Three roads from tensors models to continuous geometry Nicolas Delporte & Vincent Rivasseau IJCLab, Universit´ e Paris-Saclay Workshop ”Higher Structures Emerging from Renormalisation” Schr¨ odinger Institute, Vienna October 15, 2020
Introduction 1 Motivation Renormalization Tensor Models: a survey First Road: Double and Multiple Scaling 2 Double Scaling for Matrices and Tensors Multiple Scaling and Topological Recursion Second Road: Flowing from Trees to New Fixed Points 3 Breaking the Propagator Finding New Fixed Points Third road: Random Geometry from Trees 4 QFT on Random Trees Our results Conclusions and Futur Prospectives 5 Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 2 / 39
Introduction Introduction 1 Motivation Renormalization Tensor Models: a survey First Road: Double and Multiple Scaling 2 Double Scaling for Matrices and Tensors Multiple Scaling and Topological Recursion Second Road: Flowing from Trees to New Fixed Points 3 Breaking the Propagator Finding New Fixed Points Third road: Random Geometry from Trees 4 QFT on Random Trees Our results Conclusions and Futur Prospectives 5 Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 3 / 39
Introduction Motivation Motivation We use the perspective Quantizing Gravity ≃ Randomizing Geometry Functional integral quantization, in Euclidean setting � � � − S A EH ( g ) Z ≃ Dg e S where Dg and even S are to be defined... Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 4 / 39
Introduction Motivation Motivation We use the perspective Quantizing Gravity ≃ Randomizing Geometry Functional integral quantization, in Euclidean setting � � � − S A EH ( g ) Z ≃ Dg e S where Dg and even S are to be defined... A fundamental difficulty is that the theory on a four dimensional flat space is perturbatively not renormalisable = ⇒ non-UV complete. In two dimensions, random matrix models are among the most successful ways to explore quantum gravity non perturbatively & ab initio. The Tensor Track generalizes this success to use tensors to explore to quantum gravity in higher dimensions [VR ’11, ’12, ’13, ’16, ’18, ’20] . Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 4 / 39
Introduction Renormalization Renormalization Physics is mathematics plus scales. Since 1930’s, the idea that physics also depends on the probing scale was independently exploited in particle physics and condensed matter: - [Gell-Mann, Low, Dyson] “dress” an elementary particle with an effective (renormalized) charge; - [Stueckelberg, Petermann, Kadanoff] block spin transformations to recover scaling laws near critical point. Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 5 / 39
Introduction Renormalization Renormalization Physics is mathematics plus scales. Since 1930’s, the idea that physics also depends on the probing scale was independently exploited in particle physics and condensed matter: - [Gell-Mann, Low, Dyson] “dress” an elementary particle with an effective (renormalized) charge; - [Stueckelberg, Petermann, Kadanoff] block spin transformations to recover scaling laws near critical point. Wilson fused both points of view [Wilson ’71] : � e − S k [ φ < k ] = D φ k ′ e − S Λ [ φ k ′ < k + φ k ′ > k ] . k < k ′ < Λ Fluctuations of higher energy scales are integrated out, generates a flow of the effective action in theory space. Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 5 / 39
Introduction Renormalization Renormalization Group Given a QFT defined by a set of (dimensionless) couplings { g i } i =1 ,... , after regularization, they flow with the probing scale µ as dg i β i := d log µ = f ( g 1 , . . . ) . UV/IR fixed points form universality classes of QFTs, characterized by • symmetries, • spacetime dimensions, • number of degrees of freedom. Relevant, irrelevant, marginal directions. Asymptotic freedom: UV Gaussian fixed point. [Credit: David Tong] A theory is renormalizable if it has a finite number of relevant couplings. Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 6 / 39
