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Partition Functions and Topology-Changing Amplitudes

arXiv:hep-th/9112072v1 24 Dec 1991RIMS-851December 1991Partition Functions and Topology-Changing Amplitudesin the 3D Lattice Gravity of Ponzano and ReggeHirosi Ooguri⋆Research Institute for Mathematical SciencesKyoto University, Kyoto 606, JapanABSTRACTWe define a physical Hilbert space for the three-dimensional lattice gravity of Ponzanoand Regge and establish its isomorphism to the one in the ISO(3) Chern-Simons theory.It is shown that, for a handlebody of any genus, a Hartle-Hawking-type wave-function ofthe lattice gravity transforms into the corresponding state in the Chern-Simons theoryunder this isomorphism. Using the Heegaard splitting of a three-dimensional manifold, apartition functions of each of these theories is expressed as an inner product of such wave-functions.

Since the isomorphism preserves the inner products, the partition functions ofthe two theories are the same for any closed orientable manifold. We also discuss on aclass of topology-changing amplitudes in the lattice gravity and their relation to the onesin the Chern-Simons theory.⋆e-mail addresses : ooguri@jpnrifp.bitnet and ooguri@kekvax.kek.ac.jp

01IntroductionIn 1968, Ponzano and Regge derived the following asymptotic form of the Racah-Wigner 6j-symbol for large angular momenta ji’s [1]. (−1)P6i=1 ji(j1j2j3j4j5j6)∼1√12πVcos(SRegge + π/4)(ji ∈Z≥0).

(1)To explain the notations in the right hand side, it is useful to imagine a tetrahedron andassociate ji’s to its edges as in Fig. 1.

In the following, we call this as coloring of thetetrahedron. Since the 6j-symbol has the tetrahedral symmetry, we can uniquely associateit to the colored tetrahedron.

Now we regard (ji + 12) as a length of the i-th edge of thetetrahedron. The factor V in the right hand side of (1) is defined as a volume of such atetrahedron, and SRegge is given bySRegge =6Xi=1θi(ji + 12),(2)where θi is the angle between the outward normals of the two faces separated by the i-thedge.What is remarkable about this formula is that SRegge is nothing but the Regge action[2]for the single tetrahedron.Suppose there is a three-dimensional manifold M which isdecomposed into a collection of tetrahedra.

If we assume that each tetrahedron is filled inwith flat space and the curvature of M is concentrated on the edges of the tetrahedron, ametric gµν on M is specified once the length (j + 12) of each edge is fixed. The Einstein-Hilbert actionRd3x√gR is then a function of j’s on the edges and it is given by summingthe Regge action (2) over all the tetrahedra in M. Thus, as a model for the three-dimensioalEinstein gravity, Ponzano and Regge considered a lattice statistical model whose dynamicalvariables are the angular momenta j’s on the edges and whose weight is given by a productof the 6j-symbols over all the tetrahedra in M (including the sign-factor (−1)Pi ji in theleft hand side of (1)).In the lattice gravity, we sum over geometries of M based on its simplicial decom-position.

In one approach, size and shape of each simplex are fixed, and the quantum– 2 –

flctuation of the geometry is evaluated by summing over all the possible ways of gluingthe simplices together. The recent studies on two-dimensional gravity are mostly based onthis approach[3].

In the other approach, one fixes the lattice structure and sums over thelattice lengths [2]. The lattice model of Ponzano and Regge belongs to the latter approach.In both of these approaches, it is important to know if the lattice model has a nicecontinuum limit.

In this respect, it has already been pointed out by Ponzano and Reggethat their lattice model can be made scale-invariant with appropriate modification of thestatistical weight. Let us take the tetrahedron in Fig.

1, and decompose it into four smalltetrahedra as in Fig. 2.

There are four edges inside of the original tetrahedron, and weput angular momenta l1, ..., l4 on them. Corresponding to the four tetrahedra, we considerthe following product of the 6j-symbols.

(−1)Pi li(j1j2j3l1l2l3) (j4j6j2l3l1l4) (j3j4j5l4l2l1) (j1j5j6l4l3l2)(3)Now we are going to sum this weight over the coloring li on the internal edges.Thesummation can be performed analytically if we multiply an additional factor Q4i=1(2li +1)to the summand (3). By using the identity due to Biedenharn and Elliot, the summationover l1 can be done asXl1(2l1 + 1)(−1)Pi li(j4j6j2l3l1l4) (j3j4j5l4l2l1) (j1j5j6l4l3l2)= (−1)Pi ji(j1j2j3j4j5j6) (j1j5j6l4l3l2).We can then sum over l4 using the orthonormality of the 6j-symbolXl4(2l4 + 1)(j1j5j6l4l3l2)2=12j1 + 1.– 3 –

Thus we are left with the sum over l2 and l3 asXl1,...,l4(−1)Pi li(2l1 + 1)(2l2 + 1)(2l3 + 1)(2l4 + 1)×(j1j2j3l1l2l3) (j4j6j2l3l1l4) (j3j4j5l4l2l1) (j1j5j6l4l3l2)= (−1)Pi ji(j1j2j3j4j5j6)12j1 + 1X|l2−l3|≤j1≤l2+l3(2l2 + 1)(2l3 + 1). (4)However, the sum over l2 and l3 in the right hand side is divergent.

In order to regularizeit, we cut offthe summation by li ≤L and rescale the summand of (4) by multiplying afactor Λ(L)−1, whereΛ(L) =12j1 + 1Xl2,l3≤L|l2−l3|≤j1≤l2+l3(2l2 + 1)(2l3 + 1). (5)For a sufficiently large value of L, Λ(L) becomes independ on l1 and behaves as Λ(L) ∼4L3/3 for L →∞.

After multiplying this factor, we can take L to ∞and the divergenceis removed. Thus, with the additional factor Λ(L)−1 Qi(2li + 1), the sum of (3) over thecoloring li on the internal edges in Fig.

2 reproduces the weight(−1)Pi ji(j1j2j3j4j5j6)for the original tetrahedron in Fig. 1.Based on this observation, Ponzano and Regge defined a partition function ZM for themanifold M byZM = limL→∞X{j:j≤L}YverticesΛ(L)−1 Yedges(2j + 1)Ytetrahedra(−1)Pi ji(j1j2j3j4j5j6).

(6)Due to the identity (4), ZM is invariant under the refinement of any tetrahedron in M intofour smaller tetrahedra. Namely this lattice model is at a fixed point the renormalizationgroup transformation⋆.⋆Although the divergence due to the scale-invariance of the model is regularized in (6) by multiplyingthe factor Λ(L)−1, it is not obvious that ZM defined in the above is finite.

