Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorphism on the Bruhat-Tits Tree
BBending the Bruhat-Tits Tree IIthe p-adic BTZ Black hole and Local Diffeomorephismon the Bruhat-Tits Tree
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
In this sequel to [1], we take up a second approach in bending the Bruhat-Tits tree. Inspiredby the BTZ black hole connection, we demonstrate that one can transplant it to the Bruhat-Titstree, at the cost of defining a novel “exponential function” on the p-adic numbers that is hintedby the BT tree. We demonstrate that the PGL(2 , Q p ) Wilson lines [2] evaluated on this analogueBTZ connection is indeed consistent with correlation functions of a CFT at finite temperatures.We demonstrate that these results match up with the tensor network reconstruction of the p-adic AdS/CFT with a different cutoff surface at the asymptotic boundary, and give explicitcoordinate transformations that relate the analogue p-adic BTZ background and the “pure”Bruhat-Tits tree background. This is an interesting demonstration that despite the purportedlack of descendents in p-adic CFTs, there exists non-trivial local Weyl transformations in theCFT corresponding to diffeomorphism in the Bruhat-Tits tree. ∗ Chen and Liu are co- first authors of the manuscript. a r X i v : . [ h e p - t h ] F e b ontents Q p “plane” and its three dimensional bulk dual ? . . . . . . . . . . . . . 52.2 BTZ black hole and the Black Hole coordinates . . . . . . . . . . . . . . . . . . . . . 72.2.1 An alternative p-adic “logarithm” and “exponential” . . . . . . . . . . . . . . 92.2.2 A black hole coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Local diffeomorphism connecting black hole and Poincare coordinates . . . . . . . . 122.4 Correlation functions in the BTZ background . . . . . . . . . . . . . . . . . . . . . . 142.5 Weyl transformation in the p-adic CFT . . . . . . . . . . . . . . . . . . . . . . . . . 16 connection vs Bruhat-Tits connection . . . . . . . . . . . . . . . . . 193.2 Transplanting the BTZ connection to the Bruhat-Tits tree . . . . . . . . . . . . . . . 213.3 Evaluating the Wilson line at fixed representations . . . . . . . . . . . . . . . . . . . 223.4 Open Wilson line expectation values in p-adic black hole . . . . . . . . . . . . . . . . 23 p-adic AdS/CFT as a toy version of the AdS/CFT was proposed in [3, 4]. An n -dimensionalCFT lives on a number field Q p n , and the purported dual bulk geometry whose isometry is theconformal group of the CFT SL (2 , Q p n ) is a discrete graph called the Bruhat-Tits (BT) tree. TheBT tree is thus the analogue of the pure AdS space. As in the usual AdS/CFT dictionary wherecomputing correlation functions of a small number of operator insertions can be done by consideringquantum field theories in the pure AdS background without worrying about back-reaction to thegeometry, the p-adic AdS/CFT works in the same way. Witten diagrams on the BT tree have beenstudied extensively [3, 5–10], and the results satisfy all the expected features of the p-adic CFT [11].2 natural question would be to ask whether one can include dynamics of the background so that thebulk theory becomes an analogue of gravitational theory on AdS. Some attempts have been madein [12], where each edge of the graph is assigned a dynamical length and that an action is proposedthat governs the dynamics of the edge lengths. The action is the analogue of the Einstein Hilbertaction, given by the sum of local graph curvatures. The matter action was also covariantized bycoupling the matter fields to these edge lengths.This actually poses a puzzle. It is well known that the p-adic CFT does not accommodate de-scendants. This does not mean that the representations of the conformal group is finite dimensional– in fact it is known in the Mathematics literature that all finite dimensional representations of thep-adic conformal group SL (2 , Q p ) is in fact trivial (i.e. one dimensional). See for example [13]. It isjust that Lie algebra is not well defined because the exponential map could take an exponent withfinite p-adic norm to infinity. Moreover, functions taking p-adic numbers to the reals are locallyconstant. Therefore usual derivatives of correlation functions vanish. It would thus appear that thestress tensor is not well-defined in the p-adic CFTs [11]. The stress tensor is however the centralingredient in the AdS/CFT correspondence that is identified with the bulk metric. Therefore it isnot entirely clear what the edge dynamics correspond to from the perspective of the p-adic CFT.In our prequel [1], the tensor network reconstruction of the p-adic CFT partition function isshown to naturally recover a bulk action and a covariantized matter action. The edge lengths did notshow up as an independent set of operators in the CFT, but rather as a Fisher information metricbetween states with various non-local operator insertion. This is not in any contradiction withknown AdS/CFT dictionary, since the Ryu-Takayanagi formula would also imply that distancesin the bulk are related to entanglement of the boundary CFT [14]. Yet, for technical reasons weworked in the perturbative limit away from the pure BT background. It would be interesting tounderstand how more exotic and non-perturbative backgrounds such as black holes can be describedin the BT bulk. Moreover we have not yet resolved the mystery about descendents, and relatedly,whether local conformal transformation and Weyl transformations are well defined in the CFT, andif they do, how they manifest themselves in the BT bulk.In this paper, we revisit the problem of the BTZ black hole in the BT background. It hasbeen noted that similar to the AdS case the BT tree can be orbifolded to produce structures thatcarry higher genus and they appear to be analogues of the handle-bodies studied in the AdS case.In particular, one can generate genus curves that appear to be analogues of the BTZ black holes[4, 13, 15]. We were initially inspired by the tensor network formulation that was shown to bereproduced by a Wilson line network in a purported SL (2 , Q p ) Chern-Simons gauge theory [2]. Welooked into Wilson lines evaluated on flat connections that mimic results obtained from the BTZ3lack hole formulated in terms of SL (2 , C ) connections. The study however led us to observationsthat apply generally, independently of the tensor network construction. Our presentation willhowever start with generalities before zooming in on the Wilson line interpretation. What we wouldlike to demonstrate in this paper is that the p-adic version of the BTZ black hole actually carriesa way more intimate semblance with the AdS sister. There exists a set of natural coordinates onthe BTZ black hole that is the analogue of black hole coordinates in AdS . Moreover, there existsa non-trivial coordinate transformation that connects the black hole coordinates with the analogue Poincare coordinates advocated in [3]. To do so, a novel notion of “exponential” is introduced.This would be the first example of a local Weyl transformation defined in the p-adic CFT that isexplicitly demonstrated to be equivalent to bulk diffeomorphism in the BT tree. These results aredescribed in section 2.Then we would describe how the p-adic Chern-Simons formulation admits also the analogueof the BTZ SL (2 , Q p ) connection, and that correlation functions would be correctly reproducedagain using the Wilson line network that is again in complete agreement with the tensor networkcovering the BT tree now with a different cutoff surface – now the constant ” r ” surface in black holecoordinates. Moreover, with the explicit form of the gauge connection written down, we are ableto make comparison with AdS case and identify concretely the black hole horizon, temperatureand black hole entropy of the BTZ black hole. These results are presented in section 3 and 4.We will conclude in section 5. In this section, we will describe a novel coordinates system that is most natural for the BT treewhen it is arranged to describe the analogue BTZ black hole. We will also describe how this blackhole coordinates can be related to the