Post on 21-Jul-2020
The full Schwinger-Dyson towerfor coloured tensor models
Carlos. I. Perez-Sanchez
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Mathematics Institute,University of Munster
SFB 878
Corfu 17. 19 September
COST Training School, Qspace (MP 1405)
QSPACE Summer School 2016
QUANTUM STRUCTURE OF SPACETIMEAND GRAVITY
21 – 28 August, 2016Belgrade, Serbia
Topics and lecturers:
Random matrices Francois David Institute for Theoretical Physics, Paris
3D gravity Catherine Meusburger Friedrich-Alexander University, Erlangen
Geometry with fluxes Peter Schupp Jacobs University, Bremen
Fuzzy spaces and applications Harold Steinacker University of Vienna, Vienna
Spectral triples and related Dmitri Vassilevich Federal University ABC, Sao Paulo
The school consists of lectures accompanied by tutorial sessions, as well as some slots reserved for studentpresentations. The aim is to provide the students with the necessary background to pursue original re-search in these and related topics. Students and postdocs from countries participating in COST QSPACEAction are eligible for financial support.
Organizers:
Paolo Aschieri (Italy) Larisa Jonke (Croatia) Harold Steinacker (Austria)John Barrett (UK) Catharine Meusburger (Germany) Richard Szabo (UK)Maja Buric (Serbia) Vincent Rivasseau (France) Marko Vojinovic (Serbia)
Marija Dimitrijevic Ciric (Serbia) Mairi Sakellariadou (UK)
Contact: Info: Application deadline:
qssg16@ipb.ac.rs http://www.qssg16.ipb.ac.rs May 31 2016
OutlineOutlineNon-perturbative approach to quantum (coloured) tensor fields
ϕ4-interaction Λ0 = 1 Λ = N � 1
I correlation functions G(2k)B
G• : {coloured graphs} function space
I full Ward-Takahashi IdentitiesI Schwinger-Dyson equations (joint work with Raimar Wulkenhaar)
C. I. Perez Sanchez (Math. Munster) Outline 7 July 2 / 19
MotivationMotivationTensor models generalize the random 2D geometry of random matrices(“Quantum Gravity”)
Z =∫∑
topologiesgeometries
D[g]e−SEH [g] ∼∫∑
( )∈ topologiesgeometries
µ
(k
1
2
DD
1
2
k
k
1
)
{1 1 , 2 2 , 3 3
}=
{ }
23
1
2 3
1
3
1
23
1
21
2
3
1
2
3
=
tensor models also useful in AdS2/CFT1 (Gurau-WittenSachdev-Ye–Kitaev-like model; course by V. Rivaseeau)
C. I. Perez Sanchez (Math. Munster) Motivation 7 July 3 / 19
MotivationMotivationTensor models generalize the random 2D geometry of random matrices(“Quantum Gravity”)
Z =∫∑
topologiesgeometries
D[g]e−SEH [g] ∼∫∑
( )∈ topologiesgeometries
µ
(k
1
2
DD
1
2
k
k
1
){
1 1 , 2 2 , 3 3
}=
{ }
23
1
2 3
1
3
1
23
1
21
2
3
1
2
3
=
tensor models also useful in AdS2/CFT1 (Gurau-WittenSachdev-Ye–Kitaev-like model; course by V. Rivaseeau)
C. I. Perez Sanchez (Math. Munster) Motivation 7 July 3 / 19
MotivationMotivationTensor models generalize the random 2D geometry of random matrices(“Quantum Gravity”)
Z =∫∑
topologiesgeometries
D[g]e−SEH [g] ∼∫∑
( )∈ topologiesgeometries
µ
(k
1
2
DD
1
2
k
k
1
){
1 1 , 2 2 , 3 3
}=
{ }
23
1
2 3
1
3
1
23
1
21
2
3
1
2
3
=
tensor models also useful in AdS2/CFT1 (Gurau-WittenSachdev-Ye–Kitaev-like model; course by V. Rivaseeau)
C. I. Perez Sanchez (Math. Munster) Motivation 7 July 3 / 19
Matrix modelsC as “Rank-2 tensor models”
For complex matrix models∫D[M, M]e−Tr(MM†)−λV(M, M†)
M
M
M
M
M
M
M
...
