[PPT] - Random tensors, a functional integral point of view R azvan Gur PowerPoint Presentation

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Random tensors, a “functional integral” point of view

R˘ azvan Gur˘ au IHP, 2016

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Introduction Tensor Models The quartic model The continuum limit is a phase transition

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Why random discretized spaces?

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Why random discretized spaces?

Try to make sense of:

topologies
Dg(metrics) DXmatter e−S

S ∼ κR √gR − κV √g + κmSm .

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Why random discretized spaces?

Try to make sense of:

topologies
Dg(metrics) DXmatter e−S

S ∼ κR √gR − κV √g + κmSm . Idea: Replace the sum over topologies and metrics with a sum over random discretizations

topologies
Dg(metrics) →
random discretizations

.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Why random discretized spaces?

Try to make sense of:

topologies
Dg(metrics) DXmatter e−S

S ∼ κR √gR − κV √g + κmSm . Idea: Replace the sum over topologies and metrics with a sum over random discretizations

topologies
Dg(metrics) →
random discretizations

. But what measure should one use over the random discretizations?

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Matrix and tensor models

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Matrix and tensor models

Matrix and tensor models are generating functions for random discrete geometries

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Matrix and tensor models

Matrix and tensor models are generating functions for random discrete geometries Field theories with no kinetic term (probability measures) invariant under conjugation by the unitary group U(N)

◮ “path integrals” for matrix (tensor) “fields” ◮ Feynman graphs dual to discretized D-dimensional spaces ◮ the weight of a discretization is fixed by the Feynman rules: canonical

measures over random discretizations.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Matrix and tensor models

Matrix and tensor models are generating functions for random discrete geometries Field theories with no kinetic term (probability measures) invariant under conjugation by the unitary group U(N)

◮ “path integrals” for matrix (tensor) “fields” ◮ Feynman graphs dual to discretized D-dimensional spaces ◮ the weight of a discretization is fixed by the Feynman rules: canonical

measures over random discretizations.

◮ bonus: the perturbative expansion can be reorganized in powers of 1/N. ◮ the perturbative series diverges, but the series at fixed order in 1/N converges.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

From Matrix to Tensor Models

Invariant action for a matrix Mab Invariant action for a complex, generic tensor Ta1...aD ribbon graphs ↔ surfaces colored graphs ↔ D dimensional spaces

1 2 1 2 3

g(G) ≥ 0 genus ω(G) ≥ 0 degree 1/N expansion A(G) = N2−2g(G) 1/N expansion A(G) = ND−

2 (D−1)! ω(G)

leading order: g(G) = 0, spheres. leading order: ω(G) = 0, spheres.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Introduction Tensor Models The quartic model The continuum limit is a phase transition

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Tensor invariants as Edge Colored Graphs

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Tensor invariants as Edge Colored Graphs

Building blocks: tensors with no symmetry transforming as T ′

b1...bD =

U(1)

b1a1 . . . U(D) bDaDTa1...aD ,

¯ T ′

p1...pD =

¯ U(1)

p1q1 . . . ¯

U(D)

pDqD ¯

Tq1...qD

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Tensor invariants as Edge Colored Graphs

Building blocks: tensors with no symmetry transforming as T ′

b1...bD =

U(1)

b1a1 . . . U(D) bDaDTa1...aD ,

¯ T ′

p1...pD =

¯ U(1)

p1q1 . . . ¯

U(D)

pDqD ¯

Tq1...qD Invariants: colored graphs TrB(T, ¯ T) =

v

Ta1

v...aD v

¯

v

¯ Tq1

¯ v...qD ¯ v

D

c=1
ec=(w, ¯

w)

δac

wqc ¯ w T T 1 2 3 1 3 2 2 1 3 1 2 2 2 1 1 3 1 2 1 1 2 2 3 3 3 1 2 2 3 1 3 2 3 1

◮ White (black) vertices for T ( ¯

T).

◮ Edges for δacqc colored by c, the position of the index.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Invariant Actions for Tensor Models

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Invariant Actions for Tensor Models

The most general single trace tensor model S(T, ¯ T) = 1 λ

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) ZTM(tB) =

[d ¯

TdT] e−ND−1S(T, ¯

T)

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Invariant Actions for Tensor Models

The most general single trace tensor model S(T, ¯ T) = 1 λ

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) ZTM(tB) =

[d ¯

TdT] e−ND−1S(T, ¯

T)

Feynman graphs: “vertices” B.

