Holographic Operator Product Expansion of Loop Operators in $\mathcal{N}=4$ $SO(N)$ Super Yang-Mills Theory

In this paper, we compute the correlation functions of Wilson(-'t~Hooft) loops with chiral primary operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory with $SO(N)$ gauge symmetry, which has a holographic dual description of Type IIB superstring theory on the $AdS_{5}\times\mathbf{RP}^{5}$ background. Specifically, we compute the coefficients of the chiral primary operators in the operator product expansion of Wilson loops in the fundamental representation, Wilson-'t Hooft loops in the symmetric representation, Wilson loops in the anti-fundamental representation and the spinor representation. We also compare these results to the $\mathcal{N}=4$ $SU(N)$ super Yang-Mills theory.


Introduction
The holographic duality between the maximally supersymmetric Yang-Mills theory (SYM) with SU (N ) gauge group and Type IIB string theory on the AdS 5 × S 5 background is a most studied example of the AdS/CFT correspondence [1]. The vacuum expectation values of Wilson loops are natural observables in gauge theories and they are also calculable from the AdS side. In the string theory description, a Wilson loop 1 in the fundamental representation are related to a fundamental string whose worldsheet ending on the AdS boundary along the contour of this Wilson loop [2,3]. The on-shell action, with the boundary terms from Legendre transformation included [4], give the prediction for the vacuum expectation value (vev) of this Wilson loop at large N and large 't Hooft coupling λ ≡ g 2 YM N , in which case the classical string theory is a good approximation with large string tension and small curvature. This holographic prediction matches with the field theory results in the large N and λ limit. The field theory results were obtained based on the conjecture that the computations can be reduced to the ones in a Gaussian matrix model [5]. Later this conjecture was proved using supersymmetric localization [6]. This match provided a highly non-trivial check to the AdS/CFT conjecture since the vev of the Wilson loop is a non-trival function of λ and N . Higher rank Wilson loops in gauge theories are dual to D-branes carrying electric flux on the their worldvolume [7][8][9][10]. When the rank of the representation is sufficiently large, the back reaction from the D-branes must be considered. The Wilson loop in the higher rank representation with mixed symmetries is dual to a certain bubbling supergravity solution [11][12][13]. We will not discuss such supergravity solutions in this paper.
Specifically, half-BPS circular Wilson loops in the rank-k symmetric representation of the gauge group corresponds to D3-brane with AdS 2 × S 2 worldvolume and k units of fundamental string charge [7]. While half-BPS circular Wilson loops in the rank-k antisymmetric representation of the gauge group have a bulk description in terms of AdS 2 × S 4 D5-brane with k units of fundamental string charge [8]. These D-branes are 1/2-BPS and preserve the same SO(2, 1)× SO(3)× SO (5) isometries. On the other hand, the 't Hooft loop, which is the magnetic dual of Wilson loop, can be obtained using S-duality in N = 4 SYM. General SL(2, Z) transformation maps a Wilson loop to a Wilson-'t Hooft (WH) loop [14]. It was proposed in [15] that a WH loop in symmetric representations of both the gauge group and its Goddard-Nuyts-Olive (GNO) dual group [16] (the Langlands dual group) is dual to a D3-brane carrying both F-string and D-string charges. More details of such WH loops will be provided later in this section. A circular Wilson loop can be expanded in a series of local operators with different conformal dimensions, when the probing distance is much larger than the radius of this loop. Half-BPS chiral primary operators (CPOs) are an important class of operators with protected dimensions appearing in this operator product expansion (OPE). The OPE coefficient can be extracted from the correlation function of the Wilson loop and the local operators [17]. In the large N and λ limit, the correlation function of a Wilson loop in the fundamental representation with a CPO can be derived by calculating the coupling of the supergravity modes dual to this CPO to the string worldsheet [17]. Similar procedure can be used to compute the correlator of a higher rank Wilson loop with a CPO using D3 k and D5 k branes and replacing string worldsheet by the brane worldvolume [18]. These results were confirmed by the results in the field theory side using the matrix model [18,19]. The reduction to this matrix model computations was later proved by supersymmetric localization [20].