1
= dual triangulation to...
1
.
For V(M, M) = λTr((MM†)2), different connected O(λ2)-graphs are
12
1 2
1 2
12
0
0
0
0
1
12
1 2
1 2
12
0
0
0
0
rectangular matrices, M ∈MN1×N2(C) and M W(1)M(W(2))t
(W(a) ∈ U(Na)). U(N1)×U(N2)-invariants are Tr((MM†)q), q ∈ Z≥1
C. I. Perez Sanchez (Math. Munster) Ribbon graphs as coloured ones 7 July 4 / 19
Matrix modelsC as “Rank-2 tensor models”
For complex matrix models∫D[M, M]e−Tr(MM†)−λV(M, M†)
M
M
M
M
M
M
M
...
1
= dual triangulation to...
1
.
For V(M, M) = λTr((MM†)2), different connected O(λ2)-graphs are
12
1 2
1 2
12
0
0
0
0
1
12
1 2
1 2
12
0
0
0
0
rectangular matrices, M ∈MN1×N2(C) and M W(1)M(W(2))t
(W(a) ∈ U(Na)). U(N1)×U(N2)-invariants are Tr((MM†)q), q ∈ Z≥1
C. I. Perez Sanchez (Math. Munster) Ribbon graphs as coloured ones 7 July 4 / 19
Coloured Tensor ModelsColoured Tensor Models
a QFT for tensors ϕa1 ...aD and ϕa1 ...aD, whose indices transform
independently under each factor of G = U(N1)×U(N2)× . . .×U(ND)
for each g = (W(1), . . . , W(D)) ∈ G, W(a) ∈ U(Na),
ϕa1a2 ...aD
g(ϕ′)a1a2 ...aD = W(1)
a1b1W(2)
a2b2. . . W(D)
aDbDϕb1 ...bD
ϕa1a2 ...aD
g(ϕ′)a1a2 ...aD = W(1)
a1b1W(2)
a2b2. . . W(D)
aDbDϕb1b2 ...bD
S0[ϕ, ϕ] = Tr2(ϕ, ϕ) = is the kinetic term and higher G-invariants
serve as interaction vertices. For instance, the ϕ43-theory has:
Sint[ϕ, ϕ] = λ
(1
+
1
+
1
)
Z =∫D[ϕ, ϕ] e−N2(S0+Sint)[ϕ,ϕ] (with D[ϕ, ϕ] = ∏
a
dϕadϕa2πi
)
C. I. Perez Sanchez (Math. Munster) Coloured Tensors 7 July 5 / 19
An example of O(λ4) Feynman graph of the ϕ43-model, where is the
propagator:
1C. I. Perez Sanchez (Math. Munster) Coloured Tensors 7 July 6 / 19
1
=
1
2
3
1
2
3
2
3
12
3
11
2
3
1
2
3
3
1
23
1
2
0 0
0
0
00
0
0
0 0
0
0
00
0
0
0 0
0
0
00
0
0
1
Vertex bipartite regularly edge–D-coloured graphs
(Vacuum) Feynman graphs of a model V, Feynvac.D (V), are (D + 1)-coloured
graphs. PL-manifolds can be crystallized [Pezzana, ‘74] by such graphs
1/N-expansion
A(G) = λV(G)/2N F(G)−D(D−1)4 V(G)︸ ︷︷ ︸
=: D− 2(D−1)! ω(G) generalizes g; not topol. invariant
= exp(−SRegge[N, D, λ])
[Gurau, ’09], [Bonzom, Gurau, Riello, Rivasseau, ’11]
C. I. Perez Sanchez (Math. Munster) Coloured Tensors 7 July 7 / 19
. . .
1
12
1 2
21
2 1
0
0
0
0
1
. . .
a
b
c
d ΓC
C / ∼(Objects mod ∼)
Manifolds GraphsPezzana
Theorem
Hbubble∗
π1
Physical γC
On the low-dim topology of colored tensor models without Pezzana’stheorem [CP 2017, J.Geom.Phys. 120 (2017) (arXiv:1608.00246)]
C. I. Perez Sanchez (Math. Munster) Coloured Tensors 7 July 8 / 19
Correlation functionsCorrelation functionsreplace the expansion of log ZQFT[J] by the respective CTM-one
log ZQFT[J] =∞
∑n=0
1n!
