JHEP05(2017)142 ld Springer h defects. amiltonian May 16, 2017 May 29, 2017 : : March 29, 2017 : November 23, 2016 : for non-perturbative es are a spinor and a at the speed of light. Revised Accepted Published ical field theory for the rfaces, the gravitational 10.1007/JHEP05(2017)142 ere similar variables have field is trivial everywhere Received doi: Published for SISSA by . 3 1611.02784 The Authors. Lattice Models of Gravity, Models of Quantum Gravity, Topological Fie c

I present a model of discrete gravity as a topological field theory wit , [email protected] Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L2Y5,E-mail: Canada with light-like curvature defects Discrete gravity as a topological field theory Article funded by SCOAP Open Access Abstract: Wolfgang Wieland Keywords: Theories ArXiv ePrint: spinor-valued two-surface density, whichLorentz are connection coupled in to the aapproaches bulk. to topolog quantum I gravity, discuss such as therecently loop appeared relevance quantum as of gravity, wh well. the model The theory has no localexcept degrees at of a freedom number andfield of the gravitational can intersecting null be surfaces.The singular, underlying At representing these action a null is curvatureformulation. su local defect The and propagating canonically it conjugate is variables studied on in the both null its surfac Lagrangian and H JHEP05(2017)142 3 5 7 1 3 7 e- 32 12 16 19 22 26 27 34 39 22 (1.1) ing the bulk ge- hes and only the ]. For a manifold t the most natural 1 fundamental configu- tion [ cile the variational prob- is the gauge covariant exte- B ℓ B A ., A + + cc A ℓ A Dℓ = d ∧ A – 1 – A Dℓ π M ∂ Z . M ∂ is a spinor, whose square returns the null generators of the thre A ℓ is a spinor-valued two-form, A π We will then use this boundary term to discretise gravity by truncat 4.2 Phase space,4.3 symplectic structure, constraints Constraint algebra4.4 and gauge symmetries Relevance for quantum gravity 3.2 Definition of3.3 the action Equations of3.4 motion Special solutions: plane-fronted gravitational waves 4.1 Space-time decomposition of the boundary action 2.1 Self-dual area2.2 two-form on a Boundary null term surface on a null surface 3.1 Glueing flat four-volumes along null surfaces ometries to field configurations that are locally flat. The bulk action vanis rior derivative and where lem with the boundary conditions.such If boundary the term boundary is is null, given we by will the see tha three-dimensional integral with boundaries, the action acquires boundary terms, which recon 1 Introduction We will consider gravity asration gauge variables theory in for the the bulk Lorentz are group. the The tetrad and the self-dual connec A Spinors and world tensors 5 Summary, outlook and conclusion 4 Hamiltonian formulation, gauge symmetries 3 Discretised gravity with impulsive gravitational waves Contents 1 Introduction 2 Boundary spinors in GR dimensional null boundary JHEP05(2017)142 (1.3) (1.2) ivity is a dif- e some indica- ns of the theory is distributional. ], but the precise h possible higher ravity, it becomes 18 ative approaches to aper presents a pro- , ributional geometries ]. The resulting semi- other words, the Ein- ree-dimensional quan- geometries in terms of ea, outside of this one- of a network of three- 17 given by the covariant of motion derived from of light. 13 more difficult question. , instein’s equations with , 9 ational field, and we can hose developed for Regge
towards non-perturbative etries, whose Hamiltonian ) sents a potential curvature 12 s [ ( metric) glued among bounding p, l , (3) c δ , and all quantum numbers are sent ) m ]. In loop quantum gravity, geometry 0 (0++) s e ( b 11 l i l → – e E limit of the amplitudes will define a clas- 7 ~ ) ilm 0 s ǫ s ( ] may provide useful tools for the future. a d 6 l → dual to a cellular decomposition of the spatial abc , – 2 – d ˆ ǫ 5 ~ L s 1 2 d l = Z a i , . . . , l l 1 E X l ) = ] or Thiemann’s canonical program [ p ( null boundaries and two-dimensional corners contribute non- 16 a i – E 14 , 11 internal ] and related approaches cf. [ 4 – 2 Whether the continuum limit exists and brings us back to general relat The main motivation concerns possible applications for non-perturb represent four-dimensional Lorentzian geometries,dimensional which null surfaces are (equipped built with a signature ficult question. Thistions paper in does favour not of provide our an proposal: answer. first of We all, can we only will giv show that the solutio sical theory of discrete,dynamics or is rather formulated distributional, in spacetime termsposal geom of for gauge such connection a variables. theory,quantum and This gravity. may, p therefore, open up a new road stein equations are satisfiedorder locally. curvature Whether corrections this holds andSophisticated on running coarse all graining coupling scales, and constants, wit averagingcalculus is techniques, [ such a as t two-surfaces. We will then finddefect. that every We such will null also surface show repre the that discretised there exist action solutions that ofnon-vanishing the represent Weyl equations curvature distributional in solutions the of neighbourhood E of a defect. In to infinity, while keeping fixedThe semi-classical the electric eigenvalues field of geometric operators — expect, therefore, that the semi-classical transition amplitudes are unknown.tum We geometries know, represent however, that distributional the excitations th of the gravit classical geometry — in the naive limit, where quantum gravity, such as loopis quantum described gravity [ in terms of the inverse and densitised triad has support only along the links manifold. There is no geometry,dimensional no fabric notion of of volume, space. distanceat and What ar is the then quantum the level? dynamicsspinfoam for approach There these [ are dist several proposals, such as those trivially. The resulting action definesa a topological theory field of theory distributional with four- defects that propagate at the speed three-dimensional which is the gravity analogue of thean Yang-Mills operator, electric whose field. flux across In a quantum g surface has a discrete spectrum JHEP05(2017)142 , . ) b 1 2 A β , dle, ℓ can e orm (0 (2.5) (2.4) (2.3) (2.1) a ) and , and A (2.2a) (2.2b) α ℓ e M ed tensor +++ αβ η − into = . N ab A g ℓ 2 explain it in more , the spinor a φ/ miliar to the reader, i ℓ e in a four-dimensional r . It always exists then a 7→ N N A . . This is a key observation ℓ β referring to the fundamental spin A to e η r bundle over either all of spacetime to be parallizable, hence there ∧ denote its future pointing null , , he metric signature is ( AB β β α e e e Σ M ] N . β ∧ ∧ ¯ T A . ¯ belong to the complex conjugate spin ℓ ) α α , γ , A, B, C, . . . e e A B ∈ . We can then always find a spinor α β ℓ ℓ a e γ a ¯ [ ℓ Bβ Bβ A a diagonalising the metric ¯ that map spinors into internal Minkowski ( such that A η ∧ ¯ ℓ ¯ 1 8 C,... C C e η a A α ¯ α σ σ − ¯ α ¯ σ e B, N A e = ¯ 7→ 2 ¯ A Aα Cα = ¯ 1 – 3 – A, = σ a A C √ on . We assume ℓ ¯ ! AB σ σ ) phase transformations ) − ¯ αβ 4 4 1 1 B 2 Σ ab ¯ ∗ Σ A C = ∅ , g = = ϕ ¯ : Σ a are internal Minkowski indices. For the associated spinor bun U(1) ¯ ℓ B M ¯ N ( A AB ( B ¯ Σ Σ 2 1 for further details on the notation. of the self-dual two-form ∅ − A Ω with a spinor-valued two-form Σ A is null. Let then . Primed indices ∈ AB