The N = 4 SYM theory with gauge group SO(N ) has some features different from the SU (N ) theory. When N is odd, the group is non-simply-laced and the S-dual theory has gauge algebra sp( N −1 2 ) [16]. In this case, the gauge algebras before and after the S-duality transformation are different. This is distinct from the S-duality transformation of the theory with gauge group SU (N ). When N is even, the group SO(N ) is simply-laced and the dual theory still has the gauge algebra spin(N ). Another notable feature regarding the Wilson loops in SO(N ) theories is the presence of Wilson loops in spinor representations.
In string theory, N = 4 SO(N ) SYM can be realized as the low energy effective theory of coincident D3-branes on top of a suitable O3 plane. Based on this, Witten proposed that the N = 4 SO(N ) SYM is holographically dual to string theory on the AdS 5 ×RP 5 orientifold [21]. The five-dimension real projective space RP 5 is obtained by the five-dimensional sphere S 5 by identifying antipodal points, RP 5 = S 5 /Z 2 . This correspondence was recently studied in [22]. It has been demonstrated that the expectation value of the Wilson loop in the spinor representation of the gauge group, calculated through supersymmetric localization [22,23], precisely matches the result obtained from the D5-brane, with its worldvolume including the RP 4 subspace of RP 5 . The holographic description of Wilson loops in the fundamental, symmetric, and anti-symmetric representations were also studied, and the holographic prediction of their vevs matches exactly with the result from supersymmetric localization [22,23]. In this paper, we compute the correlation functions of Wilson(-'t Hooft) loops with chiral primary operators of N = 4 SYM with SO(N ) gauge symmetry. The line operators we will consider include: • The half-BPS circular Wilson loop in the fundamental representation of Lie algebra g = spin(N ), W .
• The half-BPS circular Wilson loops in the k-th anti-symmetric representation of g, W A k .
• The half-BPS circular Wilson loops in the spinor representation of g, W sp .
Here Λ w and Λ mw are weight lattices of g and L g, respectively. L g is the GNO dual group [16] of g 2 , W is the Weyl group of g and L g. We focus on the case that the W -orbit [λ elec. ] corresponds to the n-th symmetric representation of g and the W -orbit [λ mag. ] corresponds to the m-th symmetric representation of L g. We label these WH loops by W H Sn,Sm 's.
The paper is organized as follows. In Section 2 and 3, we will briefly review the dual string description of the N = 4 SO(N ) theory and the half-BPS CPOs with their gravity duals. In Section 4, 74, 6 and 7, we will compute the OPE coefficients of these CPOs in the OPE expansion of the Wilson loops in the fundamental representation, the WH loops in the symmetric representation, the Wilson loops in the anti-fundamental representation and the spinor representation, respectively. The final section is devoted to the conclusion and the discussions. In Appendix A, we briefly discuss the coefficient of the bulk-to-boundary propagator of a certain mode in AdS 5 .
2 The string theory description of the N = 4 SO(N ) theory Four-dimensional N = 4 SYM with gauge group SO(N ) is dual to Type IIB superstring theory on the AdS 5 × RP 5 background with Ramond-Ramond (RR) 5-form fluxes F 5 [21]. We should make also a choice of "discrete torsion" of RR 2-form B RR . We will describe this discrete torsion later. In the large N and large 't Hooft coupling limit, the IIB supergravity on AdS 5 × RP 5 is a good approximation of this superstring theory. We choose the radius of AdS 5 , L AdS 5 to be 1, then the metric of AdS 5 × RP 5 is The RR 5-form fluxes is where ω 5 andω 5 are the volume forms on AdS 5 and RP 5 with unit radius, respectively. From L AdS 5 = 1, one can get that [22] in the large N limit, which leads to by using the relation g 2 YM = 8πg s in the SO(N ) case [22] and the definition of the 't Hooft coupling λ ≡ g 2 YM N . The discrete torsion for the Neveu-Schwarz 2-form B N S and the RR 2-form B RR are defined through where we pick up a RP 2 inside RP 5 . When (θ N S , θ RR ) = (0, 0) the gauge group of the dual theory is SO(2n). When (θ N S , θ RR ) = (0, 1 2 ) the gauge group of the dual theory is SO(2n+1).