∫dx1 · · ·dxn G(n)
conn.(x1, x2, . . . , xn)J(x1)J(x2) · · · J(xn)
FeynD(V) = {open Feynman diagrams of the (rank-D) tensor model V}.The boundary ∂G of G ∈ FeynD(V) has as vertex-set the external legs of Gand as a-coloured edges 0a-paths in G between them.
2 2
1
12
2
1 1
2
2 1
1
∂V2
1
2
1
2
1
33
3
1
=1
Also
F =2
12
12
1
333
has as boundary1
, but F /∈ Feyn(ϕ43).
Correlation functionsCorrelation functionsreplace the expansion of log ZQFT[J] by the respective CTM-one
log ZQFT[J] =∞
∑n=0
1n!
∫dx1 · · ·dxn G(n)
conn.(x1, x2, . . . , xn)J(x1)J(x2) · · · J(xn)
FeynD(V) = {open Feynman diagrams of the (rank-D) tensor model V}.The boundary ∂G of G ∈ FeynD(V) has as vertex-set the external legs of Gand as a-coloured edges 0a-paths in G between them.
2 2
1
12
2
1 1
2
2 1
1
∂V2
1
2
1
2
1
33
3
1
=1
Also
F =2
12
12
1
333
has as boundary1
, but F /∈ Feyn(ϕ43).
Correlation functionsCorrelation functionsreplace the expansion of log ZQFT[J] by the respective CTM-one
log ZQFT[J] =∞
∑n=0
1n!
∫dx1 · · ·dxn G(n)
conn.(x1, x2, . . . , xn)J(x1)J(x2) · · · J(xn)
FeynD(V) = {open Feynman diagrams of the (rank-D) tensor model V}.The boundary ∂G of G ∈ FeynD(V) has as vertex-set the external legs of Gand as a-coloured edges 0a-paths in G between them.
2 2
1
12
2
1 1
2
2 1
1
∂V2
1
2
1
2
1
33
3
1
=1
Also F =2
12
12
1
333
has as boundary1
, but F /∈ Feyn(ϕ43).
Expansion of the free energy
im ∂V = ∂ FeynD(V) is the boundary sector of the model V
W[J, J] =∞
∑k=1
∑B∈∂FeynD(V(ϕ,ϕ))
2k=#(B(0))
1|Autc(B)|
G(2k)B ? J(B) .
Coloured automorphisms of B
Jx1
Jx2
Jxk
......
Jy1
Jy2
Jyk
G
(J(B))(x1, . . . , xk︸ ︷︷ ︸(ZD)k
) = Jx1 · · · Jxk Jy1 · · · Jyk
Correlation function G(2k)B = ∂2kW[J, J]/∂J(B)|J=J=0
F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a
F(a) · J(B)(a)
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 10 / 19
Expansion of the free energy
im ∂V = ∂ FeynD(V) is the boundary sector of the model V
W[J, J] =∞
∑k=1
∑B∈ im ∂V
2k=#(Vertices ofB)
1|Autc(B)|
G(2k)B ? J(B) .
Coloured automorphisms of B
Jx1
Jx2
Jxk
......
Jy1
Jy2
Jyk
G
(J(B))(x1, . . . , xk︸ ︷︷ ︸(ZD)k
) = Jx1 · · · Jxk Jy1 · · · Jyk
Correlation function G(2k)B = ∂2kW[J, J]/∂J(B)|J=J=0
F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a
F(a) · J(B)(a)
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 10 / 19
Expansion of the free energy
im ∂V = ∂ FeynD(V) is the boundary sector of the model V
W[J, J] =∞
∑k=1
∑B∈∂FeynD(V(ϕ,ϕ))
2k=#(B(0))
1|Autc(B)|
G(2k)B ? J(B) .
Coloured automorphisms of B
Jx1
Jx2
Jxk
......
