Constructive Tensor Field Theory through an example

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Constructive Tensor Field Theory through an example

Fabien Vignes-Tourneret

To cite this version:

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Constructive Tensor Field Theory

through an example

Vincent Rivasseau1 _{Fabien Vignes-Tourneret}2 1_{Université Paris-Saclay}

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Outline

Random tensors, random spaces Loop Vertex Expansion

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Random tensors, random spaces

Why? How?

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Random surfaces

• 2D quantum gravity and matrix models:

• Matrix models provide a theory of random discrete surfaces weighted by a discretized Einstein-Hilbert action.

• Evidence for matrices being the right discretization of 2D quantum gravity.

• In the last 10 years, much progress from probabilists:

• Random metric surfaces (e.g. Brownian map).

• Universal limit of large planar maps.

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Why tensor fields?

1.

Generalize matrix models to higher dimensions

• w.r.t. their symmetry properties

• provide a theory of random spaces

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Why tensor fields?

1.

Generalize matrix models to higher dimensions

• w.r.t. their symmetry properties

• provide a theory of random spaces

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Invariant actions

Symmetry

Consider T , T : ZD → 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).

Tn₁n2···nD −→ X m Un(1)1m1U (2) n2m2· · · U (D) nDmDTm1m2···mD

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Invariant actions

Symmetry

Tn₁n2···nD−→ X m Un(1)1m1U (2) n2m2· · · U (D) nDmDTm1m2···mD

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Invariant actions

Symmetry

• Invariants as D-coloured graphs

1 2 4 3

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Invariant actions

Symmetry

• Invariants as D-coloured graphs (bipartite D-reg. properly edge-coloured)

1 2 4 3

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Invariant actions

Symmetry

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Invariant actions

Symmetry

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Invariant actions

Symmetry

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Invariant actions

Symmetry

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Invariant actions

Symmetry

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Invariant actions

Symmetry

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Invariant actions

Feynman graphs

• Action of a tensor model

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

I⊂ {D-coloured graphs of order > 4} interaction vertices

• Feynman graphs

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

• Feynman graphs

= (D + 1)-coloured graphs = D-Triang. spaces

1 1 2

0 0

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

• Feynman graphs

= (D + 1)-coloured graphs = D-Triang. spaces

1 1 2

0 0

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

• Feynman graphs =(D + 1)-coloured graphs

= D-Triang. spaces

1 1 2

0 0

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

• Feynman graphs = (D + 1)-coloured graphs=D-Triang. spaces

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

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Invariant actions

Feynman graphs

S(T , T ) = T · T +X

B∈I

gBTrB[T , T ],

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Loop Vertex Expansion

Why? How?

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Originally designed for random matrices. [Rivasseau 2007]

• Initial goals:

1. Constructive φ∗44 ,

2. Simplify Bosonic constructive theory.

1. Classical constructive techniques unsuited to matrices/tensors.

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

1. Constructive φ∗44 ,

1. Classical constructivetechniquesunsuited to matrices/tensors.

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

n + 2 faces at order n

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

4n indices

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

2n + 2 free indices

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

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Loop Vertex Expansion

Motivations

LVE = main constructive tool for matrices and tensors

• Initial goals:

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The BKAR forest formula

• _{Fix an integer n > 2.}

• f a function of n(n−1)₂ variables x`, sufficiently differentiable. • Kn, complete graph on {1, 2, . . . , n}. #E (Kn) = n(n−1)₂ Then, f (1, 1, . . . , 1) =X F Z dwF∂Ff (XF(wF)) where

• the sum is over spanning forests of Kn, • R dwF:=Q`∈E (F ) R1 0 dw`, • ∂F :=Q`∈E (F ) ∂ ∂x`, • XF = (x_`F)`∈E (Kn) with x F

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The BKAR forest formula

• _{Fix an integer n > 2.}

• f a function of n(n−1)₂ variables x`, sufficiently differentiable. • Kn, complete graph on {1, 2, . . . , n}. #E (Kn) = n(n−1)₂ Then, f (1, 1, . . . , 1) =X F Z dwF∂Ff (XF(wF)) where

• the sum is over spanning forests of Kn, • R dwF:=Q`∈E (F ) R1 0 dw`, • ∂F :=Q`∈E (F ) ∂ ∂x`, • XF = (x_`F)`∈E (Kn) with x F

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Loop Vertex Expansion

Analyticity of the free energy of φ4 0 Z (λ) = Z R e−λ2φ 4 d µ(φ), d µ(φ) = √d φ 2πe −1 2φ 2

Theorem

log Z is analytic in the cardioid domain_{λ ∈ C : |λ| <} 1 2cos

2₍1

2arg λ) .

