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Holographic operator product expansion of loop operators in the N=4SO(N) super Yang-Mills theory

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Hong-Zhe Zhang, Wan-Zhe Feng and Jun-Bao Wu. Holographic Operator Product Expansion of Loop Operators in N=4SO(N) Super Yang-Mills Theory[J]. Chinese Physics C. doi: 10.1088/1674-1137/acd364
Hong-Zhe Zhang, Wan-Zhe Feng and Jun-Bao Wu. Holographic Operator Product Expansion of Loop Operators in N=4SO(N) Super Yang-Mills Theory[J]. Chinese Physics C.  doi: 10.1088/1674-1137/acd364 shu
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Holographic operator product expansion of loop operators in the N=4SO(N) super Yang-Mills theory

    Corresponding author: Wan-Zhe Feng, vicf@tju.edu.cn
    Corresponding author: Jun-Bao Wu, junbao.wu@tju.edu.cn
  • Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China

Abstract: In this study, we compute the correlation functions of Wilson(-'t Hooft) loops with chiral primary operators in the N=4 supersymmetric Yang-Mills theory with SO(N) gauge symmetry, which has a holographic dual description of the Type IIB superstring theory on the AdS5×RP5 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 Wilson loops in the spinor representation. We also compare these results to those of the N=4 SU(N) super Yang-Mills theory.

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    I.   INTRODUCTION
    • The holographic duality between the maximally supersymmetric Yang-Mills theory (SYM) with the SU(N) gauge group and Type IIB string theory on the AdS5×S5 background is the 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 is related to a fundamental string with the worldsheet ending on the AdS boundary along the contour of this Wilson loop [2, 3]. The on-shell action, with the boundary terms from the Legendre transformation [4], yields the prediction for the vacuum expectation value (vev) of this Wilson loop at large N and large 't Hooft coupling λg2YMN, when the classical string theory becomes a good approximation with large string tension and small curvature. This holographic prediction matches 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 the Gaussian matrix model [5]. Later, this conjecture was proved using supersymmetric localization [6]. This match provided a highly non-trivial check of the AdS/CFT conjecture since the vev of a 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 [710]. When the rank of the representation is sufficiently high, the back reaction from the D-branes must be considered. A Wilson loop in the higher-rank representation with mixed symmetries is dual to a certain bubbling supergravity solution [1113]. 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 correspond to a D3-brane with the AdS2×S2 worldvolume and k units of fundamental string charge [7]. Half-BPS circular Wilson loops in the rank-k anti-symmetric representation of the gauge group have a bulk description in terms of the AdS2×S4 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. While a 't Hooft loop, which is the magnetic dual of a Wilson loop, can be obtained using S-duality in N=4 SYM. A 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 on 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 a Wilson loop and 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 D3k and D5k branes and replacing the string worldsheet by the brane worldvolume [18]. These results were confirmed by the field theory side using the matrix model [18, 19]. The reduction to this matrix model computations was later confirmed by supersymmetric localization [20].

      The N=4 SYM theory with the gauge group SO(N) has some features different from the SU(N) theory. For odd N, the group is non-simply-laced, and the S-dual theory has the gauge algebra sp(N12) [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 the gauge group SU(N). For even N, the group SO(N) is simply-laced and the dual theory still has the gauge algebra spin(N). Another notable feature regarding Wilson loops in SO(N) theories is the presence of Wilson loops in spinor representations.

      In the string theory, N=4 SO(N) SYM can be realized as the low energy effective theory of coincident D3-branes atop a suitable O3 plane. Based on this, Witten proposed that the N=4 SO(N) SYM is holographically dual to the string theory on the AdS5×RP5 orientifold [21]. The five-dimensional real projective space RP5 is obtained by the five-dimensional sphere S5 by identifying antipodal points, RP5=S5/Z2. 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 RP4 subspace of RP5. The holographic descriptions of Wilson loops in the fundamental, symmetric, and anti-symmetric representations were also studied, and the holographic predictions of their vevs exactly matched the results of supersymmetric localization [22,23]. In this study, we compute the correlation functions of Wilson(-'t Hooft) loops with CPOs of N=4SYM with SO(N)gauge symmetry. The considered line operators include the following:

      ● Half-BPS circular Wilson loops in the fundamental representation of the Lie algebra g=spin(N), W.

      ● Half-BPS circular Wilson loops in the k-th anti-symmetric representation of g, WAk.

      ● Half-BPS circular Wilson loops in the spinor representation of g, Wsp.

