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Local spin foams Elena Magliaro∗ and Claudio Perini† Institute for Gravitation and the Cosmos, Physics Department, Penn State, University Park, PA 16802-6300, U.S.A. (Dated: October26, 2010) Thecentralobject ofthispaperisanholonomyformulation forspinfoams. Withinthisnewrepre- sentation,weanalyzethreegeneral requirements: locality, composition law, cylindricalconsistency. In particular, cylindrical consistency is shown to fix the arbitrary normalization of the vertex amplitude. Dedicated to our families, overseas. sectionIII we discuss the locality and compositionprop- erties(studied in[15])inthis formulation,andintroduce 0 the requirement of cylindrical consistency in spin foams. 1 0 I. INTRODUCTION Inparticularweanalyzeinvarianceunderfaceorientation 2 reversal, face splitting, and face erasing. Similar invari- t In this paper we consider the holonomy representation ance properties were considered in [18, 19], as an imple- c for spin foams [1]. This representation allows to write mentation of diffeomorphism invariance and in analogy O spin foams in a Feynman path-integral form, where the with cylindrical consistency. Here our perspective is dif- 5 configuration variables are SU(2) group elements. ferent: we require consistency to make a deeper contact 2 Spin foam models [2–7] provide the transition ampli- with Loop Quantum Gravity. As a technical byproduct, tude from an ‘in’ state to an ‘out’ state of 3-geometry: we are able to extend the projection map on the solu- ] they give a mathematical and physical meaning to the tion of simplicity constraints to the full Hilbert space. c q formal expression Two subsections are dedicated to the specific cases of - Ponzano-Regge and EPRL spin foams. Section IV is a gr go(3u)t short discussion on the relationship between continuum [ W[gi(n3),go(3u)t]= Dg(4) expiS[g(4)] (1) limit and cylindrical consistency. Zgi(n3) 1 for the Misner-Hawking transition amplitude of 3- v II. HOLONOMY FORMULATION 7 geometries in terms of a sum over 4-geometries [8, 9]. 2 The recent convergence between covariant and canoni- 2 calapproachestoquantumgravitystrengthenedtheidea In Loop Quantum Gravity [20–24], the kinemati- 5 that the spin foam theory constitutes a good alternative cal Hilbert space is attached to graphs Γ embedded 0. framework for the dynamics of Loop Quantum Gravity. in a 3-dimensional space-like Cauchy hypersurface Σ. 1 In particular: i) in the ’new’ spin foam models [4–6, 10] For a given graph, it is the Hilbert space HΓ = 0 the correct weak imposition of linearized simplicity con- L2(SU(2)L/SU(2)N) where L is the number of links of 1 straints provides a kinematical boundary Hilbert space the graph and N the number of nodes, so a state is a : v isomorphic to the one of Loop Quantum Gravity. Hence gauge invariant function of SU(2) group elements hl i we can interpret a spin foam amplitude as a transition (l = 1...L) that is invariant under SU(2) gauge trans- X amplitude between Loop Quantum Gravity kinematical formations at nodes, r a states. This is also the philosophy behind the computa- Ψ(h )=Ψ(g h g−1). (2) tion of n-point functions [11–14]. ii) the SU(2) formula- l s(l) l t(l) tion ofEPRL spin foammodel [1, 5] respects a composi- Here s(l) and t(l) are respectively the nodes which are tion rule, studied in [15], typical of canonical dynamics. source/target of the link l, according to the orientation iii)herewemakeafurtherstep: weintroducecylindrical of the link. The full Hilbert space [25] of loop gravity consistency in spin foams. This is required for the inter- H=L2(A,dµ ) (3) pretation of covariant amplitudes as transitions between AL wave-functionsofaconnection. Thelaststepgoesinthe becomes separable [26] after imposing spatial diffeomor- directionofdefiningspinfoamdynamicsforthefullLQG phism invariance1, and decomposes into the orthogonal Hilbert space [16,17], andnotfor a truncationof it ona sum fixed (simplicial) graph. H= H , (4) The paper is organized as follows. Section II is a re- Γ viewoftheholonomylocalformulationforspinfoams. In MΓ 1 The group of diffeomorphisms must be extended to allow for ∗ [email protected] isolated points in which maps are not differentiable, but still † [email protected] continuous. 