Triangle-hinge models for unoriented membranes

Triangle-hinge models [arXiv:1503.08812] are introduced to describe worldvolume dynamics of membranes. The Feynman diagrams consist of triangles glued together along hinges and can be restricted to tetrahedral decompositions in a large N limit. In this paper, after clarifying that all the tetrahedra resulting in the original models are orientable, we define a version of triangle-hinge models that can describe the dynamics of unoriented membranes. By regarding each triangle as representing a propagation of an open membrane of disk topology, we introduce a local worldvolume parity transformation which inverts the orientation of triangle, and define unoriented triangle-hinge models by gauging the transformation. Unlike two-dimensional cases, this local transformation generally relates a manifold to a nonmanifold, but still is a well-defined manipulation among tetrahedral decompositions. We further show that matter fields can be introduced in the same way as in the original oriented models. In particular, the models will describe unoriented membranes in a target spacetime by taking matter fields to be the target space coordinates.


Introduction
The worldvolume theory of membranes in a spacetime is equivalent to a system of threedimensional quantum gravity coupled to matter fields corresponding to the target space coordinates. One approach to treating such class of systems is the use of models that generate three-dimensional random volumes. Triangle-hinge models [1] are proposed as such models. The dynamical variables are given by a pair of N × N symmetric matrices, A and B, and the Feynman diagrams consist of triangles glued together along their edges. We can restrict diagrams such that they represent only three-dimensional tetrahedral decompositions by taking a large N limit. The simplest model thus obtained corresponds to discretized three-dimensional pure quantum gravity with a bare cosmological constant. We can further introduce extra degrees of freedom representing the target space coordinates. A prescription to introduce such matter degrees of freedom to triangle-hinge models is given in [2]. The prescription also enables us to describe various spin systems such as the q-state Potts models coupled to quantum gravity, and to realize colored tensor models [3,4] in terms of trianglehinge models.
As pointed out in [1], the original triangle-hinge models generate only (and all of the) orientable tetrahedral decompositions. In this paper, we generalize the models such that unoriented membranes can be treated. We will call the obtained models unoriented triangle-hinge models. In the context of string theory, to consider unoriented models is not just interesting as a mathematical generalization, but has a physically important meaning. Actually, an unoriented superstring theory, type I superstring, is one of (perturbatively) consistent superstring theories. We expect that unoriented membrane theory is also physically important. 1 In two-dimensional cases, an unoriented theory is obtained by gauging the worldsheet parity of an oriented theory. If we discretize worldsheets by triangular decompositions, the gauging procedure is to treat equally two ways to identify an edge of a triangle with that of another triangle; one way preserves the local orientations of two triangles and the other does not. We will define unoriented membrane theories by generalizing the prescription to three-dimensional tetrahedral decompositions. Roughly speaking, the unoriented models equally treat two possible ways to identify a triangle of a tetrahedron with that of another tetrahedron; one way preserves the orientation and the other does not.
Here, we make comments on the treatment of orientability in other three-dimensional random volume theories, tensor models. Tensor models [5,6,7] are natural generalizations of matrix models to higher dimensions. One can introduce various kinds of tensor models [6] depending on how indices of rank-3 tensors are assigned to triangles in tetrahedral decompositions. 2 A class of models, where each index is assigned to a vertex of a triangle, generate only orientable tetrahedral decompositions [6]. Although, by modifying the models, we may be able to construct unoriented models, it is difficult to solve them. Analytical treatment of 1 We should comment that the low energy effective theory of unoriented supermembranes is not the eleven-dimensional supergravity, because the 3-form fields in the supergravity multiplet cannot couple to unoriented membranes. Nevertheless, we expect that unoriented membrane theory serves as a toy model to obtain a better understanding of dynamics of membranes. 2 There is another type of tensor model (called the canonical tensor model) which realizes the constraints in the canonical quantization of gravity [8,9,10]. An interesting connection to random tensor networks is studied in [11,12].
tensor models is improved in colored tensor models [3,4]. The models only generate tetrahedral decompositions belonging to a specific class. 3 This restriction enables us to take a 1/N expansion of the free energy [14,15]. Furthermore, in the so-called invariant models [16] one can take the double scaling limit [17,18]. It can be shown that tetrahedral decompositions generated by colored tensor models are orientable. However, if we try to modify the models to unoriented ones, the solvability of the models may be lost.
Original triangle-hinge models are expected to be solvable because the dynamical variables are matrices. In fact, for simple models (such as the models characterized by matrix rings), the interaction terms in the action can be rewritten to the traces of powers of matrices [19]. Thus, in order to reduce the systems to those of eigenvalues, we only need to integrate the exponential of the quadratic term in the action over the angular parts of matrices. Moreover, numerical integrations [19] show that the eigenvalue distributions of matrices A and B have a similar structure to those of one-matrix models with double-well potentials, and that the effective theory of eigenvalues for either of matrix A or B has critical points. Since there are integration contours for which the matrix integrations are finite [19], it is highly expected that the original oriented triangle-hinge models have well-defined continuum limits. We will see that the actions of unoriented triangle-hinge models have a similar structure to the original ones, and thus we expect that unoriented triangle-hinge models are also solvable and might be easier to solve due to the higher symmetry they have. This paper is organized as follows. In section 2, we review triangle-hinge models and show that the tetrahedral decompositions generated by the models are orientable. In section 3, after reviewing matrix models for unoriented strings, we define unoriented membrane theories in terms of tetrahedral decompositions. In section 4, we give a version of trianglehinge models that realize unoriented membrane theories. In section 5, we show that matter fields can be introduced also to unoriented triangle-hinge models by the same procedure as [2]. Section 6 is devoted to conclusion.

