Surface/State Correspondence and Bulk Local Operators in Pp-Wave

Physics Letters B 751 (2015) 508–511

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Physics Letters B

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Surface/state correspondence and bulk local operators in pp-wave holography ∗ Nakwoo Kim a, , Hyeonjoon Shin b a Department of Physics and Research Institute of Basic Science, Kyung Hee University, Seoul 02447, Republic of Korea b School of Physics, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea a r t i c l e i n f o a b s t r a c t

Article history: We apply the surface/state correspondence proposal of Miyaji et al. to IIB pp-waves and propose that Received 24 August 2015 the bulk local operators should be instantonic D-branes. In line with ordinary AdS/CFT correspondence, Received in revised form 19 October 2015 the bulk local operators in pp-waves also create a hole, or a boundary, in the dual gauge theory as Accepted 4 November 2015 pointed out by H. Verlinde, and by Y. Nakayama and H. Ooguri. We also present simple calculations Available online 10 November 2015 which illustrate how to extract the spacetime metric of pp-waves from instantonic D-branes in boundary Editor: M. Cveticˇ state formalism. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction like the flat background the pp-wave does not have a clear boundary and the notion of holography has remained rather ambiguous Recently there appeared an interesting proposal called sur- [11,12]. Thus we reckon the pp-wave/BMN relation provides an face/state correspondence [1–3] which aims to generalize the idea excellent setup to test the surface/state correspondence in more and applicability of holography [4]. It is partly motivated by the detail and also in the full string theory, potentially including even observation that a particular tensor network prescription, called the string interaction through string field theory [13]. MERA (Multiscale Entanglement Renormalization Ansatz) [5], ap- Our main point is that instantonic D-branes are the “surfaces” parently realizes the emergence of holographic spacetime through which are mapped to gauge theory “states” through the BMN re- a real space renormalization [6]. The idea of surface/state corre- lation to gauge operators. It is probably important to emphasize spondence is based on the holographic entanglement entropy con- here that the brane configurations we will consider are not defined jecture [7], and extends it by assigning a density matrix not only in Wick-rotated euclidean bulk spacetime. The D-brane worldvol- to a spacelike minimal surface ending on the boundary but also to any open or closed surface in the bulk. Moving and deform- ume is euclidean, and they occupy a spacelike subspace in pp-wave ing the surface in the bulk should correspond to performing the background. Back to the surface/state correspondence, in particu- coarse-graining procedure: dubbed as “disentangler” and “isom- lar for the bulk local operators we use D-instantons. D-instantons etry” in MERA literature. If possible, this generalized holography probing the bulk geometry is a very well known idea in AdS/CFT: relation can be also applied to bulk geometries without a bound- they are dual to Yang–Mills instantons, of which the moduli space ary, including the Minkowski and de Sitter spacetime. In particular, contains AdS5 precisely [14]. As usual, it is expected that the su- when the surface shrinks to a point the dual state should be bulk persymmetry and modular invariance of 1/2-BPS D-instantons will local operators which have been studied in e.g. Ref. [8]. protect them against perturbative corrections and string interac- In this note we consider the maximally supersymmetric IIB tions. pp-wave [9], and its gauge/gravity correspondence by Berenstein, We take a modest first step in this note. Following mainly the Maldacena and Nastase (BMN) [10]. The advantage is obviously discussions in Refs. [3,15,16], we first discuss how the pp-wave twofold. The string spectrum is exactly solvable in the lightcone symmetry algebra should constrain the candidate bulk local op- gauge, and the proposed mapping to large R-charge operators has