Matter fields in triangle-hinge models

The worldvolume theory of membrane is mathematically equivalent to three-dimensional quantum gravity coupled to matter fields corresponding to the target space coordinates of embedded membrane. In a recent paper [arXiv:1503.08812] a new class of models are introduced that generate three-dimensional random volumes, where the Boltzmann weight of each configuration is given by the product of values assigned to the triangles and the hinges. These triangle-hinge models describe three-dimensional pure gravity and are characterized by semisimple associative algebras. In this paper, we introduce matter degrees of freedom to the models by coloring simplices in a way that they have local interactions. This is achieved simply by extending the associative algebras of the original triangle-hinge models, and the profile of matter field is specified by the set of colors and the form of interactions. The dynamics of a membrane in $D$-dimensional spacetime can then be described by taking the set of colors to be $\mathbb{R}^D$. By taking another set of colors, we can also realize three-dimensional quantum gravity coupled to the Ising model, the $q$-state Potts models or the RSOS models. One can actually assign colors to simplices of any dimensions (tetrahedra, triangles, edges and vertices), and three-dimensional colored tensor models can be realized as triangle-hinge models by coloring tetrahedra, triangles and edges at a time.


Introduction
Recently, the authors constructed a new class of models generating three-dimensional random volumes [1]. We call these models the triangle-hinge models since the fundamental constituent parts of diagrams are "triangles and hinges." 4 This is in sharp contrast to the setup in tensor models [3], [4], [5] or in group field theory [6], [7], where tetrahedra play the role of fundamental constituent parts. The approach of the triangle-hinge models has an intrinsic problem that one cannot assign a priori three-dimensional volumes to general diagrams. However, it was further shown in [1] that one can reduce the set of possible diagrams such that they represent only and all of the tetrahedral decompositions of threedimensional manifolds, by introducing specific interaction terms and taking an appropriate limit of parameters in the models. Therefore, these models can be considered as discretized models of three-dimensional quantum pure gravity.
In this letter, we introduce local matter degrees of freedom to the triangle-hinge models by assigning colors to simplices in those diagrams representing tetrahedral decompositions. Such coloring can be realized by slightly modifying the interaction terms in the original triangle-hinge models and can be made for simplices of arbitrary dimensions (tetrahedra, triangles, edges and vertices). For example, we can realize the Ising model on random volumes by assigning either of two colors (±) to each tetrahedron. We can also setup the qstate Potts models, the RSOS models [8] and even more generic models on random volumes. We further show that the three-dimensional colored tensor models [9] 5 can be obtained by putting specific matters on both tetrahedra and triangles. This paper is organized as follows. In section 2, we review the basic structure of the triangle-hinge models. In section 3, we give a general prescription to introduce matter degrees of freedom to the models. In section 4, we review the Feynman rules of the colored tensor models and show that they can be obtained in the triangle-hinge models by putting specific matters on both tetrahedra and triangles. Section 5 is devoted to conclusion.

Review of the triangle-hinge models
In this section, we give a brief review on the triangle-hinge models (see the original paper [1] for details). 4 A similar approach was taken for three-dimensional topological lattice field theories [2]. 5 Although the original tensor models can generate diagrams not homeomorphic to pseudomanifolds, the colored tensor models are free from this issue [10]. Furthermore, it is known that the colored tensor models have good analytical properties (see, e.g., [11] for a review).

