$\mathcal{N}=3$ Harmonic Super-Wilson Loop

We study supersymmetric Wilson loops in $d=3$, $\mathcal{N}=3$ harmonic superspace, leading to a construction of a supersymmetrized generalization of the $\frac{1}{3}$-BPS Wilson loop for $\mathcal{N}=3$ gauge theories. This also includes a generalization of the $\frac{1}{6}$-BPS loop for ABJM theory. We perform a 'one-loop' computation of the vacuum expectation value of this operator directly in superspace and compare with the known $\mathcal{N}=2$ localization results at large $N$. This comparison also lets us identify certain fermionic contributions that do not receive any subleading corrections.


Introduction
The power of supersymmetry to simplify computations and gain insights cannot be overstated. It sheds light on hidden structures and illuminates relationships among seemingly different objects. A perfect example of this power is given by the Wilson loops/Scattering amplitudes duality in d = 4, N = 4 super-Yang Mills (SYM) theory. Even though evidence for such a duality existed [1][2][3][4][5][6], only after the construction of a supersymmetrized Wilson loop (WL) in superspace [7,8] has the duality been confirmed for all helicity sectors of the amplitude. In three-dimensions, while similar evidence in the case of four-point amplitude/four-gon Wilson loop for N = 6 ABJM theory [9] exists [10,11], extending beyond four-points immediately forces us into the remaining sectors (in terms of R-symmetry instead of helicity) of the theory. This motivates us to construct supersymmetric Wilson loops in superspace.
After the introduction of ABJM theory, various Wilson loop operators with different amounts of preserved supersymmetry were studied extensively. Earlier efforts dealt with construction and perturbative computations of 1 6 -BPS WL [12][13][14][15]. Localization was applied to evaluate the vacuum expectation value (vev) of this WL in [16] and the results were found to match the perturbative calculations at large N limit. 1 2 -BPS operators were constructed later in [17] and more calculations followed in [18,19] where even finite N contributions were computed. Being 'cohomologically equivalent' to the 1 6 -BPS operator, the localization results do not differ for these two operators. In [20], a classification was given for Wilson loops preserving various amounts of supersymmetry in N = 2, · · · , 6 Chern-Simons (CS) matter theories. New Wilson loops in N = 4 theories have been constructed recently in [21].
In this ever-expanding literature of construction, classification and computation involving Wilson loops, we present here a supersymmetrization of the simplest WL operator in threedimensional CS matter theories including ABJ(M) theories. Such an attempt has been made in [22] for ABJM theory in the framework of 'ordinary' N = 6 superspace. It was also pointed out that there are at least three reasons why such a WL cannot be dual to the scattering amplitudes of ABJM theory. The main issue is the non-chiral nature of the superspace that leads to torsion, which does not allow a straightforward identification of the kinematics on the two sides of the duality [23]. So we content ourselves with the 'well-studied' framework of N = 3 harmonic superspace [24,25] to construct the supersymmetrized Wilson loop 1 . This is to have as much manifest (off-shell) supersymmetry as possible along with a notion of chirality (or 'harmonic analyticity') built-in.
In the next section we consider a warm-up exercise of constructing a supersymmetrized WL in N = 2 superspace and a sample localization computation. Then we review the d = 3, N = 3 harmonic superspace in Section 3 before constructing the supersymmetrized 1 3 -BPS WL in Section 4. This leads to a generalization for 1 6 -BPS WL in ABJ(M) theories. In Section 5, we compute perturbatively the 'one-loop' vev of this new WL operator directly in harmonic superspace. Finally, we compare the perturbative result with localization computation and comment on future outlook in Section 6.