      ● Special half-BPS circular WH loops. Recall that WH loops [14] are labelled by (λelec.,λmag.)Λw×Λmw with the identification

      (λelec.,λmag.)(wλelec.,wλmag.),wW.

      (1)

      Here, Λw and Λmw are the weight lattices of g and Lg, respectively, Lg is the GNO dual group [16] of g 2, and W is the Weyl group of g and Lg. We focus on the case in which the W-orbit [λelec.] corresponds to the n-th symmetric representation of g and the W-orbit [λmag.] corresponds to the m-th symmetric representation of Lg. We label these WH loops by WHSn,Sm's.

      The paper is organized as follows. In Sections II and III, we briefly review the dual string description of the N=4SO(N) theory and the half-BPS CPOs with their gravity duals. In Sections IV, V, VI, and VII, we 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 Wilson loops in the spinor representation, respectively. The final section lists our conclusions and provides a discussion. In Appendix A, we briefly discuss the coefficient of the bulk-to-boundary propagator of a certain mode in AdS5.

    II.   THE STRING THEORY DESCRIPTION OF THE N=4 SO(N) THEORY
    • Four-dimensional N=4 SYM with the gauge group SO(N) is dual to the Type IIB superstring theory on the AdS5×RP5 background with Ramond-Ramond (RR) 5-form fluxes F5 [21]. We also choose "discrete torsion'' of the RR 2-form BRR. We will describe this discrete torsion later. In the large N and large 't Hooft coupling limit, the IIB supergravity on AdS5×RP5 is a good approximation of this superstring theory. We set the radius of AdS5, LAdS5 to1; then, the metric of AdS5×RP5 is

      ds2=ds2AdS5+ds2RP5.

      (2)

      The RR 5-form flux is

      F5=4(ω5+˜ω5),

      (3)

      where ω5 and ˜ω5 are the volume forms on AdS5 and RP5 with unit radius, respectively.

      From LAdS5=1, one obtains that [22] in the large N limit,

      4πgsNα=1,

      (4)

      which leads to

      α=2λ,

      (5)

      by using the relation g2YM=8πgs in the SO(N) case [22] and the definition of the 't Hooft coupling λg2YMN.

      The discrete torsions for the Neveu-Schwarz 2-form BNS and the RR 2-form BRR are defined through

      e2πiθNSexp(iRP2BNS)=±1,

      (6)

      e2πiθRRexp(iRP2BRR)=±1.

      (7)

      where we use RP2 inside RP5. When (θNS,θRR)=(0,0) the gauge group of the dual theory is SO(2n). When (θNS,θRR)=(0,12), the gauge group of the dual theory is SO(2n+1).

    III.   CPOs AND THE CORRESPONDING SUPERGRAVITY MODES
    • We plan to compute the correlation functions of half-BPS CPOs and various loop operators. These CPOs are constructed using the six scalar fields Φi,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

      OI=CIi1ilTr(Φi1Φil),

      (8)

      with l2. Here, the trace is taken in the fundamental representation of SO(N), and CI is in the traceless l-th totally symmetric representation of SO(6)R. We choose CI to satisfy

      CIi1ilCJi1il=δIJ,

      (9)

      here, CJi1il is defined as CJi1il=δi1j1δiljlCJj1jl. Since Φi's are N×N anti-symmetric matrices, l should be even for non-vanishing OI. This constraint is new compared with the case in which the gauge group is SU(N).

      For lN, the holographic description of OI is expressed in terms of fluctuations of the background fields in the IIB supergravity on AdS5×RP5, 3

      Gmn_=gmn_+hmn_,

      (10)

      Fm1m5_=fm1m5_+δfm1m5_,δfm1m5_=5[m_1am2m5_],

      (11)

      where gmn_ and fm1m5_ are the background fields (2) and (3), and hmn_ and δfm1m5_ are fluctuations.

      The fluctuations dual to half-BPS CPOs are [24]

      hμν_=65Δsgμν_+4Δ+1(μ_ν_)s,

      (12)

      hαβ_=2Δsgαβ_,

      (13)

      aμ1μ4_=4ϵμ1μ5_μ5_s,

      (14)

      aα1α4_=4Iϵαα1α4_sI(x)α_YI(y).

      (15)

      Here, s(x,y)=IsI(x)YI(y) with x,y being coordinates in the AdS5 part and RP5 part, respectively. (μν_) in (12) assumes the traceless symmetric part. YI(y) is the "scalar spherical harmonics'' on RP5 satisfying,

      α_α_YI=Δ(Δ+4)YI.

