Simplicial graviton from selfdual Ashtekar variables

In perturbative gravity, it is straight-forward to characterize the two local degrees of freedom of the gravitational field in terms of a mode expansion of the linearized perturbation. In the non-perturbative regime, we are in a more difficult position. It is not at all obvious how to construct Dirac observables that can separate the gauge orbits. Standard procedures rely on asymptotic boundary conditions or formal Taylor expansions of relational observables. In this paper, we lay out a new non-perturbative lattice approach to tackle the problem in terms of Ashtekar’s self-dual formulation. Starting from a simplicial decomposition of space, we introduce a local kinematical phase space at the lattice sites. At each lattice site, we introduce a set of constraints that replace the generators of the hypersurface deformation algebra in the continuum. We show that the discretized constraints close under the Poisson bracket. The resulting reduced phase space describes two complex physical degrees of freedom representing the two radiative modes at the discretized level. The paper concludes with a discussion of the key open problems ahead and the implications for quantum gravity.


Introduction
Many non-perturbative approaches to quantum gravity start from discrete truncations, for example a lattice [1][2][3][4][5][6].When the lattice is finite, the truncation yields a finite-dimensional mechanical system that approximates the dynamics of general relativity in the continuum.In principle, such mechanical systems can be quantized at the full non-perturbative level.A smooth semi-classical geometry, so it exists, can then only emerge after infinite refining [7][8][9][10][11].The main difficulties with this approach arise from the gauge symmetries of the theory in the continuum.In the continuum, the symmetries of Einstein's equations gives rise to a vast gauge redundancy.The corresponding gauge generators are the scalar and vector constraints for the initial data.Gauge-invariant physical states are represented by gauge orbits on the constraint hypersurface.Although we have no explicit construction of such gauge-invariant physical states on a lattice, 1 there is a simple proposal for how to proceed to quantum theory.To characterize physical states, we would start from a discrete representation of the constraint algebra.Once we have such a representation, we turn to quantum theory to establish a suitable kinematical Hilbert space that carries a representation of the constraints.On this auxiliary Hilbert space, the constraints turn into the generators of gauge symmetries.The physical Hilbert space is the kernel of the now discretized constraints.A concrete realization of this idea can be found in the loop gravity and spinfoam approaches to quantum gravity [2,3,[12][13][14][15].
One of the main difficulties with this idea is the issue of anomalies [16,17].It is not at all obvious how to put the constraints on a lattice without violating the gauge symmetries.If the gauge symmetry is broken, the constraints no longer form a closed algebra and we can no longer impose them strongly.The best that we could do is to impose them weakly, e.g. by restricting ourselves to a subspace of the Hilbert space, where all matrix elements of the constraints vanish.The fundamental problem with this idea is that there is no reason to believe that such a subspace would be stable under the dynamics.If it is not, transition amplitudes will no longer be unitary.Processes could occur where initial states that are sharply peaked on a classical configuration would decay into final states that violate gauge invariance strongly.
Thus there seems to be a fundamental conflict between diffeomorphism invariance and lattice truncations.To resolve the problem, we could either resort to an infinite lattice refinement or work with an appropriately deformed constraint algebra.In the first case, diffeomorphism invariance is restored in the continuum limit [7,9].In the second case, the lattice stays, but the gauge symmetries are deformed in such a way that there is no longer an anomaly.In this paper, we consider this second possibility [13,[18][19][20].By introducing a new discretization scheme, we obtain an anomaly-free lattice regularization for selfdual gravity [21,22].Compared to the situation in the continuum, we have two additional constraints.Besides the Gauss, vector and Hamilton constraints, there is a new closure constraint and a cocylce condition for certain magnetic fluxes.Otherwise the algebra does not close.We will argue below that the additional constraints vanish in the continuum and are necessary to remove otherwise unphysical lattice artefacts.A simple counting demonstrates that the resulting reduced phase space has 2 × 2 complex dimensions per lattice site.This is an encouraging result, for it agrees with the situation in the continuum.In the continuum, self-dual gravity has two complex propagating modes.
Our construction consists of two steps with two main results.The first result provides a new lattice regularization of the constraints on a fixed triangulation.The second, and more important result concerns the resulting constraint algebra, which closes under the Poisson bracket without any unwanted anomalies.At the same time, there are two important caveats.The first caveat is that we restrict ourselves in this paper to the classical level.At the quantum level, new anomalies may arise through operator ordering ambiguities.The second caveat is more severe.Our construction assumes that the constraints can be solved locally.The basic idea is to split the initial hypersurface into fundamental building blocks, then solve the discretized constraints in each building block separately.It is only in a second step that the fundamental cells are glued back together such that a new and more complicate solution of the constraint equations is found.This entire gluing procedure requires auxiliary boundary modes [23][24][25][26][27][28] to regularize the constraints in each fundamental building block.In the present framework, these boundary modes are given by a flat boundary connection A = g −1 dg at the two-dimensional boundary of each cell on the three-dimensional Cauchy surface.
Outline of the paper.In section 2, we give a concise review of selfdual gravity as a constrained Hamiltonian system.The two main results are developed in section 3 and section 4. In section 3, we introduce the quasi-local regularization of the constraints in a single building block.It is in this section that we also identify the additional closure constraints and cocycle conditions that remove otherwise unphysical lattice artefacts.The constraint algebra and corresponding structure functions are computed in section 4. The paper concludes with an outlook and discussion about the relevance of our results for non-perturbative approaches to quantum gravity such as loop quantum gravity and group field theory.
Notation.In the following, indices a, b, c, . . .are abstract (co)tangent indices on the spatial manifold M , greek indices α, β, γ, • • • = 1, 2, 3 refer to a fiducial coordinate system on M , indices I, J, K, • • • = 1, . . ., 4 (from the center of the alphabet) refer to the four sides of a tetrahedron T ⊂ M (see figure 1), indices A, B, C, . . .are left-handed spinor indices, which are raised and lowered by the skew symmetric ǫ-tensor2 and indices i, j, k, • • • = 1, 2, 3 refer to a complexified basis in sl(2, C), which consists of the Pauli matrices that satisfy the familiar Pauli identity (2).With respect to this complexified basis, the structure constants of sl(2, C) are given by the components of the usual three-dimensional internal Levi-Civita tensor ǫ i jk .Finally, δ ij denotes the corresponding internal three-metric, i.e. δ ij = − 1 2 ǫ l im ǫ m jl .

