PoS(FFP14)177 http://pos.sissa.it/ † ∗ [email protected] Speaker. based on work in collaboration with Daniele Oriti and James Ryan [1] Group field theories are a generalization ofreformulation matrix models of which loop provide both quantum a secondWhile gravity quantized states as in well canonical as loopon generating quantum graphs functions gravity, with in for vertices the spin of traditionalsimplicial foam arbitrary continuum setting models. valence, setting, such group that are field states based the have theories question support have whether only been group on defined field graphs so theorygravity. of can far In fixed indeed in this valency. cover contribution a the This based whole has onsatisfy state led [1] this space I to objective: of present i) loop two a quantum new straightforward, classesfor but of rather each group formal valency field generalization and theories to ii) which multiplespace a fields, through simplicial one a group dual field weighting, theoryI a which will technique effectively covers further common the in discuss larger matrix inthe state and some group tensor detail field models. the theory combinatorial To partition this structurethe end links function. of between the the The mentioned complexes quantum new generated gravitygroup group by approaches field field but, theories, broadening theories they the might do theory also space not prove of useful only in strengthen the investigation of renormalizability. ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Frontiers of Fundamental Physics 14 -15-18 FFP14, July 2014 Aix Marseille University (AMU) Saint-Charles Campus, Marseille Johannes Thürigen Max-Planck Institute for Gravitational Physics (Albert-EinsteinE-mail: Institute), Potsdam, Germany Group field theories generating polyhedral complexes PoS(FFP14)177 (1.1) (1.2) pairing the . for finite sets k 2 ) | i j J G -regular graphs, | g k ( Johannes Thürigen D G φ i J -dimensional spatial ∈ ) j ∏ 1
i − J } D j ( g { i is a function on V ) are functions on ] k i g 2 λ d g [ , , k ] Z 1 φ i [ g λ S ; I − ∈ e i ∑ ... ; -simplicial building blocks. Boundary states are φ 21 ) + D D 2 g 2 , and the GFT is defined by a partition function g Z ( with couplings 11 G i φ g = ( ) V 2 K g GFT , Z 1 g ) = ( 2 g K , ) 1 1 -manifold such that boundary states are based on g g ( D ( K φ ] g d [ copies of a Lie group Z -valent. But SF models can also be defined on other cellular decompositions N 1 2 D ∈ k -regular graphs. ] = D φ of [ S R denotes a (formal) measure on the space of group fields and the action is of the form φ → D k × Here I would like to address this difference in the combinatorial structure and show that there Explicating in detail the combinatorial structure of standard GFT in section 1 will lay the The common notion of GFT is that of a quantum field theory on group manifolds with a Loop quantum gravity (LQG) [2, 3] , spin foam (SF) models [4] and group field theory G : where are also classes of GFTs combinatorially fullyis compatible to with LQG find and GFTs SF which models. generate The complexes challenge that allow for boundary graphsground of to arbitrary present valencies. two strategies forto such a an extension: set i) of Invarious section fields valency. Though 2, with rather I formal, various will such extend numbers multi-fieldextension the GFTs of of field are spin the space group foams direct counterpart [8]. arguments, to ii)in thus the In KKL terms creating section of 3, boundary a I dual-weighting vertices will mechanism.to therefore of distinguish An present real a extra from more label efficient virtual on generalization cells aof and standard the the virtual simplicial dual structures GFT resulting weighting field in implements allows effectivefield dynamically amplitudes GFT. a which contraction are exactly the same as in the multi- 1. The combinatorial structure of group field theory particular kind of non-local interactionφ vertices [5–7]. More precisely, a group field is a function thus based on manifold and related by cylindrical consistency.lations Spin foam of models a are given topological originally basedi.e. on triangu- all vertices are and the amplitudes usually dependgeneral only 2-complexes on with their