}\right)\,.$
(124)

We find the following algebra

$\displaystyle\anticommutator*{Q^{(\ell+)}_{\alpha}}{Q^{(\ell-)}_{\beta}}$
$\displaystyle=\frac{i}{r}\left[J_{\alpha\beta}^{(\ell)}+\frac{1}{2}\epsilon_{\alpha\beta}R_{\ell}\right]\,,$
(125) $\displaystyle\anticommutator*{Q^{(r+)}_{\alpha}}{Q^{(r-)}_{\beta}}$
$\displaystyle=\frac{i}{r}\left[J_{\alpha\beta}^{(r)}+\frac{1}{2}\epsilon_{\alpha\beta}R_{r}\right]\,,$
(126)

where we have defined

$R_{\ell}=R_{C}-R_{H}\,,\qquad R_{r}=R_{C}+R_{H}$ (127)

The even/odd commutators are

$\displaystyle\commutator*{R_{\ell}}{Q_{\alpha}^{(\ell\pm)}}$
$\displaystyle=\pm Q_{\alpha}^{(\ell\pm)}$
$\displaystyle\commutator*{R_{r}}{Q^{(r\pm)}_{\alpha}}$ $\displaystyle=\pm
Q^{(r\pm)}_{\alpha}$ (128)
$\displaystyle\commutator*{{{J^{(\ell)}_{\alpha}}^{\beta}}}{Q^{(\ell\pm)}_{\gamma}}$
$\displaystyle=\delta_{\gamma}^{\beta}Q^{(r\pm)}_{\alpha}-\frac{1}{2}\delta_{\alpha}^{\beta}Q^{(r\pm)}_{\gamma}$
$\displaystyle\commutator*{{{J^{(r)}_{\alpha}}^{\beta}}}{Q^{(\ell\pm)}_{\gamma}}$
$\displaystyle=\delta_{\gamma}^{\beta}Q^{(r\pm)}_{\alpha}-\frac{1}{2}\delta_{\alpha}^{\beta}Q^{(r\pm)}_{\gamma}$
(129)

All the other commutators are vanishing. Finally, to recover the choice
${h_{a}}^{b}=-{(\sigma_{2})_{a}}^{b}$,
${\bar{h}_{\dot{a}}}^{\;\;\dot{b}}=-{(\sigma_{3})_{\dot{a}}}^{\;\;\dot{b}}$ we
choose, like in Dedushenko:2016jxl , the following R-symmetry vectors

$u^{a}_{+}=\frac{1}{2}\begin{pmatrix}1-i\\\ 1+i\end{pmatrix}\,,\quad
u^{a}_{-}=\frac{1}{2}\begin{pmatrix}1-i\\\
-1-i\end{pmatrix}\,,\quad\bar{u}^{\dot{a}}_{+}=\begin{pmatrix}0\\\
1\end{pmatrix}\,,\quad\bar{u}^{\dot{a}}_{-}=\begin{pmatrix}1\\\
0\end{pmatrix}\,.$ (130)

These $\mathcal{N}=4$ algebras are realized by the transformations which
leaves invariant the actions introduced in Sec. 2.1. For the vector multiplet
we have

$\displaystyle\delta A_{\mu}$
$\displaystyle=\frac{i}{2}\xi^{a\dot{b}}\gamma_{\mu}\lambda_{a\dot{b}}\,,$
(131a) $\displaystyle\delta\lambda_{a\dot{b}}$
$\displaystyle=-\frac{i}{2}\epsilon^{\mu\nu\rho}\gamma_{\rho}\xi_{a\dot{b}}F_{\mu\nu}-{D_{a}}^{c}\xi_{c\dot{b}}-i\gamma^{\mu}{\xi_{a}}^{\dot{c}}\mathcal{D}_{\mu}\Phi_{\dot{c}\dot{b}}+2i{\Phi_{\dot{b}}}^{\dot{c}}\xi^{\prime}_{a\dot{c}}+$
(131b)
$\displaystyle+\frac{i}{2}\xi_{a\dot{d}}\commutator*{{\Phi_{\dot{b}}}^{\dot{c}}}{{\Phi_{\dot{c}}}^{\dot{d}}}\,,$
$\displaystyle\delta\Phi_{\dot{a}\dot{b}}$
$\displaystyle={\xi^{c}}_{(\dot{a}}\lambda_{|c|\dot{b})}\,,$ (131c)
$\displaystyle\delta D_{ab}$
$\displaystyle=-i\mathcal{D}_{\mu}({\xi_{(a}}^{\dot{c}}\gamma^{\mu}\lambda_{b)\dot{c}})-2i{\xi^{\prime}_{(a}}^{\dot{c}}\lambda_{b)\dot{c}}+i\commutator*{{\xi_{(a}}^{\dot{c}}\lambda_{b)}^{\dot{d}}}{\Phi_{\dot{c}\dot{d}}}\,.$
(131d)