erators, and also illustrate how to extract the information of the been extensively studied and confirmed. On the other hand, just pp-wave metric from D-instantons. Along the way we also argue that the D-instantons as bulk local operators are “boundary creat- * Corresponding author. ing” operators on the boundary, just as proposed in Refs. [15,16]. E-mail addresses: [email protected] (N. Kim), [email protected] (H. Shin). We conclude with discussions on possible future avenues. http://dx.doi.org/10.1016/j.physletb.2015.11.007 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. N. Kim, H. Shin / Physics Letters B 751 (2015) 508–511 509

2. IIB pp-wave and BMN correspondence plex scalar fields in N = 4SYM with = J = 1. These are the only kind of N = 4operators which satisfy = J classically and also In this section we give a brief review of IIB pp-wave geometry in the interacting theory. The eight bosonic fields on the world- [9] and its gauge/gravity duality relation [10]. The metric of IIB sheet correspond to: 4 remaining scalar fields φi (i = 1, ··· , 4) of maximally supersymmetric pp-wave is given as (I = 1, 2, ··· , 8) SYM, and covariant derivatives Dμ along 4 Euclidean spacetime directions. And the eight fermionic directions come from the half of 2 = + − − 2 I 2 + 2 + I I ds 2dx dx μ (x ) (dx ) dx dx . (1) the set of fermion fields in N = 4SYM with = 3/2, J = 1/2. Note that the apparent SO(8) symmetry of the metric is broken The BMN excited states are long operators where the aforemen- into SO(4) × SO(4) due to the nonvanishing RR 5-form F+1234 = tioned “impurity” are inserted in the “string” of Z fields. Excitation 5 F+5678 = 2μ. This background is related to AdS5 × S through the by zero modes are constructed as totally symmetric combinations: Penrose limit, where one takes a null geodesic, and blows up the for instance (up to a normalization constant), R4 spacetime around it. Morally speaking the first is understood as i† j † + − 4 | ; ←→ l J l j Euclidean spacetime, and the second R is the (infinitely magni- a0 a0 0 p Tr(Z (Di Z)Z φ ) (5) fied) four-dimensional space around a great circle in S5. One com- l ment is that to obtain this nontrivial pp-wave the null geodesic and in an analogous way with fermions. These totally symmetrized 5 should be extended in both AdS5 and S . If the null curve is totally combinations are also 1/2-BPS just as Tr Z J , so the conformal di- inside AdS5, the resulting background is just the trivial Minkowski mensions are protected. It agrees with the string side computation background without any interesting gauge/gravity correspondence. that − J is exactly integral. For non-zero modes, the BMN pro- The total bosonic symmetry group is 30 dimensional: there are + + posal is that one should now consider impurities with “momen- kinematical generators P , P I , J I , Mij, Mi j and a dynamical − tum”. It goes like for instance generator P (here and below i, j = 1, 2, 3, 4 and i , j = 5, 6, 7, 8). ± + = ∓ i † j † More explicitly P i∂ , and an an |0; p I + + I ←→ 2πinl/ J l i k j J−k P =−i(cos(μx )∂I + μ sin(μx )x ∂−), (2) e Tr(Z φ Z φ Z ). (6) +I −1 + + I l J =−i(μ sin(μx )∂I − cos(μx )x ∂−). (3) Then the string spectrum predicts that the SYM operator should ij i j × M , M are the usual rotation generators of SO(4) SO(4). The receive quantum corrections for conformal dimension so that − relevant commutation relations are J = 2 1 + n2/(α p+μ)2. This is justified from the explicit pertur- − + − + [P , P I ]=−iμ2 J I , [P , J I ]=iPI , bative computations in SYM, upon the duality identification [10] + + [P I , J J ]=−iδIJP . (4) 1 g2 N = YM . (7) + + 2 2 P is a central element and commutes with all other generators. (α p μ) J − I +I It is easy to see that P , P , J comprise Heisenberg algebras This proposal was extensively analyzed and confirmed to higher I +I I + I† +I when we define a = (μ J + iP )/ 2μP and a = (μ J − orders in perturbation theory, for a review see e.g. [18]. − iPI )/ 2μP +. In that case −P /μ becomes the number opera- − tor, since [P , aI ] = μaI . The isometry generators can be of course 3. Bulk local operators as D-instantons more explicitly written in terms of string modes in lightcone gauge