usual Poincare coordinates via a coordinate transformationanalogous to the corresponding transformation in AdS [16]. There is a full set of coordinates on the BT tree used in the mathematics literature. This isreviewed in detail in [2, 17] that would allow the action of the isometry on the tree completelyexplicit. This is the analogue of the Poincare coordinates. This coordinate system was taken andsomewhat simplified in [3] that hides certain detail. We will briefly review it here. Recall that theasymptotic boundary of the tree is the Q p line, and every boundary point has a unique path that4onnects it to a main branch, which encodes the p-adic number. There is a horizontal coordinate x ( n +1) ∈ Q p which is the analogue of x in AdS poincare coordinates. The “radial coordinate” p n +1 encodes the “accuracy” of x ( n +1) . i.e. x ( n +1) is defined up to identification by adding to it anarbitrary y with norm | y | p ≤ p − ( n +1) . We might as well set all coefficients of p l +1 , l ≥ n in x ( n +1) to be zero. Then the coordinates ( p n +1 , x ( n +1) ) would take the following form x ( n +1) = p v n − v (cid:88) i =0 a i p i (2.1)as shown in Fig. 1. It may appear somewhat unusual that the x and radial coordinate p n appearsFigure 1: The coordinates on the BT tree.somewhat intertwined. Also, the fact that it has two labels would make it appear very differentfrom its AdS sister.Let us massage this coordinate to make it look more familiar. Q p “plane” and its three dimensional bulk dual ? As we have seen, the coordinates in [3] reviewed above involve two parameters – the radialcoordinate p n and a p-adic number x ( n ) . It is thus often regarded as a two dimensional space, whileits boundary (the p-adic number) is regarded as a one dimensional Q p line. Here, we would like toarrange the coordinates so that the BT tree looks more analogous to AdS , and its boundary, tothe complex plane C .A p-adic number is written as x = p v ∞ (cid:88) i =0 a i p i , (2.2)5ith a i ∈ { , . . . , p − } and a (cid:54) = 0. And we have | x | p = | p v | p , (2.3) | ∞ (cid:88) i =0 a i p i | p = 1 . (2.4)If we consider the complex plane, the holomorphic complex coordinate ζ can be written as ζ ≡ ρe iθ , (2.5)with | ζ | R = | ρ | R , (2.6) | e iθ | R = 1 . (2.7)So p v plays the role of ρ , while (cid:80) ∞ i =0 a i p i plays the role of e iθ . Then the boundary of the BT treecan be regarded as a 2 d plane with 0 sitting on the origin, | x | p characterizing the distance from x to origin, and x with the same norm sitting on the same circle centered on the origin as shown inFig. 2. Since the boundary is a 2 d plane, the BT tree itself is a 3 d object.Figure 2: The boundary of the BT tree can be viewed as a 2 d plane. The boundary points of asame color brunch have the same norm and are on the same circle.The p-adic Poincare coordinates can thus be taken as( z, ρ, e iθ ) ≡ ( p n +1 , p v , n − v (cid:88) i =0 a i p i ) , (2.8)where we have artificially taken out the unit norm piece in x ( n +1) and take it as a third coordinate.This now looks more similar to the AdS picture.6 .2 BTZ black hole and the Black Hole coordinates In AdS , it is well known that the BTZ black hole can be understood as an orbifold of AdS byan appropriate Schottky group [16, 18], since higher genus Riemann surfaces can be constructedas quotients of the projective complex plane P ( C ). One can replace the complex numbers C bythe p-adic numbers Q p . By orbifolding the Bruhat-Tits tree by a PGL(2 , Q p ) subgroup S with n generators, one generates a genus n graph. The n = 1 case would be the closest cousin of the BTZblack hole. This was considered in [4, 13], and also in [15] where we obtained the partition functionof a p-adic CFT by covering the genus one graph with the tensor network that we have proposed.In the case of a genus one graph, it is generated by quotienting the tree by a singly generatedsubgoup of PGL(2 , Q p ). This subgroup actually has a very simple action on the tree. Considerpicking a branch of the tree (such as the main-branch as shown in Fig. 3. ). The BT tree hastranslation invariance along this branch (when there is no cutoff surface). Each singly generatedsubgroup corresponds to translation on the chosen branch. The quotient is carried out by identifyingevery t steps, and t would play the role of a moduli of the resultant genus 1 curve. The resultantgraph contains a loop and it is shown in Fig. 3. That closed loop has the same flavour as a horizonin the Euclidean BTZ black hole. As we are going to see the parallels run far deeper.Figure 3: Orbifold of the BT tree (at p = 2) that converts it into a genus 1 graph. Here theperiodicity t =3.An important point is that as soon as we consider any practical calculations, it is necessary tointroduce a cutoff surface. The natural cutoff surface in this case appears to be one that is parallelto the horizon. This cutoff surface is thus very different from what is customarily used in say [3].We will call the coordinates there the Poincare coordinates and those cutoff surface a constant z cutoff surface.Here we would like to assign a different set of coordinates so that the horizon and lines parallelto the horizon have constant radial coordinate. 7o proceed, it is convenient to consider the case where the periodicity of the loop approachesinfinity. Without the cutoff surface, this would be the original BT tree, but now we are moti-vated to consider coordinates whose constant radial “surfaces” are parallel to the horizon. Thisuncompactified tree is shown in Fig. 4(b). (a) (b) Figure 4: (a): The p-adic BTZ BH, (b): The cylinder type p-adic BH. The red curve is the horizon.In the case of the usual BTZ black hole, when we uncompactify the spatial circle, the asymptoticboundary is a cylinder. Here, we will also refer to the uncompactified black hole as cylinder type.The cylinder type BH has translation invariance along its horizon. At every vertex along thehorizon an identical subtree emanates from the vertex. Our black hole coordinates are chosen asfollows. We label every vertex on the horizon by an integer k . Supposing each branch emanatingfrom each vertex is a semi-infinite tree if not for the cut-off surface. Here, consider that it is cutfrom the BT tree at ( p h , p h + l , y ( h + l ) ) = ( p h + l , p h l − (cid:88) i =0 a i p i ) . ( a (cid:54) = 0 , l ≥
1) (2.9)Then the coordinates on the cylinder type BH is given by ( k, p h + l , y ( h + l ) ) as shown in Fig. 5.i.e. the second and third entries came from Poincare labels originally assigned to that branch.This same branch is repeated an infinite number of times on the horizon and they are distinguishedby the label k . See Fig. 5, where the branch colored green on the horizon can be thought of asbeing taken from the BT tree with Poincare labels. This gives a unique label to every vertex onthe tree.One must have noticed that a parameter h has been introduced. As we are going to show whenwe compute correlation functions, that h controls the temperature of the p-adic CFT. We will8igure 5: The coordinates on the cylinder type BH. Each branch of the BH is taken from the BTtree emanating from the vertex labeled ( p h ,
0) in Poincare coordinates, which is colored green inthe figure. The coordinates of v is ( k, p h +2 , y ( h +2) ).give further support to this claim when we consider the Chern-Simons formulation of the tensornetwork, and show that h would appear precisely as a moduli in the connection that plays the samerole of parametrising the horizon radius.These labels do not look similar to the black hole coordinates on AdS yet. To make themlook more familiar, we need to introduce a function that we call Θ. It is essentially the analogueof logarithm that takes a p-adic number of unit norm to another p-adic number that is howeverperiodically identified. Before proceeding to give the precise definition of the black hole coordinates, we were inspiredby the BT tree itself, which inspired us to consider an alternative notion of “logorithm” for p-adicnumbers that we denote as Θ. From which it is also going to inspire an alternative notion of“exponential”.
