Asymptotic analysis of the Ponzano-Regge model for handlebodies
aa r X i v : . [ g r- q c ] S e p Asymptotic analysis of the Ponzano-Regge model forhandlebodies
Richard J. Dowdall ∗ , Henrique Gomes † , Frank Hellmann ‡ School of Mathematical SciencesNottingham UniversityUniversity ParkNottingham NG7 2RDUKOctober 31, 2018
Abstract
Using the coherent state techniques developed for the analysis of the EPRL model we givethe asymptotic formula for the Ponzano-Regge model amplitude for non-tardis triangulationsof handlebodies in the limit of large boundary spins. The formula produces a sum over allpossible immersions of the boundary triangulation and its value is given by the cosine of theRegge action evaluated on these. Furthermore the asymptotic scaling registers the existenceof flexible immersions. We verify numerically that this formula approximates the 6j-symbolfor large spins.
In [1], Ponzano and Regge gave a formula for the large spin limit of the 6j symbol. It wasfound to be related to the Regge action for discrete general relativity and with this motivationthey constructed the first spin foam model of 3d gravity. Their asymptotic formula was firstproved in [2] then more recently using different methods in [3, 4] and the square of the 6j symbolwas also studied in the context of relativistic spin networks [5, 6]. The next to leading orderapproximation was recently considered in [7]. The precise formulation of the full state sum wasstudied in [8].In [9, 10], the semiclassical limit of some recent spin foam models [11, 12, 13] was analysedusing the coherent state techniques introduced in [13]. In particular the boundary there wasformulated in term of coherent tetrahedra. Here we apply the same techniques in the 3d caseusing coherent triangles and, instead of a single vertex amplitude, we analyze triangulations ofarbitrary genus handlebodies. This finally opens up the possibility of a continuum limit andrenormalization analysis of the model for this restricted class of 3-manifolds. In particular theresulting formula is well suited for studying the graviton propagator as introduced for Ponzano-Regge in [14].We begin the paper by describing the formulation of the Ponzano-Regge model in terms of asingle spin network diagram dual to the boundary triangulation. We then describe the boundarystate choice in detail and proceed to give the asymptotic formula in terms of immersions of the ∗ [email protected] † [email protected] ‡ [email protected] The Ponzano Regge amplitude was originally defined in terms of 6j symbols with a cutoff regular-ization on the interior vertices. More recently, it was shown in [8] that the cutoff regularizationfor sums over representations in some cases disallows a 2-3 Pachner move (the Biedenharn-Elliotidentity does not hold for a restricted sum over representations.) This meant topological in-variance of the partition function can not be proved with Pachner moves in this form. Thealternative formulation in terms of delta functions and integrals over SU(2) regularized with agauge fixing tree is both finite and invariant under Pachner moves. Another regularization usingrepresentations of a quantum group is given by the Turaev-Viro model, however it was also shownin [8] that the limiting procedure that gives the Ponzano-Regge model is only known to existfor so-called non-tardis triangulations - i.e. a triangulation whose edge lengths are restricted toa finite range by the boundary edge lengths. In order to avoid discussing regularization, in thispaper we will restrict to only considering these non-tardis triangulations which are by definitionfinite. Slightly extending the terminology of [8], we will call a manifold Σ a non-tardis manifold if there exists a ‘non-tardis’ triangulation of Σ .For a 3-manifold Σ with orientable 2-boundary ∂ Σ its boundary state space is then givenby the possible geometric triangulations of the 2-boundary with half integer edge lengths. Theamplitude for such a non-tardis manifold is given in terms of a non-tardis triangulation T of Σ that extends the boundary triangulation, and some boundary state Ψ: Z P R (Ψ , T ) = X j e Y e dim( j e ) Y ∆ h Theta i Y σ h Tet i . (1)Here e is an edge, ∆ a triangle and σ a tetrahedron of the triangulation of the interior, j are halfintegers labelling the irreps of SU(2). The amplitudes h Theta i and h Tet i are the spin networkevaluation of the theta graph and the planar tetrahedral spin network respectively. These spinnetworks are the two dimensional duals to the interior triangles ∆ and, respectively, to thesurface of the tetrahedra σ in the triangulation. The labelling of the spin network surface dualsof the ∆ and σ is given by assigning the j associated to each edge to each dual edge that crossesit. Finally dim( j ) = ( − j (2 j + 1) is the (super)-dimension of the j th SU(2) irrep in graphicalcalculus.In the interior the normalisation and phase of the intertwiners cancels. However, at theboundary these are arbitrary normalisation for each face. This information is in the boundarystate Ψ which consists of the boundary edge length data and the particular intertwiner chosenat each face. In some cases it is possible to reformulate the Ponzano-Regge model defined above as a spinnetwork evaluation on the 2-boundary of the manifold. In fact, Ponzano and Regge originally2onstructed the state sum model such that it agreed with the evaluation of a planar spin net-work associated to the boundary of a 3-ball. An algorithm to construct a non-tardis interiortriangulation given an arbitrary triangulation of the boundary of B , was given using recouplingtheory by Moussouris in [15]. This algorithm consists of reducing the boundary spin networkto a product of 6j symbols (which is always possible for a planar diagram) using the recouplingidentity and Schur’s Lemma and then reconstructing the interior triangulation from these 6jsymbols. Since the boundary spin network is finite, this procedure gives a manifestly finitedefinition of the partition function.In this paper we will extend this result to spin networks on the boundary of handlebodies ofarbitrary genus. A non-tardis triangulation of a handlebody of genus g can be constructed asfollows. Start with a triangulation of the boundary of B with g distinct pairs of triangles thatdo not share a common vertex. The boundary of the handlebody can be formed by identifyingthese triangles and a non-tardis triangulation of the interior is given by applying the Moussourisalgorithm. This procedure may result in a degenerate triangulation of the handlebody even ifthe triangulation of the ball is non-degenerate.We will begin our analysis by reformulating the amplitude for the 3-ball B on a non-tardistriangulation as a spin network on the boundary by a “reverse Moussouris algorithm.” We thendescribe how this procedure is altered for handlebodies of arbitrary genus. From now on, Σ denotes a handlebody. Lemma 1.
