Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations
BBending the Bruhat-Tits Tree ITensor Network and Emergent Einstein Equations
Lin Chen, Xirong Liu ∗ , and Ling-Yan Hung State Key Laboratory of Surface Physics,Fudan University,200433 Shanghai, China Shanghai Qi Zhi Institute,41st Floor, AI Tower, No. 701 Yunjin Road,Xuhui District, Shanghai, 200232, China Department of Physics and Center for Field Theory and Particle Physics,Fudan University,200433 Shanghai, China Institute for Nanoelectronic devices and Quantum computing,Fudan University,200433 Shanghai , China
Abstract
As an extended companion paper to [1], we elaborate in detail how the tensor network con-struction of a p-adic CFT encodes geometric information of a dual geometry even as we deformthe CFT away from the fixed point by finding a way to assign distances to the tensor network.In fact we demonstrate that a unique (up to normalizations) emergent graph Einstein equationis satisfied by the geometric data encoded in the tensor network, and the graph Einstein tensorautomatically recovers the known proposal in the mathematics literature, at least perturbativelyorder by order in the deformation away from the pure Bruhat-Tits Tree geometry dual to pureCFTs. Once the dust settles, it becomes apparent that the assigned distance indeed correspondsto some Fisher metric between quantum states encoding expectation values of bulk fields in onehigher dimension . This is perhaps a first quantitative demonstration that a concrete Einsteinequation can be extracted directly from the tensor network, albeit in the simplified setting ofthe p-adic AdS/CFT. ∗ Chen and Liu are co- first authors of the manuscript. a r X i v : . [ h e p - t h ] F e b ontents λ . . . . . . . . . . . . . . . . . . . . . . . 154.4 Solving the Einstein constraint to order λ . . . . . . . . . . . . . . . . . . . . . . . . 194.4.1 The interaction term in the semi-classical limit . . . . . . . . . . . . . . . . . 234.4.2 Fisher metric and the edge lengths . . . . . . . . . . . . . . . . . . . . . . . . 23 The Ryu-Takanayagi formula provides very deep insight into the underlying physics of theAdS/CFT correspondence [2]. It suggests that the gravity dual is encoding the entanglementstructure of the CFT in a geometrical way [3]. This is strongly reminiscent of tensor networks,which captures the entanglement of a given many body state it approximates through a prudentchoice of graph the network covers. This led to suggestions that the tensor network is perhaps themicroscopic explanation of the AdS/CFT correspondence [4].2any models have been proposed since, capturing different aspects of the AdS/CFT, in additionto the RT formula that first inspired the analogy. The perfect tensor and the random tensornetworks [5, 6] capture the error correcting properties [7], and explicitly realized the notion ofentanglement wedge reconstruction. More recently there are different algorithms developed togrow a bulk tensor network. See for example [8–11].The tensor network also makes ready connection with notions of complexities [12–19], allowingvisualisation of complexity growth of the wavefunction simply through the counting of tensors,giving extra support to the idea of volume/action of the AdS bulk being a measure of complexity.The tensor network also gives an intuitive picture to the island formula [20], the latest break-through that potentially explains the black hole information paradox [21, 22].The list of successes give strong support that the tensor network picture is capturing someimportant essence of the AdS/CFT correspondence. One very important challenge is therefore toturn these beautiful qualitative pictures into quantitative ones. That has proved a very formidabletask. There are two parts in this challenge. On the one hand, we need to be able to accuratelyread off the CFT and its observables from the tensor network. On the other, we have to identifygravitational observables in the tensor network. The former is hard if we build the tensor networkusing tensors engineered to recover nice bulk properties, such as the perfect/random tensor net-works. In those constructions, it is unclear what the boundary CFT is. The latter task is alwayshard, whether we start with a tensor network with known or unknown CFT connections. Oneneeds to identify matter excitations in the bulk, and also the metric of the background space-time.Identification of matter content can be done in some toy models [23, 24] so that Witten diagramsemerge to some extent. But these results are far from satisfactory in most cases where the tensornetwork as a discretization breaks most of the symmetries of AdS space. The hint for geometricaldata lies mostly in the Ryu-Takayanagi formula that connects entanglement with areas of mini-mal surfaces. But practically, to extract the metric from it remains cumbersome, particularly ifthe theory deviates from the fixed point, and that the corresponding tensor network describes abackground that deviates from the pure AdS space. Some attempts based on the Fisher metricbetween density matrices and correlation functions of the CFT [17, 25–27] have been made, andthey resemble known gravitational solutions to different extent.To make progress, one needs to put these proposals to further tests by showing that the dynamicsof gravity should emerge from the tensor network description of the CFT as well. All the difficultiesdescribed above mingle in this task. There are some progresses based on extremalising relativeentropies [28–32] and complexities, but the results either require various assumptions (such as theapplicability of the RT formula [31], or certain behaviour of the modular hamiltonian of the matter3elds [32]), or that they are not explicitly covariant, which is a common issue that plague tensornetwork reconstruction of CFT states.There is an ideal testing ground where quantitative computations can be more readily made. Ifmaking overly simplistic choice of tensors has little hope of recovering well-behaved CFT’s and AdSbulks, then perhaps the second best option is to work with a simplified version of the AdS/CFTcorrespondence. The p-adic AdS/CFT [33, 34] is such a perfect arena where the correspondencepreserves much of the essence of the AdS/CFT. The boundary CFT is sufficiently simple that canbe precisely reconstructed by a tensor network covering the dual bulk – which is naturally discrete in this case [23, 24]. We note that this construction recovers the CFT partition function rather thana state at a given time slice, and so it naturally avoids issues of covariance mentioned above. Manyof the well known items in the AdS/CFT dictionary, such as the correspondence between primaryoperators and bulk fields, their bulk-boundary propagators, the HKLL formula [35, 36] and Wittendiagrams quantitatively emerge from the tensor network representation. It remains to show thatthis tensor network can describe space-times deviated from the pure
Bruhat-Tits tree geometry, thep-adic analogue of pure AdS space, and that the interplay between matter and geometry followsrules that can be interpreted as the Einstein equation on a discrete graph.In this paper, which is an extended companion to [1], we will take the tensor network construc-tion of the p-adic CFT partition function in [24], and achieve precisely this goal, perturbativelyaway from the pure Bruhat Tits geometry. Our procedure comes in 6 steps.1. Define the notion of bulk operator insertion in the tensor network so that it is consistent withthe AdS/CFT dictionary, including the correct bulk-bulk propagator and the HKLL formula.The bulk and boundary correlation functions where the boundary conditions correspond tothe CFT fixed points are consistent with a quantum field theory living on the Bruhat-Titstree. We can read off the bulk action as well. This is reviewed and expanded in section 2.2. Obtain a deformed geometry by picking appropriate boundary conditions of the tensor net-work. This has been defined and partially studied in our previous sequel [24]. These boundaryconditions are the direct analogue of choosing non-trivial boundary conditions of bulk scalarfields which corresponds to turning on non-trivial relevant operators in the CFT to drive anRG flow. We can then read off the expectation value of bulk matter fields from the tensornetwork when the boundary conditions of the tensor network have been deformed. This isexplained in section 2.3. In the deformed geometry, one needs to define the notion of distance. Rather than makingguesses, we assume that it has to depend on tensor network data locally and homogenously.4n the case where deviations from the CFT fixed point boundary condition is small, thisdependence can be expressed as a power series of the deformation parameters. We treat theexpansion coefficients, with symmetries following from the locality and homogeneity assump-tions, as unknowns to be determined. This is discussed in section 3.4. The notion of graph curvature is generically a local function of edge distances on the graph[37–39]. We also stay agnostic about the precise definition of the graph curvature, and consideran expansion of the curvature as a power series of deviation of edge distances from the pureBruhat-Tits space. The coefficients are again constrained by locality and homogeneity, butotherwise kept as unknowns. This is discussed in section 3.5. Covariantize the matter action previousely read off from the tensor network when the bulk isa pure Bruhat-Tits geometry. Again we will expand their dependence on the edge lengths asa power series. This is discussed in section 4.6. With the graph curvature and covariant matter action in place, we can obtain a graph Einsteinequation by varying the edge lengths. We substitute the expectation values of the bulk scalarfields and also the edge lengths as a power series of boundary conditions into the equations ofmotion. Since the boundary conditions are chosen to be arbitrary, assuming that the Einsteinequation is satisfied becomes a very stringent constraint on the unknown parameters that wehave introduced thus far. In fact this is an over-determined system that is not guaranteed tohave any solutions at all. Amazingly, a solution exists, and that the unknown coefficients canbe determined uniquely (up to some overall normalizations). The resultant graph curvatureas a function of edge lengths recovers the proposal in the mathematics literature! This isdiscussed in section 4.A review of p-adic CFT has appeared in many places. Since the current paper is focussed onbulk physics, we will relegate a brief review to the appendix A for completeness and for settingnotations for the tensor network.We will conclude in section 5.
