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

# Interpolating Wilson loops and enriched RG flows

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

We study new $1/24$ BPS circular Wilson loops in ABJ(M) theory, which are
defined in terms of several parameters that continuously interpolate between
previously known $1/6$ BPS loops (both bosonic and fermionic) and $1/2$ BPS
fermionic loops. We compute the expectation value of these operators up to
second order in perturbation theory using a one-dimensional effective field
theory approach. Within dimensional regularization, we find non-trivial
$\beta$-functions for the parameters, which are marginally relevant
deformations triggering RG flows from a UV fixed point represented by the
$1/6$ BPS bosonic loop to an IR fixed point represented by a $1/2$ BPS
fermionic loop. Generically, along all flows at least one supercharge of the
theory is preserved, so that we refer to them as enriched RG flows. In
particular, fixed points are connected through $1/6$ BPS fermionic operators.
This holds at framing zero, which is a consequence of the regularization
scheme employed. We also establish a g-theorem, relating the expectation
values of the Wilson loops corresponding to the UV and IR fixed points of the
flow, and discuss the one-dimensional defect SCFT living on the Wilson loop
contour.

###### Keywords:

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

##  1 Introduction

Over the past few years, three-dimensional Chern-Simons-matter theories have
been shown to display rich moduli spaces of supersymmetric line operators,
starting with the discovery of $1/6$ BPS bosonic Wilson loops Drukker:2008zx ;
Chen:2008bp ; Kluson:2008zrv ; Rey:2008bh , vortex loops Drukker:2008jm  and
the $1/2$ BPS fermionic Wilson loop Drukker:2009hy  of the ABJ(M) theory
Aharony:2008ug ; Aharony:2008gk . These studies have been subsequently
extended to less supersymmetric settings like ${\mathcal{N}}\geq 2$ quiver
theories Gaiotto:2008sd ; Imamura:2008dt ; Hosomichi:2008jd ; Hama:2010av  in,
for example, Gaiotto:2007qi ; Ouyang_2015 ; Cooke_2015 ; Ouyang:2015iza ;
Ouyang:2015bmy ; Mauri:2017whf ; Mauri_2018 , to continue with more recent
attempts at a full classification of so-called hyperloop operators in
drukker2020bps ; Drukker:2020dvr ; Drukker:2022ywj ; Drukker:2022bff . See
Drukker:2019bev  for a review.

A characteristic feature in the construction of the BPS Wilson loops in these
theories is the appearance of parametric families of operators interpolating
between different amounts of preserved supersymmetries. One can in fact start
from a given operator, be it bosonic as in Drukker:2020dvr  or fermionic as in
Drukker:2022ywj , choose a combination of supercharges it preserves and write
down a deformation of that operator built out of the matter fields, which
still preserves that supercharge. For special values of the parameters
entering the definition of the deformation, supersymmetry enhancement is
possible. This allows to interpolate continuously among different operators,
preserving a varying number of supercharges of the theory.

Given this plethora of BPS Wilson loops, it is natural to study the
interpolations among them from the point of view of RG flows on defects,
following the seminal work by Polchinski-Sully Polchinski_2011  and the
subsequent literature, see for example Beccaria:2017rbe ; Beccaria:2018ocq ;
Correa:2019rdk ; Cuomo:2021rkm ; Beccaria:2021rmj ; Beccaria:2022bcr ;
Garay:2022szq ; Aharony:2022ntz . In those cases, one has typically one
parameter interpolating between BPS and non-supersymmetric operators, like the
prototypical example of the $\zeta$-deformed operator introduced in
Polchinski_2011  for ${\mathcal{N}}=4$ super Yang-Mills in four dimensions,
which interpolates between the ordinary, non-supersymmetric Wilson loop for
$\zeta=0$ and the $1/2$ BPS Wilson-Maldacena loop Maldacena_1998  for
$\zeta=1$.

In this paper we initiate a study of such RG flows between Wilson loop
operators in ABJ(M) theory. One main difference with respect to the cases
mentioned above is that our flows are between operators that always preserve
some supercharge, being therefore ‘enriched’ flows: symmetries (in particular
supersymmetries) are not completely broken along the flow, similarly to what
has been considered in Cordova:2022lms . Moreover, the flow spaces we consider
are multi-dimensional,111For an example in four dimensions see
Beccaria:2018ocq , in which the deformation of the latitude Wilson loop in
${\mathcal{N}}=4$ super Yang-Mills is considered. as these Wilson loops are
defined in terms of more than one parameter undergoing renormalization.

More specifically, we consider ABJ(M) theory on $\mathbb{R}^{3}$ and introduce
a new BPS circular Wilson loop that preserves, generically, only one
supercharge of the theory and is therefore $1/24$ BPS. We call it
${\mathcal{W}}_{1/24}$. This operator has not been discussed before in ABJ(M),
but its equivalent has appeared in the context of hyperloops in
${\mathcal{N}}=4$ Chern-Simons-matter theories Drukker:2020dvr  and can be
mapped to a corresponding operator in ABJ(M). This Wilson loop is defined in
terms of a superconnection containing a coupling to the scalars and the
fermions of the theory through 8 dimensionless parameters that we call
$\alpha_{i},\bar{\alpha}^{i}$ (with $i=1,2$) and $\beta^{j},\bar{\beta}_{j}$
(with $j=3,4$). For generic values of these parameters the operator is $1/24$
BPS, as already mentioned. Selecting either $\alpha_{i}=\bar{\alpha}^{i}=0$ or
$\beta^{j}=\bar{\beta}_{j}=0$, there is supersymmetry enhancement and the loop
from $1/24$ BPS becomes a $1/6$ BPS fermionic operator. In fact there are two
different $1/6$ BPS operators that can be obtained in this way (one with
vanishing alphas and one with vanishing betas), which we denote
${\mathcal{W}}^{\textrm{I}}_{1/6}$ and ${\mathcal{W}}^{\textrm{II}}_{1/6}$
Ouyang:2015iza ; Ouyang:2015bmy . If, moreover, the remaining parameters are
set to a specific value, $\alpha_{i}\bar{\alpha}^{i}=1$ or
$\bar{\beta}_{j}\beta^{j}=-1$, respectively, the $1/6$ BPS fermionic operators
become the $1/2$ BPS fermionic Wilson loops ${\mathcal{W}}_{1/2}^{\textrm{I}}$
and ${\mathcal{W}}_{1/2}^{\textrm{II}}$ Drukker:2009hy . These two loops
differ by an overall sign in the scalar coupling, with
${\mathcal{W}}_{1/2}^{\textrm{I}}$ being the loop with a mostly plus coupling
originally introduced in Drukker:2009hy . In this paper we shall be mainly
interested in ${\mathcal{W}}_{1/2}^{\textrm{I}}$. On the other hand, when all
the parameters are turned off at the same time,
$\alpha_{i}=\bar{\alpha}^{i}=\beta^{j}=\bar{\beta}_{j}=0$, one has the bosonic
$1/6$ BPS Wilson loop ${\mathcal{W}}^{\textrm{bos}}_{1/6}$ of Drukker:2008zx ;
Chen:2008bp ; Kluson:2008zrv ; Rey:2008bh . This network of interpolations is
summarized in figure 1.

Figure 1: The interpolations among the various Wilson loops considered in this
paper.

At the classical level all Wilson loops in figure 1 are cohomologically
equivalent, i.e. their expressions differ by a ${\cal Q}$-exact term, where
${\cal Q}$ is one of the preserved supercharges. In principle, this would
imply that their vacuum expectation values (VEVs) should be all equal and
independent of the parameters. However, this is true for instance for
operators supported along straight lines, but it is no longer true on the
circle, due to the well-known conformal anomaly gross  and framing effects. In
fact, while supersymmetric localization requires framing 1, the regularization
scheme we employ, namely dimensional regularization, is alternative to framing
regularization and corresponds to framing 0. This is the reason why the
circular VEVs that we are going to compute carry a non-trivial dependence on
the parameters, so providing interpolating BPS (enriched) flows.

We compute the vacuum expectation value of this $1/24$ BPS circular loop, for
generic values of the parameters, up to two loops in perturbation theory at
weak coupling. The way we do it is by mapping this problem to the computation
of the two-point function of certain auxiliary fields living on the one-
dimensional theory along the Wilson loop contour. This is something that has
been done in the past Samuel:1978iy ; Gervais:1979fv  for ordinary Wilson
loops, but we extend it to the case at hand, namely for Wilson loops defined
in terms of superconnections. In particular, compared to the previous
applications, in which the one-dimensional theory only contained a Fermi
field, we have a theory with both commuting and anticommuting fields. The
final planar-limit result for the Wilson loop VEV is given by

$\langle{\mathcal{W}}_{1/24}\rangle=1-\frac{\pi^{2}}{6k^{2}}\left[N_{1}^{2}+N_{2}^{2}-4N_{1}N_{2}-3N_{1}N_{2}(\bar{\alpha}^{i}\alpha_{i}-\beta^{j}\bar{\beta}_{j}-1)^{2}\right]+{\cal
O}\left(\frac{1}{k^{3}}\right)\,.$ (1.1)

Here $N_{1}$ and $N_{2}$ are the ranks of the two gauge fields of the
$U(N_{1})_{k}\times U(N_{2})_{-k}$ ABJ(M) theory, and $k$ is the Chern-Simons
level. From this expression one obtains the VEVs of the $1/6$ BPS bosonic (all
parameters equal to zero) and $1/6$ BPS fermionic operators (alphas or betas
equal to zero), as well as of the $1/2$ BPS fermionic operator (last term in
the square bracket equal to zero).

The coupling parameters undergo a non-trivial renormalization, leading to non-
vanishing $\beta$-functions

$\beta_{\alpha_{k}}=\frac{g^{2}}{4\pi}(N_{1}+N_{2})\,(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}-1)\alpha_{k}\,,\qquad\beta_{\beta^{k}}=\frac{g^{2}}{4\pi}(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}+1)\beta^{k}\,,$
(1.2)

with similar expressions for the barred quantities. This shows that the Wilson
loop parameters can be seen as marginally relevant deformations, triggering an
RG flow from a UV fixed point represented by the $1/6$ BPS bosonic Wilson loop
of ABJ(M) towards the $1/2$ BPS loop ${\mathcal{W}}_{1/2}^{\text{I}}$. Such a
flow is presented in figure 13.

This has a nice interpretation in terms of defects. In fact, it is well known
that the bosonic $1/6$ BPS and fermionic $1/6$ and $1/2$ BPS operators
describe one-dimensional superconformal theories (SCFTs) given by local
operators inserted on the Wilson loop contour. Instead, the new $1/24$ BPS
operator supports a defect which is no longer (super)conformal, as it does not
preserve enough supersymmetries and the contour dependence of the scalar
couplings breaks conformality.

In this framework the RG flows depicted in figure 13 can be interpreted as
connecting different (super)conformal defects seated at the fixed points.
Flowing along the green line of that figure we reach a non-trivial IR fixed
point. In the defect theory at the UV fixed point we compute the anomalous
dimension of the parametric perturbation and consistently find a small
negative value, thus confirming that it corresponds to a marginally relevant
deformation.

Finally, from (1.1) and the $\beta$-functions we also establish a g-theorem,
relating the VEVs of the Wilson loops corresponding to the UV and IR fixed
points of the flows

$\log\langle{\mathcal{W}}_{1/6}^{\textrm{bos}}\rangle>\log\langle{\mathcal{W}}_{1/2}^{\textrm{I}}\rangle\,,$
(1.3)

similarly to what has been done for Wilson loops in four dimensions in
Cuomo:2021rkm ; Beccaria:2021rmj , with the main difference being that our
flows are BPS, as stressed above.

As mentioned already above, the comparison of the Wilson loop VEV (1.1) with
the result coming from a matrix model computation Kapustin:2009kz ;
Marino:2009jd  requires taking into account framing issues, see chapter 6 of
Drukker:2019bev  for a review. The regularization scheme employed in our
perturbative computation amounts in fact to computing the VEV at framing
$f=0$, whereas the matrix model computation yields a result valid for $f=1$.
Moreover, at framing one the VEVs of all loops of figure 1 coincide, as these
are all cohomologically equivalent operators. This is clearly not true for
(1.1), which is obtained at framing zero.222Perturbative computations at
framing 1 represent a hard open problem, especially regarding the evaluation
of fermionic diagrams. Note, in particular, how this VEV depends explicitly on
the alpha and beta parameters of the deformation, which are not present in the
definition of the matrix model insertion corresponding to these operators. The
relation between the VEVs of ${\mathcal{W}}_{1/24}$ at different framings is
encoded in a phase, which we find empirically from our two-loop results to be
given by

$\langle{\cal
W}_{1/24}\rangle_{f=1}=\frac{N_{1}e^{\frac{i\pi}{k}(N_{1}-(\bar{\alpha}^{i}\alpha_{i}-\beta^{j}\bar{\beta}_{j})N_{2})}+N_{2}\,e^{\frac{i\pi}{k}((\bar{\alpha}^{i}\alpha_{i}-\beta^{j}\bar{\beta}_{j})N_{1}-N_{2})}}{N_{1}+N_{2}}\langle{\cal
W}_{1/24}\rangle_{f=0}\,.$ (1.4)

We expect this phase to receive corrections at higher order in perturbation
theory, similarly to what happens for the $1/6$ BPS bosonic operator
Bianchi:2016yzj .

In this paper we also introduce a new operator: a $1/12$ BPS latitude Wilson
loop defined in terms of an extra parameter, a latitude angle, along the lines
of what has been done in ${\mathcal{N}}=4$ super Yang-Mills in Drukker:2007dw
; Drukker:2007yx ; Drukker:2007qr  and in three-dimensional theories in
Cardinali:2012ru ; Bianchi:2014laa  and Drukker:2020dvr . In a forthcoming
publication CPTT , we will generalize to this new setting the investigation of
the present paper.

This paper is organized as follows. In section 2 we introduce the $1/24$ BPS
circular Wilson loop, which is going to be the main character of our analysis,
as well as the $1/12$ BPS latitude Wilson loop to be considered in the future.
These operators are defined in terms of either traces or supertraces of
superconnections. The former formulation simplifies the perturbative analysis
while the latter is more natural for superconnections, so we discuss how to go
from one to the other. In section 3 we consider an auxiliary problem in terms
of one-dimensional fields which is suitable for studying the renormalization
of the parameters of the $1/24$ BPS Wilson loop. This allows us to compute the
$\beta$-functions of the Wilson loop parameters. In section 4 we finally
compute the vacuum expectation value of the $1/24$ BPS circular Wilson loop up
to two loops in perturbation theory. This is the main result of this paper,
together with the evaluation of the $\beta$-functions. In section 5 we collect
and discuss our results. Specifically, we describe the RG flows among the
different operators of figure 1 and plot an explicit example, we study the
defect SCFT living on the Wilson loop, we establish the g-theorem mentioned
above, and we compare the Wilson loop VEV with a matrix model computation.
Finally, we conclude that section with some outlook. We collect some technical
aspects in a series of appendices. In appendix A we discuss our notation and
conventions. In appendix B we derive the generalization of the one-dimensional
auxiliary field method to Wilson loops defined in terms of superconnections.
In appendix C we detail the computation of the various Feynman diagrams
considered in the main text.

The reader who is not interested in technical details may skip section 3 and
go directly to section 4.3.

##  2 Theory and supersymmetric loops

The field content of ABJ(M) can be depicted in terms of a quiver diagram as
the one shown in figure 2. It includes two gauge fields, $A$ and $\hat{A}$,
with respective gauge groups $U(N_{1})$ and $U(N_{2})$. The matter sector has
$SU(4)$ R-symmetry and is composed of scalars $C_{I}$ and fermions
$\bar{\psi}^{I}$, $I=\\{1,2,3,4\\}$, in the $(\Box,\bar{\Box})$ representation
of $U(N_{1})\times U(N_{2})$. By conjugation there are also $\bar{C}^{I}$ and
$\psi_{I}$ in $(\bar{\Box},\Box)$.

$A$$\hat{A}$${C_{I}\,\,\bar{\psi}^{I}}$$\bar{C}^{I}\,\,\psi_{I}$$k$$-k$ Figure
2: Quiver diagram of ABJ(M) theory. Below each node we include the level of
the respective copy of the Chern-Simons action.

Wilson loops preserving some amount of the 24 supercharges of the theory can
be constructed by allowing for couplings to scalar bilinears. In this case the
usual gauge connection $A$ is promoted to a bosonic connection
${\mathcal{A}}$. Besides that, there is also the possibility of adding fermi
fields, in which case the bosonic connection is further promoted to a
superconnection ${\mathcal{L}}$ Drukker:2009hy . Recently, the structure of
these operators started being unravelled Drukker:2019bev  through the
understanding that they are related via

${\mathcal{L}}={\mathcal{L}}_{0}+i{\mathcal{Q}}G+G^{2}\,.$ (2.1)

The quantity ${\mathcal{L}}_{0}$ is a composite bosonic connection
complemented by a constant shift in one of the entries333In what follows $c$
will be either $\frac{1}{2}$ or $\frac{\cos\theta}{2}$, but this is not
necessarily always the case Drukker:2022ywj .

${\mathcal{L}}_{0}=\begin{pmatrix}{\mathcal{A}}+c&&0\\\
0&&\hat{\mathcal{A}}\end{pmatrix}\,.$ (2.2)

The supercharge ${\mathcal{Q}}$ is a suitable linear combination of
supercharges preserved by ${\mathcal{L}}_{0}$, and $G$ is an off-diagonal
matrix comprised of scalars. These appear through a set of constant complex
parameters that we denote as
$\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j}$ (with $i=1,2$ and
$j=3,4$), though they are not complex conjugates of each other. The
construction (2.1) is such that ${\mathcal{Q}}$ is always preserved by
${\mathcal{L}}$, but one may find extra preserved supercharges depending on
particular values of the parameters in $G$.

Below we consider two possible operators built from different choices of
${\mathcal{L}}_{0}$. The first one is the $1/24$ BPS circular loop, the
protagonist of the present analysis, and the second one is the $1/12$ BPS
latitude loop of ABJ(M), which is going to be studied in detail in a future
publication CPTT . The construction of the latter is to a great extent
parallel to the $\theta$-deformation considered in Drukker:2020dvr , with the
difference that what we mean by ‘latitude’ here is an actual geometric
latitude of the contour $x^{\mu}$ of the loop, instead of simply an internal
$\theta$-deformation in the space of the couplings.

###  2.1 1/24 BPS circular Wilson loop

The first operator that we are going to consider is supported along the circle

$x^{\mu}=(0,\cos\tau,\sin\tau)\,.$ (2.3)

Its bosonic components can be separately charged under each node of the quiver

$\displaystyle{\mathcal{W}}^{\textrm{bos}}$
$\displaystyle=\operatorname{Tr}{\mathcal{P}}\exp\left(-i\oint{\mathcal{A}}\,d\tau\right)\,,\qquad{\mathcal{A}}=A_{\mu}\dot{x}^{\mu}-\frac{2\pi
i}{k}|\dot{x}|M_{J}^{\ I}C_{I}\bar{C}^{J}\,,$ (2.4)
$\displaystyle\hat{\mathcal{W}}^{\textrm{bos}}$
$\displaystyle=\operatorname{Tr}{\mathcal{P}}\exp\left(-i\oint\hat{\mathcal{A}}\,d\tau\right)\,,\qquad\hat{\mathcal{A}}=\hat{A}_{\mu}\dot{x}^{\mu}-\frac{2\pi
i}{k}|\dot{x}|M_{J}^{\ I}\bar{C}^{J}C_{I}\,.$

When $M_{J}^{\ I}=\operatorname{diag}(-1,-1,1,1)$, they preserve the set of
supercharges

$Q_{12}^{+}-iS_{12}^{+}\,,\quad Q_{12}^{-}+iS_{12}^{-}\,,\quad
Q_{34-}+iS_{34-}\,,\quad Q_{34+}-iS_{34+}\,,$ (2.5)

and are therefore $1/6$ BPS operators Drukker:2008zx ; Chen:2008bp ;
Kluson:2008zrv ; Rey:2008bh .

