PoS(CNCFG2010)004 LPT-20XX-xx 1 Abstract March 11, 2011 Vincent Rivasseau We review some aspects of non commutative quantum field theory Universit´eParis XI, F-91405 Orsay Cedex, France and group fieldatic theory, study in of particular the recenttheory. scaling and progress We renormalization thank on properties G. theWorkshop of Zoupanos on group system- and Noncommutative field the Field Theory organizersaging and of us Gravity to the for write Corfu encour- this 2010 review. Laboratoire de Physique Th´eorique,CNRS UMR 8627, At the same time Wilson and followers understood the true meaning Towards Renormalizing Group Field Theory of renormalization. Theyinfinities under explained the rug it but no simply as longer the natural as change of a the mysterious laws of hiding physics of 1 Introduction Renormalization has been the soulfrom and the driving force early of computationslous quantum of magnetic field theory, moment the in Lamb70’s, quantum shift electrodynamics when and to renormalization of thekey golden of the to era electron the of their anoma- non-Abelian the interactions. successful gauge adoption theories as became theories the of the electroweak and strong MSC: 81T08, Pacs numbers:Key 11.10.Cd, words: 11.10.Ef Group field theory, renormalization expansion. PoS(CNCFG2010)004 . 1 theory on the 4 ? 4 φ 2 This proposal was presented in some detail eg at the Nottingham conference in July In spite of these achievements, many physicists still dream of a funda- It is also often suggested that absence of infinities would mean that there But we might even prefer if quantum gravity after all was just like the The main encouragement which lead to such a program came from the 1 2008, and at the Beijing LQG 09 conference. non commutative Moyal space. For abecause while of it was the dubbedturned non so-called renormalizable ultraviolet-infrared out mixing to problem. become Ultimately renormalizable it and even asymptotically safe under mental theory of physicsthe start. that It would would besince have to free gravity include from is gravity, themainstream any not “rebellious proposal infinities interaction”. perturbatively is right But to renormalizable from which include in includes it new into the aand unobserved naive framework which symmetries sense, such would (in as make the every string particular computation theory supersymmetry) finite. is a fundamental scale,nothing, eg not the justnothing Planck would because scale, exist. of beyond In whichgroup this temporary view would we physics experimental not could and exist limits, the probe beyond flow but that of the scale. because renormalization rest of the standard model.divergences of In a particular we renormalizabledynamics. would type like These like if divergences it could electroweakgroup then has and drive flow perturbative chromo an upstream quantum interesting to renormalization thethe Planck QCD scale flow (or “before”upstream drives to the the the big hadronic asymptotically bang), scale. justenlarged free These like features and physics could reinterpreted appear of quantum withinsuch quarks a field suitably theory a and formalism. formalism, gluons In weloop order should quantum to adopt gravity find a communityprinciple the guiding is feeling principle. that background Weinvention the independence, share of most with the general appealing the promising one such relativity quantum principle in field that the theoryfield theory first lead formalism (GFT). All based place to these on considerationsstudy [1]. the lead of these us the to criteria A propose scaling is the and most systematic group renormalization natural properties and of“happy group end” field story theory of a simpler quantum field theory, the under change of the observation scale.progress in This other brilliant domains step immediately ofmisleading) lead physics. name to It of became the known renormalization under group. the (somewhat PoS(CNCFG2010)004 3 Power counting. Locality principle, Scale decomposition, The theory is renormalizable when these three ingredients nicely fit to- So right at the start lies the scale decomposition. It is perhaps both the We will subsequently review the modest but already promising results • • • gether. Roughly speakingtheory the to key particular recipeexternal subgraphs is scales. to whose These relate internal graphs theing