Testing the imposition of the Spin Foam Simplicity Constraints
Abstract
We introduce a threedimensional Plebanski action for the gauge group SO(4). In this model, the field satisfies quadratic simplicity constraints similar to that of the fourdimensional Plebanski theory, but with the difference that the field is now a oneform. We exhibit a natural notion of “simple oneform”, and identify a gravitational sector, a topological sector and a degenerate sector in the space of solutions to the simplicity constraints. Classically, in the gravitational sector, the action is shown to be equivalent to that of threedimensional first order Riemannian gravity. This enables us to perform the complete spin foam quantization of the theory once the simplicity constraints are solved at the classical level, and to compare this result with the various models that have been proposed for the implementation of the constraints after quantization. In particular, we impose the simplicity constraints following the prescriptions of the socalled BC and EPRL models. We observe that the BC prescription cannot lead to the proper vertex amplitude. The EPRL prescription allows to recover the expected result when, in this threedimensional model, it is supplemented with additional secondary second class constraints.
Introduction
Spin foam models (1); (2); (3); (4) constitute an exciting proposal for the definition of a background independent and nonperturbative quantization of general relativity. They were introduced originally (5) as a way to implement the dynamics of loop quantum gravity (6); (7), and can be thought of as representing a sum over histories of the gravitational field (8).
In three spacetime dimensions, pure gravity being a topological theory (in the sense that it has no local degrees of freedom), spin foam models can be obtained as an exact simplicial path integral for the firstorder Palatini action, where the gauge group is taken to be in Riemannian signature (the whole symmetry group is larger than and depends on the sign of the cosmological constant ). This corresponds to the celebrated PonzanoRegge (9) and TuraevViro (10) models when and respectively. Their vertex amplitudes are simply given by and (for a root of unity) coefficients. It has been shown (11) that the PonzanoRegge amplitudes can be obtained from a canonical quantization of gravity in the spirit of loop quantum gravity, in which the kinematical states are given by spin networks, and the dynamics leads to a spin foam evolution. The TuraevViro amplitudes can also be obtained as the scalar product between physical states which are in turn given in terms of unitary representations of some quantum groups closely related to (12); (13). Therefore, in three dimensions, there is a clear understanding of the relationship between the socalled canonical and covariant approaches to quantum gravity (at least when the signature is Riemannian).
In four spacetime dimensions, the construction of such a correspondence between the canonical and spin foam quantizations is still an open problem (14); (15); (16). Since it is technically too involved to follow the construction of the threedimensional theory and perform the straight simplicial path integral quantization of the fourdimensional Palatini action (see (17) for an attempt to follow this direction), spin foam models have to be derived using an alternative strategy. More precisely, the approach to fourdimensional spin foam models is based on the fact that gravity can be formulated as a constrained topological field theory, by virtue of the socalled Plebanski action (18). This latter is defined as the sum of a wellknown topological action (19), plus a set of constraints on the field ensuring onshell the condition that it comes from the exterior product of two tetrad oneform fields
The three spin foam models for fourdimensional quantum gravity that have been studied the most are the BarrettCrane (BC) model (20), the EnglePereiraRovelliLivine (EPRL) model (21); (22); (23), and the FreidelKrasnov (FK) model (24); (25). They all rely on the general strategy outlined above, even though their construction requires to impose the simplicity constraints in drastically different ways. In the BC model, which was the first fourdimensional model to be introduced, the simplicity constraints are imposed as strong operator relations. This has the result of assigning simple representations of the gauge group to faces of the dual triangulation , and a specific unique intertwiner, known as the BC intertwiner (20); (26), to edges of . Due to difficulties in reproducing the correct semiclassical limit (27); (28); (29) and the structure of the graviton propagator (30), this model was partly discarded