      (16)

      They are in the [0,Δ,0] representation of SO(6)R, and we choose the normalization of YI to be the same as the one in [24]. Since RP5=S5/Z2, locally YI is the same as the scalar spherical harmonics on S5. Δ is dual to the conformal dimension of the CPO. For the case at hand, we have Δ=l since it is protected by supersymmetry. Recall that l should be even. In the supergravity side, this is owing to the fact that the Z2 projection of the fields on AdS5×S5 gives the fields on AdS5×RP5. ϵμ1μ5_ and ϵα1α5_ are the anti-symmetric tensors corresponding to the volume form of AdS5 and RP5, respectively. The background five-form field strength can then be expressed as

      Fμ1μ5_=4ϵμ1μ5_,Fα1α5_=4ϵα1α5_.

      (17)
    IV.   OPE OF WILSON LOOPS IN THE FUNDAMENTAL REPRESENTATION
    • We consider the half-BPS Wilson loop in the SO(N) theory in Euclidean space R4,

      W[C]=1NTrPexp[Ci(Aμ(x)˙xμ+i|˙x|ΘjΦj(x))ds],

      (18)

      where the contour C is xμ(s)=(acoss,asins,0,0), ˙xμ=xμs, and Θj is a constant unit 6-vector. The trace is taken in the fundamental representation. For the dual description, we use the Euclidean AdS5 (EAdS5) in the Poincarè coordinates, such that the metric is

      ds2=1z2(dz2+dxi_dxi_).

      (19)

      The action of the fundamental string (F-string) is

      S=12παd2σdetgμν,

      (20)

      with the induced metric gμν being

      gμν=xρ_σμxκ_σμgρκ_.

      (21)

      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]

      x1=a2z2coss,x2=a2z2sins,x3=x4=0.

      (22)

      The worldsheet of this F-string has the topology of EAdS2 and is entirely embedded within the EAdS5 region of the background geometry. 4

      Taking into account the boundary terms from the Legendre transformation [4], the on-shell action of this F-string is given by [4, 17]

      SF1=12πα(2π)=1α.

      (23)

      Using (5), we get [22]

      SF1=λ2.

      (24)

      Thus, the holographic prediction for the vev of the Wilson loop is

      W[C]=expλ2,

      (25)

      in the large N and large λ limit.

      When probing W[C] from a distance L much larger than its radius a, the operator product expansion (OPE) of W[C] is

      W[C]=W[C](1+i,nCniaΔniOni),

      (26)

      where Δni are the conformal weights of the operator Oni, O0i is the i-th primary field, and Oni's with n>0 are its conformal descends.

      To extract the OPE coefficients of the half-BPS CPOs OI with normalized two-point functions, we can compute the normalized correlation of this Wilson loop and the half-BPS CPO OI, 5

      OI(x)W[C]W[C]OI(x)NOIW[C],

      (27)

      where NOIis defined by the two point function ofOI,

      OI(y)OJ(z)=δIJNOI|yz|2ΔOI.

      (28)

      Taking the OPE limit where L=x2a, we have

      OI(x)W[C]=C,OaΔL2Δ.

      (29)

      The goal is to compute C,O holographically, which is the OPE coefficient of the primary operator OI in the expansion (26).

      To achieve this, we need to calculate the change in the F-string action owing to the fluctuations of the background fields dual to OI [17],

      δSF1=12παd2σdetgμν12gμνxρ_σμxκ_σνhρκ_,

      (30)

      where σμ's are the worldsheet coordinates and xρ_=xρ_(σμ) expresses how the string worldsheet is embedded in the spacetime.

      Then, we write sI as sI(x,z)=d4xGΔ(x;x,z)sI0(x); here, sI0 is a source for OI on the boundary, and

      GΔ(x;x,z)=c(zz2+|xx|2)Δ,

      (31)

      is the boundary-to-bulk propagator with the constant c being 6

      c=Δ+12(3Δ)/2NΔ.

      (32)

      Then, the correlation function is given by

      OI(x)W[C]=δSF1δsI0(x)|sI0=0.

      (33)

      In the OPE limit, we have

      GΔ(x,x,z)czΔL2Δ,

      (34)

      μ_sIδzμ_ΔzsI,

      (35)

      μ_ν_sIδzμ_δzν_Δ(Δ1)z2sI.

