Asymptotic symmetries, holography and topological hair

Journal of High Energy Physics, Jan 2018

Asymptotic symmetries of AdS4 quantum gravity and gauge theory are derived by coupling the holographically dual CFT3 to Chern-Simons gauge theory and 3D gravity in a “probe” (large-level) limit. Despite the fact that the three-dimensional AdS4 boundary as a whole is consistent with only finite-dimensional asymptotic symmetries, given by AdS isometries, infinite-dimensional symmetries are shown to arise in circumstances where one is restricted to boundary subspaces with effectively two-dimensional geometry. A canonical example of such a restriction occurs within the 4D subregion described by a Wheeler-DeWitt wavefunctional of AdS4 quantum gravity. An AdS4 analog of Minkowski “super-rotation” asymptotic symmetry is probed by 3D Einstein gravity, yielding CFT2 structure (in a large central charge limit), via AdS3 foliation of AdS4 and the AdS3/CFT2 correspondence. The maximal asymptotic symmetry is however probed by 3D conformal gravity. Both 3D gravities have Chern-Simons formulation, manifesting their topological character. Chern-Simons structure is also shown to be emergent in the Poincare patch of AdS4, as soft/boundary limits of 4D gauge theory, rather than “put in by hand” as an external probe. This results in a finite effective Chern-Simons level. Several of the considerations of asymptotic symmetry structure are found to be simpler for AdS4 than for Mink4, such as non-zero 4D particle masses, 4D non-perturbative “hard” effects, and consistency with unitarity. The last of these in particular is greatly simplified because in some set-ups the time dimension is explicitly shared by each level of description: Lorentzian AdS4, CFT3 and CFT2. Relatedly, the CFT2 structure clarifies the sense in which the infinite asymptotic charges constitute a useful form of “hair” for black holes and other complex 4D states. An AdS4 analog of Minkowski “memory” effects is derived, but with late-time memory of earlier events being replaced by (holographic) “shadow” effects. Lessons from AdS4 provide hints for better understanding Minkowski asymptotic symmetries, the 3D structure of its soft limits, and Minkowski holography.

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Asymptotic symmetries, holography and topological hair

HJE Asymptotic symmetries, holography and topological hair Rashmish K. Mishra 0 1 Raman Sundrum 0 1 0 College Park, MD 20742 , U.S.A 1 Maryland Center for Fundamental Physics, Department of Physics, University of Maryland , USA Asymptotic symmetries of AdS4 quantum gravity and gauge theory are derived by coupling the holographically dual CFT3 to Chern-Simons gauge theory and 3D gravity in a \probe" (large-level) limit. Despite the fact that the three-dimensional AdS4 boundary as a whole is consistent with only nite-dimensional asymptotic symmetries, given by AdS isometries, in nite-dimensional symmetries are shown to arise in circumstances where one is restricted to boundary subspaces with e ectively two-dimensional geometry. A canonical example of such a restriction occurs within the 4D subregion described by a Wheeler-DeWitt wavefunctional of AdS4 quantum gravity. An AdS4 analog of Minkowski \super-rotation" asymptotic symmetry is probed by 3D Einstein gravity, yielding CFT2 structure (in a large central charge limit), via AdS3 foliation of AdS4 and the AdS3=CFT2 correspondence. The maximal asymptotic symmetry is however probed by 3D conformal gravity. Both 3D gravities have Chern-Simons formulation, manifesting their topological character. Chern-Simons structure is also shown to be emergent in the Poincare patch of AdS4, as soft/boundary limits of 4D gauge theory, rather than \put in by hand" as an external probe. This results in a nite e ective Chern-Simons level. Several of the considerations of asymptotic symmetry structure are found to be simpler for AdS4 than for Mink4, such as non-zero 4D particle masses, 4D non-perturbative hard" e ects, and consistency with unitarity. The last of these in particular is greatly simpli ed because in some setups the time dimension is explicitly shared by each level of description: Lorentzian AdS4, CFT3 and CFT2. Relatedly, the CFT2 structure clari es the sense in which the in nite asymptotic charges constitute a useful form of \hair" for black holes and other complex 4D states. An AdS4 analog of Minkowski \memory" e ects is derived, but with late-time memory of earlier events being replaced by (holographic) \shadow" e ects. Lessons from AdS4 provide hints for better understanding Minkowski asymptotic symmetries, the 3D structure of its soft limits, and Minkowski holography. ArXiv ePrint: 1706.09080 AdS-CFT Correspondence; Chern-Simons Theories; Gauge Symmetry; Global Symmetries 1 Introduction 2 Lightning review of CS/GR3 AS and CFT2 currents 2.1 Non-abelian CS gauge theory 2.2 GR3 3 Holographic matter coupled to CS/GR3 on AdS3 3.1 3.2 CS and GR3 coupled to CFT3 4 The large-level \probe" limit Abelian CS Non-abelian CS and GR3 Abelian gauge theory Non-abelian gauge theory and gravity 5 Non-standard @AdS4=2 correlators as CFT2 correlators 5.3 Compatibility with 4D quantum loops and masses 6 Evading the no-go for in nite-dimensional AS in AdS4=2 7 Maximal spacetime AS from 3D conformal gravity 7.1 A \super-translation"-like KM AS for AdS4=2 7.2 Non-unitary nature of CGR3 8 AdS4Poincare: AS from holography and holography from AS 4.1 4.2 5.1 5.2 8.1 8.2 8.3 9.1 9.2 9.3 { i { XBMS3 from Vir+ CGR3 on Mink3 Vir Holographic grammar from AS 9 Emergent CS and \shadow" e ects from boundary/soft limits Set-up CS on @AdS4Poincare and a \holographic shadow" e ect Emergent CS level 9.4 The soft limit, CS on the Poincare horizon, and a bulk \shadow" e ect 10.1 CS gauge theory on @AdS4global 10.2 Time-evolution from AS algebra via CGR3 on @AdS4global 11.1 (A)dS4 11.2 AS as \hair" 11.3 Mink4 1 Introduction In gravitational and gauge theories, Asymptotic Symmetries (AS) are di eomorphisms and gauge transformations that preserve the asymptotic structure of spacetime while still acting non-trivially on asymptotic dynamical data. They include isometries of spacetime and the standard global charges arising from gauge theory, but they can be larger. Famously, 4D Minkowksi spacetime (Mink4) has an in nite-dimensional spacetime AS algebra (see [1] for a recent review). This was originally identi ed as the BMS algebra of super-translations [2, 3], but has been extended more recently to include super-rotations as a subalgebra [4]. We refer to this extended algebra in 4D as XBMS4. The ongoing challenge since discovery of these symmetries has been to understand their physical signi cance and utility. Considerable progress has been made in this regard by the discovery that the associated large di eomorphisms and gauge transformations arise as soft limits of physical gravitational and gauge elds emerging from scattering processes [5{15], as captured by the Weinberg Soft Theorems [16{20]. The in nite-dimensional AS then describe the soft eld dressing of a hard process, and are sensitive to the passage of charge/energy-momentum as a function of angle, through \memory" e ects [21{30]. This generalization of the usual overall charge/energy-momentum conservation laws has led to the suggestion that AS charges can act as a new subtle form of \hair" that can characterize black holes (or other complex states), giving a ner understanding of black hole entropy and information puzzles [31{34]. The fact that the super-rotation subalgebra of Mink4 AS has a Virasoro Virasoro form (Vir Vir), while gauge theory gives rise to Kac-Moody (KM) subalgebras, is highly reminiscent of Euclidean two-dimensional conformal eld theories (ECFT2) [4]. Indeed such a ECFT2-like structure living on the celestial sphere was discovered [35, 36], AS charges arising as Laurent expansion coe cients of a 2D holomorphic stress tensor and other currents. A straightforward derivation [36] follows by foliating Mink4 by 3D de Sitter spacetimes (dS3) and hyperbolic spaces [37{41], more suggestively considered as the Euclidean continuation of 3D anti-de Sitter (EAdS3). 4D elds can then be \Kaluza-Klein" (KK) reduced by separation of variables into 3D (EA)dS3 elds, with a continuum of 3D masses, m3KK > 0. In this language, 4D S-matrix elements map to boundary (EA)dS3 correlators [37], the associated 4D LSZ-reduced Feynman diagrams mapping to 3D Witten diagrams (modulo superpositions). Most importantly, the 3D massless limit, m3 ! 