Their fermionic counterpart can be derived using the prescription outlined
above. In this case, we take the supercharge ${\mathcal{Q}}$ to be given by
the linear combination

${\mathcal{Q}}\equiv(Q_{12}^{+}-iS_{12}^{+})+(Q_{34+}-iS_{34+})\,.$ (2.6)

The constant shift is implemented in the composite bosonic connection
${\mathcal{L}}_{0}$ such that $c=\frac{1}{2}$ and the $G$ matrix includes all
four scalars of the theory as444The parameters appears in $G$ sticking to the
notation of drukker2020bps , so that unbarred (barred) parameters accompany
(anti-)chiral fields in the chiral decomposition of the theory in
${\mathcal{N}}=2$ language. We stress that barred/unbarred parameters are not
complex conjugates of each others.

$G=\begin{pmatrix}0&&\bar{\alpha}^{1}C_{1}+\bar{\alpha}^{2}C_{2}+e^{-i\tau}(\beta^{3}C_{3}+\beta^{4}C_{4})\\\
\alpha_{1}\bar{C}^{1}+\alpha_{2}\bar{C}^{2}+e^{i\tau}(\bar{\beta}_{3}\bar{C}^{3}+\bar{\beta}_{4}\bar{C}^{4})&&0\end{pmatrix}\,.$
(2.7)

Plugging this in (2.1) we find that the resulting superconnection
${\mathcal{L}}$ can be explicitly written as555To write ${\mathcal{L}}$ we
suitably scaled couplings so to recover the $1/2$ BPS loop of Drukker:2009hy
when $\alpha^{2}=\bar{\alpha}_{2}=-1$ and all other parameters are zero. Also,
whenever omitted, spinorial indices are meant to be contracted up-down, i.e.
$\lambda\chi\equiv\lambda^{\alpha}\chi_{\alpha}$.

${\mathcal{L}}=\begin{pmatrix}{\mathcal{A}}^{\prime}+\frac{1}{2}&&\eta\,(\bar{\alpha}^{1}\bar{\psi}^{2}-\bar{\alpha}^{2}\bar{\psi}^{1})+e^{-i\tau}\xi\,(\beta^{3}\bar{\psi}^{4}-\beta^{4}\bar{\psi}^{3})\\\
\xi\,(\alpha_{1}\psi_{2}-\alpha_{2}\psi_{1})+e^{i\tau}\eta\,(\bar{\beta}_{3}\psi_{4}-\bar{\beta}_{4}\psi_{3})&&\hat{{\mathcal{A}}}^{\prime}\end{pmatrix}\,,$
(2.8)

where the commuting spinors $\eta$ and $\xi$ are

$\eta^{\alpha}=\sqrt{\frac{2\pi
i}{k}}(1,-ie^{-i\tau})^{\alpha}\,,\qquad\xi^{\alpha}=\sqrt{\frac{2\pi
i}{k}}(-ie^{i\tau},1)^{\alpha}\,.$ (2.9)

The diagonal entries are primed because now the scalar coupling matrix
$M_{J}^{\ I}$ is such that it receives the contribution coming from $G^{2}$,
i.e. it is

$M_{J}^{\
I}=\begin{pmatrix}-1+2\bar{\alpha}^{1}\alpha_{1}&&2\bar{\alpha}^{1}\alpha_{2}&&2e^{i\tau}\bar{\alpha}^{1}\bar{\beta}_{3}&&2e^{i\tau}\bar{\alpha}^{1}\bar{\beta}_{4}\\\
2\bar{\alpha}^{2}\alpha_{1}&&-1+2\bar{\alpha}^{2}\alpha_{2}&&2e^{i\tau}\bar{\alpha}^{2}\bar{\beta}_{3}&&2e^{i\tau}\bar{\alpha}^{2}\bar{\beta}_{4}\\\
2e^{-i\tau}\beta^{3}\alpha_{1}&&2e^{-i\tau}\beta^{3}\alpha_{2}&&1+2\beta^{3}\bar{\beta}_{3}&&2\beta^{3}\bar{\beta}_{4}\\\
2e^{-i\tau}\beta^{4}\alpha_{1}&&2e^{-i\tau}\beta^{4}\alpha_{2}&&2\beta^{4}\bar{\beta}_{3}&&1+2\beta^{4}\bar{\beta}_{4}\end{pmatrix}\,.$
(2.10)

The resulting operator,

${\cal
W}=\operatorname{sTr}{\mathcal{P}}\exp\left(-i\oint{\mathcal{L}}\,d\tau\right)\,,$
(2.11)

preserves ${\mathcal{Q}}$. Following the proposal of drukker2020bps , it can
be represented in terms of a quiver diagram as the one shown in figure 3.

$\alpha_{1}\,\alpha_{2}$${\bar{\alpha}}^{1}\,\bar{\alpha}^{2}$$\beta^{3}\,\beta^{4}$${\bar{\beta}}_{3}\,\bar{\beta}_{4}$
Figure 3: The quiver diagram of the $1/24$ BPS Wilson loop in ABJ(M).
Following the notation of drukker2020bps , a squiggly circle, like the one on
the left here, represents a node whose bosonic connection is shifted by the
constant $c$. The couplings to (anti-)chiral fields are denoted by solid
(dashed) arrows.

As to the best of our knowledge this is the first time that such an operator
is presented,666The quiver representation of figure 3 has already appeared in
drukker2020bps , see their figure 7, but the corresponding operator was not
written down explicitly. we find it enlightening to stop and make a few
comments about it before proceeding. First of all we notice that in the quiver
of figure 3 solid arrows point both into and out of the squiggly node. From
the analysis in Drukker:2020dvr , one can conclude that only ${\mathcal{Q}}$
is preserved, generically, and the loop is therefore 1/24 BPS.

Particular subcases of supersymmetry enhancement can be read off directly from
the quiver structure. We know Drukker:2020dvr  that when solid arrows point
only into the squiggly node, all supercharges originally preserved by
${\mathcal{L}}_{0}$ are preserved by ${\mathcal{L}}$ and the resulting
operator is $1/6$ BPS. To be explicit, when only $\alpha_{i},\bar{\alpha}^{i}$
parameters appear in $G$, see (2.7), the corresponding operator can be
depicted in terms of a quiver diagram as in figure 4. In this case $G$ breaks
only one $SU(2)$ R-symmetry subgroup of ${\mathcal{L}}_{0}$. Moreover, at the
particular point $\bar{\alpha}^{i}\alpha_{i}=1$ the resulting loop enjoys
extra $SU(3)$ symmetry and becomes $1/2$ BPS. On the other hand, for the case
where only the $\beta^{j},\bar{\beta}_{j}$ parameters appear in $G$, it is
useful to employ the gauge where the constant shift (and therefore the
squigglyness of the corresponding quiver diagram) lies in the second node. In
this case $G$ loses the awkward $e^{\pm i\tau}$ phases and the resulting
operator can be depicted as in figure 4. This corresponds to $G$ breaking the
other $SU(2)$ R-symmetry subgroup of ${\mathcal{L}}_{0}$ and at the particular
point where $\beta^{j}\bar{\beta}_{j}=-1$, $SU(3)$ symmetry is restored and
the loop is $1/2$ BPS. This is summarized in figure 1.

The particular cases outlined above recover the original analysis proposed in
the second chapter of Drukker:2019bev , where the authors propose $G$’s that
can be comprised of $\\{C_{1},\bar{C}^{1},C_{2},\bar{C}^{2}\\}$ or of
$\\{C_{3},\bar{C}^{3},C_{4},\bar{C}^{4}\\}$. Our construction is therefore a
generalization of that and corresponds to the most generic BPS operator one
can build out of ${\mathcal{L}}_{0}$. All previously known examples can be
derived from it through appropriate choices of the parameters.

${\bar{\alpha}}^{1}\,{\bar{\alpha}}^{2}$$\alpha_{1}\,\alpha_{2}$

${\bar{\beta}}_{3}\,\bar{\beta}_{4}$$\beta^{3}\,\beta^{4}$

Figure 4: Branches of $1/6$ BPS loops breaking different $SU(2)$ R-symmetries
of ${\mathcal{L}}_{0}$.

Finally, seen from a different perspective, by viewing ABJ(M) as the orbifold
of ${\mathcal{N}}=4$, the operator outlined above corresponds to the $1/16$
BPS operator appearing in figure 5 of Drukker:2020dvr , now specialized to
ABJ(M).

###  2.2 1/12 BPS latitude Wilson loops

The latitude operators are supported along

$x^{\mu}=(\sin\theta,\cos\theta\cos\tau,\cos\theta\sin\tau)\,.$ (2.12)

Their bosonic representatives are still written as (2.4), but this time with

$M_{J}^{\ I}=\begin{pmatrix}-\cos\theta&&0&&e^{-i\tau}\sin\theta&&0\\\
0&&-1&&0&&0\\\ e^{i\tau}\sin\theta&&0&&\cos\theta&&0\\\
0&&0&&0&&1\end{pmatrix}\,.$ (2.13)

This form of $M$ is such that the resulting loops are invariant under

$\displaystyle\cos\frac{\theta}{2}\bigg{(}Q_{12}^{+}-ie^{-i\theta}S_{12}^{+}\bigg{)}$
$\displaystyle-i\sin\frac{\theta}{2}\bigg{(}Q_{23}^{-}+ie^{-i\theta}S_{23}^{-}\bigg{)}\,,$
(2.14)
$\displaystyle\cos\frac{\theta}{2}\bigg{(}Q_{34+}-ie^{i\theta}S_{34+}\bigg{)}$
$\displaystyle-i\sin\frac{\theta}{2}\bigg{(}Q_{14-}+ie^{i\theta}S_{14-}\bigg{)}\,.$

As before, we follow the prescription (2.1) to construct the fermionic
counterparts. We take the supercharge ${\mathcal{Q}}$ to be given by the sum
of the supercharges above. Then the analysis of possible scalars to include in
$G$ is parallel to the $\theta\neq 0$ discussion of Drukker:2020dvr . We find
that $C_{2},\bar{C}^{2}$ and $C_{4},\bar{C}^{4}$ can not be included
simultaneously due to the non-periodicity of boundary conditions that can not
be fixed by means of a gauge transformation. To be precise, the
superconnection would transform as the supercovariant derivative of

$\begin{pmatrix}0&&\bar{\alpha}^{2}C_{2}+e^{-i\tau\cos\theta}\beta^{4}C_{4}\\\
\alpha_{2}\bar{C}^{2}+e^{i\tau\cos\theta}\bar{\beta}_{4}\bar{C}^{4}&&0\end{pmatrix}\,,$
(2.15)

which does not have well-behaved boundary conditions. As for the inclusion of
$C_{1},\bar{C}^{1}$ and $C_{3},\bar{C}^{3}$, we find that it requires
promoting the superconnection to a $3\times 3$ supermatrix and taking a cover
of the quiver of the theory. Since this goes beyond the scope of our present
discussion, we leave such possibility to the future.

We consider, therefore, two possible loops built out of $G$ coupling either to
$C_{2},\bar{C}^{2}$ or to $C_{4},\bar{C}^{4}$. Both options are represented in
figure 5, where the squigglyness of the nodes now stands for a constant shift
of $c=\frac{\cos\theta}{2}$.

$\bar{\alpha}^{2}$$\alpha_{2}$

$\bar{\beta}_{4}$$\beta^{4}$

Figure 5: Branches of $1/12$ BPS latitude loops. Points where supersymmetry is
enhanced correspond to
$\alpha^{2}\bar{\alpha}_{2}=-\bar{\beta}^{4}\beta_{4}=1$, where an $SU(2)$
subgroup of R-symmetry is restored and the operators become $1/6$ BPS.

For brevity, we focus here on the explicit construction of the first branch.
Its composite bosonic connection has a constant shift lying in the first node
and the final form of the superconnection is

${\mathcal{L}}^{\theta}=\begin{pmatrix}{\mathcal{A}}^{{}^{\prime}}+\frac{\cos\theta}{2}&&-\bar{\alpha}^{2}\eta\left(\cos\frac{\theta}{2}\bar{\psi}^{1}-\sin\frac{\theta}{2}e^{i\tau}\bar{\psi}^{3}\right)\\\
-\alpha_{2}\xi\left(\cos\frac{\theta}{2}\psi_{1}-\sin\frac{\theta}{2}e^{-i\tau}\psi_{3}\right)&&\hat{\mathcal{A}}^{{}^{\prime}}\end{pmatrix}\,,$
(2.16)

with the scalar coupling now given by

$M_{J}^{\ I}=\begin{pmatrix}-\cos\theta&&0&&e^{-i\tau}\sin\theta&&0\\\
0&&-1+2\bar{\alpha}^{2}\alpha_{2}&&0&&0\\\
e^{i\tau}\sin\theta&&0&&\cos\theta&&0\\\ 0&&0&&0&&1\end{pmatrix}\,.$ (2.17)

At the particular point where $\bar{\alpha}^{2}\alpha_{2}=1$, an $SU(2)$
subgroup of R-symmetry is preserved. The loop is invariant under the
supercharges (2.14) and the ones obtained by swapping the $2\leftrightarrow 4$
indices. This is the $1/6$ BPS latitude operator introduced in Bianchi:2014laa
and further studied in Bianchi:2018bke ; Griguolo_2021 .

###  2.3 Removing the constant shift

The constant shift $c$ in (2.2) is useful in the definition of the operators
(see chapter 2 of Drukker:2019bev ). Its presence gives rise to a manifestly
reparametrisation invariant operator. Moreover, Wilson loops with this shift
are (super)gauge invariant without the need for an additional twist matrix
Cardinali:2012ru  and can be defined as the supertrace of a superconnection,
as in (2.11), rather than with a trace, as in the original construction of
Drukker:2009hy . However, the presence of this shift makes the perturbative
calculation more intricate (see chapter 5 of Drukker:2019bev ), so we find it
helpful to remove it before proceeding to the next section.

To illustrate the procedure we will take the latitude operator. The
$\theta\rightarrow 0$ limit of the analysis below reproduces the circular
case. We make a $U(N_{1})$ gauge transformation in order to remove the
constant shift from the first node,

$A_{\mu}\dot{x}^{\mu}+\frac{\cos\theta}{2}\quad\longrightarrow\quad
A_{\mu}\dot{x}^{\mu}+\frac{d\Lambda}{d\tau}+\frac{\cos\theta}{2}=A_{\mu}\dot{x}^{\mu}+\text{boundary
terms}\,,$ (2.18)

where boundary terms may arise from the discontinuity of $\Lambda$ on the
circle. Precisely, we choose

$\Lambda=-\frac{\cos\theta}{2}\tau+\Delta\sum_{n\in\mathbb{Z}}\theta(\tau-2\pi
n)\,,$ (2.19)

where a constant $\Delta$ has been introduced, so to insure that
$\int_{0}^{2\pi}d\tau\frac{d\Lambda}{d\tau}$ vanishes. Requiring

$0=\int_{0}^{2\pi}d\tau\
\frac{d\Lambda}{d\tau}=\int_{0}^{2\pi}d\tau\left(-\frac{\cos\theta}{2}+\Delta\,\delta(\tau-2\pi)\right)=-\pi\cos\theta+\Delta\,,$
(2.20)

we obtain $\Delta=\pi\cos\theta$. Therefore, the original gauge term
$(A_{\mu}\dot{x}^{\mu}+\frac{\cos\theta}{2})$ in the superconnection is now
replaced by $(A_{\mu}\dot{x}^{\mu}+\pi\cos{\theta}\,\delta(\tau-2\pi))$.

Taking this delta function contribution into account, we recover the twist
matrix ${\mathcal{T}}$

${\mathcal{P}}\exp\left(-i\int_{2\pi-\epsilon}^{2\pi+\epsilon}{\mathcal{L}}\,d\tau\right)\rightarrow\exp\left(-i\begin{pmatrix}\pi\cos\theta&0\\\
0&0\end{pmatrix}\right)=\begin{pmatrix}e^{-i\pi\cos\theta}&0\\\
0&1\end{pmatrix}\equiv{\mathcal{T}}\,.$ (2.21)

In the circular case this is simply ${\mathcal{T}}=\operatorname{diag}(-1,1)$.
In the latitude case, in order to follow the same conventions of
Bianchi:2014laa , we rescale it such that

${\mathcal{T}}\equiv\begin{pmatrix}e^{-i\pi(\cos\theta)/2}&0\\\
0&e^{i\pi(\cos\theta)/2}\end{pmatrix}\,.$ (2.22)

The gauge transformation we have performed in order to remove the constant
shift acts on the matter fields as

$\begin{split}&\bar{\psi}^{I}\to\bar{\psi}^{I}e^{-i\Lambda}=\bar{\psi}^{I}e^{i\cos\theta\,\tau/2}\,,\qquad\qquad\psi_{I}\to\psi_{I}e^{i\Lambda}=\psi_{I}e^{-i\cos\theta\,\tau/2}\,,\\\
&\bar{C}^{I}\to\bar{C}^{I}e^{-i\Lambda}=\bar{C}^{I}e^{i\cos\theta\,\tau/2}\,,\qquad\qquad
C_{I}\to C_{I}e^{i\Lambda}=C_{I}e^{-i\cos\theta\,\tau/2}\,.\end{split}$ (2.23)

The diagonal elements of the superconnection remain unchanged, while the
fermionic entries gain extra phases. For the circular $1/24$ BPS operator
these are

$\begin{split}\bar{f}&=e^{i\tau/2}\,\eta\left(\bar{\alpha}^{1}\bar{\psi}^{2}-\bar{\alpha}^{2}\bar{\psi}^{1}\right)+e^{-i\tau/2}\xi\,(\beta^{3}\bar{\psi}^{4}-\beta^{4}\bar{\psi}^{3})\,,\\\
f&=e^{-i\tau/2}\,\xi\,\left(\alpha_{1}\psi_{2}-\alpha_{2}\psi_{1}\right)+e^{i\tau/2}\eta\,(\bar{\beta}_{3}\psi_{4}-\bar{\beta}_{4}\psi_{3})\,,\end{split}$
(2.24)

while for the latitude $1/12$ BPS operator they are

$\displaystyle\bar{f}^{\theta}$
$\displaystyle=-e^{i\cos\theta\,\tau/2}\,\eta\,\bar{\alpha}^{2}\left(\cos\frac{\theta}{2}\bar{\psi}^{1}-\sin\frac{\theta}{2}e^{i\tau}\bar{\psi}^{3}\right),$
(2.25) $\displaystyle f^{\theta}$
$\displaystyle=-e^{-i\cos\theta\,\tau/2}\,\xi\,\alpha_{2}\left(\cos\frac{\theta}{2}\psi_{1}-\sin\frac{\theta}{2}e^{-i\tau}\psi_{3}\right)\,.$

Therefore, the final form of the superconnection is

${\mathcal{L}}=\begin{pmatrix}{\mathcal{A}}&\bar{f}\\\
f&\hat{\mathcal{A}}\end{pmatrix}\,,$ (2.26)

without the constant shift in the first diagonal block, unlike (2.2), and
similarly for the case with $\theta\neq 0$.777We keep the same symbol ${\cal
L}$ for this superconnection without the shift, hoping that it will not be
confusing. From now on, ${\cal L}$ will refer to this expression.

The Wilson loop operator is now written as

${\mathcal{W}}={\mathcal{R}}^{-1}\,\operatorname{sTr}{\cal
P}\left(e^{-i\oint{\cal L}d\tau}{\mathcal{T}}\right)\,,$ (2.27)

where we have introduced the normalization factor
${\mathcal{R}}=\operatorname{sTr}({\mathcal{T}})$. In particular, from now on,
we will refer to the circular Wilson loop as

${\mathcal{W}}=\frac{W}{N_{1}+N_{2}},\qquad W\equiv\text{Tr}\,{\cal
P}\exp\left(-i\oint{\cal L}\,d\tau\right),$ (2.28)

with the ${\cal L}$ in (2.26).

