Loop Quantum Gravity Boundary Dynamics and SL(2,C) Gauge Theory
LLoop Quantum Gravity Boundary Dynamics and
SL(2 , C ) Gauge Theory
Etera R. Livine ∗ Univ. Lyon, Ens de Lyon, Univ. Claude Bernard, CNRS, LPENSL, 69007 Lyon, France (Dated: January 20, 2021)In the context of the quest for a holographic formulation of quantum gravity, we investigatethe basic boundary theory structure for loop quantum gravity. In 3+1 space-time dimensions, theboundary theory lives on the 2+1-dimensional time-like boundary and is supposed to describe thetime evolution of the edge modes living on the 2-dimensional boundary of space, i.e. the space-time corner. Focusing on “electric” excitations -quanta of area- living on the corner, we formulatetheir dynamics in terms of classical spinor variables and we show that the coupling constants ofa polynomial Hamiltonian can be understood as the components of a background boundary 2+1-metric. This leads to a deeper conjecture of a correspondence between boundary Hamiltonian andboundary metric states. We further show that one can reformulate the quanta of area data interms of a SL(2 , C ) connection, transporting the spinors on the boundary surface and whose SU(2)component would define “magnetic” excitations (tangential Ashtekar-Barbero connection), therebyopening the door to writing the loop quantum gravity boundary dynamics as a 2+1-dimensionalSL(2 , C ) gauge theory. Contents
I. Loop Quantum Gravity on Space-Time Corners
II. Lorentz Connection on the Boundary , C )-holonomies between spinors 11B. Non-trivial stabilizer and (relative) locality on the boundary 14C. SL(2 , C ) boundary theory 17D. Recovering local SU(2) gauge invariance on the boundary: magnetic excitations 19 Outlook & Conclusion Acknowledgement A. SL(2 , C ) -holonomy between 3-vectors References ∗ Electronic address: [email protected] a r X i v : . [ h e p - t h ] J a n in loop quantum gravity, whether or not the standard framework of loop quantum gravity should be extended andquantum states of geometry enriched with more information, as proposed for instance in [6, 13–17], and what typeof boundary dynamics could one hope to translate from the classical setting to the quantum realm of loop gravity’sspin network states.In the canonical framework of loop quantum gravity based on a 3+1 splitting of space-time in terms of a 3dspatial hypersurface evolving in time, we focus on the space-time corner defined by the hypersurface’s boundary.Loop quantum gravity defines quantum states of the bulk geometry as spin networks, which are polymer structuresconsisting of (embedded) graphs dressed with algebraic data from the representation theories of the Lie groups SU(2)(and possibly of the SL(2 , C ) Lorentz group and their quantum group deformations). Assuming the spatial boundaryto have the topology of a 2-sphere, we consider a spin network state puncturing the corner, as drawn on fig.1, similarto the lightning within a plasma globe, thus leading to the boundary data of the algebraic data carried by the linkscut by the boundary. The corner thus carries a certain number of SU(2) spin states | j i , m i (cid:105) at the quantum level, or ofspinors z i ∈ C at the classical level. From this starting point, section I reviews the spinor phase space on space-timecorners, according to the holomorphic reformulation of loop quantum gravity [18–26], describes what type of spinortheory one could expect on the boundary for a given number of punctures and discusses the possible continuum limitinto a spinor field theory living on the 2+1-d time-like boundary of space-time, which could be considered as a secondquantization allowing to vary the number of punctures.Section II is dedicated to re-writing the dynamics of boundary spinors as a SL(2 , C ) gauge theory, having in mind theidea of a gravity-gauge duality linking the bulk and boundary dynamics for finite space-time regions. More precisely,we show how one can reformulate the kinematics and dynamics of a collection of spinors in terms of a discrete SL(2 , C )connection. This leads us to the proposal of templates for loop quantum gravity boundary theories in the continuumin terms of SL(2 , C ) gauge connections living on the 2+1-d time-like boundary of space-time.FIG. 1: Loop quantum gravity boundary data defined by the spin network puncturing the two-dimensionalspace-time corner S = ∂ Σ : the spin network edges puncturing the boundary of the 3d space Σ carry fluxexcitations, encoded as spin states | j i , m i (cid:105) and defining quanta of area on the boundary surface. I. LOOP QUANTUM GRAVITY ON SPACE-TIME CORNERS
We consider here the simplest setting in loop quantum gravity: a region of the 3d space with the topology ofa 3-ball bounded by a 2-sphere, whose quantum state is a spin network state whose links puncture the 2-spheretransversally (without looking into the subtle possibility of links tangential to the boundary). The boundary datawill thus consist in the fluxes carried by the open links going through the boundary. This means that we are focusingon flux excitations on the boundary, i.e. “electric” excitations, as illustrated in fig.1, and discarding (as for now) thepossibility of magnetic excitations and momentum excitations as presented in [6, 7, 10, 11].We do not assume any further a priori boundary data, such as a graph-like linking the punctures. For instance,such a boundary graph was introduced in [29] as a notion of nearest neighbour (and thus of locality) on the boundaryinherited from the bulk graph of the spin network (i.e. a notion of boundary locality as the projection of the near-boundary bulk locality). Taking into the magnetic excitations on the boundary also led to another notion of boundary Such a description of the boundary data in terms of a continuous spinor field is of course reminiscent of the formulation of generalrelativity’s boundary data on null surfaces in terms of spinors, as in e.g. [27, 28]. A possible relation between the two frameworks shouldclearly be investigated but is beyond the scope of the present work. graph, as a spin network living on the 2d boundary itself and carrying non-trivial holonomies of the connection alongcurves tangential to the boundary. This is a recurring idea in recent investigation of a necessary extension of the loopquantum gravity formalism, as when representing twisted geometries as a moduli of SL(2 , C ) flat connections [30, 31],double spin networks [32], Drinfeld tube networks [15, 33], Poincar´e charge networks [6]. This should definitely bekept in mind in future investigation. We will nevertheless see below in section I B that a notion of boundary graphnaturally arises when defining the dynamics for the boundary degrees of freedom.Within those limitations of the present approach, the boundary data on a spatial boundary (or space-time corner)induced by the bulk geometry state in loop quantum gravity consists in the spin states carried by the puncturesrepresenting the flux excitations on the boundary. Each puncture state lives in the Hilbert space, H (1) = (cid:77) j ∈ N V j , (1)where V j is the (2 j + 1)-dimensional Hilbert space carrying the spin- j representation of the SU(2) Lie group. TheHilbert space for boundary states with N punctures is simply the tensor product of N copies of the one-punctureHilbert space: H ( N ) = N (cid:79) i =1 H (1) i . (2)And our goal is to describe the dynamics of the boundary flux excitations represented as the spin states carried by thepunctures. This is actually the same setting as if one would like endow an isolated horizon with microscopic dynamicsin loop quantum gravity in the standard description of quantum horizons [34–38]. Indeed, a horizon state is entirelydescribed as a tensor product of spin states . The boundary theory is usually to be trivial, i.e. the spin states areusually assumed not to interact, although the interested reader can see [29] for a proposal of bulk-induced interactionon the horizon through a Bose-Hubbard exchange hamiltonian. Here we do not place ourselves in the restrictivesetting of an (isolated) horizon but consider a general boundary surface without assuming any specific boundarycondition. The present work would nevertheless be relevant when investigating the quantum gravity dynamics aroundblack hole horizons.Interestingly, the spin states can be seen as wave-functions over a complex 2-vector, or spinor z = ( z , z ) ∈ C . This is a standard construction in mathematics for semi-simple Lie groups. The vector space V j consistsin homogeneous polynomials in z and z of degree 2 j , with the standard basis state | j, m (cid:105) corresponding to themonomial ( z ) j + m ( z ) j − m . The SU(2) action on those polynomials results from the natural action of SU(2) groupelements as 2 × z : ∀ g ∈ SU(2) , z ∈ C , g (cid:46) z = (cid:18) a bc d (cid:19) (cid:18) z z (cid:19) ∈ C . (3)The scalar product is then given by the integration of the polynomials with respect to the Gaussian measure on C (up to a j -dependent factor) or equivalently to the integration over the 3-sphere of unit spinors satisfying (cid:104) z | z (cid:105) = | z | + | z | = 1. This simple correspondence is the foundation for the holomorphic reformulation of loop quantumgravity in terms of spinorial coherent states as proposed in [21–25]. As a consequence, we work with boundary dataon the space-time corner given as a collection of spinors z i ∈ C , one spinor variable per puncture. In this section, weexplain how to endow those spinors with a dynamics. In the next section, we will show how to reformulate collectionof spinors in terms of flat SL(2 , C ) connections on the boundary. A. Punctures and Spinors on the Boundary
The phase space for N spin network punctures on the boundary consists in N spinors z i ∈ C with i = 1 ..N provided with a canonical Poisson bracket: { z Ai , ¯ z Bj } = − iδ AB δ ij , { z Ai , z Bj } = { ¯ z Ai , ¯ z Bj } = 0 . (4)Each spinor z i ∈ C defines a 3-vector (cid:126)X i ∈ R , identified as the flux associated to the puncture. Using bra-ketnotation, the flux vectors are defined as: X ai = 12 (cid:104) z i | σ a | z i (cid:105) , (5)where the σ a , with a = 1 ..
