Instanton Corrections of 1/6 BPS Wilson Loops in ABJM Theory

We study instanton corrections to the vacuum expectation value (VEV) of 1/6 BPS Wilson loops in ABJM theory from the Fermi gas approach. We mainly consider Wilson loops in the fundamental representation and winding Wilson loops, but we also initiate the study of Wilson loops with two boundaries. We find that the membrane instanton corrections to the Wilson loop VEV are determined by the refined topological string in the Nekrasov-Shatashvili limit, and the pole cancellation mechanism between membrane instantons and worldsheet instantons works also in the Wilson loop VEVs as in the case of the partition functions.


Introduction
The AdS/CFT correspondence [1] has been tested in many examples and we believe that holography is one of the fundamental principles of quantum gravity and string theory. The localization computation in supersymmetric field theories [2] opened a door to a new era and we are now in a stage to study the holographic duality at a precise quantitative level. In particular, in the past few years we have witnessed a tremendous progress in understanding the holographic duality between M-theory on AdS 4 × S 7 /Z k and 3d N = 6 U (N ) k × U (N ) −k Chern-Simons-matter theory, known as the ABJM theory [3]. From the analysis of the partition function of ABJM theory on a three-sphere obtained by the localization technique [4], the N 3/2 scaling law of the degrees of freedom on N M2-branes predicted from the gravity side [5] is correctly reproduced from the first principles computation on the field theory side [6].
Moreover, there are exponentially small corrections to the free energy [6,7], corresponding to the M2-brane instantons wrapping some three-cycles inside S 7 /Z k in the bulk M-theory side. The Fermi gas formalism [8] is a powerful technique to study such instanton corrections. In this approach, we consider the grand partition function of ABJM theory by introducing the chemical potential µ and summing over N . It turns out that the grand partition function is written as a Fredholm determinant which in turn is interpreted as a system of free fermions on a real line. The grand potential of ABJM Fermi gas system receives two types of instanton corrections: worldsheet instanton corrections of order O(e −4µ/k ) and membrane instanton corrections of order O(e −2µ ). There are also bound states of worldsheet instantons and membrane instantons, but they can be removed by introducing the "effective" chemical potential [9]. Building on a series of works [9][10][11][12], it is finally realized that the grand potential of ABJM theory is completely determined by the refined topological string on local P 1 × P 1 [13]. In particular, the membrane instanton corrections are given by the refined topological free energy in the Nekrasov-Shatashvili limit (NS free energy), while the worldsheet instanton corrections correspond to the standard, un-refined topological string. Although the membrane instantons and the worldsheet instantons separately have poles at rational values of k, those poles are actually canceled by adding the two contributions. This pole cancellation mechanism guarantees that the grand partition function is well-defined for all k. For the special values k = 1, 2, 4, 8, the grand partition function can be written in closed forms in terms of the Jacobi theta functions [14][15][16].
One can also compute exactly the vacuum expectation value (VEV) of 1/6 BPS Wilson loops in ABJM theory using the localization technique [4]. The study of such BPS Wilson loops from the Fermi gas approach was initiated in [17] and further developed in [18] particularly for the 1/2 BPS Wilson loops. It is found that 1/2 BPS Wilson loops in arbitrary representations are given by a determinant of hook representations [18][19][20], and 1/2 BPS Wilson loops are closely related to the open topological string on local P 1 × P 1 [20][21][22][23]. Interestingly, it is observed that there are no "pure" membrane instanton corrections in 1/2 BPS Wilson loops [18] which turns out to be a consequence of the open-closed duality between the 1/2 BPS Wilson loops and the grand partition functions of ABJ theory [20]. In particular, there is no pole cancellation between worldsheet instantons and membrane instantons in the VEV of 1/2 BPS Wilson loops.
On the other hand, 1/6 BPS Wilson loops in ABJM theory have no direct connection to open topological strings and the instanton corrections of 1/6 BPS Wilson loops have not been fully explored in the literature. In this paper, we initiate a study of the instanton corrections of 1/6 BPS Wilson loops mainly using the numerical analysis. We will show that the VEV of 1/6 BPS Wilson loops can be computed numerically with high precision and we will study the instanton corrections of 1/6 BPS Wilson loops from numerical fitting. We first consider the 1/6 BPS Wilson loops in the fundamental representation. We find that the membrane instanton corrections are given by the NS free energy, and the pole cancellation mechanism works also for the 1/6 BPS Wilson loops. We find that, up to regular terms, the 1/6 BPS Wilson loops in the fundamental representation is essentially determined by the quantum volume appearing in the exact quantization condition of the spectrum [15,[24][25][26][27][28].
We also study the 1/6 BPS Wilson loops with winding number n ≥ 2 and the 1/6 BPS Wilson loops with two boundaries. Our numerical study shows that the perturbative part of the 1/6 BPS Wilson loops with winding number n ≥ 2 is different from the expression obtained in [17]. We also study the instanton corrections to those Wilson loops and find evidence that the membrane instanton corrections are again related to the NS free energy. This paper is organized as follows. In section 2, we first review the Fermi gas approach to the 1/6 BPS Wilson loops, and explain our algorithm to compute them numerically. In section 3, we consider the perturbative part (or zero-instanton part) of the grand canonical VEV of winding Wilson loops. Our result (3.11) is different from the one found in [17] for winding number n ≥ 2. In section 4, we study the instanton corrections to the fundamental Wilson loop numerically. We find that the grand canonical VEV of fundamental Wilson loop is closely related to the quantum volume (4.33). In section 5, we study the WKB expansion of the fundamental Wilson loop. Using the Padé approximation we confirm that the membrane instanton corrections are correctly reproduced from the WKB expansion. In section 6, we consider winding Wilson loops with winding number n ≥ 2. We find that the membrane instanton corrections for the winding number n = 2 are also related to the NS free energy. We also find the first few worldsheet instanton corrections for the winding numbers n = 3, 4. In section 7, we study the Wilson loops with two boundaries. We show that the VEV of Wilson loops with two boundaries can be systematically computed by constructing a sequence of functions, in a similar manner as the winding Wilson loops. Then we study the instanton corrections of Wilson loops with two boundaries. Again, the membrane instanton corrections to the imaginary part of Wilson loop is related to the NS free energy. Finally, we conclude in section 8. In Appendix A, we summarize the instanton corrections for some integer values of k.

