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Surface/State Correspondence and Bulk Local Operators in Pp-Wave Physics Letters B 751 (2015) 508–511 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 bound- ary 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 background. Back to the surface/state correspondence, in particu- ing the surface in the bulk should correspond to performing the 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 di- + − + rections. And the eight fermionic directions come from the half of ds2 = 2dx dx − μ2(xI )2(dx )2 + dxI dxI . (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) 5 fied) four-dimensional space around a great circle in S . 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 AdS and S . If the null curve is totally J 5 combinations are also 1/2-BPS just as Tr Z , 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- + I +I ij i j kinematical generators P , P , J , M , M 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) + (α p+μ)2 J 2 P is a central element and commutes with all other generators. − 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]. I + − iP )/ 2μP . In that case −P /μ becomes the number opera- − I I tor, since [P , a ] = μa . 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.
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