0, corresponds to 4D soft limits, in particular the soft limit of 4D gauge theory yielding 3D Chern-Simons (CS) gauge elds, and the subleading soft limit of 4D General Relativity (GR4) elds yielding GR3 (which also has a CS formulation [42]) on (EA)dS3. The basic { 1 { grammar of (EA)dS3/ECFT2 [43] then yields the ECFT2-like structure. The 3D CS elds \live" on the boundary of 4D spacetime. Despite these recent developments, several important questions and puzzles remain: A central question is how fully the axioms of CFT2 are realized in the structure underlying AS. In particular, it has not been clear what the values of the associated central charge and KM levels are, whether zero, in nite or nite. This question is not answerable at the AS level of discussion which focuses on external CS/soft elds, since the central charge and levels are probed by internal CS/soft lines (at tree level). It was argued in ref. [36], that a central charge would be IR sensitive to the HJEP01(28)4 experimental delineation between \soft" and \hard", but this was not fully clari ed. The ECFT2 structure is not consistent with being the Euclidean continuation of a unitary CFT2, much as in the dS/ECFT context. It is an open question as to how the unitarity of the Mink4 quantum gravity (QG) S-matrix is encoded in the ECFT2 correlators. The subleading soft limit of GR4 leads to the super-rotation subalgebra of Mink4 AS, and is elegantly encoded in GR3, which has a SO(3; 1) CS formulation, but the leading soft limit and the associated super-translations do not have a CS formulation [36]. Naively, the SO(3; 1) Lorentz gauge group should be extended to the full Poincare group ISO(3; 1) as the CS gauge group in order to include (super)translations, but ISO(3; 1) lacks the requisite quadratic invariant to construct a CS action. Relatedly, ref. [36] found that the ECFT2 current, whose Laurent expansion yields super-translations, is non-primary. Therefore there is an open question as to what the 3D characterization of subleading and leading soft GR4 elds is that leads to XBMS4 in a uni ed way. Previous discussions of memory e ects describe them in classical terms, while the hallmark of CS theories are quantum mechanical topological e ects that generalize the Aharonov-Bohm e ect [44{46]. These two views of memories need to be better reconciled. been spelled out. The connection of AS to 3D CS characterization of soft elds hints at a possible connection to a 3D holographic duality with Mink4 QG, but this connection has not It is very attractive to contemplate AS charges as a new rich form of \hair" for black holes or other 4D states. But such a role is still unclear, and being debated [47, 48]. In this paper, we make some progress on all these fronts within a more transparent context, by generalizing the notion of AS to AdS4 QG and gauge theory. Primarily this is because we know the 3D holographic dual of AdS4 is CFT3 [49{55], and there is a natural way to connect this to CS and GR3, and from this to CFT2 and in nite-dimensional AS. Yet by standard analysis the AS of asymptotically AdS4 GR only consist of the nitedimensional isometries [56], SO(3; 2), in sharp contrast to the in nite-dimensional AS of asymptotically Mink4 GR. Let us sketch why this is the case. { 2 { First consider Mink4, ds2Mink4 = 1 cos2 u+ cos2 u =2 and u = =2, 1 4 du+ du sin2(u+ where d 2 is the usual metric of the angular sphere. We see that at the boundary of Mink4, u+ = where Weyl In particular, these di eomorphisms include those reducing to conformal isometries on the boundary geometry, namely the in nite-dimensional conformal symmetries of the 2D angular sphere, and correspond to the super-rotations. But in AdS4global, the boundary at = =2 has a fully three-dimensional geometry d 2 d 22 : The conformal isometries of this boundary S2 R, and hence AS of AdS4, are just nite-dimensional SO(3; 2). By contrast, in the case of AdS3, @AdS3 is obviously twodimensional, famously with in nite-dimensional conformal isometries and AS [57]. Nevertheless, there is a loop-hole to this no-go argument for in nite-dimensional AdS4 AS if one is restricted to subspaces of @AdS4 with two-dimensional geometry, which we will see can happen for di erent physical reasons. Most straightforwardly, this is illustrated by the subregion of AdS4 described by a Wheeler-DeWitt QG wavefunctional, holographically dual to a quantum state of CFT3 at some xed time, as depicted in gure 1. Its 3D boundary resembles the null boundary of Mink4, with e ectively 2-dimensional geometry, re ecting the two-dimensional holographic geometry of @AdS4 at = 0. This has in nitedimensional conformal isometries, leading to in nite-dimensional AS. The basic strategy of this paper will be to study CS gauge theory and GR3 coupled to CFT3, where the CFT3 is (in isolation) the holographic dual of AdS4 QG, on a variety of 3D spacetimes M3: S = SCS + SGR3 + SCFT3 + UV-completion: (1.5) The CFT3 global internal symmetries are gauged by the CS sector, and the CFT3 spacetime symmetries are gauged by GR3. Such GR3 and CS + matter theories are well-known to have in nite-dimensional AS [57{65]. In particular, when M3 = AdS3, AdS3/CFT2 implies this setup is dual to CFT2, where there is a standard connection of the 2D chiral currents and stress-tensor with in nite-dimensional KM and Vir+ Vir symmetries (brie y reviewed { 3 { r); (1.1) (1.2) (1.3) (1.4) HJEP01(28)4 τ 4 Boundary of AdS4 at τ = 0 : S2 S2 Spatial hypersurface Boundary of subregion is spanned by all spacelike hypersurfaces ending on this boundary S2. An example of such a hypersurface is shown in green. A vertical cross-section is shown on the right. in section 2). The in nity of (AS) charges of CFT2 (AdS3) form a well-known type of 2D (3D) \hair", operating on and nely diagnosing quantum states, in a manner generalizing the action of ordinary conserved global charges. But now the CFT3=AdS4 duality of the 3D matter \lifts" the AS charges and their utility to 4D. This construction yields three layers of description of the dynamics. The quarks and gluons of some large-Ncolor formulation of CFT3 will be called for brevity, \quarks". The dual AdS4 gravitons and matter are the \hadrons", composites of the 3D \quarks", the 4D elds being equivalent to KK towers of 3D \hadronic" states related by 3D conformal symmetry. The well-de ned @AdS3 correlators will involve external lines of these \hadrons", rather than \quarks" (as discussed in section 3). This is in complete analogy to the wellde ned nature of hadronic S-matrix elements in Minkowski spacetimes, as compared with the provisional nature of the quark/gluon S-matrix. Even more fundamentally, the CFT3 \quarks" and the CS + GR3 elds themselves are composites of the CFT2 degrees of freedom, which we call \preons". AS charges are simple moments of these local \preon" degrees of freedom. A nice feature here is that time persists at each layer of description, and hence unitarity is manifest at each stage. The 4D loop expansion (controlled by the expansion parameter 1=Ncolor in 3D) can be done to all orders without spoiling these results. Including 4D massive particles is straightforward, captured automatically by the CFT3 description. We will show that even in the large-level limit, in which the CS and GR3 elds are decoupled, these AS remain as subtle charges of the matter sector, CFT3 (see section 4). Because the CFT3 on AdS3 is dual to (half of) AdS4, the 3D AS are inherited as AS of AdS4 QG and gauge theory. From the 4D perspective (section 5), the \hadronic" @AdS3 correlators which manifest the in nite-dimensional AS are also @AdS4 correlators, but not { 4 { of the standard form. In particular, the @AdS3 endpoints are restricted to a submanifold of @AdS4 with two-dimensional geometry, one natural realization of the loop-hole mentioned earlier in the no-go argument for in nite-dimensional AdS4 AS (section 6). The AS of AdS4 are in fact closely analogous to those of Mink4, in particular the Vir+ can be viewed as analogous to Mink4 super-rotations. The analog of Mink4 super-translations is subtler. We will show (section 7) that these can be incorporated by replacing GR3 by 3D conformal gravity (CGR3) [66, 67], which also has a SO(3; 2) CS formulation [42, 68]. In the case of M3 = AdS3, this leads to an extension of the AS by a KM algebra [69]. But the full AS of AdS4 is even larger, because CFT3 on AdS3 only projects half of AdS4. The technically simplest approach to the full AS structure is taken by switching to M3 = Mink3, where the dual of the CFT3 is given by the Poincare patch, AdS4Poincare (section 8). While not the entirety of AdS4global, it shares all of its (in nitesimal) isometries, and hence exhibits the full AS algebra. This full AS structure allows us to run the connection to AdS4=CFT3 duality in reverse: if one begins by identifying the AS of AdS4 in CGR3 (SO(3; 2) CS) form, the only form of compatible matter that can couple to CGR3, respecting its Weyl invariance, is CFT3. In this sense, the holographic grammar follows from the AS structure. The Poincare patch provides other simpli cations. It gives the most straightforward 4D dual picture when GR3 is not yet decoupled from CFT3, namely a lower-dimensional Randall-Sundrum 2 (RS2) construction [70], with a 3D \Planck brane" in a 4D bulk [71]. The GR3 then incarnates as the localized gravity of RS2. The familiarity of RS2 helps to make an important contrast. We have argued above, and in the body of this paper, that the in nite-dimensional AS are most readily recognized as coming from 3D GR3/CS