Mass-deformed ABJM theory and LLM geometries: exact holography

Journal of High Energy Physics, Apr 2017

We present a detailed account and extension of our claim in arXiv:​1610.​01490. We test the gauge/gravity duality between the \( \mathcal{N} \) = 6 mass-deformed ABJM theory with U k (N )×U−k (N ) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(4)/ℤ k × SO(4)/ℤ k isometry, in the large N limit. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of mass-deformed ABJM theory and from the implementation of Kaluza-Klein holography to the LLM geometries. We focus on the chiral primary operator with conformal dimension Δ = 1. We show that \( \left\langle {\mathcal{O}}^{\left(\varDelta =1\right)}\right\rangle ={N}^{\frac{3}{2}}{f}_{\left(\varDelta =1\right)} \) for all supersymmetric vacuum solutions and LLM geometries with k = 1, where the factor f (Δ) is independent of N. We also confirm that the vacuum expectation value of the energy momentum tensor is vanishing as expected by the supersymmetry. We extend our results to the case of k ≠ 1 for LLM geometries represented by rectangular-shaped Young-diagrams. In analogy with the Coulomb branch of the \( \mathcal{N} \) = 4 super Yang-Mills theory, we argue that the discrete Higgs vacua of the mABJM theory as well as the corresponding LLM geometries are parametrized by the vevs of the chiral primary operators.

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Mass-deformed ABJM theory and LLM geometries: exact holography

Received: December Mass-deformed ABJM theory and LLM geometries: exact holography Dongmin Jang 0 1 3 Yoonbai Kim 0 1 3 O-Kab Kwon 0 1 3 D.D. Tolla 0 1 2 3 with SO 0 1 =Zk 0 1 Open Access 0 1 c The Authors. 0 1 0 Suwon 440-746 , South Korea 1 Institute of Basic Science, Sungkyunkwan University 2 University College, Sungkyunkwan University 3 Department of Physics, BK21 Physics Research Division We test the gauge/gravity duality between the N = 6 mass-deformed ABJM theory with Uk(N ) U k(N ) gauge symmetry and the 11-dimensional supergravity on LLM geometries SO(4)=Zk isometry, in the large N limit. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of mass-deformed ABJM theory and from the implementation of Kaluza-Klein holography to the LLM geometries. We focus on the chiral primary operator with convanishing as expected by the supersymmetry. We extend our results to the case of k 6= 1 for Coulomb branch of the N = 4 super Yang-Mills theory, we argue that the discrete Higgs vacua of the mABJM theory as well as the corresponding LLM geometries are parametrized by the vevs of the chiral primary operators. exact; holography; AdS-CFT Correspondence; Brane Dynamics in Gauge Theories; M-Theory - = 1. We show that hO( =1) Contents 1 Introduction 2 Vacua of the mABJM theory and the LLM geometries Supersymmetric vacua of the mABJM theory 2.2 LLM geometries and their droplet picture 3 Vevs of CPOs in mABJM theory 3.1 CPOs in ABJM theory 3.2 vevs of CPO in mABJM theory 4 KK reduction and gauge invariant modes 11-dimensional gravity equations of motion Fluctuations on AdS4 Expansion in S7 spherical harmonics Gauge invariant uctuations KK reduction The equations for spin zero elds The equations for spin one elds The equations for spin two elds 5 Exact KK holography for LLM geometries Asymptotic expansion of the LLM