Holography in the BMN limit of AdS/CFT Correspondence

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1 Holography in the BMN limit of AdS/CFT Correspondence Tamiaki Yoneya (University of Tokyo, Komaba) BMN limit from the viewpoint of correlation functions How to reconcile the BMN conjecture with holographic principle? One of open problems in the pp-wave limit In particular, we show how the GKP-W relation is realized in the BMN limit, and give a concrete holographic relation for 3-point OPE coefficients. Based on S. Dobashi and T. Yoneya, hep-th/ , hep-th/ (also previous work, S. Dobashi, H. Shimada and T. Yoneya hep-th/ )

2 1 Contents : Motivation : puzzles, and resolution Holographic relation for 3-point correlators Holographic string-field theory Explicit checks in the large-µ limit Impurity-preserving case Impurity non-preserving case Concluding remarks Nonconformal Dp-brane backgrounds M-theory holography AdS 4 S 7

3 Motivation 2 AdS holography : Typically, AdS 5 S 5 N = 4 SYM 4 provides a concrete example of holographic correspondence between gravity (string theory) [ bulk] and gauge theory [ boundary] opens an entirely new perspective both for gauge theories and for string theories GKP-W relation (conjecture) : Basic holographic relation bulk (AdS 5 or EAdS 5 ) boundary (M 3,1 or R 4 )(z 0) Z[Φ 0 ] string gravity = exp( d 4 x i Φ i 0(x)O i (x)) SYM, lim z 0 Φi (z,x) z 4 i Φ i 0(x) Has been checked for protected supergravity modes (chiral primary fields) for 3-point functions

4 However, whether this relation is also valid for nonprotected stringy degrees of freedom has been an open question. BMN proposal as an important first step: proposed a set of local gauge-invarinat operators on the gauge theory side (CFT) corresponding to stringy massive states of bulk (closed) string theory in the so-called pp-wave limit. pp-wave limit approximating the AdS geometry by a neighborhood around a null geodesic which describes a ray of center-of-mass motion of strings with large angular momentum (J 1) along a large cirlce of S 5 φ i (i = 1,...,6): scalar fields ( collective coordinates of D3-branes) of SYM 4 ground state p + : O Tr[Z J ], Z = φ 5 + iφ 6 stringy excitation modes (n world-sheet momentum): 3 a i+4, n Z l φ i e 2πinl/J Z J l, a i, n Z l (D i Z)e 2πinl/J Z J l (i = 1, 2,3, 4) etc The scalar fields φ i s and derivatives D i are called impurities. BMN operators near BPS states.

5 4 8 transverse string excitations in the bulk 4 (=SO(4) R-charge directions) + 4 (=SO(4) base space directions of SYM) J, keeping J/R 2 fixed Anomalous conformal dimensions are given by J = {i} N i 1 + R4 n 2 i J 2 P R, P + J/R ( and are proposed to be identified with energy spectrum for strings in the light-cone gauge R = (g 2 Y MN) 1/4 α scale parameter of AdS geometry )

6 Puzzles related to holography 5 The null geodesics dominating the pp-wave limit do not reach the conformal boundary of AdS space Impossible to apply the GKP-W relation to the BMN operators!? τ τ r, x 4 = e τrˆx 4, of radial quantization on the boundary? 4 base-space directions of SYM 8 transverse directions of bulk (irrespectively of the identification of the global time and the target time on the boundary): light-cone time x + = τ mixed with transverse directions!? The null geodesic requires Minkowski metric, while the boundary theory must be assumed to be Euclidean!? These puzzles must be resolved: The spectrum of conformal dimensions alone are not sufficient for defining CFT. What about the correlation functions, in particular, OPE coefficients?