elds, and yet are interesting because we can \lift" them beyond three dimensions. But there appears to be an even easier way to arrange this, by just considering gravitational theories in higher-dimensional product spacetimes of the form Mink3 X or AdS3 X, where X is some compact manifold. Under Kaluza-Klein reduction to Mink3 or AdS3, such theories would have a GR3 3D-massless mode, which would again yield in nite-dimensional symmetries. The distinction with what we are doing here is that such product theories would not have a non-trivial decoupling limit for the GR3 elds. That is, we cannot sensibly remove the GR3 subsector in some limit while keeping the rest of the physics xed. But RS2 with 4D bulk is dual to GR3 + CFT3, and there is a limit in which the 3D gravitational coupling vanishes, leaving a xed limiting CFT3, dual to AdS4Poincare QG. In other words, we will argue that GR3/CS has a tight connection with AS structure on the one hand and with 3D holography of the 4D QG on the other. But this only takes place in higherdimensional theories where the GR3 subsector has a decoupling limit. Higher-dimensional product spacetimes are not of this type. The Poincare patch also provides the stage to simply derive the emergence of CS gauge elds as helicity-cut soft/boundary limits of AdS4 gauge elds (CFT3 composites), which couple to charged modes (section 9). In this way, the CS structure is not put in \by hand" and then removed by a large-level limit, but rather describes a subsector of the pure CFT3/AdS4, with a nite but subtle type of CS level. We will see that the e ective CS gauge elds mediate analogs of the \memory" e ects identi ed in Mink4, which we call { 5 { \shadow" e ects since their relationship to the holographically emergent spatial direction is analogous to the relationship of memory e ects to time. For the CFT3 to project all of AdS4global, we must choose M3 = S2 R (section 10), but this closed universe does not have an asymptotic region or boundary to straightforwardly display AS. The AS arise by cutting at some point in time (say zero), so that the wavefunctional is given by functional integration up to that point, that is on M3 = S2 R (where the last factor refers to only negative values of time). This yields precisely the holographic dual of the Wheeler-DeWitt wavefunctional in AdS4, brie y discussed above. Finally, it is obviously of interest to ask how to translate the insights of AdS4 AS back to Mink4 (section 11). A strategy is suggested by the argument of section 8 for deriving the holographic grammar of AdS4 from its AS structure. In Mink4 we are ignorant of the former but know the latter, so the analogous steps should yield new insight into Mink4 holography. The rst step is to give the 3D characterization of the full AS and soft elds of Mink4 QG, in analogy to identifying CGR3 for AdS4. Currently this is not known for the Mink4 super-translations, although super-rotations take a simple GR3 form. We will provide some concrete guesses as to how to obtain the full 3D structure, which will then form the \mold" for a compatible holographic form of (hard) matter. 2 Lightning review of CS/GR3 AS and CFT2 currents CS theories, including GR3 in CS form, are famously gauge invariant and topological, insensitive to the geometry on 3D spacetime M3, except at the boundary @M3 where local degrees of freedom emerge, exhibiting in nite-dimensional AS. We brie y review how this happens for M3 = AdS3, where the boundary structure and AS are just those of the dual CFT2. Concretely, we write the metric in the form ds2AdS3global = d 2 d 2 cos2 d 2 sin2 ; RAdS3 1: A point y in AdS3 is represented by the coordinates ( ; ; ), where 0 < 2 and 0 < =2. The space of AdS3 is conformally equivalent to S2=2 1 < and the boundary @AdS3 is at = 0 in these coordinates. (2.1) < 1, R, 2.1 Non-abelian CS gauge theory We begin with internal CS gauge theory, ; (2.2) Aa ta, ta are the generators of the CS gauge group, Tr(tatb) = ab, and is where A the CS level. This action is metric-independent and gauge-invariant in the AdS3 \bulk", but since gauge-invariance depends on integration by parts it is violated on the boundary, @AdS3. This implies that \gauge orbit" degrees of freedom \live" on this 2D boundary, = 0, { 6 { This implies boundary conditions, A ( = 0) = 0. Further, bulk gauge invariance can be used to go to the axial gauge: A = 0. With this the boundary conditions are too stringent, giving A = 0 throughout AdS3 as the only solution to the rst order equations. We can modify the boundary conditions to constrain just one linear combination of boundary components of A, say A . To accomplish this we can add a boundary term to the action, 2 Z dz+ dz Tr A ( = 0) A+( = 0) : (While this explicitly violates gauge invariance, recall the bulk action is already not gaugeinvariant on the boundary.) In the presence of this term, the total boundary contribution to the variation of the action is given by Stotal = Z dz+ dz Tr A+ A ; implying the boundary condition A (z+; z ; = 0) = 0. But now A+(z+; z ; = 0) is unconstrained, consistent with non-trivial solutions (in the presence of matter). Even though we have xed axial gauge A = 0, we must retain the A equation of motion, F+ = 0; away from any matter sources, where F is the non-abelian eld strength. Evaluating this on the boundary, and using the boundary condition A = 0, which is the root of the equivalence of the CS gauge sector to a 2D Wess-Zumino-Witten (WZW) current-algebra sector on the boundary [62{65]. It is convenient to use light-cone coordinates in the boundary directions, z : The equations of motion read The dual CFT2 current, is therefore chirally conserved, The Fourier components de ne AS charges, F+ !0 j+(z+; z ) = li!m0A+(z+; z ; ); j+(z+) = X Qan+( )ta ein ; n2Z { 7 { (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) which are angle-dependent \harmonics" of the conserved global charges, Q0a+. The dependence of Qan+( ) follows by the fact that j+ is a function of z+ = + only, simply given by Qan+( ) / ein : In the Qn+ basis, the simple structure of j correlators within @AdS3 Witten diagrams takes the form of a Kac-Moody algebra (at = 0), h Qan+; Qbm+ i = X i f abc Qcn++m + n ab n+m;0 c where f abc are the structure constants, and the CS level sets the central extension. The non-abelian rst term on the right-hand side re ects the non-abelian CS interaction, while the central extension second term on the right-hand side re ects the CS \propagation". (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) HJEP01(28)4 grav = M3Pl RAdS3 : The dreibein VEVs lock the six SO(2; 2) global generators Ln= 1;0;1 to the AdS3 isometries. The action of these generators at the boundary of AdS3 is given by n = 1; 0 : Analogous to the case of internal CS gauge symmetries, the stress tensor components are chiral, and their Fourier modes give angle-dependent \harmonics" of the above SO(2; 2) global symmetries, 2.2 GR3 Consider next the case of 3D gravity on AdS3, which can be formulated as a SO(2; 2) CS theory in terms of dreibein and spin connection variables [42], with level t++(z+) = X Ln+( )ein ; n2Z t (z ) = X Ln ( )e in ; n2Z where the dependence of Ln+( ) is xed: These AS charges now form a Vir+ Vir algebra [57] generalizing the SO(2; 2) isometries, as opposed to a KM algebra if there had been no VEVs (as reviewed in ref. [72]), The central charge is given by Again, the two terms on the right-hand side are the 2D re ection of the non-abelian interaction of GR3 and the free \propagation". Ln+( ) / ein : c 12 c = grav = M3PlRAdS3 : { 8 { The charges Ln have a non-zero commutator with internal KM charges Qn, while the commutator between + and charges vanishes. Given that m = 0 measures the energy corresponding to translational symmetry: E = L0+ + L0 , it follows that matching our earlier observation that Qn+ / ein . 