Warm-up
We construct here a supersymmetrized Wilson loop operator in d = 3, N = 2 superspace with coordinates {x µ (x (αβ) ), θ α ,θ α }, where the vector index µ = 0, 1, 2 and spinorial index α = 1, 2 correspond to the SO(2, 1) ≃ SL(2, R) group 2 . Though it is rather straightforward, we think this analysis has not appeared in the literature in this form so we discuss it as a warm-up exercise leading to the less trivial N = 3 superspace in the next section.
The N = 2 supersymmetry algebra has the following set of gauge-covariant superspace derivatives: {D µ (D (αβ) ), D α ,D α }. These satisfy the following algebra: (2.1) The Jacobi identities give further relations among the field strengths W, W α and F µν . One such relation is D α W = −ιW α along with the chirality constraint D αWβ = 0 [27,28]. The supersymmetrization of the familiar 1 2 -BPS Wilson loop in chiral superspace then looks like . We can do the component analysis of the connections and field strengths, leading to the fields of N = 2 vector multiplet {a αβ , σ, λ α ,λ α , D} along with the field strength f αβ : Here | denotes that all θ's are set to vanish. Also relevant is D (αD 2 A β) | = f αβ . It is now trivial to verify that the θ-independent piece of the exponent in (2.2) reduces to the well-known bosonic expression: dτ (ιẋ µ A µ + |ẋ|σ).
It can be easily checked that the W(x, θ,θ) preserves 'some' supersymmetry: In arriving at the last step, we have used the algebra (2.1) to convert covariant derivatives acting on connections into the corresponding field strengths, and terms that look like field-dependent gauge transformations of the connections, i.e.ẋ A,αβ D αβ (ǫ γ A γ ), are dropped as W(x, θ,θ) is gauge invariant. The BPS condition for the purely bosonic WL requires x µ (τ ) to be an infinite line in Minkowski space or a great circle on S 3 and one can choose it to satisfy |ẋ| = 1 [16,20]. Since (2.4) for the supersymmetrized case results in a similar equation, we will also consider |ẋ A | = 1. This does not determine θ(τ ) completely but only up to a function of τ : θ(τ ) = f (τ )θ 0 ,θ(τ ) = f −1 (τ )θ 0 . 3 Hence, constant solutions for ǫ can still be found for 2 The vector x µ can be traded for a real second-rank symmetric tensor x αβ ≡ x µ (γ µ ) αβ with the help of d = 3 "gamma"-matrices. We do not need the explicit basis but the relation x αβ x αβ = −2x µ x µ ≡ −2|x| 2 will be quite useful to know. 3 It is most likely that one needs to consider superconformal transformations of the WL operator to fully these configurations, where the condition ǫ α ẋ A αβ + |ẋ A |ǫ αβ = 0 projects half of the degrees of freedom, thus preserving two real degrees of freedom, i.e. 1 2 -BPS. Given the Lagrangian and propagators of [27,28], one should be able to compute the vev of the WL (2.2) perturbatively in superspace as well as in components at different θ-orders. However, we will skip this analysis here and comment on the non-perturbative analysis instead. Using the localization results of [16] where a N = 2 theory on S 3 (of radius r) is considered, we can obtain an 'exact' result for the vev of the supersymmetrized Wilson loop. Since the path integral is localized on the vector multiplet's scalar field σ = constant and D = − σ r , we have, Even though we do not know f (τ ) explicitly, we can evaluate W(x, θ,θ) formally. Let us denote everything in the exponent by Θσ, with Θ = 1 2π dτ 1 +˙ι 2ẋ αβ A θ αθβ +θ ·θ − θ ·θ +θ · θθ 2 (also set r = 1). The path integral reduces to a matrix model in terms of eigenvalues λ i of σ (we choose ABJM for concreteness, which has two U(N)'s as gauge groups and ±k as the two Chern-Simons levels): We refer the reader to [16] for the definitions of various factors in the above result as we are interested in its perturbative limit only. To obtain a perturbative α expansion, we can expand W in λ and compute the vev using the orthogonal polynomials method note that λ 2k = O(α k ) : (2.7) Rewriting Θ = 1 + 1 2 ϑ, we get (note ϑ 3 = 0) In the above expression, we have removed the bosonic term at O(α) by multiplying the result by an overall phase e − 1 2 α , which is necessary in matching the perturbative computation [16]. Note that we do not remove the whole ϑ-dependent term at O(α), since as we will see later there are indeed fermionic contributions at O(α) in perturbative computation. We will return back to this result in Section 6.

Review of N = Harmonic Superspace
Now, we turn to N = 3 supersymmetry. We collect here the necessary ingredients from threedimensional N = 3 harmonic superspace literature along with a few explicitly worked out details that will be relevant for us in later sections.
determine the θ(τ ) profile consistent with the circular bosonic WL. We do not pursue this exercise here. Thus, we will not evaluate the τ -integrals explicitly and leave all the τ -dependence of the coordinates intact.

