Supersymmetric Wilson loops in N=4 super Chern-Simons-matter theory

We investigate the supersymmetric Wilson loops in $d=3$ $\mathcal{N}=4$ super Chern-Simons-matter theory obtained from non-chiral orbifold of ABJM theory. We work in both Minkowski spacetime and Euclidean space, and we construct 1/4 and 1/2 BPS Wilson loops. We also provide a complete proof that the difference between 1/4 and 1/2 Wilson loops is $Q$-exact with $Q$ being some supercharge that is preserved by both the 1/4 and 1/2 Wilson loops. This plays an important role in applying the localization techniques to compute the vacuum expectation values of Wilson loops. We also study the M-theory dual of the 1/2 BPS circular Wilson loop.


Introduction
After the discovery of AdS 5 /CF T 4 correspondence [1][2][3], people have also been interested in the AdS 4 /CF T 3 correspondence.A superconformal field theory (SCFT) that is dual to M-theory on AdS 4 ×S 7 spacetime is needed, or equivalently an SCFT that describes coinciding M2-branes is needed.
The theory was finally constructed in [4] and is known as Aharony-Bergman-Jafferis-Maldacena (ABJM) theory.The ABJM theory is an N = 6 super Chern-Simons-matter (SCSM) theory with gauge group U (N ) × U (N ) and levels (k, −k), and it is dual to M-theory in AdS 4 × S 7 /Z k spacetime, or type IIA superstring theory in AdS 4 × CP3 spacetime.When k = 1, 2 the supersymmetries are enhanced non-perturbatively to N = 8.
Like the study of Bogomol'nyi-Prasad-Sommerfield (BPS) Wilson loops in d = 4 N = 4 super Yang-Mills theory in AdS 5 /CF T 4 correspondence [5][6][7][8][9], there are also many studies of Wilson loops in ABJM theory in the AdS 4 /CF T 3 correspondence.The simplest BPS Wilson loop in ABJM theory is the 1/6 BPS one that was constructed in [10][11][12], but the simplest fundamental string solution dual to a Wilson loop in type IIA superstring theory in AdS 4 × CP 3 spacetime is 1/2 BPS.With more efforts the 1/2 BPS Wilson loop was constructed in [13]. 1 When M theory or IIA superstring theory is weakly coupled, ABJM theory is strongly coupled.Then in order to compare with available results in gravity side, one needs to compute the vacuum expectation values (VEVs) of Wilson loops at strong coupling which is usually a hard task.However, using localization techniques [19] 2 one can compute these VEVs in ABJM theory at both weak and strong couplings [19,22,23].In applying the localization techniques to 1/2 BPS Wilson loops one needs to check that 1/2 and 1/6 BPS Wilson loops difference by a Q-exact term with Q being some supercharge that is preserved by both the 1/2 and 1/6 BPS Wilson loops and being used in localization [13].The result of localization techniques for the 1/2 BPS Wilson loop has also been checked up to two loops in [24][25][26].
We will call these Wilson loops Gaiotto-Yin (GY) type.When 2 ≤ N ≤ 6, the GY type Wilson loop usually preserves four real supercharges, including the Poncaré supercharges and superconformal ones, if the loop is along a straight line or a circle 3 .The construction of 1/2 BPS Wilson loops in [13] includes essentially the fermionic matter fields and the supergroup structure appearing in the ABJM theory.This type of Wilson loops will be called Drukker-Trancanelli (DT) type.It is an interesting question whether DT type BPS Wilson loops exist in theories with fewer supersymmetries.In N = 5 theories [29,30], such Wilson loops do exist and they are 2/5 BPS [31].The situation in N = 3 theories is interesting but not completely clear.Based on studies of the dual M2-brane solutions, strong evidence was given in [32] to support the conjecture that there are no Wilson loops in N = 3 theories that preserve more that four supercharges.If this conjecture is right, there are two possibilities about the fate of the DT type BPS Wilson loops: one is that there are no DT type BPS Wilson loops in such theories, and the other is that DT type BPS Wilson loops exist in some N = 3 theories and they are at most 1/3 BPS.This direction is interesting to be further studied.
