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The 1 / N Expansion in Colored Tensor Models R azvan Gur au Laboratoire dInformatique de Paris-Nord, 2011 The 1 / N Expansion in Colored Tensor Models, Laboratoire dInformatique de Paris-Nord, 2011 R azvan Gur au, Introduction

  1. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs as Feynman Graphs 4

  2. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs as Feynman Graphs Consider the partition function. � � � � φ a 1 a 2 δ a 1 b 1 δ a 2 b 2 φ ∗ b 1 b 2 + λ � φ a 1 a 2 φ a 2 a 3 φ a 3 a 1 1 − N 2 Z ( Q ) = [ d φ ] e 4

  3. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs as Feynman Graphs Consider the partition function. � � � � φ a 1 a 2 δ a 1 b 1 δ a 2 b 2 φ ∗ b 1 b 2 + λ � φ a 1 a 2 φ a 2 a 3 φ a 3 a 1 1 − N 2 Z ( Q ) = [ d φ ] e The vertex is a ribbon vertex because the field φ has two arguments. φ φ a a a a 1 2 3 1 a 1 a a 3 2 φ a a 2 3 4

  4. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs as Feynman Graphs Consider the partition function. � � � � φ a 1 a 2 δ a 1 b 1 δ a 2 b 2 φ ∗ b 1 b 2 + λ � φ a 1 a 2 φ a 2 a 3 φ a 3 a 1 1 − N 2 Z ( Q ) = [ d φ ] e The vertex is a ribbon vertex because the field φ has two arguments. The lines conserve the two arguments (thus having two strands). φ φ a a a a 1 2 b 3 1 a 1 a 1 1 b a a 3 2 2 a 2 φ a a 2 3 4

  5. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs as Feynman Graphs Consider the partition function. � � � � φ a 1 a 2 δ a 1 b 1 δ a 2 b 2 φ ∗ b 1 b 2 + λ � φ a 1 a 2 φ a 2 a 3 φ a 3 a 1 1 − N 2 Z ( Q ) = [ d φ ] e The vertex is a ribbon vertex because the field φ has two arguments. The lines conserve the two arguments (thus having two strands). The strands close into faces. φ φ a a b a a 1 2 3 1 a a 1 1 1 b a a 3 2 2 a 2 φ a a 2 3 4

  6. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs as Feynman Graphs Consider the partition function. � � � � φ a 1 a 2 δ a 1 b 1 δ a 2 b 2 φ ∗ b 1 b 2 + λ � φ a 1 a 2 φ a 2 a 3 φ a 3 a 1 1 − N 2 Z ( Q ) = [ d φ ] e The vertex is a ribbon vertex because the field φ has two arguments. The lines conserve the two arguments (thus having two strands). The strands close into faces. φ φ a a b a a 1 2 3 1 a a 1 1 1 b a a 3 2 2 a 2 φ a a 2 3 Z ( Q ) is a sum over ribbon Feynman graphs. 4

  7. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Amplitude of Ribbon Graphs 5

  8. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Amplitude of Ribbon Graphs The Amplitude of a graph with N vertices is A = λ N N −L + N � � δ a 1 b 1 δ a 2 b 2 lines 5

  9. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Amplitude of Ribbon Graphs The Amplitude of a graph with N vertices is A = λ N N −L + N � � δ a 1 b 1 δ a 2 b 2 lines � δ a 1 b 1 w 1 a b 1 1 b 3 a 3 a b 2 2 5

  10. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Amplitude of Ribbon Graphs The Amplitude of a graph with N vertices is A = λ N N −L + N � � δ a 1 b 1 δ a 2 b 2 lines � δ a 1 b 1 δ b 1 c 1 w 1 a b 1 1 b 3 a 3 a b 2 2 5

  11. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Amplitude of Ribbon Graphs The Amplitude of a graph with N vertices is A = λ N N −L + N � � δ a 1 b 1 δ a 2 b 2 lines � δ a 1 b 1 δ b 1 c 1 . . . δ w 1 a 1 w 1 a b 1 1 b 3 a 3 a b 2 2 5