N = 3 Harmonic Superspace
The 'ordinary' d = 3, N = 3 superspace with coordinates {x αβ , θ α ij } has the following algebra of superspace derivatives: To obtain constrained superfields in the form of D ij α Φ = 0, it is useful to consider the case where D ij α is given by a simple partial derivative, indicating the independence of Φ on certain variables. The obstacle to having a representation of D ij α as a partial derivative is its anticommutator algebra. This can be overcome by the introduction of SU(2)/U(1) harmonics u ± i . These bosonic variables satisfy where the raising and lowering of the SU(2) index i is done by contracting with the invariant tensor ǫ ij . (The contracted i among u's will be suppressed most of the time.) These new variables are to be integrated away using the following rules: In other words, only the SU(2) invariant polynomial with vanishing U(1) charge survives the integration. The harmonic variables allow us to linearly recombine the 3 × 2 fermionic coordi- The upshot is that doing the same for the covariant derivatives, the supersymmetry algebra now reads, where one finds that we can SU(2) covariantly isolate a doublet of commuting fermionic derivatives, for example D ++ α . This implies that we can have a representation for the covariant derivatives where D ++ α is a simple partial derivative. This is referred to as the "analytic basis", and it is given as the following: In the analytic basis, we obtain constrained superfields by imposing the 'analytic' constraint D ++ α Φ = 0, which now implies that Φ does not depend on θ −− α : The introduction of harmonic variables also introduces R-symmetry covariant derivatives, and are given by 4 : These have non-trivial commutator algebra with the fermionic derivatives: 5

Chern-Simons Matter Theories
To study gauge theories, we gauge-covariantize the full superspace derivatives D → D = D +A, which define the relevant field strengths: The covariant derivatives, and the field-strengths, transforms as D → e τ De −τ . Choosing a suitable 'gauge-frame' (from τ → λ) such that A ++ α = 0, allows us to define analytic super fields covariantly while maintaining its implication of independence on Note that choosing such a gauge generates (new) harmonic connections A ±± , from which all other connections can be obtained through Bianchi identities. In particular, A ++ turns out to be the unique analytic (D ++ α A ++ = 0) prepotential in this formalism. The prepotential transforms under a gauge variation as usual: where λ is an analytic gauge parameter. A convenient gauge is the Wess-Zumino gauge in which the prepotential has the following component expansion [25]: This is clearly the N = 3 vector multiplet with fields a µ , λ α , χ It is now possible to write every other connection and field strength in terms of the analytic prepotential A ++ . We start with the connections: (3.13) 4 D 0 is strictly speaking not a covariant derivative on SU (2)/U (1). It should be treated as the subgroup generator that defines the U (1) charge for a given operator or field, as in D 0 Φ (q) = qΦ (q) . 5 For completeness, their explicit forms in the analytic basis is given as Then the covariant field strengths can be derived from the connections as follows: (3.14) The N = 3 matter multiplet consists of two complex scalars f i transforming as a doublet under SU(2) and their fermionic partners ψ i α , which are encoded in the following hypermultiplet superfield: i , and similarly for the fermions. For the ABJM theory, we have two sets of q +a , with a = 1, 2. In this representation, the SO(6) R-symmetry is broken: where A ∈ U(N),Ā ∈ U(M), andq +a have 'opposite' gauge charges under the two gauge groups. From the action, one finds the following equations of motion: with proper ordering ofqq to match the gauge indices of W ++ L,R . The latter equation of motion implies that scalars from the vector multiplet get equated to bi-scalars of the matter multiplet. One such relation will be relevant for later use: This crucial relation is responsible for generating the well-known sextic potential involving f 's once φ's are integrated out from the ABJM action. The CS theories coupled to matter can be quantized directly in superspace [25] and the resulting propagators read:

21)
where (3.23) The ρ αβ has quite a complicated expression but in the presence of δ 2 (θ ++ 12 )δ(u 1 , u 2 ), it simplifies in the vector propagator to the following: The vertices are easily read from the relevant actions.