1/6 BPS Wilson loops from Fermi gas approach
In this paper, we will consider the VEV of 1/6 BPS Wilson loops in ABJM theory on S 3 , first constructed in [29][30][31] Here x µ (s) parametrizes a great circle of S 3 , C I (I = 1, 2, 3, 4) are the scalar fields in the bi-fundamental chiral multiplets, and M I J is a constant matrix which can be brought to the form M = diag(1, 1, −1, −1) in a suitable choice of basis. A µ is the gauge field of one of the U (N ) factor of the gauge group U (N ) k × U (N ) −k of ABJM theory. On the other hand, the construction of 1/2 BPS Wilson loops involves the gauge fields in both of the U (N ) factors and utilizes the supergroup structure U (N |N ) [32].
The VEV of such 1/6 BPS Wilson loops can be reduced to a matrix model by the supersymmetric localization [4] where U corresponds to the holonomy in one of the gauge group U (N ) One can also consider the Wilson loop in the other U (N ) factor, but it is related to (2.2) by the complex conjugation (or k → −k). In this paper we consider the un-normalized VEV, i.e. we do not divide (2.2) by the partition function Z(N, k). As shown in [17], the VEV of 1/6 BPS Wilson loops in the grand canonical picture can be written as a Fermi gas system. Let us first recall the Fermi gas description of the partition function of ABJM theory on S 3 [8]. Introducing the fugacity κ = e µ with chemical potential µ, we define the grand canonical partition function by summing over N It turns out that the grand partition function can be written as a Fredholm determinant Ξ(κ, k) = Det(1 + κρ), (2.5) where the density matrix ρ is given by (2 (2.9) The crucial observation in [17,18]  for some function f is written as For instance, setting f to f (e µ i ) = 1 + εe nµ i (2.12) and picking up the term of order O(ε), we find that the grand canonical VEV of 1/6 BPS Wilson loops with winding number n is given by where we defined R as (2.14) Once we know the grand canonical VEV, one can easily find the canonical VEV with fixed N by For the special case n = 1, Tr U corresponds to the 1/6 BPS Wilson loop in the fundamental representation.