An alternative “logarithm”
For the usual logarithm that acts on complex numbers, Θ takes a pure phase y with norm | y | R = 1 to a number θ that is periodically identified every 2 π (alternatively we call that a branch-cut): Θ : y = e iθ → θ, (2.10)satisfying Θ( y ) − Θ( y ) = Θ ( y /y ) , (2.11)9ince e i ( θ − θ ) = e iθ e iθ . (2.12)Now analogously, let’s define the p-adic version of the logarithm Θ as a map from a p-adicnumber y with norm | y | p = 1 to a periodic number θ :Θ : y = ∞ (cid:88) i =0 a i p i → θ, ( a (cid:54) = 0) (2.13)satisfying Θ( y ) − Θ( y ) = Θ ( y /y ) . (2.14)Here we consider a simple example to illustrate what Θ does. Let’s choose p = 5 and truncate y so that we keep only the leading term y = a p + O ( p ). Then y ∈ { , , , } . We find that1 , , , ÷ −−→ , , , , (2.15)1 , , , ÷ −−→ , , , , (2.16)1 , , , ÷ −−→ , , , , (2.17)1 , , , ÷ −−→ , , , . (2.18)Note that the division computed above are p-adic divisions where the result has to be arranged interms of the expansion (2.2) and then from which one extracts a .To satisfy (2.14), we can chooseΘ(1) = 0 , Θ(2) = π , Θ(3) = 3 π , Θ(4) = π, (2.19)with the period 2 π as shown in Fig. 6.Figure 6: The Θ( i ) on the circle.10or a general p and any truncation of y , there always exists a map Θ satisfying (2.14), thoughthe explicit expression of Θ hasn’t been worked out.And we can further define a map e p as the inverse of Θ, i.e. e p (Θ( y )) = y. (2.20)This function is almost an exponential. Now we would like to introduce a notion of the expo-nential that acts on an arbitrary p-adic number. An alternative “exponential”
Now we can define the p-adic exponential map E p as E p : w = ( φ, τ ) → p φ e p ( τ ) . (2.21)where φ is an integer, and τ a periodic number in the image of Θ. The calculation rules of w aresimply ( φ , τ ) − ( φ , τ ) = ( φ − φ , τ − τ ) , (2.22) a ( φ, τ ) = ( aφ, aτ ) . (2.23)The map E p has following properties: E p ((0 , , (2.24) E p ( w ) E p ( w ) = p φ p φ e p (Θ( y )) e p (Θ( y )) (2.25)= p φ + φ e p (Θ( y y )) (2.26)= p φ + φ e p (Θ( y ) + Θ( y )) (2.27)= E p ( w + w ) , (2.28)where we have used (2.14),(2.20).We further define sinh p , cosh p assinh p ( w ) ≡ E p ( w ) − E p ( − w )2 , (2.29)cosh p ( w ) ≡ E p ( w ) + E p ( − w )2 . (2.30)And we have cosh p ( w ) − sinh p ( w ) = 1 . (2.31)Staring at the requirements (2.12, 2.11, 2.24, 2.28) one would have been reminded of solutions ofcharacters of representations in the principal series of PGL(2 , Q p ). We note that one very important11ifference is that the current map takes p-adic numbers to an ordered series where we have borrowedlabels from the real line 2 π/k to rename the results and to act as a mnemonic of the number ofsteps i.e. Θ( y ) − Θ( y ) between the images in the ordered series. This is vastly different from theusual exponential whose action is defined based on its power series expansion. Note in particularthat there is no divergence here that otherwise plagued the usual exponential (or logarithm for thatmatter). As we are going to see, these definitions are inspired by the computation of correlationfunctions that first follows from the BT tree with these different choices of cut-off surface. In the previous sub-section, we argued that the p-adic tree can be naturally treated as a 3dobject that mimics AdS closely. In the case of the cylinder type BH one can also arrange theBruhat-Tits tree into a cylinder like object as shown in Fig. 7. Recall that we introduced analternative coordinates to label vertices on the cylinder.( k, p h + l , y ( h + l ) ) = ( k, p h + l , p h l − (cid:88) i =0 a i p i ) . ( a (cid:54) = 0 , l ≥
1) (2.32)The black hole coordinates we would like to introduce that makes use of the logarithm introducedin the previous sub-section is related to the above as follows( r, φ, τ ) = ( p h + l , p h k, p h Θ( l − (cid:88) i =0 a i p i )) . (2.33)Here r is the radial coordinate, and φ, τ are coordinates on the cylinder with τ the compactifieddirection as shown in Fig. 7. The coordinate transformation between black hole coordinate (2.33) and Poincare coordinate(2.8) is simply given by θ = τ /p h , (2.34) ρ = p φ/p h , (2.35) x ≡ ρe p ( θ ) = p φ/p h e p ( τ /p h ) , (2.36) z = rp φ/p h /p h . (2.37)Here we have used l − n − v , and chosen k = v . Note that θ is indeed always in the imageof the function Θ from the definition (2.33). Let’s introduce w = ( φ, τ ), and the p-adic black hole12igure 7: The cylinder type BH looks like a cylinder ( p = 3). The red line is its horizon. The rightmost picture shows the cross-section of the cylinder.coordinate can be written as ( r, w ). Using the definition of the “exponential” defined in (2.21), wecan write x = E p ( wp h ) . (2.38)To illustrate the coordinate transformations more clearly, see Fig. 8.Figure 8: A marked vertex in Black Hole (in black) and Poincare (in red) coordinates. The redarrow gives the constant Poincare radial coordinate z surface. The horizon here is the ”mainbranch” in Poincare coordinates and on which marks the radial z values.This coordinate transformation can be contrasted with the transformation between Poincare13oordinates and the BTZ coordinates [16].There, we have ζ = (cid:115) r − r h r exp ( r h l φ + iτ ) (2.39) z = rr h exp ( r h l φ ) , (2.40)where here we have already set the angular momentum of the BTZ black hole to zero keeping onlythe outer-horizon, which is the situation analogous to the cylinder type Bruhat Tits tree black hole.The metrics corresponding to these coordinates take the form ds P oincare = l dz z + z dωd ¯ ω, (2.41) ds BH = ( r − r h ) r dτ + r l r − r h dr + r dφ . (2.42)Apart from the choice of norms of ζ being somewhat different from (2.35), the relations arestep by step identical. Moreover, this distinction in the choice of normalization along the ”CFT”directions disappears towards the asymptotic boundary r → ∞ , where the extra factor of r reducesto unity. This means that the interpretation as far as the CFT is concerned would be identical.In the following we will show that, indeed, the form of the correlation functions following from thebulk computations with the black hole cut-off surface can be interpreted as a finite temperaturecorrelator in analogy to the usual CFT on a cylinder. Having assigned a new black hole coordinate to the Bruhat Tits tree and pick a correspondingcutoff surface parallel to the purported horizon, we can now compute boundary CFT correlationfunctions using the bulk. Here, we will compute the CFT correlation functions using the tensornetwork we have proposed in [15]. A very brief review of p-adic CFT and our tensor network canbe found also in the first of this series [1, 19].