7 6 Our approach: Study the large-j limit of GKP-W relation directly! we propose a simple resolution of all these puzzles related to the identification of the boundary in the pp-wave limit a general hologographic relation such that OPE coefficeints C ijk of BMN operators on the CFT side interaction vertex λ ijk of string theory in the bulk pp-wave geometry

8 Our results are based on a definite spacetime tunneling picture Dobashi-Shimada-T.Y., hep-th/ To preserve the GKP-W relation in the PP-wave limit, we should consider tunneling geodesics V eff (z) tunneling trajectory real trajectory boundary horizon boundary horizon (z = 0) (z ) (z = 0) (z = )

9 8 Though this tunnuling trajectory is apparently different from the null geodesic of the usual interpretation of the pp-wave limit, we have the same bulk Hamiltonian with the ordinary Minkowski treatment: At a purely formal level, [real picture tunneling picture] is equivalent to a double Wick rotation: affine time : τ iτ target time : t it Tunneling geodesic : z = 1 cosh τ, t = tanhτ in terms of the Poincaré coordinate of Euclideanized AdS spacetime: ds 2 = R 2 ( dz2 z 2 + dx2 3 + dt 2 z 2 + ) Reaches the boundary z 0 as τ = ±T, T : z 2e T

10 This explains why 9 free asymptotic Hamiltonian dilatation is correct, by way of renormalization group, in spite of the fact that Hamiltonian cannot be directly identified with dilatation in our approach. boundary z=0 final state initial state

11 10 Affine time direction near the boundary is manifestly orthogonal to the boundary. : τ should not be identified directly with the radial time boundary boundary infinite affine time interval: τ = T +T T (no periodicity) Boundary theory must be treated as Euclidean, because we are considering tunneling amplitudes solves all of the puzzles! observables = Euclideanized S-matrix elements [ bulk] (in the sense of 1( τ) + 1 ( ψ) dimensions ) correlators of local gauge invariant BMN operators [ boundary]

12 Derivation from Witten diagrams 11 Boundary correlation functions are computed diagramatically by Witten diagrams using the bulk-to-boundary (z, x) (0, y) propagator. 2-pt function: K (z, y; x) = ( z ) z 2 + (x y) 2 1 x x = lim 2 ɛ 0 ɛ N( ) 2 d 4 ydz z 5 z ɛ K (z, y; x)k (z, y; x ) In the limit of large = J + k (k=finite), the integral is dominated by a one-dimensional integral along the tunneling null trajectory. K K (z; y) boundary z=0 boundary z=0

13 The tunneling trajectory connecting points from boundary to boundary (uniquely) solves the saddle-point eq. [ ] ln K (z, y; x) + lnk (z, y; x ) z [ ] lnk y µ (z, y; x) + lnk (z, y; x ) = 0, = 0 12 z(τ) = x x 2cosh τ, yµ (τ) = 1 2 (x + x ) µ 1 2 (x x ) µ tanhτ d 4 ydz z 5 z ɛ K (z, y; x)k (z, y; x ) N( )2 x x 2 τ collective coordinate with measure affine time along the tunneling trajectory T T dτ d 4 ydz z 5 dτ

14 We can extend this picture to 3-pt functions : 13 C 123 x 1 x x2 x x3 x = 1 N( 1, 2, 3 ) d 4 xdz K 1 (z, x; x 1 )K 2 (z, x; x 2 )K 3 (z, x; x 3 )V 123 (z; x) z 5 For generic configurations of 3 points x 1, x 2, x 3, there is no saddle point. However, in an appropriate short-distance limit on SYM side, the integral is dominated by a single tunneling trajectory parametrized by the collective coordinate τ with appropriate cutoff. K K δ x 2 x 3 0 K boundary z=0 boundary z=0

15 14 precise holographic relation between the 3-point vertex V 123 λ 123 and the CFT coefficient C 123. For general stringy modes, O 1 O 2 O 3 λ 123 x 1 x c dτ e ( )τ exp[ f J 2J 3 (2δ) 2 ] J 1 x 1 x c 2e2τ boundary-bulk propagators cutoff factor The factor f takes into account the nonlocal nature of string interactions: f J 2J 3 J 1 J 1 4πµ α (1) = R2 4πα = g sn for large µ

