Some ideas about constructive tensor field theory Vincent Rivasseau 1 - PowerPoint PPT Presentation

Some ideas about constructive tensor field theory Vincent Rivasseau 1 Fabien Vignes-Tourneret 2 1 Université Paris-Saclay 2 CNRS & Université de Lyon

Outline Random tensors, random spaces Loop Vertex Expansion A constructive result for tensors

Random tensors, random spaces Random tensors, random spaces Why? How? Loop Vertex Expansion A constructive result for tensors

Why tensor fields? 1. Generalize matrix models to higher dimensions • w.r.t. their symmetry properties, • provide a theory of random spaces. 2. Define a canonical way of summing over spaces 3. Implement a geometrogenesis scenario • spacetime from scratch, • background independent.

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry .

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m • Invariants as D -coloured graphs 4 3 � n i T n 1 n 2 n 3 n 4 T n 1 n 2 n 3 n 4 =: T · T T T 2 1

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m • Invariants as D -coloured graphs (bipartite D -reg. properly edge-coloured) 4 3 � n i T n 1 n 2 n 3 n 4 T n 1 n 2 n 3 n 4 =: T · T T T 2 1

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m • Invariants as D -coloured graphs (bipartite D -reg. properly edge-coloured) 1 T T 3 4 � m i , n j T m 1 m 2 m 3 m 4 T m 1 n 2 m 3 m 4 T n 1 n 2 n 3 n 4 T n 1 m 2 n 3 n 4 =: Tr 4 [ T , T ] 1 2 2 3 2 2 4 3 4 1 3 1 T T

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m • Invariants as D -coloured graphs (bipartite D -reg. properly edge-coloured) 1 T T 3 4 � m i , n j T m 1 m 2 m 3 m 4 T m 1 n 2 m 3 m 4 T n 1 n 2 n 3 n 4 T n 1 m 2 n 3 n 4 =: Tr 4 [ T , T ] 1 2 2 3 2 2 4 3 4 1 3 1 T T

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m • Invariants as D -coloured graphs (bipartite D -reg. properly edge-coloured) 1 T T 3 4 � m i , n j T m 1 m 2 m 3 m 4 T m 1 n 2 m 3 m 4 T n 1 n 2 n 3 n 4 T n 1 m 2 n 3 n 4 =: Tr 4 [ T , T ] 1 2 2 3 2 2 4 3 4 1 3 1 T T

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m • Invariants as D -coloured graphs (bipartite D -reg. properly edge-coloured) 1 T T 3 4 � m i , n j T m 1 m 2 m 3 m 4 T m 1 n 2 m 3 m 4 T n 1 n 2 n 3 n 4 T n 1 m 2 n 3 n 4 =: Tr 4 [ T , T ] 1 2 2 3 2 2 4 3 4 1 3 1 T T

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m • Invariants as D -coloured graphs (bipartite D -reg. properly edge-coloured) 1 T T 3 4 � m i , n j T m 1 m 2 m 3 m 4 T m 1 n 2 m 3 m 4 T n 1 n 2 n 3 n 4 T n 1 m 2 n 3 n 4 =: Tr 4 [ T , T ] 1 2 2 3 2 2 4 3 4 1 3 1 T T

Invariant actions Symmetry Consider T , T : Z D → C , complex rank D tensors with no symmetry . • Matrix models: invariant under (at most) two copies of U ( N ). Tensor models (rank D ): invariant under D copies of U ( N ). � (1) (2) (D) T n 1 n 2 ··· n D − → U n 1 m 1 U n 2 m 2 · · · U n D m D T m 1 m 2 ··· m D m • Invariants as D -coloured graphs (bipartite D -reg. properly edge-coloured) 1 T T 2 � n 1 n 2 n 3 n 4 T n 1 n 2 m 3 m 4 =: Tr m i , n j T m 1 m 2 m 3 m 4 T m 1 m 2 n 3 n 4 T 4 [ T , T ] 1 4 3 3 4 2 4 3 3 2 1 2 1 T T

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs 0 0 1 1 2 0 0

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs 0 0 1 1 2 0 0

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs 0 0 1 1 2 0 0

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs = D -Triang. spaces

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs = D -Triang. spaces D -simplex vertex 1 1 3 0 2 0 0 2 2 1 3 3

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs = D -Triang. spaces half-edge ( D − 1)-face 1 1 3 0 2 0 0 2 2 1 3 3

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs = D -Triang. spaces half-edge ( D − 1)-face 1 1 3 0 2 0 0 2 2 1 3 3

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs = D -Triang. spaces half-edge ( D − 1)-face 1 1 3 0 2 0 0 2 2 1 3 3

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs = D -Triang. spaces half-edge ( D − 1)-face 1 1 3 0 2 0 0 2 2 1 3 3

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs = D -Triang. spaces D -simplex vertex 1 1 3 0 2 0 0 2 2 1 3 3

Invariant actions Feynman graphs • Action of a tensor model � S ( T , T ) = T · T + g B Tr B [ T , T ] , B∈ I I ⊂ { D -coloured graphs of order � 4 } interaction vertices • Feynman graphs = ( D + 1)-coloured graphs = D -Triang. spaces edge gluing 1 3 2 3 0 2 1 0 0 1 2 3

Loop Vertex Expansion Random tensors, random spaces Loop Vertex Expansion Why? How? A constructive result for tensors

Constructive field theory A functional integral point of view • Aim: get some control on connected quantities via the derivation of tractable formulas for the logarithm of correlation functions (say).

Constructive field theory The classical approach • Aim: get some control on connected quantities via the derivation of tractable formulas for the logarithm of correlation functions (say). • How? By finding an expansion which interpolates between the functional integral and the perturbative series.