Introduction Tensor Models: a survey Tensor Models in 0 dimensions Generalising vector and matrix models, tensor models are: Field: T a 1 ... a r rank r (unsymmetrized) tensor, transforms under G ⊗ r ( G of rank N ): � U ( i ) ∈ G . T ′ U (1) b 1 a 1 . . . U ( r ) b 1 ... b r = b r a r T a 1 ... a r , a Action and Observables: G ⊗ r -invariants ( B “bubbles”). S = S 0 + S int ; � � S 0 ( T , ¯ T a 1 ... a r ¯ t B Tr B ( T , ¯ T ) = T a 1 ... a r ; S int = T ) . a r -colored graphs B � �� � � �� � propagator interaction This action is invariant under the symmetry G ⊗ r . Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 7 / 39
Introduction Tensor Models: a survey Tensor invariants as Colored Graphs Example ( r = 3 , G = U ( N )): � δ a 1 p 1 δ a 2 q 2 δ a 3 r 3 δ b 1 r 1 δ b 2 p 2 δ b 3 q 3 δ c 1 q 1 δ c 2 r 2 δ c 3 p 3 T a 1 a 2 a 3 T b 1 b 2 b 3 T c 1 c 2 c 3 ¯ T p 1 p 2 p 3 ¯ T q 1 q 2 q 3 ¯ T r 1 r 2 r 3 White (black) vertices for T ( ¯ T ). Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 8 / 39
Introduction Tensor Models: a survey Tensor invariants as Colored Graphs Example ( r = 3 , G = U ( N )): � δ a 1 p 1 δ a 2 q 2 δ a 3 r 3 δ b 1 r 1 δ b 2 p 2 δ b 3 q 3 δ c 1 q 1 δ c 2 r 2 δ c 3 p 3 T a 1 a 2 a 3 T b 1 b 2 b 3 T c 1 c 2 c 3 ¯ T p 1 p 2 p 3 ¯ T q 1 q 2 q 3 ¯ T r 1 r 2 r 3 White (black) vertices for T ( ¯ T ). Edges for δ a c q c Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 8 / 39
Introduction Tensor Models: a survey Tensor invariants as Colored Graphs Example ( r = 3 , G = U ( N )): � δ a 1 p 1 δ a 2 q 2 δ a 3 r 3 δ b 1 r 1 δ b 2 p 2 δ b 3 q 3 δ c 1 q 1 δ c 2 r 2 δ c 3 p 3 T a 1 a 2 a 3 T b 1 b 2 b 3 T c 1 c 2 c 3 ¯ T p 1 p 2 p 3 ¯ T q 1 q 2 q 3 ¯ T r 1 r 2 r 3 White (black) vertices for T ( ¯ T ). Edges for δ a c q c colored by c , the position of the index. Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 8 / 39
Introduction Tensor Models: a survey Tensor invariants as Colored Graphs Example ( r = 3 , G = U ( N )): � δ a 1 p 1 δ a 2 q 2 δ a 3 r 3 δ b 1 r 1 δ b 2 p 2 δ b 3 q 3 δ c 1 q 1 δ c 2 r 2 δ c 3 p 3 T a 1 a 2 a 3 T b 1 b 2 b 3 T c 1 c 2 c 3 ¯ T p 1 p 2 p 3 ¯ T q 1 q 2 q 3 ¯ T r 1 r 2 r 3 White (black) vertices for T ( ¯ T ). Edges for δ a c q c colored by c , the position of the index. Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 8 / 39
Introduction Tensor Models: a survey Tensor invariants as Colored Graphs Example ( r = 3 , G = U ( N )): � δ a 1 p 1 δ a 2 q 2 δ a 3 r 3 δ b 1 r 1 δ b 2 p 2 δ b 3 q 3 δ c 1 q 1 δ c 2 r 2 δ c 3 p 3 T a 1 a 2 a 3 T b 1 b 2 b 3 T c 1 c 2 c 3 ¯ T p 1 p 2 p 3 ¯ T q 1 q 2 q 3 ¯ T r 1 r 2 r 3 White (black) vertices for T ( ¯ T ). Edges for δ a c q c colored by c , the position of the index. Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 8 / 39
Introduction Tensor Models: a survey Tensor invariants as Colored Graphs Example ( r = 3 , G = U ( N )): r � � � � � Tr B ( T , ¯ ¯ T ) = T a 1 T q 1 δ a c w q c v ... a r v ... q r w ¯ v ¯ v ¯ e c =( w , ¯ v v ¯ c =1 w ) T 1 1 2 2 2 2 3 3 3 T White (black) vertices for T ( ¯ T ). 3 1 1 1 1 2 2 Edges for δ a c q c colored by c , the 1 1 position of the index. 3 2 2 2 2 3 3 3 3 1 1 1 1 3 2 2 Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 8 / 39
Introduction Tensor Models: a survey Feynman expansion r � � � S ( T , ¯ T b 1 ... b r ¯ t B Tr B ( T , ¯ δ b c q c + T ) = T q 1 ... q r T ) , c =1 r -colored graphs B � TdT ] e − N r − 1 S ( T , ¯ [ d ¯ T ) Z ( t B ) = Feynman expansion: • Taylor expand the interactions ( r -colored graphs) � � T 1 1 e − N r − 1 T ¯ T t B 1 Tr B 1 ( T , ¯ T ) t B 2 Tr B 2 ( T , ¯ 2 2 2 Z ( { t B i } ) = T ) . . . 2 3 3 3 T 3 T , ¯ T 1 1 1 1 2 2 1 1 3 2 2 2 2 3 3 3 3 1 1 1 1 3 2 2 Nicolas Delporte & Vincent Rivasseau (IJCLab) Tensors models and Random Geometry October 13, 2020 9 / 39
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