The Ponzano-Regge amplitude for a non-tardis triangulation of B can be expressedin terms of a single spin network evaluation Z P R (Ψ , B ) = h ( ∂B ) ∗ i (2) where ∗ is the two dimensional dual of the surface triangulation with each dual edge labelled withthe SU(2) irrep corresponding to the length of the edge it is dual to, and the spin network isevaluated as the planar projection without crossings, with the intertwiner normalisation given by Ψ . Proof : In order to reexpress the 3-ball with a given triangulation T and the amplitude Z (Ψ , B ) as the spin network evaluation of its boundary we proceed inductively. Note first thata triangulation of B given by a single tetrahedron is already of the form we want to put it in: by(1) its amplitude is given exactly by the evaluation of the spin network dual to its boundary 2-geometry with an intertwiner normalisation chosen at each surface triangle. This establishes thebase case. We now need to show that the statement remains true when one glues tetrahedra onto a ball amplitude already expressed in this manner, and thus reconstruct arbitrary non-tardistriangulations of the 3-ball. To glue we add the necessary face and edge amplitudes for the newinterior faces and edges with the same the normalisation and phase choice of the intertwinerschosen in the boundary state before. These boundary choices will therefore cancel. This is inaccordance with the observation above that the phase choice and normalisation on the interiorare left arbitrary. A tetrahedron can be glued onto a ball non-degenerately with one or twofaces:1. If we glue one face of the tetrahedron with one face of the ball we create an inner triangle.The PR amplitude of the new ball differs from the old one by a h Theta i and a tetrahedralnet. In the spin network evaluation the vertices of the 3-ball and the tetrahedral amplitudecorresponding to the glued face, together with the face amplitude, are the normalizedprojector on the invariant subspace of the irreps on the edges. As both amplitudes beingglued already are invariant we can simply replace them with parallel strands, see Figure 1.This changes the spin network graph being evaluated by changing a vertex to a triangle.This is the dual to the change of the surface triangulation, and the resulting amplitudestill satisfies the lemma. 3Sfrag replacements j j j j j j = θ ( j , j , j )PSfrag replacements j j j Figure 1: Case 1: Reduction of the PR amplitude for two tetrahedra to the spin network on theboundary.PSfrag replacements h PSfrag replacements h Figure 2: By replacing the gluing along the 4 faces with a group averaging on edges crossing thedashed line (left) we reexpress the T amplitude on the boundary (right).2. If we glue two faces of the tetrahedron onto the ball, we create an inner edge and twoinner faces, the PR amplitude changes by adding a tetrahedral net, two thetas and onedimension factor. However, these nets correspond exactly to the 6 j symbol for changingthe ball amplitude from being connected along the dual of the old boundary to the dualof the new boundary.Note that for any non-tardis triangulation of B we can always build it up from a single tetra-hedron by gluing on one or two faces. Furthermore the two operations described above do notintroduce crossings and respect the planar projection chosen.This establishes that one can express the Ponzano Regge amplitude of an arbitrary triangu-lation of B as a spin network evaluation on its boundary S . This proves the lemma (cid:3) .Consider next the case of a solid torus D × S , which we call T . Take a disc D ⊂ T suchthat D = D × { p } ∈ D × S . For future purposes, note that it intersects a non-contractibleloop in T . We can now always move this disc by a homotopy that keeps ∂ D on ∂ T such that itsboundary is the union of at least three edges of the boundary triangulation. Due to triangulationinvariance we can then choose a triangulation such that D has no internal vertex. We can thencut the Ponzano Regge amplitude along this surface, the resulting space is topologically B andwe can apply the previous lemma. This yields a ball where two discs on the boundary are gluedby identifying edges and using the PR face and edge weights. Call n the number of edges thatmake up ∂ D . As we chose a disc with no internal vertex, the spin network dual to it has to bean n − ∂ D . This projection can then be replaced by a group averaging on the strands dualto ∂ D : Z P R (Ψ , T ) = Z SU(2) d h h ( ∂ T ) ∗ h i (3)4here ( ∂ T ) ∗ h is the spin network dual to the surface of the torus with h inserted along dual edgescrossing ∂ D . The diagram is defined by first cutting along this circle, choosing the planar nocrossing diagram of the graph and then connecting up along the identified edges. See figure 2,and Appendix A for an explicit example.We can easily generalize this example to arbitrary genus handlebodies. By definition, ahandlebody of genus g comes equipped with a set of g standard cuts that reduce the handlebodyto the 3-ball. We call these cuts D i with an index i ∈ C where C is a set of labels for the standardcuts. For later use, note that it is always possible to define a complete set of generators c i ofthe homology group H (Σ ) such that each c i is transversal to the cut D i and does not intersectthe other cuts. We can choose an equivalent set of cuts that are related to the standard cuts byboundary preserving homotopy as long as the cuts remain non-intersecting. In particular fromnow on we will choose the cuts so as to lie on the triangulation. This implies a restriction onthe class of triangulations considered as such a choice may not exist for small triangulations.Now we can state: Lemma 2. Z P R (Ψ , Σ ) = Z SU(2) Y i ∈ C d h i h ( ∂ Σ ) ∗ h i i (4) where Σ is a handlebody, ∂ Σ is its triangulated boundary which carries half integer labels onits edges and C labels the cuts. Choose a set of cuts D i that lie on the triangulation. h ( ∂ Σ ) ∗ h i i isthe spin network evaluation of the dual of the triangulation of the surface, with the links labelledby the half integer lengths of the edges they cross and a h i ∈ SU(2) inserted on every link thatcrosses a cut ∂ D i ∈ ∂ Σ , i ∈ C . The spin network is evaluated in the planar projection of theboundary of the cut manifold. That is, with all crossings being due to the links crossing a cut. Proof: Cutting Σ along the discs D i reduces it to a 3-ball. The spin network evaluation isdefined by taking the planar no crossing representation of the graph cut along the circles ∂ D i and then connecting the identified open ends. If we choose a triangulation that triangulateseach disc D i without internal vertices and reexpress the resulting amplitude as a spin networkevaluation, then the gluing of the faces corresponds to a projection onto invariant subspaces.Replace the projection onto the invariant subspace by a group integration and we get the lemma. (cid:3) Note that due to the intertwining property of the spin network the choice of D i does notmatter, it merely moves the h i insertion in the intertwiner around. In order to have a clear geometric picture of the amplitude we will choose the intertwinersin the boundary state Ψ by using coherent states α k ( n , θ ) [16]. These are the highest weighteigenstates of the normalized Lie algebra elements, that is for L i = i σ i the Lie algebra generatorsand n ∈ S , a coherent state α k ( n , θ ) in the k representation satisfies: L. n α k ( n , θ ) = ikα k ( n , θ ) (5)The parameter θ describes a choice of representative of the U(1) equivalence class of statesthat correspond to the same n . These states transform with a phase under the group elementsgenerated by L. n and the label n transforms covariantly under the SO(3) action of SU(2). Thatis for g ∈ SU(2) with corresponding SO(3) element ˆ g : gα k ( n , θ ) = e ikφ α k (ˆ g n , θ ) (6)For the asymptotic analysis three further properties will be crucial:5 The k representation can be constructed as the symmetric subspace of 2 k copies of thefundamental representation. In this picture coherent states decompose into a tensor prod-uct of coherent states in the fundamental representation. Consequently the group actionfactorizes: gα k ( n , θ ) = g k O i =1 α ( n , θ ) = k O i =1 e i φ α (ˆ g n , θ ) . (7) • The modulus squared of the Hermitian inner product of coherent states is given by: |h α k ( n , θ ) , α k ( n , θ ) i| = (cid:18)
