The full Schwinger-Dyson tower for coloured tensor models

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% Yellow) auf weißem Untergrund Variante (siehe nächste Seite) verwendet Diese Definition gilt für sämtliche hochwer- werden Außerdem sollte das Logo weder tigen Printerzeugnisse wie zum eispiel mit einem harten bzw weichen Schatten riefbogen Premium, Visitenkarte, Flyer, hinterlegt werden Photoshop-Spielereien, Folder, roschüren, Einladungskarten, wie zum eispiel die Verwendung von Relief- Magazine, auf Fahnen und Universitäts- Filtern sind ausdrücklich nicht gestattet > fahrzeugen, Schildern etc > The full Schwinger-Dyson tower for coloured tensor models Auf Faxbögen, dem iefbogen Standard und Sollte dieser Kontrast in der Wirkung in Aus- dem Kurzbrief wird das Logo in % Schwarz nahmefällen nicht ausreichend sein, kann angelegt, um einen Qualitätsverlust bei der das Logo negativ (Weiß) auf dunklen Farb- Wiedergabe des Logos auf herkömmlichen hintergründen platziert werden, die zu Arbeitsplatzdruckern zu vermeiden > dem verbindlichen Farbspektrum (siehe Kapitel Farben) gehören Carlos I Pe rez-sa nchez Logo in Pantone lack 7 auf weißem Hintergrund Mathematics Institute, University of Mu nster Logo negativ (Weiß) auf Hintergrund in Pantone lack 7 SF 878 Corfu 7 9 September Logo negativ (Weiß) auf Hintergrund in Pantone 5 COST Training School, Qspace (MP 45)

Outline Non-perturbative approach to quantum (coloured) tensor fields ϕ 4 -interaction Λ = Λ = N correlation functions G (k) G : {coloured graphs} function space full Ward-Takahashi Identities Schwinger-Dyson equations (joint work with Raimar Wulkenhaar) C I Pérez Sánchez (Math Münster) Outline 7 July / 9

Motivation Motivation Tensor models generalize the random D geometry of random matrices ( Quantum Gravity )! Z Z Z = topologies geometries D[g]e SEH [g] ( topologies ) geometries µ k D k k D = tensor models also useful in AdS /CFT (Gurau-Witten Sachdev-Ye Kitaev-like model; course by V Rivaseeau) C I Pe rez Sa nchez (Math Mu nster) Motivation 7 July / 9

Motivation Motivation Tensor models generalize the random D geometry of random matrices ( Quantum Gravity )! Z Z Z = topologies geometries D[g]e SEH [g], (, topologies ) geometries µ k D k k D = = tensor models also useful in AdS /CFT (Gurau-Witten Sachdev-Ye Kitaev-like model; course by V Rivaseeau) C I Pe rez Sa nchez (Math Mu nster) Motivation 7 July / 9

Motivation Motivation Tensor models generalize the random D geometry of random matrices ( Quantum Gravity )! Z Z Z = topologies geometries D[g]e SEH [g] topologies ) geometries (,, µ k D k k D = = tensor models also useful in AdS /CFT (Gurau-Witten Sachdev-Ye Kitaev-like model; course by V Rivaseeau) C I Pe rez Sa nchez (Math Mu nster) Motivation 7 July / 9

Matrix models C as Rank- tensor models For complex matrix models D[M, M]e Tr(MM ) λv(m, M ) M M M M M M = dual triangulation to For V(M, M) = λtr((mm ) ), different connected O(λ )-graphs are rectangular matrices, M M N N (C) and M W () M(W () ) t (W (a) U(N a )) U(N ) U(N )-invariants are Tr((MM ) q ), q Z C I Pérez Sánchez (Math Münster) Ribbon graphs as coloured ones 7 July 4 / 9

