aainstitutetext: School of Phyiscs and Shing-Tung Yau Center
Southeast University, Nanjing 210096, Chinabbinstitutetext: Center for Joint
Quantum Studies and Department of Physics, School of Science,
Tianjin University, 135 Yaguan Road, Tianjin 300350, P. R.
Chinaccinstitutetext: Peng Huangwu Center for Fundamental Theory, Hefei, Anhui
230026, P. R. China

# Wilson-loop One-point Functions in ABJM Theory

Yunfeng Jiang b,c Jun-Bao Wu111Corresponding author b Peihe Yang
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we initiate the study of correlation functions of a single trace
operator and a circular supersymmetric Wilson loop in ABJM theory. The single
trace operator is in the scalar sector and is an eigenstate of the planar two-
loop dilatation operator. The Wilson loop is in the fundamental representation
of the gauge group or a suitable (super-)group. Such correlation functions at
tree level can be written as an overlap of the Bethe state corresponding to
the single trace operator and a boundary state which corresponds to the Wilson
loop. There are various type of supersymmetric Wilson loops in ABJM theory. We
show that some of them correspond to tree-level integrable boundary states
while some are not. For the tree-level integrable ones, we prove their
integrability and obtain analytic formula for the overlaps. For the non-
integrable ones, we give examples of non-vanishing overlaps for Bethe states
which violate selection rules.

††preprint: CJQS-2023-nnn, USTC-ICTS/PCFT-23-11

##  1 Introduction

Integrable structure of four-dimensional $\mathcal{N}=4$ super Yang-Mills
(SYM) theory enables us to compute many physical observables non-
perturbatively in the planar limit.222For a collection of reviews, see
Beisert:2010jr . The study of integrability in AdS/CFT was initiated by the
discovery that the planar one-loop dilatation operator in the scalar sector is
identical to the Hamiltonian of an integrable spin chain Minahan:2002ve .
Later this result was generalized to the full sector and all-loop order. In
the asymptotic regime, the spectrum of local operators can be computed by the
all-loop asymptotic Bethe ansatz, which was first proposed in Beisert:2005fw
and derived more rigorously in Beisert:2005tm . For operators with finite
length, the so-called finite size corrections should be taken into account. To
solve this challenging problem, different approaches such as the Lüscher
formula Bajnok:2008bm  and the thermodynamic Bethe ansatz (TBA) Ambjorn:2005wa
; Bombardelli:2009ns ; Gromov:2009tv ; Arutyunov:2009ur  have been developed,
and finally culminated in the quantum spectral curve (QSC) method
Gromov:2013pga .

Single-trace operators in $\mathcal{N}=4$ SYM are mapped to _closed_ spin
chain states. It turns out that integrable _open_ spin chains also play
important roles in AdS/CFT. There are at least two ways that open chains could
emerge. The first is by changing the theory. Examples include theories with
matters in the fundamental representation of the gauge group, such as four-
dimensional $\mathcal{N}=2$ $Sp(N)$ theory Chen:2004mu ; Chen:2004yf  and
$\mathcal{N}=2$ theory obtained by adding flavors to $\mathcal{N}=4$ SYM
Erler:2005nr . The second way is considering specific objects within
${\mathcal{N}}=4$ SYM theory which play the role of integrable boundaries.
Such objects include domain walls DeWolfe:2004zt , determinant operators which
are dual to giant gravitons Berenstein:2005vf  and Wilson lines Drukker:2006xg
; Correa:2018fgz .

More recently, the study of domain wall one-point functions in defect
$\mathcal{N}=4$ SYM deLeeuw:2015hxa ; Buhl-Mortensen:2015gfd  introduced
integrable boundary states into AdS/CFT integrability333In DeLeeuw:2018cal ,
it was proved that the boundary states from the domain wall in the D3-D5 case
satisfy the condition for the integrable boundary states in Piroli:2017sei ..
Integrable boundary states are specific states in the Hilbert space which are
annihilated by odd conserved charges. Later integrable boundary states also
appear in the computation of the correlation function of two determinant
operators and a single trace operator Jiang:2019xdz ; Jiang:2019zig , ’t Hooft
loop one-point functions Kristjansen:2023ysz  and Wilson-loop one-point
functions JKV .444A class of fermionic BPS Wilson loops in four-dimensional
$\mathcal{N}=2$ quiver theories and $\mathcal{N}=4$ SYM were constructed last
year Ouyang:2022vof . it is interesting to study whether they lead to
integrable open chains and/or integrable boundary states. Although the
integrable boundary states appear quite naturally in the cases of domain
walls, ’t Hooft loops and Wilson loops, their emergence in the correlation
functions involving two determinant operators are less obvious. The integrable
boundary states only show up after some non-trivial computations, such as
using large-$N$ effective field theory or performing partial Wick contractions
between the giant gravitons.

Three-dimensional $\mathcal{N}=6$ Chern-Simons-matter theory (ABJM theory)
Aharony:2008ug  is another important example of supersymmetric gauge theories
which are integrable in the planar limit.555The review Klose:2010ki
summarised the related achievement till the end of $2010$. Compared to
$\mathcal{N}=4$ SYM theory, almost every aspect of integrability gets more
complicated and challenging, due to its smaller symmetry. Similar to
$\mathcal{N}=4$ SYM theory, the first hint of integrability of ABJM theory
comes from the fact that the planar two-loop dilatation operator in the scalar
sector is integrable Minahan:2008hf ; Bak:2008cp . All-loop asymptotic Bethe
ansatz equations were proposed in Gromov:2008qe . But there is a to-be-
determined interpolating function $h(\lambda)$ appearing in the dispersion
relation of the magnons. A conjecture for the exact expression of $h(\lambda)$
was proposed Gromov:2014eha , based on the computation of the planar slope
function using QSC Cavaglia:2014exa  and the result on the vacuum expectation
value of $1/6$-BPS bosonic circular Wilson loop Drukker:2008zx ; Chen:2008bp ;
Rey:2008bh  computed using supersymmetric localization Kapustin:2009kz ;
Marino:2009jd .

Integrable open spin chains are relatively less studied from the perspective
of 3d super-Chern-Simons theories. Planar two-loop reflection matrices of open
chains from $\mathcal{N}=3$ flavored ABJM theory was shown to satisfy boundary
Yang-Baxter equations (BYBEs) in Bai:2017jpe , this is a strong evidence for
these chains to be integrable at two-loop. Quite recently, it was proved
Correa:2023lsm  that all-loop reflection matrices of open chains from half-BPS
Wilson lines in ABJM theory satisfy BYBEs, under certain assumptions Bargheer
; Ahn ; Drukker:2019bev . TBA equations for composite operators inserted in
cusped Wilson lines were also obtained Correa:2023lsm . Solutions of the TBA
equations can be used to confirm the conjectured interpolating function
$h(\lambda)$. As for the open chain from the determinant operators in ABJM
theory, its two-loop integrability was checked using coordinate Bethe ansatz
(CBA) Chen:2018sbp  and proved by algebraic Bethe ansatz (ABA) Bai:2019soy . A
proposal for the asymptotic all-loop Bethe ansatz equations was given in
Chen:2019igg . The boundary reflection exchanges A-type magnons and B-type
magnons in the flavored ABJM case, while the type of the magnon is preserved
during the reflection in the determinant operator and Wilson line cases.

The study of integrable boundary states in ABJM theory started with the
computation of three-point functions involving two determinant operators and a
single trace operator Yang:2021hrl . Later it was shown that certain domain
walls in ABJM theory also lead to integrable boundary states
Kristjansen:2021abc 666It was shown in Linardopoulos:2021rfq ;
Linardopoulos:2022wol  that certain D-branes dual to domain walls in both
$\mathcal{N}=4$ SYM and ABJM theory indeed provide the integrable boundary
conditions for open string attached to them. . In both cases, the integrable
boundary states satisfy the twisted integrability condition Gombor:2020kgu  at
two-loop level. The aim of this paper is to initiate the study of correlation
functions of a single trace operator and a BPS Wilson loop in ABJM theory. As
we will see, this is another important set-up where integrable boundary states
emerges naturally.

There are various types of BPS Wilson loops in ABJM theory (see the review
Drukker:2019bev ). The first BPS Wilson loops which were constructed are the
bosonic $1/6$-BPS ones Drukker:2008zx ; Chen:2008bp ; Rey:2008bh . The
construction is based on the bosonic $1/2$-BPS ($1/3$-BPS) Wilson loops in
general $\mathcal{N}=2$ ($\mathcal{N}=3$) super-Chern-Simons theory
Gaiotto:2007qi  and is similar to the half-BPS Maldacena-Wilson loop
Maldacena:1998im ; Rey:1998ik  in $\mathcal{N}=4$ SYM. These Wilson loops
correspond to F-strings smearing in a $\mathbf{CP}^{1}\subset\mathbf{CP}^{3}$
in the dual $AdS_{4}\times\mathbf{CP}^{3}$ background Drukker:2008zx ;
Rey:2008bh . In another word, the worldsheet theory has Neumann boundary
conditions for the directions along the $\mathbf{CP}^{1}$ subspace
Lewkowycz:2013laa . The existence of certain half-BPS probe F-string solutions
Drukker:2008zx ; Rey:2008bh  with Dirichlet boundary conditions in all
directions of $\mathbf{CP}^{3}$ indicates the existence of half-BPS Wilson
loops which are invariant under a subgroup $SU(3)\times U(1)$ of $SU(4)_{R}$.
Such Wilson loops were constructed by Drukker and Trancanelli Drukker:2009hy ,
who introduced fermions in the construction of the Wilson loops. Fermionic
$1/6$-BPS Wilson loops were constructed in Ouyang:2015iza ; Ouyang:2015bmy ,
based on the construction of fermionic half-BPS Wilson loops in generic
$\mathcal{N}=2$ super-Chern-Simons theories. These fermionic $1/6$-BPS Wilson
loops in ABJM theory include the above half-BPS Wilson loops and bosonic
$1/6$-BPS ones as special cases. A subclass of these fermionic $1/6$-BPS
Wilson lines are shown to dual to F-strings with complicated mixed boundary
conditions Correa:2019rdk . In this paper, we will study the Wilson-loop one-
point function within a subclass of circular fermonic $1/6$-BPS Wilson
loops,777More precisely speaking, this subclass was chosen here to insider the
Class I of the classification in Ouyang:2015iza ; Ouyang:2015bmy . The
situation for Class II should be similar.  and $1/3$-BPS Wilson loops
constructed based on the $1/3$-BPS Wilson lines in Drukker:2022txy .

We will see that the corresponding structure constant can be calculated as the
overlap of a boundary state and a Bethe state. The tree-level computation of
this one-point function demands the single-trace operator to be an eigenvector
of two-loop dilatation operator, which is identical to an integrable
Hamiltonian of an alternating SU(4) spin chain Minahan:2008hf ; Bak:2008cp .
The eigenvectors can be constructed by Bethe ansatz, with additional zero
momentum condition. A particularly interesting question for us is that, which
supersymmetric Wilson loops correspond to _integrable_ boundary states?

We find that the boundary states corresponding to generic $1/6$-BPS Wilson
loops in this subclass are not integrable. Only two special cases, namely the
bosonic $1/6$-BPS and half-BPS Wilson loops give rise to tree-level integrable
boundary states. This result implies that the boundary states corresponding to
$1/3$-BPS Wilson loops are also integrable at tree level. All the
aforementioned tree-level integrable boundary states satisfy twisted
integrability condition which leads to selection rule
$\mathbf{u}=-\mathbf{v},\mathbf{w}=-\mathbf{w}$ where $\mathbf{u}$ and
$\mathbf{v}$ are the two sets of momentum carrying Bethe roots and
$\mathbf{w}$ are the auxiliary roots. For these tree-level integrable boundary
states, we obtain analytic formulas for the Wilson loop one-point functions
(normalized overlaps) in terms of Bethe roots up to an unimportant phase
factor.

The remaining part of this paper is organized as follows. In section 2, we
review the construction of various supersymmetric Wilson loops in ABJM theory.
In section 3 we compute the Wilson loop one-point functions and find the
circular Wilson loops which correspond to integrable boundary states. We prove
the tree-level integrability of such states by algebraic Bethe ansatz and
derive the selection rules for the exact overlap. In section 4 and 5, we
derive the exact overlap formula for the bosonic 1/6-BPS and 1/2-BPS Wilson
loops respectively. We conclude in section 6 and discuss some future
directions. Two appendices include our conventions for ABJM theory and
numerical solutions of the Bethe equations.

##  2 Various BPS Wilson loops in ABJM theory

In this section, we list Wilson loops that will be studied in this paper.
Among these Wilson loops, the $1/3$-BPS circular Wilson loops are new, and
they are constructed based on $1/3$-BPS Wilson lines in Drukker:2022txy . We
consider the ABJM theory in three-dimensional Euclidean space $\mathbf{R}^{3}$
and adopt the notations in Ouyang:2015bmy . The spinor convention, the
Lagrangian and the supersymmetry transformation are listed in Appendix A.