12 (1 + n . n ) (cid:19) k , (8) • Under the action of the standard antilinear structure on SU(2) (see [9]) the coherent statechanges as: L. n J α k ( n , θ ) = − ikJ α k ( n , θ ) (9)The antilinear map J is given by multiplication by the epsilon tensor in the spin k repre-sentation followed by complex conjugation. J commutes with SU(2) elements.Note that given a set of three edge labels k i there is a non zero intertwiner exactly if theysatisfy the triangle inequalities. Therefore there is a set of n i , unique up to O(3) such that P i =1 k i n i = 0. We can then choose our intertwiner in the boundary state Ψ as ι = Z SU(2) d X O i ( Xα k ( n i , θ i )) (10)This state is clearly an SU(2) invariant state. As we noted that SU(2) acts covariantly as SO(3)on the labels n i this choice is only dependent on an unspecified phase as we left open whicheigenstates of L. n i we are using. In particular it does not depend on the remaining parity P = O(3) / SO(3) as this acts on the plane of the triangle as an SO(3) element.Thus choosing normalized α k ( n , θ ) compatible with the boundary spin labels fixes the inter-twiner states up to a parity choice and up to a phase. These two data will be fixed by consideringthe gluing of the boundary. Let V be a set of labels for the boundary faces. Then we label the boundary edges by pairs ab | a, b ∈ V and call the set of such pairs E . Let φ a : ∆ a → N ⊥ be an orientation preservingmap from the a -th triangle on ∂ Σ to the plane orthogonal to the north pole of S (which wedenote N = (0 , , N ⊥ to be the one inherited from R bytaking N to be the outward surface normal. As the boundary of Σ is orientable, we can define n ab = φ a ( e ab ) where ab ∈ E .The requirement that φ a be orientation preserving implies that the triangles with the edgevectors given by k i n i all have the same orientation in N ⊥ . In particular we can require themto have the same orientation as we have chosen for N ⊥ . In particular this implies that we canglue up any two triangles a , b with a common edge in N ⊥ in an orientation preserving way.Thus there exists an element ˆ g ab ∈ SO(3) such that: − n ba = ˆ g ab n ab N = ˆ g ab N (11)Where n ab is the edge vector of triangle a that gets glued to triangle b . As in [9], ˆ g ab is the Levi-Civita parallel translation from triangle a to triangle b , according to the bases provided by φ a φ b . Again, given a choice of spin structure for Σ , a choice of a spin frame for each triangledefines the SU(2) lift g ab as the parallel translation of the spin connection in these frames.Next we will describe a canonical choice of phase for the boundary state Ψ. From (6), α k ( − n ab , θ ab ) is proportional to g ab α k ( n ba , θ ba ). Then we fix the relative phase of the coherentstates on the boundary: J α k ( n ba , θ ba ) = g ab α k ( n ab , θ ab ) (12)We call coherent states with the above relative state choice Regge states, and denote them | n , k i Their image under the antilinear structure is |− n , k i = J | n , k i , and states in the fundamentalrepresentation are denoted | n i .The total boundary state is then given by:Ψ( k i , n i ) = Z Y a ∈ V d X a ! O cd ∈ E X c | n cd , k cd i (13)Due to the presence of the antilinear map in the definition of the relative phase the overallambiguity not fixed by (12) cancels in the overall state. At each triangle a we have a signfreedom as adding a sign contributes ( − P b,ab ∈ E k ab = 1 by the admissibility conditions onintertwiners. This shows that as in [9] the possible lifts of ˆ g ab are defined by the spin structureson the boundary, and do not rely on the arbitrary spin frame covering chosen to define the lift.Finally note that inverting the orientation of N ⊥ would have the same effect as turning thestate into Ψ ′ = J Ψ. We begin with B . To evaluate the spin network defining our amplitude in terms of thesecoherent intertwiners we choose a particular diagrammatic representation of the planar graph.To obtain the spin network evaluation of this graph we then contract the intertwiners chosenusing the epsilon inner product defined in terms of the Hermitian inner product by ( α, β ) = h J α | β i . Number the triangles in the graph from left to right. We then assume that the coherentintertwiners have been specified with respect to this planar representation of the graph as well.Then we have no crossings in the diagram and we can now explicitly write the contraction ofcoherent intertwiners as: Z P R (Ψ , B ) = Z Y a ∈ V d X a Y bc ∈ E ( X b | n bc , k bc i , X c | n cb , k bc i )= Z Y a ∈ V d X a Y bc ∈ E h− n bc , k bc | X † b X c | n cb , k cb i = Z Y a ∈ V d X a Y bc ∈ E h− n bc | X † b X c | n cb i k bc (14)Where we have written | n cb i for | n cb , i .For general manifolds we need to make sure that, after we have chosen the circles, the dualedges crossing a circle all have the same orientation relative to the circle. This can be doneby using a planar representation that has all the glued discs strictly left or right of each other.Call ˜ E the set of edges not crossing circles and E j the set of edges crossing circle j ∈ C . Theamplitude is then given by Z P R (Ψ , Σ ) = ( − χ Z Y a ∈ V d X a Y j ∈ C d h j Y bc ∈ ˜ E h− n bc | X † b X c | n cb i k bc Y l ∈ C Y de ∈ E l h− n de | X † d h l X e | n ed i k de (15)7here ( − χ is a sign factor incurred in the spin network evaluation when connecting up theglued edges in the spin network evaluation. This can then be written as Z P R (Ψ , Σ ) = ( − χ Z Y i ∈ V dX i Y j ∈ C dh j e S (16)with the action given by S = X ab ∈ ˜ E k ab ln h n ab | J X † a X b | n ba i + X l ∈ C X de ∈ E l k de ln h n de | J X † d h l X e | n ed i . (17)Note that the ambiguity in the logarithm of a complex number does not affect the amplitude. The action (17) has the following symmetries (up to 2 πi ) • Continuous.
A global rotation Y ∈ SU(2) acting on each X a and h i as X a → Y X a and h i → Y h i Y − . This represents a rigid motion of the whole manifold. • Discrete.
At each triangle a the transformation X a → ǫ a X a with ǫ a = ± ǫ P b,ab ∈ E k ab a . As the admissibility conditions are satisfied on each triangle, this factorequals one. Similarly we have an arbitrary sign ǫ i on h i as the edges on which h i actsatisfy the admissibility condition for intertwiners.This latter symmetry will be used to compensate for the ambiguity of the lifts of SO(3) to SU(2).We can now state the theorem on the asymptotic formula. The standard choice of phase for an intertwiner, defined by chromatic evaluation [17], gives realnumbers for a spin network evaluation. We will now show that with the Regge phase choice theamplitude is real so can only differ from the chromatic evaluation by ± n ab orthogonal to e z , the rotation e − iπ e z · σ rotates n cb to − n cb and leaves e z invariant. Under this rotation, the coherent state | n cb i willtransform as e − iπ e z · σ | n cb i = e iφ J | n cb i (18)for some phase φ .Consider a single term in the amplitude (15), and rewrite it inserting the identity: h− n bc | X † b X c | n cb i = h− n bc | e iπ e z · σ ( e − iπ e z · σ X † b )( X c e iπ e z · σ ) e − iπ e z · σ | n cb i = h n bc | J † e iπ e z · σ ˜ X † b ˜ X c e − iπ e z · σ g cb J | n bc i = h n bc | J † e − iφ J † ˜ X † b ˜ X c g cb J e iφ J | n bc i = h n bc | J † J † ˜ X † b ˜ X c J | n cb i = h− n bc | ˜ X † b ˜ X c | n cb i (19)where we have defined the transformation ˜ X c = X c e iπ e z · σ , which can be absorbed on the groupintegration in (15) and the fact that J | n cb i = |− n cb i . We have used the Regge phase choice (12)from going from the first to the second line and the fact that the SU(2) transformations are allin fact in the same U(1) subgroup (and hence commute). From going to the second to the thirdline we have noted that we are acting with opposite rotations on the same state. Hence we getthat Z P R (Ψ , Σ ) = Z P R (Ψ , Σ ) which is thus real.8Sfrag replacements ˆ h ∈ SO(3)Figure 3: A cut immersion for a particular boundary triangulation of a torus. The cutting circlesare shown in bold and there is an ˆ h ∈ SO(3) that identifies them.