T T 1 2 3 1 3 2 2 1 3 1 2 2 2 1 1 3 1 2 1 1 2 2 3 3 3 1 2 2 3 1 3 2 3 1

¯

T,T

e−ND−1

1 λ

Ta1...aD ¯ Tq1...qD D

c=1 δac qc

TrB1( ¯ T, T)TrB2( ¯ T, T) . . .

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Invariant Actions for Tensor Models

The most general single trace tensor model S(T, ¯ T) = 1 λ

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) ZTM(tB) =

[d ¯

TdT] e−ND−1S(T, ¯

T)

Feynman graphs: “vertices” B.

T T 1 2 3 1 3 2 2 1 3 1 2 2 2 1 1 3 1 2 1 1 2 2 3 3 3 1 2 2 3 1 3 2 3 1

¯

T,T

e−ND−1

1 λ

Ta1...aD ¯ Tq1...qD D

c=1 δac qc

δ...
Ta1a2a3 ¯

Tp1p2p3 . . .

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Invariant Actions for Tensor Models

The most general single trace tensor model S(T, ¯ T) = 1 λ

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) ZTM(tB) =

[d ¯

TdT] e−ND−1S(T, ¯

T)

Feynman graphs: “vertices” B. Gaussian integral: Wick contractions of T and ¯ T (“propagators”) → dashed edges to which we assign the fictitious color 0.

T T 1 2 3 1 3 2 2 1 3 1 2 2 2 1 1 3 1 2 1 1 2 2 3 3 3 1 2 2 3 1 3 2 3 1

¯

T,T

e−ND−1

1 λ

Ta1...aD ¯ Tq1...qD D

c=1 δac qc

δ...
Ta1a2a3 ¯

Tp1p2p3

∼

λ ND−1 δa1p1δa2p2δa3p3

. . .

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Invariant Actions for Tensor Models

The most general single trace tensor model S(T, ¯ T) = 1 λ

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) ZTM(tB) =

[d ¯

TdT] e−ND−1S(T, ¯

T)

Feynman graphs: “vertices” B. Gaussian integral: Wick contractions of T and ¯ T (“propagators”) → dashed edges to which we assign the fictitious color 0.

T T 1 2 3 1 3 2 2 1 3 1 2 2 2 1 1 3 1 2 1 1 2 2 3 3 3 1 2 2 3 1 3 2 3 1

Graphs G with D + 1 colors.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Invariant Actions for Tensor Models

The most general single trace tensor model S(T, ¯ T) = 1 λ

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) ZTM(tB) =

[d ¯

TdT] e−ND−1S(T, ¯

T)

Feynman graphs: “vertices” B. Gaussian integral: Wick contractions of T and ¯ T (“propagators”) → dashed edges to which we assign the fictitious color 0.

T T 1 2 3 1 3 2 2 1 3 1 2 2 2 1 1 3 1 2 1 1 2 2 3 3 3 1 2 2 3 1 3 2 3 1

Graphs G with D + 1 colors. Represent triangulated D dimensional spaces.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The general framework

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The general framework

Observables = invariants TrB encoding boundary triangulations.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The general framework

Observables = invariants TrB encoding boundary triangulations. Expectations =

TrB1TrB2 . . . TrBq
=

1 ZTM(tB)

[d ¯

TdT] TrB1TrB2 . . . TrBq e−ND−1S(T, ¯

T)

correlations between boundary states given by sums over all bulk triangulations compatible with the boundary states

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The general framework

Observables = invariants TrB encoding boundary triangulations. Expectations =

TrB1TrB2 . . . TrBq
=

1 ZTM(tB)

[d ¯

TdT] TrB1TrB2 . . . TrBq e−ND−1S(T, ¯

T)

correlations between boundary states given by sums over all bulk triangulations compatible with the boundary states

◮

TrB
: B to vacuum amplitude

◮

TrB1TrB2
c =
TrB1TrB2
−
TrB1
TrB2
: transition amplitude between

the boundary states B1 and B2

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Amplitudes and Dynamical Triangulations

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Amplitudes and Dynamical Triangulations

Set tB ∼ (−1)N

−

2 (D−2)! ω(B) and expand in Feynman graphs ln ZTM(λ; N) = G AG(λ; N):

AG(λ; N) = λ#0−edgesN

D− 2 (D−1)! ω(G) 10

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Amplitudes and Dynamical Triangulations

Set tB ∼ (−1)N

−

2 (D−2)! ω(B) and expand in Feynman graphs ln ZTM(λ; N) = G AG(λ; N):

AG(λ; N) = λ#0−edgesN

D− 2 (D−1)! ω(G)

= e

κD−2(λ,N)

#(D−2)−simplices
−κD (λ,N)
#D−simplices
.

