aainstitutetext: Dipartimento di Fisica, Università degli Studi di
Milano–Bicocca and INFN, Sezione di Milano–Bicocca, Piazza della Scienza 3,
20126 Milano, Italybbinstitutetext: Dipartimento di Scienze Fisiche,
Informatiche e Matematiche, Università di Modena e Reggio Emilia, via G. Campi
213/A, 41125 Modena, Italyccinstitutetext: INFN Sezione di Bologna, via
Irnerio 46, 40126 Bologna, Italyddinstitutetext: Institute of Physics,
University of São Paulo, 05314-970 São Paulo, Brazil

# Wilson loops and defect RG flows in ABJM

Luigi Castiglioni a Silvia Penati a Marcia Tenser b,c,d Diego Trancanelli
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We continue our study of renormalization group (RG) flows on Wilson loop
defects in ABJM theory, which we have initiated in arXiv:2211.16501. We
generalize that analysis by including non-supersymmetric fixed points and RG
trajectories. To this end, we first determine the “ordinary”, non-
supersymmetric Wilson loops, which turn out to be two and to include an
R-symmetry preserving coupling to the scalar fields of the theory, contrary to
their four-dimensional counterpart defined solely in terms of the gauge field
holonomy. We then deform these operators by turning on bosonic and/or
fermionic couplings, which trigger an elaborate, multi-dimensional network of
possible RG trajectories connecting a large spectrum of fixed points
classified in terms of the amount (possibly zero) of supersymmetry and
R-symmetry preserved. The $\beta$-functions are computed to leading order in
the ABJM coupling but exactly in the deformation parameters, using an
auxiliary one-dimensional theory on the defect and a dimensional
regularization scheme. A striking result is the different behavior of the two
ordinary Wilson loops, of which one turns out to be a UV unstable point while
the other is IR stable. The same is true for the two 1/2 BPS Wilson loops. We
interpret our results from a defect CFT (dCFT) point of view, computing the
anomalous dimensions of the operators associated to the deformations and
establishing appropriate g-theorems. In particular, the fermionic unstable
fixed point is associated to a dCFT which is not reflection positive.

###### Keywords:

Chern-Simons theories, Wilson, ’t Hooft and Polyakov loops, Renormalization
Group

##  1 Introduction

Three-dimensional supersymmetric Chern-Simons-matter theories, like ABJ(M)
Aharony:2008ug ; Aharony:2008gk  and other ${\mathcal{N}}\geq 2$ quiver
theories Gaiotto:2008sd ; Imamura:2008dt ; Hosomichi:2008jd ; Hama:2010av ,
have a rich moduli space of BPS Wilson loops discovered along the years in,
for example, Drukker:2008zx ; Chen:2008bp ; Kluson:2008zrv ; Rey:2008bh ;
Drukker:2009hy ; Gaiotto:2007qi ; Ouyang_2015 ; Cooke_2015 ; Ouyang:2015iza ;
Ouyang:2015bmy ; Mauri:2017whf ; Mauri_2018 ; drukker2020bps ; Drukker:2020dvr
; Drukker:2022ywj ; Drukker:2022bff ; Castiglioni:2022yes .111See
Drukker:2019bev  for a fairly recent review. Such Wilson loops typically come
in families related by a certain number of parameters, that allow to
interpolate continuously among representatives preserving varying amounts of
supercharges of the theory.

These interpolations can be studied from the point of view of RG flows on
defects, as initiated recently in Castiglioni:2022yes  for ABJM, building on
similar analyses done in four dimensions in Polchinski_2011  and subsequent
literature, see for example Beccaria:2017rbe ; Beccaria:2018ocq ;
Correa:2019rdk ; Cuomo:2021rkm ; Beccaria:2021rmj ; Beccaria:2022bcr ;
Garay:2022szq ; Aharony:2022ntz . The main idea is to start from a certain
loop operator – a UV fixed point – and deform it with a marginally relevant
parameter, that either leads to another loop operator – an IR fixed point – or
to infinity along runaway directions.

The most studied and best understood example is the interpolation, initially
proposed in Polchinski_2011 , between the non-BPS Wilson loop of four-
dimensional ${\cal N}=4$ super Yang-Mills (SYM) and the 1/2 BPS operator of
that theory. This interpolation is controlled by a single parameter $\zeta$
that acts as a marginally relevant deformation of the non-BPS operator defined
only in terms of the gauge field (for $\zeta=0$) and triggers a flow toward
the supersymmetric operator Maldacena_1998  which is also coupled to a scalar
of the theory (for $\zeta=1$).

Performing a similar investigation in the ABJM theory, one expects to find a
much richer spectrum of flows, given the larger number of Wilson loops that
can be defined. In Castiglioni:2022yes  our focus has been on interpolations
and RG trajectories associated to operators in ABJM that always preserve some
amount of supersymmetry, at least one supercharge, something we dubbed as
‘enriched flows’. Here we continue that analysis by including also fixed
points and RG trajectories corresponding to non-supersymmetric operators, in
the original spirit of Polchinski_2011 .

The first step consists in identifying what are such non-BPS operators in the
case of ABJM. Differently from the four-dimensional counterpart, these
operators are not just defined in terms of the gauge fields, but they also
include an R-symmetry preserving coupling to the scalars. This can be
justified as follows. First of all, scalar bilinears have classical dimension
1 in three dimensions and allow for the contraction of the R-symmetry indices
to produce the $SU(4)$ singlet operator $C_{I}\bar{C}^{I}$ (with
$I=1,\ldots,4$). It follows that when we perform the renormalization of the
gauge field on the defect theory, nothing prevents it from mixing with the
singlet. In fact, one discovers that, due to the interaction with the bulk
theory, this new effective vertex is produced. It is divergent and can only be
cancelled if the defect theory defined by the Wilson loop includes a coupling
to the scalars as the one above. It turns out there are two such operators,
which we call $W^{\pm}$, corresponding to the two possible signs in front of
$C_{I}\bar{C}^{I}$. We demonstrate this in section 2.222Similar results were
found in Gabai:2022vri ; Gabai:2022mya , where a classification of line
operators in Chern-Simons theories with matter was obtained.

Having identified the non-BPS, or “ordinary”, Wilson loops we then proceed to
deforming them in several ways and in computing the associated
$\beta$-functions at leading order in the ABJM coupling and in the planar
limit. We start in section 3 with deforming the scalar couplings only. We
restrict these deformations to be diagonal, thus generally given by four
independent parameters that allow to either break (completely or in part) the
R-symmetry group or to preserve it. We obtain the $\beta$-functions associated
to these deformations using the same one-dimensional effective approach
originally developed in Samuel:1978iy ; Gervais:1979fv  for QCD and then
extended to ABJM in Castiglioni:2022yes .

As is well-known, BPS Wilson loops in three-dimensional quiver theories are
defined in terms of superconnections including couplings to the fermions of
the theory. It is then natural to also consider fermionic deformations, which
we do in section 4 for purely fermionic deformations and in section 5 for more
generic deformations of both fermions and scalars.

This whole setup allows for a very rich, multi-dimensional space of possible
RG flows, which we depict in a series of plots. In these plots we denote fixed
points according to their R-symmetry structure: $SU(4)$ invariants as squares,
$SU(3)$ as triangles and $SU(2)$ as circles. When they are black no
supersymmetry is preserved, whereas when they are colored they preserve some
amount of supersymmetry: blue points are $1/6$ BPS and red points are $1/2$
BPS. There are multiple facets in these series of plots and we leave a
detailed presentation for the body of the text.

Nevertheless, in figure 1 we show a glimpse of a schematic representation of
some of our results. Along what we depict as a horizontal line we find bosonic
flows. In particular, connecting the two $SU(4)$ symmetric “ordinary”
operators $W^{-}$ and $W^{+}$ at the end points ( ) we have two $SU(3)$ non-
BPS bosonic loops ( ) and/or a bosonic $1/6$ BPS operators ( ), which is
$SU(2)\times SU(2)$ symmetric. Along the vertical direction we represent
fermionic flows and among these we find in particular two $1/2$ BPS fixed
points ( ). In this representation, arrows go from the UV to the IR. We hope
the reader carries this general picture in mind throughout the paper.

One of the most interesting outputs of this web of flows is the qualitatively
different behavior between $W^{+}$ and $W^{-}$, and between $W_{1/2}^{+}$ and
$W_{1/2}^{-}$. Though classically they are equivalent (they simply differ by
harmless signs in the couplings, which do not affect symmetries), at quantum
level they describe completely different defect theories. In fact, while
$W^{-}$ is a UV unstable fixed point, its counterpart $W^{+}$ is IR stable.
Similarly, including fermionic deformations we find that UV instability lies
at $W_{1/2}^{-}$ while IR stability corresponds to $W_{1/2}^{+}$.

Figure 1: Representation of flows connecting different Wilson loop operators
in ABJM theory. Along the horizontal line we have purely bosonic deformations
whereas in the vertical direction we turn on fermions. Mixed deformations
correspond to compositions of these two.

After unravelling this multi-dimensional space of possible flows, in section 6
we interpret our results from the point of view of a defect CFT. First of all,
computing anomalous dimensions at the first non-trivial order, we find that
the perturbations are marginally relevant operators for the fermionic $W^{-}$
and $W^{-}_{1/2}$ defects, consistently with the direction of the flows. Then,
through the evaluation of the defect entropy Affleck:1991tk , we manage to
test the validity of a g-theorem Affleck:1991tk ; Friedan:2003yc ;
Casini:2016fgb ; Cuomo:2021rkm ; Casini:2022bsu .

A striking result arises, which concerns the g-theorem along the flows
connecting $W^{\pm}_{1/2}$ defect theories with non-BPS bosonic $SU(3)$
invariant defects. We find that, while $W^{+}_{1/2}$ satisfies reflection-
positivity and the g-theorem Cuomo:2021rkm  holds, the $W^{-}_{1/2}$ defect
does not because of a crucial sign change in the two-point function of its
stress-tensor. As a consequence, the RG flow is still irreversible, but the
defect entropy is increasing rather than decreasing from the UV unstable to
the IR stable fixed points. However, this seems to be consistent with the dual
description of defects at strong coupling and the Higgsing procedure used to
construct these two operators Lietti:2017gtc . In fact, in field theory they
arise by Higgsing either with particle or antiparticle modes, while in
M-theory $W^{+}_{1/2}$ is dual to a M2-brane configuration and $W^{-}_{1/2}$
is dual to an anti-M2 brane. This might explain why one defect is reflection-
positive, whereas the other one is not.

We conclude with section 7, where we further discuss our findings and address
a few relevant open questions. The technical details of our calculations are
reported in four appendices.

As a final remark, we stress that the same caveats of Castiglioni:2022yes
regarding the employed regularization scheme also apply here. All our
computations are in fact performed in dimensional regularization, which is
alternative to introducing framing and can then be thought as giving results
which correspond to framing zero. Restricting to BPS Wilson loops, this is of
course not what one obtains from localization, which works at framing one.
However, it is well-known that the results from the two different schemes
differ simply by an overall phase, the famous framing factor Witten:1988hf ;
Guadagnini:1989am ; Alvarez:1991sx  or framing function Bianchi:2016yzj .
Since dimensional regularization breaks conformal invariance, while
localization requires and preserves superconformal invariance Pestun_2012 ;
Kapustin:2009kz , the framing function has the interpretation of a conformal
(or framing) anomaly. It is ultimately responsible for the BPS RG flows that
we find at framing zero, thus we conclude that enriched flows are anomaly
driven. In the more general case of non-BPS flows, the absence of
cohomological equivalence does not guarantee that this is the case.

##  2 Ordinary Wilson loops in ABJM

Following what has been done in ${\mathcal{N}}=4$ SYM theory, see for instance
Polchinski_2011 ; Beccaria:2017rbe ; Beccaria:2018ocq ; Beccaria:2019dws ;
Beccaria:2021rmj ; Beccaria:2022bcr , here we want to study how to generalize
to ABJM theory the idea of interpolating between the non-supersymmetric Wilson
loop (WL) given by the holonomy of the gauge field (usually referred to as
“ordinary”) and BPS loops that include extra couplings to the matter fields of
the theory333A brief summary of the ABJM Wilson loops entering our study is
given in appendix B. We will always refer to this appendix for details on
their explicit realization..

In the ${\mathcal{N}}=4$ SYM case, the interpolation takes the form
Polchinski_2011

$W^{(\zeta)}(C)=\frac{1}{N}\operatorname{Tr}{\mathcal{P}}\exp\oint_{C}d\tau\left[\,i{A}_{\mu}(x)\dot{x}^{\mu}+\zeta\Phi_{m}(x)\theta^{m}|\dot{x}|\,\right],\qquad\theta^{m}\theta^{m}=1,$
(2.1)

where $m$ runs over the six scalar fields of the theory.

This is a one-parameter family of Wilson loops that connects the ordinary WL
at $\zeta=0$ and the 1/2 BPS Wilson-Maldacena loop Maldacena_1998  at
$\zeta=1$. Since $\langle W^{(\zeta)}(C)\rangle$ is invariant under
$\zeta\to-\zeta$, one can restrict to $\zeta\geq 0$.

In Polchinski_2011  it was found perturbatively at first order in the ’t Hooft
coupling $\lambda$ that a non-trivial RG flow connecting these two operators
exists, which is dictated by the following $\beta$-function

$\beta_{\zeta}=\mu\frac{d\zeta}{d\mu}=-\frac{\lambda}{8\pi^{2}}\zeta(1-\zeta^{2})+{\mathcal{O}}(\lambda^{2}).$
(2.2)

Around the $\zeta=0$ UV fixed point the scalar perturbation
$\Phi_{m}\theta^{m}$ becomes marginally relevant and drives the system towards
the 1/2 BPS IR fixed point at $\zeta=1$.

We note that within the one-parameter family (2.1) the ordinary Wilson loop
corresponds to the operator preserving the maximal amount of R-symmetry, that
is $SO(6)$. The $\zeta$-deformation breaks $SO(6)$ to $SO(5)$. The Polchinski-
Sully flow can then be interpreted as connecting the maximally R-symmetric
operator to the maximally supersymmetric one.

In order to investigate the existence of an analogous pattern in ABJM theory,
we focus on the study of parametric deformations that connect non-BPS with BPS
Wilson loops. The logic that we are going to apply is the following: We start
with the maximally R-symmetric operator, that is the “ordinary” WL preserving
the maximal amount of R-symmetry, we then add marginally relevant, partial
R-symmetry breaking deformations and study the corresponding RG flows.

Naively, one would expect the “ordinary” WL in the ABJM theory to be the same
as for ${\cal N}=4$ SYM, that is the gauge field holonomy given by (2.1) with
$\zeta=0$. In fact, $A_{\mu}\dot{x}^{\mu}$ is invariant under the action of
the full R-symmetry group, $SO(6)\simeq SU(4)$. However, this is not correct,
since in this case there are other R-symmetry preserving operators with the
same dimension, which necessarily mix with $A_{\mu}\dot{x}^{\mu}$ and need to
be included.

In order to prove that, we make use of the auxiliary one-dimensional method
originally proposed for QCD in Dorn:1986dt ; Samuel:1978iy ; Gervais:1979fv ;
Arefeva:1980zd ; CRAIGIE1981204  and suitably generalized to the ABJM case in
Castiglioni:2022yes . As we are going to show, in this approach the appearance
of further R-symmetry preserving operators is signaled by the fact that the
$A_{\mu}\dot{x}^{\mu}$ operator itself leads to a non-renormalizable one-
dimensional auxiliary theory.

