Bending the Bruhat-Tits Tree I
Tensor Network and Emergent Einstein Equations

The asymptotic boundary of the Bruhat-Tits tree is the $Q_{p}$ line. The tensor network has to be cutoff near the asymptotic boundary of the Bruhat-Tits tree – analogous to the cutoff introduced in AdS space. The dangling legs of the tensors are contracted with a reference vector

$$|V_{f}\rangle\equiv\sum_{a}V_{f}^{a}|a\rangle,\qquad V_{f}^{a}\equiv\delta^{a}_{1}.$$

(2.2)

i.e. it projects the dangling legs to the identity operator label. The tensor network evaluates to a number for such boundary conditions. This number is interpreted as the (normalized) partition function of the p-adic CFT. To insert operator $\mathcal{O}_{a}(x)$ into the partition function, the dangling leg located 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

$$G_{a}(x,v)=\frac{\zeta_{p}(2\Delta_{a})}{p^{\Delta_{a}}}p^{-{\Delta_{a}}d(x,v)},$$

(2.3)

where $d(x,v)$ counts the number of edges connecting the boundary vertex linked to $x$ and the bulk vertex $v$. Let us emphasize here that the result of the correlation functions of the CFT is read-off directly from the tensor network without reference to any action. And yet, it has the appearance of being composed of this propagators which are solutions to the graph Klein-Gordon equation

		$\displaystyle(\Box_{v}+m_{a}^{2})G_{a}(x,v)=\delta_{x,v},\qquad m_{a}^{2}=-\frac{1}{\zeta_{p}(\Delta_{a}-1)\zeta_{p}(-\Delta_{a})},\qquad\zeta_{p}(s)\equiv\frac{1}{1-p^{-s}}.$		(2.4)
		$\displaystyle\Box_{v}\phi(v)\equiv\sum_{u\sim v}(\phi(v)-\phi(u)).$		(2.5)

The result of the tensor network thus implies the existence of a massive bulk field $\phi_{a}$ in 1-1 correspondence with each CFT primary $\mathcal{O}_{a}$. We can define the notion of bulk operator insertion at a bulk vertex $v$ by fusing an extra leg with label $a$ to the bulk vertex. For fusion rules that are commutative such that $C^{abc}$ is completely symmetric in the three indices, we can simply stick an extra bulk leg to the vertex $v$ without worrying about the order of the fusion. This is illustrated in Fig. 1. The computation of a three point correlation function of the CFT is also illustrated there. When these bulk insertions are pushed to the boundary they become boundary operator insertions. i.e. This definition of bulk operator insertion ensures that the extrapolation dictionary – where boundary operator corresponds to 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 Witten diagrams constructed from bulk-bulk propagators meeting at other bulk vertices. The bulk-bulk propagator is simply given by the same $G_{a}$ in (2.4), with the boundary vertex label $x$ moved to the bulk.

By construction, each bulk vertex where propagators meet gains a factor of the OPE coefficients. A three point vertex is weighted by $C^{abc}$.

These suggest that at least where the operator insertion at the boundary is sparse, the tensor network can be described by an emergent bulk quantum field theory with action given by

$\displaystyle S_{m}$

$\displaystyle=$

$\displaystyle\sum_{\langle xy\rangle}\frac{1}{2}(\phi^{a}_{x}-\phi^{a}_{y})^{2}+\sum_{x}\frac{1}{2}m_{a}^{2}(\phi^{a}_{x})^{2}+\mathcal{O}(\phi^{3}),$

(2.6)

where $x$ denotes the vertices, and $a$ denotes different fields whose mass is $m_{a}$. The $a$ labelling the a primary is summed over though not shown explicitly. More accurately speaking, since the vertex where propagators meet in the tensor network is unique, it exactly matches with the semi-classical limit of the field theory, one where the masses $m_{a}$ approach infinity, or equivalently, $\Delta_{a}\to\infty$, so that the interaction vertex would be fixed at the intersection of geodesics.

To set the normalization rigorously, simple insertion of legs at a bulk vertex $x$ corresponds to inserting a field $\tilde{\phi}_{x}$ that is related to the canonically normalized $\phi_{a}(x)$ appearing in the above action by

$\displaystyle\tilde{\phi}^{a}_{x}$

$\displaystyle\equiv$

$\displaystyle\left(\frac{\zeta_{p}(2\Delta_{a})}{p^{\Delta_{a}}}\right)^{-\frac{1}{2}}\phi^{a}_{x},$

(2.7)

so that the two point function read off from the insertion of extra bulk legs is simply given by

$\displaystyle\langle\tilde{\phi}^{a}_{x}\tilde{\phi}^{a}_{y}\rangle=p^{-\Delta_{a}d(x,y)}$