Let us summarize the situation. At the classical level, it is clear that the Plebanski action is equivalent to the Palatini action once the field is forced to be simple. But since we do not know how to compute the simplicial path integral for the Palatini action, we try to compute that of the Plebanski action by first quantizing the unconstrained topological theory and then imposing the simplicity constraints at the quantum level. In fact, it is quite clear that we are restricted in our understanding of how to properly perform this last step, by the fact that we do no know what to “expect” from the spin foam quantization of the fourdimensional Plebanski action. For this reason, it would be nice to have a model in which we could work out explicitly the two alternative methods. By this, we mean that we would like to perform the spin foam quantization of the action in which the simplicity constraints have already been solved at the classical level, and then compare this result to the spin foam quantization in which the simplicity constraints are solved in the quantum theory. This would be a way to test the various proposals to deal with the simplicity constraints. It has already been argued in the literature (36) that dealing with second class constraints at the quantum level can lead to inconsistencies, and is not compatible with the quantization program à la Dirac. However, the argument alone is not fully convincing, since it was formulated on a finite dimensional model with no clear analogy with gravity, and since it does not give a clear explanation for how the various impositions of the constraints in the fourdimensional spin foam models should lead to a model with inconsistent physical predictions. It might very well be that despite the fact that we are dealing with second class constraints at the quantum level, there exists a preferred scheme in which the simplicity constraints can be imposed in a (yettobe defined) robust way to lead to an acceptable model for quantum gravity. In the end, the only way to discriminate between the various proposals is either to extract physical predictions to compare with experiments, or at least to test the strategies on toy models which bear a close analogies with gravity. Let us also point out that a model was introduced in (34) in order to illustrate the claim that additional secondary second class constraints should be taken into account in the spin foam models. This model is based on the idea of reducing a fourdimensional theory to an one by means of simplicity constraints, and demonstrates that this is only possible if certain secondary second class constraints are imposed. The weak point of this model is that the constraints that it imposes are not derived from the Hamiltonian analysis of a given Plebanski theory, but are instead put in by hand.
The aim of this paper is to formulate a robust model which allows to test the imposition of the spin foam simplicity constraints. Since pure gravity is always topological in three spacetime dimensions, it makes a priori no sense to have a Plebanski formulation in which the field is forced to be simple. In fact threedimensional theory is already equivalent to gravity, when is a oneform field valued in the Lie algebra of . However, if one replaces the gauge group by (or its double cover ), the threedimensional theory admits simplicity constraints similar to that of the fourdimensional theory. In the gravitational sector of solution to these simplicity constraint, the action becomes that of first order Riemannian gravity. In the topological sector, the theory is trivial. Therefore, at the classical level, the Plebanski theory that we introduce reduces to the HilbertPalatini action, whose spin foam quantization naturally leads to the PonzanoRegge amplitudes. Now, it is also possible to perform the spin foam quantization of the threedimensional theory, and to impose the simplicity constraints at the quantum level, mimicking in particular the prescription of the BC and EPRL
We start in section I by introducing the classical framework for our analysis. In particular, we show how the threedimensional action is reduced by the simplicity constraints to the usual action for threedimensional gravity, and we perform the Hamiltonian analysis of the Plebanski theory to support this fact. Section II is devoted to the study of the quantum theory. We first recall the state sum model corresponding to the constrained classical action, and then perform the spin form quantization of the Plebanski theory. For this, we start by writing down the simplicial path integral for the theory, and then study the imposition of the simplicity constraints. In particular, we study the BC and EPRL prescriptions, in which the constraints are imposed strongly and weakly (in a sense that we will make more precise), respectively. We also study the socalled Warsaw modification (37); (38); (39) of the EPRL model in order to illustrate its signification on the threedimensional model at hand. Our conclusion is that the BC prescription cannot lead to the proper vertex amplitude, whereas the EPRL prescription, when supplemented with the imposition of the additional secondary second class constraints, enables one to recover the expected result. This might be a strong indication of the fact that a weak imposition of the simplicity constraints in the quantum theory is an important ingredient in order to derive the vertex amplitude of fourdimensional quantum gravity. Nevertheless, we want to emphasize that in our threedimensional model, none of abovementioned proposals lead by themselves to the proper vertex amplitude, which is that of the PonzanoRegge model. The reason for this is clear: the current spin foam models miss the imposition of the secondary second class constraints that are exhibited in the canonical analysis of the Plebanski theory. Indeed, we show how these constraints can be solved by imposing conditions on the spatial components of the connection, and how their imposition in the path integral with the EPRL prescription allows to recover the proper vertex amplitude.