3 Holographic matter coupled to CS/GR3 on AdS3 We now couple CS and GR3 to 3D matter in the form of CFT3, all living on asymptotic AdS3. The CFT3 is chosen such that when living (in isolation) on @AdS4 = S2 R it is holographically dual to some AdS4 QG and gauge theory. HJEP01(28)4 3.1 CFT3 in isolation on AdS3 We begin by noting that AdS3 Weyl S2=2 R; (2.20) (2.21) (3.1) (3.2) < 2 , where = S2 Weyl R is only de ned up to Weyl equivalence, this suggests that CFT3 on AdS3 is holographically dual to half of AdS4, as follows. It is useful to use AdS4 coordinates exhibiting an AdS3 foliation [73], The AdS3 coordinates ( ; ; ) have the ranges while the fourth dimension coordinate r takes all real values. Ref. [74] argued (translating their analysis down a dimension to the set-up of interest here) that CFT3 states on AdS3, re ecting o @AdS3 are dual to AdS4 particles in the region r > 0 re ecting o the r = 0 surface. The speci c boundary condition at r = 0 is determined by the whether or not the CFT3 ground state on AdS3 preserves or spontaneously breaks the CFT3 global symmetry. We will consider the case where the global symmetry is preserved, in which case we must choose Neumann boundary condition at r = 0. We denote the region r > 0, holographically projected by CFT3, by \AdS4=2". In the original AdS4 global coordinates the AdS3 foliation by constant r hypersurfaces is depicted in gure 2, where the restriction to AdS4/2 corresponds to keeping only the northern half of the coordinate ball, r = 0 being the equatorial disc. The CFT3 lives on the boundary of this region, the upper hemisphere. @AdS correlators are the classic di eomorphism and gauge invariant observable in AdS QG, just as the S-matrix is in Mink QG. Here we are preparing to couple CFT3 on AdS3 to GR3 and CS, so we are interested in @AdS3 correlators. In this subsection however we L+m; Qn+ = nQ+m+n L0+ + L0 ; Qn+ = n Qn+; Fixed τ fixed r AdS 3 are not yet including the gauging by GR3 and CS, focusing therefore on @AdS3 correlators of just the CFT3. In standard Minkowski QCD we have a provisional meaning for the Smatrix elements of quarks and gluons. But strictly speaking this is ill-de ned because they are not asymptotic states. Instead we should more properly consider S-matrix elements of hadrons such as protons and pions. Similarly, with the CFT3, instead of \quark" @AdS3 correlators, we consider \hadron" @AdS3 correlators. Of course these \hadrons" are given precisely by the AdS4 dual. But now each AdS4 eld contains many \hadronic" AdS3 mass eigenstates, which we can isolate by KK decomposition based on the AdS3 foliation. We illustrate this for the simple case of tree-level AdS4 Yang-Mills theory, with 4D eld A. For this purpose it is convenient to adopt what we call \product-space" coordinates. CFT3 \hadrons"). External lines are AdS3 bulk-boundary propagators for masses proportional to `. Blob consists of AdS3 KK interactions and bulk-bulk lines (Fourier transformed in from AdS4). Using the change of variables from r to 2 tan 1 (tanh(r=2)), the metric changes to ds2AdS4 = f 2( ) d 2 + ds2AdS3 = 2 tan 1 tanh ; f ( ) = cosh r ; displaying Weyl-equivalence to the product geometry AdS3 Interval. The restricted region < =2 (in AdS units). Figure 3 shows this AdS4/2, 0 < r < 1 corresponds to 0 < \product-space" representation of AdS4. Because of the Weyl invariance of classical 4D Yang-Mills, the factor f 2( ) is irrelevant and the spacetime is e ectively of product form. (Non-Weyl-invariant theories can also be KK-decomposed, but less straightforwardly.) In standard KK fashion, in axial gauge A = 0, the 4D Maxwellian eld decomposes as A ( ; ; ; ) = X A` ( ; ; ) cos((2` + 1) ) ; `2Z where the A` ( ; ; ) are a tower of 3D Proca elds with AdS3 masses 2` + 1 in units of AdS radius. It is AdS3 Witten diagrams of these KK elds that correspond to \hadron" 3.2 CS and GR3 coupled to CFT3 We now switch on CS and GR3, gauging any internal global symmetries of CFT3 (dual to 4D gauge symmetries) and the global spacetime symmetries and stress tensor of CFT3 (dual to 4D gravity), so that @AdS3 correlators include the dual 2D chiral currents j , and stress tensor t . The 3D KK modes discussed above are dual to local 2D primary operators O2`D. All the CFT2 operators are composites of some 2D \preon" elds. A typical Witten diagram is shown in gure 5, with CS and GR3 lines decorating the earlier purely \hadronic" diagrams. Such 3D Witten diagrams yield general CFT2 dual correlators, now including stress tensor and chiral currents, h0j T fj : : : t As we reviewed in section 2, this CFT2 has in nite-dimensional symmetries associated with : : : O` : : : g j0i. its chiral currents and stress tensor. (3.3) (3.4) HJEP01(28)4 The decoupling of the CS and GR3 sectors from CFT3 is accomplished by simply taking the large CS-level limit, ! 1. (For decoupling the GR3 sector this is equivalent to the large-central-charge limit of the Virasoro symmetry of the CFT2 dual.) We will show that in this limit there is a remnant of the AS algebra that survives for CFT3 alone, providing a new form of \hair" for the dual AdS4/2 states and black holes. 4.1 The diagrammatics are very simple in the abelian CS case. The factor of 1= suppresses CS propagators, so the large level limit ! 1 naively eliminates AdS3 correlators involving CS lines altogether. However, choosing the normalization for the dual CFT2 current according to j+ = !0 lim A+(z+; z ; ) ; (4.1) we e ectively multiply CS endpoints in @AdS3 Witten diagrams by , canceling the 1= of bulk-boundary propagators, so these survive the limit. Only bulk-bulk CS lines are suppressed. The surviving diagrams have the form shown in gure 6. We see that correlators with the CFT3 are O( 0) ( gure 6a). However, the pure CS diagram shown in gure 6b corresponding to the correlator hj+ j+i is special. While the propagator scales as 1= , there are two factors of for the two end points, making this O( ), dominating all other correlators as ! 1. But, if we restrict our attention to correlations with CFT3 \matter" (AdS4 particles), then obviously this purely CS correlator drops out and we have a nite limit as seen in Mink4 AS. For ! 1. This explains a puzzle regarding the CS level rst nite but large , appears in the central extension of the KM algebra as the KM face of the hj+ j+i correlator. But if we are only tracking correlations that involve 4D particles (CFT3), then we are blind to the purely CS correlator hj+ j+i j j ~ κ0 j j ~ κ1 (a) Leading Witten diagrams including external (b) Leading Witten diagram without external HJEP01(28)4 CFT3 lines. CFT3 lines. (a) Leading diagrams with external CFT3 lines. (b) Leading diagrams without external CFT3 and may mistakenly conclude that we are in the limit of vanishing KM level, when in fact we are in the limit of in nite KM level! 4.2 Non-abelian CS and GR3 For the case of non-abelian CS and GR3, ! 1 correlators with CFT3 hadrons have only tree like CS branches dressing KK @AdS3 Witten diagrams, such as in gure 7a. This is very similar to the CS/soft dressing of Mink4 hard S-matrix elements [36]. While these diagrams are O( 0) for large , again there are O( ) correlators given by the pure CS tree diagrams, such as in gure 7b. And again, focusing on correlations with the CFT3 matter eliminates these, and gives a nite limit as The fact that the CS/GR3 branches attach externally to CFT3 subdiagrams (blobs), rather than connecting di erent CFT3 subdiagrams as in gure 5, means that the surviving diagrams are e ectively purely CFT3 correlators, with the branches just smearing the correlator point for CFT3 currents/stress-tensor where they attach. It is these smeared correlators that manifest the CFT2 and AS structure (in large-level limit). That is, in this limit the CS/GR3 are just probes of the dynamical CFT3, with no backreaction on it. We Smeared by ∫dξ cos((2l2+1)ξ) relators. Here the external lines correspond to superpositions of 3D o -shell \hadrons". discuss the structure and signi cance of the non-abelian branches as smearing functions in the next section, from the 4D viewpoint. correlator. External lines correspond to 3D onshell \hadrons" (KK modes) of 3D mass / `. HJEP01(28)4 5 Non-standard @AdS4=2 correlators as CFT2 correlators A standard @AdS4 correlator is a gauge invariant correlator of local composite operators made of CFT3 \quarks", but from the viewpoint of AdS3 \hadron" mass eigenstates, they are o -shell correlators. Instead we are considering @AdS3 correlators of the \hadron" mass eigenstates. In 4D \product-space" coordinates (eq. (3.3), see gure 3) the distinction is shown in gure 8. These illustrate two alternative means of probing the bulk physics. In standard @AdS4 correlators we are putting sources and detectors on the ceiling and oor of AdS4 (generic points on the @AdS4 in standard global coordinates) while having signals re ect o the walls with Dirichlet boundary conditions. In the KK-reduced @AdS3 correlators we have sources and detectors on the walls (only on @AdS3 \equator" of @AdS4) with signals re ecting o the ceiling and oor with Dirichlet boundary conditions. (In the case of AdS4/2 we simply put the oor at = 0, mid-level in the AdS4 \product-space", with Neumann boundary conditions as discussed earlier.) Either way, no probability or energy is lost through the regions without sources because of the re ecting boundaries. We stress again that the reason we must insist on the non-standard form of @AdS4 correlators is because when CS/GR3 \emissions" are added, it is these that become CS/GR3 gauge/di eomorphism invariant \on-shell" @AdS3 correlators. This is in contrast to the non-gauge/di eomorphism invariance of standard @AdS4 correlators, which are \o -shell" from the AdS3 viewpoint. The situation is entirely analogous to the gauge/di eomorphism invariance of the Minkowski on-shell S-matrix in contrast to the non-invariance of o -shell Minkowski correlators in quantum eld theory. For simplicity let us begin by considering U(1) CS coupled to a U(1) symmetry current of CFT3, in turn dual to an AdS4 U(1) gauge eld. We focus on a 2D chiral current correlator of CFT2 with other 2D operators in the large- limit. The 2D current of course contains the