Discrete gravity as a topological field theory with light-like curvature defects

Journal of High Energy Physics, May 2017

I present a model of discrete gravity as a topological field theory with defects. The theory has no local degrees of freedom and the gravitational field is trivial everywhere except at a number of intersecting null surfaces. At these null surfaces, the gravitational field can be singular, representing a curvature defect propagating at the speed of light. The underlying action is local and it is studied in both its Lagrangian and Hamiltonian formulation. The canonically conjugate variables on the null surfaces are a spinor and a spinor-valued two-surface density, which are coupled to a topological field theory for the Lorentz connection in the bulk. I discuss the relevance of the model for non-perturbative approaches to quantum gravity, such as loop quantum gravity, where similar variables have recently appeared as well.

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Discrete gravity as a topological field theory with light-like curvature defects

Received: November Discrete gravity as a topological field theory with light-like curvature defects 0 Open Access , c The Authors 1 31 Caroline Street North , Waterloo, ON N2L 2Y5 , Canada 2 Perimeter Institute for Theoretical Physics I present a model of discrete gravity as a topological field theory with defects. The theory has no local degrees of freedom and the gravitational field is trivial everywhere except at a number of intersecting null surfaces. At these null surfaces, the gravitational field can be singular, representing a curvature defect propagating at the speed of light. The underlying action is local and it is studied in both its Lagrangian and Hamiltonian formulation. The canonically conjugate variables on the null surfaces are a spinor and a spinor-valued two-surface density, which are coupled to a topological field theory for the Lorentz connection in the bulk. I discuss the relevance of the model for non-perturbative approaches to quantum gravity, such as loop quantum gravity, where similar variables have recently appeared as well. ArXiv ePrint: 1611.02784 with; light-like; curvature; defects; Lattice Models of Gravity; Models of Quantum Gravity; Topological Field - Boundary spinors in GR Self-dual area two-form on a null surface Boundary term on a null surface Discretised gravity with impulsive gravitational waves Glueing flat four-volumes along null surfaces Definition of the action Equations of motion Special solutions: plane-fronted gravitational waves Hamiltonian formulation, gauge symmetries Space-time decomposition of the boundary action Phase space, symplectic structure, constraints Constraint algebra and gauge symmetries Relevance for quantum gravity Summary, outlook and conclusion A Spinors and world tensors 1 Introduction 2 3 2.1 We will consider gravity as gauge theory for the Lorentz group. The fundamental configuration variables in the bulk are the tetrad and the self-dual connection [1]. For a manifold with boundaries, the action acquires boundary terms, which reconcile the variational problem with the boundary conditions. If the boundary is null, we will see that the most natural such boundary term is given by the three-dimensional integral ∂M rior derivative and ℓA is a spinor, whose square returns the null generators of the threedimensional null boundary ∂M. We will then use this boundary term to discretise gravity by truncating the bulk geometries to field configurations that are locally flat. The bulk action vanishes and only the three-dimensional internal null boundaries and two-dimensional corners contribute nontrivially. The resulting action defines a theory of distributional four-geometries in terms of a topological field theory with defects that propagate at the speed of light. Whether the continuum limit exists and brings us back to general relativity is a difficult question. This paper does not provide an answer. We can only give some indications in favour of our proposal: first of all, we will show that the solutions of the theory represent four-dimensional Lorentzian geometries, which are built of a network of threedimensional null surfaces (equipped with a signature (0++) metric) glued among bounding two-surfaces. We will then find that every such null surface represents a potential curvature defect. We will also show that there exist solutions of the equations of motion derived from the discretised action that represent distributional solutions of Einstein’s equations with non-vanishing Weyl curvature in the neighbourhood of a defect. In other words, the Einstein equations are satisfied locally. Whether this holds on all scales, with possible higher order curvature corrections and running coupling constants, is a more difficult question. Sophisticated coarse graining and averaging techniques, such as those developed for Regge calculus [2–4] and related approaches cf. [5, 6] may provide useful tools for the future. The main motivation concerns possible applications for non-perturbative approaches to quantum gravity, such as loop quantum gravity [7–11]. In loop quantum gravity, geometry is described in terms of the inverse and densitised triad Eia = which is the gravity analogue of the Yang-Mills electric field. In quantum gravity, it becomes an operator, whose flux across a surface has a discrete spectrum [12, 13]. The resulting semiclassical geometry — in the naive limit, where ~ → 0, and all quantum numbers are sent to infinity, while keeping fixed the eigenvalues of geometric operators — is distributional. The semi-classical electric field Eia(p) = has support only along the links l1, . . . , lL dual to a cellular decomposition of the spatial manifold. There is no geometry, no notion of volume, distance and area, outside of this onedimensional fabric of space. What is then the dynamics for these distributional geometries at the quantum level? There are several proposals, such as those given by the covariant spinfoam approach [11, 14–16] or Thiemann’s canonical program [9, 17, 18], but the precise transition amplitudes are unknown. We know, however, that the three-dimensional quantum geometries represent distributional excitations of the gravitational field, and we can expect, therefore, that the semi-classical ~ → 0 limit of the amplitudes will define a classical theory of discrete, or rather distributional, spacetime geometries, whose Hamiltonian dynamics is formulated in terms of gauge connection variables. This paper presents a proposal for such a theory, and may, therefore, open up a new road towards non-perturbative Boundary spinors in general relativity Self-dual area two-form on a null surface On a null surface N, the pull-back of the self-dual component of the Plebański two-form and it is important for the further development of this paper. Let us explain it in more Consider thus an oriented three-dimensional null surface N in a four-dimensional Lorentzian spacetime manifold1 (M, gab). We assume M to be parallizable, hence there We define the Plebański two-form and split it into its self-dual and anti-self-dual components vectors. In terms of the Dirac gamma matrices, which may be more familiar to the reader, we could also write See the appendix A for further details on the notation. which squares to Clearly, there is an additional gauge symmetry: given the null vector ℓa, the spinor ℓA can M be the canonical embedding of the null surface N into M, and a, b, c, . . . are abstract tensor indices labelling the sections of the tensor bundle over either all of spacetime we use an index notation as well: its sections carry indices A, B, C, . . . referring to the fundamental spin ( 21 , 0) representation of SL(2, C). Primed indices A¯, B¯, C¯, . . . belong to the complex conjugate spin (0, 21 ) representation. See the appendix A for further details on the notation. This can be seen as follows: first of all we choose a second linearly independent spinor kA ϕ∗ΣAB = µ ℓAℓB + νkAkB + iεk(AℓB). Next, we also know that ℓa is the null generator of N. The pull-back of ℓa to N thus vanishes, which implies, in turn This is the same as to say We contract the free indices with all possible combinations of the spinors ℓA, kA and their which proves the desired equation (2.5). dimensional spatial submanifold C of N. Indeed ± Area[C] = Whether this integral coincides with the metrical surface area Area[C] = dx dy g(∂x, ∂x)g(∂y, ∂y) − g(∂x, ∂y)2 depends on whether ℓa, which is future pointing, is an outgoing or incoming null generator with respect to the induced orientation2 on C. 2That C inherits an orientation from N is immediate. We can say, in fact, that a pair of tangent vectors choose an arbitrary future oriented time-like vector ta, which is based on C, and say that ℓa is outgoing and this definition will not depend on the choice of ta. A straightforward calculation shows then that the The last section gave a parametrisation (2.5) of the