Discretized Einstein Hilbert action on the equilateral triangulation dual to G.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Amplitudes and Dynamical Triangulations

Set tB ∼ (−1)N

−

2 (D−2)! ω(B) and expand in Feynman graphs ln ZTM(λ; N) = G AG(λ; N):

AG(λ; N) = λ#0−edgesN

D− 2 (D−1)! ω(G)

= e

κD−2(λ,N)

#(D−2)−simplices
−κD (λ,N)
#D−simplices
.

Discretized Einstein Hilbert action on the equilateral triangulation dual to G.

topologies
[Dg] e−

1 16πG dD x√g(2Λ−R) →

Triangulations

edge length a

e−Sdiscr.

EH (G,Λ;a) =

1 ND ln ZTM(λ; N) ,

G aD−2 ≡ ˜ G = c1 1 ln N , Λa2 ≡ ˜ Λ = c2 ˜ G ln 1 λ

+ c3 ,

c1, c2, c3 > 0 (∼ O(1)) . 10

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Amplitudes and Dynamical Triangulations

Set tB ∼ (−1)N

−

2 (D−2)! ω(B) and expand in Feynman graphs ln ZTM(λ; N) = G AG(λ; N):

AG(λ; N) = λ#0−edgesN

D− 2 (D−1)! ω(G)

= e

κD−2(λ,N)

#(D−2)−simplices
−κD (λ,N)
#D−simplices
.

Discretized Einstein Hilbert action on the equilateral triangulation dual to G.

topologies
[Dg] e−

1 16πG dD x√g(2Λ−R) →

Triangulations

edge length a

e−Sdiscr.

EH (G,Λ;a) =

1 ND ln ZTM(λ; N) ,

G aD−2 ≡ ˜ G = c1 1 ln N , Λa2 ≡ ˜ Λ = c2 ˜ G ln 1 λ

+ c3 ,

c1, c2, c3 > 0 (∼ O(1)) .

˜ G ˜ Λ ˜ Λ( ˜ G)

c1 ln N

α λ → 0 N → ∞ 10

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Introduction Tensor Models The quartic model The continuum limit is a phase transition

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The quartic tensor model

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The quartic tensor model

S(T, ¯ T) =

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T)

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The quartic tensor model

S(T, ¯ T) =

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) The simplest quartic invariants correspond to “melonic” graphs with four vertices B(4),c

Ta1...aD ¯

Tq1...qD

c′=c

δac′ qc′

δac pc δbc qc
Tb1...bD ¯

Tp1...pD

c′=c

δbc′ pc′

c

c

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The quartic tensor model

S(T, ¯ T) =

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) The simplest quartic invariants correspond to “melonic” graphs with four vertices B(4),c

Ta1...aD ¯

Tq1...qD

c′=c

δac′ qc′

δac pc δbc qc
Tb1...bD ¯

Tp1...pD

c′=c

δbc′ pc′

c

c

The simplest interacting theory: coupling constants tB =

g

2 , B = B(4),c

0 , otherwise

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The quartic tensor model

S(T, ¯ T) =

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc +

B

tBTrB( ¯ T, T) The simplest quartic invariants correspond to “melonic” graphs with four vertices B(4),c

Ta1...aD ¯

Tq1...qD

c′=c

δac′ qc′

δac pc δbc qc
Tb1...bD ¯

Tp1...pD

c′=c

δbc′ pc′

c

c

The simplest interacting theory: coupling constants tB =

g

2 , B = B(4),c

0 , otherwise The simplest observable: K2 = 1 N

Ta1...aD ¯

Tq1...qD

D

c=1

δacqc

=
G

(−g)

#0−edges 2

N−

2 (D−1)! ω(G) 12

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The 1/N expansion

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The 1/N expansion

Two parameters: g and N.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The 1/N expansion