  12. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Amplitude of Ribbon Graphs The Amplitude of a graph with N vertices is A = λ N N −L + N � � δ a 1 b 1 δ a 2 b 2 lines � � δ a 1 b 1 δ b 1 c 1 . . . δ w 1 a 1 = δ a 1 a 1 = N w 1 a b 1 1 b 3 a 3 a b 2 2 A = λ N N N −L + F = λ N N 2 − 2 g ( G ) with g G is the genus of the graph. 5

  13. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Amplitude of Ribbon Graphs The Amplitude of a graph with N vertices is A = λ N N −L + N � � δ a 1 b 1 δ a 2 b 2 lines � � δ a 1 b 1 δ b 1 c 1 . . . δ w 1 a 1 = δ a 1 a 1 = N w 1 a b 1 1 b 3 a 3 a b 2 2 A = λ N N N −L + F = λ N N 2 − 2 g ( G ) with g G is the genus of the graph. 1 / N expansion in the genus. 5

  14. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Amplitude of Ribbon Graphs The Amplitude of a graph with N vertices is A = λ N N −L + N � � δ a 1 b 1 δ a 2 b 2 lines � � δ a 1 b 1 δ b 1 c 1 . . . δ w 1 a 1 = δ a 1 a 1 = N w 1 a b 1 1 b 3 a 3 a b 2 2 A = λ N N N −L + F = λ N N 2 − 2 g ( G ) with g G is the genus of the graph. 1 / N expansion in the genus. Planar graphs ( g G = 0) dominate in the large N limit. 5

  15. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs are Dual to Discrete Surfaces 6

  16. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs are Dual to Discrete Surfaces w 1 a b 1 1 b 3 a 3 b a 2 2 6

  17. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs are Dual to Discrete Surfaces Place a point in the middle of each face. w 1 a b 1 1 b 3 a 3 b a 2 2 6

  18. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs are Dual to Discrete Surfaces Place a point in the middle of each face. Draw a line crossing each ribbon line. w 1 a b 1 1 b 3 a 3 b a 2 2 6

  19. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs are Dual to Discrete Surfaces Place a point in the middle of each face. Draw a line crossing each ribbon line. The ribbon w 1 a b 1 vertices correspond to triangles. 1 b 3 a 3 b a 2 2 6

  20. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs are Dual to Discrete Surfaces Place a point in the middle of each face. Draw a line crossing each ribbon line. The ribbon w 1 a b 1 vertices correspond to triangles. 1 b 3 a 3 b a 2 2 A ribbon graph encodes unambiguously a gluing of triangles. 6

  21. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs are Dual to Discrete Surfaces Place a point in the middle of each face. Draw a line crossing each ribbon line. The ribbon w 1 a b 1 vertices correspond to triangles. 1 b 3 a 3 b a 2 2 A ribbon graph encodes unambiguously a gluing of triangles. Matrix models sum over all graphs (i.e. surfaces) with canonical weights (Feynman rules). 6

  22. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Ribbon Graphs are Dual to Discrete Surfaces Place a point in the middle of each face. Draw a line crossing each ribbon line. The ribbon w 1 a b 1 vertices correspond to triangles. 1 b 3 a 3 b a 2 2 A ribbon graph encodes unambiguously a gluing of triangles. Matrix models sum over all graphs (i.e. surfaces) with canonical weights (Feynman rules). The dominant planar graphs represent spheres. 6

  23. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models 7

  24. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models surfaces ↔ ribbon graphs 7

  25. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs 7

  26. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs Matrix M ab , � � M ab ¯ S = N M ab + λ M ab M bc M ca 7

  27. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs Tensors T i a 1 ... a D with color i Matrix M ab , � � S = N D / 2 � � M ab ¯ ... ¯ S = N M ab + λ M ab M bc M ca T i T i ... + λ T 0 ... T 1 ... . . . T D ... 7

  28. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs Tensors T i a 1 ... a D with color i Matrix M ab , � � S = N D / 2 � � M ab ¯ ... ¯ S = N M ab + λ M ab M bc M ca T i T i ... + λ T 0 ... T 1 ... . . . T D ... g ( G ) ≥ 0 genus 7