Faster than lightening review of the tensor network
Here, all we will need in the following is that we are covering the BT tree with a tensor network,with every vertex hosting a tensor and each edge emanating from a vertex is an index of the vertextensor. Being a p + 1-valent tree, each tensor also has p + 1 indices. It is explicitly given by 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 , (2.43)14here C abc are OPE coefficients of the boundary CFT, with a collection of primary fields labeledby { a, b, c · · · } and corresponding conformal dimensions { ∆ a , ∆ b , ∆ c · · · } . Two vertices connectedby an edge denote a contracted index between two vertex tensors. The contraction is a weightedsum, with each term with label a weighted by p − ∆ a . The CFT partition function is defined as theevaluation of the tensor network covering the BT tree, such that the un-contracted dangling legsat the cutoff surface is projected to the identity label 1, the unique identity operator with zeroconformal dimension.Each operator O a inserted at point x would then correspond to projecting the leg x to label a .The evaluation of the tensor network with appropriate boundary conditions thus returns for us thecorrelation functions of the CFT.One can also define bulk operator φ a inserted at a bulk vertex v . This is achieved by fusinga label a to the fusion tree at v defined by the OPE coefficients C abc . For further details one canrefer to volume one of our current series [1]. Correlation functions
Now we are ready to compute correlation functions. Let us remark that while the result hereis based on the tensor network, the form of correlation functions up to three point functions arecompletely universal.Consider two points v = ( r , φ , τ ) and v = ( r , φ , τ ) in black hole coordinates. Their twopoint function is given by: (cid:104) φ a ( v ) φ a ( v ) (cid:105) = p − ∆ a d ( v ,v ) , (2.44)with d ( v , v ) the distance between v , v . If we would like to compute correlation of the p-adicCFT, we should consider putting these operators on the cutoff surface at constant r , and thenfinally pushing r to infinity. (cid:104) O a ( φ , τ ) O a ( φ , τ ) (cid:105) ≡ lim r →∞ r a (cid:104) φ a ( v ) φ a ( v ) (cid:105) , (2.45)where ( φ , τ ) , ( φ , τ ) are the boundary points in black hole coordinate. This is illustrated in Fig.9 when r = r . The factors of r are normalizations that take away the divergence in the geodesicdistance as one pushes the cutoff surface to infinity. This is analogous to the usual practice inAdS/CFT, and has been discussed in detail before in [15].When the dust settles, we find (cid:104) O a ( w ) O a ( w ) (cid:105) = 1 | p h sinh p ( w − w p h ) | a p . (2.46)The result now appears uncannily similar to the correlation functions of real CFTs on a cylinder.15igure 9: Computing 2 point functions in the cylinder type black hole background.In fact it is in the process of understanding this result that had inspired the definition of the“exponential”.It may appear somewhat mysterious how (2.45) reduces to (2.46). Let us make it explicit byinspecting its relationship with correlation functions computed in Poincare coordinates (where theasymptotic boundary is “plane like” as opposed to “cylinder like”). Recall that in Poincare coordinate, we have ˜ v = ( z , x ) and ˜ v = ( z , x ). Recall that inPoincare coordinate, we have [3] (cid:104) φ a (˜ v ) φ a (˜ v ) (cid:105) = p − ∆ a d (˜ v , ˜ v ) = z − ∆ a z − ∆ a | x − x | a p , (2.47) (cid:104) ˜ O a ( x ) ˜ O a ( x ) (cid:105) ≡ lim z i →∞ z ∆ a z ∆ a (cid:104) φ a (˜ v ) φ a (˜ v ) (cid:105) = 1 | x − x | a p , (2.48)with x , x the boundary points in Poincare coordinate. We have denoted operators inserted alongthe constant radial cutoff surface in Poincare coordinates by ˜ O a , with an extra tilde, to distinguishthem from operators inserted at constant r surfaces in the black hole coordinates. It is evidentthat d ( v , v ) = d (˜ v , ˜ v ). Now when we want to compute correlation functions in the boundary”cylinder frame”, we have to insert operators at the cutoff surface r as opposed to constant z surface. Therefore we need to “rectify” the difference in the geodesic distance when we push theoperators at the constant z surface to the constant r surface. This is illustrated in Fig 9. Amoment’s thought suggests that the rectification corresponds to change in the normalizations, andwe have (cid:104) O a ( φ , τ ) O a ( φ , τ ) (cid:105) = r ∆ a r ∆ a z − ∆ a z − ∆ a (cid:104) ˜ O a ( x ) ˜ O a ( x ) (cid:105) = p h ∆ a | x | ∆ a p | x | ∆ a p | x − x | a p , (2.49)16here we have used (2.36),(2.37). Using the w coordinate and x = E p ( w/p h ), (2.49) becomes (cid:104) O a ( w ) O a ( w ) (cid:105) = | p − h | a p |E p ( w /p h ) | ∆ a p |E p ( w /p h ) | ∆ a p |E p ( w /p h ) − E p ( w /p h ) | a p = 1 | p h sinh p ( w − w p h ) | a p . (2.50)Using (2.50),(2.45), we note also that the bulk two point function (2.44) can also be expressedas (cid:104) φ a ( v ) φ a ( v ) (cid:105) = r − ∆ a r − ∆ a | p h sinh p ( w − w p h ) | a p = p − (2 h + l + l )∆ a | p h sinh p ( w − w p h ) | a p . (2.51)These results are very suggestive that the black hole frame is describing a finite temperatureboundary p-adic CFT. This can be made most transparent when we compare with correlationfunctions of real 2d CFT at temperature T ≡ π/β . This is equivalent to evaluating correlationfunctions on a cylinder coordinate. We suggestively take the holomorphic complex coordinate onthe cylinder also as w . The planar holomorphic complex coordinate ζ is related to the cylindercoordinate w by ζ = e πw/β . Correlation functions on the cylinder are then related to correlationfunctions on the plane by (cid:104) O ( w ) O ( w ) (cid:105) = | z (cid:48) | ∆ | z (cid:48) | ∆ (cid:104) ˜ O ( z ) ˜ O ( z ) (cid:105) = | ( e πw /β ) (cid:48) | ∆ | ( e πw /β ) (cid:48) | ∆ | e πw /β − e πw /β | = | πβ | | e πw /β | ∆ | e πw /β | ∆ | e πw /β − e πw /β | = 1 (cid:12)(cid:12)(cid:12) βπ sinh (cid:16) π ( w − w ) β (cid:17)(cid:12)(cid:12)(cid:12) . (2.52)with | . . . | the real norm | . . . | R . Comparing (2.50) and (2.52), we findexp( . . . ) → E p ( . . . ) , (2.53)2 πβ → T = p − h , (2.54) | . . . | R → | . . . | p . (2.55)Note that we denote the p-adic temperature which is a p-adic number, by