Loop Operators in Three-Dimensional $\mathcal{N}=2$ Fishnet Theories

In this work, we study the line and loop operators in three-dimensional ${\mathcal N}=2$ fishnet theories in detail. We construct the straight line and circular loop operators which are at least classically half-BPS. We develop a new regularization scheme at frame $-1$ which is suitable for the study of the fermionic BPS loops in general super-Chern-Simons-matter theories. We initialize the perturbative computation for the vacuum expectation values of the circular BPS loop operators based on this scheme. We construct the cusped line operators as well, and compute the vacuum expectation values of these cusped line operators up to two-loop order. We find that the universal cusp anomalous dimension vanishes, if we put aside the fact that the generalized potential has a double pole in the $1/\epsilon$ expansion.


Introduction
Wilson loop operators play an important role in the study of dynamics of gauge theory. The vacuum expectation value (VEV) of the Wilson loop provides the criteria for color confinement [1]. In supersymmetric gauge theory, it is natural to consider the Wilson loop operators preserving part of the supersymmetries, which have better ultraviolet behavior. The first kind of BPS Wilson loop operator was constructed [2,3] [2,3] in the frame work of the AdS/CFT correspondence. This opened a new window to study the AdS/CFT correspondence via the loop operators.
Due to its supersymmetric nature, the study of the Bogomol'nyi-Prasad-Sommerfield (BPS) Wilson loop operators benefits from new techniques in supersymmetric field theories. One typical example is the supersymmetric localization. Supersymmetric localization reduces the path integral computing the VEV of the circular half-BPS Wilson loop in SYM 4 to a finite-dimensional integral [5]. And this finite-dimensional integral turns out be a Gaussian matrix model. It was known before the localization computations that the results from this Gaussian matrix model are consistent with the prediction of the AdS/CFT correspondence [6]. Moreover the localization provides a very powerful tool to calculate the VEVs or certain correlators of the BPS Wilson loops even at finite N (for a collection of reviews, see [7]).
Besides the smooth line and loop operators, there are the line/loop operators with a cusp, possibly with the insertion of local composite operators. The cusp anomalous dimension is related to many important quantities like the ultraviolet divergences of cusped Wilson loops, the infrared divergences of gluon amplitudes and the anomalous dimension of twist-two operators in gauge theories. The study of the cusp anomaly in the BPS line operator could be helped by using the integrability in the AdS 5 /CF T 4 duality [8,9]. When we insert the composite operators into the Wilson loops, the computation of the anomalous dimensions leads to integrable open spin chains. This happens for both the BPS Maldacena-Wilson loops [10] and the usual Wilson loops [11]. The cusp anomalous dimension can be computed based on this open chain firstly using the open asymptotic Bethe ansatz equations and the boundary thermal Bethe ansatz [12,13] and later using very powerful tools like the Y-system [14] and quantum spectral curve [15].
The obtained results are consistent with the ones from the localization [16]. Further studies on comparing results from localization and integrability can be found in [17,18].
Some details of the development on the Wilson loops and integrable structure can be found in a short review [19].
The supersymmetric localization has also been applied to the study of the BPS Wilson loop operators in the AdS 4 /CFT 3 correspondence [20], which states that a three dimensional N = 6 Chern-Simons-matter theory is dual to the IIA string theory on AdS 4 × CP 3 background. In this Chern-Simons-matter theory proposed by Aharony, Bergman, Jafferis and Maldacena (ABJM), the construction of the BPS Wilson loops is subtle [21,22,23]. The half BPS one requires the introduction of superconnection [24].
The VEV's of both bosonic 1/6-BPS [21,22,23] and fermionic half-BPS Wilson loops [24] can be computed using the localization [25] which leads to the results [26] consistent with string theory prediction. Some fermionic BPS Wilson loops with less supersymmetries in the ABJM theory were constructed in [27,28]. The ABJM theory is also integrable in the planar limit [29,30]. There is an interpolating function in the magnon dispersion relation [31]. The computation using the integrability of the cusp anomalous dimension of the Wilson loops was hoped to exactly confirm a conjecture [32] about this interpolating function [13]. For a recent progress report on various aspects on the BPS Wilson loops in Chern-Simons-matter theory, see [33] and [34]. It is certainly interesting to study the BPS Wilson loops, localizations and integrable structures in the theories with less supersymmetries. The BPS circular Wilson loops in four dimensional theories need that the theories have at least N = 2 supersymmetries.