We wish to study the semiclassical limit of the amplitude Z P R (Ψ , Σ ). In order to do this, wehomogeneously rescale the spin labels by a factor λ . The corresponding boundary state Ψ λ isgiven by Ψ λ = Ψ( λk i , n i ).Given a set B = { n ab , k ab } a = b of boundary data we denote as I the set of cut immersions ofthe polyhedral surface ∂ Σ with edge lengths k ab in R up to rigid motion.A cut immersion i ∈ I is an immersion of the manifold obtained from ∂ Σ by the trivializingcuts ∂ D i , i ∈ C , i.e. it is an immersion ı ( ∂ Σ − {∪ i ∈ C ∂ D i } ) ֒ → R . Furthermore, we requirethe existence of SO(3) elements that identify the two sides of the cut, i.e. ˆ h i ∈ SO(3) suchthat ˆ h i ( ı ( ∂ D + i )) = ı ( ∂ D − i ), where ∂ D − i and ∂ D + i are the elements of the boundary ∂ ( ∂ Σ − ∂ D i )created by the removal of ∂ D i from ∂ Σ .Any two cut immersions of Σ are defined to be equivalent, and can be obtained from eachother if the cuts are related by a homotopy on the surface. Therefore, different choices of cuts D i lead to equivalent cut immersions. An example of a cut immersion is given in Figure 3.Such an immersion is called rigid if every continuous deformation of it requires changing theedge lengths, and flexible otherwise. We denote the subset of rigid immersions I r ⊂ I . Throughevery immersion in I passes at least one manifold (with dimension d ) of immersions that can becontinuously deformed into each other. We call these flexifolds and denote them f , we denotethe set of flexifolds F . We then define F max to be the set of flexifolds in F of maximal dimension d max . With this definition the rigid immersions are a special case of a flexifold with dimension d = 0. We assume from now on that the flexifolds f do not intersect.In the limit λ → ∞ we have the following theorem: Theorem 1. (Asymptotic formula)
1. If I is not empty we have that: Z P R ( ψ λ , Σ ) = (cid:18) πλ (cid:19) | V | + | C |− − d max X f ∈ F max N f cos λ X ab ∈ E k ab Θ f ab + φ f ab ! + O (cid:18) λ (cid:19) | V | + | C | ) − d max (20) Note that since D i is transversal to generators of H (Σ ), its removal changes the connectivity of Σ andcreates two new boundaries, D − i and D + i . he coefficient N f , the dihedral angle Θ f ab and the phase φ f ab are independent of λ . The φ f ab and the dihedral angle Θ f ab are evaluated on an arbitrary immersion i in f . It can beshown that these are independent of the cuts. Thus for any particular edge we can evaluatethe dihedral angle by moving the cut away from it. | V | is the number of triangles (orequivalently vertices in the set V ) and | C | is the number of cutting circles. d max is thedimension of the flexifolds f ∈ F max , and N f now also contains an integral over the unionof flexifolds in f .2. If no immersions in R exist the amplitude is exponentially suppressed: Z P R ( ψ λ , Σ ) = o ( λ − n ) ∀ n (21)Note that in the simple case where the boundary data only admits rigid immersions, ie if d max = 0, then the sum becomes a sum over the rigid immersions i ∈ I r and we have that: Z P R ( ψ λ , Σ ) = (cid:18) πλ (cid:19) | V | + | C |− X i ∈ I r N i cos λ X ab ∈ E k ab Θ i ab + φ i ab ! + O (cid:18) λ (cid:19) | V | + | C |− +1 (22)since d max = 0. Since the immersions are now rigid, the coefficient N i , the dihedral angles Θ i ab and the phase φ i ab are evaluated on the cut immersion i . We now prove the above theorem. We begin by describing the methods used to give the asymp-totic form of the amplitude, this will require finding the so-called stationary and critical pointsof the action. We can then interpret these points geometrically and give the asymptotic formula.Much of this section is similar to [9] but one dimension lower so the analysis will be as brief aspossible.
To find the asymptotic form of the amplitude I , we will use the complex stationary phase formula[18]. The details of this are recalled below.Let D be a closed manifold of dimension n , and let S and a be smooth, complex valuedfunctions on D such that the real part Re S ≤
0. Consider the function f ( λ ) = Z D dx a ( x ) e λS ( x ) . (23)The Hessian of S is the n × n matrix of second derivatives of S denoted H . For now let us assumethat the stationary points are isolated, which, by the Morse lemma is a condition equivalent tothe statement that the Hessian is non-degenerate at the critical points; det H = 0. Such functionsare called Morse functions.In the extended stationary phase, the key role is played by critical points , that is, points x ,which are not only stationary: δS ( x ) = 0 but for which Re S ( x ) = 0 as well. So to computethe dominant terms in the asymptotics for large spins, we need to find the stationary pointsof the action S and restrict to those with zero real part. If S has no critical points then for10arge parameter λ the function f decreases faster that any power of λ − . In other words, for all N ≥ f ( λ ) = o ( λ − N ) , (24)If there are isolated critical points, then each critical point contributes to the asymptotics of f by a term of order λ − n/ . For large λ the asymptotic expansion of the integral yields for eachcritical point a ( x ) (cid:18) πλ (cid:19) n/ p det( − H ) e λS ( x ) [1 + O (1 /λ )] . (25)At a critical point, the matrix − H has a positive definite real part, and the square root ofthe determinant of this matrix is the unique square root which is continuous on matrices withpositive definite real part, and positive on real ones. If S admits several isolated critical pointswith non-degenerate Hessian, we obtain a sum of contributions of the form (25) from each ofthem.If there are also degenerate critical points, more care is needed to compute the asymptotics(see eg [19]). Let C := { y ∈ D | δS ( y ) = 0 , Re S ( y ) = 0 } denote the set of critical points. For theaction S : SU(2) | V | + | C | → C we will show that C is the set of immersions I . Note that we cannota priori assume C to be a disjoint union of manifolds as the flexifolds in I can intersect. However,here we have restricted ourselves to this case so that the following generalized stationary phasetheorem applies.For a smooth function S whose critical set C is a disjoint union of closed manifolds , eachcritical manifold C x of dimension p , labelled by some x on the critical manifold, contributesthe following to the asymptotic formula [19]: (cid:18) πλ (cid:19) ( n − p ) / e λS ( x ) Z C x dσ C x ( y ) a ( y ) p det( − H ⊤ ( y )) [1 + O (1 /λ )] . (26)where H ⊤ ( y ) is the restriction of the matrix to the directions normal to C x with respect to someRiemannian metric on the domain , and dσ C x is the canonical measure induced on the criticalsubmanifold by the same Riemannian measure on the domain space. This extends to the casewhere C is a manifold-with-boundary. As described above, we must find the points of the action (17) such that Re S = 0 as these arethe only points that contribute in the limit λ → ∞ . First, we introduce some more notation.The action of the elements X b on the coherent states will produce a new coherent state | n ′ ab i = X a | n ab i (27)We will denote the corresponding rotated three vectors by n ′ ab = ˆ X a n ab (28)where ˆ X a is the SO(3) element corresponding to X a .We will first consider critical points for edges that are not on one of the cutting circles. Using(8), we can see that the real part of the action is given byRe S = X ab ∈ ˜ E k ab ln 12 (1 − n ′ ab · n ′ ba ) . (29) In the literature this is called a Morse-Bott function. A Morse function is the special case where the criticalmanifolds are zero-dimensional (so the Hessian at critical points is non-degenerate in every direction, i.e., has nokernel). S = 0 when n ′ ab = − n ′ ba for all ab , or explicitly in terms of SO(3) rotationsˆ X a n ab = − ˆ X b n ba . (30)The critical points for an edge that crosses a cutting circle i differ by the inclusion of the h i ˆ X a n ab = − ˆ h i ˆ X b n ba . (31) The stationary points are found by varying the action with respect to each of the group variables X a . The variation of an SU(2) group variable and its inverse is δX = T X δX − = − X − T (32)for an arbitrary su (2) Lie algebra element T = iT j σ j . The stationary points are given by δS = 0 and lead to the following equation X b : b = a k ab V ab = 0 (33)where V ab = h− n ab | X − a σσσ X b | n ba ih− n ab | X − a X b | n ba i , (34)These equations can then be evaluated at the critical points to give X b : b = a k ab n ab = 0 (35)which is the closure constraint for an immersed triangle.The stationary phase condition for the h i variables is the same but in this case we obtain X ab ∈ C i k ab n ′ ab = 0 (36)Which is the closure condition for edges on the circle i immersed in R . Note that unlike theclosure condition for the triangle, this relation involves the n ′ ab as each edge belongs to a differenttriangle.If the critical points are not isolated but form a manifold of critical points, then we denotethis manifold by C X = { ( X , ..., X | V | , h , ..., h | C | ) ∈ SU(2) | V | + | C | : δS = 0 , Re( S ) = 0 } (37) We will now describe how the critical/stationary points can be given a geometric interpretation.We first consider the case where the critical points are isolated.