I Classical theory – actions for threedimensional gravity
To introduce our model, let us start with the familiar action, written in three dimensions and for the gauge group . In terms of an valued oneform field (our notations are defined in appendix A) and the spacetime connection , it is given by
(1) 
where denotes a trace in the Lie algebra , and is the curvature twoform associated to the connection. Clearly, this action is not that of Riemannian threedimensional first order gravity, but this latter (or at least two copies of it) can be obtained if we use a decomposition of the fields into selfdual and anti selfdual components. Here we are not interested in doing this decomposition, because our goal is to mimic the constrained fourdimensional action which is the starting point for the spin foam quantization. To see that this is indeed possible, let us proceed in two steps. First, let us assume that the field is of the form
where is a zeroform in , and a Lie algebravalued oneform. This is the threedimensional analogue of the notion of simple twoform field that is used in fourdimensional Plebanski theory. Second, let us choose a gauge (which we will refer to as the timegauge) in which . This gauge choice reduces the symmetry group to its subgroup. A key observation is that it is always possible to fix due to the presence of the “scaling” symmetry
(2) 
for . With this choice of gauge, the action (1) becomes the usual action for gravity
(3) 
We have included the first equality to make explicit the fact that the threedimensional HilbertPalatini action is simply a constrained action. This argument suggests that it is possible to obtain the action for gravity starting from (1) if we impose a simplicitylike condition on the field, and pick up a rotational subgroup of by making a gauge choice for (notice that this choice is not canonical).
We are going to see in this section that the simplicity of the field can be enforced with a Plebanski action whose Hamiltonian analysis will be detailed.
i.1 Plebanski action and simplicity constraints
As we mentioned above, the first things that we would like to do is to impose the simplicity of the field in the theory. Following what is done in four dimensions, we are going to introduce the Plebanski action as the sum of the action (1) plus a set of constraints on the field. Our aim is to reproduce simplicitylike conditions ensuring that we are indeed dealing with threedimensional gravity. The action that we consider is
(4) 
where denotes the Hodge dual operator in , and is a symmetric Lagrange multiplier used to enforce the constraints
(5) 
We want to understand the exact meaning of these constraints in the present context. The key difference with fourdimensional Plebanski theory is that we are dealing here with vectors, and not bivectors. Therefore, we need another notion of simplicity than the usual one used in spin foam models. As we have already mentioned earlier, we are going to say that a vector is simple if it can be written as
where is a zeroform in , and a oneform. The fact that the constraint defined by (5) can be used to obtain simple vectors is ensured by the following proposition:
Proposition 1.
When the simplicity constraints are satisfied, there are three possible solutions for the field. They correspond respectively to:

A gravitational sector, in which there exists an valued one form and a vector in such that

A topological sector, in which there exists an valued one form and a vector in such that

A degenerate sector, in which and .
Proof.
Let us introduce the boost and rotational components of the field . They are given respectively by and . Each of these oneforms can be interpreted as a vector in , and the simplicity constraints simply mean that the vectors are orthogonal to the vectors , i.e.
(6) 
Let us first assume that the three vectors form a basis of . Due to the simplicity constraints (6), the three remaining vectors are linked, and therefore they lie in the same plane, whose (nonnecessarily unit) normal is denoted by . More precisely, there exist three vectors such that
(7) 
Furthermore, since the vectors and are orthogonal, we can write to
(8) 
where is a (nonvanishing) scalar, and is a realvalued one form. Due to the fact that the vectors form a basis, the three vectors form a basis as well. The solution given by (7) and (8) corresponds to the gravitational sector.
If we assume on the contrary that the three vectors form a basis, the same construction can be used to obtain the topological sector described in proposition 1.