As done in Castiglioni:2022yes  (see also Dorn:1986dt ; CRAIGIE1981204 ), we
take the loop to be supported along the circular curve parametrized as

$x^{\mu}(\tau)=(0,\cos\tau,\sin\tau)\,,\quad\tau\in[0,2\pi)\,.$ (2.3)

For the ordinary Wilson loop we introduce the one-dimensional fermionic field
$z$($\bar{z}$) in the (anti)fundamental representation of $U(N)$ and define
the Wilson loop vacuum expectation value (VEV) on a contour
${\mathcal{C}}_{12}$ connecting two points $\tau_{1},\tau_{2}$ as444This
prescription is briefly reviewed in appendix C. It is easy to realize that for
the ordinary Wilson loop the $\Psi$ auxiliary fermion defined in (C.2) boils
down to $z$.

$\langle W\rangle=\langle z(\tau_{2})\bar{z}(\tau_{1})\rangle\,,$ (2.4)

where the correlation function is computed with the effective action

$S_{eff}=S_{ABJM}+\int
d\tau\operatorname{Tr}(\bar{z}{\mathcal{D}}_{\tau}z)\,.$ (2.5)

Here, the $\tau$-integration is along the Wilson loop contour and the
covariant derivative is defined as
${\mathcal{D}}_{\tau}=\partial_{\tau}+i{\cal L}(\tau)$, where ${\cal L}(\tau)$
is the connection of the Wilson operator under investigation. Since we are
interested in studying the ordinary Wilson loop, we set
${\mathcal{D}}_{\tau}=\partial_{\tau}+iA_{\mu}\dot{x}^{\mu}$.

Figure 2: The $\bar{z}C_{I}\bar{C}^{I}z$ vertex arising at one-loop. Dashed
lines correspond to ABJM scalars while double blue straight lines corresponds
to the one-dimensional fermion $z$. Wavy lines are gauge fields.

The renormalization properties of this operator can be studied by looking at
the renormalization of the one-dimensional QFT defined by the action in (2.5).
As shown in Castiglioni:2022yes , using the Feynman rules in appendix C, at
one loop the $\bar{z}\dot{z}$ kinetic term and the
$\bar{z}A_{\mu}\dot{x}^{\mu}z$ vertex do not renormalize. Instead, a divergent
term arises from the diagram in figure 2, which corresponds to a new
$\bar{z}C_{I}\bar{C}^{I}z$ vertex, not present in the original action (2.5).
Its explicit expression is given by

$\begin{split}\Gamma^{(\ref{fig:diagram1})}&=\int d\tau_{1}\int\tau_{2}\int
d^{d}x\,\big{(}i\bar{z}A_{\mu}\dot{x}^{\mu}z\big{)}(x_{1})\big{(}i\bar{z}A_{\nu}\dot{x}^{\nu}z\big{)}(x_{2})\big{(}A_{\rho}C_{I}\bar{C}^{I}A^{\rho}\big{)}(x)\\\
&=\frac{g^{2}N}{8\pi\epsilon}\int
d\tau\,\bar{z}C_{I}\bar{C}^{I}z\,.\end{split}$ (2.6)

The appearance of this UV divergent term spoils renormalizability of the one-
dimensional auxiliary theory, unless we include such a new term already at the
classical level. This amounts to modifying the covariant derivative in (2.5)
as
${\mathcal{D}}_{\tau}=\partial_{\tau}+i\left(A_{\mu}\dot{x}^{\mu}\mp\tfrac{2\pi
i}{k}|\dot{x}|C_{I}\bar{C}^{I}\right)$, which in turn amounts to stating that
the correct WLs to consider are

$W^{\pm}=\frac{1}{N}\operatorname{Tr}{\mathcal{P}}\exp\left[-i\oint
d\tau\left(A_{\mu}\dot{x}^{\mu}-\tfrac{2\pi
i}{k}|\dot{x}|(\pm\delta_{I}^{J})C_{J}\bar{C}^{I}\right)\right]\,.$ (2.7)

In principle, there are no restrictions on the choice of the sign in front of
$\delta_{I}^{J}$. We have denoted the two possible options as $W^{\pm}$. Since
there is no field redefinition in the path integral that can compensate for
this sign, $W^{+}$ and $W^{-}$ are genuine different operators. In the next
sections we will further discuss the implications associated with the choice
of this sign.

The scalar couplings in (2.7) preserve the whole $SU(4)$ R-symmetry group,
thus the two $W^{\pm}$ operators also do. This implies that in the ABJM theory
these are the “ordinary” maximally R-symmetric WLs.

This result is simply the manifestation of the general property according to
which any operator preserving the symmetries of the conformal theory at the
fixed point gets turned on along the RG flow. We emphasize that in the present
case this is triggered by a bulk-defect interaction. In fact, the ABJM vertex
$A_{\mu}C_{I}\bar{C}^{I}A^{\mu}$ plays a central role in constructing the
diagram of figure 2. As already mentioned, a similar pattern does not arise in
${\mathcal{N}}=4$ SYM where, for dimensional reasons, any scalar coupling in
the WL must be linear, thus necessarily breaking $SO(6)$.

Finally, we mention that $W^{\pm}$ does not preserve any supersymmetry. In
fact, supersymmetry invariance requires the scalar coupling matrix $M$ of
bosonic loops (see its definition in appendix B) to be traceless
Drukker:2022bff . This is clearly not the case for these operators, for which
the scalar coupling matrix is simply $M=\pm{\mathbb{1}}$.

##  3 Bosonic flows

We now begin the study of RG flows which involve fixed points corresponding to
the ordinary Wilson loops defined above. We choose to perturb around $W^{-}$
because, as it will become evident, this corresponds to a UV unstable fixed
point. As the simplest case, we consider perturbing with a bosonic operator.

Referring to (2.7), we consider a four-parameter deformation of $W^{-}$

$W(\zeta)=\frac{1}{N}\operatorname{Tr}{\mathcal{P}}\exp\bigg{[}-i\oint
d\tau\left(A_{\mu}\dot{x}^{\mu}+\frac{2\pi i}{k}C_{I}\bar{C}^{I}-\frac{2\pi
i}{k}\Delta M_{I}^{\ J}(\zeta_{i})C_{J}\bar{C}^{I}\right)\bigg{]},$ (3.1)

where

$\Delta M(\zeta_{i})=2\begin{pmatrix}\zeta_{1}&0&0&0\\\ 0&\zeta_{2}&0&0\\\
0&0&\zeta_{3}&0\\\ 0&0&0&\zeta_{4}\end{pmatrix}\,.$ (3.2)

For generic values of the parameters this perturbation breaks the $SU(4)$
R-symmetry completely.

According to the calculation presented in appendix C, at one loop the
$\beta$-functions parameterising this flow are given by

$\beta_{\zeta_{i}}=\frac{g^{2}N}{2\pi}\zeta_{i}(\zeta_{i}-1)\,,\qquad
i=1,2,3,4,$ (3.3)

where $g^{2}=2\pi/k$. Since the RG equations are decoupled, for the running
coupling constants $\zeta_{i}$ we find

$\zeta_{i}(\mu)=\frac{1}{1+e^{c_{i}}\mu^{\frac{g^{2}N}{2\pi}}}\,,\qquad
i=1,2,3,4,$ (3.4)

where $c_{i}$ are arbitrary real constants. The fixed points can be classified
and denoted as follows:

  1. $SU(4)$ invariant fixed points. The origin $\zeta_{i}=0,\ i=1,2,3,4,$ trivially corresponds to the ordinary $W^{-}$ Wilson loop. Similarly, the point $\zeta_{i}=1,\ i=1,2,3,4,$ corresponds to the ordinary $W^{+}$ operator. At these two fixed points $SU(4)$ R-symmetry is restored.

  2. $SU(3)$ invariant fixed points. A first set is obtained by setting one $\zeta_{i}$ equal to zero and the others equal to one. They correspond to four equivalent Wilson loops with $M={\text{diag}}(-1,1,1,1)$ and permutations, which exhibit a $SU(3)$ R-symmetry invariance and no supersymmetry. A second set is obtained by setting three $\zeta_{i}$ equal to zero and one equal to one. In this case the corresponding scalar matrix is $M={\text{diag}}(1,-1,-1,-1)$ and permutations. Again, the WL is $SU(3)$ invariant, but no supersymmetry is preserved.

  3. $SU(2)\times SU(2)$ invariant fixed points. These are obtained by setting two $\zeta_{i}$ equal to zero and two equal to one. They correspond to six equivalent 1/6 BPS bosonic Wilson loops defined in (B.1).

We can study the RG flows among these fixed points by looking at line, surface
and volume projections of the four-dimensional parameter space. In all our
pictures arrows go from the UV to the IR.

To begin with, we consider turning on only the $\zeta_{1}$ deformation. This
leads to the one-dimensional RG flow depicted in figure 3. Already in this
one-dimensional projection the UV instability of the $W^{-}$ operator emerges
clearly. Instead, the point $\zeta_{1}=1$ corresponding to one of the $SU(3)$
invariant fixed points appears as attractive along the line.

Figure 3: The RG flow along the $\zeta_{1}$-line. Arrows go from the UV to the
IR. The black square represents $W^{-}$, whereas the triangle corresponds to a
$SU(3)$ invariant bosonic Wilson loop.

We now move to a two-dimensional section of the parameter space by turning on
also the $\zeta_{2}$ perturbation. The corresponding flows are presented in
figure 4. We see that under the $\zeta_{2}$ perturbation the $SU(3)$ fixed
point corresponding to the $(1,0)$ point becomes unstable and the RG flow
drives the system towards the point $(1,1)$ marked in blue. This is the $1/6$
BPS bosonic loop corresponding to $M={\text{diag}}(1,1,-1,-1)$.

Figure 4: The RG flow in the $(\zeta_{1},\zeta_{2})$ plane corresponding to
$\zeta_{3}=\zeta_{4}=0$. The blue dot is the bosonic $1/6$ BPS Wilson loop,
while the two black triangles correspond to bosonic $SU(3)$ invariant Wilson
loops.

Going one step further, we turn on for instance $\zeta_{3}$, still keeping
$\zeta_{4}=0$. The resulting RG flows are now depicted in three dimensions,
see figure 5. We find that as soon as we leave the $\zeta_{3}=0$ plane, the
system is driven towards the $SU(3)$ invariant $(1,1,1)$ point. Moreover, in
addition to the $1/6$ BPS operator at $(1,1,0)$ already marked in figure 4,
two other $1/6$ BPS bosonic loops appear, which correspond to the blue dots at
$(1,0,1)$ and $(0,1,1)$. They are simply R-symmetry rotations of each other.
The pattern of the flows in figure 4 is nothing but the projection of the
three-dimensional flow on one of the three planes containing the origin.

Figure 5: The RG flow in the $(\zeta_{1},\zeta_{2},\zeta_{3})$ space, with
$\zeta_{4}=0$. Blue spheres describe 1/6 BPS bosonic Wilson loops, the black
pyramids correspond to $SU(3)$ invariant operators.

Finally, we turn on also the $\zeta_{4}$ deformation, thus disclosing the
whole spectrum of fixed points. In particular, the ordinary Wilson loop
$W^{+}$ now appears, which turns out to be an IR stable fixed point in any
direction. The other 14 fixed points, eight $SU(3)$ invariant bosonic WLs and
six $1/6$ BPS bosonic operators, are all saddle points and correspond to the
vertices of a four-dimensional hypercube with side 1.

We have considered only the portion of the parameter space within the interval
$[0,1]$, because this should correspond to the isolated invariant set
Gukov:2016tnp ; 10.2307/1995770  of the RG space. Had we considered values
outside this interval, we would flow to infinity along runaway directions.
This is analogue to what has been found in ${\cal N}=4$ SYM Polchinski_2011 .

###  $SU(2)\times SU(2)$ flows

One interesting subset of deformations corresponds to $SU(2)\times SU(2)$
preserving $\Delta M$ matrices. This necessarily restricts the spectrum of
fixed points to a subset of $SU(2)\times SU(2)$ invariant CFTs.

To this end we consider perturbing $W^{-}$ as in (3.1) with

$\Delta M(\zeta_{1},\zeta_{2})=2\begin{pmatrix}\zeta_{1}&0&0&0\\\
0&\zeta_{1}&0&0\\\ 0&0&\zeta_{2}&0\\\ 0&0&0&\zeta_{2}\end{pmatrix}\,.$ (3.5)

Such a deformation gives rise to a two-parameter family of interpolating
Wilson loops. The one-loop $\beta$-functions and their solutions can be easily
read from (3.3) and (3.4) and the corresponding flows are depicted in figure
6. They connect $W^{-}$ (for $\zeta_{1}=\zeta_{2}=0$), $W^{+}$ (for
$\zeta_{1}=\zeta_{2}=1$) and the bosonic BPS Wilson loop $W_{1/6}$ (for
$\zeta_{1}=1,\,\zeta_{2}=0$ or $\zeta_{1}=0,\,\zeta_{2}=1$). The
$W(\zeta_{1},\zeta_{2})$ operators possess a $\mathbb{Z}_{2}$ symmetry under
the $\zeta_{1}\leftrightarrow\zeta_{2}$ exchange, which comes from the freedom
to exchange the two $SU(2)$ factors in the R-symmetry group. This symmetry is
also manifest in the behavior of the RG trajectories.

Figure 6: RG flows in the $(\zeta_{1},\zeta_{2})$ plane. The blue dots
correspond to the $SU(2)\times SU(2)$ $1/6$ BPS bosonic loops fixed points,
while the black squares correspond to the ordinary $SU(4)$ invariant Wilson
loops. The white square is the Wilson loop with purely gauge connection
$A_{\mu}\dot{x}^{\mu}$.

We note that although the deformation is not the same and thus fixed points
are different, the flows in figure 4 and 6 are equal. In fact, they both share
the same behavior as the RG flow of the $O(N)$ model (see for instance figures
4 and 7 of Gukov:2016tnp ).

###  $SU(4)$ flows

A further interesting subset of flows is the one induced by a single parameter
deformation $\Delta M_{I}^{\ J}(\zeta)=2\zeta\,\delta_{I}^{J}$. In the same
spirit as ${\mathcal{N}}=4$ SYM, this deformation produces a one-parameter
family of interpolating Wilson loops, which connects $W^{-}$ for $\zeta=0$ and
$W^{+}$ for $\zeta=1$.

Figure 7: The RG flow along the $\zeta$-line. The black squares represent
$W^{-}$ and $W^{+}$ at $\zeta=0$ and $\zeta=1$, respectively. The white square
at $\zeta=1/2$ represents the Wilson loop with connection
$A_{\mu}\dot{x}^{\mu}$.

As is evident from figure 7, the ordinary Wilson loop $W^{-}$ corresponding to
a negative scalar coupling $M=-{\mathbb{1}}$ is a UV unstable fixed point. On
the other hand, the ordinary loop $W^{+}$ with $M=+{\mathbb{1}}$ is an IR
stable point. We note that the naive analogue of the ordinary Wilson loop in
${\mathcal{N}}=4$ SYM, i.e. the Wilson loop corresponding to a pure gauge
connection $A_{\mu}\dot{x}^{\mu}$, is located at point $\zeta=1/2$.
Consistently with our previous findings, it does not correspond to any fixed
point, rather it belongs to the RG straight line that connects the two “true”
ordinary WLs, $W^{-}$ and $W^{+}$.

The main conclusion we can draw from the present investigation is that at
quantum level the $W^{+}$ and $W^{-}$ operators exhibit a very different
behavior under renormalization, though classically they simply differ by the
overall sign in front of the scalar coupling. More evidence on their deep
different nature will emerge in the next sections.

