Exact results for Wilson loops in orbifold ABJM theory

We investigate the exact results for circular 1/4 and 1/2 BPS Wilson loops in the $d=3$ ${\mathcal N}=4$ super Chern-Simons-matter theory that could be obtained by orbifolding Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. The partition function of the ${\mathcal N}=4$ orbifold ABJM theory has been computed previously in the literature. In this paper, we re-derive it using a slightly different method. We calculate the vacuum expectation values of the circular 1/4 BPS Wilson loops in fundamental representation and of circular 1/2 BPS Wilson loops in arbitrary representations. We use both the saddle point approach and Fermi gas approach. The results for Wilson loops are in accord with the available gravity results.

When string theory is weakly coupled and the supergravity approximation is a good one, the dual d = 4 N = 4 super Yang-Mills theory is strongly coupled. To compare with the gravity results, one has to know the vacuum expectation values of the Wilson loops at strong coupling. To do this it was proposed in [8,9] that d = 4 N = 4 super Yang-Mills theory is related to the Gaussian matrix model, and this was proved in [10] using localization techniques.
There is a similar but more complicated story in the AdS 4 /CFT 3 correspondence. M-theory in AdS 4 ×S 7 /Z k spacetime, or type IIA string theory in AdS 4 ×CP 3 spacetime, is dual to the d = 3 N = 6 super Chern-Simons-matter (SCSM) theory with gauge group U (N ) × U (N ) and levels (k, −k), which is known as Aharony-Bergman-Jafferis-Maldacena (ABJM) theory [11]. In ABJM theory there are 1/6 BPS [12][13][14] and 1/2 BPS [15] Wilson loops. The 1/6 BPS Wilson loops are closely related to the 1/2 BPS Wilson loops in N = 2 SCSM theory in [16]. Localization techniques have been applied to ABJM theory and other SCSM theories with fewer supersymmeties [17][18][19] and lead to matrix models that are more complicated than the Gaussian matrix model. By using localization, one can calculate the partition function and vacuum expectation values of Wilson loops at both weak coupling and strong coupling [15,17,[20][21][22]. The computations in [22] are based on the saddle point solution of the ABJM matrix model at large N limit with finite k, and we will call such a method the saddle point approach. Furthermore, the ABJM matrix model could be reformulated as an ideal Fermi gas with a complicated potential [23], and one can calculate the vacuum expectation values of BPS Wilson loops with fixed winding number using the Fermi gas approach [24]. One can also use the Fermi gas approach to calculate the vacuum expectation values of the 1/2 BPS Wilson loops in arbitrary representations [25].
By Z r orbifolding the U (rN ) × U (rN ) ABJM theory, one can obtain a d = 3 N = 4 SCSM theory with gauge group U (N ) 2r and levels (k, −k, · · · , k, −k). This theory is dual to M-theory in AdS 4 ×S 7 /(Z r ×Z rk ) spacetime [26][27][28][29]. The partition function of the orbifold ABJM theory has been calculated using Fermi gas approach in [30], and in this paper we will re-derive it using a slightly different way. In the orbifold ABJM theory there are 1/4 and 1/2 BPS Wilson loops, and the 1/2 BPS Wilson loops in fundamental representation should be dual to M2-branes with one dimension wrapping on the M-theory circle [31,32]. In this paper, we will calculate the leading contributions of vacuum expectation values of the Wilson loops using the saddle point approach in the large N limit with k and r being finite. We will also calculate the perturbative part 1 of the vacuum expectation values of the Wilson loops using the Fermi gas approach. The results are in agreement with the available gravity results.
In the N = 4 orbifold ABJM theory with gauge group U (N ) 2r , there are 2r linearly independent 1/2 BPS Wilson loops that preserve the same supersymmetries, but there are not so many 1/2 BPS branes in M-theory in AdS 4 ×S 7 /(Z r ×Z rk ) spacetime. It was conjectured that these Wilson loops are 1/2 BPS classically, and only a special linear combination of them is 1/2 BPS quantum mechanically [32] The rest of the paper is arranged as follows. In Section 2 we review the results in ABJM theory, including the partition function and vacuum expectation values of Wilson loops. In Section 3 we investigate the partition function of the N = 4 orbifold ABJM theory. In Section 4 we review the circular 1/4 and 1/2 BPS Wilson loops in the N = 4 orbifold ABJM theory in Euclidean space. In Section 5 we calculate vacuum expectation values of Wilson loops with fixed winding number using the saddle point approach. In Section 6 we calculate vacuum expectation values of Wilson loops in arbitrary representations using the Fermi gas approach. We end with conclusions and discussions in 1 By this we mean to include all of the 1/N corrections, putting aside the non-perturbative contributions. Section 7. In Appendix A we investigate if there are more general 1/2 BPS Wilson loops in N = 4 orbifold ABJM theory other than the ones found in [31,32]. We find no new ones.

