Discretization of 4d Poincar\'e BF theory: from groups to 2-groups

We study the discretization of a Poincar\'e/Euclidean BF theory. Upon the addition of a boundary term, this theory is equivalent to the BFCG theory defined in terms of the Poincar\'e/Euclidean 2-group. At an intermediate step in the discretization, we note that there are multiple options for how to proceed. One option brings us back to recovering the discrete variables and phase space of BF theory. Another option allows us to rediscover the phase space related to the G-networks given in [2]. Indeed, our main result is that we are now able to relate the continuum fields with the discrete variables in [2]. This relation is important to determine how to implement the simplicity constraints to recover gravity using the BFCG action. In fact, we show that such relation is not as simple as in the BF discretization: the discretized variables on the triangles actually depend on several of the continuum fields instead of solely the continuum B-field. We also compare and contrast the discretized BF and BFCG models as pairs of dual 2-groups. This work highlights (again) how the choice of boundary term influences the resulting symmetry structure of the \textit{discretized theory} -- and hence ultimately the choice of quantum states.


Introduction
Recent developments have pointed out the importance of boundary terms in the construction of a quantum theory of gravity [17,19,18,16]. Indeed we expect the quantum states of geometry to be given as representations of the symmetries at hand. The symmetries depend on the choice of variables used, as can be seen when comparing the metric and Palatini formalism. What was under-appreciated before is how adding boundary terms to the action might greatly influence the resulting symmetries relevant to the construction of quantum states.
We illustrate this point by considering the 4d "Euclidean" BF theory, i.e. a BF theory where the underlying gauge group is the Euclidean (or similarly the Poincaré) group. Depending on whether or not we introduce a (very simple) boundary term to the theory, the symmetries of the quantum theory are given by two different types of (strict) Lie 2-groups -a categorified version of a group, also called a crossed module [6]. From a pragmatic point of view, we can understand the difference between a group and a 2-group in the following way. If elements of a (1-)group G can be naturally associated to paths (holonomies), then elements in a 2-group are elements of a pair of groups G and H -with some additional maps between themwhich decorate paths and faces. It is widely expected that such structure (or their deformation) is the right structure to probe 4d topology just like (quantum) groups are the relevant structure to probe 3d topology [3].
4d BF theory for a d dimensional Lie group G with Lie algebra g is a theory which can be seen as a 2-gauge theory, i.e. a gauge theory built on a 2-group [6,24]. However upon discretization, it is well known that we recover a copy of T * G ∼ = G g * ∼ = G R d associated to each link/triangle pair. The usual quantum states, called spin networks, are built from representations of G so that in the end there is not much left of the 2-group picture. Indeed, a 1-group can be seen as a trivial 2-group.
Following the key insights by Dittrich and Geiller [13,9,12], it was realized that we could instead build the quantum theory for a BF theory from representations of the (quantum) group R d (which is still a trivial 2-group). This amounts to a change of polarization [14], which can actually be obtained by the addition of a simple boundary term to the initial action. This new representation pioneered by Dittrich and Geiller is important as it allows us to discuss quantum states of geometry defined in terms of the frame field (as opposed to functions of the connection) in the quantum gravity context.
Most approaches to construct a 4d quantum gravity model rely on using a 4d BF theory [35]. The usual one consists of taking the underlying group to be the Lorentz group SO(1, 3), however, it was shown that one could also use the Poincaré group ISO (1,3) [32,31,33,11]. In either case, gravity can be recovered by imposing the simplicity constraints on the 2-form B. The new interesting part in the Poincaré formulation is that the translational part is associated to the frame field. Hence the frame field is already present at the BF level, contrary to the Lorentz formulation where it appears only through the simplicity constraints 1 .
Interestingly, Poincaré BF theory can be seen to be equivalent -up to a boundary term -to another topological theory based on a non-trivial 2-group, the Poincaré 2-group [32,31,33]. This boundary term now implies only a partial change of polarization. It is only in the translational that the configuration and configuration variables are swapped with respect to the BF formulation. In a sense, we only do half of the dualization done by Dittrich and Geiller 2 . The new action is called the BFCG action [22] and was introduced as the continuum counterpart of the topological invariant built on 2-groups, the Yetter model [38]. In terms of actions, the symmetries are of the Poincaré BF theory or the BFCG theory are obviously the same, they are just packaged differently since the what is called configuration or momentum differs between the two actions. This is what will create the difference at the discrete/quantum level between the two theories.
Since 2-groups (or 2-categories) seem to be a natural tool with which to probe the topology of 4d spaces [3], it is expected that they should be useful in building a quantum theory for gravity. With this in mind, a partition function has been constructed based on the representation theory of the 2-Euclidean group, the KBF model [10,28,26,27]. It was recently shown how such model could actually be related to the partition function of the BFCG action for the 2-Euclidean group [2]. Part of the argument relied on the proposal (by B. Dittrich [2]) of a phase space structure for the polyhedron which seemed to be naturally related to the BFCG variables. This discretization was based on an educated guess and the correspondence with the continuum variables was not identified. One of the key results of this paper is to provide such relation. This relation is especially important to build the quantum gravity amplitude. Indeed, it is naturally expected that the discretized B field should be constrained by the discrete simplicity constraint to obtain the quantum gravity amplitude. However doing this led to some possibly overly simplified model [31,33]. Imposing the simplicity constraint on the discretized B field is justified provided that the discretized B field is only dependent on the continuum B field. We will show here that the discretized B field is actually a function not only of the B field but also of some other fields. This implies that implementing a naive simplicity constraint on such discretized B field is probably not the right thing to do to recover the gravity regime.
The discretization procedure first introduced in [20] has already shown its power in several instances. The first step is to decompose the manifold of interest into cells. The next step is to determine what happens to the degrees of freedom in these cells. We use a truncation: in the bulk we only consider the fields on-shell. At least for the cases of interest, we can then express the symplectic data solely in terms of fields living on the boundary. By gluing the cells together (while imposing continuity between neighbouring cells), we obtain a way to explicitly evaluate the symplectic potential and hence identify both the discretized variables and their phase space structure. In particular we recover, in the BF case based on gauge group G, the usual T * G phase space, associated to each link of the dual complex. This approach allows one to recover the particle degrees of freedom in the 3d gravity context [21] or the quantum group symmetry for 3d gravity with a cosmological constant [15]. Another interesting outcome of this discretization is the emphasis that a choice of polarization must be made [14,37]. In particular there is always the choice on where to discretize the different variables. While at the continuum level this might seem benign, at the quantum level it is important. Indeed, at the discrete level this choice restricts the set of symmetries that are imposed first. For example in the BF case, we can have the Lorentz symmetries or the translational symmetries, as we alluded before. The polarisation choice in that case affects which of these two symmetries is imposed on the spin networks, and which arise in the dynamics.
In present work, we show how this discretization approach allows us to recover the discrete phase space proposed in [2] starting from the BFCG action. The main outcome is the expression of the discretized variables in terms of the (smeared) continuum fields which arise in the discreization procedure. Another outcome we find interesting is that the nature of the symmetries at play, which is very important for the construction of the quantum states, is very much dependent on the type of boundary terms one considers. While we discretize BF, a very well known theory, we are able to find non-trivial 2-group symmetries. We discussed above how changing the polarization was impacting the symmetry structure, dealing either with the gauge symmetries or the translational symmetries. Now with a partial change of polarization we will have either some standard group symmetries (the aforementioned gauge/translational symmetries) or a non trivial 2-group symmetry.
Outline In section 2, we review the basics of constructing a BF theory and its symmetries. We then go on to show how to get BFCG theory from BF theory by choosing the underlying group to be ISO(4) and adding a boundary term. For both theories, we recall the symplectic structure and charge structure.
In section 3, we describe the discretization process allowing to recover the standard BF discretization, which can be reinterpreted as recovering spinning tops for the group G interacting through conservation of angular momenta.
In section 4, we identify an alternate way of simplifying the symplectic potential which leads to a natural definition of variables defined on edges and triangles of the cellular decomposition as well as on faces and links of its dual. We use a specific triangulation of the 3-sphere in order to identify the constraints. We show also how to recover the standard BF discretization, starting from the BFCG variables: in a sense we can "cancel" the boundary term transforming BF into BFCG in the discretization process by a proper choice of polarization.
In section 5, we discuss how the 2-gauge theory structure applies to the current context and how we have different 2-groups at hand, according to the choice of boundary term, or polarization.