3, are the three Pauli matrices, normalized such that ( σ a ) = I . Each flux vector forms a su (2) Lie algebra: { X ai , X bj } = δ ab δ ij . (6)This is simply the Schwinger presentation of the su (2) algebra. Indeed, upon quantization, we promote each spinorcomponent to a quantum harmonic oscillator: (cid:12)(cid:12)(cid:12)(cid:12) z Ai → a Ai ¯ z Ai → a † Ai with [ a Ai , a † Bj ] = 1 . (7)Then the flux vectors become the su (2) Lie algebra generators for the SU(2) representation attached to each punctureconsidered as an open end of the spin network links: X ai → J ai , J ai = J † ai with [ J ai , J bj ] = iδ ij (cid:15) abc J ci . (8)These SU(2) representations are all independent and we recover the Hilbert space H ( N ) for N spin network punctureson the boundary. The SU(2) Casimir of each puncture, C i = J ai J ai , corresponds to the quantization of the squaredflux vector norm | (cid:126)X i | and gives the spin j i carried by the puncture by the usual Casimir expression, C i = j i ( j i + 1).Actually, in the Schwinger presentation, there is an operator corresponding to the quantization of the flux vectornorm | (cid:126)X i | and giving directly the spin j i : | (cid:126)X i | = 12 (cid:104) z i | z i (cid:105) → j i = 12 (cid:88) A =0 , a † Ai a Ai . (9)This provides the spin j i with the interpretation as a quantum of area A i = j i (cid:96) P lanck on the boundary surface inPlanck units (suitably renormalized by the Immirzi parameter). For details on the Schwinger presentation and thederivation of the standard spin basis | j, m (cid:105) , the interested reader can refer to [39]. Furthermore, for details on how theclassical spinor variables naturally label a system of coherent states minimizing the uncertainty relations associatedto the su (2) commutators and transforming consistently under the SU(2) action, the interested reader can refer to[40, 41]. These SU(2) coherent states are a basic tool in the definition and construction of EPRL-like spin foamamplitudes for the dynamics of spin networks [23, 42–48].A final subtlety is that a spinor z ∈ C , with four real components, contains more information than the flux vector (cid:126)X ∈ R , with three real components. The extra data is the phase of the spinor. Indeed the flux vector is invariantunder phase shift of the spinor: ∀ φ ∈ R , (cid:126)X ( e iφ z ) = (cid:126)X ( z ) . (10)This phase degree of freedom is called the twist angle and identified as a measure of the extrinsic curvature (of thecanonical hypersurface) at the puncture [18, 19]. It is the variable canonically conjugate to the flux vector norm (cid:126)X ,i.e. to the puncture area, and thus to the spin at the quantum level. Without this phase freedom, one can not definea phase space with varying puncture area, i.e. quantized into a Hilbert space allowing arbitrary spin superpositions(see e.g. [41] for more mathematical details).In the rest of the paper, we will not work at the quantum level with spin states but will work in the simpler classicalsetting of the boundary spinorial phase space. This allows to describe the boundary theory in the space-time cornerthrough action principles defined in terms of the spinor variables. These action principles can then be quantized atfinite number of punctures, for fixed N , or quantized and renormalized as quantum field theories in the continuumlimit of infinite refinement N → ∞ . B. Spinor dynamics on the 2+1-d time-like boundary
The present goal is to define dynamics for the degrees of freedom on the spatial boundary of spin network states,that is for the spinor variables attached to the punctures on the boundary. These boundary degrees of freedom areinitially defined on the 2d boundary of the canonical 3d hypersurface and then evolve along the 2+1-d time-likeboundary of the 3+1-d space-time.Let us start with the case of a fixed number of punctures N . A natural action principle for the boundary dynamicsis, first to take into account the canonical Poisson bracket (4) between spinor components, second define a relativisticdynamics through a Hamiltonian constraint accounting for the invariance of the theory under time reparametrizations.This yields the general ansatz for the boundary action: S ∂ [ { z k ( t ) } k ] = (cid:90) d t (cid:34) − i (cid:88) k (cid:104) z k | d t z k (cid:105) − N H [ { z k } ] (cid:35) , (11)where N is a lapse variable and H [ { z k } ] is the to-be-specified Hamiltonian constraint. At that point, there are twostandard strategies: • A. one can analyze the Hamiltonian framework of general relativity with boundary terms, understand theboundary symmetry algebra and derive the pool of possible boundary theories depending on the chosen boundaryconditions, discretize and/or quantize the resulting boundary dynamics in order to translate it in terms of discretegeometry, spin network states and boundary surface excitations; • B. one can alternatively work within the already defined mathematical framework of loop quantum gravity,derive the natural boundary variables, understand the pool of possible dynamics that can be defined for thisboundary data and investigate the resulting physics.Ultimately, one of course wish for a convergence of those approaches. Here we will focus on the latter strategy (B)and postpone the comparison with the first strategy (A) to future (but necessary) investigation. From this viewpoint,the natural path to follow is to investigate what kind of Hamiltonian H [ { z k } ] we should, or could, define for theboundary dynamics, and then, later, analyze the coarse-graining, renormalization flow, possible fixed points anduniversality classes dynamics. Going step by step, let us understand what makes a good ansatz for the Hamiltonianconstraint. Once we have chosen basic variables, here the spinors, it is natural to proceed to a Taylor expansionof the functionals and thus consider polynomial ansatz (of increasing power) for the Hamiltonian H [ { z k } ]. If thequadratic ansatz, corrected by potential higher order terms, does not yield expected or realistic physics, then thisusually indicates that we have made the wrong choice of basic variables and that we should very likely consider adifferent phase for spin networks (for example, consider a type of condensate of spin networks, as proposed in thegroup field theory approach [49–51]) and the corresponding different boundary data for loop quantum gravity onspace-time corners.We naturally require the action to be invariant to be invariant under global SU(2) transformations acting simulta-neously on all the spinor: | z k (cid:105) (cid:55)→ g | z k (cid:105) , g ∈ SU(2) . (12)This descends from the SU(2) local gauge invariance of canonical general relativity (formulated in terms of vier-bein/connection variables) translated to a single overall SU(2) reference frame for the whole spatial boundary. Thiscan be seen as a gauge-fixing of the SU(2) local gauge invariance of the bulk spin network state down to a SU(2)gauge invariance for the boundary data rooted at a chosen boundary vertex (see [52] for more details on the bulk-to-boundary projection of spin network states from a coarse-graining perspective). We will discuss later in the sectionII D the possibility of imposing a local SU(2) gauge invariance on the boundary and its relation to magnetic boundarydegrees of freedom. As for now, imposing this global SU(2) gauge invariance on the boundary automatically removesthe possibility of linear terms in the boundary Hamiltonian.At the quadratic level, we can have local potential terms in (cid:104) z k | z k (cid:105) for each puncture. As there is not an obviousreason to prefer one puncture over another, this lead to a single term for the Hamiltonian constraint: H = β (cid:88) k (cid:104) z k | z k (cid:105) + . . . , (13)where β is a to-be-specified coupling constant. This first term of the Taylor expansion is simply the total boundaryarea A [ { z k } ] = (cid:80) k (cid:104) z k | z k (cid:105) . Then we can introduce diffusion/propagation terms as non-local terms coupling puncturestogether. There are two types of such terms, scalar products (cid:104) z k | z l (cid:105) and scalar products [ z k | z l (cid:105) , where we have usedthe notation introduced in [19, 40] to denote the dual spinor: | z (cid:105) = (cid:18) z z (cid:19) , | z ] = (cid:18) − ¯ z ¯ z (cid:19) = (cid:18) − (cid:19) (cid:18) ¯ z ¯ z (cid:19) , (14) (cid:104) z | w (cid:105) = ¯ z w + ¯ z w , [ z | w (cid:105) = z w − z w . (15)The dual spinor transforms under the same SU(2) group action as the original spinor: ∀ g ∈ SU(2) , g | z (cid:105) = (cid:18) α − ¯ ββ ¯ α (cid:19) (cid:18) z z (cid:19) = (cid:18) αz − ¯ βz ¯ αz + βz (cid:19) = | gz (cid:105) (16) ⇒ g | z ] = (cid:18) α − ¯ ββ ¯ α (cid:19) (cid:18) − ¯ z ¯ z (cid:19) = (cid:18) − α ¯ z − ¯ β ¯ z ¯ α ¯ z − β ¯ z (cid:19) = | gz ] . (17)A direct consequence is that the scalar products (cid:104) z k | z l (cid:105) and [ z k | z l (cid:105) are the only quadratic polynomials which areinvariant under the global SU(2) action on the spinors. The combinations (cid:104) z k | z l (cid:105) commute with the total area A [ { z i } ] = (cid:80) i (cid:104) z i | z i (cid:105) and form a u ( N ) Lie algebra as shown in [39, 53]: {(cid:104) z k | z l (cid:105) , A} = 0 , {(cid:104) z k | z l (cid:105) , (cid:104) z m | z n (cid:105)} = i (cid:0) δ kn (cid:104) z m | z l (cid:105) − δ lm (cid:104) z k | z n (cid:105) (cid:1) . (18)Therefore the Hamiltonian flow that they generate can be integrated as U( N ) transformations that preserves the totalsurface area of the boundary [39, 41, 53].On the other hand, the combinations [ z k | z l (cid:105) do not commute with the total area. There Hamiltonian flow actu-ally decreases the boundary area, while their complex conjugates (cid:104) z k | z l ] generate a Hamiltonian flow increasing theboundary area: { [ z k | z l (cid:105) , A} = + i (cid:104) z k | z k (cid:105) + i (cid:104) z l | z l (cid:105) , {(cid:104) z k | z l ] , A} = − i (cid:104) z k | z k (cid:105) − i (cid:104) z l | z l (cid:105) . (19)These are thus creation and annihilation of entangled area quanta between the two punctures k and l . The scalarproduct [ z k | z l (cid:105) is holomorphic in the spinors, while its complex conjugate (cid:104) z k | z l ] is anti-holomorphic. Their Poissonbrackets do not close on their own. Nevertheless, together with the area-preserving scalar products, they form a closed so ∗ (2 N ) Lie algebra [40, 54]: { [ z k | z l (cid:105) , [ z m | z n (cid:105)} = {(cid:104) z k | z l ] , (cid:104) z m | z n ] } = 0 , { [ z k | z l (cid:105) , (cid:104) z m | z n ] } = − i (cid:0) δ lm (cid:104) z n | z k (cid:105) − δ ln (cid:104) z m | z k (cid:105) − δ km (cid:104) z n | z l (cid:105) + δ kn (cid:104) z m | z l (cid:105) (cid:1) , {(cid:104) z k | z l (cid:105) , [ z m | z n (cid:105)} = i (cid:0) δ kn [ z m | z l (cid:105) − δ km [ z n | z l (cid:105) (cid:1) . (20)The Hamiltonian flow of all those quadratic couplings between punctures can thus be integrated as a SO ∗ (2 N ) Liegroup flow.The resulting quadratic ansatz for the boundary Hamiltonian constraint, including the homogeneous local potentialterm and the puncture interaction terms, reads: H = β (cid:88) k (cid:104) z k | z k (cid:105) + γ (cid:88) k,l C kl (cid:104) z k | z l (cid:105) + ˜ γ (cid:88) k,l (cid:16) D kl [ z k | z l (cid:105) + D lk (cid:104) z k | z l ] (cid:17) + . . . , (21)where we have added the coupling constants γ and ˜ γ for respectively the area-preserving interactions and thearea-changing interactions. The matrix C can be assumed symmetric in order to ensure that the Hamiltonian bereal. It generates an exchange of boundary area quanta between punctures. We refer to this term as the “exchangeHamiltonian”. The coefficients C kl give the strength of the coupling between the two punctures k and l . As suggestedin [29], one could restrict this exchange Hamiltonian to nearest neighbour interactions. This is done by drawing aboundary graph between the punctures and taking the matrix C to be the adjacency matrix of this graph. Theboundary graph does not necessarily need to be planar, although this is a natural choice if the boundary topology isthat of a 2-sphere. This boundary graph is interpreted as a notion of locality on the boundary, which can be thoughtas given a priori as induced by the bulk spin network structure (as in [29]), or as a choice of boundary backgroundstructure, or defined a posteriori from the choice of Hamiltonian. The local potential term can be considered simplyas the diagonal terms of exchange matrix C .The matrix D encodes the coupled creation and annihilation of quanta of area on the boundary. We refer tothis term as the “expansion Hamiltonian”. Together the matrices C and D define the quadratic truncation of theHamiltonian constraint. The exponentiated flow of this quadratic Hamiltonian can be integrated in terms of SO ∗ (2 N )group elements. If we remove the expansion term and solely focus on the exchange Hamiltonian, the flow can be thenmore simply integrated in terms of U( N ) group elements, which describe the deformation of the boundary surface atconstant area [39, 41].This quadratic ansatz already seems rich enough, especially for the purpose of comparing the importance of bound-ary wave propagation (generated by the exchange Hamiltonian) versus the expansion/shrinking of the boundarysurface. However, since the local potential term can be entirely re-absorbed in the quadratic exchange term, it seemsnatural to push its Taylor expansion to the next order and introduce quartic local terms (cid:104) z k | z k (cid:105) for each puncture,thus yielding a more complete ansatz for the boundary Hamiltonian constraint: H = β (cid:88) k (cid:104) z k | z k (cid:105) + γ (cid:88) k,l C kl (cid:104) z k | z l (cid:105) + ˜ γ (cid:88) k,l (cid:16) D kl [ z k | z l (cid:105) + D lk (cid:104) z k | z l ] (cid:17) + β (cid:88) k (cid:104) z k | z k (cid:105) + . . . , (22)This new term moves the Hamiltonian outside of the so ∗ (2 N ) Lie algebra and the exponentiated flow can not besimply expressed in terms of SO ∗ ( N ) group elements. Putting the expansion term aside by setting ˜ γ to zero, andassuming the matrix C to be the adjacency matrix of a boundary graph, the truncation to area-preserving termsdefines a Bose-Hubbard
Hamiltonian, with a local potential (cid:80) k (cid:104) z k | z k (cid:105) balanced against an exchange term of quantabetween nearest neighbours (cid:80) ( k,l ) (cid:104) z k | z l (cid:105) . We call this the Bose-Hubbard truncation for the boundary dynamics.The quartic potential term will clearly affect the propagation of waves on the boundary. Beside possible Andersonlocalization phenomena, this will allow to study the propagation of perturbations on the boundary surface -ballisticversus diffusive- and the resulting relaxation towards homogeneous equilibrium (or, even, the possible instability ofthe homogeneous configuration). It was even speculated in [29] that such a Bose-Hubbard model applied to blackhole horizons should lead to a phase transition between classical black hole with fast relaxation of the event horizonunder local perturbations and a quantum regime for microscopic black holes where horizon perturbations will diffusein a slower fashion making the in-falling information more visible to the exterior observer. More generally, it wouldbe interesting to identify which phase of the generic ansatz given above could correspond to an isolated horizon andto a black hole or to other type of boundary conditions studied in general relativity.We believe that this ansatz (22) for a boundary Hamiltonian constraint at fixed number of punctures N is a goodstarting point for a systematic study of the boundary dynamics in loop quantum gravity. It could already lead toexciting new phenomena and predictions for quantum gravity. C. Boundary Field Dynamics
We have provided a natural ansatz for the loop quantum gravity boundary dynamics of flux excitations for a finitefixed number N of punctures on the space-time corner. Geometrically this corresponds to a deep quantum regimewhere the boundary surface is made of N elementary surface patches carrying quanta of area. In order to compare withthe classical analysis of boundaries in general relativity (see the recent work in e.g. [8–12]), it is appropriate to lookat the classical regime for these boundary degrees of freedom. To this purpose, we consider the na¨ıve continuum limitdefined by the infinite refinement of the boundary surface, sending the number of punctures to infinity N → + ∞ . Wecall this “na¨ıve” because we will not look at the dynamics of coherent states and seek to define a semi-classical regimeat high energy and high action, but we will instead consider the boundary action (11)-(22) at fixed N described in theprevious section as the discretization of a continuous field theory. From a 2nd quantization viewpoint, the puncturescan be thought of as “boundary particles” and the field theory considered as describing the regime where the numberof punctures can (arbitrarily) fluctuate.Introducing a 2d coordinate system x k =1 , on the boundary sphere, the sum over punctures becomes an integralover the 2-sphere. Assuming that the exchange and expansion terms in the finite N Hamiltonian (22) are definedon terms of nearest neighbours on a (planar) lattice living on the 2-sphere, and taking the infinite refinement of thisboundary lattice, we get an action for a continuous spinor field z ( x k ) ∈ C living on the 2+1 time-like boundary ofspace-time parametrized by the time coordinate t and the 2d coordinates x k : S ∂ [ z, ¯ z, N ] = (cid:90) d t (cid:90) d x (cid:20) i (cid:104) z | ∂ t z (cid:105) − N (cid:104) iγ k (cid:104) z | ∂ k z (cid:105) + i (cid:16) ˜ γ k [ z | ∂ k z (cid:105) + ˜ γ k (cid:104) z | ∂ k z ] (cid:17) + (cid:88) n β n (cid:104) z | z (cid:105) n (cid:105)(cid:21) . (23)This is a first order action principle with kinetic terms and local potential. It is real (up to boundary terms ) and hascoupling constants β n for the Taylor expansion of the potential and γ k , ˜ γ k defining the geometric background. Havingusual derivative terms such as (cid:104) z | ∂z (cid:105) = ¯ z ∂z + ¯ z ∂z and holomorphic derivative terms such as [ z | ∂z (cid:105) = z ∂z − z ∂z can feel a little awkward. It could nevertheless become more natural if writing it as a four-dimensional spinor Boundary terms of this boundary action live on the corner of the 2+1-dimensional time-like boundary of space-time, i.e. on 1d boundarieson the 3d spatial slices, i.e. on the contour around the punctures. Such terms will ultimately be relevant and should be studied inmore details. From a topological field theory point of view, cells of every dimension carry algebraic data whose type depend on thedimensionality and represent the boundary charges generated by the corresponding boundary terms in the action.
Ψ = ( | z (cid:105) , | z ]) encompassing both the 2-spinor and its dual spinor. This 4-spinor does not however have any specialmeaning or role, so we do not pursue this possibility.In order to better understand the meaning of this field theory, an interesting step is go back to a Lagrangianformulation. Since the field here is complex, we first separate its real and imaginary parts and then compute theinverse Legendre transform. This will yield a second order Lagrangian for the boundary theory. To illustrate theprocedure, we start by presenting the example of a complex field instead of the spinor field. Let us consider thefollowing 1+1-d action in its Hamiltonian form, truncated to quadratic terms: S [ z ( x )] = (cid:90) d t d x (cid:104) i ¯ z∂ t z − N β z ¯ z − N iγ ¯ z∂ x z (cid:105) . (24)Writing z = ( q + ip ) / √
2, this action reads (up to total derivative terms, which we set aside): S [ z ( x )] = (cid:90) d t d x (cid:104) p∂ t q − N β q + p ) − N γp∂ x q (cid:105) . (25)We can consider q as the configuration field, while p is its conjugate momentum. Solving for the field p yields: N βp = ∂ t q − N γ∂ x q . (26)Plugging this expression into the action gives its Lagrangian formulation: S [ q ( x ) , ∂ t q ( x )] = 1 β (cid:90) d t d x (cid:104) N ( ∂ t q ) − γ∂ t q∂ x q + N γ ∂ x q ) − N β q (cid:105) . (27)Here we immediately recognise the action for a (real) massive scalar field living on a curved 1+1-d metric writtenin its ADM form with the lapse factor in the time direction. Indeed, if we write a d+1-dimensional metric h for aspace-time foliation in terms of space-like slices according its ADM form,d s = h µν d x µ d x ν = ( N − N k N k )d t − N k d t d x k − c kl d x k d x l , (28)in terms of the lapse N , shift vector N k and the d-dimensional space metric c , the action for a (real) massive scalarfield φ reads: S h [ φ ] = (cid:90) d t d d x √ c (cid:20) N ( ∂ t φ ) − N k N ∂ t φ∂ k φ − (cid:18) c kl − N k N l N (cid:19) ∂ k φ∂ l φ − m N φ (cid:21) . (29)For a 1+1-dimensional field, this leads to the identification of the coupling constants of the spinor field hamiltonian interms of the space-time metric components and the field properties: the mass m is identified to the quadratic potentialcoupling β and the exchange coupling γ is identified (up to a numerical factor) to the normalized shift vector N x / N .Following the same procedure from the spinor field action (23) in its Hamiltonian form truncated to quadratic terms(i.e. discarding the quartic potential and higher order terms), we decompose the two spinor components in their realand imaginary parts, z A = ( q A + ip A ) / √
2, leading to a second order action for two coupled scalar fields, φ A = q A with A = 0 ,