2 where Γ is a diff-equivalence class of graphs Γ ⊂ Σ. By canbethoughtasassociatedtothelinksintheboundary the Peter-Weyl theorem, an orthonormal basis of H is graphsofvertices. Weshallcallthesegraphslocalbound- Γ given by spin-network functions: ary graphs, or simply local boundaries, in order to dis- tinguishthemfromtheglobal boundary ofthe spinfoam. ψΓjlin(gl)=⊗nvin ·⊗l 2jl+1Djl(gl) (5) Theboundarygraphofavertexisdefinedasfollows: the links and the nodes of the boundary graph result from p labeled by spins jl (one per each link l) and intertwiners the intersectionbetweenthe faces andthe edgesmeeting in(onepereachnoden);inisthelabelofanorthonormal atthevertexwiththeboundaryofasmall4-ballcontain- basis vin in the space of intertwiners. The pattern of ing the vertex. The orientationof the boundary graphis the contractionmap“·” is determinedfromthe graphΓ. inherited from the orientation of the 2-complex. Finally Dj is the Wigner SU(2) representation matrix. For spin foams with boundary graph Γ, the partition ForagivenHΓ,theconfigurationvariableshlareinter- functionbecomesafunctionofboundaryholonomiesand pretedasholonomiesoftheAshtekar-Barberoconnection generalizes to [27, 28] Z(h )= W (h ), (9) l σ l Aia =Γia+γKai (6) ∂Xσ=Γ wherethesumisover2-complexesboundedbyΓandthe along the link l of the graph (Γi is the spin-connection, a amplitude associated to each 2-complex is Ki theextrinsiccurvatureofthehypersurfaceΣ,andthe a real number γ 6=0 is the Barbero-Immirziparameter). W (h )= dh W (h )× σ l vf v vf The partition function of a spin foam model takes the Z v form of a sum2 over partition functions for 2-complexes Y × δ( h ) δ(h h ). (10) σ vf l vf fYint vY∈f fYext vY∈f Z = Z . (7) σ To simplify the notation, the face amplitudes have been Xσ split in internal times external. The ones associated to external faces (faces cutting the boundary surface Σ A 2-complex is a collection of 2-dimensional faces, 1- throughalinkl)containtheboundaryholonomyh . The dimensional edges and 0-dimensional vertices, with spe- l cific adjacency relations and orientations. The general form of Z we consider in this paper is σ Z = dh W (h ) δ( h ), (8) σ vf v vf vf Z v f v∈f Y Y Y namelyanholonomy formulationofspinfoams[1],where thevariablesh areanalogoustothecanonicalvariables vf in (2). The partition function Z is local in space-time, σ i.e. it is given by a product of elementary vertex amplitudes W (h ), and ’face amplitudes’ which impose a v vl local condition on holonomies. FIG.1. Labelingofspinfoamintheholonomyrepresentation. The ordered product inside the face amplitude is over An internal face f and the external face fãre shown. The holonomies h are associated to wedges, and the holonomy a cyclic sequence of vertices, according to the face ori- vf h totheexternalfacef˜,orequivalentlytotheboundarylink entation. The bulk holonomies h have a vertex label l vf l. v and a face label f, so that they can be uniquely associated to wedges3. Alternatively, the bulk holonomies amplitude W (h ) defines a linear functional: σ l W :H −→ (11) σ Γ C where H = L2(SU(2)L)/ ∼ is the boundary Hilbert 2 The sum over 2-complexes can be generated by an auxiliary Γ space associated to the boundary graph Γ (we have di- Group Field Theory [29]. It is usually not well-defined (divergent), and requires suitable gauge-fixing or regularization. For videdbythegaugeactionofSU(2)atnodes). Thisfunc- a fixed 2-complex, divergencies are associated to “bubbles”. In tional is topological (i.e. unphysical) theories, topological invarianceim- plies that those “bubbles” can be removed up to a divergent (W |ψi= dh W(h )ψ(h ), (12) σ l l l overallfactor thatdepends onlyonthecutoff. Inquantum gen- Z eral relativity the situation is different: “bubble” divergencies and assigns a quantum amplitude to the kinematical aretrueradiativecorrectionswhichcarryinformationaboutthe infraredbehavior ofthetheory[30–33]. states. This amplitude (its modulus) gives the probabil- 3 A wedge labeled by vf is a portion of a face f adjacent to the ityforajointsetofmeasurements,codedintheboundary vertexv. state ψ performed on the boundary of σ [23, 34]. 