Theorem 2. (Geometry)
Given a set of boundary data B satisfying the closure constraint oneach triangle, the solutions X a , h i to the critical and stationary point equations (30) , (31) and (36) correspond to the oriented geometric immersions i of a geometric triangulated 2-manifoldwith boundary S i ∈ C ∂ D + i ∪ ∂ D − i obtained by a suitable cutting of the boundary manifold. Thisimmersion is subject to the constraint that a set of ˆ h i ∈ SO(3) exist that map the immersion of D + i to the parity flipped P ( ∂ D − i ) , that is, the immersion of ∂ D + i is congruent and oppositelyoriented to the immersion of ∂ D − i .The geometric vectors of the immersion are given by v ab ( i ) = k ab ˆ X a n ab . Its orientation the one induced by the vectors on each face.Conversely, an immersed surface i determines a set of k ab , n ab and a set of SO(3) elements ˆ X a ( i ) . Proof: Start somewhere on the surface of Σ that is not in a circle. Since Σ has connectedboundary, the entire surface is now contractible to this point if cut along the circles. Take thetriangle ∆ a you are on as embedded in N ⊥ and rotate it according to the stationary pointequations to X a ∆ a . Embed the next triangle and rotate it, according to the stationary pointequations it’s edges are now antiparallel to already immersed edges. As they are geometricallyglued a translation exists that identifies all its edges with already immersed ones. Thus iterativelythe whole immersed surface can be built up and the closure conditions on the cuts now implythat the surface closes up. Finally the stationarity equations on the circles imply that the P ˆ h i identifies the circles where we cut the surface.Conversely given an oriented immersion with the right edge lengths we can choose a set ofedge vectors compatible with the orientation on the surface. On each triangle there are twolinearly independent edge vectors. The map from these to the corresponding boundary elementsdefines a rotation in SO(3). On the boundary circles we explicitly have SO(3) elements. Thecomplete set of these solves the critical point equations. (cid:3) If there is a manifold of dimension d > C X . Since these critical points lie on a manifold, there is a continuous deformationof the immersed surface that does not change the edge lengths. Hence these critical pointsreconstruct flexible immersions and we arrive at the flexifolds f described in section 3. We willnow label the critical manifolds by C f , where f is the flexifold that it describes. Geometrical interpretation
We will now describe in some more detail the geometric struc-ture of these surfaces. Given an oriented surface in R the standard orientation automaticallygives us a consistent set of normals N a . By our choice of boundary data we have automaticallyensured that these are given simply by N a = ˆ X a N . We can define the dihedral rotation for an oriented surface unambiguously as the rotation ˆ D ab ∈ SO(3) around the geometric edge vector v ab ( i ) that takes N a to N b , that is N b = ˆ D ab N a and v ab ( i ) = ˆ D ab v ab ( i ) . the lift of this rotation can thus be written as D ab = exp (cid:18) Θ i ab v ab ( i ) | v ab ( i ) | .L (cid:19) = exp (Θ i ab n ′ ab .L ) (38)where we require − π < Θ i ab ≤ π . We then call Θ i ab the dihedral angle. As v ab ( i ) = − v ba ( i ) thisdefinition clearly implies Θ i ab = Θ i ba . If we have a surface defining a convex subspace of R this13efinition reduces to the usual definition. In particular the consistent choice of orientation thenensures that we have 0 < Θ i ab ≤ π and cos (Θ i ab ) = N a . N b for outward facing normals.On the boundaries of the surface we can also define an analogue of the dihedral rotation byrequiring ˆ h i N b = ˆ D ab N a and − ˆ h i v ba ( i ) = v ab ( i ) = ˆ D ab v ab ( i )Geometrically this makes sense as it corresponds to the angle obtained by gluing on the twoidentified boundaries. Commuting diagram
To fully connect the geometry of these surfaces to the boundary wecan relate the dihedral rotation to the gluing of the boundary g ab defined above.Consider the following diagram which applies to two adjacent triangles that are not on acutting circle: t ag ab (cid:15) (cid:15) X a / / τ a ( − νab D ab (cid:15) (cid:15) t b X b / / τ b (39)Here t a is the boundary triangle at N ⊥ with edge vectors given by k ab n ab and τ a is thetriangle rotated according to its location in the surface, which according to the reconstructiontheorem 2 has edge vectors given by v ab ( i ) = k ab ˆ X a n ab . The SO(3) action of this diagramimmediately commutes, as can be seen by acting on n ab and N . As an SU(2) diagram thisequation defines a sign ( − ν ab that makes it commute. The discrete sign symmetry X a → ǫ a X a of the action can be seen as acting on this sign by ( − ν ab → ǫ a ǫ b ( − ν ab .Now, for two triangles whose common edge is on a cutting circle i , in the same way we havea commuting diagram as t ag ab (cid:15) (cid:15) X a / / τ a ( − νab D ab (cid:15) (cid:15) t b h i X b / / τ b (40)and additionally have ( − ν ab → ǫ a ǫ b ǫ i ( − ν ab .We now show that the dihedral rotation is unchanged by moving the cut. Suppose that ab ∈ i ∈ C , in the sense that the edge v ab ( i ) crosses the cutting circle labelled i . Now choosea different, yet homotopic, cutting circle ∂ e D i ∼ ∂ D i , such that now ba / ∈ i , but cb ∈ i . Thiscorresponds to sliding the cut from one edge of τ b to another, and as any other change in thecut, it will modify wether labels a and b satisfy (30) or (31). By our orientation convention, wenow have X a n ab = − e X b n ba , − e h i f X c n cb = X b n bc (41)By just looking at the first set of equations above, and comparing it to X a n ab = − h i X b n ba wehave that h i X b n ba = e X b n ba . This shows that h i X b and e X b are equal up to a phase, and by thereconstruction theorem in fact h i X b = e X b . Hence, comparing the diagrams (39) and (40) we getthat moving the cut does not affect the dihedral angle as here defined.14 arity Given an oriented surface immersed in R all surfaces related to it by O(3) clearlyhave the same boundary data. Those related by SO(3) also have the same dihedral angle asdefined above. However if we act by parity P : n → − n we switch the dihedral angle. This isbecause the two equations defining D ab are invariant under parity, and the dihedral rotation isunchanged. Thus by the definition of the dihedral angle we have D ab = exp (cid:18) Θ i ab ( − v ab | v ab | ( P σ )) .L (cid:19) and so Θ P i ab = − Θ i ab . Now note that given a geometric surface and associated solution X a ( i ), we can obtain the solution X a ( P i ) corresponding to the parity flipped surface explicitly. That is, by equation (38) we havethat ( X a ( i )) − D ab X a ( i ) = exp (Θ i ab n ab .L ) (42)Acting with X − a by the left of the commuting diagram equations, using the notation X ab ( i ) =( X a ( i )) − X b ( i ) if not on a circle and X ab ( i ) = ( X a ( i )) − h i X b ( i ) if on, we get X ab ( i ) g ab = ( − ν ab ( X a ( i )) − D ab X a ( i ) = ( − ν ab exp (Θ ab ( n ab .L ) . (43)where we have used (42).It is now straightforward to evaluate the matrix elements in the amplitude. These are of theform h− n ab | X ab | n ba i . Using the gluing condition this becomes h− n ab | X ab g ab |− n ab i . Finally,by (43) this is just ( − ν ab e i Θ i ab . Thus we have overall that: h− n ab | X ab | n ba i = ( − ν ab e i Θ i ab (44) Spin Structure
Now we will fix the ambiguity in signs emerging from the spin lifts of thedihedral angle by exploring the discrete sign symmetry in h i and X a . Recall that the discretesign freedom of the action X a → ǫ a X a emerged from a different choice of spin frame for eachtriangle. Now, we show that the discrete sign symmetry related to the cuts h i → ǫ i h i correspondsto different choices of spin structures for the manifold Σ . Then, using the fact that ( − νab D ab is a gauge transform of the connection g ab we can fix the symmetries by adjusting the spin framesand the spin structure such that ( − ν ab = 1. Thus we will show that: Lemma 3.