Chiral primary operators and the corresponding supergravity modes
We plan to compute the correlation functions of half-BPS chiral primary operators (CPOs) and various loop operators. These CPOs are constructed using the six scalar fields Φ i , i = 1, · · · , 6, which are in the adjoint representation of SO(N ) and the vector representation of SO(6) R , the R-symmetry group of this theory. Such CPOs are with l ≥ 2. Here the trace is taken in the fundamental representation of SO(N ) and C I is in the tracelass l-th totally symmetric representation of SO(6) R . We choose C I to satisfy here C Ji 1 ···i l is defined as C Ji 1 ···i l = δ i 1 j 1 · · · δ i l j l C J j 1 ···j l . Since Φ i 's are N × N anti-symmetric matrices, l should be even for O I being non-vanishing. This constraint is new compared to the case where the gauge group is SU (N ).
When l ≪ N , the holographic description of O I is expressed in terms of fluctuations of the background fields in IIB supergravity on AdS 5 × RP 5 , 3 where g mn and f m 1 ···m 5 are the background fields (2) and (3), h mn and δf m 1 ···m 5 are fluctuations.
The fluctuations dual to half-BPS CPOs are [24] h µν = − Here s(x, y) = I s I (x)Y I (y) with x, y being coordinates in AdS 5 part and RP 5 part, respectively. (µν) in (12) means to take the traceless symmetric part. Y I (y) is the "scalar spherical harmonics" on RP 5 satisfying, They are in the [0, ∆, 0] representation of SO(6) R and we choose the normalization of Y I to be the same as the one in [24]. Since RP 5 = S 5 /Z 2 , locally Y I is the same as the scalar spherical harmonics on S 5 . ∆ is dual to the conformal dimension of the CPO. For the case at hand, we have ∆ = l since it is protected by supersymmetry. Recall that l should be even. In the supergravity side, this is from the fact that the Z 2 projection of the fields on AdS 5 × S 5 gives the fields on AdS 5 × RP 5 . ǫ µ 1 ···µ 5 and ǫ α 1 ···α 5 are the anti-symmetric tensors corresponding to the volume form of AdS 5 and RP 5 , respectively. The background five-form field strength can then be expressed as

OPE of Wilson loops in the fundamental representation
We consider half-BPS Wilson loop in the SO(N ) theory in Euclidean space R 4 , where the contour C is x µ (s) = (a cos s, a sin s, 0, 0),ẋ µ = ∂x µ ∂s , and Θ j is a constant unit 6-vector. The trace is taken in the fundamental representation. For the dual description, we use the Euclidean AdS 5 (EAdS 5 ) in the Poincarè coordinates, such that the metric is The action of the fundamental string (F-string) is with the induced metric g µν being As for the F-string solution dual to the circular Wilson loop, we choose the worldsheet coordinates to be (z, s). The corresponding classical F-string solution can be parameterized as [4,17] x 1 = a 2 − z 2 cos s , The worldsheet of this F-string has the topology of EAdS 2 and is entirely embedded within the EAdS 5 region of the background geometry. 4 Taking into account the boundary terms from Legendre transformation [4], the on-shell action of this F-string is given by [4,17] Using (5), we get [22] Thus the holographic prediction for the vev of the Wilson loop is in the large N and large λ limit. When probing W [C] from a distance L much larger than its radius a. The operator where ∆ n i are the conformal weight of the operator O n i . O 0 i is the i-th primary field and O n i 's with n > 0 are its conformal descends.
To extract the OPE coefficients of the half-BPS CPOs O I with normalized two-point functions, we can compute the normalized correlation of this Wilson loop and the half-BPS where N O I is defined by the two point function of O I 's, The goal is to compute C ,O holographically, which is the OPE coefficient of the primary operator O I in the expansion (26). For this goal, we need to calculate the change of F-string action due to the fluctuations of the background fields dual to O I [17], where σ µ 's are worldsheet coordinates and x ρ = x ρ (σ µ ) expresses how the string worldsheet is embedded in the spacetime.
Then we write s I as is the boundary-to-bulk propagator with the constant c being 6 Then the correlation function is given by In the OPE limit, we have Using these and the fact that in the Poincarè coordinates Then from (12), we get The induce metric is .
(39) 6 This constant c is √ 2 times the corresponding constant in the AdS5 × S 5 case given in [24,25], due to the fact that RP 5 = S 5 /Z2. For more details, see Appendix A.