Proof. LVE is done in 3 steps: 1. Intermediate field representation,

2. Replication of fields,

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Loop Vertex Expansion

Analyticity of the free energy of φ4 0 Z (λ) = Z R e−λ2φ 4 d µ(φ), d µ(φ) = √d φ 2πe −1 2φ 2

Theorem

log Z is analytic in the cardioid domain_{λ ∈ C : |λ| <} 1 2cos

2₍1

2arg λ) .

Proof. LVE is done in 3 steps: 1. Intermediate field representation,

2. Replication of fields,

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Loop Vertex Expansion

Analyticity of the free energy of φ4 0

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Loop Vertex Expansion

1. Intermediate field representation:

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Loop Vertex Expansion

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Loop Vertex Expansion

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Loop Vertex Expansion

Analyticity of the free energy of φ4 0 log Z =X n>1 1 n! X T ⊂Kn Z dwT Z _h Y `∈E (T ) ∂ ∂σi (`) ∂ ∂σj(`) n Y i =1 V (σi) i d µCT_{(w )}(~σ) =1 2 X n>1 (−λ/2)n−1 n! X T ⊂Kn Z dwT Z _h n Y i =1 (di− 1)! (1 − ıσi √ λ)di i d µCT_{(w )}(~σ). Using 1 − ıσ √ λ >cos( 1 2arg λ), we get | log Z | 61 2 ∞ X n=1 1 n! |λ| 2 cos2₍1 2arg λ) n−1 X T ⊂Kn n Y i =1 (di− 1)! 6 2 ∞ X n=1 2|λ| cos2₍1 2arg λ) n−1

which is convergent for all λ ∈ C such that |λ| <1 2cos

2₍1

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Loop Vertex Expansion

Analyticity of the free energy of φ4 0 log Z =X n>1 1 n! X T ⊂Kn Z dwT Z _h Y `∈E (T ) ∂ ∂σi (`) ∂ ∂σj(`) n Y i =1 V (σi) i d µCT_{(w )}(~σ) =1 2 X n>1 (−λ/2)n−1 n! X T ⊂Kn Z dwT Z _h n Y i =1 (di− 1)! (1 − ıσi √ λ)di i d µCT_{(w )}(~σ). Using 1 − ıσ √ λ >cos( 1 2arg λ), we get | log Z | 61 2 ∞ X n=1 1 n! |λ| 2 cos2₍1 2arg λ) n−1 X T ⊂Kn n Y i =1 (di− 1)! 6 2 ∞ X n=1 2|λ| cos2₍1 2arg λ) n−1

which is convergent for all λ ∈ C such that |λ| <1 2cos

2₍1

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An example at work

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The T

₄4

field theory

• Tensor fields: T : Z4→ C, Tn, Tn with n, n ∈ Z4. • Free action: • Interactions: V (T , T ) = g₂ 4 X c=1 Vc(T , T ), Vc(T , T ) = c T T T T

• Formal partition function:

Z0(g ) = Z

e−g2 P

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The T

₄4

field theory

• Tensor fields: T : Z4→ C, Tn, Tn with n, n ∈ Z4. • Free action: d µC(T , T ) = Y n,n dTnd Tn 2i π det(C−1) e− P n,nTnC −1 nn Tn_, Cn,n= (1_6jmax)nn n2_{+ 1} δn,n, n 2 := n12+ n 2 2+ n 2 3+ n 2 4, 1_6jmax= 1n2_+16M2jmaxδn,n. • Interactions: V (T , T ) = g₂ 4 X c=1 Vc(T , T ), Vc(T , T ) = c T T T T

Z0(g ) = Z

e−g2 P

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The T

₄4

field theory

• Tensor fields: T : Z4→ C, Tn, Tn with n, n ∈ Z4. • Free action: Cn,n= δn,n (1_6jmax)nn n2_{+ 1} , n 2 := n21+ n 2 2+ n 2 3+ n 2 4. • Interactions: V (T , T ) = g₂ 4 X c=1 Vc(T , T ), Vc(T , T ) = c T T T T

Z0(g ) = Z

e−g2 P

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The T

₄4

field theory

• Tensor fields: T : Z4→ C, Tn, Tn with n, n ∈ Z4. • Free action: Cn,n= δn,n (1_6jmax)nn n2_{+ 1} , n 2 := n21+ n 2 2+ n 2 3+ n 2 4. • Interactions: V (T , T ) = g₂ 4 X c=1 Vc(T , T ), Vc(T , T ) = c T T T T

Z0(g ) = Z

e−g2 P

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The T

₄4

field theory

Perturbative renormalisation

Proposition

T4

4 is renormalisable to all orders of perturbation.