Putting the situations in N = 3 theories aside, we would like now to study the construction of DT type BPS Wilson loops in N = 4 Chern-Simons-matter theories.Among these theories [33,34], we will focus on the theories that are obtained from the non-chiral orbifold of ABJM theory.From ABJM theory with gauge group U (nN ) × U (nN ), one can perform a Z n orbifolding and get an N = 4 SCSM theory [35], and it is dual to M-theory in AdS 4 × S 7 /(Z n × Z nk ) background [35][36][37].The Wilson loops in this N = 4 theory in fundamental representation are dual to M2-branes in AdS 4 × S 7 /(Z n × Z nk ) spacetime.We consider the simplest embedding of such a membrane.The topology of the membrane worldvolume is AdS 2 × S 1 , where AdS 2 ⊂ AdS 4 and S 1 is along the M-theory circle direction in S 7 /(Z n × Z nk ).We find that there exists such an M2-brane that preserves half of the supersymmetries of the M-theory in AdS 4 × S 7 /(Z n × Z nk ) spacetime.This indicates that there should be half-BPS Wilson loops in such N = 4 SCSM theory.Based on experience in ABJM case, this Wilson loop may not be GY type, and may be DT type.One of the main results in this paper is the construction of such DT type half-BPS Wilson loops.We get the Poncaré and conformal supersymmetry (SUSY) transformation of the N = 4 SCSM theory from that of ABJM theory.As a warm-up, we firstly construct the 1/4 BPS GY type Wilson loops.Then we give the details of the construction of the 1/2 BPS DT type Wilson loops.In Minkowski spacetime, we have the 1/4 and 1/2 BPS Wilson loops along a timelike infinite straight line.In Euclidean space, we have the 1/4 and 1/2 BPS Wilson loops along a infinite straight line, as well as 1/4 and 1/2 BPS Wilson loops along a circle.We also provide a complete proof that 1/2 and 1/4 BPS Wilson loops difference by a Q-exact term, with Q being some supercharge that is preserved by both the 1/4 and 1/2 BPS Wilson loops.
The rest of the paper is arranged as follows.In Section 2, we study the simplest M2-brane solution dual to a circular Wilson loop in orbifold ABJM theory.We compute its on-shell action with boundary terms included and we also find that this M2-brane can be half BPS.In Section 3 we review the basics of this N = 4 SCSM theory and derive its SUSY transformation.In Section 4 we consider the 1/4 and 1/2 BPS Wilson loops along a timelike infinite straight line in Minkowski spacetime.In Section 5 we consider the 1/4 and 1/2 BPS Wilson loops along an infinite straight line in Euclidean space.In Section 6 we construct the circular 1/4 and 1/2 BPS Wilson loops.We conclude with conclusion and discussion in Section 7. We review the 1/6 and 1/2 BPS Wilson loops in ABJM theory in Appendix A.
In Appendix B we provide a simple proof of gauge covariance of Wilson loops, from which we also prove a useful statement that has appeared in [13].In Appendix C we explore alternative definitions of Wilson loops for a super connection, but we find no nontrivial ones.In Appendix D there are the calculation details of Subsection 6.3, and we give a complete proof that the difference between 1/2 and 1/4 BPS Wilson loops is Q-exact.

Note added.
After the paper appears in ArXiv, there appears another paper [28] that has some overlaps with ours.There are more general 1/2 BPS Wilson loops in the orbifold ABJM theory, as well as in other N = 4 SCSM theories.According to terminology of [28], the 1/2 BPS Wilson loops in this paper are ψ 1 -loops.For a timelike straight line x µ = τ δ µ 0 in Minkowski space if we change the first two equations of the ansatz (4.16) to η(2ℓ we would get the ψ 2 -loops.In this case we would have m = n = −1, as well as The ψ 1 -and ψ 2 -loops have the same conserved supersymmetries (4.19) and (4.20).There are similar stories for 1/2 BPS Wilson loops along straight lines and circles in Euclidean space.There are a large number of 1/2 BPS Wilson loops, but there are not so many dual 1/2 BPS objects in M-theory.And so it is expected in [28] that these Wilson loops are 1/2 BPS classically, and only some special linear combination of them is 1/2 BPS quantum mechanically.