##  3 Renormalization

###  3.1 1D effective field theory for the Wilson loop VEV

At weak coupling, the standard procedure for computing the vacuum expectation
value of a Wilson loop is ordinary perturbation theory. In the functional
approach, this amounts to expanding the exponential of the interaction part of
the bulk action in powers of the coupling constant and performing contractions
with the Wilson loop expansion using Feynman rules for the bulk theory.

In the QCD context, in the 80’s Samuel Samuel:1978iy , Gervais and Neveu
Gervais:1979fv  proposed an alternative method to study Wilson loop operators,
based on the formulation of a one-dimensional effective field theory.
Subsequently, this method was further developed and heavily exploited to study
the renormalization of composite operators Arefeva:1980zd ; CRAIGIE1981204 ;
Dorn:1986dt .

The method makes use of auxiliary one-dimensional fermions and can be briefly
summarized as follows. Suppose that in a given gauge theory one wants to
evaluate a generic Wilson loop supported along a contour ${\mathcal{C}}$,

$W[{\mathcal{C}}]=\operatorname{Tr}{\mathcal{P}}\exp\left(-i\int_{\mathcal{C}}{\mathcal{L}}\,d\tau\right)\,.$
(3.1)

In this expression ${\mathcal{L}}$ may be the ordinary gauge connection $A$ or
one of the bosonic connections ${\mathcal{A}}$ given in (2.4). In any case,
one can write the perturbative expansion of the operator as

$\begin{split}W[{\cal
C}]&=\operatorname{Tr}\left(1+\sum_{k=1}^{\infty}\frac{(-i)^{k}}{k!}{\mathcal{P}}\int_{{\mathcal{C}}}d\tau_{1}d\tau_{2}\dots
d\tau_{k}\,{\mathcal{L}}(\tau_{1}){\mathcal{L}}(\tau_{2})\dots\mathcal{L}(\tau_{k})\right)\\\
&=\operatorname{Tr}\left(1+\sum_{k=1}^{\infty}(-i)^{k}\int_{{\mathcal{C}}}d\tau_{1}d\tau_{2}\dots
d\tau_{k}\,\theta(\tau_{k}-\tau_{k-1})\dots\theta(\tau_{2}-\tau_{1}){\mathcal{L}}(\tau_{1}){\mathcal{L}}(\tau_{2})\dots\mathcal{L}(\tau_{k})\right)\,.\end{split}$
(3.2)

The idea is to interpret $\theta(\tau_{i}-\tau_{j})$ as the propagator of an
auxiliary fermionic field $z$ living on the Wilson loop contour, whose
interaction with the rest of the fields is dictated by ${\mathcal{L}}$. Taking
the $z$ field in the fundamental representation of the gauge group, its action
is chosen to be

$S_{\textrm{eff}}=S+\int
d\tau\,\bar{z}(\tau)\left[\partial_{\tau}+i{\mathcal{L}}\right]z(\tau)\,,$
(3.3)

where $S$ is the action of the underlying gauge theory. Performing the
Gaussian $z$-integral in the generating functional, it can be shown that for a
contour ${\mathcal{C}}_{12}$ connecting two points parametrized by
$\tau_{1},\tau_{2}$ one has CRAIGIE1981204 ; Dorn:1986dt

$\langle
W[{\mathcal{C}}_{12}]\rangle=\left\langle\operatorname{Tr}{\mathcal{P}}\exp\left(-i\int_{\tau_{2}}^{\tau_{1}}{\mathcal{L}}\,d\tau\right)\right\rangle=\langle
z(\tau_{2})\bar{z}(\tau_{1})\rangle\,,$ (3.4)

where

$\langle
z(\tau_{2})\bar{z}(\tau_{1})\rangle=\int[{\mathcal{D}}z{\mathcal{D}}\bar{z}]\,z(\tau_{2})\bar{z}(\tau_{1})\,e^{-S_{\textrm{eff}}}.$
(3.5)

Therefore, the expectation value of $W$ is nothing but the two-point function
of the one-dimensional theory defined on it.

This method could be naturally generalized to study renormalization properties
of Wilson loops in supersymmetric theories. Here, we propose a generalization
that captures the expectation value of operators in the ABJ(M) theory.

Since in this theory Wilson loops are defined in terms of supermatrices, the
natural way to proceed is to replace the one-dimensional auxiliary $z$ fermion
with a Grassmann odd supermatrix

$\Psi=\begin{pmatrix}z&\varphi\\\
\tilde{\varphi}&\tilde{z}\end{pmatrix}\,,\qquad\bar{\Psi}=\begin{pmatrix}\bar{z}&\bar{\tilde{\varphi}}\\\
\bar{\varphi}&\bar{\tilde{z}}\end{pmatrix}\,,$ (3.6)

where $z$ ($\tilde{z}$) and $\varphi$ ($\tilde{\varphi}$) are a spinor and a
scalar, respectively, in the fundamental representation of $U(N_{1})$
($U(N_{2})$). We then look for an effective theory such that the ABJ(M) Wilson
loop VEV (2.27) can be computed as a two-point function of the one-dimensional
fields. To this end, we consider the following action

$S_{\textrm{eff}}=S_{\textrm{ABJ(M)}}+\int
d\tau\,\operatorname{Tr}\left(\bar{\Psi}{\mathcal{D}}_{\tau}\Psi\right)\,,$
(3.7)

where $S_{\textrm{ABJ(M)}}$ is the ABJ(M) action (see (A.2)) and
${\mathcal{D}}_{\tau}=\partial_{\tau}+i{\mathcal{L}}(\tau)$, ${\mathcal{L}}$
being the Wilson loop superconnection. It is then easy to prove that

$\langle
W[{\mathcal{C}}_{12}]\rangle=\,\langle\operatorname{Tr}\Psi(\tau_{2})\bar{\Psi}(\tau_{1})\rangle\,,$
(3.8)

where the vacuum functional on the right-hand side includes the integrations
over both the bulk fields and the one-dimensional $\Psi,\bar{\Psi}$
supermatrices, weighted by the action (3.7).888Here it is sufficient to assume
that a consistent definition of integration over supermatrices exists, which
leads to well-defined, finite and non-vanishing results for Gaussian
integrals. We provide more details about this derivation in appendix B.

We focus here on the circular Wilson loop defined in section 2.1, while
postponing the investigation of the latitude operator of section 2.2 to a
future publication CPTT . Expanding the matrix product and defining for
simplicity $g=\sqrt{\frac{2\pi}{k}}$, the effective action can be written
explicitly as

$\begin{split}S_{\textrm{eff}}=S_{\textrm{ABJ(M)}}+\int
d\tau\Big{[}&\bar{\varphi}D_{\tau}\varphi+\bar{\tilde{\varphi}}\hat{D}_{\tau}\tilde{\varphi}+\bar{z}D_{\tau}z+\bar{\tilde{z}}\hat{D}_{\tau}\tilde{z}\\\
&+i(\bar{\tilde{z}}f\varphi+\bar{\varphi}\bar{f}\tilde{z}+\bar{\tilde{\varphi}}fz+\bar{z}\bar{f}\tilde{\varphi})\Big{]}\,,\end{split}$
(3.9)

where we have defined $D_{\tau}=\partial_{\tau}+i{\mathcal{A}}$ and
$\hat{D}_{\tau}=\partial_{\tau}+i\hat{\mathcal{A}}$. ${\mathcal{A}}$,
$\hat{{\mathcal{A}}}$, $f$ and $\bar{f}$ are the even and odd elements of the
Wilson loop superconnection, see (2.26). The covariant $\tau$-derivatives give
rise to the usual minimal coupling between the one-dimensional fields and the
bulk gauge vectors, plus quartic interactions with bulk scalar bilinears. We
have not inserted the explicit expressions of $f,\bar{f}$, which can be found
in (2.25). At this stage, it is only important to take into account that these
couplings are proportional to one power of $g$. The tree-level propagators of
the one-dimensional fields are

$\displaystyle=\langle z^{i}(\tau_{1})\bar{z}_{j}(\tau_{2})\rangle$
$\displaystyle=\delta_{j}^{i}\,\theta(\tau_{1}-\tau_{2})\,,$ (3.10)
$\displaystyle=\langle{\tilde{z}}^{\hat{i}}(\tau_{1})\bar{\tilde{z}}_{\hat{j}}(\tau_{2})\rangle$
$\displaystyle=\delta_{\hat{j}}^{\hat{i}}\,\theta(\tau_{1}-\tau_{2})\,,$
$\displaystyle=\langle\varphi^{i}(\tau_{1})\bar{\varphi}_{j}(\tau_{2})\rangle$
$\displaystyle=\delta_{j}^{i}\,\theta(\tau_{1}-\tau_{2})\,,$
$\displaystyle=\langle\tilde{\varphi}^{\hat{i}}(\tau_{1})\bar{\tilde{\varphi}}_{\hat{j}}(\tau_{2})\rangle$
$\displaystyle=\delta_{\hat{j}}^{\hat{i}}\,\theta(\tau_{1}-\tau_{2})\,.$

###  3.2 Renormalization scheme

We now focus on the perturbative evaluation of the two-point function (3.8)
for the one-dimensional theory. This first requires investigating whether the
one-dimensional fields and the couplings undergo a non-trivial
renormalization, due to short distance divergences arising on the loop.

For each one-dimensional field
$\phi=\\{\varphi,\tilde{\varphi},z,\tilde{z}\\}$ the corresponding
renormalization functions are defined as
$\phi=Z_{\phi}^{-\frac{1}{2}}\phi_{0}$, where $\phi_{0}$ stands for the bare
quantity. We note that since the action (3.9) is invariant under the formal
exchanges $z\leftrightarrow\varphi$ and
$\tilde{z}\leftrightarrow\tilde{\varphi}$, we can set $Z_{z}=Z_{\varphi}$ and
$Z_{\tilde{z}}=Z_{\tilde{\varphi}}$. As we are going to prove, the field
function renormalization is sufficient to cancel UV divergent contributions to
both the kinetic terms and the interaction vertices between auxiliary fields
and gauge connections, i.e. the $\bar{\phi}A_{\mu}\dot{x}^{\mu}\phi$ vertices.
This is consistent with the expectation that the addition of the auxiliary
action in (3.7) does not affect the UV finiteness of the ABJ(M) theory
($A_{\mu}$ does not renormalize).

As follows from the definition of $f,\bar{f}$ and $M_{I}^{\ J}$ in (2.24) and
(2.10), respectively, the fermionic interactions (as for instance
$\bar{\tilde{z}}f\varphi$) and the quartic couplings with the scalar bilinears
contain the $g$ coupling and the
$\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j}$ parameters as further
couplings.

For the renormalization of the fermionic interactions we define

$\begin{split}&(\bar{\alpha}^{i})_{0}\,Z^{1/2}_{\tilde{z}}Z_{\varphi}^{1/2}=(\bar{\alpha}^{i})_{0}\,Z^{1/2}_{z}Z_{\tilde{\varphi}}^{1/2}=Z_{\bar{\alpha}^{i}}\,\bar{\alpha}^{i}\,,\\\
&(\alpha_{i})_{0}\,Z^{1/2}_{\tilde{z}}Z_{\varphi}^{1/2}=(\alpha_{i})_{0}\,Z^{1/2}_{z}Z_{\tilde{\varphi}}^{1/2}=Z_{\alpha_{i}}\,\alpha_{i}\,,\\\
&(\beta^{j})_{0}\,Z^{1/2}_{\tilde{z}}Z_{\varphi}^{1/2}=(\beta^{j})_{0}\,Z^{1/2}_{z}Z_{\tilde{\varphi}}^{1/2}=Z_{\beta^{j}}\,\beta^{j}\,,\\\
&(\bar{\beta}_{j})_{0}\,Z^{1/2}_{\tilde{z}}Z_{\varphi}^{1/2}=(\bar{\beta}_{j})_{0}\,Z^{1/2}_{z}Z_{\tilde{\varphi}}^{1/2}=Z_{\bar{\beta}_{j}}\,\bar{\beta}_{j}\,,\end{split}$
(3.11)

where the subscript 0 denotes bare parameters and we have used that the ABJ(M)
coupling does not renormalize, i.e. $g_{0}=g$. The scalar vertices of the form
$M_{I}^{\ J}\bar{\phi}C_{J}\bar{C}^{I}\phi$ deserve more attention since the
parametric dependence is hidden inside the scalar coupling matrix $M_{I}^{\
J}$. We set

$\displaystyle Z_{\varphi}({M}_{I}^{\ J})_{0}=Z_{\varphi C}M_{I}^{\
J}\,,\qquad$ $\displaystyle Z_{\tilde{\varphi}}({M}_{I}^{\
J})_{0}=Z_{\tilde{\varphi}C}M_{I}^{\ J}\,,$ (3.12) $\displaystyle
Z_{z}({M}_{I}^{\ J})_{0}=Z_{zC}M_{I}^{\ J}\,,\qquad$ $\displaystyle
Z_{\tilde{z}}\,({M}_{I}^{\ J})_{0}=Z_{\tilde{z}C}M_{I}^{\ J}\,,$

where $({M}_{I}^{\ J})_{0}$ is the scalar coupling matrix expressed in terms
of the bare parameters.

Using the standard BPHZ renormalization procedure, we write all
renormalization functions as $Z=1+\delta$, where $\delta$ are the
corresponding countertems. We then extract the Feynman rules from the one-
dimensional Lagrangian written as the sum of a renormalized Lagrangian plus
the counterterm part

${\mathcal{L}}_{\textrm{1D}}={\mathcal{L}}^{\textrm{ren}}_{\textrm{1D}}+{\mathcal{L}}_{\textrm{1D}}^{\text{ct}}\,,$
(3.13)

where ${\mathcal{L}}^{\textrm{ren}}_{\textrm{1D}}$ is given by (3.9) written
in terms of renormalized quantities and the counterterms read

$\begin{split}{\mathcal{L}}^{\text{ct}}_{\textrm{1D}}&=\sum_{\phi=\varphi,z}\left(\delta_{\phi}\,\bar{\phi}\partial_{\tau}\phi+\delta_{\phi}\,ig\bar{\phi}A_{\mu}\dot{x}^{\mu}\phi+\delta_{\phi
C}\,g^{2}|\dot{x}|M_{I}^{\ J}\bar{\phi}C_{J}\bar{C}^{I}\phi\right)\\\
&+\sum_{\tilde{\phi}=\tilde{\varphi},\tilde{z}}\left(\delta_{\tilde{\phi}}\,\bar{\tilde{\phi}}\partial_{\tau}\tilde{\phi}+\delta_{\tilde{\phi}}\,ig\bar{\tilde{\phi}}\hat{A}_{\mu}\dot{x}^{\mu}\tilde{\phi}+\delta_{\tilde{\phi}C}\,g^{2}|\dot{x}|M_{I}^{\
J}\bar{\tilde{\phi}}\bar{C}^{I}C_{J}\tilde{\phi}\right)\\\
&+i\bar{\tilde{z}}\left(e^{-\frac{i\tau}{2}}\xi\left(\delta_{\alpha_{1}}\,\alpha_{1}\psi^{2}-\delta_{\alpha_{2}}\,\alpha_{2}\psi^{1}\right)+e^{\frac{i\tau}{2}}\eta\,(\delta_{\bar{\beta}_{3}}\,\bar{\beta}_{3}\psi^{4}-\delta_{\bar{\beta}_{4}}\,\bar{\beta}_{4}\psi^{3})\right)\varphi\\\
&+i\bar{\tilde{\varphi}}\left(e^{-\frac{i\tau}{2}}\xi\left(\delta_{\alpha_{1}}\,\alpha_{1}\psi^{2}-\delta_{\alpha_{2}}\,\alpha_{2}\psi^{1}\right)+e^{\frac{i\tau}{2}}\eta\,(\delta_{\bar{\beta}_{3}}\,\bar{\beta}_{3}\psi^{4}-\delta_{\bar{\beta}_{4}}\,\bar{\beta}_{4}\psi^{3})\right)z\\\
&+i\bar{\varphi}\left(e^{\frac{i\tau}{2}}\eta\left(\delta_{\bar{\alpha}^{1}}\,\bar{\alpha}^{1}\bar{\psi}^{2}-\delta_{\bar{\alpha}^{2}}\,\bar{\alpha}^{2}\bar{\psi}^{1}\right)+e^{-\frac{i\tau}{2}}\xi\,(\delta_{\beta^{3}}\,\beta^{3}\bar{\psi}^{4}-\delta_{\beta^{4}}\,\beta^{4}\bar{\psi}^{3})\right)\tilde{z}\\\
&+i\bar{z}\left(e^{\frac{i\tau}{2}}\eta\left(\delta_{\bar{\alpha}^{1}}\,\bar{\alpha}^{1}\bar{\psi}^{2}-\delta_{\bar{\alpha}^{2}}\,\bar{\alpha}^{2}\bar{\psi}^{1}\right)+e^{-\frac{i\tau}{2}}\xi\,(\delta_{\beta^{3}}\beta^{3}\bar{\psi}^{4}-\delta_{\beta^{4}}\,\beta^{4}\bar{\psi}^{3})\right)\tilde{\varphi}\,,\end{split}$
(3.14)

with obvious meanings of the $\delta$’s.

###  3.3 Evaluation of one-loop counterterms

We begin by investigating the structure of the counterterms at one loop. We
tame short distance divergences arising from the evaluation of Feynman
integrals by using dimensional regularization in $D=1-2\epsilon$ and a minimal
subtraction scheme. We work in the large $N_{1},N_{2}$ limit.

Since we want to study the UV behavior of our one-dimensional theory, we work
in the $\tau_{2}\to\tau_{1}$ limit, where $\tau$ parameterizes the curve on
which the theory is defined. Therefore, any regular contour can be
approximated, around a point, by a straight segment, such that $|\dot{x}|=1$
and $\dot{x}\cdot\ddot{x}=0$. In this limit, for a generic one-dimensional
field $\phi$ we use the following approximation999We use the notation
$\phi(\tau_{i})\equiv\phi_{i}$ and $x_{i}\equiv x(\tau_{i})$.

$\phi_{2}\simeq\phi_{1}+(\tau_{2}-\tau_{1})\dot{\phi}_{1}\,,$ (3.15)

as well as the following expansion for the coordinates on the contour

$\begin{split}&x^{\mu}_{2}\simeq
x^{\mu}_{1}+(\tau_{2}-\tau_{1})\dot{x}^{\mu}_{1}\,,\qquad\dot{x}^{\mu}_{2}\simeq\dot{x}^{\mu}_{1}+(\tau_{2}-\tau_{1})\ddot{x}^{\mu}_{1}\,,\\\
&\left(x_{2}-x_{1}\right)^{2}\simeq(\tau_{2}-\tau_{1})^{2}\,\dot{x}_{1}^{2}\,\left(1+(\tau_{2}-\tau_{1})\frac{\dot{x}_{1}\cdot\ddot{x}_{1}}{\dot{x}_{1}^{2}}\right)\,,\end{split}$
(3.16)

To keep the discussion as clear as possible, we provide here details for the
first few diagrams and collect the rest of the calculations in appendix C.

#### Corrections to the kinetic term

We begin by considering one-loop self-energy corrections to the
$\langle\bar{z}z\rangle$ propagator of the one-dimensional theory. The
contributing diagrams are drawn in figure 6 (we neglect tadpole diagrams, as
they vanish in dimensional regularization).