arbitrary boundary 2-skeleton graphs whichof [8]. has complex-dependent GFTs SF lead can models, to be providing extensions understood acomplexes to as sum of more a of various completion topologies, SF generated amplitudes from over a class of combinatorial arguments; the vertex (interaction) kernels (GFT) [5–7] are threethat closely the related degrees of approaches freedommodels to and are a GFT algebraic can quantum data be theorythe based understood as approaches on of providing consists certain gravity. dynamics in combinatorial for thetraditional structures LQG. They A role and LQG share major and SF are difference structure arbitrary between of closed these graphs combinatorics. embedded in The a quantum smooth states of GFTs generating polyhedral complexes Motivation Therein, the kinetic kernel PoS(FFP14)177 2 1 ) 1 ¯ v 2 − kb g p .A ( g j (14) φ ¯ (1.4) (1.3) v ja ) → p 2 (12) g 1 g 1 , ¯ v 1 (see fig. g p ( : in a unique b K ) γ v 1 in 4 do not impose g ( ) 2 i ˆ φ v Johannes Thürigen ] ¯ V v field arguments is g , a perturbative ex- ( and denoted d ] | 3 . [ i R φ J -dimensional extension [ . 2 1 i k · | O − V k (34) e patch i , c ) have the structure of two- (23) Γ # ) j 1.4 g ( induces a bijection φ for every edge i J ) , coined a K ˆ ∈ v j (spin foam) atom ∏
¯ jk v ˆ } v v (14)  1 ( i J − kb (12) ) by identifying for each argument } g 1 with a single internal vertex j ,..., . One may further understand the graph as ja g commonly have kernels of the same type. a 1 1.3 i j j g { can be represented by a graph consisting of a ] v { v leads to a series of Gaussian integrals evaluated  ) is called a φ i and interaction kernels I [ (13) jk } a 1 V V 4 3 in the series. These weights are constructed by i O g ] − 2 λ but that the totality of the g = Γ { K (24) d k
[ × J 3 = ,..., } Z G 1 j j i P and adding a face g λ g { ( b (34) " ! φ i 1 V (23) c I . Observables , univalent vertices ˆ ) = ∈ } j i ) ∏ k k I I g } } ( into a bivalent vertex ˆ i i c ∑ is the weight of φ { λ ,..., kb ] { ˆ ) 1 v ; I φ 1 , [ } Γ i ja ( O ∈ { -regular graph, its corresponding spin foam atom, and a (12) λ v to all vertices in of a two-dimensional complex A k (14) { b φ ) v ; , a Γ a D Γ ∂ connected to ( 2 ( 1 represents then the convolutions in ( j (24) Z A = v 4 are the combinatorial factors related to the automorphism group of the Feynman b and b sym ) 1 J GFT Γ and Γ A bisected Z ( ∑ ∈ (23) (34) Γ k (13) = = , j The essential point for further clarification of the structure of these diagrams is to notice that In this way, the GFT Feynman diagrams in the perturbative sum ( For the evaluation of expectation values of such quantum observables GFT ), each group field term i and have a combinatorially non-local structure. This means that the kernels O -valent vertex ¯ i 1.2 3 h convolving (in group space) propagators diagram the specific non-locality of each vertex( is captured by a boundary graph. In the interaction term in where sym k bisected graph for illustration). Such a one-vertex two-complex complexes because Wick contractions effectpicture, bondings of the such pairing atoms of along group patches. arguments in In the the graph kinetic kernel the univalent vertices ˆ the boundary way, i.e. connecting Figure 1: GFTs generating polyhedral complexes which, in this example, is the dual of the tetrahedron. J coincidence of points in the group space partitioned into pairs convoluted by the kernel, where pansion with respect to the coupling constants through Wick contraction which are catalogued by Feynman diagrams PoS(FFP14)177 ). 2 . Since ev- . L , k . γ M Johannes Thürigen ⊂ . Let the set of all S . The combinatorial , B ) k 2 g -simplex, it is natural to M ( k φ and ) (spin foam) molecule 1 g ( φ is the bisected version of the complete S , k b ) whose vertex atoms have all an extension γ ] ), generate a peculiar kind of complexes de- 1.2 -regular, loopless graphs are possible for the k 1.2 are possible since they are defined in terms of the of 4 ) is then a collection of spin foam atoms, one for L B , k 1.4 ⊂ B . L , L , k k of two atoms along an identification of patches B -dimensional discrete spacetimes. But note that in general M γ ] D as GFT defined by the single interaction vertex based on the