Two parameters: g and N. 1) Feynman expansion: K2 = 1 − Dg −

1 ND−2 Dg + G AG (N) , AG (N) ∼ g2 13

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The 1/N expansion

Two parameters: g and N. 1) Feynman expansion: K2 = 1 − Dg −

1 ND−2 Dg + G AG (N) , AG (N) ∼ g2

2) 1/N expansion: K2 = (1+4Dg)

1 2 −1 2Dg

+

G AG(N), AG (N) ≤ 1 ND−2 13

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The 1/N expansion

Two parameters: g and N. 1) Feynman expansion: K2 = 1 − Dg −

1 ND−2 Dg + G AG (N) , AG (N) ∼ g2

2) 1/N expansion: K2 = (1+4Dg)

1 2 −1 2Dg

+

G AG(N), AG (N) ≤ 1 ND−2

3) non perturbative: K2 = (1+4Dg)

1 2 −1 2Dg

+ . . . + R(p)

N (g), |R(p) N (g)| ≤ 1 Np(D−2) |g|p

cos arg g

2 2p+2 p! A Bp analytic

in |g| ≤

1 4D

cos arg g

2 2 . 13

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The 1/N expansion

Two parameters: g and N. 1) Feynman expansion: K2 = 1 − Dg −

1 ND−2 Dg + G AG (N) , AG (N) ∼ g2

2) 1/N expansion: K2 = (1+4Dg)

1 2 −1 2Dg

+

G AG(N), AG (N) ≤ 1 ND−2

3) non perturbative: K2 = (1+4Dg)

1 2 −1 2Dg

+ . . . + R(p)

N (g), |R(p) N (g)| ≤ 1 Np(D−2) |g|p

cos arg g

2 2p+2 p! A Bp analytic

in |g| ≤

1 4D

cos arg g

2 2 .

4) critical behavior: limN→∞ K2 = (1+4Dg)

1 2 −1 2Dg

becomes critical for g → −(4D)−1.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The 1/N expansion

Two parameters: g and N. 1) Feynman expansion: K2 = 1 − Dg −

1 ND−2 Dg + G AG (N) , AG (N) ∼ g2

2) 1/N expansion: K2 = (1+4Dg)

1 2 −1 2Dg

+

G AG(N), AG (N) ≤ 1 ND−2

3) non perturbative: K2 = (1+4Dg)

1 2 −1 2Dg

+ . . . + R(p)

N (g), |R(p) N (g)| ≤ 1 Np(D−2) |g|p

cos arg g

2 2p+2 p! A Bp analytic

in |g| ≤

1 4D

cos arg g

2 2 .

4) critical behavior: limN→∞ K2 = (1+4Dg)

1 2 −1 2Dg

becomes critical for g → −(4D)−1. 5) double scaling (D = 3, 4, 5): g = − 1

4D + x ND−2 , K2 = N1− D 2 p≥0 cp xp− 1 2

+ Rest, Rest < N1−D/2 .

Send N → ∞, g → − 1

4D while keeping x fixed: explore terms subleading in 1/N.

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Introduction Tensor Models The quartic model The continuum limit is a phase transition

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Continuum limit

15

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Continuum limit

critical point ⇔ continuum limit

15

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Continuum limit

critical point ⇔ continuum limit limN→∞ K2 is the sum of an infinite family of triangulations of the sphere (“melons”):

lim

N→∞ K2 = (1 + 4Dg) 1 2 − 1

2Dg =

melons

g

#D−simplices 4 15

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Continuum limit

critical point ⇔ continuum limit limN→∞ K2 is the sum of an infinite family of triangulations of the sphere (“melons”):

lim

N→∞ K2 = (1 + 4Dg) 1 2 − 1

2Dg =

melons

g

#D−simplices 4

Physical volume: V = aD (#D − simplices) ∼ aDg∂g ln K2 ∼

aD g−(−4D)−1 15

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Continuum limit

critical point ⇔ continuum limit limN→∞ K2 is the sum of an infinite family of triangulations of the sphere (“melons”):

lim

N→∞ K2 = (1 + 4Dg) 1 2 − 1

2Dg =

melons

g

#D−simplices 4

Physical volume: V = aD (#D − simplices) ∼ aDg∂g ln K2 ∼

aD g−(−4D)−1

g → −(4D)−1 ⇒ (#D − simplices) → ∞

15

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Continuum limit

critical point ⇔ continuum limit limN→∞ K2 is the sum of an infinite family of triangulations of the sphere (“melons”):

lim

N→∞ K2 = (1 + 4Dg) 1 2 − 1

2Dg =

melons

g

#D−simplices 4

Physical volume: V = aD (#D − simplices) ∼ aDg∂g ln K2 ∼

aD g−(−4D)−1

g → −(4D)−1 ⇒ (#D − simplices) → ∞

Continuum limit: send g → (−4D)−1, a → 0 keeping the physical volume fixed: infinitely refined triangulations dominate.