Orientability in triangle-hinge models
In this section, we clarify the fact that the original triangle-hinge model [1] generates the set of oriented tetrahedral decompositions.

Brief review of triangle-hinge models
Triangle-hinge models [1] are designed to generate Feynman diagrams each of which can be regarded as a collection of triangles glued together along multiple hinges and will eventually give a three-dimensional tetrahedral decomposition in a large N limit (see Fig. 1). 4 Figure 1: A part of a configuration consisting of triangles glued together with multiple hinges [1].
The dynamical variables are given by a pair of N × N real symmetric matrices, A = (A ij = A ji ) and B = (B ij = B ji ), and the action takes the form 5 The free energy is given by and if we expand F with respect to λ and µ k , each term is expressed by a group of Wick contractions as usual. There are two types of interaction vertices; one (coming from λ [CA 3 ]) corresponds to a triangle and the other (coming from µ k [Y k B k ]) to a multiple hinge (see Fig. 2). The coefficients C ijklmn and Y i 1 j 1 ...i k j k are real constant tensors, and we will not assume any symmetry for the indices of C i 1 j 1 i 2 j 2 i 3 j 3 and Y i 1 j 1 ...i k j k until we give their explicit 4 Here, a (multiple) hinge is an object connecting edges of triangles. A hinge with k edges is called a k-hinge. 5 Note that we have included in the action the interaction term corresponding to 1-hinges.
forms later [see (2.12) and (2.10)]. 6 They are connected by a free propagator The two terms on the right-hand side in (2.3) express that there are two ways to connect an edge of a hinge to an edge of a triangle as shown in Fig. 3. We regard the two types of pairing as representing two independent Wick contractions and write the first type as AB (+) or simply as AB (+) and the second type as AB (−) = AB (−) , where we have omitted the indices i, j, . . .. 7 Then, a group of Wick contractions (denoted by x) can be specified uniquely in a form such as where the subscript I of A I and B (±) I indicates that they belong to the I-th contraction of type AB (±) . As explained above, we do not impose a symmetry for the coefficients C and Y , and think that changing the order of the labels in [ leads to a different group of Wick contractions.
In [1], we investigated in detail the case where the interaction terms µ k [Y k B k ] are characterized by a semisimple associative algebra A. Let a basis of A be {e i } (i = 1, . . . , N), 6 In fact, when multiplied by A i1j1 A i2j2 A i3j3 (A ij = A ji ), only fully symmetric part of C i1j1i2j2i3j3 survive that are invariant under interchanges of indices i α and j α (α = 1, . . . , 3) and under permutations of three pairs of indices (i 1 j 1 ), (i 2 j 2 ), (i 3 j 3 ), so we could have assumed that the tensor C in (2.1) have the symmetry C i1j1i2j2i3j3 = C i2j2i3j3i1j1 = C j1i1i2j2i3j3 = C i2j2i1j1i3j3 . We, however, do not assume this symmetry and regard contractions using C i1j1i2j2i3j3 , C i2j2i3j3i1j1 , C j1i1i2j2i3j3 or C i2j2i1j1i3j3 as giving independent Wick contractions [1,2]. Note that only the fully symmetric part is actually left when all the diagrams are summed. The same argument is applied to the hinge parts.
where N is the dimension of A as a linear space. The multiplication × of A is then specified by the structure constants y k ij as e i × e j = y k ij e k . (2.5) We further introduce a rank k tensor from the structure constants as 8 which enjoys the cyclic symmetry, y i 1 ...i k = y i 2 ...i k i 1 . Then the coupling constants Y i 1 j 1 ...i k j k associated with hinges are defined to be which enjoy the symmetry properties In order to restrict configurations so as to represent only tetrahedral decompositions, we consider the case where the algebra A is a matrix ring A = M n=3m (R) with n being a multiple of three [1]. The dimension of A is then given by N = n 2 = (3m) 2 . We take a basis {e i } to be {e ab } (a, b = 1, . . . n), where e ab are the matrix units whose (c, d) elements are (e ab ) cd = δ ac δ bd . Note that indices i are replaced by double indices ab. Then, the rank k tensors (2.6) are given by which in turn give the k-hinge tensor Y i 1 j 1 i 2 j 2 ...i k j k as (2.10) We further introduce a permutation matrix ω of the following form: and set the tensor C i 1 j 1 i 2 j 2 i 3 j 3 in (2.1) to be (2.12) Note that C + enjoys the symmetry properties (2.13) The action then takes the form 14) The interaction vertices can be represented by thickened triangles and hinges (see Fig. 4), and are connected with the use of two types of Wick contractions between A abcd and B ef gh , Note that arrows are assigned to index lines on triangles as in Fig. 4, and that their directions We here introduce some terminology to discriminate between a group of Wick contractions and a Feynman diagram. We have already clarified our rule about Wick contractions. We now introduce an equivalence relation to the set {x} of groups of Wick contractions, saying that a group x of Wick contractions is equivalent to another group y (and writing x ∼ y) if x is obtained from y by repetitive use of the relations (2.8) and (2.13) and by permuting interaction vertices of the same type. We denote the equivalence class of x by [x] = {y | y ∼ x}, and call γ = [x] a Feynman diagram. The perturbative expansion of the free energy is then given by a sum over connected Feynman diagrams γ, F = γ F (γ) . In a connected diagram, every index line makes a loop since all the indices are contracted.