A tensor network that recovers the p-adic CFT partition function was introduced in [24]. (Forcompleteness a very brief review of p-adic CFTs is included in Appendix A.) It is a tensor networkthat covers the Bruhat-Tits tree. The Bruhat-Tits tree is an infinite tree graph whose isometry is5igure 1: The tensor network representation of a p = 2-adic CFT. The diagram depicts three bulkoperator insertion. When these bulk insertions are pushed to the asymptotic boundary they areequivalent to boundary insertions. The boundary condition V Λ i here are chosen to be the fixedpoint tensor V af = δ a . Each vertex tensor is given by C abc and each edge of the tensor with index a is weighted by p − ∆ a .given by the conformal group of the p-adic CFT. Each vertex has p + 1 legs. The case for p = 2 isdepicted in Fig. 1.For concreteness, we will focus on a 1-dimensional p-adic CFT that lives in the p-adic line x ∈ Q p , where Q p is the p-adic number fields. Although as emphasized before [33], generalizationto n dimensions corresponds to replacing Q p by a field extension Q p n . The bulk becomes a p n + 1valent tree, and all expressions basically change by replacing p → p n . Therefore without loss ofgenerality and to avoid clutter we will take n = 1 in the rest of this paper.The tensor network constructed in [24] puts a tensor at every vertex of the Bruhat-tits tree.The indices of these tensors are labeled by the primary operators of the CFTs. The value of thetensor T a ··· a p +1 is given by the fusion tree of p + 1 primaries, with fusion coefficients given by theOPE coefficients of the CFT (see appendix A). i.e. T a a ··· a p +1 = (cid:88) b ··· b p − C a a b C b a b · · · C b p − a p a p +1 . (2.1)In the special case where p = 2, we simply have T a a a = C a a a . Tensors sitting at two verticesconnected by an edge have one paired index contracted, with the sum over the index weighted by p − ∆ a , where ∆ a is the conformal dimension of the primary operator O a in the p-adic CFT. The asymptotic boundary of the Bruhat-Tits tree is the Q p line. The tensor network has to becutoff near the asymptotic boundary of the Bruhat-Tits tree – analogous to the cutoff introduced6n AdS space. The dangling legs of the tensors are contracted with a reference tensor | V f (cid:105) ≡ (cid:88) a V af | a (cid:105) , V af ≡ δ a . (2.2)i.e. it projects the dangling legs to the identity operator label. The tensor network evaluates toa number for such boundary conditions. This number is interpreted as the (normalized) partitionfunction of the p-adic CFT. To insert operator O a ( x ) into the partition function, the dangling leglocated at x is projected to label a instead.One can see that the computation of CFT correlation functions naturally reduces to sums of Witten diagrams , constructed from bulk-boundary propagators G a ( x i , v ) meeting at bulk vertices v . These bulk-boundary propagators are given by G a ( x, v ) = ζ p (2∆ a ) p ∆ a p − d ( x,v ) , (2.3)where d ( x, v ) counts the number of edges connecting the boundary vertex linked to x and the bulkvertex v . This is indeed a solution to the graph Klein-Gordon equation( (cid:3) v + m a ) G a ( x, v ) = δ x,v , m a = − ζ p (∆ a − ζ p ( − ∆ a ) , ζ p ( s ) ≡ − p − s . (2.4) (cid:3) v φ ( v ) ≡ (cid:88) u ∼ v ( φ ( v ) − φ ( u )) . (2.5)This suggests that there is a massive bulk field φ a corresponding to each CFT primary O a .We can define the notion of bulk operator insertion at a bulk vertex v by fusing an extra leg withlabel a to the bulk vertex. For fusion rules that are commutative such that C abc is completelysymmetric in the three indices, we can simply stick an extra bulk leg to the vertex v withoutworrying about the order of the fusion. This is illustrated in Fig. 1. The computation of a threepoint correlation function of the CFT is also illustrated there. When these bulk insertions arepushed to the boundary they become boundary operator insertions. i.e. This definition of bulkoperator insertion ensures that the extrapolation dictionary – where boundary operator correspondsto moving a bulk operator towards the asymptotic boundary – is automatically realized.One can thus compute bulk correlation functions, which would be given by sums of Wittendiagrams constructed from bulk-bulk propagators meeting at other bulk vertices. The bulk-bulkpropagator is simply given by the same G a in (2.4), with the boundary vertex label x moved to thebulk.By construction, each bulk vertex where propagators meet gains a factor of the OPE coefficients.A three point vertex is weighted by C abc . 7hese suggest that at least where the operator insertion at the boundary is sparse, the tensornetwork can be described by an emergent bulk quantum field theory with action given by S m = (cid:88) (cid:104) xy (cid:105)
12 ( φ ax − φ ay ) + (cid:88) x m a ( φ ax ) + O ( φ ) , (2.6)where x denotes the vertices, and a denotes different fields whose mass is m a . The a index is summedover though not shown explicitly. More accurately speaking, since the vertex where propagatorsmeet in the tensor network is unique, it exactly matches with the semi-classical limit of the fieldtheory, one where the masses m a approach infinity, or equivalently, ∆ a → ∞ , so that the interactionvertex would be fixed at the intersection of geodesics.To set the normalization rigorousely, simple insertion of legs at a bulk vertex x correspondsto inserting a field ˜ φ x that is related to the canonically normalized φ a ( x ) appearing in the aboveaction by ˜ φ ax ≡ (cid:18) ζ p (2∆ a ) p ∆ a (cid:19) − φ ax , (2.7)so that the two point function read off from the insertion of extra bulk legs is simply given by (cid:104) ˜ φ ax ˜ φ ay (cid:105) = p − ∆ a d ( x,y ) (2.8)In this normalization, the action is expressed as S m = (cid:88) (cid:104) xy (cid:105) ζ p (2∆ a )2 p ∆ a ( ˜ φ ax − ˜ φ ay ) + (cid:88) x ζ p (2∆ a )2 p ∆ a m a ( ˜ φ ax ) + O ( ˜ φ ) . (2.9)One can rewrite the summation over vertices to a sum over edges, which gives S m = (cid:88) (cid:104) xy (cid:105) ζ p (2∆ a )2 p ∆ a ( ˜ φ ax − ˜ φ ay ) + (cid:88) (cid:104) xy (cid:105) ζ p (2∆ a )2 p ∆ a p + 1 m a (( ˜ φ ax ) + ( ˜ φ ay ) ) + O ( ˜ φ ) . (2.10)The factor 1 / ( p + 1) is to cancel the p + 1 times overcounting since each vertex attaches to p + 1edges. RG flow of the p-adic CFT was considered in [24]. In the tensor network, the RG flow of theCFT can be driven by changing the boundary conditions at the asymptotic boundary. Instead ofprojecting the boundary leg to the vector V af defined in (2.2), we pick instead | V Λ (cid:105) ≡ (cid:88) a V a Λ | a (cid:105) (2.11)8hich generically turns on other primaries at the boundary. The RG flow considered in [24] respectstranslation invariance along Q p . In that case, all boundary legs are projected to the same | V Λ (cid:105) .More generally, one can pick different | V Λ i (cid:105) for every boundary leg x i .We note that p such boundary vectors would be fed to the same vertex tensor at the cutoffsurface, which would then return a new vector | V Λ − (cid:105) : | V Λ − (cid:105) ≡ (cid:88) a p +1 V a p +1 Λ − | a p +1 (cid:105) = (cid:88) a ··· a p ,a p +1 p − (∆ a + ··· ∆ ap ) T a ··· a p a p +1 V a Λ · · · V a p Λ p | a p +1 (cid:105) (2.12)Note that the vector | V f (cid:105) is in fact a fixed point under this flow, explaining how a CFT partitionfunction is recovered as fixed point vectors in the tensor network. For other choices of boundaryvectors | V Λ (cid:105) , a flow is driven down the network, so that different parts of the network carry differentweights. Homogenous boundary conditions have been studied quite generally in [24], where we showthat the flow would eventually lead to a new fixed point vector, corresponding to a CFT driven byrelevant perturbations that eventually reaches another new CFT. The geometry in the interior ofthe network would thus resemble pure Bruhat-Tits geometry again, where the