combina- S ) and the resulting molecules are a subset , 1 D M γ is denoted ) since these have to be constructed from the single patch corresponding arguments. Accordingly, the class of molecules which can be obtained L , The bonding k k 1.2 B simplicial 1 vertices (fig. , only graphs in the subset group arguments. To match the larger space of states of LQG based on the whole of Figure 2: -simplex. The corresponding vertex graph + B k k k and understand , new classes of theories have to be introduced. This is the topic of the next two sections. D with B = φ Even though the GFTs described provide already a broad class of theories, in any case the The most straightforward way to generalize GFT to include observables and states on arbitrary Since a patch corresponds to a group field in the GFT, one has to extend the field space to a Standard GFTs, as described by the action ( A special case is -polytopes due to their effective construction as triangulations [12]. -regular loopless boundary graphs k k D combinatorial boundary structure is still simplicial.on In particular, only observables and states based 2. Multi-field group field theory graphs is to extend thestrategy, I field present space it here [13]. to point Though out from the a possibility of QFT such pointgroup a theory of field and view for discuss a each its rather combinatorics. kind unattractive of patch needed to construct all boundary graphs to field graphs between the patches representing thestructure two of group a field term terms ineach the vertex perturbative kernel, sum quotiented ( byBecause of a this set construction of such a bonding two-complex maps, will one be called for a eachtermined Wick by contraction the (fig. regular-graph structuremultigraphs of the interaction vertices. In the setto of the closed group bisected field with from atoms based on GFTs generating polyhedral complexes interaction kernels in ( torics of the graph with ery atom in such aset simplicial molecule has a canonicalsimplicial extension molecules to are the not abstractWell-behaved simplicial simplicial complexes complexes preventing due such toversion degeneracies various of can kinds simplicial GFT, be of distinguished loops obtained byone [9, in even a 10]. arrives a colouring at a [9, modified 11]. GFT of Integrating the out general all action type but ( one colour PoS(FFP14)177 . v p MF . ∈ is in B (2.1) ) ). To ¯ ) v ˆ v 4 g B ¯ v ( ( p ∈ -polytope. φ (fig. b D V is necessary. L ∈ , ∏ ¯ v k P ) } e B p V Johannes Thürigen φ } ¯ v { g { = ( b MF V to the case where both are ] Φ g , one for each edge kb the set of all but the bisecting ˆ v ˆ d v ¯ v [ , g V ja Z has a natural extension to a map v b L λ , , it generates the amplitudes of the and k p MF π MF K B ∑ ∈ P b , the molecules where bisecting vertices ˆ L , k ) + 2 ¯ v f M . g 3 ( p ⊂ 5 even the image consists of all graphs with vertices φ k ) NB - 2 ¯ defined as contraction of all virtual edges in a labelled v L , g k 3 , B 1 π ¯ v f M g → resulting from bonding atoms induced by ( p L L , , k , the action is then with such an edge labelling are obtained from labelled patches K k e ) B B e 1 B f M ¯ : v ∈ odd (while for g L , ( b k k p π -complex for each molecule in the expansion, though in analogy to the φ , which are functions of group elements ] D g P d [ ⊂ Z the set of all graphs constructed from , and, with appropriate kinetic kernels MF 2 1 . A multi-field (MF) group field theory is then determined by a set of group fields B M P MF P , A vertex in an arbitrary graph obtained from a labelled 3-valent graph via contraction. ⊂ P ∑ MF ∈ p P MF } p B ] = φ { MF on the labelled molecules = Figure 3: For a construction based on those GFTs which are currently analytically tractable it is nec- The crucial idea to create arbitrary boundary graphs in a more efficient way is to distinguish In terms of these contractions, any spin foam molecule can be obtained from a molecule con- To cover the whole LQG state space, the infinite number of fields Φ [ L , by restricting the allowed identifications of univalent vertices ˆ S MF k -regular, loopless graph for e of even valency). An example is illustrated in fig. essary to rearrange theof molecules the discussed preceding in section. this