charges of a U(1) KM algebra by Laurent expansion. There are two equivalent ways of reading such CFT2 correlators in the large- limit: (i) at face value, as a 3D \hadronic" correlator involving CS \emission" (see gure 9), or (ii) as a purely CFT3 correlator involving a CFT3 conserved current at a point y in the AdS3 bulk (see gure 10), where this bulk point is \smeared" by a function of y given by the AdS3 CS bulk-boundary propagator: h0jT fj+(z0) : : :gj0iCFT2 = d3ypgAdS3 K+CS(z0; y)h0jT fJCFT3 (y) : : :gj0iCFT3 : By standard AdS4/CFT3 diagrammatics, this lifts to 4D: h0jT fj+(z0) : : :gj0iCFT2 = d3ypgAdS3 K+CS(z0; y) d4Xp GAdS4 KN (y; X)J N (X); Z Z Z (5.1) (5.2) (5.3) (5.4) (5.5) HJEP01(28)4 where K is an AdS4 bulk-boundary propagator corresponding to the 4D photon line in gure 10, while J (X) is the bulk 4D current to which it couples, set up by the 4D matter. We can write this compactly as where h0jT fj+(z0) : : :gj0iCFT2 = Z d4Xp GAdS4 AN (X)J N (X); Z AN (X) d3ypgAdS3 K+CS(z0; y)KN (y; X): By the de ning properties of K in -axial gauge, A (X) is that solution to the sourceless 4D Maxwell equations with boundary limit, A (y; ) ! ! =2 K+CS(z0; y): ξ l1 l3 4D photon X l2 l4 HJEP01(28)4 external 3D CS line (red) \smearing" a 4D photon @AdS4 correlator point. lifts to the simple 4D solution, That is, we deviate from the default Dirichlet boundary condition A corresponding to the unperturbed CFT3, because KCS acts as a perturbing source for the = 0 at = =2, CFT3 current. It is straightforward to identify this A. Since K+CS(z0; y) is a solution to the free CS equation of motion as a function of y, it must be purely a (large) 3D gauge transformation, (y), speci ed by its non-trivial boundary limit (at z0). This then clearly A (y) = K+CS(z0; y); A = 0: The 3D large gauge transformation of CS is thereby lifted to a large 4D gauge transformation, such pure gauge con gurations being at the root of traditional 4D AS analyses. Here, substituting eq. (5.6) into eq. (5.3) we see that h0jT fj+(z0) : : :gj0iCFT2 = d3ypgAdS3 K+CS(z0; y)Je (y); pgAdS3 Je (y) d p GAdS4 J (y; ): where bulk, Je (y). In this way, we see that we can compute hj+(z0)i via a CS gauge eld coupled either to the holographic CFT3 current hJCFT3 (y)i or the e ective \soft" current made from the 4D 5.2 Non-abelian gauge theory and gravity Note that in non-abelian gauge theory and gravity, there will be non-abelian CS or GR3 external branches in correlator diagrams, such as gure 11. Again, such correlators can Z Z (5.6) (5.7) (5.8) K(z’, yint) yint K(z’’, yint) Non abelian branch K(z, y): Bulk-Boundary propagator G(y, y’): Bulk-Bulk propagator G(y, yint) y l1 ! 1 with non-abelian CS branch, smearing CFT3 KK be viewed as purely CFT3 correlators, but with CFT3 currents/stress-tensor in the AdS3 bulk, at points y smeared by the non-abelian branch. These branches as functions of y are a non-abelian generalization of abelian CS bulk-boundary propagators, in that they just describe (large) gauge-transformations/di eomorphisms, because they add up to solutions to the sourceless CS/GR3 equations of motion (with non-trivial boundary limits). The non-abelian interactions in the branches are just a diagrammatic representation of nding such large gauge-transformations/di eomorphisms, which is a non-linear problem for non-abelian gauge/di eomorphism symmetry. As for the abelian case, these are straightforwardly lifted into 4D large gauge-transformations/di eomorphisms (as was done in Mink4 [36]). Thus, once again we see that large gauge-transformations/di eomorphisms are central to isolating the AS, by suitably smearing @AdS4=CFT3 correlators into the canonical form of CFT2 correlators. 5.3 Compatibility with 4D quantum loops and masses Note that while there are only tree-like CS and GR3 branches dressing CFT3/AdS4 diagrams surviving in the large limit, the CFT3 hadron (AdS4) diagrams can be at full loop level, controlled by a separate parameter such as 1=NCFT3 . In this sense, the AS we derive are an all-loop feature, in fact a non-perturbative feature, of AdS4 QG. Furthermore, while it is technically easier to explicitly consider massless 4D elds, there is absolutely no obstruction to massive 4D elds, dual to high-dimension CFT3 operators. 6 Evading the no-go for in nite-dimensional AS in AdS4=2 We have derived CFT2 correlators for 2D currents/stress-tensor in the large level limit from purely CFT3 (AdS4/2) correlators of \hadronic" (KK) modes and CFT3 currents/stresstensor, smeared by large gauge-transformations/di eomorphisms. This gives rise to in nite dimensional AS of Vir+ Vir and KM type. The Virasoro symmetries are analogous to the super-rotations of Mink4. Here, we show from the 4D viewpoint how we have evaded the no-go argument sketched in the introduction for such in nite-dimensional symmetries of AdS4, which would equally apply to AdS4=2. To understand this, note that in \product-space" coordinates (eq. (3.3)), there are two = =2 and the round wall at = 0 (refer to gure 3). These two boundary regions have di erent conformal structure, < 8>ds2AdS3 ; ! ! 0: =2 (6.1) Vir+ S1 In standard global coordinates standard @AdS4 correlators only have sources on the ceiling/ oor, and in this boundary region the geometry is fully three-dimensional, with only nite-dimensional conformal isometries as candidate AS. This is the no-go argument in \product-space" coordinates. However, we see that when we put sources only on the wall boundary region, as we have been led to do by the sca olding of GR3 and CS on AdS3, the bulk geometry degenerates as we approach this boundary region to the 2D geometry R, which has in nite-dimensional conformal isometries, corresponding to Thus far SO(2; 2) has played the analogous role of Lorentz transformations SO(3; 1) in Mink4, being extended to Vir+ Vir AS of AdS4 analogously to the super-rotations Vir Vir of Mink4. The analog of Mink4 translation generators are the extra four generators of the AdS4 isometries SO(3; 2) which lie outside SO(2; 2), just as Mink4 translations are the Poincare generators outside SO(3; 1). We would like to identify these extra generators and the full set of AS of AdS4 that follows from them, in analogy to XBMS4 incorporating translations to go beyond just the super-rotations in Mink4. The problem is that the strategy we used to identify CFT2 structure forced us to consider AdS4/2 (rather than AdS4) and asymptotically AdS3 GR3, both of which respect only the SO(2; 2) subgroup of the global SO(3; 2). How can we recover some analog of \super-translations", and more generally the complete AS analog of XBMS4? 7 Maximal spacetime AS from 3D conformal gravity The in nite-dimensional extension of SO(2; 2) isometry arose in our approach by gauging the CFT3 by SO(2; 2) CS = GR3 on AdS3. This suggests that we may get the larger in nite-dimensional extension of SO(3; 2) by gauging the CFT3 by SO(3; 2) CS instead. Remarkably, this is simply equivalent to 3D conformal gravity (CGR3) [68]. 7.1 A \super-translation"-like KM AS for AdS4=2 CGR3 is compatible with asymptotically AdS3 spacetime, even though AdS3 does not have full SO(3; 2) conformal isometry. CGR3 is not only di eomorphism invariant, but also Weyl invariant. The Weyl invariance shares much in common with an internal U(1) gauge invariance (not coincidentally given Weyl's original gauging of scale symmetry in the history of gauge theory and its similarity to QED's gauging of rephasing invariance). Therefore it is not surprising that the AS of CGR3 (+ CFT3) matter on AdS3 are of the form Vir along with an abelian KM, the latter associated with Weyl symmetry [69]: L+m; Ln+ Lm; Ln L+m; Jn+ Jm+; Jn+ = (m = (m = nJm++n = 2 grav m m+n;0 ; n)L+m+n grav n)Lm+n + grav m3 m 1 12 m3 m m+n;0 m+n;0 where grav is the level of the CGR3 theory in CS form, in the same manner as for GR3. Note, it is critical that the \quark" sector is compatible with being gauged by CGR3, precisely because it is 3D conformally invariant, so that it can be made Weyl-invariant once coupled to gravity. We can interpret the KM resulting from the Weyl invariance as an AdS4/2 analog of the Mink4 super-translation KM. Non-unitary nature of CGR3 For large we see that the two Virasoro sub-algebras in eq. (7.1) have opposite sign central charges [69], c c+, incompatible with unitarity [75]! This may be surprising because it only pertains to the Vir+ Vir subalgebra and might be thought to be the same as in GR3. But crucially, GR3 is not a truncation of CGR3. They employ di erent quadratic invariants of the generators to de ne the trace in their CS formulations. Note that for SO(2; 2) there are two distinct quadratic invariants, IJKLJIJ JKL and JIJ J IJ ; I; J; K; L = 0; : : : ; 3: The standard GR3 formulation uses the rst of these and it corresponds to c+ = c , so they may both be positive. But the second alternative instead has c+ = c , at odds with that positivity. For the SO(3; 2) CS formulation of CGR3, there is clearly only a single option, JMN J MN ; M; N = 0; : : : ; 4; and the truncation to SO(2; 2) is then the non-positive choice for central charge. Nevertheless since we take ; c ! 