  29. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs Tensors T i a 1 ... a D with color i Matrix M ab , � � S = N D / 2 � � M ab ¯ ... ¯ S = N M ab + λ M ab M bc M ca T i T i ... + λ T 0 ... T 1 ... . . . T D ... g ( G ) ≥ 0 genus ω ( G ) ≥ 0 degree 7

  30. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs Tensors T i a 1 ... a D with color i Matrix M ab , � � S = N D / 2 � � M ab ¯ ... ¯ S = N M ab + λ M ab M bc M ca T i T i ... + λ T 0 ... T 1 ... . . . T D ... g ( G ) ≥ 0 genus ω ( G ) ≥ 0 degree 1 / N expansion in the genus A ( G ) = N 2 − 2 g ( G ) 7

  31. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs Tensors T i a 1 ... a D with color i Matrix M ab , � � S = N D / 2 � � M ab ¯ ... ¯ S = N M ab + λ M ab M bc M ca T i T i ... + λ T 0 ... T 1 ... . . . T D ... g ( G ) ≥ 0 genus ω ( G ) ≥ 0 degree 1 / N expansion in the degree 1 / N expansion in the genus 2 A ( G ) = N 2 − 2 g ( G ) A ( G ) = N D − ( D − 1)! ω ( G ) 7

  32. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs Tensors T i a 1 ... a D with color i Matrix M ab , � � S = N D / 2 � � M ab ¯ ... ¯ S = N M ab + λ M ab M bc M ca T i T i ... + λ T 0 ... T 1 ... . . . T D ... g ( G ) ≥ 0 genus ω ( G ) ≥ 0 degree 1 / N expansion in the degree 1 / N expansion in the genus 2 A ( G ) = N 2 − 2 g ( G ) A ( G ) = N D − ( D − 1)! ω ( G ) leading order: g ( G ) = 0, spheres. 7

  33. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion From Matrix to COLORED Tensor Models D dimensional spaces ↔ colored stranded graphs surfaces ↔ ribbon graphs Tensors T i a 1 ... a D with color i Matrix M ab , � � S = N D / 2 � � M ab ¯ ... ¯ S = N M ab + λ M ab M bc M ca T i T i ... + λ T 0 ... T 1 ... . . . T D ... g ( G ) ≥ 0 genus ω ( G ) ≥ 0 degree 1 / N expansion in the degree 1 / N expansion in the genus 2 A ( G ) = N 2 − 2 g ( G ) A ( G ) = N D − ( D − 1)! ω ( G ) leading order: g ( G ) = 0, spheres. leading order: ω ( G ) = 0, spheres. 7

  34. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Introduction Colored Tensor Models Colored Graphs Jackets and the 1 / N expansion Topology Leading order graphs are spheres Conclusion 8

  35. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Colored Stranded Graphs 9

  36. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Colored Stranded Graphs Clockwise and anticlockwise turning colored vertices (positive and negative oriented D simplices). 9

  37. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Colored Stranded Graphs 1 1 2 (0,1) (0,1) Clockwise and anticlockwise turning colored 0 0 2 vertices (positive and negative oriented D 3 4 simplices). 3 (0,1) 1 (0,1) 1 0 0 9

  38. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Colored Stranded Graphs 1 1 2 (0,1) (0,1) Clockwise and anticlockwise turning colored 0 0 2 vertices (positive and negative oriented D 3 4 simplices). 3 (0,1) 1 (0,1) 1 0 0 Lines have a well defined color and D parallel strands ( D − 1 simplices). 9

  39. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Colored Stranded Graphs 1 1 2 (0,1) (0,1) Clockwise and anticlockwise turning colored 0 0 2 vertices (positive and negative oriented D 3 4 simplices). 3 (0,1) 1 (0,1) 1 0 0 1 1 1 1 (0,1) (0,1) Lines have a well defined color and D parallel 0 0 strands ( D − 1 simplices). (0,1) (0,1) 1 1 0 0 1 1 9