1. Assuming for the sake of simplicity for the couplings ˜ γ k are real, the inverse Legendre transform ofthe action (23) gives the following lagrangian: S ∂ [ φ , φ ] = β − (cid:90) d t d x (cid:34) N ( ∂ t φ ) + 12 N ( ∂ t φ ) − γ k ∂ t φ ∂ k φ − γ k ∂ t φ ∂ k φ − ˜ γ k ∂ t φ ∂ k φ + ˜ γ k ∂ t φ ∂ k φ + N γ k ˜ γ k (cid:48) + γ k (cid:48) ˜ γ k )( ∂ k φ ∂ k (cid:48) φ + ∂ k φ ∂ k (cid:48) φ ) + 2 N ( γ k ˜ γ k (cid:48) − γ k (cid:48) ˜ γ k ) ∂ k φ ∂ k (cid:48) φ − N β ( φ ) − N β ( φ ) (cid:35) . (30)We recognise the action for a pair of coupled massive scalar field -or equivalently a 2-component scalar field φ A - onthe 2+1-dimensional time-like boundary written in its ADM form, with an identification of the shift N k and cornermetric c with the spinor hamiltonian coupling constants γ k , ˜ γ k . This little exercise of going back to the lagrangian One might wonder about the corner metric factor √ c . It does not appear in the continuum limit of the presently studied discrete spinoraction. Nevertheless, it can easily be made to appear by adding a puncture-dependent weight factor to the sum over punctures. Thismight indicate that this is a necessary ingredient of the discrete formulation in order to obtain a properly covariant continuum limit.Another possibility, which we won’t study here, is to extract the 2d metric density from the norm of the spinors themselves, as hintedby the balance equation between 2d metric and flux norm derived in [2, 16]. from the postulated hamiltonian boundary dynamics shows that the spinor variables associated to the boundarypunctures do not simply lead to an actual spinor field but can be written as scalar fields in the continuum limit. Herewe do not refer to the statistics of the field, but to the type of covariant derivative to which it couples. Indeed, due tothe intricate coupling between its two components, the field φ could still acquire non-trivial statistics (at the quantumlevel), but this would require a detailed analysis of the physics of this boundary action.Although the derivation of the continuum field theory in its hamiltonian form from the spinor dynamics at finitenumber of punctures, and then the derivation of the lagrangian field theory by an inverse Legendre transform, arestraightforward analytical steps, this procedure faces a few hurdles: • the role of higher order potential terms:In the spinor dynamics for a fixed number of boundary punctures N , the higher order potential terms, such thequartic Bose-Hubbard coupling, typically balance the exchange of quanta between punctures and modulate thepropagation of waves on the boundary surface. Such terms, of the type (cid:104) z | z (cid:105) n for n ≥
2, involve higher powersof both the scalar field components q A and their conjugate momenta p A . These terms not only lead to a morecomplicated relation between momenta and derivatives of the field, making much harder to perform explicitlythe inverse Legendre transform, but they also involve high powers of the field derivative, leading most likely toa higher order lagrangian. One should probably seek inspiration from the analysis of the semi-classical regimeand continuum limit of the Bose-Hubbard model in order to better understand the physics created by thosehigher order potential terms. • the SU(2) invariance and the coupling between the two field components φ and φ :The spinor hamiltonian (22) for fixed number of puncture N , as well as the continuous spinor action (23), areboth obviously invariant under global SU(2) transformations acting on the whole spatial slice, | z (cid:105) (cid:55)→ g | z (cid:105) for g ∈ SU(2). On the other hand, once splitting the spinor variables into real and imaginary part, respectively definingthe scalar field φ A and its conjugate momentum, this obvious character of the SU(2) action is lost. Indeed,SU(2) transformations do not act simply on the scalar field components, but become Bogoliubov canonicaltransformations mixing the scalar field and its momentum field. Nonetheless, the action is still invariant underthose transformations and this gauge invariance somehow reflects into the intricate structure of the couplingbetween the two field components φ and φ . It would definitely be interesting to investigate further thissymmetry, for instance understand the expression of the induced Noether charges in terms of the boundaryscalar field. • the unknown physical meaning of the boundary scalar field φ :Although the initial spinor variables attached to the punctures on the space-time corner have a clear inter-pretation in terms of geometrical flux vectors (representing the triad field) at the classical level and then as(SU(2)-covariant) creation and annihilation operators of quanta of area at the quantum level, the scalar fielddefined as the real part of the continuum limit of those spinors do not yet have an enlightening physical meaning.On the one hand, it is not clear to which boundary field of general relativity the scalar field φ A should corre-spond to. On the other hand, its behaviour under SU(2) transformations and its action with a specific couplingbetween its two components should be analysed in detail in order. to understand the defining properties of thisboundary field. • the apparently asymmetric role of the boundary lapse and boundary shift:We have started from an action for a discrete set of configurations with a postulated lapse variable N enforcinga Hamiltonian constraint and, after taking the continuum limit of the 2d boundary and performing an inverse SU(2) group elements naturally act as 2 × (cid:18) α − ¯ ββ ¯ α (cid:19) (cid:18) z z (cid:19) = (cid:18) e iϕ cos θ − e − iψ sin θe iψ sin θ e − iϕ cos θ (cid:19) (cid:18) q + ip q + ip (cid:19) = (cid:18) ˜ q + i ˜ p ˜ q + i ˜ p (cid:19) , with (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ q = cos θ cos ψq − cos θ sin ψp − sin θ cos ψq − sin θ sin ψp , ˜ p = cos θ sin ψq + cos θ cos ψp + sin θ sin ψq − sin θ cos ψp , ˜ q = sin θ cos ψq − sin θ sin ψp + cos θ cos ψq + cos θ sin ψp , ˜ p = sin θ sin ψq + sin θ cos ψp − cos θ sin ψq + cos θ cos ψp . This is a canonical transformation, leaving the canonical Poisson brackets invariant { ˜ q , ˜ p } = { ˜ q , ˜ p } = 1 and { ˜ q , ˜ p } = 0, but mixinglinearly the configuration variables and their momenta. It is thus a Bogoliubov transformation. N under some specific exchange of area quanta between punctures , which wouldbecome the invariance under spatial boundary diffeomorphisms in the continuum limit and whose Noethercharges should carry a discrete version of the boundary charges for general relativity. One should neverthelesskeep in mind that the boundary symmetry depend on the chosen boundary conditions in the continuum theory,and that this begs the question of which type of boundary conditions do boundary flux excitations correspondto. This leads us to the second point: the role of boundary conditions. More precisely, should the lapse andshift be dynamical fields on the boundary? Or should they be held fixed on the boundary? In the latter case, weshould not work with a Hamiltonian constraint for the spinor dynamics but directly with a Hamiltonian, i.e. wedo not require that the Hamiltonian vanishes but it could have a non-zero value, and we do not consider that itgenerates gauge transformations but more simply boundary symmetries (see [3] for a discussion of gauge versussymmetry). In fact, it is a more general question: which component(s) of the boundary 2+1-d metric should befixed and which component(s) should be dynamical? There are several apparent possibilities, playing around thelapse, shift, 2d corner metric, the time-like extrinsic curvature, the spatial extrinsic curvature,... This is deeplyrelated to the study of edge modes in general relativity [10, 11]. The third path is a more drastic change: if spaceacquires a discrete nature at the quantum level, then time could/should do. For instance, that’s what happensin ’t Hooft’s polygonal quantization of 2+1-d gravity [55, 56], and that’s what is postulated in causal dynamicaltriangulations [57]. And that’s what is worked out in spinfoam models for a loop quantum gravity path integral(see e.g. [58, 59] for reviews): space and time intervals become discrete geometrical objects (represented as2d cells). Then the whole 2+1-dimensional boundary need to be described in discrete terms and not onlyas a discrete 2D spatial boundary evolving in continuous time. This was investigated in the context of thePonzano-Regge topological state-sum for 2+1-d gravity with time-like boundaries defined at the quantum levelin terms of a 1+1-d network structure [60–62]. However it is then not obvious to come back to a Hamiltonianformulation, which needs to be derived a posteriori from a notion of boundary transfer matrix similarly to whatis done with quantum integrable spin systems.Despite these gaps in our understanding, the main lesson to draw here is that the polynomial spinor Hamiltonianfor N punctures on the 2d spatial boundary in loop quantum gravity can be written as a σ -model for a massivetwo-component scalar field living on the 2+1-d time-like boundary of space-time, with an action roughly of the type S [ φ ] = (cid:82) √ h (cid:2) h µν K AB ∂ µ φ A ∂ ν φ B − m φ (cid:3) . This new correspondence allows to translate the couplings constants ofthe spinor Hamiltonian into the components of the 2+1 boundary metric. This boundary metric plays the role of abackground geometry in which the flux excitations evolve, just as the spinor couplings dictate the dynamics of thequanta of area living on the punctures in the quantum regime.In other words, we seek to describe a boundary theory, in which flux excitations evolve over a vacuum state onthe space-time boundary. This boundary dynamics effectively depend on the chosen vacuum state, which translatesinto particular values for the coupling constants determining the boundary Hamiltonian. In the (na¨ıve) continuumlimit that we have investigated, these coupling constants translate into a background metric on the 2+1-dimensionaltime-like boundary. This naturally leads to the conjecture of a correspondence (at the leading polynomial order)between the vaccuum state on loop quantum gravity’s corner of space-time (on which the spin network punctures It was hinted in [39, 40, 52, 53] that the boundary diffeomorphisms in the discrete loop quantum gravity setting should be related tothe U( N ) transformations generated by the (cid:104) z k | z l (cid:105) observables. It looks more realistic that they might be realised as field-dependentU( N ) transformations (i.e. whose transformation parameters depend on the spinors themselves) (in a similar way that diffeomorphismscan be written as field-dependent translations, e.g. [7]), or that they need to be defined, one level higher, as fusion operators changingthe number of punctures N (as when trying to define a discrete equivalent of the Virasoro operators on tensor networks). One should distinguish the two different notions (and thereby sources) of discreteness: working with discrete elements of geometry(cells of various dimensions) which we glue together to make the space-time and providing those cells with geometric operators andobservables with discrete spectrum. For instance, various area operators for 2d cells in loop quantum gravity can be defined either witha continuous or discrete spectrum. This is similar to the discreteness of particles versus the continuity of fields and the discretenessversus the continuity of their momenta.
1- a.k.a. the area quanta- evolve) and the 2+1 boundary metric. This conjectured correspondence deserves furtherstudy, in order to determine how far it can be pushed and to which extent it can be made explicit and/or exact.There remains the (deeper) question of whether the boundary metric, and thus the coupling constants of thespinor dynamics, should become dynamical, for instance by adding a gravitational term governing the evolution ofthe boundary metric and thus of the coupling constant. This is the question usually summarised as “what boundaryconditions should we choose?” This is deeply intertwined with the definition and fate of gravitational edge modes andwith the possibility (or impossibility) of representing (a suitable discrete version of) the boundary charges of generalrelativity on discrete quantum geometry states.