4d BF Theory
Action We consider the Lie group D which has a pair of d-dimensional Lie subgroups G and G * , with G * abelian, such that D ∼ = G G * ∼ = G * G.
We denote by g and g * the Lie algebra of G and G * with generators e i and e * i for i ∈ {1 . . . d}. The 2d dimensional Lie algebra of D is noted d ∼ = g g * ∼ = g * g, with Lie brackets where index summation notation is used here and for the rest of this paper. The Lie bracket structure can be translated into an action of g on g * .
with the action given by hence the Lie bracket can be written as There is a natural pairing between g and g * which is compatible with the Lie algebra brackets of g and g * that extends to d.
This pairing gives the relation d is called the (Drinfeld) double.
Let B be a g * valued 2-form and let A be a g valued 1-form. The curvature of A is denoted F and is defined as The BF theory on the four dimensional manifold M is defined by the action Symmetries The symmetries of the action (10) are given by the gauge transformation parametrized by the group element G ∈ G, and the translation/shift symmetries parametrized by a g * valued 1-form η.
Gauge transformations: Shift transformations: where The action is invariant under the gauge transformation due to the invariance of ·, · given in (8). Under the shift transformation, the action is invariant up to a boundary term, where we used the Bianchi identity, d A F = 0. As we will recall in the next section, the shift symmetry can be interpreted as a 2-gauge transformation.
Equations of motion and potential The equations of motion associated with S G are obtained by varying the fields, A and B: The first two terms are the equations of motion: The third term in (14) is the symplectic potential and is responsible for the Poisson structure. Denoting the boundary of M by ∂M := M , the potential is The associated symplectic 2-form is We note that the equations of motion imply two other sets of condition, namely The first relation is the standard Bianchi identity. The condition [F, B] = 0 is weaker than the flatness constraint.
Charges/momentum maps. We consider the infinitesimal versions of the gauge transformations and want to identify the charges generating such transformations. Let the infinitesimal version of (11) be parametrized by χ ∈ g, and the infinitesimal version of the shift transformation (12) be parametrized by β ∈ g * , Infinitesimal gauge transformations: Infinitesimal shift transformations: We think of δ χ and δ β as vector fields in the field space. We use notation such as δ χ δA = δ χ A to express the interior product of a vector and a 1-form in field space. We define the charges as Some manipulation reveals (provided we assume δα = 0 and δβ = 0, ie the parameters are not field dependent) where ≈ means we went on-shell. We note that the charges are essentially given by the corner charges specified by the variables B and A, a 2-form and a 1-form respectively. When the parameters are constant, we will call the charges global. We have used in each case the pull-back to M of the equations of motion. These pull-backs are also interpreted as constraints. A momentum map is a function on phase space generating the symmetry transformations. As such the constraints are momentum maps. Table 1: Summarizing the charges, momentum maps and associated symmetries.
To anticipate a bit our results, we note that the global charges J and P are simply generated by the configuration/momentum variables. This implies that where the phase space variables are discretized will directly influence how the charges will be discretized. This means that we might get different types of symmetry structure (1-group versus 2-group) according to the choice of discretization of the variables.
The subalgebra R 4 is generated by P µ and the rotation algebra so(4) is generated by J µν . Greek indices range from 0 to 3. The Lie brackets are where η is the flat metric. We will sometimes write [J µν , J σρ ] = f µνσρ αβ J αβ where f is the structure constant of so (4). The bilinear pairing is where the generators in the dual are distinguished by lowered indices and an asterix. The second term in the second pairing is to account for the antisymmetry of the indices. We will identify the subspace generated by P * as R 4 * , the subspace generated by J * as so(4) * , and iso * (4) ∼ = R 10 .
The Lie brackets of the double is constructed according to 3 (7) The non-zero brackets [P µ , P * ν ] in (30) will play an important role in the discretization procedure. We decompose the fields A and B into their rotation and translation components: The curvature is also rewritten in terms of its projections into subalgebras.
Hence the action is The quantity G := d A Σ is called the 2-curvature. The BFCG action is obtained from S ISO(4) up to a boundary term [32,31], Equations of motion and potential The equations of motion can be determined by varying S BF CG with respect to A, B, C, and Σ. Alternatively, since a boundary term in the action does not change the equations of motion we can decompose the expression in (15) into translation and rotation components. Either way, we get We remind the reader that though C and Σ are both elements of a Lie subalgebra with trivial brackets, the bracket between them is given by (30) which is not zero but in so * (4). Similarly the potential is decomposed The potential from S BF CG is slightly different, The two potentials differ by a total functional derivative and therefore give the same 2-form 4 : We note that we also have the analogue of the Bianchi identity and its companion (18) which are still valid.
Breaking them into components we have Symmetries Now let's review the symmetries introduced in the previous section. We have now different names as the fields are interpreted in the 2-gauge theory picture [22].
Gauge : Shift : Charge and momentum maps. As before, we define the charges 5 After some algebra, we find that, still assuming a non-dependence of the parameters in terms of the fields, If the coefficients ζ, X, Y, α are constant on the boundary, we deal then with "global" charges. There is then no central extension in their Poisson algebra (see Appendix A). As before we have a set of constraints, which are the pull-back of the equations of motion to M .
These constraints are momentum maps generating the symmetry transformations, we list them in table 2. Once again anticipating, we can see that that the discretized symmetries will different whether we consider the pair (A, C) (ie A) discretized on the dual complex as would be done for regular BF theory (hence obtaining a 1-gauge theory), or if we consider the pair (A, Σ) discretized on the dual complex as would be done for the BFCG theory (hence considering a 2-gauge theory).
While the continuum theories are equivalent up to a boundary term, the discrete symmetry will be different.