Restriction to tetrahedral decompositions
Although configurations generated in triangle-hinge models do not generally represent tetrahedral decompositions, the set of Feynman diagrams can be reduced such that they represent only (and all of the) tetrahedral decompositions if we take a large n limit with n/λ and n 2 µ k fixed [1]. 9 The point is the following. We have tr ω ℓ when ω appears ℓ times in an index loop (see Fig. 5), and thus F (γ) is given by Here, t ℓ 2 (γ) denotes the numbers of index ℓ-gons in diagram γ. 10 s 2 (γ) and s k 1 (γ) denote the number of triangles and k-hinges in diagram γ, respectively, and S(γ) is the symmetry factor. Due to the definition of matrix ω, we have Thus, there can survive only the diagrams with ℓ a multiple of three (ℓ ≡ 3ℓ ′ ), and we can assume (2.16) to take the form One can show that only the index polygons with ℓ = 3 (i.e. ℓ ′ = 1) survive in the limit n → ∞ with n/λ and n 2 µ k fixed [1]. We here give a proof in a form slightly different from 9 The set of tetrahedral decomposition can be further restricted so as to represent manifolds by extending the algebra A as having a center to count the number of vertices [1]. 10 An index loop is called an index ℓ-gon if it consists of ℓ intervals, each living on a side of an intermediate triangle [1].
the original one such that it can be applied to unoriented models. We first note that the relation ℓ ′ ≥1 3ℓ ′ t 3ℓ ′ 2 (γ) = 6s 2 (γ) holds because the left-hand side counts the number of ω in diagram γ and each thickened triangle has six insertions of ω. Then, if we introduce a nonnegative quantity . Thus, λ s 2 (γ) can be rewritten as Substituting this expression to (2.18), F (γ) is expressed as . (2.20) Therefore, in the limit n → ∞ with n/λ and n 2 µ k fixed (and thus λ → ∞), only the diagrams satisfying d(γ) = 0 can give nonzero contributions to the free energy. Since d(γ) = 0 means that all the index polygons in γ are triangles, we conclude that the large n limit reduces the set of diagrams so that all the index polygons are triangles. One can further prove that such diagrams represent tetrahedral decompositions [1], as may be understood intuitively from the fact that if there is an index triangle, then sides of thickened triangles must be attached as in Fig. 6. 11 Figure 6: An index triangle made on three sides of thickened triangles, which form a corner of a tetrahedron.
We end this subsection with a comment. The above argument can also be applied to unoriented models to be defined in the next section. Namely, if a set of diagrams is reduced such that all the index polygons are triangles, then the diagrams represent tetrahedral decompositions even for unoriented models.

Orientability
It is pointed out in [1] that all the tetrahedral decompositions generated by the action (2.14) are orientable. We here give a detailed proof of this statement, by clarifying the definition of orientation for Feynman diagrams in a triangle-hinge model.
We first recall that a thickened triangle has two triangular sides, on each of which directed index lines are drawn [see Fig. 4 (a)]. Given a tetrahedron T formed by four triangular sides (each coming from a thickened triangle), we embed it to a three-dimensional Euclidean space E 3 as a regular tetrahedron of unit volume. Note that there can be two embeddings f + and f − (up to rotations and translations in E 3 ), depending on whether the directions of index lines are counterclockwise or clockwise when seen from the center of the embedded tetrahedron (see Fig. 7). 12 We say the former embedding to be positive and the latter Figure 7: A positively oriented tetrahedra T + , corresponding to a positive embedding of T in E 3 .
negative. We then define an oriented tetrahedron T ± to be the pair of tetrahedron and embedding, T ± ≡ (T, f ± ).
When two positively oriented tetrahedra T + 1 and T + 2 are glued at a triangle ∆, we say that the orientation is preserved if the two positive embeddings f + 1 and f + 2 can be extended (with the use of rotations and translations) to a common embedding f of T + 1 ∪ T + 2 such that the images of two tetrahedra are in opposite positions with respect to the intermediate triangle ∆ (see Fig. 8). We then say that a tetrahedral decomposition Γ is orientable if the orientation is preserved for any two adjacent tetrahedra of positive orientation.
The above orientability condition actually holds for tetrahedral decompositions discussed in the previous subsection. In fact, the index lines on the two sides of a thickened triangle are drawn in opposite directions as in Fig. 4 (a), and thus, for any two adjacent tetrahedra there always exists a natural extension of their positive embeddings such that the images of two tetrahedra are in opposite positions with respect to the triangle. Since it holds for every two adjacent tetrahedra, we conclude that all the tetrahedral decompositions are orientable.