geometries Asymptotic expansions for the physical modes in 4 dimensions 5.3 Comparison with eld theory results 6 Conclusion A Spherical harmonics on S7 A.1 Scalar spherical harmonics A.2 Vector spherical harmonics A.4 CI1=2 and C( =1) B Asymptotic expansions C Proof of (5.48) A.3 Scalar spherical harmonics on S7 with SO(4) SO(4) symmetry Introduction ity in (d+1)-dimensional spacetime and a quantum eld theory (QFT) on the d-dimensional string/M theory on AdSd+1 X , with a compact internal manifold X , and conformal eld X . However, rank of the gauge group. AdSd+1 operator on the eld theory side there is a corresponding gravity eld among these KK gravity theory [13{15]. For a CPO of conformal dimension , the vev which is obtained which is asymptotically AdS5 S5 [13, 14]. N = 6 supersymmetry and Uk(N ) U k(N ) gauge symmetry, where k is the Chern-Simons due to the existence of these discrete vacua. SO(4)=Zk obtained [22, 23]. See also [24{29] for related works. with conformal dimension motion on AdS4 S7 background in 11-dimensional supergravity. In order to obtain the When we are interested only in the CPO with conformal dimension = 1, the linearized equations are su cient. However, for CPOs with 2, one has to consider non-linear ground. Here we focus on the S5 backobtain an exact holographic relation which is given by for some speci c types of the LLM solutions. 1For the linearized equations of motion on AdS4 S7 background in de Donder gauge, see [30, 31]. It is in the de Donder gauge is not straightforward. the vevs of the operators in the case renormalization to read the vevs of CPO with the LLM solutions and compare the eld theory and the gravity results. In section 6, We also include of equation (5.48). Vacua of the mABJM theory and the LLM geometries U(N ) k gauge group is a superconformal CS M2-branes on the C4=Zk orbifold xed point [19]. One interesting feature of the ABJM which break the global SU(4) symmetry to SU(2) U(1). Solving the classical section, we brie y review the correspondence. Supersymmetric vacua of the mABJM theory as follows Y A = (Za; W ya); Lbos = Vbos = 0, is written as ZaZyZb ZbZyZa = WaZbWb WbZbWa = 0; W yaWbW yb W ybWbW ya = ZbWbZa ZaWbZb = 0; Z0a = W0ya = (n+1) matrices, GRVV matrices (n) = BB (n) = BB where n = 0; 1; 1. The vacuum solutions are given by remains to be supersymmetric [22]. LLM geometries and their droplet picture The LLM solution with SO(2; 1) SO(4) isometry in 11-dimensional supergravity and W ya are N following two constraints, The solution contains Nn rectangular matrices of the type Ma (n) and N n0 rectangular matri N matrices, the occupation numbers, Nn and N n0, should satisfy the N = Nn + N n0 X Nn = as the dual gravity theory [23]. The LLM geometry with Zk orbifold is given by ds2 = Gtt( dt2 + dw12 + dw22) + Gxx(dx~2 + dy~2) + G ds2S3=Zk + G~~ds2S~3=Zk ; are given by where ds2S3=Zk and ds2S~3=Zk are metrics of the two S3's with Zk orbifold and the warp factors Gtt = q 12 + Z 12=3 Z2 12=3 Gxx = @ Z2 12=3 Z 12=3 given by f (x~; y~) = p1 0 = i=1 2p(x~ x~i)2 + y~2 i=1 2p(x~ x~i)2 + y~2 ?2 dZ with y~x~ = 1. The corresponding 4-form eld strength is given by F4 = ^ dt ^ dw1 ^ dw2 + 0 1 V d(y~2e2G) + h2e3G ?2 d(y~2e 2G) ^ d 3 1 V d(y~2e 2G) h2e 3G ?2 d(y~2e2G) ^ d ~ 3; system.2 The Zk quotient acts as where d 3 = (sin =8)d ^ d ^ d , d ~ 