-
The holographic duality between the maximally supersymmetric Yang-Mills theory (SYM) with the
SU(N) gauge group and Type IIB string theory on theAdS5×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 loop1 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 couplingλ≡g2YMN , 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 [7−10]. 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 [11−13]. 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 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 theAdS2×S4 D5-brane with k units of fundamental string charge [8]. These D-branes are1/2 -BPS and preserve the sameSO(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 inN=4 SYM. A generalSL(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 D3
k 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 groupSO(N) has some features different from theSU(N) theory. For odd N, the group is non-simply-laced, and the S-dual theory has the gauge algebrasp(N−12) [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 groupSU(N) . For even N, the groupSO(N) is simply-laced and the dual theory still has the gauge algebraspin(N) . Another notable feature regarding Wilson loops inSO(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 theN=4 SO(N) SYM is holographically dual to the string theory on theAdS5×RP5 orientifold [21]. The five-dimensional real projective spaceRP5 is obtained by the five-dimensional sphereS5 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 theRP4 subspace ofRP5 . 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 ofN=4 SYM withSO(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.),w∈W.
(1) Here,
Λw andΛmw are the weight lattices of g andLg , respectively,Lg is the GNO dual group [16] of g2 , and W is the Weyl group of g andLg . 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 ofLg . We label these WH loops byWHSn,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 inAdS5 . -
Four-dimensional
N=4 SYM with the gauge groupSO(N) is dual to the Type IIB superstring theory on theAdS5×RP5 background with Ramond-Ramond (RR)5 -form fluxesF5 [21]. We also choose "discrete torsion'' of the RR2 -formBRR . We will describe this discrete torsion later. In the large N and large 't Hooft coupling limit, the IIB supergravity onAdS5×RP5 is a good approximation of this superstring theory. We set the radius ofAdS5 ,LAdS5 to1 ; then, the metric ofAdS5×RP5 isds2=ds2AdS5+ds2RP5.
(2) The RR
5 -form flux isF5=4(ω5+˜ω5),
(3) where
ω5 and˜ω5 are the volume forms onAdS5 andRP5 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 theSO(N) case [22] and the definition of the 't Hooft couplingλ≡g2YMN .The discrete torsions for the Neveu-Schwarz
2 -formBNS and the RR2 -formBRR are defined throughe2πiθNS≡exp(i∫RP2BNS)=±1,
(6) e2πiθRR≡exp(i∫RP2BRR)=±1.
(7) where we use
RP2 insideRP5 . When(θNS,θRR)=(0,0) the gauge group of the dual theory isSO(2n) . When(θNS,θRR)=(0,12) , the gauge group of the dual theory isSO(2n+1) . -
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 ofSO(N) and the vector representation ofSO(6)R , the R-symmetry group of this theory. Such CPOs areOI=CIi1⋯ilTr◻(Φi1⋯Φil),
(8) with
l≥2 . Here, the trace is taken in the fundamental representation ofSO(N) , andCI is in the traceless l-th totally symmetric representation ofSO(6)R . We chooseCI to satisfyCIi1⋯ilCJi1⋯il=δIJ,
(9) here,
CJi1⋯il is defined asCJi1⋯il=δi1j1⋯δiljlCJj1⋯jl . SinceΦi 's areN×N anti-symmetric matrices, l should be even for non-vanishingOI . This constraint is new compared with the case in which the gauge group isSU(N) .For
l≪N , the holographic description ofOI is expressed in terms of fluctuations of the background fields in the IIB supergravity onAdS5×RP5 ,3 Gmn_=gmn_+hmn_,
(10) Fm1⋯m5_=fm1⋯m5_+δfm1⋯m5_,δfm1⋯m5_=5∇[m_1am2⋯m5_],
(11) where
gmn_ andfm1⋯m5_ are the background fields (2) and (3), andhmn_ andδfm1⋯m5_ are fluctuations.The fluctuations dual to half-BPS CPOs are [24]
hμν_=−65Δsgμν_+4Δ+1∇(μ_∇ν_)s,
(12) hαβ_=2Δsgαβ_,
(13) aμ1⋯μ4_=−4ϵμ1⋯μ5_∇μ5_s,
(14) aα1⋯α4_=4∑Iϵαα1⋯α4_sI(x)∇α_YI(y).