Jy1
Jy2
Jyk
G
(J(B))(x1, . . . , xk︸ ︷︷ ︸(ZD)k
) = Jx1 · · · Jxk Jy1 · · · Jyk
Correlation function G(2k)B = ∂2kW[J, J]/∂J(B)|J=J=0
F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a
F(a) · J(B)(a)
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 10 / 19
Expansion of the free energy
im ∂V = ∂ FeynD(V) is the boundary sector of the model V
W[J, J] =∞
∑k=1
∑B∈∂FeynD(V(ϕ,ϕ))
2k=#(B(0))
1|Autc(B)|
G(2k)B ? J(B) .
Coloured automorphisms of BJx1
Jx2
Jxk
......
Jy1
Jy2
Jyk
G (J(B))(x1, . . . , xk︸ ︷︷ ︸(ZD)k
) = Jx1 · · · Jxk Jy1 · · · Jyk
Correlation function G(2k)B = ∂2kW[J, J]/∂J(B)|J=J=0
F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a
F(a) · J(B)(a)
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 10 / 19
Expansion of the free energy
im ∂V = ∂ FeynD(V) is the boundary sector of the model V
W[J, J] =∞
∑k=1
∑B∈∂FeynD(V(ϕ,ϕ))
2k=#(B(0))
1|Autc(B)|
G(2k)B ? J(B) .
Coloured automorphisms of BJx1
Jx2
Jxk
......
Jy1
Jy2
Jyk
G (J(B))(x1, . . . , xk︸ ︷︷ ︸(ZD)k
) = Jx1 · · · Jxk Jy1 · · · Jyk
Correlation function G(2k)B = ∂2kW[J, J]/∂J(B)|J=J=0
F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a
F(a) · J(B)(a)
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 10 / 19
Expansion of the free energy
im ∂V = ∂ FeynD(V) is the boundary sector of the model V
W[J, J] =∞
∑k=1
∑B∈∂FeynD(V(ϕ,ϕ))
2k=#(B(0))
1|Autc(B)|
G(2k)B ? J(B) .
Coloured automorphisms of BJx1
Jx2
Jxk
......
Jy1
Jy2
Jyk
G (J(B))(x1, . . . , xk︸ ︷︷ ︸(ZD)k
) = Jx1 · · · Jxk Jy1 · · · Jyk
Correlation function G(2k)B = ∂2kW[J, J]/∂J(B)|J=J=0
F : MD×k(B)(Z) C; ? : (F, J(B)) F ? J(B) = ∑a
F(a) · J(B)(a)
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 10 / 19
1
n = 2
n = 4
n = 6
n = 8
n = 10 . . . . .....
G(2)
1
G(4)1 1
G(4)2 2
G(4)3 3
G(4)|
1
|1
|
G(6) G(6)1
G(6)2
G(6)3
G(6)
32
G(6)
31
G(6)
21
. . .
G(8)j iiij
ll
G(8)j il
1
G(8)
i
jG(8)
il lj
G(8)l
j i
G(8) G(8)i
G(8)a
ba
c
ca
b
. . .SDE
SDE
SDE
SDE
WTI
WTI
WTI
WTI
[CP](arXiv:1608.08134) [CP, R. Wulkenhaar](arXiv:1706.07358)
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 11 / 19
WD=3[J, J]
WD=3[J, J] = G(2)
1
? J(1
) +
12!
G(4)|
1
|1
| ? J(1
t2) +
12 ∑
cG(4)
c c ? J( c )
+13 ∑
cG(6)
c ?
J( c )
+13
G(6)? J( )
+ ∑i
G(6)
i? J(
i
)+
13!
G(6)|
1
|1
|1
| ? J(1
t3) +
12 ∑
cG(6)|
1
| c c | ? J(
1
t c )+
12! · 22 ∑
cG(8)| c c | c c | ? J
( c t c )+
122 ∑
c<iG(8)| c c | i i | ?
J( c t i
)+
14!
G(8)|
1
|1
|1
|1
| ? J(1
t4) +
12 · 2! ∑
cG(8)|
1
|1
| c c | ? J(
1
t1
t c )+
13
G(8)|
1
| | ? J(
1
t)+
13 ∑
cG(8)
|1
|c|? J(
1
tc )
+ ∑i
G(8)
|1
| i |? J(
1
t
i
)+ ∑
j ; l<iG(8)
j il1
? J(
l jliij
j
j i
i
l
l
)+ ∑
j 6=iG(8)
j iiij
ll
? J(
j iiiij
j
j l
l
l
l
)+
14 ∑
jG(8)
j?