We will examine this pointin Section 3.– 4 –

Because of this property, one may suspect that the lattice gravity of Ponzano and Reggecan be related to some quantum field theory in the continuum. Although there have beensome works on physical interpretation of this model[4], little progress had been made onthe continuum limit of this model until recently.

Last year, Turaev and Virostudied theq-analogue of the Ponzano-Regge model, and found that its partition function is invariantunder a class of transformations larger than the renormalization group in the above. More-over they have shown that any two tetrahedral decompositions of M can be related by asequence of such transformations.

Therefore the partition is independent of the tetrahedraldecomposition and depends only on the topology of M. Although they have studied the q-analogue, their argument is directly applicable to the original model of Ponzano and Regge.Thus it is natural to expect that the model of Ponzano and Regge and its q-analogue byTuraev and Viro are equivalent to some topological field theories. Indeed, in the paper [5],Turaev and Viro have conjectured that the partition function of their q-analogue modelis equal to the absolute value square of the partition function of the SU(2) Chern-Simonstheory[6] of level k (q = e2πi/(k+2)) when the manifold M is orientable.In the previous paper[7], the author and Sasakura have examined physical states inthe lattice gravity of Ponzano and Regge and suggested that they are related to physicalstates of the ISO(3) Chern-Simons theory whose action is given bySCS(e, ω) =Zd3xea ∧(dωa + ǫabcωb ∧ωc),(7)where ea and ωa (a = 1, 2, 3) are one-forms on M with adjoint indices of SO(3).

If weidentify them as a dreibein and a spin-connection following the observation by Witten,the action SCS may be regarded as the Einstein-Hilbert actionRe ∧R in the first orderformalism.In this paper, we extend the analysis of [7] and show that the partition function ofthe lattice gravity of Ponzano and Regge agrees with the one of the ISO(3) Chern-Simonstheory for any orientable manifold. This result corresponds to the k →∞limit of thecojecture by Turaev and Viro.

In Section 2, we define a physical Hilbert space for the latticegravity and establish its isomorphism to the physical Hilbert space of the Chern-Simonstheory. We show later in Section 4 that this isomorphism preserves the inner products ofthe two Hilbert spaces.

In Section 3, we compute the Hartle-Hawking-type wave-functions– 5 –

of the lattice gravity for a handlebody of any genus and show that it transformes intothe corresponding state in the Chern-Simons theory under the isomorphism. By gluingHartle-Hawking-type wave-functions, one can compute a partition function for any closedorientable manifold.

We check in Section 4 that this gluing procedure is compatible withthe isomorphism. Therefore the partition functions of these two theories are the same,as far as they are finite.

We also study some class of topology-changing amplitudes inthe lattice gravity and their relation to the ones in the Chern-Simons theory. In the lastsection, we discuss on interpretations of these results and their extensions.In the course of this work, the author was informed of a paper by Turaev[9] where heannounces to have proven the equivalence of the q-analogue lattice model and the Chern-Simons theory for finite k. Details of his derivation not being available, it is not clear tothe author how his approach is related to the one presented here.02Wave-FunctionsIn the lattice gravity of Ponzano and Regge, one can define a discretized verion of theWheeler-DeWitt equation which characterizes physical states in the theory.

On the otherhand, in the ISO(3) Chern-Simons theory, a physical state is given by a gauge-invarianthalf-density Φ(ω) on the moduli space of a flat SO(3) connection ω on a two-dimensionalsurface Σ. In this section, we establish a correspondence between physical states in thelattice gravity and in the continuum Chern-Simons theory.

This correspondence will beused in the later sections to compare partition functions and topology-changing amplitudesin those two theories.First we should clarify what we mean by physical states in the lattice gravity. To mo-tivate our definition of physical states, let us consider a closed three-dimensional manifoldM and decompose it into three parts, M1, M2 and N, as in Fig.

3, where N has a topologyof Σ × [0, 1] with Σ being a closed orientable two-dimensional surface, and Mi (i = 1, 2)has a boundary which is isomorphic to Σ. The manifold M is reconstructed by gluing theboundaries of N with ∂M1 and ∂M2.Corresponding to this decomposition of M, the partition function ZM of the manifoldM can be expressed as a sum of products of three components each of which is associatedto M1, M2 and N. To find such an expression, we note that the partition function ZM– 6 –

is independent of a choice of tetrahedral decomposition of M. Therefore we can placetetrahedra in M in such a way that M1, M2 and N do not share a tetrahedron, namelytheir boundaries are triangulated by the faces of the tetrahedra. Corresponding to thistetrahedral decomposition, we can express ZM asZM =Xc1∈C(∆1)c2∈C(∆2)ZM1,∆1(c1)Λ−n(∆1)P∆1,∆2(c1, c2)Λ−n(∆2)ZM2,∆2(c2),(8)Here ∆i (i = 1, 2) denotes the triangulations of the boundary ∂Mi, C(∆i) is a set of all thepossible colorings on ∆i, and n(∆i) is a number of vertices on ∆i⋆.

The factor ZMi,∆i(ci)is given by the sum over all the possible coloring on the edges interior of MiZMi,∆(ci) =Yedges on ∆i(−1)2jp2j + 1×XcoloringYverticesinterior of MiΛ−1Yedgesinterior of Mi(2j + 1)×Ytetrahedra in Mi(−1)Pi ji(j1j2j3j4j5j6),where we keep fixed the coloring ci on the edges on ∂Mi (Fig. 4).

Similarly P∆1,∆2(c1, c2)is given by a sum over all the possible colorings on the interior edges of N with fixedcolorings c1 and c2 on ∂N ≃Σ + Σ.Since P∆1,∆2 is independent of the tetrahedral decomposition of the interior of N, itsatisfies the following remarkable property,Xc2∈C(∆2)P∆1,∆2(c1, c2)Λ−n(∆2)P∆2,∆3(c2, c3) = P∆1,∆3(c1, c3). (9)Therefore we can define an operator PP[φ∆](c) =Xc′∈C(∆)P∆,∆(c, c′)Λ−n(∆)φ∆(c′),which acts as a projection operator (P ·P = P) on a space of functions on C(∆).