Generalities
The triangle-hinge models generate random diagrams consisting of triangles glued together along multiple hinges. The dynamical variables are N × N real symmetric matrices A and B: and the action is given by where the coupling constants C ijklmn , y i 1 ...i k , λ and µ k are real-valued and have the following symmetry properties: The Feynman diagrams are obtained by expanding the action (2.2) around the kinetic term (1/2)A ij B ij . As in Fig. 1, we represent by triangles the interaction vertices corresponding to λ C i 1 j 1 i 2 j 2 i 3 j 3 , and by k-hinge the interaction vertices corresponding to µ k y i 1 ...i k y j k ...j 1 . The where the two terms on the right-hand side correspond to two ways in gluing an edge of a triangle to that of a hinge (in the same or opposite direction).
The free energy of the model is given by the summation of the Boltzmann weights w(γ) over all possible connected diagrams γ: where S(γ) denotes the symmetry factor of diagram γ, s 2 (γ) the number of triangles, and s k 1 (γ) the number of k-hinges. F (γ) is a function of C ijklmn and y i 1 ...i k and is called the index function of diagram γ.
The triangle-hinge models are characterized by semisimple associative algebras A of linear dimension N. With a basis {e i } (i = 1, . . . , N) of A: A = N i=1 R e i , the multiplication is expressed as Then, the cyclically symmetric rank k tensor y i 1 ...i k is constructed from the structure constants y k ij as (2.9) The rank two tensor y ij is especially denoted by g ij and is called metric; g ij ≡ y ij = y ℓ ik y k jℓ . 6 A possible choice of C ijklmn is   [1]. Two hinges share the same vertex v, around which a connected index network is formed.
of γ have one-to-one correspondence with the vertices of γ. Moreover, one can show that each connected index network around every vertex represents a polygonal decomposition of a closed two-dimensional surface enclosing the vertex (see Fig. 3). 7 6 An associative algebra A is semisimple (i.e. a direct sum of matrix rings) if and only if the metric g = (g ij ) has its inverse g −1 ≡ (g ij ) [12]. In this paper, the indices will be lowered or raised always with g ij or g ij . 7 As is argued in [1], a two-dimensional surface can be uniquely assigned to each connected index network by carefully following the contraction of indices. The index function ζ(v) of vertex v is the product of y i 1 i 2 i 3 and g ij with appropriate contractions of indices, and can be evaluated by repeatedly using the topology-preserving local moves (fusion move and bubble move [12]) which represent the associativity, y l ij y m lk = y l jk y m il , and the definition of the metric, y l ik y k jl = g ij , respectively (see Fig. 4). This Figure 4: Fusion move and bubble move [12]. The index function can be evaluated by repeatedly using these topology-preserving local moves [1].
implies that the index function ζ(v) is the two-dimensional topological invariant defined by the algebra A [12] and is characterized only by the genus g(v) of the connected index network around vertex v : ζ(v) = I g(v) . Therefore, the free energy of the model takes the form

Matrix ring
The simplest semisimple algebra is matrix ring M n (R) = Re ab (with linear dimension N = n 2 ). Here, we take the basis to be {e ab } (a, b = 1, . . . n), where e ab is a matrix unit whose (c, d) element is (e ab ) cd = δ ac δ bd . Note that the index i becomes double index, i = (a, b). When we take A = M n (R) as the defining associative algebra of a triangle-hinge model, the choice of (2.9) and (2.10) gives the action of the form [1] Here, the variables A and B satisfy and we have used the fact that the tensor Thus, the index lines on a triangle are expressed with double lines as in Fig. 5. In this case, Figure 5: Index lines on the triangle in the case of matrix ring [1].
index lines in Fig. 3 become double lines as in Fig. 6. Each polygon formed by an index Figure 6: A connected index network with double lines [1]. This represents a polygonal decomposition of a closed surface. Each polygon is called an index polygon.
loop will be called index polygon. One can show that I g is given by n 2−2g for the connected index network with genus g [1]. Furthermore, the model with A = M n (R) has a strong-weak duality which interchanges the roles of triangles and hinges [1]. In fact, with the variables dual to A and B, the action (2.13) can be rewritten to the form The way to contract the indices ofÃ (orB) in the dual action (2.17) is the same as that of B (or A) in the original action (2.13). Thus, in the dual picture, the diagrams consist of polygons and 3-hinges, which are actually the dual diagrams to the original ones.