The signs ( − ν ab arising from the spin lift on each face not on the cut obey ( − ν ab = κ ab = κ a κ b for some κ a = ± . The signs for a face on the cut, i.e. ab ∈ i ∈ C obey ( − ν ab = κ ab = κ a κ b κ i where κ i parametrizes the spin structures of Σ .Proof. First of all, by (4.2) a lift of the dihedral rotations, κ ab D ab , are just a gauge trans-formation of the g ab . Now recall that ˆ g ab ∈ SO(3) are parallel translations on the boundarytriangles according to the Levi-Civita connection of the associated metric, with g ab being theparallel translation of the respective spin connection (a lift of ˆ g ab to SU(2)).But when the geometry around a vertex is continuously deformed to the flat geometry, the g ab holonomy of a trivial cycle around said vertex has to go to the identity rotation, as opposedto a 2 π rotation. This implies that for the holonomy around a vertex through triangles a, b and c (which of course consists of a trivial cycle), we have κ ca κ bc κ ab D ca D bc D ab = κ ca κ bc κ ab = 115hich implies that locally we must have κ ab = κ a κ b . The problem now is that if there arenon-trivial cycles, i.e. g = 0, we may not be able to extend this globally, i.e. κ ab may not beglobally pure gauge.In other words, for trivial cycles the lift of the holonomy given by the κ ab D ab is fixed tobe the same as that given by D ab . But not so for the holonomy of a non-trivial cycle; thereexist inequivalent spin structures on a manifold. These have a one-to-one correspondence withthe elements of H (Σ , Z ), and so are 2 g in number. Hence for a non-trivial cycle, dual to thesequence of triangles ∆ a · · · ∆ a n ∆ a crossing the circle i ∈ C , we have κ a n a κ a n − a n · · · κ a a D a n a D a n − a n · · · D a a = κ i D a n a D a n − a n · · · D a a (45)where a κ i is introduced whenever there is an implicit choice of spin structure; i.e. it parametrizesthe different spin structures associated with the cut.We reconcile this case with the g = 0 one by keeping the form κ ab = κ a κ b for all the edges ab that do not lie on a circle, i.e. ab / ∈ i for any i ∈ C . Then by (45) immediately we must havefor ab ∈ i , κ ab = κ i κ a κ b . Since our chosen basis for H (Σ , Z ) generates all cycles, we can seethat this form of κ ab has all the right properties demanded by our equations and accounts forthe different spin structures.Therefore, taking advantage of the discrete sign symmetry, we can choose the spin structure tobe compatible with the one chosen for the lift of g ab and thus we will have ( − ν ab → ǫ i ǫ a ǫ b ( − ν ab makes ( − ν ab = 1. (cid:3) .Finally, we obtain that the action evaluated at the critical points is the Regge action for theimmersed surface i . S = X ab ∈ E k ab Θ i ab (46)For the flexible immersions, the action is the same for all points on the critical manifold so weevaluate it on an arbitrary immersion in the flexifold.This concludes the derivation of the Regge action. The stationary phase formula requires us to calculate the Hessian of the action S to determinethe weights with which the stationary points contribute to the action. This will be a 3( | V | + | C | ) × | V | + | C | ) matrix defined by H = (cid:18) H XX H Xh H hX H hh (cid:19) . (47)Where ( H XX ) ijcd = ∂ S∂X ic ∂X jd ! (48)( H hX ) ijpd = ∂ S∂h ip ∂X jd ! (49)( H Xh ) ijcq = ∂ S∂X ic ∂h jq ! (50)( H hh ) ijpq = ∂ S∂h ip ∂h jq ! (51) This is actually a particular case of the “strong bellows conjecture” that was shown in []. I .This will cause the determinant of the Hessian to be zero unless it is gauge fixed. To solve this,we make the change of variables X a → X b X a for some b ∈ { , ..., | V |} . This has the effect ofremoving the X b variables and the integral gives a volume of SU(2) which can be normalised toone as it is compact. The remaining Hessian is now a 3( | V | + | C | − × | V | + | C | −
1) matrix.The submatrix H XX at the critical points, is given by ∂ S∂X ic ∂X jc !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δS =0 , Re S =0 = 12 X b = c,bc ∈ E k cb (cid:16) δ ij − n ′ icb n ′ jcb (cid:17) (52)for the diagonal terms. The off diagonal part is ∂ S∂X ic ∂X jd !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δS =0Re S =0 = − X e ( δ c s ( e ) δ d t ( e ) + δ d s ( e ) δ c t ( e ) ) (cid:16) δ ij − iǫ ijk n ks ( e ) t ( e ) − n ′ is ( e ) t ( e ) n ′ js ( e ) t ( e ) (cid:17) . (53)So one can see that only the off-diagonal elements that represent two neighbouring triangles arenon zero. The ( H hh ) submatrix will be diagonal since each term in the action only contains one h p term (ie, each dual edge only crosses one cut.) ∂ S∂h ip ∂h jp !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δS =0 , Re S =0 = 12 X b = c,bc ∈ C p k cb (cid:16) δ ij − n ′ icb n ′ jcb (cid:17) (54)The mixed terms H Xh , H Xh will be non zero only for triangles with an edge on the cut ∂ S∂X ic ∂h jq !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δS =0Re S =0 = − X ab ∈ E q c = a,b k ab (cid:16) δ ij − iǫ ijk n kab − n ′ iab n ′ jab (cid:17) (55)Note that( X a ( i )) − D ab X a ( i ) = exp ( − Θ i ab ( n ab ( i )) .L ) − = (cid:0) ( X a ( P i )) − D ab X a ( P i ) (cid:1) − (56)where we used that D ab = exp (Θ i ab ( − v ab ( P i ) | v ab ( P i ) | ) .L ) on the second equality. By (43) we then have( X ab ( i ) g ab ) = ( X ab ( P i ) g ab ) − , so if we replace the X ab ( i ) in h− n ab | X ab ( i ) | n ba i with the parityrelated one we now obtain the complex conjugate: h− n ab | X ab ( i ) g ab |− n ab i = h− n ab | ( X ab ( P i ) g ab ) − |− n ab i = h− n ab | X ab ( P i ) g ab |− n ab i = h− n ab | X ab ( P i ) | n ba i (57)Thus we can see that the action of parity on the Hessian matrix will also result in complexconjugation when evaluated at the critical points. We can now evaluate the stationary phase approximation to the amplitude Z P R (Ψ λ , Σ ) definedin (15). We begin by fixing the symmetries of the action. This can be achieved by taking anarbitrary vertex and dropping the group integration associated to it. As shown in section 4.1.1the critical point equations are the equations for the immersion of a polyhedral surface withthe geometry specified in the boundary data. If a particular immersion is rigid no infinitesimal17eformation taking it to another such immersion exists and therefore it is an isolated criticalpoint of the amplitude.For the isolated critical points in I r we can explicitly evaluate the stationary phase approx-imation. Having fixed one group integration we are left with a 3( | V | + | C | −