#### Bosonic $1/6$-BPS circular WLs.

These loops were first constructed in Drukker:2008zx ; Chen:2008bp ;
Rey:2008bh . We consider the loops along
$x^{\mu}=(R\cos\tau,R\sin\tau,0),\tau\in[0,2\pi]$. The construction is the
following,

$\displaystyle W_{1/6}^{B}=$
$\displaystyle\,\mathrm{Tr}\mathcal{P}\,\exp\left(-i\oint
d\tau\mathcal{A}_{1/6}^{B}(\tau)\right)\,,\,\,$
$\displaystyle\hat{W}_{1/6}^{B}=$
$\displaystyle\,\mathrm{Tr}\mathcal{P}\,\exp\left(-i\oint
d\tau\hat{\mathcal{A}}_{1/6}^{B}(\tau)\right)\,,$ (1)
$\displaystyle\mathcal{A}_{1/6}^{B}=$
$\displaystyle\,A_{\mu}\dot{x}^{\mu}+\frac{2\pi}{k}R_{I}^{\,\,\,J}Y^{I}Y^{\dagger}_{J}|\dot{x}|\,,\,$
$\displaystyle\hat{\mathcal{A}}_{1/6}^{B}=$
$\displaystyle\,\hat{A}_{\mu}\dot{x}^{\mu}+\frac{2\pi}{k}R_{I}^{\,\,\,J}Y^{\dagger}_{J}Y^{I}|\dot{x}|\,,$
(2)

where $\dot{x}^{\mu}=\frac{dx^{\mu}}{d\tau}$, and
$R_{I}^{\,\,\,J}=\mathrm{diag}(i,i,-i,-i)$.

These two Wilson loops preserve the same supersymmetries,

$\displaystyle\vartheta_{12}=iR^{-1}\gamma_{3}\theta_{12}\,\,\vartheta_{34}=-iR^{-1}\gamma_{3}\theta_{34}\,,$
$\displaystyle\theta_{13}=\theta_{14}=\theta_{23}=\theta_{24}=0\,,\vartheta_{13}=\vartheta_{14}=\vartheta_{23}=\vartheta_{24}=0\,.$
(3)

We can combine the above two connections into a big one,

$\displaystyle L^{B}_{1/6}=\left(\begin{array}[]{cc}\mathcal{A}_{1/6}^{B}&\\\
&\hat{\mathcal{A}}_{1/6}^{B}\\\ \end{array}\right)\,,$ (6)

and construct the following Wilson loops

$W_{1/6}^{B,\mathrm{big}}=\mathrm{Tr}\mathcal{P}\,\exp\left(-i\oint d\tau
L^{B}_{1/6}(\tau)\right)\,.$ (7)

Obviously the preserved supersymmetries are still the ones in (2).

#### Fermionic $1/6$-BPS circular WLs.

These loops were constructed in Ouyang:2015iza ; Ouyang:2015bmy . We focus on
the Class I loops according to the classification in these papers. Let us
consider the loops along the same contour
$x^{\mu}(\tau)=(R\cos\tau,R\sin\tau,0)$ as the bosonic $1/6$-BPS Wilson loops,

$\displaystyle W^{F}_{1/6}=\mathrm{Tr}\mathcal{P}\,\exp\left(-i\oint d\tau
L^{F}_{1/6}(\tau)\right)\,,\,\quad
L^{F}_{1/6}=\left(\begin{array}[]{cc}\mathcal{A}&\bar{f}_{1}\\\
f_{2}&\hat{\mathcal{A}}\end{array}\right)\,,$ (10)
$\displaystyle\mathcal{A}=A_{\mu}\dot{x}^{\mu}+\frac{2\pi}{k}U_{I}^{\,\,\,J}Y^{I}Y^{\dagger}_{J}|\dot{x}|\,,\,\,\quad\bar{f}_{1}=\sqrt{\frac{2\pi}{k}}\bar{\alpha}^{I}\bar{\zeta}\psi_{I}|\dot{x}|\,,$
(11)
$\displaystyle\hat{\mathcal{A}}=\hat{A}_{\mu}\dot{x}^{\mu}+\frac{2\pi}{k}U_{I}^{\,\,\,J}Y^{\dagger}_{J}Y^{I}|\dot{x}|\,,\,\quad
f_{2}=\sqrt{\frac{2\pi}{k}}\psi^{\dagger I}\eta\beta_{I}|\dot{x}|\,,$ (12)

with

$\displaystyle\bar{\alpha}^{I}$ $\displaystyle=$
$\displaystyle(\bar{\alpha}^{1},\bar{\alpha}^{2},0,0)\,,\,\quad\beta_{I}=(\beta_{1},\beta_{2},0,0)\,,$
(13) $\displaystyle\bar{\zeta}^{\alpha}$ $\displaystyle=$
$\displaystyle(e^{i\tau/2},e^{-i\tau/2})\,,\,\quad\eta_{\alpha}=\left(\begin{array}[]{c}e^{-i\tau/2}\\\
e^{i\tau/2}\end{array}\right)\,,$ (16) $\displaystyle U_{I}^{\,\,\,J}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}i-2\bar{\alpha}^{2}\beta_{2}&2\bar{\alpha}^{2}\beta_{1}&0&0\\\
2\bar{\alpha}^{1}\beta_{2}&i-2\bar{\alpha}^{1}\beta_{1}&0&0\\\ 0&0&-i&0\\\
0&0&0&-i\end{array}\right)\,.$ (21)

The preserved supersymmetric is the same as the ones in (2) for generic
$\bar{\alpha}^{I},\beta_{I}$.

It was later noticed that we have the equivalence relation Drukker:2019bev ;
Drukker:2020opf ,

$(\bar{\alpha}^{I},\beta_{J})\sim(\lambda\bar{\alpha}^{I},\lambda^{-1}\beta_{J})\,,\,\lambda\in\mathbf{C}^{\ast}=\mathbf{C}-\\{0\\}\,.$
(22)

One can set $\bar{\alpha}^{2}=\beta_{2}=0$ to get a subclass of Wilson loops.
Then in this subclass, we have
$U_{I}^{\,\,\,J}=\mathrm{diag}(i,i-2\bar{\alpha}^{1}\beta_{1},-i,-i)$, and

$\bar{f}_{1}=\sqrt{\frac{2\pi}{k}}\bar{\alpha}^{1}\bar{\zeta}\psi_{1}|\dot{x}|\,,\,\quad
f_{2}=\sqrt{\frac{2\pi}{k}}\psi^{\dagger 1}\eta\beta_{1}|\dot{x}|\,.$ (23)

Similar subclass in the fermionic $1/6$-BPS Wilson lines was considered in the
study of the dual string theory prescription Correa:2019rdk .

#### Half-BPS circular WLs.

Half-BPS Wilson loops were first constructed in Drukker:2009hy . A class of
them, $W_{1/2}$, appears among the above class of fermionic $1/6$-BPS Wilson
loops when the parameters $\bar{\alpha}^{I},\beta_{I}$ satisfy the following
constraints,

$\beta_{I}=\frac{i\alpha_{I}}{\bar{\alpha}^{J}\alpha_{J}}\,,$ (24)

and at least one of $\bar{\alpha}^{1},\bar{\alpha}^{2}$ is non-zero. Here
$\alpha_{I}$ is defined by $\alpha_{I}=(\bar{\alpha}^{I})^{\ast}$. The
preserved supersymmetries are now enhanced to

$\bar{\alpha}^{I}\vartheta_{IJ}=iR^{-1}\gamma_{3}\bar{\alpha}^{I}\theta_{IJ}\,,\,\quad\epsilon^{IJKL}\alpha_{J}\vartheta_{KL}=-iR^{-1}\gamma_{3}\epsilon^{IJKL}\alpha_{J}\theta_{KL}\,.$
(25)

Using a suitable R-symmetry transformation acting only on $I=1,2$, we can
choose888Notice that this transformation is inside a $SU(2)$ subgroup of
$SU_{R}(4)$ and keeps ${\mathrm{diag}}(-i,-i,i,i)$ invariant. So its action on
the Wilson loop only changes $\bar{\alpha}^{I},\beta_{I}$.

$\bar{\alpha}^{I}=(\bar{\alpha},0,0,0)\,,\,\quad\beta_{I}=(\beta,0,0,0)\,,$
(26)

with the constrains $\bar{\alpha}\beta=i$. Now
$U_{I}^{\,\,\,J}=\mathrm{diag}(i,-i,-i,-i)$, $\bar{f}_{1}$ and $f_{2}$ become,

$\bar{f}_{1}=\sqrt{\frac{2\pi}{k}}\bar{\alpha}\bar{\zeta}\psi_{1}|\dot{x}|\,,\,\quad
f_{2}=\sqrt{\frac{2\pi}{k}}\psi^{\dagger 1}\eta\beta|\dot{x}|\,.$ (27)

The equivalence relation (22) now becomes,

$(\bar{\alpha},\beta)\sim(\lambda\bar{\alpha},\lambda^{-1}\beta)\,,\,\lambda\in\mathbf{C}^{\ast}\,.$
(28)

We will denote this half-BPS Wilson loops by $W^{1+}_{1/2}$ and the
corresponding super-connection by $L^{1+}_{1/2}$. The supersymmetries
preserved by $W^{1+}_{1/2}$ are,

$\vartheta_{1I}=iR^{-1}\gamma_{3}\theta_{1I}\,,\,\quad\epsilon^{1IJK}\vartheta_{JK}=-iR^{-1}\gamma_{3}\epsilon^{1IJK}\theta_{JK}\,.$
(29)

####  $1/3$-BPS circular WLs.

In the construction of $1/3$-BPS Wilson loops, we start with the following
super-connection $L^{4-}_{1/2}$ in another half-BPS Wilson loop
$W^{4-}_{1/2}$,

$\displaystyle
L^{4-}_{1/2}=\left(\begin{array}[]{cc}\mathcal{A}&\bar{f}_{1}\\\
f_{2}&\hat{\mathcal{A}}\end{array}\right)\,,$ (32)

with

$\displaystyle\mathcal{A}=A_{\mu}\dot{x}^{\mu}+\frac{2\pi}{k}|\dot{x}|\tilde{U}_{I}^{\,\,\,J}Y^{I}Y^{\dagger}_{J}\,,\,\,\quad\bar{f}_{1}=\frac{2\pi}{k}\bar{\rho}\bar{\mu}\psi_{4}|\dot{x}|\,,$
$\displaystyle\hat{\mathcal{A}}=\hat{A}_{\mu}\dot{x}^{\mu}+\frac{2\pi}{k}|\dot{x}|\tilde{U}_{I}^{\,\,\,J}Y^{\dagger}_{J}Y^{I}\,,\,\,\quad
f_{2}=\frac{2\pi}{k}\psi^{\dagger 4}\nu\delta|\dot{x}|\,,$
$\displaystyle\tilde{U}_{I}^{\,\,\,J}=\left(\begin{array}[]{cccc}i&0&0&0\\\
0&i&0&0\\\ 0&0&i&0\\\ 0&0&0&-i\end{array}\right)\,,$ (37)
$\displaystyle\bar{\mu}^{\alpha}=(e^{i\tau/2},-e^{-i\tau/2})\,,\,\quad\nu_{\alpha}=\left(-e^{-i\tau/2},e^{i\tau/2}\right)\,.$
(38)

Here $\bar{\rho},\delta$ are two complex numbers satisfying
$\bar{\rho}\delta=-i$, and we have the equivalence relation

$(\bar{\rho},\delta)\sim(\lambda\bar{\rho},\lambda^{-1}\delta)\,,\,\lambda\in\mathbf{C}^{\ast}\,.$
(39)

The corresponding Wilson loop

$W^{4-}_{1/2}=\mathrm{Tr}\mathcal{P}\,\exp\left(-i\oint d\tau
L^{4-}_{1/2}(\tau)\right)\,,$ (40)

preserve the following supersymmetries,

$\vartheta_{I4}=-i\gamma_{3}R^{-1}\theta_{I4}\,,\,\epsilon^{4IJK}\vartheta_{JK}=i\gamma_{3}R^{-1}\epsilon_{4IJK}\vartheta_{JK}.$
(41)

This Wilson loop belong to the class II of the fermionic $1/6$-BPS Wilson
loops in Ouyang:2015iza ; Ouyang:2015bmy .

The $1/3$-BPS Wilson loops are constructed by $L^{1+}_{1/2}$ and
$L^{4-}_{1/2}$,

$\displaystyle W_{1/3}=\mathrm{Tr}\mathcal{P}\,\exp\left(-i\oint d\tau
L_{1/3}(\tau)\right)\,,$ $\displaystyle
L_{1/3}={\mathrm{diag}}(\underbrace{L_{1/2}^{1+},\cdots,L_{1/2}^{1+}}_{n_{1}},\underbrace{L^{1/2}_{4-},\cdots,L_{1/2}^{4-}}_{n_{4}})\,.$
(42)

Notice that $L_{1/3}$ is a $((n_{1}+n_{4})N|(n_{1}+n_{4})N)$ supermatrix,
since both $L^{1+}_{1/2}$ and $L^{4-}_{1/2}$ are $(N|N)$ supermatrices.

The supersymmetries preserved by $W_{1/3}$ are given by the following ones
shared by $L^{1+}_{1/2}$ and $L^{4-}_{1/2}$,

$\displaystyle\vartheta_{12}=i\gamma_{3}\theta_{12}\,,\,\vartheta_{13}=i\gamma_{3}\theta_{13}\,,\,\vartheta_{24}=-i\gamma_{3}\theta_{24}\,,\,\vartheta_{34}=-i\gamma_{3}\theta_{34}\,,$
(43) $\displaystyle\theta_{14}=\theta_{23}=\vartheta_{14}=\vartheta_{23}=0\,.$
(44)

##  3 Wilson loop one-point function in ABJM theory

###  3.1 The boundary states from Wilson loops

The main goal of this paper is to study the tree-level correlation function of
the BPS Wilson loops reviewed in the previous section and the single-trace
operator,

$\mathcal{O}_{C}=C^{J_{1}\cdots J_{L}}_{I_{1}\cdots
I_{L}}\mathrm{Tr}(Y^{I_{1}}Y^{\dagger}_{J_{1}}\cdots
Y^{I_{L}}Y^{\dagger}_{J_{L}})\,,$ (45)

in the scalar sector. The coefficients $C^{J_{1}\cdots J_{L}}_{I_{1}\cdots
I_{L}}$ are chosen such that this single-trace operator is the eigenstate of
the planar two-loop dilatation operator. The single-trace operator is put at
the origin of the three-dimensional space. The Wilson loops considered in this
paper are in the fundamental representation of a suitable (super-)group. More
precisely, the bosonic $1/6$-BPS Wilson loop (1) is in the fundamental
representation of $U(N)$. $W^{B,\mathrm{big}}_{1/6}$ is in the fundamental
representation of the gauge group $U(N)\times U(N)$. The fermionic $1/6$-BPS
Wilson loop (10) and the half-BPS Wilson one are in the fundamental
representation of the supergroup $U(N|N)$. Finally, the $1/3$-BPS Wilson loop
(42) is in the fundamental representation of the supergroup
$U((n_{1}+n_{4})N|(n_{1}+n_{4})N)$.