Restriction to tetrahedral decompositions
The diagrams generated in the model (2.13) consist of triangles whose edges are randomly glued with others, and do not generally represent tetrahedral decompositions. However, one can setup the models such that the leading contributions represent tetrahedral decompositions. This can be carried out by (i) taking A to be M 3m (R), (ii) modifying the tensor and (iii) taking an appropriate limit of parameters in the model [1].
In fact, with this modification of the tensor C, each index polygon with ℓ segments gives a factor tr ω ℓ , which vanishes unless ℓ ≡ 0 (mod 3). Thus, the index function ζ(v) = I g(v) at each vertex v takes a nonvanishing value (= (3m) 2−2g(v) ) only when the index network around every vertex gives a polygonal decomposition such that the number of segments of every index polygon is a multiple of three. One can further reduce the possible number of segments of every index polygon to three by taking an appropriate large N limit (or equivalently, large m limit). To see this, setting ℓ = 3ℓ ′ , we denote the numbers of 3ℓ ′gons, segments and k-junctions of the index network around vertex v by t 3ℓ ′ 2 (v), t 1 (v) and t k 0 (v), respectively. Then, by using the relations s 2 = 1 , the Boltzmann weight of each diagram can be rewritten as where the function d(v) is given by Thus, if one takes the limit λ → ∞ with λ 2 µ k and 3m/λ being fixed, the leading contribution comes from diagrams where every index network represents a triangular decomposition (ℓ = 3ℓ ′ = 3), that is, every index polygon is an index triangle. Furthermore, it is shown in [1] that such diagrams whose index networks all form triangular decompositions of two-dimensional surfaces represent tetrahedral decompositions. 9

Putting matters
The above prescription to reduce the configurations to tetrahedral decompositions is also effective when A is a tensor product of the form A = A grav ⊗ A mat , where A grav is again M 3m (R) and A mat is another semisimple associative algebra, to be characterizing matter degrees of freedom. In fact, since the structure constants of A are given by the product of the structure constants of A grav = M 3m (R) and those of A mat , the index function F (γ) of each diagram γ is factorized to the product of the contributions from A grav and A mat if we assume that the tensor C has a factorized form C = C grav C mat : Then, by setting C grav to the form (2.18), and by taking the limit λ → ∞ with λ 2 µ k and 3m/λ being fixed, we can again reduce the set of possible diagrams to tetrahedral decompositions, independently of the choice of A mat . 10 In this section, assuming that this reduction is already made, we show that colors representing matter degrees of freedom can be assigned to simplices of arbitrary dimensions (tetrahedra, triangles, edges and vertices) by choosing A mat carefully and by modifying interaction terms appropriately. The number of colors will be denoted by q.
Note that for A = A grav ⊗ A mat the dynamical variables should take the form A abcd,ij and B abcd,ij . In the rest of paper, we omit for brevity the indices a, b, . . . with respect to A grav = M 3m (R).

Coloring tetrahedra
Although the action (2.2) does not have interaction terms corresponding to tetrahedra, we can assign colors to tetrahedra by modifying the interaction terms corresponding to triangles. To show this, we set A mat = M q (R) = q α,β=1 R e αβ and set the interaction terms to be 11 Although this does not ensure the manifoldness of the diagrams, it is also shown in [1] that diagrams can be further reduced to all of the three-dimensional manifolds by introducing a parameter to control the number of vertices. 10 Note that the set of possible diagrams is generally smaller than that of the case A = A grav = M 3m (R). 11 Recall that we are only looking at the matter part. where λ αβ = λ βα , and p α is the projection matrix onto the α-th block of M q (R): The interaction terms can be illustrated as in Fig. 7. If we look at a tetrahedron in diagram γ generated by this modified model, then we find that the four index triangles inside the tetrahedron has the following form (see Fig. 8): Figure 8: Index triangles inside a tetrahedron with triangles colored as in (3.2).

(3.4)
That is, the index function F (γ) can take a nonvanishing value only when the four index triangles of the tetrahedron have the same color α. We thus succeed in assigning colors to tetrahedra in γ. The parameters λ αβ in (3.2) represent the coupling constants for local interactions among the matters on tetrahedra. In fact, the index function F (γ) has the factor λ αβ when an α-colored tetrahedron resides next to a β-colored one.
This colored model describes a q-state system on three-dimensional random volumes. In particular, if we consider the case q = 2 (with colors α = +, −), then the model represents the Ising model coupled to three-dimensional quantum gravity. The system is ferromagnetic when λ ++ ≥ λ +− and λ −− ≥ λ +− . If the global Z 2 symmetry is explicitly broken by setting λ ++ = λ −− , then the model describes a system under an external magnetic field. For general q, we can obtain the q-state Potts models or the RSOS models [8] on random volumes by appropriately choosing λ αβ .