II. LORENTZ CONNECTION ON THE BOUNDARY
In view of the unclear physical meaning of the boundary scalar field derived from the spinors, we would like toinvestigate an alternative reformulation of the boundary spinor dynamics as a theory of a SL(2 , C ) connection, whichcould offer further possibilities of comparison with gauge theory reformulations of general relativity boundaries.This section is thus dedicated to showing how the boundary data defined by the spinors living at the punctures canbe described in terms of discrete SL(2 , C ) connections. These connections are required to be flat up to a stabilisergroup. The special case of flat connections are also provided with a geometrical interpretation. This hints towards aformulation of boundary theories for loop quantum gravity on space-time corners in terms of SL(2 , C ) gauge theories. A. SL(2 , C ) -holonomies between spinors Consider two spinors z i and z j living on two punctures. Two arbitrary spinors can not be related by a SU(2)transformation, since they are not constrain to have equal norm. So mapping a quantum of area onto another can notmade solely by a SU(2) holonomy. This would simply change the (normal) direction of the elementary surface but cannot change the size of the quantum of area. To change the norm of the spinor requires using enlarging the possibletransformations by allowing for dilatations, for instance by moving up from SU(2) to SL(2 , C ) group elements. Indeed,two spinors can be mapped onto each other by SL(2 , C ) group elements. From the perspective of a bulk-to-boundarycoarse-graining as discussed in [17, 52], such a Lorentz transformation would account for both the non-trivial SU(2)transport within the bulk from one boundary puncture to another, but also for the changes of spin occurring at everybulk vertex, as illustrated on fig.2, leading overall to the propagation from one boundary puncture to another througha SL(2 , C ) holonomy.FIG. 2: Transport between punctures on the boundary (in red) mapping a quantum of area onto anotherversus transport along a curve (in bold purple) diving into the bulk following the spin network graph (in blue)spanning the 3d geometry: the boundary transport between punctures, formalized as
SL(2 , C ) group elements,can be considered as a coarse-graining of the bulk geometry, projected onto the boundary surface. More precisely, an arbitrary SL(2 , C ) group element is defined by 6 real parameters, while a spinor is determinedby 4 real parameter. So the Lorentz holonomy G ij ∈ SL(2 , C ) relating the two spinors z i and z j on the boundary isnot uniquely determined by those spinors. It is determined up to a stabilizer group element of the initial spinor z i ,or equivalently of the target spinor z j .Let us choose a reference spinor, the up complex vector | ↑(cid:105) = (1 , T ⊂
SL(2 , C ) of upper triangular matrices with trivial diagonal: | ↑(cid:105) = (cid:18) (cid:19) , G | ↑(cid:105) = | ↑(cid:105) ⇐⇒ G = (cid:18) µ (cid:19) ∈ T with µ ∈ C . (31)Then SL(2 , C ) group elements mapping the reference spinor to an arbitrary spinor is simply given by the Iwasawadecomposition. Indeed a SL(2 , C ) group element can be uniquely decomposed as the product of an upper triangularmatrix and a SU(2) group element : G = g ∆ λ t µ with g ∈ SU(2) , ∆ λ = (cid:18) λ λ − (cid:19) ∈ D , t µ = (cid:18) µ (cid:19) ∈ T , (32)where λ ∈ R + defines a dilatation and µ ∈ C defines a translation. This decomposition describes the space of spinors C as a section of the coset SL(2 , C ) / T . Indeed all the SL(2 , C ) transformations mapping the reference spinor toa given spinor z ∈ C are given by a group element Λ z ∈ SU(2) × D uniquely determined by z times an arbitrarytranslation t ∈ T : z = Λ z t | ↑(cid:105) , with Λ z = (cid:32) z − ¯ z (cid:104) z | z (cid:105) z z (cid:104) z | z (cid:105) (cid:33) and t ∈ T , (33)where the section Λ z more precisely decomposes into a dilatation to adjust the norm composed with a SU(2) rotation: z = Λ z | ↑(cid:105) = g ˆ z ∆ λ | ↑(cid:105) , λ = (cid:112) (cid:104) z | z (cid:105) , ˆ z = z (cid:112) (cid:104) z | z (cid:105) , g ˆ z = | z (cid:105)(cid:104)↑ | + | z ][ ↑ | (cid:112) (cid:104) z | z (cid:105) , g ˆ z | ↑(cid:105) = ˆ z . (34)Since the stabilizer group of the reference spinor is G ↑ = T , the stabilizer group for a non-vanishing spinor z is thusobtained by conjugation as G z = Λ z T Λ − z .Now we would like to trade the information of the N spinors z i living on the boundary punctures with the transportinformation between punctures given SL(2 , C ) holonomies G ij ∈ SL(2 , C ) such that G ij | z i (cid:105) = | z j (cid:105) . These SL(2 , C )group elements define a discrete Lorentz connection on the boundary.These holonomies clearly can not be arbitrary. Indeed, if we go around a loop from punctures to punctures on theboundary to come back to the initial puncture, the overall holonomy bring map the initial spinor to itself and musttherefore lay in its stabilizer. This means that the discrete Lorentz connection is almost flat, in the sense that it musteffectively project down to a T connection. More precisely, looking at a loop of punctures i → i → .. → i n → i ,the holonomy around the loop is not constrained to the identity but lays in the stabilizer of the initial spinor: G = G i n i ..G i i , G | z i (cid:105) = | z i (cid:105) , G ∈ G z i = Λ z i T Λ − z i . (35)In more details, each Lorentz holonomy can be decomposed in Λ and triangular matrices: G ij = Λ z j t ij (Λ z i ) − with t ij ∈ T , G i n i ..G i i = Λ z i ( t i n i ..t i i ) Λ − z i . (36)The fact that a SL(2 , C ) group element G belongs to one of those stabilizer groups is equivalent to requiring that itstrace is equal to 2: ∃ z ∈ C , G ∈ G z ⇐⇒ Tr G = 2 . (37)Indeed, if G has its trace equal to 2, and if it belongs to SU(2) or if it is a dilatation, then it is necessarily theidentity group element. More technically, Tr G = 2, combined with det G = 1, implies that the determinant det( G − I )vanishes, which means that G is a triangular matrix with trivial diagonal up to a change of orthonormal basis.This means that we can replace the boundary data of N spinors z i by the data of a discretized SL(2 , C ) connection,defined by group elements G ij between punctures, with a holonomy constraint around each loop. That holonomyconstraint requires that the trace of the holonomy around each loop is equal to 2 and is equivalent to requiring thatall those holonomies around loops live in a spinor stabilizer. This can be set as a mathematical proposition: One could also assume λ ∈ R , in which case g belongs to SU(2) / Z ∼ SO(3). Or even assume that λ ∈ C , in which case g ∈ SU(2) / U(1)where we remove from SU(2) the rotations generated by σ z and included them in the dilatations. Proposition II.1.
We call a discrete Lorentz connection on the boundary a collection of N ( N − / group elements G ij ∈ SL(2 , C ) (linking punctures), with the orientation convention that G ji = G − ij . We introduce a holonomyconstraint for every cycle of puncture on the boundary: ∀C cycle between boundary punctures , Tr (cid:20) ←− (cid:89) (cid:96) ∈C G (cid:96) (cid:21) = 2 . (38) This constraint ensures that every Lorentz holonomy around a closed loop is conjugated to a unique group element in T (also referred to as a translation). Then there is a one-to-one correspondence between constrained discrete Lorentzconnections G ij and collections of N spinors z i ∈ C (up to a global sign) times discrete T -connections, i.e. collectionof N ( N − / group elements t ij ∈ T , such that the spinors are transported by the SL(2 , C ) holonomies, G ij | z i (cid:105) = | z j (cid:105) ,and G ij = Λ z j t ij Λ − z i . The sign ambiguity corresponds to a global flip of the signs, z i → − z i , which doesn’t change the SL(2 , C ) holonomies. Proof.
From the definitions given above, if we start from N spinors z i , we define the group elements Λ z i and combinethem with the discrete T -connection t ij ∈ T to define the SL(2 , C ) holonomies between punctures as G ij = Λ z j t ij Λ − z i .The Λ z ’s were defined to automatically imply that G ij | z i (cid:105) = | z j (cid:105) . Moreover the (ordered) product of the G ij aroundcycles between punctures automatically gives group elements conjugated to a translation, thus with trace equal to 2.What remains is to prove the reverse. Let us start with a collection of SL(2 , C ) group elements G ij satisfyingthe holonomy trace constraint around every cycle on the boundary. We need to reconstruct the spinors and thetranslations. Let us choose a root puncture i and consider a cycle C starting and finishing at i (for example, atriangle i → i → i → i ). The holonomy G C around that cycle has a trace equal to 2 and thus is conjugated to atranslation t C : G C = Gt C G − , (39)for some group element G ∈ SL(2 , C ). We Iwasawa decompose this group element, G = g ∆ t , so that G C =( g ∆) t C ( g ∆) − . We define the spinor at the puncture i as the image of the up spinor by g ∆: z i = g ∆ | ↑(cid:105) , which implies that Λ z i = g ∆ . (40)We can then transport this spinor to define the spinors at every other puncture, z i = G i i z i . We still have to checkthat this is a consistent definition, i.e. that z j = G ij z i . This is equivalent to checking that z i is stabilized by theholonomy around the triangle i → i → j → i : z j = G ij z i ⇔ G i j z i = G ij G i i z i ⇔ z i = G − i j G ij G i i z i . (41)This means that we would have obtained the same spinor z i if we had used the cycle i → i → j → i instead of C .To show this, it is enough to consider two cycles C and C , with SL(2 , C ) holonomies G and G , starting at i and finishing at i , and prove that the spinor stabilized by G is also stabilized by G . As explained above, since G is conjugated to a translation, we can write: G = Λ ω t µ Λ − ω = I + µ | ω (cid:105) [ ω | , (42)for a spinor ω ∈ C , and the same for G . The holonomy for the concatenated cycle ( C ; C ) then reads: G G = I + µ | ω (cid:105) [ ω | + µ | ω (cid:105) [ ω | + µ µ [ ω | ω (cid:105) | ω (cid:105) [ ω | , Tr G G = 2 − µ µ [ ω | ω (cid:105) . (43)The holonomy constraint around the concatenated cycle ( C ; C ) thus implies that the holomorphic scalar productbetween the two spinors vanishes, [ ω | ω (cid:105) = 0, which in turn implies that ω is proportional to ω . In particular, ω is also stabilized by G and vice-versa ω is also stabilized by G .We would like to point out a variation about the reconstruction of SL(2 , C ) holonomies from the boundary spinordata. It is traditional in standard works in loop quantum gravity to split the spinor z ∈ C in its flux vector (cid:126)X ∈ R (carrying information about the embedding of the surface patch within the 3d space) and its phase (interpreted asthe twist angle, carrying information about the 2+1 embedding of the surface patch in the time direction). Puttingthe phase aside and focusing on the flux vector, one can introduce SL(2 , C ) group elements mapping the flux vector (cid:126)X i at one boundary puncture onto the flux vector (cid:126)X j at another puncture. As we show in the appendix A, this boostaction has a SU(1 ,
1) stabilizer group, thereby realizing a 3+3 splitting of SL(2 , C ) instead of the 4+2 splitting used4above when acting on spinors. The holonomy condition is then looser. Instead of enforcing that the trace of theSL(2 , C ) around boundary cycles is necessarily 2, it can now be an arbitrary real number strictly larger than 2. Wedo not pursue in this direction and focused instead of the spinor variables.Here, we will not study in details the possible symplectic structures that one can endow the space of discrete(constrained) Lorentz connections with and the question of their matching with the canonical Poisson bracket on thespinors. There is not a straightforward obvious answer. On the one hand, there are (well-known) ambiguities on thedefinition of symplectic structures on discrete Lorentz connections and T connections, and on the other hand, it isnot clear if the Λ z ’s are the best choice of section and what is supposed to be the brackets of the spinor z with thetranslation parameter µ of the upper triangular matrix. On top of these ambiguities, there is also the question ofthe precise role of the holonomy constraints. For instance, should we define the symplectic structure before or afterimposing the holonomy constraints? Do the holonomy constraints form a set of first class constraints generating agauge invariance and leading to a symplectic quotient? Following previous work on the combinatorial quantization ofChern-Simons theory [63] and the related phase space of 2+1-d loop quantum gravity with a cosmological constant[64–68], it is tempting to hope that the holonomy constraints would generate a kind of translations, but we postponesuch analysis to future investigation.We will focus instead on the geometrical interpretation of discrete Lorentz connections, on the formulation of adynamical boundary theory and their possible continuum limit. B. Non-trivial stabilizer and (relative) locality on the boundary