When and , i.e. neither the nor the form a basis, we say that the field belongs to the degenerate sector.
It is clear from the expression of in the gravitational sector that in the time gauge, where we have and , the boost component is vanishing, while the rotational component reduces to the triad . This supports the heuristic argument that we have given at the beginning of this section in order to derive the action (3) from the theory. Note also that in the gravitational sector we have , while in the topological sector we have .
Let us conclude with a remark. There are 6 simplicity constraints acting on the 18 components of the field. Therefore we expect the simple field to be written in terms of 12 components only. Here, we have expressed in terms of the 16 components and . However, notice that there are 4 redundant components due to the presence of the following symmetries:

A rescaling symmetry, given by (2), which allows to remove one component (for instance, we can fix ).

Three translational symmetries acting in the nondegenerate sector. They are generated by a vector , and act like
As a consequence, we can always choose , but here we do not make this choice in order not to break the symmetry of the theory.
i.2 Gravitational and topological sectors
Now that we have a good understanding of how the simplicity constraints can be used to define a sector of the Plebanski theory in which the vector is simple, let us see how it is possible to recover the action for threedimensional gravity. For this, we are going to study separately the gravitational and topological sectors introduced in proposition (1).
Using the decomposition of into selfdual and anti selfdual generators (see appendix B), we can write the field as
where , and the action (1) therefore becomes
(9) 
As usual, the equations of motion with respect to the connection lead to the torsionfree condition
and if , this relation can be inverted to find the torsionfree spin connection. This latter, when plugged back into the original action (9), leads to the sum of two second order EinsteinHilbert actions,
(10) 
each being defined with respect to a twodimensional Urbantkelike metric (40) (in the sense that it is constructed with the field). In this expression, denotes the sign of (we refer the reader to appendix C for the calculation of ).
In the gravitational and topological sectors, the selfdual and anti selfdual components of the field are given by:
(11b)  

gravitational sector: ,

topological sector: .
In each of these two sectors, we can now compute the two Urbantke metrics defined by the expressions above. In fact, from the simplicity constraints, we know already that . It is however less trivial to see that the metrics in the topological and gravitational sector are identical. A simple calculation shows that we have
(12) 
where . In the time gauge, this metric reduces to .
Gathering these results on the Urbantke metric and the sign factors , we can conclude that the action (1) reduces in the gravitational sector to
while in the topological sector it becomes simply .
This shows that the gravitational sector to the solutions of the simplicity constraints corresponds indeed to threedimensional gravity, whereas in the topological sector the action vanishes onshell. This is exactly what happens in the fourdimensional Plebanski theory. Interestingly, the presence of a topological sector in this threedimensional theory allows for the introduction of a BarberoImmirzi parameter, but we shall come back to this point in section III.
i.3 Canonical analysis of the Plebanski action
In this subsection we perform the canonical analysis of the threedimensional Plebanski theory. It is similar to the study of the fourdimensional Plebanski action (41). Using a decomposition of the spacetime manifold, (4) becomes
where the curvature and torsion twoforms are defined by
(13) 
We can see from this formula for the action that , and are nondynamical variables, since the Lagrangian does not feature their time derivatives. The time component of the connection and are true Lagrange multipliers, enforcing respectively the torsionfree and simplicity constraints. However, since is involved quadratically in the simplicity constraint, it cannot be treated as a Lagrange multiplier. For this reason, we add to the Lagrangian the term
where and are new auxiliary valued fields that do not affect the dynamics of the theory.
The basic variables of the theory are then the 12 spatial components of the connection, their 12 canonical momenta , and the 6 components with their momenta . The Poisson structure is given by
(14) 
where . The total Hamiltonian is
(15) 
We can now identify the primary constraints and compute their Poisson bracket with this Hamiltonian in order to study their evolution in time.