M2-branes in
The N = 4 theory obtained from orbifolding ABJM theory is dual to M-theory in AdS 4 ×S 7 /(Z n ×Z nk ) spacetime.We will denote the action of Γ n,k on S 7 is generated by [36] ( and where ω m ≡ exp 2πi m .We parameterize z i as ) and (2.6) The IIA limit of M-theory is obtained by taking k → ∞ while keeping n fixed.In this limit the circle along ζ direction will shrink.So this circle is the M-theory circle, and its circumference is 8π nk .The metric of unit S 7 is (2.7) The metric of AdS 4 × S 7 /Γ n,k is For Lorentzian signature we choose the following global coordinate on AdS 4 , (2.9) The four-form field strength on this background is Flux quantization gives where ℓ p is the eleven-dimensional Planck length, and we have used 12) The radius of the ζ circle in Planck unit is of order R/(nkℓ p ) ∝ (n 2 N k) 1/6 /(nk), and so the M-theory description is a good one when N ≫ n 4 k 5 .
We consider the probe M2-brane solution in this background.In Lorentzian signature the bosonic part of the M2-brane action is Here g mn is the induced metric of the membrane worldvolume, T M 2 is the tension of the M2-brane and P [C 3 ] is the pullback of the bulk 3-form gauge potential to the worldvolume of the membrane.
The gauge choice for the background 3-form gauge potential C 3 is From the action, one can obtain the membrane equation of motion as4 Note that ǫ mnp is a tensor density on the world-volume of the membrane.
Since we want to find the simplest membrane embedding corresponding to a Wilson loop in the dual field theory, we take the topology of the membrane worldvolume to be AdS 2 × S 1 .The AdS 2 is embedded in AdS 4 , while S 1 is along the M-theory circle.So we consider the ansatz where σ µ , µ = 0, 1, 2 is the coordinates on the worldvolume of M2-brane.One can find that the equations of motion only lead to the constraint that u = 0. Then the induced metric of M2-brane is To compute the on-shell action of the M2-brane whose boundary at infinity is S 1 , we work in Euclidean signature and choose the AdS 4 coordinates where ψ ∈ [0, 2π].In Euclidean signature the M2-brane action becomes5 For the M2-brane that is put at the on-shell action is After adding boundary terms to regulate the action as in [8], we get where ǫ 0 is a constant eleven-dimensional Majorana spinor which has 32 real degrees of freedom.The definitions of ξ i with i = 1, 2, 3, 4 are , γ ♯ are eleven-dimensional gamma matrices, and γ ≡ γ 0123 .Note that the elevendimensional gamma matrices are chosen such that To obtained the Killing spinor of M-theory on AdS 4 × S 7 /Γ n,k , we need to impose the conditions where K 1,2 are the following two Killing vectors and they are related to the generators of Γ n,k .The definition of L K ǫ is After some computations, we find that the two conditions (2.29) are equivalent to So the background is half BPS compared to the maximal possibility, i.e. there are 16 real supercharges.This is consistent with the fact that the dual three-dimensional SCFT is an N = 4 theory.
The supercharges preserved by the probe membrane are determined by the following equation with For the membrane we just found, we have So the supercharges preserved by this probe membrane correspond to the solution of At the positions with α = θ 1 = 0, this is equivalent to [10] γ 01♯ ǫ 0 = ǫ 0 . (2.37) Since it is compatible with the projection condition in (2.32), we arrive at the conclusion that the probe M2-brane put at α = θ 1 = 0 is half BPS compared to the supersymmetries of M-theory in