We have From these, we obtain Then the variation of the F-string action is Using (31), we get Now by using (5) and (32), we obtain Thus the OPE coefficient is 7 We use the convention that the factor Y I (y) is not included in the OPE coefficient which leads to The above result expressed in terms of λ, N and ∆, is identical to the result obtained in the SU (N ) case [17]. Since the string worldsheet is an AdS 2 subspace completely embedded inside the AdS 5 part of the background geometry, the change from S 5 to RP 5 does not impact the calculation of the coupling between the supergravity modes and the string worldsheet. The relation between α ′ and λ in the SO(N ) case is α ′ = 2/λ, which has an extra factor of √ 2, compared with the relation α ′ = √ λ in the SU (N ) case. While the coefficient inside the bulk-to-boundary propagator, c, is also changed as c SO = √ 2c SU . These two effects cancel with each other, thus the results of the OPE coefficients in terms of λ, N, ∆ are identical in the cases of both SO(N ) and SU (N ). But one should keep in mind that ∆ should be even for the case of SO(N ).

OPE of WH loops in the symmetric representation
In this section, we will compute the OPE coefficients of the half-BPS circular Wilson loop-'t Hooft loops in the symmetric representation. The WH loop comes from the worldline of a dyon which carry both electric and magnetic charges of the gauge theory. In this section, we will only consider the case when the dyon is in the n-th symmetric representation of g and the m-th symmetric representation of L g. 8 When m = 0, we get the following Wilson loops in the n-th symmetric representation, where S n denotes the n-th symmetric representation of SO(N ) and dimS n denotes its dimension. Non-trivially generalizing the results in [7], it was proposed in [15] that for SU (N ) case, the Wilson-'t Hooft (WH) loop is dual to D3-branes in AdS 5 × S 5 . In [22], D3-brane dual to Wilson loop in the symmetric representation for the SO(N ) case was given. We expect the generalization of the solution in [15] to AdS 5 × RP 5 case will provide the dual description of the WH loop in the symmetric representation for the SO(N ) case.
The boundary of EAdS 5 is now at r → ∞ and η = 0. In this coordinate the AdS 5 part of the RR 4-form potential is chosen to be We place the WH loop on the boundary at r 1 = a, r 2 = 0. We make the following coordinate transformation, The metric on EAdS 5 in this coordinate system is ds 2 = 1 sin 2 η dη 2 + cos 2 ηdψ 2 + sinh 2 ρ(dθ 2 + sin 2 θdφ 2 ) .
We only consider the case when the theta angle in the field theory is zero. This corresponds to set the background RR zero form potential (the axion), C 0 , to be zero. Then the action of the D3-brane in AdS 5 × RP 5 background is where Here g is the induced metric on the D3-brane, F the electromagnetic field on the D3-brane worldvolume, P [C 4 ] is the pull-back of C 4 to the worldvolume and the D3-brane tension reads where the relations α ′ = 2/λ and g 2 YM = 8πg s in SO(N ) case have been used. For the D3-brane dual to the above WH loop, we take the worldvolume coordinates to be ρ, ψ, θ, φ, and η = η(ρ) on the worldvolume. We also need to turn on the components F ψρ and F θφ of the electromagnetic field strength on the D3-brane worldvolume.
The D3-brane solution, obtained by adjusting the solution in [15] to the SO(N ) case, is given by Let us introduce dual 't Hooft coupling 9 ,λ = 16π 2 N 2 ng λ , where n g = 1 for g = spin(2n), and n g = 2 for g = spin(2n + 1). Then we can express κ as κ = 1 4N Taking into account the boundary terms, the on-shell action of the D3-brane is Thus the holographic prediction of vacuum expectation value of WH is When we take m = 0, this D3-brane solution is the same as the one in [22], though in different coordinates. Furthermore, the holographic prediction for W Sn is consistent with the results from localization [22] in the large λ limit with κ fixed. Now we holographically compute the correlator of W H Sn,Sm [C] and O I (x) 9 This results is fromλ = L g 2 YM N , with the dual Yang-Mills coupling L gYM = 4π √ ng g YM when θYM = 0 [26][27][28].