scales satisfy their divergences are a local of locality higher parts the principle.of keep than the the If their form model, by of then power theof count- the terms the model already theory is present does renormalizable. inconstants not the move. It change action means with that the the observation structure scale, onlymost the fundamental coupling and the mostcan technical indeed ingredient. be Atimately scale realized decomposition be in equivalent. manythrough For different block instance technical spinning a ways, ormentum scale which through cutoffs. decomposition should the can use ul- Atspace be of and first created many distance. sight different Small scale shapesshort scales seems distances; of correspond mo- large forever to the scales linkeddistances. ultraviolet correspond to The regime to names and the of the to and notion infrared infrared and electrodynamics. of regime ultraviolet come and Smallenergetic from to Fourier probes details analysis, large that are can indeedwave read lengths. detected them thanks by to sending their highly high momenta or short 2 Scales In all quantum field theoryon models three we basic know ingredients the renormalization group relies that have been obtainedin the on last group two field years. theory scaling and renormalization a quite natural modificationscenario and of main the source propagatorquantum of [3, gravity. inspiration 4, for We 5].section, will our own recall This views some provides and key the hopes points about of this story in the next PoS(CNCFG2010)004 ) 2 cut x 2 + Ω 2 p ( gently / with 1 4 R 4 slice of eigenvalues of a propagator model on the Moyal space 4 ? φ is a does not look scientific but should not be suppressed. It means eg scale A gently A new scale decompositioninfrared which mixes scales, the traditional ultraviolet and The word However a background-independent quantum theory of space-time for The problem lies in our deeply rooted and often subconscious identifica- Based on this experience in condensed matter and non commutative field So we turned the problem into that of finding the right propagator with • 2 propagator is just renormalizableing, [3, since 4]. all Its threerenormalization renormalization main of is ingredients that very are model interest- subtly relies changed. on More precisely the if the scale isthan defined sharp ones through in momentum order cutoffs to it ensure the is corresponding good dual to decays of use the smooth sliced cutoffs propagator. rather gravity should startideally without not assuming even anyquestion any arises: fixed particular how background space-time toa define topology. metric, renormalization a group and background-independent So flow notion inof the of the any scale absence first particular and of puzzling topology? any ordinary metric, and even tion of the notionmodels of at scale our with disposalcondensed that of to matter distance. disentangle scale thattum Fortunately is we distance identification. not have to some In given anWulkenhaar the extended model by is singularity, models ordinary the a of pedagogical Fermi distancemuch surface toy but further model [2]. with which by The allows a momen- short us Grosse- renormalization to and group experiment long that distance mixes scales. the usual notionstheory of we propose the following more abstractDefinition definition 1. of scale: 3 The Grosse-WulkenhaarThe Grosse-Wulkenhaar model according to a geometric progression. the right non-trivial spectrumfact for that quantum such gravity. abstract Thein propagators progress absence can lies of exist in any and the space-time have manifold. non-trivial spectrum PoS(CNCFG2010)004 [5] (3.1)  , ) space ) 2 2 k k x ) y y 4 k k x + 2 + k 2 − x k k 3 ( x i x 2 k ( − + 2 α M graphs do diverge. 