Primary and secondary constraints
We have the following 18 primary constraints:
The first set is obtained by varying the action with respect to , the second set by varying the action with respect to , and the third one by varying with respect to
Before going any further, let us introduce smeared variables in order to deal with the delta functions appearing in the Poisson brackets. To this end, we define the quantities
for any valued smooth test function on . This enables us to compute
and
where stands for the covariant derivative with respect to the connection . Now we can compute the Poisson bracket of the primary constraints with the total Hamiltonian, to see if the requirement that they be preserved under the time evolution gives rise to secondary constraints.
a. Evolution of the constraints
First, it is easy to show that the conservation of does not lead to any secondary constraints. To see this, we can introduce the new constraint
(16) 
and notice that it is in fact the generator of the infinitesimal gauge transformations on the phase space. Indeed, on the action of on the phase space variables (14) is given by
Since the Hamiltonian contains only terms with traces, it is left invariant under the action of this constraint, which therefore satisfies . The modification (16) of the constraint is permissible since it amounts to adding a term which is itself proportional to a constraint.
b. Evolution of the constraints
The requirement that the primary constraint be preserved in time leads to the following relation:
Among these 6 equations, 3 are in fact fixing the components (if we assume that the nondegeneracy condition holds), and the remaining 3 have to be imposed as secondary constraints. Projecting onto the vector , these secondary constraints can be written as
c. Evolution of the simplicity constraints
To study the evolution of the simplicity constraint, it is convenient to compute separately the Poisson bracket of its various components with the Hamiltonian. Requiring that the constraints and be preserved under time evolution leads to
(17a)  
(17b) 
These 3 equations determine 3 of the 6 components of the multiplier , and imply no secondary constraints on the dynamical variables of the phase space. Finally, a direct computation shows that the requirement that leads to the 3 secondary constraints
(18) 
The Dirac algorithm stops here, and there are no tertiary constraints.
d. Summary of the analysis of the primary constraints
To summarize, we have a theory with 18 primary constraints consisting of 6 constraints , 6 constraints , and 6 constraints . They generate the 6 secondary constraints comprising the 3 constraints and the 3 constraints .
First and second class constraints
In the previous subsection, we have derived the primary and secondary constraints of the Plebanski theory. We are now going to split them between first class and second class constraints.
a. The first class constraints
First, it is easy to see that the constraints are first class. Indeed, they commute with all the other constraints since they generate the infinitesimal gauge symmetries.
The analysis of equation (17) suggests that amongst the 6 constraints , 3 are first class and 3 are second class. To see that this is indeed the case, let us decompose the set of constraints into
If the field does not belong to the degenerate sector, the previous set of constraints is equivalent to the requirement that . Furthermore, a direct computation (see the algebra of constraints in appendix D) shows that the constraints are first class. They are associated with the 3 components of that remain undetermined after taking (17) into account.
Finally, the last first class constraints are given by a new set , which is obtained by adding to a linear combination of constraints. In particular, is defined as being equal (up to a factor 2) to the Hamiltonian density (15):
where the values of the Lagrange multipliers , and , are those determined by the constraint analysis. When the Lagrange multipliers are not fixed by the Hamiltonian analysis, it is possible to fix them to the value zero in the expression for . Concerning the constraints , a direct calculation shows that they are given by
and that they generate as expected space diffeomorphisms on the phase space variables:
where is a vector field on , and denotes the smearing of the with .
b. The second class constraints
The remaining constraints, , and , are the second class constraints of the theory. In order to prove this, it is possible to compute the (square 12dimensional) Dirac matrix , whose elements are given by Poisson brackets between the various (candidate) second class constraints, and show that its determinant is nonvanishing. This is indeed the case, since the Dirac matrix is given by

and it satisfies clearly . Here the elements of the matrix are given by
and we have , where denotes the threedimensional volume. The matrix has elements determined by the Poisson brackets
and satisfies . The explicit form of the matrices and is not important in order to prove the invertibility of the Dirac matrix, and it is in fact easy to see that its determinant is given by
Summary
To summarize, the first class constraints of the system are given by , and , and the constraints , and , are of second class. Therefore, we have 36 phase space variables and that are subject to 12 first class constraints (generating gauge symmetries) and 12 second class constraints, and the usual counting shows that there are no phase space degrees of freedom. This is of course to be expected in threedimensional gravity.