self-dual area two-form on a null surface boundary action with spinors as the fundamental boundary variables. Working in a first-order formalism, we write the gravitational action as a functional of is then independent of the triad, which means that the torsionless condition does not hold equations of motion, which are the Einstein equations plus the torsionless condition. With SM[A, e] = We then write the bulk action in terms of self-dual variables, which are the self-dual area two-form (2.2a) and the SL(2, C) spin connection AAB, whose curvature is the self-dual SM[A, e] = − 6 where cc. denotes the complex conjugate of all preceding terms. Consider then the variation of the action. The Einstein equations follow from the variation of the tetrad, the variation of the connection yields the torsionless condition ∂M at the boundary. This boundary integral must cancel against the variation of the boundary action, otherwise the entire action is not functionally differentiable. We have assumed that form of equation (2.5). What is then the right boundary action? The remainder (2.15) is linear in the connection, and we expect, therefore, that the boundary action is linear gauge covariant and linear in AABa, which suggests that the boundary term is built from the gauge covariant derivative of some boundary fields. The only available fields at the boundary, which are functionally independent of the connection, are the boundary spinors complex-valued three-forms, whose boundary integrals define the most obvious candidates to the latter by a total derivative). We are left to determine the coupling constants in front, ∂M may seem rather dull, but it will be important for us later, when we will learn how to glue causal regions across a bounding null surface. reality conditions (2.9), which now turn into If (2.21) is satisfied, the area is real and we can define the oriented area two-form on a null surface N simply by which can be read off the remainder (2.15) of the connection variation at the boundary. The resulting boundary term is ∂M where Da denotes the gauge covariant derivative DaℓA = ∂aℓA + AABaℓB. Notice that there is now an additional U(1)C gauge symmetry appearing: the spinors where the gauge element z : N → C generates both boost and rotations preserving the null normal iℓAℓ¯A¯. The boundary action (2.16) is not invariant under this symmetry, but we The entire action for the gravitational degrees of freedom in a region M bounded by a null surface N is therefore given by the expression A are not varied in the action, they are kept fixed in the variational principle, because they determine the boundary value (2.5) of connection in the bulk and boundary. In the literature, other boundary terms have been used on null surfaces as well. For a recent survey in the metric formalism, we refer to [19] and references in there. The boundary term (2.19) is a generalisation — it is formulated in terms of spinors, and does not assume that the connection is torsionless, which explains the implicit appearance of the Barbero-Immirzi parameter, which enters the action (2.19) through the definition of the momentum spinor (2.20). Notice also that additional corner terms may be necessary as well. We will introduce them below. Discretised gravity with impulsive gravitational waves Glueing flat four-volumes along null surfaces In Regge calculus [20], the Einstein equations are discretised by cutting the spacetime manifold M into four-simplices, and truncating the metric to field configurations that are locally flat. The gravitational action for the entire manifold is then a sum over all such foursimplices, each one of which contributes a bulk and boundary term. The bulk contribution reorganised into a sum over triangles, with every triangle contributing its area times the surrounding deficit angle. The task is then to generalise Regge calculus and find a theory of discretised gravity in the connection formalism, whose action is still simple enough to admit a Hamiltonian quantisation. We will propose such a theory by dropping the assumption that the elementary building blocks are flat four-simplices. We work instead with four-dimensional regions, which are flat or constantly curved inside (depending on the value of the cosmological constant), and whose boundary is null. The theory is then specified by the matching conditions that determine the discontinuity of the gravitational field in the vicinity of the interjacent null surface. In Regge calculus, it is the intrinsic three-dimensional geometry at the interface that is matched between the two sides. We require this condition as well, and thus impose that qab = qab, where qab denotes the intrinsic three-metric from below3 the interjacent null surface N, while qab determines the geometry from the other side. It has signature (0++), and the null vector ℓa defines the single degenerate direction ℓa : 3The null surface N is oriented, and the null vectors ℓa are future pointing. In a neighbourhood of N we can thus distinguish points sitting below the null surface from those lying above. The quantities describe the boundary from above. and ǫˆabc is the metric-independent Levi-Civita density on N, which is defined for any Area-matching condition. We now have to convince ourselves that the matching conditions for the spinors (3.3) are indeed equivalent to the conditions (3.1) and (3.2) for ℓa and qab. We start with the area-matching condition (3.3a). Going back to equation (2.10) either side, and write In our formalism, the fundamental configuration variables are the boundary spinors ℓA would certainly be sufficient, but they are too strong, for they also match unpheysical gauege degrees of freedom, which are absent in (3.1). We should thus only match SL(2, C) gauge We contract both (3e.5a) aend (3.5b) with ℓA and ℓA, going back to e(3.3a) we then obtain the area-matching condition of the null generators ℓa (resp. ℓa), and the matching condition e(3.3a) implies that they both point into the same directioen, hence ℓa ∼ ℓa as desired. Having shown that the area-matching conedition (3.3a) implies the matching (3.2) of the null vectors, we are now left to show that the shape-matching conditions (3.3b) are on the two sides. This requires some preparation: first of all, we have to understand how 4We can always extend ℓA (resp. ℓA) with a second linearly independent spinor kA (resp. kA) into a local spin basis such that kAℓA = 1 =ekAℓA. To reconstruct the three-metric from the spinors, it is more intuitive to work with densitised vectors rather than three-forms on N. then also have tehe component two-form µ tangent vector µ a in the complexified tangent space (T N)C through ∈ Ω2(N : C). Its densitised dual defines a m¯aµ a = ±1. = ±1. ℓaka = ∓1, µ aka = 0, µ a? The aneswer is simple: it defines a dual dyad {ma, m¯a}, which diagonalises the intrinsic The normalisation N will be determined in a moment. The co-vector ma may vanish, but this is a singular case. It implies µ a ∝ ℓa, which is the same as to say that there exists a normalised spin dyad {kA, ℓA} such that the pull-back of the self-dual two-form to N, i.e. the null-surface N becomes effectively two-dimensional — it has no affine extension along its degenerate as well, for it implies that the triple {ℓa, µ a, µ¯a} is linearly dependent. It then notes the pull-back to N. This is incompatible with the existence of a non-degenerate tetrad tetrad, we proceed as follows: first of all, we fix the normalisation N of ma by demanding The sign depends on the orientation of {ℓa, µ a, µ¯a} with respect to the fiducial volume form, a dual basis of (T ∗N)C, such that with all signs in (3.10), (3.11) and (3.12) matching according to the indicated pattern. This allows us to write the pull-back of the tetrad to N as ϕ∗eAA¯ = ∓iℓAℓ¯A¯ka ± iℓAk¯A¯ m¯a ± ikAℓ¯A¯ma. qab = 2m(a m¯b), of the self-dual two-form to N, and get A short calculation reveals that this parametrisation is indeed compatible with the boundary ϕ∗eAC¯ (ϕ∗eBC¯ )b] = = +2ℓAℓBk[a m¯b] + 2ℓ(AkB) m¯[amb], By duality, equation (3.16b) is the same as to say that the area element is the wedge product metric qab = (ϕ∗eAA¯)a(ϕ∗eAA¯)b. A short calculation gives of the two-dimensional co-dyad {ma, m¯a}. We can then, finally, also compute the induced which concludes the reconstruction of the induced geometry of the null surface N from the Shape-matching conditions. We are now left to show that the shape-matching conditions (3.3b) imply that the intrinsic three-metrics qab and qab match between the two sides. To show this, we first extend ℓA (resp. ℓA) with a seconde linearly independent spinor kA ηAaηAb = ηˆ2 (−iℓAµ¯a + ikAℓa) −iℓAµ¯b + ikAℓb a fixed fiducial volume element, we can remove the density weights and the area-matching ∈ Ω2(Ne: C2) admits the decomposition We can now replace kA by ηAab = µ ab − iεeabζ¯ ℓA + iεabkA. without actually changing the canonical enormalisation kAℓA = 1 of the spin dyad {kA, ℓA}. e If the glueing conditions (3.3) are