##  4 Fermionic flows

In three-dimensional Chern-Simons-matter theories - ABJM theory included - a
more general class of Wilson loops can be considered, which involves coupling
to fermions Drukker:2009hy ; Cardinali:2012ru . In this case the connection
appearing in the exponent of the Wilson operator is promoted to a supermatrix
${\mathcal{L}}$ where, in addition to gauge fields and scalar bilinears in the
diagonal elements, fermi fields appear linearly in the off-diagonal
entries555For longer quiver gauge theories, this construction can be
generalized to build loops which couple to more than two nodes. See for
instance Mauri_2017 ; Drukker:2020dvr ; Drukker:2022txy ..

In the same spirit as in the previous section, we can study flows driven by
adding fermionic deformations. As a first case, here we will keep the diagonal
structure of ${\mathcal{L}}$ fixed, i.e. gauge fields and scalar bilinears
with a fixed structure, and discuss possible flows obtained by simply adding
fermionic fields to the off-diagonal entries of ${\mathcal{L}}$.

We start with a composite non-BPS bosonic loop with superconnection

${\mathcal{L}}=\begin{pmatrix}A_{\mu}\dot{x}^{\mu}-\frac{2\pi
i}{k}|\dot{x}|M_{I}^{\ J}C_{J}\bar{C}^{I}&0\\\
0&\hat{A}_{\mu}\dot{x}^{\mu}-\frac{2\pi i}{k}|\dot{x}|M_{I}^{\
J}\bar{C}^{I}C_{J}\end{pmatrix},$ (4.1)

where $A,\hat{A}$ are charged under the two nodes of the ABJM theory and $M$
is a fixed - not better specified - scalar coupling matrix. We take the
fermionic perturbation to be of the following form

$\Delta{\mathcal{L}}_{F}=-ig\begin{pmatrix}0&\eta\left(\chi_{1}\bar{\psi}^{1}+\chi_{2}\bar{\psi}^{2}+\chi_{3}\bar{\psi}^{3}+\chi_{4}\bar{\psi}^{4}\right)\\\
\left(\chi_{1}\psi_{1}+\chi_{2}\psi_{2}+\chi_{3}\psi_{3}+\chi_{4}\psi_{4}\right)\bar{\eta}&0\end{pmatrix}\,,$
(4.2)

where $\chi_{i},\ i=1,2,3,4$, are four arbitrary real parameters, the bosonic
spinorial couplings $\eta^{\alpha},\bar{\eta}_{\alpha}$ on the circle are
defined as in Cardinali:2012ru

$\begin{split}&\eta=ie^{\frac{i}{2}\ell\tau}\left[\cos\left((1-\ell)\tfrac{\pi}{4}\right)-e^{i\tau}\sin\left((1-\ell)\tfrac{\pi}{4}\right)\right]\begin{pmatrix}1&\,-i\ell
e^{-i\tau}\end{pmatrix}\,,\\\
&\bar{\eta}=ie^{-\frac{i}{2}\ell\tau}\left[\cos\left((1-\ell)\tfrac{\pi}{4}\right)-e^{-i\tau}\sin\left((1-\ell)\tfrac{\pi}{4}\right)\right]\begin{pmatrix}-i\\\
\ell e^{i\tau}\end{pmatrix}\,,\end{split}$ (4.3)

and the spinorial products are always meant to be
$\lambda\bar{\rho}\equiv\lambda^{\alpha}\bar{\rho}_{\alpha}$. The constant
parameter $\ell$ in (4.3) can take only the two values $\pm 1$. Therefore, we
have two branches of fermionic deformations, which differ by the sign of
$\ell$. In principle, the structure of $\eta,\bar{\eta}$ is totally arbitrary,
however we fix them as in (4.3) in order to generate flows that can reach
$1/2$ BPS operators (see appendix B for their definition).

For arbitrary parameters $\chi_{i}$, the fermionic deformation in (4.2) breaks
completely the R-symmetry group. However, if the scalar matrix $M$ is
originally chosen to preserve a subgroup of the R-symmetry, whichever fermions
we plan to add to ${\mathcal{L}}$, they have to preserve the same R-symmetry
structure. In fact, if this were not the case the R-symmetry mismatch between
the fermion and scalar couplings would give rise to UV divergent contributions
that would necessarily turn on a deformation of $M$ along the flow. In the
auxiliary one-dimensional field approach, this is made manifest by the fact
that, as explained in appendix C.2, already at one loop a divergent triangle
fermionic diagram with external fields $\bar{z}C\bar{C}z$ arises, see figure
16, which is proportional to the fermionic deformation. It follows that in
order to cancel this divergence we have to add a parametric deformation in the
scalar coupling matrix, as well. This general observation has interesting
consequences.

First of all, since $SU(4)$ invariant fermionic couplings do not exist, it is
not possible to deform a $SU(4)$ invariant ordinary WL by adding purely
fermionic deformations. A fermionic perturbation of the ordinary $W^{\pm}$
operators considered above would necessarily require adding also a bosonic
perturbation. This kind of perturbations will be discussed in section 5.

Sticking to pure fermionic deformations, the most we can do is to start with a
$SU(3)$ invariant superconnection and add a $SU(3)$ invariant fermionic
coupling. For instance, we can perturb around the bosonic WL corresponding to
superconnection (4.1) with scalar coupling matrix
$M=\tilde{\ell}\,\text{diag}(-1,1,1,1)$, with $\tilde{\ell}=\pm 1$ independent
of $\ell$ (see definition (B.3)). In order to preserve $SU(3)$ R-symmetry
along the RG flows, it is crucial to choose in (4.2) only one non-vanishing
fermion, precisely the one carrying the same R-symmetry index of the scalar
bilinear which appears in $MC\bar{C}$ with opposite sign. Therefore, given
$M=\tilde{\ell}\,\text{diag}(-1,1,1,1)$, we are forced to take $\chi_{1}\neq
0$ and $\chi_{2}=\chi_{3}=\chi_{4}=0$ in (4.2). If we were to choose a
different scalar coupling matrix, for instance
$M=\tilde{\ell}\,\text{diag}(1,-1,1,1)$, we should perturb with $\chi_{2}$.

The $\beta$-functions for single fermion deformations can be computed
following Castiglioni:2022yes . Turning on the $\chi_{i}$ deformation and
evaluating the corresponding $\beta$-function for generic $\ell$, we find

$\beta_{\chi_{i}}=\frac{g^{2}N}{2\pi}\ell\left(\chi_{i}^{2}-1\right)\chi_{i}\,.$
(4.4)

The effect of $\ell$ on the $\beta$-function is a relevant overall sign.

From (4.4) we see that there is one fixed point at $\chi_{i}=0$, corresponding
to the undeformed operator, _i.e._ the non-BPS bosonic $SU(3)$ Wilson loop,
and two other ones at $\chi_{i}=\pm 1$. However, since the deformed WL is
invariant under $\chi_{i}\to-\chi_{i}$, without losing generality we restrict
our study to $\chi_{i}\geq 0$.

In the study of these flows we have two different possibilities corresponding
to the sign of $\ell=\tilde{\ell}$ 666When $\ell=-\tilde{\ell}$ the theory is
not renormalizable, since divergent contributions in the bosonic sector arise.
This means turning on a mixed bosonic-fermionic flow, as we study in section
5.. We interpolate between $SU(3)$ bosonic operators ( ) and $1/2$ BPS ones (
), see (B.2). In particular, for $\ell=1$ the bosonic operator is unstable and
flows to the stable $W_{1/2}^{+}$. In contrast, for $\ell=-1$ the bosonic
operator is stable while $W_{1/2}^{-}$ is unstable. This pattern is
represented in figure 8.

Figure 8: $SU(3)$ symmetric fermionic flows for 8 $\ell=1$ and 8 $\ell=-1$.
The black triangles at the origin represent non-BPS bosonic $SU(3)$ operators,
whereas the red triangles are $1/2$ BPS Wilson loops.

So far we have considered only single fermion deformations. One might wonder
whether a more general pattern exists, where more than one fermion parametric
deformation is turned on. The answer is no. In fact, as one can infer from the
calculations in appendix C.2, as soon as we turn on a second $\chi_{j}$
coupling without a suitable accompanying bosonic deformation, the one-
dimensional auxiliary theory is not renormalizable. In other words, deforming
with more than one fermionic coupling necessarily turns on also a bosonic
deformation along the RG flows.

##  5 Mixed flows

As a last step, we consider moving out from the ordinary $W^{-}$ fixed point
by adding a combination of both bosonic (3.2) and fermionic (4.2) deforming
operators. In general, the perturbation will depend on eight parameters
$\zeta_{i},\chi_{i}$, with $i=1,2,3,4$.

The $\beta$-functions for the fermionic deformations are the straightforward
generalization of the ones evaluated in (4.4). At the order we are working,
they are not affected by the presence of bosonic deformations and read

$\beta_{\chi_{i}}=\frac{g^{2}N}{2\pi}\left(\ell_{1}\chi_{1}^{2}+\ell_{2}\chi_{2}^{2}+\ell_{3}\chi_{3}^{2}+\ell_{4}\chi_{4}^{2}-\ell_{i}\right)\chi_{i}\,,\qquad
i=1,2,3,4,$ (5.1)

where $\ell_{i}=\pm 1$ distinguishes between the two possible branches of
fermionic deformations, as described in section 4.

Instead, the $\beta$-functions of the scalar couplings $\zeta_{i}$ get
affected by the presence of the fermionic deformations, in fact they acquire a
non-trivial dependence on $\chi_{i}$. To simplify the discussion, in the
following sections we investigate in details the cases where only one or two
fermions appear in the deformation. Turning on more than two fermionic
parameters does not add much to the discussion. The results, although more
involved, exhibit the same qualitative behavior.

###  5.1 Bosonic plus one-fermion deformations

We start discussing RG flows driven by the bosonic deformation $\Delta
M(\zeta_{i})$ in (3.2) plus the fermionic one in (4.2) where we set
$\chi_{2}=\chi_{3}=\chi_{4}=0$. As long as we are interested in single fermion
perturbations, this choice is completely general, as deformations generated by
$\chi_{2},\chi_{3}$ or $\chi_{4}$ can be obtained by simply applying an
R-symmetry rotation.

The $\beta$-function for $\chi_{1}$ can be easily read from (5.1). For
$\chi_{2}=\chi_{3}=\chi_{4}=0$ it reduces to (4.4) with $i=1$.

The $\beta$-functions for the bosonic deformations are computed in appendix
C.2. More precisely, they can be read from (C.17) setting $\chi_{a}=\chi_{1}$
and $\chi_{b}=0$

$\begin{split}\beta_{\zeta_{1}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{1}-1+\ell\chi_{1}^{2}+\frac{(1-\ell)}{2}\frac{\chi_{1}^{2}}{\zeta_{1}}\right)\zeta_{1}\,,\\\
\beta_{\zeta_{k}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{k}-1+\ell\chi_{1}^{2}-\frac{(1+\ell)}{2}\frac{\chi_{1}^{2}}{\zeta_{k}}\right)\zeta_{k}\,,\qquad\qquad
k=2,3,4\,.\end{split}$ (5.2)

These $\beta$-functions together with $\beta_{\chi_{1}}$ describe RG flows in
a five-dimensional space. As already shown in section 4, the non-trivial
dependence of $\beta_{\chi_{1}}$ on $\ell$ leads to two qualitatively
different classes of flows. Therefore, we discuss the two cases, $\ell=1$ and
$\ell=-1$, separately.

####  $\boldsymbol{\ell=1}$ case.

In this case the $\beta$-functions reduce to

$\begin{split}\beta_{\chi_{1}}&=\frac{g^{2}N}{2\pi}(\chi_{1}^{2}-1)\chi_{1}\,,\\\
\beta_{\zeta_{1}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{1}-1+\chi_{1}^{2}\right)\zeta_{1}\,,\\\
\beta_{\zeta_{k}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{k}(\zeta_{k}-1+\chi_{1}^{2})-\chi_{1}^{2}\right)\,,\qquad\qquad
k=2,3,4\,.\end{split}$ (5.3)

The nature of the RG flows and the stability of the fixed points in the five
dimensional space can be understood by plotting the solutions to the
$\beta$-function equations $\mu\frac{\partial\Xi}{\partial\mu}=\beta_{\Xi}$,
for any coupling $\Xi=(\chi_{1},\zeta_{1},\zeta_{k})$. The solutions depend on
five arbitrary parameters and read explicitly

$\begin{split}\chi_{1}(\mu)&=\frac{1}{\sqrt{e^{2c_{0}}\mu^{2}+1}}\,,\\\
\zeta_{1}(\mu)&=\frac{1}{\sqrt{e^{2c_{0}}\mu^{2}+1}}\left(\operatorname{arctanh}\left(\sqrt{e^{2c_{0}}\mu^{2}+1}\right)+c_{1}\right)^{-1}\,,\\\
\zeta_{k}(\mu)&=1-\frac{\mu^{2}}{e^{-2c_{0}}+\mu^{2}-c_{k}\sqrt{1+e^{2c_{0}}\mu^{2}}}\,.\end{split}$
(5.4)

In order to have a visual description of the RG flows, in figures 9 we provide
projections on a few planes where fixed points can be detected. To this end,
we note that as long as $\chi_{1}\neq 0$ \- which is our case of interest -
the last term in $\beta_{\zeta_{k}}$ prevents from projecting on planes where
$\zeta_{k}=0$, $k=2,3,4$. We then choose to project on $\zeta_{1}=0$ setting
either $\zeta_{2}=\zeta_{3}=\zeta_{4}\equiv\zeta$ (figure 9) or $\zeta_{2}=1$,
$\zeta_{3}=\zeta_{4}\equiv\zeta$ (figure 9) or $\zeta_{3}=\zeta_{4}=1$,
$\zeta_{2}\equiv\zeta$ (figure 9).

Figure 9: The $\ell=1$ RG flows 9 in the
$(\zeta_{1}=0,\zeta_{2}=\zeta_{3}=\zeta_{4}\equiv\zeta,\chi_{1})$ plane, 9 in
the $(\zeta_{1}=0,\zeta_{2}=1,\zeta_{3}=\zeta_{4}\equiv\zeta,\chi_{1})$ plane
and 9 in the
$(\zeta_{1}=0,\zeta_{2}\equiv\zeta,\zeta_{3}=\zeta_{4}=1,\chi_{1})$ plane. All
cases have similar flows but with fixed points corresponding to different
operators.

In figure 9 we recognize the $W^{-}$ unstable point at the origin ( ) and the
IR stable fixed point $W^{+}_{1/2}$ ( ). The other two saddle points
correspond to non-BPS $SU(3)$ invariant operators. The point
$(\zeta,\chi_{1})=(1,0)$ is a bosonic WL with scalar coupling matrix
$M=\text{diag}(-1,1,1,1)$ ( ), whereas $(\zeta,\chi_{1})=(-1,1)$ corresponds
to a fermionic WL with $M=\text{diag}(-1,-3,-3,-3)$ and non-BPS coupling to
$\psi_{1},\bar{\psi}^{1}$ (still indicated with ).

In figure 9 the origin and the point $(\zeta,\chi_{1})=(1,0)$ correspond to
non-BPS SU(3) invariant bosonic WL ( ). The flow still reaches the fermionic
$1/2$ BPS $W_{1/2}^{+}$ ( ) and a novel non-BPS SU(2) invariant fermionic
operator with $M=\text{diag}(-1,1,-3,-3)$ and $\psi_{1},\bar{\psi}^{1}$
couplings ( ).