Unoriented membrane theories
In this section, we define unoriented membrane theories in terms of tetrahedral decompositions. A realization of unoriented membrane theories within the framework of triangle-hinge models will be given in the next section.

Matrix models for unoriented strings
As a warm-up before discussing unoriented membrane theories, we review the definition of unoriented string theories and how some of them are realized in terms of real symmetric matrix models.
We first recall that an oriented open string is an oriented one-dimensional object with two ends. If we forget about the target-space degrees of freedom, the scattering processes of oriented open strings are represented by Feynman diagrams of Hermitian matrix models: In fact, the propagator and the interaction vertex are expressed as 13 propagator : Each Feynman diagram can also be thought of as a triangular decomposition of an orientable two-dimensional surface by representing it with the dual diagram.
We now introduce a transformation Ω which acts on one-string states and inverts the worldsheet parity (the orientation of string). Unoriented open string theories are then defined as theories where the transformation Ω is gauged (see, e.g., [20]). Namely, we demand that every propagator in the open-string channel be invariant under the action of Ω . This is realized by inserting the projector (1 + Ω)/2 to every propagator. If we do not change the form of interaction, the Feynman rules are then expressed as follows (we have rescaled the projector for later convenience): propagator : It is easy to see that the above Feynman rules are obtained from a real symmetric matrix model: A Feynman diagram for the above unoriented string theory can also be represented as a collection of triangles glued together along two-hinges. In fact, if we express the vertex C ijklmn (only with a cyclic symmetry) by an oriented triangle, the two contractions in (3.4) can be illustrated as in Fig. 9. The first contraction leads to a gluing of two oriented triangles 13 Note that in tr(M 3 ) = C ijklmn M ij M kl M mn , only such components of C ijklmn survive that are totally symmetric under the permutation of three pairs of indices, (ij), (kl), (mn). However, when we write C ijklmn = δ jk δ lm δ ni , we intensionally think that C ijklmn are only cyclically symmetric for three pairs of indices, and distinguish two diagrams, one coming from C ijklmn and the other from C klijmn . Of course, by summing two ways of Wick contractions in calculating the free energy, the two diagrams appear in a combined way and C ijklmn will be automatically symmetrized. This "trick" enables us to identify a Feynman diagram with a triangulated surface, and is widely and implicitly adopted in the study of matrix models. Figure 9: Two ways to identify edges of triangles of positive orientation. The orientation is preserved for (a): (i 1 , j 1 ) = (j 2 , i 2 ), but is not for (b): (i 1 , j 1 ) = (i 2 , j 2 ). The local two-dimensional orientations of triangles induce one-dimensional orientations of the edges to be identified. The identification (b) can also be expressed as (b1) or (b2). The expression (b2) is necessarily accompanied by the flip of the right triangle, which means that the local two-dimensional orientation is not preserved when one moves from the left triangle to the right triangle across the identified edge.
with the orientation being preserved, while the second contraction to a gluing for which the orientation is not preserved.

Unoriented membrane theories
In the previous subsection, we have seen that unoriented string theories are obtained from oriented theories by gauging the worldsheet parity transformation Ω. We now apply the same prescription to membrane theories in order to define unoriented membrane theories; We first prepare oriented models and introduce the worldvolume parity transformation Ω that inverts the orientation of open membrane, and then gauge the transformation Ω by inserting (1 + Ω)/2 to every propagator of open membrane in the original oriented models. In the rest of this paper, we assume that worldvolumes in oriented models are already represented as tetrahedral decompositions.

Open membranes of disk topology as fundamental objects
We first argue that the worldvolume dynamics of oriented closed membranes of various topologies can also be regarded as that of oriented open membranes of disk topology. In fact, tetrahedra in a tetrahedral decomposition can be thought of as interaction vertices that are connected with propagators of membrane of disk topology (i.e. triangles). One thus may say that a worldvolume theory of closed membranes of arbitrary topologies has a dual picture where open membranes of disk topology play fundamental roles, despite the fact that open membranes can have topologies other than disk (such as disks with handles).