satisfied, we have tehues shown that there alwayes exeists normalised spin dyads {kA, ℓA}, {kA, ℓA} on either side of the interface, such that In other words, there are always spin dyads {kA, ℓA} and {kA, ℓA} such that the component two sides. But now wee ealso know that the intrinsic three-geometrye qab is already uniquely the resulting three-metrics qab and qab must agree as well. Thies ceoncludes the argument, for it implies that the intrinsic threee-metric is the same whether we compute it from the spinors on one side or the other. In other words qab = qab, which is the desired constraint (3.1) as derived from both the area-matching and shapeis a Lorentz transformation, which is given explicitly by hAB = ℓAkB − kAℓB : N ℓA = hABℓB, Before we go on to the next section, let me briefly summarise: in this section, we have studied the conditions to glue two adjacent regions along a null surface N. In terms of metric variables, equations (3.2) and (3.1) match the null generator and the intrinsic threestraints: the area-matching condition (3.3a) and the shape-matching condition (3.3b). The terminology should be clear: equation (3.3a) matches the two-dimensional area elements angles dreawn on N are the same whether we compute them from the boundary spinors on one side or the other. Notice also, that the number of constraints is the same for both variables: the induced three-metric qab has signature (0++), hence there are five independent matching constraints in the metric formalism. In terms of spinors we have five constraints to three real constraints. The shape-matching conditions (3.3b), on the other hand, add only one additional complex constraint. This is not obvious from equation (3.3b), but it is immediate when we look at (3.19). In both formalisms, we are thus dealing with the same five number of constraints. Definition of the action (as introduced in e.g. equation (2.19) above), we first introduce a cellular decomposition and cut the four-dimensional oriented manifold M into a finite family5 of closed cells {M1, M2, . . . MN }, which are flat or constantly curved inside, i.e. ∀p ∈ Mi : FAB(p) − 3 We require, in addition, that the intersection of any two such regions Mi and Mj is at most three-dimensional. If it is three-dimensional, we give it a name and call M i ∩ M j =: Nij an interface, whose orientation is chosen so as to match the induced orientation from Mi. In other words N−1 = Nji. If, on the other hand, M ij j intersect in a two-dimensional surface, we call it a corner C, and we shall also assume, for further consistency, that all corners in the interior of M are adjacent to four definite such regions — four and not three or five, simply because we require that the internal boundaries are null, in which case all such corners arise from the intersection of two such null surfaces. See figure 1 for an The requirement that the boundary ∂M ij of all four-dimensional building such internal boundary N ij there exists a spinor6 ℓiAj : N ij → C2 and a spinor-valued twoin (2.2a)) admits the decomposition 5The orientation of every Mi matches the orientation of M, and every Mi is homeomorphic to a closed four-ball in R4. The vertical position of the (ij)-indices has no geometrical significance. Simplifying our notation, we which contain no local degrees of freedom inside. Non-trivial curvature is confined to threeas to say that N M is the canonical embedding of N ij into M. That this is the same ij is null, has been shown in section 2.1 above following the discussion of The discontinuity of the metric across the null surface will be encoded in a discontinuity of the spinors and the connection. Along a given null surface Nij , we will have two kinds of [Aij ]ABa denotes the pullback of the SL(2, C) connection from the bulk Mi to the boundary component N mon interface N ij ⊂ ∂Mi, and [Aji]ABa is the pull-back from M j to Nji ⊂ ∂Mj . The comij between Mi and M j carries then two independent SL(2, C) connections [Aij ]ABa and [Aji]ABa. What is the relation between the two? Consider first the SL(2, C) transformation [hij ]AB, ℓiAj = [hji]ABℓjBi, which brings us from one frame to the other. Such an SL(2, C) gauge transformation exists provided the matching conditions (3.3) are satisfied, which has been shown in (3.26) above. The spinors are gauge equivalent, but the connections may not: there is, in general, a nonvanishing difference tensor [Cij ]ABa between the two SL(2, C) connections, and we define it as follows: Aij = hjidhij + hjiAjihij + hjiCjihij . Construction of the action. The action will consists of a contribution from every fourdimensional cell Mi, a boundary term from every interjacent null surface, and a corner term from any two such null surfaces intersecting in a two-dimensional face. The bulk which is an sl(2, C)-valued two-form with dimensions of ~. The bulk contribution to the action is therefore nothing but the integral the integrability condition DΠAB = dΠAB − 2AC (A ∧ ΠB)C = 0. The next term to add is the three-dimensional boundary term from the interface between two adjacent regions. This boundary term has a twofold job: it cancels the connection variation from the bulk, and it imposes that the intrinsic boundary geometry is null. It consists of the covariant symplectic7 potential A ∧ DjiℓjAii + cc., plus additional constraints: the glueing conditions (3.3), which match the spinors from the ∧ πiAj ℓiAj − πjAiℓjAi + Ψiajb πiAj aπiAj a − πjAiaπjAia arising from the ℓA-variation on N. We sum the bulk action (3.32) with the boundary action for the spinors (3.34) and the constraints (3.35) and also add the right corner term, which we will discuss below. The resulting action is then given by the expression = 2 X Z − 2 − 2 α ℓiAm ℓiAn − ℓjAm ℓjAn + ℓAmj ℓAmi − ℓnAj ℓnAi + cc. is the pullback of the self-dual connection from Mi to Nij ⊂ ∂Mi. M denoting the canonical embedding. Next, there is the reality condition: arising from the ℓA-variation of the coupled boundary plus corner terms. The resulting boundary conditions are Let us summarise and briefly explain the role of each term in the action (3.36). The tion. Next, there are the integrals over the internal boundaries Nij . The variation of the connection in the bulk yields a remainder at the boundary. The variation of the boundary connection couples then the boundary with the bulk, yielding the constraint The first sum P i goes over all four-dimensional bulk regions M second sum goes over all ordered pairs (Mi, Mj ), which share an interface Nij . This sum crucially contains both possible orientations, i.e. P Nij · · · = R The last integral is the corner term, which is a sum over all quadruples (Mi, Mj , Mm, Mn) that share a two-dimensional corner such that (Mi, Mm), (Mi, Mn) and (Mj , Mm) and and (Mm, Mn) only meet in the corner itself: M (Mj , Mn) each share a three-dimensional interface (e.g. Mi∩ for an illustration. We write P Mm = Nim), whereas (Mi, Mj ) i ∩ M j = Mm ∩ M n = Cimj n. See figure 1 [ijmn] to say that any such corner appears with only one possible orientation in the sum. This orientation is chosen arbitrarily and can be absorbed are obtained from the two-dimensional remainder for the area two-form (2.22) as in equation (2.21) above. Finally, there are the glueing Notice also that we have used a condensed notation in (3.36), we dropped all wedge products, and suppressed all (ij)-indices in the boundary variables. The second line in (3.36) has to be understood, therefore, in the following sense A ∧ Dij pendent Levi-Civita density ǫˆabc. Equations of motion In this section, we study the equations of motion as derived from the action (3.36). This is a preparation for the next section, where we will find a family of explicit solutions representing plane fronted gravitational waves, which are exact solutions of both the discretised theory and general relativity as well. Some of the equations of motion derived from the action (3.36) have already been constraint (3.27), i.e. ∀p ∈ Mi : FAB(p) = Finally, a word on the Lagrange multipliers: the continuity conditions (3.41) and (3.42) of the momentum density which is a spinor-valued vector density of weight one, with ǫˆabc denoting the metric indeWe then also have the variation of the self-dual connection AAB, which gives the integrathe internal null boundaries { ij }, where we find a number of additional constraints. The they are obtained from the stationary points of the action (3.36) with respect to variations added to the action in order to impose the glueing conditions across the interface, namely: the area-matching condition (3.3a) and the shape-matching condition (3.3b). Finally, we have the relation between the bulk and the boundary, which is provided by the glueing to the boundary. As we have seen previously in section 2.1 above, this is the same as to say that the boundary is null. The action (3.36) contains the boundary spinors as additional configuration variables, and the action is stationary with respect to them provided