At tree level, the correlator $\langle
W(\mathcal{C})^{B}_{1/6}\mathcal{O}_{C}(0)\rangle$ only gets contributions
from

$\displaystyle\oint\cdots\oint
d\tau_{1>2>\cdots>L}\left(\frac{2\pi}{k}\right)^{L}\langle{\mathrm{tr}}(R^{\tilde{J}_{1}}_{\,\,\,\,\,\tilde{I}_{1}}Y^{\tilde{I}_{1}}(x_{1})Y^{\dagger}_{\tilde{J}_{1}}(x_{1})\cdots
R^{\tilde{J}_{L}}_{\,\,\,\,\,\tilde{I}_{L}}Y^{\tilde{I}_{L}}(x_{L})Y^{\dagger}_{\tilde{J}_{L}}(x_{L}))$
$\displaystyle C^{J_{1}\cdots J_{L}}_{I_{1}\cdots
I_{L}}{\mathrm{tr}}(Y^{I_{1}}(0)Y^{\dagger}_{J_{1}}(0)\cdots
Y^{I_{L}}(0)Y^{\dagger}_{J_{L}}(0))\rangle\,,$ (46)

where $x_{i}=(R\cos\tau_{i},R\sin\tau_{i},0),\,i=1,\cdots,L$, and

$\oint\cdots\oint
d\tau_{1>2>\cdots>L}=\int_{0}^{2\pi}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\cdots\int_{0}^{\tau_{L-1}}d\tau_{L}\,.$
(47)

In the large $N$ limit, we only take into account planar Wick contractions, as
is shown in figure 1.

Figure 1: Planar Wick contractions between the local operator and the Wilson
loop. Here each blue or red dot indicates a pair of scalar fields and each
straight line indicates a pair of contractions. The outer circle indicates the
Wilson loop, and the inner circle indicates the single-trace operator.
Although the single-trace operator is local in space-time, we draw it as a
circle to make the color structure of the operator and the contraction more
clear.

One can easily obtain

$\langle
W(\mathcal{C})^{B}_{1/6}\mathcal{O}_{C}(0)\rangle=\frac{\lambda^{2L}k^{L}}{(L-1)!(2R)^{2L}}C^{J_{1}\cdots
J_{L}}_{I_{1}\cdots I_{L}}R^{I_{L}}_{\,\,\,\,J_{L}}\cdots
R^{I_{1}}_{\,\,\,\,J_{1}}\,,$ (48)

where $\lambda\equiv\frac{N}{k}$ is the ’t Hooft coupling of ABJM theory and
the tree-level propagators of the scalar fields (174) have been used. For
later convenience, we introduce the following two-site boundary state which is
specified by a $4\times 4$ matrix ${M}$ as

$\displaystyle\langle\mathcal{B}_{M}|$ $\displaystyle\equiv$ $\displaystyle
M^{I_{1}}_{\,\,\,\,J_{1}}M^{I_{2}}_{\,\,\,\,J_{2}}\cdots
M^{I_{L}}_{\,\,\,\,J_{L}}\langle I_{1},J_{1},\cdots,I_{L},J_{L}|$ (49)
$\displaystyle=$ $\displaystyle\left(M^{I}_{\,\,J}\langle I,J|\right)^{\otimes
L}\,.$

Then

$|\mathcal{B}_{M}\rangle=\left((M^{I}_{\,\,J})^{\ast}|I,J\rangle\right)^{\otimes
L}$ (50)

Our convention for the Hermitian conjugation of the spin chain states is

$\displaystyle\left(\langle
I_{1},J_{1},\cdots,I_{L},J_{L}|\right)^{\dagger}=|I_{1},J_{1}\cdots,I_{L},J_{L}\rangle\,.$
(51)

We define the 1/6-BPS boundary state as999Notice that ${R^{I}}_{J}$ should not
be confused with $R$ which is the radius of the circular Wilson loop.

$|\mathcal{B}_{1/6}^{B}\rangle=|\mathcal{B}_{R}\rangle\,,$ (52)

where ${R^{I}}_{J}=\text{diag}(i,i,-i,-i)$. Then the above correlation
function can be expressed as

$\langle
W(\mathcal{C})^{B}_{1/6}\mathcal{O}_{C}(0)\rangle=\frac{\lambda^{2L}k^{L}}{(L-1)!(2R)^{2L}}\langle\mathcal{B}_{1/6}^{B}|\mathcal{O}_{C}\rangle\,,$
(53)

where $|\mathcal{O}_{C}\rangle{\equiv C^{J_{1}\cdots J_{L}}_{I_{1}\cdots
I_{L}}|I_{1},J_{1},\cdots,I_{L},J_{L}\rangle}$ is the spin chain state
corresponding to the operator $\mathcal{O}_{C}$. Our convention for the
overlap of two spin chain states is

$\displaystyle\langle
I_{1},J_{1},\cdots,I_{L},J_{L}|M_{1},N_{1},\cdots,M_{L},N_{L}\rangle=\delta_{I_{1}}^{M_{1}}\delta^{J_{1}}_{N_{1}}\cdots\delta_{I_{L}}^{M_{L}}\delta^{J_{L}}_{N_{L}}\,.$
(54)

Let us define the normalization factor $\mathcal{N}_{\mathcal{O}}$ using the
two-point function of $\mathcal{O}$ and $\mathcal{O}^{\dagger}$ as

$\langle\mathcal{O}(x)\mathcal{O}^{\dagger}(y)\rangle=\frac{\mathcal{N}_{\mathcal{O}}}{|x-y|^{2\Delta_{\mathcal{O}}}}\,,$
(55)

where $\Delta_{\mathcal{O}}$ is the conformal dimension of $\mathcal{O}$. At
tree level and the planar limit, we have

$\mathcal{N}_{\mathcal{O}}=\left(\frac{N}{4\pi}\right)^{2L}L\langle\mathcal{O}|\mathcal{O}\rangle\,.$
(56)

We define the Wilson-loop one-point function as

$\langle\\!\langle\mathcal{O}\rangle\\!\rangle_{W(\mathcal{C})}\equiv\frac{\langle
W(\mathcal{C})\mathcal{O}\rangle}{\sqrt{\mathcal{N}_{\mathcal{O}}}}\,.$ (57)

Then for $W^{B}_{1/6}$ we have

$\langle\\!\langle\mathcal{O}\rangle\\!\rangle_{W(\mathcal{C})^{B}_{1/6}}=\frac{\pi^{L}\lambda^{L}}{R^{2L}(L-1)!\sqrt{L}}\frac{\langle\mathcal{B}_{1/6}^{B}|\mathcal{O}\rangle}{\sqrt{\langle\mathcal{O}|\mathcal{O}\rangle}}\,.$
(58)

The computation of the Wilson loop one-point function thus amounts to the
calculation of

$\frac{\langle\mathcal{B}_{1/6}^{B}|\mathcal{O}\rangle}{\sqrt{\langle\mathcal{O}|\mathcal{O}\rangle}}\,.$
(59)

Similar computations for other Wilson loops studied in this paper can also be
reduced to the computation of overlaps of the form in (59), with the
corresponding boundary states. For $\hat{W}(\mathcal{C})^{B}_{1/6}$, the
boundary state is

$\langle\hat{\mathcal{B}}^{B}_{1/6}|=R^{I_{1}}_{\,\,\,\,J_{L}}R^{I_{2}}_{\,\,\,\,J_{1}}\cdots
R^{I_{L}}_{\,\,\,\,J_{L-1}}\langle I_{1},J_{1},\cdots,I_{L},J_{L}|\,.$ (60)

We can rewrite $|\hat{\mathcal{B}}_{1/6}^{B}\rangle$ as

$|\hat{\mathcal{B}}_{1/6}^{B}\rangle=U_{\rm
even}|\mathcal{B}_{1/6}^{B}\rangle\,$ (61)

where $U_{\rm even}$ is the shift operator which shifts all even site to the
left by two units and leave the odd sites untouched,

$U_{\rm
even}|I_{1},J_{1},I_{2},J_{2},\cdots,I_{L-1},J_{L-1},I_{L},J_{L}\rangle=|I_{1},J_{2},I_{2},J_{3},\cdots,I_{L-1},J_{L},I_{L},J_{1}\rangle.$
(62)

Combining (52) and (60), we obtain the boundary state of
$W(\mathcal{C})^{B,\mathrm{big}}_{1/6}$

$\displaystyle|\mathcal{B}_{1/6}^{B,\mathrm{big}}\rangle=|\mathcal{B}_{1/6}^{B}\rangle+|\hat{\mathcal{B}}_{1/6}^{B}\rangle=(1+U_{\rm
even})|\mathcal{B}_{R}\rangle\,.$ (63)

For $W(\mathcal{C})^{F}_{1/6}$, we simply replace ${R^{I}}_{J}$ by
${U^{I}}_{J}$ given in (21),

$|\mathcal{B}_{1/6}^{F}\rangle=(1+U_{\rm even})|\mathcal{B}_{U}\rangle\,.$
(64)

The boundary state corresponding to $W(C)_{1/2}$, which will be denoted by
$|\mathcal{B}_{1/2}\rangle$ is given by $|\mathcal{B}_{1/6}^{F}\rangle$ with
the additional constraints (24).

In particular, the boundary state $|\mathcal{B}_{1/2}^{1+}\rangle$
corresponding to $W(C)_{1/2}^{1+}$ is

$|\mathcal{B}_{1/2}^{1+}\rangle=(1+U_{\rm even})|\mathcal{B}_{U}\rangle\,,$
(65)

with ${U^{I}}_{J}=\mathrm{diag}(i,-i,-i,-i)$. Similarly, the boundary state
$|\mathcal{B}_{1/2}^{4-}\rangle$ corresponding to $W(C)_{1/2}^{4-}$ is

$|\mathcal{B}_{1/2}^{4-}\rangle=(1+U_{\rm
even})|\mathcal{B}_{\tilde{U}}\rangle\,,$ (66)

with ${\tilde{U}^{I}}_{\,\,J}=\mathrm{diag}(i,i,i,-i)$.

Finally, for $W(\mathcal{C})_{1/3}$, we have

$|\mathcal{B}_{1/3}\rangle=n_{1}|\mathcal{B}_{1/2}^{1+}\rangle+n_{4}|\mathcal{B}_{1/2}^{4-}\rangle\,.$
(67)

###  3.2 Integrable and non-integrable boundary states

In this subsection, we will prove that the boundary states
$|\mathcal{B}_{1/6}^{B}\rangle,|\hat{\mathcal{B}}_{1/6}^{B}\rangle,|\mathcal{B}_{1/2}\rangle$
and $|\mathcal{B}_{1/3}\rangle$ are integrable at tree level by employing the
method proposed in Piroli:2017sei . We will also show that the
$|\mathcal{B}_{1/6}^{F}\rangle$ with $\bar{\alpha}^{2}=\beta_{2}=0$ is not
integrable unless $\bar{\alpha}^{1}\beta_{1}=0,i$.

Let us consider the boundary state $|\mathcal{B}_{M}\rangle$ defined by a
matrix $M$ as in (49). In what follows, we will encounter several examples in
which $M$ is a diagonal matrix. In this case, the overlap
$\langle\mathcal{B}_{M}|\mathbf{u},\mathbf{v},\mathbf{w}\rangle$ is nonzero
only if the numbers of the Bethe roots,
$K_{\mathbf{u}},K_{\mathbf{v}},K_{\mathbf{w}}$, and the length of the spin
chain $2L$, satisfy $K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=L$. Notice
that this selection rule has nothing to do with integrability of the boundary
state.

In the algebraic Behte ansatz approach, the $SU(4)$ sector $R$-matrices of the
ABJM theory at two-loop level are given by

$\displaystyle\begin{array}[]{l}R_{12}^{\bullet\bullet}(u)=R_{12}^{\circ\circ}(u)=u+P_{12}\equiv
R_{12}(u)\,,\\\
R_{12}^{\bullet\circ}(u)=R_{12}^{\circ\bullet}(u)=-u-2+K_{12}\equiv\bar{R}_{12}(u)\,,\end{array}$
(70)

where $\bullet$ denotes the states in the $\mathbf{4}$ representation of
$SU(4)_{R}$, while $\circ$ denotes the states in the $\bar{\mathbf{4}}$
representation. The $R$-matrices satisfy the following crossing symmetry
relations

$\displaystyle
R_{12}(u)^{t_{1}}=\bar{R}_{12}(-u-2),\qquad\bar{R}_{12}(u)^{t_{1}}=R_{12}(-u-2)\,,$
(71)

and the relation

$\displaystyle R_{12}(u)^{t_{1}t_{2}}$ $\displaystyle=$ $\displaystyle
R_{12}(u),\qquad\bar{R}_{12}(u)^{t_{1}t_{2}}=\bar{R}_{12}(u),$ $\displaystyle
P_{12}R_{12}(u)P_{12}$ $\displaystyle=$ $\displaystyle R_{12}(u),\qquad
P_{12}\bar{R}_{12}(u)P_{12}=\bar{R}_{12}(u)\,,$ (72)

where $t_{i}$ denotes transposition in the $i$-th space. One key feature of
algebraic Bethe ansatz approach of the ABJM spin chain is that it requires two
$R$-matrices, $R(u)$ and $\bar{R}(u)$, due to its alternating nature. This is
different from the case when the spin on each site is in the same
representation.