  40. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Colored Stranded Graphs 1 1 2 (0,1) (0,1) Clockwise and anticlockwise turning colored 0 0 2 vertices (positive and negative oriented D 3 4 simplices). 3 (0,1) 1 (0,1) 1 0 0 1 1 1 1 (0,1) (0,1) Lines have a well defined color and D parallel 0 0 strands ( D − 1 simplices). (0,1) (0,1) 1 1 0 0 1 1 Strands are identified by a couple of colors ( D − 2 simplices). 9

  41. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action 10

  42. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . 10

  43. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i 10

  44. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs 10

  45. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs Amplitude of the graphs: 10

  46. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs Amplitude of the graphs: ◮ the N = 2 p vertices of a graph bring each N D / 2 10

  47. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs Amplitude of the graphs: ◮ the N = 2 p vertices of a graph bring each N D / 2 ◮ the L lines of a graphs bring each N − D / 2 10

  48. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs Amplitude of the graphs: ◮ the N = 2 p vertices of a graph bring each N D / 2 ◮ the L lines of a graphs bring each N − D / 2 ◮ the F faces of a graph bring each N 10

  49. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs Amplitude of the graphs: ◮ the N = 2 p vertices of a graph bring each N D / 2 ◮ the L lines of a graphs bring each N − D / 2 ◮ the F faces of a graph bring each N A G = ( λ ¯ λ ) p N −L D 2 + N D 2 + F 10

  50. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs Amplitude of the graphs: ◮ the N = 2 p vertices of a graph bring each N D / 2 ◮ the L lines of a graphs bring each N − D / 2 ◮ the F faces of a graph bring each N A G = ( λ ¯ λ ) p N −L D 2 + N D 2 + F But N ( D + 1) = 2 L ⇒ L = ( D + 1) p 10

  51. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs Amplitude of the graphs: ◮ the N = 2 p vertices of a graph bring each N D / 2 ◮ the L lines of a graphs bring each N − D / 2 ◮ the F faces of a graph bring each N A G = ( λ ¯ λ ) p N −L D 2 + F = ( λ ¯ λ ) p N − p D ( D − 1) 2 + N D + F 2 But N ( D + 1) = 2 L ⇒ L = ( D + 1) p 10

  52. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Action a 1 ... a D , ¯ Let T i T i a 1 ... a D tensor fields with color i = 0 . . . D . S = N D / 2 � � � � � ¯ a ii − 1 ... a i 0 a iD ... a ii +1 + ¯ ¯ T i a 1 ... a D T i T i T i a 1 ... a D + λ λ a ii − 1 ... a i 0 a iD ... a ii +1 i i i Topology of the Colored Graphs Amplitude of the graphs: ◮ the N = 2 p vertices of a graph bring each N D / 2 ◮ the L lines of a graphs bring each N − D / 2 ◮ the F faces of a graph bring each N A G = ( λ ¯ λ ) p N −L D 2 + F = ( λ ¯ λ ) p N − p D ( D − 1) 2 + N D + F 2 But N ( D + 1) = 2 L ⇒ L = ( D + 1) p Compute F ! 10

  53. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Introduction Colored Tensor Models Colored Graphs Jackets and the 1 / N expansion Topology Leading order graphs are spheres Conclusion 11

  54. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 12

  55. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. 12

  56. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 12

  57. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 1 0 2 3 1 2 0 3 12

  58. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. 1 0 2 3 1 2 0 3 12

  59. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. 1 0 2 3 1 2 0 3 12

  60. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 1 2 0 3 12

  61. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 12

  62. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 1 2 0 12

  63. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 1 2 0 0 , 1 , 2 , . . . 12

  64. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 π(0) 1 2 π (0) 2 0 0 0 , 1 , 2 , . . . 12

  65. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 π(0) 1 2 π (0) 2 0 0 0 , π (0) , π 2 (0) , . . . 0 , 1 , 2 , . . . 12

  66. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 π(0) 1 2 π (0) 1 2 D ! jackets. 2 0 0 0 , π (0) , π 2 (0) , . . . 0 , 1 , 2 , . . . 12