16 15 This leads to the Holographic relation for the OPE coefficients of BMN operators: C 123 = λ 123 µ( ) ( λ 123 = f J ) 2J ( )/ Γ( + 1)λ 123 J 1 2 (f = 1 4µα (1) α (2) α (3) K takes into account the extendedness of strings) J1 J 2 J 3 λ 123 = (1) 1 (2) 2 (3) 3 N H 3 h,

17 Remarks: 16 normalization of 2-point correlators O 1 ( x 1 )O 2 ( x 2 ) = δ 12 x H 3 h is the 3-point interaction vertex of the holographic string field theory S 3 = 1 2 J1 J 2 J 3 dτ (1) ψ (2) ψ (3) ψ N H 3 h + h.c.. For the special case of impurity-preserving sector, where O(1/µ 2 ), 1/µ 2 R 4 /J 2 this reduces to the simple form C λ/( ) in the limit of large µ, But this particular form is only valid with our unique holographic string-field theory vertex. Our relation is valid for much more general case of impurity-non-preserving processes.

18 17 Large-µ expansion : ω (r) n = (µα (r) ) 2 + n 2 µα (r) + n2 2 µα (r) + (α (r) = α p + r ) large-α expansion for fixed R 2 /J tensionless limit short-distance ( high-energy ) limit or for fixed α (r) ( string length parameter) weak t Hooft-coupling expansion on SYM side small AdS-scale (large curvature) on bulk side Thus in the large µ limit, we can compare both sides of AdS and CFT by perturbative methods.

19 18 Properties of the holographic string-field theory satisfyies the SUSY algebra to the first order in the string coupling The prefactor consists of two parts H 3 h = 1 2 (P D + P SV ) E 3 E 3 = standard overlap vertex of light-cone SFT P D = (h (2) + h (3) h (1) ) energy (h (r) ) difference Di Vecchia et al P SV Brink-Green-Schwarz type prefactor Spradlin-Volovich,... Both of these prefactors have so far been discussed separately. However, our holographic relation demands that they must be combined with the uniquely fixed coefficients as above.

20 19 For purely bosonic states, H 3 h ( 3 8 r=1 i=5 m=0 ω (r) m a m (r)i α (r) a (r)i m + 4 i=1 m=1 ) ω m (r) a (r)i m a (r)i m E 3 α (r) cos-modes sin-modes scalar directions ( S 5 ) vector directions ( AdS 5 ) In particular, for bosonic supergravity modes (chiral primary states), h (r) s = µ 8 i=5 H 3 (h (1) s + h (2) s h (3) 3 ) E 3 a (r) i,0 a(r) i,0 excitation number of only scalar directions

21 20 This corresponds to the known fact in supergravity that the derivative 3-point interaction terms can be eliminated by a field redefinition in the bulk of AdS 5 geometry : Lee-Minwalla-Rangamani-Seiberg, hep-th/ After reducing to the single tunneling trajectory, this means that the derivative part of interaction vertex is contained in the form V der ( τ + h(1) + h (2) h (3) )φ (1) φ (2) φ (3) 0 eq. of motion with h (i) being the total free Hamiltonian. Note: Total derivative cannot be ignored: energies (as conformal dimensions) are not conserved in our Euclideanized S-matrix picture. The existence of the P D part is crucial for the holographic relation.

22 21 Explicit checks in the large-µ limit Impurity-preserving case: We have checked our predictions by direct computation for two-impurity cases with general vector and fermionic excitations, to the leading order in the large-µ expansion. We can also use previously known results for 3-point functions and explain the compatibility of seemingly contradictory results in the literature. It is known that at the first order in λ the stringy single-trace operators are not operators with definite conformal dimensions. To obtain operators with definite conformal dimensions, mixing with double-trace operators with degenerate (in the lowest (λ ) 0 order) conformal dimensions must be taken into account. Beisert et. al., hep-th/ , Constable et. al., hep-th/ Let the order g 2 λ part of the two-point functions for O 1 and normal-product (local) operator : O 2 O 3 : be (J 1 = J 2 + J 3 ) O 1 : O 2 O 3 : g2 λ = g 2λ ( b 123 )C log(x 12Λ) 2 λ 1 r = first order anomalous dimension