In the following, we will show that when there exists a four-dimensional
matrix $K(u)$ satisfying the boundary Yang-Baxter equation (BYBE)101010Notice
that here we only need to use the BYBE involving one of the $R$-matrices,
$R(u)$.

$\displaystyle
R_{12}(u-v)K_{1}(u)R_{12}(u+v)K_{2}(v)=K_{2}(v)R_{12}(u+v)K_{1}(u)R_{12}(u-v)\,,$
(73)

then the boundary state $|\mathcal{B}_{M}\rangle$ with $M=K(-1)^{\ast}$ is
integrable in the sense that it satisfies the following twisted integrability
condition Gombor:2020kgu ; Gombor:2022deb ,

$\displaystyle\tau(-u-2)|\mathcal{B}_{M}\rangle=\tau(u)|\mathcal{B}_{M}\rangle\,,$
(74)

or equivalently Yang:2022dlk ,
$\Pi\bar{\tau}(u)\Pi|\mathcal{B}_{M}\rangle=\tau(u)|\mathcal{B}_{M}\rangle$ ,
where

$\displaystyle\tau(u)$
$\displaystyle=\operatorname{Tr}_{0}\left(R_{01}(u)\bar{R}_{02}(u)\cdots
R_{0,2L-1}(u)\bar{R}_{0,2L}(u)\right)\,,$ (75) $\displaystyle\bar{\tau}(u)$
$\displaystyle=\operatorname{Tr}_{0^{\prime}}\left(\bar{R}_{0^{\prime}1}(u)R_{0^{\prime}2}(u)\cdots\bar{R}_{0^{\prime},2L-1}(u)R_{0^{\prime},2L}(u)\right)\,,$
(76)

are the transfer matrices. Here $0,0^{\prime}$ denote two auxiliary spaces and
$\Pi$ is the parity operator

$\Pi|I_{1},J_{1},\cdots,J_{2L}\rangle=|J_{2L},\cdots,J_{2},I_{1}\rangle\,.$
(77)

Using the explicit forms of eigenvalues of $\tau(u)$ and $\bar{\tau}(u)$
Yang:2022dlk , we conclude that for integrable boundary states, the overlap
$\langle\mathcal{B}_{M}|\mathbf{u},\mathbf{v},\mathbf{w}\rangle$ is non-zero
only if the selection rules

$\displaystyle\mathbf{u}=-\mathbf{v}\,,\qquad\mathbf{w}=-\mathbf{w}\,$ (78)

are satisfied.

Let us define the state

$\displaystyle|\phi(u-1)\rangle_{ab}=K_{J}^{I}(u-1)|I\rangle_{a}\otimes|J\rangle_{b}\,.$
(79)

The boundary Yang-Baxter equation (73) leads to

$\displaystyle\begin{aligned}
&\check{R}_{34}(v-u)\check{\bar{R}}_{23}(-u-v)\left|\phi_{0}(u-1)\right\rangle_{12}\otimes\left|\phi_{0}(v-1)\right\rangle_{34}\\\
=&\check{R}_{12}(v-u)\check{\bar{R}}_{23}(-u-v)\left|\phi_{0}(v-1)\right\rangle_{12}\otimes\left|\phi_{0}(u-1)\right\rangle_{34}\,,\end{aligned}$
(80)

where $\check{R}_{12}=P_{12}R_{12}$ and
$\check{\bar{R}}_{12}=P_{12}\bar{R}_{12}$. This relation can be shown
pictorially as in figure 2.

Figure 2: A pictorial representation of eq. (LABEL:eq:RR) from the boundary
Yang-Baxter equation. Here the red node denotes $\check{\bar{R}}(-u-v)$ and
the blue node denotes $\check{R}(v-u)$.

Notice that $\bar{R}(u)$ also appears here, since we have used one of the
crossing symmetry relations (71).

Introducing two four-dimensional auxiliary spaces $h_{0}$ and $h_{2L+1}$ and
following the derivation in the Section 4 and Appendix C of Piroli:2017sei ,
we can prove that

$\displaystyle\begin{aligned}
&\check{R}_{2L+1,2L}(v-u)\check{\bar{R}}_{2L,2L-1}(-v-u)\cdots\check{R}_{32}(v-u)\check{\bar{R}}_{21}(-v-u)\\\
&|\phi(u-1)\rangle_{01}\otimes|\phi(v-1)\rangle_{23}\otimes\cdots\otimes|\phi(v-1)\rangle_{2L,2L+1}\\\
=&\check{R}_{2L+1,2L}(v-u)\check{\bar{R}}_{2L,2L-1}(-v-u)\cdots\check{R}_{01}(v-u)\check{\bar{R}}_{12}(-v-u)\\\
&|\phi(v-1)\rangle_{01}\otimes|\phi(u-1)\rangle_{23}\otimes|\phi(v-1)\rangle_{45}\otimes\cdots\otimes|\phi(v-1)\rangle_{2L,2L+1}\\\
=&\cdots\\\
=&\check{R}_{01}(v-u)\check{\bar{R}}_{12}(-v-u)\cdots\check{R}_{2L-2,2L-1}(v-u)\check{\bar{R}}_{2L-1,2L}(-v-u)\\\
&|\phi(v-1)\rangle_{01}\otimes|\phi(v-1)\rangle_{23}\otimes\cdots\otimes|\phi(v-1)\rangle_{2L-2,2L-1}\otimes|\phi(u-1)\rangle_{2L,2L+1}\,.\end{aligned}$
(81)

This in turn implies

$\displaystyle\begin{aligned}
&\operatorname{Tr}_{0}\left(R_{01}(-u-2)\bar{R}_{02}(-u-2)\cdots
R_{0,2L-1}(-u-2)\bar{R}_{0,2L}(-u-2)\right)|\mathcal{B}_{M}\rangle\\\
=&\operatorname{Tr}_{0}\left(R_{01}(u)\bar{R}_{02}(u)\cdots
R_{0,2L-1}(u)\bar{R}_{0,2L}(u)\right)|\mathcal{B}_{M}\rangle\,.\end{aligned}$
(82)

Here the condition $K(-1)=M^{\ast}$ has been used. Pictorially the above
derivation is shown in figure 3.

Figure 3: A pictorial derivation of (81).

In terms of the transfer matrices $\tau(u)$ and $\bar{\tau}(u)$ given in (75),
(82) can be written as

$\displaystyle\tau(-u-2)|\mathcal{B}_{M}\rangle=\tau(u)|\mathcal{B}_{M}\rangle.$
(83)

As mentioned above, this equation is equivalent Yang:2022dlk  to the twisted
integrability condition Gombor:2020kgu ; Gombor:2022deb ,
$\Pi\bar{\tau}(u)\Pi|\mathcal{B}_{M}\rangle=\tau(u)|\mathcal{B}_{M}\rangle$
111111In fact, there exist untwisted integrability condition Gombor:2020kgu ;
Gombor:2022deb
$\Pi\tau(u)\Pi|\mathcal{B}_{M}\rangle=\tau(u)|\mathcal{B}_{M}\rangle,\Pi\bar{\tau}(u)\Pi|\mathcal{B}_{M}\rangle=\bar{\tau}(u)|\mathcal{B}_{M}\rangle$,
which leads to the selection rule
$\mathbf{u}=-\mathbf{u},\mathbf{v}=-\mathbf{v},\mathbf{w}=-\mathbf{w}$. But
such case has not appeared in the boundary states in the ABJM theory yet
Yang:2021hrl ; Kristjansen:2021abc .. This finishes the proof about the
integrability of the boundary state $|\mathcal{B}_{M}\rangle$ assuming that
there exists the matrix $K(u)$ satisfying the BYBE, (73), and
$K(-1)=M^{\ast}$. We can similarly prove that for such $M$, the boundary state
$|\hat{\mathcal{B}}_{M}\rangle\equiv U_{\rm even}|{B}_{M}\rangle$ is also
integrable and leads to the same selection rule121212This result can be also
obtained from the general classification of integrable boundary states in
Gombor:2021hmj ..

Now we turn to boundary states from some BPS Wilson loops we list above. For
$|\mathcal{B}_{1/6}^{B}\rangle(=|\mathcal{B}_{R}\rangle)$ and
$|\hat{\mathcal{B}}_{1/6}^{B}\rangle(=|\hat{\mathcal{B}}_{R}\rangle)$, one can
check that the matrix $K(u)=R^{\ast}$ is the solution of BYBE (73), so these
two boundary states are both tree-level integrable. Then
$|\mathcal{B}_{1/6}^{B,\mathrm{big}}\rangle$ is tree-level integrable as well.
As for the boundary state $|\mathcal{B}_{1/2}^{1+}\rangle(=(1+U_{\rm
even})|\mathcal{B}_{U}\rangle)$, we can just choose
$K(u)=U^{\ast}=\mathrm{diag}(-i,i,i,i)$ since it also satisfies the BYBE (73).
Then we get that $|\mathcal{B}_{1/2}^{1+}\rangle$ is integrable at tree level.
Since the tree-level integrability of $|\mathcal{B}_{M}\rangle$ is preserved
when we perform a $SU(4)_{R}$ transformation on $M$ or multiply $M$ by a
constant, we get that the generic $|\mathcal{B}_{1/2}\rangle$ and
$|\mathcal{B}_{1/2}^{4-}\rangle$ are also tree-level integrable. This leads to
the conclusion that $|\mathcal{B}_{1/3}\rangle$ is tree-level integrable.

Now we turn consider the boundary state

$\displaystyle|\mathcal{B}_{1/6}^{F}\rangle$ $\displaystyle=$
$\displaystyle|\mathcal{B}_{U}\rangle+|\hat{\mathcal{B}}_{U}\rangle=(1+U_{\rm
even})|\mathcal{B}_{U}\rangle\,,$ (84)

with $U=\mathrm{diag}(i,i-2\epsilon,-i,-i)$. This boundary state corresponds
to the generic fermionic $1/6$-BPS Wilson loops with
$\bar{\alpha}^{2}=\beta_{2}=0$ and $\epsilon$ is given by
$\epsilon=\bar{\alpha}^{1}\beta_{1}$.

In the following we will show that when $\epsilon\neq 0,i$, this state is not
integrable. The idea is to employ the following set of Bethe roots with
$L=3,K_{\mathbf{u}}=K_{\mathbf{w}}=1,K_{\mathbf{v}}=2$,

$u_{1}=0.866025,\,\quad w_{1}=0.866025,\,\quad v_{1}=-0.198072,\,\quad
v_{2}=0.631084\,,$ (85)

which does not satisfy the selection rule
$\mathbf{u}=-\mathbf{v},\mathbf{w}=-\mathbf{w}$. Notice that the set of roots
also satisfy the zero momentum condition. However these roots do not satisfy
the first selection rule $K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=L$.
This fact leads to the result that the overlap
$\langle\mathcal{B}_{1/6}^{F}|\mathbf{u},\mathbf{v},\mathbf{w}\rangle=0$ for
this set of Bethe roots and whether the $|\mathcal{B}_{1/6}^{F}\rangle$ is
integrable or not can not be detected by this result. The way out is to
perform the following $SO(4)\subset SU(4)_{R}$ transformation Gombor:2022aqj ,

$U_{\theta}=g(\theta)Ug(\theta)^{-1}\,,$ (86)

with

$\displaystyle
g(\theta)=\left(\begin{array}[]{cccc}\cos^{2}\theta&\sin\theta&0&\sin\theta\cos\theta\\\
-\sin\theta\cos^{2}\theta&\cos^{2}\theta&\sin\theta&-\sin^{2}\theta\cos\theta\\\
\sin^{2}\theta\cos\theta&-\sin\theta\cos\theta&\cos\theta&\sin^{3}\theta\\\
-\sin\theta&0&0&\cos\theta\\\ \end{array}\right)\,,$ (91)

where $\theta$ satisfies $0<\theta<\frac{\pi}{2}$. Due to $SU(4)_{R}$
invariance, $|\mathcal{B}_{1/6}^{F}\rangle$ is integrable if and only if
$|\mathcal{B}_{1/6,\,\theta}^{F}\rangle\equiv(1+U_{\rm
even})|\mathcal{B}_{U_{\theta}}\rangle$ is. Through direct computation, we
found that $\langle\mathcal{B}_{1/6,\,\theta}^{F}|\bf{u},\bf{v},\bf{w}\rangle$
is zero if and only if $\epsilon=0$ or $\epsilon=i$. This shows that generic
fermionic $1/6$-BPS Wilson loop gives non-integrable boundary state. More
precisely, such boundary state satisfies neither the twisted condition nor the
untwisted one. Notice that, when $\epsilon=i$, there is supersymmetric
enhancement for this fermionic $1/6$-BPS Wilson loop and it in fact becomes
half-BPS. When $\epsilon=0$, the bosonic part of this fermionic $1/6$-BPS
Wilson loop is the same as the big bosonic $1/6$-BPS Wilson loop
$W_{1/6}^{B,\mathrm{big}}$.131313If we consider the vacuum expectation value
of the Wilson loop or the correlators of the Wilson loop with operators out of
it, this fermionic $1/6$-BPS Wilson loop with $\epsilon=0$ is identidical to
the bosonic $1/6$-BPS Wilson loop $W_{1/6}^{B,\mathrm{big}}$ Drukker:2020opf .
We have already shown that $\epsilon=0$ and $\epsilon=i$ lead to integrable
boundary states.