In this setting with the correspondence between spinors and discrete SL(2 , C )-connections, the next natural questionis whether we can identify a geometrical or physical meaning to the extra data carried by the Lorentz connectionscompared to the spinors, i.e. provide an interpretation to the stabilizer group data and the discrete T -connection.Since the stabilizer group T is a two-dimensional abelian Lie group, isomorphic to C (provided with the addition),we propose to interpret these group elements as translation on the 2d boundary. More precisely, assuming that the 2dboundary has a spherical topology, we see it as a complex manifold and parametrize it in terms of a complex variable ζ locating points on the boundary. We propose to interpret the extra data contained in the SL(2 , C ) holonomies ontop of the spinors as position coordinates ζ i ∈ C for each puncture.Let us take a step back and reflect on the structure of (boundary) surfaces in loop quantum gravity. A surface consistsin a set of quanta of areas carried by the spin network punctures. These quanta of areas are defined mathematicallyas spin states resulting from the (canonical) quantization of spinors. Each spin state is geometrically interpreted asan elementary surface patch carrying a (quantized) vector, whose norm gives the quantized area in Planck unit andwhose direction is the normal direction to the surface, and a U(1) phase called the twist angle, which indicates theextrinsic curvature integrated over the surface patch. In this traditional setting for loop quantum gravity, there isabsolutely no information on where a surface patch is located with respect to other patches on the overall (boundary)surface.Indeed, in the background independent framework of loop quantum gravity, it is the spin network state - itsunderlying graph and the algebraic data dressing it- which defines the 3d space (quantized) geometry. Reconstructingthe overall 3d geometry of a (large) region of the spin network is not straightforward, it is a non-local reconstructionand recognising two graph nodes as close or far is a hard question . In fact, considering a region with a 3-balltopology (and its boundary with a 2-sphere topology), it is the bulk spin network state data that determines the 3dgeometry and thus the notion of locality on its 2d boundary. If we focus (too much) on the algebraic data induced onthe boundary - the spin states- and discard all the other bulk information, we lose the possibility to localise pointson the boundary.This issue is circumvented for 3d regions with a single node. This corresponds to a single quantum of volume, definedby the intertwiner state carried by the node. The intertwiner is interpreted as a quantized convex polyhedron. Indeed,considering the 3-vectors induced by the spin states around the node, there exists a unique convex polyhedron such thatthese are the normal vectors of the polyhedron’s faces. This is ensured by Minkowski’s theorem for convex polytopes:in this algorithm assuming the convexity of the surface, the normal vectors play the double role of determining thedirection of the boundary face and determining each face position with respect to the other faces. A spin network isthen interpreted as a discrete 3d geometry resulting from gluing those quantized convex polyhedra together [18, 70]. There is actually no rigorous theorem proving that this reconstruction is systematically possible and unique. The reader will find a veryinteresting discussion of possible non-locality effects resulting from the background independence of spin network states in [69]. , C ) holonomies. More precisely, we showbelow that there is an enticing one-to-one correspondence between punctures provided with spinor and position andflat discrete Lorentz connections in the boundary.Assuming that 2d space boundary has a spherical topology, we view the 2-sphere as a complex manifold andparametrize it in terms of a complex variable ζ locating points on the boundary. We provide every punctures on theboundary with position coordinates, given by an extra complex variable ζ i associated to each puncture. Then weshow below that we can define a unique Lorentz holonomy between two punctures, which transports the extendeddata defined y the spinor-position pair ( z i , ζ i ) from one puncture onto another.Drawing inspiration from previous works on SU(2) twisted geometries and SL(2 , C ) spin networks [19, 23, 71], weembed the complex position ζ i as a spinor w i ∈ C up to complex rescaling: w = (cid:18) w w (cid:19) ∼ λw , ∀ λ ∈ C −→ equivalence class [ w ] = [ λw ] defined by ratio ζ = w w . (44)Then we define the unique SL(2 , C ) group element between two punctures which maps ( z i , w i ) onto ( z j , w j ): G ij = | z j (cid:105) [ w i | − | w j (cid:105) [ z i | [ w i | z i (cid:105) ∈ SL(2 , C ) , G ij | z i (cid:105) = | z j (cid:105) , G ij | w i (cid:105) = | w j (cid:105) . (45)This Lorentz holonomy transports as wanted the spinor and position from one puncture to another, G ij (cid:46) ( z i , ζ i ) =( z j , ζ j ). Its definition is possible and valid if and only if the (holomorphic) scalar product between spinors remainsconstant [ w i | z i (cid:105) = [ w j | z j (cid:105) . This gives a unique prescription for the SL(2 , C ) holonomy in terms of the spinors ( z i , z j )and positions ( ζ i , ζ j ). Indeed this necessary condition fixes the spinors w ’s in terms of the complex coordinates ζ ’s, upto an arbitrary overall scale fixed for the whole boundary network. Let’s indeed choose a global value [ w i | z i (cid:105) = σ ∈ C for all the punctures i = 1 ..N . Then if we are given the spinors z i ∈ C and the complex positions ζ i , we canreconstruct the spinors w i : w i = (cid:18) λ i ζ i λ i (cid:19) , σ = [ w i | z i (cid:105) = λ i (cid:0) ζ i z i − z i (cid:1) ⇒ λ i = σζ i z i − z i ∈ C . (46)This leads to unique SL(2 , C ) group elements, which are actually independent of the specific value chosen for σ : G ij = 1 σ (cid:18) λ j z i ζ j − λ i z j λ i ζ i z j − λ j z i ζ j λ j z i − λ i z j λ i ζ i z j − λ j z i (cid:19) . (47)We can recast this SL(2 , C ) holonomy in the factorized form involving the T -holonomy t ij and the section groupelements Λ z i , G ij = Λ z j (cid:18) µ ij (cid:19) Λ − z i = | z j (cid:105)(cid:104) z i |(cid:104) z i | z i (cid:105) + | z j ][ z i |(cid:104) z j | z j (cid:105) + µ ij | z j (cid:105) [ z i | . (48)Matching this with the formulas above gives the expression of the T -holonomy coefficient µ ij between punctures interms of the complex coordinates ζ i of the punctures: µ ij = µ j − µ i , with µ i = 1 (cid:104) z i | z i (cid:105) (cid:18) ζ i ¯ z i + ¯ z i z i − ζ i z i (cid:19) . (49)Since the group elements G ij are uniquely determined by each pair of source and target spinor-position ( z i , ζ i , z j , ζ j ),this construction defines a flat discrete SL(2 , C ) connection on the boundary surface, G i n i ..G i i = I for every cycleof punctures on the boundary i → i → i → · · · → i n → i . This leads to the following proposition: Proposition II.2.
Considering the boundary data defined by a collection of N punctures each dressed with a spinor z i ∈ C and a complex coordinate ζ i with i labelling the punctures and running from 1 to N , this defines a unique discrete Lorentz connection with SL(2 , C ) group elements G ij transporting the boundary data from the puncture i tothe puncture j : G ij z i = z j , G ij (cid:18) ζ i (cid:19) ∝ (cid:18) ζ j (cid:19) , (50) where G ij acts by multiplication by the corresponding 2 × SL(2 , C ) holonomy around anycycle C on the boundary is trivial: ←− (cid:89) (cid:96) ∈C G (cid:96) = I . (51) Reciprocally, a discrete flat Lorentz connection defines boundary data ( z i , ζ i ) i =1 ..N up to a global SL(2 , C ) transfor-mation.Proof. We have already constructed the SL(2 , C ) holonomies from the spinor-position data ( z i , ζ i ) i =1 ..N . We simplyhave to describe the reverse procedure. Considering a discrete flat Lorentz connection G ij ∈ SL(2 , ; C ), we start fromone arbitrarily chosen puncture, i and we choose arbitrary data ( z i , ζ i ) ∈ C × C . Then we define all the remainingspinor-position variables by transport from i as ( z i , ζ i ) = G i i (cid:46) ( z i , ζ i ). The flatness condition for the connection, G ij = G i j G − i i around a triangle, ensures that ( z j , ζ j ) = G ij (cid:46) ( z i , ζ i ) for all pairs of punctures. The global SL(2 , C )freedom corresponds to the choice of the initial data ( z i , ζ i ).FIG. 3: The spinor z , living at each puncture, encodes the (semi-classical) information about the quantumof area carried by the puncture. It contains both the flux vector (cid:126)X giving the normal vector to the surfacepatch and the twist angle θ representing the measure of extrinsic curvature to the spatial slice. We define SL(2 , C ) boundary holonomies G ij that transport the spinor information from puncture to puncture, such that G ij | z i (cid:105) = | z j (cid:105) . Either we consider the spinors as the whole boundary data and the SL(2 , C ) holonomies aredetermined by the spinors up to stabilizer group elements given by triangular matrices t ij ∈ T (or the lefthand side). Or we supplement the spinors z i with complex variables ζ i indicating the position of the punctureon the 2d boundary and then the SL(2 , C ) holonomies are entirely determined by the spinors and complexcoordinates (on the right hand side). In the first case, the holonomies of the discrete SL(2 , C ) connection livein the stabilizer subgroup T . While, in the latter case, the resulting discrete SL(2 , C ) connection is exactlyflat. The geometric picture is that the complex variables ζ i , parametrizing the spinor stabilizer and interpreted as 2dcoordinates, play the role of discrete 2d metric data on the boundary. From this point of view, they describe theintrinsic geometry of the boundary. On the other hand, the spinors z i define the normal 3d vectors to the surface andthus describe the extrinsic geometry of the boundary (within the 3d slice). Together, they describe the whole discreteembedded geometry of the boundary surface within the spatial slice, as illustrated on fig.3.If we compare the spinor dynamics ansatz introduced in the previous section I B, the complex coordinates ζ i aremeant to play a similar role than the coupling constants of the spinor Hamiltonian, which were shown to give the7boundary metric. In this formulation in terms of SL(2 , C ) connections, the intrinsic geometry, encoded by the ζ i ’s,are put on the same footing than the extrinsic geometry, encoded by the z i ’s. Both will naturally acquire dynamics.There is obvious reason to keep the ζ i ’s or z i ’s fixed while the other evolve and fluctuate, except if we make a specialchoice of boundary conditions. For instance, we could keep the intrinsic 2d metric fixed on the boundary surface andlet the embedding data evolve, which would correspond to fixing the complex coordinates ζ i ’s and the spinor norms (cid:104) z i | z i (cid:105) .Since we have identified field configurations corresponding to flat Lorentz connections and understood them interms of boundary geometry, it is natural to think about curvature defects and seek a geometrical interpretation forSL(2 , C ) connections with non-trivial curvature. For instance, in the more general setting where we focus solely onthe spinors, as above in section II A, we do not require the SL(2 , C ) connection to be flat, but allow for a non-trivialstabilizer. This means that, if we start at a puncture dressed with spinor z and complex position ζ and go arounda cycle on the boundary back to that initial puncture, the non-trivial holonomy G ∈ G z = Λ z T Λ − z will shift the 2dcoordinate from ζ to a new position ˜ ζ . Or in short, if we go around a loop, the position of the puncture has changed.This is reminiscent of the framework of relative locality [72–74], where non-trivial connection and torsion in phasespace lead to a relativity of the position of an event with respect to the observer and its history. That would providea geometrical interpretation for translational curvature defects of the Lorentz connection. It is tempting to try tointerpret SU(2) curvature defects as magnetic excitations of the Ashtekar-Barbero connection along the directionstangent to the boundary (and not transversally) and dilatation curvature defects as some conformal