α -expansion of antisymmetric Wilson loops in N = 4 SYM from Fermi gas

...................................................................................................................Westudythelarge’tHooftcouplingexpansionof1/2BPSWilsonloopsintheantisymmetricrepresentationin N = 4 super Yang–Mills (SYM) theory at the leading order in the 1 / N expansion.ViaAdS/CFT correspondence, this expansion corresponds to the α (cid:3) expansion in bulk type IIB string theory. We show that this expansion can be systematically computed by using the low temperature expansion of the Fermi distribution function, known as the Sommerfeld expansion in statistical mechanics. We check numerically that our expansion agrees with the exact result of antisymmetric Wilson loops recently found by Fiol and Torrents.


Introduction
1/2 BPS circular Wilson loops in 4D N = 4 super Yang-Mills (SYM) theory are interesting observables that can be computed exactly by a Gaussian matrix model [1][2][3]. Via AdS/CFT correspondence, 1/2 BPS Wilson loops in the fundamental representation correspond to the fundamental string in type IIB string theory on AdS 5 × S 5 [4,5]. When the rank of the representations becomes large, the corresponding dual objects in the bulk are not fundamental strings but D-branes, and such Wilson loops are sometimes called "giant Wilson loops". In particular, 1/2 BPS Wilson loops in the rank-k symmetric and antisymmetric representations correspond to D3-branes and D5-branes, respectively, with k units of electric flux on their world volumes [6][7][8][9]. The leading term in the 't Hooft expansion of N = 4 SYM side is successfully matched with the DBI action of D-branes in the bulk side. For more general representations, a dictionary between Wilson loops in higher-rank representations and bulk D-brane pictures was proposed in Refs. [10,11].
We are interested in the subleading corrections in this correspondence. Recently, there has been some progress in the computation of one-loop corrections in the 1/N expansion of giant Wilson loops [12][13][14][15]. Here we will focus on the subleading corrections in the large λ expansion (or 1/λ expansion) with λ being the 't Hooft coupling λ = g 2 YM N , and we will restrict ourselves to the leading order in the 1/N expansion. From the holographic dictionary R 2 AdS /α = √ λ, the large λ expansion on the SYM side corresponds to the α expansion in the bulk string theory side.
In this paper, we consider the large λ expansion of the 1/2 BPS Wilson loops in the antisymmetric representation. Using the fact that the generating function of antisymmetric representations can be written as a system of fermions, one can systematically compute the subleading corrections in the large λ expansion by a low-temperature expansion of the Fermi distribution function, known as the Sommerfeld expansion. Here the role of temperature is played by 1/ √ λ. We have checked PTEP 2016, 113B05 M. Horikoshi and K. Okuyama numerically that the subleading corrections agree with the exact expression of the antisymmetric Wilson loops recently found in Ref. [16].
The rest of the paper is organized as follows. In Sect. 2, we find a systematic large λ expansion of antisymmetric Wilson loops using the Sommerfeld expansion of Fermi distribution function. Our main result is Eq. (2.10). In Sect. 3, we compare our result (2.10) with the exact expression in Ref. [16], and find nice agreement. We conclude in Sect. 4 and discuss some future directions.

Large λ expansion of Wilson loops in the antisymmetric representation
We consider the vacuum expectation value (VEV) of 1/2 BPS circular Wilson loops in N = 4 SYM with gauge group U (N ). After applying the supersymmetric localization [3], the Wilson loop VEV is reduced to a Gaussian matrix model Here x μ (s) parametrizes a great circle of S 4 on which N = 4 SYM lives, and I (I = 1, . . . , 6) denote the adjoint scalar fields in N = 4 SYM and θ I ∈ S 5 is a constant unit vector. In Eq. (2.1), g s denotes the string coupling, which is related to the Yang-Mills gauge coupling g YM by In this paper, we will focus on the Wilson loop VEV in the kth antisymmetric representation R = A k , It is convenient to define the VEV of SU(N ) part by removing the U (1) contribution: We are interested in the behavior of the Wilson loop VEV W A k in the limit In the large λ limit together with Eq. (2.6), the antisymmetric Wilson loop W A k is holographically dual to a D5-brane in AdS 5 × S 5 , whose world volume has the form AdS 2 × S 4 [7]. From the computation of the DBI action of the D5-brane, the leading behavior of W A k is found to be where θ k is given by M. Horikoshi and K. Okuyama From the bulk D5-brane picture, the angle θ k parametrizes the position of the S 4 part of the world volume inside the S 5 of bulk geometry AdS 5 × S 5 . We are interested in the subleading corrections of W A k . There are two expansion parameters g s and 1/λ. In Ref. [12], it was reported that the one-loop correction in the g s expansion has the form where c is an order 1 constant. In this paper, we will consider subleading corrections of the 1/λ expansion while we focus on the leading order in the g s -expansion. As we will show below, the 1/λ expansion of W A k can be computed as This is our main result. Let us explain how we obtained Eq. (2.10). To study the antisymmetric Wilson loops systematically, it is convenient to introduce the generating function of W A k by summing over k with fugacity e μ , where O mm denotes the expectation value in the Gaussian matrix model, Using the large N factorization we find det(1 + e μ e M ) mm ≈ exp Tr log(1 + e μ e M ) mm (2.13) up to 1/N corrections, and the right-hand side of Eq. (2.13) in the planar limit becomes where ρ(m) is the Wigner semicircle distribution of the Gaussian matrix model Then, as discussed in Ref. [8], the Wilson loop VEV in the kth antisymmetric representation is written as an integral over the chemical potential μ: where μ * is determined by the saddle point equation Note that the fluctuation of the μ-integral around the saddle point gives rise to a subleading correction in g s , as in the case of the ABJM Fermi gas [17], and hence we can safely ignore such corrections for our purpose of studying the leading-order behavior in the g s expansion. 1 Noticing that the Fermi distribution function naturally appears in the derivative of J (μ), one can easily compute the 1/λ expansion by the standard Sommerfeld expansion in statistical mechanics, where 1/ √ λ plays the role of temperature. The large λ expansion of the Fermi distribution function reads from which the expansion of J (μ) is found to be