.1 Notation
Let us set up the notations for the cellular decomposition once and for all.
We divide the spatial slice M into subregions forming a cellular decomposition. The 3 dimensional cells will be tetrahedra for simplicity, but other cases can be considered as well. Within each tetrahedron we identify a center point c, which we refer to as a node. The tetrahedron dual to c is denoted c * . The oriented segment between two nodes, c and c is denoted by the ordered pair (cc ), which we will call a link. The ordering of the nodes determines the orientation of the link. We refer to the first node in the pair (cc ) as the source and the second node as the target. The vertices v of tetrahedra are denoted with an overline. The edges between two vertices, v and v is then denoted [vv ]. For each vertex v, there is a set of tetrahedra containing v. The centers of these tetrahedra generate a polyhedron in the dual cellular complex. This polyhedron will be denoted v * . The faces of the polyhedron, called dual faces are labelled by the edge it intersects, [vv ] * or by the set of nodes it contains. Similarly, the triangles making up the surface of the tetrahedra are either labelled by the three vertices they contain such as [v 1 v 2 v 3 ] or by the link intercepting it, (cc ) * . Some of the structures are shown in figure 1.

Recovering the standard discretization of 4d Euclidean BF theory
We use the discretization approach that has been used in several works [20,14,15]. Restricting the fields to subregions First, let's consider the symplectic potential of BF theory The second equality makes it explicit that we are dividing M into 3-cells c * . Furthermore, we label the restrictions of the fields A and B within each cell with a subscript. The fields in neighbouring 3-cells will be related through some continuity relations on the boundary of the cell.
Truncation. Until now we have only rewritten the symplectic potential to account for the fact we broke up M into subregions. The next step is to truncate the theory, by going on shell in the interior of each cell. Since we are dealing with a topological theory, we could argue that we push all curvature defects or "torsion" defect (ie such that d A B = 0 to respectively the edges and the vertices of the triangulation. Then we should regularize properly them and treat them accordingly. This would be going beyond the scope of the present paper, so we just assume that there are no such defects at all and leave their study for later investigations. The equations of motion (15) imply that A is a flat connection and therefore pure gauge. For an ISO(4) group element H c (x), interpreted as an ISO(4) holonomy connecting c to a point x in the cell, is a solution for F = 0. Then, for a iso(4) * valued 1-form χ, is a solution to d A B = 0. Hence both A c and B c are pure gauge.
Continuity equations. As mentioned above, the fields inside each cell c may be considered separately so long as there is continuity between cells. This puts conditions on the fields H and χ.
The continuity relations for A and B on the interior of the triangle shared by c * and c * are expressed as where G cc ∈ ISO(4) does not depend on x. We denote the inverse G −1 cc = G c c . The above continuity equations are valid on the interior of the triangle. If there is no curvature concentrated on the edges (we will assume this later on), the equations would be valid there as well.
We emphasize that we also have the induced equation [F, B] = 0 and the Bianchi identity d A F = 0. We should also assess how they can be realized in the truncated scheme.
Since they are expressed in terms of the notion of curvature, they should be obtained by considering several continuity relations concatenated together to generate a loop. To this aim, let us consider the loop ∂e * , which is the boundary of the dual face e * (dual to the edge e). This loop can be described by the links relating the nodes (c i c i+1 ).
The version of the condition [F, B] = 0 in terms of continuity equation is then As we could infer that the curvature is given in terms of the holonomy around the loop ∂e * , the Bianchi identity is naturally obtained by demanding that the (dual) polyhedron made of loops ∂e * i is closed so that We will not use this constraint in the BF discretization. However we demand as a requirement for the fields χ c and G cici+1 to satisfy (59).
Evaluation of the symplectic potential. In order to write the potential in terms of H and χ, we should express the variation δA in terms of H: The potential evaluated on-shell in the cells reads where ≈ means we went on-shell in the cells. We can then use the continuity equations (57) and (58) to simplify its expression and recover the well-known results.
We note that the integrand in (62) is a total derivative so we can use Stokes theorem to recast it as an integral over triangles bounding each tetrahedron. However, there is a choice to be made as to which variable keeps the derivative when dealing with the integral on the boundary. A similar choice arises when dealing with (3d) gravity, and we have the LQG or dual LQG picture [14]. For now we will deal with the case where the derivative is kept on the 1-form χ.
The standard discretization of the 4d BF theory is summarized in the following proposition. Proposition 1. The symplectic potential is given as a sum of symplectic potentials associated to the phase space T * ISO(4).
which we construct from the solutions of the continuity equations (57) and (58), Table 3 provides the geometric structure which the discretized fields are attached to. These discrete variables satisfy by definition two kinds of constraints, the so-called Gauss and "face simplicity" constraints