2 2 c x 3 − 2 − k tanh y 1 non trivial fixed point − x θ Ω x ( k δ i 2 ) − 4 2 M x 1 k c . y − −  e 3 i − ) 2 x 4 α x θ x 5 k 2 0 + 4Ω µ ∧ 2 α KM − 3 x x 6 Ω − sinh + 1 dα e θ 2 x ··· ∞ ( x 0 δ − Z ∧ dα ) i  1)  1 x − i x ( Ω ( πθ 2( i φ exp 1 i 2 −  − − x 2 M 4 ) θ M ıθ d 4Ω α Z 2 1  4 =1 ]. This propagator has parametric representation in i Y 2 ) ) = ribbon graph is a planar graph with all external legs on the outer boundary. (sinh x exp Z (˜ ) = is a fixed number, namely the ratio of the geometric slicing pro- 2 x, y ( i x, y M ( A new power counting under which only regular A new locality principle, called Moyality, C regular + Ω C 2 A A big unexpected bonus is the existence of a A convenient slice decomposition is made through the parametric repre- • • p 3 [ / sentation which makes that GWfor model a fully essentially consistent the four prime dimensional and quantum simplest field theory. candidate 3.1 The newThe multiscale scale analysis decomposition relies1 on slicing the Grosse-Wulkenhaar propagator where gression. The correspondingultraviolet scales and correspond infrared to notions a [4]. mixture of the ordinary 3.2 The MoyalityThe principle Moyal vertex in direct space is proportional to involving the Mehler kernel rather than the heat kernel. PoS(CNCFG2010)004 . renormalizable 6 and oscillates. It has a parallelogram shape, and Figure 1: The Moyal vertex Figure 2: The Moyality principle non-local In Figure 2 it is shown how moving up the scales of the inner lines of a This Moyality principle applies only to regular, high subgraphs, where This vertex is the phase of the oscillation is proportional tobubble its glues area. its two parallelogram-shapednal vertices legs into what point fromoscillations of the add view exter- up appears toarea as of the a the correct big new one parallelogram parallelogram, is for and the the sum how new ofhigh the the parallelogram, means inner areas that since of all the theerwise internal two eg smaller scales the ones. are oscillations higher won’t than add all up correctly. external scales.3.3 Oth- Noncommutative Renormalization Renormalizability of the GWscale model decomposition combines the definescounting three the tells elements. us “high” that The only and regular new be graphs “low” renormalized. with two scales. The and Moyality four principle externallook says legs The that must like such new “high” Moyal regular products. power graphs the form The of corresponding the counterterms initial are theory, thus therefore the of theory is PoS(CNCFG2010)004 theory, hence 4 4 φ 7 matrix models really triangulate 2Dlate Riemann much surfaces, more GFT’s singular triangu- objectssingularities). (not even pseudo-manifolds with local perturbative meaning of the theory is unclear. An unexpected bonus is that it has a non-trivial ultraviolet fixed point See [6] for further references on noncommutative field theory. There are several enticing factors about GFT. The spin foams of loop The main objections to GFT’s that I heard during the last two years were 2. There is no analog of the famous3. 1/N expansionThe for group natural field theories. GFT’s action does not look positive. Hence the non- 1. The GFT’s are tensor generalizations of matrix models; but whereas [5]. Hencecutoff. the This theory is should not expected be to fully be consistent the case and of non-trivial the without ordinary noncommutativity has indeed improved renormalizability butsense. in a very subtle 4 Group FieldGroup field Theory theory (GFT) [7,tion 8] of lies at Cartan’s the firstdimensional crossroads tensorial order between the extension formalism discretiza- of bytum the loop gravity matrix and quantum the models gravity, “Regge” relevant the or for higher simplicial 2D approach quan- gravity to are quantum exactly gravity theever Feynman spin amplitudes foam of formalism groupnamely alone field how still theory misses to [1]. whatwhy weight How- we the correctly feel spin different is foam topologiestize a formalism key gravity: of alone ingredient, space is thethem still time. amplitudes up. an are In incomplete That’s there, contrast, proposalsumming GFT but to is at not an the quan- attempt the same tospace-time correct time quantize dimension over weights completely metrics space-time, and to and doescould sum topologies. provide not a It scenario privilege assumes foras emergent any only large, the a manifold manifold-like classical condensed at space-time phase thecase, of there start. more is elementary It nothrough space-time reason a quanta. that suitably some adapted If GFT’s multiscale that RG. cannot is hopefully the be renormalized PoS(CNCFG2010)004 γ group field theory [9, 10, 11, colored 8 with a connection Γ (eg Levi-Civita of a metric). The M can be interpreted either as ultraviolet limit (on the group) → ∞ ables in GFT’s. or can be absorbed in a renormalization process; j . However at least some of the objections have been partly answered