Spin Foam Perturbation Theory for ThreeDimensional Quantum Gravity
Abstract
We formulate the spin foam perturbation theory for threedimensional Euclidean Quantum Gravity with a cosmological constant. We analyse the perturbative expansion of the partition function in the dilutegas limit and we argue that the Baez conjecture stating that the number of possible distinct topological classes of perturbative configurations is finite for the set of all triangulations of a manifold, is not true. However, the conjecture is true for a special class of triangulations which are based on subdivisions of certain 3manifold cubulations. In this case we calculate the partition function and show that the dilutegas correction vanishes for the simplest choice of the volume operator. By slightly modifying the dilutegas limit, we obtain a nonvanishing correction which is related to the second order perturbative correction. By assuming that the dilutegas limit coupling constant is a function of the cosmological constant, we obtain a value for the partition function which is independent of the choice of the volume operator.
Contents
1 Introduction
Spin foam state sum models can be understood as the path integrals for BF topological field theories [Ba1]. Since General Relativity in 3 and 4 dimensions can be represented as a perturbed BF theory, see [FK, M2], then, in order to find the corresponding Quantum Gravity theory, one would need a spin foam perturbation theory. Baez has analysed the spin foam perturbation theory from a general point of view in [Ba2], and he was able to show that, under certain reasonable assumptions, the perturbed spin foam state sum can be calculated in the dilute gas limit. In this limit, the number of tetrahedra of a manifold triangulation tends to infinity and , the perturbation theory parameter, tends to zero, in a way such that the effective coupling constant is finite. By assuming that the number of topologically inequivalent classes of perturbed configurations at a given order of perturbation theory is limited when , Baez showed that the perturbation series
(1) 
where is the manifold, is dominated by the contributions from the dilute configurations in the dilute gas limit. The dilute configurations of order are the configurations where nonintersecting simplices carry a single perturbation insertion. Let , then
(2) 
where
does not depend on the chosen manifold .
In this paper, we are going to study in detail the Baez approach on the example of threedimensional (3d) Euclidean Quantum Gravity with a cosmological constant. In this case it is possible to construct explicitly the perturbative corrections, and we will show how to do it. Therefore one can check all the assumptions and the results from the general approach. We will show that the conjecture that there are only finitely many topological classes of perturbative configurations at a given order of the perturbation theory is not true. However, if the triangulations are restricted to those corresponding to special subdivisions of certain acceptable manifold cubulations, then the number of these topological classes is finite. In this case we show that the formula (2) is still valid and we calculate . A surprising feature of 3d gravity is that and consequently the dillute gas limit has to be modified in order to obtain a nonzero contribution. We will show that for
(3) 
where
The value of , which we conjecture to be nonzero, is independent of the triangulation and the manifold . Recall that denotes the number of tetrahedra of the triangulation of .
In order to construct the perturbative corrections , we will use the pathintegral expression for the partition function of Euclidean 3d gravity with a cosmological constant
(4) 
where is an principal bundle connection, is the corresponding curvature 2form, is a oneform taking values in the Lie algebra and are the structure constants. This path integral can be defined as a finite spin foam state sum when , , see [FK], and in this case it is given by the TuraevViro (TV) state sum [TV]. However, if then it is not obvious how to define . A natural approach is to use the generating functional technique [FK], and in [HS] the first order perturbation theory spin foam amplitudes were studied for the PonzanoRegge (PR) model [PR]. However, the problem with the PR model is that it is not finite, so that the state sums in (1) are not well defined. Since the TV model can be considered as a quantum group regularisation of the PR model, we are going to use the TV model to define the perturbation series (1). Physically this means expanding the path integral (4) by using as the perturbation theory parameter instead of .
