Four-Dimensional Spin Foam Perturbation Theory
Four-Dimensional Spin Foam Perturbation Theory
João FARIA MARTINS and Aleksandar MIKOVIĆ
J. Faria Martins and A. Miković
Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologia,
Av do Campo Grande, 376, 1749-024 Lisboa, Portugal \EmailDamikovic@ulusofona.pt
Grupo de Física Matemática da Universidade de Lisboa,
Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal
Received June 03, 2011, in final form September 23, 2011; Published online October 11, 2011
We define a four-dimensional spin-foam perturbation theory for the -theory with a potential term defined for a compact semi-simple Lie group on a compact orientable 4-manifold . This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group where is the Lie algebra of and is a root of unity. The Chain–Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners , where is the adjoint representation of , is 1-dimensional for each irrep . We calculate the partition function in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold . We prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. We show that is an analytic continuation of the Crane–Yetter partition function. Furthermore, we relate to the partition function for the theory.
spin foam models; BF-theory; spin networks; dilute-gas limit; Crane–Yetter invariant; spin-foam perturbation theory
81T25; 81T45; 57R56
Spin foam models are state-sum representations of the path integrals for theories on simplicial complexes. Spin foam models are used to define topological quantum field theories and quantum gravity theories, see . However, there are also perturbed theories in various dimensions, whose potential terms are powers of the field, see . The corresponding spin-foam perturbation theory generating functional was formulated in , but further progress was hindered by the lack of the regularization procedure for the corresponding perturbative expansion and the problem of implementation of the triangulation independence.
The problem of implementation of the triangulation independence for general spin foam perturbation theory was studied in , and a solution was proposed, in the form of calculating the perturbation series in a special limit. This limit was called the dilute-gas limit, and it was given by , , such that is a fixed constant, where is the perturbation theory parameter, also called the coupling constant, is the number of -simplices in a simplical decomposition of a -dimensional compact manifold and is the effective perturbation parameter, also called the renormalized coupling constant. However, the dilute-gas limit could be used in a concrete example only if one knew how to regularize the perturbative contributions.
The regularization problem has been solved recently in the case of three-dimensional (3d) Euclidean quantum gravity with a cosmological constant , following the approach of [4, 9]. The 3d Euclidean classical gravity theory is equivalent to the -theory with a perturbation, and the corresponding spin foam perturbation expansion can be written by using the Ponzano–Regge model. The terms in this series can be regularized by replacing all the spin-network evaluations with the corresponding quantum spin-network evaluations at a root of unity. By using the Chain–Mail formalism  one can calculate the quantum group perturbative corrections, and show that the first-order correction vanishes . Consequently, the dilute-gas limit has to be modified so that is the effective perturbation parameter .
Another result of  was to show that the dilute gas limit cannot be defined for an arbitrary class of triangulations of the manifold. One needs a restricted class of triangulations such that the number of possible isotopy classes of a graph defined from the perturbative insertions is bounded. In 3d this can be achieved by using the triangulations coming from the barycentric subdivisions of a regular cubulation of the manifold .
In this paper we are going to define the four-dimensional (4d) spin-foam perturbation theory by using the same approach and the techniques as in the 3d case. We start from a BF-theory with a potential term defined for a compact semi-simple Lie group on a compact 4-manifold . In Section 2 we define the formal spin foam perturbative series by using the spin-foam generating functional method. We then regularize the terms in the series by passing to the category of representations for the quantum group where is the Lie algebra of and is a root of unity. In Sections 4 and 5 we then use the Chain–Mail formalism to calculate the perturbative contributions. The first-order perturbative contribution vanishes, so that we define the dilute-gas limit in Section 6 by using the second-order contribution. We calculate the partition function in the dilute-gas limit for a class of triangulations of a 4-dimensional manifold which are arbitrarily fine and have a controllable local complexity. We conjecture that such a class of triangulations always exists for any 4-dimensional manifold, and can be given by the triangulations corresponding to the barycentric subdivisions of a fixed cubulation of the manifold. We then show that is given as an analytic continuation of the Crane–Yetter partition function. In Section 7 we relate the path-integral for the theory with the spin foam partition function and in Section 8 we present our conclusions.
2 Spin foam perturbative expansion
Let be the Lie algebra of a semisimple compact Lie group . The action for a perturbed -theory in 4d can be written as
where is a -valued two-form, is a basis of , is the curvature 2-form for the -connection on a principal -bundle over , and is a symmetric -invariant tensor. Here if and are vector fields in the manifold then .