Matrix models C as Rank- tensor models For complex matrix models D[M, M]e Tr(MM ) λv(m, M ) M M M M M M = dual triangulation to For V(M, M) = λtr((mm ) ), different connected O(λ )-graphs are rectangular matrices, M M N N (C) and M W () M(W () ) t (W (a) U(N a )) U(N ) U(N )-invariants are Tr((MM ) q ), q Z C I Pérez Sánchez (Math Münster) Ribbon graphs as coloured ones 7 July 4 / 9

Coloured Tensor Models a QFT for tensors ϕ a a D and ϕ a a D, whose indices transform independently under each factor of G = U(N ) U(N ) U(N D ) for each g = (W (),, W (D) ) G, W (a) U(N a ), ϕ a a a D g (ϕ ) a a a D = W () a b W () a b W (D) a D b D ϕ b b D ϕ a a a D g (ϕ ) a a a D = W () a b W () a b W (D) a D b D ϕ b b b D S [ϕ, ϕ] = Tr (ϕ, ϕ) = is the kinetic term and higher G-invariants serve as interaction vertices For instance, the ϕ 4 -theory has: S int [ϕ, ϕ] = λ ( + + ) Z = D[ϕ, ϕ] e N (S +S int )[ϕ,ϕ] (with D[ϕ, ϕ] = a dϕ a dϕ a ) πi C I Pérez Sánchez (Math Münster) Coloured Tensors 7 July 5 / 9

An example of O(λ 4 ) Feynman graph of the ϕ 4 -model, where propagator: is the C I Pérez Sánchez (Math Münster) Coloured Tensors 7 July 6 / 9

= Vertex bipartite regularly edge D-coloured graphs (Vacuum) Feynman graphs of a model V, Feyn vac D (V), are (D + )-coloured graphs PL-manifolds can be crystallized [Pezzana, 74] by such graphs /N-expansion A(G) = λ V(G)/ N F(G) D(D ) 4 V(G) }{{} =: D (D )! ω(g) = exp( S Regge [N, D, λ]) generalizes g; not topol invariant [Gurău, 9], [onzom, Gurău, Riello, Rivasseau, ] C I Pérez Sánchez (Math Münster) Coloured Tensors 7 July 7 / 9

Manifolds Pezzana Theorem H bubble Graphs π C / (Objects mod ) c b Physical γ C d Γ C a On the low-dim topology of colored tensor models without Pezzana s theorem [CP 7, JGeomPhys (7) (arxiv:6846)] C I Pérez Sánchez (Math Münster) Coloured Tensors 7 July 8 / 9

Correlation functions replace the expansion of log Z QFT [J] by the respective CTM-one log Z QFT [J] = dx n! dx n G conn(x (n), x,, x n )J(x )J(x ) J(x n ) n= Feyn D (V) = {open Feynman diagrams of the (rank-d) tensor model V} The boundary G of G Feyn D (V) has as vertex-set the external legs of G and as a-coloured edges a-paths in G between them Also

Correlation functions replace the expansion of log Z QFT [J] by the respective CTM-one log Z QFT [J] = dx n! dx n G conn(x (n), x,, x n )J(x )J(x ) J(x n ) n= Feyn D (V) = {open Feynman diagrams of the (rank-d) tensor model V} The boundary G of G Feyn D (V) has as vertex-set the external legs of G and as a-coloured edges a-paths in G between them V = Also

Correlation functions replace the expansion of log Z QFT [J] by the respective CTM-one log Z QFT [J] = dx n! dx n G conn(x (n), x,, x n )J(x )J(x ) J(x n ) n= Feyn D (V) = {open Feynman diagrams of the (rank-d) tensor model V} The boundary G of G Feyn D (V) has as vertex-set the external legs of G and as a-coloured edges a-paths in G between them V = Also F = has as boundary, but F / Feyn(ϕ 4 )

Expansion of the free energy im V = Feyn D (V) is the boundary sector of the model V W[J, J] = k= Feyn D (V(ϕ, ϕ)) k=#( () ) Coloured automorphisms of Aut c () G (k) J() (J())(x,, x }{{} k ) = J x J xk J y J yk (Z D ) k Correlation function G (k) = k W[J, J]/ J() J= J= F : M D k() (Z) C; : (F, J()) F J() = F(a) J()(a) a C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