A be the canonical embedding of the null surface ) α, β, γ, . . . ℓ C Σ A N , ∗ η M ϕ , the pull-back of the self-dual component of the Plebański two-f SL(2 for further details on the notation. → ֒ N A N : ϕ ) of a spinor 2.5 are abstract tensor indices labelling the sections of the tenso admits a very simple algebraic description: it turns into the symmetris β representation of e Let now Consider thus an oriented three-dimensional null surface The three-manifold Concerning the notation, the conventions are the following: t ∧ 0) 1 , α 1 2 spinor-valued two-form consider the pull-back we could also write vectors. In terms of the Dirac gamma matrices, which may be more fa Lorentzian spacetime manifold be determined only up to residual Clearly, there is an additional gauge symmetry: given the null vecto generator, which is unique up to boosts and it is importantdetail. for the further development of this paper. Let us we use an index notation as well: its sections carry indices which squares to a, b, c, . . . ( which are built from the soldering forms product ( e or some submanifold in there. See the appendix 2.1 Self-dual areaOn two-form on a a null surface null surface 2 Boundary spinors in general relativity representation. See the appendix is a spin structure and a global frame field and split it into its self-dual and anti-self-dual components We define the Plebański two-form JHEP05(2017)142 , ) A k AB thus (2.6) (2.9) (2.7) (2.8) ( (2.10) (2.11) (2.12) reality S N = and their is outgoing to a A ℓ . We can then . This dyad k a AB ℓ N , S . We then have A = 1 ℓ endent spinor N . , we can thus say that A 0 0 ℓ ors 2 < ) A > y ) k ) y y , ∂ , and say that , ∂ , otherwise inwardly oriented, = x , ∂ x C ∂ . x ∂ M with ) ( B ( ∂ ℓ ( g ε B . A ε ℓ . A A η − k A . ℓ ( ) A k y = 0 . The pull-back of AB is positively oriented in ε η of the self-dual area two-form into b = 0 ǫ ) , ∂ ℓ N C , and if a y : ¯ y , b ) B + i Z vanishes, but it also implies the ¯ ∂ β a ℓ y A AB , ∂ i } ( e B k a x B ν Σ , ∂ g A ) − k ℓ ε ∗ ) a x , ∂
A x . ϕ a , k . αβ = , ∂ ℓ k + i , which is based on ¯ ε A a ( . A straightforward calculation shows then that the . , ∂ Σ a ε ν ℓ has an immediate geometrical interpretation: a ) and identify t ∗ x A , ℓ t { C = ℓ C a ∂ + cc ϕ + t A ( is positively oriented in – 4 – , and it is incoming if Z ( µ ε 2.6 ℓ g ) B , which measures the oriented area of any two- C on A . Indeed a y ℓ AB ) = q 2 η A ] = Σ ) = ( R ). in the space of symmetric bispinors , ∂ N y ℓ A ℓ ∗ a x C d : is immediate. We can say, in fact, that a pair of tangent vectors } µ η ϕ ∧ of 2.5 ) , ∂ x , if the triple ¯ a N B B d α = C ¯ N ℓ C A , ℓ e ( , which are complex-valued two-forms on C ¯ ǫ ( a A 2 Area[ t ( ∗ ε Z ( AB Ω is the null generator of ϕ ± , k Σ ∗ a B ∈ ] = and ℓ ϕ k C ε A ν , , which is future pointing, is an outgoing or incoming null generator , k a µ B ℓ ℓ Area[ A ℓ { , if the quadruple coincides with the orientation of C is positively oriented in ) y , we can go back to equation ( C inherits an orientation from , ∂ T x C = 0 ∂ ( in ε ν ) a y The Lorentz invariant contraction That , ∂ 2 is an outgoing null generator with respect to a x a ∂ ℓ sign of and this definition will not depend on the choice of which we then usecomponent to functions decompose the pull-back Next, we also know that such that we have a normalised spin dyad with respect to induces a triad with respect to the induced orientation choose an arbitrary future oriented time-like vector ( Whether this integral coincides with the metrical surface area vanishes, which implies, in turn condition depends on whether Since dimensional spatial submanifold This is the same as to say complex conjugate. This immediately implies that This can be seen as follows: first of all we choose a second linearly indep it defines the two-form We contract the free indices with all possible combinations of the spin which proves the desired equation ( JHEP05(2017)142 ) are 2.15 ¯ A (2.13) (2.15) (2.14) ¯ ℓ is both D ] · ∧ are related A, ¯ A A . ℓ ¯ llow from the has the simple η e the self-dual β ∧ ., s a functional of A AB ]+ ilable fields at the and η Σ A orsionless condition + cc = d + [ [ D A nless condition. With  ndary action is linear We have assumed that ondition does not hold αβ D les. re the boundary spinors Dℓ F most obvious candidates AB upling constants in front, variation of the boundary Σ ∧ ∧ . ns like ∧ A  η β e AB + cc , whose curvature is the self-dual ∧ Σ B α to the action without changing the A AB Λ e 6 A β , 1 β αβ δA  − F β − ∧ e ) ∧ β AB ∧ β . At the level of the action, the connection AB e e F α α Σ . We now use them to build a gravitational ∧ e e ∧ ∧ A ) of the self-dual area two-form on a null surface α – 5 – M ∧ ℓ α , we thus work with the bulk action e ∂ ) e ( Λ Z 2.5 AB β ∗ e and spin connection and the remainder ∧  ) + i) , which suggests that the boundary term is built from A α is a two-form, such that is called the Barbero-Immirzi parameter C is the field strength of the Lorentz connection, and the M + i) Σ β η e , ( Ba Z = 0 ( ν β A A ∗ αβ ( µ C η )  A i F 6 A Λ πG 1 and the tetrad B SL(2 πβG M ∧ ∧ − 8 Σ 16 β Z β µ α ∧ e α A C ] = ∧ i A ). What is then the right boundary action? The remainder ( A πβG α ( + 8 e A A, e ) and the 2.5 β [ is null, which implies that the self-dual area two-form 2 α M ] = − A S M 2.2a ∂ A, e AB themselves. Now, [ = d . We have denotes the complex conjugate of all preceding terms. M β A αβ . ℓ α S . This allows us to add the term F = dΣ F cc We then write the bulk action in terms of self-dual variables, which ar Working in a first-order formalism, we write the gravitational action a Consider then the variation of the action. The Einstein equations fo and AB in terms of boundary spinors Σ A equations of motion, which arethe the inclusion Einstein of equations a plus cosmological the constant torsio area two-form ( boundary action with spinors as the fundamental boundary variab the Lorentz connection form of equation ( is then independent of theoff-shell triad, which means that the torsionless c complex-valued three-forms, whose boundary integralsfor define the gravitational the boundary term at a null surface (expressio constant in front of to the latter by a total derivative). We are left to determine the co where boundary, which are functionally independentη of the connection, a the gauge covariant derivative of some boundary fields. The only ava is linear in thein connection, the and connection we as expect, well. therefore, Only that the the exterior bou covariant derivative N gauge covariant and linear in The last section gave a parametrisation ( 2.2 Boundary term on a null surface the boundary at the boundary. This boundaryaction, integral otherwise must the cancel entire against action the is not functionally differentiable. part of where variation of theD tetrad, the variation of the connection yields the t JHEP05(2017)142 ) of (2.20) (2.23) (2.18) (2.19) (2.17) (2.22) (2.16) (2.21) 2.5 and writing ω form on a null arn how to glue bounded by a null undary value ( M ) fully gauge invariant ., ion . ] 2.19 + cc ., , ℓ, ω A , gauge connection ℓ π ) |  C . + cc ω . A A [ B . ℓ A A ℓ − 2 z to render ( . N η U(1) A − S Ba ω Dℓ D = 0 ℓ e are not varied in the action, they are ( gauge symmetry appearing: the spinors A . , A ∧ + i) A ] + A ∧ A C π as an abbreviation for A β ℓ η A ( + 2 z η + cc A π A, e e + i A [ are not independent, for we have to satisfy the π A M U(1)  ℓ πβG i ) of the connection variation at the boundary. ℓ ∂ β – 6 – M ) is not invariant under this symmetry, but we and a M 8 A πβG ∂ A Z ∂ ℓ S 8 connection Z − A π −→ 2.15 = 2.16 π C = ) + i) := ] = i and + i A ] = generates both boost and rotations preserving the null A ε ℓ β A β a A ( , ℓ C U(1) π π D A , ℓ, ω , ℓ, ω η → π i as an equation of motion derived from the variation of the π ( | | πβG 8 N A [ AB : A, e [ Σ M z ∗ ∂ S S ϕ ), which now turn into 2.9 . The boundary action ( ¯ A are unique only up to transformations ¯ ℓ is therefore given by the expression simply by denotes the gauge covariant derivative A A ℓ a ℓ i N N ) is satisfied, the area is real and we can define the oriented area two- D Notice also, that the spinors Notice that there is now an additional and 2.21 A normal connection in the bulk and boundary. kept fixed in the variational principle, because they determine the bo reality conditions ( The introduction of an additional At this point, the boundary spinors where the gauge element where we introduced the momentum spinor surface the self-dual two-form surface may seem rather dull, butcausal it regions will across be a important bounding for null us surface. later, when we will le can easily make it invariant by introducing a fiducial The resulting boundary term is The entire action for the gravitational degrees of freedom in a reg which can be read off the remainder ( where η If ( JHEP05(2017)142 , : a N ℓ ,... (3.2) (3.1) A e , k hat are A e η , A e , ℓ is degenerate. etised gravity a a Hamiltonian e , ℓ ab he cosmological g ab faces as well. For ∗ e the discontinuity q e licit appearance of ) implies that they ϕ s m over all such four- dimensional regions, he bulk contribution tion that the elemen- e. In Regge calculus, = 3.1 ing its area times the is matched between the ab q ) through the definition of 2.19 the interjacent null surface ] and references in there. The 3 rom those lying above. The quantities . 19 a ℓ e η are future pointing. In a neighbourhood of , and equation ( a a ℓ e ℓ = e , a defines the single degenerate direction ab ℓ q e : a ℓ = R – 7 – ab → q N , which is the same as to say : a ℓ e η with null vector ∃ ∼ a ab ℓ q e ). Notice also that additional corner terms may be necessary 2.20 , and the null vector ], the Einstein equations are discretised by cutting the spacetime is oriented, and the null vectors ) is a generalisation — it is formulated in terms of spinors, and does 20 ), and we are left with a distributional boundary term, which can be describe the boundary as seen from below, the tilde quantitie N 2.19 (0++) ,... Λ = 0 A into four-simplices, and truncating the metric to field configurations t , k denotes the intrinsic three-metric from below A determines the geometry from the other side. M . There is then also η , ab ab q A q e = 0 , ℓ The task is then to generalise Regge calculus and find a theory of discr In our case, the interface is null, and the induced three-metric In the literature, other boundary terms have been used on null sur The theory is then specified by the matching conditions that determin a b The null surface ℓ we can thus distinguish points sitting below the null surface f 3 , ℓ ab ab q reorganised into a sumsurrounding over deficit triangles, angle. with every triangle contribut locally flat. The gravitational actionsimplices, for each the one entire manifold of is whichvanishes then contributes a (for a su bulk and boundary term. T while match from the two sides, i.e. In Regge calculusmanifold [ as well. We will introduce them below. 3 Discretised gravity with impulsive3.1 gravitational waves Glueing flat four-volumes along null surfaces a recent survey inboundary term the ( metricnot formalism, assume we that refer thethe to connection Barbero-Immirzi [ is parameter, torsionless, whichthe which enters momentum explains the spinor the ( action imp ( It has signature two sides. We require this condition as well, and thus impose that of the gravitational fieldit in is the the vicinity intrinsic of three-dimensional the geometry interjacent at null the surfac interface that where tary building blocks are flatwhich four-simplices. are We work instead flat withconstant), or four- and constantly whose curved boundary is inside null. (depending on the value of t in the connection formalism,quantisation. whose We action will propose is such still a simple theory enough by to dropping the admit assump describe the boundary from above. N q JHEP05(2017)142 ) ) a A A ab . ℓ ℓ on C e η c g k ) of : ∗ ∂ } (3.4) (3.6) ) are 2.10 b j ϕ = gauge (3.5a) (3.3a) would (3.5b) (3.3b) A ∂ 3.2 ) into a N e ) for ) a A i ( = A ) above. , ℓ ab 2 ∂ k e η C 3.3b A ε k ( , ab 3.2 Ω , k e x q { A 2.10 d ∈ ℓ e (resp. ∧ ε SL(2 = j alone. and A ) we then obtain x ) and ( A k undary spinors as in ( ℓ d A } to understand how ) implies that they ℓ A 3.1 A ∧ 3.3a ) defines the direction i e η , ℓ se otherwise x 3.3a ab A match unphysical gauge and ε e k { and = d , which is defined for any Aab A N η η abc (resp. ˆ ǫ . How do we then express the ) implies the matching ( a ). Going back to equation ( ℓ ab ε 3.3a , , , 3.3a , spin dyad shape-matching conditions , ( , going back to ( A A Ab 4 A A k k e Abc e η ℓ ) that the area two-form e as desired. ℓ e ε ε e . ) for the intrinsic three-metric a a η simply by a A A ab 2.9 ℓ e + i + i abc e e 3.1 ε η η e N ˆ ǫ denotes the densitised dual of the two-form and ∼ A A ). We should thus only match 2 1 = = = ℓ ℓ from the spinors e a a A – 8 – on e ℓ µ µ ℓ A A ab 3.1 ab Ab ℓ := } η ε g a i = = η ∗ a a A x A A A ϕ { is a real-valued two-form in three-dimensions, it has A . η . In other words, we ought to impose and the null vector e η η η A η A = ) with ℓ e ), and the matching condition ( ab ) with a second linearly independent spinor η a A ab into a normalised ε A ℓ ab q k e e ℓ e q We now have to convince ourselves that the matching con- A 3.5b e η and are the component functions of area-matching condition A ) are indeed equivalent to the conditions ( = 1 = ε e (resp. ℓ (resp. A . Since and A ℓ 3.3 ε e a ) and ( ℓ A ) in terms of the new variables? The conditions ℓ A and k η e 3.3 µ 3.5a and ε is the metric-independent Levi-Civita density on and rather than the metric . We start with the µ A abc ab ˆ ǫ , namely η q In our formalism, the fundamental configuration variables are the bo Having shown that the area-matching condition ( area-matching condition We can always extend ab 4 A of the null generators vanish identically. This single degenerate eigenvector of must be real and so is at least one degenerate direction. We assume it has only one, becau would certainly be sufficient, butdegrees they of are too freedom, strong, which for are they absent also in ( Area-matching condition. ditions for the spinors ( local spin basis such that three-dimensional coordinate system either side, and write equivalent to the remaining glueing conditions ( both point into the same direction, hence the null vectors, we are now left to show that the the We contract both ( we decompose both and where and glueing conditions ( on the two sides.to This reconstruct requires the some three-metric preparation: first of all, we have We then also know from the reality condition ( and η invariant combinations of where the spinor-valued vector-density JHEP05(2017)142 . , . its N de- into N = 0 , i.e. (3.9) (3.7) (3.8) ∗ , cor- (3.12) (3.10) (3.11) b ϕ C N ℓ N ) a ∗ µ T N b ∗ ) ) is equally ] ∈ such that T β ( a e a , where by demanding ¯ 6= 0 k ∗ ℓ may vanish, but ϕ a a = ( a = 0 a m m a ) b k m of the spinor-valued α ℓ [ of . To reconstruct this an three-forms on a e ε ∗ µ N N ϕ . Hence we say ab ( ) ab and on ε being invertible around αβ , which is the same from the ¯ µ A is linearly dependent. It then Σ a through A ∗ N ) α . Its densitised dual defines a e } C e , which diagonalises the intrinsic ϕ ) ∗ a ) ( } . ¯ 3.5 C µ ϕ on , a 1 , ¯ N A : ) ¯ , a m ± T A a , R ( = 0 . , α ¯ would imply a , µ µ : = a N is linearly independent in a σ a . ˆ ba with respect to the fiducial volume form, η ( ℓ m ℓ i k 2 1  2 ε } ˆ η { } { a c N b To reconstruct the three-metric from the a − √ ± Ω ( a = 0 ¯ µ ¯ 3 µ / ¯ = b µ b , is linearly dependent (but . = , . What is then the geometric interpretation of ∈ Ω Nµ ℓ . This defines a degenerate geometry, because µ =: a . This can be seen as follows: we define the a bc ) , µ a a A } a ) = i ε ℓ b 1 – 9 – N ∈ B ℓ a bc µ µ µ , µ , which is the same as to say that there exists a and hence also , µ ℓ ¯ := i α a a ¯ ∓ µ m ˆ a a ab m η abc abc A e ℓ ℓ ) a ℓ ¯ , ( a ˆ ǫ m ∗ ( ˆ { η , otherwise the geometry would be degenerate. We = { a k abc i 2 1 ϕ m αβ = 0 ˆ and ∝ ǫ ε m a  m by lowering an index with 1 2 Σ b k a { 6= 0 ℓ ∗ a A = i µ a ℓ ϕ a = 2 sgn η N ( µ ℓ such that the pull-back of the self-dual two-form to AB ab ab on } , which is incompatible with q Σ µ a A ∗ a , for . We can then always rescale the null generator with a third linearly independent co-vector µ from ϕ as , ℓ m a A } N ) A k . This is incompatible with the existence of a non-degenerate tetrad a that α ab , such that k ε ab ¯ e g m { will be determined in a moment. The co-vector C q N ∗ ∗ ) , from the other side of + i a ϕ ϕ = 0 N ( N becomes effectively two-dimensional — it has no affine extension along in the complexified tangent space A . We assume m N b ∗ . The case where and dualise the component functions ( ℓ a = { ∝ ¯ a a µ T ℓ T e N µ ℓ ( ˆ µ e η a ab ab ∈ ℓ = q µ a a = ) A e µ α ˆ η e η ∗ ϕ , assumes the form ( The generic case, where the triple AB Σ ? The answer is simple: it defines a dual dyad ∗ a ϕ the null-surface this is a singular case. It implies normalised spin dyad complex-valued one-form three-metric degenerate as well, for it implies that the triple notes the pull-back to tangent vector null generators follows from then also have the component two-form µ The normalisation spinors, it is more intuitive to work with densitised vectors rather th Hence, we fix a fiducial volume element and equally for two sides equally for a dual basis of tetrad, we proceed as follows: first of all, we fix the normalisation hence Reconstruction of The sign depends on the orientation of in the neighbourhood of Next, we extend two-form defined as responds to a non-degenerate tetrad JHEP05(2017)142 ) A A ), a k k k ε η 3.8 ) into − (3.15) (3.17) (3.19) (3.18) (3.13) (3.14) e + i (3.16a) (3.16b) , from the A a 3.3b ℓ ℓ ( ) and ( η N µ (e 3.7 = → A η ) on the two sides. ) a = , k = 1 . , a tible with the boundary  a ] ℓ b b A ( ℓ e null surface m ℓ e m ¯ A A A a ), we compute the pull-back ¯ ℓ k [ k e , as written in ( match between the two sides. A ¯ m k ) + i i 3.14 ab Aab B b q e = k ¯ η ± µ as ] (resp. A b A a ( ) of ℓ . ℓ ¯ i N ¯ . c m and C , c ¯ , ab , − A c ] B = 1 m ) , ε ¯ b ab ¯ k e  m b + 2 b ] ¯ q m b b ∗ ] ¯ ) A A ¯ m ¯ b b m ¯ m e µ a ℓ ℓ ϕ a m k i a a [ ℓ a ( ¯ A and ( [ m k A a ℓ e k a m ± [ [ abc abc 2 m k ab : ˆ ˆ ǫ ǫ  k a abc ˆ η ¯ µ ) with a second linearly independent spinor k ˆ ǫ C } B – 10 – ) matching according to the indicated pattern. i + i ¯ . We can then, finally, also compute the induced ℓ A = 2 = 2i A A A = i = i } ! ¯ ℓ a ℓ ± e A e = 2 a a a ¯ , ℓ ℓ ∗ µ ab A ab ) of the spinor-valued two-form ] ℓ ¯ 3.12 µ ¯ b ℓ q = A ε ϕ A m i ˆ η ˆ ¯ ℓ µ η , k  i ˆ 3.5 η ∓ a a { [ − We are now left to show that the shape-matching condi- ℓ (resp. = +2 = m ( = 2 on the null surface: given ( { 2 ˆ A η . A short calculation gives a ab , and get ˆ ℓ b ) and ( A ) η  ) alone. ℓ ¯ ¯ A N A = = 2 AB A A ) is the same as to say that the area element is the wedge product A 3.11 ℓ e Σ e and ∗ Ab ∗ ∗ η ϕ ϕ ϕ A ), ( a ( (  k and 3.16b a A ε ) . We remove this ambiguity by demanding that the fiducial volume A η ¯ A 3.10 η η +i A e A ∗ ℓ ϕ µ = ) imply that the intrinsic three-metrics = ( ) into a normalised basis ) into the normalised basis spinors, which brings the glueing conditions A ab A A η q k e 3.3b e η which agrees with the component functions We then have the decomposition ( (resp. By duality, equation ( (resp. for a boost angle element equals the form since indeed of the self-dual two-form to We then have the additional freedom to perform the rescaling which concludes the reconstruction of the induced geometry of th Shape-matching conditions. tions ( This allows us to write the pull-back of the tetrad to To show this, we first extend with all signs in ( metric boundary spinors A short calculation reveals that this parametrisationspinors is indeed compa of the two-dimensional co-dyad JHEP05(2017)142 . ) ), } A A e , ℓ e , ℓ 3.19 A (3.22) (3.20) (3.21) (3.25) (3.24) A (3.23a) (3.26a) (3.23b) (3.26b) e η k e ( { to ) . In terms of A ction, we have ) how to write , ℓ N he area-matching A 3.1 mpute it from the η is already uniquely ( ace are the same on the . ab a agree on the two sides, q A ). Going back to ( ζℓ ε e e η such that the component , of the spin dyad + } 3.7 e µ , ) above. Having introduced a A ) admits the decomposition e and . µ C ) , ℓ = 1 3.7 A 2 , and A A = k e A C k e ε η ℓ e a ab { : . , , A e µ ε A A SL(2 µ k e : k k e as in ( N ( + i and C → 2 ˆ ab ab η . , A } ε ε Ω B ℓ e B , , → A N ℓ η