In figure 9 the flows connect the $1/6$ BPS bosonic WL ( ) at the origin with
a non-BPS SU(3) invariant bosonic point ( ), the fermionic $1/2$ BPS
$W^{+}_{1/2}$ ( ) and a novel non-BPS $SU(2)$ invariant fermionic operator
with $M=\text{diag}(-1,-3,1,1)$ and $\psi_{1},\bar{\psi}^{1}$ couplings ( ).

As a further case, in figure 10 we consider the $\chi_{1}=1$ section, taking
$\zeta_{1}$ generically non-vanishing and setting
$\zeta_{2}=\zeta_{3}=\zeta_{4}\equiv\zeta$. We see that it exhibits a flow
between the non-BPS SU(3) invariant fermionic operator with
$M=\text{diag}(-1,-3,-3,-3)$ ( ) and the $1/2$ BPS WL $W^{+}_{1/2}$ ( ).

Figure 10: Section $\chi_{1}=1$ in the $(\zeta_{1},\zeta)$ plane for
$\zeta_{2}=\zeta_{3}=\zeta_{4}\equiv\zeta$.

####  $\boldsymbol{\ell=-1}$ case.

A similar analysis can be done for $\ell=-1$. In this case the
$\beta$-functions in (4.4, 5.2) reduce to

$\begin{split}\beta_{\chi_{1}}&=-\frac{g^{2}N}{2\pi}(\chi_{1}^{2}-1)\chi_{1}\,,\\\
\beta_{\zeta_{1}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{1}(\zeta_{1}-1-\chi_{1}^{2})+\chi_{1}^{2}\right)\,,\\\
\beta_{\zeta_{k}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{k}-1-\chi_{1}^{2}\right)\zeta_{k}\,,\qquad\qquad
k=2,3,4\,.\end{split}$ (5.5)

and solving the corresponding flow equations we find

$\begin{split}\chi_{1}(\mu)&=\frac{\mu}{\sqrt{e^{2c_{0}}+\mu^{2}}}\,,\\\
\zeta_{1}(\mu)&=1-\frac{\mu}{\sqrt{e^{2c_{0}}+\mu^{2}}}\left(\operatorname{arctanh}\left(\frac{\mu}{\sqrt{e^{2c_{0}}+\mu^{2}}}\right)-c_{1}\right)^{-1}\,,\\\
\zeta_{k}(\mu)&=\frac{1}{e^{-2c_{0}}\mu^{2}+1+c_{k}\mu\sqrt{e^{2c_{0}}+\mu^{2}}}\,.\end{split}$
(5.6)

The main differences with the previous case are the crucial sign change in
$\beta_{\chi_{1}}$ and somehow the exchange of the roles of $\zeta_{1}$ with
$\zeta_{k}$. In particular, in order to project on planes containing fixed
points, this time $\zeta_{1}=0$ is not a consistent choice, whereas we can
project on $\zeta_{k}=0,(k=2,3,4)$ hyperplanes.

In figure 11 we plot the flows in the $(\zeta_{1}\equiv\zeta,\chi_{1})$ plane,
setting $\zeta_{2}=\zeta_{3}=\zeta_{4}=0$, whereas figure 11 describes the
flows in the $\chi_{1}=1$ section, taking $\zeta_{1}$ and
$\zeta_{2}=\zeta_{3}=\zeta_{4}\equiv\zeta$ free. The pattern is similar to the
$\ell=1$ case, the main difference being that the red triangle now corresponds
to the 1/2 BPS fermionic loop $W^{-}_{1/2}$. Moreover, the black triangle in
the second figure corresponds to a $SU(3)$ invariant non-BPS fermionic
operator with scalar coupling $M=\text{diag}(1,3,3,3)$.

Figure 11: The $\ell=-1$ RG flows 9 in the $(\zeta_{1}\equiv\zeta,\chi_{1})$
plane with $\zeta_{2}=\zeta_{3}=\zeta_{4}=0$ and 9 in the $(\zeta_{1},\zeta)$
plane with $\chi_{1}=1$ and $\zeta_{2}=\zeta_{3}=\zeta_{4}\equiv\zeta$.

It is now interesting to compare the qualitative behavior of the RG flows in
the $\ell=1$ and $\ell=-1$ cases. To this end, in figure 12 we present a
three-dimensional picture of the flows in the $(\zeta_{1},\zeta,\chi_{1})$
space, where $\zeta_{2}=\zeta_{3}=\zeta_{4}\equiv\zeta$, for both cases. The
points along the $\chi_{1}=0$ plane, highlighted in green in both graphs,
correspond to bosonic operators. The origin is the repulsive ordinary Wilson
loop $W^{-}$. The two neighbour points along this plane are non-BPS $SU(3)$
invariant bosonic operators with $M=\pm\text{diag}(1,-1,-1,-1)$. The fourth
vertex of the green plane, diagonally opposite to the origin, is the
attractive (on this plane) ordinary Wilson loop $W^{+}$.

Figure 12: 12 The RG flow with $\ell=1$ in the
$(\zeta_{1},\zeta_{k}=\zeta_{2}=\zeta_{3}=\zeta_{4},\chi_{1})$ space. We can
see that in the $\chi_{1}=0$ plane (no fermions), the most attractive point is
the ordinary WL $W^{+}$. As soon as we turn on fermions, the most attractive
point is the $1/2$ BPS WL with mostly positive scalar coupling matrix. 12 The
RG flow with $\ell=-1$ in the
$(\zeta_{1},\zeta_{k}=\zeta_{2}=\zeta_{3}=\zeta_{4},\chi_{1})$ space. We can
see that the $1/2$ BPS Wilson loop with a mostly negative scalar coupling
matrix acts as a repulsive fixed point.

Leaving the $\chi_{1}=0$ plane, the behavior of the flows in the two graphs
start deviating. In 12 all flows point towards the $1/2$ BPS Wilson loop
$W^{+}_{1/2}$ (red triangle), confirming that this point is stable in any
direction. In 12 the $1/2$ BPS Wilson loop (red triangle) now corresponding to
$W^{-}_{1/2}$ is a repulsive point, thus the most stable point is the ordinary
Wilson loop $W^{+}$.

We can understand the different nature of the fixed points in the whole five
dimensional parameter space by looking at the RG flow solutions (5.4) and
(5.6). The most stable point will be the IR fixed point at $\mu=0$

$\begin{split}\lim_{\mu\to
0}\begin{pmatrix}\zeta_{1}(\mu),&\zeta_{k}(\mu),&\chi_{1}(\mu)\end{pmatrix}=\left\\{\begin{aligned}
(0,1,1)\,,\qquad&\ell=+1\,.\\\
(1,1,0)\,,\qquad&\ell=-1\,.\end{aligned}\right.\end{split}$ (5.7)

which corresponds to the $1/2$ BPS operator $W^{+}_{1/2}$ for $\ell=1$ and to
the ordinary Wilson loop $W^{+}$ for $\ell=-1$. Similarly, the most unstable
point will be the UV fixed point at $\mu\to\infty$

$\begin{split}\lim_{\mu\to\infty}\begin{pmatrix}\zeta_{1}(\mu),&\zeta_{k}(\mu),&\chi_{1}(\mu)\end{pmatrix}=\left\\{\begin{aligned}
(0,0,0)\,,\qquad&\ell=+1\,.\\\
(1,0,1)\,,\qquad&\ell=-1\,.\end{aligned}\right.\end{split}$ (5.8)

which corresponds to the ordinary Wilson loop $W^{-}$ for $\ell=1$ and to the
$1/2$ BPS operator $W^{-}_{1/2}$ for $\ell=-1$. Therefore, the nature of the
fixed points enlightened in figure 12 for three-dimensional subspaces remains
unchanged in five dimensions.

###  5.2 Bosonic plus two-fermion deformations

Now we consider deformations that involve two fermions sourced by two pairs of
$(\eta,\bar{\eta})$. For concreteness, we specialize to the case where
$\psi_{1}$ and $\psi_{2}$ deformations are turned on. The corresponding
$\beta$-functions can be read in (5.1) setting $\chi_{3}=\chi_{4}=0$, whereas
the ones for the bosonic parameters are computed in detail in appendix C. They
can be obtained from (C.17) setting $a=1$ and $b=2$.

As comes out from these results, we have to distinguish between deformations
with $\ell_{1}=\ell_{2}$ or ones with $\ell_{1}=-\ell_{2}$. In the first case,
the fermionic diagram which contributes to the renormalization of the
$\bar{z}MC\bar{C}z$ scalar vertex gives rise to off-diagonal terms in the
scalar coupling matrix $M$. Instead, a deformation with opposite $\ell$’s does
not require a non-diagonal scalar coupling matrix. Since in what follows we
restrict to the study of diagonal scalar deformations we will only consider
the $\ell_{1}=-\ell_{2}$ case.

Setting for instance $\ell_{1}=1$ and $\ell_{2}=-1$, the $\beta$-functions are
explicitly given by

$\begin{split}\beta_{\chi_{1}}&=\frac{g^{2}N}{2\pi}\bigg{(}\chi_{1}^{2}-\chi_{2}^{2}-1\bigg{)}\chi_{1}\,,\\\
\beta_{\chi_{2}}&=\frac{g^{2}N}{2\pi}\bigg{(}\chi_{1}^{2}-\chi_{2}^{2}+1\bigg{)}\chi_{2}\,,\\\
\beta_{\zeta_{1}}&=\frac{g^{2}N}{2\pi}\bigg{(}\zeta_{1}-1+(\chi_{1}^{2}-\chi_{2}^{2})\bigg{)}\zeta_{1}\,,\\\
\beta_{\zeta_{2}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{2}-1+(\chi_{1}^{2}-\chi_{2}^{2})\left(1-\frac{1}{\zeta_{2}}\right)\right)\zeta_{2}\,,\\\
\beta_{\zeta_{c}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{c}-1+(\chi_{1}^{2}-\chi_{2}^{2})-\frac{\chi_{1}^{2}}{\zeta_{c}}\right)\zeta_{c}\,,\qquad
c=3,4\,.\end{split}$ (5.9)

In this case the RG flow occurs in a six-dimensional space, therefore it is
quite difficult to visualize it. In order to grasp some partial information,
we note that at this order the fermionic $\beta$-functions do not depend on
the bosonic parameters, so one can study their flows independently of the
$\zeta_{i}$, $i=1,\dots,4$ behavior. The fermionic flows in the
$(\chi_{1},\chi_{2})$ plane are depicted in figure 13, where we focus only on
the positive sector. As already stressed, the deformed theory is invariant
under $\chi_{i}\to-\chi_{i}$, therefore the flows are simply mirrored with
respect to the two axes. We highlight in orange the boundaries of the region
where flows start and end at the fixed points $(1,0)$ and $(0,1)$. Since for
particular choices of the $\zeta_{i}$ the $(1,0)$ point corresponds to
$W_{1/2}^{+}$ while $(0,1)$ corresponds to $W_{1/2}^{-}$, once more we find
that within the isolated invariant set $W_{1/2}^{+}$ is stable whereas
$W_{1/2}^{-}$ is unstable.

Figure 13: The RG flow in the $(\chi_{1},\chi_{2})$ plane for $\ell_{1}=+1$
and $\ell_{2}=-1$. The region between the two curves highlighted in orange
delimits flows that start and end at fixed points $(1,0)$ and $(0,1)$.

This pattern is also very clear from figure 14, where we plot the flows on the
$(\zeta_{i},\chi_{1},\chi_{2})$ subspace for different $\zeta_{i}$. In orange
we highlight the planes corresponding to the orange lines of figure 13.
Selecting flows that lie within these two hyperplanes one easily realizes that
the upper plane is repulsive whereas the lower one is attractive.

Figure 14: The RG flow 14 in the $(\zeta_{1},\chi_{1},\chi_{2})$ space, 14 in
the $(\zeta_{2},\chi_{1},\chi_{2})$ space and 14 in the
$(\zeta_{c},\chi_{1},\chi_{2})$ space, $c=3,4$.

##  6 Defect theory interpretation

The fixed points appearing in the RG flow figures of the previous sections
describe defect conformal field theories (dCFTs) living on the corresponding
Wilson loops, in interaction with the ABJM bulk theory. BPS fixed points give
rise to superconformal defects, whereas non-BPS points correspond to ordinary
dCFTs. The RG flows have then the interesting interpretation of flows in the
space of one-dimensional defect theories triggered by dynamical interactions
with the bulk. From this perspective, they can be interpreted as being driven
by marginally relevant perturbations of the defect.

In general, in the presence of deformations driven by a set of local operators
$\hat{d}_{i}$, a defect stress tensor $T_{D}$ turns on, which is given by the
product Cuomo:2021rkm ; Giombi:2022vnz

$T_{D}=\beta_{i}\,\hat{d}_{i}\,,$ (6.1)

where $\beta_{i}$ is the $\beta$-function of the coupling associated to the
$i$-th deformation. The $T_{D}$ operator affects the ABJM stress tensor
conservation law as

$\nabla_{\mu}T^{\mu\nu}=-\delta_{D}^{(2)}\left(\dot{x}^{\nu}\dot{T}_{D}+n_{i}^{\nu}D^{i}\right)\,,$
(6.2)

where $\delta_{D}^{(2)}$ is the Dirac delta localized at the defect, $x^{\mu}$
is the embedding function describing the defect location (2.3), $n_{i}^{\mu}$
is a unit vector normal to the defect and $D^{i}$ is the displacement
operator.

From the explicit expression of $T_{D}$ one can compute physical quantities,
as for instance the anomalous dimension of the $\hat{d}_{i}$ operators, and
investigate the validity of a g-theorem Affleck:1991tk ; Friedan:2003yc ;
Casini:2016fgb ; Cuomo:2021rkm ; Casini:2022bsu .

In fact, recalling that for a set of generic operators $\hat{O}_{i}$ the
matrix of anomalous dimensions is defined as
$\mu\frac{\partial\hat{O}_{i}}{\partial\mu}=-\Delta_{i}^{\,j}\hat{O}_{j}$, and
taking into account that $T_{D}$ has protected mass dimension equal to one, we
can apply $\mu\frac{\partial}{\partial\mu}$ to both sides of (6.1), obtaining

$\Delta_{i}^{\,j}=\delta_{i}^{\,j}+\frac{\partial\beta^{j}}{\partial\zeta_{i}}\quad\Longrightarrow\quad\Gamma_{i}^{\,j}=\frac{\partial\beta^{j}}{\partial\zeta_{i}}\,,$
(6.3)

where $\Gamma_{i}^{\,j}$ is the anomalous dimension matrix and $\zeta_{i}$ is
the coupling associated to $i$-th deforming operator.

Furthermore, using the definition (6.1) for the defect stress tensor one can
establish a one-dimensional version of the g-theorem, that is the statement
that the free energy of a $\zeta$-deformed defect, ${\rm g}\equiv\log\langle
W(\zeta)\rangle$, should be a monotonically decreasing function along the RG
flows. In particular, it should satisfy ${\rm g}_{\rm UV}>{\rm g}_{\rm IR}$,
where ${\rm g}_{\rm UV}$ and ${\rm g}_{\rm IR}$ are the values of ${\rm g}$ at
the UV and the IR fixed points, respectively. In Cuomo:2021rkm  this has been
proven for line defects in any dimension by studying the $\zeta$-induced flow
of a related observable, which is the defect entropy

$s(\zeta)=\left(1+\beta_{\zeta}\frac{\partial}{\partial\zeta}\right)\log\langle
W(\zeta)\rangle\,,$ (6.4)

that at the fixed points coincides with ${\rm g}$. From the relevant identity
Cuomo:2021rkm

$\mu\frac{\partial s}{\partial\mu}=-\int
d\tau_{1>2}\,\operatorname{\big{\langle}\big{\langle}}T_{D}(\tau_{1})T_{D}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}(1-\cos(\tau_{1}-\tau_{2}))\,,$
(6.5)

which provides the mass scaling of $s$, one can conclude that the ${\rm
g}$-theorem holds whenever the defect theory is reflection positive (in
Euclidean signature) or unitary (in Minkowski), that is whenever
$\operatorname{\big{\langle}\big{\langle}}T_{D}(\tau_{1})T_{D}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}>0$.