1 in our analysis of CFT3/AdS4 AS, this does not obstruct the unitarity of the target theory. It does however seem strangely at odds with our development so far, which has made physical sense for nite ; c. Possibly, we must restrict to a single CS sector (4D helicity), say \+", with c+ > 0 [75]. Although CGR3 has led us to identify a KM \super-translation"-like extension of AdS4/2 AS, this extended algebra still does not contain all of global SO(3; 2), presumably because we are still explicitly breaking AdS4 isometries by working with AdS4/2. We rectify this by rst switching to the Poincare patch of AdS4 in the next section, and then later to all of global AdS4. 8 AdS4Poincare: AS from holography and holography from AS We have accumulated a number of questions. Is there an AS algebra of AdS4 that contains the isometry SO(3; 2) as a subgroup? While we are taking the large limit, what does the (7.1) (7.2) (7.3) nite- set-up look like in the 4D dual prior to the limit? So far the CS and (C)GR3 are added \by hand", even if then removed by ! 1. Is there a sense in which such CS elds emerge as soft limits of the AdS4 (hence CFT3) elds themselves, as was the case in Mink4? within the Poincare patch of AdS4, AdS4Poincare. The AdS4Poincare metric is given by If so, do we get a nite emergent level, < 1? These questions are most simply addressed ds2 = dx dx w2 dw2 ; 0 < w < 1 ; (8.1) which manifests a Mink3 foliation, where is the Mink3 metric. Although only a portion of AdS4global, it has the full AdS4 isometry algebra of SO(3; 2), unlike AdS4/2. We also know its holographic dual, namely CFT3 on Mink3, where SO(3; 2) are the conformal isometries. Now we can couple this CFT3 to GR3. GR3 on Mink3 again has a CS formulation with gauge group ISO(2; 1), the 3D Poincare group. For nite grav the 4D dual of GR3 + CFT3 (+ UV completion) is well known, namely it is the (UV completion of the) RandallSundrum 2 (RS2) model [70], but in one dimension lower than the originally formulated [71]. That is, the AdS4Poincare boundary is cut o by a \Planck brane" whose 3D geometry is dynamical, dual to GR3, and coupled to the 4D dynamical bulk geometry (dual to CFT3).1 Rather than dwelling on nite grav, we proceed with the strategy for the 4D theory to inherit the 3D AS of GR3 in the large grav limit. This AS of Mink3 is XBMS3 [58]. Here we review its derivation by a \contraction" of the Vir+ Vir AS of AdS3, essentially getting at 3D by taking the RAdS3 ! 1 limit [59, 76{81]. It is clear in what sense the \vacuum" geometry of AdS3 approaches Mink3 in the limit of large RAdS3 , but we must study the GR3 dynamics as well in this limit in order to understand the relationship of the two AS algebras. GR3 on asymptotically AdS3 can formulated in terms of SO(2; 2) SO(2; 1)+ from the dreibein e and spin connection ! as [42] SO(2; 1) Chern-Simons gauge elds made A a ea =RAdS3 : If we plug this into the AdS3 gravity action in CS form SCS(A+) SCS(A ), and keep the leading terms for large RAdS3 , we nd straightforwardly that it is the CS form of the gravity action in Mink3 (with gauge group ISO(2; 1)) written in terms of e and !. Staying in AdS3, the asymptotic expansion of A in terms of L (reviewed in section 2) translates into an expansion for e given by RAdS3 Pn(Ln+ Pn(Ln+ + L n)ein . That is, the AS charges for e and ! respectively are L n)ein , and for ! given by RAdS3 ln = RAdS3 (Ln+ RAdS3 Tn = (Ln+ + L n); L n ) 1The analogous dual in the case of M3 = AdS3 is less familiar, a 3D Planck brane in AdS4global/2. It is important to distinguish this from the Karch-Randall model [73], in this dimensionality a 3D Planck brane in all of AdS4global. (8.2) (8.3) where the overall normalization of RAdS3 on the left-hand side does not a ect relative sizes of terms in the charge algebra, but does give a nite limit as RAdS3 ! 1. Indeed, algebra in these variables and taking RAdS3 ! 1 yields the As in AdS3, this XBMS3 AS is symptomatic of the topological character of GR3, the non-trivial topology arising from the \holes" drilled out by the matter world lines, where GR3 reacts by introducing conical-type singularities. We will think of XBMS3 as an AdS4Poincare analog of \super-rotations" in Mink4 since they are the contraction of Vir+ Vir AS. The global subalgebra of XBMS3 is the Poincare isometry ISO(2; 1). But now AdS4Poincare (Mink3) has the larger (conformal) isometry algebra of SO(3; 2), containing ISO(2; 1) as a subalgebra. Therefore the extra generators of SO(3; 2) can be (repeatedly) commuted with XBMS3 to generate the full AS of AdS4Poincare, with global subgroup SO(3; 2)! This strategy was analogously followed in Mink4 as one of the ways to (re-)derive super-translations by commuting ordinary translations with superrotations [36]. Above we outlined a strategy for nding the full AS of AdS4Poincare by starting with its subalgebra, XBMS3, arising from gauging with GR3. It would be more elegant and insightful if the entire AS emerged by the same procedure. This can now be done by replacing GR3 by CGR3 on Mink3, coupled to CFT3. Since Mink3 has SO(3; 2) conformal isometries, and CGR3 is SO(3; 2) CS, and our \quark" matter is also conformally invariant CFT3, SO(3; 2) is respected by each component, and therefore the in nite dimensional symmetries that arise from the CS structure must contain all of SO(3; 2) as a global subalgebra. We will pursue the explicit form of this AS algebra elsewhere, just observing here that it is implicitly completely characterized by CGR3 on Mink3. 8.3 Holographic grammar from AS While we have used the holographic grammar of AdS4/CFT3 in this paper to clarify the nature and utility of AS, we can run our arguments in a di erent order. Suppose that one did not know the holographic dual of AdS4 QG, but was given the full AS structure of AdS4 and learned to characterize it in terms of CGR3 elds to capture the associated large gauge transformations. Then by the fact that matter compatible with coupling to 3D gravity must be a 3D local quantum eld theory in order to have the requisite local stress tensor to source gravity, we can deduce that the holographic dual of AdS4 must be such a 3D QFT. The fact that the 3D gravity is speci cally conformal gravity implies that the dual 3D QFT must also be conformally invariant, that is CFT3! It is just such a set of steps that awaits to be performed in the case of nding a holographic grammar behind Mink4 QG. Emergent CS and \shadow" e ects from boundary/soft limits In Mink4 gauge theory, it was shown that AS and memory e ects arise from considering same-helicity gauge boson emissions in the soft limit [6, 7]. Ref. [36] showed that these features were captured by an emergent 3D CS description of the soft elds, \living" at @Mink4, as well as on Rindler/Milne horizons. Here, we will demonstrate that analogous phenomena emerge within AdS4Poincare U(1) gauge theory. (If AdS4Poincare GR4 is added, we can think of these phenomena as emerging from within the dual CFT3 with U(1) global symmetry, even though the gravity will play no explicit role in our analysis.) a discrete spectrum, AdS4Poincare has a continuous spectrum and a natural generalization of \soft" limit. We nd emergent CS gauge elds localized on @AdS4Poincare as well as on the Poincare horizon, connected by this soft limit. These CS elds connect to analogs of electromagnetic memory e ects in Mink4 [21{23, 36], which we will refer to as \shadow" e ects, since they relate to the holographically emergent spatial direction rather than time. We will also see a sense in which a nite CS level emerges. While our approach here parallels similar steps in Mink4 [36], the emergent CS structure in AdS4 is closely related to \mirror" symmetry in the dual CFT3 [82{84]. This aspect will be explored elsewhere [85]. While AdS4global has We consider an AdS4Poincare U(1) Maxwell gauge eld AN , coupled to a bulk 4D conserved source current JN , which is taken to implicitly describe interacting charged matter. The 4D gauge coupling is g. Because of the Weyl invariance of the 4D Maxwell action, AdS4Poincare (eq. (8.1)) is e ectively just Mink4/2, ds2 Weyl dx dx dw2 ; w > 0; ; = f0; 1; 2g: The natural notion of \soft" in AdS4Poincare/CFT3 is m23 ! 0, where m23 is 3D invariant mass-squared in the x directions. This is obviously analogous to the observation of ref. [36] that Mink4 soft limits correspond to m3 ! 