Results in ABJM theory
In this section we review some results in ABJM theory. This includes the partition function and vacuum expectation values of circular 1/6 and 1/2 BPS Wilson loops. We focus on what will be used in the following sections, so this is merely a brief review.

Partition function
The partition function of ABJM theory with gauge group U (N ) × U (N ) and levels (k, −k) can be localized to be the ABJM matrix model [17] The partition function of the matrix model (2.1) can be written as the canonical partition function Z(N ) of N -particle free Fermi gas with the one-particle density matrix being [23] ρ = e −Ĥ , (2.2) whose explicit form will not be used in this paper. Note thatĤ is the one-particle Hamiltonian operator. To calculate Z(N ), one can firstly calculate the grand partition function with Z(0) = 1, z = e µ being the fugacity and µ being the chemical potential. The grand potential is defined as And then one gets One can define j(µ) according to [33] and then One adopts the phase space formulation of quantum mechanics, and defines The quantity n(µ) counts the number of one-particle states whose energy is less than µ. Using the Sommerfeld expansion one can get the expectation value of particle number N (µ) in the grand canonical ensemble N (µ) = π∂ µ csc(π∂ µ )n(µ). (2.9) It is standard in the grand canonical ensemble that and so we get We find that when µ → −∞, This results is very useful for us. Note that the way from n(µ) to N (µ) and then to J(µ) is a slightly different method of getting J(µ) to the one in the original paper [23].
In the large µ (i.e. large N ) limit, one can split a quantity into the perturbative part and nonperturbative part. The perturbative part is denoted as pt. The non-perturbative part is exponentially suppressed in the large µ (i.e. large N ) limit, and it is denoted as np. In this paper we will mainly focus on the perturbative part. It turns out that [23] n pt (µ) = Cµ 2 + n 0 , (2.14) One then gets Here A appears as an integral constant, and its exact form depends on the full form of N (µ). One can find the result for A in [34,35]. One has j pt (µ) = J pt (µ), (2.17) and then one gets the perturbative part of the partition function [23,36] Z pt (N ) with Ai(x) being the Airy function.

Wilson loops
The representations of group U (N ) and supergroup U (N |N ) can be denoted by Young diagrams. We write a general Young diagram as R, and it can be a representation of U (N ) or U (N |N ).
We consider the hook representation R = (a|b) with a + 1 boxes in the first row and one box in each of the remaining b rows. For both the 1/6 BPS and 1/2 BPS cases, a Wilson loop with winding number n is related to Wilson loops in the hook representations by When n = 1, it is just the fundamental representation. In the matrix model (2.1), the circular 1/6 and 1/2 BPS Wilson loops with winding number n can be written as [15,17] W n with n being the winding number of the loop and the right hand sides being the expectation values in the matrix model. For their expectation values one has the relation with * being the complex conjugate.
In the large N limit with finite k, i.e. the M-theory limit, the values µ i and ν i at the saddle point can be denoted as a continuous distribution [22] µ(x) = with the uniform density In the saddle point approach the Wilson loop vacuum expectation values can be calculated as The exponentially suppressed terms are omitted here. Note that one can only get the correct leading contribution of large N in the saddle point approach.
The vacuum expectation values of circular Wilson loops can also be calculated in the Fermi gas approach [24]. One firstly calculates Then the 1/6 BPS Wilson loop expectation value with winding n in the canonical ensemble is Similar to the partition function, one has with non-perturbative contributions being neglected. It turns out that Here H n is the harmonic number, with H 0 being 1. Then the 1/6 BPS Wilson loop vacuum expectation value is