Figure 6: One-loop corrections to the fermionic propagator
$\langle\bar{z}z\rangle$. Double straight lines represent $z$ and $\bar{z}$,
blue double dashed lines are the one-dimensional $\tilde{\varphi}$ scalars,
simple straight lines are ABJ(M) fermions, whereas wavy lines describe the
ABJ(M) $U(N_{1})$ gauge field. Diagram (c) is the $\delta_{z}$ counterterm in
(3.14).

The first diagram is the gauge field correction and gives rise to the
following contribution

$\Sigma^{\rm\ref{subfig:vertexcorrectiona}}_{z}=\int d\tau_{1}\int
d\tau_{2}\,\bar{z}_{1}\,z_{2}\,\dot{x}_{1}^{\mu}\,\dot{x}_{2}^{\nu}\,\theta(\tau_{1}-\tau_{2})\langle
A_{\mu}(\tau_{1})A_{\nu}(\tau_{2})\rangle\,.$ (3.17)

However, inserting the explicit expression (A.4) for the gauge propagator and
using the expansions (3.16), it is easy to see that in dimensional
regularization this integral vanishes, due to the antisymmetry of the
$\epsilon_{\mu\nu\rho}$ tensor.

The second diagram gives (we define $\tau_{12}\equiv\tau_{1}-\tau_{2}$)

$\begin{split}\Sigma^{\rm\ref{subfig:vertexcorrectionb}}_{z}&=\int
d\tau_{1}\int
d\tau_{2}\left(i\bar{z}\bar{f}\tilde{\varphi}\right)(\tau_{1})\left(i\bar{\tilde{\varphi}}fz\right)(\tau_{2})\\\
&=\bar{\alpha}^{i}\alpha_{i}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\,\bar{z}_{1}\,z_{2}\,e^{\frac{i\tau_{12}}{2}}\eta_{\alpha}(\tau_{1})\xi^{\beta}(\tau_{2})\langle\bar{\psi}^{\alpha}(\tau_{1})\psi_{\beta}(\tau_{2})\rangle\\\
&+\beta^{j}\bar{\beta}_{j}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\,\bar{z}_{1}\,z_{2}\,e^{-\frac{i\tau_{12}}{2}}\xi_{\alpha}(\tau_{1})\eta^{\beta}(\tau_{2})\langle\bar{\psi}^{\alpha}(\tau_{1})\psi_{\beta}(\tau_{2})\rangle\,,\end{split}$
(3.18)

where we have already used
$\langle\tilde{\varphi}_{1}\bar{\tilde{\varphi}}_{2}\rangle=\theta(\tau_{1}-\tau_{2})$.
Inserting the fermionic propagator (A.4), it explicitly reads

$\begin{split}\Sigma^{\rm\ref{subfig:vertexcorrectionb}}_{z}&=i\bar{\alpha}^{i}\alpha_{i}\frac{\Gamma(\frac{3}{2}-\epsilon)}{2\pi^{\frac{3}{2}-\epsilon}}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\,\bar{z}_{1}\,z_{2}\,e^{i\frac{\tau_{12}}{2}}\xi^{\alpha}(\tau_{2})(\gamma_{\mu})_{\alpha}^{\
\beta}\eta_{\beta}(\tau_{1})\frac{(x_{2}-x_{1})^{\mu}}{|x_{2}-x_{1}|^{3-2\epsilon}}\\\
&+i\beta^{j}\bar{\beta}_{j}\frac{\Gamma(\frac{3}{2}-\epsilon)}{2\pi^{\frac{3}{2}-\epsilon}}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\,\bar{z}_{1}\,z_{2}\,e^{-i\frac{\tau_{12}}{2}}\eta^{\alpha}(\tau_{2})(\gamma_{\mu})_{\alpha}^{\
\beta}\xi_{\beta}(\tau_{1})\frac{(x_{2}-x_{1})^{\mu}}{|x_{2}-x_{1}|^{3-2\epsilon}}\,.\end{split}$
(3.19)

In the $\tau_{2}\to\tau_{1}$ limit, using the explicit expression for the
$\xi,\eta$ spinors and the gamma matrices in (A.1), we can write

$\begin{split}e^{i\frac{\tau_{12}}{2}}\xi^{\alpha}(\tau_{2})\,(\gamma_{\mu})_{\alpha}^{\
\beta}\,\eta_{\beta}(\tau_{1})\,(x_{2}-x_{1})^{\mu}&=-4ig^{2}\sin\frac{\tau_{12}}{2}\,\simeq\,-2ig^{2}(\tau_{1}-\tau_{2})\,,\\\
e^{-i\frac{\tau_{12}}{2}}\eta^{\alpha}(\tau_{2})\,(\gamma_{\mu})_{\alpha}^{\
\beta}\,\xi_{\beta}(\tau_{1})\,(x_{2}-x_{1})^{\mu}&=-4ig^{2}\sin\frac{\tau_{12}}{2}\,\simeq\,-2ig^{2}(\tau_{1}-\tau_{2})\,.\end{split}$
(3.20)

Expanding the rest of the integrand with (3.15), (3.16), the integral reduces
to

$\begin{split}\Sigma^{\rm\ref{subfig:vertexcorrectionb}}_{z}&=-g^{2}N_{2}\,(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\frac{\Gamma(\frac{3}{2}-\epsilon)}{\pi^{\frac{3}{2}-\epsilon}}\int
d\tau_{1}\bar{z}_{1}\,\dot{z}_{1}\int^{\tau_{1}}d\tau_{2}\,(\tau_{1}-\tau_{2})^{-1+2\epsilon}+\cdots\\\
&=-g^{2}\frac{N_{2}}{4\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\int
d\tau\,\bar{z}\,\partial_{\tau}z+\text{finite terms}\,,\end{split}$ (3.21)

where in the first line dots indicate terms of the expansion which give rise
to finite integrals, and in the second line we have extracted the divergent
part for $\epsilon\to 0$.

Finally, the counterterm contribution is

$\Sigma^{\rm\ref{subfig:vertexcorrectiond}}_{z}=-\delta_{z}\int
d\tau\,\bar{z}\partial_{\tau}z\,.$ (3.22)

Therefore the total correction to the $z,\bar{z}$ propagator $\Sigma_{z}$,
given by the sum of all diagrams above, is

$\Sigma_{z}=\left(-g^{2}\frac{N_{2}}{4\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})-\delta_{z}\right)\int
d\tau\bar{z}\partial_{\tau}z\,.$ (3.23)

Requiring the counterterm to cancel the divergence, we eventually find

$Z_{z}=Z_{\varphi}=1-g^{2}\frac{N_{2}}{4\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\,.$
(3.24)

The same procedure can be applied to the tilde fields, obtaining similar
contributions

$Z_{\tilde{z}}=Z_{\tilde{\varphi}}=1-g^{2}\frac{N_{1}}{4\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\,.$
(3.25)

#### Corrections to the gauge-fermion vertex

We now consider one-loop corrections to the gauge-fermion vertex
$S_{\bar{z}Az}=i\int d\tau\,\bar{z}A_{\mu}\dot{x}^{\mu}z$. The corresponding
diagrams are summarized in figure 7. In the following we simply list the final
result of each integral, referring to appendix C.1 for the details of the
computation.

Figure 7: One-loop corrections of the $V_{\bar{z}Az}$ vertex. The blue double-
dashed line describes the $\tilde{\varphi}$ propagator, while the single-
dashed line describes the ABJM scalar fields $C_{I}$. The last diagram
represents the $\delta_{z}$ counterterm.

To begin with, it is easy to see that diagram 7 does not contribute, due to
planarity. In fact, using the expansions (3.16), the associated divergence
turns out to be proportional to
$\epsilon_{\mu\nu\rho}\dot{x}_{1}^{\mu}\dot{x}_{1}^{\nu}$.

For the same reason, as discussed in appendix C.1, the divergent contribution
of diagram 7 also vanishes. This diagram contains the three gauge field vertex
coming from the ABJ(M) action, $\frac{1}{3g^{2}}\epsilon^{\mu\nu\rho}\int
d^{d}x\,\operatorname{Tr}(A_{\mu}A_{\nu}A_{\rho})$ (here $d=3-2\epsilon$).
Therefore, it is proportional to the product of three epsilon tensors, one
from the vertex and two from the gauge propagators. Using ordinary epsilon
tensor algebra, this product can be reduced to a single epsilon, but
eventually the remaining tensor is contracted with the same vector twice.

Diagram 7 is built using the gauge-fermion-fermion vertex from the original
ABJ(M) action $-\int
d^{d}x\operatorname{Tr}(g\bar{\psi}^{J}\gamma^{\mu}A_{\mu}\psi_{J})$. Its
divergent contribution reads

$\Gamma^{\rm\ref{subfig:gauge1c}}_{\text{gauge}}=-\frac{g^{2}N_{2}}{4\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\int
d\tau\ i\bar{z}A_{\mu}\dot{x}^{\mu}z\,.$ (3.26)

Finally, diagram 7 contains the gauge-scalar vertex coming from minimal
coupling in the ABJ(M) action, $i\int
d^{d}x\,(A^{\mu}C_{I}\partial_{\mu}\bar{C}^{I}-\partial_{\mu}C_{I}\,\bar{C}^{I}A^{\mu})$.
In Lorentz gauge this diagram turns out to be equal to zero, as shown in
appendix C.1.

Summing all the contributions, the correction to the gauge vertex
$\Gamma_{\text{gauge}}$ is eventually given by

$\Gamma_{\text{gauge}}=\left(-\delta_{z}-\frac{g^{2}N_{2}}{4\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\right)\int
d\tau\,i\bar{z}A_{\mu}\dot{x}^{\mu}z\,.$ (3.27)

Comparing with (3.24), we see that $\delta_{z}$ cancels exactly the
divergence.

Following the same procedure for the $\bar{\tilde{z}}\hat{A}_{\mu}\tilde{z}$
vertex, we find that the result changes only by a color factor. Precisely, we
obtain

${\hat{\Gamma}}_{\text{gauge}}=\left(-\delta_{\tilde{z}}-\frac{g^{2}N_{1}}{4\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\right)\int
d\tau\,i\bar{\tilde{z}}\hat{A}_{\mu}\dot{x}^{\mu}{\tilde{z}}\,,$ (3.28)

and $\delta_{\tilde{z}}$ in (3.25) cancels exactly this vertex divergence.

The same pattern holds also for the remaining gauge-boson vertices, i.e.
$\bar{\varphi}A_{\mu}\dot{x}^{\mu}\varphi$ and
$\bar{\tilde{\varphi}}A_{\mu}\dot{x}^{\mu}{\tilde{\varphi}}$.

#### Fermion vertex corrections

To compute the counterterm associated with the fermion vertex correction (last
four lines in (3.14)), we first consider the coupling
$i\bar{\tilde{z}}f\varphi$. Inserting the explicit expression (2.24) for $f$,
this amounts to evaluating four different vertex structures, precisely

$i\alpha_{1}e^{-i\tau/2}\xi\,\bar{\tilde{z}}\psi^{2}\varphi-i\alpha_{2}e^{-i\tau/2}\xi\,\bar{\tilde{z}}\psi^{1}\varphi+i\bar{\beta}_{3}e^{i\tau/2}\eta\,\bar{\tilde{z}}\psi^{4}\varphi-i\bar{\beta}_{4}e^{i\tau/2}\eta\,\bar{\tilde{z}}\psi^{3}\varphi\,,$
(3.29)

with $\xi,\eta$ given in (2.9).

For all the structures, the typologies of diagrams to be considered are shown
in figure 8. The ABJ(M) vertices $-\bar{\psi}^{I}\gamma^{\mu}\psi_{I}A_{\mu}$
and $\bar{\psi}^{I}\gamma^{\mu}\hat{A}_{\mu}\psi_{I}$ appear in 8 and 8,
respectively. Since these vertices are diagonal in the fermion colors, the
correction to the $\alpha_{1}$ and $\alpha_{2}$ vertices will be the same, as
well as the ones for $\bar{\beta}_{3}$ and $\bar{\beta}_{4}$.

Figure 8: One-loop corrections to the $V_{\bar{\tilde{z}}f\varphi}$ vertex.
Wavy lines correspond to $A_{\mu}$ propagators, whereas wiggly lines represent
$\hat{A}_{\mu}$ propagators.

Considering first the corrections to the $\alpha_{i}$ vertices, from diagram 8
we obtain (the details are in appendix C.2)

$\Gamma^{\rm\ref{subfig:fermion1a}}_{\text{fermion}}=\frac{g^{2}N_{1}}{8\pi\epsilon}i\int
d\tau\bar{\tilde{z}}\left(\alpha_{1}e^{-i\tau/2}\xi\,\psi^{2}-\alpha_{2}\,e^{-i\tau/2}\xi\,\psi^{1}\right)\varphi\,.$
(3.30)

For diagram 8 we find the same contribution with $N_{1}$ replaced by $N_{2}$.

Summing up the three diagrams, the one-loop correction to the $\alpha_{i}$
fermion vertices is given by

$i\int
d\tau\bar{\tilde{z}}\left[\left(\delta_{\alpha_{1}}+\frac{g^{2}(N_{1}+N_{2})}{8\pi\epsilon}\right)\alpha_{1}e^{-i\tau/2}\xi\,\psi^{2}-\left(\delta_{\alpha_{2}}+\frac{g^{2}(N_{1}+N_{2})}{8\pi\epsilon}\right)\alpha_{2}\,e^{-i\tau/2}\xi\,\psi^{1}\right]\varphi\,.$
(3.31)

It is easy to check that performing the same computation for the fermionic
vertices proportional to $\bar{f}$ in (2.24), we obtain the same corrections
to the $\bar{\alpha}^{i}$ couplings. Consequently, we find

$Z_{\bar{\alpha}^{i}}=Z_{\alpha_{i}}=1-\frac{g^{2}(N_{1}+N_{2})}{8\pi\epsilon},\qquad\quad
i=1,2\,.$ (3.32)

Similarly, for the $\beta^{j},\bar{\beta}_{j}$ couplings in (3.29) we obtain

$Z_{\beta^{j}}=Z_{\bar{\beta}_{j}}=1+\frac{g^{2}(N_{1}+N_{2})}{8\pi\epsilon},\qquad\quad
j=3,4\,.$ (3.33)

The different sign compared with (3.32) comes from the different couplings
accompanying $\bar{\alpha}^{i},\alpha_{i}$ and $\beta^{j},\bar{\beta}_{j}$
parameters.

#### Scalar vertex corrections

Now we study the corrections to the scalar vertex $S_{\varphi C}=g^{2}M_{I}^{\
J}\int d\tau\bar{\varphi}C_{J}\bar{C}^{I}\varphi$, as the prototype of the
four-point vertices in (3.9). This vertex requires particular attention since
the components of $M_{I}^{\ J}$ are functions of the
$\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j}$ parameters. In the
most general case, the $1/24$ BPS matrix of (2.10), the parameters appear in
all the components, and this renders the computation rather involved. However,
considering the particular case
$\bar{\alpha}^{2}\alpha_{2}=\beta^{3}\bar{\beta}_{3}=\beta^{4}\bar{\beta}_{4}=0$
is sufficient to compute the desired corrections, while simplifying
considerably the calculations. We will then stick to this case.

At leading order in the gauge colors, the diagrams that contribute to the
four-point vertex are depicted in figure 9. Further non-vanishing diagrams
could be drawn, which however lead to subleading corrections proportional to
double-trace vertices. Since we work at large $N_{1},N_{2}$, we neglect them.

Details on the computation of each diagram are presented separately in
appendix C.3. Here, we list only the results.

Figure 9: Leading one-loop corrections of the
$V_{\bar{\varphi}C\bar{C}\varphi}$ vertex. The last diagram corresponds to the
$\delta_{\varphi C}$ counterterm.

From the first two diagrams we obtain

$\begin{split}&\Gamma^{\rm\ref{subfig:scalarvertexa}}_{\text{scalar}}=-g^{4}\frac{N_{1}}{8\pi\epsilon}M_{I}^{\
K}M_{K}^{\ J}\int d\tau\,\bar{\varphi}\,C_{J}\bar{C}^{I}\varphi\,,\\\
&\Gamma^{\rm\ref{subfig:scalarvertexb}}_{\text{scalar}}=g^{4}\frac{N_{1}}{8\pi\epsilon}\int
d\tau\,\bar{\varphi}\,C_{I}\bar{C}^{I}\varphi\,.\end{split}$ (3.34)

Diagram 9 involves the Yukawa couplings appearing in the last two lines of the
ABJ(M) action in (A.2). It can be built either using the
$2g^{2}C_{I}\bar{C}^{J}\bar{\psi}^{I}\psi_{J}$ vertex or the
$-g^{2}C_{I}\bar{C}^{I}\bar{\psi}^{J}\psi_{J}$ one. The two corresponding
contributions read respectively

$\begin{split}&\Gamma^{\rm\ref{subfig:scalarvertexc},1}_{\text{scalar}}=-g^{4}\frac{N_{2}}{2\pi\epsilon}\bar{\alpha}^{1}\alpha_{1}\int
d\tau\bar{\varphi}\,C_{2}\bar{C}^{2}\varphi\,,\\\
&\Gamma^{\rm\ref{subfig:scalarvertexc},2}_{\text{scalar}}=g^{4}\frac{N_{2}}{4\pi\epsilon}\bar{\alpha}^{1}\alpha_{1}\int
d\tau\bar{\varphi}\,C_{I}\bar{C}^{I}\varphi\,.\end{split}$ (3.35)

Therefore, we can summarize the result from this diagram as

$\Gamma^{\rm\ref{subfig:scalarvertexc}}_{\text{scalar}}=g^{4}\frac{N_{2}}{4\pi\epsilon}\bar{\alpha}^{1}\alpha_{1}\begin{pmatrix}1&0&0&0\\\
0&-1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}_{\\!\\!\,I}^{\ J}\int
d\tau\,\bar{\varphi}C_{J}\bar{C}^{I}\varphi\,.\,$ (3.36)

Finally, we move on to diagrams 9 and 9. Working in Lorentz gauge, it is easy
to see that the corresponding contributions are not divergent. In fact, we can
always integrate by parts the $\partial_{\mu}$ derivatives coming from the
ABJ(M) vertices on the external $C_{I},\bar{C}^{I}$ lines. As a consequence,
the integrand is finite for dimensional reasons.

Summing all the contributions, we eventually obtain

$0=g^{2}\left[\delta_{\varphi C}M_{I}^{\
J}-g^{2}\frac{N_{1}}{8\pi\epsilon}\left(M_{I}^{\ K}M_{K}^{\
J}-\delta_{I}^{J}\right)+g^{2}\frac{N_{2}}{4\pi\epsilon}\bar{\alpha}^{1}\alpha_{1}(\delta_{I}^{J}-2\delta_{I}^{2}\delta^{J}_{2})\right]\int
d\tau\,\bar{\varphi}C_{J}\bar{C}^{I}\varphi\,.$ (3.37)

This implies that

$Z_{\varphi C}M_{I}^{\ J}=(1+\delta_{\varphi C})M_{I}^{\ J}=M_{I}^{\
J}+\frac{g^{2}}{4\pi\epsilon}\bar{\alpha}^{1}\alpha_{1}\left[-N_{2}(\delta_{I}^{J}-2\delta_{I}^{2}\delta^{J}_{2})+2N_{1}(\bar{\alpha}^{1}\alpha_{1}-1)\delta_{I}^{1}\delta^{J}_{1}\right]\,.$
(3.38)

From the definition (LABEL:eq:Mren), for the scalar coupling renormalization
we can write

$({M}_{I}^{\ J})_{0}=\frac{Z_{\varphi C}}{Z_{\varphi}}M_{I}^{\ J}=M_{I}^{\
J}+\frac{g^{2}}{4\pi\epsilon}\bar{\alpha}^{1}\alpha_{1}\left[-N_{2}(\delta_{I}^{J}-2\delta_{I}^{2}\delta^{J}_{2})+2N_{1}(\bar{\alpha}^{1}\alpha_{1}-1)\delta_{I}^{1}\delta^{J}_{1}+N_{2}M_{I}^{\
J}\right]\,.$ (3.39)

The term at order $g^{2}$ on the right-hand side is zero for $I=J=2,3,4$,
whereas for $I=J=1$ we obtain

$(\bar{\alpha}^{1}\alpha_{1})_{0}=\bar{\alpha}^{1}\alpha_{1}\left[1+\frac{g^{2}}{4\pi\epsilon}(N_{1}+N_{2})(\bar{\alpha}^{1}\alpha_{1}-1)\right]\,,$
(3.40)

where on the left-hand side the subscript indicates the product of the two
bare parameters. This result is consistent with the
$\alpha_{1},\bar{\alpha}^{1}$ renormalization that we have already discussed.