or Concerning the fourth item, one could say that to identify ordinary space- Concerning the fifth item it is exactly the point that our program would Finally it is true that GFT’s don’t predict space-time dimension, nor 4. It seems difficult to identify ordinary space-time and5. physical observ- GFT’s have infinities and it is not clear whether these6. infinitiesGFT’s should don’t predict space-time dimension, nor the standard model. M are under detailed investigation.already The solved first through two the items invention have of been more or less 12]. The third oneGFT may fields also be were eg solved if, Fermionic as [9]. colored field theorytime suggests, and the physical observablestifying in hadrons GFT’s out islimit of a non-trivial quarks task, and just gluons as is. iden- In particular the ultraspin like to investigate intopological detail. group Progress has field beenconclusions theories made of for essentially the for more ordinary renormalizable BF general with type theories, a and suitable that it choice could is of well the too propagator. turnthe early out standard to model, to draw but bemaking progress the just [14, issue 15, 16], ofreally and completing fares it GFT is any not better with clear at matter to us this fields whether point. is the competition 4.1 GFT asConsider a theories manifold of Holonomies in or as an infraredmight limit therefore for be a amore large complicated theory “chunk” than with of in some space-time.trivial the kind GW 4d Quantum model. of space gravity ultraviolet/infrared However timerelativity once mixing, is on large, it obtained topologically should asbackground follow an independence, naturally effective which if phase shouldance we of then of have the the respected imply GFT, theory the diffeomorphism on general principle invari- any of preferred specific manifold. geometric information is encoded in the holonomies along all closed curves PoS(CNCFG2010)004 . The ) is the 0 M in Fourier X factors into dimensional , φ ) F n 1 , g 3 g , 2 dimensional cells. ( 0 ) φ . We associate to this ) 2 − X with flat 3 g n , g . , g ) 1 , M 2 2 g ( g (independent of ( ∗ , g (0) = mn 1 g φ φ g φ ) ) ( 2 2 2 ). are associated to the sides of the K , g , g ıng = 1 and developing ,X 1 1 (0) − g γ g g L e (1). To all vertices (points) in our ( ( Y 1 , encoding the deficit angle along any , and fix a triangulation of 1 φ ) U g = 0 − ) 3 9 TM 3 M ımg K,V ( σ ). Quantization of geometry should sum K e = , g ) , g dt 2 Z 2 2 dγ 1 dimensional) is also flat. The curvature × GL G ν , g ,... Z X , g , g 0 1 1 1 − ∈ X g g g ( ( ( n g µ νσ g, g ) = V φ ( 2 . Suppose we discretize V G γ ). Taking F G V , g 1 + Γ × Y 1 × G µ g G , g ( Z = 2 × X are associated to blocks of codimension 2. φ g 1 2 G ν ( F for some ∗ Z ∂ h ν = φ 0 λ dt + S dγ gX ) = GFT as Matrix Models 2 ) = , g 1 is the vector field solution of the parallel transport equation D T g ( 2 ( X φ X If The discrete information about the triangulation of the surfaceIn can the be dual graph the group elements with series then holonomy along the curve simplices. Their boundary ( Then a 2-stranded graphintegrand of is the a Feynman amplitude) ribbon of Feynman a matrix graph model (and defined its by the weight action is the is located at the “joints” of these blocks, that is at Hence holonomies surface a weight function 4.2 Consider a two dimensional surface triangulation we associate a holonomy small curve encircling theby the vertex. gluing of The the surface triangles and and by itsboth the metric over holonomies metrics are compatible with specified a triangulation and over all triangulations. encoded in agraph ribbon are (2-stranded) dual to dual triangles graph. and the ribbon Theribbons, lines also ribbon are called vertices dual strands. to of Suppose edges. that the the weight function holonomy group of a surface is contributions of dual vertices and dual lines: PoS(CNCFG2010)004 = 0, , F ) n 2 , g gives 1 − e n ) fix a boundary 2 2 g ( , g 1 . g gravity. ( . . . φ φ km ) φ 2 pure nk , g 1 φ i g ) (3) Lie algebra ( = 0. Varying mn ω . φ ω. so ( a a φ S ∧ F − dx dx e De ) ) ω ] ∧ mnk X x x i ( ( λ + e dφ i 10 a i a [ e ω + Z dω Z ∗ mn = ) = ) = φ ) = : x ) to any point-particle’s mass. ) = x > ( ( π i ω i ) ω mn ( e n ω e, ω φ 2 ( F with values in the S , g mn 1 X ω − 1 2 n 2 g = ( S gives Cartan’s equation . . . φ ) space-time. 3D gravity is a topological theory with only global ω 2 , g is the curvature of 1 flat g ( F Varying A complete correlation function of a matrix model < φ hence a observables and no propagatingis degrees of physically freedom. interesting. Nevertheless Matterspace the can but theory induce be an added;3D angular point gravity deficit particles there proportional do is to not a their limit curve mass. (2 Therefore in where and a spin connection The action is 4.3 General RelativityIn Cartan’s in formalism, 3D three quantum gravity dimensions is expressed in terms of a dreibein triangulation and geometry. Each FeynmanThe graph amplitude fixes of a a graph bulkthe is triangulation. the triangulation. sum over The allof bulk correlation geometries all function compatible triangulations automatically with and sumsthe geometries the matrix compatible weights models with provide the a boundary! quantization As of such 2D admits the following interpretation. The insertions the action takes the more familiar form PoS(CNCFG2010)004 . ) 03 . The . , g ) if it has 1 13 ”). It has − 12 g , g , g 1 0 (2), the uni- g 23 ( g δ ( simple , ) is associated to g, g ) SU ) 6 φ ( 1 . ) . φ ) 12 φ − 11 , g ) = c 23 g 5 1 ) 5 − 3 , g 3 G , g 0 g , ..g , g b ( 1 4 02 ”) = δ g g , g hg flatness condition ) ( ( , hg 3 a , g 1 delta functions matching 2 φ g g Q − 10 ) , hg ( ( 12 ) 0 3 g ∗ p δ g 9 12 ( , hg φ ) g , g 1 ) 1 φ ( 2 c , g ) − 2 δ 0 hg ) 11 , g 12 , g ( hg, hg 1 1 b ( hg g − , g 7 , g 2 φ ( , g g g 10 13 a 6 φ dhφ ( g g g δ ( ( ( , g term, namely ! 11 ) 1 Z φ i δ 1 01 ) strands in GFT is called ) 4 − 9 g 1 − 1 dg φ 0 ( p − K 8 ) = , g 2 φ ) g 3 8 c ) =1 12 hg 2 i Y 1 g 01 , g , g , g g (

2 b 7 ( δ , g g ) ( , g , g 1 Z 02 1 a φ (3) rotations group. Group elements are associated − 4 dh δ g λ g g , g ( ]( 3 φ 03 SO g Z g ( 3 ( Cφ δ G = [ φ Z ] = ) C φ 2 1 [ ) = A vertex joining is the inverse of the quadratic part V 12 = 6) as it writes p K , ..g g, . . . , g 1 ( g ( V 6 Q G Z The 3D GFT propagator should implement the The natural 3D GFT tetrahedron vertex in 3 dimensions is simple in this λ kernel is a projector onto gauge invariant fields propagator with a kernel What characterizes vertices inlarly QFT we is propose some kind of localityDefinition property. 2. Simi- for kernel instrands direct two group by two space in different a half-lines. product of sense (with This vertex istriangles dual therefore to the vertex a is tetrahedron. a A tetrahedron is bounded by four 4.4 Group FieldIn three Theory dimensions in we three must dimensions use the holonomy group versal covering of the to edges in thea triangulation triangular (codimension face 2).propagator of Each a field tetrahedron, therefore it has three arguments. The PoS(CNCFG2010)004 , 4 ∧ e B K e ∧ [ ? e R field, the B are considered ω on the F, ∧ )] . Therefore 3D GFT with e f ) , involves an arbitrary two f ∧ g F e ( constraints ( δ ∧ 1 γ f B Y e R ) + e dh 12 ∧ e e = 0. in four dimenions are of the form Y and the spin connection ( ? [ e F B Z Z = G 1 πG Z 8 − theory, of action = ’s than just is the holonomy along the face S F e BF h as propagator implements exactly the correct flatness condi- f at zero external data, if Λ is the ultraviolet cutoff on the size ∈ e C 3 ~ Q elements unintegrated gives for each triangulation, or 3-stranded = a Feynman amplitude h f . As not all two-forms where again the vierbein G g B theory. The F ’s. Regularization by going to a quantum group at a root of unity leads to An approach to gravity in 4 dimensions is to consider it as a constrained Still another action classically equivalent to the Einstein-Hilbert action This 3D GFT is topological and the amplitude of a graph changes through j ∧ ] Performing the integrations over alling group the elements of the vertices and keep- graph e independent variables. BF is the Holst action: one needs to implementso a called certain Plebanski number constraints. of more complicated These constraints andpropagating render interesting degrees 4D of since gravity freedom, they the much allow gravitational richer are waves, set as responsible of constraints on for the local form well-defined topological invariants ofViro the invariants. triangulation, However namely theas the theory the Turaev- seems propagator unsuited for hasused a spectrum to true define limited RG