Review: Phase space of selfdual gravity
Selfdual (complex) gravity admits a remarkably simple Hamiltonian formulation [21,22].In the following, we briefly summarize the formalism.More recent developments on the subject can be found in [29][30][31][32].This section will be useful for the coherence of the presentation and the discussion of our results in the conclusion.
In selfdual gravity, the fundamental configuration variable is the self-dual and sl(2, C)-valued Ashtekar connection A i a on the initial hypersurface.Its conjugate momentum is the densitized triad Ẽ a i , which is an sl(2, C)-valued vector density.At the kinematical level, the fundamental Poisson brackets are given by where G is Newton's constant and δ(3) (x, y) denotes the Dirac distribution (a scalar density of weight one).All other Poisson brackets among A i a and Ẽ a i vanish.Tensor indices a, b, c, . . .are abstract tangent indices on the spatial hypersurface M .The internal indices i, j, k, • • • = 1, 2, 3 refer to a three-dimensional basis in the sl(2, C) gauge fibres over every point of the manifold.This basis satisfies the usual Pauli identity where A, B, C, . . .are spinor indices for two-component Weyl spinors.The basis is complex, which is to say that sl(2, C) is treated as the complexification of su (2).Real (hermitian) and imaginary (anti-hermitian) parts are computed with respect to a hermitian metric δ ĀA .Given an embedding of the initial hypersurface M into the four-dimensional manifold, this metric can be uniquely fixed as follows.If σ ĀAα are the four-dimensional Infeld-Van der Waerden symbols, and n a is the time-like normal vector to M , the metric in the spin bundle is given by δ ĀA = σ ĀAα e α a n a , where e α a is the co-tetrad.It is with respect to this metric that the Pauli matrices (2) are anti-hermitian, i.e.
To characterize initial data for selfdual gravity, we have to impose the Gauss, vector, and scalar constraints on phase space, which are the Hamiltonian generators of SL(2, C) frame rotations, spatial diffeomorphisms and hypersurface deformations.In terms of the Ashtekar variables ( Ẽ a i , A i a ), the constraints are given by where D a is the SL(2, C) gauge covariant derivative, b is its curvature, and Λ i , N a and Ñ are smearing functions of compact support ( Ñ is a inverse density of weight minus one).Its density weight will play an important role below.
The resulting constraint algebra is first-class.The only non-trivial Poisson commutators among the constraints are given by All other commutators among the constraints vanish identically on the entire kinematical phase space.In here, [Λ, M] i = ǫ i jk Λ j M k is the Lie bracket with respect to the sl(2, C) Lie algebra and the Lie bracket between vector fields is denoted by is the Lie derivative along the vector field N a ∈ T M of the density weight −1 lapse function Ñ .For any two such scalar densities M and Ñ of weight −1, the bracket [ Ñ, M ] a denotes the vector field Notice that [ Ñ, M ] a is independent of the choice of covariant derivative operator D a , i.e.
Notice also that the constraint algebra remains well-defined even for those field configurations for which the densitized triads are degenerate, i.e.
3. New lattice regularization of the constraints