N = 4 SCSM theory
Orbifolding the ABJM theory with gauge group U (nN )× U (nN ) and levels (k, −k) by Z n , one can get the N = 4 SCSM theory with gauge group U (N ) 2n and Chern-Simons levels We can get the SUSY transformation of this N = 4 theory from that of ABJM theory (A.1) by the orbifolding.The result is Here There are no summations of ℓ here, and would not be summations of ℓ later unless it is given out explicitly.Indices i, j, R-symmetry.The definitions of covariant derivatives are Also χ iî = θ iî + x µ γ µ ϑ iî and χiî = θiî − θiî x µ γ µ , and θ iî , θiî , ϑ iî , θiî are Dirac spinors with constraints Symbols ǫ ij and ǫ î are antisymmetric with ǫ 12 = ǫ 12 = 1.Note that for δ in (3.1) we have with P iî , Piî and S iî , S iî being Poncaré and conformal supercharges that satisfy In Euclidean space, the SUSY transformation is formally identical to (3.1), with Note the eight spinors θ iî , ϑ iî with i = 1, 2, î = 1, 2 are independent Dirac spinors.Now equations

Straight line in Minkowski spacetime
In Minkowski spacetime there are BPS Wilson loops along null and timelike infinite straight lines [38].
It is easy to construct a null Wilson loop, and it is 1/2 BPS.We only consider the timelike BPS Wilson loops here.

1/4 BPS Wilson loop
We consider the Wilson loop along a timelike straight line x µ = τ δ µ 0 as For Poncaré SUSY transformation we can get We work in the basis of diagonal M i j = m i δ i j and M î = m îδ î , and then we get Supposing θ 1 1 = 0, we choose without loss of generality Using (3.3) we know (θ 1 1) * = θ 2 2, and then we get We can check that the equations (4.3) are consistent.It is similar for conformal SUSY transformation.
Thus we get a 1/4 BPS Wilson loop.
Similarly we can construct the 1/4 BPS Wilson loop along x µ = τ δ µ 0 that preserves the same supersymmetries Also we can combine (4.1) and (4.7) and get the 1/4 BPS Wilson loop A = diag(A (1) , Note that we can also construct the 1/4 BPS Wilson loop in this subsection using the consideration in [27] for general N = 2 theories.So this Wilson loop is GY type.

1/2 BPS Wilson loop
We consider the timelike Wilson loop along where L 1/2 is a supermatrix Here we have definitions ), as well as Note that A and Â are Grassmann even.Also η(2ℓ+1) i , η î (2ℓ+1) and η i (2ℓ) are Grassmann even, and so F1 and F 2 are Grassmann odd.To make W 1/2 SUSY invariant, we need [31] for some Grassmann odd supermatrix Concretely, we need As in [13], we can use symmetry to guide the search for a 1/2 BPS Wilson loop.We break the SU (2) × SU (2) R-symmetry to U (1) × SU (2) by writing (i, î) = (1, 2, î).We wish to get a BPS Wilson loop with the SU (2) subgroup intact, and so we choose ) Then we need ), 2 , • • • , g ). (4.17) And then equations (4.15) become Without loss of generality, we suppose that and then from (3.3) we have For ψ 2 , ψ î and ψ2 , ψî not appearing in δA (2ℓ+1) and δ Â(2ℓ) , we have to choose m 1 = n 1 = −1 and , and then we get For δ f (2ℓ) and δf satisfying the form of (4.18), we must choose Then we get One can show that, given6 equations (4.18) are satisfied.It is similar for conformal SUSY transformation.Thus we get the 1/2 BPS Wilson loop along a timelike infinite straight line.