To sum up, we find that only some of the supersymmetric Wilson loops
correspond to tree-level integrable boundary states, they are listed as
follows

  * •

The bosonic $1/6$-BPS Wilson loop corresponding to the state

$\displaystyle|\mathcal{B}_{1/6}^{B,\mathrm{big}}\rangle=(1+U_{even})|\mathcal{B}_{R}\rangle,\qquad{R^{I}}_{J}=\text{diag}(i,i,-i,-i)$
(92)

  * •

The $1/2$-BPS Wilson loop corresponding to the state

$\displaystyle|\mathcal{B}_{1/2}^{1+}\rangle=(1+U_{\text{even}})|\mathcal{B}_{U}\rangle,\qquad{U^{I}}_{J}=\text{diag}(i,-i,-i,-i)$
(93)

  * •

The $1/2$-BPS Wilson loop corresponding to the state

$\displaystyle|\mathcal{B}_{1/2}^{4-}\rangle=(1+U_{\text{even}})|\mathcal{B}_{\tilde{U}}\rangle,\qquad{{\tilde{U}}^{I}}_{\phantom{I}J}=\text{diag}(i,i,i,-i)\,.$
(94)

  * •

The $1/3$-BPS Wilson loop corresponding to the state

$\displaystyle|\mathcal{B}_{1/3}\rangle=n_{1}|\mathcal{B}_{1/2}^{1+}\rangle+n_{4}|\mathcal{B}_{1/2}^{4-}\rangle$
(95)

Notice that in the above states, both $|\mathcal{B}_{M}\rangle$ and
$U_{\text{even}}|\mathcal{B}_{M}\rangle$ ($M=R,U,\tilde{U}$) are integrable at
tree level. In the next section, we will derive the exact overlap formula of
the tree-level integrable boundary states and the on-shell Bethe states of the
SU(4) alternating spin chain.

##  4 Overlap of $1/6$ BPS Wilson loop

In this section, we derive the exact overlap formula for the $1/6$-BPS Wilson
loop $|\mathcal{B}^{B,\text{big}}_{1/6}\rangle$ in (92). We will derive the
formula for $|\mathcal{B}_{R}\rangle$ and
$U_{\text{even}}|\mathcal{B}_{R}\rangle$ separately and then take the sum.

We will use the method developed in Gombor:2020kgu . For a two-site state
$|\mathcal{B}\rangle$ with the selection rule (78), one expects that the
overlap takes the following form

$\displaystyle\dfrac{\langle\mathcal{B}|\textbf{u,v,w}\rangle}{\sqrt{\langle\textbf{u,v,w}|\textbf{u,v,w}\rangle}}=\prod_{j=1}^{K_{\mathbf{u}}}h^{(1)}(u_{j})\prod_{k=1}^{K_{\mathbf{w}}/2}h^{(2)}(w_{k})\times\sqrt{\dfrac{\text{det}G_{+}}{\text{det}G_{-}}}.$
(96)

where $\det G_{\pm}$ are the Gaudin-like determinants whose definitions were
given in Yang:2021hrl . The prefactors $h^{(1)}(u)$ and $h^{(2)}(w)$ can be
calculated by a nesting procedure in the sparse limit where $L\to\infty$ and
the number of excitations are kept finite Gombor:2020kgu ; Gombor:2020auk ;
Gombor:2022aqj . In this limit, the ratio of determinants $\det G_{+}/\det
G_{-}\to 1$ and we are left with the contribution from the prefactors.
However, this method can not be applied directly in the current situation, for
the following two reasons.

First, the Bajnok-Gombor nesting procedure starts with evaluating the overlap
$\langle\mathcal{B}|0\rangle$, where $|0\rangle$ is the pseudovacuum state.
However, it is easy to see this overlap is vanishing for our Wilson loop
boundary states. Second, from $R$-charge conservation, the overlap
$\langle\mathcal{B}|\textbf{u,v,w}\rangle$ for the Wilson loop boundary states
are non-zero only if $K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=L$.
Therefore we cannot take the limit $L\to\infty$ while keeping the excitation
numbers finite.

###  4.1 Rotating boundary state

To address these two issues, one can rotate the boundary state by a certain
angle $\theta$ Gombor:2022aqj . The $K$-matrix (73) still satisfies the BYBE
under a $SO(4)$ rotation and hence integrability is preserved. The overlap for
the rotated boundary state $|\mathcal{B}_{\theta}\rangle$ is no longer
constrained by the selection rule
$K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=L$ and we can apply Bajnok-
Gombor approach to obtain the prefactor. Assuming the $\theta\rightarrow 0$
limit is smooth, we then obtain the prefactors of the original boundary state
by taking $\theta=0$. We will see that this method indeed gives the correct
result for the $1/6$-BPS Wilson loop.

We first consider the boundary state $\langle\mathcal{B}_{R}|$ in (49) with
the following $SO(4)$ rotation Gombor:2022aqj

$\displaystyle
g(\theta)=\left(\begin{array}[]{cccc}\cos\theta&0&0&-\sin\theta\\\
0&\cos\theta&-\sin\theta&0\\\ 0&\sin\theta&\cos\theta&0\\\
\sin\theta&0&0&\cos\theta\end{array}\right)$ (101)

and define

$\displaystyle R(\theta)=g(\theta)Rg(-\theta).$ (102)

The rotated dual boundary state is given by141414For the computation of the
overlap, we consider the dual boundary state.

$\displaystyle\langle\mathcal{B}_{R(\theta)}|=\left({{R(\theta)}^{I}}_{J}\langle
I,J|\right)^{\otimes L}$ (103)

Similarly, we define

$\displaystyle\langle\widehat{\mathcal{B}}_{R(\theta)}|=\langle\mathcal{B}_{R(\theta)}|U_{\text{even}}^{\dagger}\,.$
(104)

We find that

$\displaystyle{R(\theta)^{I}}_{J}\langle I,J|=$
$\displaystyle\,i\cos(2\theta)\left(\langle 1\bar{1}|+\langle
2\bar{2}|-\langle 3\bar{3}|-\langle 4\bar{4}|\right)$ (105)
$\displaystyle\,+i\sin(2\theta)\left(\langle 1\bar{4}|+\langle
2\bar{3}|+\langle 3\bar{2}|+\langle 4\bar{1}|\right)$

where the second line which breaks the selection rule
$K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=L$. The pseudovacuum state is

$\displaystyle|0\rangle=(|1\bar{4}\rangle)^{\otimes L}.$ (106)

We thus have

$\displaystyle\langle\mathcal{B}_{R(\theta)}|0\rangle=(i\sin
2\theta)^{L},\qquad$
$\displaystyle\langle\widehat{\mathcal{B}}_{R(\theta)}|0\rangle=(i\sin
2\theta)^{L}.$ (107)

We will perform the Bajnok-Gombor procedure for the state
$|\mathcal{B}_{R(\theta)}\rangle$, the state
$|\widehat{\mathcal{B}}_{R(\theta)}\rangle$ can be treated similarly. We first
define the renormalized state

$\displaystyle\langle\mathcal{B}_{R(\theta)}^{(1)}|=\frac{\langle\mathcal{B}_{R(\theta)}|}{(i\sin
2\theta)^{L}}\,,\qquad\langle\mathcal{B}_{R(\theta)}^{(1)}|0\rangle=1.$ (108)

###  4.2 Two-particle state

Now we consider the excited state with two particles with rapidities $u$ and
$v$. We denote the type of the particles by $a$ and $b$, where $a,b=1,2$. When
a type-$a$ sites on an odd (even) site, the field $Y^{1}$ ($\bar{Y}_{4}$) is
replaced by $Y^{1+a}$ ($\bar{Y}_{4-a}$). It is also possible for two particles
with different labels to occupy the same site, leading to the composite
excitations $Y^{4}$ on odd sites and $\bar{Y}_{1}$ on even sites. In what
follows, we will denote a state with two particles of type-$a$ and -$b$ at
sites $2n-1$ and $2m$ by $|2n-1,2m\rangle\\!\rangle_{a,b}$. If the two
particles are on the same site $n$, we denote the state by
$|n\rangle\\!\rangle_{\bullet}$. The asymptotic two-particle Bethe state of
the SU(4) alternating spin chain has been constructed in Gombor:2022aqj  and
reads

$\displaystyle\begin{aligned}
|\\{u\\},\\{v\\}\rangle_{a,b}=&\sum_{m,n=1}^{L}e^{ipn+iqm}\sum_{c,d=1}^{2}(-1)^{d-1}\chi_{a,b}^{c,d}(u-v)|2n-1,2m\rangle\\!\rangle_{c,d}\\\
&+\sum_{n=1}^{L}e^{i(p+q)n}\left(\zeta_{a,b}^{(1)}(u,v)|2n-1\rangle\\!\rangle_{\bullet}+\zeta_{a,b}^{(2)}(u,v)|2n\rangle\\!\rangle_{\bullet}\right)\,.\end{aligned}$
(109)

where

$\displaystyle e^{ip}=\frac{u+i/2}{u-i/2},\qquad
e^{iq}=\frac{v+i/2}{v-i/2}\,.$ (110)

The coefficients are given by

$\displaystyle\chi_{a,b}^{c,d}(u)=\left\\{\begin{array}[]{ll}\delta_{a}^{c}\delta_{b}^{d},&2n-1<2m\\\
R_{ab}^{cd}(u),&2n-1>2m\end{array}\right.$ (113)

with

$\displaystyle
R_{ab}^{cd}(u)=\frac{u}{u-i}\delta_{a}^{c}\delta_{b}^{d}-\frac{i}{u-i}\delta_{a}^{d}\delta_{b}^{c}$
(114)

The coefficients for the double occupation factor read

$\displaystyle\zeta_{ab}^{(1)}(u,v)=\epsilon_{ab}\frac{-v+i/2}{u-v-i},\qquad\zeta_{ab}^{(2)}(u,v)=\epsilon_{ab}\frac{u+i/2}{u-v-i}.$
(115)

Two comments are in order for the two particle state (109). Firstly, this
state is an eigenstate of the Hamiltonian in the asymptotic sense. This means
it is only an eigenstate for $L\to\infty$. Secondly, when constructing the
asymptotic two-particle state, in principle we should also take into account
the possibilities where the two particles are on two different even or odd
sites. Nevertheless such states will not contribute to the overlap and can be
ignored.

###  4.3 First level nesting

The first level $K$-matrix is given by

$\displaystyle
K_{ab}^{(1)}(u)=\lim_{L\rightarrow\infty}\frac{1}{A^{L}}\frac{\left\langle\mathcal{B}_{R(\theta)}|\\{u\\},\\{-u\\}\right\rangle_{a,b}}{\sqrt{\langle\\{u\\},\\{-u\\}|\\{u\\},\\{-u\\}\rangle_{a,b}}},$
(116)

where $A^{L}=\langle\mathcal{B}_{R(\theta)}|0\rangle=(-i\sin 2\theta)^{L}$.
The overlap is given by

$\displaystyle\left\langle\mathcal{B}_{R(\theta)}|u,-u\right\rangle_{a,b}=\left(\begin{array}[]{cc}A_{11}&A_{12}\\\
A_{21}&A_{22}\\\ \end{array}\right)$ (119)

Taking $v=-u$, the two-particle state is simplified further to

$\displaystyle|\\{u\\},\\{-u\\}\rangle_{a,b}=$
$\displaystyle\,\sum_{n,m}e^{ip(n-m)}\sum_{c,d=1}^{2}(-1)^{d-1}\chi_{a,b}^{c,d}(2u)|2n-1,2m\rangle\\!\rangle_{c,d}$
(120)
$\displaystyle\,+\frac{\epsilon_{ab}}{2}\frac{u+i/2}{u-i/2}\sum_{n}\left(|2n-1\rangle\\!\rangle_{\bullet}+|2n\rangle\\!\rangle_{\bullet}\right)$

We have

$\displaystyle\langle\mathcal{B}_{R(\theta)}|2m-1,2n\rangle\\!\rangle_{1,1}=\langle\mathcal{B}_{R(\theta)}|2m-1,2n\rangle\\!\rangle_{2,2}=(i\sin
2\theta)^{L}\,\delta_{m,n}\,,$ (121)
$\displaystyle\langle\mathcal{B}_{R(\theta)}|2m-1,2n\rangle\\!\rangle_{1,2}=i\cos
2\theta(i\sin 2\theta)^{L-1}\delta_{m,n}\,,$
$\displaystyle\langle\mathcal{B}_{R(\theta)}|2m-1,2n\rangle\\!\rangle_{2,1}=-i\cos
2\theta(i\sin 2\theta)^{L-1}\delta_{m,n}\,.$

and

$\displaystyle\langle\mathcal{B}_{R(\theta)}|2n-1\rangle\\!\rangle_{\bullet}=-(i\cos
2\theta)(i\sin
2\theta)^{L-1},\qquad\langle\mathcal{B}_{R(\theta)}|2n\rangle\\!\rangle_{\bullet}=(i\cos
2\theta)(i\sin 2\theta)^{L-1}\,.$ (122)

From (122), it is clear that the boundary state has vanishing overlap with the
second line in (120) for all $a,b=1,2$. Using (121), the matrix components of
(119) and (122) can be computed straightforwardly, yielding