1) dimensionalintegration. The overall scaling of these points is thus (cid:0) πλ (cid:1) | V | + | C |− / . Further we obtain aset of 2 | V | + | C |− critical points for each immersion from the spin lift of each SU(2). Finallythe derivatives in the Hessian as defined above are taken with respect to a parametrization ofSU(2) with volume (4 π ) , so we need to rescale by this factor. Using equation (44) and lemma3 the amplitude itself evaluates to the Regge action of the cut immersion:ln h− n ab | X ab | n ba i k ab = ik ab Θ i ab . Taking all these factors together and using the fact that we know parity related immersions tobe the complex conjugate of each other we can approximate the contributions of the isolatedcritical points to the partition function as: Z P R (Ψ λ , Σ ) = ( − χ (cid:18) πλ (cid:19) | V | + | C |− (cid:18) π ) (cid:19) | V | + | C |− X i ∈ I r √ det H i exp iλ X ab ∈ E k ab Θ i ab ! + 1 p det H i exp − iλ X ab ∈ E k ab Θ i ab ! + O (cid:18) λ (cid:19) | V | + | C |− +1 (58)where Θ i ab is the dihedral angle of the edge ab in the cut immersion i ∈ I r . Since the Hessianmatrix changes to its complex conjugate with parity, we can absorb the phase of the determinantinto the exponentials and combine the terms Z P R (Ψ λ , Σ ) = 2( − χ (cid:18) πλ (cid:19) | V | + | C |− X i ∈ I r p | det H i | cos iλ X ab ∈ E k ab Θ i ab −
12 Arg(det H i ) ! + O (cid:18) λ (cid:19) | V | + | C |− +1 . (59)If there are any flexible immersions of the boundary data then there will be a manifold ofcritical points. Since the critical points extremize the action, it must have the same value onevery point of the critical manifold. The Hessian therefore has zero modes along the directions ofthe flexifold and we must treat the integral as having further symmetries in the neighbourhood ofthe flexifold. Factoring out these changes the scaling of the contribution of these critical pointsby λ d max / , where d max is the dimension of the flexifold. Therefore these immersions dominatethe rigid immersions if they exist, their contribution is given by: Z P R (Ψ λ , Σ ) = ( − χ (cid:18) πλ (cid:19) | V | + | C |− − d max2 (cid:18) π ) (cid:19) | V | + | C |− − d max2 × X f ∈ F max L f exp iλ X ab ∈ E k ab Θ f ab ! + L f exp − iλ X ab ∈ E k ab Θ f ab ! + O (cid:18) λ (cid:19) | V | + | C |− − d max2 +1 (60)18 f ab is the dihedral angle of the edge ab of a particular cut immersion i in the flexifold f . As theaction is constant along the flexifold it does not matter where we evaluate it. L f is given by L f = Z C f dσ C f ( y ) a ( y ) q det H ⊤ f ( y ) (61)where H ⊤ f is the Hessian matrix for the transverse directions which we can not give a generalformula for. Combining the exponentials into cosines as above we obtain part one of the maintheorem.Finally, if no immersions of the boundary data exist then there are no solutions to the criticalpoint equations and the stationary phase formula gives that the amplitude is suppressed. (cid:3) Here we apply the above results to the well known case of the asymptotics of the amplitudefor a single tetrahedron which, with an appropriate choice of normalisation for the boundaryintertwiners, will correspond to the 6j symbol. This is a special case of theorem 1 so the proof isthe same as above. In particular, the critical and stationary point equations are the same andthe action evaluated at these points reduces to the Regge action for a tetrahedron. Since theasymptotic formula for the tetrahedron is already known, we must verify that our formula agreeswith this result. This also provides further evidence that the asymptotic formula for the 4d casederived using the same methods in [9] is correct. We begin by noting that, up to parity, theboundary data of a tetrahedron has only one immersion so the sum in the asymptotic formuladisappears.We will also require an explicit formula for the Hessian matrix in order to perform thenumerical calculations. For the tetrahedron, the Hessian is the 12 ×
12 matrix defined by H ijcd = ∂ S∂X ic ∂X jd ! . (62)The global SU(2) symmetry of the action implies that there is a redundant integration in I .This will cause the determinant of the Hessian to be zero unless it is gauge fixed. To solve this,we make the change of variables X a → X X a for a = 1 , ,
3. This has the effect of removing the X variables and the integral gives a volume of SU(2) which can be normalised to one as it iscompact. The remaining Hessian is now a 9 × ∂ S∂X ic ∂X jc !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δS =0 , Re S =0 = 12 X b = c k cb (cid:16) δ ij − n ′ icb n ′ jcb (cid:17) (63)for the diagonal terms. The off diagonal part is ∂ S∂X ic ∂X jd !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δS =0 , Re S =0 = − k cd (cid:16) δ ij − iǫ ijk n kcd − n ′ icd n ′ jcd (cid:17) . (64) We can now compare our theorem with the Ponzano-Regge asymptotic formula for the 6j symbol.The Ponzano-Regge formula is (cid:26) λk λk λk λk λk λk (cid:27) → √ π Vol cos X a
12 Arg(det H ) ! (71)Note that we have the correct scaling behaviour once the additional scaling factors from theintertwiners are included. The normalisation terms are real so do not contribute any additionalphase.The formula for the equilateral tetrahedron with both the exact and approximate intertwinernormalisation was compared to the 6j symbol and the Ponzano-Regge asymptotic formula usingMathematica in Figure 4. We see that our formula differs from the Ponzano Regge formulafor low spins. The only point where our formula differs from Ponzano Regge is in the factthat the Ponzano Regge asymptotics are given in terms of the dihedral angles and volume ofthe tetrahedron with edge lengths λk ab + . Therefore the dihedral angles and volume changenontrivially with λ . A stationary phase approximation extracts only the scaling behaviour withrespect to lambda in the asymptotic regime and cannot register this type of low spin behaviour.This agrees as well as the PR formula for larger spins, however the agreement for very low valuesis not as good - Figure 4. 21Sfrag replacements 5 55555 66 6666 66 66 2 . . . . . . . Here we discuss an example for which the second part of theorem 1 is relevant, that is we describea set of boundary data that admits a flexible immersion. This particular example is taken froma flexible polyhedron with half integer edge lengths consisting of fourteen boundary triangleswhich was found by K. Steffen [20]. A net for constructing this polyhedron is given in Figure5 and the corresponding spin network in Figure 6. Since Steffen’s polyhedron admits a flex inone direction, we know that the flexifold is at least one dimensional. As a polyhedron, it is notallowed to self intersect but there may be other immersions with flexibility in more than onedimension. Applying the asymptotic formula with the same intertwiner normalisation as thetetrahedron in section 5, we would expect the scaling to be λ − / . As discussed, the asymptotic formula produces a sum over all possible immersions of the bound-ary data in R , including flexible ones. These flexible immersions scale differently and thusdominate the rigid immersions. The question of whether a particular polyhedron is rigid is adifficult long standing problem in mathematics. A classic result is that convex polyhedra are infact rigid, however this does not extend to immersions and non convex polyhedra where counterexamples, like Steffen’s polyhedron discussed above, are known.If the boundary data is topologically S then a theorem by