Expansion of the free energy im V = Feyn D (V) is the boundary sector of the model V W[J, J] = k= im V k=#(vertices of ) Coloured automorphisms of Aut c () G (k) J() (J())(x,, x }{{} k ) = J x J xk J y J yk (Z D ) k Correlation function G (k) = k W[J, J]/ J() J= J= F : M D k() (Z) C; : (F, J()) F J() = F(a) J()(a) a C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

Expansion of the free energy im V = Feyn D (V) is the boundary sector of the model V W[J, J] = k= Feyn D (V(ϕ, ϕ)) k=#( () ) Coloured automorphisms of Aut c () G (k) J() (J())(x,, x }{{} k ) = J x J xk J y J yk (Z D ) k Correlation function G (k) = k W[J, J]/ J() J= J= F : M D k() (Z) C; : (F, J()) F J() = F(a) J()(a) a C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

Expansion of the free energy im V = Feyn D (V) is the boundary sector of the model V W[J, J] = k= Feyn D (V(ϕ, ϕ)) k=#( () ) Coloured automorphisms of Aut c () G (k) J() J x J y J x J y J x k G J y k (J())(x,, x }{{} k ) = J x J xk J y J yk (Z D ) k Correlation function G (k) = k W[J, J]/ J() J= J= F : M D k() (Z) C; : (F, J()) F J() = F(a) J()(a) a C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

Expansion of the free energy im V = Feyn D (V) is the boundary sector of the model V W[J, J] = k= Feyn D (V(ϕ, ϕ)) k=#( () ) Coloured automorphisms of Aut c () G (k) J() J x J y J x J y J x k G J y k (J())(x,, x }{{} k ) = J x J xk J y J yk (Z D ) k Correlation function G (k) = k W[J, J]/ J() J= J= F : M D k() (Z) C; : (F, J()) F J() = F(a) J()(a) a C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

Expansion of the free energy im V = Feyn D (V) is the boundary sector of the model V W[J, J] = k= Feyn D (V(ϕ, ϕ)) k=#( () ) Coloured automorphisms of Aut c () G (k) J() J x J y J x J y J x k G J y k (J())(x,, x }{{} k ) = J x J xk J y J yk (Z D ) k Correlation function G (k) = k W[J, J]/ J() J= J= F : M D k() (Z) C; : (F, J()) F J() = F(a) J()(a) a C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

n = SDE WTI n = 8 SDE WTI n = 6 SDE WTI n = 4 SDE WTI G (8) j i i j i l l G (8) j l G (6) G (6) i G (4) G (8) G (8) j i G (6) G (4) l j i G (6) l G (8) l j i G (6) G (4) G (8) G (6) G (4) G (8) i G (6) G (8) b c a a b a c n = G () [CP](arXiv:6884) [CP, R Wulkenhaar](arXiv:76758) C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

W D= [J, J] W D= [J, J] = G () J( ) +

W D= [J, J] W D= [J, J] = G () J( ) +! G(4) J( ) + G (4) J( c ) c c c

W D= [J, J] W D= [J, J] = G () J( ) +! G(4) ( c ) J + ( ) G(6) J c G (6) + G (6) i i c c J( c ) J( ) + G (4) J( c ) c c + c G (6) c c ( ) J i +! G(6) J( ) +