Link (cc )
Dual face e * Edges e Triangles (cc ) * G cc ∈ ISO(4) --β (cc ) * ∈ iso * (4) where in the second constraint, the loop being considered begins and ends at the node c. Furthermore, if we assume there is no curvature, then we have the flatness constraint and the discretized Bianchi identity.
We note that the flatness constraint implies the face simplicity as well as the discretized Bianchi identity (as it should).
Proof. Let us evaluate the symplectic potential with the given choice of application of Stokes theorem.
The boundary of the tetrahedra c * is made up of four triangles. Since each triangle is shared by two tetrahedra, the contribution to the potential from each triangle contains two terms with a relative minus sign to account for the opposite orientation: The last equality is obtained by using the continuity equations and defining ∆G c c = δG cc G c c . We can identify the factors with structures of the cellular decomposition and its dual graph. We define discrete variables G (cc ) = G cc to be the discrete variable associated to the link (cc ) and β (cc ) * = (cc ) * dχ c is the discrete variable associated to the triangle (cc ) * . Thus, as a function of discrete variables, the potential is From this potential, we can determine the Poisson brackets, which are the canonical ones associated with the cotangent bundle T * ISO(4). We will review this in the following section.
Gauss constraint. By construction the phase space variables satisfy some constraints. For a given tetrahedron, if we perform the sum over the triangles by Stokes theorem. This constraint is the discretization of the (pull-back of the) continuum constraint In order to accommodate the different possible orientations of the links connecting c * to its neighbours, we point out that the base point of the variable β can be changed according to Face simplicity. Since we have by the continuity equations that where the product of links begins and ends on the node c, we can just perform the integration over (cc ) * and get the face simplicity constraint.
Flatness constraint. The definition of the discretized field does not imply that the holonomies G cici+1 should be flat, so we implement it by hand.
This constraint is the discretization of the (pull-back of the) continuum constraint F = 0. One can check that they generate the discretized version of the BF symmetries. We note that this is a non-abelian group valued momentum map [1].
Bianchi identity. This condition is naturally discretized by demanding that concatenating all the holonomies on the dual faces of a (dual) polyhedron v * (as dual to a vertex v) gives the identity. This is automatically satisfied if each face is flat. The constraint then reads for every dual polyhedron v e with faces e * , e * ∈∂v * G e * = 1. (77)

Relativistic spinning top phase space
We can decompose the discrete variables into subalgebra components. For simplicity we will omit the indices (cc ). The iso(4) * holonomy is decomposed as β = b + V for b ∈ so(4) * and V ∈ R 4 * . This is equivalent to decomposing the continuous variable as dχ c = db c + dσ c for b c ∈ so(4) * and σ c ∈ R 4 * . We write the ISO(4) holonomy as G = e x h where h ∈ SO(4) and x is a constant element of the Lie algebra of R 4 (which also happens to be R 4 ). Since R 4 is abelian, in calculations we use the representation such that 6 e x = 1 + x. The variation then reads ∆G = δx + ∆h + [x, ∆h]. Associated to a link (cc ), the potential is This is the symplectic potential for the phase space T * ISO(4). The Poisson brackets corresponding to this potential are then where f is the structure constant of so(4): The discrete variables are summarized in table 4 and Fig. 2. 6 We use the following representation highlighting that the abelian group product of R 4 is isomorphic to the the Lie algebra R 4 seen as an abelian group. The generators P of the Lie algebra R 4 are such that P µ P ν = 0. As a consequence the group element is e P µ = 1 + P µ . We recover in this way the addition as the product of the group R 4 , since e p·P e q·P = (1 + p · P )(1 + q · P ) = 1 + p · P + q · P = 1 + (p + q) · P = e (p+q)·P .
Note that the phase space we have recovered is the isomorphic to the phase space of the relativistic spinning top [25], as one could expect. We define a new variable S = b + [V, x]. The potential (for a given link/face pair) now reads The Poisson brackets of these new variables are those of the relativistic spinning top: Constraints/charges. The total phase space we have obtained for the triangulation can therefore be interpreted as a set of relativistic spinning tops, which satisfy some constraints. This is consistent with Penrose's idea of spin networks [34]. We have the constraints encoding the conservation of (relativistic) angular momentum The curvature constraints are given by To get the components of G e , we write G cc = e x (cc ) h cc .
where n in the above is the total number of nodes around the edge e and we define c n+1 = c 1 .
Finally we also have the Bianchi identity for F and its companion. The Bianchi identity d A F = 0 implies the constraints The face simplicity constraints follows from the constraint d A B = 0:

Discretization of BFCG theory
In this section we will go through the same procedure starting from the BFCG potential. To illustrate that the theory is similar to the ISO(4) BF theory, we will first show how to recover the results of the previous subsection (the discrete potential (78)). We will then proceed and obtain the proper discretized BFCG potential, the main result of this section. As a consequence we will recover the classical picture behind the G-networks introduced in [2].
Restricting the fields to subregions. We start by expressing the integral Θ BF CG as a sum of integrals over each cell.
Truncation. As before, we go on-shell inside the cells. Since BFCG and BF theory differ by a boundary term, they share the same equations of motion so we can either decompose the solutions of (15) or directly solve (38). We will take the former approach: we solve the equations of motion from BF theory to get, for a ISO(4) holonomy H c (x) connecting c to x in the cell and a iso(4) * valued 1-form χ, We decompose H = e c g, where g is a rotation and e c is a translation. χ is decomposed into χ = b + σ for b ∈ so(4) * and σ ∈ R 4 * . Thus, (still using the convenient representation e c = 1 + c) giving where the different fields are defined in table 5.
Continuity equations. The continuity of the field between neighbouring cells c * and c * is expressed as The solutions of these continuity equations are given in Table 5.
If we apply the continuity equations consecutively around a loop ∂e * , we also get the equations where again, the product along the loop of links begins and ends at node c. The condition (97) can be seen as the discretization of [F, C] = 0, while the second ones (98) come from d A B = 0. The solutions of the continuity equations are given in the table 5. Anticipating a bit, we will see that in the BFCG discretization, Continuity eqs Solutions to continuity eqs Fields y c c function in so(4) * if we allow for some curvature on the edges, we will still need to assume that on the edges of the triangulation This will ensure that we can integrate the symplectic potential. While in the construction given in [2] particular emphasis was put on the edge simplicity, that is something of the type to recover the KBF amplitude, it seems that for the discretization, the key property will be more (99).
Evaluating the symplectic potential: before making the choice. In order to express the potential in terms of the fields g, σ, b, and c we need to express the variation of the fields A and Σ in these variables. We define then, Using these expressions, the potential in a cell is where ≈ means we went on-shell. We see that Θ c is a total derivative and can be written as an integral over the boundary ∂c * by Stokes theorem. As in the previous section, there is a choice to make regarding which variable keeps the derivative when we perform Stokes' theorem.

Choice 1: recovering the BF discretization
At this time, we make the following choice (we note that the first term is the same polarization as the LQG case [14]).
The boundary ∂c * is made up of four triangles. Each triangle is shared by two tetrahedra. The contribution from each triangle, (cc ) * , is Θ (cc ) * , Proposition 2. The symplectic potential is given as a sum of symplectic potential associated to the phase space T * ISO(4).
where the discrete variables are obtained from the continuity equations from table 5, Table 6 provides the geometric structure which they are attached to.

Links Dual faces Edges
Triangles We note that we almost recover the same potential as in the BF standard discretization (78). The difference comes from a minus sign in the (x, V ) sector. The reason is the following. By adding the boundary term to go to the BFCG action, we have swapped the polarization, we have exchanged the configuration and momentum variables. If we consider a symplectic form δq ∧ δp, this form is not invariant under the exchange q ↔ p, which leads to δp ∧ δq = −δq ∧ δp. This is what we have done by changing the polarization. The symplectic transformation is instead given by q → − p and p → q. Hence the V in the BF discretization and the V in the BFCG discretization are related by a minus sign.
Proof. We can use the continuity relations from table 5 to simplify Θ (cc ) * .
where we have defined ∆h c c := δh cc h c c . The total potential is now The factors in Θ BF CG can be associated to structures in the cellular decomposition. We already saw that h cc is related to the the links in the dual cellular decomposition. Similarly x c c is also associated to the links. The factors involving integrals over triangles are associated to the triangles in the cellular decomposition, which are dual to the links. The discrete variables and where they live in the cellular decomposition is summarized in table 6 which is the same as table 3.

Choice 2: recovering the phase space of [2]
As we emphasized already, when using Stokes' theorem, there is a choice which is made on which variable will keep the differential. We recall that we obtained the symplectic potential for a given tetrahedron c * , In the previous section we used Stokes' theorem to write the potential on the triangles bounding c * . In particular, the last term was expressed as and when we determined the discrete variables, (cc ) * dσ was assigned to a triangular face. We can alternatively write this term as In performing Stokes' theorem in this way, we will have different discrete variables living on different structures in the cellular decomposition. Let us first identify the discretized variables and then the constraints associated to them.

Identifying the discrete variables.
After performing Stokes theorem and applying the continuity equations the potential on the triangles is now The last term is indeed a problematic one. In contrast to the previous section, neither dc c nor δς c c are constant and so we must do some work to perform this integration. We shall once again use Stokes' theorem on each triangle and deal with integrals over edges bounding triangles instead.
The first sum and the integral are over edges e. The second sum is over the links (cc ) which make up the polygon e * dual to the e. The factor e (cc ) is either 1 or −1, depending on whether the orientation of (cc ) * is aligned with e or not.
To illustrate we take the example edge we have in figure 3. The contribution of this edge to the potential is In the second line, we were able to use the continuity equation in the variable c to base each term at the center 1 (an arbitrary choice). The second and third term involve something proportional to ∆h 2 1 and ∆h 3 1 (the value of the superscript and subscript are a result of the arbitrary choice made to base everything at 1) and can therefore be absorbed in the first term of (114) (since the total potential involves summing over the links). This will be the source of a non-trivial closure constraint for the tetrahedron.
The first term involves a combination of the continuity variables ς.
Here, as we alluded earlier, we need to take a specific assumption on the behaviour of σ c under consecutive change of frames, as in (99).
Consider the three continuity equations for σ which are satisfied on e = (12) * ∩ (23) * ∩ (31) * : Putting these together we have and putting together (121) and (122), we get And so we have that for some constant V e * 1 , which then decorates the dual face e * . This is exactly the expression which appears in the first term of (119). Since V e * is a constant, we are able to consider e dc 1 as our discrete variable associated to e and V e * 1 as the discrete variable associated to the polygon e * . The potential due to e is then Summarizing, the symplectic potential takes now the shape A lot has been concealed in writing equation (126). The label c e is the choice of base point in the polygon dual to e (in the example edge we took c e to be the node 1). We also introducedb c c . Simply put, this is shorthand notation for everything which appears in Θ next to ∆h c c . The explicit form of such a term depends on the choices of c e and so we won't write it out in general. We will defineb c c in an explicit example shortly.  Table 7: Summary of discretization of BFCG theory. The key result is that b (cc ) * depends on many variables, namely c, ς and b, as illustrated in (130). In particular, we have integrations both on the triangle and some of the edges forming its boundary.
The associated forms are
ms are  Figure 4: A link in red is decorated by a SO(4) holonomy, a dual face is decorated by an element in R 4 * ∼ = R 4 . An edge in blue is decorated by an element in R 4 , the triangle in blue is decorated by a so * (4) ∼ = R 6 element. The building blocks to construct the discrete phase space are still isomorphic to T * ISO(4) since We can now determine the discrete variables. The discrete variables are b (cc ) * = (cc ) * b c c ∈ so(4) * on triangles, h cc ∈ SO(4) on links dual to triangles, e = e dc ce ∈ R 4 on edges, and V e * = V e * ce on polygons dual to an edge. The variables are summarized in table 7 and Fig. 4.
The discrete potential looks just like that of (82), but with the translation sector on the edges and dual faces instead of the links and triangles.
For each edge e of the triangulation we have the decorations e and V e * which decorate respectively the edge and the dual face, with Poisson bracket (127). For each link l, we have the decorations h l and b l which decorate respectively the link and the triangle l * , with Poisson brackets (128).
These variables satisfy a set of constraints different than in the ones in the BF discretization. In order to justify these constraints and see that they follow from the definitions of the discrete variables in terms of the continuous functions, we need to explicitly defineb c c and make choices about where to base the edge variables. In order to simplify these expressions and be exhaustive, we will consider an explicit example.