Fundamental triplets for oriented membranes
Given an oriented model, we focus on two adjacent, positively oriented tetrahedra T + 1 and T + 2 in a tetrahedral decomposition Γ, where T + 1 and T + 2 are glued by identifying a triangle ∆ 1 in T + 1 with a triangle ∆ 2 in T + 2 (the resulting identified triangle will be denoted by ∆). Note that the orientation of a tetrahedron naturally induces the positive orientation for four triangles belonging to the tetrahedron, and we represent them by arrows as in Fig. 10 (a). We express the identification of edges at ∆ as Figure 10: Two ways to identify triangles of tetrahedra of positive orientation. The orientation is preserved for (a): . The local three-dimensional orientations of tetrahedra induce two-dimensional orientations of the triangles to be identified. The identification (b) can also be expressed as (b1) or (b2). The expression (b2) is necessarily accompanied by the orientation change of the right tetrahedron, which means that the local three-dimensional orientation is not preserved when one moves from the left tetrahedron to the right tetrahedron across the identified triangle. where are the edges of ∆ 1 (or ∆ 2 ). Note that the orientations of ∆ 1 and ∆ 2 must be opposite in order to form an oriented tetrahedral decomposition, and thus the three-dimensional orientation is preserved when one moves from the inside of T 1 to that of T 2 through the identified triangle ∆ [ Fig. 10 (a)]. Three-dimensional orientation is also preserved for the two other tetrahedral decompositions that are obtained from (3.7) by cyclically permuting the edges (E ′ 1 , E ′ 2 , E ′ 3 ). We denote by Γ 1 (= Γ), Γ 2 , Γ 3 , respectively, the tetrahedral decompositions corresponding to the three edge-identifications that preserve the orientation, We will call (Γ 1 , Γ 2 , Γ 3 ) the fundamental triplet associated with triangle ∆.

Definition of unoriented membrane theories
In addition to the edge-identifications (3.8) [leading to the fundamental triplet (Γ 1 , Γ 2 , Γ 3 )] , we introduce another triplet (Γ 1 ,Γ 2 ,Γ 3 ) that are obtained, respectively, by the following edge-identifications at the same triangle ∆ : Note that, in contrast to (3.8), three-dimensional orientation is not preserved across ∆ [see Fig. 10 (b)]. We introduce a transformation Ω that interchanges two triplets (Γ 1 , Γ 2 , Γ 3 ) and (Γ 1 ,Γ 2 ,Γ 3 ), and define unoriented membrane theories to be those that are obtained from the oriented theories by acting the projection operator (1 + Ω)/2 on every triangle. We will call the set (Γ 1 , Γ 2 , Γ 3 ,Γ 1 ,Γ 2 ,Γ 3 ) the fundamental sextet associated with triangle ∆ . So far we have assumed that the tetrahedral decomposition Γ 1 is orientable, but one can easily see that Γ 1 is not necessarily orientable for the above definition of a sextet to make sense because we focus only on local configurations around triangle ∆. In the rest of paper, we understand that the domain of definition for Ω is extended so as to include nonorientable tetrahedral decompositions.
Note that each sextet (Γ 1 , . . . ,Γ 3 ) consists of both manifolds and nonmanifolds, unlike the two-dimensional cases where Ω always relates a manifold to another manifold. In fact, suppose that a tetrahedral decomposition Γ 1 represents a three-dimensional manifold. Then, the change of the edge-identification at ∆ from ( gives rise to a singularity at the midpoint of edge E 2 = E ′ 2 inΓ 1 around which we cannot define a local orientation. The appearance of singularity will be demonstrated explicitly when we consider an example in subsection 4.3.