3 The formalism admits boundaries with two connected respectively. We have components Σ and Σ . In this case, W belongs to in out σ HevΓ∗ooluvtin⊗gHthΓeininacnodmcianngbsteattheoψughttoatsheaopurotgpoaignagtsiotantkeeψrnel dhextWσ1(h,hext)Wσ2(hext,h)=Wσ1∪σ2(h) (15) in out Z where the integration is over the boundary ψout(hl′)= dglW(hl′,gl)ψin(gl) (13) holonomies hext. This follows easily from the fol- Z lowing simple property of the face amplitudes so we recoverthe more standardinterpretationof W as σ a transition amplitude. dh δ(h...h )δ(h′...h )=δ(h...h′...), (16) ext ext ext Z namely the two external face amplitudes collapse III. LOCAL SPIN FOAMS into a single internal face amplitude, after integration. If we change the face amplitude in (14), Inthissectionweanalyzethepropertiesofalocalspin the composition property does not hold anymore foam of the form (8), and argue that the requirement of [15]. In particular, this fixes the face amplitude cylindrical consistency can fix the arbitrary normaliza- of EPRL model to be dj, and not the SO(4) one tion of the vertex amplitude. At the conceptual level, dj+dj−, or even worse the SL(2, ) dimension in C cylindrical consistency is an important step for the in- the Lorentzian case, which is infinite. terpretation of a spin foam as a Feynman path-integral 3. Cilindrical consistency (face reversal). As a first over histories of the connection A (~x,t). The type of µ step, we require face reversalinvarianceofthe spin consistencies we discuss in this section are ’canonical’, foammodel(Fig.2). Underaflipinthe orientation in the following sense; they are a sort of time-evolution of the well-known cylindrical consistency for LQG op- erators (e.g. the volume operator [24]): link reversal, link splitting and link erasing are lifted to the spin foam → framework. 1. Locality. As we already mentioned, the spin foam partitionfunction(8)islocal. Actually,formula(8) FIG. 2. Theface reversal. defines a notion of locality. The standard locality in terms of colorings of the 2-complex is recovered of a face f, the face amplitude has the transforma- using the Peter-Weyl decomposition: tion rule Zσ = (2jf +1) Ae(jf,ie) Av(jf,ie) (14) δ( hvf)→δ( h−vf1) (17) jXfieYf Ye Yv vY∈f vY∈f where the two products have the same ordering. Here the spins j label the spin foam faces. No- f It follows that the partition function is invariant if ticethatthefaceamplitude isthe dimensionofthe weassumethefollowingtransformationruleforthe SU(2)representationd =2j+1,foranyspinfoam j vertexamplitudesofverticesbelongingtothesame model of the form (8). face f: The edge and vertex amplitudes A , A are lo- e v cal: A depends only on the intertwiner labeling W (h ,...)→W (h−1,...) (18) e v vf v vf the edge and the spins of the faces meeting at the edge,Avdependsonlyonthethequantumnumbers 4. Cilindricalconsistency(facesplitting). Consideran of edges and faces meeting at the vertex. Finally, holonomyh associatedtoalinkinthelocalbound- l the edge amplitude can be absorbed in a redefini- ary of a vertex, or in the global boundary of a 2- tion of the vertex amplitude. As a consequence, complex. Ifwe splitthe link intwopartsl′∪l′′ =l possible normalizationambiguities are absorbedin and associate to each part an holonomy, we would the vertex amplitude. like to regard the product hl′hl′′ as equivalent to the single holonomyh (Fig.3). This picture comes l 2. Compositionproperty(facecutting). Thespinfoam fromthecompositionlawofholonomiesinthecon- amplitude W (h ) satisfies a composition prop- σ vf nection representation of Loop Quantum Gravity, erty under face cutting, emphasized