Super-Wilson Loop
There are two main types of Wilson loop operators that can be considered for d = 3 Chern-Simons theories [12,16,17,20]: GY-type ( 1 N -BPS for N = 2, 3, 4, 6) and DT-type (still 1 N -BPS for N = 2, 3 but 1 2 -BPS for N = 4, 6). We will focus only on the former case here. The 1 3 -BPS Wilson loop is usually written for N = 3 CS theory as follows: where y ij = y ji are 3 SU(2) 'coordinates'. For this operator to locally preserve any supersymmetry, the susy parameter ǫ ij α needs to be a solution oḟ provided that |ẋ| = |ẏ|. To incorporate the condition on |ẏ|, we can rewrite the scalar term in WL as dτ |ẋ|(u + i u − j )φ (ij) using the harmonic coordinates on SU(2). Now, we are ready to write down the most general supersymmetrized expression for a Wilson loop (such that (4.1) is its bosonic component): The usual BPS condition on the bosonic WL (ǫ (ij)γ Q (ij) γ W1 /3 (x) = 0), which results in (4.2), translates to ǫ (ij)γ D (ij) γ W1 /3 = 0 (along withẋ →ẋ A ) in superspace for obvious reasons (see [29] for an explicit proof). Let us see what that implies for (4.3): where we use F A,B to represent the field strength arising from the (anti-)commutator of {D A , D B ]. As we did for the case of N = 2 WL, we have ignored here terms that look like field-dependent gauge transformations. Since we know that only F 0,0 γ,α = F ±±,±± γ,α = 0, we can have only one of theθ terms above in the Wilson loop. This means either ǫ ++ or ǫ 0 can be the only unbroken susy. However, choosing ǫ 0 , we find that F 0 γ,αβ [25] contains not only the D 0 α W 0 term but also D ++ α W −− so the above variation cannot vanish. Thus we are left with ǫ ++ and the remaining couple of terms do vanish in this case because This expression vanishes (for arbitrary W 0 ) in a way similar to the N = 2 case, and we preserve half of the complex spinor ǫ ++γ . Thus, the final result for the supersymmetric generalization of the 1 3 -BPS Wilson loop is: To compare with the usual bosonic WL operator, we write the above in component fields The difference starts at terms of order θ containing fermionic fields (χ ij α ) and at θ 2 order with bosonic fields (φ ij ). Higher-order terms will contain λ α , X ij fields too.
With this construction, we can readily give the supersymmetrized generalization of the 1 6 -BPS WL operator for N = 6 ABJM theory in N = 3 harmonic superspace: To show this, we need to use the equation of motion for A ++ (3.19) and (3.20) along with a change of notation from f i → C I as discussed in [24]. (Without the constraint on u, this operator has more content due to W 0 containing not only φ 12 ≡ M J I C JC I but also φ 11 and φ 22 .)

Computation
In this section, we will compute the 'one-loop' vacuum expectation value of the Wilson loop W1 /3 . The constraint on u will also be imposed so the operator and the expected vev slightly simplify (with R being the fundamental representation of U(N) gauge group): An important subtlety that occurs repeatedly in the computation is when D ++ α (u) = u + i u + j D ij α acts on an analytic superfield which depends on another harmonic variable, say (θ ++ α , θ 0 α , u ′ ). The result is not the naive zero since using (u + u − ) = 1 and repeated Schouten identities, one can rewrite: where we have converted the harmonic dependence of the derivative from u to u ′ . Note that the charges match on both sides separately for u's and u ′ 's. Thus for any analytic superfield Φ(θ ++ α , θ 0 α , u ′ ), we have: Similar manipulations lead to the following list of identities: For the sake of convenience, we list generating expressions for component expansions of some connections and field strengths below that is, keeping only a single A ++ in (3.13) and (3.14) : Note that all the fields depend on the same θ-coordinate. The components can be obtained from the above expressions by using (3.12) and performing not only the D-algebra but some harmonic algebra too. The simplest component to obtain is the vector: A αβ | = 2ιa αβ . To get the scalars, we need to perform slightly more involved algebra: Other components can be similarly obtained, which we leave as an exercise and refer the reader to [30] for useful identities involving harmonic variables. Now we turn to evaluating various contributions to W1 /3 . First let us consider the contribution from the vector connection. In general, we have from (3.13): Using (5.6), we finḋ A,2 x p A,12 → 0, and expanded 1 ρ 2 in powers of θ 0 's. The next term (quadratic in A ++ ) in the expansion of A −− also contributes: Let us sketch how we got this result. We require that all δ 2 (θ ++ 12 ) be cancelled so higher orders of A ++ cannot contribute as there are not enough D −− α derivatives in A 1,αβ A 2,γδ to cancel more than two such δ-functions. After expanding D 0 1α D ++ 1β using the identities given above, doing two harmonic integrals using the harmonic δ-functions in the two propagators and then hitting the two δ 2 (θ ++ 12 ) with correct D −− 's, we are left with: Keeping track of various signs and numerical factors above, we get (5.10).
Let us now evaluate the second contribution to W due to the 'charged' fermionic connection. Using the fact that we need enough D −− α to get relevant terms, we ignore terms with D 0 α in the expansion of A −− in (5.6): The u-factor in parentheses might look divergent upon imposing the constraint on u-matrix discussed in the previous section but using an explicit parameterization, one can show that it instead limits to unity up to a U(1) 'charge factor'. We will, however, leave this factor as it is to account for the correct U(1) charges along with an understanding that there is no non-trivial u-dependence.
The third contribution to W due to mixed contraction of the two connections vanishes: The fourth contribution to W due to the scalar field strength is This is a contribution from the linear term in A −− and is straightforward to compute. Like A αβ A γδ , we get a second contribution from the contraction of quadratic terms in A −− here too: . (5.14) This computation proceeds very similarly to the case of the vector connection but there are more terms; we sketch them below (again, various signs and numerical factors need to be tracked): Similarly, we can compute two more mixed contractions between the two connections and W 0 , but only one is non vanishing: One more contribution to W needs to be considered (at the order being studied) and this one includes a 3-point vertex insertion: The second to last line is obtained after performing the d 6 θ 0 in the previous line with the help of three δ 2 (θ ++ 0i )'s, cancelling the divergent harmonic denominator. The last line then follows by converting D 0 i → D ++ i+1 in cyclic order and acting on the numerator, thus picking out eight terms. The d 3 v integral produces only a numerical factor. Note that the first term in the integral d 3 x 0 is the only 'bosonic' piece given by the well-known integral (6.12) of [15].
Finally, collecting all the results at 'one-loop' order (we suppress u-dependent factors from W 0 W 0 to keep the expression below manageable), we have