additional equations of motion are satisfied along the system of interfaces. For simplicity, consider only a single such from, say, below and above the interefacee: if ϕ : M ֒ N−1) is the Goingebacek to the definition of the action (3.36), we then see that the variation of the boundary spinors yields the equations of motion DaℓA = Da ℓA = ωa − β + i λa ℓA + ΨabπAb, where Da (and Da) is the covariant derivative with respect to the SL(2, C) connection in M (and M) to the inteerface N ebetween M and M. Three immediate observations: integrability, geodesity and the expansion of the Before we proceed, we need to develop some better intuition for this system of equations (3.47). First of all, we can see that they are consistent with the torsionelss geometrical interpretation. It measures the expansion ϑ(ℓ) of the null surface. This can be seen as follows. The expansion ϑ(ℓ) can be defined by of ϑ(ℓ) is gauge dependent — it depends on a representative ℓa of the equivalence class of intrinsic to N. In terms of the boundary spinors, the two-dimensional area element is Going back to the equations of motion (3.47), we find Finally, let us turn to the null generator ℓa itself. It is geodesic, and this can be seen as follows. First of all, we write this vector field (modulo an overall normalisation) in terms of the boundary spinors, obtaining which follows from the area-two from as written in terms of the boundary spinors, and iℓAℓ¯A¯, see for instance equation (3.14) above. Now, the covariant derivative of ℓA along the null generators is proportional to ℓA itself, which follows from ∝ ℓ ⇔ ℓAℓaDaℓA = 0, and equation (3.47a) by noting that ∝ iℓAℓ¯A¯, which means that the integral curves of ℓa to the SL(2, C) connection. The SL(2, C) connection is torsionless (3.33), hence the null ≡ iℓAℓ¯A¯ are auto-parallel curves with respect generators of N are geodesics. Difference tensor. The null surface N bounds two bulk regions M (from above) and M (from below). There are then two SL(2, C) connections on N, one (namely AABa) from bfelow the interface, the other (namely AABa) from above. Their relative strength is given by a difference tensor CABa, whose algeebraic form is determined as follows. First of all, gauge equivalent, which means that they are related by an SL(2, C) gauge transeforme ation. We can thus write We then have the difference tensor (3.30) on N, which is defined as ≈ ℓA, AABa ≈ AABa + CABa. Subtracting the covariant differential of ℓA from the differential of ℓA, i.e. subtracting (3.47c) This implies that CABa admits the decomposition CABa = − β + i ℓ(AkB)λa − β + i ℓAℓBΓa, algebraic constraint which is a consequence of (3.57b). Finally, the difference tensor is subject to one additional constraint: the field strengths as induced from the two sides are gauge equivalent, which is sourced by the boundary spinors (as in (3.46)). The boundary spinors are gauge equivalent DCAB +eCAC ∧ CC B = 0. In the next section, we will demonstrate that explicit solutions to these equations exist: plane gravitational waves solve the system of equations (3.47), (3.58), (3.60), (3.59) for certain boundary spinors and a definite difference tensor CAB at the interface. Special solutions: plane-fronted gravitational waves This section is dedicated to finding explicit solutions to the equations of motion in the neighbourhood of an interface. Rather than exploring the entire solution space, we study only a single family of solutions, thus giving a constructive proof of existence: there are non-trivial8 solutions to the equations of motion derived from the action (3.36), and the particular solutions thus constructed are distributional solutions of Einstein’s equations Consider thus a single interface N, bounding the four-dimensional regions M and M from above and below. We set the cosmological constant to zero, hence M and M are flaft, where {xµ } are inertial coordinates in M. The glueing conditions (3.1) imply that the intrinsic geometry of N is the same from the two sides. The discontinuity in the metric can, therefore, only be in the transversal direction, which motivates the following ansatz for the tetrad across the interface 8We will find solutions with a non-vanishing distributional Weyl tensor at the interface. Killing vector. Having solved the equations of motion for v < 0, and matched the geometries across the interface, we are now left to solve the equations of motion for v > 0. The only missing condition is to impose that the curvature vanishes for v > 0. A straightforward calculation reveals that this is possible if and only if ∂uf = 0. ∂z¯f − ∂zf¯ = 0. It is also instructive to have a look at the boundary spinors and compute them explicitly. Going back to our initial ansatz (3.61) for the tetrad and taking also into account the parametrisation (3.64) of the null tetrad, we immediately see that we are in a gauge for straightforward exercise returns the momentum spinors The two-dimensional volume element on N, as given by equation (2.11), is then simply In order to satisfy the equations of motion, this difference tensor must be equal to CABa as derived from the equations of motion for the action (3.36). Going back to the decomposition of CABa into ℓA and kA, as in (3.58) above, we thus find the conditions complex. Unless the Barbero-Immirzi parameter goes to zero, which is a singular limit in the original selfdual action (2.14), the equation (3.66a) can be only imposed, therefore, for Going back to the equations of motion for the spinors, i.e. equations (3.47) above, we can Finally, we can now also compute the distributional curvature tensor across the interface. Following Penrose’s conventions [23], we compute the irreducible components of the field strength of the SL(2, C) connection, namely is the traceless part of the Ricci tensor, which is the irreducible spin (1, 1) component of the The resulting geometry is a solution to Einstein’s equation with a distributional source field of a massless point particle [22]. Hamiltonian formulation, gauge symmetries Space-time decomposition of the boundary action The main purpose of this paper is to open up a new road towards non-perturbative quantum gravity. We have defined the action (3.36) and demonstrated that explicit solutions exist, which have a non-vanishing (yet distributional) Weyl curvature (3.72) in the neighbourhood of a null surface. The next logical step is to study the Hamiltonian formulation of the theory. First of all, we note that the bulk action (3.32) is topological. All physical degrees of freedom sit, therefore, either at the system of null surfaces { ij } or at the two-dimensional Cmn . As long as we are concerned with the canonical analysis on only one such null surface N alone, we can then also work with a simplified action, which is found We insert the equations of motion (3.45) for the self-dual two-form back into the bulk action, thus obtaining − 2 M FAB ∧ F AB = 2 ∂M Tr A ∧ dA + A ∧ A ∧ A 2 SCS, ∂M[A], which is the self-dual Chern-Simons action with complex-valued coupling constant Riemann curvature tensor. Both components vanish everywhere except at the null surface and the traceless part of the Ricci tensor is determined to be Tab = The appearence of the SL(2, C) Chern-Simons action in a four-dimensional theory may come as a surprise, but it has been anticipated by several authors, who have suggested that the so-called Kodama state, which is the exponential of the Chern-Simons functional for the underlying gauge group, plays a significant role for quantum gravity in four-dimensions [24– 29]. This paper confirms these early expectations. dimensional cells { M1, M2, . . . }. Each one of them has has the topology of a four ball, ij : ∂M i = S Nij . For definiteness, consider only one such three-surface N at the boundary between two bulk regions, say M and M. Integrating out the self-dual two-form action (4.1) alongfthe interjacent null surface N. Each one of these SL(2, C) Chern-Simons either side. The resulting coupled action is therefore given by 2 SCS, N[A] − 2 SCS, N[A]+ All terms in this boundary action have a straightforward geometrical interpretation: variation with respect to the self-dual connection imposes that the field strength FAB on N geometric, i.e. compatible with the existence of a signature (0++) null metric qab on N. Finally, we have the glueing conditions (3.3), which are obtained by demanding that the conditions impose11 continuity across the interface: the intrinsic geometry of N is the same whether we compute it from the boundary spinors on either side of the interface. Next, we write the action in a Hamiltonian form. This requires a clock — a foliation N ≃ [0, 1] × vector field ta. This should strike us as a surprise: a three-dimensional null surface always has a preferred time direction: the direction of its null generators. So how can it be that our action (4.3), which is meant to be an action for a null surface N, lacks such a preferred structure? The answer is simple: in our