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Deformed Spinor Networks for Loop Gravity: Towards Hyperbolic Twisted Geometries

In the context of a canonical quantization of general relativity, one can deform the loop gravity phase space on a graph by replacing the T*SU(2) phase space attached to each edge by SL(2,C) seen as a phase space. This deformation is supposed to encode the presence of a non-zero cosmological constant. Here we show how to parametrize this phase space in terms of spinor variables, thus obtaining deformed spinor networks for loop gravity, with a deformed action of the gauge group SU(2) at the vertices. These are to be formally interpreted as the generalization of loop gravity twisted geometries to a hyperbolic curvature.


Introduction
Loop quantum gravity is an approach to quantum gravity, based on a canonical quantization of general relativity formulated as a gauge field theory (typically SU (2) or SL(2, C) depending on the precise formulation). It defines quantum states of geometry and encodes the theory's dynamics in quantum Hamiltonian constraints generating spacetime diffeomorphisms. In this context, the cosmological constant Λ is a mere coupling constant (for the 3-volume term) entering the Hamiltonian.
There are nevertheless many claims that we could encode it in a quantum deformation of the gauge group. Such claims are backed up by rigorous analysis in 2+1-dimensional quantum gravity, for which there are clear relations between the cosmological constant and quantum group deformation (see e.g. [1] for a summary). These studies are mostly based on the Chern-Simons reformulation of 2+1d gravity, seen from the perspectives of path integral quantization [2,3] and Hamiltonian or combinatorial quantization [4][5][6][7], but also on the Turaev-Viro state-sum model [8] defining a topological spinfoam path integral for 3d quantum gravity based on U q (su (2)) and believed to account for a non-vanishing cosmological constant (based on the asymptotics of the quantum-deformed 6j-symbol) [9,10]. We refer also to [11] for the link between the spinfoam quantization and the combinatorial quantization and the more recent works [12][13][14][15] for attempts to relate the spinfoam framework and the canonical loop gravity quantization.
The previous work [16] showed how to deform at the classical level the loop gravity phase space (on a fixed graph) and analyzed in details its deformed symmetries. It was then shown in [17] that its quantization yields to spin networks states based on the quantum gauge group U q (su (2)). The results in [16] were based on the deformation of the T * SU(2) phase space identified as the (double cover of the) ISO(3) Poincaré group to the SL(2, C) phase space with Poisson brackets defined in terms of the classical r-matrix of sl 2 . The deformed loop gravity phase space on a given graph was then defined as many copies of this SL(2, C) phase space together with a deformed action of the SU(2) gauge group on the data at the graph vertices. We further discussed the geometrical interpretation of this deformed loop gravity phase space on (3-valent) graphs within the context of 3d gravity. More precisely, we proved that a graph dressed with SL(2, C) group elements along the edges, provided with appropriate gauge invariant SU (2) flatness constraints, could be interpreted as defining a discrete hyperbolic surface, built from hyperbolic triangles (dual to the graph vertices) glued together consistently within the 3d one-sheet-hyperboloid (defined as set of unit time-like vectors in 4d Minkowski space-time).
Here we introduce and investigate a reformulation of this hyperbolically-deformed phase space in terms of spinor variables. Indeed in standard loop quantum gravity (with Λ = 0), there has been a growing interest in the reformulation of the loop gravity phase space in terms of twisted geometries and spinor networks [18][19][20][21][22][23]. These tools brought a clarification of the geometrical meaning of the holonomy-flux observables on a graph at the classical level and of spin network states at the quantum level. A big advantage of this formalism is a straightforward quantization of the spinor variables parametrizing the phase space and a straightforward construction of coherent states of geometry [21,24,25].
They also led to new insights on spinfoam amplitudes (for the dynamics of spin networks in loop quantum gravity) and on the associated 3nj-symbols of spin recoupling, providing a new light on the recursion relations and generating functions for the amplitudes and allowing for some exact analytical evaluations [26][27][28][29]. This spinorial formalism also turned out to be relevant from a purely mathematical point of view, see e.g. [30].