W D= [J, J] W D= [J, J] = G () J( ) +! G(4) ( c ) J + ( ) G(6) J c J ( c G(8) J ( i G (8) a c a c j G (6) j j + G (6) i i J( ) + G (4) J( c ) c c + c G (6) c c ( ) J i +! G(6) J( ) + c c J( c ) +! G (8) c c c c J( c c ) + c G (8) c c i i c<i i ) + 4! G(8) J( 4 ) +! G (8) c c J( c ) + c ( ) J + ( G (8) c ) ( c J + G (8) c i i ) + G (8) J ( l j i ) j i i l l j l j + G (8) J ( l j l j i i i j i j i i j i j ; l<i j =i l l l j l ) i + 4 G (8) j j j J ( ) + G (8) J ( i j ij j =i i a a c b ) + G (8) b J ( b c c a a b c a l j i ) + i b c a a a b a b G (8) J ( i l i l j j l l j l i l i i ) + O() ) + l =i =j G (8) l j i J ( i l l i j i l l i ) +

oundary graphs and bordisms interpretation for quartic melonic theories, the boundary sector is the set of all (possible disconnected) coloured graphs since = G represents the boundary of a simplicial complex that G triangulates, one can give a bordism-interpretation to the Green s functions for instance, if () = S (S S ) L, = M, then G = W[J, J]/ describes the bulk compatible with the triangulation of M Each correlation function supports also a /N-expansion, G (k) = G (k,ω) ω C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

oundary graphs and bordisms interpretation for quartic melonic theories, the boundary sector is the set of all (possible disconnected) coloured graphs since = G represents the boundary of a simplicial complex that G triangulates, one can give a bordism-interpretation to the Green s functions for instance, if () = S (S S ) L, = M, then G = W[J, J]/ describes the bulk compatible with the triangulation of M S S S ϕ 4 -generated 4D-bulk + ϕ 4 -generated 4D-bulk + ϕ 4 -generated 4D-bulk + L, S S L, S S L, S S Each correlation function supports also a /N-expansion, G (k) = G (k,ω) ω C I Pérez Sánchez (Math Münster) oundary-graph-expansion of W[J, J] 7 July / 9

Theorem (CP) (Full Ward-Takahashi Identity for arbitrary tensor models) The partition function Z[J, J] of a tensor model with S = Tr ( ϕ, Eϕ) such that E p p a m a p a+ p D E p p a n a p a+ p D = E(m a, n a ) for each a =,, D, satisfies, as consequence of the U(N) invariance of the path-integral measure, p i Z = p i Z δ Z[J, J] δj p p a m a p a+ p D δ J p p a n a p a+ p D ( E(m a, n a ) ( δ ma n a Y m (a) ) a [J, J] Z[J, J] J p m a p D δ δ J p n a p D J p n a p D δ δj p m a p D ) Z[J, J] where Y (a) m a [J, J] = C f (a) C,m a J(C) e r a i j x ξ(r,i,a) x ξ(r,j,a) e r a C I Pérez Sánchez (Math Münster) The WTI for tensor models 7 July 4 / 9

Schwinger-Dyson equations SDEs for the ϕ 4 mel,d-model (k ) [CP, R Wulkenhaar] Let D and let be a connected boundary graph G = Grph cl D Feyn D(ϕ 4 m,d ) Let X = (x,, x k ) and s = y Then G (k) obeys ( + λ E s = ( λ) E s D a= D a= qâ { ) (s a, qâ) G (k) (X) σ f (a),s a (X) + ˆσ Aut c () ρ> b a E(s a, b a ) [ G (k) Z[J, J] E(ya, ρ s a ) Z ς a (;, ρ)(x) (X) G(k) (X s a b a ) ] } where (s a, q a ) = (q, q,, q a, s a, q a+,, q D ) C I Pérez Sánchez (Math Münster) Schwinger-Dyson equations 7 July 5 / 9

Schwinger-Dyson equations SDEs for the ϕ 4 mel,d-model (k ) Let D and let be a connected boundary graph G = Grph cl D Feyn D(ϕ 4 m,d ) Let X = (x,, x k ) and s = y Then G (k) obeys ( + λ E s = ( λ) E s D a= D a= qâ { G () ) (s a, qâ) G (k) (X) σ f (a),s a (X) + ˆσ Aut c () ρ> b a E(s a, b a ) [ G (k) [CP, R Wulkenhaar] J x J x k J x G J y J y J y k Z[J, J] E(ya, ρ s a ) Z ς a (;, ρ)(x) (X) G(k) (X s a b a ) ] } where (s a, q a ) = (q, q,, q a, s a, q a+,, q D ) C I Pérez Sánchez (Math Münster) Schwinger-Dyson equations 7 July 5 / 9