in [15]. Sup- where the holonomy of a connection A satisfies µ pose we cut the 2-complex in two pieces σ and 1 σ2 (the cut is realized by intersecting with a 3- Hl=l′∪l′′(A)=Hl′(A)Hl′′(A) (19) surface), in such a way that each face which has beencut turns into twoexternalfaces ofσ and σ if the path l′′ starts where the path l′ ends. 1 2 4 FIG. 3. The composition low of holonomies. Hl=l′∪l′′(A) = Hl′(A)Hl′′(A). The fact that in the spin foam picture the two ob- jects are different has important consequences for thedynamics. ConsideranN-valentspinfoamver- FIG. 5. On the left: the vertex-to-vertex face splitting. A tex and its local boundary whose links are labeled face f issplit in twofaces f′,f′′ byanewdummy(2-valent) by vf. The vertex amplitude is W (h ). Let v vf edge. Ontheright: thefacesplittingendingontheboundary. us split one of the boundary links in two pieces, with associated holonomies hvf′ and hvf′′. Since we added a dummy (i.e. 2-valent) node, we have kinds of splitting). The cylindrical requirement split the face f in two parts (Fig.4) by adding a (20) is trivially met for the Ponzano-Regge model, while in the case of EPRL it fixes the residual am- biguity in the definition of vertex amplitude. This is done in the following two subsections. 5. Cylindrical consistency (face erasing) As a last requirement,wedemandthespinfoamamplitudeW σ for a 2-complex σ bounded by the graph Γ to satisfy: (W |ψ i=(W |ψ i (22) σ Γ,j σ˜ Γ˜,j where σ˜ is obtained from σ by erasing some of the externalfaces,thenewboundaryΓ˜ isthesubgraph of Γ obtained by erasing the corresponding links, andthespins ofthe spin-networkψ labelingΓ− Γ,j FIG.4. Thefacesplitting. Afacef issplitintwofacesf′,f′′ Γ˜ have been set to zero (Fig.6). The cylindrical byanewdummy(2-valent)edge. Theholonomylabelingthe original face is split in two independentholonomies h′, h′′. dummy (2-valent) edge that comes out from the vertex. Now we have actually increased the va- lenceofthevertexby1. Thenewvertexamplitude readsWv(hvf′,hvf′′,...). Ingeneralthetwovertex amplitudes (the N and the (N +1)-valent) can be different, so we require: Wv(hvf′,hvf′′,...)=Wv(hvf′hvf′′,...). (20) This requirement provides a cylindrical consistency for spin foam theory: if we use redundant FIG. 6. The face erasing. holonomies in the description of the kinematical space, the quantum amplitudes should not depend consistency requirement (22) can be translated in on this choice. Physics must be independent of it. a requirement on the colorings of the 2-complex. Itisimportanttonoticethatthesplittingedgecan In fact, (22) implies the following thing. Consider end in another vertex or in the (global) boundary, the Peter-Weyl expansion of Z over colorings of σ ifthereisone. Itisstraightforwardtoshowthatfor the 2-complex, as in (14), and consider a generic a generalmodel (8), equation(20) implies that the term of this sum such that some of the spins van- partition function is invariant under this splitting. ish. Thenthistermwillcoincidewithananalogous More precisely, Wσ is invariant under a vertex-to- term in the partition function Zσ˜, where the sub- vertex splitting, and satisfies foam σ˜ ⊂ σ is derived from σ by erasing all the faces labeled by vanishing spins and glueing faces Wσ˜(hl′,hl′′,...)=Wσ(hl′hl′′,...) (21) along trivial (2-valent) edges. for a splitting edge that ends on the boundary Consistencyunderfaceerasingisstrictlyrelatedto (Fig.5 gives a pictorial representation of the two consistency under face splitting. Indeed even if we 5 start with a n-valent edge (n > 2) we can erase so the two face factors (25) collapse into a single face n − 2 faces and end up with a trivial edge (see factor. This shows that the requirement (20) holds. Fig.6). Thissituationisclearlyequivalenttoaface As a last step, we prove cylindrical consistency under splitting, so property (20) ensure that the trivial faceerasing. Inparticular,weprove(22)forasinglever- edge can be safely removed. tex, the generalizationbeing straightforward. A