Basic holonomies and fluxes
In the following, we consider the discretization of the constraints (4a, 4b, 4c) in a small tetrahedron T ⊂ M , whose four corners are located at coordinate values X µ I : 4 I=1 X µ I = 0 with respect to some fiducial coordinate system {x µ } (see figure 1 for an illustration).The origin of this coordinate system is put into the centroid of the tetrahedron.In these coordinates, the tetrahedron is the point set {x µ ∈ R 3 : x µ = 4 I=1 p I X µ I , 4 I=1 p I ≤ 1, 0 ≤ p I }.In the limit in which the coordinate lengths X µ I shrink to zero, i.e.X µ I → 0, the tetrahedron shrinks into its centroid.
To discretize the constraints, we first need to discretise the basic variables on phase space, namely the Ashtekar connection A i a and the electric field Ẽ a i .To discretize the connection, consider the path γ I from the centroid c of the tetrahedron to the centroid of the I-th face.With respect to the fiducial coordinate system, this path is given by the map The discretized connection is now simply given by the SL(2, C) parallel transport with respect to the Ashtekar connection along γ I .This is the bulk holonomy To discretize the densitized triad Ẽ a i , it is useful to assume that the curvature is located PSfrag replacements inside the tetrahedron and the boundary is flat.This implies that there exists a gauge element g : ∂T → SL(2, C) such that the pull-back of the connection to the boundary is where ϕ * ∂T denotes the pull-back to the boundary.If c I denotes the centroid of the I-th face, the parallel transport along the boundary from c I to c J is now simply the product where g I = g(c I ).In ( 8), the bulk connection A i a is smeared over the links of the tetrahedron.Its conjugate momentum, which is the densitized triad Ẽ a i , is smeared over the dual faces, which are the four boundary triangles f I .To maintain local gauge invariance, we use the boundary parallel transport and map all free gauge indices into the centre of the tetrahedron.In this way, we obtain the electric flux, where E j denotes the dual two-form, which is obtained by dualizing the densitized triad with respect to the metric-independent and inverse Levi-Civita tensor density 3 ǫabc .Our conventions are In addition, where τ j are the anti-hermitian Pauli matrices (2).
PSfrag replacements aligned to three sides of the tetrahedron.The triad is based at one of its corners.In here, this is the fourth corner p4.

Regularization of the Gauss constraint
The next step is to regularize the constraints.First of all, we consider the Gauss constraint (4a).Using the non-abelian version of Stokes's theorem, we integrate Gauss constraint (4a) against a smearing function Λ i inside the tetrahedron and obtain (for small curvature) the closure constraint, To regularize the other two constraints, it is useful to introduce an adapted triad (a righthanded basis (u a 1 , u a 2 , u a 3 ) of T * M ) at a marked point of the tetrahedron.The simplest choice is to put the origin of the coordinate system at one of the vertices of the tetrahedron (e.g. the 3 Given local coordinates {x µ }, the tensor densities ǫabc and ǫabc are defined as fourth) and use the three edge vectors pointing to the other three vertices as our basis elements, see figure 2. With respect to our fiducial coordinate system {x µ } the basis vectors are now given by their coordinate expressions Given the triadic basis u a α = u µ α ∂ a µ , for α = 1, 2, 3, we now also have a dual cotriad {u α a } α=1,2,3 such that In addition, we may now also introduce the scalar densities Consider now the flux through the first triangle f 1 .To leading order in the coordinate size of the tetrahedron, 4 we then have where all fields are evaluated at the centroid of the tetrahedron (or at any other point inside T ).Equation ( 19) must be true for all three faces f 1 , f 2 and f 3 , hence Notice also that the forth missing flux can be expressed in terms of the triad E α i .Going back to (14), we have