###  3.4 $\beta$-functions

Having determined the renormalization functions, we can now compute the one-
loop $\beta$-functions for the parameters. To this end, we first recall that
the definition of the bare parameters are given in (LABEL:eq:parren).
Collecting the results for the renormalization functions found in the previous
sections,

$\begin{split}Z_{z}^{1/2}&=1-g^{2}\frac{N_{2}}{8\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\,,\\\
Z_{\tilde{\varphi}}^{1/2}&=1-g^{2}\frac{N_{1}}{8\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\,,\\\
Z_{\bar{\alpha}^{i}}=Z_{\alpha_{i}}&=1-\frac{g^{2}(N_{1}+N_{2})}{8\pi\epsilon}\,,\\\
Z_{\beta^{j}}=Z_{\bar{\beta}_{j}}&=1+\frac{g^{2}(N_{1}+N_{2})}{8\pi\epsilon}\,,\end{split}$
(3.41)

and plugging them there, we find

$\begin{split}&(\alpha_{k})_{0}=\left(1+\frac{g^{2}}{8\pi\epsilon}\,(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}-1)\right)\alpha_{k}\qquad
k=1,2\,,\\\
&(\bar{\alpha}^{k})_{0}=\left(1+\frac{g^{2}}{8\pi\epsilon}\,(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}-1)\right)\bar{\alpha}^{k}\,,\\\
&(\bar{\beta}_{l})_{0}=\left(1+\frac{g^{2}}{8\pi\epsilon}\,(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}+1)\right)\bar{\beta}_{l}\;\,\qquad
l=3,4\,,\\\
&(\beta^{l})_{0}=\left(1+\frac{g^{2}}{8\pi\epsilon}\,(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}+1)\right)\beta^{l}\,.\end{split}$
(3.42)

As already mentioned, if we set
$\bar{\alpha}^{2}=\alpha_{2}=\bar{\beta}_{j}=\beta^{j}=0$ and consider the
product $(\bar{\alpha}^{1}\alpha_{1})_{0}$, we obtain exactly the expression
(3.40) coming from the renormalization of the four-point scalar vertices. This
is a non-trivial check of our renormalization procedure.

The one-dimensional theory under investigation possesses nine dimensionless
coupling constants
$g_{a}=(g^{2},\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j})$, with
the new indices $a,b,\ldots$ running over these nine couplings. In dimensional
regularization with $d=3-2\epsilon$, the
$\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j}$ parameters remain
dimensionless, while $g^{2}$ acquires dimension $\Delta_{g^{2}}=2\epsilon$.

Expressing the bare coupling constants $(g_{a})_{0}$ as a function of the
renormalized ones as

$(g_{a})_{0}=\mu^{u_{a}\epsilon}\left[g_{a}+\frac{1}{\epsilon}K_{a}+{\mathcal{O}}\left(\tfrac{1}{\epsilon^{2}}\right)\right]\,,$
(3.43)

with $u_{g^{2}}=2$ and the others vanishing, the corresponding
$\beta$-functions are given by

$\beta_{a}=\mu\frac{dg_{a}}{d\mu}=-\epsilon
u_{a}g_{a}-u_{a}K_{a}+\sum_{b}u_{b}g_{b}\frac{\partial K_{a}}{\partial
g_{b}}\,.$ (3.44)

Specializing this to the $\alpha_{k}$ parameters, we find

$(\alpha_{k})_{0}=\alpha_{k}+\frac{1}{\epsilon}K_{\alpha_{k}}\,,\qquad\qquad\beta_{\alpha_{k}}=2g^{2}\frac{\partial
K_{\alpha_{k}}}{\partial g^{2}}\,,$ (3.45)

and similarly for the other parameters.

From (3.42) we can read off the explicit expressions of the $K$’s, which lead
to the following one-loop $\beta$-functions

$\begin{split}\beta_{\alpha_{k}}&=\frac{g^{2}}{4\pi}(N_{1}+N_{2})\,(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}-1)\alpha_{k}\,,\qquad\beta_{\bar{\alpha}^{k}}=\frac{g^{2}}{4\pi}(N_{1}+N_{2})\,(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}-1)\bar{\alpha}^{k},\\\
\beta_{\bar{\beta}_{l}}&=\frac{g^{2}}{4\pi}(N_{1}+N_{2})\,(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}+1)\bar{\beta}_{l}\,,\qquad\,\,\,\beta_{\beta^{l}}=\frac{g^{2}}{4\pi}(N_{1}+N_{2})\,(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}+1)\beta^{l}\,.\end{split}$
(3.46)

These are the analogues of the Polchinski-Sully $\beta$-functions for the
parameter $\zeta$ of the interpolating Wilson loop in ${\mathcal{N}}=4$ super
Yang-Mills theory Polchinski_2011 .

To conclude this section we observe that the results we have obtained for the
renormalization functions and the $\beta$-functions are path independent,
since short distance divergences should be blind to the actual form of the
Wilson loop contour. Therefore, we expect them to be valid also for the
renormalization of the parametric latitude Wilson loops of section 2.2. In
fact, as it will be discussed in CPTT , the renormalization functions that
remove UV divergences in that case are independent of the latitude angle and
coincide with the present ones.

##  4 Wilson loop expectation value

In this section we compute the two-loop VEV for the circular $1/24$ BPS Wilson
loop. In the auxiliary field approach this is given by (see appendix B for the
proof of this identity)

$\displaystyle\langle
W_{1/24}(\bar{\alpha}^{i},\alpha_{i},\beta^{j},\bar{\beta}_{j})\rangle$
$\displaystyle=\frac{1}{2}\langle\operatorname{Tr}\left(\Psi_{0}(2\pi)\bar{\Psi}_{0}(0)\right)\rangle$
$\displaystyle=\frac{1}{2}\Big{(}\langle
z_{0}(2\pi)\bar{z}_{0}(0)\rangle+\langle\varphi_{0}(2\pi)\bar{\varphi}_{0}(0)\rangle+\langle\tilde{z}_{0}(2\pi)\bar{\tilde{z}}_{0}(0)\rangle+\langle\tilde{\varphi}_{0}(2\pi)\bar{\tilde{\varphi}}_{0}(0)\rangle\Big{)}$
$\displaystyle=(1+\delta_{z})\langle
z(2\pi){\bar{z}}(0)\rangle+(1+\delta_{\tilde{z}})\langle{\tilde{z}}(2\pi)\bar{\tilde{z}}(0)\rangle\,.$
(4.1)

where in the last line we have taken into account the relation between bare
and renormalized fields, and the fact that $z$ and $\varphi$ in the auxiliary
matrix (3.6) have the same two-point function, as well as $\tilde{z}$ and
$\tilde{\varphi}$. We recall that the two counterterms
$\delta_{z},\delta_{\tilde{z}}$ can be read off from (3.24) and (3.25),
respectively.

Since the one-dimensional auxiliary field method is analogous to the
conventional way of computing Wilson loops VEVs, there are straightforward
relations between diagrams of the one-dimensional theory and diagrams coming
from the perturbative expansion of the Wilson loop. In fact, if in the
diagrams contributing to the two-point functions we identify the end points,
and identify the one-dimensional propagators with the Wilson loop contour, we
formally reproduce the one- and two-loop diagrams from the expansion of the
Wilson loop. It then follows that the typologies of integrals are the same in
the two cases, so we can exploit the results already present in the literature
for two-loop integrals of Wilson loops. We refer in particular to
Bianchi:2013zda ; Bianchi:2013rma  for details on the evaluation of the
integrals in the same set of conventions.

We evaluate the $\langle z(2\pi){\bar{z}}(0)\rangle$,
$\langle{\tilde{z}}(2\pi)\bar{\tilde{z}}(0)\rangle$ correlators at two loops
using the Lagrangian (3.13), that is using the Feynman rules for renormalized
quantities. According to (4) the result for the Wilson loop VEV is then
obtained by multiplying by the renormalization factors $(1+\delta)$ and
keeping the correct order in loops.

For instance, focusing on the $\langle z{\bar{z}}\rangle$ correlator, we
organize the perturbative expansion as

$\begin{split}\langle
W\rangle&=(1+\delta_{z}^{(1)}+\delta_{z}^{(2)}+\cdots)\left(\langle
z(2\pi){\bar{z}}(0)\rangle^{(0)}+\langle
z(2\pi){\bar{z}}(0)\rangle^{(1)}+\langle
z(2\pi){\bar{z}}(0)\rangle^{(2)}+\cdots\right)\\\
&=1+\left[\delta_{z}^{(1)}+\langle
z(2\pi){\bar{z}}(0)\rangle^{(1)}\right]+\left[\delta_{z}^{(2)}+\delta_{z}^{(1)}\langle
z(2\pi){\bar{z}}(0)\rangle^{(1)}+\langle
z(2\pi){\bar{z}}(0)\rangle^{(2)}\right]+\cdots\end{split}$ (4.2)

where $\delta^{(L)}_{z}$ indicates the counterterm at order $L$. The presence
of counterterms properly grouped according to their loop order is crucial to
remove short distance divergences from the integrals and make the expansion
order by order finite. In the next two sections we evaluate the finite
contributions corresponding to the two square brackets in (4.2).

###  4.1 One-loop analysis

At one-loop, the diagrams contributing to the two-point functions are the ones
depicted in figures 6 and 6. We can then exploit part of the previous
calculations, except that now we have to evaluate the finite part of the
integrals, having removed already the short distance divergence.

Diagram 6 still vanishes for planarity, as the epsilon tensor coming from the
vector propagator is contracted with three vectors lying on the plane of the
circular contour.

The contribution from 6 is given in (3.19). The integrals appearing there were
computed in dimensional regularization in Bianchi:2013zda ; Bianchi:2013rma ;
Griguolo_2013a . Using those results and taking into account that at this
order we find $\langle
z\bar{z}\rangle=\langle\tilde{z}\bar{\tilde{z}}\rangle$, the one-loop
expectation value of the 1/24 BPS operator reads

$\langle
W_{1/24}(\bar{\alpha}^{i},\alpha_{i},\beta^{j},\bar{\beta}_{j})\rangle^{(1)}=-(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})g^{2}N_{1}N_{2}\frac{4^{\epsilon}\pi^{\epsilon+1}\sec\pi\epsilon}{\Gamma(\epsilon)}=-(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})g^{2}N_{1}N_{2}\pi\epsilon.$
(4.3)

In the $\epsilon\to 0$ limit this contribution vanishes. However, since it
will enter later at two loops multiplied by the counterterms, it is necessary
to keep it for finite $\epsilon$.

###  4.2 Two-loop analysis

We now move on to the evaluation of the two-point functions in (4) at two
loops. In what follows we focus separately on bosonic and fermionic diagrams,
as well as on contributions due to the counterterms of the one-dimensional
theory.

Figure 10: Two-loop corrections to the one-dimensional fermionic propagator
$\langle z\bar{z}\rangle$.

#### Bosonic diagrams

The bosonic diagrams which contribute non-trivially are reported in the first
line of figure 10.

Diagram 10 contains the gauge propagator corrected at one loop. Using its
explicit expression (A.5), the corresponding contribution to $\langle
z\bar{z}\rangle^{(2)}$ reads

${\rm\ref{subfig:2-loop_bosa}}=-\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\langle
A_{\mu}(\tau_{1})A_{\nu}(\tau_{2})\rangle^{(1)}\dot{x}_{1}^{\mu}\dot{x}^{\nu}_{2}=\frac{g^{4}}{4}N^{2}_{1}N_{2}\,.$
(4.4)

The result for $\langle\tilde{z}\bar{\tilde{z}}\rangle^{(2)}$ is the same with
$N_{1}$ and $N_{2}$ exchanged.

Diagram 10 contains the ABJ(M) pure gauge vertex. Exploiting the results in
Bianchi:2013zda ; Bianchi:2013rma  for the corresponding integral, this gives

$\displaystyle{\rm{\ref{subfig:2-loop_bosb}}}=-\frac{i}{3g^{2}}\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}$
$\displaystyle
d\tau_{2}\int_{0}^{\tau_{2}}d\tau_{3}\,\epsilon^{\alpha\beta\gamma}\dot{x}_{1}^{\mu}\,\dot{x}_{2}^{\nu}\,\dot{x}_{3}^{\rho}$
(4.5) $\displaystyle\times\langle
A_{\mu}(\tau_{1})A_{\nu}(\tau_{2})\rangle\langle
A_{\rho}(\tau_{3})A_{\alpha}(x)\rangle\langle
A_{\beta}(x)A_{\gamma}(x)\rangle=-\frac{g^{4}}{24}N_{1}^{3}\,.$

The result for $\langle\tilde{z}\bar{\tilde{z}}\rangle^{(2)}$ is the same with
$N_{1}^{3}$ replaced by $N_{2}^{3}$.

Diagram 10 deserves more attention, since it is the only diagram which
contributes to the 1/24 BPS operator, but is absent in the more supersymmetric
cases. Its contribution to $\langle z\bar{z}\rangle^{(2)}$ is

${\rm\ref{subfig:2-loop_bosc}}=g^{4}N^{2}_{1}N_{2}\frac{\Gamma^{2}(\frac{1}{2}-\epsilon)}{16\pi^{3-2\epsilon}}\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\,\operatorname{Tr}\left(M(\tau_{1})M(\tau_{2})\right)\frac{1}{|x_{1}-x_{2}|^{2-4\epsilon}}\,.$
(4.6)

As long as the trace of two $M$ matrices is $\tau$-independent, this integral
is identically zero (see for instance Bianchi:2013rma ). This is what happens
in the 1/6 and 1/2 BPS cases. However, in the present case this trace acquires
a non-trivial $\tau$-dependence proportional to the loop parameters,

$\operatorname{Tr}\left(M(\tau_{1})M(\tau_{2})\right)\,\to\,8(\bar{\alpha}^{i}\alpha_{i})(\beta^{j}\bar{\beta}_{j})\cos\tau_{12}\,.$
(4.7)

This modifies the nature of the integral leading to a non-vanishing result. In
fact, the resulting integral is the same as the one-loop correction to the
gauge field propagator 10. Exploiting that result, we obtain

$\begin{split}{\rm\ref{subfig:2-loop_bosc}}&=g^{4}N_{1}N_{2}(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i})(\beta^{j}\bar{\beta}_{j})\frac{\Gamma^{2}(\frac{1}{2}-\epsilon)}{2\pi^{3-2\epsilon}}\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\,\frac{\cos\tau_{12}}{|x_{12}|^{2-4\epsilon}}\\\
&=-\frac{g^{4}}{2}(\bar{\alpha}^{i}\alpha_{i})(\beta^{j}\bar{\beta}_{j})N_{1}N_{2}(N_{1}+N_{2})\,.\end{split}$
(4.8)

Summarizing, the bosonic contribution to the $1/24$ BPS Wilson loop in (4) is

${\mathcal{B}}=\frac{g^{4}}{4}N_{1}N_{2}(N_{1}+N_{2})-\frac{g^{4}}{24}\left(N_{1}^{3}+N_{2}^{3}\right)-\frac{g^{4}}{2}(\bar{\alpha}^{i}\alpha_{i})(\beta^{j}\bar{\beta}_{j})N_{1}N_{2}(N_{1}+N_{2})\,.$
(4.9)

#### Fermionic diagrams

Fermionic diagrams contributing to the two-point functions are depicted in the
second and third lines of figure 10. The first diagram 10 contains the one-
loop corrected fermion propagator given in (A.5). Since this is proportional
to $(N_{1}-N_{2})$, when in (4) we sum up the contribution of $\langle
z(2\pi)\bar{z}(0)\rangle$ with the one from
$\langle\tilde{z}(2\pi)\bar{\tilde{z}}(0)\rangle$ obtained by exchanging
$N_{1}$ with $N_{2}$, they cancel each other. Therefore, this diagram does not
contribute to the Wilson loop VEV.

Moving to the double fermion-exchange diagrams illustrated in figures 10-10,
using known integrals from the literature Bianchi:2013zda ; Bianchi:2013rma ,
we obtain

$\begin{split}{\rm\ref{subfig:2-loop_fer2a}}+{\rm\ref{subfig:2-loop_fer2b}}&=\frac{1}{24}\int
d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}\,\langle
z(2\pi)\bar{z}(0)\big{(}\bar{z}\bar{f}\tilde{\varphi}\big{)}(\tau_{1})\big{(}\bar{z}\bar{f}\tilde{\varphi}\big{)}(\tau_{2})\\\
&\qquad\qquad\times\big{(}\bar{\tilde{\varphi}}fz\big{)}(\tau_{3})\big{(}\bar{\tilde{\varphi}}fz\big{)}(\tau_{4})\rangle+(\tau_{1},\tau_{2},\tau_{3},\tau_{4}\text{
permutations})\\\
&=\frac{3g^{4}}{8}N_{1}N_{2}(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})^{2}\end{split}$
(4.10)

Finally, the three diagrams in the last line of figure 10 correspond to the
three different ways of contracting the fields that exit the fermion-vector
vertex. Their sum reads

$\begin{split}&{\rm\ref{subfig:2-loop_fer3a}}+{\rm\ref{subfig:2-loop_fer3b}}+{\rm\ref{subfig:2-loop_fer3c}}\\\
&=i\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\int_{0}^{\tau_{2}}d\tau_{3}\Big{[}\langle\bar{f}(\tau_{1})\hat{A}_{\mu}(\tau_{2})\dot{x}^{\mu}_{2}f(\tau_{3})\rangle\\\
&\hskip
156.49014pt\langle\bar{f}(\tau_{1})f(\tau_{2})A_{\mu}(\tau_{3})\dot{x}^{\mu}_{3}\rangle+\langle
A_{\mu}(\tau_{1})\dot{x}^{\mu}_{1}\bar{f}(\tau_{2})f(\tau_{3})\rangle\Big{]}\,.\end{split}$
(4.11)

Inserting the explicit expressions (2.24) for the $f,\bar{f}$ functions, we
obtain a linear combination of integrals which are the same ones appearing in
the ordinary Wilson loop expansion. Therefore, exploiting known results in the
literature Bianchi:2013zda ; Bianchi:2013rma  and combining the contributions
from $\langle z(2\pi)\bar{z}(0)\rangle$ and
$\langle\tilde{z}(2\pi)\bar{\tilde{z}}(0)\rangle$, we eventually obtain

${\rm\ref{subfig:2-loop_fer3a}}+{\rm\ref{subfig:2-loop_fer3b}}+{\rm\ref{subfig:2-loop_fer3c}}=-\frac{g^{4}}{2}N_{1}N_{2}(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}-\beta^{j}\bar{\beta}_{j})\,.$
(4.12)

In conclusion, the total sum of fermionic diagrams reads

${\mathcal{F}}=\frac{3g^{4}}{8}N_{1}N_{2}(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})^{2}-\frac{g^{4}}{2}N_{1}N_{2}(N_{1}+N_{2})(\bar{\alpha}^{i}\alpha_{i}-\beta^{j}\bar{\beta}_{j})\,.$
(4.13)

#### Counterterms

As seen in section 3.3, we also need to include diagrams with vertices coming
from the counterterms. In particular, for the $\langle
z(2\pi)\bar{z}(0)\rangle$ two-loop correction we obtain the four diagrams in
figure 11. For all the other one-dimensional fields we have analogous diagrams
and the results extend straightforwardly.