which will be explained below. Let us now consider how the isometry algebra should restrict The proposal of Ref. [10] by Berenstein, Maldacena and Nastase the putative BMN operator assignment in the bulk of IIB pp-wave. (BMN) relates a certain class of long operators in N = 4 supersym- We will employ the closed string theory states here mainly be- metric Yang–Mills (SYM) theory to the string spectrum in pp-wave cause of notational simplicity: thanks to BMN proposal one can background. The relevant operators, which we will call BMN oper- immediately write down the dual gauge theory operators. The ators, carry a large U (1)R charge J , so the conformal dimension maximally supersymmetric pp-wave is homogeneous, so we may − + − is also large. The BMN correspondence equates −P /μ = − J as start with any convenient point: let us choose x = x = xI = 0. + a duality relation. Obviously the little group is then generated by J I , Mij, Mi j . The The Green–Schwarz superstring theory of pp-waves in lightcone candidate string states are easily constructed. For the Fock vacuum gauge turns out to be a free theory of 8 massive scalars and 8 |0 (in string theory realization of course this is the unique string + massive fermions [17]. The quantum spectrum starts with a unique ground state |0; p ) satisfying aI |0 = 0, the “position” eigenstate + + + vacuum state |0; p , where p is lightcone momentum and a free J I |B = 0is given by parameter. There is a tower of string modes: first the zero modes 1 I , θ and their conjugates. For n ≥ 1, I , ˜ I are the right and the (aI†)2 α0 0 αn αn |B=e 2 I |0 . (8) left moving bosonic modes in SO(8) vector representation, while ˜ θn, θn are the right and the left moving fermionic oscillators in In the string theory realization of course the symmetry re- + spinor representation of SO(8). The string vacuum |0; p is the quirement alone does not lead to a unique string state. There − Fock vacuum with lightcone energy P = 0. Raising the level of a are fermionic modes as well, but more importantly the kinemati- bosonic or a fermionic zero mode by one will increase − J by cal generators only involve the zero-modes and we may consider one precisely. On the other hand, adding the occupation number of excitation by higher string modes. This is akin to the conformal the n-th mode by one increases − J by 1 + n2/(α p+μ)2. The field theory consideration in Refs. [3,15,16], that in general the usual level matching condition also applies, and the total occupa- symmetry requirement for boundary states is satisfied by a linear tion number of left- and right-moving modes should be the same combination of Ishibashi states. It is a natural guess that in string for physical states. theory the bulk local state |B should be D-instantons constructed The string spectrum beautifully matches that of the large R- as coherent states in closed string theory. Indeed, IIB pp-waves do charge operators in N = 4SYM. The vacuum state of the string allow 1/2-BPS D-instantons [19] which do include fermionic and + theory |0; p corresponds to Tr(Z J ), where Z is one of the com- other higher stringy mode excitations, and they are in general dual 510 N. Kim, H. Shin / Physics Letters B 751 (2015) 508–511 to long operators in N = 4super Yang–Mills with fermion and sis in [19], there exist supersymmetric instantonic D-branes of type gauge field insertions. We propose the operators thus obtained to (0, 0), (2, 2), (4, 0), (0, 4), (4, 4), (1, 1), (3, 3) where (p, q) refers to be the bulk local operators. Summarizing, the idea is that we have the number of legs in each R4. In Ref. [19] the authors studied bulk local operator expressions in closed string theory constructed explicitly (0, 0), (4, 0), (0, 4) branes. Once they are constructed in in [19], which then can be mapped SYM operators through BMN closed string theory the identification as operators in BMN sector proposal. is of course straightforward. A couple of comments are in order here. We first note that the Let us here consider specifically the case of (0, 4) (D3) and BMN dual of (8) are in general highly non-local in the dual field (4, 4) (D7) since they are easier to understand holographically. The i † (0, 