vertices eventuallycontribute equally to the partition function. That the actual weights of the tensors contributingdifferently is suggestive of a varying metric in the tree. How the metric should depend on thetensors is the crux of reading off geometry and subsequently gravity from the tensor network. Wewill pick up this problem in the next section.This is a highly non-linear flow and it is difficult to keep analytic control. To make furtherprogress, we will consider small deviations from the fixed point vector | V f (cid:105) . i.e. In the following,we will consider V a Λ i = δ a + λv a Λ i , (2.13)and treat λ as a small parameter in a power series expansion around the CFT fixed point. Consider the p -adic tensor network as shown in Fig. 3(a). Expanded in the contributing tensors,it takes the form Z = (cid:88) ...,a,b,c,d,e,... . . . p − ∆ a p − ∆ b C abc p − ∆ c C cde p − ∆ d p − ∆ e . . . . (2.14)We can rewrite Z as Z = (cid:88) c V c p − ∆ c ˜ V c , (2.15)9igure 2: Vector V axy and ˜ V axy following from the contraction of tensors above (colored red) and below (colored green) the edge (cid:104) xy (cid:105) respectively. The boundary conditions V Λ i generically deviate fromthe fixed point vector. The curvature of the patch centered at x depends on the graph distances d (cid:104) xy i (cid:105) symmetrically.defining V c ≡ (cid:88) ...,a,b . . . p − ∆ a p − ∆ b C abc , (2.16)˜ V c ≡ (cid:88) d,e,... C cde p − ∆ d p − ∆ e . . . . (2.17)When one is restricted on the red edge, V a and ˜ V a carry all the information in the tree. Theyfollow from contracting all the tensors above or below the edge. This is illustrated in Fig. 2.Using these definitions, the two point function on this edge regardless of boundary conditioncan always be written in terms of the pair of V a and ˜ V a (cid:104) ˜ φ a ˜ φ b (cid:105) = (cid:88) c,d,e V d C acd p − ∆ c C bce ˜ V e , (2.18)as shown in Fig. 3(b).In the p -adic vacuum where the boundary legs are projected to the fixed point vector | V f (cid:105) , asexpected, V a and ˜ V a are both equal to the fixed point vector. V a = δ a , ˜ V a = δ a , (2.19)Now, anticipating what will be needed when we begin considering curvatures, let us in particularconsider the following local patch of the p -adic tensor network as shown in Fig. 4. The edge xy is connected with { xy i } and { x i y } . When one is restricted within this part, we can express all10 a) (b) Figure 3: (a):Here we demonstrate the p = 2 case for simplicity. (b):The two black legs are bulklegs. We insert them near the vertices to make the labelling more transparent. They should beunderstood as inserted on the verticesrelevant information describing this patch from the collection { V ai } and { ˜ V ai } located at each ofthe out-stretching legs. One can see that for edge xy i , its V a is V ai , while for edge x i y , its ˜ V a is˜ V ai . Perturbed around the vacuum under generic boundary conditions (2.13), we have V ai = δ a + ω ai , ω ai ≡ λ ai + η ai + O ( λ ) , ˜ V ai = δ a + ˜ ω ai , ˜ ω ai ≡ ˜ λ ai + ˜ η ai + O ( λ ) , (2.20)where λ ai , ˜ λ ai ∼ O ( λ ), η ai , ˜ η ai ∼ O ( λ ), and λ (cid:28) xy is connected with { xy i } and { x i y } . When one is restricted within this part, { V ai } and { ˜ V ai } encode all the information needed.Reading off from the tensor network, we have (cid:104) ˜ φ ax (cid:105) = λ a p − ∆ a + ˜ λ a p − a + O ( λ ) , (2.21) (cid:104) ˜ φ ay (cid:105) = ˜ λ a p − ∆ a + λ a p − a + O ( λ ) , (2.22)11here λ a , ˜ λ a are defined by λ a ≡ p (cid:88) i =1 λ ai , (2.23)˜ λ a ≡ p (cid:88) i =1 ˜ λ ai . (2.24) We have discussed the local data that is encoded in the tensor network in detail in the previoussection. In this section, we would like to explain how one should assign distances to edges andgraph curvatures to patches based on the local data obtained from the tensor network.
As discussed in the previous section using Fig. 3(b), all the information carried by an edge e is completely captured by V ae and ˜ V ae at the two ends of the edge. So it’s natural to propose thatthe edge length d e is also determined by V ae and ˜ V ae . It is a bit challenging to obtain the completedependence of the edge length on V ae and ˜ V ae . However, if we are working with the perturbativelimit, where V ae and ˜ V ae admit a perturbation expansion of the form (2.20): V ae = δ a + w ae = δ a + λ ae + η ae + O ( λ ) , (3.1)˜ V ae = δ a + ˜ w ae = δ a + ˜ λ ae + ˜ η ae + O ( λ ) . (3.2)then the edge distance d e also admits an expansion in λ ae , ˜ λ ae , η ae , ˜ η ae .But first, we note that when the boundary condition corresponds to the fixed point vector, everyvertex and thus every edge gives equal contribution to the partition function. Therefore preciselyat the fixed point, all edge distances are equal, and we can conveniently set it to 1. Therefore,under perturbation away from the fixed point, the edge length at each edge is also admitting smalldeviation from unity as follows: d e = 1 + j e (3.3)with j e (cid:28)
1, and this deviation should depend on the local data on the tensor network. Thereforewe can also expand it in powers of λ as follows: j e = A a ( ω ae + ˜ ω ae ) + B ab ( ω ae ω be + ˜ ω ae ˜ ω be ) + C ab ω ae ˜ ω be + O ( ω )= A a ( λ ae + ˜ λ ae ) + B ab ( λ ae λ be + ˜ λ ae ˜ λ be ) + C ab λ ae ˜ λ be + A a ( η ae + ˜ η ae ) + O ( λ ) , (3.4)12ith B ab , C ab symmetric matrices that parametrise this expansion. Here we have used the symmetryof ω ae ↔ ˜ ω ae , i.e. V ae ↔ ˜ V ae , since the edge should depend symmetrically on the information at itstwo ends. These unknown parameters, as we are going to find out, are almost uniquely fixed if aconsistent Einstein equation is to exist at all. We would like to define graph Ricci curvatures given the set of edge distances we have nowdefined on the tensor network. Graph curvatures have been considered in the mathematics literature[37–40]. Inspired by them, we consider the Ricci curvature R ( d xy , d xy , . . . , d xy p +1 ) of a patcharound a vertex x to be a symmetric function of the lengths d xy i of edges emanating from thevertex x . For the precise dependence however, we first take the agnostic approach, and considerthe most general expansion of the curvature on the edge distances perturbed away from unity. Weexpect that there is a regular expansion since the curvature should evaluate to a constant when alledges have equal lengths.We parametrize the expansion of the graph Ricci curvature R ( d xy , d xy , . . . , d xy p +1 ) as a powerseries in { j xy i } as: R ( d xy , d xy , . . . , d xy p +1 ) = a + a (cid:88) i j xy i + b (cid:88) i j xy i + c (cid:88) i (cid:54) = k j xy i j xy k + O ( j ) . (3.5)Here we have imposed symmetric dependence of all edges connected to x . These coefficients a , b, c are dependence of curvatures on distances and they should be universal independent of the tensornetwork.Now we would like to write down the analogue of the Einstein-Hilbert action on the graph. OnRiemann manifold the Einstein-Hilbert action takes the form: S EH [ g µν ] = (cid:90) d d x √− g ( R + Λ) , (3.6)with Λ the cosmological constant and R the Ricci scalar which is a function of the metric g µν .Having defined the graph scalar curvature in (3.5), the Einstein-Hilbert action on the BT treeshould take the general form: S EH = (cid:88) x R ( d xy , d xy , . . . , d xy p +1 ) + (cid:88) (cid:104) xy (cid:105) d xy Λ , (3.7)where the first sum runs over all bulk vertices x , and (cid:80) (cid:104) xy (cid:105) indicates a sum over edges. The secondterm is proposed to mimic the cosmological constant term. In the continuous case, the volume formis given by d d x √− g . Therefore it is natural to propose that d d x √− g ∼ d xy , (3.8)13hich leads to the second term in (3.7). Having obtained an ansatz of edge distances and graph curvature, we would like to considercovariantizing the matter action, and eventually obtain a graph Einstein equation.
