Correlation functions for colored tensor models Schwinger-Dyson Equations Carlos. I. P´ erez-S´ anchez Mathematics Institute, University of M¨ unster LQP 40, Max-Planck-Institut f¨ ur Mathematik MIS Leipzig, 23 June
Motivation Motivation Random Geometry framework (“Quantum Gravity”) � � � � 1 1 D [ g ] e − S EH [ g ] ∼ k ∑ ∑ Z = µ D k k D 2 2 topologies ) ∈ topologies ( geometries geometries Random matrices do that successfully for 2 D. Random tensor models is a higher-dimensional arena, together with QFT-techniques, based on this idea Gurau-Witten model based on SYK-model (Sachdev-Ye–Kitaev). Random tensor methods useful in AdS 2 /CFT 1 (Maldacena, Stanford) C. I. P´ erez S´ anchez (Math. M¨ unster) Motivation 23 June 2 / 30
Outline Outline matrix and random tensor models Non-perturbative approach to quantum (coloured) tensor fields ϕ 4 -intearction Λ 0 Λ ≫ Λ 0 ◮ graph-calculus: correlation functions ◮ full Ward-Takahashi Identities: non-perturbative, systematic approach ◮ Schwinger-Dyson equations: equations for the multiple-point functions (joint work with Raimar Wulkenhaar) C. I. P´ erez S´ anchez (Math. M¨ unster) Outline 23 June 3 / 30
Random matrix theory: ensembles Nuclear physics (Wigner). Stochastics: E ⊂ M N ( K ) : � Z = E d µ Statistics of random eigenvalues; study limit N ∞ ; universality, µ -independence (tensor models too: book by R. Gur˘ au) usually, for certain polynomial P ( x ) = Nx 2 /2 + N V ( x ) , � � � E d M e − Tr P ( M ) = − N Tr V ( M ) = E d M e − N 2 Tr M 2 E d µ 0 e − N Tr V ( M ) Z = � �� � d µ 0 Kontsevich, Grosse-Wulkenhaar, Barrett-Glaser, . . . models V ( M ) = M p ( p = 4, 6, 8 ) C. I. P´ erez S´ anchez (Math. M¨ unster) Matrix models 23 June 4 / 30
Random matrix theory: ensembles Nuclear physics (Wigner). Stochastics: E ⊂ M N ( K ) : � Z = E d µ Statistics of random eigenvalues; study limit N ∞ ; universality, µ -independence (tensor models too: book by R. Gur˘ au) usually, for certain polynomial P ( x ) = Nx 2 /2 + N V ( x ) , � � � E d M e − Tr P ( M ) = − N Tr V ( M ) = E d M e − N 2 Tr M 2 E d µ 0 e − N Tr V ( M ) Z = � �� � d µ 0 Kontsevich, Grosse-Wulkenhaar, Barrett-Glaser, . . . models V ( M ) = M p ( p = 4, 6, 8 ) , , , . . . C. I. P´ erez S´ anchez (Math. M¨ unster) Matrix models 23 June 4 / 30
“Rank- 2 tensor models” � D [ M , M ] e − Tr ( MM † ) − λ V ( M , M † ) For complex matrix models M M = dual triangulation to . M M M . . . . . . M M M ) = λ Tr (( MM † ) 2 ) , different connected O ( λ 2 ) -graphs are For V ( M , ¯ 0 0 0 0 2 1 1 2 2 1 2 1 0 0 1 2 2 1 1 2 1 2 0 0 t . U ( 1 ) M ( U ( 2 ) ) rectangular matrices, M ∈ M N 1 × N 2 ( C ) and M U ( N 1 ) × U ( N 2 ) -invariants are Tr (( MM † ) q ) , q ∈ Z ≥ 1 C. I. P´ erez S´ anchez (Math. M¨ unster) Ribbon graphs as coloured ones 23 June 5 / 30