Regularization of the scalar constraint
The Gauss, vector and Hamiltonian constraints (4a,4b,4c) of selfdual gravity are constructed from tensor densities.To regularize such densities, we proceed as follows.First of all, we note that the integral of the fiducial density d 3 u (see (17a)) over T is nothing but This is so, because the corners of the tetrahedron are located at the coordinate values (1, 0, 0), (0, 1, 0) and (0, 0, 1) with respect to the three basis vectors u a 1 , u a 2 and u a 3 .Given the integral of the fiducial density d 3 u along T , we obtain the following approximation for the integral of a varying scalar density f of weight one within T , namely where c is again a point inside T (such as the centroid of the tetrahedron). 4Provided the fields do not vary much over the extension of the tetrahedron.Accordingly, the symbol "≈" denotes equality up to subleading terms in a Taylor expansion around the centroid of the tetrahedron, e.g.
In Ashtekar's self-dual formulation, the scalar constraint is a density of weight two.The corresponding smearing function Ñ is an inverse density of compact support.To regularize Ñ , we take its inverse and integrate the resulting density over the tetrahedron.In other words, Given ( 22) and ( 23), we obtain as the leading contribution of the expansion of the smeared scalar constrained in the tetrahedron T .The next step is to replace in this expression the densitized triad Ẽ a i by the electric fluxes over the four sides of the tetrahedron, see (11).We insert the completeness relation (16a) and return to our expression for the electric fluxes E α i , α = 1, 2, 3 as given in (19).We obtain The curvature term has a neat geometric interpretation in terms of magnetic fluxes through the dual facets {f IJ }, see figure 1.Any such surface f IJ has the geometry of a kite: its four corners are the centroid of the tetrahedron, which is the origin of our fiducial coordinate system {x µ }, the centroids , which is the centroid of the edge5 connecting X µ K and X µ L .The integral of a slowly varying two-form ω over f IJ can be approximated, therefore, by the following expression, where X a I are the position vectors X a I = X µ I ∂ a µ of the four vertices of the tetrahedron.To regularize the scalar constraint (25), let us then notice that where we reintroduced the triad u a α adapted to the sides of the tetrahedron, see (15).Let now F i [f IJ ] denote the integral of the curvature two-form over f IJ , i.e.
Going back to (27), we obtain the important identity We insert this expression back into our regularization for the scalar constraint (25), which leads us to where the sum is taken over all indices α, β = 1, 2, 3.The closure constraint (20) finally brings this equation into the following neat form where we now sum over all repeated indices I, J, K, • • • = 1, . . ., 4 labelling the four sides of the tetrahedron.
On the lattice, the fundamental configuration variables are the parallel propagators (8, 10) and fluxes (11).The integral of the curvature two-form along the dual faces f IJ must be approximated, therefore, by parallel propagators.This can be achieved by using the non-abelian Stokes's theorem, which states that the holonomy around the perimeter of a surface f is the surface ordered exponential of the field strength, On the other hand, we now also know from ( 8) and ( 10) that the holonomy around the perimeter of f IJ is given by the magnetic fluxes, Replacing in (31) the integral of F i over f IJ by the holonomy around the boundary, we obtain as a regularization of the scalar constraint smeared over the tetrahedron T , where we sum over all repeated indices I, J, = 1, . . ., 4 and i, j = 1, 2, 3.

Regularization of the vector constraint
In the previous section, we considered the scalar constraint.Next, we turn to the vector constraint.First of all, we decompose the corresponding vector-valued Lagrange multiplier N a with respect to the triadic basis (15), obtaining Using our approximation scheme for scalar densities (22) and the completeness relations (16a) for the triadic basis {u a α } α=1,2,3 , we then have If we now use the approximation for the electric fluxes (19) together with the identity (29) for the magnetic fluxes, we can rewrite this expression as follows, where we sum again over all indices α, β = 1, 2, 3.The final expression (37) suggests to introduce a four-vector N I ∈ R 4 , whose components are given by In other words, the discretized Lagrange multiplier N I ∈ R 4 is not arbitrary, but satisfies a closure constraint as well, namely If we now also take into account the closure constraint for the electric fluxes, namely (20), we can recast the regularized vector constraint into the following form, Finally, we have to replace the integral F i [f IJ ] = f IJ F i by the SL(2, C) group valued magnetic fluxes (33).This leads us to our final expression for the regularized vector constrained in a single tetrahedron, namely, where we sum over all repeated indices I, J = 1, . . ., 4 for a smearing function N I that satisfies the closure constraint (39).