theory, there is no metric formulation to begin with, the metric is a derived or composite field. It exists only on-shell — only if the reality condi11See section 3.1 above.e 10The same is true for F AB and −π(AℓB)/γ. taking into account also the corner terms and boundary conditions at the two-dimensional intersections (see figure 1) will be left for the future. qab on N. Only for those configurations that satisfy the reality conditions (2.21) can such a metric be defined, according to the construction that has been given in section 3.1 above. We then choose a time function t, which is a mere coordinate, and a transversal vector ponents. With a slight abuse of notation, we denote the pull-back of the SL(2, C) bulk connection AAB ∈ Ω1(M : sl(2, C)) to the t = const. slices simply by the three-boundary N. Equation (4.4) defines the spatial components of the connection. Its t-component13 defines the Lagrange multiplier Finally, we also have the velocity with Lt denoting the Lie derivative along the vector field ta. AABa = ϕt∗[AAB]a, A˙ ABa = ϕt∗[LtAAB]a, ℓ˙A = LtℓA, In the following, we restrict ourselves to those parts of configuration space, where the area element (2.11) is non-degenerate on S, hence we assume in terms of components Finally, we define the velocities Notice, the component functions U a and Va have different density weights: Va is a one-form is a two-form (hence a density) on S. Both U a and Va are Lagrange multipliers, since the boundary action (4.3) contains no derivatives of them. Yet they are not completely arbitrary: the matching conditions (3.3) and the reality conditions (2.21) impose constraints on them. The reality conditions (2.21), which follows from the variation of the action (4.3) U a = U¯ a. On the other hand, there are the matching conditions (3.3), which follow from the variation the other side oef the interface, i.e. Va = V a, U a = U a, χeAa = ϕt∗[etyπA]a = Ue aπA + V aℓA. It is then useful to dualise the component functions U a and Va. We take the canonical Levi-Civita density ǫˆab on S, and define N a := ǫˆabU b, J a := ǫˆabVb, ∈ T S is a tangent vector, while J a ∈ Ω2(S : T S) is a vector-valued density. We insert the 2 + 1 decompositions for both the connection (i.e. (4.4), (4.5)) and for the boundary spinors (i.e. (4.7), (4.13)) back into the boundary action, and get γ ǫˆab AABaA˙ ABb − ΛABFABab + π − ϕℓA γ ǫˆab AABaA˙ ABb − ΛABF ABab + π − ϕℓA − ℓDaℓ S∂N = are the time components of the Lagrange multipliers imposing the area-matching constraint (3.3a), and the reality conditions (2.21). where ˇǫab is the inverse Levi-Civita density on S, implicitly defined through ǫˆacˇǫbc = δb . a Poisson brackets (4.17) and (4.18) and their complex conjugate, e.g. { canonical variables vanish. Next, we have the constraints. The variation with respect to the sl(2, C) Lie algebra Variation with respect to N yields the scalar constraint The Lagrange multiplier N a gives rise to the vector constraint S[N ] = ΛAB h γ ǫˆabFABab + πAℓBi =! 0, ΛAB h γ ǫˆabF ABab + πAℓBi =! 0. Going back to the 2+1 split (4.15) of the action, we can immediately read off the symplectic structure. First of all, we see, that the spinors ℓA, ℓ A are canonical conjugate to the A = ϕt∗[πA] and π = eϕ∗[πA]. The fundamental Poisson which is given by πA(p), ℓB(q) = − ǫABδ(2)(p, q), AABa(p), ACDb(q) = + AABa(p), ACDb(q) = − γˇǫabδC(AδDB)δ(2)(p, q), Ha[N a] = Finally, we have the matching constraints M [ϕ] = Ma[J a] = − ℓADaℓ A =! 0, which are obtained from the variation of the action (4.15) with respect to the Lagrange multipliers ϕ and J a. Thus far concerning the constraints. The evolution equations, on the other hand, assume a Hamiltonian form as well: for any phase space functional F , its time evolution is governed by the Hamilton equations F = {H, F } , where the corresponding Hamiltonian is the sum over all constraints of the system H = S[N ] + Ha[N a] + Constraint algebra and gauge symmetries Having defined the Hamiltonian, we proceed to calculate the Poisson algebra among the constraints and check whether the constraints are preserved under the Hamiltonian flow (4.23). For the Gauss constraints (4.19) the situation is straightforward. We recover two functions on S, we find where we have defined the sl(2, C) Lie bracket GAB[Λ1AB], GCD[Λ2CD] = GAB [Λ1, Λ2]AB , GAB[Λ1AB], GCD[Λ2CD] = GAB [Λ1, Λ2]AB , [Λ1, Λ2]AB = [Λ1]AC [Λ2]C B − [Λ2]AC [Λ1]C B. The vector constraint, on the other hand, gives rise to two-dimensional diffeomorphisms modulo SL(2, C) gauge transformations. If N a and M a denote vector-valued test functions on S, we find after a straightforward calculation that Ha[N a], Hb[M b]o = −Ha[LN M a]+ N M a = [N, M ]a is the Lie derivative. of two-dimensional diffeomorphism and local SL(2, C) gauge transformations. First of all, {Ha[N a], πA} = Da(N aπA) = LN↑ πA, Hb[N b], AABao = of the Lie deerivaetive intoethe spin bundle. The third Poisson bracket (4.28b), on the other hand, returns a diffeomorphism only on-shell — only if, in fact, the Gauss constraint (4.19) is satisfied, in which case N bˇǫabǫˆcdF ABcd = N bF ABba = LN↑ AABa, For the Gauss constraint, the situation is easier. There we get the fundamental transwhich are the generators of right translations along the fibres of the SL(2, C) principal A and AABa is completely analogous. We then also have the matching consetraient (4.22ae), which Poisson commutes with the SL(2, C) connections AABa and AABa, but generates the U(1)C transformations = +ϕℓA, {M [ϕ], Ma[J a]} = 2Ma[ϕJ a]. immediately, as feor instanece Concerning the algebra of constraints, we only need to consider two further Poisson {S[N ], Ma[J a]} = {Ha[N a], Mb[J a]} = − Da (J aℓA) N bDbℓA+ If we perform a partial integration (N a and J a all have compact support) and bring all covariant derivatives on one side, this can be simplified to − N aDaℓAJ bDbℓA + Da J aℓA N bDbℓAi. {Ha[N a], Mb[J a]} = − Ma [LN J a] + where the Lie derivative of the vector-valued density J a on S is defined by LN J a = 2Db N [bJ a] + N aDbJ b. The equations (4.25), (4.27), (4.32), (4.33) and (4.35) give already all relevant Poisson brackets. All other Poisson brackets among the constraints can be inferred trivially from either (4.28) or (4.30). So as for e.g. {Ha[N a], S[M ]} = −S[LN M ], which is an immediate consequence of (4.28). First class and second class constraints. We have now collected all Poisson brackets that are necessary to identify the first class and second class constraints of the system. The vector constraint Ha[N a] (see (4.28)) generating two-dimensional diffeomorphisms on S is SL(2, C) gauge transformations (4.30) on either side of the ineterfaece. Equally, for the matching constraint: M [ϕ] is first class and generates the U(1)C transformations (4.31) of the boundary spinors. We are then left with the scalar constraint S[N ] (as in (4.20)) and the matching condition Ma[J a] (as in (4.22b)). The scalar constraint is second class, which is a consequence of (4.35). For the matching condition, the situation is more complicated. The constraint them is second class, all others are again first class. This can be seen as follows: in general, we will have that ℓADaℓA does not vanish14 on phase space. We can then parametrise the densitised vector J a as follows = z()ǫˆabℓADbℓA + z()d2v qab(β + i)ℓ¯A¯Dbℓ¯A¯, where qab is a fiducial signature (++) two-metric on S and d2v denotes the corresponding area element. The only relevant Poisson bracket for determining the second class component z() and z() for which the Poisson bracket {Ma[J a(z(), z())], S[N ]} vanishes on-shell 15 for ≈ 0, vant Poisson brackets involving Ma[J a(z(), 0)] weakly vanish as well, which is an immediate consequence of (4.28), (4.30) and (4.31). Hence the constraint straints (4.19), (4.20), (4.21), (4.22a), (4.22b) first-class. 