“Rank- 2 tensor models” � D [ M , M ] e − Tr ( MM † ) − λ V ( M , M † ) For complex matrix models M M = dual triangulation to . M M M . . . . . . M M M ) = λ Tr (( MM † ) 2 ) , different connected O ( λ 2 ) -graphs are For V ( M , ¯ 0 0 0 0 2 1 1 2 2 1 1 2 0 1 2 2 1 1 2 2 1 0 0 0 t . U ( 1 ) M ( U ( 2 ) ) rectangular matrices, M ∈ M N 1 × N 2 ( C ) and M U ( N 1 ) × U ( N 2 ) -invariants are Tr (( MM † ) q ) , q ∈ Z ≥ 1 C. I. P´ erez S´ anchez (Math. M¨ unster) Ribbon graphs as coloured ones 23 June 5 / 30
“Rank- 2 tensor models” � D [ M , M ] e − Tr ( MM † ) − λ V ( M , M † ) For complex matrix models M M = dual triangulation to . M M M . . . . . . M M M ) = λ Tr (( MM † ) 2 ) , different connected O ( λ 2 ) -graphs are For V ( M , ¯ 0 0 0 0 2 1 1 2 2 1 1 2 0 1 2 2 1 1 2 2 1 0 0 0 t . U ( 1 ) M ( U ( 2 ) ) rectangular matrices, M ∈ M N 1 × N 2 ( C ) and M U ( N 1 ) × U ( N 2 ) -invariants are Tr (( MM † ) q ) , q ∈ Z ≥ 1 C. I. P´ erez S´ anchez (Math. M¨ unster) Ribbon graphs as coloured ones 23 June 5 / 30
Coloured Tensor Models Coloured Tensor Models a quantum field theory for tensors ϕ a 1 ... a D and ϕ a 1 ... a D the indices transform under different representations of G = U ( N 1 ) × U ( N 2 ) × . . . × U ( N D ) for g ∈ G , g = ( U ( 1 ) , . . . , U ( D ) ) , U ( a ) ∈ U ( N a ) , g ( ϕ ′ ) a 1 a 2 ... a D = U ( 1 ) a 1 b 1 U ( 2 ) a 2 b 2 . . . U ( D ) ϕ a 1 a 2 ... a D a D b D ϕ b 1 ... b D the complex conjugate tensor ϕ a 1 a 2 ... a D transforms as g ( ϕ ′ ) a 1 a 2 ... a D = U ( 1 ) ( 2 ) ( D ) ϕ a 1 a 2 ... a D a 1 b 1 U a 2 b 2 . . . U a D b D ϕ b 1 b 2 ... b D G -invariants serve as interaction vertices S [ ϕ , ϕ ] = ∑ τ i Tr B i ( ϕ , ϕ ) = Tr B 2 ( ϕ , ϕ ) + ∑ λ α Tr B α ( ϕ , ϕ ) α i C. I. P´ erez S´ anchez (Math. M¨ unster) Coloured Tensors 23 June 6 / 30
Feynman diagrams: Choose an action, for instance, the ϕ 4 3 -theory, S [ ϕ , ¯ ϕ ] = Tr B 2 ( ϕ , ϕ ) + λ ( Tr V 1 ( ϕ , ϕ ) + Tr V 2 ( ϕ , ϕ ) + Tr V 3 ( ϕ , ϕ )) and V 1 = , V 2 = , V 3 = , � D [ ϕ , ¯ ϕ J ) − N 2 S [ ϕ , ϕ ] ϕ ] e Tr B 2 ( J ϕ )+ Tr B 2 ( ¯ Z [ J , ¯ J ] = � D [ ϕ , ¯ , with Tr B 2 ↔ ϕ ] e − N 2 S [ ϕ , ϕ ] d ϕ a d ϕ a ϕ ] e − N 2 S 0 [ ϕ , ϕ ] : = ∏ e − N 2 Tr B 2 ( ϕ , ϕ ) d µ C ( ϕ , ϕ ) : = D [ ϕ , ¯ 2 π i a • Write for Wick’s contractions w.r.t. the Gaußian measure � d µ C ( ϕ , ϕ ) ϕ a ϕ p = C ( a , p ) = δ ap = a p C. I. P´ erez S´ anchez (Math. M¨ unster) Feynman diagrams 23 June 7 / 30
Feynman diagrams: Choose an action, for instance, the ϕ 4 3 -theory, S [ ϕ , ¯ ϕ ] = Tr B 2 ( ϕ , ϕ ) + λ ( Tr V 1 ( ϕ , ϕ ) + Tr V 2 ( ϕ , ϕ ) + Tr V 3 ( ϕ , ϕ )) and V 1 = , V 2 = , V 3 = , � D [ ϕ , ¯ ϕ J ) − N 2 S [ ϕ , ϕ ] ϕ ] e Tr B 2 ( J ϕ )+ Tr B 2 ( ¯ Z [ J , ¯ J ] = � D [ ϕ , ¯ , with Tr B 2 ↔ ϕ ] e − N 2 S [ ϕ , ϕ ] d ϕ a d ϕ a ϕ ] e − N 2 S 0 [ ϕ , ϕ ] : = ∏ e − N 2 Tr B 2 ( ϕ , ϕ ) d µ C ( ϕ , ϕ ) : = D [ ϕ , ¯ 2 π i a • Write for Wick’s contractions w.r.t. the Gaußian measure � d µ C ( ϕ , ϕ ) ϕ a ϕ p = C ( a , p ) = δ ap = a p C. I. P´ erez S´ anchez (Math. M¨ unster) Feynman diagrams 23 June 7 / 30