Computation of trace
From the small κ expansion of the grand canonical VEV (2.13) one can see that the Wilson loop VEV Tr U n N at fixed N can be computed from the traces Tr(ρ W n ) and Tr ρ with = 1, · · · , N . Note that the trace Tr ρ appears in the computation of the partition function as well. One can compute Tr ρ by applying the Tracy-Widom lemma [33]. Thus our remaining task is to compute the trace Tr(ρ W n ) with W n insertion systematically. As we will show below, one can also apply the Tracy-Widom lemma to the computation of the trace Tr(ρ W n ).
First we observe that, using the Baker-Campbell-Hausdorf formula, W n is written as where the matrix element of ρ is given by (2.20) This is exactly the form of kernel to which we can apply the Tracy-Widom lemma [33]. As shown in [33], the matrix element x|ρ |y can be written as Since the trace Tr(ρ W n ) with insertion of W n is given by the integral of x|ρ |y with the replacement y → x + 2πin Tr(ρ W n ) = dx x|ρ |x + 2πin e n(x+nπi) k , (2.23) 1 We use the following normalization of quantum mechanical states (2.18) -6 -this trace can also be written in terms of the functions φ j (x) in (2.22) Tr(ρ W n ) = e πin(n+1) k To summarize, the computation of trace Tr(ρ W n ) boils down to the construction of a sequence of functions φ j (x) (j = 0, 1, · · · ). We should stress that φ j (x) is the same function appearing in the computation of trace Tr ρ . Note that for the integral (2.24) to converge, k should satisfy the condition k > 2n. (2.25) For some integer values of k, the integral in (2.22) and (2.24) can be evaluated exactly by closing the contour and picking up the residue of poles [10,11,34,35]. For instance, we have computed the exact values of Tr(ρ W ) with the winding number n = 1, for k = 3, 4, 6 up to certain . For example, for k = 3 we find , . (2.26) For k = 4 we find (2.27) For k = 6 we find , . (2.28) We note in passing that Tr(ρW n ) can be found in a closed form for general k, n Tr(ρW n ) = e πin 2 k 4k cos 2 πn k . (2.29)

Numerical computation
It is useful to develop a technique to compute the trace Tr(ρ W n ) numerically with high precision. We use the observation that the integral appearing in the computation of φ j (x) in (2.22) is written schematically as This can be easily done numerically by disretizing the continuous variable x ∈ R to some finite number of points in the range −L/2 < x ≤ L/2 where L is an IR cut-off. Then we can approximate the Fourier transformation by the discrete Fourier transformation 2 . By taking L and m sufficiently large, we can numerically evaluate the integral (2.32) with very high precision. We will use the parameters L = 3000 and m = 2 16 in the numerical computations below. For integer value of k, we can estimate the error of the above numerical computation by comparing with the exact values of traces obtained in the previous subsection. Let us consider the imaginary part of Tr(ρ W n ) for k = 4, n = 1, as an example. For k = 4, we have computed the exact values of Tr(ρ W ) up to = 20 (see (2.27) for the first three terms). In Fig. 1, we show the plot of the relative error e between the exact values and the numerical values of the imaginary part of Tr(ρ W ) As we can see from On the other hand, the order of worldsheet 1-instanton correction and membrane 1-instanton correction for k = 4, N = 20 are given by (see (4.9)) (2.36) Comparing (2.35) and (2.36), we conclude that our numerical calculation has enough accuracy to study instanton corrections.