Triangle-hinge models for unoriented membranes 4.1. Action and Feynman rules
In this section, we realize unoriented membrane theories as triangle-hinge models. We show that they are obtained simply by replacing C = C + in the original oriented models (2.14) with C = C + + C − : Here, C + is again given by eq. (2.12), and C − by We first note that the interaction vertices corresponding to [C + AAA] and [C − AAA] can be expressed by thickened triangles with directed index lines as in Fig. 11. Contractions If C + represents triangle-identifications which preserve three-dimensional local orientation, C − represents triangle-identifications which do not preserve orientation.
using [C + AAA] yield the identification of a triangle belonging to a tetrahedron with another triangle belonging to an adjacent tetrahedron so that the orientation is preserved [see Fig. 11 (a)]. The two positively oriented tetrahedra thus reside in opposite positions with respect to the thickened triangle, and the edges (a 1 , d 1 ), (a 2 , d 2 ), (a 3 , d 3 ) will be identified with the edges (b 1 , c 1 ), (b 2 , c 2 ), (b 3 , c 3 ), respectively, when we deflate the triangle to get a tetrahedral decomposition. On the contrary, contractions using [C − AAA] yield an identification of triangles where the orientation is not preserved [see Fig. 11 (b1,b2)]. In fact, the indices of C − [see (4.2)] can be expressed as Fig. 11 (b1) or (b2). We use the expression (b1) in a Feynman diagram where the triangle is connected to hinges, but we exploit the other expression (b2) when the thickened triangle is interpreted as representing two triangles to be identified in gluing two tetrahedra of positive orientation. Then, the edges (a 1 , d 1 ), (a 2 , d 2 ), (a 3 , d 3 ) will be identified with the edges (c 1 , b 1 ), (c 2 , b 2 ), (c 3 , b 3 ), respectively, when we deflate triangles to get a tetrahedral decomposition. It is easy to see that the two positively oriented tetrahedra are now in the same position with respect to the triangle and thus will take a configuration of Fig. 8 (b1) after the triangle is deflated. This means that the orientation is not preserved for this gluing of tetrahedra.
Note that the direction of arrows on index lines is still preserved for diagrams using C − . Thus, taking the same large n limit as in the oriented models, we can reduce the set of diagrams such that all their index polygons are triangles, 14 and can conclude that they represent tetrahedral decompositions.
We first note that, while the total number of triangles (as well as that of tetrahedra) is the same among the sextet (Γ 1 , . . . ,Γ 3 ), this is not the case for those numbers around each edge of triangle ∆ . For example, let us consider the case where the three edges of ∆ in Γ 1 [denoted by ] are connected to three different hinges. If we change the identification at ∆ from ( ) to obtain Γ 2 , all the three edges E 1 , E 2 , E 3 must be the same due to triangle-identifications at other triangles. 15 Therefore, Feynman diagrams γ 1 and γ 2 in a triangle-hinge model must have different numbers and different types of hinges if they correspond to Γ 1 and Γ 2 , respectively. This means that the constructions of sextets are not so straightforward in triangle-hinge models compared to other models (such as tensor models).
Let us make the above consideration to a more concrete form, considering a triangle ∆ in a tetrahedral decomposition Γ 1 , at which two positively oriented tetrahedra glued with the orientation being preserved. We first note that there are the following three cases for 14 As in the original models, a tetrahedron has one index triangle at each corner (see Fig. 6). 15 Since we assume that Γ is a tetrahedral decomposition without boundaries, other triangle-identifications in Γ ensure the edge-identifications E 1 = E ′ 1 , E 2 = E ′ 2 and E 3 = E ′ 3 . See discussions below (4.12) for more details.
the three edges I, J, K of triangle ∆: (1) Three edges I, J, K are connected to three different hinges.
(2) Two and only two of them are connected to the same hinge.
(3) All of them are connected to the same hinge. (4.3) We suppose that Γ 1 is of the type (1) at ∆, and that edges I, J, K are connected to (p + 1)-, (q + 1)-, (r + 1)-hinges, respectively. Including p other edges connected to the (p + 1)-hinge, we label the edges around the (p + 1)-hinge as [I, I 1 , . . . , I p ] in a cyclic order. Here, we define the cyclic ordering of edges around a k-hinge as follows (see Fig. 12): We first pick Figure 12: Labeling of edges around a k-hinge.
up two neighboring triangles t 1 and t 2 belonging to the same tetrahedron T + 1 of positive orientation, and label their edges connected to the hinge as L 1 and L 2 , respectively. Next to T + 1 there is another positively oriented tetrahedron T + 2 determined by triangle t 2 and another triangle t 3 sharing the same hinge, and we label as L 3 the edge of t 3 that is connected to the hinge. Repeating this procedure, we obtain a sequence [L 1 , L 2 , . . . , L k ] around the khinge. Since another choice (t 2 , t 3 ) is possible as the initial pair for the same configuration of edges around the hinge, we should regard the above sequence as being cyclically symmetric, [L 1 , L 2 , . . . , L k ] = [L 2 , L 3 , . . . , L k , L 1 ]. Note that, if we take (t 2 , t 1 ) as the initial pair, the edges around the k-hinge will be represented as a sequence in reverse order, [L k , . . . , L 2 , L 1 ].
Labeling similarly the edges around the (q + 1)-and (r + 1)-hinges by [J, J 1 , . . . , J q ] and [K, K 1 , . . . , K r ], respectively, we have Equations (4.4)-(4.9) can be understood in the following way. We begin with (4.4), which is simplest and obvious. We first split the triangle ∆ in Γ 1 to two triangles as in Fig. 13 in order to realize the configuration before the edge-identification ( is made. This splitting is accompanied by that of edge I to two edges E 1 and E ′ 1 , and Figure 13: The splitting of ∆ corresponding to Γ 1 . Edge I becomes two edges E 1 and E ′ 1 , and edge J (or K) becomes E 2 and E ′ 2 (or to E 3 and E ′ 3 ). The (p+1)-, (q + 1)-, (r + 1)-hinges accordingly become (p + 2)-, (q + 2)-, (r + 2)-hinges, respectively. Γ 1 is restored by the edge-identification ( for the split triangles. that of edge J (or K) to E 2 and E ′ 2 (or to E 3 and E ′ 3 ). Accordingly, the (p + 1)-, (q + 1)-(r + 1)-hinges are transformed to (p + 2)-, (q + 2)-(r + 2)-hinges, respectively. Now we follow the sequence of the edges connected to each hinge in the other way around. If we start from the edge E 1 connected to the (p + 2)-hinge, we then pass through the edges I 1 , . . . , I p following the original sequence [I, I 1 , . . . , I p ], and reach the edge E ′ 1 , which will be identified with the starting edge E 1 (i.e., E ′ 1 = E 1 = I) under the edge-identification for Γ 1 , . Let us write the total path schematically as a cycle, Similarly, if we start from the edge E 2 connected to the (q + 2)-hinge or from the edge E 3 connected to the (r + 2)-hinge, we then have the following paths: Now we consider the tetrahedral decomposition Γ 2 , which was obtained from Γ 1 by changing the edge-identification from ( Fig. 14 (a)]. If we start from the edge E 1 connected to the (p + 2)-hinge, we then again for the split triangles.
pass through the edges I 1 , . . . , I p and reach the edge E ′ 1 . However, this is not the end of journey because E ′ 1 will be identified with E 3 in the edge-identification, and we need to follow another sequence of edges, K 1 , . . . , K r , to reach E ′ 3 . Since E ′ 3 will be identified with E 2 , we need to continue the journey; we pass through the edges J 1 , . . . , J q to reach E ′ 2 , which finally will agree with the starting edge E 1 . The total path thus can be written as the following cycle: Since E ′ 1 = E 3 = K, E ′ 3 = E 2 = J and E ′ 2 = E 1 = I under the edge-identification, the path (4.13) can be written as (4.5). Similarly, (4.6) can be understood from the path: (4.14) Equation (4.7) can be understood in a similar way, by recalling thatΓ 1 is obtained from Γ 1 by changing the edge-identification from ( [see Fig. 14 (b)], which gives the following two disconnected cycles: Similarly, the cycles forΓ 2 [edge-identification ( ] are given by and those forΓ 3 [edge-identification ( ] are given by Now that we understand in detail the configurations of the fundamental sextet (Γ 1 , . . . ,Γ 3 ) associated with triangle ∆, it is easy to translate (4.4)-(4.9) in terms of unoriented trianglehinge models, and we obtain the sextet of Feynman diagrams (γ 1 , . . . ,γ 3 ) as the following groups of Wick contractions: 16 Kr ] , (4.20) Here, we have used the abbreviation for Wick contractions introduced in subsection 2.1, and the superscript σ takes (+) or (−). We have written explicitly only for the part of the interaction vertices corresponding to ∆ (expressed by [C ± A I A J A K ]) and the hinges connected to ∆. The remaining part (denoted by X I 1 ...Ip,J 1 ...Jq,K 1 ...Kr ) is common among the sextet (γ 1 , . . . ,γ 3 ) and represents the other interaction vertices and their contractions. 17 As for diagrams γ 1 , γ 2 , γ 3 , the identification at ∆ preserves the orientation as in Fig. 10 (a), and thus we have used the vertex [C + A I A J A K ]. On the other hand, as for diagramsγ 1 ,γ 2 ,γ 3 , the identification at ∆ does not preserve the orientation as in Fig. 10 (b), and thus we have used the vertex [C − A I A J A K ]. In Appendix A we prove that the diagrams γ 2 , . . .,γ 3 represent tetrahedral decompositions if γ 1 does.
So far we have assumed that the orientation is preserved at ∆ in Γ 1 and also that Γ 1 is of the type (1) in (4.3). For other cases, one can also obtain the corresponding sextets (γ 1 , . . . ,γ 3 ) in a similar way.