3 = [23, 36]. The 4-form relations [37{39] h2 = 14 d 2 + sin2 d 2 + (d + cos d )2 . sin d + sin cos d ; 2 = + sin sin d ; 3 = d + cos d with ranges of the angles, 0 e2G = 211 + Z a value 12 denoted by representation. See the gure 1. Since the function Z(x~; 0) is 12 if x~ < x~1 and it is 12 x~F = x~1 + X(x~2i+1 starting at the than k these parameters should satisfy the condition 0 k, which is the same fNn; N n0g, the one-to-one correspondence is given by fln; ln0g () fNn; N n0g: gure 1 for the parametrization of droplet picture and Young diagram. Vevs of CPOs in mABJM theory to AdS5 SO(4)=Zk The Young diagram corresponding to the droplet picture (a). theory in the large N limit were tested in a systematic way [13, 14]. theories are away from the UV xed point and they preserve the full supersymmetry with renormalization procedure. In this section, we construct the CPO with vacua of mABJM theory for general k in the large N limit. CPOs in ABJM theory The gauge invariant CPOs of conformal dimension in the ABJM theory are given by ( ) = CA1; ;An where A; B = 1; ( )B1; ;Bn are symmetric in lower as well as upper indices It is well known fact that the coe cients CA1; ;An ( )B1; ;Bn are identi ed with the similar coe cients CI cal harmonics; In particular, the coe cients of the CPO with the operator is given by , we obtain im = O where h theory, respectively, and i0 denote the vevs of operators in the mABJM theory and the ABJM ( ) is an operator containing at least one Y^ A or Y^ yA. The N1 ( =1) = vevs of CPO in mABJM theory Here we calculate the vevs of CPO with expanded as Y A = Y0A + Y^ A; ( )(Y A)im = O For CPO with ( =1) im = N n0)n(n + 1): nite k in the large N limit. KK reduction and gauge invariant modes on AdS4 11-dimensional gravity equations of motion M(an), while W0ya contains N n0 of the matrix M(an), we have Tr(Z0aZay0 W0yaWa0) = N n0)Tr(Ma M(an)) (n) N n0)n(n + 1); 2 gpqR = 2 gpqFrstuFrstu + 4FpstuFqstu ; @p(eFpqrs) + 3The ABJM theory is dual to M-theory on AdS4 We will extend our results to general k case eventually. S = 16 G11 ~p1p2p3q1 q4r1 r4 Cp1p2p3 Fq1 q4 Fr1 r4 ; G11 = where we used the index notation p; q; r; = 0; ; 10, ~0123 10 = 1 is the Levi-Civita symbol. The 11-dimensional Newton's gravitational constant is equations of motion for the metric and the 4-form eld strength: radius of S7. = pjgAdS4 j ~ is the Levi-Civita tensor for the AdS4 space, and L is the where e write the AdS4 g. Using the index notation ( ; ; ; = 0; S7 solution of the equations of motion in (4.3) as follows ds2 = dt2 + dw12 + dw22 + d 2 + L2ds2S7 ; ; and it is zero otherwise: Fluctuations on AdS4 We consider a solution which is asymptotically AdS4 S7 so that we can write it as gpq = gpq + hpq; Cpqr = Cpqr + cpqr Fpqrs = Fpqrs + fpqrs; where hpq; cpqr; fpqrs represent deviations from the AdS4 S7 geometry and they become following equations of motion for hpq and fpqrs up to linear order, r r r rphqr +r rqhpr Rrshrs +rrrshrs FrstuF rstuhpq 4gpqhrsF rtuvF stuv +rar h a +rar h a (r r +rara)h +rarbhab +(r ra +rar )h a (r r +rara)(h +hbb) = 0; r r ha + r rah b b + r r hab + r rah b = 0; whereas the AdS4 S7 values of those objects are denoted by normal font symbols. = 0; combination which is de ned as T(ab) = (Tab + Tba) where gab is a metric on S7. The notation [abc indices, a; b; c; , for