Perturbative part of winding Wilson loops
We will study the large µ expansion of the grand canonical VEV of 1/6 BPS winding Wilson loops. First we consider the perturbative part (or zero-instanton sector) of this expansion.
In the large µ limit, the grand canonical VEV of 1/6 Wilson loops with winding number n behaves as [17] Tr U n GC ∼ e 2nµ k . (3.1) We would like to understand the large µ behavior of 1/6 Wilson loops in more detail. It is shown in [17] that Tr U n GC can be decomposed as The first term is the absolute value of the 1/2 BPS Wilson loop which was studied extensively in the literature [17,18]. The perturbative part of 1/2 BPS Wilson loop is given by [17] |W (1/2),pert It is known that the 1/2 BPS Wilson loops are related to the open topological string on local [20][21][22][23], and the worldsheet instanton corrections to the 1/2 BPS Wilson loops can be readily obtained from the results of open topological string.
On the other hand, W n in (3.2) is less understood. Note that W n is written as We will loosely call W n "the imaginary part of 1/6 BPS Wilson loop" with the understanding that the precise meaning is (3.4). In this paper, we will study the instanton corrections in W n .
To do that, first we compute the canonical VEV W n (N, k) numerically with high precision using the algorithm in section 2.2. Then we can find the instanton corrections by numerically fitting with the expansion where a j,w is the coefficient in the instanton expansion of the grand canonical VEV Here J pert (µ, k) denotes the perturbative part of the grand potential of ABJM theory and the coefficients C(k), B(k) and A(k) are given by It is found [36][37][38] that A(k) is a certain resummation of the constant map contribution of topological string on local P 1 × P 1 . The expansion (3.5) in the canonical picture and the grand canonical picture (3.6) are related by the integral transformation where C is the contour in the µ-plane from e − πi 3 ∞ to e πi 3 ∞, and J(µ, k) is known as the modified grand potential which is related to the grand partition function by [11] Ξ(µ, k) = n∈Z e J(µ+2πin,k) . (3.10) Note that the original integral in (2.15) is along the finite segment, but we can extend the contour to an infinite path C in (3.9) because of the summation over the 2πi-shift (3.10). From the numerical fitting, we find that the perturbative part of the imaginary part of winding Wilson loop is given by Using (3.5), the corresponding canonical VEV is written as In Fig. 2, we show the plot of "free energy" F = − log W n (N, k) for n = 1, 2 as a function of N 3/2 . One can clearly see that the numerical value of W n (N, k) computed from the algorithm in section 2.2 exhibits a nice agreement with the perturbative part given by the Airy function and its derivative (3.12).
Comparison with KMSS. In [17] (which we refer to as KMSS), the perturbative part of winding Wilson loop was obtained as Here H n denotes the harmonic number. We can recast (3.13) in the form of our decomposition in (3.2) Tr U n GC,pert = i n−1 1 2 The first term is the same as (3.3) while the second term is given by which is different from our result (3.11) for general n ≥ 2. For the fundamental representation n = 1, (3.16) agrees with (3.11) since H 0 = 0. In the 't Hooft limit the genus-zero part of (3.11) is equal to (3.16) for general n, but the higher genus parts are different. Note that, for n ≥ 2 the agreement of (3.16) and the matrix model computation has been checked in [17] only for the genus-zero part 3 . To see whether our conjecture (3.11) is correct or not, we should therefore consider the higher genus corrections. It is likely that the approximation used in [17] to derive (3.16) misses some 1/k corrections 4 . Our numerical study strongly suggests that (3.11) is the correct perturbative part.