4) case can be understood as a local operator in SYM space- theory point of view, since for instance acting on an means inser- 5 tion of a covariant derivative. And D-branes are non-perturbative time, but in the bulk it covers a four-dimensional subspace in S objects (solitons) in string theory, so the bulk local operators transverse to a great circle in terms of the original AdS/CFT cor- should be understood as non-perturbative states in SYM. Sec- respondence. In other words, it is a local operator in AdS5 but a hypersurface in S5, before we take the Penrose limit. The pre- ondly, in string theory there is a distinction between D-branes +i i ij i j and anti-D-branes. In the original proposal of surface/state corre- served symmetries are J , P in addition to M , M . On the →∞ + = spondence [1] the bulk local operators are given as closed surface holographic screen r at x 0, the same symmetry is pre- 3 R4 shrunk to a point. The lesson here is: as one tries to embed it in served for a space which is a product of: S in the first , the R4 − string theory we need to consider not only the shape but also the entire second , and the real line of x . In short, an operator R4 R4 orientation of the surface which can be related to the charge of which is local in the first but non-local in the second in R4 the D-instantons. the bulk is boundary creating operator only in the first This is For the standard AdS/CFT correspondence, it was argued that a natural result. The causal cone (in the sense of AdS/MERA [6]) of the local bulk operators are mapped to boundary creation operators a point in the bulk extended to UV is obviously a ball in bound- through a unitary transformation [3,15,16]. It was in particular em- ary, but the causal cone of the entire bulk should cover the entire phasized in [16] that the mapping is mediated through dilatation boundary spacetime. And there is no boundary to be created for (i.e. Hamiltonian in the radial quantization) by the imaginary unit. (4, 4) D7-branes obviously. The spherical boundaries of the holes created on the CFT side are Let us now go back to the bulk local state and the D-instanton, put at a constant time slice. and check they have the properties which are naturally expected For the holography of pp-waves, we recall that the light-cone from the particle dynamics in pp-waves. The first result we would + coordinate x becomes time both on the string worldsheet and like to reproduce is the solution of Klein–Gordon equation in pp- I 2 + 2 presumably also for the dual BMN sector. One way to see it is to wave. The (x ) (dx ) term in the metric introduces a harmonic R8 rewrite the metric as follows [12]. potential in , and one can easily obtain the eigenmodes of the D = 10 massless scalar wave equation 2 = 0as follows (see + − + 2 = + 2 + 2 − 2 2 + 2 e.g. [12]) ds 2dx dx dr1 r1( μ (dx ) d 3(1)) + − − + + 2 + 2 − 2 + 2 + 2 = ip x iP (nI )x I dr2 r2( μ (dx ) d 3(2)). (9) nI e e ψnI (x ) (12) I It is implied here that the radial coordinates of R4 × R4 are the √ 1 −μx2/2 holographic coordinates, and the boundary includes two copies ψn(x) = √ e Hn( μx) (13) 1/4 n ! of R × S3. We note that in the surface/state correspondence [1, π 2 n − 3] the holes are codimension-two spacelike hypersurfaces in the P (nI ) =−μ (nI + 1/2) (14) bulk. From the rearrangement in (9) we infer that it is a natural + − I guess that the hole is at x = 0, extended along x , and at r =∞. On the other hand, one can compute the overlap of local states at This subspace is invariant under the same symmetry algebra of the +I two different bulk points when we recall |B is basically a position point of our interest: J and SO(4) ×SO(4) rotations. It is amusing + ( J I ) eigenstate. to see that, in general, + − − + i(p x +P x ) −ixI P I + + − B|e e |B | = = ix0 P | Aholeatx x e B . (10) + − − + √ 0 ip x iP (nI )x = e e ψn (0)ψn ( μx). (15) This is the central point of this short note: namely, the bulk local I I I operator and the spherical boundary are again mapped through even n the time evolution in the boundary theory in pp-wave holography. Being a linear combination of the eigenmodes (12) it is clear that Thanks to the relative simplicity of pp-wave