∈ A : + i + i of the boundary spinors (as shown in the ab ¯ , ℓ ζ B B e q N e ¯ A ζℓ A A A A A B ab ε : ℓ ℓ e ℓ e k η h h ε = − , i { ζ A ab ab – 11 – on either side of the interface, such that µ k e = = A ab − µ µ ) match the null generator and the intrinsic three- q k e } must agree as well. This concludes the argument, A A − ⇔ ∃ ab ℓ A e = = e η 3.1 B ). Notice also that the map from e µ ab , ℓ k , which is a consequence of ( Ab ) as derived from both the area-matching and shape- q e A a A e η Aab Aab ℓ e ℓ k e e = a 3.3b e from the two sides. We then saw in ( η η 3.1 { A = = , and ) and ( e η ab } a Aab ) are satisfied, we have thus shown that there always exists B g ℓ A ab e η ∗ = 3.2 A q ϕ , ℓ 3.3 h ) and ( of the spinor valued two-forms Ab A by k = η ε e a { 3.3a A , A ab k e e q µ η ) above). If the component functions and 3.18 ) boils down to ε , 3.6 µ Before we go on to the next section, let me briefly summarise: in this se derivation of ( If the glueing conditions ( functions two sides. But now we also know that the intrinsic three-geometry without actually changing the canonical normalisation In other words, there are always spin dyads dimensional metric matching constraints ( normalised spin dyads a fixed fiducial volume element,condition we ( can remove thewe density thus weights have and t We can now replace This shows that the spinor valued two-from such that the resulting three-metrics where we have used the same fiducial three-volume determined by the component functions is a Lorentz transformation, which is given explicitly by for it implies thatspinors the on intrinsic one three-metric side is or the the other. same In whether other we words co studied the conditions to glue two adjacent regions along a null surf which is the desired constraint ( metric variables, equations ( JHEP05(2017)142 ) A ℓ an on (as . In i ij 3.3a (3.28) (3.27) for an ion, we N ). The AB M and 1 Σ =: A is at most ), but it is 3.3b j π j M of closed cells M 3.3b ∩ 5 ) imply that all i and M ave five constraints i 3.3b tion ( M rientation from s s are null, in which case ces. See figure is homeomorphic to a closed and a spinor-valued two- is the same for both vari- i ), on the other hand, add 2 M C . . There are two kinds of con- intersect in a two-dimensional 3.3b A → as boundary variables: for any j ℓ of all four-dimensional building ) ) = 0 ij , hence there are five independent M into a finite family ij p A N ( N and , ℓ , : , and every M ) j A AB and A ij B A ij M η S ℓ ℓ i Σ η -indices altogether. (0++) ( ) A 3 6 Λ ij ( M = ij reduce the area-matching constraints ( ( η i − ) matches the two-dimensional area elements ab ) = M ¯ ε p ) and the shape-matching condition ( ( ∂ – 12 – = ij AB 3.3a AB Σ 3.3a ab ) above), we first introduce a cellular decomposi- F ∗ ij ε : ϕ has signature ) i 2.19 ). In both formalisms, we are thus dealing with the same ab M q 2.21 ∈ -indices has no geometrical significance. Simplifying our notat are adjacent to four definite such regions — four and not ) , and we shall also assume, for further consistency, that all 3.19 p ij there exists a spinor C matches the orientation of ∀ ( M i ij , such that the pull-back of the self-dual area two-form M N ) . If, on the other hand, 2 ji C N or, wherever possible, drop the are the same whether we compute them from the boundary spinors : , which are flat or constantly curved inside, i.e. ij B = To now discretise gravity with the new boundary variables ℓ ij } N 1 N N − AB ( ij ǫ 2 M N . from the two sides, while the shape-matching conditions ( = Ω 4 R A ij ab ∈ ℓ ,... ε e 2 )) admits the decomposition A ij η M , The requirement that the boundary and 2.2a The vertical position of the The orientation of every 1 6 5 ab M surface, we call it a corner corners in thethree interior or of five, simply becauseall we require such that corners the arise internal boundarie from the intersection of two such null surfa other words thus write straints: the area-matching condition ( four-ball in three-dimensional. If it is three-dimensional, we give it a name and call in ( blocks is null, allows us then to use the spinors form illustration. tion and cut the{ four-dimensional oriented manifold five number of constraints. 3.2 Definition of theGeneral action idea. to three real constraints.only one The additional shape-matching complex conditions constraint.immediate ( when This we is look not obvious at from ( equa terminology should be clear:ε equation ( angles drawn on (as introduced in e.g. equation ( such internal boundary these equations in terms of the boundary spinors We require, in addition, that the intersection of any two suchinterface, region whose orientation is chosen so as to match the induced o matching constraints in the metricas formalism. well. The In reality terms conditions of ( spinors we h ables: the induced three-metric one side or the other. Notice also, that the number of constraints JHEP05(2017)142 , ) C . ner , ,... 2 (3.30) ,... (3.29a) (3.29b) M ) above. 34 , SL(2 12 1 C M connections . The com- 3.26 j ) M C , to the boundary ∂ i ⊂ M nal face. The bulk ed in a discontinuity SL(2 ji N ere is, in general, a non- . That this is the same to , we will have two kinds of M connections, and we define . gauge transformation exists 2 ij j ij ) . Equally for the connection: ) N M h 42 j M C C l curvature is confined to three- , ji into N , M above following the discussion of C ij ji or N h SL(2 SL(2 i 2.1 3 , , + B ji M M B ji ℓ π ij h B B ji A A ] ] A connection from the bulk ji ji ji ) h h h – 13 – C , = [ = [ + carries then two independent from either , which intersect in two-dimensional corners 34 12 4 is the pull-back from A ij A ij ij j ) ) are satisfied, which has been shown in ( splits into a union of four-dimensional cells ) of the connection. We allow for the presence of an ℓ C between the two h π M SL(2 A ji M ,... d Ba 3.3 3 M , ℓ ji A Ba 3.27 The action will consists of a contribution from every four- ] 13 ji A h M A ] ji and N π ∩ = ( ij i A 1 1 [ C ij [ M M M A and = ) is the canonical embedding of -flatness ( , and , 13 A ij i . What is the relation between the two? Consider first the Λ B N , ℓ , a boundary term from every interjacent null surface, and a cor M M Ba A i ] ij A is null, has been shown in section ∂ between A ] ij π M → ij ֒ ( h ij ji ⊂ [ N N A ij [ ij ). N N The spacetime manifold : 2.5 and denotes the pullback of the ij ϕ Ba Ba A A The discontinuity of the metric across the null surface will be encod ] ] ij ij A A which brings us from one frame to the other. Such an provided the matching conditions ( transformation [ component mon interface [ spinors, namely The spinors are gauge equivalent,vanishing but difference the tensor connections may not: th equation ( of the spinors and the connection. Along a given null surface which contain nodimensional local interfaces degrees of freedom inside. Non-trivia Figure 1. it as follows: Construction of the action. dimensional cell as to say that term from any twocontribution imposes such null surfaces intersecting in a two-dimensio where JHEP05(2017)142 A δℓ A (3.36) (3.31) (3.35) (3.32) (3.33) (3.34) , which π is real. twisted ij ε N A ) and the ∂ R +  3.34  + . Ab  . π the interface be- a ometry is null. It connection + cc A . ) + cc π  C dary term A ji , ab  ℓ a + cc Ψ . ij A A ji imposes the desired con-  1 2 : it cancels the connection SL(2 π π ., . a − AB + + cc ji A AB Π 
= 0 A ji + π . + cc Π . The bulk contribution to the ℓ ∧ A ni  , i ~ C ij A − ℓ ) A ji a π B . AB nj A ℓ AB + cc AB i ), which match the spinors from the and add the terms  ℓ A ij Π ji Π A Π M π λ ℓ − ∧ ∂ D 3.3 a i A + i AB + i ij A 1 A ⊂ + i)Σ ∧ A mi ( π β Π π ℓ β and ij β ji A C   ( N i π + i A mj A G + i 1 ∧ ab ℓ 2 ij ab Λ β to − β Ψ ij 3 i β i + − – 14 –  i , πβG λ ) above), which amounts to work with the A ij ), which impose that the area two-from G M ℓ π + Ψ ω 2 λ ij A jn 16 Λ 8 ij AB
ℓ N 3 β 2.14 − A ji i Z 2.21 D potential = + ℓ jm A π A 7 + ∧ ℓ . ji A 8 ℓ ) ) with the boundary action for the spinors ( = dΠ AB AB π ij A − N ] = + ω Π F π , and the variation with respect to the connection implies − A in h AB − , α ℓ ∧ 3.32 A ij AB AB ij Π Ψ ℓ D F N im A Π ij A ( D AB ℓ Z is the covariant derivative with respect to the A π Π ∝ ] AB · π  α ω, λ, , mn | ij i Π  ∧ (as in equation ( ij C  ij , ℓ M AB Z i A -valued two-form with dimensions of ij N β Z -variation on ] π ) F | ω M Z A C  Z ) and also add the right corner term, which we will discuss below. The i ℓ , ,A ij X ij ijmn h i X [ (2 = d + [ N [Π X 3.35 Z sl ), i.e. + + ij S D = 2 3.27 We sum the bulk action ( The next term to add is the three-dimensional boundary term from Notice: 7 bulk momentum straint ( the integrability condition constraints ( two sides and the reality conditions ( which is an action is therefore nothing but the integral arising from the resulting action is then given by the expression to the action. Finally, we also need a corner term to cancel the boun We thus introduce Lagrange multipliers Clearly, variation with respect to the self-dual two-form Immirzi parameter tween two adjacent regions. This boundaryvariation term from has a the twofold job bulk,consists and of it the covariant imposes symplectic that the intrinsic boundary ge is the pullback of the self-dual connection from plus additional constraints: the glueing conditions ( JHEP05(2017)142 . ) ) g 1 = j n ion: and ... M ab M , the , , ) (3.40) (3.42) (3.41) (3.37) (3.39) (3.38) i } Ψ + ). The , which m m ive sign M ( ··· M M ,... , and , 3.36 2 . . This sum j 21 j in N ij ω . See figure M M R M N N , ( , i 1 mn + ij M C M ( nd can be absorbed and ··· ), whereas ion of the boundary s straint = ) 12 im ∈ { . The variation of the . n n N N ij ji i R ) of the self-dual connec- M M N N | , = M . This allows us to restrict i = ∩ ab ji m 3.27 M m λ ( ··· M + M , = Ψ ij ∩ ) , its variation imposes the reality
i ij N ij = ij m R A in λ . N M j i , | N , M ) δℓ ji ij , ab h M ij B N -flatness ( i im A | ℓ im A Λ ) above. Finally, there are the glueing λ ∩ P A M , αℓ ij ( i , which share an interface ) αℓ ( relative to the null surface − = Ψ ) ab π ∓ j M ± ¯ ε 2.21 1 2 = ij ab mn = ij 3.31 M A in Ψ = ij , = C i in A N δℓ – 15 – | ab π , in A M λ ε AB ( ji π to say that any such corner appears with only one mn ij Π = N | ∗ C ] ∗ ij ij ω ϕ ϕ mn ij λ C = ijmn [ Z ij P N | by a term proportional to ) as in equation ( ω ij such that its sign flips across the interface, in other words is a real-valued one-form on ω = λ λ 2.22 ij , which is a complex-valued two-form at the corner. The variation ω α denoting the canonical embedding. Next, there is the reality condit -variation of the coupled boundary plus corner terms. The resultin goes over all four-dimensional bulk regions A i ℓ turns out to vanish by taking into account the boundary conditions M only meet in the corner itself: α P ), which follow from the variation of the Lagrange multipliers → ֒ ) n denoting the pull-back to the two-dimensional corner and the relat 3.3 ij M each share a three-dimensional interface (e.g. N , : ) mn ij m n ∗ C ij ϕ , which are continuous across the interface: M M ϕ ( Let us summarise and briefly explain the role of each term in the action ( , ba j Ψ M condition for the area two-form ( the Lagrange multiplier connection in the bulk yieldsconnection a couples remainder then at the the boundary boundary. with The the variat bulk, yielding the con are obtained from the two-dimensional remainder We can then always shift arising from the − the Lagrange multiplier with conditions ( first line is the bulktion. action. Next, It imposes there with are ( the integrals over the internal boundaries depending on the orientation of the corner for an illustration.possible We orientation write in the sum.into the This orientation definition is of chosen arbitrarily a The first sum with respect to with boundary conditions are and ( second sum goes over all ordered pairs crucially contains both possible orientations, i.e. The last integral is the corner term, which is a sum over all quadruple that share a two-dimensional corner such that JHEP05(2017)142 ) ) ). AB are Π 3.36 3.42 3.36 ab (3.43) (3.45) (3.44) (3.46) Ψ is real, and ). This is a and ) and ( A ℓ ). Finally, we ω 3.36 A 3.41 π ed by the glueing 3.3b ... ) have already been olutions representing +i) he interface, namely: + β action ( ( B ij . th the discretised theory 3.36 ℓ ) , which gives the integra- denoting the metric inde- p
is a complex-valued tensor B ( ) with respect to variations ij A πβG/ abc above, this is the same as to ω condition ( ji ab AB 8 ), we dropped all wedge prod- ˆ A ǫ in the bulk yields the flatness 3.36 − Π − 2.1 ij = Ψ 3.36 = can be inferred from the definition AB + i 1 D ε , Π , ij ba β ab ) B Ψ ∧ Ψ G ℓ Abc − ij A Λ A π ( connection along the internal boundaries, π 3i = π abc πβ ij 1 2 C ˆ ǫ N ij ab shape-matching 16 1 2 are part of the definition of the action ( Z to the pull-back of the self-dual two-form = Ψ – 16 – λ U(1) A := ℓ ) = AB a p , ( Π · · · ≡ A ∗ and π is a π + ϕ AB ω ) and the , where we find a number of additional constraints. The ji A F } , ℓ ω : . ) ij ab i ω 3.3a = abc N Ψ ( . In the same way, the Lagrange multipliers -indices in the boundary variables. The second line in ( ˆ M ǫ { − ) λ . This is essentially the torsionless condition. Next, there are ij ∈ ij ω D ( p ( = 0 ∀ A ) impose that the area two-form π AB ij 2.21 N Π ( Z D ), i.e. is a real-valued one-form, and 3.27 ji λ − Some of the equations of motion derived from the action ( area-matching condition = ij density of weight minus one. The density weight of of the momentum density added to the action inthe order to impose the glueing conditions across t λ the internal null boundaries 3.3 Equations of motion In this section, we study the equations of motion as derived from the Finally, a word on the Lagrange multipliers: the continuity conditions ( reality conditions We then also have the variation of the self-dual connection bility condition constraint ( condition say that the boundary is null. of the Lagrange multiplier have the relation between the bulk and the boundary, which is provid mentioned. The variation of the self-dual two-form preparation for the next section, whereplane we will fronted find gravitational a waves, family which ofand explicit are general s exact relativity solutions as of well. bo for the Lagrange multipliers The Lagrange multiplier ucts, and suppressed all has to be understood, therefore, in the following sense which is a spinor-valued vector density of weight one, with they are obtained from the stationary points of the action ( pendent Levi-Civita density to the boundary. As we have seen previously in section Notice also that we have used a condensed notation in ( which couples the boundary spinors JHEP05(2017)142 s, ba lss ant ) = Ψ and − ) (3.48) (3.49) AB A (3.47c) (3.47a) (3.47d) (3.47b) ) is the ℓ = 1 nt is (Π is system A , which is ( ∗ − ab connection 1 π Ψ . This inter- N ) − and contract Dϕ . We can then, C = ansion of the M b = , → ֒ ∂ Aa . The definition a ) = π ℓ AB N ⊂ b l equations of mo- a f and M Π SL(2 A k AB ∗ : a N π : Π ider only a single such ϕ − ) on e ϕ D ( N ∗ ∗ of the equivalence class of , = , ϕ T 2.11 a , with Ab Ab ℓ (and Ab ∈ e π π f M , π of the null surface. This can be ) a , ab a ab , N ) k R A A ℓ A ( : π and π e π ϑ → + Ψ ֒ + Ψ is symmetric in . . ∧ . Indeed, N ∧ b A A A M ( ) ), we then see that the variation of the λ ℓ f ℓ M e ℓ 2 λ M B AB A Ω   : π i Π π + i a a i + i 3.36 a ∈ λ and λ β ϕ A β ( ε , which represent the induced three-geometry + i ) i − π i + i + i M πβG + ∧ ), which is the pull-back of the bulk connection β A 8 – 17 – A a β β k e A ] , ℓ ) − π e π B ℓ − A + ( A ∧ = a ∧ e ϑ a π ( A ω ω ω ε can be defined by − ω ∗ between   − ) e ϕ − ℓ = ( = and = = N ϑ ε = , which vanishes, since the Lagrange multiplier denoting a dual one-form = [ ) d A A A A ℓ a e ℓ A a e π k Ba = 0 π a , ℓ A e e D D D D A ab e A π Ψ ( b for the self-dual two-form ) , with ) contains the boundary spinors as additional configuration variable B ) is the covariant derivative with respect to the (and π respectively. a N ). First of all, we can see that they are consistent with the torsione a = 0 a ) e 3.36 ] D A Before we proceed, we need to develop some better intuition for th B ( B ℓ e 3.47 . In terms of the boundary spinors, the two-dimensional area eleme ) to the interface π AB A A Π A ( . The resulting density does not vanish in general and it has an import N , bounding the four-dimensional regions f M ∗ (and e π D ab ) = ϕ N ) a Ψ = B denoting the canonical two-dimensional area element ( is gauge dependent — it depends on a representative D ℓ = [ (and ) ε A ℓ ( Going back to the definition of the action ( The action ( AB ( Ba π ϑ M Π ( A ∗ e with seen as follows. The expansion geometrical interpretation. It measures the expansion Three immediate observations: integrability, geodesitynull and surface. the exp it with intrinsic to in D however, also form the anti-symmetric tensor density A is an anti-symmetric tensor density, whereas where null generators of of equations ( condition boundary spinors yields the equations of motion and the action istion are stationary satisfied with along respect theinterface system to of them interfaces. provided For simplicity, additiona cons and ϕ of from, say, below and above the interface: if canonical embedding of the boundary into the bulk, we will have face will carry spinors JHEP05(2017)142 are ) (3.50) (3.56) (3.52) (3.54) (3.55) (3.51) (3.53) A ) from e ) above. , ℓ Ba along the A A of the null e π A A 3.29 ( ℓ ) ℓ ( ), ( ϑ and ), hence the null (from above) and dary spinors, and ) alisation) in terms 3.26 A 3.33 M , ℓ gauge transformation. A , ) ε π C ( , , one (namely ∧ , k N ) = 0 ℓ ( , b SL(2 ϑ ℓ a ℓ = 0 . , , ab ¯ ≡ − A A , Ba A ¯ ℓ are auto-parallel curves with respect ℓ Ψ A a A π A Ab ℓ ¯ A ℓ C a ∝ D i ¯ ℓ π ≈ a A a A itself. It is geodesic, and this can be seen ℓ A + ∝ ℓ , which is defined as ℓ A A π A i a b connections on ℓ e  π π ℓ Ba N A ) ) from above. Their relative strength is given ¯ A ), we find ≡ ab A + i) ¯ π ℓ bounds two bulk regions is a measure for the expansion connection is torsionless ( C ⇔ , Ba a , Ψ – 18 – A A ab β ) ℓ A A ℓ A ( ) imply that the spinors ) on i 3.47 N Ab ℓ ℓ Ψ C ≈ e A  a , + i π ∝ ℓ b ≈ ∝ 3.3 SL(2 a πβG itself, which follows from ) above. Now, the covariant derivative of a β Ba 3.30 8 A D ℓ A A = A b A ℓ ℓ e π − ℓ SL(2 ℓ e a A A 3.14 ℓ ab = D a Ψ a ε ℓ D , whose algebraic form is determined as follows. First of all, a D ℓ Ba A = ℓ A is anti-symmetric. We have thus shown that ε C The null surface d ). If we contract this vector field with the soldering form, we get back ab ”, means equality up to gauge transformation as in ( ) by noting that Ψ ≈ 3.7 are geodesics. connection. The 3.47a ) N C as in ( , a . ℓ N ∝ SL(2 , see for instance equation ( bc ¯ Finally, let us turn to the null generator A ε (from below). There are then two ¯ ℓ A abc ℓ f ˆ we know that the glueing conditions ( by a difference tensor below the interface, the other (namely M and equation ( null generators is proportional to We then have the difference tensor ( We can thus write generators of Difference tensor. i which means that the integral curves of gauge equivalent, which means that they are related by an surface to the Going back to the equations of motion ( which follows from the area-two from as written in terms of the boun ǫ which is zero since as follows. First of all,of we the write boundary this spinors, vector field obtaining (modulo an overall norm which shows that the density where the symbol “ JHEP05(2017)142 , ) 3 f M x ) for being = (3.59) (3.58) (3.61) (3.60) 3.47c (3.57a) and (3.57b) 0 are flat, x 3.59 AB ), and the M Π f M ∗ ), ( ϕ motion in the 3.36 and 3.60 ) imply that the istence: there are se equations exist: M ) is subject to the ), ( 3.1 )), and + N v of Einstein’s equations , i.e. subtracting ( d olution space, we study uge equivalent, which is ence 3.58 tes the following ansatz A 3.45 , ℓ
 e a α ), ( at the interface. Γ ¯ fℓ B B ℓ f is the flat hyperplane 3.47 A . A (as in ( . ℓ A C + , N π α A eyl tensor at the interface. AB + i ℓ 4 = 0 ¯ ∧ m a Π ¯ β f , which is the same as to say B λ λ , C ε − + AB C + i + i ∧ 2i 2i a α F λ β β ∧ ) ≈ B C fm − )). The boundary spinors are gauge equivalent = Γ k A we then find the conditions . The glueing conditions ( ) – 19 – A from the differential of = = + µ AB C ( v A ℓ z. e M 3.46 A F ∧ B B e π d¯ + ℓ , bounding the four-dimensional regions ℓ π λ α e D + i B 4i N Ba + Θ( ∧ (a complex-valued one-form on m A β being sourced by A α B a + ) that − k C A Γ  ). Finally, the difference tensor is subject to one additional from z DC AB N = C d F + A α π u m d 3.57b ABa D α is the same from the two sides. The discontinuity in the metric + ¯ C ℓ admits the decomposition = N Ba α A e C are inertial coordinates in solutions to the equations of motion derived from the action ( ), and equally } 8 )), which implies (on µ x { 3.47a 3.55 Consider thus a single interface We will find solutions with a non-vanishing distributional W 8 where from above and below. We set the cosmological constant to zero, h This implies that and we assume, in addition, that the lightlike interface intrinsic geometry of particular solutions thus constructedas well. are distributional solutions certain boundary spinors and a definite difference tensor (see ( constraint: the field strengths asan induced immediate from consequence the of two sides are ga from ( neighbourhood of an interface.only Rather a than single exploring familynon-trivial the of entire solutions, s thus giving a constructive proof of ex 3.4 Special solutions:This plane-fronted section gravitational is waves dedicated to finding explicit solutions to the equations of which is a consequence of ( where the component function algebraic constraint for the tetrad across the interface can, therefore, only be in the transversal direction, which motiva Subtracting the covariant differential of sourced by the boundary spinors (as in ( plane gravitational waves solve the system of equations ( In the next section, we will demonstrate that explicit solutions to the JHEP05(2017)142 ), for 2.2 , i.e. β (3.64) (3.65) (3.62) (3.63) e . The , while M 2 ∧ √ α , and they = 0 / ). We thus e ) 2 α = = 1 x , which defines m 3.58 , we decompose α + i k αβ ventions of ( . Ba 1 = 0 ], it includes plane α Σ = d ℓ
A x in the canonical way v ¯ 21 surface. To compare f        . We then introduce − C α ) describes a class of } , which is a complex- z e . ) and ( erence tensor in terms direction, which yields en the two sides. The k = ( α ¯ f Ba th of this discontinuity and = , . z∂ ional field of a massless v ¯ ¯ z ¯ ¯ A m A A 3.61 α , ¯ ℓ 3.56 ¯ ¯ k ∆ = d α m A + d¯ A A B = d α ℓ k and α δ f = const ℓ , α α , m ¯ ¯ z 2 m ¯ ¯ D d α  A A + v . v √ A A 9 z ∂ , k / − σ σ