0 in the (EA)dS3 foliation of Mink4. Maxwell radiation can be decomposed into positive and negative helicity components, A . More generally, away from charged matter (away from the support of J ), we will decompose the electromagnetic eld strength FMN into self-dual and anti-self-dual components, 1 2 FMN (x; w = 0) FMN We will focus on the soft limit of A+. Let us rst imagine that we are in full Mink4 instead of Mink4=2. In momentum space, (q ; qw), m23 = q q , so that for 4D on-shell radiation, m23 = qw2. More precisely, the leading soft limit would be given by qlwim!0 qw A+(qw) = 1 { 22 { (9.1) (9.2) (9.3) where the 3D argument is implicit and can be either q or x . Within Mink4=2, the analogous soft limit is truncated to2 0 A+(w = 0): We will take this as our \soft limit". and A+(x; w = 0) obeys an interesting CS-type equation. In what follows, we will see that each of the 3D elds in this soft limit, A+(x; w = 1) CS on @AdS4Poincare and a \holographic shadow" e ect Consider that the source current J emits radiation towards @AdS4Poincare. The positive HJEP01(28)4 F + (x; w = 0) = 2 Fe (x; w = 0) i 4 F w (x; w = 0); because the standard AdS4 Dirichlet boundary condition, A (x; w = 0) = 0, implies (x; w = 0) = 0. In terms of the standard AdS4/CFT3 dictionary for the holographic (9.4) (9.5) (9.6) (9.7) (9.8) (9.9) JCFT3 (x) = w (x; w = 0); F + (x; w = 0) = We can view this as the equation of motion for an emergent CS gauge eld coupled to where the CS gauge eld is identi ed with the helicity-cut boundary limit of the 4D gauge (This does not vanish since only A = A + + A obeys the AdS Dirichlet boundary condition.) It was just such a CS eld coupled to CFT3 (but on AdS3 instead of Mink3) which was invoked in earlier sections to derive AS for AdS4. It is useful to cast the CS equation in integrated form, using Stokes' Theorem, F symmetry current, we obtain CFT3 charged matter, eld in w-axial gauge, I dx ACS(x) = g d 2 Z JCF T3 : (9.10) Here is a nite two-dimensional surface in the @AdS4Poincare x-spacetime, with boundary , the right-hand side is the total CFT3 \quark" charge lying inside , a holographic \shadow" of the 4D bulk state. 2We can think of Mink4=2 as the quotient space of Mink4 under the identi cation w $ w. If we imagine a \polarizer" projecting onto positive helicity in the physical region, w > 0, and its \mirror image" projecting onto negative helicity for w < 0, then the de nition of leading soft limit in the Mink4 covering space reduces to the truncated expression in Mink4=2. 9.3 As explained earlier, in CS theory, sensitivity to the CS level (in correlators with external matter lines) arises from diagrams with internal CS lines. In the present context, we have considered radiation emitted by a source J . The CS gauge eld is the boundary limit of the positive helicity component of this 4D radiation, A+(w = 0). To measure the associated CS level we imagine \detecting" this eld with a probe charge localized near or at the boundary, w = 0. The subtlety is that physical charges couple to both positive and negative helicity components. We straightforwardly see that the boundary limit of the negative helicity That is, while the probe charge couples to the sum of the helicity components in the form, gA = gA+ + gA , the two helicities couple with opposite strength to the holographic current, in the form gJCFT3 . Therefore the CS exchanges mediated by A + and A have strengths g2, yielding a net cancelation. However, we can formally focus on just the A+(w = 0) CS exchange with strength +g2, corresponding to CS level, 4 g 1 g2 : (9.11) (9.12) (9.13) A similar result was anticipated in ref. [36] for Mink4. The soft limit, CS on the Poincare horizon, and a bulk \shadow" e ect The -component of the 4D Maxwell equations reads = gJ : to the three-volume, R01 dw R d2 . . . , to get We again look at an integrated form of these equations, on a two-dimensional surface in @AdS4Poincare x-spacetime, and in our \soft limit" in w. That is, we integrate with respect Z d 2 hF w (w = 1) F w (w = 0) i + 0 I F = g 0 dw Z d 2 J ; (9.14) where we have used Stokes' Theorem to get the second line of the left-hand side. For the simple case of purely spatial this is nothing but Gauss' Law, the right-hand side being just the total bulk charge lying inside the three-volume, while the left-hand side is the total electric ux through its boundary. Taking the source current J to be localized at nite w at nite times, and to only span nite times, we can drop the eld strength at w = 1 on the rst line, by causality. The eld strength at w = 0 on the rst line is just the holographic current again, so we have I F = g Z d 2 0 JCFT3 + dwJ (w) : (9.15) This is closely analogous to the electromagnetic memory e ect in Mink4 for purely spatial , with w now playing the role of time there. The total charge passing through regardless of when in the memory e ect is replaced here by the total charge in regardless of where in w. We will refer to this as a \bulk shadow" e ect. As was done for the memory e ect in ref. [36], we can write the bulk shadow e ect in CS form. First note that the left-hand side can be re-expressed in terms of the dual eld strength Fe to give 2 I Z 1 0 dwFew = g Z JCFT3 + Je ; where we have de ned a second 3D \shadow" current by taking the soft limit of the bulk 4D current, 0 Z 1 0 dwF w dwJ (x; w): I dx A (w = 1) 2i dx A (w = 0); We add zero to the bulk shadow e ect in the form, 0 = 2i = 2i I I I dx A+(w = 1) dx A+(w = 0) = ig Z 4 d 2 h JCFT3 + Je : i (9.16) (9.17) (9.18) (9.20) (9.21) where the term at w = 1 is by Stokes' Theorem = i R d2 F (w = 1), which vanishes by causality, and the term at w = 0 vanishes by the standard AdS4 Dirichlet boundary conditions on A. Therefore we can write the bulk shadow e ect in the form to avoid the support of J for all w, so that the self-dual component of the eld strength on the left-hand side can be expressed in terms of the gauge potential A+. By Stokes' Theorem, 2i I Z 1 0 dw F w + iFew = g Z d 2 JCFT3 + Je : (9.19) We see that the term on the left at w = 0 and the JCFT3 term on the right are equal by the last subsection, so we isolate a new CS-type relation on the Poincare horizon, dx A+(w = 1) = ig Z 4 d 2 Je : This is the ( -integrated) CS form of the bulk shadow e ect, where the role of CS current is played by the shadow current, Je . In subsection 5.1, with CFT3 on AdS3, we saw that AS (CFT2 chiral current j+) could be derived by CS coupled to either JCFT3 or Je . But this equivalence required going to the boundary of AdS3. For in the \bulk" of Mink3, the two CS relations at w = 0 and w = 1, with CS currents JCFT3 and Je respectively, are distinct. In the same sense as for the CS gauge eld localized at the boundary, the CS gauge eld on the Poincare horizon also has level e 1=g2. Spatial hypersurfaces spanning entire Mink4 surfaces) are just related by di eomorphisms within a single \timeless" Wheeler-DeWitt wavefunctional. 10 AS of Wheeler-DeWitt wavefunctionals on @AdSg4lobal The choices of M = AdS3 and Mink3 have given an approach to AS on portions of AdS4global, but here we return to the full AdS4global. It is natural then to consider CS coupled to CFT3 on the global boundary S2 R. However, space is then closed and there is no obvious asymptotic region to get AS or 2D chiral currents. Yet, it is well known from the CS viewpoint that there are e ectively in nite-dimensional symmetries still at play, and these are revealed by cutting at a time slice to reveal a state [62]. Technically, this is clear if we consider the wavefunctional, say at time = 0, to be determined by a 3D functional integral over all earlier times < 0 and all of space, that is e ectively M3 R is the negative- half-line. Once again, this spacetime has a boundary, the S2 space at = 0, on which AS appear in standard CS fashion. They act on the states of the theory. S2 R , where Let us return to the no-go argument for in nite-dimensional AS of AdS4 and the loop-hole pointed out in the introduction. CFT3 states are dual to AdS4 di eomorphisminvariant Wheeler-DeWitt wavefunctionals. In particular they describe the state at = 0 on the boundary, but on any interpolating spacelike hypersurface in the bulk. The collection of such hypersurfaces gives the 4D subregion of AdS4 described by the quantum state, as depicted in gure 1. Its boundary geometry is e ectively two-dimensional, compatible with in nite dimensional AS. Such a restriction to a subregion does not occur for Mink4. The quantum state at Minkowski time = 0 on the boundary describes the 4D region foliated by all interpolating spacelike hypersurfaces, as for AdS4, but unlike AdS4 this foliation covers all of Mink4. See gure 12. 