T .Now we have more confidence to regard E p as the appropriate exponential function here. InCFT , ζ = e πw/β maps a CFT with temperature | π/β | R to a zero temperature CFT, while x = E p ( w/p h ) maps a p-adic CFT with temperature | p − h | p to a zero temperature p-adic CFT.17he extra parameter that is introduced along with the black hole coordinates in (2.33) acquires ameaning of a temperature! We will comment further about black hole thermodynamics after ourdiscussion of the Wilson lines.Now we are ready to connect correlation functions on the “planar frame” with correlationfunctions on the “finite temperature cylinder frame” more generally. As discussed above, theyare related by the “correction” that one has to make because of the change in the cutoff surface.Therefore, each operator insertion would be corrected by the same set of factors | z (cid:48) i | ∆ as discussedabove. We can thus summarise the general rule of transformations of correlation functions undergeneral Weyl transformation under coordinate transformation x → f p ( x ): O ( x ) = | f Jp ( x ) | ∆ p ˜ O ( f ( x )) , (2.56)where f Jp is fully determined by f p , with the caveat that its general expression for arbitrary cutoffis yet to be found. Similar to e ( w ) ≡ e w satisfying e J ( aw ) = a e ( aw ), our E p satisfies E Jp ( aw ) = a E p ( aw ) , (2.57)which can be obtained by observing (2.50).This is completely parallel to the story in usual two dimensional CFTs, where under Weyltransformation f ( w ), a primary operator O with conformal dimension ∆ transforms as O ( w ) = | f J ( w ) | ∆ ˜ O ( f ( w )) , (2.58)with f J ( w ) ≡ ∂f /∂w . In [2], we showed that it is possible to formulate a Chern-Simons like theory on the Bruhat-Titstree. While it is not immediately obvious what is the appropriate action for a topological gaugetheory on a tree graph, it is quite apparent that if it existed, one would also impose gauge invarianceat every vertex where Wilson lines meet. While the tree graph has no loops and therefore it isnot clear how to define generic “flat connections”, a pure gauge connection that can be writtenas g ( v ) g − ( v ) for some gauge group valued scalar function g ( v ) would certainly qualify for beingflat.Taking these two assumptions, we constructed a Wilson line network of a purported PGL(2 , Q p )gauge group with a connection that can be written as a pure gauge [2]. The form of the flat18onnection ended up looking completely parallel to the SL(2 , C ) flat connection that follows fromthe Euclidean AdS metric . Moreover the Wilson line junctions evaluated in this flat connectioncoincides with the tensor network that we constructed to recover the p-adic partition function.It is well known that 2d CFT states with finite energy density and momenta are dual to BTZblack holes whose metrics can be expressed as flat connections of SL(2 , C ) in the Chern-Simonsformulation. It would immediately suggest that the BTZ connection perhaps can be transplantedto the p-adic Chern-Simons theory as well.This is indeed possible – which is the main subject of the section. In fact it is this computationthat inspired us to look for a different set of coordinates so that the parallel between the p-adicAdS/CFT and AdS /CFT is more complete. connection vs Bruhat-Tits connection To introduce the p-adic BTZ connection it is helpful to start with the EAdS BTZ black holeexpressed as a flat connection in Chern-Simons gauge theory, a classic story instigated in [20]. Inthis paper we will make use of the notations and expressions set up in [21].For a generic asymptotically AdS spacetime, the (Euclidean) metric can be expressed as ds = − Gl ( L ( ζ ) dζ + ¯ L (¯ ζ ) d ¯ ζ ) + l dr + ( l e r + 16 G L ( ζ ) ¯ L (¯ ζ ) e − r ) dζd ¯ ζ. (3.1)Here ζ is related to the cylinder coordinates ( φ, τ ) by ζ = φ + iτ . The vierbein e aµ satisfying g µν = e aµ e bν δ ab together with its derived spin connection ω abµ can be expressed as a SL(2 , C ) Chern-Simons connection which takes the form A = (cid:32) − dr e r dζ Gl L ( ζ ) e − r dζ dr (cid:33) . (3.2)So we have A ζ = (cid:32) e r Gl L ( ζ ) e − r (cid:33) , A r = (cid:32) − (cid:33) . (3.3)A finite Wilson line stretching from v = ( r , ζ ) to v = ( r , ζ ) is given by W ( v → v ) = P exp (cid:18)(cid:90) v v A µ ( ξ ) dξ µ (cid:19) = P exp (cid:18)(cid:90) ζ ζ A ζ ( r , ζ ) dζ (cid:19) · P exp (cid:18)(cid:90) r r A r ( r, ζ ) dr (cid:19) . (3.4) In the Lorentzian case the gauge group is SL(2 , R ) × SL(2 , R ). We have converted the SL (2 , R ) × SL (2 , R ) connection into an SL(2 , C ) connection. We presented the “holomor-phic” part of the connection which is given as in (3.2). L ( ζ ) = 0, the connection describes pure AdS space. It is known that thisconnection takes a simple form A ( r, ζ ) = g − dg, g ∈ SL (2 , C ) , g ( r, ζ ) = (cid:32) e − r ζ (cid:33) (3.5)Then (3.4) evaluates to W ( v → v ) | L ( ζ )=0 = (cid:32) e − δr e r δζ (cid:33) e δr/ . (3.6)where k = l/ G , δζ = ζ − ζ , δr = r − r . And we set k = 1 for convenience.The “pure” BT tree connection was obtained through a series of guess. But when the dustsettles, it can be very simply summarised most transparently as follows. We can define an analogousPGL(2 , Q p ) valued scalar function g ( v ), defined at each vertex, such that g ( v ) = (cid:32) p n x ( n ) (cid:33) , (3.7)where ( p n , x ( n ) ) is the coordinate of the vertex v . Then a Wilson line connecting two vertices v , v is given by W padic ( v → v ) = g ( v ) − g ( v ) . (3.8)It evaluates to W padic ( v → v ) = (cid:32) p n − n p − n ( x ( n )2 − x ( n )1 )0 1 (cid:33) , (3.9)with v = ( p n , x ( n )1 ) , (3.10) v = ( p n , x ( n )2 ) . (3.11)This result would agree with the AdS Wilson lines given in (3.6) if we blindly make the replace-ment e − r ↔ p n , ζ ↔ x ( n ) , W ( v → v ) ↔ W p ( v → v ) (3.12)The expectation value of a Wilson line p + 1 point junction in this connection evaluated atrepresentations corresponding to primaries of the p-adic CFT agrees precisely with the tensornetwork we proposed in [15] that is mentioned in the previous section. We would like to see if sucha correspondence continues to hold when we consider the BTZ connection.20 .2 Transplanting the BTZ connection to the Bruhat-Tits tree Consider the AdS metric (3.1) where L ( ζ ) does not vanish. Consider