instance, ] means anti-symmetrization among T[ab] = r f abc + rdf dabc + Expansion in S7 spherical harmonics h (x; y) = hI1 (x)Y I1(y); haa(x; y) = I1(x)Y I1(y); (x; y) = s~I1 (x)Y I1(y); = 0; ) = 0; r f ab + rcf c ab = 0; (x; y) = 4r[ sI1 ](x)Y I1 (y); a(x; y) = 3r[ vI7 ](x)YaI7 (y) f abc(x; y) = r tI35 (x)Y[aI3b5c](y) 3tI21 (x)r[aYbIc2]1 (y); fabcd(x; y) = 4tI35 (x)r[aYbIc3d5](y); c3 ! d (2). However, the 4-form eld strength is invariant under this transformation. binations which are de ned as t~I21 (x) t~I35 (x): s~I1 (x) v~I7 (x) + r[ v~I7] (x); and projecting onto the scalar harmonics Y I1 , we obtain L2 hI1 + r r hI1 + r r hI1 6 I1 + = 0; r r . We have used the AdS4 in (4.20) gives the following equation for scalar elds S7 solutions in (4.4) and the results of 8 I1 + = 0: raY I1 and gabYbI7 , respectively 5 I1 + 6 I1 + 1 I7 + = 0; = 0: elements: gabY I1 , r (aY b)I7 and Y (ab)I27 , r r hI1 = 0; 2r sI1 = 2r vI7 = 6 I1 + 5 I1 sI1 + hI1 + 5 I1 ; 2r v^I7 = 0; following set of equations r sI1 = r s^I1 = 0; I35 = (I35+3)2 and we have used the relation ra1 YaI23a53a4 = to obtain the two equations in (4.34). sI1 = 0; r vI7 = 0; tI21 = 0; r vI21 = 0; r vI7 = 0; t~I35 = 0; h^I1 = hI1 Since the only non-vanishing component of the 4-form eld strength is F ~hI1 = r ~tI27 = 0; ~vI7 = 2 (Iv7); ~vI7 = r ~sI1 = 2 (Is1); ~sI1 = r ~ I1 = 2 I1 (Is1): ^I1 = v^I7 = vI7 where s~I1 = sI1 non-trivial equations in second part of (4.36) are Then, we obtain the following results a = I1 F r[ ~sI1 ] = = 0; Gauge invariant uctuations are . Up to linear order, these gauge-dependent degrees of freedom transform as, ~hpq = (rp q + rq p); ~fpqrs = parameter p(x; y) in terms of the spherical harmonics on S7 as (x; y) = I1 (x)Y I1 (y); gauge-dependent coe cients of the spherical harmonics linear order. KK reduction motion for physical modes. The equations for spin zero of motions for two gauge invariant scalar elds 14 ^I1 = 0; 5 I1 ^I1 = 0; where the gauge invariant scalar elds are ^I1 = (18hI1 ^I1 = ( I1 where we have introduced (I1 + 6)(I1 + 12) I1 = 0; I1 = 0; I1 = I1 = (I1 + 7) 18(I1 1) ^I1 + 7 ^I1 14(I1 + 3) 18(I1 + 7) ^I1 + 7 ^I1 14(I1 + 3) three more elds which are already diagonal and gauge invariant I27(I27 + 6) tI27 = 0; (I35 + 3)(I35 3) tI35 = 0; where tI27 introduced in [13, 14] SI = sI + JIJnJm tJn tJm + LIJnJm r tJn s s where s I represent any of the 11-dimensional elds, and SI is the corresponding a pseudotensor), we note that the rst three elds last two elds T+I35 ; T I35 are pseudoscalar elds. The equations for spin one same. Therefore, the elds I1 ; I1 ; tI27 ; tI+35 ; tI35 are the correct 4-dimensional spin-zero elds at the linear order and are denoted as I1 ; I1 ; T I27 ; T+I35 ; T I35 , respectively. Based I1 ; I1 ; T I27 are scalar elds while the u^I7 + 12 3 u^I7 = 0; v^I7 = 0; I7(I7+6) 1 , and we have introduced the following gauge invariant combinawith v^I7 = v~I7 + r[ v~I7] : I72 + 12I7 + 23 vI7 = 0; uI7 = 0; vI7 = 4(I7 + 3) uI7 = 2(I7 + 7)v^I7 + u^I7 4(I7 + 3) tI21 = 0; where I7 = vector mode where tI21 is equivalent to tI21 in (4.19). third one is a pseudovector eld. The equations for spin two following linear order equations for gauge invariant tensor elds ^I1 = 0; ^I1 = 0; where we have introduced the following gauge invariant tensor elds and we have de ned We note that ^I1 in (4.44). of