J(
j
j
j
j
)+ ∑
j 6=iG(8)
i
j? J(
i
i
j)+ ∑
iG(8)
il lj
? J( il
l i
lj
i
i
jl)+ ∑
l 6=i 6=jG(8)
lj i
? J( ll
l l
j i
i
i
i
)+
G(8)ca
ca
? J( ca
ca
b
b )+ G(8)
a
ba
c
ca
b
? J(
a
b
ba
c
a
b
a)+O(10) .
WD=3[J, J]
WD=3[J, J] = G(2)
1
? J(1
) +12!
G(4)|
1
|1
| ? J(1
t2) +
12 ∑
cG(4)
c c ? J( c )
+13 ∑
cG(6)
c ?
J( c )
+13
G(6)? J( )
+ ∑i
G(6)
i? J(
i
)+
13!
G(6)|
1
|1
|1
| ? J(1
t3) +
12 ∑
cG(6)|
1
| c c | ? J(
1
t c )+
12! · 22 ∑
cG(8)| c c | c c | ? J
( c t c )+
122 ∑
c<iG(8)| c c | i i | ?
J( c t i
)+
14!
G(8)|
1
|1
|1
|1
| ? J(1
t4) +
12 · 2! ∑
cG(8)|
1
|1
| c c | ? J(
1
t1
t c )+
13
G(8)|
1
| | ? J(
1
t)+
13 ∑
cG(8)
|1
|c|? J(
1
tc )
+ ∑i
G(8)
|1
| i |? J(
1
t
i
)+ ∑
j ; l<iG(8)
j il1
? J(
l jliij
j
j i
i
l
l
)+ ∑
j 6=iG(8)
j iiij
ll
? J(
j iiiij
j
j l
l
l
l
)+
14 ∑
jG(8)
j?
J(
j
j
j
j
)+ ∑
j 6=iG(8)
i
j? J(
i
i
j)+ ∑
iG(8)
il lj
? J( il
l i
lj
i
i
jl)+ ∑
l 6=i 6=jG(8)
lj i
? J( ll
l l
j i
i
i
i
)+
G(8)ca
ca
? J( ca
ca
b
b )+ G(8)
a
ba
c
ca
b
? J(
a
b
ba
c
a
b
a)+O(10) .
WD=3[J, J]
WD=3[J, J] = G(2)
1
? J(1
) +12!
G(4)|
1
|1
| ? J(1
t2) +
12 ∑
cG(4)
c c ? J( c )
+13 ∑
cG(6)
c ?
J( c )
+13
G(6)? J( )
+ ∑i
G(6)
i? J(
i
)+
13!
G(6)|
1
|1
|1
| ? J(1
t3) +
12 ∑
cG(6)|
1
| c c | ? J(
1
t c )
+1
2! · 22 ∑c
G(8)| c c | c c | ? J
( c t c )+
122 ∑
c<iG(8)| c c | i i | ?
J( c t i
)+
14!
G(8)|
1
|1
|1
|1
| ? J(1
t4) +
12 · 2! ∑
cG(8)|
1
|1
| c c | ? J(
1
t1
t c )+
13
G(8)|
1
| | ? J(
1
t)+
13 ∑
cG(8)
|1
|c|? J(
1
tc )
+ ∑i
G(8)
|1
| i |? J(
1
t
i
)+ ∑
j ; l<iG(8)
j il1
? J(
l jliij
j
j i
i
l
l
)+ ∑
j 6=iG(8)
j iiij
ll
? J(
j iiiij
j
j l
l
l
l
)+
14 ∑
jG(8)
j?
J(
j
j
j
j
)+ ∑
j 6=iG(8)
i
j? J(
i
i
j)+ ∑
iG(8)
il lj
? J( il
l i
lj
i
i
jl)+ ∑
l 6=i 6=jG(8)
lj i
? J( ll
l l
j i
i
i
i
)+
G(8)ca
ca
? J( ca
ca
b
b )+ G(8)
a
ba
c
ca
b
? J(
a
b
ba
c
a
b
a)+O(10) .