the specific case where L ( ζ ) is a constant, which reduces to the metric of the BTZ black hole. We can obtain the SL(2 , C )connection that follows from this metric, and subsequently a Wilson line connecting two points v , v as follows: W ( v → v ) = e − δr cosh (cid:16) δζ √ L √ k (cid:17) e r √ k sinh (cid:16) δζ √ L √ k (cid:17) √ Le − r √ L sinh (cid:16) δζ √ L √ k (cid:17) √ k cosh (cid:16) δζ √ L √ k (cid:17) e δr/ , (3.13)Of course (3.1) corresponds to pure AdS space for all L ( ζ ). On the other hand they would berelated to the standard Poincare frame differently as we pick different L ( ζ ). Locally, the coordinatescan be taken as r, ζ for all these different frames.Equipped with (3.12), we are now ready to transplant the BTZ black hole connection to theBruhat-Tits tree. As in the AdS case, the local coordinates describe a particular branch k inequation (2.32), and looks like ( p h + l , y ( h + l ) ). They are related to ( r, φ, τ ) by (2.33). We nowtransplant (3.13) and treat it as a PGL(2 , Q p ) Wilson lines, and replace the coordinates by localcoordinates on the Bruhat-Tits tree. This gives W padic ( v → v ) = p l − l cosh p (cid:16) δx √ L (cid:17) p − h − l √ L − sinh p (cid:16) δx √ L (cid:17) p h + l √ L sinh p (cid:16) δx √ L (cid:17) cosh p (cid:16) δx √ L (cid:17) , (3.14)where v = ( r , x ) = ( p h + l , x ) ,v = ( r , x ) = ( p h + l , x ) ,δx = x − x . (3.15)There is a mystery here. Exponentials appear in the connection. As we have emphasisedmany times in the previous section, exponentials of p-adic numbers with finite norms could leadto divergence. As we are going to show in the next sub-section, if we take these exponentials andsubsequently sinh and cosh to be those defined in (2.29),(2.30), we will recover complete agreementwith the result of the tensor network.One might wonder – what happens had we insisted upon using the usual exponential in thecomputation of expectations of Wilson line networks? It turns out that it would lead to the same result unless the Wilson lines touches the horizon, at which point all the network vanishes! Theblack hole horizon would appear so dark that branches emanating from the horizon essentially21ecomes disconnected. The deformed definition preserves most of the results except to connect thebranches together at the horizon exactly like what happens in the tensor network. These will bemade explicit in the next sub-section. Having determined a connection, evaluating a Wilson line requires putting the connection intosome given representations. It is too lengthy to give a complete review of constructing representa-tions of PGL(2 , Q p ). For complete details, please refer to [2].Here we only collect the necessary expressions. The basis we work with is based on the fact thatfunctions of p-adic numbers form representations of the group PGL(2 , Q p ). For a given conformalprimary of conformal ∆, we can construct a basis | X ; ∆ (cid:105) ≡ ˜ O ∆ ( X ) | (cid:105) , X ∈ Q p (3.16)where ˜ O ∆ ( X ) = N ( d, ∆) (cid:90) Q p dY | X − Y | − dq O ∆ ( Y ) . (3.17) N ( d, ∆) = ζ p (2 d − ζ p (2∆) ζ p ( d − ζ p (2∆ − d ) , ζ p ( s ) = 11 − p − s . (3.18)The state | (cid:105) is invariant under PGL(2 , Q p ) transformations, which is analogous to the vaccuumstate of a CFT and ˜ O ∆ ( X ) is referred to as a shadow operator in the CFT literature. The parameter d is the dimension of the CFT, which we can also take to d = 1 for simplicity.The corresponding bra basis states are given by (cid:104) X ; ∆ | ≡ (cid:104) |O ∆ ( X ) (3.19)Under the action of a group element g ∈ PGL(2 , Q p ), where g = (cid:32) a bc d (cid:33) , a, b, c, d ∈ Q p , (3.20)the states transform according to how the primary state transforms. i.e. (cid:104) X ; ∆ | → (cid:12)(cid:12)(cid:12)(cid:12) ad − bc ( cX + d ) (cid:12)(cid:12)(cid:12)(cid:12) ∆ p (cid:28) aX + bcX + d ; ∆ (cid:12)(cid:12)(cid:12)(cid:12) (3.21)There is also a set of bra and ket states constructed from the primaries (cid:104) ∆ | ≡ (cid:104) |O ∆ ( Z ) , | ∆ (cid:105) = O ∆ (0) | (cid:105) . (3.22)22n usual CFT , Z is often taken to be infinity. We showed in [2] that the value of Z drops outfor any finite Z when the end points of the Wilson lines are pushed to the asymptotic boundary.Therefore we can also think of Z as a kind of gauge choice.The definition of these basis states ensure that (cid:104) ∆ i | X ; ∆ j (cid:105) = δ ij δ ( X − Z ) , (cid:104) X ; ∆ i | ∆ j (cid:105) = δ ij | X | p . (3.23)We can then evaluate a component of the open Wilson line operator using these identities above.This gives (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) ≡ (cid:90) Q p dX (cid:104) ∆ | X ; ∆ (cid:105) W ( v → v ) (cid:104) X ; ∆ | ∆ (cid:105) = p − ( l + l )∆ (cid:12)(cid:12)(cid:12) p l cosh p (cid:16) δx √ L (cid:17) Z + p − h √ L − sinh p (cid:16) δx √ L (cid:17)(cid:12)(cid:12)(cid:12) p , (3.24)where we have used (2.31) and v , v are still given by (3.15).In the following, we will choose Z = 0. This choice ensures that at least for L = 0, (3.24)reduces to the two point function that follows from the tensor network (A.8). The details of thederivation is relegated to the appendix. After choosing Z = 0, the expectation value of the open Wilson line in p-adic black holeconnection becomes (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) = p − ( l + l )∆ (cid:12)(cid:12)(cid:12) p − h √ L − sinh p (cid:16) δx √ L (cid:17)(cid:12)(cid:12)(cid:12) p = p − (2 h + l + l )∆ (cid:12)(cid:12)(cid:12) √ L − sinh p (cid:16) δx √ L (cid:17)(cid:12)(cid:12)(cid:12) p . (3.25)One can see that (3.25) and (2.51) are equal if we identify √ L = 12 p h . (3.26)Of course we know that (2.44) depends only on the number of edges separating the two vertices v , v , and so this result locally is really the same whether in the “pure” BT tree or the black holebackground. Therefore locally, the black hole background is indeed identical to the “pure” BT tree.This is parallel to the fact that locally the BTZ black hole is the same as pure AdS space.23et us also comment that one can see here that if the exponentials in the sinh , cosh in (3.25)are replaced by the usual exponential, the denominator diverges whenever | δx √ L | p ≥ , (3.27)and so these two point functions vanish – signifying that the branches are disconnected as claimedin