The matter action read off from the tensor network when the boundary conditions are taken tobe the fixed point vectors except at isolated locations of operator insertion is given by (2.10). Thisis essentially the action for pure BT space.We would like to covariantize this action, to couple edge lengths to the matter fields. Thishas been considered in [39]. However we find the ansatz quite restrictive there, and we would liketo write down a more general ansatz that mimics closely the covariant coupling of matter to thebackground in the continuous case, although very much like in our treatment of the edge lengthsand graph curvature, the ansatz allows for a collection of parameters that cannot be fixed purelyby symmetry and locality considerations.Recall (3.8), then (cid:90) d d x √− gf ( x ) → (cid:88) (cid:104) xy (cid:105) d xy f xy . (4.1)A natural ansatz for the covariantized matter action S covm therefore takes the following form: S covm = (cid:88) (cid:104) xy (cid:105) d kxy ζ p (2∆ a )2 p ∆ a ( ˜ φ ax − ˜ φ ay ) + (cid:88) (cid:104) xy (cid:105) d xy ζ p (2∆ a )2 p ∆ a p + 1 m a (( ˜ φ ax ) + ( ˜ φ ay ) ) + O ( ˜ φ ) , (4.2)where k is some constant we can’t determine now since ( ˜ φ ax − ˜ φ ay ) may contribute additional 1 /d xy .Writing down the general O ( ˜ φ ) term explicitly, S covm becomes S covm = (cid:88) (cid:104) xy (cid:105) d kxy ζ p (2∆ a )2 p ∆ a ( ˜ φ ax − ˜ φ ay ) + (cid:88) (cid:104) xy (cid:105) d xy ζ p (2∆ a )2 p ∆ a p + 1 m a (( ˜ φ ax ) + ( ˜ φ ay ) )+ (cid:88) (cid:104) xy (cid:105) (cid:16) h ( d xy ) H abc ( ˜ φ ax ˜ φ bx ˜ φ cx + ˜ φ ay ˜ φ by ˜ φ cy ) + r ( d xy ) R abc ( ˜ φ ax ˜ φ bx ˜ φ cy + ˜ φ ay ˜ φ by ˜ φ cx ) (cid:17) + O ( ˜ φ ) , (4.3)where h ( d xy ) , r ( d xy ) are functions of d xy , and H abc is totally symmetric, while R abc is symmetricin a, b . 14imilar to our previous treatment of the graph curvature, we can expand h ( d xy ) , r ( d xy ) in powersof perturbations j xy of the edge lengths : h ( d xy ) = h + h j xy + h j xy + . . . , (4.4) r ( d xy ) = r + r j xy + r j xy + . . . . (4.5) We have proposed the ansatz for an Einstein-Hilbert action and a covariant matter action. Weare now ready to obtain a graph Einstein equation by varying the total action S EH + S covm withrespect to the edge lengths. This gives δS tot δd xy = δS EH δd xy + δS covm δd xy = 0 , i.e. G + T = 0 , (4.6)with G ≡ δS EH δd xy = Λ + 2 a + 4 bj xy + c ( (cid:88) i ( y i (cid:54) = y ) j xy i + (cid:88) i ( x i (cid:54) = x ) j x i y ) + O ( j ) , (4.7) T ≡ δS covm δd xy = k ζ p (2∆ a )2 p ∆ a ( ˜ φ ax − ˜ φ ay ) + ζ p (2∆ a )2 p ∆ a p + 1 m a (( ˜ φ ax ) + ( ˜ φ ay ) )+ (cid:16) h H abc ( ˜ φ ax ˜ φ bx ˜ φ cx + ˜ φ ay ˜ φ by ˜ φ cy ) + r R abc ( ˜ φ ax ˜ φ bx ˜ φ cy + ˜ φ ay ˜ φ by ˜ φ cx ) (cid:17) + O ( ˜ φ ) , (4.8)One very important fact to note is that by considering the equations following from variationof d xy , the edge data involved in the equation span a patch that is precisely given by Fig. 4, whichwas anticipated there. λ We have obtained from the tensor network expectation values of some scalar field ˜ φ a . We havealso read off edge lengths and edge curvatures based on data of the tensor network. The issue athand is that – we have a solution in search of an equation relating these data. Using very fewassumptions, based on locality and also correlation functions of these scalar fields at the CFT fixedpoint, we have written down an ansatz of an action up to some undetermined coefficients, whichled to an Einstein equation. Here we would like to ask the following question: if the expectationvalues and geometrical data that we have read off from the tensor network are indeed related by thepurported graph Einstein equation, what are the allowed values of the undetermined parametersso that the Einstein equation can indeed be satisfied, if at all?We are therefore going to treat the Einstein equation as a constraint on the undeterminedparameters. The surprising result is that this over-determined system does have a unique solution15up to some overall normalizations) that naturally recovers results in the mathematics literatureand satisfies all other consistency conditions.We will demonstrate this order by order in the λ expansion.Plugging (2.21), (2.22) into (4.8), we find that T = p − a (cid:16) k ( p + 1) (cid:0) p ∆ a − (cid:1) + m a (cid:0) p a + 1 (cid:1)(cid:17) p + 1) ( p ∆ a −
1) ( p ∆ a + 1) ( λ a λ a + ˜ λ a ˜ λ a ) (4.9)+ p − a (cid:16) m a p ∆ a − k ( p + 1) (cid:0) p ∆ a − (cid:1) (cid:17) ( p + 1) ( p ∆ a −
1) ( p ∆ a + 1) λ a ˜ λ a + O ( λ ) . (4.10)For the edge xy i , its V a and ˜ V a can also be read off from the tensor network: V a = V ai = δ a + λ ai + O ( λ ) , (4.11)˜ V a = δ a + ( λ a − λ ai ) p − ∆ a + ˜ λ a p − a + O ( λ ) . (4.12)Similarly, for the edge x i y , its V a and ˜ V a are given by V a = δ a + (˜ λ a − ˜ λ ai ) p − ∆ a + λ a p − a + O ( λ ) , (4.13)˜ V a = ˜ V ai = δ a + ˜ λ ai + O ( λ ) . (4.14)For the edge xy , its V a and ˜ V a are given by V a = δ a + λ a p − ∆ a + O ( λ ) , (4.15)˜ V a = δ a + ˜ λ a p − ∆ a + O ( λ ) . (4.16)Plugging them into (3.4) and recalling that d xy = 1 + j xy , we can obtain j xy , { j xy i } , { j x i y } .Plugging all the j into (4.7), we find that G = Λ + 2 a + A a ( λ a + ˜ λ a ) p − a (cid:0) bp ∆ a + c (cid:0) p a + ( p − p ∆ a + p (cid:1)(cid:1) + O ( λ ) . (4.17)To satisfy G + T = 0 order by order in λ , we should haveΛ + 2 a = 0 , (4.18) A a = 0 , (4.19)since T ∼ O ( λ ) and (cid:0) bp ∆ a + c (cid:0) p a + ( p − p ∆ a + p (cid:1)(cid:1) is zero for at most two ∆ a . The two solutions are ∆ a and d − ∆ a , where d is the dimension of the CFT which is conveniently taken to beone here. d e = 1 + B ab ( λ ae λ be + ˜ λ ae ˜ λ be ) + C ab λ ae ˜ λ be + O ( λ ) , (4.20)which means we can work out the O ( λ ) term only with λ ae , ˜ λ ae . Now plugging all the V a , ˜ V a aboveinto (4.20), we can work out j xy , { j xy i } , { j x i y } to O ( λ ). Plugging them into (4.7), this time wefind that G = p − a +∆ b ) (cid:16) bB ab p ∆ a +∆ b + B ab c (cid:0) − p ∆ a +∆ b + p ∆ a +∆ b +1 + p (cid:1) + cC ab p a +∆ b (cid:17) ( λ a λ b + ˜ λ a ˜ λ b )+ p − a +∆ b ) (cid:16) bC ab p ∆ a +∆ b + 2 B ab c ( p − (cid:0) p ∆ b + p ∆ a (cid:1) + cC ab (cid:0) p a + p b (cid:1)(cid:17) λ a ˜ λ b + (cid:88) i c (cid:16) B ab (1 + p − ∆ a − ∆ b ) − C ab ( p − ∆ a + p − ∆ b ) (cid:17) ( λ ai λ bi + ˜ λ ai ˜ λ bi ) + O ( λ ) . (4.21)To satisfy G + T = 0, firstly we should have (cid:88) i c (cid:16) B ab (1 + p − ∆ a − ∆ b ) − C ab ( p − ∆ a + p − ∆ b ) (cid:17) ( λ ai λ bi + ˜ λ ai ˜ λ bi ) = 0 , (4.22)since other terms in G + T only depend on λ a , ˜ λ a . It is obvious that c = 0 is one solution for it.That is too stringent. We are interested in c (cid:54) = 0 case which is more non-trivial, and this gives aconstraint on the parameters2 B ab (1 + p − ∆ a − ∆ b ) − C ab ( p − ∆ a + p − ∆ b ) = 0 , (4.23)which is already made symmetric in a, b since it was contracted with a symmetric tensor λ ai λ bi . Sowe have C ab = 2 B ab (1 + p − ∆ a − ∆ b ) p − ∆ a + p − ∆ b . (4.24)Now comparing G with T , we notice that B ab should be diagonal. And this gives G = B aa p − a (cid:18) bp a + 12 c (cid:0) − p a + 2 p a + 2 p ∆ a +1 + 2 p a +1 (cid:1) p − ∆ a (cid:19) ( λ a λ a + ˜ λ a ˜ λ a )+2 B aa p − a (cid:0) b (cid:0) p a + 1 (cid:1) + c (cid:0) p a + 2 p − (cid:1)(cid:1) λ a ˜ λ a + O ( λ ) . (4.25)Then G + T = 0 gives B aa p − a (cid:18) bp a + 12 c (cid:0) − p a + 2 p a + 2 p ∆ a +1 + 2 p a +1 (cid:1) p − ∆ a (cid:19) = − p − a (cid:16) k ( p + 1) (cid:0) p ∆ a − (cid:1) + m a (cid:0) p a + 1 (cid:1)(cid:17) p + 1) ( p ∆ a −