We set on introducing a similar parametrization of the SL(2, C) phase space in terms of spinor variables. Given an oriented graph, we consider two spinors (complex 2-vectors) on each edge, one at each end of the edge. We will reconstruct the whole SL(2, C) phase space for these spinors, both the vector variables (the generalization of the fluxes), identified as triangular matrices, and the SU(2) holonomy along the edge. We will identify canonical spinor variables, as Darboux coordinates for the phase space, that do not transform simply under SU(2) transformations, and we will also define non-canonical spinors, that transform properly as spin-1 2 vectors for the fundamental representation of SU(2), from which we can directly reconstruct the vector variables as quadratic polynomials in their components.
The paper goes as follows. In a first section, we review the standard construction of the T * SU(2) ∼ ISO(3) phase space from spinor variables. We review in the second section the definition of the SL(2, C) phase space, parametrized in terms of triangular matrices and SU(2) group elements. The third and fourth sections introduce the new spinor variables and shows how to define the triangular matrices from them. The fifth section describes how to reconstruct the SU(2) group elements and thus how to recover the whole SL(2, C) phase space from the spinors. The sixth section summarizes the new spinorial parametrization of the hyperbolically-deformed loop gravity phase space and the deformed action of SU(2) transformations on all the variables. We conclude with a discussion on the possibilities opened by this formalism.

I. STANDARD SPINOR PHASE SPACE AND HOLONOMY RECONSTRUCTION
Let us consider a spinor |ζ ∈ C 2 and its conjugate ζ| ∈ C 2 , provided with the canonical Poisson brackets: We also introduce the dual spinor: We will also write N A = ζ AζA for the modulus of the spinor components for A = 0, 1 and N = N 0 + N 1 for their sum. It is easy to check that these generate dilatations on the complex variables: Considering the rank-1 Hermitian matrix X ≡ |ζ ζ|, one defines its projection onto the identity and the Pauli matrices σ i for i = 1..3: These satisfy the obvious equality X 0 = | X| and their Poisson brackets form a su(2) algebra: We can also decompose this in self-dual and anti-self-dual components: Note that [ζ| σ|ζ] = − ζ| σ|ζ hence the dual spinor |ζ] defines the opposite vector − X. The vector X actually generates SU(2) transformations on the spinor: These finite SU(2) transformations act on the spinor ζ as 2×2 matrices as in the fundamental representation while they simply induce 3d rotations on the vector X: where the spinor ζ and its dual both transform properly under SU(2) transformations. We can also give the details of the infinitesimal transformations of the spinor components: We now consider one oriented edge of a twisted geometry. We have two vectors at each end of the edge, X and X, and a SU(2) group element g. We assume the standard T * SU(2) Poisson brackets: Now we further impose the condition that the holonomy g sends one vector onto the other, which can be equivalent written in terms of the corresponding 2×2 Hermitian matrices X and X (see Fig. 1): The sign flip is important for the consistency of this condition with the Poisson brackets. Indeed it compensates the sign difference in the brackets of X and X with the holonomy g in (9) so that the brackets of this parallel transport condition with g, X and X vanish. This condition gXg −1 = X s can be re-written in terms of 3d Poincaré transformations or equivalently as ISO(3) ∼ SU(2) R 3 group elements represented as pairs consisting of a rotation and a translation : (g, X) = (I, X)(g, 0) = (g, 0)(I, − X) = (g, −g X) .
Noticing that the parallel transport condition gX = X s g does not entirely fix the holonomy g in terms of the two vectors X and X but leaves the possibility of an arbitrary 3d rotation around the X axis for instance. It is possible to replace this condition by a slightly stronger one using spinor variables. This parametrization of twisted geometries in terms of spinors allows to reconstruct both vectors variables X, X and SU(2) holonomies g as composite observables on the spinor phase space. These structures are called the spinor networks.