Ba ) α + = 0 2 2 3 β ℓ A 1 1 + d¯
x { α ) is compatible with the equations of β √ √ ¯ ∆ f e gives a rather lengthy expression. For A ←−− f ¯ − ¯ A u B − − z 0 ) ∧ δ . The ansatz ( − 3.61 ¯ ) z x = = z ∂ β z ∂ v ¯ z = , which is covariantly constant in

α α α β = ( A ¯ + d¯ α m ¯ ) of the difference tensor u, z, Ba v ( + 2d u, z, = 1 + A ←−− f B , ( – 20 – f u , ¯  f α ). Clearly, the equations of motion in the bulk, A 2 , m f ) are satisfied for ¯ . The result is A 3.56 0 A u e ¯ A A ℓ } √ ¯ k z ∂ ¯ ) ℓ ց . ≡ A / ∆ d v A B u ∂ 0 A ) 3.36 k 3.45 = d ℓ k ℓ 3 d f ) : α α A x α := lim ( ¯ ¯ B v < A A B k + β } A A ℓ , k De α 0 A A , with σ σ ( A B are a normalised null tetrad: x in terms of as in ( ℓ ℓ ∆ 2 2 , ℓ k ¯ Ba alone: 1 1 1 2 2 1 } β A A ¯ √ √ A α = ( α k u − + , k ¯ − − { ¯ ∆ = 0 A m u B , ℓ = = = α αβ A and α α ℓ and Σ ℓ k , m { z AB ) with the definition ( α D ∆ Ba , k A , and which is related to the null tetrad α 3.63 is the pull-back of the spin connection to a ∆ ℓ A and { k v