truly to analysis, scales 0 in and the 1. abstract sense It4.5 of cannot Definition therefore 1. be The 4In Dimensional the Case first order Cartan formalism, the action is proportional to where the projector a global multiplicative factor undermay the so-called be Pachner moves. infinite, Amplitudes diverges for as instance Λ theof tetrahedron graph or complete graph tions of 3D gravityRegge [13]! amplitudes. Fourier transforming the model, one gets Ponzano- PoS(CNCFG2010)004 the- (4.2) BF , ) 4 , hg 3 DB Dω. ) , hg e 2 ∧ , hg e 1 = hg ( B ( δ (called the Barbero-Immirzi parame- dhφ F γ 13 ∧ ] Z B 1 γ + ) = B 4 ? [ R , g 3 1 πG , g 8 Figure 3: The pentachore 2 − e , g 1 1 Z g ]( = Cφ [ dν GFT with 4-stranded graphs called the Ooguri group field theory 5 φ A first attempt to implement the Plebanski constraints in this language The corresponding graphs are made of vertices dual to the pentachores, of we get a [17]. It isPlebanski not constraints are 4D missing. gravity, Againis but such unsuited a a topological for discretization version RG of of analysis. gravity the 4D BFis theory, due ie to the Barrett and Crane. They suggested one should restrict to simple ter) plays no rˆoleat theof classical 4D level gravity but [1]. is Weory can crucial plus then to rewrite constraints) the this and loop action getmeasure, `ala Plebanski quantization the (i.e. loosely Palatini-Holst-Plebanski functional written integral as: where the firstterm) term is is topological. the The Palatini parameter action, and the second one (the Holst 4.5.1 4D GFTThe Vertex vertex is thethe five easy tetrahedra part which as join it into pentachores. should befour-stranded again edges dual given to by tetrahedra,dual gluing to and triangles. of rules faces If for we or keep closed the thread same propagator circuits than in the Boulatov theory PoS(CNCFG2010)004 C (5.3) , hence . However S − j = : has non-trivial = . γ 2 1 + K S j and < γ < C divergence for the graph (2) residual gauge invari- 6 = SU for 0 2 C j γ 2 − with 1 at both ends like in Ooguri theory 14 = C − CSC j = (2), namely those satisfying ; K j SU γ averaging on a single × 2 S 1 + (2) (2) averaging = SU SU + j × (2) SU are two projectors, but they do not commute, hence A simplicity constraint which depends on the value of A A new projector ance in the middle of the propagator. S A first step in this direction has been performed in 2008 [22]. In some • • • 2 5 The EPR/FKParallel works theory by Engle-Pereira-Rovelli andbutions by of Freidel-Krasnov, Livine with and contri- Speziale,[18, lead 19, in 20] 2007 which toangular may an degrees better of improved freedom incorporate spin of foam thefor the model Plebanski graviton. a constraints 4D It and translates GFT into the [21] an with new proposal an improved propagator which incorporates this does not seemment to the Plebanski work constraints because toothe the strongly. correct Barrett-Crane In angular proposal particular degrees it may of may imple- freedom not of lead to the graviton. 5.1 Renormalization group,The at main last? newcan feature therefore of be the written EPR/FK as models is that the full propagator representations of and spectrum! This model isalong therefore our a program. natural starting point for a RGnormalization analysis of the theorytive a correction tantalizing logarithmic to divergence theG for coupling a constant radia- and a Λ PoS(CNCFG2010)004 in the case of 2 divergence for the G 9 15 divergence of the EPR/FK graph 6 We need to establish resultsnot such only as in the the BF dominance but of also spherical in graphs the EPR/FK colored models. The jacket theory, and found how to relate them to the number of bubbles of the This section mostly summarizes joint work with J. Ben Geloun, R. Gurau, T. Kra- We have started the study of the general superficial degree of divergence We have written still a more compactFinally let representation us of mention the recent GFT results with which establish the 1/N expansion have been found. We have computed in the Abelian case the power counting of graphs for • 4 We have investigated the power counting of the Boulatov model, and found jewski, J. Magnen, K. Noui, M. Smerlak, A. Tanasa and P. Vitale. of EPR/FK graphs through saddleniques point [30]. analysis using We coherentcorrections recover states in tech- to many it caseseties, in the degenerate