  67. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 π(0) 1 2 π (0) 1 2 D ! jackets. Contain all the vertices and 2 0 0 all the lines of G . 0 , π (0) , π 2 (0) , . . . 0 , 1 , 2 , . . . 12

  68. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 π(0) 1 2 π (0) 1 2 D ! jackets. Contain all the vertices and 2 0 0 all the lines of G . A face belongs to ( D − 1)! jackets. 0 , π (0) , π 2 (0) , . . . 0 , 1 , 2 , . . . 12

  69. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 1 Define simpler graphs. Idea: forget the interior strands! Leads to a ribbon graph. 02 and 13: opposing edges of the tetrahedron. But 01, 23 and 12, 03 are perfectly equivalent. Three jacket (ribbon) 1 graphs. 0 2 3 0 0 0 1 2 1 1 1 3 3 3 0 2 2 2 3 π(0) 1 2 π (0) 1 2 D ! jackets. Contain all the vertices and 2 0 0 all the lines of G . A face belongs to ( D − 1)! jackets. 0 , π (0) , π 2 (0) , . . . 0 , 1 , 2 , . . . The degree of G is ω ( G ) = � J g J . 12

  70. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 2: Jackets and Amplitude 13

  71. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 2: Jackets and Amplitude Theorem F and ω ( G ) are related by F = 1 2 2 D ( D − 1) p + D − ( D − 1)! ω ( G ) 13

  72. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 2: Jackets and Amplitude Theorem F and ω ( G ) are related by F = 1 2 2 D ( D − 1) p + D − ( D − 1)! ω ( G ) Proof: N = 2 p , L = ( D + 1) p 13

  73. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 2: Jackets and Amplitude Theorem F and ω ( G ) are related by F = 1 2 2 D ( D − 1) p + D − ( D − 1)! ω ( G ) Proof: N = 2 p , L = ( D + 1) p For each jacket J , 2 p − ( D + 1) p + F J = 2 − 2 g J . 13

  74. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 2: Jackets and Amplitude Theorem F and ω ( G ) are related by F = 1 2 2 D ( D − 1) p + D − ( D − 1)! ω ( G ) Proof: N = 2 p , L = ( D + 1) p For each jacket J , 2 p − ( D + 1) p + F J = 2 − 2 g J . Sum over the jackets: ( D − 1)! F = � 2 D !( D − 1) p + D ! − 2 � J F J = 1 J g J 13

  75. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Jackets 2: Jackets and Amplitude Theorem F and ω ( G ) are related by F = 1 2 2 D ( D − 1) p + D − ( D − 1)! ω ( G ) Proof: N = 2 p , L = ( D + 1) p For each jacket J , 2 p − ( D + 1) p + F J = 2 − 2 g J . Sum over the jackets: ( D − 1)! F = � 2 D !( D − 1) p + D ! − 2 � J F J = 1 J g J The amplitude of a graph is given by its degree A G = ( λ ¯ λ ) p N − p D ( D − 1) + F = ( λ ¯ λ ) p N D − 2 ( D − 1)! ω ( G ) 2 13

  76. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Introduction Colored Tensor Models Colored Graphs Jackets and the 1 / N expansion Topology Leading order graphs are spheres Conclusion 14

  77. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Topology 1: Colored vs. Stranded Graphs 15

  78. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Topology 1: Colored vs. Stranded Graphs THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D -dimensional piecewise linear orientable manifold admits a colored triangulation. 15

  79. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Topology 1: Colored vs. Stranded Graphs THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D -dimensional piecewise linear orientable manifold admits a colored triangulation. We have clockwise and anticlockwise turning vertices. 15

  80. The 1 / N Expansion in Colored Tensor Models, Laboratoire d’Informatique de Paris-Nord, 2011 R˘ azvan Gur˘ au, Introduction Colored Tensor Models Conclusion Topology 1: Colored vs. Stranded Graphs THEOREM: [M. Ferri and C. Gagliardi, ’82] Any D -dimensional piecewise linear orientable manifold admits a colored triangulation. We have clockwise and anticlockwise turning vertices. Lines connect opposing vertices and have a color index. 15

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