The VEV of the BPS Wilson loop in the N = 2 * theory, which comes from a mass deformation of the N = 4 SYM, was computed using the localizations and the results are consistent with the predictions from the AdS/CFT correspondence [35]. This provided an important test of the holographic correspondence in the non-conformal case. However, the integrable β-deformed SYM 4 has only N = 1 supersymmetry and it only admits the BPS Wilson line along a null straight line. The situation is different in three dimensions.
The β-deformed ABJM theory has N = 2 supersymmetries so it admits both bosonic half-BPS Wilson loops [56] and fermionic half-BPS Wilson loops [28] beyond the lightlike case. Other studies of the BPS Wilson loops in three-dimensional Chern-Simons-matter theories with 2 ≤ N ≤ 4 supersymmetries can be found in [36,37,38,40]. It was raised in [37] that some classically BPS fermionic Wilson loops may not be truly supersymmetric at the quantum level. Though there were efforts to resolve this issue either by perturbative calculation [41,42,43] or by studying the precise map between the BPS Wilson loops and the probe M2-brane solutions in dual M-theory backgrounds [38], it is fair to say that this big question is still to be answered. It should be interesting to study this issue in some simple theories with key properties left.
Few years ago, Kazakov et. al. proposed a new double scaling limit on the γ-twisted N = 4 SYM theory and ABJM theories [44,45]. The novel feature in this limit is to take the twisting parameters q i to zero (or infinity) and the 't Hooft coupling λ to zero but keep the ratio q i /λ (or the product q i λ) of the parameters and the coupling to be finite such that the resulting actions with the gauge field (and gluinos in the 4D case) being decoupled include only the complex scalars and complex fermions. Under this limit, the theories are not unitary and generically break supersymmetry. However, the resulting new theories are still integrable in the planar limit, presenting the remarkable features in the fishnet theories [46,47]. Various aspects [48] of these so-called conformal fishnet theories have been studied in the past few years.
In this work, we would like to study the BPS Wilson loop in the three-dimensional (3D) N = 2 fishnet theories. A special double-scaling limit of the γ-deformed ABJM theory leads to N = 2 fishnet theory with only scalars and fermions [45]. We construct the fermionic BPS line and loop operators in this N = 2 fishnet theory and compute the VEV of circular loop operators perturbatively after developing a new regularization scheme. Our computation suggests that one-loop and two-loop contributions to the VEV are vanishing. Furthermore we investigate the aforementioned line operators with a cusp, which are not supersymmetric anymore. We find that the universal anomalous dimension of the cusp is vanishing at two loop order.
In the perturbative computations of the BPS Wilson loops in the Chern-Simons(matter) theory, one important issue is on the effects of the framing. The perturbative computations should be performed at framing −1 to compare with the prediction from the localization. In this work, our regularization scheme at framing −1 has taken into account of the effects on the spinors in the definition of the fermionic Wilson loops appropriately. Such improvement could be useful in other settings.
The remaining parts of this paper are organized as follows: in section 2 and 3, we briefly review the three-dimensional N = 2 fishnet theories and construct the BPS Wilson loop operators in them respectively. In section 4, we present our detailed perturbative computation on the VEVs of the BPS circular loops operators. In section 5, we discuss the line operator with a cusp. We conclude this paper with some discussions in section 6. We collect some technical details into the appendices.
2 Three-dimensional N = 2 fishnet theories Integrable field theories in higher than two dimensions are very rare. In three dimensional spacetime, the ABJM theory [20] was known to be integrable in the planar limit [29,30,31]. This theory admits various marginal deformations such as β-and γ-deformations [52], like the four dimensional N = 4 SYM theory [49,50]. The β-/γ-deformed or twisted theories have less/no supersymmetries while the integrable structures are preserved [51,53,54]. 1 Other studies on the spin chain from three dimensional Chern-Simons-matter theories with less or no supersymmetries include the ones in [55].
In [44], a double scaling limit of the β-and γ-deformed theories was proposed and the resulting theories were called as the fishnet theories due to their fishnet-like Feynman diagrams. One remarkable feature of the fishnet theories is that gauge fields and possible some matter fields in original theories are decoupled so that the resulted theories are substantially simplified. The fishnet structures of the Feynman diagrams indicate the integrability of these higher dimensional theories [46]. In this section we will review the construction of a three-dimensional fishnet theory with N = 2 supersymmetries.