WD=3[J, J]
WD=3[J, J] = G(2)
1
? J(1
) +12!
G(4)|
1
|1
| ? J(1
t2) +
12 ∑
cG(4)
c c ? J( c )
+13 ∑
cG(6)
c ?
J( c )
+13
G(6)? J( )
+ ∑i
G(6)
i? J(
i
)+
13!
G(6)|
1
|1
|1
| ? J(1
t3) +
12 ∑
cG(6)|
1
| c c | ? J(
1
t c )+
12! · 22 ∑
cG(8)| c c | c c | ? J
( c t c )+
122 ∑
c<iG(8)| c c | i i | ?
J( c t i
)+
14!
G(8)|
1
|1
|1
|1
| ? J(1
t4) +
12 · 2! ∑
cG(8)|
1
|1
| c c | ? J(
1
t1
t c )+
13
G(8)|
1
| | ? J(
1
t)+
13 ∑
cG(8)
|1
|c|? J(
1
tc )
+ ∑i
G(8)
|1
| i |? J(
1
t
i
)+ ∑
j ; l<iG(8)
j il1
? J(
l jliij
j
j i
i
l
l
)+ ∑
j 6=iG(8)
j iiij
ll
? J(
j iiiij
j
j l
l
l
l
)+
14 ∑
jG(8)
j?
J(
j
j
j
j
)+ ∑
j 6=iG(8)
i
j? J(
i
i
j)+ ∑
iG(8)
il lj
? J( il
l i
lj
i
i
jl)+ ∑
l 6=i 6=jG(8)
lj i
? J( ll
l l
j i
i
i
i
)+
G(8)ca
ca
? J( ca
ca
b
b )+ G(8)
a
ba
c
ca
b
? J(
a
b
ba
c
a
b
a)+O(10) .
Boundary graphs and bordisms interpretation
for quartic melonic theories, the boundary sector is the set of all (possibledisconnected) coloured graphs
since B = ∂G represents the ‘boundary of a simplicial complex that Gtriangulates’, one can give a bordism-interpretation to the Green’s functions
for instance, if |∆(B)| = S3 t (S2 × S1) t L3,1 =M, then GB = ∂W[J, J]/∂Bdescribes the bulk compatible with the triangulation of M
ϕ4-generated
4D-bulk
S3
L3,1
S2 × S1
+ϕ4-generated
4D-bulk
S3
L3,1
S2 × S1
+ϕ4-generated
4D-bulk
S3
L3,1
S2 × S1
+ . . .
Each correlation function supports also a 1/N-expansion, G(2k)B = ∑
ω
G(2k,ω)B .
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 13 / 19
Boundary graphs and bordisms interpretation
for quartic melonic theories, the boundary sector is the set of all (possibledisconnected) coloured graphs
since B = ∂G represents the ‘boundary of a simplicial complex that Gtriangulates’, one can give a bordism-interpretation to the Green’s functions
for instance, if |∆(B)| = S3 t (S2 × S1) t L3,1 =M, then GB = ∂W[J, J]/∂Bdescribes the bulk compatible with the triangulation of M
ϕ4-generated
4D-bulk
S3
L3,1
S2 × S1
+ϕ4-generated
4D-bulk
S3
L3,1
S2 × S1
+ϕ4-generated
4D-bulk
S3
L3,1
S2 × S1
+ . . .
Each correlation function supports also a 1/N-expansion, G(2k)B = ∑
ω
G(2k,ω)B .
C. I. Perez Sanchez (Math. Munster) Boundary-graph-expansion of W[J, J] 7 July 13 / 19
Theorem (CP) (Full Ward-Takahashi Identity for arbitrary tensor models)
The partition function Z[J, J] of a tensor model with S0 = Tr2(ϕ, Eϕ) such that
Ep1 ...pa−1mapa+1 ...pD − Ep1 ...pa−1napa+1 ...pD = E(ma, na) for each a = 1, . . . , D,
satisfies, as consequence of the U(N) invariance of the path-integral measure,
∑pi∈Z
δ2Z[J, J]δJp1 ...pa−1mapa+1 ...pD δJp1 ...pa−1napa+1 ...pD
−(
δmanaY(a)ma [J, J]
)· Z[J, J]
= ∑pi∈Z
1E(ma, na)
(Jp1 ...ma ...pD
δ
δJ p1 ...na ...pD
− Jp1 ...na ...pD
δ
δJ p1 ...ma ...pD
)Z[J, J]
where
Y(a)ma [J, J] = ∑
Cf(a)C,ma
? J(C) .