in the OPE limit L ≫ a, and extract the OPE coefficient C W H Sn,Sm ,O . The change of the S D3 DBI due to the fluctuations of the background field is where we have defined the matrix M = g + 2πα ′ F , σ µ 's are worldvolume coordinates. By using the result of h ρκ in the OPE limit given in (38) and the above D3-brane solution, we obtain Now we turn to compute the change of S WZ due to the fluctuations of the background fields, From (14), we have Thus From the coordinate transformation (50)-(52), we obtain Then the total change of the action is Using In the OPE limit, we have Performing the two integral we get Thus Here V n (x) = sin(n cos −1 x) is one type of the Chebyshev polynomials and we have use the fact that ∆ is even. The result for Wilson loop (m = 0) in terms of κ is √ 2 times the results in [18] for SU (N ) case due to the change of c. 10 Here we provide a brief explanation on this point. Since the worldvolume of D3-brane is completely inside AdS 5 , the calculations of the coupling between the supergravity modes and the D3-brane worldvolume for both SU (N ) and SO(N ) cases are the same. For the SO(N ) case, the relation between α ′ and λ reads α ′ = 2/λ, while the relation g 2 YM = 8πg s in the SO(N ) case is also changed compared with the SU (N ) case. However, their effects on T D3 cancel with each other. The relation between T D3 and N , i.e., T D3 = N/(2π 2 ) is unchanged. Formally, when we express the results in terms of κ and ∆, the only change is from the coefficient of the bulk-to-boundary propagator c SO = √ 2c SU . This leads to the above conclusion about the OPE coefficients. However, the relation between κ and λ is changed for the case of SO(N ), which is while for the Wilson loop in the n-th symmetric representation of spin(N ) in the SU (N ) case, the relation reads Hence, the result in terms of λ and ∆ for the SO(N ) case is not just a constant multiplying the result in the SU (N ) case. Finally, to compare with the result about in C ,O in (46), we set m = 0 in (74) and take the κ → 0 limit. Using κ = n 4N λ/2 in this case, we obtain which is just nC ,O , as expected.

OPE of Wilson loops in the anti-symmetric representation
Let us consider the half-BPS circular Wilson loops in the rank-k anti-symmetric representation of the gauge group SO(N ), They have a bulk description in terms of D5-brane with k units of fundamental string charge. The worldvolume of this D5-brane has topology AdS 2 × S 4 . The D5 description of Wilson loop is valid in the large N large λ limit with k/N fixed.
We can parameterize the unit S 5 , 6 i=1 z 2 i = 1 as z 1 = cos θ, z j+1 = sin θw j , j = 1, · · · , 5, with 5 j=1 w 2 j = 1. Then the metric of unit S 5 can be written as with dΩ 2 4 the metric of unit S 4 . RP 5 can be obtained from S 5 by identifying antipodal points z i ∼ −z i . One way to realize this is to view RP 5 as the upper hemisphere of S 5 (0 ≤ θ ≤ π/2) with antipodal points on the equator (θ = π/2) identified. The metric of RP 5 is thus given by where ds ′2 4 = dΩ 2 4 when θ < π/2 and the ds ′2 4 is the metric of RP 4 when θ = π/2. Hence, the metric of AdS 5 × RP 5 reads ds 2 = cosh 2 u(dζ 2 + sinh 2 ζdψ 2 ) + du 2 + sinh 2 u(dϑ 2 + sin 2 ϑdφ 2 ) + dθ 2 + sin 2 θds ′2 4 , (82) with the radius of AdS 5 and RP 5 set to be 1. The AdS 5 part of the above metric is written in the form of an AdS 2 × S 2 fibration for computation convenience and these coordinates are related to the one in (48) by the following coordinate transformation, where a is the radius of the Wilson loop.