15Given all constraints (4.19), (4.20), (4.21), (4.22a), (4.22b) are satisfied. That (4.41) defines another first class constraint follows from (4.33) and ≈ 0. All other constraints Poisson commute with (4.41). This can be inferred already from the infinitesimal gauge transformations (4.28), (4.30) and (4.31). We are thus left to identify the single second class component of Ma[J a]. It is given by the expression from (4.33) through MahJ a 0, iy() i, S[N ]o + cc. ≈ ≈ 4 6= 0, which does not vanish unless ℓADaℓA = 0. Gauge symmetries. We have now identified all first class constraints of the system: vector constraint (4.21) and the matching ceonditeion (4.22a), which generate two copies of internal SL(2, C) gauge transformations (on either side of the interface), two-dimensional diffeomorphisms of S and U(1)C gauge transformations (4.31). We then have additional first-class constraints, which can be identified with those components (4.40) and (4.41) of Ma[J a] that Poisson commute with the scalar constraint S[N ]. The geometric meaning of local SL(2, C) gauge transformations, U(1)C transformations and two-dimensional diffeomorphisms is clear,16 but what kind of gauge transformations are generated by those components of Ma[J a] that are first class? The answer is hidden in an additional gauge symmetry, which appears in the bulk. The bulk action (3.32) has in fact more symmetries than just four-dimensional diffeomorphisms and local SL(2, C) transformations. It enjoys a further gauge symmetry, which renders the entire theory topological. The action is invariant under the infinitesimal shifts equation (2.5) and (2.20). A → ez/2ℓA and ηA → e−z/2, see of the boundary action breaks this shift symmetry, but only partially. To understand this more explicitly, let us first define illustrate that Ma[J a] generates a version of the shift symmetry (4.45) on the boundary. Using the Poisson brackets (4.17) and (4.18), a short calculation gives 1 J bˇǫbaℓAℓB, o = o = = J a(Daℓ(A)ℓB) + Notice that the last line is nothing but the covariant exterior derivative of the first. Comfor the shift symmetry in the bulk, with the gauge parameter J a at the boundary by The matching constraint Ma[J a] therefore generates a shift transformation (4.45) with gauge parameter (4.49). Notice, however, that this shift symmetry is broken partially by the addition of the reality conditions (4.20) at the boundary. Only if J a is of the particular form of (4.40) or (4.41), do we get a symmetry preserving the constraint hypersurface. For generic values of J a, the conditions (4.40) and (4.41) will be violated, and the Hamiltonian vector field {Ma[J a], ·} will lie transversal to the constraint hypersurface. Dimension of the physical phase space. In summary, the system admits four types of gauge constraints, and two second class constraints. The scalar constraint S[N ] and the y()-component (4.43) of the matching constraint Ma[J a] are second class. We then have the first class constraints, which are the vector constraint Ha[N a], generating diffeomorphisms side of the interface, the matching coenstraeint M [ϕ], generating U(1)C transformations and the remaining components (4.40) and (4.41) of Ma[J a], which generate the residual shift symmetry (4.48) at the boundary. The situation is summarised in the table 1 below. The counting proceeds as follows: we have two second class constraints and nineteen first class constraints; there are two independent components of the vector constraint, two Dirac classification two first class constraints three C-valued first class constraints three C-valued first class constraints one C-valued first class constraint one is second class, three are first class one first class constraint 2 × 2 = 4 2 × 6 = 12 2 × 6 = 12 2 × 2 = 4 1 + 2 × 3 = 7 directions. This renders the boundary theory topological. There are no local degrees of freedom. times six independent constraints generating the SL(2, C) transformations on either side of the interface, two independent components of the U(1)C generators M [ϕ] (the smearing function ϕ : S → C is complex) and three additional first class constraints, namely (4.40) and (4.41) generating the residual shift symmetry (4.48) at the boundary. The kinematical dimensions, every first class constraint removes two degrees of freeedom,ewhiech leaves us the scalar constraint S[N ] and the second class component (4.43) of Ma[J a], which leaves us with no local degrees of freedom along the three-dimensional interface. This renders the boundary theory topological. All physical degrees of freedom can only appear at the two-dimensional corners. Relevance for quantum gravity So far, we have only been studying the classical theory. The main motivation concerns, however, non-perturbative approaches to quantum gravity, such as, in particular, loop quantum gravity. Let me explain and justify this expectation, without going into the mathematical Loop quantum gravity can be based either on the phase space [1, 7] for an SL(2, C) connection or the phase space for an SU(2) connection [30, 31]. The complex variables have the advantage that local Lorentz invariance is manifest, though we then also need to impose additional reality conditions, which are otherwise already solved implicitly (see e.g. [32, 33] for a recent analysis on the issue). On the phase space for the complex variables, the symplectic structure is determined by the fundamental Poisson brackets for the self-dual variables which can be derived from the topological bulk action (3.32) as well. It was then noted [34, 35] that the theory can be discretised, or rather truncated, by requiring that the connection 17The symplectic structure is determined by (4.17) and (4.18). be flat everywhere except along the one-dimensional edges {Ei} of a cellular decomposition to the system of edges {Ei}, modulo SL(2, C) gauge invariance at the nodes p1, . . . , pN of = ǫAB, = −ǫAB, One then postulates Poisson brackets and shows that the symplectic reduction with respect to the area-matching constraint One of them is vector to the face fl dual to the link. This normal is often required to be timelike, if it is, however, null rather than timelike, the condition simplifies: there is then always a A on fl, such that nAA¯ = iℓAℓ¯A¯, which implies that either ω Al are interchangeable, ℓA is unique modulo U(1)C transformations, and we can, therefore, always restrict ourselves to the case e.g. [36–38], a generalisation to null surfaces was proposed as well cf. [39]. 19By convention ΠAB is assigned to the fibre over the initial point of the underlying link. l transport along the link going from the intersection p towards either endpoint l(0) or l(1). This in turn suggests to identify the conjugate spinors with the two-dimensional surface implies to view the faces fl dual to the links as two-dimensional cross sections of threedimensional null surfaces Nij . For any such face fl there is then a three-dimensional internal side of a null interface N(ij)(l) shining out of fl. shining out of the faces fl dual to the links of the graph. The Poisson brackets (4.53), which were previously postulated, can be then derived from the Poisson brackets (4.17) of the discrete spinors is analogous to equation (3.46). The same happens for the link holonomy; the SL(2, C) parallel transport (4.52b) along a link is analogous to the SL(2, C) gauge transformtion (3.25) across the interface. Equally for the constraints: the areamatching condition and the reality conditions appear both in the discrete theory on a graph (as in (4.54) and (4.55)) and in the three-dimensional boundary theory (as in (4.22a) and (4.20)). There is no doubt that this correspondence must be worked out in more detail. So far, I find the analogy encouraging. It suggests that the kinematical structure of loop quantum gravity — graphs, operators and spin-network functions — can be all lifted along null surfaces obtaining a fully covariant picture of the dynamics in terms of a topological field theory with defects. Summary, outlook and conclusion This paper developed a model for discrete gravity in four spacetime dimensions where the only excitations of geometry are carried along curvature defects propagating at the speed of light. The resulting theory has no local degrees of freedom in the bulk, nontrivial curvature is confined to three-dimensional internal boundaries, which represent a system of colliding null surfaces. The theory is similar to Regge calculus [20] and other discrete approaches, such as ’t Hooft’s model of locally finite gravity [40], causal dynamical triangulations and causal sets [41, 42], but there are fundamental differences. First of all, and most crucially, we have a field theory for the Lorentz connection rather than a lattice model for the metric. This field theory is topological and the underlying spacetime manifold splits into a union of four-dimensional cells { Mi} : M = SiN=1 Mi, whose geometry is either flat or constantly cell, there are no local degrees of freedom. Non-trivial curvature is confined to internal boundaries N ij = M i ∩ Mj , which are three-dimensional. The underlying action (3.36) is local, and splits into a sum over all four-dimensional building blocks, inner three-boundaries and two-dimensional corners. The internal boundary terms are necessary to have a well-posed variational principle. The problem of finding the correct dynamics for the curvature defects boils then down to finding the right boundary action, which cancels the connection variation from the bulk and consistently glues the Mi} across their boundaries. What is then the right boundary term? We are viewing gravity as a Yang-Mills gauge theory for the Lorentz group. At a boundary, a Yang-Mills gauge connection couples naturally to its boundary charges. Consider, for example, a configuration where the Yang-Mills electric field is squeezed into a Wilson line. Wherever this Wilson line ends and hits a two-dimensional boundary, a colour charge appears that cancels the gauge symmetry from the bulk. For an SL(2, C) Lorentz connection, the relevant charge is spin, which suggests to look for an action with spinors as the