23 The operators must then be replaced by 22 Then, 3-point correlator is O 1,single O 1 O 1,single + g 2 D 123 : O 2 O 3 : 1 + b 123 D 123 = C O 1O 2O 3 = C 123 standard spacetime factors + higher orders C 123 = g 2 λ b 123 C O(g 2 ) O(λ ) so that the correct 3-point OPE coefficient to the leading order in the large-µ expansion is this C 123, not g 2 C 0 123

24 On the other hand, it has also been proposed that the wrong coefficient satisfies (in the large-µ limit) C J1 J 2 J 3 N 1, 2,3 1 2 P D E 3 = g 2 µ( )C Di Vecchia et. al., hep-th/ Our prediction for the impurity-preserving case C 123 = J1 J 2 J 3 N 1,2, 3 1 2µ (P D+P SV ) E 3 +higher orders with respec to 1/µ allows us to compute the matrix elements of the vertex P SV E 3, using these two observations. The results explain the known proposal which relates the mixing matrix of SYM in a particular basis to P SV E 3 Gomis-Moriyama-Park, hep-th/ , hep-th/ Previously, it was not clear how to justify their special choice of the basis. All these discussions has been restricted to impurity preserving sector.

25 Impurity non-preserving case: 24 This case has not been treated before at least in the context of the BMN limit. When impurities are not preserved, the correction factor ( f J ) ( 2J ) 3 2 Γ( J ) in our holographic relation plays a crucial role, since in the large-µ limit ( f J ) ( 2J ) 3 2 J 1 ( J 1 4πµα (1) ) ( ) 2 = (gymn) 2 ( ) 2 in explaining the correct order of 3-point correlation functions. Also, the Gamma-function factor could be singular when negative integer in the classical approximation. Γ( ) = n=0 2( 1) n /n! n + nonsigular terms

26 Exchanges of arbitrary number of scalars with non-singlet representation (SO(4)) can be treated in a general way: 25 Outer circle = 1 Two inner cirlces = 2 and 3 dotted lines : Z-Z contractions solid lines : φ-φ contractions... Non-singlet vector exchanges can also be treated in a general way inductively. Mixings with double-trace operators must be taken into account in this case, when there is no propagators between 2 and 3.

27 Exchanges of singlet impurities : This case is somewhat subtle and more interesting. 26 Consider two simple but typical examples: 1. Scalar singlet operator with > 0 O (1) = 1 J1 N J 1Tr[Z J1 ], O (3) = 1 J3 N J 3Tr[Z J 3 ], O (2) s(p; p) = 1 [ J2 2 J 2 N Tr J 2+2 a=0 φ i Z a φ i Z J 2 a e 2πi J 2 ap 4ZZ J 2 +1 ]. The leading contribution comes from the following diagram with a Z-Z exchange

28 The CFT coefficient is simply 27 J1 J 2 J 3 1 C 123 = 2. N J 2 On the string side, the matrix element of the interaction vertex is (Ñ rs mn = Neumann coefficient) J1 J 2 J 3 4µ N J1 J 2 J 3 ( Ñ22 p, p) = 4µ N The multiplying factor in the holographic relation is 1 4πµ α (1) y 1 ( f J ) 1J Γ( µ( ) J ) 1 4πµ α (1) 2µ J 1 This preciesely gives the above CFT coefficient. (notation: y = J 2 /J 1,1 y = J 3 /J 1 )

29 2. Singlet vector operator with < 0 28 O (1) vs(p; p) = 1 4 J 1 N J 1 J 1 2 a=0 Tr [(D j Z)Z a (D j Z)Z J 2πi 1 2 a ap] e J 1 1 with O (2) and O (3) are both ground-state operators. The correlation function on SYM side consists of 3 different types of Feynman diagrams.