Motivation for a second closure constraint
In the continuum, the Gauss, vector and scalar constraints (4a, 4b, 4c) satisfy a closed algebra.
The kinematical phase space of field configurations ( Ẽ a i , A i a ) has 3 × 3 × 2 complex dimensions (per point), and there are 3 + 3 + 1 = 7 complex constraints, which are all first class.The resulting reduced phase space has 18 − 2 × 7 = 4 complex dimensions, which describe the two complex degrees of freedom of self-dual gravity (the two polarisations of the self-dual graviton).
In the previous section, we introduced a lattice regularization for the Gauss (14), scalar (34) and vector constraint (41) in a single tetrahedron.Do we have a chance to recover the right degrees of freedom at the discretized level?Suppose, for example that the discrete constraints (14,34,41) are all first-class, which would give us seven (complex) constraints for each tetrahedron.How many dimensions would the resulting reduced phase space have?In the continuum, the phase space is equipped with the symplectic potential At the discretized level, the contribution to the symplectic potential from each tetrahedron is 6 see e.g.[33].The fundamental configuration variables of the discretized theory are the bulk holonomies h I , which are parametrized by 4 × 3 = 12 complex numbers, the resulting lattice phase space (see section section 3.1) has 24 complex dimensions.The naive counting would then give 24 − 2 × 7 = 10 physical degrees of freedom per tetrahedron, which does not match the theory in the continuum.There seem to be three unphysical lattice degrees of freedom.
That there are such spurious lattice modes has a simple explanation.The symplectic potential for the lattice phase space (43) treats the variables assigned to all four directions I = 1, . . ., 4 6 As fas as the Hamiltonian analysis for a single tetrahedron is concerned, the four edge modes (boundary holonomies) gI ∈ SL(2, C) are mere spectators, and Poisson commute with all bulk variables (E I i , hI ) ∈ sl(2, C) × SL(2, C).Their own phase space and canonical momenta become relevant only once we consider the glueing of adjacent tetrahedra, see section 4.
in the tetrahedron as independent.This contradicts the theory in the continuum: for small curvature, we can approximate the parallel propagators as see ( 8) and (8).In our derivations of the scalar and vector constraint ( 41) and ( 34), we implicitly assumed that 4
Therefore, already at the kinematical level, the bulk holonomies h I and fluxes E I i should not be seen as completely independent variables.Going from ( 42) to ( 43) there are 2 × 3 additional spurious modes appearing.An additional closure constraint seems to be missing that should impose an appropriate version of (45).
We now need a candidate for the missing constraint.Consider the following dressed closure constraint where [h] i j denotes again the adjoint representation (13) of SL(2, C) and F IJ is the magnetic flux (33).For small curvature, the SL(2, C)-valued magnetic flux F IJ can be replaced to good approximation by the holonomy For slowly varying curvature, the integral of the curvature two-form over the dual faces f IJ can be approximated by the components of the curvature two-form F i ab contracted with the position vectors X a I and see (26).This in turn allows us to expand equation ( 46) for small curvature.We obtain The leading order of this expression returns the usual closure constraint (14).For generic configurations of F i ab the next to leading order vanishes if and only if (45) is satisfied.This observation is a first indication to add the dressed closure constraint (46) to our set of constraints ( 14), ( 34), (41).In the next section, we will find another such indication.The resulting constraint algebra closes under the Poisson bracket. 7At the kinematical level, there are 4 × 3 × 2 = 24 complex phase space dimensions.In addition, there are 3 + 3 + 1 + 3 = 10 constraints removing 2 × 10 = 20 complex directions in phase space.The resulting reduced phase space describes, therefore, 2 complex degrees of freedoms that we identify with the two complex degrees of freedom of the selfdual graviton at the discretized level.Notice also that in those special Regge-like configurations in which the holonomy F IJ = Pexp(− f IJ A) around each facet f IJ preserves the flux E I , i.e. [F IJ ] j i E J j = E J i , the dressed closure constraint (46) reduces to the usual discretized Gauss constraint (14).Thus, for Regge-like configurations, no additional constraint appears.

Holonomy flux algebra
The Poisson commutation relations for the Ashtekar variables (1) determine the commutation relations for the corresponding smeared variables, namely the bulk holonomies (8) and electric fluxes (11).A straight-forward calculation gives,8 which may be derived also directly from (43), see [33,34].On the resulting 2 × 3 × 4 = 24 complex-dimensional phase space, we need to impose the regularized versions of the Gauss, vector and scalar constraint, see ( 14), ( 34), (41), in addition to the dressed closure constraint (46).To this goal, we now need to calculate the algebra of constraints.Before we continue, a few further comments.When introducing the holonomies, see ( 8) and ( 10), we made an implicit restriction on the allowed boundary data: the pull-back of the connection to the boundary is flat.This restriction implies a relation among the magnetic fluxes, namely ∀I, J, K : which means that only three out of the six group elements F IJ ∈ SL(2, C) are functionally independent. 9If we take this condition into account, we see that the dressed closure constraint simplifies (46).One of the sums can be dropped, and the constraint simplifies to for some arbitrary K ∈ {1, . . ., 4}.
Notice also that the bulk holonomies h I , I = 1, . . ., 4 enter the definition of the constraints only through the magnetic fluxes F IJ ∈ SL(2, C).It is therefore more convenient to use the commutation relations between electric and magnetic fluxes rather than (50b), which are given by Finally, let us briefly summarize what we have done so far.In a tetrahedron, which is small against the curvature scale at which the fields fluctuate, the Gauss (4a), vector (4b) and scalar (4c) constraints of selfdual gravity can be written to good approximation in terms of electric and magnetic fluxes alone, see ( 14), ( 34) and ( 41).In addition, there is one more constraint, that does not appear in the continuum, namely the dressed closure constraint (46).The role of this additional constraint is to remove unphysical degrees of freedom that appeared by introducing the discretization.That there are such unphysical lattice degrees of freedom follows from a simple counting argument: the kinematical phase space in the continuum (42) has 2 × 3 × 3 = 18 complex dimensions per point, the lattice phase space for a tetrahedron has 2 × 3 × 4 = 24 complex dimensions, hence we should add three additional complex (first class) constraints to remove the spurious directions from phase space.As a candidate for the missing constraint, we consider the dressed closure constraint.In summary, the list of constraints is given by closure constraint: dressed closure: vector constraint: scalar constraint: Tr The next task ahead is to demonstrate that the corresponding constraint algebra closes under the Poisson brackets (50a), (50c) and (53).