traceless tensor modes are the gauge invariant scalar elds de ned where ^(I1 ) = ^I1 ^I1 and ^(I1 ) = ^I1 ^I1 . Then we introduce the transverse ( ) = 0, and diagonalized linear equation, 30(I1 +2)(I1 +4) r( r ) ( ) = 0; where MI21 = I1(I1+6) 8 L2 T I21 , and one tower of spin-two mode HI1 . order, HI1 ) are the correct 4-dimensional spin two KK modes. ity yields, three towers of scalar modes I1 ; I1 ; T I27 , two towers of pseudoscalar modes Exact KK holography for LLM geometries = 1, are Asymptotic expansion of the LLM geometries of the Legendre polynomials as follows Z( ; ) = V ( ; ) = + X [(n + 1) Pn+1 ( ) 2 nPn ( ) + (n 1) Pn 1 ( )] Cn have de ned [39] = L4r~3 , Cn = where A = kN the parameters Cn's satisfy an identity: where G = L166G4xx . ds2 = dt2 + dw12 + dw22 + G + G ds2S3 + G~~ds2S~3 ; (5.4) C12 = 2: direction is well de ned. In a geometry which is asymptotically AdSd+1 the FG coordinate system is given by X , the metric in ds2 = transformations = (z; ); = (z; ); which should satisfy two conditions Then we obtain = 0: ds2 = dt2 + dw12 + dw22 + g2 (z; ) d 2 + g3 (z; ) ds2S3 + g4 (z; ) ds2S~3 ; g1 (z; ) = Gtt ( (z; ) ; (z; )) ; g2 (z; ) = Gxx ( (z; ) ; (z; )) g3 (z; ) = G ( (z; ) ; (z; )) ; g4 (z; ) = G~~ ( (z; ); (z; )) : tions (5.7) in the asymptotic region, we use the following ansatze, (z; ) = 8C2C14 + 12C3C13 + 3 C22 4C2C3C1 + C23 4C32 + 3C2C4 60C2C14 100C3C13 + 9 11C22 + 5C4 C12 12C2C3C1 27C23 + 56C32 45C2C4 60C2C14 100C3C13 + 9 11C22 + 5C4 C12 12C2C3C1 27C23 + 56C32 45C2C4 (1 where the ai and bi are determined from (5.7), a1 = a2 = b1 = b2 = for I1 = 0; 2; 4; hi0j = hi2j = hi4j = h0 = h2 = h4 = 0 17C16 51C2C14 28C3C13 +72C22C12 +42C2C3C1 45C23 7C32 +O L2 0 2C13 3C2C1 +C3 +O 0 28C16 84C2C14 +28C3C13 +9 7C22 15C4 C12 +228C2C3C1 135C23 128C32 +135C2C4 +O gijhiIj1 = 4Lz22 ijhiIj1, 51C2C14 28C3C13 + 72C22C12 + 42C2C3C1 3C2C1 + C3 + O( 03); 84C2C14 + 28C3C13 + 9 7C22 15C4 C12 + 228C2C3C1 128C32 + 135C2C4 + O( 04): The values of the scalars sI1 are obtained from pendix B, 0 = 4 = by expanding, 14C16 42C2C14 +4C3C13 +39C22C12 6C2C3C1 10C23 +C32 124C16 +372C2C14 124C3C13 +9 15C4 31C22 C12 84C2C3C1 +104C32 +135 C23 C2C4 r r h(ab) = sI1rarbr(arb)Y I1 = 6sI1 I1 a b Then using appendix B we obtain s0 = s2 = s4 = u~0 = u~2 = u~4 = 21C2C14 + 12C22C12 + 6 2C12 3C2 C3C1 + 5C23 + 3C32 3C2C1 + C3 + O( 03); 12C2C14 + 4C3C13 + 9 C22 + 15C4 C12 276C2C3C1 + 136C32 + 135 C23 In order to read the values of the scalars I1, we take the trace of hab, haa = I1Y I1 = gab gab gab = gabgab 14C16 42C2C14 +4C3C13 +39C22C12 6C2C3C1 10C23 +C32 +O( 04); (5.22) 2C13 3C2C1 +C3 +O( 03); 124C16 +372C2C14 124C3C13 +9 15C4 31C22 C12 84C2C3C1 +104C32 +135 C23 C2C4 To determine the graviton mode, in addition to h we also need the values of the tensor elds u~ de ned in (4.58). We can rewrite the de nition in (4.58)as u~I1 I1Y I1 = a = and noting that, for the LLM geometry, the only non zero f following results I1 Y I1 . Using this into (5.30) a is ftw1w2 , we obtain the u~iIj1 I1 Y I1 = 3! u~Iz1z I1 Y I1 = @ @zFtw1w2 + 3 @ Ftw1w2 @zFtw1w2 + 3 z Ftw1w2 0 = 0, we can not read u~0 . 