Fundamental Wilson loop
In this section, we will consider the instanton corrections of the imaginary part of 1/6 BPS Wilson loop W 1 (µ, k) in the fundamental representation, i.e. the winding number n = 1.
Using the instanton expansion (3.5) in terms of the Airy function and its derivatives, one can find the coefficient a j,w in (3.5) by fitting the value of canonical VEV W 1 (N, k) computed either exactly (see section 2.1) or numerically (see section 2.2). The expansion coefficients a j,w becomes simple numbers for some integer k, and one can guess the exact value of the coefficients from the numerical fitting. The results are summarized in Appendix A.1. We have also computed the instanton expansion for fractional k using the numerical computation of the trace Tr(ρ W ) explained in section 2.2. It turns out that the grand canonical VEV of the imaginary part of 1/6 BPS Wilson loop in the fundamental representation has the following expansion 3 For the fundamental representation, the agreement between (3.16) and the matrix model was checked for higher genus corrections as well. While for n > 1, the authors of [17] used (3.16) to predict the higher genus corrections for general winding. 4 We would like to thank Marcos Mariño for raising this possibility.
where W ( ,m) 1 (µ, k) denotes the contribution of the bound state of membrane -instanton and worldsheet m-instanton Worldsheet instantons. First, let us consider the worldsheet instanton corrections (and the perturbative part) From the numerical fitting, we find that the worldsheet instanton corrections are given by Note that the worldsheet instanton in (4.4) has poles at k = 4, 6, which should be canceled by the membrane instanton and the bound state. Thus, we expect that there are membrane instanton corrections in the imaginary part of 1/6 BPS Wilson loop. This is in contrast to the 1/2 BPS Wilson loops where the "pure" membrane instanton corrections are absent except for the bound state corrections coming from the quantum mirror map µ → µ eff [18,20].
Membrane instantons and Bound states. From the numerical fitting, we find that the membrane 1-instanton and the (1, 1) bound state are given by (4.5) As expected, one can show that the poles at k = 4, 6 are canceled between worldsheet instantons (4.4) and membrane instantons or bound states (4.5), and the remaining finite part This pole cancellation mechanism was originally found in the grand potential of ABJM theory [11]. Here we find that the similar pole cancellation mechanism works also in the VEV of 1/6 BPS Wilson loops.
Let us see that our conjecture of instanton corrections (4.4) and (4.5) correctly reproduces the behavior of W 1 (N, k) for non-integer k as well. We take k = 2.7 as an example. For k = 2.7, the instanton factors have the following ordering Once we know the instanton correction in the grand canonical picture W (4.8) We have included the exponential factor e 2 µ * + 4mµ * k in the definition of δ ,m with µ * being the saddle point value of the chemical potential 5 (4.10) The canonical VEV W 1 (N, k) in (4.8) can be evaluated numerically with high precision using the method in section 2.2 even for a non-integer value of k = 2.7. We expect that δ ,m is approximated by As shown in Fig. 3, we find a nice agreement between (4.8) and (4.11), as expected. This confirms the validity of our conjecture of instanton corrections (4.4) and (4.5), for k = 2.7.
We have performed similar checks for various values of k.
Rewriting in terms of µ eff . Let us rewrite W 1 (µ, k) in terms of the "effective" chemical potential µ eff , which was first introduced in [11] µ eff = µ − 2 cos πk 2 e −2µ − (4 + 5 cos πk)e −4µ + · · · . (4.12) This is interpreted as the quantum mirror map of local P 1 × P 1 [13] and the coefficient of this expansion can be easily obtained from the formal solution of the wavefunction annihilated by the quantized mirror curve of local P 1 × P 1 [39].
After rewriting the instanton corrections in (4.4) and (4.5) in terms of µ eff , somewhat miraculously, we find that W 1 (µ, k) is completely factorized up to the (1, 1) bound state where we have introduced the worldsheet instanton factor Q w and the membrane instanton factor Q m by (4.14) Assuming that this factorized structure holds for higher instanton numbers, we can continue the numerical fitting of instanton coefficients. In this way, we find that W 1 is written as 5 Note that this relation (4.10) implies that the worldsheet instanton and the membrane instanton factors in the canonical picture are given by We have already used this relation in (2.36).  where We observe that the coefficient of V m is proportional to the membrane instanton coefficient b (k) in the modified grand potential of ABJM theory [11] From this observation, we conjecture that V m is given by We also observe that f m in (4.18) is equal to the derivative of the effective chemical potential Relation to quantum volume. As shown in [13], the membrane instanton part J b of the modified grand potential of ABJM theory is given by the NS free energy on local P 1 × P 1 . From this relation, V m in (4.20) is further rewritten as 6 where we defined Q m = e −t , t = 2µ eff , q = e i top , top = πk. Note that the normalization of the Planck constant top in topological string is different from the of ABJM Fermi gas (2.8) by a factor of 2, which comes from rewriting the operator ρ −1 into the canonical form of the mirror curve of local P 1 × P 1 [24]. Appearance of the NS free energy in (4.23) suggests that the imaginary part of 1/6 BPS Wilson loop is closely related to the quantum volume Ω of the phase space, which determines the exact quantization condition of the operator ρ Ω = 2π n + 1 2 .
(4.25) 6 For a general local Calabi-Yau, the NS free energy is written as sin w 2 (2jL + 1) sin w 2 (2jR + 1) 2w 2 sin 3 w It is found in [24,[26][27][28] that the quantum volume Ω has two pieces Ω = Ω WKB + Ω np , (4.26) where the WKB part Ω WKB is given by the NS free energy while the non-perturbative part Ω np of quantum volume is given by the "S-dual" of Ω WKB where the "S-dual" variables are given by We find that the membrane instanton part of W 1 (µ, k) is related to Ω WKB by It is found that the pole cancellation at rational value of k is guaranteed by the combination of Ω WKB + Ω np . Since the pole cancellation also occurs in the imaginary part of 1/6 BPS Wilson loop W 1 (µ, k), it is natural to expect that the worldsheet instanton correction V w in (4.17) is related to the S-dual of NS free energy in (4.28). In fact, if we define we find that the ∂ t D Ω np and ∂ t D Ω np have the same singularity structure, and their difference is regular in the convergence region k > 2 of W 1 (µ, k) Putting all together, we arrive at a surprisingly simple expression of the imaginary part of the 1/6 BPS Wilson loop in the fundamental representation (Ω WKB + Ω np ).  where the string coupling g s and the parameter s are defined by From the planar solution of the resolvent of ABJM matrix model, the genus-zero part of the fundamental Wilson loop is written as [6,21] lim N,k→∞ where the end-point of the cut a is given by iκ = 4 sinh 2 a 2 , (4.37) and λ and κ are related by We find that the imaginary part of 1/6 BPS Wilson loop at genus-zero can be written in a simple form where K(k 2 ) and E(k 2 ) denote the complete elliptic integrals of the first and the second kinds, respectively. The worldsheet instanton corrections of W 1 (µ, k) in (4.4) in the grand canonical picture can be easily translated to the canonical picture in the 't Hooft limit One can show that the genus-zero and the genus-one amplitudes in (4.40) are consistent with the matrix model results [6]. Interestingly, from (4.40) we observe that the genus-zero and the genus-one amplitudes are proportional to each other It would be interesting to prove this relation directly from the matrix model.