normal- ized spin-network4 function ψ on the local boundary Γj graph Γ has the property of being equal to the spin- Ponzano-Regge model network ψ on a sub-graph Γ˜ ⊂ Γ when the spins of Γ˜j Γ − Γ˜ are set to zero. The cylindrical consistency of The Ponzano-Regge model for three-dimensional Eu- the spin foam model requires the corresponding vertex clideanquantumgravity[35–37]isdefinedbythegeneral amplitudes to be equal. This is true, due to the follow- formula (8) with the following vertex amplitude ing simple observation: when evaluating (W |ψ i, some v j ofthe face factorsinthe vertexamplitude areintegrated WPR(h )= dg δ(h , g−1 g ) (23) againsttrivial(spinzero)Wignermatrices,andgivetriv- v vf ve vf t(vf) s(vf) Z vf ial contribution to the evaluation: Y where the integration is over gauge group variables gve dh δ(h , g−1 g )Djvf=0(h )=1. (27) (one per each edge e coming out fromthe vertex v), and vf vf t(vf) s(vf) vf Z s, t specify if the edge is ’source’ or ’target’ according to the orientation of faces. Clearly, the interpretation of variables is slightly different from the four-dimensional EPRL model models. Here the Ponzano-Regge model is viewed as a covariant path-integral formulation of 2+1 Loop Quan- The EPRL model [5] is a candidate model for quantum Gravity [38]. Therefore the SU(2)variables h are tum general relativity in four dimensions. Here we are vf interpreted as holonomies of the SU(2) spin-connection interestedin amanifestly SU(2)-invariantholonomyfor- ω . Moreover,local boundaries of spin foam vertices are mulationofthis model,whichappearedinthelocalform µ defined by intersection with a small 3-ball. In the fol- (8) inreference [1]. The model is specified by the follow- lowingwereviewwell-knowninvariancepropertiesofthe ing vertex amplitude: Ponzano-Reggemodel in the language of the new holonomy formulation for local spin foams. The proofs will WEPRL(h )= dG δ (h , G−1 G ) (28) v vf ve γ vf t(vf) s(vf) be similar in the four-dimensional model, therefore we Z vf consider 2+1 gravity as a warm-up. Y As a first step, we show that the Ponzano-Regge par- where the integral is over the SO(4) (SL(2, ) in C tition function is left invariant under face reversal. To the Lorentzian theory) gauge group variables and the achieve this, it suffices to prove the transformation rule source/target group elements are defined according to of vertex amplitude (18). So consider a vertex v belong- the orientation of faces. We have introduced the following to the face f, and consider the face factor (δ distri- ing distribution: bution) in the vertex amplitude associated to this face. Under a flip in the orientation of f, the source and tar- j SU(2)dcNjχj(ch)χ(j+,j−)(cG) (e) get gauge variables must be interchanged. But from the δγ(h,G)= simple identity δ(g)=δ(g−1) we have PjRSU(2)dcNjχj(ch)χ(n,ρ)(cG) (l) (29) δ(hvf, gs−(1vf)gt(vf))=δ(h−vf1, gt−(v1f)gs(vf)) (24) P R where the two lines refer to the Euclidean (e) and hence we prove (18). Lorentzian (l) versions of the model respectively, and The second cylindrical consistency we consider is face γ 6=1istheBarbero-Immirziparameter. Theirreducible splitting invariance, in particular we prove (20). Con- representations labeling the characters χ of SO(4) (or sider a vertex v of a face f and let us split f in two SL(2, )) in (29) satisfy the simplicity constraint: new faces f′ and f′′. To simplify the notation, call C gs(vf′) =gt(vf′′) =g the gaugeassociatedto the 2-valent j+ = 1+γj, j− = |1−γ|j (e) pedlygehw′ haenrdehf′′′arnedspfe′c′tmiveeelyt,. gCt(avlfl′a)ls=ohtvafn′dangds(hvfv′f′)′′ =sims-. (n=2j,2 ρ=2γj 2 (l) (30) Then the new vertex will contain the two face factors The simple form (28, 29) of the vertex is derived from δ(h′,t−1g)δ(h′′,g−1s). (25) the one of reference [1]. Basically, the embedding maps Performing the integral in g, we have dgδ(h′,t−1g)δ(h′′,g−1s)=δ(h′h′′,t−1s) (26) 4 We have omitted the intertwiner labels in the notation, since theyplaynoroleinouranalysis. Z 6 SU(2) → SO(4) (or SU(2) → SL(2, )) are