For the hypermultiplet, we have

$\displaystyle\delta q^{a}$ $\displaystyle=\xi^{a\dot{b}}\psi_{\dot{b}}\,,$
$\displaystyle\delta_{\xi}\psi_{\dot{a}}$
$\displaystyle=i\gamma_{\mu}\xi_{a\dot{a}}\mathcal{D}_{\mu}q^{a}+i\xi^{\prime}_{a\dot{a}}q^{a}-i\xi_{a\dot{c}}\Phi\indices{{}^{\dot{c}}_{\dot{a}}}q^{a}\,,$
(132a) $\displaystyle\delta\tilde{q}^{a}$
$\displaystyle=\xi^{a\dot{b}}\tilde{\psi}_{\dot{b}}\,,$
$\displaystyle\delta_{\xi}\tilde{\psi}_{\dot{a}}$
$\displaystyle=i\gamma_{\mu}\xi_{a\dot{a}}\mathcal{D}_{\mu}\tilde{q}^{a}+i\xi^{\prime}_{a\dot{a}}\tilde{q}^{a}-i\xi_{a\dot{c}}\Phi\indices{{}^{\dot{c}}_{\dot{a}}}\tilde{q}^{a}\,.$
(132b)

We take $\xi_{a\dot{b}}$ to be conformal Killing spinors

$\nabla_{\mu}\xi_{a\dot{a}}=\gamma_{\mu}\xi^{\prime}_{a\dot{a}}\,,\qquad\nabla_{\mu}\xi^{\prime}_{a\dot{a}}=-\frac{1}{4r^{2}}\gamma_{\mu}\xi^{\prime}_{a\dot{a}}$
(133)

On $S^{3}$, in stereographic coordinates $x^{i}$ with $i=1,2,3$ and in the
stereographic frame 161616 The stereographic frame is defined as
$e^{i}_{\mu}=e^{\Omega}\,\delta_{\mu}^{i}$, with
$e^{\Omega}=\frac{1}{1+\frac{x_{i}^{2}}{4r^{2}}}=\frac{1+\sin\theta\cos\varphi}{2}$
being the conformal factor. We refer to the App. A of Dedushenko:2016jxl  for
details), the explicit solution is

$\displaystyle\xi_{a\dot{a}}$
$\displaystyle=e^{\Omega/2}\left(\theta_{a\dot{a}}-x^{i}\gamma_{i}\eta_{a\dot{a}}\right)$
(134a) $\displaystyle\xi_{a\dot{a}}$
$\displaystyle=e^{\Omega/2}\left(\eta_{a\dot{a}}+\frac{1}{4r^{2}}x^{i}\gamma_{i}\theta_{a\dot{a}}\right)$
(134b)

In the flat space limit $r\to\infty$, we recover the familiar form of flat
space conformal Killing spinor of Eq. (81)

$\xi_{a\dot{a}}=\theta_{a\dot{a}}-x_{i}\gamma_{i}\eta_{a\dot{a}}\,,\qquad\xi^{\prime}_{a\dot{a}}=-\eta_{a\dot{a}}$
(135)

The Killing spinors associated with the
$\mathfrak{su}(2|1)_{\ell}\oplus\mathfrak{su}(2|1)_{r}$ superalgebra on
$S^{3}$ satisfy the additional constraint

$\xi_{a\dot{a}}=-\frac{i}{2r}{h_{a}}^{b}\,\xi^{\prime}_{b\dot{b}}\,\bar{h}_{\dot{a}}^{\;\;\dot{b}}$
(136)

We can recover the explicit action of the bosonic generators from the
following definition

$\delta\Phi=\commutator*{\theta^{a\dot{a}}Q_{a\dot{a}}-i\eta^{a\dot{a}}S_{a\dot{a}}}{\Phi}$
(137)

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