By using⋆For conciseness of equations, here and in the following, we do not write the cut-offparameter Lexplicitly.– 7 –

(9), we can rewrite (8) asZM =Xc1,c2P[ZM1,∆1](c′1)Λ−n(∆1)P∆1,∆2(c1, c2)Λ−n(∆2)P[ZM2,∆2](c′2).One sees that “states” propagating from M1 to M2 through N are projected out by P.Thus it is natural to define a physical Hilbert space H(∆) for the triangulated surface Σas a subspace projected out by P, i.e.φ∆(c) ∈H(∆)⇐⇒φ∆= P[φ∆](10)Since P∆,∆is associated to the topology Σ × [0, 1], we may regard it as a time evolutionoperator in the lattice gravity. Therefore it should be appropriate to call the physical statecondition (10) as a discretized version of the Wheeler-DeWitt equation.

We define an innerproduct in H(∆) by(φ∆, φ′∆) =Xc,c′∈C(∆)φ∆(c)Λ−n(∆)P∆,∆(c, c′)Λ−n(∆)φ′∆(c′). (11)It is easy to see that ZM1,∆(c) and ZM2,∆(c) are real solutions to the Wheeler-DeWittequation (10) and the partition function ZM is given by their inner productZM = (ZM1,∆, ZM2,∆).

(12)Although this definition of H(∆) depends of the triangulation ∆of Σ, there is anatural isomorphism given by the map P∆1,∆2 between H(∆1) and H(∆2) for any twotriangulations ∆1, ∆2.Due to the equation (9), the map P∆1,∆2 preserves the innerproduct defined by (11). It also follows from (9) that the map P∆1,∆2 has an inverse and itis given by P∆2,∆1.

Thus we may choose an arbitrary triangulation in defining the physicalHilbert space for Σ.On the other hand, the physical Hilbert space HCS of the ISO(3) Chern-Simons theoryconsists of half-densities on the moduli space of a flat SO(3) connection on Σ. To see this,– 8 –

we consider the topology N = Σ × [0, 1] again, and decompose the dreibein ea and thespin-connection ωa (a = 1, 2, 3) asea =Xi=1,2eai dxi + ea0dt ,ωa =Xi=1,2ωai dxi + ωa0dt(x1, x2) ∈Σ,t ∈[0, 1].Corresponding to this decomposition, the Chern-Simons action (7) takes the form,SCS(e, ω) =Zdtd2xǫij(eaj∂tωai + ea0F aij −ωa0Dieaj),where Di is a covariant derivative given by ωai and F aij is its curvature. From this ex-pression, one sees that (ωi, ǫijej) are cannonically cojugate to each other, while e0 and ω0are Lagrange multipliers and impose constraints, Fij = 0 and Diej −Djei = 0.

Thusa wave-function of the theory can be represented by a function Φ(ω) of ωi, a SO(3)connection on Σ. The constraint Fij = 0 implies that Φ(ω) should vanish unless ω isflat, and ǫijDiejΦ(ω) = iDi δδωiΦ(ω) = 0 means that Φ(ω) is invariant under the gauge-transformation ωi →ωi + Diλ.

The inner product in HCS is given by the integral(Φ1, Φ2)CS =Z[dω]δ(Fij)Φ∗1(ω)Φ2(ω). (13)Thus a physical wave-function is a half-density on the moduli space of a flat SO(3) con-nection.Now we would like to show that there is a natural isomorphism between H(∆) andHCS.

To interpolate between the two Hilbert spaces, we introduce the following (over-complete) basis for HCS constructed from Wilson-lines Uj(x, y) (x, y ∈Σ, j = 0, 1, 2, ...),Uj(x, y) = P exp(yZxωataj),where P exp denotes the path-ordered exponential and taj (a = 1, 2, 3) is the spin-j gen-erator of SO(3). Under a gauge transformation ω →Ω−1ωΩ+ Ω−1dΩ, the Wilson-linebehaves as U(x, y) →Ω(x)−1U(x, y)Ω(y).

Now consider their tensor product ⊗iUji(xi, yi).– 9 –

To make this gauge-invariant, we need to contract group indices of Uj’s so that the gaugefactor Ωcancels out. In the case of the group SO(3), invariant tensors we can use tocontract group indices are the Clebsch-Gordan coefficient ⟨j1j2m1m2|j3m3⟩and the metricgjmm′ = (−1)j−m1√2j + 1δm+m′,0gmm′j= (−1)j+mp2j + 1δm+m′,0.Actually it is more convenient to use the cyclic-symmetric 3j-symbol given by j1j2j3m1m2m3!= (−1)j1−j2−m3√2j3 + 1⟨j1j2m1m2|j3 −m3⟩,rather than the Clebsch-Gordan coefficient.

We regard mi’s in the 3j-symbol as lowerindices which can be raised by the metric gmim′iji. When three Wilson-lines meet togetherat the same point on Σ, we can use the 3j-symbol and the metric gjmm′ to contract theirgroup indices.

We can also connect two Wilson-lines by the metric if they carry the samespin. The gauge-invariant function constructed this way corresponds to a colored trivalentgraph Y on Σ, where a contour from x to y in Y with color j corresponds to a Wilson-lineUj(x, y), and a three-point vertex in Y represents the 3j-symbol⋆.Due to the cyclicsymmetry of the 3j-symbol, to each graph Y on the orientable surface Σ, we can associatesuch a gauge-invariant function uniquely.A physical wave-function of the Chern-Simons theory is obtained from such a networkof Wilson-lines by restricting the support of the function on flat SO(3) connections.

Thisrestriction however gives rise to linear dependence among the Wilson-line networks. Specif-ically, if two graphes Y and Y ′ are homotopic, the corresponding gauge-invariant functionshave the same value on a flat connection.

Since there is one to one correspondence betweena homotopy class of colored trivalent graphes on Σ and a triangulation of Σ with coloringon their sides, we may parametrize the gauge-invariant function by a colored triangulation⋆There may be a pair of Wilson-lines intersecting with each other, which cannot be described as a partof a trivalent graph as it is. In such a case, we may cut the Wilson-lines at the intersecting point anduse the orthonormality of the 3j-symbols,δm1,m′1δm2,m′2 =Xj3,m3(2j3 + 1) j1j2j3m1m2m3 j1j2j3m′1m′2m3to replace the intersection by two vertices and an infinitesimal Wilson-line connecting the vertices.– 10 –

defined by a pair (∆, c) (c ∈C(∆)) rather than a trivalent graph Y . In this way, to eachcolored triangulation, we can associate a physical wave-function Ψ∆,c of the Chern-Simonstheory.

An arbitrary wave-function Φ(ω) is expanded in terms of them asΦ(ω) =X∆Xc∈∆ϕ∆(c)Λ−n(∆)Ψ∆,c(ω). (14)Now we are in a position to establish a correspondence between a solution to the dis-cretized Wheeler-DeWitt equation (10) and a physical state in the Chern-Simons theory.To understand the correspondence, the following fact is most important.