Figure 11: Counterterm contributions to the self-energy of the one-dimensional
fermion $z$.

We start from diagram 11, which corresponds to the insertion of a one-loop
$\delta_{z}$ counterterm. We obtain

$\begin{split}{\rm\ref{subfig:diagcounta}}&=\delta_{z}\int d\tau\int
d\tau_{1}\int d\tau_{2}\langle
z(2\pi)\bar{z}(0)\big{(}\bar{z}\bar{f}\bar{\tilde{\varphi}}\big{)}(\tau_{1})\big{(}\varphi
fz\big{)}(\tau_{2})\big{(}\bar{z}\partial_{\tau}z\big{)}(\tau)\rangle+(\tau_{1}\leftrightarrow\tau_{2})\\\
&=2\delta_{z}\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\langle\bar{f}(\tau_{1})f(\tau_{2})\rangle\,,\end{split}$
(4.14)

where we have used $\partial_{\tau}\theta(\tau-x)=\delta(\tau-x)$. Similarly,
from 11 we have

${\rm\ref{subfig:diagcountb}}=\delta_{\tilde{\varphi}}\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\langle\bar{f}(\tau_{1})f(\tau_{2})\rangle\,.$
(4.15)

Diagrams 11 and 11 correspond to insertions of a fermionic counterterm vertex

$\displaystyle{\rm\ref{subfig:diagcountc}}+{\rm\ref{subfig:diagcountd}}$
$\displaystyle=-\frac{1}{2}\int d\tau_{1}\int d\tau_{2}\langle
z(2\pi)\bar{z}(0)\big{(}\bar{z}\bar{f}\bar{\tilde{\varphi}}\big{)}(\tau_{1})\big{(}\varphi\,(\delta_{f}f)z\big{)}(\tau_{2})\rangle$
(4.16) $\displaystyle\quad-\frac{1}{2}\int d\tau_{1}\int d\tau_{2}\langle
z(2\pi)\bar{z}(0)\big{(}\bar{z}\,(\delta_{\bar{f}}\bar{f})\bar{\tilde{\varphi}}\big{)}(\tau_{1})\big{(}\varphi
fz\big{)}(\tau_{2})\rangle+(\tau_{1}\leftrightarrow\tau_{2})$
$\displaystyle=-\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\bigg{(}\langle\bar{f}(\tau_{1})\delta_{f}f(\tau_{2})\rangle+\langle\delta_{\bar{f}}\bar{f}(\tau_{1})f(\tau_{2})\rangle\bigg{)}\,,$

where we have defined

$\delta_{f}f=e^{-\frac{i\tau}{2}}\xi\left(\delta_{\bar{\alpha}^{1}}\bar{\alpha}^{1}\psi^{2}-\delta_{\bar{\alpha}^{2}}\bar{\alpha}^{2}\psi^{1}\right)+e^{\frac{i\tau}{2}}\eta(\delta_{\beta^{3}}\beta^{3}\psi^{4}-\delta_{\beta^{4}}\beta^{4}\psi^{3})\,.$
(4.17)

and similarly for $\delta_{\bar{f}}\bar{f}$.

The same calculation can be reproduced for the
$\langle\tilde{z}\bar{\tilde{z}}\rangle$ two-point function. Taking into
account that $\delta_{\tilde{\varphi}}=\delta_{\tilde{z}}$ and
$\delta_{\varphi}=\delta_{z}$ the total contribution is

$\left(\langle
z(2\pi)\bar{z}(0)\rangle+\langle\tilde{z}(2\pi)\bar{\tilde{z}}(0)\rangle\right)^{\text{ct}}=\left(\delta_{f}+\delta_{\bar{f}}-3\delta_{z}-3\delta_{\tilde{z}}\right)\langle
W\rangle^{(1)}\,,$ (4.18)

where $\langle W\rangle^{(1)}$ is the one-loop contribution to circular Wilson
loops (4.3).

According to expansion (4.2), at two loops we have extra finite contributions
coming from the product of the one-loop counterterms (set
$\delta^{(1)}_{z}\equiv\delta_{z},\delta_{\tilde{z}}$ there) multiplying the
one-loop ${\cal O}(\epsilon)$ two-point functions

$\delta_{z}\langle
z(2\pi)\bar{z}(0)\rangle^{(1)}+\delta_{\tilde{z}}\langle\tilde{z}(2\pi)\bar{\tilde{z}}(0)\rangle^{(1)}=\left(\delta_{z}+\delta_{\tilde{z}}\right)\langle
W\rangle^{(1)}\,.$ (4.19)

Therefore, summing (4.18) and (4.19), the final contribution to the Wilson
loop VEV from the counterterms is

${\mathcal{C}}=-g^{4}\frac{N_{1}N_{2}}{4}(N_{1}+N_{2})\left[(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})^{2}-\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j}\right]\,.$
(4.20)

###  4.3 The final result for the Wilson loop VEV

Combining all the previous results, ${\cal B}+{\cal F}+{\cal C}$, the large
$N_{1},N_{2}$ expectation value for the parametric circular $1/24$ BPS Wilson
loop at two loops reads

$\langle{\mathcal{W}}_{1/24}(\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j})\rangle=1-\frac{g^{4}}{24}\left[N_{1}^{2}+N_{2}^{2}-4N_{1}N_{2}-3N_{1}N_{2}(\bar{\alpha}^{i}\alpha_{i}-\beta^{j}\bar{\beta}_{j}-1)^{2}\right]+{\mathcal{O}}(g^{6})\,,$
(4.21)

where we recall that we have defined $g=\sqrt{2\pi/k}$.

By setting $\beta^{j}=\bar{\beta}_{j}=0$ or $\bar{\alpha}^{i}=\alpha_{i}=0$,
one recovers two branches of interpolating $1/6$ BPS fermionic Wilson loops,
which have the following VEVs

$\begin{split}\langle{\mathcal{W}}_{1/6}^{\textrm{I}}(\alpha_{i},\bar{\alpha}^{i})\rangle&=1-\frac{g^{4}}{24}\left[N_{1}^{2}+N_{2}^{2}-4N_{1}N_{2}-3N_{1}N_{2}(\bar{\alpha}^{i}\alpha_{i}-1)^{2}\right]+{\mathcal{O}}(g^{6})\,,\\\
\langle{\mathcal{W}}_{1/6}^{\textrm{II}}(\beta^{j},\bar{\beta}_{j})\rangle&=1-\frac{g^{4}}{24}\left[N_{1}^{2}+N_{2}^{2}-4N_{1}N_{2}-3N_{1}N_{2}(\beta^{j}\bar{\beta}_{j}+1)^{2}\right]+{\mathcal{O}}(g^{6})\,.\end{split}$
(4.22)

We recall that, according to the classification in Ouyang:2015iza ;
Ouyang:2015bmy , “type I” and “type II” $1/6$ BPS fermionic Wilson loops
differ by the preserved $SU(2)\subset SU(4)$ R-symmetry group.

If in particular we choose $\bar{\alpha}^{i}\alpha_{i}=1$ in
${\mathcal{W}}_{1/6}^{\textrm{I}}(\alpha_{i},\bar{\alpha}^{i})$ or
$\beta^{j}\bar{\beta}_{j}=-1$ in
${\mathcal{W}}_{1/6}^{\textrm{II}}(\beta^{j},\bar{\beta}_{j})$, we recover the
known result for 1/2 BPS operators Bianchi:2013zda ; Bianchi:2013rma ;
Griguolo_2013a , at two loops and in the large $N_{1},N_{2}$ limit:

$\begin{split}\langle{\mathcal{W}}^{\textrm{I}}_{1/2}\rangle=\langle{\mathcal{W}}^{\textrm{II}}_{1/2}\rangle=1-\frac{g^{4}}{24}\left(N_{1}^{2}+N_{2}^{2}-4N_{1}N_{2}\right)+{\mathcal{O}}(g^{6})\,.\end{split}$
(4.23)

Finally, if we set all the parameters to zero we obtain the two-loop
expectation value of the bosonic operator Drukker:2008zx ; Chen:2008bp ;
Kluson:2008zrv ; Rey:2008bh

$\langle{\mathcal{W}}_{1/6}^{\rm bos}\rangle\equiv\left\langle\frac{N_{1}{\cal
W}^{\rm bos}+N_{2}\hat{\cal W}^{\rm
bos}}{N_{1}+N_{2}}\right\rangle=1-\frac{g^{4}}{24}\left(N_{1}^{2}+N_{2}^{2}-7N_{1}N_{2}\right)+{\mathcal{O}}(g^{6}).$
(4.24)

where ${\cal W}^{\rm bos},\hat{\cal W}^{\rm bos}$ are the bosonic operators
defined in (2.4).

We have evaluated the Wilson loop VEVs exploiting the one-dimensional
auxiliary field formulation. Alternatively, one could use the ordinary
procedure of expanding $W$ in powers of the superconnection and evaluate
correlation functions of ${\cal L}$. We have checked that proceeding in this
way, once we replace bare parameters with their renormalized expressions found
above, the final result coincides with (4.21). This is a non-trivial check of
our procedure.

##  5 Discussion

###  5.1 Renormalization Group flows

In section 3.4 we have shown that the introduction of the weakly relevant
couplings $\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j}$ triggers a
RG flow driven by the eight $\beta$-functions (3.46). Here we study this flow
by focusing on the “type I” operators defined above. The study of RG flows
involving “type II” Wilson loops will be presented elsewhere CPTT .

In order to give an intuitive visual description of the RG flow we define
$\bar{\alpha}^{1}=\alpha_{1}=x$ and $\beta^{3}=\bar{\beta}_{3}=y$ and set the
other parameters to zero. The relevant $\beta$-functions are then

$\begin{split}\beta_{x}(x,y)&=\mu\frac{\partial
x}{\partial\mu}=\frac{g^{2}}{4\pi}(N_{1}+N_{2})(x^{2}+y^{2}-1)\,x\,,\\\
\beta_{y}(x,y)&=\mu\frac{\partial
y}{\partial\mu}=\frac{g^{2}}{4\pi}(N_{1}+N_{2})(x^{2}+y^{2}+1)\,y\,.\end{split}$
(5.1)

In figure 12 we plot $\beta_{x}(x,y=0)$ and highlight the zeros of the
$\beta_{x}$-function.

Figure 12: Plot of $\beta_{x}$ when $y=0$. To make contact with figure 13, we
have highlighted the fixed points.

In figure 13 we draw the RG flow trajectories in the $(x,y)$ plane, for $x,y$
real and non-negative. For negative $x$ we would obtain an equivalent fixed
point. The blue point in figures 12, 13 corresponds to the $x=y=0$ fixed point
where the associated operator is the bosonic $1/6$ BPS Wilson loop ${\cal
W}_{1/6}^{\rm bos}$ defined in (4.24). This is a UV fixed point where the
parameters trigger an outgoing flow. Moving along the horizontal green line in
figure 13 we reach an IR fixed point (highlighted in red) corresponding to
“type I” fermionic $1/2$ BPS Wilson loop obtained from
${\mathcal{W}}^{\textrm{I}}_{1/6}$ by selecting
$\bar{\alpha}^{i}\alpha_{i}=1$.

Figure 13: The RG flow in the $(x,y)$ plane. Arrows go from the UV to the IR.
Arrows on the $x,y$ axes correspond to $1/6$ BPS flows, while arrows outside
the $x,y$ axes correspond to $1/24$ BPS flows. Horizontal green arrows
correspond to the RG flow between the bosonic $1/6$ BPS Wilson loop (blue dot)
and “type I” fermionic $1/2$ BPS Wilson loop described by
$\langle{\mathcal{W}}^{\textrm{I}}_{1/6}(\alpha_{i},\bar{\alpha}^{i})\rangle$
at $\bar{\alpha}^{i}\alpha_{i}=1$ (red dot).

The green line describes an enriched RG flow between UV and IR fixed points,
along which supersymmetry is partially preserved. In fact, all the points on
the two axes, even those not highlighted in green, correspond to operators
which preserve four supercharges.

Similarly, flows along a generic direction in the plane preserve one
supercharge, corresponding to the $1/24$ BPS operator, whose VEV is given in
(4.21). In this sense, they can still be interpreted as enriched RG flows.

More generally, still setting $y=0$, we relax the condition
$\bar{\alpha}^{1}=\alpha_{1}=x$, and consider the flow in the
$(\alpha_{1},\bar{\alpha}^{1})$ plane, as presented in figure 14. As expected,
the $1/2$ BPS curve $\alpha_{1}\bar{\alpha}^{1}=1$ corresponds to a set of
attractive points. This is in agreement with the analysis of figure 13 where
the red dot is also attractive and green trajectories connect fixed points.

Figure 14: The RG flow in the $(\alpha_{1},\bar{\alpha}^{1})$ plane. Arrows go
from the UV to the IR. The red curve corresponds to
$\alpha_{1}\bar{\alpha}^{1}=1$ and the blue dot is the bosonic $1/6$ BPS
Wilson loop.

###  5.2 The defect SQFT

Non-local operators like Wilson loops can be used to describe one-dimensional
defect quantum field theories (dQFTs).101010For a quite exhaustive list of
references on linear defects, see for instance Penati:2021tfj . In particular,
if the operator preserves the one-dimensional conformal algebra
$sl(2,{\mathbb{R}})$, it defines a defect conformal field theory (dCFT). In
addition, if the operator is BPS and preserves a sufficient amount of
supersymmetry, the corresponding defect is a superconformal field theory
(dSCFT).

Focusing on the set of ABJ(M) circular Wilson loops considered in this paper,
it is well known that the 1/6 and 1/2 BPS fermionic operators preserve the
$su(1,1|1)$ and $su(1,1|3)$ one-dimensional superconformal algebras,
respectively. Therefore, they describe superconformal defects. Instead, the
new 1/24 BPS operator defined in (2.8)-(2.10) describes a supersymmetric, but
not (super)conformal defect, as it preserves only one supercharge and the
dependence of the scalar couplings on the contour coordinate breaks conformal
invariance.

Regardless of their superconformal or only supersymmetric nature, the dQFTs
supported by ABJ(M) fermionic Wilson loops are generated by local operators
defined by $U(N_{1}|N_{2})$ supermatrices localized on the Wilson loop. The
dQFT is featured by the set of correlation functions of these local operators,
inserted on the Wilson loop vacuum. Precisely, the defect $n$-point function
of a set of local operators inserted at points $\tau_{1},\ldots,\tau_{n}$
along the circle is defined as

$\operatorname{\big{\langle}\big{\langle}}\operatorname{Tr}\left(O_{n}O_{n-1}\ldots
O_{1}\right)\operatorname{\big{\rangle}\big{\rangle}}\equiv\frac{\left\langle\operatorname{Tr}{\mathcal{P}}\left(e^{-i\int_{\tau_{n}}^{2\pi}d\tau{\mathcal{L}}(\tau)}O_{n}e^{-i\int_{\tau_{n-1}}^{\tau_{n}}d\tau{\mathcal{L}}(\tau)}O_{n-1}\ldots
O_{1}e^{-i\int_{0}^{\tau_{1}}d\tau{\mathcal{L}}(\tau)}\right)\right\rangle}{\langle{\cal
W}\rangle}\,,$ (5.2)

where in the right hand side the expectation value is on the ABJ(M) vacuum.
The insertion of Wilson links ensures gauge invariance. In the one-dimensional
auxiliary field formalism introduced in section 3.1 the defect $n$-point
function can be written entirely in terms of ABJ(M) expectation values of
products of supermatrices localized on the contour

$\operatorname{\big{\langle}\big{\langle}}\operatorname{Tr}\left(O_{n}O_{n-1}\ldots
O_{1}\right)\operatorname{\big{\rangle}\big{\rangle}}\equiv\frac{1}{2^{n+1}}\frac{\langle\operatorname{Tr}\left(\Psi(2\pi)\bar{\Psi}(\tau_{n})O_{n}\Psi(\tau_{n})\bar{\Psi}(\tau_{n-1})O_{n-1}\ldots
O_{1}\Psi(\tau_{1})\bar{\Psi}(0)\right)\rangle}{\langle\operatorname{Tr}\Psi(2\pi)\bar{\Psi}(0)\rangle}\,.$
(5.3)

In this context, the $(x,y)$ plane depicted in figure 13 has the nice
interpretation of describing different defect theories, with fixed points
corresponding to theories at their critical point. The blue point is a UV
unstable fixed point corresponding to a one-dimensional ${\cal N}=1$ SCFT. The
red point represents an interacting IR critical theory where supersymmetry
gets enhanced to ${\cal N}=3$.

The Wilson loop RG flows that we have found are interpreted as flows in the
space of one-dimensional defects. The two axes describe a continuum of one-
dimensional SCFTs. Along these two directions the enriched flow preserves
${\cal N}=1$ superconformal invariance. As soon as we move out of the two
axes, superconformal invariance is broken, although one supercharge is still
preserved. Nevertheless along all of these flows the system is driven towards
the IR fixed point corresponding to ${\mathcal{W}}_{1/2}^{\textrm{I}}$.

It is important to give a closer look at the weakly relevant operators which
perturb the system and drive it away from the UV fixed point. To this end, we
recall that the UV fixed point corresponds to the bosonic Wilson operator
${\cal W}_{1/6}^{\rm bos}$ obtained by setting all the parameters to zero.
Moving along the two green lines in figure 13 amounts to adding a deforming
operator as ${\cal L}_{1/6}^{\rm bos}\rightarrow{\cal L}_{1/6}^{\rm bos}+{\cal
L}^{\rm def}$. For instance, if we move along the horizontal axis, which
amounts to setting $\beta^{j}=\bar{\beta}_{j}=0$, and choose for simplicity
$\alpha_{1}=\bar{\alpha}^{1}=0$, we have

${\cal L}^{\rm def}=\bar{\alpha}^{2}\alpha_{2}\begin{pmatrix}-\frac{4\pi
i}{k}C_{2}\bar{C}^{2}&0\\\ 0&-\frac{4\pi
i}{k}\bar{C}^{2}C_{2}\end{pmatrix}-\bar{\alpha}^{2}\,e^{i\tau/2}\begin{pmatrix}0&\eta\bar{\psi}^{1}\\\
0&0\end{pmatrix}-\alpha_{2}\,e^{-i\tau/2}\begin{pmatrix}0&0\\\
\xi\psi_{1}&0\end{pmatrix}\,.$ (5.4)

We are interested in computing the anomalous dimensions of these operators.
This amounts to computing their two-point functions in the bosonic $1/6$ BPS
defect.