replaced Therefore,imposingthefacesplittinginvariance,wehave C by an auxiliary integration over the c variable (SU(2)- N = 1 and prove the full cylindrical consistency of j=0 averaging),whichisresponsibleforthecouplingbetween EPRL model with vertex normalization (35). the little group SU(2) with the 4-dimensional gauge Before concluding, we observe that cylindrical consis- group. The positive constant N in (29) parametrizes tency of EPRL model allows to extend to the full Loop j thenormalizationofthevertex(andedge)amplitude. In Quantum Gravity kinematical space (3) the embedding the Euclidean model, the vertex normalization N can mapf ofSU(2)spin-networksinto(simple)SO(4)spin- j γ be fixed from the requirement of cylindrical consistency. networks. This map is the key ingredient for the defini- This could be done in principle also in the Lorentzian tionofthemodel[5]. Withournotations,theembedding theory, but we need to handle potential divergencies re- map sulting from the non-compactness of SL(2, ), and will be discussed elsewhere. C fΓ :H →HSO(4), (38) γ Γ Γ Following the three steps as for the Ponzano-Regge model,wefirststudytheeffectofaflipintheorientation with HSO(4) =L2(SO(4)L/∼), is defined as Γ of a face. The identity we have to prove, analogous to (24) is the following: (fΓψ)(H )= dG dh δ (h ,G−1H G )ψ(h ). γ l n l γ l t(l) l s(l) l δγ(hvf, G−s(1vf)Gt(vf))=δγ(h−vf1, G−t(v1f)Gs(vf)). (31) Z Yl (39) This is easily done using the formalism (29): To our knowledge, this map has been defined only for δγ(h,G−1)= an arbitrary, but fixed, graph. In order to define it con- sistently on all graphs, we need to check its cylindrical = dcN χj(cg)χj+(c(g+)−1)χj−(c(g−)−1)= j properties. Basically, the action on the space H have Γ Xj Z to be consistent with the action on HΓ˜, where Γ is a = dcN χj(c−1g−1)χj+(c−1g+)χj−(c−1g−)= sub-graph Γ˜ ⊂ Γ. A sub-graph can be obtained from a j larger graph with a finite number of elementary opera- j Z X tions, which constist in flipping the orientation of links, =δ (h−1,G) (32) γ splitting a link, or erasing links. It is not hard to under- where we used the simple property δ(g) = δ(g−1) and stand that the three cylindrical requirements discussed inthis paper imply that (atleastwith the normalization the cyclic invariance of traces. (35)) the embedding map f extends to the full Hilbert Oursecondgoalistoprovethefacesplittingrule(20). γ space After a face splitting (as in (25)), one face factor in the corresponding vertex splits into two parts f :L2(A,dµ )→L2(ASO(4),dµSO(4)) (40) γ AL AL δ (h′,T−1G)δ (h′′,G−1S). (33) γ γ where Next,integrationofthepreviousformulaoverG∈SO(4) gives L2(ASO(4),dµSO(4))≃L2(A ×A ,dµ+ dµ− ) (41) AL + − AL AL N2 dc j χj(ch′h′′)χ(j+,j−)(cT−1S). (34) is built up from two copies (the ’left’ one denoted by +, j ZSU(2) djdj+dj− the ’right’ one by a −) of the kinematical state space of X loop gravity. Clearly, a basis for the image of f is given γ Therefore, choosing the normalization5 by simple spin-networks(see [40] for a recentreview and analysis). Nj =djdj+dj−, (35) expression (33) collapses to the single δ distribution γ IV. TAKING THE CONTINUUM LIMIT δ (h′h′′,T−1S). (36) γ Consider the full partition function As a final step, we provethe cylindricalconsistency of EPRL model under face erasing. Here the analogous of Z(h )= W (h ) (42) equation (27) is, after a simple calculation, l σ l σ X dhvfδγ(hvf, G−t(v1f)Gs(vf))Djvf=0(h)=Nj=0. (37) withsumover2-complexeswhichareboundedbyagraph Z Γ. Thephysicalintuitionsuggeststhatagiven2-complex actsasaregulatinglattice. Infact,spinfoammodelsare generallydefinedasaquantizationofatruncationofclas- 5 Thesamenormalizationisconsideredin[39],fortheGroupField sical General Relativity to a finite number of degrees of Theoryformulation. freedom, e.g. by first discretizing it over a piecewise flat 7 simplicial manifold, then quantizing it [5, 41–43]. Differ- trivialedge,wehavetoensurethattheyhavecompatible entlyfromalatticegaugetheory,expression(42)getsrid orientations (they must induce opposite orientations on oftheregulatordependencebysummingoverallpossible the common edge). 