Comparison with the exact result
Let us compare our result (2.10) with the exact result of antisymmetric Wilson loops at finite N and k found in Ref. [16]. It is found in Ref. [16] that the generating function of W A k , Eq. (2.11), is exactly written as a characteristic polynomial of the N × N matrix A, N k=0 e kμ W A k = det(1 + e μ A), (3.1) where the matrix element A i,j is given by the generalized Laguerre polynomial From expression (3.1), one can extract the exact value of the Wilson loop VEV W A k at arbitrary values of N , k, and g s . In Fig. 1, we show the plot of log W A k as a function of k/N for N = 300 and λ = 100, corresponding to the value of string coupling g s = λ/4πN = 1/12π . The blue dots are the exact values obtained from Eq. (3.1) while the red dots are the plot of our result (2.10) for the leading term ( Fig. 1(a)) and the leading + next-to-leading terms ( Fig. 1(b)). One can clearly see that the leading + next-to-leading terms in Fig. 1(b) exhibit nice agreement with the exact result (3.1). Interestingly, the leading term alone is not enough to reproduce the behavior of the exact result (3.1), and the next-to-leading correction has a rather large contribution for this choice of parameters N = 300, λ = 100. Note that the leading and the next-to-leading terms in Eq. (2.10) are of order λ 3/2 /g s and λ 1/2 /g s , respectively, while the higher-order terms have negative powers of λ; hence in the large λ limit higher-order corrections in Eq. (2.10) are suppressed. Indeed we have checked that the inclusion of higher-order corrections does not change the plot significantly, and the exact result (3.1) is well approximated already at the next-to-leading order. We have performed similar numerical checks for various values of N and λ (N , λ 1) and find good agreement for all cases.

Conclusion
We have computed the 1/λ expansion (or the α -expansion of bulk type IIB string theory) of the Wilson loop VEV in the antisymmetric representation using the Sommerfeld expansion of the Fermi distribution function. It would be very interesting to reproduce this result from the computation of α -correction of the D5-brane action in the AdS 5 × S 5 background.
There are many things to be studied further. It is important to develop a method to compute both the 1/λ expansion and the g s expansion systematically. In particular, it would be interesting to find the 1/λ expansion of the Wilson loop VEV in the symmetric representation by the low-temperature expansion of the Bose distribution. However, the integrand of the μ-integral might have a singularity corresponding to the onset of Bose-Einstein condensation. It would be interesting to understand the analytic structure of the integrand in the case of symmetric representation (see Ref. [8] for a discussion).
Also, it is interesting to understand the convergence property of the expansion. For the 1/2 BPS Wilson loop in the fundamental representation, it is observed that the α -expansion is not Borel summable [2], reflecting the fact that there are corrections of order exp − √ λ , which is nonperturbative in α . It would be very interesting to understand the Borel summability of α -corrections at fixed g s for Wilson loops in various representations. On the other hand, the g s expansion of 1/2 BPS Wilson loops with fixed λ seems to have a finite radius of convergence, 2 which is consistent with the absence of Yang-Mills instanton corrections to Wilson loop VEV in N = 4 SYM [3]. We hope that the study of Wilson loops in various representations will provide us with precious information on the nonperturbative structure of string theory. Finally, we would like to emphasize the importance of our findings. It was reported in Ref. [12] that there is a discrepancy between the computation on the field theory side and the string theory side of the one-loop correction in 1/N (2.9) (see [19] for the current status of this problem). 3 However, PTEP 2016, 113B05 M. Horikoshi and K. Okuyama before settling the issue of this problem of one-loop correction, we have to compute the leading term at large N as a function of λ, including all 1/λ corrections (2.10). After having found the leading term (2.10), one can try to study the one-loop correction in 1/N , either numerically or analytically, by subtracting the leading term from the exact result (3.1). In this computation, it is important to subtract the subleading terms in 1/λ since they are rather large and cannot be neglected, as we saw in Sect. 3. We believe that our result (2.10) is an important first step to resolving the issue of the one-loop discrepancy. We leave the study of the one-loop correction as an interesting future problem.