]. β α 22 = 0 A ←− into its self-dual and anti-self-dual components. Following the con denotes the Heaviside step function. The only freedom is in . At the interface, we have a discontinuity in the transversal αβ = 0 = d We now have to make sure that our ansatz ( 0 ) F This is the covariant derivative for βa v 9 A α ℓ d ∆ coordinate functions are a normalised spin dyad A straightforward and ratherof lengthy calculation the triadic gives basis then the diff v < we call them have to compute thespin spin connection connection is and the take unique the solution difference to the betwe torsionless condition fronted gravitational waves, whichparticle may [ describe e.g. the gravitat motion as derived fromi.e. the action ( impulsive gravitational wave solutions widely known in the literature [ valued function of Θ( the moment, we are only interested in the pull-back to the hyperplan The vectors a discontinuity in theis connection measured across by the a interface. difference tensor, The which streng must be matched to ( The unique solution for are all constant with respect to the ordinary derivative the difference tensor where this equation ( JHEP05(2017)142 ¯ B is as ¯ A is a and Γ a Ba AB (3.69) (3.71) (3.70) (3.67) (3.68) ℓ A (3.66a) (3.66b) Φ C = = 0 a u a . The only ∂ 0 ω )), while les. ) above, we can v > and 2.21 component of the te them explicitly. r across the inter- ), is then simply 3.47 1) . A straightforward . , ). The result of this ( , 0 ] = 0 ¯ D z h is a singular limit in (1 ¯ 2.11 C component, and z. d ust be equal to ion for ab 2.2a d¯ ∧ v > Ψ AB 0)
¯ , z f Φ , and matched the geometries d¯ z (2 0 ∂ A CD k ǫ . + 2 + v < − f x ). Going back to the decomposition ¯ z d u ) can be only imposed, therefore, for ∂ d ∧ . 3.36 ∧ 1 + x ABCD z . z ¯ 3.66a d¯ = 0 f, Ψ d u A ¯ ¯ D f = d f ℓ ¯ z = 0 z C z z ∂ ∂ ¯ ǫ ∂ , as given by equation ( f d − – 21 – u − ], we compute the irreducible components of the + i) [ N ∧ ∂ + d¯ + 2 f = ) for the tetrad and taking also into account the z β 23 ¯ z f u ( ¯ ∂ u D id¯ d f ) above, we thus find the conditions − 3.61 ¯ z ∂ i u CD ∂ , on the other hand, we need to consider the Plebański πβG = 8 ), the equation ( is real (it imposes the reality conditions ( A 3.58 ε = d represents the irreducible spin ) for the tetrad, we thus see that the null vector e ABC π λ = connection, namely λ Γ = F 2.14 ) A ≡ 3.61 C e + i + i π 8 and 8i , , and determine its self-dual part as in ( ABCD β , as in ( β = A β A Ψ e − + π ) of the null tetrad, we immediately see that we are in a gauge for ABcd A k SL(2 ∧ F π α 3.64 e and . For A = A ℓ ℓ e αβ = Σ into A ℓ . This fully determines the geometry in terms of the boundary variab Ba . This implies, in turn, A It is also instructive to have a look at the boundary spinors and compu Finally, we can now also compute the distributional curvature tenso = 0 C = 0 a Going back to our initial ansatz ( derived from the equations of motion for the action ( where the Weyl spinor Killing vector. Having solved the equations of motion for field strength of the then also determine the missing Lagrange multipliers: we get parametrisation ( which straightforward exercise returns the momentum spinors complex. Unless the Barbero-Immirzithe parameter original goes selfdual action to ( λ zero, whic of two-form across the interface, we are now left to solve the equations of mot λ calculation reveals that this is possible if and only if missing condition is to impose that the curvature vanishes for Going back to the equations of motion for the spinors, i.e. equations In order to satisfy the equations of motion, this difference tensor m face. Following Penrose’s conventions [ Now, the Lagrange multiplier is the traceless part of the Ricci tensor, which is the irreducible spin Going back to our ansatz ( The two-dimensional volume element on JHEP05(2017)142 – 24 (4.1) (4.2) (3.74) (3.72) (3.73) , ] A ing [ M , ∂ utional source CS eyl spinor is S four-dimensions [ ling constant 2 γ perturbative quantum ) in the neighbourhood rmulation of the theory. = = Simons functional for the except at the null surface or at the two-dimensional , who have suggested that  3.72 . } AB A ¯ B ij ¯ ℓ F . ∧ ¯ b N A , ℓ ∧ ¯ { ℓ A a D B ℓ ℓ ∧ , which generates the gravitational ℓ
AB , we have a solution of the vacuum C ) . ¯ A ℓ z f A F ℓ z B πz 3 2
∂ ℓ M ¯ ) is topological. All physical degrees of + i) f (4 A Z + z + / β in the bulk. We insert the equations of ∂ o 2 γ ( A f f ℓ for all f 3.32 ¯ z z d − + G ∂ ∂ AB ∧ Λ f ) ¯ ) = z Π ) 3 β = 0 v = A z ∂ – 22 – ( i v ) and demonstrated that explicit solutions exist, EOM ( (  f δ π ¯ f z δ ) 8  ∂ v = Tr ( 3.36 1 = δ πG AB M ]. 8 ∂ γ Π = Chern-Simons action in a four-dimensional theory may Z 22 = ∧ ¯ ABCD B 2 ) γ ¯ A Ψ ab C AB , = T AB Π Φ 1 γ SL(2 + alone, we can then also work with a simplified action, which is found AB is holomorphic, i.e. F N f ∧ . As long as we are concerned with the canonical analysis on only one } ) for the self-dual two-form back into the bulk action, thus obtain AB mn ij Π C  3.45 { M Z , where there occurs a distributional curvature singularity. The W First of all, we note that the bulk action ( 2 ]. This paper confirms these early expectations. = 0 motion ( by integrating out the self-dual two-form such null surface corners Einstein equations, the other extreme is If, in particular, of a null surface. The next logical step is to studyfreedom the Hamiltonian sit, fo therefore, either at the system of null surfaces 4.1 Space-time decompositionThe of main the purpose of boundary this action papergravity. is We to have open defined up a thewhich new have action road a towards ( non-vanishing non- (yet distributional) Weyl curvature ( 4 Hamiltonian formulation, gauge symmetries which is the self-dual Chern-Simons action with complex-valued coup field of a massless point particle [ 29 v Riemann curvature tensor. Both components vanish everywhere come as a surprise, butthe it so-called has Kodama been state, anticipated which by isunderlying several the gauge authors exponential group, of plays the a Chern- significant role for quantum gravity in The appearence of the and the traceless part of the Ricci tensor is determined to be The resulting geometry is a solution to Einstein’s equation with a distrib JHEP05(2017)142 , . , ) is 3 + + A N / N nal   π 2.5 Λ from (4.3) AB Ab Ab on at the on metric ) e Σ π π ), which A foliation a a N ab e ce always AB A A q , ℓ times F is the same e π π of the four- A 2.21 i e ab ab π (0++) . The glueing N Chern-Simons Chern-Simons ( AB Ψ Ψ M ) ab ) he reality condi- 2 2 1 1 Σ ∂ C Ψ C , − − , and how can it be that strength   ) . . he interface. boundary spinors null metric and A al interpretation: vari- , lacks such a preferred SL(2 SL(2 rmulation to begin with, , ℓ ω N + cc + cc A A A π ( ℓ ℓ e , imposes then that (0++) A A λ e π π s at the two-dimensional intersections i i + i + i β β . A more complete Hamiltonian analysis ]+ ∅   e A . Integrating out the self-dual two-form [ = 2 2 λ λ N f M , − + . Each one of these S ∂ A A , which is transversal to the two-dimensional CS N ℓ ℓ e ) ) S ), which are obtained by demanding that the and ω ω 2 γ = 1 – 23 – 3.3 − − t M − a . e ] D D ∂ ( ( a /γ A t [ ) A A : B . At the level of the action, there is no preferred such e π π N ℓ ) is evaluated over the three-boundary e , a   S . A t ( . Each one of them has has the topology of a four ball, N N CS 4.1 e π } Z Z S − } × + cc 2 γ + − t , which is consisted with what we have seen before: by ( { . For definiteness, consider only one such three-surface ,... and 2 ij /γ = ) N M as in pure (anti-)de Sitter space. The reality condition ( t Ψ] = AB B , j is homeomorphic to a three-sphere tessellated into three-dimensio ℓ S e 1 F i A S continuity across the interface: the intrinsic geometry of ( AB , yields, therefore, two copies of the self-dual M 12 M λ, ω, π { = 11 | f ∂ M above. i e 3 Σ − , ℓ / and a vector field ), which is meant to be an action for a null surface M e , and a generic configuration of them does not define a signature π . This should strike us as a surprise: a three-dimensional null surfa 3.1 by ∂ a S A and t 4.3 : ℓ e = Λ surfaces , ℓ, ) along the interjacent null surface × , ) will be left for the future. 10 ), this source is itself nothing but the self-dual area two-form . ij 1 π A | M 1] AB 4.1 N e π , e F in 2.20 [0 a preferred time direction: the direction of its null generators. So The Chern-Simons action ( Next, we write the action in a Hamiltonian form. This requires a clock — a A, A [ See section The same is true for For the moment, we assume them to be closed ≃ and 11 10 12 AB N = const A S Π boundary between two bulk regions, say action ( regions dimensional cells whose boundary conditions impose ℓ has vector field structure? The answer is simple:the in metric our is theory, a there is derived notions or metric composite are fo field. satisfied. It Our exists fundamental only configuration on-shell — variables only are if the t is obtained from the variation of the Lagrange multiplier action be stationary with respect to variations of the multipliers and ( actions is itself coupled to the respective boundary spinors our action ( is sourced either side. The resulting coupled action is therefore given by whether we compute it from the boundary spinors onN either side of t All terms in thisation boundary with action have respect a to straightforward the geometric self-dual connection imposes that the field geometric, i.e. compatible with the existence of a signature t Finally, we have the glueing conditions ( taking into account also(see the figure corner terms and boundary condition hence JHEP05(2017)142 , A A π π ∗ t bulk , i.e. (4.9) (4.5) (4.6) (4.4) ϕ (4.7a) (4.8a) (4.10) ω (4.8b) (4.7b) ) above. C = , slices into 3.1 A ) can such . π -form is a one-form p SL(2 and write, ac- 2.21 a V A he π n space, where the = const t with a en in section V . , a are Lagrange multipliers, since t ab ] a A V , . , such that we can write the spinor- π a , . 2 A ] ∗ t ] slices simply by a ℓ . ] B C ] ϕ B . a [ and B A A V A . ab A A π a A ˆ ǫ t , t + A y [ 2 1 A . U t L L A 6= 0 ∗ t [ ℓ a [ [ ] t ∗ t ∗ t π ∗ t ϕ = const A have different density weights: A a ϕ ϕ ϕ L , hence we assume ℓ ] = t a – 24 – = π A A U S = V = y = = π π t [ [ as a basis in = Ba B A A ) defines the spatial components of the connection. . Both ∗ t ∗ t ˙ ℓ Ba ˙ ) A A , which is a mere coordinate, and a transversal vector π ϕ ϕ S and t A A Aa to the Λ A is the canonical embedding of the 4.4 ˙ a χ A = = , ℓ , which is the time-space component of the two-form ) )) ) U A A 2 C Aa π t, p π , C ( ( χ : . (2 ) 7→ sl S ( : decomposition for the spinor-valued two-form 1 ” denotes the interior product of a vector , p Ω y . Equation ( M M ( ∈ 1 N and decompose the configuration variables into space and time com- 2 + 1 a ⊂ Ω V,X,Y,... defines the Lagrange multiplier ) is non-degenerate on ( A ω ∈ χ N 13 is an inverse density one-form, which follows from the fact that = 1 B 2.11 a t ) = → A ֒ a U A ∂ S a t : . Only for those configurations that satisfy the reality conditions ( denoting the Lie derivative along the vector field t : t N ϕ X,Y,... a L t )( , while We repeat the We then choose a time function In the following, we restrict ourselves to those parts of configuratio -component on ω The hook operator “ t S y 13 ab V is a two-form (hence a density) on This allows us to use the pair in terms of components valued one-form on with Finally, we also have the velocity ( Its a metric be defined, according to the construction that has been giv Notice, the component functions area element ( cordingly Finally, we define the velocities q the three-boundary connection where field ponents. With a slight abuse of notation, we denote the pull-back of t JHEP05(2017)142 ) 4.3 (4.14) (4.12) (4.11) (4.15) (4.16) + (4.13a) (4.13b)  +   )) and for ., A  e 4.5 A ϕℓ + cc − -matching con- ϕℓ ), ( 
B − ℓ e A 4.4 ℓ B ) impose constraints B e ℓ a A B e e D A 2.21 A . We take the canonical + Λ e π a A V + Λ ˙ − ℓ e .  A A A is a vector-valued density. ˙ ℓ ℓ ℓ e A ) , a  a and b e π ), which follow from the variation S e V A D V a T , , and they simply yield + π A + a ab + 3.3 a U : ω  λ ˆ π ǫ  + , . e a A U S t a e  π ( := 2 a = Aab . and N = ABab a a + cc Ω e e a π U e F with respect to the boundary spinors from J + ¯
ab ABab U ab ∈ U ).
A = ˆ ǫ Ψ F AB ℓ a e A = Aa , a 1 2 ,N ℓ e ] A e b Λ , J e χ a a – 25 – AB a 2.21 a A e , implies π = U ω − Λ e λ U e e D π V a b + ℓ ab e t A y ) and the reality conditions ( − ˆ ǫ t )) back into the boundary action, and get ), which follows from the variation of the action ( e = A π [ − b = AB ℓ ∗ t ∗ t a 3.3 ˙ A e := A ϕ ϕ A V ϕ AB ℓ 4.13 2.21 π a ˙ a = = A N D ABa a ), ( A A e i A A + i e π ℓ , and define ABa e  χ ) with respect to 4.7 β S A a ab ) contains no derivatives of them. Yet they are not completely   decompositions for both the connection (i.e. ( J 4.3 ˆ ǫ on 2 ab 2 N γ 4.3 + ˆ ǫ   ab 2 γ ˆ ǫ S S are the components of  2 + 1 Z Z S a t t is a tangent vector, while Z d d e V t S d Z Z T ), and the reality conditions ( − − Z and ∈ a a 3.3a = = e U N N ∂ We insert the S Levi-Civita density the other side of the interface, i.e. It is then useful to dualise the component functions where the boundary spinors (i.e. ( of the boundary action ( arbitrary: the matching conditions ( on them. The reality conditionswith ( respect to the Lagrange multiplier where On the other hand, there are the matching conditions ( where the boundary action ( are the timestraint components ( of the Lagrange multipliers imposing the area JHEP05(2017)142 , . a b = S δ } ) = on (4.20) (4.21) . The q (4.17a) (4.19a) (4.18a) (4.22a) ( (4.17b) (4.19b) (4.18b) (4.22b) β a bc ¯ ] B ǫ ˇ ¯ ℓ B , ac Lie algebra A ) ˆ ǫ p ) A ts among the ( [ ¯ C ∗ t A , . Next, we have rameter ϕ ¯ π { (2 S = sl . . , . Ba , , ) ) ), which has dimensions A ! ! ! , = 0 ! = 0 = 0 A = 0  p, q p, q 4.2 i i .  ( ( ! . The fundamental Poisson . = 0 ] B B , , ℓ ℓ e (2) (2) ) ) A
, δ δ A A are canonical conjugate to the + cc A e π ) ) [ e ℓ + cc π π e p, q p, q B B ∗ D D A a ! A
( ( δ δ ℓ = 0 e ϕ + + ℓ e e a D A A A
ℓ ( ( (2) (2) C C e , A e connection D A = δ δ δ δ ℓ A e A ℓ e A ) ℓ e π ab ab ABab ABab e A π − e π , implicitly defined through AB AB ǫ ǫ C ˇ ˇ e F ǫ ǫ e π + , F 1 1 γ γ A S − ℓ ab ab A − − a ˆ ǫ A ℓ ˆ − ǫ and ℓ A 2 γ D = = + A 2 γ ] SL(2 a ℓ = = + h ) yields the Gauss constraints A π h A