When evalu-ated on a flat connection ω, Ψ∆,c(ω) are not yet linearly independent, but they obey thefollowing relations,Ψ∆,c(ω) =Xc′∈C(∆′)P∆,∆′(c, c′)Λ−n(∆′)Ψ∆′,c′(ω). (15)Furthermore they are the only linear relations on Ψ∆,c.Before proving (15), let us examine its consequences.

By substituting (15) into (14),we obtainΦ(ω) =Xc∈C(∆)φ∆(c)Λ−n(∆)Ψ∆,c(ω),(16)where φ∆(c) is defined byφ∆(c) =X∆′Xc′∈C(∆′)P∆,∆′(c, c′)Λ−n(∆′)ϕ∆′(c′)for an arbitrary fixed triangulation ∆of Σ.It follows from (9) that φ∆(c) solves theWheeler-DeWitt equation (10) of the lattice gravity.φ∆= P[φ∆].Thus, to each solution φ∆(c) of the Wheeler-DeWitt equation, there is a physical stateΦ(ω) of the Chern-Simons theory given by (16). Since (15) are the only relations amongΨ∆,c’s, this correspondence between φ∆and Φ is one to one.

In Section 4, we will show– 11 –

that the inner product of Wilson-line networks (Ψ∆1,c1, Ψ∆2,c2)CS in the Chern-Simonstheory is equal to P∆1,∆2(c1, c2) for the lattice gravity, upto a constant factor. Thereforethe map from H(∆) to HCS defined by (16) preserves their inner products.

Thus (16)gives the isomorphism between the physical Hilbert spaces of the lattice gravity and theChern-Simons theory.Now we would like to prove that the relations (15) indeed hold, and that they are theonly relations among Ψ∆,c’s. We will show this by mathematical induction with respect tothe number of tetrahedra in N = Σ × [0, 1].

When the number is zero, the triangulations∆and ∆′ must be identical and they are attached to each other. In this case, (15) is anobvious identity.

Now we are going to pile tetrahedra one on another and increase thenumber of tetrahedra in N. Since one tetrahedra has four faces, there are three ways toattach one on another. (i) Choose one of the faces of the tetrahedron and attach it to one of the triangles on thesurface Σ of N (Fig.

5). (ii) Attach two faces of the tetrahedron to two neighbouring triangles on Σ (Fig.

6). (iii) Attach three faces of the tetrahedron to three neighbouring triangles on Σ (figureobtained by inverting the arrows in Fig.

5).Let us first check that the induction holds in the second move in the above list. Considera part of the Wilson-line network of Ψ∆,c which looks like the diagram in the right handside of Fig.

6. Because of the flatness of ω, we can take the Wilson-line colored by k andmake its length to be arbitrary small without changing the value of Ψ∆,c(ω).

When itsend-points meet with each other, the Wilson-line can be replace by an identity. Since thegroup indices of the Wilson-line Uk(x, y) at the end-points x and y are contracted withthe 3j-symbols, in the limit x →y when Uj(x, y) becomes an identity, the function Ψ∆,cshould contain a sum of product of these 3j-symbols.

Now there is a formula which relatetwo different ways of summing 3j-symbols,– 12 –

Xmm′gmm′k j2j3km2m3m! j4j1km4m1m′!=Xl(−1)j1+j2+j3+j4p(2k + 1)(2l + 1)(j1j2lj3j4k)×Xnn′gnn′l j1j2lm1m2n!

j3j4lm3m4n′! (17)The left hand side of this equation corresponds to the diagram in the left hand side of Fig.6.

These four external Wilson-lines are recombined in the right hand side; the Wilson-linesof j1 and j2 make a pair and they are connected to j3 and j4 by an infinitesimal Wilson-linewith color-l. This is exactly the right hand side of Fig.

6. Therefore we obtainΨ∆,ck =Xl(−1)j1+j2+j3+j4p(2k + 1)(2l + 1)(j1j2lj3j4k)Ψe∆,ecl,(18)where the triangulation (∆, ck) contains two triangles colored as in the left hand side ofFig.

6, and it is replaced by the ones in the dual position in (e∆, ecl).To prove (15) inductively, suppose that we have used n-tetrahedra in constructing theprojection operator P for the topology N = Σ×[0, 1]. When we add one more tetrahedronto N, as is prescribed in (ii), the corresponding projection operator P ′ is obtained from Pby multiplying to it an appropriate factor involving the 6j-symbol, and by summing overcoloring on the common side of two neighbouring triangles to which the new tetrahedronis attached.

This operator P ′ is obtained exactly by substituting (18) into the right handside of (15). Therefore the inductive proof of (15) holds when we add one tetradedron inthe second move in the list.We can also add a tetrahedron as in (i) or (iii) in the list.

If the network containsa contractable loop with several external Wilson-lines attached, by repeatedly using theidentity (17), the loop can be recombined into a tree-like diagram with a one-loop tadpole.The tadpole can be made arbitrarily small, and the infinitesimal tadpole can be removedby usingXmm′gmm′j jjJmm′M!= δJ,0δM,0.For example, if the network defined by (∆, c) contains a loop with three external lines j1,– 13 –

j2 and j3 as in the right hand side of Fig. 5, we man shrink the loop to obtain anothernetwork (∆′, c′) where the three lines meet at one point.

By using the formula,Xnijgn12n′12l12gn23n′23l23gn31n′31l31 l12j2l23n′12m2n23! l23j3l31n′23m3n31!

l31j1l12n′31m1n12!= (−1)j1+j2+j3p(2l12 + 1)(2l23 + 1)(2l31 + 1)×(j1j2j3l23l31l12) j1j2j3m1m2m3!,we can relate the corresponding functions Ψ∆,c and Ψ∆′,c′ asΨ∆,c = (−1)j1+j2+j3p(2l12 + 1)(2l23 + 1)(2l31 + 1)(j1j2j3l23l31l12)Ψ∆′,c′,(19)where lij is the color of the segment of the loop in (∆, c) connecting ji and jj.Thiscorresponds to (iii) in the list, and the inductive proof holds in this move. The inductionfor the move (i) is also guaranteed by the same equation (19).In this way, we have proved that the identities (15) holds forn arbitrary pair of ∆and∆′.