geometry, it is easy the above expression satisfies the scalar equation in the bulk. to consider planar hypersurface operators in the bulk. If the sur- The overlap function of a D-instanton vs. an anti-D-instanton face is localized along I ∈ D and extended along I ∈ N directions, can be of course calculated in the full lightcone superstring theory, + it is invariant under P I for I ∈ N and J I for I ∈ D. The reduc- and the result can be found in Ref. [19]. Adjusting to our notation, tion of the rotation group M , M should be done in the obvious + + ij i j A(x ; x , x ) = h (x ; x , x ) way. Then the associated bulk operators in the bosonic zero sector 1 2 0 1 2 should be × (non-zero mode contribution), (16) 1 1 + (aI†)2− (aI†)2 where the zero-mode part contribution h0 is given by |B=e 2 I∈D 2 I∈N |0 . (11) + h (x , x , x ) The ambiguity with respect to stringy modes and fermions should 0 1 2 m 2 2 m/2 be fixed using worldsheet modular invariance and supersymmetry. m(1 + q )(x + x ) 2mq x1 · x2 = exp − 1 2 + . (17) As far as we know the analysis of modular invariance for pp-wave 2(1 − qm) (1 − qm) instantons is not complete, although it is worked out for a number of cases in Refs. [19,20]. But it is known how unbroken supersym- The parameters here are defined as follows, metry restricts the possible D-brane configurations, depending on + + = + = −2x /p how many legs they have in each R4 [21]. According to the analy- m p μ , q e . (18) N. Kim, H. Shin / Physics Letters B 751 (2015) 508–511 511

+ + In the above x is the difference of x coordinate, and x1, x2 References are position vectors in the transverse R8. We expect to be able to + “read off” the bulk metric when we take the limit x → 0 and [1] Masamichi Miyaji, Shinsei Ryu, Tadashi Takayanagi, Xueda Wen, Boundary x ≡ x2 − x1 → 0. In that limit, setting x ≡ (x1 + x2)/2, states as holographic duals of trivial spacetimes, J. High Energy Phys. 1505 (2015) 152, arXiv:1412.6226 [hep-th]. + 2 p x 2 2 + [2] Masamichi Miyaji, Tadashi Takayanagi, Surface/state correspondence as a gen- 1 − h0 + μ x x . (19) 2 x+ eralized holography, PTEP 2015 (2015) 073B03, arXiv:1503.03542 [hep-th]. 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It will be very interest- superYang–Mills and holography, arXiv:hep-th/0205048, 2002. ing if we can relate the gravitational backreaction of D-instantons [14] Massimo Bianchi, Michael B. Green, Stefano Kovacs, Giancarlo Rossi, Instan- using the recent proposals on holographic Fisher information met- tons in supersymmetric Yang–Mills and D instantons in IIB superstring theory, J. High Energy Phys. 08 (1998) 013, arXiv:hep-th/9807033. ric [22]. [15] Herman Verlinde, Poking holes in AdS/CFT: bulk fields from boundary states, The study of M-theory pp-waves and their relation to a massive arXiv:1505.05069 [hep-th], 2015. matrix theory [10] is also an interesting subject. The kinematical [16] Yu Nakayama, Hirosi Ooguri, Bulk locality and boundary creating operators, consideration should straightforwardly carry through, in terms of arXiv:1507.04130 [hep-th], 2015. the abelian part of the BMN matrix model. On the other hand, [17] R.R. Metsaev, Arkady A. 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This work was supported by NRF grants [21] Kostas Skenderis, Marika Taylor, Branes in AdS and pp wave space-times, funded by the Korea government with Grant Nos. 2010-0023121, J. High Energy Phys. 0206 (2002) 025, arXiv:hep-th/0204054. 2012046278 (N.K.) and NRF-2012R1A1A2004203, 2012-009117, [22] Masamichi Miyaji, Tokiro Numasawa, Noburo Shiba, Tadashi Takayanagi, Kento NRF-2015R1A2A2A01004532 (H.S.). This work was completed Watanabe, Gravity dual of quantum information metric, arXiv:1507.07555 while NK was visiting CERN for “CERN-CKC TH Institute on Du- [hep-th], 2015; Nima Lashkari, Mark Van Raamsdonk, Canonical energy is quantum Fisher in- ality Symmetries in String and M-Theories”. NK is grateful to CERN formation, arXiv:1508.00897 [hep-th], 2015. for hospitality, and to CERN-Korea theory collaboration committee [23] Nakwoo Kim, Jung-Tay Yee, Supersymmetry and branes in M theory plane for supporting the workshop. waves, Phys. Rev. D 67 (2003) 046004, arXiv:hep-th/0211029.