(2.8)

In this normalization, the action is expressed as

$\displaystyle S_{m}$

$\displaystyle=$

$\displaystyle\sum_{\langle xy\rangle}\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}(\tilde{\phi}^{a}_{x}-\tilde{\phi}^{a}_{y})^{2}+\sum_{x}\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}m_{a}^{2}(\tilde{\phi}^{a}_{x})^{2}+\mathcal{O}(\tilde{\phi}^{3}).$

(2.9)

One can rewrite the summation over vertices to a sum over edges, which gives

$\displaystyle S_{m}$

$\displaystyle=$

$\displaystyle\sum_{\langle xy\rangle}\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}(\tilde{\phi}^{a}_{x}-\tilde{\phi}^{a}_{y})^{2}+\sum_{\langle xy\rangle}\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}\frac{1}{p+1}m_{a}^{2}((\tilde{\phi}^{a}_{x})^{2}+(\tilde{\phi}^{a}_{y})^{2})+\mathcal{O}(\tilde{\phi}^{3}).$

(2.10)

The factor $1/(p+1)$ is to cancel the $p+1$ times overcounting since each vertex attaches to $p+1$ edges.

RG flow of the p-adic CFT was considered in [24]. In the tensor network, the RG flow of the CFT can be driven by changing the boundary conditions at the asymptotic boundary. Instead of projecting the boundary leg to the vector $V_{f}^{a}$ defined in (2.2), we pick instead

$$|V_{\Lambda}\rangle\equiv\sum_{a}V_{\Lambda}^{a}|a\rangle$$

(2.11)

which generically turns on other primaries at the boundary. The subscript $\Lambda$ labels the cutoff surface on the tensor network at which these reference vectors are inserted. The RG flow considered in [24] respects translation invariance along $Q_{p}$. In that case, all boundary legs are projected to the same $|V_{\Lambda}\rangle$. More generally, one can pick different $|V_{\Lambda_{i}}\rangle$ for every boundary leg $x_{i}$.

We note that $p$ such boundary vectors would be fed to the same vertex tensor at the cutoff surface, which would then return a new vector $|V_{\Lambda-1}\rangle$:

$$|V_{\Lambda-1}\rangle\equiv\sum_{a_{p+1}}V^{a_{p+1}}_{\Lambda-1}|a_{p+1}\rangle=\sum_{a_{1}\cdots a_{p},a_{p+1}}p^{-(\Delta_{a_{1}}+\cdots\Delta_{a_{p}})}T^{a_{1}\cdots a_{p}a_{p+1}}V_{\Lambda_{1}}^{a_{1}}\cdots V_{\Lambda_{p}}^{a_{p}}|a_{p+1}\rangle$$

(2.12)

i.e. $\Lambda-1$ labels the legs one step away from the cutoff surface.

Note that the vector $|V_{f}\rangle$ is in fact a fixed point under this flow, explaining how a CFT partition function is recovered as fixed point vectors in the tensor network. For other choices of boundary vectors $|V_{\Lambda}\rangle$, a flow is driven down the network, so that different parts of the network carry different weights. Homogenous boundary conditions have been studied quite generally in [24], where we show that the flow would eventually lead to a new fixed point vector, corresponding to a CFT driven by relevant perturbations that eventually reaches another new CFT. The geometry in the interior of the network would thus resemble pure Bruhat-Tits geometry again, where the vertices eventually contribute equally to the partition function. That the actual weights of the tensors contributing differently is suggestive of a varying metric in the tree. How the metric should depend on the tensors is the crux of reading off geometry and subsequently gravity from the tensor network. We will 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 further progress, we will consider small deviations from the fixed point vector $|V_{f}\rangle$. i.e. In the following, we will consider

$$V^{a}_{\Lambda_{i}}=\delta^{a}_{1}+\lambda v_{\Lambda_{i}}^{a},$$

(2.13)

and treat $\lambda$ as a small parameter in a power series expansion around the CFT fixed point. Note that the subscript “$\Lambda_{i}$” denotes the position $i$ at the cutoff surface. At first sight, it may appear that a translation invariant boundary condition is simpler. But as we are going to see below, by assuming the most general perturbations $\lambda_{\Lambda_{i}}$ at the boundary, it gives the strongest constraint of the emergent Einstein equation.

Consider the $p$-adic tensor network as shown in Fig. 3. Expanded in the contributing tensors, it takes the form

$\displaystyle Z=\sum_{\dots,a,b,c,d,e,\dots}\dots p^{-\Delta_{a}}p^{-\Delta_{b}}C^{abc}p^{-\Delta_{c}}C^{cde}p^{-\Delta_{d}}p^{-\Delta_{e}}\dots.$

(2.14)

We can rewrite $Z$ as

$\displaystyle Z$

$\displaystyle=$

$\displaystyle\sum_{c}V^{c}p^{-\Delta_{c}}\tilde{V}^{c},$

(2.15)

defining

	$\displaystyle V^{c}$	$\displaystyle\equiv$	$\displaystyle\sum_{\dots,a,b}\dots p^{-\Delta_{a}}p^{-\Delta_{b}}C^{abc},$		(2.16)
	$\displaystyle\tilde{V}^{c}$	$\displaystyle\equiv$	$\displaystyle\sum_{d,e,\dots}C^{cde}p^{-\Delta_{d}}p^{-\Delta_{e}}\dots.$		(2.17)

When one is restricted on the red edge, $V^{a}$ and $\tilde{V}^{a}$ carry all the information in the tree. They follow 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 condition can always be written in terms of the pair of $V^{a}$ and $\tilde{V}^{a}$

$\displaystyle\langle\tilde{\phi}_{\alpha}\tilde{\phi}_{\beta}\rangle$

$\displaystyle=$

$\displaystyle\sum_{h,c,g}V^{h}C^{hc\alpha}p^{-\Delta_{c}}C^{cg\beta}\tilde{V}^{g},$

(2.18)

as shown in Fig. 3.

In the $p$-adic vacuum where the boundary legs are projected to the fixed point vector $|V_{f}\rangle$, as expected, $V^{a}$ and $\tilde{V}^{a}$ are both equal to the fixed point vector.

$\displaystyle V^{a}=\delta_{1}^{a},\;\tilde{V}^{a}=\delta_{1}^{a},$

(2.19)

Now, anticipating what will be needed when we begin considering curvatures, let us in particular consider 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 all relevant information describing this patch from the collection $\{V_{i}^{a}\}$ and $\{\tilde{V}_{i}^{a}\}$ located at each of the out-stretching legs. One can see that for edge $xy_{i}$, its $V^{a}$ is $V_{i}^{a}$, while for edge $x_{i}y$, its $\tilde{V}^{a}$ is $\tilde{V}_{i}^{a}$. Perturbed around the vacuum under generic boundary conditions (2.13), we have

	$\displaystyle V_{i}^{a}$	$\displaystyle=$	$\displaystyle\delta_{1}^{a}+\omega_{i}^{a},\qquad\omega_{i}^{a}\equiv\lambda_{i}^{a}+\eta_{i}^{a}+\mathcal{O}(\lambda^{3}),$
	$\displaystyle\tilde{V}_{i}^{a}$	$\displaystyle=$	$\displaystyle\delta_{1}^{a}+\tilde{\omega}_{i}^{a},\qquad\tilde{\omega}_{i}^{a}\equiv\tilde{\lambda}_{i}^{a}+\tilde{\eta}_{i}^{a}+\mathcal{O}(\lambda^{3}),$		(2.20)

where $\lambda_{i}^{a},\tilde{\lambda}_{i}^{a}\sim\mathcal{O}(\lambda)$, $\eta_{i}^{a},\tilde{\eta}_{i}^{a}\sim\mathcal{O}(\lambda^{2})$, and $\lambda\ll 1$.

Reading off from the tensor network, we have

	$\displaystyle\langle\tilde{\phi}_{x}^{a}\rangle$	$\displaystyle=$	$\displaystyle\lambda^{a}p^{-\Delta_{a}}+\tilde{\lambda}^{a}p^{-2\Delta_{a}}+\mathcal{O}(\lambda^{2}),$		(2.21)
	$\displaystyle\langle\tilde{\phi}_{y}^{a}\rangle$	$\displaystyle=$	$\displaystyle\tilde{\lambda}^{a}p^{-\Delta_{a}}+\lambda^{a}p^{-2\Delta_{a}}+\mathcal{O}(\lambda^{2}),$		(2.22)

where $\lambda^{a},\tilde{\lambda}^{a}$ are defined by

	$\displaystyle\lambda^{a}$	$\displaystyle\equiv$	$\displaystyle\sum_{i=1}^{p}\lambda^{a}_{i},$		(2.23)
	$\displaystyle\tilde{\lambda}^{a}$	$\displaystyle\equiv$	$\displaystyle\sum_{i=1}^{p}\tilde{\lambda}^{a}_{i}.$		(2.24)

As discussed in the previous section using Fig. 3, all the information carried by an edge $e$ is completely captured by $V_{e}^{a}$ and $\tilde{V}_{e}^{a}$ at the two ends of the edge. So it’s natural to propose that the edge length $d_{e}$ is also determined by $V_{e}^{a}$ and $\tilde{V}_{e}^{a}$. It is a bit challenging to obtain the complete dependence of the edge length on $V_{e}^{a}$ and $\tilde{V}_{e}^{a}$. However, if we are working with the perturbative limit, where $V_{e}^{a}$ and $\tilde{V}_{e}^{a}$ admit a perturbation expansion of the form (2.20):

	$\displaystyle V_{e}^{a}$	$\displaystyle=$	$\displaystyle\delta^{a}_{1}+w^{a}_{e}=\delta^{a}_{1}+\lambda^{a}_{e}+\eta^{a}_{e}+\mathcal{O}(\lambda^{3}),$		(3.1)
	$\displaystyle\tilde{V}_{e}^{a}$	$\displaystyle=$	$\displaystyle\delta^{a}_{1}+\tilde{w}^{a}_{e}=\delta^{a}_{1}+\tilde{\lambda}^{a}_{e}+\tilde{\eta}^{a}_{e}+\mathcal{O}(\lambda^{3}).$		(3.2)

then the edge distance $d_{e}$ also admits an expansion in $\lambda^{a}_{e},\tilde{\lambda}^{a}_{e},\eta^{a}_{e},\tilde{\eta}^{a}_{e}$.

But first, we note that when the boundary condition corresponds to the fixed point vector, every vertex and thus every edge gives equal contribution to the partition function. Therefore precisely at 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 small deviation from unity as follows:

$\displaystyle d_{e}$

$\displaystyle=$

$\displaystyle 1+j_{e}$

(3.3)

with $j_{e}\ll 1$, and this deviation should depend on the local data on the tensor network. Therefore we can also expand it in powers of $\lambda$ as follows:

	$\displaystyle j_{e}$	$\displaystyle=$	$\displaystyle A^{a}(\omega_{e}^{a}+\tilde{\omega}_{e}^{a})+B^{ab}(\omega_{e}^{a}\omega_{e}^{b}+\tilde{\omega}_{e}^{a}\tilde{\omega}_{e}^{b})+C^{ab}\omega_{e}^{a}\tilde{\omega}_{e}^{b}+\mathcal{O}(\omega^{3})$		(3.4)
		$\displaystyle=$	$\displaystyle A^{a}(\lambda_{e}^{a}+\tilde{\lambda}_{e}^{a})+B^{ab}(\lambda_{e}^{a}\lambda_{e}^{b}+\tilde{\lambda}_{e}^{a}\tilde{\lambda}_{e}^{b})+C^{ab}\lambda_{e}^{a}\tilde{\lambda}_{e}^{b}+A^{a}(\eta_{e}^{a}+\tilde{\eta}_{e}^{a})+\mathcal{O}(\lambda^{3}),\;\;\;\;\;$

with $B^{ab},C^{ab}$ symmetric matrices that parametrise this expansion. Here we have used the symmetry of $\omega_{e}^{a}\leftrightarrow\tilde{\omega}_{e}^{a}$, i.e. $V_{e}^{a}\leftrightarrow\tilde{V}_{e}^{a}$, since the edge should depend symmetrically on the information at its two ends. These unknown parameters, as we are going to find out, are almost uniquely fixed if a consistent Einstein equation is to exist at all.

We would like to define graph Ricci curvatures given the set of edge distances we have now defined on the tensor network. Graph curvatures have been considered in the mathematics literature [44, 45, 46, 47]. Inspired by them, we consider the Ricci curvature $R(d_{xy_{1}},d_{xy_{2}},\dots,d_{xy_{p+1}})$ of a patch around a vertex $x$ to be a symmetric function of the lengths $d_{xy_{i}}$ of edges emanating from the vertex $x$. For the precise dependence however, we first take the agnostic approach, and consider the most general expansion of the curvature on the edge distances perturbed away from unity. We expect that there is a regular expansion since the curvature should evaluate to a constant when all edges have equal lengths.

We parametrize the expansion of the graph Ricci curvature $R(d_{xy_{1}},d_{xy_{2}},\dots,d_{xy_{p+1}})$ as a power series in $\{j_{xy_{i}}\}$ as:

$\displaystyle R(d_{xy_{1}},d_{xy_{2}},\dots,d_{xy_{p+1}})$

$\displaystyle=$

$\displaystyle a_{0}+a_{1}\sum_{i}j_{xy_{i}}+b\sum_{i}j^{2}_{xy_{i}}+c\sum_{i\neq k}j_{xy_{i}}j_{xy_{k}}+\mathcal{O}(j^{3}).$

(3.5)

Here we have imposed symmetric dependence of all edges connected to $x$. These coefficients $a_{0},b,c$ are dependence of curvatures on distances and they should be universal independent of the tensor network.

Now we would like to write down the analogue of the Einstein-Hilbert action on the graph. On Riemann manifold the Einstein-Hilbert action takes the form:

$\displaystyle S_{EH}[g_{\mu\nu}]$

$\displaystyle=$

$\displaystyle\int d^{d}x\sqrt{-g}(R+\Lambda),$

(3.6)

with $\Lambda$ the cosmological constant and $R$ the Ricci scalar which is a function of the metric $g_{\mu\nu}$.

Having defined the graph scalar curvature in (3.5), the Einstein-Hilbert action on the BT tree should take the general form:

$\displaystyle S_{EH}$

$\displaystyle=$

$\displaystyle\sum_{x}R(d_{xy_{1}},d_{xy_{2}},\dots,d_{xy_{p+1}})+\sum_{\langle xy\rangle}d_{xy}\Lambda,$

(3.7)

where the first sum runs over all bulk vertices $x$, and $\sum_{\langle xy\rangle}$ indicates a sum over edges. The second term is proposed to mimic the cosmological constant term. In the continuous case, the volume form is given by $d^{d}x\sqrt{-g}$. Therefore it is natural to propose that

$\displaystyle d^{d}x\sqrt{-g}$

$\displaystyle\sim$

$\displaystyle d_{xy},$

(3.8)

which leads to the second term in (3.7).

The matter action read off from the tensor network when the boundary conditions are taken to be the fixed point vectors except at isolated locations of operator insertion is given by (2.10). This is essentially the action for pure BT space.

We would like to covariantize this action, to couple edge lengths to the matter fields. This has been considered in [46]. However we find the ansatz quite restrictive there, and we would like to write down a more general ansatz that mimics closely the covariant coupling of matter to the background in the continuous case, although very much like in our treatment of the edge lengths and graph curvature, the ansatz allows for a collection of parameters that cannot be fixed purely by symmetry and locality considerations.

Recall (3.8), then

$\displaystyle\int d^{d}x\sqrt{-g}f(x)$

$\displaystyle\rightarrow$

$\displaystyle\sum_{\langle xy\rangle}d_{xy}f_{xy}.$

(4.1)

A natural ansatz for the covariantized matter action $S^{cov}_{m}$ therefore takes the following form:

$\displaystyle S^{cov}_{m}$

$\displaystyle=$

$\displaystyle\sum_{\langle xy\rangle}d^{k}_{xy}\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}(\tilde{\phi}^{a}_{x}-\tilde{\phi}^{a}_{y})^{2}+\sum_{\langle xy\rangle}d_{xy}\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}\frac{1}{p+1}m_{a}^{2}((\tilde{\phi}^{a}_{x})^{2}+(\tilde{\phi}^{a}_{y})^{2})+\mathcal{O}(\tilde{\phi}^{3}),$

(4.2)

where $k$ is some constant we can’t determine now since $(\tilde{\phi}^{a}_{x}-\tilde{\phi}^{a}_{y})^{2}$ may contribute additional $1/d^{2}_{xy}$. Writing down the general $\mathcal{O}(\tilde{\phi}^{3})$ term explicitly, $S^{cov}_{m}$ becomes

	$\displaystyle S^{cov}_{m}$	$\displaystyle=$	$\displaystyle\sum_{\langle xy\rangle}d^{k}_{xy}\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}(\tilde{\phi}^{a}_{x}-\tilde{\phi}^{a}_{y})^{2}+\sum_{\langle xy\rangle}d_{xy}\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}\frac{1}{p+1}m_{a}^{2}((\tilde{\phi}^{a}_{x})^{2}+(\tilde{\phi}^{a}_{y})^{2})$		(4.3)
			$\displaystyle+\sum_{\langle xy\rangle}\left(h(d_{xy})H^{abc}(\tilde{\phi}^{a}_{x}\tilde{\phi}^{b}_{x}\tilde{\phi}^{c}_{x}+\tilde{\phi}^{a}_{y}\tilde{\phi}^{b}_{y}\tilde{\phi}^{c}_{y})+r(d_{xy})R^{abc}(\tilde{\phi}^{a}_{x}\tilde{\phi}^{b}_{x}\tilde{\phi}^{c}_{y}+\tilde{\phi}^{a}_{y}\tilde{\phi}^{b}_{y}\tilde{\phi}^{c}_{x})\right)+\mathcal{O}(\tilde{\phi}^{4}),\;\;\;\;$

where $h(d_{xy}),r(d_{xy})$ are functions of $d_{xy}$, and $H^{abc}$ is totally symmetric, while $R^{abc}$ is symmetric in $a,b$.

Similar to our previous treatment of the graph curvature, we can expand $h(d_{xy}),r(d_{xy})$ in powers of perturbations $j_{xy}$ of the edge lengths :

	$\displaystyle h(d_{xy})$	$\displaystyle=$	$\displaystyle h_{0}+h_{1}j_{xy}+h_{2}j^{2}_{xy}+\dots,$		(4.4)
	$\displaystyle r(d_{xy})$	$\displaystyle=$	$\displaystyle r_{0}+r_{1}j_{xy}+r_{2}j^{2}_{xy}+\dots.$		(4.5)

We have proposed the ansatz for an Einstein-Hilbert action and a covariant matter action. We are now ready to obtain a graph Einstein equation by varying the total action $S_{EH}+S^{cov}_{m}$ with respect to the edge lengths. This gives

$\displaystyle\frac{\delta S_{tot}}{\delta d_{xy}}=\frac{\delta S_{EH}}{\delta d_{xy}}+\frac{\delta S^{cov}_{m}}{\delta d_{xy}}=0,\;\;\textrm{i.e.}\;\;G+T=0,$

(4.6)

with

	$\displaystyle G\equiv\frac{\delta S_{EH}}{\delta d_{xy}}$	$\displaystyle=$	$\displaystyle\Lambda+2a_{1}+4bj_{xy}+c(\sum_{\begin{subarray}{c}i\\ (y_{i}\neq y)\end{subarray}}j_{xy_{i}}+\sum_{\begin{subarray}{c}i\\ (x_{i}\neq x)\end{subarray}}j_{x_{i}y})+\mathcal{O}(j^{2}),$		(4.7)
	$\displaystyle T\equiv\frac{\delta S^{cov}_{m}}{\delta d_{xy}}$	$\displaystyle=$	$\displaystyle k\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}(\tilde{\phi}^{a}_{x}-\tilde{\phi}^{a}_{y})^{2}+\frac{\zeta_{p}(2\Delta_{a})}{2p^{\Delta_{a}}}\frac{1}{p+1}m_{a}^{2}((\tilde{\phi}^{a}_{x})^{2}+(\tilde{\phi}^{a}_{y})^{2})$		(4.8)
			$\displaystyle+\left(h_{1}H^{abc}(\tilde{\phi}^{a}_{x}\tilde{\phi}^{b}_{x}\tilde{\phi}^{c}_{x}+\tilde{\phi}^{a}_{y}\tilde{\phi}^{b}_{y}\tilde{\phi}^{c}_{y})+r_{1}R^{abc}(\tilde{\phi}^{a}_{x}\tilde{\phi}^{b}_{x}\tilde{\phi}^{c}_{y}+\tilde{\phi}^{a}_{y}\tilde{\phi}^{b}_{y}\tilde{\phi}^{c}_{x})\right)+\mathcal{O}(\tilde{\phi}^{4}),$

One very important fact to note is that by considering the equations following from variation of $d_{xy}$, the edge data involved in the equation span a patch that is precisely given by Fig. 4, which was anticipated there.

We have obtained from the tensor network expectation values of some scalar field $\tilde{\phi}_{a}$. We have also read off edge lengths and edge curvatures based on data of the tensor network. The issue at hand is that – we have a solution in search of an equation relating these data. Using very few assumptions, based on locality and also correlation functions of these scalar fields at the CFT fixed point, we have written down an ansatz of an action up to some undetermined coefficients, which led to an Einstein equation. Here we would like to ask the following question: if the expectation values and geometrical data that we have read off from the tensor network are indeed related by the purported graph Einstein equation, what are the allowed values of the undetermined parameters so 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 undetermined parameters. The surprising result is that this over-determined system does have a unique solution (up to some overall normalizations) that naturally recovers results in the mathematics literature and satisfies all other consistency conditions.

We will demonstrate this order by order in the $\lambda$ expansion.

Plugging (2.21), (2.22) into (4.8), we find that

	$\displaystyle T$	$\displaystyle=$	$\displaystyle\frac{p^{-3\Delta_{a}}\left(k(p+1)\left(p^{\Delta_{a}}-1\right)^{2}+m_{a}^{2}\left(p^{2\Delta_{a}}+1\right)\right)}{2(p+1)\left(p^{\Delta_{a}}-1\right)\left(p^{\Delta_{a}}+1\right)}(\lambda^{a}\lambda^{a}+\tilde{\lambda}^{a}\tilde{\lambda}^{a})$		(4.9)
			$\displaystyle+\frac{p^{-3\Delta_{a}}\left(2m_{a}^{2}p^{\Delta_{a}}-k(p+1)\left(p^{\Delta_{a}}-1\right)^{2}\right)}{(p+1)\left(p^{\Delta_{a}}-1\right)\left(p^{\Delta_{a}}+1\right)}\lambda^{a}\tilde{\lambda}^{a}+\mathcal{O}(\lambda^{3}).$		(4.10)

For the edge $xy_{i}$, its $V^{a}$ and $\tilde{V}^{a}$ can also be read off from the tensor network:

	$\displaystyle V^{a}$	$\displaystyle=$	$\displaystyle V^{a}_{i}=\delta_{1}^{a}+\lambda_{i}^{a}+\mathcal{O}(\lambda^{2}),$		(4.11)
	$\displaystyle\tilde{V}^{a}$	$\displaystyle=$	$\displaystyle\delta_{1}^{a}+(\lambda^{a}-\lambda^{a}_{i})p^{-\Delta_{a}}+\tilde{\lambda}^{a}p^{-2\Delta_{a}}+\mathcal{O}(\lambda^{2}).$		(4.12)

Similarly, for the edge $x_{i}y$, its $V^{a}$ and $\tilde{V}^{a}$ are given by

	$\displaystyle V^{a}$	$\displaystyle=$	$\displaystyle\delta_{1}^{a}+(\tilde{\lambda}^{a}-\tilde{\lambda}^{a}_{i})p^{-\Delta_{a}}+\lambda^{a}p^{-2\Delta_{a}}+\mathcal{O}(\lambda^{2}),$		(4.13)
	$\displaystyle\tilde{V}^{a}$	$\displaystyle=$	$\displaystyle\tilde{V}^{a}_{i}=\delta_{1}^{a}+\tilde{\lambda}_{i}^{a}+\mathcal{O}(\lambda^{2}),$		(4.14)

where we recall that the definition of $\lambda^{a}$ and $\tilde{\lambda}^{a}$ first appeared in (2.21).

For the edge $xy$, its $V^{a}$ and $\tilde{V}^{a}$ are given by

	$\displaystyle V^{a}$	$\displaystyle=$	$\displaystyle\delta_{1}^{a}+\lambda^{a}p^{-\Delta_{a}}+\mathcal{O}(\lambda^{2}),$		(4.15)
	$\displaystyle\tilde{V}^{a}$	$\displaystyle=$	$\displaystyle\delta_{1}^{a}+\tilde{\lambda}^{a}p^{-\Delta_{a}}+\mathcal{O}(\lambda^{2}).$		(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

$\displaystyle G=\Lambda+2a_{1}+A^{a}(\lambda^{a}+\tilde{\lambda}^{a})p^{-2\Delta_{a}}\left(4bp^{\Delta_{a}}+c\left(p^{2\Delta_{a}}+(p-1)p^{\Delta_{a}}+p\right)\right)+\mathcal{O}(\lambda^{2}).$

(4.17)

To satisfy $G+T=0$ order by order in $\lambda$, we should have

	$\displaystyle\Lambda+2a_{1}$	$\displaystyle=$	$\displaystyle 0,$		(4.18)
	$\displaystyle A^{a}$	$\displaystyle=$	$\displaystyle 0,$		(4.19)

since $T\sim\mathcal{O}(\lambda^{2})$ and $\left(4bp^{\Delta_{a}}+c\left(p^{2\Delta_{a}}+(p-1)p^{\Delta_{a}}+p\right)\right)$ is zero for at most two $\Delta_{a}$. ³³3The two solutions are $\Delta_{a}$ and $d-\Delta_{a}$, where $d$ is the dimension of the CFT which is conveniently taken to be one here.

Then (3.4) becomes

$\displaystyle d_{e}=1+B^{ab}(\lambda_{e}^{a}\lambda_{e}^{b}+\tilde{\lambda}_{e}^{a}\tilde{\lambda}_{e}^{b})+C^{ab}\lambda_{e}^{a}\tilde{\lambda}_{e}^{b}+\mathcal{O}(\lambda^{3}),$

(4.20)

which means we can work out the $\mathcal{O}(\lambda^{2})$ term only with $\lambda_{e}^{a},\tilde{\lambda}_{e}^{a}$. Plugging them into (4.7), this time we find that

	$\displaystyle G$	$\displaystyle=$	$\displaystyle p^{-2(\Delta_{a}+\Delta_{b})}\left(4bB^{ab}p^{\Delta_{a}+\Delta_{b}}+B^{ab}c\left(-2p^{\Delta_{a}+\Delta_{b}}+p^{\Delta_{a}+\Delta_{b}+1}+p\right)+cC^{ab}p^{2\Delta_{a}+\Delta_{b}}\right)(\lambda^{a}\lambda^{b}+\tilde{\lambda}^{a}\tilde{\lambda}^{b})$		(4.21)
			$\displaystyle+p^{-2(\Delta_{a}+\Delta_{b})}\left(4bC^{ab}p^{\Delta_{a}+\Delta_{b}}+2B^{ab}c(p-1)\left(p^{\Delta_{b}}+p^{\Delta_{a}}\right)+cC^{ab}\left(p^{2\Delta_{a}}+p^{2\Delta_{b}}\right)\right)\lambda^{a}\tilde{\lambda}^{b}$
			$\displaystyle+\sum_{i}\frac{c}{2}\left(2B^{ab}(1+p^{-\Delta_{a}-\Delta_{b}})-C^{ab}(p^{-\Delta_{a}}+p^{-\Delta_{b}})\right)(\lambda^{a}_{i}\lambda^{b}_{i}+\tilde{\lambda}^{a}_{i}\tilde{\lambda}^{b}_{i})+\mathcal{O}(\lambda^{3}).$

To satisfy $G+T=0$, firstly we should have

$\displaystyle\sum_{i}\frac{c}{2}\left(2B^{ab}(1+p^{-\Delta_{a}-\Delta_{b}})-C^{ab}(p^{-\Delta_{a}}+p^{-\Delta_{b}})\right)(\lambda^{a}_{i}\lambda^{b}_{i}+\tilde{\lambda}^{a}_{i}\tilde{\lambda}^{b}_{i})$

$\displaystyle=$

$\displaystyle 0,$

(4.22)

since other terms in $G+T$ only depend on $\lambda^{a},\tilde{\lambda}^{a}$. It is obvious that $c=0$ is one solution for it. That is too stringent. We are interested in $c\neq 0$ case which is more non-trivial, and this gives a constraint on the parameters

$\displaystyle 2B^{ab}(1+p^{-\Delta_{a}-\Delta_{b}})-C^{ab}(p^{-\Delta_{a}}+p^{-\Delta_{b}})=0,$

(4.23)

which is already made symmetric in $a,b$ since it was contracted with a symmetric tensor $\lambda^{a}_{i}\lambda^{b}_{i}$. So we have

$\displaystyle C^{ab}$

$\displaystyle=$

$\displaystyle\frac{2B^{ab}(1+p^{-\Delta_{a}-\Delta_{b}})}{p^{-\Delta_{a}}+p^{-\Delta_{b}}}.$

(4.24)

Now comparing $G$ with $T$, we notice that $B^{ab}$ should be diagonal. And this gives

	$\displaystyle G$	$\displaystyle=$	$\displaystyle B^{aa}p^{-4\Delta_{a}}\left(4bp^{2\Delta_{a}}+\frac{1}{2}c\left(-2p^{3\Delta_{a}}+2p^{5\Delta_{a}}+2p^{\Delta_{a}+1}+2p^{3\Delta_{a}+1}\right)p^{-\Delta_{a}}\right)(\lambda^{a}\lambda^{a}+\tilde{\lambda}^{a}\tilde{\lambda}^{a})$		(4.25)
			$\displaystyle+2B^{aa}p^{-3\Delta_{a}}\left(2b\left(p^{2\Delta_{a}}+1\right)+c\left(p^{2\Delta_{a}}+2p-1\right)\right)\lambda^{a}\tilde{\lambda}^{a}+\mathcal{O}(\lambda^{3}).$

Then $G+T=0$ gives

			$\displaystyle B^{aa}p^{-4\Delta_{a}}\left(4bp^{2\Delta_{a}}+\frac{1}{2}c\left(-2p^{3\Delta_{a}}+2p^{5\Delta_{a}}+2p^{\Delta_{a}+1}+2p^{3\Delta_{a}+1}\right)p^{-\Delta_{a}}\right)$		(4.26)
		$\displaystyle=$	$\displaystyle-\frac{p^{-3\Delta_{a}}\left(k(p+1)\left(p^{\Delta_{a}}-1\right)^{2}+m_{a}^{2}\left(p^{2\Delta_{a}}+1\right)\right)}{2(p+1)\left(p^{\Delta_{a}}-1\right)\left(p^{\Delta_{a}}+1\right)},$

and also

			$\displaystyle 2B^{aa}p^{-3\Delta_{a}}\left(2b\left(p^{2\Delta_{a}}+1\right)+c\left(p^{2\Delta_{a}}+2p-1\right)\right)$		(4.27)
		$\displaystyle=$	$\displaystyle-\frac{p^{-3\Delta_{a}}\left(2m_{a}^{2}p^{\Delta_{a}}-k(p+1)\left(p^{\Delta_{a}}-1\right)^{2}\right)}{(p+1)\left(p^{\Delta_{a}}-1\right)\left(p^{\Delta_{a}}+1\right)}.$		(4.28)

And $m_{a}^{2}$ can be solved by $b,c,k$:

$\displaystyle m_{a}^{2}$

$\displaystyle=$

$\displaystyle-k(p+1)-\frac{ck(p+1)\left(p^{1-\Delta_{a}}+p^{\Delta_{a}}\right)}{2b-c}.$

(4.29)