Relation between 1/4 and 1/2 BPS Wilson loops
We check for some supercharge Q preserved by both W 1/4 and W 1/2 and some operator V .This is similar to the ABJM case in [13].In the N = 4 SCSM theory we have the 1/4 and 1/2 BPS Wilson loops (4.8) and (4.9) In this subsection it is convenient to rearrange the rows and columns and rewrite Note that for L 1/4 there are , φî (2ℓ) , but also φ . In some sense, this construction is a kind of hybrid of 1/2 BPS Wilson loop in ABJM theory and 1/4 BPS Wilson loop in this N = 4 theory.
We also define (4.28) Explicitly there are .
Then we get We define and a Grassmann odd operator with P 1 1 and P1 1 being Poncaré charges in (3.4).It can be checked that Now we have the SUSY transformation with have been derived in (4.23).And then we have Note that from (4.24) we have η(2ℓ) η (2ℓ) = −2i with no summation of ℓ.
We define We can show that For the infinite straight line, we have s → ∞ and t → −∞, and we also assume Λ(±∞) = 0. Then we get Operator S 1 or S 4 is just the V we are looking for.

Straight line in Euclidean space
There are BPS Wilson loops along spacelike infinite straight lines in Euclidean space.Since they are similar to BPS Wilson loops along timelike infinite straight lines in Minkowski spacetime, and so it will be brief in this section.

1/4 BPS Wilson loop
We use coordinates x µ = (x 1 , x 2 , x 3 ) in Euclidean space.We have the 1/4 BPS Wilson loop along the infinite straight line x µ = τ δ µ 1 the same as (4.8) except that The preserved Poncaré and conformal supersymmetries are

Relation between 1/4 and 1/2 BPS Wilson loops
The check of W 1/2 − W 1/4 = QV for a straight line in Euclidean space is similar to the case of a timelike straight line in Minkowski spacetime.The only differences are that (5.5)

Circle in Euclidean space
The N = 4 SCSM theory is a superconformal theory, and a conformal transformation can change an infinite straight line to a circle.So there would be BPS circular Wilson loops if there exist BPS Wilson loops along infinite straight lines.

Relation between 1/4 and 1/2 BPS Wilson loops
The check of W 1/2 − W 1/4 = QV for a circle in Euclidean space is different to the case of a straight line in Minkowski spacetime.Firstly we have the differences Then, the construction of V is also different, since we have to treat the boundary terms carefully.The calculation is very involved, and so we collect them in Appendix D.

Conclusion and discussion
In this paper, we have investigated the supersymmetric Wilson loops in N = 4 SCSM theory.In Minkowski spacetime we have 1/2 BPS Wilson loops along null infinite straight lines, and 1/4 and 1/2 BPS Wilson loops along timelike infinite straight lines.In Euclidean space we have 1/4 and 1/2 Wilson loops along infinite straight lines, as well as circular 1/4 and 1/2 Wilson loops.We also gave a complete proof that the difference between 1/4 and 1/2 Wilson loops is Q-exact with Q being some supercharge that is preserved by both the 1/4 and 1/2 Wilson loops.On the gravity side, we also studied the probe M2-branes dual to half BPS circular Wilson loops in the fundamental representation, and give the holographic prediction of the VEV of this 1/2 BPS Wilson loops in the M-theory limit.
The VEV of the half-BPS circular Wilson loop in fundamental representation can be calculated using localization in the M-theory limit (N → ∞ with k and n being fixed) based on results in [39].It will be also interesting to compute the vacuum expectation values of these BPS Wilson loops beyond the M-theory limit.We can use the fermi gas approach [40,41] to include all of the 1/N corrections.
Similar to the ABJM case [42], it is interesting to study these Wilson loops in arbitrary representations.
These results are in accordance with the gravity ones, and they will be presented in [43].
We think that the construction of DT type BPS Wilson loops here could be easily generalized to similar ones in N = 4 theories obtained from orbifolding ABJ theory or N = 5 theories in [29,30].