The TV model perturbation series can be constructed by using the PR model perturbation series and then replacing all the weights in the PR amplitudes with the corresponding quantum group spin network evaluations. The calculation of the corresponding state sums is substantially simplified if the ChainMail technique is used, see [R, BGM, FMM]. In section 2 we review the PR perturbation theory. In section 3 we review the ChainMail technique, while in section 4 we define the perturbative corrections. In section 5 we discuss the dilute gas limit, while in section 6 we present our conclusions.
2 Perturbative expansion for the PR model
Given a triangulation of with tetrahedra, let us associate to each edge of a source current which belongs to the Lie algebra with a basis . One can then write
(5) 
where is a differential operator associated with the volume of a tetrahedron and is the generating functional, given by the PonzanoRegge state sum with the insertions at the edges of the spin networks, where is the matrix of the group element in the representation of spin , see [FK, HS]. The operator can be chosen to be
where is a totally antisymmetric tensor and are tetrahedron edges sharing a common vertex^{2}^{2}2One can choose a more general expression for , involving all possible triples of the edges, see [HS], but in this paper we will study the simplest possible choice..
Since
where
then the result of the action of on a tetrahedron’s vertex will be given by the grasping insertion
where and are the spins of the three edges sharing the vertex.
By using
where is the intertwiner tensor for ( symbol) and is a normalisation factor given by
(6) 
one obtains
This equation implies that the evaluation of a tetrahedral spin network with a grasping insertion is proportional to the evaluation of a spin network based on a tetrahedron graph with an additional trivalent vertex whose edges carry the spin one representations and connect the 3 edges carrying the spins and , see Figure 1. The which follows from (5) will be then given by a sum of terms where each term corresponds to the PR state sum with graspings distributed among the tetrahedra. The weight of a tetrahedron with graspings is given by an analogous evaluation of the spin network from Figure 1 with insertions.
In order to make all the PR state sums finite, we will replace all the spin networks associated with a with the corresponding quantum spin networks at a root of unity. In the following sections we will show how to do this.
3 Quantum invariants of links and threemanifolds
We gather some well known facts about quantum invariants which we will need in this paper.
3.1 Spin network calculus
Consider an integer parameter (fixed throughout this article), and let . Define the quantum dimensions , where . If the edges of a trivalent framed graph embedded in are assigned spins , then we can consider the value obtained by using the quantum spin network calculus at ; we will use the normalisation of [KL]. We can also consider the case in which the edges of are assigned linear combination of spins, with multilinear dependence on the colourings of each edge of . A very important linear combination of spins is the “element” given by:
where denotes the representation of spin .
A sample of the properties satisfied by the element appears in [R, L, KL]. In Figure 3 we display a special case of the Lickorish Encircling Lemma, of which we will make explicit use.
Define and , the evaluation of the 0 and 1framed unknots coloured with the element. Therefore we have that
On the other hand , and ; see [R].
3.2 Generalised Heegaard diagrams
Let be a closed oriented piecewiselinear 3manifold. Choose a handle decomposition of ; see [RS, GS]. Let be the union of the 0 and 1handles of . Let also be the union of the 2 and 3handles of . Both and have natural orientations induced by the orientation of . There exist two nonintersecting naturally defined framed links and in ; see [R]. The second one is given by the attaching regions of the 2handles of in , pushed inside , slightly. On the other hand, is given by the beltspheres of the 1handles of , living in . The sets of curves and in have natural framings, parallel to . The triple will be called a generalised Heegaard diagram of the oriented closed 3manifold .
3.3 The ChainMail Invariant
We now recall the definition of J. Robert’s ChainMail invariant of closed oriented manifolds. This construction will play a fundamental role in this article. Let be a connected 3dimensional closed oriented piecewise linear manifold. Consider a generalised Heegaard diagram , associated with a handle decomposition of . Give the orientation induced by the orientation of . Let be an orientation preserving embedding. Then the image of the links and under defines a link in , called the “ChainMail Link”. J. Roberts proved that the evaluation of the ChainMail Link coloured with the element is independent of the orientation preserving embedding ; see [R], Proposition .