15

SLIDE 52

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Continuum limit

critical point ⇔ continuum limit limN→∞ K2 is the sum of an infinite family of triangulations of the sphere (“melons”):

lim

N→∞ K2 = (1 + 4Dg) 1 2 − 1

2Dg =

melons

g

#D−simplices 4

Physical volume: V = aD (#D − simplices) ∼ aDg∂g ln K2 ∼

aD g−(−4D)−1

g → −(4D)−1 ⇒ (#D − simplices) → ∞

Continuum limit: send g → (−4D)−1, a → 0 keeping the physical volume fixed: infinitely refined triangulations dominate.

This continuum limit is a phase transition associated to a symmetry breaking in the tensor model

15

SLIDE 53

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Phase transitions in field theory

16

SLIDE 54

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Phase transitions in field theory

Phase transition ⇔ symmetry breaking

16

SLIDE 55

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Phase transitions in field theory

Phase transition ⇔ symmetry breaking Z =

[d ¯

φdφ] e−[

∂ ¯

φ∂φ+m2 ¯ φφ+ g

2

( ¯

φφ)2]

16

SLIDE 56

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Phase transitions in field theory

Phase transition ⇔ symmetry breaking Z =

[d ¯

φdφ] e−[

∂ ¯

φ∂φ+m2 ¯ φφ+ g

2

( ¯

φφ)2]

invariant under complex rotations φ = eıαφ, ¯ φ = e−ıα ¯ φ

16

SLIDE 57

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Phase transitions in field theory

Phase transition ⇔ symmetry breaking Z =

[d ¯

φdφ] e−[

∂ ¯

φ∂φ+m2 ¯ φφ+ g

2

( ¯

φφ)2]

invariant under complex rotations φ = eıαφ, ¯ φ = e−ıα ¯ φ

m >0

2

m =0

2

m <0

2

16

SLIDE 58

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Phase transitions in field theory

Phase transition ⇔ symmetry breaking Z =

[d ¯

φdφ] e−[

∂ ¯

φ∂φ+m2 ¯ φφ+ g

2

( ¯

φφ)2]

invariant under complex rotations φ = eıαφ, ¯ φ = e−ıα ¯ φ

m >0

2

m =0

2

m <0

2

Broken phase → VEV: ¯ φφ

= −m2

g

≡ v 2.

16

SLIDE 59

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Phase transitions in field theory

Phase transition ⇔ symmetry breaking Z =

[d ¯

φdφ] e−[

∂ ¯

φ∂φ+m2 ¯ φφ+ g

2

( ¯

φφ)2]

invariant under complex rotations φ = eıαφ, ¯ φ = e−ıα ¯ φ

m >0

2

m =0

2

m <0

2

Broken phase → VEV: ¯ φφ

= −m2

g

≡ v 2. Expand around the VEV φ = (v + ρ)eı θ

v

Sbroken ∼

1 + ρ

v 2 ∂θ∂θ + ∂ρ∂ρ + 2|m2|ρ2 + 2|m2|ρ3 + g 2 ρ4

16

SLIDE 60

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Phase transitions in field theory

Phase transition ⇔ symmetry breaking Z =

[d ¯

φdφ] e−[

∂ ¯

φ∂φ+m2 ¯ φφ+ g

2

( ¯

φφ)2]

invariant under complex rotations φ = eıαφ, ¯ φ = e−ıα ¯ φ

m >0

2

m =0

2

m <0

2

Broken phase → VEV: ¯ φφ

= −m2

g

≡ v 2. Expand around the VEV φ = (v + ρ)eı θ

v

Sbroken ∼

1 + ρ

v 2 ∂θ∂θ + ∂ρ∂ρ + 2|m2|ρ2 + 2|m2|ρ3 + g 2 ρ4 Phase transition: zero eigenvalue of the “mass matrix” δ2Snotkinetic δ ¯ φδφ