Example
To understand the meaning of the above sextet (4.18)-(4.23), let us consider a simple example. We take a tetrahedral decomposition Γ 1 of a three-sphere, consisting of two tetrahedra glued together at their faces as shown in Fig. 15. Diagram γ 1 representing Γ 1 is realized by (4.24) Here, the subscripts ∆, 1, 2, 3 specify the triangles corresponding to the interaction vertices; ∆ specifies the triangle at which we change the edge-identification to obtain γ 2 , . . . ,γ 3 , and 1, 2, 3 specify the triangles belonging to the rest part X, which consists of three triangles (1, 2, 3) and three 2-hinges (see Fig. 16). Fig. 16 (a1) depicts the part corresponding to X, while Fig. 16 (a2) depicts the part corresponding to ∆, which consists of a triangle and three 2-hinges. In Fig. 16 (a1) and (a2), edges with the same indices are connected by contractions. Recalling that the edge-identification of Γ 1 is expressed by ( Fig. 16 Since edges E i and E ′ i are expressed as in Fig. 16 (b) when (a1) and (a2) are combined, we see that the edge-identification ( will certainly be realized after triangle ∆ is deflated. Now we consider diagram γ 2 representing the tetrahedral decomposition Γ 2 that is obtained from Γ 1 by changing the edge-identification from ( Fig. 17). Note that Γ 2 has the topology of a three-dimensional lens space L(3, 1), as can be seen from Fig. 17 (c). Diagram γ 2 is given by (4.19), that is, Since the part given by X is common among the sextet, we have the same labeling of edges, 1, 2, 3), and diagram γ 2 takes the form shown in Fig. 17 (a1) and (a2). In Fig. 17 (a2), the ordered indices (b 1 , c 1 ) are connected to (g, f ) by index lines, and (a 2 , d 2 ) are connected to (h, e). Thus, as can be seen from Fig. 17 (b), edge E 1 (b 1 , c 1 ) = (g, f ) will be identified with edge E ′ 2 (a 2 , d 2 ) = (h, e) after triangle ∆ is deflated. Similarly, edges E 2 (b 2 , c 2 ) and E 3 (b 3 , c 3 ) will be identified with edges E ′ 3 (a 3 , d 3 ) and E ′ 1 (a 1 , d 1 ), respectively. We thus obtain the edge-identification ( In a similar way, we can realize Γ 3 [resulting from the edge-identification ( at ∆] as a diagram γ 3 [eq. (4.20)] of a triangle-hinge model. Γ 3 has the topology of a lens space L(3, 2) = L(3, 1). Thus, in this simple example, two diagrams γ 2 and γ 3 represent the same tetrahedral decomposition, Γ 2 = Γ 3 .
As for the tetrahedral decompositionΓ 1 [resulting from the edge-identification ( , the corresponding diagramγ 1 is obtained from the following group of Wick contractions [eq. (4.21)]: The diagram is depicted in Fig. 18. In Fig. 18 (a2), the ordered indices (b 1 , c 1 ) are con- nected to (h, e) by index lines, and (d 3 , a 3 ) are connected to (f, g). Thus, due to the edgeidentification for C − explained below (4.2), edge E 1 (b 1 , c 1 ) = (h, e) will be identified with edge E ′ 3 (d 3 , a 3 ) = (f, g) after triangle ∆ is deflated. Similarly, edges E 2 (b 2 , c 2 ) and E 3 (b 3 , c 3 ) will be identified with edges E ′ 2 (d 2 , a 2 ) and E ′ 1 (d 1 , a 1 ), respectively. We thus obtain the edge-identification (E 1 E 2 E 3 ) = (E ′ 3 E ′ 2 E ′ 1 ) ofΓ 1 . AlthoughΓ 1 consists of two tetrahedra, it is not a three-dimensional manifold. In fact, there is a singularity at the midpoint of edge E 2 = E ′ 2 of ∆, around which we cannot define a local orientation. It is easy to see that the other diagramsγ 2 andγ 3 are realized by (4.22) and (4.23), respectively, and represent the same tetrahedral decomposition asΓ 1 .