$\displaystyle A_{11}=L\,(i\sin 2\theta)^{L}\,,\qquad A_{22}=-L\,(i\sin
2\theta)^{L}\,,$ (123) $\displaystyle A_{12}=A_{21}=-L(i\cos 2\theta)(i\sin
2\theta)^{L-1}.$

The norm of the Bethe state is more involved, but the $L\to\infty$ the leading
term is simply

$\displaystyle\lim_{L\to\infty}\langle\\{u\\},\\{-u\\}|\\{u\\},\\{-u\\}\rangle_{a,b}=L^{2}+\cdots$
(124)

where the ellipsis denote subleading terms. Therefore in the $L\to\infty$
limit, we obtain

$\displaystyle K_{ab}^{(1)}(u)=\lim_{L\to\infty}\frac{1}{L(-i\sin
2\theta)^{L}}\left(\begin{array}[]{cc}A_{11}&A_{12}\\\ A_{21}&A_{22}\\\
\end{array}\right)=\left(\begin{array}[]{cc}1&-\cot 2\theta\\\ -\cot
2\theta&-1\\\ \end{array}\right)$ (129)

for $1/6$-BPS Wilson loop. Following Bajnok-Gombor approach, (129) implies
that

$\displaystyle h^{(1)}(u)=K^{(1)}_{1,1}(u)=1$ (130)

###  4.4 Second level nesting

At the second level, we can take a shortcut. It has been shown Gombor:2020kgu
that for a boundary state described by the following $K$-matrix

$\displaystyle
K=\left(\begin{array}[]{cc}1&-ie^{-\gamma}(\cosh\beta+2\alpha\sinh\beta)\\\
-ie^{-\gamma}(\cosh\beta-2\alpha\sinh\beta)&-e^{-2\gamma}\end{array}\right)$
(133)

the absolute value of the prefactor is given by

$\displaystyle
h^{(2)}(w)=e^{-2\gamma}(\sinh\beta)^{2}\frac{w^{2}+\alpha^{2}}{w(w-i/2)}$
(134)

Comparing (134) with (129), we find that they are identical if we take

$\displaystyle\gamma=0,\qquad\alpha=0,\qquad\cosh\beta=-i\cot(2\theta)$ (135)

Therefore we conclude that

$\displaystyle h^{(2)}(w)=-\frac{1}{(\sin 2\theta)^{2}}\frac{w}{w-i/2}\,.$
(136)

One comment is that in principle at the second level nesting we should
consider the inhomogeneous spin chain and the resulting prefactors contains
contributions from the inhomogeneities — Bethe roots from the first level
nesting. At the same time, we should renormalize the second level boundary
state similar to (108). This cancels precisely the contributions from the
inhomogeneities. Therefore our result is valid. Plugging $h^{(1)}(u)=1$ and
(136) into (96), we obtain the general overlap formula151515This overlap
formula can be also obtained using the recursion method in Gombor:2021hmj .

$\displaystyle\frac{|\langle\mathcal{B}_{R(\theta)}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}=(\sin
2\theta)^{2(L-K_{\mathbf{w}})}\prod_{i=1}^{K_{\mathbf{w}}/2}\frac{4w_{i}^{2}}{4w_{i}^{2}+1}\times\frac{\det
G^{+}}{\det G^{-}}.$ (137)

We have $L\leq K_{\mathbf{u}}=K_{\mathbf{v}}\leq K_{\mathbf{w}}$. From (137),
we find that the limit $\theta\rightarrow 0$ is none vanishing only if
$L=K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}$, in which case we obtain

$\displaystyle\frac{|\langle\mathcal{B}_{R}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}=\prod_{i=1}^{K_{\mathbf{w}}/2}\frac{w_{i}^{2}}{w_{i}^{2}+1/4}\times\frac{\det
G^{+}}{\det G^{-}}\,.$ (138)

We have tested (138) numerically up to
$L=K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=4$, which is already quite
non-trivial.161616The Bethe roots used in this numerical test are listed in
Appendix B. We exploit the coordinate Bethe ansatz in Yang:2021hrl  to
construct the Bethe states. Notice that the above result is derived for
$K_{\mathbf{w}}$ being even. Numerical computation shows that the overlap
vanishes for odd $K_{\mathbf{w}}$.

###  4.5 Exact overlap for the shifted boundary state

Now we move to compute the exact overlap for
$\langle\widehat{\mathcal{B}}_{R}|=\langle\mathcal{B}_{R}|U_{\text{even}}^{\dagger}$
where $U_{\text{even}}$ is defined in (62) and it shifts all the even sites to
the left by two units. We make the same assumption (96) about the exact
overlap for
$\langle\widehat{\mathcal{B}}_{R}|\mathbf{u},\mathbf{v},\mathbf{w}\rangle$. To
determine the prefactors, we need to compute the overlap with the two-particle
state

$\displaystyle\langle\widehat{\mathcal{B}}_{R}|\\{u\\},\\{-u\\}\rangle_{a,b}=\langle\mathcal{B}_{R}|U_{\text{even}}^{\dagger}|\\{u\\},\\{-u\\}\rangle_{a,b}$
(139)

From the definition of $U_{\text{even}}$, it is clear that171717The following
computation is precise in the asymptotic sense, which is enough for our
purpose.

$\displaystyle U_{\text{even}}^{\dagger}|\\{u\\},\\{-u\\}\rangle_{a,b}=$
$\displaystyle\,\sum_{n,m}e^{ip(n-m)}\sum_{c,d=1}^{2}(-1)^{d-1}\chi_{a,b}^{c,d}(2u)|2n-1,2(m+1)\rangle\\!\rangle_{c,d}$
(140)
$\displaystyle\,+\frac{\epsilon_{ab}}{2}\frac{u+i/2}{u-i/2}\sum_{n}\left(|2n-1\rangle\\!\rangle_{\bullet}+|2n\rangle\\!\rangle_{\bullet}\right)$
$\displaystyle=$
$\displaystyle\,\left(\frac{u+i/2}{u-i/2}\right)\sum_{n,m}e^{ip(n-m)}\sum_{c,d=1}^{2}(-1)^{d-1}\chi_{a,b}^{c,d}(2u)|2n-1,2m\rangle\\!\rangle_{c,d}$
$\displaystyle\,+\frac{\epsilon_{ab}}{2}\frac{u+i/2}{u-i/2}\sum_{n}\left(|2n-1\rangle\\!\rangle_{\bullet}+|2n\rangle\\!\rangle_{\bullet}\right)\,.$

Namely, after the action of $U_{\text{even}}^{\dagger}$, the first line is
multiplied by a global factor while the second line is left invariant. As we
have shown that the second line does not contribute to the overlap. Therefore,
$\langle\widehat{\mathcal{B}}_{R(\theta)}|\\{u\\},\\{-u\\}\rangle_{a,b}$ is
simply proportional to
$\langle{\mathcal{B}}_{R(\theta)}|\\{u\\},\\{-u\\}\rangle_{a,b}$. The
corresponding first level $K$-matrix is given by

$\displaystyle\widehat{K}_{ab}^{(1)}(u)=\lim_{L\rightarrow\infty}\frac{1}{A^{L}}\frac{\left\langle\widehat{\mathcal{B}}_{R(\theta)}|\\{u\\},\\{-u\\}\right\rangle_{a,b}}{\sqrt{\langle\\{u\\},\\{-u\\}|\\{u\\},\\{-u\\}\rangle_{a,b}}}=\left(\frac{u+i/2}{u-i/2}\right)K^{(1)}_{ab}(u)\,.$
(141)

Therefore we arrived at the following exact overlap formula for
$|\widehat{\mathcal{B}}_{R}\rangle$

$\displaystyle\frac{\langle\widehat{\mathcal{B}}_{R}|\mathbf{u},\mathbf{v},\mathbf{w}\rangle}{\sqrt{\langle\mathbf{u},\mathbf{v},\mathbf{w}|\mathbf{u},\mathbf{v},\mathbf{w}\rangle}}=\prod_{j=1}^{K_{\mathbf{u}}}\frac{u_{j}+i/2}{u_{j}-i/2}\frac{{\langle\mathcal{B}}_{R}|\mathbf{u},\mathbf{v},\mathbf{w}\rangle}{\sqrt{\langle\mathbf{u},\mathbf{v},\mathbf{w}|\mathbf{u},\mathbf{v},\mathbf{w}\rangle}}.$
(142)

Hence there is a relative phase between these two boundary state. From this we
get

$\displaystyle\frac{|\langle\mathcal{B}_{1/6}^{B,\mathrm{big}}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}=\Bigg{|}1+\prod_{j=1}^{K_{\mathbf{u}}}\frac{u_{j}+i/2}{u_{j}-i/2}\Bigg{|}^{2}\frac{|\langle\mathcal{B}_{R}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}\,.$
(143)

##  5 Overlap of 1/2-BPS Wilson loop

We derive the overlap formula for the 1/2-BPS Wilson loop in this section. The
procedure is basically the same as for the 1/6-BPS case. There are two types
of 1/2-BPS Wilson loops (93) and (94), we will consider them in the following
two subsections.

###  5.1 1/2-BPS Wilson loop $|\mathcal{B}_{1/2}^{1+}\rangle$

Similar to the 1/6-BPS approach, we need to perform an SO(4) rotation in order
to apply the Bajnok-Gombor approach. We take the same rotation as in (101) and
define

$\displaystyle
U(\theta)=g(\theta)Ug(-\theta),\qquad\langle\mathcal{B}_{U(\theta)}|=\left({U(\theta)^{I}}_{J}\langle
I,J|\right)^{\otimes
L},\qquad\langle\widehat{\mathcal{B}}_{U(\theta)}|=\langle\mathcal{B}_{U(\theta)}|U^{\dagger}_{\text{even}}\,.$
(144)

More explicitly, we have

$\displaystyle{U(\theta)^{I}}_{J}\langle I,J|=i\cos(2\theta)\left(\langle
1\bar{1}|-\langle 4\bar{4}|\right)-i\left(\langle 2\bar{2}|+\langle
3\bar{3}|\right)+i\sin(2\theta)\left(\langle 1\bar{4}|+\langle
4\bar{1}|\right)$ (145)

We start with the first level nesting. We have

$\displaystyle\langle\mathcal{B}_{U(\theta)}|0\rangle=(i\sin 2\theta)^{L}\,.$
(146)

We then compute the overlap
$\langle\mathcal{B}_{U(\theta)}|\\{u\\},\\{-u\\}\rangle_{a,b}$ where
$|\\{u\\},\\{-u\\}\rangle_{a,b}$ is defined in (120). Using

$\displaystyle\langle\mathcal{B}_{U(\theta)}|2m-1,2n\rangle\\!\rangle_{1,1}=\langle\mathcal{B}_{U(\theta)}|2m-1,2n\rangle\\!\rangle_{2,2}=0\,,$
(147)
$\displaystyle\langle\mathcal{B}_{U(\theta)}|2m-1,2n\rangle\\!\rangle_{1,2}=-i(i\sin
2\theta)^{L-1}\,\delta_{m,n}\,,$
$\displaystyle\langle\mathcal{B}_{U(\theta)}|2m-1,2n\rangle\\!\rangle_{2,1}=i(-i\sin
2\theta)^{L-1}\,\delta_{m,n}\,,$
$\displaystyle\langle\mathcal{B}_{U(\theta)}|2n-1\rangle\\!\rangle_{\bullet}=-(i\cos
2\theta)(i\sin 2\theta)^{L-1}\,,$
$\displaystyle\langle\mathcal{B}_{U(\theta)}|2n\rangle\\!\rangle_{\bullet}=(i\cos
2\theta)(i\sin 2\theta)^{L-1}\,,$

We find that

$\displaystyle K_{ab}^{(1)}(u)=\lim_{L\to\infty}\frac{1}{L(-i\sin
2\theta)^{L}}\langle\mathcal{B}_{U(\theta)}|\\{u\\},\\{-u\\}\rangle_{a,b}=\frac{\epsilon_{ab}}{\sin
2\theta}\,,$ (148)

which is nothing but the dimer state. For the second level nesting, we can
directly apply the result of the dimer state Gombor:2020kgu , which leads to

$\displaystyle\frac{|\langle\mathcal{B}_{U(\theta)}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}=\frac{(-1)^{L}}{(\sin
2\theta)^{2(K_{\mathbf{w}}-L)}}\prod_{i=1}^{K_{\mathbf{u}}}\big{(}u_{i}^{2}+\frac{1}{4}\big{)}\prod_{j=1}^{[K_{\mathbf{w}}/2]}\frac{1}{w_{i}^{2}(w_{i}^{2}+1/4)}\,\frac{\det
G_{+}}{\det G_{-}}\,.$ (149)

Again, we find that in the $\theta\to 0$ limit, we must have
$K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=L$ and the finite result reads

$\displaystyle\frac{|\langle\mathcal{B}_{U}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}=(-1)^{L}\prod_{i=1}^{K_{\mathbf{u}}}\big{(}u_{i}^{2}+\frac{1}{4}\big{)}\prod_{j=1}^{[K_{\mathbf{w}}/2]}\frac{1}{w_{i}^{2}(w_{i}^{2}+1/4)}\,\frac{\det
G_{+}}{\det G_{-}}\,.$ (150)

This result has been tested numerically. Our numerical results also reveal
that this formula is also correct when $L$ is odd although the above
derivation was performed for even $L$. This aspect is different from the
bosonic $1/6$-BPS case in the previous section.