3C1C2 + C3 + O( 03) 12C2C14 + 4C3C13 + 9 C22 + 15C4 C12 276C2C3C1 u~i2j = u~i4j = u~z2z = u~z4z = p10 discussed in subsection 4.5. totic expansions for I1 and 3C1C2 + C3: + 135C23 + 136C32 135C2C4 + O( 04) 3C1C2 + C3 + O( 03); 12C2C14 + 4C3C13 + 9 C22 + 15C4 C12 276C2C3C1 + 135C23 + 136C32 135C2C4 + O( 04): because for the LLM geometry h a is zero. only non-vanishing 0 = O( 02); 0 = O( 02); 2 = O( 03); 2 = 4 = O( 02); 4 = O( 02); on the gravity side is related to the conformal dimension of the dual gauge invariant operator by m2RA2dSd+1 = m2L2 = 2 (2 spin zero elds are as follows: For the scalar eld I1 we have 6) = 2 (2 fI1 = 2; 4; 6; (I1 + 12)(I1 + 6) = 2 (2 fI1 = 0; 2; 4; (I27 + 6)I27 = 2 (2 fI27 = 2; 4; 6 For the pseudoscalar eld T I35 we have For the pseudoscalar eld T+I35 we have (I35 + 3)(I35 3) = 2 (2 (I35 + 9)(I35 + 3) = 2 (2 For the scalar eld I1 we have For the scalar eld T I27 we have = 1; 2; I1 and T I27 are also not the candidates. The only scalar I1 , with I1 = subsection 4.5.2. I1 + 12 I27 + 6 I35 + 3 I35 + 9 ( ) = O( 03); ( ) = O( 04): Comparison with eld theory results conformal dimension is determined by the coe cient in the asymptotic expansion of a dual gauge invariant scalar elds on the gravity side, i.e. im = p N ( ) 11-dimensional supergravity, and is de ned as dual relation for the M2-brane theory [19, 46, 47]. trivial asymptotic expansions at linear order in 0. At quadratic order or higher, more of We have obtained the asymptotic expansion of the scalar eld 2 in (5.36) while the vev In order to x the normalization factor N, we use the identity hO(1)im = p N (1) = ln0)i = N 3=2 4 2 n=0 N n0)i = 72N 0 X1 hn(n + 1)(ln Now recalling that with conformal dimension = 1 is given by 1442 . Therefore, the vev of the CPO im = large N limit [47]. dual relation is then given by im = 3 = = 1 CPO with 0 expansion of the because the theory is supersymmetric. CPO with the identity (5.3) while C3 is determined by the vevs of CPO with = 1. Therefore, the vevs of CPO with and k ( these issues for future study [32]. with those of mABJM theory on S3. Acknowledgments and NRF-2014R1A1A2059761 (O.K.). Spherical harmonics on S7 The spherical harmonics on S7 are de ned as follows r2Y I1 = I1 Y I1 = I7 YaI7 = I21 Y[aI2b1] = I35 Y[aI3b5c] = 2)Y(Ia2b7); 3)Y[aI3b5c]; raY I7 = raY(Ia2b7) = raY[aI2b1] = raY[aI3b5c] = 0; a gabY(Ia2b7) = gabY[aI2b1] = gabY[aI3b5c] = 0: antiSymmetric 2 antiSymmetric 3 Now consider 1 Z Scalar spherical harmonics The scalar spherical harmonics on S7 are the restriction of Y I1 = ; 8) are the Cartesian coordinates of IR8 and the coe cients CI1 i1 iI1 are totally symmetric and traceless. In order to evaluate integrals 1 Z xi2m = 2m 1(m + 3)! all possible pairing ; Y I1 Y J1 = !7LI1+J1 S7 i1 iI1 CjJ11 jJ1 xi1 CI1 only when I1 = J1 Y I1 Y J1 = !7L2I1 S7 i1 iI1 CjJ11; jJ1 xi1 CI1 2I1 1(I1 + 3)! CiI11; iI1 CjJ11 jJ1 all possible pairing : total number of such terms is I1!. Therefore we get Y I1 Y J1 = 2I1 1(I1 + 3)! hCI1 CJ1 i; where hCI1 CJ1 i = CI1 i1 iI1 . Actually, we normalize the scalar harmonics such that by parts. Vector spherical harmonics Consider a vector eld in IR8 p = 1; VagabVb = Vpe^pagabe^bqVq: pq = pq VagabVb = npnq)Vq: Now we can make the following replacement Recalling that xpY I7 = 0, the second piece is zero. In general we can drop the second p 1 Z gabYaI7 YbJ7 = npnq)VpI;7i1 iI7 VqJ;j71 jI7 xi1 xjI7 : (A.14) with V I7 traceless and totally symmetric in the i1; p ; iI7 indices. It also satisfy xpY I7 = p Y I7 = e^pY I7 = a a p a = 1; e^pa = harmonics we use the following procedure. Lets consider in (A.8) we can write Therefore we have 1 Z gabYaI7 YbJ7 = !7L2I7 S7 VpI;7i1 iI7 VpJ;j71 jI7 xi1 V I7 V J7 : spherical harmonics. 