1) ( p ∆ a + 1) , (4.26)2 B aa p − a (cid:0) b (cid:0) p a + 1 (cid:1) + c (cid:0) p a + 2 p − (cid:1)(cid:1) = − p − a (cid:16) m a p ∆ a − k ( p + 1) (cid:0) p ∆ a − (cid:1) (cid:17) ( p + 1) ( p ∆ a −
1) ( p ∆ a + 1) . (4.27)17nd m a can be solved by b, c, k : m a = − k ( p + 1) − ck ( p + 1) (cid:0) p − ∆ a + p ∆ a (cid:1) b − c . (4.28)Recall that for scalar field φ a on the BT tree, its mass is related to the conformal dimension by m a = − ζ p (∆ a − ζ p ( − ∆ a ) = − p − p − ∆ a + p ∆ a , (4.29)which is obtained by analyzing its Green’s function. Comparing (4.28) with (4.29), we find that k = 1 , (4.30)2 bc = − p, (4.31)which ensures that k, b, c do not depend on ∆ a as they are universal geometric dependence inde-pendent of the tensor network.This result is amazing! Firstly, since k, b, c do not depend on ∆ a , there may be no solutionto match (4.28) with (4.29) if (4.28) were not satisfied. The existence of the solution is a strongevidence that Einstein equation is indeed encoded in the tensor network. Secondly, supposing k, b, c are known at the beginning, the mass-dimension relation can be derived from the Einstein equation.We discover a known relation with a new perspective. Thirdly, 2 b/c = − p means that to leadingorder G ≡ δS EH δd xy = 4 bj xy + c ( (cid:88) i ( y i (cid:54) = y ) j xy i + (cid:88) i ( x i (cid:54) = x ) j x i y ) = − c (cid:3) j xy , (4.32)where (cid:3) j e is defined by (cid:3) j e ≡ (cid:88) f ∼ e ( j e − j f ) . (4.33)Here (cid:80) f ∼ e denotes the sum over all edges f that share a vertex with a fixed edge e . Our resultis similar to the one that appears in [39]. The authors in [39] start with a reasonable definition ofRicci curvature on graphs and then obtain a result similar to (4.32). Assuming only locality andhomogeneity in our ansatz and requiring that the resultant Einstein equation is satisfied leads toa unique expression that matches with the mathematics literature for G in the perturbative limitaway from the pure BT geometry.Having k = 1 , b/c = − p , we further get B aa = 12 c ( p + 1) (1 − p a ) . (4.34)18 .4 Solving the Einstein constraint to order λ To order O ( λ ), we have no restriction on the interaction term ˜ φ in the action since ˜ φ ∼ O ( λ ).While a solution for Einstein equation to this order is non-trivial – as we have seen above it involvedvarious non-trivial constraints on the mass that happens to be consistent with the AdS/CFTdictionary – we note that the variation of the relative entropy also led to an Einstein equation tothis order. In this section we would show that such constraints can also be solved in the λ orderleading to a self-consistent result. This is certainly beyond kinematics and provide strong evidencethat the correct dynamics are indeed encoded in the tensor network.Now let’s work out G and T to order O ( λ ) and see what kinds of interaction terms we willget. The procedure is the same as before, but the calculation becomes more complicated.We still consider the graph shown in Fig. 4. The setup is the same as before. Now we have˜ φ ax = λ a p − ∆ a + ˜ λ a p − a + η a p − ∆ a + ˜ η a p − a + γ a + ˜ γ a p − ∆ a + λ b ˜ λ c p − ∆ b p − c C abc + O ( λ ) , (4.35)˜ φ ay = ˜ λ a p − ∆ a + λ a p − a + ˜ η a p − ∆ a + η a p − a + ˜ γ a + γ a p − ∆ a + ˜ λ b λ c p − ∆ b p − c C abc + O ( λ ) , (4.36)where η a ≡ (cid:88) i η ai , (4.37)˜ η a ≡ (cid:88) i ˜ η ai , (4.38) γ a ≡ (cid:88) i (cid:54) = j λ bi λ cj C abc p − ∆ b p − ∆ c , (4.39)˜ γ a ≡ (cid:88) i (cid:54) = j ˜ λ bi ˜ λ cj C abc p − ∆ b p − ∆ c . (4.40)Here (cid:80) i (cid:54) = j denotes the sum of all possible ways of picking two edges from p edges. We can alsoexpress γ a , ˜ γ a as γ a = 12 ( λ b λ c C abc p − ∆ b p − ∆ c − (cid:88) k λ bk λ ck C abc p − ∆ b p − ∆ c ) , (4.41)˜ γ a = 12 (˜ λ b ˜ λ c C abc p − ∆ b p − ∆ c − (cid:88) k ˜ λ bk ˜ λ ck C abc p − ∆ b p − ∆ c ) . (4.42)19lugging (4.35), (4.36) into (4.8), we get T λ = r R abc p − ( ∆ a +∆ b +∆ c ) (cid:16)(cid:16) ˜ λ a p ∆ a + λ a (cid:17) (cid:16) ˜ λ b p ∆ b + λ b (cid:17) (cid:16) ˜ λ c + λ c p ∆ c (cid:17) + (cid:16) ˜ λ a + λ a p ∆ a (cid:17) (cid:16) ˜ λ b + λ b p ∆ b (cid:17) (cid:16) ˜ λ c p ∆ c + λ c (cid:17)(cid:17) + h H abc p − ( ∆ a +∆ b +∆ c ) (cid:16)(cid:16) ˜ λ a + λ a p ∆ a (cid:17) (cid:16) ˜ λ b + λ b p ∆ b (cid:17) (cid:16) ˜ λ c + λ c p ∆ c (cid:17) + (cid:16) ˜ λ a p ∆ a + λ a (cid:17) (cid:16) ˜ λ b p ∆ b + λ b (cid:17) (cid:16) ˜ λ c p ∆ c + λ c (cid:17)(cid:17) + (cid:16) λ a − ˜ λ a (cid:17) p − a (cid:16) p ∆ a − (cid:17) (cid:16)(cid:16) p ∆ a − (cid:17) p − a (cid:16) − ˜ η a + ( γ a − ˜ γ a ) p ∆ a + η a (cid:17) + C abc (cid:16) λ b ˜ λ c − λ c ˜ λ b (cid:17) p − ∆ b − c (cid:17) − p − a + (cid:16) p ∆ a − p (cid:17) p − a − ∆ b − c ( p + 1) (cid:0) p ∆ a + 1 (cid:1) (cid:104) (cid:16) ˜ γ a ˜ λ a + 2 λ a ˜ η a + 2 η a ˜ λ a + γ a λ a (cid:17) p ∆ a +∆ b +2∆ c + (cid:16) ˜ γ a ˜ λ a + γ a λ a (cid:17) p a +∆ b +2∆ c + (cid:16) λ a ˜ γ a + 2 γ a ˜ λ a + ˜ η a ˜ λ a + η a λ a (cid:17) p a +∆ b +2∆ c + (cid:16) ˜ η a ˜ λ a + η a λ a (cid:17) p ∆ b +2∆ c + C abc p a (cid:16) λ a λ c ˜ λ b + λ b ˜ λ a ˜ λ c (cid:17) + C abc p a (cid:16) λ c ˜ λ a ˜ λ b + λ a λ b ˜ λ c (cid:17) (cid:105) . (4.43) Here T λ is the O ( λ ) term in T .For edge xy i , after careful analysis its V a and ˜ V a are V a = V ai = δ a + λ ai + η ai + O ( λ ) , (4.44)˜ V a = δ a + ( λ a − λ ai ) p − ∆ a + ˜ λ a p − a + ( η a − η ai ) p − ∆ a + ˜ η a p − a + ˜ γ a p − ∆ a +( λ b − λ bi )˜ λ c C abc p − ∆ b p − c + γ a − λ bi ( λ c − λ ci ) C abc p − ∆ b p − ∆ c + O ( λ ) . (4.45)Similarly for edge x i y , its V a and ˜ V a are˜ V a = ˜ V ai = δ a + ˜ λ ai + ˜ η ai + O ( λ ) , (4.46) V a = δ a + (˜ λ a − ˜ λ ai ) p − ∆ a + λ a p − a + (˜ η a − ˜ η ai ) p − ∆ a + η a p − a + γ a p − ∆ a +(˜ λ b − ˜ λ bi ) λ c C abc p − ∆ b p − c + ˜ γ a − ˜ λ bi (˜ λ c − ˜ λ ci ) C abc p − ∆ b p − ∆ c + O ( λ ) . (4.47)For edge xy , its V a and ˜ V a are V a = δ a + λ a p − ∆ a + η a p − ∆ a + γ a + O ( λ ) , (4.48)˜ V a = δ a + ˜ λ a p − ∆ a + ˜ η a p − ∆ a + ˜ γ a + O ( λ ) . (4.49)This time the precision of (3.4) is not enough since it is only of order O ( λ ). Now we persevereto recover the order O ( λ ) term, which is given by d e = 1 + B ab ( ω ae ω be + ˜ ω ae ˜ ω be ) + C ab ω ae ˜ ω be + D abc ( ω ae ω be ω ce + ˜ ω ae ˜ ω be ˜ ω ce ) + E abc ( ω ae ω be ˜ ω ce + ˜ ω ae ˜ ω be ω ce ) + O ( ω )= 1 + B ab ( λ ae λ be + ˜ λ ae ˜ λ be ) + C ab λ ae ˜ λ be + 2 B ab ( λ ae η be + ˜ λ ae ˜ η be ) + C ab ( λ ae ˜ η be + η ae ˜ λ be )+ D abc ( λ ae λ be λ ce + ˜ λ ae ˜ λ be ˜ λ ce ) + E abc ( λ ae λ be ˜ λ ce + ˜ λ ae ˜ λ be λ ce ) + O ( λ ) , (4.50)where D abc is fully symmetric and E abc is symmetric in a, b . Here we have used A a = 0, whichmeans expanding V a , ˜ V a to O ( λ ) is enough to obtain d e to O ( λ ).Plugging V a , ˜ V a above into (4.50), we can obtain j xy , { j xy i } , { j x i y } to order O ( λ ). Plugging allthe j into (4.7), we can obtain G to order O ( λ ). Since j ∼ O ( λ ), we have j ∼ O ( λ ). Thereforeit is consistent to ignore the O ( j ) term in G . 