30 Each contribution is 29 F = F = F 1 (2,3) 123 = 2 N J 2 J 3 J 2 sin 2 πyp 1 1 J 1 J 2 J 3 (πp) 2 x 12 2J 2+2 x 13, 2J 3 2 N J 2 J 3 J 2 sin 2 π(1 y)p 1 1 J 1 J 2 J 3 (πp) 2 x 12 2J 2 x13, 2J N J 1 J 2 J 3 J 2 J 3 J1( 1) 2 psin πyp sinπ(1 y)p ( x 12 x 13 ) (πp) 2 x 12 2J 2+2 x 13 2J 3+2. This leads to the CFT coefficient C 123 = 2 J1 J 2 J 3 N J sin 2 πyp 1 (πp) 2 as the coefficient in front of the correct spacetime factor.

31 On the string side, the matrix element of the interaction vertex consists of 3 terms 30 J1 J 2 J 3 4 N µ 2 ( 2Ñ11 p, p + Ñ11 p, p + Ñ11 pp) = J1 J 2 J 3 N 8 π α (1) sin2 (πµp) The multiplying factor for this case is 1 µ fj 2J 3 (µα (1) ) 2 J 1 p 2 = 1 µ J 1 4πµ α (1) by which we find again precise matching. (µα (1) ) 2 p 2 = J 1 α (1) 4πp 2 The following specific characteristics of our holographic string-field theory and of the holographic relation, respectively, are crucial for the exact matching. Prefactor: scalar cos-modes only vector sin-modes only Multiplying factor: depends on the sign of , singular (for p = 0) in the latter case.

32 Concluding Remarks 31 We have proposed a natural spacetime picture for holography in the BMN limit proposed a precise and general holographic relation for OPE coefficients [boundary] and 3-point string vertex [bulk] checked the matching of bulk and boundary to the leading order in perturbation theory in 1/µ Remaining problems higher orders both in g s and λ higher-point functions extension to more general operators (spin chain,...)... derivation of the holographic relation within the logic of SYM origin of fifth dimension? extension to nonconformal case of general Dp-brane backgrounds see Asano-Sekino-T.Y. hep-th/ extension to M-theory holography, AdS 4(7) S 7(4)...

33 Extension to Dp-brane backgrounds : 32 Our picture does not depends on exact conformal symmetry, and hence can be applied to the plane-wave limit of a more general Dp-brane backgrounds (p 3). Since the tunneling trajectories do not go into the singular horizons, we can in principle use the same strategy as for p = 3. Even at the level of two-point functions for sugra modes, we have quite nontrivial predictions. Asano-Sekino-T.Y. hep-th/ For example, the two-point functions for scalar excitations behave as (p < 5) x 1 x p J+c(p), c(p) = J-independent const. We can extract effective spacetime dimensions for scalar excitations from this result: d eff (p) = p, eff = 2 5 p integer value effective spacetime dimensions when p = 1, 3,4 d eff (1) = 3, d eff (3) = 4, d eff (4) = 6,

34 This is consistent with the possible existence of nontrivial conformal field theories (with maximal susy) related to AdS holography (such as AdS 4 S 7 ) in M-theory (d eff = 3 M2 and d eff = 6 M5 branes). 33 (p = 1 ) 2D SYM with g YM 0, g 2 YM N Matrix-string theory in the weak(g YM 0)-coupling limit (Recall perturbative string theory strong-coupling limit) Sekino-T.Y. hep-th/ , T.Y. hep-th/ (Wrapped) supermembrane in the decompactification limit R 11 g s ( 1/g 2 YM )? 3D conformal field theory Scale dimension of scalar fields = 1/2 for CFT (AdS 4 S 7 case)

35 Scales for AdS 4 S D Plack scale: l 11 = gs 1/3 l s Compactification radius: R 11 = g s l s D2 brane scale: L D2 = g 2 YM g 1 s l s M2 scale (scale of AdS 4 with flux N): L AdS = l 11 N 1/6 M-theory limit: R 11 keeping l 11 fixed i. e. g s, l s 0 L D2 l 4 s/l Expected entropy: S L 9 AdS N 3/2 The boundary CFT dual to AdS 4 S 7 is an extreme IR limit of (N = 8) 3D SYM, but its degrees of freedom must be of order N 3/2 (not naive N 2 ) in the large N limit. No lagrangian description? (for a related discussion, see J. H. Schwarz, hepth/ ))