ABJM theory and γ-deformation
We start with the ABJM theory whose Lagrangian is shown in the Appendix A. The We perform the γ-deformation through replacing the ususal product of any two fields A and B with the product defined as 1 In [53], a non-standard γ-deformation was considered. There it was mentioned that this deformation degenerates to β-deformation. This statement is incorrect since these two deformations lead to different actions. Nevertheless, the planar two-loop anomalous dimension matrices in the scalar sector are the same for these two deformations, and the Hamiltonians on the alternating spin chains are the same.
where the antisymmetric product of the two charge vectors q A and q B is given by The U (1) charges of the fields are given by the table below: where ± ≡ ± 1 2 (Note that the gauge fields A µ and B µ are neutral under these three U (1)'s.). This replacement should be performed on every product appearing in the Lagrangian. The γ-deformed ABJM action in our convention is presented in the Appendix A.

Double scaling limit and fishnet theory
Starting from the γ-deformed ABJM theory, there are several ways of taking double scaling limits [45] to get a new theory with only scalars and fermions. The one preserves supersymmetries is the following: By setting γ 1 = γ 2 = 0 without taking the double scaling limit, the resulting theory is the well known β-deformed theory with N = 2 supersymmetry [52]. After taking the double scaling limit we find that the N = 2 supersymmetry is preserved. Under this limit, the gauge sector is decoupled and the Lagrangian reads where the kinetic term is the scalar potential term is and the Yukawa-like terms involving the interactions among the scalars and the fermions are This fishnet theory is invariant under the N = 2 supersymmetries 2 whose transformation rules are summarized as follows, where The supersymmetry parameters are In the Euclidean space, the constraints on the supersymmetry parameters are relaxed to The Lagrangian of this fishnet theory has been derived in [45] as a particular example of the plethora of three dimensional fishnet theories. Even though this fishnet theory is not the simplest one which contains the least fields, the residue supersymmetries provide us another handle to study three dimensional integrable theory. In this note we focus on the supersymmetric (BPS) non-local operators such as the BPS line and loop operators in this fishnet theory. The above discussions can be also applied to the ABJ theory [57] in which the only difference from the ABJM theory is that the gauge group is now chosen to be U (N 1 ) × U (N 2 ). Here we focus on the case where N 1 and N 2 are of the same order. Then the γdeformation is performed as before. The fishnet limit preserving N = 2 supersymmetries now becomes, The above discussion on the ABJM theory can be carried through straightforwardly with √ ξ N in eqs. (8) and (10) being replaced by

BPS Line/Loop Operators
The three-dimensional N = 2 fishnet theory obtained from the ABJ(M) theory can be used as a prototype to study various problems in general super-Chern-Simons-matter theories. We start the investigation with the construction of the BPS line/loop operators in the fishnet theory. As in N ≥ 2 Chern-Simons-matter theories, the construction of the fermionic loop operators starts with a superconnection L which combines the bosonic and fermionic field contents together. The supersymmetric (BPS) condition is given by requiring that, under the preserved supercharges, the supersymmetry transformation of the superconnection can be written as for some G [24,58]. The BPS conditions are relaxed comparing with δL = 0 for the bosonic loop operators. In the following, we still use W to denote the loop operators and reserve L for the superconnection.