WKB expansion
In this section, we study the membrane instanton corrections of 1/6 BPS Wilson loops in the fundamental representation from the WKB expansion of Fermi gas. As discussed in [41], the WKB expansion can be systematically analyzed using the spectral trace and the Mellin-Barnes representation. First, we rewrite the small κ expansion of Tr U GC in (2.16) into the Mellin-Barnes type integral representation where c is a constant in the range 2/k < c < 1. Picking up the pole at s = ∈ N, we recover the small κ expansion (2.16). On the other hand, closing the contour in the direction Re(s) < 0 and picking up the poles along the negative real s-axis we find the large µ expansion of Tr U GC . Hence the non-trivial information of instanton corrections is contained in the pole structure of the spectral trace Tr(ρ s W ). In the classical limit → 0, the spectral trace Tr(ρ s W ) can be approximated by the phase space integral Note that Z 0 (s) has poles at s = 2/k − 2 ( ∈ N) corresponding to the membrane instanton corrections. We consider the WKB expansion of the imaginary part of Tr(ρ s W ) where Z 0 (s) is given by ( where the star-product is defined by From this expression, we can easily compute the -expansion of Tr(ρ W ). In this way, we have computed D 2n−1 (s) up to n max = 18. The first few terms of D 2n−1 (s) read  Perturbative part. The perturbative part of W 1 comes from the pole at s = 2/k. The residue of the pole at s = 2/k is given by We do not have an analytic proof of this relation, but we can check this using the diagonal Padé approximation As we can see from Fig. 4, the Padé approximation of D (2/k) nicely agrees with the expected function (5.14).
Membrane 1-instanton. The membrane 1-instanton term comes from the pole at s = 2/k − 2. In order to reproduce the result in (4.5), we find that D(2/k − 2) and D (2/k − 2) should satisfy As in the case of perturbative part, we can check these relations numerically by the Padé approximation. From Fig. 5, one can see that the Padé approximation of D(2/k − 2) and D (2/k − 2) agrees with the exact functions (5.16).
Membrane 2-instanton. We can repeat the same analysis for the membrane 2-instanton. In order to reproduce the result in the previous section, we need the following relations Again, the Padé approximation of the WKB expansion of D(2/k−4) and D (2/k−4) correctly reproduces the expected analytic functions on the right hand side of (5.17) (see Fig. 6).

Comment on winding Wilson loops
One can in principle compute the WKB expansion of winding Wilson loops W n (µ, k) with n ≥ 2 in a similar manner as the n = 1 case studied above. Then W n (µ, k) is written as a Mellin-Barnes type integral where the classical trace Z (a) Figure 6. We show the plot of (a) . The orange dots are the numerical values obtained from the Padé approximation, while the solid curves represent the exact functions in (5.17). and D (n) (s) represents the -corrections. Let us consider the perturbative part coming from the pole at s = 2n/k (5.20) One can easily show that D (n) (2n/k) = 1 as in the case of n = 1, and hence the coefficient of µ in the perturbative part (3.11) is correctly reproduced from the WKB expansion. In order to reproduce the O(µ 0 ) term in (3.11), we have to show that However, it is not so straightforward to reproduce this behavior from the Padé approximation of WKB expansion, since the function on the right hand side of (5.21) has a dense set of poles at rational values of k, accumulating around k = 0. Note that this problem does not occur for n = 1. At present we could not succeed in reproducing the perturbative part of winding Wilson loop from the WKB expansion for n ≥ 2. It would be interesting to derive the perturbative part (3.11) of winding Wilson loop analytically.