In the rest of this section we will apply eqs. (6.3) and (6.5) to our flows in
order to better investigate their nature from a defect theory perspective.

We stress that our computations, despite being perturbative in the Chern-
Simons coupling $g$, are exact in the running coupling constants
$\zeta_{i},\chi_{i}$. Therefore, from the point of view of the one-dimensional
theory, the $\beta$-functions and the general behavior of the RG flows are
reliable at any scale.

###  6.1 Bosonic defects

We begin by focusing on the bosonic deformations described in section 3. They
can be interpreted as deformations of the $W^{-}$ bosonic dCFT777We call
“bosonic defects” the one-dimensional theories defined on bosonic WL (no
fermion fields turned on), and “fermionic defects” the ones defined on
fermionic WLs. that drive the system towards IR fixed points still given by
bosonic dCFTs.

In the simplest case of one-parameter deformations (3.2) with
$\zeta_{1}=\zeta_{2}=\zeta_{3}=\zeta_{4}\equiv\zeta$, the defect stress tensor
for the deformed theory is simply given by

$T_{D}=\beta_{\zeta}\,\hat{d}_{\zeta}=-2g^{2}\beta_{\zeta}C_{I}\bar{C}^{I}\,,$
(6.6)

where $\beta_{\zeta}$ is the $\beta$-function evaluated in (3.3).

In this case the $SU(4)$ symmetry ensures that the four operators
$C_{1}\bar{C}^{1},\cdots,C_{4}\bar{C}^{4}$ have the same anomalous dimension.
The anomalous dimension matrix in (6.3) is then proportional to the identity
matrix. Using (3.3) for $\beta_{\zeta}$, its value at $\zeta=0$ is
$\Gamma=-\frac{g^{2}N}{2\pi}\ \mathbb{1}$. A negative anomalous dimension
signals the fact that around $W^{-}$ the perturbation is a weakly relevant
operator that drives the defect CFT away from the fixed point. On the other
hand, evaluating the anomalous dimension matrix for deformations around the
$W^{+}$ fixed point in figure 7, which amounts to evaluate (6.3) at $\zeta=1$,
we find a positive anomalous dimension, thus the perturbation is marginally
irrelevant and, consistently with the RG flow behavior, $W^{+}$ corresponds to
an IR stable fixed point.

Moving to two-parameter deformations (3.2) with
$\zeta_{1}=\zeta_{2}(\equiv\zeta_{1})$ and
$\zeta_{3}=\zeta_{4}(\equiv\zeta_{2})$, the preserved $SU(2)\times SU(2)$
symmetry this time insures that the $C_{1}\bar{C}^{1}$ and $C_{2}\bar{C}^{2}$
deforming operators share the same anomalous dimension $\gamma_{1}$, and
$C_{3}\bar{C}^{3}$ and $C_{4}\bar{C}^{4}$ share the same $\gamma_{2}$, as
well. In principle, there might be mixing between the two pairs of operators.
However, at the order we are working this is not the case. In fact, the two
$\beta$-functions that can be read from (3.3) are decoupled, so that the
anomalous dimension matrix (6.3) is diagonal,
$\Gamma=\text{diag}(\gamma_{1},\gamma_{1},\gamma_{2},\gamma_{2})$.

Referring to figure 6, we compute the values of $\gamma_{1}$ and $\gamma_{2}$
at the four fixed points. At the two black squares points, corresponding to
$W^{-}$ (at the origin) and $W^{+}$, we still find that
$\gamma_{1}(W^{\pm})=\gamma_{2}(W^{\pm})=\pm\frac{g^{2}N}{2\pi}$, in agreement
with the $W^{+}(W^{-})$ fixed point being attractive (repulsive).

If we consider instead the $1/6$ BPS Wilson loop associated to the scalar
coupling matrix $M=\text{diag}(-1,-1,1,1)$ and corresponding to the upper blue
dot in figure 6, the anomalous dimension for the $\zeta_{1}$ deformation is
negative, $\gamma_{1}=-\frac{g^{2}N}{2\pi}$, whereas for the $\zeta_{2}$
deformation it is positive, $\gamma_{2}=\frac{g^{2}N}{2\pi}$. Therefore the
$\zeta_{1}$ deformation of the $1/6$ BPS Wilson loop is weakly relevant, while
the $\zeta_{2}$ one is weakly irrelevant. A similar pattern arises for the
other 1/6 BPS Wilson loop associated to $M=\text{diag}(1,1,-1,-1)$ (lower blue
point in figure 6), simply with the signs of $\gamma_{1}$ and $\gamma_{2}$
interchanged. This is in agreement with the saddle point behavior and the
directions of the flows shown in figure 6.

This analysis can be easily generalized to deformations featured by (3.2) with
four different parameters. In that case the is no symmetry constraining the
anomalous dimension matrix. However, at the order we are working the
$\beta$-functions in (3.3) are decoupled and the anomalous dimension matrix
turns out to be of the form
$\Gamma={\text{diag}}(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$, with four
independent entries. A careful analysis reveals that $\Gamma$ is negative-
definite at the $W^{-}$ fixed point, it is positive-definite at $W^{+}$,
whereas at all the other fixed points it does not have a definite sign, thus
confirming that they are all saddle points.

As anticipated above, we now establish the validity of a g-theorem for the RG
flows under consideration. We focus for simplicity on the one-parameter
deformation that drives the RG flow between $W^{-}$ and $W^{+}$ (figure 7).
Defining ${\rm g}_{\rm UV}=\log\langle W^{-}\rangle$ and ${\rm g}_{\rm
IR}=\log\langle W^{+}\rangle$, we want to check whether ${\rm g}_{\rm UV}>{\rm
g}_{\rm IR}$.

In principle, one could try a direct check by computing perturbatively ${\rm
g}_{\rm UV}$ and ${\rm g}_{\rm IR}$. However, this would require a three-loop
calculation, as up to two loops $W^{\pm}$ share the same expectation value.
Alternatively, we can use the prescription of Cuomo:2021rkm . If we insert the
explicit expression (6.6) for $T_{D}$ in the scaling equation (6.5) for the
defect entropy, at one loop we find ($\tau_{12}\equiv\tau_{1}-\tau_{2}$)

$\mu\frac{\partial s}{\partial\mu}=-4g^{4}\beta_{\zeta}^{2}\int
d\tau_{1>2}\operatorname{\big{\langle}\big{\langle}}(C_{I}\bar{C}^{I})(\tau_{1})(C_{J}\bar{C}^{J})(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}(1-\cos\tau_{12})=-g^{4}N^{2}\beta_{\zeta}^{2}<0\,.$
(6.7)

This means that the defect entropy decreases monotonically along the flow,
leading to the expected result

${\rm g}_{\rm UV}>{\rm g}_{\rm IR}\,.$ (6.8)

This analysis can be generalized straightforwardly to the $SU(2)\times SU(2)$
and generic bosonic flows, always leading to inequality (6.8). We can then
conclude that within the set of bosonic deformations of the form (3.2), the
${\rm g}$-theorem is always respected.

###  6.2 Fermionic defects

We now move to the case of fermionic deformations described in section 4.
Without loss of generality, we turn on only the $\chi_{1}$ deformation and
consider the flows connecting a $SU(3)$ invariant bosonic WL ( ) and a 1/2 BPS
one ( ) (see figure 8). As is clear from this figure, the direction of the
flow depends on the sign of $\ell$ entering the fermionic couplings (4.3),
thus if the ${\rm g}$-theorem is still at work, we should expect an opposite
inequality between ${\rm g}(\leavevmode\hbox to7.51pt{\vbox
to7.51pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ and ${\rm
g}(\leavevmode\hbox to7.51pt{\vbox
to7.51pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ in the two cases.
Moreover, since the sign of $\ell$ also discriminates between different
defects at the fixed points,888It determines the overall sign of the scalar
coupling matrix at the two fixed points and the $(\eta,\bar{\eta})$ couplings
at the 1/2 BPS fixed point. the two cases $\ell=\pm 1$ have to be studied
separately.

####  $\boldsymbol{\ell=1}$ case.

In this case the $SU(3)$ invariant bosonic defect corresponds to the scalar
coupling matrix $M={\text{diag}}(-1,1,1,1)$. As is clear from figure 8, it is
a UV unstable fixed point under the supermatrix deformation

$\hat{d}=-g\chi_{1}\begin{pmatrix}0&\eta\bar{\psi}^{1}\\\
\psi_{1}\bar{\eta}&0\end{pmatrix}\,.$ (6.9)

In fact, a straightforward application of prescription (6.3) with
$\beta_{\chi_{1}}$ as in (5.3) leads to a negative anomalous dimension for
$\hat{d}$ at the $\chi_{1}=0$ fixed point, signaling that this is indeed a
marginally relevant perturbation around the ( ) point. An analogous
calculation reveals that around the BPS fixed point $W^{+}_{1/2}$ this is
instead an irrelevant operator.

In order to check the ${\rm g}$-theorem, we evaluate the scaling behavior of
the defect entropy inserting in (6.5) the expression
$T_{D}=\beta_{\chi_{1}}\hat{d}$, with $\hat{d}$ given in (6.9).999Since
coupling to fermions are defined in terms of $U(N|N)$ superconnections, the
operator insertions on the defect theory are $U(N|N)$ supermatrices.
Therefore, in this case
$\operatorname{\big{\langle}\big{\langle}}T_{D}(\tau_{1})T_{D}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}$
in (6.5) includes also a trace on supermatrices. Evaluating the resulting
expression at one loop, we find

$\displaystyle\mu\frac{\partial s}{\partial\mu}$ $\displaystyle=$
$\displaystyle-g^{2}\beta_{\chi_{1}}^{2}\int
d\tau_{1>2}\bigg{[}\operatorname{\big{\langle}\big{\langle}}\big{(}\eta\bar{\psi}^{1}\big{)}(\tau_{1})\big{(}\psi_{1}\bar{\eta}\big{)}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}_{\ell=1}+\operatorname{\big{\langle}\big{\langle}}\big{(}\psi_{1}\bar{\eta}\big{)}(\tau_{1})\big{(}\eta\bar{\psi}^{1}\big{)}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}_{\ell=1}\bigg{]}(1-\cos\tau_{12})$
$\displaystyle=-\frac{\pi}{2}g^{2}N^{2}\beta_{\chi_{1}}^{2}$ (6.10)

This expression is manifestly negative, therefore the defect entropy is a
monotonically decreasing function from the UV to the IR and the ${\rm
g}$-theorem holds.

####  $\boldsymbol{\ell=-1}$ case.

From figure 8 it is clear that in this case we have to consider perturbing
around the 1/2 BPS operator ( ) given by a a scalar matrix
$M={\rm{diag}}(1,-1,-1,-1)$ and fermionic couplings

$-i{\cal L}_{F}=-g\begin{pmatrix}0&\eta\bar{\psi}^{1}\\\
\psi_{1}\bar{\eta}&0\end{pmatrix}\,,$ (6.11)

This corresponds to the $W^{-}_{1/2}$ fixed point. We now add the supermatrix
deformation

$\hat{d}=g\tilde{\chi}_{1}\begin{pmatrix}0&\eta\bar{\psi}^{1}\\\
\psi_{1}\bar{\eta}&0\end{pmatrix}\,,$ (6.12)

such that the 1/2 BPS point corresponds to $\tilde{\chi}_{1}=0$ and the IR
stable bosonic WL is obtained for $\tilde{\chi}_{1}=1$. A simple application
of prescription (6.3) reveals that at $\tilde{\chi}_{1}=0$ this deformation is
indeed marginally relevant.

Evaluating the corresponding $\beta$-function, in this case we find

$\beta_{\tilde{\chi}_{1}}=-\frac{g^{2}N}{2\pi}\tilde{\chi}_{1}(\tilde{\chi}_{1}-2)(\tilde{\chi}_{1}-1)$
(6.13)

while the defect stress tensor is given by
$T_{D}=\beta_{\tilde{\chi}_{1}}\hat{d}$, with $\hat{d}$ in (6.12).

Inserting $T_{D}$ in (6.5) and evaluating the expectation values at one loop,
we find

$\displaystyle\mu\frac{\partial s}{\partial\mu}$ $\displaystyle=$
$\displaystyle-g^{2}\beta_{\tilde{\chi}_{1}}^{2}\int
d\tau_{1>2}\bigg{[}\operatorname{\big{\langle}\big{\langle}}\big{(}\eta\bar{\psi}^{1}\big{)}(\tau_{1})\big{(}\psi_{1}\bar{\eta}\big{)}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}_{\ell=-1}+\operatorname{\big{\langle}\big{\langle}}\big{(}\psi_{1}\bar{\eta}\big{)}(\tau_{1})\big{(}\eta\bar{\psi}^{1}\big{)}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}_{\ell=-1}\bigg{]}(1-\cos\tau_{12})$
$\displaystyle=\frac{\pi}{2}g^{2}N^{2}\beta_{\tilde{\chi}_{1}}^{2}$ (6.14)

In contrast with the $\ell=1$ case, this result is positive. Therefore, in the
$\ell=-1$ class of deformations the ${\rm g}$-theorem is not respected.
According to the general formulation of the theorem Cuomo:2021rkm , this
signals the lack of reflection positivity of this class of defects.

Summarizing, for the class of one-fermion deformations we have explicitly
found that

$\mu\frac{\partial s}{\partial\mu}=\left\\{\begin{aligned}
&\ell=+1\quad-g^{2}\beta_{\chi_{1}}^{2}N^{2}\frac{\pi}{2}\qquad\Rightarrow\quad{\rm
g}_{\rm UV}(\leavevmode\hbox to7.51pt{\vbox
to7.51pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})>{\rm g}_{\rm
IR}(\leavevmode\hbox to7.51pt{\vbox
to7.51pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})\,,\\\
&\ell=-1\qquad\,g^{2}\beta_{\tilde{\chi}_{1}}^{2}N^{2}\frac{\pi}{2}\qquad\Rightarrow\quad{\rm
g}_{\rm UV}(\leavevmode\hbox to7.51pt{\vbox
to7.51pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
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{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})<{\rm g}_{\rm
IR}(\leavevmode\hbox to7.51pt{\vbox
to7.51pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})\,.\end{aligned}\right.$
(6.15)

Since this result can appear quite surprising, we provide the technical
explanation of why the $\ell=1$ and $\ell=-1$ flows behave so differently.

First, we note that possible sign differences cannot come from the
$\beta$-functions since in the variation of the entropy they always appear
squared. The difference comes directly from the integrated two-point
functions, in particular from the different definition of the
$\eta,\bar{\eta}$ couplings entering $T_{D}$.

In fact, looking at the first term in the integrals (6.2), (6.2) (for the
second term the argument works similarly) we see that, as the result of
contracting the two fermions with the propagator (A.3), one obtains

$\operatorname{\big{\langle}\big{\langle}}T_{D}(\tau_{1})T_{D}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}_{\ell=\pm
1}\sim(\eta_{1}\gamma^{\mu}\,\bar{\eta}_{2})_{\ell=\pm 1}\,(x_{12})_{\mu}$
(6.16)

where $\eta_{i},\bar{\eta}_{i}$ stands for
$\eta(\tau_{i}),\bar{\eta}(\tau_{i})$ and
$x_{12}^{\mu}\equiv(x^{\mu}(\tau_{1})-x^{\mu}(\tau_{2}))$.