– 26 – A ℓ D ) )

π π ) ) A q q [ AB AB a ( ( q q ∗ t 4.15 π i + i ( ( Λ e  Λ B B ϕ ϕ J e β a S S S S , ℓ , ℓ Z  ) ) = Z ) and their complex conjugate, e.g. CDb Z CDb Z N p p e ( ( ) of the action, we can immediately read off the symplectic N S A ,A ,A A A ] = Z ] = ) ) π ] = ] = S 4.18 e π π a p p Z ϕ gives rise to the vector constraint ( ( [ 4.15 yields the scalar constraint   AB AB J a a ] = [ a e a a [Λ M [Λ N ] = N AB AB in the action ( N M [ N e AB A A AB [ ) and ( a B   G e S G H A e Λ 4.17 is the Dirac delta distribution — a scalar density — on denotes the complex-valued coupling constant ( and ) γ B , determine the phase-space completely, all other Poisson bracke ) p, q A is the inverse Levi-Civita density on ( Λ p, q ab (2) ( ǫ δ ˇ (2) Next, we have the constraints. The variation with respect to the . Its real and imaginary parts are related by the Barbero-Immirzi pa δ ~ ¯ B ¯ A ¯ where brackets are ǫ densitised momentum spinors the canonical symplectic structure for the Furthermore, canonical variables vanish. of structure. First of all, we see, that the spinors Variation with respect to where The Lagrange multiplier Finally, we have the matching constraints Poisson brackets ( Going back to the 2+1 split ( 4.2 Phase space, symplectic structure, constraints which is given by elements JHEP05(2017)142 ). 4.23 (4.26) (4.24) (4.27) (4.23) (4.28c) (4.25a) (4.28a) (4.25b) (4.28b) -valued test ) , C , i . the other hand, (2 sl ebra among the con- + cc , its time evolution is of the system are indeed generators i are ] , , sional diffeomorphisms F B . .   ℓ e B  B AB . A A C , ] AB AB e π ] i ] ] [Λ A 1 2 2 ab e r the Hamiltonian flow ( . π [Λ , ǫ e Λ Λ ˇ ) [Λ AB b ↑ ] + cc , , A N ) with respect to the Lagrange B C 1 1 ℓ a G ℓ ]+ L M A ↑ e J N and ] A a [Λ [Λ [ denote vector-valued test functions ( 2   gauge transformations. First of all, a 4.15 AB L B π N a and ) ) = [Λ h A M , [Λ ] ab = ] AB AB A C M i a ǫ } ˇ − , e b π A G G AB AB ] + a ℓ [Λ N B [ e ,F G N a ]+ G ϕ = = a C N [ and a 1 ] γ  D ( + H SL(2 2

H a a a ] ] { M M i = [Λ D N N ] + B N = – 27 – a CD 2 CD 2 C ℓ o ] + L = a = e A A [ N F ] [Λ [Λ [ a } π 1 t

AB a d AB A d ) the situation is straightforward. We recover two H A ab e CD CD H π ǫ [Λ ˇ − , ℓ Lie bracket e b ,A , ] ] ] = [Λ ,G ,G b a ) a ] ] 4.19 AB = ] + M B C a e N N G N , o [ [ N [ A AB 1 AB 1 ] [ b ] a a N b + e 2 (2 S h [Λ [Λ H H H Λ sl M  { n , [ = AB is the Lie derivative. b 1 AB AB G e a gauge transformations: if G G [Λ H ] h ,H   ) ] . 1 γ a C a , N + J [ N,M gauge transformations. If a ) H , we find SL(2 = [ C and n S , a ϕ M N SL(2 L , we find after a straightforward calculation that Thus far concerning the constraints. The evolution equations, on It is instructive to work out and see that For the Gauss constraints ( S where we have defined the functions on of two-dimensional diffeomorphism and local we have modulo on The vector constraint, on the other hand, gives rise to two-dimen 4.3 Constraint algebra andHaving defined gauge the Hamiltonian, symmetries we proceedstraints to and calculate the check Poisson whether alg the constraints are preserved unde where copies of local multipliers where the corresponding Hamiltonian is the sum over all constraints which are obtained from the variation of the action ( governed by the Hamilton equations assume a Hamiltonian form as well: for any phase space functional JHEP05(2017)142 ) ↑ L 4.19 (4.35) (4.34) (4.29) (4.33) (4.32) (4.30c) (4.31a) (4.30a) (4.31b) (4.30b) principal ) , C , + . ), on the other ii , i a A B ℓ A ℓ e o further Poisson b , ℓ e SL(2 A AB b fundamental trans- D ℓ e 4.28b
 e uss constraint ( e A b D ab A b ↑ N ǫ ℓ N e ˇ a a N ) L ). The situation on the
e A J D A transformations ℓ a = A A π a ℓ e ℓ C e N a ba J , h , ( ) define the horizontal lift + J B . is completely analogous. , B a ] and AB A ℓ π AB a B a U(1) ℓ D F A B e A Ba A J G a e b D , ℓ A B 4.28b − A , Λ ϕ D ), which Poisson commutes with the [ A + N a Λ A + e A a A A π i ℓ ℓ − D = A b ϕℓ ϕ M B ℓ e ℓ all have compact support) and bring all b = = +Λ = a D − 4.22a cd and A b e ) and ( J a D ℓ

= 2 b = + = J A AB A A J ab N } A e J π Ba } ]

 ǫ π F ˇ ℓ , ℓ a A , A A a , ] A a S ] ℓ 4.28a cd e – 28 – A J π J ) alone, many Poisson brackets can be inferred Z a ˆ [ ǫ and , ℓ D ℓ e , ,A a ] a CD ] a ] + ] e CD D ab a ϕ a ϕ N a [ ǫ ˇ [ N [Λ + i , but generates the 2i 4.31 b [Λ h J N ,M CD h N ] β M N M Ba N S CD ϕ [Λ  AB CD { [ − A Z 1 γ L = G [ G G e A  M CD a h }  = ] { G . Equation ( a 1 γ M ≈ − } ]  J ) and . From ( a [ Ba + − B a A J A ℓ [ e π Ba b e A = A ( ,M A ] } π ] A ,M ” denotes equality modulo terms constrained to vanish. a ] N and ab [ and a ≈ J ǫ ˇ [ S b A b A N { [ ℓ e e N π a 1 , γ ,M H ] A { a ℓ e N [ connections a ) H C { , Concerning the algebra of constraints, we only need to consider tw For the Gauss constraint, the situation is easier. There we get the We then also have the matching constraint ( brackets, namely and equally for If we perform a partial integration ( immediately, as for instance SL(2 where the symbol “ hand, returns a diffeomorphism onlyis on-shell satisfied, — in only if, which in case fact, the Ga and covariant derivatives on one side, this can be simplified to Equally for of the Lie derivative into the spin bundle. The third Poisson bracket ( formations which are the generators of right translations along the fibres of th bundle and the associated spin bundle (with sections other side of the interface with variables JHEP05(2017)142 is le- for ) of S 15 (4.38) (4.39) (4.40) (4.36) (4.37) 4.31 constraint tem of con- on-shell , . In other words ¯ , which generate A ] ¯ ℓ b = 0 AB D ) as follows: in general, ¯ e  vanishes A ( [Λ ¯ ℓ z } ] e second class component transformations ( be inferred trivially from )) and the matching con- AB denotes the corresponding traints of the system. The N + i) e C [ G  is defined by . v β b 4.20 2 ( ,S S , J , d ] )] ab b U(1) and 0 ) q   ] on M ) give already all relevant Poisson D ( ≈ a  such conditions per point. One of and N a v , z AB o ) (as in ( N 2 J L ]  S 4.35 ) are satisfied. [ d ] ( i [Λ ) + N S z  [ N ( ( 0) [  − a AB , ] z 2 = 4 ) 4.22b S a ,S We have now collected all Poisson brackets J  G = i [ weakly vanish as well, which is an immediate ( + J × ) and ( it then suffices to determine those values for a ), ( b z } [ on phase space. We can then parametrise the 0) A ( ] 2 , ). ℓ M 0)] a ) N b , { – 29 – 14  M 4.33 J )  ( [ 6= 0 4.22a D  h z b ( 4.28 a A ( A z ). Hence the constraint ,S ), ( two-metric on ℓ ), ( D a ) on either side of the interface. Equally, for the ℓ ] ( ) first-class. M a a a ab J ˆ ǫ )) generating two-dimensional diffeomorphisms on h J D N 4.21 4.31 ) = 2 [ 4.32 a [ 4.30  A a 4.22b a ( a ℓ (++) M ), ( z 4.28 ), ( H M J ), ( n { N = 4.20 ) and ( is first class and generates the L  4.27 )). The scalar constraint is second class, which is a consequence ) does not vanish ] 4.22a ), we see that this is certainly true for all ), ( (see (  ( ϕ ] A [ ), ( 4.30 ), ( a ℓ , z 4.19 a ) 4.33 M 4.22b N  as follows ” denotes equality modulo terms constrained to vanish. All other re ). So as for e.g. [ ( ), ( D 4.25 4.21 a z a ≈ , we have one further constraint, which renders the entire sys A ). For generic ℓ  H J ), ( 4.30 a 4.28 J = 0 (as in ( 4.35 4.20 ] for which the Poisson bracket a A ) or ( ) ), ( ℓ is ( is a fiducial signature J  a [ ( gauge transformations ( ] a z ab D a is complex-valued, hence there are 4.19 ). For the matching condition, the situation is more complicated. The ) 4.28 q A ] J M  ℓ [ C a . Going back to ( a , We are then left with the scalar constraint The equations ( and J 4.35 Given all constraints ( If [ N ) M a  15 14 ( consequence of ( SL(2 them is second class, allwe others will are have again that first class. This can be seen vant Poisson brackets involving where the symbol “ where area element. The only relevant Poissonof bracket for determining th all densitised vector the boundary spinors. dition of ( M that are necessary to identifyvector the constraint first class and secondfirst class cons class. And so are thematching Gauss constraint: constraints First class and second class constraints. brackets. All other Poissoneither brackets ( among the constraints can z where the Lie derivative of the vector-valued density straints ( which is an immediate consequence of ( JHEP05(2017)142 ) s B ℓ ) of , see A ( 2 (4.42) (4.41) (4.44) (4.43) η z/ phisms (4.45a) (4.45b) − 4.41 ders the ensional = e → AB . It is given ] Σ A ∗ a , η ) and ( ϕ J [ in the bulk. The a 6= 0 i.g. and l shifts 4.40 M ¯ A A ¯ ℓ ℓ s ( 2 b ) and z/ D e ¯ A . The geometric meaning ¯ ℓ ] → 4.33 A ). We then have additional ℓ N A [ . a ℓ 0 S ). ), which generate two copies of D 4.31 A ≈ ℓ . ., 4.31 on either side of the interface, the o , ab ] 4.22a ] q ,  N AB + cc + cc B [  AB v A ). This can be inferred already from the e 2 ,S φ transformations and two-dimensional dif- Dφ  i [Λ  i ) and ( d ) ) 1 γ 2 1   C transformations S ( (  i 4.41 AB ) − Z y C  i 4.30 e G , x ( 4 , = + = 0 – 30 – 0 U(1) . , x  ), ( ≈ ≈  B U(1) 0 is real, we get an additional first class constraint, a . a AB and A  ) J = 0 J ] Π A a  h 4.28 ( h φ φ A a J a + cc x δ δ ℓ AB h gauge transformations ( a M a o M ] = [Λ D C M under the ) A N  [ n ℓ AB ( z AB G ,S U(1) Σ but what kind of gauge transformations are generated by those = ¯ gauge symmetry is the invariance of the parametrisation We have now identified all first class constraints of the system:  i ) ) that are first class? C   ) and the matching condition ( and 16 ( ( ] z y a ). i S U(1) , J transformations. It enjoys a further gauge symmetry, which ren 4.21 is purely imaginary. That this defines a second class constraint, follow [ 0 gauge transformations (on either side of the interface), two-dim gauge transformations, a ) 2.20 ) ) has in fact more symmetries than just four-dimensional diffeomor  )  ) C ( M a , C C y J , , 3.32 h a ) and ( = i ) defines another first class constraint follows from ( ) through SL(2 ) M 2.5 SL(2  SL(2 that Poisson commute with the scalar constraint ( n ] z 4.41 4.33 a The answer is hidden in an additional gauge symmetry, which appears We are thus left to identify the single second class component of J The origin of the [ a 16 bulk action ( of the self-dual area two-form equation ( namely and local first-class constraints, which canM be identified with those component components of internal infinitesimal gauge transformations ( diffeomorphisms of of local feomorphisms is clear, All other constraints Poisson commute with ( is first class. When which does not vanish unless That ( entire theory topological. The action is invariant under the infinitesima by the expression from ( where vector constraint ( Gauge symmetries. the two Gauss constraints JHEP05(2017)142 )) C , . For in the (4.49) (4.46) (4.47) ) with (2 (4.48a) (4.48b) t and the sl S ] : 4.45 N [ below. M S ( slice 1 1 . Ω is of the particular ∈ aints and nineteen . We now want to ) on the boundary. are eigen-spinors of t a transformations and at the boundary by S . ive of the first. Com- A Ba J C a ℓ a = const A 4.45 J t J φ a the vector constraint, two slice and U(1) D lar constraint . B , generating diffeomorphisms , to a A ℓ ] = a π A ab gauge transformations on either ] o ℓ AB ) N [ 2 1 ) are second class. We then have the B . Π a AB , which generate the residual shift ℓ C ] ] = const B + , , a H Π A a ℓ ( ) t ∗ t , B J A J ) [ B π ℓ [ ) will be violated, and the Hamiltonian ϕ ℓ [ , generating ℓ a , B a , but only partially. To understand this A ] ] ) ℓ SL(2 ba ℓ b ϕ ab M A ǫ M A ˇ [ ( ˆ ǫ J b ( ba 4.41 In summary, the system admits four types valued one-form in the bulk. The addition of [ ℓ ǫ ˇ 2 1 b π J a ) above). We thus also know that ] M b ) 2 1 D J M ) of C ), a short calculation gives ≡ = ( AB , 1 γ n – 31 – = a ) at the boundary. Only if a 3.46 e ] ) and ( [Λ (2 1 2 ) down to a − J AB 4.41 4.18 sl AB Π AB 4.20 = = = AB ∗ t φ Π 4.40 3.46 e G ∗ t ϕ o o shift symmetry therefore generates a shift transformation ( ϕ [ = is an Ba ] AB ) and ( ) and ( a A and Π B J ] . Now, the boundary spinors , AB [ ] A ,A a b 4.40 4.17 ] Π φ M b AB J (see equation ( M [ J b [ [Λ of N b will lie transversal to the constraint hypersurface. M generates a version of the shift symmetry ( ). Notice, however, that this shift symmetry is broken partially by AB ), do we get a symmetry preserving the constraint hypersurface M N to n ] ·} ) of the matching constraint n a G , , the conditions ( ] J a a 4.49 ) allows us then to match the gauge parameter [ AB 4.41 a J J 4.43 Π [ ) at the boundary. The situation is summarised in the table a M 4.45 M ) or ( { 4.48 4.40 The counting proceeds as follows: we have two second class constr , the generators -component ( ) S  ( the remaining components ( side of the interface, the matching constraint demanding that of The matching constraint first class constraints, which are the vector constraint Dimension of the physicalof phase gauge constraints, space. and two second class constraints. The sca for the shift symmetry in the bulk, with the gauge parameter which is the pull-back of equation ( form of ( first class constraints; there are two independent components of gauge parameter ( the addition of the reality conditions ( generic values of vector field Using the Poisson brackets ( y of the boundary action breaks this boundary component parison with ( Notice that the last line is nothing but the covariant exterior derivat symmetry ( more explicitly, let us first define where the difference tensor illustrate that the pull-back of which is the pull-back of the canonical bulk momentum JHEP05(2017)142 ] , ) ) C 33 34 2 + , , mely 4.40 × (4.50) 32 (6 SL(2 × , has forty 2 A ℓ e (the smearing , which leaves , ] ] A a ϕ e ] for an π re just pure gauge J [ [ 7 , 3 = 7 ture is determined a o the mathematical , M itly (see e.g. [ × 1 Ba erface. This renders M A 6 = 12 6 = 12 2 = 4 2 = 4 tivation concerns, how- can only appear at the e eedom, which leaves us , A then also need to impose e of the theory. The phase × × × × ) ) of 40 1 2 2 2 2 1 + 2 quiring that the connection p, q and ( 4.43 ]. The complex variables have s, in particular, loop quantum , which parametrise A transformations on either side generators (3) A ) as well. It was then noted [ ℓ 31 ℓ δ ) e , C , a b , C A δ A , ) at the boundary. The kinematical ) 30 3.32 π e π ). There are no local degrees of freedom. D B , U(1) , δ 4.48 C Ba SL(2 Ba 4.18 ( A A δ A 2 1 e A A = ) and ( and connection [