The 1/2 BPS Wilson loop vacuum expectation value is
Now we turn to Wilson loops in hook representations based on [25]. There the density matrix for Fermi gas dual to ABJM theory was obtained aŝ (2.35) Though it is the same as the matrix in [23] but different from the one in [24], it gives the same partition functions and vacuum expectation values of BPS Wilson loops. One of the key steps in [25] is the following result The density matrixρ f with f (W ) = (1 + tW )/(1 − sW ) can be written as [25] where |a and b| are defined in the coordinate q representation as For the half BPS Wilson loop in a hook representation (a|b), the generating function is given . Therefore, the grand canonical ensemble expectation value of 1/2 BPS Wilson loop generating function in ABJM theory becomes One gets the relation As discussed in [25], the perturbative part of the half BPS hook Wilson loop in ABJM theory is determined by the topological vertex of C 3 in [38] Let us consider the circular half BPS Wilson loops in non-hook representations. One can decompose the Young diagram for a non-hook representation into hooks from the upper left to the lower right to get (a 1 |b 1 ), · · · , (a s |b s ). This general representation will be denoted as R = (a 1 · · · a s |b 1 · · · b s ). The Giambelli formula states that (2.44) The authors of [25] considered the following generating function The computations in [25] give The multiplication between boldface variables is understood as matrix multiplication with indices µ, ν and summation being replaced by integration with measures [dµ], [dν]. Then by introducing one can get Taking t = 0 in the above results, one gets and then one has The coefficient of t s in both sides of the above equation gives Restricted to hook representation cases, one has So finally one gets When r = 1 it is reduced to the ABJM matrix model (2.1). It can be written as canonical ensemble partition function of an N -particle Fermi gas with one-particle density matrix [23] withĤ being the same as that of ABJM theory in (2.2).
We calculate the partition function in the Fermi gas approach. We firstly have with n(µ) being the same function as (2.8). Then Then using (2.12) we can get Note that we have the following expansion Formula (3.5) is a convenient way to get the grand potential J r (µ) with Hamiltonian rĤ from the grand potential J(µ) with HamiltonianĤ, including both the perturbative and non-perturbative parts.
From the results of ABJM theory we have with Then we have the perturbative part of the partition function This is in accordance with the result in [30], and here we re-derive it in a different way.
The non-perturbative part of the grand potential for ABJM theory J np (µ) is a summation of terms of the form [23] ( with a, b, c, d being constants and d > 0. Correspondingly in the grand potential of the N = 4 SCSM theory J np r (µ) there is the term sin(rπ∂ µ ) sin(π∂ µ ) (3.12) Here we have defined the function Note that when x = l is an integer we have (3.14)

Circular BPS Wilson loops
In this section we review the circular 1/4 and 1/2 BPS Wilson loops for the N = 4 orbifold ABJM theory in Euclidean space [31,32]. This theory is an SCSM theory with gauge groups U (N ) 2r and levels (k, −k, · · · , k, −k). 4 In d = 3 Euclidean space we use the convention of spinors in [39], and especially we have the coordinates x µ = (x 1 , x 2 , x 3 ) and the gamma matrices with σ 1,2,3 being the Pauli matrices. The circle is parameterized as x µ = (cos τ, sin τ, 0).
It has been checked that the difference of 1/4 and 1/2 BPS Wilson loops is Q-exact with Q being some supercharge preserved by both the 1/4 and 1/2 BPS Wilson loops [31,32]. This applies to both the ψ 1 -loop (4.6) and ψ 2 -loop (4.8), and explicitly one has In this case, equations (4.9) would also be spoiled. We expect that with c being (5.4). If (4.12) is not true, we only have some matrix model results.
We expand the above results in the limit N k 1 with r being fixed, and now for the 't Hooft We make expansion of large λ and large k. For the 1/4 BPS Wilson loop we have For the 1/2 BPS Wilson loops we have These are in accord with the results in saddle point approach (5.7). Note that for the leading contribution of large k, i.e. the genus zero part, there is no r dependence.