Focusing on the two fermionic operators, it is easy to see that their
integrated two-point function can be expressed as

$\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\left[\frac{e^{i\frac{\tau_{12}}{2}}\operatorname{\big{\langle}\big{\langle}}(\eta\bar{\psi}^{1})(\tau_{1})(\xi\psi_{1})(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}+e^{-i\frac{\tau_{12}}{2}}\operatorname{\big{\langle}\big{\langle}}(\xi\psi_{1})(\tau_{1})(\eta\bar{\psi}^{1})(\tau_{2})\operatorname{\big{\rangle}\big{\rangle}}}{N_{1}+N_{2}}\right]_{\alpha_{2},\bar{\alpha}^{2}=0}=\\\
\qquad=-\frac{\partial^{2}}{\partial\alpha_{2}\partial\bar{\alpha}^{2}}\log\langle{\cal
W}^{I}_{1/6}(\alpha_{2},\bar{\alpha}^{2})\rangle\Bigg{|}_{\alpha_{2},\bar{\alpha}^{2}=0}\,.$
(5.5)

If the operators develop an anomalous dimension $\gamma$, the left hand side
of this equation formally becomes

$\begin{split}-\frac{g^{2}}{\pi}\frac{N_{1}N_{2}}{N_{1}+N_{2}}\int_{0}^{2\pi}\\!\\!\\!\\!d\tau_{1}\int_{0}^{\tau_{1}}\\!\\!\\!\\!d\tau_{2}\,\frac{1}{|4\sin^{2}\frac{\tau_{12}}{2}|^{1+\gamma}}&=-\frac{\sqrt{\pi}g^{2}}{2}\frac{N_{1}N_{2}}{N_{1}+N_{2}}\frac{\Gamma(-\frac{1}{2}-\gamma)}{\Gamma(-\gamma)}\\\
&\underset{|\gamma|\ll
1}{\simeq}-g^{2}\pi\frac{N_{1}N_{2}}{N_{1}+N_{2}}\gamma\,.\end{split}$ (5.6)

On the other hand, the right hand side of (5.5) can be easily evaluated
observing that the two-loop result satisfies the following identity

$\frac{\partial}{\partial\bar{\alpha}^{2}}\log\langle{\cal
W}^{I}_{1/6}(\alpha_{2},\bar{\alpha}^{2})\rangle=\kappa\,\beta_{\alpha_{2}}\,,\qquad\text{with
}\qquad\kappa=\pi g^{2}\frac{N_{1}N_{2}}{N_{1}+N_{2}}\,.$ (5.7)

Therefore, comparing the two expressions we finally obtain

$\gamma=\frac{\partial\beta_{\alpha_{2}}}{\partial\alpha_{2}}\Bigg{|}_{\alpha_{2},\bar{\alpha}^{2}=0}=-\frac{g^{2}}{4\pi}(N_{1}+N_{2})\,.$
(5.8)

This result can also be checked by explicitly computing the first order
correction to the two-point function in (5.5).

We have found that the first contribution to the anomalous dimension of the
fermionic fields is negative. With a similar reasoning, one can check that
also the bi-scalar operator $C_{2}\bar{C}^{2}$ acquires negative anomalous
dimension. This confirms that the deformation (5.4) is a weakly relevant
operator.

More generally, we can compute the $\psi_{1},\bar{\psi}^{1}$ anomalous
dimension in the ${\cal W}^{I}_{1/6}(\alpha_{2},\bar{\alpha}^{2})$ defect.
This amounts to evaluating the derivative of the $\beta$-function without
fixing the values of the parameters. We easily find

$\gamma(\alpha_{2},\bar{\alpha}^{2})=\frac{\partial\beta_{\alpha_{2}}}{\partial\alpha_{2}}=\frac{g^{2}}{4\pi}(N_{1}+N_{2})(2\bar{\alpha}^{2}\alpha_{2}-1)\,.$
(5.9)

This interpolates between the dimension of the weakly relevant operator in the
UV and its dimension in the IR.

###  5.3 A g-theorem

Focusing on $1/6$ BPS flow along the green line we now prove the validity of a
g-theorem. In order to keep the discussion simpler we again set
$\alpha_{1}=\bar{\alpha}^{1}=x$ and $\beta^{3}=\bar{\beta}_{3}=y$ with other
parameters set to zero.

Referring to the $\beta$-function as written in (5.1), in the $x\in[0,1]$
region we can write

$\frac{\partial}{\partial
x}\log\langle{\mathcal{W}}^{\textrm{I}}_{1/6}(x)\rangle=2\kappa\,\beta_{x}\Big{|}_{y=0}\,.$
(5.10)

where $\kappa$ has been defined in (5.7). First of all, this implies that the
blue and the red conformal fixed points in figure 13 are extrema of
$\langle{\mathcal{W}}_{1/24}(\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j})\rangle$.
Moreover, it is easy to show that the (red) $1/2$ BPS point is a minimum while
the (blue) bosonic $1/6$ BPS point is a maximum. Therefore, comparing the two
fixed points connected by the horizontal green line in figure 13, we can write

$\log\langle{\mathcal{W}}_{1/6}^{\textrm{bos}}\rangle\equiv\log\langle{\mathcal{W}}_{1/6}^{\textrm{I}}(x=0)\rangle>\log\langle{\mathcal{W}}_{1/6}^{\textrm{I}}(x=1)\rangle\equiv\log\langle{\mathcal{W}}^{\textrm{I}}_{1/2}\rangle\,.$
(5.11)

This result can be interpreted as a g-theorem Cuomo:2021rkm  for the one-
dimensional defect. In fact, defining the interpolating functions
$g^{\textrm{I}}=\langle{\mathcal{W}}^{\textrm{I}}_{1/6}\rangle$, we find
$g^{\textrm{I}}_{UV}>g^{\textrm{I}}_{IR}$. Recalling that
$\log\langle{\mathcal{W}}\rangle$ is nothing but the partition function of the
one-dimensional defect, monotonicity is consistent with the decreasing of
degrees of freedom from the UV to the IR fixed point. Our result is in line
with what has been already found in ${\mathcal{N}}=4$ super Yang-Mills
Beccaria:2017rbe ; Beccaria:2018ocq , although in a different setup, as it
consists of BPS flows.

###  5.4 Comparison with the localization result

It is well known that in supersymmetric theories defined on compact manifolds
BPS Wilson loops can be computed using supersymmetric localization Pestun_2012
. This provides a representation of the path integral evaluating the Wilson
loop VEV as a matrix integral.

For the ABJ(M) theory on $S^{3}$ localization allows to exactly compute the
VEV of the bosonic Wilson loops in (2.4) as the expectation values

$\langle{\mathcal{W}}^{\rm
bos}\rangle_{1}=\left\langle\frac{1}{N_{1}}\sum_{i=1}^{N_{1}}e^{2\pi\lambda_{i}}\right\rangle_{{}_{\mathcal{Z}}}\,,\qquad\langle\hat{\cal
W}^{\rm
bos}\rangle_{1}=\left\langle\frac{1}{N_{2}}\sum_{i=1}^{N_{2}}e^{2\pi\hat{\lambda}_{i}}\right\rangle_{{}_{\mathcal{Z}}}\,,$
(5.12)

where the average is evaluated and normalized using the following non-Gaussian
matrix model Kapustin:2009kz

${\mathcal{Z}}=\int\prod_{i=1}^{N_{1}}d\lambda_{i}e^{i\pi
k\lambda^{2}_{i}}\prod^{N_{2}}_{j=1}d\hat{\lambda}_{j}e^{-i\pi
k\hat{\lambda}^{2}_{j}}\frac{\prod_{i<j}^{N_{1}}\sinh^{2}(\pi(\lambda_{i}-\lambda_{j}))\prod_{i<j}^{N_{2}}\sinh^{2}(\pi(\hat{\lambda}_{i}-\hat{\lambda}_{j}))}{\prod_{i=1}^{N_{1}}\prod_{j=1}^{N_{2}}\cosh^{2}(\pi(\lambda_{i}-\hat{\lambda}_{j}))}\,.$
(5.13)

Here the integrations are on two complete sets of eigenvalues
$\\{\lambda_{i}\\}$, $\\{\hat{\lambda}_{j}\\}$ of the Cartan subalgebras of
$U(N_{1})$ and $U(N_{2})$, respectively. The “1” subscript in (5.12) indicates
that the matrix model computes the expectation values at framing $f=1$
Kapustin:2009kz .111111See also Bianchi:2013rma ; Bianchi:2016yzj  and chapter
6 of Drukker:2019bev  for an introductory discussion to framing in three-
dimensional Chern-Simons-matter theories. As we have emphasized throughout
this paper, the dimensional regularization used in section 4 is alternative to
framing regularization, therefore the perturbative results obtained in this
paper correspond to the $f=0$ scheme.

The main observation is that the prescription (5.12) automatically provides
exact results for the whole class of BPS Wilson loops that we have considered
in this paper. This stems from the fact that, classically, the 1/24 BPS, the
1/6 BPS fermionic and the 1/2 BPS Wilson loops are all cohomologically
equivalent to the linear combination ${\cal W}_{1/6}^{\rm bos}$ defined in
(4.24). In other words, they differ from ${\cal W}_{1/6}^{\rm bos}$ by a
${\cal Q}$-exact term, where ${\cal Q}$ is one of the supercharges preserved
by all the operators in the game. Therefore, if cohomological equivalence is
preserved at the quantum level, one can in principle use this ${\cal Q}$ to
localize the path integral for the Wilson loop VEV. As a consequence, the
following identities hold

$\langle{\mathcal{W}}_{1/24}(\alpha_{i},\bar{\alpha}^{i},\beta^{j},\bar{\beta}_{j})\rangle_{1}=\langle{\mathcal{W}}_{1/6}^{\textrm{I}}(\alpha_{i},\bar{\alpha}^{i})\rangle_{1}=\langle{\mathcal{W}}_{1/6}^{\textrm{II}}(\beta^{j},\bar{\beta}_{j})\rangle_{1}=\langle{\mathcal{W}}_{1/2}^{\textrm{I,II}}\rangle_{1}=\langle{\cal
W}_{1/6}^{\rm bos}\rangle_{1}\,.$ (5.14)

At framing one, all the VEVs must equal $\langle{\cal W}_{1/6}^{\rm
bos}\rangle_{1}=(N_{1}\langle{\cal W}^{\rm bos}\rangle+N_{2}\langle\hat{\cal
W}^{\rm bos}\rangle)/(N_{1}+N_{2})$, which can be easily evaluated from
(5.12). At weak coupling, this quantity is known both from the matrix model
expansion Kapustin:2009kz  and from perturbation theory Mauri_2018 . Up to two
loops it reads

$\langle{\cal W}_{1/6}^{\rm
bos}\rangle_{1}=1+\frac{i\pi(N_{1}-N_{2})}{k}-\frac{\pi^{2}}{6k^{2}}\left[4(N_{1}^{2}+N_{2}^{2})-10N_{1}N_{2}-1\right]\,.$
(5.15)

Therefore, as a consequence of identities (5.14), at $f=1$ the VEVs loose any
dependence on the alpha and beta parameters. In other words, all the points of
the plot in figure 13 correspond to the same quantum operator. It is then
interesting to understand how the parameter dependence arises when the
expectation values are evaluated at $f\neq 1$, in particular at framing zero.

For 1/6 BPS bosonic and 1/2 BPS fermionic Wilson loops, a relation between
their expectation values at framing zero computed perturbatively and the ones
at framing one coming from the matrix model has been found Drukker:2009hy ;
Bianchi_2016 . Up to two loops, these read

$\langle{\cal W}^{\rm bos}\rangle_{1}=e^{\frac{i\pi N_{1}}{k}}\langle{\cal
W}^{\rm bos}\rangle_{0}\,,\quad\langle\hat{\cal W}^{\rm
bos}\rangle_{1}=e^{-\frac{i\pi N_{2}}{k}}\langle\hat{\cal W}^{\rm
bos}\rangle_{0}\,,\quad\langle{\cal
W}^{\textrm{I,II}}_{1/2}\rangle_{1}=e^{\frac{i\pi(N_{1}-N_{2})}{k}}\langle{\cal
W}_{1/2}\rangle_{0}\,.$ (5.16)

They can be generalized to generic (also non-integer) framing $f$ in a rather
simple way, see the discussion in Bianchi:2014laa .

Since the interpolating operators under investigation have a non-trivial
parametric dependence at framing zero, but loose this dependence at framing
one, we expect the parameter dependence to be carried by “phase” factors, in
analogy with (5.16). Perturbative analysis reveals that up to two loops and in
the large $N_{1},N_{2}$ limit, the matrix model results (5.14)-(5.15) are
related to the perturbative ones in section 4.3 as follows. For the fermionic
1/6 BPS operators we are led to the following identities

$\begin{split}&\langle{\cal
W}_{1/6}^{\textrm{I}}(\bar{\alpha}^{i},\alpha_{i})\rangle_{1}=\frac{N_{1}e^{\frac{i\pi}{k}(N_{1}-\bar{\alpha}^{i}\alpha_{i}N_{2})}+N_{2}\,e^{\frac{i\pi}{k}(\bar{\alpha}^{i}\alpha_{i}N_{1}-N_{2})}}{N_{1}+N_{2}}\langle{\cal
W}_{1/6}^{\textrm{I}}(\bar{\alpha}^{i},\alpha_{i})\rangle_{0}\,,\\\
&\langle{\cal
W}_{1/6}^{\textrm{II}}(\beta^{j},\bar{\beta}_{j})\rangle_{1}=\frac{N_{1}e^{\frac{i\pi}{k}(N_{1}+\beta^{j}\bar{\beta}_{j}N_{2})}+N_{2}\,e^{\frac{i\pi}{k}(-\beta^{j}\bar{\beta}_{j}N_{1}-N_{2})}}{N_{1}+N_{2}}\langle{\cal
W}_{1/6}^{\textrm{II}}(\beta^{j},\bar{\beta}_{j})\rangle_{0}\,,\end{split}$
(5.17)

whereas for the more general $1/24$ BPS operator the relation it reads

$\displaystyle\langle{\cal
W}_{1/24}\rangle_{1}=\frac{N_{1}e^{\frac{i\pi}{k}(N_{1}-(\bar{\alpha}^{i}\alpha_{i}-\beta^{j}\bar{\beta}_{j})N_{2})}+N_{2}\,e^{\frac{i\pi}{k}((\bar{\alpha}^{i}\alpha_{i}-\beta^{j}\bar{\beta}_{j})N_{1}-N_{2})}}{N_{1}+N_{2}}\langle{\cal
W}_{1/24}\rangle_{0}\,.$ (5.18)

In the ABJM limit, $N_{1}=N_{2}$, it boils down to

$\langle{\cal
W}_{1/24}(\bar{\alpha}^{i},\alpha_{i},\beta^{j},\bar{\beta}_{j})\rangle_{1}=\cos\left(\tfrac{\pi
N}{k}(1-\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\right)\langle{\cal
W}_{1/24}(\bar{\alpha}^{i},\alpha_{i},\beta^{j},\bar{\beta}_{j})\rangle_{0}\,.$
(5.19)

Similar relations come from (LABEL:eqn:MMrelation2) for $N_{1}=N_{2}$.

We note that the exponentials carrying the parameter dependence are no longer
pure phases as in (5.16), since the parameters can be generically complex and
the barred parameters are not the complex conjugates. However, for the special
values $\bar{\alpha}^{i}\alpha_{i}=1$ and $\beta^{j}\bar{\beta}_{j}=0$ they
reduce to the last phase in (5.16).

These identities have been empirically inferred from the two-loop results and
are not expected to be true in general. In fact, the parametric exponents are
likely to be corrected at higher orders, as already happens at three loops for
the ${\cal W}^{\rm bos},\hat{\cal W}^{\rm bos}$ phases Bianchi:2016yzj .

We close this section with a technical observation on the integrals in the two
schemes, framing or dimensional regularization ($f=0$). The parameter
independence of the framing-one results indicates that a genuine perturbative
calculation done at $f=1$ should sensibly differ from our present calculation
done using dimensional regularization. In particular, new non-vanishing
contributions should arise to compensate the parameter dependence carried by
diagrams that are framing independent. For instance, let us focus on the
contributions proportional to
$\bar{\alpha}^{i}\alpha_{i}\beta^{j}\bar{\beta}_{j}$. At two loops and framing
zero, they come from the scalar diagram 10 and the two fermion ones, 10 and
10. As shown in Mauri_2018 , the scalar integral is framing independent, thus
its dependence on the parameters should survive also at framing one. On the
other hand, it has been argued in Bianchi_2016  that at $f=1$ the fermionic
diagrams should be identically vanishing. Therefore, at framing one some new
parameter dependent contribution should arise, which eventually cancels the
scalar diagram. A proof of this statement would require performing a genuine
two-loop calculation at framing one, though this might be obstructed by the
difficulty of computing fermionic diagrams at non-trivial framing.

###  5.5 Outlook

There are several natural directions in which these investigations could be
extended.

As stressed repeatedly, the flows considered in this paper are BPS, with at
least one supercharge preserved at all points of the flow. It would of course
be interesting to introduce a $\zeta$-parameter in the 1/24 BPS Wilson loop,
to interpolate to a fully non-supersymmetric limit, like it has been done in
Polchinski_2011  for the 1/2 BPS circular Wilson loop of ${\mathcal{N}}=4$
super Yang-Mills in four dimensions. This could be achieved by rescaling the
overall couplings to the scalars and the fermions in (2.26) and would make the
RG flow space even richer, opening up a new direction corresponding to the
renormalization of the new parameter. We plan to address this in a future
investigation.

As usual, there is always the question of the holographic dual description in
terms of minimal surfaces in M-theory or type IIA superstring theory. The 1/24
BPS Wilson loop considered in this paper should be described by mixed boundary
conditions in $AdS_{2}$, generalizing what has been done in Correa:2019rdk ;
Garay:2022szq  for deformations of the 1/6 BPS bosonic operator defined on a
straight line. For instance, it would interesting to understand whether the
set of boundary conditions that do not preserve conformal invariance – the
reason why they were not further discussed in Correa:2019rdk ; Garay:2022szq
– may play a role in this context.

We should stress that the generalization of this approach to the present case
would require to first adapt it to the circular case, where a conformal
anomaly gross  makes operators, which are cohomologically equivalent at the
classical level, no longer equivalent at the quantum level. In particular, as
we have discussed, this causes a non-trivial dependence of the VEVs on the
deformations. Therefore, in this case the mixing of Neumann and Dirichlet
boundary conditions should entail a non-trivial parametric dependence in the
interpolating string solutions. More generally, the question of what is, if
any, the holographic counterpart of framing is quite important.

Finally, it is interesting to repeat this analysis for the 1/12 BPS latitude
Wilson loop of section 2.2. This is going to be addressed in CPTT . In that
case the one-dimensional effective field theory on the Wilson loop is modified
by the presence of non-trivial shifts in the superconnection
${\mathcal{L}}^{\theta}$, which result in mass terms for the one-dimensional
fields.

## Acknowledgements

We are grateful to Luca Griguolo and Domenico Seminara for discussions at the
early stage of this work. We also thank Diego Correa, Alberto Faraggi and
Guillermo Silva for useful correspondence. 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

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 $U(N_{1})_{k}\times
U(N_{2})_{-k}$ ABJ(M) theory is

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

with covariant derivatives defined as

$\begin{split}&D_{\mu}C_{I}=\partial_{\mu}C_{I}+iA_{\mu}C_{I}-iC_{I}\hat{A}_{\mu}\,,\qquad
D_{\mu}\bar{C}^{I}=\partial_{\mu}\bar{C}^{I}-i\bar{C}^{I}A_{\mu}+i\hat{A}_{\mu}\bar{C}^{I}\,,\\\
&D_{\mu}\bar{\psi}^{I}=\partial_{\mu}\bar{\psi}^{I}+iA_{\mu}\bar{\psi}^{I}-i\bar{\psi}^{I}\hat{A}_{\mu}\,,\qquad
D_{\mu}\psi_{I}=\partial_{\mu}\psi_{I}-i\psi_{I}A_{\mu}+i\hat{A}_{\mu}\psi_{I}\,.\end{split}$
(A.3)

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.4)

while the one-loop propagators are

$\begin{split}\langle(A_{\mu})_{p}^{\
q}(x)(A_{\nu})_{r}^{s}(y)\rangle^{(1)}&=\delta_{p}^{s}\delta_{r}^{q}\left(\frac{2\pi}{k}\right)^{2}N_{1}\frac{\Gamma^{2}(\frac{1}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\left[\frac{\delta_{\mu\nu}}{|x-y|^{2-4\epsilon}}-\partial_{\mu}\partial_{\nu}\frac{|x-y|^{2\epsilon}}{4\epsilon(1+2\epsilon)}\right],\\\
\langle(\hat{A}_{\mu})_{\hat{p}}^{\
\hat{q}}(x)(\hat{A}_{\nu})_{\hat{r}}^{\hat{s}}(y)\rangle^{(1)}&=\delta_{\hat{p}}^{\hat{s}}\delta_{\hat{r}}^{\hat{q}}\left(\frac{2\pi}{k}\right)^{2}N_{2}\frac{\Gamma^{2}(\frac{1}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\left[\frac{\delta_{\mu\nu}}{|x-y|^{2-4\epsilon}}-\partial_{\mu}\partial_{\nu}\frac{|x-y|^{2\epsilon}}{4\epsilon(1+2\epsilon)}\right],\\\
\langle(\psi_{I}^{\alpha})_{\hat{i}}^{j}(x)(\bar{\psi}^{J}_{\beta})_{k}^{\hat{l}}(y)\rangle^{(1)}&=i\delta_{I}^{J}\delta_{\hat{i}}^{\hat{l}}\delta_{\hat{k}}^{\hat{j}}\delta^{\alpha}_{\beta}\left(\frac{2\pi}{k}\right)(N_{1}-N_{2})\frac{\Gamma^{2}(\frac{1}{2}-\epsilon)}{16\pi^{3-2\epsilon}}\frac{1}{|x-y|^{2-4\epsilon}}.\end{split}$
(A.5)

The latin indices are color indices. For instance, $(A_{\mu})_{p}^{\ q}\equiv
A_{\mu}^{a}(T^{a})_{p}^{\ q}$ where $T^{a}$ are $U(N_{1})$ generators in
fundamental representation.

