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Introduction to Group Field Theory Sylvain Carrozza University of Bordeaux, LaBRI The Helsinki Workshop on Quantum Gravity, 01/06/2016 Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 1 / 21 Group Field Theory: what is it? Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 2 / 21 Group Field Theory: what is it? It is an approach to quantum gravity at the crossroad of loop quantum gravity (LQG) and matrix/tensor models. A simple definition: A Group Field Theory (GFT) is a non-local quantum field theory defined on a group manifold. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 2 / 21 Group Field Theory: what is it? It is an approach to quantum gravity at the crossroad of loop quantum gravity (LQG) and matrix/tensor models. A simple definition: A Group Field Theory (GFT) is a non-local quantum field theory defined on a group manifold. The group manifold is auxiliary: should not be interpreted as space-time! Rather, the Feynman amplitudes are thought of as describing space-time processes → QFT of space-time rather than on space-time. Specific non-locality: determines the combinatorial structure of space-time processes (graphs, 2-complexes, triangulations...). Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 2 / 21 Group Field Theory: what is it? It is an approach to quantum gravity at the crossroad of loop quantum gravity (LQG) and matrix/tensor models. A simple definition: A Group Field Theory (GFT) is a non-local quantum field theory defined on a group manifold. The group manifold is auxiliary: should not be interpreted as space-time! Rather, the Feynman amplitudes are thought of as describing space-time processes → QFT of space-time rather than on space-time. Specific non-locality: determines the combinatorial structure of space-time processes (graphs, 2-complexes, triangulations...). Recommended reviews: L. Freidel, ”Group Field Theory: an overview”, 2005 D. Oriti, ”The microscopic dynamics of quantum space as a group field theory”, 2011 Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 2 / 21 From Loop Quantum Gravity to Group Field Theory 1 From Loop Quantum Gravity to Group Field Theory 2 Group Field Theory Fock space and physical applications 3 Group Field Theory renormalization programme 4 Summary and outlook Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 3 / 21 Loop Quantum Gravity proposes kinematical states describing (spatial) quantum geometry [Ashtekar, Rovelli, Smolin, Lewandowski... ’90s; Dittrich, Geiller, Bahr ’15]: Dynamics? Define the (improper) projector P : Hkin → Hphys on physical states Hphys 3 |siphys ≡ P|si , hs|s 0 iphys ≡ hs|P|s 0 i Spin Foams [Reisenberger, Rovelli... ’00s] are a path-integral formulation of the dynamics → amplitudes As,C associated to a 2-complex C with boundary spin-network state s. XY Y Y As,C = Af Ae Av j Sylvain Carrozza (Univ. Bordeaux) f Introduction to GFT e v Univ. Helsinki, 01/06/2016 4 / 21 Structural incompleteness of Spin Foams: How one should interpret and organize the 2-complexes? How to extract As from the family {As,C | ∂C = s}? Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 5 / 21 Structural incompleteness of Spin Foams: How one should interpret and organize the 2-complexes? How to extract As from the family {As,C | ∂C = s}? Three interpretations of C found in the literature: (i) a convenient way of writing up the amplitudes, but amplitudes independent of it from the outset: As = As,C ; (ex: Turaev-Viro model) (ii) a regulator, analogous to the lattice of lattice gauge theory; (iii) a specific quantum history compatible with the boundary state, analogous to a Feynman diagram in QFT. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 5 / 21 Structural incompleteness of Spin Foams: How one should interpret and organize the 2-complexes? How to extract As from the family {As,C | ∂C = s}? Three interpretations of C found in the literature: (i) a convenient way of writing up the amplitudes, but amplitudes independent of it from the outset: As = As,C ; (ex: Turaev-Viro model) (ii) a regulator, analogous to the lattice of lattice gauge theory; (iii) a specific quantum history compatible with the boundary state, analogous to a Feynman diagram in QFT. First interpretation seems very hard to realize in 4d (→ construction of 4d invariants of manifolds), and the other two hinge on renormalization theory: 1 Lattice interpretation: refining and coarse-graining C (and s) ⇒ As ≡ lim As,C [Dittrich, Bahr, Steinhaus, Martin-Benito... C →∞ 2 ’10s] QFT interpretation: amplitudes of a Group Field Theory, to be summed over P ⇒ As ≡ wC As,C [De Pietri, Rovelli, Freidel, Oriti... ’00s, ’10s] C |∂C =s Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 5 / 21 In the two interpretations, renormalization is central and allows in principle to address some other open challenges: Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 6 / 21 In the two interpretations, renormalization is central and allows in principle to address some other open challenges: 1 consistency of the quantum dynamics under coarse-graining? 2 quantization / discretization ambiguities inherent to spin-foams: what are the universal features of the known models? [EPRL, DL, BO, ...] 3 macro-physics from microscopic dynamics: how do we extract the low-energy limit of LQG? are there several quantum phases? compatibility with general relativity? Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 6 / 21 In the two interpretations, renormalization is central and allows in principle to address some other open challenges: 1 consistency of the quantum dynamics under coarse-graining? 2 quantization / discretization ambiguities inherent to spin-foams: what are the universal features of the known models? [EPRL, DL, BO, ...] 3 macro-physics from microscopic dynamics: how do we extract the low-energy limit of LQG? are there several quantum phases? compatibility with general relativity? Refining framework ⇒ background independent generalization of direct space renormalization methods: scale = lattice itself consistency over scales ⇔ dynamical cylindrical consistency Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 6 / 21 In the two interpretations, renormalization is central and allows in principle to address some other open challenges: 1 consistency of the quantum dynamics under coarse-graining? 2 quantization / discretization ambiguities inherent to spin-foams: what are the universal features of the known models? [EPRL, DL, BO, ...] 3 macro-physics from microscopic dynamics: how do we extract the low-energy limit of LQG? are there several quantum phases? compatibility with general relativity? Refining framework ⇒ background independent generalization of direct space renormalization methods: scale = lattice itself consistency over scales ⇔ dynamical cylindrical consistency Summing framework ⇒ background independent generalization of momentum shell renormalization methods: scale = spectrum of a specific 1-particle operator (e.g. spin labels) consistency over scales ⇔ renormalization group flow of a (non-local) field theory defined on internal space (e.g. SU(2)). Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 6 / 21 General structure of a GFT and long-term objectives Typical form of a GFT: field ϕ(g1 , . . . , gd ), g` ∈ G , with partition function Z X X Y n V tV V · ϕ = (tVi )kVi {SF amplitudes} Z = [Dϕ]Λ exp −ϕ · K · ϕ + {V} Sylvain Carrozza (Univ. Bordeaux) kV1 ,...,kV Introduction to GFT i i Univ. Helsinki, 01/06/2016 7 / 21 General structure of a GFT and long-term objectives Typical form of a GFT: field ϕ(g1 , . . . , gd ), g` ∈ G , with partition function Z X X Y n V tV V · ϕ = (tVi )kVi {SF amplitudes} Z = [Dϕ]Λ exp −ϕ · K · ϕ + {V} kV1 ,...,kV i i Main objectives of the GFT research programme: 1 Model building: define the theory space. e.g. spin foam models + combinatorial considerations (tensor models) → d, G , K, {V} and [Dϕ]Λ . 2 Perturbative definition: prove that the spin foam expansion is consistent in some range of Λ. e.g. perturbative multi-scale renormalization. 3 Systematically explore the theory space: effective continuum regime reproducing GR in some limit? e.g. functional RG, constructive methods, condensate states... Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 7 / 21 Group Field Theory Fock space and physical applications 1 From Loop Quantum Gravity to Group Field Theory 2 Group Field Theory Fock space and physical applications 3 Group Field Theory renormalization programme 4 Summary and outlook Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 8 / 21 GFT Hilbert space No embedding in a continuum manifold and no cylindrical consistency imposed. Instead: Fock construction through decomposition of spin network states in terms of elementary building blocks. h1 g1 g2 g1 h−1 1 h4 h2 h3 g3 g4 Elementary excitations over a vacuum |0i interpreted as a ’no-space vacuum’. Creation/annihilation operators ϕ(g b i )† /ϕ(g b i ). HGFT = Fock(Hv ) = +∞ M Sym Hv(1) ⊗ · · · ⊗ Hv(n) Hv = L2 (G ×d /G ) with n=0 (rem: bosonic statistics, arbitrary at this stage) ϕ̂(g1 , g2 , g3 , g4 )|0i = 0 , Sylvain Carrozza (Univ. Bordeaux) † g1 ϕ̂ (g1 , g2 , g3 , g4 )|0i = | Introduction to GFT g4 g2 g3 i, ... Univ. Helsinki, 01/06/2016 9 / 21 Dynamics Dynamics expressed as a projection in the Fock Hilbert space b |Ψi ≡ P b − 1l |Ψi = 0 F It turns out that current GFT models do not correspond to a ’micro-canonical’ ensemble X b )|si Z= hs|δ(F s but a kind of ’grand-canonical’ ensemble X b b Z= hs|e−β(F −µN ) |si [Oriti ’13] s ⇒ the GFT genuinely contains more information than the LQG projector on physical states [Freidel ’05] Open questions: how to extract the LQG physical projector? what is the role of topology changing processes? Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 10 / 21 Physical applications The Fock representation permits the construction of simple condensate states e.g. Z |σi ∝ exp [dgi ]4 σ(g1 , g2 , g3 , g4 )ϕ̂† (g1 , g2 , g3 , g4 ) |0i → arbitrary number of spin-network vertices excited with the same 1-particle wave-function σ(g1 , g2 , g3 , g4 ). Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 11 / 21 Physical applications The Fock representation permits the construction of simple condensate states e.g. Z |σi ∝ exp [dgi ]4 σ(g1 , g2 , g3 , g4 )ϕ̂† (g1 , g2 , g3 , g4 ) |0i → arbitrary number of spin-network vertices excited with the same 1-particle wave-function σ(g1 , g2 , g3 , g4 ). Such states have been successfully used to describe symmetric quantum geometries directly at the GFT level, hence without recourse to classical symmetry reduction: Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 11 / 21 Physical applications The Fock representation permits the construction of simple condensate states e.g. Z |σi ∝ exp [dgi ]4 σ(g1 , g2 , g3 , g4 )ϕ̂† (g1 , g2 , g3 , g4 ) |0i → arbitrary number of spin-network vertices excited with the same 1-particle wave-function σ(g1 , g2 , g3 , g4 ). Such states have been successfully used to describe symmetric quantum geometries directly at the GFT level, hence without recourse to classical symmetry reduction: Cosmology: [Gielen, Oriti, Sindoni, Calcagni, Wilson-Ewing, Pithis,...] EPRL model coupled to a scalar field −→ condensate in the hydrodynamic approximation −→ Friedmann equations with quantum gravity corrections −→ bounce at the Planck scale. [Oriti, Sindoni, Wilson-Ewing ’16] Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 11 / 21 Physical applications The Fock representation permits the construction of simple condensate states e.g. Z |σi ∝ exp [dgi ]4 σ(g1 , g2 , g3 , g4 )ϕ̂† (g1 , g2 , g3 , g4 ) |0i → arbitrary number of spin-network vertices excited with the same 1-particle wave-function σ(g1 , g2 , g3 , g4 ). Such states have been successfully used to describe symmetric quantum geometries directly at the GFT level, hence without recourse to classical symmetry reduction: Cosmology: [Gielen, Oriti, Sindoni, Calcagni, Wilson-Ewing, Pithis,...] EPRL model coupled to a scalar field −→ condensate in the hydrodynamic approximation −→ Friedmann equations with quantum gravity corrections −→ bounce at the Planck scale. [Oriti, Sindoni, Wilson-Ewing ’16] Black Holes: [Pranzetti, Sindoni, Oriti ’15] Condensates encoding spherically symmetric quantum geometry −→ reduced density matrix associated to a horizon −→ horizon entanglement entropy −→ Bekenstein-Hawking entropy formula for any value of the Immirzi parameter. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 11 / 21 Summary up to now GFT can be understood as a version of LQG with: no embedding in a continuous manifold; organization of LQG states in ’space atoms’; b a new fundamental observable: N. Provides statistical techniques to explore the many-body sector of quantum geometry: condensate states used for e.g. quantum cosmology and black holes The construction seems quite general ⇒ other choices of ’building blocks’ ? Useful for construction of GFT analogues of new kinematical vacua? [Dittrich, Geiller ’15 ’16] Quantization ambiguities are encoded in free coupling constants for the various spin foam vertices compatible with the dynamics one would like to implement ⇒ renormalization has to tell us which of these are more relevant. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 12 / 21 Group Field Theory renormalization programme 1 From Loop Quantum Gravity to Group Field Theory 2 Group Field Theory Fock space and physical applications 3 Group Field Theory renormalization programme 4 Summary and outlook Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 13 / 21 Importance of combinatorics Mathematical objective: step-by-step generalization of standard renormalization techniques, until we are able to tackle 4d quantum gravity proposals. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 14 / 21 Importance of combinatorics Mathematical objective: step-by-step generalization of standard renormalization techniques, until we are able to tackle 4d quantum gravity proposals. Two main aspects in the definition of a group field theory: Algebraic content and type of dynamics implemented: from LQG and Spin Foams Combinatorial structures: Which types of spin-network boundary states? In general, restriction on the valency. Which type of spin foam vertices? In general, restriction on the valency too. Which types of 2-complexes are summed over? Local restrictions on gluing rules to avoid too pathological topologies. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 14 / 21 Importance of combinatorics Mathematical objective: step-by-step generalization of standard renormalization techniques, until we are able to tackle 4d quantum gravity proposals. Two main aspects in the definition of a group field theory: Algebraic content and type of dynamics implemented: from LQG and Spin Foams Combinatorial structures: Which types of spin-network boundary states? In general, restriction on the valency. Which type of spin foam vertices? In general, restriction on the valency too. Which types of 2-complexes are summed over? Local restrictions on gluing rules to avoid too pathological topologies. Requirement: the GFT theory space should be stable enough under renormalization / coarse-graining. We currently know of only one such combinatorial structure: tensorial interactions initially introduced in the context of tensor models. [Gurau, Bonzom, Rivasseau, Ben Geloun... ’11...] Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 14 / 21 Trace invariants Trace invariants of fields ϕ(g1 , g2 , . . . , gd ) labelled by d-colored bubbles b: 3 Z 1 2 2 1 Trb (ϕ, ϕ) = [dgi ]6 ϕ(g6 , g2 , g3 )ϕ(g1 , g2 , g3 ) ϕ(g6 , g4 , g5 )ϕ(g1 , g4 , g5 ) 3 Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 15 / 21 Trace invariants Trace invariants of fields ϕ(g1 , g2 , . . . , gd ) labelled by d-colored bubbles b: 3 Z 1 2 1 2 Trb (ϕ, ϕ) = [dgi ]6 ϕ(g6 , g2 , g3 )ϕ(g1 , g2 , g3 ) ϕ(g6 , g4 , g5 )ϕ(g1 , g4 , g5 ) 3 ··· (d = 2) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 15 / 21 Trace invariants Trace invariants of fields ϕ(g1 , g2 , . . . , gd ) labelled by d-colored bubbles b: 3 Z 1 2 1 2 Trb (ϕ, ϕ) = [dgi ]6 ϕ(g6 , g2 , g3 )ϕ(g1 , g2 , g3 ) ϕ(g6 , g4 , g5 )ϕ(g1 , g4 , g5 ) 3 ··· (d = 2) ··· (d = 3) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 15 / 21 Trace invariants Trace invariants of fields ϕ(g1 , g2 , . . . , gd ) labelled by d-colored bubbles b: 3 Z 1 2 1 2 Trb (ϕ, ϕ) = [dgi ]6 ϕ(g6 , g2 , g3 )ϕ(g1 , g2 , g3 ) ϕ(g6 , g4 , g5 )ϕ(g1 , g4 , g5 ) 3 ··· (d = 2) ··· (d = 3) ··· (d = 4) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 15 / 21 Feynman amplitudes of TGFTs Perturbative expansion in the bubble coupling constants tb : ! X Y nb (G) Z= (−tb ) AG G b∈B Feynman graphs G: g1 g2 = g3 g g2 g̃ g1 g̃1 g3 g̃3 R dg1 dg2 dg3 . . . = δ(gg̃ −1 ) g̃2 = C(g1 , g2 , g3 ; g̃1 , g̃2 , g̃3 ) Covariances associated to the dashed, color-0 lines. Face of color ` = connected set of (alternating) color-0 and color-` lines. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 16 / 21 Perturbative renormalization: overview Goal: check that the perturbative expansion - and henceforth the connection to spin foam models - is consistent. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 17 / 21 Perturbative renormalization: overview Goal: check that the perturbative expansion - and henceforth the connection to spin foam models - is consistent. Types of models considered so far: ’combinatorial’ models on G = U(1)D : C =( X ∆` )-1 , CΛ (g` ; g`0 ) = ` Z +∞ dα Λ−2 d Y KαG (g` g`0−1 ) `=1 [Ben Geloun, Rivasseau ’11; Ben Geloun, Ousmane Samary ’12; Ben Geloun, Livine ’12...] models with ’gauge invariance’ on G = U(1)D or SU(2): Z +∞ Z d X Y C = P( ∆` )-1 P , CΛ (g` ; g`0 ) = dα dh KαG (g` hg`0−1 ) Λ−2 ` G `=1 [SC, Oriti, Rivasseau ’12 ’13; Ousmane Samary, Vignes-Tourneret ’12; SC ’14 ’14; Lahoche, Oriti, Rivasseau ’14...] Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 17 / 21 Perturbative renormalization: overview Goal: check that the perturbative expansion - and henceforth the connection to spin foam models - is consistent. Types of models considered so far: ’combinatorial’ models on G = U(1)D : C =( X ∆` )-1 , CΛ (g` ; g`0 ) = ` Z +∞ dα Λ−2 d Y KαG (g` g`0−1 ) `=1 [Ben Geloun, Rivasseau ’11; Ben Geloun, Ousmane Samary ’12; Ben Geloun, Livine ’12...] models with ’gauge invariance’ on G = U(1)D or SU(2): Z +∞ Z d X Y C = P( ∆` )-1 P , CΛ (g` ; g`0 ) = dα dh KαG (g` hg`0−1 ) Λ−2 ` G `=1 [SC, Oriti, Rivasseau ’12 ’13; Ousmane Samary, Vignes-Tourneret ’12; SC ’14 ’14; Lahoche, Oriti, Rivasseau ’14...] Methods: multiscale analysis: allows to rigorously prove renormalizability at all orders in perturbation theory; Connes–Kreimer algebraic methods [Raasakka, Tanasa ’13; Avohou, Rivasseau, Tanasa ’15]. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 17 / 21 Quasi-locality of divergences The divergent subgraphs must be quasi-local, i.e. look like trace invariants at high scales. Always the case in known models, but non–trivial! Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 18 / 21 Quasi-locality of divergences The divergent subgraphs must be quasi-local, i.e. look like trace invariants at high scales. Always the case in known models, but non–trivial! ϕ(g1 ) 3 h1 , α1 1 ϕ(g3 ) ϕ(g2 ) h2 , α2 Z dα1 dα2 ϕ(g3 ) ∼ K× 2 Z ϕ(g1 ) ϕ(g4 ) 2 dh1 dh2 Kα1 +α2 (h1 h2 ) ϕ(g2 ) + ··· ϕ(g4 ) Z Y −1 −1 [ dgij ] Kα1 (g11 h1 g31 )Kα2 (g21 h2 g41 ) i<j −1 −1 −1 −1 δ(g12 g22 )δ(g13 g22 )δ(g42 g32 )δ(g43 g33 ) ϕ(g1 ) ϕ(g2 ) ϕ(g3 ) ϕ(g4 ) Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 18 / 21 Quasi-locality of divergences The divergent subgraphs must be quasi-local, i.e. look like trace invariants at high scales. Always the case in known models, but non–trivial! ϕ(g1 ) 3 h1 , α1 1 ϕ(g3 ) ϕ(g2 ) h2 , α2 Z dα1 dα2 ϕ(g3 ) ∼ K× 2 Z ϕ(g1 ) ϕ(g4 ) 2 dh1 dh2 Kα1 +α2 (h1 h2 ) ϕ(g2 ) + ··· ϕ(g4 ) Z Y −1 −1 [ dgij ] Kα1 (g11 h1 g31 )Kα2 (g21 h2 g41 ) i<j −1 −1 −1 −1 δ(g12 g22 )δ(g13 g22 )δ(g42 g32 )δ(g43 g33 ) ϕ(g1 ) ϕ(g2 ) ϕ(g3 ) ϕ(g4 ) This property is not generic in TGFTs → ”traciality” criterion. Nice interplay between structure of divergences and topology → renormalizable interactions are spherical. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 18 / 21 Current developments 1 Non-perturbative renormalization: Wetterich equation applied to: matrix and tensor models; TGFT without gauge-invariance; gauge-invariant models. [Eichhorn, Koslowski ’13 ’14] [Benedetti, Ben Geloun, Oriti ’14] [Lahoche, Benedetti ’15; Lahoche, SC wip] Polchinski equation [Krajewski, Toriumi ’15] Constructive methods such as the loop-vertex expansion (intermediate field) applied to: tensor models; TGFTs without gauge invariance; TGFTs with gauge invariance. Sylvain Carrozza (Univ. Bordeaux) [Gurau ’11 ’13; Delepouve, Gurau, Rivasseau ’14...] [Delepouve, Rivasseau ’14...] [Lahoche, Oriti, Rivasseau ’15] Introduction to GFT Univ. Helsinki, 01/06/2016 19 / 21 Current developments 1 Non-perturbative renormalization: Wetterich equation applied to: matrix and tensor models; TGFT without gauge-invariance; gauge-invariant models. [Eichhorn, Koslowski ’13 ’14] [Benedetti, Ben Geloun, Oriti ’14] [Lahoche, Benedetti ’15; Lahoche, SC wip] Polchinski equation [Krajewski, Toriumi ’15] Constructive methods such as the loop-vertex expansion (intermediate field) applied to: tensor models; TGFTs without gauge invariance; TGFTs with gauge invariance. [Gurau ’11 ’13; Delepouve, Gurau, Rivasseau ’14...] [Delepouve, Rivasseau ’14...] [Lahoche, Oriti, Rivasseau ’15] Lesson: non-trivial fixed points seem generic. Phase transition to a condensed phase? Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 19 / 21 Current developments 1 Non-perturbative renormalization: Wetterich equation applied to: matrix and tensor models; TGFT without gauge-invariance; gauge-invariant models. [Eichhorn, Koslowski ’13 ’14] [Benedetti, Ben Geloun, Oriti ’14] [Lahoche, Benedetti ’15; Lahoche, SC wip] Polchinski equation [Krajewski, Toriumi ’15] Constructive methods such as the loop-vertex expansion (intermediate field) applied to: tensor models; TGFTs without gauge invariance; TGFTs with gauge invariance. [Gurau ’11 ’13; Delepouve, Gurau, Rivasseau ’14...] [Delepouve, Rivasseau ’14...] [Lahoche, Oriti, Rivasseau ’15] Lesson: non-trivial fixed points seem generic. Phase transition to a condensed phase? 2 Towards renormalizable models with simplicity constraints: GFT on SU(2)/U(1); [Lahoche, Oriti ’15] 4d GFT on Spin(4) with Barrett-Crane simplicity constraints. CΛ (g` ; g`0 ) = Sylvain Carrozza (Univ. Bordeaux) Z +∞ Z Z dα Λ−2 Z dh Spin(4) dk SU(2) Introduction to GFT Hk [dl` ] d Y [Lahoche, Oriti, SC wip] Spin(4) Kα (g` hl` g`0−1 ) . `=1 Univ. Helsinki, 01/06/2016 19 / 21 Summary and outlook 1 From Loop Quantum Gravity to Group Field Theory 2 Group Field Theory Fock space and physical applications 3 Group Field Theory renormalization programme 4 Summary and outlook Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 20 / 21 Summary and outlook GFT is a QFT completion of spin foam models. It allows to (define and) explore the many-body sector of LQG. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 21 / 21 Summary and outlook GFT is a QFT completion of spin foam models. It allows to (define and) explore the many-body sector of LQG. Two parallel lines of investigations: Construction of effective geometries from condensate states and approximations of the full GFT dynamics → some aspects of quantum cosmology and black holes recovered from 4d quantum gravity models! See talks by Wilson-Ewing and Pithis Development of suitable renormalizable tools to check the overall consistency of GFTs and explore more systematically their phase diagrams → applicable to simplified toy-models, not yet to 4d quantum gravity. Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 21 / 21 Summary and outlook GFT is a QFT completion of spin foam models. It allows to (define and) explore the many-body sector of LQG. Two parallel lines of investigations: Construction of effective geometries from condensate states and approximations of the full GFT dynamics → some aspects of quantum cosmology and black holes recovered from 4d quantum gravity models! See talks by Wilson-Ewing and Pithis Development of suitable renormalizable tools to check the overall consistency of GFTs and explore more systematically their phase diagrams → applicable to simplified toy-models, not yet to 4d quantum gravity. Can we define a renormalizable 4d quantum gravity model and prove the existence of a condensed phases with the right properties? Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 21 / 21 Summary and outlook GFT is a QFT completion of spin foam models. It allows to (define and) explore the many-body sector of LQG. Two parallel lines of investigations: Construction of effective geometries from condensate states and approximations of the full GFT dynamics → some aspects of quantum cosmology and black holes recovered from 4d quantum gravity models! See talks by Wilson-Ewing and Pithis Development of suitable renormalizable tools to check the overall consistency of GFTs and explore more systematically their phase diagrams → applicable to simplified toy-models, not yet to 4d quantum gravity. Can we define a renormalizable 4d quantum gravity model and prove the existence of a condensed phases with the right properties? Thank you for your attention Sylvain Carrozza (Univ. Bordeaux) Introduction to GFT Univ. Helsinki, 01/06/2016 21 / 21