Winding Wilson Loops
In this section, we will consider the instanton corrections of the imaginary part of 1/6 BPS winding Wilson loops W n (µ, k) with the winding number n ≥ 2. We can study the the instanton corrections of W n (µ, k) for n ≥ 2 by the same method of numerical fitting we used for the fundamental representation in section 4.

Winding number n = 2
Let us first consider the Wilson loop with winding number n = 2. After rewriting W 2 (µ, k) in terms of the effective chemical potential µ eff , we find that the factorized structure (4.15) found in the fundamental representation also appears in the n = 2 case Interestingly, the membrane instanton correction V m in (6.2) is exactly the same as that of the fundamental Wilson loop. From this observation, we conjecture that V m for n = 2 is again given by the NS free energy (4.23). One can show that poles in the convergence region k > 4 are canceled between worldsheet instantons and membrane instantons, and the remaining finite part reproduces the result in (A.3). This implies that the worldsheet instanton corrections are essentially given by the S-dual of NS free energy (4.28), up to regular terms.

Winding number n ≥ 3
Let us consider the Wilson loops with winding number n ≥ 3. From the convergence condition k > 2n, as the winding number increases the membrane instanton correction e −2µ becomes highly suppressed relative to the worldsheet instanton correction Because of this, it becomes harder and harder to study membrane instanton corrections numerically as the winding number increases. We were unable to find the analytic expressions of the membrane instanton coefficients for n ≥ 3.
On the other hand, we can determine the worldsheet instanton corrections of W n (µ, k) from numerical fitting. For n = 3, the worldsheet instanton corrections are given by and for n = 4, worldsheet instanton corrections are given by We find that the worldsheet 1-instanton correction in the coefficient of µ can be written in a closed form for general n ≥ 2 It would be very interesting to understand the structure of instanton correction of W n (µ, k) and see if it is related to the quantum volume for the general winding number n.

Wilson loops with two boundaries
As pointed out in [17], 1/6 BPS Wilson loops in higher rank representation involves multiparticle interaction in the Fermi gas picture, hence it is not so easy to study the 1/6 BPS Wilson loops in general representations. Here we initiate the study of Wilson loop with two boundaries (Tr U ) 2 , which is related to the second rank symmetric and anti-symmetric representations by Tr U 2 . (7.1) Note that Tr U 2 is the 1/6 BPS Wilson loop with winding number n = 2 which we have studied in the previous section.
From the general formula (2.11), the generating function of 1/6 BPS Wilson loops in the anti-symmetric representations is given by n t n W An = det(1 + κρ(1 + tW )) det(1 + κρ) , (7.2) while the generating function of 1/6 BPS Wilson loops in the symmetric representations is given by Picking up the coefficient of t 2 in the above expressions, we find that W A 2 and W S 2 are written in terms of R defined in (2.14): Note that Tr RW and Tr RW 2 appearing in (7.4) are the grand canonical VEV of 1/6 BPS winding Wilson loops with the winding numbers n = 1, 2, respectively The last term Tr(RW ) 2 in (7.4) is the new contribution which we should compute. It turns out that the grand canonical VEV of (Tr U ) 2 , which we denote by W (1,1) , has a simple expansion in the large µ limit To study the 1/6 BPS Wilson loop with two boundaries (7.7), we have to compute Tr(RW ) 2 . The first two terms in (7.7) have been already obtained in the previous sections. Now let us expand Tr(RW ) 2 in κ as where t is given by Tr(ρ j W ρ −j W ). (7.9) To compute t , we rewrite the trace in (7.9) as Then one can show that t is written as √ k x|ρ |πi . (7.14) The integral I ( ) 1/2 has appeared in the computation of 1/2 BPS Wilson loops in a hook representation [18]. As shown in [18], I Again, we can compute I ( ) 1/6 numerically using the algorithm in section 2.2. For some values of integer k, we can compute the trace t exactly by closing the contour and picking up the residue of poles. For k = 6 we find (7.17) For k = 8 we find (7.18) For k = 12 we find We have checked that the numerical computation using the method in section 2.2 correctly reproduces the exact values of t for k = 6, 8, 12 with high precision. We should stress that we can compute t numerically at arbitrary k in the convergence region k > 4.