Steinitz [21] applies that statesthat any simplicial complex with underlying space homeomorphic to a 2-sphere admits a sim-plexwise linear embedding into R whose image is strictly convex. This embedding will indeedbe rigid and we can conclude that for the ball I r will always be non-empty.To our knowledge the only more general results on rigidity of immersions are those givingconditions on bar frameworks, that is a graph immersed in R d , to be generically rigid. A barframework is considered generic if the coordinates of the vertices are algebraically independent22Sfrag replacements 5555 55 66 666 66 65 / / / / / / / With the asymptotic analysis performed above, we explicitly obtain a sum over immersions of theboundary data weighted by the cosine of the Regge action for the immersed surface. Previously,asymptotics of the Ponzano-Regge model for larger triangulations could only be considered bytaking the product of the asymptotic formula for each 6j symbol. We now illustrate schematicallythat, in a simple example, that this is in fact equivalent to the asymptotic formula above.We will consider the case of two tetrahedra σ , σ glued along a common triangle ∆ and usethe boundary normalisation that agrees with the 6j symbol. The partition function then reads Z P R (Ψ λ , σ ∪ ∆ σ ) = (cid:26) λk λk λk λk λk λk (cid:27) (cid:26) λk λk λk λk λk λk (cid:27) (72)We write the asymptotic formula for the 6j in terms of the Regge action S σ for a tetrahedron σ (cid:26) λk λk λk λk λk λk (cid:27) = N (exp( iλS σ ) + exp( − iλS σ )) (73)Where several of the factors have been absorbed into the amplitude N . Asymptotically, thisgives Z P R (Ψ λ , σ ∪ ∆ σ ) = N N (exp( iλ ( S σ + S σ )) + exp( − iλ ( S σ + S σ ))+ N N (exp( iλ ( S σ − S σ )) + exp( − iλ ( S σ − S σ ))= N N exp( X e ⊂ ∆ k e π ) (cos ( λ ( S σ ∪ t σ ) + cos ( λ ( S σ ∪ ∆ P σ )) (74)23Sfrag replacements ∆ σ P σ σ PSfrag replacements ∆ σ P σ σ Figure 7: Two different possible immersions of the boundary data for two tetrahedra σ , σ gluedon a common triangle ∆.Where P σ is the parity related tetrahedron and we have used the fact that the Regge action fortwo tetrahedra becomes S σ + S σ = S σ ∪ ∆ σ + X e ⊂ ∆ k e π. (75)Thus the formula gives a sum over the two different ways of immersing the boundary triangles in R , see Figure 7 . If we now consider larger triangulations, possibly with interior vertices, edgesor faces, then the immersed boundary surfaces we obtain are related to immersions of the entiretriangulation by adding the immersions of the interior triangulation. Note that the internalvertices can be placed anywhere in R which leads to the divergent factors that appear in thePonzano-Regge state sum for these kind of triangulations. The two parts of the cosine appearbecause these immersions can be orientation reversing on particular tetrahedra. These willthen contribute with negative dihedral angles to the interior action. Thus the integration overpossible interior geometries can be split into parts corresponding to particular orientations on theinterior tetrahedra. The terms of this sum will then correspond to the sum of terms obtained bymultiplying out the sums in the interior cosines. On shell the interior action on these immersionsis zero so the overall amplitude only registers the boundary contribution to the action. Inparticular interior configurations corresponding to different geometries and orientations on theinterior, and thus to different terms in the cosine, contribute with the same phase. We also note that it is possible to select a particular immersion in the sum by choosing aboundary state peaked around a particular set of dihedral angles, see for example [24, 14, 25].This boundary state also selects one overall orientation of the immersion which removes theparity related term in the asymptotic formula. For non-rigid immersions, the boundary statewould also have the ability to select a particular configuration of the immersed surface whichwould stop these immersions dominating the integral.A possible problem with the boundary state is that while it selects an orientation for theboundary, it was not clear if the orientations of the interior tetrahedra behaved consistently.This was considered in [26] and our result also suggests that these do not cause a problem asthe asymptotic formula does not register these orientations.
In this paper we addressed the problem of asymptotics of larger triangulations for the Ponzano-Regge model. By reformulating the partition function as a spin network on the boundary andthen rewrote this amplitude using SU(2) coherent states. While this particular feature will notbe available for non topological theories one could expect that in general boundary data will benot suppressed if it can be continued to a solution of the equations of motion on the interior.24he asymptotic formula contains a sum over immersions of the boundary data weighted by thecosine of the Regge action. Interestingly, Ponzano and Regge point out in [1] that the differentpossible immersions corresponding to 3-nj symbols should contribute to the asymptotics but didnot obtain a concrete formula. The presented work opens up the possibility to do an exhaustiveanalysis of the classical limit of the Ponzano Regge model including correlation functions on theboundary. As such it can serve as a toy model and proof of concept for conceptual issues likelyto arise in all background independent theories.Of further interest would be to consider in more detail how the asymptotics obtained herecan be obtained from the “product of cosines” picture. In particular to shed light on the issueof causality and orientation in spin foam models.Interestingly, and unexpectedly, we found that spin networks contain some information aboutthe rigidity properties of surfaces. The scaling properties of a spin network correspond to themaximum dimension of flexibility if the geometry to which it corresponds has any non-rigidimmersions.
RD and FH are funded by EPSRC doctoral grants. We would like to thank John Barrett fordiscussions and comments on a draft of this paper.
A Example of the Ponzano-Regge amplitude as a spin networkon the boundary of the solid torus
Here we give a simple example of Lemma 2 on the solid torus T . A non-tardis (degenerate)triangulation of the solid torus with three tetrahedra is given byPSfrag replacements k k k k k k k k k k k k The two triangles with edges k , k , k are identified. The Ponzano-Regge amplitude is given by Z P R (Ψ , T ) = (cid:26) k k k k k k (cid:27) (cid:26) k k k k k k (cid:27) (cid:26) k k k k k k (cid:27) . (76)We choose the cutting disc D to be the triangle k , k , k and perform the cut that reduces T tothe 3-ball. A net for constructing the triangulation on the boundary is given byPSfrag replacements k k k k k k k k k k k k k D .PSfrag replacements k k k k k k k k k hhh Z P R (Ψ , T ) = R SU(2) dh Expressing this spin network in terms of 6j symbols gives equation (76).
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