Using a variation of the analysis in Appendix D of[11], one can show that all otherrelations among Ψ∆,c on a flat connection ω are generated from (18) and (19). Therefore(15) are the only relations among Ψ∆,c’s.03The Hartle-Hawking-Type Wave-FunctionIn the previous section, we defined the isomorphism between the physical Hilbert spacesof the lattice gravity and the Chern-Simons theory.

In this section, we will show that thisisomorphism indeed identifies wave-functions associated to the same geometry of the three-dimensional manifold. The geometry we consider here is a handlebody M. To describeM, we embed a closed orientable two-dimensional surface Σ into R3.

The handlebody Mis taken as the interior of Σ. Associated to such a geometry, we can construct physicalstates in both the lattice gravity and the Chern-Simons theory.In the Chern-Simons theory, the physical wave-function ΦM for M is defined asΦM(ω|Σ)δ(Fij|Σ)=Zω|Σ:fixed[de, dω] exp(iZMe ∧(dω + ω ∧ω)),(20)where we perform the functional integral over e and ω in the interior of M with a fixed– 14 –

boundary condition of ω on ∂M = Σ. The integration over e|Σ gives rise to δ(Fij|Σ) whichis explicitly written in the left hand side of (20).

This ensures that the functional integralin the right hand side gives a physical state of the Chern-Simons theory. Such a wave-function may be regarded as a generalization of the Hartle-Hawking wave-function (Theoriginal wave-function of Hartle and Hawking[12] corresponds to the case when Σ is S2 andthe handlebody M is a three-dimensional ball.).

The wave-function for the lattice gravityis defined in a similar fashion by fixing a triangulation ∆and its coloring c of Σ, and bysumming over all possible coloring in the interior of M. This is nothing but ZM,∆(c) wehave introduced in Section 2.In the previous section, we have found that, to each physical state φ∆(c) of the latticegravity, there is a corresponding state in the Chern-Simons theory defined by (16). There-fore it is natural to expect that the wave-functions ΦM(ω|Σ) and Z∆,M(c) associated tothe same handlebody M are related asΦM(ω|Σ) = AgXc∈C(∆)ZM,∆(c)Λ−n(∆)Ψ∆,c(ω|Σ),(21)when ω|Σ is a flat connection.

Here Ag is a constant depending only on the genus of thehandlebody M. This indeed is the case as we shall see below.As was shown by Witten in the case of the Lorentzian Einstein gravity[13], there is afairly explicity expression for the Hartle-Hawking-type wave-function ΦM(ω|Σ). As well asthe constraint Fij|Σ = 0 written explicitly in (20), the integration over e in (20) imposesthat ΦM should vanish unless ω|Σ have a flat extension ω interior of the handlebody M.This condition can be rephrased as follows.

If the boundary Σ of M is of genus g, it has 2ghomology cycles. Among these, there are g cycles which are contractable in M while otherg cycles are not.

The necessary and sufficient condition for ω|Σ to have a flat extensionin M is that its holonomies U(a) (a = 1, ..., g) around these contractable cycles are trivial.ThereforeΦM(ω|Σ) = A′ggYa=1δ(U(a) −1),(22)where A′g is a constant independent of ω|Σ, and δ(U −1) is a δ-function with respect tothe Haar measure of SO(3). Thus in order to prove the identity (21), we need to show– 15 –

that the sum over coloring in the right hand side of the equation imposes the constraintU(a) = 1 on ω|Σ. In the following we will show that, by recombining the Wilson-lines, thesum in the right hand side of (21) reduces to sums over colorings of the contractable cyclesasXc∈C(∆)ZM,∆(c)Λ−n(∆)Ψ∆,c(ω|Σ) =gYa=1∞Xj=0(2j + 1) Tr(U(a)j).

(23)The orthonormality and the completeness of the irreducible SO(3) characters[14] implythat the right hand side of this equation gives the product of the δ-functions as in (22).Now we would like to prove the equation (23). Suppose we have used n-tetrahedra incomputing the wave-function Z∆,M(c) for the handlebody M. The tetrahedra must havebeen placed in such a way that the boundary Σ of M is triangulated as (∆, c).

Let uschoose one of the tetrahedra attached on the boundary surface. Since ω|Σ is flat, we canuse (15) to remove this tetrahedron, i.e.Xc∈C(∆)ZM,∆(c)Λ−n(∆)Ψ∆,c(ω|Σ) =Xc′∈C(∆′)ZM,∆′(c)Λ−n(∆′)Ψ∆′,c′(ω|Σ),where ∆is the original triangulation of Σ in (23), and ∆′ is the one which is obtained byremoving the tetrahedron attached on Σ.

In computing ZM,∆′(c), the number of tetrahedrawe use is (n −1). By repeating this procedure, we can eliminate all the tetrahedra in M.To visualize this process, it is useful to imagine the handlebody M as a balloon whosesurface is of genus g.For example, when Σ is a torus, we consider a tube of a tire.Removing the tetrahedra is then like reducing the air from the balloon.

After graduallydecreasing its volume, the balloon will eventually be flattened. To describe the flattenedballoon, we note that the surface Σ can be constructed from two discs with g holes, S+gand S−g , by gluing their boundaries together as shown in Fig.

7. We call S+g and S−gas upper and lower parts of Σ.

The boundaries of the g holes in S±g correspond to thehomology cycles on Σ which are not contractable in M. In the limit when the balloonis flattened, the upper and the lower parts of Σ overlap one on another. Reflecting theoriginal tetrahedral decomposition of M, S±g are covered by triangles.

It is not difficult tosee that the triangulations of S+g and S−g must be identical and that they must have thesame coloring. Namely the Wilson-line network in the upper part of Σ is the mirror imageof the one in the lower part as shown in Fig.

8.– 16 –

Let us perform the sum over colorings of the Wilson-lines across the boundaries of S+gand S−g (for example the Wilson-line j3 in Fig. 8).

As we did in the previous section,we may take the lengths of these Wilson-lines arbitrarily small and replace them by 1.Because of the reflection symmetry of the Wilson-lines, we may use the orthonormality ofthe 3j-symbolsXj3,m3,m′3(−1)j3p2j3 + 1gm3m′3j3 j1j2j3m1m2m3! j3j2j1m′3m′2m′1!= (−1)j1+j2p(2j1 + 1)(2j2 + 1)gj1m1m′1gj2m2m′2(24)to emilinate the trivalent vertices at the end-points of the Wilson-lines (as shown in Fig.9).

Repeating this procedure, we can remove the vertices on Σ one by one.To understand how the resulting Wilson-line network looks like, let us examine thecase when Σ is a torus, in detail. In this case, its upper and lower parts are topologicallythe same as annuli, and each of them can be decomposed into two triangles as shown inFig.