Line operators along timelike straight line
Here and in the following, we will use the spinor conventions in [40]. For the Minkowski spacetime with the coordinates x µ = (x 0 , x 1 , x 2 ) and the metric η µν = diag(−1, 1, 1), the γ matrices are chosen to be γ µ β α = (iσ 2 , σ 1 , σ 3 ). Following the discussions in [40], we can construct BPS line operator along the timelike straight line x µ (τ ) = (τ, 0, 0). The line operator, which is at least classically half-BPS, is where with As in [40], we define the following bosonic spinors and decompose a spinor as For the above line operators to be BPS, the following constraints should be satisfied. Then the corresponding preserved supercharges are θ 12 + , θ 34 − , ϑ 12 + , ϑ 34 − . There are two classes of solutions to the BPS constraints in eq. (21) Class II: leading to nontrivial BPS line operators.

Circular loop operators
To study the circular loop operator, we put the theory on the Euclidean space 3 with the coordinates x µ = (x 1 , x 2 , x 3 ), the metric δ µν = diag(1, 1, 1) and the γ matrices We search for the BPS loop operators along the circle x µ (τ ) = (cos τ, sin τ, 0).
The resulting operator, which is at least classically half-BPS , is, where with the same constants U I J ,ᾱ I , β I ,γ I , δ I and the BPS constraints as the ones in the previous subsection. Then the corresponding preserved supercharges are the ones satisfying Notice that now the definition of u ± is changed to And for the circular loop operator, we use the decomposition of the spinor θ α with these u ± .
Based on the discussions in [24,36,40], we can show that classically is Q-exact, where the supercharge Q can be used to perform supersymmetric localization and N 1 , N 2 are the ranks of the gauge groups. If this relation is preserved at quantum level, we will have exactly.

N = 2 notations
Later on we will compute the expectation values of the circular and cusped loop/line operators. For the perturbative computations, it is convenient to switch from the ABJM notation to the N = 2 notation [40], As for the N ab 's, we have In the N = 2 notations, the line operators are, Note that ζ i (ab) u − with u ± given in eq. (19). The circular loop operators now in the N = 2 notations read, Note that ζ i (ab) which lead to nontrivial line/loop operators.
We can easily switch between the ABJM notations and the N = 2 notations by just finding possibly nonzero components 4 ofm ab i , n i ab . For the line operators, they arē And for the circular loop operators, they arē It is then easy to see that the solutions to the BPS constraints in these two notations map to each other.

Perturbative computations
In this section we will evaluate the vacuum expectation value ( It is reasonable to modify the regularization scheme by demanding that the spinors corresponding to these points should also be changed to the solutions of the Killing spinor equations on these contours. This is the key of our proposal for a new regularization scheme for the computations of the fermionic BPS Wilson loops. In the following subsection, we will first give the details of our new regularization scheme and later we will use this scheme to compute the VEV of the circular loop operator in the three-dimensional N = 2 fishnet theory.
The first equation is a Killing spinor equation along the auxiliary contour. Demanding that when η → 0 these spinors go back to the spinors in eq. (27), we get that These spinors can be also obtained without directly solving the BPS conditions. Notice that the auxiliary contour x µ η (τ ) can be got from the Wilson loop contour x µ W L by an affine transformationx followed by a reparametrization of τ = τ (τ ) where is a rotation matrix, and From with u(τ ) ± being the ones given in eq. (27) and by keeping only the terms up to the linear order of η, we find as claimed in eq. (49). This procedure can be performed for a finite η with a finite φ as well, but we would not give the details here. Now, consider an unregularized integral from the computations of VEV of the loop operator, where x(τ m ) = (cos τ m , sin τ m , 0) and u(τ m ) being given in eq. (27), and in this section To regularize it, we replace x(τ m ), u(τ m ) with x m (τ m ) = (cos τ m +(m−1)δ cos 2 τ m , sin τ m +(m−1)δ sin τ m cos τ m , (m−1)δ sin τ m ), (60) and with δ the regularization parameter 5 . This is our proposal of the regularization scheme for the circular BPS loop operators at framing −1.