excitations, butwe leave this for future analysis. C. SL(2 , C ) boundary theory Now that we have reformulated the flux excitations (or area quanta) of loop quantum gravity on space-time cornersin terms of (discrete) flat Lorentz connections, it is natural to think about its continuum limit as a theory of a Lorentzconnection field on the 2+1-d time-like boundary of space-time. If we want to discuss the specifics of the boundarytheory and dynamics, we need to first address two questions: • As it is natural with a Lorentz connection, should we impose gauge invariance under local Lorentz transforma-tions? • As boundary flux excitations correspond to flat Lorentz connections, should we focus on solely on flat connectionsor more broadly consider theories of arbitrary Lorentz connections whose equations of motion in vaccuum (i.e.without defects or sources) nevertheless impose flatness?Starting with the issue of enforcing or not local gauge invariance, one needs to keep in mind that there is a singleflat Lorentz connection on the 2-sphere up to gauge transformations. So, if one focuses on purely flux excitations(thus only area quanta and not dual defects on the boundary) and thus considers only flat connections, imposing localSL(2 , C ) gauge invariance on the boundary renders the boundary theory trivial and empty. Indeed local SL(2 , C )transformations at a puncture means that one can arbitrarily shift the spinor | z (cid:105) ∈ C living at the puncture. Sincethe spinor indicates both the size of the area quantum and its (normal) direction, enforcing SL(2 , C ) gauge invariancewould mean that all quanta of area are physically equivalent. This would amount to requiring the invariance ofthe boundary theory under both 2d diffeomorphisms and conformal transformations. Although this seems perfectlyacceptable, reasonable and even preferable, that goes against the traditional view of boundaries in loop quantumgravity where the spins carried by each boundary puncture is thought of as a physical observable.So let us start with the case where we do not require a local gauge invariance under boundary SL(2 , C ) transforma-tions. We will consider the case of gauge-invariant dynamics later in this section. So the boundary variables are theSL(2 , C ) holonomies G ij between punctures. In fact, these determine the spinors (and positions) only up to a globalSL(2 , C ) transformation, which correspond to the choice of spinor (and position) at a chosen root puncture i . So wewould like to define a theory, its action principle and Hamiltonian, in terms of group elements G ij ∈ SL(2 , C ) betweenpunctures and one spinor z i ∈ C . From this spinor at the root puncture, one immediately reconstructs all the spinorsby transporting it to the other punctures, assuming that z i = G i i z i . Then we can immediately reformulate thespinor action (11) and Hamiltonian (22) defined in the previous section by expressing the scalar product betweenspinors in terms oif the SL(2 , C ) holonomies: (cid:104) z i | z j (cid:105) = (cid:104) z i | G † i i G i j | z i (cid:105) , [ z i | z j (cid:105) = [ z i | G − i i G i j | z i (cid:105) = [ z i | G ij | z i (cid:105) , (52)8where we have used the flatness condition of the discrete Lorentz connection. This leads to an action written in itscanonical form which reads: S [ { G kl } , z i ] = (cid:90) d t (cid:20) − i (cid:104) z i | (cid:88) k G † i k G i k | d t z i (cid:105) − i (cid:88) k (cid:104) z i | G † i k d t G i k | z i (cid:105) − N H (cid:21) , (53)with H = (cid:88) n β n (cid:88) k (cid:104) z i | G † i k G i k | z i (cid:105) n + γ (cid:88) k,l C kl (cid:104) z i | G † i k G i l | z i (cid:105) +˜ γ (cid:88) k,l D kl (cid:104) z i | G kl | z i (cid:105) + ¯ D kl (cid:104) z i | G kl | z i (cid:105) + . . . This is the exact equivalent of the boundary spinor action ansatz (22) proposed earlier. Here we have implicitlyassumed the flatness of the Lorentz connection, through the triangular relation between SL(2 , C ) group elements G kl = G − i k G i l . The flatness condition allows to transport all the spinors back to the root puncture. We could putboth the flatness condition and the use of a root spinor aside and generalize the Hamiltonian above to define an actionprinciple in terms of solely the discrete SL(2 , C ) connection. Indeed, forgetting about z i , the natural proposal isto replace the projection on | z i (cid:105)(cid:104) z i | by the trace of the group elements, thus writing the Hamiltonian as a linearcombination of terms Tr G † kl G kl and Tr G kl and their powers: S SL(2 , C ) [ { G kl } ] = (cid:90) d t (cid:20) − i (cid:88) k,l Tr G † kl d t G kl − N H SL(2 , C ) (cid:21) , (54)with H SL(2 , C ) = (cid:88) n β n (cid:88) k,l (Tr G † kl G kl ) n + γ (cid:88) k,l C kl Tr G † kl G kl + ˜ γ (cid:88) k,l D kl Tr G kl + . . . The next step would be to analyze the physics predicted by such dynamics. This would require 1. understand thesolution to the equations of motion; 2. check the stability or not of flat connections; 3. work out the continuum limitof this discrete ansatz as a field theory.This conclude the proposal for a boundary theory non gauge-invariant under local SL(2 , C ) transformations. Wenow turn to the possibility of defining gauge invariant boundary dynamics. Let us first point out that the bulk(loop) quantum gravity is invariant under two kinds of gauge transformations: SU(2) gauge transformations and(space-time) diffeomorphisms. Since SL(2 , C ) is (much) larger than SU(2), requiring the SL(2 , C ) gauge invarianceof the boundary is a non-trivial extension of the SU(2) gauge transformations generated by the Gauss law of theAshtekar-Barbero connection: it could involve boundary diffeomorphisms or other types of transformations (e.g. con-formal transformations,. . . ). It should thus correspond to a very specific class of boundary conditions with enhancedsymmetry.As we underlined earlier, since there is a unique flat SL(2 , C ) connection on the boundary 2-sphere up to SL(2 , C )gauge transformations, enforcing the gauge invariance of the boundary theory under local SL(2 , C ) transformationsleads to a trivial theory with a single physical state. This amounts to considering all possible area quanta on thespatial boundary as physically equivalent, whatever their direction and size. This would become interesting only ifwe venture away from the flatness condition of the SL(2 , C ) connection. This follows a perfect natural logic: tradingthe spinors representing boundary flux excitations for Lorentz connections, thus identifying the flux excitations as flatLorentz connections, naturally leads to contemplating the meaning of non-flat connections, which should representdifferent types of boundary excitations. In the continuum limit, we are therefore looking for a gauge field theory ofa connection, whose equations of motion in vaccuum (i.e. without source or defect) amount to the flatness of theconnection. Natural candidates are coset SL(2 , C ) Chern-Simons theories (see e.g. [75]) or a SL(2 , C ) Yang-Millstheory on the 2+1-d boundary. What needs to be understood is 1. the (quantum) boundary conditions that theygenerate on the 2+1-d time-like boundary of space-time; 2. if those boundary conditions are compatible with the bulkdynamics of (loop) quantum gravity (i.e. generated by the bulk Hamiltonian constraints for some choice of space-timefoliation).One should nevertheless keep in mind that we are not yet searching for a unique boundary theory, but more forclasses of boundary theories. They should correspond to classes of boundary conditions for the bulk fields. Theywill likely admit for a non-trivial renormalization group flow corresponding to the changes of (quantum) boundaryconditions under dilatations and deformations of the boundary.9 D. Recovering local
SU(2) gauge invariance on the boundary: magnetic excitations
We have discussed the possibility of defining boundary dynamics invariant or not under SL(2 , C ) local gaugetransformations. However, the natural set of gauge transformations in the loop quantum gravity consists in SU(2)transformations. These act as local 3d rotations on the flux. The question is then if it makes sense to impose a SU(2)gauge invariance on the boundary (and not a SL(2 , C ) gauge invariance anymore).The SU(2) action on the boundary spinors is the straightforward multiplication by 2 × | z (cid:105) ∈ C (cid:55)→ h | z (cid:105) ∈ C , h ∈ SU(2) . (55)The boundary spinor dynamics (11)-(22) that we have studied up to now has already been assumed to be (gauge)invariant under global SU(2) transformations, leading to an expansion of the Hamiltonian in terms of scalar productsbetween the spinors. Now we would like to upgrade this invariance to local SU(2) transformations. This is naturallyachieved by introducing extra degrees of freedom on the boundary : a (discrete) SU(2) connection defining thetransport of the flux excitations -the spinors- on the boundary.We thus consider SU(2) group elements g kl associated to (oriented) pairs of punctures ( k, l ). While local SU(2) trans-formations acts locally on the spinors at each puncture, they act at both source and target of the SU(2) holonomies: z k (cid:55)→ h k z k , g kl (cid:55)→ h l g kl h − k . (56)Thus the quadratic SU(2)-invariant combination of spinors are the transported scalar products, (cid:104) z l | g kl | z k (cid:105) . We candefine the boundary theory as before in section I B, with a polynomial Hamiltonian in (cid:104) z l | g kl | z k (cid:105) and [ z l | g kl | z k (cid:105) . Theaction principle would now depend on both the spinors z k and the SU(2) holonomies g kl as independent variables.In the continuum limit, these SU(2) holonomies would become a SU(2) connection field, defining a SU(2) covariantderivative on the time-like boundary of space-time.It is very tempting to interpret this boundary SU(2) connection as the (pull-back of the) Ashtekar-Barbero con-nection on the space-time corner. This means that we do not consider only the Ashtekar-Barbero connection in thedirection transversal to the boundary surface -carried by the spin network edges puncturing the surface- but alsoits components tangential to the boundary surface -thus to be carried by spin network links running along the sur-face . These are interpreted as magnetic excitations of the boundary supplementing the electric excitations definedby the flux living at the boundary punctures (see e.g. [7] for a discussion of the various edge modes of loop quantumgravity). This type of extended spin network structure, with links running transversally and tangentially to surfaces,was already proposed in e.g. [6, 15, 32], and it would be interesting to investigate further the corresponding possibleboundary dynamics.The last point we would to discuss is the relation between the newly introduced SU(2) connection g kl on theboundary and the SL(2 , C ) connection G kl that we have discussed up to now. The essential difference is that theSU(2) connection is a field independent from the spinors while the SL(2 , C ) connection depends on the spinors andis actually a reformulation of the spinors. Thus, the SL(2 , C ) transport between punctures exactly maps the spinorsonto each other, G kl | z k (cid:105) = | z l (cid:105) , while the SU(2) transport between punctures simply allows to write down locallySU(2)-invariant scalar products (cid:104) z l | g kl | z k (cid:105) . Nevertheless, depending on the precise boundary action and Hamiltonian,the two connections could be related on-shell. Indeed, if as an example the Hamiltonian consisted in − (cid:80) k,l (cid:104) z l | g kl | z k (cid:105) ,its minimal value for fixed spinors z k would be obtained if and only if the SU(2) group elements exactly transport thespinors, i.e. | z l (cid:105) = g kl | z k (cid:105) . Then the SU(2) connection could be identified as the SU(2) part of the SL(2 , C ) connection.We postpone a more in-depth analysis of the possible coupled dynamics of boundary spinors and boundary SU(2)connection, i.e. of both electric and magnetic excitations on the boundary- to future investigation. Another -na¨ıve- way to impose invariance under local SU(2) transformations is to restrict to the SU(2)-invariant part of each fluxexcitation. This means keeping only the norm of each spinor (cid:104) z k | z k (cid:105) , or equivalently the spin j k of the corresponding quanta of area,discarding any information about the direction and phase of the spinor z k , or equivalently about the magnetic moment and phase of thespin state living in V j k . A boundary state would simply be labelled by the spins, without further data (i.e. no spin state or intertwiner).The dynamics would then couple those spins j k together and let them evolve. From the point of view of the 2d geometry of the boundarysurface, this corresponds to keeping only the density factor (determinant of the induced 2d metric) as boundary data and discardingthe rest of the 2d metric and all the extrinsic data about e embedding of the 2d boundary surface in the 3d space. Although this couldmake sense mathematically, we do not see any physical reason for such drastic reduction of the boundary modes. The special role fo tangential spin network links was already discussed in early loop quantum gravity work, especially for their non-trivialcontribution to the area spectrum and to black hole entropy computations [76, 77]. This Hamiltonian (cid:80) k,l (cid:104) z l | g kl z k (cid:105) is actually real if we assume the natural orientation condition that g lk = g − kl . Indeed, this means that (cid:104) z k | g lk z l (cid:105) = (cid:104) z k | g † kl | z l (cid:105) = (cid:104) z l | g kl | z k (cid:105) . Outlook & Conclusion