Explicit case: example of the sphere
The triangulation we choose is that of the 4-simplex. The space is divided into 5 tetrahedra. The centers of the 5 tetrahedra will be labelled by integers {1, 2, . . . , 5}. The vertices of the tetrahedra will be labelled  figure 5. The calculation on the edges is just like in the example we did previously since each edge is dual to a triangle. The resulting dual face variables are In the above, we have always based the variables at the lowest node in numerical order. This is a choice made arbitrarily, any node which is a vertex of [ij] * would be equally valid. The resulting expressions for the so(4) * variables are: [dc 1 , h 12 ς 3 2 h 21 ] + [34] [dc 1 , h 12 ς 5 [12] [dc 3 , h 34 ς 5 [34] [dc 5 , ς 1 5 ] − [23] [dc 5 , ς [dc 4 , ς 2 4 ] − [31] [dc 4 , ς 5 4 ] + [31] [dc 4  [dc 5 , ς 3 5 ] − [12] [dc 5 , ς 3 5 ] Clearly there is a lack of symmetry due to the orientation choices for the links.

Constraints
The discrete variables in this new polarisation give rise to a new set of constraints. We follow the terminology of [2] to name the constraints.
Compared to the BF case, there are two sets of new constraints, the 2-Gauss constraints encoding the triangles are closed and the 2-flatness encoding that the dual polyhedra close.
We then have the more usual set of constraints, the 1-Gauss constraints and the 1-flatness constraints. The latter encodes that the holonomies along the links forming a closed loops should be trivial. The former encodes that the triangle decoration should be equal to a specific quantity.
Finally, there is also the edge simplicity constraints. This constraint is actually implied if we have flatness, but can hold without having flatness.
Let us review the explicit shape of the constraints.
Such constraint is the discrete analogue of the constraint d A C = 0, since In general, the edge variables corresponding to the edges of a triangle sum to zero. The sum can only be performed after each variable is transported to the appropriate node. The full list of triangle constraints in the 4-simplex are given below The relative sign differences between terms comes from the e (cc ) factor introduced above.

2-flatness.
This set of constraints is interpreted as the closure of the dual polyhedra (tetrahedra in our illustrating example). By Minkowski's theorem, the sum of vectors normal to faces of a polyhedron with magnitude equal to the face area is zero. In this system, the analogous quantities are the variables on the faces, V . For example the closure of the polyhedron dual to 1 is This follows directly from the definitions of the face variables (also using V [ij] = −V [ji] ). In our example, there are five such polyhedron constraints shown below 1-Gauss. Due to the less trivial continuity equations for the so(4) * variables, the expression for the tetrahedron constraints appears more cumbersome: The explicit form of the sum depends on where we choose to base our variables and V as well as the orientation of the links. The first sum is a sum over the links which have the source point at c and the third sum is over links which have their target node at c. The second sum is over the edges such that e and V e * is based at c. If no is based at c, this is just the empty sum. With the variables defined above, we have the five constraints We emphasize that these constraints are realized by definition of the fields.We expect them to be the discretization of the constraint d A B + [C ∧ Σ] = 0.
We have used the observation that the four surfaces we are integrating over in the first four terms of the first line are the boundary of the tetrahedron 1 * . We then use Stokes theorem to write it as derivative in the bulk of the tetrahedron. We can then use that d 2 = 0 to say that the first integral after the second equality vanishes. Next we write out the integral ∂[125] and ∂[245] as integrals over edges. We then collect terms: which is precisely T 3 = 0.
Edge simplicity and 1-flatness Now let's consider the edge simplicity constraint. We consider this condition instead of the usual flatness constraint because it appears as the less strict condition we must impose in order to discretize the potential.
The edge simplicity constraint for a given edge e follows directly from the property of the fields we introduced in (97). By integrating this expression on the edge, we just get This constraint can be viewed as the discretization of the constraint [F ∧ C] = 0.
Similar constraints hold when V replaces , which is inherited from (122). This can be seen as a dual face simplicity.
The 1-flatness constraint appears by constraining the constants h cc to form a flat closed holonomy along the loop ∂e * .

2-group structures
In this section, we put our findings in light of the higher gauge theory context and its discretization. We first recall quickly the concept of higher gauge theory mostly relying on [6].