Starting with two spinors ζ andζ as above, we define the vectors X and X as previously: We assume the norm matching condition, ζ|ζ − ζ |ζ = 0, which implies that X and X have equal norms. We then define the holonomy g as the unique SU(2) group element mapping the C 2 orthonomal basis |ζ , |ζ] to |ζ], −|ζ : This properly implements the required parallel transport: Finally, it is straightforward to check that the components of this SU(2) group element weakly commute with each other assuming the norm-matching-condition: which provides us with the correct T * SU(2) Poisson structure.
Note that it is possible to avoid the sign flip of X in this formalism. Imposing gX = Xg would require switching the sign of the brackets { X, g} and { X, X}. Modifying the latter would create an asymmetry between the X and X sectors.
The SL(2, C) phase space is defined from the SL(2, C) group element D provided with the following Poisson bracket: with the standard convention D 1 = D ⊗ I and D 2 = I ⊗ D and the classical r-matrix: in terms of the Pauli matrices σ i and We are using here the usual notation for the tensor product of two 2×2 matrices as a 4×4 matrix: The key to our analysis of the phase space structure is the (left) Iwasawa decomposition SL(2, C) = SB(2, C) SU (2): The SL(2, C) bracket (16) reads in this parametrization: To get the full symplectic structure, one also computes the brackets with the complex conjugate variables † . For more details, the reader can refer to [16]. Defining l = †−1 , we get: This formula contains all the brackets between the components of and their complex conjugate. More precisely, we explicitly parametrize the triangular matrix as: and compute the Poisson algebra: Let us underline that these brackets are invariant under the exchange ←→¯ −1 , or explicitly (λ, z) ←→ (λ −1 , z).
Here we have used the left Iwasawa decomposition. We will also need the right Iwasawa decomposition, which corresponds to the left Iwasawa decomposition for the inverse of the SL(2, C) group element D: We use a slightly different convention from the one previously used in [16], to account for the sign flip. The Poisson brackets for the components of the inverse matrix are easy to compute: which amounts in the end to switching the classical r-matrix with its Hermitian conjugate. This leads to the following brackets between the triangular matrix and the SU(2) holonomy: We complete this algebra with the bracket with the complex conjugate of the triangular matrix,l =˜ †−1 : Parametrizing the triangular matrix˜ as before:˜ we get exactly the same brackets for the tilded sector as for the original sector: We can also calculate the Poisson brackets between the two different decompositions.
Our goal will be first to reproduce the Poisson brackets (27) from some spinor variables and then to reconstruct the whole SL(2, C) phase space, with in particular the SU(2) group elements u andũ, from those spinors.

III. NEW SPINOR VARIABLES AND DEFORMED ACTION OF ROTATIONS
We now define κ-deformed spinors |ζ κ , ζ κ | with components We do not change the definition of the N 's, keeping N κ A ≡ N A = ζ AζA for both A = 0, 1, but the norm ζ κ Aζ κ A of the deformed spinor has a N A -dependent factor: These κ-deformed spinors satisfy simple Poisson brackets among themselves We check the limit κ → 0 + when we should recover the undeformed quantities and we get as expected at leading order in the deformation parameter: Then, from these κ-deformed spinors, we build the lower triangular matrix ∈ SB(2, C) with It is easy to check that they satisfy the expected Poisson brackets, Let us point out that the components of the triangular matrix all commute with N = N 0 + N 1 : which simply expresses invariance of λ and z under U(1) phase transformation of the spinors |ζ κ → e iθ |ζ κ . An important remark to underline is that the parameter z is not holomorphic in the spinors ζ κ and that the notion of holomorphicity depends here on the considered variables (for instance, ζ or ζ κ or ).