3.4 The TuraevViro Invariant
Let be 3dimensional closed connected oriented piecewise linear manifold. Consider a piecewise linear triangulation of . We can consider a handle decomposition of where each simplex of generates a handle of ; see for example [R]. Applying the ChainMail construction to this handle decomposition, yields the following combinatorial picture for the calculation of the ChainMail Invariant , which, in this form, is called the TuraevViro Invariant .
A colouring of is an assignment of a spin to each edge of . Each colouring of a simplex gives rise to a weight , in the way shown in Figure 2.
Using the identity shown in Figure 3 together with Figure 4, it follows that:
Note that we apply Lickorish Encircling Lemma to the 0framed unknot defined from each face of the triangulation of ; see Figure 5. The last expression for is the usual definition of the TuraevViro Invariant. For a complete proof of the fact that , see [R].
3.5 The WittenReshetikhinTuraev Invariant
The main references now are [L], [RT] and [R]. Let be an oriented connected closed manifold. Then can be presented by surgery on some framed link , up to orientation preserving diffeomorphism. Any framed graph in can be pushed away from the areas where the surgery is performed, and therefore any pair , where is a trivalent framed graph in the oriented closed 3manifold , can be presented as a pair , where is a framed trivalent graph in , not intersecting .
The WittenReshetikhinTuraev Invariant of a pair , where the framed graph is coloured with the spins , is defined as:
Here is the signature of the linking matrix of the framed link , and is the number of components of . This is an invariant of the pair , up to orientation preserving diffeomorphism. In contrast with the TuraevViro Invariant, the WittenReshetikhinTuraev Invariant is sensitive to the orientation of . If is an oriented 3manifold, we represent the manifold with the reverse orientation by .
With the normalisations that we are using, the TuraevWalker theorem reads:
Some other well known properties of the WittenReshetikhinTuraev Invariant are the following:
Theorem 3.1
We have:
Here , and are oriented closed connected 3manifolds. In addition, and are coloured graphs embedded in and . It is understood that the connected sum is performed away from and .
Given oriented closed connected 3manifolds and , we define in the following way; see [BGM]. Remove 3balls from and , and glue the resulting manifolds and along their boundary in the obvious way, so that the final result is an oriented manifold. We denote it by:
It is immediate that , and that is diffeomorphic to , if . By using Theorem 3.1 it follows that:
(7) 
Here and are closed oriented 3manifolds. In addition, and are graphs in and , coloured with the spins and , respectively. As before, it is implicit that the multiple connected sum is performed away from and .
4 Perturbative expansion
In this section we are going to define the ’s considered in the introduction.
4.1 Graspings
Let be a simplicial complex whose geometric realisation is a piecewiselinear closed dimensional manifold. Recall that we can define the dual cell decomposition of , where each simplex of generates a dual cell of the dual cell decomposition of , see [TV, 3.3], for example. This is very easy to visualise in three dimensions.
Definition 4.1 (grasping)
Let be the standard tetrahedron. For a positive integer , an grasping is a sequence , where is a vertex of for each .
Any grasping , where is a vertex of , naturally defines a trivalent graph on the boundary of (usually called a grasping itself), by doing the transition shown in Figure 6. The graph is therefore the union , where is the dual graph to the 1skeleton of the obvious triangulation of , and is homeomorphic to the graph made from a trivalent vertex and three openends.
We want to define, in an analogous fashion, a trivalent graph from an grasping . This is not possible unless further information is given. We want to be the union of and a disjoint union , where each graph is homeomorphic to the graph . To describe we need to specify where the ends of each intersect , as well as the crossing information. To this end we give the following definition:
Definition 4.2 (Space ordering of an grasping)
Let again be the standard tetrahedron. Let be the dual graph to the obvious triangulation of the boundary of . Any edge of therefore defines a dual edge of the graph . Let be an grasping. A space preordering of is given by an assignment of a subset to each such that:

For each , and belong to different edges of , and each of these points belongs to the dual edge of an edge incident to .

if .
A space ordering of is given by a space preordering of considered up to ambient isotopy of inside .
There exists therefore a unique space ordering of a 1grasping .
Let now be an grasping with a certain space ordering . We define an associated graph in , with crossing information, as being:
where:

is homeomorphic to (see above) for each .

for each .

If then is placed above , with respect to the boundary of .
See Figure 7 for the description of the graph for two different space orderings of , where .
Given a grasping living in the tetrahedron , define as being given by the set made from the three edges of incident to . Given an grasping living in , define .
4.2 The firstorder volume expectation value
Let be a piecewise linear oriented closed 3manifold (from now on called simply a manifold). Consider a triangulation of . Let and be the set of vertices, edges, triangles and tetrahedra of .
A colouring of is an assignment of a spin to each edge of . Each colouring of a simplex of gives rise to a weight , in the way shown in Figure 2, exactly the same fashion as in the definition of the TuraevViro Invariant .
Consider a tetrahedron of (whose edges are coloured), with some grasping , provided with a space ordering . Choose an orientation preserving embedding of into , which is defined up to isotopy. Then the weight is defined as being the factor (see below) times the evaluation of the spin network , where has the colouring given by the colouring of (recall that each edge of is dual to a unique edge of ), and all edges of each are assigned the spin . Note that the graph has a natural framing parallel to the boundary of . If is an grasping, the factor is, by definition:
where for each we let , and denote the colourings of the three edges of incident to and the ’s are given by (6).
The firstorder volume expectation value can be represented as
where is the volume of . It corresponds to , which can be defined as
(8) 
where, by definition:
(9) 
Recall that any grasping has a unique space ordering .
Let , where is the number of tetrahedra of .
Theorem 4.3
For any triangulation of we have .
Proof. Each term can be presented in a ChainMail way. As in [R], consider the natural handle decomposition of for which each simplex of generates a handle of . This handle decomposition is dual to the one where each simplex of is thickened to an handle of .
Let us consider the ChainMail formula
for , obtained from this handle decomposition of ; see 3.4 and [R]. Here is the number of handles of , and therefore it equals the number of simplices of . From the same argument that shows that , follows that:
where:

All components of are coloured with .

Recall that each circle of the link corresponds to a certain edge of . The components and of which correspond to the edges , and incident to , where , should be coloured with the spins , whereas the remaining components (which form the link ) should be coloured with .

The component , where , is a trivalent vertex with three open ends, each of which is incident to either , or , with no repetitions, with framing parallel to the surface of ; see Figure 8. The three edges of are to be assigned the spin .

Finally, is an orientation preserving embedding . As in the case where no graspings are present, the final result is independent of this choice; see [R, Proof of Proposition 3.3].
By cancelling some pairs of  and handles, we can reduce the handle decomposition of to one with a single handle. Similarly, by eliminating pairs of  and handles, we can reduce the handle decomposition of to one having four handles, each of which corresponds to one of the vertices of the triangulation of which are endpoints of the edges of incident to , where , and so that the 2handles corresponding to the three edges of incident to are still in the handle decomposition. The chainmail link of the new handle decomposition of will then be , where is obtained from by removing some circles, and the same for . Let be the number of handles of the new handle decomposition of .
Let . By the same argument as in [R, Proof of Theorem 3.4] follows:
Given a compact manifold with border embedded in the oriented 3manifold , define as being the manifold obtained from and by removing the interior of from each of them and gluing the resulting manifolds along the identity map .
Let be the graph in made from the edges of incident to , together with their endpoints. Each edge of will induce a handle of and its four vertices will induce handles of . The union of these handles will be a regular neighbourhood of . Consider the graph in made from the attaching spheres of these handles, with a graph inserted, as in Figure 9.
By the using either the argument in [BGM, Proof of Theorem 1]) or in [FMM, Proof of Lemma 3.3], the pair is a surgery presentation of the manifold , with the graph inserted in .
The signature of the link is zero, given that it is a Kirby diagram for the manifold ; see [R, Proof of Theorem 3.7] or [BGM, Proof of Theorem 1]. Therefore it follows that:
We have used the calculation , which follows from the fact that the Euler characteristic of a closed orientable manifold is zero. Note that and , by construction.
Since is a closed ball embedded in it follows that . On the other hand, the graph is obviously trivially embedded in , in the sense that there exists an embedding sending the graph of Figure 10 to . This leads to:
Since the graph consists of three tadpole graphs, and the evaluation of a tadpole quantum spin network is zero, see [CFS, Theorem 3.7.1] or [KL, Chapter 9], then , which implies .
Given that , we have
where
and is the number of tetrahedra of .
4.3 Higherorder corrections
Let be a 3manifold with a triangulation . Let be a positive integer. An grasping of is a set of tetrahedra of (where if ), each of which is provided with a space ordered grasping , where , such that . The set is said to be an grasping support and the grasping of is said to be supported in .
Recall the definition of the weights , where is a coloured tetrahedra, with a space ordered grasping living in . This appears in the beginning of Subsection 4.2, to which we refer for the notation below.
Define:
(10) 
which can also be written as