From 2-group to higher gauge theory in a nutshell
There is a natural 2-group interpretation for the discretized symmetries we have obtained. We recall that a (strict) 2-group can be seen as a crossed module, (G, H, t, ), which consists in a pair of (Lie) groups (G, H), with a group homomorphism t : H → G called the target map and an action of G on H. The target map and the action must satisfy some compatibly relations.
The crossed module is equipped with two product laws. The first one is inherited from considering H > G.
The other multiplication comes from the product of H.
We note that demanding that the t map is trivial, the vertical composition implies that the holonomy g 2 g −1 1 must be flat.
The notion of 2-Lie algebra which can be exponentiated to the Lie 2-group [5] is relevant to discuss the BF or BFCG theories.
Equipped with the notion of Lie 2-group and Lie 2-algebra, we can define the notion of higher gauge theory [4]. It is specified in terms of a 1-connection, the usual gauge connection, denoted A, a 1-form with value in Lie G, and a 2-connection, often noted B or Σ, a 2-form with value in Lie H. Together with these connections, we get the associated curvatures. We have the 1-curvature, which is the usual curvature found in gauge theory, F (A) = dA+ 1 2 [A∧A]. We also have the 2-curvature which is given by G(Σ, A) = dΣ+α(A)(Σ), where we used the action of Lie G on Lie H. We note that the 1-curvature is actually is specified in terms of τ (Σ), Finally, we have the 1-gauge transformations and the 2-gauge transformations, parameterized by a group element g ∈ G and a 1-form a ∈ LieH respectively.

BF case.
The four dimensional BF action can related to higher gauge symmetries, more specifically to the (co-)tangent 2-group, and at the continuum level to the associated (co-)tangent 2-Lie algebra [6].

Continuum level and 2-Lie algebra
The B-field, a 2-form with value in g * ∼ R d is usually interpreted as a Lagrange multiplier implementing the fact we are dealing with a flat connection. In the higher gauge theory picture, we interpret it instead as a 2-connection, while A with value in g is the 1-connection. The relevant 2-Lie group is given by the (co-)tangent 2-group, with H = Lie G * ∼ R d , G, with G acting with the coadjoint action on Lie G * and the t-map is constant, t = 1. The connections are valued in the associated Lie 2-algebras.
The translational symmetry of the BF action is then interpreted as the 2-gauge transformation.

Discrete level and 2-Lie group
While the relevant 2-group is (co-)tangent 2-group, we have obtained at the discrete level that this 2-group can actually be seen as a pair of trivial 2-groups, ie a pair of 1-groups.
Firstly on the dual complex, we have solely a decoration of the links, by the group elements in G. There are no decoration on the dual faces. This means that the 2-group is actually trivial, we have 8 We note that the target map being trivial is also equivalent to the dynamical constraint that the ISO(4) holonomy on the links must be flat.
Secondly, on the triangulation, we have no decoration on the edges (so we can set equivalently any decoration to the identity), but the triangles decorated by the abelian group R 10 ∼ iso(4) * .
Finally, the two (trivial) 2-groups can be seen as dual to each other, one being configuration, the other one momentum space, and together form the (co-)tangent 2-group. Put together, we can set up a phase space for each (link, trivial face decoration)/(trivial edge decoration, triangle), with symplectic form given by (78).
Let us reformulate the previous statement. Locally the cotangent bundle T * G ∼ R d G has a simple cross module structure (with a trivial t-map) hence can be interpreted as a 2-group. Since the cotangent bundle T * G is naturally equipped with a symplectic form, the 2-group is also equipped with a symplectic form. This points towards a possible generalization of the Heisenberg double [36] to the 2-group context.

Continuum level and 2-Lie algebra
While the BFCG action we considered is actually equivalent to a BF action, the 2-gauge structure is actually different. Now the 2-connection is not given by the full B = B + Σ field with value in Lie G ∼ iso(4) * , but by the translational sector Σ only. The B-field component with value in so(4) * is now seen as a Lagrange multiplier. The 1-gauge connection is not the full connection A = A + C but is A lying in the so(4) sector while the C component is seen as a Lagrange multiplier. As a consequence this means that the relevant gauge symmetry is the Euclidean 2-group SO(4) R 4 . While this might look as a cosmetic change with respect to the original Euclidean BF theory, it is actually deeper than that. Indeed, the addition of the boundary term modifying the Euclidean BF theory into the BFCG theory can be seen as dualization, more exactly semi-dualization, since we only dualize "half" of the Lie algebra iso(4), namely the translational part. While the charge algebra is not modified, the place where these charges will be discretized is modified.
Said otherwise, in the BF formulation, the B-field, the 2-connection, is seen solely as the momentum, and the 1-connection is solely configuration variable. In contrast, in the BFCG case, we have that the momentum variables are a mix of 1-connection C (in R 4 ) and 2-form B (in so(4) * ), while the configuration variables are given in terms of the 1-connection A (in so (4)) and the 2-connection Σ in R 4 .
Nevertheless due to the equivalence with the Euclidean BF theory, we still have the (co)-tangent 2-group structure present. To see how they still connect together we can move to the discrete picture.

Discrete level and 2-Lie groups
Firstly, on the dual complex, we have SO(4) holonomies decorating on the links and elements in R 4 decorating the dual faces. We recognize this as the Euclidean 2-group.
Once again the target map being trivial is also equivalent to the dynamical constraint stating that the SO(4) holonomy on the links must be flat.
Secondly, on the triangulation, we have the edges are decorated by elements in R 4 * ∼ R 4 , and the triangles decorated by the abelian group R 6 ∼ so(4) * . Hence we have the trivial cross module Finally, once again, the two 2-groups can be seen as dual to each other. Put together, we can set up a phase space for each (link, face )/(edge, triangle), with symplectic form given by (126). The total phase space is again based on T * ISO(4). This is where the equivalence with the usual Euclidean BF formulation appears. As we recalled in the previous section T * ISO(4) can be seen as a cross module equipped with a symplectic structure. Unlike in the BF case, now the components of this 2-group are themselves non-trivial 2-groups in the sense that both contain decoration on the face/triangle. This puts further arguments that the generalization of a 2-Heisenberg double, defined as a 2-group seen as a phase space with both configuration and momentum spaces being 2-groups might exist.