Up to now, we have merely introduced a re-parametrization of the same phase space as in the undeformed case. The big difference will come in the action of the SU(2) transformations. As derived in [16], the infinitesimal variations generated by the (left) SU(2) rotations on the SL(2, C) phase space are given by the following re-scaled Poisson brackets with an arbitrary observable f : This comes from finite SU(2) transformations on triangular matrices: where v ∈ SU(2) is a priori different from v and compensates for the fact that triangular matrices are not stable under conjugation by SU(2) group elements. For infinitesimal transformations, these group elements read: We look for a new spinor t 1/2 , or in short t, that transforms covariantly under SU (2), that is that has the correct infinitesimal variations under left SU(2) rotations as a spin-1 2 vector: This is translated in the following required Poisson bracket identities: One can solve these equations explicitly and one finds two independent spinor solutions, which are simply a rescaled ζ κ spinor and its dual spinor: These new spinor components satisfy the following Poisson brackets: Let us underline that this algebra does not completely close since there are the e ± κ 2 N terms 1 . We compute the new rank-one Hermitian matrix |t t| and compare it to the matrix T = † : This means that the two spinors, |t and its dual |t], are the two eigenvectors of the Hermitian matrix T = † . We usually project the T matrix onto the Pauli matrices to get the SU(2)-covariant 3-vector T as follows: Moreover, due to the fact that belongs to SL(2, C) and that its determinant is normalized to one, we have a very simple expression for the inverse of the T -matrix: which we recognize as simply switching the sign of the 3-vector T and thus implementing the orientation flip introduced in the undeformed case. With these conventions, we can express the components of |t t| in terms of the vector T : We summarize the relation between the spinor t and the matrix T by the formulas: All these objects all transform as follows under SU(2) transformations: The 3-vector T could be seen as the analogue of the flux X in the flat case, given in (4). Indeed, the vector T appears naturally in the analysis of the deformed phase space in [16]. It allows to recover some information about the 1 We can obtain a closed algebra if we rescale our spinors t A ,t A by e − κ 4 N . Such rescaled components are then the exact classical analogue of the quantum spinor variables used in [13], provided we identify t 0 ≡ t − , t 1 ≡ t + and so on. discrete geometry, such as the hyperbolic cosine law or the flatness constraint [16]. However, it is not sufficient, we also need another vector T op to recover this information. T op is a vector that transforms under the deformed SU (2) transformations v as given in (38). To reconstruct T op , we need therefore to introduce another type of spinor, a "braided-covariant" spinor |τ , which transforms with v and not v under the SU(2) action: This allows to decompose this matrix and express it in terms of the 3-vector T op : And we give all the relations: Let us point out the useful identity relating the dual of both covariant and braided-covariant spinors: This new spinor clearly transforms under v : and one can check that we recover the correct infinitesimal variations as prescribed by the expression of the modified parameter given in (38).
Finally, let us conclude this section with the brackets between the components of the spinor τ and its complex conjugate:

IV. THE TILDED SECTOR -AN EXACT COPY OF THE STRAIGHT SECTOR
We repeat the same spinor construction for the tilded sector˜ since the Poisson algebra ofλ andz is exactly the same as the one of λ and z. We thus introduce a new spinorζ and its κ-deformed versionζ κ , from which we define the components of the new tilded triangular matrix˜ : These satisfy the same Poisson brackets as in (34): We define the new spinort as: The spinors live at the vertices of the ribbon. We can read off the mappings between spinors, |t = |τ and |t =˜ |τ respectively for the straight and tilded sectors separately, and |τ ∝ u |t] and |τ ∝ũ |t] for the SU(2) holonomies relating the two sectors. Notice that we have to take the dual spinors, which we haven't represented in the diagram. This graphical representation is consistent with the realization of the symmetries (left rotation in this case) as described in Fig. 3.