Comments
We have constructed a 1 3 -BPS supersymmetrized Wilson loop operator in d = 3, N = 3 harmonic superspace for CS theories. This operator readily generalizes the 1 6 -BPS operator for ABJM theories. We were also able to use the power of harmonic superspace to compute the 'one-loop' perturbative corrections directly in superspace.
Using the component expansion of N = 3 connections and field strengths, and just focussing on the localization locus discussed in Section 2 (σ ≡ φ 0 and D ≡ X 0 = − σ r ), we can see that the W1 /3 given in (5.1) reduces to (2.5) once we identify θ,θ with θ ++ , θ −− . Then one can expect that At order α 2 , we can directly compare the 'bosonic' factor − 5α 2 24 ≡ 5π 2 N 2 6k 2 above to the corresponding perturbative expression in (5.17). They exactly match once we perform the integrals in the latter case for a circular WL, i.e., x µ = (0, sin(τ ), cos(τ )) as one might expect 6 .
Formally, both (5.17) and (6.1) have nonvanishing 'fermionic' contributions at O(α) and O(α 2 ). However, without knowing the explicit profile functions of θ(τ ) and u(τ ) we cannot proceed further. However, we can identify a contribution at O(α 2 ) that does not receive any O( 1 N ) corrections! 7 These are the O(θθ) terms in the ϑ piece of (6.1) and comparing with (5.17), we can give an explicit expression for these terms: ϑ| O(θθ) = 2ι π 2 dτ 1 dτ 2 ẋ 1 ·ẋ 2 − |ẋ 1 ||ẋ 2 | x αβ 12 θ 1,αθ1,β − θ 2,αθ2,β |x 12 | 4 −ι 16π 3 dτ 1,2,3ẋ αβ 1ẋ γδ 2ẋ κλ 3 d 3 x 0 (x 01 ) βγ (x 02 ) δκ θ 3,λθ3,α + 2 more terms |x 01 | 3 |x 02 | 3 |x 03 | 3 · (6.2) The fermionic pieces from the term proportional to ẋ A,1 ·ẋ A,2 − |ẋ A,1 ||ẋ A,2 | do not contribute above because the combination θθ is independent of τ as discussed in Section 2. Such fermionic contributions to the Wilson loop operators do not seem to have been considered in d = 3 but similar terms have appeared in the d = 4, N = 4 SYM literature, specifically in the study of supersymmetrized Maldacena-Wilson loops [29,31]. Thus, a careful study of the τ -dependence of the θ and u coordinates that is consistent with the 'bosonic' circular WL is required to understand how the general perturbative result (5.17) reduces to the simpler localization result (6.1) at various θ orders. 8 We leave this exercise for future work. One can also ask whether the construction of a 1 2 -BPS WL with 'supermatrix' structure [20] is feasible in harmonic superspace. Our preliminary analysis suggests that the U(1) charge structure of the supersymmetry parameters (ǫ ±± , ǫ 0 ) and the matter superfield q + is an obstruction for constructing a straightforward generalization. As mentioned in the Introduction, a motivation to study such supersymmetrized WL operators is to probe Wilson loops/Scattering amplitudes duality in ABJM theory. The expectation is that polygonal WL operators with certain bifundamental vertex insertions would be dual to the ABJM scattering amplitudes. The matter superfield (q +a ) B A in bifundamental representation provides a natural candidate for such insertions. However, this leads to some superficial divergences that need to be tamed. Progress on these aspects will be reported elsewhere. 6 We refer the readers to [15] for evaluation of the relevant integrals. 7 We do not have O( 1 N ) terms at O(α) either, but that could still be treated as a phase factor. The striking feature of the term at O(α 2 ) is that it remains unchanged even after the removal of the O(α) phase: