Group field theory and tensor networks: towards a Ryu–Takayanagi formula in full quantum gravity

Let G denote an arbitrary semi-simple Lie group; in the following, we assume for simplicity that G is compact, but the framework can easily be generalized to the non-compact case. A group field φ is a complex function defined on a number of copies of the group manifold G:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \displaystyle \varphi&:G^d\rightarrow \mathbb{C} \nonumber \\ \nonumber & {g_i} \mapsto \varphi({g}_i) \nonumber \end{align} \tag{ 1 }$$

where we use the shorthand notation g_i for the set of d group elements $\{g_1, g_2, \cdots, g_d\}$.

The GFT field can be also seen as an infinite-dimensional tensor, transforming under the action of some (unitary) group $U^{\times d}$, as:

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\dd}{{\rm d}} \displaystyle \varphi(g_1,..,g_d) \rightarrow \int [\dd g_i]\, U(g'_1, g_1)\cdots U(g'_d, g_d)\, \varphi(g_1,...,g_d), \nonumber \end{align*}$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\dd}{{\rm d}} \displaystyle &\varphi^*(g_1,..,g_d) \rightarrow \int [\dd g_i]\, U^*(g_1, g'_1)\cdots U^*(g_d, g'_d)\, \varphi^*(g_1,...,g_d)\nonumber \\ & \quad {\rm{for}} \quad \int \dd g_i\, U(g'_i,g_i)^* U(g_i, \tilde{g}_i) = \delta(g'_i, \tilde{g}_i). \nonumber \end{align} \tag{ 2 }$$

This requires the d arguments of the GFT field to be labeled and ordered. We will see in the following how one can decompose the same field into finite-dimensional tensors; in this finite-dimensional case, the correspondence with tensor network formalism will be evident, and it will also be evident then in which sense GFTs provide a generalization of it.

The GFT dynamics is defined by an action, at the classical level, and a partition function at the quantum level. The combinatorial structure of the pairing of field arguments in the GFT interactions is part of the definition of a GFT model. An interesting class of models [6–12] is defined by the requirement that the interaction monomials are tensor invariants, i.e. that GFT fields are convoluted in such a way as to produce an invariant under the above mentioned (unitary) transformations¹.

Another class of GFT models is instead based on the requirement that the Feynman diagrams of the theory are simplicial complexes, which in turn requires the interaction kernels to have the combinatorial structure of d-simplices. This class of models is also the one on which model building for 4d quantum gravity has focused on, producing models whose Feynman amplitudes have the form of simplicial gravity path integrals and spin foam models [6–9], and, more generally, lattice gauge theories. This involves an additional symmetry requirements on the GFT fields and interactions, which will play a crucial role in the following.

In this simplicial case, the GFT action has the general form

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\dd}{{\rm d}} \newcommand{\f}{\frac} \newcommand{\V}{\mathcal{V}} \newcommand{\m}{\mu} \newcommand{\K}{\mathcal{K}} \newcommand{\vv}{\varphi} \displaystyle S_d[\varphi]&=\frac{1}{2} \int \dd g_i \dd g'_i \, \vv(g_i)\, \K(g_i g_i'^{-1})\, \vv(g_i')\,\nonumber \\ \nonumber &+\frac{\lambda}{d+1} \int \prod_{i\neq j =1}^{d+1} \dd g_{ij}\, \V(g_{ij} g_{ji}'^{-1})\, \vv(g_{1j})\cdots \vv(g_{d+1j}), \nonumber \end{align} \tag{ 3 }$$

where $\newcommand{\dd}{{\rm d}} \dd g_i$ is an invariant measure on G and we use the notation $\newcommand{\vv}{\varphi} \vv(g_{1j}) =\vv(g_{12}, \cdots, g_{1d+1})$. $\newcommand{\m}{\mu} \newcommand{\K}{\mathcal{K}} \K$ is the kinetic kernel, $\newcommand{\m}{\mu} \newcommand{\V}{\mathcal{V}} \V$ the interaction kernel, λ a coupling constant for the d  +  1-degree homogeneous interaction. The two kernels satisfy the invariance properties

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\f}{\frac} \newcommand{\V}{\mathcal{V}} \newcommand{\m}{\mu} \newcommand{\K}{\mathcal{K}} \displaystyle \K(h\,g_i g_i'^{-1}\,h') &= \K(g_i g_i'), \nonumber \\ \nonumber \V(h_i\,g_{ij} g_{ji}'^{-1},h_j^{-1}) &= \V(g_{ij} g_{ji}'^{-1}) \qquad \forall h,h',h_i \in G. \nonumber \end{align} \tag{ 4 }$$

This implies that the action is invariant under the gauge transformations $\newcommand{\vv}{\varphi} \delta \vv(g_i) = \tilde{\vv}(g_i)$, where $\newcommand{\vv}{\varphi} \tilde{\vv}$ is any function satisfying

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \newcommand{\dd}{{\rm d}} \newcommand{\vv}{\varphi} \displaystyle \label{gauge} \int_G \dd h \, \tilde{\vv}(hg_1, \cdots, hg_d)=0. \nonumber \end{align} \tag{ 5 }$$

This symmetry is gauge fixed if one restricts the field $\newcommand{\vv}{\varphi} \vv$ to satisfy

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \newcommand{\vv}{\varphi} \displaystyle \label{symme} \vv(hg_i) = \vv(g_i). \nonumber \end{align} \tag{ 6 }$$

The action is also invariant under the global symmetry

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\vv}{\varphi} \displaystyle \vv(g_1,\cdots ,g_d) \rightarrow \vv(g_1h,\cdots,g_dh). \nonumber \end{align} \tag{ 7 }$$

GFT's Feynman diagrams define cellular complexes $\newcommand{\m}{\mu} \mathcal{F}$ weighted by amplitudes assigned to the faces, edges and vertices of the dual two-skeleton of a chosen triangulation of a d dimensional topological spacetime $\newcommand{\m}{\mu} \mathcal{M}_{\mathcal{F}}$. As mentioned, their Feynman diagram evaluations reproduce the associated amplitudes of a spin foam model, or, in different variables, of a simplicial gravity path integral [56–58], providing a generalisation of the lattice formulation of gravity à la Regge, with an accompanying sum over lattices, generalising matrix models for 2d gravity to any dimension [6–12].

Let us give some more detail on the construction, to clarify the above points. A specific theory, with a specific related Feynman cellular complex, is completely defined by the choice of the kernels. Lets consider the simplest case, consisting in the choice

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \newcommand{\K}{\mathcal{K}} \newcommand{\m}{\mu} \displaystyle \K(g_i,g_i')=\int_G \dd h \prod_i \delta(g_ig_i'^{-1}\,h), \nonumber \end{align} \tag{ 8 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\dd}{{\rm d}} \newcommand{\V}{\mathcal{V}} \newcommand{\m}{\mu} \displaystyle \V(g_{ij} g_{ji}'^{-1})=\int_G \prod_i \dd h_i \prod_{i < j} \delta(h_i\,g_{ij} g_{ji}'^{-1},h_j^{-1})\label{V} \nonumber \end{align} \tag{ 9 }$$

where $\delta(\cdot)$ is the delta function on G and the integrals ensure the gauge invariance defined in (5), and let us restrict to the case of dimension d  =  3. To keep track of the combinatorics of field arguments in the kernels, it is useful to represent the Feynman diagram as a stranded graph. The field $\newcommand{\vv}{\varphi} \vv$ has three arguments, so each edge of a Feynman diagram comprises three strands running parallel to it. Four edges meet at each vertex and the form of the interaction $\newcommand{\m}{\mu} \newcommand{\V}{\mathcal{V}} \V$ in (9) forces the strands to recombine as in figure 1.

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The three strands running along the edges can be understood to be dual to a triangle and the propagator $\newcommand{\m}{\mu} \newcommand{\K}{\mathcal{K}} \K$ gives a prescription for the gluing of two triangles. At the vertex, four triangles meet and their gluing via $\newcommand{\m}{\mu} \newcommand{\V}{\mathcal{V}} \V$ form a tetrahedron. With this interpretation the Feynman diagram of a GFT is clearly dual to a triangulated 3d simplicial complex (which will be generically a singular pseudo-manifold) and this is true in any dimension [59–61].

The quantum states of the theory can be given a similar combinatorial characterization in terms of graphs and dual cellular complexes, as it should be already intuitive in the above example, in which GFT fields themselves are associated to triangles. We will not detail this aspect of the formalism.

As a function on a group G, the field $\newcommand{\vv}{\varphi} \vv$ can be decomposed in terms of unitary irreducible representations $(\rho, V_{\rho})$ of G using the Peter–Weyl theorem, $\newcommand{\s}{\sigma} \newcommand{\bi}{\boldsymbol}L^2(G)\simeq \bigoplus\nolimits_{\rho}V^{\rho} \otimes V^{*{\rho}}$, giving

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \newcommand{\vv}{\varphi} \displaystyle \vv(g)=\sum_{\rho} d_{\rho} {\rm Tr}[\hat{\vv}_{ab}\, \rho^{ab}(g)]. \nonumber \end{align} \tag{ 10 }$$

Here, $\newcommand{\m}{\mu} d_\rho \in \mathbb{N}$ is the dimension of the representation $\newcommand{\m}{\mu} \rho : G\rightarrow {\rm{Aut}} (V_{\rho})$, the indices $a, b = 1, \dots, d_\rho$ are matrix indices associated to the matrix $\rho(g)$ representing the group element g, and $\newcommand{\vv}{\varphi} \newcommand{\h}{{{\mathcal H}}} \newcommand{\s}{\sigma} \newcommand{\m}{\mu} \hat{\vv}_{\rho} \in V^{\rho} \otimes V^{*\rho}\simeq {\rm{End}}(V_{\rho})$ is the matrix Fourier coefficient of the function $\newcommand{\vv}{\varphi} \vv$. In other words, each $\newcommand{\vv}{\varphi} \newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \hat{\vv}_{\rho}$ is a rank $d_{\rho} = N$ matrix.

Let us consider, as a specific example, the same decomposition for the case of d  =  3, with $G=SU(2)$. The unitary irreps of $SU(2)$, $\newcommand{\m}{\mu} \mathbb{V}^{\,j}$, are labeled by the spin $\newcommand{\m}{\mu} j\in \mathbb{N}/2$. Using the right invariance property of the field, one obtains the following decomposition

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\la}{\langle} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\vv}{\varphi} \displaystyle \label{fourier} \,\vv(g_1,g_2,g_3)= \sum_{\{\,j\}} {\rm Tr}\left[\vv^{\{\,j\}}_{m_1,m_2,m_3}\,\left(\prod_i \sqrt{d_{j_i}}D^{\,j_i}_{m_i,n_i}(g_i)\right)\,\bar{i}^{\{\,j\}}_{n_1,n_2,n_3}\right] \nonumber \end{align} \tag{ 11 }$$

where d_j is the dimension, $\newcommand{\m}{\mu} D^{\,j}(g) \in End(\mathbb{V}^{\,j})$ the group matrix element and $\newcommand{\m}{\mu} {i}^{\{\,j\}}_{n_1, n_2, n_3} \in {\rm{Hom}}_G(\mathbb{V}^{\,j_1}\otimes \mathbb{V}^{\,j_2} \otimes \mathbb{V}^{\,j_3}, \mathbb{C})$ is the three-valent intertwiner operator (related to the Clebsch–Gordan map $\newcommand{\m}{\mu} \Psi^{\,j_3}_{j_1 j_2}: \mathbb{V}^{\,j_1}\otimes \mathbb{V}^{\,j_2} \rightarrow \mathbb{V}^{\,j_3}$). We used the shorthand notation { j } for the set of spin labels $(\,j_1, j_2, j_3)$.

The fields $\newcommand{\vv}{\varphi} \vv^{\{\,j\}}_{m_1, m_2, m_3}$ result from the contraction of the Fourier transformed GFT fields $\newcommand{\vv}{\varphi} \newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \hat{\vv}_{\{\,j\}}$ with the intertwiner tensor imposing the gauge symmetry at the vertex².

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \newcommand{\vv}{\varphi} \displaystyle \vv^{\{\,j\}}_{\{m\}}=\sum_{\{k\}}\hat{\vv}^{\,\{\,j\}}_{\{m\};\{k\}}\,i_{\{\,j\}}^{\{k\}}\,\prod_i \sqrt{d_{j_i}}. \nonumber \end{align} \tag{ 12 }$$

The Fourier transformed fields depend on the (discrete) representation space labels of the Lie group in question. Thus, generically Fourier transformed GFT fields are tensors of some rank d, $\newcommand{\vv}{\varphi} \vv_{\{m_j\}}$ with discrete indices $\vec{m}_j=\{m_1, \dots, m_d\}$³.

In (11), such tensors are contracted with the spin network basis tensors

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\la}{\langle} \displaystyle \label{see} S^{\{\,j\}}_{\{m\}}=\left(\prod_i \sqrt{d_{j_i}}D^{\,j_i}_{m_i,n_i}(g_i)\right)\,\bar{i}^{\{\,j\}}_{\{n\}} , \nonumber \end{align} \tag{ 13 }$$

encoding the properties of the vertex of the spin network graph dual to the (d-1)-dimensional triangulation that can be associated to the GFT states.

Functions $\newcommand{\vv}{\varphi} \vv(g_i)$ can also be understood as single particle wave functions for quanta corresponding to single open vertices of a spin network graph (in fact, they also label coherent states of the GFT field operator, which define the simplest condensate states of the theory [55, 62, 63]).

Let us define these 'single-particle' quantum states as

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\dd}{{\rm d}} \newcommand{\re}{{\rm Re}} \displaystyle \label{gmod} |\varphi \rangle=\int_{G^d} \dd{g_i}~\varphi({g_i})\ket{{g_i}} \nonumber \end{align} \tag{ 14 }$$

where $\newcommand{\dd}{{\rm d}} \newcommand{\e}{{\rm e}} \dd {g_i}\equiv \dd g_1\dd g_2\dots\dd g_d$ is the Haar measure on the group manifold G^d, invariant under the gauge transformation, and the vectors $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{g_1} \dots \ket{g_d}$ provide a basis on the respective infinite dimensional spaces $\newcommand{\s}{\sigma} \newcommand{\m}{\mu} \mathbb{H}\simeq L^2[G]$.

The single particle state $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{\varphi}$ is then defined in $\newcommand{\m}{\mu} \mathbb{H}^{\otimes d}$. Moreover we require $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{\varphi}$ to be normalized (this is of course not the case for the classical GFT fields or the GFT condensate wavefunctions):

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\braket}[2]{\langle {#1}|{#2}\rangle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\dd}{{\rm d}} \displaystyle \braket{\varphi}{\varphi}=\int \dd {g_i} ~\overline{\varphi({g_i})}\varphi({g_i})=1. \nonumber \end{align} \tag{ 15 }$$

Considering the case of $G=SU(2)$, we can decompose the basis $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{g}$ into the unitary irreducible representation of $SU(2)$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\k}{\kappa} \newcommand{\s}{\sigma} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\ra}{\rangle} \newcommand{\re}{{\rm Re}} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \displaystyle \ket{g}\equiv \sum_{j,m,n}\sqrt{d_j} \overline{D^{\,j}_{mn}(g)}\ket{j,n,m^{\dagger}} \nonumber \end{align} \tag{ 16 }$$

and viceversa

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\k}{\kappa} \newcommand{\s}{\sigma} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\ra}{\rangle} \newcommand{\dd}{{\rm d}} \newcommand{\re}{{\rm Re}} \displaystyle \ket{j,n,m^{\dagger}} = \int_{SU(2)}\dd g \sqrt{d_j}D_{mn}^{\,j}(g)\ket{g}. \nonumber \end{align} \tag{ 17 }$$

In particular, the tensor decomposition given in (11) holds at the quantum level, hence defining the quantum fields $\newcommand{\vv}{\varphi} \vv^{\{\,j\}}_{m_1, m_2, m_3}$ as actual tensors states.

Tensors in (13) defines the $SU(2)$-invariant single vertex spin network wave functions (in group representation)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \displaystyle \psi_\chi (g_i)=\langle{\chi}|g_i\rangle=\left(\prod_i \sqrt{d_{j_i}}D^{\,j_i}_{m_i,n_i}(g_i)\right)\,\bar{i}^{\{\,j\}}_{\{n\}} . \nonumber \end{align} \tag{ 18 }$$

The basis vector $\newcommand{\ra}{\rangle} |\chi\rangle=|j, m, i\rangle$ denotes the standard $SU(2)$ spin network basis (labelled by spins and angular momentum projections associated to their d open edges, and intertwiner quantum numbers).

We now describe the quantum states of the formalism, emphasizing their many-body structure, following [64].

Consider a d-valent graph formed by V disconnected components, each corresponding to a single gauge invariant d-valent vertex and d 1-valent vertices, thus having d edges⁴. We refer to this type of disconnected components as open spin network vertices.

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To such a graph we can associate a generic wavefunction given by a function of $d\times V$ group elements,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \displaystyle \label{multi:1} \Phi(g_a^i)=\Phi(g_1^1, ...,g_1^d, g_2^1, ...,g_2^d, \cdots, g_V^1, ...,g_V^d) \nonumber \end{align} \tag{ 19 }$$

defined on the group space ${G^{d \times V}}/{G^V}$ (V copies of G^d, quotiented by the isotropy group of the single particle function $\newcommand{\vv}{\varphi} \vv_{(v)}(g_i)$ at the each vertex); here the index a runs over the set of vertices, while the index i still runs over the links attached to each vertex).

These functions are exactly like many-particles wave functions for point particles living on the group manifold G^d, and having as classical phase space $\newcommand{\h}{{{\mathcal H}}} \newcommand{\m}{\mu} (\mathcal{T}^*G){}^d$ (which is also the classical phase space of a single open spin network vertex or polyhedron).

Accordingly, a state $\newcommand{\ra}{\rangle} \newcommand{\s}{\sigma} \newcommand{\m}{\mu} |\Phi \rangle \in \mathbb{H}_V \simeq L^2[{G^{d \times V}}/{G^V}]$ can be conveniently decomposed into products of single-particle (single-vertex) states,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\vv}{\varphi} \displaystyle \Phi(g_i^a)= \langle g_i^a |\Phi \rangle=\sum_{\chi_i, i=1...V} \vv^{\chi_1...\chi_V}\, \psi_{\chi_1} (g_i)\cdots \psi_{\chi_V} (g_i). \nonumber \end{align} \tag{ 20 }$$

While the above decomposition is completely general, a special class of states can be constructed in direct association with a graph or network Γ. The association works as follows. Start from the d-valent graph with $V$ disconnected components (open spin network vertices) to which a generic $V$-body state of the theory is associated. A partially connected d-valent graph can be constructed by choosing at least one edge i in a vertex a and gluing it to one edge j of the vertex b, i.e. joining the two edges along their 1-valent vertices. The final graph will be fully connected if all edges have been glued. Each pair of glued edges $\{a i, b j \}$ will identify a link L of the resulting (partially) connected graph. In the spin representation, i.e. in terms of the basis of functions $\newcommand{\p}{\partial} \psi_{\chi_1} (g_i)\cdots \psi_{\chi_V} (g_i)$, the gluing is implemented by the identification of the spin labels $j^a_i$ and $j_j^b$ associated to the two edges being glued and by the contraction of the corresponding vector indices $m_i^a$ and $m_j^b$. In other words, the corresponding wave functions for closed graphs can be decomposed in a basis of closed spin network wave functions, obtained from the general product basis by means of the same contractions:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \displaystyle \Phi_\Gamma(g_i^a)= \langle g_i^a |\Phi_\Gamma \rangle =\sum_{\chi_a, a=1...V} \Phi_{\Gamma}^{\,j_i^1...j_i^V}\,\left[\left(\prod_{L\in \Gamma} \delta_{j_i^a , j_j^b} \delta_{m_i^a , m_j^b}\right) \psi_{\chi_1} (g_i)\cdots \psi_{\chi_V} (g_i)\right] \nonumber \end{align} \tag{ 21 }$$

where the coefficients of the wave function can in turn be understood as the resulting of considering generic coefficients $\newcommand{\vv}{\varphi} \vv^{\chi_1...\chi_V}$ and contracting them with some choice of functions $M_{n_i^a n_j^b}^{\,j^a_i j^b_j} \delta_{j^a_i , j^b_j}$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\vv}{\varphi} \displaystyle \Phi_\Gamma^{\,j_i^1...j_i^V} \, = \vv^{\chi_1...\chi_V} \, \left(\prod_{L\in \Gamma} \delta_{j_i^a , j_j^b}\, M^{\,j^a_i j^b_j}_{n_i^a , n_j^b}\right) , \nonumber \end{align} \tag{ 22 }$$

where the contraction is left implicit.

For fixed { j }, each resulting contraction scheme of tensors (each identified by a set of labels χ) defines a tensor network state.

In the group representation, the gluing amounts to considering wave functions with a specific symmetry under simultaneous group translation of the arguments associated to the edges being glued:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \displaystyle \label{multi} \Phi_\Gamma(g_a^i)=\Phi_\Gamma(g_1^1, ...,g_1^d h_{1V}^{d1}, g_2^1, ...,g_2^d, \cdots, g_V^1 h_{1V}^{d1}, ...,g_V^d) . \nonumber \end{align} \tag{ 23 }$$

In the end, given a tensor network with graph Γ, the $\Phi^{\,j_i^1...j_i^V}$ defined above will contain all the information about the combinatorics of the quantum geometry state.

A further special case corresponds to those states for which the coefficients $\newcommand{\vv}{\varphi} \vv^{\chi_1...\chi_V}$ themselves have a product form, i.e. can be decomposed in terms of tensors. In this case, as it is for the spin network wave functions, the coefficients $\Phi^{\,j_i^1...j_i^V}$ can be obtained as a tensor trace

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \newcommand{\vv}{\varphi} \displaystyle \label{tt} \Phi^{\,j_i^1...j_i^V} = {\rm Tr}[\bigotimes_L M \bigotimes_v \vv^{\{\,j\}\,(v)}_{\{m\}}] , \nonumber \end{align} \tag{ 24 }$$

again, in the case of fully connected graphs Γ (otherwise, some angular momentum labels will remain on the left had side, corresponding to the edges that have not been glued). In lattice theory, we would say that the network Γ (fixed { j }) provides a tensor network decomposition of the tensor state $\Phi^{\,j_i^1...j_i^V}$.

The equivalence of a special class of GFT states with the lattice tensor network states, and the sense in which GFT states generalise them, can be further elucidated by the following example.

A tensor $\newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \hat{T}$ is a multidimensional array of complex numbers $\newcommand{\h}{{{\mathcal H}}} \newcommand{\la}{\langle} \newcommand{\m}{\mu} \hat{T}_{\lambda_1, \dots, \lambda_d} \in \mathbb{C}$. The rank of tensor $\newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \hat{T}$ is the number d of indices. The size of an index λ, denoted $\newcommand{\la}{\langle} d_{|\lambda|}$, is the number of values that the index $\newcommand{\la}{\langle} \newcommand{\m}{\mu} \lambda \in \mathbb{N}$ takes [65].

Analogously, at the quantum level, to each leg of the tensor one associates a Hermitian inner product space $\newcommand{\m}{\mu} \mathbb{H}_D$, with dimension D given by the size of the indices $\newcommand{\la}{\langle} \lambda \in \{1, 2, ..., d_{|\lambda|}=D\}$. Given an orthonormal basis $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{\lambda}$, in $\newcommand{\m}{\mu} \mathbb{H}_D$, a covariant tensor of rank d is a multilinear form on the Hilbert space of the vertex $\newcommand{\m}{\mu} {T} \colon \mathbb{H}_D^{\otimes d} \rightarrow \mathbb{C}$. Hence a tensor state is written as

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\k}{\kappa} \newcommand{\s}{\sigma} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\re}{{\rm Re}} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \displaystyle |T\rangle = \sum_{\lambda_1,\dots \lambda_d} \hat{T}_{\lambda_1\cdots \lambda_d}\ket{\lambda_1}\otimes\cdots\otimes\ket{\lambda_d} \nonumber \end{align} \tag{ 25 }$$

where $\newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \newcommand{\la}{\langle} \newcommand{\e}{{\rm e}} \hat{T}_{\lambda_1\cdots \lambda_d} \equiv {T} (\lambda_{1}, \dots, \lambda_{d})$ denote the components in the canonical dual tensor product basis.

A tensor network is generally given by a set of d-valent vertices $v$, corresponding to rank d tensors. In particular, a state corresponding to a set of unconnected vertices is written as a tensor product of individual vertex states

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\ra}{\rangle} \newcommand{\re}{{\rm Re}} \displaystyle \ket{\mathcal{T}_{\mathcal{N}}}\equiv \bigotimes_n \ket{T_n}. \nonumber \end{align} \tag{ 26 }$$

Individual vertex states are glued by links. To each end of a link we associate a Hilbert space $\newcommand{\m}{\mu} \mathbb{H}_D$. The Hilbert space of the link $\newcommand{\e}{{\rm e}} \ell$ is then $\newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \mathbb{H}_{\ell}=\mathbb{H}_D^{\otimes 2}$ and a link state can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\re}{{\rm Re}} \displaystyle \label{tlink} \ket{M}=M_{\lambda_1\lambda_2}\ket{\lambda_1}\otimes\ket{\lambda_2} \nonumber \end{align} \tag{ 27 }$$

where we choose to take the link states $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{M}$ to be generically entangled⁵. In general, the entanglement of the links will encode the information on the connectivity of the graph. Two nodes are connected if their corresponding states contract with a link state,

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\re}{{\rm Re}} \newcommand{\h}{{{\mathcal H}}} \displaystyle \hat{\mathcal{T}}_{12}\equiv \bra{M}\ket{T_1}\ket{T_2} =T^{(1)}_{\lambda_1\cdots\lambda_a\cdots\lambda_v}\overline{M}_{\lambda_a\lambda_b}T^{(2)}_{\lambda'_1\cdots\lambda_b\cdots\lambda_u'} \bigotimes_{i\neq a}^v\ket{\lambda_i}\otimes\bigotimes_{j\neq b}^u\ket{\lambda'_i}. \nonumber \end{align} \tag{ 29 }$$

Notice that if $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{M}$ was a non-entangled state, the connection would be trivial, i.e. the two nodes would be practically disconnected and the corresponding state could be written as a tensor product of two states,

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\re}{{\rm Re}} \newcommand{\h}{{{\mathcal H}}} \displaystyle \nonumber \hat{\mathcal{T}}_{12}&= T^{(1)}_{\lambda_a\lambda_1\cdots\lambda_v}\overline{A}_{\lambda_a}\bigotimes_{i=1}^v\ket{\lambda_i} \otimes \overline{B}_{\lambda_b}T^{(2)}_{\lambda_b\lambda'_1\cdots\lambda_u'}\bigotimes_{j=1}^u\ket{\lambda'_i}\nonumber \\ &=\ket{T'_1}\otimes\ket{T_2'}. \nonumber \end{align} \tag{ 30 }$$

Then given a network $\newcommand{\m}{\mu} \mathcal{N}$ with N nodes and L links, the corresponding state is

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\re}{{\rm Re}} \displaystyle \label{eq:TNS} \ket{\Psi_{\mathcal{N}}}\equiv \bigotimes_\ell^L\bra{M_{\ell}}\bigotimes_n^N\ket{T_n}. \nonumber \end{align} \tag{ 31 }$$

Because all links are contracted with nodes, $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \ket{\Psi_{\mathcal{N}}}$ is then in the Hilbert space associated to the boundary links of the network, which is denoted as $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathbb{H}_{\partial\mathcal{N}}$. $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \ket{\Psi_{\mathcal{N}}}$ is a state in $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathbb{H}_{\partial\mathcal{N}}$.

The above structure can be identified also for the special GFT states mentioned at the end of the previous subsection, which are formed by generalised $L^2(G^d)$ functions associated to the nodes of the network. In this case, the analogous of the generic link state in (27), which is also the group counterpart of the gluing operators associated in the spin representation to the matrices M, can be defined in as the convolution functional

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\dd}{{\rm d}} \displaystyle \bra{M_{g_{\ell}}}\equiv \int\dd g_1\dd g_2~M(g_1^{\dagger}g_{\ell}g_2)~\bra{g_1}\otimes\bra{g_2}~\in \mathbb{H}^{*\otimes 2}, \nonumber \end{align} \tag{ 32 }$$

where the functions $M(g)$ are assumed to be invariant under conjugation M(g)  =  M(hgh⁻¹). When a link $\newcommand{\e}{{\rm e}} \ell$ connects two nodes, say a and b, the corresponding state $\newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\e}{{\rm e}} \bra{M_{g_{\ell}}}$ contracts with states $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{\varphi_a}$ and $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{\varphi_b}$

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\dd}{{\rm d}} \newcommand{\re}{{\rm Re}} \displaystyle \,\bra{M_{g_{\ell}}}\ket{\varphi_a}\ket{\varphi_b}= \int\dd g_1\dd g_2\dd g_i^a\dd g_i^b~M(g_1^{\dagger}g_{\ell}g_2)~\varphi_a(g_{1},g_i^{a})\varphi_b(g_{2},g_i^{b})\ket{g_i^{a}}|{g_i^{b}}\rangle , \nonumber \end{align} \tag{ 33 }$$

where we have singled out, among the arguments of the vertex wave functions $\newcommand{\vv}{\varphi} \vv$ the ones affected by the gluing operation. In these terms, the open d-valent tensor network graph Γ with $V$ vertices, can be written as

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\dd}{{\rm d}} \newcommand{\re}{{\rm Re}} \displaystyle \label{TS} \ket{\Phi_{\Gamma}^{g_\ell}} \equiv \bigotimes_{\ell \in \Gamma} \bra{M_{g_{\ell}}}\bigotimes_{n}^V \ket{\varphi_n}=\int \dd {g}_\p\,\Phi_{\Gamma} ({g}_{\ell},{g}_\p)\ket{{g}_{\p}} \nonumber \end{align} \tag{ 34 }$$

where the $\newcommand{\p}{\partial} \{{g}_{\p}\}$ denote the group elements on the open links.

The role of the link state in tensor network, thus, is naturally generalised by the convolution function, defined for the group field variables. This is due to the fact that the group fields $\newcommand{\vv}{\varphi} \vv(g_i)$ on G^d can be interpreted as rank d tensors, with indices spanning the group space G, and associated Hilbert space (for each index) being L²(G)⁶. The multiparticle state given in (23) can then be interpreted as a tensor state with indices $\newcommand{\p}{\partial} {g}_\p$ and rank given by the number of open links of the spin network graph.

As showed in section 2.4, many-body state can also be decomposed into spin representations. Suppose $\newcommand{\e}{{\rm e}} M(g_1^{\dagger} g_{\ell}g_2)$ can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \displaystyle M(g_1^{\dagger} g_{\ell}g_2)=\sum_{jmn}d_jM_{m n}^{\,j}D_{mn}^{\,j}(g_1^{\dagger} g_{\ell}g_2). \nonumber \end{align} \tag{ 37 }$$

Then, as a simple example, the state $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\e}{{\rm e}} \bra{M_{g_{\ell}}}\ket{\varphi_a}\ket{\varphi_b}$ can be written in terms of $\varphi_{{\bf k}{\bf n}}^{{\bf j}}$, $i_{{\bf m}}$ and $M_{m n}^{\,j}$ as⁷

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\k}{\kappa} \newcommand{\s}{\sigma} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\dd}{{\rm d}} \newcommand{\re}{{\rm Re}} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \displaystyle \nonumber &\bra{M_{g_{\ell}}}\ket{\varphi_a}\ket{\varphi_b} = \int\dd g_1\dd g_2\dd {g}_i^a\dd {g}_i^b~M(g_1^{\dagger}g_{\ell}g_2)\, \varphi_a(g_{1},{g}_i^{a})\varphi_b(g_{2}, {g}_i^{b})\ket{{g}_i^{a}}|{{g}_i^{b}}\rangle\nonumber \\ &= \sum_{jmnklpq}\sum_{i_ai_b}\overline{[i_{a}]_{{\bf p}}}[\varphi_{a}]_{{\bf p}{\bf n}_an}^{\,j{\bf j}_a}M_{nm}^{\,j}\overline{[i_{b}]_{{\bf q}}}[\varphi_{b}]_{{\bf q}{\bf m}_b(-m)}^{\,j{\bf j}_b} (-)^m \ket{{\bf j}_b,{\bf m}_b}\ket{{\bf j}_a,{\bf n_a}} [i_a]_{{\bf k}_ak}D_{kl}^{\,j}(g_{\ell})[i_b]_{{\bf l}_b(-l)} \nonumber \\ &\times (-)^l ~|{\bf j}_a,{\bf k}_a^\dagger\rangle |{\bf j}_b,{\bf l}_b^{\dagger} \rangle. \nonumber \end{align} \tag{ 38 }$$

Graphically, the last line can be presented as

From the graphic equation, one can immediately observe that the upper part is an open tensor network $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{\Phi^{{\bf ji}}}$, given by the tensor trace of a collection of tensors

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \displaystyle \phi_{{\bf m}}^{\,{\bf j}i}\equiv \sum_{{\bf n}}\overline{i_{{\bf n}}}\varphi_{{\bf nm}}^{\,{\bf j}} \nonumber \end{align} \tag{ 40 }$$

for each node and matrices $M_{mn}^{\,j}$ for each link.

We summarize the established dictionary between group field theory states and generalized random tensor networks in terms of two synthetic tables. The correspondence between group field theory and tensor network description is summarized in table A:

Table A	Group fields	Tensors
Group basis	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\s}{\sigma} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \ket{{g_i}\,} \in \mathbb{H} \simeq L^2[G]$	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{\lambda_i}$, $\newcommand{\la}{\langle} \lambda_i=1, \dots , D$ in $\newcommand{\m}{\mu} \mathbb{H}_D$	Index basis
One particle state	$\newcommand{\re}{{\rm Re}} \newcommand{\dd}{{\rm d}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} |\varphi \rangle=\int_{G^d} \dd{g_i}~\varphi({g_i})\ket{{g_i}}$	$\newcommand{\su}{{\mathfrak{su}}} \newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\s}{\sigma} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \ket{T_n} = \sum\nolimits_{\{\lambda_i\}}T_{\{\lambda_i\}}$$\newcommand{\su}{{\mathfrak{su}}} \newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\s}{\sigma} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} |{{\lambda}_i}\rangle \in \mathbb{H}_n=\mathbb{H}_D^{\otimes d}$	Tensor state
Gluing functional	$\newcommand{\dd}{{\rm d}} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \bra{M_{g_{\ell}}}= \int\dd g_1\dd g_2~M(g_1^{\dagger}g_{\ell}g_2)$$\newcommand{\dd}{{\rm d}} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \bra{g_1}\, \bra{g_2}\, \in \mathbb{H}^{*\otimes 2}$	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \ket{M}=M_{\lambda_1\lambda_2}\ket{\lambda_1}$$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}}\otimes\ket{\lambda_2} \in \mathbb{H}_{\ell}=\mathbb{H}_D^{\otimes 2}$	Link state
Multiparticle State	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\s}{\sigma} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \ket{\Phi_{\Gamma}} \in \mathbb{H}_V \simeq L^2[{G^{d \times V}}/{G^V}]$	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \ket{\Psi_{\mathcal{N}}}$	Tensor network state
Product state convolution	$\newcommand{\re}{{\rm Re}} \newcommand{\dd}{{\rm d}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\boldsymbol}\ket{\Phi_{\Gamma}^{g_\ell}} \equiv \bigotimes\nolimits_{\ell \in \Gamma} \bra{M_{g_{\ell}}}\bigotimes\nolimits_{n}^V \ket{\varphi_n}$$\newcommand{\re}{{\rm Re}} \newcommand{\dd}{{\rm d}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\boldsymbol}=\int \dd {g}_\p\, \Phi_{\Gamma} ({g}_{\ell}, {g}_\p)\ket{{g}_{\p}}$	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\boldsymbol}\ket{\Psi_{\mathcal{N}}}\equiv \bigotimes\nolimits_\ell^L\bra{M_{\ell}}$$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\boldsymbol}\bigotimes\nolimits_n^N\ket{T_n} \in \mathbb{H}_{\partial\mathcal{N}}$	Tensor network decomposition
Randomness	$\newcommand{\vv}{\varphi} \newcommand{\f}{\frac} \frac{1}{Z}{\rm d}\nu(\vv)$ Field theory probability measure	$\newcommand{\U}{{\rm U}} \newcommand{\la}{\langle} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} T^U_{\mu}\equiv (UT^0)_{\mu} \\ T^0_\mu\equiv$$\newcommand{\U}{{\rm U}} \newcommand{\la}{\langle} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}}v T^0_{\lambda_1\cdots \lambda_d} \in \mathbb{H}_T,$$\newcommand{\U}{{\rm U}} \newcommand{\la}{\langle} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} U \in \U({\rm{dim}}(\mathbb{H}_T))$	Random tensor state

The generalisation of tensor networks in terms of GFT states is evident in the spin-j decomposition of the latter $\newcommand{\vv}{\varphi} \newcommand{\m}{\mu} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\s}{\sigma} \newcommand{\p}{\partial} \vv(g_i)= \sum\nolimits_{{\bf j}} {\rm Tr}[\, \vv^{\, {\bf j}}_{\{m\}}\, \left(\prod\nolimits_i \sqrt{d_{j_i}}D^{\,j_i}_{m_i, n_i}(g_i)\right)\, \bar{i}^{\, {\bf j}}_{\{n\}}\,]$.

Once we turn off the sum over all possible js, fix the representation labels and ask them to be equal, generically Fourier transformed GFT fields $\newcommand{\vv}{\varphi} \vv^{\, {\bf j}}_{\{m\}}$, are tensors of single rank d, with discrete indices $m_i=\{m_1, \dots, m_d\}$ spanning a finite dimensional space. The equivalence is summarized in table B:

Table B	GFT network	Spin tensor network	Tensor network
Node	$\newcommand{\vv}{\varphi} \newcommand{\e}{{\rm e}} \vv(\vec{g}) \equiv \vv(g_1, g_2, g_3, g_4)$	$\newcommand{\vv}{\varphi} \newcommand{\h}{{{\mathcal H}}} \newcommand{\m}{\mu} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\s}{\sigma} \newcommand{\p}{\partial} \vv^{\, {\bf j}}_{\{m\}} \propto \sum\nolimits_{\{k\}}\hat{\vv}^{\, {\bf j}}_{\{m\}\{k\}}\, i^{{\bf j} \{k\}}$	$\newcommand{\m}{\mu} T_{\{\mu\}}$
Link	$\newcommand{\e}{{\rm e}} M(g_1^{\dagger} g_\ell g_2)$	$M^{\,j}_{mn}$	$\newcommand{\la}{\langle} M_{\lambda_1\lambda_2}$
Sym	$\newcommand{\vv}{\varphi} \vv(h \vec{g})=\vv(\vec{g})$	$\newcommand{\p}{\partial} \prod\nolimits_s^v D^{\,j}_{m_sm'_s}(g)i_{m'_1\cdots m'_v}^i =i_{m_1\cdots m_v}^i$	$\newcommand{\p}{\partial} \newcommand{\m}{\mu} \prod\nolimits_{s}^v U_{\mu_s\mu'_s}T_{\mu'_1\cdots\mu'_v}=T_{\mu_1\cdots\mu_v}$
State	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\boldsymbol}\ket{\Phi_{\Gamma}^{g_\ell}}\equiv \bigotimes\nolimits_\ell\bra{M_{g_{\ell}}}\bigotimes\nolimits_{n}\ket{\psi_n}$	$\newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\boldsymbol}|\Psi_{\Gamma}^{{\bf ji}}\rangle \equiv \bigotimes\nolimits_\ell \langle{M^{\,j_\ell}}|\bigotimes\nolimits_{n}|\phi_n^{~{\bf j}_ni_n}\rangle$	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\la}{\langle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\boldsymbol}\ket{\Psi_{\mathcal{N}}}\equiv \bigotimes\nolimits_\ell^L\bra{M_{\ell}}\bigotimes\nolimits_n^N\ket{T_n}$
Indices	$\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\s}{\sigma} \renewcommand{\k}{\kappa} \newcommand{\m}{\mu} g_i \in {G}, \, \, \ket{{g_i}\,} \in \mathbb{H} \simeq L^2[G]$	$\newcommand{\SU}{{\rm SU}} \newcommand{\m}{\mu} m_i\in\mathbb{H}_j, \, \SU(2)$ spin-j irrep.	$\newcommand{\m}{\mu} \mu_i\in\mathbb{Z}_n$, nth cyclic group
dim	$\infty$	$\newcommand{\m}{\mu} \dim\mathbb{H}_j=2j+1$	$\newcommand{\m}{\mu} \dim\mathbb{Z}_n=n$

In the following sections, with the longer-term goal of a full understanding and computation of the Ryu–Takayanagi (RT) formula [20] in the field-theoretic GFT context, we are going to use the inputs provided by the established dictionary to investigate the holographic RT formula for the case of networks of combinatorial tensor group fields described by means of the GFT formalism and spin network techniques, along the lines proposed for the case of random tensor networks by [23].

In the tensor network generalisation of the gauge gravity duality [21], the RT formula strongly supports a general relation between entanglement and geometry, in turn leading to the suggestion that the whole of spacetime geometry can be understood as emergent from (quantum information-theoretic) properties of non-spatiotemporal quantum building blocks. Of course, this last suggestion has a life on its own and it has been brought forward in many different contexts [63, 66–73]. In this sense, our work provides further steps towards the calculation of the RT formula within a complete quantum gravity setting, a concrete and general indication of the holographic character of gravity, which goes beyond the AdS/CFT gauge gravity duality framework.

We focus now on the case of the Nth Rényi entropy for a bipartite GFT state $\newcommand{\re}{{\rm Re}} \newcommand{\ra}{\rangle} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \renewcommand{\k}{\kappa} \ket{\Psi_{\Gamma}}$ with support on a generic open graph Γ. We divide the boundary $\newcommand{\p}{\partial} \p\Gamma$ of the graph $\newcommand{\p}{\partial} \Gamma(V_\Gamma, L_{\Gamma}, L_{\p\Gamma})$, with $V_{\Gamma}$ nodes, $L_{\Gamma}$ internal links and $\newcommand{\p}{\partial} L_{\p\Gamma}$ boundary links, into two parts, called A and B. The Nth Rényi entropy between A and B is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\ex}{{\rm e}} \newcommand{\f}{\frac} \displaystyle \ex^{(1-N)S_N}=\frac{Z_N}{Z_0^N} \nonumber \end{align} \tag{ 47 }$$

with $\newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \newcommand{\e}{{\rm e}} \nonumber Z_0^{N} \equiv ({\rm Tr}\rho){}^N$, $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} Z_N \equiv {\rm Tr}\rho_A^N = {\rm Tr} [\rho^{\otimes N}\mathds{P}(\pi^0_A;N, d)]$ and the network density matrix ρ defined as

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\re}{{\rm Re}} \newcommand{\tr}{{\rm Tr}} \displaystyle \rho = \ket{\Psi_{\Gamma}}\bra{\Psi_{\Gamma}} = \tr_\ell\left[\bigotimes_{\ell}\ket{M_\ell}\bra{M_\ell}\bigotimes_n\ket{\psi_n}\bra{\psi_n}\right]\equiv \tr_\ell\left[\bigotimes_{\ell}\rho_\ell\bigotimes_n\rho_n\right]. \nonumber \end{align} \tag{ 48 }$$

Here, for convenience, we use the equivalence of the trace of the reduced density with the result of the trace over the action of the permutation operator $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathds{P}(\pi^0_A;N, d)$ on the full $\rho^N$⁸, for

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \displaystyle \mathds{P}(\pi^0_A;N,d)=\prod_{s=1}^N\delta_{\mu_A^{([s+1]_D)}\mu_A^{(s)}} \nonumber \end{align} \tag{ 49 }$$

with d is the dimension of the Hilbert space in the same region A.

Given the random nature of the tensor network, we look for the typical value of the entropy. Analogously to the case considered in [23], the variables Z_N and Z₀ are easier to average than the entropy, since they are quadratic functions of the network density matrix ff. In particular, the entropy average can be expanded in powers of the fluctuations $\newcommand{\m}{\mu} \delta Z_N = Z_N - \mathbb{E}({Z_N})$ and $\newcommand{\m}{\mu} \delta Z_0^{N} = Z_0^{N} - \mathbb{E}({Z_0^{N}})$, so that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\f}{\frac} \displaystyle \mathbb{E}({S_N(A)})&=-\mathbb{E}\left({{\log\frac{\mathbb{E}({Z_N})+\delta Z_N}{\mathbb{E}({Z_0^{N}})+\delta Z_0^N}}}\right)\nonumber \\ \nonumber &=-\log \frac{\mathbb{E}({Z_N})}{\mathbb{E}({Z_0^{N}})}+{\rm fluctuations}. \nonumber \end{align} \tag{ 50 }$$

As showed in [23], for large enough bond dimensions D, as a direct consequence of the concentration of measure phenomenon [74], the statistical fluctuations around the average value are exponentially suppressed. Therefore, it is possible to approximate the entropy with high probability by the averages of Z_N and $Z_0^{N}$,

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\ex}{{\rm e}} \newcommand{\f}{\frac} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \ex^{(1-N)\mathbb{E}(S_N)}&\simeq \frac{\mathbb{E}(\tr\rho_A^N)}{\mathbb{E}(\tr\rho)^N}=\frac{\mathbb{E}\tr [\rho^{\otimes N}\mathds{P}(\pi^0_A;N,d)]}{\mathbb{E}(\tr\rho)^N}\nonumber \\ &= \frac{\tr \left[\bigotimes\nolimits_{\ell}\rho^N_\ell\bigotimes\nolimits_n\mathbb{E}(\rho^N_n)\mathds{P}(\pi^0_A;N,d)\right]}{\tr \left[\bigotimes\nolimits_{\ell}\rho^N_\ell\bigotimes\nolimits_n\mathbb{E}(\rho^N_n)\right]} . \nonumber \end{align} \tag{ 51 }$$

In order to get the typical Rényi entropy one needs then to compute $\newcommand{\m}{\mu} \mathbb{E}({Z_N})$ and $\newcommand{\m}{\mu} \mathbb{E}({Z_0^{N}})$ separately. The average over the tensor fields can be carried out before taking the partial trace, since the latter is a linear operation. Therefore, the key step consists in computing the quantity

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \renewcommand{\k}{\kappa} \newcommand{\ket}[1]{\left|{#1}\right\rangle} \newcommand{\bra}[1]{\left\langle {#1}\right|} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\dd}{{\rm d}} \newcommand{\re}{{\rm Re}} \displaystyle \mathbb{E}(\rho^N_n)=\mathbb{E}[(\ket{\psi_n}\bra{\psi_n})^N]=\mathbb{E}\left[\left(\int\prod_{a}^N\dd{\bf g}_a\dd\underline{{\bf g}_{a}}~\psi_n({\bf g}_{a})\overline{\psi_n(\underline{{\bf g}_{a}})}\ket{{\bf g}_a}\bra{\underline{{\bf g}_{a}}}\right)\right] , \nonumber \end{align} \tag{ 52 }$$

hence, eventually, the expectation value of N copies of the network wavefunction,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \displaystyle \label{eq:Npoint} \mathbb{E}\left[\prod_a^N\psi_n({\bf g}_{a})\overline{\psi_n(\underline{{\bf g}_{a}})}\right] , \nonumber \end{align} \tag{ 53 }$$

where $\newcommand{\dd}{{\rm d}} \newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \dd{\bf g}\equiv \prod\nolimits_i\dd g_i$, $\newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \psi({\bf g})\equiv \psi(g_1, \cdots, g_4)$ and $\underline{{\bf g}}$ is independent from ${\bf g}$, which denotes the arguments of $\newcommand{\p}{\partial} \overline{\psi}$.

Now, we define the averaging operation $\newcommand{\m}{\mu} \mathbb{E}[\cdots]$ via the path integral of a generic group field theory model

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\ex}{{\rm e}} \displaystyle \label{eq:E} \mathbb{E}\left[\,f[\psi,\overline{\psi}]\right]\equiv \int[\mathcal{D}\psi][\mathcal{D}\overline{\psi}]~f[\psi,\overline{\psi}]~\ex^{- S[\psi,\overline{\psi}]} \nonumber \end{align} \tag{ 54 }$$

where $\newcommand{\p}{\partial} S[\psi, \overline{\psi}]$ is the action of the given model of interest,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\dd}{{\rm d}} \displaystyle S[\psi,\overline{\psi}]=\int\dd{\bf g}\dd\underline{{\bf g}}~\overline{\psi({\bf g})}\mathcal{K}({\bf g},\underline{{\bf g}})\psi(\underline{{\bf g}})+ \lambda S_{\rm{int}}[\psi,\overline{\psi}]+{\rm cc}, \nonumber \end{align} \tag{ 55 }$$

the first term on the right hand side defining the kinetic term of the model. In the following calculation, we consider the particular case where

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \displaystyle \mathcal{K}({\bf g},\underline{{\bf g}})=\delta({\bf g}^\dagger\underline{{\bf g}}) , \nonumber \end{align} \tag{ 56 }$$

which thus implies a free part of the action of the simple form⁹

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \displaystyle S_0[\psi,\overline{\psi}]=\int\dd{\bf g}~\overline{\psi({\bf g})}\psi({\bf g}) . \nonumber \end{align} \tag{ 57 }$$

We further assume that the coupling constant λ is much smaller than 1, so the path integral $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathbb{E}\left[\,f[\psi, \overline{\psi}]\right]$ can be perturbatively expanded in powers of λ

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\ex}{{\rm e}} \displaystyle \label{eq:EExp} \mathbb{E}\left[\,f[\psi,\overline{\psi}]\right] &= \int[\mathcal{D}\psi][\mathcal{D}\overline{\psi}]~f[\psi,\overline{\psi}]~\ex^{- S_0[\psi,\overline{\psi}]}\left(1+\lambda S_{\rm{int}}[\psi,\overline{\psi}]+\mathcal{O}(\lambda^2)\right)\nonumber \\ &\equiv \mathbb{E}_0\left[\,f[\psi,\overline{\psi}]\right] + \mathcal{O}(\lambda) . \nonumber \end{align} \tag{ 58 }$$

This is the regime of validity of the so-called spin foam expansion, seen from within the GFT formalism [6–9]. In the following calculation, we will only focus on the leading term $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathbb{E}_0\left[\,f[\psi, \overline{\psi}]\right]$¹⁰.

Because of the gauge symmetry $\newcommand{\p}{\partial} \psi(h{\bf g})=\psi({\bf g})$, the gauge equivalent paths in the above path integral have to be removed (via gauge fixing). In order to do so, we first introduce the following notation: if ${\bf g}=(g_1, g_2, g_3, g_4)$, then

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \displaystyle [{\bf g}]\equiv g_1^{-1}{\bf g}=(\mathds{1},~g_1^{-1}g_2,~g_1^{-1}g_3,~g_1^{-1}g_4) . \nonumber \end{align} \tag{ 59 }$$

Then, we insert the delta functional $\newcommand{\p}{\partial} \delta[\psi({\bf g})-\psi([{\bf g}])]$ constraint into the path integral, so that the average becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \newcommand{\ex}{{\rm e}} \displaystyle \mathbb{E}_0\left[\,f[\psi,\overline{\psi}]\right]\equiv \int[\mathcal{D}\psi][\mathcal{D}\overline{\psi}]~f[\psi,\overline{\psi}]~\delta[\psi({\bf g})-\psi([{\bf g}])]~\ex^{-\int\dd{\bf g}~\overline{\psi({\bf g})}\psi({\bf g})} . \nonumber \end{align} \tag{ 60 }$$

Since this equation is simply the expectation value of $\newcommand{\p}{\partial} f[\psi, \overline{\psi}]$ in the free group field theory, we can immediately give the expectation value of (53) via Wick theorem:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\la}{\langle} \newcommand{\dd}{{\rm d}} \newcommand{\su}{{\mathfrak{su}}} \displaystyle \label{eq:gaugeReIntro} \mathbb{E}_0\left[\prod_a^N\psi({\bf g}_{a})\overline{\psi(\underline{{\bf g}_{a}})}\right]&=\mathcal{C}\sum_{\pi\in\mathcal{S}_N}\prod_a^N\delta\left([{\bf g}_a][\underline{{\bf g}_{\pi(a)}}]^\dagger\right)\nonumber \\ &=\mathcal{C}\sum_{\pi\in\mathcal{S}_N}\int\prod_a^N\dd h_a \prod_a^N\delta\left(h_a {\bf g}_a\underline{{\bf g}_{\pi(a)}}^{\dagger}\right) , \nonumber \end{align} \tag{ 61 }$$

where $\underline{{\bf g}}$ is independent from ${\bf g}$, $\newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \delta([{\bf g}][\underline{{\bf g}}]^\dagger)\equiv \prod\nolimits_{s=2}^4\delta\left(g_{1}^{\dagger}g_s\underline{g_s}^{\dagger}\underline{g_1}\right)$ and $\newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \delta\left(h{\bf g}\underline{{\bf g}}\right)\equiv \prod\nolimits_{s=1}^4\delta\left(h g_s\underline{g_s}^{\dagger}\right)$.

In the second equality, we re-introduce the gauge symmetry by inserting integrals of $\newcommand{\SU}{{\rm SU}} h_a\in \SU(2), N=1, 2, \cdots N$ into the delta functions such that g_sa on each leg of the node are on an equal footing, unlike $\newcommand{\m}{\mu} g_1=\mathds{1}$ in the gauge fixing procedure. So in the following calculation, the network is without gauge fixing, i.e. all integrals of ${\bf g}$ have to be performed.

Denote now $\newcommand{\p}{\partial} \prod\nolimits_a^N\delta\left(h_a {\bf g}_a\underline{{\bf g}_{\pi(a)}}^{\dagger}\right)$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \displaystyle \label{eq:Ph} \mathds{P}_{\bf h}(\pi)\equiv \prod_a^N\delta\left(h_a {\bf g}_a\underline{{\bf g}_{\pi(a)}}^{\dagger}\right)=\prod_{s=1}^4\prod_a^N\delta\left(h_a g_{sa}\underline{g_{s\pi(a)}}^{\dagger}\right)\equiv \prod_s^4\mathds{P}^s_{\bf h}(\pi) , \nonumber \end{align} \tag{ 62 }$$

where ${\bf h}$ denotes the set of h_a, $a=1, \cdots, N$. When $\newcommand{\m}{\mu} h_a=\mathds{1}$ for all a from 1 to N,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \displaystyle \mathds{P}_\mathds{1}(\pi)=\prod_a^N\delta\left({\bf g}_a\underline{{\bf g}_{\pi(a)}}^{\dagger}\right)= \mathds{P}(\pi;N,D^4)=\prod_s^4 \mathds{P}^s(\pi;N,D^4) \nonumber \end{align} \tag{ 63 }$$

where $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathds{P}(\pi;N, D^4)$ and $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathds{P}^s(\pi;N, D^4)$ are the representations of $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi\in\mathcal{S}_N$ on $\newcommand{\m}{\mu} \mathbb{H}^{\otimes 4}$ and $\newcommand{\m}{\mu} \mathbb{H}$, respectively.

Then, Z_N and $Z_0^N$ become

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\dd}{{\rm d}} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\bm}{\boldsymbol} \displaystyle Z_N & \approx \mathcal{C}^{V_\Gamma}\sum_{\bm{\pi_n}\in\mathcal{S}_N}\int\prod_n\dd{\bf h}_n~\tr \left[\bigotimes_{\ell}\rho^N_\ell\bigotimes_n\mathds{P}_{{\bf h}_n}(\pi_n)\mathds{P}(\pi^0_A;N,d)\right]\nonumber \\ & \equiv \mathcal{C}{V_\Gamma}\sum_{\bm{\pi_n}\in\mathcal{S}_N}\int\prod_n\dd{\bf h}_n~\mathcal{N}_A({\bf h_n},\bm{\pi_n}) \nonumber \end{align} \tag{ 64 }$$

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\dd}{{\rm d}} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\bm}{\boldsymbol} \displaystyle Z^N_0 &= \mathcal{C}^{V_\Gamma}\sum_{\bm{\pi_n}\in\mathcal{S}_N}\int\prod_n\dd{\bf h}_n~\tr \left[\bigotimes_{\ell}\rho^N_\ell\bigotimes_n\mathds{P}_{{\bf h}_n}(\pi_n)\right]\nonumber \\ & \equiv \mathcal{C}^{V_\Gamma}\sum_{\bm{\pi_n}\in\mathcal{S}_N}\int\prod_n\dd{\bf h}_n ~\mathcal{N}_0({\bf h_n},\bm{\pi_n}) , \nonumber \end{align} \tag{ 65 }$$

which means that Z_N and $Z_0^N$ correspond to summations of the networks $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{N}_A({\bf h_n}, \bm{\pi_n})$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{N}_0({\bf h_n}, \bm{\pi_n})$ where at each node n we have a contribution $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathds{P}_{{\bf h}_n}(\pi_n)$ and at each link $\newcommand{\e}{{\rm e}} \ell$ we have a contribution $\newcommand{\e}{{\rm e}} \rho_\ell^N$. The only difference between these two networks is the boundary condition: where Z_N is defined with $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathds{P}(\pi^0_A;N, d)$ on A of $\newcommand{\p}{\partial} \p\Gamma$ and $\newcommand{\m}{\mu} \mathds{P}(\mathds{1};N, d)$ on $\overline{A}$ of $\newcommand{\p}{\partial} \p\Gamma$, and $Z_0^N$ is defined with $\newcommand{\m}{\mu} \mathds{P}(\mathds{1};N, d)$ for all boundary region $\newcommand{\p}{\partial} \p\Gamma$.

Since at each node $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathds{P}_{{\bf h}_n}(\pi_n)$ is decoupled among the incident legs, because of (62), the value of the networks $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{N}_A({\bf h_n}, \bm{\pi_n})$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{N}_0({\bf h_n}, \bm{\pi_n})$ can be written as products factorised over links:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\bm}{\boldsymbol} \displaystyle \mathcal{N}_A({\bf h_n},\bm{\pi_n})=\prod_{\ell\in\Gamma} \mathcal{L_\ell}(\pi_{n},\pi_{n'};{\bf h}_{n},{\bf h}_{n'})\prod_{\ell\in A}\mathcal{L}_\ell(\pi_n,\pi_A^0;{\bf h}_n)\prod_{\ell\in\overline{A}}\mathcal{L}_{\ell}(\pi_n,\mathds{1};{\bf h}_n) \nonumber \end{align} \tag{ 66 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\bm}{\boldsymbol} \displaystyle \mathcal{N}_0({\bf h_n},\bm{\pi_n})=\prod_{\ell\in\Gamma} \mathcal{L_\ell}(\pi_{n},\pi_{n'};{\bf h}_{n},{\bf h}_{n'})\prod_{\ell\in \p\Gamma}\mathcal{L}_{\ell}(\pi_n,\mathds{1};{\bf h}_n) . \nonumber \end{align} \tag{ 67 }$$

Because the $\newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \mathcal{L}_{\ell}$ on the boundary are special cases of the $\newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \mathcal{L}_{\ell}$ in the graph Γ, it is enough to calculate the $\newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \mathcal{L}_{\ell}$ on the internal links. In general, $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{L}(\pi, \pi', {\bf h}, {\bf h}')$ can be written as a trace of a modified representation of a permutation group element $\newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \varpi\equiv (\pi'){}^{-1}\pi$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \mathcal{L}(\pi,\pi';{\bf h},{\bf h}')=\tr\left[\mathds{P}_{{\bf h}}(\pi)\rho_{\ell}^N\mathds{P}_{{\bf h}'}(\pi)\right] = \tr\left[\mathds{P}_{{\bf H}}\left((\pi')^{-1}\pi\right)\right]\equiv \tr\left[\mathds{P}_{{\bf H}}\left(\varpi\right)\right] \qquad , \nonumber \end{align} \tag{ 68 }$$

where

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \newcommand{\f}{\frac} \displaystyle \label{eq:Hdef} {\bf H}=\left\{H_a ~\big| ~ H_a\equiv \left(h'_{\varpi(a)}\right)^{\dagger} h_a,~\forall a=1,\cdots,N\right\}. \nonumber \end{align} \tag{ 69 }$$

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When $\newcommand{\p}{\partial} \pi=\pi'$, we have $\newcommand{\m}{\mu} \varpi=\mathds{1}$ and $\newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} {\bf H}=({\bf h}'){}^\dagger{\bf h}$, and then

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \label{eq:LpipiD} \mathcal{L}(\pi,\pi;{\bf h},{\bf h}') = \tr\left[\mathds{P}_{{\bf h}}(\pi)\rho_{\ell}^N\mathds{P}_{{\bf h}'}(\pi)\right]=\tr\left[\mathds{P}_{({\bf h}')^\dagger{\bf h}}\left(\mathds{1}\right)\right] \nonumber \end{align} \tag{ 70 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \displaystyle &= \prod_a^N \int\dd g_a \dd g'_a\dd\underline{g_{\pi(a)}}\dd\underline{g'_{\pi(a)}}~\delta\left(h_{a}g_{a}\underline{g_{\pi(a)}}^\dagger\right)\delta\left(\underline{g_{\pi(a)}}\underline{g'_{\pi(a)}}^{\dagger}\right) \nonumber \\ &\quad\times\delta\left(h'_{a}g'_{a}\underline{g'_{\pi(a)}}^{\dagger}\right)\delta\left(g'_{a}g_{a}^\dagger\right)\nonumber \\ &= \prod_a^N\delta\left((h'_a)^{\dagger} h_a\right)= \prod_a^N\delta(H_a) . \nonumber \end{align} \tag{ 71 }$$

The above equation can be depicted graphically as in figure 5

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When $\newcommand{\p}{\partial} \pi\neq\pi'$, we have

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \mathcal{L}(\pi,\pi';{\bf h},{\bf h}')=\tr\left[\mathds{P}_{{\bf H}}\left(\varpi\right)\right]. \nonumber \end{align} \tag{ 72 }$$

In order to perform the computation, it is necessary to use some facts about the permutation group $\newcommand{\m}{\mu} \mathcal{S}_N$, which we recall briefly, before proceeding.

• Any element $\newcommand{\m}{\mu} \varpi\in\mathcal{S}_N$ can be expressed as the product of disjoint cycles $\newcommand{\m}{\mu} \mathcal{C}_i$
  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \displaystyle \varpi\equiv \prod_i^{\chi(\varpi)}\mathcal{C}_i \nonumber \end{align} \tag{ 73 }$$
  where $1 \leqslant\chi(\varpi)\leqslant N$ is the number of cycles in ϖ, which is 1 when ϖ is a 1-cycle and is N only when $\newcommand{\m}{\mu} \varpi=\mathds{1}$. For instance, the permutation $\varpi=\{3241\}$ can be expressed as a product of two cycles $(134)(2)$, in which $\varpi(1)=3, \varpi(3)=4, \varpi(4)=1$ and $\varpi(2)=2$. $(132)$ is a 3-cycle, because there are three elements in the cycle. We denote the number of elements in the cycle $\newcommand{\m}{\mu} \mathcal{C}_i$ as r_i, which is also called the length of the cycle. We also have $\newcommand{\m}{\mu} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\s}{\sigma} \sum\nolimits_i r_i= N$. Although the cycles $\newcommand{\m}{\mu} \mathcal{C}_i$ commute with each other, we order the cycles such that
  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle 1\leqslant \cdots \leqslant r_i \leqslant r_{i+1}\leqslant \cdots \leqslant N . \nonumber \end{align} \tag{ 74 }$$
  We denote $a^i_k$, where k is from 1 to r_i, the elements of $\newcommand{\m}{\mu} \mathcal{C}_i$, and then we furthermore assume that
  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \varpi(a^i_k)=a^i_{[k]_{r_i}+1} . \nonumber \end{align} \tag{ 75 }$$
  Thus, the cycle can be written as
  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \displaystyle \mathcal{C}_i=\left(a^i_1a^i_2\cdots a^i_{r_i}\right) . \nonumber \end{align} \tag{ 76 }$$
• The trace of $\newcommand{\m}{\mu} \mathds{P}_{{\bf H}}\left(\varpi\right)$ can be expressed as the product of the traces of the individual cycles $\newcommand{\m}{\mu} \mathcal{C}_i$
  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \tr\left[\mathds{P}_{{\bf H}}\left(\varpi\right)\right]=\prod_i\tr\left[\mathds{P}_{{\bf H}}\left(\mathcal{C}_i\right)\right]. \nonumber \end{align} \tag{ 77 }$$
  Using the definition of $\newcommand{\m}{\mu} \mathds{P}_{{\bf H}}$, one can immediately obtain the trace of the cycle $\newcommand{\m}{\mu} \mathcal{C}_i$ as
  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \tr\left[\mathds{P}_{{\bf H}}\left(\mathcal{C}_i\right)\right] = \int\prod_{k=1}^{r_i}\dd g_{a^i_k} ~\delta\left(H_{a^i_k}g_{a^i_k}g^\dagger_{a^i_{[k]_{r_i}+1}}\right)=\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{a^i_k}\right) , \nonumber \end{align} \tag{ 78 }$$
  where
  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \displaystyle \overleftarrow{\prod_{k=1}^{r_i}}H_{a^i_k} \equiv H_{a^i_{r_i}}\cdots H_{a^i_2}H_{a^i_1} . \nonumber \end{align} \tag{ 79 }$$
  Then the trace of $\newcommand{\m}{\mu} \mathds{P}_{{\bf H}}\left(\varpi\right)$ is
  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \mathcal{L}(\pi,\pi';{\bf h},{\bf h}')=\tr\left[\mathds{P}_{{\bf H}}\left(\varpi\right)\right]= \prod_i^{\chi(\varpi)}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{a^i_k}\right) . \nonumber \end{align} \tag{ 80 }$$
• On the boundary of $\newcommand{\m}{\mu} \mathcal{N}_0$ and B of $\newcommand{\m}{\mu} \mathcal{N}_A$, $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{L}(\pi, \mathds{1};{\bf h})$ is a very special case of $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{L}(\pi, \pi';{\bf h}, {\bf h}')$ where $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi'=\mathds{1}$ and $\newcommand{\m}{\mu} {\bf h}'=\mathds{1}$
  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \mathcal{L}(\pi,\mathds{1};{\bf h})\equiv \mathcal{L}(\pi,\mathds{1};{\bf h},\mathds{1}) = \tr\left[\mathds{P}_{{\bf h}}\left(\pi\right)\right] = \prod_i^{\chi(\pi)}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}h_{a^i_k}\right). \nonumber \end{align} \tag{ 81 }$$
  On the boudnary A of $\newcommand{\m}{\mu} \mathcal{N}_A$, $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{L}(\pi, \pi_A^0;{\bf h})$ corresponds also to a special case of $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{L}(\pi, \pi';{\bf h}, {\bf h}')$, where $\newcommand{\m}{\mu} {\bf h'}=\mathds{1}$ and $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi'=\pi_A^0 = \mathcal{C}_0$, which is the N-cycle that for any integer k from 1 to N, $\newcommand{\m}{\mu} \mathcal{C}_0(k)=[k]_N+1$
  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \mathcal{L}(\pi,\pi_A^0;{\bf h})\equiv \mathcal{L}(\pi,\mathcal{C}_0;{\bf h},\mathds{1})=\tr\left[\mathds{P}_{{\bf h}}\left(\mathcal{C}_0^{-1}\pi\right)\right] = \prod_i^{\chi(\mathcal{C}_0^{-1}\pi)}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}h_{a^i_k}\right) . \nonumber \end{align} \tag{ 82 }$$

Altogether, for a given network $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{N}({\bf h_n}, \bm{\pi_n})$, defining the new variables $\newcommand{\m}{\mu} \newcommand{\h}{{{\mathcal H}}} \newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \varpi\equiv (\pi'){}^{-1}\pi$ and ${\bf H}$ given by (69) for each link, the corresponding link value is a product of $\chi(\varpi)$ delta function

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \label{eq:L} \mathcal{L}(\pi,\pi';{\bf h},{\bf h}')\equiv \mathcal{L}(\varpi;{\bf H}) =\tr\left[\mathds{P}_{{\bf H}}\left(\varpi\right)\right]= \prod_i^{\chi(\varpi)}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{a^i_k}\right). \nonumber \end{align} \tag{ 83 }$$

In particular, when $\newcommand{\p}{\partial} \pi=\pi'$, the link value $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{L}(\pi, \pi;{\bf h}, {\bf h}')$ is given by a product of N delta functions as shown in (70) and we re-present it here

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \displaystyle \label{eq:LpipiS} \mathcal{L}(\pi,\pi;{\bf h},{\bf h}') =\prod_a^N\delta\left((h'_a)^{\dagger} h_a\right)=\prod_a^N\delta(H_a) , \nonumber \end{align} \tag{ 84 }$$

which is non-zero only when ${\bf h}={\bf h}'$.

So in the end the network is divided into several regions, in each of which $\newcommand{\p}{\partial} \pi_n$ and ${\bf h}_n$ are the same. The links which connect different regions identify boundaries between each pair of different regions, called again domain walls. Corresponding to different domain walls and different assignments of permutation groups to each region, we have different patterns for the given network. We introduce pattern functions $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_A(\bm{\pi_n})$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_0(\bm{\pi_n})$ such that

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \newcommand{\bm}{\boldsymbol} \displaystyle \mathcal{P}_A(\bm{\pi_n})\equiv \int\prod_n\dd{\bf h}_n~\mathcal{N}_A({\bf h_n},\bm{\pi_n}) \nonumber \end{align} \tag{ 85 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \newcommand{\bm}{\boldsymbol} \displaystyle \mathcal{P}_0(\bm{\pi_n})\equiv \int\prod_n\dd{\bf h}_n~\mathcal{N}_0({\bf h_n},\bm{\pi_n}) . \nonumber \end{align} \tag{ 86 }$$

Given a set of $\newcommand{\p}{\partial} \{\pi_n\}$, $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_A(\bm{\pi_n})$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_0(\bm{\pi_n})$ correspond to a certain network pattern with fixed boundary conditions, illustrated in the following figure.

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More explicitly,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \newcommand{\bm}{\boldsymbol} \displaystyle \mathcal{P}_A(\bm{\pi_n})=\int\prod_n\dd{\bf h}_n~\prod_{\ell\in\Gamma}\left[\prod_{i}^{\chi(\varpi_{\ell})}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{\ell a^i_k}\right)\right]\prod_{\ell\in A}\left[\prod_i^{\chi(\mathcal{C}_0^{-1}\pi_{n\ell})}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}h_{\ell a^i_k}\right)\right]\prod_{\ell\in\overline{A}}\left[\prod_i^{\chi(\pi_{n\ell})}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}h_{\ell a^i_k}\right)\right] \nonumber \end{align} \tag{ 87 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \newcommand{\bm}{\boldsymbol} \displaystyle \mathcal{P}_0(\bm{\pi_n})=\int\prod_n\dd{\bf h}_n~\prod_{\ell\in\Gamma}\left[\prod_{i}^{\chi(\varpi_{\ell})}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{\ell a^i_k}\right)\right]\prod_{\ell\in\p\Gamma}\left[\prod_i^{\chi(\pi_{n\ell})}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}h_{\ell a^i_k}\right)\right]. \nonumber \end{align} \tag{ 88 }$$

They are exactly the amplitudes of a topological BF field theory, with given boundary condition, discretized on a specific 2-complex among the N replica of networks, with each different pattern $\newcommand{\m}{\mu} \mathcal{P}$ corresponding to a different 2-complex. Each edge of the 2-complex is associated with a holonomy h_na that is on node n and the ath replica. The two ends of the holonomy are the vertices of the 2-complex. The h_na inside a delta function form a loop holonomy, the corresponding edges of which form the face of the 2-complex. Then Z_N and $Z_0^N$ are sum of BF amplitudes with different 2-complexes.

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\bm}{\boldsymbol} \displaystyle Z_N\equiv \mathcal{C}^{V_\Gamma}\sum_{\bm{\pi_n}\in\mathcal{S}_N}\mathcal{P}_A(\bm{\pi_n}),\quad Z_0^N\equiv \mathcal{C}^{V_\Gamma}\sum_{\bm{\pi_n}\in\mathcal{S}_N}\mathcal{P}_0(\bm{\pi_n}). \nonumber \end{align} \tag{ 89 }$$

It is important to notice that this simple form of the various functions entering the calculation of the entropy, with the emergence of BF-like amplitudes, is not generic. It follows from the choice of GFT kinetic term, from the approximation used in the calculation of expectation values (neglecting GFT interactions) and from the special type of tensor network, in GFT language, that we have chosen (with simple delta functions associated to the links of the network). More involved, and interesting, cases could be considered.

What we are interested in is the leading term of Z_N and $Z_0^N$, while the dimension D of Hilbert space $\newcommand{\m}{\mu} \mathbb{H}$ is much larger than 1. This leads us to seek the most divergent term of $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_A(\bm{\pi_n})$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_0(\bm{\pi_n})$. In other words, we need to know the degree of divergence of $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_A(\bm{\pi_n})$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_0(\bm{\pi_n})$. The divergence degree of BF amplitudes discretized on a lattice has been the subject of a number of works, both in the spin foam an GFT literature (see for example [76–78]), the most complete analysis being [79–81].

Let us first focus on a sub-region R of the network such that $\newcommand{\p}{\partial} \pi_n=\pi$ for all nodes n inside of R. Suppose that there are L_i links inside R and L_e links connecting with other regions. Since we only consider 4-valent nodes, the number of nodes inside R is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\f}{\frac} \displaystyle V \equiv \frac{1}{4} \left(2 L_i + L_e \right) = \frac{L_i}{2} + \frac{L_e}{4}. \nonumber \end{align} \tag{ 90 }$$

A minimum spanning tree (MST) T, which contains $\#_{T}=V-1$ links, can be found in R.

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \displaystyle T\equiv \{\ell|\ell\in {\rm{MST}}\}. \nonumber \end{align} \tag{ 91 }$$

According to (84), since $\newcommand{\p}{\partial} \pi_n=\pi$, there are N delta functions on each link. The integrals over ${\bf h}_n$ would eliminate the $(V-1)N$ deltas associated to the MST and leave only one set of N integrals over ${\bf h}=\{h_a\}$ and $(L_i/2 - L_e/4+1)N$ $\newcommand{\m}{\mu} \delta(\mathds{1})$'s. Here we keep indicating the divergent factor as the delta function evaluation originating it, but of course it should be understood more properly as a function of the cut-off used to regularize it. The pattern function of region R is then

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \newcommand{\f}{\frac} \displaystyle \mathcal{P}_R(\pi) & \equiv \int\prod_{n\in R}\dd {\bf h}_n ~ \prod_{\ell}^{L_i}\prod_a^N\delta(H_{\ell a})\prod_{\ell}^{L_e}\left[\prod_{i}^{\chi(\varpi_{\ell})}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{\ell a^i_k}\right)\right]\nonumber \\ &=\int\prod_{n\in R}\dd {\bf h}_n \prod_{\ell\in{\rm{MST}}}\prod_a^N\delta(H_{\ell a})\prod_{\ell\notin{\rm{MST}}}\prod_a^N\delta(H_{\ell a}) \prod_{\ell}^{L_e}\left[\prod_{i}^{\chi(\varpi_{\ell})}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{\ell a^i_k}\right)\right]\nonumber \\ &= \left[\delta(\mathds{1})\right]^{\left(\frac{L_i}{2}-\frac{L_e}{4}+1\right)N} \int\dd {\bf h} ~\prod_{\ell}^{L_e}\left[\prod_{i}^{\chi(\varpi_{\ell})}\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{\ell a^i_k}\right)\right]. \nonumber \end{align} \tag{ 92 }$$

In the calculation, we have used

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\dd}{{\rm d}} \displaystyle \int\prod_{n\in R}\dd {\bf h}_n \prod_{\ell\in{\rm{MST}}}\prod_a^N\delta(H_{\ell a}) = \int\dd {\bf h} \nonumber \end{align} \tag{ 93 }$$

and (${\bf h}_n={\bf h}$)

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\f}{\frac} \displaystyle \prod_{\ell\notin{\rm{MST}}}\prod_a^N\delta(H_{\ell a})=\left[\delta(\mathds{1})\right]^{\left(\frac{L_i}{2}-\frac{L_e}{4}+1\right)N} . \nonumber \end{align} \tag{ 94 }$$

The above calculation shows that we can coarse-grain the region R into one single L_e-valent node which is colored by π and ${\bf h}$.

So the degree of divergence in region R is: the number of internal links $\#_i=L_i$ subtracted the number of links in the MST $\#_{T}=V-1$, and then times the number of replica N,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\la}{\langle} \newcommand{\f}{\frac} \displaystyle \label{eq:DivPi} \#_R = (\#_i-\#_{T})N= (L_i-V+1)N = \left(\frac{L_i}{2}-\frac{L_e}{4}+1\right)N . \nonumber \end{align} \tag{ 96 }$$

Since the boundary condition of $\newcommand{\m}{\mu} \mathcal{N}_0$ is $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi=\mathds{1}$ and $\newcommand{\m}{\mu} {\bf h}=\mathds{1}$, the boundary of $\newcommand{\m}{\mu} \mathcal{N}_0$ can be coarse-grained into a single node with $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi=\mathds{1}$ and $\newcommand{\m}{\mu} {\bf h}=\mathds{1}$. The same consideration holds for $\newcommand{\m}{\mu} \mathcal{N}_A$: its boundary can be coarse-grained into two nodes, one of which corresponds to A with $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi=\mathcal{C}_0$, $\newcommand{\m}{\mu} {\bf h}=\mathds{1}$ and the other to B with $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi=\mathds{1}$ and $\newcommand{\m}{\mu} {\bf h}=\mathds{1}$. The corresponding closed graphs are denoted as $\Gamma_0$ and $\Gamma_{AB}$. A certain pattern $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}(\bm{\pi_n})$ divides $\Gamma_0$ and $\Gamma_{AB}$ into M regions that can be coarse-grained into M nodes, each of which is colored with permutation group $\newcommand{\p}{\partial} \pi_m$ and N integrals over ${\bf h}_m$. Denote the graph with pattern $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}(\bm{\pi_n})$ as $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_0(\bm{\pi_m})$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_{AB}(\bm{\pi_m})$, and denote the corresponding coarse-grained graphs as $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma^{c}_0(\bm{\pi_m})$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma^c_{AB}(\bm{\pi_m})$.

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One can show that, for $\Gamma_0$, the pattern in which all nodes have assigned the same permutation group $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi=\mathds{1}$ has the highest degree of divergence $\#_0$.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \#_0 = (\#_{\ell\in\Gamma_0}-\#_{T_{\Gamma_0}})N \nonumber \end{align} \tag{ 97 }$$

where $\newcommand{\e}{{\rm e}} \#_{\ell\in\Gamma_0}$ is the number of links in graph $\Gamma_0$. Let us consider a coarse-grained graph $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma^c_0(\bm{\pi_m})$. Denote the number of links in region m, between regions m and $m'$, and between region m and boundary $\newcommand{\p}{\partial} \p\Gamma$ are L_m, $L_{mm'}$ and L_m0, respectively. The proof goes as follows:

• 1.  
  The permutation group on links between coarse-grained nodes m and $m'$ is $\newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \varpi_{mm'}\equiv \pi_m^{-1}\pi_{m'}$. As given by (83), the number of the delta functions on one of the links is the number of the disjoint cycles in ϖ, which is $\chi(\varpi_{mm'})<N$. Since all links between m and $m'$ are identical, having the same link value, which is given by (83), when one integrate over ${\bf h}_m$ and ${\bf h}_{m'}$, only $\chi(\varpi_{mm'})$ deltas will be eliminated and left with $\newcommand{\m}{\mu} \delta(\mathds{1})$ to the order of $\chi(\varpi_{mm'})(L_{mm'}-1)$ and $2N-\chi(\varpi_{mm'})$ ${\bf h}$ integrals. In fact

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\dd}{{\rm d}} \newcommand{\tr}{{\rm Tr}} \newcommand{\re}{{\rm Re}} \displaystyle \label{eq:step1} \int \dd{\bf h}\dd{\bf h}'\left(\tr\left[\mathds{P}_{{\bf H}}\left(\varpi\right)\right]\right)^{L} &= \int \dd{\bf h}\dd{\bf h}'\prod_i^{\chi(\varpi)}\left[\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{a^i_k}\right)\right]^{L}\nonumber \\ &= \int\dd{\bf h} \prod_i^{\chi(\varpi)}\left[\delta\left(\overleftarrow{\prod_{k=1}^{r_i}}H_{a^i_k}\overrightarrow{\prod_{k=1}^{r_i}}H_{a^i_k}^\dagger\right)\right]^{L-1}\nonumber \\ &= \left[\delta \left(\mathds{1}\right)\right]^{\chi(\varpi)(L-1)}\int\dd{\bf h}. \nonumber \end{align} \tag{ 98 }$$
• 2.  
  MST can be chosen for $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_0(\bm{\pi_m})$, $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma^c_0(\bm{\pi_m})$ and M regions. It is obvious that, given a MST T_m for each of the M regions and a MST $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} T_{\Gamma^c_0(\bm{\pi_m})}$ for $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma^c_0(\bm{\pi_m})$, rooting from the coarse-grained boundary node $\newcommand{\p}{\partial} \p\Gamma$, a MST $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} T_{\Gamma_0(\bm{\pi_m})}$ of $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_0(\bm{\pi_m})$ can be constructed.

  $$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\bm}{\boldsymbol} \displaystyle T_{\Gamma_0(\bm{\pi_m})} = \bigcup_m^M T_m\cup T_{\Gamma^c_0(\bm{\pi_m})}. \nonumber \end{align} \tag{ 99 }$$

  The number of branches of the trees is

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\bm}{\boldsymbol} \displaystyle \#_{T_{\Gamma_0(\bm{\pi_m})}} = \sum_m^M \#_{T_m}+ \#_{T_{\Gamma_0(\bm{\pi_m})}}. \nonumber \end{align} \tag{ 100 }$$
• 3.  
  The degree of divergence of region m is given by (96)

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \#_m = (L_m - \#_{T_m})N. \nonumber \end{align} \tag{ 101 }$$

  Similarly, for the divergence degree $\newcommand{\m}{\mu} \#_{\Gamma^c_0(\mathds{1})}$ of the pattern where all coarse-grained nodes have the same permutation $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \pi_m=\mathds{1}$ is

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\bm}{\boldsymbol} \displaystyle \#_{\Gamma^c_0(\mathds{1})} = \left(\sum_{0\leqslant m<m'\leqslant M}L_{mm'}-\#_{T_{\Gamma^c_0(\bm{\pi_m})}}\right)N. \nonumber \end{align} \tag{ 102 }$$

  The degree of divergence of $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma^c_0(\bm{\pi_m})$ is smaller than $\newcommand{\m}{\mu} \#_{\Gamma^c_0(\mathds{1})}$

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\bm}{\boldsymbol} \displaystyle \#_{\Gamma^c_0(\bm{\pi_m})} < \#_{\Gamma^c_0(\mathds{1})}. \nonumber \end{align} \tag{ 103 }$$

  This is because, after evaluating the delta functions on the MST $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma^c_0(\bm{\pi_m})$ in accordance with (98), there are still $\newcommand{\bm}{\boldsymbol} \newcommand{\m}{\mu} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\s}{\sigma} \newcommand{\p}{\partial} MN-\sum\nolimits_{(mm')\in T_{\Gamma^c_0(\bm{\pi_m})}}\chi(\varpi_{mm'})$ integrals over ${\bf h}$. Performing these integrals makes the degree of divergence of $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma^c_0(\bm{\pi_m})$ not bigger than the following quantity

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\bm}{\boldsymbol} \displaystyle \#_{\Gamma^c_0(\bm{\pi_m})} \leqslant \sum_{0\leqslant m<m'\leqslant M}L_{mm'}\chi(\varpi_{mm'})-\sum_{(mm')\in T_{\Gamma^c_0(\bm{\pi_m})}}\chi(\varpi_{mm'}) \nonumber \end{align} \tag{ 104 }$$

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\bm}{\boldsymbol} \displaystyle = \sum_{(mm')\notin T_{\Gamma^c_0(\bm{\pi_m})}}L_{mm'}\chi(\varpi_{mm'})\nonumber \end{align} \tag{ 105 }$$

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\bm}{\boldsymbol} \displaystyle +\sum_{(mm')\in T_{\Gamma^c_0(\bm{\pi_m})}}(L_{mm'}-1)\chi(\varpi_{mm'}) \nonumber \end{align} \tag{ 106 }$$

  which is definitely smaller than $\newcommand{\m}{\mu} \#_{\Gamma_0(\mathds{1})}$ because $\chi(\varpi_{mm'})<N$.
• 4.  
  So the divergence degree of $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_0(\bm{\pi_m})$ is smaller than the divergence degree $\#_0$ for the pattern where all nodes have the same permutation.

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\bm}{\boldsymbol} \displaystyle \#_{\Gamma_0(\bm{\pi_m})}=\#_{\Gamma^c_0(\bm{\pi_m})}+\sum_m^M\#_m<\#_{\Gamma^c_0(\mathds{1})}+\sum_m^M\#_m = (\#_{\ell\in\Gamma_0}-\#_{T_{\Gamma_0}})N = \#_0. \nonumber \end{align} \tag{ 107 }$$

  The leading term of $Z_0^N$ is $\newcommand{\m}{\mu} \mathcal{P}_0(\mathds{1})$, whose divergence degree is $\#_0$.

  $$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\la}{\langle} \displaystyle Z_0^N = \mathcal{C}^{V_{\Gamma}}[\delta(\mathds{1})]^{\#_0}\left[1+\mathcal{O}(\delta^{-1}(\mathds{1}))+\mathcal{O}(\lambda)\right]. \nonumber \end{align} \tag{ 108 }$$

For Z_N, since the boundary is separated into two parts, the most divergent pattern $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_A(\bm{\pi_n})$ is the one such that its corresponding coarse-grained graph has only two coarse-grained nodes A and B, which are connected by the minimum number of links $\newcommand{\p}{\partial} \newcommand{\m}{\mu} \newcommand{\e}{{\rm e}} \min(\#_{\ell\in\p_{AB}})$, whose divergence degree is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \displaystyle \#_{AB} &= \#_{A}+\#_{B}+\min(\#_{\ell\in\p_{AB}})\nonumber \\ &= \left(\#_{\ell\in\Gamma_{AB}}-\min(\#_{\ell\in\p_{AB}})-\#_{T_{A}}-\#_{T_{B}}\right)N+\min(\#_{\ell\in\p_{AB}})\nonumber \\ &= \left(\#_{\ell\in\Gamma_{AB}}-\#_{T_{A}}-\#_{T_{B}}\right)N+(1-N)\min(\#_{\ell\in\p_{AB}})\nonumber \\ &= \#_0 + (1-N)\min(\#_{\ell\in\p_{AB}}) \nonumber \end{align} \tag{ 109 }$$

where the second equality is in terms of (96) and the forth equality is because $\newcommand{\e}{{\rm e}} \#_{\ell\in\Gamma_{AB}}=\#_{\ell\in\Gamma_{0}}$ and $\#_{T_{A}}+\#_{T_{B}}=\#_{T_{\Gamma_0}}$¹¹.

Let us consider a graph $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_{AB}(\bm{\pi_m})$ and its corresponding coarse-grained graph $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_{AB}^c(\bm{\pi_m})$. The divergence degree of $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_{AB}(\bm{\pi_m})$ is given as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\m}{\mu} \newcommand{\bm}{\boldsymbol} \displaystyle \#_{\Gamma_{AB}(\bm{\pi_m})} = \#_{\Gamma_{AB}^c(\bm{\pi_m})} + \sum_{m = \{1,\cdots M,A,B\}}\#_m \nonumber \end{align} \tag{ 112 }$$

where $\#_m$ is given by (96)

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \#_m = (L_m - \#_{T_m})N. \nonumber \end{align} \tag{ 113 }$$

Adapting the same argument as for $Z_0^N$, because of the integral over ${\bf h_n}$, $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \#_{\Gamma_{AB}^c(\bm{\pi_m})}$ should not be bigger than the following quantity

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\bm}{\boldsymbol} \displaystyle \#_{\Gamma_{AB}^c(\bm{\pi_m})} & \leqslant \sum_{(mm')\notin T^A_{\Gamma_{AB}^c(\bm{\pi_m})},T^B_{\Gamma_{AB}^c(\bm{\pi_m})}}L_{mm'}\chi(\varpi_{mm'})\nonumber \\ &+ \sum_{(mm')\in T^A_{\Gamma_{AB}^c(\bm{\pi_m})}~ {\rm{or}} ~T^B_{\Gamma_{AB}^c(\bm{\pi_m})}}(L_{mm'}-1)\chi(\varpi_{mm'}) \nonumber \end{align} \tag{ 114 }$$

where we assume $m<m'$ in order to avoid double counting, and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} T^A_{\Gamma_{AB}^c(\bm{\pi_m})}$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} T^B_{\Gamma_{AB}^c(\bm{\pi_m})}$ are the MST rooting from coarse-grained nodes A and B, respectively. The right hand side of the above formula corresponds to the divergence degree of pattern $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_A(\bm{\pi_m})$ on a graph $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \overline{\Gamma_{AB}^c(\bm{\pi_m})}$ with all $\newcommand{\m}{\mu} {\bf h_n}=\mathds{1}$, which differs from $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \Gamma_{AB}^c(\bm{\pi_m})$ by $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} T^A_{\Gamma_{AB}^c(\bm{\pi_m})}$ and $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} T^B_{\Gamma_{AB}^c(\bm{\pi_m})}$, i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\bm}{\boldsymbol} \displaystyle \overline{\Gamma_{AB}^c(\bm{\pi_m})} \equiv \Gamma_{AB}^c(\bm{\pi_m})\setminus \{T^A_{\Gamma_{AB}^c(\bm{\pi_m})},~T^B_{\Gamma_{AB}^c(\bm{\pi_m})}\}. \nonumber \end{align} \tag{ 115 }$$

As presented in section 2, the major difference between [23, 27] and our paper is that we are considering the gauge transformation ${\bf h}_n$ on each node n. When all ${\bf h_n}$ are set to be the identity, our Z_N and $Z_0^N$ simplify to the ones in [23, 27] up to overall normalization. In this case, as shown in [23, 27], the patterns which gives only one domain wall for $\Gamma_{AB}$ have higher divergence degree than the divergent degree of multi-domain walls, which in our language means that the patterns whose corresponding coarse-grained graph contains only two coarse-grained nodes are more divergent than the patterns $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_A(\bm{\pi_m})$, which give more than two coarse-grained nodes. So the divergence degree of the pattern $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \newcommand{\m}{\mu} \mathcal{P}_A(\bm{\pi_m})$ on the graph $\newcommand{\bm}{\boldsymbol} \newcommand{\p}{\partial} \overline{\Gamma_{AB}(\bm{\pi_m})}$ is not bigger than the pattern $\newcommand{\m}{\mu} \mathcal{P}_A(\mathds{1}, \mathcal{C}_0)$. So we have

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\s}{\sigma} \newcommand{\su}{{\mathfrak{su}}} \newcommand{\bm}{\boldsymbol} \displaystyle \#_{\Gamma_{AB}(\bm{\pi_m})} &= \#_{\Gamma_{AB}^c(\bm{\pi_m})} + \sum_{m = \{1,\cdots M,A,B\}}\#_m\nonumber \\ & \leqslant \#_{\overline{\Gamma_{AB}^c(\bm{\pi_m})}} + \sum_{m = \{1,\cdots M,A,B\}}\#_{\overline{m}}\nonumber \\ & \leqslant \#_{A}+\#_{B}+\#_{\ell\in\p_{AB}}= \#_0 + (1-N)\#_{\ell\in\p_{AB}}\nonumber \\ & \leqslant \#_{AB} = \#_0 + (1-N)\min(\#_{\ell\in\p_{AB}}). \nonumber \end{align} \tag{ 116 }$$

It follows that the amplitude Z_N is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \displaystyle Z_N = \mathcal{C}^{V_{\Gamma}}[\delta(\mathds{1})]^{\#_0 + (1-N)\min(\#_{\ell\in\p_{AB}})}\left[1+\mathcal{O}(\delta^{-1}(\mathds{1}))+\mathcal{O}(\lambda)\right]. \nonumber \end{align} \tag{ 117 }$$

Finally, the Nth order Rényi entropy S_N is then:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \newcommand{\la}{\langle} \newcommand{\ex}{{\rm e}} \newcommand{\f}{\frac} \displaystyle \ex^{(1-N)S_N} = \frac{Z_N}{Z_0^N} = [\delta(\mathds{1})]^{(1-N)\min(\#_{\ell\in\p_{AB}})}\left[1+\mathcal{O}(\delta^{-1}(\mathds{1}))+\mathcal{O}(\lambda)\right]. \nonumber \end{align} \tag{ 118 }$$

When N goes to 1, S_N becomes the entanglement entropy $S_{\rm{EE}}$. The leading term of the entanglement entropy $S_{\rm{EE}}$ is therefore

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\m}{\mu} \newcommand{\p}{\partial} \displaystyle S_{\rm{EE}} = \min(\#_{\ell\in\p_{AB}})\ln\delta(\mathds{1}) , \nonumber \end{align} \tag{ 119 }$$

which can be understood a the discrete tensor network analogue of the Ryu–Takayanagi formula in a GFT context.

Before moving on to a different derivation of the same result, we want to clarify the interpretation of this calculation.

The definition of the expectation value (54) in the GFT language shows that the exponential of S_N can be interpreted as a GFT 2N-point function, at least within the limits of the approximation made, focusing on the average over group field functions at each node, without recasting the whole generalized tensor network as a GFT correlation function. As shown in previous sections, the GFT amplitudes can in turn be written, by standard perturbative expansion, as a sum of Feynman amplitudes associated to Feynman diagrams, each of which corresponds to a different discretized 'space-time' with fixed boundary, with the Feynman amplitude defining (for quantum gravity models) a lattice path integral for gravity discretised on the corresponding cellular complex. This allows a tentative (and partial) interpretation of the entropy formula we have derived, in geometric spatiotemporal terms. It implies, in fact, that, in the calculation of the entropy, not only the information of a time-slice of a space-time is considered, as encoded in a given network, but also its full quantum dynamics. This, at least, is true when the complete GFT partition function (for quantum gravity models) is employed in the computation of the entropy. The leading term, the free GFT amplitude, captures only a sector of that full quantum dynamics. With the specific (trivial) choice of kinetic term we have used, the quantum dynamics can at best correspond to (summing over) static space-times. When N goes to 1, in particular, the amplitude becomes the trivial propagation of GFT states, with any given network propagating to itself. This corresponds exactly to the context (static space-time) in which the Ryu–Takayanagi formula is usually derived. In other words, our calculation provides a realization of the Ryu–Takayanagi formula, at least in one extremely simple case, within the full dynamics of a non-perturbative approach to quantum gravity, the group field theory formalism, which can also be seen as a different definition of loop quantum gravity. Our result also shows that the same formalism allows to compute non-perturbative quantum gravity corrections to the Ryu–Takayanagi formula, by including the contributions from the GFT interaction term into the amplitude (as well as considering different choices for the GFT kinetic term).