10. Corresponding to this triangulations, there are six Wilson-lines on Σ which areconnected by four vertices (Fig.

11a). We can choose one of the Wilson-lines, say j2 in Fig.11a, and remove a pair of vertices at its end-points by using (24).

As the result, we obtaina diagram as shown in Fig. 11b.

Because of the flatness of ω|Σ, we can move around theWilson-line j1 homotopically, and the network in Fig. 11b can be brought into the one inFig.

11c. Now the Wilson-loop consisting of j1 and j3 is contractable on Σ, and we endup in Fig.

11d. In this way, the Wilson-line network on the torus is deformed into a singleWilson-loop around its homology cycle contractable in M, as shown in Fig.

11e. Takinginto account the weight ZM,∆(c), we have checked that the resulting summation over j4reproduces the right hand side of (23) for g = 1.For g ≥2, we can, for example, choose a triangulation of S±g as in Fig.

12a. Thecorresponding Wilson-line network is shown in Fig.12b.As in the case of the torusdescribed in the above, one can follow the deformation of the network and show thatPc ZM,∆(c)Ψ∆,c for this triangulation ∆reduces to the right hand side of (23).Thisproves the identity (21), and we found the factor Ag is equal to A′g which is related to thenormalization of the path integral (20).– 17 –

04Partition Functions and Topology-Changing AmplitudesWe have found that the Hartle-Hawking-type wave-functions in the lattice gravity andthe Chern-Simons theory are related by the isomorphism (14) between the physical Hilbertspaces of the two theories. In this section, we will exploit this result to show that, for anyclosed orientable manifold M, the partition functions of the two theories agree with eachother.

The idea is to use the Heegard splitting of M[15]. Consider two handlebodies M1and M2 whose boundaries are of the same topology Σ.

Since M1 and M2 differ only by themarkings of the homology cycles on their boundaries, we can glue the boundaries togetherby their diffeomorphism and obtain a closed three-dimensional manifold. Moreover it isknown that any closed manifold can be realized in this way.

In this construction, thetopology of M is encoded into the topology of ∂M1 and ∂M2 and how they are gluedtogether.Corresponding to this splitting of M, the partition function of the Chern-Simons theoryis expressed as an inner product of the Hartle-Hawking-type wave functions ΦM1 and ΦM2,Z(CS)M= (ΦM1, ΦM2)CS,(25)as far as M is orientable. This formula is derived from the functional integral expressionfor Z(CS)M; the functional integrals over M1 and M2 result in the Hartle-Hawking-typewave-functions ΦM1 and ΦM2, and the functional integral on the boundary ∂M1 ≃∂M2corresponds to taking their inner product.

On the other hand, the partition function forthe lattice gravity has also the expressionZM = (ZM1,∆, ZM2,∆),(26)as we saw in Section 2. Since the Hartle-Hawking-type wave-functions in the Chern-Simonstheory and the lattice gravity are related by (21), Z(CS)Mand ZM are the same provided theisomorphism (14) preserves the inner products in the two Hilbert spaces, HCS and H(∆).Thus, in order to establish the equivalence Z(CS)M= ZM, we want to show(Ψ∆1,c1, Ψ∆2,c2)CS = A2g · P∆1,∆2(c1, c2)(27)– 18 –

or equivalentlyXc1∈C(∆1)c2∈C(∆2)Ψ∆1,c1(ω1)Λ−n(∆1)P∆1,∆2(c1, c2)Λ−n(∆2)Ψ∆2,c2(ω2)= A−2g· K(ω1, ω2),(28)where K(ω1, ω2) is a kernel for the inner product(Φ, Φ′)CS =Z[dω1]δ(F1,ij)Z[dω2]δ(F2,ij)Φ(ω1)K(ω1, ω2)Φ(ω2),and it is given in term of the functional integralK(ω1, ω2)δ(F1,ij)δ(F2,ij) =Zω(t=0)=ω1ω(t=1)=ω2[de, dω] exp(iSCS(e, ω))(29)for the topology N = Σ × [0, 1].Now we are going to show that the left hand side of (28) is proportional to the righthand side. The factor A−2gwill be fixed later.

Since Ψ∆i,ci is evaluated on a flat connectionωi, we may use (15) to rewrite the left hand side of (28) asXc∈C(∆)Λ−n(∆)Ψ∆,c(ω1)Ψ∆,c(ω2). (30)On the other hand, it follows from the functional integral expression (29) that the kernelK(ω1, ω2) vanishes unless ω1 and ω2 has a flat extension in N. For N = Σ × [0, 1], the flatextension exists if and only if ω1 and ω2 are gauge-equivalent.

Thus we need to show thatthe sum over coloring in (30) imposes the the constraint, ω1 ≃ω2Let us study the case when Σ is a torus, in detail. In this case, the surface Σ canbe decomposed into two triangles as shown in Fig.

13a. The corresponding network ofWilson-lines is shown in Fig.

13b. A flat connection ω on the torus can be specified byholonomies U and V around the two homology cycles on Σ.

The wave-function Ψ∆,c forthe network can then be regarded as a function of U and V . In the network in Fig.

13b,– 19 –

the Wilson-line j3 can be made arbitrarily short using the flatness of ω and be replaced byan identity. In this case, the wave-function Ψ∆,c is expressed as a function of U and V asΨ∆;c(U, V ) =Xmi,m′i,m′′iU m1j1m′1V m2j2m′2× gm′1m′′1j1gm′2m′′2j2gm3m′′3j3 j1j2j3m1m2m3!

j1j2j3m′′1m′′2m′′3!. (31)Here we marked the homology cycles on Σ in such a way that the Wilson-lines j1 and j2wind around cycles corresponding to the holonomies U and V .

The holonomies U and Vcommute with each other, so they can be diagonalized simultaneously. Since the wave-function Ψ∆,c is invariant under the simultaneous conjugation, U →Ω−1UΩ, V →Ω−1V Ω,we can substitute diagonal matrices U mjm′ = eimθδmm′ and V mjm′ = eimϕδmm′.

into U and V in(31).Now we would like to perform the summation,Xj1,j2,j3Λ−1Ψ∆,c(θ1, ϕ1)Ψ∆,c(θ2, ϕ2),(32)where (θ1, ϕ1) and (θ2, ϕ2) are phases of the holonomies (U1, V1) and (U2, V2) for ω1 andω2. Although it is possible to do the summation for generic values of the phases, it ismore instructive to study the cases when two among the four phases vanish.