Note on the weights of diagrams
We comment that a sextet (γ 1 , γ 2 , γ 3 ,γ 1 ,γ 2 ,γ 3 ) appears in the free energy with the same coefficients. By "the same coefficients" we mean that the numerical factors of these diagrams are the same except for powers of n, λ, µ k if we treat the common part X as a set of distinguished external vertices and sum over all diagrams representing the same tetrahedral decomposition.
We note that, if a group x of Wick contractions represents a tetrahedral decomposition and if all the interaction vertices are distinguished, then x contributes to the free energy as 18 (4.28) Here, s 2 and s k 1 denote the numbers of triangles and k-hinges, respectively, in diagram γ = [x]. Thus, there arise 1/(s 2 ! 6 s 2 ) and 1/(s k 1 ! (2k) s k 1 ) in the free energy as numerical factors.
If there are n k internal k-hinges in a diagram, 19 there are n k ! different contractions corresponding to the permutation of these hinges, since the external vertices are distinguished. For each k-hinge, there are 2k ways to give the same diagram due to the symmetry of k-hinge vertices. Thus, the numerical factor of each contraction, 1/(n k !(2k) n k ), is compensated if we sum these contributions. The numerical factor 1/6 of triangle ∆ is also canceled. Actually, since C + has the symmetry (2.13) there are six ways to give the same diagram. The above computation ensures that three diagrams γ 1 , γ 2 , γ 3 are generated with unit coefficient in the original triangle-hinge model. Furthermore, since C − also has the symmetry (4.29) γ 1 ,γ 2 ,γ 3 are also generated with unit coefficient in an unoriented model. 18 The n dependence comes from the assumption that all the index polygons are triangles. 19 Internal hinges mean the parts not in X but connected to ∆.
If two tetrahedra with color α and β are glued at their faces, the face (corresponding to C ±mat ) gives the factor λ αβ . In this sense the coupling constants λ αβ define a local interaction between the color α and β [2]. If we take the set of colors, J , to be R D = {x} and let the coupling constants λ x,y (x, y ∈ R D ) take nonvanishing values only around y as a function of x, then x can be interpreted as the target space coordinates of a tetrahedron in R D . Since neighboring tetrahedra are locally connected in R D , the model can describe the dynamics of unoriented membranes embedded in R D .
An unoriented membrane theory is realized as a triangle-hinge model with the action (4.1). It generates Feynman diagrams representing unoriented tetrahedral decompositions. We gave explicitly in (4.18)-(4.23) the sextet of Feynman diagrams (γ 1 , . . . ,γ 3 ) corresponding to (Γ 1 , . . . ,Γ 3 ), and showed that these six diagrams appear with unit coefficient up to factors of coupling constants if we treat the common part X as a set of distinguished external vertices and sum over all Wick contractions giving the same diagram. We further showed that matter degrees of freedom can be introduced to unoriented triangle-hinge models by coloring tetrahedra as carried out in [2]. Although we only discussed the coloring of tetrahedra in this paper, we can set matter degrees of freedom to simplices of any dimensions (i.e. tetrahedra, triangles, edges and/or vertices) as in [2].
We expect that unoriented triangle-hinge models are solvable at least at the same level of the original oriented models [19], since the dynamical variables are the same type of matrices and the actions have almost the same structure as the original oriented triangle-hinge models. The unoriented models actually might be easier to solve than the original oriented models, because the interaction term corresponding to a triangle has higher symmetry, which may help us to carry out the path-integrals more analytically. It is interesting to study critical behaviors of the models in both analytical and numerical ways.