Closure constraint
Given the commutation relations for the electric fluxes (50c), it immediately follows that the closure constraint ( 14) is the regenerator of SL(2, C) gauge transformations at the centre of the tetrahedron.The relevant Poisson brackets are Thus, the closure constraint (54a) weakly commutes will all other constraints.

Dressed closure constraint
Next, we consider the dressed closure constraint (54b).To simplify the calculation, it is useful to consider first its Hamiltonian vector field as it acts on the electric and magnetic fluxes.The Poisson brackets between the constraint and the electric fluxes is given by which follows immediately from (50c) and (53).The magnetic fluxes, on the other hand, Poisson commute with the dressed closure constraint, where we used in the last step the product identity (51) for the magnetic fluxes and [h] i j denotes again the adjoint representation of SL(2, C), see (13).
The equations ( 56) and (57) tell us that the electric and magnetic fluxes weakly Poisson commute 10 with the dressed closure constraint (54b).All our constraints can be expressed in terms of polynomials of the electric and magnetic fluxes alone.Therefore, the dressed closure constraint weakly commutes with all other constraints.In particular, To summarise, the dressed closure constraint (54b) commutes with all other constraints as well.

Vector -vector bracket
The basic commutation relations for the electric and magnetic fluxes (50c) and (53) imply the Poisson bracket between two vector constraints, To simplify our notation, we have adopted a summation convention, where we sum over all repeated indices I, J, K, • • • = 1, . . ., 4 and i, j, k, • • • = 1, 2, 3.For any complex matrix X ∈ C 2 ⊗ (C 2 ) * and h ∈ SL(2, C), we now have the identity This idenity brings the expression for the Poisson bracket (59) into the following form, We now need to show that the right hand side is again a sum of constraints.To this goal, consider first the following algebraic identity, which follows from the Pauli identity (2) and the definition of H I , namely11 If we now also use the product identity (51) for the magnetic fluxes and take into account that the smearing functions N I and M I satisfy I N I = I M I = 0, we obtain The first and second term on the right hand side return the dressed and undressed closure constraints, (54a) and (54b).The last two terms are again proportional to the vector constraint.This can be seen as follows.Define the following vector, Since Tr(F IJ ) = Tr(F JI ), we also have 4 I=1 [N, M ] I = 0, which implies that the bracket [N, M ] I defines again a discrete smearing function for the vector constraint, since it satisfies the closure condition that any such smearing function ought to obey, see (54c).Going back to the definition of the constraints, we thus see that the right hand side of (64) is again a sum of constraints.More explicitly,