S
era
.
.
....
.
.
.
ixξ(r,i,a)
xξ(r,j,a)j
• era7→ .
.
....
C. I. Perez Sanchez (Math. Munster) The WTI for tensor models 7 July 14 / 19
Schwinger-Dyson equationsSchwinger-Dyson equations
SDEs for the ϕ4mel,D-model (k ≥ 2) [CP, R. Wulkenhaar]
Let D ≥ 3 and let B be a connected boundary graph∂G = B ∈ Grphcl
D ⊂ ∂FeynD(ϕ4m,D).
Let X = (x1, . . . , xk) and s = y1. Then G(2k)B obeys
Jx1
Jx2
Jxk
......
Jy1
Jy2
Jyk
G
(1 +
2λ
Es
D
∑a=1
∑qa
G(2)...
1
(sa, qa)
)G(2k)B (X)
=(−2λ)
Es
D
∑a=1
{∑
σ∈Autc(B)σ∗f(a)B,sa
(X) + ∑ρ>1
1E(yρ
a , sa)Z−1
0∂Z[J, J]
∂ςa(B; 1, ρ)(X)
−∑ba
1E(sa, ba)
[G(2k)B (X)−G(2k)
B (X|sa→ba)]}
where (sa, qa) = (q1, q2, . . . , qa−1, sa, qa+1, . . . , qD).
C. I. Perez Sanchez (Math. Munster) Schwinger-Dyson equations 7 July 15 / 19
Schwinger-Dyson equationsSchwinger-Dyson equations
SDEs for the ϕ4mel,D-model (k ≥ 2) [CP, R. Wulkenhaar]
Let D ≥ 3 and let B be a connected boundary graph∂G = B ∈ Grphcl
D ⊂ ∂FeynD(ϕ4m,D).
Let X = (x1, . . . , xk) and s = y1. Then G(2k)B obeys
Jx1
Jx2
Jxk
......
Jy1
Jy2
Jyk
G
(1 +
2λ
Es
D
∑a=1
∑qa
G(2)...
1
(sa, qa)
)G(2k)B (X)
=(−2λ)
Es
D
∑a=1
{∑
σ∈Autc(B)σ∗f(a)B,sa
(X) + ∑ρ>1
1E(yρ
a , sa)Z−1
0∂Z[J, J]
∂ςa(B; 1, ρ)(X)
−∑ba
1E(sa, ba)
[G(2k)B (X)−G(2k)
B (X|sa→ba)]}
where (sa, qa) = (q1, q2, . . . , qa−1, sa, qa+1, . . . , qD).
C. I. Perez Sanchez (Math. Munster) Schwinger-Dyson equations 7 July 15 / 19
A simple quartic modelA simple quartic model
Proposal of a model with V[ϕ, ϕ] = λ · 1 1
S0[ϕ, ϕ] = Tr2(ϕ, Eϕ) = ∑x∈Z3
ϕx(m2 + |x|2)ϕx , |x|2 = x21 + x2
2 + x23 .
The boundary sector is:
∂ Feyn3( 1 1 ) = {3-coloured graphs with connected components in Θ} ,
being
Θ ={
1
, 1 1 ,1
, 1 , 1 ,1
, . . .}
.
Let X2k be the graph in Θ with 2k vertices, and G(2k) = G(2k)X2k
:
G(2) = G(2)
1
, G(4) = G(4)1 1
, G(6) = G(6)1
, G(8) = G(8)1
, G(10) = G(10)1
.