#### Shifted state

We find that the shifted state overlap is again proportional to the un-shifted
one as

$\displaystyle\frac{\langle\widehat{\mathcal{B}}_{U}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}{\sqrt{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}}=\prod_{j=1}^{K_{\mathbf{u}}}\left(\frac{u_{j}+i/2}{u_{j}-i/2}\right)^{2}\frac{{\langle\mathcal{B}}_{U}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}{\sqrt{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}}.$
(151)

Notice that the phase factor is different from the 1/6-BPS case. We have
tested this result numerically up to
$L=K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=4$. Naively, one might expect
that according to the same argument of 1/6-BPS case, one should obtain the
same phase factor (142). However, generalizing this argument to the Dimer
state seems a bit subtle, as $\widehat{K}_{11}=0$ in this case.

Then we have

$\displaystyle\frac{|\langle\mathcal{B}_{1/2}^{1+}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}=\Bigg{|}1+\prod_{j=1}^{K_{\mathbf{u}}}\left(\frac{u_{j}+i/2}{u_{j}-i/2}\right)^{2}\Bigg{|}^{2}\frac{|\langle\mathcal{B}_{U}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}\,.$
(152)

###  5.2 1/2-BPS Wilson loop $|\mathcal{B}_{1/2}^{4-}\rangle$

We take the same rotation as in (101) and define

$\displaystyle\tilde{U}(\theta)=g(\theta)\tilde{U}g(-\theta),\qquad\langle\mathcal{B}_{\tilde{U}(\theta)}|=\left({\tilde{U}(\theta)^{I}}_{J}\langle
I,J|\right)^{\otimes
L},\qquad\langle\widehat{\mathcal{B}}_{\tilde{U}(\theta)}|=\langle\mathcal{B}_{\tilde{U}(\theta)}|\tilde{U}^{\dagger}_{\text{even}}\,.$
(153)

More explicitly, we have

$\displaystyle{\tilde{U}(\theta)^{I}}_{J}\langle
I,J|=i\cos(2\theta)\left(\langle 1\bar{1}|-\langle
4\bar{4}|\right)+i\left(\langle 2\bar{2}|+\langle
3\bar{3}|\right)+i\sin(2\theta)\left(\langle 1\bar{4}|+\langle
4\bar{1}|\right)$ (154)

The rest of the computations are almost identical to the state
$\langle\mathcal{B}_{1/2}^{1+}|$. The only difference is that at the first
level nesting we have an additional minus sign, this leads to the following
relative phase between the two overlaps

$\displaystyle\langle\mathcal{B}_{\tilde{U}}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle=(-1)^{L}\langle\mathcal{B}_{{U}}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle$
(155)

The norm of the overlap is thus the same as $\langle\mathcal{B}_{1/2}^{1+}|$

$\displaystyle\frac{|\langle\mathcal{B}_{\tilde{U}}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}=(-1)^{L}\prod_{i=1}^{K_{\mathbf{u}}}\big{(}u_{i}^{2}+\frac{1}{4}\big{)}\prod_{j=1}^{[K_{\mathbf{w}}/2]}\frac{1}{w_{i}^{2}(w_{i}^{2}+1/4)}\,\frac{\det
G_{+}}{\det G_{-}}\,.$ (156)

#### Shifted state

The overlap for the shifted state leads to the same phase factor as for
$\langle\mathcal{B}_{1/2}^{1+}|$ case

$\displaystyle\frac{\langle\widehat{\mathcal{B}}_{\tilde{U}}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}{\sqrt{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}}=\prod_{j=1}^{K_{\mathbf{u}}}\left(\frac{u_{j}+i/2}{u_{j}-i/2}\right)^{2}\frac{{\langle\mathcal{B}}_{\tilde{U}}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}{\sqrt{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}}.$
(157)

So as in the previous case, we have,

$\displaystyle\frac{|\langle\mathcal{B}_{1/2}^{4-}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}=\Bigg{|}1+\prod_{j=1}^{K_{\mathbf{u}}}\left(\frac{u_{j}+i/2}{u_{j}-i/2}\right)^{2}\Bigg{|}^{2}\frac{|\langle\mathcal{B}_{\tilde{U}}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}\,.$
(158)

Finally for $|\mathcal{B}_{1/3}\rangle$, we have

$\displaystyle\frac{|\langle\mathcal{B}_{1/3}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}$
$\displaystyle=\Bigg{|}(n_{1}+(-1)^{L}n_{4})\left(1+\prod_{j=1}^{K_{\mathbf{u}}}\left(\frac{u_{j}+i/2}{u_{j}-i/2}\right)^{2}\right)\Bigg{|}^{2}$
$\displaystyle\times\frac{|\langle\mathcal{B}_{U}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle|^{2}}{\langle\mathbf{u},-\mathbf{u},\mathbf{w}|\mathbf{u},-\mathbf{u},\mathbf{w}\rangle}\,.$
(159)

##  6 Conclusion

In this paper, we computed the correlation function of a circular BPS Wilson
loop and a single-trace operator in ABJM theory. This correlation function is
proportional to the overlap of a boundary state from the Wilson loop and the
Bethe state corresponding to the single-trace operator. We proved that among a
sub-class of the fermionic $1/6$-BPS Wilson loops, only two special cases,
bosonic $1/6$-BPS Wilson loops and half-BPS Wilson loops can lead to tree-
level integrable boundary states. The boundary state from the $1/3$-BPS Wilson
loop is integrable at tree level as well. Our result for the subclass of
fermionic $1/6$-BPS Wilson loops is in some sense similar to the results on
integrability of the open chains from bosonic (non-)supersymmetric Maldacena-
Wilson lines in the $\mathcal{N}=4$ SYM Correa:2018fgz . There it was found
that only the two special cases, half-BPS Maldacena-Wilson loops and the usual
Wilson loops, lead to integrable open chains. We also obtained the exact
overlap formulae up to a phase for all the tree-level integrable boundary
states corresponding to $W^{B}_{1/6}$, $\hat{W}^{B}_{1/6}$,
$W^{B,\mathrm{big}}_{1/6}$, $W_{1/2}^{1+}$, $W_{1/2}^{4-}$ and $W_{1/3}$.

There are various directions that deserve further investigations. One
immediate problem is to fix the phase of the overlaps for $W^{B}_{1/6}$, and
$W_{1/2}^{1+}$. Although as long as the Wilson loop one-point function is
concerned, the phase is unimportant, as a spin chain problem it is still nice
to have a method which also gives the phase factor. It is valuable to
generalize our result to other closed sectors even to the full sector at two
loop level. If some BPS Wilson loops give integrable boundary states in the
full sector, the constraints from the bosonic and fermionic duality
Kristjansen:2020vbe ; Kristjansen:2021xno ; Kristjansen:2021abc  of the Bethe
ansatz equations may help us to pin down the full sector overlap formulas. The
next step is to obtain all loop overlap in the asymptotic sense using
integrable bootstrap method, as done for some integrable boundary states in
$\mathcal{N}=4$ SYM Jiang:2019xdz ; Jiang:2019zig ; Komatsu:2020sup  case. An
even more ambitious goal would be computing the finite-size corrections using
the worldsheet $g$-function approach Jiang:2019xdz ; Jiang:2019zig .

In this paper, we only consider certain BPS Wilson loop in the fundamental
representation of a suitable group or super-group. It is interesting to study
whether similar Wilson loops in higher dimensional representations can also
lead to integrable boundary states. Generating functions of Wilson loops in
various representations Hartnoll:2006is  and the method of introducing one-
dimensional scalars and/or fermions along the contour of the Wilson loop JKV ;
Gomis:2006sb ; Gomis:2006im ; Castiglioni:2022yes  should be very helpful
here.

One common feature of the Wilson loops considered here is that the scalar
coupling is constant. There are also other BPS Wilson loops whose scalar
couplings are $\tau$-dependent Cardinali:2012ru ; Bianchi:2014laa ;
Castiglioni:2022yes , where $\tau$ is used to parameterise the Wilson loop
contour. The correlator of such Wilson loops and a single-trace operator in
the $SU(4)$ sector will lead to boundary states involving integration of
$\tau$ along the circle. It is interesting to seek integrable boundary states
among them.

Comparing with the $\mathcal{N}=4$ SYM case, the study of correlators of BPS
Wilson loops and single-trace operators is much more preliminary and there are
far less results. One direction complementary to the study here is using
localization Kapustin:2009kz  to compute the correlation functions of BPS
Wilson loops and certain BPS local operators and comparing the results at
strong coupling in the large $N$ limit with holographic computations. Some
computations in $\mathcal{N}=4$ SYM case in this direction can be found in
Giombi:2009ds ; Giombi:2012ep ; Giombi:2020amn ; Berenstein:1998ij ;
Giombi:2006de ; Zarembo:2002ph , however the localization computation in the
ABJM case seems more challenging and calls for new developments. Some
important progress in this direction was made recently in Guerrini:2023rdw .

## Acknowledgement

We would like to thank Bin Chen for collaboration at early stages of this
project, Hong-Fei Shu, Jiaju Zhang for help discussions, and Zhi-Xin Hu for
helps on using the computer cluster at CJQS, TJU. Y.J. would like to thank
Center for Joint Quantum Studies of Tianjin University for hospitality during
the final stage of the work. The work of Y.J. is partly supported by Startup
Funding no. 3207022217A1 of Southeast University. The work of J.W. and P.Y. is
partly supported by the National Natural Science Foundation of China, Grant
No. 11975164, 11935009, 12247103, 12047502, and Natural Science Foundation of
Tianjin under Grant No. 20JCYBJC00910.

##  Appendix A The Lagrangian and supersymmetry transformations of ABJM theory

Spinor convention. The circular BPS Wilson loops in ABJM theory can only exist
when the theory is put in the Euclidean space $\mathbf{R}^{3}$ Ouyang:2015ada
. We follow the spinor convention in Ouyang:2015ada . The metric on
$\mathbf{R}^{3}$ is $\delta_{\mu\nu}=\mathrm{diag}(1,1,1)$, the coordinates
are $x^{\mu}=(x^{1},x^{2},x^{3})$. The $\gamma$ matrices are

$\gamma^{\mu\,\,\,\beta}_{\,\,\alpha}=(-\sigma^{2},\sigma^{1},\sigma^{3})\,.$
(160)

They satisfy

$\gamma^{\mu}\gamma^{\nu}=\delta^{\mu\nu}+i\epsilon^{\mu\nu\rho}\gamma_{\rho}\,,$
(161)

where $\epsilon^{\mu\nu\rho}$ is the rank-$3$ antisymmetric tensor with
$\epsilon^{123}=1$. We raise or lower the spinor indices using anti-symmetric
tensor $\epsilon^{\alpha\beta}$ and $\epsilon_{\alpha\beta}$

$\theta^{\alpha}=\epsilon^{\alpha\beta}\theta_{\beta}\,,\,\theta_{\alpha}=\epsilon_{\alpha\beta}\theta^{\beta}\,,$
(162)

with $\epsilon^{12}=-\epsilon_{12}=1$. We will use the shorthand notation,

$\theta\psi=\theta^{\alpha}\psi_{\alpha}\,,\,\theta\gamma^{\mu}\psi=\theta^{\alpha}\gamma^{\mu\,\,\,\beta}_{\,\,\alpha}\psi^{\beta}.$
(163)

Field content. ABJM theory is the three-dimensional $\mathcal{N}=6$ super-
Chern-Simons theory with gauge group $U(N)\times U(N)$. The Chern-Simons
levels are $k$ and $-k$, respectively. Besides the gauge fields
$A_{\mu},\hat{A}_{\mu}$ in the adjoint representation of each $U(N)$. The
matter fields include four complex scalars $Y^{I}$ and four Dirac spinors
$\psi_{I}$ in the bi-fundamental representation of the gauge group. $Y^{I}$’s
are in the $\bf{4}$ representation of R-symmetry group $SU_{R}(4)$ and
$\psi_{I}$’s are in the $\bar{\bf{4}}$ representation.

#### Lagrangian.

The Lagrangian of ABJM theory can be written as the sum of four parts,

$\mathcal{L}_{\mathrm{ABJM}}=\mathcal{L}_{\mathrm{CS}}+\mathcal{L}_{k}+\mathcal{L}_{k}+\mathcal{L}_{Y}\,,$
(164)

with

$\displaystyle\mathcal{L}_{\mathrm{CS}}$ $\displaystyle=$
$\displaystyle-\frac{k}{4\pi}\epsilon^{\mu\nu\rho}\mathrm{Tr}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}-\hat{A}_{\mu}\partial_{\nu}\hat{A}_{\rho}+\frac{2i}{3}\hat{A}_{\mu}\hat{A}_{\nu}\hat{A}_{\rho}\right)\,,$
$\displaystyle\mathcal{L}_{p}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left(-D_{\mu}Y^{\dagger}_{I}D^{\mu}Y^{I}+i\psi^{\dagger
I}\gamma^{\mu}D_{\mu}\psi_{I}\right)\,,$ $\displaystyle\mathcal{L}_{p}$
$\displaystyle=$
$\displaystyle\frac{4\pi^{2}}{3k^{2}}\mathrm{Tr}\left(Y^{I}Y^{\dagger}_{I}Y^{J}Y^{\dagger}_{J}Y^{K}Y^{\dagger}_{K}+Y^{I}Y^{\dagger}_{J}Y^{J}Y^{\dagger}_{K}Y^{K}Y^{\dagger}_{I}+4Y^{I}Y^{\dagger}_{J}Y^{K}Y^{\dagger}_{I}Y^{J}Y^{\dagger}_{K}\right.$
$\displaystyle\left.-6Y^{I}Y^{\dagger}_{J}Y^{J}Y^{\dagger}_{I}Y^{K}Y^{\dagger}_{K}\right)\,,$
$\displaystyle\mathcal{L}_{Y}$ $\displaystyle=$ $\displaystyle-\frac{2\pi
i}{k}\mathrm{Tr}\left(Y^{I}Y^{\dagger}_{I}\psi_{J}\psi^{\dagger
J}-2Y^{I}Y^{\dagger}_{J}\psi_{I}\psi^{\dagger
J}-Y^{\dagger}_{I}Y^{I}\psi^{\dagger
J}\psi_{J}+2Y^{\dagger}_{I}Y^{J}\psi^{\dagger I}\psi_{J}\right.$ (165)
$\displaystyle\left.+\epsilon_{IJKL}Y^{I}\psi^{\dagger J}Y^{K}\psi^{\dagger
L}-\epsilon^{IJKL}Y^{\dagger}_{I}\psi_{J}Y^{\dagger}_{K}\psi_{L}\right)\,.$

Here the covariant derivatives are defined as

$\displaystyle D_{\mu}Y^{I}$ $\displaystyle=$
$\displaystyle\partial_{\mu}Y^{I}+iA_{\mu}Y^{I}-iY^{I}\hat{A}_{\mu}\,,$
$\displaystyle D_{\mu}Y^{\dagger}_{I}$ $\displaystyle=$
$\displaystyle\partial_{\mu}Y^{\dagger}_{I}+i\hat{A}_{\mu}Y^{\dagger}_{I}-iY^{\dagger}_{I}A_{\mu}\,,$
$\displaystyle D_{\mu}\psi_{I}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\psi_{I}+iA_{\mu}\psi_{I}-i\psi_{I}\hat{A}_{\mu}\,.$
(166)

and $\epsilon_{IJKL},\epsilon^{IJKL}$ is totally anti-symmetric tensor with
$\epsilon_{1234}=\epsilon^{1234}=1$.