As usual, the first class constraints of the theory are the infinitesimal generators of the gauge symmetries. The constraint is the generator of the internal symmetries. The 3 primary constraints appear because we have treated the lapse and the shift encoded in as dynamical variables. The constraints and therefore encode the vanishing of the momenta and conjugated to the lapse and the shift. The remaining constraints, and , are related to the scalar and vector constraints, i.e. to the spacetime diffeomorphisms.
Let us conclude this canonical analysis with an important remark concerning the second class constraint. The 3 constraints constrain 3 out of the 6 components of the variables , which are the conjugate momenta to . As a consequence, only 3 components of are left, and they correspond to the momenta and associated with the lapse and shift variables. Moreover, the lapse and the shift are the 3 independent components of that are left after imposing the 3 components of the simplicity constraints. The spatial part of the simplicity constraints ensures that the components can be expressed in terms of the zeroform and the oneform . Finally, the meaning of the 3 constraints is that 3 of the components of the connection can be expressed in terms of and . It is important to stress that these are secondary constraints which have been obtained from the requirement that be preserved in time. In other words, the usual simplicity constraints are of second class because there are in fact secondary constraints which do not commute with the primary ones. This point will be very important in the spin foam quantization that we perform in the next section. We are going to show the importance of imposing the secondary second class constraints in the spin foam models.
Ii Spin foam quantization
In this section, we study the quantization of the threedimensional Plebanski theory. From the classical analysis of the previous section, it should be clear that the action (4) is equivalent in the gravitational sector to that of first order gravity, whose spin foam quantization is the PonzanoRegge model. We can now try to reproduce the strategy of fourdimensional spin foam models, and see if the spin foam quantization of the full Plebanski theory leads to the PonzanoRegge model as well. This is a wellposed question which will enable us to clarify the issue of imposing the simplicity constraints at the quantum level.
ii.1 The PonzanoRegge model
We have seen in the previous section that when the simplicity constraints are imposed at the classical level, the gravitational sector of solutions to the Plebanski action (4) is equivalent to the Palatini action. It is therefore possible to compute explicitly the simplicial path integral of this theory, and we know that it leads to the PonzanoRegge state sum model, whose vertex amplitude is given by an symbol. We recall some basic facts about the derivation of this result, and also introduce the notations that we will use in the next subsections.
The partition function of threedimensional Riemannian gravity is formally given by
where denotes a trace in the Lie algebra . To compute the simplicial path integral, we introduce a simplicial decomposition of the spacetime manifold , along with its dual twocomplex (42). This latter is consisting of vertices (dual to tetrahedra ), edges (dual to triangles ) and faces (dual to links ). Since the triad is a oneform, it is natural to integrate it along the onecells (links) of to obtain valued elements . The connection is discretized by computing its holonomy along the edges of . We can then discretize the curvature by taking the product of holonomies along the edges lying on the boundary of a face to form
With these variables, the discretized version of the action (3) can be written as
and the partition function becomes
(19) 
where is the Lebesgue measure on , and the Haar measure on . It is now possible to perform the integral over to obtain
where the delta distribution over imposes the flatness of the connection. Using the PeterWeyl decomposition, we can write
(20)  
where the sum is taken over all the possible representations labeling the set of faces . For arbitrary cellular decompositions , let us call the number of faces meeting at every edge . In (20) we will have an integral over of products of representation matrices for the group element . We can therefore use the fact that
(21) 
projects onto the space of intertwiners between the representations coloring the faces meeting at the edge . Then, all the intertwiners meeting at a vertex can be contracted to define a vertex amplitude . Finally, the partition function can be written as a sum over spin foam amplitudes
To clarify the meaning of this formula, let us assume that the cellular decomposition is simplicial. In this case, vertices in are fourvalent, while edges are threevalent. The vertex amplitude is therefore given by a contraction of four threevalent intertwiners, which is the socalled symbol. The final result of this calculation is
(22) 
where the labels are associated to the six faces dual to the links a tetrahedron . Notice that the sum over intertwiners has now disappeared since there is only a unique (up to normalization) threevalent intertwiner. In other words, the intertwiner space
is onedimensional when satisfy the triangular inequalities, and zerodimensional otherwise.