      (36)

      We use these and the fact that in the Poincarè coordinates

      Γzμν_=zgμν_2zδzμ_δzν_.

      (37)

      Then, from (12), we get

      hμν_2Δgμν_sIYI+4Δz2δzμ_δzν_sIYI,

      (38)

      The induced metric is

      gss=a2z2z2,gsz=0,gzz=a2z2(a2z2).

      (39)

      We have

      det(gμν)=a2z4.

      (40)

      From these, we obtain

      gμνxρ_σμxρ_σνhρκ_=2Δz2a2sIYI.

      (41)

      Then, the variation of the F-string action is

      δSF1=ΔYIπαad2σsI.

      (42)

      Using (31), we get

      OI(x)W[C]=δSF1δsI0(x)|sI0=0=ΔYI(y)cπαaL2Δd2σzΔ=ΔYI(y)cπαaL2Δπ0dψa0dzzΔ=YI(y)2cΔα(Δ+1)aΔL2Δ.

      (43)

      Now, using (5) and (32), we obtain

      OI(x)W[C]=YI(y)2Δ/21λΔNaΔL2Δ.

      (44)

      Thus, the OPE coefficient is 7

      C,O=YI(y)2Δ/21λΔN.

      (45)

      We use the convention that the factor YI(y) is not included in the OPE coefficient, which leads to

      C,O=2Δ/21λΔN.

      (46)

      The above result expressed in terms of λ,N, and Δ is identical to the result obtained in the SU(N) case [17]. Since the string worldsheet is an AdS2 subspace completely embedded inside the AdS5 part of the background geometry, the change from S5 to RP5 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. The coefficient inside the bulk-to-boundary propagator, c, is cSO=2cSU. These two effects cancel each other; thus, the results of the OPE coefficients in terms of λ,N,Δ are identical for both SO(N) and SU(N). However, one should keep in mind that Δ should be even in the case of SO(N).

    V.   OPE OF WH LOOPS IN THE SYMMETRIC REPRESENTATION
    • In this section, we compute the OPE coefficients of half-BPS circular WH loops in the symmetric representation. WH loops appear owing to the worldlines of dyons that carry both electric and magnetic charges of the gauge theory. In this section, we only consider the case in which the dyons are in the n-th symmetric representation of g and the m-th symmetric representation of Lg. 8 When m=0, we get the following Wilson loops in the n-th symmetric representation,

      WSn[C]=1dimSnTrSnPexp[Ci(Aμ(x)˙xμ+i|˙x|ΘjΦj(x))ds],

      (47)

      where Sn denotes the n-th symmetric representation of SO(N) and dimSn denotes its dimensionality.

      Non-trivially generalizing the results in [7], it was proposed in [15] that for the SU(N) case, WH loops are dual to D3-branes in AdS5×S5. In [22], a D3-brane dual to a Wilson loop in the symmetric representation for the SO(N) case was given. We expect that generalizing the solution in [15] to the AdS5×RP5 case will provide the dual description of WH loops in the symmetric representation, in the SO(N) case.

      We start with the coordinate system in EAdS5, such that the metric takes the form [7]

      ds2=1z2(dz2+dr21+r21dψ2+dr22+r22dϕ2).

      (48)

      The boundary of the EAdS5 is now at r and η=0. In this coordinate, the AdS5 part of the RR 4-form potential is

      CAdS4=r1r2z4dr1dψdr2dϕ.

      (49)

      We place the WH loop on the boundary at r1=a,r2=0. We make the following coordinate transformation:

      r1=acosηcoshρsinhρcosθ,

      (50)

      r2=asinhρsinθcoshρsinhρcosθ,

      (51)

      z=asinηcoshρsinhρcosθ.

      (52)

      The metric on EAdS5 in this coordinate system is

      ds2=1sin2η[dη2+cos2ηdψ2+sinh2ρ(dθ2+sin2θdϕ2)].

      (53)

      We only consider the case in which the theta angle in the field theory is zero. This corresponds to setting the background RR zero form potential (the axion), C0, to zero. Then, the action of the D3-brane on the AdS5×RP5 background is

      SD3=SD3DBI+SD3WZ,

      (54)

      where

      SD3DBI=TD3d4σdet(g+2παF),

      (55)

      SD3WZ=TD3P[C4].