Outlook
We have revisited the discretization of 4d BF theory and highlighted that this discretization (and in turns the quantum theory) is very sensitive on the choice of boundary terms. This is yet another example of importance such boundary data to probe the quantum regime of a gauge theory [18,19,17]. Here the boundary term just implemented a partial change of polarization which allowed to rewrite the Euclidean BF theory as the BF CG theory. It was already known that a full change of polarization leads at the discretized level to the notion of "BF vacuum" [13], so that the kinematical symmetries are not the Lorentz ones (through the Gauss constraint) but instead the translational ones (through the flatness constraint).
Here we performed only a partial change of polarization, in the case where the gauge group is not the Lorentz group but the Euclidean group. At the level of the action, this implemented a change from a regular BF action to the BF CG action. At the discrete level, one gets the usual phase space built out of T * ISO(4) for each link of the dual 1-complex, which is the classical phase space behind the standard notion of spin networks. With the partial change of polarization, we obtain the phase space of the so-called G-networks, This is just changing the ribbon subcomponents order. At the geome Heisenberg double is now made of a pair of 2-grou (SO(3,1) ⋉ ℝ 4 ) ⋉ (SO*(3,1) × ℝ 4* ) → (SO(3,1) ⋉ ℝ 4* ) ⋉ (ℝ 4 g 1 C g 2 =g 1g2 = g 1 . g 2 g 1 g 2g 1 1 =g 2 , g 2 Bg 1 = g 1 g 2 =g 2 Cg 1 g 1 g 2g 1 1 =g 2 , g 2 = g 1 ( g 1 + g 2 g 2 ( g 1 )g 1 2 )g 1 The associated forms are  which are based on 2-group structures. We were therefore able to recover the expression of the G-network discrete variables in terms of the continuum variables.
Such identification is actually very important to relate gravity to the BF or BFCG theory. Indeed, gravity can be obtained at the continuum level by constraining the B field, through the simplicity constraints [32,31,11].
In the BFCG case, what stands for the discretized B field is actually a function of several of the continuous fields. Hence demanding that such discretized B field satisfies the simplicity constraint is not equivalent to the continuum simplicity constraint. This might explain why the discrete simplicity constraint leads to such drastic reduction of the model [32,31]. It would be interesting to see what is the actual relevant discretization of the simplicity constraint for the BFCG case, we leave this question for the future.
The Euclidean BF theory or the BFCG theory are equivalent up to a boundary term, so we expect them to have the same physical content. We note that each theory corresponds to different symmetries, both given in terms of 2-groups. In the BF case, we have a pair of trivial 2-groups, where one of the groups is the trivial group, so that the 2-group is really a 1-group (decorating the links or the triangles). In the BFCG case, we deal with the Euclidean 2-group and its dual. At the quantum level, amplitudes can be constructed in terms of the representations of each symmetry structure, be it the Euclidean 1-group or the Euclidean 2-group. Since the two descriptions describe the same theory, this means there must be some relation between the amplitudes expressed in terms of the different representation theories. Being able to identify explicitly the relation would be extremely interesting.
In fact some ideas on how to proceed could rely on the concept of semi-dualization [29,30]. The algebraic structure underlying the discretized BF structure, the cotangent bundle T * G, is the Drinfeld double d ∼ iso(4) < iso(4) * ∼ (R 4 > so(4)) < (R 4 * > so(4) * ) as a Lie algebra. To get to the BFCG formulation, we swapped the sector R 4 and R 4 * , which amounts to slicing d in a different manner. Still as a Lie algebra, the relevant Drinfeld double for the BFCG formulation is still d but now given by d ∼ (R 4 * > so(4)) < (R 4 > so(4) * ).
Let us recall quickly what is the semi-dualization. If we consider a pair of Lie algebras g i (equipped with some cocycle), acting on each other so that we can consider the big Lie algebra g 1 g 2 , then roughly the semi-dualization is defined via the map where encodes a co-action. However since we are dealing still with an abelian group R 4 , there is no kickback action and no co-action upon semi-dualization. Note however that the non trivial 2-group appeared thanks to the semi-dualization! We expect that a deformation of R 4 into a non abelian group such as AN (3) for example would lead to non trivial interesting structures related to Majid's bicrossproduct construction [29].
This later consideration brings us to a natural extension of the current work to include the cosmological constant. Instead of working with ISO(4), we could work with a SO(4, 1) BF theory. Under the semidualization we would expect to recover the Poincaré 2-group decorating the dual complex and a non-trivial 2-group on the triangulation given by the crossed module AN R 6 , which has a trivial t map. Interestingly to be consistent with the Poisson structure, these 2-groups should have some non-trivial Poisson structure (inherited from the coaction we just mentioned) so that they would be really quantum 2-groups under quantization. Preliminary studies point to the apparition of the κ-Poincaré group and the κ-Poincaré algebra. This is work in progress [23].
During our discussion, we did not consider excitations of any kind. We could decorate the vertices with 2-curvature excitation, or the edges with 1-curvature excitation. While a 4d BF theory is naturally coupled with string like excitations [7,8] it would be interesting to see how these are expressed when using the non-trivial 2-gauge picture, ie the BFCG formulation. We leave this for later investigations.
Finally, we have discussed that BF theory has two main discretizations, the standard one and the dual one where we swap the full group with the dual abelian group. We argued that we can do a partial dualization where we swap only the translational sector to recover the BFCG discretization. There is therefore a fourth case, where we do a partial dualization on the Lorentz sector. This would amount to a "dual" BFCG discretization. We leave this case for later study.

A.2 BFCG case
We note that these charges follow directly from decomposing the fields in the charges of BF theory into the subalgebra components. The resulting Poisson brackets are We find again the central extension The Poisson brackets form a closed algebra.