We will consider the case when , where is the Kronecker delta symbol. In the case of a simple Lie group, this is the only possibility, while in the case of a semisimple Lie group one can also have which are not proportional to . For example, in the case of the group one can put , where is the totally antisymmetric tensor and . We will also use the notation and .
Consider the path integral
It can be evaluated perturbatively in by using the generating functional
(where is an arbitrary 2-form valued in ) and the formula
The path integrals (2.3) and (2.4) can be represented as spin foam state sums by discretizing the 4-manifold , see . This is done by using a simplicial decomposition (triangulation) of , . It is useful to introduce the dual cell complex  (a cell decomposition of ), and we will denote the vertices, edges and faces of as , and , respectively. A vertex of is dual to a 4-simplex of , an edge of is dual to a tetrahedron of and a face of is dual to a triangle of .
The action (2.1) then takes the following form on
where and are pairs of triangles in a four-simplex whose intersection is a single vertex of and . The variable is defined as
where is the face dual to a triangle , ’s are the edges of the polygon boundary of and are the dual edge holonomies.
One can then show that
where the amplitude , also called the weight for the 4-simplex , is given by the evaluation of the four-simplex spin network whose edges are colored by ten irreps and five intertwiners, while each edge has a insertion. Here is the representation matrix for a group element in the irreducible representation (irrep) , see [11, 19]. Note that a vertex is dual to a 4-simplex , so that the set of faces intersecting at is dual to the set of ten triangles of . Similarly, the set of five dual edges intersecting at is dual to the set of five tetrahedrons of . The sum in (2.5) is over all colorings of the set of faces of by the irreps of , as well as over the corresponding intertwiners for the dual complex edges . Equivalently, label the triangles of , while label the tetrahedrons of .
which after quantum group regularization (by passing to a root of unity) becomes a manifold invariant known as the Crane–Yetter invariant .
The formula (2.4) is now given by the discretized expression
where denotes the triangulation of . The equation (2.7) can be rewritten as
where the operator is given by
This operator acts on the -spin network evaluation by inserting the Lie algebra basis element for an irrep into the spin network edge carrying the irrep . The expression (2.9) follows from (2.5), (2.7) and the relation
Following (2.9), let us define a -edge in a 4-simplex spin network as a line connecting the middle points of two edges of the spin network, such that this line is labelled by the tensor . We associate to a -edge the linear map
where and are the labels of the spin network edges connected by the -edge and denotes the action of the basis element of in the representation .
The action of the operator in a single 4-simplex of the spin foam state sum (2.5) can be represented as the evaluation of a spin network obtained from the 4-simplex spin network by adding a -edge between the two edges of labeled by and .
When and the intertwiners , from to , where is the adjoint representation, form a one-dimensional vector space, becomes the 4-simplex spin network with an insertion of an edge labeled by the adjoint irrep, see Fig. 1. This simplification happens because the matrix elements of can be identified with the components of the intertwiner , since these intertwiners are one-dimensional vector spaces, i.e.
Then the right-hand side of the equation (2.11) represents the evaluation of the spin network in Fig. 2. The condition (2.10) is not too restrictive since it includes the and groups. We need to consider this particular case in order to be able to use the Chain–Mail techniques.
The action of in is given by the evaluation of a spin network which is obtained from the spin network by inserting g-edges labeled by the adjoint irrep. These additional edges connect the edges of which correspond to the triangles of the 4-simplex where the operators from act.
The state sum is infinite, unless it is regularized. The usual way of regularization is by using the representations of the quantum group at a root of unity, which, by passing to a finite-dimensional quotient, yields a modular Hopf algebra . There are only finitely many irreps with non-zero quantum dimension in this case, and the corresponding state sum has the same form as in the Lie group case, except that the usual spin network evaluation used for the spin-foam amplitudes has to be replaced by the quantum spin network evaluation. In this way one obtains a finite and triangulation independent , usually known as the Crane–Yetter invariant . This 4-manifold invariant is determined by the signature of the manifold [24, 27]. The same procedure of passing to the quantum group at a root of unity can be applied to the perturbative corrections , but in order to obtain triangulation independent results, the dilute gas limit has to be implemented [3, 10].