20et us denote the order O ( λ ) term of G by G λ . After careful calculation, we can work out G λ + T λ . After using the B ab , C ab , b = − pc/ G λ + T λ the termwhich depends on { η ai } , { ˜ η ai } automatically vanishes. Finally we find G λ + T λ = p (cid:88) j =1 M abc ( λ aj λ bj λ cj + ˜ λ aj ˜ λ bj ˜ λ cj ) + p (cid:88) k =1 M abc (cid:16) λ ak λ bk ( p ∆ c λ c + ˜ λ c ) + ˜ λ ak ˜ λ bk ( p ∆ c ˜ λ c + λ c ) (cid:17) + M abc ( λ a λ b λ c + ˜ λ a ˜ λ b ˜ λ c ) + M abc ( λ a λ b ˜ λ c + ˜ λ a ˜ λ b λ c ) , (4.51)where M abc = p − ∆ a − ∆ b − ∆ c (cid:0) c ( p + 1) (cid:0) D abc (cid:0) p ∆ a +∆ b +∆ c − (cid:1) + E abc (cid:0) p ∆ c − p ∆ a +∆ b (cid:1)(cid:1) − C abc (cid:1) p + 1) , (4.52) M abc = 14 p − ∆ a − ∆ b − c (cid:32) c (cid:0) E abc p ∆ a +∆ b − E bca p ∆ a − E cab p ∆ b + 3 D abc (cid:1) + C abc p +1) p c − + p + 1 (cid:33) , (4.53) M abc = 14 p − a +∆ b +∆ c ) (cid:32) r R abc (cid:0) p ∆ a +∆ b + p ∆ c (cid:1) + 4 h H abc (cid:0) p ∆ a +∆ b +∆ c + 1 (cid:1) + 4 cE abc p ∆ a +∆ b +2∆ c +4 cD abc (cid:0) p − ( p + 3) p ∆ a +∆ b +∆ c (cid:1) + C abc (cid:0) p a + 3 (cid:1) p ∆ a +∆ b +∆ c ( p + 1) ( p a − (cid:33) , (4.54) M abc = 14 p − a +∆ b +∆ c ) (cid:32) cD abc (cid:0) ( p − p ∆ a +∆ b + ( p − p ∆ c (cid:1) + C abc (cid:0) ( p + 1) (cid:0) p a − (cid:1) (cid:0) p c − (cid:1)(cid:1) − (cid:104) (cid:0) p a +1 − (cid:1) p ∆ c − (cid:0) p a +1 − (cid:1) p c + (cid:0) p a − p + 1 (cid:1) p ∆ a +∆ b +2∆ c + (cid:0) p a + 4 p − (cid:1) p ∆ a +∆ b (cid:105) + 4 (cid:104) h H abc p ∆ a +∆ b + cE abc p ∆ c (cid:0) p ∆ c − p ∆ a +∆ b +1 (cid:1) + cE bca p a +∆ b + cE cab p ∆ a +2∆ b + r R cab p ∆ a + r R bca p ∆ b + r R abc + p ∆ c (cid:0) r p ∆ b (cid:0) R abc p ∆ a + R cab (cid:1) + r R bca p ∆ a + 3 h H abc (cid:1) (cid:105)(cid:33) . (4.55) Here we have used (4.41),(4.42), so γ a , ˜ γ a do not appear in our final expression.It is obvious that the Einstein equation is G λ + T λ = 0 , (4.56)for any { λ ai } , { ˜ λ ai } . So we should have M ( abc )1 = 0 , M ( ab ) c = 0 , M ( abc )3 = 0 , M ( ab ) c = 0 , (4.57)where ( ab ) , ( abc ) means the symmetrization of the matrix. The unknown matrices are D abc = D ( abc ) , E abc = E ( ab ) c , H abc = H ( abc ) , R abc = R ( ab ) c . They can be fully determined by (4.57) sincethe number of unknown independent matrix elements is the same as the number of independentequations. 21sing M ( abc )1 = 0, we can get D abc = 3 C abc + 2 c (1 + p ) (cid:0) ( p ∆ a +∆ b − p ∆ c ) E abc + ( p ∆ b +∆ c − p ∆ a ) E bca + ( p ∆ a +∆ c − p ∆ b ) E cab (cid:1) c (1 + p )( − p ∆ a +∆ b +∆ c ) . (4.58) Plugging it into M ( ab ) c = 0, we can get ( p a +∆ b ) − E abc − (cid:0) p a − (cid:1) p ∆ b E bca + p ∆ a (cid:0) − (cid:0) p b − (cid:1)(cid:1) E cab = − C abc p − ∆ c (cid:0) − p ∆ a +∆ b +∆ c + p ∆ a +∆ b +3∆ c + 5 p c − (cid:1) c ( p + 1) ( p c − . (4.59) By solving it, we obtain E abc = C abc ( − p ∆ a +∆ b +2∆ c − p ∆ a − ∆ b − p ∆ b − ∆ a + 3 p ∆ a +∆ b − p a +∆ c − p b +∆ c − p − ∆ c + 3 p ∆ c )4 c ( p + 1) ( p a −
1) ( p b −
1) ( p c − . (4.60) Using M ( abc )3 = 0, we can get H abc = − h ( p ∆ a +∆ b +∆ c + 1) (cid:16) p ∆ a +∆ b +∆ c C abc ( p + 1) ( p a −
1) ( p b −
1) ( p c − × (cid:0) p a +2∆ b +2∆ c + p a +2∆ b + p a +2∆ c + p b +2∆ c − p a − p b − p c + 9 (cid:1) +4 (cid:2) cD abc (cid:0) p − ( p + 3) p ∆ a +∆ b +∆ c (cid:1) + cE abc p ∆ a +∆ b +2∆ c + cE bca p a +∆ b +∆ c + cE cab p ∆ a +2∆ b +∆ c + r R abc p ∆ a +∆ b + r R cab p ∆ a +∆ c + r R bca p ∆ b +∆ c + r R bca p ∆ a + r R cab p ∆ b + r R abc p ∆ c (cid:3)(cid:17) . (4.61) Plugging (4.58),(4.60),(4.61) into M ( ab ) c = 0, we can get2 R abc (cid:16) p a +∆ b ) − (cid:17) + 2 R bca (cid:0) p a − (cid:1) p ∆ b + 2 R cab p ∆ a (cid:0) p b − (cid:1) = C abc p ∆ a +∆ b r . (4.62)By solving it, we obtain R abc = C abc (cid:0) p ∆ a +∆ c − p ∆ b (cid:1) (cid:0) p ∆ b +∆ c − p ∆ a (cid:1) r ( p a −
1) ( p b −
1) ( p c − . (4.63)Having E abc , R abc , we can further get D abc = − C abc p − ∆ a − ∆ b − ∆ c c ( p + 1) ( p a −
1) ( p b −
1) ( p c − (cid:16) − p ∆ a +∆ b +∆ c − p a +∆ b +∆ c ) + p a +∆ b +∆ c + p ∆ a +3∆ b +∆ c + p ∆ a +∆ b +3∆ c + p a +∆ b ) + p a +∆ c ) + p b +∆ c ) (cid:17) , (4.64) H abc = C abc p − ∆ a − ∆ b − ∆ c (cid:0) p ∆ a +∆ b +∆ c + p (cid:1) h ( p + 1) ( p a −
1) ( p b −
1) ( p c − (cid:16) − p ∆ a +∆ b +∆ c − p a +∆ b +∆ c ) + p a +∆ b +∆ c + p ∆ a +3∆ b +∆ c + p ∆ a +∆ b +3∆ c + p a +∆ b ) + p a +∆ c ) + p b +∆ c ) (cid:17) . (4.65) As we can see, R abc is simple, while E abc , D abc , H abc are a bit complicated. And we notice that D abc H abc = − h c ( p ∆ a +∆ b +∆ c + p ) . (4.66)It is not trivial that D abc , E abc , R abc , H abc can be exactly solved. It means that to satisfy theEinstein equation the interaction term and the definition of edge length are fixed!22 .4.1 The interaction term in the semi-classical limit As we have noted in section 2, the field theory exactly recovers the correlation functions in thetensor network in the semi-classical limit where the masses approach infinity. Therefore we wouldlike to take this limit in the results obtained in the previous section.Recall that (cid:104) ˜ φ ax ˜ φ by ˜ φ cz (cid:105) = (cid:104) O ax O by O cz (cid:105) . This is consistent with an action with cubic term. S cubic = 13! (cid:88) x C abc ˜ φ ax ˜ φ bx ˜ φ cx . (4.67)At first glance our H abc , R abc do not agree with (4.67). However, when we take the classical limit,i.e. ∆ (cid:29)
1, we have H abc = − C abc h (1 + p ) , (4.68) R abc = 0 , (4.69)where we have used ∆ a ∼ ∆ b ∼ ∆ c (cid:29)