(15) Here,
s(x,y)=∑IsI(x)YI(y) withx,y being coordinates in theAdS5 part andRP5 part, respectively.(μν_) in (12) assumes the traceless symmetric part.YI(y) is the "scalar spherical harmonics'' onRP5 satisfying,∇α_∇α_YI=−Δ(Δ+4)YI.
(16) They are in the
[0,Δ,0] representation ofSO(6)R , and we choose the normalization ofYI to be the same as the one in [24]. SinceRP5=S5/Z2 , locallyYI is the same as the scalar spherical harmonics onS5 . Δ is dual to the conformal dimension of the CPO. For the case at hand, we haveΔ=l since it is protected by supersymmetry. Recall that l should be even. In the supergravity side, this is owing to the fact that theZ2 projection of the fields onAdS5×S5 gives the fields onAdS5×RP5 .ϵμ1⋯μ5_ andϵα1⋯α5_ are the anti-symmetric tensors corresponding to the volume form ofAdS5 andRP5 , respectively. The background five-form field strength can then be expressed asFμ1⋯μ5_=−4ϵμ1⋯μ5_,Fα1⋯α5_=−4ϵα1⋯α5_.
(17) -
We consider the half-BPS Wilson loop in the
SO(N) theory in Euclidean spaceR4 ,W◻[C]=1NTr◻Pexp[∮Ci(Aμ(x)˙xμ+i|˙x|ΘjΦj(x))ds],
(18) where the contour C is
xμ(s)=(acoss,asins,0,0) ,˙xμ=∂xμ∂s , andΘj is a constant unit6 -vector. The trace is taken in the fundamental representation. For the dual description, we use the EuclideanAdS5 (EAdS5 ) in the Poincarè coordinates, such that the metric isds2=1z2(dz2+dxi_dxi_).
(19) The action of the fundamental string (F-string) is
S=12πα′∫d2σ√detgμν,
(20) with the induced metric
gμν beinggμν=∂xρ_∂σμ∂xκ_∂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=√a2−z2coss,x2=√a2−z2sins,x3=x4=0.
(22) The worldsheet of this F-string has the topology of
EAdS2 and is entirely embedded within theEAdS5 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]⟩=exp√λ2,
(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) ofW◻[C] isW◻[C]=⟨W◻[C]⟩(1+∑i,nCniaΔniOni),
(26) where
Δni are the conformal weights of the operatorOni ,O0i is the i-th primary field, andOni 's withn>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 CPOOI ,5 ⟨⟨OI(x)⟩⟩W◻[C]≡⟨W◻[C]OI(x)⟩√NOI⟨W◻[C]⟩,
(27) where
NOI is defined by the two point function ofOI ,⟨OI(y)OJ(z)⟩=δIJNOI|y−z|2ΔOI.
(28) Taking the OPE limit where
L=√x2≫a , 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 operatorOI 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=12πα′∫d2σ√detgμν12gμν∂xρ_∂σμ∂xκ_∂σνhρκ_,
(30) where
σμ 's are the worldsheet coordinates andxρ_=xρ_(σμ) expresses how the string worldsheet is embedded in the spacetime.Then, we write
sI assI(x,z)=∫d4x′GΔ(x′;x,z)sI0(x′) ; here,sI0 is a source forOI on the boundary, andGΔ(x′;x,z)=c(zz2+|x−x′|2)Δ,
(31) is the boundary-to-bulk propagator with the constant c being
6 c=Δ+12(3−Δ)/2N√Δ.
(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_sI≃δzμ_ΔzsI,
(35) ∂μ_∂ν_sI≃δzμ_δzν_Δ(Δ−1)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=a2−z2z2,gsz=0,gzz=a2z2(a2−z2).
(39) We have
det(gμν)=a2z4.
(40) From these, we obtain
gμν∂xρ_∂σμ∂xρ_∂σνhρκ_=−2Δz2a2sIYI.
(41) Then, the variation of the F-string action is
δSF1=−ΔYIπα′a∫d2σsI.
(42) Using (31), we get
⟨⟨OI(x)⟩⟩W◻[C]=−δSF1δsI0(x)|sI0=0=ΔYI(y)cπα′aL2Δ∫d2σzΔ=ΔYI(y)cπα′aL2Δ∫π0dψ∫a0dzzΔ=YI(y)2cΔα′(Δ+1)aΔL2Δ.