We have investigated the basic structure of boundary theories for loop quantum gravity, meaning the dynamicsof the degrees of freedom induced by the fluctuations and evolution of the bulk geometry on the 2+1-dimensionaltime-like boundary of space-time. This 2+1-dimensional boundary is seen as the time evolution of the 2d spaceboundary surface. As spin network states span and create the quantum geometry of the 3d space, the spin networklinks puncture the boundary surface. The spin states carried by those boundary punctures define quanta of area ofthe surface. These are also referred to as flux excitations or “electric” excitations. They are the basic boundary dataof loop quantum gravity.We showed that the dynamics of those quanta of area on the boundary surface can be mathematically formulated interms of complex 2-vectors -spinors- attached to the boundary punctures. Considering a Hamiltonian polynomial inthose excitations, the lowest order Hamiltonian (with up to quartic terms) corresponds to a Bose-Hubbard model (withtwo species). Such condensed matter models are known for their phase transitions, which could lead to interestingnew physics and predictions for loop quantum gravity phenomenology. Furthermore, we showed that the couplingconstants of the quadratic terms of the Hamiltonian can be understood in the continuum limit (as the number ofpunctures is sent to infinity, thereby creating a lattice structure on the boundary surface) as the components of abackground metric on the 2+1-d time-like boundary. We consider this result as the hint of a deeper correspondencebetween the coupling constants of the boundary Hamiltonian for area quanta and (quantum) states of the 2+1-dboundary metric (or 2+1-d dressed metric in semi-classical terms).We further showed for to reformulate the boundary spinor data in terms of discrete flat SL(2 , C ) connections.Curvature of this boundary SL(2 , C ) connection should be understood as novel boundary excitations for loop quantumgravity, which we speculate to correspond to the “translational” and “magnetic” edge modes envisioned in [7]. Thisopens the door to formulating boundary theories for loop quantum gravity as SL(2 , C ) gauge theories (such as e.g.coset Chern-Simones theories).In the present work, our starting point was the formalism of loop quantum gravity and its spin network statesfor the quantum geometry. We have tried to clarify what could/would be a boundary theory in this framework andone should now analyze the physics of those models and compare them with the (many) recent works on boundarytheories and edge modes in classical general relativity. In order to tackle this next step, there is first the broaderquestion of what is meant by a boundary theory for (quantum) gravity. We see three levels of sophistication:0. boundary terms to the bulk action: One usually needs to add a boundary term to a field theory action principle in order to ensure the differentiabilityof the action with respect to field variations and cleanly derive the equations of motion for the bulk field whilerespecting chosen boundary conditions. For instance, the Gibbons-Hawking-York term in the metric formulationof general relativity, defined as the boundary integral of the extrinsic curvature, ensures the differentiability ofthe corrected Einstein-Hilbert action (space-time integral of the scalar curvature) when fixing the inducedmetric on the boundary. This typically further ensures the gauge invariance of the overall action - bulk actionplus boundary term- under gauge transformations consistent with the boundary conditions. This can be doneconsistently in a covariant phase space approach.1. dynamical boundary conditions and edge modes:
The covariant phase space goes further and allows to define dynamical boundary conditions. Indeed, in acanonical setting, we distinguish the canonical boundary - the 3d space for general relativity - and the time-likeboundary - the 2+1-d boundary, which describes the evolution in time of the 2d boundary of the 3d space.We usually refer to the 2d spatial boundary surface as the corner (of space-time). Instead of fixing boundaryconditions on the whole 2+1-d time-like boundary, one seeks to identify boundary field degrees of freedomliving on the corner - typically, in general relativity’s metric formulation, the 2d metric on the corner plus extrafields (e.g. a radial expansion scalar and a shear vector)- and let them evolve in time. The time-like boundaryconditions are thus generated as the time evolution of initial boundary conditions on the corner. One usuallyrequires that this boundary evolution satisfies a suitably extended gauge-invariance of the original bulk theory.The obvious question is whether the time evolution of the boundary data on the corner can be derived as aHamiltonian dynamics defined on its own, i.e. solely in terms of the boundary data without any reference to thebulk fields . For such classes of boundary conditions, this edge mode dynamics defines the boundary theory. This seems to be the natural (simplest) setting for (quasi-local) holography (from a canonical/hamiltonian viewpoint). In the case effective boundary theory:
A more intricate setting is when the boundary dynamics does not decouple from the bulk theory and thus cannot be defined on its own. One can nevertheless hope to integrate over the bulk fields and dynamics and extractan effective theory for the edge modes, i.e. an effective boundary theory. As a coarse-graining of the bulktheory , this procedure can be understood as studying the renormalisation group flow for the 2+1-d boundarytheory . This totally reverse the original logic of the field theory: instead of fixing boundary conditions andanalysing the resulting bulk field and (quantum) fluctuations, one integrate over the bulk degrees of freedomand focus on the induced dynamics of the boundary conditions.The present work, discussing the possible dynamics of quanta of area on the 2+1-d time-like boundary of space-time, is naturally set at the level 1 of this hierarchy. The next step in this direction would thus be to compare theboundary theory ansatz which we introduced with edge mode hamiltonians derived from general relativity in the casethat the boundary conditions admit a proper hamiltonian formulation (i.e. if the edge mode dynamics can be definedwithout referring to the bulk degrees of freedom). From this perspective, the natural questions to face are: • how do the presently introduced templates for the dynamics of boundary area quanta compare the edge modephase space and dynamical boundary conditions in general relativity? • do those models of boundary dynamics carry a representation of the boundary charge algebra (i.e. of thesymmetry algebra) of general relativity or of a suitably deformed version to be implemented i quantum gravity? • do they define renormalizable quantum field theories in the continuum limit? does this renormalization flowcorrectly represent deformation and rescaling of the boundary surface? What are the fixed points (as conformalfield theories) of this flow?Beside connecting the present work to the study of edge modes in gravity theories, we would like to conclude onthe fact that it also opens new perspectives. Indeed the boundary dynamics models we introduced can naturallybe written as condensed matter models with non-trivial phase diagrams, and thus with possible interesting phasetransitions, which would enrich the phenomenology of (loop) quantum gravity. Acknowledgement
I am very grateful for the regular discussions on edge modes in quantum field theory and holography in quantumgravity with Laurent Freidel, Marc Geiller and Christophe Goeller.
Appendix A:
SL(2 , C ) -holonomy between 3-vectors A variation of our new framework for LQG boundaries in terms of SL(2 , C ) holonomies is to define a more equalsplitting of the SL(2 , C ) data, as a pair of 3-vectors instead of a 4-dimensional spinor and a 2-dimensional complexnumber. This can be thought as natural. In the context of twisted geometries, a spinor z ∈ C encodes a 3-vector (cid:126)X ∈ R plus a phase θ . The 3-vector is interpreted as the normal vector to the boundary surface, while the twist angle θ is canonically conjugate to the 3-vector norm and usually interpreted as encoding the extrinsic curvature across theboundary surface. A splitting in terms of pairs of 3-vectors would amount to bundling the twist angle together withthe complex position. that one can identify classes of boundary conditions such that the boundary decouples from the bulk, in the sense that the edge modedynamics can be defined entirely in terms of boundary fields, there is a holographic duality when the resulting boundary theory carries arepresentation of the conserved charges of the bulk dynamics: then the boundary and bulk theories have the same symmetry algebra andone can seek a correspondence between the observables of the two theories. A weaker form is when the boundary and bulk theories areMorita-equivalent, i.e. do not necessarily have the same symmetry algebra but their algebra of observables have isomorphic categoriesof representations. Thus, there is no a priori reason for this effective boundary theory to be unitary. One should instead account for possible dissipation inthe bulk or in other words, flux across the boundary. Holography seems to be achieved at a fixed point of this renormalisation flow, when the effective theory boundary is given by the originalboundary theory -the part of the dynamics proper to the boundary without the interaction terms with the bulk- up to possible shifts ofthe coupling constants (i.e. without any new interaction terms). (cid:126)X i , which can be defined mathematically as a spinor z i up to a phase: (cid:126)X = (cid:104) z | (cid:126)σ | z (cid:105) , | (cid:126)X | = (cid:104) z | z (cid:105) , | z (cid:105)(cid:104) z | = 12 (cid:16) | (cid:126)X | I + (cid:126)X · (cid:126)σ (cid:17) , X = (cid:126)X · (cid:126)σ = 2 | z (cid:105)(cid:104) z | − (cid:104) z | z (cid:105) I , (A1)where the σ ’s are the Pauli matrices. Considering two 3-vectors (cid:126)X i and (cid:126)X j , or equivalently the corresponding twotraceless Hermitian matrices X i and X j , we can map one onto the other with a Lorentz transformation G ij ∈ SL(2 , C ): G ij X i G † ij = X j . (A2)Let us start with the unit 3-vector along the z -axis and see how to boost it to an arbitrary 3-vector X , i.e. whatSL(2 , C ) realizes Gσ z G † = X . We consider a spinor z ∈ C such that (cid:126)X = (cid:104) z | (cid:126)σ | z (cid:105) and identify a group element G mapping the reference spinor to z : G | ↑(cid:105) = z ⇒ Gσ z G † = X . (A3)The stabilizer group of σ z under the action by conjugation obviously contains all the transformations generated by σ z , that is all the complex dilatations ∆ λ with λ ∈ C , but is actually larger. It is the SU(1 ,
1) subgroup of SL(2 , C ): Gσ z G † = σ z ⇐⇒ G = (cid:18) a b ¯ b ¯ a (cid:19) with | a | − | b | = 1 . (A4)So the stabilizer group for the reference 3-vector X = σ z is S = SU(1 ,
1) and we can rotate it to get the stabilizergroup for an arbitrary vector X ∈ R obtained as S X = g ˆ X S g − X , where g ˆ X is a SU(2) group element rotating the z -axis unit vector to the direction of the 3-vector ˆ X = (cid:126)X/ | (cid:126)X | .Similarly to what we have done in the spinorial case, we can now trade the set of 3-vectors (cid:126)X i living at thepunctures on the boundary for a network of SL(2 , C ) holonomies G ij between punctures on the boundary. Requiringthat the holonomies map the 3-vectors from one puncture to another, G ij (cid:46) X i = G ij X i G † ij = X j means that allSL(2 , C )-holonomies G around loops on the boundary have to live in a SU(1 ,
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