Actually itis enough to study these cases as we shall see below.Let us consider the case when V1 = V2 = 1. In this case, the wave-function Ψ∆,c(Ui, Vi)is simplified asΨ∆,c(Ui, Vi = 1) =s(2j1 + 1)(2j2 + 1)(2j3 + 1)2j1 + 1Tr(Uij1).The summation (32) is then performed asXj1,j2,j3Ψ∆,c(U1, V1 = 1)Ψ∆,c(U2, V2 = 1)=Xj1Tr(U1j1)Tr(U2j1) · Λ−1 ·12j1 + 1X|j2−j3|≤j1≤j2+j3(2j2 + 1)(2j3 + 1)=Xj1Tr(U1j1)Tr(U2j1) = δ(U1 −U2).Here we have used the definition (5) of Λ and the orthonormality of the irreducible char-– 20 –

acters Tr(Uj1). Thus the sum over the coloring in the left hand side of (28) indeed imposesthe constraint U1 = U2 when V1 = V2.

It is straightforward to do the computations inother cases when U2 = V2 = 1 or V1 = U2 = 1, and we have found that the sum in (32)imposes U1 = U2 and V1 = V2 in both of these cases.We have seen that the left hand side of (28) is proportional to K(ω1, ω2) as far as twoamong the four phases are equal to zero. Let us relax this condition and suppose that theyare not necessarily zero, but their ratios θi/ϕi (i = 1, 2) are rational numbers.

Since (32) isinvariant under the modular transformation of Σ, we can change the basis of the homologycycles in such a way that two among the four phases around the cycles become equal tozero. The summation in (32) then reduces to the computation in the above and we seethat the constraints ω1 ≃ω2 arises upon the summation.

In general, when the ratios arenot necessarily rational, we can find a series of rational numbers which converges to θi/ϕi.At each step in the series, the sum over the coloring in (32) gives the constraints ω1 ≃ω2.Thus it should also be the case in the limit of the series.This result is extended to surfaces of higher genera as follows. A genus-g surface Σ canbe constructed from a 4g-sided polygon by gluing its sides together as is indicated in Fig.14.

Correspondingly the surface is decomposed into 4g triangles. As in the case of the torus,we can parametrize the flat connection ω on Σ by its holonomies U(a), V(a) (a = 1, ..., g)around the homology cycles αa and βa as marked in Fig.

14. These holonomies are subjectto the constraint,U(1)V(1)U−1(1) V −1(1) · · · U(g)V(g)U−1(g)V −1(g) = 1(33)In this case, the wave-function Ψ∆,c(ω) is a product of U(a)’s and V(a)’s connected by the3j-symbols.

Especially it depends on U(1) asΨ∆,c(ω) =Xmi,m′i,m′′iUm1(1)j1m′1W m3m4× gm′1m′′1j1gm′2m′′2j2 j1j2j3m1m2m3! j1j2j4m′′1m′2m4!,where W m3m4 is independent of U(1).

The sum of Ψ∆,c(ω1)Ψ∆,c(ω2) over j1 and j2 thenimposes the constraint U(1)1 = U(1)2 as in the case of the torus. The rest of the summationcan be done inductively, and we obtain the constraint ω1 ≃ω2.– 21 –

We have found that the left hand side of (28) is equal to K(ω1, ω2) upto a constantfactor BgXc1∈C(∆1)c2∈C(∆2)Ψ∆1,c1(ω1)Λ−n(∆1)P∆1,∆2(c1, c2)Λ−n(∆2)Ψ∆2,c2(ω2)= Bg · K(ω1, ω2).Equivalently(Ψ∆1,c1, Ψ∆2,c2)CS = B−1g· P∆1,∆2(c1, c2).By combining this with (21) and by using the expressions (25) and (26), we obtainZ(CS)M= A2gB−1g ZM. (34)Now we would like to show that Bg is equal to A2g.

Since Ag comes from the normalizationof the functional integral (20), we must specify it in order to relate Bg to Ag. Here wedefine the normalization in such a way that the Chern-Simons partition function for S3 isequal to 1.

To show A2gB−1g= 1, we note that S3 can be constructed from two handlebodiesof any genus g. Let us take a closed surface Σ of genus g and embed it into S3. It is easy tosee that both the interior and the exterior of Σ are handlebodies of genus g. In this setting,the left hand side of (34) is equal to 1 due to the normalization convention of Z(CS)M. Onthe other hand, it follows from the definition of the lattice gravity that ZM for M = S3 isalso 1.

In this way, we have shown A2gB−1g= 1 for any value g. This proves the equalityof the partition functions of the lattice gravity and the Chern-Simons theory.Actually, there is a cavear here. In the above, we have assumed that the integral (25)and the sum (26) are convergent.

This is not always the case. For example, when M is ofthe topology Σ × S1, the partition function ZM of the lattice gravity is given by a traceover physical statesXc∈C(∆)Λ−n(∆)P∆,∆(c, c),where ∆is a triangulation of Σ. Namely the partition function ZM counts the number ofphysical states for Σ which is infinite if g ≥1.

In this case, the partition function Z(CS)Mfor the Chern-Simons theory also diverges as was pointed out by Witten [13]. There hehas shown that the divergence occurs when e →∞, namely when the size of M is large,and thus it is infrared in nature.– 22 –

So far, we have considered the case of a closed three-dimensional manifold M. It is alsopossible to consider a manifold with boundaries and discuss transition amplitudes betweeninitial and final states.We have already studied some of such processes in this paper. For example, P∆1,∆2(c1, c2)corresponds to the geometry N ≃Σ × [0, 1] for a transition of Σ into Σ.

The Hartle-Hawking-type wave-function Z∆,M(c) can also be viewed as describing a transition of apoint into a closed surface Σ (or a creation of Σ from nothing). In both of these cases,we have found that the amplitudes of the lattice gravity and the Chern-Simons theory arerelated by the isomorphis defined in Section 2.We can extend this analysis and study more elaborated transition processes involvinga topology-change of Σ.

In [13], Witten has examined the following situation in the caseof the Chern-Simons theory. Consider Σinitial consisting of two components Σ1 and Σ2 ofrespective genus g1 and g2.

One can construct a manifold M which interpolates Σinitial toanother surface Σfinal of genus g = g1 + g2 consisting of a single component, as follows.We first embed Σfinal into R3 to obtain a handlebody M0 of genus g. We then removefrom M0 handlebodies of genus g1 and g2 whose boundaries are Σ1 and Σ2. The remainingprotion of M0 gives the manifold M whose boundaries are Σ1, Σ2 and Σfinal.