– 32 – ) A q 4.17 ℓ ( b , A SU(2) π CD coordinates , ,A 17 Ba ) p A ( -valued first class constraints -valued first class constraints A a -valued first class constraint C C and the second class component ( C AB ] Π  N [ three one is second class, three are first class two first class constraints three one one first class constraint S dimensions. There are two additional second class constraints, na is complex) and three additional first class constraints, namely ( real dimensions. Therefore, all directions in phase space a ] ] C AB AB ] ] → a a e 19 = 2 [Λ [Λ ] ] J S N [ ϕ [ × 2) = 40 : [ a N a AB AB [ 2 The constraints remove forty dimensions from the phase spac ) generating the residual shift symmetry ( × ϕ e S H G G M M Constraints Dirac classification DOF removed − 4.41 40 On the phase space for the complex variables, the symplectic struc Loop quantum gravity can be based either on the phase space [ 2 + 2 The symplectic structure is determined by ( ] that the theory can be discretised, or rather truncated, by re 17 × function of the interface, two independent components of the directions. This renders the boundary theory topological. times six independent constraints generating the the scalar constraint 2 dimensions, every first classwith constraint removes two degrees of fr the advantage that local Lorentz invarianceadditional is reality manifest, conditions, though which we arefor otherwise a already recent solved implic analysis on the issue). and ( phase space, which has canonical connection or the phase space for an ever, non-perturbative approaches to quantum gravity,gravity. such a Let me explaindetails. and justify this expectation, without going int the boundary theory topological.two-dimensional corners. All physical degrees of freedom 4.4 Relevance forSo quantum far, gravity we have only been studying the classical theory. The main mo us with no local degrees of freedom along the three-dimensional int which can be derived from the topological bulk action ( 35 Table 1. space has canonical coordinates by the fundamental Poisson brackets for the self-dual variables JHEP05(2017)142 ) ) of A l C , of a dual N , ω ) ¯ (4.54) (4.51) (4.53) (4.56) (4.55) l A Γ (4.52a) (4.52b) C being a π , SL(2 must be (¯ is unique for which Σ (2 ∈ , . . . , p l A ) A 1 l sl π ℓ C p h , × constraint lves to the case or ) is then always a C l A , SL(2 ng link. ω ∗ of the graph T , ) L SL(2 p ( ≃ B , of a cellular decomposition ℓ e ) B } ]. , . . . , l C i , area-matching AB A 1 , ] ǫ l 39 E N , which have been introduced in { ) − (1) . A l are interchangeable, C ℓ , D l = SL(2 → l , B , l ∗ A , and a link holonomy p e ω )

ω A l ω is the spinor equivalent of a normal h T l D A l l B C e ω and = 0 , which is incompatible with e π ¯ , SL(2 e π e A l A . , = [ Γ A , ω A √ ) e π (2 / and − l A π n l A l = 0 B C l sl L e − π gauge invariance at the nodes l e B is often required to be a Cauchy hypersurface as in l ω A
ω on each link + cc  ) ∈ π π A ) A l l C AB l l ( ) Σ A l l C A l is incompatible with a four-dimensional metric. π . Finally, one needs to impose additional reality ω C , ω π , Π C ω , which are those parts of e , ω , which implies that either ) Π , – 33 – l A ) 1 2 √ , ¯ l A p A π l AB ( , which is a finite-dimensional phase space ¯ C 6= 0 ℓ π l AB 19 , where AB = , = } B Π A ǫ = i Π SL(2 can be parametrised by a pair of bi-spinors ℓ SL(2 ℓ . This results in the cotangent bundle of the moduli ∗ l i SL(2 B E + i AB l ) B = ∗ { l l AB = 0 Σ T A i A M β Π ] SL(2 T ] l

Π = i Π ¯ ∗ S l , A h 18 l B l [ = l AB (0) ¯ ¯ π T A l h − l . The roles of A Π Γ , ω ( A → ω A l A P n p Σ , modulo ℓ ¯ π A h } i  A n E = [ { dual to the link. This normal is often required to be timelike, if l A is assigned to the fibre over the initial point of the underlyi Poisson brackets ω l f , or more precisely l AB A Π ℓ , such that l transformations, and we can, therefore, always restrict ourse f on either end of the link. The parametrisation is the following = C ) A A l on postulates ω e ], a generalisation to null surfaces was proposed as well cf. [ , ω U(1) A ¯ ℓ 38 l A – e π (¯ The relation to the new boundary variables 36 By convention In a Hamiltonian approach, the three-manifold 19 18 One of them is conditions, otherwise the self-dual flux the flux is non-degenerate, i.e. and connecting the two endpoints. Unless returns the relevant portion of spatial hypersurface, the pair of the underlying three-manifold built from the cotangent bundle space of flat connections on and shows that the symplectic reduction with respect to the One then e.g. [ to the system of edges for which modulo be flat everywhere except along the one-dimensional edges themselves proportional to the other requires that vector to the face it is, however, nullspinor rather than timelike, the condition simplifies: there the graph. this paper, is as follows. A point in the phase space link is labelled by a Lie algebra element JHEP05(2017)142 . ) ) ) ), C (1) , l 4.17 4.53 (4.57) 4.22a dimen- or SL(2 in terms on either cially, we ) (0) l A l AB e is the parallel , ℓ Π ) A on a graph with ǫ t in more detail. at the endpoints ete theory on a lits into a union ( e l π ) ) ( i A l → A l , son brackets ( p odel for the metric. e ) rms of a topological -dimensional surface e . h , ω ical structure of loop ). In every such four- A , ω i B ¯ l A Λ l A ries, which represent a e , ℓ π e π e π reedom in the bulk, non- (¯ A B ature defects propagating ( s — can be all lifted along A π , on a family of null surfaces ] ( ) A l and A (1) l ℓ ) , ω i , . Adopting these identifications, → A l p l A A Γ h π , ω [ π ( ). The same happens for the link i ¯ l l A towards either endpoint f π Z p ) of the self-dual flux (¯ 3.46 = there is then a three-dimensional internal ] and other discrete approaches, such as l ) along a link is analogous to the l A f 4.52a with its dual face, and 20 . e , whose geometry is either flat or constantly l i l f of the graph ], causal dynamical triangulations and causal M , π 4.52b L 40 – 34 – B =1 π N i B S A , . . . , l ] 1 = l (1) l M → : p shining out of h } dual to the links as two-dimensional cross sections of three- [ ) . For any such face i )) and in the three-dimensional boundary theory (as in ( l dual to the links of the graph. The Poisson brackets ( l l f ij )( ) across the interface. Equally for the constraints: the area- l f M , and the two pairs of spinors Z l ij f N { ( 4.55 f parallel transport ( = N 3.25 ) ⊃ l A C ) π l , dual to the links )( ) and ( ij } ( l is the intersection of the link f mirror the continuous boundary spinors SL(2 l N { This paper developed a model for discrete gravity in four spacetime 4.54 L ∩ l ], but there are fundamental differences. First of all, and most cru f )). There is no doubt that this correspondence must be worked ou 42 = , , . . . , l p 1 l 41 4.20 The theory is similar to Regge calculus [ In other words, the discrete loop gravity spinors null boundary shining out of the faces of the link dual to the face correspond to the boundary spinors dimensional null surfaces transport along the link going from the intersection links side of a null interface implies to view the faces This in turn suggestsintegrals to identify the conjugate spinors with the two over the faces curved inside (depending on the value of the cosmological constant sions where the only excitations ofat geometry the are speed carried of along curv light.trivial The curvature resulting theory is has confinedsystem no local to of degrees colliding three-dimensional of null f internal surfaces. bounda gauge transformtion ( sets [ have a field theoryThis for field the theory Lorentz is connectionof topological rather four-dimensional and than cells the a underlying lattice spacetime m manifold sp ’t Hooft’s model of locally finite gravity [ 5 Summary, outlook and conclusion Summary. and ( So far, I findquantum the gravity — analogy graphs, encouraging. operatorsnull and It surfaces spin-network suggests function obtaining that a thefield fully kinemat theory covariant with picture defects. of the dynamics in te on the null surface, and theholonomy; decomposition the ( matching condition andgraph the (as reality in ( conditions appear both in the discr which were previously postulated, can be then derived from the Pois of the discrete spinors is analogous to equation ( where JHEP05(2017)142 ) ey s a 5.1 (5.1) ]. alued i two- ), that 22 , where 3.4 2.21 ) metric at -Mills gauge action assumes ic structure for (0++ l notion of causality: iables. We proposed consistently glues the point particle [ e neighbourhood of an nnection couples natu- conditions ( ffect: given a boundary The problem of finding n where the Yang-Mills rs. The internal bound- are explicit solutions of ity is in the transversal e is confined to internal ified in section the gauge symmetry from ) that match the intrinsic heets of curvature defects o finding the right bound- defines the two-dimensional transformations and residual 3.3 have an immediate geometric A ) ℓ C A A , π π . Clearly, the boundary term ( A . ), which represent impulsive gravita- SL(2 π 2 z/ + cc 3.36 − A e Dℓ component → ∧ is compatible with a signature A – 35 – 0) ) , A π B π (0 ℓ A N ( and Z π connection is trivial everywhere except at the internal A defines the null generator of the interface, the spin (1,0) ℓ ) ) is local, and splits into a sum over all four-dimensional ¯ 2 A C ¯ ℓ are unique only up to z/ , A e and the momentum spinor 3.36 A ℓ for a boundary that is null. That the internal boundaries are i Lorentz connection, the relevant charge is spin, which suggests , which are three-dimensional. π A → 2 j ℓ SL(2 ) A ), and the spin M C ℓ , and ∩ i A 3.31 returns the pull-back of the self-dual component of the Plebańsk ℓ ). The spinors at the interface are not completely independent, th M ) SL(2 B ), ( = ℓ 2.22 A ij ( . The resulting boundary action is and its canonical momentum, which is (in three dimensions) a spinor-v 3.46 π N A across their boundaries. A π ), ( } ℓ i transformations 2.3 M { C What is then the right boundary term? We are viewing gravity as a Yang The model is specified by the action for the internal boundaries. This The underlying action ( to look for ansuch action an with action spinors in section as the fundamental boundary var the interface. Furthermore, there are the glueing conditions ( two-dimensional boundary, a colour chargethe appears bulk. that cancels For an theory for the Lorentz group.rally At to a its boundary, boundary a charges.electric Yang-Mills gauge Consider, field co for is example, a squeezed configuratio into a Wilson line. Wherever this Wilson line ends and hit three-geometry across the interface.direction. Adding the The glueing conditions onlymetric, to the metric the action spinors discontinu has a further e are subject to certainensure constraints. that the First self-dual of two-form all, we have the reality tional waves. These are exactinterface, solutions and of may describe Einstein’s equations e.g. in the th gravitational field of a massless the field strength of the we gave a constructivethe proof equations of of existence: motion derived we from showed the action that ( there null rather than space-like or time-like is well desired, it imposes a loca ary action, which cancels thecells connection variation from the bulk and building blocks, inner three-boundariesary and terms two-dimensional corne are necessary tothe correct have a dynamics well-posed for variational the principle. curvature defects boils then down t boundaries, which are nullpropagating and at represent, the therefore, universal speed the of world light. s This was further just boundaries cell, there are no local degrees of freedom. Non-trivial curvatur area element ( interpretation: the bilinear form ( The configuration spinor component U(1) two-form a surprisingly simple form.a spinor It defines, in fact, nothing but the symplect JHEP05(2017)142 ) to 4.1 A π . First has the , which 4 ], a new j A ℓ 45 M , where the – ω 43 ). All of these connection be- , and − ) of the glueing section ) . The symplectic i ative approaches 38 D ) above). Finally, 4.48 C – M ications supporting , 4.41 its dynamical struc- e phase space on a he continuum. This 36 ries, which represent e theory has no local canonical structure is ), ( ] = 0 4.18 split of the boundary transformations of the a damental variables ap- ), ( ation spinor loop gravity spinors are sional diffeomorphisms, al cells, whose boundary SL(2 al internal boundaries. J C [ pinors as the fundamental 4.45 ion amplitudes, which are a Chern-Simons action ( tical phase space, which is 4.40 field on a null surface. ibutional spacetime geome- ) M derivative 2 + 1 U(1) ) and ( C C , gauge transformations on either gauge symmetry. The symmetry 4.17 ) C U(1) SL(2 C , ), on the other hand, is second class, × ) U(1) as in ( ) is topological. We then integrated out C SL(2 4.20 , A ), which is added to the action by replacing e π 3.32 bounds two bulk regions SL(2 3.3a ij and – 36 – ), which is a sum over the constraints of the sys- N A ℓ e The proposal defines a topological gauge theory with 4.24 by the ) of the glueing condition , which is the pull-back of the momentum spinor D A 4.43 π in the bulk obtaining the acts as a Lagrange multiplier for the constraint to be imposed. AB ω derivative Π ) C Gauss constraints generating , ), which generate the residual shift symmetry ( ) C , SL(2 hypersurface (equally for 4.22b . invariant, but it violates this additional ) SL(2 C , gauge connection C A more thorough analysis of the gauge symmetries was performed in The most interesting indication in favour of our proposal concerns Our main motivation concerns possible applications for non-perturb = const SL(2 t boundary spinors, and finally the three first-class components ( canonically conjugate variable comes Poisson non-commutative at the boundary, while the configur side of the interface, the area-matching condition generating a induce boundary variables from either side. We found that the action and identified the symplecticpear structure twice, of because the every theory. such All interface fun U(1) graph. An interpretation waspaper closes missing this for gap what andthe these provides canonical a spinors boundary continuum are interpretation: degrees in of the freedom t of the gravitational ture. The theory is topological and this suggests that the transit representation of loop quantum gravity has beenconfiguration introduced with variables. s This construction was bound to the discret conditions ( constraints are first-class, theand reality so condition is ( the fourth component ( we found the canonical Hamiltoniantem, ( which consist ofa the pair vector of constraint generating two-dimen the self-dual two-form at the three-dimensional interfaces. Next, we performed a the covariant of all, we noticed that the action in the bulk ( is is restored by the area-matching condition ( to quantum gravity, such asthis loop idea: quantum first gravity. of all,extremely We the have close model a to has few recent a ind developments kinematical in phase loop space, whose quantum gravity. In [ defects. Solutions of the equationstries, where of the motivation gravitational represent field distr isis trivial in null. four-dimensional caus Thecurvature geometry defects is propagating discontinuous at the across speed these of internal light. bounda reduction removes, therefore, forty dimensionsforty-dimensional from as the well. kinema Thisdegrees brought of us freedom, to neither the in the conclusion bulk that nor th at the three-dimension Relevance for quantum gravity. JHEP05(2017)142 . AB (5.2) (5.3) (such Dφ ∝ M is invariant . It is also i AB Σ M × φ at the boundary. δ ain conceptual efinition of the  ) as an external . . This combina- s assigned to the } ugh a continuum = 0 ij e scale entering the exist and turn into B 3.36 + cc ℓ N up flow, which would ch turns into a boundary with different combi- uld be reminiscent of { A i our-manifold ) for the gauge parameter M plitudes can be written π } nt hint that the formal ∂ i delta functions imposing B AB × ℓ tion ( H 4.49 ] re two possible answers to M Π A ℓ { ., ∝ A, ℓ AB SU(2) Π , AB + cc 1 γ ) Ψ[ ! Σ . ), implies that the gauge variation of the B + ℓ AB t preserve the bulk region holonomy around the perimeter of a A AB Σ ℓ ) + cc Σ AB ∧ ∝ C A F ∝ , AB AB Dℓ AB AB constraints Σ Σ ( A Π Π i δ SL(2 π  AB M i φ – 37 – ij FP Z ). A shift symmetry generates a variation M N ) for i Z Z 3! ] ∆ ) are defined for a given and fixed family of four- i 4.45 ] for a recent derivation). Such moduli exist, and the ~ ~ 3.33 2i , ℓ 20 -symbol defines the vertex amplitude for the Ponzano-