Coloring triangles
We can also assign colors to triangles by making an argument similar to the one in subsection 3.1. We set A mat = M s (R) = s α,β=1 R e αβ , 12 and let the interaction terms take the form The model with (3.5) generates diagrams where a color µ (= 1, . . . , q) is assigned to each triangle. If three triangles (with colors µ, ν, ρ) meet to construct an index triangle in an index network, then the index function has the factor tr(u µ u ν u ρ ). We note that when four triangles meet to construct a tetrahedron as in Fig. 9, F (γ) should have a factor tr(u µ u ν u ρ ) tr(u ν u µ u σ ) tr(u µ u ρ u σ ) tr(u ρ u ν u σ ). (3.6) Such factors give the coupling constants for the local interactions among the matters on Figure 9: Index triangles inside a tetrahedron formed by colored triangles. triangles.
There is another prescription to assign colors to triangles. Suppose that there are q 12 The size s of matrix can be taken arbitrarily as long as coupling constants in a desired form can be obtained from the expression (3.6).

copies of variables A and B, and the action is given by
Then, this model also generates tetrahedral decompositions with colored triangles. The index function F (γ) of the model (3.7) has the factor µ r 1 ...r k k with respect to an edge shared by k triangles with color r 1 , . . . , r k , and thus has a form different from (3.6). Therefore, although both the models (3.5) and (3.7) describe matters on triangles, they give different local interactions.

Coloring edges
Next let us consider the models with colored edges. Similarly to the case of coloring triangles, there are two prescriptions to assign colors to edges.
As in the first prescription in subsection 3.2, we choose A mat = M s (R) = s α,β=1 R e αβ . We now let the interaction terms corresponding to hinges take the form q m=1 k≥2 This generates diagrams where each edge has a color m (m = 1, . . . , q), and each index triangle gives the factor tr(u m 1 u m 2 u m 3 ) depending on the colors of the edges. They give the coupling constants for the local interactions among the matters on edges.
Another prescription to assign colors to edges can be given by modifying the action (2.2) to the following form: Similarly to the case of subsection 3.2, the contribution to the free energy is different from that of the former prescription. In the latter prescription, each triangle gives the factor λ r 1 r 2 r 3 if three hinges (with colors r 1 , r 2 , r 3 ) meet there .

Coloring vertices
We can also assign colors to vertices although the action (2.2) does not have interaction terms corresponding to vertices.
In order to color vertices, we set the matter associative algebra to be A mat = A 1 ⊕. . .⊕A q , and let the interaction terms corresponding to hinges take the form q α,β=1 k≥2 where y (α) i 1 ...i k take nonvanishing values only when all the indices i 1 , . . . , i k belong to A α . Accordingly, all the junctions in the same connected index network should have the same color α so that the index function F (γ) takes nonvanishing values. Thus, we can assign a color to the index network of each vertex in diagram γ, and can say that the model generates diagrams with colored vertices. If two vertices with color α and β are next to each other in diagram γ, the index function F (γ) has the factor µ αβ k . This coloring of vertices can also be given by setting A mat = M q (R) and letting the interaction terms corresponding to hinges take the form q α,β=1 k≥2 where p α is the projection matrix onto the α-th block of M q (R) [the same one given in (3.3)]. In this case, by using the duality transformation (2.16), we find that this model is dual to the model with (3.2). That is, the action with the interaction term (3.11) can be regarded as a q-state system on the dual lattice of γ.
We thus conclude that the triangle-hinge models admit the introduction of matters which can live on simplices of any dimensions and interact with themselves locally.

Relations to the colored tensor models
In this section we show that the three-dimensional colored tensor models [11] can be obtained from the triangle-hinge models by putting specific matters on both tetrahedra and triangles.