1. Then the interaction term in the action (4.3) becomes (cid:88) (cid:104) xy (cid:105) − h C abc h (1 + p ) ( ˜ φ ax ˜ φ bx ˜ φ cx + ˜ φ ay ˜ φ by ˜ φ cy ) = (cid:88) x − h C abc h ˜ φ ax ˜ φ bx ˜ φ cx . (4.70)Here we consider the vacuum case so that h ( d xy ) = h and we have changed the sum over edgesinto vertices. We have noted in section 2 that at every vertex where propagators meet, the tensornetwork dictates that it is weighted by a factor of the fusion coefficients. In particular when thereare exactly three propagators meeting at the vertex, the interaction coupling is precisely C abc .Up to some ∆ a independent normalizations that is not fixed by the Einstein constraints, thelatter led to (4.70) that is in complete agreement with results consistent with the tensor network. In the definition of d e , D abc , E abc are a bit complicated. We find a way to simplify the expressionof d e . Originally it is given by d e = 1 + B ab ( λ ae λ be + ˜ λ ae ˜ λ be ) + C ab λ ae ˜ λ be + 2 B ab ( λ ae η be + ˜ λ ae ˜ η be ) + C ab ( λ ae ˜ η be + η ae ˜ λ be )+ D abc ( λ ae λ be λ ce + ˜ λ ae ˜ λ be ˜ λ ce ) + E abc ( λ ae λ be ˜ λ ce + ˜ λ ae ˜ λ be λ ce ) + O ( λ ) . (4.71)Supposing the two ends of edge e are u, v , V a is on u side and ˜ V a is on v side. Then˜ φ au = λ ae + ˜ λ ae p − ∆ a + η ae + ˜ η ae p − ∆ a + C abc λ b ˜ λ c p − ∆ c + O ( λ ) , (4.72)˜ φ av = ˜ λ ae + λ ae p − ∆ a + ˜ η ae + η ae p − ∆ a + C abc ˜ λ b λ c p − ∆ c + O ( λ ) . (4.73)23e find that to order O ( λ )1 + p ∆ a c ( p + 1) (1 − p a ) ˜ φ au ˜ φ av + r R abc c ( p + 1) ( ˜ φ au ˜ φ bu ˜ φ cv + ˜ φ av ˜ φ bv ˜ φ cu ) = d e . (4.74)The nonlocal term in T is given by T ♦ xy = − ζ p (2∆ a )2 p ∆ a ˜ φ ax ˜ φ ay + r R abc ( ˜ φ ax ˜ φ bx ˜ φ cy + ˜ φ ay ˜ φ by ˜ φ cx )= p ∆ a − p a ˜ φ ax ˜ φ ay + r R abc ( ˜ φ ax ˜ φ bx ˜ φ cy + ˜ φ ay ˜ φ by ˜ φ cx ) . (4.75)So it turns out that 1 + T ♦ uv c ( p + 1) = d e . (4.76)The edge length is closely related to the nonlocal part of T .We can also introduce some states on the vertices u, v : | u (cid:105) ≡ (cid:88) a N a ( ˜ φ au + ˜ φ bu ˜ φ cu ˜ R bca ) | a (cid:105) , (4.77) | v (cid:105) ≡ (cid:88) a N a ( ˜ φ av + ˜ φ bv ˜ φ cv ˜ R bca ) | a (cid:105) , (4.78)where N a = (cid:115) p ∆ a c ( p + 1)( p a − ) , (4.79)˜ R abc ≡ r R abc (1 − p c ) p ∆ c . (4.80)Then we have 1 − (cid:104) u | v (cid:105) = d e . (4.81)So d e may also be understood as some information metric. In particular, let us consider the semi-classical limit that is of direct relevance in the comparison with the tensor network. One readilyfinds that lim ∆ a →∞ | u (cid:105) = 1 (cid:112) p + 1) (cid:88) a ˜ φ au | a (cid:105) . (4.82)The wavefunction of this state is obtained by sticking a dangling leg to the vertex u of the tensornetwork. The edge distance turns out to be given by the Fisher information distance between twostates each with a leg stuck at one of the two ends of the edge!24 Summary and Discussion
In this paper we study in detail the tensor network proposed in [24] with deformed boundaryconditions. We show that there is virtually a unique way of assigning geometrical data to thetensor network based on its local data so that it satisfies a graph Einstein equation that is self-consistent with the field theory that is known to be encoded in the tensor network. The emergentEinstein tensor also naturally recovers the proposals in the mathematics literature, at least in theperturbative expansion that we are considering [37–39]. This is arguably the first such quantita-tive demonstration of an emergent Einstein equation in a holographic tensor network, albeit in asimplified setting of the p-adic AdS/CFT.There are several interesting observations and future problems that we would like to commentupon. First of all, in the covariant tensor network reconstruction of the CFT partition functionrather than a wavefunction, the same choices of tensors can describe different geometries. Theimportant aspect is the sub-vector space that is explored – which is controlled by the boundaryconditions in the current construction. Near a fixed point vector, only vectors close to the fixed pointvector is actually contributing to the partition function. When a different sub-space is explored,the same tensor network can describe a vastly different geometry. This is in fact very close in spiritwith the gravity path-integral, where different saddle points are merely exploring different vectorsubspaces.Second, the tensor network construction here bears uncanny resemblance to strange correlators[41–43]. PEPS tensor networks produce (minimal) CFT partition function by projecting each ofthe boundary leg to a particular fixed point vector. Here, something almost identical happens. Wecan take the partition function we have constructed as the overlap of a direct product state and aMERA like tensor network state that covers the BT tree. i.e. Z CF T = ( (cid:89) i (cid:104) V f | ) | Ψ (cid:105) , (5.1)where | Ψ (cid:105) is a “state” corresponding to the tensor network covering the BT tree and whose “physicallegs” are the dangling ones at the cutoff surface. We are currently pushing this analogy to constructholographic tensor networks that describe real CFTs. Moreover, we find our tensor network also adiscrete realization of the recent consideration of [44], where the partition function of a generic QFTcan always be thought of as the overlap between a “fixed point” state (which should be some directproduct or gapped state) and another wavefunction in one higher dimensions. By considering RGtransformation of the wavefunctions (which keeps the fixed point state invariant), one generates aholographic radial direction – which is precisely what our BT tree tensor network achieved. Our25onstruction of geometries and Einstein equations therefore could inspire more general constructionsin real CFTs by making better use of this connection.Our tensor network is also known to be equivalent to a Wilson line network with gauge groupgiven by SL (2 , Q p ) [45]. This is parallel to the Chern-Simons formulation of the 3d Einstein-Hilbert gravity [46]. Recently, it is noticed that Wilson line networks provide more direct linkwith complexities. It would be very interesting to translate these results here, which might give analternative way of assigning curvatures to the tensor network.Indeed given our tensor network’s connection with Wilson lines, it inspired an alternative wayof bending the BT tree – by deforming the connection upon which the Wilson line network isevaluated. In the process, one discovers a formulation of the BTZ black hole in the p-adic Wilsonline network that bears profound resemblance to the AdS BTZ black hole, but unique in its way.We have also found a new set of coordinates on the BT tree that is the analogue of black holecoordinates in AdS space and opens the door to more general diffeomorphism transformation onthe p-adic tree. The p-adic CFT, known for its lack of descendants, should still contain intricatestructures and accommodate local conformal transformation. These exciting developments will bediscussed in detail in our sequel to appear together! Acknowledgements.—
LYH acknowledges the support of NSFC (Grant No. 11922502, 11875111)and the Shanghai Municipal Science and Technology Major Project (Shanghai Grant No.2019SHZDZX01),and Perimeter Institute for hospitality as a part of the Emmy Noether Fellowship programme. Partof this work was instigated in KITP during the program qgravity20. LC acknowledges support ofNSFC (Grant No. 12047515). We thank Bartek Czech, Ce Shen, Gabriel Wong, Qifeng Wu andZhengcheng Gu for useful discussions and comments. We thank Si-nong Liu and Jiaqi Lou forcollaboration on related projects.