or general. Abelian not. counting,We and recovered This But find the richness saddle also isnon-degenerate Λ points interesting saddle but point may a configurations, comemaximally but major degenerate also in challenge. saddle a points many Λ [30].external vari- Our spins method . also works at non zero EPR/FK propagator in terms of traces [21]. for colored models12, and 31]. prove This that isas spheres encouraging we dominate for know generating in it. adiffeomorphism all large In invariance dimensions topologically seems colored trivial [11, also theory world more the promising issue [32]. of respecting suitable5.3 discretized What remainsEssentially everything! to be done graphs in the colored case,lead along to the new lines class of ofalso [26, topological the 9]. polynomials related for The stranded Abelian workpolynomials. graphs amplitudes of [24, 25]. Bonzom and See Smerlak [27, 28] and [29] for colored 5.2 What4 has been donethe so first far uniform boundsusing for the its loop Freidel-Louapre vertex constructive expansion regularization, technique [23]. BF PoS(CNCFG2010)004 = 1 create fixed points of the RG γ 16 . This fixed point might be characterized 5 expansion [11, 12, 31] seem a promising starting /N symmetry recovered at according to their spectrum. Such a slice requires two cut- BF slices We need tocondensed get phase a can better lead to glimpse emergence of of whether a large a smooth phase space-time. transition to a analysis. We need to findlogical out whether symmetries such as the tantalizing topo- We need to cut propagatorsinto with non trivialoffs, spectra like one the for EPR/FK perform the a top true onewhich multiscale for should RG obey the analysis, therect bottom study new power of the locality counting. the principle, “high”models and slice. subgraphs, are Only establish renormalizable then their or Then cor- can not. we we can find whether some of these bounds of thepoint 1 in this direction. Of course the next mystery would be to understand the vicinity of this fixed point Suppose some group field theory with the simple pentachore vertex and a The physics of the universe from far “before” the big bang to the current • • • 5 by the valueture. 1 of Our the world Immirzi could parameter then and be might called be asymptotically topological topological, in just na- like 6 Conclusion andLet us further warn Outlook thatcourse none this of section my iswill collaborators of turn are out a responsible to for purely be these speculative speculations wrong! nature which andyet that to of be specifiedviolet propagator fixed turns point out and toeffective a be world low renormalizable of energy with general phasewould an relativity it transition ultra- on mean leading a for to four physics? the dimensional ordinary space-time. What cosmos could be summarizedtrajectory. as It a would flow singlehigh out giant “transplanckian” from renormalization fixed very point group near (RG) a simple very high or infinitely and why the world emerged at all out of it... PoS(CNCFG2010)004 γ of λ term to 1 is a way of fixing 2 p 17 which in loop quantum gravity is usually 2 [33]. √ γ /π However summarizing the transplanckian physics as a simple flow of This transplanckian era of the universe would be characterized by the flow would not be quite right yet. We also expect the coupling constant of the Immirzi parameter fromPlanck 1 scale to and some geometrogenesis, non trivial semiclassicalsical value. computations geometry involving Slightly coupled clas- below the toBekenstein-Hawking quantized radiation, matter, such would as beginareas the and to computation horizons make of start sense, to the sically appear. quantum as geometric, Therefore macroscopic could theso Immirzi freeze parameter, there, as intrin- at to a recoverrenormalized value the which effective we usual value might for Bekenstein-Hawking fix entropy. This would give a the energy scales ofhave the non-trivial renormalization logarithmic group.term RG Then which two flows, measures other the namely parameters rate Ω, of the mixing of harmonic infrared potential and ultraviolet, and the Yang-Mills theory is asymptotically free.planckian We fixed hope point the could vicinitydo be of not that fully require trans- understood computerand with simulations, probably analytical for like tools the is Grosse-Wulkenhaarbang which