(43) Now, using (5) and (32), we obtain
⟨⟨OI(x)⟩⟩W◻[C]=YI(y)2Δ/2−1√λΔNaΔL2Δ.
(44) Thus, the OPE coefficient is
7 C◻,O=YI(y)2Δ/2−1√λΔN.
(45) We use the convention that the factor
YI(y) is not included in the OPE coefficient, which leads toC◻,O=2Δ/2−1√λΔN.
(46) The above result expressed in terms of
λ,N , and Δ is identical to the result obtained in theSU(N) case [17]. Since the string worldsheet is anAdS2 subspace completely embedded inside theAdS5 part of the background geometry, the change fromS5 toRP5 does not impact the calculation of the coupling between the supergravity modes and the string worldsheet. The relation betweenα′ and λ in theSO(N) case isα′=√2/λ , which has an extra factor of√2 , compared with the relationα′=√λ in theSU(N) case. The coefficient inside the bulk-to-boundary propagator, c, iscSO=√2cSU . These two effects cancel each other; thus, the results of the OPE coefficients in terms ofλ,N,Δ are identical for bothSO(N) andSU(N) . However, one should keep in mind that Δ should be even in the case ofSO(N) . -
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 Whenm=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 ofSO(N) anddimSn 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 inAdS5×S5 . In [22], a D3-brane dual to a Wilson loop in the symmetric representation for theSO(N) case was given. We expect that generalizing the solution in [15] to theAdS5×RP5 case will provide the dual description of WH loops in the symmetric representation, in theSO(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 atr→∞ andη=0 . In this coordinate, theAdS5 part of the RR4 -form potential isCAdS4=r1r2z4dr1∧dψ∧dr2∧dϕ.
(49) We place the WH loop on the boundary at
r1=a,r2=0 . We make the following coordinate transformation:r1=acosηcoshρ−sinhρcosθ,
(50) r2=asinhρsinθcoshρ−sinhρcosθ,
(51) z=asinηcoshρ−sinhρcosθ.
(52) The metric on
EAdS5 in this coordinate system isds2=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 theAdS5×RP5 background isSD3=SD3DBI+SD3WZ,
(54) where
SD3DBI=TD3∫d4σ√det(g+2πα′F),
(55) SD3WZ=−TD3∫P[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 ofC4 to the worldvolume, and the D3-brane tension readsTD3=1(2π)3α′2gs=N2π2,
(57) where the relations
α′=√2/λ andg2YM=8πgs in theSO(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 , andη=η(ρ) on the worldvolume. We also need to consider the componentsFψρ andFθϕ 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 bysinη=κ−1sinhρ,κ=√n2λ32N2+2π2m2λ,
(58) Fψρ=inλ16πNsinh2ρ,Fθϕ=msinθ2.
(59) Let us introduce a dual 't Hooft coupling
9 ,˜λ=16π2N2ngλ , whereng=1 forg=spin(2n) , andng=2 forg=spin(2n+1) . Then, we can express κ asκ=14N√n2λ2+2m2˜λ2ng.
(60) Taking into account the boundary terms, the on-shell action of the D3-brane is
SD3=−2N(κ√1+κ2+sinh−1κ).
(61) Thus, the holographic prediction of the vacuum expectation value of the WH is
⟨WHSn,Sm[C]⟩=exp[2N(κ√1+κ2+sinh−1κ)].
(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] andOI(x) ⟨⟨OI(x)⟩⟩WHSn,Sm[C]≡⟨WHSn,Sm[C]OI(x)⟩√NO⟨WHSn,Sm[C]⟩,
(63) in the OPE limit
L≫a , and extract the OPE coefficientCWHSn,Sm,O . The change inSD3DBI due to the fluctuations of the background field isδSD3DBI=TD3∫d4σ√detM12(M−1)μν∂xρ_∂σμ∂xκ_∂σνhρ_κ_,
(64) where we have defined the matrix
M=g+2πα′F , andσμ 's are worldvolume coordinates. By using the result ofhρκ_ in the OPE limit given in (38) and the above D3-brane solution, we obtainδSD3DBI=4Nκ2YI∫dρdθsinθsinh2ρ×(−1−2κ2+1−sinh2ρ(κ−2−sin2θ)(coshρ−sinhρcosθ)2)sI.
(65) Now, we compute the change of
SWZ due to the fluctuations of the background fields,δSD3WZ=−TD3∫P[a4].