2.1 The Chain–Mail formalism and observables of the Crane–Yetter invariant
The Chain–Mail formalism for defining the Turaev–Viro invariant and the Crane–Yetter invariant was introduced by Roberts in . In the four-dimensional case, the construction of the related manifold invariant had already been implemented by Broda in . However, the equality with the Crane–Yetter invariant, as well as the relation of with the signature of appears only in the work of Roberts .
We will follow the conventions of . Let be a four-dimensional manifold. Suppose we have a handle decomposition [13, 15, 25] of , with a unique 0-handle, and with handles of order (where ). This gives rise to the link in the three-sphere , with components (the “Chain–Mail link”), which is the Kirby diagram of the handle decomposition [13, 15]. Namely, we have a dotted unknotted (and 0-framed) circle for each 1-handle of , determining the attaching of the 1-handle along the disjoint union of two balls, and we also have a framed knot for each 2-handle, along which we attach it. This is the four-dimensional counterpart of the three-dimensional Chain–Mail link of Roberts, see [24, 4].
The Crane–Yetter invariant , which coincides with the invariant , defined in the introduction, see equation (2.6), can be represented as a multiple of the spin-network evaluation of the chain mail link , colored with the following linear combination of quantum group irreps (the -element):
Roberts also proved in  that , where is the evaluation of a 1-framed unknot colored with the -element and denotes the signature of .
Given a triangulated manifold , consider the natural handle decomposition of obtained from thickening the dual cell decomposition of ; see [24, 25]. Then a handle decomposition of (with a single 0-handle), such that (2.12) explicitly gives the formula for , appearing in equation (2.5), is obtained from this handle decomposition by canceling pairs of 0- and 1-handles [13, 15, 25], until a single 0-handle is left; in this case, in the vicinity of each -simplex, the Chain–Mail link has the form given in Fig. 3. This explicit calculation appears in [24, 4] and essentially follows from the Lickorish encircling lemma [14, 16]: the spin-network evaluation of a graph containing a unique strand (colored with the representation ) passing through a zero framed unknot colored with vanishes unless is the trivial representation.
A variant of the Crane–Yetter model in (2.12) is achieved by inspecting its observables, addressed in . Consider a triangulated 4-manifold . Consider the handle decomposition of obtained from thickening the dual complex of the triangulation, and eliminating pairs of - and -handles until a single 0-handle is left. Any triangle of the triangulation of therefore yields a 2-handle of .
Now choose a set with triangles of , which will span a (possibly singular) surface of . Color each with a representation . The associated observable of the Crane–Yetter invariant is:
where denotes the spin-network evaluation of the Chain–Mail link , where the components associated with the triangles are colored by and the remaining components with . We can see as a pair , where denotes the components of the Chain–Mail link given by the triangles of and the remaining components of the Chain–Mail link. We thus have .
Here is an open regular neighborhood of in , denotes the signature of the manifold and denotes the Euler characteristic. The link is the link in along which the 2-handles associated to the triangles of would be attached, in order to obtain . This theorem of  essentially follows from the fact that the pair is a surgery presentation of the pair , a link embedded in a manifold, apart from connected sums with .
3 The first-order correction
Recall that there are two possible ways of representing the Crane–Yetter invariant : as a state sum invariant (2.6) and as the evaluation of a Chain–Mail link (2.12). It follows from (2.8) that can be written as where is the number of 4-simplices of and is the state sum given by a modification of the state sum where a single 4-simplex is perturbed by the operator .
In order to calculate consider a 4-manifold with a triangulation whose dual complex is . Given a 4-simplex we define an insertion as being a choice of a pair of triangles of which do not belong to the same tetrahedron of and have therefore a single vertex in common (following (2.7) we will distinguish the order in which the triangles appear). Given the colorings of the triangles of (or of the dual faces ) and the colorings of the tetrahedrons of (or of the dual edges ), then is the evaluation of the spin network of Fig. 1. We then have:
where is the vertex of corresponding to . This sum is over the set of all 4-simplices of , as well as over the set of all insertions of and over the set of all colorings of the faces and the edges of (or equivalently, a sum over the colorings of the triangles and the tetrahedrons of .)
The infinite sum in (3.1) is regularized by passing to the category of representations of the quantum group , where is a root of unity. In order to calculate in this case, let us represent it as an evaluation of the Chain–Mail link  in the three-sphere .