As before, all these objects transform covariantly under the suitable SU(2) transformations, that is for w ∈ SU (2): We also introduce the "braided-covariant" spinorτ , The Poisson commutation relations between the different spinor components are essentially the same as in (42) and (56), putting some tilde everywhere. We can also consider the cross terms, that is the Poisson commutation relations between the spinors and their tilde alter ego.

V. Q-DEFORMED HOLONOMY RECONSTRUCTION
Now starting from and˜ , i.e the 3-vectors T and T , we would like to reconstruct the SU(2) holonomies u andũ. However, as in the undeformed case, it is impossible to fully reconstruct the holonomy g from the two 3-vectors X and X. Indeed, we have an undetermined phase, corresponding to a free rotation around the X axis. Nevertheless, using the spinor variables freezes that extra phase and we can explicitly reconstruct the whole holonomy g from z andz as shown in section I. Similarly here, we will use the spinors t andt, from which we have already shown how to reconstruct the triangular matrices and˜ , and we will see below how to derive the SU(2) holonomies u,ũ and thus ultimately the SL(2, C) group element D.
More precisely, from the triangular matrices and˜ , one builds the following Hermitian matrices: whose projection on the Pauli matrices define the 3-vectors T and T , that live on the two ends of one edge. The fact that and˜ both come from the same SL(2, C) group element D decomposed following the left and right Iwasawa decomposition as D = u =ũ −1˜ −1 with u,ũ ∈ SU(2) implies that those 3-vectors are related by the SU(2) holonomies (obtained from the expression of DD † and D † D ): We now consider both left and right SU(2) transformations: The left transformations read: They affect but not˜ . On the other hand, right transformations read: and likewise affect only˜ while leaving unchanged. The SU(2) holonomies have slightly skewed transformations since they are covariant on one side and braided-covariant on the other: We are now ready to describe how to define the SU(2) holonomies u andũ from the spinor variables, using their transformation properties under the SU(2) action as guide. First, one must impose a norm-matching condition, N =Ñ , or equivalently T 0 =T 0 or in terms of spinors t|t = t |t . Starting with u, its natural definition is mapping the spinors |t , |t] to |τ ], −|τ : We first check that this ensures the correct parallel transport: Similarly, we define the SU(2) holonomyũ for the tilded sector as the unitary group element mapping the basis |t , |t] to −|τ ], |τ :ũ Once again, this ensures the expected parallel transport: We can further check that −1ũ−1˜ −1 u −1 = I, by using the maps between the covariant spinors and dual and their braided-covariant counterparts, and that the straight and tilded sectors indeed define the same SL(2, C) group element Finally, one can compute the Poisson brackets of the components of u andũ, and check that it provides the correct phase space structure. Let us consider the u holonomy: In order to compute the Poisson brackets of the components of u, we need the Poisson algebra 2 of the spinor components given earlier in (42), (56) and (63). Being careful to the norm factors, which are actually rather important and not to be neglected, we compute the Poisson brackets of u with itself and find: which are weak equalities up to assuming the norm-matching constraint N −Ñ = 0. This is valid since all these variables ,˜ , u,ũ have a vanishing Poisson bracket with that constraint. These are the expected Poisson brackets for the SU(2) holonomy u from the SL(2, C) phase space formulas. Proceeding similarly, it is straightforward to check that we recover the whole phase space with the correct brackets between the left decomposition variables, between the right decomposition variables (i.e. the tilded variables) as well between these two sets of variables (63). We have therefore the following theorem.