(66) From (14), we have
aIμ⋯μ4_=−4Δzϵμ⋯μ4_zsIYI=−4ΔCμ⋯μ4_sIYI.
(67) Thus,
δSD3WZ=4TD3ΔYI∫P[C4]sI.
(68) From the coordinate transformation (50)−(52), we obtain
δSD3WZ=8NΔκ4YI∫dρdθsinθsinh2ρ(1−1κ2sinhρcosθcoshρ−sinhρcosθ).
(69) Then, the total change of the action is
δSD3=δSD3DBI+δSD3WZ=−4NΔYI∫sinh−1κ0dρ×∫π0dθsinθ(coshρ−sinhρcosθ)2sI.
(70) Using
sI(x,z)=∫d4x′GΔ(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)κΔ∫sinh−1ρ0dρ×∫π0dθsinhΔρsinθ(coshρ−sinhρcosθ)2+Δ.
(72) Taking the two integrals, we get
CWHSn,Sm,O=2(Δ+3)/2√ΔYI(y)sinh(Δsinh−1κ).
(73) Thus,
CWHSn,Sm,O=2(Δ+3)/2√Δsinh(Δsinh−1κ)=i(−1)Δ/22(Δ+3)/2√ΔVΔ(iκ).
(74) Here,
Vn(x)=sin(ncos−1x) 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 theSU(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 insideAdS5 , the calculations of the coupling between the supergravity modes and the D3-brane worldvolume for bothSU(N) andSO(N) cases are the same. In theSO(N) case, the relation betweenα′ and λ readsα′=√2/λ , while the relationg2YM=8πgs in theSO(N) case is also changed compared with theSU(N) case. However, their effects onTD3 cancel each other. The relation betweenTD3 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 propagatorcSO=√2cSU . This leads to the above conclusion about the OPE coefficients. However, the relation between κ and λ changes in the case ofSO(N) , becomingκ=n4N√λ2,
(75) while for the Wilson loop in the n-th symmetric representation of
spin(N) in theSU(N) case, the relation readsκ=n√λ4N.
(76) Hence, the result in terms of λ and Δ in the
SO(N) case is not just a constant multiplying the result in theSU(N) case.Finally, to compare with the result for
C◻,O in (46), we setm=0 in (74) and take theκ→0 limit. Usingκ=n4N√λ/2 in this case, we obtainCWSn,O≃2(Δ+3)/2√Δκ=2Δ/2−1√ΔλNn,
(77) which is just
nC◻,O , as expected. -
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 withk/N fixed.We can parameterize the unit
S5 ;∑6i=1z2i=1 asz1=cosθ,zj+1=sinθwj,j=1,j=2,⋯,5,
(79) with
∑5j=1w2j=1 . Then, the metric of the unitS5 can be written asdΩ25=dθ2+sin2θdΩ24,
(80) with
dΩ24 as the metric of the unitS4 .RP5 can be obtained fromS5 by identifying antipodal pointszi∼−zi . One way to realize this is to viewRP5 as the upper hemisphere ofS5 (0≤θ≤π/2 ) with antipodal points on the equator (θ=π/2 ) identified. The metric ofRP5 is thus given byds2RP5=dθ2+sin2θds′24,0≤θ≤π/2
(81) where
ds′24=dΩ24 whenθ<π/2 , andds′24 is the metric ofRP4 whenθ=π/2 .Hence, the metric of
AdS5×RP5 readsds2=cosh2u(dζ2+sinh2ζdψ2)+du2+sinh2u(dϑ2+sin2ϑdϕ2)+dθ2+sin2θds′24,
(82) with the radius of
AdS5 andRP5 set to1 . TheAdS5 part of the above metric is written in the form of anAdS2×S2 fibration for computational convenience, and these coordinates are related to the one in (48) by the following coordinate transformation:r1=acoshusinhζcoshucoshζ−cosϑsinhu,
(83) r2=asinhusinϑcoshucoshζ−cosϑsinhu,
(84) z=acoshucoshζ−cosϑsinhu,
(85) where a is the radius of the Wilson loop.
Using the
SO(6)R transformation, we setΘI in (78) toΘI=(1,0,⋯,0) . Then, inAdS5×RP5 , the D5-brane dual to this antisymmetric Wilson loop occupies theAdS2 in the above metric withu=0 and wraps anS4 submanifold ofRP5 at a constant polar angleθk (on the upper hemisphere ofS5 ) [22]. The D5-brane worldvolume isAdS2×S4⊂AdS5×RP5 , and its metric readsd˜s2=dζ2+sinh2ζdψ2+sin2θkdΩ24.