#### Supersymmetry transformations.

The ABJM action is invariant under the following supersymmetry transformations
Gaiotto:2008cg ; Hosomichi:2008jb ; Terashima:2008sy ; Bandres:2008ry :

$\displaystyle\delta A_{\mu}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{k}(Y^{I}\psi^{\dagger
J}\gamma_{\mu}\varepsilon_{IJ}+\bar{\varepsilon}^{IJ}\gamma_{\mu}\psi_{J}Y^{\dagger}_{I})\,,$
(167) $\displaystyle\delta\hat{A}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{k}(\psi^{\dagger
J}Y^{I}\gamma_{\mu}\varepsilon_{IJ}+\bar{\varepsilon^{IJ}}Y^{\dagger}_{I}\gamma_{\mu}\psi_{J})\,,$
(168) $\displaystyle\delta Y^{I}$ $\displaystyle=$ $\displaystyle
i\bar{\varepsilon}^{IJ}\psi_{J}\,,\,\,\delta Y^{\dagger}_{I}=i\psi^{\dagger
J}\varepsilon_{IJ}\,,$ (169) $\displaystyle\delta\psi_{I}$ $\displaystyle=$
$\displaystyle\gamma^{\mu}\varepsilon_{IJ}D_{\mu}Y^{J}+\vartheta_{IJ}Y^{J}-\frac{2\pi}{k}\varepsilon_{IJ}(Y^{J}Y^{\dagger}_{K}Y^{K}-Y^{K}Y^{\dagger}_{K}Y^{J})$
(170)
$\displaystyle-\frac{4\pi}{k}\varepsilon_{KL}Y^{K}Y^{\dagger}_{I}Y^{L}\,,$
$\displaystyle\delta\psi^{\dagger I}$ $\displaystyle=$
$\displaystyle-\bar{\varepsilon}^{IJ}\gamma^{\mu}D_{\mu}Y^{\dagger}_{J}+\bar{\vartheta}^{IJ}Y^{\dagger}_{J}+\frac{2\pi}{k}\bar{\varepsilon}^{IJ}(Y^{\dagger}_{J}Y^{K}Y^{\dagger}_{K}-Y^{\dagger}_{K}Y^{K}Y^{\dagger}_{J})$
(171)
$\displaystyle+\frac{4\pi}{k}\bar{\varepsilon}^{KL}Y^{\dagger}_{K}Y^{I}Y^{\dagger}_{L}.$

The supersymmetry parameters are
$\varepsilon_{IJ}=\theta_{IJ}+x^{\mu}\gamma_{\mu}\vartheta_{IJ}$ and
$\bar{\varepsilon}^{IJ}=\bar{\theta}^{IJ}-\bar{\vartheta}^{IJ}x^{\mu}\gamma_{\mu}$.
Here $\theta$’s give the Ponicaré supersymmetry, and $\vartheta$’s give the
conformal supersymmetry. They satisfy the following constraints,

$\displaystyle\theta_{IJ}$ $\displaystyle=$
$\displaystyle-\theta_{JI}\,,\,\bar{\theta}^{IJ}=\frac{1}{2}\epsilon^{IJKL}\theta_{KL}\,,$
(172) $\displaystyle\vartheta_{IJ}$ $\displaystyle=$
$\displaystyle-\vartheta_{JI}\,,\bar{\vartheta}^{IJ}=\frac{1}{2}\epsilon^{IJKL}\vartheta_{KL}\,.$
(173)

Notice that for the theory in the Euclidean signature we do not impose that
$\bar{\theta}^{IJ}$ ($\bar{\vartheta}^{IJ}$) is the complex conjugation of
$\theta_{IJ}$ ($\vartheta_{IJ}$) Nicolai:1978vc .

#### Propagators of the scalar fields.

The tree-level propagators of the scalar fields are,

$\langle
Y^{I\alpha}_{\,\,\,\,\,\,\,\,\bar{\beta}}(x)Y^{\dagger\,\,\bar{\gamma}}_{J\,\,\,\,\,\,\rho}(y)\rangle=\frac{\delta^{I}_{J}\delta^{\alpha}_{\rho}\delta^{\bar{\gamma}}_{\bar{\beta}}}{4\pi|x-y|}\,,$
(174)

where $\alpha,\bar{\beta},\bar{\gamma},\rho$ are color indices.

##  Appendix B Numerical solutions of the Bethe equations

In this appendix, we present a collection of numerical solutions for the Bethe
equations in the $SU(4)$ sector of the ABJM theory

$\displaystyle
1=e^{i\phi_{u_{j}}}=\left(\frac{u_{j}+\frac{i}{2}}{u_{j}-\frac{i}{2}}\right)^{L}\prod_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{K_{\mathrm{u}}}S\left(u_{j},u_{k}\right)\prod_{k=1}^{K_{\mathrm{w}}}\tilde{S}\left(u_{j},w_{k}\right)\,,$
(175) $\displaystyle 1=e^{i\phi_{w_{j}}}=\prod_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{K_{\mathrm{w}}}S\left(w_{j},w_{k}\right)\prod_{k=1}^{K_{\mathrm{u}}}\tilde{S}\left(w_{j},u_{k}\right)\prod_{k=1}^{K_{\mathrm{v}}}\tilde{S}\left(w_{j},v_{k}\right)\,,$
(176) $\displaystyle
1=e^{i\phi_{v_{j}}}=\left(\frac{v_{j}+\frac{i}{2}}{v_{j}-\frac{i}{2}}\right)^{L}\prod_{\begin{subarray}{c}k=1\\\
k\neq
j\end{subarray}}^{K_{\mathrm{v}}}S\left(v_{j},v_{k}\right)\prod_{k=1}^{K_{\mathrm{w}}}\tilde{S}\left(v_{j},w_{k}\right)\,,$
(177)

where the S-matrices $S(u,v)$ and $\tilde{S}(u,v)$ are given by

$S(u,v)\equiv\frac{u-v-i}{u-v+i},\quad\tilde{S}(u,v)\equiv\frac{u-v+\frac{i}{2}}{u-v-\frac{i}{2}}\,.$
(178)

Here the numbers of rapidities $\mathbf{u,v,w}$ are denoted by
$K_{u},K_{v},K_{w}$.

The cyclicity property of the single trace operator is equivalent to the zero
momentum condition

$1=\prod_{j=1}^{K_{\mathbf{u}}}\frac{u_{j}+\frac{i}{2}}{u_{j}-\frac{i}{2}}\prod_{j=1}^{K_{\mathbf{v}}}\frac{v_{j}+\frac{i}{2}}{v_{j}-\frac{i}{2}}\,.$
(179)

In the following, we present a collection of solutions that fulfill both Bethe
ansatz equations and the zero momentum condition. Rational Q-system
Marboe:2016yyn ; Gu:2022dac  plays an important role here.

$L$ |  $K_{\bf w}$ |  $[{\bf u},{\bf v},{\bf w}]$
---|---|---
1 |  1 |  $[\\{0\\},\\{0\\},\\{0\\}]$
2 |  2 |  $[\\{\sqrt{\tfrac{3}{20}},-\sqrt{\tfrac{3}{20}}\\},\\{-\sqrt{\tfrac{3}{20}},\sqrt{\tfrac{3}{20}}\\},\\{\tfrac{1}{\sqrt{5}},-\tfrac{1}{\sqrt{5}}\\}]$
3 |  3 |  $[\\{-0.61842989257770833,\,0\,,0.61842989257770833\\},$ $\\{0.61842989257770833,\,0\,,-0.61842989257770833\\},$ $\\{0.71410132990930250,-0.71410132990930250,0\\}]$
|  |  $[\\{\,0.36628143864284446,\,-0.18314071932142223-0.5006211833519472i$
|  |  $-0.18314071932142223+0.5006211833519472i\\},$
|  |  $\\{\,-0.36628143864284446,\,0.18314071932142223+0.5006211833519472i$
|  |  $0.18314071932142223-0.5006211833519472i\\},$
|  |  $\\{\,-0.4472135954999579i\,,\,0\,,\,0.4472135954999579i\\}]$
|  |  $[\\{\,-0.9393910431943004i,\,0\,,\,0.9393910431943004i\\},$ $\\{\,0.9393910431943004i,\,0\,,\,-0.9393910431943004i\\},$ $\\{\,-1.0847153433251056i\,,\,0\,,\,1.0847153433251056i\\}]$
4 |  4 |  $[\\{\,-0.30330564928014186-0.4984162134634997i,\,-0.30330564928014186+0.4984162134634997i\,,\,0.02628462005210284\,,0.5803266785081809\\},$
|  |  $\\{\,0.30330564928014186+0.4984162134634997i,\,0.30330564928014186-0.4984162134634997i\,,\,-0.02628462005210284\,,-0.5803266785081809\\},$
|  |  $\\{\,-0.5162715680301216\,,\,-0.5001222335995396i\,,\,0.5162715680301216\,$
|  |  $0.5001222335995396i\\}]$
|  |  $[\\{\,-0.16030976462353738-0.9768494810075854i,\,-0.16030976462353738+0.9768494810075854i\,,\,-0.14056546652006302\,,0.46118499576713784\\},$
|  |  $\\{\,0.16030976462353738+0.9768494810075854i,\,0.16030976462353738-0.9768494810075854i\,,\,0.14056546652006302\,,-0.46118499576713784\\},$
|  |  $\\{\\-0.2416681458566768\,,\,-1.0684026594894636i\,,\,0.24166814585667684\,$
|  |  $1.0684026594894636i\\}]$
|  |  $[\\{\,-0.7905846950242429,\,-0.18184585032628545\,,$
|  |  $0.18184585032628545\,,0.7905846950242429\\},$
|  |  $\\{\,0.7905846950242429,\,0.18184585032628545\,,$
|  |  $-0.18184585032628545\,,-0.7905846950242429\\},$
|  |  $\\{\,-0.9018353804885377\,,\,-0.25327661600652046\,,$
|  |  $0.9018353804885376\,,\,0.25327661600652046\\}]$
|  |  $[\\{\,-0.4913865158293109-0.5254890261600584i,\,-0.4913865158293109+0.5254890261600584i\,,\,0.49138651582931087-0.5254890261600584i\,$
|  |  $,0.49138651582931087+0.5254890261600584i\\},$
|  |  $\\{\,0.4913865158293109+0.5254890261600584i,\,0.4913865158293109-0.5254890261600584i\,,\,-0.49138651582931087+0.5254890261600584i\,$
|  |  $,-0.49138651582931087-0.5254890261600584i\\},$
|  |  $\\{\\-0.53872049128905-0.5800492329412736i\,,\,-0.538720491289058+0.5800492329412736i\,,\,0.538720491289058+0.5800492329412736i\,,$
|  |  $\,0.538720491289058-0.5800492329412736i\\}]$
|  |  $[\\{\,-0.14792308614892888i,\,0.14792308614892888i\,$
|  |  $,\,-0.8757024703174519i\,,0.8757024703174519i\\},$
|  |  $\\{\,0.14792308614892888i,\,-0.14792308614892888i\,$
|  |  $,\,0.8757024703174519i\,,-0.8757024703174519i\\},$
|  |  $\\{\\-0.1144398362148366\,,\\-1.0318645990119506i\,,$
|  |  $\,0.1144398362148366\,,\,1.0318645990119506i\\}]$

Notice that all of the above sets of Bethe roots satisfy the selection rules
$K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=L$ and (78).

Moreover, we have also found a set of Bethe roots with
$L=3,K_{\mathbf{u}}=K_{\mathbf{w}}=1,K_{\mathbf{w}}=2$ that satisfies the zero
momentum condition and Bethe equations, however it does not satisfy the
selection rules $K_{\mathbf{u}}=K_{\mathbf{v}}=K_{\mathbf{w}}=L$ and (78).
This set of Bethe roots is

$\displaystyle u_{1}=0.866025\,\qquad w_{1}=0.866025,$ (180) $\displaystyle
v_{1}=-0.198072\,\qquad v_{2}=0.631084\,.$

## Index

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