A simple quartic model Proposal of a model with V[ϕ, ϕ] = λ S [ϕ, ϕ] = Tr ( ϕ, Eϕ) = x Z ϕ x (m + x )ϕ x, x = x + x + x The boundary sector is: Feyn ( ) = {-coloured graphs with connected components in Θ}, being Θ = {,,,,,, } Let X k be the graph in Θ with k vertices, and G (k) = G (k) X k : G () = G (), G (4) = G (4), G(6) = G (6), G(8) = G (8), G () = G () Full tower of exact equations obtained [CP, R Wulkenhaar] C I Pérez Sánchez (Math Münster) Schwinger-Dyson equations 7 July 6 / 9

The exact pt-function equation Melonic (planar) limit, conjecturally ( m + x + λ q,p Z ) G () mel (x, q, p) G () mel (x) [ () = + λ G q Z x q mel (x, x, x ) G () mel (q, x, x ) ] The exact k-pt-function equation Melonic limit, conjecturally ( + λ m + s = ( λ) [ k m + s ρ= ) G () mel (x, q, p) G (k) mel (x,, x k ) q,p Z ( (x ρ ) (x ) G (ρ ) mel (x,, x ρ ) G (k ρ+) ) mel (x ρ,, x k ) G (k) mel (x, x, x, x,, x k ) G (k) mel (q, x, x, x,, x k ] ) q Z (x ) q C I Pérez Sánchez (Math Münster) Schwinger-Dyson equations 7 July 7 / 9

The exact pt-function equation Melonic (planar) limit, conjecturally ( m + x + λ q,p Z ) G () mel (x, q, p) G () mel (x) [ () = + λ G q Z x q mel (x, x, x ) G () mel (q, x, x ) ] The exact k-pt-function equation Melonic limit, conjecturally ( + λ m + s = ( λ) [ k m + s ρ= ) G () mel (x, q, p) G (k) mel (x,, x k ) q,p Z ( (x ρ ) (x ) G (ρ ) mel (x,, x ρ ) G (k ρ+) ) mel (x ρ,, x k ) G (k) mel (x, x, x, x,, x k ) G (k) mel (q, x, x, x,, x k ] ) q Z (x ) q C I Pérez Sánchez (Math Münster) Schwinger-Dyson equations 7 July 7 / 9

Conclusions & outlook (Coloured) tensor field theories [en Geloun, onzom, Carrozza, Gurău, Krajewski, Oriti, Ousmane-Samary, Rivasseau, Ryan, Tanasa, Toriumi, Vignes-Tourneret, ] provide a framework for D-dimensional random geometry A bordism interpretation of the correlation functions was given A (non-perturbative) Ward-Takahashi identity [CP] based that for matrix models has been found It has been used to derive the full tower of SDE [CP-Wulkenhaar] Closed equations: hope of a solvable theory for the simple -model (spherical -geometries) Outlook: Apply these techniques SYK-like (Sachdev-Ye-Kitaev) models [Witten] Applications to GFT Gauge fields on colored graphs (random spaces) by representing graphs on finite spectral triples C I Pérez Sánchez (Math Münster) The WTI for tensor models 7 July 8 / 9

References M Disertori, R Gurău, J Magnen, and V Rivasseau Phys Lett, 649 95, 7 arxiv:hep-th/65 onzom, V and Gurău, R and Riello, A and Rivasseau, V Nucl Phys 85, arxiv:5 R Gurău, Commun Math Phys 4, 69 () arxiv:9758 [hep-th] C I Pérez-Sánchez, R Wulkenhaar arxiv:76758 C I Pérez-Sánchez JGeomPhys 6-89 (7) (arxiv:6846) and arxiv:6884 E Witten arxiv:69758v [hep-th] Thank you for your attention!

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