Vector -scalar bracket
Finally, let us complete the calculation of the constraint algebra and turn to the Poisson brackets between the scalar and vector constraints.The elementary Poisson brackets for the electric and magnetic fluxes are given in (50c) and (53).A straightforward calculation yields where we sum again over all repeated indices I, J, • • • = 1, . . ., 4 and i, j, k = 1, . . ., 3. Consider then the following identity, which is satisfied for any h ∈ SL(2, C) and any complex 2 × 2 matrices which is a consequence of the fundamental Pauli identity (2).With both (60) and , we now get Next, we use the product identity for the magnetic fluxes, (51), in addition to the algebraic identity (62) to simplify this expression further.If we also take into account that the vector N I satisfies the closure constraint 4 I=1 N I = 0, we arrive at the following expression where [h] i j denotes again the adjoint representation (13) and we sum over all repeated indices.We now need to convince ourselves that the right hand side vanishes provided our set of constraints (54a, 54b, 54c, 54d) is satisfied.First of all we note that the first term returns the dressed closure consteraint (54a) and the last term is proportional to the ordinary closure constraint (54a).The second term, on the other hand, is again proportional to the vector constraint.To see that this is indeed the case, we have to first convince ourselves that the corresponding multiplier satisfies the closure constraint (39).Equation (70) suggests to introduce the following field-dependent multiplier H ⊲ N I , which is defined for any N I : 4 I=1 N I = 0 by A short moment of reflection reveals that H ⊲ N I satisfies again the closure constraint (39) for the smearing functions N I provided the vector constraint is satisfied, i.e.
where the symbol " c =" denotes equality up to terms that vanish provided the constraints (54a, 54b, 54c, 54d) are satisfied.
Going back to the definition of the discretized vector (54c), closure (54a) and dressed closure constraint (54b), we can now give our final expression for the Poisson bracket between the discretized vector and scalar constraints, The right hand side is a sum over constraints, hence the Poisson bracket {H I [N I ], H} weakly vanishes.
Let us briefly summarize the section before discussing the main open task ahead.In this section, we considered the discretized Gauss, vector and scalar constraint in addition to the dressed closure constraint, and proved that all the constraints (there are 3 + 3 + 1 + 3 = 10 of them) commute among themselves.Since there are structure functions rather than structure constants, the resulting Poiusson commutation relations do not define a Lie algebra.This becomes particularly obvious when considering the field dependent smearing functions (65) and (71) that enter the right hand side of the Poisson brackets between the vector and scalar constraints, see (66) and (73).Our result neatly mirrors the situation in the continuum, where the Poisson brackets among the scalar and vector constraint closes, but does not define a Lie algebra, since there are non-trivial structure functions, see e.g.(5c).
The main open problem ahead is to generalize the results beyond a single simplex, to introduce, in other words, a prescription for how to glue neighbouring tetrahedra in such a way that we generate a triangulation of the spatial manifold, see figure 3. Solving this problem amounts to constructing a Hamiltonian description for self-dual gravity on a simplicial lattice.Although the problem may be difficult and tedious, the basic strategy is clear.The starting point is the discretized action on a single tetrahedron, which is the sum of the symplectic potential (43) and the constraints (54a), (54b), (54c), (54d), where Θ(δ) = 16πi G Tr(E I h I δh −1 I ) is the symplectic potential and N A is the collection of Lagrange multipliers for the constraints C A ≡ (G i , G (K) i , H, H I ).Notice that the constraints depend as functions not only on the discretized phase space variables, namely fluxes E = (E 1 , . . . ) ∈ sl(2, C) 4 and holonomies h = (h 1 , . . . ) ∈ SL(2, C) 4 , but they also depend on additional group variables g = (g 1 , . . . ) ∈ SL(2, C) 4 .These auxiliary variables, which are reminiscent of gravitational edge modes, parametrize the flat boundary connection (9).The introduction of these boundary fields was necessary to replace the curvature tensor in a single tetrahedron by the magnetic fluxes F IJ ∈ SL(2, C), see (33).To obtain the equations of motion for an isolated tetrahedron, the boundary condition δg I = 0 is fixed.When we glue adjacent regions, the boundary conditions are relaxed and replaced by a gluing condition.For each edge e that connects two adjacent tetrahedra, we impose the constraint g I s(e) = g I t(e) between the source and target nodes of the underlying edge e = γ −1 I t(e) • γ I s(e) , which is dual to the face f I s(e) = [f I t(e) ] −1 .Assuming a triangulation ∆ without a boundary, the resulting constrained action is given by where the sum is taken over tetrahedra T and edges e connecting adjacent nodes.Each node is dual to a tetrahedron and the second sum goes over next neighbours.Although the action looks fairly innocent, we expect it to define a highly non-trivial mechanical system with second-class constraints.This expectation is justified, because there are no canonical momenta conjugate to the boundary variables g.Hence there are additional constraints besides the Gauss, closure, vector and Hamiltonian constraints C A = 0 at each lattice site, which, taken by themselves, would weakly Poisson commute among themselves.The additional constraints are the gluing conditions g I s(e) = g It(e) for each edge and further secondary constraints.Common experience with similar mechanical systems suggest that there will be both first-class and second-class constraints.The additional constraints arise from the stationary points of the action with respect to the variation of the now internal boundary fields g.The presence of second-class constraints would alter the canonical commutation relations replacing the Poisson bracket by the Dirac bracket.It would be a surprise if this alteration would not affect the commutation relations between the Hamiltonian and vector constraints at neighbouring lattice sites, rendering the algebra local rather than ultra-local.This would mirror the situation in the continuum, where the Poisson bracket between localized scalar and vector constraints involves first derivatives of the Dirac delta distribution rather than the bare delta distribution itself, see e.g.(5c), (5d).