fundamental boundary variables. We proposed such an action in section 2 for a boundary that is null. That the internal boundaries are null rather than space-like or time-like is well desired, it imposes a local notion of causality: the field strength of the SL(2, C) connection is trivial everywhere except at the internal boundaries, which are null and represent, therefore, the world sheets of curvature defects propagating at the universal speed of light. This was further justified in section 3.4, where we gave a constructive proof of existence: we showed that there are explicit solutions of the equations of motion derived from the action (3.36), which represent impulsive gravitational waves. These are exact solutions of Einstein’s equations in the neighbourhood of an interface, and may describe e.g. the gravitational field of a massless point particle [22]. The model is specified by the action for the internal boundaries. This action assumes a surprisingly simple form. It defines, in fact, nothing but the symplectic structure for a spinor ℓA and its canonical momentum, which is (in three dimensions) a spinor-valued interpretation: the bilinear iℓAℓ¯A¯ defines the null generator of the interface, the spin (1,0) area element (2.22). The spinors at the interface are not completely independent, they are subject to certain constraints. First of all, we have the reality conditions (2.21), that the interface. Furthermore, there are the glueing conditions (3.3) that match the intrinsic three-geometry across the interface. The only metric discontinuity is in the transversal direction. Adding the glueing conditions to the action has a further effect: given a boundary is SL(2, C) invariant, but it violates this additional U(1)C gauge symmetry. The symmetry is restored by the area-matching condition (3.3a), which is added to the action by replacing A more thorough analysis of the gauge symmetries was performed in section 4. First of all, we noticed that the action in the bulk (3.32) is topological. We then integrated out at the three-dimensional interfaces. Next, we performed a 2 + 1 split of the boundary action and identified the symplectic structure of the theory. All fundamental variables appear twice, because every such interface N ij bounds two bulk regions M induce boundary variables from either side. We found that the SL(2, C) connection becomes Poisson non-commutative at the boundary, while the configuration spinor ℓA has the we found the canonical Hamiltonian (4e.24), wheich is a sum over the constraints of the system, which consist of the vector constraint generating two-dimensional diffeomorphisms, a pair of SL(2, C) Gauss constraints generating SL(2, C) gauge transformations on either side of the interface, the area-matching condition generating U(1)C transformations of the boundary spinors, and finally the three first-class components (4.40), (4.41) of the glueing conditions (4.22b), which generate the residual shift symmetry (4.45), (4.48). All of these constraints are first-class, the reality condition (4.20), on the other hand, is second class, reduction removes, therefore, forty dimensions from the kinematical phase space, which is forty-dimensional as well. This brought us to the conclusion that the theory has no local degrees of freedom, neither in the bulk nor at the three-dimensional internal boundaries. Relevance for quantum gravity. The proposal defines a topological gauge theory with defects. Solutions of the equations of motivation represent distributional spacetime geometries, where the gravitational field is trivial in four-dimensional causal cells, whose boundary is null. The geometry is discontinuous across these internal boundaries, which represent curvature defects propagating at the speed of light. Our main motivation concerns possible applications for non-perturbative approaches to quantum gravity, such as loop quantum gravity. We have a few indications supporting this idea: first of all, the model has a kinematical phase space, whose canonical structure is extremely close to recent developments in loop quantum gravity. In [36–38, 43–45], a new representation of loop quantum gravity has been introduced with spinors as the fundamental configuration variables. This construction was bound to the discrete phase space on a graph. An interpretation was missing for what these spinors are in the continuum. This paper closes this gap and provides a continuum interpretation: the loop gravity spinors are the canonical boundary degrees of freedom of the gravitational field on a null surface. The most interesting indication in favour of our proposal concerns its dynamical structure. The theory is topological and this suggests that the transition amplitudes, which are formally given by the path integral ordinary integrals (or sums) over the moduli of the theory. This would be reminiscent of quantum gravity in three dimensions, where the Ponzano-Regge amplitudes can be written as a product over SU(2) group integrals for each edge times SU(2) delta functions imposing the flatness of the connection (see [46] for a recent derivation). Such moduli exist, and the simplest example is the four-volume20 4Vol[Mi] = of a given four-cell Mi, or the trace of the SL(2, C) holonomy around the perimeter of a corner. The existence of such non-local observables is an important hint that the formal definition of the path integral (5.2) has a mathematical precise meaning and defines a socalled spinfoam model, which is given by certain fundamental amplitudes assigned to the adjacency relations of the underlying cellular decomposition of the four-manifold M (such as in three dimensions where the 6j-symbol defines the vertex amplitude for the Ponzano Perspectives. The amplitudes (5.2) are defined for a given and fixed family of fourdimensional cells { Mi}, which are glued among bounding interfaces { ij }. This combinaN torical structure is an ad-hoc input, which enters the classical action (3.36) as an external background structure. How are then different discretizations { Mi} with different combinatorial structures supposed to be taken into account? There are two possible answers to this question: in the first scenario, the full theory will be defined through a continuum limit, which sends the number of four-dimensional cells to infinity. The definition of the theory would then most likely include some sort of renormalisation group flow, which would give a prescription for how to take this limit in a rigorous manner. The main conceptual difficulty with such an approach is that there is no fundamental lattice scale entering the 20That the four-volume 4Vol[Mi] defines an observables is straightforward to see: clearly, it is invariant under local Lorentz transformations and diffeomorphisms that preserve the bulk region Mi. It is also integral ∝ R action (3.36). Indeed, it is the gravitational field itself that determines the size of the individual building blocks, and this makes it difficult to identify the correct variables and the correct notion of scale to study the renormalization group flow. Therefore, more sophisticated tools and techniques such as those developed for Regge calculus [2–4] and so-called spinfoam models [5, 6, 47, 48] may be required. The second possibility, which I find more appealing, is a more radical idea. In this scenario, the amplitudes for a given and fixed configuration of four-cells would be seen as Feynman amplitudes for an auxiliary quantum field theory. To define the entire theory, one would then sum over an infinite, but most likely very preferred class of combinatorical structures, which would arise from the perturbative expansion of the auxiliary field theory. The approach would be conceptually very similar to group field theory [49–51], where the gravitational path integral on a given simplicial discretisation arises from the perturbative expansion of a quantum field theory over a group manifold. Finally, there is one obvious open question that I have avoided altogether, namely how the two physical degrees of freedom of general relativity should come out of the model. This question is certainly related to the previous question regarding the continuum limit, but some hints of an answer should already appear at the level of the microscopic theory, which is defined by the action (3.36). This action was constructed such that the solutions of the equations of motion represent four-dimensional distributional geometries, where the curvature is trivial in four-dimensional cells, which are glued among bounding null-surfaces. The geometry is described in terms of SL(2, C) gauge variables (an SL(2, C) connection in the bulk coupled to spinors at the internal null boundaries). If we then take the quotient by the internal SL(2, C) gauge transformations, we are left with a theory that can only be described by a metric and a connection. The connection satisfies the torsionless condition (3.33), hence we expect that the only relevant degrees of freedom are captured by the metric, which is now locally flat. We then saw in section 3.2 that special solutions of the equations of motion exist that have a non-trivial distributional curvature tensor at the defect: the resulting Weyl tensor is of Petrov type IV, thus describing transverse gravitational radiation. But we then also saw that there are solutions where both the traceless part of the Ricci tensor (3.73) and the Weyl tensor (3.72) are non-vanishing, the Ricci tensor being tributional matter (as in e.g. string theory), with a distributional energy momentum tensor gravity with more than just two propagating degrees of freedom. A minimal example for such a theory is given by the Starobinsky model [52] of inflation, which has three propagating degrees of freedom (which are given by the two polarisations of gravitational radiation, and one additional spin-0 scalar mode). I find this idea very promising and exciting, and it is, in fact, the line of reasoning that I am currently investigating. A more rigorous analysis will be presented in an upcoming article, which is currently under preparation. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research and Innovation. Spinors and world tensors Following Penrose’s notation, we write ℓA with A, B, C, . . . to denote a two-component spinor that transforms under the fundamental representation of SL(2, C), primed indices A¯, B¯, C¯, . . . refer to the complex conjugate representation. The indices are raised and group action. Our conventions are ℓA = ǫBAℓB, ℓA = ǫABℓB, ℓ¯A¯ = ǫ¯B¯A¯ℓ¯B¯ , ℓ¯A¯ = ǫ¯A¯B¯ ℓ¯¯ , B with ǫAC ǫBC = δA. The relation between spinors and internal Minkowski vectors vα is B matrix representation the generalised Pauli identity α maps an internal Lorentz vector vα ∈ R4 into an anti-hermitian21 map. This isomorphism can be generalised to any world tensor. It maps the Lorentz cover of the restricted Lorentz group) into a proper orthochronous Lorentz transformation = 1, σAC¯ ασ¯C¯Bβ = −δBAηαβ − 2ΣABαβ, are the self-dual generators of SL(2, C). Equation (A.3) implies that the matrices relations of the Lorentz group. Indeed, we have Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Phys. Rev. Lett. 57 (1986) 2244 [INSPIRE]. Limit, Nucl. Phys. B 245 (1984) 343 [INSPIRE]. Calculus, Class. Quant. Grav. 5 (1988) 1543 [INSPIRE]. [5] B. Bahr and B. Dittrich, Breaking and restoring of diffeomorphism symmetry in discrete gravity, AIP Conf. Proc. 1196 (2009) 10 [arXiv:0909.5688] [INSPIRE]. Phys. Rev. D 80 (2009) 124030 [arXiv:0907.4323] [INSPIRE]. Class. Quant. Grav. 21 (2004) R53 [gr-qc/0404018] [INSPIRE]. [13] A. Ashtekar and J. Lewandowski, Quantum theory of geometry. 1: Area operators, Class. Quant. Grav. 14 (1997) A55 [gr-qc/9602046] [INSPIRE]. Phys. Rev. D 56 (1997) 3490 [gr-qc/9612035] [INSPIRE]. Gen. Rel. Grav. 2 (1971) 303 [INSPIRE]. Fields. Volumes 1 and 2, Cambridge University Press, Cambridge (1984). [26] L. Smolin, Unimodular loop quantum gravity and the problems of time, Phys. Rev. D 84 (2011) 044047 [arXiv:1008.1759] [INSPIRE]. non-planar graph operator and 4D quantum gravity with a cosmological constant: Semiclassical geometry, Nucl. Phys. B 900 (2015) 1 [arXiv:1412.7546] [INSPIRE]. with a Cosmological Constant from Three-dimensional Holomorphic Blocks, [30] J.F. Barbero G., Real Ashtekar variables for Lorentzian signature space times, Phys. Rev. D 51 (1995) 5507 [gr-qc/9410014] [INSPIRE]. [31] G. Immirzi, Real and complex connections for canonical gravity, Class. Quant. Grav. 14 (1997) L177 [gr-qc/9612030] [INSPIRE]. [32] W. Wieland, Complex Ashtekar variables and reality conditions for Holst’s action, Annales Henri Poincaré 13 (2012) 425 [arXiv:1012.1738] [INSPIRE]. Foams, in Proceedings of the 3rd Quantum Gravity and Quantum Geometry School, representation of loop quantum gravity, Phys. Rev. D 94 (2016) 086009 [arXiv:1609.02219] theory, in Loop Quantum Gravity, The First Thirty Years, A. Abhay and J. Pullin eds., volume 4, World Scientific, 2017 [arXiv:1409.1450] [INSPIRE]. spin foam models, New J. Phys. 14 (2012) 035008 [arXiv:1109.4927] [INSPIRE]. First Thirty Years, A. Abhay and J. Pullin eds., volume 4, World Scientific, 2017 Holst-Plebanski gravity, Phys. Rev. D 85 (2012) 044003 [arXiv:1111.5842] [INSPIRE]. [1] A. Ashtekar , New Variables for Classical and Quantum Gravity , [2] G. Feinberg , R. Friedberg , T.D. Lee and H.C. Ren , Lattice Gravity Near the Continuum [3] J.W. Barrett and R.M. Williams , The Convergence of Lattice Solutions of Linearized Regge [4] J.W. Barrett , A convergence result for linearised Regge calculus , Class. Quant. Grav . 5 [6] B. Bahr and B. Dittrich , Improved and Perfect Actions in Discrete Gravity, [7] A. Ashtekar , Lectures on Non-Pertubative Canonical Gravity, World Scientific ( 1991 ). [8] C. Rovelli , Quantum Gravity, Cambridge University Press, Cambridge ( 2008 ). [9] T. Thiemann , Modern Canonical Quantum General Relativity , Cambridge University Press, [10] A. Ashtekar and J. Lewandowski , Background independent quantum gravity: A Status report , [11] A. Perez , The Spin Foam Approach to Quantum Gravity , Living Rev. Rel . 16 ( 2013 ) 3 [14] M.P. Reisenberger and C. Rovelli , ' Sum over surfaces' form of loop quantum gravity , [15] J. Engle , R. Pereira and C. Rovelli , Flipped spinfoam vertex and loop gravity , [16] J. Engle , E. Livine , R. Pereira and C. Rovelli , LQG vertex with finite Immirzi parameter , [17] T. Thiemann , Quantum spin dynamics (QSD), Class . Quant. Grav. 15 ( 1998 ) 839 [18] T. Thiemann , Quantum spin dynamics . VIII. The Master constraint, [19] L. Lehner , R.C. Myers , E. Poisson and R.D. Sorkin , Gravitational action with null [20] T. Regge , General relativity without coordinates , Nuovo Cim . 19 ( 1961 ) 558 [INSPIRE]. [21] J.B. Griffiths , Colliding plane waves in general relativity , Oxford University Press ( 1991 ). [22] P.C. Aichelburg and R.U. Sexl, On the Gravitational field of a massless particle , [23] R. Penrose and W. Rindler , Spinors and Space-Time , Two-Spinor Calculus and Relativistic [24] H. Kodama , Holomorphic Wave Function of the Universe , Phys. Rev. D 42 (1990) 2548 [25] L. Freidel and L. Smolin , The Linearization of the Kodama state, [27] W. Wieland , Complex Ashtekar variables, the Kodama state and spinfoam gravity , [28] H.M. Haggard , M. Han , W. Kamiński and A. Riello , SL(2, C) Chern-Simons theory, a [29] H.M. Haggard , M. Han , W. Kamiński and A. Riello , Four-dimensional Quantum Gravity [33] S. Alexandrov and K. Krasnov , Hamiltonian Analysis of non-chiral Plebanski Theory and its [34] E. Bianchi , Loop Quantum Gravity a la Aharonov-Bohm , Gen. Rel. Grav. 46 ( 2014 ) 1668 [35] L. Freidel , M. Geiller and J. Ziprick , Continuous formulation of the Loop Quantum Gravity phase space , Class. Quant. Grav . 30 ( 2013 ) 085013 [arXiv:1110.4833] [INSPIRE]. [36] L. Freidel and S. Speziale , From twistors to twisted geometries, [37] W.M. Wieland , Twistorial phase space for complex Ashtekar variables , Class. Quant. Grav . 29 ( 2012 ) 045007 [arXiv:1107.5002] [INSPIRE]. [38] S. Speziale and W.M. Wieland , The twistorial structure of loop-gravity transition amplitudes , Phys. Rev. D 86 ( 2012 ) 124023 [arXiv:1207.6348] [INSPIRE]. [39] S. Speziale and M. Zhang , Null twisted geometries , Phys. Rev. D 89 (2014) 084070 [40] G. 't Hooft, A Locally finite model for gravity, Found. Phys . 38 ( 2008 ) 733 [41] R. Loll , Discrete approaches to quantum gravity in four-dimensions , Living Rev. Rel . 1 ( 1998 ) 13 [gr-qc /9805049] [INSPIRE]. [42] R.D. Sorkin, Causal sets: Discrete gravity , in Proceedings of the Valdivia Summer School , A. Gomberoff and D. Marolf eds., 2003 , pp. 305 - 327 [gr-qc/0309009] [INSPIRE]. [43] M. Dupuis , S. Speziale and J. Tambornino , Spinors and Twistors in Loop Gravity and Spin [44] E. Bianchi , J. Guglielmon , L. Hackl and N. Yokomizo , Loop expansion and the bosonic [45] E.F. Borja , L. Freidel , I. Garay and E.R. Livine , U(N) tools for Loop Quantum Gravity: The Return of the Spinor , Class. Quant. Grav. 28 ( 2011 ) 055005 [arXiv:1010.5451] [INSPIRE]. [46] W.M. Wieland , One-dimensional action for simplicial gravity in three dimensions , Phys. Rev. D 90 ( 2014 ) 044008 [arXiv:1402.6708] [INSPIRE]. [47] B. Dittrich , The continuum limit of loop quantum gravity - a framework for solving the [ 48] B. Dittrich , F.C. Eckert and M. Martin-Benito , Coarse graining methods for spin net and [49] D. Oriti , Group Field Theory and Loop Quantum Gravity , in Loop Quantum Gravity, The [50] D. Oriti , Group field theory as the 2nd quantization of Loop Quantum Gravity , [51] A. Baratin and D. Oriti , Group field theory and simplicial gravity path integrals: A model for [52] A. Kehagias , A.M. Dizgah and A. Riotto , Remarks on the Starobinsky model of inflation and

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Wolfgang Wieland. Discrete gravity as a topological field theory with light-like curvature defects, Journal of High Energy Physics, 2017, 1-43, DOI: 10.1007/JHEP05(2017)142