Theorem V.1. Consider S κ ∼ C 2 the phase space given in terms of the following (non-canonical) Poisson brackets and its subspace S κ * ≡ C 2 /{t a ∈ C 2 , t|t = 0}. The symplectic reduction P κ = (S κ * × S κ * )//M of the phase space S κ * × S κ * (parametrized by (t A ,t A )) by the matching norm condition M = t|t − t |t = 0 is isomorphic to the phase space SL(2, C) which Poisson structure is . 2 We summarize there all the required Poisson brackets. First, the straight spinors Poisson-commute with the tilded spinors by definition. Then, we recall the brackets for the spinors t, τ and their complex conjugates: Finally, the contribution of the norm-factors is computed using the explicit expression of the norms t|t = (e ) and τ |τ = (e κN − 1): This theorem is the classical analogue of the Schwinger-Jordan representation of U q (su(2)) in terms of harmonic oscillators [31,32]. Note furthermore that we can relate S κ to the undeformed phase space C 2 , as we have recalled in (29). Hence there is a symplectomorphism between the undeformed phase space C 2 and S κ . This symplectomorphism can then be naturally extended between the spaces P = (C 2 * × C 2 * )//M and P κ = (S κ * × S κ * )//M. We recall that P is the phase space behind the (flat) twisted geometries construction and is such that P ∼ T * SU(2) [18,20]. Hence we have recovered in a new manner the well-known fact that T * SU(2) and SL(2, C) are symplectomorphic as phase spaces (while still obviously different as groups).

VI. TOWARDS HYPERBOLIC TWISTED GEOMETRIES
We would like to use the spinor phase space developed above to parametrize the hyperbolic deformation of twisted geometries for loop gravity. We consider a graph, with oriented edges and chosen order around each vertex (i.e all the attached edges are ordered). We attach spinor variables to each half-edge: around each vertex ϑ, for each attached edge e ϑ, we define a spinor ζ e,ϑ ∈ C 2 with canonical Poisson brackets and then as developed previously spinors ζ κ e,ϑ , t e,ϑ and τ e,ϑ satisfying deformed Poisson brackets. Then for each edge we have a pair of spinors, one at each end. We naturally associate the source vertex to the straight sector and the target vertex to the tilded sector: we write for short ζ e = ζ e,s(e) at the target vertex ϑ = s(e) andζ e = ζ e,t(e) at the source vertex ϑ = t(e). Up to now, we have simply introduce a re-parametrization of the standard phase space (C 2 ) 2E //U(1) E , where we are taking the symplectic quotient by the matching conditions along every edge, in terms of the non-trivial and noncanonical variables e ,˜ e , u e ,ũ e . The big difference with the standard spinor networks and twisted geometries is the deformed action of the SU(2) gauge group at the vertices. Indeed the initial canonical spinors ζ e ,ζ e do not transform simply as spin-1/2 vectors under SU(2) transformations. It is instead the non-canonical spinors t e ,t e that transform linearly under the action of SU (2). This SU(2) action can be in principle transposed on the canonical spinors ζ e ,ζ e , it is highly non-linear and the explicit expression is rather cumbersome. Taking this into account, it is not obvious that the matching conditions M e = ζ e |ζ e − ζ e |ζ e = 0 are invariant under the SU(2) action. There are nevertheless exactly equivalent to the obviously SU(2)-invariant constraint written in terms of the non-canonical spinors t's, M e = t e |t e − t e |t e = 0 .
Actually the new setting is even more subtle. In order to define the SU(2) action at a vertex ϑ, we need to consider the chosen ordering of the edges around it. Let's number these edges e i with i = 1..n. As shown and explained in [16], assuming that all the edges are oriented outward, the SU(2) transformations are generated by the deformed closure constraints (or Gauss law in the context of loop gravity) G ϑ = 1 .. n = I. The product is obviously non-abelian and the ordering of the triangular matrices matters. More precisely, the SU(2) transformation for one (half-)edge e i will get braided by all the previous k for k < i. This reads for a transformation v ∈ SU(2); where the last v factor is equal to the initial SU(2) transformation v n = v due to the condition 1 .. n = I itself. As shown in [16], these SU(2) transformations are Poisson-generated by the Hermitian square of the closure constraints G † ϑ G ϑ as: for infinitesimal transformation v ∼ I with ∼ 0. For n = 1, we recover the infinitesimal transformations already given in (36). If an edge e k is incoming instead of outgoing, one simply replaces k by˜ k in the closure constraint and nothing else changes. The interested reader will find detailed explanations and proofs for this in [16]. We can summarize all these structures by a graphical representation of our deformed spin network as ribbon graphs, as illustrated on Fig.  4.