      (56)

      Here, g is the induced metric on the D3-brane, F is the electromagnetic field on the D3-brane worldvolume, P[C4] is the pull-back of C4 to the worldvolume, and the D3-brane tension reads

      TD3=1(2π)3α2gs=N2π2,

      (57)

      where the relations α=2/λ and g2YM=8πgs in the 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 consider the components Fψρ and Fθϕ 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

      sinη=κ1sinhρ,κ=n2λ32N2+2π2m2λ,

      (58)

      Fψρ=inλ16πNsinh2ρ,Fθϕ=msinθ2.

      (59)

      Let us introduce a dual 't Hooft coupling 9, ˜λ=16π2N2ngλ, where ng=1 for g=spin(2n), and ng=2 for g=spin(2n+1). Then, we can express κ as

      κ=14Nn2λ2+2m2˜λ2ng.

      (60)

      Taking into account the boundary terms, the on-shell action of the D3-brane is

      SD3=2N(κ1+κ2+sinh1κ).

      (61)

      Thus, the holographic prediction of the vacuum expectation value of the WH is

      WHSn,Sm[C]=exp[2N(κ1+κ2+sinh1κ)].

      (62)

      When we take m=0, this D3-brane solution becomes the same as the one in [22], though in different coordinates. Furthermore, the holographic prediction for WSn is consistent with the results from localization [22] in the large λ limit with κ fixed.

      Now, we holographically compute the correlator of WHSn,Sm[C] and OI(x)

      OI(x)WHSn,Sm[C]WHSn,Sm[C]OI(x)NOWHSn,Sm[C],

      (63)

      in the OPE limit La, and extract the OPE coefficient CWHSn,Sm,O. The change in SD3DBI due to the fluctuations of the background field is

      δSD3DBI=TD3d4σdetM12(M1)μνxρ_σμxκ_σνhρ_κ_,

      (64)

      where we have defined the matrix M=g+2παF, and σμ's are worldvolume coordinates. By using the result of hρκ_ in the OPE limit given in (38) and the above D3-brane solution, we obtain

      δSD3DBI=4Nκ2YIdρdθsinθsinh2ρ×(12κ2+1sinh2ρ(κ2sin2θ)(coshρsinhρcosθ)2)sI.

      (65)

      Now, we compute the change of SWZ due to the fluctuations of the background fields,

      δSD3WZ=TD3P[a4].

      (66)

      From (14), we have

      aIμμ4_=4Δzϵμμ4_zsIYI=4ΔCμμ4_sIYI.

      (67)

      Thus,

      δSD3WZ=4TD3ΔYIP[C4]sI.

      (68)

      From the coordinate transformation (50)−(52), we obtain

      δSD3WZ=8NΔκ4YIdρdθsinθsinh2ρ(11κ2sinhρcosθcoshρsinhρcosθ).

      (69)

      Then, the total change of the action is

      δSD3=δSD3DBI+δSD3WZ=4NΔYIsinh1κ0dρ×π0dθsinθ(coshρsinhρcosθ)2sI.

      (70)

      Using sI(x,z)=d4xGΔ(x;x,z)sI0(x), we can compute O(x)WHSn,Sm[C] as

      O(x)WHSn,Sm[C]=δSD3δs0(x).

      (71)

      In the OPE limit, we have

      O(x)WHSn,Sm[C]=aΔL2Δ4NΔcYI(y)κΔsinh1ρ0dρ×π0dθsinhΔρsinθ(coshρsinhρcosθ)2+Δ.

      (72)

      Taking the two integrals, we get

      CWHSn,Sm,O=2(Δ+3)/2ΔYI(y)sinh(Δsinh1κ).

      (73)

      Thus,

      CWHSn,Sm,O=2(Δ+3)/2Δsinh(Δsinh1κ)=i(1)Δ/22(Δ+3)/2ΔVΔ(iκ).

      (74)

      Here, Vn(x)=sin(ncos1x) is one type of the Chebyshev polynomials, and we have used the fact that Δ is even.

      The result for the Wilson loop (m=0) in terms of κ is 2 times the results in [18] for the SU(N) case due to the change of c. 10 Here, we provide a brief explanation on this point. Since the worldvolume of the D3-brane is completely inside AdS5, 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. In the SO(N) case, the relation between α and λ reads α=2/λ, while the relation g2YM=8πgs in the SO(N) case is also changed compared with the SU(N) case. However, their effects on TD3 cancel each other. The relation between TD3 and N, i.e., TD3=N/(2π2), is unchanged. Formally, when we express the results in terms of κ and Δ, the only change is from the coefficient of the bulk-to-boundary propagator cSO=2cSU. This leads to the above conclusion about the OPE coefficients. However, the relation between κ and λ changes in the case of SO(N), becoming

      κ=n4Nλ2,

      (75)

      while for the Wilson loop in the n-th symmetric representation of spin(N) in the SU(N) case, the relation reads

      κ=nλ4N.