A Brief review of p-adic CFTs
Let’s begin with the brief introduction of padic number field Q p which is the field extension ofthe rational numbers alternative to the real number R . For a given prime number p , any rationalnumber r ∈ Q can be written as r = p k ab , ( k ∈ Z ) (A.1)where a and b are integers relatively prime to p . And the p-adic norm of r is defined as | r | p = p − k , | | p = 0 . (A.2)26fter the field extension by using the p-adic norm | . . . | p , the resulting number field is called p-adicnumber field Q p . A p-adic number x ∈ Q p is usually expressed as an infinite series x = p v ∞ (cid:88) i =0 a i p i , ( v ∈ Z ) (A.3)where a i ∈ F p = { , , . . . , p − } and a (cid:54) = 0. And its p-adic norm is | x | p = p − v , (A.4)which satisfies various axioms of norms: | x | ≥ , (A.5) | x | = 0 ↔ x = 0 , (A.6) | xy | = | x || y | , (A.7) | x + y | ≤ | x | + | y | . (A.8)Actually the Euclidean norm and the p-adic norm are the only two types of norms that obey thefour axioms above. Other than the real number R , the padic number Q p is the only possible fieldextension of rational number Q .The padic CFT is a field theory living in the p-adic number field Q p , i.e. its coordinates x ∈ Q p .The global conformal symmetry of the padic CFT is defined as the transformation x → x (cid:48) = ax + bcx + d , ( a, b, c, d ∈ Q p ) . (A.9)It furnishes the matrix group PGL(2 , Q p ), the direct analogue of SL(2 , R ) in 1 d conformal trans-formation in real space-time. Similar to the CFT, there are two pieces of algebraic data requiredto specify completely a p-adic CFT: • First the spectrum of primary operators O a with conformal dimensions ∆ a , which transformunder conformal symmetry PGL(2 , Q p ) as O a ( x ) → ˜ O a ( x (cid:48) ) = (cid:12)(cid:12)(cid:12)(cid:12) ad − bc ( cx + d ) (cid:12)(cid:12)(cid:12)(cid:12) − ∆ a p O a ( x ) , (A.10) • Second, OPE coefficients C abc defined as O a ( x ) O b ( x ) = (cid:88) c C abc | x − x | ∆ c − ∆ a − ∆ b p O c ( x ) . (A.11)27he form of (A.11) follows from the fact that the p-adic CFT has no descendant which is a con-sequence of the fact that on padic number field a locally-constant function has zero-derivative. Asusual, there must be a unique identity operator I = O of dimension ∆ = 0 whose OPE coefficientssatisfy C b a = C ba = δ ba . The totally symmetric OPE coefficients C abc ≡ (cid:80) d C abd C cd is most com-monly used. After suitable orthogonalization and normalization, we can always make C ab = δ ab .Then C ab = C a b = C ab = δ ab . And inserting the OPE into a 2-point function implies that (cid:104)O a ( x ) O b ( x ) (cid:105) = δ ab | x − x | a p . (A.12)For the 3-point function we will obtain (cid:104)O a ( x ) O b ( x ) O c ( x ) (cid:105) = C abc | x | ∆ a +∆ b − ∆ c p | x | ∆ b +∆ c − ∆ a p | x | ∆ c +∆ a − ∆ b p . (A.13)When considering the 4-point function, the crossing symmetry will lead to following constraint (cid:88) c C abc C cde = (cid:88) c C bdc C cae , (A.14)which is equivalent to the associativity of the fusion algebra defined by C abc . B Relation between cubic term and quadratic term
To avoid confusion, the summation of a, b, c in (4.8) and (4.50) is understood as (cid:80) a (cid:54) =1 ,b (cid:54) =1 ,c (cid:54) =1 .Then our result is valid for a (cid:54) = 1 , b (cid:54) = 1 , c (cid:54) = 1. Now let’s consider (cid:88) a,b,c h H abc ( ˜ φ ax ˜ φ bx ˜ φ cx + ˜ φ ay ˜ φ by ˜ φ cy ) + (cid:88) a,b,c r R abc ( ˜ φ ax ˜ φ bx ˜ φ cy + ˜ φ ay ˜ φ by ˜ φ cx ) . (B.1) When one of a, b, c is 1, it will contribute (cid:88) a (cid:54) =1 (cid:16) h H aa ( ˜ φ ax ˜ φ ax + ˜ φ ay ˜ φ ay ) + r R aa ( ˜ φ ax ˜ φ ax + ˜ φ ay ˜ φ ay ) + 2 r R aa ( ˜ φ ax ˜ φ ay + ˜ φ ay ˜ φ ax ) (cid:17) . (B.2)Here we have used ˜ φ x = 1 + . . . , C ab = δ ab and the symmetry of H abc , R abc . Plugging (4.63),(4.65)into (B.2), we will get (cid:88) a (cid:54) =1 (cid:18) − p a + p p + 1) ( p a −
1) ( ˜ φ ax ˜ φ ax + ˜ φ ay ˜ φ ay ) + p ∆ a p a − φ ax ˜ φ ay ) (cid:19) , (B.3)which is exactly − (cid:88) a (cid:54) =1 (cid:18) k ζ p (2∆ a )2 p ∆ a ( ˜ φ ax − ˜ φ ay ) + ζ p (2∆ a )2 p ∆ a p + 1 m a (( ˜ φ ax ) + ( ˜ φ ay ) ) (cid:19) = − T ˜ φ . (B.4)Here T ˜ φ is the quadratic term in T as shown in (4.8). It implies that regarding the identity operator I as an elementary field, in the action the quadratic term can be included in the interaction termin a compact way. 28 Some identities in the flow of the boundary conditions
As described in section 2.2, the authors of [24] assume translation invariance along Q p . Following2.12, we also find some interesting relationship between the Fisher metric and the matter field. Letthe cutoff be denoted by Λ, without normalization, | V Λ (cid:105) = ( a Λ + (cid:88) k b ( k )Λ v k ) | (cid:105) (C.1)These are the boundary vectors, and p such vectors will return a new vector sharing the samevertex tensor with them, which is | V Λ − (cid:105) = ( a Λ − + (cid:88) k b ( k )Λ − v k ) | (cid:105) (C.2)in which v k represents some primary field. According to 2.12, we have | V Λ − (cid:105) = ( a Λ + (cid:88) k p − ∆ k b ( k )Λ v k ) p | (cid:105) (C.3) a Λ − = a p Λ + C p (cid:88) i,j N ij a p − b i Λ b j Λ p − ∆ i − ∆ j + C p (cid:88) i,j,k,w C wk C ijw a p − b i Λ b j Λ b k Λ p − ∆ i − ∆ j − ∆ k + . . . (C.4) b ( k )Λ − = p − ∆ k a p − b ( k )Λ + C p (cid:88) i,j C ijk b ( i )Λ b ( j )Λ p − ∆ i − ∆ j + . . . (C.5)And λ k = b ( k )Λ a Λ (C.6)plays the role of a source, which satisfies, λ k << p ∆ k − (C.7)The fisher metric is 1 − |(cid:104) V Λ − n | V Λ − n − (cid:105)| (cid:104) V Λ − n | V Λ − n (cid:105)(cid:104) V Λ − n − | V Λ − n − (cid:105) (C.8)Let n be the steps from the cutoff surface to the layer we are considering, and expand the fishermetric between the nth layer and n+1th layer to first order, the recurrence relation tells us thatthe fisher metric should be1 − ( a Λ − n − a Λ − n + (cid:80) k b ( k )Λ − n − b ( k )Λ − n ) ( a − n + (cid:80) k b ( k )Λ − n )( a − n − + (cid:80) k b ( k )Λ − n − ) = (cid:88) k p n − n ∆ k ( p − ∆ k − b ( k )Λ 2 a Λ2 (C.9)29ow consider the expectation value of the operator inserted into some vertex x which is n stepsfrom the boundary of the tree, to first order it is (cid:104) φ x (cid:105) = (cid:88) r J G ( x, r ) (C.10)First ignore the normalization factor of the bulk-boundary propagator defined in [33], which is C = 1 − p s − k − p − k (C.11)Here s=1.We have φ ( k ) x = [ p n p − n ∆ k + ( p n +1 − p n ) p − n ∆ k − k ) + · · · + ( p n + w − p n + w − ) p − n ∆ k − w ∆ k ] λ k (C.12)Similarly, φ ( k ) y = [ p n +1 − ( n +1)∆ k (1 − p − k ) 1 − p ( w − − w − k − p − k + p n + w p − ( n +1)∆ k − w − k ] λ k (C.13)y is the point just below x, and w represents the distance between x and the IR limit, whichgoes to infinity. 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