• Power counting similar to the one of φ4 3.

• 2 divergent 2-point graphs:

• 7 divergent melonic vacuum graphs

• 3 divergent non melonic vacuum graphs:

N1

c c

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The T

₄4

field theory

Proposition

T4

N1

c c

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The T

₄4

field theory

Proposition

T4

c Mc 1 c Mc 2

N1

c c

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The T

₄4

field theory

Proposition

T4

• 2 divergent 2-point graphs: Mc₁, Mc₂ • 7 divergent melonic vacuum graphs

N1

c c

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The T

₄4

field theory

Proposition

T4

• 2 divergent 2-point graphs: Mc₁, Mc₂

• 7 divergent melonic vacuum graphs: Vi, i = 1, 2, . . . , 7 • 3 divergent non melonic vacuum graphs:

N1

c c

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The T

₄4

field theory

Analyticity

• Renormalised partition function:

Zjmax(g ) = N Z e−g2 P cVc(T ,T )+T · T P G∈M (−g )|G| SG δG d µC(T , T ), M = {Mc 1, M c 2, c = 1, 2, 3, 4} ,

log N = (finite) sum of the counterterms of the divergent vacuum connected graphs.

Theorem (Rivasseau, V.-T. 2017)

There exists ρ > 0 such that limjmax→∞log Zjmax(g ) is analytic in the cardioid domain defined by |arg g | < π and |g | < ρ cos2₍1

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The T

₄4

field theory

Analyticity

• Renormalised partition function:

Zjmax(g ) = N Z e−g2 P cVc(T ,T )+T · T P G∈M (−g )|G| SG δG d µC(T , T ), M = {Mc 1, M c 2, c = 1, 2, 3, 4} ,

log N = (finite) sum of the counterterms of the divergent vacuum connected graphs.

Theorem (Rivasseau, V.-T. 2017)

There exists ρ > 0 such that limjmax→∞log Zjmax(g ) is analytic in the cardioid domain defined by |arg g | < π and |g | < ρ cos2₍1

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The general strategy

0. Renormalised partition function:

Zjmax(g ) = N Z e− g 2 P cVc(T ,T )+T · T P G∈M (−g )|G| SG δG d µC(T , T ).

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The general strategy

Zjmax(g ) = N Z e− g 2 P cVc(T ,T )+T · T P G∈M (−g )|G| SG δG d µC(T , T ). 1. Intermediate field representation: σc ∈ HermMjmax, c = 1, 2, 3, 4

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The general strategy

Zjmax(g ) = N Z e− g 2 P cVc(T ,T )+T · T P G∈M (−g )|G| SG δG d µC(T , T ). 1. Intermediate field representation: σc ∈ HermMjmax

Zj_max(g ) = N eδt Z e− Tr log(I −Σ)−ıλPcδ c mTrcσc_{d ν} I(~σ). 2. Renormalised action: Zjmax(g ) = Z

e− Tr log3(I −U)−Tr(D1Σ2)−λ22 :σ · Qσ:_{d ν} I(~σ)

U = Σ + D1+ D2

D1= −λ2C1/2ArM1C1/2, D2= λ4C1/2ArM2C1/2 log₃_{(I −U) = log(I −U) + U +}U

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The general strategy

Zj_max(g ) = N eδt Z e− Tr log(I −Σ)−ıλPcδ c mTrcσc_{d ν} I(~σ). 2. Renormalised action: Zjmax(g ) = Z

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The general strategy

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The general strategy

Zj_max(g ) = N eδt Z e− Tr log(I −Σ)−ıλPcδ c mTrcσc_{d ν} I(~σ). 2. Renormalised action: Zjmax(g ) = Z

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The general strategy

4. Multiscale Loop Vertex Expansion: [Gurau, Rivasseau 2014] • 2 forest formulas on top of each other (2-jungle formula) • First, a Bosonic forest then a Fermionic one

log Zjmax(g ) = ∞ X n=1 1 n! X J tree jmax X j1=1 · · · jmax X jn=1 Z d wJ Z d νJ∂J hY B Y a∈B −χB_jaWj_a(σa)χB_jai • wJ = weakening parameters w`, ` ∈ E (J )