0 0 0 2 1 1 3 3 2 3 2 0 0 0 1 3 2 1 = 0 0 0 0 0 0 0 0 0 0 0 0 1 3 3 2 0 0 0 2 1 2 1 3 2 1 3 0 0 0 1 1 Vertex bipartite regularly edge– D -coloured graphs Feynman graphs of a model V , Feyn D ( V ) are ( D + 1 ) -coloured. Crystallization theory or GEMs [ Pezzana , ‘74] says all PL-manifolds of dimension D can be represented as D + 1 -coloured graphs, Grph D + 1 . C. I. P´ erez S´ anchez (Math. M¨ unster) Feynman diagrams 23 June 8 / 30
The complex ∆ ( G ) for each vertex v ∈ G ( 0 ) , add a D -simplex σ v to ∆ ( G ) with colour-labelled vertices { 0, 1, . . . , D } 0 ϕ , ¯ ϕ v D k 1 for each edge e k ∈ G ( 1 ) of arbitrary colour k , one identifies the two k ( D − 1 ) -simplices σ s ( e k ) and σ t ( e k ) that do not contain the colour k . 0 0 k s ( e k ) t ( e k ) , D k D k e k 1 1 ϕ p 1 ... p k ... p D ( k � = 0 ) or ϕ a ¯ edges come from either ϕ a 1 ... a k ... a D δ a k p k ¯ ϕ p ( k = 0 ). au, ’09] and [Bonzom, Gur˘ au, Riello, Rivasseau, ’11] ; [Gur˘ F ( G ) − D ( D − 1 ) A ( G ) = λ V ( G ) /2 N V ( G ) = exp ( − S Regge [ N , D , λ ]) 4 � �� � 2 = : D − ( D − 1 ) ! ω ( G ) � generalizes g ; not topol. invariant C. I. P´ erez S´ anchez (Math. M¨ unster) Graph-encoded topology 23 June 9 / 30
The complex ∆ ( G ) for each vertex v ∈ G ( 0 ) , add a D -simplex σ v to ∆ ( G ) with colour-labelled vertices { 0, 1, . . . , D } 0 ϕ , ¯ ϕ v D k 1 for each edge e k ∈ G ( 1 ) of arbitrary colour k , one identifies the two k ( D − 1 ) -simplices σ s ( e k ) and σ t ( e k ) that do not contain the colour k . 0 0 k s ( e k ) t ( e k ) , D k D k e k 1 1 ϕ p 1 ... p k ... p D ( k � = 0 ) or ϕ a ¯ edges come from either ϕ a 1 ... a k ... a D δ a k p k ¯ ϕ p ( k = 0 ). au, ’09] and [Bonzom, Gur˘ au, Riello, Rivasseau, ’11] ; [Gur˘ F ( G ) − D ( D − 1 ) A ( G ) = λ V ( G ) /2 N V ( G ) = exp ( − S Regge [ N , D , λ ]) 4 � �� � 2 = : D − ( D − 1 ) ! ω ( G ) � generalizes g ; not topol. invariant C. I. P´ erez S´ anchez (Math. M¨ unster) Graph-encoded topology 23 June 9 / 30
Ward-Takahashi Identity Ward-Takahashi Identity motivated by the WTI for matrix models by [Disertori-Gurau-Magnen-Rivasseau] ; WTI fully exploited by [Grosse-Wulkenhaar] for T α a a hermitian generator of the a -th summand of Lie ( U ( N ) D ) , δ log Z [ J , ¯ J ] = 0. δ ( T α a ) m a n a this implies a relation of the type δ 2 Z [ J , ¯ J ] J Z [ J , ¯ ∑ E ( m a , n a ) = D J ,¯ J ] δ J p 1 ... p a − 1 m a p a + 1 ... p D ¯ J p 1 ... p a − 1 n a p a + 1 ... p D p i ∈ Z where E ( m a , n a ) = − E ( n a , m a ) anihilates δ m a n a -terms. Aim: find them. C. I. P´ erez S´ anchez (Math. M¨ unster) The full WTI for tensor models 23 June 10 / 30
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