Renormalizable Tensorial Field Theories as Models of Quantum Geometry Sylvain Carrozza University of Bordeaux, LaBRI Universit¨ at Potsdam, 8/02/2016 ”Paths to, from and in renormalization” Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 1 / 24
Purpose of this talk Give you an impression of what are Tensorial Field Theories, and why people study them. Figure : ”Potsdamer Platz bei Nacht”, Lesser Ury, 1920s Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 2 / 24
Research context and motivations Research context and motivations 1 Tensorial locality and combinatorial representation of pseudo-manifolds 2 Tensorial Group Field Theories 3 Perturbative renormalizability 4 Summary and outlook 5 Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 3 / 24
Group Field Theory: what is it? Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 4 / 24
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 (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 4 / 24
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 (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 4 / 24
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 (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 4 / 24
General structure of a GFT and long-term objectives Typical form of a GFT: field ϕ ( g 1 , . . . , g d ), g ℓ ∈ G , with partition function � � � � t V V · ϕ n V = ( t V i ) k V i { SF amplitudes } Z = [ D ϕ ] Λ exp − ϕ · K · ϕ + {V} k V 1 ,..., k V i i Main objectives of the GFT research programme: Model building: define the theory space . 1 e.g. spin foam models + combinatorial considerations (tensor models) → d, G, K and {V} . Perturbative definition: prove that the spin foam expansion is consistent in some 2 range of Λ. e.g. perturbative multi-scale renormalization. Systematically explore the theory space: effective continuum regime reproducing 3 GR in some limit? e.g. functional RG, constructive methods, condensate states... Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 5 / 24
Tensorial locality and combinatorial representation of pseudo-manifolds Research context and motivations 1 Tensorial locality and combinatorial representation of pseudo-manifolds 2 Tensorial Group Field Theories 3 Perturbative renormalizability 4 Summary and outlook 5 Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 6 / 24
Matrix models and random surfaces [t’Hooft, Polyakov, Kazakov, David,... ’70s ’80s] Partition function for N × N symmetric matrix : � � � − 1 λ 2 Tr M 2 + N 1 / 2 Tr M 3 Z ( N , λ ) = [ d M ] exp Large N expansion → ensembles of combinatorial maps : λ n ∆ N 2 − 2 g Z g ( λ ) � � Z ( N , λ ) = s (∆) A ∆ ( N ) = triangulation ∆ g ∈ ◆ Continuum limit of Z 0 : tune λ → λ c ⇒ very refined triangulations dominate. ( Z 0 ( λ ) ∼ | λ − λ c | 2 − γ ) Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 7 / 24
Matrix models and random surfaces [t’Hooft, Polyakov, Kazakov, David,... ’70s ’80s] Partition function for N × N symmetric matrix : � � � − 1 λ 2 Tr M 2 + N 1 / 2 Tr M 3 Z ( N , λ ) = [ d M ] exp Large N expansion → ensembles of combinatorial maps : λ n ∆ N 2 − 2 g Z g ( λ ) � � Z ( N , λ ) = s (∆) A ∆ ( N ) = triangulation ∆ g ∈ ◆ Continuum limit of Z 0 : tune λ → λ c ⇒ very refined triangulations dominate. ( Z 0 ( λ ) ∼ | λ − λ c | 2 − γ ) ⇒ definition of universal 2d random geometries : do not depend on the details of the discretization, i.e. on the type of trace invariants used in the action; similarly, Brownian map rigorously constructed as a scaling limit of infinite triangulations and 2 p -angulations of the sphere. [Le Gall, Miermont ’13] Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 7 / 24
Colored cell decompositions of surfaces Gluing of 2 p -angles: 0 0 1 2 2 0 1 1 2 0 Duality: 3 − colored graph ← → colored triangulation node ← → triangle line ← → edge bicolored cycle ← → vertex Any orientable surface with boundaries can be represented by such a 3-colored graph . Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 8 / 24
Colored cell decompositions of pseudo-manifolds Works in any dimension, e.g. in 3d: 0 3 0 3 1 2 1 2 Colored structure ⇒ unambiguous prescription for how to glue d -simplices along their sub-simplices. Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 9 / 24
Colored cell decompositions of pseudo-manifolds Works in any dimension, e.g. in 3d: 0 3 0 3 1 2 1 2 Colored structure ⇒ unambiguous prescription for how to glue d -simplices along their sub-simplices. ( d + 1) − colored graph ← → colored triangulation of dimension d node ← → d − simplex connected component with k colors ← → ( d − k ) − simplex Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 9 / 24
Colored cell decompositions of pseudo-manifolds Works in any dimension, e.g. in 3d: 0 3 0 3 1 2 1 2 Colored structure ⇒ unambiguous prescription for how to glue d -simplices along their sub-simplices. ( d + 1) − colored graph ← → colored triangulation of dimension d node ← → d − simplex connected component with k colors ← → ( d − k ) − simplex ✎ ☞ Theorem: [Pezzana ’74] Any PL manifold can be represented by a colored graph. In general, a ( d + 1)-colored graph represents a triangulated pseudo-manifold of dimension d . ✍ ✌ ⇒ Crystallisation theory [Cagliardi, Ferri et al. ’80s] Only recently introduced in GFTs / tensor models [Gurau ’09...] Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 9 / 24
Trace invariants Trace invariants of fields ϕ ( g 1 , g 2 , . . . , g d ) labelled by d -colored bubbles b : 3 � [ d g i ] 6 ϕ ( g 6 , g 2 , g 3 ) ϕ ( g 1 , g 2 , g 3 ) Tr b ( ϕ, ϕ ) = 2 1 1 2 ϕ ( g 6 , g 4 , g 5 ) ϕ ( g 1 , g 4 , g 5 ) 3 Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 10 / 24
Trace invariants Trace invariants of fields ϕ ( g 1 , g 2 , . . . , g d ) labelled by d -colored bubbles b : 3 � [ d g i ] 6 ϕ ( g 6 , g 2 , g 3 ) ϕ ( g 1 , g 2 , g 3 ) Tr b ( ϕ, ϕ ) = 2 1 1 2 ϕ ( g 6 , g 4 , g 5 ) ϕ ( g 1 , g 4 , g 5 ) 3 ( d = 2) · · · Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 10 / 24
Trace invariants Trace invariants of fields ϕ ( g 1 , g 2 , . . . , g d ) labelled by d -colored bubbles b : 3 � [ d g i ] 6 ϕ ( g 6 , g 2 , g 3 ) ϕ ( g 1 , g 2 , g 3 ) Tr b ( ϕ, ϕ ) = 2 1 1 2 ϕ ( g 6 , g 4 , g 5 ) ϕ ( g 1 , g 4 , g 5 ) 3 ( d = 2) · · · ( d = 3) · · · Sylvain Carrozza (Uni. Bordeaux) Renormalizable Tensorial Field Theories Paths to, from and in renormalization 10 / 24
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