5We will use the index notation where p; q; are the IR8 indices and a; b; are the S7 indices. Scalar spherical harmonics on S7 with SO(4) which satisfy the following harmonic equation I1(I1 + 6) Y I1 = I1(I1 + 6) Y I1 = 0; ds2S7 = L 2 d 2 + cos2 ds2S3 + sin2 ds2S~3 : of S7 is related to the = cos . Then we have coordinate of LLM as ds2S7 = L Imposing the SO(4) SO(4) symmetry, the spherical harmonics depend only on the coordinate, which implies ), the second solution is a polynomial and the rst few terms are which gives I1(I1 + 6) =) I1(I1 + 6) Y I1 = 0 Y I1 ( ) = 0: 2 = Y 2 = = 2 2 Y 2; Y I1 ( ) = N I1 2F1 Y I1 ( ) = N I1 2F1 For I1 = 4i, (i = 0; 1; 2; ), the rst solution is a polynomial and the rst few terms are Y 0 = 1; Y 4 = For later convenience lets invert these relation and write the following In subsection 3.1, we have stated the fact that the coe cients CI1 scalar spherical harmonics in (A.2) are related to the coe cients CB1 with conformal dimension the solution in (A.22) is obtained by imposing the SO(4) SO(4) symmetry, we rewrite the scalar harmonics in (A.2) in a form that manifests this symmetry i1 iI1 , which de nes the ( )A1 An of the CPOs with the R8 coordinates restricted to S7 are written as follows Y 2 = Ci2j xixj + x2 = L x4 = L x6 = L ; x8 = L : (A.25) x1 = L x3 = x5 = L x7 = ment gives gives C121 = 2p1 2 . From (A.22) we notice that Y 2 depends only on the coordinate of S7. Therefore, the C121 = C222 = C323 = C424; C525 = C626 = C727 = C828; Ci2j = 0; for i 6= j: C525 while the orthonormality condition spherical harmonics in terms of C4 coordinates as Y 2 = where the C4 coordinates are given by Comparing (A.24) and (A.27) we obtain y1 = x1 + ix2; y2 = x3 + ix4; y3 = x5 + ix6; y4 = x7 + ix8: C(1)1 = C(1)3 = C121 + C222 = C525 + C626 = C(1)2 = C(1)4 = C323 + C424 = C727 + C828 = Hence the CPO of conformal dimension = 1 is given by (3.3). Asymptotic expansions as well as for the various components of the 4-form eld strength Fpqrs in (2.11) using the but we keep only up to 20 for our purpose. Applying the ansatze (5.10) to the de ning the warp factors in FG coordinate system, ( 0z)2 28C16 84C2C14+28C3C13+9 7C22 15C4 C12+228C2C3C1 9C23+20C32 27C2C4 135C23 128C32+135C2C4 2+ 12C16+36C2C14+4C3C13 3 13C22+3C4 C12+12C2C3C1 +7C23 8C32+9C2C4 + (2C13 3C2C1+C3)(1+ ) (2C13 3C2C1+C3)(1 ) 3C23+8C32 9C2C4 63C23 8C32+15C2C4 135C23 104C32+135C2C4 2+ same manner Ftw1w2z (z; ) = 2 2C13 3C2C1 +C3 Ftw1w2 (z; ) = 2C13 3C2C1 +C3 16C16 48C2C14 +28C3C13 +27 C22 C4 C12 +12C2C3C1 15C23 20C32 +27C2C4 4C16 12C2C14 +4C3C13 +9 C22 +15C4 C12 276C2C3C1 +135C23 +136C32 135C2C4 4C16 12C2C14 +4C3C13 +9 C22 +15C4 C12 276C2C3C1 +135C23 +136C32 135C2C4 z (z; ) = (1+ )2 sin (z; ) = (1+ ) sin F~~ ~z (z; ) = F~~ ~ (z; ) = 2C13 3C2C1 +C3 (1+6 )+ 2C13 3C2C1 +C3 (4+9 )+ 2C13 3C2C1 +C3 (1 6 )+ 2C13 3C2C1 +C3 (4 9 )+ Proof of (5.48) C1 = p we rewrite C1 as C1 = p m2i) = x~1 + X(x~2i+1 x~2i) : 3 = region. See gure 1. 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Dongmin Jang, Yoonbai Kim, O-Kab Kwon, D.D. Tolla. Mass-deformed ABJM theory and LLM geometries: exact holography, Journal of High Energy Physics, 2017, 104, DOI: 10.1007/JHEP04(2017)104