¯

φ=φ=0 = m2 = 0

16

SLIDE 61

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The intermediate field representation

17

SLIDE 62

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The intermediate field representation

A Hubbard Stratanovich transformation leads to a coupled multi-matrix model

Z(g) = c [dHc ]

e− 1

2

c ND−1Trc [Hc Hc ]−TrD
ln
1⊗D−ı√g

c Hc ⊗1⊗(D\c) , 17

SLIDE 63

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The intermediate field representation

A Hubbard Stratanovich transformation leads to a coupled multi-matrix model

Z(g) = c [dHc ]

e− 1

2

c ND−1Trc [Hc Hc ]−TrD
ln
1⊗D−ı√g

c Hc ⊗1⊗(D\c) ,

The classical e.o.m. admit a unique invariant solution Hc = A · 1, A = ı√g

√1+4Dg−1 2Dg

.

17

SLIDE 64

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The intermediate field representation

A Hubbard Stratanovich transformation leads to a coupled multi-matrix model

Z(g) = c [dHc ]

e− 1

2

c ND−1Trc [Hc Hc ]−TrD
ln
1⊗D−ı√g

c Hc ⊗1⊗(D\c) ,

The classical e.o.m. admit a unique invariant solution Hc = A · 1, A = ı√g

√1+4Dg−1 2Dg

. Translate to the vacuum (not a phase transition as the vacuum is invariant!):

Z(g) ∼

[dMc ]e− 1

2 ND−1(1−A2) D c=1 Trc

Mc Mc

+ 1 2 ND−2 D−1 D A2D c=1 Trc [Mc ] 2+O(M3) 17

SLIDE 65

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The intermediate field representation

A Hubbard Stratanovich transformation leads to a coupled multi-matrix model

Z(g) = c [dHc ]

e− 1

2

c ND−1Trc [Hc Hc ]−TrD
ln
1⊗D−ı√g

c Hc ⊗1⊗(D\c) ,

The classical e.o.m. admit a unique invariant solution Hc = A · 1, A = ı√g

√1+4Dg−1 2Dg

. Translate to the vacuum (not a phase transition as the vacuum is invariant!):

Z(g) ∼

[dMc ]e− 1

2 ND−1(1−A2) D c=1 Trc

Mc Mc

+ 1 2 ND−2 D−1 D A2D c=1 Trc [Mc ] 2+O(M3)

Mass matrix for the fluctuation:

ND−1(1 − A2)

δcc′ δαδδβγ −

1 DN δαβδγδ

+ ND−1(1 − DA2)

1 DN δαβδγδ

17

SLIDE 66

Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

The intermediate field representation

A Hubbard Stratanovich transformation leads to a coupled multi-matrix model

Z(g) = c [dHc ]

e− 1

2

c ND−1Trc [Hc Hc ]−TrD
ln
1⊗D−ı√g

c Hc ⊗1⊗(D\c) ,

The classical e.o.m. admit a unique invariant solution Hc = A · 1, A = ı√g

√1+4Dg−1 2Dg

. Translate to the vacuum (not a phase transition as the vacuum is invariant!):

Z(g) ∼

[dMc ]e− 1

2 ND−1(1−A2) D c=1 Trc

Mc Mc

+ 1 2 ND−2 D−1 D A2D c=1 Trc [Mc ] 2+O(M3)

Mass matrix for the fluctuation:

ND−1(1 − A2)

δcc′ δαδδβγ −

1 DN δαβδγδ

+ ND−1(1 − DA2)

1 DN δαβδγδ

At criticality g = (−4D)−1, A2 =

1 D the effective mass matrix

develops a flat direction ⇒ Symmetry breaking!

17

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Random tensors, a “functional integral” point of view, IHP, 2016 R˘ azvan Gur˘ au, Introduction Tensor Models The quartic model The continuum limit is a phase transition

Conclusion

Tensor models generalize matrix models in higher dimensions and generate random D-dimensional spaces. Like matrix models, tensor models:

◮ admit a 1/N expansion ◮ triangulations (graphs) with spherical topology dominate in the large N limit ◮ exhibit a critical behavior and a continuum limit ◮ admit a double scaling limit

Unlike in matrix models, in tensor models:

◮ the number of invariants at fixed order in T and ¯

T is very large

◮ the double scaling limit is summable for D = 3, 4, 5 ◮ the continuum limit is a phase transition associated to a symmetry breaking

18