      (76)

      Hence, the result in terms of λ and Δ in the SO(N) case is not just a constant multiplying the result in the SU(N) case.

      Finally, to compare with the result for C,O in (46), we set m=0 in (74) and take the κ0 limit. Using κ=n4Nλ/2 in this case, we obtain

      CWSn,O2(Δ+3)/2Δκ=2Δ/21ΔλNn,

      (77)

      which is just nC,O, as expected.

    VI.   OPE OF WILSON LOOPS IN THE ANTI-SYMMETRIC REPRESENTATION
    • Let us consider half-BPS circular Wilson loops in the rank-k anti-symmetric representation of the gauge group SO(N),

      WAk[C]=1dimAkTrAkPexp[Ci(Aμ(x)˙xμ+i|˙x|ΘjΦj(x))ds].

      (78)

      They have a bulk description in terms of the D5-brane with k units of fundamental string charge. The worldvolume of this D5-brane has topology AdS2×S4. The D5 description of Wilson loops is valid in the large N and large λ limit with k/N fixed.

      We can parameterize the unit S5; 6i=1z2i=1 as

      z1=cosθ,zj+1=sinθwj,j=1,j=2,,5,

      (79)

      with 5j=1w2j=1. Then, the metric of the unit S5 can be written as

      dΩ25=dθ2+sin2θdΩ24,

      (80)

      with dΩ24 as the metric of the unit S4.

      RP5 can be obtained from S5 by identifying antipodal points zizi. One way to realize this is to view RP5 as the upper hemisphere of S5 (0θπ/2) with antipodal points on the equator (θ=π/2) identified. The metric of RP5 is thus given by

      ds2RP5=dθ2+sin2θds24,0θπ/2

      (81)

      where ds24=dΩ24 when θ<π/2, and ds24 is the metric of RP4 when θ=π/2.

      Hence, the metric of AdS5×RP5 reads

      ds2=cosh2u(dζ2+sinh2ζdψ2)+du2+sinh2u(dϑ2+sin2ϑdϕ2)+dθ2+sin2θds24,

      (82)

      with the radius of AdS5 and RP5 set to 1. The AdS5 part of the above metric is written in the form of an AdS2×S2 fibration for computational convenience, and these coordinates are related to the one in (48) by the following coordinate transformation:

      r1=acoshusinhζcoshucoshζcosϑsinhu,

      (83)

      r2=asinhusinϑcoshucoshζcosϑsinhu,

      (84)

      z=acoshucoshζcosϑsinhu,

      (85)

      where a is the radius of the Wilson loop.

      Using the SO(6)R transformation, we set ΘIin (78) to ΘI=(1,0,,0). Then, in AdS5×RP5, the D5-brane dual to this antisymmetric Wilson loop occupies the AdS2 in the above metric with u=0 and wraps an S4 submanifold of RP5 at a constant polar angle θk (on the upper hemisphere of S5) [22]. The D5-brane worldvolume is AdS2×S4AdS5×RP5, and its metric reads

      d˜s2=dζ2+sinh2ζdψ2+sin2θkdΩ24.

      (86)

      Turning on the worldvolume U(1) gauge field Fψζ to account for the k units of fundamental brane charge, the action of the D5-brane on the AdS5×RP5 background can be written as

      SD5=SD5DBI+SD5WZ,

      (87)

      where

      SD5DBI=TD5d6σdet(g+2παF),

      (88)

      SD5WZ=2παiTD5FP[C(4)].

      (89)

      In the above equations, the tension of the D5-brane reads

      TD5=1gs(2π)5(α)3=N8π4λ2,

      (90)

      and the self-dual 4-form potential is [8]

      C(4)=4(u8sinh4u32)dH2dΩ2(32θsin2θ+18sin4θ)dΩ4.

      (91)

      Here, dH2 is the volume form of the unit AdS2, sinhζdζdψ, dΩ2 is the volume form of the unit S2, sinϑdϑdϕ, and dΩ4 is the volume form of the unit S4.