On the other hand, from (4.3) it is easy to see that the $\eta,\bar{\eta}$
couplings satisfy the following identity

$(\eta_{i}\gamma^{\mu}\bar{\eta}_{j})_{\ell}\,(x_{ij})_{\mu}=4i\ell\sin{\frac{\tau_{ij}}{2}}.$
(6.17)

Since the rest of the factors in
$\operatorname{\big{\langle}\big{\langle}}T_{D}(\tau_{1})T_{D}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}_{\ell}$
do not depend on $\ell$, from identity (6.17) we immediately conclude that the
two two-point function is proportional to $\ell$, thus it has opposite sign in
the two cases

$\operatorname{\big{\langle}\big{\langle}}T_{D}(\tau_{1})T_{D}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}_{\ell=-1}=-\operatorname{\big{\langle}\big{\langle}}T_{D}(\tau_{1})T_{D}(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}_{\ell=1}\,.$
(6.18)

Finally, since the results of the circle integrations are the same in two
cases, we are easily led to the conclusions in (6.15).

##  7 Discussion

We have studied defect RG flows in ABJM theory induced by bosonic and/or
fermionic marginally relevant deformations. Specifically, we have focused on
the subset of Wilson loops and deformations featured by diagonal scalar matrix
couplings. Exploiting the one-dimensional auxiliary field description of
Wilson loops to perform renormalization on the defect, we managed to compute
the $\beta$-functions exactly in the deforming parameters, and at first order
in the bulk ABJM coupling. Relevant features of these flows arise, which are
worth recapping.

First, starting from defects with only scalar bilinears, also referred to as
bosonic defects, we have shown that generic diagonal, scalar bilinear
deformations are always well-defined marginally relevant operators on the
defect, as no further constraint on their structure arises along the flow.

Regarding deformations of $SU(4)$ invariant defects, it is always possible to
turn on a deformation made by a single fermion, thus necessarily breaking
$SU(4)$ R-symmetry to $SU(3)$, as long as this reminiscent $SU(3)$ is also
preserved by the scalar bilinears.

Progressing towards deformations containing more than one fermion, we have
found in particular that turning on a two-fermion operator can be achieved
only if this is accompanied by a bosonic deformation, which necessarily turns
on along the flow due to the interaction of the two-fermion deformation with
the bulk fields. Therefore, marginally relevant operators containing at least
two fermions are necessarily given by $U(N|N)$ supermatrices with both
diagonal and off-diagonal entries. Moreover, if the two fermionic deformations
correspond to $(\eta,\bar{\eta})$ couplings with the same $\ell$ the bosonic
deformation turns out to be non-diagonal. This is not a problem in general, as
$M$ is generically non-diagonal (see for instance equation (2.10) in
Castiglioni:2022yes ). However, since here we have restricted our
investigation to diagonal $M$’s, we have disregarded these cases. The
generalization to non-diagonal bosonic deformations should be worth
investigating in the future.

Our results share some common features with similar results in
${\mathcal{N}}=4$ SYM Polchinski_2011 ; Beccaria:2017rbe , where the flow
connects the ordinary loop sitting at the UV fixed point to the $1/2$ BPS
fixed point in the IR. In ABJM theory the analogous pattern is realised by the
mixed bosonic plus one-fermion deformation connecting $W^{-}$ to
$W_{1/2}^{+}$. However, as usual, ABJM offers a much broader spectrum of fixed
points. Indeed this particular realisation is only one of the many
possibilities depicted in figure 1.

One of the most striking results we have obtained is that the ordinary
$W^{\pm}$ operators, which differ only by an apparently harmless overall sign
in the scalar coupling matrix $M$, exhibit a very different nature at quantum
level, being one an IR stable fixed point and the other a UV unstable one,
respectively. Similarly, the $W^{\pm}_{1/2}$ BPS Wilson loops which differ
simply by the choice of $\ell=1$ or $-1$ in the bosonic and fermionic
couplings, related to the two possible ways of solving the supersymmetry
preserving constraints Cardinali:2012ru , turn out to behave in an opposite
way under deformations. In addition, the $W^{-}_{1/2}$ WL seems to describe a
non-unitary dCFT. This is consistent with what was found in Gabai:2022vri ;
Gabai:2022mya  for mesonic line operators in Chern-Simons-matter theories.
While we have provided a perturbative explanation of these features, it would
be nice to have a more physical interpretation.

Focusing on 1/2 BPS operators, we recall that in field theory $W_{1/2}^{+}$
and $W_{1/2}^{-}$ can be obtained by a Higgsing construction that makes use of
heavy $W$-particles and $W$-antiparticles, respectively Lietti:2017gtc  ($W$
and $\tilde{W}$ there). This provides already some indication that they could
give rise to physically different dCFTs. This seems to be confirmed also
holographically. In fact, as shown in Lietti:2017gtc , the two $1/2$ BPS fixed
points have different classical M-theory duals:101010A first quantum
perturbative calculation around this classical configuration can be found in
Giombi:2023vzu . while $W_{1/2}^{+}$ is dual to a M2-brane configuration in
${\rm AdS}_{4}\times{\rm S}^{7}/{\mathbb{Z}}_{k}$, $W_{1/2}^{-}$ is described
by an anti-M2-brane. Although this seems to be consistent with our findings,
it is not clear how to exactly relate the opposite behavior under RG flow
found in this paper to the two different holographic descriptions. This is a
very interesting question that deserves further investigation.

A less clear picture exists for ordinary WLs. As is well-known, the
holographic descriptions of different Wilson loops with different amount of
supersymmetry differ by the set of boundary conditions of the corresponding
string configuration in the internal space being Dirichlet or Neumann. A naive
comparison with the ${\mathcal{N}}=4$ SYM setting suggests that our “ordinary”
$SU(4)$ Wilson loop operators should be described by strings satisfying pure
Neumann boundary conditions, that is strings smeared over $\mathbb{CP}^{3}$.
It is not clear, however, how a distinction between $W^{+}$ and $W^{-}$ should
arise from the dual point of view. Of course, a holographic input would allow
for a better understanding of our results.

Figure 15: Representation of enriched flows connecting $W_{1/6}$ and
$W_{1/2}^{\pm}$.

Our current discussion includes the enriched flows studied in
Castiglioni:2022yes , when we focus on the particular case of BPS RG
trajectories connecting $W_{1/6}$ and $W_{1/2}^{\pm}$. These are depicted as
green lines in figure 15. The green curve connecting $W_{1/6}$ and
$W^{+}_{1/2}$ corresponds to the parabolic RG trajectory in figure 9 given by
$\zeta=\chi_{1}^{2}$. It consists of a smooth interpolation made by a
continuum of $1/6$ BPS fermionic loops, thus the name “enriched flow”. It
corresponds exactly to the green trajectory in figure 13 of
Castiglioni:2022yes . The enriched trajectory connecting $W^{-}_{1/2}$ and
$W_{1/6}$, instead, is not directly included in the analysis of section 5, as
the $W_{1/6}$ fixed point should correspond to the one considered there, but
with $M\to-M$. Of course, it is straightforward to duplicate the analysis of
section 5 in this case and pinpoint the enriched trajectory. The $W^{-}_{1/2}$
operator, which turns out to be a repulsive fixed point, coincides with ${\cal
W}^{II}_{1/2}$ of Castiglioni:2022yes  upon an R-symmetry rotation. We thus
complete the analysis of Castiglioni:2022yes  with new information about
enriched RG flows connecting the repulsive ${\cal W}^{\rm II}_{1/2}$. For the
interested reader, we discuss details in appendix D.

Our results are perturbative in the ABJM coupling, so higher order corrections
to the $\beta$-functions might potentially modify the spectrum of fixed
points. Checking what happens to them at higher loops is an interesting open
question that would be worth addressing.

## Acknowledgements

We are grateful to Diego Correa and Guillermo Silva for discussions. LC, SP
and MT are partially supported by the INFN grant Gauge Theories, Strings and
Supergravity (GSS). DT is supported in part by the INFN grant Gauge and String
Theory (GAST). DT would like to thank FAPESP’s partial support through the
grants 2016/01343-7 and 2019/21281-4.

##  Appendix A Conventions and Feynman rules

For ABJM theory we follow the conventions in Bianchi:2014laa . We work in
three-dimensional Euclidean space with coordinates
$x^{\mu}=(x^{0},x^{1},x^{2})$. The three-dimensional gamma matrices are
defined as

$(\gamma^{\mu})^{\
\beta}_{\alpha}=(-\sigma^{3},\sigma^{1},\sigma^{2})_{\alpha}^{\ \beta}\,,$
(A.1)

with $(\sigma^{i})^{\ \beta}_{\alpha}$ ($\alpha,\beta=1,2$) being the Pauli
matrices, such that
$\gamma^{\mu}\gamma^{\nu}=\delta^{\mu\nu}+i\epsilon^{\mu\nu\rho}\gamma_{\rho}$,
where $\epsilon^{123}=\epsilon_{123}=1$ is totally antisymmetric. Spinorial
indices are lowered and raised as $(\gamma^{\mu})^{\alpha}_{\
\beta}=\epsilon^{\alpha\gamma}(\gamma^{\mu})^{\
\delta}_{\gamma}\epsilon_{\beta\delta}$, with
$\epsilon_{12}=-\epsilon^{12}=1$. The Euclidean action of $U(N)_{k}\times
U(N)_{-k}$ ABJM theory is

$\begin{split}S_{\textrm{ABJM}}=&\frac{k}{4\pi}\int
d^{3}x\,\epsilon^{\mu\nu\rho}\Big{\\{}-i\text{Tr}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}\right)+i\text{Tr}\left(\hat{A}_{\mu}\partial_{\nu}\hat{A}_{\rho}+\frac{2i}{3}\hat{A}_{\mu}\hat{A}_{\nu}\hat{A}_{\rho}\right)\\\
&+\text{Tr}\left[\frac{1}{\xi}(\partial_{\mu}A^{\mu})^{2}-\frac{1}{\xi}(\partial_{\mu}\hat{A}^{\mu})^{2}+\partial_{\mu}\bar{c}D^{\mu}c-\partial_{\mu}\bar{\hat{c}}D^{\mu}\hat{c}\right]\Big{\\}}\\\
&+\int
d^{3}x\text{Tr}\left[D_{\mu}C_{I}D^{\mu}\bar{C}^{I}+i\bar{\psi}^{I}\gamma^{\mu}D_{\mu}\psi_{I}\right]\\\
&\begin{split}-\frac{2\pi i}{k}\int
d^{3}x\text{Tr}\Big{[}&\bar{C}^{I}C_{I}\psi_{J}\bar{\psi}^{J}-C_{I}\bar{C}^{I}\bar{\psi}^{J}\psi_{J}+2C_{I}\bar{C}^{J}\bar{\psi}^{I}\psi_{J}\\\
&-2\bar{C}^{I}C_{J}\psi_{I}\bar{\psi}^{J}-\epsilon_{IJKL}\bar{C}^{I}\bar{\psi}^{J}\bar{C}^{K}\bar{\psi}^{L}+\epsilon^{IJKL}C_{I}\psi_{J}C_{K}\psi_{L}\Big{]}+S^{\text{bos}}_{\text{int}}\,,\end{split}\end{split}$
(A.2)

with covariant derivatives defined as

$\displaystyle D_{\mu}C_{I}$
$\displaystyle=\partial_{\mu}C_{I}+iA_{\mu}C_{I}-iC_{I}\hat{A}_{\mu}\,,\qquad
D_{\mu}\bar{C}^{I}$
$\displaystyle=\partial_{\mu}\bar{C}^{I}-i\bar{C}^{I}A_{\mu}+i\hat{A}_{\mu}\bar{C}^{I}\,,$
$\displaystyle D_{\mu}\bar{\psi}^{I}$
$\displaystyle=\partial_{\mu}\bar{\psi}^{I}+iA_{\mu}\bar{\psi}^{I}-i\bar{\psi}^{I}\hat{A}_{\mu}\,,\qquad
D_{\mu}\psi_{I}$
$\displaystyle=\partial_{\mu}\psi_{I}-i\psi_{I}A_{\mu}+i\hat{A}_{\mu}\psi_{I}\,.$

We work in Landau gauge for vector fields and in dimensional regularization
with $d=3-2\epsilon$. The tree-level propagators are (with $g=\sqrt{2\pi/k}$)

$\begin{split}\langle(A_{\mu})_{p}^{\ q}(x)(A_{\nu})_{r}^{\
s}(y)\rangle^{(0)}&=\delta_{p}^{s}\delta_{r}^{q}\,ig^{2}\,\frac{\Gamma(\frac{3}{2}-\epsilon)}{2\pi^{\frac{3}{2}-\epsilon}}\frac{\epsilon_{\mu\nu\rho}(x-y)^{\rho}}{|x-y|^{3-2\epsilon}},\\\
\langle(\hat{A}_{\mu})_{\hat{p}}^{\ \hat{q}}(x)(\hat{A}_{\nu})_{\hat{r}}^{\
\hat{s}}(y)\rangle^{(0)}&=-\delta_{\hat{p}}^{\hat{s}}\delta_{\hat{r}}^{\hat{q}}\,ig^{2}\,\frac{\Gamma(\frac{3}{2}-\epsilon)}{2\pi^{\frac{3}{2}-\epsilon}}\frac{\epsilon_{\mu\nu\rho}(x-y)^{\rho}}{|x-y|^{3-2\epsilon}},\\\
\langle(\psi_{I}^{\alpha})_{\hat{i}}^{j}(x)(\bar{\psi}_{\beta}^{J})_{k}^{\hat{l}}(y)\rangle^{(0)}&=-i\delta_{I}^{J}\delta_{i}^{\hat{l}}\delta_{k}^{j}\frac{\Gamma(\frac{3}{2}-\epsilon)}{2\pi^{\frac{3}{2}-\epsilon}}\frac{(\gamma_{\mu})^{\alpha}_{\
\beta}(x-y)^{\mu}}{|x-y|^{3-2\epsilon}}\\\
&=i\delta_{I}^{J}\delta_{i}^{\hat{l}}\delta_{k}^{j}(\gamma_{\mu})^{\alpha}_{\
\beta}\partial_{\mu}\left(\frac{\Gamma(\frac{1}{2}-\epsilon)}{4\pi^{\frac{3}{2}-\epsilon}}\frac{1}{|x-y|^{1-2\epsilon}}\right),\\\
\langle(C_{I})_{i}^{\hat{j}}(x)(\bar{C}^{J})_{\hat{k}}^{l}(y)\rangle^{(0)}&=\delta_{I}^{J}\delta_{i}^{l}\delta_{\hat{k}}^{\hat{j}}\frac{\Gamma(\frac{1}{2}-\epsilon)}{4\pi^{\frac{3}{2}-\epsilon}}\frac{1}{|x-y|^{1-2\epsilon}},\end{split}$
(A.3)

while the one-loop propagators read

$\begin{split}\langle(A_{\mu})_{p}^{\
q}(x)(A_{\nu})_{r}^{s}(y)\rangle^{(1)}&=\delta_{p}^{s}\delta_{r}^{q}\,g^{4}N\,\frac{\Gamma^{2}(\frac{1}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\left[\frac{\delta_{\mu\nu}}{|x-y|^{2-4\epsilon}}-\partial_{\mu}\partial_{\nu}\frac{|x-y|^{2\epsilon}}{4\epsilon(1+2\epsilon)}\right],\\\
\langle(\hat{A}_{\mu})_{\hat{p}}^{\
\hat{q}}(x)(\hat{A}_{\nu})_{\hat{r}}^{\hat{s}}(y)\rangle^{(1)}&=\delta_{\hat{p}}^{\hat{s}}\delta_{\hat{r}}^{\hat{q}}\,g^{4}N\,\frac{\Gamma^{2}(\frac{1}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\left[\frac{\delta_{\mu\nu}}{|x-y|^{2-4\epsilon}}-\partial_{\mu}\partial_{\nu}\frac{|x-y|^{2\epsilon}}{4\epsilon(1+2\epsilon)}\right]\,.\end{split}$
(A.4)

The latin indices are color indices. For instance, $(A_{\mu})_{p}^{\ q}\equiv
A_{\mu}^{a}(T^{a})_{p}^{\ q}$ where $T^{a}$ are $U(N)$ generators in the
fundamental representation.