Outlook and discussion
There are two main results in this article.The first result was developed in section 3, where we found a new lattice regularization of the constraints for self-dual gravity.The construction was motivated by the relatively simple form that the constraints assume in terms of Ashtekar's connection variables.The resulting expressions for the discretized Gauss, vector and Hamilton constraint were given in ( 14), (34) and (41).At the discretized level, the kinematical phase space for each tetrahedron is T * SL(2, C) 4 , which has 2 × 3 × 4 = 24 complex dimensions.This observation creates a tension with the situation in the continuum.In the continuum, the densitized and complexified triad Ẽ a i together with the Ashtekar connection A i a span 2×3×3 = 18 dimensions over each point on the initial hypersurface.Hence, there is an apparent mismatch between the continuum and the discrete.The additional six complex dimensions are an artefact of working with holonomy flux variables.At the discretized level, the holonomy-flux variables (h I , E I ) ∈ SL(2, C) × sl(2, C) are attached to all four directions I = 1, . . ., 4 of the tetrahedron and they are treated as functionally independent directions of the kinematical phase space.In the continuum, they are not.The tangent space indices a, b, c . . . of the Ashtekar PSfrag replacements variables ( Ẽ a i , A i a ) run over a three-dimensional vector space.Hence there seem to be additional spurious directions at the discretized level.To remove the spurious directions, we argued that there are three additional closure constraints missing that would reduce T * SL(2, C) 4 to a 18dimensional phase space.A proposal for such a constraint was given in (46).The second result was to demonstrate that the resulting constraint algebra closes (section 4).It is at this point that the additional closure constraints become crucial.Without the additional constraints, there would be an anomaly.The discretized Gauss, vector and Hamiltonian constraints do not form a closed algebra among themselves.If we add, however, the dressed closure constraint (46), we end up with an algebra of first-class constraints for each tetrahedron.
There are many open questions.In our model, there is no matter, the cosmological constant is set to zero and we completely ignored the issue with the reality conditions that reduce the complexified theory to general relativity with a real Lorentzian metric.However, at this stage, the main limitation of our results appears at a more basic level.The Hamiltonian analysis was conducted on a single tetrahedron.The main open problem is to generalize the results to an arbitrary triangulation.A proposal for how to address this problem was given at the end of section 4. For a triangulation with N building blocks, the resulting action (75) describes a theory of N coupled particles that carve out a trajectory on the superspace of discrete and selfdual geometries on the lattice.Strikingly similar ideas can be found in the group field theory (GFT) approach to quantum gravity [35][36][37][38].Group field theory is an approach to quantum gravity where the kinematical wave function for an individual simplicial building block is promoted into a second-quantized field operator.In this way, transition amplitudes for simplicial boundary states turn into Feynman amplitudes for an auxiliary quantum field theory on the underlying configuration space, which is a sort-of mini-superspace of geometry (in three spatial dimensions, this is usually G 4 /G for gauge groups G = SU (2) or G = SL(2, C)).A large macroscopic geometry is to be modelled from the collective and average behaviour of a large number of GFT quanta (atoms of geometry) [39].The results of this paper resonate with this idea.If the analogy is correct, the action (75) governs a semi-classical sector of a selfdual GFT at fixed particle number with interactions between nearest neighbours.Quantum cosmology would provide a good test for this scenario.In this way, a link would be established between loop quantum gravity [2,3,12], loop quantum cosmology [40][41][42] and the GFT cosmological sector [43][44][45][46][47].At the present stage, it is unclear whether the model can support this conjecture.Only future research can tell.

Figure 1 :
Figure 1: Points, paths and surfaces in a tetrahedron: the bulk links γI connect the centroid of the tetrahedron with the centroid of the I-th face f I .The boundary links γIJ lie tangential to the boundary and connect the centroid of f I with the centroid of f J .The four triangles f 1 , . . ., f 4 form the boundary of the tetrahedron, the dual faces fJK , on the other hand, lie inside the tetrahedron and are bounded by the path γ −1 K • γJK • γJ (this choice implicitly fixes the orientation of fJK, while the orientation of f I is induced from the interior of the tetrahedron).

4 I=1H ⊲ N I = 4 I=1H
I N I = H I [N I ]

Figure 3 :
Figure 3: Two adjacent tetrahedra T and T ′ are glued along the triangle f I = (f I ′ ) −1 between.The connection at the interface is flat.The transition function from T to T ′ is given by the SL(2, C) gauge element h T T ′ = g −1 I ′ gI .