46 5.2 ] =  π j in a boundary Hilbert space ) has a mathematical precise meaning and defines a so- i 6 ] M 5.2 exp , A, exp (as in ( defines an observables is straightforward to see: clearly, it } } ] A, ℓ i [Π i ij ) building blocks, glued and traced together according to the Vol[ N M Y Y 4 = 0 4 Ψ[ M D { { group integrals for each edge times × × that vanishes due to the boundary conditions ( Z AB Vol[ , which are glued among bounding interfaces , or the trace of the 4 Σ i ,..., } , which implies, in turn, i AB D yields a bulk integral over a total exterior derivative, whi B M Σ ] ℓ i M = 0 A SU(2) ∧ { [Ψ] = The amplitudes ( ℓ M } i n ∝ AB M φ Vol[ { M 4 i ∂ φ | Z M δ ∂ R AB φ ∝ , i.e. That the four-volume 20 -dimensional ( AB the flatness of the connection (see [ as a product over ordinary integrals (or sums)quantum over gravity the in moduli three dimensions, of where the the theory. Ponzano-Regge am This wo Regge model). Perspectives. as in three dimensions where the corner. The existencedefinition of of such the non-local path observables integral is ( an importa integral invariant under the residual shift symmetry ( for fixed boundary states limit, which sends thetheory number would then of most four-dimensional likely includegive cells some a to sort prescription of infinity. renormalisation for gro Thedifficulty how with d to such take an this approach limit is in that a there rigorous is manner. no fundamental The lattic m natorial structures supposed tothis be question: taken into in account? the There first a scenario, the full theory will be defined thro φ dimensional cells background structure. How are then different discretizations n called spinfoam model, which is given by certain fundamental amplitude simplest example is the four-volume four-volume under local Lorentz transformations and diffeomorphisms tha of a given four-cell The equation of motion formally given by the path integral adjacency relations of the underlying cellular decomposition of the f torical structure is an ad-hoc input, which enters the classical ac JHEP05(2017)142 ry of t of dis- the indi- . In this ], where the connection in entire theory, 51 ] and so-called ) – 4 captured by the C – variables and the ther, namely how 49 , 2 s of combinatorical aration. the traceless part of he torsionless condi- ore rigorous analysis en take the quotient the continuum limit, ells would be seen as e microscopic theory, refore, more sophisti- oretical Physics. Re- ounding null-surfaces. from the perturbative ravitational radiation, SL(2 anada through the De- ome out of the model. A minimal example for ulus [ vature tensor at the de- nergy momentum tensor the auxiliary field theory. g transverse gravitational nal geometries, where the that special solutions of the 3.2 ] of inflation, which has three propagat- 52 ) are non-vanishing, the Ricci tensor being gauge variables (an ) C 3.72 , – 38 – SL(2 ). This action was constructed such that the solutions 3.36 ] may be required. 48 scalar mode). I find this idea very promising and exciting, and it gauge transformations, we are left with a theory that can only , 0 ) 47 C , ) and the Weyl tensor ( , 6 , 5 3.73 SL(2 ). Indeed, it is the gravitational field itself that determines the size of . This could now either mean that the model already includes some sor , or — and I find this more likely — that the model describes a metric theo b ), hence we expect that the only relevant degrees of freedom are b ℓ ℓ 3.36 a a ℓ ℓ 3.33 Finally, there is one obvious open question that I have avoided altoge The second possibility, which I find more appealing, is a more radical idea ∼ ∼ ab ab by the internal the bulk coupled to spinors at the internal null boundaries). If we th be described by ation metric ( and a connection. The connection satisfies t T tributional matter (as in e.g. string theory), with a distributional e search at Perimeter Institute is supported by the Government of C will be presented in an upcoming article, whichAcknowledgments is currently under prep This research was supported in part by Perimeter Institute for The is, in fact, the line of reasoning that I am currently investigating. A m ing degrees of freedom (whichand are one given additional by spin- the two polarisations of g fect: the resulting Weyl tensorradiation. is of But Petrov type we IV, thenthe thus describin also Ricci tensor saw ( that thereR are solutions where both equations of motion exist that have a non-trivial distributional cur of the equations of motioncurvature is represent trivial four-dimensional in distributio four-dimensional cells,The which geometry are is glued among described b in terms of expansion of a quantum field theory over a group manifold. scenario, the amplitudes forFeynman a amplitudes given for and fixed anone configuration auxiliary would of then quantum four-c sum field overstructures, theory. an which infinite, would To but arise define most fromThe likely the the approach very perturbative would preferred expansion be clas of gravitational conceptually path very integral similar on to a group given field simplicial theory discretisation [ arises vidual building blocks, andcorrect this notion makes of it scale difficultcated to to tools study identify the and the correct renormalization techniquesspinfoam such group models as flow. [ those The developed for Regge calc action ( gravity with more thansuch just a two theory propagating is given degrees by of the freedom. Starobinsky model [ which is defined by the action ( the two physical degreesThis of question freedom is certainly ofbut related general some to relativity hints the of should previous an c question answer regarding should already appear at the level of th metric, which is now locally flat. We then saw in section JHEP05(2017)142 is 21 α v (A.3) (A.6) (A.4) (A.5) (A.1) (A.2) aised and the Lorentz for the inverse ¯ , primed indices A , , ′ ) A ¯ ν B ′ C v ¯ ℓ , α ¯ B Bα ¯ he Province of Ontario A ¯ A AA ǫ Σ σ invariant epsilon tensors into an anti-hermitian SL(2 ′ ] 2¯ = ¯ ! ν , ν ) . It is then straightforward 4 (an element of the double i δ ous Lorentz transformation α ¯ of √ 1 C ′ A , which commutes with the σ R to denote a two-component ¯ µ ¯ / ℓ B , µ g [ , ∅ αβ δ A ) for the metric signature. = ′ BA ∈ η σ ǫ = i µ , ′ 2 Bαβ ! α ¯ β − α SL(2 B i i ] − v ¯ A η ℓ ¯ 1 ¯ 1 v +++ β ′ ] ¯ ) implies that the matrices 0 1 ∅ ¯ A = β ABα β − = σ σ ¯ 2Σ δ B ¯ ¯ σ CB ′ } ǫ i i α A.3 α ¯ [ σ β − ¯ 0 ¯ 0

AB δ 0 1 A, B, C, . . . α = ¯ ǫ [ σ σ , γ = ¯ αβ ¯ C A α η

¯ ℓ = 4 A γ ! are the three-dimensional spin matrices. The provide a representation of the commutation transformation A B σ { . We have α α ) δ with ) 1 2 ¯ α ¯ , 3 , B B − v – 39 – ¯ Bαβ 1 C A A B A Bαβ ] α ] , σ , = ℓ ℓ C ¯ = 2 γ γ A A = [ [ Σ . Equation ( A Σ ) αβ AB , σ . The generalised Pauli matrices provide an explicit ! ǫ σ 1 α C SL(2 0 0 ¯ 2 Bα CBβ σ ¯ Cµν , ¯ 1 ¯ 1 AB A ¯ = ( ¯ 0 1 ABα A σ A √ A ] ] Σ α σ σ into the product of the / A Σ γ σ γ ¯ . [ [ satisfy important algebraic identities. First of all, we have C 0 0 SL(2 ↑ + A ¯ 0 ¯ 0 −

= i αβ α 0 1 L σ ¯ η , ℓ A σ σ ≡ and any ¯ B A ∈ AA

Bµν ℓ α σ v ¯ ¯ C B B γ ¯ ¯ A A BA Σ ¯ ǫ 7→ maps an internal Lorentz vector ǫ ¯ g B α α . The relation between spinors and internal Minkowski vectors = AB Cαβ v ¯ A A ǫ A B g A A : A δ ℓ ¯ Σ A σ = = A = ¯ v ¯ B B refer to the complex conjugate representation. The indices are r is anti-hermitian is a consequence of our choice ( B ¯ A ¯ ¯ BC A AB A ǫ A A is the identity matrix and v η Λ 1 AC ¯ matrix C,... ǫ ≡ = The soldering forms 2 That ¯ β B, 21 × α αβ ¯ where the generalised Pauli identity cover of the restricted LorentzΛ group) into a proper orthochron relations of the Lorentz group. Indeed, we have η to check that the self-dual generators invariant Minkowski metric map. This isomorphism can be generalised to any world tensor. It maps are the self-dual generators of spinor that transforms under the fundamental representation A, define a representation of the Clifford algebra 2 through the Ministry of Research and Innovation. A Spinors and worldFollowing tensors Penrose’s notation, we write partment of Innovation, Science and Economic Development and by t provided by the soldering form matrix representation with where group action. Our conventions are soldering form lowered using the anti-symmetric epsilon tensor JHEP05(2017)142 , ]. , (A.7) 5 SPIRE , IN , ][ (2013) 3 , , , 16 . = 1 ]. ed. gr-qc/9411005 0123 ǫ Class. Quant. Grav. SPIRE , , , ]. ]. IN , Cambridge University Press, ]. , World Scientific (1991). ][ ]. ]. Living Rev. Rel. ABαβ ]. , SPIRE SPIRE SPIRE IN IN ]. SPIRE SPIRE (1995) 753] [ IN = iΣ ][ ][ Lattice Gravity Near the Continuum IN IN ][ SPIRE ][ ][ IN SPIRE LQG vertex with finite Immirzi parameter ]. 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