##  Appendix B Wilson loops in ABJM theory

#### BPS Wilson loops.

In ABJM theory there is a wide set of BPS operators one can construct, which
may carry some parametric dependence that allows for a continuous
interpolation between different observables. Such set was the main character
of Castiglioni:2022yes . Here we focus instead on non-BPS flows and only a few
BPS operators within the aforementioned set appear in our discussion. We list
these below, together with the pictorial representation used in the figures
throughout the paper.

We start from the simplest realization where the loops are charged under a
single node and are bosonic. In this case we may have operators separately
charged under the first node, associated with $A$, or the second node,
associated with $\hat{A}$, of the ABJM quiver. In Euclidean signature they are
explicitly given by

$\leavevmode\hbox to6.4pt{\vbox to6.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.2pt\lower-3.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@curveto{3.0pt}{1.65688pt}{1.65688pt}{3.0pt}{0.0pt}{3.0pt}\pgfsys@curveto{-1.65688pt}{3.0pt}{-3.0pt}{1.65688pt}{-3.0pt}{0.0pt}\pgfsys@curveto{-3.0pt}{-1.65688pt}{-1.65688pt}{-3.0pt}{0.0pt}{-3.0pt}\pgfsys@curveto{1.65688pt}{-3.0pt}{3.0pt}{-1.65688pt}{3.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\,\,\left\\{\begin{aligned}
W_{1/6}&=\operatorname{Tr}{\mathcal{P}}\exp\bigg{(}-i\oint
d\tau\,{\mathcal{A}}\bigg{)}\,,\quad{\mathcal{A}}\equiv
A_{\mu}\dot{x}^{\mu}-ig^{2}|\dot{x}|{M_{I}}^{J}C^{I}\bar{C}_{J}\,,\\\
\hat{W}_{1/6}&=\operatorname{Tr}{\mathcal{P}}\exp\bigg{(}-i\oint
d\tau\,\hat{{\mathcal{A}}}\bigg{)}\,,\quad\hat{{\mathcal{A}}}\equiv\hat{A}_{\mu}\dot{x}^{\mu}-ig^{2}|\dot{x}|{M_{I}}^{J}\bar{C}_{J}C^{I},\end{aligned}\right.$
(B.1)

where $g^{2}=2\pi/k$. In both cases for $M=\pm\operatorname{diag}(-1,-1,1,1)$
(plus any other permutation of the diagonal entries) these operators become
$1/6$ BPS. Whenever present in the figures, these $SU(2)\times SU(2)$
symmetric objects are represented as blue dots/balls.

In addition, we also have a pair of $1/2$ BPS operators that are charged under
both nodes of the quiver and are therefore written in terms of a
superconnection ${\mathcal{L}}$. Explicitly, we have

$\leavevmode\hbox to7.51pt{\vbox to7.51pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\,\,\;\begin{aligned}
W^{\pm}_{1/2}&=\operatorname{Tr}{\mathcal{P}}\exp\bigg{(}-i\oint
d\tau\,{\mathcal{L}}^{\pm}\bigg{)}\,,\quad{\mathcal{L}}^{\pm}\equiv\begin{pmatrix}{\mathcal{A}}&&-ig\eta\bar{\psi}^{1}\\\
-ig\psi_{1}\bar{\eta}&&\hat{{\mathcal{A}}}\end{pmatrix}\,,\end{aligned}$ (B.2)

where ${\mathcal{A}}$ and $\hat{\mathcal{A}}$ are as in (B.1) but with scalar
coupling matrix $M=\ell\operatorname{diag}(-1,1,1,1)$. Correspondingly the
commuting spinors $\eta$ and $\bar{\eta}$ are those in (4.3). $W^{+}_{1/2}$ is
then obtained by setting $\ell=1$, while $W^{-}_{1/2}$ corresponds to
$\ell=-1$. In our figures these $SU(3)$ symmetric BPS objects are represented
as red triangles/pyramids.

#### Non-BPS Wilson loops.

We find fixed points corresponding to non-BPS operators that are $SU(3)$
invariant. As in the BPS case, for bosonic operators these can be defined
separately for each node of the ABJM quiver,

$\leavevmode\hbox to7.51pt{\vbox to7.51pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\,\,\left\\{\begin{aligned}
W&=\operatorname{Tr}{\mathcal{P}}\exp\bigg{(}-i\oint
d\tau\,{\mathcal{A}}\bigg{)}\,,\quad{\mathcal{A}}=A_{\mu}\dot{x}^{\mu}-ig^{2}|\dot{x}|{M_{I}}^{J}C^{I}\bar{C}_{J}\,,\\\
\hat{W}&=\operatorname{Tr}{\mathcal{P}}\exp\bigg{(}-i\oint
d\tau\,\hat{{\mathcal{A}}}\bigg{)}\,,\quad\hat{{\mathcal{A}}}=\hat{A}_{\mu}\dot{x}^{\mu}-ig^{2}|\dot{x}|{M_{I}}^{J}\bar{C}_{J}C^{I}.\end{aligned}\right.$
(B.3)

with $M=\pm\operatorname{diag}(-1,1,1,1)$ or permutations. Such options should
be equivalent up to R-symmetry rotations.

In addition, we also find $SU(3)$ fixed points that are fermionic, in which
case they are defined as

$\leavevmode\hbox to7.51pt{\vbox to7.51pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{0.0pt}\pgfsys@lineto{3.5566pt}{7.11319pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\,\,\;\begin{aligned}
W&=\operatorname{Tr}{\mathcal{P}}\exp\bigg{(}-i\oint
d\tau\,{\mathcal{L}}\bigg{)}\,,\quad{\mathcal{L}}\equiv\begin{pmatrix}{\mathcal{A}}&&-ig\eta\bar{\psi}^{1}\\\
-ig\psi_{1}\bar{\eta}&&\hat{{\mathcal{A}}}\end{pmatrix}\,,\end{aligned}$ (B.4)

with $\cal A,\hat{\cal A}$ given as in (B.1) but now with
$M=\ell\operatorname{diag}(-1,-3,-3,-3)$, and $\eta$, $\bar{\eta}$ given in
(4.3). Such points are shown in figures 9 and 11.

Finally, there are also $SU(2)$ fermionic fixed points, denoted as black
circles. These are defined as

$\leavevmode\hbox to6.4pt{\vbox to6.4pt{\pgfpicture\makeatletter\hbox{\hskip
3.2pt\lower-3.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{3.0pt}{0.0pt}\pgfsys@curveto{3.0pt}{1.65688pt}{1.65688pt}{3.0pt}{0.0pt}{3.0pt}\pgfsys@curveto{-1.65688pt}{3.0pt}{-3.0pt}{1.65688pt}{-3.0pt}{0.0pt}\pgfsys@curveto{-3.0pt}{-1.65688pt}{-1.65688pt}{-3.0pt}{0.0pt}{-3.0pt}\pgfsys@curveto{1.65688pt}{-3.0pt}{3.0pt}{-1.65688pt}{3.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\,\,\;\begin{aligned}
W&=\operatorname{Tr}{\mathcal{P}}\exp\bigg{(}-i\oint
d\tau\,{\mathcal{L}}\bigg{)}\,,\quad{\mathcal{L}}\equiv\begin{pmatrix}{\mathcal{A}}&&-ig\eta\bar{\psi}^{1}\\\
-ig\psi_{1}\bar{\eta}&&\hat{{\mathcal{A}}}\end{pmatrix}\,,\end{aligned}$ (B.5)

with $M=\operatorname{diag}(-1,1,-3,-3)$ or $M=\operatorname{diag}(-1,-3,1,1)$
in $\cal A,\hat{\cal A}$. They appear in figures 9 and 9.

##  Appendix C Renormalization computations

In this section we report the explicit calculation of the $\beta$-functions.
We follow what was done for the $1/24$ BPS interpolating Wilson loop in
Castiglioni:2022yes . We study the renormalization of the quantum field theory
defined by the following effective action

$\begin{split}S_{\textrm{eff}}=S_{\textrm{ABJM}}+\int
d\tau\bar{\Psi}\left(\partial_{\tau}+i{\mathcal{L}}\right)\Psi\,,\end{split}$
(C.1)

where ${\mathcal{L}}$ is the deformed Wilson loop (super)connection and we
have defined the one-dimensional Grassmann odd superfield

$\Psi=\begin{pmatrix}z&\varphi\\\
\tilde{\varphi}&\tilde{z}\end{pmatrix}\,,\qquad\bar{\Psi}=\begin{pmatrix}\bar{z}&\bar{\tilde{\varphi}}\\\
\bar{\varphi}&\bar{\tilde{z}}\end{pmatrix}\,,$ (C.2)

where $z$ ($\tilde{z}$) and $\varphi$ ($\tilde{\varphi}$) are a spinor and a
scalar, respectively, in the fundamental representation of $U(N)$.

The tree-level propagators of the one-dimensional fields are

$\displaystyle=\langle z^{i}(\tau_{1})\bar{z}_{j}(\tau_{2})\rangle$
$\displaystyle=\delta_{j}^{i}\,\theta(\tau_{1}-\tau_{2})\,,$ (C.3)
$\displaystyle=\langle{\tilde{z}}^{\hat{i}}(\tau_{1})\bar{\tilde{z}}_{\hat{j}}(\tau_{2})\rangle$
$\displaystyle=\delta_{\hat{j}}^{\hat{i}}\,\theta(\tau_{1}-\tau_{2})\,,$
$\displaystyle=\langle\varphi^{i}(\tau_{1})\bar{\varphi}_{j}(\tau_{2})\rangle$
$\displaystyle=\delta_{j}^{i}\,\theta(\tau_{1}-\tau_{2})\,,$
$\displaystyle=\langle\tilde{\varphi}^{\hat{i}}(\tau_{1})\bar{\tilde{\varphi}}_{\hat{j}}(\tau_{2})\rangle$
$\displaystyle=\delta_{\hat{j}}^{\hat{i}}\,\theta(\tau_{1}-\tau_{2})\,.$

In order to renormalize the theory, for each one-dimensional field
$\phi=\\{\varphi,\tilde{\varphi},z,\tilde{z}\\}$ we introduce the
corresponding renormalization function as
$\phi=Z_{\phi}^{-\frac{1}{2}}\phi_{0}$, where $\phi_{0}$ stands for the bare
quantity.

Focusing on the one-loop renormalization of the $\zeta_{i}$ parameters in the
bosonic deformation (3.2), we need to consider the scalar vertex
$\bar{z}C\bar{C}z$ (similarly for the other one-dimensional fields). We define
the renormalization functions $Z_{\zeta_{i}}$ such that

$(\zeta_{i})_{0}=Z_{\zeta_{i}}\zeta_{i}=(1+\delta_{\zeta_{i}})\zeta_{i}\,.$
(C.4)

We write the action (C.1) as a function of the renormalized parameters adding
the counterterm $g^{2}\delta M_{I}^{\ J}\bar{z}C_{J}\bar{C}^{I}z$ with

$\delta M_{I}^{\ J}=2\begin{pmatrix}\delta_{\zeta_{1}}\zeta_{1}&0&0&0\\\
0&\delta_{\zeta_{2}}\zeta_{2}&0&0\\\ 0&0&\delta_{\zeta_{3}}\zeta_{3}&0\\\
0&0&0&\delta_{\zeta_{4}}\zeta_{4}\end{pmatrix}\,.$ (C.5)

For the other one-dimensional scalar field vertices the calculations are
similar.

###  C.1 Bosonic deformation

Figure 16: One-loop corrections to the $\bar{z}C\bar{C}z$ vertex. 16 and 16
are purely bosonic diagrams, while 16 contains fermions.

The divergent scalar one-loop diagrams related to the $\bar{z}C\bar{C}z$
vertex are depicted in figures 16 and 16. We refer to Castiglioni:2022yes  for
the explicit computation. We find

$\begin{split}&\Gamma^{\rm\ref{subfig:diaga}}=-\frac{g^{4}N}{8\pi\epsilon}M_{I}^{\
K}M_{K}^{\ J}\int d\tau\,\bar{z}\,C_{J}\bar{C}^{I}z\,,\\\
&\Gamma^{\rm\ref{subfig:diagb}}=\frac{g^{4}N}{8\pi\epsilon}\int
d\tau\,\bar{z}\,C_{I}\bar{C}^{I}z\,,\end{split}$ (C.6)

where $M$ is the deformed scalar matrix.

These are the only divergent contributions to the vertex, as in the absence of
fermions there is no field function renormalization at one loop. In minimal
subtraction scheme the counterterm $\delta M_{I}^{\ J}$ is then obtained by
imposing

$0=g^{2}\left[\delta M_{I}^{\ J}-\frac{g^{2}N}{8\pi\epsilon}M_{I}^{\
K}M_{K}^{\ J}+\frac{g^{2}N}{8\pi\epsilon}\delta_{I}^{J}\right]\int
d\tau\bar{z}C_{J}\bar{C}^{I}z\,.$ (C.7)

From each diagonal element of (C.7) we find the corresponding counterterm

$\delta_{\zeta_{i}}\zeta_{i}=\frac{g^{2}N}{4\pi\epsilon}(\zeta_{i}-1)\zeta_{i}\,.$
(C.8)

In dimensional regularization, $d=3-2\epsilon$ with minimal subtraction, the
$\zeta_{i}$ parameters are dimensionless while $g^{2}$ has dimension
$\Delta_{g^{2}}=2\epsilon$. We can then write the $\beta$-functions for
$\zeta_{i}$ as

$\beta_{\zeta_{i}}=2g^{2}\frac{\partial K_{\zeta_{i}}}{\partial g^{2}}\,,$
(C.9)

where $K_{\zeta_{i}}$ are the coefficients of the divergent part of
$(\zeta_{i})_{0}$ as a function of $\zeta_{i}$ (see for instance equation
(3.43) in Castiglioni:2022yes ). Therefore, combining (C.4) with (C.8), we
eventually find

$\beta_{\zeta_{i}}=\frac{g^{2}N}{2\pi}(\zeta_{i}-1)\zeta_{i}\,.$ (C.10)

###  C.2 Fermionic deformation

The presence of fermions in the Wilson loop superconnection gives rise to one
extra divergent diagram, figure 16, which contributes to the
$\bar{z}C\bar{C}z$ vertex renormalization. Moreover we also have to take into
consideration the renormalization of the $z$ field.

Diagram 16 gives two different contributions. The first one arises from the
ABJM vertex $C_{I}\bar{C}^{I}\bar{\psi}^{J}\psi_{J}$. This contribution is not
affected by the two possible values of $\ell$ in the definition of
$\eta,\bar{\eta}$. Turning on only the $i$th deformation we find

$\Gamma^{\rm\ref{subfig:diagc}}_{1}=\frac{g^{2}N}{4\pi\epsilon}\chi_{i}^{2}\int
d\tau\,\bar{z}C_{I}\bar{C}^{I}z\,,\qquad i=1,2,3,4\,.$ (C.11)

This is a divergent contribution to the diagonal scalar coupling matrix
$M_{I}^{\;J}$ to be included in (C.7). More generally, if we turn on more than
one $\chi$-deformation the overall contribution will be the sum of
$\Gamma^{\rm\ref{subfig:diagc}}_{1}$, one for each $\chi_{i}$.