      The fact that the flux of the worldvolume gauge field equals k, together with the brane equations of motion, gives rise to the condition [8] 11

      θksinθkcosθk=πkN,

      (92)

      and the worldvolume gauge field is

      Fψζ=iλ/2sinhζcosθk2π.

      (93)

      The on-shell D5-brane DBI and WZ action are

      SD5DBI=2N3πλ2dζsinhζsin5θk.

      (94)

      SD5WZ=4iN3dζFψζ(32θsin2θ+18sin4θ),

      (95)

      Adding appropriate boundary terms [8], the on-shell action for the D5-brane is

      SD5=SD5DBI+SD5WZ+SD5bdy=2N3πλ2sin3θk.

      (96)

      Thus, the holographic prediction for the expectation value of the Wilson loop in the rank k antisymmetric representation is given by

      WAk=exp(2N3πλ2sin3θk).

      (97)

      The variation of the DBI part of the action to the first order in the fluctuation hμν and hαβ is

      δSD5DBI=TD5d6σdet(g+2παF)((g+2παF)1)mn×12(hμν_mXμ_nXν_+hαβ_mXα_nXβ_)=N3πλ2sin5θkdζsinhζ×(4Δcosh2ζsin2θk+8Δ)sΔYΔ,0(θk),

      (98)

      where we have used the D5 solution z=a/coshζ, c.f., (85). The variation of the WZ part of the action to the first order in the fluctuation is given by 12

      δSD5WZ=2παiTD5FP[a(4)]=2παiTD5dψdζdσ1dσ2dσ3dσ4μ(Ω4)Fψζ×4sin4θSIθYI=8N3πλ2cosθksin4θkdζsinhζsΔθkYΔ,0(θk),

      (99)

      where the 4-form fluctuation is given by

      aσ1σ2σ3σ4=4sin4θμ(Ω4)sIθYI

      (100)

      with σ1,σ2,σ3,σ4 being the coordinates on S4 and the corresponding measure μ(Ω4) is

      μ(Ω4)=sin3σ1sin2σ2sinσ3.

      (101)

      Thus, the variation of the D5 action to the first order is given by

      δSD5=δSD5DBI+δSD5WZ.

      (102)

      The normalized correlation function between the Wilson loop and the CPO is evaluated as

      WAk(C)OΔ(L)NOW(C)=δSD5δs0

      (103)

      Recall that

      sI(x,z)=d4xGΔ(x,x,z)sI0(x),

      (104)

      where the bulk-to-boundary propagator

      GΔ(x,x,z)=c(zz2+|xx|2)ΔczΔL2Δ

      (105)

      and the D5 solution z=a/coshζ. The only integral operation one needs to perform is

      0dζsinhζcoshΔ+2ζ=1Δ+1.

      (106)

      Hence, we obtain

      WAkOΔ(L)NOWAk=aΔL2Δ2Δ/2NΔ3πλΔΔ+3Δ1sin3θk×[2(Δ+1)cosθkC(2)Δ1ΔC(2)Δ],

      (107)

      where we have used the following results for the SO(5) invariant harmonics [18] 13

      YΔ,0(θk)=NΔC(2)Δ(cosθk),

      (108)

      and

      ΔYΔ,0(θk)+cosθksinθkθkYΔ,0(θk)=NΔ[Δsin2θkC(2)Δ(Δ+3)cosθksin2θkC(2)Δ1].

      (109)

      The normalization factor NΔ is obtained from

      YΔ,0(0)=NΔC(2)Δ(1)=NΔ(Δ+3)!6Δ!,

      (110)

      yielding

      NΔ=6Δ!(Δ+3)!YΔ,0(0).

      (111)

      We then obtain

      WAkOΔ(L)NOWAk=aΔL2ΔYΔ,0(0)2Δ/23πΔλ6(Δ2)!(Δ+1)!sin3θk×[2(Δ+1)cosθkC(2)Δ1ΔC(2)Δ].

      (112)

      Using the recurrence relation [29, 30]

      ΔC(λ)Δ(x)=2(Δ+λ1)xC(λ)Δ1(x)(Δ+2λ2)C(λ)Δ2(x),

      (113)

      we finally arrive at

      CWAk,O=2Δ/23πΔλsin3θk6(Δ2)!(Δ+1)!C(2)Δ2(cosθk).