A Review of Wilson loops in ABJM theory
The ABJM theory is an N = 6 SCSM theory, and it was constructed in [4].ABJM theory has gauge group U (N )×U (N ) and Chern-Simons levels (k, −k), and the gauge fields are A µ and Âµ respectively.
In Minkowski spacetime, one has the 1/6 BPS Wilson loop along the timelike infinite straight line Here W 1/6 and Ŵ1/6 can be combined to give the 1/6 BPS Wilson loop The preserved Poncaré and conformal supersymmetries are The preserved Poncaré and conformal supersymmetries are with i, j = 2, 3, 4. One can use localization techniques to calculate the vacuum expectation value of the 1/6 BPS Wilson loop [19], and in order to generalize this to the 1/2 BPS Wilson loops [22,23] one needs the relation between 1/6 and 1/2 BPS Wilson loops [13] W 1/2 − W 1/6 = QV, (A.10) for some supercharge Q preserved by both W 1/6 and W 1/2 and some operator V .Operator V has been given in [13], and W 1/2 − W 1/6 = QV has been checked for the first several orders. 8Note that there is no spacelike BPS Wilson loop in Minkowski spacetime [38].One has 1/2 BPS Wilson loops along null infinite straight lines.
In Euclidean space, we use coordinates x µ = (x 1 , x 2 , x 3 ).One has the 1/6 BPS Wilson loop along the infinite straight line x µ = τ δ µ 1 the same as (A.6) except that The preserved Poncaré and conformal supersymmetries are θ 13 = θ 14 = θ 23 = θ 24 = 0, (A.12) Also one has the 1/2 BPS Wilson loop along the infinite straight line x µ = τ δ µ 1 the same as (A.8) except that The preserved supersymmetries are with i, j = 2, 3, 4. The check of W 1/2 − W 1/6 = QV for a straight line in Euclidean space is similar to the previous case.

B A simple proof of gauge covariance of Wilson lines
We have a general line in spacetime parameterized by τ ∈ [t, s].For gauge field A(τ ) we define the Wilson line with P being path-ordering.For a general infinitesimal gauge transformation the Wilson line transforms as This gauge covariance of Wilson lines is well-known, one can see a complete proof in, for example, the textbook [54].Here we give a simple proof using induction.
For n ≥ 0 we define the symbols with the shorthand Note that we have X 0 = 1 and With the recursive relations for n ≥ 1 we can use induction to prove This leads to the infinitesimal version of the gauge covariance of the Wilson line (B.3).
We rewrite (B.3) as Then it follows that more generally for [t ′ , s ′ ] ⊂ [t, s] we can easily get This is just the statement in [13] that one can integrate out the covariant derivative D = d + iA term in the presence of path ordered exp(−i A).

C Alternative Wilson loops for a super connection
In this appendix we explore alternative definitions of Wilson loops for a super connection.The result is that we find no nontrivial ones.
A super connection L can be written as L = B + F with Grassmann even part B being block diagonal and Grassmann odd part F being block off-diagonal When defining the path-ordering for the Grassmann odd part F of the supermatrix L, we have ambiguities.We can do it as an ordinary matrix or we can define the super path-ordering as Note that when acting on two Grassmann even matrices, or one even matrix and one odd matrix, SP is no different with P.Only when acting on two odd matrices, SP is different from P as shown above.
We can define the Wilson loop along a super connection L as that of an ordinary connection W (s, t) = P exp −i and so it is not covariant under δ ′ L. with where S n denotes order n permutation group.It is easy to see that T n = 0 for n ≥ 2. And then we get SW (s, t) = P exp −i • Quantum mechanically, the Grassmann odd part will not contribute to the vacuum expectation value, • Furthermore, it is not covariant under the transformation (C.5) or (C.8).
D A complete proof of 1/2 and 1/4 BPS Wilson loops differencing by a Q-exact term The result here is general and applies not only to the N = 4 SCSM case, but also to the ABJM case.
A complete proof of 1/2 and 1/6 BPS Wilson loops difference in ABJM theory being Q-exact is the same as what is presented here.