Indynamically section using the in a following dual terms section weighting I of in show the the how GFT simplicial action. this molecules can be done3. and implemented Dually-weighted group field theory between virtual and real edges andvirtual obtain edges. arbitrary Boundary graphs graphs from regular ones by contraction of the KKL extension [8] of the EPRL [14] and similar spin foam models [15, 16] (see [1] for details). adjacent to an edge with the same label. Then onek can easily check [1] that any graph P Such a theory will likelyneeded remain for on non-trivial a dynamics for formal eachfoam level. field. molecules In Still, particular, this infinitely is many the interactions theory are generating all possible spin the image of the projection avoid branching one further restricts to structed from labelled regular graphs. The contraction map simplicial case these complexes are abstract polyhedral complexes only in a generalized sense [10]. Like in corresponding spinmolecules foam generated models in [8], the there perturbativefor sum is interactions and for no observables this direct in action.In spacetime which this each interpretation One way atom for might one can obtains choose the be a a extended to subset the of dual of a Π Φ GFTs generating polyhedral complexes such patches be Denoting vertices in a boundary graph PoS(FFP14)177 . } M M A (3.1) are a { ) ) = ¯ v 2 , k m NB is enough - δ ; ¯ L v , NB g k = - ( [1]. S ,  φ f M k k ( B ) L Johannes Thürigen , f M M k ∈ 0 the corresponding A ∈ Π b ( = e  m ˆ v ¯ v tr ) of the theory is a sum ∞ m 1.4 → M . If } , or maybe more meaningful on M M for any graph b which distinguishes exactly whether ,..., follows then that arguments of the field where lim = 1 j K L k 2 , , k 0 m m j π 1 ∂ incident to two virtual edges in a molecule. m ∈ { v ˆ v ! ¯ v 1 m − M 0 6 l A Π amplitudes only depend on the contracted molecules 1 0 limit [11,12,17] renormalizability properties [18,19] M

N 1 k = with boundary j ∏ ) 2 M limit the perturbative expansion ( and an integer m ∈ ˆ , v M ¯ 1 v ) , one would expect a genuine polyhedral version of the constraints g e m m ; NB ( - 2 S . With an appropriate kernel S , , g k k , NB - 1 Π S g f M , ( k = K f M m ) = 2 m and match exactly the ones of the multi-field GFT [1], and thus the amplitudes of Contraction move with respect to a vertex ˆ , 1 ) m NB ; - is real, otherwise virtual. A kinetic kernel with a family of dual-weighting matrices 2 S , ) g k ˆ v , ¯ v 1 f M ( g In this contribution I have addressed the goal of a generalization of GFT to be compatible with Improving the relations between GFT, SF models and LQG and broadening the GFT theory Based on these combinatorics facts it is possible to define a labelled simplicial GFT which ( Figure 4: ( S , k K guarantees then that in the large- only over molecules an edge is real or virtual, the effective large- the simplicial molecules to be more appropriateGFT on perspective, these it will molecules, be which mostclasses of interesting has theories, to not in investigate particular been the their field-theoreticand addressed large- properties phase so of structure the far. [17, 20]. new ii) From the LQG in three steps. First, Iperturbative have GFT clarified using the the combinatorial notion structure ofspacetime underlying spin interpretation. the foam amplitudes Then atoms of I and have molecules laidmulti-field theory out and which the discussed can details their cover of on possible a thehow formal straightforward to level generalization obtain arbitrary to such the molecules. a same Finally dynamics I using have a shown dual-weighting mechanism on aspace, simplicial these GFT. results introduceare two based obvious on research a questions. simplicialbe version i) implemented of in While the simplicity all presented constraints gravitational GFTs and models either the on resulting the molecules edge amplitudes can the KKL extension [8] of spin foam models [14–16]. Conclusions effectively covers observables and states on arbitraryimplement graphs, the using contractions a dual-weighting dynamically. mechanism To to this end, the to obtain a molecule Π are bivalent also after bonding. 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