Imaginary part of W (1,1)
Let us first consider the imaginary part of W (1,1) , which we denote by W Im (1,1) . From the numerical fitting, we find that the perturbative part of W Im (1,1) is given by We conjecture that the instanton corrections have the following structure where the membrane instanton correction V m is given by the NS free energy (4.23), and the worldsheet instanton corrections are given by (7.23) We have checked that the membrane 1-instanton and (1, 1)-boundstate are correctly reproduced from (7.22). Also, one can check that the pole cancellation between worldsheet instantons and membrane instantons works for k = 6, 8, 12, and the remaining finite terms correctly reproduce the result in Appendix A.5.

Real part of W (1,1)
Next consider the real part of W (1,1) , which we denote by W Re  (1,1) . Again, from the numerical fitting, we find that the perturbative part of W Re (1,1) is given by π 2 k 2 µ 2 eff + b 1 (k)µ eff + c 1 (k) e −2µ eff . (7.26) In the limit k → 6, we find that the membrane 1-instanton (7.26) and the worldsheet 3instanton in (7.25)  The O(µ eff ) term in the worldsheet 3-instanton has a pole at k = 6 which should be canceled by the membrane 1-instanton. From this pole cancellation condition, we conjecture that the membrane 1-instanton coefficient of µ eff in (7.26) has the structure b 1 (k) = cos πk 2 cot πk 2 πk + (regular). (7.28) We leave the further study of membrane instanton corrections of W Re (1,1) as an interesting future problem.

Connected part of W (1,1)
We expect that the connected part of (Tr U ) 2 has a simpler behavior However, this cancellation of µ 2 term does not hold for the worldsheet instanton corrections. It would be interesting to study the structure of the connected part (Tr U ) 2 conn further.

Conclusions
In this paper we have studied the instanton corrections to the VEV of 1/6 BPS Wilson loops in ABJM theory from Fermi gas approach. For the fundamental representation, we find that the grand canonical VEV of the imaginary part of Wilson loop has a remarkably simple form (4.33), and it is closely related to the quantum volume given by the NS free energy on local P 1 × P 1 . The poles at rational value of k in the membrane instanton corrections are canceled by the worldsheet instanton corrections and the singular part of worldsheet instantons is essentially given by the "S-dual" of membrane instantons. This is reminiscent of the exact quantization condition of the spectrum. It would be very interesting to understand the physical meaning of this relation between the quantum volume and the 1/6 BPS Wilson loops. It is curious that the NS free energy is determined by the BPS invariant of closed topological string, while we naively expect that the Wilson loops are related to some open string amplitudes. This type of open-closed duality was observed for the 1/2 BPS Wilson loops in ABJ(M) theory [20]. We speculate that the relation (4.33) between 1/6 BPS Wilson loops and the quantum volume is a "1/6 BPS version" of the open-closed duality. It would be interesting to study 1/6 BPS Wilson loops in ABJ theory along the lines of [20,42,43], and see if the relation to quantum volume also appears in Wilson loops in ABJ theory.
We have also studied 1/6 BPS winding Wilson loops and initiated the study of 1/6 BPS Wilson loops with two boundaries. For both of the imaginary part of winding Wilson loop W n and the imaginary part of Wilson loop with two boundaries W Im (1,1) , we find that the membrane instanton correction V m is again given by the NS free energy (4.23). We also find that the perturbative part of winding Wilson loop is different from the expression in [17]. It would be interesting to derive our result of W pert n in (3.11) analytically. Our method in section 7.1 can in principle be generalized to arbitrary number of boundaries, but the computation becomes cumbersome as the number of boundaries increases. It would be important to develop a more efficient method to compute the multi-boundary Wilson loops both numerically and analytically. From our result of W (1,1) , it is natural to expect that as the number of boundaries increases, the order of polynomial of µ eff in the grand canonical VEV also increases. It would be interesting to find the general structure of the grand canonical VEV of 1/6 BPS Wilson loops with h boundaries.