the previous section. The deformed exponential however allows the Wilson line computation tomatch with the tensor network exactly. i.e. the effect of local diffeomorphism leading to the changein the cutoff surface on the BT tree is reproduced by the “deformed” exponential in the connection,and not the usual exponential.When p h → p −∞ , the points v , v will live in the same branch since a single branch on thehorizon already contains the entire Bruhat-Tits tree. Then, φ = φ . Let’s write ( r , x ) , ( r , x )explicitly as ( r , x ) = ( p h + l , ( p h k , p h Θ( y ))) , (3.28)( r , x ) = ( p h + l , ( p h k , p h Θ( y ))) , (3.29)with k = k = k and y = l − (cid:88) i =0 a (1) i p i , ( a (1)0 (cid:54) = 0) (3.30) y = l − (cid:88) i =0 a (2) i p i . ( a (2)0 (cid:54) = 0) (3.31)Recall that our choice of Z = 0 now matches the Wilson line expectation values in the bulk withthe two-point correlation functions computed using the tensor network. Therefore we can rewritethe open Wilson line component as in (2.50), and get (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) = r − ∆1 r − ∆2 | p − h | p |E p ( x /p h ) | ∆ p |E p ( x /p h ) | ∆ p |E p ( x /p h ) − E p ( x /p h ) | p = p − (2 h + l + l )∆ | p − h | p | p k y | ∆ p | p k y | ∆ p | p k y − p k y | p = p − ( l + l )∆ | y | ∆ p | y | ∆ p | y − y | p = p − ( l + l )∆ | y − y | p . (3.32)Here we have used | y | p = | y | p = 1. And we notice that we actually recover the result (A.8) of the“pure” Bruhat Tits connection at L = 0. 24 Comments on p-adic black hole thermodynamics
By making comparisons with real CFT , we have established that correlation functions com-puted in the black hole coordinates following from the tensor network describe some p-adic CFTat “finite temperatures”.Separately, we worked with a p-adic BTZ black hole connection transplanted from the BTZblack hole connection in AdS and computed the expectation value of an open Wilson line, andshow that it agrees with the tensor network results with careful choice of the ambiguous parameter Z . It shows that indeed the p-adic BTZ connection encodes a finite temperature CFT exactly as inthe AdS case. Now, we had the identification (3.26), and recall also (2.54), we land on the followingrelation √ L = T . (4.1)This is formally in complete agreement with the relation obtained from the BTZ black hole, althoughnow √ L and T are both p-adic numbers.Now then we can work backwards and ask for the horizon radius. Formally, in accordance withthe BTZ black hole, the horizon radius r h is related to this parameter √ L by [16, 21] √ L = r h l , (4.2)where we have already used the fact that we have set l/ (4 G ) = 1. Using (4.2), |T | p = | p − h | p , (4.3)we finally have (cid:12)(cid:12)(cid:12) r h l (cid:12)(cid:12)(cid:12) p = p h . (4.4)As promised, the transplanted BTZ connection does end up implying that the parameter we haveintroduced in the black hole coordinates back in (5), is indeed giving us the radius of the black holehorizon. Let us also note that to make comparison with the radial coordinates in (5), p h appearsas a real number and so the comparison (4.4) only makes sense if we take the p-adic norm to map r h /l which is a p-adic number, to a real number.When | p − h | p = p h → p −∞ , we have |T | p → p h → p −∞ considered in the computation (3.32), which indeed recovers the original BT tree with vanishing |T | p .Without an action, it is less clear how to define black hole entropy. On the other hand, thehorizon radius p h is giving a scale to the density of branches emanated from the horizon. Recallthe left most picture in Fig. 7. Therefore, it is very intuitive to interpret this density of branches25s being proportional to the entropy S , which we take as a measure of information and thus a realnumber here. If we take S ∝ p h = | p − h | p , = ⇒ S ∼ |T | p ∼ | r h | p (4.5)The p-adic BTZ black hole satisfies the area law just as its AdS sister.A determination of the constant of proportionality would require some extra input such asdetermining the Chern-Simons action defined on the tree exactly. This is beyond the scope of thecurrent paper. This is the second instalment of our series to bend the Bruhat-Tits tree – that is to describegeometries that is deformed away from the “pure” Bruhat Tits tree. In our first instalment, thedeformation is achieved by an RG flow that takes the p-adic CFT away from the fixed point. In thecurrent instalment, we were inspired by the BTZ black hole, and considered transplanting the BTZconnection to a PGL(2 , Q p ) connection so that it describes a black hole in the Bruhat-Tits tree.Making use of the relation between these PGL(2 , Q p ) Wilson lines and the p-adic tensor network[15], we demonstrated not only that the expectation values of the Wilson lines can indeed be relatedto a p-adic CFT at finite temperatures, but more surprisingly that this black hole background canindeed be related to the “pure” BT tree by a local diffeomorphism, reflecting a non-trivial Weyltransformation performed in the p-adic CFT. Along the way, the tree inspired us to define a p-adic “deformed” exponential function that shares many properties with the usual exponential thatcaptures this local diffeomorphism.The results are quite surprising. It is known for a long time that the p-adic CFT does not appearto have descendents. But it is at the same time well known that non-trivial representations of the p-adic conformal group PGL(2 , Q p ) cannot be finite dimensional [13]. This paper is, to our knowledge,a first demonstration that it is possible to accommodate local conformal transformation in a wayparallel to real 2d CFTs. This raises many questions, particularly whether Virasoro symmetry canin fact be defined in these p-adic CFTs after all. At first sight – if it is possible to recover analoguesof thermal correlators in the p-adic CFT, whether one could find cousins of the stress tensors thattestify to some finite energy density at finite temperatures. Descendents and the stress tensor haveeluded the p-adic CFT community for a long time. It would be very interesting as a first step tolook for more examples of Weyl transformations connected to non-trivial diffeomorphism of the BTtree to gain insight of the problem.From the geometric perspective, we have got a glimpse of “black