A wave-function for the initial surface Σinitial is spanned by products of functions ΦΣ1 and ΦΣ2 offlat connections on Σ1 and Σ2. The transition amplitude for M then relates ΦΣ1ΦΣ2 toΦΣfinal for the final surface.

It is shown in [13] that the relation is as follows.ΦΣfinal ∼δ(A −1)ΦΣ1ΦΣ2. (35)Here A is an element of SO(3) given in terms of holonomies U(a) and V(a) (a = 1, .., g) onΣfinal asA = U(1)V(1)U−1(1) V −1(1) · · · U(g1)V(g1)U−1(g1)V −1(g1),and the holonomies U(1), ..., U(g1) and V(1), ..., V(g1) correspond to the homology cycles onΣfinal which are homotopically equivalent to the cycles on Σ1 through the manifold M.For the lattice gravity, the transition amplitude Z∆1,∆2;∆final(c1, c2; cfinal) for M isgiven by summing over coloring of tetrahedra interior of M while keeping fixed the coloringsc1, c2 and cfinal on Σ1, Σ2 and Σfinal.

To see its relation to the transition amplitude (35)– 23 –

in the Chern-Simons theory, we multiply the Wilson-line networks to Z∆1,∆2;∆final and sumover the colorings asXc1∈C(∆1),c2∈C(∆2)Cfinal∈C(∆final)Z∆1,∆2;∆final(c1, c2; cfinal)× Λ−n(∆1)Ψ∆1,c1Λ−n(∆2)Ψ∆2,c2Λ−n(∆final)Ψ∆fianl,cfinal.The computation is essentiall the same as we did in (28), and the result agrees with (35).Thus the transition ampltudes of this type are also equivalent in the two theories.05DiscussionsWe have found that the partition functions and some topology-changing amplitudesof the lattice gravity of Ponzano and Reggeare equal to the ones of the ISO(3) Chern-Simons theory. This result supports the original conjecture by Ponzano and Regge thatthe statistical sum of the 6j-symbols describes the fluctuating geometry with the weightQx∈M cos(√gR).

Indeed, if we integrate over ω first in the Chern-Simons functional inte-gral, we are left with an integral over the dreibein e with a weight exp(Re ∧R) where Ris a curvature two-form constructed from e. Although we seem to have gotten exp ratherthan cos, we should note thatRe ∧R changes its sign if we flip the orientation of e. Sincewe integrate over e as a part of the ISO(3) gauge field, at each point in M, both orienta-tions of e contribute to the functional integral. If one tries to integrate over e of a fixedorientation, one would need to replace the exponential by the cosine to compensate forthe restriction.

Therefore it appears that the lattice gravity of Ponzano and Regge gives afunctional integral of Qx∈M cos(√gR) with the correct measure for the fluctuating metricon M.To regard this system as the Euclidean Einstein gravity, the factor i in front of theaction is disturbing. One cannot eliminate it by rotating the contour of the e-integral sincethe resulting functional integral would be divergent.

To address this issue, it would be morefruitful to study a Lorentzian version of the lattice model based on the infinite dimensionalrepresentations of SO(2, 1). Since the representation theory of SO(2, 1) is far richer thanthat of SO(3), we must have good criteria in choosing a class of representations we put ofthe edges of the tetrahedron.

One of the criteria would be that a sum of characters over– 24 –

such representations should give the invariant δ-function δ(U −1), in order for the latticemodel to have the same partition function as in the Lorentzian Chern-Simons gravity.Such study will be useful in understanding the structure of physical observables in theLorentzian gravity.Recently Mizoguchi and Tada[16] have studied the q-analogue of the 6j-symbol andfound the asymptotic formula for q = e2πi/(k+2)(−1)Pi ji(j1j2j3j4j5j6)q∼c√Vcos(SRegge −λk2 V + π./4),where c is some constant and λk = (4π/k)2. Thus it is natural to expect that the q-analogueof the Ponzano and Regge model introduced by Turaev and Viro would be related to thegravity with the cosmological constant λk or the SO(3) × SO(3) Chern-Simons theory.The recent paper by Turaev [9] supports the latter possibility.Durhuus, Jakobsen and Ryszard[17] have constructed a large class of topological latticemodels extending the model of Turaev and Viro.

The method developed in this paper couldalso be applicable to study those models.AcknowledgmentsThe author would like to thank N. Sakakura for discussions.He is also thankfulto T. Maskawa for explaining him some group theoretical facts and to L. Kauffman forencouragements.REFERENCES1. G. Ponzano and T. Regge, in Spectroscopic and Group Theoretical Methods in Physics,ed.

F. Block (North-Holland, Amsterdam, 1968).2. T. Regge, Nuovo Cimento 19 (1961) 558.3.

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Phys. B335 (1990) 635;D.J.

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5. V.G.

Turaev and O.Y. Viro, “State Sum Invariants of 3-Manifolds and Quantum6j-Symbols,” preprint (1990).6.

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FIGURE CAPTIONS1. A tetrahedron colored by angular momenta ji’s.2.

One tetrahedron can be decomposed into four small tetradedra.3. A manifold M is decomposed into three parts, M1, M2 and N.4.

ZMi,∆i(ci) is defined by the summation over colorings on edges interior of Mi with thefixed coloring ci on the boundary.5. Attaching a tetrahedron to a triangle on a surface, as seen from the above, and thecorresponding Wilson-line networks.6.

Attaching a tetrahedron to two neighbouring triangles on a surface, as seen from theabove, and the corresponding Wilson-line networks.7.A genus-g surface can be constructed from two discs with g-holes by gluing theirboundaries together. The case of g = 3 is shown in the figure.8.

Part of Wilson-lines on S+g and S−g .9. The Wilson-line j3 in Fig.

8 can be removed using the orthogonality of the 3j-symbols.10. Triangulation of S±g=1 in the case of a torus.11a.

Wilson-line network corresponding to the triangulation in Fig. 10.11b.

The Wilson-line j2 is removed using the orthogonality of the 3j-symbols.11c. The Wilson-line j1 can be moved around using the flatness of ω|Σ.11d.

The homotopically trivial loop is removed.11e. The network is transformed into a single Wilson-loop around the homology cyclecontractable in M. The sum over j4 restricts the holonomy around this cycle to be trivial.12a.

Triangulation of S±g .12b. The corresponding Wilson-line network on S±g .13a.

Torus can be decomposed into two triangles.13b. The corresponding Wilson-line network.14.

A genus-g surface can be constructed from a 4g-sided polygon. Correspondingly, thesurface is decomposed into 4g triangles.– 27 –

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