1/2 BPS Wilson loops in hook representations
Now we turn to half BPS Wilson loops 5 in hook representations based on [25], where the density matrix for Fermi gas dual to ABJM theory was obtained as (2.35). For the N = 4 orbifold ABJM theory, we haveρ r =ρ r . (6.10) Similar to the calculation in [25], we obtain the following result in N = 4 orbifold ABJM theory, 11) 5 We have the ψ1-loops W ( ) 1/2 (4.6) and ψ2-loopsW ( ) 1/2 (4.8) with = 0, 1, · · · , r−1. If all of them are half BPS quantum mechanically, we can calculate their vacuum expectation values in the matrix model as shown in this subsection. Due to the Zr symmetry of the theory, the results are independent of . From now on we can omit the index and subscript 1/2. We add subscript r to some quantities of the N = 4 orbifold ABJM theory to distinguish them from their counterparts in ABJM theory. Also the results are the same for the ψ1-loops and ψ2-loops, and so we will not write the same results twice. If only a special combination of the ψ1-loops and ψ2-loops (4.10) is half BPS quantum mechanically and (4.12) holds, the following calculations still apply provided that a constant c (5.4) is added to the result. In the worst condition (4.12) does not hold quantum mechanically, and the calculations here are just matrix model results.
The generating function for the half BPS Wilson loop in hook representations (a|b) was given in [37] 1 + (s + t) ∞ a,b=0 Therefore, the grand canonical ensemble expectation value of a circular 1/2 BPS Wilson loop W r, (a|b) in the N = 4 orbifold ABJM theory in Euclidean space becomes s a t b zρ r−1 1 + zρ r |b a|   (6.14) with the states |a and b| being defined the same as (2.39). We get the relation W r,(a|b) GC = a| zρ r−1 1 + zρ r |b (6.15) = Tr zρ r−1 1 + zρ r |b a| = Tr 1 e rĤ−µ + 1 eĤ |b a| .
Note that for a circular half BPS Wilson loop in ABJM theory W (a|b) there is In the large N limit, the expectation values scale as W r,(a|b) pt ∼ e nπ √ 2λ . (6.22)

1/2 BPS Wilson loops in non-hook representations
Let us consider the half BPS Wilson loops in general representations R = (a 1 · · · a s |b 1 · · · b s ). The Giambelli formula states that As in [25], we consider the following generating function W r (N ) = det p,q (δ pq + tW r,(ap|bq) (e µ i , e ν j )) . (6.24) Similar to computations in [25], with the definitions (2.47) and (2.49) we can get Note that the multiplication between boldface variables is understood as matrix multiplication with indices µ, ν and summation being replaced by integration with measures [dµ], [dν] in eq. (2.47). We then have

Conclusions and discussions
In this paper, we have calculated the vacuum expectation values of the circular BPS Wilson loops in arbitrary representations in the N = 4 orbifold ABJM theory. We used both the saddle point approach in [22] and the Fermi gas approach in [23,24], and the results agree with the available gravity results in [31,32]. It will be quite interesting to study the string/M theory dual of the Wilson loops in the higher dimensional representations.
Note that θ 1î and θ 2î are nonvanishing, general and linearly independent, and similarlyθ 1î andθ 2î are nonvanishing, general and linearly independent. First of all,ḡ (2 )î 1 and g (2 ) 2î cannot be vanishing, otherwise there would be no solutions for the matrix Mî. Then we must have η 1 (2 ) = 0 or η 2 (2 ) = 0, as well asη   In summary we have no choices other than the ψ 1 -loop and ψ 2 -loop that satisfies the following conditions.
• It is constructed by two adjacent gauge fields A