(86) Turning on the worldvolume
U(1) gauge fieldFψζ to account for the k units of fundamental brane charge, the action of the D5-brane on theAdS5×RP5 background can be written asSD5=SD5DBI+SD5WZ,
(87) where
SD5DBI=TD5∫d6σ√det(g+2πα′F),
(88) SD5WZ=−2πα′iTD5∫F∧P[C(4)].
(89) In the above equations, the tension of the D5-brane reads
TD5=1gs(2π)5(α′)3=N8π4√λ2,
(90) and the self-dual 4-form potential is [8]
C(4)=4(u8−sinh4u32)dH2∧dΩ2−(32θ−sin2θ+18sin4θ)dΩ4.
(91) Here,
dH2 is the volume form of the unitAdS2 ,sinhζdζ∧dψ ,dΩ2 is the volume form of the unitS2 ,sinϑdϑ∧dϕ , anddΩ4 is the volume form of the unitS4 .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 θk−sinθkcosθk=πkN,
(92) and the worldvolume gauge field is
Fψζ=i√λ/2sinhζcosθk2π.
(93) The on-shell D5-brane DBI and WZ action are
SD5DBI=2N3π√λ2∫dζsinhζsin5θk.
(94) SD5WZ=4iN3∫dζFψζ(32θ−sin2θ+18sin4θ),
(95) Adding appropriate boundary terms [8], the on-shell action for the D5-brane is
SD5=SD5DBI+SD5WZ+SD5bdy=−2N3π√λ2sin3θk.
(96) Thus, the holographic prediction for the expectation value of the Wilson loop in the rank k antisymmetric representation is given by
⟨WAk⟩=exp(2N3π√λ2sin3θk).
(97) The variation of the DBI part of the action to the first order in the fluctuation
hμν andhαβ isδSD5DBI=TD5∫d6σ√det(g+2πα′F)((g+2πα′F)−1)mn×12(hμν_∂mXμ_∂nXν_+hαβ_∂mXα_∂nXβ_)=N3π√λ2sin5θk∫dζsinhζ×(−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 by12 δSD5WZ=−2πα′iTD5∫F∧P[a(4)]=−2πα′iTD5∫dψdζdσ1dσ2dσ3dσ4μ(Ω4)Fψζ×4sin4θSI∂θYI=8N3π√λ2cosθksin4θk∫dζsinhζsΔ∂θkYΔ,0(θk),
(99) where the 4-form fluctuation is given by
aσ1σ2σ3σ4=4sin4θμ(Ω4)∑sI∂θYI
(100) with
σ1,σ2,σ3,σ4 being the coordinates onS4 and the corresponding measureμ(Ω4) isμ(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)⟩√NO⟨W(C)⟩=−δSD5δs0
(103) Recall that
sI(→x,z)=∫d4→x′GΔ(→x′,→x,z)sI0(→x),
(104) where the bulk-to-boundary propagator
GΔ(→x′,→x,z)=c(zz2+|→x−→x′|2)Δ≃czΔL2Δ
(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)⟩√NO⟨WAk⟩=aΔL2Δ2Δ/2NΔ3π√λΔΔ+3Δ−1sin3θk×[2(Δ+1)cosθkC(2)Δ−1−ΔC(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)+cosθksinθk∂θkYΔ,0(θk)=NΔ[Δsin2θkC(2)Δ−(Δ+3)cosθksin2θkC(2)Δ−1].
(109) The normalization factor
NΔ is obtained fromYΔ,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)⟩√NO⟨WAk⟩=aΔL2ΔYΔ,0(0)2Δ/23π√Δλ6(Δ−2)!(Δ+1)!sin3θk×[2(Δ+1)cosθkC(2)Δ−1−ΔC(2)Δ].
(112) Using the recurrence relation [29, 30]
ΔC(λ)Δ(x)=2(Δ+λ−1)xC(λ)Δ−1(x)−(Δ+2λ−2)C(λ)Δ−2(x),
(113) we finally arrive at
CWAk,O=2Δ/23π√Δλsin3θk6(Δ−2)!(Δ+1)!C(2)Δ−2(cosθk).