10.1 Consider a U(1) CS eld for simplicity. The CS eld sees the U(1) charged CFT3 state at = 0 via an Aharonov-Bohm(AB) phase in Wilson loops. One can de ne associated d` A; measuring the total \quark" charge inside subregion of the spatial S2 at = 0, by the integrated form of equations of motion for CS coupled to CFT3. These contour-associated AS charges are related to the standard KM charges as follows. We use complex coordinates z; z on S2 via sterographic projection. Out of the two boundary components of the CS gauge eld, Az; Az, one component is removed by a CS boundary condition, say Az = 0, while the other component is holomorphically conserved, @zAz = 0 (refer to section 2). This holomorphic Az(z) is then completely determined by the poles at the location of charged CFT3 \quarks", so that (10.1) (10.2) (10.3) as a complex contour integral. Laurent expanding about z = 0 say, dzAz(z) Az(z) = X n zQn+n1 ; then determines the KM charges. Note that even as the CS is decoupled at Q ; Qn remain as non-gauged charges registering the location of charged \quarks", and = 1, the therefore the holographic boundary \shadows" of 4D particles. While this form of \hair" for 4D charges is amusing, the key question is whether it is useful, say in the sense that it makes time-evolution algebraic in terms of the charges, as opposed to having to solve complicated dynamics. We have already seen how such simple To see the analogous form of time-evolution of charges in the S2 time-evolution of charges arises for M = AdS3 in the context of AdS4=2 (see eq. (2.12)). R setting we need the full power of CGR3. Time-evolution from AS algebra via CGR3 on @AdS4global But to capture all of AdS4global we want CFT3 on all of S2 Given the CS form of 3D gravity, one might think to just repeat the above steps performed for internal CS gauge symmetries. But now S2 R geometry must be a solution to dy namical gravity. And yet, for standard GR3 (with or without a cosmological constant) it is not a solution to 3D Einstein equations. The closest is GR3 with positive cosmological constant, which has dS3 solution. This is Weyl equivalent to S2 timelike-interval. The Weyl equivalence is acceptable because @AdS4 is only de ned within such Weyl rescaling. R, not just a time interval. Fortunately if we switch to CGR3, then by its Weyl invariance, Weyl rescalings of GR3 solutions are also solutions of CGR3 [68]. In particular S2 solution. By locality of CGR3 equations of motions, this means S2 time-interval must be a R is also a solution. We can now couple CGR3 to CFT3 on S2 R. Given the CS form of CGR3, we expect states at xed time = 0 to transform under AS charges arising on the S2 boundary at = 0 from CGR3 = SO(3; 2) CS structure, and to persist in the ! 1 limit. This gives AS charges acting on CFT3 states. The full AS will contain the spacetime AS associated to CGR3 as well as any related to internal (CS) symmetries. The former has SO(3; 2) global subalgebra. The SO(2) subgroup of SO(3,2) is just time translation in , that is, the global AdS4/CFT3 Hamiltonian H. In particular all AS charges will have commutation relations with H, determining their -dependence by the AS charge algebra. 11 Mink4 and future directions In this paper, we generalized the notion of asymptotic symmetries (AS) applied to AdS4, so that in nite-dimensional symmetries arise, analogous to the AS of Mink4. We found a tight connection between these AS and the 3D holographic dual, in this case CFT3, coupled to 3D gravity and Chern-Simons topological sectors. In turn, the combined 3D theory is dual to a CFT2 structure in the sense of the AdS3/CFT2 correspondence, whose chiral currents and stress tensor house the AS charges. Several issues remain in order to ll out this story. Also, having seen these interconnections in AdS4 quantum gravity and gauge theory, it is worth seeing if a parallel understanding can be gained for other 4D spacetimes where holography is less well understood, including Mink4. It remains an important task to explore how 3D gravity emerges from AdS4Poincare gravity as a (helicity-cut) soft or boundary limit in analogy to our discussion of U(1) CS emerging from AdS4Poincare U(1) gauge theory. It will be interesting to see what type of 3D gravity emerges, GR3 or CGR3, or whether this depends on leading or subleading soft limits in some way. It will again be interesting to see if, and under what conditions, a nite e ective level or central charge emerges. These gauge and gravitational exercises should be repeated for AdS4global. Here we do not have the notion of soft limit since the spectrum is discrete, but the helicity-cut boundary limit continues to make sense. The connection between the emergent CS gauge elds and AS with 3D \mirror" symmetry recast in dual 4D form [82{ 84] will be explored later [85]. We have explicitly given some in nite-dimensional subalgebras of the AS algebra acting on AdS4 states, while we have argued that the full AS algebra is implicitly captured by CGR3 on @AdS4. It remains to explicitly describe this algebra of AS charges acting on states of CFT3 living on the S2 boundary space. By comparison with Mink4, where IR divergences at loop level a ect and complicate the soft limit [86{90], it is possible that AdS4 curvature IR-regulates and simpli es the considerations. This remains to be explored. It appears feasible to do a similar analysis in dS4 as done here for AdS4, and thereby discover AS in that case. It would be interesting to compare this approach to that of ref. [91]. The approach suggested here would be compatible with the Poincare patch of dS. 11.2 AS as \hair" We have argued that in nite-dimensional AdS4 AS are a useful form of \hair" for 4D black holes and other complex states, very much in the manner that the in nite-dimensional symmetry charges of CFT2 characterize 2D states. Given how explicit this is in AdS4, it would be interesting to explore whether the AS fully characterize any AdS4 quantum state. Even a partial but still rich characterization may be relevant to the information puzzles of quantum black holes. The fact that AdS4 gives a new 4D example of how soft elds take a 3D CS topological form, suggests that this phenomenon is more general, and should be understood on less symmetric (black hole) 4D spacetimes. Vir super-rotations from the subleading soft limit of GR4 were shown to be captured by SO(3,1) CS = GR3 on (EA)dS3 [36]. But this CS description does not capture super-translations and the leading soft limit of GR4, or even just Minkowski translations. It remains to nd the 3D representation of all the soft limiting elds underlying the full XBMS4 AS. Doing so would be the analog of nding CGR3 = SO(3,2) CS for AdS4. The most obvious guess would be to try CS gauging of the 4D Poincare group, ISO(3; 1). But there is a simple no-go argument for this approach, in that there is no quadratic invariant to de ne the CS trace. The analogous GR3 on Mink3 is given by ISO(2; 1) CS, where the quadratic invariant is given by J P [42], obviously lacking 4D generalization. Nevertheless it is possible that a non-CS 3D characterization of Mink4 soft elds exists, reducing to SO(3; 1) CS for the subleading soft limit and super-rotations. The fact that the 2D conserved current housing super-translation KM charges in Mink4 was found to be a ECFT2 descendent operator of a partially conserved operator [36] suggests a role for partially massless gauge elds [92, 93] in 3D, in turn coupled to partially conserved currents of a 3D holographic dual of Mink4 QG. One strategy to nd this 3D characterization begins with the recently considered case of CGR4 on Mink4 [94]. Here we may guess that the soft elds are characterized by SO(4,2) CS on (EA)dS3, which does have the requisite quadratic invariant J J , ; 0; 1; 2; 3; 4; 5. This suggests that the 3D SO(4; 2) CS theory might be truncated (\Higgsed") to the 3D characterization of just 4D Poincare symmetric soft elds in terms of massless and partially massless 3D elds. Another strategy is to see if there is a \contraction" procedure for the SO(3; 2) CS description of AdS4 AS found here that yields the 3D description of Mink4 AS, in rough analogy to the contraction of SO(2; 2) CS governing AS of AdS3 to the ISO(2; 1) CS governing AS of Mink3. A full 3D characterization of the soft Mink4 elds would strongly constrain the form of a 3D holographic dual of 4D Mink QG, since the latter would have to be able to be coupled to the soft elds. This is in analogy to the neat compatibility of CFT3 with coupling to CGR3 in the AdS4 context. One can view such a connection in Mink4 as a modern extension of Weinberg's classic derivation of consistency conditions on the S-matrix involving massless spin-1 and spin-2 particles. He showed [95{97] by studying soft limits that matter necessarily has to couple to soft spin-1 through conserved charges and to soft spin-2 through gravitational-form charges satisfying the Equivalence Principle. But the full soft eld structure may in fact be strong enough to prescribe the full holographic grammar of the dynamics. Such a grammar would e ectively have to force the precise vanishing of the 4D cosmological constant, perhaps in a novel way. HJEP01(28)4 Acknowledgments RS would like to thank Cli ord Cheung and Anton de la Fuente for earlier collaboration, insights and discussions related to this paper. In addition, the authors are grateful to Hamid Afshar, Nima Arkani-Hamed, Christopher Brust, Jared Kaplan, Juan Maldacena, Arif Mohd, Massimo Porrati and John Terning for helpful discussions and correspondence. This research was supported in part by the NSF under Grant No. PHY-1620074 and by the Maryland Center for Fundamental Physics (MCFP). Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] A. 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Rashmish K. Mishra, Raman Sundrum. Asymptotic symmetries, holography and topological hair, Journal of High Energy Physics, 2018, 14, DOI: 10.1007/JHEP01(2018)014