It is convenient at this point to introduce the diagrammatic notations (43), in which lines are used to represent unitary irreducible representations, and boxes to represent integrals over the group defining a projector on the intertwiner space following (21). With this notation, the partition function defining the PonzanoRegge model can be represented as follows:
Once the integrations are performed, we obtain a contraction a four threevalent intertwiners for each vertex dual to a tetrahedron, and the partition function becomes (22).
ii.2 Quantization of the sector
Spin foam models in four dimensions take as a starting point the fourdimensional Plebanski action, and are derived by first quantizing the topological part of the action, and then imposing the simplicity constraints at the quantum level.
Here we follow the same procedure, and start by quantizing the part of the Plebanski action (4). This can be done very easily along the lines of the standard construction introduced in the previous subsection, with the only difference that is now replaced by . Therefore, the field is discretized by assigning an element to each face dual to a onecell in . Once we perform the integration over in the simplicial path integral, we are left with
where now we have a product of delta distributions on the group , which can again be evaluated with the PeterWeyl decomposition. Using the fact that , it is possible to write an irreducible representation of as a tensor product of two irreducible representations of . The discretized partition function for the threedimensional theory then takes the form
Now, we have to implement the simplicity constraints in this state sum in order to ensure that it describes quantum three dimensional gravity. Indeed, since at the classical level the gravitational sector of the Plebanski action corresponds to the action for gravity, whose spin foam quantization corresponds to (22), we expect that a proper imposition of the simplicity constraints will lead to the same result.
ii.3 Imposition of the simplicity constraints
In the simplicial path integral that we have written for the threedimensional theory, the oneforms have been discretized by assigning an element to the faces of , which are dual to the links defining the boundary of the twocells (triangles) . These vectors in fact determine the geometry of the triangles, and they satisfy the closure constraint
This is just the discrete analogue of the continuous Gauss law which ensures gauge invariance under the action of . Additionally, the discretized field is required to satisfy the discretized version of the simplicity constraints , which are given by
diagonal simplicity:  (23)  
offdiagonal simplicity:  (24) 
These simplicity constraints do not distinguish between the topological and gravitational sectors, since they are left unchanged if we change for .
Just like in the fourdimensional theory (24); (35), the simplicity constraints can be linearized, and are equivalent to the requirement that
(25) 
Indeed, in the gravitational sector where we have , it is clear that the linear simplicity constraint is satisfied. This linearized version has therefore the advantage of selecting the gravitational sector.
The quantization of the theory is based on the symplectic structure of the topological part of the action. In the action, the variable is canonically conjugated to the connection , and the quantization rule is therefore simply to identify the discretized field with the generators of the Lie algebra . The next step is to impose the simplicity constraints on these generators following the different spin foam models that have been introduced in the literature.
The imposition of the constraints is done before the integration over the connection components defining the intertwiners. Schematically, we can represent this imposition of the constraints as a white box acting on the group generators between two neighboring triangles. Our task is therefore to implement the simplicity constraints in such a way that the partition function reduces to
(26) 
Here we have represented the implementation of the constraints via on one triangle only for the sake of clarity. More explicitly the meaning of this graphical notation is that we have an integral over the holonomies, together with a yettobe defined implementation of the simplicity constraints . In other words, we have
(27) 
and the implementation of the simplicity constraints is done before the integration over the holonomies that defines the intertwiners leaving on the edges.
The BC prescription
We are going to start by solving the diagonal simplicity constraint. It is known that this constraint is equivalent to the requirement that the pseudoscalar quadratic Casimir operator of defined by vanishes, which constrains the representations to be simple. Indeed, in terms of the selfdual and anti selfdual generators, (23) takes the form
which implies the restriction to simple representations of