The second contribution comes from using the ABJM vertex
$C_{I}\bar{C}^{J}\bar{\psi}^{I}\psi_{J}$. Also in this case, if we turn on
only one fermion the result is not affected by the choice of $\ell$. For the
$i$th deformation we find

$\Gamma^{\rm\ref{subfig:diagc}}_{2}=-\frac{g^{2}N}{2\pi\epsilon}\chi_{i}^{2}\int
d\tau\,\bar{z}C_{I}\bar{C}^{I}z\,,\qquad i=1,2,3,4\,.$ (C.12)

However, from the point of view of the $\ell$ dependence this time things
drastically change if we turn on more than one fermion. In fact, if we
consider a two-fermion deformation corresponding to $\chi_{a}$ with $\ell_{a}$
and $\chi_{b}$ with $\ell_{b}$ (with $a\neq b$) we find

$\Gamma^{\rm\ref{subfig:diagc}}_{2}=-\frac{g^{2}N}{2\pi\epsilon}\bigg{(}\chi_{a}^{2}\,\delta_{I}^{a}\delta_{a}^{J}+\chi_{b}^{2}\,\delta_{I}^{b}\delta_{b}^{J}-\frac{|\ell_{a}+\ell_{b}|}{2}\chi_{a}\chi_{b}\,\big{(}\delta_{I}^{a}\delta_{b}^{J}+\delta_{I}^{b}\delta_{a}^{J}\big{)}\bigg{)}\int
d\tau\,\bar{z}C_{J}\bar{C}^{I}z\,,$ (C.13)

where there is no implicit sum over $a$ and $b$. Therefore, if we turn on two
fermions with the same $\ell$, off-diagonal contributions arise in the scalar
coupling matrix. On the contrary, turning on two fermions with opposite $\ell$
leaves the scalar coupling matrix diagonal.

Since in the main text we restricted our analysis to diagonal scalar matrices,
we consider only the case of double fermionic deformations with opposite
$\ell$’s, let’s say $\ell_{a}=-\ell_{b}=\ell$.

In the present case extra divergent contributions to the vertex come from the
field function renormalization. In fact, we recall that in the presence of
fermions the scalar coupling matrix renormalizes as Castiglioni:2022yes

$(M_{I}^{\ J})_{0}={Z_{z}}^{-1}\left(M_{I}^{\ J}+\delta M_{I}^{\
J}\right)\simeq(1-\delta_{z})M_{I}^{\ J}+\delta M_{I}^{\ J}\,,$ (C.14)

with $\delta_{z}=-\frac{g^{2}N}{4\pi\epsilon}\ell(\chi_{a}^{2}-\chi_{b}^{2})$.

Proceeding as done in C.1, we impose the matrix counterterm $(\delta
M-\delta_{z}M)$ to cancel divergent contributions (C.7, C.12, C.13). We write

$\begin{split}0=g^{2}\bigg{[}\delta M_{I}^{\
J}-\frac{g^{2}N}{8\pi\epsilon}\big{(}M_{I}^{\ K}M_{K}^{\
J}-\delta_{I}^{J}\big{)}&+\frac{g^{2}N}{4\pi\epsilon}\big{(}\chi_{a}^{2}+\chi_{b}^{2}\big{)}\delta_{I}^{J}\\\
&-\frac{g^{2}N}{2\pi\epsilon}\big{(}\chi_{a}^{2}\delta_{I}^{a}\delta_{a}^{J}+\chi_{b}^{2}\delta_{I}^{b}\delta^{J}_{b}\big{)}\bigg{]}\int
d\tau\,\bar{z}C_{J}\bar{C}^{I}z\,,\end{split}$ (C.15)

from which we find

$\displaystyle(\zeta_{a})_{0}$
$\displaystyle=\left[1+\frac{g^{2}N}{4\pi\epsilon}\left(\zeta_{a}-1+\ell(\chi_{a}^{2}-\chi_{b}^{2})+\frac{(1-\ell)}{2}\frac{(\chi_{a}^{2}-\chi_{b}^{2})}{\zeta_{a}}\right)\right]\zeta_{a}\,,$
(C.16) $\displaystyle(\zeta_{b})_{0}$
$\displaystyle=\left[1+\frac{g^{2}N}{4\pi\epsilon}\left(\zeta_{b}-1+\ell(\chi_{a}^{2}-\chi_{b}^{2})-\frac{(1+\ell)}{2}\frac{(\chi_{a}^{2}-\chi_{b}^{2})}{\zeta_{b}}\right)\right]\zeta_{b}\,,$
$\displaystyle(\zeta_{c})_{0}$
$\displaystyle=\left[1+\frac{g^{2}N}{4\pi\epsilon}\left(\zeta_{c}-1+\ell(\chi_{a}^{2}+\chi_{b}^{2})-\frac{(1+\ell)}{2}\frac{\chi_{a}^{2}}{\zeta_{c}}-\frac{(1-\ell)}{2}\frac{\chi_{b}^{2}}{\zeta_{c}}\right)\right]\zeta_{c}\,,\quad
c\neq a,b\,.$

Therefore, the $\beta$-functions read

$\begin{split}\beta_{\zeta_{a}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{a}-1+\ell(\chi_{a}^{2}-\chi_{b}^{2})+\frac{(1-\ell)}{2}\frac{(\chi_{a}^{2}-\chi_{b}^{2})}{\zeta_{a}}\right)\zeta_{a}\,,\\\
\beta_{\zeta_{b}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{b}-1+\ell(\chi_{a}^{2}-\chi_{b}^{2})-\frac{(1+\ell)}{2}\frac{(\chi_{a}^{2}-\chi_{b}^{2})}{\zeta_{b}}\right)\zeta_{b}\,,\\\
\beta_{\zeta_{c}}&=\frac{g^{2}N}{2\pi}\left(\zeta_{c}-1+\ell(\chi_{a}^{2}-\chi_{b}^{2})-\frac{(1+\ell)}{2}\frac{\chi_{a}^{2}}{\zeta_{c}}-\frac{(1-\ell)}{2}\frac{\chi_{b}^{2}}{\zeta_{c}}\right)\zeta_{c}\,,\quad
c\neq a,b\,.\end{split}$ (C.17)

##  Appendix D Enriched flows realization

As highlighted in figure 15, the three fixed points $W_{1/6}$ and
$W_{1/2}^{\pm}$ can be connected both through enriched flows previously
explored in Castiglioni:2022yes  or through mixed flows as we study here.
Naturally, we should be able to recover the former from the latter by
specifically tuning our mixed bosonic plus one fermion deformation. In figure
18 we propose a picture that captures the connecting features of each
construction.

Concretely, $W_{1/2}^{+}$ can be obtained from $W_{1/6}$ through a mixed flow
in the $(\zeta_{2},\chi_{1})$ plane and $\ell=1$, as described in section 5.1.
An enriched flow, i.e. a trajectory that is BPS along all of its points (and
not only at the fixed points), is then obtained imposing the constraint
$\zeta_{2}=\chi_{1}^{2}$. For generic $\zeta_{2}$ the corresponding operators
are the $1/6$ BPS fermionic loops in green in figure 13 of Castiglioni:2022yes
, whereas at $\zeta_{2}=1$ it becomes $W_{1/2}^{+}$ (see figure 18).

On the other hand, $W_{1/6}$ can be obtained from $W_{1/2}^{-}$ through a
mixed bosonic plus one-fermion flow with $\ell=-1$. We have not considered
such a construction explicitly in the main text and for completeness we
include it here. The realization involves replicating (6.11)-(6.13) with the
additional bosonic contribution from $\zeta_{2}$. Starting with $W_{1/2}^{-}$
in eq. (B.2) with $\ell=-1$, we add the mixed deformation

$\hat{d}=-2g^{2}\zeta_{2}\begin{pmatrix}C_{2}\bar{C}^{2}&0\\\
0&\bar{C}^{2}C_{2}\end{pmatrix}+g\tilde{\chi}_{1}\begin{pmatrix}0&\eta\bar{\psi}^{1}\\\
\psi_{1}\bar{\eta}&0\end{pmatrix}\,.$ (D.1)

The corresponding RG flows are then described by the fermionic
$\beta$-function in (6.13) together with

$\beta_{\zeta_{2}}=\frac{g^{2}N}{2\pi}\big{(}(\zeta_{2}-2)-\tilde{\chi}_{1}(\tilde{\chi}_{1}-2)\big{)}\zeta_{2}\,.$
(D.2)

We depict the flow in figure 17. In particular, the enriched flow is obtained
imposing the constraint $\zeta_{2}=-\tilde{\chi}_{1}(\tilde{\chi}_{1}-2)$ and
at the particular point where $\zeta_{2}=1$ we recover $W_{1/6}$ with $M={\rm
diag}(1,1,-1,-1)$.

Figure 17: Flows starting from $W_{1/2}^{-}$. Fixed points correspond to
bosonic operators, either $SU(3)$ non-BPS or a $1/6$ BPS $W_{1/6}$. The
trajectory in green corresponds to an enriched flow, made by a continuum of
$1/6$ BPS fermionic loops.

Even though by imposing these constraints, either between $\zeta_{2}$ and
$\chi_{1}$ or between $\zeta_{2}$ and $\tilde{\chi}_{1}$, we do recover
enriched flows, these are not precisely the ones studied in
Castiglioni:2022yes , see figure 1 there. Indeed, $W_{1/2}^{+}$ is nothing but
${\mathcal{W}}_{1/2}^{\rm I}$ introduced there, while $W_{1/2}^{-}$ coincides
with ${\mathcal{W}}_{1/2}^{\rm II}$ of Castiglioni:2022yes  only upon an
R-symmetry transformation that interchanges the $1$ and $4$ indices. In fact,
if we look at the fermionic couplings, for instance, $W_{1/2}^{-}$ couples to
$\psi_{1},\bar{\psi}^{1}$ whereas ${\mathcal{W}}_{1/2}^{\rm II}$ couples to
$\psi_{4},\bar{\psi}^{4}$. Nevertheless, consistently, under RG flows both
$W_{1/2}^{-}$ and ${\mathcal{W}}_{1/2}^{\rm II}$ share the same nature, in the
sense that both are repulsive.

The R-symmetry difference between $W_{1/2}^{-}$ and ${\mathcal{W}}_{1/2}^{\rm
II}$ has its origin on a quite technical but nevertheless interesting aspect:
from the point of view of enriched flows they are built upon different (but
R-equivalent) $1/6$ BPS operators.

To clarify this statement, we recall that the construction for enriched flows
is based on the deformation of a $1/6$ BPS bosonic operator. Such a
deformation is written in terms of a matrix $G$ that may include all four
scalars of the theory via $\alpha$ and $\beta$ parameters,111111This
construction was originally proposed in the second chapter of Drukker:2019bev
and then generalized in drukker2020bps ; Drukker:2020dvr ; Drukker:2022ywj ;
Kong:2022yib .

$G=\begin{pmatrix}0&&\bar{\alpha}^{1}C_{1}+\bar{\alpha}^{2}C_{2}+e^{-i\tau}(\beta^{3}C_{3}+\beta^{4}C_{4})\\\
\alpha_{1}\bar{C}^{1}+\alpha_{2}\bar{C}^{2}+e^{i\tau}(\bar{\beta}_{3}\bar{C}^{3}+\bar{\beta}_{4}\bar{C}^{4})&&0\end{pmatrix}\,.$
(D.3)

In Castiglioni:2022yes , the $W_{1/6}$ fixed point around which the
deformation is constructed is characterized by a scalar coupling matrix
$M=\operatorname{diag}(-1,-1,+1,+1)$ that breaks $SU(4)$ R-symmetry to
$SU(2)\times SU(2)$. One can construct BPS deformations which interpolate
between $W_{1/6}$ and either ${\mathcal{W}}_{1/2}^{\rm I}$ or
${\mathcal{W}}_{1/2}^{\rm II}$, according to which $SU(2)$ subgroup the
scalars in $G$ belong to. For instance, as shown in figure 18, if we restrict
the parameters in $G$ to be $\alpha_{2},\bar{\alpha}^{2}$ we obtain a family
of 1/6 BPS fermionic WLs interpolating between $W_{1/6}$ and
${\mathcal{W}}_{1/2}^{\rm I}$. Instead, if in $G$ we turn on only
$\beta^{3},\bar{\beta}_{3}$ we flow from ${\mathcal{W}}_{1/2}^{\rm II}$ to
$W_{1/6}$. It turns out that the upper triangle ${\mathcal{W}}_{1/2}^{\rm I}$
coincides with $W_{1/2}^{+}$, instead, as one would naively expect, the lower
triangle is not exactly $W_{1/2}^{-}$.

If we were to find $W_{1/2}^{-}$ from the enriched flow interpolation, we
should consider as one of the two fixed points the 1/6 BPS operator with
opposite $M$, as in Castiglioni:2022yes . This other flow is depicted in
figure 18. In this case, the BPS interpolation obtained via
$\alpha_{2},\bar{\alpha}^{2}$ would give rise exactly to $W_{1/2}^{-}$. Note
that, in accordance with the map outlined above, from the point of view of
mixed deformations, $W_{1/2}^{+}$ and $W_{1/2}^{-}$ are both obtained by
turning on $\zeta_{2}$.

Naturally, the flows mentioned above are particular realizations and do not
cover all possibilities. In particular, we could also have their $\zeta_{3}$
counterparts in both 18 and 18, as well as the $\beta^{3},\bar{\beta}_{3}$
flow in 18. We include these flows as light curves to remind that there is
such a connection without overloading the picture.

Figure 18: Comparison between enriched RG flows and flows from mixed
deformations.

This technical discussion addresses subtleties that should be taken into
account when investigating the strong coupling setting. In particular, it
follows from the construction of each type of flow that, whereas
${\mathcal{W}}_{1/2}^{\rm I,II}$ necessarily share a subset of preserved
supercharges, the pair $W_{1/2}^{\pm}$ does not share any. Thus when studying
the corresponding brane/anti-brane description in the dual setting along the
lines of Lietti:2017gtc , we should keep in mind that mixed flows, and not the
enriched ones, are the ones to be considered.

We have discussed one possible scenario of complementary choices of $M$,
suitable for turning on enriched flows realized by
$\alpha_{2},\bar{\alpha}^{2}$ (alias $\zeta_{2},\chi_{1}$ mixed flows in this
paper) or $\beta^{3},\bar{\beta}_{3}$ ($\zeta_{3},\chi_{4}$ here).
Alternatively, we could consider complementary $M$ pairs suitable for
$\alpha_{1},\bar{\alpha}^{1}$ or $\beta^{4},\bar{\beta}_{4}$ ($\zeta_{1}$ or
$\zeta_{4}$) deformations. In this case we would unravel four extra $1/2$ BPS
points that, instead of coupling to fermions with R-symmetry index 1 or 4,
would couple to those of index 2 or 3.

The eight possible realizations of 1/2 BPS fixed points are for instance
collected in figure 2(a) of Lietti:2017gtc . Referring to that diagrammatic
representation, enriched flows interpolate between operators connected by
solid red lines (operators that share 2/3 of preserved supercharges), whereas
mixed flows connect neighbour operators sharing no common supercharges,
denoted there as $W_{i}$ and $\tilde{W}_{i}$.

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