Turning on the worldvolume U (1) gauge field F ψζ to account for the k units of fundamental brane charge, the action of the D5-brane in AdS 5 × RP 5 background can be written as where In the above equations the tension of D5-brane reads and the self-dual 4-form potential is chosen to be [8] Here dH 2 is the volume form of the unit AdS 2 , sinh ζdζ ∧ dψ, dΩ 2 is the volume form of the unit S 2 , sin ϑdϑ ∧ dφ, and dΩ 4 is the volume form of the unit S 4 . The fact that the flux of the worldvolume gauge field equal to k together with the brane equations of motion give rise to the condition [8] 11 and the worldvolume gauge field is The on-shell D5-brane DBI and WZ action is Adding appropriate boundary terms [8], the on-shell action for the D5-brane is Thus the holographic prediction for the expectation value of the Wilson loop in the rank k antisymmetric representation is given by The variation of the DBI part of the action to the first order in the fluctuation h µν and h αβ is where we have used the D5 solution z = a/ cosh ζ, c.f., (85). The variation of the WZ part of the action to the first order in the fluctuation is given by 12 where 4-form fluctuation is given by with σ 1 , σ 2 , σ 3 , σ 4 the coordinates on the S 4 and the corresponding measure µ(Ω 4 ) is Thus the variation of the D5 action to the first order is given by The normalized correlation function between the Wilson loop and the chiral primary operator is evaluated as Recall that where the bulk-to-boundary propagator and the D5 solution z = a/ cosh ζ. The only integral one needs to perform iŝ Hence we obtain where we have used the following results about the SO(5) invariant harmonics [18] 13 and The normalization factor N ∆ is obtained by and thus We then obtain ∆ .
(112) Using the recurrence relation [29,30] ∆C (λ) we finally arrive This SO(N ) result is identical to the SU (N ) case obtained in [18]. The D5-brane worldvolume has topology AdS 2 × S 4 with AdS 2 in the AdS 5 part of the background geometry and S 4 in the RP 5 part. Since θ k < π/2, the S 4 we consider in this case is the same as the S 4 embedded in S 5 determined by θ = θ k in the parametrization given in (79)

OPE of Wilson loops in the spinor representation
Now we turn to the half-BPS circular Wilson loop in the spinor representation S of SO(N ), The dual description of such Wilson loop is in term of D5-brane whose worldvolume has topology AdS 2 × RP 4 [21]. If we still chose the Φ I to be Θ I = (1, 0, · · · , 0). The embedding of the D5-brane is given by u = 0, θ = π/2 in the coordinates used in the previous section [22]. In this case, the field strength of the worldvolume U (1) gauge field vanishes. Taken into account the boundary terms, the total on-shell action of this D5-brane is so the holographic prediction for the expectation value of the Wilson loop in the spinor representation is [22] W S = exp N 3π As observed in [22], F ψζ , given by (93), vanishes when θ k = π/2. A shortcut to compute the OPE coefficient C S,O using the result obtained in the previous section is by setting θ k = π/2 in C A k ,O and dividing the result by 2 to take into account the change of D5-brane worldvolume from AdS 2 × S 4 into AdS 2 × RP 4 , C S,O = 1 2 C W A k ,O θ k =π/2 = 2 ∆/2 3π √ ∆λ 6(∆ − 2)! (∆ + 1)! C ∆−2 (0) = (−2) ∆/2−1 √ ∆λ π(∆ 2 − 1) .
Here we have used that, for even ∆, C obtained from the following generating function of the Gegenbauer polynomials C (λ) ∆ (x),

Conclusion
In this paper we studied the holographic duality of the N = 4 SO(N ) SYM theory and the Type IIB string theory on the AdS 5 × RP 5 background in the large N and λ limit. To this end, we investigated the OPE coefficients of half-BPS circular Wilson loops in various representations. The Wilson loop can be expanded in terms of local operators when the probing distances are much larger than the size of the Wilson loop. The coefficients can be extracted from the expansion for the operators we consider. Our focus is on the half-BPS CPOs and their corresponding gravity duals. Specifically, we computed the correlation functions of local CPOs and the Wilson loop in the fundamental representation, the symmetric representation, the anti-symmetric representation and the spinor representation. We studied the SO(N ) Wilson loops in the symmetric/anti-symmetric representations through their dual D3/D5-brane descriptions. The appearance of Wilson loops in the spinor representation is a new feature in the SO(N ) theories. In addition, we discussed the WH loops in the symmetric representation using a D3-brane with both electric and magnetic charges. The N = 4 SYM theory with gauge group SO(N ) has some features different from the SU (N ) theory. We compared our results with the N = 4 SU (N ) SYM theory.
which equals to half of the result in the SU (N ) case since the integration is over RP 5 = S 5 /Z 2 . Using the above result, we obtain The coefficient of the bulk-to-boundary propagator is where [17] which is identical for both SO(N ) and SU (N ) cases. Finally we arrive which equals √ 2 times the result in the SU (N ) case.