the fixed and case point below, the [34]. for RG integrabletransitions Cooling trajectory until of systems to our the the large universe big and couldemergence undergo smooth of many world phase space appears time, to called usdensation geometrogenesis, as of would it the occur is vertices through today. ofThrough a The group con- later field transitions theory into space-timethe a particles cools non and perturbative further phase. fieldsgalaxies and of we becomes actually the filled observe standardnesis with at model that large at scales. we small can Itultraviolet scales really is scales and only distinguish in with after between theonly the the these the traditional geometroge- abstract large sense; “ultraspin” infrared scales. beforeand and Ordinary geometrogenesis time small distances there itself cannot are becould should measured rather be imagine replaced thesort by world of this atoms as of abstract madeno space notion of space-time time loosely of at that interacting “scale”. would all.not float pentachores, bound These in We into atoms an any have abstract specific not vacuum manifold. which condensed is yet, that is they are considered to be close to log 3 the group field theoryas vertex well, to which might move,the be and standard related probably model. to afixing the few the future In other wave matter the function operators and factor model radiation in case front fields of of of the the Grosse-Wulkenhaar model, PoS(CNCFG2010)004 we expect n P should simply K at large order, might perhaps c n S c λ λ K is in = n λ , two main things are reaches a critical value 6 λ = 1, at which the beta function . The value of S γ K = 1 fixed point towards our effective 18 γ This review benefitted from discussions with many peo- . At Ω = 1, hence perfect mixing, the beta function λ to flow from a fixed point at λ . and S γ /K We would like also this manifold or at least its main four-dimensional part to be The vicinity of the phase transition being around Running the RG flow from that at which the sum of all ‘spherical graphs” would diverge. A spherical 6 c topologically trivial, i.e.full spherical, of macroscopic because holes our or four handles. dimesnional universe does not look required. First perturbationthat theory pentachores should should beto pullulate; at glue second the in they vergesay a should of a nice sphere. energetically diverging and so Both prefer λ coherent phenomena pattern should so occurgraph when as is to the creategraphs a analog smooth dominate of manifold, a inmodels, planar power spherical graph counting graphs in inasymptotic dominate matrix behavior matrix in of models. colored and thethe 4D planar in Just asymptotic series GFTs behavior Grosse-Wulkenhaar like at [12]. of large planar thesewith order spherical Just an graphs analytically like to computable the bebe value in of 1 be probed non perturbativelyphase with transition a should double bevector scaling like, more limit. complex or thanlike” In quark BCS any hence confinement case condensation which a this whichmurky is new is and matrix step it like; may incomputer it be simulations, complexity. is just that like “4d-manifold- for This quarkstill some confinement studied part and time today. hadron of we formation have the are toAcknowledgments scenario investigate is it through quite ple. Let meand thank also more Joseph particularlyNoui, Razvan Daniele Ben Oriti, Gurau Geloun, Carlo and Rovelli, Harald Matteo Jacques Smerlak, Magnen, Grosse, Simone Speziale, Thomas Adrian Krajewski, Karim coupling constant vanishes because of theboth enhanced symmetry [5]. By analogy we expect also should vanish because ofpoint. the purely topological nature of the GFT at that world, the coupling constant mightwhere grow a from non-perturbative a small phasethe value transition to BCS occurs. a theory larger This ofoccur value is superconductivity and what or some happens in manifold-like in QCD. space For time this to condensation emerge to PoS(CNCFG2010)004 , , 44 256 , 235023 (2010) [arXiv:1006.0714 [hep- 19 27 , 515 (2006) [arXiv:hep-th/0512271]. 267 , 95 (2007) [arXiv:hep-th/0612251]. 649 [hep-th]. cations to Non-Relativistic Quantum[math-ph]. Electron Gases,” arXiv:1102.5117 noncommutative R**4 in the matrix305 base,” (2005) Commun. Math. [arXiv:hep-th/0401128]. 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