hole” geometries in the context26f the Bruhat-Tits tree. The construction of the genus 1 tree via orbifolding was considered before[4, 13, 15], but its connection to finite temperature physics is explored in depth only now. Therelation between black hole temperature and horizon radius is carried through exactly to the BTtree. The black hole does appear to satisfy some notion of the area law, although a precise definitionof the black hole entropy is still lacking, partially because much like anything p-adic, derivativesof the real valued partition function with respect to temperatures would not be well defined. Oneneeds to be more creative to look for a proper measure of the entropy.We note that everything that is observed in this paper was inspired by the tensor networkconstruction of the p-adic CFT partition function. Indeed it has guided us in recovering piecesof the p-adic AdS/CFT dictionary that is more universal than the tensor network reconstruction.The geometry considered in this paper is essentially locally a “pure” BT tree, and it would thusautomatically satisfy the emergent Einstein equation discussed in [1, 19]. A very natural nextstep would thus be to synergise results of our two distinct approaches in bending the BT tree– RG flow and diffeomorphism, to obtain more complicated scenarios that hopefully should helpus understand the emergent Einstein equation we found in the prequel [1, 19]. Nonetheless, thepresent work should give further support that the tensor network does capture some very importantessence of the p-adic AdS/CFT both qualitatively and quantitatively. The tensor network shouldhave more in supply in understanding new insights such as wormholes and islands [22, 23]. Thetensor network should also give further insight in understanding complexities. Our tensor networkhas very precise relation with Wilson lines for example, which is strongly reminiscent of recentdiscussions of complexities based on a Wilson line network [24].Last but not least, it may be confusing that we have, in our first and second instalment ofthe series, presented two apparently different interpretations of the tensor network. In the firstpaper, we have directly read off a metric from the tensor network, while in the current secondinstalment, we have made use of the Wilson line interpretation of the tensor network. One wouldhave wondered whether the gauge connection and the edge distance read off from the tensor networkcan be related to each other in some way. The usual relation between vierbein and metric wouldnot work, because we do not have an infinitesimal version of the gauge connection. What we havepresented are two self-consistent complementary interpretations of the tensor network. It would beimportant to explore their connections further in the future.We hope that this success in the p-adic AdS/CFT reconstruction can ultimately be replicatedto some extent in AdS/CFTs in general. Acknowledgements.—
LYH acknowledges the support of NSFC (Grant No. 11922502, 11875111)and the Shanghai Municipal Science and Technology Major Project (Shanghai Grant No.2019SHZDZX01),27nd 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 Matching the tensor network bulk correlation functions to theWilson line computation
When the end points of the Wilson lines are not all pushed to the boundary, the componentshave dependence of the parameter Z . We show below that there is a choice such that the result(3.24) reduces to two point correlation functions in the bulk as well.When L = 0 (“pure Bruhat-Tits connection”), we have (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) = p − ( n + n )∆ | p n Z + δx | p . (A.1)For p-adic number, when | a | p > | b | p , we have | a + b | p = | a | p . (A.2)So when | Z | p <
1, we have | p n Z + δx | p = | δx | p , (A.3)since | δx | p ≥ | p n | p > | p n Z | p . Let us check what happens when we take Z = 0. (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) = p − ( n + n )∆ | δx | p . (A.4)It is explicitly symmetric in v and v as desired. All the possible relative positions of v and v are shown in Fig10.In case (a), the two point function is (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) = p − ( n + n )∆ p − n ∆ = p − ( n + n − n )∆ , (A.5)where n + n − n is the distance between v and v .In case (b), the two point function is (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) = p − ( n + n )∆ p − n ∆ = p − ( n − n )∆ , (A.6)28 a) (b) (c) Figure 10: There are three types of relative positions. In (a), | δx | p = p − n . In (b), | δx | p = p − n .In (c), | δx | p = p − n .where n − n is the distance between v and v .In case (c), the two point function is (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) = p − ( n + n )∆ p − n ∆ = p − ( n − n )∆ , (A.7)where n − n is the distance between v and v .In conclusion, we find that (cid:104) ∆ | W ∆ ( v → v ) | ∆ (cid:105) = p − ( n + n )∆ | dx | p = p − ∆ d ( v ,v ) , (A.8)where d ( v , v ) is the distance between v and v . So after neglecting the term p n Z , the two pointfunction is the same as the two point function that would have followed from the tensor networkdiscussed in section 2. Similarly, we can prove that after choosing Z = 0, the three point functionis the same as the one that follows from the tensor network. References [1] L. Chen, X. Liu, and L.-Y. Hung, “Bending the Bruhat-Tits Tree I Tensor Network andEmergent Einstein Equations,”.[2] L.-Y. Hung, W. Li, and C. M. Melby-Thompson, “Wilson line networks in p -adicAdS/CFT,” JHEP (2019) 118, arXiv:1812.06059 [hep-th] .[3] S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk, “ p -adic AdS/CFT,” Commun. Math. Phys. no. 3, (2017) 1019–1059, arXiv:1605.01061 [hep-th] .294] M. Heydeman, M. Marcolli, I. Saberi, and B. Stoica, “Tensor networks, p -adic fields, andalgebraic curves: arithmetic and the AdS /CFT correspondence,” Adv. Theor. Math. Phys. (2018) 93–176, arXiv:1605.07639 [hep-th] .[5] S. Chapman, H. Marrochio, and R. C. Myers, “Complexity of Formation in Holography,” JHEP (2017) 062, arXiv:1610.08063 [hep-th] .[6] S. S. Gubser and S. Parikh, “Geodesic bulk diagrams on the Bruhat–Tits tree,” Phys. Rev. D no. 6, (2017) 066024, arXiv:1704.01149 [hep-th] .[7] F. Qu and Y.-h. Gao, “Scalar fields on p AdS,”
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