Review of the colored tensor models
We first review the basics of the three-dimensional colored tensor models (see, e.g., [11] for a review). The dynamical variables are given by a pair of rank-three tensors φ µ IJK andφ µ IJK , which represent two kinds of colored triangles. Here, {I} is the set of indices assigned to edges, and {µ} = {1, 2, 3, 4} are the colors assigned to triangles. 13 The action takes the form Looking at the way of contraction of index I, one easily sees that this action generates the Feynman diagrams where the interaction vertices can be identified with tetrahedra which are glued at their faces through the propagator. Since there are two types of interaction terms λ φ 4 andλφ 4 , the set of tetrahedra can also be decomposed to two different classes, which we label with α = ±, respectively. Each tetrahedron has four triangles, to which four different colors µ = 1, . . . , 4 are assigned. We say that the tetrahedron has positive (or negative) orientation if triangles 1, 2, 3 are located clockwise (or counterclockwise) when seen from triangle 4 (see Fig. 10). Since the kinetic term has the form φφ (not including φ 2 orφ 2 ), two adjacent tetrahedra must have different orientations as in Fig. 10. Figure 10: A part of a Feynman diagram in the colored tensor models. There are two tetrahedra, one corresponding to an interaction vertex proportional to λ and the other toλ. The two adjacent tetrahedra have opposite orientations.
The Feynman rules for the colored tensor models (4.1) thus can be summarized as follows: 1. Interaction vertices are represented by two types of tetrahedra, α = ±, and any two adjacent tetrahedra have different types.
2. To the four triangles of each tetrahedron are assigned four different colors µ = 1, . . . , 4, and this assignment of colors agrees with the orientation of the tetrahedron when α = +, while it is opposite when α = −.

Realization of the colored tensor models from the trianglehinge models
The above Feynman rules for the there-dimensional colored tensor models can be obtained from the triangle-hinge models by putting specific matters on both tetrahedra and triangles. To see this, we set the defining associative algebra to be a matrix ring M 2s (R) and further let the interaction terms corresponding to triangles take the form Here we assume that the matrices u ±µ (µ = 1, . . . , 4) satisfy the relations where ǫ µνρσ is the totally antisymmetric tensor with ǫ 1234 = 1. The corresponding interaction vertices can be expressed as in Fig. 11. This is a hybrid of two prescriptions described Figure 11: Triangle vertex realizing the colored tensor models.
in subsections 3.1 and 3.2; colors are assigned to both tetrahedra and triangles. Each tetrahedron has a type (orientation) α = ± and each triangle has a color µ = 1, . . . , 4. Note that we particularly set λ αβ as λ ++ = λ −− = 0 (and λ +− = λ −+ = λ), so that any two adjacent tetrahedra have different types. As can be seen from (3.6), a tetrahedron of type α = + (or α = −) gives the factor tr u αµ u αν u αρ tr u αν u αµ u ασ tr u αµ u αρ u ασ tr u αρ u αν u ασ (α = ±), (4.7) which takes a nonvanishing value (= 1) only when the four colors µ, ν, ρ, σ correspond to the positive (or negative) orientation, and thus the tetrahedra in the nonvanishing Feynman diagrams are all positively oriented for α = + and are all negatively oriented for α = −. This is exactly the same as the Feynman rules of the colored tensor models, which was briefly reviewed in the previous subsection.
The matrices u µ can be chosen arbitrarily as long as they satisfy the relations (4.6). For example, one can take the following 6 × 6 matrices: where σ i (i = 1, 2, 3) are Pauli matrices.

Conclusion
In this letter, we give a general prescription to put matters on the triangle-hinge models. We show that color degrees of freedom can be assigned to simplices of any dimensions (tetrahedra, triangles, edges and vertices), by setting the associative algebra A to be a matrix ring M n (R) and by appropriately modifying the interaction terms of the models. Since a colored simplex can only interact with its neighbors, the matters corresponding to the colors have local interactions. We also show that there exists a duality between matters on a tetrahedral lattice and those on its dual lattice, which can be realized by making the duality transformation (2.16) for the variables.
Simple and interesting examples obtained by the above coloring include the Ising model, the q-state Potts models and the RSOS models coupled to three-dimensional quantum gravity, which are obtained by assigning colors to tetrahedra as in (3.2).
We can also construct various kinds of models by combining several prescriptions given in section 3. For example, the three-dimensional colored tensor models can be obtained by assigning colors to both tetrahedra and triangles, as shown in subsection 4.2 by explicitly demonstrating that the same Feynman rules are obtained.
As a future direction of the present work, it should be interesting to investigate the critical behaviors of the triangle-hinge models with matters. It should be especially important to study the case where the matters correspond to the target space coordinates of an embedded membrane. A study in this direction is now in progress and will be communicated elsewhere.