2-complexes. Thissum,includingpossiblesymmetryfac- Hencethesub-foamsρareobtainedviaafinitenumber tors,canbe generatedby a GroupField Theory[29,44]. of elementary operations. These are the face orientation However, a closer look at formula (42) suggests that reversal, the face splitting, and the face erasing. So if there is a large amount of redundancy in this sum. Let the cylindrical requirements discussed in this paper are us expand (42) in a sum over colored 2-complexes: satisfied,asfortheEPRLspinfoammodel,thepartition functions (45) and (46) are the same. Z(h )= W (h ). (43) More difficult is to show that a sum like (46) corre- l σ,j,i l sponds to a sum over equivalence classes of 2-complexes σ j,i XX similarlytoGroupFieldTheories(see[49]foracomplete A term W where some of the j’s are vanishing can analysis in 2+1 gravity). To this regard a possible diffi- σ,j,i be naturally interpreted as the amplitude for a sub 2- culty could come from trivial vertices, that is vertices of complex σ˜ obtained by erasing the corresponding faces. valence two, bounding at least three faces. Preliminary Sincewearesummingoverthesub2-complexesthisterm investigations indicate that, in the case of EPRL model is counted at least twice. The degeneracy is clearly pro- (atleast its SU(2)version)the trivialvertices cannotbe portional to the number of vanishing spins in the most erasedwithoutaffectingthepartitionfunction. We leave refined2-complexσ. Toavoidthisovercounting,thesum this as an open problem. over colorings (43) should be restricted to nonvanishing spins. A different way for recovering the infinite number of V. CONCLUSIONS degrees of freedom of General Relativity is to consider the partition function for a very fine 2-complexand take In this paper a new holonomy formulation for spin the limit of infinite refinement: foams was shown to be an appropriate tool to deal with general features of spin foam models. Within the Z(hl)= lim Wσ(hl). (44) holonomy representation, we introduced cylindrical con- σ→σ∞ sistency for spin foams as a natural step towards a con- This approachis muchmoresimilarto lattice gaugethe- tact with the (Ashtekar) connection representation of ories or dynamical triangulations [45, 46]. Despite the canonical Loop Quantum Gravity. We discussed an im- two partition functions (42) and (44) look very differ- portant consequence of cylindrical consistency: it fixes ent, they are likely to be related, or even identical. This the arbitrary normalization of the vertex amplitude of observation was pointed out recently by Rovelli [47, 48]. EPRL model. Furthermore, it provides key insights on thecontinuumlimit. Theextensionofouranalysistothe An insight on the the relation between the refinement Lorentzian signature is in progress. limit (44) and a sum over 2-complexes comes from the consistency requirements discussed in this paper. Let us put a cut-off on the theory, namely consider a very fine ACKNOWLEDGMENTS 2-complex σ. The quantity Z (h )=W (h ) (45) Theideaofimplementingcylindricalconsistencyinthe (σ) l σ l ’new’spinfoammodelscamesometimeagoinMarseille. can be interpreted as a cut-off ’a la lattice gauge theory’ At that time we lacked a simple formalism (the local of the partition function. An alternative definition of holonomy formulation) to handle this problem. Our in- partition function as a sum over 2-complexes with the terest was renewed by a stimulating exchange of ideas cut-off induced by the choice of σ is with Jerzy Lewandowski, who gave a series of three lectures “Canonical Loop Quantum Gravity and Spin Z′ (h )= W (h ) (46) Foams” in Penn State, during August 2010. A warm (σ) l ρ,j,i l thank goes to Carlo Rovelli, Matteo Smerlak and An- ρ⊂σj6=0,i X X tonino Marcianòfor useful discussions and comments on where the sum is over sub 2-complexes and over non- the manuscript. 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