      (114)

      This SO(N)result is identical to that for the SU(N) case obtained in [18]. The D5-brane worldvolume has topology AdS2×S4 with AdS2 in the AdS5 part of the background geometry and S4 in the RP5 part. Since θk<π/2, the S4 we consider in this case is the same as the S4 embedded in S5 determined by θ=θk in the parameterization given in (79). Thus, the computation of the coupling of the supergravity modes to the D5-brane is the same as that for the SU(N) case, although the expression of TD5 in terms of λ and N for the SO(N) case is different from that for the SU(N) case.

      In the SO(N) case, we have TD5=N8π4λ2, while for the SU(N) case the relation is TD5=Nλ8π4. Taking this change and the relation cSO=2cSU into account, we arrive at the conclusion that the OPE coefficients in the SO(N) case are the same as the ones in the SU(N) case. Thus, in the kN limit, the relation CAk,O=kC,O remains the same as that in the SU(N) case. This can be obtained from the result θ3k3πk/2N in this limit and

      C(2)Δ2(1)=(Δ+1)!6(Δ2)!.

      (115)
    VII.   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),

      WS[C]=1dimSTrSPexp[Ci(Aμ(x)˙xμ+iΘiΦi(x))ds].

      (116)

      The dual description of this Wilson loop is in terms of the D5-brane whose worldvolume has topology AdS2×RP4 [21]. If we still chose the ΦI to be ΘI=(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. Taking into account the boundary terms, the total on-shell action of this D5-brane is

      SD5=N3πλ2,

      (117)

      so the holographic prediction for the expectation value of the Wilson loop in the spinor representation is [22]

      WS=exp(N3πλ2).

      (118)

      As observed in [22], Fψζ, given by (93), vanishes when θk=π/2. A shortcut to compute the OPE coefficient CS,O using the result obtained in the previous section is by setting θk=π/2 in CAk,O and dividing the result by 2 to take into account the change of the D5-brane worldvolume from AdS2×S4 into AdS2×RP4,

      CS,O=12CWAk,O|θk=π/2=2Δ/23πΔλ6(Δ2)!(Δ+1)!C(2)Δ2(0)=(2)Δ/21Δλπ(Δ21).

      (119)

      Here, we have used the fact that, for even Δ,

      C(2)Δ(0)=(1)Δ/2Δ+22,

      (120)

      obtained from the following generating function of the Gegenbauer polynomials C(λ)Δ(x):

      1(12xt+t2)λ=Δ=0C(λ)Δ(x)tΔ.

      (121)
    VIII.   CONCLUSION
    • In this study, we investigated the holographic duality of the N=4 SO(N) SYM theory and the Type IIB string theory on the AdS5×RP5 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. Wilson loops were expanded in terms of local operators when the probing distances were much larger than the sizes of the Wilson loops. The coefficients were extracted from the expansion for the operators we considered. Our focus was on the half-BPS CPOs and their corresponding gravity duals. Specifically, we computed the correlation functions of local CPOs and the Wilson loops 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 the 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 the gauge group SO(N) has some features different from the SU(N) theory. We compared our results with those of the N=4 SU(N) SYM theory.

    APPENDIX A: THE COEFFICIENT c OF THE BULK-TO-BOUNDARY PROPAGATORS
    • In this appendix, we compute the coefficient c of the bulk-to-boundary propagator of the modes sI. The action for sI, obtained from the full "actual" action of IIB supergravity [31] is [24]

      S=AdS5d5xdet(gAdS5)12BI[μsIμsI+Δ(Δ4)(sI)2],

      where BI is given by

      BI=16κ2Δ(Δ1)(Δ+2)Δ+1z(Δ),

      where κ is the coupling constant of type IIB supergravity, and z(Δ) is explained below. Using

      2κ2=(2π)7g2α4,

      and the relations α=2/λ and gs=g2YM/(8π) for the SO(N) case, we obtain

      κ2=(2π)58N2,

      which is same as the one for the SU(N) case. z(Δ) is defined by

      RP5d5ydet(gRP5)YIYJ=δIJz(Δ).

      The expression for z(Δ) is

      z(Δ)=π32Δ(Δ+1)(Δ+2),

      which equals to half of the result in the SU(N) case since the integration is over RP5=S5/Z2. Using the above result, we obtain

      BI=22ΔN2Δ(Δ1)π2(Δ+1)2.

      The coefficient of the bulk-to-boundary propagator is

      c=α0BI,

      where [17]

      α0=Δ12π2,

      which is identical for both SO(N) and SU(N) cases. Finally, we obtain

      c=Δ+12(3Δ)/2NΔ,

      which equals 2 times the result for the SU(N) case.

Reference (31)

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