D.1 Some simplifications
First of all, let us repeat the problem that we are going to tackle and make some simplifications.We have a circle parameterized by τ ∈ [0, 1] with x µ (1) = x µ (0). 9 We have the 1/4 and 1/2 BPS Wilson loops with Here L 1/2 is a supermatrix, with L 1/4 + LB being its Grassmann even block diagonal part and LF being its Grassmann odd block off-diagonal part.As shown in (6.8), we can find a Grassmann odd operator Q and a Grassmann even block off-diagonal matrix Λ satisfying Here the factor α has been redefined as κ.Note that [ LB , Λ] = 0 has been used.We want to find some operators V and U that satisfy We call this task I.
To avoid cluster of factors and indices, we make the following redefinitions Then we have with We have the relations QΛ = F, κΛ 2 = B, κ(1) = κ(0), Λ(1) = −Λ(0), (D.9) Our task is still to find some operators V and U that satisfy Our task is to find some operators V and U that satisfy We call it task III.It is a special case of task II, and so is easier.

D.2 Some definitions
Before tacking task III we make some formal definitions.We have a circle parameterized by τ ∈ [0, 1] with x µ (1) = x µ (0), and we will also use s, t, τ 1 , τ 2 , • • • to denote the parameter of the circle.We define two kinds of quantities on the circle.
• We call the first kind type 1, and a type 1 quantity has only one argument.We can prove and so we can write directly For the shorthand (D.17) we have useful relations

D.3 The main part
We tackle task III in this subsection and Subsections D.4 and D.5.As done in [13], we expand (D.11) in powers of B and F .We set F to be of order one, and B to be of order two.And then we have with the first few orders being It can be seen that our definitions simplify these formulas significantly.There is recursive relation for Note that we can define W −1 = 0 to make the above equations apply to n ≥ 1.
We also define And then task III (D.13) becomes to find V n and U n for n ≥ 1 to satisfy For m ≥ −1 and n ≥ −1 we define Note that we have Λ mn = 0 for m = −1 or n = −1, and it is similar to B mn and F mn .Because of (D.26), for m ≥ 1 we have and for n ≥ 1 we have And we can show Then we claim that the V n and U n with n ≥ 1 we want in (D.28) are that for k ≥ 0 We will prove this claim using two different methods in Subsections D. 4  Note that V 1,2,3 are just the ones given in [13].Here we have gone further, and give a general expression of V n and U n for all integers n ≥ 1. Besides, we will give a general proof of (D.28).There are two methods of doing so, and then we split the main part to two branches.

D.4 The first branch
In the first branch we use (D.36) and get for k ≥ 0 This is indeed just (D.28).However, we note that neither V nor U can be written directly as combinations of S 5,6,7,8 .

D.6 Back to the main part
Now we turn back to the main part, with task III being completed.From task III to task II we need to use (B.9) and just make some simple changes in the proof of task III.Thus task II is completed following task III and (B.9).Since task I is equivalent to task II, task I is competed too.
In summary we have given a complete proof that the difference of circular 1/2 and 1/4 BPS Wilson loops in this N = 4 SCSM theory is Q-exact, with Q being some supercharge that is preserved by the both the 1/2 and 1/4 BPS Wilson loops.As we have stated, this proof also applies to the 1/2 and 1/6 BPS Wilson loops in ABJM theory.

•
The second kind is type 2, and a type 2 quantity has two arguments.We denote them generally by lowercase Greek letters α(s, t), β(s, t), γ(s, t), • • • , or simply α, β, γ, • • • .Note that s ≥ t is required.Type 2 quantities below will include W , W n , V , V n , U , U n , Λ mn , B mn , F mn , S 5,6,7,8 , et al.We also define the identity type 2 quantity I(s, t) = 1.For one type 1 quantity a and one type 2 quantity α we define type 2 quantities Note that α * a = a * α.For two type 2 quantities α, β we do NOT define α * β, and so it is illegal.