As explained in Subsection 2.1, the invariant can be represented as a multiple of the evaluation of the chain mail link colored with the linear combination of the quantum group irreps , see equation (2.12). Analogously, by extending the 3-dimensional approach of , a chain-mail formulation for the equation (3.1) can be given. Consider the handle decomposition of obtained by thickening the dual cell decomposition of associated to the triangulation of . One can cancel pairs of 0- and 1-handles, until a single 0-handle is left. Let be the associated chain-mail link (the Kirby diagram of the handle decomposition). We then have
where, as before, an insertion is the choice of a pair of triangles and in some 4-simplex of , such that and have only one vertex in common. Given an insertion , the graph is obtained from the chain-mail link by inserting a single edge (colored with the adjoint representation of ) connecting the components of (colored with and ) corresponding to and ; see Fig. 4. can be considered as a pair where denotes the components of not incident to the insertion (which are exactly ) and denotes the component of containing the insertion . Hence we use the notation to mean the evaluation of the pair where all components of are colored with and the two circles of are colored with and , with an extra edge connecting them, colored with the adjoint representation .
Consider an insertion connecting the triangles and , which intersect at a single vertex. Equation (3.2) coincides, apart from the inclusion of the single insertion, with the observables of the Crane–Yetter invariant  (Subsection 2.1) for the pair of triangles colored with and ; see equation (2.13). By using the discussion in Section 6 and Theorem 2 of  (see equation (2.14)) one therefore obtains, for each insertion and each pair of irreps , :
where is the colored link of Fig. 5. In addition denotes the Witten–Reshetikhin–Turaev invariant [28, 23] of graphs in manifolds, in the normalization of [4, 9]. Note that in the notation of equation (2.14), is two triangles which intersect at a vertex, thus and also its regular neighborhood is homeomorphic to the 4-disk, thus .
To see this, note that (after turning the circles associated with the 1-handles of into dotted circles) the link is a Kirby diagram for the manifold minus an open regular neighborhood of the 2-complex made from the vertices and edges of the triangulation of , together with the triangles and . Since and intersect at a single vertex, any regular neighborhood of the (singular) surface spanned by and is homeomorphic to the 4-disk . Therefore is certainly homeomorphic to the boundary connected sum whose boundary is , for some positive integer . Here denotes the connected sum of manifolds and denotes the boundary connected sum of manifolds. The circles associated with the triangles and define a link which lives in .
The two circles and define a 0-framed unlink in , with each individual component being unknotted. Let us see why this is the case. We will turn the underlying handle decomposition of upside down, by passing to the dual handle decomposition of , where each -simplex of the triangulation of yields an -handle of ; see [13, p. 107]. Consider the bit of the handle-body yielded by the 2-complex , thus is (like ) a regular neighborhood of . Maintaining the 0-handle generated by the vertex , eliminate some pairs of - and -handles, in the usual way, until a single -handle of is left. Clearly , where denotes the orientation reversal. The circles and , in , correspond now (since we considered the dual handle decomposition) to the belt-spheres of the 2-handles of (attached along and ) and associated with the triangles and . Since and are 0-framed meridians going around and (see [13, Example 1.6.3]) it therefore follows that these circles are unlinked and are also, individually, unknotted; see Fig. 6. Given this and the fact that the insertion colored with also lives in , it follows that is a surgery presentation of the pair , apart from the connected sums, distant from , with .
Since the evaluation of the tadpole spin network is zero, it follows that , and consequently
For any triangulation of we have
4 The second-order correction
Since , we have to calculate in an appropriate limit such that the partition function is different from and such that is independent of the triangulation . Let be the number of -simplices. From (2.8) we obtain
if is dual to , see Fig. 1. On the other hand, if is not dual to . In order to to solve the possible framing and crossing ambiguities arising from the equation (4.1), a method analogous to the one used in  can be employed. Note that there are exactly 30 insertions in a -simplex , corresponding to pairs of triangles of with a single vertex in common. This is because there are exactly three triangles of having only one vertex in common with a given triangle of .
Analogously to the first-order correction, can be written as
where the first sum denotes the contributions from two insertions in the same 4-simplex and the second sum represents the contributions when the two insertions act in different 4-simplices and . As in the previous section, we will use the handle decomposition of with an unique 0-handle naturally obtained from the thickening of .
Note that each corresponds to a sum over all the possible choices of pairs of insertions in the 4-simplex . The value of is obtained from the evaluation of the chain-mail link colored with , which contains g-edges carrying the adjoint representation, as in the calculation of the first-order correction.