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The general strategy

4. Multiscale Loop Vertex Expansion: [Gurau, Rivasseau 2014] • 2 forest formulas on top of each other (2-jungle formula) • First, a Bosonic forest then a Fermionic one

log Zjmax(g ) = ∞ X n=1 1 n! X J tree jmax X j1=1 · · · jmax X jn=1 Z d wJ Z d νJ ∂J hY B Y a∈B −χB_jaWj_a(σa)χB_jai = ∞ X n=1 1 n! X J tree jmax X j1=1 · · · jmax X jn=1 Z d wJ Y B Y a,b∈B a6=b (1 − δjajb) IB IB= Z ∂B Y a∈B Wj_a(σa) d νB= X G Z Y a∈B e−Vja(~σa)_A G(σ) d νB(σ)

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The general strategy

Bosonic bounds |IB| 6 Z Y a∈B e2|Vja(~σa)|_{d ν} B | {z } INP B , non-perturbative 1/2 X G Z |AG(σ)|2d νB | {z } IP B, perturbative 1/2 5. Non-perturbative bound:

Theorem

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The general strategy

Bosonic bounds |IB| 6 Z Y a∈B e2|Vja(~σa)|_{d ν} B | {z } INP B , non-perturbative 1/2 X G Z |AG(σ)|2d νB | {z } IP B, perturbative 1/2 6. Perturbative bound:

Theorem

Let B be a Bosonic block. Then there exists K ∈ R∗+ such that

I_BP_{(G) 6 K}|B|−1 (|B| − 1)!37/2

ρe(G) Y a∈B

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The general strategy

0. Renormalised partition function

1. Intermediate field representation

2. Renormalised action

3. Multiscale decomposition

4. Multiscale Loop Vertex Expansion

5. Non-perturbative Bosonic bound

6. Perturbative Bosonic bound

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The general strategy

0. Renormalised partition function

1. Intermediate field representation

2. Renormalised action

3. Multiscale decomposition

4. Multiscale Loop Vertex Expansion

5. Non-perturbative Bosonic bound

6. Perturbative Bosonic bound

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Conclusion and perspectives

• Tensor field theory provides a combinatorial theory of random D-spaces.

• In the last 8 years, a lot of results with tensors:

1

N-expansion, perturbative and constructive renormalisation, continuum

limit of the dominant triangulations, double scaling limit, uniform random complexes etc.

• Regarding T4

4, one could also prove Borel summability and analyticity of the connected correlation functions.

• T₅4(just renormalisable, asymptotically free)

• New Loop Vertex Representation [Rivasseau 2017]

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Conclusion and perspectives

• Tensor field theory provides a combinatorial theory of random D-spaces.

• In the last 8 years, a lot of results with tensors:

1

N-expansion, perturbative and constructive renormalisation, continuum

limit of the dominant triangulations, double scaling limit, uniform random complexes etc.

• Regarding T4

4, one could also prove Borel summability and analyticity of the connected correlation functions.

• T₅4(just renormalisable, asymptotically free)

• New Loop Vertex Representation [Rivasseau 2017]

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Non-perturbative bounds

Theorem

For ρ small enough and for any value of the w interpolating parameters, I_BNP= Z Y a∈B e2|Vja(~σa)|_{d ν} B6 O(1)|B|eO(1)ρ 3/2_|B| . Proof.

1. Expand each node:

e2|Vja|₌ pa X k=0 (2|Vja|) k k! + Z 1 0 dtj_a(1 − tja) pa(2|Vja|) pa+1 pa! e 2tja|Vja|_.

2. Crude non-perturbative bound: (Quadratic bound)

Z Y a∈B e2|Vja(~σ a_)| d νB6 K|B|eK 0_ρMj1 .

3. Power counting (via quartic bound)beats both combinatorics and

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Perturbative bound

Theorem

Let B be a Bosonic block. Then there exists K ∈ R∗+ such that

I_BP(G) = Z |AG(σ)|2d νB6 K|B|−1 (|B| − 1)! 37/2 ρe(G) Y a∈B M−481ja_.

• AG(σ) depends on σ (essentially) through resolvents.

• If not for resolvents, AG would be the amplitude of a convergent graph.