##  Appendix B The auxiliary field method for fermionic Wilson loops

In this section we prove that the ABJ(M) Wilson loop VEV

$\langle W\rangle=\langle\operatorname{Tr}{\mathcal{P}}e^{-i\int
d\tau{\mathcal{L}}}\rangle\,,$ (B.1)

can be written as the two-point function of the one-dimensional field
supermatrix, i.e.

$\langle
W\rangle=\frac{1}{2}\langle\operatorname{Tr}\Psi_{0}\bar{\Psi}_{0}\rangle_{1D}\,,$
(B.2)

where the one-dimensional fields on the r.h.s. of this equation are the bare
ones. In order to simplify the notation, in what follows we will neglect the
subscript 0 under the assumption that all the fields have to be meant as bare
fields.

We start by defining

${\mathcal{Z}}[\eta,\bar{\eta}]=\int D\Psi D\bar{\Psi}e^{-\int
d\tau\operatorname{Tr}(\bar{\Psi}D_{\tau}\Psi-\bar{\Psi}\eta-\bar{\eta}\Psi)}\,,$
(B.3)

where ${\mathcal{D}}_{\tau}=\partial_{\tau}+i{\mathcal{L}}$ and

$\eta=\begin{pmatrix}\chi&\phi\\\
\tilde{\phi}&\tilde{\chi}\end{pmatrix}\,,\qquad\qquad\bar{\eta}=\begin{pmatrix}\bar{\chi}&\bar{\tilde{\phi}}\\\
\bar{\phi}&\bar{\tilde{\chi}}\end{pmatrix}$ (B.4)

are odd supermatrices with $\chi$ ($\tilde{\chi}$) and $\phi$ ($\tilde{\phi}$)
one-dimensional fermion and scalar fields in the fundamental representation of
$U(N_{1})$ ($U(N_{2})$). The one-dimensional fields two-point function can be
written as

$\langle\Psi\bar{\Psi}\rangle_{1D}=\frac{\delta^{2}\log{\mathcal{Z}}[\eta,\bar{\eta}]}{\delta^{L}\bar{\eta}\,\delta^{R}\eta}\Big{|}_{\eta=\bar{\eta}=0}$
(B.5)

where $\delta^{L}\bar{\eta}$ and $\delta^{R}\eta$ are the left and right
derivative respectively, defined as

$\frac{\delta}{\delta^{L}\bar{\eta}}=\begin{pmatrix}\frac{\delta}{\delta{\bar{\chi}}}&\frac{\delta}{\delta\bar{\phi}}\\\
\frac{\delta}{\delta\bar{\tilde{\phi}}}&\frac{\delta}{\delta\bar{\tilde{\chi}}}\end{pmatrix}\,,\qquad\frac{\delta}{\delta^{R}\eta}=\begin{pmatrix}\frac{\delta}{\delta{\chi}}&\frac{\delta}{\delta\tilde{\phi}}\\\
\frac{\delta}{\delta\phi}&\frac{\delta}{\delta\tilde{\chi}}\end{pmatrix}\,,$
(B.6)

such that
$\frac{\delta(\theta_{1}\theta_{2})}{\delta^{R}\theta_{2}}=\theta_{1}$, where
$\theta_{1},\theta_{2}$ are Grassmann numbers.

Assuming that a consistent definition of integration over supermatrices
exists, and that it leads to a well-defined and non-vanishing results for
Gaussian integrals, we can solve the path integral in (B.3) with the standard
technique of completing the square at the exponent. In particular, we find

${\mathcal{Z}}[\eta,\bar{\eta}]\propto\exp\int
d\tau\,\bar{\eta}D_{\tau}^{-1}\eta\,,$ (B.7)

where the overall coefficient associated to the result of the supermatrix
Gaussian integration is irrelevant, since the two-point function is defined as
the derivative of the logarithm of ${\mathcal{Z}}$.

Now, taking the double derivative of (B.7), one can easily check that

$\frac{\delta^{2}\log{\mathcal{Z}}[\eta,\bar{\eta}]}{\delta^{L}\bar{\eta}\,\delta^{R}\eta}\Big{|}_{\eta=\bar{\eta}=0}=2D_{\tau}^{-1}\,.$
(B.8)

On the other hand, the following identity holds Arefeva:1980zd ;
CRAIGIE1981204 ; Dorn:1986dt

$D_{\tau}^{-1}=\left(\partial_{\tau}+i{\mathcal{L}}\right)^{-1}=\theta(\tau){\mathcal{P}}e^{-i\int_{0}^{\tau}d\tau^{\prime}{\mathcal{L}}(\tau^{\prime})}\,.$
(B.9)

In conclusion, combining (B.5) with (B.8) and inserting (B.9), we find

$\frac{1}{2}\langle\Psi(\tau)\bar{\Psi}(0)\rangle_{1D}=\theta(\tau){\mathcal{P}}e^{-i\int_{0}^{\tau}d\tau^{\prime}{\mathcal{L}}(\tau^{\prime})}\,.$
(B.10)

Finally, taking the trace and the ABJ(M) expectation value we reproduce (B.1).

##  Appendix C Perturbative computations

###  C.1 Gauge-fermion vertex corrections

Here we provide details on the evaluations of the gauge vertex corrections.

We start from diagram in figure 7 that corresponds to the following integral

$\Gamma^{\rm\ref{subfig:gauge1b}}_{\text{gauge}}=\frac{1}{18g^{2}}\int
d\tau_{1}\int d\tau_{2}\int
d^{d}x\,\langle\left(\bar{z}_{1}A_{\mu}(\tau_{1})\dot{x}_{1}^{\mu}z_{1}\right)\left(\bar{z}_{2}A_{\nu}(\tau_{2})\dot{x}_{2}^{\nu}z_{2}\right)\left(\epsilon^{\rho\sigma\tau}A_{\rho}A_{\sigma}A_{\tau}\right)(x)\rangle.$
(C.1)

It is easy to see that this contribution is vanishing due to the antisymmetry
of the $\epsilon$ tensor. In fact, to begin with, we perform the contractions
using the one-dimensional and gauge fields propagators. We obtain

$\Gamma^{\rm\ref{subfig:gauge1b}}_{\text{gauge}}\sim\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\int
d^{d}x\bar{z}_{1}\dot{x}^{\mu}_{1}\dot{x}^{\nu}_{2}z_{2}A_{\rho}\,\epsilon^{\rho\sigma\tau}\epsilon_{\mu\sigma\omega}\epsilon_{\nu\tau\eta}\,\frac{(x_{1}-x)^{\omega}(x_{2}-x)^{\eta}}{|x_{1}-x|^{d}|x_{2}-x|^{d}}.$
(C.2)

Then, we take the $\tau_{2}\to\tau_{1}$ limit focusing only on the potentially
divergent terms. The numerator of (C.2) turns out to be proportional to

$\epsilon^{\rho\sigma\tau}\epsilon_{\mu\sigma\omega}\epsilon_{\nu\tau\eta}\,\dot{x}^{\mu}_{1}\dot{x}^{\nu}_{1}(x_{1}-x)^{\omega}\left[(x_{1}-x)^{\eta}+(\tau_{2}-\tau_{1})\dot{x}^{\eta}_{1}\right].$
(C.3)

By using the following relation

$\epsilon^{\rho\sigma\tau}\epsilon_{\mu\sigma\omega}\epsilon_{\nu\tau\eta}=(\delta^{\rho}_{\mu}\delta^{\tau}_{\omega}-\delta^{\rho}_{\omega}\delta^{\tau}_{\mu})\epsilon_{\nu\tau\eta}=\delta^{\rho}_{\mu}\epsilon_{\nu\omega\eta}-\delta^{\rho}_{\omega}\epsilon_{\nu\mu\eta}\,,$
(C.4)

it can be reduced to contractions between symmetric and antisymmetric tensors,
which eventually lead to a vanishing result.

From the diagram in figure 7 we have

$\begin{split}\Gamma^{\rm\ref{subfig:gauge1c}}_{\text{gauge}}&=\int
d\tau_{1}\int d\tau_{2}\int d^{d}x\langle\left(\bar{\psi}\gamma^{\mu}\psi
A_{\mu}\right)(x)\left(\bar{z}_{1}\bar{f}_{1}\tilde{\varphi}_{1}\right)\left(\bar{\tilde{\varphi}}_{2}f_{2}z_{2}\right)\rangle\\\
&\,\,\begin{split}=-\int d\tau_{1}\int^{\tau_{1}}d\tau_{2}\int
d^{d}x&\left(\bar{\alpha}^{i}\alpha_{i}e^{i\frac{\tau_{12}}{2}}\eta_{1}\xi_{2}+\beta^{j}\bar{\beta}_{j}e^{-i\frac{\tau_{12}}{2}}\xi_{1}\eta_{2}\right)\\\
&\times\langle\psi(\tau_{2})\bar{\psi}(x)\rangle\gamma^{\mu}\langle\psi(x)\bar{\psi}(\tau_{1})\rangle
z_{2}\bar{z}_{1}A_{\mu}(x)\,.\end{split}\end{split}$ (C.5)

Inserting the propagators and exploiting the properties of $\eta$ and $\xi$,
we find

$\Gamma^{\rm\ref{subfig:gauge1c}}_{\text{gauge}}=N_{2}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\frac{\Gamma^{2}(\frac{3}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\\\
\times\int d\tau_{1}\int^{\tau_{1}}d\tau_{2}\int
d^{d}x\xi_{2}\gamma^{\nu}\gamma^{\mu}\gamma^{\rho}\eta_{1}e^{i\frac{\tau_{12}}{2}}\frac{(x_{2}-x)_{\nu}(x-x_{1})_{\rho}}{|x_{2}-x|^{d}|x-x_{1}|^{d}}\bar{z}_{1}z_{2}A_{\mu}(x).$
(C.6)

We then use the spinorial relation
$\gamma^{\nu}\gamma^{\mu}\gamma^{\rho}=\delta^{\nu\mu}\gamma^{\rho}+\delta^{\mu\rho}\gamma^{\nu}-\delta^{\nu\rho}\gamma^{\mu}+i\epsilon^{\rho\mu\nu}$
in order to write the integrand as

$\begin{split}\frac{(x_{2}-x)^{\mu}\gamma^{\rho}(x-x_{1})_{\rho}+(x_{2}-x)_{\nu}\gamma^{\nu}(x-x_{1})^{\mu}-(x_{2}-x)_{\rho}(x-x_{1})^{\rho}\gamma^{\mu}}{|x_{2}-x|^{d}|x-x_{1}|^{d}}A_{\mu}(x)\bar{z}_{1}z_{2},\end{split}$
(C.7)

where we dropped the $\epsilon^{\rho\mu\nu}$ term since it will not contribute
in the $\tau_{1}\to\tau_{2}$ limit. By using the following integral
Dorn:1986dt

$\int\\!\\!d^{d}y\,\frac{f(y)(y-x_{1})_{\mu}(y-x_{2})_{\nu}}{|x_{1}-y|^{d}|x_{2}-y|^{d}}=\frac{2\pi^{d/2}}{(d-2)^{2}\Gamma(\frac{d}{2}-1)}(f(x_{1})+O(|x_{1}-x_{2}|))\\\
\times\left[\delta_{\mu\nu}|x_{1}-x_{2}|^{2-d}+(x_{1}-x_{2})_{\mu}(x_{1}-x_{2})_{\nu}(2-d)|x_{1}-x_{2}|^{-d}\right],$
(C.8)

we obtain

$\Gamma^{\rm\ref{subfig:gauge1c}}_{\text{gauge}}=N_{2}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\frac{2\pi^{\frac{3}{2}-\epsilon}}{(1-2\epsilon)\Gamma(\frac{1}{2}-\epsilon)}\frac{\Gamma^{2}(\frac{3}{2}-\epsilon)}{2\pi^{3-2\epsilon}}\\\
\times\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}e^{i\frac{\tau_{12}}{2}}\Big{[}(x_{1}-x_{2})^{\mu}(x_{1}-x_{2})_{\nu}\xi_{2}\gamma^{\nu}\eta_{1}|\tau_{12}|^{-3+2\epsilon}\Big{]}A_{\mu}(x_{1})\bar{z}_{1}z_{2}\,.$
(C.9)

If we focus on the $\tau_{2}\to\tau_{1}$ limit we find

$\displaystyle\Gamma^{\rm\ref{subfig:gauge1c}}_{\text{gauge}}$
$\displaystyle=-ig^{2}N_{2}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\frac{2\Gamma^{2}(\frac{3}{2}-\epsilon)}{(1-2\epsilon)\pi^{\frac{3}{2}-\epsilon}\Gamma(\frac{1}{2}-\epsilon)}\int\\!\\!d\tau_{1}\int^{\tau_{1}}\\!\\!\\!\\!d\tau_{2}(\tau_{12})^{-1+2\epsilon}\dot{x}^{\mu}_{1}A_{\mu}(x_{1})\bar{z}_{1}z_{1}$
$\displaystyle=-\frac{g^{2}N_{2}}{4\pi\epsilon}(\bar{\alpha}^{i}\alpha_{i}+\beta^{j}\bar{\beta}_{j})\int
d\tau\ i\bar{z}A_{\mu}\dot{x}^{\mu}z.$ (C.10)

Finally, from diagram 7 we obtain

$\begin{split}\Gamma^{\rm\ref{subfig:gauge1d}}_{\text{gauge}}&\sim g^{2}\int
d\tau_{1}\int
d^{d}x\langle(\bar{z}_{1}C_{J}(x_{1})\bar{C}^{J}(x_{1})z_{1})(\partial^{\mu}C_{I}\,\bar{C}^{I}A_{\mu})(x)\rangle\\\
&\sim g^{2}\int d\tau_{1}\bar{z}_{1}z_{1}\int
d^{d}x\,A_{\mu}(x)\,\partial^{\mu}_{x}\left(\langle
C(x)\bar{C}(x_{1})\rangle\right)^{2},\end{split}$ (C.11)

where in the second line we have exploited the identity $\langle
C(x)\bar{C}(x_{1})\rangle=\langle\bar{C}(x)C(x_{1})\rangle$. Since in Lorentz
gauge the integrand is a total $x$-derivative, the final result is zero.

###  C.2 Fermion vertex corrections

In this section we perform the explicit calculation of the one-loop
corrections to the fermion vertices (3.29), shown in figure 8.

Focusing on the $\alpha_{i}$ corrections, the algebraic expression
corresponding to diagram 8 reads

$\begin{split}\Gamma^{\rm\ref{subfig:fermion1a}}_{\text{fermion}}&=\int
d\tau_{1}\int d\tau_{2}\int
d^{d}x\langle(\bar{\tilde{z}}_{1}f_{1}\varphi_{1})(\bar{\varphi}_{2}A_{\mu}(x_{2})\dot{x}^{\mu}_{2}\varphi_{2})(\bar{\psi}\gamma^{\nu}\psi
A_{\nu})(x)\rangle\\\ &\to\,(\alpha_{1}-\alpha_{2})\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\int
d^{d}x\bar{\tilde{z}}_{1}\varphi_{2}e^{-i\frac{\tau_{1}}{2}}\xi_{1}\langle\psi(x_{1})\bar{\psi}(x)\rangle\langle
A_{\mu}(x_{2})A_{\nu}(x)\rangle\dot{x}^{\mu}_{2}\gamma^{\nu}\psi(x).\end{split}$
(C.12)

Here $\psi$ stands for any fermion component and we have taken into account
that the contributions to $\alpha_{1}$ and $\alpha_{2}$ are the same, apart
from a different overall sign.

Reading the propagators from equation (A.4), we obtain

$\displaystyle\Gamma^{\rm\ref{subfig:fermion1a}}_{\text{fermion}}$
$\displaystyle=g^{2}N_{1}(\alpha_{1}-\alpha_{2})\frac{\Gamma^{2}(\frac{3}{2}-\epsilon)}{4\pi^{3-2\epsilon}}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\int
d^{d}x\,\bigg{[}\bar{\tilde{z}}_{1}\varphi_{2}e^{-i\frac{\tau_{1}}{2}}\xi_{1}\gamma^{\sigma}\epsilon_{\mu\nu\rho}\frac{(x_{1}-x)_{\sigma}}{|x_{1}-x|^{d}}$
$\displaystyle\times\frac{(x_{2}-x)^{\rho}}{|x_{2}-x|^{d}}\dot{x}^{\mu}_{2}\gamma^{\nu}\psi(x)\bigg{]}.$
(C.13)

The $d$-dimensional integral can be evaluated using (C.8). This leads to

$\displaystyle\Gamma^{\rm\ref{subfig:fermion1a}}_{\text{fermion}}$
$\displaystyle=g^{2}N_{1}(\alpha_{1}-\alpha_{2})\frac{\Gamma^{2}(\frac{3}{2}-\epsilon)}{2(1-2\epsilon)^{2}\pi^{\frac{3}{2}-\epsilon}\Gamma(\frac{1}{2}-\epsilon)}\int
d\tau_{1}\int^{\tau_{1}}d\tau_{2}\,\bigg{[}\bar{\tilde{z}}_{1}\varphi_{2}e^{-i\frac{\tau_{1}}{2}}\xi_{1}\epsilon_{\mu\nu\rho}\gamma^{\sigma}\gamma^{\nu}\dot{x}^{\mu}_{2}$
$\displaystyle\times\psi(x_{1})\bigg{(}(\tau_{1}-\tau_{2})^{-1+2\epsilon}\delta^{\rho}_{\sigma}-(1-2\epsilon)(x_{1}-x_{2})_{\sigma}(x_{1}-x_{2})^{\rho}(\tau_{1}-\tau_{2})^{-3+2\epsilon}\bigg{)}\bigg{]}.$
(C.14)

Using $(\gamma^{\sigma})_{\alpha}^{\ \beta}(\gamma^{\nu})_{\beta}^{\
\delta}=\delta^{\sigma\nu}\delta_{\alpha}^{\delta}+i\epsilon^{\sigma\nu\tau}(\gamma_{\tau})_{\alpha}^{\
\delta}$ we see that some terms drop out due to anti-symmetry. Eventually, in
the $\tau_{2}\to\tau_{1}$ limit we obtain

$\displaystyle\Gamma^{\rm\ref{subfig:fermion1a}}_{\text{fermion}}$
$\displaystyle=ig^{2}N_{1}(\alpha_{1}-\alpha_{2})\frac{\Gamma^{2}(\frac{3}{2}-\epsilon)}{(1-2\epsilon)\pi^{\frac{3}{2}-\epsilon}\Gamma(\frac{1}{2}-\epsilon)}\int