(114) This
SO(N) result is identical to that for theSU(N) case obtained in [18]. The D5-brane worldvolume has topologyAdS2×S4 withAdS2 in theAdS5 part of the background geometry andS4 in theRP5 part. Sinceθk<π/2 , theS4 we consider in this case is the same as theS4 embedded inS5 determined byθ=θk in the parameterization given in (79). Thus, the computation of the coupling of the supergravity modes to theD5 -brane is the same as that for theSU(N) case, although the expression ofTD5 in terms of λ and N for theSO(N) case is different from that for theSU(N) case.In the
SO(N) case, we haveTD5=N8π4√λ2 , while for theSU(N) case the relation isTD5=N√λ8π4 . Taking this change and the relationcSO=√2cSU into account, we arrive at the conclusion that the OPE coefficients in theSO(N) case are the same as the ones in theSU(N) case. Thus, in thek≪N limit, the relationCAk,O=kC◻,O remains the same as that in theSU(N) case. This can be obtained from the resultθ3k∼3πk/2N in this limit andC(2)Δ−2(1)=(Δ+1)!6(Δ−2)!.
(115) -
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 theΦI to beΘI=(1,0,⋯,0) , the embedding of the D5-brane is given byu=0,θ=π/2 in the coordinates used in the previous section [22]. In this case, the field strength of the worldvolumeU(1) gauge field vanishes. Taking into account the boundary terms, the total on-shell action of this D5-brane isSD5=−N3π√λ2,
(117) so the holographic prediction for the expectation value of the Wilson loop in the spinor representation is [22]
⟨WS⟩=exp(N3π√λ2).
(118) As observed in [22],
Fψζ , given by (93), vanishes whenθk=π/2 . A shortcut to compute the OPE coefficientCS,O using the result obtained in the previous section is by settingθk=π/2 inCAk,O and dividing the result by2 to take into account the change of the D5-brane worldvolume fromAdS2×S4 intoAdS2×RP4 ,CS,O=12CWAk,O|θk=π/2=2Δ/23π√Δλ6(Δ−2)!(Δ+1)!C(2)Δ−2(0)=(−2)Δ/2−1√Δλπ(Δ2−1).
(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(1−2xt+t2)λ=∞∑Δ=0C(λ)Δ(x)tΔ.
(121) -
In this study, we investigated the holographic duality of the
N=4 SO(N) SYM theory and the Type IIB string theory on theAdS5×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 theSO(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 theSO(N) theories. In addition, we discussed the WH loops in the symmetric representation using a D3-brane with both electric and magnetic charges. TheN=4 SYM theory with the gauge groupSO(N) has some features different from theSU(N) theory. We compared our results with those of theN=4 SU(N) SYM theory. -
In this appendix, we compute the coefficient c of the bulk-to-boundary propagator of the modes
sI . The action forsI , obtained from the full "actual" action of IIB supergravity [31] is [24]S=∫AdS5d5x√det(gAdS5)12BI[∂μsI∂μsI+Δ(Δ−4)(sI)2],
where
BI is given byBI=16κ2Δ(Δ−1)(Δ+2)Δ+1z(Δ),
where κ is the coupling constant of type IIB supergravity, and
z(Δ) is explained below. Using2κ2=(2π)7g2α′4,
and the relations
α′=√2/λ andgs=g2YM/(8π) for theSO(N) case, we obtainκ2=(2π)58N2,
which is same as the one for the
SU(N) case.z(Δ) is defined by∫RP5d5y√det(gRP5)YIYJ=δIJz(Δ).
The expression for
z(Δ) isz(Δ)=π32Δ(Δ+1)(Δ+2),
which equals to half of the result in the
SU(N) case since the integration is overRP5=S5/Z2 . Using the above result, we obtainBI=22−ΔN2Δ(Δ−1)π2(Δ+1)2.
The coefficient of the bulk-to-boundary propagator is
c=√α0BI,
where [17]
α0=Δ−12π2,
which is identical for both
SO(N) andSU(N) cases. Finally, we obtainc=Δ+12(3−Δ)/2N√Δ,
which equals
√2 times the result for theSU(N) case.
Holographic operator product expansion of loop operators in the N=4∼SO(N) super Yang-Mills theory
- Received Date: 2023-05-10
- Available Online: 2023-08-15
Abstract: In this study, we compute the correlation functions of Wilson(-'t Hooft) loops with chiral primary operators in the