A configuration is, by definition, a choice of insertions distributed along a set of 4-simplices of . Given a positive integer and a set of 4-simplices of , we denote by the set of configurations with insertions distributed along . By expanding each into a sum of insertions, the equation (4.3) can be written as:
Note that each graph splits naturally as , where the first component contains the circles non incident to any insertion of .
The second sum in equation (4.4) vanishes, because it is the sum of terms proportional to:
These spin networks arise from the cases when the pair is a surgery presentation of the disjoint union and of , respectively, apart from connected sums with ; see Theorem 3 below. The former case corresponds to a situation where the two insertions act in pairs of triangles without a common triangle, and the latter corresponds to a situation where the two pairs of triangles have a triangle in common, which necessarily is a triangle in the intersection of the 4-simplices and . The evaluations in (4.5) and (4.6) vanish since the corresponding spin networks have tadpole subdiagrams.
The first sum in (4.4) also gives the terms proportional to the ones in equations (4.5) and (4.6). These terms correspond to two insertions connecting two pairs made from four distinct triangles of and to two insertions connecting two pairs of triangles made from three distinct triangles of , respectively. All these terms vanish.
The non-vanishing terms in equation (4.4) arise from a pair of insertions connecting the same two triangles in a 4-simplex. There are exactly 30 of these. Therefore, by using Theorem 3 of Section 5, we obtain:
where is a two-handle dumbbell spin network, see Fig. 8. We thus have:
The second-order perturbative correction divided by the number of of -simplices of the manifold is triangulation independent. In fact:
Here denotes the spin-network evaluation of the colored graph . Note that
which is is obviously non-zero, and therefore .
5 Higher-order corrections
For , the contributions to the partition function will be of the form
where . By using the equation (4.2), each of these terms splits as a sum of terms of the form:
where is a set of insertions (a “configuration”) distributed among the 4-simplices of the chosen triangulation of , such that the 4-simplex has insertions. Insertions are added to the chain-mail link as in Fig. 4, forming a graph (for framing and crossing ambiguities we refer to ).
Note that each graph splits as where contains the components of not incident to any insertion. As in the and cases, equation (5.1) coincides, apart from the extra insertions, with the observables of the Crane–Yetter invariant defined in ; see Subsection 2.1. Therefore, by using the same argument that proves Theorem 2 of  we have:
Given a configuration consider the -complex spanned by the triangles of incident to the insertions of . Let be a regular neighborhood of in . Then can be obtained from by adding the -handles corresponding to the faces of and some further - and -handles, corresponding to the edges and vertices of . These -handles attach along a framed link in , a manifold diffeomorphic to with the reverse orientation. The insertions of can be transported to this link defining a graph in . We have:
where denotes the signature of the manifold and denotes the Euler characteristic.
Note that, up to connected sums with , the pair is a surgery presentation of Unlike the and cases, it is not possible to determine the pair for without having an additional information about the configuration. In fact, considering the set of all triangulations of , an infinite number of diffeomorphism classes for is in general possible for a fixed ; see  for the three dimensional case. This makes it complicated to analyze the triangulation independence of the formula for for
in order to resolve the triangulation dependence of , let us introduce the quantities
see [10, 3]. The limit is to be extended to the set of all triangulations of , with being the number of 4-simplices of , in a sense to be made precise; see . From Sections 3, 4 and Theorem 2 it follows that
Note that the values of and are universal for all compact 4-manifolds. The expression for is finite because there are only finitely many irreps for the quantum group , of non-zero quantum dimension, when is a root of unity.
6 Dilute-gas limit
We will now show how to define and calculate the limit in the equation (5.2). Let be a 4-manifold and let us consider a set of triangulations of , such that for any given there exists a triangulation such that the diameter of the biggest 4-simplex is smaller than , i.e. the triangulations in can be chosen to be arbitrarily fine. We want to calculate the limit in equation (5.2) only for triangulations belonging to the set .
Furthermore, we suppose that is such that (c.f. ):
Restriction 1 (Control of local complexity-I).
Together with the fact that the triangulations in are arbitrarily fine we suppose that:
There exists a positive integer such that any -simplex of any triangulation intersects at most -simplices of .
Let us fix and consider when . The value of will be given as a sum of contributions of configurations such that insertions of