The vacuum expectation values of circular loop operators
In the perturbative computations, we assume 6mab i and n i (ab) to be of order O(ξ 1/4 / √ N 1 ) . We make this choice partly because this is similar for what happens for the half-BPS Wilson loops in the ABJM theory. Then up to two-loop order, there are no contribution from the diagrams with vertices. All the relevant diagrams have appeared in the previous computations [21,22,23,62,40], but here for the diagrams with fermionic propagators we would apply our new regularization scheme. The only contributing one-loop diagram with the integral being 7 There are numerical evidences that the results for the integral in the above equation is the same as the one from the dimensional regularization with dimensional reduction 5 it is just the η in the previous discussions. 6 Recall that we always assume that N 1 and N 2 are at the same order. 7 The factor |ẋ i (τ i )| is introduced to make the integral be reparametrization invariant of τ i . This could be understood as part of the regularization scheme at framing −1. They do not appear in the construction of the loop operators since we have chosen τ such that we always have |ẋ i | = 1 there. at framing 0 when > 1/2. We take the continuation of the later to = 0 as the results in our new supersymmetic regularization, then we are led to the claim that One part of the two-loop contributions comes from the diagrams with two scalar prop- agators ( fig. 2(a)), which give with It can be shown that it vanishes in any framing.
The other part coming from the diagrams with two fermion propagators ( fig. 2(b)) gives where It is easy to get that We expect them to be zero, but we have only weak numerical evidence for this. If it is the case, we have up to two-loop order.

Line operator with a cusp
The cusp anomalous dimension plays an important role in gauge theories. It is related to many important quantities like the ultraviolet divergences of the cusped Wilson loops, the infrared divergences of the gluon amplitudes and the anomalous dimension of the twist-two operators. The generalized Wilson loops in ABJ(M) have been constructed in [63] and their VEV's have been computed there up to two loops. 8 Here we construct the cusped line operators in three-dimensional N = 2 fishnet theory and compute their VEV's.

The construction
After doing Wick rotation to the Euclidean space, the BPS line in the N = 2 notations with 8 Studies on related Bremsstrahlung functions can be found in [64].
We consider a cusp which is parametrized by The constant spinors in the definition of the cusped line operator is given by applying the proper rotation to the constant spinors u ±α in eq. (19). The results are where we have defined s ± = exp(±iφ/2). It is easy to check that along both the left and the right half the following BPS conditions are satisfied Finally, the cusp operators are defined by taking the trace 9 of eq. (75) with the superconnection in eq. (76), the contour in eq. (77) and the spinors in eq. (78). We also need to keep in mind the BPS constraints in eq. (39) whose solutionsm 21 i = n i 12 = 0 and m 12 i = n i 21 = 0 lead to nontrivial cusp operators. Since the left and right part of the cusp operators preserves different supersymmetry the whole cusp configuraton is not BPS. One may try to search for the BPS generalized cusp as in [63] in the fishnet theory. But we find that a direct generalization of the construction introduced in [63] does not lead to the BPS configuration. This is reasonable considering the fact that the fishnet theory has less supersymmetries than the ABJM theory.