Full tower of exact equations obtained [CP, R. Wulkenhaar]
C. I. Perez Sanchez (Math. Munster) Schwinger-Dyson equations 7 July 16 / 19
The exact 2pt-function equation. Melonic (planar) limit, conjecturally
(m2 + |x|2 + 2λ ∑
q,p∈Z
G(2)mel(x1, q, p)
)·G(2)
mel(x)
= 1 + 2λ ∑q∈Z
1x2
1 − q2
[G(2)
mel(x1, x2, x3)−G(2)mel(q, x2, x3)
]
The exact 2k-pt-function equation. Melonic limit, conjecturally
(1 +
2λ
m2 + |s|2 ∑q,p∈Z
G(2)mel(x
11, q, p)
)·G(2k)
mel (x1, . . . , xk)
=(−2λ)
m2 + |s|2[ k
∑ρ=2
1(xρ
1)2 − (x1
1)2
(G(2ρ−2)
mel (x1, . . . , xρ−1) ·G(2k−2ρ+2)mel (xρ, . . . , xk)
)
− ∑q∈Z
G(2k)mel (x
11, x1
2, x13, x2, . . . , xk)−G(2k)
mel (q, x12, x1
3, x2, . . . , xk)
(x11)
2 − q2
]C. I. Perez Sanchez (Math. Munster) Schwinger-Dyson equations 7 July 17 / 19
The exact 2pt-function equation. Melonic (planar) limit, conjecturally
(m2 + |x|2 + 2λ ∑
q,p∈Z
G(2)mel(x1, q, p)
)·G(2)
mel(x)
= 1 + 2λ ∑q∈Z
1x2
1 − q2
[G(2)
mel(x1, x2, x3)−G(2)mel(q, x2, x3)
]
The exact 2k-pt-function equation. Melonic limit, conjecturally
(1 +
2λ
m2 + |s|2 ∑q,p∈Z
G(2)mel(x
11, q, p)
)·G(2k)
mel (x1, . . . , xk)
=(−2λ)
m2 + |s|2[ k
∑ρ=2
1(xρ
1)2 − (x1
1)2
(G(2ρ−2)
mel (x1, . . . , xρ−1) ·G(2k−2ρ+2)mel (xρ, . . . , xk)
)
− ∑q∈Z
G(2k)mel (x
11, x1
2, x13, x2, . . . , xk)−G(2k)
mel (q, x12, x1
3, x2, . . . , xk)
(x11)
2 − q2
]C. I. Perez Sanchez (Math. Munster) Schwinger-Dyson equations 7 July 17 / 19
Conclusions & outlookConclusions & outlook(Coloured) tensor field theories [Ben Geloun, Bonzom, Carrozza, Gurau, Krajewski,
Oriti, Ousmane-Samary, Rivasseau, Ryan, Tanasa, Toriumi, Vignes-Tourneret,. . .] providea framework for 3 ≤ D-dimensional random geometry
I A bordism interpretation of the correlation functions was givenI A (non-perturbative) Ward-Takahashi identity [CP] based that for
matrix models has been foundI It has been used to derive the full tower of SDE [CP-Wulkenhaar]
I Closed equations: hope of a solvable theory for the simple 1 1 -model(spherical 3-geometries)
Outlook:
I Apply these techniques SYK-like (Sachdev-Ye-Kitaev) models [Witten]
I Applications to GFTI Gauge fields on colored graphs (random spaces) by representing graphs
on finite spectral triples
C. I. Perez Sanchez (Math. Munster) The WTI for tensor models 7 July 18 / 19
ReferencesReferencesM. Disertori, R. Gurau, J. Magnen, and
V. Rivasseau.Phys. Lett. B, 649 95–102, 2007.arXiv:hep-th/0612251.
Bonzom, V. and Gurau, R. and Riello, A.and Rivasseau, V.Nucl. Phys. B 853, 2011arXiv:1105.3122
R. Gurau,Commun. Math. Phys. 304, 69 (2011)arXiv:0907.2582 [hep-th] .
C. I. Perez-Sanchez, R. WulkenhaararXiv:1706.07358
C. I. Perez-Sanchez.J.Geom.Phys. 120 262-289 (2017) (arXiv:1608.00246)and arXiv:1608.08134
E. WittenarXiv:1610.09758v2 [hep-th].
Thank you for your attention!