Moreover, the previous work [16] further provided a geometrical interpretation for the closure constraints at least in a (2+1)-dimensional setting. Considering trivalent graphs, one can map uniquely a triplet of triangular matrices satisfying the closure 1 2 3 = I onto a hyperbolic triangle (up to translations) such that the 's are interpreted as the "hyperbolic normals" to the triangle's edges. Then, assuming a (SU(2)-invariant) flatness condition around every loop (in terms of the SU(2) holonomies u e andũ e ), one could glue these hyperbolic triangles consistently together within the 3-hyperboloid (of unit time-like vectors in the 4d Minkowsi space-time R 4 ). However, a (3+1)-d geometric interpretation of the closure constraints, and thus of these deformed spinor networks, in terms of hyperbolic tetrahedra and polyhedra has not yet been worked out. This is a highly non-trivial mathematical issue and we leave for future investigation. This would provide us with hyperbolic twisted geometries as a generalization to non-vanishing cosmological constant of the standard twisted geometries describing the kinematical configurations of loop gravity.

Discussion
We have shortly reviewed the deformed phase space for loop gravity and the associated deformed action of SU (2) introduced in [16] in order to represent discrete hyperbolic surfaces and to yield directly spin networks for U q (su(2)) upon canonical quantization [17]. Here, we have introduced spinor variables with canonical Poisson brackets as Darboux coordinates for this phase space and we have defined non-canonical spinor variables that allow to represent the deformed phase space on a given graph as a spinor network. The resulting structures on the graph are a pair of spinors on each edge, one attached to each of its ends, provided with norm-matching constraints for every edge and deformed closure constraints for every vertex. This ensures that one can reconstruct the full SL(2, C) phase space on each edge, given in terms of a group element D e = e u e ∈ SL(2, C) ∼ SB(2, C) SU(2) together with a Poisson bracket defined in terms of the classical r-matrix of sl 2 , and that we have a local SU(2) invariance at each vertex.
Just like the use of spinor variables for twisted geometries in standard loop quantum gravity (e.g. [20,21]), we hope that this spinorial formalism for the hyperbolically-deformed loop gravity phase space will similarly shed light on the construction of observables and the interpretation of the deformed phase space as 2d and 3d hyperbolic geometries, at the classical level. We expect that it should provide some insights on the definition of coherent states at the quantum level. This work should be considered as setting up the framework for a larger program whose next steps would be: • Identify the SU(2)-invariant observables at the vertices in terms of suitable scalar products between spinors and investigate the deformed u(N ) structure of their Poisson algebra. This should be the classical analogue of the U q (u(N )) structure found in the quantum case [13].
• Build on the work [12,13] and investigate the relation between the classical flatness constraints and the spinfoam dynamics for 3d quantum gravity with negative cosmological constant through the analysis of recursion relations for the q-deformed 6j-symbols using spinor operators, as done in [27].
• Study the SU(2) flatness holonomy constraint defined as − → u e = I in the previous work [16], analyze its expansion in terms of the deformation parameter κ and understand the deviations from the standard flat spinor network case and the origin of the (hyperbolic) curvature in the continuum limit.
• Extend the general construction to the spherical case, where quasi-Poisson Lie group structures will likely be the right tool [33].
• Understand the geometrical meaning of the deformed spinor network in terms of 3d geometry embedded in a 3+1d space-time and extend the construction to a deformed twistor phase space and appropriate simplicity constraints as done in the standard flat case [34,35].
At the end of the day, we believe spinorial variables to be a very promising and relevant tool to investigate the quantization of loop gravity with a non-vanishing cosmological constant and its relation to the quantum deformation of the gauge group.