Perturbative calculations
In this subsection, we compute the VEV's of the cusp operators constructed in the previous subsection. For simplicity, we do the computations in the framing 0 using the dimensional regularization with dimensional reduction and take the internal datam ab i , n i ab 9 In this paper, we only consider taking trace in the fundamental representation of U (N 1 |N 2 ).
being the same along the cusp. As in the circular loop case, we assumem ab i and n i ab to be of order O(ξ 1/4 / √ N ). Almost all of the needed integrals have already been computed in [63], so we skip the details of the calculations for these integrals. Notices that we should keep the finite piece in the one-loop result. The one-loop contributions come from two kinds of diagrams with a single fermion propagator ending on the same line ( fig. 3(a)) or the different lines ( fig. 3(b)). The sum of their contributions is with and Here we have used the identities The two-loop contributions are from the diagrams with two scalar propagators ( fig. 4(a  with and Here we have defined a shorthand variable a ≡ cos(2φ). For details we evaluate the last integral in the following way, with where we have defined s(x, y) = (∂ x − ∂ y )(x 2 + y 2 − 2xy cos(φ)) −1/2 .
Summing up all the contributions, we end up with the result up to two-loop level as 10 Notice that the contributions from exchanging two scalars are cancelled by a part of the contributions from exchanging two fermions. As the case of the ABJM theory, we can get the divergent part of V N as Notice that there are 1/ 2 terms in (µL) 4 even in the straight line limit. This is different from the ABJM case and is related to the fact that no diagrams with vertices appears at two loops in the fishnet theory. It indicates that there may be a better way to extract V n from W . 11 We leave this problem for further studies.
As discussed in [63], the suitable renormalization condition is that when φ = 0 the renormalized V should vanish 12 . From this condition, we get the renormalized generalized potential The so-called universal cusp anomalous dimension, γ cusp , is obtained from the large imaginary φ limit [66]: If we still use prescription despite of the existence of 1/ 2 terms, we get γ cusp = 0 at two-loop order in our fishnet theory.

Conclusion and Discussions
In this work, we studied the loop operators in the three-dimensional N = 2 fishnet theories. This kind of theories was obtained by taking appropriate double scaling limit of the γ-deformed ABJM theory. The action of this 3D N = 2 fishnet theory includes the kinetic terms and potential terms of the scalars and fermions, with the gauge sector got decoupled. It is not the simplest fishnet theory, but the presence of supersymmetries makes it attractive in studying the BPS loop operators in a setup simpler than the Chern-Simons-matter theories. We constructed the BPS line and loop operators, and 11 It was found in [65] that the prescription in [63] fails starting at three-loop order. An alternative prescription which works better at higher-loop order in the ladder limit was provided in [65]. This prescription is identical to the one in [63] up to two-loop order. We would like to thank Michelangelo Preti for discussions on this point. 12 Recall that we take the samem ab i 's and n i ab for the left and right part of the cusp.
computed the VEV of circular BPS operators perturbatively. We got some numerical evidence to suggest that the one-loop and two-loop contributions are vanishing. This is in consistency with the prediction from the localization, under the assumption that our loop operators are truly BPS at the quantum level.
During Furthermore, we discussed the line operators with a cusp and studied their universal cusp anomalous dimensions. We found that the two-loop generalized potential defined as in [63] has a double pole in the expansion of 1/ . It will be interesting to see if there are other prescription for the generalized potential which leads to the results with only a simple pole. Put this issue aside, we found that the generalized cusp anomalous dimension vanishes.
As we mentioned in the introduction, there has been little progress in the study of open chain from the (cusped) Wilson lines in the ABJM theory. It would be valuable to study this hard problem in the simple setup here. That means to insert composite operator in the (cusped) line operators in the 3D N = 2 fishnet theory and then investigate whether the open spin chain obtained here is integrable or not. We leave this problem for future study.
A The Lagrangian of ABJM theory and its γ-deformation In our convention the Lagrangian of ABJM theory with gauge group U (N ) × U (N ) reads L k = Tr(−D µφ I D µ φ I + iψ I γ µ D µ ψ I ), Here After the γ-deformation, some of the interaction terms will acquire non-trivial phases.

B The propagators of the scalars and fermions
In the Minkowski spacetime, we have the following kinematic terms for the scalars and the spinors in the N = 2 notation Tr(−∂ µZ where the dimensional regularization with d = 3 − 2 has been used.